text
stringlengths 3
1.04M
| lang
stringclasses 4
values | len
int64 3
1.04M
|
|---|---|---|
\begin{document}
\title{Subwavelength resonances of encapsulated bubbles hanks{ ootnotesize The work of Hyundae Lee was supported by National Research Fund of Korea (NRF-2015R1D1A1A01059357, NRF-2018R1D1A1B07042678).}
\begin{abstract}
The aim of this paper is to derive a formula for the subwavelength resonance frequency of an encapsulated bubble with arbitrary shape in two dimensions. Using Gohberg-Sigal theory, we derive an asymptotic formula for this resonance frequency, as a perturbation away from the resonance of the uncoated bubble, in terms of the thickness of the coating.
The formula is numerically verified in the case of circular bubbles, where the resonance can be efficiently computed using the multipole method.
\end{abstract}
\def
{\textbf{ Keywords.}~\,\relax}2{
{\textbf{ Mathematics Subject Classification
(MSC2000).}~\,\relax}}
\def\par{\partialar}
{\textbf{ Keywords.}~\,\relax}2{35R30, 35C20.}
\def
{\textbf{ Keywords.}~\,\relax}{
{\textbf{ Keywords.}~\,\relax}}
\def\par{\partialar}
{\textbf{ Keywords.}~\,\relax}{bubble, subwavelength resonance, encapsulated bubble, thin coating.}
\section{Introduction}
A gas bubble in a liquid is an acoustic scatterer which possesses a subwavelength resonance called the Minnaert resonance \cite{first, Minnaert}. A remarkable feature of this resonance is its subwavelength scale; the size of the bubble can be several orders of magnitude smaller than the wavelength at the resonant frequency. This is due to the high contrast in density between the bubble and the surrounding medium and it opens up a wide range of applications, some examples being the creation of subwavelength phononic crystals \cite{DesignOfBandgap} or to achieve super-resolution in medical ultrasound-imaging \cite{ultrafastultrasound}.
Despite having interesting properties with regards to the creation of subwavelength scale metamaterials \cite{metasurface, superfocusing, effectivemedium, defect, bandgap}, bubbly media comprised of air bubbles inside water is highly unstable. There exist various approaches to stabilizing such structures. One approach is to replace the background medium, water, with a soft elastic matrix, and it has been demonstrated that this technique results in metamaterials having properties similar to those of metamaterials comprised of air bubbles in water \cite{DesignOfBandgap, DesignOfBubbles}. Another approach is to encapsulate the bubbles in a thin coating \cite{reviewDoinikov, reviewFaez}, the aim being to prevent the fast dissolution and coalescence of the bubbles. Encapsulated bubbles have long been used as ultrasound contrast agents, whereby the gas is trapped inside a coating of an albumin, polymer or lipid. However, the effect of such coating on the acoustic properties of the bubbly media has not yet been fully described.
Clearly, the introduction of a coating will affect the resonance frequency of the bubble, with the thin coating causing a slight perturbation of the Minnaert resonance. Through the application of layer potential techniques, asymptotic analysis and Gohberg-Sigal theory, we derive an original formula for the subwavelength resonance of an encapsulated bubble in two dimensions. Our results are complemented by several numerical examples which serve to validate them.
The paper is organized as follows. In Section \ref{sec-prelim}, we introduce some basic results regarding layer potentials and review the subwavelength resonance of an uncoated bubble in two dimensions. We also provide a correction to the formula for the Minnaert resonance in two dimensions, given in \cite{first}. In Section \ref{sec-problem}, we state the resonance problem for the encapsulated bubble. In Section \ref{sec:analysis}, we perform an asymptotic analysis in terms of the thickness of the coating, and use this to derive the resonance frequency in terms of the Minnaert frequency of the uncoated bubble. The main result is stated in Theorem \ref{thm:main} and equation \eqnref{eq:thm}. In Section \ref{sec-numerics}, we perform numerical simulations to illustrate the main findings of this paper. We make use of the multipole expansion method to validate our asymptotic formula for the subwavelength resonance of the encapsulated bubble in terms of its thickness. The paper ends with some concluding remarks.
\section{Preliminaries} \label{sec-prelim}
In this section we state some well-known results about layer potentials. We also provide a correction to the Minnaert resonance formula in two dimensions given in \cite{first}.
\subsection{Layer potentials} \label{sec:layerpot}
For $k>0$ and $k=0$, let $\Gamma^k$ be the fundamental solution of the Helmholtz and Laplace equations in dimension two, respectively, \textit{i.e.}{},
\begin{equation*}
\begin{cases}
\displaystyle \Gamma^k(x,y) = -\frac{i}{4}H_0^{(1)}(k|x-y|), \ & k>0, \\
\noalign{
}
\displaystyle \Gamma^0(x,y) = \frac{1}{2\partiali}\ln|x-y|, & k=0,
\end{cases}
\end{equation*}
where $H_0^{(1)}$ is the Hankel function of the first kind of order zero. In the following, we will omit the superscript and denote this function by $H_0$.
Let $\mathcal{S}_{D}^k: L^2(\partialartial D) \rightarrow H_{\textrm{loc}}^1(\mathbb{R}^2)$ be the single layer potential defined by
\begin{equation*}
\mathcal{S}_D^k[\partialhi](x) = \int_{\partialartial D} \Gamma^k(x,y)\partialhi(y) \: \mathrm{d} \sigma(y), \quad x \in \mathbb{R}^2.
\end{equation*}
Furthermore, let $\mathcal{D}_{D}^k: L^2(\partialartial D) \rightarrow H_{\textrm{loc}}^1(\mathbb{R}^2\setminus \partialartial D)$ be the double layer potential defined by
\begin{equation*}
\left(\mathcal{D}_D^k\right)[\partialhi](x) = \int_{\partialartial D} \frac{\partialartial }{\partialartial \nu_y}\Gamma^k(x,y) \partialhi(y) \: \mathrm{d} \sigma(y), \quad x \in \mathbb{R}^2\setminus \partialartial D.
\end{equation*}
We also define the Neumann-Poincar\'e operator $\left(\mathcal{K}_D^k\right)^*: L^2(\partialartial D) \rightarrow L^2(\partialartial D)$ by
\begin{equation*}
\left(\mathcal{K}_D^k\right)^*[\partialhi](x) = \int_{\partialartial D} \frac{\partialartial }{\partialartial \nu_x}\Gamma^k(x,y) \partialhi(y) \: \mathrm{d} \sigma(y), \quad x \in \partialartial D.
\end{equation*}
In the case when $k=0$, we will omit the superscripts and write $\mathcal{S}_D$, $\mathcal{D}_D$ and $\mathcal{K}_D^*$, respectively. The following so-called jump relations of $\mathcal{S}_D^k$ and $\mathcal{D}_D^k$ on the boundary $\partialartial D$ are well-known (see, for instance, \cite{MaCMiPaP}):
\begin{equation*}
\mathcal{S}_D^k[\partialhi]\big|_+ = \mathcal{S}_D^k[\partialhi]\big|_-, \qquad \mathcal{D}_D^k[\partialhi]\big|_\partialm = \left(\mp\frac{1}{2} I + \mathcal{K}_D^k\right) [\partialhi],
\end{equation*}
and
\begin{equation*}
\frac{\partialartial }{\partialartial \nu}\mathcal{S}_D^k[\partialhi] \bigg|_{\partialm} = \left(\partialm\frac{1}{2} I + \left(\mathcal{K}_D^k\right)^*\right) [\partialhi], \qquad \frac{\partialartial }{\partialartial \nu}\mathcal{D}_D^k[\partialhi]\bigg|_+ = \frac{\partialartial }{\partialartial \nu}\mathcal{D}_D^k[\partialhi]\bigg|_-.
\end{equation*}
Here, $\partialartial/\partialartial \nu$ denotes the outward normal derivative, and $|_\partialm$ denotes the limits from outside and inside $D$.
We now state some basic properties of the single-layer potential in two dimensions, given in \cite{MaCMiPaP}. The operator $-\frac{1}{2}I + \mathcal{K}_D^*$ is known to have a kernel of dimension 1. Let $\ker(-\frac{1}{2}I + \mathcal{K}_D^*) =\mathrm{span}(\partialsi_0)$ with $||\partialsi_0||=1$. Also, denote by $\partialhi_0 = \chi_{\partialartial D}$. Then
$$\mathcal{S}_D[\partialsi_0] = \gamma_0 \partialhi_0,$$
for some constant $\gamma_0$. It can be shown that $\mathcal{S}_D$ is invertible if and only if $\gamma_0 \neq 0$. In two dimensions, the fundamental solution of the free-space Helmholtz equation has a logarithmic singularity. Indeed, we have the following expansion \cite{MaCMiPaP}
\begin{equation}\label{eq:hankel}
-\frac{i}{4}H_0(k|x-y|) = \frac{1}{2\partiali} \ln |x-y| + \eta_k + \sum_{j=1}^\infty\left( b_j \ln(k|x-y|) + c_j \right) (k|x-y| )^{2j},
\end{equation}
where $\ln$ is the principal branch of the logarithm and
$$ \eta_k = \frac{1}{2\partiali}(\ln k+\gamma-\ln 2)-\frac{i}{4}, \quad b_j=\frac{(-1)^j}{2\partiali}\frac{1}{2^{2j}(j!)^2}, \quad c_j=b_j\left( \gamma - \ln 2 - \frac{i\partiali}{2} - \sum_{n=1}^j \frac{1}{n} \right),$$
and $\gamma$ is the Euler constant. Define
\begin{equation*}
\hat{S}_D^k[\partialhi](x) = \mathcal{S}_D[\partialhi](x) + \eta_k\int_{\partialartial D} \partialhi\: \mathrm{d} \sigma.
\end{equation*}
Then the following expansion holds:
\begin{equation} \label{eq:Sexpansion}
\mathcal{S}_D^k = \hat{\mathcal{S}}_{D}^k +k^{2}\ln k \mathcal{S}_{D, 1}^{(1)} +k^{2} \mathcal{S}_{D, 1}^{(2)} + \mathcal{O}(k^4 \ln k),
\end{equation}
where
\begin{align*}
\mathcal{S}_{D, j}^{(1)} [\partialsi](x) &= \int_{\partial D} b_j|x-y|^{2j} \partialsi(y)d\sigma(y),\\
\mathcal{S}_{D, j}^{(2)} [\partialsi](x) &= \int_{\partial D} |x-y|^{2j}(b_j\ln|x-y|+c_j)\partialsi(y)d\sigma(y).
\end{align*}
Turning to the expansion of $\left(\mathcal{K}_D^k\right)^*$, we have
\begin{equation} \label{eq:Kexpansion}
\left(\mathcal{K}_{D}^k\right)^* = \mathcal{K}_{D}^* +k^{2}\ln k \mathcal{K}_{D, 1}^{(1)}+k^{2} \mathcal{K}_{D, 1}^{(2)} + \mathcal{O}(k^4 \ln k),
\end{equation}
where
\begin{align*}
\mathcal{K}_{D, j}^{(1)} [\partialsi](x) &= \int_{\partial D} b_j\dfrac{\partialartial |x-y|^{2j}}{\partialartial \nu(x)}\partialsi(y)d\sigma(y),\\
\mathcal{K}_{D, j}^{(2)} [\partialsi](x) &= \int_{\partial D} \dfrac{\partialartial \left( |x-y|^{2j}(b_j\ln|x-y|+c_j)\right)}{\nu(x)}\partialsi(y)d\sigma(y).
\end{align*}
The operator $\hat{S}_D^k$ is known to be invertible for any $k$ \cite{MaCMiPaP}. From this follows that $\mathcal{S}_D^k$ is invertible for $k$ small enough. For later reference, we conclude this section by defining the constant $a$ as
$$a = \frac{\gamma_0+\langle \partialsi_0, \partialhi_0\rangle \eta_{k_b}}{\gamma_0+\langle \partialsi_0, \partialhi_0\rangle \eta_{k_w}}.$$
\subsection{Subwavelength resonance of a bubble}\label{subsec:bandgap}
Here, we briefly review the subwavelength resonance of a bubble as described in \cite{first}.
Assume that the uncoated bubble occupies the bounded and simply connected domain $D$ with $\partialartial D \in C^{1,s}$ for some $0<s<1$. We denote by $\rho_b$ and $\kappa_b$ the density and the bulk modulus of the air inside the bubble, respectively. We let $\rho_w$ and $\kappa_w$ be the corresponding parameters for the water. We introduce the variables
\begin{equation*}
v_w = \sqrt{\frac{\kappa_w}{\rho_w}}, \quad v_b = \sqrt{\frac{\kappa_b}{\rho_b}}, \quad k_w= \frac{\omega}{v_w} \quad \text{and} \quad k_b= \frac{\omega}{v_b}
\end{equation*}
which represent the speed of sound outside and inside the bubble, and the wavenumber outside and inside the bubble, respectively. Also, $\omega$ means the operating frequency of acoustic waves.
We also introduce the dimensionless contrast parameter
\begin{equation*}
\delta = \frac{\rho_b}{\rho_w}.
\end{equation*}
By choosing proper physical units, we may assume that the size of the bubble is of order one. We assume that the wave speeds outside and inside the bubbles are comparable to each other and that there is a large contrast in the density, that is, $$\delta \ll 1 \quad \text{and} \quad v_b, v_w = \mathcal{O}(1).$$
We consider the problem
\begin{equation} \label{eq-scattering-single}
\left\{
\begin{array} {ll}
&\displaystyle \nabla \cdot \frac{1}{\rho_w} \nabla u+ \frac{\omega^2}{\kappa_w} u = 0 \quad \text{in} \quad \mathbb{R}^2 \backslash D, \\
\noalign{
}
&\displaystyle \nabla \cdot \frac{1}{\rho_b} \nabla u+ \frac{\omega^2}{\kappa_b} u = 0 \quad \text{in} \quad D, \\
\noalign{
}
&\displaystyle u|_{+} -u|_{-} =0 \quad \text{on} \quad \partialartial D , \\
\noalign{
}
& \displaystyle \frac{1}{\rho_w} \frac{\partialartial u}{\partialartial \nu} \bigg|_{+} - \frac{1}{\rho_b} \frac{\partialartial u}{\partialartial \nu} \bigg|_{-} =0 \quad \text{on} \quad \partialartial D, \\
& u \ \text{satisfies the Sommerfeld radiation condition.}
\end{array}
\right.
\end{equation}
A resonance frequency to this problem is a complex number $\omega$ with negative imaginary part, such that a nonzero solution to equation \eqnref{eq-scattering-single} exists. In \cite{first}, it is proved that there exists a resonance frequency of subwavelength scale for this problem. The solution of \eqref{eq-scattering-single} has the following form:
\begin{equation} \label{Helm-solution}
u =
\begin{cases}
\mathcal{S}_{D}^{k_w} [\varphi]\quad & \text{in} ~ \mathbb{R}^2 \setminus \overline{D},\\
\mathcal{S}_{D}^{k_b} [\partialsi] &\text{in} ~ {D},
\end{cases}
\end{equation}
for some densities $\varphi, \partialsi \in L^2(\partial D)$. Using the jump relations for the single layer potentials, one can show that~\eqref{eq-scattering-single} is equivalent to the boundary integral equation
\begin{equation} \label{eq-boundary}
M_0(\omega, \delta)[\Phi] =0,
\end{equation}
where
\[
M_0(\omega, \delta) =
\begin{pmatrix}
\mathcal{S}_D^{k_b} & -\mathcal{S}_D^{k_w} \\
-\frac{1}{2}+ \left(\mathcal{K}_D^{k_b}\right)^*& -\delta\left( \frac{1}{2}+ \left(\mathcal{K}_D^{k_w}\right)^*\right)
\end{pmatrix},
\,\, \Phi =
\begin{pmatrix}
\varphi \\
\partialsi
\end{pmatrix}.
\]
Since it can be shown that $\omega=0$ is a characteristic value for the operator-valued analytic function $M_0(\omega,0)$, we can conclude the following result by the Gohberg-Sigal theory \cite{MaCMiPaP, Gohberg1971}.
\begin{lem}\label{lem:GSchar}
For any $\delta$ sufficiently small, there exists a characteristic value
$\omega_M= \omega_M(\delta)$ to the operator-valued analytic function
$M_0(\omega, \delta)$
such that $\omega_M(0)=0$ and
$\omega_M$ depends on $\delta$ continuously.
\end{lem}
In \cite{first}, an asymptotic formula for this characteristic value is computed. The formula in two dimensions is corrected in the following theorem.
\begin{thm} \label{thm:single}
In the quasi-static regime, there exists resonances for a single bubble. Their leading order terms are given by the roots of the following equation:
\begin{equation*}
\omega^2 \ln \omega +
\left[\left(1 + \frac{c_1}{b_1}-\ln v_b\right) + \frac{2\partiali\gamma_0 }{(\partialsi_0, \partialhi_0)} \right] \omega^2
- \frac{v_b^2}{4Vol(D)} \frac{a \delta}{b_1} =0,
\end{equation*}
where the constants $b_1, c_1, \gamma_0$ and $a$ are defined in Section \ref{sec:layerpot}.
\end{thm}
The root with positive real part is known as the \emph{Minnaert resonance} frequency, and will be denoted by $\omega_M = \omega_M(\delta)$.
\section{Encapsulated bubble: problem formulation} \label{sec-problem}
Consider now an encapsulated bubble, in which case $D$ is coated by a thin layer $D_l$ with a characteristic thickness $\varepsilon$. Let $D_d = D_l \cup \overline{D}$ be the encapsulated bubble. We consider the following problem:
\begin{equation} \label{eq:scattering}
\left\{
\begin{array} {ll}
&\displaystyle \nabla \cdot \frac{1}{\rho_w} \nabla u+ \frac{\omega^2}{\kappa_w} u = 0 \quad \text{in} \quad \mathbb{R}^2 \backslash D_d, \\
\noalign{
}
&\displaystyle \nabla \cdot \frac{1}{\rho_b} \nabla u+ \frac{\omega^2}{\kappa_b} u = 0 \quad \text{in} \quad D, \\
\noalign{
}
&\displaystyle \nabla \cdot \frac{1}{\rho_l} \nabla u+ \frac{\omega^2}{\kappa_l} u = 0 \quad \text{in} \quad D_l, \\
\noalign{
}
&\displaystyle u|_{+} -u|_{-} =0 \quad \text{on} \quad \partialartial D \cup \partialartial D_d , \\
\noalign{
}
& \displaystyle \frac{1}{\rho_l} \frac{\partialartial u}{\partialartial \nu} \bigg|_{+} - \frac{1}{\rho_b} \frac{\partialartial u}{\partialartial \nu} \bigg|_{-} =0 \quad \text{on} \quad \partialartial D, \\
& \displaystyle \frac{1}{\rho_w} \frac{\partialartial u}{\partialartial \nu} \bigg|_{+} - \frac{1}{\rho_l} \frac{\partialartial u}{\partialartial \nu} \bigg|_{-} =0 \quad \text{on} \quad \partialartial D_d, \\
& u \ \text{satisfies the Sommerfeld radiation condition.}
\end{array}
\right.
\end{equation}
Here, $\kappa_l$ and $\rho_l$ are the bulk modulus and density of the thin layer. Furthermore, define the two density contrast parameters $\delta_{bl}$ and $\delta_{lw}$ as
$$\delta_{bl} = \frac{\rho_b}{\rho_l}, \qquad \delta_{lw} = \frac{\rho_l}{\rho_w}.$$
Observe that $\delta = \delta_{bl}\delta_{lw}$. We will consider the case when $\delta_{bl}$ is small while $\delta_{lw}$ is of order 1, and that all wave speeds are of order one, that is,
$$
\delta_{bl} \ll 1, \ \delta_{lw} = \mathcal{O}(1) \quad \text{and} \quad v_b, v_l, v_w = \mathcal{O}(1).
$$
In this paper, we want to show that, by encapsulating the bubble $D$, there is a specific frequency $\omega_\varepsilon$ at which a non-trivial solution to the problem \eqnref{eq:scattering} exists. Moreover, we want to find an asymptotic formula for the frequency $\omega_\varepsilon$ when $\varepsilon$ is small.
\subsection{Integral representation of the solution}
We seek a solution $u(x)$ of the form
\begin{equation} \label{eq:sol}
u(x) = \begin{cases}
\mathcal{S}_{D}^{k_b}[\partialhi_1](x) \quad &x\in D, \\
\mathcal{S}_{D}^{k_l}[\partialhi_2](x) + S_{D_d}^{k_l} [\partialhi_3](x) & x\in D_l, \\
\mathcal{S}_{D_d}^{k_w}[\partialhi_4](x) & x \in \mathbb{R}^2\setminus \overline{D_d}.
\end{cases}
\end{equation}
A solution of this form satisfies the differential equation in \eqnref{eq:scattering}. Using the boundary conditions and the jump relations, it can be shown that the problem \eqnref{eq:scattering} admits a nonzero solution if and only if the layer densities $\partialhi_1,...,\partialhi_4$ are a nonzero solution to
\begin{equation} \label{eq:inteq}
\mathcal{A}(\omega, \varepsilon, \delta)\Phi = 0,
\end{equation}
where $\Phi= \left(\begin{smallmatrix}\partialhi_1\\\partialhi_2 \\\partialhi_3\\\partialhi_4\end{smallmatrix}\right)$ and
\begin{equation*}
\mathcal{A}(\omega, \varepsilon, \delta) =
\begin{pmatrix}
\mathcal{S}_{D}^{k_b} & -\mathcal{S}_{D}^{k_l} & -\mathcal{S}_{D,D_d}^{k_l} & 0 \\
0 & \mathcal{S}_{D_d,D}^{k_l} & \mathcal{S}_{D_d}^{k_l} & -\mathcal{S}^{k_w}_{D_d} \\
-\frac{1}{2}I+ \left(\mathcal{K}_{D}^{k_b}\right)^*& -\delta_{bl}\left( \frac{1}{2}I+ \left(\mathcal{K}_{D}^{k_l}\right)^*\right) & -\delta_{bl} \frac{\partialartial \mathcal{S}_{D,D_d}^{k_l}}{\partialartial \nu} & 0 \\
0 & \frac{\partialartial \mathcal{S}_{D_d,D}^{k_l}}{\partialartial \nu} & -\frac{1}{2}I+ \left(\mathcal{K}_{D_d}^{k_l}\right)^* & -\delta_{lw}\left( \frac{1}{2}I+ \left(\mathcal{K}_{D_d}^{k_w}\right)^*\right)
\end{pmatrix}.
\end{equation*}
Here the operator $\mathcal{S}_{D_d,D}^{k_w} = \mathcal{S}_{D}^{k_w}|_{x\in \partialartial D_d}$ is the restriction of $\mathcal{S}_{D}^{k_w}$ onto $\partialartial D_d$ and similarly for $\mathcal{S}_{D,D_d}^{k_w}$. Define $\mathcal{H} = L^2(\partialartial D) \times L^2(\partialartial D_d)\times L^2(\partialartial D) \times L^2(\partialartial D_d)$ and $\mathcal{H}_1 = H^1(\partialartial D) \times H^1(\partialartial D_d)\times L^2(\partialartial D) \times L^2(\partialartial D_d)$. It is clear that $\mathcal{A}$ is a bounded linear operator from $\mathcal{H}$ to $\mathcal{H}_1$, \textit{i.e.}{}, $\mathcal{A} \in \mathcal{L}(\mathcal{H},\mathcal{H}_1)$.
\section{Asymptotic analysis} \label{sec:analysis}
In this section we expand the operator $\mathcal{A}(\omega, \varepsilon, \delta)$ in terms of the small parameters $\varepsilon,\omega$ and $\delta$. Using these expansions, we derive a formula for the perturbation $\omega_\varepsilon-\omega_M$, which represents the shift of
the resonance of the encapsulated bubble $\omega_\varepsilon$ away
from the resonance of the uncoated bubble, that
is, the Minnaert resonance frequency $\omega_M$. The key idea involves the use of a pole-pencil decomposition of the leading order term in the asymptotic expansion of $\mathcal{A}$ in terms of $\varepsilon$, followed by the application of the generalized argument principle to find the characteristic value.
\subsection{Expansions as $\varepsilon \rightarrow 0$} \label{sec:expansions}
Observe that the mapping $p: \partialartial D \rightarrow \partialartial D_d, \ p(x) = x+\varepsilon \nu_x$ is bijective. Let $x,y\in \partialartial D$ and let $\widetilde{x} = p(x) \in \partialartial D_d$ and $\widetilde{y} = p(y) \in \partialartial D_d$. Define $f: L^2( \partialartial D) \rightarrow L^2(\partialartial D_d), \ f(\partialhi)(\widetilde x) = \partialhi ( p^{-1}(\widetilde x)) $, and for a surface density $\partialhi$ on $\partialartial D$, define $\widetilde{\partialhi} = f(\partialhi)$ on $\partialartial D_d$.
We define the signed curvature $\tau = \tau(x), x\in \partialartial D$ in the following way. Let $x = x(t)$ be a parametrization of $\partialartial D$ by arc length. Then define $\tau$ by
$$
\frac{d^2}{dt^2}x(t) = -\tau \nu_x.
$$
Recall that $\nu_x$ is defined as the \emph{outward} normal, and observe that $\tau$ is independent of the orientation of $\partialartial D$. The following proposition gives the expansion of the operators $\mathcal{S}^k_{D_d}, \mathcal{S}^k_{D,D_d}$ and $\mathcal{S}^k_{D_d,D}$ for small $\varepsilon$.
\begin{prop}\label{prop:asympsingle}
Let $k>0$. Let $\partialhi \in L^2(\partialartial D)$ and let $x,y,\widetilde{x},\widetilde{y},\widetilde{\partialhi}$ be as above. Then
\begin{equation} \label{eq:asympSdD}
\mathcal{S}_{D_d,D}^{k}[\partialhi](\widetilde{x}) = \mathcal{S}_D^k[\partialhi](x) +\varepsilon \left(\frac{1}{2}I + \left(\mathcal{K}_D^k\right)^*\right)[\partialhi](x) + o(\varepsilon),
\end{equation}
\begin{equation} \label{eq:asympSd}
\mathcal{S}_{D_d}^k[\widetilde{\partialhi}](\widetilde{x}) = \mathcal{S}_D^k[\partialhi](x) + \varepsilon \left(\mathcal{K}_D^k + \left(\mathcal{K}_D^k\right)^*\right)[\partialhi](x) + \varepsilon\mathcal{S}_D^k[\tau\partialhi](x) + o(\varepsilon),
\end{equation}
\begin{equation} \label{eq:asympSDd}
\mathcal{S}_{D,D_d}^k[\widetilde{\partialhi}](x) = \mathcal{S}_D^k[\partialhi](x) + \varepsilon \left(\frac{1}{2}I+ \mathcal{K}_D^k \right)[\partialhi](x) + \varepsilon\mathcal{S}_D^k[\tau\partialhi](x) + o(\varepsilon),
\end{equation}
\end{prop}
Here the $o(\varepsilon)$ terms are in the pointwise $L^2$ sense, \textit{i.e.}{}, for any fixed $\partialhi$ we have
\begin{equation*}
\lim_{\varepsilon \rightarrow 0} \frac{1}{\varepsilon}\left\| \mathcal{S}_{D_d,D}^k[\partialhi](\widetilde{x}) -\left( \mathcal{S}_D^k[\partialhi](x) -\varepsilon \left(-\frac{1}{2}I + \left(\mathcal{K}_D^k\right)^*\right)[\partialhi]\right) \right\|_{L^2(\partialartial D)} = 0,
\end{equation*}
and similarly for the other expansions.
\begin{proof}
The proof is given in \cite{asymptotics}, but in our case with the Taylor expansions in the $L^2$ sense (as given in \cite{weakdifffcn}, Theorem 3.4.2).
\end{proof}
\begin{prop} \label{prop:asympK}
Let $\partialhi \in L^2(\partialartial D)$ and let $x,y,\widetilde{x},\widetilde{y},\widetilde{\partialhi}$ be as above. Then
\begin{equation} \label{eq:asympK}
\left(\mathcal{K}_{D_d}^k\right)^*[\widetilde{\partialhi}](\widetilde{x}) = \left(\mathcal{K}_D^k\right)^*[\partialhi](x) + \varepsilon\mathcal{K}_1^k[\partialhi](x) + o(\varepsilon).
\end{equation}
Let $\tau$ be the curvature of $\partialartial D$. Then $\mathcal{K}_1^k$ is given by
\begin{equation*}
\mathcal{K}_1^k = \left(\mathcal{K}_D^k\right)^*[\tau\partialhi](x) - \tau(x) \left(\mathcal{K}_D^k\right)^*[\partialhi](x) + \frac{\partialartial \mathcal{D}_D^k}{\partialartial \nu}[\partialhi](x) - \frac{\partialartial^2}{\partialartial T^2}\mathcal{S}_D^k[\partialhi](x) - k^2\mathcal{S}_D^k[\partialhi](x),
\end{equation*}
where $\frac{\partialartial^2}{\partialartial T^2}$ denotes the second tangential derivative, which is independent of the orientation of $\partialartial D$.
\end{prop}
\begin{proof}
The explicit expansion of $\mathcal{K}_{D_d}^*$ is derived in \cite{MaCMiPaP} for the Laplace case. We compute this in our case using similar arguments. As derived in \cite{MaCMiPaP}, we have
\begin{equation} \label{eq:dsigma}
\: \mathrm{d} \sigma(\widetilde{y}) = \left( 1+\varepsilon \tau(y) \right) \: \mathrm{d} \sigma(y),
\end{equation}
Because the shapes of $\partialartial D$ and $\partialartial D_d$ are the same, we have $\widetilde \nu_{\widetilde x} = \nu_x$ for all $x\in \partialartial D$. Furthermore, we have
\begin{equation}\label{eq:exp1}
\frac{\partialartial}{\partialartial \widetilde{\nu}_{\widetilde{x}}} H_0(k|\widetilde{x}-\widetilde{y}|) = kH_0'(k|\widetilde{x}-\widetilde{y}|)\frac{\langle \widetilde{x}-\widetilde{y},\nu_x \rangle}{|\widetilde{x}-\widetilde{y}|}.
\end{equation}
We have
$$ |\widetilde{x}-\widetilde{y}|^2 = |x-y|^2 +2\varepsilon \langle x-y, \nu_x-\nu_y\rangle + \varepsilon^2|\nu_x-\nu_y|^2,$$
and therefore the following expansions hold as $\varepsilon \rightarrow 0$,
$$ |\widetilde{x}-\widetilde{y}| = |x-y| + \varepsilon\frac{\langle x-y, \nu_x-\nu_y \rangle}{|x-y|} + \mathcal{O}(\varepsilon^2),$$
and
$$ \frac{1}{|\widetilde{x}-\widetilde{y}|} = \frac{1}{|x-y|} - \varepsilon\frac{\langle x-y, \nu_x-\nu_y \rangle}{|x-y|^3} + \mathcal{O}(\varepsilon^2).$$
We therefore expand
\begin{align} \label{eq:exp2}
H_0'(k|\widetilde{x}-\widetilde{y}|) &= H_0'\left(k|x-y|\right) + \varepsilon k H_0''\left(k|x-y|\right)\frac{\langle x-y,\nu_x-\nu_y\rangle}{|x-y|} + \mathcal{O}(\varepsilon^2),
\end{align}
and
\begin{align} \label{eq:exp3}
\frac{\langle \widetilde{x}-\widetilde{y},\nu_x \rangle}{|\widetilde{x}-\widetilde{y}|} = \frac{\langle {x}-{y},\nu_{{x}} \rangle}{|{x}-{y}|} + \varepsilon \left( \frac{\langle \nu_x-\nu_y, \nu_x \rangle}{|x-y|}-\frac{\langle x-y,\nu_x-\nu_y \rangle \langle x-y,\nu_x\rangle }{|x-y|^3} \right) + \mathcal{O}(\varepsilon^2).
\end{align}
Using the expansions \eqnref{eq:dsigma}, \eqnref{eq:exp1}, \eqnref{eq:exp2} and \eqnref{eq:exp3} we obtain
\begin{align*}
\left(\mathcal{K}_{D_d}^k\right)^*[\widetilde{\partialhi}](\widetilde{x}) &= -\frac{i}{4}\int_{D_d} \frac{\partialartial}{\partialartial \widetilde{\nu}_{\widetilde{x}}} H_0'(k|\widetilde{x}-\widetilde{y}|) \widetilde{\partialhi}({\widetilde{y}}) \: \mathrm{d}\sigma(\widetilde{y}) \\
&= -\frac{i}{4}\int_{\partialartial D} k H_0'\left(k|x-y|\right)\frac{\langle x-y,\nu_{x} \rangle}{|x-y|} \partialhi(y) \: \mathrm{d} \sigma(y) \\
& \quad + \varepsilon \mathcal{B}igg[-\frac{i}{4}\int_{\partialartial D} k H_0'\left(k|x-y|\right)\frac{\langle x-y,\nu_{x} \rangle}{|x-y|} \tau(y) \partialhi(y) \: \mathrm{d} \sigma(y) \\
& \qquad \qquad -\frac{i}{4}\int_{\partialartial D} k^2H_0''(k|x-y|) \frac{\langle x-y,\nu_x-\nu_y\rangle \langle x-y,\nu_x \rangle }{|x-y|^2}\partialhi(y)\: \mathrm{d} \sigma(y) \\
& \qquad \qquad -\frac{i}{4}\int_{\partialartial D} k H_0'\left(k|x-y|\right)\left( \frac{\langle \nu_x-\nu_y,\nu_{x} \rangle}{|x-y|} -\frac{\langle x-y,\nu_x-\nu_y\rangle \langle x-y,\nu_x \rangle }{|x-y|^3} \right) \partialhi(y) \: \mathrm{d} \sigma(y) \mathcal{B}igg] \\
& \quad + \mathcal{O}(\varepsilon^2),
\end{align*}
giving us the intermediate result
\begin{multline} \label{eq:intermediate}
\left(\mathcal{K}_{D_d}^k\right)^*[\widetilde{\partialhi}](\widetilde{x}) = \left(\mathcal{K}_d^k\right)^*[\partialhi](x) + \varepsilon\mathcal{B}igg[\left(\mathcal{K}_d^k\right)^*[\tau\partialhi](x) \\
-\frac{i}{4}\int_{\partialartial D} k^2H_0''(k|x-y|) \frac{\langle x-y,\nu_x-\nu_y\rangle \langle x-y,\nu_x \rangle }{|x-y|^2}\partialhi(y)\: \mathrm{d} \sigma(y) \\
-\frac{i}{4}\int_{\partialartial D} k H_0'\left(k|x-y|\right)\left( \frac{\langle \nu_x-\nu_y,\nu_{x} \rangle}{|x-y|} -\frac{\langle x-y,\nu_x-\nu_y\rangle \langle x-y,\nu_x \rangle }{|x-y|^3} \right) \partialhi(y) \: \mathrm{d} \sigma(y) \mathcal{B}igg]+ \mathcal{O}(\varepsilon^2).
\end{multline}
Observe that
\begin{align*}
\frac{\partialartial \mathcal{D}_D^k}{\partialartial \nu}[\partialhi](x) = &-\frac{i}{4}\int_{\partialartial D} k^2H_0''(k|x-y|) \frac{\langle x-y,-\nu_y\rangle \langle x-y,\nu_x \rangle }{|x-y|^2}\partialhi(y)\: \mathrm{d} \sigma(y) \\
&-\frac{i}{4}\int_{\partialartial D} k H_0'\left(k|x-y|\right)\left( \frac{\langle -\nu_y,\nu_{x} \rangle}{|x-y|} -\frac{\langle x-y,-\nu_y\rangle \langle x-y,\nu_x \rangle }{|x-y|^3} \right) \partialhi(y) \: \mathrm{d} \sigma(y), \quad x\in \partialartial D,
\end{align*}
and that
\begin{align*}
\frac{\partialartial^2}{\partialartial T^2}\mathcal{S}_D^k[\partialhi](x) = &-\frac{i}{4}\int_{\partialartial D} k^2H_0''(k|x-y|) \frac{\langle x-y,T_x\rangle^2}{|x-y|^2}\partialhi(y)\: \mathrm{d} \sigma(y) \\
&-\frac{i}{4}\int_{\partialartial D} k H_0'\left(k|x-y|\right)\left( \frac{1 - \tau(x)\langle x-y,\nu_{x} \rangle}{|x-y|} -\frac{\langle x-y,T_x\rangle^2}{|x-y|^3} \right)\partialhi(y) \: \mathrm{d} \sigma(y), \quad x\in \partialartial D.
\end{align*}
Using these expressions in equation \eqnref{eq:intermediate}, together with the identity $|x-y|^2 = \langle x-y,\nu_x \rangle^2 + \langle x-y, T_x\rangle^2$, we obtain
\begin{multline} \label{eq:almostthere}
\left(\mathcal{K}_{D_d}^k\right)^*[\widetilde{\partialhi}](\widetilde{x}) = \left(\mathcal{K}_d^k\right)^*[\partialhi](x) + \varepsilon\mathcal{B}igg[\left(\mathcal{K}_d^k\right)^*[\tau\partialhi](x) - \tau(x) \left(\mathcal{K}_D^k\right)^*[\partialhi](x) + \frac{\partialartial \mathcal{D}_D^k}{\partialartial \nu}[\partialhi](x) - \frac{\partialartial^2}{\partialartial T^2}\mathcal{S}_D^k[\partialhi](x) \\
-\frac{i}{4}\int_{\partialartial D} k^2H_0''(k|x-y|) \partialhi(y)\: \mathrm{d} \sigma(y)
-\frac{i}{4}\int_{\partialartial D} \frac{k H_0'\left(k|x-y|\right)}{|x-y|} \partialhi(y) \: \mathrm{d} \sigma(y) \mathcal{B}igg]+ \mathcal{O}(\varepsilon^2).
\end{multline}
Applying standard relations for Bessel functions, we have the relation
$$\frac{H_0'(k|x-y|)}{k|x-y|} + H_0''(k|x-y|) = -H_0(k|x-y|).$$
Using this in equation \eqnref{eq:almostthere}, we find
\begin{multline*}
\left(\mathcal{K}_{D_d}^k\right)^*[\widetilde{\partialhi}](\widetilde{x}) = \left(\mathcal{K}_d^k\right)^*[\partialhi](x) + \varepsilon\mathcal{B}igg[\left(\mathcal{K}_d^k\right)^*[\tau\partialhi](x) - \tau(x) \left(\mathcal{K}_D^k\right)^*[\partialhi](x) + \frac{\partialartial \mathcal{D}_D^k}{\partialartial \nu}[\partialhi](x) \\
- \frac{\partialartial^2}{\partialartial T^2}\mathcal{S}_D^k[\partialhi](x) - k^2\mathcal{S}_D^k[\partialhi](x) \mathcal{B}igg]+ \mathcal{O}(\varepsilon^2),
\end{multline*}
which is the desired result.
\end{proof}
\begin{prop} \label{prop:asympderiv}
Let $\partialhi \in H^1(\partialartial D)$ and let $x,y,\widetilde{x},\widetilde{y},\widetilde{\partialhi}$ be as above. Then
\begin{equation} \label{eq:asymppSdD}
\frac{\partialartial \mathcal{S}_{D_d,D}^k[\partialhi]}{\partialartial \widetilde{\nu}} (\widetilde{x}) = \left(\frac{1}{2}I + \left(\mathcal{K}_D^k\right)^*\right)[\partialhi](x) + \varepsilon \mathcal{R}_D^k[\partialhi](x) + o(\varepsilon),
\end{equation}
\begin{equation} \label{eq:asymppSDd}
\frac{\partialartial \mathcal{S}_{D,D_d}^k[\widetilde{\partialhi}]}{\partialartial \nu} (x) = \left(-\frac{1}{2}I + \left(\mathcal{K}_D^k\right)^*\right)[\partialhi](x) + \varepsilon \mathcal{L}_D^k[\partialhi](x) + o(\varepsilon),
\end{equation}
where $\mathcal{R}_D^k$ and $\mathcal{L}_D^k$ are given by
\begin{equation*}
\mathcal{R}_D^k[\partialhi](x) = -k^2\mathcal{S}_D^k[\partialhi](x) - \tau(x)\left(\frac{1}{2}I + \left(\mathcal{K}_D^k\right)^*\right)[\partialhi](x) -\frac{\partialartial^2}{\partialartial T^2}\mathcal{S}_D^k[\partialhi](x),
\end{equation*}
\begin{equation*}
\mathcal{L}_D^k[\partialhi](x) = \left(-\frac{1}{2}I + \left(\mathcal{K}_D^k\right)^*\right)[\tau\partialhi](x) +\frac{\partialartial \mathcal{D}_D^k}{\partialartial \nu}[\partialhi](x).
\end{equation*}
\end{prop}
\begin{proof}
The proof is similar to the one given in \cite{asymptoticsderivative}, but adjusted for the Helmholtz case. Because $\partialhi\in H^1(\partialartial D)$ we have that $\mathcal{S}_D^k[\partialhi]\in H^2(\partialartial D)$. Because the normals $\widetilde{\nu}_{\widetilde{x}}$ and $\nu_x$ coincide, we have
\begin{align*}
\frac{\partialartial \mathcal{S}_{D_d,D}^k[\partialhi]}{\partialartial \widetilde{\nu}} (\widetilde{x}) &= \nu \cdot \nabla \mathcal{S}_{D_d,D}^k[\partialhi](\widetilde{x}) \\
&= \frac{\partialartial \mathcal{S}_D^k[\partialhi]}{\partialartial \nu} \bigg|_{+}(x) + \varepsilon\left(\frac{\partialartial^2}{\partialartial \nu^2}\mathcal{S}_D^k[\partialhi]\bigg|_{+}(x) \right) + o(\varepsilon).
\end{align*}
Using the Laplacian in the curvilinear coordinates defined by $T_x,\nu_x$ for $x\in \partialartial D$,
\begin{equation*}\label{eq:lapcurve}
\mathcal{D}elta = \frac{\partialartial^2}{\partialartial \nu^2} + \tau\frac{\partialartial}{\partialartial \nu} + \frac{\partialartial^2}{\partialartial T^2},
\end{equation*}
we find
\begin{equation*}
\frac{\partialartial^2}{\partialartial \nu^2}\mathcal{S}_D^k[\partialhi]\bigg|_{+}(x) = -k^2\mathcal{S}_D^k[\partialhi](x) -\tau(x)\frac{\partialartial \mathcal{S}_D^k[\partialhi]}{\partialartial \nu}\bigg|_{+}(x) - \frac{\partialartial^2 \mathcal{S}_D^k[\partialhi]}{\partialartial T^2}(x),
\end{equation*}
so equation \eqnref{eq:asymppSdD} follows using the jump relations. To derive equation \eqnref{eq:asymppSDd}, pick a function $f\in H^1(\partialartial D)$. Then there exists a solution $u$ to the Dirichlet problem
\begin{equation*}
\begin{cases}
\mathcal{D}elta u + k^2 u = 0 \quad &\text{in } D ,\\
u = f & \text{in } \partialartial D.
\end{cases}
\end{equation*}
Using duality and integration by parts in the interior region, we obtain that
\begin{align*}
\int_{\partialartial D} \frac{\partialartial \mathcal{S}_{D,D_d}^k[\widetilde{\partialhi}]}{\partialartial \nu} (x) f(x) \: \mathrm{d} \sigma(x) &=\int_{D}\left(u\mathcal{D}elta \mathcal{S}_{D,D_d}-\mathcal{S}_{D,D_d}\mathcal{D}elta u \right)\: \mathrm{d} x + \int_{\partialartial D} \mathcal{S}_{D,D_d}^k[\widetilde{\partialhi}](x)\frac{\partialartial f }{\partialartial \nu} (x) \: \mathrm{d} \sigma(x) \\
&=\int_{\partialartial D} \mathcal{S}_{D,D_d}^k[\widetilde{\partialhi}](x)\frac{\partialartial f }{\partialartial \nu} (x) \: \mathrm{d} \sigma(x) \\
&= \int_{\partialartial D_d} \mathcal{S}_{D_d,D}^k\left[\frac{\partialartial f }{\partialartial \nu}\right](\widetilde{x})\widetilde{\partialhi}(\widetilde{x}) \: \mathrm{d} \sigma(\widetilde{x}).
\end{align*}
Combining Proposition \ref{prop:asympsingle} together with \eqnref{eq:dsigma}, we find
\begin{align*}
\int_{\partialartial D_d} \mathcal{S}_{D_d,D}^k\left[\frac{\partialartial f }{\partialartial \nu}\right](\widetilde{x})\widetilde{\partialhi}(\widetilde{x}) \: \mathrm{d} \sigma(\widetilde{x}) &= \int_{\partialartial D}\left( \mathcal{S}_D^k\left[\frac{\partialartial f }{\partialartial \nu}\right] (x) +\varepsilon \left(\frac{1}{2}I + \left(\mathcal{K}_D^k\right)^*\right)\left[\frac{\partialartial f }{\partialartial \nu}\right](x)\right)\partialhi(x)\left(1+\varepsilon \tau(x)\right)\: \mathrm{d} \sigma(x) + o(\varepsilon) \\
&= \int_{\partialartial D}\mathcal{S}_D^k\left[\frac{\partialartial f }{\partialartial \nu}\right] \partialhi\: \mathrm{d} \sigma +\varepsilon \int_{\partialartial D}\left( \tau(x)\mathcal{S}_D^k + \frac{1}{2}I + \left(\mathcal{K}_D^k\right)^*\right)\left[\frac{\partialartial f }{\partialartial \nu}\right]\partialhi\: \mathrm{d} \sigma + o(\varepsilon) \\
& = \int_{\partialartial D} \mathcal{S}_D^k\left[\partialhi\right]\frac{\partialartial f }{\partialartial \nu} \: \mathrm{d} \sigma +\varepsilon \int_{\partialartial D}\left(\mathcal{S}_D^k[\tau\partialhi] +\left(\frac{1}{2}I + \mathcal{K}_D^k\right)\left[\partialhi\right]\right)\frac{\partialartial f }{\partialartial \nu}\: \mathrm{d} \sigma + o(\varepsilon) \\
&= \int_{\partialartial D} \frac{\partialartial \mathcal{S}_D^k }{\partialartial \nu}[\partialhi]\bigg|_{-}f \: \mathrm{d} \sigma +\varepsilon \int_{\partialartial D}\left(\frac{\partialartial \mathcal{S}_D^k }{\partialartial \nu}\bigg|_{-}[\tau\partialhi] +\frac{\partialartial \mathcal{D}_D^k}{\partialartial \nu} \left[\partialhi\right]\right)f\: \mathrm{d} \sigma + o(\varepsilon).
\end{align*}
Therefore, \eqnref{eq:asymppSDd} follows using the jump formulas.
\end{proof}
\subsection{Expansion of $\mathcal{A}$}
Observe that Proposition \ref{prop:asympderiv} assumes $\partialhi \in H^1(\partialartial D)$. Define by $\mathcal{H}_2 = H^1(\partialartial D) \times H^1(\partialartial D_d) \times H^1(\partialartial D) \times H^1(\partialartial D_d)$. We seek the solution to equation \eqnref{eq:inteq}, and it is clear that this solution satisfies $\Phi \in \mathcal{H}_2$. In the following, we will consider $\mathcal{A}$ as an operator on the space $\mathcal{H}_2$.
Define the bijection $F: \left(H^1(\partialartial D)\right)^4 \rightarrow \mathcal{H}_2, F = (id, f, id, f)$, where $f$ is defined as in Section \ref{sec:expansions}. Using the asymptotic expansions \eqnref{eq:asympSdD}, \eqnref{eq:asympSd}, \eqnref{eq:asympSDd}, \eqnref{eq:asympK}, \eqnref{eq:asymppSdD} and \eqnref{eq:asymppSDd}, we can expand the operator $\mathcal{A}$ as
\begin{equation*}
\mathcal{A}(\omega, \varepsilon, \delta) = F \circ \left(\hat{\mathcal{A}}_0(\omega,\delta) + \varepsilon \hat{\mathcal{A}}_1(\omega,\delta) + o(\varepsilon)\right)\circ F^{-1},
\end{equation*}
where
\begin{equation*} \label{eq:A0}
\hat{\mathcal{A}}_0(\omega,\delta) =
\begin{pmatrix}
\mathcal{S}_{D}^{k_b} & -\mathcal{S}_{D}^{k_l} & -\mathcal{S}_{D}^{k_l} & 0 \\
0 & \mathcal{S}_{D}^{k_l} & \mathcal{S}_{D}^{k_l} & -\mathcal{S}_D^{k_w} \\
-\frac{1}{2}I+ (\mathcal{K}_{D}^{k_b})^*& -\delta_{bl}\left( \frac{1}{2}I+ (\mathcal{K}_{D}^{k_l})^*\right) & -\delta_{bl}\left( -\frac{1}{2}I+ (\mathcal{K}_{D}^{k_l})^*\right) & 0 \\
0 & \frac{1}{2}I+ (\mathcal{K}_D^{k_l})^* & -\frac{1}{2}I+ (\mathcal{K}_D^{k_l})^* & -\delta_{lw}\left( \frac{1}{2}I+ \left(\mathcal{K}_D^{k_w}\right)^*\right)
\end{pmatrix},
\end{equation*}
and
\begin{equation*} \label{eq:A1}
\hat{\mathcal{A}}_1(\omega,\delta) =
\begin{pmatrix}
0 & 0 & -\left(\frac{1}{2}I+ \mathcal{K}_D^{k_l} +\mathcal{S}_D^{k_l}[\tau\cdot] \right) & 0 \\
0 & \frac{1}{2}I + \left(\mathcal{K}_D^{k_l}\right)^* & \mathcal{K}_D^{k_l} + \left(\mathcal{K}_D^{k_l}\right)^* + \mathcal{S}_D^{k_l}[\tau\cdot] & -\left(\mathcal{K}_D^{k_w} + \left(\mathcal{K}_D^{k_w}\right)^* + \mathcal{S}_D^{k_w}[\tau\cdot]\right) \\
0 & 0 & -\delta_{bl} \mathcal{L}_D^{k_l} & 0 \\
0 & \mathcal{R}_D^{k_l} & \mathcal{K}_1^{k_l} & -\delta_{lw}\mathcal{K}_1^{k_w}
\end{pmatrix}.
\end{equation*}
It is clear that $\omega_\varepsilon$ is a characteristic value for $\mathcal{A}$ if and only if $\omega_\varepsilon$ is a characteristic value for $F^{-1}\circ\mathcal{A}\circ F = \hat{\mathcal{A}}_0(\omega,\delta) + \varepsilon \hat{\mathcal{A}}_1(\omega,\delta) + o(\varepsilon)$. Using elementary row reductions, it is clear that this operator has the same characteristic values as
\begin{equation*}
\mathcal{A}_0 + \varepsilon \mathcal{A}_1 + o(\varepsilon),
\end{equation*}
where
\begin{equation*} \label{eq:Ahat0}
\mathcal{A}_0(\omega,\delta) =
\begin{pmatrix}
\mathcal{S}_{D}^{k_b} & 0 & 0 & -\mathcal{S}_D^{k_w} \\
\mathcal{S}_{D}^{k_b} & -\mathcal{S}_{D}^{k_l} & -\mathcal{S}_{D}^{k_l} & 0 \\
0 & \frac{1}{2}I+ (\mathcal{K}_D^{k_l})^* & -\frac{1}{2}I+ (\mathcal{K}_D^{k_l})^* & -\delta_{lw}\left( \frac{1}{2}I+ \left(\mathcal{K}_D^{k_w}\right)^*\right) \\
-\frac{1}{2}I+ (\mathcal{K}_{D}^{k_b})^* & 0 & 0 & -\delta\left( \frac{1}{2}I+ \left(\mathcal{K}_D^{k_w}\right)^*\right)\\
\end{pmatrix},
\end{equation*}
and
\begin{equation*} \label{eq:Ahat1}
\mathcal{A}_1(\omega,\delta) =
\begin{pmatrix}
0 & \frac{1}{2}I + \left(\mathcal{K}_D^{k_l}\right)^* & -\frac{1}{2}I + \left(\mathcal{K}_D^{k_l}\right)^* & -\left(\mathcal{K}_D^{k_w} + \left(\mathcal{K}_D^{k_w}\right)^* + \mathcal{S}_D^{k_w}[\tau\cdot]\right) \\
0 & 0 & -\left(\frac{1}{2}I+ \mathcal{K}_D^{k_l} +\mathcal{S}_D^{k_l}[\tau\cdot] \right) & 0 \\
0 & \mathcal{R}_D^{k_l} & \mathcal{K}_1^{k_l} & -\delta_{lw}\mathcal{K}_1^{k_w} \\
0 & \delta_{bl} \mathcal{R}_D^{k_l} & \delta_{bl}\left( \tau(x)I + \mathcal{R}_D^{k_l} \right) & -\delta\mathcal{K}_1^{k_w} \\
\end{pmatrix}.
\end{equation*}
In these equations, recall that the three contrast parameters are related by $\delta = \delta_{bl}\delta_{lw}$. The following proposition is one of the key steps in computing the resonance frequency.
\begin{prop}\label{prop:polepencil}
Let $\omega_M$ be the Minnaert resonance for the uncoated bubble. For $\omega$ in a punctured neighbourhood of $\omega_M$ and for $\delta$ small enough, $\mathcal{A}_0(\omega,\delta)$ is an injective operator. Furthermore, the following pole-pencil decomposition holds
\begin{equation} \label{pencil}
\big(\mathcal{A}_0(\omega,\delta)\big)^{-1} = \frac{L}{\omega-\omega_M} + R(\omega),
\end{equation}
where $R(\omega)$ is holomorphic, $L: \left(L^2(\partialartial D)\right)^4 \rightarrow \ker(\mathcal{A}_0(\omega_M,\delta))$ and $\dim\ker(\mathcal{A}_0(\omega_M,\delta)) = 1$.
\end{prop}
\begin{proof}
The first and the fourth row of $\mathcal{A}_0$ decouples, which leads to the matrices
\begin{equation*}
M_0=\begin{pmatrix}
\mathcal{S}_{D}^{k_b} & -\mathcal{S}_D^{k_w} \\
-\frac{1}{2}I+ (\mathcal{K}_{D}^{k_b})^* & -\delta\left( \frac{1}{2}I+ \left(\mathcal{K}_D^{k_w}\right)^*\right)\\
\end{pmatrix}
\end{equation*}
and
\begin{equation*}
M_d=\begin{pmatrix}
-\mathcal{S}_{D}^{k_l} & -\mathcal{S}_{D}^{k_l} \\
\frac{1}{2}I+ \left(\mathcal{K}_D^{k_l}\right)^* & -\frac{1}{2}I+ \left(\mathcal{K}_D^{k_l}\right)^*\\
\end{pmatrix}.
\end{equation*}
$M_0$ is the operator which corresponds to the uncoated bubble, and is known to have a discrete set of characteristic values \cite{first}. Hence there is a punctured neighbourhood of $\omega_M$ where $M_0$ is invertible. It is easily shown that $M_d$ is invertible if and only if $\mathcal{S}_D^{k_w}$ is invertible. Because $\omega_M$ is of subwavelength scale, and tends to zero as $\delta \rightarrow 0$, $\mathcal{S}_D^{k_w}$ is invertible for $\delta$ small enough. It follows that $\mathcal{A}_0$ is invertible for $\omega$ in a punctured neighbourhood of $\omega_M$. Because $\ker M_0(\omega_M,\delta)$ is one-dimensional \cite{first} and because $M_d$ is invertible, it follows that $\ker(\mathcal{A}_0(\omega_M,\delta))$ is one-dimensional. Finally, because $\omega_M$ is a pole of order one for $M_0$ \cite{first}, and because $M_d$ is invertible, it follows that $\omega_M$ is a pole of order one for $\mathcal{A}_0$.
\end{proof}
Using the expansion of $\mathcal{A}$, Lemma \ref{lem:GSchar} and the observation that $\mathcal{A}_0$ has the same characteristic values as $M_0$, Gohberg-Sigal theory implies the following result.
\begin{lem} \label{lem:GScharA}
For any $\varepsilon$ and $\delta$ sufficiently small, there exists a characteristic value
$\omega_\varepsilon= \omega_\varepsilon(\varepsilon,\delta)$ to the operator-valued analytic function
$\mathcal{A}(\varepsilon,\omega, \delta)$
such that $\omega_\varepsilon(0,\delta) = \omega_M(\delta)$ and
$\omega_\varepsilon$ depends continuously on $\varepsilon$ and $\delta$.
\end{lem}
Since $\ker \mathcal{A}_0(\omega_M,\delta)$ is one-dimensional, define $\Psi$ and $\Phi$ by
\begin{align*}
\ker \mathcal{A}_0(\omega_M,\delta) &= \mathrm{span}(\Psi), \\
\ker \mathcal{A}_0^*(\omega_M,\delta) &= \mathrm{span}(\Phi).
\end{align*}
In the next sections, we compute $\Psi$, $\Phi$ and $L$.
\subsection{Computation of $\Psi$ and $\Phi$} \label{sec:psi}
We make use of the computations in \cite{first}, and asymptotically expand $\mathcal{A}_0(\omega,\delta)$ in terms of $\omega$ and $\delta$. We are interested in the case when the contrast parameter $\delta$ is small, while $\delta_{lw} = \mathcal{O}(1)$, \textit{i.e.}{} the contrast between the layer of coating and the water is of order one. Taking into consideration Lemma \ref{lem:GScharA}, we will assume $\omega$ is close to $\omega_M$, and Theorem \ref{thm:single} shows that this gives
\begin{equation} \label{eq:regime}
\omega^2\ln\omega = \mathcal{O}(\delta).
\end{equation}
Define $\mathcal{A}_0^0$ as
\begin{equation*}
\mathcal{A}_0^0 =
\begin{pmatrix}
\hat\mathcal{S}_{D}^{k_b} & 0 & 0 & -\hat\mathcal{S}_D^{k_w} \\
\hat\mathcal{S}_{D}^{k_b} & -\hat\mathcal{S}_{D}^{k_l} & -\hat\mathcal{S}_{D}^{k_l} & 0 \\
0 & \frac{1}{2}I+ (\mathcal{K}_D)^* & -\frac{1}{2}I+ (\mathcal{K}_D)^* & -\delta_{lw}\left( \frac{1}{2}I+ \left(\mathcal{K}_D\right)^*\right) \\
-\frac{1}{2}I+ (\mathcal{K}_{D})^* & 0 & 0 & 0
\end{pmatrix}.
\end{equation*}
In light of the expansion \eqnref{eq:Sexpansion}, we have $\mathcal{A}_0(\omega,\delta) = \mathcal{A}_0^0 + \mathcal{B}(\omega,\delta)$ with $\mathcal{B}(\omega,\delta) = \mathcal{O}(\delta)$. Let $\Psi_0$ be such that $\mathrm{span}\{\Psi_0\}= \ker (\mathcal{A}_0^0)$. Then $\Psi(\omega,\delta) = \Psi_0 + \mathcal{O}(\delta)$. Let us write $$\Psi_0 = \alpha_0 \left(\begin{smallmatrix}\partialsi_1 \\\partialsi_2 \\\partialsi_3 \\\partialsi_4 \end{smallmatrix}\right),$$
where $\alpha_0$ is a normalization constant. Observe that the equation $\mathcal{A}_0^0\Psi_0=0$ is equivalent to
\begin{equation*}
\begin{pmatrix}
\hat\mathcal{S}_{D}^{k_b} & -\hat\mathcal{S}_D^{k_w} \\
-\frac{1}{2}I+ \mathcal{K}_{D}^* & 0\\
\end{pmatrix}
\begin{pmatrix}
\partialsi_1 \\ \partialsi_4
\end{pmatrix}
=0 \ \ \text{and} \
\begin{pmatrix}
\hat\mathcal{S}_{D}^{k_l} & \hat\mathcal{S}_{D}^{k_l} \\
\frac{1}{2}I+ \mathcal{K}_D^* & -\frac{1}{2}I+ \mathcal{K}_D^*
\end{pmatrix}
\begin{pmatrix}
\partialsi_2 \\
\partialsi_3
\end{pmatrix}
=
\begin{pmatrix}
\hat\mathcal{S}_D^{k_b}[\partialsi_1] \\
\delta_{lw}\left(\frac{1}{2}+ \mathcal{K}_D^*\right)[\partialsi_4]
\end{pmatrix}.
\end{equation*}
As before, let $\partialhi_0 = \chi_{\partialartial D}$ and let $\partialsi_0$ be a solution to
\begin{equation*}
\left(-\frac{1}{2}I+ \mathcal{K}_{D}^*\right)\partialsi_0=0, \ \ \int_{\partialartial D} |\partialsi_0|^2 \: \mathrm{d} \sigma = 1.
\end{equation*}
Observe that $\partialsi_0$ is unique up to sign. Define by $c := \langle \partialsi_0, \partialhi_0 \rangle$. Clearly, we can choose $\partialsi_1 = \partialsi_0$. In \cite{first}, it is shown that $\partialsi_4 = a\partialsi_0$, where
$$a = \frac{\gamma_0+ c \eta_{k_b}}{\gamma_0+ c \eta_{k_w}},$$
as defined in Section \ref{sec:layerpot}.
Defining
$$a_l = \frac{\gamma_0+ c \eta_{k_b}}{\gamma_0+ c \eta_{k_l}},$$
it is easily shown that
\begin{equation}\label{eq:psi1234}
\Psi_0 = \alpha_0 \begin{pmatrix}
\partialsi_0 \\
\delta_{lw}a\partialsi_0\\
(a_l-a\delta_{lw})\partialsi_0\\
a\partialsi_0
\end{pmatrix}
\end{equation}
We now turn to $\Phi$. Define $\Phi_0$ by $\left(\mathcal{A}_0^0\right)^* \Phi_0 = 0$. Then $\Phi = \Phi_0 + \Phi_1 $, where $\Phi_1 = \mathcal{O}(\delta)$. A direct computation gives
$$
\Phi_0 = \beta_0
\begin{pmatrix}
0 \\ 0 \\ 0 \\ \partialhi_0
\end{pmatrix},
$$
where $\beta_0$ is a normalization constant. We will also need the leading order term of $\Phi_1$. It is easily seen that $\Phi_1$ has the form
$$\Phi_1 =
\begin{pmatrix}
u_1 \\ 0 \\ 0 \\ u_4
\end{pmatrix},
$$
where $u = \left(\begin{smallmatrix} u_1 \\ u_4 \end{smallmatrix} \right)$. Using the methods from \cite{first}, it is easily shown that $u$ is given by
\begin{equation} \label{eq:u}
u = -\beta_0\left(\tilde{M}_0^*\right)^{-1}\mathcal{B}_0^* \begin{pmatrix} 0\\ \partialhi_0 \end{pmatrix}.
\end{equation}
Here, $\mathcal{B}_0$ is the $2\times 2$ matrix given as the first and fourth rows and columns of $\mathcal{B}$, and $\tilde{M}_0$ is given by
$$
\tilde{M}_0^* = M_0^* + \big\langle \cdot , \left(\begin{smallmatrix} 0\\ \partialhi_0 \end{smallmatrix}\right)
\big\rangle \left(\begin{smallmatrix} \partialsi_0\\ a\partialsi_0 \end{smallmatrix}\right).
$$
From the expansions given in \eqnref{eq:Sexpansion} and \eqnref{eq:Kexpansion}, we have
$$
\mathcal{B}_0^* =
\bar{\omega}_M^2\ln\bar{\omega}_M\begin{pmatrix}
v_b^{-2}\mathcal{S}_{D,1}^{(1)} & v_b^{-2}\left(K_{D,1}^{(1)}\right)^* \\
-v_w^{-2}\mathcal{S}_{D,1}^{(1)} & 0
\end{pmatrix}
+ \delta \begin{pmatrix}
0 & 0 \\
0 & -\left(\frac{1}{2}+\mathcal{K}_D\right)
\end{pmatrix} + \mathcal{O}(\omega^2).
$$
In \cite{first}, it is shown that $\left(\mathcal{K}_{D,1}^{(1)}\right)^*[\partialhi_0] = 4b_1Vol(D)\partialhi_0$. It follows that
$$
\mathcal{B}_0^*\begin{pmatrix}
0 \\
\partialhi_0
\end{pmatrix}
=
\frac{4b_1Vol(D)}{v_b^2}\bar{\omega}_M^2\ln\bar{\omega}_M \begin{pmatrix}
\partialhi_0 \\
0
\end{pmatrix} - \delta\begin{pmatrix}
0 \\
\partialhi_0
\end{pmatrix} + \mathcal{O}(\omega^2).
$$
Now, from \eqnref{eq:u} we know that the leading order term $u^{(1)}$ of $u$ is the solution to the equation
$$
\tilde{M}_0^*u^{(1)} = -\beta_0\frac{4b_1Vol(D)}{v_b^2}\bar{\omega}_M^2\ln\bar{\omega}_M \begin{pmatrix}
\partialhi_0 \\
0
\end{pmatrix} + \delta\begin{pmatrix}
0 \\
\partialhi_0
\end{pmatrix}.
$$
Observe that
\begin{align*}
\left(\hat{\mathcal{S}}_D^{k}\right)^*[\partialsi_0] &= \mathcal{S}_D[\partialhi_0] + \bar{\eta}_k\int_{\partialartial D}\partialsi_0 \: \mathrm{d} \sigma \\
&= \left(\gamma_0 + \bar{\eta}_kc\right)\partialhi_0.
\end{align*}
In this equation, recall that $c = \langle \partialsi_0, \partialhi_0 \rangle$. Assume now that $u^{(1)}$ has the form $u^{(1)} = \left(\begin{smallmatrix} y_1\partialsi_0\\ 0 \end{smallmatrix}\right)$. Then
$$
\tilde{M}_0^*u^{(1)} = M_0^*u^{(1)} = y_1\begin{pmatrix}
-\left(\gamma_0 + \bar{\eta}_{k_b}c\right) \partialhi_0\\
\left(\gamma_0 + \bar{\eta}_{k_w}c\right) \partialhi_0
\end{pmatrix}.
$$
Finally, by Theorem \ref{thm:single} we have
$$
\delta = \frac{4b_1Vol(D)}{a_wv_b^2}\omega_M^2\ln\omega_M + \mathcal{O}(\omega^2),
$$
which shows that $y_1 = -\beta_0\frac{\delta}{\left(\gamma_0 + \bar{\eta}_{k_w}c\right)}$, and that
$$\Phi_1 = -\beta_0\frac{\delta}{\left(\gamma_0 + \bar{\eta}_{k_w}c\right)}
\begin{pmatrix}
\partialsi_0 \\ 0 \\ 0 \\ 0
\end{pmatrix} + \mathcal{O}(\omega^2).
$$
\subsection{Computation of $L$}
Let $L$ be defined by (\ref{pencil}).
From \eqnref{pencil} we obtain
$$
L \mathcal{A}_0(\omega_M)= 0 \quad \mbox{ and } \quad \mathcal{A}_0(\omega_M) L = 0.
$$
Therefore, $L$ maps $L^2(\partialartial D)$ into $\ker (\mathcal{A}_0(\omega_M))$ and $L^*$ maps $L^2(\partialartial D)$ into $\ker (\mathcal{A}_0^*(\omega_M))$. These facts, together with the Riesz representation theorem show that $L = l\langle \cdot, \Phi \rangle \Psi$ for some constant $l$. To compute $l$, we use the generalized argument principle for operator valued functions. The operator $\mathcal{A}_0$ is known to have a discrete spectrum \cite{first}, so we can find a small neighbourhood $V$ of $\omega_M$ that contains no characteristic values other than $\omega_M$. Then we have
\begin{equation*}
1 = \frac{1}{2\partiali i}\text{tr}\int_{\partialartial V} \mathcal{A}_0(\omega)^{-1}\frac{d}{d\omega}\mathcal{A}_0(\omega)\: \mathrm{d} \omega,
\end{equation*}
This, together with Proposition \ref{prop:polepencil} gives
\begin{align*}
1 &= \frac{1}{2\partiali i}\text{tr}\int_{\partialartial V} \frac{L\frac{d}{d\omega}\mathcal{A}_0}{\omega-\omega_M} \: \mathrm{d} \omega \\
&= \frac{l}{2\partiali i}\int_{\partialartial V} \frac{\langle \frac{d}{d\omega}\mathcal{A}_0\Psi,\Phi\rangle}{\omega-\omega_M} \: \mathrm{d} \omega,
\end{align*}
and using Cauchy's integral formula we obtain that $l = \frac{1}{\langle \frac{d}{d\omega}\mathcal{A}_0(\omega_M)\Psi,\Phi\rangle}$, so
\begin{equation*}
L = \frac{\langle \cdot,\Phi\rangle}{\langle \frac{d}{d\omega}\mathcal{A}_0(\omega_M)\Psi,\Phi\rangle}.
\end{equation*}
\subsection{Computation of resonance perturbation}
Again, let $V$ be a neighbourhood of $\omega_M$, this time containing only one characteristic value of $\mathcal{A}$, that is,
$V$ contains only the characteristic value $\omega_\varepsilon$ corresponding to a perturbation of the characteristic value $\omega_M$ of $\mathcal{A}_0$. Using the eigenvalue perturbation theory found in \cite{Ammari2009_book}, the leading order term of $\omega_\varepsilon - \omega_M$ is given by
\begin{equation*}
\omega_\varepsilon-\omega_M = -\frac{\varepsilon}{2\partiali i}\text{tr}\int_{\partialartial V} \mathcal{A}_0(\omega)^{-1}\mathcal{A}_1(\omega) \: \mathrm{d} \omega + \mathcal{O}(\varepsilon^2),
\end{equation*}
which gives
\begin{align*}
\omega_\varepsilon-\omega_M &= -\frac{\varepsilon}{2\partiali i}\text{tr}\int_{\partialartial V} \frac{L\mathcal{A}_1(\omega)}{\omega-\omega_M} \: \mathrm{d} \omega + \mathcal{O}(\varepsilon^2) \\
&= -\frac{\varepsilon}{2\partiali i}\int_{\partialartial V} \frac{l\langle\mathcal{A}_1(\omega)\Psi,\Phi\rangle}{\omega-\omega_M} \: \mathrm{d} \omega +\mathcal{O}(\varepsilon^2).
\end{align*}
Because $\langle\mathcal{A}_1(\omega)\Phi,\Psi\rangle$ is holomorphic in $\omega$, Cauchy's integral formula yields
\begin{equation} \label{eq:main}
\omega_\varepsilon-\omega_M = -\varepsilon\frac{\langle\mathcal{A}_1(\omega_M)\Psi,\Phi\rangle}{\langle\frac{d}{d\omega}\mathcal{A}_0(\omega_M)\Psi,\Phi\rangle} + \mathcal{O}(\varepsilon^2).
\end{equation}
We now state the main result of this paper, which gives the leading order term in the expansion of the encapsulated bubble resonance frequency.
\begin{thm} \label{thm:main}
In the quasi-static regime, for any $\varepsilon$ sufficiently small, there exists a resonance frequency $\omega_\varepsilon=\omega_\varepsilon(\varepsilon,\delta)$ for the encapsulated bubble such that $\omega_\varepsilon(0,\delta) = \omega_M(\delta)$ and
\begin{equation} \label{eq:thm}
\omega_\varepsilon =\omega_M + \varepsilon\frac{2\partiali\omega_Ma\left( \delta_{lw}-1\right) }{4\partiali c\left(\gamma_0+\eta_{k_b}c \right) -c^2\left(1-a\right)} + \mathcal{O}(\varepsilon\omega^2) + \mathcal{O}(\varepsilon^2).
\end{equation}
\end{thm}
\begin{proof}
We compute the expression \eqnref{eq:main}. We use subscripts to denote a specific component of a vector. Furthermore, as in \eqnref{eq:regime}, we will work in the regime $\omega^2\ln\omega = \mathcal{O}(\delta)$. Using the expressions for $\Psi$ and $\Phi$ from Section \ref{sec:psi}, we find that the numerator in \eqnref{eq:main} is given by
\begin{equation}\label{eq:nom}
\big\langle\mathcal{A}_1(\omega_M)\Psi,\Phi\big\rangle = \big\langle (\mathcal{A}_1(\omega_M)\Psi_0)_1,u_1\big\rangle + \big\langle (\mathcal{A}_1(\omega_M)\Psi_0)_4, \partialhi_0\big\rangle +\mathcal{O}(\delta^2).
\end{equation}
We begin with the first term of the numerator. Using the low-frequency expansions \eqnref{eq:Sexpansion} and \eqnref{eq:Kexpansion}, we find
\begin{align*}
\left(\mathcal{A}_1(\omega_M,\delta)\Psi\right)_1 &= a\delta_{lw}\left(\frac{1}{2}I + \mathcal{K}_D^*\right)[\partialsi_0] + \left(a_l-a\delta_{lw}\right) \left(-\frac{1}{2}I + \mathcal{K}_D^*\right)[\partialsi_0] - a\left(\mathcal{K}_D + \mathcal{K}_D^*\right)[\partialsi_0] -a\hat{\mathcal{S}}_D^{k_w}[\tau\partialsi_0] + \mathcal{O}(\delta)\\
&= a\delta_{lw}\partialsi_0 - a\left(\frac{1}{2}I + \mathcal{K}_D\right)[\partialsi_0] -a\hat{\mathcal{S}}_D^{k_w}[\tau\partialsi_0] + \mathcal{O}(\delta).
\end{align*}
From this, we can compute
\begin{align*}
\big\langle (\mathcal{A}_1(\omega_M)\Psi_0)_1,u_1\big\rangle &= \bar{y}_1\left( a\delta_{lw}\langle\partialsi_0,\partialsi_0\rangle - a\big\langle\left(\frac{1}{2}I + \mathcal{K}_D\right)[\partialsi_0],\partialsi_0\big\rangle -a\big\langle\hat{\mathcal{S}}_D^{k_w}[\tau\partialsi_0],\partialsi_0\big\rangle \right) + \mathcal{O}(\delta^2) \\
&= \bar{y}_1\left( a\left(\delta_{lw} - 1\right) -a\big\langle\hat{\mathcal{S}}_D^{k_w}[\tau\partialsi_0],\partialsi_0\big\rangle \right) + \mathcal{O}(\delta^2).
\end{align*}
Define $c_\tau$ as $c_\tau = \langle \tau\partialsi_0,\partialhi_0 \rangle$. Then we can compute the term $\big\langle\hat{\mathcal{S}}_D^{k_w}[\tau\partialsi_0],\partialsi_0\big\rangle$ as
$$
\langle \hat\mathcal{S}_D^k[\tau\partialsi_0], \partialsi_0 \rangle = \langle \tau\partialsi_0, \left(\hat\mathcal{S}_D^k\right)^*[\partialsi_0] \rangle = \langle \tau\partialsi_0, (\gamma_0 + \bar{\eta}_kc)\partialhi_0 \rangle = \left(\gamma_0 + \eta_kc\right)c_\tau.
$$
Using this expression, we find that
\begin{align}
\big\langle (\mathcal{A}_1(\omega_M)\Psi_0)_1,u_1\big\rangle &= \bar{y}_1\left( a\left(\delta_{lw} - 1\right) -a\left(\gamma_0 + \eta_kc\right)c_\tau \right) +\mathcal{O}(\delta^2) \nonumber \\
&= -\frac{\delta}{\gamma_0 + \eta_{k_w}c}\left( a\left(\delta_{lw} - 1\right) -a\left(\gamma_0 + \eta_kc\right)c_\tau \right) +\mathcal{O}(\delta^2) \nonumber \\
&=-\delta a\left( \frac{\delta_{lw}-1}{\gamma_0+\eta_{k_w}c} - c_\tau\right) +\mathcal{O}(\delta^2). \label{eq:first}
\end{align}
We now turn to the second term of the numerator. It is easily shown that $\mathcal{S}^{k}_D[\partialsi_0] = a\partialhi_0 + \mathcal{O}(\omega^2\ln \omega)$ for some constant $a$, so $\frac{\partialartial^2}{\partialartial T^2}\mathcal{S}^{k}_D[\partialsi_0] = \mathcal{O}(\omega^2\ln \omega)$. Hence
\begin{align*}
\left(\mathcal{A}_1(\omega_M,\delta)\Psi\right)_4 &= a\delta_{lw}\delta_{bl}\mathbb{R}c_D^{k_l}[\partialsi_0] + (a_l - a\delta_{lw})\delta_{bl}\left(\tau I + \mathbb{R}c_D^{k_l}\right)[\partialsi_0] - a\delta\mathcal{K}_1^{k_w}[\partialsi_0] +\mathcal{O}(\delta^2)\\
&= -a\delta \tau\partialsi_0-a\delta \left(\mathcal{K}_D^*[\tau\partialsi_0]-\frac{\tau}{2}\partialsi_0 + \frac{\partialartial D_D^{k_w}}{\partialartial \nu}[\partialsi_0] \right) +\mathcal{O}(\delta^2) \\
&= -a\delta \left( \mathcal{K}_D^*[\tau\partialsi_0] +\frac{\tau}{2}\partialsi_0 + \frac{\partialartial D_D^{k_w}}{\partialartial \nu}[\partialsi_0] \right) +\mathcal{O}(\delta^2).
\end{align*}
It follows that
\begin{align*}
\big\langle (\mathcal{A}_1(\omega_M)\Psi_0)_4, \partialhi_0\big\rangle &= -a\delta\left( \langle \tau\partialsi_0,K_D[\partialhi_0]\rangle + \frac{1}{2}\langle \tau\partialsi_0,\partialhi_0\rangle + \big\langle \frac{\partialartial D_D^{k_w}}{\partialartial \nu}[\partialsi_0],\partialhi_0 \big\rangle\right) +\mathcal{O}(\delta^2) \\
&= -a\delta\left( c_\tau + \big\langle \frac{\partialartial D_D^{k_w}}{\partialartial \nu}[\partialsi_0],\partialhi_0 \big\rangle \right) +\mathcal{O}(\delta^2).
\end{align*}
Next we compute the term $\big\langle \frac{\partialartial D_D^{k_w}}{\partialartial \nu}[\partialsi_0],\partialhi_0 \big\rangle.$ We know from the expansion \eqnref{eq:hankel} that $\frac{\partialartial D_D^{k_w}}{\partialartial \nu} = \frac{\partialartial D_D}{\partialartial \nu} + \mathcal{O}(\omega^2\ln\omega)$. Furthermore, because $\frac{\partialartial D_D}{\partialartial \nu}$ is self-adjoint, we have
$$
\big\langle \frac{\partialartial D_D^{k_w}}{\partialartial \nu}[\partialsi_0],\partialhi_0 \big\rangle =
\big\langle \partialsi_0,\frac{\partialartial D_D}{\partialartial \nu}[\partialhi_0] \big\rangle + \mathcal{O}(\delta) = \mathcal{O}(\delta),
$$
where the last step follows from the well-known fact that $\mathcal{D}_D[\partialhi_0](x) = 1$ for $x\in D$ \cite{MaCMiPaP}. In total, the second term of the numerator is
\begin{align}
\big\langle (\mathcal{A}_1(\omega_M)\Psi_0)_4, \partialhi_0\big\rangle = -a\delta c_\tau +\mathcal{O}(\delta^2). \label{eq:second}
\end{align}
Next consider the denominator $\langle \frac{d}{d\omega}\mathcal{A}_{0} \Psi,\Phi \rangle$. As with equation \eqnref{eq:nom}, we have
\begin{equation}\label{eq:denom}
\big\langle\frac{d}{d\omega}\mathcal{A}_0(\omega_M)\Psi,\Phi\big\rangle = \big\langle \frac{d}{d\omega}(\mathcal{A}_0(\omega_M)\Psi_0)_1,y_1\big\rangle + \big\langle \frac{d}{d\omega}(\mathcal{A}_0(\omega_M)\Psi_0)_4, \partialhi_0\big\rangle +\mathcal{O}(\delta^2).
\end{equation}
We begin with the first term of the denominator. Using the asymptotic expansion of the fundamental solution for small $\omega$ given in equation \eqnref{eq:hankel}, one can see that the following approximation holds:
\begin{align*}
\frac{d}{d\omega}(\mathcal{A}_0(\omega_M)\Psi_0)_1 &= \frac{1}{2\partiali\omega_M} \left( \int_{\partialartial D} \partialsi_0 - a\int_{\partialartial D} \partialsi_0 \right)\partialhi_0 + \mathcal{O}(\omega\ln\omega) \\
&= \frac{c}{2\partiali\omega_M}\left(1-a\right)\partialhi_0 + \mathcal{O}(\omega\ln\omega).
\end{align*}
It follows that
\begin{align}
\big\langle \frac{d}{d\omega}(\mathcal{A}_0(\omega_M)\Psi_0)_1,y_1\big\rangle &= \bar{y}_1 \frac{c}{2\partiali\omega_M}\left(1-a\right)\langle\partialhi_0,\partialsi_0\rangle \nonumber \\
&= -\frac{\delta}{\gamma_0 + \eta_{k_w}c}\frac{c^2}{2\partiali\omega_M}\left(1-a\right) + \mathcal{O}(\omega^3\ln\omega). \label{eq:third}
\end{align}
To compute the second term of the denominator, we use the expansion \eqnref{eq:Kexpansion} to find that
\begin{align*}
\frac{d}{d\omega}(\mathcal{A}_0(\omega_M)\Psi_0)_4 &= 2\frac{\omega_M\ln\omega_M}{v_b^2} \mathcal{K}_{D,1}^{(1)}[\partialsi_0] + \mathcal{O}(\omega).
\end{align*}
It follows that
\begin{align}
\big\langle \frac{d}{d\omega}(\mathcal{A}_0(\omega_M)\Psi_0)_4, \partialhi_0\big\rangle &= 2\frac{\omega_M\ln\omega_M}{v_b^2}\big\langle \partialsi_0,\left(\mathcal{K}_{D,1}^{(1)}\right)^*[\partialhi_0]\big\rangle + \mathcal{O}(\omega) \nonumber \\
&= \frac{8b_1Vol(D) c}{v_b^2}\omega_M\ln\omega_M + \mathcal{O}(\omega), \label{eq:fourth}
\end{align}
where we have used the fact that $\left(\mathcal{K}_{D,1}^{(1)}\right)^*[\partialhi_0] = 4b_1Vol(D)\partialhi_0$ \cite{first}.
In total, combining \eqnref{eq:first}, \eqnref{eq:second}, \eqnref{eq:third} and \eqnref{eq:fourth} we have
\begin{align*}
\omega_\varepsilon-\omega_M &= -\varepsilon\frac{-\delta a\left( \frac{\delta_{lw}-1}{\gamma_0+\eta_{k_w}c} - c_\tau\right) - a\delta c_\tau}{\frac{8b_1Vol(D) c}{v_b^2}\omega_M\ln\omega_M -\frac{\delta}{\gamma_0 + \eta_{k_w}c}\frac{c^2}{2\partiali\omega_M}\left(1-a\right)} + \mathcal{O}(\varepsilon\omega^2) + \mathcal{O}(\varepsilon^2) \\
&=\varepsilon\frac{2\partiali\omega_Ma\left( \delta_{lw}-1\right) }{4\partiali c\left(\gamma_0+\eta_{k_b}c \right) -c^2\left(1-a\right)} + \mathcal{O}(\varepsilon\omega^2) + \mathcal{O}(\varepsilon^2),
\end{align*}
which proves the theorem.
\end{proof}
\begin{rmk}
From formula \eqnref{eq:thm}, we see that, to leading order, $\omega_\varepsilon = \omega_M$ if $\delta_{lw} = 1$, \textit{i.e.}{} there is no shift in the resonance if there is no density contrast between the layer of coating and the water. This is expected in the case when $\kappa_l = \kappa_w$, \textit{i.e.}{} if there is no contrast in the bulk modulus. In this case the layer of coating and the water have identical wave properties, but the formula shows that this is true even if we have a contrast in the bulk modulus.
\end{rmk}
\begin{rmk}
All the terms in equation \eqnref{eq:thm} can be numerically computed using standard methods. Moreover, the equation simplifies when $D$ is a circle with radius $r$. In this case we have
$$
\gamma_0 = \frac{\ln(r)}{2\sqrt{\partiali}} , \quad a = \frac{\eta_{k_b}}{\eta_{k_w}}, \quad c = \sqrt{2\partiali r}.
$$
For the unit circle with $r=1$, we get
$$
\omega_\varepsilon-\omega_M =\varepsilon\frac{\omega_M\eta_{k_b}\left( \delta_{lw}-1\right) }{4\partiali \eta_{k_b}\eta_{k_w} -\left(\eta_{k_w}-\eta_{k_b}\right)} + \mathcal{O}(\varepsilon\omega^2) + \mathcal{O}(\varepsilon^2).
$$
\end{rmk}
\section{Numerical illustration} \label{sec-numerics}
Here we give numerical examples to verify the formula \eqnref{eq:thm} for the encapsulated bubble frequency in the specific case of a circular bubble. In this case, the resonance frequency is easily computed using the multipole method.
Consider again the equation \eqnref{eq:scattering}. If $D$ is a circle with radius $R$, the encapsulated bubble $D_d$ will also be a circle with radius $R+\varepsilon$. Using polar coordinates $(r,\theta)$, is clear that the solution $u$ can be written as
\begin{equation*}
u(x) = \begin{cases}
\sum_{n=-\infty}^\infty a_n J_n(k_br)e^{in\theta} \quad &\text{if } r<R ,\\
\sum_{n=-\infty}^\infty \big(b_n J_n(k_lr) + c_n H_n(k_lr)\big) e^{in\theta} &\text{if } R<r<R+\varepsilon. \\
\sum_{n=-\infty}^\infty d_n H_n(k_wr) e^{in\theta} &\text{if } R+\varepsilon < r,
\end{cases}
\end{equation*}
for some set of constants $a_n, b_n, c_n, d_n, \ n\in \mathbb{Z}$. Here, $J_n$ is the Bessel function of the first kind and of order $n$. Using the boundary conditions, we find that the constants satisfy
\begin{equation*}
\begin{pmatrix}
J_n(k_bR) & -J_n(k_lR) & -H_n(k_lR) & 0 \\
0 & J_n(k_l(R+\varepsilon)) & H_n(k_l(R+\varepsilon)) & -H_n(k_w(R+\varepsilon)) \\
k_bJ_n'(k_bR) & -\delta_{bl} kJ_n'(k_lR) & -\delta_{bl} k H'_n(k_lR) & 0 \\
0 & k_lJ_n'(k_l(R+\varepsilon)) & k_lH'_n(k_l(R+\varepsilon)) & -\delta_{lw}k_wH'_n(k_w(R+\varepsilon)) \\
\end{pmatrix}
\begin{pmatrix}
a_n \\ b_n \\ c_n \\ d_n
\end{pmatrix} = 0,
\end{equation*}
for all $n\in \mathbb{Z}$. We seek $\omega$ such that for some $n$, the corresponding
system is not invertible. In particular, we seek the
encapsulated bubble resonance, which corresponds
to the lowest resonance of the system. It is clear that at the lowest resonant frequency this system features a
factor with $n=0$, because the lowest resonance has the least number of oscillations. Thus, at the lowest resonant frequency the matrix
\begin{equation} \label{eq:multipole}
A(\omega) = \begin{pmatrix}
J_0(k_bR) & -J_0(k_lR) & -H_0(k_lR) & 0 \\
0 & J_0(k_l(R+\varepsilon)) & H_0(k_l(R+\varepsilon)) & -H_0(k_w(R+\varepsilon)) \\
k_bJ_0'(k_bR) & -\delta_{bl} k_lJ_0'(k_lR) & -\delta_{bl} k_lH'_0(k_lR) & 0 \\
0 & k_lJ_0'(k_l(R+\varepsilon)) & k_lH'_0(k_l(R+\varepsilon)) & -\delta_{lw}k_wH'_0(k_w(R+\varepsilon)) \\
\end{pmatrix}
\end{equation}
becomes singular. Approaching this as a root-finding problem and setting $f(\omega) = \det(A(\omega))$, we have
\begin{equation*}
\hat{\omega}_\varepsilon = \min_{\omega \in \mathcal{C}} \{ \omega \mid f(\omega) = 0 \}.
\end{equation*}
The equation $f(\omega) = 0$ can be solved numerically using Muller's method \cite{Ammari2009_book}, and by choosing initial values
in the vicinity of the Minnaert resonance of the uncoated bubble we can ensure that we find the lowest resonance frequency.
The numerically computed resonance $\hat{\omega}_\varepsilon$ is compared against the encapsulated bubble resonance
${\omega}_\varepsilon$ given by equation \eqnref{eq:thm} in Figure \ref{fig:solPos}, for the case $v_w = v_l = v_b = 1$, radius $R=0.5$, $\delta =10^{-3}$ and $\delta_{lw} = 0.5$. In this case, the coating shifts the frequency to a higher value.
We observe a good agreement between the numerical resonance $\hat{\omega}_\varepsilon$, computed using \eqnref{eq:multipole}, and
the ``Gohberg-Sigal'' resonance ${\omega}_\varepsilon$, computed using \eqnref{eq:thm}, as the thickness of the coating decreases.
In Figure \ref{fig:solNeg}, we again plot the encapsulated bubble resonance $\hat{\omega}_\varepsilon$, using the same parameters as in Figure \ref{fig:solPos} apart from the contrast parameter $\delta_{lw}$ which this time we set to be $\delta_{lw} = 1.5$ instead. It can be seen that the presence of the coating shifts the frequency to a lower value. In summary, if $\delta_{lw} > 1$, the layer of coating increases
the effective density contrast between the gas and the liquid, resulting in a lower resonance frequency, while on the other hand, if $\delta_{lw} < 1$, the effective density contrast is reduced which leads to a higher resonance frequency.
\begin{figure}
\caption{Comparison between the numerically
computed resonance $\hat{\omega}
\label{fig:solPos}
\end{figure}
\begin{figure}
\caption{Comparison between the numerically
computed resonance $\hat{\omega}
\label{fig:solNeg}
\end{figure}
In Figure \ref{fig:err}, we plot the relative error of the $\omega_\varepsilon$ when $\varepsilon$ is fixed and $\delta \in [10^{-8}, 10^{-2}]$. As expected, the error reduces with decreasing $\delta$, and for $\delta = 10^{-3}$ (approximately the value for water and air) the error is $\approx 0.04\%$.
\begin{figure}
\caption{Relative error of $\omega_\varepsilon$ as a function of $\delta$, for the case of a disk with radius $R= 0.5$, $\delta = 10^{-3}
\label{fig:err}
\end{figure}
\section{Concluding remarks} \label{sec-remarks}
In this paper, we have proved an original asymptotic formula for the resonance shift that occurs
when a gas bubble in water is encased in a thin layer of coating. The formula is valid for an arbitrarily shaped bubble, and we have numerically verified it in the case of a circular bubble. The findings are of interest for the application of encapsulated bubbles as ultrasound contrast agents. Furthermore, the subwavelength nature of the encapsulated bubble resonance implies that the encapsulation of bubbles can be a useful approach when synthesizing bubbly phononic crystals. In future work, we plan to study wave scattering
by encapsulated bubbles in the full elastic case, thereby providing an even more realistic description of encapsulated bubbles as ultrasound contrast agents.
{}
\end{document} | math | 69,109 |
\begin{document}
\title{Approaching the Quantum Speed Limit with Global-Local Optimization}
\author{J. J. W. H. S\o rensen}
\author{M. O. Aranburu}
\author{T. Heinzel}
\author{J. F. Sherson}
\affiliation{Aarhus University}
\email{[email protected]}
\date{\today}
\begin{abstract}
We propose a Global-Local optimization algorithm for quantum control that combines standard local search methodologies with evolutionary algorithms. This allows us to find faster solutions to a set of problems relating to ultracold control of Bose-Einstein condensates.
\end{abstract}
\maketitle
\section{Introduction}
Quantum engineering aims to control and steer quantum dynamics in order to realize specific quantum states or operations. This has numerous applications in e.g. femtosecond lasers \cite{assion1998control,meshulach1998coherent}, quantum gate synthesis \cite{schulte2005optimal} and quantum many-body systems \cite{doria2011optimal}. These applications often rEquire tailored control pulses that precisely manipulate the quantum dynamics. Due to experimental limitations such as decoherence, the control pulses must typically be as fast as possible \cite{van2016optimal}. The search for the most time-optimal control or Quantum Speed Limit (QSL) has attracted much attention in the literature \cite{levitin2009fundamental,taddei2013quantum,caneva2009optimal,deffner2017quantum}.
Quantum Optimal Control (QOC) is a tool that finds control pulses by reformulating the control problem as an optimization problem \cite{werschnik2007quantum}. If there are few constraints and full controllability, then these optimization problems are benign in the sense that all local maxima are also global maxima \cite{russell2016quantum,rabitz2004quantum}. These optimization problems can be solved with local "hill climber" type algorithms, since they converge towards to a local and thereby global maxima. However, when we seek fast solutions one must introduce a low bound on the total process duration (\textit{T}). This constraint removes the benign properties of the control problem and local algorithms are no longer guaranteed to find global maxima \cite{zhdanov2015role,bukov2017machine}.
These considerations show that finding the precise location of the QSL can be a difficult optimization problem. Solving such problems require consideration of three main aspects: exploration, exploitation and problem parametrization - see Fig. \ref{fig:cartoon}\textbf{a}. Exploration is searching for new candidate solutions in sparsely probed parts of the control space, whereas exploitation is intense analysis of a small portion of the control space enhancing the best solution \cite{neri2012memetic}. Finally, the control space is often high dimensional. This high dimension can be reduced by a proper problem parametrization, which eases the search for optimal solutions \cite{localPaper}.
\begin{figure}
\caption{(\textbf{a}
\label{fig:cartoon}
\end{figure}
Traditionally, in QOC emphasis is placed on purely exploitative local search algorithms like \textsc{grape} and Nelder-Mead with \textsc{crab} that respectively find solutions using derivative-information or search in a reduced control basis \cite{khaneja2005optimal,caneva2011chopped}. Recently, we introduced the \textsc{group} optimization algorithm that does a gradient descent in a reduced basis and thereby combines the advantages of \textsc{crab} and \textsc{grape} \cite{localPaper}. For these local algorithms, basic exploration is typically added on top of the local search using multistarting \cite{brouzos2015quantum,doria2011optimal}. Modern algorithms from computer science like Differential Evolution (\textsc{de}) dynamically adjust the balance between exploration and exploitation to increase performance. However, they lack domain specific features such as analytic gradient expressions built into the standard algorithms.
Here we propose a hybrid Global-Local algorithm that combines \textsc{de} with local algorithms. This gives a better balance between exploration and exploitation while retaining the domain specific features. We parametrize the control with sinusoidal functions as in the \textsc{crab}-method. As shown in Fig. \ref{fig:cartoon}\textbf{a} this algorithm better balances all three main aspects. We apply this method to control of Bose-Einstein condensates (BECs) in Condensate Splitting (CS) and Condensate Driving (CD). Here we observe improvements in the estimate of the QSL. Below the quantum speed limit there is a conjectured universal $\sin^2$-behavior of the fidelity as a function of duration ($F(T)$) \cite{caneva2009optimal,gajdacz2015time}. In recent work in Refs. \cite{sorensen2016exploring,gajdacz2015time} we cast doubt on the generality of this conjecture and the $F(T)$-curves presented in this work further strengthen this doubt.
This paper is organized as follows: In section \ref{sec:CostFunLocalOpt} we briefly present the local optimization used in the combined Global-Local algorithm. In section \ref{sec:GloLloOpt} we present our proposed combined Global-Local algorithm and we apply it in two controls problems in section \ref{sec:Results}. In section \ref{sec:Learning} we compare our proposed algorithm with conventional multistarting algorithms.
\begin{figure*}
\caption{Condensate Driving (CD). (\textbf{a}
\label{fig:antonioCondensate}
\caption{Condensate Splitting (CS). (\textbf{a}
\label{fig:lesanovskyCondensate}
\end{figure*}
\section{Cost Functional and Local Optimization} \label{sec:CostFunLocalOpt}
Before we discuss the in-depth structure of the Global-Local algorithm we briefly present the BEC control problems. The dynamics of a BEC can be described in a mean-field by the Gross-Pitaevskii Equation (GPE)
\begin{align}
i\diff{\psi}{t}&=-\frac{1}{2m}\diff{^2\psi}{t^2}+V(x,u)\psi+\beta | \psi|^2 \psi \\
&=\bigl(\hat{H}+\beta |\psi|^2\bigr) \psi, \label{GPE}
\end{align}
where $\hbar=1$, $\beta$ is the non-linear self-interaction and $\hat{H}$ is the Hamiltonian. Here we assume the system is one-dimensional since the two other spatial directions can be frozen out \cite{van2016optimal}. The potential depends on the control \textit{u}, and the specific expressions for the different potentials are presented in section \ref{sec:Results}. The objective is to transfer the initial state $\psi_0$ into the target state $\psi_t$, which are both eigenstates of the GPE for $u(t=0)$ and $u(t=T)$. This is a so-called state-to-state problem \cite{schirmer2011efficient}.
In QOC such control problems are expressed as a minimization of the cost functional
\begin{equation}
\hat{J}(u)=\frac{1}{2}(1-F)+\frac{\gamma}{2}\int_0^T \dot{u}(t)^2 \text{d}t, \label{costFun}
\end{equation}
where $F=|\langle \psi_T|\psi(T)\rangle|^2$ is the fidelity and $1-F$ is the infidelity, which characterizes the deviation of the final state $\psi(T)$ from the target state $\psi_t$. Here $\hat{J}(u)$ is the so-called reduced cost functional so the states $\psi(t)$ depend implicitly on $u(t)$ through the GPE \cite{localPaper}. The final state $\psi(T)$ is found by solving the GPE with the potential defined by $u(t)$. The second term enforces regularization, which smooths out the control. This accounts for the fact that arbitrarily fast changes in the control cannot be realized experimentally. Typically, a weight $\gamma$ of about $1\cdot 10^{-6}$ is sufficient to achieve an acceptable regularization.
Analytically calculating the minimum of Eq. (\ref{costFun}) is typically not feasible. Instead the standard approach in QOC is to use local iterative optimization algorithms in order to find solutions for Eq. (\ref{costFun}) \cite{hohenester2007optimal,mennemann2015optimal}.
In order to perform the optimization numerically the control $u(t)$ is typically discretized in steps of $\Delta t$, where $\Delta t$ is set by the required accuracy when numerically solving the GPE. This gives an effective dimension for the simulation at $N= \floor{T/\Delta t} $. Often, this dimension is larger than the required dimension for the control problem, since the optimal controls can be expressed in a basis with a smaller dimension \cite{lloyd2014information,localPaper,caneva2011chopped}. This motivates expanding the control in a chopped basis,
\begin{equation}
u(t) = u_0(t) + S(t) \sum_{n=1}^M c_n f_n(t), \label{crabExpansion}
\end{equation}
where \textit{M} is the size of the basis. As an example we here use $f_n=\sin((n+r_n)\pi t)/T))$ are the basis function with $-0.5\leq r_n \leq 0.5$ being a randomly selected frequency shifts. Here $0\leq S(t) \leq 1$ is a shape function that ensures $u(t=0)=u_0$ and $u(t=T)=u_T$ so $S(0)=S(T)=0$. Within QOC using a random chopped basis was originally introduced in the \textsc{crab} methodology \cite{caneva2011chopped,doria2011optimal}. With this expansion the optimization is performed over the expansion coefficients, so the optimization is for $\hat{J}(\mathbf{c})$ where $\mathbf{c}=(c_1,c_2,...,c_M)$ \cite{localPaper}. Optimization is typically done in this basis using the derivative-free method Nelder-Mead \cite{caneva2011chopped,doria2011optimal}. In Ref. \cite{localPaper} we introduced Gradient Optimization Using Parametrization (\textsc{group}) method that performs a gradient descent within this basis and showed it is competitive with standard QOC algorithms. Here the gradient is calculated using the analytic expression,
\begin{equation}
\diff{\hat{J}(\mathbf{c})}{c_n}=-\int_0^T \biggl(\Re \biggl\langle \chi\biggl|\diff{\hat{H}}{u}\biggr|\psi\biggr\rangle + \gamma \ddot{u} \biggr)S(t)f_n(t)\text{d}t, \label{groupGRAD}
\end{equation}
where $\chi$ is a Lagrange multiplier, which satisfies the equation of motion
\begin{equation}
i \dot{\chi} = \bigl(\hat{H}+2\beta|\psi|^2\bigr)\chi+ \beta \psi^2\chi^*.
\end{equation}
When the gradient is calculated using Eq. (\ref{groupGRAD}) the control can be iteratively updated along the gradient $u^{(i+1)}=u^{(i)}-\alpha^{(i)} \nabla \hat{J}\bigl(u^{(i)}\bigr)$ with $i=0,1,2,...$ where \textit{i} is the iteration index \cite{khaneja2005optimal,jager2014optimal,mennemann2015optimal}. An appropriate value for $\alpha^{(i)}$ is found using a step-size algorithm. Instead of searching along the negative gradient we use a quasi-Newton method \cite{localPaper}. An in-depth discussion of \textsc{group} is presented in Ref. \cite{localPaper}.
Although this method is competitive with standard methods in QOC, it is still a linesearch algorithm so it cannot escape local optima with $F<1$ \cite{localPaper}. In order to add such a capability we combine \textsc{group} with a global optimization algorithm.
\section{Global-Local Optimization} \label{sec:GloLloOpt}
If a local optimization algorithm like \textsc{group} converges to a local suboptimum (a control with $F<1$) then the result may be improved by simply optimizing another initial control. This straightforward way of exploring the optimization landscape is known as multistarting \cite{ugray2007scatter}. Multistarting has a constant probability of success given by how likely it is to randomly select a control that optimizes to $F=1$. Close to the QSL there may be only few global maxima so this probability can be very low \cite{zahedinejad2015high,zahedinejad2014evolutionary}.
An alternative to multistarting is using a global optimization algorithm like Particle Swarm Optimization, Simulated annealing, Differential Evolution (\textsc{de}) and Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) \cite{kennedy2011particle,kirkpatrick1983optimization,das2011differential,hansen2003reducing}. However, these are domain general algorithms and they do not have access to domain specific features like the analytic gradients and good parametrization used in \textsc{group}, which are important for finding high fidelity solutions.
In order to combine these two approaches we propose a combined Global-Local algorithm. Here the global algorithm is a replacement for the multistarting strategy. Based on the performance of past solutions the global algorithm proposes new seeds for the local optimization algorithm. In principle, this type of combination could be done with any global optimization algorithm, but here we focus on \textsc{de} due to its good performance in quantum optimal control problems and general optimization contests \cite{zahedinejad2015high,zahedinejad2014evolutionary,das2011differential}. The algorithm updates a population of points (members) in the optimization landscape $P=\{\mathbf{x}_1,\mathbf{x}_2,....,\mathbf{x}_N\}$. These points must fully characterize the control in the chopped random basis (Eq. (\ref{crabExpansion})) so they consists of both the expansion coefficients and the random frequencies $\mathbf{x}_n = (c_1,c_2,...,c_M,r_1,r_2,...,r_M)$. The algorithm iterates in three main steps being evolution, local optimization, and selection, which are illustrated graphically in Fig. \ref{fig:cartoon}\textbf{b}. Completing all three steps is one generation. First a new trial population ($P_t$) is formed using the evolution strategy from \textsc{de}. We outline the evolution strategy below. In the next step some of the members in $P_t$ are locally optimized. The probability for being optimized ($p(n)$) is given as a sigmoid and the members with the lowest cost have the highest probability of being optimized. If a member $\mathbf{x}_n$ is selected for optimization then it is replaced by the optimized member in $P_t$. The local optimization is \textsc{group} as outlined in the previous section. The optimization is only performed on the expansion coefficients ($c_n$). Finally, in the last selection step the $\mathbf{x}_n$ member in $P$ is replaced by the $\mathbf{x}_n$ member in $P_t$ if the trial member has a lower cost. A pseudocode for the algorithm is shown in Fig. \ref{fig:GloLlo}.
Before discussing the results from the Global-Local algorithm we give a brief account of the evolution strategy used in \textsc{de}. \textsc{de} randomly selects two distinct members ($\mathbf{x}_{j_1},\mathbf{x}_{j_2}$) and the current best member ($\mathbf{x}_{j_b}$). From these three members a donor vector is given as $\mathbf{v}_n = \mathbf{x}_{j_b}+\mathcal{F}(\mathbf{x}_{j_1}-\mathbf{x}_{j_2})$ where $\mathcal{F}$ is a scaling factor. From this donor vector the \textit{n}'th member in the trial population is found by replacing \textit{L} consecutive values of $\mathbf{x}_n$ with values from $v_n$. The length of \textit{L} is given by a Possion distribution with mean \textit{Cr} and minimum length of one. The starting point of this replacement is random. In our simulations, $\mathcal{F}$ is linearly decreased from 0.4 to 0.1 over the simulation in order to promote early exploration and later exploitation. We also use $Cr = 0.97$.
\begin{figure}
\caption{A pseudo-code for the combined Global-Local algorithm.}
\label{fig:GloLlo}
\end{figure}
\section{Results} \label{sec:Results}
Optimal control of BECs of $^{87}\text{Rb}$ atoms trapped atom-chips has been explored by several authors and realized experimentally \cite{bucker2013vibrational,schumm2005matter,van2016optimal,jager2014optimal,hohenester2007optimal} . The dynamics of the two problems are shown in Fig. \ref{fig:antonioCondensate}\textbf{a} and \ref{fig:lesanovskyCondensate}\textbf{a}. We discuss each of the two control problems separately.
\subsection{Condensate Driving}
In Condensate Driving (CD) a BEC must be transferred from the initial ground state of an anharmonic well into the first excited state. This state can be used as a source for stimulated emission of matter waves \cite{bucker2011twin}. The transfer is completed by shaking the trap. Previously, in CD an $F(T)$-curve demonstrating a conjectured double-$\sin^2$-behavior has been reported \cite{van2016optimal}. The potential is well described by the polynomial,
\begin{equation*}
V(x,u(t))=p_2\bigl(x-u(t)\bigr)^2+p_4\bigl(x-u(t)\bigr)^4+p_6\bigl(x-u(t)\bigr)^6,
\end{equation*}
where the control $u(t)$ is the trap displacement \cite{van2016optimal}. The coefficients are given by $p_2 = 2\pi \hbar \cdot 310/r_0^2 \text{J}/\text{m}^2$,$p_4=2\pi\hbar \cdot 13.6/r_0^4 \text{J}/\text{m}^4$ and $p_6=-2\pi\hbar \cdot 0.0634/r_0^6 \text{J}/\text{m}^6$ with $r_0=172 \text{nm} $\cite{van2016optimal}. The nonlinear coupling constant is $\beta=2.61 \hbar \, \mu \text{m}\,\text{Hz}$ for 700 atoms, which takes corrections for going from the three-dimensional to the one-dimensional case into account \cite{van2016optimal,gerbier2004quasi}. A further complication arises in this control problem, which is the fact that it is necessary to include the finite bandwidth of the control electronics. This effect causes the control to become convolved into the new control $v(t)$ and the atoms experience the potential $V(x,v(t))$. This correction must also be included in the expression for the gradient (Eq. (\ref{groupGRAD})). The procedure for including this effect into local optimization is discussed in Ref. \cite{localPaper}.
The result of our Global-Local algorithm on the CD-problem is presented in Fig. \ref{fig:antonioCondensate}\textbf{b}. The resulting numerical estimate for CD is $T_{\text{QSL}}^{\text{num}}=0.89\text{ms}$, which is the shortest duration with $F\geq 0.99$. This result is lower than the $1.09\text{ms}$ report in Ref. \cite{van2016optimal} where traditional multistarting and gradient-free optimization was used. The Global-Local algorithm also finds better fidelities than Ref. \cite{van2016optimal} below the estimated QSL ($T\leq T_{\text{QSL}}^{\text{num}}$) - see the blue dots in Fig. \ref{fig:antonioCondensate}\textbf{b}. This highlights that the landscape has become highly complex due to the strong duration constraint and traditional multistarting is no longer sufficient. In Ref. \cite{van2016optimal} a double $\sin^2$-behavior is found, which was interpreted to indicate that the solutions had mapped out the true QSL. Surprisingly, the Global-Local algorithm breaks this QSL and finds a new $F(T)$-curve that does not follow a $\sin^2$-behavior. This indicates that a $\sin^2$-behavior does not always imply, that the numerical results have identified the QSL. The Global-Local algorithm will theoretically identify the true QSL in the infinite time limit, due to our finite optimization time even better solutions could possibly exist. Therefore we cannot exclude the existence of a better $F(T)$-curve, which could have a $\sin^2$-behavior. Previous $\sin^2$-results from e.g. \cite{caneva2009optimal, caneva2011speeding} were all for a linear Schr\"{o}dinger equation, so it is not directly clear that these results generalize to the nonlinear dynamics studied here. The $F(T)$-curve has a distinct kink around $T=0.2 \text{ms}$. At short durations below $T_{\text{QSL}}^{\text{num}}$ the best control only displaces the condensate and at longer durations the control does a partial transfer of the wavefunction into the first excited state - see the insert in Fig. \ref{fig:antonioCondensate}\textbf{b}. These two processes scale differently with respect to \textit{T}, so at some durations a displacement is better than a partial transfer. This gives an explanation for the kink in Fig. \ref{fig:antonioCondensate}\textbf{b}. A similar kink was observed in Ref. \cite{van2016optimal}. The optimal control curve and $\langle \hat{x}(t)\rangle$ is shown in Fig. \ref{fig:antonioOptimalControl}. The control is highly complex. However, a Fourier transformation of $\langle \hat{x}(t)\rangle$ reveals that the main oscillation frequency in $\langle \hat{x}(t)\rangle$ is close to the energy difference between the ground state and the first excited state, so the resulting control can partially be understood as resonant driving.
\begin{figure*}
\caption{(\textbf{a}
\label{fig:antonioOptimalControl}
\caption{(\textbf{a}
\label{fig:LesanovskyOptimalControl}
\end{figure*}
\subsection{Condensate Splitting}
In Condensate Splitting (CS) a BEC is split into two seperate BECs with the same phase. This splitting procedure can be used in a Mach-Zehnder type interferometer for matter waves \cite{schumm2005matter}. The quality of this interferometer depends on the quality of the splitting, which is optimized. The atoms are trapped in an Ioffe-Pritchard field configuration on the atom chip. The potential is created by applying RF-dressing, which causes mixing of the Zeeman levels of the $F=1$ manifold coupling them to the dressed states \cite{bucker2013vibrational}. In Ref. \cite{lesanovsky2006adiabatic} it is shown within the rotating wave approximation that this gives the potential
\begin{equation*}
V(x,u(t))=g_F\mu_B \sqrt{\biggl(B_S(x)-\frac{\hbar \omega}{g_F\mu_B}\biggr)^2+\biggl(\frac{B_{\text{RF}}B_I}{2B_S(x)}\biggr)^2},
\end{equation*}
where $g_F$ is the \textit{g}-factor, $\mu_B$ is the Bohr magneton. $\omega = 1.26 \cdot 2\pi \text{MHz}$ is the field detuning. $B_I=1.0 \text{G}$ and $B_\text{RF}=(0.5+0.3u(t))\text{G}$ are a magnetic field component related to the inhomogenous Ioffe field and the experimentally adjustable RF-field. $B_S^2(x)=(Gx)^2+B_I^2$ is a static field where \textit{G} is the gradient of the Ioffe-field trap\cite{lesanovsky2006adiabatic,hohenester2007optimal,bucker2013vibrational}. For $u=0$ the potential is a single well and it dynamically changes into a double well as \textit{u} changes into $u=1$, which is shown in Fig. \ref{fig:lesanovskyCondensate}\textbf{a}.
The result of our Global-Local optimization on the CS-problem is presented in Fig. \ref{fig:lesanovskyCondensate}\textbf{b}. The resulting numerical estimate for
CS is $T_{\text{QSL}}^{\text{num}}=1.0\text{ms}$. To our knowledge there is no $F(T)$-curve in the literature for comparison but our results are faster than the $2.0\text{ms}$ report in Ref. \cite{jager2014optimal,hohenester2007optimal}. Neither in the case of CS do we find a $\sin^2$-behavior as seen on Fig. \ref{fig:lesanovskyCondensate}\textbf{b}. The optimal control curve and the population in the instantaneous linear energy eigenstates are shown in Fig. \ref{fig:LesanovskyOptimalControl}. The optimal control first excites the condensate as much as possible by applying the maximally allowed splitting for the first 1/5 of the control duration. This excites the BEC and by Heisenberg's time-energy uncertainty relation allows for fast motion in the Hilbert Space. The importance of constraints on the QSL is especially clear here, since the excitation process could be completed faster if a larger double-well splitting was allowed. After the initial excitation period the condensate is transferred into the target state. This optimal control is highly diabatic and clearly differs from adiabatically inspired solutions where the splitting is gradually turned on. The control found here is quite different from that in e.g. Ref \cite{jager2014optimal}. This highlights the Global-Local algorithm's ability to search control subspaces far from the adiabatic regime, which is important when approaching the QSL where the optimization landscape is complex.
\section{Learning} \label{sec:Learning}
\begin{figure}
\caption{Learning in the Global-Local algorithm in CS at $T_{\text{QSL}
\label{fig:learning}
\end{figure}
Finally we discuss the learning in the Global-Local optimization. In Fig. \ref{fig:learning} the distribution of cost values within each generation is shown as a function of generations in the optimization in CS at $T_{\text{QSL}}^{\text{num}}=1.0\text{ms}$. The figure shows that the Global-Local algorithm gradually decreases the median infidelity and thus learns a better solution strategy. In comparison, a multistarting algorithm would have a constant distribution set by the seeding strategy. We also did a traditional multistarting on the seeds in the initial population and none of the solutions achieved a similar value of the cost. The initial population is wide and it narrows as a function of generations. This shows that the optimization does an early exploration phase and subsequently spends successive generations on refining the current best members. However, this plot also suggest that the learning in the later stages of the algorithm could be substantially improved. With these early results we also cannot rule out that with sufficient fine tuning of the local optimization algorithms it might be possible to achieve similar results. The relative merit of improving local and global search methodologies represents an interesting avenue of future research.
\section{Conclusion}
We have implemented a combined a Global-Local optimization algorithm that combines evolutionary algorithms like \textsc{de} with local quantum control methods like \textsc{group}. This combination has allowed the improvement of existing estimates of the QSL in problems related to control of BECs.
This scheme is directly applicable to other problems in quantum control where it might also give improvements in the QSL.
It would be possible to further tune the balance between exploration and exploitation by modifying the \textsc{DE} algorithm or using another explorative algorithm like CMA-ES. Within \textsc{DE}, exploration can be promoted by changing the scheme for the generation of the donor vector using a method like SaDE \cite{das2011differential}. Finally, it would also be very interesting to do a more in depth comparison with multistarting in order to better quantify the advantages of the Global-Local algorithm.
\end{document} | math | 25,538 |
{{\mathfrak{m}}athfrak{b}}egin{document}
\title[Fano 5-folds with nef tangent bundles]{Fano 5-folds with nef tangent bundles and Picard numbers greater than one}
{{\mathfrak{m}}athfrak{a}}uthor{Kiwamu Watanabe}
\date{\today}
{{\mathfrak{m}}athfrak{a}}ddress{Course of Mathematics, Programs in Mathematics, Electronics and Informatics,
Graduate School of Science and Engineering, Saitama University.
Shimo-Okubo 255, Sakura-ku Saitama-shi, 338-8570, Japan.}
{{\mathfrak{m}}athfrak{e}}mail{[email protected]}
\subjclass[2010]{Primary~14J40, 14J45, 14M17.}
\keywords{Fano manifold with nef tangent bundle, homogeneous manifold}
{\mathfrak{m}}aketitle
{{\mathfrak{m}}athfrak{b}}egin{abstract}
We prove that smooth Fano 5-folds with nef tangent bundles and Picard numbers greater than one are rational homogeneous manifolds.
{{\mathfrak{m}}athfrak{e}}nd{abstract}
\section{Introduction}
Characterization problems of special projective manifolds in terms of positivity properties of the tangent bundle have been considered by several authors. One of the most important results is S. Mori's solution of the Hartshorne-Frankel conjecture \cite{Mori}: a projective manifold with ample tangent bundle is a projective space.
As a generalization of Mori's theorem, F. Campana and T. Peternell \cite{CP} proposed to study complex projective manifolds with nef tangent bundles and gave the classification in case of dimension $3$. After that, a structure theorem of such manifolds in arbitrary dimension was provided by J. P. Demailly, T. Peternell and M. Schneider \cite{DPS}: a projective (or more generally, compact K${\rm \ddot{a}}$lher) manifold $X$ with nef tangent bundle admits a finite \'etale cover $\tilde{X} \rightarrow X$ such that the Albanese map $\tilde{X} \rightarrow {\rm Alb}(\tilde{X})$ is a smooth morphism whose fibers are Fano manifolds with nef tangent bundles.
Hence, we obtain the complete picture of projective manifolds with nef tangent bundles if the following conjecture due to Campana and Peternell is solved:
{{\mathfrak{m}}athfrak{b}}egin{conj}[{\cite{CP}}]\label{CPC} A Fano manifold $X$ with nef tangent bundle is rational homogeneous.
{{\mathfrak{m}}athfrak{e}}nd{conj}
By the classification theory of Fano manifolds, one can check that this conjecture holds when $\dim X \leq 3$. Furthermore, Campana and Peternell \cite{CP2} gave an affirmative answer when $\dim X=4$ and the Picard number $\rho_X>1$. After that, via the works of \cite{CMSB}, \cite{Mi} and \cite{Mok}, the case when $\dim X=4$ was finally completed by J. M. Hwang \cite{Hwang}. However this conjecture remains open in $\dim X {\mathfrak{g}}eq 5$. Our main purpose of this article is to treat the case when $\dim X=5$ and $\rho_X>1$.
{{\mathfrak{m}}athfrak{b}}egin{them}[=Theorem~\ref{MT2}]\label{MT}
Let $X$ be a complex Fano manifold of dimension $5$ with nef tangent bundle and Picard number $\rho_X>1$. Then $X$ is a rational homogeneous manifold.
{{\mathfrak{m}}athfrak{e}}nd{them}
The proof proceeds as follows. Let $X$ be a Fano $5$-fold with nef tangent bundle of $\rho_X>1$. For any contraction $f: X \rightarrow Y$ of an extremal ray, $f$ is smooth, and $Y$ and the fibers $X_y$ are Fano manifolds with nef tangent bundles (Theorem~\ref{sm}). Furthermore, we see that $\rho_{X_y}=1$. Since Conjecture~\ref{CPC} holds for Fano manifolds of dimension $\leq 4$, it is easy to see that $X$ is a holomorphic fiber bundle over a rational homogeneous manifold $Y$ whose fibers are projective spaces or quadrics (Lemma~\ref{l}). Since $\rho_X>1$, $X$ admits at least two different fiber bundle structures. Studying these bundle structures, we get the complete classification.
This paper is organized as follows: In Section~$2$, we recall some known results on Fano manifolds. Section~$3$ is dedicated to study properties of Fano manifolds with nef tangent bundles. Furthermore, we shall determine if some concrete examples of Fano manifolds with projective bundle structures have nef tangent bundles. In Section~$4$, we prove our main result Theorem~\ref{MT}. In the final section, we deal with Fano $5$-folds with nef tangent bundles of $\rho=1$.
In this paper, we use notation as in \cite{Ha} and every point on a variety we deal with is a closed point. Denote the $m$ times product of ${{\mathfrak{m}}athbb{P}}^n$ by $({{\mathfrak{m}}athbb{P}}^n)^m$. A {\it ${{\mathfrak{m}}athbb{P}}^m$-bundle} means the Grothendieck projectivization of a rank $(m+1)$ vector bundle, whereas a smooth morphism whose fibers are isomorphic to ${{\mathfrak{m}}athbb{P}}^m$ will be called a {\it smooth ${{\mathfrak{m}}athbb{P}}^m$-fibration}. We work over the field of complex numbers. \\
\section{Known results on Fano manifolds}
A {\it Fano manifold} means a projective manifold $X$ with ample anticanonical divisor $-K_X$. For a Fano manifold $X$, the {\it pseudoindex} is defined as the minimum $i_X$ of the anticanonical degrees of rational curves on $X$.
Given a projective manifold $X$, we denote by $N_1(X)$ the space of $1$-cycles with real coefficients modulo numerical equivalence. The dimension of $N_1(X)$ is the Picard number $\rho_X$ of $X$. The convex cone of effective $1$-cycles in $N_1(X)$ is denoted by $NE(X)$.
By the Contraction Theorem, given a $K_X$-negative extremal ray $R$ of the Kleiman-Mori cone $\overline{NE}(X)$, we obtain the contraction of the extremal ray $\varphi_R :X \rightarrow Y$.
We say that $\varphi_R$ is {\it of fiber type} if $\dim X > \dim Y$, otherwise it is {\it of birational type}.
{{\mathfrak{m}}athfrak{b}}egin{pro}[{\cite[Lemma~3.3, Remark~3.7]{Casa}}]\label{Casa} Let $X$ be a Fano manifold, $f: X \rightarrow Y$ a contraction of an extremal ray of fiber type, and $X_y$ a fiber of $f$. Suppose that $f$ is smooth. Then $X_y$ is a Fano manifold of $\rho_{X_y}=1$.
{{\mathfrak{m}}athfrak{e}}nd{pro}
{{\mathfrak{m}}athfrak{b}}egin{pro}[{\cite[Lemma~4.1]{NO}}]\label{NO2} Let $X$ be a Fano manifold admitting a ${{\mathfrak{m}}athbb{P}}^r$-bundle structure $f: X \rightarrow Y$ and $R$ the extremal ray corresponding to $f$. If there exists a proper morphism $g: X \rightarrow Z$ onto a variety $Z$ of dimension $r$ which does not contract curves of $R$. Then $X \cong {{\mathfrak{m}}athbb{P}}^r \times Y$
{{\mathfrak{m}}athfrak{e}}nd{pro}
{{\mathfrak{m}}athfrak{b}}egin{pro}[See {\cite[Proposition~5.1]{NO}} and {\cite[Proposition~2.4]{BCDD}}]\label{NO} Let $X$ be a Fano manifold of dimension $n$ and pseudoindex ${\mathfrak{g}}eq 2$ which has only contractions of fiber type.
Then $\rho_X \leq n$. Moreover,
{{\mathfrak{m}}athfrak{b}}egin{enumerate}
\item if $\rho_X=n$, then $X=({{\mathfrak{m}}athbb{P}}^1)^n$;
\item if $\rho_X=n-1,$ then $X$ is either $({{\mathfrak{m}}athbb{P}}^1)^{n-2} \times {{\mathfrak{m}}athbb{P}}^2$ or $X=({{\mathfrak{m}}athbb{P}}^1)^{n-3} \times {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$.
{{\mathfrak{m}}athfrak{e}}nd{enumerate}
{{\mathfrak{m}}athfrak{e}}nd{pro}
{{\mathfrak{m}}athfrak{b}}egin{rem} \rm The above result {\cite[Proposition~5.1]{NO}} was obtained by applying {\cite[Lemma~2.13]{NO}}. As one of referees pointed out to the author, the proof of {\cite[Lemma~2.13]{NO}} contains a gap. To be more precise, it is based on a result in algebraic topology due to A. Borel \cite[Expos$\rm {{\mathfrak{m}}athfrak{a}}cute{e}$ IX, Remark 2 after Theorem 6]{Borel}, and it seems that the Borel's proof works when both $H^{{{\mathfrak{m}}athfrak{a}}st}(F, {{\mathfrak{m}}athscr{Z}}Z)$ and $H^{{{\mathfrak{m}}athfrak{a}}st}(B, {{\mathfrak{m}}athscr{Z}}Z)$ are torsion-free.
However it does not affect {\cite[Proposition~5.1]{NO}}. Under the notation as in {\cite[Proposition~5.1]{NO}}, the Borel's result was applied to prove that every elementary contraction $\varphi_j: X \rightarrow Y_j$ with one-dimensional fibers is given by the projectivization of a rank $2$ vector bundle. Without using \cite[Expos$\rm {{\mathfrak{m}}athfrak{a}}cute{e}$ IX, Remark 2 after Theorem 6]{Borel}, by the same way as in the proof of {\cite[Proposition~5.1]{NO}}, we see that $Y_j \cong ({{\mathfrak{m}}athbb{P}}^1)^{n-3} \times {{\mathfrak{m}}athbb{P}}^2$ or $({{\mathfrak{m}}athbb{P}}^1)^{n-4} \times {{\mathfrak{m}}athbb{P}}(T_{{\mathfrak{m}}athbb{P}}^2)$. Then it follows from Proposition~\ref{Br} below that $\varphi_j$ is a ${{\mathfrak{m}}athbb{P}}^1$-bundle. As a consequence, we obtain {\cite[Proposition~5.1]{NO}} by the same argument.
{{\mathfrak{m}}athfrak{e}}nd{rem}
{{\mathfrak{m}}athfrak{b}}egin{pro}\label{Br} Let $f: X \rightarrow Y$ be a smooth ${{\mathfrak{m}}athbb{P}}$-fibration over a projective manifold $Y$. If $Y$ is rational or a curve, then there exists a rank $(d+1)$ vector bundle ${{\mathfrak{m}}athscr{E}}$ on $Y$ such that $X={{\mathfrak{m}}athbb{P}}_Y({{\mathfrak{m}}athscr{E}})$.
{{\mathfrak{m}}athfrak{e}}nd{pro}
{{\mathfrak{m}}athfrak{b}}egin{proof} Consider an exact sequence of algebraic groups over $Y$:
{{\mathfrak{m}}athfrak{b}}egin{eqnarray}
1 \rightarrow {{\mathfrak{m}}athscr{G}}G_m \rightarrow GL(d+1) \rightarrow PGL(d) \rightarrow 1. \nonumber
{{\mathfrak{m}}athfrak{e}}nd{eqnarray}
Then we have an exact sequence of $\rm {{\mathfrak{m}}athfrak{a}}cute{e}$tale cohomologies:
{{\mathfrak{m}}athfrak{b}}egin{eqnarray}\label{exact}
H^1_{{\it {{{\mathfrak{m}}athfrak{a}}cute{e}}t}}(Y, GL(d+1)) \rightarrow H^1_{{\it {{{\mathfrak{m}}athfrak{a}}cute{e}}t}}(Y, PGL(d)) \rightarrow H^2_{{\it {{{\mathfrak{m}}athfrak{a}}cute{e}}t}}(Y, {{\mathfrak{m}}athscr{G}}G_m).
{{\mathfrak{m}}athfrak{e}}nd{eqnarray}
Here ${\rm Br}'(Y):=H^2_{\it {{{\mathfrak{m}}athfrak{a}}cute{e}}t}(Y, {{\mathfrak{m}}athscr{G}}G_m)$ is called the {\it Brauer-Grothendieck group} of $Y$. The Brauer-Grothendieck group is birational invariant of complex projective manifolds \cite[III, Corollary~7.3]{Gr}. Furthermore, it is well-known that ${\rm Br}'(Y)$ is trivial when $Y$ is a complex projective space or a curve. Hence, in the above sequence (\ref{exact}), the first arrow is surjective.
On the other hand, a smooth ${{\mathfrak{m}}athbb{P}}$-fibration $f$ defines a cocycle $[f] \in H^1_{{\it {{{\mathfrak{m}}athfrak{a}}cute{e}}t}}(Y, PGL(d))$. Then $f$ is given by the projectivization of a vector bundle if and only if there exists a preimage of $[f]$ in $H^1_{{\it {{{\mathfrak{m}}athfrak{a}}cute{e}}t}}(Y, GL(d+1))$. Since the first arrow of the above sequence (\ref{exact}) is surjective, we obtain our assertion.
{{\mathfrak{m}}athfrak{e}}nd{proof}
{{\mathfrak{m}}athfrak{b}}egin{pro}[{\cite[Theorem~2]{OW}}]\label{WO} Let $X$ be a projective manifold of dimension $n$, endowed with two different smooth ${{\mathfrak{m}}athbb{P}}$-fibration structures $f: X \rightarrow Y$ and $g: X \rightarrow Z$ such that $\dim Y + \dim Z = n+1$. Then either $n=2m-1$, $Y =Z={{\mathfrak{m}}athbb{P}}^m$ and $X={{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^m})$ or $Y$ and $Z$ have a ${{\mathfrak{m}}athbb{P}}$-bundle structure over a smooth curve $C$ and $X = Y \times_C Z$.
{{\mathfrak{m}}athfrak{e}}nd{pro}
{{\mathfrak{m}}athfrak{b}}egin{proof} See {\cite[Theorem~2]{OW}}. According to Proposition~\ref{Br}, a smooth ${{\mathfrak{m}}athbb{P}}$-fibration over a curve is a ${{\mathfrak{m}}athbb{P}}$-bundle.
{{\mathfrak{m}}athfrak{e}}nd{proof}
\section{Fano manifolds with nef tangent bundles}
{{\mathfrak{m}}athfrak{b}}egin{them}[{See \cite[Theorem~4.2]{Hwang}}]\label{4} Let $X$ be a Fano manifold with nef tangent bundle of dimension $n \leq 4$. Then the following holds.
{{\mathfrak{m}}athfrak{b}}egin{enumerate}
\item If $n=1$, then $X$ is ${{\mathfrak{m}}athbb{P}}^1$.
\item If $n=2$, then $X$ is ${{\mathfrak{m}}athbb{P}}^2$ or $({{\mathfrak{m}}athbb{P}}^1)^2$.
\item If $n=3$, then $X$ is one of the following: \\
${{\mathfrak{m}}athbb{P}}^3$, $Q^3$, ${{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}^2$, ${{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$, $({{\mathfrak{m}}athbb{P}}^1)^3$.
\item If $n=4$, then $X$ is one of the following: \\
${{\mathfrak{m}}athbb{P}}^4$, $Q^4$, ${{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}^3$, ${{\mathfrak{m}}athbb{P}}^1 \times Q^3$, $({{\mathfrak{m}}athbb{P}}^2)^2$, ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})$, where ${{\mathfrak{m}}athscr{N}}$ is the null-correlation bundle over ${{\mathfrak{m}}athbb{P}}^3$ (see Example~\ref{spe} below), $({{\mathfrak{m}}athbb{P}}^1)^2 \times {{\mathfrak{m}}athbb{P}}^2$, ${{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$, $({{\mathfrak{m}}athbb{P}}^1)^4$.
{{\mathfrak{m}}athfrak{e}}nd{enumerate}
{{\mathfrak{m}}athfrak{e}}nd{them}
{{\mathfrak{m}}athfrak{b}}egin{proof} When $n \leq 2$, it is easy to prove our assertion. When $n=3$, this is in \cite[Theorem~5.1, Theorem~6.1]{CP}. Of course, this also follows from the classification theory of Fano manifolds of $n \leq 3$. If $n=4$ and $\rho_X > 1$, then our assertion is dealt in \cite[Theorem~3.1]{CP2}. However we should remark that the tangent bundle of ${{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})\times_{{{\mathfrak{m}}athbb{P}}^2} {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$, which is listed in \cite[Theorem~3.1~(4)-(d)]{CP2}, is not nef, see Lemma~\ref{non} below. If $n=4$ and $\rho_X =1$, we see that $X$ is isomorphic to ${{\mathfrak{m}}athbb{P}}^4$ or $Q^4$. This follows from \cite{CMSB}, \cite{Mi} and \cite[Theorem~4.3]{Hwang} (see also Section~\ref{1}).
{{\mathfrak{m}}athfrak{e}}nd{proof}
{{\mathfrak{m}}athfrak{b}}egin{lem}\label{pi} Let $X$ be a Fano manifold with nef tangent bundle. Then the pseudoindex of $X$ is at least $2$.
{{\mathfrak{m}}athfrak{e}}nd{lem}
{{\mathfrak{m}}athfrak{b}}egin{proof} Let $C$ be a rational curve on $X$ and $f: {{\mathfrak{m}}athbb{P}}^1 \rightarrow C \subset X$ its normalization. Since $T_X$ is nef, so is $f^{{{\mathfrak{m}}athfrak{a}}st}T_X$. This implies that $f^{{{\mathfrak{m}}athfrak{a}}st}T_X \cong {{\mathfrak{m}}athfrak{b}}igoplus^n_{i=1} {{\mathfrak{m}}athscr{O}}_{{{\mathfrak{m}}athbb{P}}^1}(a_i)$, where $a_i {\mathfrak{g}}eq 0$. Furthermore we have an injection ${{\mathfrak{m}}athscr{O}}_{{{\mathfrak{m}}athbb{P}}^1}(2) \rightarrow f^{{{\mathfrak{m}}athfrak{a}}st}T_X$. This implies that $a_i {\mathfrak{g}}eq 2$ for some $i$. Consequently, $-K_X.C= \sum a_i {\mathfrak{g}}eq 2$. This means the pseudoindex of $X$ is at least $2$.
{{\mathfrak{m}}athfrak{e}}nd{proof}
{{\mathfrak{m}}athfrak{b}}egin{lem}\label{non} The tangent bundle of ${{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})\times_{{{\mathfrak{m}}athbb{P}}^2} {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$ is not nef.
{{\mathfrak{m}}athfrak{e}}nd{lem}
{{\mathfrak{m}}athfrak{b}}egin{proof} For $X:={{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})\times_{{{\mathfrak{m}}athbb{P}}^2} {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$, consider the commutative diagram: \[\xymatrix{
&& X {{\mathfrak{m}}athfrak{a}}r[dd]_{{\mathfrak{p}}i} {{\mathfrak{m}}athfrak{a}}r[dl]_{{\mathfrak{p}}i_1} {{\mathfrak{m}}athfrak{a}}r[dr]^{{\mathfrak{p}}i_2} && \\
&{{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2}) {{\mathfrak{m}}athfrak{a}}r[dl]_{p_1} {{\mathfrak{m}}athfrak{a}}r[dr]^{p_2} & & {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2}) {{\mathfrak{m}}athfrak{a}}r[dl]_{p_2} {{\mathfrak{m}}athfrak{a}}r[dr]^{p_1} & \\
{{\mathfrak{m}}athbb{P}}^2 & &{{\mathfrak{m}}athbb{P}}^2& & {{\mathfrak{m}}athbb{P}}^2 \\
}\]
Then we have
{{\mathfrak{m}}athfrak{b}}egin{eqnarray} \nonumber
-K_X&=& {\mathfrak{p}}i^{{{\mathfrak{m}}athfrak{a}}st}(-K_{{{\mathfrak{m}}athbb{P}}^2})+(-K_{X/{{\mathfrak{m}}athbb{P}}^2})\\
\nonumber &=& {\mathfrak{p}}i^{{{\mathfrak{m}}athfrak{a}}st}(-K_{{{\mathfrak{m}}athbb{P}}^2})+{\mathfrak{p}}i_1^{{{\mathfrak{m}}athfrak{a}}st}(-K_{{{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})/{{\mathfrak{m}}athbb{P}}^2})+ {\mathfrak{p}}i_2^{{{\mathfrak{m}}athfrak{a}}st}(-K_{{{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})/{{\mathfrak{m}}athbb{P}}^2})\\
\nonumber &=& {\mathfrak{p}}i^{{{\mathfrak{m}}athfrak{a}}st}(-K_{{{\mathfrak{m}}athbb{P}}^2})+{\mathfrak{p}}i_1^{{{\mathfrak{m}}athfrak{a}}st}(-K_{{{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})}+p_2^{{{\mathfrak{m}}athfrak{a}}st}(K_{{{\mathfrak{m}}athbb{P}}^2}))+ {\mathfrak{p}}i_2^{{{\mathfrak{m}}athfrak{a}}st}(-K_{{{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})}+p_2^{{{\mathfrak{m}}athfrak{a}}st}(K_{{{\mathfrak{m}}athbb{P}}^2}))\\
\nonumber &=& {\mathfrak{p}}i^{{{\mathfrak{m}}athfrak{a}}st}(K_{{{\mathfrak{m}}athbb{P}}^2})+{\mathfrak{p}}i_1^{{{\mathfrak{m}}athfrak{a}}st}(-K_{{{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})})+ {\mathfrak{p}}i_2^{{{\mathfrak{m}}athfrak{a}}st}(-K_{{{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})}).
{{\mathfrak{m}}athfrak{e}}nd{eqnarray}
Let $l \subset {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$ be a fiber of $p_1$. Then ${p_2}_{{{\mathfrak{m}}athfrak{a}}st}(l)$ is a line in ${{\mathfrak{m}}athbb{P}}^2$. Furthermore, $l$ can be regarded as a curve in $X$ via the diagonal embedding ${{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2}) \subset X$. Then
{{\mathfrak{m}}athfrak{b}}egin{eqnarray} \nonumber
-K_X.l= ({\mathfrak{p}}i^{{{\mathfrak{m}}athfrak{a}}st}(K_{{{\mathfrak{m}}athbb{P}}^2})+{\mathfrak{p}}i_1^{{{\mathfrak{m}}athfrak{a}}st}(-K_{{{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})})+ {\mathfrak{p}}i_2^{{{\mathfrak{m}}athfrak{a}}st}(-K_{{{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})})).l=1.
{{\mathfrak{m}}athfrak{e}}nd{eqnarray}
Thus, Lemma~\ref{pi} concludes that the tangent bundle of $X$ is not nef.
{{\mathfrak{m}}athfrak{e}}nd{proof}
{{\mathfrak{m}}athfrak{b}}egin{rem}\rm We see that ${{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})\times_{{{\mathfrak{m}}athbb{P}}^2} {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$ is the blow-up of ${{\mathfrak{m}}athbb{P}}^2 \times {{\mathfrak{m}}athbb{P}}^2$ along the diagonal. Lemma~\ref{non} also follows from this fact (see Theorem~\ref{sm}~$\rm (i)$ below).
{{\mathfrak{m}}athfrak{e}}nd{rem}
{{\mathfrak{m}}athfrak{b}}egin{them}\label{sm} Let $X$ be a Fano manifold with nef tangent bundle, $f: X \rightarrow Y$ a contraction of an extremal ray and $X_y$ a fiber of $f$. Then the following holds.
{{\mathfrak{m}}athfrak{b}}egin{enumerate}
\item $f$ is smooth, in particular, of fiber type.
\item $Y$ is a Fano manifold with nef tangent bundle of $\rho_Y=\rho_X-1$.
\item $X_y$ is a Fano manifold with nef tangent bundle of $\rho_{X_y}=1$.
{{\mathfrak{m}}athfrak{e}}nd{enumerate}
{{\mathfrak{m}}athfrak{e}}nd{them}
{{\mathfrak{m}}athfrak{b}}egin{proof} $\rm (i)$ This is in \cite[Theorem~5.2]{DPS} (see also \cite[Theorem~4.4]{SolW}).
$\rm (ii)$ An image of a Fano manifold by a smooth morphism is again Fano (see \cite[Corollary~2.9]{KMM}). Furthermore, it follows from \cite[Proposition~2.11 (2)]{CP} that $T_Y$ is nef.
$\rm (iii)$ From Proposition~\ref{Casa}, it follows that $X_y$ is a Fano manifold of $\rho_{X_y}=1$. Moreover \cite[Proposition~2.11 (1)]{CP} implies that $T_{X_y}$ is nef.
{{\mathfrak{m}}athfrak{e}}nd{proof}
{{\mathfrak{m}}athfrak{b}}egin{pro}\label{normal} Let $X$ be a Fano manifold with nef tangent bundle, $f: X \rightarrow Y$ a contraction of an extremal ray and $F$ a projective submanifold of $Y$ whose normal bundle is trivial, i.e., $N_{F/Y} \cong {{\mathfrak{m}}athscr{O}}_F^{\oplus l}$. Then the preimage $W:=f^{-1}(F)$ is a Fano manifold with nef tangent bundle.
{{\mathfrak{m}}athfrak{e}}nd{pro}
{{\mathfrak{m}}athfrak{b}}egin{proof} By \cite[II.~Proposition~8.10]{Ha}, we see that $T_{W/F} \cong T_{X/Y}|_W$. So we have the following exact commutative diagram:
\[\xymatrix{ &&0&0& \\
& 0 & N_{W/X} {{\mathfrak{m}}athfrak{a}}r[u] & f_W^{{{\mathfrak{m}}athfrak{a}}st}(N_{F/Y}) {{\mathfrak{m}}athfrak{a}}r[u] & \\
0 {{\mathfrak{m}}athfrak{a}}r[r] & T_{X/Y}|_W {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & T_X|_W {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & f^{{{\mathfrak{m}}athfrak{a}}st}(T_Y)|_W \cong f_W^{{{\mathfrak{m}}athfrak{a}}st}(T_Y|_F) {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & 0 \\
0 {{\mathfrak{m}}athfrak{a}}r[r] & T_{W/F} {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & T_W {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & f_W^{{{\mathfrak{m}}athfrak{a}}st}(T_F) {{\mathfrak{m}}athfrak{a}}r[r] {{\mathfrak{m}}athfrak{a}}r[u] & 0 \\
&0 {{\mathfrak{m}}athfrak{a}}r[u]&0{{\mathfrak{m}}athfrak{a}}r[u]&0{{\mathfrak{m}}athfrak{a}}r[u]& \\
} \]
Thus the snake lemma implies that $N_{W/X} \cong f_W^{{{\mathfrak{m}}athfrak{a}}st}(N_{F/Y})$. By our assumption, we obtain $N_{W/X} \cong {{\mathfrak{m}}athscr{O}}_W^{\oplus l}$. Then it follows in a similar way to Theorem~\ref{sm}~$\rm (iii)$ that $T_W$ is nef. Furthermore, the adjunction formula tells us that $-K_W=(-K_X)|_W$. This means that $W$ is also a Fano manifold.
{{\mathfrak{m}}athfrak{e}}nd{proof}
{{\mathfrak{m}}athfrak{b}}egin{exa}[Spinor bundle and Null-correlation bundle]\label{spe} \rm Let denote the null-correlation bundle on ${{\mathfrak{m}}athbb{P}}^3$ by ${{\mathfrak{m}}athscr{N}}$ (see \cite[Chapter~1, Section~4.2]{OSS} for the definition). Denote by ${{\mathfrak{m}}athscr{S}}$ the spinor bundle on $Q^3$, by ${{\mathfrak{m}}athscr{S}}_1$ and ${{\mathfrak{m}}athscr{S}}_2$ the two spinor bundles on $Q^4$ (see \cite[Definition~1.3]{Ot}).
Then it is known that ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})$ and ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{S}})$ coincides with the full-flag manifold of type $B_2$. In particular, ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})={{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{S}})$ is a homogeneous manifold.
On the other hand, the two spinor bundles ${{\mathfrak{m}}athscr{S}}_1$ and ${{\mathfrak{m}}athscr{S}}_2$ on $Q^4$ are the universal bundle and the dual of the quotient bundle (see \cite[Example~1.5]{Ot}). Thus, ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{S}}_1)$ and ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{S}}_2)$ are isomorphic to the flag manifold $F(1, 2, {{\mathfrak{m}}athbb{P}}^3)$ parametrizing pairs $(l, P)$, where $l$ is a line in a plane $P \subset {{\mathfrak{m}}athbb{P}}^3$. In particular, ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{S}}_1) \cong {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{S}}_2)$ is a homogeneous manifold.
{{\mathfrak{m}}athfrak{e}}nd{exa}
For a smooth quadric $Q^4$ of dimension $4$, let $H$ be a hyperplane section, and let $P_1$ and $P_2$ be planes in $Q^4$ whose numerical classes are different. Then we have $H^2(Q^4, {{\mathfrak{m}}athscr{Z}}Z)={{\mathfrak{m}}athscr{Z}}Z[H]$ and $H^4(Q^4, {{\mathfrak{m}}athscr{Z}}Z)= {{\mathfrak{m}}athscr{Z}}Z[P_1] \oplus {{\mathfrak{m}}athscr{Z}}Z[P_2]$. By these descriptions, we regard an element of $H^2(Q^4, {{\mathfrak{m}}athscr{Z}}Z)$ (reap. $H^4(Q^4, {{\mathfrak{m}}athscr{Z}}Z)$) as one of ${{\mathfrak{m}}athscr{Z}}Z$ (reap. ${{\mathfrak{m}}athscr{Z}}Z \oplus {{\mathfrak{m}}athscr{Z}}Z$).
{{\mathfrak{m}}athfrak{b}}egin{lem}\label{f}
Let ${{\mathfrak{m}}athscr{F}}$ be a rank $2$ stable vector bundle on $Q^4$ with Chern classes $c_1=-1$ and $c_2=(1, 1)$.
Then the tangent bundle of ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{F}})$ is not nef.
{{\mathfrak{m}}athfrak{e}}nd{lem}
{{\mathfrak{m}}athfrak{b}}egin{proof} According to \cite[Remark~3.4]{Ota}, ${{\mathfrak{m}}athscr{F}}$ extends to $Q^5$ to a Cayley bundle ${{\mathfrak{m}}athscr{C}}$. Cayley bundles are characterized by their Chern classes among rank $2$ stable bundles on $Q^5$ (see \cite[Main Theorem]{Ota}). Let $K(G_2)$ be the $5$-dimensional contact homogeneous manifold of type $G_2$. It is known that $K(G_2)$ is a linear section of the Grassmannian $G(1, {{\mathfrak{m}}athbb{P}}^6)$ with a ${{\mathfrak{m}}athbb{P}}^{13}$. For the restriction of the universal quotient bundle ${{\mathfrak{m}}athscr{Q}}$ on $G(1, {{\mathfrak{m}}athbb{P}}^6)$, we see that ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{Q}}|_{K(G_2)})$ coincides with ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{C}})$. Then it follows from \cite[1.3]{Ota} that $K(G_2)$ is the variety of special lines in $Q^5$ and ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{C}})={{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{Q}}|_{K(G_2)})$ is its flag variety $\{(p, l)| p \in l, l~{\rm special~line~in~}Q^5\}$:
\[\xymatrix{
& {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{C}})={{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{Q}}|_{K(G_2)}) {{\mathfrak{m}}athfrak{a}}r[dl]_{p_1} {{\mathfrak{m}}athfrak{a}}r[dr]^{p_2} & \\
Q^5 & & K(G_2) \\
}\]
Since $Q^4$ is a hyperplane section of $Q^5$, the restriction map $p_2|_{{{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{F}})}: {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{F}}) \rightarrow K(G_2)$ is surjective.
Furthermore, by \cite[Theorem~3.5]{Ota} and its proof, it turns out that $Q^4 \subset Q^5$ contains a special line $l_0$ in $Q^5$. It implies that $p_2|_{{{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{F}})}$ has a positive-dimensional fiber. By taking the Stein factorization, one can factor $p_2|_{{{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{F}})}$ into $g \circ f$, where $f$ is a projective morphism with connected fibers, and $g$ is a finite morphism.
Since $p_2|_{{{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{F}})}$ has a positive-dimensional fiber and ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{F}})$ is a Fano manifold (see \cite[Example~2.2]{APW}), $f$ is a contraction of an extremal face.
If the tangent bundle of ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{F}})$ would be nef, then it follows from Theorem~\ref{sm} that $f$ is of fiber type. However it contradicts to $\dim {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{F}})=\dim K(G_2)$.
{{\mathfrak{m}}athfrak{e}}nd{proof}
{{\mathfrak{m}}athfrak{b}}egin{pro}\label{fb} Let $X$ be a Fano $5$-fold with nef tangent bundle which admits a ${{\mathfrak{m}}athbb{P}}^1$-bundle structure $f: X \rightarrow Y$. Let ${{\mathfrak{m}}athscr{N}}$ be the null-correlation bundle on ${{\mathfrak{m}}athbb{P}}^3$, ${{\mathfrak{m}}athscr{S}}$ the spinor bundle on $Q^3$ and ${{\mathfrak{m}}athscr{S}}_i$ $(i=1, 2)$ the spinor bundles on $Q^4$ as in Example~\ref{spe}. Then the following holds.
{{\mathfrak{m}}athfrak{b}}egin{enumerate}
\item If $Y$ is ${{\mathfrak{m}}athbb{P}}^4$, then $X$ is ${{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}^4$.
\item If $Y$ is $Q^4$, then $X$ is ${{\mathfrak{m}}athbb{P}}^1 \times Q^4$ or ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{S}}_i)$.
\item If $Y$ is ${{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}^3$ (resp. ${{\mathfrak{m}}athbb{P}}^1 \times Q^3$), then $X$ is $({{\mathfrak{m}}athbb{P}}^1)^2 \times {{\mathfrak{m}}athbb{P}}^3$ or ${{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})$ (resp. $({{\mathfrak{m}}athbb{P}}^1)^2 \times Q^3$ or ${{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{S}})$).
\item If $Y$ is ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})$, then $X$ is ${{\mathfrak{m}}athbb{P}}^1 \times{{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})$.
\item If $Y$ is $({{\mathfrak{m}}athbb{P}}^2)^2$, then $X$ is ${{\mathfrak{m}}athbb{P}}^1 \times ({{\mathfrak{m}}athbb{P}}^2)^2 $ or ${{\mathfrak{m}}athbb{P}}^2 \times {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$.
{{\mathfrak{m}}athfrak{e}}nd{enumerate}
In particular, every manifold appeared in the above list is rational homogeneous.
{{\mathfrak{m}}athfrak{e}}nd{pro}
{{\mathfrak{m}}athfrak{b}}egin{proof} Let ${{\mathfrak{m}}athscr{E}}$ be a rank $2$ vector bundle on $Y$ such that $X={{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{E}})$.
$\rm (i)$ If $Y$ is ${{\mathfrak{m}}athbb{P}}^4$, then it follows from \cite[Main Theorem~2.4]{APW} that ${{\mathfrak{m}}athscr{E}}$ splits into a direct sum of line bundles as ${{\mathfrak{m}}athscr{O}}_Y(a) \oplus {{\mathfrak{m}}athscr{O}}_Y(b)$. If $a$ is not equal to $b$, then $Y$ has a contraction of birational type. However this contradicts to Theorem~\ref{sm}~{\rm (i)}. Hence $X$ is ${{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}^4$.
$\rm (ii)$ If $Y$ is $Q^4$, then \cite[Main Theorem~2.4]{APW} and Lemma~\ref{f} imply that $X$ is ${{\mathfrak{m}}athbb{P}}^1 \times Q^4$ or ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{S}}_i)$, via the same argument as in $\rm (i)$.
$\rm (iii)$ Let $Y$ be ${{\mathfrak{m}}athbb{P}}^1 \times V$, where $V$ is ${{\mathfrak{m}}athbb{P}}^3$ or $Q^3$. Let $p_1$ be the first projection $Y \rightarrow {{\mathfrak{m}}athbb{P}}^1$ and $p_2$ the second projection $Y \rightarrow V$:
\[\xymatrix{
& Y {{\mathfrak{m}}athfrak{a}}r[dl]_{p_1} {{\mathfrak{m}}athfrak{a}}r[dr]^{p_2} & \\
{{\mathfrak{m}}athbb{P}}^1 & & V \\
}\]
Let $l$ be a fiber of $p_2$.
According to Proposition~\ref{normal}, ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{E}}|_{l})$ is a Fano surface with nef tangent bundle. Thus, by Theorem~\ref{4}, we see that ${{\mathfrak{m}}athscr{E}}|_{l} \cong {{\mathfrak{m}}athscr{O}}_{{{\mathfrak{m}}athbb{P}}^2} \oplus {{\mathfrak{m}}athscr{O}}_{{{\mathfrak{m}}athbb{P}}^2}$ up to a twist by a line bundle. Thus, by tensoring a line bundle, we may assume that ${{\mathfrak{m}}athscr{E}}|_{l} \cong {{\mathfrak{m}}athscr{O}}_{{{\mathfrak{m}}athbb{P}}^2} \oplus {{\mathfrak{m}}athscr{O}}_{{{\mathfrak{m}}athbb{P}}^2}$ for every fiber $l$ of $p_2$. By applying Grauert's theorem \cite[III. Corollary~12.9]{Ha}, we see that ${p_2}_{{\mathfrak{m}}athfrak{a}}st({{\mathfrak{m}}athscr{E}})$ is a rank $2$ vector bundle on $V$. Furthermore, there is a natural map ${p_2}^{{{\mathfrak{m}}athfrak{a}}st}({p_2}_{{\mathfrak{m}}athfrak{a}}st({{\mathfrak{m}}athscr{E}})) \rightarrow {{\mathfrak{m}}athscr{E}}$. For $y \in l$, we have ${p_2}^{{{\mathfrak{m}}athfrak{a}}st}({p_2}_{{\mathfrak{m}}athfrak{a}}st({{\mathfrak{m}}athscr{E}})) \otimes k(y) \cong H^0(l, {{\mathfrak{m}}athscr{E}}|_l)$. Again, this follows from Grauert's theorem \cite[III. Corollary~12.9]{Ha}. Hence ${p_2}^{{{\mathfrak{m}}athfrak{a}}st}({p_2}_{{\mathfrak{m}}athfrak{a}}st({{\mathfrak{m}}athscr{E}})) \otimes k(y) \rightarrow {{\mathfrak{m}}athscr{E}} \otimes k(y)$ is surjective. By Nakayama's lemma, ${p_2}^{{{\mathfrak{m}}athfrak{a}}st}({p_2}_{{\mathfrak{m}}athfrak{a}}st({{\mathfrak{m}}athscr{E}}))_y \rightarrow {{\mathfrak{m}}athscr{E}}_y$ is also surjective, hence, so is ${p_2}^{{{\mathfrak{m}}athfrak{a}}st}({p_2}_{{\mathfrak{m}}athfrak{a}}st({{\mathfrak{m}}athscr{E}})) \rightarrow {{\mathfrak{m}}athscr{E}}$.
As a consequence, it turns out that
{{\mathfrak{m}}athfrak{b}}egin{eqnarray}
{p_2}^{{{\mathfrak{m}}athfrak{a}}st}({p_2}_{{\mathfrak{m}}athfrak{a}}st({{\mathfrak{m}}athscr{E}})) \cong {{\mathfrak{m}}athscr{E}}. \nonumber
{{\mathfrak{m}}athfrak{e}}nd{eqnarray}
For a fiber $F$ of $p_1$, ${p_2}_{{{\mathfrak{m}}athfrak{a}}st}({{\mathfrak{m}}athscr{E}})\cong {p_2}^{{{\mathfrak{m}}athfrak{a}}st}({p_2}_{{\mathfrak{m}}athfrak{a}}st({{\mathfrak{m}}athscr{E}}))|_F \cong {{\mathfrak{m}}athscr{E}}|_F$. This implies that ${{\mathfrak{m}}athscr{E}} \cong {p_2}^{{{\mathfrak{m}}athfrak{a}}st}({p_2}_{{\mathfrak{m}}athfrak{a}}st({{\mathfrak{m}}athscr{E}})) \cong {p_2}^{{{\mathfrak{m}}athfrak{a}}st}({{\mathfrak{m}}athscr{E}}|_F)$. Thus, we see that $X \cong {{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{E}}|_F)$. By Proposition~\ref{normal}, ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{E}}|_F)$ is a Fano $4$-fold with nef tangent bundle. According to Theorem~\ref{4}, if $F \cong {{\mathfrak{m}}athbb{P}}^3$ (resp. $F \cong Q^3$), then ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{E}}|_F)$ is ${{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}^3$ or ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})$ (resp. ${{\mathfrak{m}}athbb{P}}^1 \times Q^3$ or ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{S}}))$. Hence our assertion holds.
$\rm (iv)$ Let $Y$ be ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})$ and $p: {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}}) \rightarrow {{\mathfrak{m}}athbb{P}}^3$ the bundle projection. By a similar argument to $\rm (iii)$, one can show that ${{\mathfrak{m}}athscr{E}}|_l={{\mathfrak{m}}athscr{O}}_{{{\mathfrak{m}}athbb{P}}^1} \oplus {{\mathfrak{m}}athscr{O}}_{{{\mathfrak{m}}athbb{P}}^1}$ for a fiber $l$ of $p$, and ${{\mathfrak{m}}athscr{E}}=p^{{{\mathfrak{m}}athfrak{a}}st}({{\mathfrak{m}}athscr{E}}_0)$ for ${{\mathfrak{m}}athscr{E}}_0:=p_{{{\mathfrak{m}}athfrak{a}}st}({{\mathfrak{m}}athscr{E}})$. Now we have a base change diagram
\[\xymatrix{
{{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{E}}) {{\mathfrak{m}}athfrak{a}}r[r]^{} {{\mathfrak{m}}athfrak{a}}r[d] & {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{E}}_0) {{\mathfrak{m}}athfrak{a}}r[d] \\
{{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}}) {{\mathfrak{m}}athfrak{a}}r[r]^{p} & {{\mathfrak{m}}athbb{P}}^3 \\
} \]
Since $X={{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{E}})$ is a ${{\mathfrak{m}}athbb{P}}^1$-bundle over ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{E}}_0)$, ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{E}}_0)$ is a Fano $4$-fold with nef tangent bundle. Moreover ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{E}}_0)$ is a ${{\mathfrak{m}}athbb{P}}^1$-bundle over ${{\mathfrak{m}}athbb{P}}^3$. Thus, by Theorem~\ref{4}, ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{E}}_0)$ is ${{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}^3$ or ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})$. This implies that $X$ is ${{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})$ or ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})\times_{{{\mathfrak{m}}athbb{P}}^3} {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})$. In the later case, we can show that the tangent bundle of $X$ is not nef in a similar way to Lemma~\ref{non}. Indeed, ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})$ admits a ${{\mathfrak{m}}athbb{P}}^1$-bundle structure over $Q^3$ and denote its fiber by $l$. Remark that $l$ can be regarded as a curve in $X:={{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})\times_{{{\mathfrak{m}}athbb{P}}^3} {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})$ via the diagonal embedding ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}}) \subset X$. Then we see that $-K_X.l=0$. This implies that $X$ is not Fano. Hence our assertion holds.
$\rm (v)$ Let $Y$ be $({{\mathfrak{m}}athbb{P}}^2)^2$ and $p_i$ the $i$-th projection $Y \rightarrow {{\mathfrak{m}}athbb{P}}^2$ ($i=1$, $2$). Let $F_i$ be a fiber of $p_i$. According to Proposition~\ref{normal}, ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{E}}|_{F_i})$ is a Fano manifold with nef tangent bundle. Thus, by Theorem~\ref{4}, we see that ${{\mathfrak{m}}athscr{E}}|_{F_i} \cong {{\mathfrak{m}}athscr{O}}_{{{\mathfrak{m}}athbb{P}}^2} \oplus {{\mathfrak{m}}athscr{O}}_{{{\mathfrak{m}}athbb{P}}^2}$ or $T_{{{\mathfrak{m}}athbb{P}}^2}(-1)$, up to a twist by a line bundle. If ${{\mathfrak{m}}athscr{E}}|_{F_i} \cong {{\mathfrak{m}}athscr{O}}_{{{\mathfrak{m}}athbb{P}}^2} \oplus {{\mathfrak{m}}athscr{O}}_{{{\mathfrak{m}}athbb{P}}^2}$ for some $i$, then we see that ${p_i}^{{{\mathfrak{m}}athfrak{a}}st}({p_i}_{{\mathfrak{m}}athfrak{a}}st({{\mathfrak{m}}athscr{E}})) \cong {{\mathfrak{m}}athscr{E}}$ in a similar way to $\rm (iii)$. Furthermore, ${{\mathfrak{m}}athscr{E}} \cong {p_i}^{{{\mathfrak{m}}athfrak{a}}st}({{\mathfrak{m}}athscr{E}}|_{F_j})$ and ${{\mathfrak{m}}athscr{E}}|_{F_j} \cong {{\mathfrak{m}}athscr{O}}_{{{\mathfrak{m}}athbb{P}}^2} \oplus {{\mathfrak{m}}athscr{O}}_{{{\mathfrak{m}}athbb{P}}^2}$ or $T_{{{\mathfrak{m}}athbb{P}}^2}(-1)$ for $j \neq i$. As a consequence, $X$ is ${{\mathfrak{m}}athbb{P}}^1 \times ({{\mathfrak{m}}athbb{P}}^2)^2 $ or ${{\mathfrak{m}}athbb{P}}^2 \times {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$. On the other hand, assume that ${{\mathfrak{m}}athscr{E}}|_{F_i} \cong T_{{{\mathfrak{m}}athbb{P}}^2}(-1)$ for $i=1, 2$. Then $c_1({{\mathfrak{m}}athscr{E}})=(1, 1)$. This implies that ${{\mathfrak{m}}athscr{O}}_X(-K_X) \cong {{\mathfrak{m}}athscr{O}}_{{{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{E}})}(2) \otimes f^{{{\mathfrak{m}}athfrak{a}}st}{{\mathfrak{m}}athscr{O}}_{{{\mathfrak{m}}athbb{P}}^2 \times {{\mathfrak{m}}athbb{P}}^2}(2,2)$, where ${{\mathfrak{m}}athscr{O}}_{{{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{E}})}(1)$ is the tautological invertible sheaf of $X={{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{E}})$. This implies that the Fano index of $X$ is $2$. According to Theorem~\ref{sm}, $X$ has only contractions of fiber type. Thus,
it follows from \cite[Proposition~7.1]{NO} that $X$ is a product with ${{\mathfrak{m}}athbb{P}}^1$ as a factor. However, this contradicts to ${{\mathfrak{m}}athscr{E}}|_{F_i} \cong T_{{{\mathfrak{m}}athbb{P}}^2}(-1)$.
{{\mathfrak{m}}athfrak{e}}nd{proof}
{{\mathfrak{m}}athfrak{b}}egin{pro}\label{P} Let $X$ be a Fano $5$-fold with nef tangent bundle. Then $\rho_X \leq 3$ or $X$ is one of the following:\\
$({{\mathfrak{m}}athbb{P}}^1)^5$, $({{\mathfrak{m}}athbb{P}}^1)^3 \times {{\mathfrak{m}}athbb{P}}^2$, $({{\mathfrak{m}}athbb{P}}^1)^2 \times {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$.
{{\mathfrak{m}}athfrak{e}}nd{pro}
{{\mathfrak{m}}athfrak{b}}egin{proof} By Lemma~\ref{pi}, the pseudoindex of $X$ is at least $2$. Moreover, $X$ has only contractions of fiber type because of Theorem~\ref{sm}. Thus, by applying Proposition~\ref{NO}, we get our assertion.
{{\mathfrak{m}}athfrak{e}}nd{proof}
\section{Proof of Theorem~\ref{MT}}
Let ${{\mathfrak{m}}athscr{N}}$ be the null-correlation bundle on ${{\mathfrak{m}}athbb{P}}^3$, ${{\mathfrak{m}}athscr{S}}$ the spinor bundle on $Q^3$ and ${{\mathfrak{m}}athscr{S}}_i$ $(i=1, 2)$ the spinor bundles on $Q^4$ as in Example~\ref{spe}. In this section, we prove Theorem~\ref{MT}:
{{\mathfrak{m}}athfrak{b}}egin{them}[=Theorem~\ref{MT}]\label{MT2}
Let $X$ be a Fano manifold of dimension $5$ with nef tangent bundle and Picard number $\rho_X>1$. Then $X$ is one of the following:\\
${{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}^4$, ${{\mathfrak{m}}athbb{P}}^1 \times Q^4$, ${{\mathfrak{m}}athbb{P}}^2 \times {{\mathfrak{m}}athbb{P}}^3$, ${{\mathfrak{m}}athbb{P}}^2 \times Q^3$, ${{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^3})$, ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{S}}_i)$, ${{\mathfrak{m}}athbb{P}}^1 \times ({{\mathfrak{m}}athbb{P}}^2)^2$, $({{\mathfrak{m}}athbb{P}}^1)^2 \times {{\mathfrak{m}}athbb{P}}^3$, $({{\mathfrak{m}}athbb{P}}^1)^2 \times Q^3$, ${{\mathfrak{m}}athbb{P}}^2 \times {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$, ${{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})={{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{S}})$,
$({{\mathfrak{m}}athbb{P}}^1)^3 \times {{\mathfrak{m}}athbb{P}}^2$, $({{\mathfrak{m}}athbb{P}}^1)^2 \times {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$, $({{\mathfrak{m}}athbb{P}}^1)^5$.
In particular, $X$ is a rational homogeneous manifold.
{{\mathfrak{m}}athfrak{e}}nd{them}
Let $X$ be a Fano $5$-fold with nef tangent bundle of $\rho_X {\mathfrak{g}}eq 2$. Then there exist two different contractions $f: X \rightarrow Y$ and $g: X \rightarrow
Z$ of extremal rays:
\[\xymatrix{
X {{\mathfrak{m}}athfrak{a}}r[r]^{f} {{\mathfrak{m}}athfrak{a}}r[d]_{g} & Y \\
Z & \\
} \]
Denote by $X_y$ (resp. $X_z$) a fiber of $f$ (resp. one of $g$). We may assume that $\dim Z {\mathfrak{g}}eq \dim Y ({\mathfrak{g}}eq 1)$.
{{\mathfrak{m}}athfrak{b}}egin{lem}\label{l} Under the above setting, the following holds.
{{\mathfrak{m}}athfrak{b}}egin{enumerate}
\item $\rho_Y=\rho_Z$.
\item $Y$ and $Z$ are rational homogeneous manifolds listed in Theorem~\ref{4}. Furthermore, $X_y$ and $X_z$ are either ${{\mathfrak{m}}athbb{P}}^d$ {\rm (}$1 \leq d \leq 4${\rm )} or $Q^d$ {\rm (}$d=3$ or $4${\rm )}.
\item $5>\dim Y {\mathfrak{g}}eq \dim X_z$ and $5>\dim Z {\mathfrak{g}}eq \dim X_y$.
\item If $\dim Z = \dim X_y$ and $X_y \cong {{\mathfrak{m}}athbb{P}}^d$ (resp. $\dim Y = \dim X_z$ and $X_z \cong {{\mathfrak{m}}athbb{P}}^d$), then we have $X \cong {{\mathfrak{m}}athbb{P}}^d \times Y$ (resp. ${{\mathfrak{m}}athbb{P}}^d \times Z$).
\item If $\dim Z = \dim X_y$ and $X_y \cong Q^d${\rm (}$d=3$ or $4${\rm )} (resp. $\dim Y = \dim X_z$ and $X_z \cong Q^d$), then $Z$ (resp. $Y$) is either ${{\mathfrak{m}}athbb{P}}^d$ or $Q^d$ and $X$ is a ${{\mathfrak{m}}athbb{P}}^{5-d}$-bundle over $Z$ (resp. $Y$).
{{\mathfrak{m}}athfrak{e}}nd{enumerate}
{{\mathfrak{m}}athfrak{e}}nd{lem}
{{\mathfrak{m}}athfrak{b}}egin{proof} $\rm (i)$ Since $f$ and $g$ are contractions of extremal rays, $\rho_Y=\rho_X-1=\rho_Z$.
$\rm (ii)$ From Theorem~\ref{sm}, $Y$, $Z$, $X_y$ and $X_z$ are Fano manifolds with nef tangent bundles, and $\rho_{X_y}=\rho_{X_z}=1$. Hence Theorem~\ref{4} implies our assertion.
$\rm (iii)$ Since $f$ and $g$ are different contractions, $X_y$ and $X_z$ are not contracted by $g$ and $f$, respectively. Furthermore, we have $\rho_{X_y}=\rho_{X_z}=1$. This implies that $\dim Y {\mathfrak{g}}eq \dim X_z$ and $\dim Z {\mathfrak{g}}eq \dim X_y$.
$\rm (iv)$ If $\dim Z = \dim X_y$ and $X_y \cong {{\mathfrak{m}}athbb{P}}^d$, then our claim follows from Proposition~\ref{Br} and Proposition~\ref{NO2}.
$\rm (v)$ We see that $Z \cong {{\mathfrak{m}}athbb{P}}^d$ or $Q^d$ by \cite[Proposition~8]{PS}, and it follows from $\rm (ii)$ and Proposition~\ref{Br} that $X$ is a ${{\mathfrak{m}}athbb{P}}^{5-d}$-bundle over $Z$.
{{\mathfrak{m}}athfrak{e}}nd{proof}
\subsection{Case where $\dim Y=1$ }
{{\mathfrak{m}}athfrak{b}}egin{pro}\label{l1} If $\dim Y=1$, then $X$ is $ {{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}^4$ or ${{\mathfrak{m}}athbb{P}}^1 \times Q^4$.
{{\mathfrak{m}}athfrak{e}}nd{pro}
{{\mathfrak{m}}athfrak{b}}egin{proof} By Lemma~\ref{l}~$\rm (ii)$, $Y \cong {{\mathfrak{m}}athbb{P}}^1$ and $X_y \cong {{\mathfrak{m}}athbb{P}}^4$ or $Q^4$. Furthermore, it follows from Lemma~\ref{l}~$\rm (iii)$ that $\dim Z=\dim X_y =4$. If $X_y \cong {{\mathfrak{m}}athbb{P}}^4$, then Lemma~\ref{l}~$\rm (iv)$ concludes that $X \cong {{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}^4$. On the other hand, if $X_y \cong Q^4$, then Lemma~\ref{l}~$\rm (v)$ tells us that $X$ is a ${{\mathfrak{m}}athbb{P}}^{1}$-bundle over $Z$. Then, using Proposition~\ref{NO2}, we see that $X \cong {{\mathfrak{m}}athbb{P}}^1 \times Q^4$.
{{\mathfrak{m}}athfrak{e}}nd{proof}
\subsection{Case where $\dim Y=2$ }
{{\mathfrak{m}}athfrak{b}}egin{pro} If $\dim Y=2$, then $X \cong {{\mathfrak{m}}athbb{P}}^2 \times {{\mathfrak{m}}athbb{P}}^3$, ${{\mathfrak{m}}athbb{P}}^2 \times Q^3$, $({{\mathfrak{m}}athbb{P}}^1)^2 \times {{\mathfrak{m}}athbb{P}}^3$ or $({{\mathfrak{m}}athbb{P}}^1)^2 \times Q^3$.
{{\mathfrak{m}}athfrak{e}}nd{pro}
{{\mathfrak{m}}athfrak{b}}egin{proof}
By Lemma~\ref{l}, we see that $Y \cong {{\mathfrak{m}}athbb{P}}^2$ or $({{\mathfrak{m}}athbb{P}}^1)^2$, $X_y \cong {{\mathfrak{m}}athbb{P}}^3$ or $Q^3$ and $\dim Z=3$ or $4$, in a similar way to Proposition~\ref{l1}.
If $Y \cong {{\mathfrak{m}}athbb{P}}^2$ and $\dim Z=3$, then we have $\dim Y= \dim X_z$ and it follows from Lemma~\ref{l}~$\rm (ii)$ that $X_z \cong {{\mathfrak{m}}athbb{P}}^2$. Therefore Lemma~\ref{l}~$\rm (iv)$ implies that $X \cong {{\mathfrak{m}}athbb{P}}^2 \times {{\mathfrak{m}}athbb{P}}^3$ or ${{\mathfrak{m}}athbb{P}}^2 \times Q^3$.
If $Y \cong {{\mathfrak{m}}athbb{P}}^2$ and $\dim Z=4$, then $X$ is a ${{\mathfrak{m}}athbb{P}}^1$-bundle over ${{\mathfrak{m}}athbb{P}}^4$ or $Q^4$ by Lemma~\ref{l}~$\rm (ii)$ and Proposition~\ref{Br}. Therefore we are in the situation of Proposition~\ref{fb}~$\rm (i)$ and $\rm (ii)$. However every manifold appeared there has no contractions to ${{\mathfrak{m}}athbb{P}}^2$. Hence we get a contradiction.
If $Y \cong ({{\mathfrak{m}}athbb{P}}^1)^2$, then it follows from Lemma~\ref{l}~$\rm (i)$ that $\rho_Z=\rho_Y=2$. By virtue of Lemma~\ref{l}~${\rm (ii)}$, $X_y \cong {{\mathfrak{m}}athbb{P}}^3$ or $Q^3$. If $\dim Z=3$, then $Z$ would be isomorphic to ${{\mathfrak{m}}athbb{P}}^3$ or $Q^3$ by Lemma~\ref{l}~$\rm (iv)$ and $\rm (v)$. This contradicts to $\rho_Z=2$. Hence $\dim Z=4$. Then, it follows from Lemma~\ref{l}~$\rm (ii)$ and Proposition~\ref{Br} that $X$ is a ${{\mathfrak{m}}athbb{P}}^1$-bundle over $Z$, where $Z \cong {{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}^3, {{\mathfrak{m}}athbb{P}}^1 \times Q^3$, $({{\mathfrak{m}}athbb{P}}^2)^2$ or ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})$. Thus we are in the situation of Proposition~\ref{fb}~$\rm (iii)-\rm (v)$. Since $X$ admits a contraction of an extremal ray to $Y \cong ({{\mathfrak{m}}athbb{P}}^1)^2$, we see that $X \cong ({{\mathfrak{m}}athbb{P}}^1)^2 \times {{\mathfrak{m}}athbb{P}}^3$ or $({{\mathfrak{m}}athbb{P}}^1)^2 \times Q^3$.
{{\mathfrak{m}}athfrak{e}}nd{proof}
\subsection{Case where $\dim Y=3$ }
{{\mathfrak{m}}athfrak{b}}egin{pro} If $\dim Y=3$, then $X \cong {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^3})$, ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{S}}_i)$, ${{\mathfrak{m}}athbb{P}}^1 \times ({{\mathfrak{m}}athbb{P}}^2)^2$, ${{\mathfrak{m}}athbb{P}}^2 \times {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$ or $({{\mathfrak{m}}athbb{P}}^1)^3 \times {{\mathfrak{m}}athbb{P}}^2$.
{{\mathfrak{m}}athfrak{e}}nd{pro}
{{\mathfrak{m}}athfrak{b}}egin{proof} According to Proposition~\ref{P}, $\rho_X \leq 3$ if $X$ is not isomorphic to $({{\mathfrak{m}}athbb{P}}^1)^5$, $({{\mathfrak{m}}athbb{P}}^1)^3 \times {{\mathfrak{m}}athbb{P}}^2$ or $({{\mathfrak{m}}athbb{P}}^1)^2 \times {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$. Since $({{\mathfrak{m}}athbb{P}}^1)^5$ and $({{\mathfrak{m}}athbb{P}}^1)^2 \times {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$ have no contractions of extremal rays to $3$-dimensional manifolds, we have $X \cong ({{\mathfrak{m}}athbb{P}}^1)^3 \times {{\mathfrak{m}}athbb{P}}^2$ or $\rho_X \leq 3$. So it is enough to consider the case where $\rho_X \leq 3$. Then it follows from Lemma~\ref{l}~$\rm (i)$ that $\rho_Y=\rho_Z \leq 2$. By our assumption, we see that $5> \dim Z {\mathfrak{g}}eq \dim Y =3$.
If $\dim Z=3$, then it follows from Lemma~\ref{l}~$\rm (ii)$ and Proposition~\ref{Br} that $X$ admits two different ${{\mathfrak{m}}athbb{P}}^2$-bundle structures. By Proposition~\ref{WO}, $X ={{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^3})$ or $Y \times_CZ$, where $Y$ and $Z$ are ${{\mathfrak{m}}athbb{P}}^2$-bundles over a smooth curve $C$. In the latter case, since $Y$ and $Z$ are projective bundles of $\rho = 2$, it follows from Theorem~\ref{4}~$\rm (iii)$ that $Y \cong Z \cong {{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}^2$ and $C \cong {{\mathfrak{m}}athbb{P}}^1$. Therefore, $X \cong ({{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}^2) \times_{{{\mathfrak{m}}athbb{P}}^1}({{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}^2) \cong {{\mathfrak{m}}athbb{P}}^1 \times ({{\mathfrak{m}}athbb{P}}^2)^2$.
If $\dim Z=4$, then Lemma~\ref{l}~$\rm (ii)$ and Proposition~\ref{Br} imply that $X$ is a ${{\mathfrak{m}}athbb{P}}^1$-bundle over ${{\mathfrak{m}}athbb{P}}^4$, $Q^4$, ${{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}^3$, ${{\mathfrak{m}}athbb{P}}^1 \times Q^3$, $({{\mathfrak{m}}athbb{P}}^2)^2$ or ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})$. Therefore we are in the situation of Proposition~\ref{fb}~$\rm (i)-(v)$. Since $X$ admits a contraction of an extremal ray to a $3$-dimensional manifold $Y$, $X$ is ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{S}}_i)$, ${{\mathfrak{m}}athbb{P}}^1 \times ({{\mathfrak{m}}athbb{P}}^2)^2$ or ${{\mathfrak{m}}athbb{P}}^2 \times {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$.
{{\mathfrak{m}}athfrak{e}}nd{proof}
\subsection{Case where $\dim Y=4$ }
{{\mathfrak{m}}athfrak{b}}egin{pro} If $\dim Y=4$, then $X$ is isomorphic to one of the following:\\
$({{\mathfrak{m}}athbb{P}}^1)^5$, $({{\mathfrak{m}}athbb{P}}^1)^3 \times {{\mathfrak{m}}athbb{P}}^2$, $({{\mathfrak{m}}athbb{P}}^1)^2 \times {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$, $({{\mathfrak{m}}athbb{P}}^1)^2 \times {{\mathfrak{m}}athbb{P}}^3$, $({{\mathfrak{m}}athbb{P}}^1)^2 \times Q^3$, ${{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})={{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{S}})$, ${{\mathfrak{m}}athbb{P}}^2 \times {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$.
{{\mathfrak{m}}athfrak{e}}nd{pro}
{{\mathfrak{m}}athfrak{b}}egin{proof}
According to Proposition~\ref{P}, $\rho_X \leq 3$ if $X$ is not isomorphic to $({{\mathfrak{m}}athbb{P}}^1)^5$, $({{\mathfrak{m}}athbb{P}}^1)^3 \times {{\mathfrak{m}}athbb{P}}^2$ or $({{\mathfrak{m}}athbb{P}}^1)^2 \times {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$. So it is enough to consider the case where $\rho_X \leq 3$. Then it is equivalent to $\rho_Y=\rho_Z \leq 2$.
Lemma~\ref{l}~$\rm (ii)$ and Proposition~\ref{Br} imply that $X$ admits two different ${{\mathfrak{m}}athbb{P}}^1$-bundle structures over $4$-folds $Y$ and $Z$ of $\rho \leq 2$. By Lemma~\ref{l}~$\rm (ii)$, $Y$ and $Z$ are ${{\mathfrak{m}}athbb{P}}^4$, $Q^4$, ${{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}^3$, ${{\mathfrak{m}}athbb{P}}^1 \times Q^3$, $({{\mathfrak{m}}athbb{P}}^2)^2$ or ${{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})$. Therefore we are in the situation of Proposition~\ref{fb}~$\rm (i)-(v)$. Since $X$ admits two different ${{\mathfrak{m}}athbb{P}}^1$-bundle structures over $4$-folds $Y$ and $Z$ of $\rho \leq 2$, $X$ is $({{\mathfrak{m}}athbb{P}}^1)^2 \times {{\mathfrak{m}}athbb{P}}^3$, $({{\mathfrak{m}}athbb{P}}^1)^2 \times Q^3$, ${{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{N}})={{\mathfrak{m}}athbb{P}}^1 \times {{\mathfrak{m}}athbb{P}}({{\mathfrak{m}}athscr{S}})$ or ${{\mathfrak{m}}athbb{P}}^2 \times {{\mathfrak{m}}athbb{P}}(T_{{{\mathfrak{m}}athbb{P}}^2})$.
{{\mathfrak{m}}athfrak{e}}nd{proof}
\section{Case where $\rho_X=1$}\label{1}
Finally, we deal with Fano manifolds with nef tangent bundles of $\rho_X=1$. All the results in this section are well-known for experts.
{{\mathfrak{m}}athfrak{b}}egin{them}\label{} Let $X$ be a smooth Fano $n$-fold with nef tangent bundle of $\rho_X=1$. Then the pseudoindex $i_X$ satisfies $3 \leq i_X \leq n+1$. Furthermore, the following holds.
{{\mathfrak{m}}athfrak{b}}egin{enumerate}
\item If $i_X=n+1$, then $X$ is ${{\mathfrak{m}}athbb{P}}^n$.
\item If $i_X=n$, then $X$ is $Q^n$.
\item If $i_X=3$, then $X$ is ${{\mathfrak{m}}athbb{P}}^2$, $Q^3$ or $K(G_2)$, where $K(G_2)$ is the $5$-dimensional contact homogeneous manifold of type $G_2$.
{{\mathfrak{m}}athfrak{e}}nd{enumerate}
{{\mathfrak{m}}athfrak{e}}nd{them}
{{\mathfrak{m}}athfrak{b}}egin{proof} By virtue of Lemma~\ref{pi}, we see that $2 \leq i_X$. Furthermore, it follows from the argument as in \cite[Before Theorem~4.3, P. 623]{Hwang} that $i_X$ is not $2$. On the other hand, if $i_X {\mathfrak{g}}eq n+1$, then $X$ is ${{\mathfrak{m}}athbb{P}}^n$. This is dealt in \cite{CMSB}. If $i_X=n$, then our assertion follows from \cite{Mi}. The case where $i_X=3$ is treated in \cite[Theorem~4.3]{Hwang}.
{{\mathfrak{m}}athfrak{e}}nd{proof}
As a consequence, we have the following:
{{\mathfrak{m}}athfrak{b}}egin{cor}\label{} Let $X$ be a smooth Fano $5$-fold with nef tangent bundle of $\rho_X=1$. Then one of the following holds.
{{\mathfrak{m}}athfrak{b}}egin{enumerate}
\item $X$ is ${{\mathfrak{m}}athbb{P}}^5$, $Q^5$ or $K(G_2)$.
\item $i_X=4$
{{\mathfrak{m}}athfrak{e}}nd{enumerate}
{{\mathfrak{m}}athfrak{e}}nd{cor}
{{\mathfrak{m}}athfrak{b}}egin{rem} \rm Let $X$ be a smooth Fano $5$-fold with nef tangent bundle of $\rho_X=1$. For the ample generator $H$ of ${\rm Pic}(X)$, if there exists a rational curve $l$ such that $H.l=1$, then we see that the Fano index coincides with the pseudoindex $i_X=4$. Hence, it turns out that $X$ is a Fano $5$-fold with index $4$. In other words, $X$ is a del Pezzo $5$-fold.
On the other hand, a rational homogeneous manifold of $\rho=1$ contains a line (see for instance \cite[V.1.15]{Ko}). Furthermore, we see that there is no rational homogeneous $5$-fold of $\rho=1$ with $i_X=4$.
{{\mathfrak{m}}athfrak{e}}nd{rem} \
{{{\mathfrak{m}}athfrak{b}}f Acknowledgements}
The author would like to thank Dr. Kazunori Yasutake for reading this paper and his comments. He also would like to express his gratitude to referees for their careful reading of the text and useful suggestions and comments. The author is partially supported by the Grant-in-Aid for Research Activity Start-up $\sharp$24840008 from the Japan Society for the Promotion of Science.
{{\mathfrak{m}}athfrak{b}}egin{thebibliography}{7}
{{\mathfrak{m}}athfrak{b}}ibitem{APW} V. Ancona, T. Peternell, J. A. Wi\'sniewski, {\it Fano bundles and splitting theorems on projective spaces and quadrics}, Pacific J. Math. 163 (1994), no. 1, 17-42.
{{\mathfrak{m}}athfrak{b}}ibitem{Borel} A. Borel, Cohomologie des espaces localement compacts d'aprs J. Leray. Lecture Notes in Mathematics, Vol. 2 Springer-Verlag, Berlin-G${\rm \ddot{o}}$ttingen-Heidelberg, 1964.
{{\mathfrak{m}}athfrak{b}}ibitem{BCDD} Bonavero, L., Casagrande, C., Debarre, O., Druel, S., {\it Sur une conjecture de Mukai}, Com- ment. Math. Helv. 78 (2003), 601-626.
{{\mathfrak{m}}athfrak{b}}ibitem{Casa} C. Casagrande, {\it Quasi-elementary contractions of Fano manifolds}, Compos. Math. 144 (2008), no. 6, 1429-1460.
{{\mathfrak{m}}athfrak{b}}ibitem{CP} F. Campana, T. Peternell, {\it Projective manifolds whose tangent bundles are numerically effective}, Math. Ann. 289 (1991), 169-187.
{{\mathfrak{m}}athfrak{b}}ibitem{CP2} F. Campana, T. Peternell, {\it 4-folds with numerically effective tangent bundles and second Betti numbers greater than one}, Manuscripta Math. 79 (1993), no. 3-4, 225-238.
{{\mathfrak{m}}athfrak{b}}ibitem{CMSB} K. Cho, Y. Miyaoka and N. I. Shepherd-Barron, {\it Characterizations of projective space and applications to complex symplectic manifolds}, in {\it Higher dimensional birational geometry}, Kyoto, 1997, Advanced Studies in Pure Mathematics, vol. 35, (Mathematical Society of Japan, Tokyo, 2002), 1-88.
{{\mathfrak{m}}athfrak{b}}ibitem{DPS} J. P. Demailly, T. Peternell, M. Schneider, {\it Compact complex manifolds with numerically effective tangent bundles}, J. Algebraic Geom. 3 (1994), no. 2, 295-345.
{{\mathfrak{m}}athfrak{b}}ibitem{Gr} Grothendieck, A.: Le groupe de Brauer I, II, III. Exemples et compl{\'e}ments. In: Dix Expos{\'e}s sur la Coho- mologie des Sch{\'e}mas, North-Holland, Amsterdam, 1968, pp.46-188.
{{\mathfrak{m}}athfrak{b}}ibitem{Ha} R. Hartshorne, Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
{{\mathfrak{m}}athfrak{b}}ibitem{Hwang} J.M. Hwang, {\it Rigidity of rational homogeneous spaces}, Proceedings of ICM. 2006 Madrid, volume II, European Mathematical Society, 2006, 613-626.
{{\mathfrak{m}}athfrak{b}}ibitem{Ko} J. Koll\'ar, {\it Rational curves on algebraic varieties}, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 32 (Springer, Berlin, 1996).
{{\mathfrak{m}}athfrak{b}}ibitem{KMM} J. Koll\'ar, Y. Miyaoka and S. Mori, {\it Rational connectedness and boundedness of Fano manifolds}, J. Differential Geom. 36 (1992), no. 3, 765-779.
{{\mathfrak{m}}athfrak{b}}ibitem{Mi} Y. Miyaoka, {\it Numerical characterisations of hyperquadrics}, Complex analysis in several variables-Memorial Conference of Kiyoshi Oka's Centennial Birthday, Advanced Studies of Pure Mathematics, vol. 42, (Mathematical Society of Japan, Tokyo, 2004), 209-235.
{{\mathfrak{m}}athfrak{b}}ibitem{Mok} N. Mok, {\it On Fano manifolds with nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents}, Trans. Amer. Math. Soc. 354 (2002), no. 7, 2639-2658.
{{\mathfrak{m}}athfrak{b}}ibitem{Mori} S. Mori, {\it Projective manifolds with ample tangent bundles}, Ann. of Math. (2) 110 (1979), no. 3, 593-606.
{{\mathfrak{m}}athfrak{b}}ibitem{NO} C. Novelli, G. Occhetta, {\it Ruled Fano fivefolds of index two}, Indiana Univ. Math. J. 56 (2007), no. 1, 207-241.
{{\mathfrak{m}}athfrak{b}}ibitem{OW} G. Occhetta, J. A. Wi\'sniewski, {\it On Euler-Jaczewski sequence and. Remmert-Van de Ven problem for toric varieties}, Math. Z. 241 (2002), 35-44.
{{\mathfrak{m}}athfrak{b}}ibitem{OSS} C. Okonek, M. Schneider and H. Spindler, Vector bundles over complex projective space, Progress in Math., vol. 3, Birkh$\rm \ddot{a}$user, Boston, Basel, Stuttgart, 1980.
{{\mathfrak{m}}athfrak{b}}ibitem{Ot} G. Ottaviani, {\it Spinor bundles on quadrics}, Trans. Amer. Math. Soc. 307 (1988), no. 1, 301-316.
{{\mathfrak{m}}athfrak{b}}ibitem{Ota} G. Ottaviani, {\it On Cayley bundles on the five-dimensional quadric}, Boll. Un. Mat. Ital. A (7) 4 (1990), 87-100.
{{\mathfrak{m}}athfrak{b}}ibitem{PS} K. H. Paranjape, V. Srinivas, {\it Self-maps of homogeneous spaces}, Invent. Math. 98 (1989), no. 2, 425-444.
{{\mathfrak{m}}athfrak{b}}ibitem{SolW} L. Sol\'a Conde, J. A. Wi\'sniewski, {\it On manifolds whose tangent bundle is big and 1-ample}, Proc. London Math. Soc. (3) 89 (2004), no. 2, 273-290.
{{\mathfrak{m}}athfrak{b}}ibitem{SW} M. Szurek, J. A. Wi\'sniewski, {\it Fano bundles over ${{\mathfrak{m}}athbb{P}}^3$ and $Q^3$}, Pacific J. Math. 141 (1990), no. 1, 197-208.
{{\mathfrak{m}}athfrak{e}}nd{thebibliography}
{{\mathfrak{m}}athfrak{e}}nd{document} | math | 58,011 |
\begin{document}
\begin{center}
{\bf {\large On the $SU(3)$ Parametrization of
Qutrits}\footnote{Paper presented at
The 12th Central European Workshop on Quantum Optics, 6-9 June 2005, Ankara, Turkey.}\\
A. T. B\"{o}l\"{u}kba\c{s}{\i}\footnote{E.mail:
[email protected]} ,
\underline{T. Dereli}\footnote{E.mail: [email protected]}}
Department of Physics, Ko\c{c} University\\
34450 Sar{\i}yer, \.{I}stanbul, Turkey\\
{\bf Abstract}
\end{center}
\noindent {\small Parametrization of qutrits on the complex
projective plane $\mathcal{C}P^{2}=SU(3)/U(2)$ is given
explicitly. A set of constraints that characterize mixed state
density matrices is found.}
\vskip 8mm
\noindent Many recent ideas of quantum information theory are
based on the notion of qubits. A qubit may be represented by a
point on the Poincar\'{e} sphere $S^2$ that is homeomorhic to the
complex projective line $ \mathcal{H}^{(2)}= \mathcal{C}P^{1} =
SU(2)/U(1)$. A similar parametrization in the case of higher
dimensional quantum systems is desirable both from theoretical [1]
and technical points of view [2], [3]. A qutrit may be represented
by a point on the complex projective plane
$\mathcal{H}^{(3)}=\mathcal{C}P^{2}=SU(3)/U(2)$ . Such a
representation is given explicitly in terms of Gell-Mann matrices
[4]. We determine a set of constraints that characterize mixed
states of qutrits below.
A qutrit $|\psi>=\alpha_{0}|0>+\alpha_{1}|1>+\alpha_{2}
|2>$where $\alpha_{0},\alpha_{1},\alpha_{2}\in\mathbf{{C}}$ , $|\alpha
_{0}|^{2}+|\alpha_{1}|^{2}+|\alpha_{2}|^{2}=1$ , is a state vector in the
Hilbert space of states $\mathcal{H}^{(3)}$ of a 3-level system. It is spanned
by an orthonormal basis $\{|0>,|1>,|2>\}$ which in matrix notation reads
\[
|0>\rightarrow\left(
\begin{array}
[c]{c}
1\\
0\\
0
\end{array}
\right) ,|1>\rightarrow\left(
\begin{array}
[c]{c}
0\\
1\\
0
\end{array}
\right) ,|2>\rightarrow\left(
\begin{array}
[c]{c}
0\\
0\\
1
\end{array}
\right) .
\]
Therefore
\[
|\psi>\rightarrow\left(
\begin{array}
[c]{c}
\alpha_{0}\\
\alpha_{1}\\
\alpha_{2}
\end{array}
\right) \in\mathbf{C}^{3}\simeq\mathbf{R}^{6}.
\]
Since $|\alpha_{0}|^{2}+|\alpha_{1}|^{2}+|\alpha_{2}|^{2}=1$ and since
$|\psi>$ is determined up to a multiplicative phase factor, dim$\mathcal{H}
^{(3)}=4$. \newline Any $3\times3$ density matrix can be written
as
\[
\rho=\frac{1}{3}(I+\sqrt{3}\vec{n}\cdot\vec{\lambda})
\]
where $\vec{n}$ is a real 8-vector, and components of $\vec{\lambda}$ are the
(Hermitian, traceless) Gell-Mann matrices
\[
\lambda_{1}=\left(
\begin{array}
[c]{ccc}
0 & 1 & 0\\
1 & 0 & 0\\
0 & 0 & 0
\end{array}
\right) , \text{ \ \ }\lambda_{2}=\left(
\begin{array}
[c]{ccc}
0 & -i & 0\\
i & 0 & 0\\
0 & 0 & 0
\end{array}
\right) , \text{ \ }\lambda_{4}=\left(
\begin{array}
[c]{ccc}
0 & 0 & 1\\
0 & 0 & 0\\
1 & 0 & 0
\end{array}
\right) , \text{\ }
\]
\[
\text{ \ \ }\lambda_{5}=\left(
\begin{array}
[c]{ccc}
0 & 0 & -i\\
0 & 0 & 0\\
i & 0 & 0
\end{array}
\right) , \text{ \ }\lambda_{6}=\left(
\begin{array}
[c]{ccc}
0 & 0 & 0\\
0 & 0 & 1\\
0 & 1 & 0
\end{array}
\right) , \text{\ }\lambda_{7}=\left(
\begin{array}
[c]{ccc}
0 & 0 & 0\\
0 & 0 & -i\\
0 & i & 0
\end{array}
\right) , \text{ }
\]
\[
\lambda_{3}=\left(
\begin{array}
[c]{ccc}
1 & 0 & 0\\
0 & -1 & 0\\
0 & 0 & 0
\end{array}
\right) , \text{ \ \ }\lambda_{8}=\frac{1}{\sqrt{3}}\left(
\begin{array}
[c]{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & -2
\end{array}
\right) .
\]
The product of two Gell-Mann matrices is given by
\[
\lambda_{j}\lambda_{k}=\frac{2}{3}\delta_{jk}+\sum_{l}d_{jkl}\lambda_{l}
+i\sum_{l}f_{jkl}\lambda_{l}
\]
where $j,k=1,2,\dots,8$. The $f$-symbols (structure constants of
the Lie algebra $\textsf{su}(3)$) are totally anti-symmetric :
\begin{align}
f_{123} & =1,f_{458}=f_{678}=\frac{\sqrt{3}}{2},\nonumber\\
f_{147} & =f_{246}=f_{257}=f_{345}=f_{516}=f_{637}=\frac{1}{2},\nonumber
\end{align}
and the $d$-symbols are totally symmetric:
\begin{align}
d_{118} & =d_{228}=d_{338}=-d_{888}=\frac{1}{\sqrt{3}},\text{
\ \ \ \ \ \ \ \ }d_{448}=d_{558}=d_{668}=d_{778}=-\frac{1}{2\sqrt{3}
},\nonumber\\
d_{146} & =d_{157}=-d_{247}=d_{256}=d_{344}=d_{355}=-d_{366}=-d_{377}
=\frac{1}{2}.\nonumber
\end{align}
\noindent Given two real 8-vectors $\vec{a}$ and $\vec{b}$, we
define their Euclidean inner product
\[
\vec{a} \cdot\vec{b}= \sum_{k} a_{k}b_{k} \quad ,
\]
skew-symmetric vector $\wedge$-product
\[
(\vec{a} \wedge\vec{b})_{j}= \sqrt{3} \sum_{k,l}
f_{jkl}~a_{k}b_{l} \quad ,
\]
and symmetric vector $\star$-product
\[
(\vec{a}\star\vec{b})_{j}=\sqrt{3}\sum_{k,l}d_{jkl}~a_{k}b_{l}
\quad .
\]
The pure states that satisfy $\rho^{2}=\rho$ are therefore
characterized by
\[
|\vec{n}|^{2}=1 \quad \text{ \ and } \quad \vec{n}\star\vec{n}=\vec{n} \quad \text{.}
\]
Suppose that
$\rho=\frac{1}{3}(I+\sqrt{3}\vec{n}\cdot\vec{\lambda})$ \ is the
density matrix of a mixed state. It is Hermitian, positive with
trace equal to $1$. Therefore all the eigenvalues
$x_{1},x_{2},x_{3}$ are positive and add to one:
$x_{1}+x_{2}+x_{3} = 1$. The Cayley-Hamilton equation satisfied
by $\rho$ reads
\[
\rho^{3}-\rho^{2}+(x_{1}x_{2}+x_{2}x_{3}+x_{1}x_{3})\rho-x_{1}x_{2}x_{3}I
= 0 \quad .
\]
The following inequalities hold:
\[
\frac{1}{3}\geq x_{1}x_{2}+x_{2}x_{3}+x_{1}x_{3}\geq 0\quad , \quad\frac{1}
{27}\geq x_{1}x_{2}x_{3}\geq 0 \quad .
\]
Starting from these, a straightforward computation shows that the
necessary and sufficient conditions for $\rho =
\frac{1}{3}(I+\sqrt{3}n\cdot\lambda)$ to be a density matrix of a
mixed state are given by
\[
1 \geq |\vec{n}|^{2} \geq 0 \quad \text{ \ and } \quad 1 \geq 3|\vec{n}|^{2}-2\vec{n}\cdot(\vec{n}
\star\vec{n}) \geq 0 \quad .
\]
\noindent An arbitrary diagonal density matrix of a 3-level system
will be
\[
\rho=\frac{1}{3}(I+\sqrt{3}(n_{3}\lambda_{3}+n_{8}\lambda_{8})).
\]
In this case, the mixed-state density matrix constraints reduce to
\[
0 \leq n_{3}^{2}+n_{8}^{2} \leq 1 \quad \text{ \ and } \quad 0 \leq 2n_{8}^{3}-6n_{3}^{2}n_{8}+3n_{3}
^{2}+3n_{8}^{2} \leq 1 \quad .
\]
The region in the $n_{3}n_{8}$-plane where both the constraints
are satisfied is bound by an equilateral triangle with vertices at
the points
\[
(n_{3},n_{8})_{R}=(\frac{\sqrt{3}}{2},\frac{1}{2})\leftrightarrow\left(
\begin{array}
[c]{ccc}
1 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}
\right) ,
(n_{3},n_{8})_{B}=(-\frac{\sqrt{3}}{2},\frac{1}{2})\leftrightarrow\left(
\begin{array}
[c]{ccc}
0 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 0
\end{array}
\right) ,
\]\[
(n_{3},n_{8})_{G}=(0,-1)\leftrightarrow.\left(
\begin{array}
[c]{ccc}
0 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 1
\end{array}
\right) .
\]
\begin{center}
\epsfig{figure=vertices.eps}
\end{center}
Vertices of the above triangle correspond to three mutually
orthogonal pure-states. We labeled them Red, Blue, Green in
analogy with colored quarks [2]. In fact two pure-state vectors
$|\psi>$ and $|\psi^{\prime}>$ are orthogonal if and only if
$<\psi|\psi^{\prime}> = 0$, so that $Tr(\rho \rho^{\prime}) = 0$.
This implies $\vec{n}\cdot{\vec{n}}^{\prime} = -\frac{1}{2}$. Then
$\arccos(\vec{n}\cdot{\vec{n}}^{\prime}) = \pm\frac{2\pi}{3}$.
This is equal to the geodesic distance between two orthogonal
pure-states as measured by the standard Fubini-Study metric on
$\mathcal{C}P^{2}$.
Points on the edges of the triangle correspond to mixing of two orthogonal
pure-states of qutrits. In particular, at mid-points where bi-sectors
intersect with the edges we have
\[
(n_{3},n_{8})_{C} = ( 0,\frac{1}{2}) \leftrightarrow\frac{1}{2}\left(
\begin{array}
[c]{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 0
\end{array}
\right) , (n_{3},n_{8})_{B} = (\frac{\sqrt{3}}{4}, -\frac{1}{4})
\leftrightarrow\frac {1}{2}\left(
\begin{array}
[c]{ccc}
1 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 1
\end{array}
\right) ,
\]
\[
(n_{3},n_{8})_{A} = (-\frac{\sqrt{3}}{2},-\frac{1}{4}) \leftrightarrow\frac
{1}{2}\left(
\begin{array}
[c]{ccc}
0 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{array}
\right) .
\]
Triply mixed-states correspond to points inside the triangle. In particular
the origin
\[
(n_{3},n_{8})_{O} = (0,0) \leftrightarrow\frac{1}{3} \left(
\begin{array}
[c]{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{array}
\right)
\]
corresponds to the maximally mixed state.
For 2-level systems, the orbit of any diagonal $ 2 \times 2$
density matrix of qubits
\[
\rho= \frac{1}{2}\left(
\begin{array}
[c]{cc}
1 + n_{3} & 0\\
0 & 1-n_{3}
\end{array}
\right) \leftrightarrow(0,0,n_{3})
\]
under the action of the unitary group \textsf{SU(2)},
\[
\rho\rightarrow U \rho U^{\dag}
\]
where $U \in\mathsf{SU(2)}$ (i.e. adjoint representation of
\textsf{SU(2)}, which is \textsf{SO(3)} applied on $n_{3}$) sweeps
the whole Poincar\'{e} sphere $S^{2}$. Adjoint representation
\textsf{Ad }of a given \textsf{U }$\in$ \textsf{SU(2)} is
explicitly
\[
\mathsf{Ad(U)}_{ij}\mathsf{=}\frac{1}{2}\mathsf{Tr}(\sigma_{i}\mathsf{U}
\sigma_{j}\mathsf{U}^{\dag})\in\mathsf{SO(3)} ,
\]
so that $n_{j} \rightarrow\mathsf{Ad(U)}_{jk} n_{k} .$
In a similar way, for 3-level systems the orbit of each point
$(n_{3},n_{8})$ of the above triangle under the unitary action of
\textsf{SU(3)} (i.e. adjoint representation of \textsf{SU(3)})
will provide a generalization of the Poincar\'{e} sphere to
3-level systems. Adjoint representation \textsf{Ad }of a given
\textsf{U }$\in$ \textsf{SU(3) } is found as follows:
\[
\text{\textsf{Ad(U)}}_{ij}\mathsf{=}\frac{1}{2}\text{\textsf{Tr}}(\lambda
_{i}\mathsf{U}\lambda_{j}\mathsf{U}^{\dag})\in\text{\textsf{SO(8)}}
\]
so that
\[
n_{j} \rightarrow\text{\textsf{Ad(U)}}_{jk} n_{k} .
\]
In fact \textsf{Ad(SU(3))} is an 8-parameter subgroup of the
28-parameter rotation group \textsf{SO(8)}.
We also consider the \textit{entropy of mixing} of $\rho$ defined
as
\[
E(\rho)= - x_{1} \log_{3} (x_{1})- x_{2} \log_{3} (x_{2})- x_{3}
\log_{3} (x_{3}) \quad .
\]
Since in diagonal form
\[
\rho= \frac{1}{3}\left(
\begin{array}
[c]{ccc}
1+\sqrt{3}n_{3}+n_{8} & & \\
& 1-\sqrt{3}n_{3}+n_{8} & \\
& & 1-2n_{8}
\end{array}
\right) ,
\]
the entropy of mixing of $\rho$ becomes
\begin{align}
E(\rho)= &
-(\frac{1+\sqrt{3}n_{3}+n_{8}}{3})\log_{3}(\frac{1+\sqrt{3}
n_{3}+n_{8}}{3}) - (\frac{1-\sqrt{3}n_{3}+n_{8}}{3})\log_{3}(\frac{1-\sqrt{3}n_{3}+n_{8}}
{3})\nonumber\\
& - (\frac{1-2n_{8}}{3})\log_{3}(\frac{1-2n_{8}}{3}) \quad .
\nonumber
\end{align}
The equi-mixing curves in the $n_{3}n_{8}$-plane are shown on the
following diagram:
\begin{center}
\epsfig{figure=entropy.eps}
\end{center}
\noindent \textbf{Acknowledgement}
\noindent We thank Professor A. Shumovsky for bringing Ref.[2] to
our attention and Professor V. Manko for comments.
\noindent \textbf{References}
\begin{description}
\item {\small {[1]} C. M. Caves, G. J. Milburn, Opt. Commun. \textbf{179}, 439
(2000)}
\item {\small {[2]} A. V. Burlankov, D.
N.Klyshko, JETP Letters \textbf{69}, 839 (1999)} \item
{\small{[3]} M. V. Chekova, L. A. Krivitsky, S. P. Kulik, G. A.
Maslennikov, Phys. Rev. A\textbf{70}, 053801 (2004)} \item {\small
{[4]} G. Khanna, S. Mukhopadhyay, R. Simon, N. Mukunda, Ann. Phys.
\textbf{253},55 (1997) }
\end{description}
\end{document} | math | 11,072 |
\begin{document}
\author{Dongseok Kim}
\address{Department of Mathematics \\Kyonggi University
\\ Suwon, 443-760 Korea}
\email{[email protected]}
\subjclass[2000]{Primary 57M27; Secondary 57M25, 57R56}
\keywords{Links, Seifert surfaces, Flat plumbing basket number}
\title[Links of the flat plumbing basket numbers $4$ or less]
{A classification of links of the flat plumbing basket numbers $4$ or less}
\begin{abstract}
Flat plumbing basket surfaces of links were introduced to study the geometry of the complement of the links.
In present article, we study links of the flat plumbing basket numbers $4$ or less using
a special presentation of the flat plumbing basket surfaces. We find a
complete classification theorem of links of the flat plumbing basket numbers $4$ or less.
\end{abstract}
\maketitle
\section{Introduction}
A \emph{link} $L$ is an embedding of $n$ copies of $\mathbb{S}^1$ in
$\mathbb{S}^3$. If the number of components of the link $L$ is $1$,
a link is called a \emph{knot}.
Throughout the article, we will assume all links are \emph{tame}
which means all links can be in a form of a finite union of line segments.
Two links are \emph{equivalent}
if there is an isotopy between them.
In the case of prime knots, this equivalence is the same as
the existence of an orientation preserving homeomorphism on $\mathbb{S}^3$,
which sends a knot to the other knot. Although the equivalent
class of a link $L$ is called a \emph{link type},
throughout the article, a link really means the equivalent class of the link.
A compact orientable surface $\mathcal{F}$ is called a \emph{Seifert surface} of a link $L$ if the boundary of $\mathcal{F}$ is isotopic to
$L$. The existence of such a surface was first proven by Seifert
using an algorithm on a diagram of $L$, this algorithm was named after him as
\emph{Seifert's algorithm}~\cite{Seifert:def}. A Seifert surface $\mathcal{F}_L$ of
an oriented link $L$ which is produced by applying Seifert's
algorithm to a link diagram and
is called a \emph{canonical Seifert surface}.
Some Seifert surfaces feature extra structures. Seifert
surfaces obtained by flat annuli plumbings are the main subjects of this
article. Even though higher dimensional plumbings can be defined,
but we will only concentrate on \emph{annuli plumbings}.
This is often called a \emph{Murasugi sum} and it
has been studied extensively for the fibreness of links and surfaces
\cite{Gabai:genera, Stallings:const}.
The definition of flat plumbing basket surfaces~\cite{Rudolph:plumbing} is very technical
and so it is difficult to handle but the work in~\cite{FHK:openbook} provided
a tangible equivalent definition of a flat plumbing basket surface using
an open book decomposition. In a recent article by Hirose and Nakashima~\cite{HN},
the flat plumbing basket numbers of knots of $9$ crossings or less
were studied using this definition. Using the results in~\cite{KKL:string} and in~\cite{Kim:flat},
Choi, Do and the author~\cite{CDK} are working on a new knot tabulation with respect
to the flat plumbing basket number of knots using Dowker-Thistlethwaite notation
and computer program {\tt{knotscape}}.
However, none of these methods can be directly applied for links with more than one components.
In present article, we study links of the flat plumbing basket numbers $4$ or less using
a special presentation of the flat plumbing basket surfaces. We find a
complete classification theorem of links of the flat plumbing basket numbers $4$ or less.
The outline of this paper is as follows. We first provide some preliminary definitions and results
in Section~\ref{prelim}.
We provide a special presentation of the flat plumbing basket surfaces with $n$-annuli where $n\le 4$.
Using this presentation, we prove complete classification theorem
of links of the flat plumbing basket numbers $4$ or less in Section~\ref{fpbs}.
\section{Preliminaries} \label{prelim}
The followings are the exact definitions of the flat
plumbing basket surface given by Rudolph~\cite{Rudolph:plumbing}.
Spaces, maps, etc., are piecewise smooth unless stated differently.
Let $M$ be an oriented manifold. ${-}M$ denotes $M$ with its orientation reversed
and when notation requires it, $+M$ denotes $M$. For a suitable subset
$S \subset M$, $N_M (S)$ denotes a closed regular neighborhood of $S$ in $(M, \partial M)$
where an ordered pair $(S, T)$ stands a condition $T \subset S$ and a
map between ordered pairs $f: (S, T) \rightarrow (U,V)$ is a
map $f: S \rightarrow U$ which requires to preserve
subsets so that $f(T) \subset V$.
For a suitable codimension-$1$ submanifold $S \subset M$ (resp.,
submanifold pair
$(S, \partial S) \subset (M, \partial M))$, a emph{collaring}
is an orientation-preserving embedding
$S \times [0, 1] \rightarrow M$ (resp., $(S, \partial S)
\times[0, 1] \rightarrow (M, \partial M))$ extending $id_S = id_{S \times\{0\}}$;
a \emph{collar} of $S$ in $M$ (resp., of $(S, \partial S)$
in $(M, \partial M)$) is the image col$_M (S)$
(resp., col$_{(M, \partial M)} (S, \partial S)$) of a collaring.
The push-off of $S$ determined by
a collaring of $S$ or $(S, \partial S)$, denoted by $S^+$,
is the image by the collaring of
$S \times \{1\}$ with the orientation of $S$;
let $S^{-} :=$ ${-}$ $S^{+}$ such that $S$ and $S^{-}$ are
oriented submanifolds of the boundary of col$_M (S)$.
\begin{figure}
\caption{$(a)$ A geometric shape of $\alpha, B_{\alpha}
\label{topfig}
\end{figure}
\begin{defi}(\cite{Rudolph:plumbing}) \label{topdef}
Let $\alpha$ be a proper arc on a Seifert surface $S$.
Let $C_{\alpha}$ be \emph{col}$_{(S, \partial S)} (\alpha, \partial \alpha)$ which is called
\emph{the gluing region}.
Let $B_{\alpha}$ be \emph{col}$_{(S, \partial S)} (\alpha, \partial \alpha)$ (so $B_{\alpha}$
is a $3$-cell in \emph{top}$(S)$, that is the positive normal to $S$
along $C_{\alpha}=S \cap B_{\alpha} \subset \partial B_{\alpha}$
points into $B_{\alpha}$) as depicted in Figure~\ref{topfig}.
Let $A_n\subset B_{\alpha}$ be an $n$-full twisted annulus such that
$A_n \cap \partial B_{\alpha}= C_{\alpha}$. Then \emph{top plumbing} on $S$ along a path $\alpha$
is the new surface $\overline{S}= S \cup A_n$ where $A_n, C_{\alpha}, B_{\alpha}$ satisfy
the previous conditions.
\end{defi}
\begin{defi}(\cite{Rudolph:plumbing}) \label{fpbdef}
A Seifert surface $S$ is a \emph{flat plumbing basket surface} if either it is
$2$-disc $D^2$ or it can be constructed by plumbing $A_0$ to a
flat plumbing basket surface $S_0$ along a proper arc $\alpha \subset D^2\subset S_0$.
We say that a link $L$ admits a \emph{flat plumbing basket
presentation} if there exists a flat plumbing basket $S$ such that
$\partial S$ is equivalent to $L$.
The \emph{flat plumbing basket number} of $L$, denoted
by $fpbk(L)$, is the minimal number of flat annuli to obtain a
flat plumbing basket surface of $L$.
\end{defi}
An alternative definition of the flat plumbing basket
surfaces is given in~\cite{FHK:openbook}
and it is very easy to follow. The \emph{trivial
open book decomposition} of $\mathbb{R}^3$
is a decomposition of $\mathbb{R}^3$ into the half planes
in the following form. In a cylindrical coordinate, it can be presented
$$ \mathbb{R}^3 = \bigcup_{\theta \in [0, 2\pi)}
\{(r, \theta, z) | r \ge 0, z \in \mathbb{R} \}$$
where $\{(r, \theta, z) | r \ge 0, z \in \mathbb{R} \}$
is called a \emph{page} for $\theta \in [0, 2\pi)$.
Let $\mathcal{O}$ be the \emph{trivial open book decomposition} of the
$3$-sphere $\mathcal{B}S^3$ which is obtained from the trivial
open book decomposition of $\mathbb{R}^3$
by the one point compactification. A Seifert surface
is said to be a flat plumbing basket surface
if it consists of a single page of $\mathcal{O}$ as a $2$-disc $D^2$ and
finitely many bands which are
embedded in distinct pages~\cite{FHK:openbook}.
\begin{figure}
\caption{$(a)$ The knot $5_2$ as a closed braid, $(b)$ Seifert surface of $5_2$ in order to apply the algorithm in~\cite{FHK:openbook}
\label{52complete}
\end{figure}
In~\cite{FHK:openbook}, it is
shown that every link admits a flat plumbing basket representation by setting up the link
as a special closed braid form. So
we can define the \emph{flat plumbing basket number} of $L$, denoted
by $fpbk(L)$, to be the minimal number of flat annuli to obtain a
flat plumbing basket surface of $L$.
The author proved that every link $L$ admits a flat plumbing basket representation
by modifying the Seifert surface $S_L$ of the link $L$ to have a property that the Seifert graph
$\Gamma(S_L)$ holds the property described in~\cite[Theorem 3.3]{Kim:flat}.
Furthermore, an algorithm provided in~\cite{FHK:openbook} and in~\cite{CDK},
every links admit a flat plumbing basket presentation
$(a_1, a_2, \ldots, a_{2n})$ where $a_i \in \{1, 2, \ldots, n\}$
and each $i\in \{1, 2, \ldots, n\}$ appears exactly twice.
In stead of explaining the exact algorithm, let us
provide a concrete example as follows.
\begin{exa}
The knot $5_2$ has a flat plumbing basket presentation
$(1,2,3,4,5,6,4,5,1$, $2$, $3$, $6)$ as
illustrated in Figure~\ref{52complete}.
\begin{proof}
From the braid representative $\sigma_2\sigma_1^{-1} (\sigma_2^{-1})^3 \sigma_1^{-1}$ of the knot $5_2$,
we choose the $2$-disc $\mathcal{D}$ the union of three disc bounded by three Seifert circle and two
half twisted bands represented by $\sigma_2\sigma_1^{-1}$ as depicted as the dashed purple line in
Figure~\ref{52complete} $(a)$. To make flat plumbing
we first change the crossing presented by $\sigma_1^{-1}$ by adding extra two flat annuli
as shown in Figure~\ref{52complete} $(b)$. Fix a starting point and an orientation coming from the braid
as indicated the red ball and arrow in Figure~\ref{52complete} $(b)$.
Put numbering for each bands when we move around the $2$-disc $\mathcal{D}$ from the starting point
in the direction indicated as given in Figure~\ref{52complete} $(b)$. Isotop the $2$-disc $\mathcal{D}$
into the standard $2$-disc $\mathcal{D}$ as illustrated in Figure~\ref{52complete} $(c)$.
Now we are ready to find a flat plumbing basket presentation of the flat plumbing basket surface
which is in the position of the trivial open book decomposition of $\mathcal{B}S^3$.
From the top annulus, a pair of two points in the boundary of the annulus will receive $1$ and so on.
In Figure~\ref{52complete} $(c)$, there is no distinction of order between two groups annuli
connecting $\{(1,7), (2,8),(3,9)\}$ and $\{(4,7),(5,8)\}$. If we consider $\{(1,7), (2,8),(3,9)\}$
are in front of $\{(4,7),(5,8)\}$, we get a flat plumbing basket presentation
$(1$, $2$, $3,4,5,6,4,5,1$, $2$, $3$, $6)$. Otherwise, one may get
$(3$, $4$, $5,1,2,6,1,2,3$, $4$, $5$, $6)$.
\end{proof}
\end{exa}
\begin{figure}
\caption{Flat plumbing basket surfaces of $(a)$ the trefoil knot and $(b)$ the figure eight knot.}
\label{figure34}
\end{figure}
\begin{exa} (\cite{CDK})
Flat plumbing basket numbers of the trefoil knot and the figure eight knot are $4$.
\begin{proof}
Flat plumbing basket surfaces of the trefoil knot and the figure eight knot with four annuli are depicted in Figure~\ref{figure34}.
By Theorem~\cite[Theorem 3.3]{CDK}, the flat plumbing basket number is bigger than or equal to the flat band index of the link.
The flat band index of the trefoil knot and the figure eight knot are $4$~\cite{KKL:string}. It complete the proof.
\end{proof}
\end{exa}
One may notice that these flat plumbing basket surfaces of three knots $3_1$, $4_1$ and $5_2$
have a common property that all of numbers in the set $\{ 1, 2, \ldots, n\}$ appear in the first half and
the last half of the flat plumbing basket presentation. Thus, we may
rewrite it as a \emph{permutations presentation} $(\sigma : \mu)$,
the first permutation $\sigma$ presents the order of
the annuli in the flat plumbing basket surface are connected, which is called the
\emph{connection permutation} and the second permutation $\mu$
presents the order of annuli from the top to the bottom
which is called the \emph{order permutation}. For example,
the flat plumbing basket surfaces of knot $5_2$ in Figure~\ref{52complete}
has permutations presentation $(3,4,5,1,2,6 : 1,2,3,4,5,6)$
and $3_1$ has $(1,2,3,4:1,2,3,4)$ while $4_1$ has
$(1,2,3,4: 1,2,4,3)$.
\section{Classification} \label{fpbs}
Since the flat plumbing basket number is defined to be the minimal number of flat annuli to obtain a
flat plumbing basket surface of $L$. If some annuli in flat plumbing basket presentations
can be removed, we should not consider such presentation from the beginning.
If there is a part of the form $iji$ in a flat plumbing basket presentation,
two annuli presented by $i$ and $j$ can be removed by a simple isotopy as illustrated in Figure~\ref{twobridge}
and such a flat plumbing basket presentation is said to be \emph{reducible}.
If a permutations presentation $(a_1, a_2, \ldots, a_n : b_1, b_2 , \ldots, b_n)$
admits such a removal of annuli, we say that the permutations presentation is \emph{reducible}.
\begin{figure}
\caption{A fundamental move which decreases the flat plumbing basket number by $2$.}
\label{twobridge}
\end{figure}
First we show that any flat plumbing basket surface with $4$ or less
admits a permutations presentation as follows.
\begin{figure}
\caption{A handle slide the band which is labeled by $j$ along the
band labeled by $i$.}
\label{slide}
\end{figure}
\begin{thm} \label{twopermutation}
The flat plumbing basket surfaces with $4$ or less annuli can be
presented by a permutations presentation.
\begin{proof}
For $n \le 1$, is obvious. For $n=2$, there are six flat plumbing basket presentations
$(i,i,j,j)$ $(i,j,i,j)$ where $i,j=1,2$.
The first one $(i,i,j,j)$ can be changed to $(i,j,j,i)$ by a
slide of annulus presented by $i$ along the annulus presented by $j$ as illustrated in Figure~\ref{slide}.
The second $(i,j,i,j)$ flat plumbing basket presentations
is already written as a permutations presentation without changing anything.
For $n=3$, if two numbers are in the first half, there are three possibilities :
either $i,i,j$, $j,i,i$ or $i,j,i$ where $i, j=1, 2, 3$.
But the last one is reducible to a flat plumbing basket presentation $(1,1)$
by the move described in Figure~\ref{twobridge}.
For the first one, it is either $(i,i,j,k,k,j)$ or $(i,i,j,j,k,k)$ for some $k\in \{1,2,3\}$.
By sliding the annulus presented by $i$ along the
annulus presented by $j$ for $(i,i,j,k,k,j)$, we get $(i,j,k,k,j,i)$ which admits a permutations presentation.
Cyclically rotating $(i,i,j,j,k,k)$ by changing the starting point, we get $(i,j,j,k,k,i)$ we reduce to a previous case.
For the second, it is either $(j,i,i,j,k,k)$ or $(j,i,i,k,k,j)$ and each of them can be
transformed to one admits a permutations presentation by a handle slide.
For $n=4$, now we divide cases depending on the number of the same letter in the first half, say $m$.
If $m=0$, we are done. If $m=1$ and the flat plumbing basket presentation is not reducible,
it is either $(i,i,j,k,*,*,*,*)$, $(i, j, j, k, *,*,*,*)$, $(i, j, k, k, *,*,*,*)$ or
$(i, j, k, i, *,*,*,*)$ where $i,j,k=1,2,3,$.
For $(i,i,j,k,*,*,*,*)$, the fifth components must be either $k$ or $l$ for some $l\in\{1,2,3,4\}$.
If it is $l$, by sliding the annulus presented by $i$ along the
annulus presented by $j$ for $(i,i,j,k,l,*,*,*)$, we have $(i,j,k,l,*,*,*,*)$ which admits a permutations presentation.
If it is $k$, there are two possibilities as follows and each can be changed to
one which admits a permutations presentation by sliding twice,
$$
\begin{matrix}
(1)& (i,i,j,k,k,l,l,j) & \rightarrow & (i,j,k,k,l,l,j,i) & \rightarrow & (i,j,k,l,l,k,j,i) \\
(2)& (i,i,j,k,k,j,l,l) & \rightarrow & (i,j,k,k,j,i,l,l) & \rightarrow & (l,i,j,k,k,j,i,l)
\end{matrix}
$$
For $(i, j, j, k, *,*,*,*)$, we only need to look at $(i,j,j,k,i,l,l,k)$ which can be changed
to $(i,j,k,i,l,l,k,j)$ and then to $(l,i,j,k,i,l,k,j)$ by two handle slide.
For $(i, j, k, k, *,*,*,*)$, there are six
possibilities but all can be changed to one which admits a permutations presentation by sliding
the annulus presented by $l$ along one annulus which is the right before $l$.
For $(i, j, k, i, *,*,*,*)$, we look the second half. By the previous argument, the only cases we have
to take care of is $(i, j, k, i, l, j, k, l)$ or $(i, j, k, i, l, k, j, l)$. But both cases
can be changed to one which admits a permutations presentation by sliding
the annulus presented by $i$ along the annulus presented by $l$.
If $m=2$, there are four possibilities, $(i,j,j,i,k,l,l,k)$, $(i,j,j,i,k,k,l,l)$,
$(i,i,j,j,k,l,l,k)$ and $(i,i,j,j,k,k,l,l)$. But any sliding of an annulus along an adjacent annulus
will decrease $m$ by $1$, it returns to the previous cases. Therefore, it completes the proof of theorem.
\end{proof}
\end{thm}
Now, we are set to prove the main theorem which completely classifies the links
of the flat plumbing basket numbers $4$ or less.
\begin{table}
\begin{tabular}{|c|c|c|c|c|}\hline
\rm{Name ~of ~link} & $\begin{matrix} \rm{Connection} \\ \rm{Permutation} \end{matrix}$
& $\begin{matrix} \rm{Order} \\ \rm{Permutation} \end{matrix}$ \\
\hline $3_1$ & $1234$ & $\begin{matrix} 1234, 1432, 2143, 2341,\\ 3214, 3412, 4123, 4321\end{matrix}$\\
\hline $4_1$ & $1234$ & all~other \\
\hline $L2a1 \sqcup O$ & $1243$ & $\begin{matrix} 1324, 1342, 2413, 2431,\\ 3124, 3142, 4213, 4231\end{matrix}$ \\
\hline $L2a1 \# L2a1$ & $1243$ & all~other \\
\hline $L2a1 \sqcup O$ & $2341$ & all \\
\hline $L6a5$ &$2143$ & $\begin{matrix} 1324, 1342, 2413, 2431,\\ 3124, 3142, 4213, 4231\end{matrix}$ \\
\hline $O \sqcup O\sqcup O \sqcup O\sqcup O$ & $4321$ & all\\
\hline
\end{tabular}
\vskip .2cm
\caption{Links of flat plumbing basket number $4$.} \label{t1}
\end{table}
\begin{thm} \label{class}
\begin{enumerate}
\item[{\rm (1)}] A link $L$ has the flat plumbing basket number $0$ if and only if $L$ is the trivial knot.
\item[{\rm (2)}] A link $L$ has the flat plumbing basket number $1$ if and only if $L$ is the trivial link of two components.
\item[{\rm (3)}] A link $L$ has the flat plumbing basket number $2$ if and only if $L$ is the trivial link of three components.
\item[{\rm (4)}] A link $L$ has the flat plumbing basket number $3$ if and only if $L$ is either the trivial link of four components or
the Hopf link which is denoted by $L2a1$.
\item[{\rm (5)}] A link $L$ has the flat plumbing basket number $4$ if and only if $K$ is
either the trefoil knot, the figure eight knot, $L2a1 \sqcup O$, $L2a1 \# L2a1$, or $L6a5$.
\end{enumerate}
\begin{proof}
(1) The flat plumbing basket surface with $0$ flat plumbing must be a $2$-disc.
Thus one can easily see that a link $L$ has the flat plumbing basket number $0$ if and only if $L$ is the trivial knot.
(2) The flat plumbing basket surface with $1$ flat plumbing must be a $2$-disc with a single flat annulus.
A link $L$ has the flat plumbing basket number $1$ if and only if $L$ is the trivial link of two components.
(3) Consider a link of the flat plumbing basket number $2$. There are only two possible such surfaces and
one's boundary is the trivial knot and the others is the trivial link of three components. Therefore, we have that
a link $L$ has the flat plumbing basket number $2$ if and only if $L$ is the trivial link of three components.
(4) By considering all possible flat $3$ band diagrams which are $36$ cases,
we find they are either the trivial link of two components, the trivial link of four components or
the Hopf link. However, the trivial link of two components has the flat plumbing basket number $1$.
(5) Suppose a link $L$ has the flat plumbing basket number $4$.
Before we consider all possible permutation presentation $(a_1 a_2 a_3 a_4 : b_1 b_2 b_3 b_4)$,
we first found that all $(a_1 a_2 a_3 a_4 : b_1 b_2 b_3 b_4)$ are reducible except
$(1234:b_1 b_2 b_3 b_4)$, $(1243:b_1 b_2 b_3 b_4)$, $(1342:b_1 b_2 b_3 b_4)$, $(2143: b_1 b_2 b_3 b_4)$,
$(2341:b_1 b_2 b_3 b_4)$ and $(4321:b_1 b_2 b_3 b_4)$. Let us remark that
there are more irreducible permutation presentation but
$(1324:b_1 b_2 b_3 b_4)$ and $(2134:b_1 b_2 b_3 b_4)$ can be obtained from $(1243:b_1 b_2 b_3 b_4)$,
$(1423:b_1 b_2 b_3 b_4)$ and $(3124:b_1 b_2 b_3 b_4)$ can be obtained from $(1423:b_1 b_2 b_3 b_4)$,
$(4123:b_1 b_2 b_3 b_4)$ can be obtained from $(2341:b_1 b_2 b_3 b_4)$ by cyclic relabeling.
Then we find a complete list of links which can be obtained by permutation presentations
of flat plumbing basket surfaces in Table~\ref{t1}. Let us remark that this link corresponding to
permutation presentations $(1342:b_1 b_2 b_3 b_4)$ are all unknot.
Therefore, it complete the proof.
\end{proof}
\end{thm}
Although, we have found the classification theorem of links
of the flat plumbing basket number $4$ or less, it can be used
to determine the flat plumbing basket number of a link which
is either $5$ or $6$.
In fact, if one find
a flat plumbing basket surface of $5$ annuli whose boundary
is a link $L$ which is not listed in the classification theorem,
then the flat plumbing basket number of the link $L$ must be $5$. The following
is a such a example.
\begin{figure}
\caption{A flat plumbing basket surface of the link $L4a1$ with $5$ flat plumbings.}
\label{412fig}
\end{figure}
\begin{cor} \label{link42}
The flat plumbing basket number of the link $L4a1$ is $5$.
\begin{proof}
Since $L4a1$ is not listed in Theorem~\ref{class} and
it is the boundary of the flat plumbing basket surface
$(1,2,3,4,5,1,4,5,2,3)$ as depicted in Figure~\ref{412fig}. It completes the proof
of the corollary.
\end{proof}
\end{cor}
In the case of a knot, the flat plumbing basket number of a knot must be even because
the boundary of a flat plumbing basket surface of $n$ annuli has at most $n+1$ components,
and the number of components is always congruent to $n+1$ modulo $2$.
\begin{cor} \label{52cor}
The flat plumbing basket number of the knot $5_2$ is $6$.
\begin{proof}
The knot $5_2$ is not listed in Theorem~\ref{class} and it admits
a flat plumbing basket surface of $6$ annuli as illustrated in Figure~\ref{52complete}.
Therefore, the flat plumbing basket number of the knot $5_2$ is $6$.
\end{proof}
\end{cor}
Let us remark that the flat plumbing basket number $6$ of the knot $5_2$ in Corollary~\ref{52cor}
is independently found by Hirose and Nakashima~\cite{HN} using the lower bound by the genus
and Alexander polynomial and by Y. Choi, Y. Do and D. Kim~\cite{CDK} using
a complete classification of knots of flat plumbing basket presentation with
$6$ annuli.
\section*{Acknowledgments}
The \TeX\, macro package PSTricks~\cite{PSTricks} was essential for
typesetting the equations and figures. This work was supported by Kyonggi University Research Grant 2011.
\end{document} | math | 22,709 |
\begin{document}
\author{Muhammet Boran}
\address{Department of Mathematics, Yıldız Technical University, 34220 Esenler, Istanbul, TURKEY}
\email{\bburl{[email protected]}}
\author{Garam Choi}
\address{Department of Mathematics, Colby College, Waterville, ME 04901, USA}
\email{\bburl{[email protected]}}
\author{Steven J.\ Miller}
\address{Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267, USA}
\email{\bburl{[email protected]}}
\author{Jesse Purice}
\address{School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA}
\email{\bburl{[email protected]}}
\author{Daniel Tsai}
\address{Department of Mathematics, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan (R.O.C.)}
\email{\bburl{[email protected]}}
\title{A characterization of prime $v$-palindromes}
\begin{abstract}
An integer $n\geq 1$ is a $v$-palindrome if it is not a multiple of $10$, nor a decimal palindrome, and such that the sum of the prime factors and corresponding exponents larger than $1$ in the prime factorization of $n$ is equal to that of the integer formed by reversing the decimal digits of $n$. For example, if we take 198 and its reversal 891, their prime factorizations are $198 = 2\cdot 3^2\cdot 11$ and $891 = 3^4\cdot 11$ respectively, and summing the numbers appearing in each factorization both give 18. This means that $198$ and $891$ are $v$-palindromes. We establish a characterization of prime $v$-palindromes: they are precisely the larger of twin prime pairs of the form $(5 \cdot 10^m - 3, 5 \cdot 10^m - 1)$, and thus standard conjectures on the distribution of twin primes imply that there are only finitely many prime $v$-palindromes.
\end{abstract}
\subjclass[2010]{Primary 11A45, 11A63.}
\keywords{Prime $v$-palindromes, Iverson bracket, Cram\'er model}
\thanks{This work was supported in part by the 2022 Polymath Jr REU program.}
\maketitle
\section{Introduction}
\subsection{Related Work}
There have been many papers studying properties shared by numbers and their reversals. We first set some notation.
\begin{defn}
Let $b\geq2$, $L\geq1$, and $0\leq a_0,a_1,\ldots,a_{L-1}<b$ be any integers. We denote
\begin{equation}
(a_{L-1}\cdots a_1a_0)_b \ :=\ \sum^{L-1}_{i=0}a_ib^i.
\end{equation}
We also write $(a_{L-1},\ldots,a_1,a_0)_b$ to make it clear which are each digit.
\end{defn}
\begin{defn}
Let the base $b\geq2$ representation of an integer $n\geq1$ be\\ $(a_{L-1}\cdots a_1a_0)_b$, where $a_{L-1}\neq0$. The $b$-\emph{reverse} of $n$ is defined to be
\begin{equation}
r_b(n) \ :=\ (a_0a_1\cdots a_{L-1})_b.
\end{equation}
We write $r(n)$ for $r_{10}(n)$.
\end{defn}
So for example $r(198) = 891$.
In \emph{A Mathematician's Apology} \cite{Har}, G.\ H.\ Hardy states that ``8712 and 9801 are the only 4-digit numbers which are integral multiples of their decimal reversal'':
\begin{equation}
8712\ =\ 4\cdot r(8712),\ \ 9801\ = \ 9\cdot r(9801).
\end{equation}
In 1966, A.\ Sutcliffe \cite{Su} generalized this observation and studied all integer solutions of the equation
\begin{equation}
k\cdot r_{b}(n)\ = \ n,
\end{equation} where $b\geq2$ is the base and $0 < k < n$. In \cite{KS}, numbers $n$ such that $n$ divides $r(n)$ are mentioned. In particular, numbers of the form
\begin{equation}
2178,\ \ \ 21978,\ \ \ 219978,\ \ \ 2199978,\ \ \ \dots,
\end{equation}
with any number of $9$'s in the middle, all satisfy $4n=r(n)$.
Suppose that the prime factorization of an integer $n\geq1$ is
\begin{equation}
n\ =\ p_{1}^{\alpha_1}\cdots p_{k}^{\alpha_k},
\end{equation}
where $p_1<\cdots <p_k$ are primes and $\alpha_1,\ldots,\alpha_k\geq1$ integers. In 1977, P.\ Erd\H{o}s and K.\ Alladi \cite{AE} studied the function
\begin{equation} \label{prfctr}
A(n)\ =\ \sum_{i=1}^{k} p_i \cdot \alpha_i.
\end{equation}
The entries A008474 and A000026 from the OEIS \cite{OEIS} are
\begin{align}
F(n) &\ =\ \sum_{i=1}^{k} \left(p_i +\alpha_i\right), \\
G(n) & \ =\ \prod_{i=1}^k p_i \cdot \alpha_i ,
\end{align}
respectively. These functions are somehow similar in expression. We introduce an arithmetic function denoted by $v(n)$ which is obtained from $F(n)$ by replacing $\alpha_i$ with 0 when $\alpha_i = 1$. In other words,
\begin{equation}\label{defofv}
v(n) \ =\ \sum_{\substack{1\leq i\leq k,\\ \alpha_i = 1}}p_i + \sum_{\substack{1\leq i\leq k,\\ \alpha_i \geq 2}} \left(p_i +\alpha_i\right).
\end{equation}
\subsection{$v$-palindromes}
The concept of $v$-palindromes was introduced by Tsai in \cite{tsai0,tsai} and explored further in four later manuscripts \cite{tsai6,tsai3,tsai4,tsai5}. As in the abstract, consider the number $198$ whose digit reversal is $891$. Their prime factorizations are
\begin{align}
198 &\ =\ 2\cdot 3^2\cdot 11,\label{rei1}\\
891 &\ =\ 3^4\cdot 11\label{rei2},
\end{align}
and we have
\begin{equation}
2+(3+2)+11 \ = \ (3+4) + 11.
\end{equation}
In other words, the sum of the numbers ``appearing'' on the right-hand side of \eqref{rei1} equals that of \eqref{rei2}. We first give the following definitions.
\begin{defn}
The additive function $v\colon \mathbb{N}\to\mathbb{Z}$ is defined by setting $v(p):=p$ for primes $p$ and $v(p^\alpha):=p+\alpha$ for prime powers $p^\alpha$ with $\alpha\geq2$.
\end{defn}
Notice that this definition of $v(n)$ agrees with \eqref{defofv}. We define $v$-palindromes as follows.
\begin{defn}\label{def.4}
Let $n\geq1$ and $b\geq2$ be integers. Then $n$ is a $v$-\emph{palindrome in base} $b$ if $b\nmid n$, $n\neq r_b(n)$, and $v(n) = v(r_b(n))$. A $v$-palindrome in base $10$ is simply called a $v$-\emph{palindrome}.
\end{defn}
Thus $198$ and $891$ are $v$-palindromes. The following are infinite sequences of $v$-palindromes \cite{tsai0,tsai}:
\begin{gather}
18,\ 198,\ 1998,\ 19998,\ 199998,\ 1999998,\ \dots,\label{ten1}\\
18,\ 1818,\ 181818,\ 18181818,\ 1818181818,\ 181818181818,\ \dots\label{ten2}.
\end{gather}
In \eqref{ten1}, we simply keep increasing the number of $9$'s in the middle; in \eqref{ten2}, we simply keep appending another $18$. The sequences \eqref{ten1} and \eqref{ten2} are actually parts of a larger family of $v$-palindromes derived in \cite[Theorem 3]{tsai3}. In particular, there are infinitely many $v$-palindromes. According to \cite{tsai0}, the $v$-palindromes $n\leq 10^5$ with $n<r(n)$ are
\begin{align}
&18,\ 198,\ 576,\ 819,\ 1131,\ 1304,\ 1818,\ 1998,\ 2262,\ 3393,\ 4154,\ 4636,\ 8749,\ 12441,\nonumber \\
&14269,\ 14344,\ 15167,\ 15602,\ 16237,\ 18018,\ 18449,\ 18977,\ 19998,\ 23843,\ 24882,\nonumber\\
&26677,\ 26892,\ 27225,\ 29925,\ 31229,\ 36679,\ 38967,\ 39169,\ 42788,\ 45694,\ 46215,\nonumber \\
&46655,\ 47259,\ 48048,\ 52416,\ 56056,\ 60147,\ 62218,\ 66218,\ 79689,\ 97999.\nonumber
\end{align}
The sequence of $v$-palindromes $n$ (whether $n<r(n)$ or not) is \seqnum{A338039} in the OEIS \cite{OEIS}. In \cite{tsai0}, it is said that extensive computer calculations suggest the following.
\begin{conj}\label{1.conj}
There are no prime $v$-palindromes.
\end{conj}
\subsection{New Results}
We are able to make significant progress towards a possible proof of Conjecture \ref{1.conj} by proving the following characterization of prime $v$-palindromes.
\begin{theorem}\label{thm:main}
The prime $v$-palindromes are precisely the primes of the form
\begin{equation}
5\cdot 10^m - 1 \ =\ 4\underbrace{9\cdots 9}_{m},\label{p1}
\end{equation}
for some integer $m\geq4$, such that
\begin{equation}
5\cdot 10^m-3 \ =\ 4\underbrace{9\cdots 9}_{m-1}7 \label{p2}
\end{equation}
is also prime.
\end{theorem}
Here, the $m\geq4$ is not very significant and only means that it has been checked that there are no prime $v$-palindromes of fewer than $5$ decimal digits, and thus conceivably can be improved with more checking for small values of $m$. From this characterization of prime $v$-palindromes, it is a consequence of standard models for primes (such as the Cram\'er model, though weaker assumptions suffice) that there are only finitely many prime $v$-palindromes. In particular, we have the following.
\begin{theorem}\label{thm:heuristicfinite} Assume that the probability $n$ and $n+2$ are both prime is bounded by $C / \log^2 n$ for some $C$. Then there are only finitely many prime $v$-palindromes. \end{theorem}
The main purpose of this paper is to prove Theorem \ref{thm:main}. In Section \ref{sec:lemmas}, we give various definitions and lemmas to be used throughout the rest of the paper. The proof of the forward direction of Theorem \ref{thm:main} consists of Sections \ref{sec:setup} to \ref{sec:otherdec}. The proof of the converse consists of just Section \ref{sec:converse}. In Section \ref{heuristic}, we elaborate on the above-mentioned heuristics that there are only finitely many prime $v$-palindromes.
\section{Preliminaries}\label{sec:lemmas}
We start with some useful definitions.
\begin{defn}
Let $P$ denote any mathematical statement. Then the \emph{Iverson bracket} is defined by
\begin{equation}
\label{eq:000}
[ P ] \ :=\ \begin{cases}
1,& \text{if $P$ is true}, \\
0, &\text{if $P$ is false}.
\end{cases}
\end{equation}
\end{defn}
\begin{defn}
For integers $\alpha\geq1$, denote $\iota(\alpha):=\alpha[\alpha>1]$. That is,
\begin{equation}
\label{eq:000}
\iota(\alpha) \ :=\ \begin{cases}
0,& \text{if $\alpha\ =\ 1$}, \\
\alpha, &\text{if $\alpha\ >\ 1$}.
\end{cases}
\end{equation}
\end{defn}
With this notation, the additive function $v\colon\mathbb{N}\to\mathbb{Z}$ can be defined by setting in one stroke
\begin{equation}\label{v-p^a}
v(p^\alpha) \ :=\ p + \iota(\alpha)
\end{equation}
for all prime powers $p^\alpha$.
\begin{defn}
Let the decimal representation of an integer $n\geq1$ be\\ $(a_{L-1}\cdots a_1a_0)_{10}$, where $a_{L-1}\neq0$. Then we denote
\begin{gather}
a_i(n) \ :=\ a_i \quad \text{for $i\ =\ 0,1,\ldots,L-1$},\\
L(n) \ :=\ L,
\end{gather}
to indicate dependence on $n$. We also by convention denote $L(0):=0$.
\end{defn}
For example,
\begin{gather}
a_0(198) \ =\ 8,\quad a_1(198) \ =\ 9,\quad a_2(198) \ =\ 1,\\
L(198) \ =\ 3.
\end{gather}
Hence we have defined a function $L\colon\mathbb{N}\cup\{0\}\to\mathbb{Z}$, where $L$ stands for length. We then have the following lemmas the first two of which are obvious and follow immediately from definitions.
\begin{lemma}\label{lem:ob}
Let $0\leq m\leq n$ be integers. Then $L(m)\leq L(n)$.
\end{lemma}
\begin{lemma}\label{lem:ran}
Let $n,\ell\geq1$ be integers. Then $L(n) = \ell$ if and only if $10^{\ell-1}\leq n<10^\ell$.
\end{lemma}
\begin{lemma}\label{lem:jessie}
Let $m,n\geq1$ be integers. Then
\begin{equation}\label{eq:exp}
L(mn) \ = \ L(m)+ L(n) - [mn\ <\ 10^{L(m)+L(n)-1}].
\end{equation}
In particular,
\begin{equation}\label{eq:ineqL}
L(m)+L(n)-1\ \le \ L(mn)\ \le \ L(m) + L(n).
\end{equation}
\end{lemma}
\begin{proof}
By Lemma \ref{lem:ran}, $10^{L(m)-1}\leq m<10^{L(m)}$ and $10^{L(n)-1}\leq n<10^{L(n)}$. Therefore
\begin{equation}
10^{L(m)+L(n)-2}\ \le \ mn\ <\ 10^{L(m)+L(n)}.
\end{equation}
If $mn<10^{L(m)+L(n)-1}$, then by Lemma \ref{lem:ran},
\begin{equation}
L(mn) \ = \ L(m) + L(n) - 1 \ = \ L(m) + L(n) - [mn\ <\ 10^{L(m)+L(n)-1}].
\end{equation}
If $10^{L(m)+L(n)-1}\leq mn$, then by Lemma \ref{lem:ran},
\begin{equation}
L(mn) \ = \ L(m) + L(n) \ = \ L(m) + L(n) - [mn\ <\ 10^{L(m)+L(n)-1}].
\end{equation}
This proves \eqref{eq:exp}. Because an Iverson bracket is always $0$ or $1$, clearly \eqref{eq:ineqL} holds.
\end{proof}
\begin{lemma}\label{lem:longineq}
Let $n_1,\ldots,n_k\geq1$ be integers. Then
\begin{equation}
L(n_1\cdots n_k)\ \geq\ L(n_1) +\cdots+ L(n_k) - (k-1).
\end{equation}
\end{lemma}
\begin{proof}
This follows by repeated application of the left inequality in \eqref{eq:ineqL} in Lemma \ref{lem:jessie}.
\end{proof}
The following is an elementary inequality which essentially says that the sum is no greater than the product and which we do not prove.
\begin{lemma}\label{lem:stack}
Let $x_1,\ldots,x_k\geq2$ be real numbers. Then
\begin{equation}
x_1 +\cdots+x_k\ \le \ x_1\cdots x_k.
\end{equation}
\end{lemma}
\begin{lemma}\label{lem:preless}
Let $p$ be a prime and $\alpha\geq0$ an integer. Then
\begin{itemize}
\item[{\rm (i)}] if $\alpha\in\{0,1\}$, then $v(p^\alpha)\leq p^\alpha\leq p+\alpha$;
\item[{\rm (ii)}] if $\alpha>1$, then $v(p^\alpha)\ =\ p+\alpha\leq p^\alpha$.
\end{itemize}
\end{lemma}
\begin{proof}
(i) can be easily checked. For (ii), we have, using Lemma \ref{lem:stack} twice,
\begin{equation}
v(p^\alpha) \ =\ p +\alpha\ \le \ p\alpha\ =\ \underbrace{p +\cdots + p}_{\alpha} \ \le \ p^\alpha.
\end{equation}
\end{proof}
\begin{lemma}
\label{lem:ineq}
Let $n\geq1$ be an integer. Then $v(n)\leq n$ and $L(v(n))\leq L(n)$.
\end{lemma}
\begin{proof}
We first prove that $v(n)\leq n$. We have $v(1) = 0 \leq 1$. Now assume that $n>1$ has prime factorization $n=p^{\alpha_1}_1\cdots p^{\alpha_k}_k$, where $p_1<\cdots<p_k$ are primes and $\alpha_1,\ldots,\alpha_k\geq1$ integers. Then using the fact that $v$ is additive and Lemmas \ref{lem:preless} and \ref{lem:stack},
\begin{align}
v(n) &\ =\ v(p^{\alpha_1}_1\cdots p^{\alpha_k}_k) \ =\ v(p^{\alpha_1}_1) +\cdots+ v(p^{\alpha_k}_k)\nonumber\\
&\ \le \ p^{\alpha_1}_1 +\cdots + p^{\alpha_k}_k \ \le \ p^{\alpha_1}_1\cdots p^{\alpha_k}_k\ =\ n.
\end{align}
Now $L(v(n))\leq L(n)$ follows from $v(n)\leq n$ with Lemma \ref{lem:ob}.
\end{proof}
\begin{lemma}\label{lem:twoineq}
We have the following inequalities.
\begin{itemize}
\item[{\rm (i)}] If $x>-1$ is real, then $\log_2(10^{x+1}-1) < (x+1)\log_210$.
\item[{\rm (ii)}] If $n\geq2$ is an integer, then $(n+1)\log_210< 10^{n-1}$.
\end{itemize}
\end{lemma}
\begin{proof}
\begin{itemize}
\item[{\rm (i)}] Since $\log_2$ is strictly increasing, we have
\begin{equation}
\log_2(10^{x+1}-1)\ <\ \log_2(10^{x+1}) \ =\ (x+1)\log_210.
\end{equation}
\item[{\rm (ii)}] Define the function
\begin{equation}
q(n) \ =\ \frac{10^{n-1}}{n+1},\quad\text{for integers $n\geq0$}.
\end{equation}
Then for any $n\geq0$,
\begin{equation}
q(n+1) \ =\ \frac{10^n}{n+2} \ =\ \frac{10^{n-1}}{n+1}\cdot \frac{10(n+1)}{n+2} \ =\ q(n)\cdot \frac{10(n+1)}{n+2}.
\end{equation}
As $10(n+1)/(n+2)>1$ for $n\geq0$, we see that $q(n)$ is strictly increasing. Now because
\begin{equation}
q(2) \ =\ \frac{10}{3} \ =\ 3.\overline{3}\ >\ 3.32\cdots \ =\ \log_210,
\end{equation}
we see that $q(n)>\log_210$ for $n\geq2$, which is exactly what is required.
\end{itemize}
\end{proof}
\section{Setup}\label{sec:setup}
It can be checked by a computer that there are no prime $v$-palindromes of fewer than $5$ decimal digits. Therefore assume that $p$ is a prime $v$-palindrome of $m+1$ decimal digits, where $m\geq4$. In particular, according to Definition \ref{def.4}, $p\neq r(p)$ and $p=v(p) = v(r(p))$. Consequently,
\begin{equation}\label{eq:prange}
10^m+3\ \leq\ p\ \leq\ 10^{m+1}-3.
\end{equation}
By the end of Section \ref{sec:otherdec}, we will have deduced that $p=5\cdot 10^m-1$ and a bit more.
In the case $r(p)<p$, by Lemma \ref{lem:ineq}, $v(r(p))\leq r(p)<p$, and thus $p = v(r(p))$ cannot hold. Therefore we may assume that $r(p)>p$. Further, in the case $r(p)$ is prime, $v(r(p))=r(p)>p$, and thus again $p = v(r(p))$ cannot hold. Therefore we may assume that $r(p)$ is composite. Suppose that
\begin{equation}\label{eq:fact}
r(p)\ =\ fq^\beta,
\end{equation}
where $q$ is the largest prime factor of $r(p)$ and $q^\beta$ the highest power of $q$ dividing $r(p)$, namely, $q^\beta\parallel r(p)$. Let the number of decimal digits of $q$ be denoted by $\ell$, i.e., $L(q) =\ell$.
\ \\
\textbf{\emph{We shall assume the conditions and notation laid out in this section throughout the rest of this paper, without explicitly stating such assumptions in each lemma below. }} \ \\
\begin{lemma}\label{lem:basic}
We have the following:
\begin{itemize}
\item[{\rm (i)}] $v(f)\ =\ p-v(q^{\beta})\ =\ p-q-\iota(\beta)$,
\item[{\rm (ii)}] $L(v(f))\ =\ L(p-q-\iota(\beta))$,
\item[{\rm (iii)}] $L(v(f))\ \geq\ L(p-q-\beta)$.
\end{itemize}
\end{lemma}
\begin{proof}
\begin{itemize}
\item[{\rm (i)}] This follows by applying $v$ to \eqref{eq:fact}.
\item[{\rm(ii)}] This follows by applying $L$ to part (i).
\item[{\rm(iii)}] If $\beta>1$, then (ii) becomes $L(v(f))=L(p-q-\beta)$. If $\beta = 1$, then because $r(p)$ is composite, $f>1$, and so part (i) implies $2\leq v(f) = p-q$. Therefore
\begin{equation}
v(f)\ >\ p - q - 1\ \geq\ 1,
\end{equation}
and by Lemma \ref{lem:ob}, $L(v(f))\geq L(p-q-1)$.
\end{itemize}
\end{proof}
\begin{lemma}\label{lem:srange}
We have $\beta\leq \log_2(10^{m+1}-1) < 10^{m-1}$.
\end{lemma}
\begin{proof}
Since $r(p)=fq^\beta$ in equation \eqref{eq:fact}, we have $r(p)\geq q^\beta$. Also, because $r(p)$ has $m+1$ decimal digits, we have $r(p)\leq 10^{m+1}-1$. Consequently,
\begin{equation}
\beta\log q\ \le \ \log r(p)\ \le \ \log (10^{m+1}-1),
\end{equation}
and so
\begin{equation}
\beta\ \le \ \log_q(10^{m+1}-1)\ \le \ \log_2(10^{m+1}-1).
\end{equation}
That $\log_2(10^{m+1}-1) < 10^{m-1}$ follows from Lemma \ref{lem:twoineq}, using both parts.
\end{proof}
\begin{lemma}
\label{lem:pqs}
If $\ell\leq m-1$, then
\begin{itemize}
\item[{\rm (i)}] $L(p-q-\beta)\ \geq\ m$,
\item[{\rm (ii)}] $L(p-q-\iota(\beta))\ \geq\ m$,
\item[{\rm (iii)}] $L(v(f))\ \geq\ m$.
\end{itemize}
\end{lemma}
\begin{proof}
\begin{itemize}
\item[{\rm (i)}] Since $q$ has $\ell$ digits, $q\leq 10^{\ell}-1$. Together with \eqref{eq:prange} and Lemma \ref{lem:srange}, we have
\begin{align}
p-q-\beta&\ \geq\ 10^m+3-10^{\ell}+1-10^{m-1}+1 \nonumber\\
&\ \geq\ 10^m+3-10^{m-1}+1-10^{m-1}+1 \nonumber\\
&\ \geq\ 8\cdot 10^{m-1}+5.
\end{align}
Therefore $p-q-\beta$ has at least $m$ decimal digits.
\item[{\rm (ii)}] This is because $p-q-\iota(\beta)\geq p-q-\beta$.
\item[{\rm (iii)}] This is by combining Lemma \ref{lem:basic}(iii) and part (i).
\end{itemize}
\end{proof}
\begin{lemma}
\label{lem:ob2}
We have
\begin{itemize}
\item[{\rm (i)}]
$L(f) \ =\ m+1-L(q^{\beta})+[r(p)\ <\ 10^{L(f)+L(q^{\beta})-1}]$,
\item[{\rm (ii)}] $L(q^{\beta}) \ =\ m+1-L(f)+[r(p)\ <\ 10^{L(f)+L(q^\beta)-1}]$,
\item[{\rm (iii)}] $L(q^{\beta})\ \geq\ \ell$,
\item[{\rm (iv)}] $L(f)\ \leq\ m+1$,
\item[{\rm (v)}] $L(f)\ \leq\ m+2-\ell$,
\item[{\rm (vi)}] $L(f)\ \leq\ m+1-\beta(\ell-1)$, and
\item[{\rm (vii)}] $L(p-q-\beta)\ \leq\ m+2-\ell$.
\end{itemize}
\end{lemma}
\begin{proof}
\begin{itemize}
\item[{\rm (i)}] By \eqref{eq:fact} and Lemma \ref{lem:jessie}, we have
\begin{align}
m+1&\ =\ L(r(p)) \ =\ L(f q^\beta) \ =\ L(f)+L(q^\beta)-[fq^\beta\ <\ 10^{L(f)+L(q^\beta)-1}]\nonumber\\
& \ =\ L(f)+L(q^\beta)-[r(p)\ <\ 10^{L(f)+L(q^\beta)-1}].
\end{align}
The required equality then follows by rearranging.
\item[{\rm (ii)}] This follows by rearranging part (i).
\item[{\rm (iii)}] Since $q^\beta\geq q$, by Lemma \ref{lem:ob} we have $L(q^\beta)\geq L(q)=\ell$.
\item[{\rm (iv)}] By \eqref{eq:fact}, we have $f\leq r(p)$. Thus by Lemma \ref{lem:ob}, we have $L(f)\leq L(r(p)) = m+1$.
\item[{\rm (v)}] By parts (i) and (iii) and the fact that an Iverson bracket must be no greater than $1$, we have
\begin{align}
L(f) &\ =\ m+1-L(q^{\beta})+[r(p)\ <\ 10^{L(f)+L(q^{\beta})-1}] \nonumber\\
& \ \le \ m+1-\ell+1 \ =\ m+2-\ell.
\end{align}
\item[{\rm (vi)}] By Lemma \ref{lem:longineq}, we have
\begin{equation}
L(q^\beta)\ \geq\ \beta L(q)-(\beta-1) \ =\ \beta \ell - (\beta - 1) \ =\ \beta(\ell-1)+1.
\end{equation}
Consequently from part (i),
\begin{align}
L(f) &\ =\ m+1-L(q^{\beta})+[r(p)\ <\ 10^{L(f)+L(q^{\beta})-1}]\nonumber\\
&\ \le \ m+1-\beta(\ell-1)-1+1 \ =\ m+1-\beta(\ell-1).
\end{align}
\item[{\rm (vii)}] By Lemma \ref{lem:basic}(iii), Lemma \ref{lem:ineq}, and part (v),
\begin{equation}
L(p-q-\beta)\ \le \ L(v(f))\ \le \ L(f)\ \le \ m+2-\ell.
\end{equation}
\end{itemize}
\end{proof}
\section{The case $\ell\leq m$}\label{sec:leqm}
Since $r(p) = fq^\beta$ and the number of decimal digits of $r(p)$ and $q$ are $m+1$ and $\ell$, respectively, clearly $m+1\geq \ell$. In this section we consider the case $\ell\leq m$, dividing it into four cases corresponding to the four subsections below, and in each case show that a contradiction results. This means that necessarily $\ell = m+1$, which we consider in the next section.
\ \\
\subsection{Case $\ell=1$} Since $\ell=1\leq m-1$, by Lemma \ref{lem:pqs}(iii), $L(v(f))\geq m$. By Lemma \ref{lem:ineq} and Lemma \ref{lem:ob2}(iv),
\begin{equation}
m\ \le \ L(v(f))\ \le \ L(f)\ \le \ m+1.
\end{equation}
By Lemma \ref{lem:ob2}(ii),
\begin{equation}
L(q^{\beta}) \ =\ m+1-L(f)+[r(p)\ <\ 10^{L(f)+L(q^{\beta})-1}]\ \le \ 1+[r(p)\ <\ 10^{L(f)+L(q^{\beta})-1}]\ \le \ 2.
\end{equation}
So we have $L(q^{\beta})\leq 2$. If $q=2$, then because $q$ is the largest prime factor of $r(p)$, necessarily $f=1$. This implies that $r(p)=2^{\beta}$ has at most $2$ decimal digits, which is a contradiction. Hence $q\in\{3,5,7\}$. There remains only $8$ possibilities for $q^{\beta}$ and by checking one by one, it can be seen that $3\leq v(q^{\beta})\leq 9$. By Lemma \ref{lem:basic}(i) and \eqref{eq:prange},
\begin{equation}
10^m-6\ =\ 10^m+3-9\ \le \ v(f)\ =\ p-v(q^\beta)\ \le \ 10^{m+1}-3-3 \ =\ 10^{m+1}-6.
\end{equation}
Thus we have
\begin{equation}\label{eq:cons}
10^m-6 \ \le \ v(f)\ \le \ 10^{m+1}-6.
\end{equation}
In the remainder of this subsection we discuss the cases $q=3$, $q=5$, and $q=7$, one by one, showing that each case leads to a contradiction. This means that the whole case $\ell=1$ leads to a contradiction.
\ \\
\underline{Sub case $q=3$}: We must have $r(p) = 2^{\gamma} 3^{\beta}$, where $\gamma\geq0$ is an integer. Since $L(3^\beta)\leq2$, we have $3^\beta\leq 81$. Therefore $10^4\leq r(p) \leq 2^\gamma \cdot 81$, and so $\gamma\geq7$. Thus because $f=2^{\gamma}$, \eqref{eq:cons} and Lemma \ref{lem:preless} implies
\begin{equation}
10^m-6\ \le \ v(f) \ = \ v(2^\gamma)\ \le \ 2+\gamma.
\end{equation}
Consequently,
\begin{equation}
10^m-8 \ \le \ \gamma.
\end{equation}
Hence
\begin{equation}\label{eq:just}
r(p) \ =\ 2^{\gamma} 3^{\beta}\ \geq\ 2^{10^m-8}\cdot 3\ \geq\ 10^{m+1}
\end{equation}
(the last inequality can be shown to hold for $m\geq2$). This contradicts the fact that $L(r(p)) = m+1$.
\ \\
\underline{Sub case $q=5$}: We must have $r(p) = 2^{\delta}3^{\gamma}5^{\beta}$, where $\gamma,\delta\geq0$ are integers. Thus because $f=2^{\delta}3^{\gamma}$, by \eqref{eq:cons} and Lemma \ref{lem:preless},
\begin{equation}\label{eq:tov}
10^m-6\ \le \ v(f) \ = \ v(2^\delta)+ v(3^\gamma) \ \le \ 2+\delta+3+\gamma.
\end{equation}
Consequently,
\begin{equation}
\delta+\gamma\ \geq\ 10^m-11.
\end{equation}
Hence
\begin{equation}
r(p) \ = \ 2^{\delta}3^{\gamma}5^{\beta}\ \geq\ 2^{\delta+\gamma}\cdot 5\ \geq\ 2^{10^m-11}\cdot 5\ \geq\ 10^{m+1}
\end{equation}
(the last inequality can be shown to hold for $m\geq2$). This contradicts the fact that $L(r(p)) = m+1$.
\ \\
\underline{Sub case $q=7$}: We must have $r(p) = 2^{\eta}3^{\delta}5^{\gamma}7^{\beta}$, where $\gamma, \delta, \eta \geq0$ are integers. Thus because $f=2^{\eta}3^{\delta}5^{\gamma}$, by \eqref{eq:cons} and Lemma \ref{lem:preless},
\begin{equation}\label{eq:tovs}
10^m-6\ \le \ v(f) \ = \ v(2^\eta)+v(3^\delta)+v(5^\gamma) \ \le \ 10+\eta+\delta+\gamma.
\end{equation}
Consequently,
\begin{equation}
\eta+\delta+\gamma\ \geq\ 10^m-16.
\end{equation}
We then have
\begin{equation}
r(p)\ \geq\ 2^{\eta+\delta+\gamma}\cdot 7\ \geq\ 2^{10^m-16}\cdot 7\ \geq\ 10^{m+1}
\end{equation}
(the last inequality can be shown to hold for $m\geq2$). This contradicts the fact that $L(r(p)) = m+1$.
\ \\
\subsection{Case $\ell=2$} By Lemma \ref{lem:ob2}(iii), $L(q^{\beta})\geq 2$, and by Lemma \ref{lem:ob2}(v), $L(f)\leq m$. Since $\ell=2\leq m-1$, by Lemma \ref{lem:pqs}(iii), $L(v(f))\geq m$. By Lemma \ref{lem:ineq},
\begin{equation}
m\ \le \ L(v(f))\ \le \ L(f)\ \le \ m.
\end{equation}
Therefore $L(f) = L(v(f)) = m$. By Lemma \ref{lem:ob2}(vi),
\begin{equation}
m\ =\ L(f)\ \le \ m+1-\beta,
\end{equation}
and thus $\beta=1$. Hence \eqref{eq:fact} simplifies to $r(p)=fq$. Since $\ell=2$, we have $11\leq q\leq 97$. Therefore because $L(r(p)) = m+1$,
\begin{equation}
f\ =\ \frac{r(p)}{q}\ <\ \frac{10^{m+1}}{11}\ \le \ 10^m-100
\end{equation}
(it can be shown that the rightmost inequality holds for $m\geq4$). By taking the $v$ of $r(p) = fq$, we have
\begin{equation}
p \ =\ v(f) + q \ \le \ f+97\ <\ 10^m-100 + 97\ =\ 10^m-3.
\end{equation}
This implies that $L(p)<m+1$, which is a contradiction.
\ \\
\subsection{Case $3\leq \ell\leq m-1$ }Since $\ell\leq m-1$, by Lemma \ref{lem:pqs}(i) and Lemma \ref{lem:ob2}(vii),
\begin{equation}
m\ \le \ L(p-q-\beta)\ \le \ m+2-\ell.
\end{equation}
This implies that $\ell\leq 2$, a contradiction. Hence this case is impossible.
\ \\
\subsection{Case $\ell=m$} By Lemma \ref{lem:ob2}(iii), $L(q^{\beta})\geq m$, and by Lemma \ref{lem:ob2}(v), $L(f)\leq 2$.
Therefore $L(f)\in\{1,2\}$. By Lemma \ref{lem:ob2}(vi),
\begin{equation}
1\ \le \ L(f)\ \le \ m+1-\beta(m-1).
\end{equation}
This implies that
\begin{equation}
\beta\ \le \ \frac{m}{m-1}\ =\ 1+\frac{1}{m-1},
\end{equation}
and so $\beta=1$. Therefore $r(p) = fq$. If $q=2$, then $f = 1$ and so $r(p)=2$, which contradicts $L(r(p)) = m+1\geq5$. Hence $q$ is an odd prime. In addition, if $f=1$, then $r(p)=q$ is prime, contrary to our assumption that $r(p)$ is composite. Hence $f>1$. By Lemma \ref{lem:basic}(i), $v(f) = p-q$, and so $v(f)$ is even. In the following we consider the cases $L(f) = 1$ and $L(f) = 2$ separately, showing that each leads to a contradiction and so ultimately this case $\ell=m$ is also impossible.
\ \\
\underline{Sub case $L(f)=1$}: Since $v(f)$ is even and $2\leq f\leq 9$, we have $f\in\{2,4\}$. In the case $f=2$, we have $2=p-q$. Then by \eqref{eq:prange},
\begin{equation}
q\ =\ p-2\ \geq\ 10^m+3-2\ =\ 10^m+1,
\end{equation}
contradicting that $L(q) = m$. In the case $f=4$, we have $4=p-q$. Similarly by \eqref{eq:prange},
\begin{equation}
q\ =\ p-4\ \geq\ 10^m+3-4\ =\ 10^m-1.
\end{equation}
As $L(q) = m$, we have $q=10^m-1$, contradicting the primeness of $q$.
\ \\
\underline{Sub case $L(f)=2$}: Since $v(f)$ is even and $10 \leq f\leq 99$, we see that $f$ must be one of
\begin{align}
&15,\
16,\
21,\
24,\
27,\
30,\
33,\
35,\
39,\
40,\
42,\
45,\
51,\
54,\
55,\
56,\
57,\
60,\
63,\nonumber\\
&64,65,\
66,\
69,70, 72,\
75,\
77,\
78,\
84,\
85,\
87,\
88,\
90,\
91,\
93,\
95,\
96,\
99,
\end{align}
with $v(f)$ being one of
\begin{align}
6,\ 8,\ 10,\ 12,\ 14,\ 16,\ 18,\ 20,\ 22,\ 24,\ 26,\ 32,\ 34.
\end{align}
Consequently, by \eqref{eq:prange},
\begin{equation}
q\ =\ p-v(f)\ \geq\ 10^m+3 -34 \ =\ 10^m-31,
\end{equation}
and so
\begin{equation}
r(p) \ =\ fq \ \geq\ 15 (10^m-31)\ \geq\ 10^{m+1}
\end{equation}
(it can be shown that the rightmost inequality holds for $m\geq2$). This contradicts the fact that $L(r(p)) = m+1$.
\section{The case $\ell\ =\ m+1$}\label{sec:mp1}
In this section we consider the case $\ell=m+1$ and narrow down the potentially possible values of $p$ more, i.e., deduce more necessary conditions.
By Lemma \ref{lem:ob2}(v), $L(f)=1$.
By Lemma \ref{lem:ob2}(vi),
\begin{equation}
1 \ =\ L(f)\ \le \ m+1-\beta m.
\end{equation}
This implies that $\beta\leq 1$, and so $\beta=1$. Therefore $r(p) = fq$. If $q=2$, then $f = 1$ and so $r(p)=2$, which contradicts $L(r(p)) = m+1\geq5$. Hence $q$ is an odd prime. In addition, if $f=1$, then $r(p)=q$ is prime, contrary to our assumption that $r(p)$ is composite. Hence $f>1$. By Lemma \ref{lem:basic}(i), $v(f) = p-q$, and so $v(f)$ is even. As $L(f) = 1$, we see that $v(f) = f\in\{2,4\}$. Consequently, $r(p)$ must be even.
Let the decimal representations of $p$, $r(p)$, and $q$ be
\begin{align}
p &\ =\ (a_m\cdots a_0)_{10},\label{eq:reprp}\\
r(p) &\ =\ (a_0\cdots a_m)_{10},\label{eq:reprrp}\\
q & \ =\ (b_m\cdots b_0)_{10}, \label{eq:repq}
\end{align}
where $a_m,b_m\neq0$.
As $p$ is odd and prime, $a_0\in\{1,3,7,9\}$, and as $r(p)$ is even, $a_m\in\{2,4,6,8\}$. Since $v(f) = p-q$, we have $q = p-v(f)$, and so
\begin{equation}\label{eq:long}
(b_m\cdots b_0)_{10} \ =\ (a_m\cdots a_0)_{10}-v(f).
\end{equation}
Consequently, because $v(f)\in\{2,4\}$,
\begin{equation}
a_m\cdot 10^m-4 \ \le \ (b_m\cdots b_0)_{10}\ <\ (a_m\cdots a_0)_{10},
\end{equation}
and so $b_m\in\{a_m-1,a_m\}$. Since $r(p) = fq$,
\begin{equation}\label{eq:rfq}
(a_0\cdots a_m)_{10} \ =\ f(b_m\cdots b_0)_{10}.
\end{equation}
This implies that
\begin{equation}\label{eq:forcon}
fb_m\ \le \ a_0.
\end{equation}
Hence
\begin{equation}\label{eq:erange}
a_m-1\ \le \ b_m\ \le \ \frac{a_0}{f},\quad\text{which implies that}\quad a_m\ \le \ 1+\frac{a_0}{f}\ \le \ 1+\frac{9}{f}.
\end{equation}
Notice that from \eqref{eq:long} we have
\begin{equation}\label{eq:mod1}
b_0\ \equiv\ a_0-v(f)\pmod{10},
\end{equation}
and that from \eqref{eq:rfq} we have
\begin{equation}\label{eq:mod2}
a_m\ \equiv\ fb_0 \pmod{10}.
\end{equation}
In the following we consider the cases $f=2$ and $f=4$ separately, corresponding to two subsections. For $f=2$, we show that necessarily $a_m=4$, $a_0=9$, $b_m=4$, and $b_0=7$; while for $f=4$, we show that a contradiction results.
\ \\
\subsection{Case $f=2$}\eqref{eq:mod1} and \eqref{eq:mod2} become respectively
\begin{eqnarray}
b_0 & \ \equiv\ & a_0-2\pmod{10}, \label{eq:mod12}\\
a_m & \ \equiv\ & 2b_0 \pmod{10} \label{eq:mod22}.
\end{eqnarray}
By \eqref{eq:erange}, $a_m\leq 1+9/2 = 5.5$ and so as $a_m\in\{2,4,6,8\}$, we have $a_m\in\{2,4\}$. In the following we consider the cases $a_m=2$ and $a_m=4$ separately. For $a_m=2$, we show that a contradiction results; while for $a_m=4$, we show that necessarily $a_0=9$, $b_m=4$, and $b_0=7$.
\ \\
\underline{Sub case $a_m=2$}: \eqref{eq:mod22} becomes $2\equiv 2b_0 \pmod{10}$, or equivalently, $b_0\equiv 1\pmod{5}$. Thus as $0\leq b_0<10$, we have $b_0\in\{1,6\}$. By \eqref{eq:mod12}, we have modulo $10$,
\begin{equation}
a_0\ \equiv\ b_0+2\ \equiv\ \begin{cases}
3,& \text{if $b_0\ =\ 1$}, \\
8, &\text{if $b_0\ =\ 6$}.
\end{cases}
\end{equation}
Thus as $0\leq a_0<10$,
\begin{equation}
a_0\ =\ \begin{cases}
3,& \text{if $b_0\ =\ 1$}, \\
8, &\text{if $b_0\ =\ 6$}.
\end{cases}
\end{equation}
As $a_0\in\{1,3,7,9\}$, we have $b_0=1$ and $a_0=3$. Consequently, \eqref{eq:long} becomes
\begin{equation}
(b_m\cdots 1)_{10} \ =\ (2\cdots 3)_{10}-2,
\end{equation}
which means that $b_m=2$. Then however, \eqref{eq:forcon} becomes $2\cdot 2\leq 3$, which is false and we have a contradiction.
\ \\
\underline{Sub case $a_m=4$}: \eqref{eq:mod22} becomes $4\equiv 2b_0 \pmod{10}$, or equivalently, $b_0\equiv 2\pmod{5}$. Thus as $0\leq b_0<10$, we have $b_0\in\{2,7\}$. By \eqref{eq:mod12}, we have modulo $10$,
\begin{equation}
a_0\ \equiv\ b_0+2\ \equiv\ \begin{cases}
4,& \text{if $b_0\ =\ 2$}, \\
9, &\text{if $b_0\ =\ 7$}.
\end{cases}
\end{equation}
Thus as $0\leq a_0<10$,
\begin{equation}
a_0\ =\ \begin{cases}
4,& \text{if $b_0\ =\ 2$}, \\
9, &\text{if $b_0\ =\ 7$}.
\end{cases}
\end{equation}
As $a_0\in\{1,3,7,9\}$, we have $b_0=7$ and $a_0=9$. Consequently, \eqref{eq:long} becomes
\begin{equation}
(b_m\cdots 7)_{10} \ =\ (4\cdots 9)_{10}-2,
\end{equation}
which means that $b_m=4$. Then \eqref{eq:forcon} becomes $2\cdot 4\leq 9$, which is true and so we do not have a contradiction like we just did in the case $a_m=2$.
\ \\
\subsection{Case $f=4$} \eqref{eq:mod1} and \eqref{eq:mod2} become respectively
\begin{eqnarray}
b_0 & \ \equiv\ & a_0-4\pmod{10}, \label{eq:mod14}\\
a_m & \ \equiv \ & 4b_0 \pmod{10} \label{eq:mod24}.
\end{eqnarray}
By \eqref{eq:erange}, $a_m\leq 1+9/4 = 3.25$ and so as $a_m\in\{2,4,6,8\}$, we have $a_m=2$.
Thus \eqref{eq:mod24} becomes $2\equiv 4b_0 \pmod{10}$, or equivalently, $2b_0\equiv 1\pmod{5}$, or equivalently, $b_0\equiv 3\pmod{5}$. Thus as $0\leq b_0<10$, we have $b_0\in\{3,8\}$. By \eqref{eq:mod14}, we have modulo $10$,
\begin{equation}
a_0\ \equiv\ b_0+4\ \equiv\ \begin{cases}
7,& \text{if $b_0\ =\ 3$}, \\
12, &\text{if $b_0\ =\ 8$}.
\end{cases}
\end{equation}
Thus as $0\leq a_0<10$,
\begin{equation}
a_0\ =\ \begin{cases}
7,& \text{if $b_0\ =\ 3$}, \\
2, &\text{if $b_0\ =\ 8$}.
\end{cases}
\end{equation}
As $a_0\in\{1,3,7,9\}$, we have $b_0=3$ and $a_0=7$. Consequently, \eqref{eq:long} becomes
\begin{equation}
(b_m\cdots 3)_{10} \ =\ (2\cdots 7)_{10}-4,
\end{equation}
which means that $b_m=2$. Then however, \eqref{eq:forcon} becomes $4\cdot 2\leq 7$, which is false and we have a contradiction.
\section{The other decimal digits of $p$}\label{sec:otherdec}
As a result of Sections \ref{sec:setup} through \ref{sec:mp1}, we see that if $p$ is a prime $v$-palindrome, then the following are true:
\begin{itemize}
\item[{\rm (i)}] $p$ has $m+1$ decimal digits, for some $m\geq4$,
\item[{\rm (ii)}] $p$ is of the form $(4\cdots 9)_{10}$, i.e., its leftmost decimal digit is $4$ and its rightmost decimal digit is $9$,
\item[{\rm (iii)}] $p-2$ is prime, and
\item[{\rm (iv)}] $r(p) = 2(p-2)$.
\end{itemize}
In this section we show further that all other decimal digits of $p$ must be $9$'s as well, and thus $p = 5\cdot 10^m - 1$. Filling in what we know into \eqref{eq:reprp}, \eqref{eq:reprrp}, and \eqref{eq:repq}, we have
\begin{align}
p &\ =\ (4,a_{m-1},\ldots,a_1,9)_{10},\label{eq:reprpN}\\
r(p) &\ =\ (9,a_1,\ldots,a_{m-1},4)_{10},\label{eq:reprrpN}\\
p-2 & \ =\ (4,a_{m-1},\ldots ,a_1,7)_{10}. \label{eq:repqN}
\end{align}
Since $r(p) = 2(p-2)$,
\begin{equation}\label{eq:rfqN}
(9,a_1,\ldots,a_{m-1},4)_{10} \ =\ 2(4,a_{m-1},\ldots ,a_1,7)_{10}.
\end{equation}
We need to prove that
\begin{equation}\label{eq:need}
a_i \ =\ a_{m-i} \ =\ 9,\quad \text{for $1\ \leq\ i\ \leq\ \left\lfloor\frac{m}{2}\right\rfloor$}.
\end{equation}
For integers $0\leq I\leq \left\lfloor\frac{m}{2}\right\rfloor$, let $S(I)$ be the statement that
\begin{equation}
a_i \ =\ a_{m-i} \ =\ 9,\quad \text{for $1\ \leq\ i\ \leq\ I$}.
\end{equation}
We prove that $S(I)$ holds for all $0\leq I\leq \left\lfloor\frac{m}{2}\right\rfloor$ inductively, which will imply in particular that $S(\left\lfloor\frac{m}{2}\right\rfloor)$, i.e., \eqref{eq:need}, holds. Firstly, notice that $S(0)$ holds vacuously. Next, suppose that $S(I)$ holds for some $0\leq I< \left\lfloor\frac{m}{2}\right\rfloor$. We shall proceed to prove $S(I+1)$, which amounts to proving
\begin{equation}
a_{I+1} \ =\ a_{m-I-1} \ =\ 9.
\end{equation}
We have
\begin{align}
p &\ =\ (4,\{9\}^{I},a_{m-I-1},\ldots,a_{I+1},\{9\}^{I+1})_{10},\\
r(p) &\ =\ (\{9\}^{I+1},a_{I+1},\ldots,a_{m-I-1},\{9\}^{I},4)_{10},\\
p-2 &\ =\ (4,\{9\}^{I},a_{m-I-1},\ldots,a_{I+1},\{9\}^{I},7)_{10},
\end{align}
where $\{9\}^j$ for some integer $j\geq0$ means that there are $j$ digits of $9$ consecutively; $\{9\}^0$ means that there is nothing. \eqref{eq:rfqN}
becomes
\begin{equation}\label{eq:crucial}
(\{9\}^{I+1},a_{I+1},\ldots,a_{m-I-1},\{9\}^{I},4)_{10} \ =\ 2(4,\{9\}^{I},a_{m-I-1},\ldots,a_{I+1},\{9\}^{I},7)_{10}.
\end{equation}
If $a_{m-I-1}\leq 4$, then the right-hand side of \eqref{eq:crucial} must be of the form
\begin{equation}
(\underbrace{9,\ldots,9}_{I},8,\ldots)_{10},
\end{equation}
which cannot equal the left-hand side. Therefore necessarily $5\leq a_{m-I-1}\leq 9$. Notice that the congruence
\begin{equation}\label{eq:crucialcong}
2a_{I+1}+1\ \equiv\ a_{m-I-1}\pmod{10}
\end{equation}
follows from \eqref{eq:crucial}. Reducing this congruence to modulo $2$, we see that $a_{m-I-1}$ must be odd. Therefore necessarily $a_{m-I-1}\in\{5,7,9\}$. In the following we consider each such possible value of $a_{m-I-1}$, corresponding to three subsections.
\ \\
\subsection{Case $a_{m-I-1} = 5$} \eqref{eq:crucial} becomes
\begin{equation}\label{eq:crucial5}
(\{9\}^{I+1},a_{I+1},\ldots,5,\{9\}^{I},4)_{10} \ =\ 2(4,\{9\}^{I},5,\ldots,a_{I+1},\{9\}^{I},7)_{10}
\end{equation}
and \eqref{eq:crucialcong} becomes
\begin{equation}
2a_{I+1} + 1\ \equiv\ 5\pmod{10},
\end{equation}
which forces $a_{I+1} \in \{2,7\}$. However, in view of integer multiplication, we see that the digit of $10^{m-I-1}$ of the right-hand side of \eqref{eq:crucial5} must be $0$ or $1$. This means that we need to have $a_{I+1}\in\{0,1\}$, which is impossible.
\ \\
\subsection{Case $a_{m-I-1} = 7$} \eqref{eq:crucial} becomes
\begin{equation}\label{eq:crucial7}
(\{9\}^{I+1},a_{I+1},\ldots,7,\{9\}^{I},4)_{10} \ =\ 2(4,\{9\}^{I},7,\ldots,a_{I+1},\{9\}^{I},7)_{10}
\end{equation}
and \eqref{eq:crucialcong} becomes
\begin{equation}
2a_{I+1} + 1\ \equiv\ 7\pmod{10},
\end{equation}
which forces $a_{I+1} \in \{3,8\}$. However, in view of integer multiplication, we see that the digit of $10^{m-I-1}$ of the right-hand side of \eqref{eq:crucial7} must be $4$ or $5$. This means that we need to have $a_{I+1}\in\{4,5\}$, which is impossible.
\ \\
\subsection{Case $a_{m-I-1} = 9$} \eqref{eq:crucial} becomes
\begin{equation}\label{eq:crucial9}
(\{9\}^{I+1},a_{I+1},\ldots,\{9\}^{I+1},4)_{10} \ =\ 2(4,\{9\}^{I+1},\ldots,a_{I+1},\{9\}^{I},7)_{10}
\end{equation}
and \eqref{eq:crucialcong} becomes
\begin{equation}
2a_{I+1}+ 1\ \equiv\ 9\pmod{10},
\end{equation}
which forces $a_{I+1} \in \{4,9\}$.
Assume that $a_{I+1} = 4$, then \eqref{eq:crucial9} becomes
\begin{equation}\label{eq:crucial94}
(\{9\}^{I+1},4,\ldots,\{9\}^{I+1},4)_{10} \ =\ 2(4,\{9\}^{I+1},\ldots,4,\{9\}^{I},7)_{10}.
\end{equation}
However, in view of integer multiplication, we see that the digit of $10^{m-I-1}$ of the right-hand side of \eqref{eq:crucial94} must be $8$ or $9$, in contrary to the left-hand side. Hence we must have $a_{I+1}= 9$.
Notice that this completes the induction because we are in the final case of $a_{m-I-1} = 9$.
\section{Proof of the converse}\label{sec:converse}
Sections \ref{sec:setup} through \ref{sec:otherdec} proved the forward direction of Theorem \ref{thm:main}. In this section we prove the converse.
Let $p = 5\cdot 10^m - 1 = 4\underbrace{9\cdots 9}_{m}$, for some integer $m\geq4$, be a prime such that $p-2 = 5\cdot 10^m - 3$ is also prime. We show that $p$ is a $v$-palindrome. Firstly, clearly $10\nmid p$ and $p\neq r(p)$. We have
\begin{align}
r(p) \ =\ r(4\underbrace{9\cdots 9}_{m}) \ =\ \underbrace{9\cdots 9}_{m}4 \ =\ 2\cdot 4\underbrace{9\cdots 9}_{m-1}7 \ =\ 2(p-2).
\end{align}
Consequently, as $p-2$ is an odd prime,
\begin{equation}
v(r(p)) \ =\ v(2(p-2)) \ =\ 2+(p-2) \ =\ p.
\end{equation}
This completes the proof.
\section{Number Of Prime $v$-palindromes}\label{heuristic}
Theorem \ref{thm:heuristicfinite} follows from standard models for prime numbers; we sketch below how a slightly weakened Cram\'er model, combined with our characterization of the form of prime $v$-palindromes, implies that there can only be finitely many.
For the standard Cram\'er model, one assumes that each integer $n$ is prime with probability on the order of $1/\log n$, and the probability any two numbers are both prime is simply the product of the probabilities. This of course is clearly false, as we know if $n \ge 2$ is even then it cannot be prime, and if $n \equiv -2 \bmod p$ for any prime $p < n$ then $n+2$ cannot be prime. However, our goal is simply to provide support, and thus we ignore the more refined arguments one can do (see for example \cite{Rub}). We assume instead that the probability $n$ and $n+2$ are both prime is bounded by $C / \log^2 n$ for some fixed $C$; as we are only trying to prove there are at most finitely many prime $v$-palindromes, we are fine with a slightly larger but still finite upper bound.
Let $T_n$ be the event that $5 \cdot 10^n - 3$ and $5 \cdot 10^n - 1$ are both prime, then the expected number of prime $v$-palindromes at most $10^{N+1}$ is \begin{equation} \sum_{n=1}^N 1 \cdot {\rm Prob}(T_n). \end{equation} As we are just concerned with supporting the conjecture that there are only finitely many, let us over-estimate and say
\begin{equation}\label{eq:heuristicsum} {\rm Prob}(T_n) \ \le \ \frac{C}{\log^2(5 \cdot 10^n - 3)} \ \le \ \frac{400C}{n^2 \log^2 10} \ \le
\ \frac{100C}{n^2}. \end{equation} As the sum of $1/n^2$ converges, the expected number of prime $v$-palindromes is finite.
\begin{rek} The Cram\'er model suggests we can take $C$ to be around 1. With such an assumption, given that there are no prime $v$-palindromes for the first several candidates of the form $5 \cdot 10^m - 1$, the expected number of numbers of this form that are the larger in a twin prime pair is less than 1/2, and thus we do not expect there to be any prime $v$-palindromes. \end{rek}
\begin{rek} While standard models predict the probability two integers of size $x$ differing by 2 are both prime is on the order of $1/\log^2 x$, a significantly larger bound would still imply there are only finitely many $v$-primes. For example, if we instead had the probability bounded by a quantity of size $1/\log^{1+\epsilon} x$ for any $\epsilon > 0$ we would still get a finite sum in \eqref{eq:heuristicsum}. \end{rek}
\ \\
\end{document} | math | 41,891 |
\begin{document}
\title{On links with locally infinite {K}akimizu complexes}
\author{Jessica E. Banks}
\date{}
\maketitle
\begin{abstract}
We show that the Kakimizu complex of a knot may be locally infinite, answering a question of Przytycki--Schultens. We then prove that if a link $L$ only has connected Seifert surfaces and has a locally infinite Kakimizu complex then $L$ is a satellite of either a torus knot, a cable knot or a connected sum, with winding number 0.
\end{abstract}
\section{Introduction}
The Kakimizu complex $\ms(L)$ of a non-split, oriented link $L$ in $\mathbb{S}^3$ records the structure of the set of minimal genus Seifert surfaces for $L$. When every minimal genus Seifert surface for $L$ is connected, $\ms(L)$ has the following description, which mirrors the definition of the curve complex of a compact surface.
\begin{definition}[\cite{MR1177053} p225]
$\ms(L)$ is a simplicial complex, the vertices of which are the ambient isotopy classes of minimal genus Seifert surfaces for $L$. Vertices $R_0,\cdots,R_n$ span an $n$--simplex exactly when they can be realised disjointly.
\end{definition}
In \cite{przytycki-2010}, Przytycki and Schultens generalise this definition as follows.
\begin{definition}
Let $M$ be a compact, connected, orientable, irreducible, $\partial$--irreducible 3--manifold. Let $\gamma$ be a union of disjoint, oriented, simple closed curves on $\partial M$ such that $\gamma$ does not separate any component of $\partial M$. Let $\alpha\in H_2(M,\partial M;\mathbb{Z})$ with $\partial\alpha=[\gamma]$.
Call an oriented surface $S$ properly embedded in $M$ a \textit{$(\gamma,\alpha)$--surface} if $[S]=\alpha$ and $\partial S$ is homotopic to $\gamma$.
The flag simplicial complex $\ms(M,\gamma,\alpha)$ is defined as follows. The set $\V(\ms(M,\gamma,\alpha))$ of vertices is defined to be the set of isotopy classes of {$(\gamma,\alpha)$--surfaces} with maximal Euler characteristic $\chi$ in their homology class. Two such surfaces $S,S'$ are joined by an edge if they can be isotoped such that a lift of $M\setminus S'$ to the infinite cyclic cover of $M$ associated to $\alpha$ intersects exactly two lifts of $M\setminus S$.
\end{definition}
\begin{remark}
Using this definition, $\ms(L)$ is ${\ms(\mathbb{S}^3\setminus\Int\nhd(L),\partial R, [R])}$, where $R$ is any Seifert surface for $L$.
\end{remark}
Viewing $\ms(L)$ in terms of the infinite cyclic cover of its complement in this way has proved especially useful when considering questions about distances in $\ms(L)$. In particular, the following results are proved using this viewpoint.
\begin{theorem}[\cite{MR1177053} Theorem A]
Let $L$ be a non-split link. Then $\ms(L)$ is connected.
\end{theorem}
\begin{theorem}[\cite{MR2531146} Theorem 1.1]\label{diametertheorem}
Let $K$ be a knot in $\mathbb{S}^3$ that is not a satellite. Then the diameter of $\ms(K)$ is bounded above by $2g(K)(3g(K)-2)+1$, where $g(K)$ denotes the genus of $K$.
\end{theorem}
\begin{theorem}[\cite{przytycki-2010} Theorem 1.1]
If $M,\gamma,\alpha$ are as above, $\ms(M,\gamma,\alpha)$ is contractible.
\end{theorem}
It is known that any knot that is not a satellite has only finitely many minimal genus Seifert surfaces (see, for example, \cite{MR0440528} p329). Contrasting with this and Theorem \ref{diametertheorem}, Kakimizu has shown (\cite{MR1177053} Theorem B) that there are knots $K$ such that $\ms(K)$ has infinite diameter.
Przytycki and Schultens raise the question of whether the complex $\ms(M,\gamma,\alpha)$ can be locally infinite. In Section \ref{examplesection} we give an example that answers this question with the following result.
\begin{theorem}
$\ms$ can be locally infinite even for a knot.
\end{theorem}
In Section \ref{conversesection} we prove the following condition on the types of links that might have a locally infinite Kakimizu complex, under the additional assumption that all minimal genus Seifert surfaces for the link are connected. Note that such a link cannot be split.
\begin{restatable}{theorem}{satellitethm}\label{satellitetheorem}
Let $L$ be an oriented link such that every minimal genus Seifert surface for $L$ is connected. If $\ms(L)$ is locally infinite then $L$ is a satellite of either a torus knot, a cable knot or a connected sum, with winding number 0.
\end{restatable}
\noindent This, in particular, includes all links with non-zero Alexander polynomial.
I wish to thank Marc Lackenby for helpful conversations, particularly with regard to the proof of Theorem \ref{satellitetheorem}.
\section{A knot with locally infinite $\ms$}\label{examplesection}
\begin{definition}[\cite{przytycki-2010} Section 3]
Let $M$ be a connected 3--manifold, and let $S,S'$ be (possibly disconnected) surfaces properly embedded in $M$ in general position.
$S$ and $S'$ \textit{bound a product region} if the following holds. There is a compact surface $T$, a finite collection $\rho\subseteq \partial T$ of arcs and simple closed curves and an embedding of $T^*=(T\times\mathrm{I})/\!\sim$ into $M$ with the following properties.
\begin{itemize}
\item $T\times\{0\}=S\cap T^*$ and $T\times\{1\}=S'\cap T^*$.
\item $\partial T^*\setminus (T\times\partial\mathrm{I})\subseteq\partial M$.
\end{itemize}
Here $\sim$ collapses $x\times\mathrm{I}$ to a point for each $x\in\rho$.
Say $S$ and $S'$ have \textit{simplified intersection} if they do not bound a product region.
\end{definition}
\begin{proposition}[\cite{MR1315011} Proposition 4.8(2)]\label{productregionsprop}
Let $M$ be a $\partial$--irreducible Haken manifold.
Let $S,S'$ be incompressible, $\partial$--incompressible surfaces properly embedded in $M$ in general position. Suppose $S\cap S'\neq\emptyset$, but $S$ can be isotoped to be disjoint from $S'$. Then there is a product region between $S$ and $S'$.
\end{proposition}
\begin{theorem}
Let $K_{\alpha}$ be the twisted Whitehead double of the trefoil shown in Figure \ref{locallyinfinitepic1}.
\begin{figure}
\caption{\label{locallyinfinitepic1}
\label{locallyinfinitepic1}
\end{figure}
Then $\ms(K_{\alpha})$ is not locally finite.
\end{theorem}
\begin{proof}
Let $R$ be the genus 1 Seifert surface for $K_{\alpha}$ shown in Figure \ref{locallyinfinitepic1} (note that every Whitehead double has such a Seifert surface). We construct an infinite family of genus 1 Seifert surfaces for $K_{\alpha}$ that are disjoint from $R$.
Let $M=\mathbb{S}^3\setminus \Int\nhd(K_{\alpha})$. Let $T$ be the torus that bounds the trefoil knot companion of $K_{\alpha}$, such that $K_{\alpha}$ lies in the solid torus bounded by $T$. In addition, let $M_1$ be the part of $M$ outside of $T$ as drawn in Figure \ref{locallyinfinitepic2} (that is, the side away from the knot), and $M_0$ the part on the inside. Let $\mu$ be a meridian of $T\subset\mathbb{S}^3$. There is a M\"obius band properly embedded in $M_1$, the boundary of which is a longitude $\lambda$ of the solid torus bounded by $T$. Then $\lambda$ and $\mu$ are as shown in Figure \ref{locallyinfinitepic2}.
\begin{figure}
\caption{\label{locallyinfinitepic2}
\label{locallyinfinitepic2}
\end{figure}
Let $S_1$ be the annulus properly embedded in $M_1$ that is contained in the boundary of a regular neighbourhood of this M\"obius band in $M_1$. Then $\partial S_1$ is two copies of $\lambda$, with opposite orientations. Let $S_T$ be one of the two annuli into which $T$ is divided by $\partial S_1$.
$R$ is a plumbing of two annuli $S_0$ and $S'_0$ in $M_0$, where $S_0$ is parallel to $S_T$ in $\mathbb{S}^3\setminus\Int(M_1)$. Isotope $R$ in $M$ so that $R\cap T=S_T$, keeping $\partial R$ fixed. Let $R_0$ be the Seifert surface for $K_{\alpha}$ given by removing $S_T$ from $R$ and replacing it with $S_1$. Then $|R_0\cap T|=2$. In addition, $R_0$ can be made disjoint from $R$.
Express a regular neighbourhood $\nhd(T)$ of $T$ as $\mathbb{S}^1\times\mathrm{I}\times\mathbb{S}^1$, where $\mathbb{S}^1\times\{\frac{1}{2}\}\times\{1\}=\mu$ and $\{1\}\times\{\frac{1}{2}\}\times\mathbb{S}^1=\lambda$, and let $S$ be the annulus $\mathbb{S}^1\times\mathrm{I}\times\{1\}$.
Let $\psi\colon S\to S$ be a Dehn twist. Define $\Psi\colon \mathbb{S}^3\setminus\nhd(K_{\alpha})\to \mathbb{S}^3\setminus\nhd(K_{\alpha})$ by
\[
\Psi(x)=
\begin{cases}
(\psi(y),z) & \textit{ if }x=(y,z)\in S\times\mathbb{S}^1=\nhd(T)\\
x & \textit{ else.}
\end{cases}
\]
For $n\in\mathbb{Z}$ let $R_n=\Psi^n(R_0)$. Then, for each $n$, $R_n$ is a minimal genus Seifert surface for $K_{\alpha}$ that can be made disjoint from $R$.
It remains to show that $R_n\neq R$ and $R_n\neq R_m$ for $m\neq n$ when viewed as vertices of $\ms(K_{\alpha})$.
Fix $n\in\mathbb{Z}$. To show that $R_n\neq R$ we will show that $R_n$ cannot be made disjoint from $T$. In this case we may assume $n=0$. First note that $M$ is $\partial$--irreducible, $R_0$ and $T$ are incompressible, and $T$ is obviously $\partial$--incompressible. $R_0$ is also $\partial$--incompressible as it is orientable, incompressible and not $\partial$--parallel and $\partial M$ is a torus. $M\setminus \Int\nhd(R_0\cup T)$ has three components. One of these is $M_0\setminus\Int\nhd(R_0)$. This is not a product manifold between $R_0$ and $T$ since $R_0$ meets $K_{\alpha}$ in $M_0$ whereas $T$ does not. The other two components lie in $M_1$. One is homeomorphic as a sutured manifold to that shown in Figure \ref{locallyinfinitepic3}, and the other is homeomorphic to its complement.
\begin{figure}
\caption{\label{locallyinfinitepic3}
\label{locallyinfinitepic3}
\end{figure}
Neither of these is a product manifold. By Proposition \ref{productregionsprop}, $R_n$ cannot be isotoped to be disjoint from $T$.
Now fix $m\in\mathbb{Z}$. Again we may assume $n=0$. Let $R'_0$ be a copy of $R_0$, isotoped to be disjoint from $R_0$ (except along its boundary). Then $R'_m=\Psi^m(R'_0)$ is isotopic to $R_m$. Figure \ref{locallyinfinitepic4} shows a cross-section of $\nhd(T)$ in the case $m=2$,
\begin{figure}
\caption{\label{locallyinfinitepic4}
\label{locallyinfinitepic4}
\end{figure}
where $K_{\alpha}$ lies on the inside of $T$ as shown. The components of $M\setminus(R_0\cup R'_m)$ are of five types, as marked. Outside $\nhd(T)$, those marked $M_{0,b}$ and $M_{1,b}$ are each part of the parallel region between $R_0$ and $R'_0$. It is therefore clear that neither of $M_{0,b},M_{1,b}$ is a product region as they each have disconnected intersection with $R_0$. For the same reason, the components of the same type as $M_T$ are not product regions, and neither is $M_{0,a}$. The manifolds $M_{1,a}$ and $M'_{1,a}$ are sutured manifolds and are the same as the components of $M\setminus(R_0\cup T)$ in $M_1$. Hence, again by applying Proposition \ref{productregionsprop}, we see that {$R_0\neq R_m$}.
Thus $\ms(K_{\alpha})$ is locally infinite at $R$.
\end{proof}
\begin{remark}
In \cite{MR1123342}, Kakimizu constructs incompressible Seifert surfaces for a Whitehead double of a knot $K$ using two copies of a Seifert surface for $K$.
Although expressed differently, the above construction is very similar to that used by Kakimizu, with the two Seifert surfaces replaced by the annulus $S'$.
\end{remark}
\section{A restriction on links with locally infinite $\ms$}\label{conversesection}
In this section we prove Theorem \ref{satellitetheorem}. Our proof relies heavily on the work of Wilson in \cite{MR2420023}, to which we refer the reader for definitions not given here. We will also need the following proposition.
\begin{proposition}[\cite{MR808776} 15.26]\label{knotannulusprop}
Let $K$ be a knot, and let $M=\mathbb{S}^3\setminus\Int\nhd(K)$. Suppose there is an annulus $S$ properly embedded in $M$ that is not $\partial$--parallel. If neither component of $\partial S$ bounds a disc in $\partial M$ then $K$ is a torus knot, a cable knot, or a connected sum.
\end{proposition}
\begin{definition}
A compact surface $S$ embedded in $\mathbb{S}^3$ with no closed components is a \textit{spanning surface} for an unoriented link $L$ if $\partial S=L$. We will call $S$ an \textit{unoriented Seifert surface} for $L$ if $S$ is orientable.
\end{definition}
\begin{remark}
An unoriented Seifert surface $R$ for an unoriented link $L$, together with a fixed orientation on $R$, is a Seifert surface for $L$ with the orientation induced by $R$.
\end{remark}
\begin{definition}
Let $S$ be a normal surface in a triangulated 3--manifold.
Its \textit{weight} is the number of times it meets the 1--skeleton of the triangulation. Call $S$ \textit{minimal} if it has minimal weight among normal surfaces isotopic to $S$ by an isotopy fixing $\partial S$.
\end{definition}
\begin{definition}
Let `$+$' denote the usual addition on normal surfaces.
Given normal surfaces $S,S_1,S_2$ with $S=S_1+S_2$, say that $S_1$ and $S_2$ are in \textit{reduced form} if they have been isotoped to minimise $|S_1\cap S_2|$ while maintaining the equation $S=S_1+S_2$.
\end{definition}
In \cite{MR2420023}, Wilson states the following.
\begin{theorem}[\cite{MR2420023} Main Theorem 1.1]\label{wilsonmainthm}
Let $K$ be a non-trivial knot, and let $M=\mathbb{S}^3\setminus\nhd(K)$. Then there is a finite set $\{R_1,\cdots,R_m\}$ of incompressible Seifert surfaces for $K$ and a finite set $\{S_1,\cdots,S_n\}$ of closed surfaces in $M$ that are not boundary parallel such that any incompressible Seifert surface $R$ is isotopic to a Haken sum $R=R_i+a_1 S_1+\cdots+a_n S_n$, where $a_1,\ldots,a_n$ are non-negative integers.
\end{theorem}
The surfaces $R_1,\ldots,R_m$ that arise from Wilson's proof are spanning surfaces for $K$. However, he does not consider the orientability of these surfaces, which is necessary to conclude, as he does, that they are in fact Seifert surfaces. With some further work it can be shown that it is possible to require these surfaces to be orientable. We will not need this.
It is also worth noting the nature of the isotopy referred to in Theorem \ref{wilsonmainthm}. In his proof, Wilson isotopes the chosen Seifert surface $R$ into normal form based on the following lemma.
\begin{lemma}[\cite{MR2420023} Lemma 3.3]
Let $K$ be a knot, let $M=\mathbb{S}^3\setminus\nhd(K)$ and let $R$ be an incompressible Seifert surface for $K$ in $M$. Suppose that $M$ is triangulated, and $\partial R$ meets each 2--simplex of the triangulation in at most one normal arc. Then $R$ can be put into normal form by an isotopy fixing $\partial R$.
\end{lemma}
The proof of this lemma gives the stronger conclusion that the isotopy puts the surface into minimal normal form. This is important because minimality is a key hypothesis of \cite{MR744850} Theorem 2.2, which is used in the proof of Theorem \ref{wilsonmainthm}.
Aside from these points, Wilson's proof is actually stronger than the statement of Theorem \ref{wilsonmainthm} suggests. In particular, by following the proof with $M$ the complement of a minimal genus Seifert surface for a link, it gives the following.
\begin{theorem}\label{wilsontheorem}
Let $L$ be an oriented link such that every minimal genus Seifert surface for $L$ is connected. Let $R$ be a minimal genus Seifert surface for $L$, let $M=\mathbb{S}^3\setminus\Int\nhd(R)$, and fix a set $\rho_1,\cdots,\rho_k$ of core curves of the annuli $\partial M\cap\partial\!\nhd(L)$, one for each link component. There is a triangulation of $M$ such that every Seifert surface $R'$ for $L$ disjoint from $R$ can be put into normal form with $\partial R'=\bigcup_{i=1}^k{\rho_i}$.
Furthermore, there is a finite set $\{R_1,\cdots,R_m\}$ of surfaces in $M$ with non-empty boundary contained in $\bigcup_{i=1}^k{\rho_i}$, and a finite set $\{S_1,\cdots,S_n\}$ of closed surfaces in $M$, such that all these surfaces are incompressible and in normal form, and the following holds. Any minimal genus Seifert surface $R'$ for $L$ in $M$ with $\partial R'=\bigcup_{i=1}^k{\rho_i}$ and in minimal normal form can be expressed as $a_1 R_1+\cdots+a_m R_m +b_1 S_1+\cdots+b_n S_n$ for some $a_i,b_i\in\mathbb{Z}_{\geq 0}$.
\end{theorem}
If $L$ has more than one component, it is possible that, for a given $j\leq m$, $\partial R_j$ is a strict subset of $\bigcup_{i=1}^k{\rho_i}$. However, only finitely many combinations of $R_1,\cdots,R_m$ will yield the correct boundary. Hence we may assume that $\partial R_j=\bigcup_{i=1}^k{\rho_i}$. Then $\sum_{i=1}^{m}{a_i}=1$.
If $K$ is an oriented knot, any unoriented Seifert surface for $K$ can be oriented to make it a Seifert surface. For a link $L$ with more than one component this might not be the case in general.
The presence of the Seifert surface $R$ for the oriented link $L$ that is disjoint from the spanning surfaces $R_i$ allows us to say more in this case. Suppose that, for some $j$, $R_j$ cannot be oriented to make it a Seifert surface for $L$. Combining it with $R$ then gives a closed, non-orientable surface in $\mathbb{S}^3$, which is not possible. Hence each $R_i$ is a Seifert surface for $L$.
\satellitethm*
\begin{proof}
Let $R$ be a minimal genus Seifert surface for $L$ such that $\ms(L)$ is locally infinite at $R$. That is, there are infinitely many minimal genus Seifert surfaces for $L$ that can be made disjoint from $R$. Let $M={\mathbb{S}^3\setminus\Int\nhd(R)}$, and fix a set $\rho_1,\cdots,\rho_k$ of core curves of the annuli $\partial M\cap\partial\!\nhd(L)$, one for each link component.
Then Theorem \ref{wilsontheorem} applies. In addition, it is clear that none of the $R_i$ is a disc and that, since $R$ is connected, $M$ is irreducible.
By discarding surfaces if necessary, we may ensure that, for any $j\leq n$, the sets $\{R_1,\cdots,R_m\}$ and $\{S_1,\cdots,S_n\}\setminus\{S_j\}$ do not satisfy the conclusions of Theorem \ref{wilsontheorem}. We may also assume that $S_1$ has minimal genus among the $S_i$.
Let $R'$ be a minimal genus Seifert surface in minimal normal form such that $R'=R_1+b_1 S_1 + \cdots + b_n S_n$ with $b_1>0$, and set $T=S_1$.
Let $R^-=R_1+(b_1-1) S_1 +b_2 S_2+ \cdots + b_n S_n$, so that $R'=R^- + T$, and isotope $R^-$ and $T$ into reduced form. Since the isotopy keeps $\partial R'$ fixed and $T$ is closed, this will leave $\partial R^-$ unchanged.
Then, by \cite{MR744850} Lemma 2.1, no curve of $R^-\cap T$ bounds a disc in either $R^-$ or $T$. Note that although \cite{MR744850} Lemma 2.1 is proved only for closed surfaces, the same proof works in this case because $T$ is closed.
Suppose that $T$ is a 2--sphere. Then, after the isotopy, it must be disjoint from $R^-$. This contradicts that $R'$ is connected.
Since there are infinitely many minimal genus Seifert surfaces in minimal normal form in $M$, it follows that $T$ is a torus.
Let $M_0$ be the component of $M\setminus\Int\nhd(T)$ containing $\partial M$, and $M_1$ the other component.
The orientation that $R'$ inherits from $L$ induces an orientation on each component of $R'\cap M_0$ and hence on each curve of $R^-\cap T$. Let $\rho$ be a curve on $T$ that meets each curve of $R^-\cap T$ once. Because $T$ is disjoint from $R$, the algebraic intersection $\rho.R$ of $\rho$ and $R$ is 0. As $[R']=[R]$ in $\mathbb{S}^3\setminus\Int\nhd(L)$, this gives that $\rho.R'=0$, and so $\rho.(R^-\cap T)=0$ on $T$. Therefore half the curves of $R^-\cap T$ are oriented in one direction, and half are oriented in the other direction. In particular, $|R^-\cap T|$ is even. Find adjacent curves with opposite orientations, and surger $R^-$ along the subannulus of $T$ between them. Repeating this to remove all curves of $R^-\cap T$ gives a new Seifert surface $R''$ for $L$, together with a closed, possibly disconnected, surface $S''$. Note that $R''\subset M_0$ and $S''$ is orientable.
As $R'$ is minimal genus, $\chi(R')\geq\chi(R'')=\chi(R^-)-\chi(S'')=\chi(R')-\chi(T)-\chi(S'')$, so $\chi(S'')\geq 0$.
The components of $(R^-\cup T)\setminus(R^-\cap T)$ from which $S''$ is constructed each have boundary, and none of them is a disc. Therefore each of these components is an annulus, and in particular this includes every component of $R^-\cap M_1$.
Let $S$ be one such annulus in $M_1$, and suppose it is parallel to a subannulus $S_T$ of $T$. If there are other curves of $R^-\cap T$ in $S_T$, they must also bound annuli parallel to $T$. Hence we may assume $R^-\cap\Int(S_T)=\emptyset$. At each of the two boundary curves of $S_T$, the cut-and-paste operation that creates $R'$ from $R^-$ and $T$ might go one of two ways (see Figure \ref{satellitepic1}).
\begin{figure}
\caption{\label{satellitepic1}
\label{satellitepic1}
\end{figure}
If both join together $S$ and $S_T$ then this creates a torus component of $R'$, contradicting that $R'$ is connected. If both go the other way, we see that an isotopy of $R^-$ and $T$ could reduce $R^-\cap T$ without changing $R'$, contradicting the choice of $R^-$ and $T$. If only one joins the two annuli, an isotopy along the product region reduces the weight of $R'$, again giving a contradiction.
Thus $S$ is not $\partial$--parallel in $M_1$.
Note that the part of $\mathbb{S}^3\setminus \Int\nhd(T)$ containing $L$ is a solid torus $V$.
Let $K$ be the core curve of $V$.
Since $R\subset V$ and $T$ is incompressible, $L$ is a satellite of $K$ with winding number 0.
Because $S$ is not parallel to $T$, the knot $K$ satisfies the hypotheses of Proposition \ref{knotannulusprop}.
\end{proof}
\noindent
Mathematical Institute
\noindent
University of Oxford
\noindent
24--29 St Giles'
\noindent
Oxford OX1 3LB
\noindent
England
\noindent
\textit{jessica.banks[at]lmh.oxon.org}
\end{document} | math | 21,566 |
\begin{document}
\title{
On the Euler-Maruyama scheme for spectrally one-sided L\'evy driven SDEs with H\"older continuous coefficients\\
}
\vskip 40pt
\author{Libo Li\footnote{Corresponding Author}\\
School of Mathematics and Statistics,\\
University of New South Wales,\\
NSW 2006, Australia,\\
Email: [email protected]\\
\and
Dai Taguchi\\
Graduate School of Engineering Science,\\
Osaka University,\\
1-3, Machikaneyama-cho, Toyonaka,\\
Osaka, Japan, \\
Email: [email protected], \\
}
\date{}
\maketitle
\begin{abstract}
We study in this article the strong rate of convergence of the Euler-Maruyama scheme and associated with the jump-type equation introduced in Li and Mytnik \cite{LiMy}. We obtain the strong rate of convergence under similar assumptions for strong existence and pathwise uniqueness.
Models of this type can be considered as a generalization of the CIR (Cox-Ingersoll-Ross) process with jumps.
\\\\
\textbf{2010 Mathematics Subject Classification}:
60H35; 41A25; 60H10; 65C30
\\\\
\textbf{Keywords}:
Euler-Maruyama scheme $\cdot$
$\alpha$-CIR models $\cdot$
L\'evy driven SDEs $\cdot$
H\"older continuous coefficients $\cdot$
Spectrally positive L\'evy process
\mathbb{E}nd{abstract}
\section{Introduction}
In mathematical finance, a popular model for short term interest rates is the Cox-Ingersoll-Ross (CIR) model, which is the solution to the one-dimensional stochastic differential equation (SDE)
\begin{align}\label{SDE-cir}
X_t=
x_0
+\int_{0}^{t} a(c-X_s) ds
+\int_{0}^{t} \sqrt{X_{s}}dW_s\quad
~x_0 \in \mathbb{R},
~t \in [0,T],
\mathbb{E}nd{align}
where $a,c> 0$ and $W=(W_t)_{0\leq t \leq T}$ is a standard one-dimensional Brownian motion. There has been a push in the financial mathematics literature to generalize the CIR models to include jumps. The most noteable works in this direction are the affline jump-diffusion models proposed in Duffie et al. \cite{DFS, DPS}.
Motivated by the recent developments of continuous-state branching processes. It was shown in Fu and Li \cite{FL}, and later extended in Li and Mytnik \cite{LiMy} to more general jump type equations, that is if $b$, $\sigma$ and $h$ are H\"older continuous and $h$ is non-decreasing then existence and pathwise uniqueness of solution holds for SDEs of the form
\begin{align}\label{SDE_1}
X_t& = x_0
+\int_{0}^{t} b(X_{s-})ds
+\int_{0}^{t} \sigma(X_{s-})dW_s
+\int_{0}^{t} h(X_{s-})dL_s,\quad
~x_0 \in \mathbb{R},
~t \in [0,T].\\
L_t & =\int_{0}^{t} \int_{0}^{\infty} z \widetilde{N}(ds,dz).\label{ll2}
\mathbb{E}nd{align}
where the process $W=(W_t)_{0\leq t \leq T}$ is a standard one-dimensional Brownian motion and $\widetilde N$ is a compensated Poisson random measure with intensity or L\'evy measure $\mathbb{N}u$ satisfying the condition $\int^\infty_0 \{z^2\wedge z\}\, \mathbb{N}u(dz) < \infty$. In the recent paper of Jiao et al. \cite{JMS, JMSS}, in order to capture the persistency of low interest rate, self-exciting and large jump behaviours exhibited by sovereign interest rates and power markets, a version of the model considered in \cite{FL, LiMy} was introduced to the financial mathematics literature as the $\alpha$-CIR process.
In pracitice, the solution to equation (\ref{SDE_1}) is rarely analytically tractable, the goal of this article is to study under similar assumptions to those of \cite{LiMy}, the strong rate of convergence for Euler-Maruyama scheme associated with the SDE \mathbb{E}qref{SDE_1}. From the point of view of strong existence and pathwise uniqueness of a solution, the fact that the L\'evy measure $\mathbb{N}u$ is stable plays (as chosen in Jiao et al. \cite{JMS}) very little role (see Theorem 2.3 in \cite{LiMy}). One can consider any spectrally positive L\'evy process of the form given in \mathbb{E}qref{ll2} and produce a wide range of {\it generalized CIR processes} with different jump structures.
Given $n\in \mathbb{N}$ and a time grid $0=t_0<t_1\dots< t_n = T$, the Euler-Maruyama scheme associated with equation \mathbb{E}qref{SDE_1} is given by $X_0 := x_0$ and
\begin{align*}
X^{(n)} _{t_i} := x_0 +\int_{(0,t_i]} \sum_{j=0}^{n-1} b(X^{(n)} _{t_j}){\bf 1}_{(t_j,t_{j+1}]}(s) ds + \int_{(0,t_i]} \sum_{j=0}^{n-1} \sigma(X^{(n)} _{t_j}){\bf 1}_{(t_j,t_{j+1}]}(s) dW_s + \int_{(0,t_i]} \sum_{j=0}^{n-1} h(X^{(n)} _{t_j}){\bf 1}_{(t_j,t_{j+1} ]}(s) dL_s
\mathbb{E}nd{align*}
and one can extend the definition of the Euler-Maruyama scheme to continuous time by setting
\begin{align*}
X^{(n)}_{t} = x_0 + \int_{(0,t]} b(X^{(n)} _{\mathbb{E}ta_n(s)})ds+ \int_{(0,t]} \sigma(X^{(n)} _{\mathbb{E}ta_n(s)})dW_s + \int_{(0,t]} h(X^{(n)} _{\mathbb{E}ta_n(s)})dL_s
\mathbb{E}nd{align*}
where $\mathbb{E}ta_n(s):=t_j$ if $s\in (t_j,t_{j+1}]$. The process $(X^{(n)}_{\mathbb{E}ta_n(t)})_{0\leq t\leq T}$ is left continuous and for the purpose of this paper, we take equally spaced time grid of size $T/n$.
Using techniques from Yamada and Watanabe \cite{YaWa}, Gy\"ongy and R\'asonyi \cite{GyRa} proved that if the drift coefficient $b$ is the sum of a Lipschitz and a non-increasing $\rho$-H\"older continuous, the diffusion coefficient $\sigma$ is $\gamma$-H\"older continuous with $\gamma \in [1/2,1]$ and the jump coefficient $h=0$, then
\begin{align*}
\mathbb{E}[|X_T-X_{T}^{(n)}|]
\leq \left\{ \begin{array}{ll}
Cn^{-\frac{\rho}{2} \wedge (\gamma-\frac{1}{2})} &\text{ if } \gamma \in (1/2,1],\\
C(\log n)^{-1}, &\text{ if } \gamma =1/2.
\mathbb{E}nd{array}\right.
\mathbb{E}nd{align*}
In \cite{Y}, Yan proved similar results when $\gamma>1/2$ by using Tanaka's formula. These results are later extended, in for exmaple \cite{MeTa, NT}, to SDEs with irregular drift and diffusion coefficients. In the case where $h\mathbb{N}eq 0$, $L$ is a symmetric $\alpha$-stable process with $\alpha \in (1,2)$ and $b=\sigma=0$, Hashimoto and Tsuchiya \cite{HaTsu} shown using the method of Komatsu \cite{Komatsu}, if the coefficient jump $h$ is bounded $\gamma$-H\"older continuous with $\gamma \in [1/\alpha,1]$, then
\begin{align*}
\mathbb{E}[|X_T-X_{T}^{(n)}|^{\alpha-1}]
\leq \left\{ \begin{array}{ll}
Cn^{-(\gamma-\frac{1}{\alpha})} &\text{ if } \gamma \in (1/\alpha,1],\\
C(\log n)^{-(\alpha-1)}, &\text{ if } \gamma =1/\alpha.
\mathbb{E}nd{array}\right.
\mathbb{E}nd{align*}
We mention here also the works of Hashimoto \cite{Ha}, Mikulevi\v cius and Xu \cite{MiXu}, Qiao \cite{Qiao} for strong convergence and Mikulevi\v cius and Zhang \cite{MiZh} for weak convergence. However there is little in the current literature on the Euler-Maruyama scheme for jump-type equation with H\"older continuous coefficients and drift. To the best of our knowledge, there is no result on the strong rate of convergence for equation of the form \mathbb{E}qref{SDE_1}.
The structure of the current work is as follows. In section \ref{notation} we introduce the necessary notations and our standing assumptions. In section \ref{YW}, we introduce the Yamada-Watanabe approximation technique and give two auxiliary results in Lemma \ref{key_lem_0} and Lemma \ref{key_lem12}, which are used in controlling the jump part of the approximation. In section \ref{L1}, under boundedness assumption on the coefficients $\sigma$ and $h$, we obtain in Theorem \ref{main_1} the strong rate of convergence of the Euler-Maruyama scheme for driving L\'evy processes which are non-square integrable. In section \ref{L2}, we consider the case of square integrable L\'evy processes and obtain in Theorem \ref{l2t} the strong rate of convergence without any boundedness assumption on the coefficients.
\subsection{Notations and Assumptions}\label{notation}
We work on the usual filtered probability space $(\Omega, \mathcal{F}, \mathbb{P})$ endowed with a filtration $\mathbb{F}:=(\mathcal{F}_t)_{t\geq 0}$ which satisfies the usual conditions and $\mathcal{F}_\infty \subset \mathcal{F}$. We denote the sup-norm by $\| \cdot \|_\infty$ and set
\begin{align*}
\alpha_{\mathbb{N}u}:=\inf\{\widehat{\alpha}>1; \lim_{x \to 0+} x^{\widehat{\alpha}-1} \int_{x}^{\infty}z \mathbb{N}u(dz)=0\}.
\mathbb{E}nd{align*}
\begin{Ass}\label{Ass_1}
We assume that the L\'evy measure $\mathbb{N}u$, and the coefficients $b$, $\sigma$ and $h$ satisfies the following conditions:
\begin{itemize}
\item[(i)]
There exist $\mathbb{Z}eta \in [1/2,1]$ and $K_0>0$ such that
\begin{align*}
\sup_{t,s \in [0,T]}\mathbb{E}[|L_t-L_s|] \leq K_0|t-s|^{\mathbb{Z}eta}.
\mathbb{E}nd{align*}
Note that examples of $L$ include compensated $\alpha$-stable L\'evy process for $\alpha \in [1,2]$, compensated square integrable L\'evy processes and compensated compound Poisson process with integrable jump size.
\item[(ii)] The L\'evy measure $\mathbb{N}u$ is such that $\mathbb{N}u((-\infty,0))=0$ and $\displaystyle \int_{0}^{\infty} \{z \wedge z^2\} \mathbb{N}u(dz)<\infty$.
\item[(iii)] The drift coefficient $b$ is of the form $b=b_1+b_2$ where $b_1$ is a Lipschitz continuous function, and $b_2$ is a non-increasing $\rho$-H\"older continuous function with $\rho \in (0,1)$, that is,
\begin{align*}
K_1:=\sup_{x,y \in \mathbb{R},x\mathbb{N}eq y} \frac{|b_1(x)-b_1(y)|}{|x-y|}
+\sup_{x,y \in \mathbb{R},x\mathbb{N}eq y} \frac{|b_2(x)-b_2(y)|}{|x-y|^{\rho}}
<\infty.
\mathbb{E}nd{align*}
\item[(iv)] The diffusion coefficient $\sigma$ is an $\gamma$-H\"older continuous function with $\gamma \in [1/2,1)$ and the coefficient $h$ is an $\beta$-H\"older continuous function with $\beta \in (1-1/\alpha_{\mathbb{N}u},1)$, that is,
\begin{align*}
K_2:=\sup_{x,y \in \mathbb{R},x\mathbb{N}eq y} \frac{|\sigma(x)-\sigma(y)|}{|x-y|^{\gamma}}
+\sup_{x,y \in \mathbb{R},x\mathbb{N}eq y} \frac{|h(x)-h(y)|}{|x-y|^{\beta}}
<\infty.
\mathbb{E}nd{align*}
\item[(v)] The coefficient $h$ is a non-decreasing function.
\mathbb{E}nd{itemize}
\mathbb{E}nd{Ass}
By Assumption \ref{Ass_1} (iii) and (iv), there exists $K_3$ such that for any $x \in \mathbb{R}$, $|b(x)|+|\sigma(x)|+|h(x)|\leq K_3(1+|x|)$ and we denote $K:=\max\{K_0,K_1,K_2,K_3\}$.
\begin{Rem}\label{remark1.2} We list now some consequences of Assumption \ref{Ass_1}.
\begin{itemize}
\item[(i)]
From Lemma 2.1 of Li and Mytnik \cite{LiMy}, if $\int_{0}^{\infty} \{z \wedge z^2\} \mathbb{N}u(dz)<\infty$ then $\alpha_{\mathbb{N}u} \in [1,2]$ and for any $\widehat{\alpha}>\alpha_{\mathbb{N}u}$, $\lim_{x \to 0+} x^{\widehat{\alpha}-2} \int_{0}^{x} z^2 \mathbb{N}u(dz)=0$.
\item[(ii)]
From Theorem 25.3 and Theorem 25.18 of Sato \cite{SK}, we know that for any $p>0$, $\mathbb{E}[|L_t|^p]$ and $\mathbb{E}[\sup_{s\leq t} |L_s|^p]$ are finite for all $t \geq 0$ if and only if $\int_1^\infty z^p \mathbb{N}u(dz) < \infty$.
\mathbb{E}nd{itemize}
\mathbb{E}nd{Rem}
\subsection{Yamada and Watanabe Approximation Technique}\label{YW}
To deal with the H\"older continuity of the coefficients $\sigma$ and $h$, we introduce below the Yamada and Watanabe approximation technique (see for example \cite{GyRa,LiMy,YaWa}).
For each $\delta \in (1,\infty)$ and $\varepsilon \in (0,1)$, we select a continuous function $\mathbb{P}si _{\delta, \varepsilon}: \mathbb{R} \to \mathbb{R}^+$ with support of $\mathbb{P}si _{\delta, \varepsilon}$ belongs to $[\varepsilon/\delta, \varepsilon]$ and is such that
\begin{align*}
\int_{\varepsilon/\delta}^{\varepsilon} \mathbb{P}si _{\delta, \varepsilon}(z) dz
= 1 \quad \text{ and } \quad 0 \leq \mathbb{P}si _{\delta, \varepsilon}(z) \leq \frac{2}{z \log \delta}, \:\:\:z > 0.
\mathbb{E}nd{align*}
We define a function $\mathbb{P}hi_{\delta, \varepsilon} \in C^2(\mathbb{R};\mathbb{R})$ by setting
\begin{align*}
\mathbb{P}hi_{\delta, \varepsilon}(x)&:=\int_0^{|x|}\int_0^y \mathbb{P}si _{\delta, \varepsilon}(z)dzdy.
\mathbb{E}nd{align*}
It is straight forward to verify that $\mathbb{P}hi_{\delta, \varepsilon}$ has the following useful properties:
\begin{align}
&|x| \leq \varepsilon + \mathbb{P}hi_{\delta, \varepsilon}(x), \text{ for any $x \in \mathbb{R} $}, \label{phi3}\\
&0 \leq |\mathbb{P}hi'_{\delta, \varepsilon}(x)| \leq 1, \text{ for any $x \in \mathbb{R}$} \label{phi2}, \\
&\mathbb{P}hi'_{\delta, \varepsilon}(x) \geq 0, \text{ for } x\geq 0 \text{ and } \mathbb{P}hi'_{\delta, \varepsilon}(x) < 0, \text{ for } x< 0, \label{phi1}\\
&\mathbb{P}hi''_{\delta, \varepsilon}(\mathbb{P}m|x|)=\mathbb{P}si_{\delta, \varepsilon}(|x|)
\leq \frac{2}{|x|\log \delta}{\bf 1}_{[\varepsilon/\delta, \varepsilon]}(|x|)
\leq \frac{2\delta }{\varepsilon \log \delta},
\text{ for any $x \in \mathbb{R} \setminus\{0\}$}. \label{phi4}
\mathbb{E}nd{align}
We present below two auxiliary lemmas, which are used to control the jumps in the estimation of the strong error. Lemma \ref{key_lem_0} below is analogues to Lemma 3.2 given in \cite{LiMy}.
\begin{Lem}\label{key_lem_0}
Suppose that the L\'evy measure $\mathbb{N}u$ satisfies $\int_0^{\infty} \{z \wedge z^2\} \mathbb{N}u(dz) < \infty$.
Let $\varepsilon \in (0,1)$ and $\delta \in (1,\infty)$.
Then for any $x \in \mathbb{R}$, $y \in \mathbb{R} \setminus\{0\}$ with $xy \geq 0$ and $u>0$, it holds that
\begin{align*}
&\int_{0}^{\infty}
\{\mathbb{P}hi_{\delta,\varepsilon}(y+xz)-\mathbb{P}hi_{\delta,\varepsilon}(y)-xz\mathbb{P}hi_{\delta,\varepsilon}'(y)\} \mathbb{N}u(dz)\mathbb{N}otag
\leq 2 \cdot {\bf 1}_{(0,\varepsilon]}(|y|)
\left\{
\frac{|x|^2}{\log \delta} \left(\frac{1}{|y|} \wedge \frac{\delta}{\varepsilon}\right) \int_{0}^{u} z^2 \mathbb{N}u(dz)
+ |x| \int_{u}^{\infty} z \mathbb{N}u(dz)
\right\}.
\mathbb{E}nd{align*}
\mathbb{E}nd{Lem}
\begin{proof}
Let $x \in \mathbb{R}$, $y \in \mathbb{R} \setminus\{0\}$ with $xy \geq 0$ and $z >0$.
By the second order Taylor's expansion for $\mathbb{P}hi_{\delta,\varepsilon}$, it follows from \mathbb{E}qref{phi4} that
\begin{align*}
\mathbb{P}hi_{\delta,\varepsilon}(y+xz)-\mathbb{P}hi_{\delta,\varepsilon}(y)-xz\mathbb{P}hi_{\delta,\varepsilon}'(y)
&=|xz|^2 \int_{0}^{1} \theta \mathbb{P}hi_{\delta,\varepsilon}''(y+ \theta xz)d \theta
\leq \frac{2|xz|^2}{\log \delta} \int_{0}^{1} \frac{\theta {\bf 1}_{[\varepsilon/\delta,\varepsilon]}(|y+ \theta xz|) }{|y+ \theta xz|} d \theta.
\mathbb{E}nd{align*}
Since $xy \geq 0$, we have $|y| \leq |y+\theta xz|$ and ${\bf 1}_{[\varepsilon/\delta,\varepsilon]}(|y+ \theta xz|) \leq {\bf 1}_{(0,\varepsilon]}(|y|)$.
Hence we obtain
\begin{align}\label{key_lem_2}
\mathbb{P}hi_{\delta,\varepsilon}(y+xz)-\mathbb{P}hi_{\delta,\varepsilon}(y)-xz\mathbb{P}hi_{\delta,\varepsilon}'(y)
& \leq \frac{2 |xz|^2 {\bf 1}_{(0,\varepsilon]}(|y|)}{ \log \delta} \left(\frac{1}{|y|} \wedge \frac{\delta}{\varepsilon}\right).
\mathbb{E}nd{align}
Moreover, since $xy \geq 0$, by \mathbb{E}qref{phi1} we have $x \mathbb{P}hi_{\delta,\varepsilon}'(y) \geq 0$. This together with the fact that the right hand side of \mathbb{E}qref{key_lem_2} has ${\bf 1}_{(0,\varepsilon]}(|y|)$, we obtain
\begin{align}\label{key_lem_3}
\mathbb{P}hi_{\delta,\varepsilon}(y+xz)-\mathbb{P}hi_{\delta,\varepsilon}(y)-xz\mathbb{P}hi_{\delta,\varepsilon}'(y)
&\leq {\bf 1}_{(0,\varepsilon]}(|y|) \{\mathbb{P}hi_{\delta,\varepsilon}(y+xz)-\mathbb{P}hi_{\delta,\varepsilon}(y) \} \mathbb{N}otag\\
&={\bf 1}_{(0,\varepsilon]}(|y|) xz \int_{0}^{1}\mathbb{P}hi_{\delta,\varepsilon}'(y+\theta xz) d \theta
\leq {\bf 1}_{(0,\varepsilon]}(|y|) |xz|.
\mathbb{E}nd{align}
The result then follows from \mathbb{E}qref{key_lem_2} and \mathbb{E}qref{key_lem_3}.
\mathbb{E}nd{proof}
\begin{Lem}\label{key_lem12}
Suppose that the L\'evy measure $\mathbb{N}u$ satisfies $\int_0^{\infty} \{z \wedge z^2\} \mathbb{N}u(dz) < \infty$.
Let $\varepsilon \in (0,1)$ and $\delta \in (1,\infty)$.
Then for any $x,x' \in \mathbb{R}$, $y \in \mathbb{R}$ and $u \in (0,\infty]$, it holds that
\begin{align}\label{key_lem_122}
&\int_{0}^{\infty}\left|
\mathbb{P}hi_{\delta,\varepsilon}(y+xz)-\mathbb{P}hi_{\delta,\varepsilon}(y+x'z)-(x-x')z\mathbb{P}hi_{\delta,\varepsilon}'(y)
\right| \mathbb{N}u(dz) \mathbb{N}otag\\
&\leq 2 \left\{ \frac{\delta ( |x-x'|^2+|x'||x-x'|)}{\varepsilon\log \delta}\int_{0}^{u} z^2 \mathbb{N}u(dz)
+ |x-x'| \int_{u}^{\infty} z \mathbb{N}u(dz)
\right\}.
\mathbb{E}nd{align}
In particular, if $x'=0$, then
\begin{align}\label{key_lem_123}
\int_{0}^{\infty}
\{\mathbb{P}hi_{\delta,\varepsilon}(y+xz)-\mathbb{P}hi_{\delta,\varepsilon}(y)-xz\mathbb{P}hi_{\delta,\varepsilon}'(y)\} \mathbb{N}u(dz)
\leq 2 \left\{ \frac{\delta |x|^2}{\varepsilon\log \delta}\int_{0}^{u} z^2 \mathbb{N}u(dz)
+ |x| \int_{u}^{\infty} z \mathbb{N}u(dz)
\right\}.
\mathbb{E}nd{align}
\mathbb{E}nd{Lem}
\begin{proof}
For $z \in (0,u)$, from the second order Taylor's expansion for $\mathbb{P}hi_{\delta,\varepsilon}$ and mean value theorem applied to $\mathbb{P}hi_{\delta,\varepsilon}'$, we obtain from \mathbb{E}qref{phi4},
\begin{align*}
&\left| \mathbb{P}hi_{\delta,\varepsilon}(y+xz)-\mathbb{P}hi_{\delta,\varepsilon}(y+x'z)-(x-x')z\mathbb{P}hi_{\delta,\varepsilon}'(y) \right|\\
&\leq \left| \mathbb{P}hi_{\delta,\varepsilon}(y+xz)-\mathbb{P}hi_{\delta,\varepsilon}(y+x'z)-(x-x')z\mathbb{P}hi_{\delta,\varepsilon}'(y+x'z) \right|
+|x-x'||z|\left| \mathbb{P}hi_{\delta,\varepsilon}'(y)-\mathbb{P}hi_{\delta,\varepsilon}'(y+x'z) \right|\\
&\leq |x-x'|^2 |z|^2 \int_{0}^{1} \theta \mathbb{P}hi_{\delta,\varepsilon}''(y+ \theta xz+(1-\theta)x'z)d \theta
+|x'| |x-x'||z|^2 \int_{0}^{1} \mathbb{P}hi_{\delta,\varepsilon}''(y+ \theta x'z)d \theta\\
&\leq \left\{ |x-x'|^2 |z|^2+|x'| |x-x'||z|^2 \right\}\frac{2\delta }{\varepsilon \log \delta}.
\mathbb{E}nd{align*}
For the $z \in [u,\infty)$, apply mean value theorem to $\mathbb{P}hi_{\delta, \varepsilon}$,
\begin{align*}
& \left| \mathbb{P}hi_{\delta,\varepsilon}(y+xz)-\mathbb{P}hi_{\delta,\varepsilon}(y+x'z)-(x-x')z\mathbb{P}hi_{\delta,\varepsilon}'(y)
\right|\\
& \qquad = |x-x'| |z| \int_{0}^{1} \left|\mathbb{P}hi'_{\delta,\varepsilon}(y+\theta xz+(1-\theta)x'z)-\mathbb{P}hi'_{\delta,\varepsilon}(y) \right| d\theta
\leq 2 |x-x'| |z|.
\mathbb{E}nd{align*}
This concludes the proof of \mathbb{E}qref{key_lem_122}. In the case where $x'=0$, then since $\mathbb{P}hi''_{\delta,\varepsilon} \geq 0$, we have
\begin{align*}
\mathbb{P}hi_{\delta,\varepsilon}(y+xz)-\mathbb{P}hi_{\delta,\varepsilon}(y)-xz\mathbb{P}hi_{\delta,\varepsilon}'(y)
&=|xz|^2 \int_{0}^{1} \theta \mathbb{P}hi_{\delta,\varepsilon}''(y+ \theta xz)d \theta\
\geq 0,
\mathbb{E}nd{align*}
which concludes the proof of \mathbb{E}qref{key_lem_123}.
\mathbb{E}nd{proof}
\vskip 5pt
\begin{Rem}\label{key_lem13}
Suppose that the L\'evy measure $\mathbb{N}u$ satisfies the condition $\int_1^{\infty} z^2 \mathbb{N}u(dz) < \infty$ then one can take $u = \infty$ in Lemma \ref{key_lem12} and the right hand side of \mathbb{E}qref{key_lem_122} and \mathbb{E}qref{key_lem_123} are still finite.
\mathbb{E}nd{Rem}
\section{Strong Rate of Convergence}\label{L1}
\subsection{The Non-Square Integrable Case}\label{L1}
In this subsection, we compute the strong rate of convergence in the case where $L$ is a non-square integrable. The typical example one should keep in mind is when the L\'evy measure $\mathbb{N}u$ is spectrally positive $\alpha$-stable with $\alpha \in [1,2]$.
\begin{Lem}\label{EM_esti_1}
Suppose that Assumption \ref{Ass_1} holds and $h$ is bounded.
\begin{itemize}
\item[(i)]
There exists $C_1>0$ depend on $x_0$, $K$, $T$ and $\|h\|_{\infty}$ such that for any $t \in [0,T]$,
\begin{align}\label{EM_esti_2}
\mathbb{E}\big[\, \sup_{t\leq T}|{X}_{t}^{(n)}|\,\big]
\leq C_1.
\mathbb{E}nd{align}
\item[(ii)]
There exists $C_2>0$ depend on $x_0$, $C_1$, $K$, $T$ and $\|h\|_{\infty}$ such that for any $t \in [0,T]$,
\begin{align}
\mathbb{E}[|{X}_{t}^{(n)}-{X}_{\mathbb{E}ta_{n}(t)}^{(n)}|]
\leq C_2 \left( \frac{1}{n}\right)^{1/2}. \mathbb{N}otag
\mathbb{E}nd{align}
\mathbb{E}nd{itemize}
\mathbb{E}nd{Lem}
\begin{proof}
To prove $(i)$, we aim to apply Lemma 3.2 of Gy\"ongy and R\'asonyi \cite{GyRa}. To bound the stochastic integral against $L$, we note that by Theorem 7.30 of He et al. \cite{HWY}, there exists a localizing sequence of stopping times $(T_m)_{m\in \mathbb{N}}$ with $T_m\uparrow \infty$ such that
$\int_{0}^{t} h({X}_{\mathbb{E}ta_{n}(s)}^{(n)})dL^{T_m}_s \in \mathcal{H}^1$, where $\mathcal{H}^1$ is the martingale Hardy space. By applying the Burkholder-Davis-Gundy inequality, see for example Theorem 10.36 of He et al. \cite{HWY}, we obtain
\begin{align*}
\mathbb{E}\Big[\,\sup_{t\leq T} \, \Big|\int_{0}^{t} h({X}_{\mathbb{E}ta_{n}(s)}^{(n)})dL^{T_m}_s \Big| \,\Big]
& \leq c_1 \mathbb{E}\Big[\, \Big\{ \, \int_{0}^{T} |h({X}_{\mathbb{E}ta_{n}(s)}^{(n)})|^2d[L]^{T_m}_s \,\Big\}^{1/2}\, \Big]\\
& \leq c_1^2 \|h\|_{\infty}\mathbb{E}\Big[ \sup_{s\leq T\wedge T_m} |L_{s}|\,\Big],
\mathbb{E}nd{align*}
for some $c_1>0$. The right hand side above is bounded by $\lambda:= c_1^2 \|h\|_{\infty} \mathbb{E}\big[ \sup_{s\leq T} |L_{s}|\big]<\infty$ for all $m \in \mathbb{N}$. To take the limit as $m\rightarrow \infty$ in the above inequalities we note that
\begin{gather*}
\mathbb{E}\Big[\,\sup_{t\leq T} \, \Big|\int_{0}^{t} h({X}_{\mathbb{E}ta_{n}(s)}^{(n)})dL^{T_m}_s \Big| \,\Big] = \mathbb{E}\Big[\,\sup_{t\leq T\wedge {T_m}} \, \Big|\int_{0}^{t} h({X}_{\mathbb{E}ta_{n}(s)}^{(n)})dL_s \Big| \,\Big]
\mathbb{E}nd{gather*}
and monotone convergence theorem can be applied.
To estimate the time integral and the Brownian integral we proceed similarly to Remark 3.2 of \cite{GyRa}, however we have to pay extra attention as $X^{(n)}$ is not continuous. Using left continuity of ${X}_{-}^{(n)}$, there exists a localizing sequence $(T_m)_{m\in \mathbb{N}}$ such that $|{X}_{-}^{(n)}|$ when stopped at $T_m$ is bounded and the Brownian integral is a martingale.
By applying the Burkholder-Davis-Gundy inequality, linear growth condition on $\sigma$ and Jensen's inequality, we obtain
\begin{align*}
\mathbb{E}\Big[\,\sup_{t\leq T} \, \Big| \int_{0}^{t} \sigma({X}_{\mathbb{E}ta_{n}(s)}^{(n)}) dW_s^{T_m} \Big| \Big]
\leq
c_0 \Big\{
\mathbb{E} \Big[\int_{0}^{T\wedge T_m} \Big( 1+ \sup_{u\leq s}|{X}_{\mathbb{E}ta_{n}(u)}^{(n)}|^2\,\,\Big)ds\Big]
\Big\}^\frac{1}{2}.
\mathbb{E}nd{align*}
Using the linear growth condition on $b$ and the fact that for each $m\in \mathbb{N}$, there exists a constant $C_m$ such that $\sup_{u\leq s\wedge T_m} |{X}_{\mathbb{E}ta_n(u)}^{(n)}| \leq \sup_{u< s\wedge T_m} |{X}_{u}^{(n)}|\leq C_m$, we obtain
\begin{align}
\mathbb{E} \Big[\, \sup_{t\leq T}|X_{t\wedge T_m}^{(n)}|\,\Big]
& \leq |x_0| +\lambda + KT
+ K\mathbb{E}\Big[ \int_{0}^{T\wedge T_m}\,\sup_{u\leq s} |{X}_{\mathbb{E}ta_n(u)}^{(n)}|\,ds \, \Big]
+ c_0 \Big\{\mathbb{E}\Big[\int_{0}^{T\wedge T_m} \big( 1+ \sup_{u\leq s}|{X}_{\mathbb{E}ta_{n}(u)}^{(n)}|^2\,\,\big)ds\Big]\Big\}^\frac{1}{2}\mathbb{N}onumber \\
& \leq C_{T,x_0}
+ K\mathbb{E}\Big[\int_{0}^{T\wedge T_m} \,\sup_{u< s} |{X}_{u}^{(n)} |\,ds \, \Big]
+ c_0 \Big\{\mathbb{E}\Big[\int_{0}^{T\wedge T_m} \sup_{u< s}|{X}_{u}^{(n)}|^2\,ds\,\Big]\Big\}^\frac{1}{2}
< \infty\label{l2.1}
\mathbb{E}nd{align}
where $C_{T,x_0}:=|x_0|+\lambda+ KT + c_0 \sqrt{T}$. Using the fact that $X^{(n)}$ is a c\` adl\`ag process and we replace $\sup_{u< s}|{X}_{u}^{(n)}|$ by $\sup_{u\leq s}|{X}_{u}^{(n)}|$ in the Lebesgue integral, equation \mathbb{E}qref{l2.1} can be estimated by
\begin{align*}
\mathbb{E} \Big[\, \sup_{t\leq T}|X_{t\wedge T_m}^{(n)}|\,\Big]
& \leq C_{T,x_0} + K\mathbb{E}\Big[\int_{0}^{T} \,\sup_{u \leq s} |{X}_{u\wedge T_m}^{(n)} |\,ds \, \Big]
+ c_0\Big\{\mathbb{E}\Big[\int_{0}^{T} \sup_{u\leq s}|{X}_{u\wedge T_m}^{(n)}|^2\,ds\,\Big]\Big\}^\frac{1}{2}.
\mathbb{E}nd{align*}
Then it follows from Lemma 3.2 (i) of \cite{GyRa} with $p =1$, $q = 2$ and $V(t)=Z(t)=\sup_{u \leq t} |X_{u\wedge T_m}^{(n)}|$ that there exists $C_T$ such that
\begin{align*}
\mathbb{E} \Big[\, \sup_{t\leq T}|X_{t\wedge T_m}^{(n)}|\,\Big]
& \leq C_{T,x_0} C_T.
\mathbb{E}nd{align*}
Hence the result follows from an application of the monotone convergence theorem.
To prove $(ii)$, we note that the coefficients $b$, $\sigma$ satisfies the linear growth condition and $h$ is bounded, then
\begin{align*}
|{X}_{t}^{(n)}-{X}_{\mathbb{E}ta_{n}(t)}^{(n)}|
& \leq
K(1+|{X}_{\mathbb{E}ta_{n}(t)}^{(n)}|)
\left(|t-\mathbb{E}ta_{n}(t)|
+ |W_{t}-W_{\mathbb{E}ta_{n}(t)}|
\right)
+ \|h\|_{\infty}|L_{t}-L_{\mathbb{E}ta_{n}(t)}|.
\mathbb{E}nd{align*}
From \mathbb{E}qref{EM_esti_2} and Assumption \mathbb{E}qref{Ass_1}-(i), we have
\begin{align*}
\mathbb{E}[|{X}_{t}^{(n)}-{X}_{\mathbb{E}ta_{n}(t)}^{(n)}|]
&\leq M_1 \big(|t-\mathbb{E}ta_{n}(t)| + |t-\mathbb{E}ta_{n}(t)|^{1/2} + |t-\mathbb{E}ta_{n}(t)|^{\mathbb{Z}eta} \big),\\
& \leq 3T M_1 \left(\frac{1}{n}\right)^\frac{1}{2},
\mathbb{E}nd{align*}
where the constant $M_1$ is given by
\begin{align*}
M_1
:=\max\left\{K(1+C_1)(1 \vee \sqrt{2 \mathbb{P}i^{-1}}), \|h\|_{\infty}K_0 \right\}.
\mathbb{E}nd{align*}
This concludes the proof.
\mathbb{E}nd{proof}
From Theorem 2.2 in \cite{LiMy}, under Assumption \ref{Ass_1} (and the assumption that $\sigma$ and $h$ are bounded) there exists a unique strong solution to the SDE \mathbb{E}qref{SDE_1}. We now present our first result on the rate of convergence for the Euler-Maruyama scheme.
\begin{Thm}\label{main_1}
Suppose that Assumption \ref{Ass_1} holds and $\sigma$, $h$ are bounded. Then there exists $C_3>0$ depending on $x_0$, $K$, $T$, $\rho$, $\gamma$, $\beta$, $\|\sigma\|_{\infty}$ and $\|h\|_{\infty}$ such that for any $\varepsilon \in (0,\frac{1}{1-\beta}-\alpha_{\mathbb{N}u})$,
\begin{align*}
\sup_{0 \leq t \leq T}\mathbb{E}[|X_{t}-X_{t}^{(n)}|]
\leq C_3
\left\{ \begin{array}{ll}
\displaystyle
n^{-\rho/2}
+ n^{-\frac{\beta}{2} \left(1-\frac{1}{2\gamma}\right)}
&\gamma \in (1/2,1],\,\alpha_{\mathbb{N}u}<\frac{2(1-\gamma)}{1-\beta},\\
\displaystyle
n^{-\rho/2}
+ n^{-\frac{\beta}{2} \left(1-\frac{1}{2-(\alpha_{\mathbb{N}u}+\varepsilon)(1-\beta)}\right)}
&\gamma \in (1/2,1],\,\alpha_{\mathbb{N}u}\geq \frac{2(1-\gamma)}{1-\beta},\\
\displaystyle
(\log n)^{-1}
&\gamma=1/2.
\mathbb{E}nd{array}\right.
\mathbb{E}nd{align*}
Moreover, if $\mathbb{N}u(dz)$ is defined by
\begin{align}\label{stable_meas}
\mathbb{N}u(dz)=\frac{{\bf 1}_{(0,\infty)} (z) \mu(z)}{z^{1+\alpha}} dz,
\mathbb{E}nd{align}
for some $\alpha \in (1,2)$ and bounded measurable function $\mu$ then the above $\varepsilon$ can be chosen as zero and $\alpha_{\mathbb{N}u}=\alpha$.
\mathbb{E}nd{Thm}
\begin{Rem}
We set $\alpha_*:=\sup\{\widehat{\alpha}>1;\int_1^\infty z^{\widehat \alpha}\mathbb{N}u(dz) < \infty\}$.
We point out that if $\gamma \in [\frac{1}{2}, \frac{\alpha_*}{2}]$ then the boundedness assumption on $\sigma$ can be removed. The rate of convergence can be retrieve by performing similar computations as in Theorem \ref{l2t} and we leave this to reader.
\mathbb{E}nd{Rem}
\begin{proof}
Define ${Z}_t^{(n)}:=X_t-X_t^{(n)}$ and let $\varepsilon \in (0,1)$ and $\delta \in (1,\infty)$.
By using \mathbb{E}qref{phi3} and It\^o's formula,
\begin{align*}
&|{Z}_t^{(n)}|
\leq \varepsilon
+\mathbb{P}hi_{\delta,\varepsilon}({Z}_t^{(n)})
=\varepsilon
+{M}_t^{n,\delta,\varepsilon}
+{I}_t^{n,\delta,\varepsilon}
+{J}_t^{n,\delta,\varepsilon}
+{K}_t^{n,\delta,\varepsilon},
\mathbb{E}nd{align*}
where we set
\begin{align*}
{M}_t^{n,\delta,\varepsilon}
:=& \int_{0}^{t} \mathbb{P}hi_{\delta,\varepsilon}'(Z_{s}^{(n)})\{\sigma(X_{s})-\sigma(X_{\mathbb{E}ta_n(s)}^{(n)})\}dW_{s}\\
&+\int_{0}^{t} \int_{0}^{\infty}
\left\{
\mathbb{P}hi_{\delta,\varepsilon}(Z_{s-}^{(n)}+\{h(X_{s-})-h(X_{\mathbb{E}ta_n(s)}^{(n)})\}z)-\mathbb{P}hi_{\delta,\varepsilon}(Z_{s-}^{(n)})
\right\}
\widetilde{N}(ds,dz),\\
{I}_t^{n,\delta,\varepsilon}
:=&\int_{0}^{t} \mathbb{P}hi_{\delta,\varepsilon}'(Z_{s}^{(n)})\{b(X_{s})-b(X_{\mathbb{E}ta_n(s)}^{(n)})\}ds,
\quad\\
{J}_t^{n,\delta,\varepsilon}
:=& \frac{1}{2} \int_{0}^{t} \mathbb{P}hi_{\delta,\varepsilon}''(Z_{s}^{(n)}) |\sigma(X_{s})-\sigma(X_{\mathbb{E}ta_n(s)}^{(n)})|^2ds,\\
{K}_t^{n,\delta,\varepsilon}
:=&
\int_{0}^{t} \int_{0}^{\infty}
\Big\{
\mathbb{P}hi_{\delta,\varepsilon}(Z_{s-}^{(n)}+\{h(X_{s-})-h(X_{\mathbb{E}ta_n(s)}^{(n)})\}z)-\mathbb{P}hi_{\delta,\varepsilon}(Z_{s-}^{(n)})\\
&-\{h(X_{s-})-h(X_{\mathbb{E}ta_n(s)}^{(n)})\}z \mathbb{P}hi_{\delta,\varepsilon}'(Z_{s-}^{(n)})
\Big\}
\mathbb{N}u(dz)ds.
\mathbb{E}nd{align*}
By localization arguments, we can take ${M}_t^{n,\delta,\varepsilon}$ to be a martingale and can be removed after taking the expectation. Therefore we only estimate the terms ${I}_{t}^{n,\delta,\varepsilon}$, ${J}_{t}^{n,\delta,\varepsilon}$ and ${K}_{t}^{n,\delta,\varepsilon}$.
The coefficient $b_1$ is Lipschitz continuous and $b_2$ is non-increasing, we have for $x, y \in \mathbb{R}$ with $x \mathbb{N}eq y$,
\begin{align*}
\mathbb{P}hi'_{\delta,\varepsilon}(x-y)(b(x)-b(y))
=\frac{\mathbb{P}hi'_{\delta,\varepsilon}(x-y)}{x-y} (x-y)(b(x)-b(y))
\leq K_1\frac{|\mathbb{P}hi'_{\delta,\varepsilon}(x-y)|}{|x-y|} |x-y|^2
\leq K|x-y|,
\mathbb{E}nd{align*}
where in the first inequality, we used \mathbb{E}qref{phi1} and the fact that $(x-y)(b_2(x)-b_2(y)) \leq 0$ and in the last inequality, we used \mathbb{E}qref{phi2} and Lipschitz continuity of $b_1$.
From the above we have
\begin{align}\label{EP_6}
I_t^{n,\delta,\varepsilon}
&\leq \int_{0}^{t} \mathbb{P}hi_{\delta,\varepsilon}'(Z_{s}^{(n)}) (b(X_{s})-b(X_{s}^{(n)}))ds
+ \int_{0}^{t} \mathbb{P}hi_{\delta,\varepsilon}'(Z_{s}^{(n)}) (b(X_{s}^{(n)})-b(X_{\mathbb{E}ta_n(s)}^{(n)}))ds \mathbb{N}otag\\
&\leq K \int_{0}^{t} |Z_s^{(n)}| ds
+K\int_{0}^{t} |X_{s}^{(n)}-X_{\mathbb{E}ta_n(s)}^{(n)}|+|X_{s}^{(n)}-X_{\mathbb{E}ta_n(s)}^{(n)}|^{\rho}ds.
\mathbb{E}nd{align}
Using the fact that $\sigma$ is bounded and \mathbb{E}qref{phi4}, we have
\begin{align}
J_t^{n,\delta,\varepsilon}
&\leq \int_{0}^{t} \mathbb{P}hi_{\delta,\varepsilon}''(Z_{s}^{(n)}) |\sigma(X_{s})-\sigma(X_{s}^{(n)})|^2ds
+(2\|\sigma\|_{\infty})^{2-1/\gamma} \int_{0}^{t} \mathbb{P}hi_{\delta,\varepsilon}''(Z_{s}^{(n)}) |\sigma(X_{s}^{(n)})-\sigma(X_{\mathbb{E}ta_n(s)}^{(n)})|^{1/\gamma}ds \mathbb{N}otag\\
&\leq 2K^2\int_{0}^{t} \frac{{\bf 1}_{[\varepsilon/\delta,\varepsilon]}(|Z_s^{(n)}|) |Z_s^{(n)}|^{2\gamma}}{|Z_s^{(n)}|\log \delta}ds
+ 2 K^{1/\gamma} (2\|\sigma\|_{\infty})^{2-1/\gamma} \int_{0}^{t} \frac{{\bf 1}_{[\varepsilon/\delta,\varepsilon]}(|Z_s^{(n)}|) |X_{s}^{(n)}-X_{\mathbb{E}ta_n(s)}^{(n)}|}{|Z_s^{(n)}|\log \delta}ds \mathbb{N}otag\\
&\leq \frac{2TK^2\varepsilon^{2\gamma-1}}{\log \delta}
+ \frac{2K^{1/\gamma}(2\|\sigma\|_{\infty})^{2-1/\gamma} \delta}{\varepsilon \log \delta} \int_{0}^{t} |X_{s}^{(n)}-X_{\mathbb{E}ta_n(s)}^{(n)}|ds.\label{EP_7}
\mathbb{E}nd{align}
Finally, to estimate $K_t^{n,\delta,\varepsilon}$, we write it into two terms
\begin{align*}
K_t^{n,\delta,\varepsilon}
=K_t^{n,\delta,\varepsilon,1}+K_t^{n,\delta,\varepsilon,2},
\mathbb{E}nd{align*}
where $K_t^{n,\delta,\varepsilon,1}$ and $K_t^{n,\delta,\varepsilon,2}$ are given by
\begin{align*}
K_t^{n,\delta,\varepsilon,1}
&:=
\int_{0}^{t} \int_{0}^{\infty}
\Big\{
\mathbb{P}hi_{\delta,\varepsilon}(Z_{s}^{(n)}+\{h(X_{s})-h(X_{s}^{(n)})\}z)-\mathbb{P}hi_{\delta,\varepsilon}(Z_{s}^{(n)})
-\{h(X_{s})-h(X_{s}^{(n)})\}z \mathbb{P}hi_{\delta,\varepsilon}'(Z_{s}^{(n)})
\Big\}
\mathbb{N}u(dz)ds\\
K_t^{n,\delta,\varepsilon,2}
&:=
\int_{0}^{t} \int_{0}^{\infty}
\Big\{
\mathbb{P}hi_{\delta,\varepsilon}(Z_{s}^{(n)}+\{h(X_{s})-h(X_{\mathbb{E}ta_n(s)}^{(n)})\}z)-\mathbb{P}hi_{\delta,\varepsilon}(Z_{s}^{(n)}+\{h(X_{s})-h(X_{s}^{(n)})\}z)\\
& \quad -\{h(X_{s}^{(n)})-h(X_{\mathbb{E}ta_n(s)}^{(n)})\}z \mathbb{P}hi_{\delta,\varepsilon}'(Z_{s}^{(n)})
\Big\}
\mathbb{N}u(dz)ds.
\mathbb{E}nd{align*}
We observe that for each $s \in [0,t]$, if $Z_{s}^{(n)}=0$ then $h(X_{s})-h(X_{s}^{(n)})=0$. Therefore we can apply Lemma \ref{key_lem_0} with $y=Z_{s}^{(n)}$ and $x=h(X_{s})-h(X_{s}^{(n)})$ since $h$ is non-decreasing. That is for any $u>0$,
\begin{align}
&\int_{0}^{\infty}
\left\{
\mathbb{P}hi_{\delta,\varepsilon}(Z_{s}^{(n)}+\{h(X_{s})-h(X_{s}^{(n)})\}z)-\mathbb{P}hi_{\delta,\varepsilon}(Z_{s}^{(n)})
-\{h(X_{s})-h(X_{s}^{(n)})\}z \mathbb{P}hi_{\delta,\varepsilon}(Z_{s}^{(n)})
\right\}
\mathbb{N}u(dz) \mathbb{N}otag\\
&\leq
\frac{2|h(X_{s})-h(X_{s}^{(n)})|^2 {\bf 1}_{(0,\varepsilon]}(|Z_{s}^{(n)}|)}{|Z_{s}^{(n)}| \log \delta} \int_{0}^{u} z^2 \mathbb{N}u(dz)
+ 2|h(X_{s})-h(X_{s}^{(n)})| {\bf 1}_{(0,\varepsilon]}(|Z_{s}^{(n)}|) \int_{u}^{\infty} z \mathbb{N}u(dz)\mathbb{N}otag\\
& \leq \frac{2K^{2} |Z_{s}^{(n)}|^{2 \beta} {\bf 1}_{(0,\varepsilon]}(|Z_{s}^{(n)}|)}{|Z_{s}^{(n)}| \log \delta} \int_{0}^{u} z^2 \mathbb{N}u(dz)
+ 2K |Z_{s}^{(n)}|^{\beta} {\bf 1}_{(0,\varepsilon]}(|Z_{s}^{(n)}|) \int_{u}^{\infty} z \mathbb{N}u(dz) \mathbb{N}otag \\
& \leq \frac{2K^{2}}{\log \delta} \varepsilon^{2 \beta-1} \int_{0}^{u} z^2 \mathbb{N}u(dz)
+ 2K \varepsilon^{\beta} \int_{u}^{\infty} z \mathbb{N}u(dz),\label{EP_2}
\mathbb{E}nd{align}
where in the second last inequality, we used the fact that $h$ is a $\beta$-H\"older continuous function with $\beta \in (1-1/{\alpha_{\mathbb{N}u}},1)$.
We recall that $\alpha_{\mathbb{N}u}=\inf\{\widehat{\alpha}>1; \lim_{x \to 0+} x^{\widehat{\alpha}-1} \int_{x}^{\infty}z \mathbb{N}u(dz)=0\}$.
From Lemma 2.1 in \cite{LiMy}, we know that $\alpha_{\mathbb{N}u} \in [1,2]$ and for any $\widehat{\alpha}>\alpha_{\mathbb{N}u}$, $\lim_{x \to 0+} x^{\widehat{\alpha}-2} \int_{0}^{x} z^2 \mathbb{N}u(dz)=0$.
Also by the definition of $\alpha_{\mathbb{N}u}$, $\lim_{x \to 0+} x^{\widehat{\alpha}-1} \int_{x}^{\infty}z \mathbb{N}u(dz)=0$.
Let $u=\varepsilon^{q}$ for some $q >0$, which we will choose later.
Since $\beta \in (1-1/\alpha_{\mathbb{N}u},1)$, we can take $\widehat{\alpha}$ such that $\alpha_{\mathbb{N}u}<\widehat{\alpha}<\frac{1}{1-\beta}$.
Then for sufficiently small $\varepsilon$, equation \mathbb{E}qref{EP_2} can be further bounded as follows
\begin{align*}
&\frac{2K^{2}}{\log \delta} \varepsilon^{2 \beta-1} \int_{0}^{\varepsilon^{q}} z^2 \mathbb{N}u(dz)
+ 2K \varepsilon^{\beta} \int_{\varepsilon^{q}}^{\infty} z \mathbb{N}u(dz) \\
& = \frac{K^{2}}{\log \delta} \varepsilon^{2 \beta-1-q(\widehat{\alpha}-2)} \varepsilon^{q(\widehat{\alpha}-2)}\int_{0}^{\varepsilon^{q}} z^2 \mathbb{N}u(dz)
+ 2K \varepsilon^{\beta-q(\widehat{\alpha}-1)} \varepsilon^{q(\widehat{\alpha}-1)}\int_{\varepsilon^{q}}^{\infty} z \mathbb{N}u(dz) \\%\label{EP_3} \\
&\leq \frac{2K^{2}}{\log \delta} \varepsilon^{2 \beta-1-q(\widehat{\alpha}-2)}
+ 2K \varepsilon^{\beta-q(\widehat{\alpha}-1)}
=2\left(\frac{K^{2}}{\log \delta}+ K\right)\varepsilon^{1-\widehat{\alpha}(1-\beta)} \mathbb{N}otag,
\mathbb{E}nd{align*}
where in the last equality, we have chosen $q>0$ such $2 \beta-1-q(\widehat{\alpha}-2)=\beta-q(\widehat{\alpha}-1)$, that is, $q=1-\beta$. From the above computation we have
\begin{align}\label{EP_5}
K_t^{n,\delta,\varepsilon,1}
\leq 2T \left\{ \frac{K^{2}}{\log \delta}+ K\right\}\varepsilon^{1-\widehat{\alpha}(1-\beta)}.
\mathbb{E}nd{align}
By applying \mathbb{E}qref{key_lem_122} in Lemma \ref{key_lem12} with $u=1,
y=Z_{s}^{(n)},
x=h(X_{s})-h(X_{\mathbb{E}ta_n(s)}^{(n)})$, $x'=h(X_{s})-h(X_{s}^{(n)})$
and using the fact that $h$ is bounded, $K_t^{n,\delta,\varepsilon,2}$ can be bounded above by
\begin{align}\label{EP_8}
&K_t^{n,\delta,\varepsilon,2}
\leq |K_t^{n,\delta,\varepsilon,2}| \mathbb{N}otag\\
&\leq 2 \int_{0}^{1} z^2 \mathbb{N}u(dz) \int_{0}^{t}
\frac{\delta}{\varepsilon \log \delta}
\left(
|h(X_{s}^{(n)})-h(X_{\mathbb{E}ta_n(s)}^{(n)})|^2
+|h(X_{s})-h(X_{s}^{(n)})| |h(X_{s}^{(n)})-h(X_{\mathbb{E}ta_n(s)}^{(n)})|
\right)
\,ds \mathbb{N}otag\\
&\quad + 2 \int_{1}^{\infty} z \mathbb{N}u(dz) \int_{0}^{t} |h(X_{s}^{(n)})-h(X_{\mathbb{E}ta_n(s)}^{(n)})| ds \mathbb{N}onumber \\
&\leq 2
\left\{
\frac{4\|h\|_{\infty} \delta}{\varepsilon \log \delta}
\int_{0}^{1} z^2 \mathbb{N}u(dz) \,ds
+ \int_{1}^{\infty} z \mathbb{N}u(dz)
\right\}
\int_{0}^{t}
|h(X_{s}^{(n)})-h(X_{\mathbb{E}ta_n(s)}^{(n)})| ds \mathbb{N}onumber \\
&\leq 2K
\left\{
\left( 4\|h\|_{\infty} \int_{0}^{1} z^2 \mathbb{N}u(dz)\right) \vee \int_{1}^{\infty} z \mathbb{N}u(dz)
\right\}
\left(\frac{\delta}{\varepsilon \log \delta} + 1\right)
\int_{0}^{t} |X_{s}^{(n)}- X_{\mathbb{E}ta_n(s)}^{(n)}|^{\beta} ds.
\mathbb{E}nd{align}
\mathbb{N}oindent By taking the expectation in \mathbb{E}qref{EP_6}, \mathbb{E}qref{EP_7}, \mathbb{E}qref{EP_5} and \mathbb{E}qref{EP_8}, we obtain for any $t \in [0,T]$,
\begin{align*}
\mathbb{E}[|Z_t^{(n)}|]
& \leq \varepsilon
+\mathbb{E}[{I}_t^{n,\delta,\varepsilon}]
+\mathbb{E}[{J}_t^{n,\delta,\varepsilon}]
+\mathbb{E}[K_t^{n,\delta,\varepsilon}] \mathbb{N}otag\\
&\leq \varepsilon
+K \int_{0}^{t} \mathbb{E}[|Z_{s-}^{(n)}|] ds
+\frac{2TK^2\varepsilon^{2\gamma-1}}{\log \delta}
+2T\left\{ \frac{K^{2}}{\log \delta}+ K\right\}\varepsilon^{1-\widehat{\alpha}(1-\beta)} \mathbb{N}otag\\
&\quad +K\int_{0}^{t} \mathbb{E}[|X_{s-}^{(n)}-X_{\mathbb{E}ta_n(s)}^{(n)}|]+\mathbb{E}[|X_{s-}^{(n)}-X_{\mathbb{E}ta_n(s)}^{(n)})|^{\rho}]ds\\
&\quad +\frac{2K^{1/\gamma}(2\|\sigma\|_{\infty})^{2-1/\gamma} \delta}{\varepsilon \log \delta} \int_{0}^{t} \mathbb{E}[|X_{s-}^{(n)}-X_{\mathbb{E}ta_n(s)}^{(n)})|] ds \mathbb{N}otag \\
&\quad+2K
\left\{
\left( 4\|h\|_{\infty} \int_{0}^{1} z^2 \mathbb{N}u(dz)\right) \vee \int_{1}^{\infty} z \mathbb{N}u(dz)
\right\} \left( \frac{\delta}{\varepsilon \log \delta} + 1\right) \int_{0}^{t}
\mathbb{E}[|X_{s}^{(n)}-X_{\mathbb{E}ta_n(s)}^{(n)}|^{\beta}]ds.
\mathbb{E}nd{align*}
Using (ii) of Lemma \ref{EM_esti_1}, we have
\begin{align*}
\mathbb{E}[|Z_t^{(n)}|]
&\leq \varepsilon
+K \int_{0}^{t} \mathbb{E}[|Z_s^{(n)}|] ds
+\frac{2TK^2\varepsilon^{2\gamma-1}}{\log \delta}
+2T\left\{ \frac{K^{2}}{\log \delta}
+K\right\}\varepsilon^{1-\widehat{\alpha}(1-\beta)} \mathbb{N}otag\\
&\quad+KT \left\{ \frac{C_2}{n^{1/2}}+\frac{C_2^{\rho} }{n^{\rho/2}} \right\}
+2K^{1/\gamma}T (2\|\sigma\|_{\infty})^{2-1/\gamma} \frac{\delta}{\varepsilon \log \delta} \frac{C_2}{n^{1/2}} \mathbb{N}otag\\
&\quad+2KT
\left\{
\left( 4\|h\|_{\infty} \int_{0}^{1} z^2 \mathbb{N}u(dz)\right) \vee \int_{1}^{\infty} z \mathbb{N}u(dz)
\right\} \left( \frac{\delta}{\varepsilon \log \delta} + 1\right) \frac{C_2^{\beta}}{n^{\beta/2}}.
\mathbb{E}nd{align*}
By using Gronwall's inequality, we have
\begin{align*}
e^{-KT} \mathbb{E}[|Z_t^{(n)}|]
&\leq
\varepsilon
+\frac{2TK^2\varepsilon^{2\gamma-1}}{\log \delta}
+2T\left\{ \frac{K^{2}}{\log \delta}
+K\right\}\varepsilon^{1-\widehat{\alpha}(1-\beta)} \mathbb{N}otag\\
&\quad+KT \left\{ \frac{C_2}{n^{1/2}}+\frac{C_2^{\rho} }{n^{\rho/2}} \right\}
+2K^{1/\gamma}T (2\|\sigma\|_{\infty})^{2-1/\gamma} \frac{\delta}{\varepsilon \log \delta} \frac{C_2}{n^{1/2}} \mathbb{N}otag\\
&\quad+2KT
\left\{
\left( 4\|h\|_{\infty} \int_{0}^{1} z^2 \mathbb{N}u(dz)\right) \vee \int_{1}^{\infty} z \mathbb{N}u(dz)
\right\} \left( \frac{\delta}{\varepsilon \log \delta} + 1\right) \frac{C_2^{\beta}}{n^{\beta/2}}.
\mathbb{E}nd{align*}
To optimize the above bound, if $\gamma \in (1/2,1]$, then we choose $\delta =2$ and obtain
\begin{align*}
\mathbb{E}[|Z_t^{(n)}|]
\leq M_2
\left\{
\varepsilon
+\varepsilon^{2\gamma-1}
+\varepsilon^{1-\widehat{\alpha}(1-\beta)}
+\frac{1}{n^{\rho/2}}
+\frac{1}{\varepsilon n^{1/2}}
+\left(\frac{1}{\varepsilon}+1\right)\frac{1}{n^{\beta/2}}
\right\},
\mathbb{E}nd{align*}
where the constant $M_2$ given by
\begin{align*}
M_2
:=e^{KT} \max\Bigg\{
1,\frac{2TK^2}{\log 2},T\left\{ \frac{K^{2}}{\log 2}+ K\right\},
2KT\{C_2+C_2^{\rho}\},
\frac{4 K^{1/\gamma}T (2\|\sigma\|_{\infty})^{2-1/\gamma} }{\log 2},\\
2KT
\left\{
\left( 2\|h\|_{\infty} \int_{0}^{1} z^2 \mathbb{N}u(dz)\right) \vee \int_{1}^{\infty} z \mathbb{N}u(dz)
\right\} \frac{2C_2^{\beta}}{\log 2}
\Bigg\}.
\mathbb{E}nd{align*}
We let $\varepsilon=n^{-q}$, where the optimal $q>0$ is chosen later. There are two cases to consider. If $\alpha_{\mathbb{N}u}<\frac{2(1-\gamma)}{1-\beta}$, then we choose $\widehat{\alpha}=\frac{2(1-\gamma)}{1-\beta}$ and we have $2\gamma-1=1-\widehat{\alpha}(1-\beta)$.
Hence by choosing $q$ such $q(2\gamma-1)=\beta/2-q$, that is $q=\frac{\beta}{4\gamma}$, we have
\begin{align*}
\mathbb{E}[|Z_t^{(n)}|]
\leq
6M_2\left\{
n^{-\rho/2}
+ n^{-\frac{\beta}{2}\left(1-\frac{1}{2\gamma}\right)}
\right\}.
\mathbb{E}nd{align*}
If $\alpha_{\mathbb{N}u} \geq \frac{2(1-\gamma)}{1-\beta}$, then we choose $\widehat{\alpha}=\alpha_{\mathbb{N}u}+\varepsilon$ for any $\varepsilon \in (0,\frac{1}{1-\beta}-\alpha_{\mathbb{N}u})$ and then $2\gamma-1>1-(\alpha_{\mathbb{N}u}+\varepsilon)(1-\beta)$.
Hence by choosing $q$ such that $q(1-(\alpha_{\mathbb{N}u}+\varepsilon)(1-\beta))=\beta/2-q$, that is $q=\frac{\beta}{2}\frac{1}{2-(\alpha_{\mathbb{N}u}+\varepsilon)(1-\beta)}$, we have
\begin{align*}
\mathbb{E}[|Z_t^{(n)}|]
\leq
6M_2\left\{
n^{-\rho/2}
+ n^{-\frac{\beta}{2}\left(1-\frac{1}{2-(\alpha_{\mathbb{N}u}+\varepsilon)(1-\beta) }\right)}
\right\}.
\mathbb{E}nd{align*}
This concludes the proof for $\gamma \in (1/2,1]$.
If $\gamma = 1/2$, then we choose $\varepsilon=n^{-q}$ and $\delta =n^{p}$ with $p,q>0$ and $p+q<\beta/2$, we have
\begin{align*}
e^{-KT} \mathbb{E}[|Z_t^{(n)}|]
\leq &
\frac{1}{n^{q}}
+\frac{2TK^2}{p \log n}
+2T\left\{ \frac{K^{2}}{p \log n}
+K\right\}\frac{1}{n^{q-q\widehat{\alpha}(1-\beta)}} \mathbb{N}otag\\
&\quad+KT \left\{ \frac{C_2}{n^{1/2}}+\frac{C_2^{\rho} }{n^{\rho/2}} \right\}
+K^{2}T \frac{n^{p+q}}{p \log n} \frac{C_2}{n^{1/2}} \mathbb{N}otag\\
&\quad+2KT
\left\{
\left( 4\|h\|_{\infty} \int_{0}^{1} z^2 \mathbb{N}u(dz)\right) \vee \int_{1}^{\infty} z \mathbb{N}u(dz)
\right\} \left( \frac{n^{p+q}}{p \log n} + 1\right) \frac{C_2^{\beta}}{n^{\beta/2}}.
\mathbb{E}nd{align*}
Hence we can conclude that
\begin{align*}
\mathbb{E}[|Z_t^{(n)}|]
\leq \frac{M_3}{\log n},
\mathbb{E}nd{align*}
where the constant $M_3$ is given by
\begin{align*}
M_3=
e^{KT} \max\Bigg\{
1,
&\frac{2TK^2}{p},T\left\{\frac{K^2}{p}+K\right\},
2KT\left\{C_2+C_2^{\rho} \right\},
p^{-1}K^{2}C_2,\\
&2KT
\left\{
\left( 4\|h\|_{\infty} \int_{0}^{1} z^2 \mathbb{N}u(dz)\right) \vee \int_{1}^{\infty} z \mathbb{N}u(dz)
\right\} \left( p^{-1} + 1\right) C_2^{\beta}
\Bigg\}.
\mathbb{E}nd{align*}
This concludes the proof for $\gamma =1/2$.
We consider now the L\'evy measure $\mathbb{N}u(dz)$ defined by
\begin{align*}
\mathbb{N}u(dz)=\frac{{\bf 1}_{(0,\infty)} (z) \mu(z)}{z^{1+\alpha}} dz,
\mathbb{E}nd{align*}
for some $\alpha \in (1,2)$ and bounded measurable function $\mu$.
Then since
\begin{align*}
\int_{x}^{\infty} z \mathbb{N}u(dz)
\leq \|\mu\|_{\infty} \int_{x}^{\infty} z^{-\alpha} dz
= \frac{\|\mu\|_{\infty} x^{1-\alpha}}{\alpha-1},
\mathbb{E}nd{align*}
we have $\alpha_{\mathbb{N}u}=\alpha$.
To conclude the statement, it is suffices to estimate the upper bounded of $K^{n,\delta,\varepsilon,1}$.
From \mathbb{E}qref{EP_2}, with $u=\varepsilon^{q}$ and $q >0$, we have
\begin{align*}
K_t^{n,\delta,\varepsilon,1}
&\leq \frac{2K^{2}T}{\log \delta} \varepsilon^{2 \beta-1} \int_{0}^{\varepsilon^{q}} z^2 \mathbb{N}u(dz)
+ 2KT \varepsilon^{\beta} \int_{\varepsilon^{q}}^{\infty} z \mathbb{N}u(dz) \mathbb{N}otag\\
&\leq \frac{2K^{2} \|\mu\|_{\infty} }{\log \delta} \varepsilon^{2 \beta-1} \int_{0}^{\varepsilon^{q}} z^{1-\alpha} dz
+ 2K\|\mu\|_{\infty} \varepsilon^{\beta} \int_{\varepsilon^{q}}^{\infty} z^{-\alpha}dz \mathbb{N}otag \\
&= \frac{2K^{2} \|\mu\|_{\infty}}{(2-\alpha) \log \delta} \varepsilon^{2 \beta-1-q(\alpha-2)}
+ \frac{2K\|\mu\|_{\infty}}{\alpha-1} \varepsilon^{\beta-q(\alpha-1)}\mathbb{N}otag \\
&=\left(\frac{2K^{2}}{(2-\alpha)\log \delta}+ \frac{2K}{(\alpha-1)}\right) \|\mu\|_{\infty} \varepsilon^{1-\alpha(1-\beta)}
\mathbb{E}nd{align*}
where in the last equality, we have chosen $q=1-\beta$.
This upper bound concludes the proof.
\mathbb{E}nd{proof}
\subsection{The Square Integrable Case}\label{L2}
In this subsection we compute the strong rate of convergence in the case where $L$ is a square integrable.
In this case, the boundedness condition on the coefficients $\sigma$ and $h$ can be lifted.
Examples of square integrable L\'evy process which can be simulated include compensated Poisson process, spectrally positive tempered stable processes or spectrally positive truncated stable processes.
\begin{Lem}\label{lem_moment}
Suppose that Assumption \ref{Ass_1} holds and $\int_{1}^{\infty}z^2 \mathbb{N}u(dz)<\infty$.
\begin{itemize}
\item[(i)] Then there exists a constant $C_3>0$ such that
\begin{align}
\mathbb{E}\big[\,\sup_{t \leq T} |X^{(n)}_t|^2\,\big]
&\leq C_3, \label{lem_moment_eq1}
\mathbb{E}nd{align}
\item[(i)] Then there exists a constant $C_4>0$ such that and for any $t \in [0,T]$,
\begin{align}\label{EP_esti_3}
\mathbb{E}\big[|{X}_t^{(n)}-{X}^{(n)}_{\mathbb{E}ta_{n}(t)}|^2\big]
\leq \frac{C_4}{n}.
\mathbb{E}nd{align}
\mathbb{E}nd{itemize}
\begin{proof}
The proof is similar to Lemma \ref{EM_esti_1}. It is sufficient to apply It\^o's isometry and linear growth condition on the coefficients.
\mathbb{E}nd{proof}
\mathbb{E}nd{Lem}
\begin{Thm}\label{l2t}
Suppose that Assumption \ref{Ass_1} holds and $\int_{1}^{\infty}z^2 \mathbb{N}u(dz)<\infty$. Then there exists $C_{5}>0$ such that for any $\varepsilon \in (0,\frac{1}{1-\beta}-\alpha_{\mathbb{N}u})$,
\begin{align*}
\sup_{t \leq T}\mathbb{E}[|X_{t}-{X}_{t}^{(n)}|]
\leq C_{5}
\left\{ \begin{array}{ll}
\displaystyle
n^{-\rho/2}
+ n^{-\frac{\beta}{2} \left(1-\frac{1}{2\gamma}\right)}
&\gamma \in (1/2,1],\,\alpha_{\mathbb{N}u}<\frac{2(1-\gamma)}{1-\beta},\\
\displaystyle
n^{-\rho/2}
+ n^{-\frac{\beta}{2} \left(1-\frac{1}{2-(\alpha_{\mathbb{N}u}+\varepsilon)(1-\beta)}\right)}
&\gamma \in (1/2,1],\, \alpha_{\mathbb{N}u}\geq \frac{2(1-\gamma)}{1-\beta},\\
\displaystyle
(\log n)^{-1}
&\gamma=1/2.
\mathbb{E}nd{array}\right.
\mathbb{E}nd{align*}
\mathbb{E}nd{Thm}
\begin{proof}
The proof is similar to that of Theorem \ref{main_1}. We recall that ${Z}_t^{(n)}:=X_t-X_t^{(n)}$ and in the proof of Theorem \ref{main_1}, the boundedness of $\sigma$ and $h$ were only used in the estimation of ${J}_t^{n,\delta,\varepsilon}$ and ${K}_t^{n,\delta,\varepsilon,2}$. Therefore, we present here only the estimates of ${J}_t^{n,\delta,\varepsilon}$ and ${K}_t^{n,\delta,\varepsilon,2}$.
Using the fact that $\sigma$ is $\gamma$-H\"older continuous, we have
\begin{align}\label{EP_L2_0}
J_t^{n,\delta,\varepsilon}
&\leq \int_{0}^{t} \mathbb{P}hi_{\delta,\varepsilon}''(Z_{s}^{(n)}) |\sigma(X_{s})-\sigma(X_{s}^{(n)})|^2ds
+\int_{0}^{t} \mathbb{P}hi_{\delta,\varepsilon}''(Z_{s}^{(n)}) |\sigma(X_{s}^{(n)})-\sigma(X_{\mathbb{E}ta_n(s)}^{(n)})|^{2}ds \mathbb{N}otag\\
&\leq 2K^2\int_{0}^{t} \frac{{\bf 1}_{[\varepsilon/\delta,\varepsilon]}(|Z_s^{(n)}|) |Z_s^{(n)}|^{2\gamma}}{|Z_s^{(n)}|\log \delta}ds
+ 2K^2 \int_{0}^{t} \frac{{\bf 1}_{[\varepsilon/\delta,\varepsilon]}(|Z_s^{(n)}|) |X_{s}^{(n)}-X_{\mathbb{E}ta_n(s)}^{(n)})|^{2\gamma}}{|Z_s^{(n)}|\log \delta}ds \mathbb{N}otag\\
&\leq \frac{2TK^2\varepsilon^{2\gamma-1}}{\log \delta}
+ \frac{2K^2 \delta}{\varepsilon \log \delta} \int_{0}^{t} |X_{s}^{(n)}-X_{\mathbb{E}ta_n(s)}^{(n)})|^{2\gamma}ds.
\mathbb{E}nd{align}
Next, we estimate the $K_t^{n,\delta,\varepsilon,2}$ term. By applying \mathbb{E}qref{key_lem_122} in Lemma \ref{key_lem12} with
\begin{align*}
u=+\infty,~
y=Z_{s}^{(n)},~
x=h(X_{s})-h(X_{\mathbb{E}ta_n(s)}^{(n)})\quad \text{and}\quad
x'=h(X_{s})-h(X_{s}^{(n)}),
\mathbb{E}nd{align*}
the term $K_t^{n,\delta,\varepsilon,2}$ can be bounded above by (see Remark \ref{key_lem13}),
\begin{align*}
&{K}_t^{n,\delta,\varepsilon,2}
\leq |K_t^{n,\delta,\varepsilon,2}| \mathbb{N}otag\\
&\leq 2 \int_{0}^{t}
\frac{\delta}{\varepsilon \log \delta}
\left( |h(X_{s}^{(n)})-h(X_{\mathbb{E}ta_n(s)}^{(n)})|^2+|h( X_{s})-h(X_{s}^{(n)})| |h(X_{s}^{(n)})-h(X_{\mathbb{E}ta_n(s)}^{(n)})| \right)
\int_{0}^{\infty} z^2 \mathbb{N}u(dz) \,ds.
\mathbb{E}nd{align*}
Hence by taking the expectation of both hand sides and using the H\"older inequality, we have
\begin{align*}
\mathbb{E}[K_t^{n,\delta,\varepsilon,2}] \mathbb{N}otag
&\leq \frac{2 \delta}{\varepsilon \log \delta} \int_{0}^{\infty} z^2 \mathbb{N}u(dz)
\int_{0}^{t}
\mathbb{E}[|h(X_{s}^{(n)})-h( X_{\mathbb{E}ta_n(s)}^{(n)})|^2]ds \mathbb{N}otag\\
&\quad
+\frac{2 \delta}{\varepsilon \log \delta} \int_{0}^{\infty} z^2 \mathbb{N}u(dz)
\int_{0}^{t}\mathbb{E}[|h(X_{s})-h( X_{s}^{(n)})|^2]^{1/2} \mathbb{E}[|h(X_{s}^{(n)})-h(X_{\mathbb{E}ta_n(s)}^{(n)})|^2]^{1/2}
\,ds.
\mathbb{E}nd{align*}
Next, by using the fact that $h$ is of linear growth and $\beta$-H\"older continuous,
\begin{align}\label{EP_L2_2}
\mathbb{E}[K_t^{n,\delta,\varepsilon,2}] & \leq \frac{2 K^2 \delta}{\varepsilon \log \delta} \int_{0}^{\infty} z^2 \mathbb{N}u(dz)
\int_{0}^{t}
\mathbb{E}[|X_{s}^{(n)}- X_{\mathbb{E}ta_n(s)}^{(n)}|^{2\beta}]ds \mathbb{N}otag\\
&\quad
+\frac{2 \cdot 3^{1/2} K^{3/2} \delta }{\varepsilon \log \delta} \int_{0}^{\infty} z^2 \mathbb{N}u(dz)
\int_{0}^{t}\mathbb{E}[(4+|X_{s}|^2+|X_{s}^{(n)} |^2)]^{1/2} \mathbb{E}[|X_{s}^{(n)}-X_{\mathbb{E}ta_n(s)}^{(n)}|^{2\beta}]^{1/2}
\,ds \mathbb{N}otag\\
&\leq 2 K^2 T C_4^{\beta} \int_{0}^{\infty} z^2 \mathbb{N}u(dz)
\frac{\delta}{\varepsilon \log \delta}
\left(\frac{1}{n}\right)^{\beta}
\mathbb{N}otag\\
&\quad
+2 \cdot 3^{1/2} K^{3/2}T C_4^{\beta/2}\int_{0}^{\infty} z^2 \mathbb{N}u(dz)
\left\{ 4+\sup_{t \leq T} \mathbb{E}[|X_{s}|^2]+C_3\right\}^{1/2}
\frac{\delta}{\varepsilon \log \delta}
\left(\frac{1}{n}\right)^{\beta/2}.
\mathbb{E}nd{align}
Take the expectation in \mathbb{E}qref{EP_6}, \mathbb{E}qref{EP_L2_0}, \mathbb{E}qref{EP_5} and \mathbb{E}qref{EP_L2_2}, we obtain from \mathbb{E}qref{EP_esti_3} and the Gronwall's inequality, for any $t \in [0,T]$,
\begin{align*}
e^{-KT}\mathbb{E}[|Z_t^{(n)}|]
&\leq \varepsilon
+\frac{2TK^2\varepsilon^{2\gamma-1}}{\log \delta}
+2T\left\{ \frac{K^{2}}{\log \delta}+ K\right\}\varepsilon^{1-\widehat{\alpha}(1-\beta)}\\
& \quad +KT \left\{
\left( \frac{C_4}{n}\right)^{1/2}
+\left( \frac{C_4}{n}\right)^{\rho/2}
\right\}
\mathbb{N}otag \\
&\quad
+K^2T C_4^{\gamma} \frac{ \delta}{\varepsilon \log \delta}
\left(\frac{1}{n}\right)^{\gamma}
+2 K^2 T C_4^{\beta} \int_{0}^{\infty} z^2 \mathbb{N}u(dz)
\frac{\delta}{\varepsilon \log \delta}
\left(\frac{1}{n}\right)^{\beta}
\mathbb{N}otag\\
&\quad
+2 \cdot 3^{1/2} K^{3/2}T C_4^{\beta/2}\int_{0}^{\infty} z^2 \mathbb{N}u(dz)
\left\{ 4+\sup_{ t \leq T} \mathbb{E}[|X_{s}|^2]+C_3\right\}^{1/2}
\frac{\delta}{\varepsilon \log \delta}
\left(\frac{1}{n}\right)^{\beta/2}.
\mathbb{E}nd{align*}
To optimize the above bound, if $\gamma \in (1/2,1]$, then we choose $\delta =2$ and obtain
\begin{align*}
\mathbb{E}[|Z_t^{(n)}|]
\leq M_4
\Bigg\{
\varepsilon
+\varepsilon^{2\gamma-1}
+\varepsilon^{1-\widehat{\alpha}(1-\beta)}
+\frac{1}{n^{\rho/2}}
+\frac{1}{\varepsilon n^{\gamma}}
+\frac{1}{\varepsilon n^{\beta}}
+\left(\frac{1}{\varepsilon}+1\right)\frac{1}{n^{\beta/2}}
\Bigg\},
\mathbb{E}nd{align*}
where $M_4$ is some constant defined by
\begin{align*}
M_4
:=e^{KT} \max\Bigg\{
1,
\frac{2TK^2}{\log 2},
2T\left\{ \frac{K^{2}}{\log 2}+ K\right\},
KT\{C_4^{1/2}+C_4^{\rho/2}\},
\frac{2K^2T C_4^{\gamma}}{\log 2},
\frac{4K^2 T C_4^{\beta}}{\log 2} \int_{0}^{\infty} z^2 \mathbb{N}u(dz),
\\
\frac{4 \cdot 3^{1/2} K^{3/2}T C_4^{\beta/2}}{\log 2} \int_{0}^{\infty} z^2 \mathbb{N}u(dz)
\left\{ 4+\sup_{ t \leq T} \mathbb{E}[|X_{s}|^2]+C_3\right\}^{1/2}
\Bigg\}.
\mathbb{E}nd{align*}
We choose $\varepsilon=n^{-q}$ and then we choose the optimal $q>0$.
There are again two cases to consider, if $\alpha_{\mathbb{N}u}<\frac{2(1-\gamma)}{1-\beta}$, then we choose $\widehat{\alpha}=\frac{2(1-\gamma)}{1-\beta}$ and then $2\gamma-1=1-\widehat{\alpha}(1-\beta)$.
Hence by choosing $q$ as $q(2\gamma-1)=\beta/2-q$, that is $q=\frac{\beta}{4\gamma}$, we have
\begin{align*}
\mathbb{E}[|Z_t^{(n)}|]
\leq
7M_4\left\{
\left(\frac{1}{n}\right)^{\rho/2}
+ \left(\frac{1}{n}\right)^{\frac{\beta}{2}\left(1-\frac{1}{2\gamma}\right)}
\right\}.
\mathbb{E}nd{align*}
If $\alpha_{\mathbb{N}u} \geq \frac{2(1-\gamma)}{1-\beta}$, then we choose $\widehat{\alpha}=\alpha_{\mathbb{N}u}+\varepsilon$ for any $\varepsilon \in (0,\frac{1}{1-\beta}-\alpha_{\mathbb{N}u})$ and then $2\gamma-1>1-(\alpha_{\mathbb{N}u}+\varepsilon)(1-\beta)$.
Hence by choosing $q$ such that $q\{1-(\alpha_{\mathbb{N}u}+\varepsilon)(1-\beta)\}=\beta/2-q$, that is $q=\frac{\beta}{2}\frac{1}{2-(\alpha_{\mathbb{N}u}+\varepsilon)(1-\beta)}$, we have
\begin{align*}
\mathbb{E}[|Z_t^{(n)}|]
\leq
7M_4\left\{
\left(\frac{1}{n}\right)^{\rho/2}
+ \left(\frac{1}{n}\right)^{\frac{\beta}{2} \left( 1-\frac{1}{2-(\alpha_{\mathbb{N}u}+\varepsilon)(1-\beta)}\right) }
\right\}.
\mathbb{E}nd{align*}
This concludes the proof for $\gamma \in (1/2,1]$.
If $\gamma = 1/2$, then we choose $\varepsilon=n^{-q}$ and $\delta =n^{p}$ with $p,q>0$ and $p+q<\beta/2<1/2=\gamma$. Then
\begin{align*}
e^{-KT}\mathbb{E}[|Z_t^{(n)}|]
&\leq \frac{1}{n^{q}}
+\frac{2TK^2}{p \log n}
+2T\left\{ \frac{K^{2}}{p \log n}+ K\right\} \frac{1}{n^{q-q\widehat{\alpha}(1-\beta)}}
+KT \left\{
\left( \frac{C_4}{n}\right)^{1/2}
+\left( \frac{C_4}{n}\right)^{\rho/2}
\right\}
\mathbb{N}otag \\
&\quad
+K^2T C_4^{1/2} \frac{ n^{p+q}}{p \log n}
\left(\frac{1}{n}\right)^{1/2}
+2 K^2 T C_4^{\beta} \int_{0}^{\infty} z^2 \mathbb{N}u(dz)
\frac{n^{p+q}}{p\log n}
\left(\frac{1}{n}\right)^{\beta}
\mathbb{N}otag\\
&\quad
+2 \cdot 3^{1/2} K^{3/2}T \int_{0}^{\infty} z^2 \mathbb{N}u(dz)
\left\{ 4+\sup_{s \leq t} \mathbb{E}[|X_{s}|^2]+C_3\right\}^{1/2}
\frac{n^{p+q}}{p\log n}
\left(\frac{1}{n}\right)^{\beta/2}.
\mathbb{E}nd{align*}
Hence we can conclude that
\begin{align*}
\mathbb{E}[|Z_t^{(n)}|]
\leq \frac{M_5}{\log n},
\mathbb{E}nd{align*}
where the constant $M_5$ is given by
\begin{align*}
M_5=
e^{KT} \max\Bigg\{
1,
\frac{2TK^2}{p},T\left\{\frac{K^2}{p}+K\right\},
2KT\left\{C_4^{1/2}+C_4^{\rho/2} \right\},
\frac{K^2T C_4^{1/2}}{p},
\frac{2 K^2 T C_4^{\beta}}{p} \int_{0}^{\infty} z^2 \mathbb{N}u(dz),\\
\frac{2 \cdot 3^{1/2} K^{3/2}TC_4^{\beta/2}}{p} \int_{0}^{\infty} z^2 \mathbb{N}u(dz)
\left\{ 4+\sup_{s \leq t} \mathbb{E}[|X_{s}|^2]+C_3\right\}^{1/2}
\Bigg\}.
\mathbb{E}nd{align*}
This concludes the proof for $\gamma =1/2$.
\mathbb{E}nd{proof}
\begin{thebibliography}{99}
\bibitem{DA}
Applebaum, D.:
{\it L\'evy Process and Stochastic Calculus, second edition,}
Cambridge University Press, (2009).
\bibitem{DFS}
Duffie, D., Filipovi\'c, D. and Schachermayer, W.:
{\it Affine processes and applications in finance,}
Ann. Appl. Probab.
{\bf13},
984--1053
(2003).
\bibitem{DPS}
Duffie, D., Pan, J. and Singleton, K.:
{\it Transform analysis and asset pricing for affine jump-diffusions,}
Econometrica
{\bf 68},
1343--1376
(2000).
\bibitem{F}
Fournier, N.:
{\it On pathwise uniqueness for stochastic differential equations driven by stable L\'evy processes,}
Ann. Inst. H. Poincar\'e Probab. Statist.
{\bf 49}(1)
138--159
(2013).
\bibitem{FL}
Fu, Z. and Li, Z.:
{\it Stochastic equations of non-negative processes with jumps,}
Stoch. Process. Appl.
{\bf 120}(3)
306--330
(2010).
\bibitem{Ha}
Hashimoto, H.:
{\it Approximation and stability of solutions of SDEs driven by a symmetric a stable process with non-Lipschitz coefficients,}
{\it S\'eminaire de probabilit\'es} XLV.
\bibitem{HaTsu}
Hashimoto, H. and Tsuchiya, T.:
{\it On the convergent rates of Euler-Maruyama schemes for SDEs driven by rotation invariant $\alpha$-stable processes,}
RIMS Kokyuroku,
229--236
(2013), in Japanese.
\bibitem{JMS}
Jiao, Y., Ma, C. and Scotti, S.:
{\it Alpha-CIR model with branching processes in sovereign interest rate modelling,}
Finance Stoch.
{\bf 21}(3)
789--813
(2017).
\bibitem{JMSS}
Jiao, Y., Ma, C. and Scotti, S. and Sgarra, C.:
{\it A Branching Process Approach to Power Markets,}
to appear in Energy Econ. (2018).
\bibitem{Komatsu}
Komatsu, T.:
{\it On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations of jump type,}
Proc. Japan Acad.,
{\bf 58},
Ser. A
353--356
(1982).
\bibitem{GyRa}
Gy\"ongy, I. and R\'asonyi, M.:
{\it A note on Euler approximations for SDEs with H\"older continuous diffusion coefficients,}
Stochastic. Process. Appl.
{\bf 121}
2189--2200
(2011).
\bibitem{HWY}
He, S.W., Wang, J.G. and Yan. J.A.:
{\it Semimartingale theory and stochastic calculus},
Science Press. Boca Raton, FL: CRC Press Inc..
(1992).
\bibitem{LiMy}
Li, Z. and Mytnik, L.:
{\it Strong solutions for stochastic differential equations with jumps,}
Ann. Inst. H. Poincar\'e Probab. Statist.
{\bf 47},
1055--1067
(2011).
\bibitem{MeTa}
Menoukeu Pamen, O. and Taguchi, D.:
{\it Strong rate of convergence for the Euler-Maruyama approximation of SDEs with H\"older continuous drift coefficient,}
Stochastic. Process. Appl.
{\bf 127},
2542--2559
(2017).
\bibitem{MiZh}
Mikulevi\v cius, R. and Zhang, C.:
{\it On the rate of convergence of weak Euler approximation for nondegenerate SDEs driven by L\'evy processes,}
Stochastic. Process. Appl.
{\bf 121},
1720--1748
(2011).
\bibitem{MiXu}
Mikulevi\v cius, R. and Xu, F.:
{\it On the rate of convergence of strong Euler approximation for SDEs driven by L\'evy processes,}
Stochastics. {\bf 90}(4), 569--604 (2018)
\bibitem{NT}
Ngo, H-L. and Taguchi, D.:
{\it Strong rate of convergence for the Euler-Maruyama approximation of stochastic
differential equations with irregular coefficients,}
Math. Comp.
{\bf 85}(300)
1793--1819
(2016).
\bibitem{PE}
Priola, E.:
{\it Pathwise uniqueness for singular SDEs driven by stable processes,}
Osaka J. Math.
{\bf 49}(2)
421--447
(2012).
\bibitem{Qiao}
Qiao, H.:
{\it Euler-Maruyama approximation for SDEs with jumps and non-Lipschitz coefficients,}
Osaka J. Math.
{\bf 51},
47--66
(2014).
\bibitem{SK}
Sato, K.:
{\it L\'evy Processes and Infinitely Divisible Distributions,}
Cambridge University Press
(2011).
\bibitem{YaWa}
Yamada, T. and Watanabe, S.:
{\it On the uniqueness of solutions of stochastic differential equations,}
J. Math. Kyoto Univ.
{\bf 11},
155--167
(1971).
\bibitem{Y}
Yan, B. L.:
{\it The Euler scheme with irregular coefficients,}
Ann. Probab.
{\bf 30}(3)
1172--1194
(2002).
\mathbb{E}nd{thebibliography}
\mathbb{E}nd{document} | math | 58,495 |
\begin{document}
\title[Real Representatives of Equisingular Strata of Simple Quartics]{Real Representatives of Equisingular Strata of Simple Quartic Surfaces}
\author[\c{C}.~G\"{u}ne\c{s} ~Akta\c{s} ]{\c{C}\.{i}sem G\"{u}ne\c{s} Akta\c{s}}
\address{
Department of Mathematics, Bilkent University\\
06800 Ankara, Turkey} \email{[email protected]}
\thanks{The author was supported by the \textit{T\"{U}B\.{I}TAK} grant $116$F$211$}
\subjclass[2010]{Primary 14J28; Secondary 14J10, 14J17}
\keywords{Projective model, $K3$-surface, complex quartic, singular quartic}
\date{}
\dedicatory{}
\begin{abstract}
We develop an algorithm detecting real representatives in equisingular strata of projective models of $K3$-surfaces. We apply this algorithm to spatial quartics and find two new examples of real strata without real representatives.
As a by-product, we also give a new proof for the only previously known example of plane sextics.
\end{abstract}
\maketitle
\section{Introduction}\label{introduction}
Throughout the paper, all varieties are over the field $\mathbb{C}$ of complex numbers.
\subsection{Motivation and historical remarks}
It is a wide open problem what kind of singularities a projective surface or a curve of a given degree can have. In general, this problem seems hopeless. However, in the case of $K3$-surfaces, thanks to the global Torelli theorem~\cite{K3} and subjectivity of the period map~\cite{periodmap}, the equisingular deformation classification of surfaces with any given polarization becomes a mere computation. Various deformation classification problems for $K3$-surfaces have been intensively studied in the literature. Any model of $K3$-surfaces is necessarily \emph{simple}, \emph{i.e.}, has at worst simple singularities. The most popular projective models of $K3$-surfaces are sextic curves $C\subset\mathbb{P}^2$ (where a $K3$-surface appears as the double plane ramified along $C$) and quartic surfaces $X\subset\mathbb{P}^3$. A complete list of all possible combinations of simple singularities realized by sextics and quartics was found by Yang~\cite{Yang.sextics,Yang}. Using the arithmetical reduction~\cite{Alex1}, Shimada~\cite{Shimada.Maximizing} gave a complete description of the moduli spaces of \emph{maximizing} sextics. Later, based on the same approach, Degtyarev and Akyol~\cite{Alex2} completed the equisingular deformation classification of plane sextics. This approach was also used by G\"{u}ne\c{s} Akta\c{s}~\cite{Cisem1} to obtain a complete description of the equisingular strata of the so-called \emph{nonspecial} simple quartics (see section~\ref{simple.quartics} for the definition). In the meanwhile, Shimada~\cite{Shimada.connEllK3} has listed the connected components of the moduli space of the Jacobian elliptic $K3$-surfaces (which can be regarded as $K3$-surfaces with a $\mathbf{U}$-polarization).
Also worth mentioning is the vast literature on the deformation classification problems in the \emph{real} case, see,\emph{ e.g.}, the classification of real (nonsingular) quartics by Kharlamov~\cite{Kharlamov}, the study on moduli space of real $K3$-surfaces by Nikulin~\cite{Niku4} or the recent
work on quartic spectrahedra by Degtyarev and Itenberg~\cite{AI} and Ottem \emph{et al.}~\cite{Sturmfels}.
Typically, over $\mathbb{C}$, one deals with singular models, whereas, over $\mathbb{R}$, with very few exceptions (see, \emph{e.g.}~\cite{AI} and~\cite{Sturmfels}), one usually confines oneself to the smooth ones. Certainly, one could have considered singular real models up to equisingular equivariant deformations, but the results in lists would be huge. (More importantly, one would have to check, on a case by case basis, the validity of the lattice theoretical reduction, which does not hold automatically, see, \emph{e.g.}~\cite{Degtyarev:finiteness}.) In this paper, we make an attempt to bridge this gap from a slightly different perspective, namely, we discuss the existence of real representatives in the real equisingular strata of complex singular models.
\subsection{Principal results}
This paper originates from my paper~\cite{Cisem1}, where I started a systematic equisingular deformation classification of simple quartics. In this paper, based on a more general perspective, we study all models of $K3$-surfaces of a certain fixed \emph{kind} (see
\autoref{projective.models.of.K3surfaces}).
Denote by $\mathcal{M}$ the space of all models $f\colon X\rightarrow \mathbb{P}^n$; it is divided into equisingular strata $\mathcal{M}(S)$ according to sets $S$ of simple singularities. Each stratum $\mathcal{M}(S)$ splits further into its connected components, which are the equisingular deformation classes. We will mainly work with the subspace $\mathcal{M}_1\subset\mathcal{M}$ consisting of the nonspecial models and the respective strata $\mathcal{M}_1(S)=\mathcal{M}(S)\cap\mathcal{M}_1$; however, having further applications in mind, we will also discuss the general case whenever possible.
Fix a \emph{real structure} (\emph{i.e.}, an antiholomorphic involution) $\operatorname{conj}\colon \mathbb{P}^n\rightarrow\mathbb{P}^n$, then sending $f\colon X\rightarrow \mathbb{P}^n$ to $\operatorname{conj}\circ f\colon \bar{X}\rightarrow \mathbb{P}^n$ induces a real structure $\operatorname{c}: \mathcal{M}\rightarrow\mathcal{M}$. This real structure $\operatorname{c}$ depends on the choice of $\operatorname{conj}$; however the induced action on the connected components of equisingular strata is well defined. A connected component $\mathcal{D}\subset\mathcal{M}(S)$ is called \textit{real} if $\operatorname{c}(\mathcal{D})=\mathcal{D}$. Clearly, each stratum $\mathcal{M}(S)$ consists of real and pairs of complex conjugate components; this classification of components is given in~\cite{Alex2} for sextics and in~\cite{Cisem1} for (nonspecial) quartics.
Although it is quite common that a real variety may have no real points, very few examples of equsingular deformation classes with this property are known. Clearly, any class $\mathcal{D}\subset\mathcal{M}(S)$ containing a real model is real. However, the converse is not true, but the only known counterexample is the stratum $\mathcal{M}_1(\mathbf{A}_7\oplus\mathbf{A}_6\oplus\mathbf{A}_5)$ of the space of sextics found in \cite{Alex2}.
In the present paper, we study phenomena of this kind in the space of simple quartics; in particular, we find two more examples as above. Our principal result is the following theorem.
\begin{theorem}\label{principal.result}
Let $\mathcal{M}$ be the space of spatial quartics, and let $S_1=\mathbf{A}_7\oplus\mathbf{A}_6\oplus\mathbf{A}_3\oplus\mathbf{A}_2$ and $S_2=\mathbf{D}_7\oplus\mathbf{A}_6\oplus\mathbf{A}_3\oplus\mathbf{A}_2$. Any real component of any stratum $\mathcal{M}_1(S)$ other than $\mathcal{M}_1(S_1)$ or $\mathcal{M}_1(S_2)$ contains a real surface. The strata $\mathcal{M}_1(S_1)$ and $\mathcal{M}_1(S_2)$ consist of one real component each but they contain no real surfaces.
\end{theorem}
Theorem~\ref{principal.result} is proved in section~\ref{Applications}. As an important by-product of our approach, we give a simpler proof for the following example of Degtyarev and Akyol~\cite{Alex2}.
\begin{proposition}[Proposition 2.6 in~\cite{Alex2}]\label{byproduct}
The stratum $\mathcal{M}_1(\mathbf{A}_7\oplus\mathbf{A}_6\oplus\mathbf{A}_5)$ in the space $\mathcal{M}$ of plane sextics contains no real curves.
\end{proposition}
\subsection{Contents of the paper}
In $\S2$, we recall a few facts of Nikulin's theory of discriminant forms which is the principal technical tool of the paper. In $\S3$, we discuss projective models of $K3$-surfaces, introduce the abstract homological types, and recall the arithmetical reduction of the classification problem (see Theorem~\ref{def.class}). In $\S4$, we restate the existence of real models in arithmetical terms (see Theorem~\ref{real.model}) and suggest two approaches to find such models: via perturbations and via reflections. In particular, in Proposition~\ref{rankT=2}, we assert that the reflections suffice to detect all real representatives in the submaximal case $\operatorname{rk}\operatorname{NS}(X)=19$. Thus, in $\S5$, we develope a new algorithm listing all involutive skew-automorphism of an abstract homological type inducing a reflection on the transcendental lattice. In $\S6$, this algorithm is applied to two polarizations: spatial quartics (to prove Theorem~\ref{principal.result}) and to plane sextics (to prove Proposition~\ref{byproduct}).
\section{Integral lattices}
\subsection{Finite quadratic forms}
A finite quadractic form is a finite abelian group $\mathcal{L}$ equipped with a map $q\colon \mathcal{L}\rightarrow\mathbb{Q}/2\mathbb{Z}$ such that $q(x+y)=q(x)+q(y)-2b(x,y)$ for all $x,y\in\mathcal{L}$, where $b\colon \mathcal{L}\otimes\mathcal{L}\rightarrow\mathbb{Q}/\mathbb{Z}$ is a symmetric bilinear form (which is determined by $q$) and $2$ is the isomorphism $\times2\colon \mathbb{Q}/\mathbb{Z}\rightarrow\mathbb{Q}/2\mathbb{Z}$. We write $x^2$ and $x\cdot y$ for $q(x)$ and $b(x,y)$, respectively. A finite quadratic form is \emph{nondegenerate} if the homomorphism
\begin{align*}
\mathcal{L}\rightarrow \operatorname{Hom}(\mathcal{L},\mathbb{Q}/\mathbb{Z}),\quad x\mapsto(y\mapsto x\cdot y)
\end{align*}
is an isomorphism. We denote by $\mathcal{A}ut (\mathcal{L})$ the group of automorphisms of $\mathcal{L}$ preserving the form $q$. A subgroup $\mathcal{K}\subset \mathcal{L}$ is called \emph{isotropic} if the restriction of the quadratic form $q$ on $\mathcal{L}$ to $\mathcal{K}$ is identically zero. If this is case $\mathcal{K}^{\bot}/\mathcal{K}$ also inherits from $L$ a nondegenerate quadratic form.
Each finite quadratic form can be decomposed into the orthogonal direct sum $\mathcal{L}=\bigoplus_p\mathcal{L}_{[p]}$ of its $p$-primary components $\mathcal{L}_{[p]}:=\mathcal{L}\otimes \mathbb{Z}_p$, where the summation runs over all primes $p$. We denote by $\ell(\mathcal{L})$ the minimal number of generators of $\mathcal{L}$ and we put $\ell_p(\mathcal{L})=\ell(\mathcal{L}_{[p]})$. A finite quadratic form $\mathcal{L}$ is called \emph{even} if there is no element $x\in \mathcal{L}_{[2]}$ of order $2$ with $x^2=\pm\frac{1}{2}\bmod 2\mathbb{Z}$.
Given coprime integers $(m,n)$ such that $mn=0\bmod 2$, we denote by $[ \frac{m}{n}]$ the nondegenerate finite quadratic form on $\mathbb{Z}/n\mathbb{Z}$ sending the generator to $\frac{m}{n}\bmod 2\mathbb{Z} $. For a positive integer $k$, we use the notation $\mathcal{U}(2^k)$ and $\mathcal{V}(2^k)$ for the quadratic forms on $\mathbb{Z}/2^k\mathbb{Z} \times \mathbb{Z}/2^k\mathbb{Z}$, defined by the matrices
\begin{align*}
\mathcal{U}(2^k):=\frac{1}{2^k}\left[
\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}
\right],\quad \mathcal{ V}(2^k):=\frac{1}{2^k}\left[
\begin{array}{cc}
2& 1\\
1 & 2\\
\end{array}\right].
\end{align*}
(A finite quadratic form can be described by means of the Gram matrix $[\varepsilon_{ij}]$ such that $\varepsilon_{ij}=\beta_i\cdot\beta_j \bmod\mathbb{Z}$ and $\varepsilon_{ii}=\beta_i^2\bmod 2\mathbb{Z}$ where $\beta_k$'s are the basis vectors ). Nikulin \cite{Niku2} proved that, any finite nondegenerate quadratic form decomposes into an orthogonal direct sum of cyclic forms $[\frac{m}{n}]$ and length 2 forms $\mathcal{U}(2^k)$, $\mathcal{V}(2^k)$. We use the notation $\langle\alpha\rangle$ for the the cyclic subgroup generated by $\alpha$.
\begin{definition}\label{detp}
Let $\mathcal{L}$ be a nondegenerate quadratic form. Given a prime $p$, the determinant of the Gram matrix (in some basis) of $\mathcal{L}_{[p]}$ has the form $u/|{\mathcal{L}_{[p]}}|$ for some unit $u\in\mathbb{Z}_p^{\times}$, and this unit is independent of the basis modulo $(\mathbb{Z}_p^{\times})^2$ (if $p$ is odd or $\mathcal{L}_{[p]}$ is even) or modulo $(\mathbb{Z}_2^{\times})^2\times\{1,5\}$ (if $p=2$ and $\mathcal{L}_{[2]}$ is odd). We define $\det_p \mathcal{L}=u/|{\mathcal{L}_{[p]}}|$ where $u\in\mathbb{Z}_p^{\times}/(\mathbb{Z}_p^{\times})^2$ or ${u\in\mathbb{Z}_2^{\times}/(\mathbb{Z}_2^{\times})^2\times\{1,5\}}$ is as above (see, \cite{MM3}).
\end{definition}
\begin{remark}\label{NikuDef}
According to Nikulin~\cite{Niku2}, given a prime $p$ and a quadratic form $\mathcal{L}$ on a $p$ group, there is a unique $p$-adic lattice $L$ such that $\operatorname{rk} L= \ell_p(\mathcal{L})$ and $\operatorname{disc} L = \mathcal{L}_{[p]}$. One has $\det L= \det_p\mathcal{L}|{\mathcal{L}_{[p]}}|^2= u |{\mathcal{L}_{[p]}}|$ for some unit $u$ as in the Definition \ref{detp} (Nikulin uses this equality as a definition of $\det_p \mathcal{L}$).
\end{remark}
\begin{proposition}\label{KpMp}
Let $p$ be a prime and assume that $\mathcal{M}$ is a quadratic form on a $p$-group and $\mathcal{K}\subset\mathcal{M}$ an isotropic subgroup. If $\ell_p(\mathcal{K}^{\bot}/\mathcal{K})=\ell_p(\mathcal{M})$, then $\operatorname{det}_p(\mathcal{K}^{\bot}/\mathcal{K})=\det_p(\mathcal{M})\bmod (\mathbb{Q}_p^{\times})^{2}$.
\end{proposition}
\begin{remark}\label{detcongruence}
The equality in Proposition \ref{KpMp} holds in the groups where both determinants are well-defined, \emph{i.e.}, typically in $\mathbb{Q}_p^{\times}/(\mathbb{Q}_p^{\times})^2$; however if $p=2$ and at least one of the forms is odd then the equality holds in $\mathbb{Q}_2^{\times}/(\mathbb{Q}_2^{\times})^2\times\{1,5\}$ (\emph{cf}., Definition \ref{detp}). More precisely, $\operatorname{det}_p(\mathcal{K}^{\bot}/\mathcal{K})=|{\mathcal{K}}|^2\det_p(\mathcal{M})\bmod (\mathbb{Z}_p^{\times})^{2}$; in fact, we are comparing the ``essential parts", \emph{i.e.}, units $u$ as in Definition \ref{detp}.
\end{remark}
\begin{proof}
Let $M$ be the $p$-adic lattice as in Remark~\ref{NikuDef} and consider the finite index extension $M'\supset M$ given by the kernel $\mathcal{K}$ (see Proposition~\ref{L-K}). Since $\operatorname{rk}M=\operatorname{rk}M'$ and $\ell_p(\mathcal{M}')=\ell_p(\mathcal{M})$, the extension $M'$ can be used to compute $\det \mathcal{K}^{\bot}/\mathcal{K}$. Since $\operatorname{det}(M)=\operatorname{det}(M')|{\mathcal{K}}|^2$, we have the statement.
\end{proof}
The following proposition holds for $p=2$ only.
\begin{proposition}\label{K2M2}
Let $\mathcal{M}$ be a finite quadratic form on a $2$-group and $\mathcal{K}\subset\mathcal{M}$ a cyclic isotropic subgroup. Assume that $\ell_2(\mathcal{K}^{\bot}/\mathcal{K})<\ell_2(\mathcal{M})$, then $\ell_2(\mathcal{K}^{\bot}/\mathcal{K})=\ell_2(\mathcal{M})-2$ and $\operatorname{det}_2(\mathcal{K}^{\bot}/\mathcal{K})=-\det_2{\mathcal{M}} \bmod(\mathbb{Q}_2^{\times})^2$ (cf. Remark \ref{detcongruence}).
\end{proposition}
We preceed the proof of Proposition \ref{K2M2} with the following two Lemmas.
\begin{lemma}\label{sequences}
Let $\mathcal{M}$ be a finite quadratic form on a $2$-group and $\mathcal{C}\subset\mathcal{M}$ a cyclic subgroup. Then there exist sequences of integers
\begin{equation*}
0<m_1<m_2<\cdots<m_N=\log_2|{\mathcal{C}}|
\end{equation*}
and
\begin{equation*}
0\leq r_1<r_2<\cdots<r_N
\end{equation*}
such that $\mathcal{M}\cong\mathcal{ N}_0\oplus \bigoplus \mathcal{N}_s$, where $\mathcal{N}_s$ is a nondegenerate finite quadratic form generated by either one element $u_s$ or two elements $u_s, v_s$, all of order $2^{m_s+r_s}$, and whose Gram matrix is
\begin{align*}
\frac{1}{2^{m_s+r_s}}\left[
\begin{array}{c}
\mu_s\\
\end{array}
\right]
\mbox{ or }\frac{1}{2^{m_s+r_s}}\left[ \begin{array}{cc}
\mu_s & 1 \\
1 & \nu_s \\
\end{array}
\right]
\end{align*}
respectively, where $\mu_s$ is odd in the former case and even in the latter case.
Furthermore, the cyclic subgroup $\mathcal{C}$ is generated by $\kappa=\bigoplus\kappa_s$, where $\kappa_s=2^{r_s}u_s$.
\end{lemma}
\begin{proof}
Using the \emph{partial normal form} (see Lemma 4.2 in \cite{MM3}), we can decompose the quadratic form $\mathcal{M}$ into orthogonal sum $\mathcal{M}=\bigoplus \mathcal{M}_i$, where $\mathcal{M}_i$ is a homogenous group of exponent $2^i$. We construct the sequences $\{m_s\}$, $\{r_s\}$ and $\{\mathcal{N}_s\ni\kappa_s\}$ inductively, starting with $\bar{\kappa}_1:=\kappa$, where $\kappa$ is a generator of the cyclic group $\mathcal{C}$. At step $s$, let $\bar{\kappa}_s=\bigoplus_i\kappa_i'$, $\kappa_i'\in\mathcal{M}_i$, be the corresponding decomposition. Take
\begin{equation}\label{r1}
r_s=\operatorname{max}\{r: \bar{\kappa}_s=2^r\alpha,\mbox{ $\alpha\in\mathcal{M}$}\}
\end{equation}
and let
\begin{equation}\label{n.max}
n=\operatorname{max}\{i:\operatorname{ord}(\kappa'_i)=2^{i-r_s} \}\mbox{ and } m_s=n-r_s=\operatorname{log}_2\operatorname{ord}(\kappa_n').
\end{equation}
Choose $\kappa_s$ as
\begin{equation}\label{kappa1}
\kappa_s=\bigoplus_{i=\operatorname{ord}(\kappa'_i)\leq {2^{m_s}}}\kappa'_i
\end{equation}
We have $\kappa_s=2^{r_s}u_s$ for some $u_s\in \mathcal{M}$ with $\operatorname{ord}(u_s)=2^n$ and $u_s^2=\lambda/2^{n}$. If $\lambda$ is odd then we take for $\mathcal{N}_s$ the cyclic group generated by $u_1$ which is an orthogonal summand. If $\lambda$ is even, since $\mathcal{M}_n$ is non-degenerate, there exists $v_s \in\mathcal{M}_n$ such that $u_s\cdot v_s=\frac{1}{2^{n}}$ and we take for $\mathcal{N}_s$ the group generated by $u_s,v_s$.
Now, consider $\bar{\kappa}_{s+1}=\bar{\kappa}_s-\kappa_s$. If $\bar{\kappa}_{s+1}\neq0$, pass to the next step. Eventually, we obtain sequences $\{m_s\}$, $\{r_s\}$ and $\{\mathcal{N}_s\ni\kappa_s\}$ as in the statement and there remains to let $\mathcal{N}_0=(\bigoplus\mathcal{N}_s)^{\bot}$.
\end{proof}
\begin{lemma}\label{r1=0}
In the notation of Lemma~\ref{sequences}, if $\mathcal{C}$ is isotropic and $\ell_2(\mathcal{C}^{\bot}/\mathcal{C})<\ell_2(M)$, then $r_1=0$.
\end{lemma}
\begin{proof}
Clearly, if $r_s>0$ for some $s$, then all elements of order $2$ in $\mathcal{N}_s$ are in $\mathcal{C}^{\bot}$. Hence, if $r_1>0$, we have $\ell_2(\mathcal{C}^{\bot})=\ell_2(\mathcal{M})$ and $\ell_2(\mathcal{C}^{\bot}/\mathcal{C})\geq\ell_2(\mathcal{M})-1$. Then the assumption $\ell_2(\mathcal{C}^{\bot}/\mathcal{C})<\ell_2(\mathcal{M})$ implies that $\ell_2(\mathcal{C}^{\bot}/\mathcal{C})=\ell_2(\mathcal{M})-1$, which contradicts to the congruence $\ell_2(\mathcal{C}^{\bot}/\mathcal{C})=\ell_2(\mathcal{M})\bmod 2$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{K2M2}]
We apply Lemma~\ref{sequences} and Lemma~\ref{r1=0} to $\mathcal{C}=\mathcal{K}$. Consider the sequences $r_s, m_s$ and the decomposition $\mathcal{M}\cong\mathcal{ N}_0\oplus \bigoplus \mathcal{N}_s$ given by Lemma \ref{sequences}, we have $r_1=0$ by Lemma \ref{r1=0}. Our goal is to reduce this decomposition to its shortest form. Assuming $N>1$, define $\tilde{\mathcal{K}}:=m_1\mathcal{K}$ and consider $\mathcal{M}'=\tilde{\mathcal{K}}^{\bot}/\tilde{\mathcal{K}}$ and $\mathcal{K}'=\mathcal{K}/\tilde{\mathcal{K}}$. By Lemma \ref{r1=0}, we have $\ell_2(\mathcal{M}')=\ell_2(\mathcal{M})$. Hence by Proposition \ref{KpMp}, we get $\operatorname{det}_2(\mathcal{M}')=\det_2(\mathcal{M})$. (Strictly speaking, one should adjust the proof of Lemma \ref{sequences} to show that $\mathcal{M}'$ is even whenever $\mathcal{M}$ is and, hence we do not loose any information, \emph{cf.} Remark \ref{detcongruence}). On the other hand, $\mathcal{K}'^{\bot}/\mathcal{K}'=\mathcal{K}^{\bot}/\mathcal{K}$ and we can replace the pair $(\mathcal{M},\mathcal{K})$ by the pair $(\mathcal{M}', \mathcal{K}')$ and apply Lemma \ref{sequences} again. Note that $|{\mathcal{K}'}|=2^{m_1}<|{\mathcal{K}}|$, hence the process bound to converge and we end up with a single essential term decomposition, \emph{ i.e.}, $\mathcal{M}\cong\mathcal{N}_0\oplus\mathcal{N}_1$. Since still $r_1=0$ (by Lemma \ref{r1=0} again), the essential term $\mathcal{N}_1$ is the group generated by $u=\kappa$ and $v$, with the quadratic form given by the Gram matrix
\begin{equation*}
\frac{1}{2^m} \left[
\begin{array}{cc}
0 & 1 \\
1 & \tau \\
\end{array}
\right].
\end{equation*}
Then, clearly, $\mathcal{K}^{\bot}/\mathcal{K}=\mathcal{N}_0$, and it is obvious that $\ell_2(\mathcal{K}^{\bot}/\mathcal{K})=\ell_2(\mathcal{M})-2$ and $\operatorname{det}_2(\mathcal{K}^{\bot}/\mathcal{K})=-\det_2{\mathcal{M}}$.
\end{proof}
\subsection{Integral lattices and discriminant forms}
An \emph{(integral) lattice} is a finitely generated free abelian group $L$ equipped with a symmetric bilinear form $b\colon L\otimes L\rightarrow \mathbb{Z}$. Whenever the form is fixed, we use the abbreviation $x^2=b(x,x)$ and $x\cdot y:=b(x,y)$. A lattice $L$ is called \emph{even} if $x^2:=0\mod 2$ for all $x\in L$; it is called \emph{odd} otherwise. The \emph{determinant} $\det L \in \mathbb{Z}$ is the determinant of the Gram matrix of $b$ in any basis of $L$. Since the transition matrix between any two integral bases has determinant $\pm 1$, the determinant $\det L \in \mathbb{Z}$ is well-defined. A lattice $L$ is called \emph{unimodular} if $\det L=\pm 1$; it is called \emph{nondegenerate} if $\det L \neq 0$, or equivalently, the \emph{kernel}
\begin{align*}
\operatorname{ker}L=L^{\bot} :=\{x\in L \mid\text{ $x\cdot y= 0$ for all $y\in L$}\}
\end{align*}
is trivial.
Given a lattice $L$, the bilinear form on $L$ can be extended by linearity to a $\mathbb{Q}$-valued bilinear form on $L\otimes\mathbb{Q}$. The inertia indices $\sigma_{\pm}$ of $L$ are the classical inertia indices of $L\otimes\mathbb{Q}$ and the signature $\sigma L$ is the pair $\sigma L=(\sigma_+L,\sigma_-L)$
If $L$ is nondegenerate, then the dual group $L^{\vee}:=\operatorname {Hom}(L,\mathbb{Z})$ can be identified with the subgroup
\begin{align*}
\{x \in L\otimes\mathbb{Q} \mid \text{$x\cdot y \in \mathbb{Z}$ for all $y \in L$}\}
\end{align*}
There is an obvious canonical inclusion $L=L\otimes \mathbb{Z}\subset L^{\vee}$ and the finite quotient group $\operatorname{disc} L :=L^{\vee}/L$ is called the \emph{discriminant} group. The order of $\operatorname{disc} L$ is equal to $\mathopen|{\det L}\mathclose|$. In particular, $L$ is unimodular if and only if $\operatorname{disc} L=0$.
The discriminant group inherits from $L\otimes\mathbb{Q}$ a nondegenerate symmetric bilinear form
\begin{align*}
b\colon \operatorname{disc} L\otimes \operatorname{disc} L\rightarrow \mathbb{Q}/\mathbb{Z},\quad (x\bmod \mathbb{Z})\otimes(y\bmod \mathbb{Z})\mapsto(x\cdot y)\bmod \mathbb{Z}
\end{align*}
called the \emph{discriminant bilinear form}, and, if $L$ is even, its quadratic extension
\begin{align*}
q\colon \operatorname{disc} L\rightarrow \mathbb{Q}/2\mathbb{Z}, \quad (x \bmod L) \mapsto x^2 \bmod2\mathbb{Z},
\end{align*}
called the \emph{discriminant quadratic form}. Note that the discriminant group of an even lattice is a finite quadratic form. We use the notation $\operatorname{disc}_p L$ for the $p$-primary part of $\operatorname{disc} L$.
According to Nikulin \cite{Niku2}, two nondegenerate even lattices $L',L''$ are in the same \emph{genus} if and only if $\sigma L'=\sigma L''$ and $\operatorname{disc} L'\cong\operatorname{disc} L''$ (Here, we skip the original definition of a genus, instead, we use this criterion). We denote by $g(L)$ the set of all isomorphism classes of nondegenerate even lattices in the genus of $L$. This set is finite (see \cite{Milnor.Sym.bilinear}, the result is due to Milnor but the proof is given in \cite{Serre.Cours}).
The group of autoisometries of a nondegenerate lattice $L$ is denoted by $O(L)$. The action of $O(L)$ extends to $L\otimes\mathbb{Q}$ by linearity, restricts to the dual $L^{\vee}$ and factors to $\operatorname{disc} L$. Therefore, there is a natural homomorphism $O(L)\rightarrow \mathcal{A}ut(\operatorname{disc} L)$. If this does not lead to a confusion, we use the same notation for an autoisometry of $L$ and the induced autoisometry of $\operatorname{disc} L$.
The orthogonal projection of any maximal positive definite subspace in $L\otimes\mathbb{R}$ to any other such subspace is an isomorphism of vector spaces. Hence, all maximal positive definite subspaces in $L\otimes\mathbb{R}$ can be oriented in a coherent way.
A choice of such coherent orientations is called a \textit{positive sign structure} on $L$. We denote by $O^+(L)$ (as opposed to $SO(L)$) the subgroup of $O(L)$ consisting of the isometries preserving a positive sign structure. Either one has $O^{+}(L)=O(L)$ or $O(L)^+$ is a subgroup of $O(L)$ of index $2$. In the latter case, each element of $O(L)\smallsetminus O^+(L)$ is called a \emph{skew-autoisometry} of L,\emph{ i.e.}, skew-autoisometries of $L$ are the autoisometries of $L$ that reverse the positive sign structure.
An important examples of autoisometries are reflections. For a vector $a\in L$, the reflection
\begin{align}
t_a: x\mapsto x- \frac{2a(x\cdot a)}{a^2} \label{ta}
\end{align}
is well defined if and only if
\begin{align}
\frac{2a}{a^2}\in L^{\vee}\label{wdfta}.
\end{align}
Note that $t_a$ is an involutive isometry of $L$. If $a^2=\pm1$ or $a^2=\pm2$, then $t_a$ acts identically on $\operatorname{disc} L$ and extends to any overlattice of $L$. Moreover, $a^2>0$ if and only if $t_a$ reverses the positive sign structure.
The \emph{hyperbolic plane} is the lattice $\mathbf{U}:=\mathbb{Z}u\oplus\mathbb{Z}v$, with $u^2=v^2=0$ and $u\cdot v$=1.
\subsection{Root Lattices}\label{root.lattices}
A \emph{root} in an even lattice is a vector of square $(-2)$. A \emph{root lattice} is a negative definite lattice generated by its roots. Each root lattice admits a unique decomposition into an orthogonal direct sum of irreducible ones which are of type $\textbf{A}_n$, $n\geq1$, $\textbf{D}_n$, $n\geq4$, or $\textbf{E}_n$, $n=6,7,8$. For further details on irreducible root systems see \cite{Bour}.
Given a root lattice $S$, we have $O(S)=R(S)\rtimes\operatorname{Sym}(\Gamma)$, where $R(S)\subset O(S)$ is the group generated by reflections against roots and $\operatorname{Sym}(\Gamma)$ is the group of symmetries of the Dynkin graph $\Gamma_S:=\Gamma$. Let $\operatorname{Sym}'(\Gamma)$ be the group of symmetries of $\textbf{E}_8$-type components. Then the kernel of the map $d\colon O(S)\rightarrow\mathcal{A}ut (\operatorname{disc} S)$ is $R(S)\rtimes\operatorname{Sym}'(\Gamma)$ and, hence,
$d$ admits a partial section,\emph{ i.e.}, an isomorphism
\begin{align}\label{Sym0}
\operatorname{Im}d \cong\operatorname{Sym}_0(\Gamma)\subset\operatorname{Sym}(\Gamma)\subset O(S)
\end{align}
where $\operatorname{Sym}_0(\Gamma)$ is the group of symmetries acting identically on the union of $\textbf{E}_8$-type components.
\subsection{Lattice extensions}\label{lattice.extensions}
From now on, unless specified otherwise, all lattices considered are nondegenerate and even.
An \emph{extension} of a lattice $S$ is an over lattice $L\supset S$. An \emph{isomorphism} between two extensions $L',L''$ is a bijective isometry $L'\rightarrow L''$ identical on $S$. More generally, for a given subgroup $G\subset O(S)$, we define $G$-\emph{isomorphisms} of extensions of $S$ as those which restrict to an element of $G$ on $S$.
Given a \emph{finite index extension} $L\supset S$ (\emph{i.e.}, $S$ is a finite index subgroup of $L$), there is a unique embedding $L\subset S\otimes \mathbb{Q}$. Then we have a chain of inclusions
\begin{align*}
S\subset L\subset L^{\vee}\subset S^{\vee}.
\end{align*}
The subgroup $\mathcal{K}:=L/S\subset S^{\vee}/S=\operatorname{disc} S$ is called the \emph{kernel} of the finite index extension $L\supset S$. Since $L$ is an even integral lattice, the restriction to $\mathcal{K}$ of the quadratic form $q$ on $\operatorname{disc} S$ is trivial, \emph{i.e.}, $\mathcal{K}$ is \emph{isotropic}. Conversely, given an isotropic subgroup $\mathcal{K}\subset \operatorname{disc} S$, the lattice $L:=\{x\in S\otimes \mathbb{Q} \mid x\bmod S\in \mathcal{K}\}$ is an extension of $S$. Hence, we have the following result.
\begin{proposition}[Nikulin \cite{Niku2}]\label{L-K}
Let $S$ be a nondegenerate even lattice, and fix a subgroup $G\subset
O(S)$. The map $L\mapsto \mathcal{K}=L/S \subset \operatorname{disc} S$
establishes a one-to-one correspondence between the set of
$G$-isomorphism classes of finite index extensions $L\supset S$
and the set of $G$-orbits of isotropic subgroups
$\mathcal{K}\subset \operatorname{disc} S$. Under this correspondence one
has $\operatorname{disc} L=\mathcal{K}^{\bot}/\mathcal{K}$. Furthermore an autoisometry of $S$ extends to a finite index extension $L\supset S$ if and only if it preserves $\mathcal{K}$.
\end{proposition}
An extension $L\supset S$ is called \emph{primitive} if $L/S$ is torsion free. Clearly, $L$ is a finite index extension of $S\oplus N$, where $N:=S^{\bot}$ is also primitive in $L$, and by Proposition \ref{L-K}, it is described by its kernel
\begin{equation*}
\mathcal{K}\subset \operatorname{disc} (S\oplus N)=\operatorname{disc} S\oplus \operatorname{disc} N.
\end{equation*}
Since $S$ and $N$ are both primitive in $L$, the kernel $\mathcal{K}$ does not intersect with any of $\operatorname{disc} S$ and $\operatorname{disc} N$. It follows that the projection maps
\begin{equation*}
\operatorname{proj}_{S}\colon \mathcal{K} \rightarrow \operatorname{disc} S\mbox{ and } \operatorname{proj}_{N}\colon \mathcal{K} \rightarrow \operatorname{disc} N
\end{equation*}
are both monomorphisms. Since $\mathcal{K}$ is isotropic, it is the graph of a bijective anti-isometry $\psi\colon \mathcal{S}'\rightarrow\mathcal{N}'$, where $\mathcal{S}'= \operatorname{proj}_{S}(\mathcal{K})$ and $\mathcal{N}'= \operatorname{proj}_{N}(\mathcal{K})$. Conversely, given a bijective anti-isometry $\psi\colon\mathcal{S}'\rightarrow\mathcal{N}'$ where $\mathcal{S}'\subset\operatorname{disc} S$ and $\mathcal{N}'\subset \operatorname{disc} N$, the graph of $\psi$ is an isotropic subgroup $\mathcal{K}\subset\operatorname{disc} S\oplus \operatorname{disc} N$ and the corresponding finite index extension $L\supset S\oplus N$ is a primitive extension whose kernel is $\mathcal{K}$. Thus, we have the following statement (\emph{cf.} Nikulin~\cite{Niku2}).
\begin{lemma}\label{genlemma}
Given two nondegenerate even lattices $S$, $N$ and a subgroup $G\subset
O(S)\times O(N)$, there is a one-to-one correspondence between the set of $G$-isomorphism classes of finite index extensions $L\supset S\oplus N$ in which both $S$ and $N$ are primitive and that of $G$-conjugacy classes of bijective anti-isometries
\begin{equation}\label{isodisc}
\psi\colon \mathcal{S}'\rightarrow \mathcal{N}'
\end{equation}
where $\mathcal{S}'\subset \operatorname{disc} S$ and $\mathcal{N}'\subset \operatorname{disc} N$. Furthermore, a pair of isometries $f_1\in O(S)$ and $f_2\in O(N)$ extends to $L$ if and only if $f_1|_{\mathcal{S}'}=\psi^{-1}f_2|_{\mathcal{N}'}\psi$ in $\mathcal{A}ut (\mathcal{S}')$.
\end{lemma}
If $L$ above is unimodular, $\operatorname{disc} L=0$, we have
$\mathopen|\operatorname{disc} S\mathclose|\mathopen|\operatorname{disc} N\mathclose|=\mathopen|\mathcal{S}'\mathclose|\mathopen|\mathcal{N}'\mathclose|$. Hence, $\mathcal{S}'=\operatorname{disc} S$ and $\mathcal{N}'=\operatorname{disc} N$ and $\psi$ in \eqref{isodisc} is an anti-isomorphism $\operatorname{disc} S\rightarrow\operatorname{disc} N$. Since also $\sigma_{\pm}N=\sigma_{\pm}L-\sigma_{\pm}S$, it follows that the genus $g(N)$ is determined by the genera $g(S)$ and $g(L)$; we will denote this common genus by $g(S^{\bot}_L)$ (We emphasize that $g(S^{\bot}_L)$ merely encodes a ``local data" composed formally from $g(S)$ and $g(L)$; apriori, it may even be empty, \emph{cf.} Theorem \ref{th.N.existence} below). If $L$ is also indefinite, it is unique in its genus (see, \emph{e.g.}, Siegel~\cite{SiegelI,SiegelII,SiegelIII}). Then, given a subgroup $G\subset O(S)$ and unimodular even indefinite lattice $L$, a $G$-isomorphism class of a primitive extension $L\supset S$ is determined by a choice of
\begin{enumerate}
\item an even lattice $N\in g(S^{\bot}_L)$, and
\item a bi-coset in $G\backslash \mathcal{A}ut (\operatorname{disc} N)/O(N)$.
\end{enumerate}
In particular the extension $L\supset S$ exists if and only if the genus $g(S^{\bot}_L)$ is nonempty.
From now on we fix the notation $\textbf{L}:= 2\textbf{E}_8\oplus3\textbf{U}$. Note that $2\textbf{E}_8\oplus3\textbf{U}$ is the unique even unimodular lattice of signature $(3,19)$. We are concerned about this lattice, since it is the intersection index form of a $K3$-surface. More precisely, we are interested in the embeddings to this lattice $\mathbf{L}$. For the ease of the references, we recast a special case of Nikulin's existence theorem as a criterion for $g(S^{\bot}_{\textbf{L}})\neq\emptyset$.
\begin{theorem}[Nikulin~\cite{Niku2}]\label{th.N.existence}
Given a nondegenerate even lattice~$S$, a primitive extension $\textbf{L}\supset S$ exists
if and only if
the following conditions hold
\begin{enumerate}
\item $\sigma_+ S\leq 3$, $\sigma_-S\leq 19$ and $\ell(\mathcal{S})\leq 22- \operatorname{rk} S$, where $\mathcal{S}=\operatorname{disc} S$;
\item one has $|{\mathcal{S}}|\det_p (\mathcal{S}) =(-1)^{\sigma_+S-1} \bmod (\mathbb{Z}_p^{\times})^2$ for each odd prime $p$ such that $\ell_p(\mathcal{S})= 22- \operatorname{rk} S$;
\item If $\ell_2(\mathcal{S})= 22- \operatorname{rk} S$, and $\mathcal{S}_2$ is even then $|{\mathcal{S}}|\det_2 (\mathcal{S}) =\pm 1 \bmod (\mathbb{Z}_2^{\times})^2$.
\end{enumerate}
\end{theorem}
\section{Projective models of K3-Surfaces}\label{projective.models.of.K3surfaces}
In this section, we consider the \emph{projective models} of a smooth $K3$-surface $X$, \emph{i.e.}, the morphisms $f_h\colon X\rightarrow \mathbb{P}^{n+1}$ defined by a complete linear system $|h|$ without fixed components and such that $h\in NS(X)\subset H_2(X;\mathbb{Z})$ and $h^2=2n>0$. Given a projective model $f_h\colon X\rightarrow \mathbb{P}^{n+1}$, the class $h$ is called the \emph{polarization}. Note that $\operatorname{dim}f_h(X)=2$. It can be found in \cite{Donat} that only the following two cases can happen:
\begin{enumerate}
\item either $f_h$ is birational mapping of $X$ onto a surface of degree $2n$,
\item or $f_h$ is two-to-one mapping of $X$ onto a surface of degree $n$.
\end{enumerate}
A projective model $f_h$ as in $(1)$ is called \emph{birational} and it is is called \emph{hyperelliptic} if it is as in $(2)$. By Saint-Donat~\cite{Donat}, we have the following result about hyperelliptic models of K3-surfaces:
\begin{proposition}
The projective model $f_h\colon X\rightarrow \mathbb{P}^{n+1}$ with $h^2=2n$ is hyperelliptic if and only if
\begin{itemize}
\item [(i)] either $n=1$ or,
\item [(ii)] $n=4$ and $h=2h'$ for some vector $h'\in NS(X)$ ($h$ is imprimitive) or,
\item [(iii)] there is a vector $e\in NS(X)$ such that $e^2=0$ and $e\cdot h=2$ .
\end{itemize}
\end{proposition}
Note that the intersection lattice $L_X:=H_2(X;\mathbb{Z})$ is of the form
\begin{equation*}
L_X=H_2(X;\mathbb{Z})\cong \mathbf{L}=2\textbf{E}_8\oplus3\textbf{U}.
\end{equation*}
Given a projective model $f_h\colon X\rightarrow \mathbb{P}^{n+1}$, we introduce the following notations:
\begin{itemize}
\item $S_X\subset L_X$: the sublattice generated by the curves contracted by $f_h$;
\item $S_{X,h}:=S_X\oplus \mathbb{Z}h_X\subset L_X$ where $h_X=h\in NS(X)$ is the class of the pull-back of a generic plane section of $X$;
\item $\tilde{S}_X \subset \tilde{S}_{X,h}\subset L_X$: the primitive hulls of $S_X$ and $S_{X,h}$, respectively, \textit{i.e}, $\tilde{S}_X:=(S_X\otimes\mathbb{Q})\cap L_X$ and $\tilde{S}_{X,h}:=(S_{X,h}\otimes\mathbb{Q})\cap L_X$;
\item $\omega_X \subset L_X\otimes \mathbb{R}$: the oriented $2$-subspace spanned by the real and imaginary parts of the class of a holomorphic $2$-form on $X$ (the \emph{period} of $X$).
\end{itemize}
Recall that all singularities are \emph{simple}
and, hence, $S_X$ is a \emph{root} lattice, \emph{i.e.}, a negative definite lattice generated by vectors of square $(-2)$ (\emph{roots}).
The triple $(S_X,h_X,L_X)$ is called the \emph{homological type} of the projective model $f_h\colon X\rightarrow \mathbb{P}^{n+1}$. This triple has certain properties depending on the ``kind" of the model. Here, fixing the ``kind of the model" includes
\begin{itemize}
\item [(i)] fixing the degree,
\item [(ii)]deciding whether the model is birational or hyperelliptic and
\item [(iii)]sometimes, assuming some additional geometric properties detectable homologically, \emph{e.g.}, presence or absence of certain classes, \emph{cf}. Definition~\ref{badvectors}
\end{itemize}
To capture those properties of a homological type of certain kinds of models, we have the following definition:
\begin{definition}\label{badvectors}
Let $L$ be a lattice isomorphic to $\mathbf{L}$. Depending on the geometric problem, we define ``bad vectors'' of a polarized sublattice $S_h\subset L$ containing a distinguished vector $h$ with $h^2=2n$, as the vectors $e\in \tilde{S}_h:=(S_h\otimes\mathbb{Q})\cap L$ satisfying one of the following properties:
\begin{enumerate}
\item $e^2=0$ and $e\cdot h= 1$ (fixed components);
\item $e^2=0$ and $e\cdot h= 2$ (linear generatricies);
\item $e^2=0$ and $e\cdot h= 3$ (cubic equations);
\item $n=4$ and $2e=h$ (Veronese polarization).
\end{enumerate}
\end{definition}
\begin{remark}
Note that the existence of a vector as in Definition \ref{badvectors}(2) implies the existence of a vector as in (1) in the definition. Hence, usually only the vectors as in (2) are mentioned whereas the vectors as in (3) are excluded if $n=4$ and we want to consider thriquadrics rather than all octic surfaces.
\end{remark}
The definition below is intended to capture the necessary arithmetical properties of the homological types of models of K3-surfaces. Therefore it depends on the kinds of models considered which is assumed to be fixed in advance.
\begin{definition}\label{abstract.homological.type}
Let $S$ be a root lattice and $n$ be an integer such that $n\geq1$. An \emph{abstract homological type (extending $S$)} \emph{associated} to a given kind of model is an extension of $S_h:=S\oplus\mathbb{Z} h$, $h^2=2n$, to a lattice $L$ isomorphic to $\mathbf{L}$ satisfying the following conditions:
\begin{enumerate}
\item each vector $e\in(S\otimes\mathbb{Q})\cap L$ with $e^2=-2$ and $e\cdot h=0$ is in $S$;
\item depending on the kind of the models considered, the primitive hull $\tilde{S}_h:=(S_h\otimes\mathbb{Q})\cap L$ should not contain the ``\emph{bad vectors}" (specified on the case by case basis, see below). Most notably,
\begin{itemize}
\item vectors as in Definition \ref{badvectors}(1) are always excluded,
\item vectors as in Definition \ref{badvectors}(2) and (4) are excluded if and only if the model is birational.
\end{itemize}
\end{enumerate}
\end{definition}
Commonly used birational projective models/polarizations are as follows (where we mention also precise type of \emph{bad vectors}, by referring to the names introduced in the Definition \ref{badvectors}, that are to be excluded in Definition \ref{abstract.homological.type}).
\begin{enumerate}
\item $h^2=4$: The image of $f_h\colon X\rightarrow \mathbb{P}^3$ is a quartic (spatial model); the excluded bad vectors are fixed components as in (1) and linear generatrices as in (2).
\item $h^2=6$: The image of $f_h\colon X\rightarrow \mathbb{P}^4$ is a sextic given by a complete intersection of a quadric and cubic (sextic model); the excluded bad vectors are fixed components as in (1) and linear generatrices as in (2).
\item $h^2=8$: The image of $f_h\colon X\rightarrow \mathbb{P}^5$ is an octic (octic model); in the most general case the excluded bad vectors are fixed components as in (1), linear generatrices as in (2) and Veronese polarizations as in (4). We can distinguish triquadric vs. all octics; in the former case the bad vectors as in (3) are also to be excluded.
\end{enumerate}
Commonly used hyperelliptic projective models are as follows:
\begin{enumerate}
\setcounter{enumi} 3 \item $h^2=2$: The map $f_h\colon X\rightarrow \mathbb{P}^2$ is a degree $2$ map ramified at a sextic curve $C\subset \mathbb{P}^2$ (planar model); the excluded bad vectors are fixed components as in (1).
\item $h^2=4$: The map $f_h\colon X\rightarrow \mathbb{P}^1\times\mathbb{P}^1$ is a degree $2$ map ramified at a curve $C\subset \mathbb{P}^1\times\mathbb{P}^1$ of bidegree $(4,4)$; the excluded bad vectors are the fixed components as in (1), whereas at least one linear generatrice as in (2) are \textit{assumed}.
\end{enumerate}
In section~\ref{Applications}, we consider examples of the planar model with $h^2=2$ and the spatial model with $h^2=4$.
\marginpar{\em{\color{red}}}
We use the notation $\mathcal{H}=(S\oplus \mathbb{Z} h\subset L)$ for an abstract homological type extending the root lattice $S$. An abstract homological type $\mathcal{H}=(S\oplus\mathbb{Z} h\subset L)$ is called \emph{maximizing} if $\operatorname{rk}S=19$ (the maximal possible), \emph{i.e.}, $\operatorname{rk}S^{\bot}_h=2$.
An \textit{isomorphism} between two abstract homological types $\mathcal{H}_i=(S_i\oplus \mathbb{Z} h_i\subset L_i)$, $i=1,2$, is an isometry $L_1\rightarrow L_2$, taking $h_1$ to $h_2$ and $S_1$ to $S_2$ (as a set). A \emph{skew-automorphism} of an abstract homological type $\mathcal{H}=(S\oplus\mathbb{Z} h\subset L)$ is a skew-autoisometry of $L$ preserving $S$ (as a set) and $h$.
Given an abstract homological type $\mathcal{H}=(S\oplus \mathbb{Z} h\subset L)$, the group of autoisometries of the primitive hull $\tilde{S}_h=(S_h\otimes\mathbb{Q})\cap L$ preserving $h$ is denoted by $O_h(\tilde{S}_h)$. Obviously we have
\begin{equation*}
O_h(\tilde{S}_h)\subset O_h(S_h)=O(S).
\end{equation*}
Note that $S_h^{\bot}$ is a nondegenerate lattice with $\sigma_+S_h^{\bot}=2$, hence a choice of an orientation of one positive definite $2$-subspace in $S_h^{\bot}\otimes\mathbb{R}$ defines a coherent orientation of any other.
\begin{definition}
An \emph{orientation} of an (abstract) homological type $\mathcal{H}=(S\oplus \mathbb{Z} h\subset L)$ is a positive sign structure $\theta$ on $S_h^{\bot}$. Oriented abstract homological types $(\mathcal{H}_i,\theta_i)$, $i=1,2$, are \emph{isomorphic} if there is an isomorphism $ \mathcal{H}_1\rightarrow\mathcal{H}_2$ taking $\theta_1$ to $\theta_2$. The type $\mathcal{H}$ is called \emph{symmetric} if it admits a skew-automorphism, \emph{i.e.}, $(\mathcal{H},\theta)\cong(\mathcal{H},-\theta)$ for some orientation $\theta$ of $\mathcal{H}$.
\end{definition}
Due to Saint-Donat~\cite{Donat} and Urabe~\cite{Urabe2}, a homological type $\mathcal{H}_X=(S_X,h_X,L_X)$ of a projective model $f_h\colon X\rightarrow \mathbb{P}^{n+1}$ is an abstract homological type (as in the Definition \ref{abstract.homological.type}). Then, the period $\omega_X$ of $X$ defines an orientation of $\mathcal{H}_X$.
\begin{theorem}[\emph{cf.} Theorem 2.3.1 in \cite{AI}]\label{def.class}
The map sending a projective model $f_h\colon X\rightarrow\mathbb{P}^{n+1}$ to its oriented homological type establishes a one-to-one correspondence between equisingular deformation classes of models of a certain fixed kind $\mathfrak{K}$ (with a fixed set of simple singularities $S$) and orientation preserving isomorphism classes of oriented abstract homological types (extending $S$) associated to $\mathfrak{K}$. The homological types of complex conjugate strata differ by orientations.
\end{theorem}
\section{Real structures}
\subsection{Real models}
Let X be a complex analytic variety. A \emph{real structure} on X is an anti-holomorphic map $c\colon X\rightarrow X$ which is an involution. The fixed point set $X_{\mathbb{R}}:=\operatorname{Fix} c$ is called the \emph{real part} of $X$. A subvariety $Y\subset X$ is called \emph{real} if $c(Y)=Y$. A \emph{real model} is a pair $(f_h,c)$, where $f_h \colon X\rightarrow \mathbb{P}^n$ is a projective model of a smooth $K3$-surface $X$ and $c$ is a real structure on $X$ preserving $h$, \emph{i.e.}, such that $c(h)=-h$. Note that for a projective model $f_h \colon X\rightarrow \mathbb{P}^n$, we have $\mathbb{P}^n=|h|^{\vee}$, hence the real structure $c$ gives a real structure on $\mathbb{P}^n=|h|^{\vee}$.
Recall that, if $n$ is even, there is one \emph{standard} real structure up to isomorphism (and up to deformation equivalence) on $\mathbb{P}^n$ given by the standard complex conjugation $\operatorname{conj}\colon\mathbb{P}^n\rightarrow\mathbb{P}^n$, $z=(z_0:z_1:\ldots:z_{n})\mapsto \bar{z}=(\bar{z}_0:\bar{z}_1:\ldots:\bar{z}_{n})$ in appropriate homogeneous coordinates. If $n=2k+1$ is odd, there are two real structures up to isomorphism (and up to deformation equivalence) on $\mathbb{P}^n$.
One of them is the standard one $\operatorname{conj}\colon\mathbb{P}^n\rightarrow\mathbb{P}^n$, $z\mapsto\bar{z}$ mentioned above and the second one is given by $c_2\colon\mathbb{P}^{2k+1}\rightarrow\mathbb{P}^{2k+1}$, $(z_0:z_1:\ldots:z_{2k}:z_{2k+1})\mapsto (\bar{z}_1:-\bar{z}_0:\ldots:\bar{z}_{2k+1}:-\bar{z}_{2k})$. Note that the nonstandard real structure $c_2$ on $\mathbb{P}^{2k+1}$ has empty real part, \emph{i.e.}, $\operatorname{Fix}(c_2)=\emptyset$.
\begin{remark}
Given a real projective model $f_h \colon X\rightarrow \mathbb{P}^n$ where $n$ is odd, it is not obvious which one of the real structures $\operatorname{conj}$, $c_2$ (as above) on $\mathbb{P}^n=|h|^{\vee}$ is induced by the real structure on $X$. This question is answered by Kharlamov \cite{Kharlamov}, in terms of the induced action $c_{*}$ on the polarized lattice $(H_2(X),h)$.
\end{remark}
The following theorem is the arithmetical reduction of the main problem of finding real models.
\begin{theorem}[see Theorem $6.1$ in \cite{Alex2}]\label{real.model}
An abstract oriented homological type $\mathcal{H}$ is realized by a real model if and only if $\mathcal{H}$ admits an involutive skew-automorphism.
\end{theorem}
\subsection{Finding real representatives}
By Theorem \ref{real.model}, to obtain a real model, we will attempt to find involutive skew-automorphisms of the abstract homological type $\mathcal{H}=(S\oplus\mathbb{Z} h\subset L)$.
This problem is straightforward for the maximizing abstract homological types; for the others, we discuss two approaches, via perturbations of abstract homological types and via reflections.
\emph{A (formal) perturbation} of an abstract homological type $\mathcal{H}=(S\oplus\mathbb{Z} h\subset L)$ is any abstract homological type $\mathcal{H}'=(S'\oplus\mathbb{Z} h\subset L)$ such that $S'\subset S$ is primitive in $S$ and the embedding $S'\subset L$ is the restriction of the embedding $S\subset L$. Note that any perturbation of a primitive abstract homological type is also primitive. Most abstract homological types can be obtained by a perturbation from maximizing homological types and this phenomena can be used by means of the following proposition.
\begin{proposition}\label{real.pert}
If an abstract homological type $\mathcal{H}$ is realized by a real birational model $(f_h,c)$, then any $c$-invariant perturbation $\mathcal{H}'$ is also realized by a real model.
\end{proposition}
\begin{remark}
In the computations in chapter~\ref{Applications}, the above proposition holds for birational models; for the hyperelliptic case a stronger statement can be found in \cite{Alex2}.
\end{remark}
The following simple sufficient condition is in fact also necessary for the existence of a real model in which all exceptional divisors are real.
\begin{proposition}
Let $\mathcal{H}=(S\oplus\mathbb{Z} h\subset L)$ be an abstract homological type. If the transcendental lattice $T:=S_h^{\bot}$ contains a sublattice isomorphic to $[2]$ or $\mathcal{U}(2)$ then $\mathcal{H}$ is realized by a real model. Conversely, if $\mathcal{H}$ admits a real model under which all exceptional divisors are real, $T$ contains a sublattice isomorphic to $[2]$ or $\mathcal{U}(2)$.
\end{proposition}
\begin{proof}
If $A=[2]$ or $\mathcal{U}(2)$ is contained in $T=S_h^{\bot}$, then the pair $(-\operatorname{id}, \operatorname{id})$ on $A\oplus A^{\bot}$ extends to $L$ by Lemma \ref{genlemma}; this extension is an involutive skew-automorphism which implies the statement by Theorem~\ref{real.model}. Conversely, let
$c\colon L\rightarrow L$ be an automorphism induced by a real structure as in the statement ($-c$ is an involutive skew-automorphism) and denote by $L_{\pm c}$ its eigenlattices. Note that $\sigma_+L_{-c}=2$ and $h\in L_{-c}$, hence $S\subset L_{-c}$ which implies $T\supset L_{+c}$. Then by Nikulin's classification of real structures on $K3$-surfaces (see \cite{Niku2}), $L_{+c}$ contains $[2]$ or $\mathcal{U}(2)$ and, hence, so does $T$.
\end{proof}
Let $\mathcal{H}=(S\oplus\mathbb{Z} h\subset L)$ be an abstract homological type and $T:=S_h^{\bot}$ be the transcendental lattice. We discuss the existence of real models realizing $\mathcal{H}$ case by case in terms of rank of $T$, explaining the role of reflections (the existence of which is established in the next section).
\subsubsection{The case $\operatorname{rk}T=2$}
The lattice $T$ is a positive definite lattice of rank $2$. By a classical and well known result of Gauss \cite{Gauss}, the group of isometries $O(T)$ is a finite group, which is easily computable. In fact, it turns out that any skew-autoisometry of $T$ is a reflection. (Note, though, that our approach for finding reflections in section \ref{section.reflection} does not apply here since $T$ is often not unique in its genus)
\begin{proposition}\label{rankT=2}
Any symmetric maximizing abstract homological type (extending $S$) admits an involutive skew-automorphism; equivalently, any real component of the strata $\mathcal{M}_1(S)$ contains a real model.
\end{proposition}
\begin{proof}
Let $\mathcal{H}=(S\oplus\mathbb{Z} h\subset L)$ be a maximizing symmetric abstract homological type. Since any skew-autoisometry $r$ of $T=S^{\bot}_h$ is a reflection ($\operatorname{rk}T=2$), it acts as an involution on $\operatorname{disc} T\cong -\operatorname{disc} S$. Then by \eqref{Sym0}, the resulting involution in $\mathcal{A}ut(\operatorname{disc} S)$ is realized by an involution $r'\in \operatorname{Sym}_0(\Gamma_S)\subset O(S)$ (see section \ref{root.lattices}) and $r\oplus r'$ extends to an involution on $L$.
\end{proof}
\subsubsection{The case $\operatorname{rk}T=3$}
Typically, the group $O(T)$ is infinite; however we have the following simple characterization of involutive skew-autoisometries.
\begin{proposition}\label{rankT=3}
Let $T=S_h^{\bot}$ be a lattice of rank $3$. Then any involutive skew-autoisometry $r$ of $T$ is of the form $\pm r'$ where $r'$ is a reflection on $T$ .
\end{proposition}
\begin{proof}
We denote by $T_{\pm r}$ the $(\pm 1)$-eigenspaces of $r$. Recall that the involution $r$ is a reflection if and only if $\operatorname{dim}(T_{-r})=1$. Since here $\operatorname{rk}T=3$, one can have either $\operatorname{dim}(T_{-r})=1$ or $\operatorname{dim}(T_{+r})=1$ or $T_{-r}=0$ (and $r=\operatorname{id}$) or $T_{+r}=0$ (and $r=-\operatorname{id}$). In the last two cases $r$ is not a skew-autoisometry. If $\operatorname{dim}(T_{-r})=1$, the involution $r$ itself is a reflection and if $\operatorname{dim}(T_{+r})=1$, the map $-r$ is a reflection.
\end{proof}
\begin{remark}\label{plusminus.h}
If necessary, we multiply all the maps by $-1$ to make sure that the involutive skew-autoisometry of $T$ is a reflection. Hence, in this case, we have to extend the group $O_h(S_h)$ to $O_{\pm h}(S_h):=O_h(S_h)\times\{\pm \operatorname{id}_h\}$ allowing the involution $h \mapsto -h$.
\end{remark}
Thus, by Proposition \ref{rankT=3} (and Remark~\ref{plusminus.h}), reflections on $T$ are enough to prove both the existence and non-existence of a real model realizing the strata in this case.
\subsubsection{The case $\operatorname{rk} T\geq 4$}
In this case, we can no longer guarantee that any involutive skew-autoisometry of $T$ is a reflection. However, in all examples considered in the paper, it turns out that each symmetric homological type with $\operatorname{rk}T\geq4$ does admit an involutive skew-autoisometry which is a reflection on $T$; thus, it appears that reflections still suffice to conclude the realizability of real strata by real models.
\section{Real structures via reflections}\label{section.reflection}
\subsection{The set-up}
Fix a primitive sublattice $M\subset L\cong\mathbf{L}$ and let $N:= M^\bot$ be its orthogonal complement. Fix also a subgroup $G\subset O(M)$. Consider the finite index extension
\begin{equation}\label{L2}
M\oplus N \subset L,
\end{equation}
both $M$ and $N$ being primitive in $L$. Our aim is to search for
\begin{align}\label{aim}
\text{an involution $\varphi\in G$ on $M$ such that $\varphi\oplus t_a$ extends to $L$}
\end{align}
where $a\in N$ is a primitive vector satisfying \eqref{wdfta} such that $a^2>0$ (since we want a map that reverses positive sign structure) and $t_a$ is a reflection as in \eqref{ta}. By \eqref{wdfta}, we have an apriori bound
\begin{align}\label{adivides2exp}
a^2\mathrel|2\operatorname{exp}(\operatorname{disc} N)
\end{align}
where $\operatorname{exp}(\operatorname{disc} N)$ is the exponent of the group $\operatorname{disc} N$.
Let $N'$ be the orthogonal complement of the primitive vector $a$, \emph{i.e.,} $N':= a^{\bot}\subset N$. Then $N$ is a finite index extension of $\mathbb{Z} a\oplus N'$ and we have
\begin{equation}\label{L3}
M\oplus\mathbb{Z} a\oplus N'\subset L.
\end{equation}
Hence to study finite index extension $L \supset M\oplus N$ as in \eqref{L2}, one can first study the finite index extension
\begin{equation*}
N\supset\mathbb{Z} a\oplus N'.\label{fieT}
\end{equation*}
However, there is another approach: We start by analyzing finite index extension
\begin{equation}\label{fieL}
\widetilde{M}_a:=(M\oplus\mathbb{Z} a)\otimes \mathbb{Q} \cap L\supset M\oplus\mathbb{Z} a=:M_a
\end{equation}
Then we have $L\supset \widetilde{M}_a\oplus N'$. Note that
\begin{align}\label{tildeprimitive}
\text{$M\subset \widetilde{M}_a$ and $\mathbb{Z} a\subset \widetilde{M}_a$ are both primitive}.
\end{align}
Furthermore,
\begin{align}\label{inducesid}
\text{$\varphi\oplus t_a$ should induce $\operatorname{id}$ on $\operatorname{disc} \widetilde{M}_a$}.
\end{align}
\begin{remark}
Strictly speaking, the approach above gives us the reflections $t_a$ in lattices which are in the genus of $N$. However, in our calculations usually $N$ is unique in its genus (see, section \ref{Applications}).
\end{remark}
Let $\mathcal{A}:=\operatorname{disc}\mathbb{Z} a\cong[\frac{1}{a^2}]$ generated by $\alpha:=a/a^2$. By Proposition \ref{L-K}, there is a one-to-one correspondence between the set of isomorphism classes of finite index extensions $\widetilde{M}_a\supset M_a$ satisfying certain properties and that of isotropic subgroups
\begin{equation*}
\mathcal{K} \subset \operatorname{disc} (M\oplus \mathbb{Z} a)=\operatorname{disc}M\oplus \mathcal{A},
\end{equation*}
and we have $\operatorname{disc} \widetilde{M}_a=\mathcal{K}^{\bot}/\mathcal{K}$ (see Lemma~\ref{genlemma}).
\begin{lemma}\label{main.lemma}
A pair $\mathcal{K}\subset \operatorname{disc} M \oplus \mathcal{A}$ and $\varphi\in O(M)$, $\varphi^2= \operatorname{id}$ satisfy conditions \eqref{tildeprimitive} and \eqref{inducesid} above if and only if
\begin{enumerate}
\item $\mathcal{K}$ is the cyclic group generated by $\vartheta:=\kappa\oplus na/a^2$ where $n=1$ or $2$ and $\kappa\in \operatorname{disc} M$ is such that
$\operatorname{order}(\kappa)=a^2/n$ and $\kappa^2=-n^2/a^2$,
\item $\varphi(\kappa)=-\kappa$ in $\operatorname{disc} M$ and,
\item $\varphi\oplus t_a$ induces $\operatorname{id}$ on $\mathcal{K}^{\bot}/\mathcal{K}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Since $\mathcal{A}$ is cyclic, by Lemma \ref{genlemma}, the extension $\widetilde{M}_a\supset M\oplus \mathbb{Z} a$ gives rise to an anti-isometry
\begin{equation*}
\psi'\colon \langle\kappa\rangle\rightarrow \langle n\alpha\rangle\subset\mathcal{A}
\end{equation*}
where $n$ divides $a^2=\operatorname{order}(\alpha)$; hence, $\operatorname{order}(\kappa)=\operatorname{order}(n\alpha)=a^2/n$ and $\kappa^2=(n\alpha)^2=-n^2/a^2$.
Let $m=a^2/n$. Then $m\alpha\in\mathcal{K}^{\bot}$ and $t_a(m\alpha)=-m\alpha$. Hence, the condition $t_a(m\alpha)=m\alpha \bmod \mathcal{K}$ implies $2m\alpha=0$, \emph{i.e.}, $n=\operatorname{order}(\alpha)/m$ is $1$ or $2$.
In view of Lemma \ref{genlemma} again, statements $(2)$ and $(3)$ are a paraphrase of the condition that $\varphi\oplus t_a$ should extend $\operatorname{id}$ on $\operatorname{disc} \widetilde{M}_a$.
\end{proof}
Lemma \ref{main.lemma}, gives pairs $(\varphi,a)$ such that the involution $\varphi\oplus t_a$ extends to unimodular primitive extensions of $\widetilde{M}_a$. The only question remaining is whether such an extension $L\supset \widetilde{M}_a\oplus N'$ exists. The answer is given by Nikulin's existence theorem applied to the genus with discriminant $-(\mathcal{K}^{\bot}/\mathcal{K})$ and signature $(2,19-\sigma_-M)$. We denote this genus depending on $M$ and the pair $(\kappa,n)$ by $\tilde{g}_n(M,\kappa)$ (see, section \ref{The.embedding}).
We have the following corollaries applied to $M=\tilde{S}_h$, for which instead of $O(M)$ we restrict to $\varphi\in O_{\pm h}(\tilde{S}_h)$
\begin{corollary}\label{corlem1}
Let a pair $(n,\kappa)$ and $\varphi\in O_{\pm h}(\tilde{S}_h)$ be as in the conclusion of Lemma \ref{main.lemma} and assume $\tilde g_n(\tilde{S}_h,\kappa)\neq\emptyset$, then $\tilde{S}_h$ extends to an abstract homological type admitting an involutive skew-automorphism.
\end{corollary}
\begin{corollary}\label{corlem2}
Let $\operatorname{rk} \tilde{S}_h=18$. Then, $\tilde{S}_h$ extends to an abstract homological type admitting an involutive skew-automorphism if and only if there exists a pair $(n,\kappa)$ and $\varphi\in O_{\pm h}(\tilde{S}_h)$ satisfying Lemma \ref{main.lemma} and $\tilde g_n(\tilde{S}_h,\kappa)\neq\emptyset$.
\end{corollary}
\begin{warning}
Corollary \ref{corlem1} and Corollary \ref{corlem2} do not say anything about any particular abstract homological type. However, typically those corollaries will be applied in the cases when the abstract homological type extending $\tilde{S}_h$ is unique.
\end{warning}
Now we consider $p$-primary components $\kappa_{[p]}$ of the vector $\kappa\in \operatorname{disc} M$ as in the conclusion of Lemma \ref{main.lemma}. The cyclic group $\langle\kappa\rangle$ generated by $\kappa$ decomposes into orthogonal sum $\langle\kappa\rangle=\bigoplus_p\langle\kappa_{[p]}\rangle$ of its $p$-primary components. Note that if $p$ is odd then the group $\langle\kappa_{[p]}\rangle$ is nondegenerate.
\begin{lemma}\label{kappaperb}
Given $n$ and $\kappa$ as in the conclusion of Lemma \ref{main.lemma}, if $p\neq 2$ or $n=1$ then $(\mathcal{K}^{\bot}_{[p]}/\mathcal{K}_{[p]})\cong\kappa_{[p]}^{\bot} $ where $\kappa_{[p]}^{\bot}$ is the orthogonal complement in $\operatorname{disc}_p M$.
\end{lemma}
\begin{proof}
Since $\mathcal{K}\cap\operatorname{disc} S=0$, we have $\mathcal{K}_{[p]}\cap\kappa_{[p]}^{\bot}=0$. Hence the projection map from $\kappa_{[p]}^{\bot}$ to $\mathcal{K}^{\bot}_{[p]}/\mathcal{K}_{[p]}$ is injective. Note that $|\mathcal{K}^{\bot}_{[p]}||\mathcal{K}_{[p]}|=|\operatorname{disc}_p M\oplus \mathcal{A}_{[p]}|$ which implies $|\mathcal{K}^{\bot}_{[p]}/\mathcal{K}_{[p]}|=|\operatorname{disc}_p M||\mathcal{A}_{[p]}|/|\mathcal{K}_{[p]}|^2$. Since $|\mathcal{A}_{[p]}|=|\mathcal{K}_{[p]}|$ for $p\neq 2$ or $n=1$, we obtain $|\mathcal{K}^{\bot}_{[p]}/\mathcal{K}_{[p]}|=|\operatorname{disc}_p M|/|\mathcal{K}_{[p]}|$ . We also have $|\kappa_{[p]}||\kappa_{[p]}^{\bot}|=|\operatorname{disc}_p M|$. Since $|\kappa_{[p]}|=|\mathcal{K}_{[p]}|$, we get $|\kappa_{[p]}^{\bot}|=|\operatorname{disc}_p M|/|\mathcal{K}_{[p]}|$. It follows that $(\mathcal{K}^{\bot}_{[p]}/\mathcal{K}_{[p]})\cong\kappa_{[p]}^{\bot} $.
\end{proof}
\begin{corollary}
Let a pair $(n,\kappa)$ and $\varphi\in O_{\pm h}(\tilde{S}_h)$ be as in the conclusion of Lemma \ref{main.lemma} and assume $\tilde g_n(\tilde{S}_h,\kappa)\neq\emptyset$, and $p\neq 2$ or $n=1$. Then the involution $\varphi\oplus t_a$ extends to $L$ if and only if
\begin{enumerate}
\item $\varphi(\kappa)=-\kappa$ in $\operatorname{disc} \tilde{S}_h$,
\item $\varphi$ induces $\operatorname{id}$ on $\kappa_{[p]}^{\bot}$.
\end{enumerate}
\end{corollary}
\subsection{The embedding $M\oplus\mathbb{Z} a\hookrightarrow L$}\label{The.embedding}
In the previous section we discussed the conditions for the involution $\varphi\oplus t_a$ to extend to a unimodular primitive extensions $L\supset\widetilde{M}_a$, provided that the latter exists, \emph{i.e.} $\tilde{g}_n(M,\kappa)\neq\emptyset$. Now, we analyze this existence.
The isotropic subgroup $\mathcal{K}\subset\operatorname{disc} M\oplus\mathcal{A}$ given as in the conclusion of Lemma \ref{main.lemma} decomposes into orthogonal sum of its $p$-primary components $\mathcal{K}_{[p]}\subset \operatorname{disc}_p M\oplus\mathcal{A}_{[p]}$ generated by $\kappa_{[p]}+n\alpha_{[p]}$ where $\alpha_{[p]}$ is a generator of $\mathcal{A}_{[p]}$ and $n=1$ or $2$.
\begin{lemma}\label{orth.summand}
Given a primitive extension $L\supset M$ and a pair $(n,\kappa)$ as in the conclusion of Lemma \ref{main.lemma}, if $p$ is odd or $n=1$, then the group generated by $\kappa_{[p]}$ is an orthogonal direct summand, i.e., $\operatorname{disc} M\cong\bar{\mathcal{M}}\oplus\langle\kappa_{[p]}\rangle$.
\end{lemma}
\begin{proof}
Let $p$ be an odd prime or $n=1$, then we have $\operatorname{order}(\kappa_{[p]})=\operatorname{order}(\kappa_{[p]}^2)$. Then the form generated by $\kappa_{[p]}$ is nondegenerate and hence an orthogonal direct summand in any form.
\end{proof}
\begin{corollary}\label{coronNiku}
Given a primitive extension $L\supset M$ and a pair $(n,\kappa)$ as in the conclusion of Lemma \ref{main.lemma}, the hypotheses of Theorem \ref{th.N.existence} for the extension $L\supset M_a$ hold automatically for
\begin{itemize}
\item all odd primes $p\mathrel |a^2$;
\item $p=2$ provided that $n=1$ and parity does not change, \emph{i.e.}, $\operatorname{disc} M$ and $\kappa^{\bot} \subset \operatorname{disc} M$ are either both even or both odd.
\end{itemize}
\end{corollary}
\subsection{The $2$-primary part}
By Lemma~\ref{orth.summand}, the only nontrivial case (\emph{i.e.}, $\langle\kappa_{[p]}\rangle$ is not an orthogonal summand) is when $p=2$ and $n=2$.
From now on, to avoid more than one subscript, we often abbreviate $\mathcal{M}:=\operatorname{disc}_2 M$, $\mathcal{K}:=\mathcal{K}_{[2]}$, $\kappa:=\kappa_{[2]}$ and $\alpha:=\alpha_{[2]}$ for the corresponding $2$-primary parts. In this notation, we have $\operatorname{ord}(\alpha)=2^{m+1}$, $\alpha^2=\frac{\delta}{2^{m+1}}$ for some positive integer $m$, and, hence,
\begin{align}\label{kappasquare}
\text{$\operatorname{ord}(\kappa)=2^m$, $\kappa^2=\frac{\xi}{2^{m-1}}$, $\xi$ is odd}.
\end{align}
By applying Lemma \ref{sequences} to the $2$-subgroup $\mathcal{C}=\langle\kappa\rangle$ of $\mathcal{M}=\operatorname{disc}_2 M$, we arrive at the decomposition $\mathcal{M}\cong\bar{\mathcal{M}}\oplus \bigoplus \mathcal{N}_s$.
\begin{observation}\label{observation1}
Recall that $\mathcal{M}=\bigoplus\mathcal{M}_i$, where $\mathcal{M}_i$ is the homogenous group of exponent $2^i$. For any $\sigma_i'\in 2^r\mathcal{M}_i$, we have
\begin{equation}\label{observation1.1}
(\sigma'_i)^2=\frac{\mu'}{2^{i-2r}},\, \mu'\in\mathbb{Z}.
\end{equation}
Equivalently, for any $\sigma\in 2^{r}\mathcal{M}$ such that $\operatorname{ord}(\sigma)\leq 2^d$, we have
\begin{equation}\label{observation1.2}
(\sigma)^2=\frac{\mu}{2^{d-r}},\,\mu\in\mathbb{Z}.
\end{equation}
\end{observation}
\begin{observation}\label{observation2}
In view of \eqref{kappasquare}, only the following three homogenous components contribute to $\kappa^2$:
$\kappa'_{m-1}$, $\kappa'_{m}$ and $\kappa'_{m+1}=2\bar{u}_{m+1}$ where $\kappa'_{m-1}\in\mathcal{M}_{m-1}$ and $\bar{u}_{m+1}\in\mathcal{M}_{m+1}$ are orthogonal direct summands whereas $\kappa'_{m}$ is \emph{not}: $\operatorname{ord}(\kappa'_m)=2^m$ and $(\kappa')^2=\xi'/2^{m-1}$.
\end{observation}
For $r_1$ as in \eqref{r1}, by Observation \ref{observation1}, we have $\kappa^2=\bar{\kappa}^2_1=\bar{\xi}/2^{m-r_1}$, $\bar{\xi}\in \mathbb{Z}$, and hence, either $r_1=0$ or $r_1=1$ by \eqref{kappasquare}.
\noindent\textbf{{The case $r_1=0$}}: Following the construction in Lemma \ref{sequences}, let
\begin{equation*}
n=\operatorname{max}\{i:\operatorname{ord}(\kappa'_i)=2^{i} \}\leq m\mbox{ and } m_1=n=\operatorname{log}_2\operatorname{ord}(\kappa_n').
\end{equation*}
We have $2^0u_1=\kappa_1=\Sigma_{\operatorname{ord}(\kappa'_i)\leq2^n}\kappa'_i$. We consider three cases: $m_1=m$ or $m_1= m-1$ or $m_1\leq m-2$.
$\bullet$\textit{ The case $n:=m_1=m$}:
Then we have $\kappa=\bar{\kappa}_1=\kappa_1=u_1$ (see \eqref{kappa1}) and the process terminates: $\mathcal{M}=\bar{\mathcal{M}}\oplus\mathcal{N}_1$. By \eqref{kappasquare}, we have $\ell(\mathcal{N}_1)=2$, \emph{i.e.},
\begin{equation}\label{N1.1}
\mathcal{M}\cong\bar{\mathcal{M}}\oplus\frac{1}{2^{m}}\left[ \begin{array}{cc}
\mu_1 & 1 \\
1 & \nu_1 \\
\end{array}
\right],\,\mu_1\in2\mathbb{Z};\quad\kappa_{[2]}=u_1.
\end{equation}
$\bullet$ \textit{The case $n:=m_1=m-1$}: Then we have $\kappa_1=u_1$, $\operatorname{ord}(\kappa_1)=2^{m-1}$ and $\kappa_1^2=\mu_1/2^{m-1}$.
Consider $\bar{\kappa}_2=\bar{\kappa}_1-\kappa_1$. By construction, $\bar{\kappa}_2=\bigoplus_i\kappa'_i$ where $\kappa_i'\in\mathcal{M}_{i}$ with $i>m-1$ for all $i$. We have $r_2\geq 1$, and the inequality $m_1=m-1<m_2\leq m$ implies that $m_2=m$. Hence $\kappa_2=\bar{\kappa}_2$ (see \eqref{kappa1}) and the algorithm terminates: $\kappa=\kappa_1\oplus\kappa_2$.
It follows that
$$\kappa_1^2+\kappa_2^2=\frac{\mu_1}{2^{m-1}}+ \frac{\mu_2}{2^{m-r_2}}$$
and, essentially by Observation \eqref{observation2}, we have the following possibilities:
either $r_2=1$, $\mu_1$ is odd and $\mu_2$ is even; then
\begin{equation}\label{N1N2.1}
\mathcal{M}\cong\bar{\mathcal{M}}\oplus\frac{1}{2^{m-1}}\left[
\begin{array}{c}
\mu_1\\
\end{array}
\right]\oplus\frac{1}{2^{m+1}}\left[ \begin{array}{cc}
\mu_2 & 1 \\
1 & \nu_2 \\
\end{array}
\right];\quad\kappa_{[2]}=u_1\oplus2u_2,
\end{equation}
or $r_2=1$, $\mu_1$ is even and $\mu_2$ is odd; then
\begin{equation}\label{N1N2.2}
\mathcal{M}\cong\bar{\mathcal{M}}\oplus\frac{1}{2^{m-1}}\left[ \begin{array}{cc}
\mu_1 & 1 \\
1 & \nu_1 \\
\end{array}
\right]\oplus\frac{1}{2^{m+1}}\left[
\begin{array}{c}
\mu_2\\
\end{array}
\right];\quad\kappa_{[2]}=u_1\oplus2u_2,
\end{equation}
or $r_2>1$, then $\mu_1$ is odd and $\mu_2$ can be odd or even; then
\begin{equation}\label{N1N2.3}
\mathcal{M}\cong\bar{\mathcal{M}}\oplus\frac{1}{2^{m-1}}\left[
\begin{array}{c}
\mu_1\\
\end{array}
\right]\oplus\mathcal{N}_2;\quad\kappa_{[2]}=u_1\oplus2^{r_2}u_2,
\end{equation}
where $\mathcal{N}_2$ is either
\begin{equation}\label{N1N2.4}
\frac{1}{2^{m+r_2}}\left[
\begin{array}{c}
\mu_2\\
\end{array}
\right]\quad\mbox{or}\quad\frac{1}{2^{m+r_2}}\left[ \begin{array}{cc}
\mu_2 & 1 \\
1 & \nu_2 \\
\end{array}
\right].
\end{equation}
$\bullet$\textit{ The case $n:=m_1\leq m-2$}: By Observation \ref{observation2}, at the next step we have $r_2=1$ and $n_2\geq m+1$, hence again $m_2=m$, and the algorithm terminates: $\kappa=\kappa_1\oplus\kappa_2$, hence we have
$$\kappa_1^2+\kappa_2^2=\frac{\mu_1}{2^{m-2}}+ \frac{\mu_2}{2^{m-1}}$$
and by Observation \ref{observation2} again, $\mu_2$ is odd and $\mu_1$ can be odd or even. Then, we obtain
\begin{equation}\label{N1N2.5}
\mathcal{M}\cong\bar{\mathcal{M}}\oplus\mathcal{N}_1\oplus\frac{1}{2^{m+1}}\left[
\begin{array}{c}
\mu_2\\
\end{array}
\right]\quad\kappa_{[2]}=u_1\oplus2u_2
\end{equation}
where $\mathcal{N}_1$ is either
\begin{equation}\label{N1N2.6}
\frac{1}{2^{n}}\left[
\begin{array}{c}
\mu_1\\
\end{array}
\right], \, n\leq m-2\quad\mbox{or}\quad\frac{1}{2^{n}}\left[ \begin{array}{cc}
\mu_1 & 1 \\
1 & \nu_1 \\
\end{array}
\right],\, n\leq m-2.
\end{equation}
\noindent\textbf{The case $r_1=1$}: Then we have
\begin{equation*}
n=\operatorname{max}\{i:\operatorname{ord}(\kappa'_i)=2^{i-1} \}\leq m+1.
\end{equation*}
Since $n>m$, by Observation \ref{observation2}, we conclude that $n=m+1$ and $m_1=m$, \emph{i.e.}, the algorithm terminates at the first step and we arrive at
\begin{equation}\label{N1N2.7}
\mathcal{M}\cong\bar{\mathcal{M}}\oplus\frac{1}{2^{m+1}}\left[ \begin{array}{c}
\mu_1
\end{array}
\right];\quad\kappa_{[2]}=2u_1
\end{equation}
\subsection{The group $\mathcal{K}^{\bot}/\mathcal{K}$}\label{description.KperbmodK}
For the decompositions $\mathcal{M}\cong\bar{\mathcal{M}}\oplus \bigoplus_{s=1}^N \mathcal{N}_s$ given above of length $N\leq2$, the corresponding groups $\mathcal{K}^{\bot}/\mathcal{K}$ which are the orthogonal direct sum of $\bar{\mathcal{M}}$ and a subgroup generated by certain elements $\{w_i\}$ are described below (in terms of the notation given in Lemma \ref{sequences}) on a case by case basis. We also indicate
the ``\emph{ambiguous}" cases where $\mathcal{M}$ is odd and $\mathcal{K}^{\bot}/\mathcal{K}$ is even (when describing the \emph{ambiguous} cases, we assume $\bar{\mathcal{M}}$ is even since otherwise both $\mathcal{M}$ and $\mathcal{K}^{\bot}/\mathcal{K}$ are odd forms ):
The case $r_1=0$ :
\begin{itemize}
\item In case \eqref{N1.1}, $\mathcal{K}=\langle u_1+2\alpha\rangle$ and $\mathcal{K}^{\bot}/\mathcal{K}\cong\bar{\mathcal{M}}\oplus\mathbb{Z}/2^{m+1}$, generated by $w_1=\alpha-\delta v_1$. This case is \emph{ambiguous} if $m=1$ and $\nu_1$ is odd.
\item In case \eqref{N1N2.1}, $\mathcal{K}=\langle u_1+2u_2+2\alpha\rangle$ and $\mathcal{K}^{\bot}/\mathcal{K}\cong\bar{\mathcal{M}}\oplus\mathbb{Z}/2^{m+1}\oplus\mathbb{Z}/2^{m+1}$, generated by $w_1=\delta v_2-\alpha$ and $w_2=\delta u_2-\mu_2\alpha$. This case is \emph{ambiguous} if $m=2$.
\item In case \eqref{N1N2.2}, $\mathcal{K}=\langle u_1+2u_2+2\alpha\rangle$ and $\mathcal{K}^{\bot}/\mathcal{K}\cong\bar{\mathcal{M}}\oplus\mathbb{Z}/2^{m}\oplus\mathbb{Z}/2^{m}$, generated by $w_1=v_1-\frac{2}{\mu_2}u_2$ and $w_2=\frac{\mu_1}{2}v_1+u_2+\alpha$. This case is \emph{ambiguous} if $m=1$ and $\nu_1$ is odd.
\item In case \eqref{N1N2.3} with the former case of \eqref{N1N2.4}, $\mathcal{K}=\langle u_1+2^{r_2}u_2+2\alpha\rangle$ and $\mathcal{K}^{\bot}/\mathcal{K}\cong\bar{\mathcal{M}}\oplus\mathbb{Z}/2^{m+r_2}$, generated by $w_1=\delta u_2-\mu_2\alpha$. This case is \emph{ambiguous} if $m=2$.
\item In case \eqref{N1N2.3} with the latter case of \eqref{N1N2.4}, $\mathcal{K}=\langle u_1+2^{r_2}u_2+2\alpha\rangle$ and $\mathcal{K}^{\bot}/\mathcal{K}\cong\bar{\mathcal{M}}\oplus\mathbb{Z}/2^{m+r_2}\oplus\mathbb{Z}/2^{m+r_2}$, generated by $w_1=\delta v_2-\alpha$ and $w_2=\delta u_2-\mu_2\alpha$. This case is \emph{ambiguous} if $m=2$.
\item In case \eqref{N1N2.5} with the former case of \eqref{N1N2.6}, $\mathcal{K}=\langle u_1+2u_2+2\alpha\rangle$ and $\mathcal{K}^{\bot}/\mathcal{K}\cong\bar{\mathcal{M}}\oplus\mathbb{Z}/2^{n+2}$, generated by $w_1=-\delta u_2+\mu_2\alpha$. This case is \emph{ambiguous} if $m\geq3$ and $n=1$.
\item In case \eqref{N1N2.5} with the latter case of \eqref{N1N2.6}, $\mathcal{K}=\langle u_1+2u_2+2\alpha\rangle$ and $\mathcal{K}^{\bot}/\mathcal{K}\cong\bar{\mathcal{M}}\oplus\mathbb{Z}/2^{n+1}\oplus\mathbb{Z}/2^{n+1}$, generated by $w_1=\mu_2v_1-2^{m-n}u_2$ and $w_2=\frac{\mu_1}{2}v_1+u_2+\alpha$. This case is \emph{ambiguous} if $n=1$ and $\nu_1$ is odd.
\end{itemize}
The case $r_1=1$:
\begin{itemize}
\item In case \eqref{N1N2.7}, $\mathcal{K}=\langle 2u_1+2\alpha\rangle$ and $\mathcal{K}^{\bot}/\mathcal{K}\cong\bar{\mathcal{M}}\oplus\mathbb{Z}/2\oplus\mathbb{Z}/2$, generated by $w_1=u_1+\alpha$ and $w_2=2^mu_1$.
\end{itemize}
\section{Applications}\label{Applications}
\subsection{Simple Quartics}\label{simple.quartics}
In this section we consider birational projective models $f_h\colon X\rightarrow \mathbb{P}^3$ with $h^2=4$, \emph{i.e.}, spatial model.
The image is a quartic surface in $\mathbb{P}^3$.
For a simple quartic $X$, the minimal resolution of singularities $\tilde{X}$ is a smooth $K3$-surface; hence the intersection lattice is of the form $H_2(\tilde{X})\cong \mathbf{L}$.
\begin{definition}
A quartic $X$ is called \emph{nonspecial} if the abstract homological type $\mathcal{H}(S\oplus \mathbb{Z} h\subset L)$ associated to the projective model $f_h\colon X\rightarrow \mathbb{P}^3$ is primitive, \emph{i.e}, $S_h\subset L$ is a primitive extension.
\end{definition}
For a given set of simple singularities $S$, the corresponding equisingular stratum of quartics is denoted by $\mathcal{M}(S)$. Our primary interest is the family $\mathcal{M}_1(S)\subset \mathcal{M(S)}$ constituted by the nonspecial quartics with the set of singularities $S$.
A complete description of the strata $\mathcal{M}_1(S)$ of nonspecial simple quartics is given by G\"{u}ne\c{s} Akta\c{s} \cite{Cisem1}, where it is proved that
the strata $\mathcal{M}_1 (S)$ with $S=\mathbf{D}_6\oplus2\mathbf{A}_6$, $\mathbf{D}_5\oplus2\mathbf{A}_6\oplus \mathbf{A}_1$, $2\mathbf{A}_7\oplus 2\mathbf{A}_2$, $3\mathbf{A}_6$ or $2\mathbf{A}_6\oplus2\mathbf{A}_3$ split into pairs of complex conjugate components; all other non-maximizing equisingular strata are connected (i.e., they consist of one real component). The classification of the connected components of the $59$ maximizing strata is also available there. As one of the main applications of this paper, we give the proof of Theorem~\ref{principal.result}.
\begin{proof}[Proof of Theorem~\ref{principal.result}]
By Theorem \ref{real.model}, the question reduces to finding an involutive skew-automorphism of the primitive abstract homological type $\mathcal{H}=(S\oplus \mathbb{Z} h\subset L)$ extending the root lattice $S$.
\begin{remark}\label{sym}
Since the homological type is primitive we have $\tilde{S}_h=S_h$, $\operatorname{disc} \tilde{S}_h=\operatorname{disc} S\oplus [ \frac{1}{4}]$ and $O_h(\tilde{S}_h)=O(S)$. Furthermore, since we are interested in the induced action on discriminant, the group $O_{\pm h}(\tilde{S}_h)$ can be replaced with $\operatorname{Sym}(\Gamma)\times\{\pm \operatorname{id}_{h}\}$, where $\Gamma$ is the Dynking diagram of the root lattice $S$ (\emph{cf}. section \ref{root.lattices})
\end{remark}
If $\operatorname{rk} S=19$, the statement of the theorem is given by Proposition \ref{rankT=2}. Hence, throughout the rest of the proof we assume $\operatorname{rk} S\leq18$.
By computer aided computations, it is easily confirmed that most of the abstract homological types $\mathcal{H}=(S\oplus\mathbb{Z} h\subset \mathbf{L} )$ (except $100$ of them) with $\operatorname{rk} S\leq18$ are symmetric $\operatorname{c}$-invariant perturbations of the $37$ maximizing primitive homological types (see, \cite{Cisem1}) realized by a real quartic where $c$ is a real structure on the corresponding real surface. Then, due to Proposition \ref{real.pert}, these abstract homological types are also realized by a real nonspecial quartic.
The space $\mathcal{M}_1 (S)$ with $S=\mathbf{D}_6\oplus2\mathbf{A}_6$, $\mathbf{D}_5\oplus2\mathbf{A}_6\oplus \mathbf{A}_1$, $2\mathbf{A}_7\oplus 2\mathbf{A}_2$, $3\mathbf{A}_6$, $2\mathbf{A}_6\oplus2\mathbf{A}_3$ consists of two complex conjugate components (see, \cite{Cisem1}). Therefore the strata $\mathcal{M}_1(S)$ do not contain a real surface.
For each of the remaining $95$ sets of singularities $S$, we used GAP~\cite{GAP} to find a positive integer $a^2\mathrel| 2\operatorname{exp}(\operatorname{disc} \tilde{S}_h)$, integer $n=1$ or $2$, class $\kappa\in \operatorname{disc} \tilde{S}_h$ and involution $\varphi\in O_{\pm h}(\tilde{S}_h)$ satisfying the conditions in Lemma \ref{main.lemma} and such that $\tilde{g}_n(\tilde{S}_h, \kappa)\neq\emptyset$. If found, Corollary \ref{corlem1} and Theorem \ref{real.model} imply that the stratum $\mathcal{M}_1(S)$ contains a real surface. Since, on the other hand, $\mathcal{M}_1(S)$ is connected (see Corollary 4.2.4 in \cite{Cisem1}), this implies the statement.
The above algorithm fails for the set of singularities $S_1=\mathbf{A}_7\oplus\mathbf{A}_6\oplus\mathbf{A}_3\oplus\mathbf{A}_2$ and $S_2=\mathbf{D}_7\oplus\mathbf{A}_6\oplus\mathbf{A}_3\oplus\mathbf{A}_2$ (exceptional cases listed in the statement), \emph{i.e.}, computer aided calculations confirm that there does not exist a pair $(n,\kappa)$ and $\varphi\in O_{\pm h}(S_h)$ satisfying Lemma \ref{main.lemma} and $\tilde g_n(S_h,\kappa)\neq\emptyset$. Since $\operatorname{rk} S_1=\operatorname{rk} S_2 =18$, by Corollary \ref{corlem2}, the statement follows. $\qed$
Although the implemented calculations by GAP~\cite{GAP} completes the proof for the exceptional cases $S_1$ and $S_2$, in the following two subsections, we provide explicit details, just to illustrate how many things may go wrong in constructing a skew-automorphism on the abstract homological types extending $S_1$ and $S_2$.
Before continuing, we make the following observation to which we will refer in the proof for the two exceptional cases.
\begin{observation}\label{observation123}
Assume that $\operatorname{rk} S=18$ and the group $\operatorname{disc}_2(\tilde{S}_h\oplus \mathbb{Z}a)$ is the orthogonal sum of cyclic groups with generators $\alpha_i$, where $\alpha_0=\alpha_{[2]}$ is the generator of $\operatorname{disc}_2\mathbb{Z}a$, and $\operatorname{order}(\alpha_i)=4$ or $8$ for all $i$. Assume further that the action of $O_{\pm h}(\tilde{S}_h)\times\{\pm \operatorname{id}_a\}$ on $\operatorname{disc} (\tilde{S}_h\oplus\mathbb{Z}a)$ is generated by involutions $\alpha_i\mapsto\pm\alpha_i$. Let the $2$-primary part of the kernel $\mathcal{K}_{[2]}$ be generated by a single element $\vartheta=\kappa_{[2]}+n\alpha_{[2]}$ of the form $k\alpha_i+l\alpha_j+\vartheta'$, $k,l\in\mathbb{Z}$ where each $k\alpha_i$, $l\alpha_j$ and $\vartheta'$ has order at least $4$ and $\vartheta'$ is a combination of generators other than $\alpha_i,\alpha_j$. Then, in the three cases considered below, the involution $\varphi\oplus t_a$ reverses $\alpha_i$ and $\alpha_j$ (recall that $\varphi(\kappa)=-\kappa$) and the element $\nu$ described below has order at least $4$ in $\mathcal{K}^{\bot}/\mathcal{K}$ and is reversed by $\varphi\oplus t_a$, contrary to Lemma~\ref{main.lemma}(3). Hence by Corollary~\ref{corlem2}, $\tilde{S}_h$ does not extend to an abstract homological type admitting a skew-automorphism.
\begin{enumerate}
\item If $\vartheta=(\pm \alpha_i\pm\alpha_j)+\vartheta'$ with $\operatorname{order}(\alpha_j)=4$, then $\nu$ is one of $\alpha_i\pm\alpha_j$
\item If $\vartheta=(\pm 2\alpha_i\pm\alpha_j)+\vartheta'$ with $\operatorname{order}(\alpha_j)=4$, then $\nu$ is one of $\alpha_i\pm\alpha_j$
\item If $\vartheta=(\pm \alpha_i\pm\alpha_j)+\vartheta'$ with $\operatorname{order}(\alpha_j)=4$, then $\nu$ is one of $2\alpha_i\pm\alpha_j$
\end{enumerate}
\end{observation}
\subsection{The set of singularities $S=\mathbf{D}_7\oplus\mathbf{A}_6\oplus\mathbf{A}_3\oplus\mathbf{A}_2$}\label{exceptional1}\label{S1}
One has
$$\operatorname{disc} \tilde{S}_h = \operatorname{disc} S_h\cong\textstyle[\frac{1}{4}]\oplus[-\frac{6}{7}]\oplus[-\frac{3}{4}]\oplus[-\frac{2}{3}]\oplus[
\frac{1}{4}].$$
Consider an integer $a^2\mathrel| 2\operatorname{exp} (\operatorname{disc} \tilde{S}_h)= 2^3\cdot3\cdot7$, \emph{i.e.}, $a^2= 2^N\cdot3^r\cdot7^s$, where $N\in\{1,2,3\}$, $r\in\{0,1\}$ and $s\in\{0,1\}$, see \eqref{adivides2exp}. With $a^2$ fixed, consider a pair $(n,\kappa)$, where $n=1$ or $2$ and $\kappa\in \operatorname{disc}_2\tilde{ S}_h$, as in Lemma \ref{main.lemma}. Fix the generators
$$\text{$\alpha_1$ for $\operatorname{disc} \mathbf{D}_7\cong\textstyle[\frac{1}{4}]$,\quad $\alpha_2$ for $\operatorname{disc} \mathbf{A}_3\cong\textstyle[-\frac{3}{4}]$,\quad$\alpha_3$ for $\operatorname{disc} \mathbb{Z}h\cong\textstyle[\frac{1}{4}]$},$$
for the $2$-primary part. It is immediate (\emph{cf}. Remark~\ref{sym}) that the action of $O_{\pm h}(\tilde{S}_h)$ on $\operatorname{disc} \tilde{S}_h$ is generated by the involutions $\alpha_i\mapsto\pm\alpha_i$ with $i=1,2,3$, as in Observation \ref{observation123}.
\subsubsection{The case $N=1$}\label{subN1}
Then $n=2$ (as $\operatorname{disc} \tilde{S}_h$ does not contain any cyclic direct summand of order $2$). Hence, we have $\kappa_{[2]}=0$ and $\ell_2(\mathcal{K}^{\bot}/\mathcal{K})= 4$ , which implies $\tilde{g}_n(\tilde{S}_h,\kappa)=\emptyset$ by Theorem \ref{th.N.existence}.
\subsubsection{The case $N=2$ and $n=2$}\label{subN2n2}
Then we have $\ell_2(\mathcal{K}^{\bot}/\mathcal{K})= 4$ by \eqref{N1N2.7} and the respective item in section \ref{description.KperbmodK}, which implies $\tilde{g}_n(\tilde{S}_h,\kappa)=\emptyset$ by Theorem \ref{th.N.existence}.
\subsubsection{The case $N=2$ and $n=1$}\label{N2n1}
For an odd prime $p$, the group $\langle\kappa_{[p]}\rangle$ is an orthogonal summand in the cyclic group $\operatorname{disc}_p \tilde{S_h}$ (by Lemma~\ref{orth.summand}), hence, in this particular case, we have either $\kappa_{[p]}=0$, which implies $ p\nmid a^2$, or $\langle\kappa_{[p]}\rangle\cong\operatorname{disc}_p\tilde{S}_h$. Since $\operatorname{disc}_p\tilde{S}_h$ is a cyclic group, this implies an extra condition on $a^2$: one must have $a^2/3= -2 \bmod (\mathbb{Z}_2^{\times})^2$ and $a^2/7= -6 \bmod (\mathbb{Z}_7^{\times})^2$. By checking these conditions,, we rule out the cases $a^2=4\cdot3,4\cdot7,4\cdot3\cdot7$ and obtain $a^2=4$. Listing $\kappa_{[2]}\in \operatorname{disc}_2 \tilde{S}_h$ with
$$
\text{$\operatorname{order }(\kappa_{[2]})=4$ and $\kappa^2_{[2]}=-\frac{1}{4}$},
$$
we arrive at $\vartheta=(\alpha_0\pm\alpha_1)\pm\alpha_2\pm\alpha_3$ as in Observation~\ref{observation123}(1), ruling this case out.
\subsubsection{The case $N=3$}
Then $n=2$ (as $\operatorname{disc}_2 \tilde{S}_h$ does not contain any cyclic direct summand of order $8$). As in section~\ref{N2n1}, we rule out the case $a^2=8\cdot7$ and obtain $a^2=8\delta$, where $\delta=1,3$ or $21$. Then, listing all the vectors $\kappa_{[2]}\in \operatorname{disc}_2\tilde{S}_h$ satisfying
$$
\text{$\operatorname{order }(\kappa_{[2]})=4$ and $\kappa_{[2]}^2=-\frac{\delta}{2}$},
$$
we obtain $\vartheta=(2\alpha_0\pm\alpha_i)\pm\alpha_j$ or $(2\alpha_0\pm\alpha_i)\pm\alpha_j+2\alpha_k$ as in Observation~\ref{observation123}(2), eliminating this case.
\subsection{The set of singularities $S=\mathbf{A}_7\oplus\mathbf{A}_6\oplus\mathbf{A}_3\oplus\mathbf{A}_2$}\label{S2}
One has
$$\operatorname{disc} \tilde{S}_h=\operatorname{disc} S_h\cong\textstyle[-\frac{7}{8}]\oplus[-\frac{6}{7}]\oplus[-\frac{3}{4}]\oplus[-\frac{2}{3}]\oplus[\frac{1}{4}].$$
Consider an integer $a^2\mathrel| 2\operatorname{exp} (\operatorname{disc} \tilde{S}_h)= 2^4\cdot3\cdot7$, \emph{i.e.}, $a^2= 2^N\cdot3^r\cdot7^s$, where $N\in\{1,2,3,4\}$, $r\in\{0,1\}$ and $s\in\{0,1\}$. As above, fix $a^2$ and consider a pair $(n,\kappa)$ (where $n=1,2$) and $\kappa_{[2]}\in \operatorname{disc}_2\tilde {S}_h$, as in Lemma \ref{main.lemma}.
Fix the generators
$$\text{$\alpha_1$ for $\operatorname{disc} \mathbf{A}_7\cong\textstyle[-\frac{7}{8}]$,\quad $\alpha_2$ for $\operatorname{disc} \mathbf{A}_3\cong\textstyle[-\frac{3}{4}]$,\quad$\alpha_3$ for $\operatorname{disc} \mathbb{Z}h\cong\textstyle[\frac{1}{4}]$},$$
for $\operatorname{disc}_2 \tilde{S}_h$. By Remark~\ref{sym}, the group $O_{\pm h}(\tilde{S}_h)$ acts on $\operatorname{disc} \tilde{S}_h$ via the involutions $\alpha_i\mapsto\pm\alpha_i$ with $i=1,2,3$, as in Observation~\ref{observation123}.
\subsubsection{The case $N=1$ is ruled out as in section~\ref{subN1}}
\subsubsection{The case $N=2$, $n=2$ is ruled out as in section~\ref{subN2n2}}
\subsubsection{The case $N=2$ and $n=1$}
We obtain $a^2=4$ as in section~\ref{N2n1}. (Note that $3$- and $7$-primary parts of $\operatorname{disc} \tilde{S}_h$ are the same as in section~\ref{exceptional1}). Then, listing all vectors $\kappa_{[2]}\in \operatorname{disc}_2\tilde{S}_h$ with
$$\text{$\operatorname{order }(\kappa_{[2]})=4$ and $\kappa^2_{[2]}=-\frac{1}{4}$},$$
we arrive at $\vartheta=(\alpha_0\pm\alpha_3)\pm2\alpha_1\pm2\alpha_2$ or $(\alpha_0\pm\alpha_2)\pm2\alpha_1$ as in Observation~\ref{observation123}(1).
\subsubsection{The case $N=3$ and $n=1$}\label{N3n1(2)}
As in section~\ref{N2n1}, we have $a^2=8\delta$, where $\delta=1,3$ or $21$. We search for $\kappa_{[2]}\in \operatorname{disc}_2\tilde{ S}_h$ with
$$
\text{$\operatorname{order }(\kappa_
{[2]})=8$ and $\kappa_{[2]}^2=-\frac{\delta}{8}$}.
$$
\par If $\delta=1$, then there is no such element $\kappa_{[2]}$.
If $\delta=3$, then listing all such vectors $\kappa_{[2]}$, we obtain $\vartheta=(\alpha_0\pm\alpha_2)\pm3\alpha_1\pm \alpha_3$ as in Observation~\ref{observation123}(3).
If $\delta=21$, then we have $\vartheta=(\alpha_0\pm\alpha_2)\pm\alpha_1+2\alpha_3$, $(\alpha_0\pm\alpha_3)\pm3\alpha_1+2\alpha_2$, $(\alpha_0\pm\alpha_2)\pm3\alpha_1$ or $(\alpha_0\pm\alpha_3)\pm\alpha_1$ as in Observation~\ref{observation123}(3).
\subsubsection{The case $N=3$ and $n=2$}
By section~\ref{N3n1(2)}, we have $a^2=8\delta$, where $\delta=1,3$ or $21$. Then, listing $\kappa_{[2]}\in \operatorname{disc}_2\tilde{ S}_h$ such that
$$
\text{$\operatorname{order }(\kappa_{[2]})=4$ and $\kappa_{[2]}^2=-\frac{\delta}{2}$},
$$
we obtain $\vartheta=(2\alpha_0\pm \alpha_2)+4\alpha_1\pm\alpha_3$, $(2\alpha_0\pm\alpha_2)\pm\alpha_3$ or $2\alpha_0\pm2\alpha_1+2\alpha_3$, $2\alpha_0\pm2\alpha_1+2\alpha_2$. The former two cases are covered by Observation~\ref{observation123}(2). The latter two cases are ruled out as in section \ref{subN2n2}.
\subsubsection{The case $N=4$}\label{sub4}
Then $n=2$, since $\operatorname{disc}_2 \tilde{S}_h$ does not contain any cyclic direct summand of order $16$. On the other hand any order $8$ element in $\operatorname{disc}_2\tilde{S}_h$ is an orthogonal direct summand which contradicts to $n=2$.
\end{proof}
\subsection{Simple Sextics}
In this section we consider hyperelliptic projective models $f_h\colon X\rightarrow \mathbb{P}^2$ with $h^2=2$, \emph{i.e.}, planar models. Recall that $f_h$ is a degree $2$ map ramified at a sextic curve $C\subset \mathbb{P}^2$.
A complete description of the strata $\mathcal{M}_1(S)$ of nonspecial simple sextics is given in~\cite{Alex2}. Degtyarev and Akyol also showed that the space $\mathcal{M}_1(\mathbf{A}_7\oplus\mathbf{A}_6\oplus\mathbf{A}_5)$ consists of a single component, which is hence real, but it contains no real curves (see Proposition 2.6 in \cite{Alex2}). This result is similar to our main result Theorem~\ref{principal.result} for simple quartics and our approach gives a simpler and more transparent proof which is outlined below.
\begin{proposition}[Proposition 2.6 in~\cite{Alex2}]
The stratum $\mathcal{M}_1(\mathbf{A}_7\oplus\mathbf{A}_6\oplus\mathbf{A}_5)$ contains no real curves.
\end{proposition}
\begin{proof}
One has
$$\operatorname{disc} \tilde{S}_h=\operatorname{disc} S_h\cong\textstyle[-\frac{7}{8}]\oplus[-\frac{6}{7}]\oplus[\frac{2}{3}]\oplus[\frac{1}{2}]\oplus[\frac{1}{2}].$$
(Note that $\operatorname{rk}S_h=19$, hence by Proposition~\ref{rankT=3} it is enough to show there is no reflection). Consider an integer $a^2\mathrel| 2\operatorname{exp} (\operatorname{disc} \tilde{S}_h)= 2^4\cdot3\cdot7$, \emph{i.e.}, $a^2= 2^N\cdot3^r\cdot7^s$, where $N\in\{1,2,3,4\}$, $r\in\{0,1\}$ and $s\in\{0,1\}$. Similar to sections \ref{S1} and \ref{S2}, for a fixed $a^2$, we consider a pair $(n,\kappa)$ (where $n=1,2$) and $\kappa_{[2]}\in \operatorname{disc}_2\tilde {S}_h$, as in Lemma \ref{main.lemma}.
We fix the generators
$$\text{$\alpha_1$ for $\operatorname{disc} \mathbf{A}_7\cong\textstyle[-\frac{7}{8}]$,\quad $\alpha_2$ for $\operatorname{disc}_2 \mathbf{A}_5\cong\textstyle[\frac{1}{2}]$,\quad$\alpha_3$ for $\operatorname{disc} \mathbb{Z}h\cong\textstyle[\frac{1}{2}]$},$$
for $\operatorname{disc}_2 \tilde{S}_h$. By Remark~\ref{sym}, the group $O_{\pm h}(\tilde{S}_h)$ acts on $\operatorname{disc} \tilde{S}_h$ via the involutions $\alpha_i\mapsto\pm\alpha_i$ with $i=1,2,3$, as in Observation~\ref{observation123}.
\subsubsection{The case $N=1$, $n=1$ }\label{sub11}
As in section~\ref{N2n1}, we rule out the cases $a^2=2\cdot3,2\cdot7,2\cdot3\cdot7$ and obtain $a^2=2$. However, there is no element $\kappa_{[2]}\in \operatorname{disc}_2\tilde{S}_h$ such that
$$
\text{$\operatorname{order }(\kappa_{[2]})=2$ and $\kappa_{[2]}^2=-\frac{1}{2}$},
$$
eliminating this case.
\subsubsection{The case $N=1$, $n=2$ is ruled out as in section~\ref{subN1} }
\subsubsection{The case $N=2$}
Then $n=2$ (as the group $\operatorname{disc}_2\tilde{S}_h$ does not contain any cyclic direct summand of order $4$).
As in section~\ref{N2n1}, we have $a^2=4\delta$, where $\delta=1,3$ or $3\cdot7$. We search for $\kappa_{[2]}\in \operatorname{disc}_2\tilde{ S}_h$ with
$$
\text{$\operatorname{order }(\kappa_{[2]})=2$ and $\kappa_{[2]}^2=1$}.
$$
According to \ref{N1.1} and the respective item in section \ref{description.KperbmodK}, $\mathcal{K}^{\bot}/\mathcal{K}$ has a summand $\mathbb{Z}/4$ generated by $\alpha_0+$(element of order $2$) which is reversed by $\varphi\oplus t_a$, contrary to Lemma~\ref{main.lemma}(3), and hence to Corollary~\ref{corlem2}.
\subsubsection{The case $N=3$ and $n=1$}\label{N3n1(3)}
As in section~\ref{N2n1}, we have $a^2=8$ (\emph{cf}. section~\ref{sub11}). Then, there is no $\kappa_{[2]}\in \operatorname{disc} \tilde{S}_h$ with
$\operatorname{order }(\kappa_{[2]})=8$ and $\kappa_{[2]}^2=-\frac{1}{8}$.
\subsubsection{The case $N=3$ and $n=2$}
By section~\ref{N3n1(3)}, we have $a^2=8$. Then, listing all elements $\kappa_{[2]}\in \operatorname{disc} \tilde{S}_h$ with
$$
\text{$\operatorname{order }(\kappa_{[2]})=4$ and $\kappa_{[2]}^2=-\frac{1}{2}$},
$$
we obtain $\kappa_{[2]}+2\alpha_0=2\alpha_0\pm 2\alpha_1+\alpha_2+\alpha_3$. Arguing as in Observation~\ref{observation123}, we conclude that an involution $\varphi$ with $\varphi(\kappa)=-\kappa$ must send each generator $\alpha_i$, $i=1,2,3$ to $-\alpha_i$, thus inducing $-\operatorname{id}$ on $\mathcal{K}^{\bot}/\mathcal{K}$. On the other hand, one of the vectors $3\alpha_0\pm\alpha_1+\alpha_3$ is in $\mathcal{K}^{\bot}$ and has order $8$ in $\mathcal{K}^{\bot}/\mathcal{K}$. This contradicts to Lemma~\ref{main.lemma}(3), and hence to Corollary~\ref{corlem2}.
\subsubsection{The case $N=4$ is ruled out as in section~\ref{sub4}}
\end{proof}
\end{document} | math | 91,996 |
\begin{document}
\begin{abstract}
Relative trace formulas play a central role in studying automorphic forms.
In this paper, we use a relative trace formula approach to derive a Kuznetsov type formula for the group $\mathbf mathfrak{G}Sp_4$.
We focus on giving a final formula that is as explicit as possible, and we plan on returning to applications elsewhere.
{\bf{e}}nd{abstract}
\mathbf maketitle
\tableofcontents
\mathbf{s}ection{Introduction}
In this work we develop a Kuznetsov formula for the group $\mathbf mathfrak{G}Sp_4$.
To motivate our results, we first recall the Kuznetsov formula for $\mathbf mathfrak{G}L_2$, an identity relating spectral information about the quotient space
$\mathbf mathfrak{G}amma \backslash \mathbf mathbb H$ (where $\mathbf mathfrak{G}amma$ is a congruence subgroup) to some arithmetic input.
For arbitrarily chosen nonzero integers $n$ and $m$ and any reasonable test function $h$, the spectral side involves
the quantity
\begin{equation}\label{BasicQuantity}
h(t_u)a_m(u)\mathfrak{o}verline{a_n(u)},
{\bf{e}}nd{equation}
where $u$ ranges over eigenfunctions of the Laplace operator involved in the
spectral decomposition of $L^2(\mathbf mathfrak{G}amma \backslash \mathbf mathbb H)$, $a_m(u)$ is the $m$-th Fourier coefficient of $u$,
and $t_u$ is the corresponding spectral parameter.
More precisely, the spectrum of $L^2(\mathbf mathfrak{G}amma \backslash \mathbf mathbb H)$ can be described as the direct sum of the
\textit{discrete spectrum} and the \textit{continuous spectrum}.
The discrete spectrum is the direct sum of $1$-dimensional subspaces spanned by cuspidal Maa{\mathbf{s}s} forms
(the \textit{cuspidal spectrum}) plus the constant function (the \textit{residual spectrum}).
The continuous spectrum is a direct integral of $1$-dimensional subspaces spanned by the Eisenstein series.
The spectral side of the Kuznetsov formula correspondingly splits as a discrete sum over Maa{\mathbf{s}s} forms plus a
continuous integral over Eisenstein series.
The arithmetic-geometric side is a sum of two contributions, that may be seen as the contributions from the
two elements of the Weyl group of $\mathbf mathfrak{G}L_2$.
The identity contribution is given by the delta symbol $\delta_{n,m}$ times the integral of the spectral test function
$h$ against the spectral measure $\bm{f}rac{t}{\mathfrak{p}i^2} \tanh(\mathfrak{p}i t) dt$.
For this reason, the Kuznetsov formula may be viewed as a result of quasi-orthogonality for the Fourier coefficients $a_m(\cdot)$
and $a_n(\cdot)$, provided the remaining contribution can be controlled.
The latter consists of a sum of Kloosterman sums weighted by some integral transform of the test function $h$, involving Bessel
functions.
Applications of the Kuznetsov formula involve using known results about any of the two sides to derive information
about the other side.
On one hand, the flexibility allowed by the choice of the test function $h$ enables one to use known bounds about the Kloosterman
sums to study the distribution of the discrete spectrum and the size of the Fourier coefficients of Maa{\mathbf{s}s} forms.
On the other hand, understanding the Fourier coefficients of Maa{\mathbf{s}s} forms as well as the integral transform appearing on
the geometric side yields strong bounds for sums of Kloosterman sums.
Recently, Kuznetsov formulae for $\mathbf mathfrak{G}L_3$ have been developed by Blomer and Buttcane, with similar applications as described above.
It would thus be interesting to establish the corresponding formulae for other groups.
In the classical proof of the $\mathbf mathfrak{G}L_2$ Kuznetsov formula, one computes the inner product of two Poincar\'e series in two
different ways, one involving the spectral decomposition of $L^2(\mathbf mathfrak{G}amma \backslash \mathbf mathbb H)$, and the other one by computing
the Fourier coefficients of the Poincar\'e series and unfolding. This gives a ``pre-Kuznetsov formula", that one then
proceeds to integrate against the test function $h$, obtaining on the geometric side the integral transforms of $h$
mentioned above.
Another approach, which one may call the relative trace formula approach to the Kuznetsov formula, builds upon the relative trace formula introduced by
Jacquet~\cite{JacquetLai}.
In the case of $\mathbf mathfrak{G}L_2$, the relative trace formula approach to the Kuznetsov formula is apparently based on unpublished work of Zagier,
detailed in~\cite{Joyner}.
This approach is developed in the adelic framework in~\cite{KL}for the congruence subgroup $\mathbf mathfrak{G}amma=\mathbf mathfrak{G}amma_1(N)$.
It proceeds by integrating an automorphic kernel
$$K_f(x,y)=\mathbf{s}um_{\gamma \in \mathcal PGL_2(\mathbb{Q})} f(x^{-1}\gamma y),$$ where $f$ is a suitable test function.
The spectral expansion of the kernel will then involve the quantity $\tilde{f}(t_u)u(x)\mathfrak{o}verline{u(y)}$, where $u$ ranges
over the eigenfunctions involved in the spectral decomposition of $L^2(\mathbf mathfrak{G}amma(N) \backslash \mathbf mathbb H)$,
$t_u$ is the spectral parameter of $u$, and $\tilde{f}$ is the \textit{spherical transform} of $f$.
Thus integrating $K_f(x,y)$ against a suitable character on $U \times U$, where $U=\mathbf mat{1}{*}{}{1}$, one gets the
quantity~(\ref{BasicQuantity}) with $h=\tilde{f}$.
On the other hand, using the Bruhat decomposition for $\mathcal PGL_2(\mathbb{Q})$, one can decompose the integral over $U \times U$ as
a sum over elements of the Weyl group and over diagonal matrices in $\mathcal PGL_2(\mathbb{Q})$ of some orbital integrals.
In the case of the identity element, at most one diagonal matrix will have a non-zero contribution, which will turn out
to be a delta symbol times some integral transform of the function $f$. In the case of the longest element in the Weyl
group, each positive integer in $N\mathbb{Z}$ will have a nonzero contribution, given by a Kloosterman sum times a second kind of
integral transform of $f$. A more refined version is then obtained by taking the Mellin transform of the primitive formula
obtained.
Note that in this approach, one gets on the geometric side some integral transforms of the function $f$, hence one has to do
some extra work to relate these to the test function $h=\tilde{f}$ appearing in the spectral side.
A couple of remarks are in order about the choice of $f$. Firstly, the spectral expansion of the kernel involves the spectral
decomposition of $L^2(\mathbb{R}_{>0}\mathbf mathfrak{G}L_2(\mathbb{Q}) \backslash \mathbf mathfrak{G}L_2(\mathbf mathbb{A}))$ rather than $L^2(\mathbf mathfrak{G}amma \backslash \mathbf mathbb H)$.
By restricting $f$ to be left and right $K_\infty$-invariant
(where $K_\infty=\text{SO}_2$), only right-$K_\infty$-invariant automorphic forms $\mathfrak{p}hi$
(thus corresponding to adelization of functions on the homogeneous space $\mathbf mathbb H = SL_2(\mathbb{R}) / K_\infty$)
will show up in the spectral expansion of the kernel, but other choices are possible.
Also one may choose the test function $f$ at unramified places so as to get a final formula that include the Hecke eigenvalues
of a fixed Hecke operator of index coprime to the level $N$.
Our plan is to implement the relative trace formula approach in the case of $\mathbf mathfrak{G}Sp_4$.
In contrast to the case of $\mathbf mathfrak{G}L_2$, there is more than one non-conjugate unipotent subgroups~$U$.
Choosing $U$ to be the unipotent radical of the Borel subgroup (that is the minimal parabolic subgroup)
will yield Whittaker coefficients of the automorphic forms involved (instead of the Fourier coefficients).
The Whittaker coefficients have a ``multiplicity one" property, which ensures that the global Whittaker coefficients factor
into a product of local coefficients.
These local Whittaker coefficients can be written down in terms of local Satake parameters, which is important for applications.
Also in contrast to the case of $\mathbf mathfrak{G}L_2$, not every automorphic form has non-identically zero Whittaker coefficients.
For instance, Siegel modular form give rise to automorphic forms whose Whittaker coefficients vanish identically.
Thus, only \textit{generic} automorphic forms (\textit{i.e,} with non-identically zero Whittaker coefficients)
will survive the integration on $U \times U$ and contribute to the final formula.
Let us briefly sketch some similarities and differences with the Kuznetsov formula for $\mathbf mathfrak{G}L_3$, which also has
rank $2$.
On the spectral side, the continuous contribution is in both cases given on the one hand by \textit{minimal Eisenstein series},
(that is, attached to the minimal parabolic subgroup), and on the other hand by Eisenstein series induced from non-minimal parabolics by Maa{\mathbf{s}s} forms
on $\mathbf mathfrak{G}L_2$. However, in the case of $\mathbf mathfrak{G}L_3$, the two non-minimal proper standard parabolic subgroup are \textit{associated},
hence by Langlands theory their Eisenstein series are essentially the same. On the other hand, for $\mathbf mathfrak{G}Sp_4$, we have
two distinct non-associated such parabolic subgroups, giving rise to two distinct kinds of Eisenstein series.
As for as the geometric side, the Weyl group of $\mathbf mathfrak{G}L_3$ has six elements, while the Weyl goup of $\mathbf mathfrak{G}Sp_4$ has eight.
However, it seems interesting to notice that in both case, only the identity element and the longest three elements
in the Weyl group have a non-zero contribution, thus eventually giving in total four distinct terms.
Finally, let us mention that Siu Hang Man has independently derived a Kuznetsov formula for $\text{Sp}_4$ using the more classical technique
of computing the inner product of Poincar\'e series, and has derived some applications towards the Ramanujan Conjecture~\cite{SHMKuznetsov}.
However, because the techniques employed and the final formulae differ, the author believes that our works are complementary rather than redundant.
Indeed, the flexibility offered by the adelic framework enables us to treat the test function differently at each place.
As a result, by choosing an appropriate test function at finite places, our formula might incorporate the eigenvalues of an arbitrary Hecke operator.
Furthermore, at the Archimedean place, we make use of two deep theorems of functional analysis on real reductive groups (namely Harish-Chandra inversion theorem
and Wallach's Whittaker inversion theorem) in order to produce an arbitrary Paley-Wiener test function on the spectral side, and relate it explicitly to its transform
appearing on the arithmetic side.
As a last point, working with $\mathbf mathfrak{G}Sp_4$ instead of $\text{Sp}_4$ enables us to work with a central character.
The primary goal of this work is to write down the main Kuznetsov formula (in Theorem~\ref{MainTheorem}) as explicitly as possible.
Applications of Kuznetsov formulae classically include counts of the cuspidal spectrum,
bounds towards the Ramanujan conjecture and bounds for the number of automorphic forms violating the Ramanujan conjecture,
large sieve inequalities involving the Hecke eigenvalues,
estimates for moments of $L$-functions,
and distribution of the Whittaker coefficients.
Some of these have been developed in~\cite{SHMKuznetsov}, and we hope to work out further applications in future work.
\mathbf mathfrak{a}ddtocontents{toc}{\mathfrak{p}rotect\mathbf{s}etcounter{tocdepth}{1}}
\mathbf{s}ubsection*{Acknowledgments} I wish to thanks Abhishek Saha for suggesting me this problem and for helpful comments,
Ralf Schmidt for communicating a proof of Proposition~\ref{MustBePrincipalSeries} to me,
and Jack Buttcane for being in touch about the interchange of integrals conjecture.
\mathbf mathfrak{a}ddtocontents{toc}{\mathfrak{p}rotect\mathbf{s}etcounter{tocdepth}{2}}
\mathbf{s}ection{Generalities}
\begin{definition}\label{classical}
The general symplectic group of degree 2 over a field $\mathbf mathbb{F}$ is the group
$$\mathbf mathfrak{G}Sp_4(\mathbf mathbb{F})=\{M \in \mathbf mathcal Mat_4(\mathbf mathbb{F}) : {\bf{e}}xists \mathbf mu \in \mathbf mathbb{F}^\times , \trans{M}JM=\mathbf mu J\},$$
where
$J=\mathbf mat{}{I_2}{-I_2}{}$ and $\trans{M}$ denotes the transpose matrix of $M$.
{\bf{e}}nd{definition}
Note that some authors use different realizations of $\mathbf mathfrak{G}Sp_4$, for instance the realization used in~\cite{RS2}
(to which we refer, along with~\cite{RS1}, for expository details) is conjugated
in $\mathbf mathfrak{G}L_4$ to ours by the matrix $\mathbf mat{\block{}{1}{1}{}}{}{}{\block{1}{}{}{1}}$.
From now on we denote $G=\mathbf mathfrak{G}Sp$.
The {\bf Cartan involution} of $G$ is given by $\theta(M)=J^{-1}MJ=\trans{M^{-1}}$.
The scalar $\mathbf mu=\mathbf mu(M)$ in the definition is called the {\bf multiplier system}.
The centre of $G$ consists in all the invertible scalar matrices.
We fix a maximal torus in $G(\mathbf mathbb{F})$
$$T(\mathbf mathbb{F})=\left\{\mathbf mat{\block{x}{}{}{y}}{}{}{\block{tx^{-1}}{}{}{ty^{-1}}} : x,y,t \in \mathbf mathbb{F}^\times \right\}.$$
\begin{definition}
The symplectic group of degree 2 over a field $\mathbf mathbb{F}$ is the group
$$\text{Sp}_4(\mathbf mathbb{F})=\{M \in G(\mathbf mathbb{F}) : \mathbf mu(M)=1\}.$$
{\bf{e}}nd{definition}
The centre of $\text{Sp}_4$ is $\{\mathfrak{p}m 1\}$, and a maximal torus in $\text{Sp}_4(\mathbf mathbb{F})$ is given by
$$A(\mathbf mathbb{F})=T(\mathbf mathbb{F}) \cap \text{Sp}_4(\mathbf mathbb{F}) =\left\{\mathbf mat{\block{x}{}{}{y}}{}{}{\block{x^{-1}}{}{}{y^{-1}}} : x,y \in \mathbf mathbb{F}^\times \right\}.$$
\mathbf{s}ubsection{Weyl group}
Let $N(T)$ be the normalizer of $T$.
The Weyl group $\text{O}mega=N(T)/T$ is generated by (the images of)
$s_1=\mathbf mat{\block{}{1}{1}{}}{}{}{\block{}{1}{1}{}}$ and
$s_2=\left[ \begin{smallmatrix}&&1&\\ &1&&\\-1&&&\\&&&1{\bf{e}}nd{smallmatrix}\right]$, and consists of the (images of the) eight elements
$$1, s_1, s_2,
s_1s_2=\left[ \begin{smallmatrix}&1&&\\ &&1&\\&&&1\\-1&&&{\bf{e}}nd{smallmatrix}\right],
s_2s_1=\left[ \begin{smallmatrix}&&&1\\mathbf mathbbm{1}&&&\\&-1&&\\&&1&{\bf{e}}nd{smallmatrix}\right],$$
$$s_1s_2s_1=\left[ \begin{smallmatrix}1&&&\\ &&&1\\&&1&\\&-1&&{\bf{e}}nd{smallmatrix}\right],
s_2s_1s_2=\mathbf mat{}{\block{}{1}{1}{}}{\block{}{-1}{-1}{}}{},(s_1s_2)^2=J.$$
\mathbf{s}ubsection{Compact subgroups}
A choice of maximal compact subgroup of $G(\mathbb{R})$ is given by the set $K_0$ of fixed points of the Cartan involution $\theta$.
An easy computation shows
$$K_0=K_\infty \mathbf{s}qcup \diag{1}{1}{-1}{-1}K_\infty,$$
where
$$K_\infty = \left\{ \mathbf mat{A}{B}{-B}{A}:
A\trans{A}+B\trans{B}=1,
A\trans{B}=B\trans{A}. \right\}.$$
The condition
$$\begin{cases}
A\trans{A}+B\trans{B}=1\\
A\trans{B}=B\trans{A}{\bf{e}}nd{cases}$$ is equivalent to $A+iB \in U(2)$, hence $K_\infty$ is isomorphic to $U(2)$.
For each prime $p$ we also consider a (compact open) congruence subgroup $\mathbf mathfrak{G}amma_p \mathbf{s}ubset G(\mathbb{Z}_p)$,
with the properties that $\mathbf mathfrak{G}amma_p=G(\mathbb{Z}_p)$ for all but finitely many $p$ and the multiplier system $\mathbf mu$ is
surjective from $\mathbf mathfrak{G}amma_p$ to $\mathbb{Z}_p^\times$ for all $p$.
This implies we have the
{\bf strong approximation}: setting $\mathbf mathfrak{G}amma=K_\infty \mathfrak{p}rod_p \mathbf mathfrak{G}amma_p$, we have
$$G(\mathbf mathbb{A})=G(\mathbb{R})^\circ G(\mathbb{Q}) \mathbf mathfrak{G}amma,$$
where $G(\mathbb{R})^\circ$ is the connected component of the identity and $\mathbf mathbb{A}$ is the ring of ad\`eles of $\mathbb{Q}$.
Moreover we have the {\bf Iwasawa decomposition}
$G(\mathbf mathbb{A})=P(\mathbf mathbb{A})K$ for all standard parabolic subgroup $P$, where $K=K_\infty \times \mathfrak{p}rod_p G(\mathbb{Z}_p)$.
\mathbf{s}ubsection{Parabolic subgroups}
Parabolic subgroups are subgroups such that $G/P$ is a projective variety.
Given a minimal parabolic subgroup $P_0$,
{\bf standard parabolic subgroups} (with respect to $P_0$) are those parabolic subgroups that contain $P_0$.
If $P$ is a standard parabolic subgroup defined over $\mathbb{Q}$,
the {\bf Levi decomposition} of $P$ is a semidirect product $P=N_PM_P$ where $M_P$ is a reductive subgroup and
$N_P$ is a normal unipotent subgroup.
We give here the three non-trivial standard (with respect to our choice of $P_0=B$) parabolic subroups and their Levi decomposition.
\mathbf{s}ubsubsection{Borel subgroup} \label{Borelsbgp}
The {\bf Borel subgroup} is the minimal standard parabolic subgroup.
It is given by
$$B=\left[ \begin{smallmatrix}*&&*&*\\ *&*&*&*\\&&*&*\\&&&*{\bf{e}}nd{smallmatrix}\right] \cap \mathbf mathfrak{G}Sp_4$$
and has Levi decomposition $B=U T=TU$, where
$$U=\left\{\bigmat{\block{1}{}{x}{1}}{\block{c}{a-cx}{a}{b}}{}{\block{1}{-x}{}{1}},a,b,c,x \in \mathbf mathbb{F}\right\}.$$
We have the {\bf Bruhat decomposition}
\begin{align*}
G=\coprod_{\mathbf{s}igma \in \text{O}mega} B \mathbf{s}igma B=\coprod_{\mathbf{s}igma \in \text{O}mega} UT \mathbf{s}igma U.
{\bf{e}}nd{align*}
We write the Iwasawa decomposition for $\text{Sp}_4(\mathbb{R})$ as follows.
\begin{definition}
For every $g \in \text{Sp}_4(\mathbb{R})$ there is a unique element $A(g) \in \mathbf mathcal Lie{a}$, such that
$$g \in U {\bf{e}}xp(A(g))K_\infty.$$
{\bf{e}}nd{definition}
Later on we shall need the following technical lemma.
\begin{lemma}\label{A(Ju)}
With the notation $$u(x,a,b,c)=\left[
\begin{smallmatrix}
1 & & c & a-cx \\
x & 1 & a & b \\
& & 1 & -x \\
& & & 1 \\
{\bf{e}}nd{smallmatrix}
\right]$$
we have for all $a,b,c,x \in \mathbb{R}$
$$A(Ju(x,a,b,c))=A(Ju(-x,a,-b,-c)).$$
{\bf{e}}nd{lemma}
\begin{proof}
To alleviate notations, set $u=u(x,a,b,c)$.
By definition we have $u=u_1A(Ju)k$ for some $u_1 \in U(\mathbb{R})$ and $\trans{k}=k^{-1}$.
Then $$ u \trans{u} =J\trans{u}^{-1}u^{-1}J^{-1}\\
=\trans{u_1}^{-1}{\bf{e}}xp(-2A(Ju))u_1^{-1}.$$
The matrices $u_1$ and $A(Ju)$ can then be recovered from the entries of $u \trans{u}$ by an elementary calculation (see Lemma~\ref{uoppu}) below).
In particular, we find $$A(Ju)=-\bm{f}rac12\diag{a_1-a_2}{a_2}{a_2-a_1}{-a_2},$$ where
$a_1 =\log(1+x^2+a^2+b^2)$ and $a_2=\log((a(a-cx)+1-bc)^2+(x(a-cx)-b-c)^2)$.
{\bf{e}}nd{proof}
\mathbf{s}ubsubsection{Klingen subgroup}
The {\bf Klingen subgroup} is
$$P_{\text{K}}=\left[ \begin{smallmatrix}*&&*&*\\ *&*&*&*\\ *&&*&*\\&&&*{\bf{e}}nd{smallmatrix}\right] \cap \mathbf mathfrak{G}Sp_4.$$
It has Levi decomposition $P_{\text{K}}=N_{\text{K}}\mathbf mathcal Mk$, where
\begin{align*}
N_{\text{K}}=\left\{\mathbf mat{\block{1}{}{x}{1}}{\block{}{y}{y}{z}}{}{\block{1}{-x}{}{1}},x,y,z \in \mathbf mathbb{F}\right\}
\text{ and }
\mathbf mathcal Mk=\left\{\left[ \begin{smallmatrix}a&&b&\\ &t&&\\ c&&d&\\&&&t^{-1}\delta{\bf{e}}nd{smallmatrix}\right], t \in \mathbf mathbb{F}^\times,
\delta = \det\left(\mathbf mat{a}{b}{c}{d}\right) \texttt{n}eq 0\right\}.
{\bf{e}}nd{align*}
The centre of $\mathbf mathcal Mk$ is
$\mathbf mathbb{A}k=\left\{\left[ \begin{smallmatrix}u&&&\\&t&&\\&&u&\\&&&t^{-1}u^2{\bf{e}}nd{smallmatrix}\right], t,u
\in \mathbf mathbb{F}^\times \right\}.$
\mathbf{s}ubsubsection{Siegel subgroup}
The {\bf Siegel subgroup} is
$$P_{\text{S}}=\left[ \begin{smallmatrix}*&*&*&*\\ *&*&*&*\\&&*&*\\&&*&*{\bf{e}}nd{smallmatrix}\right] \cap \mathbf mathfrak{G}Sp_4.$$
It has Levi decomposition $P_{\text{S}}=N_{\text{S}}\mathbf mathcal Ms$, where
$$N_{\text{S}}=\left\{\mathbf mat{\block{1}{}{}{1}}{\block{x}{y}{y}{z}}{}{\block{1}{}{}{1}},x,y,z \in \mathbf mathbb{F}\right\}
\text{ and }
\mathbf mathcal Ms=\left\{ \mathbf mat{A}{}{}{t\trans{A^{-1}}}, A \in \mathbf mathfrak{G}L_2(\mathbf mathbb{F}), t \in \mathbf mathbb{F}^\times \right\}.$$
The centre of $\mathbf mathcal Ms$ is
$\mathbf mathbb{A}s=\left\{\left[ \begin{smallmatrix}u&&&\\&u&&\\&&tu^{-1}&\\&&&tu^{-1}{\bf{e}}nd{smallmatrix}\right], t,u \in \mathbf mathbb{F}^\times \right\}.$
\mathbf{s}ubsection{Lie algebras and characters}
Following Arthur~\cite{ArthurIntro}, we parametrize the characters of the Levi component of the parabolic subgroups by the dual of the Lie algebras of their centre.
We fix $|\cdot|_\mathbf mathbb{A}=\mathfrak{p}rod_v|\cdot|_v$ the {\bf standard adelic absolute value}.
Let $P=M_PN_P$ be a standard parabolic subgroup, and $A_P$ be the centre of $M_P$.
Then there is a surjective homomorphism
$$H_P: M_P(\mathbf mathbb{A}) \to \text{Hom}_\mathbb{Z}(X(M_P),\mathbb{R})$$ defined by
\begin{equation}\label{DefinitionOfHp}
\left(H_P(m)\right)(\chi)=\log(|\chi(m)|_\mathbf mathbb{A}),
{\bf{e}}nd{equation}
where we write $X(H)$ for the group of homomorphisms (of \textit{algebraic groups}) $H \to \mathbf mathfrak{G}L_1$ that are defined over $\mathbb{Q}$.
On the other hand, we may identify the vector space $\text{Hom}_\mathbb{Z}(X(M_P),\mathbb{R})$ with the Lie algebra
$\mathbf mathcal Lie{a}_P \mathfrak{o}plus \mathbf mathcal Lie{z}$ of $A_P(\mathbb{R})$ (where $\mathbf mathcal Lie{a}_P$ is the Lie algebra of $A_P(\mathbb{R}) \cap \text{Sp}_4(\mathbb{R})$ and $\mathbf mathcal Lie{z}$ is the Lie algebra of the centre).
If $\texttt{n}u \in \mathbf mathcal Lie{a}_P^*(\mathbf mathbb{C}) \mathfrak{o}plus \mathbf mathcal Lie{z}^*(\mathbf mathbb{C})$, then the map
\begin{equation}\label{DefOfChar}
m \mathbf mapsto {\bf{e}}xp(\texttt{n}u(H_P(m))
{\bf{e}}nd{equation}
defines a character of $M_P(\mathbf mathbb{A})$. Moreover characters of $Z(\mathbf mathbb{A})$
correspond to $\mathbf mathcal Lie{z}^*(\mathbf mathbb{C})$ while characters that are trivial on $Z(\mathbf mathbb{A})$ correspond to $\mathbf mathcal Lie{a}_P^*(\mathbf mathbb{C})$.
\mathbf{s}ection{Representations}
\mathbf{s}ubsection{Generic representations}
\mathbf{s}ubsubsection{Generic characters} \label{GenericCharacters}
A character $\mathfrak{p}si$ of $U(\mathbb{Q}) \backslash U(\mathbf mathbb{A})$ is said to be {\bf generic} if its differential is non-trivial
on each of the eigenspaces $\mathbf mathcal Lie{n}_\mathbf mathfrak{a}lpha$ corresponding to simple roots $\mathbf mathfrak{a}lpha$.
Explicitly, If $\theta$ is the standard additive character of $\mathbf mathbb{A}/\mathbb{Q}$ and $\mathbf m=(m_1, m_2) \in (\mathbb{Q}^\times)^2$,
generic characters of $U(\mathbf mathbb{A})$ are given by
\begin{equation}\label{genchar}
\mathfrak{p}si_\mathbf m\left(\left[\begin{smallmatrix}
1 & & c & a-cx \\
x & 1 & a & b \\
& & 1 & -x \\
& & & 1 \\
{\bf{e}}nd{smallmatrix}\right]\right)=\theta(m_1x+m_2c).
{\bf{e}}nd{equation}
Note that all generic characters may be obtained from each other by conjugation by an element of $T/Z$,
as we have for all $u \in U(\mathbf mathbb{A})$
\begin{equation}\label{gencharconj}
\mathfrak{p}si_\mathbf m\left(u \right)=\mathfrak{p}si_{\mathbf mathbf{1}}\left( t_\mathbf m^{-1} u t_\mathbf m\right),
{\bf{e}}nd{equation}
where
\begin{equation}\label{tm}
t_\mathbf m=\diag{m_1}{1}{m_1m_2}{m_1^2m_2}.
{\bf{e}}nd{equation}
\mathbf{s}ubsubsection{Whittaker coefficients and generic representations}
If $\mathfrak{p}hi$ is any automorphic form on $G(\mathbf mathbb{A})$ and $\mathfrak{p}si$ a generic character, the {\bf $\mathfrak{p}si$-Whittaker coefficient}
of $\mathfrak{p}hi$ is by definition
\begin{equation} \label{Whittaker}
\mathcal{W}_{\mathfrak{p}si}(\mathfrak{p}hi) (g) = \int_{U(\mathbb{Q}) \backslashslash U(\mathbf mathbb{A})} \mathfrak{p}hi(ug) \mathfrak{p}si(u)^{-1} du.
{\bf{e}}nd{equation}
$\mathfrak{p}hi$ is called $\mathfrak{p}si$-generic if $\mathcal{W}_\mathfrak{p}si$ is not identically zero as a function of $g$.
Changing variable and using the left-$G(\mathbb{Q})$-invariance of $\mathfrak{p}hi$, note that we have
$$\mathcal{W}_{\mathfrak{p}si_{\mathbf m}}(\mathfrak{p}hi) (g)=\bm{f}rac1{|m_1^4m_2^3|}\mathcal{W}_{\mathfrak{p}si_{\mathbf mathbf 1}}(\mathfrak{p}hi) (t_{\mathbf m}^{-1}g)$$
In particular, $\mathfrak{p}hi$ is $\mathfrak{p}si$-generic for some generic character $\mathfrak{p}si$ if and only if it is $\mathfrak{p}si$-generic for any
generic character $\mathfrak{p}si$, henceforth we shall just say $\mathfrak{p}hi$ is {\bf generic}.
An irreducible automorphic representation $(\mathfrak{p}i, V_\mathfrak{p}i)$ is called generic if $V_\mathfrak{p}i$ contains a generic automorphic form
$\mathfrak{p}hi$.
Equivalently, every automorphic form in the space of a generic irreducible automorphic representation $\mathfrak{p}i$ is generic,
since otherwise the kernel of the map $\mathfrak{p}hi \mathbf mapsto W_\mathfrak{p}si(\mathfrak{p}hi)$ would be lead to a non-trivial invariant subspace of $\mathfrak{p}i$,
contradicting the irreducibility of $\mathfrak{p}i$.
Since $U$ may as well be viewed as the unipotent part of the minimal parabolic subgroup of $\text{Sp}_4$,
we can define the Whittaker coefficients of automorphic forms $\mathfrak{p}hi$ on $\text{Sp}_4$ in the exact same way as~(\ref{Whittaker}), except the argument is restricted
to $\text{Sp}_4(\mathbf mathbb{A})$. This gives a similar notion of generic automorphic forms and generic representations for $\text{Sp}_4$.
Later on, we shall restrict automorphic forms on $\mathbf mathfrak{G}Sp_4$ to $\text{Sp}_4$.
Let us briefly explain the corresponding operations on automorphic representations.
\begin{definition}\label{DefOfRes}
Let $(\mathfrak{p}i,V_\mathfrak{p}i)$ be an automorphic representation of $\mathbf mathfrak{G}Sp_4(\mathbf mathbb{A})$ realized by right translation on a subspace of $L^2(G(\mathbb{Q})Z(\mathbb{R}) \backslashslash G(\mathbf mathbb{A}))$.
We define a representation $\text{res } \mathfrak{p}i$ of $\text{Sp}_4(\mathbf mathbb{A})$ as the action of $\text{Sp}_4(\mathbf mathbb{A})$ on $\left\{ \mathfrak{p}hi_{|\text{Sp}_4(\mathbf mathbb{A})}: \mathfrak{p}hi \in V_\mathfrak{p}i\right\}$.
It is a quotient of the restriction $\mathbb{R}es \mathfrak{p}i=\mathfrak{p}i_{|\text{Sp}_4(\mathbf mathbb{A})}$.
{\bf{e}}nd{definition}
The representation $\text{res } \mathfrak{p}i$ does not have finite length in general. However, the following shall be useful later on.
\begin{lemma}\label{genericrestriction}
Let $\mathfrak{p}i$ be an irreducible automorphic representation $\mathfrak{p}i$ of $G(\mathbf mathbb{A})$ that occurs discretely in $L^2(G(\mathbb{Q})Z(\mathbb{R}) \backslashslash G(\mathbf mathbb{A}))$.
Then $\mathfrak{p}i$ is generic if and only if $\text{res } \mathfrak{p}i$ has a generic constituent.
{\bf{e}}nd{lemma}
\begin{proof}
Fix a generic character $\mathfrak{p}si$.
Note that for any automorphic form $\mathfrak{p}si$ on $G(\mathbf mathbb{A})$ we have
$\mathcal{W}_{\mathfrak{p}si}(\mathfrak{p}hi_{|\text{Sp}_4})=\left(\mathcal{W}_{\mathfrak{p}si}(\mathfrak{p}hi)\right)_{|\text{Sp}_4}$.
From this, it is clear that if $\text{res } \mathfrak{p}i$ has a generic constituent then $\mathfrak{p}i$ is generic.
Let us show the converse.
Assume no constituent of $\text{res } \mathfrak{p}i$ is generic, so for all $\mathfrak{p}hi \in V_\mathfrak{p}i$,
$$\mathcal{W}_\mathfrak{p}si(\mathfrak{p}hi)_{|\text{Sp}_4(\mathbf mathbb{A})}=0.$$
Let $\mathfrak{p}hi \in \mathfrak{p}i$ and $g \in G(\mathbf mathbb{A})$. Then $\mathfrak{p}i(g)\mathfrak{p}hi\in V_\mathfrak{p}i$ hence
$$\mathcal{W}_{\mathfrak{p}si}(\mathfrak{p}hi)(g)=\mathcal{W}_{\mathfrak{p}si}(\mathfrak{p}i(g)\mathfrak{p}hi)(1)=0.$$
Thus $\mathfrak{p}i$ is not generic.
{\bf{e}}nd{proof}
We now prove a similar lemma for the restriction of non-cuspidal representations.
\begin{lemma}\label{cuspidalrestriction}
Let $\mathfrak{p}i$ be an irreducible automorphic representation $\mathfrak{p}i$ of $G(\mathbf mathbb{A})$ that occurs discretely in $L^2(G(\mathbb{Q})Z(\mathbb{R}) \backslashslash G(\mathbf mathbb{A}))$.
Then $\mathfrak{p}i$ is non-cuspidal if and only if $\text{res } \mathfrak{p}i$ has no cuspidal constituent.
{\bf{e}}nd{lemma}
\begin{proof}
Recall $\mathfrak{p}i$ is cuspidal if the constant term
$$C_\mathfrak{p}hi(g)=\int_{U(\mathbb{Q}) \backslash U(\mathbf mathbb{A})} \mathfrak{p}hi(ug)du$$
of some (equivalently, any, since $\mathfrak{p}i$ is irreducible) function $\mathfrak{p}hi$ in the space of $\mathfrak{p}i$ vanishes identically.
The exact same proof as Lemma~\ref{genericrestriction}, replacing the generic character $\mathfrak{p}si$ by $1$, shows that $\mathfrak{p}i$ is
non-cuspidal if and only if $\text{res } \mathfrak{p}i$ has a non-cuspidal component.
However, we want to show that if $\mathfrak{p}i$ is non-cuspidal, then $\text{res } \mathfrak{p}i$ has no cuspidal component.
So suppose that $\text{res } \mathfrak{p}i$ has a cuspidal component.
This means there is $\mathfrak{p}hi \in V_\mathfrak{p}i$ such that $(C_\mathfrak{p}hi)_{|\text{Sp}_4(\mathbf mathbb{A})}=0$. We want to show that $C_\mathfrak{p}hi$ is identically zero on $\mathbf mathfrak{G}Sp_4(\mathbf mathbb{A})$.
Now changing variables and using the left-invariance of $\mathfrak{p}hi$ under $\mathbf mathfrak{G}Sp_4(\mathbb{Q})$, if $t \in T(\mathbb{Q})$ then we have $C_\mathfrak{p}hi(tg)=C_\mathfrak{p}hi(g)$.
In addition, if $z\in Z(\mathbf mathbb{A})$ then $C_\mathfrak{p}hi(zg)=\mathfrak{o}mega_\mathfrak{p}i(z)C_\mathfrak{p}hi(g)$. Moreover, since $\mathfrak{p}i$ is an admissible representation, $\mathfrak{p}hi$~is right-invariant by $\mathbf mathfrak{G}Sp_4(\mathbb{Z}_p)$ for almost all prime $p$.
It follows that there exists a finite set of places $S$ such that for any $g \in \mathbf mathfrak{G}Sp_4(\mathbf mathbb{A})$,
if $\mathbf mu(g) \in \mathbb{Q}^\times (\mathbf mathbb{A}^\times)^2 \mathfrak{p}rod_{p \texttt{n}ot \in S}\mathbb{Z}_p^\times$ then $C_\mathfrak{p}hi(g)=0$.
The following lemma concludes the proof.
{\bf{e}}nd{proof}
\begin{lemma}
Let $S$ be any finite set of places containing $\infty$.
We have $\mathbb{Q}^\times (\mathbf mathbb{A}^\times)^2 \mathfrak{p}rod_{p \texttt{n}ot \in S} \mathbb{Z}_p^\times=\mathbf mathbb{A}^\times$.
{\bf{e}}nd{lemma}
\begin{proof}
Let $x \in \mathbf mathbb{A}^\times$.
By strong approximation, we have $x=qu$, with $q \in \mathbb{Q}^\times$ and $u \in \mathbb{R}_{>0}\mathfrak{p}rod_{p < \infty} \mathbb{Z}_p^\times$.
Now by the Chinese Remainders Theorem, there exists an integer $n>0$ such that for all finite $p \in S$, we have $n u_p \in (\mathbb{Z}_p^\times)^2$.
For all $p \texttt{n}ot \in S$, let ${\bf{e}}psilon_p \in \mathbb{Z}_p^\times$ such that ${\bf{e}}psilon_p n u_p \in (\mathbb{Z}_p^\times)^2$.
Define ${\bf{e}}psilon_p=1$ for $p \in S$.
Then $n {\bf{e}}psilon u \in (\mathbf mathbb{A}^\times)^2$, and $x=(qn^{-1})(n {\bf{e}}psilon u)\mathfrak{p}rod_{p \texttt{n}ot \in S}{\bf{e}}psilon_p^{-1}$.
{\bf{e}}nd{proof}
\mathbf{s}ubsection{The basic kernel}\label{basickernel}
The group $Z(\mathbb{Q})Z(\mathbb{R}) \backslashslash Z(\mathbf mathbb{A})$ is compact and acts on the Hilbert space $L^2(G(\mathbb{Q})Z(\mathbb{R}) \backslashslash G(\mathbf mathbb{A}))$ by
right translation.
Since $Z(\mathbb{Q})Z(\mathbb{R}) \backslashslash Z(\mathbf mathbb{A})$ is abelian, its irreducible representations are characters, thus
by Peter-Weyl theorem we have
$$L^2(G(\mathbb{Q})Z(\mathbb{R}) \backslashslash G(\mathbf mathbb{A}))=\bigoplus_{\mathfrak{o}mega}L^2(G(\mathbb{Q})Z(\mathbb{R}) \backslashslash G(\mathbf mathbb{A}),\mathfrak{o}mega),$$
where the orthogonal direct sum ranges characters of $Z(\mathbf mathbb{A})$ that are trivial on $Z(\mathbb{Q})Z(\mathbb{R})$, and
$L^2(G(\mathbb{Q})Z(\mathbb{R}) \backslashslash G(\mathbf mathbb{A}),\mathfrak{o}mega)$ is the subspace of $L^2(G(\mathbb{Q})Z(\mathbb{R}) \backslashslash G(\mathbf mathbb{A}))$
of functions $\mathfrak{p}hi$ satisfying $$\mathfrak{p}hi(gz)=\mathfrak{o}mega(z)\mathfrak{p}hi(g)$$ for all $z \in Z(\mathbf mathbb{A})$.
Fix such a character $\mathfrak{o}mega$. If $f:G(\mathbf mathbb{A}) \to \mathbf mathbb{C}$ is a measurable function that satisfies
\begin{itemize}
\item $f(gz)=\mathfrak{o}verline{\mathfrak{o}mega}(z)f(g)$ for all $z \in Z(\mathbf mathbb{A})$,
\item $f$ is compactly supported modulo $Z(\mathbf mathbb{A})$,
{\bf{e}}nd{itemize}
then we define an operator $R(f)$ on $L^2(G(\mathbb{Q})Z(\mathbb{R}) \backslashslash G(\mathbf mathbb{A}),\mathfrak{o}mega)$ by
$$R(f) \mathfrak{p}hi(x)=\int_{\mathbf mathfrak{G}modZ(\mathbf mathbb{A})}f(y)\mathfrak{p}hi(xy)dy,$$
where $\mathfrak{o}verline{G}$ denotes $G / Z$.
By $G(\mathbb{Q})$-invariance of $\mathfrak{p}hi$, we have
\begin{align*}
R(f) \mathfrak{p}hi(x)=\int_{\mathbf mathfrak{G}modZ(\mathbf mathbb{A})}f(x^{-1}y)\mathfrak{p}hi(y)dy
=\mathbf{s}um_{\gamma \in \mathbf mathfrak{G}modZ(\mathbb{Q})}\int_{\mathbf mathfrak{G}modZ(\mathbb{Q}) \backslashslash \mathbf mathfrak{G}modZ(\mathbf mathbb{A})}f(x^{-1}\gamma y)\mathfrak{p}hi(y)dy
{\bf{e}}nd{align*}
Hence, setting
\begin{equation}\label{TheKernel}
K_f(x,y)=\mathbf{s}um_{\gamma \in \mathbf mathfrak{G}modZ(\mathbb{Q})}f(x^{-1}\gamma y),
{\bf{e}}nd{equation}
we have
\begin{equation}\label{basicequation}
R(f) \mathfrak{p}hi(x)=\int_{\mathbf mathfrak{G}modZ(\mathbb{Q}) \backslashslash \mathbf mathfrak{G}modZ(\mathbf mathbb{A})}K_f(x,y)\mathfrak{p}hi(y)dy.
{\bf{e}}nd{equation}
Now let us argue informally to motivate the more technical actual reasoning.
Let us pretend that $K_f(x,.)$ is an element of $L^2(G(\mathbb{Q})Z(\mathbb{R}) \backslashslash G(\mathbf mathbb{A}),\mathfrak{o}mega)$, and that
$L^2(G(\mathbb{Q})Z(\mathbb{R}) \backslashslash G(\mathbf mathbb{A}),\mathfrak{o}mega)$ has a Hilbert orthonormal base $\mathbf mathcal{B}$.
Then we would have
$$K(x,.)=\mathbf{s}um_{\mathfrak{p}hi \in \mathbf mathcal{B}} \langle K(x,.) | \mathfrak{o}verline{\mathfrak{p}hi} \rangle \mathfrak{o}verline{\mathfrak{p}hi}.$$
But equation~(\ref{basicequation}) says that $\langle K(x,.) | \mathfrak{o}verline{\mathfrak{p}hi} \rangle=R(f) \mathfrak{p}hi(x)$.
Thus we might expect a spectral expansion of the kernel of the form
\begin{equation}\label{informal}
K(x,y)=\mathbf{s}um_{\mathfrak{p}hi \in \mathbf mathcal{B}}R(f) \mathfrak{p}hi(x) \mathfrak{o}verline{\mathfrak{p}hi(y)}.
{\bf{e}}nd{equation}
If moreover each element $\mathfrak{p}hi$ of our base $\mathbf mathcal{B}$ is an eigenfunction of the operator $R(f)$, say
\begin{equation}\label{evalue}
R(f)\mathfrak{p}hi=\lambda_f(\mathfrak{p}hi)
{\bf{e}}nd{equation}
then the above expansion becomes
$$K(x,y)=\mathbf{s}um_{\mathfrak{p}hi \in \mathbf mathcal{B}}\lambda_f(\mathfrak{p}hi) \mathfrak{p}hi(x) \mathfrak{o}verline{\mathfrak{p}hi(y)}.$$
Finally, integrating $K(x,y)$ on $U \times U$ against a character $\mathfrak{o}verline{\mathfrak{p}si_1(x)}\mathfrak{p}si_2(y)$ would then yield
a spectral equality involving the Whittaker coefficients and the eigenvalues $\lambda_f(\mathfrak{p}hi)$, of the form
\begin{equation}\label{informalspectral}
\int_{(U(\mathbb{Q}) \backslash U(\mathbf mathbb{A}))^{2}}K(x,y)\mathfrak{o}verline{\mathfrak{p}si_1(x)}\mathfrak{p}si_2(y)dxdy
=\mathbf{s}um_{\mathfrak{p}hi \in \mathbf mathcal{B}}\lambda_f(\mathfrak{p}hi) \mathcal{W}_{\mathfrak{p}si_1}(\mathfrak{p}hi) \mathfrak{o}verline{\mathcal{W}_{\mathfrak{p}si_2}(\mathfrak{p}hi)}.
{\bf{e}}nd{equation}
Note that in the last step we need~(\ref{informal}) to hold not only in the $L^2$ sense, but pointwise, as
$(U(\mathbb{Q}) \backslash U(\mathbf mathbb{A}))^{2}$ has measure zero.
Of course, $L^2(G(\mathbb{Q})Z(\mathbb{R}) \backslashslash G(\mathbf mathbb{A}),\mathfrak{o}mega)$ \textit{does not} have a Hilbert orthonormal base, due to the presence
of continuous spectrum. However, after adding the proper continuous contribution,
a spectral expansion of the form~(\ref{informal}) has been proved by Arthur~\cite{ArthurSpectralExpansion}*{pages 928-934},
building on the spectral decomposition of $L^2(G(\mathbb{Q})Z(\mathbb{R}) \backslashslash G(\mathbf mathbb{A}),\mathfrak{o}mega)$ by Langlands.
We may then reduce from global to local as follows.
By general theory, we may choose automorphic forms $\mathfrak{p}hi$ appearing in the spectral expansion of the kernel
to be factorizable vectors $\mathfrak{p}hi_\infty \mathfrak{o}times \bigotimes_p \mathfrak{p}hi_p$.
If moreover we take $f$ factorizable, say $f =f_\infty \mathfrak{p}rod_p f_p$, then the computation of $R(f) \mathfrak{p}hi$ reduces
to the computation of the action of each local component $f_v$ on $\mathfrak{p}hi_v$.
By choosing the local components $f_v$ appropriately, we can ensure that each $\mathfrak{p}hi_v$ is an eigenvector of the
operator corresponding to $f_v$, so that~(\ref{evalue}) holds.
The determination of $\lambda_f(\mathfrak{p}hi)$ then amounts, at the infinite place, to the study
of the spherical transform of $f_\infty$, and at finite places $p$, of the action of the local Hecke algebra.
Specifically, from now on we assume $f$ is as follows.
\begin{assumption}\label{testfunction}
From now on we assume $f = f_\infty \mathfrak{p}rod_p f_p$ where
\begin{itemize}
\item $f_\infty$ is any smooth, left and right $K_\infty$-invariant and $Z(\mathbb{R})$-invariant function on $G(\mathbb{R})$,
whose support is compact modulo the centre and contained in $G^+(\mathbb{R})=\{g \in G(\mathbb{R}) : \mathbf mu(g)>0 \}.$
\item for all prime $p$, $f_p$ is a left and right $\mathbf mathfrak{G}amma_p$-invariant function on $G(\mathbb{Q}_p)$, satisfying
$f_p(gz)=\mathfrak{o}verline{\mathfrak{o}mega_p}(z)f(g)$ for all $z \in Z(\mathbb{Q}_p)$, and compactly supported modulo the centre,
\item whenever $\mathbf mathfrak{G}amma_p \texttt{n}eq G(\mathbb{Z}_p)$, we have
$$f_p(g)=\begin{cases}
\bm{f}rac{\mathfrak{o}verline{\mathfrak{o}mega_p}(z)}{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_p})} \text{ if there exists } z \in Z(\mathbb{Q}_p) \text{ such that } g \in z \mathbf mathfrak{G}amma_p\\
0 \text{ otherwise.}
{\bf{e}}nd{cases}$$
{\bf{e}}nd{itemize}
{\bf{e}}nd{assumption}
Note that this assumption can be fulfilled if and only if we have the following compatibility condition
\begin{assumption}
For all prime $p$, the resriction of $\mathfrak{o}mega_p$ to $\mathbf mathfrak{G}amma_p \cap Z(\mathbb{Q}_p)$ is trivial.
{\bf{e}}nd{assumption}
Let us remind the following result~\cite{KL}*{Lemma~3.10}
\begin{proposition}\label{Kfixed}
Let $G$ be a locally compact group, let $K \mathbf{s}ubset G$ be a closed subgroup, and let $\mathfrak{p}i$ be a unitary representation of $G$
on a Hilbert space $V$ with central character $\mathfrak{o}mega$. Let $f$ be any left and right $K$-invariant function satisfying
\begin{itemize}
\item $f(gz)=\mathfrak{o}verline{\mathfrak{o}mega}(z)f(g)$ for all $z$ in the centre $Z$ of $G$,
\item $f$ is integrable on $G/Z$.
{\bf{e}}nd{itemize}
Then the operator $\mathfrak{o}verline{\mathfrak{p}i}(f)$ on $V$ defined by
$$\mathfrak{o}verline{\mathfrak{p}i}(f)v=\int_{G/Z}f(g)\mathfrak{p}i(g)vdg$$
has its image in the $K$-fixed subspace $V^K$ and annihilates the orthogonal complement of this subspace.
{\bf{e}}nd{proposition}
Because of Assumption~\ref{testfunction}, this result implies only $\mathbf mathfrak{G}amma$-fixed automorphic forms having central character $\mathfrak{o}mega$ will appear
in the spectral decomposition of $K_f$.
These automorphic forms come from admissible irreducible representations with central character $\mathfrak{o}mega$ and having a
$\mathbf mathfrak{G}amma$-fixed vector.
In turn, these representations factor as restricted tensor products of local representations having similar local properties.
Furthermore, only those automorphic forms $\mathfrak{p}hi$ that are generic will survive the integration against a generic character on $U$, hence we may restrict
attention to local representations that are generic.
\mathbf{s}ubsection{Non-Archimedean Hecke algebras}
Let $p$ be a prime number, and $f_p$ be the local component of the function $f$ in Assumption~\ref{testfunction}.
Let $(\mathfrak{p}i,V)$ be a unitary representation of $G(\mathbb{Q}_p)$ with central character $\mathfrak{o}mega_p$.
Throughout this section the Haar measure on $G(\mathbb{Q}_p)$ is normalised so that $K_p=G(\mathbb{Z}_p)$ has volume one.
By Proposition~\ref{Kfixed} we have an operator
\begin{equation}\label{localR}
\mathfrak{o}verline{\mathfrak{p}i}(f_p)v=\int_{\mathbf mathfrak{G}modZ(\mathbb{Q}_p)} f(g)\mathfrak{p}i(g)vdg.
{\bf{e}}nd{equation}
acting on the $\mathbf mathfrak{G}amma_p$-fixed subspace $V^{\mathbf mathfrak{G}amma_p}$ and annihilating the orthogonal complement of this subspace.
First, let us consider the case $\mathbf mathfrak{G}amma_p \texttt{n}eq G(\mathbb{Z}_p)$.
Then any $\mathbf mathfrak{G}amma_p$-fixed vector $v \in V$ is also fixed by
$\mathfrak{o}verline{\mathfrak{p}i}(f_p)$, since in this case $$\mathfrak{o}verline{\mathfrak{p}i}(f_p)v=\bm{f}rac1{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_p})}\int_{\mathfrak{o}verline{\mathbf mathfrak{G}amma}_p} \mathfrak{p}i(g)vdg=v.$$
We now turn to the situation $\mathbf mathfrak{G}amma_p=K_p=G(\mathbb{Z}_p)$ (in particular, the character $\mathfrak{o}mega_p$ must be unramified).
We have have the following~\cite{RS2}*{Theorem~7.5.4}.
\begin{proposition}
Let $(\mathfrak{p}i,V)$ be generic, irreducible, admissible, representation of $G(\mathbb{Q}_p)$.
Assume $\mathfrak{p}i$ has a non-zero $K_p$-fixed vector.
Then $V^{K_p}$ has dimension $1$.
{\bf{e}}nd{proposition}
\begin{remark}
In~\cite{RS2}*{Theorem~7.5.4} it is assumed $\mathfrak{p}i$ has trivial central character.
However, in our situation, the fact that $\mathfrak{p}i$ has a non-zero $K_p$-fixed vector forces the central character
to be unramified. We can thus twist our representation by an unramified character to reduce to the hypothesis of~\cite{RS2}.
{\bf{e}}nd{remark}
By definition, any non-zero vector $\mathfrak{p}hi$ in $V^{K_p}$ is then called the {\bf spherical vector}.
Since $\mathfrak{o}verline{\mathfrak{p}i}(f_p)$ acts on $V^{K_p}$ which is one-dimensional, the spherical vector is an eigenvector of
$\mathfrak{o}verline{\mathfrak{p}i}(f_p)$. Finally, let us relate the operator $\mathfrak{o}verline{\mathfrak{p}i}(f_p)$ to the action of the unramified Hecke algebra.
The {\bf local Hecke algebra} $\mathbf mathscr Hecke(K_p)$ is the vector space of left and right-$K_p$ invariant
compactly supported functions $f:G(\mathbb{Q}_p) \to \mathbf mathbb{C}$, endowed with the convolution product
$$(f * h) (g)=\int_{G(\mathbb{Q}_p)}f(gx^{-1})h(x)dx.$$
If $(\mathfrak{p}i,V)$ is a smooth representation of $G(\mathbb{Q}_p)$, then the Hecke algebra $\mathbf mathscr Hecke(K_p)$ acts on the $K_p$-invariant subspace $V^{K_p}$ by
$$\mathfrak{p}i(f)v=\int_{G(\mathbb{Q}_p)} f(g)\mathfrak{p}i(g)vdg.$$
\begin{lemma}\label{ModingOutCentre}
Let $f$ be a bi-$K_p$ invariant function on $G(\mathbb{Q}_p)$, with a (unramified) central character, and compactly supported modulo the centre.
There exists a compactly supported bi-$K_p$-invariant function $\tilde{f}$ on $G(\mathbb{Q}_p)$
and a complete set of representatives $\mathbf mathfrak{G}modZ$ of $G(\mathbb{Q}_p)/\mathbb{Q}_p^\times$ satisfying
$\tilde{f}(gz)=f(g)\mathbf mathbbm{1}_{\mathbb{Z}_p^{\times}}(z)$ for all $g \in \mathbf mathfrak{G}modZ$ and $z \in \mathbb{Q}_p^\times$.
{\bf{e}}nd{lemma}
\begin{proof}
By the Cartan decomposition we have $G(\mathbb{Q}_p)=\coprod_{\mathbf{s}ubstack{i \le j \in \mathbb{Z}\\ t \in \mathbb{Z}}}K_p \diag{p^i}{p^j}{p^{t-i}}{p^{t-j}}K_p$.
Thus we have
$$G(\mathbb{Q}_p) / \mathbb{Q}_p^\times=\left.
\left(\coprod_{\mathbf{s}ubstack{j \ge 0 \\ t \in \mathbb{Z}}} K_p \diag{1}{p^j}{p^t}{p^{t-j}}K_p\right) \mathbf middle/ \mathbb{Z}_p^\times. \right.$$
Fix a complete set of representatives $\mathfrak{o}verline{K_p}$ of $K_p/\mathbb{Z}_p^\times$.
Then $\mathbf mathfrak{G}modZ= \coprod_{\mathbf{s}ubstack{j \ge 0 \\ t \in \mathbb{Z}}} \mathfrak{o}verline{K_p} \diag{1}{p^j}{p^t}{p^{t-j}} \mathfrak{o}verline{K_p}$
is a complete set of representatives of $G(\mathbb{Q}_p)/\mathbb{Q}_p^\times$.
Moreover, defining $$S= \coprod_{\mathbf{s}ubstack{j \ge 0 \\ t \in \mathbb{Z}}} K_p \diag{1}{p^j}{p^t}{p^{j+t}}K_p \cap Supp(f)=(\mathbb{Z}_p^\times \mathbf mathfrak{G}modZ )\cap Supp(f),$$
the function $\tilde{f}=\mathbf mathbbm{1}_S \times f$ has the desired properties.
{\bf{e}}nd{proof}
Now the function $\tilde{f_p}$ attached to $f_p$ by Lemma~(\ref{ModingOutCentre}) is an element of the Hecke algebra,
and we have $\mathfrak{p}i(\tilde{f_p})=\mathfrak{o}verline{\mathfrak{p}i}(f_p)$, as
$$\mathfrak{p}i(\tilde{f_p})v=\int_{G(\mathbb{Q}_p)} \tilde{f}(g)\mathfrak{p}i(g)vdg
=\int_{G(\mathbb{Q}_p)/\mathbb{Q}_p^\times}\int_{\mathbb{Q}_p^\times} f_p(g)\mathbf mathbbm{1}_{\mathbb{Z}_p^{\times}}(z)\mathfrak{p}i(g)vdg
=\mathfrak{o}verline{\mathfrak{p}i}(f_p)v.$$
We summarize the above discussion in the following proposition.
\begin{proposition}\label{localevalue}
Let $p$ be a prime number, and $f_p$ be the local component of the function $f$ in Assumption~\ref{testfunction}.
Let $(\mathfrak{p}i,V)$ be a unitary representation of $G(\mathbb{Q}_p)$ with central character $\mathfrak{o}mega_p$.
Then the operator $\mathfrak{o}verline{\mathfrak{p}i}(f_p)$ from Proposition~\ref{Kfixed} acts by a scalar $\lambda_\mathfrak{p}i(f_p)$ on the $\mathbf mathfrak{G}amma_p$ fixed
subspace $V^{\mathbf mathfrak{G}amma_p}$ and annihilates the orthogonal complement of this subspace.
Moreover, if $\mathbf mathfrak{G}amma_p \texttt{n}eq G(\mathbb{Z}_p)$ then $\lambda_\mathfrak{p}i(f_p)=1$, and if $\mathbf mathfrak{G}amma_p = G(\mathbb{Z}_p)$ then $\mathfrak{o}verline{\mathfrak{p}i}(f_p)$
equals the Hecke operator $\mathfrak{p}i(\tilde{f_p})$, where $\tilde{f_p}$ is given by Lemma~\ref{ModingOutCentre}.
{\bf{e}}nd{proposition}
\mathbf{s}ubsection{The Archimedean representation}
We first show that in our situation the representation at the Archimedean place must be an irreducible principal series representation, that is full induced from the Borel subgroup.
A representation of $G(\mathbb{R})$ which has a non-zero $K_\infty$-fixed vector is called {\bf spherical}.
\begin{proposition}\label{MustBePrincipalSeries}
Any generic irreducible spherical representation $(\mathfrak{p}i, V)$ of $G(\mathbb{R})$ vector is a principal series representation.
{\bf{e}}nd{proposition}
The author wishes to thank Ralf Schmidt for communicating the following argument.
\begin{proof}
As explained at the end of~\cite{Vogan}, the generic representations are exactly the ``large" ones,
\textit{i.e.}, those with maximal Gelfand-Kirillov dimension.
The Gelfand-Kirillov dimension of all irreducible representations of $\mathbf mathfrak{G}Sp_4(\mathbb{R})$ have been calculated in~\cite{thesis}*{Appendix~A}.
In particular the maximal Gelfand-Kirillov dimension is 4, and the irreducible large representations are either discrete series
or limit of discrete series, induced from the Siegel parabolic subgroup, Langlands quotient of representation induced from the
Klingen subgroup, or principal series representations.
Now the multiplicity of each possible $K_{\infty}$-type are described in~\cite{thesis}*{Chapter~4},
and among large representations of $\mathbf mathfrak{G}Sp_4(\mathbb{R})$ only principal series representations contain the trivial $K_{\infty}$-type.
{\bf{e}}nd{proof}
It is then known that the trivial $K_\infty$-type occurs in $\mathfrak{p}i$ with multiplicity one~\cite{MiyOda}, that is to say there is
a unique $K_\infty$-fixed vector in the space $V$.
Moreover, $\mathfrak{p}i$ has a unique {\bf Whittaker model}, and the image of a non-zero $K_\infty$-fixed vector is by definition given by the
{\bf Whittaker function}. The Whittaker function is an eigenfunction of the centre of the universal enveloping algebra,
which acts as an algebra of differential operators. One may then obtain a system of partial differential equations characterizing
the Whittaker function, and compute it explicitly. The Whittaker function may also be computed by the mean of the Jacquet integral.
This has been done by Niwa~\cite{Niwa} and Ishii~\cite{Ishii}.
\mathbf{s}ubsubsection{The spherical transform}
In this section we normalize the Haar measure on $\text{Sp}_4(\mathbb{R})$ so that $K_\infty$ has measure $1$.
If $h$ is any bi-$K_\infty$-invariant compactly supported function on $\text{Sp}_4(\mathbb{R})$, its {\bf spherical transform} is the function $\tilde{h}$
defined on $\mathbf mathcal Lie{a}^*(\mathbf mathbb{C})$ by
\begin{equation}\label{SphericalTransform}
\tilde{h}(\texttt{n}u)=\int_{\text{Sp}_4(\mathbb{R})}h(g)\mathfrak{p}hi_{-\texttt{n}u}(g)dg,
{\bf{e}}nd{equation}
where
\begin{equation}\label{sphericalfunction}
\mathfrak{p}hi_{-\texttt{n}u}(g)=\int_{K_\infty}e^{(\rho-\texttt{n}u)(A(kg))}dk
{\bf{e}}nd{equation}
is the {\bf spherical function} with parameter $-\texttt{n}u$ (here $\rho$ is the half-sum of positive roots).
\begin{proposition}\label{archimedeanevalue}
Let $f_\infty$ be the Archimedean component of the function $f$ in Assumption~\ref{testfunction}.
Let $(\mathfrak{p}i,V)$ be a generic irreducible unitary representation representation of $G(\mathbb{R})$ with trivial central character.
Then the operator $\mathfrak{o}verline{\mathfrak{p}i}(f_\infty)$ from Proposition~\ref{Kfixed} acts by a scalar $\lambda_\mathfrak{p}i(f_\infty)$ on the $K_{\infty}$ fixed
subspace $V^{K_{\infty}}$ and annihilates the orthogonal complement of this subspace.
Moreover, provided this subspace $V^{K_\infty}$ is non zero, then $\mathfrak{p}i$ is a principal series representation,
and $\lambda_\mathfrak{p}i(f_\infty)=\tilde{f_\infty}(-\texttt{n}u)$, where $\tilde{f_\infty}$ is the spherical transform of $f_\infty$ and $\texttt{n}u$ is
the spectral parameter of $\mathfrak{p}i$.
{\bf{e}}nd{proposition}
\begin{proof}
If $V^{K_\infty}$ is zero then by Proposition~\ref{Kfixed} the statement is vacuous.
Assume now $\mathfrak{p}i$ has a non-zero fixed vector.
By Proposition~\ref{MustBePrincipalSeries}, $\mathfrak{p}i$ is then a principal series.
Then $V^{K_\infty}$ is one-dimensional, so if $v$ is any $K_\infty$-fixed vector in $V$ then
we have
\begin{equation}\label{sptr}
\mathfrak{p}i(f_\infty)v=\lambda_\mathfrak{p}i(f_\infty)v
{\bf{e}}nd{equation}
for some complex number $\lambda_\mathfrak{p}i(f)$.
Since $\mathfrak{p}i$ is induced by a character of the Borel subgroup,
to compute the eigenvalue $\lambda_\mathfrak{p}i(f)$, we may realize $\mathfrak{p}i$ as acting by right translation on
a space of functions $\mathfrak{p}hi$ satisfying for all $g \in G(\mathbb{R})$, $n \in U(\mathbb{R})$ and $a \in T^{+}(\mathbb{R})$
\begin{equation}\label{stdspace}
\mathfrak{p}hi(nag)=e^{(\rho+\texttt{n}u)(\log(a))}\mathfrak{p}hi(g),
{\bf{e}}nd{equation}
where $\texttt{n}u \in \mathbf mathcal Lie{a}^*(\mathbf mathbb{C})$ is the {\bf spectral parameter} of $\mathfrak{p}i$.
We may view a $Z(\mathbb{R})$-invariant function supported on $G(\mathbb{R})^+$ as a function on $\text{Sp}_4(\mathbb{R})$, so
the operator $\mathfrak{o}verline{\mathfrak{p}i}(f)$ of Proposition~\ref{Kfixed} is given by
\begin{equation}\label{infinitR}
\mathfrak{o}verline{\mathfrak{p}i}(f_\infty)v=\int_{\mathbf mathfrak{G}modZ(\mathbb{R})} f_\infty(g)\mathfrak{p}i(g)vdg=\int_{\text{Sp}_4(\mathbb{R})} f_\infty(g)\mathfrak{p}i(g)vdg.
{\bf{e}}nd{equation}
If $\mathfrak{p}hi$ is a non-zero $K_\infty$-fixed function satisfying~(\ref{stdspace}) then because of the Iwasawa decomposition we must have
$\mathfrak{p}hi(1) \texttt{n}eq 0$.
Using the integration formula~\cite{Helgason}*{Ch.~I~Corollary~5.3} and right-$K_\infty$ invariance we may compute
\begin{align*}
\mathfrak{p}i(f_\infty) \mathfrak{p}hi(1) &=\int_{\text{Sp}_4(\mathbb{R})} f_\infty(g)\mathfrak{p}i(g) \mathfrak{p}hi(1) dg \\
&=\int_{K_{\infty}}\int_{U A^+}f_\infty(an)\mathfrak{p}hi(an)dadndk
=\int_{U A^+}f_\infty(an)e^{(\rho_{\text{B}}+\texttt{n}u)(\log(a))}dadn\mathfrak{p}hi(1),
{\bf{e}}nd{align*}
where $A^+$ is the subgroup of $A(\mathbb{R})$ with positive diagonal entries.
Therefore, using the Iwasawa decomposition and left-$K_\infty$ invariance of $f_\infty$,
the eigenvalue $\lambda_\mathfrak{p}i(f)$ is given by
\begin{align*}
\lambda_\mathfrak{p}i(f)&=\int_{\text{Sp}_4(\mathbb{R})} f_\infty(g)e^{(\rho+\texttt{n}u)(A(g))}dg\\
&=\int_{K_\infty}\int_{\text{Sp}_4(\mathbb{R})} f_\infty(g)e^{(\rho+\texttt{n}u)(A(kg))}dgdk
=\int_{\text{Sp}_4(\mathbb{R})}f_\infty(g)\mathfrak{p}hi_{\texttt{n}u}(g)dg=\tilde{f_\infty}(-\texttt{n}u).
{\bf{e}}nd{align*}
{\bf{e}}nd{proof}
The spherical transform $\tilde{f_\infty}$ will thus play the role of the test function on the spectral side of our formula.
On the other hand, the geometric side will involve some different integral transform of our test function $f_\infty$.
It is therefore natural to investigate the analytic properties of $\tilde{f_\infty}$, and to seek to recover $f_\infty$ from $\tilde{f_\infty}$.
This can be achieved by the Paley-Wiener theorem and Harish-Chandra inversion theorem.
\mathbf{s}ubsubsection{The Paley-Wiener theorem and Harish-Chandra inversion theorem.}
The material in this section is taken from~\cite{Helgason}.
Let us introduce a bit of notation.
We denote by $\langle, \rangle$ the Killing form on the Lie algebra of $\text{Sp}_4(\mathbb{R})$,
and we define for each $\texttt{n}u \in \mathbf mathcal Lie{a}^*$ a vector $A_\texttt{n}u \in \mathbf mathcal Lie{a}$ by
$\texttt{n}u(H)=\langle A_\texttt{n}u, H\rangle$ for all $H \in \mathbf mathcal Lie{a}$.
We then define $\mathbf{s}cal{\lambda}{\texttt{n}u}=\mathbf{s}cal{A_\lambda}{A_\texttt{n}u}$.
We define $\mathbf mathcal Lie{a}_+$ as the subset of elements $H \in \mathbf mathcal Lie{a}$ satisfying $\mathbf mathfrak{a}lpha(H) > 0$ for all $\mathbf mathfrak{a}lpha \in \mathcal Phi_{\text{B}}$,
and $\mathbf mathcal Lie{a}^*_+=\{\texttt{n}u \in \mathbf mathcal Lie{a} : A_\texttt{n}u \in \mathbf mathcal Lie{a}_+ \}$.
Explicitly the Killing form is given by $\langle X, Y \rangle =6Tr(XY)$ and
$\mathbf mathcal Lie{a}_+=\left\{\diag{x}{y}{-x}{-y}: 0<x<y \right\}$.
Harish-Chandra's {\bf $c$-function} captures the asymptotic behaviour of the spherical function and it gives the Plancherel measure.
More precisely, by Theorem~6.14 of~\cite{Helgason}*{Chap. IV}, if $H \in \mathbf mathcal Lie{a}^+$ and $\texttt{n}u \in \mathbf mathcal Lie{a}^*_+$
then we have
$$\lim_{t \to +\infty}e^{(-\texttt{n}u+\rho)(tH)}\mathfrak{p}hi_{-i\texttt{n}u}({\bf{e}}xp(tH))=c(-i\texttt{n}u).$$
Moreover, $c(\texttt{n}u)$ is given, for $\texttt{n}u \in \mathbf mathcal Lie{a}^*_+$, by the absolutely convergent integral
\begin{equation}\label{cfunction}
c(\texttt{n}u)
=\int_{U(\mathbb{R})}e^{(\texttt{n}u+\rho)(A(Ju))}du,
{\bf{e}}nd{equation}
where the measure $du$ is normalized so that $c(\rho)=1$,
and has meromorphic continuation to $\mathbf mathcal Lie{a}^*(\mathbf mathbb{C})$ given in our situation by the expression
$$c(-i\texttt{n}u)=
c_0\mathfrak{p}rod_{\mathbf mathfrak{a}lpha \in \mathcal Phi}
\bm{f}rac{2^{-\mathbf{s}cal{i\texttt{n}u}{\mathbf mathfrak{a}lpha_0}}\mathbf mathfrak{G}amma(\mathbf{s}cal{i\texttt{n}u}{\mathbf mathfrak{a}lpha_0})}
{ \mathbf mathfrak{G}amma\left(\bm{f}rac{\bm{f}rac32+\mathbf{s}cal{i\texttt{n}u}{\mathbf mathfrak{a}lpha_0}}2\right)
\mathbf mathfrak{G}amma\left(\bm{f}rac{\bm{f}rac12+\mathbf{s}cal{i\texttt{n}u}{\mathbf mathfrak{a}lpha_0}}2\right)},$$
where $\mathcal Phi$ is the set of roots,
$\mathbf mathfrak{a}lpha_0=\bm{f}rac{\mathbf mathfrak{a}lpha}{\mathbf{s}cal{\mathbf mathfrak{a}lpha}{\mathbf mathfrak{a}lpha}}$ and the constant $c_0$
is such that $c(\rho)=1$.
Using the duplication formula $\mathbf mathfrak{G}amma(z)\mathbf mathfrak{G}amma(z+\bm{f}rac12)=\mathfrak{p}i^{\bm{f}rac12}2^{1-2z}\mathbf mathfrak{G}amma(2z)$, we can rewrite this as
$$c(-i\texttt{n}u)=
\bm{f}rac{c_0}{4\mathfrak{p}i^2}\mathfrak{p}rod_{\mathbf mathfrak{a}lpha \in \mathcal Phi}
\bm{f}rac{\mathbf mathfrak{G}amma(\mathbf{s}cal{i\texttt{n}u}{\mathbf mathfrak{a}lpha_0})}{\mathbf mathfrak{G}amma(\bm{f}rac12+\mathbf{s}cal{i\texttt{n}u}{\mathbf mathfrak{a}lpha_0})},$$
We then have the following theorems
\begin{theorem}[Paley-Wiener theorem]\label{PW}
Let $\mathcal PW^R(\mathbf mathcal Lie{a}^*_\mathbf mathbb{C})$ the set of $\text{O}mega$-invariant entire functions $h$ on $\mathbf mathcal Lie{\mathbf mathfrak{a}}^*_\mathbf mathbb{C}$ such that for all
$N \ge 0$ we have
$$h(\texttt{n}u) \ll_N (1+|\texttt{n}u|)^{-N}e^{R|\mathbb{R}e(\texttt{n}u)|}.$$
Let $$ \mathcal PW(\mathbf mathcal Lie{a}^*_\mathbf mathbb{C})=\bigcup_{R>0}\mathcal PW^R(\mathbf mathcal Lie{a}^*_\mathbf mathbb{C}).$$
Then the spherical transform $f \mathbf mapsto \tilde{f}$ is a bijection
from $C^\infty_c(K_\infty\backslash \text{Sp}_4(\mathbb{R}) / K_\infty)$ to $\mathcal PW(\mathbf mathcal Lie{a}^*_\mathbf mathbb{C})$.
{\bf{e}}nd{theorem}
\begin{theorem}[Inversion theorem]\label{sphericalinversion}
There is a constant $c$ such that for every function $f \in C^\infty_c(K\backslash \text{Sp}_4(\mathbb{R}) / K)$ we have
for all $g\in \text{Sp}_4(\mathbb{R})$
\begin{equation}\label{inversion}
cf(g)=\int_{\mathbf mathcal Lie{a}^*} \tilde{f}(-i\texttt{n}u)\mathfrak{p}hi_{-i\texttt{n}u}(g)\bm{f}rac{d\texttt{n}u}{c(i\texttt{n}u)c(-i\texttt{n}u)}.
{\bf{e}}nd{equation}
{\bf{e}}nd{theorem}
\begin{remark}
The constant $c$ may be worked out by Exercise~C.4 of~\cite{Helgason}*{Chap.~IV}.
{\bf{e}}nd{remark}
\begin{remark}
Using formulae $\mathbf mathfrak{G}amma(iz)\mathbf mathfrak{G}amma(-iz)=\bm{f}rac{\mathfrak{p}i}{z\mathbf{s}inh{\mathfrak{p}i z}}$ and $\mathbf mathfrak{G}amma(\bm{f}rac12-iz)\mathbf mathfrak{G}amma(\bm{f}rac12+iz)=\bm{f}rac{\mathfrak{p}i}{\cosh{\mathfrak{p}i z}}$,
the Plancherel measure is given by
\begin{equation}\label{Plancherel}
\bm{f}rac{d\texttt{n}u}{c(i\texttt{n}u)c(-i\texttt{n}u)}=\bm{f}rac{16\mathfrak{p}i^4}{c_0^2}\mathfrak{p}rod_{\mathbf mathfrak{a}lpha \in \mathcal Phi}\mathbf{s}cal{\texttt{n}u}{\mathbf mathfrak{a}lpha_0}\tanh(\mathfrak{p}i\mathbf{s}cal{\texttt{n}u}{\mathbf mathfrak{a}lpha_0})d\texttt{n}u.
{\bf{e}}nd{equation}
{\bf{e}}nd{remark}
\mathbf{s}ubsubsection{The Whittaker function and the Jacquet integral}
As mentioned above, the Whittaker function is a non-zero $K_\infty$-fixed vector in the Whittaker model, and it is unique up to scaling.
It is given by (meromorphic continuation of) the Jacquet integral.
Namely, if $\mathfrak{p}si$ is a generic character of $U(\mathbb{R})$, we have the {\bf Jacquet integral}
\begin{equation}\label{jacquet}
W(\texttt{n}u, g, \mathfrak{p}si)=\int_{U(\mathbb{R})} e^{(\rho+\texttt{n}u)(A(Jug))}\mathfrak{o}verline{\mathfrak{p}si(u)}du.
{\bf{e}}nd{equation}
The Jacquet integral converges absolutely for $\mathbb{R}e(\texttt{n}u) \in \mathbf mathcal Lie{a}^*_+$, as may be seen by
using the absolute convergence of~(\ref{cfunction}) and computing
\begin{equation}\label{majoration}
|W(\texttt{n}u, g, \mathfrak{p}si)| \le
\int_{U(\mathbb{R})}|e^{(\texttt{n}u + \rho)(A(Jug))}| du =
e^{(\rho-\mathbb{R}e(\texttt{n}u))(A(g))}c(\mathbb{R}e(\texttt{n}u))
{\bf{e}}nd{equation}
Moreover, it has meromorphic continuation to all $\texttt{n}u \in \mathbf mathcal Lie{a}^*_\mathbf mathbb{C}$.
Ishii~\cite{Ishii} computed explicit integral representations for the normalized Jacquet integral
$$\mathbf mathcal{W}(\texttt{n}u,g,\mathfrak{p}si)=\bm{f}rac1{4\mathfrak{p}i^2}\mathfrak{p}rod_{\mathbf mathfrak{a}lpha \in \mathcal Phi} \mathbf mathfrak{G}amma\left(\bm{f}rac12+\mathbf{s}cal{\texttt{n}u}{\mathbf mathfrak{a}lpha_0}\right)W(\texttt{n}u,g,\mathfrak{p}si),$$
namely (note the different choice of minimal parabolic subgroup) if $a=\diag{a_1}{a_2}{a_1^{-1}}{a_2^{-1}} \in A^+(\mathbb{R})$ then for any
$\texttt{n}u \in \mathbf mathcal Lie{a^*}_\mathbf mathbb{C}$
\begin{equation}\label{NiwaIntegral}
\begin{aligned}
\mathbf mathcal{W}(\texttt{n}u,a,\mathfrak{p}si)=2a_1a_2^2 &\int_0^\infty \int_0^\infty K_{\bm{f}rac{\texttt{n}u_2-\texttt{n}u_1}{2}}(2\mathfrak{p}i v_1)K_{\bm{f}rac{\texttt{n}u_1+\texttt{n}u_2}2}(2 \mathfrak{p}i v_2)\\
& \times {\bf{e}}xp\left(-\mathfrak{p}i\left(\bm{f}rac{a_2^2}{v_1v_2}+\bm{f}rac{v_1v_2}{a_1^2}+a_1^2\left(\bm{f}rac{v_1}{v_2}+\bm{f}rac{v_2}{v_1}\right)\right)\right)\bm{f}rac{dv_1dv_2}{v_1v_2}.
{\bf{e}}nd{aligned}
{\bf{e}}nd{equation}
This implies in particular that the normalized Jacquet integral satisfies the functional equations
\begin{equation}\label{whittakerftneq}
\mathbf mathcal{W}(\mathbf{s}igma \cdot \texttt{n}u,g,\mathfrak{p}si)=\mathbf mathcal{W}(\texttt{n}u,g,\mathfrak{p}si)
{\bf{e}}nd{equation}
for all $\mathbf{s}igma \in \text{O}mega$.
If $t \in A^+(\mathbb{R})$ and if we denote by $\mathfrak{p}si_t$ the character $\mathfrak{p}si_t(u)=\mathfrak{p}si(t^{-1}ut)$, then it is easy to see
(first by a change of variable in the domain where the Jacquet integral is absolutely convergent, then by meromorphic continuation)
that
\begin{equation}\label{whittakertorus}
W(\texttt{n}u,g,\mathfrak{p}si_t)=e^{(\rho-\texttt{n}u)(\log(t))}W(\texttt{n}u, t^{-1}g,\mathfrak{p}si).
{\bf{e}}nd{equation}
\begin{remark}\label{WonA}
By Lemma~\ref{A(Ju)} and changing variables $u(x,a,b,c) \mathbf mapsto u(-x,a,-b,-c)$, if $t \in A^+(\mathbb{R})$then we have
$W(\texttt{n}u, t,\mathfrak{p}si)=W(\texttt{n}u, t, \mathfrak{o}verline{\mathfrak{p}si}).$
{\bf{e}}nd{remark}
\mathbf{s}ubsubsection{Wallach's Whittaker transform}
The following theorem is a consequence of~\cite{Wallach}*{Ch.~15}.
Let $L^2(U \backslash \text{Sp}_4(\mathbb{R}) /K, \mathfrak{p}si)$ be the space of functions $f$ on $\text{Sp}_4(\mathbb{R})$ satisfying for all
$u \in U(\mathbb{R})$, for all $k \in K_\infty$
and for all $g \in \text{Sp}_4(\mathbb{R})$
$$f(ugk)=\mathfrak{p}si(u)f(g) \text{ and } \int_{U \backslash \text{Sp}_4(\mathbb{R})} |f(g)|^2dg < \infty.$$
\begin{theorem}[Wallach's Whittaker inversion]\label{WallWhit}
Define for $\mathbf mathfrak{a}lpha \in C^{\infty}_c(\mathbf mathcal Lie{a^*})$
\begin{equation*}
\mathbf mathscr{T} (\mathbf mathfrak{a}lpha)(a)=\int_{\mathbf mathcal Lie{a}^*}\mathbf mathfrak{a}lpha(\texttt{n}u)W(-i\texttt{n}u,a,\mathfrak{p}si)\bm{f}rac{d\texttt{n}u}{c(i\texttt{n}u)c(-i\texttt{n}u)}.
{\bf{e}}nd{equation*}
Then the image of the linear map $\mathbf mathscr{T}$ is a dense subset $\mathbf mathscr W \mathbf{s}ubset L^2(U \backslash \text{Sp}_4(\mathbb{R}) /K, \mathfrak{p}si)$
containing $C^\infty_c(U \backslash \text{Sp}_4(\mathbb{R})/K, \mathfrak{p}si)$, and $\mathbf mathscr{T}$ extends to a unitary operator onto
$L^2(U \backslash \text{Sp}_4(\mathbb{R}) /K, \mathfrak{p}si)$.
Moreover, the inverse of $\mathbf mathscr{T}$ is given for all $f \in \mathbf mathscr W$ by the {\bf Whittaker transform}
\begin{equation*}
W(f)(\texttt{n}u)=c \int_{A^+} f(a)W(i\texttt{n}u,a,\mathfrak{p}si)e^{-2\rho \log a}da,
{\bf{e}}nd{equation*}
where the constant $c$ is the same as in Theorem~\ref{sphericalinversion}.
{\bf{e}}nd{theorem}
\mathbf{s}ubsubsection{An integral transform}
Let $g \in G(\mathbb{R})$, $t \in A^+(\mathbb{R})$ and $\mathfrak{p}si$ a generic character of $U(\mathbb{R})$.
When dealing with the geometric side of the relative trace formula,
we shall be interested in the integral $$I(f_\infty)=\int_{U(\mathbb{R})}f_\infty(tug)\mathfrak{o}verline{\mathfrak{p}si}(u)du.$$
Using expression~(\ref{sphericalfunction}) and applying Theorem~5.20 of~\cite{Helgason}*{Ch.I} that relates integration
on $K_\infty$ to integration on $U(\mathbb{R})$, one may establish the following identity for all $\texttt{n}u \in \mathbf mathcal Lie{a}^*_\mathbf mathbb{C}$
\begin{equation}\label{sphericalU}
\mathfrak{p}hi_{\texttt{n}u}(g)=\int_{U(\mathbb{R})}
e^{(\rho+\texttt{n}u)(A(Jug)}e^{(\rho-\texttt{n}u)(A(Ju))}du.
{\bf{e}}nd{equation}
From this identity, ignoring convergence issues and treating integrals as if they were
absolutely convergent, one may heuristically expect the following
\begin{equation}\label{heuristics}
\int_{U(\mathbb{R})}\mathfrak{p}hi_\texttt{n}u(tug)\mathfrak{o}verline{\mathfrak{p}si(u)}du
=W(\texttt{n}u,g,\mathfrak{p}si)W(-\texttt{n}u,t^{-1},\mathfrak{o}verline{\mathfrak{p}si}).
{\bf{e}}nd{equation}
However, the domain of absolute convergence of the two Jacquet integral in the right hand side are complementary from each other,
and the integral in the left hand side is likely not absolutely convergent, making such a result,
where the left hand side is (optimistically) a semi-convergent integral and the right-hand side is defined by meromorphic continuation,
likely difficult to prove.
Carrying on with this heuristic and using Theorem~\ref{sphericalinversion}, let us write
\begin{align*}
cI(f_\infty)&=\int_{U(\mathbb{R})}\int_{\mathbf mathcal Lie{a}^*} \tilde{f_\infty}(-i\texttt{n}u)\mathfrak{p}hi_{-i\texttt{n}u}(tug)\bm{f}rac{d\texttt{n}u}{c(i\texttt{n}u)c(-i\texttt{n}u)}\mathfrak{o}verline{\mathfrak{p}si}(u)du \\
&=\int_{\mathbf mathcal Lie{a}^*}\tilde{f_\infty}(-i\texttt{n}u)\int_{U(\mathbb{R})} \mathfrak{p}hi_{-i\texttt{n}u}(tug)\mathfrak{o}verline{\mathfrak{p}si}(u)du\bm{f}rac{d\texttt{n}u}{c(i\texttt{n}u)c(-i\texttt{n}u)}\\
&=\int_{\mathbf mathcal Lie{a}^*}\tilde{f_\infty}(-i\texttt{n}u)W(-i\texttt{n}u,g,\mathfrak{p}si)W(i\texttt{n}u,t^{-1},\mathfrak{o}verline{\mathfrak{p}si})\bm{f}rac{d\texttt{n}u}{c(i\texttt{n}u)c(-i\texttt{n}u)}.
{\bf{e}}nd{align*}
Unlike~(\ref{heuristics}), this equality seems more reasonable.
Indeed, the left hand side is absolutely convergent because $f_\infty$ is compactly supported,
and in the right hand side $\tilde{f_\infty}$ has rapid decay.
We now give a rigorous proof of the following.
\begin{theorem}\label{GeomTransform}
Let $f_\infty$ be a smooth, bi-$K_\infty$, compactly supported function on $\text{Sp}_4(\mathbb{R})$.
Let $g \in G(\mathbb{R})$, $t \in A^+(\mathbb{R})$ and $\mathfrak{p}si$ a generic character of $U(\mathbb{R})$.
Then we have
$$c\int_{U(\mathbb{R})}f_\infty(tug)\mathfrak{o}verline{\mathfrak{p}si}(u)du=\int_{\mathbf mathcal Lie{a}^*}\tilde{f_\infty}(-i\texttt{n}u)W(-i\texttt{n}u,g,\mathfrak{p}si)W(i\texttt{n}u,t^{-1},
\mathfrak{o}verline{\mathfrak{p}si})\bm{f}rac{d\texttt{n}u}{c(i\texttt{n}u)c(-i\texttt{n}u)},$$
where $W(\texttt{n}u,\cdot,\mathfrak{p}si)$ is the $\mathfrak{p}si$-Whittaker function of the principal series with spectral parameter~$\texttt{n}u$.
{\bf{e}}nd{theorem}
\begin{proof}
Both sides transform on the left by $U(\mathbb{R})$ according to $\mathfrak{p}si$, and are $K_\infty$-invariant.
Thus by the Iwasawa decomposition, it suffices to prove it for $g=a \in A^+(\mathbb{R})$.
Also, by~(\ref{whittakertorus}), we may restrict ourself to $t=1$.
With notations of Theorem~\ref{WallWhit}, we have
$$
\int_{\mathbf mathcal Lie{a}^*}\tilde{f_\infty}(-i\texttt{n}u)W(-i\texttt{n}u,a,\mathfrak{p}si)W(i\texttt{n}u,1,\mathfrak{o}verline{\mathfrak{p}si})\bm{f}rac{d\texttt{n}u}{c(i\texttt{n}u)c(-i\texttt{n}u)}=
\mathbf mathscr{T}(\mathbf mathfrak{a}lpha)(a),$$
where $$\mathbf mathfrak{a}lpha(\texttt{n}u)= \tilde{f_\infty}(-i\texttt{n}u) W(i\texttt{n}u,1,\mathfrak{o}verline{\mathfrak{p}si}).$$
Moreover $g \mathbf mapsto \int_{U(\mathbb{R})}f_\infty(ug)\mathfrak{o}verline{\mathfrak{p}si}(u)du$ belongs to $C^\infty_c(U \backslash \text{Sp}_4(\mathbb{R}) /K, \mathfrak{p}si)$
since $f_\infty$ is smooth and compactly supported. Hence
by Wallach's Whittaker inversion it suffices to show that
for all $\texttt{n}u \in \mathbf mathcal Lie{a}^*$ we have
\begin{equation}\label{toprove}
\mathbf mathfrak{a}lpha(\texttt{n}u) = \int_{{A}^+(\mathbb{R})}e^{-2\rho \log a} \int_{U(\mathbb{R})}f_\infty(ua)\mathfrak{o}verline{\mathfrak{p}si}(u)duW(i\texttt{n}u,a,\mathfrak{p}si)da.
{\bf{e}}nd{equation}
Since both sides are meromorphic in $\texttt{n}u$, it suffices to show this for
$\mathbb{R}e(i\texttt{n}u) \in \mathbf mathcal Lie{a}^*_+$. In this region, the Jacquet integral
$W(i\texttt{n}u,a,\mathfrak{p}si)=\int_{U(\mathbb{R})} e^{(\rho+i\texttt{n}u)(A(Ju_1a))}\mathfrak{o}verline{\mathfrak{p}si(u_1)}du_1$ converges absolutely.
By Remark~\ref{WonA}, the integral in~(\ref{toprove}) may then be written as
\begin{align*}
\int_{{A}^+(\mathbb{R})}\int_{U(\mathbb{R})}f_\infty(au)\mathfrak{o}verline{\mathfrak{p}si}(aua^{-1})duW(i\texttt{n}u,a,\mathfrak{o}verline{\mathfrak{p}si})da
=\int_{{A}^+(\mathbb{R})}\int_{U(\mathbb{R})}f_\infty(au)W(i\texttt{n}u,au,\mathfrak{o}verline{\mathfrak{p}si})&duda\\
=\int_{{A}^+(\mathbb{R})}\int_{U(\mathbb{R})}f_\infty(au)\int_{U(\mathbb{R})} e^{(\rho+i\texttt{n}u)(A(Ju_1au))}\mathfrak{p}si(u_1)du_1&duda
{\bf{e}}nd{align*}
Write $Ju_1=n{\bf{e}}xp(A(Ju_1))k_0(Ju_1)$ with $n \in U(\mathbb{R})$ and $k_0(Ju_1) \in K_\infty$.
Then $A(Ju_1au)=A(Ju_1)+A(k_0au)$.
So the integral we have to evaluate becomes
\begin{align*}
\int_{{A}^+(\mathbb{R})}\int_{U(\mathbb{R})}& e^{(\rho+i\texttt{n}u)(A(Ju_1)}\mathfrak{p}si(u_1)\int_{U(\mathbb{R})}f_\infty(au)e^{(\rho+i\texttt{n}u)(A(k_0(Ju_1)au))}dudu_1da\\
&= \int_{{A}^+(\mathbb{R})}\int_{U(\mathbb{R})} e^{(\rho+i\texttt{n}u)(A(Ju_1)}\mathfrak{p}si(u_1)\int_{\text{Sp}_4(\mathbb{R})}f_\infty(g)e^{(\rho+i\texttt{n}u)(A(k_0(Ju_1)g))}dgdu_1da\\
&= \int_{{A}^+(\mathbb{R})}\int_{U(\mathbb{R})} e^{(\rho+i\texttt{n}u)(A(Ju_1)}\mathfrak{p}si(u_1)\int_{\text{Sp}_4(\mathbb{R})}f_\infty(g)e^{(\rho+i\texttt{n}u)(A(g))}dgdu_1da\\
&=W(i\texttt{n}u,1,\mathfrak{o}verline{\mathfrak{p}si})\tilde{f}(-i\texttt{n}u).
{\bf{e}}nd{align*}
{\bf{e}}nd{proof}
\mathbf{s}ubsubsection{Estimates for the Whittaker function}
We close this section with some estimates for the Whittaker function to be used later on.
We begin with recalling the following estimate for Bessel $K$ functions.
\begin{lemma}\label{EstimateBessel}
Let $\mathbf{s}igma>0$. For $\mathbb{R}e(\texttt{n}u)\in [-\mathbf{s}igma,\mathbf{s}igma]$ we have for all ${\bf{e}}psilon>0$
$$
K_\texttt{n}u(u) \ll \left\lbrace\begin{array}{ccc}
(1+|\mathbf mathcal Im(\texttt{n}u)|)^{\mathbf{s}igma+{\bf{e}}psilon} u^{-\mathbf{s}igma-{\bf{e}}psilon} & \mathbf mbox{if} & 0<u\le 1+\bm{f}rac{\mathfrak{p}i}2|\mathbf mathcal Im(\texttt{n}u)|,\\
u^{-\bm{f}rac12} e^{-u} & \mathbf mbox{if} & u > 1+\bm{f}rac{\mathfrak{p}i}2|\mathbf mathcal Im(\texttt{n}u)|.
{\bf{e}}nd{array}
\right.
$$
{\bf{e}}nd{lemma}
In the following lemma, we have only used trivial bounds and haven't seek for optimality.
\begin{lemma}\label{TrivialWhittaker}
Let $\mathbf{s}igma>0$. Let $\texttt{n}u \in \mathbf mathcal Lie{a}^*_\mathbf mathbb{C}$ with $-\mathbf{s}igma < \bm{f}rac{\mathbb{R}e(\texttt{n}u_1-\texttt{n}u_2)}2, \bm{f}rac{\mathbb{R}e(\texttt{n}u_1+\texttt{n}u_2)}{2} < \mathbf{s}igma$ and $a \in A^{+}(\mathbb{R})$.
For simplicity, set $r_1=\bm{f}rac{|\mathbf mathcal Im(\texttt{n}u_1-\texttt{n}u_2)|}2$ and $r_2=\bm{f}rac{|\mathbf mathcal Im(\texttt{n}u_1+\texttt{n}u_2)|}{2} $
Then for all ${\bf{e}}psilon>0$ we have
\begin{align*}
\mathbf mathcal{W}(\texttt{n}u,a,\mathfrak{p}si) &\ll (1+r_1)^{\mathbf{s}igma+1+{\bf{e}}psilon} (1+r_2)^{\mathbf{s}igma+1+{\bf{e}}psilon}a_1a_2^{-2\mathbf{s}igma-{\bf{e}}psilon}\\
&+(1+r_1)^{-\bm{f}rac32} (1+r_2)^{-\bm{f}rac32}a_1a_2^{2}\\
&+(1+r_1)^{\mathbf{s}igma+{\bf{e}}psilon}(1+r_2)^{-(\mathbf{s}igma+\bm{f}rac52+{\bf{e}}psilon)}a_1^{-2\mathbf{s}igma-1-{\bf{e}}psilon}a_2^2\\
&+(1+r_1)^{-(\mathbf{s}igma+\bm{f}rac52+{\bf{e}}psilon)}(1+r_2)^{\mathbf{s}igma+{\bf{e}}psilon}a_1^{-2\mathbf{s}igma-1-{\bf{e}}psilon}a_2^2.
{\bf{e}}nd{align*}
{\bf{e}}nd{lemma}
\begin{proof}
Follows trivially from the explicit integral representation~(\ref{NiwaIntegral}) and Lemma~\ref{EstimateBessel}.
{\bf{e}}nd{proof}
\begin{proposition}\label{WhittakerSpectral}
Let $a \in A^{+}(\mathbb{R})$.
Then, for $\mathbb{R}e(\texttt{n}u)$ small enough we have for all ${\bf{e}}psilon >0$
$$W(\texttt{n}u,a,\mathfrak{p}si) \ll_{\mathbb{R}e(\texttt{n}u),a} \mathfrak{p}rod_{\mathbf mathfrak{a}lpha \in \mathcal Phi} |\mathbf{s}cal{\mathbf mathcal Im(\texttt{n}u)}{\mathbf mathfrak{a}lpha_0}|^{2|\mathbf{s}cal{\mathbb{R}e(\texttt{n}u)}{\mathbf mathfrak{a}lpha_0}|+{\bf{e}}psilon}.$$
{\bf{e}}nd{proposition}
\begin{proof}
Observe that, if $\mathbb{R}e(\texttt{n}u) \in \mathbf mathcal Lie{a}^*_+$, then the claim follows by the trivial bound~(\ref{majoration}).
Next, if $\mathbb{R}e(\texttt{n}u)$ belongs to any open Weyl chamber, there is $\mathbf{s}igma \in \text{O}mega$ such that $Re(\mathbf{s}igma \cdot \texttt{n}u) \in \mathbf mathcal Lie{a}^*_+$.
The functional equation~({\ref{whittakerftneq}}) gives
$$W(\texttt{n}u,a,\mathfrak{p}si)=\mathfrak{p}rod_{\mathbf mathfrak{a}lpha \in \mathcal Phi} \bm{f}rac{\mathbf mathfrak{G}amma\left(\bm{f}rac12+\mathbf{s}cal{\mathbf{s}igma \cdot \texttt{n}u}{\mathbf mathfrak{a}lpha_0}\right)}{\mathbf mathfrak{G}amma\left(\bm{f}rac12+\mathbf{s}cal{ \texttt{n}u}{\mathbf mathfrak{a}lpha_0}\right)}W(\mathbf{s}igma \cdot\texttt{n}u,a,\mathfrak{p}si).$$
Since the Weyl group acts by permutation on the set of (positive and negative) roots, the product can be written as
$$\mathfrak{p}rod_{\mathbf mathfrak{a}lpha \in \mathcal Phi_\mathbf{s}igma} \bm{f}rac{\mathbf mathfrak{G}amma\left(\bm{f}rac12-\mathbf{s}cal{\texttt{n}u}{\mathbf mathfrak{a}lpha_0}\right)}{\mathbf mathfrak{G}amma\left(\bm{f}rac12+\mathbf{s}cal{ \texttt{n}u}{\mathbf mathfrak{a}lpha_0}\right)}
\ll \mathfrak{p}rod_{\mathbf mathfrak{a}lpha \in \mathcal Phi_\mathbf{s}igma} |\mathbf{s}cal{\mathbf mathcal Im(\texttt{n}u)}{\mathbf mathfrak{a}lpha_0}|^{-2\mathbf{s}cal{\mathbb{R}e(\texttt{n}u)}{\mathbf mathfrak{a}lpha_0}},$$
where $\mathcal Phi_\mathbf{s}igma$ is the set of positive roots whose image by $\mathbf{s}igma$ is a negative root and we have used
that $|\mathbf mathfrak{G}amma(x+iy)| \mathbf{s}im \mathbf{s}qrt{2\mathfrak{p}i}e^{-\bm{f}rac{\mathfrak{p}i}{2}|y|}|y|^{x-\bm{f}rac12}$ as $|y| \to \infty$ and that the numerator has no poles because
$\mathbb{R}e(\texttt{n}u)$ is small enough.
But if $\mathbf{s}igma \cdot \mathbf mathfrak{a}lpha$ is a negative root then we have $\mathbf{s}cal{\mathbb{R}e(\texttt{n}u)}{\mathbf mathfrak{a}lpha_0} <0$ and so we are done in this case again.
Finally, if $\mathbb{R}e(\texttt{n}u)$ belongs to a wall of a Weyl chamber, by Lemma~\ref{TrivialWhittaker} we may apply the Phragm{\'e}n-Lindel{\"o}f principle
to deduce the result.
{\bf{e}}nd{proof}
\mathbf{s}ection{Eisenstein series and the spectral decomposition}
The goal of Eisenstein series is to describe the continuous spectrum.
The latter is an orthogonal direct sum over standard parabolic subgroups $P$, each summand of which
is a direct integral parametrized by $i\mathbf mathcal Lie{a}^*_P$.
Eisenstein series will give intertwining operators from some representation induced from
$M_P$ to the corresponding part of the continuous spectrum.
One thus wants to define $E(\cdot,\mathfrak{p}hi,\texttt{n}u)$ for $\mathfrak{p}hi$ in the space $\mathbf mathscr H_P$ of the aforementioned
induced representation, and for $\texttt{n}u \in i\mathbf mathcal Lie{a}^*_P$.
Because of convergence issues, one originally defines $E(\cdot,\mathfrak{p}hi,\texttt{n}u)$ for $\mathfrak{p}hi$
lying a certain dense space of automorphic forms $\mathbf mathscr H_P^0 \mathbf{s}ubset \mathbf mathscr H_P$ and for $\texttt{n}u \in \mathbf mathcal Lie{a}^*_P(\mathbf mathbb{C})$ with large enough real part.
The definition is then extended to all $\mathfrak{p}hi$ in the completion of $\mathbf mathscr H_P^0$ and to all
$\texttt{n}u \in \mathbf mathcal Lie{a}^*_P(\mathbf mathbb{C})$. Our exposition follows Arthur, and in particular~\cite{ArthurIntro}.
\mathbf{s}ubsection{Definition of Eisenstein series}~\label{DefOfES}
Fix a standard parabolic subgroup $P=N_PM_P$ throughout this section, and let $A_P$ be the centre of $M_P$,
and $A_P^+(\mathbb{R})$ be the connected component of $1$ in $A_P(\mathbb{R})$.
Let $R_{M_P, \text{disc}}$ be the restriction of the right regular representation of $M_P(\mathbf mathbb{A})$
on the subspace of $L^2(M_P(\mathbb{Q})A_P^+(\mathbb{R})\backslash M_P(\mathbf mathbb{A}))$ that decompose discretely.
For $\texttt{n}u \in \mathbf mathcal Lie{a}_P^*(\mathbf mathbb{C}),$ consider the tensor product
$R_{M_P, \text{disc}, \texttt{n}u}(x)=R_{M_P, \text{disc}}(x)e^{\texttt{n}u (H_{M_P}(x))}$.
The continuous spectrum is described via the Eisenstein series in terms of the induced representation
$$\mathbf mathcal I_P(\texttt{n}u)=\text{Ind}_{P(\mathbf mathbb{A})}^{G(\mathbf mathbb{A})}(I_{N_P(\mathbf mathbb{A})} \mathfrak{o}times R_{M_P, \text{disc}, \texttt{n}u} ).$$
The space of this induced representation is independant of $\texttt{n}u$ and is given in the following definition.
\begin{definition}\label{defofHP}
Let $P=N_PM_P$ be the Levi decomposition of $P$,
$A_P$ be the centre of $M_P$ and $A_P^+(\mathbb{R})$ be the connected component of $1$ in $A_P(\mathbb{R})$.
We define $\mathbf mathscr H_P$ to be the Hilbert space obtained by completing
the space $\mathbf mathscr H_P^0$ of functions
\begin{equation}\label{Hp}
\mathfrak{p}hi : N_P(\mathbf mathbb{A}) M_P(\mathbb{Q}) A_P^+(\mathbb{R}) \backslashslash G(\mathbf mathbb{A}) \to \mathbf mathbb{C}
{\bf{e}}nd{equation} such that
\begin{enumerate}
\item \label{discretness}
for any $x \in G(\mathbf mathbb{A})$, the function $M_P(\mathbf mathbb{A}) \to \mathbf mathbb{C}, m \mathbf mapsto \mathfrak{p}hi(mx)$
is $\mathbf mathscr Z_{M_P}$-finite, where $\mathbf mathscr Z_{M_P}$ is the centre of the universal
enveloping algebra of $\mathbf mathfrak M_P(\mathbf mathbb{C})$,
\item \label{Kfinitness} $\mathfrak{p}hi$ is right $K$-finite,
\item \label{L2} $\| \mathfrak{p}hi\|^2 = \int_K \int_{A_P(\mathbb{R})^+ M_P(\mathbb{Q}) \backslashslash M_P(\mathbf mathbb{A})} |\mathfrak{p}hi(mk)|^2dm dk
< \infty.$
{\bf{e}}nd{enumerate}
{\bf{e}}nd{definition}
Then the representation $\mathbf mathcal I_P(\texttt{n}u)$ acts on $\mathbf mathscr H_P$ via
$$(\mathbf mathcal I_P(\texttt{n}u,y) \mathfrak{p}hi)(x)=\mathfrak{p}hi(xy){\bf{e}}xp((\texttt{n}u+\rho_P)(H_P(xy))){\bf{e}}xp(-(\texttt{n}u+\rho_P)(H_P(x))),$$
and is unitary for $\texttt{n}u \in i\mathbf mathcal Lie{a}_P(\mathbf mathbb{C})$.
We now define the Eisenstein series attached to $P$.
We may $H_P$ extends to $P(\mathbb{Q}) \backslash G(\mathbf mathbb{A})$ by setting $H_P(nmk)=H_P(m)$ ($n \in N_P, m \in M_P, k \in K$), therefore
the expression in the following proposition is well defined.
\begin{proposition}\label{defofES}
For $\texttt{n}u \in \mathbf mathcal Lie{a}_P(\mathbf mathbb{C})$ with large enough real part, if $x \in G(\mathbf mathbb{A})$ and $\mathfrak{p}hi \in \mathbf mathscr H_P^0$, the
{\bf Eisenstein series}
$$E(x, \mathfrak{p}hi, \texttt{n}u)= \mathbf{s}um_{\delta \in P(\mathbb{Q}) \backslashslash G(\mathbb{Q})} \mathfrak{p}hi(\delta x) {\bf{e}}xp((\texttt{n}u+\rho_P)(H_P(\delta x))$$
converges absolutely.
{\bf{e}}nd{proposition}
Langlands provided analytic continuation of Eisenstein series, as well as the spectral decomposition of $L^2(Z(\mathbb{R})G(\mathbb{Q}) \backslash G(\mathbf mathbb{A}))$.
The latter gives a decomposition of the right regular representation $R$ as direct sum over association classes of parabolic subgroups.
The class of $G$, viewed as a parabolic subgroup itself, gives the {\bf discrete spectrum}.
It consists on one hand of cuspidal functions on $Z(\mathbb{R})G(\mathbb{Q}) \backslash G(\mathbf mathbb{A})$ and on the other hand
of residues of Eisenstein series attached to proper parabolic subgroups.
The contribution of the other classes is given by direct integrals of corresponding induced representations and gives the
{\bf continuous spectrum}.
We now describe explicitly the Eisenstein series that are relevant for us.
\mathbf{s}ubsection{Action of the centre and of the compact \texorpdfstring{$\mathbf mathfrak{G}amma$}{Gamma}}
Since our test function $f$ is bi-$\mathbf mathfrak{G}amma$-invariant and has central character $\mathfrak{o}mega$,
Eisenstein series occurring in the spectral expansion of its kernel $K_f$ are only from the subspaces of $\mathbf mathscr H_P$ satisfying similar properties
(see Lemma~\ref{UnprovedLemma} below for a formal justification).
Using the Peter-Weyl Theorem, we can further reduce:
\begin{lemma}\label{CentralCharactersofP}
Let $P$ be a standard parabolic subgroup and $A_P$ it centre.
Let $\mathbf mathscr H_P^\mathbf mathfrak{G}amma(\mathfrak{o}mega)$ be the closed subspace of $\mathbf mathscr H_P$
consisting in functions $\mathfrak{p}hi$ such that for all $z \in Z(\mathbf mathbb{A})$ and $k \in \mathbf mathfrak{G}amma$, we have $\mathfrak{p}hi(zgk)=\mathfrak{o}mega(z)\mathfrak{p}hi(g)$.
Then
\begin{equation}\label{DecompOfCentralChar}
\mathbf mathscr H_P^\mathbf mathfrak{G}amma(\mathfrak{o}mega)=\bigoplus_{\chi} \mathbf mathscr H_P^\mathbf mathfrak{G}amma(\chi)
{\bf{e}}nd{equation}
where the $\chi$-orthogonal direct sum ranges over characters of $A^+_P(\mathbb{R})A_P(\mathbb{Q}) \backslashslash A_P(\mathbf mathbb{A})$ that coincide with
$\mathfrak{o}mega$ on $Z(\mathbf mathbb{A})$, and $\mathbf mathscr H_P^\mathbf mathfrak{G}amma(\chi)$ is the subspace of $\mathbf mathscr H_P^\mathbf mathfrak{G}amma(\mathfrak{o}mega)$
consisting in functions $\mathfrak{p}hi$ such that for all $z \in A_P(\mathbf mathbb{A})$, $\mathfrak{p}hi(zg)=\chi(z)\mathfrak{p}hi(g)$.
{\bf{e}}nd{lemma}
\mathbf{s}ubsection{Explicit description of Eisenstein series}
If $R_{M_P,\text{disc}}=\bigoplus_{\mathfrak{p}i} \mathfrak{p}i = \bigoplus_\mathfrak{p}i \left(\bigotimes_v \mathfrak{p}i_v \right)$ is the decomposition of
$R_{M_P,\text{disc}}$ into irreducible representations $\mathfrak{p}i = \bigotimes_v \mathfrak{p}i_v$ of $M_P(\mathbf mathbb{A}) / A_P(\mathbb{R})^+$, then we have
$$\mathbf mathcal I_P(\texttt{n}u)=\bigoplus_{\mathfrak{p}i} \mathbf mathcal I_P(\mathfrak{p}i_\texttt{n}u) = \bigoplus_\mathfrak{p}i \left(\bigotimes_v \mathbf mathcal I_P(\mathfrak{p}i_{v,\texttt{n}u})\right).$$
Moreover the representation space of each $\mathbf mathcal I_P(\mathfrak{p}i_\texttt{n}u)$ does not depend on $\texttt{n}u$.
Hence, to describe the spaces $\mathbf mathscr H_P^\mathbf mathfrak{G}amma(\chi)$ it suffices to describe
\begin{itemize}
\item the irreducible representations $\mathfrak{p}i$ with central character $\chi$ occurring $R_{M_P,\text{disc}}$,
\item the $\mathbf mathfrak{G}amma$-fixed subspace of each representation $\mathbf mathcal I_P(\mathfrak{p}i_\texttt{n}u)$.
{\bf{e}}nd{itemize}
By the Iwasawa decomposition, elements of this space may be viewed as families of functions indexed by $\mathbf mathfrak{G}amma \backslash K$ satisfying some
compatibility condition that we proceed to make explicit now.
We also prove that the Archimedean part of $\mathbf mathcal I_P(\mathfrak{p}i_\texttt{n}u)$ is a principal series representation, and we provide its spectral parameter.
\mathbf{s}ubsubsection{Borel Eisenstein series}
\begin{lemma}\label{BorelSpectralParameter}
The irreducible representations occurring in $R_{T,\text{disc}}$ are precisely characters $\chi$ of $T^+(\mathbb{R})T(\mathbb{Q}) \backslash T(\mathbf mathbb{A})$.
Let $\chi$ be such character and $\texttt{n}u \in i\mathbf mathcal Lie{a}^*(\mathbf mathbb{C})$.
The Archimedean part of $\mathbf mathcal I_B(\chi_\texttt{n}u)$ is an irreducible principal series representation with spectral parameter $\texttt{n}u$.
{\bf{e}}nd{lemma}
\begin{proof}
The first part is because $T^+(\mathbb{R})T(\mathbb{Q}) \backslash T(\mathbf mathbb{A})$ is abelian.
For the second part, since $\chi_\infty=1$ we have $\mathbf mathcal I_B(\chi_\texttt{n}u)_\infty= \mathbf mathcal I_B(e^\texttt{n}u)$, which is irreducible
because $\texttt{n}u \in i\mathbf mathcal Lie{a}^*(\mathbf mathbb{C})$ (see~\cite{Muic}*{Lemma~5.1}).
{\bf{e}}nd{proof}
Characters $\chi$ of $T^+(\mathbb{R})T(\mathbb{Q}) \backslash T(\mathbf mathbb{A})$ that coincide with $\mathfrak{o}mega$ on $Z(\mathbf mathbb{A})$ are in one-to-one
correspondence with triplets $(\mathfrak{o}mega_1, \mathfrak{o}mega_2, \mathfrak{o}mega_3)$ of characters of $\mathbb{R}_{>0}\mathbb{Q}^\times \backslashslash \mathbf mathbb{A}^\times$ satisfying
$\mathfrak{o}mega_1 \mathfrak{o}mega_2 \mathfrak{o}mega_3^2 = \mathfrak{o}mega$, via
$$\chi\left(\left[ \begin{smallmatrix}x&&&\\&y&&\\&&tx^{-1}&\\&&&ty^{-1}{\bf{e}}nd{smallmatrix}\right]\right)=\mathfrak{o}mega_1(x)\mathfrak{o}mega_2(y) \mathfrak{o}mega_3(t).$$
Define a character of $B$ by
$$\mathfrak{o}mega\left(\left[ \begin{smallmatrix}x&*&*&*\\ &y&*&*\\&&tx^{-1}&*\\&&&ty^{-1}{\bf{e}}nd{smallmatrix}\right]\right)
=\mathfrak{o}mega_1(x)\mathfrak{o}mega_2(y)\mathfrak{o}mega_3(t)$$
(note that this notation is sound, as it coincides with our original $\mathfrak{o}mega$ on scalar matrices.)
\begin{proposition}\label{Borelsummand}
Let $\chi=(\mathfrak{o}mega_1,\mathfrak{o}mega_2,\mathfrak{o}mega_3)$ with $\mathfrak{o}mega_1 \mathfrak{o}mega_2 \mathfrak{o}mega_3^2 = \mathfrak{o}mega$.
Consider $(\mathfrak{p}hi_k)_{k \in K / \mathbf mathfrak{G}amma}$ such that
\begin{enumerate}
\item for all $k$, $\mathfrak{p}hi_k \in \mathbf mathbb{C}$,
\item \label{Borelcompatibility} if $\gamma \in K \cap B(\mathbf mathbb{A})$ then for all $k$,
$\mathfrak{p}hi_k=\chi(\gamma^{-1})\mathfrak{p}hi_{\gamma k}$.
{\bf{e}}nd{enumerate}
Then the function on $G(\mathbf mathbb{A})$ given for $u \in U(\mathbf mathbb{A}), t \in T(\mathbf mathbb{A}), k \in K$ by
\begin{equation}\label{Borelnmk}
\mathfrak{p}hi(utk)= \chi(t) \mathfrak{p}hi_k,
{\bf{e}}nd{equation}
is well-defined and belongs to ${\mathbf mathscr H_B}^\mathbf mathfrak{G}amma(\chi)$.
Moreover, every function in ${\mathbf mathscr H_{B}}^\mathbf mathfrak{G}amma(\chi)$ has this shape.
{\bf{e}}nd{proposition}
\begin{proof}
We first prove that $\mathfrak{p}hi$ is well-defined.
Suppose $u_1t_1k_1=u_2t_2k_2$.
In particular $k_1k_2^{-1}=(u_1t_1)^{-1}(u_2t_2) \in B(\mathbf mathbb{A}) \cap K$.
Therefore
$$
\chi(t_1)\mathfrak{p}hi_{k_1}= \chi(t_1)\chi({k_1k_2^{-1}})\mathfrak{p}hi_{k_2}\\
=\chi(t_2)\mathfrak{p}hi_{k_2}.$$
Next we show that $\mathfrak{p}hi$ belongs indeed to ${\mathbf mathscr H_{B}}^\mathbf mathfrak{G}amma(\mathfrak{o}mega_1, \mathfrak{o}mega_2, \mathfrak{o}mega_3)$.
The fact that $\mathfrak{p}hi$ is invariant on the left by $U(\mathbf mathbb{A})T(\mathbb{Q})T(\mathbb{R})^+$, the right invariance by $\mathbf mathfrak{G}amma$ and the fact that $\mathfrak{p}hi$ transforms
under $T(\mathbf mathbb{A})$ according to $\chi$ are obvious from the definition. Finally,
\begin{align*}
\int_K \int_{T(\mathbb{R})^+ T(\mathbb{Q}) \backslashslash T(\mathbf mathbb{A})} |\mathfrak{p}hi(mk)|^2dmdk
&=\int_K \int_{(\mathbb{R}_{>0} \mathbb{Q}^\times \backslashslash \mathbf mathbb{A}^\times)^3} |\mathfrak{p}hi_k|^2dmdk\\
&=Vol(\mathbb{R}_{>0} \mathbb{Q}^\times \backslashslash \mathbf mathbb{A}^\times)^3Vol(\mathbf mathfrak{G}amma)\mathbf{s}um_{k \in K / \mathbf mathfrak{G}amma} |\mathfrak{p}hi_k|^2
<\infty
{\bf{e}}nd{align*}
since $\mathbb{R}_{>0}\mathbb{Q}^\times \backslashslash \mathbf mathbb{A}^\times$ is compact and $K / \mathbf mathfrak{G}amma$ is finite.
As a last point, we show that we thus exhaust all of ${\mathbf mathscr H_{B}}^\mathbf mathfrak{G}amma(\mathfrak{o}mega_1, \mathfrak{o}mega_2,\mathfrak{o}mega_3)$.
Let $\mathfrak{p}hi \in {\mathbf mathscr H_{B}}^\mathbf mathfrak{G}amma(\mathfrak{o}mega_1, \mathfrak{o}mega_2,\mathfrak{o}mega_3)$. Define
$$\mathfrak{p}hi_k=\mathfrak{p}hi(k).$$
Then it is clear that equation~(\ref{Borelnmk}) holds.
As for condition~\ref{Klingencompatibility}, note that if $\gamma=t_\gamma u_\gamma \in K \cap B(\mathbf mathbb{A})$
with $t_\gamma \in T(\mathbf mathbb{A})$ and $u_\gamma \in U(\mathbf mathbb{A})$ then
\begin{align*}
\mathfrak{p}hi_{\gamma k}&=\mathfrak{p}hi(\gamma k)\\
&=\mathfrak{p}hi (t_\gamma u_\gamma k) = \chi(t_\gamma) \mathfrak{p}hi(k)\\
&=\chi(\gamma)\mathfrak{p}hi_k.
{\bf{e}}nd{align*}
{\bf{e}}nd{proof}
\begin{remark}
Consider the action of $K \cap B(\mathbf mathbb{A})$ on $K / \mathbf mathfrak{G}amma$ by multiplication on the left.
Then the compatibility condition 2. can only be met if $\mathfrak{o}mega$ is trivial on
the stabilizer of each element of $K / \mathbf mathfrak{G}amma$.
In this case, the dimension of ${\mathbf mathscr H_{B}}^\mathbf mathfrak{G}amma(\chi)$ is the number of distinct orbits.
{\bf{e}}nd{remark}
\mathbf{s}ubsubsection{Klingen Eisenstein series}~\label{KES}
Characters $\chi$ of $\mathbf mathbb{A}k^+(\mathbb{R})\mathbf mathbb{A}k(\mathbb{Q}) \backslash \mathbf mathbb{A}k(\mathbf mathbb{A})$ that coincide with $\mathfrak{o}mega$ on $Z(\mathbf mathbb{A})$ are in one-to-one
correspondence with pairs $(\mathfrak{o}mega_1, \mathfrak{o}mega_2)$ of characters of $\mathbb{R}_{>0}\mathbb{Q}^\times \backslashslash \mathbf mathbb{A}^\times$ satisfying
$\mathfrak{o}mega_1 \mathfrak{o}mega_2 = \mathfrak{o}mega$, via
$$\chi\left(\diag{u}{t}{u}{t^{-1}u^2}\right)=\mathfrak{o}mega_1(u)\mathfrak{o}mega_2(t).$$
For convenience, if $A=\mathbf mat{a}{b}{c}{d} \in \mathbf mathfrak{G}L_2$, define
$\iota_A=\left[ \begin{smallmatrix}a&&b&\\ &1&&\\ c&&d&\\&&&\det(A){\bf{e}}nd{smallmatrix}\right]\in \mathbf mathcal Mk,$
and if
$$
p=\left[ \begin{smallmatrix}
a & &b & * \\
* &t &* & * \\
c & &d & * \\
& & & t^{-1}\det(A)
{\bf{e}}nd{smallmatrix}\right],
$$
define $\mathbf{s}igma_{\text K}(p)=A$ and $t(p)=t$.
We may extend $\mathbf{s}igma_{\text K}$ to all of $P_{\text{K}}(\mathbf mathbb{A})$ by setting $\mathbf{s}igma_{\text K}(nm)=\mathbf{s}igma_{\text K}(m)$ (and similarly for $t$),
and we may view $\mathfrak{o}mega_2$ as the character of $P_{\text{K}}(\mathbf mathbb{A})$ defined by $\mathfrak{o}mega_2(p)=\mathfrak{o}mega_2(t(p))$.
\begin{lemma}\label{KlingenInducing}
Let $\chi=(\mathfrak{o}mega_1,\mathfrak{o}mega_2)$ with $\mathfrak{o}mega_1\mathfrak{o}mega_2=\mathfrak{o}mega$.
The irreducible representations with central character $\chi$ occurring in $R_{\mathbf mathcal Mk,\text{disc}}$ are twists $\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i$, where $\mathfrak{p}i$
occurs in the discrete spectrum of $L^2(\mathbb{R}_{>0}\mathbf mathfrak{G}L_2(\mathbb{Q})\backslash \mathbf mathfrak{G}L_2(\mathbf mathbb{A}))$ and has central character $\mathfrak{o}mega_1$.
{\bf{e}}nd{lemma}
\begin{proof}
Let $\mathfrak{p}i$ be an irreducible representations with central character $\chi$ occurring $R_{\mathbf mathcal Mk,\text{disc}}$.
By definition, we may realize $\mathfrak{p}i$ in the subspace of $L^2(\mathbf mathcal Mk(\mathbb{Q})\mathbf mathbb{A}k^+(\mathbb{R})\backslash \mathbf mathcal Mk(\mathbf mathbb{A}))$ consisting of functions with central character
$\chi$.
This space identifies with $L^2(\mathbb{R}_{>0}\mathbf mathfrak{G}L_2(\mathbb{Q})\backslash \mathbf mathfrak{G}L_2(\mathbf mathbb{A}),\mathfrak{o}mega_1)$ via
$\mathfrak{p}hi \mathbf mapsto \left( \left[ \begin{smallmatrix}
a & &b & \\
&t & & \\
c & &d & \\
& & & t^{-1}\det(A)
{\bf{e}}nd{smallmatrix}\right] \mathbf mapsto \mathfrak{o}mega_2(t)\mathfrak{p}hi(\mathbf mat{a}{b}{c}{d}) \right).$
{\bf{e}}nd{proof}
\begin{proposition}\label{Klingensummand}
Let $\chi=(\mathfrak{o}mega_1,\mathfrak{o}mega_2)$ with $\mathfrak{o}mega_1\mathfrak{o}mega_2=\mathfrak{o}mega$.
Let $(\mathfrak{p}i,V_\mathfrak{p}i)$ occur in the discrete spectrum of $L^2(\mathbb{R}_{>0}\mathbf mathfrak{G}L_2(\mathbb{Q})\backslash \mathbf mathfrak{G}L_2(\mathbf mathbb{A}))$ with central character $\mathfrak{o}mega_1$.
Consider $(\mathfrak{p}hi_k)_{k \in K / \mathbf mathfrak{G}amma}$ such that
\begin{enumerate}
\item for all $k$, $\mathfrak{p}hi_k \in V_\mathfrak{p}i$,
\item \label{Klingencompatibility} if $\gamma \in K \cap P_{\text{K}}(\mathbf mathbb{A})$ then for all $k$,
$\mathfrak{p}hi_k(\cdot \mathbf{s}igma_{\text K}(\gamma))=\mathfrak{o}mega_2 \circ t(\gamma^{-1})\mathfrak{p}hi_{\gamma k}$.
{\bf{e}}nd{enumerate}
Then the function on $G(\mathbf mathbb{A})$ given for $n \in N_{\text{K}}(\mathbf mathbb{A}), m \in \mathbf mathcal Mk(\mathbf mathbb{A}), k \in K$ by
\begin{equation}\label{Klingennmk}
\mathfrak{p}hi(nmk)= \mathfrak{o}mega_2 \circ t(m) \mathfrak{p}hi_k(\mathbf{s}igma_{\text K}(m)),
{\bf{e}}nd{equation}
is well-defined and belongs to ${\mathbf mathcal I_{P_{\text{K}}}((\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i)_\texttt{n}u)}^\mathbf mathfrak{G}amma$.
Moreover, every function in ${\mathbf mathcal I_{P_{\text{K}}}((\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i)_\texttt{n}u)}^\mathbf mathfrak{G}amma$ has this shape.
{\bf{e}}nd{proposition}
\begin{remark}\label{KlingenMaassForm}
Condition~(\ref{Klingencompatibility}) implies that each $\mathfrak{p}hi_k$ is right-$SO_2(\mathbb{R})$-invariant (and hence must be an adelic Maa{\mathbf{s}s} form or a character).
Indeed, let $v \le \infty$ and let $k_v$ be a compact subgroup of $\mathbf mathfrak{G}L_2(\mathbb{Q}_v)$ such that
$$\left\{\iota_A : A \in k_v\right\} \mathbf{s}ubset K_v.$$
Assume moreover that $K_v=\mathbf mathfrak{G}amma_v$. Then $K / \mathbf mathfrak{G}amma$ is left invariant by $\mathbf mathfrak{G}amma_v$, hence for all
$A \in k_v$ we have $\mathfrak{p}hi_k( \cdot A)=\mathfrak{p}hi_{\iota_A k}=\mathfrak{p}hi_k$.
In particular, for $v=\infty$, we may take $k_v=O_2(\mathbb{R})$, hence the claim.
{\bf{e}}nd{remark}
\begin{proof}
We first prove that $\mathfrak{p}hi$ is well-defined.
Suppose $n_1m_1k_1=n_2m_2k_2$.
In particular $k_2k_1^{-1}=(n_2m_2)^{-1}(n_1m_1) \in P_{\text{K}}(\mathbf mathbb{A}) \cap K$.
Therefore $\mathbf{s}igma_{\text K}(m_1)=\mathbf{s}igma_{\text K}(n_1m_1)=\mathbf{s}igma_{\text K}(n_2m_2k_2k_1^{-1})=\mathbf{s}igma_{\text K}(m_2)\mathbf{s}igma_{\text K}(k_2k_1^{-1})$.
Then
\begin{align*}
\mathfrak{o}mega_2 \circ t(m_1) \mathfrak{p}hi_{k_1}(\mathbf{s}igma_{\text K}(m_1)) &= \mathfrak{o}mega_2 \circ t(m_1)\mathfrak{p}hi_{k_1}(\mathbf{s}igma_{\text K}(m_2)\mathbf{s}igma_{\text K}(k_2k_1^{-1})) \\
&= \mathfrak{o}mega_2 \circ t(m_1) \mathfrak{o}mega_2 \circ t(k_1k_2^{-1}) \mathfrak{p}hi_{k_2}(\mathbf{s}igma_{\text K}(m_2)) = \mathfrak{o}mega_2 \circ t(m_2)\mathfrak{p}hi_{k_2}(\mathbf{s}igma_{\text K}(m_2)).
{\bf{e}}nd{align*}
Next we show that $\mathfrak{p}hi$ belongs indeed to ${\mathbf mathcal I_{P_{\text{K}}}((\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i)_\texttt{n}u)}^\mathbf mathfrak{G}amma$.
The fact that $\mathfrak{p}hi$ is invariant on the left by $N_{\text{K}}(\mathbf mathbb{A})\mathbf mathcal Mk(\mathbb{Q})\mathbf mathbb{A}k(\mathbb{R})^\circ$ and the right invariance by $\mathbf mathfrak{G}amma$ are obvious from the definition.
The fact that $\mathfrak{p}hi$ is square integrable follows from
\begin{align*}
\int_K \int_{\mathbf mathbb{A}k(\mathbb{R})^+ \mathbf mathcal Mk(\mathbb{Q}) \backslashslash \mathbf mathcal Mk(\mathbf mathbb{A})} &|\mathfrak{p}hi(mk)|^2dmdk
=\int_K \int_{\mathbf mathbb{A}k(\mathbb{R})^+ \mathbf mathcal Mk(\mathbb{Q}) \backslashslash \mathbf mathcal Mk(\mathbf mathbb{A})} |\mathfrak{p}hi_k(\mathbf{s}igma_{\text K}(m))|^2dmdk\\
&=\mathbf{s}um_{k \in K / \mathbf mathfrak{G}amma} Vol(\mathbf mathfrak{G}amma) \int_{\mathbb{R}_{>0} \mathbf mathfrak{G}L_2(\mathbb{Q}) \backslashslash \mathbf mathfrak{G}L_2(\mathbf mathbb{A})} \int_{\mathbb{R}_{>0}\mathbb{Q}^\times \backslashslash \mathbf mathbb{A}^\times}
|\mathfrak{p}hi_k(x)|^2dtdx
<\infty
{\bf{e}}nd{align*}
since $\mathfrak{p}hi_k$ is square integrable, $\mathbb{R}_{>0}\mathbb{Q}^\times \backslashslash \mathbf mathbb{A}^\times$ is compact and $K / \mathbf mathfrak{G}amma$ is finite.
Finally, we need to show that for all $g=nmk$, the function $\mathfrak{p}hi_g : \mathbf mathcal Mk(\mathbf mathbb{A}) \to \mathbf mathbb{C}, m_1 \mathbf mapsto \mathfrak{p}hi(m_1g)$
transform under $\mathbf mathcal Mk(\mathbf mathbb{A})$ on the right according to $\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i$.
Indeed, for $m_1 \in \mathbf mathcal Mk(\mathbf mathbb{A})$ we have
$$\mathfrak{p}hi_g(m_1)=\mathfrak{p}hi(m_1nmk)=\mathfrak{p}hi(\underbrace{m_1nm_1^{-1}}_{\in N_{\text{K}}(\mathbf mathbb{A})}m_1mk)=\mathfrak{o}mega_2 \circ t (m) \mathfrak{o}mega_2 \circ t(m_1) \mathfrak{p}hi_k(m_1)$$
hence the claim since $\mathfrak{p}hi_k \in V_\mathfrak{p}i$.
As a last point, we show that ${\mathbf mathcal I_{P_{\text{K}}}((\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i)_\texttt{n}u)}^\mathbf mathfrak{G}amma$ consists exactly in such functions.
Let $\mathfrak{p}hi \in {\mathbf mathcal I_{P_{\text{K}}}((\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i)_\texttt{n}u)}^\mathbf mathfrak{G}amma$. Define
$$\mathfrak{p}hi_k(A)=\mathfrak{p}hi(\iota_A k).$$
Then it is clear that equation~(\ref{Klingennmk}) holds.
As for condition~(\ref{Klingencompatibility}), note that if $\gamma=n_\gamma m_\gamma \in K \cap P_{\text{K}}(\mathbf mathbb{A})$ then
\begin{align*}
\mathfrak{p}hi_k(A\mathfrak{p}i(\gamma))&=\mathfrak{p}hi(\iota_A\iota_{\mathfrak{p}i(\gamma)}k)\\
&=\mathfrak{p}hi \left(\iota_A \left[ \begin{smallmatrix}1&&&\\&t(\gamma)^{-1}&&\\&&1&\\&&&t(\gamma){\bf{e}}nd{smallmatrix}\right]
m_\gamma k \right)\\
&=\mathfrak{o}mega_2 \circ t (\gamma^{-1}) \mathfrak{p}hi(\iota_A n_\gamma^{-1} \gamma k)\\
&=\mathfrak{o}mega_2 \circ t (\gamma^{-1}) \mathfrak{p}hi(\underbrace{\iota_A n_\gamma^{-1} \iota_A^{-1}}_{\in N_{\text{K}}} \iota_A \gamma k)\\
&=\mathfrak{o}mega_2 \circ t (\gamma^{-1}) \mathfrak{p}hi_{\gamma k}(A).
{\bf{e}}nd{align*}
Finally, by definition of ${\mathbf mathcal I_{P_{\text{K}}}((\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i)_\texttt{n}u)}$ the function $m \mathbf mapsto \mathfrak{p}hi(mk)$ transforms
under $\mathbf mathcal Mk(\mathbf mathbb{A})$ on the right according to $\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i$, from which follows $\mathfrak{p}hi_k$ transforms according to $\mathfrak{p}i$.
{\bf{e}}nd{proof}
Finally we prove the following
\begin{proposition} \label{KlingenSpectralParameter}
Let $(\mathfrak{p}i,V_\mathfrak{p}i)$ occur in the discrete spectrum of $L^2(\mathbb{R}_{>0}\mathbf mathfrak{G}L_2(\mathbb{Q})\backslash \mathbf mathfrak{G}L_2(\mathbf mathbb{A}))$ with central character $\mathfrak{o}mega_1$.
$\mathbf mathcal I_{P_{\text{K}}}((\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i)_\texttt{n}u)$ has a $K_\infty$-fixed vector if and only if $\mathfrak{p}i$ has a $O_2(\mathbb{R})$-fixed vector.
In this case, $\mathbf mathcal I_{P_{\text{K}}}((\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i)_\texttt{n}u)_\infty$ is generic if and only if $\mathfrak{p}i_\infty$ is a principal series.
Finally if $\mathfrak{p}i_\infty$ is a spherical principal series with spectral parameter $s$ and $\texttt{n}u \in i\mathbf mathcal Lie{a}_{\mathbf mathcal Mk}^*$
then $\mathbf mathcal I_{P_{\text{K}}}((\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i)_\texttt{n}u)_\infty$
is a principal series representation with spectral parameter $\texttt{n}u+\texttt{n}u_{\text{K}}(s)$, where $\texttt{n}u_{\text{K}}(s)$ is the element of $\mathbf mathcal Lie{a}^*(\mathbf mathbb{C})$
corresponding to the character $\diag{y^{\bm{f}rac12}}{u}{y^{-\bm{f}rac12}}{u^{-1}} \mathbf mapsto |y|^s$.
{\bf{e}}nd{proposition}
\begin{proof}
The first claim follows immediately from Proposition~\ref{Klingensummand}.
By the spectral decomposition for $\mathbf mathfrak{G}L_2$, if $\mathfrak{p}i$ has a $O_2(\mathbb{R})$-fixed vector then $\mathfrak{p}i_\infty$ is either
a character or a principal series.
But representations induced from a character of the Klingen subgroup are not generic. This shows the second claim.
Finally assume $\mathfrak{p}i_\infty$ is a spherical principal series on $\mathbf mathfrak{G}L_2$ with spectral parameter $s$.
Then we might see $\mathfrak{p}i_\infty$ as the representation of $\mathcal PGL_2(\mathbb{R})$ induced from the character
$\chi_s:\mathbf mat{y^{\bm{f}rac12}}{x}{}{\mathfrak{p}m y^{{-\bm{f}rac12}}} \mathbf mapsto \left|y\right|^s$,
where $s$ is either an imaginary number or a real number with $0<|s|<\bm{f}rac12$.
Define the following subgroups:
$N_1= \left[ \begin{smallmatrix}
1 & &* & \\
&1 & & \\
& &1 & \\
& & & 1
{\bf{e}}nd{smallmatrix}\right] $,
$A_1=\left\{\diag{y^{\bm{f}rac12}}{1}{\mathfrak{p}m y^{-\bm{f}rac12}}{1}:y \texttt{n}eq 0\right\}$, $M_1=\{\iota_A : A \in \mathcal PGL_2(\mathbb{R})\}$.
Note that $N_1N_{\text{K}}=U$, $A_1\mathbf mathbb{A}k(\mathbb{R})=T(\mathbb{R})$ and $M_1\mathbf mathbb{A}k=\mathbf mathcal Mk$
We might view $\chi_s$ as a character of $A_1N_1$.
Since $\mathfrak{o}mega_2$ is trivial on $\mathbf mathbb{A}k(\mathbb{R})$, inducing in stage, we get
\begin{align*}
\mathbf mathcal I_{P_{\text{K}}}((\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i)_\texttt{n}u)_\infty &=\mathbf mathcal Ind_{P_{\text{K}}(\mathbb{R})}^{G(\mathbb{R})}\left(I_{N_{\text{K}}(\mathbb{R})} \mathfrak{o}times e^\texttt{n}u \mathfrak{o}times \mathfrak{p}i_{\infty}\right)\\
&=\mathbf mathcal Ind_{P_{\text{K}}(\mathbb{R})}^{G(\mathbb{R})}\left(I_{N_{\text{K}}(\mathbb{R})} \mathfrak{o}times e^\texttt{n}u \mathfrak{o}times \mathbf mathcal Ind_{A_1N_1}^{M_1}(\chi_s)\right)\\
&=\mathbf mathcal Ind_{P_{\text{K}}(\mathbb{R})}^{G(\mathbb{R})}\mathbf mathcal Ind_{B(\mathbb{R})}^{P_{\text{K}}(\mathbb{R})}\left(I_{N_{\text{K}}(\mathbb{R})} \mathfrak{o}times I_{N_1} \mathfrak{o}times e^{\texttt{n}u+\texttt{n}u_{\text{K}}(s)}\right)\\
&=\mathbf mathcal Ind_{B(\mathbb{R})}^{G(\mathbb{R})}(I_U \mathfrak{o}times e^{\texttt{n}u+\texttt{n}u_{\text{K}}(s)})
{\bf{e}}nd{align*}
Since $\texttt{n}u \in i \mathbf mathcal Lie{a}^*$, by Lemma~5.1 of~\cite{Muic} this representation is irreducible.
{\bf{e}}nd{proof}
\mathbf{s}ubsubsection{Siegel Eisenstein series}~\label{PES}
Characters $\chi$ of $\mathbf mathbb{A}s^+(\mathbb{R})\mathbf mathbb{A}s(\mathbb{Q}) \backslash \mathbf mathbb{A}s(\mathbf mathbb{A})$ that coincide with $\mathfrak{o}mega$ on $Z(\mathbf mathbb{A})$ are in one-to-one
correspondence with pairs $(\mathfrak{o}mega_1, \mathfrak{o}mega_2)$ of characters of $\mathbb{R}_{>0}\mathbb{Q}^\times \backslashslash \mathbf mathbb{A}^\times$ satisfying
$\mathfrak{o}mega_1 \mathfrak{o}mega_2^2 = \mathfrak{o}mega$, via
$$\chi\left(\diag{u}{u}{tu^{-1}}{tu^{-1}}\right)=\mathfrak{o}mega_1(u)\mathfrak{o}mega_2(t).$$
For convenience, if $A \in \mathbf mathfrak{G}L_2$, define
$\iota_A=\mathbf mat{A}{}{}{ \trans{A}^{-1}} \in \mathbf mathcal Ms,$
and if $p=\mathbf mat{A}{*}{}{t\trans{A}^{-1}} \in P_{\text{S}}$, define
$\mathbf{s}igma_{\text S}(p)=A$, and $\mathbf{s}igma_S(nm)=\mathbf{s}igma_S(m)$
\begin{lemma}\label{SiegelInducing}
Let $\chi=(\mathfrak{o}mega_1,\mathfrak{o}mega_2)$ with $\mathfrak{o}mega_1\mathfrak{o}mega_2^2=\mathfrak{o}mega$.
The irreducible representations with central character $\chi$ occurring in $R_{\mathbf mathcal Ms,\text{disc}}$ are twists $\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i$, where $\mathfrak{p}i$
occurs in the discrete spectrum of $L^2(\mathbb{R}_{>0}\mathbf mathfrak{G}L_2(\mathbb{Q})\backslash \mathbf mathfrak{G}L_2(\mathbf mathbb{A}))$ and has central character $\mathfrak{o}mega_1$.
{\bf{e}}nd{lemma}
\begin{proof}
Similar as Lemma~\ref{KlingenInducing} with trivial modifications where required.
{\bf{e}}nd{proof}
\begin{proposition}\label{Siegelsummand}
Let $\chi=(\mathfrak{o}mega_1,\mathfrak{o}mega_2)$ with $\mathfrak{o}mega_1\mathfrak{o}mega_2^2=\mathfrak{o}mega$.
Let $(\mathfrak{p}i,V_\mathfrak{p}i)$ occur in the discrete spectrum of $L^2(\mathbb{R}_{>0}\mathbf mathfrak{G}L_2(\mathbb{Q})\backslash \mathbf mathfrak{G}L_2(\mathbf mathbb{A}))$ with central character $\mathfrak{o}mega_1$.
Consider $(\mathfrak{p}hi_k)_{k \in K / \mathbf mathfrak{G}amma}$ such that
\begin{enumerate}
\item for all $k$, $\mathfrak{p}hi_k \in V_\mathfrak{p}i$,
\item \label{Siegelcompatibility} if $\gamma \in K \cap P_{\text{S}}(\mathbf mathbb{A})$ then for all $k$,
$\mathfrak{p}hi_k(\cdot \mathbf{s}igma_{\text S}(\gamma))=\mathfrak{o}mega_2 \circ \mathbf mu(\gamma^{-1})\mathfrak{p}hi_{\gamma k}$.
{\bf{e}}nd{enumerate}
Then the function on $G(\mathbf mathbb{A})$ given for $n \in N_{\text{S}}(\mathbf mathbb{A}), m \in \mathbf mathcal Ms(\mathbf mathbb{A}), k \in K$ by
\begin{equation}\label{Siegelnmk}
\mathfrak{p}hi(nmk)= \mathfrak{o}mega_2 \circ \mathbf mu(m) \mathfrak{p}hi_k(\mathbf{s}igma_{\text S}(m)),
{\bf{e}}nd{equation}
is well-defined and belongs to ${\mathbf mathcal I_{P_{\text{S}}}((\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i)_\texttt{n}u)}^\mathbf mathfrak{G}amma$.
Moreover, every function in ${\mathbf mathcal I_{P_{\text{S}}}((\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i)_\texttt{n}u)}^\mathbf mathfrak{G}amma$ has this shape.
{\bf{e}}nd{proposition}
\begin{remark}
Similarly as Remark~\ref{KlingenMaassForm}, condition~(\ref{Siegelcompatibility}) implies that each $\mathfrak{p}hi_k$ is right-$\text{O}_2(\mathbb{R})$-invariant
(and hence must be an adelic Maa{\mathbf{s}s} form or a character).
{\bf{e}}nd{remark}
\begin{proof}
Same proof as Proposition~\ref{Klingensummand}, with trivial modifications where required.
{\bf{e}}nd{proof}
\begin{proposition} \label{SiegelSpectralParameter}
Let $(\mathfrak{p}i,V_\mathfrak{p}i)$ occur in the discrete spectrum of $L^2(\mathbb{R}_{>0}\mathbf mathfrak{G}L_2(\mathbb{Q})\backslash \mathbf mathfrak{G}L_2(\mathbf mathbb{A}))$ with central character $\mathfrak{o}mega_1$.
$\mathbf mathcal I_{P_{\text{S}}}((\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i)_\texttt{n}u)$ has a $K_\infty$-fixed vector if and only if $\mathfrak{p}i$ has a $O_2(\mathbb{R})$-fixed vector.
In this case, $\mathbf mathcal I_{P_{\text{S}}}((\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i)_\texttt{n}u)_\infty$ is generic if and only if $\mathfrak{p}i_\infty$ is a principal series.
Finally if $\mathfrak{p}i_\infty$ is a spherical principal series with spectral parameter $s$ and $\texttt{n}u \in i\mathbf mathcal Lie{a}_{\mathbf mathcal Ms}^*$
then $\mathbf mathcal I_{P_{\text{K}}}((\mathfrak{o}mega_2 \mathfrak{o}times \mathfrak{p}i)_\texttt{n}u)_\infty$
is a principal series representation with spectral parameter $\texttt{n}u+\texttt{n}u_{\text{S}}(s)$, where $\texttt{n}u_{\text{S}}(s)$ is the element of $\mathbf mathcal Lie{a}^*(\mathbf mathbb{C})$
corresponding to the character $\diag{y^{\bm{f}rac12}u}{y^{-\bm{f}rac12}u}{y^{-\bm{f}rac12}u^{-1}}{y^{\bm{f}rac12}u^{-1}} \mathbf mapsto |y|^s$.
{\bf{e}}nd{proposition}
\begin{proof}
Same proof as Proposition~\ref{KlingenSpectralParameter}, with trivial modifications where required.
{\bf{e}}nd{proof}
\mathbf{s}ubsection{Spectral expansion of the kernel}
We now give the spectral expansion of the kernel.
\begin{definition}\label{ONBase}
For each standard parabolic $P$ we choose an orthonormal basis $\mathbf mathcal{B}_{P}$ of $\mathbf mathscr H_P(\mathfrak{o}mega)$ such that
\begin{enumerate}
\item \label{DecompIrred} if $R_{M_P,\text{disc}}=\bigoplus_\mathfrak{p}i \mathfrak{p}i$ is the decomposition of the restriction of the right regular representation of $M_P(\mathbf mathbb{A})$
on the subspace of $L^2(M_P(\mathbb{Q})A_P^0(\mathbb{R}) \backslash G(\mathbf mathbb{A}))$ that decompose discretely, then
$\mathbf mathcal{B}_P=\bigcup_\mathfrak{p}i \mathbf mathcal{B}_{\mathfrak{p}i}$, where each $\mathbf mathcal{B}_\mathfrak{p}i$ is a basis of the space of the corresponding induced representation $\mathbf mathcal I_P(\mathfrak{p}i_\texttt{n}u)$,
(note that this space does not depend on~$\texttt{n}u$).
\item \label{factor} for each representation $\mathfrak{p}i=\bigotimes_v \mathfrak{p}i_v$ as above, for each place $v$ there is an orthonormal basis $\mathbf mathcal{B}_{\mathfrak{p}i,v}$ of
the local representation $\mathfrak{p}i_v$ such that $\mathbf mathcal{B}_{\mathfrak{p}i}$ consists in factorizable vectors
$\mathfrak{p}hi=\bigotimes_{v \le \infty} \mathfrak{p}hi_v$ where each $\mathfrak{p}hi_v$ belongs to the corresponding $\mathbf mathcal{B}_{\mathfrak{p}i,v}$.
\item \label{Ktypes} for each representation $\mathfrak{p}i_v$, we have
$\mathbf mathcal{B}_{\mathfrak{p}i,v} = \bigcup_{\tau} \mathbf mathcal{B}_{\mathfrak{p}i,v, \tau}$, where the union is over the irreducible representations $\tau$ of $\mathbf mathfrak{G}amma_v$,
and $\mathbf mathcal{B}_{\mathfrak{p}i, v,\tau}$ is a basis of the space of $\mathfrak{p}i_v$ consisting of vectors $\mathfrak{p}hi$ satisfying $\mathfrak{p}i_v(\gamma)\mathfrak{p}hi=\tau(\gamma) \mathfrak{p}hi$
for all $\gamma \in \mathbf mathfrak{G}amma_v$.
{\bf{e}}nd{enumerate}
{\bf{e}}nd{definition}
Note that conditions~(\ref{factor}) and~(\ref{Ktypes}) imply in particular that elements of $\mathbf mathcal{B}_P$ are in $\mathbf mathscr H_P^0$.
\begin{definition}
For each standard parabolic $P$ and for each irreducible representation $\mathfrak{p}i$ occuring in $R_{M_P,\text{disc}}$,
define $\mathbf mathcal{B}_{\mathfrak{p}i, 1}$ to be the subset of $\mathbf mathcal{B}_{\mathfrak{p}i}$ consisting in vectors $\mathfrak{p}hi$ whose each local component
$\mathfrak{p}hi_v$ belongs to $\mathbf mathcal{B}_{\mathfrak{p}i,v, 1}$, and set $\mathbf mathcal{B}_P^\mathbf mathfrak{G}amma=\bigcup_\mathfrak{p}i \mathbf mathcal{B}_{\mathfrak{p}i, 1}$.
If $\chi$ is a character of $A_P(\mathbf mathbb{A})$, define
$$\mathbf mathfrak{G}en_P(\chi)=\bigcup_{\mathfrak{p}i} \mathbf mathcal{B}_{\mathfrak{p}i,1},$$
where the union runs over representations $\mathfrak{p}i$ with central character $\chi$ and such that the induced representations
$\mathbf mathcal I_P(\mathfrak{p}i_\texttt{n}u)$ are generic.
{\bf{e}}nd{definition}
If $u \in \mathbf mathscr H_P(\mathfrak{o}mega)$, define $$\mathbf mathcal I_P(\texttt{n}u ,f)u=\int_{\mathfrak{o}verline{G(\mathbf mathbb{A})}}f(y)\mathbf mathcal I_P(\texttt{n}u ,y)udy.$$
\begin{proposition}\label{globalevalue}
Let $\texttt{n}u \in i\mathbf mathcal Lie{a}_p^*$.
Let $u \in \mathbf mathcal{B}_P$.
Then either $\mathbf mathcal I_P(\texttt{n}u ,f)u=0$ or $u \in \mathbf mathcal{B}_P^\mathbf mathfrak{G}amma $.
In the latter case, say $u \in \mathbf mathcal{B}_\mathfrak{p}i$. Then if $\mathfrak{p}i$ is generic we have
$$\mathbf mathcal I_P(\texttt{n}u ,f)u=\lambda_f(u,\texttt{n}u)u,$$
where $\lambda_f(u,\texttt{n}u)=\lambda_{f_{\infty}}(u,\texttt{n}u)\lambda_{f_{\text{fin}}}(u,\texttt{n}u)$,
and
$$\lambda_{f_{\infty}}(u,\texttt{n}u)=
\begin{cases}
\tilde{f_\infty}(\texttt{n}u) \text{ if } P=B,\\
\tilde{f_\infty}(\texttt{n}u+\texttt{n}u_{\text{K}}(s_u)) \text{ if } P=P_{\text{K}} \text{ and } \mathfrak{p}i_\infty \text{ has spectral parameter } s_u,\\
\tilde{f_\infty}(\texttt{n}u+\texttt{n}u_{\text{S}}(s_u)) \text{ if } P=P_{\text{S}} \text{ and } \mathfrak{p}i_\infty \text{ has spectral parameter } s_u,\\
\tilde{f_\infty}(\texttt{n}u_u) \text{ if } P=G \text{ and } \mathfrak{p}i_\infty \text{ has spectral parameter } \texttt{n}u_u,\\
{\bf{e}}nd{cases}$$
and, following notations of Proposition~\ref{localevalue}, $\lambda_{f_{\text{fin}}}(u,\texttt{n}u)$ is the eigenvalue of the Hecke operator
$$\bigotimes_{\mathbf mathfrak{G}amma_p=G(\mathbb{Z}_p)}\mathfrak{o}verline{\mathfrak{p}i_{p,\texttt{n}u}}(\tilde{f_p}).$$
{\bf{e}}nd{proposition}
\begin{remark}
If $P=G$ then $\mathbf mathcal Lie{a}_P=\{0\}$ and $\mathbf mathcal I_P(\texttt{n}u ,f)=R(f)$.
{\bf{e}}nd{remark}
\begin{proof}
This is a combination of Propositions~\ref{localevalue},~\ref{archimedeanevalue}, Lemma~\ref{BorelSpectralParameter} and
Propositions~\ref{KlingenSpectralParameter} and~\ref{SiegelSpectralParameter}.
{\bf{e}}nd{proof}
The following statement~\cite{ArthurSpectralExpansion}*{pages 928-935} may be viewed as a rigorous version of the informal discussion
in Section~\ref{basickernel}.
\begin{lemma}\label{UnprovedLemma}
Let $f$ as in Assumption~\ref{testfunction}. Then we have a pointwise equality
$$K_f(x,y)=\mathbf{s}um_{P} n_P^{-1} \int_{i\mathbf mathcal Lie{a}_{P}^*}\mathbf{s}um_{u \in \mathbf mathcal{B}_P}E(x, \mathbf mathcal I_P(\texttt{n}u ,f)u,\texttt{n}u )\mathfrak{o}verline{E(y,u,\texttt{n}u)}d\texttt{n}u.$$
Here, $n_G=1,$ $n_B=8$, $n_{P_{\text{K}}}=2$ and $n_{P_{\text{S}}}=2$.
{\bf{e}}nd{lemma}
However, for the later purpose of interchanging integration order, we want to show that the above expressions for the kernel converge absolutely.
To this end, we need the following stronger statement.
\begin{proposition}\label{ACV}
Let $f$ as in Assumption~\ref{testfunction}. Then the following expression defines a continuous function in the variables $x$ and $y$
$$K_{\text{abs}}(x,y)=\mathbf{s}um_{P} n_P^{-1} \int_{i\mathbf mathcal Lie{a}_{P}^*}\mathbf{s}um_{u \in \mathbf mathcal{B}_P}|E(x, \mathbf mathcal I_P(\texttt{n}u ,f)u,\texttt{n}u )\mathfrak{o}verline{E(y,u,\texttt{n}u)}|d\texttt{n}u.$$
{\bf{e}}nd{proposition}
We do not give a proof of this proposition here, as a similar statement was proven in the setting of $\mathbf mathfrak{G}L_2$ in \S~6 of \cite{KL}, the
proof thereof can be directly adapted.
By combining it with Lemmas~\ref{Borelsummand},~\ref{Klingensummand},~\ref{Siegelsummand} and Proposition~\ref{globalevalue},
we obtain the following corollary.
\begin{corollary}\label{KernelSpectralForm}
Let $f$ as in Assumption~\ref{testfunction}.
Then we have a pointwise equality
$$K_{f}(x,y)=K_{\text{disc}}(x,y)+K_{\text{B}}(x,y)+K_{\text{K}}(x,y)+K_{\text{S}}(x,y)+K_{\text{ng}}(x,y),$$
where
$$K_{\text{disc}}(x,y)=\mathbf{s}um_{u \in \mathbf mathfrak{G}en_G(\mathfrak{o}mega)}\tilde{f_\infty}(\texttt{n}u_u)\lambda_{f_{\text{fin}}}(u)u(x)\mathfrak{o}verline{u(y)},$$
$$K_{\text{B}}(x,y)=\bm{f}rac18 \mathbf{s}um_{\mathfrak{o}mega_1\mathfrak{o}mega_2\mathfrak{o}mega_3^2=\mathfrak{o}mega}\mathbf{s}um_{u \in \mathbf mathfrak{G}en_B(\mathfrak{o}mega_1,\mathfrak{o}mega_2,\mathfrak{o}mega_3)}
\int_{i\mathbf mathcal Lie{a}^*}\tilde{f_\infty}(\texttt{n}u)\lambda_{f_{\text{fin}}}(u,\texttt{n}u)E(x, u,\texttt{n}u )\mathfrak{o}verline{E(y,u,\texttt{n}u)}d\texttt{n}u.$$
$$K_{\text{K}}(x,y)=\bm{f}rac12 \mathbf{s}um_{\mathfrak{o}mega_1\mathfrak{o}mega_2=\mathfrak{o}mega}\mathbf{s}um_{u \in \mathbf mathfrak{G}en_{P_{\text{K}}}(\mathfrak{o}mega_1,\mathfrak{o}mega_2)}
\int_{i\mathbf mathcal Lie{a}_\text{K}^*}\tilde{f_\infty}(\texttt{n}u+\texttt{n}u_{\text{K}}(s_u))\lambda_{f_{\text{fin}}}(u,\texttt{n}u)E(x, u,\texttt{n}u )\mathfrak{o}verline{E(y,u,\texttt{n}u)}d\texttt{n}u,$$
$$K_{\text{S}}(x,y)=\bm{f}rac12 \mathbf{s}um_{\mathfrak{o}mega_1\mathfrak{o}mega_2^2=\mathfrak{o}mega}\mathbf{s}um_{u \in \mathbf mathfrak{G}en_{P_{\text{S}}}(\mathfrak{o}mega_1,\mathfrak{o}mega_2)}
\int_{i\mathbf mathcal Lie{a}_\text{S}^*}\tilde{f_\infty}(\texttt{n}u+\texttt{n}u_{\text{S}}(s_u))\lambda_{f_{\text{fin}}}(u,\texttt{n}u)E(x, u,\texttt{n}u )\mathfrak{o}verline{E(y,u,\texttt{n}u)}d\texttt{n}u,$$
and all the automorphic forms involved in $K_{\text{ng}}$ are not generic.
{\bf{e}}nd{corollary}
Actually, no automorphic form from the residual spectrum is generic, as shown by the following lemma.
Thus $K_{\text{disc}}$ consists only in elements from the cuspidal spectrum.
\begin{lemma}
Let $(\mathfrak{p}i,V_\mathfrak{p}i)$ be any irreducible representation occurring in the residual spectrum of
$L^2(Z(\mathbb{R})G(\mathbb{Q}) \backslashslash G(\mathbf mathbb{A}), \mathfrak{o}mega)$. Then $\mathfrak{p}i$ is non generic.
{\bf{e}}nd{lemma}
\begin{proof}
We will rely on results of Kim that describe the residual spectrum of $\text{Sp}_4$.
Thus we first need to show that the $\text{res } \mathfrak{p}i$ given by Definition~\ref{DefOfRes} belongs to the residual spectrum of $\text{Sp}_4(\mathbf mathbb{A})$.
First, $\text{res } \mathfrak{p}i$ occurs in the discrete spectrum of $L^2(\text{Sp}_4(\mathbb{Q}) \backslashslash \text{Sp}_4(\mathbf mathbb{A}))$, because there are only finitely many possibilities
for the Archimedean component of any irreducible representation occurring in $\text{res } \mathfrak{p}i$.
Moreover $\text{res } \mathfrak{p}i$ and is not cuspidal by Lemma~\ref{cuspidalrestriction}.
Hence $\text{res } \mathfrak{p}i$ belongs to the residual spectrum of $\text{Sp}_4(\mathbf mathbb{A})$, as claimed.
In view of Lemma~\ref{genericrestriction}, it suffices to prove that the residual spectrum of $\text{Sp}_4(\mathbf mathbb{A})$ is not generic.
By Theorem~3.3 and Remark~3.2 of~\cite{Kim}, the representations occurring from poles of Siegel Eisenstein
series are non generic. Similarly, by Theorem~4.1 and Remark~4.2 of~\cite{Kim}, the representations occurring
from poles of Klingen Eisenstein series are non generic.
Finally, by~\cite{Kim}~\S~5.3, irreducible representations $\mathfrak{p}i$ occurring from the poles of Borel Eisenstein series are described as follows.
On the one hand, we have the space of constant functions, which is clearly not generic.
On the other hand, for every non-trivial quadratic gr\"ossencharacter $\mathbf mu$ of $\mathbb{Q}$ we have a representation $B(\mathbf mu)$
whose local components are irreducible subquotients of the induced representation $\mathbf mathcal Ind_B^{\text{Sp}_4}(|\cdot|_v\mathbf mu_v \times \mathbf mu_v)$.
Therefore, in the terminology of~\cite{RS2}*{\S~2.2}, for all prime~$p$, $\mathfrak{p}i_p$ belongs to Group~V if $\mathbf mu_p \texttt{n}eq 1$, and to Group~VI if $\mathbf mu_p=1$.
Now by Table~A.2 of~\cite{RS2}, we see that the only generic representations in Group~V and~VI are those from~Va and VI~a.
But Table~A.12 shows that neither of these have a $K_p$-fixed vector. Since almost all $\mathfrak{p}i_p$ contain a $K_p$-fixed vector, at least one local component of
$\mathfrak{p}i$ must be non-generic, and thus $\mathfrak{p}i$ is not globally generic.
{\bf{e}}nd{proof}
\mathbf{s}ubsection{The spectral side of the trace formula}
Let $\mathfrak{p}si_1=\mathfrak{p}si_{\mathbf m_1}, \mathfrak{p}si_2=\mathfrak{p}si_{\mathbf m_2}$ be generic characters of $U(\mathbf mathbb{A}) / U(\mathbb{Q})$.
Fix $t_1, t_2 \in A^0(\mathbb{R})$ and consider the basic integral
\begin{equation}\label{BasicIntegral}
I=\int_{(U(\mathbb{Q}) \backslashslash U(\mathbf mathbb{A}))^2} K_f(xt_1,yt_2) \mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_1}(x)}\mathfrak{p}si_{\mathbf m_2}(y)dx dy.
{\bf{e}}nd{equation}
Our goal is to compute it in two different ways -- using the spectral decomposition of the kernel
$K_f$ on the one hand, and its expression as a series together with the Bruhat decomposition on the other hand.
The latter will constitute the geometric side and will be addressed in Section~\ref{GeometricSide}. We now focus on the former.
Using the spectral expansion of the kernel $K_f$ given by Lemma~(\ref{UnprovedLemma}), we can evaluate the basic integral~(\ref{BasicIntegral}) as
$$I=\int_{(U(\mathbb{Q}) \backslashslash U(\mathbf mathbb{A}))^2} \mathbf{s}um_{P} n_P^{-1} \int_{i\mathbf mathcal Lie{a}_{P}^*}\mathbf{s}um_{u \in \mathbf mathcal{B}_P}E(xt_1, \mathbf mathcal I_P(\texttt{n}u ,f)u,\texttt{n}u )\mathfrak{o}verline{E(yt_2,u,\texttt{n}u)}d\texttt{n}u \mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_1}(x)}\mathfrak{p}si_{\mathbf m_2}\}(y)dx dy.$$
By Proposition~\ref{ACV}, this expression is absolutely integrable since $(U(\mathbb{Q}) \backslashslash U(\mathbf mathbb{A}))^2$ is compact.
Thus we may interchange integration order, thus obtaining the Whittaker coefficients of the automorphic forms involved here.
By Corollary~\ref{KernelSpectralForm}, we get a discrete contribution and a residual contribution, and a continuous contribution
-- which itself splits into the contribution of the various parabolic classes.
Thus the spectral side of the Kuznetsov formula is given as follows.
\begin{proposition}
We have $I=\bm{f}rac{1}{(\mathbf m_{1,1}\mathbf m_{2,1})^4|\mathbf m_{1,2}\mathbf m_{2,1}|^3}\left(\Sigma_{\text{disc}}+\Sigma_B+\Sigma_K+\Sigma_S\right)$, where
$$\Sigma_{\text{disc}}=\mathbf{s}um_{u \in \mathbf mathfrak{G}en_G(\mathfrak{o}mega)}\tilde{f_\infty}(\texttt{n}u_u)\lambda_{f_{\text{fin}}}(u)\mathcal{W}_{\mathfrak{p}si}(u)(t_1t_{\mathbf m_1}^{-1})\mathfrak{o}verline{\mathcal{W}_{\mathfrak{p}si}(u)}(t_2t_{\mathbf m_2}^{-1}),$$
\begin{align*}
\Sigma_{\text{B}}=\bm{f}rac18 \mathbf{s}um_{\mathfrak{o}mega_1\mathfrak{o}mega_2\mathfrak{o}mega_3^2=\mathfrak{o}mega}\mathbf{s}um_{u \in \mathbf mathfrak{G}en_B(\mathfrak{o}mega_1,\mathfrak{o}mega_2,\mathfrak{o}mega_3)} &
\int_{i\mathbf mathcal Lie{a}^*}\tilde{f_\infty}(\texttt{n}u) \lambda_{f_{\text{fin}}}(u,\texttt{n}u)\\
&\times\mathcal{W}_{\mathfrak{p}si}(E(\cdot, u,\texttt{n}u ))(t_1t_{\mathbf m_1}^{-1})\mathfrak{o}verline{\mathcal{W}_{\mathfrak{p}si}(E(\cdot,u,\texttt{n}u))}(t_2t_{\mathbf m_2}^{-1})d\texttt{n}u.
{\bf{e}}nd{align*}
\begin{align*}
\Sigma_{\text{K}}=\bm{f}rac12 \mathbf{s}um_{\mathfrak{o}mega_1\mathfrak{o}mega_2=\mathfrak{o}mega}\mathbf{s}um_{u \in \mathbf mathfrak{G}en_{P_{\text{K}}}(\mathfrak{o}mega_1,\mathfrak{o}mega_2)}&
\int_{i\mathbf mathcal Lie{a}_\text{K}^*}\tilde{f_\infty}(\texttt{n}u+\texttt{n}u_{\text{K}}(s_u))\lambda_{f_{\text{fin}}}(u,\texttt{n}u)\\
&\times \mathcal{W}_{\mathfrak{p}si}(E(\cdot, u,\texttt{n}u ))(t_1t_{\mathbf m_1}^{-1})\mathfrak{o}verline{\mathcal{W}_{\mathfrak{p}si}(E(\cdot,u,\texttt{n}u))}(t_2t_{\mathbf m_2}^{-1})d\texttt{n}u,
{\bf{e}}nd{align*}
\begin{align*}
\Sigma_{\text{S}}=\bm{f}rac12 \mathbf{s}um_{\mathfrak{o}mega_1\mathfrak{o}mega_2^2=\mathfrak{o}mega}\mathbf{s}um_{u \in \mathbf mathfrak{G}en_{P_{\text{S}}}(\mathfrak{o}mega_1,\mathfrak{o}mega_2)} &
\int_{i\mathbf mathcal Lie{a}_\text{S}^*}\tilde{f_\infty}(\texttt{n}u+\texttt{n}u_{\text{S}}(s_u))\lambda_{f_{\text{fin}}}(u,\texttt{n}u)\\
&\times \mathcal{W}_{\mathfrak{p}si}(E(\cdot, u,\texttt{n}u ))(t_1t_{\mathbf m_1}^{-1})\mathfrak{o}verline{\mathcal{W}_{\mathfrak{p}si}(E(\cdot,u,\texttt{n}u))}(t_2t_{\mathbf m_2}^{-1})d\texttt{n}u.
{\bf{e}}nd{align*}
{\bf{e}}nd{proposition}
\mathbf{s}ection{The geometric side of the trace formula}\label{GeometricSide}
Breaking the sum~(\ref{TheKernel}) over $U(\mathbb{Q}) \times U(\mathbb{Q})$ orbits
leads to a sum over representatives of the double cosets of $U \backslashslash G / U$ of
orbital integrals. Specifically, set $H=U \times U$, acting on $G$ by
$$(x,y) \cdot \delta = x^{-1} \delta y,$$
and denote by $H_\delta$ the stabilizer of $\delta$.
Since $f$ has compact support, the infinite sum $\mathbf{s}um_{\delta \in \mathbf mathfrak{G}modZ(\mathbb{Q})} |f(t_1^{-1}x^{-1} \delta yt_2)|$ is
in fact locally finite and hence
defines a continuous function in $x$ and $y$ on the compact set
$H(\mathbb{Q}) \backslashslash H(\mathbf mathbb{A})$. Thus we may interchange summation and integration order, getting
\begin{align*}
I &=
\int_{H(\mathbb{Q}) \backslashslash H(\mathbf mathbb{A})} \mathbf{s}um_{\delta \in \mathbf mathfrak{G}modZ(\mathbb{Q})} f(t_1^{-1}x^{-1} \delta yt_2)\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_1}(x)}\mathfrak{p}si_{\mathbf m_2}(y)dx dy\\
&=\mathbf{s}um_{\delta \in \mathbf mathfrak{G}modZ(\mathbb{Q})} \int_{H(\mathbb{Q}) \backslashslash H(\mathbf mathbb{A})}f(t_1^{-1}x^{-1} \delta yt_2)\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_1}(x)}\mathfrak{p}si_{\mathbf m_2}(y)dx dy\\
&= \mathbf{s}um_{\delta \in U(\mathbb{Q})\backslashslash \mathbf mathfrak{G}modZ(\mathbb{Q})/U(\mathbb{Q})} I_{\delta}(f),
{\bf{e}}nd{align*}
where
\begin{equation}\label{orbital}
I_{\delta}(f)=\int_{H_\delta(\mathbb{Q}) \backslashslash H(\mathbf mathbb{A})} f(t_1^{-1}x^{-1} \delta yt_2)\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_1}(x)}\mathfrak{p}si_{\mathbf m_2}(y)d(x,y),
{\bf{e}}nd{equation}
and $d(x,y)$ is the quotient measure on $H_\delta(\mathbb{Q}) \backslashslash H(\mathbf mathbb{A})$.
Using the Bruhat decomposition $G=B \text{O}mega B= \coprod_{\mathbf{s}igma \in \text{O}mega}U \mathbf{s}igma T U$, we have
\begin{equation} \label{BruhatmodZ}
U \backslashslash \mathbf mathfrak{G}modZ / U = \coprod_{\mathbf{s}igma \in \text{O}mega} \mathbf{s}igma \overline{T},
{\bf{e}}nd{equation}
where $\overline{T}= T / Z$.
We can then compute separately the contribution from each element from the Weyl group.
Writing $H(\mathbf mathbb{A})=H_\delta(\mathbf mathbb{A}) \times (H_\delta(\mathbf mathbb{A}) \backslashslash H(\mathbf mathbb{A}))$, we can factor out the integral
of $\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_1}}\mathfrak{o}times\mathfrak{p}si_{\mathbf m_2}$ over the compact group $H_\delta(\mathbb{Q}) \backslashslash H_\delta(\mathbf mathbb{A})$
in~(\ref{orbital}). Therefore, $I_\delta(f)$ vanishes unless the character
$ \mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_1}} \mathfrak{o}times\mathfrak{p}si_{\mathbf m_2}$ is trivial on $H_\delta(\mathbf mathbb{A})$. Following
Knightly and Li, we shall call the orbits $H \cdot \delta$ such that
$\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_1}} \mathfrak{o}times \mathfrak{p}si_{\mathbf m_2}$ is trivial on $H_\delta(\mathbf mathbb{A})$ {\bf relevant}.
\mathbf{s}ubsection{Relevant orbits}
In order to characterize the relevant orbits, let us introduce a bit of notation.
A set of representatives of $T(\mathbb{Q}) / Z(\mathbb{Q})$ is given by the elements
\begin{equation}\label{diagrepres}
\delta_1 \doteq \diag{d_1}{1}{d_2}{d_1d_2}, d_1,d_2 \in \mathbb{Q}^\times.
{\bf{e}}nd{equation}
For each $\mathbf{s}igma \in \text{O}mega$, the corresponding set of representatives of $\mathbf{s}igma \overline{T}(\mathbb{Q})$ in~(\ref{BruhatmodZ})
is given by elements of the form
\begin{equation}\label{ds}
\delta_\mathbf{s}igma=\mathbf{s}igma \delta_1,
{\bf{e}}nd{equation}
and $H_{\delta_\mathbf{s}igma}(\mathbf mathbb{A})$ consists in pairs
$(u,\delta_\mathbf{s}igma^{-1}u\delta_\mathbf{s}igma)=(u,\delta_1^{-1}\mathbf{s}igma^{-1}u\delta_1\mathbf{s}igma)$ such that both component lie in $U(\mathbf mathbb{A})$.
Since conjugation by $\delta_1$ preserves $U(\mathbf mathbb{A})$,
the condition that the second component lies in $U(\mathbf mathbb{A})$ is equivalent to $u \in U(\mathbf mathbb{A})$ and $\mathbf{s}igma^{-1} u \mathbf{s}igma \in U(\mathbf mathbb{A})$.
We accordingly make the following definition.
\begin{definition}
For $\mathbf{s}igma \in \text{O}mega,$ define
\begin{equation}\label{Usigma}
U_\mathbf{s}igma(\mathbf mathbb{A}) =\{x \in U(\mathbf mathbb{A}) :\mathbf{s}igma^{-1} x \mathbf{s}igma \in U(\mathbf mathbb{A})\},
{\bf{e}}nd{equation}
and
\begin{equation}\label{Dsigma}
D_\mathbf{s}igma(\mathbf mathbb{A}) = U_\mathbf{s}igma(\mathbf mathbb{A}) \times \mathbf{s}igma^{-1} U_\mathbf{s}igma(\mathbf mathbb{A}) \mathbf{s}igma
{\bf{e}}nd{equation}
Then we have
\begin{equation}\label{paramHd}
H_{\delta_\mathbf{s}igma}(\mathbf mathbb{A})=\{(u,\delta_\mathbf{s}igma^{-1}u\delta_\mathbf{s}igma): u \in U_\mathbf{s}igma(\mathbf mathbb{A})\} \mathbf{s}ubset D_\mathbf{s}igma(\mathbf mathbb{A}).
{\bf{e}}nd{equation}
{\bf{e}}nd{definition}
\begin{lemma}\label{relevantorbits}
The relevant orbits are the ones corresponding to the following elements:
\begin{itemize}
\item $\mathbf{s}igma=1$ with $\delta_1= {t_{\mathbf m_2}}^{-1}t_{\mathbf m_1}=
\diag{\bm{f}rac{\mathbf m_{1,1}}{\mathbf m_{2,1}}}{1}{\bm{f}rac{\mathbf m_{1,1}\mathbf m_{1,2}}{\mathbf m_{2,1}\mathbf m_{2,2}}}{\bm{f}rac{\mathbf m_{1,1}^2\mathbf m_{1,2}}{\mathbf m_{2,1}^2\mathbf m_{22}}}$,
\item $\mathbf{s}igma=s_1s_2s_1$ with $\delta_1$ satisfying $d_1\mathbf m_{1,2}=d_2\mathbf m_{2,2}$,
\item $\mathbf{s}igma=s_2s_1s_2$ with $\delta_1$ satisfying $\mathbf m_{1,1}=-d_1\mathbf m_{2,1}$,
\item $\mathbf{s}igma=s_1s_2s_1s_2=J$ with no condition on $\delta_1$.
{\bf{e}}nd{itemize}
{\bf{e}}nd{lemma}
\begin{proof}
For each representative $\delta_\mathbf{s}igma$ as in~(\ref{ds}), let us fix $u_1 \in U_\mathbf{s}igma(\mathbf mathbb{A})$, and compute
$\delta_\mathbf{s}igma^{-1}u_1\delta_\mathbf{s}igma$ in order to determine under which condition $\mathfrak{p}si_{\mathbf m_1} \mathfrak{o}times \mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}}$
is trivial on $H_{\delta_\mathbf{s}igma}(\mathbf mathbb{A})$.
For $\mathbf{s}igma=1$, we have $U_\mathbf{s}igma=U$, hence we may take $u_1=\left[
\begin{smallmatrix}
1 & & c & a-cx \\
x & 1 & a & b \\
& & 1 & -x \\
& & & 1 \\
{\bf{e}}nd{smallmatrix}
\right]$. Then we have
$\delta^{-1}u_1\delta=\left[
\begin{smallmatrix}
1 & & c\bm{f}rac{d_2}{d_1} & (a-cx)d_2 \\
xd_1 & 1 & ad_2 & bd_1d_2 \\
& & 1 & -xd_1 \\
& & & 1 \\
{\bf{e}}nd{smallmatrix}
\right]$ Thus, by~(\ref{gencharconj}), the condition that $\mathfrak{p}si_{\mathbf m_1} \mathfrak{o}times \mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}}$ be trivial on
$H_{\delta_1}(\mathbf mathbb{A})$ is equivalent to $\delta_1={t_{\mathbf m_2}}^{-1}t_{\mathbf m_1}.$
For $\mathbf{s}igma=s_1$, we have $U_\mathbf{s}igma(\mathbf mathbb{A})=\left\{\left[ \begin{smallmatrix}
1 & & c & a \\
& 1 & a & b \\
& & 1 & \\
& & & 1 \\
{\bf{e}}nd{smallmatrix}\right]: a,b,c \in \mathbf mathbb{A}\right\}$, and if $u_1=\left[ \begin{smallmatrix}
1 & & c & a \\
& 1 & a & b \\
& & 1 & \\
& & & 1 \\
{\bf{e}}nd{smallmatrix}\right]$, then $\delta^{-1}u_1\delta=\left[
\begin{smallmatrix}
1 & & b\bm{f}rac{d_2}{d_1} & ad_2 \\
& 1 & ad_2 & cd_1d_2 \\
& & 1 & \\
& & & 1 \\
{\bf{e}}nd{smallmatrix}
\right],$
hence the condition that $\mathfrak{p}si_{\mathbf m_1} \mathfrak{o}times \mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}}$ be trivial on
$H_{\delta_{s_1}}(\mathbf mathbb{A})$ is equivalent to
$\theta\left(\mathbf m_{1,2}c-\mathbf m_{2,2}\bm{f}rac{d_2}{d_1}b\right)=1$
for all $b,c \in \mathbf mathbb{A}$, which is equivalent to $\mathbf m_{1,2}=\mathbf m_{2,2}=0$ and thus contradicts the fact that $\mathfrak{p}si_{\mathbf m_1}$ and
$\mathfrak{p}si_{\mathbf m_2}$ are generic.
Similar calculations show that $\mathbf{s}igma=s_2,s_1s_2$ and $s_2s_1$ yield no relevant orbit.
For $\mathbf{s}igma=s_1s_2s_1$ we have $U_\mathbf{s}igma(\mathbf mathbb{A})=\left\{\left[ \begin{smallmatrix}
1 & & c & \\
& 1 & & \\
& & 1 & \\
& & & 1 \\
{\bf{e}}nd{smallmatrix}\right]: c \in \mathbf mathbb{A}\right\}$,
and if $u_1=\left[ \begin{smallmatrix}
1 & & c & \\
& 1 & & \\
& & 1 & \\
& & & 1 \\
{\bf{e}}nd{smallmatrix}\right]$
then we have $\delta^{-1}u_1\delta=\left[
\begin{smallmatrix}
1 & & c\bm{f}rac{d_2}{d_1} & \\
& 1 & & \\
& & 1 & \\
& & & 1 \\
{\bf{e}}nd{smallmatrix}
\right]$,
hence the condition that $\mathfrak{p}si_{\mathbf m_1} \mathfrak{o}times \mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}}$ be trivial on
$H_{\delta_{s_{121}}}(\mathbf mathbb{A})$ is equivalent to
$\theta\left(\left(\mathbf m_{1,2}-\mathbf m_{2,2}\bm{f}rac{d_2}{d_1}\right)c\right)=1$
for all $c \in \mathbf mathbb{A}$. This is equivalent to $d_1\mathbf m_{1,2}=d_2\mathbf m_{2,2}$.
The calculation for $\mathbf{s}igma=s_2s_1s_2$ is similar.
Finally, for $\mathbf{s}igma=s_1s_2s_1s_2=J$ the long Weyl element, $H_{\delta_{s_{1212}}}(\mathbf mathbb{A})$ is trivial.
{\bf{e}}nd{proof}
A case by case calculation also shows the following.
\begin{lemma}\label{Usigmarelevant}
Let $\mathbf{s}igma \in \text{O}mega$. Then there exists $\delta \in \overline{T}(\mathbb{Q})$ such that the orbit of $\delta_\mathbf{s}igma=\mathbf{s}igma\delta$ is relevant
if and only if $U_\mathbf{s}igma=\{u \in U : \mathbf{s}igma^{-1}u\mathbf{s}igma=u\}.$
{\bf{e}}nd{lemma}
In the sequel, we shall call such elements of the Weyl group {\bf relevant} as well.
In particular, by definition of the relevant orbits, and by~(\ref{paramHd}), we have the following.
\begin{corollary}\label{psionUsigma}
Suppose that the orbit of $\delta_\mathbf{s}igma=\mathbf{s}igma\delta$ is relevant.
Then for all $u \in U_\mathbf{s}igma$ we have $\mathfrak{p}si_{\mathbf m_2}(\delta^{-1}u\delta)=\mathfrak{p}si_{\mathbf m_1}(u).$
{\bf{e}}nd{corollary}
\mathbf{s}ubsection{General shape of the relevant orbital integrals}
\begin{lemma}\label{quotientmap}
For each $\delta_1 \in \overline{T}(\mathbb{Q})$ and $\mathbf{s}igma \in \text{O}mega$, the map
\begin{align*}
\varphi : D_\mathbf{s}igma(\mathbf mathbb{A}) & \to D_\mathbf{s}igma(\mathbf mathbb{A}) \\
(u_1,u_2) & \mathbf mapsto (u_1, \delta_\mathbf{s}igma^{-1}u_1^{-1}\delta_\mathbf{s}igma u_2).
{\bf{e}}nd{align*}
induces a bijective map
\begin{align*}
H_{\delta_\mathbf{s}igma} (\mathbb{Q}) \backslash D_\mathbf{s}igma(\mathbf mathbb{A}) \to & \left(U_\mathbf{s}igma(\mathbb{Q}) \times \{1\}\right) \backslash D_\mathbf{s}igma(\mathbf mathbb{A}) \\ &\cong \left(U_\mathbf{s}igma(\mathbb{Q})\backslash U_\mathbf{s}igma(\mathbf mathbb{A})\right)\times \left( \mathbf{s}igma^{-1} U_\mathbf{s}igma(\mathbf mathbb{A}) \mathbf{s}igma \right)
{\bf{e}}nd{align*}
preserving the quotient measures.
{\bf{e}}nd{lemma}
\begin{proof}
To prove $\varphi$ is well defined it is sufficient to prove that for any
$(u_1,u_2)\in U_\mathbf{s}igma(\mathbf mathbb{A}) \times \mathbf{s}igma^{-1} U_\mathbf{s}igma(\mathbf mathbb{A})$ we have
$\varphi_2(u_1,u_2)=\delta_\mathbf{s}igma^{-1}u_1^{-1}\delta_\mathbf{s}igma u_2 \in \mathbf{s}igma^{-1} U_\mathbf{s}igma(\mathbf mathbb{A})$.
This is equivalent to the condition $\mathbf{s}igma \varphi_2(u_1,u_2) \mathbf{s}igma^{-1} \in U_\mathbf{s}igma(\mathbf mathbb{A})$, which in turn is equivalent to
$$\begin{cases}
\mathbf{s}igma \varphi_2(u_1,u_2) \mathbf{s}igma^{-1} \in U(\mathbf mathbb{A}) \\
\varphi_2(u_1,u_2) \in U(\mathbf mathbb{A}).
{\bf{e}}nd{cases}$$
But $\varphi_2(u_1,u_2)=\delta_1^{-1} \mathbf{s}igma^{-1} u_1^{-1} \mathbf{s}igma \delta_1 u_2$, and since $u_1 \in U_\mathbf{s}igma(\mathbf mathbb{A})$,
we have $\mathbf{s}igma^{-1} u_1^{-1} \mathbf{s}igma \in U(\mathbf mathbb{A})$ and it follows $\varphi_2(u_1,u_2) \in U(\mathbf mathbb{A})$ as desired.
On the other hand,
\begin{align*}
\mathbf{s}igma \varphi_2(u_1,u_2) \mathbf{s}igma^{-1} &= \mathbf{s}igma \delta_1^{-1} \mathbf{s}igma^{-1} u_1^{-1} \mathbf{s}igma \delta_1 u_2 \mathbf{s}igma^{-1}\\
&=(\mathbf{s}igma \delta_1 \mathbf{s}igma^{-1})^{-1} u_1^{-1} (\mathbf{s}igma \delta_1 \mathbf{s}igma^{-1}) \mathbf{s}igma u_2 \mathbf{s}igma^{-1}
{\bf{e}}nd{align*}
By definition of the Weyl group, $\mathbf{s}igma \delta_1 \mathbf{s}igma^{-1} \in T(\mathbf mathbb{A})$ so
$(\mathbf{s}igma \delta_1 \mathbf{s}igma^{-1})^{-1} u_1^{-1} (\mathbf{s}igma \delta_1 \mathbf{s}igma^{-1}) \in U(\mathbf mathbb{A})$. Furthermore,
$ \mathbf{s}igma u_2 \mathbf{s}igma^{-1} \in U_\mathbf{s}igma(\mathbf mathbb{A}) \mathbf{s}ubset U(\mathbf mathbb{A})$ and it also follows that $\mathbf{s}igma \varphi_2(u_1,u_2) \mathbf{s}igma^{-1} \in U(\mathbf mathbb{A}).$
Next, for $h=(h_1,h_2) \in H_{\delta_\mathbf{s}igma}(\mathbb{Q})$, we clearly have $\varphi(h)=(h_1,1)$, and
\begin{align*}
\varphi(h(u_1,u_2))&=(h_1u_1, \delta_\mathbf{s}igma^{-1} u_1^{-1} \underbrace{h_1^{-1} \delta_\mathbf{s}igma h_2}_{=\delta_\mathbf{s}igma} u_2)\\
&=\varphi(h)\varphi(u_1,u_2).
{\bf{e}}nd{align*}
Finally if we define $\mathfrak{p}si(u_1,u_2)=(u_1^{-1}, u_2)$, then
$\mathfrak{p}si \circ \varphi$ is an involution, and in particular $\varphi$ is bijective, which establishes the lemma.
{\bf{e}}nd{proof}
\begin{corollary}\label{corqoutientmap}
Let $\delta_1 \in \overline{T}(\mathbb{Q})$ and $\mathbf{s}igma$ be a relevant element of the Weyl group.
We have a measure preserving map
\begin{align*}
\varphi :H_{\delta_\mathbf{s}igma}(\mathbb{Q}) \backslash H(\mathbf mathbb{A}) & \to
\left(U_\mathbf{s}igma(\mathbb{Q}) \backslash U_\mathbf{s}igma(\mathbf mathbb{A})\right) \times \left(U_\mathbf{s}igma(\mathbf mathbb{A}) \backslash U(\mathbf mathbb{A}) \right)
\times U(\mathbf mathbb{A}) \\
(x,y) & \mathbf mapsto
\left(U_\mathbf{s}igma(\mathbb{Q}) u_1, U_\mathbf{s}igma(\mathbf mathbb{A}) u_2, u_3\right)
{\bf{e}}nd{align*}
with $u_1u_2=x$ and $u_3=\delta_\mathbf{s}igma^{-1} u_1^{-1} \delta_\mathbf{s}igma y$.
{\bf{e}}nd{corollary}
\begin{remark}
The assumption that $\mathbf{s}igma$ is relevant is not really needed here, but it simplifies slightly the proof.
{\bf{e}}nd{remark}
\begin{proof}
Let $\mathfrak{o}verline{U_\mathbf{s}igma}=U \cap \mathbf{s}igma^{-1} \trans{U} \mathbf{s}igma$. Then the quotient space $U_\mathbf{s}igma \backslash U$ may be identified with $\mathfrak{o}verline{U_\mathbf{s}igma}$,
and the map $U_\mathbf{s}igma \times \mathfrak{o}verline{U_\mathbf{s}igma}, (u_\mathbf{s}igma,u_1) \mathbf mapsto u_\mathbf{s}igma u_1$ preserves the Haar measures.
Define $\mathfrak{o}verline{D_\mathbf{s}igma}=\mathfrak{o}verline{U_\mathbf{s}igma} \times \mathfrak{o}verline{U_\mathbf{s}igma}$.
Using that $\mathbf{s}igma$ is relevant and hence, by Lemma~\ref{Usigmarelevant}, that $D_\mathbf{s}igma(\mathbf mathbb{A})=U_\mathbf{s}igma(\mathbf mathbb{A}) \times U_\mathbf{s}igma(\mathbf mathbb{A})$, we obtain a measure preserving map
\begin{align*}
H_{\delta_\mathbf{s}igma}(\mathbb{Q}) \backslash H(\mathbf mathbb{A}) & \to \left(H_{\delta_\mathbf{s}igma}(\mathbb{Q}) \backslash D_\mathbf{s}igma(\mathbf mathbb{A}) \right) \times \mathfrak{o}verline{D_\mathbf{s}igma}\\
H_{\delta_\mathbf{s}igma}(\mathbb{Q})(x,y) & \mathbf mapsto (H_{\delta_\mathbf{s}igma}(\mathbb{Q}) (x_\mathbf{s}igma,y_\mathbf{s}igma),(x_1,y_1)).
{\bf{e}}nd{align*}
Composing the first coordinate with the map obtained in Lemma~\ref{quotientmap}, we get a measure preserving map
\begin{align*}
H_{\delta_\mathbf{s}igma}(\mathbb{Q}) \backslash H(\mathbf mathbb{A}) & \to \left(\left(U_\mathbf{s}igma(\mathbb{Q}) \times \{1\}\right) \backslash D_\mathbf{s}igma(\mathbf mathbb{A}) \right) \times \mathfrak{o}verline{D_\mathbf{s}igma}\\
H_{\delta_\mathbf{s}igma}(\mathbb{Q})(x,y) & \mathbf mapsto \left(\left(U_\mathbf{s}igma(\mathbb{Q}) \times \{1\}\right) (x_\mathbf{s}igma,\delta_\mathbf{s}igma^{-1}x_\mathbf{s}igma^{-1} \delta_\mathbf{s}igma y_\mathbf{s}igma),(x_1, y_1)\right).
{\bf{e}}nd{align*}
Finally, composing with $U_\mathbf{s}igma(\mathbf mathbb{A}) \times \mathfrak{o}verline{U_\mathbf{s}igma} \to U(\mathbf mathbb{A}), (y_\mathbf{s}igma,y_1) \mathbf mapsto y_\mathbf{s}igma y_1$ we obtain
\begin{align*}
H_{\delta_\mathbf{s}igma}(\mathbb{Q}) \backslash H(\mathbf mathbb{A}) & \to \left(U_\mathbf{s}igma(\mathbb{Q}) \backslash U_\mathbf{s}igma(\mathbf mathbb{A})\right) \times \mathfrak{o}verline{U_\mathbf{s}igma}(\mathbf mathbb{A}) \times U(\mathbf mathbb{A})\\
H_{\delta_\mathbf{s}igma}(\mathbb{Q})(x,y) & \mathbf mapsto \left(U_\mathbf{s}igma(\mathbb{Q})x_\mathbf{s}igma,x_1,\delta_\mathbf{s}igma^{-1}x_\mathbf{s}igma^{-1} \delta_\mathbf{s}igma y\right).
{\bf{e}}nd{align*}
{\bf{e}}nd{proof}
\begin{proposition}\label{relevantintegrals}
Let $H \cdot \delta_\mathbf{s}igma$ be a relevant orbit.
Then the integral~(\ref{orbital}) can be expressed as
\begin{align*}
I_{\delta_\mathbf{s}igma}(f)&=\int_{U_\mathbf{s}igma(\mathbf mathbb{A}) \backslash U(\mathbf mathbb{A})}\int_{U(\mathbf mathbb{A})}f(t_1^{-1} u \delta_\mathbf{s}igma u_1 t_2){\mathfrak{p}si_{\mathbf m_1}(u)}\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}(u_1)}dudu_1.
{\bf{e}}nd{align*}
Moreover, it factors as $ I_{\delta_\mathbf{s}igma}(f)=I_{\delta_\mathbf{s}igma}(f_{\infty})I_{\delta_\mathbf{s}igma}(f_{\text{fin}})$, where we have set
$f_{fin}=\mathfrak{p}rod_p f_p$.
{\bf{e}}nd{proposition}
\begin{remark}
Note that the integral is well-defined by Corollary~\ref{psionUsigma}.
{\bf{e}}nd{remark}
\begin{remark}\label{d1d2}
By Assumption~\ref{testfunction}, the support of $f_\infty$ is included in $G^+(\mathbb{R})=\{g \in G(\mathbb{R}), \mathbf mu(g)>0\}$.
Therefore, if $\delta_1=\diag{d_1}{1}{d_2}{d_1d_2}$, we have $I_{\delta_\mathbf{s}igma}(f_{\infty}) \texttt{n}eq 0$ only if $d_1d_2>0$.
{\bf{e}}nd{remark}
\begin{proof}
By Corollary~\ref{corqoutientmap} we can make the change of variable $(u_1,u_2,u_3)=\varphi(x,y)$ in~(\ref{orbital}).
So we get
\begin{equation}\label{Idelta1}
\begin{split}
I_{\delta}(f)=\int_{U_\mathbf{s}igma(\mathbb{Q}) \backslash U_\mathbf{s}igma(\mathbf mathbb{A})}\int_{U_\mathbf{s}igma(\mathbf mathbb{A}) \backslash U(\mathbf mathbb{A})}\int_{U(\mathbf mathbb{A})}
f(t_1^{-1} u_2^{-1}\delta_\mathbf{s}igma u_3 t_2)) \quad \quad \quad \quad \quad \quad \\
\quad \quad \times \mathfrak{p}si_{\mathbf m_1}(u_1u_2)\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}(\delta_\mathbf{s}igma^{-1}u_1\delta_\mathbf{s}igma u_3)}du_3du_2du_1.
{\bf{e}}nd{split}
{\bf{e}}nd{equation}
We have
\begin{align*}
\mathfrak{p}si_{\mathbf m_1}(u_1u_2)\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}(\delta_\mathbf{s}igma^{-1}u_1\delta_\mathbf{s}igma u_3)}&=
\mathfrak{p}si_{\mathbf m_1}(u_2)\mathfrak{p}si_{\mathbf m_1}(u_1)\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}(\delta_\mathbf{s}igma^{-1}u_1\delta_\mathbf{s}igma)\mathfrak{p}si_{\mathbf m_2} (u_3)}\\
&=\mathfrak{p}si_{\mathbf m_1}(u_2)\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}(u_3)}
{\bf{e}}nd{align*}since $(u_1, \delta_\mathbf{s}igma^{-1}u_1\delta_\mathbf{s}igma) \in H_\delta(\mathbf mathbb{A})$ and we assume $H \cdot \delta_\mathbf{s}igma$ is relevant orbit.
Reporting this equality in~(\ref{Idelta1}), we get
$$I_{\delta_\mathbf{s}igma}(f)=\int_{U_\mathbf{s}igma(\mathbf mathbb{A}) \backslash U(\mathbf mathbb{A})}\int_{U(\mathbf mathbb{A})}f(t_1^{-1} u_2^{-1} \delta_\mathbf{s}igma u_3 t_2)\mathfrak{p}si_{\mathbf m_1}(u_2)\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}(u_3)}du_3du_2.$$
Write $u_3=u_\mathbf{s}igma u_1$ with $u_\mathbf{s}igma \in U_\mathbf{s}igma$ and $u_1 \in U_\mathbf{s}igma \backslash U$.
Then by Lemma~\ref{Usigmarelevant} we have
$$u_2^{-1} \delta_\mathbf{s}igma u_3=u_2^{-1} \mathbf{s}igma\delta u_\mathbf{s}igma u_1
=u_2^{-1} \mathbf{s}igma\delta u_\mathbf{s}igma \delta^{-1}\delta u_1=u_2^{-1} \delta u_\mathbf{s}igma \delta^{-1} \mathbf{s}igma \delta u_1,$$
and by Corollary~\ref{psionUsigma} we have
\begin{align*}
\mathfrak{p}si_{\mathbf m_1}(u_2)\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}(u_3)}&=\mathfrak{p}si_{\mathbf m_1}(u_2)\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}(u_\mathbf{s}igma u_1)}\\
&=\mathfrak{p}si_{\mathbf m_1}(u_2)\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_1}(\delta u_\mathbf{s}igma \delta^{-1})} \mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}(u_1)}
=\mathfrak{p}si_{\mathbf m_1}(\delta u_\mathbf{s}igma^{-1} \delta^{-1}u_2) \mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}(u_1)}
{\bf{e}}nd{align*}
Setting $u=\delta u_\mathbf{s}igma^{-1}\delta^{-1} u_2$ we get the result.
{\bf{e}}nd{proof}
\mathbf{s}ubsection{The Archimedean orbital integrals}\label{ArchInt}
By Theorem~\ref{GeomTransform} and using equation~(\ref{whittakertorus}) we have the following
\begin{lemma}\label{InTermsOfW}
Let $H \cdot \delta_\mathbf{s}igma$ be a relevant orbit. Then the corresponding Archimedean orbital integral is given by
\begin{align*}
I_{\delta_\mathbf{s}igma}(f_{\infty})=\bm{f}rac1c\bm{f}rac{\mathbf mathcal Delta_\mathbf{s}igma(t_{\mathbf m_2})}{|\mathbf m_{1,1}^4\mathbf m_{1,2}^3|}
\int_{U_\mathbf{s}igma(\mathbb{R}) \backslash U(\mathbb{R})}\int_{\mathbf mathcal Lie{a}^*}&
\tilde{f_\infty}(-i\texttt{n}u)W(i\texttt{n}u, t_{\mathbf m_1}^{-1}t_1, \mathfrak{p}si_{\mathbf mathbf{1}})\\
& \times W(-i\texttt{n}u,t_{\mathbf m_1}^{-1}\delta_\mathbf{s}igma t_{\mathbf m_2}u_1 t_{\mathbf m_2}^{-1} t_2,\mathfrak{o}verline{\mathfrak{p}si_{\mathbf mathbf{1}}})
\bm{f}rac{d\texttt{n}u}{c(i\texttt{n}u)c(-i\texttt{n}u)}
{\mathfrak{o}verline{\mathfrak{p}si_{\mathbf mathbf{1}}(u_1)}}du_1,
{\bf{e}}nd{align*}
where the constant $c$ is the one appearing in the spherical inversion theorem
and $\mathbf mathcal Delta_\mathbf{s}igma$ is the modulus character of the group $U_\mathbf{s}igma(\mathbb{R}) \backslash U(\mathbb{R})$.
{\bf{e}}nd{lemma}
Note that the above integral is well-defined.
More generally, let $\mathfrak{p}si$ be a generic character and $\mathbf{s}igma$ a relevant element of the Weyl group.
Then by Lemma~\ref{Usigmarelevant}, for all $t \in G(\mathbb{R})$ the integral
$$\int_{U_\mathbf{s}igma(\mathbb{R}) \backslash U(\mathbb{R})}\int_{\mathbf mathcal Lie{a}^*}g(-i\texttt{n}u)W(-i\texttt{n}u,y \mathbf{s}igma u_1 t,\mathfrak{p}si){d\texttt{n}u}
\mathfrak{o}verline{\mathfrak{p}si(u_1)}du_1 $$
is well defined as long as the commutator $yuy^{-1}u^{-1}$ belongs to $U_0(\mathbb{R})=\left\{
\left[\begin{smallmatrix}
1 & & & a \\
& 1 & a & b \\
& & 1 & \\
& & & 1 \\
{\bf{e}}nd{smallmatrix}\right],a,b \in \mathbb{R} \right\}$ for all $u \in U_\mathbf{s}igma(\mathbb{R})$.
The following conjecture, due to Buttcane~\cite{Buttcane}, should enable to take the $\mathbf mathcal Lie{a}^*$ integral out in Lemma~\ref{InTermsOfW}.
\begin{conjecture}[Interchange of integral]\label{interchange}
Let $g$ be holomorphic with rapid decay on an open tube domain of $\mathbf mathcal Lie{a}^*_\mathbf mathbb{C}$ containing $\mathbf mathcal Lie{a}^*$,
and let $t \in G(\mathbb{R})$.
Let $\mathfrak{p}si$ be a generic character, $\mathbf{s}igma$ be a relevant element of the Weyl group.
Then for almost all $y \in \text{Sp}_4(\mathbb{R})$ satisfying $yuy^{-1}u^{-1}\in U_0(\mathbb{R})$ for all $u \in U_\mathbf{s}igma(\mathbb{R})$ we have
$$\int_{U_\mathbf{s}igma(\mathbb{R}) \backslash U(\mathbb{R})}\int_{\mathbf mathcal Lie{a}^*}g(-i\texttt{n}u)W(-i\texttt{n}u,y \mathbf{s}igma u_1 t,\mathfrak{p}si){d\texttt{n}u}
\mathfrak{o}verline{\mathfrak{p}si(u_1)}du_1=\int_{\mathbf mathcal Lie{a}^*}g(-i\texttt{n}u) \tilde{K_\mathbf{s}igma}(-i\texttt{n}u, y,t)d\texttt{n}u$$
where
$$\tilde{K_\mathbf{s}igma}(-i\texttt{n}u, y,t)=\lim_{R \to 0}\int_{U_\mathbf{s}igma(\mathbb{R}) \backslash U(\mathbb{R})}h\left(\bm{f}rac{\|u_1\|}R\right)
W(-i\texttt{n}u,y \mathbf{s}igma u_1 t,\mathfrak{p}si) \mathfrak{o}verline{\mathfrak{p}si(u_1)}du_1,$$
for some fixed, smooth, compactly supported $h$ with $h(0)=1$.
Moreover $\tilde{K_\mathbf{s}igma}$ is entire in $\texttt{n}u$ and smooth and polynomially bounded in $t$ and $y$ for $\mathbb{R}e(-i\texttt{n}u)$ in some
fixed compact set.
{\bf{e}}nd{conjecture}
Note that Conjecture~\ref{interchange} is not needed for $\mathbf{s}igma=1$ since in this case
we have $U_\mathbf{s}igma=U$ and hence $\tilde{K_\mathbf{s}igma}(-i\texttt{n}u, y,t)=W(-i\texttt{n}u,y \mathbf{s}igma t,\mathfrak{p}si)$.
Consider now the case $\mathbf{s}igma=J$ the long Weyl element. In this case $U_\mathbf{s}igma$ is trivial.
Let $u \in U(\mathbb{R})$ and $k \in K_\infty$.
Then changing variables and using the fact that $W(-i\texttt{n}u,y \cdot,\mathfrak{p}si)$ is right-$K_\infty$ invariant we have
\begin{equation}\label{TildeKIsWhittaker}
\tilde{K_\mathbf{s}igma}(-i\texttt{n}u, y,utk)=\mathfrak{p}si(u)\tilde{K_\mathbf{s}igma}(-i\texttt{n}u, y,t).
{\bf{e}}nd{equation}
Moreover $\tilde{K_\mathbf{s}igma}(-i\texttt{n}u, y, \cdot)$ is an eigenfunction of the centre of the universal enveloping algebra in each variable,
with eigenvalue matching those of $W(-i\texttt{n}u, \cdot,\mathfrak{p}si)$.
It follows from the uniqueness of the Whittaker model that $$\tilde{K_\mathbf{s}igma}(-i\texttt{n}u, y,t)=K_\mathbf{s}igma(-i\texttt{n}u,y,\mathfrak{p}si)W(-i\texttt{n}u,t,\mathfrak{p}si)$$
for some function $K_\mathbf{s}igma(-i\texttt{n}u,y, \mathfrak{p}si)$ that we call the long Weyl element {\bf Bessel function}.
$K_\mathbf{s}igma(-i\texttt{n}u, \cdot, \mathfrak{p}si)$ is itself an eigenfunction of the centre of the universal enveloping algebra
with eigenvalue matching those of $W(-i\texttt{n}u, \cdot,\mathfrak{p}si)$, and satisfies for all $u \in U(\mathbb{R})$ the transformation rule
\begin{equation}\label{BesselTransformation}
K_\mathbf{s}igma(-i\texttt{n}u, uy, \mathfrak{p}si)=\mathfrak{p}si(u)K_\mathbf{s}igma(-i\texttt{n}u, y, \mathfrak{p}si)=K_\mathbf{s}igma(-i\texttt{n}u, y \mathbf{s}igma u \mathbf{s}igma^{-1}, \mathfrak{p}si).
{\bf{e}}nd{equation}
For the remaining two relevant elements of the Weyl group
$\tilde{K_\mathbf{s}igma}(-i\texttt{n}u, y, \cdot)$ still satisfies relation~(\ref{TildeKIsWhittaker}) for all $u \in U(\mathbb{R})$.
Thus we can still factor $\tilde{K_\mathbf{s}igma}(-i\texttt{n}u, y,t)=K_\mathbf{s}igma(-i\texttt{n}u,y,\mathfrak{p}si)W(-i\texttt{n}u,t,\mathfrak{p}si)$ for appropriate $y$,
and define $K_\mathbf{s}igma(-i\texttt{n}u,y,\mathfrak{p}si)$ to be the $\mathbf{s}igma$-{\bf Bessel function}.
However because of the restriction on $y$, the conditions satisfied by $K_\mathbf{s}igma(-i\texttt{n}u,y,\mathfrak{p}si)$ are more complicated.
Buttcane has announced a proof for Conjecture~\ref{interchange} in a more general context.
This would thus yield a uniform expression for the Archimedean integrals attached to the various elements of the Weyl group.
\begin{proposition}~\label{ArchWithConj}
Assume Conjecture~\ref{interchange}
Let $H \cdot \delta_\mathbf{s}igma$ be a relevant orbit. Then the corresponding Archimedean orbital integral is given by
\begin{align*}
I_{\delta_\mathbf{s}igma}(f_{\infty})=\bm{f}rac1c\bm{f}rac{\mathbf mathcal Delta_\mathbf{s}igma(t_{\mathbf m_2})}{|\mathbf m_{1,1}^4\mathbf m_{1,2}^3|}
\int_{\mathbf mathcal Lie{a^*}}\tilde{f_\infty}(-i\texttt{n}u)
K_\mathbf{s}igma(-i\texttt{n}u,t_{\mathbf m_1}^{-1}\mathbf{s}igma \delta t_{\mathbf m_2} \mathbf{s}igma^{-1},{\mathfrak{p}si_{\mathbf mathbf{1}}})\\
\times W(-i\texttt{n}u, t_{\mathbf m_2}^{-1} t_2,{\mathfrak{p}si_{\mathbf mathbf{1}}})
W(i\texttt{n}u, t_{\mathbf m_1}^{-1}t_1, \mathfrak{p}si_{\mathbf mathbf{1}})\bm{f}rac{d\texttt{n}u}{c(i\texttt{n}u)c(-i\texttt{n}u)}
{\bf{e}}nd{align*}
where the constant $c$ is the one appearing in the spherical inversion theorem
and $\mathbf mathcal Delta_\mathbf{s}igma$ is the modular character of the group $U_\mathbf{s}igma(\mathbb{R}) \backslash U(\mathbb{R})$.
{\bf{e}}nd{proposition}
\begin{proof}
We apply the statement of Conjecture~\ref{interchange} to the integral in Lemma~\ref{InTermsOfW} for the function
defined by
$$g(-i\texttt{n}u)=\bm{f}rac{1}{c(i\texttt{n}u)c(-i\texttt{n}u)}\tilde{f_\infty}(-i\texttt{n}u)W(i\texttt{n}u, t_{\mathbf m_1}^{-1}t_1, \mathfrak{p}si_{\mathbf m_1}),$$ which has rapid decay
by the rapid decay of $\tilde{f_\infty}$ (Theorem~\ref{PW}), the explicit expression~(\ref{Plancherel}) for the spectral measure, and the estimate for the Whittaker function in the spectral aspect given by Proposition~\ref{WhittakerSpectral}.
{\bf{e}}nd{proof}
\mathbf{s}ubsection{Symplectic Kloosterman sums}
In this section, we compute the non-Archimedean part of the orbital integrals when the finite part of the test function satisfies the following.
\begin{assumption}\label{HeckeOprtr}
Recall from Assumption~\ref{testfunction} that we assume $f=f_\infty\mathfrak{p}rod_p f_p$ has central character $\mathfrak{o}mega.$
We now further assume that there are two coprime positive integers $N$ and $\texttt{n}$ such that
$\mathfrak{o}mega$ is trivial on $(1+N\hat{\mathbb{Z}}) \cap \hat{\mathbb{Z}}^\times$, and
the function $f_{fin}$ is supported on $Z(\mathbf mathbb{A}_{fin})M(\texttt{n},N)$ and satisfies
$$f_{fin}(zm)=\bm{f}rac1{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_1(N)})}\mathfrak{o}verline{\mathfrak{o}mega}(z)$$
for $z \in Z(\mathbf mathbb{A}_{fin})$ and $m\in M(\texttt{n},N)$,
where $$M(\texttt{n},N)=\left\{g \in G(\mathbf mathbb{A}_{fin}) \cap \mathbf mathcal Mat_4(\hat{\mathbb{Z}}) : g {\bf{e}}quiv \left[
\begin{smallmatrix}
* & & * & * \\
* & 1 & * & * \\
& & * & * \\
& & & *
{\bf{e}}nd{smallmatrix}\right] \mathbf mod N, \mathbf mu(g) \in \texttt{n} \hat{\mathbb{Z}}^\times \right\},$$
and $$\mathbf mathfrak{G}amma_1(N)=\left\{g \in G(\hat{\mathbb{Z}}) : g {\bf{e}}quiv \left[
\begin{smallmatrix}
* & & * & * \\
* & 1 & * & * \\
& & * & * \\
& & & *
{\bf{e}}nd{smallmatrix}\right] \mathbf mod N\right\}.$$
{\bf{e}}nd{assumption}
\begin{remark}
With this choice, $f=\bigotimes_p f_p$, and each $f_p$ is left and right $\mathbf mathfrak{G}amma_p(N)$-invariant,
where $$\mathbf mathfrak{G}amma_p(N)=\{g \in G(\mathbb{Z}_p): g {\bf{e}}quiv \left[
\begin{smallmatrix}
* & & * & * \\
* & 1 & * & * \\
& & * & * \\
& & & *
{\bf{e}}nd{smallmatrix}\right] \mathbf mod N \}.$$
In particular, if $x, c \in \mathbb{Z}_p$ then $\mathbf mathfrak{G}amma_p$ contains the matrix $\left[
\begin{smallmatrix}
1 & & c & -cx \\
x & 1 & & \\
& & 1 & -x \\
& & & 1
{\bf{e}}nd{smallmatrix}\right].$
Thus if $\mathfrak{p}hi$ is right-$\mathbf mathfrak{G}amma_p$-invariant for all prime $p$, changing variables
$u \mathbf mapsto u\left[
\begin{smallmatrix}
1 & & c & -cx \\
x & 1 & & \\
& & 1 & -x \\
& & & 1
{\bf{e}}nd{smallmatrix}\right]$ in the integral expression of the $\mathfrak{p}si_\mathbf m$-Whittaker coefficient of $\mathfrak{p}hi$,
we get $\mathcal{W}_{\mathfrak{p}si_m}(\mathfrak{p}hi)(g)=\mathfrak{o}verline{\theta(m_1x+m_2c)}\mathcal{W}_{\mathfrak{p}si_m}(\mathfrak{p}hi)(g)$ for all $g$.
Therefore $\mathcal{W}_{\mathfrak{p}si_m}(\mathfrak{p}hi)=0$ unless $m_1$ and $m_2$ are integers.
Henceforth, we shall assume $\mathbf m_1$ and $\mathbf m_2$ are two pairs of integers.
{\bf{e}}nd{remark}
\begin{remark}
Note that $\mathbf mathfrak{G}amma=K_\infty \mathfrak{p}rod_p\mathbf mathfrak{G}amma_p(N)$ is contained both in the Borel, Klingen, Siegel, and paramodular congruence subgroup of level $N$,
thus any automorphic form that is fixed by one of these groups is also fixed by $\mathbf mathfrak{G}amma$, and hence will appear in our formula.
One could fix a different choice of congruence subgroup, and accordingly define different types of Kloosterman sums.
{\bf{e}}nd{remark}
Under Assumption~\ref{HeckeOprtr}, we can give more restriction about the element $\delta_1$ in Lemma~\ref{relevantorbits}.
\begin{lemma}\label{Supportdelta}
Let $\mathbf{s}igma \in \text{O}mega$, $\delta_1=\diag{d_1}{1}{d_2}{d_1d_2}$
such that the orbit of $\delta_\mathbf{s}igma=\mathbf{s}igma \delta_1$ is relevant.
Assume $I_{\delta_\mathbf{s}igma}(f_{\text{fin}}) \texttt{n}eq 0$.
Then there is an integer $s$ such that $d_1d_2=\mathfrak{p}m\bm{f}rac{\texttt{n}}{s^2}$.
{\bf{e}}nd{lemma}
\begin{proof}
For all $u \in U(\mathbf mathbb{A})$ and $u_1 \in U_\mathbf{s}igma(\mathbf mathbb{A}) \backslash U(\mathbf mathbb{A})$
we have $\mathbf mu(u \delta_\mathbf{s}igma u_1)=d_1d_2$.
So by Assumption~\ref{HeckeOprtr}, $u \delta_\mathbf{s}igma u_1$ belongs to the support of $f$
only if $d_1d_2 \in \mathbf mathbb{A}_{fin}^2\hat{\mathbb{Z}}^\times \texttt{n}$.
Since $d_1d_2$ is a rational number, there must be a rational number $s$ such that $d_1d_2=\mathfrak{p}m \bm{f}rac{\texttt{n}}{s^2}$.
But the second diagonal entry of $s \mathbf{s}igma^{-1} u \delta_\mathbf{s}igma u_1$ is $s$ therefore $s$ must belong to $\hat{\mathbb{Z}}$,
hence $s$ is an integer.
{\bf{e}}nd{proof}
Henceforth, we shall assume $\delta$ is as in Lemma~\ref{Supportdelta}. By Remark~\ref{d1d2}, we could also assume that $d_1d_2>0$
(which would then fix the sign in the equality $d_1d_2=\mathfrak{p}m\bm{f}rac{\texttt{n}}{s^2}$ above). However, we do not need doing so for now, and we shall not,
in view of possible applications with a different choice of test function at the Archimedean place.
\begin{remark}
Consider the case $N=\texttt{n}=1$. Then $M(\texttt{n},N)=\mathbf mathfrak{G}Sp_4(\hat{\mathbb{Z}})=\mathfrak{p}rod_p \mathbf mathfrak{G}Sp_4(\mathbb{Z}_p).$
For simplicity, set ${\bf{e}}ta=\delta_\mathbf{s}igma$, and if $p$ is a prime and $x \in G(\mathbf mathbb{A})$, write $x_p$ for the $p$-th
component of $x$.
Also write $\mathfrak{p}si_{p,1}$ and $\mathfrak{p}si_{p,2}$ for the local $p$-th components of the characters $\mathfrak{p}si_{\mathbf m_1}$ and $\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}}$, respectively.
In particular, these characters are trivial on $U(\mathbb{Z}_p)$.
Then we have
\begin{align*}
I_{\delta_\mathbf{s}igma}(f_{fin})&= \bm{f}rac1{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_1(N)})}
\int_{U_\mathbf{s}igma(\mathbf mathbb{A}_{fin}) \backslash U(\mathbf mathbb{A}_{fin})}\int_{U(\mathbf mathbb{A}_{fin})}\mathbf mathbbm{1}_{\mathbf mathfrak{G}Sp_4(\hat{\mathbb{Z}})}(s u {\bf{e}}ta v){\mathfrak{p}si_{\mathbf m_1}(u)}\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}(v)}dudv\\
&=\mathfrak{p}rod_p \bm{f}rac1{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_p(N)})} \int_{U_\mathbf{s}igma(\mathbb{Q}_p) \backslash U(\mathbb{Q}_p)}\int_{U(\mathbb{Q}_p)}\mathbf mathbbm{1}_{\mathbf mathfrak{G}Sp_4(\mathbb{Z}_p)}(s_p u_p {\bf{e}}ta_p v_p){\mathfrak{p}si_{p,1}(u_p)}{\mathfrak{p}si_{p,2}(v_p)}du_pdv_p.\\
{\bf{e}}nd{align*}
For all but finitely many prime $p$, the entries of $s_p{\bf{e}}ta_p$ are in $\mathbb{Z}_p^\times$. For those primes, by the explicit Bruhat decomposition
(see Lemma~\ref{uoppu} below),
the condition $s_p u_p {\bf{e}}ta_p v_p \in \mathbf mathfrak{G}Sp_4(\mathbb{Z}_p)$ is equivalent to $u_p \in U(\mathbb{Z}_p)$ and $v_p \in U_\mathbf{s}igma(\mathbb{Z}_p) \backslash U(\mathbb{Z}_p)$, and hence
$$\int_{U_\mathbf{s}igma(\mathbb{Q}_p) \backslash U(\mathbb{Q}_p)}\int_{U(\mathbb{Q}_p)}\mathbf mathbbm{1}_{\mathbf mathfrak{G}Sp_4(\mathbb{Z}_p)}(s_p u_p {\bf{e}}ta_p v_p){\mathfrak{p}si_{p,1}(u_p)}{\mathfrak{p}si_{p,2}(v_p)}du_pdv_p
=1$$
For the remaining primes $p$, noticing that $U_\mathbf{s}igma(\mathbb{Q}_p) \backslash U(\mathbb{Q}_p)$ may be identified to the subgroup
$\mathfrak{o}verline{U_\mathbf{s}igma}(\mathbb{Q}_p)=U(\mathbb{Q}_p) \cap \mathbf{s}igma^{-1} \trans{U(\mathbb{Q}_p)} \mathbf{s}igma$,
the local integral equals the Kloosteman sum $\mathbf mathcal Kloos({\bf{e}}ta,\mathfrak{p}si_{p,1},\mathfrak{p}si_{p,2})$ as defined in~\cite{SHMKloosterman}
when ${\bf{e}}ta \in \text{Sp}_4(\mathbb{Q}_p)$ (note that we denote here by $U_\mathbf{s}igma$ what is denoted there by $\mathfrak{o}verline{U_\mathbf{s}igma}$, and conversely).
{\bf{e}}nd{remark}
We now treat separately the contribution from each relevant element of the Weyl group from a global point of view.
To alleviate notations, we shall not include $N$ and $\mathfrak{o}mega$ in the argument of the Kloosterman sums we proceed to define.
\mathbf{s}ubsubsection{The identity contribution}
\begin{definition}\label{sumid}
Let $a,b,d,N$ be integers such that $d \mathbf mid N$.
Then the following sum is well-defined
$$S(a,b,d,N)=\mathbf{s}um_{\mathbf{s}ubstack{x,y \in \mathbb{Z}/ N \mathbb{Z}\\d \mathbf mid xy}}e\left(\bm{f}rac{ax+by}N\right).$$
{\bf{e}}nd{definition}
\begin{lemma}
Let $a,b,d,N$ be integers such that $d \mathbf mid N$.
Write $a=\mathfrak{p}rod_i p_i^{a_i}$, where $p_i$ are distinct primes and $a_i$ are integers, and similarly for $b, d, N$.
Then we have
$$S(a,b,d,N)=\mathfrak{p}rod_i S(p_i^{a_i}, p_i^{b_i}, p_i^{d_i}, p_i^{N_i}).$$
Moreover if $n$ is a positive integer, $i,j,k$ are non-negative integers with $k \le n$ and $p$ is a prime, then we have
\begin{align*}
S(p^i,p^j,p^k,p^n)&=p^{2n-k}(1-p^{-1})\mathbf max(0,k+1-\mathbf max(0,n-i)-\mathbf max(0,n-j))\\
&+p^{2n-k-1}\left(\mathbf mathbbm{1}_{\mathbf{s}ubstack{i \ge n\\j \ge n}}-
\mathbf mathbbm{1}_{\mathbf{s}ubstack{i < n\\j < n\\i+j\ge 2n-k-1}}\right).
{\bf{e}}nd{align*}
In particular, it follows that $S(p^i,p^j,p^k,p^n)$ is non-zero only if
\begin{equation}\label{sumnonzero}
(n-i)+(n-j) \le k+1.
{\bf{e}}nd{equation}
{\bf{e}}nd{lemma}
\begin{proof}
The factorization is immediate from the Chinese remainders theorem.
Now let us evaluate $S=S(p^i,p^j,p^k,p^n)$.
We have (here, abusing slightly notations, we set $v_p(0)=n$)
$$S=\mathbf{s}um_{h=0}^k \mathbf{s}um_{\mathbf{s}ubstack{x \in \mathbb{Z} / p^n \mathbb{Z} \\ v_p(x)=h}} e\left(\bm{f}rac{p^ix}{p^n}\right)
\mathbf{s}um_{\mathbf{s}ubstack{y \in \mathbb{Z} / p^n \mathbb{Z} \\ v_p(y) \ge k-h}} e\left(\bm{f}rac{p^jy}{p^n}\right)
+ \mathbf{s}um_{\mathbf{s}ubstack{x \in \mathbb{Z} / p^n \mathbb{Z} \\ v_p(x) \ge k+1}} e\left(\bm{f}rac{p^ix}{p^n}\right)
\mathbf{s}um_{y \in \mathbb{Z} / p^n \mathbb{Z}} e\left(\bm{f}rac{p^jy}{p^n}\right).$$
Now if ${\bf{e}}ll$ is any non-negative integer, we have
$$ \mathbf{s}um_{\mathbf{s}ubstack{y \in \mathbb{Z} / p^n \mathbb{Z} \\ v_p(y) \ge {\bf{e}}ll}} e\left(\bm{f}rac{p^jy}{p^n}\right)=
\begin{cases}
p^{n-{\bf{e}}ll} \text{ if } j+{\bf{e}}ll \ge n \text{ and } {\bf{e}}ll \le n\\
0 \text{ otherwise.}
{\bf{e}}nd{cases}$$
Hence
$$S=\mathbf{s}um_{h=0}^{k-\mathbf max(0,n-j)}p^{n-k+h} \mathbf{s}um_{\mathbf{s}ubstack{x \in \mathbb{Z} / p^n \mathbb{Z} \\ v_p(x)=h}} e\left(\bm{f}rac{p^ix}{p^n}\right)
+p^{2n-k-1}\mathbf mathbbm{1}_{\mathbf{s}ubstack{j \ge n\\i+k+1 \ge n\\k<n}}.$$
Now $$\mathbf{s}um_{\mathbf{s}ubstack{x \in \mathbb{Z} / p^n \mathbb{Z} \\ v_p(x)=h}} e\left(\bm{f}rac{p^ix}{p^n}\right)=
\mathbf{s}um_{\mathbf{s}ubstack{x \in \mathbb{Z} / p^n \mathbb{Z} \\ v_p(x) \ge h}} e\left(\bm{f}rac{p^ix}{p^n}\right)
- \mathbf{s}um_{\mathbf{s}ubstack{x \in \mathbb{Z} / p^n \mathbb{Z} \\ v_p(x) \ge h+1}} e\left(\bm{f}rac{p^ix}{p^n}\right),$$
hence the $h$-sum becomes
$$\left(\mathbf{s}um_{h=\mathbf max(0,n-i)}^{k-\mathbf max(0,n-j)} p^{2n-k}(1-p^{-1})\right)-p^{2n-k-1}\mathbf mathbbm{1}_{0 \le n-i-1 \le k-\mathbf max(0,n-j)}
+p^{2n-k-1}\mathbf mathbbm{1}_{k-\mathbf max(0,n-j)=n},$$
so
\begin{align*}
S=p^{2n-k}(1-p^{-1})\mathbf max(0,k+1-\mathbf max(0,n-i)-\mathbf max(0,n-j))\\
+p^{2n-k-1}(\mathbf mathbbm{1}_{\mathbf{s}ubstack{j \ge n\\i+k+1 \ge n\\k<n}}-\mathbf mathbbm{1}_{0 \le n-i-1 \le k-\mathbf max(0,n-j)}+\mathbf mathbbm{1}_{k-\mathbf max(0,n-j)=n}).
{\bf{e}}nd{align*}
Finally, it can be checked by inspection of cases that
$$\mathbf mathbbm{1}_{\mathbf{s}ubstack{j \ge n\\i+k+1 \ge n\\k<n}}-\mathbf mathbbm{1}_{0 \le n-i-1 \le k-\mathbf max(0,n-j)}+\mathbf mathbbm{1}_{k-\mathbf max(0,n-j)=n}=
\mathbf mathbbm{1}_{\mathbf{s}ubstack{i \ge n\\j \ge n}}-
\mathbf mathbbm{1}_{\mathbf{s}ubstack{i < n\\j < n\\i+j\ge 2n-k-1}}.$$
{\bf{e}}nd{proof}
\begin{proposition}\label{idcontribution}
Let $\mathbf{s}igma=1$, $\delta_1=\diag{d_1}{1}{d_2}{d_1d_2}$ with $d_1d_2=\mathfrak{p}m\bm{f}rac{\texttt{n}}{s^2}$ for some integer $s$.
Then $I_{\delta_\mathbf{s}igma}(f_{fin})=0$ unless all of the following holds:
\begin{enumerate}
\item $s$ divides $\texttt{n}$,
\item $d_1=\bm{f}rac{\mathbf m_{1,1}}{\mathbf m_{2,1}}$ and $sd_1$ is an integer dividing $\texttt{n}$,
\item $d_2=\bm{f}rac{\mathbf m_{1,1}\mathbf m_{1,2}}{\mathbf m_{2,1}\mathbf m_{2,2}}$,
{\bf{e}}nd{enumerate}
If all these conditions are met, let $d=\gcd(s,sd_1s,d_2,\bm{f}rac{\texttt{n}}s)$, and $D=\gcd(sd_1,\bm{f}rac{n}s)$. Then
\begin{equation}\label{IdContribution}
I_{\delta_\mathbf{s}igma}(f_{fin})=\bm{f}rac{\mathfrak{o}verline{\mathfrak{o}mega_N(s)}}{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_1(N)})}\bm{f}rac{\texttt{n} d}{|s^3d_1|}
S\left(\mathbf m_{1,1}\bm{f}rac{\texttt{n}}D,\mathbf m_{1,2}sd_1,d,\texttt{n}\right),
{\bf{e}}nd{equation}
where $\mathfrak{o}mega_N(s)=\mathfrak{p}rod_{p \mathbf mid N} \mathfrak{o}mega_p(s)$.
{\bf{e}}nd{proposition}
\begin{remark}
The integer $s$ is only determined up to sign. However, expression~(\ref{IdContribution}) does not depend on the sign of $s$,
since $S(a,b,d,\texttt{n})=S(a,-b,d,\texttt{n})$ and $\mathfrak{o}mega_N(-1)=\mathfrak{o}mega(-1)=1$ as $\mathfrak{o}mega_p(-1)=1$ for all $p \texttt{n}mid N$.
{\bf{e}}nd{remark}
\begin{remark}
The two pair of integers $\mathbf m_1$ and $\mathbf m_2$ essentially play symmetric roles in our formula.
More precisely, for our choice of test function $f$, the operator $\mathfrak{o}mega_N(\texttt{n})^{\bm{f}rac12}R(f)$ is self-adjoint.
Thus exchanging $\mathbf m_1$ and $\mathbf m_2$ amounts to take the complex conjugate of the spectral side and multiply it by $\mathfrak{o}mega_N(\texttt{n})$.
Hence the geometric side, and in particular the identity contribution, should enjoy the same symmetries.
Proposition~\ref{idcontribution} says that the identity element has a non-zero contribution only if there is an integer $t$ dividing $\texttt{n}$
with $\bm{f}rac{\texttt{n}}t=\mathfrak{p}m\bm{f}rac{\mathbf m_{1,2}}{\mathbf m_{2,2}}t$ and such that $s=\bm{f}rac{\mathbf m_{2,1}}{\mathbf m_{1,1}}t$ is also an integer dividing $\texttt{n}$.
This condition is indeed symmetric, as interchanging $\mathbf m_1$ and $\mathbf m_2$ amounts to replace $t$ with $\bm{f}rac{\texttt{n}}t$ and $s$ with $\bm{f}rac{\texttt{n}}s$.
In addition, we have
$S\left(\mathbf m_{1,1}\bm{f}rac{\texttt{n}}{\gcd(t,\bm{f}rac{\texttt{n}}s)},\mathbf m_{1,2}t,d,\texttt{n}\right)=S\left(\mathbf m_{2,1}\bm{f}rac{\texttt{n}}{\gcd(s,\bm{f}rac{\texttt{n}}t)},\mathbf m_{2,2}\bm{f}rac\texttt{n}{t},d,\texttt{n}\right)$.
Finally, using that $|s^3d_1|=\left|\texttt{n}^3\bm{f}rac{\mathbf m_{2,1}^4\mathbf m_{2,2}^3}{\mathbf m_{1,1}^4\mathbf m_{1,2}^3}\right|^{\bm{f}rac12}$,
multiplying $\bm{f}rac{\texttt{n} }{|s^3d_1|}$ by the factor $\bm{f}rac1{|\mathbf m_{1,1}^4\mathbf m_{1,2}^3|}$ that comes from the Archimedean part in Proposition~\ref{ArchWithConj}
gives $\texttt{n}^{-\bm{f}rac12}(\mathbf m_{1,1}\mathbf m_{2,1})^{-2}|\mathbf m_{1,2}\mathbf m_{2,2}|^{-\bm{f}rac32}$.
{\bf{e}}nd{remark}
\begin{remark}
In the case $\texttt{n}=1$ we must have $s=\mathfrak{p}m1$ and hence $\mathbf m_{1,1}=\mathfrak{p}m\mathbf m_{2,1}$. Together with the condition
$d_1d_2=\mathfrak{p}m\bm{f}rac{\mathbf m_{1,1}^2\mathbf m_{1,2}}{\mathbf m_{2,1}^2\mathbf m_{2,2}}=\bm{f}rac{\texttt{n}}{s^2}$ this also gives $\mathbf m_{1,2}=\mathfrak{p}m\mathbf m_{2,2}.$
{\bf{e}}nd{remark}
\begin{remark}
Using condition~(\ref{sumnonzero}) we find that the contribution from the identity element is non-zero only if for
all prime $p \mathbf mid \texttt{n}$ we have $$v_p(s) \le v_p(\mathbf m_{2,1})+v_p(\mathbf m_{2,1})+\mathbf min(0,v_p(\mathbf m_{2,1})-v_p(\mathbf m_{1,1}))+1,$$
which in turn implies that for all prime $p$ we have
$$v_p(\texttt{n}) \le 2 \mathbf min(v_p(\mathbf m_{1,1}),v_p(\mathbf m_{2,1}))+v_p(\mathbf m_{1,2})+v_p(\mathbf m_{2,2})+1.$$
{\bf{e}}nd{remark}
\begin{proof}
The finite part of the orbital integral corresponding to the identity element reduces to
\begin{align*}
I_{\delta_\mathbf{s}igma}(f_{fin})= \int_{U(\mathbf mathbb{A}_{fin})}f( u \delta){\mathfrak{p}si_{\mathbf m_1}(u)}du
=\int_{U(\mathbf mathbb{A}_{fin})}f(s u \delta){\mathfrak{p}si_{\mathbf m_1}(u)}du.
{\bf{e}}nd{align*}
Assume it is non-zero.
Note that by Lemma~\ref{Supportdelta} we have $\mathbf mu(su\delta)=\texttt{n}$.
Then by Assumption~\ref{HeckeOprtr}, $su\delta \in Supp(f)$ if and only if $su\delta \in \mathbf mathcal Mat_4(\hat{\mathbb{Z}})$.
In particular, each entry of $s\delta$ must be an integer.
Furthermore by Lemma~\ref{relevantorbits} we must have
$\delta=\diag{d_1}{1}{d_2}{d_1d_2}$ with $d_1=\bm{f}rac{\mathbf m_{1,1}}{\mathbf m_{2,1}}$, $d_2=\bm{f}rac{\mathbf m_{1,1}\mathbf m_{1,2}}{\mathbf m_{2,1}\mathbf m_{2,2}}$.
So we learn that $sd_1= s\bm{f}rac{\mathbf m_{1,1}}{\mathbf m_{2,1}} \in \mathbb{Z}$, $s \mathbf mid \texttt{n}$, and $sd_1 \mathbf mid \texttt{n}$.
Now let us examine the non-diagonal entries of $su\delta$.
Write $u=\left[
\begin{smallmatrix}
1 & & c & a-cx \\
x & 1 & a & b \\
& & 1 & -x \\
& & & 1
{\bf{e}}nd{smallmatrix}\right]$.
Then the following conditions must hold:
\begin{enumerate}
\item $sd_1x \in \hat{\mathbb{Z}}$ and $\bm{f}rac{\texttt{n}}{s}x \in \hat{\mathbb{Z}}$,
\item $c' \doteq \bm{f}rac{\texttt{n}}{sd_1}c \in \hat{\mathbb{Z}}$,
\item $a' \doteq \bm{f}rac{\texttt{n}}{sd_1}a \in \hat{\mathbb{Z}}$,
\item $\bm{f}rac{\texttt{n}}{s}(a-cx) \in \hat{\mathbb{Z}}$,
\item $b' \doteq \bm{f}rac{\texttt{n}}{s}b \in \hat{\mathbb{Z}}$.
{\bf{e}}nd{enumerate}
Condition~(1) is equivalent to $x \in \bm{f}rac1{D}\hat{\mathbb{Z}}$, where $D=\gcd(sd_1, \bm{f}rac{\texttt{n}}s)$ (note that $sd_1 \mathbf mid sD$).
Set $x'=Dx$.
Then condition~(4) gives
$d_1a'-\bm{f}rac{d_1}{D}c'x' \in \hat{\mathbb{Z}}$.
Combined with conditions~(1),~(2) and~(3), this is equivalent to
$ c'x' {\bf{e}}quiv Da' \mathbf mod \bm{f}rac{D}{d_1}$.
Now, $\mathfrak{p}si_{\mathbf m_1}(u)=\theta_{fin}(\mathbf m_{1,1}x+\mathbf m_{1,2}c)$ and $f(su\delta)=\bm{f}rac{\mathfrak{o}verline{\mathfrak{o}mega_N(s)}}{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_1(N)})}$.
Therefore integration with respect to $b$ gives
$Vol\left(\bm{f}rac{s}{\texttt{n}}\hat{\mathbb{Z}}\right)\bm{f}rac{\mathfrak{o}verline{\mathfrak{o}mega_N(s)}}{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_1(N)})}=
\bm{f}rac{\texttt{n}}s\bm{f}rac{\mathfrak{o}verline{\mathfrak{o}mega_N(s)}}{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_1(N)})}.$
Next, changing variables $x=\bm{f}rac1Dx'$ and $c=\bm{f}rac{sd_1}\texttt{n} c'$, for fixed $a$ the $x,c$-integral is
$$I(a)=\bm{f}rac{\mathfrak{o}verline{\mathfrak{o}mega_N(s)}}{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_1(N)})}\bm{f}rac{\texttt{n}^2D}{s^2d_1}\int\int_{ c'x' {\bf{e}}quiv Da' \mathbf mod \bm{f}rac{D}{d_1}}
\theta_{fin}\left(\mathbf m_{1,1}\bm{f}rac{x'}{D}+\mathbf m_{1,2}\bm{f}rac{sd_1}\texttt{n} c'\right)dx'dc'.$$
Since $D \mathbf mid sd_1$ and $\mathbf m_{1,2}\bm{f}rac{s^2d_1^2}\texttt{n}=\mathbf m_{2,2}$, and since $\theta_{fin}$ is trivial on $\hat{\mathbb{Z}}$
the integrand is constant on cosets $x'+sd_1\hat{\mathbb{Z}}$ and $c' + sd_1\hat{\mathbb{Z}}$.
As $sd_1 \mathbf mid sD$, it is also constant on cosets $x'+sD\hat{\mathbb{Z}}$ and $c' + sD\hat{\mathbb{Z}}$.
Therefore we get
\begin{align*}
I(a)&=\bm{f}rac{\mathfrak{o}verline{\mathfrak{o}mega_N(s)}}{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_1(N)})}\bm{f}rac{\texttt{n}^2}{|Dd_1s^4|}\mathbf{s}um_{\mathbf{s}ubstack{x,y \in \mathbb{Z} / sD \mathbb{Z} \\ xy \in Da'+\bm{f}rac{D}{d_1} \hat{\mathbb{Z}}}}
e\left(\bm{f}rac{\mathbf m_{1,1}x}D+\bm{f}rac{\mathbf m_{1,2}sd_1y}{\texttt{n}}\right)
{\bf{e}}nd{align*}
Finally the $a$ integrand depends only on $a' \mathbf mod \bm{f}rac{D}{d_1} \hat{\mathbb{Z}}$, thus, setting $d=\gcd\left(D,\bm{f}rac{D}{d_1}\right)=\gcd(s,sd_1sd_2,\bm{f}rac{\texttt{n}}s)$ we get
\begin{align*}
I_{\delta_\mathbf{s}igma}(f_{fin})&=\bm{f}rac{\mathfrak{o}verline{\mathfrak{o}mega_N(s)}}{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_1(N)})}
\bm{f}rac{\texttt{n}^3 }{|s^5D^2d_1|}\mathbf{s}um_{a \in \mathbb{Z} / \bm{f}rac{D}{d_1} \mathbb{Z}}\mathbf{s}um_{\mathbf{s}ubstack{x,y \in \mathbb{Z} / sD \mathbb{Z} \\ xy \in Da+\bm{f}rac{D}{d_1} \hat{\mathbb{Z}}}}
e\left(\bm{f}rac{\mathbf m_{1,1}x}D+\bm{f}rac{\mathbf m_{1,2}sd_1y}{\texttt{n}}\right)\\
&=\bm{f}rac{\mathfrak{o}verline{\mathfrak{o}mega_N(s)}}{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_1(N)})}\bm{f}rac{\texttt{n}^3 }{|s^5D^2d_1|}d
\mathbf{s}um_{\mathbf{s}ubstack{x,y \in \mathbb{Z} / sD \mathbb{Z} \\ xy \in d \mathbb{Z}}}
e\left(\bm{f}rac{\mathbf m_{1,1}x}D+\bm{f}rac{\mathbf m_{1,2}sd_1y}{\texttt{n}}\right)\\
&=\bm{f}rac{\mathfrak{o}verline{\mathfrak{o}mega_N(s)}}{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_1(N)})}\bm{f}rac{\texttt{n} }{|s^3d_1|}d
\mathbf{s}um_{\mathbf{s}ubstack{x,y \in \mathbb{Z} / \texttt{n} \mathbb{Z} \\ xy \in d \mathbb{Z}}}
e\left(\bm{f}rac{\mathbf m_{1,1}x}D+\bm{f}rac{\mathbf m_{1,2}sd_1y}{\texttt{n}}\right)\\
{\bf{e}}nd{align*}
{\bf{e}}nd{proof}
\mathbf{s}ubsubsection{The contribution from the longest Weyl element}
The following lemma makes it explicit how to compute the Bruhat decomposition for elements in the cell of the
long Weyl element.
One could do the same for each element of the Weyl group, but, as it is straightforward calculations, we only
include this case for the sake of clarity in latter arguments.
\begin{lemma}\label{uoppu}
Let $\mathbf mathbb{F}$ be a field, and let $g \in \mathbf mathfrak{G}Sp_4(\mathbf mathbb{F})$.
Assume
$$g=\left[
\begin{smallmatrix}
1 & & c_1 & a_1 \\
x_1 & 1 & a_1+c_1x_1 & b_1 \\
& & 1 & -x_1 \\
& & & 1 \\
{\bf{e}}nd{smallmatrix}
\right]
J
\diag{t_1}{t_2}{t_3t_1^{-1}}{t_3t_2^{-1}}
\left[
\begin{smallmatrix}
1 & & c_2 & a_2-c_2x_2 \\
x_2 & 1 & a_2 & b_2 \\
& & 1 & -x_2 \\
& & & 1 \\
{\bf{e}}nd{smallmatrix}
\right]
= \mathbf mat{A}{B}{C}{D} =
\bigmat{\block{a_{11}}{a_{12}}{a_{21}}{a_{22}}}
{B}
{\block{c_{11}}{c_{12}}{c_{21}}{c_{22}}}
{\block{d_{11}}{d_{12}}{d_{21}}{d_{22}}} .$$
Set
\begin{alignat*}{1}
\mathbf mathcal Delta_1=\mathbf mat{a_{11}}{a_{12}}{c_{21}}{c_{22}},& \quad \mathbf mathcal Delta_2=\mathbf mat{c_{12}}{d_{11}}{c_{22}}{d_{21}}.
{\bf{e}}nd{alignat*}
Then
\begin{alignat*}{5}
t_3=\mathbf mu(g), & \quad t_2=- c_{22}, & \quad t_1t_2=\det(C),
{\bf{e}}nd{alignat*}
\begin{alignat*}{5}
x_1=-\bm{f}rac{c_{12}}{c_{22}},& \quad x_2=\bm{f}rac{c_{21}}{c_{22}}, & \quad c_1=\bm{f}rac{\det(\mathbf mathcal Delta_1)}{\det(C)},& \quad c_2=-\bm{f}rac{\det(\mathbf mathcal Delta_2)}{\det(C)},
{\bf{e}}nd{alignat*}
\begin{alignat*}{5}
a_{1} =\bm{f}rac{a_{12}}{c_{22}} & \quad a_2 = \bm{f}rac{d_{21}}{c_{22}}&\quad b_1 =\bm{f}rac{a_{22}}{c_{22}} & \quad b_2 =\bm{f}rac{d_{22}}{c_{22}}.
{\bf{e}}nd{alignat*}
Moreover, if $g= \mathbf mat{A}{B}{C}{D} \in \mathbf mathfrak{G}Sp_4(\mathbf mathbb{F})$ with $C=\mathbf mat{c_{11}}{c_{12}}{c_{21}}{c_{22}}$ satisfies
$\det(C) \texttt{n}eq 0$ and $c_{22} \texttt{n}eq 0$ then~$g \in UJ T U$.
{\bf{e}}nd{lemma}
\begin{proof}
The first claims follow by computing explicitly
\begin{alignat*}{3}
C&=\mathbf mat{1}{-x_1}{}{1}\mathbf mat{-t_1}{}{}{-t_2}\mathbf mat{1}{}{x_2}{1}=\mathbf mat{-t_1+t_2x_1x_2}{t_2x_1}{-t_2x_2}{-t_2},\\
\mathbf mathcal Delta_1&=\mathbf mat{c_1}{a_1}{}{1}\mathbf mat{-t_1}{}{}{-t_2}\mathbf mat{1}{}{x_2}{1}, & \quad
\mathbf mathcal Delta_2&=\mathbf mat{1}{-x_1}{}{1}\mathbf mat{-t_1}{}{}{-t_2}\mathbf mat{}{c_2}{1}{a_2},\\
D&=\mathbf mat{1}{-x_1}{}{1}\mathbf mat{-t_1}{}{}{-t_2}\mathbf mat{c_2}{a_2-c_2x_2}{a_2}{b_2},&\quad
A&=\mathbf mat{c_1}{a_1}{a_1+c_1x_1}{b_1}\mathbf mat{-t_1}{}{}{-t_2}\mathbf mat{1}{}{x_2}{1}.
{\bf{e}}nd{alignat*}
To prove the last claim, it suffices to show that provided $\det{C} \texttt{n}eq 0$ and $c_{22} \texttt{n}eq 0$,
there exist at most one $g \in \mathbf mathfrak{G}Sp_4(\mathbf mathbb{F})$ with the specified values for $\mathbf mu(g)$, $C$, $a_{12}, a_{22},$ $d_{21}, d_{22}$,
$\det(\mathbf mathcal Delta_1)$ and $\det(\mathbf mathcal Delta_2)$.
Since $c_{22} \texttt{n}eq 0$, the values of $a_{12}$, $c_{21}$ and $\det(\mathbf mathcal Delta_1)=a_{11}c_{22}-c_{21}a_{12}$ determine the value of
$a_{11}$. The equation $\trans{A}C=\trans{C}A$ then gives $a_{12}c_{11}+a_{22}c_{21}=a_{11}c_{12}+a_{21}c_{22}$,
which determines the value of $a_{21}$ hence of $A$.
The same reasoning using $\det(\mathbf mathcal Delta_1)$ and $C\trans{D}=D\trans{C}$ instead similarly fixes $D$.
Finally the equation $\trans{A}D-\trans{C}B=\mathbf mu(g)$ fixes $B$ since we are assuming $C$ is invertible.
{\bf{e}}nd{proof}
\begin{definition}\label{KloosLWE}
Let $s,d,m$ be three non-zero integers and define
$$C_J(s,d,m)= \{g=\mathbf mat{A}{B}{C}{D} : \det(C)=d, c_{22}=-s, \mathbf mu(g)=m\}$$
and
$$\mathbf mathfrak{G}amma_J(N,s,d,m)= \mathbf mathfrak{G}amma_1(N) \cap \mathbf mathcal Mat_4(\mathbb{Z}) \cap C_J(s,d,m).$$
For $g=\mathbf mat{A}{B}{C}{D} \in \mathbf mathfrak{G}Sp_4$,
let $\mathbf mathcal Delta_1=\mathbf mat{a_{11}}{a_{12}}{c_{21}}{c_{22}}$
and $\mathbf mathcal Delta_2=\mathbf mat{c_{12}}{d_{11}}{c_{22}}{d_{21}}$.
Then, for $\mathbf m_1,\mathbf m_2$ two pair of non-zero integers, we define the following generalized twisted Kloosterman sum
\begin{align*}
\mathbf mathcal Kloos_J(\mathbf m_1,\mathbf m_2,s,d,m)&=\\
\mathbf{s}um_{g \in U(\mathbb{Z}) \backslash \mathbf mathfrak{G}amma_J(N,s,d,m) / U(\mathbb{Z})}&\mathfrak{o}verline{\mathfrak{o}mega_N}(a_{22})
e\left(\bm{f}rac{\mathbf m_{11}c_{12}+\mathbf m_{21}c_{21}}{s}+\bm{f}rac{\mathbf m_{12}\det(\mathbf mathcal Delta_1)+\mathbf m_{22}\det(\mathbf mathcal Delta_2)}{d}\right).
{\bf{e}}nd{align*}
{\bf{e}}nd{definition}
\begin{remark}
Using Lemma~\ref{uoppu}, we can see that $\mathbf mathcal Kloos_J(\mathbf m_1,\mathbf m_2,s,d,m)$ is well defined.
Indeed, matrices in $\mathbf mathfrak{G}amma_J(\texttt{n},N,d,s)$ are of the form $$g=u(x_1,a_1,b_1,c_1)J\diag{\bm{f}rac{d}s}{s}{m\bm{f}rac{s}{d}}{\bm{f}rac{m}s}u(x_2,a_2,b_2,c_2).$$
Then $\bm{f}rac{c_{12}}s=x_1$,$\bm{f}rac{c_{21}}s=-x_2$, $\bm{f}rac{\det(\mathbf mathcal Delta_1)}{d}=c_1$ and $\bm{f}rac{\det(\mathbf mathcal Delta_2)}{d}=-c_2$.
Now multiplying g on the left (resp. on the right) by an element of $U(\mathbb{Z})$ does not change the classes of $x_1$ and $c_1$ (resp. $x_2$ and $c_2$)
in $\mathbb{R} / \mathbb{Z}$.
{\bf{e}}nd{remark}
\begin{proposition}\label{KloosJ}
Let $\mathbf{s}igma=J$, $\delta_1=\diag{d_1}{1}{d_2}{d_1d_2}$
with $d_1d_2=\mathfrak{p}m\bm{f}rac{\texttt{n}}{s^2}$ for some integer $s$.
Then we have $I_{\delta_\mathbf{s}igma}(f_{fin})=\bm{f}rac{1}{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_1(N)})}\mathbf mathcal Kloos_J(\mathbf m_1,\mathbf m_2,s,d_1s^2,s^2d_1d_2)$.
{\bf{e}}nd{proposition}
\begin{remark}
The set $\mathbf mathfrak{G}amma_J(N,s,d_1s^2,s^2d_1d_2)$ is non-empty only if $N$ divides $s$ and $N^2$ divides $d_1s^2$.
{\bf{e}}nd{remark}
\begin{proof}
The finite part of the orbital integral corresponding to the longest Weyl element reduces to
\begin{align*}
I_{\delta_\mathbf{s}igma}(f_{fin})&= \int_{U(\mathbf mathbb{A}_{fin})}\int_{U(\mathbf mathbb{A}_{fin})}f( u_1 J \delta u_2){\mathfrak{p}si_{\mathbf m_1}(u_1)}\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}(u_2)}du_1du_2\\
&=\int_{U(\mathbf mathbb{A}_{fin})}\int_{U(\mathbf mathbb{A}_{fin})}f(s u_1 J \delta u_2){\mathfrak{p}si_{\mathbf m_1}(u_1)}\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}(u_2)}du_1du_2.
{\bf{e}}nd{align*}
By Assumption~\ref{HeckeOprtr} $su_1 J \delta u_2 \in Supp(f)$ if and only if $su_1J\delta u_2=\mathbf mat{A}{B}{C}{D} \in Z(\mathbf mathbb{A}_{fin})M(\texttt{n},N)$.
In this case, we have $f(s u_1 J \delta u_2)=\bm{f}rac{\mathfrak{o}verline{\mathfrak{o}mega_N}(a_{22})}{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_1(N)})}$,
and Lemma~\ref{uoppu} shows that
$${\mathfrak{p}si_{\mathbf m_1}(u_1)}\mathfrak{o}verline{\mathfrak{p}si_{\mathbf m_2}(u_2)}=
e\left(-\bm{f}rac{\mathbf m_{11}c_{12}+\mathbf m_{21}c_{21}}{c_{22}}+\bm{f}rac{\mathbf m_{12}\det(\mathbf mathcal Delta_1)+\mathbf m_{22}\det(\mathbf mathcal Delta_2)}{\det(C)}\right).$$
Moreover, $f$ is left and right $U(\hat{\mathbb{Z}})$-invariant, and the characters $\mathfrak{p}si_{\mathbf m_1}$ and $\mathfrak{p}si_{\mathbf m_2}$
are trivial on~$\hat{\mathbb{Z}}$.
Therefore, if we consider the map $\varphi: U(\mathbf mathbb{A}_{fin}) \times U(\mathbf mathbb{A}_{fin}) \to G(\mathbf mathbb{A}), (u_1,u_2) \mathbf mapsto s u_1 J \delta u_2$, we have
\begin{align*}
I_{\delta_\mathbf{s}igma}(f_{fin})=
\mathbf{s}um_{U(\hat{\mathbb{Z}}) \backslash \left(M(\texttt{n},N) \cap Im(\varphi)\right) / U(\hat{\mathbb{Z}})}
&\bm{f}rac{\mathfrak{o}verline{\mathfrak{o}mega_N}(a_{22})}{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_1(N)})}\\
& \times
e\left(-\bm{f}rac{\mathbf m_{11}c_{12}+\mathbf m_{21}c_{21}}{c_{22}}+\bm{f}rac{\mathbf m_{12}\det(\mathbf mathcal Delta_1)+\mathbf m_{22}\det(\mathbf mathcal Delta_2)}{\det(C)}\right).
{\bf{e}}nd{align*}
Now by Lemma~\ref{uoppu}, $Im(\varphi)=C_J(s,d_1s^2,s^2d_1d_2)$.
Therefore, $$U(\hat{\mathbb{Z}}) \backslash (M(\texttt{n},N) \cap Im(\varphi)) / U(\hat{\mathbb{Z}})$$ may be identified to $U(\mathbb{Z}) \backslash \mathbf mathfrak{G}amma_J(N,s,d_1s^2,s^2d_1d_2) / U(\mathbb{Z})$.
{\bf{e}}nd{proof}
\mathbf{s}ubsubsection{Contribution from \texorpdfstring{$\mathbf{s}igma=s_1s_2s_1$}{s1s2s1}}
\begin{definition}
Let $s,d,m$ be three non-zero integers and define
$$C_{121}(s,d,m)= \{g=\mathbf mat{A}{B}{C}{D}: \det(C)=0, c_{22}=-s, \det(\mathbf mathcal Delta_2)=d, \mathbf mu(g)=m\}$$
and
$$\mathbf mathfrak{G}amma_{121}(N,s,d,m)= \mathbf mathfrak{G}amma_1(N) \cap \mathbf mathcal Mat_4(\mathbb{Z}) \cap C_{121}(s,d,m)$$
For $g=\mathbf mat{A}{B}{C}{D} \in \mathbf mathfrak{G}Sp_4$, let $\mathbf mathcal Delta_3=\mathbf mat{a_{12}}{b_{11}}{c_{22}}{d_{21}}$.
Then we define the following generalized twisted Kloosterman sum
\begin{align*}
\mathbf mathcal Kloos_{121}(\mathbf m_1,\mathbf m_2,s,d,m)&=\\
\mathbf{s}um_{g \in U(\mathbb{Z}) \backslash \mathbf mathfrak{G}amma_{121}(N,s,d,m) / \mathfrak{o}verline{U_\mathbf{s}igma}(\mathbb{Z})}&\mathfrak{o}verline{\mathfrak{o}mega_N}(a_{22})
e\left(\bm{f}rac{\mathbf m_{11}c_{12}+\mathbf m_{21}c_{21}}{s}+\bm{f}rac{\mathbf m_{12}\det(\mathbf mathcal Delta_3)}{d}\right).
{\bf{e}}nd{align*}
{\bf{e}}nd{definition}
By a similar argument as in the case of the long Weyl element, $\mathbf mathcal Kloos_{121}(\mathbf m_1,\mathbf m_2,s,d,m)$ is well-defined, and together with the condition on $\delta$,
from Lemma~\ref{relevantorbits} we get the following.
\begin{proposition}
Let $\mathbf{s}igma=s_1s_2s_1$, $\delta_1=\diag{d_1}{1}{d_2}{d_1d_2}$
with $d_1d_2=\mathfrak{p}m \bm{f}rac{\texttt{n}}{s^2}$ for some integer $s$ and $d_1\mathbf m_{1,2}=d_2\mathbf m_{2,2}$.
Then we have $I_{\delta_\mathbf{s}igma}(f_{fin})=\bm{f}rac{1}{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_1(N)})}\mathbf mathcal Kloos_{121}(\mathbf m_1,\mathbf m_2,s,d_2s^2,s^2d_1d_2)$.
{\bf{e}}nd{proposition}
\begin{remark}
The set $\mathbf mathfrak{G}amma_{121}(\texttt{n},N,s,d_2s^2,s^2d_1d_2)$ is non-empty only if $N$ divides $s$ and $N^2$ divides $d_2s^2$.
{\bf{e}}nd{remark}
\mathbf{s}ubsubsection{Contribution from \texorpdfstring{$\mathbf{s}igma=s_2s_1s_2$}{s2s1s2}}
\begin{definition}
Let $s,d$ be three non-zero integers and define
$$C_{212}(s,d,m)= \{g=\mathbf mat{A}{B}{C}{D} : \det(C)=-d, c_{22}=0, c_{21}=-s, \mathbf mu(g)=m \}$$
and
$$\mathbf mathfrak{G}amma_{212}(N,s,d,m)=\mathbf mathcal Mat_4(\mathbb{Z}) \cap \mathbf mathfrak{G}amma_1(N) \cap C_{212}.$$
We define the following generalized twisted Kloosterman sum
\begin{align*}
\mathbf mathcal Kloos_{212}(\mathbf m_1,\mathbf m_2,s,d,m)&=\\
\mathbf{s}um_{g \in U(\mathbb{Z}) \backslash \mathbf mathfrak{G}amma_{212}(N,s,d,m) / \mathfrak{o}verline{U_\mathbf{s}igma}(\mathbb{Z})}&\mathfrak{o}verline{\mathfrak{o}mega_N}(a_{22})
e\left(\bm{f}rac{\mathbf m_{11}c_{11}+\mathbf m_{22}d_{21}}{s}-\bm{f}rac{\mathbf m_{12}\det(\mathbf mathcal Delta_1)}{d}\right).
{\bf{e}}nd{align*}
{\bf{e}}nd{definition}
By a similar argument as above, $\mathbf mathcal Kloos_{212}(\mathbf m_1,\mathbf m_2,d,s)$ is well defined, and we have the following.
\begin{proposition}
Let $\mathbf{s}igma=s_1s_2s_1$, $\delta_1=\diag{d_1}{1}{d_2}{d_1d_2}$
with $d_1d_2=\mathfrak{p}m\bm{f}rac{\texttt{n}}{s^2}$ for some integer $s$ and $\mathbf m_{1,1}=-d_1\mathbf m_{2,1}$.
Then we have $I_{\delta_\mathbf{s}igma}(f_{fin})=\bm{f}rac{1}{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_1(N)})}\mathbf mathcal Kloos_{212}(\mathbf m_1,\mathbf m_2,sd_1,d_1s^2,sd_1d_2)$.
{\bf{e}}nd{proposition}
\begin{remark}
The set $\mathbf mathfrak{G}amma_{212}(\texttt{n},N,d_1s^2,d_1s)$ is non-empty only if $N$ divides $d_1s$ and $N^2$ divides $d_1s^2$.
{\bf{e}}nd{remark}
\mathbf{s}ection{The final formula}
We now assemble the material from previous sections to obtain our relative trace formula.
Let $N \ge 1$ be an integer.
We define the {\bf adelic congruence subgroup} $\mathbf mathfrak{G}amma_1(N)$ to be matrices of the form $g_\infty g_{fin}$
where $g_\infty \in K_\infty$ and $g_{fin} \in \{g \in G(\hat{\mathbb{Z}}): g {\bf{e}}quiv \left[
\begin{smallmatrix}
* & & * & * \\
* & 1 & * & * \\
& & * & * \\
& & & *
{\bf{e}}nd{smallmatrix}\right] \mathbf mod N \}.$
Fix a character $\mathfrak{o}mega: \mathbb{Q}^\times\mathbb{R}^\times \backslash \mathbf mathbb{A}^\times \to \mathbf mathbb{C}$, that we may see as a character of the centre of $G(\mathbf mathbb{A})$.
Assume that $\mathfrak{o}mega$ is trivial on $(1+N\hat{\mathbb{Z}}) \cap \hat{\mathbb{Z}}^\times$, and define $\mathfrak{o}mega_N(t)=\mathfrak{p}rod_{p \mathbf mid N}\mathfrak{o}mega(t_p)$.
For each standard parabolic subgroup $P=N_PM_P$ (including $G$ itself), consider the space $\mathbf mathscr H_P$ defined in Section~\ref{DefOfES}.
For each character $\chi$ of the centre of $M_P$ whose restriction to the centre of $G$ coincides with $\mathfrak{o}mega$,
let $\mathbf mathfrak{G}en_P(\chi)$ be an orthonormal basis consisting of factorizable vectors of the subspaces of functions $\mathfrak{p}hi$ in $\mathbf mathscr H_P$
that are generic, $\mathbf mathfrak{G}amma_1(N)$-fixed, and have central character $\chi$.
Specifically,
\begin{itemize}
\item If $P=G$ then $\mathbf mathfrak{G}en_P(\mathfrak{o}mega)$ consists in cuspidal elements of $L^2(Z(\mathbb{R})G(\mathbb{Q})\backslash G(\mathbf mathbb{A}), \mathfrak{o}mega)^{\mathbf mathfrak{G}amma_1(N)}$,
\item If $P=B$, each such character $\chi$ may be identified with a triplet of characters $(\mathfrak{o}mega_1,\mathfrak{o}mega_2,\mathfrak{o}mega_3)$
satisfying $\mathfrak{o}mega_1 \mathfrak{o}mega_2 \mathfrak{o}mega_3^2=\mathfrak{o}mega$.
Choose a set of representatives $S=\{k_1, \cdots, k_d\}$ of $(K \cap B(\mathbf mathbb{A})) \backslash K / \mathbf mathfrak{G}amma_1(N)$.
Then there is a basis $(e_i)_{1 \le i \le d}$
of $\mathbf mathbb{C}^S$ such that functions in $\mathbf mathfrak{G}en_P(\mathfrak{o}mega_1,\mathfrak{o}mega_2,\mathfrak{o}mega_3)$ are of the form
$$\mathfrak{p}hi_j^{\text B}(b k_i \gamma) = \chi(b)e_j(k_i)$$ for $b \in B(\mathbf mathbb{A})$, $\gamma \in \mathbf mathfrak{G}amma_1(N)$.
\item If $P=P_{\text{K}}$, each such character $\chi$ may be identified with a pair of characters $(\mathfrak{o}mega_1,\mathfrak{o}mega_2)$
satisfying $\mathfrak{o}mega_1\mathfrak{o}mega_2=\mathfrak{o}mega$.
Choose a set of representatives $S=\{k_1, \cdots, k_d\}$ of $(K \cap P_{\text{K}}(\mathbf mathbb{A})) \backslash K / \mathbf mathfrak{G}amma_1(N)$.
Keeping notations of~\S~\ref{KES}, for each $i$, consider the compact subgroup of $\mathbf mathfrak{G}L_2$ given by
$C_i=\mathbf{s}igma_{\text K}\left(Stab_{K \cap P_{\text{K}}(\mathbf mathbb{A})} (k_i)\right)$.
Then, for each cuspidal automorphic representation $\mathfrak{p}i$ of $\mathbf mathfrak{G}L_2$ with central character $\mathfrak{o}mega_1$
and whose Archimedean component is a principal series there is a basis $(u_j)_j=(u_{j,i})_{i,j}$ of $\mathfrak{p}rod_i \mathfrak{p}i^{C_i}$
such that functions in $\mathbf mathfrak{G}en_P(\mathfrak{o}mega_1,\mathfrak{o}mega_2)$ are of the form
$$\mathfrak{p}hi_{\mathfrak{p}i,j}^{\text K}(p k_i \gamma) = \mathfrak{o}mega_2(p)u_{j,i}(\mathbf{s}igma_{\text K}(p))$$ for $p \in P_{\text{K}}(\mathbf mathbb{A})$, $\gamma \in \mathbf mathfrak{G}amma_1(N)$.
In particular each $u_{i,j}$ is a $\mathbf mathfrak{G}L_2$ adelic Maa{\mathbf{s}s} forms.
\item If $P=P_{\text{S}}$, each such character $\chi$ may be identified with a pair of characters $(\mathfrak{o}mega_1,\mathfrak{o}mega_2)$
satisfying $\mathfrak{o}mega_1\mathfrak{o}mega_2^2=\mathfrak{o}mega$.
Choose a set of representatives $S=\{k_1, \cdots, k_d\}$ of $(K \cap P_{\text{S}}(\mathbf mathbb{A})) \backslash K / \mathbf mathfrak{G}amma_1(N)$.
Keeping notations of~\S~\ref{PES}, for each $i$, consider the compact subgroup of $\mathbf mathfrak{G}L_2$ given by
$C_i=\mathbf{s}igma_{\text K}\left(Stab_{K \cap P_{\text{S}}(\mathbf mathbb{A})} (k_i)\right)$.
Then, for each cuspidal automorphic representation $\mathfrak{p}i$ of $\mathbf mathfrak{G}L_2$ with central character $\mathfrak{o}mega_1$
and whose Archimedean component is a principal series there is a basis $(u_j)_j=(u_{j,i})_{i,j}$ of $\mathfrak{p}rod_i \mathfrak{p}i^{C_i}$
such that functions in $\mathbf mathfrak{G}en_P(\mathfrak{o}mega_1,\mathfrak{o}mega_2)$ are of the form
$$\mathfrak{p}hi_{\mathfrak{p}i,j}^{\text S}(p k_i \gamma) = \mathfrak{o}mega_2 \circ \mathbf mu (p)u_{j,i}(\mathbf{s}igma_{\text K}(p))$$ for $p \in P_{\text{S}}(\mathbf mathbb{A})$, $\gamma \in \mathbf mathfrak{G}amma_1(N)$.
In particular each $u_{i,j}$ is a $\mathbf mathfrak{G}L_2$ adelic Maa{\mathbf{s}s} forms.
{\bf{e}}nd{itemize}
Now fix an integer $\texttt{n}>0$ coprime to $N$. Consider
$$M(\texttt{n},N)=\left\{g \in G(\mathbf mathbb{A}_{fin}) \cap \mathbf mathcal Mat_4(\hat{\mathbb{Z}}) : g {\bf{e}}quiv \left[
\begin{smallmatrix}
* & & * & * \\
* & 1 & * & * \\
& & * & * \\
& & & *
{\bf{e}}nd{smallmatrix}\right] \mathbf mod N, \mathbf mu(g) \in \texttt{n} \hat{\mathbb{Z}}^\times \right\}.$$
Define the {\bf $\texttt{n}$-th Hecke operator of level $\mathbf mathfrak{G}amma_1(N)$} by
$$T_{\texttt{n}}\mathfrak{p}hi(g) =\int_{M(\texttt{n},N)} \mathfrak{p}hi(gx) dx.$$
Then for every standard parabolic subgroup $P$, for every element $u \in \mathbf mathfrak{G}en_P$ and for every $\texttt{n}u \in i\mathbf mathcal Lie{a}_P^*$, the Eisenstein series $E(\cdot,u,\texttt{n}u)$
is an eigenfunction of $T_{\texttt{n}}$. We shall denote the corresponding eigenvalue by $\lambda_{\texttt{n}}(u,\texttt{n}u)$.
Then we have the following.
\begin{theorem}\label{MainTheorem}
Let $\mathbf m_1$, $\mathbf m_2$ be two pairs of non-zero integers, $t_1,t_2 \in A^0(\mathbb{R})$.
Let $h$ be a Paley-Wiener function on $\mathbf mathcal Lie{a}_\mathbf mathbb{C}$ and let $c$ be the constant appearing in Theorem~\ref{sphericalinversion}.
Then we have $$c(\Sigma_{cusp}+\Sigma_{B}+\Sigma_K+\Sigma_S)=\bm{f}rac1{Vol(\mathfrak{o}verline{\mathbf mathfrak{G}amma_1(N)})}(K_1+K_{121}+K_{212}+K_J).$$
The expression $\Sigma_{cusp}+\Sigma_{B}+\Sigma_K+\Sigma_S$ is given by
$$\Sigma_{\text{cusp}}=\mathbf{s}um_{u \in \mathbf mathfrak{G}en_G(\mathfrak{o}mega)}
h(\texttt{n}u_u)\lambda_{\texttt{n}}(u)\mathcal{W}_{\mathfrak{p}si}(u)(t_1t_{\mathbf m_1}^{-1})\mathfrak{o}verline{\mathcal{W}_{\mathfrak{p}si}(u)}(t_2t_{\mathbf m_2}^{-1}),$$
\begin{align*}
\Sigma_{\text{B}}=\bm{f}rac18 \mathbf{s}um_{\mathfrak{o}mega_1\mathfrak{o}mega_2\mathfrak{o}mega_3^2=\mathfrak{o}mega}\mathbf{s}um_{u \in \mathbf mathfrak{G}en_B(\mathfrak{o}mega_1,\mathfrak{o}mega_2,\mathfrak{o}mega_3)}&
\int_{i\mathbf mathcal Lie{a}^*}h(\texttt{n}u)\lambda_{\texttt{n}}(u,\texttt{n}u)\\
&\times \mathcal{W}_{\mathfrak{p}si}(E(\cdot, u,\texttt{n}u ))(t_1t_{\mathbf m_1}^{-1})\mathfrak{o}verline{\mathcal{W}_{\mathfrak{p}si}(E(\cdot,u,\texttt{n}u))}(t_2t_{\mathbf m_2}^{-1})d\texttt{n}u,
{\bf{e}}nd{align*}
\begin{align*}
\Sigma_{\text{K}}=\bm{f}rac12 \mathbf{s}um_{\mathfrak{o}mega_1\mathfrak{o}mega_2=\mathfrak{o}mega}\mathbf{s}um_{u \in \mathbf mathfrak{G}en_{P_{\text{K}}}(\mathfrak{o}mega_1,\mathfrak{o}mega_2)}&
\int_{i\mathbf mathcal Lie{a}_\text{K}^*}h(\texttt{n}u+\texttt{n}u_{\text{K}}(s_u))\lambda_{\texttt{n}}(u,\texttt{n}u)\\
&\times \mathcal{W}_{\mathfrak{p}si}(E(\cdot, u,\texttt{n}u ))(t_1t_{\mathbf m_1}^{-1})\mathfrak{o}verline{\mathcal{W}_{\mathfrak{p}si}(E(\cdot,u,\texttt{n}u))}(t_2t_{\mathbf m_2}^{-1})d\texttt{n}u,
{\bf{e}}nd{align*}
\begin{align*}
\Sigma_{\text{S}}=\bm{f}rac12 \mathbf{s}um_{\mathfrak{o}mega_1\mathfrak{o}mega_2^2=\mathfrak{o}mega}\mathbf{s}um_{u \in \mathbf mathfrak{G}en_{P_{\text{S}}}(\mathfrak{o}mega_1,\mathfrak{o}mega_2)} &
\int_{i\mathbf mathcal Lie{a}_\text{S}^*}h(\texttt{n}u+\texttt{n}u_{\text{S}}(s_u))\lambda_{\texttt{n}}(u,\texttt{n}u)\\
& \times \mathcal{W}_{\mathfrak{p}si}(E(\cdot, u,\texttt{n}u ))(t_1t_{\mathbf m_1}^{-1})\mathfrak{o}verline{\mathcal{W}_{\mathfrak{p}si}(E(\cdot,u,\texttt{n}u))}(t_2t_{\mathbf m_2}^{-1})d\texttt{n}u,
{\bf{e}}nd{align*}
where $\texttt{n}u_u$ (resp. $s_u$) is the spectral parameter of the representation of $GSp_4(\mathbb{R})$ (resp. $GL_2(\mathbb{R})$) attached to $u$,
$\texttt{n}u_{\text{K}}$ and $\texttt{n}u_\text{S}$ are given by Propositions~\ref{KlingenSpectralParameter} and~\ref{SiegelSpectralParameter}.
On the right hand side,
\begin{itemize}
\item $K_1$ is non-zero only if there is an integer $t$ dividing $\texttt{n}$
with $\bm{f}rac{\texttt{n}}t=\bm{f}rac{\mathbf m_{1,2}}{\mathbf m_{2,2}}t$ and such that $s=\bm{f}rac{\mathbf m_{2,1}}{\mathbf m_{1,1}}t$ is also an integer dividing $\texttt{n}$, in which case,
setting $d=\gcd(s,\bm{f}rac{\texttt{n}}s,t,\bm{f}rac{\texttt{n}}t)$ and
\begin{align*}
T(\texttt{n},\mathbf m_1,\mathbf m_2)=&d \times \mathfrak{o}verline{\mathfrak{o}mega_N(s)} \texttt{n}^{-\bm{f}rac12} (\mathbf m_{1,1}\mathbf m_{2,1})^{-2}|\mathbf m_{1,2}\mathbf m_{2,2}|^{-\bm{f}rac32}
\times S\left(\mathbf m_{1,1}\bm{f}rac{\texttt{n}}{\gcd(t,\bm{f}rac{\texttt{n}}s)},\mathbf m_{1,2}t,d,\texttt{n}\right)
{\bf{e}}nd{align*} we have
$$K_1=T(\texttt{n},\mathbf m_1,\mathbf m_2)
\int_{\mathbf mathcal Lie{a}^*}h(-i\texttt{n}u)W(i\texttt{n}u, t_{\mathbf m_1}^{-1}t_1, \mathfrak{p}si_{\mathbf mathbf{1}})W(-i\texttt{n}u,t_{\mathbf m_2}^{-1} t_2,\mathfrak{o}verline{\mathfrak{p}si_{\mathbf mathbf{1}})}
\bm{f}rac{d\texttt{n}u}{c(i\texttt{n}u)c(-i\texttt{n}u)}.$$
\item The contribution of the long Weyl element is $$K_J=\mathbf{s}um_{\mathbf{s}ubstack{N \mathbf mid s\\N^2 \mathbf mid k}}\mathbf mathcal Kloos_J(\mathbf m_1,\mathbf m_2,s,k,\texttt{n})I_J(h)\left(\bm{f}rac{k}{s^2},\bm{f}rac{\texttt{n}}k\right),$$
\item The contribution of $s_1s_2s_1$ is non-zero only if $\texttt{n}\bm{f}rac{\mathbf m_{1,2}}{\mathbf m_{2,2}}=b^2$ for some rational number $b$, in which case it is given by
$$K_{121}=\mathbf m_{2,2}\mathbf{s}um_{N \mathbf mid kb}\mathbf mathcal Kloos_{121}(\mathbf m_1,\mathbf m_2,Nk,bNk,\texttt{n})I_{121}(h)\left(\bm{f}rac{\texttt{n}}{Nkb},\bm{f}rac{b}{Nk}\right),$$
\item The contribution of $s_2s_1s_2$ is given by
$$K_{212}=\mathbf m_{2,1}\mathbf{s}um_{\mathbf{s}ubstack{\mathbf m_{2,1}N \mathbf mid s\mathbf m_{1,1} \\ \mathbf m_{2,1}N^2 \mathbf mid s^2\mathbf m_{1,1}}}\mathbf mathcal Kloos_{212}\left(\mathbf m_1,\mathbf m_2,-s\bm{f}rac{\mathbf m_{1,1}}{\mathbf m_{2,1}},-s^2\bm{f}rac{\mathbf m_{1,1}}{\mathbf m_{2,1}},\texttt{n}\right)I_{212}(h)\left(-\bm{f}rac{\mathbf m_{1,1}}{\mathbf m_{2,1}},-\bm{f}rac{\texttt{n}}{s^2}\bm{f}rac{\mathbf m_{2,1}}{\mathbf m_{1,1}}\right),$$
{\bf{e}}nd{itemize}
where we have defined $I_\mathbf{s}igma(h)$
as the integral
\begin{align*}
I_\mathbf{s}igma(h)(d_1,d_2)&=\int_{\mathbf mathcal Lie{a}^*}h(-i\texttt{n}u)W(i\texttt{n}u, t_{\mathbf m_1}^{-1}t_1, \mathfrak{p}si_{\mathbf mathbf{1}})\\
&\times W\left(-i\texttt{n}u,t_{\mathbf m_1}^{-1}\mathbf{s}igma \diag{d_1}{1}{d_2}{d_1d_2}t_{\mathbf m_2}u_1 t_{\mathbf m_2}^{-1} t_2,\mathfrak{o}verline{\mathfrak{p}si_{\mathbf mathbf{1}}}\right)
\bm{f}rac{d\texttt{n}u}{c(i\texttt{n}u)c(-i\texttt{n}u)}
{\mathfrak{o}verline{\mathfrak{p}si_{\mathbf mathbf{1}}(u_1)}}du_1.
{\bf{e}}nd{align*}
Moreover, if Conjecture~1 is true then we have
\begin{align*}
I_\mathbf{s}igma(h)(d_1,d_2)=\int_{\mathbf mathcal Lie{a^*}}h(-i\texttt{n}u)
&K_\mathbf{s}igma\left(-i\texttt{n}u,t_{\mathbf m_1}^{-1}\mathbf{s}igma \diag{d_1}{1}{d_2}{d_1d_2} t_{\mathbf m_2} \mathbf{s}igma^{-1},{\mathfrak{p}si_{\mathbf mathbf{1}}}\right)\\
&\times W(-i\texttt{n}u, t_{\mathbf m_2}^{-1} t_2,{\mathfrak{p}si_{\mathbf mathbf{1}}})
W(i\texttt{n}u, t_{\mathbf m_1}^{-1}t_1, \mathfrak{p}si_{\mathbf mathbf{1}})\bm{f}rac{d\texttt{n}u}{c(i\texttt{n}u)c(-i\texttt{n}u)},
{\bf{e}}nd{align*}
where the generalised Bessel functions $K_\mathbf{s}igma$ have been defined in \S~\ref{ArchInt}.
{\bf{e}}nd{theorem}
\begin{bibdiv}
\begin{biblist}
\bib{ArthurIntro}{article}{
author={Arthur, James},
title={An introduction to the trace formula},
conference={
title={Harmonic analysis, the trace formula, and Shimura varieties},
},
book={
series={Clay Math. Proc.},
volume={4},
publisher={Amer. Math. Soc., Providence, RI},
},
date={2005},
pages={1--263},
review={\mathbf mathcal MR{2192011}},
}
\bib{ArthurSpectralExpansion}{article}{
author={Arthur, James},
title={A trace formula for reductive groups. I. Terms associated to
classes in $G({\bf Q})$},
journal={Duke Math. J.},
volume={45},
date={1978},
number={4},
pages={911--952},
issn={0012-7094},
review={\mathbf mathcal MR{518111}},
}
\bib{Buttcane}{article}{
author={Buttcane, Jack},
title={Bessel functions outside of $GL(2)$},
status={talk},
year={2020},
eprint={https://old.renyi.hu/seminars/ocaf2020/20200605_Buttcane.pdf}
}
\bib{GelbartKnapp}{article}{
author={Gelbart, S. S.},
author={Knapp, A. W.},
title={$L$-indistinguishability and $R$ groups for the special linear
group},
journal={Adv. in Math.},
volume={43},
date={1982},
number={2},
pages={101--121},
issn={0001-8708},
review={\mathbf mathcal MR{644669}},
doi={10.1016/0001-8708(82)90030-5},
}
\bib{Helgason}{book}{
author = {Helgason, Sigurdur},
title = {Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions},
series = {Mathematical Surveys and Monographs},
publisher = {Academic Press},
year = {1984},
}
\bib{Ishii}{article}{
author={Ishii, Taku},
title={On principal series Whittaker functions on ${\rm Sp}(2,{\bf R})$},
journal={J. Funct. Anal.},
volume={225},
date={2005},
number={1},
pages={1--32},
issn={0022-1236},
review={\mathbf mathcal MR{2149916}},
doi={10.1016/j.jfa.2005.03.023},
}
\bib{JacquetLai}{article}{
author={Jacquet, H.},
author={Lai, K. F.},
title={A relative trace formula},
journal={Compositio Math.},
volume={54},
date={1985},
number={2},
pages={243--310},
issn={0010-437X},
review={\mathbf mathcal MR{783512}},
}
\bib{Joyner}{article}{
author={Joyner, David},
title={On the Kuznetsov-Bruggeman formula for a Hilbert modular surface
having one cusp},
journal={Math. Z.},
volume={203},
date={1990},
number={1},
pages={59--104},
issn={0025-5874},
review={\mathbf mathcal MR{1030708}},
doi={10.1007/BF02570723},
}
\bib{Kim}{article}{
author={Kim, Henry H.},
title={The residual spectrum of ${\rm Sp}_4$},
journal={Compositio Math.},
volume={99},
date={1995},
number={2},
pages={129--151},
issn={0010-437X},
review={\mathbf mathcal MR{1351833}},
}
\bib{KL}{article}{
author={Knightly, A.},
author={Li, C.},
title={Kuznetsov's trace formula and the Hecke eigenvalues of Maass
forms},
journal={Mem. Amer. Math. Soc.},
volume={224},
date={2013},
number={1055},
pages={vi+132},
issn={0065-9266},
isbn={978-0-8218-8744-8},
review={\mathbf mathcal MR{3099744}},
doi={10.1090/S0065-9266-2012-00673-3},
}
\bib{MiyOda}{article}{
author={Miyazaki, Takuya},
author={Oda, Takayuki},
title={Principal series Whittaker functions on ${\rm Sp}(2;{\bf R})$. II},
journal={Tohoku Math. J. (2)},
volume={50},
date={1998},
number={2},
pages={243--260},
issn={0040-8735},
review={\mathbf mathcal MR{1622070}},
doi={10.2748/tmj/1178224977},
}
\bib{Muic}{article}{
author={Mui\'{c}, Goran},
title={Intertwining operators and composition series of generalized and
degenerate principal series for ${\rm Sp}(4,\mathbf mathbb R)$},
journal={Glas. Mat. Ser. III},
volume={44(64)},
date={2009},
number={2},
pages={349--399},
issn={0017-095X},
review={\mathbf mathcal MR{2587308}},
doi={10.3336/gm.44.2.08},
}
\bib{Niwa}{article}{
author={Niwa, Shinji},
title={Commutation relations of differential operators and Whittaker
functions on ${\rm Sp}_2(\mathbf mathbf R)$},
journal={Proc. Japan Acad. Ser. A Math. Sci.},
volume={71},
date={1995},
number={8},
pages={189--191},
issn={0386-2194},
review={\mathbf mathcal MR{1362994}},
}
\bib{Pitale}{book}{
author={Pitale, Ameya},
title={Siegel modular forms},
series={Lecture Notes in Mathematics},
volume={2240},
note={A classical and representation-theoretic approach},
publisher={Springer, Cham},
date={2019},
pages={ix+138},
isbn={978-3-030-15674-9},
isbn={978-3-030-15675-6},
review={\mathbf mathcal MR{3931351}},
doi={10.1007/978-3-030-15675-6},
}
\bib{RS1}{article}{
author={Roberts, Brooks},
author={Schmidt, Ralf},
title={Some results on Bessel functionals for ${\rm GSp}(4)$},
journal={Doc. Math.},
volume={21},
date={2016},
pages={467--553},
issn={1431-0635},
review={\mathbf mathcal MR{3522249}},
}
\bib{RS2}{book}{
author={Roberts, Brooks},
author={Schmidt, Ralf},
title={Local newforms for GSp(4)},
series={Lecture Notes in Mathematics},
volume={1918},
publisher={Springer, Berlin},
date={2007},
pages={viii+307},
isbn={978-3-540-73323-2},
review={\mathbf mathcal MR{2344630}},
doi={10.1007/978-3-540-73324-9},
}
\bib{SHMKloosterman}{article}{
author={Siu Hang Man},
title={Symplectic Kloosterman Sums and Poincar\'e Series},
year={2020},
eprint={https://arxiv.org/abs/2006.03036}
}
\bib{SHMKuznetsov}{article}{
author={Siu Hang Man},
title={A Density Theorem for $Sp(4)$},
year={2021},
eprint={https://arxiv.org/abs/2101.09602}
}
\bib{thesis}{thesis}{
author={VerNooy, Colin},
title={$K$-types and Invariants for the Representations of $GSp(4,\mathbb{R})$},
date={2019},
eprint={https://hdl.handle.net/11244/321122},
type={phd}
}
\bib{Vogan}{article}{
author={Vogan, David A., Jr.},
title={Gel\cprime fand-Kirillov dimension for Harish-Chandra modules},
journal={Invent. Math.},
volume={48},
date={1978},
number={1},
pages={75--98},
issn={0020-9910},
review={\mathbf mathcal MR{506503}},
doi={10.1007/BF01390063},
}
\bib{Wallach}{book}{
author={Wallach, Nolan R.},
title={Real reductive groups. II},
series={Pure and Applied Mathematics},
volume={132},
publisher={Academic Press, Inc., Boston, MA},
date={1992},
pages={xiv+454},
isbn={0-12-732961-7},
review={\mathbf mathcal MR{1170566}},
}
{\bf{e}}nd{biblist}
{\bf{e}}nd{bibdiv}
{\bf{e}}nd{document} | math | 201,026 |
\begin{document}
\setlength{\parindent}{5ex}
\begin{abstract}
Let $\mathcal{I}_{d,g,r}$ be the union of irreducible components of the Hilbert scheme whose general points correspond to smooth
irreducible non-degenerate curves of degree $d$ and genus $g$ in $\mathbb{P}^r$. We use families of curves on cones to show that
under certain numerical assumptions for $d$, $g$ and $r$, the scheme $\mathcal{I}_{d,g,r}$ acquires generically smooth
components whose general points correspond to curves that are double covers of irrational curves. In particular, in the case
$\rho(d,g,r) := g-(r+1)(g-d+r) \geq 0$ we construct explicitly a \emph{regular component} that is different from the
distinguished component of $\mathcal{I}_{d,g,r}$ dominating the moduli space $\mathcal{M}_g$. Our result implies also
that if $g \geq 57$ then $\mathcal{I}_{\frac{4g}{3}, g, \frac{g+1}{2}}$ has at least two generically smooth
components parametrizing linearly normal curves.
\end{abstract}
\subjclass[2000]{Primary 14C05; Secondary 14H10}
\keywords{Hilbert scheme of curves, Brill-Noether theory, double covering}
\title{Components of the Hilbert Scheme of smooth projective curves using ruled surfaces}
\section{Introduction}\label{Section_1}
Let $\mathcal{I}_{d,g,r}$ be the union of irreducible components of the Hilbert scheme whose general points correspond
to smooth irreducible non-degenerate complex curves of degree $d$ and genus $g$ in $\mathbb{P}^r$. A component of
$\mathcal{I}_{d,g,r}$ is called \emph{regular} if it is reduced and of expected dimension $\lambda_{d,g,r} := (r+1)d -
(r-3)(g-1)$.
Otherwise it is called \emph{superabundant}.
For $\rho(d,g,r) := g-(r+1)(g-d+r) \ge 0$, it is known that $\mathcal{I}_{d,g,r}$ has the unique component dominating
$\mathcal{M}_g$, see \cite[p. 70]{Har82}. It is usually referred to as the \emph{distinguished} component.
Historically, Severi claimed in \cite{Sev1921} that $\mathcal{I}_{d,g,r}$ is irreducible if $d\geq g+r$.
It was proved that $\mathcal{I}_{d,g,r}$ is irreducible if $d\geq g+r$ and $r = 3, 4$, see \cite{Ein86} and \cite{Ein87}.
On the other hand, for $r\geq 5$ (and $\rho(d,g,r)\geq 0$) there have been given several examples in which $\mathcal{I}_{d,g,r}$ possesses additional
non-distinguished components (\cite{MS1989}, \cite{Kee94}, \cite{CIK17}, \cite{CS89}, etc), but for none of them it has been proven to be
regular. Note that all these examples are given
by non-linearly normal curves. We remark that in \cite{CIK17} we showed the existence of a {\it non-distinguished} component $\mathcal{D}_{d,g,r}$ of
$\mathcal{I}_{d,g,r}$ parameterizing curves that are double covers of irrational curves, whereas all other known to us examples
of reducible Hilbert schemes of curves have used curves that are $m$-sheeted coverings of $\mathbb{P}^1$ with $m \geq 3$.
In \cite[Question 4.7, p. 598]{CIK17} we asked about the possibility of $\mathcal{D}_{d,g,r}$ being reduced. In the present
paper we reconstruct this component under less constrains, this time using a family of curves on cones, which are double
coverings of hyperplane sections of the cones. We construct and characterize the properties of $\mathcal{D}_{d,g,r}$ using tools
from the theory of ruled surfaces, while in \cite{CIK17} we only showed its existence using Brill-Noether theory of linear
series on curves. Our approach is motivated by the fact that for a given double covering $\varphi : X \to Y$ the curves $X$ and
$Y$ can be regarded as curves on the ruled surface $S := \mathbb{P} (\varphi_{\ast} {\mathcal O _X})$, as we explain in section
\ref{Section_2}. It allows us to construct the additional component in a more geometric way and to obtain its generic
smoothness, which gives an affirmative answer to the question raised in \cite{CIK17}.
Our main result is as follows.
\begin{theoremA}
Assume that $g$ and $\gamma$ are integers with $g \geq 4\gamma -2 \geq 38$.
Let
\[
d:= 2g - 4\gamma + 2 \quad \mbox{ and } \quad \max \left\lbrace \gamma, \frac{2(g-1)}{\gamma} \right\rbrace \leq r \leq R:= g
- 3\gamma + 2 \, .
\]
Then the Hilbert scheme $\mathcal{I}_{d, g, r}$ possesses a generically reduced component $\mathcal{D}_{d, g, r}$ for which
\[
\operatorname{dim} \mathcal{D}_{d, g, r} = \lambda_{d, g, r} + r\gamma - 2g+2 \, .
\]
Further, let $X_r \subset \mathbb{P}^r$ be a smooth curve corresponding to a general point of $\mathcal{D}_{d, g, r}$.
\begin{enumerate}
\item[{\rm (i)}] If $r = R$ then $X_R$ is the intersection of a general quadric hypersurface with a cone over a smooth
curve $Y$ of degree $g - 2\gamma + 1$ and genus $\gamma$ in $\mathbb{P}^{R-1}$ and $X_R$ is embedded in $\mathbb{P}^R$ by the
complete linear series $|R_{\varphi}|$ on $X_R$, where $R_{\varphi}$ is the ramification divisor of the natural projection
morphism $\varphi : X_R \to Y$ of degree 2 given by the ruling of cone;
\item[{\rm (ii)}] If $r < R$ then $X_r$ is given by a general projection of some $X_R$ as in {\rm (i)}, that is, $X_r$ is
embedded in $\mathbb{P}^{r}$ by a general linear subseries $g^r_d$ of $|R_{\varphi}|$.
\end{enumerate}
\end{theoremA}
In our view, one of the interesting implications of {\rm Theorem A} is that if $r = \frac{2(g-1)}{\gamma} \geq \gamma
\geq 10$ and $d = 2g - 4\gamma + 2$, then the scheme $\mathcal{I}_{d, g, r}$ acquires a second regular component
in addition to its distinguished component dominating the moduli space $\mathcal{M}_g$, see {\rm Corollary
\ref{Coro_Reg_comp_exists}}. To our best knowledge, it is the first example in which simultaneous existence of two distinct
regular components of $\mathcal{I}_{d,g,r}$ has been observed in the Brill-Noether case $\rho(d,g,r) \geq 0$. We remark also
that in the case $g = 6\gamma - 3$ and $r = R = 3\gamma - 1$, the Hilbert scheme $\mathcal{I}_{\frac{4g}{3}, g, \frac{g+1}{2}}$
has at least two generically smooth components parametrizing linearly normal curves as it is explained in {Remark
\ref{Sec3_2_LinNorm_comp_exists}}.
The remaining sections of the paper are organized as follows. In section \ref{Section_2}, we provide a motivation for the
construction of the component described in {\rm Theorem A} by reviewing the relations between double coverings of curves, ruled
surfaces and their embeddings as cones. We also prove there several statements that will be used for the construction of
$\mathcal{D}_{d, g, r}$ in section \ref{Section_4}. Possibly, some of them might be of independent interest. In section
\ref{Section_3} we briefly review several facts about the Gaussian map associated to linear series on curves and prove a
technical result facilitating the computation of the dimension of the tangent space at a general point of $\mathcal{D}_{d, g,
r}$. In section \ref{Section_4} we give the proof of {\rm Theorem A}.
We work over $\mathbb{C}$. We understand by \emph{curve} a smooth integral projective algebraic curve. We denote by $L^{\vee}$
the dual line bundle for a given line bundle $L$ defined on an algebraic variety $X$. As usual, $\omega_X$ will stand for the
canonical line bundle on $X$. We denote by $|L|$ the complete linear series $\mathbb P\left(H^0(X,L)\right)$. When $X$ is an
object of a family, we denote by $[X]$ the corresponding point of the Hilbert scheme representing the family. Throughout the
entire paper
\[
d := 2g - 4\gamma + 2 \quad \mbox{ and } \quad R := g - 3\gamma + 2 \, .
\]
For definitions and properties of the objects not explicitly introduced in the paper refer to \cite{Hart77} and \cite{ACGH}.
\noindent
{\bf Acknowledgements}
We thank KIAS for the warm hospitality when we were associate members in KIAS and the second author visited
there. We would like to thank the referees for the constructive comments and valuable suggestions, which helped to
improve the quality of our paper.
\section{Motivation and preliminary results}\label{Section_2}
Suppose that $\varphi : X \to Y$ is an $m:1$ cover, $m \geq 2$, where $X$ and $Y$ are smooth curves of genus $g$ and $\gamma$,
correspondingly. As it is well known, the covering induces a short exact sequence of vector bundles on $Y$
\[
0 \to \mathcal{O}_Y \xrightarrow{\varphi^{\sharp}} \varphi_{\ast} \mathcal{O}_X \to \mathcal{E}^{\vee} \to 0 \, ,
\]
where $\mathcal{E}^{\vee}$ is the so called \emph{Tschirnhausen module}, see \cite{Mir85}. It is a rank $(m-1)$-vector bundle on $Y$.
Since $X$ and $Y$ are curves over $\mathbb{C}$, the exact sequence splits, i.e. $\varphi_{\ast} \mathcal{O}_X \cong
\mathcal{O}_Y \oplus \mathcal{E}^{\vee}$. According to \cite[Ex. IV.2.6, p. 306]{Hart77}, $(\det \varphi_{\ast}
\mathcal{O}_X )^2 \cong \mathcal{O}_Y (-B)$, where $B$ is the branch divisor of the covering. In
particular, $\deg B = 2(g - 1) - 2m (\gamma - 1)$.
We focus on the case $m = 2$. In such a case $\mathcal{E}$ must be a line bundle on $Y$ and we can assume that
$\mathcal{E} = \mathcal{O}_Y (E)$ for some divisor $E$ on $Y$. Since
\[
\deg B = - \deg (\det \varphi_{\ast} \mathcal{O}_X )^2
= - \deg (\det ( \mathcal{O}_Y \oplus \mathcal{O}_Y (-E) ) )^2
= 2 \deg E
\]
it follows that $\deg E = g - 2\gamma + 1$.
Further we suppose that $E$ is a nonspecial and very ample divisor on $Y$. Denote by $\mathcal{F}$ the rank 2 vector bundle
$\mathcal{F} := \mathcal{O}_Y \oplus \mathcal{O}_Y (E)$ on $Y$ and let $S$ be the ruled surface $S := \mathbb{P}(\mathcal{F})$
with natural projection $f : S \to Y$. Since $\deg E > 0$,
$\mathcal{F}_0 := \mathcal{O}_Y \oplus \mathcal{O}_Y (-E)$ will be the normalization of the vector bundle $\mathcal{F}$. As it
is decomposable, $f \, : \, S \to Y$ has two canonically determined sections. They are $Y_0$ which corresponds to the
short exact sequence
\[
0 \to \mathcal{O}_Y \to \mathcal{O}_Y \oplus \mathcal{O}_Y (-E) \to \mathcal{O}_Y (-E) \to 0 \, ,
\]
and $Y_1$ which corresponds to the short exact sequence
\[
0 \to \mathcal{O}_Y \to \mathcal{O}_Y \oplus \mathcal{O}_Y (E) \to \mathcal{O}_Y (E) \to 0 \, .
\]
The section $Y_0$ is the section with minimal self-intersection on $S$ and $Y^2_0 = \deg (\mathcal{O}_Y (-E)) = -g +
2\gamma - 1$. As it well known, $\operatorname{Pic} (S) \cong \mathbb{Z}[Y_0] \oplus f^{\ast} (\operatorname{Pic} (Y))$. For a divisor $D$ on $Y$
we will denote by $D \mathfrak{f}$ the divisor $f^{\ast} (D)$ on $S$. Also, we have for the section $Y_1$ that $Y^2_1 = \deg
(\mathcal{O}_Y (E)) = g - 2\gamma + 1$ and it is not difficult to see that $Y_1 \sim Y_0 + E \mathfrak{f}$. In general,
cohomologies
like $h^{j} (S, \mathcal{O}_S (nY_0 + D \mathfrak{f}))$ are calculated using the projection formula, see \cite[Ex. III.8.3, p.
253]{Hart77}, as
\[
h^{j} (S, \mathcal{O}_S (nY_0 + D \mathfrak{f})) = h^{j} (Y, \mathcal{S}ym^{n} (\mathcal{F}_0) \otimes \mathcal{O}_Y (D)) \, ,
\]
but since $S$ is decomposable, i.e. $\mathcal{F}_0$ splits, the calculation reduces simply to
\begin{equation}\label{S_decompose_cohomologies}
h^{j} (S, \mathcal{O}_S (nY_0 + D \mathfrak{f})) = \sum^{n}_{k=0} h^{j} (Y, \mathcal{O}_Y (D - kE)) \, ,
\end{equation}
see for example \cite{FGP05}. From here
\begin{equation}\label{S_cohomologies_O_S(Y_1)}
h^{0} (S, \mathcal{O}_S (Y_1))
= h^{0} (S, \mathcal{O}_S (Y_0 + E \mathfrak{f}))
= h^{0} (Y, \mathcal{O}_Y (E)) + h^{0} (Y, \mathcal{O}_Y)
= g - 3\gamma + 3\, .
\end{equation}
Using \cite[Ex. V.2.11 (a), p. 385]{Hart77}, we obtain that the linear series $|\mathcal{O}_S (Y_1))| \equiv
|\mathcal{O}_S (Y_0 + E \mathfrak{f}))|$ is base point free. Therefore it defines a morphism
\[
\Psi := \Psi_{|\mathcal{O}_S (Y_1))|} \, : \, S \to \mathbb{P}^R \, ,
\]
where $R = g - 3\gamma + 2$. Since $E$ is very ample, it follows by \cite[Proposition 23, p. 38]{FGP05} that $\Psi$ is
isomorphism away from $Y_0$. Due to $Y_0 \cdot Y_1 = Y_0 \cdot (Y_0 + E \mathfrak{f}) = 0$, the morphism $\Psi$ contracts the
curve $Y_0$ to
a point. Therefore $F := \Psi(S) \subset \mathbb{P}^R$ is a cone of degree
\[
\deg F = Y_1 \cdot Y_1 = (Y_0 + E \mathfrak{f}) \cdot (Y_0 + E \mathfrak{f}) = \deg E = g - 2\gamma + 1
\]
over the image of a smooth integral curve from the linear series $|\mathcal{O}_S (Y_0 + E \mathfrak{f}))|$.
By Bertini's theorem, $\Psi$ maps a general element of $|\mathcal{O}_S (Y_1))|$ to a smooth integral curve of genus
$\gamma$, degree $g - 2\gamma + 1$, which is further linearly normally embedded in some hyperplane $\mathbb{P}^{R-1}$ of
$\mathbb{P}^{R}$ due to (\ref{S_cohomologies_O_S(Y_1)}). A similar fact is true about
a general element of $|\mathcal{O}_S (2Y_1))|$. Namely, a general $C \in |\mathcal{O}_S (2Y_1))| \equiv |\mathcal{O}_S
(2Y_0 + 2E \mathfrak{f}))|$ is mapped by $\Psi$ to a smooth integral curve $\Psi (C)$ of genus $g$, degree $2g - 4\gamma + 2 =
d$, which is
linearly normal in $\mathbb{P}^{R}$. Indeed, since $Y_0 \cdot Y_1 = 0$ and $\Psi$ is isomorphism away from $Y_0$, it follows by
Bertini's theorem that $\Psi (C)$ is smooth and integral. Its degree is $\deg \Psi (C) = 2 Y_1 \cdot Y_1 = 2g - 4\gamma + 2$,
while by the adjunction formula
\[
\deg C \cdot (K_S + C) = (2Y_1) \cdot (K_S + 2Y_1)
= 2(2\gamma - 2) + 2g - 4\gamma + 2
= 2g - 2
\]
we get that its genus is $g$. Finally, to see that $\Psi (C) \subset \mathbb{P}^{R}$ is linearly normal, consider the exact
sequence
\[
0 \to \mathcal{O}_S (-Y_0 - E \mathfrak{f}) \to \mathcal{O}_S (Y_0 + E \mathfrak{f}) \to \mathcal{O}_C (Y_0 + E
\mathfrak{f}) \to 0 \, .
\]
It is sufficient to see that $h^1 (S, \mathcal{O}_S (-Y_0 - E \mathfrak{f})) = 0$, which is not difficult to obtain using the
Serre duality.
The arguments above motivate the following statement.
\begin{prop}\label{Prop_ruled_surface}
Assume that $Y$ is a smooth curve of genus $\gamma$ and $E$ is a very ample non-special divisor on $Y$ of degree $e$. Let $S
:= \mathbb{P}(\mathcal{O}_Y \oplus \mathcal{O}_Y (-E))$, $Y_0$ be the section of minimal self-intersection of the
natural projection $f: S \to Y$ and $Y_1 \in |\mathcal{O}_S (Y_0 + E \mathfrak{f})|$ be a smooth integral
curve. Let $\Psi := \Psi_{|\mathcal{O}_S (Y_0 + E \mathfrak{f})|}$ be the morphism induced by the complete linear series
$|\mathcal{O}_S (Y_0 + E \mathfrak{f})|$. Then:
\begin{enumerate}
\item[{\rm (a)}] $|\mathcal{O}_S (Y_0 + E \mathfrak{f})|$ is base point free and of dimension $e - \gamma + 1$;
\item[{\rm (b)}] $\Psi$ is an isomorphism away from $Y_0$ and contracts $Y_0$ to a point in $\mathbb{P}^{e - \gamma +1} $,
in particular, $\Psi (S)$ is a cone over $\Psi (Y_1)$;
\item[{\rm (c)}] for a general $C \in |\mathcal{O}_S (2Y_0 + 2E \mathfrak{f})|$
\begin{itemize}
\item[{\rm (c.1)}] $\Psi (C)$ is a linearly normal smooth irreducible curve of genus $2\gamma + e - 1$ and degree $2e$ in
$\mathbb{P}^{e - \gamma + 1}$;
\item[{\rm (c.2)}] the linear series $|\mathcal{O}_C (R_{\varphi})|$ on $C$ is traced by the linear series $|\mathcal{O}_S
(Y_0 + E \mathfrak{f})|$ on $S$,
where $R_{\varphi}$ is the ramification divisor of the morphism $\varphi \, : \, C \to Y$ induced by the ruling of
$S$.
\end{itemize}
\end{enumerate}
\end{prop}
\begin{proof}
Statements {\rm (a)}, {\rm (b)} and {\rm (c.1)} are obtained by very similar arguments like those in the discussion preceding
the proposition. We only need to check {\rm (c.2)}.
Recall that $K_S \sim -2Y_0 + (K_Y - E) \mathfrak{f}$. On $C$ we have $K_C - \varphi^{\ast} K_{Y} \sim R_{\varphi}$, i.e.
$\mathcal{O}_C (R_{\varphi}) = \omega_C \otimes (\varphi^{\ast} \omega_{Y})^{\vee}$. The canonical divisor $K_C$ on $C$
is induced by the restriction of $K_S + C \sim K_S + (2Y_0 + 2E \mathfrak{f})$ on $C$. Similarly, the restriction of $K_S +
Y_1 \sim K_S
+ Y_0 + E \mathfrak{f}$ on $Y_1$ induces $K_{Y_1}$. Therefore
\[
R_{\varphi} \sim (K_S + (2Y_0 + 2E \mathfrak{f}) - (K_S + Y_0 + E \mathfrak{f}))_{|_C} \sim (Y_0 + E \mathfrak{f})_{|_C} \, .
\]
By {\rm (a)} and {\rm (c.1)},
$
h^0 (C, \mathcal{O}_C ({Y_0 + E \mathfrak{f}}_{|_C})) = h^0 (S, \mathcal{O}_S (Y_0 + E \mathfrak{f})) = e - \gamma + 2 \,
$.
Therefore the linear series $|\mathcal{O}_S (Y_0 + E \mathfrak{f})|$ on $S$ induces the linear series $|\mathcal{O}_C
(R_{\varphi})|$ on
$C$.
\end{proof}
\begin{remark}
When $e = g - 2\gamma + 1 \geq 2\gamma - 1$ and the divisor $E$ on $Y$ is very ample, where $\mathcal{O}_Y (-E)$ is the
Tschirnhausen module of a double covering $X \to Y$, statement {\rm (c.2)} implies that $\mathcal{O}_C (R_{\varphi})$ is very
ample and $h^0 (C, \mathcal{O}_C (R_{\varphi})) = g - 3\gamma + 3$. It improves a similar claim proved in \cite[Lemma
4.1]{CIK17} where it was assumed that $g \geq 6\gamma - 1$.
\end{remark}
\begin{remark}\label{Section_2_Remark_Fam_counting}
{\rm Proposition \ref{Prop_ruled_surface}} suggests how to give an alternative construction of the component $\mathcal{D}_{2g -
4\gamma + 2, g, r}$ constructed in \cite[Theorem 4.3, p. 594]{CIK17}. For this take $e = g - 2\gamma + 1 \geq 2\gamma - 1$ and
consider the family $\mathcal{Z}$ of surface scrolls $F \subset \mathbb{P}^R$, over a curve $Y$ of genus $\gamma$, $\deg F =
\deg Y = e = g - 2\gamma + 1$ with $h^0 (F, \mathcal{O}_F (1)) = g - 3\gamma + 3$ and $h^1 (F, \mathcal{O}_F (1)) =
\gamma$. According to \cite[Lemma 1, p. 7]{CCFM2008} such a scroll is necessarily a cone, say $F$, over a projectively normal
curve in $\mathbb{P}^{R-1}$ of genus $\gamma$ and degree $e$. Further, let $\mathcal{F}$ be the family of smooth curves in
$|\mathcal{O}_F (2)|$ on the cones $F \subset \mathbb{P}^{R}$ from the family $\mathcal{Z}$. By a counting of the parameters on
which the family
$\mathcal{Z}$ depends, similar to the one carried out in \cite[Remark 2, p. 15]{CCFM2008} and \cite[Proposition 7.1, p.
150]{CCFM2009},
\begin{itemize}
\item[$\phantom{.}$] $\operatorname{dim} \mathcal{Z} = $
\begin{itemize}
\item[$ + $] $3\gamma - 3$ \ : \ number of parameters of curves $Y \in \mathcal{M}_{ \gamma }$
\item[$ + $] $\gamma$ \ : \ number of parameters of line bundles $\mathcal{O}_Y (E) \in \operatorname{Pic} (Y)$ of degree $g - 2\gamma +
1 \geq 2\gamma - 1$ necessary to fix the geometrically ruled surface $\mathbb{P} (\mathcal{O}_Y \oplus \mathcal{O}_Y (-E))$
\item[$ + $] $(R+1)^2 - 1 = \operatorname{dim} (\operatorname{Aut} (\mathbb{P}^R))$
\item[$ - $] $((g - 2\gamma + 1) - \gamma + 2) = \operatorname{dim} G_F$, where $G_F$ is the subgroup of $\operatorname{Aut}
(\mathbb{P}^R)$ fixing the scroll $F$, see \cite[Lemma 6.4, p. 148]{CCFM2009}
\end{itemize}
\end{itemize}
one finds that
$\operatorname{dim} \mathcal{Z} = 7(\gamma-1) - g + (R+1)^2$. On the other hand computing $\operatorname{dim} |\mathcal{O}_F
(2)|$ using (\ref{S_decompose_cohomologies}) and the Riemann-Roch formula, we get easily $\operatorname{dim} |\mathcal{O}_F (2)| = 3g -
8\gamma + 5$. Therefore for the dimension of $\mathcal{F}$ we obtain
\[
\operatorname{dim} \mathcal{F} = \operatorname{dim} \mathcal{Z} + \operatorname{dim} |\mathcal{O}_F (2)| = 2g - \gamma - 2 + (g - 3\gamma + 3)^2.
\]
It is precisely the dimension of the component $\mathcal{D}_{2g-4\gamma+2, g, r}$ constructed in \cite[Theorem 4.3]{CIK17}
when $r = R = g - 3\gamma + 2$ and it improves the bound calculated in \cite[Lemma
4.1]{CIK17} where it was assumed that $g \geq 6\gamma - 1$.
\end{remark}
The above arguments do not imply yet that the family $\mathcal{F}$ gives rise to a component of the Hilbert scheme
$\mathcal{I}_{d , g, R}$. To prove this formally, we will compute in section \ref{Section_4} $h^0 (C, N_{C / \mathbb{P}^R})$ for
a general $C \in \mathcal{F}$. For the purposes of that computation we need several more formal statements about the normal
bundles of curves on cones, which we prove below.
\begin{lemma}\label{Lemma_ConeProjNormalB}
Let $X$ be a smooth non-degenerate curve in $\mathbb{P}^{r}$ and let
$H$ be a hyperplane in $\mathbb{P}^{r}$. \, Assume that $\pi_p:X\to H \subset \mathbb{P}^{r}$ is a projection from a point $p\notin H\cup X$ such that the image $Y:=\pi_p(X)$ is smooth in $\mathbb{P}^{r-1}$. Then
\begin{equation}\label{ConeProjNormalB_SES}
0 \to O_X(R_{\pi_p})\otimes\mathcal{O}_X (1) \to N_{X / \mathbb{P}^{r}} \to \pi_p^{\ast} N_{Y / \mathbb{P}^{r-1}} \to 0 \, ,
\end{equation}
where $R_{\pi_p}$ is the ramification divisor of the
covering $\pi_p : X \to Y$.
\end{lemma}
\begin{proof}
Since $\pi_p:X\to \mathbb{P}^{r-1} \subset \mathbb{P}^{r}$ is a projection from a point $p\notin X$, we have $\pi_p^{\ast}
(\mathcal O_Y(1))=\mathcal O_X(1)$. For the curves $X$ and $Y$ we have the Euler sequences
\[
0 \to \mathcal{O}_X \to \oplus^{r+1} \mathcal{O}_X (1) \to T_{\mathbb{P}^r |_X} \to 0 \,
\]
and
\[
0 \to \mathcal{O}_Y\to \oplus^{r} \mathcal{O}_Y (1) \to T_{\mathbb{P}^{r-1} |_Y} \to 0 \,
\]
\noindent Pulling the second sequence to $X$ via $\pi_p$ we obtain
{\small
\begin{equation*}\label{CommDiag_T_X_and_T_X}
\begin{array}{ccccccccccccccccccccccc}
& & & & 0 & & 0 & & \\[1ex]
& & & & \downarrow & & \downarrow & & \\[1ex]
& & & & \mathcal{O}_X(1) & \simeq & \ker(\alpha) & & \\[1ex]
& & & & \downarrow & & \downarrow & \\[1ex]
0 & \lra & \mathcal{O}_{X} & \rightarrow & \oplus^{r+1} \mathcal{O}_X (1) & \rightarrow & T_{\mathbb{P}^r |_X} &
\rightarrow & 0 \\[1ex]
& & \downarrow & & \downarrow & & \downarrow \alpha & \\[1ex]
0 & \lra & \mathcal{O}_{X} & \rightarrow & \pi^{\ast}_p \left( \oplus^{r}_{1} \mathcal{O}_{Y}(1) \right)
& \rightarrow & \pi^{\ast}_p \left( T_{\mathbb{P}^{r-1} |_{Y}}\right) & \rightarrow & 0 \\[1ex]
& & & & \downarrow & & & \\[1ex]
& & & & 0 & & & & \\[1ex]
\end{array}
\end{equation*}
}
\noindent where $\alpha$ is the induced map between the restrictions of $T_{\mathbb{P}^{r} |_X}$ and
$\pi^{\ast}_p \left( T_{\mathbb{P}^{r-1} |_{Y}} \right)$ and $\ker \left( \alpha \right)$ is its kernel. By the Snake lemma we
obtain
\[
0 \to \mathcal{O}_{X} (1)
\to T_{\mathbb{P}^r |_X}
\to \pi^{\ast}_p \left( T_{\mathbb{P}^{r-1} |_{Y}}\right)
\to 0 \, .
\]
Further, using the normal bundle sequence for $N_{X / \mathbb{P}^r}$ and $N_{{Y} / \mathbb{P}^{r-1}}$,
we get the following commutative diagram
\begin{equation*}\label{CommDiag_N_X_and_N_X}
\begin{array}{ccccccccccccccccccccccc}
& & & & 0 & & 0 & & \\[1ex]
& & & & \downarrow & & \downarrow & & \\[1ex]
& & & & \mathcal{O}_X(1) & & \ker(\beta) & & \\[1ex]
& & & & \downarrow & & \downarrow & \\[1ex]
0 & \lra & T_{X} & \rightarrow & T_{\mathbb{P}^r |_X} & \rightarrow & N_{X / \mathbb{P}^r} & \rightarrow & 0 \\[1ex]
& & \downarrow & & \downarrow & & \downarrow \beta & \\[1ex]
0 & \to & \pi^{\ast}_p(T_{Y})
& \to & \pi^{\ast}_p \left( {T_{\mathbb{P}^{r-1}}}_{|_{Y}} \right)
& \to & \pi^{\ast}_p \left( N_{Y / \mathbb{P}^{r-1}} \right)
& \to & 0
\\[1ex]
& & \downarrow & & \downarrow & & & \\[1ex]
& & \mathcal{O}_{R_{\pi_p}} & & 0 & & & & \\[1ex]
& & \downarrow & & & & & \\[1ex]
& & 0 & & & & & \\[1ex]
\end{array}
\end{equation*}
\noindent where $\beta$ is the induced map between the normal bundles $N_{X / \mathbb{P}^r}$ and $\pi^{\ast}_p \left( N_{Y /
\mathbb{P}^{r-1}} \right)$. Similarly as before, by the Snake lemma we get $\ker \beta \cong \mathcal{O}_{X}
(R_{\pi_p}) \otimes \mathcal{O}_X(1)$, and thus we
deduce the short exact sequence \color{black}
\[
0 \to \mathcal{O}_{X} (R_{\pi_p}) \otimes \mathcal{O}_X(1)
\to N_{X / \mathbb{P}^r} \to \pi^{\ast}_{p} N_{Y / \mathbb{P}^{r-1}} \to 0
\]
\end{proof}
\begin{coro}\label{Corollary_ConeProjNormalB_1}
Suppose that $Y \subset \mathbb{P}^{r-1} \subset \mathbb{P}^{r}$, $r \geq 3$, is a smooth non-degenerate curve of genus
$\gamma$. Let $p \in \mathbb{P}^{r} \setminus \mathbb{P}^{r-1}$ be and arbitrary point. Consider the cone $F \subset
\mathbb{P}^{r}$ over $Y$ with vertex $p$. Suppose that a curve $X \subset F$ is cut by a general hypersurface
$Q_m \subset \mathbb{P}^{R}$ of degree $m$, i.e. $X \in
|\mathcal{O}_F (m)|$ is general. Let $\varphi \, : \, X \xrightarrow{m:1} Y$ be the $m$-sheeted covering map induced by the
ruling of the cone. Then there is an exact sequence
\begin{equation}\label{ConeProjNormalB_SES_1}
0 \to \mathcal{O}_X (m) \to N_{X / \mathbb{P}^{r}} \to \varphi^{\ast} N_{Y / \mathbb{P}^{r-1}} \to 0 \, .
\end{equation}
\end{coro}
\begin{proof}
The line bundle $O_X(R_\varphi)$ associated to the ramification divisor $R_{\varphi}$ of the covering $\varphi : X \to Y$ has
the property $\mathcal O_X(R_\varphi) \simeq \mathcal O_X (m-1)$. To see this, recall that $R_{\varphi} \sim K_X -
\varphi^{\ast} K_Y$. The canonical divisor $K_X$ on $X$ is cut by the restriction of $K_F + X$ on $X$ and $K_Y$ is cut by the
restriction of $K_F + Y$
on $Y$. Therefore
\[
K_X - \varphi^{\ast}K_Y=(K_F + X)|_X - (K_F + Y)|_X \sim (X - Y)|_X \sim (m-1) Y|_X \, .
\]
Hence $\mathcal O_X(R_\varphi)\simeq \mathcal O_X(m-1)$ and {\rm Lemma \ref{Lemma_ConeProjNormalB}} yields the exact sequence
\eqref{ConeProjNormalB_SES_1}.
\end{proof}
\begin{coro}\label{Corollary_ConeProjNormalB_2}
Let $X, Y \subset F \subset \mathbb{P}^r$ be smooth curves on the cone $F$ with vertex $p$ as in {\rm Corollary
\ref{Corollary_ConeProjNormalB_1}}, where $r \geq 6$. Let $W \subset \mathbb{P}^r$ be a general projective subspace of
$\mathbb{P}^r$ of dimension $r - s - 1$, where \, $5 \leq s \leq r-1$. Consider the projection
$
\pi_{W} : \mathbb{P}^{r} \setminus W \to \mathbb{P}^{s}
$
with center $W$ to a general projective subspace of $\mathbb{P}^r$ of dimension $s$. Denote by $X_s$, $Y_s$ and $F_s$ the
images of $X$, $Y$ and $F$ under $\pi_{W}$. Let
$
\varphi_s \, : \, X_s \to Y_s
$
be the covering map induced by the ruling of $F_s$. Then
\begin{equation}\label{ConeProjNormalB_SES_2}
0 \to \mathcal{O}_{X_s} (m) \to N_{X_s / \mathbb{P}^{s}} \to \varphi^{\ast}_s N_{Y_s / \mathbb{P}^{s-1}} \to 0 \, .
\end{equation}
\end{coro}
\begin{proof}
Since $r \geq s+1 \geq 6$, a general projective subspace of $\mathbb{P}^r$ of dimension $r - s - 1$ does not meet the secant
variety of $F$, which is of dimension at most $5$. Therefore $X$, $Y$ and $F$ are isomorphic to their images $X_s$,
$Y_s$ and $F_s$. Also, the $m:1$ covering $\varphi : X \to Y$ induced by the ruling on $F$ goes to an $m:1$
covering $\varphi_s : X_s \to Y_s$ induced by the ruling on $F_s$ such that
$
{\pi_W}_{|_{Y}} \circ \varphi = \varphi_s \circ {\pi_W}_{|_{X}}
$.
In particular, ${\pi_W}_{|_{X}}(R_{\varphi}) = R_{\varphi_s}$. Thus the ramification divisor $R_{\varphi_s}$ is
linearly equivalent to a divisor cut on $X_s$ by a hypersurface of degree $m-1$ in $\mathbb{P}^s$.
Hence $\mathcal O_{X_s} (R_{\varphi_s}) \simeq \mathcal O_{X_s} (m-1)$ and {\rm Lemma
\ref{Lemma_ConeProjNormalB}} gives the exact sequence \eqref{ConeProjNormalB_SES_2}.
\end{proof}
\section{A short note on the Gaussian map}\label{Section_3}
Let $Y$ be a smooth curve of genus $\gamma$ and $L$ and $M$ be line bundles on $Y$. Let $\mu_{L,M}$
\begin{equation}\label{Sec3_mu_L,M}
\mu_{L,M} \, : \, H^0 (Y, L) \otimes H^0 (Y, M) \to H^0 (Y, L \otimes M)
\end{equation}
be the natural multiplication. The Gaussian map $\Phi_{L, M}$
\[
\Phi_{L, M} \, : \, \ker \mu_{L,M} \to H^0 (Y, L \otimes M \otimes \omega_Y)
\]
was introduced by Wahl in \cite{Wahl90}. Locally, $\Phi_{L, M} \, : \, s \otimes t \mapsto sdt - tds$ for sections $s \in H^0(L)$
and $t \in H^0(M)$. It has been studied by a number of authors. We refer to \cite{Wahl90} and \cite{CHM88} for its precise
definition and some properties. We recall only several notions that will be used in {\rm Proposition \ref{Prob_Gauss_NBundle}}
needed for the proof of {\rm Theorem A}.
The notation $R(L, M)$ is often used instead of $\ker \mu_{L, M}$ for the map $\mu_{L, M}$ in (\ref{Sec3_mu_L,M}). When $V
\subset H^0 (Y, L)$ is a vector subspace and $M = \omega_Y$, the map $\mu_{L, M}$ in (\ref{Sec3_mu_L,M}) restricted on $V
\otimes H^0 (Y, \omega_Y)$ will be denoted by $\mu_V$ and the Gaussian map restricted on $\ker \mu_V$ will be denoted by
$\Phi_{\omega_Y, V}$.
The proposition that follows is formulated in the specific form in which it will be used in the proof of {\rm Theorem A}.
\begin{prop}\label{Prob_Gauss_NBundle}
Let $Y$ be a smooth curve of general moduli of genus $\gamma \ge 10$, and let $E$ be a general line bundle on $Y$ of degree
$ g - 2\gamma +1 \geq 2\gamma-1$. Let $V \subseteq H^0 (Y, E)$ be general linear subspace of dimension
$
r = \operatorname{dim} V \geq \max \left\lbrace \gamma, \frac{2(g-1)}{\gamma} \right\rbrace
$.
Consider the embedding $Y \subset \mathbb{P}^{r-1} \equiv \mathbb{P} (V^{\vee})$ given by $V$. Then
\begin{itemize}
\item the restricted Gaussian mapping $\Phi_{\omega_Y, V}$ is surjective, and
\item $h^0 (N_{Y/\mathbb{P}^{r-1}}(-1)) = \operatorname{dim} V = r$.
\end{itemize}
\end{prop}
\begin{proof}
Denote by $\mu$ the cup-product map
\[
\mu: H^0(Y, E) \otimes H^0(Y, \omega_Y) \to H^0(Y, \omega_Y\otimes E) \, .
\]
Since $\deg E = g-2\gamma+1 \geq 2\gamma - 1$, so $E$ is very ample, it follows by \cite[Theorem (4.e.1) and Theorem
(4.e.4)]{Green84} and \cite{Cili83} that $\mu$ is surjective.
The linear series determined by $V$ is very ample since $Y \in \mathcal{M}_{\gamma}$ is general,
$\gamma \ge 10$ and $V \subset H^0 (Y, E)$ is also general. Consider the restriction $\mu_V$ of $\mu$ to
\[
\mu_V : V \otimes H^0(Y, \omega_Y) \to H^0(Y, \omega_Y\otimes E) \, .
\]
Let $R(\omega_Y, E)$ be the kernel of the map $\mu$ and consider the Gaussian map
$\Phi_{\omega_Y, E}$ defined on $R(\omega_Y, E)$
\[
\Phi_{\omega_Y, E} : R (\omega_Y, E) \to H^0 (\omega_Y^2 \otimes E) \, ,
\]
and similarly its restriction $\Phi_{\omega_Y, V}$ defined on the kernel $R(\omega_Y, V)$ of the map $\mu_V$
\begin{equation}\label{Gauss_map_restrict}
\Phi_{\omega_Y, V} : R (\omega_Y, V) \to H^0 (\omega_Y^2 \otimes E) \, .
\end{equation}
In the case of complete embedding, i.e. if $V = H^0 (Y, E)$, the claim follows by
\cite[Proposition 1.2]{CilMir1990}, where it is proven that
\[
h^0 (N_{Y/\mathbb{P}^{r-1}}(-1)) = h^0 (Y, E) + \operatorname{corank} \left( \Phi_{\omega_Y, E} \right) \, ,
\]
and by \cite[Proposition (2.9)]{CLM96}, where it is proven that $\Phi_{\omega_Y, E}$ is surjective for $\gamma\geq 10$ and
$\deg E = g - 2\gamma + 1 \geq 2\gamma -1$. In the case of incomplete embedding, i.e. if $V \subsetneq H^0 (Y,
E)$, exactly the same argument as in the proof of \cite[Proposition 1.2]{CilMir1990} shows that
\begin{equation}\label{Coho_dim_N(-1)_incompele}
h^0 (N_{Y/\mathbb{P}^{r-1}}(-1))
= \operatorname{dim} V + \operatorname{corank} \left( \Phi_{\omega_Y, V} \right)
= r + \operatorname{corank} \left( \Phi_{\omega_Y, V} \right) \, ,
\end{equation}
provided that $\mu_V$ is surjective. This is what we will prove next.
Since $\mu_V$ is the restriction of $\mu$ to $V \otimes H^0(Y, \omega_Y)$, we have
\[
\ker \mu_V = \ker \mu \cap \left(V \otimes H^0(Y, \omega_Y)\right) \, .
\]
Due to $\gamma \leq \operatorname{dim} V \leq \operatorname{dim} H^0 (Y, E)$, it follows from \cite[Proposition 4.3]{Bal1995} that
\begin{equation}\label{dim_ker_muV}
\operatorname{dim} \left(\ker \mu \cap \left(V \otimes H^0(Y, \omega_Y)\right) \right)
= \max \{0, \operatorname{dim} \left( \ker \mu \right) - (h^0 (Y,E) - \operatorname{dim} V) h^0 (Y, \omega_Y) \} \, .
\end{equation}
Since $\mu$ is surjective,
$
\operatorname{dim} \left( \ker \mu \right)
= (\deg(E)-\gamma+1)\gamma - (\deg(E) + \gamma - 1)
= (g-3\gamma)(\gamma-1)
$. By assumption $r = \operatorname{dim} V \geq \frac{2(g-1)}{\gamma}$, hence
\[
\begin{aligned}
\operatorname{dim} \ker \mu - (h^0 (Y, E) - \operatorname{dim} V)\gamma & = (\gamma - 1)(g - 3\gamma) - (g - 3\gamma + 2 - r)\gamma \\
& = \gamma - g + r\gamma > 0 \, .
\end{aligned}
\]
By (\ref{dim_ker_muV}) we obtain
\[
\operatorname{dim} \ker \mu_V = \gamma - g + r\gamma \, .
\]
From here we get for the dimension of its image
\[
\operatorname{dim} \left( \operatorname{Im} (\mu_V) \right) = r\gamma - \operatorname{dim} \ker \mu_V = g - \gamma = h^0 (Y, \omega_Y \otimes E) \, .
\]
This shows that $\mu_V$ is surjective, which proves (\ref{Coho_dim_N(-1)_incompele}).
It remains to show that $\Phi_{\omega_Y, V}$ is surjective. According to \cite[Theorem 4.1]{Bal1995}, the Gaussian map
$\Phi_{\omega_Y, V}$ is of maximal rank. Suppose that it is not surjective. Then it must be injective and its image in $H^0
(Y, \omega_Y^2 \otimes E)$ should be proper, hence
\[
\gamma - g + r\gamma = \operatorname{dim} \ker \mu_V < h^0 (Y, \omega_Y^2 \otimes E) = g + \gamma - 2 \, ,
\]
which implies $r < \frac{2(g-1)}{\gamma}$. The last is impossible in view of the assumption that $r = \operatorname{dim} V \geq \max
\left\lbrace \gamma, \frac{2(g-1)}{\gamma} \right\rbrace$. Therefore, $\Phi_{\omega_Y, V}$ must be surjective and from
(\ref{Coho_dim_N(-1)_incompele}) we conclude also that $h^0 (N_{Y/\mathbb{P}^{r-1}}(-1)) = \operatorname{dim} V = r$.
\end{proof}
\section{Proof of Theorem A}\label{Section_4}
Before demonstrating the proof of {\rm Theorem A} we recall a few facts concerning the Hilbert scheme of cones.
{\rm Proposition \ref{Prop_ruled_surface}} and the counting of the number of parameters in Remark
\ref{Section_2_Remark_Fam_counting} gives the idea how to construct explicitly the component $\mathcal{D}_{d, g, R}$. Recall
that $d = 2g - 4\gamma + 2$ and $R = g - 3\gamma + 2$.
Let $\gamma \geq 10$ and $g \geq 4\gamma - 2$ be integers. Consider the Hilbert scheme $\mathcal{I}_{d/2, \gamma, R-1}$ of
smooth curves of degree $d/2$ and genus $\gamma$ in $\mathbb{P}^{R - 1}$. By \cite[Theorem on p. 75]{Har82} and \cite[Theorem on
p. 26]{Ser84}, \ $\mathcal{I}_{d/2, \gamma, R - 1}$ is reduced and irreducible of dimension
$
\lambda_{d/2, \gamma, R - 1} = Rd/2 - (R-4)(\gamma - 1)
$. Denote by
$\mathcal{H}(\mathcal{I}_{d/2, \gamma, R-1})$ the family of cones in $\mathbb{P}^{R}$ over curves representing points of
$\mathcal{I}_{d/2, \gamma, R-1}$. Since $\gamma \geq 10$ it follows by \cite[Proposition 2.1]{CLM96} that for a general $[Y]
\in \mathcal{I}_{d/2, \gamma, R-1}$ the Gaussian map $\Phi_{\omega_Y, \mathcal{O}_Y (1)}$ is surjective,
hence by \cite[Proposition 2.18]{CLM96} $\mathcal{H}(\mathcal{I}_{d/2, \gamma, R-1})$ is a generically
smooth component of the Hilbert scheme of surfaces of degree $d/2$ in $\mathbb{P}^{R}$ and
\begin{equation}\label{Sec4_dim_scheme_of_cones}
\operatorname{dim} \mathcal{H}(\mathcal{I}_{d/2, \gamma, R-1})
= h^0 (Y, N_{Y / \mathbb{P}^{R - 1}}) + R
= \lambda_{d/2, \gamma, R-1} + R \, .
\end{equation}
First we give the proof of {\rm Theorem A} in the case $r = R$.
\begin{prop}\label{Prop_construct_DdgR}
Suppose that $\gamma \geq 10$ and $g \geq 4\gamma - 2$. Let $\mathcal{F}_{d, g, R}$ be the family of curves $C \subset
\mathbb{P}^R$ obtained as the intersection of a cone $F$ and a general hypersurface of degree 2 in $\mathbb{P}^R$, where $[F]
\in \mathcal{H}(\mathcal{I}_{d/2, \gamma, R-1})$. Let $\mathcal{D}_{d, g, R}$ be the closure of the set of points in
$\mathcal{I}_{d, g, R}$ corresponding to curves from the family $\mathcal{F}_{d, g, R}$. Then
\begin{itemize}
\item $\mathcal{D}_{d, g, R}$ is a generically smooth irreducible component of $\mathcal{I}_{d, g, R}$, and
\item $\operatorname{dim} \mathcal{D}_{d, g, R} = 2g - \gamma - 2 + (R+1)^2 = \lambda_{d, g, R} + R\gamma - 2g + 2$.
\end{itemize}
\end{prop}
\begin{proof}
First we compute $\operatorname{dim} \mathcal{D}_{d, g, R}$. For a general point $[F] \in \mathcal{H}(\mathcal{I}_{d/2, \gamma, R - 1})$, the
cone $F$ is projectively normal since it is a cone over a general curve $Y$ from $\mathcal{I}_{d/2, \gamma, R-1}$,
which is projectively normal by \cite[Theorem 1, p. 74]{GL1986}. Therefore the linear series $|\mathcal{O}_F (2)|$ on $F$ is
induced by $|\mathcal{O}_{\mathbb{P}^R} (2)|$. By equalities (\ref{Sec4_dim_scheme_of_cones}) and
(\ref{S_decompose_cohomologies}), $h^0 (F, \mathcal{O}_F (2)) = 3g - 8\gamma + 6$. Therefore
\[
\begin{aligned}
\operatorname{dim} \mathcal{D}_{d, g, R} & = \operatorname{dim} \mathcal{H}(\mathcal{I}_{g - 2\gamma + 1, \gamma, R - 1}) + h^0 (F, \mathcal{O}_F (2))
- 1 \\
& = \lambda_{d/2, \gamma, R - 1} + R + 3g - 8\gamma + 5 \, .
\end{aligned}
\]
\noindent Remark that since $\lambda_{d/2, \gamma, R - 1} = Rd/2 - (R-4)(\gamma - 1)$ and $\lambda_{d, g, R} = (R+1)d -
(R-1)(g-1)$, the expression for $\operatorname{dim} \mathcal{D}_{d, g, R}$ can also be written as
\[
\operatorname{dim} \mathcal{D}_{d, g, R} = \lambda_{d, g, R} + R\gamma - 2g + 2 = (R+1)^2 + 2g - \gamma - 2 \, .
\]
To prove that $\mathcal{D}_{d, g, R}$ is a generically smooth component of $\mathcal{I}_{d, g, R}$, it is sufficient to show that
for a general $[X] \in \mathcal{D}_{d, g, R}$ we have
$
h^0 (X, N_{X/\mathbb{P}^r}) = (R+1)^2 + 2g - \gamma - 2 = \operatorname{dim} \mathcal{D}_{d, g, R}.
$
Since $X \subset F$ is cut by a general quadratic hypersurface in $\mathbb{P}^R$, the ruling of $F$
induces a double covering $\varphi : X \to Y$, where $Y \subset F$ is cut by a general hyperplane in $\mathbb{P}^R$ and also
$[Y] \in \mathcal{I}_{d/2, \gamma, R-1}$ is general. It follows by {\rm Proposition \ref{Prop_ruled_surface}} and {\rm
Corollary \ref{Corollary_ConeProjNormalB_1}} that
\[
0 \to \mathcal{O}_X (2) \to N_{X / \mathbb{P}^{R}} \to \varphi^{\ast} N_{Y / \mathbb{P}^{R-1}} \to 0 \, .
\]
Since
$
\deg \mathcal{O}_X (2) = 2d = 4g - 8\gamma + 4 > 2g - 2
$
, the series $|\mathcal{O}_X (2)|$ is nonspecial, hence $h^1 (X, \mathcal{O}_X (2)) = 0$. Therefore, using projection formula, Leray's isomorphism and $\varphi_*O_X = O_Y+O_Y(-E)$, we get
\[
\begin{aligned}
h^0 (X, N_{X / \mathbb{P}^{R}})
& = h^0 (X, \mathcal{O}_X (2)) + h^0 (X, \varphi^{\ast} N_{Y / \mathbb{P}^{R-1}}) \\
& = h^0 (X, \mathcal{O}_X (2)) + h^0 (Y, N_{Y / \mathbb{P}^{R-1}}) + h^0 (Y, N_{Y / \mathbb{P}^{R-1}}(-1)) \\
& = 3g - 8\gamma + 5 + \lambda_{d/2, \gamma, R-1} + R \\
& = \operatorname{dim} \mathcal{D}_{d, g, R} \, .
\end{aligned}
\]
This implies that for a general $[X] \in \mathcal{D}_{d,g,R}$
\[
\operatorname{dim} \mathcal{D}_{d,g,R} = \operatorname{dim} \mathcal{F}_{d,g,R} = h^0 (X, N_{X / \mathbb{P}^{R}}) = \operatorname{dim} T_{[X]} \mathcal{D}_{d,g,R} \, ,
\]
therefore $\mathcal{D}_{d,g,R}$ is a generically smooth component of $\mathcal{I}_{d,g,R}$.
\end{proof}
Now we give the proof of {\rm Theorem A} for $\max \left\lbrace \gamma, \frac{2(g-1)}{\gamma} \right\rbrace \leq r < R$.
\begin{proof}[{\bf Proof of { Theorem A}}]
Let $\mathcal{F}_{d, g, r}$ be the family of curves in $\mathbb{P}^r$ obtained from the family
$\mathcal{F}_{d, g, R}$ in {\rm Proposition \ref{Prop_construct_DdgR}} by a projection $\pi_W : \mathbb{P}^R \to \mathbb{P}^r$
with center $W \subset \mathbb{P}^R$, where $W \cong \mathbb{P}^{R-r-1}$ is general.
Let $\mathcal{D}_{d, g, r}$ be the closure of the set of points in $\mathcal{I}_{d, g, r}$ corresponding to the curves from the
family $\mathcal{F}_{d, g, r}$.
Since $\operatorname{codim} (W, \mathbb{P}^R) \geq \gamma+1 \geq 10$, a cone $F$ and curves $X$ and $Y$ as in the proof of {\rm Proposition
\ref{Prop_construct_DdgR}} are isomorphic to their images $F_r = \pi_W (F)$, $X_r = \pi_W (X)$ and $Y_r = \pi_W (Y)$,
correspondingly.
Also, $\varphi_r : X_r \to Y_r$ induced by ruling of $F_r$ is a double covering as
in {\rm Corollary \ref{Corollary_ConeProjNormalB_2}}. Note that if $p$ is the vertex of $F$ then $\pi_W (p)$ is the vertex of
$F_r$. Therefore
\begin{equation}\label{TheoremA_proof_SES_Normal_bundles}
0 \to \mathcal{O}_{X_r} (2) \to N_{X_r / \mathbb{P}^{r}} \to {\varphi^{\ast}_r} N_{Y_r / \mathbb{P}^{r-1}} \to 0 \, .
\end{equation}
The embedding of $Y_r \subset \mathbb{P}^{r-1}$ is incomplete, but since $\deg Y_r = d/2 = g - 2\gamma + 1 \geq 2\gamma - 1$,
it follows by \cite{Ser84} that $\mathcal{I}_{d/2, \gamma, r-1}$ has a unique generically smooth component of the expected
dimension $\lambda_{d/2, \gamma, r-1}$. Therefore, for a general $Y_r \in \mathcal{I}_{d/2, \gamma, r-1}$ (as in our case), $h^1
(Y_r, N_{Y_r / \mathbb{P}^{r-1}}) = 0$ or equivalently $h^0 (Y_r, N_{Y_r / \mathbb{P}^{r-1}}) = \lambda_{d/2, \gamma, r-1}$.
Then we can compute $h^0 (N_{X_r / \mathbb{P}^r})$ in a similar way as before. Since the projection $\pi_W$ is general and
$
r \geq \max \left\lbrace \gamma, \frac{2(g-1)}{\gamma} \right\rbrace
$,
it follows by {\rm Proposition \ref{Prob_Gauss_NBundle}} that $h^0 (Y_r, N_{Y_r / \mathbb{P}^{r-1}}(-1)) = r$.
Since $\deg \mathcal{O}_{X_r} (2) = 2d > 2g - 2$ we have
$
h^1 (X_r, \mathcal{O}_{X_r} (2)) = h^1 (X, \mathcal{O}_{X} (2)) = 0
$
. Using (\ref{TheoremA_proof_SES_Normal_bundles}) we find
\[
\begin{aligned}
h^0 (X_r, N_{X_r / \mathbb{P}^{r}})
& = h^0 (X_r, \mathcal{O}_{X_r} (2)) + h^0 (X_r, {\varphi^{\ast}_r} N_{Y_r /
\mathbb{P}^{r-1}}) \\
& = 2d - g + 1 + h^0 (Y_r, N_{Y_r / \mathbb{P}^{r-1}}) + h^0 (Y_r, N_{Y_r /
\mathbb{P}^{r-1}} (-1)) \\
& = 4g - 8\gamma + 4 - g + 1 + \lambda_{g - 2\gamma + 1, \gamma, r - 1} + r \\
& = 3g - 8\gamma + 5 + r + (g - 2\gamma + 1)r - (r-4)(\gamma - 1) \\
& = (r+3)g - (3r+4)\gamma + 3r+1 \, .
\end{aligned}
\]
Let's compute also the dimension of the family $\mathcal{F}_{d,g,r}$. It is similar to the one carried out in the proof of
\cite[Theorem 4.3]{CIK17}. Since the curves in $\mathcal{F}_{d,g,r}$ are obtained as generic projections from $\mathbb{P}^R$ to
$\mathbb{P}^r$, we have
\[
\begin{aligned}
\operatorname{dim} \mathcal{F}_{d,g,r}
& = \operatorname{dim} \mathcal{F}_{d,g,R} - \operatorname{dim}\operatorname{Aut} \mathbb{P}^R + \operatorname{dim}\operatorname{Aut} \mathbb{P}^r + \operatorname{dim} Grass (r+1, R+1) \\
& = 2g - \gamma - 2 + (r+1)^2 + (r+1)(R-r) \\
& = 2g - \gamma - 2 + (r+1)(g - 3\gamma + 2 + 1) \\
& = (r+3)g - (3r+4)\gamma + 3r + 1 \, .
\end{aligned}
\]
Notice that this number is exactly equal to the one claimed in {\rm Theorem A} since
\[
\begin{aligned}
\lambda_{d, g, r} + r\gamma - 2g+2
& = (r+1)(2g - 4\gamma + 2) - (r-3)(g - 1) + r\gamma - 2g + 2 \\
& = (r+3)g - (3r+4)\gamma + 3r + 1 \, .
\end{aligned}
\]
Hence $\operatorname{dim} \mathcal{D}_{d,g,r} = \operatorname{dim} \mathcal{F}_{d,g,r} = h^0 (X_r, N_{X_r / \mathbb{P}^{r}}) = \operatorname{dim}
T_{[X_r]} \mathcal{D}_{d,g,r}$, which completes the proof of {\rm Theorem A}.
\end{proof}
\begin{coro}\label{Coro_Reg_comp_exists}
If
\begin{equation}\label{condition}
\gamma ~|~ 2(g-1) \quad \mbox{ and } \quad r := \frac{2(g-1)}{\gamma} \geq \gamma \geq 10,
\end{equation}
then $\mathcal{D}_{d, g, r}$ is a regular component of $\mathcal{I}_{d, g, r}$ different from the distinguished one.
\end{coro}
\begin{proof}
With the particular values of $d = 2g - 4\gamma + 2$ and $r$ we have $r = \frac{2(g-1)}{\gamma} \leq \frac{g-1}{5}$ for $\gamma
\geq 10$, hence $5r \leq g-1$. From here it is easy to see that $d - g - r \geq g+2 - 5r \geq 3$.
Therefore $\rho(d, g, r) = g - (r+1)(g-d+r) \geq g > 0$, hence the distinguished component of $\mathcal{I}_{d, g, r}$
dominating $\mathcal{M}_g$ exists. Apart
from it, {\rm Theorem A} guarantees the existence of the regular component $\mathcal{D}_{d, g, r}$ which is apparently
different from the distinguished one as the former projects properly in $\mathcal{M}_g$.
\end{proof}
\begin{remark}
It appears that the condition $r \geq \frac{2(g-1)}{\gamma}$ is essential for the family $\mathcal{F}_{d, g, r}$ giving rise to
a component of $\mathcal{I}_{d, g, r}$, because in such case $r < \frac{2(g-1)}{\gamma}$ we have $\operatorname{dim} \mathcal{F}_{d, g, r} <
\lambda_{d,g,r}$. Notice also that in such a case the Gaussian map in {\rm Proposition \ref{Prob_Gauss_NBundle}} is definitely
not surjective.
\end{remark}
\begin{remark}\label{Sec3_2_LinNorm_comp_exists}
In their paper \cite{MS1989} Mezzetti and Sacchiero constructed generically smooth irreducible components of $\mathcal{I}_{d, g,
r}$, denoted there $W^{m}_{d,g,r}$, whose general points are $m-$ sheeted coverings of $\mathbb{P}^1$, where $m \geq 3$.
In the case of $g = 6\gamma - 3$, we have $d = 2g - 4\gamma + 2 = 8\gamma - 4 = \frac{4}{3}g$, $R = g - 3\gamma + 2 = 3\gamma -
1 = \frac{g+1}{2}$, and the Hilbert scheme $\mathcal{I}_{\frac{4g}{3}, g, \frac{g+1}{2}}$ has two components parametrizing
linearly normal curves. One of them is the component $\mathcal{D}_{\frac{4g}{3}, g, \frac{g+1}{2}}$, shown to exist in our
Proposition \ref{Prop_construct_DdgR}, and the other one is the component $W^{m}_{d,g,r}$ for $m = 4$, $d = \frac{4}{3}g$,
and $r = R = \frac{g+1}{2}$ (it is easy to check that the numerical conditions for the existence of $W^{4}_{\frac{4g}{3}, g,
\frac{g+1}{2}}$ are satisfied when $\gamma \geq 10$). Notice that since $d - g - R = - \gamma < 0$, the existence
of these two components do not provide a counterexample to \emph{Severi's conjecture} claiming that $\mathcal{I}_{d, g, r}$ has
a unique irreducible component parametrizing linearly normal curves if $d \geq g+r$.
\end{remark}
\end{document} | math | 45,282 |
\begin{document}
\title{Probabilistic model of fault detection in quantum circuits}
\author{Anindita Banerjee and Anirban Pathak}
\maketitle
\begin{center}
Jaypee Institute of Information Technology University, Noida, India
\par\end{center}
\begin{abstract}
It is shown that the fault testing for quantum circuits does not follow conventional
classical techniques. If probabilistic gate like Hadamard gate is included in a circuit then the classical
notion of test vector is shown to fail. We have reported several new and distinguishing features of quantum
fault and also presented a general methodology for detection of functional faults in a quantum circuit. The
technique can generate test vectors for detection of different kinds of fault. Specific examples are given and
time complexity of the proposed quantum fault detection algorithm is reported.
\end{abstract}
\section{Introduction}
Since the introduction of quantum computation several protocols (such as quantum cryptography, quantum
algorithm, quantum teleportation) have established quantum computing as a superior future technology. Each of
these processes involves quantum circuits, which are prone to different kind of faults. Consequently it is
important to verify whether the circuit hardware is defective or not. The systematic procedure to do so is known
as fault testing. Normally testing is done by providing a set of valid input states and measuring the
corresponding output states and comparing the output states with the expected output states of the perfect
(fault less) circuit. This particular set of input vectors are known as test set~\cite{abap-nine}. If there
exist a fault then the next step would be to find the exact location and nature of the defect. This is known as
fault localization. A model that explains the logical or functional faults in the circuit is a fault model.
Conventional fault models include (i) stuck at faults, (ii) bridge faults and (iii) delay faults. These fault
models have been rigorously studied for conventional irreversible circuit. But with the advent of reversible
classical computing and quantum computing it has become important to enlarge the domain of the study on test
vectors. In the recent past people have realized this fact and have tried to provide good reversible fault
models \cite{abap-eight, abap-ten, abap-thirteen} which are independent of specific technology. The existing
reversible fault models are
\begin{enumerate}
\item{Single missing gate fault (SMF) where a single gate is missing in the circuit.} \item{Multiple missing
gate fault (MMGF) where many gates are missing in the circuit.} \item{Repeated gate fault (RGF) where same gate
is repeated consecutively many times.} \item{Partial missing gate fault (PGF) which can be understood as a
defective gate.} \item{Cross point fault \cite{abap-thirteen} where the control points disappear from a gate or
unwanted control points appear on other gate.} \item{Stuck at fault model which includes single stuck at fault
(SSF) and multiple stuck at fault (MSF) for zero and one respectively.}
\end{enumerate}
These works are concentrated on circuits composed of gates from NCT, \footnote{This gate library has NOT, CNOT
and Toffoli gates. All these gates can also be achieved in the domain of reversible classical
computing.}
and Generalized Toffoli gate libraries which are part of Maslov's benchmark \cite{abap-seven} but it all work in
the domain of classical reversible circuit. So most of the existing fault testing protocols \cite{abap-three,
abap-five, abap-eight, abap-eleven}, except \cite{abap-nine} have deterministic nature and are valid in the
domain of classical fault testing only. But advantage of quantum computing becomes prominent only when we use
superposition gates like Hadamard gate whereas hardly any effort has been made so far to include these gates in
the fault testing protocols. Further it is shown by Ito et al \cite{abap-twelve} that given a reversible circuit
C, it is NP-hard to generate a minimum complete test set for stuck at faults on the set of wires of C. These
facts have motivated us to aim to obtain an efficient algorithm for fault testing and generation of test set for
quantum circuits. Our effort in that direction yield several interesting characteristics of quantum fault. As
expected the distinguishing nature of quantum fault are only when we consider probabilistic gates and
superposition states in qubit line are considered. The fault testing models \cite{abap-eight, abap-ten,
abap-thirteen} studied so far are deterministic (D) but in quantum circuits containing superposition gates the
notion of determinism fails. To be precise if probabilistic gate like Hadamard gate is taken then the classical
notion of test vector fails. This has recently been realized by Perkowski \cite{abap-nine} and a new notion of
probabilistic test generation have been introduced by them \cite{abap-nine}. Present work follows independent
approach and reports several new and distinguishing features of quantum fault and provides a general methodology
for detection quantum fault.
To understand the basic nature of quantum fault, let us consider a Hadamard gate which maps
$\left|0\right\rangle $ to $\frac{1}{\sqrt{2}}\left|0\right\rangle +\frac{1}{\sqrt{2}}\left|1\right\rangle $ and
$\left|1\right\rangle $ to $\frac{1}{\sqrt{2}}\left|0\right\rangle -\frac{1}{\sqrt{2}}\left|1\right\rangle $.
Now if we consider $\left|x\right\rangle $ as test vector and get $\left|\bar{x}\right\rangle $ in the output
then we know that the gate exist but there is 50\% probability of getting $|x\rangle$ in the output and in that
case we shall not be able to conclude anything about the missing gate fault. Increase in number of trial will
increase the probability of detecting missing gate fault and after $n$ trials the probability of getting a
missing Hadamard fault will be $1-\frac{1}{2^{n}}.$ As the probability reduces to unity if an only if
$n=\infty$, therefore we can conclude that it is impossible to design a test vector which can always
deterministically identify a missing Hadamard gate. The conclusion remains valid for every superposition gates
and it is even valid if we do not work in the computational basis (i.e. even if you change the measurement
bases). In case of Hadamard gate $|0\rangle$ and $|1\rangle$are equally good test vectors but if we consider a 1
qubit gate $G_{1}=\left(\begin{array}{cc}
a_{11} & a_{21}\\
a_{31} & a_{41}\end{array}\right)$ in general then the probability of success of detecting a missing gate fault
after $n$ measurements by using $|0\rangle$ as test vector is $1-|a_{11}|^{2n}$ and the same with $|1\rangle$ as
test vector is $1-|a_{31}|^{2n}$. Consequently, both $|0\rangle$ and $|1\rangle$ can work as test vector but if
$|a_{11}|>|a_{31}|$ then $|1\rangle$ is a better test vector and if $|a_{31}|>|a_{11}|$ then $|0\rangle$ is a
better test vector. In general if we consider a generalized n qubit quantum gate $G_{n}$ which maps states
$|i\rangle$ (where $i$ varies from $0$ to $2^{n}-1$) as $G_{n}|i\rangle=\sum_{j=0}^{2^{n}-1}g_{ij}|j\rangle$
then we have to compare all $2^{n}-1$ values $g_{ii}$ and find out the lowest value among that. The state
$|i\rangle$ corresponding to the lowest value of $g_{ii}$ will provide the best probabilistic test vector.
Further we would like to note that in contrary to the
classical stuck at fault
\footnote{which are of only two types, namely stuck at 0 and stuck at 1 } number of possible stuck at fault in
quantum circuit is infinite as a qubit line can stuck at $\alpha\left|0\right\rangle +\beta\left|1\right\rangle
$ $\forall$$\alpha,$$\beta,$$\epsilon\mathbb{C}$ : $\left|\alpha^{2}\right|+\left|\beta^{2}\right|=1$.
Practically, in a finite size circuit we do not need to consider all such stuck at faults but number of stuck at
fault models will not remain restricted to two.
In next section we have provided an algorithm to generate test vectors for quantum circuit. We have also
considered some specific circuits as examples to find different test vectors for the same and in the end we have
obtained the time complexity of the algorithm presented here. Section 3 is dedicated for conclusions.
\section{Methodology for detection of fault}
The proposed fault testing algorithm can be logically divided into two parts. In first part we
find the total circuit matrix and in second part we find the Test vector for quantum circuit. It will comprise
of two qubit and one qubit gates and n qubit lines and is at present computable for m gates.
\begin{enumerate} \item{First let us consider an arbitrary quantum circuit having $n$ qubit lines and $m$ gates. The matrix of
each gate is expressed by a $2^{n}\times2^{n}$ matrix. This is easy because we just need to take tensor product
with identity operator for all those qubit lines which are not addressed by particular gate. For example, if we
consider a quantum circuit having three qubit lines and if the first gate (a NOT gate) is in second qubit line
then the matrix of this gate will be $I\otimes NOT\otimes I$. After this simple matrix product of all $m$
matrices (corresponding to $m$ gates) are obtained in sequence to obtain a $2^{n}\times2^{n}$ matrix which is
equivalent to the total circuit.} \item{In second part we find the matrices with all possible single and
multiple missing gate faults and relevant stuck at faults and compare them with the circuit matrix with no
faults. If any row of faulty circuit does not match with the resultant circuit matrix then the corresponding
input vector becomes the test vector. Test vector are not unique in case we obtain more than one test vector we
can follow the approach discussed in introduction in relation with the missing gate fault of a generalized $n$
qubit gate $G_{n}$. Further we would like to note that if a particular test vector appears for all faults then
it will comprise the test set. Otherwise, an optimized set of test vectors will comprise the test set. But in
this optimization process more importance should be given on the success rate than the order of the test set. To
be precise, one should choose all the test vectors which have highest possible success rate. In case two vectors
have same success rate (which is higher than all other possible test vectors) in detecting a fault then both are
initially kept in the list and then it is found how many other faults each of these vectors can detect. The one
which detects more faults get selected and become an element of the test set.}
\end{enumerate}
The algorithm discussed above can be clearly visualized through the following Fig. 1.
\begin{figure}
\caption{Flowchart showing an algorithm for detection of fault in a quantum circuit}
\label{Fig:1}
\end{figure}
\subsubsection{Specific examples}
\label{ABAPsubsec:1}
\begin{figure}
\caption{Circuits for finding test vectors}
\label{Fig:2}
\end{figure}
Consider an EPR circuit of 2 qubit lines consisting of a NOT gate in first qubit line and a CNOT gate with
target in second qubit line. The circuit is shown in Fig. 2(a). This circuit is a key component in teleportation
and all the other cases where entanglement generation is required. The total circuit matrix is given by B where
$B=B_{0}.$$B_{1}.B_{2}=\left(\begin{array}{cccc}
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0\\
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\end{array}\right)$
$B_{0}$ is the Identity matrix
$B_{1}$ is the matrix of first gate tensor product with Identity
$B_{2}$ is the matrix of second gate
If first gate is missing, then the faulty circuit matrix will be equivalent to $B_{2}=\left(\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0\end{array}\right)$ and we note that all the rows of $B_{2}$ matrix and $B$ are different,
consequently all input vectors can detect the fault and if second gate is missing then the faulty circuit matrix
will be equivalent to $B_{1}=\left(\begin{array}{cccc}
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\end{array}\right)$ and we note that the last two rows are identical and thus
$\left|00\right\rangle $ and $\left|01\right\rangle $ can only detect the fault. Now if all the gates are
missing then again all the rows of $B_{0}=\left(\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\end{array}\right)$ matrix and $B$ are not identical and all input vectors can detect the fault.
Thus we find that $\left|00\right\rangle $ and $\left|01\right\rangle $can detect all the missing gate faults.
Consider another circuit of 2 qubit line as showm in Fig. 2(b) consisting of a CNOT gate with target in second
qubit line and another CNOT gate with its target on first qubit line.
$B_{0}=I\otimes I$$=\left(\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\end{array}\right)$, $B_{1}=\left(\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0\end{array}\right)$, $B_{2}=\left(\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0\\
0 & 1 & 0 & 0\end{array}\right)$
Thus total circuit matrix
$B=B_{0}.$$B_{1}.B_{2}=\left(\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\end{array}\right)$
If first gate is missing, then the faulty circuit matrix will be equivalent to $B_{2}$ and we note that first
and second rows of $B_{2}$ matrix and $B$ are identical and thus $\left|10\right\rangle $ and
$\left|11\right\rangle $ input vectors can detect the fault and if second gate is missing then the faulty
circuit matrix will be equivalent to $B_{1}$ and we note that the second and third rows are identical and thus
$\left|01\right\rangle $ and $\left|10\right\rangle $ can detect the fault. Now if all the gates are missing
then as we can see that only first row is identical with circuit matrix and thus $\left|01\right\rangle ,$
$\left|10\right\rangle $ and $\left|11\right\rangle $ input vectors can detect the fault. Thus we find that
$\left|10\right\rangle $ can detect all the missing gate faults. A set of other examples for single missing gate
fault (SMF) and multiple missing gate fault (MMF) with their test vectors and the nature of fault which is
probabilistic (P) or deterministic (D) is \textcolor{black}{given in Table I. For large circuits as in
\cite{abap-four}} we divide it into smaller sub circuits and find test vectors for each sub circuit.
This is shown in Table II.
\begin{figure}
\caption{Table I: Test vectors for different quantum circuit}
\label{Fig:3}
\end{figure}
\begin{figure}
\caption{Circuit for non-destructive generalized orthonormal qudit Bell state discriminator}
\label{Fig:4}
\end{figure}
\begin{figure}
\caption{Table II: Test vectors for non-destructive generalized orthonormal qudit Bell state discriminator
circuit.}
\label{Fig:5}
\end{figure}
\subsection{Time complexity of the fault detection algorithm}
The first step require $m2^{2n}$ multiplications to obtain m matrices of $2^{n}\times2^{n}$ dimension which
corresponds to $m$ gates. The next step to obtain the resultant matrix require $(m-1)2^{3n}$ multiplications and
$(m-1)2^{2n}$ additions. Thus it requires $\mathcal{O}\left(m2^{3n}\right)$ steps to obtain the equivalent
matrix of the circuit similarly, it requires the same number of steps to find fault matrices corresponding to
each fault. So if we wish to check $p$ faults it requires $\mathcal{O}\left(pm2^{3n}\right)$ steps to construct
all the matrices. Now in order to compare the matrices we require $p\times2^{2n}$ steps in the worst case. Thus
the total time complexity is $\mathcal{O}\left(pm2^{3n}\right)$. As it has a linear relation with the number of
fault to be considered and the total number of stuck at fault is infinite so we can not detect all faults in
finite time but the methodology will work for all practical purposes where the number of faults of practical
interest is finite because of the physical restrictions.
\section{Conclusions}
We have followed an independent approach for generation of test set for quantum circuits and have reported
several new and distinguishing features of quantum fault. We have seen that for a quantum gate the classical
notion of test vector fails and theoretically it is impossible to determine a test set for hadamard gate.
Further we have observed that in contrary to the classical stuck at fault the number of possible stuck at fault,
in quantum circuit is infinite as the qubit line can be stuck at $\alpha\left|0\right\rangle
+\beta\left|1\right\rangle $ $\forall$$\alpha,$$\beta,$$\epsilon\mathbb{C}$ :
$\left|\alpha^{2}\right|+\left|\beta^{2}\right|=1$. It is also observed that the test set for quantum circuit,
for stuck at fault, is different from that of missing gate fault. For an odd number of repeated gate fault for
an optimized circuit is equivalent to a missing gate fault. In case of an even number it will not be detected.
It has been shown that the quantum faults are infinite in number and many of them cannot be detected
deterministically. Above observations suggested that the systematic procedure for generation of quantum test set
is would be different from the classical procedure. A methodology of generation of test set for quantum circuit
is prescribed here. Only few simple examples have been discussed here but since the algorithm is robust and
valid for any quantum or reversible circuits, following the same methodology, in future test vectors for other
useful circuits (existing and new) can be presented. Further, attempt to reduce the complexity of the method can
be made in future.
\end{document} | math | 17,594 |
\begin{document}
\centerline{\bf Remarks on Graphons}
\centerline{\bf Attila Nagy\footnote{This work was supported by the National Research, Development and Innovation Office – NKFIH, 115288.}}
\noindent
\centerline{Department of Algebra}
\centerline{Budapest University of Technology and Economics}
\centerline{1521 Budapest, Pf. 91, Hungary}
\centerline{e-mail: [email protected]}
\begin{abstract}
L. Lov\'asz and B. Szegedy proved in 2006 that
the limits of convergent graph sequences can be described by measurable symmetric functions $W: [0, 1]\times [0, 1]\to [0, 1]$ called graphons.
In our present paper we investigate the structure of the set of all graphons within the semigroup
$(\mathfrak{F}([0, 1]^2); \circ)$
of all fuzzy subsets of the unit square $[0,1]^2=[0, 1]\times [0, 1]$, where the operation $\circ$ is defined by: for every $f, g\in \mathfrak{F}([0,1]^2)$ and every $s\in [0,1]^2$,
$(f\circ g)(s)=\vee _{x\in [0,1]^2}(f(x)\wedge g(s))$.
\end{abstract}
{\bf Mathematics Subject Classification:} 20M10; 08A72; 05C99. \\
{\bf Keywords:} fuzzy subset, graphon, semigroup.
\section{Introduction and motivation}
Let $G_n$ be a sequence of finite simple graphs whose number of nodes tends to infinity. For every fixed finite simple graph $F$, let $hom(F,G_n)$ denote the number of all homomorphisms from $F$ into $G_n$, that is, the edge-preserving functions from $V(F)$ into $V(G_n)$. Put \[t(F,G_n)=\frac{hom(F,G_n)}{|V(G_n)|^{|V(F)|}}.\] Clearly, $t(F,G_n)$ is the probability that a random mapping from $V(F)$ into $V(G_n)$ should be a homomorphism. The sequence $G_n$ is called convergent if $lim_{n\to \infty}t(F,G_n)$ exists for every finite simple graph $F$. Let \[t(F)=lim_{n\to \infty}t(F,G_n).\] Then $t$ is a graph parameter, that is, a function on simple graphs that is invariant under isomorphism. In \cite{Lovasz1}, the authors given characterizations of graph parameters that arise in this manner; that is, the authors characterize the set $\mathfrak{T}$ of graph parameters $t$ for which there is a convergent sequence of simple graphs $G_n$ such that $t(F)=lim_{n\to \infty}t(F,G_n)$ for every simple graph $F$.
In the characterization of $\mathfrak{T}$, the symmetric and measurable functions $W: [0,1]^2=[0, 1]\times [0, 1]\mapsto [0,1]$ called graphons play an important role. Recall that a function $W:[0,1]^2\mapsto [0,1]$ is said to be symmetric if $W(x,y)=W(y,x)$ is satisfied for all $x, y\in [0,1]$.
A graph is said to be {\it $k$-labelled} ($k$ is a positive integer) if the graph has $k$ nodes labelled by $1, 2, \dots , k$. For a $k$-labelled simple graph $F$ and a graphon $W$, the integral
\[t(F,W)=\int _{[0, 1]^k}\prod _{ij\in E(F)}W(x_i,x_j)dx_1dx_2\cdots dx_k\] is called the \textit{density of the graph $F$ in the graphon $W$} (\cite{Lovasz11}), where $E(F)$ denotes the set of all edges of $F$.
In \cite[Theorem 2.2]{Lovasz1} it was shown that a graph parameter $t$ belongs to $\mathfrak{T}$ if and only if there is a graphon $W$ such that $t(F)=t(F,W)$ for all simple graphs $F$.
A function of a non-empty set $S$ into the real unit interval $[0, 1]$ is called a {\it fuzzy subset} of $S$ (see \cite{Zadeh}).
By \cite{Wang} and \cite{Mordeson}, if $*$ is an associative operation on a non-empty set $S$, then
the set $\mathfrak{F}(S)$ of all fuzzy subsets of $S$ form a semigroup under the operation $\circ$ defined by the following way: for arbitrary $f, g\in \mathfrak{F}(S)$ and $s\in S$,
\begin{equation}\label{circ0}
(f\circ g)(s)=\begin{cases}
\vee _{s=x*y}(f(x)\wedge g(y)), & \text{if $s\in S^2$}\\
0, & \text{otherwise.}\end{cases}
\end{equation}
As every graphon is a fuzzy subset of the unit square $[0, 1]^2$, the following problem seems interesting from a semigroup theory perspective.
\noindent
{\bf Problem}:
{\it If an associative operation $*$ is given on the unit square $[0, 1]^2$, what can we say about the structure of the set ${\cal W}_0$ of all graphons in the semigroup $(\mathfrak{F}([0, 1]^2); \circ )$? Is it true that
${\cal W}_0$ forms a substructure of $(\mathfrak{F}([0, 1]^2); \circ )$? If so, what kind of substructure is it?}
In this paper we deal with this problem in a special case: the given associative operation $*$ on $[0, 1]^2$ satisfies the identity $(x, y)*(u, v)=(u, v)$.
A semigroup $(S; *)$ is called a {\it right zero semigroup} if it satisfies the identity $a*b=b$.
With this terminology, the above problem is examined in that case when $[0, 1]^2$ is a right zero semigroup.
We note that if $S$ is a non-empty set (and so it is a right zero semigroup), then the operation $\circ$ defined in (\ref{circ0}) has the following form:
\begin{equation}\label{circle}
(f\circ g)(s)=\vee _{x\in S}(f(x)\wedge g(s)).
\end{equation}
Throughout the paper, for a non-empty set $S$, $(\mathfrak{F}(S); \circ )$ will denote the semigroup in which the operation $\circ$ is defined by (\ref{circle}). Thus the purpose of this paper is to examine the structure of the set ${\cal W}_0$ of all graphons in the semigroup $(\mathfrak{F}([0, 1]^2); \circ )$.
Our studies consist of two parts. In Section~\ref{1:1} we describe the structure of the semigroup $(\mathfrak{F}(S); \circ )$ for an arbitrary non-empty set $S$, in Section~\ref{2:2} we focus on the semigroup $(\mathfrak{F}([0, 1]^2);\circ )$ and its subset ${\cal W}_0$.
A semigroup $S$ is called a {\it band} if every element $e$ of $S$ is an idempotent element, that is, $e^2=e$. A band satisfying the identity $axa=xa$ is called a {\it right regular band} (\cite{Petrich}).
In Section~\ref{1:1} we prove that if $S$ is an arbitrary non-empty set, then the semigroup $(\mathfrak{F}(S); \circ )$ is a right regular band (Theorem~\ref{rr}). In Section~\ref{2:2}, applying the above result for the right regular band $(\mathfrak{F}([0, 1]^2);\circ )$, we show that the set
${\cal W}_0$ of all graphons is a left ideal of $(\mathfrak{F}([0, 1]^2);\circ )$. By this result, if $W$ is a graphon and $f$ is a fuzzy subset of $[0, 1]^2$, then $f\circ W$ is a graphon.
Thus, for arbitrary simple graphs $F$, we can consider the densities $t(F;W)$ and $t(F; f\circ W)$ of $F$ in $W$ and in $f\circ W$, respectively.
In Section~\ref{2:2} we give an upper bound to $|t(F; W)-t(F; f\circ W)|$. In Theorem~\ref{upperbound} we show that
$|t(F; W)-t(F; f\circ W)|\leq |E(F)|(\sup(W)-\sup(f))\Delta (\{W> \sup(f)\})$, where $\Delta (\{W> \sup(f)\})$ denotes the area of the set
$\{W> \sup(f)\}=\{ (x, y)\in [0, 1]^2:\ W(x, y)>\sup(f)\}$.
For notations and notions not defined here, we refer to the paper \cite{Lovasz1} and the books \cite{Clifford1}, \cite{Lovasz2}, \cite{Nagy}, and \cite{Petrich}.
\section{On the semigroup $(\mathfrak{F}(S); \circ )$, where $S$ is an arbitrary non-empty set}\label{1:1}
For a fuzzy subset $f$ and a subset $X$ of a non-empty set $S$, let $\sup _X(f)=\vee _{x\in X}f(x)$. Especially, let $\sup (f)=\sup _S(f)$. If $f$ and $g$ are arbitrary fuzzy subsets of $S$, then
let $g_{f}$ and $g^*_f$ denote the following fuzzy subsets of $S$: for an arbitrary $s\in S$, let
\[g_f(s)=\begin{cases}
\sup(f), & \text{if $g(s)> \sup(f)$}\\
g(s), & \text{otherwise}\end{cases}\]
and
\[g^*_f(s)=\begin{cases}
g(s)-\sup(f), & \text{if $g(s)> \sup(f)$}\\
0, & \text{otherwise.}\end{cases}\]
\begin{remark}\label{osszeg} \rm By the above definitions, $g_f+g^*_f=g$ for every fuzzy subsets $f$ and $g$ of a non-empty set $S$.
\end{remark}
\begin{remark}\label{gf=g} \rm Let $f$ and $g$ be arbitrary fuzzy subsets of a non-empty set $S$.
It is clear that $\sup(g)\leq \sup(f)$ implies $g(s)\leq \sup(f)$ for every $s\in S$ and so $g_f=g$.
In case $\sup(g)>\sup(f)$, there is an element $s\in S$ such that $g(s)>\sup(f)$ and so $g_f(s)=\sup(f)<g(s)$. Hence $g_f\neq g$.
Thus, for every fuzzy subsets $f$ and $g$ of $S$, the equation $g_f=g$ holds if and only if $\sup(g)\leq \sup(f)$.
\end{remark}
By Remark~\ref{gf=g}, the following lemma holds.
\begin{lemma}\label{sup} For arbitrary fuzzy subsets $f$ and $g$ of a non-empty set $S$, the equations $g_f=g$ and $f_g=f$ together hold if and only if $\sup(g)=\sup(f)$.
\end{lemma}
The next lemma will be used in Lemma~\ref{measurable}.
\begin{lemma}\label{supp} If $f$ and $g$ are fuzzy subsets of a non-empty set $S$ such that $\sup (f)\leq \sup (g)$ then $\sup (g_f)= \sup (f)$ and $\sup (g^*_f)=\sup (g)-\sup (f)$.
\end{lemma}
\noindent
{\bf Proof}. By the definition of $g_f$ and $g^*_f$, it is obvious.
\openbox
\begin{theorem} \label{fog} Let $S$ be a non-empty set. For every fuzzy subsets $f$ and $g$ of $S$, we have $f\circ g=g_f$.
\end{theorem}
\noindent {\bf Proof}.
Let $f$ and $g$ be arbitrary fuzzy subsets of a non-empty set $S$. By the above, $(\mathfrak{F}(S); \circ )$ is a semigroup. Let $s$ be an arbitrary element of $S$.
If $g(s)> \sup(f)$, then $f(x)\wedge g(s)=f(x)$ for every $x\in S$, and so
$(f\circ g)(s)=\vee _{x\in S}f(x)=\sup(f)$.
If $g(s)\leq \sup(f)$, then we have two subcases.
\noindent
Case 1: If $g(s)=\sup(f)$, then $f(x)\wedge g(s)=f(x)$ for all $x\in S$, and so
$(f\circ g)(s)=\vee _{x\in S}f(x)=\sup(f)=g(s)$.
\noindent
Case 2: If $g(s)<\sup(f)$, then there is an $x_0\in S$ such that $f(x_0)>g(s)$ and so $f(x_0)\wedge g(s)=g(s)$. Moreover, for arbitrary $x\in S\setminus \{ x_0\}$, we have
\[f(x)\wedge g(s)=\begin{cases}
g(s) ,& \textit{if $g(s)<f(x)$}\\
f(x) , & \textit{if $f(x)\leq g(s)$,}\end{cases}\] and so
$(f\circ g)(s)=(f(x_0)\wedge g(s))\vee(\vee _{x\in S\setminus \{ x_0\}}(f(x)\wedge g(s))=g(s)$. Summarizing our results, we get
\[(f\circ g)(s)=\begin{cases}
\sup(f), & \text{if $g(s)> \sup(f)$}\\
g(s), & \text{otherwise,}\end{cases}\] that is,
$(f\circ g)(s)=g_f(s)$, which proves our assertion.
\openbox
A commutative band is called a \textit{semilattice}.
A congruence $\alpha$ on a semigroup $A$ is said to be a \textit{semilattice congruence} if the factor semigroup $A/\alpha$ is a semilattice. A semigroup $A$ is said to be \textit{semilattice indecomposable} if the universal relation is the only semilattice congruence on $A$. It is known (\cite{Tamura}) that every semigroup has a least semilattice congruence $\eta$; the classes of $\eta$ are semilattice indecomposable. By \cite[II.3.12. Proposition]{Petrich}, a band is a right regular band if and only if its $\eta$-classes are right zero semigroups.
\begin{theorem} \label{rr} For an arbitrary non-empty set $S$,
the semigroup $(\mathfrak{F}(S); \circ )$ is a right regular band. The $\eta$-classes of $\mathfrak{F}(S)$ are right zero semigroups. Two fuzzy subsets $f$ and $g$ of $S$ are in the same $\eta$-class if and only if $\sup(f)=\sup(g)$.
\end{theorem}
\noindent
{\bf Proof}.
Let $S$ be an arbitrary non-empty set. Then $S$ is a right zero semigroup, and so $(\mathfrak{F}(S); \circ )$ is a semigroup under the operation $\circ$ defined in (\ref{circle}), that is, $(f\circ g)(s)=\vee _{x\in S}(f(x)\wedge g(s))$ for every fuzzy subsets $f$ and $g$ of $S$ and every element $s\in S$. By Theorem~\ref{fog}, it is clear that $f\circ f=f$ for every $f\in \mathfrak{F}(S)$, and so $(\mathfrak{F}(S); \circ )$ is a band. Using also Theorem~\ref{fog}, we have $g\circ f\circ g=g\circ g_f$. As $\sup(g)\geq \sup(g_f)$, we have $g\circ g_f=g_f$.
Thus $g\circ f\circ g=g_f=f\circ g$. Hence $(\mathfrak{F}(S); \circ )$ is a right regular band.
Let $\eta$ denote the least semilattice congruence on $(\mathfrak{F}(S); \circ )$.
The $\eta$-classes of $(\mathfrak{F}(S); \circ )$ are right zero semigroups by \cite[II.3.12. Proposition]{Petrich}.
Let $f$ and $g$ be arbitrary fuzzy subsets of $S$. By \cite[II.1.1. Proposition]{Petrich}, $(f, g)\in \eta$ if and only if $f\circ g\circ f=f$ and $g\circ f\circ g=g$.
As $(\mathfrak{F}(S); \circ )$ is a right regular band, we have
$f\circ g\circ f=g\circ f$ and $g\circ f\circ g=f\circ g$. Thus
$(f, g)\in \eta$ if and only if $g\circ f=f$ and $f\circ g=g$. Using Theorem~\ref{fog},
$(f, g)\in \eta$ if and only if $f_g=f$ and $g_f=g$. By Lemma~\ref{sup}, we get
$(f, g)\in \eta$ if and only if $\sup(f)=\sup(g)$.
\openbox
\section{On the structure of the set of all graphons in the semigroup $(\mathfrak{F}([0, 1]^2); \circ )$}\label{2:2}
Let $(S, {\cal A}, \mu )$ be a measurable space (\cite{Cohn}). For a fuzzy subset $h$ of $S$ and a real number $A$, let
$\{ h>A\}=\{ s\in S:\ h(s)>A\}$.
A fuzzy subset $h$ of $S$ is said to be \textit{measurable} if, for every real number $A$, the subset $\{ h>A\}$ of $S$ is measurable (that is, $\{ h>A\}\in {\cal A}$).
\begin{lemma} \label{measurable} Let $(S, {\cal A}, \mu )$ be a measurable space. Then, for an arbitrary fuzzy subset $f$ and an arbitrary measurable fuzzy subset $g$ of $S$, the fuzzy subsets $g_f$ and $g^*_f$ are measurable.
\end{lemma}
\noindent
{\bf Proof}.
Let $f$ and $g$ be arbitrary fuzzy subsets of $S$ such that $g$ is measurable. If $\sup (f)\geq \sup (g)$, then $g_f=f\circ g=g$ and $g^*_f=0$. In this case the fuzzy subsets $g_f$ and $g^*_f$ are measurable.
Consider the case when $\sup (f)< \sup (g)$. Then $\sup (g_f)=\sup (f)$ and $\sup (g^*_f)=\sup (g)-\sup (f)$ by Lemma~\ref{supp}.
Let $A$ be an arbitrary real number.
It is easy to see that
\[\{g_f>A\}=\begin{cases}
\emptyset, & \text{if $A\geq \sup(f)$}\\
\{g>A\}, & \text{otherwise}\end{cases}\]
and
\[\{g^*_f>A\}=\begin{cases}
\emptyset, & \text{if $A\geq \sup(g)-\sup(f)$}\\
\{g>A+\sup(f)\}, & \text{if $0\leq A < \sup(g)-\sup (f)$}\\
S, & \text {if $A< 0$}\end{cases}\]
from which it follows that $g_f$ and $g^*_f$ are measurable fuzzy subsets of $S$.
\openbox
A fuzzy subset $f$ of $[0, 1]^2$ is said to be \textit{symmetric} if $f(x,y)=f(y,x)$ is satisfied for all $x, y\in [0, 1]$.
\begin{lemma}\label{symleftid} If $f$ is an arbitrary fuzzy subset and $g$ is a symmetric fuzzy subset of $[0, 1]^2$, then
$g_f$ and $g^*_f$ are symmetric fuzzy subsets of $[0, 1]^2$.
\end{lemma}
\noindent
{\bf Proof}.
It is obvious by the definition of $g_f$ and $g^*_f$.
\openbox
\begin{lemma}\label{mindketto} If $W$ is a graphon and $f$ is a fuzzy subset of $[0, 1]^2$, then $W_f$ and $W^*_f$ are graphons.
\end{lemma}
\noindent
{\bf Proof}.
By Lemma~\ref{measurable} and Lemma~\ref{symleftid}, it is obvious.
\openbox
The following theorem provides an answer to the question raised in Problem in the case, where the given operation $\cdot$ on $[0, 1]^2$ satisfies the identity $a\cdot b=b$.
\begin{theorem}\label{leftideal} The set ${\cal W}_0$ of all graphons is a left ideal of the right regular band $(\mathfrak{F}([0, 1]^2); \circ )$ of all fuzzy subsets of $[0, 1]^2$. Thus the semigroup $({\cal W}_0; \circ )$ of all graphons is a right regular band, and so it is a semilattice $I$ of right zero subsemigroups $S_i$ ($i\in I$). Two graphons $W_1$ and $W_2$ are in the same $S_i$ if and only if $\sup(W_1)=\sup(W_2)$.
\end{theorem}
\noindent
{\bf Proof}.
Let $W$ be a graphon and $f$ be a fuzzy subset of $[0, 1]^2$. By Theorem~\ref{fog}, $f\circ W =W_f$. Then $f\circ W$ is a graphon by Lemma~\ref{mindketto}. Thus the set ${\cal W}_0$ of all graphons is a left ideal of the semigroup $(\mathfrak{F}([0, 1]^2); \circ )$ of all fuzzy subsets of $[0, 1]^2$. By Theorem~\ref{rr}, the semigroup $(\mathfrak{F}([0, 1]^2); \circ )$ and so its subsemigroup $({\cal W}_0; \circ )$ is a right regular band. Moreover, the $\eta$-classes of ${\cal W}_0$ are right zero semigroups; two graphons $W_1$ and $W_2$ are in the same $\eta$-class if and only if $\sup(W_1)=\sup(W_2)$.
\openbox
Let $\sigma$ denote the equivalence relation on the set ${\cal W}_0$ of all graphons defined by
$(W_1, W_2)\in \sigma$ if and only if $W_1=W_2$ almost everywhere in $[0, 1]^2$.
\begin{proposition} The equivalence relation $\sigma \cap \eta$ is a congruence on the right regular band $({\cal W}_0; \circ )$ of all graphons, where $\eta$ is the least semilattice congruence on $({\cal W}_0; \circ )$.
\end{proposition}
\noindent
{\bf Proof}.
Let $W_1$ and $W_2$ be two graphons with $(W_1, W_2)\in \sigma \cap \eta$. Then, using Theorem~\ref{leftideal}, we have
$\sup(W_1)=\sup(W_2)$ and $W_1=W_2$ almost everywhere in $[0, 1]^2$. Let $W$ be an arbitrary graphon. As
$\sup(W_1)=\sup(W_2)$, we have $W_1\circ W=W_2\circ W$. Thus $(W_1 \circ W, W_2\circ W)\in \sigma \cap \eta$. Hence $\sigma \cap \eta$ is a right congruence on $({\cal W}_0; \circ )$.
Let $T=\{(x, y)\in [0, 1]^2|\ W_1(x, y)\neq W_2(x, y)\}$. As $(W_1, W_2)\in \sigma$, the area of $T$ is $0$. It is clear that $\{ (x, y)\in [0, 1]^2: (W\circ W_1)(x, y)\neq (W\circ W_2)(x, y)\} \subseteq T$
and so $(W\circ W_1, W\circ W_2)\in \sigma$. As $(W_1, W_2)\in \eta$ and $\eta$ is a congruence on $({\cal W}_0;\circ )$, we have
$(W\circ W_1, W\circ W_2)\in \eta$. Thus $(W\circ W_1, W\circ W_2)\in \sigma \cap \eta$ and so $\sigma \cap \eta$ is a left congruence on $({\cal W}_0; \circ )$. Thus
$\sigma \cap \eta$ is a congruence on $({\cal W}_0; \circ )$.
\openbox
Let $W$ be a graphon and $f$ a fuzzy subset of $[0, 1]^2$. By Theorem~\ref{leftideal}, $f\circ W$ is a graphon.
Thus, for arbitrary simple graphs $F$, we can consider the densities $t(F;W)$ and $t(F; f\circ W)$ of $F$ in $W$ and $f\circ W$, respectively. The next theorem gives an upper bound to $|t(F; W)-t(F; f\circ W)|$.
\begin{theorem}\label{upperbound} Let $W$ be an arbitrary graphon. Then, for an arbitrary fuzzy subset $f$ of $[0, 1]^2$ and an arbitrary finite simple graph $F$,
\[|t(F; W)-t(F; f\circ W)|\leq |E(F)|(\sup(W)-\sup(f))\Delta (\{W> \sup(f)\}),\] where $E(F)$ denotes the set of all edges of $F$ and
$\Delta (\{W> \sup(f)\})$ denotes the area of the set
$\{W> \sup(f)\}=\{ (x, y)\in [0, 1]^2:\ W(x, y)>\sup(f)\}$.
\end{theorem}
\noindent
{\bf Proof}.
Let $W$ be an arbitrary graphon and $f$ an arbitrary fuzzy subset of $[0, 1]^2$. By Theorem~\ref{leftideal}, $f\circ W$ is a graphon.
If $\sup(W)\leq \sup(f)$, then
$W=f\circ W$ and $\{W> \sup(f)\}=\emptyset$. Thus
$|t(F; W)-t(F; f\circ W)|=0=|E(F)|(\sup(W)-\sup(f))\Delta (\{W> \sup(f)\})$.
Consider the case when $\sup(W)>\sup(f)$. By Remark~\ref{osszeg}, $W-(f\circ W)=W^*_f$. As $W$ is a graphon, $W_f=f\circ W$ and $W^*_f$ are graphons by Lemma~\ref{mindketto}. Thus $W$, $f\circ W$ and $W^*_f$ are integrable functions on $[0, 1]^2$. Using \cite[Lemma 4.1]{Lovasz1}, $|t(F; W)-t(F; f\circ W)|\leq |E(F)|\cdot ||W^*_f||_0$,
where $||W^*_f||_0=\sup_{A\subseteq [0, 1]\atop B\subseteq [0, 1]}\left | \int _A\int_BW^*_f(x, y)dxdy\right |$. As $W^*_f$ is a non-negative function,
$||W^*_f||_0=||W^*_f||_1$, where
$||W^*_f||_1=\int _0^1\int _0^1|W^*_f(x, y)|dxdy$. Thus
$|t(F; W)-t(F; f\circ W)|\leq |E(F)|\cdot ||W^*_f||_1$.
As $W^*_f(x, y)=0$ for all $(x, y)\in [0, 1]^2\setminus \{ W>\sup(f)\}$,
we have
$||W^*_f||_1=\int_0^1\int_0^1W^*_f(x, y)dxdy\leq (\sup(W)-\sup(f))\Delta (\{W>\sup(f)\}$,
because $\sup(W^*_f)= \sup(W)-\sup(f)$ by Lemma~\ref{supp}.
Consequently
$|t(F; W)-t(F; f\circ W)|\leq |E(F)|(\sup(W)-\sup(f))\Delta (\{W> \sup(f)\})$.
\openbox
\end{document} | math | 18,911 |
\begin{document}
\title{
Geometry of bifurcation sets: Exploring the parameter space
}
\author{R. Barrio\footnote{Departamento de Matem\'atica Aplicada and IUMA,
Computational Dynamics group,
University of Zaragoza,
E-50009 Zaragoza, Spain,
[email protected]},
S. Ib\'a\~{n}ez\footnote{Departamento de Matem\'aticas, University of
Oviedo, E-33007 Oviedo, Spain, [email protected]},
and L. P\'erez\footnote{Departamento de Matem\'aticas, University of
Oviedo, E-33007 Oviedo, Spain, [email protected]}.}
\date{\today}
\maketitle
\begin{abstract}
Inspecting a $p$-dimensional parameter space by means of $(p-1)$-dimensional slices, changes can be detected that are only determined by the geometry of the manifolds that compose the bifurcation set. We refer to these changes as geometric bifurcations. They can be understood within the framework of the theory of singularities for differentiable mappings and, particularly, the Morse Theory. Working with a three-dimensional parameter space, geometric bifurcations are discussed in the context of two models of neuron activity: the Hindmarsh-Rose and the FitzHugh-Nagumo systems. Both are fast-slow systems with a small parameter that controls the time scale of a slow variable. Geometric bifurcations are observed on slices corresponding to fixed values of this distinguished small parameter.
\end{abstract}
\section{Introduction}
\label{sec:0}
Given a family of dynamical systems, the term bifurcation is used to refer to any qualitative change in the dynamics under variation of parameters. The stratification $B$ of the parameter space by bifurcation points is said the bifurcation set. On the other hand, when one explores a $q$-dimensional parameter space with $(q-s)$-parametric families of $s$-dimensional manifolds $S_\varepsilon$, where $s < q$ and $\varepsilon \in \mathbb{R}^{q-s}$, the geometry of $B$ itself can lead to changes in the sets $B_{\varepsilon}=B \cap S_{\varepsilon}$. We are interested in these changes, that we call \textit{geometric bifurcations}. As might be expected, the study of geometric bifurcations is closely related to the Theory of Singularities.
\begin{figure*}
\caption{Spike-counting sweeping technique in the parameter plane $(b,I)$, with $\epsilon = 0.018$ (subplot (a)), $0.03$ (b) and $0.08$ (c). The color indicates the number of spikes per period, increasing from blue (0 spikes) to brown (the maximum value associated with chaotic behavior). Bifurcations: Homoclinic and period-doubling codimension-1 curves; orbit-flip (OF), inclination-flip (IF) and Belyakov codimension-2 points.}
\label{fig:general}
\end{figure*}
To illustrate the main aim of this article, the spike-counting (SC) technique (see \cite{barrio2011,Storace2008}) is used in Figure~\ref{fig:general} to show regions of periodic tonic spiking, chaotic bursting and regular bursting with different number of spikes for the classical Hindmarsh-Rose neuron model (more details in Subsection~\ref{ssA}). The SC sweeping method calculates for each set of parameter values and initial conditions the number of loops (spikes) of the stable limit cycle (when it exists), and this number is color-coded according to the number of spikes (vertical bar). Specifically, the dark blue regions correspond to stable spikings. The strips of a spike-adding cascade, ranging from blue to green, are related to fold/Hopf and fold/hom bursting (see \cite{BIPS2021} for details), which becomes chaotic in a chain of onion-like bulbs depicted in brown (see also \cite{BMSS14}). Additionally, several bifurcation lines are superimposed, such as the black one for homoclinic bifurcations and the red curves for period-doublings of periodic orbits. It is worth noting that the bifurcation curves of periodic orbits arise from homoclinic bifurcation points of codimension two, namely, the HR model exhibits orbit-flips (OF), inclination-flips (IF) and Belyakov bifurcations (see \cite{BIP2020} for a complete description). Furthermore, we highlight how the structure of the parameter plane $(b,I)$ changes as the small parameter $\varepsilon$ increases. An example would be the disappearance of some color bands and several bifurcation points of codimension two (e.g., the green and pink points) while, on the contrary, connections appear between bifurcation curves of periodic orbits (in red). Finally, we also look at how the Z-shape of the homoclinic curve (in black) evolves and disappears. Regarding these phenomena, the question that naturally arises is what is the mechanism underlying all these changes. As we will see later, the answer is that the bifurcation set goes through several geometric bifurcations as $\varepsilon$ varies.
Relevance of geometric bifurcations has been already pointed out in literature, although the terminology we are using in this paper is new. Geometric bifurcations explain the creation of isolas of homoclinic and heteroclinic bifurcations in \cite{algaba2003}. Namely, authors study Chua's equation and show how exploring the 3-parameter space with planar slices, a pair of codimension-two bifurcation points (T-points) collapse and disappear. When the collision occurs, the intersection between the slice and the codimension-two bifurcation curve is non-transversal. Through this process, spirals of codimension-one bifurcations approach together giving rise to isolas. In \cite{algaba2011}, a theoretical model explaining the behavior is provided. In \cite{algaba2015}, where authors study the distribution of homoclinic bifurcation curves in the Lorenz system, it is again evidenced how the evolution of a bifurcation set depends on the way in which the parameter space is explored. In \cite{Wieczorek05,Wieczorek2007}, authors study a three-parameter family of three-dimensional vector fields that models an optically injected laser. Planar slices provide images of the bifurcation set where codimension-two bifurcations points collapse together and codimension-one bifurcation curves contact each other. Among other phenomena, isolas of homoclinic bifurcations are exhibited.
In this paper, we provide a minimal theoretical fra\-mework regarding geometric bifurcations by appealing to the theory of singularities on differentiable manifolds, and particularly, to Morse Theory \cite{katok08,matsumoto2002}. More importantly, it is shown how all the bifurcations that we introduce are observed in rather common neuron models. As already mentioned, we provide illustrations for the Hindmarsh-Rose \cite{HR84} and FitzHugh-Nagumo systems \cite{Fitz61,Nagumo}. Both are fast-slow systems obtained as simplifications of the Hodgkin-Huxley model \cite{HH52} for the propagation of nerve impulses in axons. In both cases there are very rich bifurcation sets that, for instance, include homoclinic bifurcations of codimension one and two. In addition, we will see how geometric bifurcations arise when the parameter space is explored with slices corresponding to fixed values of a distinguished parameter $\varepsilon$, namely, the parameter that controls the time scale of the slow variable. These models illustrate a key fact, the study of geometric bifurcations only makes sense when working with families where there are distinguished parameters that determine a specific way of inspecting the parameter space; otherwise, either a reparametrization may remove singular points or tangencies can be avoided by choosing a different direction for sweepings.
In \cite{golsch1985}, Golubitsky and Schaeffer also use the term ``distinguished parameter'' to refer to a specific parameter whose variation can be seen as typical or natural in a given model. They work with scalar equations of the form
\begin{equation}
\label{escalar_equation}
G(x,\alpha,\beta)=0
\end{equation}
for a single unknown $x$ and with $\alpha\in\mathbb{R}$ and $\beta=(\beta_1,\ldots,\beta_{q-1})\in\mathbb{R}^{q-1}$. Typically, (\ref{escalar_equation}) stands for the steady-state equation of a certain dynamical problem. In this setting, they refer to $\alpha$ as a distinguished parameter and say that $\beta_1,\ldots,\beta_{q-1}$ are unfolding parameters. Golubitsky and Schaeffer wonder about how the sets
\begin{equation}
S_\beta=\{(x,\alpha)\in \mathbb{R}^2 \, : \, G(x,\alpha,\beta)=0\}
\end{equation}
change as $\beta$ varies. In this context, they characterize and classify the bifurcation points $(x_0,\alpha_0,\beta_0)$ ---singularities in their terminology--- where a change occurs. The notion of geometric bifurcation is also close to this theory of singularities. We consider families of dynamical systems depending on $q$ parameters $(\lambda,\varepsilon)\in \mathbb{R}^{q-s} \times \mathbb{R}^s$, with $s<q$ and explore the parameter space taking $(q-s)$-dimensional manifolds with $\varepsilon$ fixed. We wonder about how the intersection between these manifolds and the bifurcation set changes as $\varepsilon$ varies. In our setting, there is no state variable and $\varepsilon$ represents the distinguished parameter (or parameters in case that $s > 1$).
For the slow-fast models that we consider in this paper, the choice of distinguished parameter is indeed a ``natural'' one: the small parameter $\varepsilon$ that controls the behavior of the slow variable.
Note that the theory of singularities, as introduced in \cite{golsch1985}, allows to understand, for instance, the appearance of isola of equilibria in bifurcation diagrams (see \cite{isolas-1}).
The notion of geometric bifurcation incorporates a rather particular concept of codimension. Thus, bearing in mind a three-dimensional parameter space, we can find only three types of geometric bifurcations regarding codimension:
\begin{enumerate}
\item Codimension one-plus-one geometric bifurcations:
Those that occur on a surface of codimension-one bifurcations when the parameter space is explored with one-parametric families of two-dimensional ma\-nifolds.
\item Codimension two-plus-one geometric bifurcations:
Those that occur on a curve of codimension-two bifurcations when the parameter space is explored with one-parametric families of two-dimensional ma\-nifolds.
\item Codimension one-plus-two geometric bifurcations:
Those that occur on a surface of codimension-one bifurcations when the parameter space is explored with two-parametric families of one-dimensional ma-nifolds.
\end{enumerate}
Simple pictures that allow to illustrate the notion of geometric bifurcation are provided in Section \ref{sec:01}. They help to understand the particular way in which the codimension of a geometric bifurcation is specified (see details in \cite{Wieczorek05,Wieczorek2007}). The existence of a two-plus-one geometric bifurcation has subsidiary consequences on the codimension-one bifurcation sets that emerge from it, these effects are also discussed. In Section \ref{sec:1} we formalize some notions and provide a minimal theoretical support for dealing with geometric bifurcations. Specifically, one-plus-one and two-plus-one bifurcations are discussed. How geometric bifurcations arise in the aforementioned neural models is shown in Section \ref{sec:2}. Sketches in Section \ref{sec:01} and theoretical schemes provided in Section \ref{sec:1} should be useful to understand many of the phenomena exhibited in the neuron models. Finally, some conclusions are provided.
\section{Geometric bifurcations: illustrative examples}
\label{sec:01}
\begin{figure*}
\caption{One-plus-one geometric bifurcations exhibited in three-parameter spaces. Top: The bifurcation surface $M$ reaches a maximum value at $\varepsilon=0$. Exploring the surface with horizontal planes, an equivalence class (cyan) disappears because the limiting bifurcation curve (red circle) collapses to a point when $\varepsilon=0$ (top right panel). Bottom: The bifurcation surface has a saddle point. Passing through $\varepsilon=0$, the bifurcation curves (red branches of a hyperbola) change their position with respect to the asymptotes. As a consequence, the equivalence class around the central point switches (bottom right panel).}
\label{cod_1p1_3_parametros}
\end{figure*}
Let $X _{\lambda_1,\lambda_2,\varepsilon} $ be a three-parameter family of vector fields, and assume that a bifurcation surface $ M = \{(\lambda_1, \lambda_2, \varepsilon): \varepsilon+\lambda_1^2 + \lambda_2^2= 0 \} $ exists. When we explore the parameter space taking horizontal planes with $\varepsilon$ fixed,
we observe that, for $ \varepsilon <0$, there is a bifurcation curve, the circle $\lambda_1^2+\lambda_2^2=-\varepsilon$ (see top panel in Figure~\ref{cod_1p1_3_parametros}). In the absence of additional bifurcations, there is one equivalence class for values of the parameters inside the circle and one outside. When $\varepsilon=0$, the interior class disappears because the circle collapses to a point. Thus, when $\varepsilon>0$, bifurcations are no longer observed. If the bifurcation surface is given by
$M=\{(\lambda_1,\lambda_2,\varepsilon): \varepsilon-\lambda_1^2+\lambda_2^2 = 0 \}$, we observe a different behavior (see bottom panel in Figure~\ref{cod_1p1_3_parametros}). For
$\varepsilon<0$, there are two bifurcation curves, (the two branches of the hyperboloid $\varepsilon=\lambda_1^2-\lambda_2^2$). In the absence of additional bifurcations, there exist two equivalence classes. When $\varepsilon=0$, the bifurcation set is given by $\lambda_1^2-\lambda_2^2 = 0$, two straight lines, and there is a singular point at $(0,0)$. For $\varepsilon> 0$, we find two curves again, but placed on different sectors. Note that in the passage through $\varepsilon=0$, the class which corresponds to the neighborhood of $(0,0)$ changes. In both cases, we say that there is a one-plus-one geometric bifurcation at $\varepsilon=0$. Note that intersections between the horizontal planes and the surface $M$ are all transverse except for $\varepsilon=0$.
\begin{figure*}
\caption{Two-plus-one geometric bifurcations exhibited in three-parameter spaces and subsidiary effects. In both cases $C$ is a codimension-two bifurcation curve and $M$ is a codimension-one bifurcation surface wit a boundary along $C$. Left: Taking $\varepsilon$-slices two bifurcation curves (red) join in a unique one. Right: A bifurcation curve disappears after it collapses at the geometric bifurcation point.}
\label{cod_2p1_3_parametros}
\end{figure*}
In the previous examples, a geometric bifurcation appears because the three-parameter space is explored with planes and there is one which is tangent to a bifurcation surface. If there exists a codimension-two bifurcation curve exhibiting a point of tangency with one of planes, we say that there is a codimension--two-plus-one geometric bifurcation. Both panels in Figure \ref{cod_2p1_3_parametros} show a codimension-two bifurcation curve (blue) $C$. When the parameter space is explored with horizontal planes (fixed values of $\varepsilon$), there is a value of $\varepsilon$ for which the plane has a quadratic tangency with $C$ at a point $p$. Two codimension-two bifurcation points are exhibited for lower values of $\varepsilon$ and no one for bigger values. We say that there is a codimension--two-plus-one bifurcation at $p$.
More important, another interesting question is to wonder about the changes that occur in the bifurcation strata that arise from other strata of lower codimension. Both panels in Figure \ref{cod_2p1_3_parametros} show a surface $M$ (grey color) of codimension-one bifurcation points with a boundary given by the curve $C$. In the left panel, where the point $p$ is a saddle point on the surface, we can see how, as $\varepsilon$ increases, the horizontal slides show two curves (red color) that join together at $p$ to form a unique curve. On the other hand, looking at right panel we see that the point $p$ is a maximum on the bifurcation surface. In this case, as $\varepsilon$ increases, the horizontal slides show a unique curve (red color) that collapses at $p$ and disappear. The changes on the red curve are a subsidiary effect of the codimension two-plus-one geometric bifurcation. Note that in both cases, the bifurcation curves that are shown in each slide emerge from the codimension-two point with a well-defined tangent and we can observe how, ``generically'', they reconnect with each other or simply disappear, when the codimension-two points are no longer present. The former case is the one exhibited by the red curves in Figure \ref{fig:general} (see also examples in \cite{blackbeardthesis2012}). In Refs. \cite{algaba2003,algaba2011,Wieczorek05,Wieczorek2007} the case of spirals of bifurcations emerging at codimension-two bifurcation points was studied.
\begin{figure*}
\caption{(a) A one-plus-two geometric bifurcation. From the point $(\varepsilon_1,\varepsilon_2)=(0,0)$ two one-plus-one geometric bifurcation curves (red) emerge. They split the parameter space into two regions corresponding to non-equivalent bifurcation diagrams.
(b) Vertical one-dimensional ``slices'', obtained by fixing $\varepsilon_1$ and $\varepsilon_2$, are used to explore the bifurcation diagram exhibited in the panel (a), that is, we consider the three-parameter space as a two-parameter family of bifurcation diagrams in a one-parameter space. }
\label{cod_1p2}
\end{figure*}
New scenarios arise if we reduce the dimension of the slices.
Let $X_{\lambda_1,\varepsilon_1,\varepsilon_2}$ be a three-parameter family of vector fields and assume that the set $ M = \{(\lambda_1,\varepsilon_1,\varepsilon_2): \lambda_1^3+\lambda_1\varepsilon_2+\varepsilon_1=0\} $ is a bifurcation surface. Now, we explore the parameter space by taking vertical lines with $(\varepsilon_1,\varepsilon_2)$ fixed, see Figure~\ref{cod_1p2}. If $(\varepsilon_1,\varepsilon_2)$ belongs to the interior of region~3, i.e., the dark-green region bounded by the curve $27\varepsilon_1^2+4\varepsilon_2^3=0$ (in red), we find three bifurcation points $a$, $a'$ and $a''$ (Figure~\ref{cod_1p2}(b)---line~3). When parameters $(\varepsilon_1,\varepsilon_2)$ cross line~2 (Figure~\ref{cod_1p2}(b)), $a$ and $a'$ collapse at a codimension--one-plus-one bifurcation displayed in the two-parameter space we obtain by fixing $\varepsilon_2$. The same occurs along line $2'$, but the collision is between $a'$ and $a''$ instead. When parameters $(\varepsilon_1,\varepsilon_2)$ are in region~1 (Figure~\ref{cod_1p2}(b)---line~1), we find only one bifurcation point at $d$. We say that there is a codimension--one-plus-two geometric bifurcation at $(\varepsilon_1,\varepsilon_2)=(0,0)$. The Z-shape, exhibited by the bifurcation curves given by the intersection between $M$ and the vertical planes with $\varepsilon_2<0$ fixed, should be compared with the evolution of the black curves in Figure \ref{fig:general}. In addition, in \cite{CKKOS2007} it is illustrated the change in shape of homoclinic bifurcation curves, from Z-shape with two geometric folds to a curve without anyone (and so this is an example exhibiting a one-plus-two geometric bifurcation in between), being relevant in fast-slow neuron systems.
\section{Geometric bifurcations and Morse Theory}
\label{sec:1}
Recall that a codimension-$k$ bifurcation is generic in the set of families of vector fields dependent on a number of parameters larger than or equal to $k$, but it can be avoided in families dependent on less parameters. A codimension-$k$ bifurcation is characterized by a set of $k$ degeneracy conditions independent of each other and a set of non-degeneracy (open) conditions. Any family satisfying these conditions is said a generic family or a generic unfolding.
Let $X_{\lambda,\varepsilon}:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a $\mathcal{C}^\infty$ family of vector fields with $(\lambda,\varepsilon)\in\mathbb{R}^q \times \mathbb{R}$. Assume that the family is a generic unfolding of a codimension-$k$ bifurcation at $(\lambda,\varepsilon)=(0,0)$, with $k\leq q$. Let $M$ be the $(q+1-k)$-dimensional smooth manifold through $(0,0)$ where such bifurcation is exhibited.
\begin{definition}
\label{geometric_bifurcation}
We say that the bifurcation point $(0,0)$ in the smooth manifold $M$ is \textbf{geometrically generic} with respect to $\varepsilon$ if the hyperplane $\varepsilon=0$ is transversal to $M$ at $(0,0)$. Otherwise, we say that the bifurcation point is \textbf{geometrically degenerate with respect to $\varepsilon$}.
\end{definition}
Let us recall the notion of Morse function (see \cite{Nicolaescu2011} for additional details).
\begin{definition}{\rm (\cite{Nicolaescu2011})}
Given a smooth manifold $M$ and a smooth function $f:M\rightarrow \mathbb{R}$, we say that a critical point $p_0$ of $f$ is \textbf{nondegenerate} if its Hessian is nondegenerate. We say that $f$ is a \textbf{Morse function} if all critical points are nondegenerate.
\end{definition}
Note that if $(0,0)$ is geometrically degenerate with respect to $\varepsilon$, then $(0,0)$ is a critical point of the height function $P_\varepsilon:M \rightarrow \mathbb{R}$ defined by $P_\varepsilon(\lambda,\varepsilon)=\varepsilon$ for each $(\lambda,\varepsilon)\in M$.
\begin{definition}
We say that the geometrically degenerate bifurcation at $(0,0)$ has \textbf{geometric codimension-one} with respect to $\varepsilon$ if the height function is locally a Morse function and $(0,0)$ is a critical point.
\end{definition}
The condition for geometric degeneracy is the fact that $(0,0)$ is a critical point of the height function $P_\varepsilon$. Imposing that $P_\varepsilon$ is a Morse function, we are setting a generic geometric condition, following \cite{Wieczorek05,Wieczorek2007}.
\begin{definition}
We say that a bifurcation point has \textbf{codimension $n$-plus-$1$} if it corresponds to a bifurcation of codimension-$n$ from a dynamical point of view, but the same point has codimension-$1$ from a geometrical point of view.
\end{definition}
\begin{remark}
To provide a theoretical framework for geometric bifurcations with codimension $m$, with $m>1$, one needs to consider $m$ distinguished parameters $(\varepsilon_1,\ldots,\varepsilon_m)$. In that context, the notion of \textbf{codimension $n$-plus-$m$} bifurcation would make sense. Nevertheless, in this paper, we do not enter into this formalization. However, note that in the previous section we have illustrated the case of a codimension-one-plus-two geometric bifurcation.
\end{remark}
In the sequel we assume that $q=2$. Since the height function is a Morse function locally around $(0,0)$, it follows from the Morse Lemma that choosing convenient coordinates $\lambda=(\lambda_1,\lambda_2)$, one can write either $
\varepsilon=\lambda_1^2+\lambda_2^2
$
or
$
\varepsilon=-\lambda_1^2-\lambda_2^2
$
or
$
\varepsilon=\lambda_1^2-\lambda_2^2.
$
In the first two cases, the level curves are circles (isolas) for $\varepsilon>0$ (resp. $\varepsilon<0$) and empty sets for $\varepsilon<0$ (resp. $\varepsilon>0$) (see Figure \ref{cod_1p1_3_parametros}(top)). We say isola-type to refer to this bifurcation. In the latter case the level curves correspond to a saddle case (see Figure \ref{cod_1p1_3_parametros}(bottom)) and we say simply saddle-type bifurcation. These two cases are also discussed in \cite{Wieczorek05,Wieczorek2007}. The isola-type and saddle-type geometric bifurcations correspond to the codimension-one singularities named isola-center and simple bifurcation, respectively, in \cite{golsch1985}. Nevertheless, as we already mention in the introduction, there exist singularities which do not match any geometric bifurcation, the hysteresis point, a codimension-one singularity in \cite{golsch1985}, is one example.
Regarding codimension--two-plus-one bifurcations in a three-parameter space, in principle we only distinguish one type. Indeed, given a curve $C$ corresponding to a codimension-two bifurcation with a folding point, we can apply again the Morse Lemma and choose a convenient coordinate $\mu$ along $M$ such that $\varepsilon=\pm \mu^2$. Therefore, the level sets are given by two points that collapse and disappear (compare with the illustration provided in Figure \ref{cod_2p1_3_parametros}).
\begin{figure*}
\caption{Theoretical scheme of the different topologies (Morse classification) of the bifurcation surfaces and the possible geometric and dynamic bifurcations on them. }
\label{topologia-def}
\end{figure*}
Another relevant point is to study how the different geometric bifurcations can appear in the bifurcation diagram of a given family of dynamical systems. We need to understand how a bifurcation surface exhibiting geometric bifurcations can look like. Figure \ref{topologia-def} shows a scheme of different scenarios. A key point is the topology of the bifurcation surface, having one or several tubular structures or even maxima. We classify the different 2D manifolds using the Morse theory \cite{matsumoto2002,Nicolaescu2011}. In all cases the value of a parameter $\varepsilon$ on the surface is our Morse function.
The different possibilities for Morse manifolds shown in Figure \ref{topologia-def} are illustrated later in Section~\ref{sec:2}, namely, Cases~I to III in Subsection~\ref{ssA} and Case~IIb in Subsection~\ref{ssB}. ---\emph{Case~I}--- surface is topologically equivalent to a cylinder, that is, a two-holes sphere. All values of $\varepsilon$ are regular and hence the structure of the surface is the same, a single circle, for all level sets. ---\emph{Case II}--- surface, a three-holes sphere, is called a ``pair of pants'' surface in the context of topology. In this case the Morse function given by the value of $\varepsilon$ has a saddle point (the red point in the middle plot of Figure~\ref{topologia-def}). This point is a codimension--one-plus-one geometric bifurcation point. In this case, the passage through the saddle corresponds to a connecting transition: two isolas collide in the saddle point and give rise to a unique isola. In the case shown in ---\emph{Case III}---, the pair of pants structure is broken, giving rise to two independent surfaces that disappear finally at a maximum of the surface when $\varepsilon$ grows (see the blue points). This case is a two discs shape (two one-hole spheres). Again the points of maxima are codimension--one-plus-one bifurcation points. Finally, ---\emph{Case IIb}---, that we call ``duck-foot surface'', is obtained from the combination of a ``pair of pants'' and a ``disc'' (two Morse surfaces that meet along a curve). Note that this case requires the existence of a codimension-two bifurcation curve (magenta color) that acts as a limit of the tubular structures and enables the continuation in a disc surface. Therefore a cut on a transversal plane gives us the picture of two isolas connected with a curve. More combinations of Morse surfaces are possible, but we only discuss the one detected in the system of Subsection~\ref{ssB}.
Bifurcation surfaces include codimension-two bifurcation curves (black and blue lines) and codimension-three bifurcation points (black points). The value of $\varepsilon$ along the curves of codimension-two bifurcation plays a crucial role in the understanding of the global picture. Therefore, this parameter has a relevant ``physical'' meaning as it is assumed to be the parameter that can be changed in the system giving different phase space dynamics. All points along a curve of codimension-two bifurcation where $\varepsilon$ reaches an extremum (white and green points) are codimension--two-plus-one geometric bifurcations.
The tubular structures exhibited by the Morse surfaces allow us to distinguish between two different types of bifurcation curves of codimension two: those that appear in pairs, on opposite sides of the cylinder; and those that appear as single curves with a fold point where they cross from one side of the cylinder to the other. All curves shown in Case I and those in blue color that we observed in the pair of pants of cases II and IIb are of the first type. The second type corresponds to the bifurcation curves in black color shown in Case II and all the curves in Case III. An independent case is the bifurcation curve (blue color) which is contained in the disk component of the Morse surface in Case IIb. In all cases, the points on the curves where $\varepsilon$ reaches a maximum on the curves are codimension--two-plus-one geometric bifurcations. White points are folds with both branches contained in a unique side of the cylinder (or inside the disk for the Case IIb). Green points are folds where the two branches belong to different sides of the cylinder.
Let $d$ be the distance between both sides of a tubular structure, as depicted in Figure~\ref{topologia-def}. If $d$ is small enough, the two branches arising from green folds are indistinguishable and the folding point may be wrongly perceived as an end point. This situation happens in the fast-slow examples provided in the next section, where $d$ is exponentially small. To distinguish these ``false'' end points, green points are said ``invisible folds'', in contrast with white points that are said ``visible folds'' (note that this situation is for $d\ll 1$). In Section \ref{sec:2}, we do not include models with $d$ big, but they would not provide any other noteworthy behavior either. Note that the appearance of invisible folds is linked to the formation of the ``pants'' or the splitting of a Morse surface in different components. This allows two disconnected isolas of a codimension-two bifurcation curve to meet and create a unique isola.
\section{Some models exhibiting geometric bifurcations}
\label{sec:2}
In this section we will consider two models that exhibit geometric bifurcations, namely, the Hindmarsh-Rose model and the FitzHugh-Nagumo system, showing how geometric bifurcations permit to visualize the global parameter space.
\subsection{The Hindmarsh-Rose model}
\label{ssA}
The Hindmarsh-Rose (HR) system was introduced in
\cite{HR84} as a reduction of the Hodgkin-Huxley equations \cite{HH52} to model the neuron behavior.
The HR model is described by three coupled nonlinear ODEs:
\begin{equation}
\label{HRmodel}
\left\{\begin{array}{l}
\dot{x}=y-ax^3+bx^2-z+I, \\
\dot{y}=c-dx^2-y, \\
\dot{z}=\varepsilon[s(x-x_0)-z],
\end{array}\right.
\end{equation}
\noindent where $x$ is the membrane potential, $y$ the fast and $z$ the slow gating variables for ionic current. Typically, $b$, $I$ and $\varepsilon$ are considered as free parameters, whereas the remaining ones are set as follow: $a=1$, $c=1$, $d=5$, $s=4$, $x_0=-1.6$.
The HR model has been deeply studied in recent years (see \cite{Barrio2017,BIP2020,BIPS2020,BIPS2021,BMSS14,barrio2011,Linaro2012,Shilnikov2008,Storace2008}) using different techniques. The system is particularly useful to understand the bursting phenomena and, in particular, the mechanisms of spike-adding. Behind these dynamics there are homoclinic bifurcations \cite{BMSS14,barrio2011,Linaro2012,Shilnikov2008,Storace2008}. In \cite{Barrio2017,BIP2020} we unveiled the global homoclinic structure and also part of the tangle of bifurcations of periodic orbits that emerge from the homoclinic skeleton, which allowed us to explain the spike-adding processes (see \cite{BIP2020,BIPS2021}).
In most of fast-slow systems with explicit small parameters, these parameters play a significant role and under their variation drastic changes in the global phase space are exhibited. The HR model is a paradigmatic example of this fact. As the small parameter $\varepsilon$ increases, numerous changes occur. In what follows, we show how the HR model exhibits geometric bifurcations with respect to $\varepsilon$ that, linked with dynamic bifurcations, explain these changes. Therefore, the HR model shows how geometric bifurcations permit us to help in the visualization of the global bifurcation diagram in the parameter space.
\begin{figure*}
\caption{Three-parameter plot showing codimension-one homoclinic bifurcation surfaces. In plots (a), (b) and (c) it is presented one homoclinic bifurcation surface with different number of spikes, $hom^{(1,2)}
\label{fig:1+1-extreme-homoclinic}
\end{figure*}
From the analysis in \cite{BIP2020}, it follows that there exist many codimension-one homoclinic bifurcation surfaces which are exponentially close each other and its number grows to infinity when the small parameter tends to zero. Moreover, since these homoclinic surfaces are tubular, the intersection of each surface with horizontal planes produce isolas (closed curves). These isolas exhibit a pair of extremely sharp folds and their width is also exponentially small (compare with the scheme given in Figure~\ref{topologia-def} with $d \ll 1$). Folding points determine two different sides in the isola and also in the bifurcation surface. Typically, for parameter values on one of the ``sides'' of the homoclinic bifurcation isola, the homoclinic orbit exhibits $n$ spikes and, for parameter values on the another face, $n+1$. This explains the notation $hom^{(n,n+1)}$ to refer the different isolas and surfaces. In Figure~\ref{fig:1+1-extreme-homoclinic} we show three-parameter plots showing some codimension-one homoclinic bifurcation surfaces and codimension-two homoclinic bifurcation curves computed using the continuation AUTO software (see \cite{AUTO2,AUTO}). To construct the surfaces, a collection of codimension-one homoclinic bifurcation curves are computed for different values of $\varepsilon$. In plots (a), (b) and (c) the $hom^{(1,2)}$, $hom^{(2,3)}$ and $hom^{(11,12)}$ surfaces are given, respectively, which include Belyakov and Inclination-Flip codimension-two bifurcation curves. This partial bifurcation diagrams illustrate, respectively, Cases I, II and III shown in the theoretical scheme provided in Figure \ref{topologia-def}.
Looking for codimension--one-plus-one geometric bifurcations, we pay attention to the bifurcation surfaces themselves. In Figure~\ref{fig:1+1-extreme-homoclinic}(c) we see how $hom^{(11,12)}$ splits into two disconnected components and each of them has a maximum (with respect to $\varepsilon$). These two points are isola-type codimension--one-plus-one geometric bifurcations with respect to $\varepsilon$. Also, as shown in plot (b) for the surface $hom^{(2,3)}$, a saddle-type codimension--one-plus-one geometric bifurcation is detected on the bifurcation surface. Most likely, isola-type codimension--one-plus-one geometric bifurcations are present in all homoclinic surfaces if the small parameter $\varepsilon$ grows enough and it is no longer ``small''. This fact explains why, as the small parameter grows, fewer bands of color appear in the 2D plots of Figure~\ref{fig:general} for the largest value of $\varepsilon$, indicating that the bursting orbits have fewer spikes.
On the other hand, codimension--two-plus-one geometric bifurcations correspond to folds, with respect to $\varepsilon$, that appear along curves of codimension-two homoclinic bifurcations and, as explained in the previous section, we distinguish between ``visible'' and ``invisible'' folds. Recall that in all two-dimensional manifolds of codimension-one homoclinic bifurcations we distinguish two leaves which are exponentially close (in fact the two leaves that form the isolas) and therefore they are indistinguishable in our visualization of the numerical results. They glue together along curves of sharp folding marked with a red line in Figure~\ref{fig:1+1-extreme-homoclinic}. When a curve of codimension-two homoclinic bifurcation folds inside one of the leaves, we get a \emph{codimension--two-plus-one visible fold}, but, if the fold is along one of the curves of folding of the whole surface (that is, going from one leaf to the another), and hence it is hidden for visualization, we have a \emph{codimension--two-plus-one invisible fold}. Note that visible folds appear in pairs, one on each leaf, but invisible folds are unique points. On the surface $hom^{(1,2)}$, both the Belyakov and the Inclination-Flip bifurcation curves show ``visible'' folds. On the surface $hom^{(2,3)}$, the Belyakov bifurcation curve also undergoes a ``visible'' fold, but the Inclination-Flip curve presents now an ``invisible'' fold. And on the surface $hom^{(11,12)}$, both curves exhibit ``invisible'' folds.
The phenomenon of the ``invisible'' folds is a direct consequence of the existence of very thin tubular structures (and therefore isolas when a two-parameter section is considered) where the two leaves are infinitesimally close each other. Thus, if a codimension-two homoclinic bifurcation curve reaches the homoclinic surface folding curve, then it continues to the other side because the conditions in the phase space that are required to have the codimension-two bifurcation are still satisfied on the other side.
\begin{figure*}
\caption{Schematic processes of some codimension--two-plus-one bifurcations in the HR model increasing the parameter $\varepsilon$. On the top, SC pictures with some bifurcations illustrate the global panorama at several values of $\varepsilon$.
On the bottom pictures some numerical bifurcation curves provide an outline of different ways, through ``visible'' vs. ``invisible'' folds, to arrive to similar global situations.}
\label{fig:tabla}
\end{figure*}
In the Introduction we show in Figure~\ref{fig:general} how some codimension-two points disappear and in Section \ref{sec:01} we also explain that a subsidiary effect was the fact that different curves of periodic bifurcations may connect together, as illustrated in Figure~\ref{fig:general}(c).
Now we can connect all these phenomena with some geometric bifurcations. In Figure~\ref{fig:tabla} we show some codimension--two-plus-one bifurcations in the HR model increasing the small parameter $\varepsilon$. We illustrate schematically the global structure before and after the codimension--two-plus-one bifurcations
and how the size of the homoclinic isolas determines the type: ``visible'' or ``invisible'' fold. As $\varepsilon$ increases, the first geometric bifurcation
in Figure~\ref{fig:general} is a
codimension--two-plus-one bifurcation that occurs on the Inclination-Flip (IF) homoclinic bifurcation curve. For the $hom^{(1,2)}$ case, as the homoclinic surface is big enough, there is a maximum of each of the IF curves (one on each leaf of the tubular surface), and so we have a ``visible'' fold and we observe the geometric ``collision'' of two pairs of IF points, one pair on each of the leaves of the homoclinic surface (the white points on the Figures~\ref{topologia-def} and \ref{fig:1+1-extreme-homoclinic}). On the contrary, for the rest of homoclinic surfaces, the IF bifurcation curves have ``invisible'' folds, as they are smaller and the surfaces are composed of isolas disconnected for small values of the parameter. Moreover, when $n$ grows enough $hom^{(n,n+1)}$ has only one branch of the IF curve, that is, IF points are only present on one of the two disconnected components of the isolas for a small value of $\varepsilon$. In these cases, the IF point seems to disappear on the limit of a homoclinic curve, and what really happens is that geometrically ``collides'' with the corresponding point of the another leaf of the surface that we cannot see (the green points on the Figures~\ref{topologia-def} and \ref{fig:1+1-extreme-homoclinic}). The evolution of the Belyakov curve is similar but for higher values of the small parameter $\varepsilon$.
Figure \ref{fig:tabla} also illustrates how codimension-one bifurcation curves arising from codimension-two bifurcation points are affected by the presence of a geometric bifurcation. For instance, following the surface $hom^{(1,2)}$, first a collision of two IF points is observed (a codimension-two-plus-one geometric bifurcation) which is accompanied by the reconnection of two branches of period doubling bifurcation curves.
\begin{figure*}
\caption{One-plus-one and one-plus-two homoclinic geometric bifurcations in the HR model. (a) $(b, I)$ projections of the $hom^{(1,2)}
\label{fig:oneplustwo}
\end{figure*}
Finally, to complete the panorama of geometric bifurcations given in Section~\ref{sec:1}, we show in Figure~\ref{fig:oneplustwo} how the HR model exhibits a codimension--one-plus-two geometric bifurcation point (compare with Figure~\ref{cod_1p2}, where the $\varepsilon$ parameter of HR is the parameter $\varepsilon_2$. Recall that, in spite of the notion of geometric codimension two has not been formally introduced, we have linked that concept to conceiving the parameter space as a two-parameter family of one-parameter bifurcation diagrams. So, when we describe this codimension--one-plus-two geometric bifurcation point, geometric bifurcations must be understood as tangencies or degenerated tangencies between straight lines in the parameter space (where the value of two parameters is fixed) and a bifurcation surface. On the plot (a) we show the projection of the $hom^{(1,2)}$ bifurcation curve for different values of $\varepsilon$ on the $(b, I)$ parameter plane. The location of the codimension--one-plus-one bifurcation points ("visible folds") on this projection plane is given and we see how they form a pair of geometric bifurcation curves. These curves have a cusp-type contact giving rise to a codimension--one-plus-two geometric bifurcation point. A scheme of the geometry of the $hom^{(1,2)}$ bifurcation surface is given in the two plots (b) presenting a \emph{cusp catastrophe} geometry giving rise to this codimension--one-plus-two bifurcation point (compare with the illustration provided in Figure~\ref{cod_1p2}). Note that as the homoclinic curves finish cutting this
codimension--one-plus-one curve, the cusp catastrophe surface is incomplete in some parametric regions giving just one fold point ("C" shape" vs. "Z" shape). Plots (c) show SC pictures with the $hom^{(1,2)}$ bifurcation curves for three values of $\varepsilon$ to see more clearly the process, and illustrating the changes in the geometry of the homoclinic curves.
\subsection{The FitzHugh-Nagumo system}
\label{ssB}
The FitzHugh-Nagumo system \cite{Fitz61,Nagumo} is a simplified version of the Hodgkin-Huxley model \cite{HH52} for the propagation of nerve impulses in axons, and the reduced ODE system can be written in the form~\cite{CKKOS2007}:
\begin{equation}
\label{FNmodel-ode}
\left\{\begin{array}{rcl}
U'&=&V,
\\
V'&=&\displaystyle{\frac{1}{\Delta}}\big(sV-U(U-1)(\alpha-U)+W-p\big),
\\[6.pt]
W'&=&\displaystyle{\frac{\varepsilon}{s}}(U-\gamma W).
\end{array}\right.
\end{equation}
System (\ref{FNmodel-ode}) has been extensively studied in the literature~\cite{CS2015,CS2018,CKKOS2007,guckue2009,guckue2010,KSS1997}. In \cite{CKKOS2007}, taking $s$ and $p$ as bifurcation parameters, it was shown the existence of $C$-shaped curves of homoclinic bifurcations (travelling waves in the original PDE FitzHugh-Nagumo system correspond to homoclinic orbits in system~(\ref{FNmodel-ode})). In fact it was observed that these $C$-shaped curves were isolas of an exponentially small wide, that authors called ``homoclinic bananas''. They also showed how ``bananas'' could split due to the existence of codimension-two bifurcation points on the homoclinic curve that played the role of terminal points for the branches following after the sharp turning points. That is, a ``banana split'' consists of two isolas joined by a curve. Geometrical explanations for the sharp turns were discussed in \cite{CS2015,guckue2009,guckue2010}. In \cite{CS2018}, the transition was analytically described using geometric singular perturbation theory and blow-up techniques.
We will show that the bifurcation diagram of system~(\ref{FNmodel-ode}) with respect to parameters $(\alpha,s,\varepsilon)$ exhibits geometric bifurcations which are related with the split of the homoclinic banana.
\begin{figure*}
\caption{Three-parameter plot showing codimension-one homoclinic $hom^{(1,2)}
\label{fig:FH}
\end{figure*}
In Figure~\ref{fig:FH} we present a three-parameter plot of the codimension-one homoclinic $hom^{(1,2)}$ bifurcation surfaces for the system~(\ref{FNmodel-ode}) with $p, \gamma=0$, $\Delta=1$, and considering $\alpha$, $s$ and $\varepsilon$ as bifurcation parameters. Plot (a) shows the theoretical homoclinic structure that corresponds to ---\emph{Case IIb}---, ``duck-foot surface'', which is obtained from the combination of a ``pair of pants'' and a ``disc'' (see Figure~\ref{topologia-def}). In this case a codimension-two Belyakov bifurcation curve acts as the limit bound of the tubular structures with a secondary homoclinic (after obtaining an extra spike) that returns to the Belyakov points and enables the continuation in a disc surface. That is, a transversal cut of the structure gives an isola for large values of $\varepsilon$; for an intermediate value of $\varepsilon$, it gives a point of generation of two coupled isolas that coincides with the geometric maxima of the Belyakov curve (a codimension–two-plus-one bifurcation); and for small values of $\varepsilon$, it gives a set of two connected isolas, the ``homoclinic banana split'' that is described in the homoclinic literature. Plots (b) and (c) provide numerical
results obtained using the continuation software AUTO, and they show the homoclinic bifurcation curves for several values of the small parameter $\varepsilon$ and the codimension-two Belyakov bifurcation curve. Plot (b) is given in the three-parameter space $(\alpha, s, \varepsilon)$ and apparently we only observe just one curve for each $\varepsilon$, and no loop at all, but this is due to the very small distance between the sides of the isolas ($d \ll 1$). In order to observe the isolas and the connected isolas, we have to use the AUTO $L_2$-norm as it is shown in plot (c) using the parameters $(\alpha, \varepsilon)$ and the $L_2$-norm.
\section{Conclusions}
In this paper, partially bringing together previous studies, we introduce the concept of geometric bifurcation and illustrate how it appears in the prominent context of neural models. Of course, the Hindmarsh-Rose and FitzHugh-Nagumo systems are not unique examples. Geometric bifurcations are also expected to appear in other neural models and also in models coming from contexts other than Neuroscience (see examples in the already mentioned references \cite{algaba2003,algaba2011,algaba2015,Wieczorek2007,Wieczorek05}).
It should be noted that the study of geometric bifurcations may be of special interest in the framework of fast-slow systems. In this case, those parameters that modulate the slow dynamics are distinguished parameters. These parameters are the ones which one should fix to get 2D slices. Therefore, they specify a very concrete way of exploring the parameter space. In fact, it is possible to observe variations in the bifurcation diagrams when the distinguished parameters change, that is, signs of the presence of geometric bifurcations. In summary, if a multi-parameter space needs to be explored, there is no other option than to work with $k$-parameter slices with $k=1,2$ or, at most, $k=3$, and the different phenomena that we explain in this article could surely be present.
We formally introduce the notion of geometric bifurcation. In particular, we utilize a terminology (codimension $n$-plus-$m$) that emphasizes the distinction between bifurcations in families of dynamical systems and in ``families of families''.
Moreover, we studied the geometric bifurcations of codimension--one-plus-one and two-plus-one in three-parameter spaces. Additionally, we described a codimension--one-plus-two geometric bifurcation.
We were able to explain the appearance of different geometric bifurcations by the combined use of the Morse classification of 2D manifolds and bifurcation theory.
Finally, we conclude that this approach provides a nice tool to explain the different changes observed
in the phase plane when changing a parameter (like in real systems) as a mixture of dynamic and geometric bifurcations.
\input{BIP_Geometric.bbl}
\end{document} | math | 48,184 |
\begin{document}
\title[Congruence-simple multiplicatively idempotent semirings]{Congruence-simple multiplicatively idempotent semirings}
\author[T.~Kepka]{Tom\'{a}\v{s}~Kepka}
\address{Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovsk\'{a} 83, 186 75 Prague 8, Czech Republic}
\email{[email protected]}
\author[M.~Korbel\'a\v{r}]{Miroslav~Korbel\'a\v{r}}
\address{Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technick\'{a} 2, 166 27 Prague 6, Czech Republic}
\email{[email protected]}
\author[G.~Landsmann]{\textsc{G\"{u}nter Landsmann}}
\address{Research Institute for Symbolic Computation, Johannes Kepler University, Alten\-bergerstr. 69, A-4040 Linz, Austria}
\email{[email protected]}
\thanks{
The second and third authors acknowledge the support by the bilateral Austrian Science Fund (FWF) project I 4579-N and Czech Science Foundation (GA\v CR) project 20-09869L ``The many facets of orthomodularity''}
\keywords{congruence, simple, semiring, multiplicatively idempotent, multiplicatively absorbing}
\subseteqjclass[2010]{06D99, 16Y60}
\date{\today}
\begin{abstract}
Let $S$ be a multiplicatively idempotent congruence-simple semiring. We show that $|S|=2$ if $S$ has a multiplicatively absorbing element. We also prove that if $S$ is finite then either $|S|=2$ or $S\cong End(L)$ or $S^{op}\cong End(L)$ where $L$ is a 2-element semilattice. It seems to be an open question, whether $S$ can be infinite at all.
\end{abstract}
\maketitle
The class of finite (congruence-)simple lattices is quite opulent. Even in the case of the modular ones, there are infinitely many (non-isomorphic) examples of such simple lattices (see e.g. \cite{graetzer,schmidt}). Of course, a distribute lattice is simple if and only if it is a two-element chain. Now, distributive lattices form an (equational) subclass of the class $\mathcal{M}$ of all multiplicatively idempotent semirings.
The main aim of this paper is to show that (up to isomorphism) there exist exactly eight finite congruence-simple semirings in $\mathcal{M}$ (six of them are two-element and the remaining two are three-element). We also prove that every semiring in $\mathcal{M}$ possessing a multiplicatively absorbing element is finite (and has only two elements). It seems to be an open question, whether there are infinite congruence-simple semirings in $\mathcal{M}$ at all.
On the other hand, one can find infinite semirings in $\mathcal{M}$ that are bi-ideal-simple (ideal-simple, resp.). Moreover, there are infinitely many (non-isomorphic) examples of finite semirings in $\mathcal{M}$ that are bi-ideal-simple (ideal-simple, resp.) (see, e.g., Examples \ref{ex_1} and \ref{ex_3}).
\section{Preliminaries}
A \emph{semiring} $S$ is a non-empty set equipped with two associative binary operations (usually denoted as addition and multiplication) such that the addition is commutative and the multiplication distributes over the addition from both sides. The semiring $S$ is called \emph{commutative} if the multiplication is commutative.
For non-empty subsets $A,B\subseteq S$ we will use the usual notation of their sum and product as $A+B=\set{a+b}{a\in A, b\in B}$ and $AB=\set{a\cdot b}{a\in A, b\in B}$. If $A=\{a\}$ for some $a\in S$, we sometimes omit the brackets for a simpler notation.
A non-empty subset $I$ of $S$ is an \emph{ideal} (\emph{bi-ideal}, resp.) of $S$ if $(I+I)\cup SI\cup IS\subseteq I$ ($(S+I)\cup SI\cup IS\subseteq I$, resp.). A semiring $S$ is called \emph{congruence-simple} if $S$ has just two congruences and \emph{ideal-simple} (\emph{bi-ideal-simple}, resp.) if $|S|\geq 2$ and $I=S$ whenever $I$ is an ideal (bi-ideal, resp.) containing at least two elements.
A semiring $S$ is called \emph{multiplicatively (additively, resp.) idempotent} if $x^2=x$ ($x+x=x$, resp.) for every $x\in S$. A semiring that is both additively and multiplicatively idempotent will be called \emph{bi-idempotent}.
An element $w\in S$ is called
\begin{itemize}
\item \emph{right (left, resp.) multiplicatively absorbing} if $Sw=\{w\}$ ($wS=\{w\}$, resp.);
\item \emph{multiplicatively absorbing} if $w$ is both right and left multiplicatively absorbing;
\item \emph{additively absorbing} if $S+w=\{w\}$;
\item \emph{multiplicatively (additively, resp.) neutral} if $xw=x=wx$ ($x+w=x$, resp.) for every $x\in S$;
\item \emph{bi-absorbing} if it is both multiplicatively and additively absorbing (such an element will be denoted by $o_S$);
\item a \emph{zero} if it is multiplicatively absorbing and additively neutral (such an element will be denoted by $0_S$).
\end{itemize}
For a semiring $S(+,\cdot)$ the opposite semiring $S^{op}(+,\ast)$ is defined as $a\ast b=b\cdot a$ for every $a,b\in S^{op}=S$.
For a semiring $S$ with a multiplicatively absorbing element $w\in S$ and $T=S\setminus\{w\}$ let us denote by $\varrho_S=(T\times T)\cup\{(w,w)\}$ an equivalence on $S$.
\begin{remark}\label{remark_1}
Let $S$ be a semiring.
(i) Clearly, if $S$ is ideal-simple, then $S$ is bi-ideal-simple. Also, if $S$ is congruence-simple, then $S$ is bi-ideal-simple as well.
Indeed, if $I$ is a bi-ideal of $S$ and $|I|\geq 2$ then the equivalence relation $\varrho=(I\times I) \cup \set{(a,a)}{a\in S}$ is a congruence on $S$ such that $\varrho\neq id_S$. If $S$ is congruence-simple, we obtain that $\varrho=S\times S$ and therefore $I=S$.
(ii) Let $S$ have a multiplicatively absorbing element $w\in S$. Let $\varrho$ be a congruence on $S$ and $I=\set{a\in S}{(a,w)\in\varrho}$ be the block of this congruence. Clearly, $I=S$ if and only if $\varrho=S\times S$. If $w=o_S$ then $I$ is a bi-ideal of $S$ and, similarly, if $w=0_S$ then $I$ is an ideal of $S$.
(iii) Let $S$ have a bi-absorbing element $o_S\in S$ and $T=S\setminus\{o_S\}\neq\emptyset$. Then $\varrho_S$ is an equivalence defined on $S$ and $id_S\neq\varrho_S\neq S\times S$ if and only if $|S|\geq 3$. Besides, $\varrho_S$ is a congruence on the semiring $S$ in each of the following cases:
\begin{enumerate}
\item[(1)] $T+T=\{o_S\}$, $TT=\{o_S\}$;
\item[(2)] $T+T=\{o_S\}$, $TT\subseteq T$;
\item[(3)] $T+T\subseteq T$, $TT=\{o_S\}$;
\item[(4)] $T+T\subseteq T$, $TT\subseteq T$.
\end{enumerate}
(iv) If the semiring $S$ is non-trivial and $0_S\in S$ then $S$ is bi-ideal-simple (in fact, the only bi-ideal of $S$ is $S$ itself).
(v) Every two-element semiring is both congruence-simple and ideal-simple.
(vi) Let $S$ be multiplicatively idempotent and additively cancellative (i.e., $a+c\neq b+c$ for all $a,b,c\in S$ such that $a\neq b$). We claim that $S$ is a Boolean ring.
Indeed, as $S$ is additively cancellative, $S$ is a subsemiring of some ring $R$. For every $a,b\in S$ we have $a+b=(a+b)^2=a^2+b^2+ab+ba=a+b+ab+ba$ and therefore $0_R=ab+ba\in S$. In particular $0_S=0_R\in S$ and we obtain that $0_S=a^2+a^2=a+a$ for every $a\in S$. Therefore $S$ is a ring of characteristic $2$. Finally, we have $ab=-ba=ba$ for all $a,b\in S$ and $S$ is thus commutative. Henceforth, $S$ is a Boolean ring.
If, moreover, the (semi)ring $S$ is congruence-simple or ideal-simple then $S$ is a field and, by the multiplicative idempotency, $S$ is isomorphic to the two-element field $\mathbb{Z}_2$.
(vii) \cite[Theorem 2.2]{cornish} Let $S$ be congruence-simple and $0_S\in S$. If $a^2\neq 0_S$ for every $a\in S$, $a\neq 0_S$, then $S$ has no proper divisors of zero.
\end{remark}
\begin{remark}\label{idempotent}
Let $B$ be a band (i.e., an idempotent semigroup). If $b\in BaB$ for some $a\in B$ then $b=bab$.
Indeed, if $b=cad$ for some $c,d\in B$. Then we have $b=cad=c(ad)^2=(cad)ad=bad$ and, similarly, $b=bad=(ba)^2d=ba(bad)=bab$.
Notice that the set $BaB$ is the principal ideal of the semigroup $B$ generated by the element $a$.
If the band $B$ is ideal-simple, then $b=bab$ for all $a,b\in B$ (i.e., the band is rectangular).
\end{remark}
\section{Bi-ideal-simple multiplicatively idempotent semirings with a multiplicatively absorbing element}
Throughout this section, let $S$ be a multiplicatively idempotent bi-ideal-simple semiring possessing a multiplicatively absorbing element $w$. We put $T=S\setminus\{w\}$.
\begin{lemma}\label{2.1}
Put $A(S)=\set{a\in S}{SaS+S=\{w\}}$ and $B(S)=\set{a\in S}{SaS+S=S}$. Then:
\begin{enumerate}
\item[(i)] $2w=w$ and either $w=0_S$ or $w=o_S$.
\item[(ii)] $B(S)=S\setminus A(S)$.
\item[(iii)] Either $A(S)=\emptyset$ or $A(S)$ is a bi-ideal of $S$.
\item[(iv)] $2a=w$ for every $a\in A(S)$.
\end{enumerate}
\end{lemma}
\begin{proof}
(i) We have $2w = w + w = w^2 + w^2 =
w(w + w) = w$. Further, the set $S+w$ is a bi-ideal of $S$. If $|S+w|=1$ then $w=o_S$. Assume that $|S+w|\geq 2$. Since $S$ is bi-ideal-simple, we have $S+w=S$. Then for every $a\in S$ there is $b\in S$ such that $b+w=a$. Hence $a+w=b+w+w=b+w=a$ and therefore $w=0_S$.
(ii) For every $a\in S$ the set $SaS+S$ is a bi-ideal of $S$ and $w\in SaS+S$. As $S$ is bi-ideal simple, it follows that either $|SaS+S|=1$ or $SaS+S=S$. The rest is obvious.
(iii) Let $a\in A(S)$ and $s\in S$. Then $S(a+s)S+S\subseteq SaS+S=\{w\}$, $S(as)S+S\subseteq SaS+S=\{w\}$ and $S(sa)S+S\subseteq SaS+S=\{w\}$. Hence $s+a,sa,as\in A(S)$ and $A(S)$ is a bi-ideal of $S$.
(iv) For every $a\in A(S)$ we have $2a=a^3+a\in SaS+S=\{w\}$.
\end{proof}
\begin{proposition}\label{2.1.1}
Assume that $|S|=2$. Then $S$ is isomorphic to exactly one of the following four commutative semirings:
\begin{center}
\begin{tabular}[t]{cc}
\tabulkaa{$\mathbb{S}_1$}{$w$}{$a$}{$a$}{$w$}{$w$}{$w$}{$w$}{$a$} &
\tabulkaa{$\mathbb{S}_2$}{$w$}{$a$}{$a$}{$a$}{$w$}{$w$}{$w$}{$a$} \\
&\\[6pt]
\tabulkaa{$\mathbb{S}_3$}{$w$}{$w$}{$w$}{$a$}{$w$}{$w$}{$w$}{$a$} &
\tabulkaa{$\mathbb{S}_4$}{$w$}{$w$}{$w$}{$w$}{$w$}{$w$}{$w$}{$a$}\\
\end{tabular}
\end{center}
The semirings $\mathbb{S}_1$, $\mathbb{S}_2$ have a zero element $w=0_S$ and the semirings $\mathbb{S}_3$, $\mathbb{S}_4$ have a bi-absorbing element $w=o_S$. The only case that is a ring is $\mathbb{S}_1$.
\end{proposition}
\begin{proof}
It is easy to verify that $\mathbb{S}_1$, $\mathbb{S}_2$, $\mathbb{S}_3$ and $\mathbb{S}_4$ are semirings. Let $S=\{w,a\}$, where $a\neq w$. By Lemma \ref{2.1}(i), either $w=0_S$ or $w=o_S$. Since $S$ is multiplicatively idempotent, the multiplication is fully determined. The addition is also determined up to the case $a+a\in\{a,w\}$ (with $w$ being either a zero or a bi-absorbing element). All these four possibilities are represented by the cases $\mathbb{S}_1$, $\mathbb{S}_2$, $\mathbb{S}_3$ and $\mathbb{S}_4$.
\end{proof}
\begin{lemma}\label{2.4}
Assume that $S+S=\{w\}$. Then $w=o_S$, the relation $\varrho_S$ is a congruence on $S$ and $S/\varrho_S\cong\mathbb{S}_4$.
\end{lemma}
\begin{proof}
As $S+S=\{w\}$, the element $w$ is bi-absorbing, i.e., $w=o_S$. For every $a\in T= S\setminus\{o_S\}$, we have $o_S\in SaS$. Hence the set $SaS$ is a bi-ideal of the semiring $S$ and $|SaS|\geq 2$. Since the semiring $S$ is is bi-ideal-simple, we get $SaS=S$ and, consequently $b=bab$ for every $b\in T$, by Remark \ref{idempotent}. It follows that $o_S\neq b=b(ab)$ and therefore $ab\neq o_S$. That is, $TT\subseteq T$. Now, it follows from Remark \ref{remark_1}(iii)(2) that $\varrho_S$ is a congruence on the semiring $S$. As $|S/\varrho_S|=2$, we see readily that $S/\varrho_S\cong \mathbb{S}_4$, by Proposition \ref{2.1.1}.
\end{proof}
\begin{lemma}\label{2.6}
Assume that $S$ is additively idempotent and $w=o_S$. Then $TT\subseteq T$ and $T+T\subseteq T$.
\end{lemma}
\begin{proof}
Since $S$ is additively idempotent, it follows from Lemma \ref{2.1}(iv) and (ii), that $A(S)=\{o_S\}$ and $B(S)=S\setminus\{o_S\}=T$. Then, for $a\in T=B(S)$, we obtain that $SaS+S=S$. Let $b\in T$. Since $b\in S=SaS+S$, there are $x,y,z\in S$ such that $b=c+z$ where $c=xay\in SaS$. By Remark \ref{idempotent}, we have that $c=cac$. Now, $b=c+c+z=c+b=b+cac$.
Assume now, for contrary that $a+b=o_S$. By the previous part of the proof, $b=b+cac$ for some $c\in S$. Hence we have $cbc=c(b+cac)c=cbc+cac=c(a+b)c=o_S$. It follows that $cb=cb\cdot b=cb(b+cac)=cb+(cbc)ac=cb+o_S=o_S$. Finally, we obtain that $b=b\cdot b=(b+cac)b=b+ca(cb)=b+o_S=o_S$, a contradiction. We have shown that $a+b\in S\setminus\{o_S\}=T$. Therefore $T+T\subseteq T$.
Finally, from $b=b+cac$ it follows that $b=b\cdot b\cdot b=b(b+cac)b=b+bcacb=b+e$ where $e=bcacb\in SaS$. By Remark \ref{idempotent}, $e=eae$ and $b=b+eae$. In particular, we obtain that $b=b+eae=b+ea(bcacb)$. Now if $ab=o_S$, then $b=b+e(ab)cacb=b+o_S=o_S$, a contradiction. Hence we have shown that $ab\in S\setminus\{o_S\}=T$. Therefore $TT\subseteq T$.
\end{proof}
\begin{lemma}\label{bi-absorbing}
Assume that $S$ is additively idempotent and $w=o_S$. Then $\varrho_S$ is a congruence on $S$ and $S/\varrho_S\cong\mathbb{S}_3$.
\end{lemma}
\begin{proof}
By Lemma \ref{2.6}, we have that $T+T, TT\subseteq T$. Hence the relation $\varrho_S=(T\times T)\cup \{(o_S,o_S)\}$ is a congruence on the semiring $S$, by Remark \ref{remark_1}(iii)(4). As $|S/\varrho_S|=2$, we obtain that $S/\varrho_S\cong \mathbb{S}_3$, by Proposition \ref{2.1.1}.
\end{proof}
\begin{lemma}\label{2.7}
Assume that $S$ is additively idempotent and $w=0_S$. Then $\varrho_S$ is a congruence on $S$ and $S/\varrho_S\cong\mathbb{S}_2$.
\end{lemma}
\begin{proof}
If $a,b\in S$ are such that $a+b=0_S$, then $a=a+0_S=a+a+b=a+b=0_S$ and, similarly, $b=0_S$. Henceforth $T+T\subseteq T$. Since $S$ is multiplicatively idempotent, it follows that $a^2=a\neq 0_S$ for every $a\in S$, $a\neq 0_S$. By Remark \ref{remark_1}(vii), we have $TT\subseteq T$. Hence, the relation $\varrho_S=(T\times T)\cup\{(0_S,0_S)\}$ is a congruence on the semiring $S$, by Remark \ref{remark_1}(iii)(4). Since $|S/\varrho_S|=2$, we obtain that $S/\varrho_S\cong \mathbb{S}_2$, by Proposition \ref{2.1.1}.
\end{proof}
\begin{proposition}
Assume that either $S+S=\{w\}$ or $S$ is additively idempotent. Then $\varrho_S$ is a congruence on $S$ and $S/\varrho_S$ is isomorphic to (just) one of the two-element semirings $\mathbb{S}_2$, $\mathbb{S}_3$, $\mathbb{S}_4$.
\end{proposition}
\begin{proof}
Combine \ref{2.4}, \ref{bi-absorbing} and \ref{2.7}.
\end{proof}
The following examples show that there are infinitely many finite multiplicatively idempotent semirings that are \emph{bi-ideal-simple} (Examples \ref{ex_1} and \ref{2.10}) but neither congruence- nor ideal-simple. Similarly, there are infinitely many finite multiplicatively idempotent semirings that are \emph{ideal-simple} (Example \ref{ex_3}) but not congruence-simple.
\begin{example}\label{ex_1}
Assume that $w=o_S\in S$ ($w=0_S\in S$, resp.) for the multiplicatively idempotent bi-ideal-simple semiring $S$. Now, set $P=S\cup\{z\}$, $z\not\in S$, $z=0_P$ ($z=o_P$, resp.). In this way, $P$ becomes a multiplicatively idempotent bi-ideal-simple semiring with a zero (a bi-absorbing element, resp.). Clearly, $|P|=|S|+1$, $P$ is bi-idempotent iff $S$ is so, and $P$ is commutative iff $S$ is so. On the other hand, $P$ is \emph{not} ideal-simple as the two-element set $\{z,w\}$ is always an ideal of $P$. Also, $P$ is \emph{not} congruence-simple, as, by Lemmas \ref{bi-absorbing} and \ref{2.7}, $P/\varrho_P\cong\mathbb{S}_2$ ($P/\varrho_P\cong\mathbb{S}_3$, resp.). Of course, we can start with the two-element semirings $S=\mathbb{S}_3,\mathbb{S}_4$ ($S=\mathbb{S}_1,\mathbb{S}_2$, resp.) and, continuing by means of the ``zig-zag method``, we arrive at examples of any finite size $\geq 3$ (see also \cite{vechtomov-petrov}).
\end{example}
\begin{example}\label{2.10}
The following five-element semiring $\mathbb{P}$ is constructed in \cite[Theorem 2.1]{cornish}:
\begin{center}
\begin{tabular}{c}
$\mathbb{P}$\\
\begin{tabular}{ccc}
\begin{tabular}[t]{r|ccccc}
$+$ & $0$ & $1$ & $a$ & $b$ & $c$ \\\hline
$0$ & $0$ & $1$ & $a$ & $b$ & $c$ \\
$1$ & $1$ & $1$ & $c$ & $c$ & $c$ \\
$a$ & $a$ & $c$ & $a$ & $c$ & $c$ \\
$b$ & $b$ & $c$ & $c$ & $b$ & $c$ \\
$c$ & $c$ & $c$ & $c$ & $c$ & $c$ \\
\end{tabular}
&&
\begin{tabular}[t]{r|ccccc}
$\cdot$ & $0$ & $1$ & $a$ & $b$ & $c$ \\\hline
$0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
$1$ & $0$ & $1$ & $a$ & $b$ & $c$ \\
$a$ & $0$ & $a$ & $a$ & $0$ & $a$ \\
$b$ & $0$ & $b$ & $0$ & $b$ & $b$ \\
$c$ & $0$ & $c$ & $a$ & $b$ & $c$ \\
\end{tabular}
\end{tabular}
\end{tabular}
\end{center}
The semiring $\mathbb{P}$ is commutative, bi-idempotent and has a zero element $0=0_\mathbb{P}$. This semiring is subdirectly irreducible (where $\sigma=(J\times J)\cup id_\mathbb{P}$ for $J=\{c,1\}$ is the smallest congruence such that $\sigma\neq id_\mathbb{P}$), but neither congruence-simple nor ideal-simple (e.g., the set $I=\{0,a\}$ is an ideal of $\mathbb{P}$). On the other hand, it is easy to verify that $\mathbb{P}$ is bi-ideal-simple.
\end{example}
\begin{example}\label{ex_3}
Let $L(+)$ be a non-trivial semilattice. Define a multiplication on $L$ by $ab=b$ ($ab=a$, resp.) for all $a,b\in L$. Then $L(+,\cdot)$ becomes an ideal-simple bi-idempotent semiring. By a latter result in this paper (Theorem \ref{3.3}), if $|L|\geq 4$ then $L$ cannot be congruence-simple.
\end{example}
\begin{remark}\label{remark_5}
Let $P$ be a \emph{commutative} multiplicatively idempotent semiring.
(i) If $P$ is ideal-simple then, according to the classification in \cite[Theorem 11.2]{simple_comm}, $P$ has just two elements (and is isomorphic to one of the semirings $\mathbb{S}_1$, $\mathbb{S}_2$, $\mathbb{S}_3$, $\mathbb{S}_4$).
(ii) If $P$ is congruence-simple then it follows easily from \cite[Theorem 10.1]{simple_comm} that $P$ is also isomorphic to $\mathbb{S}_1$, $\mathbb{S}_2$, $\mathbb{S}_3$ and $\mathbb{S}_4$ (see also \cite[Lemma 3.1]{vechtomov-petrov}).
(iii) If $P$ is subdirectly irreducible then $P$ is bi-ideal-simple (and if $P$ has at least three elements then, by (i) and (ii), it is nor congruence-simple neither ideal-simple).
Indeed, by \cite[Theorem 3.1]{vechtomov-petrov}, $P$ has a unity $1\in P$ and there is $e\in P\setminus\{1\}$ such that for the smallest non-identical congruence $\tau$ on $P$ is $(1,e)\in\tau$. Now, let $I$ be a bi-ideal of $P$ such that $|I|\geq 2$. Then $\varrho=(I\times I)\cup\set{(a,a)}{a\in P}$ is a congruence on $P$ and $\varrho\neq id_P$. Hence $(1,e)\in\tau\subseteq\varrho$. Therefore $1\in I$ and we have $I=P$.
\end{remark}
\section{Congruence-simple multiplicatively idempotent semirings with a multiplicatively absorbing element}
\begin{proposition}\label{3.1}
Let $S$ be a multiplicatively idempotent congruence-simple semiring containing at least three elements. Then $S$ is additively idempotent (so that $S$ is bi-idempotent).
\end{proposition}
\begin{proof}
Firstly, the set $S+S$ is a bi-ideal of $S$. As the semiring $S$ is bi-ideal-simple, it follows that either $|S+S|=1$ or that $S+S=S$.
If $|S+S|=1$ then, by Lemma \ref{2.4}, there is a congruence $\varrho\neq id_S$ on the semiring $S$ such that $|S/\varrho|=2$, a contradiction with the congruence-simpleness of $S$. Hence $S+S=S$.
Further, consider the equivalence $\sigma$ on $S$ defined as $(x,y)\in\sigma$ if and only if $2x=2y$. This equivalence, clearly, is a congruence on the semiring $S$.
We show that $\sigma=id_S$. Assume, for contrary, that $\sigma\neq id_S$. Since $S$ is congruence-simple, it follows that $\sigma=S\times S$. Hence there is $w\in S$ such that $2x=w$ for every $x\in S$. Obviously, the element $w$ is multiplicatively absorbing and therefore, by Lemma \ref{2.1}(i), we have either $w=0_S$ or $w=o_S$.
If $w=0_S$ then $S$ is a ring and, by Remark \ref{remark_1}(vi), we obtain that $S\cong\mathbb{Z}_2$, a contradiction with $|S|\geq 3$.
Therefore we have $w=o_S$. Now, as $S+S=S$, for every $a\in S$ there are $b,c\in S$ such that $a=b+c$. Hence $ab+ca=(b+c)b+c(b+c)=2cb+b^2+c^2=o_S+b+c=o_S$. It follows that $a=a\cdot a\cdot a=a(b+c)a=aba+aca=a\cdot ab\cdot a+a\cdot ca\cdot a=a(ab+ca)a=ao_Sa=o_S$. We have obtained that $|S|=1$, a final contradiction.
Thus, we have shown that $\sigma=id_S$. This means that for every $x,y\in S$ the condition $2x=2y$ implies that $x=y$. Now, for every $a\in S$ it holds that $2a=(2a)^2=4a^2=2(2a)$. It follows that $a=2a$ and the semiring $S$ is therefore additively idempotent.
\end{proof}
\begin{theorem}\label{2.9}
Let $S$ be a multiplicatively idempotent congruence-simple semiring possessing a multiplicatively absorbing element. Then $S$ is isomorphic to one of the two-element semirings $\mathbb{S}_1$, $\mathbb{S}_2$, $\mathbb{S}_3$ or $\mathbb{S}_4$.
\end{theorem}
\begin{proof}
By Lemma \ref{2.1}(i), $S$ has either a bi-absorbing element $o_S$ or a zero element $0_S$. Further, proceeding by contradiction, assume that $|S|\geq 3$. Then, by Proposition \ref{3.1}, $S$ is bi-idempotent. By Remark \ref{remark_1}(i), the semiring $S$ is bi-ideal-simple. Therefore, by Propositions \ref{bi-absorbing} and \ref{2.7}, we obtain that there is a congruence $\varrho$ on $S$ such that $\varrho\neq id_S$ and $|S/\varrho|=2$. This is a contradiction with the fact that $S$ is congruence-simple.
We may therefore assume that $|S|=2$. The rest follows immediately from Proposition \ref{2.1.1}.
\end{proof}
\begin{corollary}\label{bi-ideal-simple}
Let $S$ be a multiplicatively idempotent semiring possessing a bi-absorbing element $w$. If
\begin{enumerate}
\item either $w=o_S$ and $S$ is bi-ideal-simple
\item or $w=0_S$ and $S$ is ideal-simple
\end{enumerate}
then $\varrho_S$ is the greatest non-trivial congruence on $S$ (i.e., the congruence $\varrho_S$ is the unique co-atom of the congruence lattice of $S$.)
\end{corollary}
\begin{proof}
Let $\sigma\neq S\times S$ be a congruence on the semiring $S$. Then there is $a\in T=S\setminus\{w\}$ such that $(a,w)\notin\sigma$. By Zorn's lemma there is a congruence $\sigma'$ of $S$ that is maximal with respect to the property $(a,w)\notin\sigma'$ and $\sigma\subseteq\sigma'$. We claim that $\sigma'$ is a maximal congruence on $S$.
Indeed, if $\tau$ is a congruence on $S$ such that $\sigma'\subseteq\tau$ and $\sigma'\neq \tau$, then $a,w\in I=\set{x\in S}{ (x,w)\in\tau}$. By Remark \ref{remark_1}(ii) the set $I$ is a bi-ideal (an ideal, resp.) of $S$ and $|I|\geq 2$. Since $S$ is bi-ideal-simple (ideal-simple, resp.), we have that $I=S$ and therefore $\tau=S\times S$.
Similarly, by Remark \ref{remark_1}(ii), the set $J=\set{x\in S}{ (x,w)\in\sigma'}$ is a bi-ideal (an ideal) of $S$. Since the factor-semiring $S/\sigma'$ is non-trivial, we have that $J\neq S$. As the semiring $S$ is bi-ideal-simple (ideal-simple, resp.), it follows that $J=\{w\}$. Now, since $\sigma'$ is a maximal congruence, we obtain, by Theorem \ref{2.9}, that the semiring $S/\sigma'$ has precisely two elements. Therefore the set $T=S\setminus\{w\}$ is a block of the congruence $\sigma'$ and $\sigma\subseteq\sigma'=(T\times T)\cup\{(w,w)\}=\varrho_S$.
It means that $\varrho_S$ is the only maximal congruence on the semiring $S$ and every proper congruence on $S$ is contained in $\varrho_S$.
\end{proof}
\begin{remark}\label{3.4.0}
Let $S$ be a semiring as in Corollary \ref{bi-ideal-simple} fulfilling the condition (1) and $T=S\setminus\{o_S\}$.
If $S/\varrho_S\cong\mathbb{S}_3$, then, by the definition of $\mathbb{S}_3$, the set $T$ is a subsemiring of $S$. If $|T|\geq 2$ then $T$ is bi-ideal-simple and has no bi-absorbing element (otherwise, for $o_T\in T$ the set $\{o_S, o_T\}$ is a bi-ideal of $S$, a contradiction).
If $S/\varrho_S\cong\mathbb{S}_4$, then $S+S=\{o_S\}$ and the semiring $S$ is ideal-simple. The band $S(\cdot)$ is ideal-simple as well. Besides, $TT\subseteq T$ and the band $T(\cdot)$ is ideal-simple and rectangular (see Remark \ref{idempotent}). If $|T|\geq 2$ then the semigroup $T(\cdot)$ has no multiplicatively absorbing element (otherwise, for such an element $w'\in T$ the set $\{o_S, w'\}$ is a bi-ideal of $S$, a contradiction).
\end{remark}
\begin{remark}\label{3.5}
Let $S$ be a semiring as in Corollary \ref{bi-ideal-simple} fulfilling the condition (2) and $T=S\setminus\{0_S\}$.
If $S/\varrho_S\cong\mathbb{S}_1$ then $S\cong\mathbb{S}_1$.
Indeed, we have that $T+T\subseteq\{0_S\}$. Hence for all $a,b\in T$ it holds that $a+a=0_S=a+b$. Therefore $a=a+0_S=a+a+b=0_S+b=b$ and it follows that $|T|=1$ and $S\cong\mathbb{S}_1$.
If $S/\varrho_S\cong\mathbb{S}_2$ then the set $T$ is a subsemiring of $S$. If $|T|\geq 2$ then $T$ is ideal-simple and has no multiplicatively absorbing element (otherwise, for such an element $w'\in T$ the set $\{0_S, w'\}$ is an ideal of $S$, a contradiction).
\end{remark}
\section{Finite congruence-simple multiplicatively idempotent semirings}
For our further consideration let $L(+)$ be a finite semilattice with the greatest element $1$. Denote by $End(L)(+,\cdot)$ the semiring of all endomorphisms of the semilattice $L$. For $\varphi,\psi\in End(L)$ the operations are defined as follows $(\varphi+\psi)(x)=\varphi(x)+\psi(x)$ and $(\varphi\cdot\psi)(x)=\varphi(\psi(x))$ for every $x\in L$.
If, moreover, $L$ has the least element $0$ (i.e., $L$ is a lattice), we set $End_0(L)=\set{\varphi\in End(L)}{\varphi(0)=0}$. Clearly, $End_0(L)$ is a subsemiring of $End(L)$.
\begin{proposition}\label{3.0}
Let $S$ be a multiplicatively idempotent semiring without a multiplicatively absorbing element. If $|S|=2$ then $S$ is bi-idempotent and isomorphic to precisely one of the following semirings $\mathbb{S}_5$ or $\mathbb{S}_6$. Moreover, $\mathbb{S}_6^{op}= \mathbb{S}_5$.
\begin{center}
\begin{tabular}[t]{cc}
\tabulkab{$\mathbb{S}_5$}{$a$}{$w$}{$w$}{$w$}{$a$}{$a$}{$w$}{$w$} &
\tabulkab{$\mathbb{S}_6$}{$a$}{$w$}{$w$}{$w$}{$a$}{$w$}{$a$}{$w$}\\
\end{tabular}
\end{center}
\end{proposition}
\begin{proof}
Let $S=\{a,w\}$ and $a\neq w$. Assume, for contrary, that $a+a\neq a$. Then $w=a+a=a^2+a^2=a(a+a)=aw$ and, similarly, $w=wa$. Since $w=w^2$, the element $w$ is multiplicatively absorbing, a contradiction. Hence $a+a=a$ and, analogously, $w+w=w$. Therefore, $S$ is additively idempotent.
Further, we may assume without loss of generality that $a<w$. Both the operations in $S$ are now determined up to the case $a\cdot w$ and $w\cdot a$. Since $S$ has no multiplicatively absorbing element, it follows that $a\cdot w\neq w\cdot a$. All the possibilities are now represented by the cases $\mathbb{S}_5$ and $\mathbb{S}_6$.
\end{proof}
The following assertion is easy to verify.
\begin{proposition}\label{3.2}
Let $L=\{0,1\}$ be a semilattice with $0<1$. Then $End(L)=\{a,b,w\}$, where $a(x)=0$, $b(x)=x$ and $w(x)=1$ for every $x\in L$.
The semirings $\mathbb{S}_7=End(L)$ and $\mathbb{S}_8=End(L)^{op}$ have three elements, are bi-idempotent, congruence-simple and are without a multiplicatively absorbing element. These two semirings are non-isomorphic and also not ideal-simple. The operations are as follows.
\begin{center}
\begin{tabular}{c}
$\mathbb{S}_7$\\
\begin{tabular}{ccc}
\begin{tabular}[t]{r|ccc}
$+$ & $a$ & $b$ & $w$ \\\hline
$a$ & $a$ & $b$ & $w$ \\
$b$ & $b$ & $b$ & $w$ \\
$w$ & $w$ & $w$ & $w$ \\
\end{tabular}
&&
\begin{tabular}[t]{r|ccc}
$\cdot$ & $a$ & $b$ & $w$ \\\hline
$a$ & $a$ & $a$ & $a$ \\
$b$ & $a$ & $b$ & $w$ \\
$w$ & $w$ & $w$ & $w$ \\
\end{tabular}
\end{tabular}
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{c}
$\mathbb{S}_8$\\
\begin{tabular}{ccc}
\begin{tabular}[t]{r|ccc}
$+$ & $a$ & $b$ & $w$ \\\hline
$a$ & $a$ & $b$ & $w$ \\
$b$ & $b$ & $b$ & $w$ \\
$w$ & $w$ & $w$ & $w$ \\
\end{tabular}
&&
\begin{tabular}[t]{r|ccc}
$\cdot$ & $a$ & $b$ & $w$ \\\hline
$a$ & $a$ & $a$ & $w$ \\
$b$ & $a$ & $b$ & $w$ \\
$w$ & $a$ & $w$ & $w$ \\
\end{tabular}
\end{tabular}
\end{tabular}
\end{center}
\end{proposition}
\begin{remark}
Notice that the semirings $\mathbb{S}_5$, $\mathbb{S}_6$, $\mathbb{S}_7$ and $\mathbb{S}_8$ have an additively neutral element $a$. Besides, the semirings $\mathbb{S}_7$ and $\mathbb{S}_8$ have a multiplicatively neutral element $b$ and none of these two semirings is ideal-simple. Finally, notice that the element $w$ is left multiplicatively absorbing in $\mathbb{S}_5$ and $\mathbb{S}_7$ and right multiplicatively absorbing in $\mathbb{S}_6$ and $\mathbb{S}_8$.
\end{remark}
\begin{theorem}\cite[Theorems 5.1 and 5.3]{zumbragel}\label{classification}
Let $S$ be a finite bi-idempotent congruence-simple semiring with the greatest element $w\in S$ (with respect to the addition) and $|S|\geq 3$.
\begin{enumerate}
\item[(i)] Let $w$ be neither left nor right multiplicatively absorbing. Then there is a finite (semi)lattice $L$ (with the least element $0$ and the greatest element $1$) such that $S$ is isomorphic to a subsemiring $R$ of $End_{0}(L)$. Further, for every $a,b\in L$ there is $e_{a,b}\in R$ such that for every $x\in L$ is $e_{a,b}(x)=0$ if $x\leq a$ and $e_{a,b}(x)=b$ otherwise.
\item[(ii)] Let $w$ be left but not right multiplicatively absorbing. Then there is a finite semilattice $L$ (with the greatest element $1$) such that $S$ is isomorphic to a subsemiring $R$ of $End(L)$. Further, for every $a\in L$ and $b\in L\setminus\{1\}$ there is $f\in R$ such that for every $x\in L$ is $f(x)=b$ if $x\leq a$ and $f(x)>b$ otherwise.
\end{enumerate}
\end{theorem}
\begin{proposition}\label{non-bi-absorbing}
Let $S$ be a finite bi-idempotent congruence-simple semiring with the greatest element $w\in S$ (with respect to the addition). If $w$ is not a bi-absorbing element then $S$ is isomorphic to one of the four two/three-element semirings $\mathbb{S}_5$, $\mathbb{S}_6$, $\mathbb{S}_7$ or $\mathbb{S}_8$.
\end{proposition}
\begin{proof}
If $|S|=2$ then, by Proposition \ref{3.0}, $S$ is isomorphic either to $\mathbb{S}_5$ or $\mathbb{S}_6$. For the rest of the proof we therefore consider that $|S|\geq 3$. Let $w\in S$ be the greatest element in $S$ with respect to the addition.
Assume, first, that $w$ is neither left nor right multiplicatively absorbing. By Theorem \ref{classification}(i), there is a finite (semi)lattice $L$ (with the least element $0$ and the greatest element $1$) such that $S$ is isomorphic to a subsemiring $R$ of $End_{0}(L)$ and for every $a,b\in L$ there is $e_{a,b}\in R$ such that for every $x\in L$ is $e_{a,b}(x)=0$ if $x\leq a$ and $e_{a,b}(x)=b$ otherwise. Assuming that there is $a\in L$ such that $0\neq a\neq 1$ we obtain that $0=e_{a,a}(a)=e_{a,a}(e_{a,a}(1))=e_{a,a}(1)=a$, a contradiction. Hence $|L|=2$ and therefore $|End_{0}(L)|=2$, by Proposition \ref{3.2}. Thus $|S|=2$ and we obtain a contradiction with the assumption that $|S|\geq 3$.
Assume further, that $w$ is left but not right multiplicatively absorbing. By Theorem \ref{classification}(ii), there is a finite semilattice $L$ (with the greatest element $1$) such that $S$ is isomorphic to a subsemiring $R$ of $End(L)$ and for every $a\in L$ and $b\in L\setminus\{1\}$ there is $f\in R$ such that for every $x\in L$ is $f(x)=b$ if $x\leq a$ and $f(x)>b$ otherwise. If there are $a\in L\setminus\{1\}$ and $b\in L\setminus\{1\}$ such that $a\not\leq b$ then we obtain that $b<f(b)=f(f(a))=f(a)=b$, a contradiction. Hence $|L|=2$ and $|End(L)|=3$, by Proposition \ref{3.2}. Therefore $S\cong End(L)=\mathbb{S}_7$.
Finally, assume that $w$ is right but not left multiplicatively absorbing. Then the semiring $S^{op}$ is finite and congruence-simple. Further, $S^{op}$ has a greatest element (with respect to the addition) that is left but not right multiplicatively absorbing (and $|S^{op}|\geq 3$). By the previous part of the proof, we have that $S^{op}\cong \mathbb{S}_7$. Hence $S\cong \mathbb{S}_7^{op}=\mathbb{S}_8$.
\end{proof}
\begin{theorem}\label{3.3}
Let $S$ be a finite multiplicatively idempotent congruence-simple semiring. Then $S$ is isomorphic to one of the eight two/three-element semirings $\mathbb{S}_1,\dots,\mathbb{S}_8$.
\end{theorem}
\begin{proof}
Let $|S|=2$. Then, by Propositions \ref{2.1.1} and \ref{3.0}, $S$ is isomorphic to one of the six semirings $\mathbb{S}_1,\dots,\mathbb{S}_6$.
Assume that $|S|\geq 3$. Then, by Propositions \ref{3.1}, $S$ is bi-idempotent. As $S$ finite, there is a greatest element $w$ with respect to the addition. If $w$ is multiplicatively absorbing then, by Theorem \ref{2.9}, $|S|=2$, a contradiction. Hence $w$ is not multiplicatively absorbing and therefore, by Proposition \ref{non-bi-absorbing}, $S$ is isomorphic to either $\mathbb{S}_7$ or $\mathbb{S}_8$.
\end{proof}
With regard to Remark \ref{remark_5} and Theorems \ref{2.9} and \ref{3.3}, it is natural to consider the following conjecture:
\begin{conjecture}
Every multiplicatively idempotent congruence-simple semiring is finite (and isomorphic to one of the semirings $\mathbb{S}_1$, $\dots$, $\mathbb{S}_8$).
\end{conjecture}
\section{Multiplicatively divisible commutative semirings}
In \cite{parasem,conj} additively divisible commutative semirings were studied. Analogous questions may be raised for the multiplicative parts of commutative semirings.
We call a semiring $S$ \emph{multiplicatively divisible} if for every $a\in S$ and every $n\in\mathbb{N}$ there is $b\in S$ such that $a=b^n$. A semiring $P$ is called a \emph{parasemifield} if the multiplicative semigroup $P(\cdot)$ is a group.
Clearly, every multiplicatively idempotent semiring is multiplicatively divisible. On the other hand, any algebraically closed field of characteristic $0$ (e.g., the countable field of algebraic real numbers) is congruence-simple and ideal-simple and both additively and multiplicatively divisible, but it is not multiplicatively idempotent. Of course, no infinite field is finitely generated as a (semi)ring. The following conjecture is now in force.
\begin{conjecture}\label{conj_2}
Let $S$ be a finitely generated commutative semiring. If $S$ is multiplicatively divisible then $S$ is multiplicatively idempotent.
\end{conjecture}
The following Remarks \ref{remark_3} and \ref{remark_4} support the plausibility of this conjecture.
\begin{remark}\label{semigroup}
Let $S$ be a divisible semigroup. Then $S$ is a finite band in each of the following two cases:
\begin{enumerate}
\item[(1)] $S$ is commutative and finitely generated.
\item[(2)] $S$ is finite.
\end{enumerate}
This result is a sort of folklore (see, e.g., \cite[Theorem 2.5]{szekely}).
\end{remark}
\begin{remark}\label{remark_3}
Every finitely generated multiplicatively divisible commutative ring is a Boolean ring. Hence Conjecture \ref{conj_2} holds in case of rings.
Indeed, let $R$ be a finitely generated commutative ring (not necessary with a unity).
First, let us recall a well-known fact (attributed sometimes to Kaplansky) that every field $F$ that is a factor of $R$ is finite. Let $R$ be multiplicatively divisible. Then, by Theorem \ref{semigroup}, the only such a field $F$ is $\mathbb{Z}_2$. Let $\mathcal{J}(R)$ be the Jacobson radical of $R$. Then $R/\mathcal{J}(R)$ is a subring of product of fields that are factors of $R$, i.e., of fields isomorphic to $\mathbb{Z}_2$. Such a product is multiplicatively idempotent and so is the ring $R/\mathcal{J}(R)$.
To prove that $R$ is multiplicatively idempotent too, it is enough to show that $\mathcal{J}(R)=0$. Since the ring of all integers $\mathbb{Z}$ is a Jacobson ring (i.e., every prime ideal is an intersection of maximal ideals) and $R$ is a finitely generated $\mathbb{Z}$-algebra, the ring $R$ is Jacobson too. Hence, the Jacobson radical and the nilradical of $R$ coincide, by the definition of the Jacobson ring. Finally, as $R$ is also noetherian, the nilradical of $R$ is nilpotent. Thus, there is $n\in\mathbb{N}$ such that $\big(\mathcal{J}(R)\big)^n=0$. Further, $\mathcal{J}(R)$ is a radical ideal and therefore the ring $\mathcal{J}(R)$ is multiplicatively divisible. Hence, for every $a\in \mathcal{J}(R)$ there is $b\in \mathcal{J}(R)$ such that $a=b^n\in \big(\mathcal{J}(R)\big)^n=0$. This proves that $\mathcal{J}(R)=0$ and that $R$ is multiplicatively idempotent.
\end{remark}
\begin{remark}\label{remark_4}
According to \cite[Corollary 4.6]{parasem} if $S$ is a commutative parasemifield, that is finitely generated as a semiring, then the multiplicative semigroup $S(\cdot)$ is finitely generated. Combining this result and \ref{semigroup} we see that the semigroup $S(\cdot)$ is idempotent. In fact, as $S(\cdot)$ is a group, the only such a parasemifield $S$ is the trivial one. Hence Conjecture \ref{conj_2} holds in case of parasemifields.
\end{remark}
\end{document} | math | 36,637 |
\ensuremath{\flat}egin{equation}gin{document}
\ensuremath{\mathfrak{a}}ddress{University of Michigan, Department of Mathematics, 2074 East Hall, 530 Church Street, Ann Arbor, MI 48109-1043}
\email{[email protected]} \ensuremath{\mathfrak{k}}eywords{Hecke algebra, Temperley-Lieb algebra, semisimple, Kronecker problem}
\ensuremath{\mathfrak{t}}ext{\rm Aut}\,hor{Jonah Blasiak}\ensuremath{\mathfrak{t}}hanks{The author was partially supported by an NSF postdoctoral fellowship.}
\ensuremath{\mathfrak{t}}itle{Representation theory of the nonstandard Hecke algebra}
\ensuremath{\mathfrak{m}}aketitle
\ensuremath{\flat}egin{equation}gin{abstract}
The nonstandard Hecke algebra $\nsbr{\ensuremath{\mathscr{H}}}_r$ was defined by Mulmuley and Sohoni to study the Kronecker problem. We study a quotient $\nsbr{\ensuremath{\mathscr{H}}}_{r,2}$ of $\nsbr{\ensuremath{\mathscr{H}}}_r$, called the nonstandard Temperley-Lieb algebra, which is a subalgebra of the symmetric square of the Temperley-Lieb algebra $\ensuremath{\mathfrak{t}}ext{TL}_r$. We give a complete description of its irreducible representations.
We find that the restriction of an $\nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducible to $\nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}$ is multiplicity-free, and as a consequence, any $\nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducible has a seminormal basis that is unique up to a diagonal transformation.
\end{abstract}
\section{Introduction}
Let $\ensuremath{\mathscr{H}}_r$ be the type $A_{r-1}$ Hecke algebra over $\ensuremath{\mathfrak{m}}athbf{A} = \ensuremath{\mathbb{Z}}[\ensuremath{u}, \ensuremath{u^{-1}}]$ and set $\ensuremath{K} := \ensuremath{\mathbb{Q}}(\ensuremath{u})$.
The nonstandard Hecke algebra $\nsbr{\ensuremath{\mathscr{H}}}_r$ is the subalgebra of $\ensuremath{\mathscr{H}}_r \ensuremath{\otimes} \ensuremath{\mathscr{H}}_r$ generated by
\[\mathcal{P}_i := \ensuremath{C^{\prime}}_{s_i} \ensuremath{\otimes}vw \ensuremath{C^{\prime}}_{s_i} + C_{s_i} \ensuremath{\otimes}vw C_{s_i}, \ i \in [r-1],\]
where $\ensuremath{C^{\prime}}_{s_i}$ and $C_{s_i}$ are the simplest lower and upper Kazhdan-Lusztig basis elements, which are proportional to the trivial and sign idempotents of the parabolic sub-Hecke algebra $\ensuremath{K} (\ensuremath{\mathscr{H}}_r)_{\{s_i\}}$.
The nonstandard Hecke algebra was introduced by Mulmuley and Sohoni in \cite{GCT4} to study the Kronecker problem. The hope was that the inclusion $ \nsbr{\ensuremath{\ensuremath{\mathfrak{m}}athcal{D}}elta} : \nsbr{\ensuremath{\mathscr{H}}}_r \ensuremath{\mathfrak{t}}o \ensuremath{\mathscr{H}}_r \ensuremath{\otimes} \ensuremath{\mathscr{H}}_r$ would quantize the coproduct $\ensuremath{\ensuremath{\mathfrak{m}}athcal{D}}elta: \ensuremath{\mathbb{Z}}\ensuremath{\mathcal{S}}_r \ensuremath{\mathfrak{t}}o \ensuremath{\mathbb{Z}}\ensuremath{\mathcal{S}}_r \ensuremath{\otimes} \ensuremath{\mathbb{Z}} \ensuremath{\mathcal{S}}_r$ of the group algebra $\ensuremath{\mathbb{Z}} \ensuremath{\mathcal{S}}_r$ and canonical basis theory could be applied to obtain formulas for Kronecker coefficients. Unfortunately, this does not work in a straightforward way since the algebra $\nsbr{\ensuremath{\mathscr{H}}}_r$ is almost as big as $\ensuremath{\mathscr{H}}_r \ensuremath{\otimes} \ensuremath{\mathscr{H}}_r$ and has $\ensuremath{\mathfrak{m}}athbf{A}$-rank much larger than $r!$, even though $\nsbr{\ensuremath{\ensuremath{\mathfrak{m}}athcal{D}}elta}$ is in a certain sense the quantization of $\ensuremath{\ensuremath{\mathfrak{m}}athcal{D}}elta$ with image as small as possible (see \cite[Remark 11.4]{BMSGCT4}). Nonetheless, as discussed in \cite[\ensuremath{\mathfrak{t}}extsection1]{BMSGCT4}, \cite{GCT7, canonical}, and briefly in this paper, the nonstandard Hecke algebra may still be useful for the Kronecker problem.
Though the nonstandard Hecke algebra has yet to prove its importance for the Kronecker problem, it is an interesting problem in its own right to determine all the irreducible representations of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r}$. This problem is difficult, but within reach. In this paper, we solve an easier version of this problem.
It is shown in \cite{BMSGCT4} that $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_r$ is semisimple.
Let $\ensuremath{\mathfrak{t}}au$ be the flip involution of $\ensuremath{\mathscr{H}}_r\ensuremath{\otimes}vw\ensuremath{\mathscr{H}}_r$ given by $h_1\ensuremath{\otimes}vw h_2\ensuremath{\mathfrak{m}}apsto h_2 \ensuremath{\otimes}vw h_1$ and let $\ensuremath{\mathfrak{t}}heta : \ensuremath{\mathscr{H}}_r \ensuremath{\mathfrak{t}}o \ensuremath{\mathscr{H}}_r$ be the $\ensuremath{\mathfrak{m}}athbf{A}$-algebra involution defined by $\ensuremath{\mathfrak{t}}heta(T_{s_i}) = - T_{s_i}^{-1},\ i \in [r-1]$. Twisting an $\ensuremath{\mathscr{H}}_r$-irreducible by $\ensuremath{\mathfrak{t}}heta$ corresponds to transposing its shape.
The algebra $\nsbr{\ensuremath{\mathscr{H}}}_r$ is a subalgebra of $(S^2 \ensuremath{\mathscr{H}}_r)^{\ensuremath{\mathfrak{t}}heta \ensuremath{\otimes}vw \ensuremath{\mathfrak{t}}heta}$, the subalgebra of $\ensuremath{\mathscr{H}}_r \ensuremath{\otimes}vw \ensuremath{\mathscr{H}}_r$ fixed by $\ensuremath{\mathfrak{t}}heta \ensuremath{\otimes}vw \ensuremath{\mathfrak{t}}heta$ and $\ensuremath{\mathfrak{t}}au$.
Based on computations for $r \leq 6$, it appears that most of the $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_r$-irreducibles are restrictions of $\ensuremath{K} (S^2 \ensuremath{\mathscr{H}}_r)^{\ensuremath{\mathfrak{t}}heta \ensuremath{\otimes}vw \ensuremath{\mathfrak{t}}heta}$-irreducibles, except for the trivial and sign representations of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_r$.
In this paper we focus on the simpler problem of determining the irreducibles of the nonstandard Temperley-Lieb algebra $\nsbr{\ensuremath{\mathscr{H}}}_{r,2}$, which is a quotient of $\nsbr{\ensuremath{\mathscr{H}}}_{r}$. The algebra $\nsbr{\ensuremath{\mathscr{H}}}_{r,2}$ is the subalgebra of $\ensuremath{\mathscr{H}}_{r,2} \ensuremath{\otimes} \ensuremath{\mathscr{H}}_{r,2}$ generated by $\mathcal{P}_i := \ensuremath{C^{\prime}}_{s_i} \ensuremath{\otimes}vw \ensuremath{C^{\prime}}_{s_i} + C_{s_i} \ensuremath{\otimes}vw C_{s_i}, \ i \in [r-1]$, where $\ensuremath{\mathscr{H}}_{r,2}$ is the Temperley-Lieb algebra (see \ensuremath{\mathfrak{t}}extsection\ref{s irreducibles of nshr2}).
The main result of this paper is a complete description of the $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducibles (Theorem \ref{t nsH irreducibles two row case}). There are no surprises here: it is fairly easy to show that $\ensuremath{K} (\ensuremath{\mathscr{H}}_{r,2} \ensuremath{\otimes} \ensuremath{\mathscr{H}}_{r,2})$-irreducibles decompose into certain $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-modules. The difficulty is showing that these modules are actually irreducible. We prove this by induction on $r$ and by computing the action of $\mathcal{P}_{r-1}$ on these modules in terms of canonical bases.
To carry out these computations, we use results from \cite{BProjected} about projecting the upper and lower canonical bases of a $\ensuremath{K} \ensuremath{\mathscr{H}}_{r}$-irreducible $M_\lambda$ onto its $\ensuremath{K} \ensuremath{\mathscr{H}}_{r-1}$-irreducible isotypic components. We also use the well-known fact that the edge weight $\ensuremath{\mathfrak{m}}u(x,w)$, $x, w \in \ensuremath{\mathcal{S}}_r$, of the $\ensuremath{\mathcal{S}}_r$-graph $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_{\ensuremath{\mathcal{S}}_r}$ is equal to 1 whenever $x$ and $w$ differ by a dual Knuth transformation (see \ensuremath{\mathfrak{t}}extsection\ref{ss upper and lower canonical basis of H(W)} and \ensuremath{\mathfrak{t}}extsection\ref{ss dual equivalence}).
One consequence of Theorem \ref{t nsH irreducibles two row case} is that the restriction
of a $ \ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducible to $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}$ is multiplicity-free.
Thus each $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducible has a seminormal basis (in the sense of \cite{RamSeminormal}---see Definition \ref{d seminormal}) that is unique up to a diagonal transformation.
This can also be used to define a seminormal basis for any $\ensuremath{K}(\ensuremath{\mathscr{H}}_{r,2}\ensuremath{\otimes}\ensuremath{\mathscr{H}}_{r,2})$-irreducible.
Even though the irreducibles of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$ are close to those of $\ensuremath{K}(\ensuremath{\mathscr{H}}_{r,2}\ensuremath{\otimes}\ensuremath{\mathscr{H}}_{r,2})$, the nonstandard Temperley-Lieb algebra offers something new:
the seminormal basis of $M_\lambda\ensuremath{\otimes} M_\ensuremath{\mathfrak{m}}u$ using the chain $\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{J_1} \subseteq \cdots \subseteq \ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{J_{r-1}} \subseteq \ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{J_r}$ is significantly different from the seminormal basis using the chain $\ensuremath{K}(\ensuremath{\mathscr{H}}_{1,2} \ensuremath{\otimes} \ensuremath{\mathscr{H}}_{1,2}) \subseteq \cdots \subseteq \ensuremath{K} (\ensuremath{\mathscr{H}}_{r-1,2} \ensuremath{\otimes} \ensuremath{\mathscr{H}}_{r-1,2}) \subseteq \ensuremath{K} (\ensuremath{\mathscr{H}}_{r,2} \ensuremath{\otimes} \ensuremath{\mathscr{H}}_{r,2})$, where $\nsbr{\ensuremath{\mathscr{H}}}_{J_k}$ is the subalgebra of $\nsbr{\ensuremath{\mathscr{H}}}_{r,2}$ generated by $\mathcal{P}_1, \dots, \mathcal{P}_{k-1}$.
We are interested in these seminormal bases primarily as a tool for constructing a canonical basis of a $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducible that is compatible with its decomposition into irreducibles at $\ensuremath{u} =1$, as described in \cite[\ensuremath{\mathfrak{t}}extsection19]{BMSGCT4}.
Thus even though the representation theory of the nonstandard Hecke algebra alone is not enough to understand Kronecker coefficients,
there is hope that the seminormal bases will yield a better understanding of Kronecker coefficients.
In fact, \cite{canonical} gives a conjectural scheme for constructing a canonical basis of a $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducible using its seminormal basis,
but this remains conjectural and we do not know how to use it to understand Kronecker coefficients.
This paper is organized as follows:
sections \ref{ss type A combinatorics preliminaries}--\ref{s the nonstandard Hecke algebra} are preparatory: \ensuremath{\mathfrak{t}}extsection\ref{s canonical bases of hecke algebras} reviews the necessary facts about canonical bases of $\ensuremath{\mathscr{H}}_r$ and their behavior under projection onto $ \ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}$-irreducibles; \ensuremath{\mathfrak{t}}extsection\ref{s the nonstandard Hecke algebra} gives some basic results about the representation theory of $\nsbr{\ensuremath{\mathscr{H}}}_r$.
Section \ref{s irreducibles of nshr2} contains the statement and proof of the main theorem. Then in \ensuremath{\mathfrak{t}}extsection\ref{s Seminormal bases}, seminormal bases of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducibles are defined, and in \ensuremath{\mathfrak{t}}extsection\ref{s enumerative consequence}, the dimension of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$ is determined.
\section{Partitions and tableaux}
\label{ss type A combinatorics preliminaries}
A \emph{partition} $\lambda$ of $r$ of length $\ell(\lambda) = l$ is a sequence $(\lambda_1, \ensuremath{\ensuremath{\mathfrak{t}}rianglelefteq}ots, \lambda_l)$ such that $\lambda_1 \ensuremath{\ensuremath{\mathfrak{m}}athfrak{g}}eq \cdots \ensuremath{\ensuremath{\mathfrak{m}}athfrak{g}}eq \lambda_l > 0$ and $r = \sum_{i=1}^l \lambda_i$. The notation $\lambda \vdash r$ means that $\lambda$ is a partition of $r$.
Let $\ensuremath{\mathfrak{m}}athscr{P}_r$ denote the set of partitions of size $r$ and $\ensuremath{\mathfrak{m}}athscr{P}'_r$ the subset of $\ensuremath{\mathfrak{m}}athscr{P}_r$ consisting of those partitions that are not a single row or column shape. The symbols $\ensuremath{\ensuremath{\mathfrak{m}}athfrak{g}}d, \ensuremath{\ensuremath{\mathfrak{m}}athfrak{g}}dneq$ will denote dominance order on partitions.
The conjugate partition $\lambda'$ of a partition $\lambda$ is the partition whose diagram is the transpose of the diagram of $\lambda$.
The set of standard Young tableaux is denoted SYT and the subset of SYT of shape $\lambda$ is denoted SYT$(\lambda)$.
Tableaux are drawn in English notation, so that entries increase from north to south along columns and increase from west to east along rows. For a tableau $T$, $\ensuremath{\mathfrak{t}}ext{\rm sh}(T)$ denotes the shape of $T$.
For a word $\ensuremath{\mathfrak{m}}athbf{k} = k_1 k_2\dots k_r$, $k_i \in \ensuremath{\mathbb{Z}}_{> 0}$, let $P(\ensuremath{\mathfrak{m}}athbf{k}), Q(\ensuremath{\mathfrak{m}}athbf{k})$ denote the insertion and recording tableaux produced by the Robinson-Schensted-Knuth (RSK) algorithm applied to $\ensuremath{\mathfrak{m}}athbf{k}$.
The notation $\ensuremath{\mathfrak{t}}ranspose{Q}$ denotes the transpose of an SYT $Q$, so that $\ensuremath{\mathfrak{t}}ext{\rm sh}(\ensuremath{\mathfrak{t}}ranspose{Q}) = \ensuremath{\mathfrak{t}}ext{\rm sh}(Q)'$.
Let $T$ be a tableau of shape $\lambda$. If $b$ is a square of the diagram of $\lambda$, then $T_b$ denotes the entry of $T$ in the square $b$. If $\nu \subseteq \lambda$, then $T_\nu$ denotes the subtableau of $T$ obtained by restricting $T$ to the diagram of $\nu$.
Let $\lambda$ and $\ensuremath{\mathfrak{m}}u$ be partitions of $r$. Throughout this paper, $a_1, \dots, a_{k_\lambda}$ (resp. $b_1, \dots, b_{k_\ensuremath{\mathfrak{m}}u}$) will denote the outer corners of the diagram of $\lambda$ (resp. $\ensuremath{\mathfrak{m}}u$) labeled so that $a_{i+1}$ lies to the east of $a_i$ (resp. $b_{i+1}$ lies to the east of $b_i$), as in the following example.
\setlength{\cellsize}{2ex}
\ensuremath{\flat}egin{equation} \label{e ai definition}
\ensuremath{\flat}egin{equation}gin{array}{c@{\ensuremath{\ensuremath{\mathfrak{m}}athfrak{h}}space{1in}}c}
{\ensuremath{\mathfrak{t}}iny \ensuremath{\mathfrak{t}}ableau{\ & \ & \ & \ & \ & \ & \ & \ & a_4 \\
\ & \ & \ & \ & \ & \ & a_3 \\
\ & \ & \ & \ \\
\ & \ & \ & a_2 \\
\ & a_1}}
&
{\ensuremath{\mathfrak{t}}iny \ensuremath{\mathfrak{t}}ableau{\ & \ & \ & \ & \ & \ & \ & \ \\
\ & \ & \ & \ & \ & \ & \ & b_2 \\
\ & \ & \ & \ & \ \\
\ & \ & \ & \ & b_1}} \\
\lambda & \ensuremath{\mathfrak{m}}u
\end{array}
\end{equation}
\section{Canonical bases of the Hecke algebra $\ensuremath{\mathscr{H}}_r$}\label{s canonical bases of hecke algebras}
Here we recall the definition of the Kazhdan-Lusztig basis elements $C_w$ and $\ensuremath{C^{\prime}}_w$ and review the connection between cells in type $A$ and tableaux combinatorics,
following \cite{BProjected}.
We then discuss dual equivalence graphs and recall some results of \cite{BProjected} about projecting canonical bases, which will make these bases fairly easy to work with in the proof of Theorem \ref{t nsH irreducibles two row case}.
We work over the ground rings $\ensuremath{\mathfrak{m}}athbf{A} = \ensuremath{\mathbb{Z}}[\ensuremath{u}, \ensuremath{u^{-1}}]$ and $\ensuremath{K} = \ensuremath{\mathbb{Q}}(\ensuremath{u})$. Define $\ensuremath{K}_0$ (resp. $\ensuremath{K}_\infty$) to be the subring of $\ensuremath{K}$ consisting of rational functions with no pole at $\ensuremath{u} = 0$ (resp. $\ensuremath{u} = \infty$).
Let $\ensuremath{\flat}r{\cdot}$ be the involution of $\ensuremath{K}$ determined by $\ensuremath{\flat}r{u} = \ensuremath{u^{-1}}$; it restricts to an involution of $\ensuremath{\mathfrak{m}}athbf{A}$.
For a nonnegative integer $k$, the $\ensuremath{\flat}r{\cdot}$-invariant quantum integer is $[k] := \ensuremath{[2]^2}rac{\ensuremath{u}^k - \ensuremath{u}^{-k}}{\ensuremath{u} - \ensuremath{u^{-1}}} \in \ensuremath{\mathfrak{m}}athbf{A}$.
We also use the notation $[k]$ to denote the set $\{1,\ensuremath{\ensuremath{\mathfrak{t}}rianglelefteq}ots,k\}$, but these usages should be easy to distinguish from context.
\subsection{The Hecke algebra $\ensuremath{\mathscr{H}}(W)$}
\label{ss the Hecke algebra H(W)}
Let $(W, S)$ be a Coxeter group with length function $\ell$ and Bruhat order $<$. If $\ell(vw)=\ell(v)+\ell(w)$, then $vw = v\cdot w$ is a \emph{reduced factorization}. The \emph{right descent set} of $w \in W$ is $R(w) = \{s\in S : ws < w\}$.
For any $L\subseteq S$, the \emph{parabolic subgroup} $W_L$ is the subgroup of $W$ generated by $L$.
The \emph{Hecke algebra} $\ensuremath{\mathscr{H}}(W)$ of $(W, S)$ is the free $\ensuremath{\mathfrak{m}}athbf{A}$-module with standard basis $\{T_w :\ w\in W\}$ and relations generated by
\ensuremath{\flat}egin{equation} \label{e Hecke algebra def} \ensuremath{\flat}egin{equation}gin{array}{ll}T_vT_w = T_{vw} & \ensuremath{\mathfrak{t}}ext{if } vw = v\cdot w\ \ensuremath{\mathfrak{t}}ext{is a reduced factorization},\\
(T_s - \ensuremath{u})(T_s + \ensuremath{u^{-1}}) = 0 & \ensuremath{\mathfrak{t}}ext{if } s\in S.\end{array}\end{equation}
\subsection{The upper and lower canonical basis of $\ensuremath{\mathscr{H}}(W)$}
\label{ss upper and lower canonical basis of H(W)}
The \emph{bar-involution}, $\ensuremath{\flat}r{\cdot}$, of $\ensuremath{\mathscr{H}}(W)$ is the additive map from $\ensuremath{\mathscr{H}}(W)$ to itself extending the $\ensuremath{\flat}r{\cdot}$-involution of $\ensuremath{\mathfrak{m}}athbf{A}$ and satisfying $\ensuremath{\flat}r{T_w} = T_{w^{-1}}^{-1}$. Observe that $\ensuremath{\flat}r{T_{s}} = T_s^{-1} = T_s + \ensuremath{u^{-1}} - u$ for $s \in S$. Some simple $\ensuremath{\flat}r{\cdot}$-invariant elements of $\ensuremath{\mathscr{H}}(W)$ are $\ensuremath{C^{\prime}}_\ensuremath{\mathfrak{t}}ext{id} := T_\ensuremath{\mathfrak{t}}ext{id}$, $C_s := T_s - \ensuremath{u} = T_s^{-1} - \ensuremath{u^{-1}}$, and $\ensuremath{C^{\prime}}_s := T_s + \ensuremath{u^{-1}} = T_s^{-1} + u$, $s\in S$.
Define the lattices $(\ensuremath{\mathscr{H}}_r)_{\ensuremath{\mathbb{Z}}[\ensuremath{u}]} := \ensuremath{\mathbb{Z}}[\ensuremath{u}] \{ T_w : w \in W \}$ and $(\ensuremath{\mathscr{H}}_r)_{\ensuremath{\mathbb{Z}}[\ensuremath{u^{-1}}]} := \ensuremath{\mathbb{Z}}[\ensuremath{u^{-1}}] \{ T_w : w \in W \}$ of $\ensuremath{\mathscr{H}}_r$.
It is shown in \cite{KL} that
\ensuremath{\flat}egin{equation}
\ensuremath{\mathfrak{p}}arbox{13cm}{for each $w \in W$, there is a unique element $C_w \in \ensuremath{\mathscr{H}}(W)$ such that $\ensuremath{\flat}r{C_w} = C_w$ and $C_w$ is congruent to $T_w \ensuremath{\mathfrak{m}}od \ensuremath{u} (\ensuremath{\mathscr{H}}_r)_{\ensuremath{\mathbb{Z}}[\ensuremath{u}]}$.}
\end{equation}
The $\ensuremath{\mathfrak{m}}athbf{A}$-basis $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_W := \{C_w: w\in W\}$
is the \emph{upper canonical basis} of $\ensuremath{\mathscr{H}}(W)$ (we use this language to be consistent with that for crystal bases).
Similarly,
\ensuremath{\flat}egin{equation}
\ensuremath{\mathfrak{p}}arbox{13cm}{for each $w \in W$, there is a unique element $\ensuremath{C^{\prime}}_w \in \ensuremath{\mathscr{H}}(W)$ such that $\ensuremath{\flat}r{\ensuremath{C^{\prime}}_w} = \ensuremath{C^{\prime}}_w$ and $\ensuremath{C^{\prime}}_w$ is congruent to $T_w \ensuremath{\mathfrak{m}}od \ensuremath{u^{-1}} (\ensuremath{\mathscr{H}}_r)_{\ensuremath{\mathbb{Z}}[\ensuremath{u^{-1}}]}$.}
\end{equation}
The $\ensuremath{\mathfrak{m}}athbf{A}$-basis
$\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_W := \{\ensuremath{C^{\prime}}_w : w \in W \}$ is the \emph{lower canonical basis} of $\ensuremath{\mathscr{H}}(W)$.
The coefficients of the lower canonical basis in terms of the standard basis are the \emph{Kazhdan-Lusztig polynomials} $P'_{x,w}$:
\ensuremath{\flat}egin{equation} \ensuremath{C^{\prime}}_w = \sum_{x \in W} P'_{x,w} T_x. \end{equation}
(Our $P'_{x,w}$ are equal to $q^{(\ell(x)-\ell(w))/2}P_{x,w}$, where $P_{x,w}$ are the polynomials defined in \cite{KL} and $q^{1/2} = \ensuremath{u}$.)
Now let $\ensuremath{\mathfrak{m}}u(x,w) \in \ensuremath{\mathbb{Z}}$ be the coefficient of $\ensuremath{u^{-1}}$ in $P'_{x,w}$ (resp. $P'_{w,x}$) if $x \leq w$ (resp. $w \leq x$).
Then the right regular representation in terms of the canonical bases of $\ensuremath{\mathscr{H}}_r$ takes the following simple forms:
\ensuremath{\flat}egin{equation}gin{equation}\label{e prime C on prime canbas}
\ensuremath{C^{\prime}}_w \ensuremath{C^{\prime}}_s =
\left\{\ensuremath{\flat}egin{equation}gin{array}{ll} [2] \ensuremath{C^{\prime}}_w & \ensuremath{\mathfrak{t}}ext{if}\ s \in R(w),\\
\displaystyle\sum_{\substack{\{w' \in W: s \in R(w')\}}} \ensuremath{\mathfrak{m}}u(w',w)\ensuremath{C^{\prime}}_{w'} & \ensuremath{\mathfrak{t}}ext{if}\ s \notin R(w).
\end{array}\right.
\end{equation}
\ensuremath{\flat}egin{equation}gin{equation}\label{e C on canbas}
C_w C_s =
\left\{\ensuremath{\flat}egin{equation}gin{array}{ll} -[2] C_w & \ensuremath{\mathfrak{t}}ext{if}\ s \in R(w),\\
\displaystyle\sum_{\substack{\{w' \in W: s \in R(w')\}}} \ensuremath{\mathfrak{m}}u(w',w)C_{w'} & \ensuremath{\mathfrak{t}}ext{if}\ s \notin R(w).
\end{array}\right.
\end{equation}
The simplicity and sparsity of this action along with the fact that the right cells of $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_W$ and $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_W$ often give rise to $\ensuremath{\ensuremath{\mathfrak{m}}athbb{C}}(\ensuremath{u}) \ensuremath{\otimes}_\ensuremath{\mathfrak{m}}athbf{A} \ensuremath{\mathscr{H}}(W)$-irreducibles are among the most amazing and useful properties of canonical bases.
We will make use of the following positivity result due to Kazhdan-Lusztig and Beilinson-Bernstein-Deligne-Gabber (see, for instance, \cite{L2}).
\ensuremath{\flat}egin{equation}gin{theorem}
\label{t positive coefficients}
If $(W,S)$ is crystallographic, then the integers $\ensuremath{\mathfrak{m}}u(x,w)$ are nonnegative.
\end{theorem}
\subsection{Cells}
\label{ss cells}
We define cells in the general setting of modules with basis, as in \cite{BProjected} (this is similar to the notion of cells of Coxeter groups from \cite{KL}).
Let $H$ be an $R$-algebra for some commutative ring $R$.
Let $M$ be a right $H$-module and $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma$ an $R$-basis of $M$. The preorder $\ensuremath{\mathfrak{k}}lo{\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma}$ (also denoted $\ensuremath{\mathfrak{k}}lo{M}$) on the vertex set $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma$ is generated by the relations
\ensuremath{\flat}egin{equation}
\label{e preorder}
\delta\ensuremath{\mathfrak{k}}locov{\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma}\ensuremath{\ensuremath{\mathfrak{m}}athfrak{g}}amma \ensuremath{\flat}egin{equation}gin{array}{c}\ensuremath{\mathfrak{t}}ext{if there is an $h\in H$ such that $\delta$ appears with nonzero}\\ \ensuremath{\mathfrak{t}}ext{coefficient in the expansion of $\ensuremath{\ensuremath{\mathfrak{m}}athfrak{g}}amma h$
in the basis $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma$}. \end{array}
\end{equation}
Equivalence classes of $\ensuremath{\mathfrak{k}}lo{\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma}$ are the \emph{right cells} of $(M, \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma)$. The preorder $\ensuremath{\mathfrak{k}}lo{M}$ induces a partial order on the right cells of $M$, which is also denoted $\ensuremath{\mathfrak{k}}lo{M}$.
We say that the right cells $\ensuremath{\mathscr{L}}ambda$ and $\ensuremath{\mathscr{L}}ambda'$ are isomorphic if $(R \ensuremath{\mathscr{L}}ambda, \ensuremath{\mathscr{L}}ambda)$ and $(R \ensuremath{\mathscr{L}}ambda', \ensuremath{\mathscr{L}}ambda')$ are isomorphic as modules with basis.
Sometimes we speak of the right cells of $M$ or right cells of $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma$ if the pair $(M, \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma)$ is clear from context.
We also use the terminology \emph{right $H$-cells} when we want to make it clear that the algebra $H$ is acting.
\subsection{Cells and tableaux} \label{ss cell label conventions C_Q C'_Q}
Let $\ensuremath{\mathscr{H}}_r = \ensuremath{\mathscr{H}}(\ensuremath{\mathcal{S}}_r)$ be the type $A$ Hecke algebra. For the remainder of the paper, set $S := \{s_1,\ensuremath{\ensuremath{\mathfrak{t}}rianglelefteq}ots,s_{r-1}\}$ and $J := \{s_1,\ensuremath{\ensuremath{\mathfrak{t}}rianglelefteq}ots,s_{r-2}\}$.
It is well known that $\ensuremath{K} \ensuremath{\mathscr{H}}_r := \ensuremath{K} \ensuremath{\otimes}_\ensuremath{\mathfrak{m}}athbf{A} \ensuremath{\mathscr{H}}_r$ is semisimple and its irreducibles in bijection with partitions of $r$; let $M_\lambda$ and $M_\lambda^{\ensuremath{\mathfrak{m}}athbf{A}}$ be the $\ensuremath{K}\ensuremath{\mathscr{H}}_r$-irreducible and Specht module of $\ensuremath{\mathscr{H}}_r$ of shape $\lambda \vdash r$ (hence $M_\lambda \cong \ensuremath{K}\ensuremath{\otimes}_\ensuremath{\mathfrak{m}}athbf{A} M_\lambda^\ensuremath{\mathfrak{m}}athbf{A}$). For any $\ensuremath{K} \ensuremath{\mathscr{H}}_r$-module $N$ and partition $\lambda$ of $r$, let $p_{M_\lambda} : N \ensuremath{\mathfrak{t}}o N$ be the $\ensuremath{K} \ensuremath{\mathscr{H}}_r$-module projector with image the $M_\lambda$-isotypic component of $N$.
The work of Kazhdan and Lusztig \cite{KL} shows that the decomposition of $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_{\ensuremath{\mathcal{S}}_r}$ into right cells is
$\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_{\ensuremath{\mathcal{S}}_r} = \ensuremath{\flat}igsqcup_{\lambda \vdash r,\, P \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_P$, where $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_P := \{C_w : P(w) = P\}$. Moreover, the right cells $\{ \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_P : \ensuremath{\mathfrak{t}}ext{\rm sh}(P) = \lambda\}$ are all isomorphic, and, denoting any of these cells by $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\lambda$, $\ensuremath{\mathfrak{m}}athbf{A}\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\lambda \cong M_\lambda^\ensuremath{\mathfrak{m}}athbf{A}$. Similarly, the decomposition of $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{\ensuremath{\mathcal{S}}_r}$ into right cells is
$\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{\ensuremath{\mathcal{S}}_r} = \ensuremath{\flat}igsqcup_{\lambda \vdash r,\, P \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_P$, where $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_P := \{\ensuremath{C^{\prime}}_w : \ensuremath{\mathfrak{t}}ranspose{P(w)} = P\}$. Moreover, the right cells $\{ \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_P : \ensuremath{\mathfrak{t}}ext{\rm sh}(P) = \lambda\}$ are all isomorphic, and, denoting any of these cells by $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda$, $\ensuremath{\mathfrak{m}}athbf{A}\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda \cong M_\lambda^\ensuremath{\mathfrak{m}}athbf{A}$.
A combinatorial discussion of left cells in type $A$ is given in \cite[\ensuremath{\mathfrak{t}}extsection 4]{B0}.
We refer to the basis $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\lambda$ of $M_\lambda^\ensuremath{\mathfrak{m}}athbf{A}$ as the \emph{upper canonical basis of $M_\lambda$} and denote it by $\{ C_Q : Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda) \}$, where $C_Q$ corresponds to $C_w$ for any (every) $w \in \ensuremath{\mathcal{S}}_r$ with recording tableau $Q$. Similarly, the basis $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda$ of $M_\lambda^\ensuremath{\mathfrak{m}}athbf{A}$ is the \emph{lower canonical basis of $M_\lambda$}, denoted $\{ \ensuremath{C^{\prime}}_Q : Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda) \}$, where $\ensuremath{C^{\prime}}_Q$ corresponds to $\ensuremath{C^{\prime}}_w$ for any (every) $w \in \ensuremath{\mathcal{S}}_r$ with recording tableau $\ensuremath{\mathfrak{t}}ranspose{Q}$. Note that with these labels the action of $C_s$ on the upper canonical basis of $M_\lambda$ is similar to \eqref{e C on canbas}, with $\ensuremath{\mathfrak{m}}u(Q',Q):= \ensuremath{\mathfrak{m}}u(w',w)$ for any $w',w$ such that $P(w')=P(w)$, $Q' = Q(w'),Q = Q(w)$, and right descent sets
\ensuremath{\flat}egin{equation}
R(C_Q) = \{ s_i : i + 1 \ensuremath{\mathfrak{t}}ext{ is strictly to the south of $i$ in $Q$}\}.
\end{equation}
Similarly, the action of $\ensuremath{C^{\prime}}_s$ on $\{ \ensuremath{C^{\prime}}_Q : Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda) \}$ is similar to \eqref{e prime C on prime canbas}, with $\ensuremath{\mathfrak{m}}u(Q',Q):= \ensuremath{\mathfrak{m}}u(w',w)$ for any $w',w$ such that $\ensuremath{\mathfrak{t}}ranspose{P(w')}=\ensuremath{\mathfrak{t}}ranspose{P(w)}$, $Q' = \ensuremath{\mathfrak{t}}ranspose{Q(w')},Q = \ensuremath{\mathfrak{t}}ranspose{Q(w)}$, and right descent sets
\ensuremath{\flat}egin{equation}
R(\ensuremath{C^{\prime}}_Q) = \{ s_i : i + 1 \ensuremath{\mathfrak{t}}ext{ is strictly to the east of $i$ in $Q$}\}.
\end{equation}
See Figure \ref{f Wgraph} for a picture of $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_{(3,2)}$ and $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{(3,2)}$ and right descent sets.
\subsection{Dual equivalence graphs}
\label{ss dual equivalence}
To work with canonical bases in the proof of Theorem \ref{t nsH irreducibles two row case}, we make use the notion of dual equivalence graphs\ensuremath{[2]^2}ootnote{We use a slightly simplified version of the dual equivalence graphs from \cite{Sami}.} from \cite{Sami}.
Given $T, T' \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)$, we say that $T$ and $T'$ are related by a \emph{dual Knuth transformation} at $i$ if
\ensuremath{\flat}egin{equation}gin{list}{(\ensuremath{\mathfrak{a}}rabic{ctr})} {\ensuremath{u}secounter{ctr} \setlength{\itemsep}{1pt} \setlength{\ensuremath{\mathfrak{t}}opsep}{2pt}}
\item $|R(\ensuremath{C^{\prime}}_T) \cap \{ s_{i - 1}, s_i \}| = |R(\ensuremath{C^{\prime}}_{T'}) \cap \{ s_{i - 1}, s_i \}| = 1,$
\item $T'$ is obtained from $T$ by swapping the entries $i$ and $i+1$ in $T$ or by swapping the entries $i-1$ and $i$ in $T$.
\end{list}
If $T$ and $T'$ are related by a dual Knuth transformation at $i$, then we also say that there is a \emph{$\ensuremath{\mathfrak{t}}ext{DKT}_i$-edge} between $T$ and $T'$ and write $T \dkt{i} T'$. We write $T \dkt{} T'$ if $T \dkt{i} T'$ for some $i$, $2 \leq i \leq r-1$.
Define the \emph{dual equivalence graph (DE graph)} on $\ensuremath{\mathfrak{t}}ext{SYT}(\lambda)$ to be the graph with vertex set $\ensuremath{\mathfrak{t}}ext{SYT}(\lambda)$ and edges given by the $\ensuremath{\mathfrak{t}}ext{DKT}_i$-edges for all $i$, $2 \leq i \leq r-1$.
We will freely use the result from \cite{KL} that $T \dkt{} T'$ implies $\ensuremath{\mathfrak{m}}u(T',T)= \ensuremath{\mathfrak{m}}u(T,T') = 1$.
Note that this means that the $\ensuremath{\mathcal{S}}_r$-graph on $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda$ (or $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\lambda$) contains the underlying simple graph of the DE graph on $\ensuremath{\mathfrak{t}}ext{SYT}(\lambda)$ (compare Figures \ref{f Wgraph} and \ref{f DE graph}).
\setlength{\cellsize}{2.2ex}
\ensuremath{\flat}egin{equation}gin{figure}
\ensuremath{\flat}egin{equation}gin{tikzpicture}[xscale = 3]
\ensuremath{\mathfrak{t}}ikzstyle{vertex}=[inner sep=0pt, outer sep=5pt, fill = white]
\ensuremath{\mathfrak{t}}ikzstyle{edge} = [draw, thick, -,black]
\ensuremath{\mathfrak{t}}ikzstyle{LabelStyleB} = [text=black, anchor=north]
\ensuremath{\mathfrak{t}}ikzstyle{LabelStyleA} = [text=black, anchor=south]
\node[vertex] (h1) at (4,0) {${\small\ensuremath{\mathfrak{t}}ableau{1&2&3 \\ 4&5}}$};
\node[vertex] (h2) at (3,0) {${\small\ensuremath{\mathfrak{t}}ableau{1&2&4 \\ 3&5}}$};
\node[vertex] (h3) at (2,0) {${\small\ensuremath{\mathfrak{t}}ableau{1&3&4 \\ 2&5}}$};
\node[vertex] (h4) at (1,0) {${\small\ensuremath{\mathfrak{t}}ableau{1&3&5 \\ 2&4}}$};
\node[vertex] (h5) at (0,0) {${\small\ensuremath{\mathfrak{t}}ableau{1&2&5 \\ 3&4}}$};
\node[vertex] (ll) at (-.6,-.8) {{\ensuremath{[2]^2}ootnotesize$ R(\ensuremath{C^{\prime}}_Q) $:}};
\node[vertex] (l1) at (4,-.8) {{\ensuremath{[2]^2}ootnotesize$\{s_1,s_2,s_4\} $}};
\node[vertex] (l2) at (3,-.8) {{\ensuremath{[2]^2}ootnotesize$ \{s_1,s_3\} $}};
\node[vertex] (l3) at (2,-.8) {{\ensuremath{[2]^2}ootnotesize$ \{s_2,s_3\} $}};
\node[vertex] (l4) at (1,-.8) {{\ensuremath{[2]^2}ootnotesize$ \{s_2,s_4\} $}};
\node[vertex] (l5) at (0,-.8) {{\ensuremath{[2]^2}ootnotesize$ \{s_1,s_3,s_4\} $}};
\node[vertex] (ss) at (-.6, -1.3) {{\ensuremath{[2]^2}ootnotesize$ R(C_Q) $:}};
\node[vertex] (s1) at (4, -1.3) {{\ensuremath{[2]^2}ootnotesize$ \{s_3\} $}};
\node[vertex] (s2) at (3, -1.3) {{\ensuremath{[2]^2}ootnotesize$ \{s_2, s_4\} $}};
\node[vertex] (s3) at (2, -1.3) {{\ensuremath{[2]^2}ootnotesize$ \{s_1,s_4\} $}};
\node[vertex] (s4) at (1, -1.3) {{\ensuremath{[2]^2}ootnotesize$ \{s_1,s_3\} $}};
\node[vertex] (s5) at (0, -1.3) {{\ensuremath{[2]^2}ootnotesize$ \{s_2\} $}};
\draw[edge] (h1) to (h2);
\draw[edge] (h2) to (h3);
\draw[edge] (h3) to (h4);
\draw[edge] (h4) to (h5);
\draw[edge, bend right=70] (h1.90) to (h4.90);
\draw[edge, bend right=70] (h2.90) to (h5.90);
\end{tikzpicture}
\caption{The $\ensuremath{\mathcal{S}}_r$-graph on $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda$ and $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\lambda$. The presence (resp. absence) of an edge means that $\ensuremath{\mathfrak{m}}u(Q',Q) = \ensuremath{\mathfrak{m}}u(Q,Q')$ is 1 (resp. 0).}
\label{f Wgraph}
\end{figure}
\ensuremath{\flat}egin{equation}gin{figure}
\ensuremath{\flat}egin{equation}gin{tikzpicture}[xscale = 3]
\ensuremath{\mathfrak{t}}ikzstyle{vertex}=[inner sep=0pt, outer sep=5pt, fill = white]
\ensuremath{\mathfrak{t}}ikzstyle{edge} = [draw, thick, -,black]
\ensuremath{\mathfrak{t}}ikzstyle{LabelStyleB} = [text=black, anchor=north]
\ensuremath{\mathfrak{t}}ikzstyle{LabelStyleA} = [text=black, anchor=south]
\node[vertex] (h1) at (4,0) {${\small\ensuremath{\mathfrak{t}}ableau{1&2&3 \\ 4&5}}$};
\node[vertex] (h2) at (3,0) {${\small\ensuremath{\mathfrak{t}}ableau{1&2&4 \\ 3&5}}$};
\node[vertex] (h3) at (2,0) {${\small\ensuremath{\mathfrak{t}}ableau{1&3&4 \\ 2&5}}$};
\node[vertex] (h4) at (1,0) {${\small\ensuremath{\mathfrak{t}}ableau{1&3&5 \\ 2&4}}$};
\node[vertex] (h5) at (0,0) {${\small\ensuremath{\mathfrak{t}}ableau{1&2&5 \\ 3&4}}$};
\draw[edge] (h1.185) to node[LabelStyleB]{\small 3} (h2.-5);
\draw[edge] (h1.-185) to node[LabelStyleA]{\small 4} (h2.5);
\draw[edge] (h2) to node[LabelStyleB]{\small 2} (h3);
\draw[edge] (h3) to node[LabelStyleB]{\small 4} (h4);
\draw[edge] (h4.185) to node[LabelStyleB]{\small 2} (h5.-5);
\draw[edge] (h4.-185) to node[LabelStyleA]{\small 3} (h5.5);
\end{tikzpicture}
\caption{The DE graph on $\ensuremath{\mathfrak{t}}ext{SYT}((3,2))$.}
\label{f DE graph}
\end{figure}
It is easy to see that (with the help of Figure \ref{f dkt example})
\ensuremath{\flat}egin{equation} \label{e DKE complete graph}
\ensuremath{\mathfrak{p}}arbox{14cm}{for any distinct $i, j \in [k_\lambda]$, there exists at least one edge $T \dkt{r-1} T'$ in the DE graph on $\ensuremath{\mathfrak{t}}ext{SYT}(\lambda)$ with $T_{a_i} = r = T'_{a_j}$ and $T_{a_j} = r-1 = T'_{a_i}$.}
\end{equation}
Here $T_{a_i}$ denotes the entry of $T$ in the square $a_i$ (see \ensuremath{\mathfrak{t}}extsection\ref{ss type A combinatorics preliminaries}).
\ensuremath{\flat}egin{equation}gin{figure}[H]
\ensuremath{\flat}egin{equation}gin{tikzpicture}[xscale = 8.1, yscale = 2]
\ensuremath{\mathfrak{t}}ikzstyle{vertex}=[inner sep=0pt, outer sep=5pt, fill = white]
\ensuremath{\mathfrak{t}}ikzstyle{edge} = [draw, thick, -,black]
\ensuremath{\mathfrak{t}}ikzstyle{LabelStyleB} = [text=black, anchor=north]
\ensuremath{\mathfrak{t}}ikzstyle{LabelStyleA} = [text=black, anchor=south]
\ensuremath{\flat}reedte=17pt;
\node[vertex] (d1) at (0,0) {
\ensuremath{\mathfrak{t}}iny \Yboxdim20pt \ensuremath{\flat}egin{equation}gin{Young}\ & \ & \ & \ & \ & \ & \ & \ & \ \cr
\ & \ & \ & \ & \ & \ & \ \cr
\ & \ & \ & \ \cr
\ & \ & $r \ensuremath{\ensuremath{\mathfrak{t}}iny \hspace{-.36mm}-\hspace{-.7mm}} 2$ & $r\ensuremath{\ensuremath{\mathfrak{t}}iny \hspace{-.36mm}-\hspace{-.7mm}}1$ \cr
\ & $r$ \cr
\end{Young}};
\node[vertex] (d2) at (1,0){
\ensuremath{\mathfrak{t}}iny \ensuremath{\flat}egin{equation}gin{Young}\ & \ & \ & \ & \ & \ & \ & \ & \ \cr
\ & \ & \ & \ & \ & \ & \ \cr
\ & \ & \ & \ \cr
\ & \ & $r\ensuremath{\ensuremath{\mathfrak{t}}iny \hspace{-.36mm}-\hspace{-.7mm}}2$ & $r$ \cr
\ & $r \ensuremath{\ensuremath{\mathfrak{t}}iny \hspace{-.36mm}-\hspace{-.7mm}} 1$ \cr
\end{Young}};
\node[vertex] (label1) at (0,-1) {$T$};
\node[vertex] (label2) at (1,-1) {$T'$};
\draw[edge] (d1) to node[LabelStyleB]{\ensuremath{\mathfrak{t}}iny $r-1$} (d2);
\end{tikzpicture}
\caption{An edge of the DE graph on $\ensuremath{\mathfrak{t}}ext{SYT}(\lambda)$ as in \eqref{e DKE complete graph} for $i = 1, j = 2$.}
\label{f dkt example}
\end{figure}
\subsection{Projected canonical bases}
\label{ss lifts}
Here we recall some results from \cite{BProjected} about projecting the upper and lower canonical bases of $M_\lambda$ onto the $\ensuremath{K} \ensuremath{\mathscr{H}}_{r-1}$-irreducible isotypic components of $M_\lambda$. These results will make it fairly easy to work with these bases in the proof of Theorem \ref{t nsH irreducibles two row case}.
For any $L \subseteq S$, define $(\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}}_Q)^L$ to be the projection of $C_Q$ onto the irreducible $\ensuremath{K} \ensuremath{\mathscr{H}}_L$-module corresponding to the right cell of $\ensuremath{\mathscr{R}}es_{\ensuremath{K} \ensuremath{\mathscr{H}}_L} \ensuremath{K}\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_{\lambda}$ containing $C_Q$, where $\ensuremath{\mathscr{H}}_L$ denotes the parabolic sub-Hecke algebra of $\ensuremath{\mathscr{H}}_r$ with $\ensuremath{\mathfrak{m}}athbf{A}$-basis $\{T_w:\ w\in (S_r)_L\}$.
Define $(\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_Q)^L$ similarly. If $L = J := \{s_1, \dots, s_{r-2} \}$, then by \cite[\ensuremath{\mathfrak{t}}extsection 4]{B0}, $(\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}}_Q)^J$ (resp. $(\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_Q)^J$) is equal to $p_{M_\ensuremath{\mathfrak{m}}u}(C_Q)$ (resp. $p_{M_\ensuremath{\mathfrak{m}}u}(\ensuremath{C^{\prime}}_Q)$), where $\ensuremath{\mathfrak{m}}u = \ensuremath{\mathfrak{t}}ext{\rm sh}(Q|_{[r-1]})$ and $p_{M_\ensuremath{\mathfrak{m}}u}$ is defined in \ensuremath{\mathfrak{t}}extsection\ref{ss cell label conventions C_Q C'_Q}. Here, for a tableau $Q$ and set $Z \subseteq \ensuremath{\mathbb{Z}}$, $Q|_Z$ denotes the subtableau of $Q$ obtained by removing the entries not in $Z$.
Maintain the notation of \eqref{e ai definition} for the outer corners of $\lambda$.
Define a partial order $\ensuremath{\ensuremath{\mathfrak{t}}rianglelefteq}neq_r$ on SYT$(\lambda)$ by declaring $Q' \ensuremath{\ensuremath{\mathfrak{t}}rianglelefteq}neq_r Q$ whenever $\ensuremath{\mathfrak{t}}ext{\rm sh}(Q'|_{[r-1]}) \ensuremath{\ensuremath{\mathfrak{m}}athfrak{g}}dneq \ensuremath{\mathfrak{t}}ext{\rm sh}(Q|_{[r-1]})$.
Recall that $\ensuremath{K}_0$ (resp. $\ensuremath{K}_\infty$) is the subring of $\ensuremath{K}$ consisting of rational functions with no pole at $\ensuremath{u} = 0$ (resp. $\ensuremath{u} = \infty$).
\ensuremath{\flat}egin{equation}gin{lemma}[\cite{BProjected}\ensuremath{[2]^2}ootnote{Lemma 7.4 of \cite{BProjected} uses a different partial order, but the proof given for this lemma also works for the partial order $\ensuremath{\ensuremath{\mathfrak{t}}rianglelefteq}neq_r$ defined here.}] \label{l lift transition matrix}
The transition matrix expressing the projected basis $\{(\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}}_Q)^J : Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda) \}$ in terms of the upper canonical basis of $M_\lambda$ is lower-unitriangular and is the identity at $\ensuremath{u} = 0$ and $\ensuremath{u} = \infty$ (i.e. $(\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}}_Q)^J = C_Q + \sum_{Q' \ensuremath{\ensuremath{\mathfrak{m}}athfrak{g}}dneq_r Q} m_{Q' Q} C_{Q'}$, $m_{Q' Q} \in \ensuremath{u}\ensuremath{K}_0 \cap \ensuremath{u^{-1}} \ensuremath{K}_\infty$). The transition matrix expressing the projected basis $\{(\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_Q)^J : Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda) \}$ in terms of the lower canonical basis of $M_\lambda$ satisfies the same properties except is upper-unitriangular instead of lower-unitriangular (i.e. $(\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_Q)^J = \ensuremath{C^{\prime}}_Q + \sum_{Q' \ensuremath{\ensuremath{\mathfrak{t}}rianglelefteq}neq_r Q} m'_{Q' Q} \ensuremath{C^{\prime}}_{Q'}$ $m'_{Q' Q} \in \ensuremath{u}\ensuremath{K}_0 \cap \ensuremath{u^{-1}} \ensuremath{K}_\infty$).
\end{lemma}
By \cite[\ensuremath{\mathfrak{t}}extsection4]{B0}, the $\ensuremath{\mathscr{H}}_J$-module with basis $(\ensuremath{\mathscr{R}}es_{\ensuremath{\mathscr{H}}_J} M_\lambda, \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda)$ decomposes into right cells as
\[\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda = \ensuremath{\flat}igsqcup_{i \in [k_\lambda]} \{\ensuremath{C^{\prime}}_Q : \ensuremath{\mathfrak{t}}ext{\rm sh}(Q|_{[r-1]}) = \lambda - a_i\},\]
and moreover, $\{\ensuremath{C^{\prime}}_Q : \ensuremath{\mathfrak{t}}ext{\rm sh}(Q|_{[r-1]}) = \lambda - a_i\} \xrightarrow{\cong} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{\lambda-a_i}, \ \ensuremath{C^{\prime}}_Q \ensuremath{\mathfrak{m}}apsto \ensuremath{C^{\prime}}_{Q|_{[r-1]}}$ is an isomorphism of right $\ensuremath{\mathscr{H}}_J$-cells.
\ensuremath{\flat}egin{equation}gin{corollary}\label{c geck relative a invariant}
Let $\ensuremath{\mathfrak{k}}lo{\ensuremath{\mathscr{R}}es_{J} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda}$ be the partial order on the right cells of the $\ensuremath{\mathscr{H}}_J$-module with basis $(\ensuremath{\mathscr{R}}es_{\ensuremath{\mathscr{H}}_J} M_\lambda, \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda)$. This partial order is a total order with $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{\lambda - a_i} \ensuremath{\mathfrak{k}}lo{\ensuremath{\mathscr{R}}es_{J}\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{\lambda - a_j}$ exactly when $i \leq j$. Similarly, $(\ensuremath{\mathscr{R}}es_{\ensuremath{\mathscr{H}}_J} M_\lambda, \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\lambda)$ has a right cell isomorphic to $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_{\lambda-a_i}$ for each $i \in [k_\lambda]$ and the partial order $\ensuremath{\mathfrak{k}}lo{\ensuremath{\mathscr{R}}es_{J}\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\lambda}$ on right cells is a total order with $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_{\lambda - a_i} \ensuremath{\mathfrak{k}}lo{\ensuremath{\mathscr{R}}es_{J}\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\lambda} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_{\lambda - a_j}$ exactly when $i \ensuremath{\ensuremath{\mathfrak{m}}athfrak{g}}eq j$.
\end{corollary}
\ensuremath{\flat}egin{equation}gin{proof}
Lemma \ref{l lift transition matrix} shows that $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{\lambda - a_i} \ensuremath{\mathfrak{k}}lo{\ensuremath{\mathscr{R}}es_{J}\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{\lambda - a_j}$ implies $i \leq j$. To prove the converse, it suffices to show the existence of certain nonzero $\ensuremath{\mathfrak{m}}u(Q',Q)$. The $\ensuremath{\mathfrak{t}}ext{DKT}_i$-edges from \eqref{e DKE complete graph} suffice.
\end{proof}
We will also need the following theorem, one of the main results of \cite{BProjected}.
\ensuremath{\flat}egin{equation}gin{theorem}[\cite{BProjected}]
\label{t transition C' to C}
The transition matrix expressing the lower canonical basis $\{ \ensuremath{C^{\prime}}_Q : Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda) \}$ of $M_\lambda$ in terms of the upper canonical basis $\{ C_Q : Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda) \}$ of $M_\lambda$ has entries belonging to $\ensuremath{K}_0 \cap \ensuremath{K}_\infty$ and is the identity matrix at $\ensuremath{u} = 0$ and $\ensuremath{u} = \infty$.
\end{theorem}
See \cite[Example 7.5]{BProjected} for an example of this transition matrix. One consequence of this theorem is that $\ensuremath{K}_0 \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda = \ensuremath{K}_0 \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\lambda$. Let $\ensuremath{\mathscr{L}}_\lambda$ denote this $\ensuremath{K}_0$-lattice.
\ensuremath{\flat}egin{equation}gin{lemma}[The projection lemma]\label{l projections are not too tricky}
Fix $i \in [k_\lambda]$. Let $x = \sum_{Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)} a_Q \ensuremath{C^{\prime}}_Q$ be an element of $M_\lambda$ such that for each $Q$ with $\ensuremath{\mathfrak{t}}ext{\rm sh}(Q|_{[r-1]}) = \lambda-a_j$ and $j \ensuremath{\ensuremath{\mathfrak{m}}athfrak{g}}eq i$, there holds $a_Q \in \ensuremath{K}_0$.
Then
\[ p_{M_{\lambda-a_i}}(x) \equiv \sum_{\substack{ Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda), \\ Q_{a_i} = r}} a_Q (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_Q)^J \ensuremath{\mathfrak{m}}od \ensuremath{u} \ensuremath{\mathscr{L}}_{\lambda-a_i}. \]
Similarly, if $x = \sum_{Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)} a_Q C_Q$ is an element of $M_\lambda$ such that for each $Q$ with $\ensuremath{\mathfrak{t}}ext{\rm sh}(Q|_{[r-1]}) = \lambda-a_j$ and $j \leq i$, there holds $a_Q \in \ensuremath{K}_0$, then
\[ p_{M_{\lambda-a_i}}(x) \equiv \sum_{\substack{ Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda), \\ Q_{a_i} = r}} a_Q (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}}_Q)^J \ensuremath{\mathfrak{m}}od \ensuremath{u} \ensuremath{\mathscr{L}}_{\lambda-a_i}. \]
\end{lemma}
\ensuremath{\flat}egin{equation}gin{proof}
This follows easily from Lemma \ref{l lift transition matrix}.
\end{proof}
\section{The nonstandard Hecke algebra $\nsbr{\ensuremath{\mathscr{H}}}_r$}
\label{s the nonstandard Hecke algebra}
The nonstandard Hecke algebra was introduced in \cite{GCT4} to study the Kronecker problem. Its role in the Kronecker problem is discussed in \cite[\ensuremath{\mathfrak{t}}extsection1]{BMSGCT4} and \cite{canonical}; some of its representation theory is discussed in \cite[\ensuremath{\mathfrak{t}}extsection11]{BMSGCT4} and \cite{GCT7}, including a complete description $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{3}$ and $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{4}$-irreducibles; the problem of constructing a canonical basis for $\nsbr{\ensuremath{\mathscr{H}}}_r$ is discussed and \cite[\ensuremath{\mathfrak{t}}extsection19]{BMSGCT4} and \cite{canonical}. The main purpose of this paper is to determine the irreducibles of the nonstandard Temperley-Lieb algebra $ \ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$, which is a quotient algebra of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r}$. Here we assemble some basic facts about $\nsbr{\ensuremath{\mathscr{H}}}_r$ from \cite{Bnsbraid, GCT4, BMSGCT4} and prove a few new ones.
\subsection{Definition of $\nsbr{\ensuremath{\mathscr{H}}}_r$}
Recall that $S$ is now defined to be $\{s_1,\ensuremath{\ensuremath{\mathfrak{t}}rianglelefteq}ots,s_{r-1}\}$. We repeat the definition of $\nsbr{\ensuremath{\mathscr{H}}}_r$ from the introduction:
\ensuremath{\flat}egin{equation}gin{definition}
\label{d nonstandard Hecke algebra}
The \emph{type $A$ nonstandard Hecke algebra} $\nsbr{\ensuremath{\mathscr{H}}}_r$ is the subalgebra of $\ensuremath{\mathscr{H}}_r \ensuremath{\otimes}vw \ensuremath{\mathscr{H}}_r$ generated by the elements
\ensuremath{\flat}egin{equation} \label{e sP definition}
\mathcal{P}_s := \ensuremath{C^{\prime}}_s \ensuremath{\otimes}vw \ensuremath{C^{\prime}}_s + C_s \ensuremath{\otimes}vw C_s, \ s \in S.
\end{equation}
We let $\nsbr{\ensuremath{\ensuremath{\mathfrak{m}}athcal{D}}elta}:\nsbr{\ensuremath{\mathscr{H}}}_r \ensuremath{\ensuremath{\mathfrak{m}}athfrak{h}}ookrightarrow \ensuremath{\mathscr{H}}_r \ensuremath{\otimes}vw \ensuremath{\mathscr{H}}_r$ denote the canonical inclusion, which we think of as a deformation of the coproduct
$\ensuremath{\ensuremath{\mathfrak{m}}athcal{D}}elta_{\ensuremath{\mathbb{Z}} \ensuremath{\mathcal{S}}_r} :\ensuremath{\mathbb{Z}} \ensuremath{\mathcal{S}}_r \ensuremath{\mathfrak{t}}o \ensuremath{\mathbb{Z}} \ensuremath{\mathcal{S}}_r \ensuremath{\otimes}vw \ensuremath{\mathbb{Z}} \ensuremath{\mathcal{S}}_r$, $w \ensuremath{\mathfrak{m}}apsto w \ensuremath{\otimes}vw w$.
\end{definition}
The nonstandard Hecke algebra is also the subalgebra of $\ensuremath{\mathscr{H}}_r \ensuremath{\otimes}vw \ensuremath{\mathscr{H}}_r$ generated by
\[ \mathcal{Q}_s := [2]^2 - \mathcal{P}_s = - \ensuremath{C^{\prime}}_s \ensuremath{\otimes}vw C_s - C_s \ensuremath{\otimes}vw \ensuremath{C^{\prime}}_s, \ s \in S. \]
We will write $\mathcal{P}_{i}$ (resp. $\mathcal{Q}_i$) as shorthand for $\mathcal{P}_{s_{i}}$ (resp. $\mathcal{Q}_{s_i}$), $i \in [r-1]$.
For a ring homomorphism $\ensuremath{K} \ensuremath{\mathfrak{t}}o \ensuremath{\mathfrak{m}}athbf{A}$, we have the specialization $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_r := \ensuremath{K} \ensuremath{\otimes}_{\ensuremath{\mathfrak{m}}athbf{A}} \nsbr{\ensuremath{\mathscr{H}}}_r$ of the nonstandard Hecke algebra.
The elements $\mathcal{P}_i$ and $\mathcal{Q}_i$ satisfy the quadratic relations $\mathcal{P}_i^2 = [2]^2 \mathcal{P}_i$ and $\mathcal{Q}_i^2 = \ensuremath{[2]^2} \mathcal{Q}_i$, and $\mathcal{P}_i$ and $\mathcal{P}_{i+1}$ satisfy a nonstandard version of the braid relation (see \cite{Bnsbraid}). For $r \ensuremath{\ensuremath{\mathfrak{m}}athfrak{g}}eq 4$, the $\mathcal{P}_i$ satisfy additional relations which seem to be extremely difficult to describe (see \cite{GCT4}).
\subsection{Representation theory of $S^2\ensuremath{\mathscr{H}}_r$}
\label{ss representation theory of S2Hr}
The representations of $\nsbr{\ensuremath{\mathscr{H}}}_r$ are related to those of $S^2 \ensuremath{\mathscr{H}}_r$ by the fact that $\nsbr{\ensuremath{\mathscr{H}}}_r \subseteq S^2 \ensuremath{\mathscr{H}}_r$ (see, e.g., \cite[Proposition 11.6]{BMSGCT4}), so any $S^2 \ensuremath{\mathscr{H}}_r$-module is an $\nsbr{\ensuremath{\mathscr{H}}}_r$-module by restriction.
We recall the description of the $\ensuremath{K} S^2\ensuremath{\mathscr{H}}_r$-irreducibles from \cite{BMSGCT4}. These irreducibles are close to those of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_r$, and even closer to
those of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$, which will be described in \ensuremath{\mathfrak{t}}extsection\ref{s irreducibles of nshr2}.
First note that we have the following commutativity property for any $\ensuremath{\mathscr{H}}_r$-modules $M$ and $M'$:
\ensuremath{\flat}egin{equation}
\label{e commutativity S2H}
\ensuremath{\mathscr{R}}es_{S^2\ensuremath{\mathscr{H}}_r} M \ensuremath{\otimes}vw M' \cong \ensuremath{\mathscr{R}}es_{S^2\ensuremath{\mathscr{H}}_r} M' \ensuremath{\otimes}vw M,
\end{equation}
where the isomorphism is given by the flip $\ensuremath{\mathfrak{t}}au$, $\ensuremath{\mathfrak{t}}au(a \ensuremath{\otimes} b) = b \ensuremath{\otimes} a$.
Recall from \ensuremath{\mathfrak{t}}extsection\ref{ss type A combinatorics preliminaries} that $\ensuremath{\mathfrak{m}}athscr{P}_{r}$ denotes the set of partitions of size $r$ and $\ensuremath{\mathfrak{m}}athscr{P}'_{r}$ the set of partitions of $r$ that are not a single row or column shape.
\ensuremath{\flat}egin{equation}gin{propdef}[\cite{BMSGCT4}]
\label{p S2Hr representations}
Define the following $S^2 \ensuremath{\mathscr{H}}_r$-modules. After tensoring these with $\ensuremath{K}$, this is the list of distinct $\ensuremath{K} S^2\ensuremath{\mathscr{H}}_r$-irreducibles
\ensuremath{\flat}egin{equation}gin{list}{\emph{(\ensuremath{\mathfrak{a}}rabic{ctr})}} {\ensuremath{u}secounter{ctr} \setlength{\itemsep}{1pt} \setlength{\ensuremath{\mathfrak{t}}opsep}{2pt}}
\item $M^\ensuremath{\mathfrak{m}}athbf{A}_{\{\lambda,\ensuremath{\mathfrak{m}}u\}} := \ensuremath{\mathscr{R}}es_{S^2\ensuremath{\mathscr{H}}_r} M^\ensuremath{\mathfrak{m}}athbf{A}_\lambda \ensuremath{\otimes}vw M^\ensuremath{\mathfrak{m}}athbf{A}_\ensuremath{\mathfrak{m}}u$, $\{\lambda,\ensuremath{\mathfrak{m}}u\}\subseteq \ensuremath{\mathfrak{m}}athscr{P}_r$, $\lambda\neq\ensuremath{\mathfrak{m}}u$,
\item $S^2 M^\ensuremath{\mathfrak{m}}athbf{A}_\lambda := \ensuremath{\mathscr{R}}es_{ S^2\ensuremath{\mathscr{H}}_r} S^2 M^\ensuremath{\mathfrak{m}}athbf{A}_\lambda$, $\lambda\in\ensuremath{\mathfrak{m}}athscr{P}_r$,
\item $\ensuremath{\ensuremath{\mathscr{O}}mega}edge^2 M^\ensuremath{\mathfrak{m}}athbf{A}_\lambda := \ensuremath{\mathscr{R}}es_{ S^2\ensuremath{\mathscr{H}}_r} \ensuremath{\ensuremath{\mathscr{O}}mega}edge^2 M^\ensuremath{\mathfrak{m}}athbf{A}_\lambda$, $\lambda\in\ensuremath{\mathfrak{m}}athscr{P}'_r$.
\end{list}
Let $ M_{\{\lambda,\ensuremath{\mathfrak{m}}u\}}$, $ S^2 M_\lambda$,
$ \ensuremath{\ensuremath{\mathscr{O}}mega}edge^2 M_\lambda$ denote the corresponding $\ensuremath{K} S^2 \ensuremath{\mathscr{H}}_r$-modules.
\end{propdef}
\subsection{Contragradients of $\ensuremath{\mathscr{H}}_r$-modules}
Any anti-automorphism $S$ of an $\ensuremath{\mathfrak{m}}athbf{A}$-algebra $H$ allows us to define contragradients of $H$-modules: let $\langle \cdot, \cdot \rangle: M \ensuremath{\otimes} M^* \ensuremath{\mathfrak{t}}o \ensuremath{\mathfrak{m}}athbf{A}$ be the canonical pairing, where $M^*$ is the $\ensuremath{\mathfrak{m}}athbf{A}$-module $\ensuremath{\ensuremath{\mathfrak{m}}athfrak{h}}om_{\ensuremath{\mathfrak{m}}athbf{A}}(M,\ensuremath{\mathfrak{m}}athbf{A})$. Then the $H$-module structure on $M^*$ is defined by
\[ \langle m, m'h\rangle = \langle m S(h),m' \rangle \ensuremath{\mathfrak{t}}ext{ for any } h \in H,\ m \in M, m' \in M^*.\]
There is an $\ensuremath{\mathfrak{m}}athbf{A}$-algebra automorphism $\ensuremath{\mathfrak{t}}heta : \ensuremath{\mathscr{H}}_r \ensuremath{\mathfrak{t}}o \ensuremath{\mathscr{H}}_r$ defined by $\ensuremath{\mathfrak{t}}heta(T_s) = - T_s^{-1},\ s \in S$.
It is not hard to show that $\ensuremath{\mathfrak{t}}heta$ is an involution and satisfies
$\ensuremath{\mathfrak{t}}heta(\ensuremath{C^{\prime}}_w) = (-1)^{\ell(w)} C_w.$
Let $1^{\ensuremath{\mathfrak{t}}ext{op}}$ be the $\ensuremath{\mathfrak{m}}athbf{A}$-anti-automorphism of $\ensuremath{\mathscr{H}}_r$ given by $1^{\ensuremath{\mathfrak{t}}ext{op}}(T_w) = T_{w^{-1}}$.
Let $\ensuremath{\mathfrak{t}}heta^{\ensuremath{\mathfrak{t}}ext{op}}$ be the $\ensuremath{\mathfrak{m}}athbf{A}$-anti-automorphism of $\ensuremath{\mathscr{H}}_r$ given by $\ensuremath{\mathfrak{t}}heta^{\ensuremath{\mathfrak{t}}ext{op}} = \ensuremath{\mathfrak{t}}heta \circ 1^{\ensuremath{\mathfrak{t}}ext{op}} = 1^{\ensuremath{\mathfrak{t}}ext{op}} \circ \ensuremath{\mathfrak{t}}heta$.
Let $\{ \dual{\ensuremath{C^{\prime}}_w} : w \in \ensuremath{\mathcal{S}}_r\} \subseteq \ensuremath{\ensuremath{\mathfrak{m}}athfrak{h}}om_{\ensuremath{\mathfrak{m}}athbf{A}}(\ensuremath{\mathscr{H}}_r,\ensuremath{\mathfrak{m}}athbf{A})$ be the basis dual to $\{\ensuremath{C^{\prime}}_w: w \in \ensuremath{\mathcal{S}}_r\}$. Let $w_0$ be the longest element of $\ensuremath{\mathcal{S}}_r$.
Let $Z_\lambda^*$ be the SYT of shape $\lambda$ with $1, \ensuremath{\ensuremath{\mathfrak{t}}rianglelefteq}ots, \lambda_1$ in the first row, $\lambda_1+1,\ensuremath{\ensuremath{\mathfrak{t}}rianglelefteq}ots,\lambda_1+\lambda_2$ in the second row, etc.
For an SYT $Q$, let $\ell(Q)$ denote the distance between $Q$ and $Z_\lambda^*$ in the DE graph on SYT$(\lambda)$.
It is not hard to show that for any $P \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)$, $\ell(Q) \equiv \ell(w)-\ell(z)\ensuremath{\mathfrak{m}}od 2$, where $w = \ensuremath{\mathfrak{t}}ext{RSK}^{-1}(P,Q),$ $z = \ensuremath{\mathfrak{t}}ext{RSK}^{-1}(P,Z_\lambda^*)$.
\ensuremath{\flat}egin{equation}gin{proposition}\label{p dual basis to C'}
\ensuremath{\flat}egin{equation}gin{list}{\emph{(\roman{ctr})}} {\ensuremath{u}secounter{ctr} \setlength{\itemsep}{1pt} \setlength{\ensuremath{\mathfrak{t}}opsep}{2pt}}
\item The right $\ensuremath{\mathscr{H}}_r$-modules $\ensuremath{\mathscr{H}}_r^\ensuremath{\diamond}$ and $\ensuremath{\mathscr{H}}_r$ are isomorphic via
\[\ensuremath{\mathfrak{a}}lpha_\ensuremath{\diamond}: \ensuremath{\mathscr{H}}_r^\ensuremath{\diamond} \xrightarrow{\cong} \ensuremath{\mathscr{H}}_r, \ \dual{\ensuremath{C^{\prime}}_w} \ensuremath{\mathfrak{m}}apsto C_{w_0 w}, \ w \in \ensuremath{\mathcal{S}}_r. \]
\item The right $\ensuremath{\mathscr{H}}_r$-modules $\ensuremath{\mathscr{H}}_r^\ensuremath{\#}$ and $\ensuremath{\mathscr{H}}_r$ are isomorphic via
\[\ensuremath{\mathfrak{a}}lpha_\ensuremath{\#}: \ensuremath{\mathscr{H}}_r^\ensuremath{\#} \xrightarrow{\cong} \ensuremath{\mathscr{H}}_r, \ \dual{\ensuremath{C^{\prime}}_w} \ensuremath{\mathfrak{m}}apsto (-1)^{\ell(w)} \ensuremath{C^{\prime}}_{w_0 w},\ w \in \ensuremath{\mathcal{S}}_r. \]
\item The restriction of $\ensuremath{\mathfrak{a}}lpha_\ensuremath{\diamond}^{-1}$ to any right cell $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\lambda$ of $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_{\ensuremath{\mathcal{S}}_r}$ yields the isomorphism
\[ M^\ensuremath{\mathfrak{m}}athbf{A}_\lambda \xrightarrow{\cong} (M^\ensuremath{\mathfrak{m}}athbf{A}_{\lambda})^\ensuremath{\diamond}, \ C_{Q} \ensuremath{\mathfrak{m}}apsto \dual{\ensuremath{C^{\prime}}_{Q}}, \ Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda). \]
\item The restriction of $\ensuremath{\mathfrak{a}}lpha_\ensuremath{\#}^{-1}$ to any right cell $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda$ of $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{\ensuremath{\mathcal{S}}_r}$ yields, up to a sign, the isomorphism
\[ M^\ensuremath{\mathfrak{m}}athbf{A}_{\lambda'} \xrightarrow{\cong} (M^\ensuremath{\mathfrak{m}}athbf{A}_{\lambda})^\ensuremath{\#}, \ (-1)^{\ell(\ensuremath{\mathfrak{t}}ranspose{Q})}\ensuremath{C^{\prime}}_{Q} \ensuremath{\mathfrak{m}}apsto \dual{\ensuremath{C^{\prime}}_{\ensuremath{\mathfrak{t}}ranspose{Q}}}, \ Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda'). \]
\end{list}
\end{proposition}
\ensuremath{\flat}egin{equation}gin{proof}
We first record the following formulae which are immediate from (\ref{e prime C on prime canbas}), (\ref{e C on canbas}), and $R(w_0 w) = S \ensuremath{\flat}ackslash R(w)$.
\ensuremath{\flat}egin{equation}gin{align}\label{e prime C on unprime canbas}
C_{w_0 w} \ensuremath{C^{\prime}}_{s} =&
\left\{ \ensuremath{\flat}egin{equation}gin{array}{ll}
[2] C_{w_0 w} + \sum_{\substack{\{w_0 w' \in \ensuremath{\mathcal{S}}_r : s \notin R(w')\}}} \ensuremath{\mathfrak{m}}u(w_0 w',w_0 w) C_{w_0 w'} & \ensuremath{\mathfrak{t}}ext{if}\ s \in R(w),\\
0 & \ensuremath{\mathfrak{t}}ext{if}\ s \notin R(w).
\end{array} \right.\\
\label{e theta prime C on prime canbas}
\ensuremath{C^{\prime}}_w \ensuremath{\mathfrak{t}}heta(\ensuremath{C^{\prime}}_{s})=&
\left\{ \ensuremath{\flat}egin{equation}gin{array}{ll} 0 & \ensuremath{\mathfrak{t}}ext{if}\ s \in R(w),\\
[2] \ensuremath{C^{\prime}}_w - \sum_{\substack{\{w' \in \ensuremath{\mathcal{S}}_r : s \in R(w')\}}} \ensuremath{\mathfrak{m}}u(w',w)\ensuremath{C^{\prime}}_{w'} & \ensuremath{\mathfrak{t}}ext{if}\ s \notin R(w).
\end{array} \right.
\end{align}
By the definition of $\ensuremath{\mathscr{H}}_r^\ensuremath{\diamond}$,
\ensuremath{\flat}egin{equation}gin{equation}
\dual{\ensuremath{C^{\prime}}_w} \ensuremath{C^{\prime}}_{s}=
\left\{\ensuremath{\flat}egin{equation}gin{array}{ll}
[2] \dual{\ensuremath{C^{\prime}}_w} + \sum_{\substack{\{w' \in \ensuremath{\mathcal{S}}_r : s \notin R(w')\}}} \ensuremath{\mathfrak{m}}u(w,w') \dual{\ensuremath{C^{\prime}}_{w'}} & \ensuremath{\mathfrak{t}}ext{if}\ s \in R(w), \\
0 & \ensuremath{\mathfrak{t}}ext{if}\ s \notin R(w).
\end{array}\right.
\end{equation}
Statement (i) then follows from (\ref{e prime C on unprime canbas}) as $\ensuremath{\mathfrak{m}}u(w,w') = \ensuremath{\mathfrak{m}}u(w_0 w',w_0 w)$ \cite[Corollary 3.2]{KL}.
By the definition of $\ensuremath{\mathscr{H}}_r^\ensuremath{\#}$ and from (\ref{e theta prime C on prime canbas}), we obtain
\ensuremath{\flat}egin{equation}gin{equation}
\dual{\ensuremath{C^{\prime}}_w} \ensuremath{C^{\prime}}_{s}=
\left\{\ensuremath{\flat}egin{equation}gin{array}{ll}
[2] \dual{\ensuremath{C^{\prime}}_w} & \ensuremath{\mathfrak{t}}ext{if}\ s \notin R(w),\\
-\sum_{\substack{\{w' \in \ensuremath{\mathcal{S}}_r : s \notin R(w')\}}} \ensuremath{\mathfrak{m}}u(w,w') \dual{\ensuremath{C^{\prime}}_{w'}} & \ensuremath{\mathfrak{t}}ext{if}\ s \in R(w).
\end{array}\right.
\end{equation}
Statement (ii) then follows from \eqref{e prime C on prime canbas} using $R(w_0 w) = S \ensuremath{\flat}ackslash R(w)$, $\ensuremath{\mathfrak{m}}u(w,w') = \ensuremath{\mathfrak{m}}u(w_0 w',w_0 w)$, and the fact that $\ensuremath{\mathfrak{m}}u(w',w) \neq 0$ implies $(-1)^{\ell(w')} = - (-1)^{\ell(w)}$.
Statements (iii) and (iv) then follow from (i) and (ii), respectively, the fact that $Q(w_0 w) = \ensuremath{\mathfrak{t}}ranspose{Q(w)}$ (see, e.g., \cite[A1.2]{F}), and the definitions in \ensuremath{\mathfrak{t}}extsection\ref{ss cell label conventions C_Q C'_Q}.
\end{proof}
As discussed in \cite{Bnsbraid, BMSGCT4}, the inclusion $\nsbr{\ensuremath{\ensuremath{\mathfrak{m}}athcal{D}}elta}:\nsbr{\ensuremath{\mathscr{H}}}_r \ensuremath{\ensuremath{\mathfrak{m}}athfrak{h}}ookrightarrow \ensuremath{\mathscr{H}}_r \ensuremath{\otimes}vw \ensuremath{\mathscr{H}}_r$ is not a good approximation of the coproduct $\ensuremath{\ensuremath{\mathfrak{m}}athcal{D}}elta_{\ensuremath{\mathbb{Z}} \ensuremath{\mathcal{S}}_r}$, though it is in a certain sense the closest approximation possible. There are a couple ways that $\nsbr{\ensuremath{\mathscr{H}}}_r$ behaves like a Hopf algebra, one of which is the following.
\ensuremath{\flat}egin{equation}gin{proposition}[\cite{Bnsbraid}] \label{p hecke algebra antipode}
The involutions $1^{\ensuremath{\mathfrak{t}}ext{op}}$ and $\ensuremath{\mathfrak{t}}heta^{\ensuremath{\mathfrak{t}}ext{op}}$ are antipodes in the following sense:
\ensuremath{\flat}egin{equation}gin{flalign*}
\ensuremath{\mathfrak{m}}u \circ (1^{\ensuremath{\mathfrak{t}}ext{op}} \ensuremath{\otimes} 1) \circ \nsbr{\ensuremath{\ensuremath{\mathfrak{m}}athcal{D}}elta} &= \eta \circ \nsbr{\epsilon}_+, \\
\ensuremath{\mathfrak{m}}u \circ (\ensuremath{\mathfrak{t}}heta^{\ensuremath{\mathfrak{t}}ext{op}} \ensuremath{\otimes} 1) \circ \nsbr{\ensuremath{\ensuremath{\mathfrak{m}}athcal{D}}elta} &= \eta \circ \nsbr{\epsilon}_-,
\end{flalign*}
where these are equalities of maps from $\nsbr{\ensuremath{\mathscr{H}}}_r$ to $\ensuremath{\mathscr{H}}_r$. Here $\ensuremath{\mathfrak{m}}u$ is the multiplication map for $\ensuremath{\mathscr{H}}_r$ and $\eta : \ensuremath{K} \ensuremath{\mathfrak{t}}o \ensuremath{\mathscr{H}}_r$ is the unit of $\ensuremath{\mathscr{H}}_r$.
\end{proposition}
\subsection{Some representation theory of $\nsbr{\ensuremath{\mathscr{H}}}_r$}
\label{ss some representation theory of nsH}
It is shown in \cite{BMSGCT4} (Proposition 11.8) that $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_r$ is semisimple.
\ensuremath{\flat}egin{equation}gin{remark}
It is reasonable to suspect that $\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_r$ is split semisimple, and indeed, our computations are consistent with this being true. In this paper we show that the nonstandard Temperley-Lieb algebra $\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r,2}$ is split semisimple by explicitly determining its irreducibles. We are curious if there is a way to show that $ \ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r}$ is split semisimple without explicitly determining its irreducibles.
\end{remark}
There are one-dimensional trivial and sign representations of $\nsbr{\ensuremath{\mathscr{H}}}_r$, which we denote by $\nsbr{\epsilon}_{+}$ and $\nsbr{\epsilon}_{-}$:
\[ \ensuremath{\flat}egin{equation}gin{array}{cccc}
\nsbr{\epsilon}_{+} : \mathcal{P}_{s} \ensuremath{\mathfrak{m}}apsto [2]^{2}, && \nsbr{\epsilon}_{-} : \mathcal{P}_s \ensuremath{\mathfrak{m}}apsto 0, & s \in S.
\end{array} \]
For $\lambda, \ensuremath{\mathfrak{m}}u \vdash r$, the $\nsbr{\ensuremath{\mathscr{H}}}_r$-module $\ensuremath{\mathscr{R}}es_{\nsbr{\ensuremath{\mathscr{H}}}_r} M^{\ensuremath{\mathfrak{m}}athbf{A}}_\lambda \ensuremath{\otimes} M^{\ensuremath{\mathfrak{m}}athbf{A}}_\ensuremath{\mathfrak{m}}u \cong \ensuremath{\mathscr{R}}es_{\nsbr{\ensuremath{\mathscr{H}}}_r} M^{\ensuremath{\mathfrak{m}}athbf{A}}_\ensuremath{\mathfrak{m}}u \ensuremath{\otimes} M^{\ensuremath{\mathfrak{m}}athbf{A}}_\lambda$ is denoted\ensuremath{[2]^2}ootnote{The more correct notation $\nsbr{M}^{\ensuremath{\mathfrak{m}}athbf{A}}_{\{\lambda, \ensuremath{\mathfrak{m}}u\}}$ is used in \cite{BMSGCT4}, but in this paper the shorter $\nsbr{M}^{\ensuremath{\mathfrak{m}}athbf{A}}_{\lambda, \ensuremath{\mathfrak{m}}u}$ is preferable for being less cumbersome.} $\nsbr{M}^{\ensuremath{\mathfrak{m}}athbf{A}}_{\lambda, \ensuremath{\mathfrak{m}}u}$. Let $\nssym{2}{M}^{\ensuremath{\mathfrak{m}}athbf{A}}_\lambda$ (resp. $\nswedge{2}{M}^{\ensuremath{\mathfrak{m}}athbf{A}}_\lambda$) denote the $\nsbr{\ensuremath{\mathscr{H}}}_r$-module $\ensuremath{\mathscr{R}}es_{\nsbr{\ensuremath{\mathscr{H}}}_r} S^2 M^{\ensuremath{\mathfrak{m}}athbf{A}}_\lambda$ (resp. $\ensuremath{\mathscr{R}}es_{\nsbr{\ensuremath{\mathscr{H}}}_r} \ensuremath{\ensuremath{\mathscr{O}}mega}edge^2 M^{\ensuremath{\mathfrak{m}}athbf{A}}_\lambda$), where $S^2 M^{\ensuremath{\mathfrak{m}}athbf{A}}_\lambda$ and $\ensuremath{\ensuremath{\mathscr{O}}mega}edge^2 M^{\ensuremath{\mathfrak{m}}athbf{A}}_\lambda$ are as in Proposition-Definition \ref{p S2Hr representations}.
Let $\nsbr{M}_{\lambda,\ensuremath{\mathfrak{m}}u}$, $\nssym{2}{M}_\lambda, \nswedge{2}{M}_\lambda$ denote the corresponding $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_r$-modules.
Let
\ensuremath{\flat}egin{equation} \label{e triv inclusion map}
\ensuremath{\mathfrak{m}}athbf{A} \xrightarrow{I} (M^\ensuremath{\mathfrak{m}}athbf{A}_{\lambda})^\ensuremath{\diamond} \ensuremath{\otimes}vw M^\ensuremath{\mathfrak{m}}athbf{A}_{\lambda}
\end{equation}
be the canonical inclusion given by sending $1 \in \ensuremath{\mathfrak{m}}athbf{A}$ to $I \in \ensuremath{\ensuremath{\mathfrak{m}}athscr{E}}nd(M^\ensuremath{\mathfrak{m}}athbf{A}_\lambda) \cong (M^\ensuremath{\mathfrak{m}}athbf{A}_{\lambda})^\ensuremath{\diamond} \ensuremath{\otimes}vw M^\ensuremath{\mathfrak{m}}athbf{A}_{\lambda}$.
Let
\ensuremath{\flat}egin{equation} \label{e triv surjection map}
M^\ensuremath{\mathfrak{m}}athbf{A}_{\lambda} \ensuremath{\otimes}vw (M^\ensuremath{\mathfrak{m}}athbf{A}_{\lambda})^\ensuremath{\diamond} \xrightarrow{\ensuremath{\mathfrak{t}}race} \ensuremath{\mathfrak{m}}athbf{A}
\end{equation}
be the canonical surjection.
We then have the following $\nsbr{\ensuremath{\mathscr{H}}}_r$-module homomorphisms
\ensuremath{\flat}egin{equation}gin{align*}
\nsbr{\epsilon}_+ &\xrightarrow{I} (M^\ensuremath{\mathfrak{m}}athbf{A}_{\lambda})^\ensuremath{\diamond} \ensuremath{\otimes}vw M^\ensuremath{\mathfrak{m}}athbf{A}_{\lambda}, \\
\ensuremath{\mathfrak{k}}er(\ensuremath{\mathfrak{t}}race) &\ensuremath{\ensuremath{\mathfrak{m}}athfrak{h}}ookrightarrow M^\ensuremath{\mathfrak{m}}athbf{A}_{\lambda} \ensuremath{\otimes}vw (M^\ensuremath{\mathfrak{m}}athbf{A}_{\lambda})^\ensuremath{\diamond} \xrightarrow{\ensuremath{\mathfrak{t}}race} \nsbr{\epsilon}_+,
\end{align*}
To see this, note that in general, if $M$ is an $H$-module and $H$ is a Hopf algebra with counit $\epsilon$, then it follows from the axiom for the antipode that $\epsilon \xrightarrow{I} M^* \ensuremath{\otimes} M$ and $M \ensuremath{\otimes} M^* \xrightarrow{\ensuremath{\mathfrak{t}}race} \epsilon$ are $H$-module homomorphisms.
The same proof works in the present setting using Proposition \ref{p hecke algebra antipode} in place of the antipode axiom.
Since $\ensuremath{[2]^2}rac{1}{|\ensuremath{\mathfrak{t}}ext{SYT}(\lambda)|} \ensuremath{\mathfrak{t}}au \circ I$ is a splitting of $\ensuremath{\mathfrak{t}}race$ and $(M^\ensuremath{\mathfrak{m}}athbf{A}_{\lambda})^\ensuremath{\diamond} \cong M^\ensuremath{\mathfrak{m}}athbf{A}_{\lambda}$ (Proposition \ref{p dual basis to C'} (iii)), we obtain the decomposition of $\nsbr{\ensuremath{\mathscr{H}}}_r$-modules
\ensuremath{\flat}egin{equation} \label{e nsH trace}
\ensuremath{\mathfrak{k}}er(\ensuremath{\mathfrak{t}}race) \oplus \nsbr{\epsilon}_+ \cong \nsbr{M}^\ensuremath{\mathfrak{m}}athbf{A}_{\lambda,\lambda}.
\end{equation}
Moreover, as a consequence of Proposition \ref{p triv in Mdual M} (i) below, $\nsbr{\epsilon}_+ \subseteq \nsbr{M}^\ensuremath{\mathfrak{m}}athbf{A}_{\lambda, \lambda}$ lies in
$\nssym{2}{M}_\lambda^\ensuremath{\mathfrak{m}}athbf{A}$.
Then define $S' \nsbr{M}^\ensuremath{\mathfrak{m}}athbf{A}_\lambda := \ensuremath{\mathfrak{k}}er(\ensuremath{\mathfrak{t}}race) \cap \nssym{2}{M}^\ensuremath{\mathfrak{m}}athbf{A}_\lambda$. The decomposition \eqref{e nsH trace} yields the decomposition
\ensuremath{\flat}egin{equation} \label{e nsH S' iso}
\nssym{2}{M}^\ensuremath{\mathfrak{m}}athbf{A}_\lambda \cong S' \nsbr{M}^\ensuremath{\mathfrak{m}}athbf{A}_\lambda \oplus \nsbr{\epsilon}_+.
\end{equation}
\ensuremath{\flat}egin{equation}gin{proposition}
\label{p triv in Mdual M}
The maps \eqref{e triv inclusion map} and \eqref{e triv surjection map} as well as the analogous maps for $ \nsbr{\epsilon}_-$ can be made explicit using upper and lower canonical bases:
\ensuremath{\flat}egin{equation}gin{list}{\emph{(\roman{ctr})}} {\ensuremath{u}secounter{ctr} \setlength{\itemsep}{1pt} \setlength{\ensuremath{\mathfrak{t}}opsep}{2pt}}
\item The inclusion $\nsbr{\epsilon}_+ \ensuremath{\ensuremath{\mathfrak{m}}athfrak{h}}ookrightarrow M_\lambda^\ensuremath{\mathfrak{m}}athbf{A} \ensuremath{\otimes} M_\lambda^\ensuremath{\mathfrak{m}}athbf{A}$ is given by
\[ 1 \ensuremath{\mathfrak{m}}apsto \sum_{Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)} C_Q \ensuremath{\otimes} \ensuremath{C^{\prime}}_Q = \sum_{Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)} \ensuremath{C^{\prime}}_Q \ensuremath{\otimes} C_Q.\]
\item The surjection $M_\lambda^\ensuremath{\mathfrak{m}}athbf{A} \ensuremath{\otimes} M_\lambda^\ensuremath{\mathfrak{m}}athbf{A} \ensuremath{\mathfrak{t}}o \nsbr{\epsilon}_+$ is given by
\[ \sum_{T, U \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)} a^{T U} \ensuremath{C^{\prime}}_T \ensuremath{\otimes} C_U \ensuremath{\mathfrak{m}}apsto \ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{1}{|\ensuremath{\mathfrak{t}}ext{SYT}(\lambda)|} \displaystyle \sum_{U \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)} a^{U U}, \ensuremath{\mathfrak{t}}ext{ for any $a^{T U} \in \ensuremath{\mathfrak{m}}athbf{A}$.}\]
\item The inclusion $\nsbr{\epsilon}_- \ensuremath{\ensuremath{\mathfrak{m}}athfrak{h}}ookrightarrow M_{\lambda'}^\ensuremath{\mathfrak{m}}athbf{A} \ensuremath{\otimes} M_\lambda^\ensuremath{\mathfrak{m}}athbf{A}$ is given by
\[ 1 \ensuremath{\mathfrak{m}}apsto \sum_{Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)} (-1)^{\ell(Q)}\ensuremath{C^{\prime}}_{\ensuremath{\mathfrak{t}}ranspose{Q}} \ensuremath{\otimes} \ensuremath{C^{\prime}}_{Q}.\]
\item The surjection $M_{\lambda}^\ensuremath{\mathfrak{m}}athbf{A} \ensuremath{\otimes} M_{\lambda'}^\ensuremath{\mathfrak{m}}athbf{A} \ensuremath{\mathfrak{t}}o \nsbr{\epsilon}_-$ is given by
\[ \sum_{T, U \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)} a^{T \ensuremath{\mathfrak{t}}ranspose{U}} \ensuremath{C^{\prime}}_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_{\ensuremath{\mathfrak{t}}ranspose{U}} \ensuremath{\mathfrak{m}}apsto \ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{1}{|\ensuremath{\mathfrak{t}}ext{SYT}(\lambda)|} \displaystyle \sum_{U \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)} (-1)^{\ell(U)} a^{U \ensuremath{\mathfrak{t}}ranspose{U}}, \ensuremath{\mathfrak{t}}ext{ for any $a^{T \ensuremath{\mathfrak{t}}ranspose{U}} \in \ensuremath{\mathfrak{m}}athbf{A}$.}\]
\end{list}
\end{proposition}
Note that since $\nsbr{\epsilon}_+ \subseteq \nssym{2}{M}^\ensuremath{\mathfrak{m}}athbf{A}_\lambda$ by (i), (ii) remains valid with $C_U \ensuremath{\otimes} \ensuremath{C^{\prime}}_T$ in place of $\ensuremath{C^{\prime}}_T \ensuremath{\otimes} C_U$.
\ensuremath{\flat}egin{equation}gin{proof}
The map $I$ of \eqref{e triv inclusion map} is given by
\[\ensuremath{\mathfrak{t}}extstyle 1 \ensuremath{\mathfrak{m}}apsto \sum_{Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)} \dual{\ensuremath{C^{\prime}}_Q} \ensuremath{\otimes} \ensuremath{C^{\prime}}_Q.\]
Applying the isomorphism of Proposition \ref{p dual basis to C'} (iii) then yields (i), except the equality. The equality in (i) follows from the fact that
$\ensuremath{\mathfrak{t}}au \circ I: \nsbr{\epsilon}_+ \ensuremath{\ensuremath{\mathfrak{m}}athfrak{h}}ookrightarrow M_\lambda^\ensuremath{\mathfrak{m}}athbf{A} \ensuremath{\otimes} (M_\lambda^\ensuremath{\mathfrak{m}}athbf{A})^\ensuremath{\diamond}$ is an $\nsbr{\ensuremath{\mathscr{H}}}_r$-module homomorphism (since $\nsbr{\ensuremath{\mathscr{H}}}_r \subseteq S^2 \ensuremath{\mathscr{H}}_r$), the multiplicity of $\ensuremath{K} \nsbr{\epsilon}_+$ in $\nsbr{M}_{\lambda,\lambda}$ is 1, and Theorem \ref{t transition C' to C}.
The map $\ensuremath{\mathfrak{t}}race$ of \eqref{e triv surjection map} is given by
\[\ensuremath{\mathfrak{t}}extstyle \sum_{T, U \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)} a^{T U} \ensuremath{C^{\prime}}_T \ensuremath{\otimes} \dual{\ensuremath{C^{\prime}}_U} \ensuremath{\mathfrak{m}}apsto \ensuremath{[2]^2}rac{1}{|\ensuremath{\mathfrak{t}}ext{SYT}(\lambda)|} \sum_{U} a^{U U}, \ensuremath{\mathfrak{t}}ext{ for any $a^{T U} \in \ensuremath{\mathfrak{m}}athbf{A}$}, \]
so (ii) also follows from Proposition \ref{p dual basis to C'} (iii).
Statements (iii) and (iv) are proved in a similar way using Proposition \ref{p hecke algebra antipode} and Proposition \ref{p dual basis to C'} (iv).
\end{proof}
\subsection{The action of $\mathcal{P}_s$ on $M_\lambda \ensuremath{\otimes} M_\ensuremath{\mathfrak{m}}u$}
\label{ss sP action on bases}
For the proof of the main theorem, it is convenient to record the action of $\mathcal{P}_s$ on $M_\lambda \ensuremath{\otimes} M_\ensuremath{\mathfrak{m}}u$ in
the bases
\[ \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda \ensuremath{\otimes} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\ensuremath{\mathfrak{m}}u = \{ \ensuremath{C^{\prime}}_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U : T \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda), \ U \in \ensuremath{\mathfrak{t}}ext{SYT}(\ensuremath{\mathfrak{m}}u) \},\]
$\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\lambda \ensuremath{\otimes} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\ensuremath{\mathfrak{m}}u$, and $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\lambda \ensuremath{\otimes} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\ensuremath{\mathfrak{m}}u$. These calculations are easily made using (\ref{e prime C on prime canbas}) and (\ref{e C on canbas}).
\refstepcounter{equation}
\ensuremath{\flat}egin{equation}gin{align*}
&(\ensuremath{C^{\prime}}_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U) \mathcal{P}_s = \ensuremath{\mathfrak{t}}ag{\ensuremath{\mathfrak{t}}heequation}\label{e sP on C' C'} \\
& \ \ensuremath{\flat}egin{equation}gin{cases}
\ensuremath{[2]^2} \ensuremath{C^{\prime}}_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U & \ensuremath{\mathfrak{t}}ext{if } s \in R(\ensuremath{C^{\prime}}_T) \ensuremath{\mathfrak{t}}ext{ and } s \in R(\ensuremath{C^{\prime}}_U) \\
[2] \displaystyle \sum_{s \in R(\ensuremath{C^{\prime}}_{U'})} \ensuremath{\mathfrak{m}}u(U', U) \ensuremath{C^{\prime}}_{T} \ensuremath{\otimes} \ensuremath{C^{\prime}}_{U'} & \ensuremath{\mathfrak{t}}ext {if } s \in R(\ensuremath{C^{\prime}}_T) \ensuremath{\mathfrak{t}}ext{ and } s \notin R(\ensuremath{C^{\prime}}_U) \\
[2] \displaystyle \sum_{s \in R(\ensuremath{C^{\prime}}_{T'})} \ensuremath{\mathfrak{m}}u(T', T) \ensuremath{C^{\prime}}_{T'} \ensuremath{\otimes} \ensuremath{C^{\prime}}_U & \ensuremath{\mathfrak{t}}ext {if } s \notin R(\ensuremath{C^{\prime}}_T) \ensuremath{\mathfrak{t}}ext{ and } s \in R(\ensuremath{C^{\prime}}_U) \\
\ensuremath{[2]^2} \ensuremath{C^{\prime}}_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U \\
-[2] \left( \displaystyle \sum_{s \in R(\ensuremath{C^{\prime}}_{T'})} \ensuremath{\mathfrak{m}}u(T', T) \ensuremath{C^{\prime}}_{T'} \ensuremath{\otimes} \ensuremath{C^{\prime}}_U + \sum_{s \in R(\ensuremath{C^{\prime}}_{U'})} \ensuremath{\mathfrak{m}}u(U', U) \ensuremath{C^{\prime}}_{T} \ensuremath{\otimes} \ensuremath{C^{\prime}}_{U'} \right) \\
+ 2 \displaystyle \sum_{s \in R(\ensuremath{C^{\prime}}_{T'}), s \in R(\ensuremath{C^{\prime}}_{U'})} \ensuremath{\mathfrak{m}}u(T', T) \ensuremath{\mathfrak{m}}u(U', U) \ensuremath{C^{\prime}}_{T'} \ensuremath{\otimes} \ensuremath{C^{\prime}}_{U'} & \ensuremath{\mathfrak{t}}ext{if } s \notin R(\ensuremath{C^{\prime}}_U) \ensuremath{\mathfrak{t}}ext{ and } s \notin R(\ensuremath{C^{\prime}}_T) \\
\end{cases}
\end{align*}
\ensuremath{\flat}egin{equation} \label{e sP on C C'}
(C_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U) \mathcal{P}_s =
\ensuremath{\flat}egin{equation}gin{cases}
0 & \ensuremath{\mathfrak{t}}ext{if } s \in R(C_T) \ensuremath{\mathfrak{t}}ext{ and } s \in R(\ensuremath{C^{\prime}}_U) \\
\ensuremath{[2]^2} C_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U - [2] \sum_{s \in R(\ensuremath{C^{\prime}}_{U'})} \ensuremath{\mathfrak{m}}u(U', U) C_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_{U'} & \ensuremath{\mathfrak{t}}ext {if } s \in R(C_T) \ensuremath{\mathfrak{t}}ext{ and } s \notin R(\ensuremath{C^{\prime}}_U) \\
\ensuremath{[2]^2} C_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U + [2] \sum_{s \in R(C_{T'})} \ensuremath{\mathfrak{m}}u(T', T) C_{T'} \ensuremath{\otimes} \ensuremath{C^{\prime}}_U & \ensuremath{\mathfrak{t}}ext {if } s \notin R(C_T) \ensuremath{\mathfrak{t}}ext{ and } s \in R(\ensuremath{C^{\prime}}_U) \\
-[2] \sum_{s \in R(C_{T'})} \ensuremath{\mathfrak{m}}u(T', T) C_{T'} \ensuremath{\otimes} \ensuremath{C^{\prime}}_U \\
+[2] \sum_{s \in R(\ensuremath{C^{\prime}}_{U'})} \ensuremath{\mathfrak{m}}u(U', U) C_{T} \ensuremath{\otimes} \ensuremath{C^{\prime}}_{U'} \\
+ 2 \sum_{s \in R(C_{T'}), s \in R(\ensuremath{C^{\prime}}_{U'})} \ensuremath{\mathfrak{m}}u(T', T) \ensuremath{\mathfrak{m}}u(U', U) C_{T'} \ensuremath{\otimes} \ensuremath{C^{\prime}}_{U'} & \ensuremath{\mathfrak{t}}ext{if } s \notin R(C_T) \ensuremath{\mathfrak{t}}ext{ and } s \notin R(\ensuremath{C^{\prime}}_U) \\
\end{cases}
\end{equation}
\refstepcounter{equation}
\ensuremath{\flat}egin{equation}gin{align*}
&(C_T \ensuremath{\otimes} C_U) \mathcal{P}_s = \ensuremath{\mathfrak{t}}ag{\ensuremath{\mathfrak{t}}heequation}\label{e sP on C C} \\
& \ \ensuremath{\flat}egin{equation}gin{cases}
\ensuremath{[2]^2} C_T \ensuremath{\otimes} C_U & \ensuremath{\mathfrak{t}}ext{if } s \in R(C_T) \ensuremath{\mathfrak{t}}ext{ and } s \in R(C_U) \\
-[2] \displaystyle \sum_{s \in R(C_{U'})} \ensuremath{\mathfrak{m}}u(U', U) C_{T} \ensuremath{\otimes} C_{U'} & \ensuremath{\mathfrak{t}}ext {if } s \in R(C_T) \ensuremath{\mathfrak{t}}ext{ and } s \notin R(C_U) \\
-[2] \displaystyle \sum_{s \in R(C_{T'})} \ensuremath{\mathfrak{m}}u(T', T) C_{T'} \ensuremath{\otimes} C_U & \ensuremath{\mathfrak{t}}ext {if } s \notin R(C_T) \ensuremath{\mathfrak{t}}ext{ and } s \in R(C_U) \\
\ensuremath{[2]^2} C_T \ensuremath{\otimes} C_U \\
+[2] \left( \displaystyle \sum_{s \in R(C_{T'})} \ensuremath{\mathfrak{m}}u(T', T) C_{T'} \ensuremath{\otimes} C_U + \sum_{s \in R(C_{U'})} \ensuremath{\mathfrak{m}}u(U', U) C_{T} \ensuremath{\otimes} C_{U'} \right) \\
+2 \displaystyle \sum_{s \in R(C_{T'}), s \in R(C_{U'})} \ensuremath{\mathfrak{m}}u(T', T) \ensuremath{\mathfrak{m}}u(U', U) C_{T'} \ensuremath{\otimes} C_{U'} & \ensuremath{\mathfrak{t}}ext{if } s \notin R(C_U) \ensuremath{\mathfrak{t}}ext{ and } s \notin R(C_T) \\
\end{cases}
\end{align*}
\section{Irreducibles of $\nsbr{\ensuremath{\mathscr{H}}}_{r,2}$}
\label{s irreducibles of nshr2}
Define the \emph{Temperley-Lieb} algebra $\ensuremath{\mathscr{H}}_{r,d}$ to be the quotient of $\ensuremath{\mathscr{H}}_r$ by the two-sided ideal
\[
\ensuremath{\flat}igoplus_{\stackrel{\lambda \vdash r,\ \ell(\lambda) > d,}{P\in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)}} \ensuremath{\mathfrak{m}}athbf{A}\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_P = \ensuremath{\mathfrak{m}}athbf{A}\{C_w:\ell(\ensuremath{\mathfrak{t}}ext{\rm sh}(P(w))) > d\}.
\]
Define the \emph{nonstandard Temperley-Lieb} algebra $\nsbr{\ensuremath{\mathscr{H}}}_{r,d}$ to be the subalgebra of $\ensuremath{\mathscr{H}}_{r,d}\ensuremath{\otimes}vw \ensuremath{\mathscr{H}}_{r,d}$ generated by the elements
$\mathcal{P}_{s} := \ensuremath{C^{\prime}}_s \ensuremath{\otimes}vw \ensuremath{C^{\prime}}_s + C_s \ensuremath{\otimes}vw C_s, \ s \in S$.
Let $\ensuremath{\mathfrak{m}}athscr{P}_{r,2}$ be the set of partitions of size $r$ with at most two parts and $\ensuremath{\mathfrak{m}}athscr{P}'_{r,2}$ be the subset of $\ensuremath{\mathfrak{m}}athscr{P}_{r,2}$ consisting of those partitions that are not a single row or column shape. Define the index set $\nsbr{\mathscr{P}}_{r,2}$ for the $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducibles as follows:
\ensuremath{\flat}egin{equation}\label{e definition nsp r2}
\nsbr{\mathscr{P}}_{r,2} = \{ \{\lambda,\ensuremath{\mathfrak{m}}u\}: \lambda, \ensuremath{\mathfrak{m}}u \in \ensuremath{\mathfrak{m}}athscr{P}_{r,2}, \, \lambda \neq \ensuremath{\mathfrak{m}}u\} \sqcup\{+\lambda: \lambda \in \ensuremath{\mathfrak{m}}athscr{P}'_{r,2}\} \sqcup \{-\lambda: \lambda \in \ensuremath{\mathfrak{m}}athscr{P}'_{r,2}\} \sqcup \{\nsbr{\epsilon}_+\}.
\end{equation}
This section is devoted to a proof of the main result of this paper:
\ensuremath{\flat}egin{equation}gin{theorem} \label{t nsH irreducibles two row case}
The algebra $\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r,2}$ is split semisimple and the list of distinct irreducibles is
\ensuremath{\flat}egin{equation}gin{list}{\emph{(\ensuremath{\mathfrak{a}}rabic{ctr})}} {\ensuremath{u}secounter{ctr} \setlength{\itemsep}{1pt} \setlength{\ensuremath{\mathfrak{t}}opsep}{2pt}}
\item $\nsbr{M}_\ensuremath{\mathfrak{a}}lpha := \nsbr{M}_{\lambda,\ensuremath{\mathfrak{m}}u} = \ensuremath{\mathscr{R}}es_{\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}}M_\lambda \ensuremath{\otimes} M_\ensuremath{\mathfrak{m}}u$, for $\ensuremath{\mathfrak{a}}lpha = \{\lambda, \ensuremath{\mathfrak{m}}u\} \in \nsbr{\mathscr{P}}_{r,2}$,
\item $\nsbr{M}_\ensuremath{\mathfrak{a}}lpha := S' \nsbr{M}_\lambda$, for $\ensuremath{\mathfrak{a}}lpha = +\lambda \in \nsbr{\mathscr{P}}_{r,2}$,
\item $\nsbr{M}_\ensuremath{\mathfrak{a}}lpha := \nswedge{2 }{M}_\lambda$, for $\ensuremath{\mathfrak{a}}lpha = -\lambda \in \nsbr{\mathscr{P}}_{r,2}$,
\item $\nsbr{M}_\ensuremath{\mathfrak{a}}lpha := \ensuremath{K}\nsbr{\epsilon}_+$, for $\ensuremath{\mathfrak{a}}lpha = \nsbr{\epsilon}_+\in \nsbr{\mathscr{P}}_{r,2}$.
\end{list}
Moreover, the irreducible $\ensuremath{K} (\ensuremath{\mathscr{H}}_{r,2} \ensuremath{\otimes} \ensuremath{\mathscr{H}}_{r,2})$-modules decompose into $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducibles as follows
\[
\ensuremath{\flat}egin{equation}gin{array}{ll}
M_{\lambda} \ensuremath{\otimes} M_{\ensuremath{\mathfrak{m}}u} \cong \nsbr{M}_{\lambda, \ensuremath{\mathfrak{m}}u} & \ensuremath{\mathfrak{t}}ext{if } \lambda \neq \ensuremath{\mathfrak{m}}u, \\
M_{\lambda} \ensuremath{\otimes} M_{\lambda} \cong S' \nsbr{M}_{\lambda} \oplus \nswedge{2}{M}_{\lambda} \oplus \ensuremath{K} \nsbr{\epsilon}_+ & \lambda \vdash r. \\
\end{array}
\]
\end{theorem}
\subsection{Gluing $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}$-irreducibles} \label{ss gluing field nsH irreducibles}
\ensuremath{\flat}egin{equation}gin{proposition} \label{p r-1 restrictions}
The four types of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r}$-modules from Theorem \ref{t nsH irreducibles two row case} decompose into $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}$-modules as follows:
\ensuremath{\flat}egin{equation}gin{list}{(\ensuremath{\mathfrak{a}}rabic{ctr})} {\ensuremath{u}secounter{ctr} \setlength{\itemsep}{1pt} \setlength{\ensuremath{\mathfrak{t}}opsep}{2pt}}
\item[\emph{(1a)}] $\ensuremath{\mathscr{R}}es_{\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r-1}} \nsbr{M}_{\lambda, \ensuremath{\mathfrak{m}}u} \cong \ensuremath{\flat}igoplus_{i \in [k_\lambda], j \in [k_\ensuremath{\mathfrak{m}}u]} \nsbr{M}_{\lambda - a_i, \ensuremath{\mathfrak{m}}u - b_j}$, if $|\lambda \cap \ensuremath{\mathfrak{m}}u| < r-1$.
\item[\emph{(1b)}] $\ensuremath{\mathscr{R}}es_{\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r-1}} \nsbr{M}_{\lambda, \ensuremath{\mathfrak{m}}u} \cong \ensuremath{\flat}igoplus_{\substack{i \in [k_\lambda], j \in [k_\ensuremath{\mathfrak{m}}u], \\ (i, j) \neq (k, l)}} \nsbr{M}_{\lambda - a_i, \ensuremath{\mathfrak{m}}u - b_j} \oplus S' \nsbr{M}_{\nu} \oplus \nswedge{2}{M}_\nu \oplus \ensuremath{K} \nsbr{\epsilon}_+$, where $\nu = \lambda - a_k = \ensuremath{\mathfrak{m}}u - b_l$.
\item[\emph{(2)}] $\ensuremath{\mathscr{R}}es_{\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r-1}} S' \nsbr{M}_{\lambda} \cong \ensuremath{\flat}igoplus_{1 \leq i < j \leq k_\lambda} \nsbr{M}_{\lambda - a_i, \lambda - a_j} \oplus \ensuremath{\flat}igoplus_{i \in [k_\lambda]} S' \nsbr{M}_{\lambda - a_i} \oplus \ensuremath{K} \nsbr{\epsilon}_+^{\oplus k_\lambda -1}$, for $\lambda \in \ensuremath{\mathfrak{m}}athscr{P}'_r$.
\item[\emph{(3)}] $\ensuremath{\mathscr{R}}es_{\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r-1}} \nswedge{2}{M}_{\lambda} \cong \ensuremath{\flat}igoplus_{1\leq i < j \leq k_\lambda} \nsbr{M}_{\lambda - a_i, \lambda - a_j} \oplus \ensuremath{\flat}igoplus_{i \in [k_\lambda]} \nswedge{2}{M}_{\lambda - a_i}$, for $\lambda \in \ensuremath{\mathfrak{m}}athscr{P}'_r$.
\item[\emph{(4)}] $\ensuremath{\mathscr{R}}es_{\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r-1}} \ensuremath{K} \nsbr{\epsilon}_+ \cong \ensuremath{K} \nsbr{\epsilon}_+$.
\end{list}
\end{proposition}
Note that if $|\ensuremath{\mathfrak{t}}ext{SYT}(\nu)| = 1$, then $S' \nsbr{M}_\nu = \nswedge{2}{M}_\nu = 0$, so some zero modules appear in the right-hand sides.
\ensuremath{\flat}egin{equation}gin{proof}
It is well known that $\ensuremath{\mathscr{R}}es_{\ensuremath{\mathscr{H}}_{r-1}} M_\lambda \cong \ensuremath{\flat}igoplus_{i \in [k_\lambda]} M_{\lambda-a_i}$. The decompositions (1a) and (1b) are clear from this and
\eqref{e nsH S' iso}. Decomposition (3) follows from the general fact that $\ensuremath{\ensuremath{\mathscr{O}}mega}edge(\ensuremath{\flat}igoplus_{i \in [k]} M_i) \cong \ensuremath{\ensuremath{\mathscr{O}}mega}edge(M_1)\ensuremath{\otimes} \ensuremath{\ensuremath{\mathfrak{t}}rianglelefteq}ots \ensuremath{\otimes}\ensuremath{\ensuremath{\mathscr{O}}mega}edge(M_k)$ is a graded isomorphism of algebras for any vector spaces $M_1,\ensuremath{\ensuremath{\mathfrak{t}}rianglelefteq}ots,M_k$, where $\ensuremath{\ensuremath{\mathscr{O}}mega}edge(M)$ is the exterior algebra of $M$.
The analogous fact holds for symmetric algebras, which implies
\[\ensuremath{\mathscr{R}}es_{\ensuremath{K} S^2\ensuremath{\mathscr{H}}_{r-1}} S^2 M_{\lambda} \cong \ensuremath{\flat}igoplus_{1 \leq i < j \leq k_\lambda} \nsbr{M}_{\lambda - a_i, \lambda - a_j} \oplus \ensuremath{\flat}igoplus_{i \in [k_\lambda]} S^2 M_{\lambda - a_i}.\]
Decomposition (2) then follows from \eqref{e nsH S' iso}.
\end{proof}
We adopt the convention that restrictions from $\nsbr{\ensuremath{\mathscr{H}}}_r$ to $\nsbr{\ensuremath{\mathscr{H}}}_{r-1}$ are considered with respect to the subalgebra of $\nsbr{\ensuremath{\mathscr{H}}}_r$ generated by $\mathcal{P}_s$, $s \in J$, where $J := \{s_1,\ensuremath{\ensuremath{\mathfrak{t}}rianglelefteq}ots,s_{r-2}\}$.
Given a $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,d}$-irreducible $N$ and a $\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r,d}$-module $M$, let $\ensuremath{\mathfrak{p}}roj_{N} : M \ensuremath{\mathfrak{t}}o M$ denote the $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,d}$ projector with image the $N$-isotypic component of $M$. Given a $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1,d}$-irreducible $N$ and a $\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_r$-module $M$, let $\ensuremath{\mathfrak{p}}rojres_{N} : M \ensuremath{\mathfrak{t}}o M$ denote the $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1,d}$ projector with image the $N$-isotypic component of $\ensuremath{\mathscr{R}}es_{\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r-1,d}}M$.
Theorem \ref{t nsH irreducibles two row case} will be proved inductively, using the list of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}$-irreducibles and the fact that the restriction of a $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducible to $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}$ is multiplicity-free.
Let $\ensuremath{\flat}igoplus_{i \in [k]} \nsbr{M}_i$ be a multiplicity-free decomposition of a $ \ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r}$-module $\nsbr{M}$ into $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}$-irreducibles.
Then any $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r}$-submodule of $ \nsbr{ M}$ is a direct sum of some of the $\nsbr{M}_i$. Suppose that $ \nsbr{M}' = \ensuremath{\flat}igoplus_{i \in I} \nsbr{M}_{i}$, for some $I \subseteq [k]$, is
contained in a $ \ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r}$-submodule of $\nsbr{M}$. We say that $\nsbr{M}'$ \emph{glues} to $\nsbr{M}_j$, $j \notin I$, if $\nsbr{M}_j \subseteq \nsbr{M}' (\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_r)$; this is equivalent to $\ensuremath{\mathfrak{p}}rojres_{\nsbr{M}_j}(x) \neq 0$ for some $x \in \nsbr{M}' (\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_r)$. Thus if we show that $\nsbr{M}_1$ glues to $\nsbr{M}_2$, $\nsbr{M}_1 \oplus \nsbr{M}_2$ glues to $\nsbr{M}_3$, $\ensuremath{\ensuremath{\mathfrak{t}}rianglelefteq}ots$, $\ensuremath{\flat}igoplus_{i \in [k-1]} \nsbr{M}_i$ glues to $\nsbr{M}_k$, then this proves that $ \nsbr{M}$ is a $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r}$-irreducible. Slight variants of this argument will be used in the propositions in the next subsection.
\subsection{Four propositions on the irreducibility of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_r$-modules} \label{ss four propositions}
In this subsection we state and prove Propositions \ref{p case lambda mu generic}, \ref{p case lambda one from mu}, \ref{p case wedge lambda}, and \ref{p case sym lambda}, which will be used inductively to show that the $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-modules in (1)--(4) of Theorem \ref{t nsH irreducibles two row case} are irreducible.
\ensuremath{\flat}egin{equation}gin{proposition} \label{p case lambda mu generic}
Maintain the setup of \ensuremath{\mathfrak{t}}extsection\ref{ss gluing field nsH irreducibles}. If $\lambda \neq \ensuremath{\mathfrak{m}}u$ and $\nsbr{M}_{\lambda-a_i, \ensuremath{\mathfrak{m}}u-b_j}$ are distinct irreducible $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}$-modules ($i \in [k_\lambda], j \in [k_\ensuremath{\mathfrak{m}}u]$), then $\nsbr{M}_{\lambda, \ensuremath{\mathfrak{m}}u}$ is an irreducible $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_r$-module.
\end{proposition}
\ensuremath{\flat}egin{equation}gin{proof}
We work with the basis $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda \ensuremath{\otimes} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda$ of $\nsbr{M}_{\lambda, \ensuremath{\mathfrak{m}}u}$.
It suffices to show that $\nsbr{M}_{\lambda - a_1, \ensuremath{\mathfrak{m}}u - b_1}$ glues to $\nsbr{M}_{\lambda - a_i, \ensuremath{\mathfrak{m}}u - b_j}$ for $(i,j) \neq (1, 1)$, which we do as follows:
choose $T \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)$ and $U \in \ensuremath{\mathfrak{t}}ext{SYT}(\ensuremath{\mathfrak{m}}u)$ so that
\ensuremath{\flat}egin{equation}
\ensuremath{\mathfrak{p}}arbox{13.5cm}{
\ensuremath{\flat}egin{equation}gin{list} {(\ensuremath{\mathfrak{a}}rabic{ctr})} {\ensuremath{u}secounter{ctr} \setlength{\itemsep}{1pt} \setlength{\ensuremath{\mathfrak{t}}opsep}{2pt}}
\item $T_{a_1} = r$, and if $i \neq 1$ then there is an edge $T \dkt{r-1} T'$ with $T'_{a_i} = r$.
\item $U_{b_1} = r$, and if $j \neq 1$ then there is an edge $U \dkt{r-1} U'$ with $U'_{b_j} = r$.
\end{list}
}
\end{equation}
Such tableaux exist by \eqref{e DKE complete graph}.
Then if $i \neq 1$ and $j \neq 1$, then $s_{r-1} \notin R(\ensuremath{C^{\prime}}_T)$, $s_{r-1} \notin R(\ensuremath{C^{\prime}}_{U})$ and $\ensuremath{C^{\prime}}_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U \mathcal{P}_{r-1}$ is computed using the last case of (\ref{e sP on C' C'}). The term $\ensuremath{C^{\prime}}_{T'} \ensuremath{\otimes} \ensuremath{C^{\prime}}_{U'}$ appears in the sum. The projection lemma (Lemma \ref{l projections are not too tricky}) and the fact that
\[
\{(\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{A})^J \ensuremath{\otimes} (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{B})^J:A \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda),\ {A_{a_i}} = r, \ B \in \ensuremath{\mathfrak{t}}ext{SYT}(\ensuremath{\mathfrak{m}}u), \ B_{b_j} = r\} \cong \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{\lambda-a_i}\ensuremath{\otimes}\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{\ensuremath{\mathfrak{m}}u-b_j}
\]
is a $\ensuremath{K}_0$-basis of $\ensuremath{\mathscr{L}}_{\lambda-a_i}\ensuremath{\otimes}\ensuremath{\mathscr{L}}_{\ensuremath{\mathfrak{m}}u-b_j}$
shows that the projection of $\ensuremath{C^{\prime}}_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U \mathcal{P}_{r-1}$ onto $\nsbr{M}_{\lambda-a_i, \ensuremath{\mathfrak{m}}u - b_j}$ is nonzero (the hypotheses of the lemma are satisfied, which depends in a somewhat delicate way on the form of the last case of (\ref{e sP on C' C'})).
Since $\ensuremath{C^{\prime}}_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U \in \nsbr{M}_{\lambda - a_1, \ensuremath{\mathfrak{m}}u - b_1}$ by Corollary \ref{c geck relative a invariant}, $\nsbr{M}_{\lambda - a_1, \ensuremath{\mathfrak{m}}u - b_1}$ glues to $\nsbr{M}_{\lambda-a_i, \ensuremath{\mathfrak{m}}u - b_j}$. If $i =1$ or $j=1$, these also glue by the same argument, possibly using the second or third case of (\ref{e sP on C' C'}) instead of the fourth.
\end{proof}
Given a vector space $M$, let $\ensuremath{\mathfrak{t}}au : M \ensuremath{\otimes} M \ensuremath{\mathfrak{t}}o M \ensuremath{\otimes} M$ denote the flip $a \ensuremath{\otimes} b \ensuremath{\mathfrak{m}}apsto b \ensuremath{\otimes} a$.
For $a,b \in M$, put $a \cdot b = \ensuremath{[2]^2}rac{1}{2} (1 + \ensuremath{\mathfrak{t}}au) (a\ensuremath{\otimes} b) = \ensuremath{[2]^2}rac{1}{2} (a \ensuremath{\otimes} b + b \ensuremath{\otimes} a)$
and $a \ensuremath{\omega}edge b = \ensuremath{[2]^2}rac{1}{2} (1 - \ensuremath{\mathfrak{t}}au) (a\ensuremath{\otimes} b) = \ensuremath{[2]^2}rac{1}{2} (a \ensuremath{\otimes} b - b \ensuremath{\otimes} a).$
Let $\ensuremath{\mathscr{L}}_\nu = \ensuremath{K}_0 \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\nu = \ensuremath{K}_0 \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\nu$ be as defined after Theorem \ref{t transition C' to C}. Let $\leq$ be a total order on $\ensuremath{\mathfrak{t}}ext{SYT}(\lambda)$. Then
\[S^2 \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\nu := \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\nu \cdot \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\nu = \{\ensuremath{C^{\prime}}_{A} \cdot \ensuremath{C^{\prime}}_{B} : A,B \in \ensuremath{\mathfrak{t}}ext{SYT}(\nu),\ A \leq B \}\]
is a basis of $S^2 M_\nu$. Let $S^2 \ensuremath{\mathscr{L}}_\nu := \ensuremath{K}_0 S^2 \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\nu$ be the corresponding $\ensuremath{K}_0$-lattice of $S^2 M_\nu$.
Similarly,
\[\ensuremath{\ensuremath{\mathscr{O}}mega}edge^2 \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\nu := \{\ensuremath{C^{\prime}}_{A} \ensuremath{\omega}edge \ensuremath{C^{\prime}}_{B} : A,B \in \ensuremath{\mathfrak{t}}ext{SYT}(\nu), \ A < B\}\]
is a basis of $\ensuremath{\ensuremath{\mathscr{O}}mega}edge^2 M_\nu$. Let $\ensuremath{\ensuremath{\mathscr{O}}mega}edge^2 \ensuremath{\mathscr{L}}_\nu := \ensuremath{K}_0 \ensuremath{\ensuremath{\mathscr{O}}mega}edge^2 \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\nu$ be the corresponding $\ensuremath{K}_0$-lattice of $\ensuremath{\ensuremath{\mathscr{O}}mega}edge^2 M_\nu$.
\ensuremath{\flat}egin{equation}gin{lemma}\label{l not in triv}
Fix some $T \in \ensuremath{\mathfrak{t}}ext{SYT}(\nu)$. The set
\[
\{\ensuremath{\mathfrak{p}}roj_{S' \nsbr{M}_\nu} (\ensuremath{C^{\prime}}_A \cdot \ensuremath{C^{\prime}}_B) : A,B \in \ensuremath{\mathfrak{t}}ext{SYT}(\nu),\ A < B\} \sqcup \{\ensuremath{\mathfrak{p}}roj_{S' \nsbr{M}_\nu} (\ensuremath{C^{\prime}}_A \cdot \ensuremath{C^{\prime}}_A) : A \in \ensuremath{\mathfrak{t}}ext{SYT}(\nu), \ A \neq T\}
\]
is a basis of $S' \nsbr{M}_\nu$.
\end{lemma}
\ensuremath{\flat}egin{equation}gin{proof}
By Proposition \ref{p triv in Mdual M} (i), $\ensuremath{K} \nsbr{\epsilon}_+ \subseteq S^2 M_\nu \subseteq M_\nu \ensuremath{\otimes} M_\nu$ is spanned by
\ensuremath{\flat}egin{equation}
\label{e not in triv2}
\sum_{Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\nu)} C_Q \ensuremath{\otimes} \ensuremath{C^{\prime}}_Q \equiv \sum_{Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\nu)}\ensuremath{C^{\prime}}_Q \ensuremath{\otimes} \ensuremath{C^{\prime}}_Q \ensuremath{\mathfrak{m}}od \ensuremath{u} S^2\ensuremath{\mathscr{L}}_{\nu},
\end{equation}
where the equivalence is by Theorem \ref{t transition C' to C}.
As $S^2 \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\nu$ is a basis of $S^2 M_\nu$, to prove the lemma, it suffices to show that the left-hand side of \eqref{e not in triv2} is not in the span of $S^2 \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\nu \setminus \{\ensuremath{C^{\prime}}_T \cdot \ensuremath{C^{\prime}}_T\}$. And this is true because the image of $\sum_{Q \in \ensuremath{\mathfrak{t}}ext{SYT}(\nu)}\ensuremath{C^{\prime}}_Q \ensuremath{\otimes} \ensuremath{C^{\prime}}_Q$ in $S^2\ensuremath{\mathscr{L}}_{\nu} / \ensuremath{u} S^2\ensuremath{\mathscr{L}}_{\nu}$ is not in the span of the image of $S^2 \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\nu \setminus \{\ensuremath{C^{\prime}}_T \cdot \ensuremath{C^{\prime}}_T\}$ in $S^2\ensuremath{\mathscr{L}}_{\nu} / \ensuremath{u} S^2\ensuremath{\mathscr{L}}_{\nu}$.
\end{proof}
\ensuremath{\flat}egin{equation}gin{proposition} \label{p case lambda one from mu}
Maintain the setup of \ensuremath{\mathfrak{t}}extsection\ref{ss gluing field nsH irreducibles} and set $\nu = \lambda - a_k = \ensuremath{\mathfrak{m}}u - b_l$. If the decomposition
\[\ensuremath{\mathscr{R}}es_{\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r-1}} \nsbr{M}_{\lambda, \ensuremath{\mathfrak{m}}u} \cong \ensuremath{\flat}igoplus_{\substack{i \in [k_\lambda], j \in [k_\ensuremath{\mathfrak{m}}u], \\ (i, j) \neq (k, l)}} \nsbr{M}_{\lambda - a_i, \ensuremath{\mathfrak{m}}u - b_j} \oplus S' \nsbr{M}_{\nu} \oplus \nswedge{2}{M}_\nu \oplus \ensuremath{K} \nsbr{\epsilon}_+\]
of Proposition \ref{p r-1 restrictions} (1b) consists of distinct irreducible $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}$-modules,
then $\nsbr{M}_{\lambda, \ensuremath{\mathfrak{m}}u}$ is an irreducible $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_r$-module.
\end{proposition}
\ensuremath{\flat}egin{equation}gin{proof}
First, if $k_\lambda = k_\ensuremath{\mathfrak{m}}u = 1,$ then $\lambda = (2)$ and $\ensuremath{\mathfrak{m}}u = (1,1)$, and the result is clear in this case. We will then assume $(k,l) \neq (1, 1)$ and prove the proposition using the basis $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda \ensuremath{\otimes} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda$; if $(k,l) = (1,1)$, the proposition can be proved in a similar way\ensuremath{[2]^2}ootnote{The main change required is that \eqref{e sP on C C} must be used in place of \eqref{e sP on C' C'}; these differ by some signs which end up being harmless.} using the argument below with $a_{k_\lambda}, b_{k_\ensuremath{\mathfrak{m}}u}$ in place of $a_1, b_1$ and the basis $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\lambda \ensuremath{\otimes} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\lambda$ in place of $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda \ensuremath{\otimes} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda$.
The $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}$-irreducible $\nsbr{M}_{\lambda - a_1, \ensuremath{\mathfrak{m}}u - b_1}$ glues to
$\nsbr{M}_{\lambda - a_i, \ensuremath{\mathfrak{m}}u - b_j}$ for $(i,j) \notin \{(k,l),(1, 1)\}$ by the same argument as in the proof of Proposition \ref{p case lambda mu generic}.
The assumption $\lambda \ensuremath{\ensuremath{\mathfrak{m}}athfrak{g}}dneq \ensuremath{\mathfrak{m}}u$ implies $k \ensuremath{\ensuremath{\mathfrak{m}}athfrak{g}}eq l$. Thus $k >1$ since we are assuming $(k,l) \neq (1,1)$. We will next show that $\ensuremath{\flat}igoplus_{i \leq l} \nsbr{M}_{\lambda - a_1, \ensuremath{\mathfrak{m}}u - b_i}$ glues to $S' \nsbr{M}_{\nu}$ and $\nswedge{2}{M}_{\nu}$. We may assume that $|\ensuremath{\mathfrak{t}}ext{SYT}(\nu)| > 1$ because this is equivalent to $S' \nsbr{M}_{\nu}$ and $\nswedge{2}{M}_{\nu}$ being nonzero.
Thus by \eqref{e DKE complete graph}, we can choose $T, T' \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)$ and $U \in \ensuremath{\mathfrak{t}}ext{SYT}(\ensuremath{\mathfrak{m}}u)$ such that
\ensuremath{\flat}egin{equation}gin{list} {(\ensuremath{\mathfrak{a}}rabic{ctr})} {\ensuremath{u}secounter{ctr} \setlength{\itemsep}{1pt} \setlength{\ensuremath{\mathfrak{t}}opsep}{2pt}}
\item $T_{a_1} = r$ and there is an edge $T \dkt{r-1} T'$ with $T'_{a_k} = r$.
\item $U_{b_l} = r$ and $U_\nu \neq T'_\nu$.
\end{list}
Here $U_\nu$ denotes the subtableau of $U$ obtained by restricting $U$ to $\nu$.
The quantity $\ensuremath{C^{\prime}}_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U \mathcal{P}_{r-1}$ is computed using the second or fourth case of \eqref{e sP on C' C'}:
if the second case applies, then the projection lemma shows that
\ensuremath{\flat}egin{equation}
\label{e nu nu case 2}
\ensuremath{\mathfrak{p}}rojres_{M_\nu \ensuremath{\otimes} M_\nu} (\ensuremath{C^{\prime}}_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U \ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{\mathcal{P}_{r-1}}{[2]}) \equiv \displaystyle \sum_{\substack{s_{r-1} \in R(\ensuremath{C^{\prime}}_{A}), \\ A_{a_k} = r}} \ensuremath{\mathfrak{m}}u(A, T) (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{A})^J \ensuremath{\otimes} (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{U})^J \ensuremath{\mathfrak{m}}od \ensuremath{u} \ensuremath{\mathscr{L}}_\nu \ensuremath{\otimes} \ensuremath{\mathscr{L}}_\nu;
\end{equation}
if the fourth case applies, then a careful application of the projection lemma shows that
\ensuremath{\flat}egin{equation}
\label{e nu nu case 4}
\ensuremath{\mathfrak{p}}rojres_{M_\nu \ensuremath{\otimes} M_\nu} (\ensuremath{C^{\prime}}_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U \ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{\mathcal{P}_{r-1}}{[2]}) \equiv -\displaystyle \sum_{\substack{s_{r-1} \in R(\ensuremath{C^{\prime}}_{A}), \\ A_{a_k} = r}} \ensuremath{\mathfrak{m}}u(A, T) (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{A})^J \ensuremath{\otimes} (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{U})^J \ensuremath{\mathfrak{m}}od \ensuremath{u} \ensuremath{\mathscr{L}}_\nu \ensuremath{\otimes} \ensuremath{\mathscr{L}}_\nu.
\end{equation}
Let $x$ (resp. $-x$) denote the right-hand side of \eqref{e nu nu case 2} (resp. \eqref{e nu nu case 4}).
Since $\ensuremath{\mathfrak{p}}m(\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{T'})^J \ensuremath{\otimes} (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{U})^J$ appears in the expression for $\ensuremath{\mathfrak{p}}m x$ and $T'_\nu \neq U_\nu$, it follows that the projection of $\ensuremath{\mathfrak{p}}m x$ to $\ensuremath{\ensuremath{\mathscr{O}}mega}edge^2 \ensuremath{\mathscr{L}}_\nu$ is nonzero. This uses that
\[
\{(\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{A})^J \ensuremath{\omega}edge (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{B})^J:A \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda),\ {A_{a_k}} = r, \ B \in \ensuremath{\mathfrak{t}}ext{SYT}(\ensuremath{\mathfrak{m}}u), \ B_{b_l} = r, \ A_\nu < B_\nu \} \cong \ensuremath{\ensuremath{\mathscr{O}}mega}edge^2 \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{\nu}
\]
is a $\ensuremath{K}_0$-basis of $\ensuremath{\ensuremath{\mathscr{O}}mega}edge^2 \ensuremath{\mathscr{L}}_{\nu}$.
The quantities $\ensuremath{\mathfrak{p}}m x$ also have nonzero projection onto $S' \nsbr{M}_{\nu}$ by Lemma \ref{l not in triv}.
Finally, we need that $\ensuremath{C^{\prime}}_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U \in \ensuremath{\flat}igoplus_{i \leq l} \nsbr{M}_{\lambda - a_1, \ensuremath{\mathfrak{m}}u - b_i}$, which holds by Corollary \ref{c geck relative a invariant}, to conclude that $\ensuremath{\flat}igoplus_{i \leq l} \nsbr{M}_{\lambda - a_1, \ensuremath{\mathfrak{m}}u - b_i}$ glues to $S' \nsbr{M}_{\nu}$ and $\nswedge{2}{M}_{\nu}$.
It remains to show that $\ensuremath{K} \nsbr{\epsilon}_+ \subseteq \nsbr{M}_{\lambda - a_k, \ensuremath{\mathfrak{m}}u - b_l}$ glues to some other $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}$-irreducible of $\ensuremath{\mathscr{R}}es_{\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}} \nsbr{M}_{\lambda,\ensuremath{\mathfrak{m}}u}$. If not, then it follows that $\nsbr{\epsilon}_+|_{\ensuremath{u} = 1}$ is a 1-dimensional $\ensuremath{\mathbb{Q}} \ensuremath{\mathcal{S}}_r$-submodule of $\nsbr{M}_{\lambda,\ensuremath{\mathfrak{m}}u}|_{\ensuremath{u} = 1} \cong M_\lambda|_{\ensuremath{u}=1} \ensuremath{\otimes} M_\ensuremath{\mathfrak{m}}u|_{\ensuremath{u}=1}$. Here, the specialization $N|_{\ensuremath{u}=1}$ of an $\ensuremath{\mathfrak{m}}athbf{A}$-module $N_\ensuremath{\mathfrak{m}}athbf{A}$ is defined to be $\ensuremath{\mathbb{Q}} \ensuremath{\otimes}_{\ensuremath{\mathfrak{m}}athbf{A}} N_\ensuremath{\mathfrak{m}}athbf{A}$, the map $\ensuremath{\mathfrak{m}}athbf{A} \ensuremath{\mathfrak{t}}o \ensuremath{\mathbb{Q}}$ given by $\ensuremath{u} \ensuremath{\mathfrak{m}}apsto 1$.
We are assuming $r \ensuremath{\ensuremath{\mathfrak{m}}athfrak{g}}eq 3$, so $\nsbr{\epsilon}_+ \mathcal{P}_1 = [2]^2 \nsbr{\epsilon}_+$. But then $\nsbr{\epsilon}_+|_{\ensuremath{u} =1}$ is the trivial $\ensuremath{\mathbb{Q}} \ensuremath{\mathcal{S}}_r$-module, which is impossible since $\lambda \neq \ensuremath{\mathfrak{m}}u$.
\end{proof}
For any $\ensuremath{K} (\ensuremath{\mathscr{H}}_r \ensuremath{\otimes} \ensuremath{\mathscr{H}}_r)$ module $M$, let $p^1_{M_{\lambda - a_i} \ensuremath{\otimes} M_{\lambda - a_j}} : M \ensuremath{\mathfrak{t}}o M$ be the $\ensuremath{K} (\ensuremath{\mathscr{H}}_{r-1} \ensuremath{\otimes} \ensuremath{\mathscr{H}}_{r-1})$ projector with image the $M_{\lambda - a_i} \ensuremath{\otimes} M_{\lambda - a_j}$-isotypic component of $M$. For any $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r}$-module $\nsbr{M}$ and $h \in \nsbr{\ensuremath{\mathscr{H}}}_r$, let $m_h : \nsbr{M} \ensuremath{\mathfrak{t}}o \nsbr{M}$ denote right multiplication by $h$.
\ensuremath{\flat}egin{equation}gin{lemma}
\label{l restriction facts}
Let $i, j \in [k_\lambda]$, $i \neq j$. There are the following equalities of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}$-module endomorphisms of $M_\lambda \ensuremath{\otimes} M_\lambda$.
\[
\ensuremath{\flat}egin{equation}gin{array}{lrcl}
\emph{(i)} & \ensuremath{\mathfrak{p}}rojres_{\nsbr{M}_{\lambda-a_i,\lambda - a_j}} \ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{1-\ensuremath{\mathfrak{t}}au}{2} & = & \ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{1-\ensuremath{\mathfrak{t}}au}{2}(p^1_{M_{\lambda-a_i} \ensuremath{\otimes} M_{\lambda - a_j}}+p^1_{M_{\lambda-a_j} \ensuremath{\otimes} M_{\lambda - a_i}})\\[2mm]
\emph{(ii)} & \ensuremath{\mathfrak{p}}rojres_{\nsbr{M}_{\lambda - a_i, \lambda - a_j}} \ensuremath{\mathfrak{p}}roj_{S' \nsbr{M}_\lambda} & = & \ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{1+\ensuremath{\mathfrak{t}}au}{2} (p^1_{M_{\lambda - a_i} \ensuremath{\otimes} M_{\lambda - a_j}} + p^1_{M_{\lambda - a_j} \ensuremath{\otimes} M_{\lambda - a_i}}) \\[2mm]
\emph{(iii)} & \ensuremath{\mathfrak{p}}rojres_{\nsbr{M}_{\lambda - a_i, \lambda - a_j}} \ensuremath{\mathfrak{p}}roj_{S' \nsbr{M}_\lambda} m_{\mathcal{P}_{r-1}} \ensuremath{\mathfrak{p}}roj_{S' \nsbr{M}_\lambda} & = &
\ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{1+\ensuremath{\mathfrak{t}}au}{2}( p^1_{M_{\lambda - a_i} \ensuremath{\otimes} M_{\lambda - a_j}} + p^1_{M_{\lambda - a_j} \ensuremath{\otimes} M_{\lambda - a_i}}) m_{\mathcal{P}_{r-1}}. \\
\end{array}
\]
\end{lemma}
\ensuremath{\flat}egin{equation}gin{proof}
First note that for any $\ensuremath{\mathscr{H}}_r \ensuremath{\otimes} \ensuremath{\mathscr{H}}_r$-module $M$, there holds
\[ \ensuremath{\mathscr{R}}es_{\nsbr{\ensuremath{\mathscr{H}}}_{r-1}} \ensuremath{\mathscr{R}}es_{\ensuremath{\mathscr{H}}_{r-1} \ensuremath{\otimes} \ensuremath{\mathscr{H}}_{r-1}} M = \ensuremath{\mathscr{R}}es_{\nsbr{\ensuremath{\mathscr{H}}}_{r-1}} M = \ensuremath{\mathscr{R}}es_{\nsbr{\ensuremath{\mathscr{H}}}_{r-1}} \ensuremath{\mathscr{R}}es_{\nsbr{\ensuremath{\mathscr{H}}}_r} M.\]
Statement (i) is immediate from the easy facts
\ensuremath{\flat}egin{equation}gin{align*}
\ensuremath{\mathfrak{p}}rojres_{\nsbr{M}_{\lambda-a_i,\lambda - a_j}} &= p^1_{M_{\lambda-a_i} \ensuremath{\otimes} M_{\lambda - a_j}}+p^1_{M_{\lambda-a_j} \ensuremath{\otimes} M_{\lambda - a_i}}, \\
\ensuremath{\mathfrak{t}}au p^1_{M_{\lambda-a_i} \ensuremath{\otimes} M_{\lambda - a_j}} &= p^1_{M_{\lambda-a_j} \ensuremath{\otimes} M_{\lambda - a_i}} \ensuremath{\mathfrak{t}}au.
\end{align*}
This also shows that (i) holds with $1+\ensuremath{\mathfrak{t}}au$ in place of $1-\ensuremath{\mathfrak{t}}au$. Then
\[
\ensuremath{\mathfrak{p}}rojres_{\nsbr{M}_{\lambda - a_i, \lambda - a_j}} \ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{1+\ensuremath{\mathfrak{t}}au}{2} = \ensuremath{\mathfrak{p}}rojres_{\nsbr{M}_{\lambda - a_i, \lambda - a_j}} \ensuremath{\mathfrak{p}}roj_{\nssym{2}{M}_\lambda} = \ensuremath{\mathfrak{p}}rojres_{\nsbr{M}_{\lambda - a_i, \lambda - a_j}} (\ensuremath{\mathfrak{p}}roj_{S'\nsbr{M}_\lambda}+\ensuremath{\mathfrak{p}}roj_{\ensuremath{K} \nsbr{\epsilon}_+}) = \ensuremath{\mathfrak{p}}rojres_{\nsbr{M}_{\lambda - a_i, \lambda - a_j}} \ensuremath{\mathfrak{p}}roj_{S' \nsbr{M}_\lambda}
\]
proves (ii).
Statement (iii) is immediate from (ii) and the fact that $\ensuremath{\mathfrak{p}}roj_{S' \nsbr{M}_\lambda}$ is a $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_r$-module homomorphism.
\end{proof}
We say that the modules in a list are \emph{essentially distinct irreducibles} if the nonzero modules in this list are distinct irreducibles.
\ensuremath{\flat}egin{equation}gin{proposition} \label{p case wedge lambda}
Maintain the setup of \ensuremath{\mathfrak{t}}extsection\ref{ss gluing field nsH irreducibles} and assume $\lambda \in \ensuremath{\mathfrak{m}}athscr{P}'_r$.
If $\nswedge{2}{M}_{\lambda - a_i},\ i \in [k_{\lambda}]$, and $\nsbr{M}_{\lambda - a_i, \lambda - a_j},\ i < j,\ i, j \in [k_{\lambda}]$, are essentially distinct irreducible $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}$-modules, then $\nswedge{2}{M}_{\lambda}$ is an irreducible $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_r$-module.
\end{proposition}
\ensuremath{\flat}egin{equation}gin{proof}
We work with the basis $\ensuremath{\ensuremath{\mathscr{O}}mega}edge^2 \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda$ of $\nswedge{2}{M}_\lambda$.
Let $i > 1$ and assume $\nswedge{2}{M}_{\lambda - a_1}$ is nonzero. We show that $\nswedge{2}{M}_{\lambda - a_1}$ glues to $\nsbr{M}_{\lambda - a_i, \lambda -a_1}$ as follows:
given the assumptions, we can choose $T, U \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)$ so that
\ensuremath{\flat}egin{equation}gin{list} {(\ensuremath{\mathfrak{a}}rabic{ctr})} {\ensuremath{u}secounter{ctr} \setlength{\itemsep}{1pt} \setlength{\ensuremath{\mathfrak{t}}opsep}{2pt}}
\item $T_{a_1} = r$ and there is an edge $T \dkt{r-1} T'$ with $T'_{a_i} = r$.
\item $U \neq T$ and $U_{a_1} = r$.
\end{list}
If $s_{r-1} \not\in R(\ensuremath{C^{\prime}}_U)$, then $\ensuremath{C^{\prime}}_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U \mathcal{P}_{r-1}$ is computed using the fourth case of \eqref{e sP on C' C'}.
Lemma \ref{l restriction facts} (i) yields the first equality and the projection lemma yields the equivalence in the following
\ensuremath{\flat}egin{equation}gin{align*}
\ensuremath{\mathfrak{p}}rojres_{\nsbr{M}_{\lambda-a_i,\lambda - a_1}} (\ensuremath{C^{\prime}}_T \ensuremath{\omega}edge \ensuremath{C^{\prime}}_U \ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{\mathcal{P}_{r-1}}{[2]}) = \ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{1-\ensuremath{\mathfrak{t}}au}{2}(p^1_{M_{\lambda-a_i} \ensuremath{\otimes} M_{\lambda - a_1}}+p^1_{M_{\lambda-a_1} \ensuremath{\otimes} M_{\lambda - a_i}}) (\ensuremath{C^{\prime}}_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U \ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{\mathcal{P}_{r-1}}{[2]})\\
\equiv -\displaystyle \sum_{\substack{s_{r-1} \in R(\ensuremath{C^{\prime}}_{A}), \\ A_{a_i} = r}} \ensuremath{\mathfrak{m}}u(A, T) (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{A})^J \ensuremath{\omega}edge (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{U})^J
- \displaystyle \sum_{\substack{s_{r-1} \in R(\ensuremath{C^{\prime}}_{B}), \\ B_{a_i} = r}} \ensuremath{\mathfrak{m}}u(B, U) (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{T})^J \ensuremath{\omega}edge (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{B})^J \\
= -\displaystyle \sum_{\substack{s_{r-1} \in R(\ensuremath{C^{\prime}}_{A}), \\ A_{a_i} = r}} \ensuremath{\mathfrak{m}}u(A, T) (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{A})^J \ensuremath{\omega}edge (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{U})^J
+\displaystyle \sum_{\substack{s_{r-1} \in R(\ensuremath{C^{\prime}}_{B}), \\ B_{a_i} = r}} \ensuremath{\mathfrak{m}}u(B, U) (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{B})^J \ensuremath{\omega}edge (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{T})^J.
\end{align*}
The equivalence is mod $\ensuremath{u} (\ensuremath{\mathscr{L}}_{\lambda-a_i} \ensuremath{\otimes} \ensuremath{\mathscr{L}}_{\lambda-a_1} \oplus \ensuremath{\mathscr{L}}_{\lambda-a_1} \ensuremath{\otimes} \ensuremath{\mathscr{L}}_{\lambda-a_i})$.
The final line is nonzero because $(\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{T'})^J \ensuremath{\omega}edge (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{U})^J$ appears in the left sum, $U \neq T$, and $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{\lambda-a_i} \ensuremath{\otimes} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{\lambda-a_1}$ is a $\ensuremath{K}_0$-basis of $\ensuremath{\mathscr{L}}_{\lambda-a_i} \ensuremath{\otimes} \ensuremath{\mathscr{L}}_{\lambda-a_1} \subseteq \nsbr{M}_{\lambda-a_i,\lambda - a_1}$.\ensuremath{[2]^2}ootnote{Throughout this proof $\nsbr{M}_{\lambda-a_i,\lambda - a_1}$ is understood as a $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}$-submodule of $\ensuremath{\mathscr{R}}es_{\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}} \nswedge{2}{M}_\lambda$.}
A similar (but easier) argument shows that $ \ensuremath{\mathfrak{p}}rojres_{\nsbr{M}_{\lambda-a_i,\lambda - a_1}} (\ensuremath{C^{\prime}}_T \ensuremath{\omega}edge \ensuremath{C^{\prime}}_U \ensuremath{[2]^2}rac{\mathcal{P}_{r-1}}{[2]})$ is nonzero in the case $s_{r-1} \in R(\ensuremath{C^{\prime}}_U)$.
Thus since $\ensuremath{C^{\prime}}_T \ensuremath{\omega}edge \ensuremath{C^{\prime}}_U \in \nswedge{2}{M}_{\lambda - a_1}$ by Corollary \ref{c geck relative a invariant}, $\nswedge{2}{M}_{\lambda-a_1 }$ glues to $\nsbr{M}_{\lambda-a_i,\lambda - a_1}$ ($i > 1$).
We next show that\ensuremath{[2]^2}ootnote{By definition, $\nsbr{M}_{\lambda-a_1, \lambda-a_2}= \nsbr{M}_{\lambda-a_2, \lambda-a_1}$; we work with the former here to keep notation more consistent with other parts of the proof of Theorem \ref{t nsH irreducibles two row case}.}
$\nsbr{M}_{\lambda-a_1, \lambda-a_2} \oplus \nswedge{2}{M}_{\lambda-a_1}$
glues to $\nswedge{2}{M}_{\lambda-a_2}$.
Since we can assume $\nswedge{2}{M}_{\lambda-a_2}$ is nonzero, we can choose $T, U \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)$ so that
\ensuremath{\flat}egin{equation}gin{list} {(\ensuremath{\mathfrak{a}}rabic{ctr})} {\ensuremath{u}secounter{ctr} \setlength{\itemsep}{1pt} \setlength{\ensuremath{\mathfrak{t}}opsep}{2pt}}
\item $T_{a_1} = r$ and there is an edge $T \dkt{r-1} T'$ with $T'_{a_2} = r$.
\item $U_{a_2} = r$, and $U \neq T'$.
\end{list}
If $s_{r-1} \not\in R(\ensuremath{C^{\prime}}_U)$, then $\ensuremath{C^{\prime}}_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U \mathcal{P}_{r-1}$ is computed using the fourth case of \eqref{e sP on C' C'}.
A careful application of the projection lemma shows that
\ensuremath{\flat}egin{equation}gin{align*}
\ensuremath{\mathfrak{p}}rojres_{\nswedge{2}{M}_{\lambda-a_2}} (\ensuremath{C^{\prime}}_T \ensuremath{\omega}edge \ensuremath{C^{\prime}}_U \ensuremath{\mathfrak{t}}extstyle\ensuremath{[2]^2}rac{\mathcal{P}_{r-1}}{[2]}) &= \ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{1-\ensuremath{\mathfrak{t}}au}{2} \ensuremath{\mathfrak{p}}rojres_{M_{\lambda-a_2} \ensuremath{\otimes} M_{\lambda - a_2}} (\ensuremath{C^{\prime}}_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U \ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{\mathcal{P}_{r-1}}{[2]})\\
&\equiv -\displaystyle \sum_{\substack{s_{r-1} \in R(\ensuremath{C^{\prime}}_{A}), \\ A_{a_2} = r}} \ensuremath{\mathfrak{m}}u(A, T) (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{A})^J \ensuremath{\omega}edge (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{U})^J
\ensuremath{\mathfrak{m}}od \ensuremath{u} \ensuremath{\ensuremath{\mathscr{O}}mega}edge^2 \ensuremath{\mathscr{L}}_{\lambda-a_2}
\end{align*}
The last line is nonzero because $(\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{T'})^J \ensuremath{\omega}edge (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{U})^J$ appears in the sum, $U \neq T'$, and $\ensuremath{\ensuremath{\mathscr{O}}mega}edge^2 \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{\lambda-a_2}$ is a $\ensuremath{K}_0$-basis of $\ensuremath{\ensuremath{\mathscr{O}}mega}edge^2 \ensuremath{\mathscr{L}}_{\lambda-a_2}$.
A similar (but easier) argument shows that $ \ensuremath{\mathfrak{p}}rojres_{\nswedge{2}{M}_{\lambda-a_2}} (\ensuremath{C^{\prime}}_T \ensuremath{\omega}edge \ensuremath{C^{\prime}}_U \ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{\mathcal{P}_{r-1}}{[2]})$ is nonzero in the case $s_{r-1} \in R(\ensuremath{C^{\prime}}_U)$.
Thus since $\ensuremath{C^{\prime}}_T \ensuremath{\omega}edge \ensuremath{C^{\prime}}_U \in \nsbr{M}_{\lambda-a_1, \lambda-a_2} \oplus \nswedge{2}{M}_{\lambda-a_1}$ by Corollary \ref{c geck relative a invariant}, $\nsbr{M}_{\lambda-a_1, \lambda-a_2} \oplus \nswedge{2}{M}_{\lambda-a_1}$ glues to $\nswedge{2}{M}_{\lambda-a_2}$.
Note that this argument still works if $\nswedge{2}{M}_{\lambda-a_1} = 0$.
Repeating the arguments of the previous two paragraphs, one shows that $\ensuremath{\flat}igoplus_{1 < i \leq k_\lambda} \nsbr{M}_{\lambda-a_1, \lambda-a_i} \oplus \ensuremath{\flat}igoplus_{i \in \{1,2\}} \nswedge{2}{M}_{\lambda-a_i}$ glues to $\nsbr{M}_{\lambda-a_2, \lambda-a_j}$ for $j > 2$, $\ensuremath{\flat}igoplus_{\substack{1 \leq i < j \leq k_\lambda, \\ i \leq 2}} \nsbr{M}_{\lambda-a_i, \lambda-a_j} \oplus \ensuremath{\flat}igoplus_{i \in \{1,2\}} \nswedge{2}{M}_{\lambda-a_i}$ glues to $\nswedge{2}{M}_{\lambda-a_3}$, etc., which shows
that all the irreducible constituents of $\ensuremath{\mathscr{R}}es_{\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}} \nswedge{2}{M}_{\lambda}$ are contained in a single $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_r$-irreducible.
\end{proof}
\ensuremath{\flat}egin{equation}gin{proposition} \label{p case sym lambda}
Maintain the setup of \ensuremath{\mathfrak{t}}extsection\ref{ss gluing field nsH irreducibles} and assume $\lambda \in \ensuremath{\mathfrak{m}}athscr{P}'_r$.
If $S' \nsbr{M}_{\lambda - a_i},\ i \in [k_{\lambda}]$ and $\nsbr{M}_{\lambda - a_i, \lambda - a_j},\ i < j,\ i, j \in [k_{\lambda}]$, are essentially distinct irreducible $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}$-modules, then $S'\nsbr{M}_{\lambda}$ is an irreducible $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_r$-module.
\end{proposition}
Note that $S'\nsbr{M}_{\lambda}$ does not necessarily have a multiplicity-free decomposition into $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}$-irreducibles, but the proof method explained in \ensuremath{\mathfrak{t}}extsection\ref{ss gluing field nsH irreducibles} still gives most of the proof. The $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}$-irreducible $\ensuremath{K} \nsbr{\epsilon}_+$ may appear with multiplicity more than one, so it is handled separately.
\ensuremath{\flat}egin{equation}gin{proof}
We work with the basis $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\lambda \ensuremath{\otimes} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda$ of $M_\lambda \ensuremath{\otimes} M_\lambda$.
If $k_\lambda =1$, then $\ensuremath{\mathscr{R}}es_{\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}} S'\nsbr{M}_\lambda \cong S'\nsbr{M}_{\lambda-a_1}$, so the result holds.
Assume $k_\lambda >1$.
First we show that $\nsbr{M}_{\lambda - a_{k_\lambda}, \lambda-a_1}$ glues to $\nsbr{M}_{\lambda - a_i, \lambda - a_j}$ for $i > j$, $(i,j) \neq (k_\lambda,1)$, as follows: choose $T, U \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)$ so that
\ensuremath{\flat}egin{equation}gin{list} {(\ensuremath{\mathfrak{a}}rabic{ctr})} {\ensuremath{u}secounter{ctr} \setlength{\itemsep}{1pt} \setlength{\ensuremath{\mathfrak{t}}opsep}{2pt}}
\item $T_{a_{k_\lambda}} = r$, and if $i \neq k_\lambda$ then there is an edge $T \dkt{r-1} T'$ with $T'_{a_i} = r$.
\item $U_{a_1} = r$, and if $j \neq 1$ then there is an edge $U \dkt{r-1} U'$ with $U'_{a_j} = r$.
\end{list}
Put $x = C_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U$. We wish to show that
\ensuremath{\flat}egin{equation} \label{e restrict two orders}
\ensuremath{\mathfrak{p}}rojres_{\nsbr{M}_{\lambda - a_i, \lambda - a_j}} \ensuremath{\mathfrak{p}}roj_{S' \nsbr{M}_\lambda} m_{\mathcal{P}_{r-1}} \ensuremath{\mathfrak{p}}roj_{S' \nsbr{M}_\lambda} x =
\ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{1+\ensuremath{\mathfrak{t}}au}{2}(p^1_{M_{\lambda - a_i} \ensuremath{\otimes} M_{\lambda - a_j}} + p^1_{M_{\lambda - a_j} \ensuremath{\otimes} M_{\lambda - a_i}}) (x\mathcal{P}_{r-1})
\end{equation}
is nonzero (the equality is by Lemma \ref{l restriction facts} (iii)). This is shown in three cases.
\noindent
The case $i \neq k_\lambda$ and $j \neq 1$: $s_{r-1} \notin R(C_T)$ and $s_{r-1} \notin R(\ensuremath{C^{\prime}}_{U})$, so $x \mathcal{P}_{r-1}$ is computed using the fourth case of (\ref{e sP on C C'}). There holds
\ensuremath{\flat}egin{equation}gin{align*}
&\ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{1+\ensuremath{\mathfrak{t}}au}{2}(p^1_{M_{\lambda-a_i} \ensuremath{\otimes} M_{\lambda - a_j}}+p^1_{M_{\lambda-a_j} \ensuremath{\otimes} M_{\lambda - a_i}}) (x \mathcal{P}_{r-1})\\
&\equiv \displaystyle \sum_{\substack{s_{r-1} \in R(C_{A}), \\ s_{r-1} \in R(\ensuremath{C^{\prime}}_B),\\ A_{a_i} = r, \ B_{a_j} = r}} \ensuremath{\mathfrak{m}}u(A, T)\ensuremath{\mathfrak{m}}u(B,U) (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}}_{A})^J \cdot (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{B})^J + \displaystyle \sum_{\substack{s_{r-1} \in R(C_{A}), \\ s_{r-1} \in R(\ensuremath{C^{\prime}}_B),\\ A_{a_j} = r, \ B_{a_i} = r}} \ensuremath{\mathfrak{m}}u(A, T)\ensuremath{\mathfrak{m}}u(B,U) (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{B})^J \cdot (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}}_{A})^J, \\
&\equiv \displaystyle \sum_{\substack{s_{r-1} \in R(C_{A}), \\ s_{r-1} \in R(\ensuremath{C^{\prime}}_B),\\ A_{a_i} = r, \ B_{a_j} = r}} \ensuremath{\mathfrak{m}}u(A, T)\ensuremath{\mathfrak{m}}u(B,U) (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{A})^J \cdot (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{B})^J + \displaystyle \sum_{\substack{s_{r-1} \in R(C_{A}), \\ s_{r-1} \in R(\ensuremath{C^{\prime}}_B),\\ A_{a_j} = r, \ B_{a_i} = r}} \ensuremath{\mathfrak{m}}u(A, T)\ensuremath{\mathfrak{m}}u(B,U) (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{B})^J \cdot (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{A})^J,
\end{align*}
where the first equivalence is by the projection lemma, the second is by Theorem \ref{t transition C' to C}, and the equivalences are mod $\ensuremath{u} \ensuremath{\mathscr{L}}_{\lambda-a_i} \ensuremath{\otimes} \ensuremath{\mathscr{L}}_{\lambda-a_j}$.
The last line is nonzero because $(\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{T'})^J \cdot (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{U'})^J$ appears in the left sum, the coefficients $\ensuremath{\mathfrak{m}}u(A,T)\ensuremath{\mathfrak{m}}u(B,U)$ are nonnegative (Theorem \ref{t positive coefficients}), and $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{\lambda-a_i} \ensuremath{\otimes} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{\lambda-a_j}$ is a $\ensuremath{K}_0$-basis of $\ensuremath{\mathscr{L}}_{\lambda-a_i} \ensuremath{\otimes} \ensuremath{\mathscr{L}}_{\lambda-a_j}$.
\noindent
The case $i \neq k_\lambda, j = 1$ (the $i = k_\lambda, j \neq 1$ case is similar): $x \mathcal{P}_{r-1}$ is computed using the third or fourth case of (\ref{e sP on C C'}).
A careful application of the projection lemma yields
\ensuremath{\flat}egin{equation}gin{align*}
&\ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{1+\ensuremath{\mathfrak{t}}au}{2}(p^1_{M_{\lambda-a_i} \ensuremath{\otimes} M_{\lambda - a_j}}+p^1_{M_{\lambda-a_j} \ensuremath{\otimes} M_{\lambda - a_i}}) (x \ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{\mathcal{P}_{r-1}}{[2]})\\
& \equiv \ensuremath{\mathfrak{p}}m\displaystyle \sum_{\substack{s_{r-1} \in R(C_{A}), \\ A_{a_i} = r}} \ensuremath{\mathfrak{m}}u(A, T) (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}}_{A})^J \cdot (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{U})^J
\ensuremath{\mathfrak{m}}od \ensuremath{u} \ensuremath{\mathscr{L}}_{\lambda-a_i} \ensuremath{\otimes} \ensuremath{\mathscr{L}}_{\lambda-a_j}
\end{align*}
The second line is nonzero because $(\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}}_{T'})^J \cdot (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{U})^J$ appears in sum and $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_{\lambda-a_i} \ensuremath{\otimes} \ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_{\lambda-a_j}$ is a $\ensuremath{K}_0$-basis of $\ensuremath{\mathscr{L}}_{\lambda-a_i} \ensuremath{\otimes} \ensuremath{\mathscr{L}}_{\lambda-a_j}$.
It follows from Proposition \ref{p triv in Mdual M} (ii) that $\ensuremath{\mathfrak{p}}roj_{S' \nsbr{M}_{\lambda}} x = C_T \cdot \ensuremath{C^{\prime}}_U$. Then by Lemma \ref{l restriction facts} (ii) and Corollary \ref{c geck relative a invariant}, $\ensuremath{\mathfrak{p}}rojres_{\nsbr{M}_{\lambda - a_{k_\lambda}, \lambda - a_1}} \ensuremath{\mathfrak{p}}roj_{S' \nsbr{M}_{\lambda}} x = C_T \cdot \ensuremath{C^{\prime}}_U$, so $\ensuremath{\mathfrak{p}}roj_{S' \nsbr{M}_{\lambda}} x \in \nsbr{M}_{\lambda - a_{k_\lambda}, \lambda-a_1} \subseteq S' \nsbr{M}_{\lambda}$. Hence the left-hand side of \eqref{e restrict two orders} being nonzero implies that $\nsbr{M}_{\lambda - a_{k_\lambda}, \lambda-a_1}$ glues to $\nsbr{M}_{\lambda - a_i, \lambda - a_j}$.
Fix $i \in [k_\lambda-1]$ and set $\nu = \lambda - a_i$.
Now we show that $\ensuremath{\flat}igoplus_{j \leq i} \nsbr{M}_{\lambda - a_{k_\lambda}, \lambda - a_j}$ glues to $S' \nsbr{M}_{\nu}$.
Choose $T,U \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda)$ so that
\ensuremath{\flat}egin{equation}gin{list} {(\ensuremath{\mathfrak{a}}rabic{ctr})} {\ensuremath{u}secounter{ctr} \setlength{\itemsep}{1pt} \setlength{\ensuremath{\mathfrak{t}}opsep}{2pt}}
\item $T_{a_{k_\lambda}} = r$ and there is an edge $T \dkt{r-1} T'$ with $T'_{a_i} = r$.
\item $U_{a_i} = r$ and $U \neq T'$.
\end{list}
This is possible since we can assume $S' \nsbr{M}_{\nu}$ is nonzero, which is equivalent to $|\ensuremath{\mathfrak{t}}ext{SYT}(\nu)| > 1$.
Then $C_T \cdot \ensuremath{C^{\prime}}_U \mathcal{P}_{r-1}$ is computed using the third or fourth case of (\ref{e sP on C C'}) with $\cdot$ in place of $\ensuremath{\otimes}$. A careful application of the projection lemma yields the first equivalence below
\ensuremath{\flat}egin{equation}gin{align*}
\ensuremath{\mathfrak{p}}rojres_{S' \nsbr{M}_{\nu}} p^1_{M_{\nu} \ensuremath{\otimes} M_{\nu}} \ensuremath{\flat}ig(C_T \cdot \ensuremath{C^{\prime}}_U \ensuremath{\mathfrak{t}}extstyle \ensuremath{[2]^2}rac{\mathcal{P}_{r-1}}{[2]}\ensuremath{\flat}ig)
&\equiv
\ensuremath{\mathfrak{p}}rojres_{S' \nsbr{M}_{\nu}} \ensuremath{\ensuremath{\mathfrak{m}}athscr{B}}ig( \ensuremath{\mathfrak{p}}m \displaystyle \sum_{\substack{s_{r-1} \in R(C_A), \\ A_{a_i} = r}} \ensuremath{\mathfrak{m}}u(A,T) (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}}_{A})^J \cdot (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_U)^J \ensuremath{\ensuremath{\mathfrak{m}}athscr{B}}ig)
\\
&\equiv \ensuremath{\mathfrak{p}}m \displaystyle \sum_{\substack{s_{r-1} \in R(C_A), \\ A_{a_i} = r}} \ensuremath{\mathfrak{m}}u(A,T) \ensuremath{\mathfrak{p}}rojres_{S' \nsbr{M}_{\nu}}\ensuremath{\flat}ig((\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_{A})^J \cdot (\ensuremath{\ensuremath{\mathfrak{t}}ilde{C}^{\prime}}_U)^J\ensuremath{\flat}ig)
\ensuremath{\mathfrak{m}}od \ensuremath{u} \ensuremath{\mathfrak{p}}roj_{S' \nsbr{M}_\nu} (\ensuremath{\mathscr{L}}_{\nu}\ensuremath{\otimes} \ensuremath{\mathscr{L}}_\nu).
\end{align*}
The second equivalence is by Theorem \ref{t transition C' to C}.
It follows from Lemma \ref{l not in triv} and $U \neq T'$ that the second line is nonzero.
By an argument similar to that in the previous paragraph, $C_T \cdot \ensuremath{C^{\prime}}_U = \ensuremath{\mathfrak{p}}roj_{S' \nsbr{M}_{\lambda}} (C_T \ensuremath{\otimes} \ensuremath{C^{\prime}}_U) \in \ensuremath{\flat}igoplus_{j \leq i} \nsbr{M}_{\lambda - a_{k_\lambda}, \lambda - a_j}$, hence $\ensuremath{\flat}igoplus_{j \leq i} \nsbr{M}_{\lambda - a_{k_\lambda}, \lambda - a_j}$ glues to $S' \nsbr{M}_{\nu}$.
By an argument similar to the $i =1$ case of the previous paragraph, $\nsbr{M}_{\lambda - a_{k_\lambda}, \lambda - a_1}$ glues to $S' \nsbr{M}_{\lambda-a_{k_\lambda}}$.
Let $X_\epsilon \subseteq S'\nsbr{M}_{\lambda}$ be the isotypic component of $\ensuremath{\mathscr{R}}es_{\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}} S'\nsbr{M}_{\lambda}$ of irreducible type $\ensuremath{K} \nsbr{\epsilon}_+$ and let $X_\epsilon^\ensuremath{\mathfrak{m}}athbf{A} := \ensuremath{\flat}igcap_{i \in [r-2]} \ensuremath{\mathfrak{k}}er(m_{\mathcal{Q}_i})$ be an integral form of $X_\epsilon$, where
$m_{\mathcal{Q}_i}: S'\nsbr{M}_{\lambda}^\ensuremath{\mathfrak{m}}athbf{A} \ensuremath{\mathfrak{t}}o S'\nsbr{M}_{\lambda}^\ensuremath{\mathfrak{m}}athbf{A}$ is right multiplication by $\mathcal{Q}_i$; there holds $\ensuremath{K} \ensuremath{\otimes}_\ensuremath{\mathfrak{m}}athbf{A} X_\epsilon^\ensuremath{\mathfrak{m}}athbf{A} \cong X_\epsilon.$ To complete the proof, it suffices to show that $x \ensuremath{\mathscr{H}}_r \not \subseteq X_\epsilon^\ensuremath{\mathfrak{m}}athbf{A}$ for any $x\in X_\epsilon^\ensuremath{\mathfrak{m}}athbf{A}$.
If $x \ensuremath{\mathscr{H}}_r \subseteq X_\epsilon^\ensuremath{\mathfrak{m}}athbf{A}$, then $\ensuremath{\mathscr{R}}es_{\ensuremath{\mathbb{Q}} \ensuremath{\mathcal{S}}_{r-1}} (x\ensuremath{\mathscr{H}}_r|_{\ensuremath{u}=1})$ is a direct sum of copies of the trivial $\ensuremath{\mathbb{Q}} \ensuremath{\mathcal{S}}_{r-1}$-module (where $N|_{\ensuremath{u}=1}$ of an $\ensuremath{\mathfrak{m}}athbf{A}$-module $N_\ensuremath{\mathfrak{m}}athbf{A}$ is defined to be $\ensuremath{\mathbb{Q}} \ensuremath{\otimes}_{\ensuremath{\mathfrak{m}}athbf{A}} N_\ensuremath{\mathfrak{m}}athbf{A}$, the map $\ensuremath{\mathfrak{m}}athbf{A} \ensuremath{\mathfrak{t}}o \ensuremath{\mathbb{Q}}$ given by $\ensuremath{u} \ensuremath{\mathfrak{m}}apsto 1$). It follows that $x\ensuremath{\mathscr{H}}_r|_{\ensuremath{u}=1}$ is a direct sum of copies of the trivial $\ensuremath{\mathbb{Q}} \ensuremath{\mathcal{S}}_r$-module. But this is impossible since there are no copies of the trivial $\ensuremath{\mathbb{Q}} \ensuremath{\mathcal{S}}_r$-module in $S'\nsbr{M}_{\lambda}|_{\ensuremath{u}=1}$.
\end{proof}
\subsection{Completing the proof}
\ensuremath{\flat}egin{equation}gin{proof}[Proof of Theorem \ref{t nsH irreducibles two row case}]
The proof is by induction on $r$. Given that the $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}$-irreducibles of the theorem are distinct, it follows from Propositions \ref{p case lambda mu generic}, \ref{p case lambda one from mu}, \ref{p case wedge lambda}, and \ref{p case sym lambda} that the $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-modules in (1)--(4) are irreducible.
The list of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducibles is complete because $\ensuremath{\mathscr{R}}es_{\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}} \ensuremath{K} (\ensuremath{\mathscr{H}}_{r,2} \ensuremath{\otimes} \ensuremath{\mathscr{H}}_{r,2})$ is a faithful $ \ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-module and all the $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducible constituents of $\nsbr{M}_{\lambda, \ensuremath{\mathfrak{m}}u}$ appear in the list.
Also, the split semisimplicity of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$ follows from the proofs of Propositions \ref{p case lambda mu generic}, \ref{p case lambda one from mu}, \ref{p case wedge lambda}, and \ref{p case sym lambda} since these work just as well over any field extension of $\ensuremath{K}$.
We now must show that the irreducibles in the list are distinct.
For this we apply Proposition \ref{p r-1 restrictions} and refine the cases as follows:
\ensuremath{\flat}egin{equation}gin{list}{} {\ensuremath{u}secounter{ctr} \setlength{\itemsep}{1pt} \setlength{\ensuremath{\mathfrak{t}}opsep}{2pt}}
\item[(1a)] $\ensuremath{\mathscr{R}}es_{\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}} \nsbr{M}_{\lambda, \ensuremath{\mathfrak{m}}u} \cong \ensuremath{\flat}igoplus_{i \in [k_\lambda], j \in [k_\ensuremath{\mathfrak{m}}u]} \nsbr{M}_{\lambda - a_i, \ensuremath{\mathfrak{m}}u - b_j}$, if $|\lambda \cap \ensuremath{\mathfrak{m}}u| < r-1$.
\item[(1b)] $\ensuremath{\mathscr{R}}es_{\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}} \nsbr{M}_{\lambda, \ensuremath{\mathfrak{m}}u} \cong \ensuremath{\flat}igoplus_{\substack{i \in [k_\lambda], j \in [k_\ensuremath{\mathfrak{m}}u], \\ (i, j) \neq (k, l)}} \nsbr{M}_{\lambda - a_i, \ensuremath{\mathfrak{m}}u - b_j} \oplus S' \nsbr{M}_{\nu} \oplus \nswedge{2}{M}_\nu \oplus \ensuremath{K} \nsbr{\epsilon}_+$, where $\nu = \lambda - a_k = \ensuremath{\mathfrak{m}}u - b_l$ and $\nu \neq (r-1)$.
\item[(1b$'$)] $\ensuremath{\mathscr{R}}es_{\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}} \nsbr{M}_{(r), (r-1,1)} \cong \nsbr{M}_{(r-1), (r-2,1)} \oplus \ensuremath{K} \nsbr{\epsilon}_+$.
\item[(2)] $\ensuremath{\mathscr{R}}es_{\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}} S' \nsbr{M}_{\lambda} \cong \ensuremath{\flat}igoplus_{1 \leq i < j \leq k_\lambda} \nsbr{M}_{\lambda - a_i, \lambda - a_j} \oplus \ensuremath{\flat}igoplus_{i \in [k_\lambda]} S' \nsbr{M}_{\lambda - a_i} \oplus \ensuremath{K} \nsbr{\epsilon}_+^{\oplus k_\lambda -1}$, for $+\lambda \in \nsbr{\mathscr{P}}_{r,2}$, $\lambda \neq (r-1,1)$.
\item[(2$'$)] $\ensuremath{\mathscr{R}}es_{\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}} S' \nsbr{M}_{(r-1,1)} \cong \nsbr{M}_{(r-1), (r-2,1)} \oplus S' \nsbr{M}_{(r-2,1)} \oplus \ensuremath{K} \nsbr{\epsilon}_+$, $r > 2$.
\item[(3)] $\ensuremath{\mathscr{R}}es_{\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}} \nswedge{2}{M}_{\lambda} \cong \ensuremath{\flat}igoplus_{1 \leq i < j \leq k_\lambda} \nsbr{M}_{\lambda - a_i, \lambda - a_j} \oplus \ensuremath{\flat}igoplus_{i \in [k_\lambda]} \nswedge{2}{M}_{\lambda - a_i}$, for $-\lambda \in \nsbr{\mathscr{P}}_{r,2}$, \newline $\lambda \neq (r-1,1)$.
\item[(3$'$)] $\ensuremath{\mathscr{R}}es_{\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}} \nswedge{2}{M}_{(r-1,1)} \cong \nsbr{M}_{(r-1), (r-2,1)} \oplus \nswedge{2}{M}_{(r-2,1)}$, $r > 2$.
\item[(4)] $\ensuremath{\mathscr{R}}es_{\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}} \ensuremath{K} \nsbr{\epsilon}_+ \cong \ensuremath{K} \nsbr{\epsilon}_+$.
\end{list}
Note that for $r=3$, $S' \nsbr{M}_{(1,1)}$ and $\nswedge{2}{M}_{(1,1)}$ are zero in the right-hand sides of (2$'$) and (3$'$), respectively.
For $r \leq 3$, we check by hand that all these irreducibles are distinct. In particular, we must check that $\nsbr{M}_{(3), (2, 1)} \not \cong S' \nsbr{M}_{(2,1)}$, which happen to have isomorphic restrictions to $\nsbr{\ensuremath{\mathscr{H}}}_{2,2}$.
Assuming that $r > 3$ we will show that this list of irreducibles does not contain repetitions by showing that the irreducibles have distinct restrictions to $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}$. We do this in two steps:
\ensuremath{\flat}egin{equation}gin{list} {({\ensuremath{\ensuremath{\mathfrak{m}}athscr{A}}}lph{ctr})} {\ensuremath{u}secounter{ctr} \setlength{\itemsep}{1pt} \setlength{\ensuremath{\mathfrak{t}}opsep}{2pt}}
\item The $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}$ restrictions of any two irreducibles of a given type above are nonisomorphic.
\item The $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}$ restriction of an irreducible of type ($\ensuremath{\mathfrak{a}}lpha$) is not isomorphic to the $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1, 2}$ restriction of an irreducible of type ($\ensuremath{\flat}egin{equation}ta$), if $\ensuremath{\mathfrak{a}}lpha \neq \ensuremath{\flat}egin{equation}ta$.
\end{list}
Claim (A) is straightforward: for example, to see that two irreducibles of type (1a) are distinct, suppose $M$ is a $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-module of type (1a) and $\ensuremath{\mathscr{R}}es_{\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}} M \cong \ensuremath{\flat}igoplus_{i \in [l]} \nsbr{M}_{\nu^{(2i-1)}, \nu^{(2i)}}$, for some $\nu^{(j)} \vdash r-1$. The set of partitions
$\{\nu^{(i)} \cup \nu^{(j)}: i, j \in [2l],\ |\nu^{(i)} \cup \nu^{(j)}| = r\}$ consists of two partitions, call them $\lambda$ and $\ensuremath{\mathfrak{m}}u$. Then $M = \nsbr{M}_{\lambda,\ensuremath{\mathfrak{m}}u}$.
For claim (B), we can look at which $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}$ restrictions have no occurrences of an irreducible of the form $S' \nsbr{M}_\nu$ and which ones have at least one occurrence of an irreducible of the form $S' \nsbr{M}_\nu$, and similarly for the forms $\nswedge{2}{M}_\nu$ and $\ensuremath{K} \nsbr{\epsilon}_+$.
This yields the claim (B) for all pairs of types except (2) and (2$'$), (3) and (3$'$), and (1b$'$) and (4) which are all easy to check directly.
The part of the theorem about the decomposition of $\ensuremath{K} (\ensuremath{\mathscr{H}}_{r,2} \ensuremath{\otimes} \ensuremath{\mathscr{H}}_{r,2})$-modules into $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducibles is immediate from the definitions in \ensuremath{\mathfrak{t}}extsection\ref{ss some representation theory of nsH}.
\end{proof}
\ensuremath{\flat}egin{equation}gin{remark}
It is possible that this proof would be easier using a Hecke algebra analog of Young's orthogonal basis (see \cite{Wenzl}) instead of the lower and upper canonical bases. However, we believe it to be important to understand the action of $\nsbr{\ensuremath{\mathscr{H}}}_{r,2}$ on the lower and upper canonical basis of $ \nsbr{M}_{\lambda,\ensuremath{\mathfrak{m}}u}$ anyway. The canonical bases also have the advantage that all computations except the projections onto $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}$-irreducible isotypic components take place over $\ensuremath{\mathfrak{m}}athbf{A}[\ensuremath{[2]^2}rac{1}{[2]}]$ rather than $\ensuremath{K}$ or some extension of $\ensuremath{K}$.
Moreover, once the results of \ensuremath{\mathfrak{t}}extsection\ref{ss lifts} are in place, the only thing we need to know about the $\ensuremath{\mathcal{S}}_r$-graphs $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma'_\lambda$ and $\ensuremath{\ensuremath{\mathfrak{m}}athscr{G}}amma_\lambda$ are the edges corresponding to dual Knuth transformations; it is likely that a proof using a Hecke orthogonal basis would amount to showing the existence of certain dual Knuth transformations in a similar way.
\end{remark}
\section{Seminormal bases}
\label{s Seminormal bases}
We recall the definition of a seminormal basis from \cite{RamSeminormal}, observe that $\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducibles have seminormal bases, and give combinatorial labels for the elements of these bases.
\ensuremath{\flat}egin{equation}gin{definition}\label{d seminormal}
Given a chain of semisimple $\ensuremath{K}$-algebras $\ensuremath{K}\cong H_1 \subseteq H_2 \subseteq \dots \subseteq H_r$ and an $H_r$-module $N_\lambda$, a \emph{seminormal basis} of $N_\lambda$ is a $\ensuremath{K}$-basis $B$ of $N_\lambda$ compatible with the restrictions in the following sense: there is a partition $B = B_{\ensuremath{\mathfrak{m}}u^1} \sqcup \dots \sqcup B_{\ensuremath{\mathfrak{m}}u^k}$ such that $N_\lambda \cong N_{\ensuremath{\mathfrak{m}}u^1} \oplus \dots \oplus N_{\ensuremath{\mathfrak{m}}u^k}$ as $H_{r-1}$-modules, where $N_{\ensuremath{\mathfrak{m}}u^i} = \ensuremath{K} B_{\ensuremath{\mathfrak{m}}u^i}$. Further, there is a partition of each $B_{\ensuremath{\mathfrak{m}}u^i}$ that gives rise to a decomposition of $N_{\ensuremath{\mathfrak{m}}u^i}$ into $H_{r-2}$-irreducibles, and so on, all the way down to $H_1$.
\end{definition}
If the restriction of an $H_i$-irreducible to $H_{i-1}$ is multiplicity-free for all $i$, then a seminormal basis of an $H_r$-irreducible is unique up to a diagonal transformation.
A consequence of Theorem \ref{t nsH irreducibles two row case} and Proposition \ref{p r-1 restrictions} is that the restriction of a $ \ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducible to $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}$ is multiplicity-free. Thus each $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducible $\nsbr{M}_\ensuremath{\mathfrak{a}}lpha$, $\ensuremath{\mathfrak{a}}lpha \in \nsbr{\mathscr{P}}_{r,2}$, has a seminormal basis $\nsbr{\ensuremath{\mathfrak{t}}ext{SN}}_\ensuremath{\mathfrak{a}}lpha$ that is unique up to a diagonal transformation. We adopt the convention to take the seminormal basis with respect to the chain $\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{J_1} \subseteq \cdots \subseteq \ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{J_{r-1}} \subseteq \ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{J_{r}}$, where $J_i = \{s_1,\ensuremath{\ensuremath{\mathfrak{t}}rianglelefteq}ots,s_{i-1}\}$ and $\nsbr{\ensuremath{\mathscr{H}}}_L$ (for $L \subseteq S$) is the subalgebra of $\nsbr{\ensuremath{\mathscr{H}}}_{r,2}$ generated by $\mathcal{P}_s$, $s \in L$.
For $\lambda, \ensuremath{\mathfrak{m}}u \vdash r$ with $\ell(\lambda), \ell(\ensuremath{\mathfrak{m}}u) \leq 2$, $M_\lambda \ensuremath{\otimes} M_\ensuremath{\mathfrak{m}}u$ has a multiplicity-free decomposition into $\nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-modules (by Theorem \ref{t nsH irreducibles two row case}). Thus we can also define a seminormal basis $\nsbr{ \ensuremath{\mathfrak{t}}ext{SN}}_{\lambda,\ensuremath{\mathfrak{m}}u}$ of $M_\lambda \ensuremath{\otimes} M_\ensuremath{\mathfrak{m}}u$ to be the union of the seminormal bases of its $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducible constituents.
We are interested in these seminormal bases primarily as a tool for constructing a canonical basis of a $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducible that is compatible with its decomposition into irreducibles at $\ensuremath{u} =1$, as described in \cite[\ensuremath{\mathfrak{t}}extsection19]{BMSGCT4}. Even though the irreducibles of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$ are close to those of $\ensuremath{K}(\ensuremath{\mathscr{H}}_{r,2}\ensuremath{\otimes}\ensuremath{\mathscr{H}}_{r,2})$,
the seminormal basis $\nsbr{\ensuremath{\mathfrak{t}}ext{SN}}_{\lambda,\ensuremath{\mathfrak{m}}u}$ of $M_\lambda\ensuremath{\otimes} M_\ensuremath{\mathfrak{m}}u$ using the chain $\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{J_1} \subseteq \cdots \subseteq \ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{J_{r-1}} \subseteq \ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{J_r}$ is significantly different from the seminormal basis using the chain $\ensuremath{K}(\ensuremath{\mathscr{H}}_{1,2} \ensuremath{\otimes} \ensuremath{\mathscr{H}}_{1,2}) \subseteq \cdots \subseteq \ensuremath{K} (\ensuremath{\mathscr{H}}_{r-1,2} \ensuremath{\otimes} \ensuremath{\mathscr{H}}_{r-1,2}) \subseteq \ensuremath{K} (\ensuremath{\mathscr{H}}_{r,2} \ensuremath{\otimes} \ensuremath{\mathscr{H}}_{r,2})$.
Thus even though the representation theory of the nonstandard Hecke algebra alone is not enough to understand Kronecker coefficients,
there is hope that the seminormal bases $\nsbr{\ensuremath{\mathfrak{t}}ext{SN}}_{\lambda,\ensuremath{\mathfrak{m}}u}$ will yield a better understanding of Kronecker coefficients.
\ensuremath{\flat}egin{equation}gin{remark}
The $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_6$-module $\nsbr{M}_{(4,1,1),(3,2,1)}$ is irreducible and its $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_5$ restriction is not multiplicity-free. However, we suspect that $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1}$ restrictions of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r}$-irreducibles are very often multiplicity-free and, if not, the multiplicities are small.
\end{remark}
\subsection{Combinatorics of seminormal bases}
For $\lambda, \ensuremath{\mathfrak{m}}u \vdash r$, $\ell(\lambda), \ell(\ensuremath{\mathfrak{m}}u) \leq 2$, define a bijection
\[\ensuremath{\mathfrak{t}}ext{SYT}(\lambda) \ensuremath{\mathfrak{t}}imes \ensuremath{\mathfrak{t}}ext{SYT}(\ensuremath{\mathfrak{m}}u) \xrightarrow{\ensuremath{\mathfrak{a}}lpha_{\lambda,\ensuremath{\mathfrak{m}}u}} \nsbr{\ensuremath{\mathfrak{t}}ext{SN}}_{\lambda,\ensuremath{\mathfrak{m}}u} \]
inductively as follows. Maintain the notation of \eqref{e ai definition} for the outer corners of $\lambda$ and $\ensuremath{\mathfrak{m}}u$.
In what follows let $(T, U) \in \ensuremath{\mathfrak{t}}ext{SYT}(\lambda) \ensuremath{\mathfrak{t}}imes \ensuremath{\mathfrak{t}}ext{SYT}(\ensuremath{\mathfrak{m}}u)$ and $i$ and $j$ be such that $T_{a_i} = r$ and $U_{b_j} = r$.
Let $Y_\lambda$ be the tableau with entries $2c-1, 2c$ in column $c$ for each column of $\lambda$ of height 2.
For convenience, we identify the basis $\nsbr{\ensuremath{\mathfrak{t}}ext{SN}}_{\lambda,\ensuremath{\mathfrak{m}}u}$ with the corresponding subset of one-dimensional subspaces of $M_\lambda \ensuremath{\otimes} M_\ensuremath{\mathfrak{m}}u$.
\ensuremath{\flat}egin{equation}gin{list}{(\roman{ctr})} {\ensuremath{u}secounter{ctr} \setlength{\itemsep}{1pt} \setlength{\ensuremath{\mathfrak{t}}opsep}{2pt}}
\item If $\lambda \neq \ensuremath{\mathfrak{m}}u$, set
\[\ensuremath{\mathfrak{a}}lpha_{\lambda,\ensuremath{\mathfrak{m}}u}(T, U) = \ensuremath{\mathfrak{a}}lpha_{\lambda-a_i,\ensuremath{\mathfrak{m}}u-a_j}(T_{\lambda-a_i},\ U_{\lambda-a_j}).\]
\item If $\lambda = \ensuremath{\mathfrak{m}}u$, then set
\[\ensuremath{\mathfrak{a}}lpha_{\lambda,\ensuremath{\mathfrak{m}}u}(T, U) =
\ensuremath{\flat}egin{equation}gin{cases}
\ensuremath{K} \nsbr{\epsilon}_+ \subseteq S^2 M_\lambda & \ensuremath{\mathfrak{t}}ext{if $(T,U) = (Y_\lambda,Y_\lambda)$}, \\
\ensuremath{\mathfrak{a}}lpha_{\lambda-a_i,\ensuremath{\mathfrak{m}}u-a_j}(T_{\lambda-a_i},\ U_{\lambda-a_j}) & \ensuremath{\mathfrak{t}}ext{otherwise},
\end{cases}
\]
where $\ensuremath{\mathfrak{a}}lpha_{\lambda-a_i,\ensuremath{\mathfrak{m}}u-a_j}(T_{\lambda-a_i},\ U_{\lambda-a_j})$ is interpreted as a seminormal basis element of
\[\left\{
\ensuremath{\flat}egin{equation}gin{array}{ll}
M_{\lambda-a_i}\ensuremath{\otimes} M_{\lambda-a_j} \subseteq \ensuremath{\mathscr{R}}es_{\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}} M_\lambda\ensuremath{\otimes} M_\lambda& \ensuremath{\mathfrak{t}}ext{if $i = j$,} \\
M_{\lambda-a_i}\ensuremath{\otimes} M_{\lambda-a_j} \subseteq \ensuremath{\mathscr{R}}es_{\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}}S' \nsbr{M}_\lambda & \ensuremath{\mathfrak{t}}ext{if $i < j$},\\
M_{\lambda-a_i}\ensuremath{\otimes} M_{\lambda-a_j} \subseteq \ensuremath{\mathscr{R}}es_{\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r-1,2}}\nswedge{2}{M}_\lambda & \ensuremath{\mathfrak{t}}ext{if $i > j$}.
\end{array}
\right.
\]
\end{list}
Given Proposition \ref{p r-1 restrictions} and Theorem \ref{t nsH irreducibles two row case}, it is clear that $\ensuremath{\mathfrak{a}}lpha_{\lambda,\ensuremath{\mathfrak{m}}u}$ is a well-defined bijection.
\ensuremath{\flat}egin{equation}gin{example}
The seminormal basis element { \setlength{\cellsize}{9pt} $\ensuremath{\mathfrak{a}}lpha_{(3,2),(3,2)}\left( \ensuremath{\mathfrak{t}}iny\ensuremath{\mathfrak{t}}ableau{1&2&4\\3&5}, \ensuremath{\mathfrak{t}}iny\ensuremath{\mathfrak{t}}ableau{1&3&4\\2&5} \right)$ } is a nonzero element of $S' \nsbr{M}_{(3,2)} \cap S' \nsbr{M}_{(3,1)} \cap S' \nsbr{M}_{(2,1)} \cap \nsbr{M}_{(2), (1,1)}$, where these are modules for $\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{J_5}$, $\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{J_{4}}$, $\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{J_{3}}$, and $\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{J_{2}}$, respectively.
The next two tables partially describe the bijection $\ensuremath{\mathfrak{a}}lpha_{(3,2),(3,2)}$; they give the $\ensuremath{K}\nsbr{\ensuremath{\mathscr{H}}}_{J_5}$ and $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{J_4}$-irreducibles that contain the seminormal basis element corresponding to each $(T,U) \in \ensuremath{\mathfrak{t}}ext{SYT}((3,2)) \ensuremath{\mathfrak{t}}imes \ensuremath{\mathfrak{t}}ext{SYT}((3,2))$, where row labels correspond to $T$ and column labels correspond to $U$; the basis element just described is in bold.
\[
\setlength{\cellsize}{10pt}\small\ensuremath{\flat}egin{equation}gin{array}{l|ccccc}
\rule[-13pt]{0pt}{0pt} & \ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&2&3\\4&5} & \ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&2&4\\3&5} & \ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&3&4\\2&5} & \ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&2&5\\3&4} & \ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&3&5\\2&4} \\
\ensuremath{\ensuremath{\mathfrak{m}}athfrak{h}}line \\
\ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&2&3\\4&5} & S' \nsbr{M}_{(3,2)} & S' \nsbr{M}_{(3,2)} & S' \nsbr{M}_{(3,2)} & S' \nsbr{M}_{(3,2)} & S' \nsbr{M}_{(3,2)} \\ \\
\ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&2&4\\3&5} & \nswedge{2}{M}_{(3,2)} & S' \nsbr{M}_{(3,2)} & \ensuremath{\mathfrak{m}}athbf{S' \nsbr{M}_{(3,2)}} & S' \nsbr{M}_{(3,2)} & S' \nsbr{M}_{(3,2)} \\ \\
\ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&3&4\\2&5} & \nswedge{2}{M}_{(3,2)} & \nswedge{2}{M}_{(3,2)} & S' \nsbr{M}_{(3,2)} & S' \nsbr{M}_{(3,2)} & S' \nsbr{M}_{(3,2)} \\ \\
\ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&2&5\\3&4} & \nswedge{2}{M}_{(3,2)} & \nswedge{2}{M}_{(3,2)} & \nswedge{2}{M}_{(3,2)} & S' \nsbr{M}_{(3,2)} & S' \nsbr{M}_{(3,2)} \\ \\
\ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&3&5\\2&4} & \nswedge{2}{M}_{(3,2)} & \nswedge{2}{M}_{(3,2)} & \nswedge{2}{M}_{(3,2)} & \nswedge{2}{M}_{(3,2)} & \ensuremath{K}\nsbr{\epsilon}_+ \\ \\
\end{array} \]
\[
\setlength{\cellsize}{10pt} \small
\ensuremath{\flat}egin{equation}gin{array}{l|ccccc}
\rule[-13pt]{0pt}{0pt} & \ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&2&3 \\ 4&5} & \ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&2&4 \\ 3&5} & \ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&3&4 \\ 2&5} & \ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&2&5 \\ 3&4} & \ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&3&5 \\2&4} \\
\ensuremath{\ensuremath{\mathfrak{m}}athfrak{h}}line \\
\ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&2&3 \\ 4&5} & S' \nsbr{M}_{(3,1)} & S' \nsbr{M}_{(3,1)} & S' \nsbr{M}_{(3,1)} & \nsbr{M}_{(3,1),(2,2)} & \nsbr{M}_{(3,1),(2,2)} \\ \\
\ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&2&4 \\ 3&5} & \nswedge{2}{M}_{(3,1)} & S' \nsbr{M}_{(3,1)} & \ensuremath{\mathfrak{m}}athbf{S' \nsbr{M}_{(3,1)}} & \nsbr{M}_{(3,1),(2,2)} & \nsbr{M}_{(3,1),(2,2)} \\ \\
\ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&3&4 \\ 2&5} & \nswedge{2}{M}_{(3,1)} & \nswedge{2}{M}_{(3,1)} & \ensuremath{K}\nsbr{\epsilon}_+ & \nsbr{M}_{(3,1),(2,2)} & \nsbr{M}_{(3,1),(2,2)} \\ \\
\ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&2&5 \\ 3&4} & \nsbr{M}_{(2,2),(3,1)} & \nsbr{M}_{(2,2),(3,1)} & \nsbr{M}_{(2,2),(3,1)} & S' \nsbr{M}_{(2,2)} & S' \nsbr{M}_{(2,2)} \\ \\
\ensuremath{[2]^2}ootnotesize\ensuremath{\mathfrak{t}}ableau{1&3&5 \\ 2&4} & \nsbr{M}_{(2,2),(3,1)} & \nsbr{M}_{(2,2),(3,1)} & \nsbr{M}_{(2,2),(3,1)} & \nswedge{2}{M}_{(2,2)} & \ensuremath{K}\nsbr{\epsilon}_+ \\ \\
\end{array}
\]
\end{example}
\section{Enumerative consequence}
\label{s enumerative consequence}
Let $C_r = \ensuremath{[2]^2}rac{1}{r+1}\ensuremath{\flat}inom{2r}{r}$ be the $r$-th Catalan number. Theorem \ref{t nsH irreducibles two row case} has the following corollary.
\ensuremath{\flat}egin{equation}gin{corollary} \label{c enumerative 2 row case}
The algebra $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r, 2}$ has dimension $\ensuremath{\flat}inom{C_r}{2} - \ensuremath{\flat}inom{r}{\ensuremath{\ensuremath{\mathfrak{m}}athfrak{h}}alfr} + \ensuremath{\ensuremath{\mathfrak{m}}athfrak{h}}alfr + 2$.
\end{corollary}
\ensuremath{\flat}egin{equation}gin{proof}
It is well known that $\dim_\ensuremath{K}(\ensuremath{K} \ensuremath{\mathscr{H}}_{r, 2}) = C_r$ and therefore $\dim_\ensuremath{K}(\ensuremath{K} S^2 \ensuremath{\mathscr{H}}_{r, 2}) = \ensuremath{\flat}inom{C_r + 1}{2}$. On the other hand, the list of irreducibles of $\ensuremath{K} S^2 \ensuremath{\mathscr{H}}_{r, 2}$ given in Proposition-Definition \ref{p S2Hr representations} and the split semisimplicity of $\ensuremath{K} S^2 \ensuremath{\mathscr{H}}_{r, 2}$ imply that
\ensuremath{\flat}egin{equation} \label{e dim S2 H}
\dim_\ensuremath{K}(\ensuremath{K} S^2 \ensuremath{\mathscr{H}}_{r,2}) = \sum_{\substack{\lambda \ensuremath{\ensuremath{\mathfrak{m}}athfrak{g}}dneq \ensuremath{\mathfrak{m}}u, \\ \ell(\lambda), \ell(\ensuremath{\mathfrak{m}}u) \leq 2}} (f_\lambda f_\ensuremath{\mathfrak{m}}u)^2 + \sum_{\ell(\lambda) \leq 2} \left( {\ensuremath{\mathfrak{t}}extstyle \ensuremath{\flat}inom{f_\lambda+1}{2}^2 + \ensuremath{\flat}inom{f_\lambda}{2}^2 } \right),
\end{equation}
where $f_\lambda = \dim_\ensuremath{K}(M_\lambda) = |\ensuremath{\mathfrak{t}}ext{SYT}(\lambda)|$.
The list of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$-irreducibles from Theorem \ref{t nsH irreducibles two row case} and the split semisimplicity of $\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}$ imply that
\ensuremath{\flat}egin{equation} \label{e dim nsH}
\dim_\ensuremath{K}(\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r,2}) = \sum_{\substack{\lambda \ensuremath{\ensuremath{\mathfrak{m}}athfrak{g}}dneq \ensuremath{\mathfrak{m}}u, \\ \ell(\lambda), \ell(\ensuremath{\mathfrak{m}}u) \leq 2}} (f_\lambda f_\ensuremath{\mathfrak{m}}u)^2 + \sum_{\ell(\lambda) \leq 2,\ \lambda \neq (r)} {\ensuremath{\mathfrak{t}}extstyle \left( \left( \ensuremath{\flat}inom{f_\lambda+1}{2} - 1 \right)^2 + \ensuremath{\flat}inom{f_\lambda}{2}^2 \right)} + 1.
\end{equation}
Taking the difference of the right-hand sides of (\ref{e dim S2 H}) and (\ref{e dim nsH}) then yields the first of the following string of equalities.
\[
\ensuremath{\flat}egin{equation}gin{array}{lrl}
\dim_\ensuremath{K}(\ensuremath{K} \nsbr{\ensuremath{\mathscr{H}}}_{r, 2}) &= & \dim_\ensuremath{K}(\ensuremath{K} S^2 \ensuremath{\mathscr{H}}_{r,2}) + \sum_{\ell(\lambda) \leq 2,\ \lambda \neq (r)} \left( -2 \ensuremath{\flat}inom{f_\lambda + 1}{2} + 1 \right) \\[2mm]
&=& \ensuremath{\flat}inom{C_r + 1}{2} - \sum_{\ell(\lambda) \leq 2} f_\lambda^2 - \sum_{\ell(\lambda) \leq 2} f_\lambda + \left( \sum_{\ell(\lambda) \leq 2} 1 \right) + 1\\[2mm]
&=& \ensuremath{\flat}inom{C_r + 1}{2} - C_r - \sum_{\ell(\lambda) \leq 2} f_\lambda + (\ensuremath{\ensuremath{\mathfrak{m}}athfrak{h}}alfr + 1)+ 1\\[2mm]
&=& \ensuremath{\flat}inom{C_r}{2} - \ensuremath{\flat}inom{r}{\ensuremath{\ensuremath{\mathfrak{m}}athfrak{h}}alfr} + \ensuremath{\ensuremath{\mathfrak{m}}athfrak{h}}alfr + 2.
\end{array}
\]
The second equality follows from $\dim_\ensuremath{K}(\ensuremath{K} S^2 \ensuremath{\mathscr{H}}_{r, 2}) = \ensuremath{\flat}inom{C_r + 1}{2}$, the third equality comes from counting the dimension of the split semisimple algebra $\ensuremath{K} \ensuremath{\mathscr{H}}_{r,2}$ in two ways, and the fourth from the fact that $\dim_\ensuremath{\ensuremath{\mathfrak{m}}athbb{C}}(\ensuremath{\mathfrak{t}}ext{\rm Ind}_{P}^{\ensuremath{\mathcal{S}}_r} \ensuremath{\mathfrak{t}}ext{triv}) = \sum_{\ell(\lambda) \leq 2} f_\lambda$, where $P$ is the maximal parabolic subgroup $(\ensuremath{\mathcal{S}}_r)_{S \setminus s_{\ensuremath{\ensuremath{\mathfrak{m}}athfrak{h}}alfr}}$ of $\ensuremath{\mathcal{S}}_r$.
\end{proof}
\section*{Acknowledgments}
I am grateful to Ketan Mulmuley for helpful conversations and to John Wood, James Courtois, and Michael Bennett for help typing and typesetting figures.
\ensuremath{\flat}ibliographystyle{plain}
\ensuremath{\flat}ibliography{mycitations}
\end{document} | math | 161,378 |
\begin{document}
\title{Variance-based uncertainty relations}
\begin{CJK}{UTF8}{gbsn}
\author{Yichen Huang (黄溢辰)\thanks{[email protected]} \thanks{Present address: Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.}}
\affil{Department of Physics, University of California, Berkeley, Berkeley, California 94720, USA}
\maketitle
\end{CJK}
\begin{abstract}
Uncertainty relations are fundamental in quantum mechanics. Here I propose state-independent variance-based uncertainty relations for two or more arbitrary observables in finite dimensional spaces. The uncertainty relations provide near-optimal lower bounds in some typical examples, and are useful for entanglement detection.
\end{abstract}
\section{Introduction}
Uncertainty relations are fundamental in quantum mechanics, underlying many conceptual differences between classical and quantum theories. They reveal by rigorous inequalities that incompatible observables cannot be measured to arbitrarily high precision simultaneously. They are useful in many areas related or even unrelated to quantum mechanics: entanglement detection \cite{HHHH09, GT09, HT03, Hua10, Hua10E}, quantum cryptography \cite{WW10}, signal processing \cite{CT06}, etc.
The Heisenberg uncertainty relation \cite{Rob29} reads
\begin{equation}
V(A)V(B)\ge\big|\langle\Psi|[A,B]|\Psi\rangle\big|^2/4,
\end{equation}
where $V(A),V(B)$ are the variances of the Hermitian operators $A,B$. The formulation of the Heisenberg uncertainty relation is not always satisfactory. For instance, the lower bound is trivial when $\langle\Psi|[A,B]|\Psi\rangle$ happens to be zero, which in finite dimensional spaces is always possible. Also, the lower bound depends on the state $|\Psi\rangle$, while in some applications a state-independent lower bound is needed \cite{HT03}. It is thus necessary to derive state-independent variance-based uncertainty relations. Note that no nontrivial state-independent lower bounds on the sum of the variances of two or more arbitrary observables are known, even if this formulation of uncertainty relations is precisely what is called for in literature \cite{HT03}.
Entropic uncertainty relations provide state-independent lower bounds on the sum of the entropies of two or more observables. They are especially useful in quantum cryptography \cite{WW10}. A lot of entropic uncertainty relations are proposed for observables in both finite \cite{WW10, MU88, dS08, BPPZ11, San95} and infinite \cite{Bec75, BM75, Hua11} dimensional spaces.
Here I propose state-independent variance-based uncertainty relations. Section \ref{sec:2} derives state-independent lower bounds on the sum of the variances of two or more arbitrary observables in finite dimensional spaces. Section \ref{sec:3} concludes with some typical examples in which the lower bounds are near-optimal and with the application to entanglement detection. The Appendix briefly surveys entropic uncertainty relations, which are useful in formulating the main theorems.
\section{Main theorems} \label{sec:2}
Let $\{\{|a_i^j\rangle|j=1,2,\ldots,n\}|i=1,2,\ldots,m\}$ be a set of $m$ orthonormal bases of an $n$-dimensional Hilbert space, and $A_i=\sum_{j=1}^na_i^j|a_i^j\rangle\langle a_i^j|$ be $m$ Hermitian operators, where $a_i^j\in\mathbb R$ and $|a_i^j\rangle$ are the eigenvalues and the eigenvectors of $A_i$, respectively. For notational simplicity, let $p_i^j=|\langle a_i^j|\Psi\rangle|^2$. The Shannon entropy of $A_i$ is
\begin{equation}
H(A_i)=-\sum_{j=1}^np_i^j\ln p_i^j.
\end{equation}
From now on, assume the entropic uncertainty relation
\begin{equation}
\sum_{i=1}^mH(A_i)\ge C
\end{equation}
Let $\alpha$ be a variational parameter. All the inequalities below hold for any $\alpha\in\mathbb R$.
\begin{lemma}
\begin{equation}
\alpha V(A_i)\ge H(A_i)-\ln\sum_{j=1}^ne^{-\alpha(a_i^j-\langle A_i\rangle)^2}.
\label{eq:lemma}
\end{equation}
\end{lemma}
\begin{proof}
Using the basic inequality $e^x\ge1+x$ with $x=-\alpha(a_i^k-\langle A_i\rangle)^2-\ln(p_i^k\sum_{j=1}^ne^{-\alpha(a_i^j-\langle A_i\rangle)^2})$,
\begin{align}
1&=\sum_{k=1}^np_i^k\times\frac{e^{-\alpha(a_i^k-\langle A_i\rangle)^2}}{p_i^k\sum_{j=1}^{n}e^{-\alpha(a_i^j-\langle A_i\rangle)^2}}\ge\sum_{k=1}^np_i^k\left(1-\alpha(a_i^k-\langle A_i\rangle)^2-\ln\left(p_i^k\sum_{j=1}^{n}e^{-\alpha(a_i^j-\langle A_i\rangle)^2}\right)\right)\nonumber\\
&=\sum_{k=1}^n p_i^k-\alpha\sum_{k=1}^n p_i^k(a_i^k-\langle A_i\rangle)^2-\sum_{k=1}^n p_i^k\ln p_i^k-\left(\sum_{k=1}^np_i^k\right)\ln \sum_{j=1}^{n}e^{-\alpha(a_i^j-\langle A_i\rangle)^2}\nonumber\\
&=1-\alpha V(A_i)+H(A_i)-\ln\sum_{j=1}^{n}e^{-\alpha(a_i^j-\langle A_i\rangle)^2}.
\end{align}
\end{proof}
Add (\ref{eq:lemma}) for $i=1,2,\ldots,m$:
\begin{theorem} [state-dependent variance-based uncertainty relation] \label{thm1}
\begin{equation}
\alpha\sum_{i=1}^mV(A_i)\ge C-\sum_{i=1}^m\ln\sum_{j=1}^ne^{-\alpha(a_i^j-\langle A_i\rangle)^2}.
\end{equation}
\end{theorem}
Note $\min_ka_i^k\le\langle A_i\rangle\le\max_ka_i^k$:
\begin{theorem} [state-independent variance-based uncertainty relation] \label{thm2}
\begin{equation} \label{eq:multi}
\alpha\sum_{i=1}^mV(A_i)\ge C-\sum_{i=1}^m\ln\max_{\min_ka_i^k\le\beta_i\le\max_ka_i^k}\sum_{j=1}^ne^{-\alpha(a_i^j-\beta_i)^2}.
\end{equation}
\end{theorem}
Usually the maximum in (\ref{eq:multi}) cannot be computed analytically, but it is easy to compute numerically. Theorems \ref{thm1} and \ref{thm2} hold more generally for positive-operator valued measures provided with the corresponding entropic uncertainty relations for positive-operator valued measures.
\section{Examples, applications, and conclusions} \label{sec:3}
A set of $m$ orthonormal bases of an $n$-dimensional Hilbert space is mutually unbiased if
\begin{equation}
\big|\langle a_i^j|a_l^k\rangle\big|=\frac{1-\delta_{il}}{\sqrt n}+\delta_{il}\delta_{jk}
\end{equation}
for $i,l=1,2,\ldots,m;j,k=1,2,\ldots,n$. Mutually unbiased bases are especially useful in quantum cryptography \cite{WW10}.
\begin{example}
The eigenvectors of the Pauli matrices are mutually unbiased bases. (\ref{eq:e2}) shows $C=2\ln 2$. The optimal choice of the variational parameter is $\alpha=0.597$:
\begin{equation} \label{eq:b1}
V(\sigma_x)+V(\sigma_y)+V(\sigma_z)>1.7243.
\end{equation}
\end{example}
\begin{example}
The eigenvectors of the matrices
\begin{gather}
\sigma_0=\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix},\quad\sigma_1=\frac{i}{\sqrt3}\begin{pmatrix}0&-1&1\\1&0&-1\\-1&1&0\end{pmatrix},\nonumber\\
\sigma_2=\frac{1}{\sqrt3}\begin{pmatrix}0&\omega^{11}&\omega^9\\\omega&0&\omega^7\\\omega^3&\omega^5&0\end{pmatrix},\quad\sigma_3=\frac{1}{\sqrt3}\begin{pmatrix}0&\omega&\omega^3\\\omega^{11}&0&\omega^5\\\omega^9&\omega^7&0\end{pmatrix}
\end{gather}
are mutually unbiased bases, where $\omega=e^{\pi i/6}$. (\ref{eq:e2}) shows $C=4\ln2$. The optimal choice of the variational parameter is $\alpha=1.92$:
\begin{equation} \label{eq:b2}
V(\sigma_0)+V(\sigma_1)+V(\sigma_2)+V(\sigma_3)>0.9083.
\end{equation}
\end{example}
The lower bounds in (\ref{eq:b1}) and (\ref{eq:b2}) are near-optimal since the optimal lower bounds are $2$ and $1$, respectively.
The state-independent variance-based uncertainty relation (\ref{eq:multi}) is precisely in the formulation that is called for in the literature of entanglement detection \cite{HT03}. Entanglement detection is an important problem as entanglement plays a key role in quantum information. Among many detection schemes \cite{HHHH09, GT09}, the entanglement criterion based on local uncertainty relations \cite{HT03} is well known. It is a necessary condition for separability:
\begin{equation}
\sum_i V(A_i+B_i)\ge U_A+U_B,
\end{equation}
where $A_i$ and $B_i$ are Hermitian operators on the first and the second subspaces, respectively, with $U_A,U_B$ given by the state-independent variance-based uncertainty relations:
\begin{equation}
\sum_i V(A_i)\ge U_A,\quad\sum_i V(B_i)\ge U_B.
\end{equation}
Section II in Ref. \cite{HT03} appreciates the importance of $U_A,U_B$ in formulating the criterion, but how to evaluate $U_A,U_B$ remains unclear, which is very inconvenient for users. The state-independent variance-based uncertainty relation (\ref{eq:multi}) precisely fills the gap.
In conclusion, I have proposed state-independent variance-based uncertainty relations for two or more arbitrary observables in finite dimensional spaces. The uncertainty relations provide near-optimal lower bounds in some typical examples, and are useful for entanglement detection.
\appendix
\section{Brief survey of entropic uncertainty relations}
Entropic uncertainty relations are useful in formulating the main theorems. See Ref. \cite{WW10} for a recent comprehensive survey.
The entropic uncertainty relation for two Hermitian operators \cite{MU88} reads
\begin{equation} \label{eq:mu}
H(A_1)+H(A_2)\ge-2\ln\max_{1\le j,k\le n}\big|\langle a_1^j|a_2^k\rangle\big|,
\end{equation}
and the lower bound is improved for $\max_{1\le j,k\le n}|\langle a_1^j|a_2^k\rangle|\ge1/\sqrt2$ \cite{dS08, BPPZ11}. (\ref{eq:mu}) reduces to
\begin{equation}
H(A_1)+H(A_2)\ge\ln n
\end{equation}
for mutually unbiased bases. The entropic uncertainty relation for $n+1$ mutually unbiased bases \cite{San95} reads
\begin{equation} \label{eq:e2}
\sum_{i=1}^{n+1}H(A_i)\ge\left[\frac{n+1}{2}\right]\ln\left[\frac{n+1}{2}\right]+\left[\frac{n+2}{2}\right]\ln\left[\frac{n+2}{2}\right],
\end{equation}
where $[\cdots]$ denotes the floor function. The entropic uncertainty relation for the position and the momentum operators $x,p$ \cite{Bec75, BM75} reads
\begin{equation}
H(x)+H(p)\ge1+\ln\pi.
\end{equation}
Its multimode generalization is given in Ref. \cite{Hua11}.
\printbibliography
\end{document} | math | 9,777 |
\begin{document}
\title{\bf Quantum secret sharing with two qubit bipartite mixed state }
\author{Satyabrata Adhikari \thanks{
[email protected]} \\
\textsl{Department of Physics, Korea Advanced Institute of Science and Technology}\\
\textsl{Daejeon 305-701, Korea}\\
} \maketitle
\begin{abstract}
\noindent Quantum secret sharing is one of the most important and
interesting quantum information processing task. In quantum secret
sharing, information is split among several parties such that only
one of them is able to recover the qubit exactly provided all the
other parties agree to cooperate. To achieve this task, all the
parties need to share entangled state. As far as my knowledge, all
the previous quantum secret sharing protocol used either pure
tripartite or pure bipartite entangled state. In this work we use
for the first time bipartite two qubit mixed state (formed due to
noisy environment) in quantum secret sharing scheme. We further
show that one party cannot extract the information without the
collaboration of other party. We also study the property of the
shared mixed state used in the quantum secret sharing scheme.
\end{abstract}
\section{Introduction}
Quantum entanglement \cite{einstein} is one of the fascinating
feature of quantum mechanics. There is no classical analog of
quantum entanglement and that makes it more fascinating than
anything else in physics. In the field of quantum information
theory entanglement plays a major role. This is also a very useful
resource in the sense that using entanglement one can do many
things in the quantum world which are usually impossible in
ordinary classical world. Some of these tasks are quantum
computing \cite{bennett1}, quantum teleportation \cite{bennett2},
quantum cryptography \cite{gisin} and quantum secret sharing
\cite{hillery}. In quantum secret sharing, quantum information
encoded in a qubit is split among several parties such that only
one of them is able to recover the qubit exactly provided all the
other parties agree to cooperate. Therefore, quantum secret
sharing is a very interesting quantum information processing task
which was introduced in \cite{hillery}. After its introduction,
Karlsson et.al. \cite{karlsson} studied the similar quantum secret
sharing protocol using bipartite pure entangled state. Many
authors studied the concept of quantum secret sharing using
tripartite pure entangled states
\cite{bandyopadhyay,bagherinezhad,lance,gordon,zheng}. Recently Q.
Li et.al. \cite{li} proposed semi-quantum secret sharing protocols
using maximally entangled GHZ state which was shown to be secured
against eavesdropping. Quantum secret sharing can also be realized
in experiment \cite{tittel, schmid, schmid1, bogdanski}. \vskip
0.1cm
\noindent In this work we discuss the quantum secret sharing
protocol in a following way: Let us suppose that a spy (Charlie)
who is working under two commanders, Alice and Bob. Charlie's job
is to sent the secret information to both the commanders. But
Charlie suspect that one of the commanders is dishonest but he
don't know who is the culprit (Alice or Bob)? i.e. he don't know
who tries to find out the secret all by himself. So he decided to
sent the secret information in such a way that one commander
cannot collect the secret information without the help of other
commander. How Charlie achieve this task is the main result of
this work. To split information among two parties we use bipartite
mixed state. Therefore, we discuss our quantum secret sharing
protocol with two qubit mixed bipartite state. In section-2, we
review generalised concurrence and quantify the maximum amount of
entanglement present in Schmidt rank $r$ pure state in $k\times
k$-dimensional system. In section-3, we study the pure state
living in $k\times k$-dimensional Hilbert space through the noisy
environment. For two qubit system, we find that the mixed state
(because of noisy environment) shared between two distant partners
remains entangled if the concurrence of the initial entangled
state greater than certain threshold value. In section-4, we use
the two qubit mixed state (discussed in section-3) in
demonstrating the quantum secret sharing protocol. In section-5,
we end with conclusion.
\section{Generalised Concurrence - A review }
Hill and Wootters \cite{hill} introduced the first measure of
entanglement for a pair of qubits and the name given to the
entanglement measure is concurrence. For $2\times 2$- dimensional
system, the concurrence for pure state is defined as
\begin{eqnarray}
C(|\Psi_{AB}^{(2)}\rangle)=|\langle\Psi_{AB}^{(2)}|\sigma_{y}\otimes
\sigma_{y}|(\Psi_{AB}^{(2)})^{*}\rangle| \label{concurrence}
\end{eqnarray}
Since $|\Psi_{AB}^{(2)}\rangle$ is a pure $2\times 2$ bipartite
state so it can be expressed in a Schmidt-decomposition form as
\begin{eqnarray}
|\Psi_{AB}^{(2)}\rangle=\sqrt{\lambda_{1}}|00\rangle+\sqrt{\lambda_{2}}|11\rangle
\label{schmidt decomposition}
\end{eqnarray}
where $\lambda_{1},\lambda_{2}$ are schmidt coefficients and
$\lambda_{1}+\lambda_{2}=1$.\vskip 0.1cm
\noindent The concurrence (\ref{concurrence}) for the state
(\ref{schmidt decomposition}) reduces to
\begin{eqnarray}
C(|\Psi_{AB}^{(2)}\rangle)=\sqrt{2(1-Tr(\rho_{A}^{2}))}=2\sqrt{\lambda_{1}\lambda_{2}}
\label{concurrence1}
\end{eqnarray}
where
$\rho_{A}=Tr_{B}|\Psi_{AB}^{(2)}\rangle\langle\Psi_{AB}^{(2)}|$
denotes the reduced density operator.\vskip 0.1cm
\noindent Rungta et.al. \cite{rungta} then generalised the
concurrence of two-qubit pure state to higher dimensional $k\times
k$ system and the generalised concurrence (or $I-concurrence$) is
defined as
\begin{eqnarray}
C_{I}(|\Psi_{AB}^{(k)}\rangle)=\sqrt{\frac{k}{k-1}(1-Tr(\rho_{A}^{2}))}
\label{I-concurrence}
\end{eqnarray}
where
$\rho_{A}=Tr_{B}|\Psi_{AB}^{(k)}\rangle\langle\Psi_{AB}^{(k)}|$,~~
$|\Psi_{AB}^{(k)}\rangle=\sum_{i=1}^{k}\sqrt{\lambda_{i}}|i_{A}\rangle|i_{B}\rangle$,~~and
~ $\sum_{i=1}^{k}\lambda_{i}=1$.\vskip 0.1cm
\noindent $I-Concurrence$ can also be expressed in terms of
Schmidt coefficients as \cite{gour}
\begin{eqnarray}
C_{I}(|\Psi_{AB}^{(k)}\rangle)=
\sqrt{\frac{S_{2}(\lambda_{1},\lambda_{2},......,\lambda_{k})}{S_{2}(\frac{1}{k},\frac{1}{k},.....,\frac{1}{k})}}\label{I-concurrence1}
\end{eqnarray}
where $S_{2}(\lambda_{1},\lambda_{2},......,\lambda_{k})$ is the
$2nd$ elementary symmetric function of
$\lambda_{1},\lambda_{2},......,\lambda_{k}$, i.e.
$S_{2}(\lambda_{1},\lambda_{2},......,\lambda_{k})=\sum_{i<j}\lambda_{i}\lambda_{j}$.
Therefore, $I-concurrence$ can be re-written as
\begin{eqnarray}
C_{I}(|\Psi_{AB}^{(k)}\rangle)=\sqrt{\frac{2k}{k-1}\sum_{i<j,i,j=1}^{k}\lambda_{i}\lambda_{j}}
\label{I-concurrence2}
\end{eqnarray}
For $2\times 2$ dimensional system, we have
$C_{I}(|\Psi_{AB}^{(2)}\rangle)=C(|\Psi_{AB}^{(2)}\rangle)$.\vskip
0.1cm
\noindent But in reality, due to decoherence or due to preparation
error, we generally have a mixed state. Therefore, the
entanglement of the mixed state $\rho_{AB}^{(k)}=\sum
p_{i}|\Psi_{i}^{(k)}\rangle_{AB}\langle\Psi_{i}^{(k)}|$ can be
measured by convex roof extension method
\begin{eqnarray}
C_{I}(\rho_{AB}^{(k)})= min\sum_{i}
p_{i}C_{I}(|\Psi_{i}^{(k)}\rangle_{AB})\label{convexroof}
\end{eqnarray}
where the minimum is taken over all possible decomposition of
$\rho_{AB}^{(k)}$.\vskip 0.1cm
\noindent It is to be noted that the maximum amount of
entanglement in $2\times 2$-dimensional pure system is unity. They
are called maximally entangled state. Now if we proceed towards
two pure qutrit entangled systems, then we can find two SLOCC
inequivalent classes of states. The two inequivalent classes are
Schmidt rank two class (SR-2) and Schmidt rank three class (SR-3).
The pure states that belong to the Schmidt rank two class can have
amount of entanglement at most
$C_{I}(|\Psi_{2}^{(3)}\rangle)=\frac{\sqrt{3}}{2}$
($|\Psi_{2}^{(3)}\rangle$ denote the pure state of Schmidt rank 2
in $3\times 3$-dimensional system) while pure SR-3 states can
achieve the maximum amount unity. Therefore, all maximally
entangled states in two qutrit system are Schmidt rank three
(SR-3) states. Therefore, a obvious conclusion is that the amount
of entanglement in any Schmidt number 2 state in two qutrit system
is at most $\frac{\sqrt{3}}{2}$. Now it is important to ask a more
general question that if we have a $k\times k$ dimensional
entangled mixed state which has schmidt number $r$ described by a
density operator $\rho_{r}^{(k)}$ then what is the upper bound of
the amount of entanglement contained in $\rho_{r}^{(k)}$? The
answer may be given as
\begin{eqnarray}
&&(i)~~~~~C_{I}(\rho_{r}^{(k)})\leq 1,~~\textit{if~ r=k
}{}\nonumber\\&& (ii)~~~~C_{I}(\rho_{r}^{(k)})\leq
[C_{I}(|\Psi_{r}^{(k)}\rangle)]_{max},~~\textit{if~ r} <
\textit{k} \label{upperbound}
\end{eqnarray}
where $|\Psi_{r}^{(k)}\rangle$ denotes the entangled pure state of
schmidt rank r in $k\times k$-dimensional system.\vskip 0.5cm
\noindent \textbf{Theorem:} If $|\Psi_{r}^{(k)}\rangle$ denotes
the entangled pure state of schmidt rank $r$ in $k\times k$
dimensional system, then
\begin{eqnarray}
[C_{I}(|\Psi_{r}^{(k)}\rangle)]_{max}=
\sqrt{\frac{k(r-1)}{r(k-1)}} \label{maximum value}
\end{eqnarray}
\textbf{Proof:} Since $|\Psi_{r}^{(k)}\rangle$ is a entangled pure
state of schmidt rank $r$, so $|\Psi_{r}^{(k)}\rangle$ can be
expressed as
\begin{eqnarray}
|\Psi_{r}^{(k)}\rangle=\sum_{i=1}^{r}\sqrt{\lambda_{i}}|i_{A}\rangle|i_{B}\rangle,~~r=2,3,....k
\end{eqnarray}
The amount of entanglement in $|\Psi_{r}^{(k)}\rangle$ is measured
by $I-concurrence$. Therefore
\begin{eqnarray}
C_{I}(|\Psi_{r}^{(k)}\rangle)=\sqrt{\frac{2k}{k-1}\sum_{i<j,i,j=1}^{r}\lambda_{i}\lambda_{j}}
\end{eqnarray}
$C_{I}(|\Psi_{r}^{(k)}\rangle)$ can be maximized using lagrange's
multiplier method subject to the constraint
$\sum_{i=1}^{r}\lambda_{i}=1$. We find that
$C_{I}(|\Psi_{r}^{(k)}\rangle)$ attains its maximum value when
$\lambda_{1}=\lambda_{2}=........=\lambda_{r}=\frac{1}{r}$.
Therefore, the maximum value is given by
\begin{eqnarray}
[C_{I}(|\Psi_{r}^{(k)}\rangle)]_{max}=
\sqrt{\frac{k(r-1)}{r(k-1)}}
\end{eqnarray}
Hence proved.\vskip 0.1cm
\noindent \textbf{Observations:}\vskip 0.1cm
\noindent(i) If r=k, then
$[C_{I}(|\Psi_{r}^{(k)}\rangle)]_{max}=1$, as expected.\vskip
0.1cm
\noindent (ii) For higher dimensional system, i.e. as
$k\rightarrow \infty $,
$[C_{I}(|\Psi_{r}^{(k)}\rangle)]_{max}\rightarrow
\sqrt{\frac{r-1}{r}}$\vskip 0.1cm
\noindent (iii) For $k\times k$- dimensional system, we have the
following ordering of maximum value of $I-concurrence$ for
different schmidt rank states
\begin{eqnarray}
[C_{I}(|\Psi_{2}^{(k)}\rangle)]_{max}<
[C_{I}(|\Psi_{3}^{(k)}\rangle)]_{max}<[C_{I}(|\Psi_{4}^{(k)}\rangle)]_{max}<.......<[C_{I}(|\Psi_{k}^{(k)}\rangle)]_{max}=1
\label{ordering}
\end{eqnarray}
\section{Pure state through noisy environment}
In this section we study the initially prepared pure state in
$k\times k$-dimensional system passing through the noisy
environment. The state can only be used in some quantum
information processing task if it is shared between two distant
partners who wishes to exchange information between them. We
assume that Charlie is the supplier of entangled states to two
users Alice and Bob. The users of the entangled states always
demand from the supplier for the maximally entangled state. But
the supplier cannot fulfill their demand. Although supplier can
prepare maximally entangled pure state in his laboratory but the
problem is that he have to send the particles to its users through
a noisy environment. In general, the noisy environment converts
pure states to mixed states and hence the entanglement decreases
in course of distributing the particles. Due to this reason, the
users Alice and Bob have to satisfy themselves with lesser
entangled mixed state compared to pure maximally entangled
state.\vskip 0.1cm
\noindent Suppose that Charlie prepare a bipartite pure state
$|\psi\rangle^{in}$ in $k\otimes k$-dimensional system. Any
bipartite pure state can be written in the Schmidt polar form as
\begin{eqnarray}
|\psi\rangle^{in}=\sum_{i=1}^{k}\sqrt{\lambda_{i}}|i\rangle_{1}\otimes|i\rangle_{2}
\label{state}
\end{eqnarray}
where $\lambda_{i}>0,~i=1,2,........k$ are the schmidt
coefficients and satisfies the condition $\sum_{i=1}^{k}\lambda_{i}=1$.\\
After creating the entanglement between two particles, Charlie
then sent the particle 1 to Alice and particle 2 to Bob through
noisy environment. In this work, the noisy environment is
described by the unitary operator \cite{buzek}
\begin{eqnarray}
|i\rangle_{a}|0\rangle_{E}|M\rangle_{x}\rightarrow
c|i\rangle_{a}|i\rangle_{E}|X_{i}\rangle_{x}+d\sum_{j\neq
i}^{k}(|i\rangle_{a}|j\rangle_{E}+|j\rangle_{a}|i\rangle_{E})|X_{j}\rangle_{x}
\label{transformation}
\end{eqnarray}
where $|0\rangle_{E}$ denote the initial state of the environment
and $|M\rangle_{x}$ and $|X_{i}\rangle_{x} (i=1,2,....k)$ denotes
the ancilla states. The ancilla state vectors $|X_{i}\rangle_{x}
(i=1,2,....k)$ form an orthonormal basis of the ancilla Hilbert
space.\vskip 0.1cm
\noindent Unitarity of the transformation (\ref{transformation})
gives the following relation between the parameters $c$ and $d$
\begin{eqnarray}
c^{2}+2(k-1)d^{2}=1 \label{unitary}
\end{eqnarray}
When both the particles 1 and 2 is being sent through the same
noisy environment (\ref{transformation}), the state (\ref{state})
transform as
\begin{eqnarray}
|\psi\rangle^{in}\rightarrow|\psi\rangle^{out}=
c^2\sum_{i=1}^{k}\sqrt{\lambda_{i}}[|i,i\rangle_{13}\otimes|i,i\rangle_{24}|X_{i}\rangle\otimes|X_{i}\rangle]
+cd\sum_{i\neq
j}^{k}\sqrt{\lambda_{i}}|i,i\rangle_{13}\otimes{}\nonumber\\(|i,j\rangle_{24}+|j,i\rangle_{24})|X_{i}\rangle\otimes|X_{j}\rangle
+ cd\sum_{i\neq
j}^{k}\sqrt{\lambda_{i}}(|i,j\rangle_{13}+|j,i\rangle_{13})\otimes|i,i\rangle_{24}
|X_{j}\rangle\otimes|X_{i}\rangle {}\nonumber\\
+d^{2}\sum_{i=1}^{k}\sqrt{\lambda_{i}}[\sum_{i\neq
j}^{k}(|i,j\rangle_{13}+|j,i\rangle_{13})\otimes \sum_{i\neq
l}(|i,l\rangle_{24}+|l,i\rangle_{24})|X_{j}\rangle\otimes|X_{l}\rangle]
\label{output}
\end{eqnarray}
where $|\rangle_{3}$ and $|\rangle_{4}$ denote the qubit of the
environment.\vskip 0.1cm
\noindent After tracing out the ancilla qubits, four qubit state
is described by the density operator $\rho_{1324}$. When the sent
qubit 1 (2) interact with its own environment qubit 3 (4), the
state described by the density operator $\rho_{13}$ $(\rho_{24})$
can be designated as local outputs. The local output is given by
\begin{eqnarray}
&&\rho_{13}^{local}=\rho_{24}^{local}=
c^{2}\sum_{i=1}^{k}\lambda_{i}|i,i\rangle\langle i,i| +
d^{2}\sum_{i\neq
j}^{k}\lambda_{i}(|i,j\rangle+|j,i\rangle)(\langle i,j|+\langle
j,i|) \label{local}
\end{eqnarray}
Since the state described by the density operator $\rho_{14}$
$(\rho_{23})$ is formed between the sent qubit 1 (2) and
environment qubit 4 (3) located at different place so they can be
treated as non-local. The non-local output is given by
\begin{eqnarray}
\rho_{14}^{non-local}=\rho_{23}^{non-local}=P\sum_{i=1}^{k}\lambda_{i}|i,i\rangle\langle
i,i|+Q\sum_{i\neq
j}^{k}\sqrt{\lambda_{i}\lambda_{j}}|i,i\rangle\langle j,j| +
{}\nonumber\\R\sum_{i\neq j}^{k}\lambda_{i}(|i,j\rangle\langle
i,j| +|j,i\rangle\langle j,i|)+ S\sum_{l,j\neq
i}\lambda_{i}|j,l\rangle\langle j,l| \label{nonlocal}
\end{eqnarray}
where $P=(c^{2}+(k-1)d^{2})^{2}$,
$Q=d^{2}(4c^{2}+4cd(k-2)+(k-2)d^{2})$,
$R=d^{2}(c^{2}+(k-1)d^{2})$, $S=d^{4}$. \vskip 0.1cm
\noindent Alice and Bob then shared a state which is described by
the density operator $\rho_{14}$ $(\rho_{23})$. Let us now
investigate the situation for $k=2$ i.e. for two qubit
systems.\vskip 0.5cm
\noindent In the computational basis
$\{|1\rangle\otimes|1\rangle,|1\rangle\otimes|2\rangle,|2\rangle\otimes|1\rangle,|2\rangle\otimes|2\rangle\}$,
the local and non-local output is given by
\begin{eqnarray}
\rho_{13}^{local}=\rho_{24}^{local}= \left(
\begin{array}{cccc}
c^{2}\lambda_{1} & 0 & 0 & 0 \\
0 & d^{2} & d^{2} & 0 \\
0 & d^{2} & d^{2} & 0 \\
0 & 0 & 0 & c^{2}\lambda_{2}\\
\end{array}
\right)
\end{eqnarray}
\begin{eqnarray}
\rho_{14}^{non-local}= \rho_{23}^{non-local}=\left(
\begin{array}{cccc}
P\lambda_{1}+S\lambda_{2} & 0 & 0 & Q\sqrt{\lambda_{1}\lambda_{2}} \\
0 & R & 0 & 0 \\
0 & 0 & R & 0 \\
Q\sqrt{\lambda_{1}\lambda_{2}} & 0 & 0 & P\lambda_{2}+S\lambda_{1}\\
\end{array}
\right)
\end{eqnarray}
where $P=(c^2+d^2)^2, Q=4c^2d^2, R=c^2d^2+d^4, S=d^4$. \vskip
0.1cm
\noindent Alice and Bob shared a mixed state described by density
operator $\rho_{14}^{non-local}~~ (\rho_{23}^{non-local})$. Since
charlie sent the two particles through the noisy environment so
the state shared by the users Alice and Bob may or may not be
entangled. It depends on the noisy environment. We will find that
the shared state is entangled if there exist a critical value of
the concurrence which measures the initial entanglement present in
the two qubit pure system. This critical value of the concurrence
depends on the parameter of the noisy environment. If the
concurrence of initially prepared state less than the critical
value then the shared state is separable. We use witness operator
to find this critical value of the concurrence.\vskip 0.1cm
\noindent The optimal witness operator for two qubit system
$W_{1}^{(2)}$ is given by \cite{bertlmann}
\begin{eqnarray}
W_{1}^{(2)}=\frac{1}{2\sqrt{3}}(I-\vartheta) \label{bwitness}
\end{eqnarray}
where $\vartheta$ can be expressed in terms of the pauli matrices
$\sigma_{x},\sigma_{y}~\textrm{and}~\sigma_{z}$ as
\begin{eqnarray}
\vartheta =
\sigma_{x}\otimes\sigma_{x}-\sigma_{y}\otimes\sigma_{y}+\sigma_{z}\otimes\sigma_{z}\label{paulimat}
\end{eqnarray}
In matrix form W can be re-expressed as
\begin{eqnarray}
W_{1}^{(2)}= \left(
\begin{array}{cccc}
0 & 0 & 0 & \frac{-1}{\sqrt{3}} \\
0 & \frac{1}{\sqrt{3}} & 0 & 0 \\
0 & 0 & \frac{1}{\sqrt{3}} & 0 \\
\frac{-1}{\sqrt{3}} & 0 & 0 & 0\\
\end{array}
\right)
\end{eqnarray}
Therefore,
\begin{eqnarray}
Tr(W_{1}^{(2)}\rho_{14})=Tr(W_{1}^{(2)}\rho_{23})=(\frac{-2}{\sqrt{3}})(Q\sqrt{\lambda_{1}\lambda_{2}}-R)\label{trace}
\end{eqnarray}
The non-local output $\rho_{14}^{non-local}=\rho_{23}^{non-local}$
is entangled if
\begin{eqnarray}
Q\sqrt{\lambda_{1}\lambda_{2}}-R>0\Rightarrow
2\sqrt{\lambda_{1}\lambda_{2}}=C_{I}(|\psi\rangle^{in})>C_{I}^{cr}(|\psi\rangle^{in})=\frac{1+c^2}{4c^2},~~~\frac{1}{\sqrt{3}}<c\leq1
\label{cond}
\end{eqnarray}
Therefore, the critical value of the concurrence depends on the
parameter of the noisy environment. Also we note that the function
of the parameter c is a decreasing function so the critical value
of the concurrence decreases as c increases. Thus, the lower value
of the concurrence of the initially prepared entangled state may
keep the non-local output shared state entangled if the noisy
parameter c tends towards unity.\vskip 0.1cm
\noindent It is clear that the local output state described by the
density matrix $\rho_{13}^{local}=\rho_{24}^{local}$ is separable
because
$Tr(W_{1}^{(2)}\rho_{13}^{local})=Tr(W_{1}^{(2)}\rho_{24}^{local})=\frac{1}{3\sqrt{3}}>0$.\vskip
0.1cm
\noindent We should note that the optimal witness operator
$W_{1}^{(2)}$ that detect the entangled mixed state described by
the density operator $\rho_{14}=\rho_{23}$ is not unique. There
exist another optimal witness operator \cite{sanpera} which
produce the same result (\ref{cond}) is of the form
\begin{eqnarray}
W_{2}^{(2)}=\frac{1}{2}(I-\vartheta) \label{bwitness1}
\end{eqnarray}
where $\vartheta$ is given by (\ref{paulimat}).\vskip 0.2cm
\noindent \textbf{Observation:} If Charlie initially prepare a
maximally entangled state, i.e. when
$\lambda_{1}=\lambda_{2}=\frac{1}{2}$, then for some specific
value of noisy parameter $c=\sqrt{2/3}$, the shared state between
Alice and Bob takes the form of maximally entangled mixed state.
The form of maximally entangled mixed state is given by
\begin{eqnarray}
\rho_{23}^{non-local}=\rho_{14}^{non-local}= \left(
\begin{array}{cccc}
\frac{13}{36} & 0 & 0 & \frac{4}{18} \\
0 & \frac{5}{36} & 0 & 0 \\
0 & 0 & \frac{5}{36} & 0 \\
\frac{4}{18} & 0 & 0 & \frac{13}{36}\\
\end{array}
\right)=\frac{4}{9}|\Phi^{+}\rangle\langle\Phi^{+}|+\frac{5}{36}I_{4}
\label{werner}
\end{eqnarray}
where
$|\Phi^{+}\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$.\vskip
0.1cm
\noindent Thus if maximally entangled pure state sent through
noisy environment defined in (\ref{transformation}) then there
exist a value of the noisy parameter which transform the maximally
entangled pure state to a maximally entangled mixed state which
belongs to the family of Werner state \cite{werner}.
\section{Application of two-qubit bipartite mixed state in a quantum secret sharing problem}
In this section, we discuss a protocol for quantum secret sharing
using two-qubit bipartite mixed state. Our protocol can be
described in a few step given below: \vskip 0.5cm
\noindent \textbf{Step-I: Maximally entangled pure state prepared
by Charlie}\vskip 0.3cm
\noindent A $\textit{secret agent}$ called Charlie want to
distribute his collected confidential secret to two senior
officers called Alice and Bob in such a way that one officer
(Alice/Bob) alone cannot gather all the confidential information
by herself/himself. To accomplish his task, Charlie prepare a two
qubit maximally entangled pure state either in the form
$|\phi^{+}\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ or in
the form
$|\phi^{-}\rangle=\frac{1}{\sqrt{2}}(|00\rangle-|11\rangle)$. He
would like to make his decision on $|\phi^{+}\rangle$ or
$|\phi^{-}\rangle$ by tossing a coin. If "head" appears then he
prepare $|\phi^{+}\rangle$, otherwise $|\phi^{-}\rangle$. We can
designate "head" as "0" and "tail" as "1". In this way he encode
one bit of information into the prepared state. Then he send one
qubit to Alice and another qubit to Bob through a noisy
environment defined by the unitary transformation
(\ref{transformation}). Because of the preparation strategy and
noisy environment, Alice and Bob shared a mixed state that
described either by the density operator
\begin{eqnarray}
\rho_{AB}^{+}=
\frac{P+S}{2}(|00\rangle\langle00|+|11\rangle\langle11|)+\frac{Q}{2}(|00\rangle\langle11|+|11\rangle\langle00|)+R(|01\rangle\langle01|
+|10\rangle\langle10|)\label{shared state1}
\end{eqnarray}
or by the density operator
\begin{eqnarray}
\rho_{AB}^{-}=
\frac{P+S}{2}(|00\rangle\langle00|+|11\rangle\langle11|)-\frac{Q}{2}(|00\rangle\langle11|+|11\rangle\langle00|)+R(|01\rangle\langle01|
+|10\rangle\langle10|)\label{shared state2}
\end{eqnarray}
where $P=(c^{2}+d^{2})^{2}$, $Q=4c^{2}d^{2}$,
$R=d^{2}(c^{2}+d^{2})$, $S=d^{4}$ and $c^2+2d^2=1$. \vskip 0.5cm
\noindent \textbf{Step-II: Single qubit measurement performed by
Alice}\vskip 0.3cm
\noindent Alice then perform measurement on her qubit in the
Hadamard basis
$B_{H}=\{\frac{|0\rangle+|1\rangle}{\sqrt{2}},\frac{|0\rangle-|1\rangle}{\sqrt{2}}\}$.
It is assumed that Bob also know about the measurement basis that
Alice used. The single qubit state received by Bob after
measurement depends on the outcome of the measurement. \vskip
0.1cm
\noindent (i) \textit{If the shared state is $\rho_{AB}^{+}$ and
the measurement outcome is
$\frac{|0\rangle+|1\rangle}{\sqrt{2}}$}, then
\begin{eqnarray}
\rho_{B}^{+0}&&=
\frac{1}{p}Tr_{1}[((\frac{|0\rangle+|1\rangle}{\sqrt{2}})(\frac{\langle0|+\langle1|}{\sqrt{2}})\otimes
I_{2}
)\rho_{AB}^{+}((\frac{|0\rangle+|1\rangle}{\sqrt{2}})(\frac{\langle0|+\langle1|}{\sqrt{2}})\otimes
I_{2}
)]{}\nonumber\\&&=\frac{1}{4p}[I_{2}+Q(|0\rangle\langle1|+|1\rangle\langle0|)]\label{outcome1}
\end{eqnarray}
(ii) \textit{If the shared state is $\rho_{AB}^{+}$ and the
measurement outcome is $\frac{|0\rangle-|1\rangle}{\sqrt{2}}$},
then
\begin{eqnarray}
\rho_{B}^{+1}&&=
\frac{1}{p}Tr_{1}[((\frac{|0\rangle-|1\rangle}{\sqrt{2}})(\frac{\langle0|-\langle1|}{\sqrt{2}})\otimes
I_{2}
)\rho_{AB}^{+}((\frac{|0\rangle-|1\rangle}{\sqrt{2}})(\frac{\langle0|-\langle1|}{\sqrt{2}})\otimes
I_{2}
)]{}\nonumber\\&&=\frac{1}{4p}[I_{2}-Q(|0\rangle\langle1|+|1\rangle\langle0|)]\label{outcome2}
\end{eqnarray}
(iii) \textit{If the shared state is $\rho_{AB}^{-}$ and the
measurement outcome is $\frac{|0\rangle+|1\rangle}{\sqrt{2}}$},
then
\begin{eqnarray}
\rho_{B}^{-0}&&=
\frac{1}{p}Tr_{1}[((\frac{|0\rangle-|1\rangle}{\sqrt{2}})(\frac{\langle0|-\langle1|}{\sqrt{2}})\otimes
I_{2}
)\rho_{AB}^{-}((\frac{|0\rangle-|1\rangle}{\sqrt{2}})(\frac{\langle0|-\langle1|}{\sqrt{2}})\otimes
I_{2}
)]{}\nonumber\\&&=\frac{1}{4p}[I_{2}-Q(|0\rangle\langle1|+|1\rangle\langle0|)]=\rho_{B}^{+1}\label{outcome3}
\end{eqnarray}
(iv) \textit{If the shared state is $\rho_{AB}^{-}$ and the
measurement outcome is $\frac{|0\rangle-|1\rangle}{\sqrt{2}}$},
then
\begin{eqnarray}
\rho_{B}^{-1}&&=
\frac{1}{p}Tr_{1}[((\frac{|0\rangle-|1\rangle}{\sqrt{2}})(\frac{\langle0|-\langle1|}{\sqrt{2}})\otimes
I_{2}
)\rho_{AB}^{-}((\frac{|0\rangle-|1\rangle}{\sqrt{2}})(\frac{\langle0|-\langle1|}{\sqrt{2}})\otimes
I_{2}
)]{}\nonumber\\&&=\frac{1}{4p}[I_{2}+Q(|0\rangle\langle1|+|1\rangle\langle0|)]=\rho_{B}^{+0}\label{outcome4}
\end{eqnarray}
where $I_{2}$ denotes the identity operator in $2\times
2$-dimensional Hilbert space and $p=\frac{1}{2}$.\vskip 0.3cm
\noindent(\ref{outcome3}) and (\ref{outcome4}) explains the fact
that it is neither possible for Alice nor for Bob alone to decode
the encoded information of Charlie. They only decode the
information of Charlie when they both agree to collaborate with
each other. If they agree to collaborate, then our protocol
proceeds further to step-III.\vskip 0.3cm
\noindent \textbf{Step-III: Alice declare the measurement
outcome}\vskip 0.3cm
\noindent After they agree to collaborate, Alice sent her
measurement outcome to Bob.\vskip 0.1cm
\noindent (i) If the measurement outcome is
$\frac{|0\rangle+|1\rangle}{\sqrt{2}}$ then she sent Bob a
classical bit "0" and \vskip 0.1cm
\noindent (ii) If the measurement outcome is
$\frac{|0\rangle-|1\rangle}{\sqrt{2}}$ then she sent classical bit
"1" to Bob.\vskip 0.3cm
\noindent \textbf{Step-IV: Positive operator valued measurement
(POVM) performed by Bob}\vskip 0.3cm
\noindent When Bob receives the classical bit from Alice, he came
to know about Alice's measurement outcome. Corresponding to each
measurement outcomes, one of the two possible single qubit state
may appear at Bob's site. To discriminate between the two possible
single qubit state, Bob have to perform POVM on his received
qubit. The constructed POVM at Bob's site is given by
\begin{eqnarray}
\Pi_{B}^{(0)}= \frac{1}{2}(I_{2}+\frac{1}{Q}\sigma_{x})
{}\nonumber\\\Pi_{B}^{(1)}=
\frac{1}{2}(I_{2}-\frac{1}{Q}\sigma_{x})\label{POVM}
\end{eqnarray}
\noindent If Bob receives the classical bit "0" then Alice's
measurement outcome should be
$\frac{|0\rangle+|1\rangle}{\sqrt{2}}$. Corresponding to the
Alice's measurement outcome
$\frac{|0\rangle+|1\rangle}{\sqrt{2}}$, Bob received either
$\rho_{B}^{+0}=\frac{1}{2}[I_{2}+Q(|0\rangle\langle1|+|1\rangle\langle0|)]$
or
$\rho_{B}^{-0}=\frac{1}{2}[I_{2}-Q(|0\rangle\langle1|+|1\rangle\langle0|)]$.
Bob then perform POVM to detect the correct received state. POVM
operators $\Pi_{B}^{(0)}$ and $\Pi_{B}^{(1)}$ discriminate the
single qubit states $\rho_{B}^{+0}$ and $\rho_{B}^{-0}$ with
certainty. \vskip 0.1cm
\noindent Similarly, if Bob receives the classical bit "1" then he
can discriminate the single qubit state using POVM operators given
in (\ref{POVM}). \vskip 0.1cm
\noindent In this way our quantum secret sharing scheme work using
two qubit mixed state.
\section{Conclusion}
Before we presented our main result, we have studied generalised
concurrence or I-concurrence. We provide a compact formula to
quantify the maximum amount of entanglement present in pure state
of Schmidt rank $r$ in $k\times k$-dimensional system. We also
have studied the $k\times k$-dimensional pure state passing
through a noisy environment. We then restrict ourselves to
$2\times 2$-dimensional pure state and found that the mixed state
(because of noisy environment) shared between two distant partners
remains entangled if the concurrence of the initial entangled
state greater than certain threshold value. Thereafter, for the
first time we discussed the quantum secret sharing protocol using
two qubit mixed state which appeared due to noisy environment. Our
quantum secret sharing protocol is very simple and may be realized
in experiment. The noisy environment used in this protocol is
nothing but can be described as a quantum cloning transformation.
This type of transformation may be used by eavesdropper to steal
information. Instead of quantum cloning transformation, one may
use amplitude damping channel or any other decoherence processes.
\end{document} | math | 28,775 |
\begin{document}
\title{Bilinear pseudo-differential operators with exotic symbols}
\author{Akihiko Miyachi \and Naohito Tomita}
\date{}
\address{Akihiko Miyachi \\
Department of Mathematics \\
Tokyo Woman's Christian University \\
Zempukuji, Suginami-ku, Tokyo 167-8585, Japan}
\email{[email protected]}
\address{Naohito Tomita \\
Department of Mathematics \\
Graduate School of Science \\
Osaka University \\
Toyonaka, Osaka 560-0043, Japan}
\email{[email protected]}
\keywords{Bilinear pseudo-differential operators,
bilinear H\"ormander symbol classes, exotic symbols}
\subjclass[2010]{42B15, 42B20, 47G30}
\begin{abstract}
The boundedness
from $L^p \times L^q$ to $L^r$, $1<p,q \le \infty$,
$0<1/p+1/q=1/r \le 1$, of
bilinear pseudo-differential operators
with symbols in the bilinear H\"ormander class
$BS^m_{\rho,\rho}$, $0 \le \rho <1$, is proved for
the critical order $m$.
Related results for the cases $p=1$, $q=1$ or $r=\infty$
are also obtained.
\end{abstract}
\maketitle
\section{Introduction}\label{section1}
Let $m \in \mathbb{R}$ and $0 \le \delta \le \rho \le 1$.
We say that a function
$\sigma(x,\xi,\eta) \in C^{\infty}(\mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}^n)$
belongs to the bilinear H\"ormander symbol class
$BS^m_{\rho,\delta}=BS^m_{\rho,\delta}(\mathbb{R}^n)$
if for every triple of multi-indices
$\alpha,\beta,\gamma \in \mathbb{N}_0^n=\{0,1,2,\dots\}^n$
there exists a constant $C_{\alpha,\beta,\gamma}>0$ such that
\[
|\partial^{\alpha}_x\partial^{\beta}_{\xi}
\partial^{\gamma}_{\eta}\sigma(x,\xi,\eta)|
\le C_{\alpha,\beta,\gamma}
(1+|\xi|+|\eta|)^{m+\delta|\alpha|-\rho(|\beta|+|\gamma|)}.
\]
For a symbol $\sigma \in BS^{m}_{\rho,\delta}$,
the bilinear pseudo-differential operator
$T_{\sigma}$ is defined by
\[
T_{\sigma}(f,g)(x)
=\frac{1}{(2\pi)^{2n}}
\int_{\mathbb{R}^n \times \mathbb{R}^n}e^{i x \cdot(\xi+\eta)}
\sigma(x,\xi,\eta)\widehat{f}(\xi)\widehat{g}(\eta)\, d\xi d\eta,
\qquad f,g \in \mathcal{S}(\mathbb{R}^n).
\]
The study of bilinear operators $T_{\sigma}$ with $\sigma$ in
the bilinear H\"ormander class $BS^m_{\rho,\delta}$
was initiated by B\'enyi, Maldonado, Naibo, and Torres in \cite{BMNT},
where in particular the symbolic calculus of the operators $T_{\sigma}$,
$\sigma \in BS^m_{\rho,\delta}$, was established.
The boundedness properties of those operators have been considered in many works,
some of which will be mentioned below.
In the present paper, we shall also consider the boundedness property of
the operators $T_{\sigma}$, $\sigma \in BS^{m}_{\rho,\delta}$.
For the boundedness of the operators $T_{\sigma}$, we shall
use the following terminology.
If $X,Y,Z$ are function spaces on $\mathbb{R}^n$
equipped with quasi-norms
$\|\cdot \|_{X},\,\|\cdot \|_{Y},\,\|\cdot \|_{Z}$
and if there exists a constant $A_{\sigma}$ such that
the estimate
\begin{equation}\label{boundedness-XYZ}
\|T_{\sigma}(f,g)\|_{Z}
\le A_{\sigma} \|f\|_{X} \|g\|_{Y},
\quad
f\in \mathcal{S} \cap X,
\quad
g\in \mathcal{S} \cap Y,
\end{equation}
holds, then we shall simply say that
$T_{\sigma}$ is bounded from $X\times Y$ to $Z$ and write
\[
T_{\sigma}: X\times Y \to Z.
\]
The smallest constant $A_{\sigma}$ of \eqref{boundedness-XYZ}
is denoted by
$\|T_{\sigma}\|_{X\times Y \to Z}$.
In the case $\rho=1$,
bilinear pseudo-differential operators
with symbols in $BS^0_{1,\delta}$, $\delta<1$,
fall into the bilinear Calder\'on-Zygmund theory
in the sense of Grafakos-Torres \cite{GT} and
their boundedness properties are well-understood;
see, e.g., Coifman-Meyer \cite{CM},
B\'enyi-Torres \cite{BT},
and B\'enyi-Maldonado-Naibo-Torres \cite{BMNT}.
In the case $\rho<1$, however,
we cannot reduce the corresponding operators
to bilinear Calder\'on-Zygmund operators
and there are some interesting features peculiar to
the bilinear case.
For example,
in contrast to the well-known
Calder\'on-Vaillancourt theorem (\cite{CV})
for linear pseudo-differential operators,
the condition $\sigma \in BS^0_{\rho,\rho}$, $0\le \rho <1$,
does not assure any boundedness of the corresponding
bilinear operator.
This gap between the linear and bilinear cases
was first pointed out by B\'enyi-Torres \cite{BT-2}
for the case $\rho=0$.
The subject of the present paper concerns with the estimate
\begin{equation}\label{boundedness-pqr}
T_{\sigma}: H^p \times H^q \to L^r,
\quad
\frac{1}{p}+ \frac{1}{q}=\frac{1}{r},
\quad
\sigma \in BS^{m}_{\rho,\rho},
\quad
0\le \rho <1,
\end{equation}
where $H^p$ denotes Hardy space and
$L^r$ denotes Lebesgue space.
In the case $p=q=r=\infty$,
instead of $L^{\infty}\times L^{\infty} \to L^{\infty}$,
we shall consider $L^{\infty}\times L^{\infty} \to BMO$.
For $0 \le \rho <1$
and for $0<p,q,r \le \infty$ satisfying $1/p+1/q=1/r$,
we define
\begin{align*}
&
m_\rho(p,q)
=(1-\rho)m_0(p,q),
\\
&
m_0(p,q)
=-n \left(\max\left\{
\frac{1}{2}, \,
\frac{1}{p}, \,
\frac{1}{q}, \,
1-\frac{1}{r}, \,
\frac{1}{r} -\frac{1}{2}
\right\}\right).
\end{align*}
Here is an expression of $m_0(p,q)$ that will be easy to see.
We divide the region of $(1/p,1/q)$ into 5 regions
$J_0,\dots,J_4$ as follows:
\begin{center}
\begin{picture}(160,150)
\thicklines
\put(20,20){\vector(1,0){120}}
\put(20,20){\vector(0,1){120}}
\put(18,120){\line(1,0){4}}
\put(120,18){\line(0,1){4}}
\put(20,70){\line(1,-1){50}}
\put(20,70){\line(1,0){110}}
\put(70,20){\line(0,1){110}}
\put(130,8){$1/p$}
\put(-3,133){$1/q$}
\put(35,30){$J_0$}
\put(53,53){$J_1$}
\put(40,100){$J_2$}
\put(100,40){$J_3$}
\put(100,100){$J_4$}
\thinlines
\put(12,10){{\tiny $0$}}
\put(65,10){{\tiny $1/2$}}
\put(118,10){{\tiny $1$}}
\put(2,68){{\tiny $1/2$}}
\put(12,118){{\tiny $1$}}
\end{picture}
\end{center}
Then
\[
m_0(p,q)
=
\begin{cases}
\frac{n}{r}-n
&\quad \text{if} \quad
\left(\frac{1}{p},\frac{1}{q}\right) \in J_0;
\\
-\frac{n}{2}
&\quad \text{if} \quad
\left(\frac{1}{p},\frac{1}{q}\right) \in J_1;
\\
-\frac{n}{q}
&\quad \text{if} \quad
\left(\frac{1}{p},\frac{1}{q}\right) \in J_2;
\\
-\frac{n}{p}
&\quad \text{if} \quad
\left(\frac{1}{p},\frac{1}{q}\right) \in J_3;
\\
\frac{n}{2}-\frac{n}{r}
&\quad \text{if} \quad
\left(\frac{1}{p},\frac{1}{q}\right) \in J_4,
\end{cases}
\]
where $1/p+1/q=1/r$.
The number $m_{\rho}(p,q)$
is the critical order as
the following proposition shows.
A proof of this proposition will be given in Appendix of this paper.
\begin{prop}\label{critical-order}
Let $0 \le \rho <1$, $0<p,q,r \le \infty$, and suppose
$1/p+1/q=1/r$.
If $r<\infty$, then
\[
m_{\rho}(p,q)
=
\sup \{m \in \mathbb{R} \,:\,
T_{\sigma}: H^p \times H^q \to L^r
\;\; \text{for all}\;\; \sigma \in BS^{m}_{\rho,\rho}
\}.
\]
When $p=q=r=\infty$, the above equality holds
if we replace
$H^p \times H^q \to L^r$ by
$L^{\infty} \times L^{\infty} \to BMO $.
\end{prop}
It should be an interesting problem
to prove the boundedness of
bilinear pseudo-differential operators
in the critical class $BS^m_{\rho,\rho}$,
$m=m_{\rho}(p,q)$.
For the case $\rho=0$, this problem was solved by
the authors in \cite{Miyachi-Tomita}.
For the case $0<\rho <1$,
to the best of the authors' knowledge,
the only known result for the problem is due to
Naibo \cite{Naibo}, which however is restricted to
the case $0<\rho <1/2$ and $p=q=r=\infty$.
The purpose of the present paper is to solve the problem
in the range $0 \le 1/p+1/q=1/r \le 1$.
The following are the main results of this paper.
\begin{thm}\label{main-thm-1}
Let $0 \le \rho <1$
and $m=-(1-\rho)n/2$.
Then all bilinear pseudo-differential operators
with symbols in $BS^{m}_{\rho,\rho}(\mathbb{R}^n)$
are bounded from $L^2(\mathbb{R}^n) \times L^{\infty}(\mathbb{R}^n)$
to $L^2(\mathbb{R}^n)$.
\end{thm}
\begin{thm}\label{main-thm-2}
Let $0 \le \rho <1$
and $m=-(1-\rho)n$.
Then all bilinear pseudo-differential operators
with symbols in $BS^{m}_{\rho,\rho}(\mathbb{R}^n)$
are bounded from $L^{\infty}(\mathbb{R}^n) \times L^{\infty}(\mathbb{R}^n)$
to $BMO (\mathbb{R}^n)$.
\end{thm}
\begin{cor}\label{main-cor}
Let $0 \le \rho <1$, $1 \le p,q,r \le \infty$,
$1/p+1/q=1/r$, and $m=m_{\rho}(p,q)$.
Then all bilinear pseudo-differential operators
with symbols in $BS^{m}_{\rho,\rho}(\mathbb{R}^n)$
are bounded from $L^p(\mathbb{R}^n) \times L^{q}(\mathbb{R}^n)$ to $L^r(\mathbb{R}^n)$,
where $L^p(\mathbb{R}^n)$ (respectively, $L^q(\mathbb{R}^n)$)
should be replaced by $H^p(\mathbb{R}^n)$ (respectively, $H^q(\mathbb{R}^n)$)
if $p=1$ (respectively, $q=1$)
and $L^r(\mathbb{R}^n)$ should be replaced by $BMO(\mathbb{R}^n)$ if $r=\infty$.
\end{cor}
Here are some comments on the previous works related
to the above results.
For the subcritical case $m<m_{\rho}(p,q)$,
the boundedness \eqref{boundedness-pqr} were
obtained by
Michalowski-Rule-Staubach \cite{MRS} (for $(1/p, 1/q)$ in the triangle
with vertices $(1/2, 1/2)$, $(1/2, 0)$, $(0, 1/2)$)
and by
B\'enyi-Bernicot-Maldonado-Naibo-Torres \cite{BBMNT}
(in the range $1/p+1/q \le 1$).
As we mentioned above, the
case $m=m_{\rho}(p,q)$ with $\rho=0$
was obtained by the authors \cite{Miyachi-Tomita}.
In fact, \cite[Theorem 1.1]{Miyachi-Tomita}
gives a sharper version of the above Corollary \ref{main-cor} for $\rho=0$
and covers the full range $0<p,q,r \le \infty$.
Naibo \cite{Naibo} has proved the claim of Theorem \ref{main-thm-2}
in the case $0<\rho<1/2$.
Theorem \ref{main-thm-1} should be one of the key estimates
to consider the critical case $m=m_{\rho}(p,q)$ in the
whole range $0<p,q \le \infty$.
Here is a comment concerning the method of proof of Theorem \ref{main-thm-1}.
As we mentioned above this theorem for the case
$\rho=0$ was already proved in \cite{Miyachi-Tomita}.
However, the method of the present paper is
totally different from that of \cite{Miyachi-Tomita}.
The method of \cite{Miyachi-Tomita} seems to work only
in the case $\rho=0$, but the method of the present paper
covers all $0\le \rho <1$.
The contents of this paper are as follows.
In Section \ref{section2},
we recall some preliminary facts.
In Sections \ref{section3}, \ref{section4} and \ref{section5},
we prove Theorems \ref{main-thm-1}, \ref{main-thm-2}
and Corollary \ref{main-cor}, respectively.
In Appendix \ref{appendix}, we prove Proposition \ref{critical-order}.
\section{Preliminaries}\label{section2}
For two nonnegative quantities $A$ and $B$,
the notation $A \lesssim B$ means that
$A \le CB$ for some unspecified constant $C>0$,
and $A \approx B$ means that
$A \lesssim B$ and $B \lesssim A$.
We denote by $\mathbf{1}_S$ the characteristic function of a set $S$,
and by $|S|$ the Lebesgue measure of a measurable set $S$ in $\mathbb{R}^n$.
Let $\mathcal{S}(\mathbb{R}^n)$ and $\mathcal{S}'(\mathbb{R}^n)$ be the Schwartz spaces of
all rapidly decreasing smooth functions
and tempered distributions,
respectively.
We define the Fourier transform $\mathcal{F} f$
and the inverse Fourier transform $\mathcal{F}^{-1}f$
of $f \in \mathcal{S}(\mathbb{R}^n)$ by
\[
\mathcal{F} f(\xi)
=\widehat{f}(\xi)
=\int_{\mathbb{R}^n}e^{-ix\cdot\xi} f(x)\, dx
\quad \text{and} \quad
\mathcal{F}^{-1}f(x)
=\frac{1}{(2\pi)^n}
\int_{\mathbb{R}^n}e^{ix\cdot \xi} f(\xi)\, d\xi.
\]
For $m \in L^{\infty}(\mathbb{R}^n)$,
the linear Fourier multiplier operator $m(D)$
is defined by
\[
m(D)f(x)
=\mathcal{F}^{-1}[m\widehat{f}](x)
=\frac{1}{(2\pi)^n}\int_{\mathbb{R}^n}
e^{ix\cdot\xi}m(\xi)\widehat{f}(\xi)\, d\xi,
\quad
f \in \mathcal{S}(\mathbb{R}^n).
\]
We recall the definition
of Hardy spaces and the space $BMO$ on $\mathbb{R}^n$
(see \cite[Chapters 3 and 4]{Stein}).
Let $0 < p \le \infty$, and let $\phi \in \mathcal{S}(\mathbb{R}^n)$ be such that
$\int_{\mathbb{R}^n}\phi(x)\, dx \neq 0$.
Then the Hardy space $H^p(\mathbb{R}^n)$ consists of
all $f \in \mathcal{S}'(\mathbb{R}^n)$ such that
\[
\|f\|_{H^p}=\left\|\sup_{0<t<\infty}|\phi_t*f|\right\|_{L^p}<\infty,
\]
where $\phi_t(x)=t^{-n}\phi(x/t)$.
It is known that $H^p(\mathbb{R}^n)$ does not depend
on the choice of the function $\phi$
and $H^p(\mathbb{R}^n)=L^p(\mathbb{R}^n)$ for $1<p \le \infty$.
The space $BMO(\mathbb{R}^n)$ consists of
all locally integrable functions $f$ on $\mathbb{R}^n$ such that
\[
\|f\|_{BMO}
=\sup_{Q}\frac{1}{|Q|}
\int_{Q}|f(x)-f_Q|\, dx<\infty,
\]
where $f_Q$ is the average of $f$ on $Q$
and the supremum is taken over all cubes $Q$ in $\mathbb{R}^n$.
It is known that the dual spaces of
$H^1(\mathbb{R}^n)$ is $BMO(\mathbb{R}^n)$.
We end this section by quoting the following,
which we shall call {\it Schur's lemma}\/.
For a proof, see, e.g., \cite[Appendix I]{Grafakos-1}.
\begin{lem}[Schur's lemma]
Let $\{A_{j,k}\}_{j,k \ge 0}$ be a sequence of nonnegative numbers
satisfying
\[
\sup_{j \ge 0}\sum_{k \ge 0}A_{j,k}\le 1
\quad \text{and}\quad
\sup_{k \ge 0}\sum_{j \ge 0}A_{j,k}
\le 1.
\]
Then
\[
\sum_{j,k \ge 0}A_{j,k}b_j c_k
\le \left(\sum_{j \ge 0}b_j^2\right)^{1/2}
\left(\sum_{k \ge 0}c_k^2\right)^{1/2}
\]
for all nonnegative sequences $\{b_j\}$ and $\{c_k\}$.
\end{lem}
\section{Proof of Theorem \ref{main-thm-1}}\label{section3}
In this section,
we shall prove Theorem \ref{main-thm-1}.
The argument is divided into three subsections.
Although the proof for general $\sigma \in BS^{m}_{\rho,\rho}$
is somewhat complicated,
the main idea already consists in the special
case that $\sigma (x,\xi,\eta)$ is independent of $x$,
namely the bilinear Fourier multiplier case.
In this case,
$\sigma_{j,k,\nu}$ to be introduced
in Subsection \ref{subsection-decomposition} reduces to
\[
\sigma_{j,k,\nu}
=
\left\{
\begin{array}{ll}
{\sigma_{j,\nu}} & {\qquad\text{if}\quad k=0}
\\
{0} & {\qquad\text{if}\quad k\ge 1,}
\end{array}
\right.
\]
and the argument will be simple.
We use the following notation and terminology.
For a finite set $\Lambda$, we write $|\Lambda|$ to denote
the number of elements of $\Lambda$.
The following are cubes in $\mathbb{R}^n$:
\begin{align*}
&Q=[-1,1]^{n}, \quad
aQ=[-a,a]^{n}, \quad a>0,
\\
&
x+aQ=\{x+ y \,:\, y\in aQ\},
\quad x\in \mathbb{R}^n.
\end{align*}
If $\sigma$ is
a function on $\mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}^n$,
then
\[
\mathrm{supp}\, ast \sigma=\;\text{closure of}\;
\{(\xi, \eta)\in \mathbb{R}^n \times \mathbb{R}^n
\,:\,
\sigma (x,\xi,\eta) \neq 0 \;\;\text{for some}\;\; x\in \mathbb{R}^n\}.
\]
The usual inner product of $f,h \in L^2=L^2 (\mathbb{R}^n)$ is denoted by
$\langle f, g\rangle$.
If $\{E_{\alpha}\}$ is a finite family of subsets of $\mathbb{R}^n$,
$L$ is a positive integer, and if
\[
|\{\beta \,:\, E_{\beta} \cap E_{\alpha}
\neq \emptyset \}|\le L
\quad \text{for all}\quad \alpha,
\]
then
we say that the {\it interaction}
of the family $\{E_{\alpha}\}$ is bounded by $L$.
\subsection{Decomposition of the symbol and some preliminaries}
\label{subsection-decomposition}
We use the following two types of partitions of unity.
One is the dyadic decomposition:
\begin{equation}\label{dyadic-decomposition}
\begin{split}
&\mathrm{supp}\, \psi_0
\subset \{\zeta \in \mathbb{R}^d \,:\, |\zeta| \le 2\},
\\
&\mathrm{supp}\, \psi_j
\subset \{\zeta \in \mathbb{R}^d \,:\, 2^{j-1} \le |\zeta| \le 2^{j+1}\},
\quad j \ge 1,
\\
&\|\partial^{\alpha}\psi_j\|_{L^{\infty}} \lesssim 2^{-j|\alpha|},
\quad \alpha \in \mathbb{N}_0^d, \ j \ge 0,
\\
&\sum_{j \ge 0}\psi_j(\zeta)=1,
\quad \zeta \in \mathbb{R}^d.
\end{split}
\end{equation}
The other is the uniform decomposition:
\begin{equation}\label{uniform-decomposition}
\begin{split}
&\mathrm{supp}\, \varphi \subset Q,
\\
&\sum_{\nu \in \mathbb{Z}^n}\varphi(\xi-\nu)=1,
\quad \xi \in \mathbb{R}^n.
\end{split}
\end{equation}
Here $\psi_j$, $j \ge 0$, and $\varphi$
are smooth real-valued functions.
We shall use \eqref{dyadic-decomposition}
with $d=2n$ and $d=n$.
We write $\Psi_j$
to denote the function $\psi_j$ of
\eqref{dyadic-decomposition} with $d=2n$
and write $\psi_j$ to denote the function
of \eqref{dyadic-decomposition}
with $d=n$.
We shall use \eqref{uniform-decomposition} only on $\mathbb{R}^n$.
In this subsection, we assume
$\sigma \in BS^{m}_{\rho,\rho}$ with $m\in \mathbb{R}$ and $0\le \rho \le 1$.
(The conditions on $m$ and $\rho$
as in Theorem \ref{main-thm-1}
are not necessary in this subsection.)
We decompose $\sigma$ as
\begin{equation}\label{symbol-dec-0}
\sigma(x,\xi,\eta)=
\sum_{j \ge 0}
\sum_{\nu=(\nu_1,\nu_2) \in \mathbb{Z}^n \times \mathbb{Z}^n}
\sigma_{j,\nu}(x,\xi,\eta)
=
\sum_{j \ge 0}\sum_{k \ge 0}
\sum_{\nu=(\nu_1,\nu_2) \in \mathbb{Z}^n \times \mathbb{Z}^n}
\sigma_{j,k,\nu}(x,\xi,\eta),
\end{equation}
where
\begin{equation}\label{def-sigmajn}
\sigma_{j,\nu}(x,\xi,\eta)
=\sigma(x,\xi,\eta)
\varphi(2^{-j\rho}\xi-\nu_1)\varphi(2^{-j\rho}\eta-\nu_2)
\Psi_j(\xi,\eta)
\end{equation}
and
\begin{equation}\label{def-sigmajkn}
\begin{split}
\sigma_{j,k,\nu}(x,\xi,\eta)
&=[\psi_k(2^{-j\rho}D_x)\sigma_{j,\nu}](x,\xi,\eta)
\\
&=
2^{j\rho n}
\int_{\mathbb{R}^n}[\mathcal{F}^{-1}\psi_k](2^{j\rho}y)
\sigma_{j,\nu}(x-y,\xi,\eta)\, dy.
\end{split}
\end{equation}
Notice the following facts.
First,
if we write the projections as
\[
\pi_1 (\xi, \eta)=\xi,
\quad
\pi_2 (\xi, \eta)=\eta,
\]
then it is obvious that
\begin{equation}\label{replace-f1-g1}
T_{\sigma_{j,k,\nu}} (f,g)
=
T_{\sigma_{j,k,\nu}} (f^{(1)},g)
=T_{\sigma_{j,k,\nu}} (f,g^{(1)})
\end{equation}
whenever $f^{(1)}$ and $g^{(1)}$ satisfy
$(f^{(1)})^{\wedge}= \widehat{f}$ on
$\pi_1 (\mathrm{supp}\, ast (\sigma_{j,\nu}))$
and $(g^{(1)})^{\wedge}= \widehat{g}$ on
$\pi_2 (\mathrm{supp}\, ast (\sigma_{j,\nu}))$.
Secondly, the Fourier transform of
$T_{\sigma_{j,k,\nu}}(f,g)$ is given by
\begin{equation}\label{Fourier-Tsjkn}
\begin{split}
&\mathcal{F} [T_{\sigma_{j,k,\nu}}(f,g)] (\zeta)
\\
&=
\frac{1}{(2\pi)^{2n}}
\int_{\mathbb{R}^n \times \mathbb{R}^n}
\psi_{k}(2^{-j\rho}(\zeta-\xi-\eta))
[\mathcal{F}_x\sigma_{j,\nu}](\zeta-\xi-\eta,\xi,\eta)
\widehat{f}(\xi)\widehat{g}(\eta)\, d\xi d\eta,
\end{split}
\end{equation}
where $\zeta \in \mathbb{R}^n$
and $\mathcal{F}_x\sigma_{j,\nu}$ denotes
the partial Fourier transform
of $\sigma_{j,\nu}(x,\xi,\eta)$ with
respect to the $x$-variable.
From this we see that
\begin{equation}\label{FouriersuppTsjkn}
\mathrm{supp}\, \mathcal{F} [T_{\sigma_{j,k,\nu}}(f,g)]
\subset
\bigcup_{(\xi, \eta)\in \mathrm{supp}\, ast (\sigma_{j,\nu})}
\{
\zeta
\,:\,
|\zeta - \xi - \eta|
\le 2^{j \rho + k+1}
\}.
\end{equation}
Hence, we have
\begin{equation}\label{replace-h1}
\big\langle T_{\sigma_{j,k,\nu}} (f,g), h \big\rangle
=
\big\langle T_{\sigma_{j,k,\nu}} (f,g), h^{(1)} \big\rangle
\end{equation}
whenever $h^{(1)}$ satisfy $(h^{(1)})^{\wedge}= \widehat{h}$
on the set on the right-hand side of \eqref{FouriersuppTsjkn}.
In the argument to follow, we shall use
\eqref{replace-f1-g1} and \eqref{replace-h1}
by choosing the functions $f^{(1)}$, $g^{(1)}$, $h^{(1)}$ according to
several different situations.
We also use the following general lemma
for nearly orthogonal functions and operators.
\begin{lem}\label{orthogonality}
{\rm (1)} If $\{f_{\alpha}\}$ is a finite family of functions in $L^2$,
$L$ is a positive integer,
and if $|\{\beta \,:\, \langle f_{\beta}, f_{\alpha}\rangle \neq 0\}|
\le L$
for all $\alpha$, then
$\|\sum_{\alpha} f_{\alpha}\|_{L^2}^2
\le L \sum_{\alpha} \|f_{\alpha}\|_{L^2}^2$.
\\
{\rm (2)} If $\{T_{\alpha}\}$ is a finite family of
bounded linear operators in $L^2$,
$L$ is a positive integer,
and if $| \{\beta \,:\, T_{\beta}^{\ast} T_{\alpha} \neq 0\} |
\le L$
for all $\alpha$, then
$\|\sum_{\alpha} T_{\alpha}\|_{L^2\to L^2}^2
\le L \sum_{\alpha} \|T_{\alpha}\|_{L^2\to L^2}^2$.
\\
{\rm (3)} If $\{T_{\alpha}\}$ is a finite family of
bounded linear operators in $L^2$,
$L$ is a positive integer,
and if $|\{\beta \,:\, T_{\beta} T_{\alpha}^{\ast} \neq 0\} |
\le L$
for all $\alpha$, then
$\|\sum_{\alpha} T_{\alpha}\|_{L^2\to L^2}^2
\le L \sum_{\alpha} \|T_{\alpha}\|_{L^2\to L^2}^2$.
\end{lem}
\begin{proof}
To prove (1), we write
\begin{equation*}
\bigg\|\sum_{\alpha} f_{\alpha}\bigg\|_{L^2}^2
=
\sum_{\alpha}\sum_{\beta}
\langle f_{\alpha}, f_{\beta} \rangle
\le
\sum_{\alpha}\sum_{\beta}
\mathbf{1} \{
\langle f_{\alpha}, f_{\beta} \rangle
\neq 0 \}
\|f_{\alpha}\|_{L^2}
\|f_{\beta}\|_{L^2}.
\end{equation*}
Applying Schur's lemma, we obtain the desired inequality.
We can prove (2) by applying (1) to $f_{\alpha}= T_{\alpha}f$.
The assertion (3) follows from (2)
since the norms of an operator and its adjoint are the same.
\end{proof}
\subsection{Basic estimates}
\label{subsection-basic-estimates}
In this subsection, except in the last lemma, Lemma
\ref{sum-on-annulus}, we only assume
$\sigma \in BS^{m}_{\rho,\rho}$ with $m\in \mathbb{R}$ and $0\le \rho \le 1$.
We shall give some basic estimates which will be used later.
We use the following notation
\[
S_{a}(f) (x)
=a^{n} \int_{\mathbb{R}^n}
\frac{|f(y)|}{(1+ a |x-y|)^{n+1}}\, dy,
\quad a>0, \;\; x\in \mathbb{R}^{n}.
\]
Let us start with the estimate for the square function of
$\widetilde{\varphi}(2^{-j\rho}D-\ell)f$
with respect to $\ell \in \mathbb{Z}^n$.
Although this is known to many people,
we shall give the proof for the reader's convenience.
\begin{lem}\label{basic-estimate-0}
Let $\widetilde{\varphi} \in \mathcal{S}(\mathbb{R}^n)$.
Then
\[
\bigg(\sum_{\ell \in \mathbb{Z}^n}
|\widetilde{\varphi}(2^{-j\rho}D-\ell)f(x)|^2\bigg)^{1/2}
\lesssim
S_{2^{j\rho}}(f^2)(x)^{1/2}
\]
holds for $j \ge 0$ and $x\in \mathbb{R}^n$.
\end{lem}
\begin{proof}
Since $\widetilde{\varphi}(2^{-j\rho}D-\ell)f(x)
=\widetilde{\varphi}(D-\ell)[f(2^{-j\rho}\, \cdot \,)](2^{j\rho}x)$,
by a scaling argument,
it is sufficient to prove the case $j=0$.
By a periodization technique,
we can write
\begin{align*}
\widetilde{\varphi}(D-\ell)f(x)
&=\int_{\mathbb{R}^n}e^{i\ell\cdot (x-y)}
\widetilde{\Phi}(x-y)f(y)\, dy
\\
&=\sum_{\widetilde{\ell} \in \mathbb{Z}^n}
\int_{2\pi\widetilde{\ell}+[-\pi,\pi]^n}e^{i\ell\cdot (x-y)}
\widetilde{\Phi}(x-y)f(y)\, dy
\\
&=e^{i \ell \cdot x}
\int_{[-\pi,\pi]^n}e^{-i\ell\cdot y}
\bigg(\sum_{\widetilde{\ell} \in \mathbb{Z}^n}
\widetilde{\Phi}(x-y-2\pi\widetilde{\ell})
f(y+2\pi\widetilde{\ell})\bigg) dy,
\end{align*}
where $\widetilde{\Phi}=\mathcal{F}^{-1}\widetilde{\varphi}$.
This means that
$|\widetilde{\varphi}(D-\ell)f(x)|$
is equal to $(2 \pi )^n$ times the absolute value of the $\ell$-th Fourier coefficient
of the $(2\pi \mathbb{Z})^n$-periodic function
$\sum_{\widetilde{\ell} \in \mathbb{Z}^n}
\widetilde{\Phi}(x-y-2\pi\widetilde{\ell})f(y+2\pi\widetilde{\ell})$
of the $y$-variable.
Hence, it follows from Parseval's identity that
\begin{align*}
\sum_{\ell \in \mathbb{Z}^n}
|\widetilde{\varphi}(D-\ell)f(x)|^2
&={(2\pi)^n}
\int_{[-\pi,\pi]^n}
\bigg|\sum_{\widetilde{\ell} \in \mathbb{Z}^n}
\widetilde{\Phi}(x-y-2\pi\widetilde{\ell})
f(y+2\pi\widetilde{\ell})\bigg|^2 dy.
\end{align*}
Since
$\sup_{z \in \mathbb{R}^n}\bigg(\sum_{\widetilde{\ell} \in \mathbb{Z}^n}|
\widetilde{\Phi}(z-2\pi\widetilde{\ell})|\bigg)<\infty$,
by Schwarz's inequality,
the right-hand side of this identity is estimated by
\begin{align*}
\int_{[-\pi,\pi]^n}
\sum_{\widetilde{\ell} \in \mathbb{Z}^n}
|\widetilde{\Phi}(x-y-2\pi\widetilde{\ell})|
|f(y+2\pi\widetilde{\ell})|^2\, dy
=\int_{\mathbb{R}^n}|\widetilde{\Phi}(x-y)||f(y)|^2\, dy.
\end{align*}
Therefore, the rapidly decreasing property of $\widetilde{\Phi}$
gives the desired estimate.
\end{proof}
\begin{lem}\label{basic-estimate-1}
For each $N \in \mathbb{N}_0$ and $\beta,\gamma \in \mathbb{N}_0^n$,
the estimate
\[
|\partial^{\beta}_{\xi}\partial^{\gamma}_{\eta}
\sigma_{j,k,\nu}(x,\xi,\eta)|
\lesssim
2^{jm-kN}2^{-j\rho(|\beta|+|\gamma|)}
\]
holds for $j,k \ge 0$ and $\nu \in \mathbb{Z}^n \times \mathbb{Z}^n$.
\end{lem}
\begin{proof}
First, suppose $k \ge 1$.
Then by the moment condition of $\mathcal{F}^{-1}\psi_k$ and
Taylor's formula,
we can write \eqref{def-sigmajkn} as
\begin{align*}
&\sigma_{j,k,\nu}(x,\xi,\eta)
\\
&=2^{j\rho n}
\int_{\mathbb{R}^n}[\mathcal{F}^{-1}\psi_k](2^{j\rho}y)
\bigg(\sigma_{j,\nu}(x-y,\xi,\eta)
-\sum_{|\alpha|<N}
\frac{\partial_x^{\alpha}\sigma_{j,\nu}(x,\xi,\eta)}{\alpha!}
(-y)^{\alpha}\bigg)dy
\\
&=2^{j\rho n}
\int_{\mathbb{R}^n}
[\mathcal{F}^{-1}\psi_k](2^{j\rho}y)
\bigg(N\sum_{|\alpha|=N}\frac{(-y)^{\alpha}}{\alpha!}
\int_0^1
(1-t)^{N-1}
[\partial_x^{\alpha}\sigma_{j,\nu}](x-ty,\xi,\eta)\, dt
\bigg)dy.
\end{align*}
Using the fact that $1+|\xi|+|\eta| \approx 2^j$
for $(\xi,\eta) \in \mathrm{supp}\, ast (\sigma_{j,\nu})$,
we have
\[
|\partial^{\alpha}_x\partial^{\beta}_{\xi}\partial^{\gamma}_{\eta}
\sigma_{j,\nu}(x,\xi,\eta)|
\lesssim 2^{jm+j\rho(|\alpha|-|\beta|-|\gamma|)}.
\]
On the other hand,
it follows from \eqref{dyadic-decomposition} that
\[
|\mathcal{F}^{-1}\psi_k(y)|
\lesssim 2^{kn}(1+2^k|y|)^{-(N+n+1)}.
\]
Hence
\begin{align*}
&|\partial^{\beta}_{\xi}\partial^{\gamma}_{\eta}
\sigma_{j,k,\nu}(x,\xi,\eta)|
\\
&\lesssim
\sum_{|\alpha|=N}
2^{j\rho n}
\int_{\mathbb{R}^n}\int_0^1
\bigg|[\mathcal{F}^{-1}\psi_k](2^{j\rho}y)y^{\alpha}
[\partial_x^{\alpha}\partial^{\beta}_{\xi}\partial^{\gamma}_{\eta}
\sigma_{j,\nu}](x-ty,\xi,\eta)\bigg| dtdy
\\
&\lesssim
2^{(j\rho+k)n}\int_{\mathbb{R}^n}(1+2^{j\rho+k}|y|)^{-(N+n+1)}
|y|^{N}2^{jm+j\rho(N-|\beta|-|\gamma|)}\, dy
\\
&
\approx
2^{jm-kN}2^{-j\rho(|\beta|+|\gamma|)}.
\end{align*}
If $k=0$, then using \eqref{def-sigmajkn}
and slightly modifying the above argument,
we obtain the desired estimate.
\end{proof}
\begin{lem}\label{basic-estimate-2}
For each $N \in \mathbb{N}_0$,
the estimate
\[
|T_{\sigma_{j,k,\nu}}(f,g)(x)|
\lesssim
2^{jm-kN}S_{2^{j\rho}}(f)(x)S_{2^{j\rho}}(g)(x)
\]
holds for $j,k \ge 0$ and $\nu \in \mathbb{Z}^n \times \mathbb{Z}^n$.
\end{lem}
\begin{proof}
We write
\[
T_{\sigma_{j,k,\nu}}(f,g)(x)
=\int_{\mathbb{R}^n \times \mathbb{R}^n}
K_{j,k,\nu}(x,x-y,x-z)f(y)g(z)\, dydz,
\]
where
\[
K_{j,k,\nu}(x,y,z)
=\frac{1}{(2\pi)^{2n}}\int_{\mathbb{R}^n \times \mathbb{R}^n}
e^{i(y\cdot\xi+z\cdot\eta)}
\sigma_{j,k,\nu}(x,\xi,\eta)\, d\xi d\eta.
\]
Since
$|\xi-2^{j\rho}\nu_1| \lesssim 2^{j\rho}$
and
$|\eta-2^{j\rho}\nu_2| \lesssim 2^{j\rho}$
for $(\xi,\eta) \in \mathrm{supp}\, ast (\sigma_{j,k,\nu})$,
it follows from Lemma \ref{basic-estimate-1}
and integration by parts that
\[
|K_{j,k,\nu}(x,y,z)|
\lesssim
2^{jm-kN}
\frac{2^{j\rho n}}{(1+2^{j\rho}|y|)^{n+1}}
\frac{2^{j\rho n}}{(1+2^{j\rho}|z|)^{n+1}}.
\]
From this the desired estimate follows.
\end{proof}
The estimate
\begin{equation}\label{single-estimate}
\|T_{\sigma_{j,k,\nu}}(f,g)\|_{L^2}
\lesssim
2^{jm -kN} \|f\|_{L^2} \|g\|_{L^{\infty}}
\end{equation}
immediately follows from Lemma \ref{basic-estimate-2}.
In the lemmas below, we shall derive finer
estimates by utilizing orthogonality.
\begin{lem}\label{sum-on-line}
{\rm (1)} For each $N \in \mathbb{N}_0$, the estimate
\[
\bigg\|\sum_{\nu_2 \in \mathbb{Z}^n}
T_{\sigma_{j,k,\nu}}(f,g)\bigg\|_{L^2}
\lesssim 2^{jm-kN}\|f\|_{L^2}\|g\|_{L^\infty}
\]
holds for all $j,k \ge 0$ and all $\nu_1 \in \mathbb{Z}^n$.
\\
{\rm (2)} For each $N \in \mathbb{N}_0$, the estimate
\[
\bigg\|\sum_{\nu_1+\nu_2=\mu}
T_{\sigma_{j,k,\nu}}(f,g)\bigg\|_{L^2}
\lesssim 2^{jm-kN}\|f\|_{L^2}\|g\|_{L^\infty}
\]
holds for all $j,k \ge 0$ and all $\mu \in \mathbb{Z}^n$.
\end{lem}
\begin{proof}
Proof of (1).
Take a function
$\widetilde{\varphi}\in C_{0}^{\infty}(\mathbb{R}^n)$
such that
$\widetilde{\varphi} (\xi)=1$ on $\mathrm{supp}\, \varphi$.
Then, by \eqref{replace-f1-g1},
\begin{equation}\label{replace-fjn-gjn}
T_{\sigma_{j,k,\nu}} (f,g)
=T_{\sigma_{j,k,\nu}} (f_{j,\nu_1},g)
=T_{\sigma_{j,k,\nu}} (f,g_{j,\nu_2})
\end{equation}
with
\begin{equation}\label{fjn-gjn}
f_{j,\nu_1}=\widetilde{\varphi} (2^{-j\rho}D - \nu_1)f,
\quad
g_{j,\nu_2}=\widetilde{\varphi} (2^{-j\rho}D - \nu_2)g.
\end{equation}
From \eqref{FouriersuppTsjkn},
we see that
\begin{equation}\label{Fouriersupp-n1n2}
\mathrm{supp}\, \mathcal{F} [T_{\sigma_{j,k,\nu}}(f,g)]
\subset
2^{j\rho}(\nu_1 + \nu_2) + 2^{j\rho + k+2}Q.
\end{equation}
Notice that for fixed $\nu_1 \in \mathbb{Z}^{n}$
the interaction of the family
$\{2^{j\rho}(\nu_1 + \nu_2)
+ 2^{j\rho + k+2}Q\}_{\nu_2 \in\mathbb{Z}^n}$
is $\lesssim 2^{kn}$.
Hence, by Lemma \ref{orthogonality} (1) and
Lemma \ref{basic-estimate-2},
we have
\begin{align}
&
\bigg\|
\sum_{\nu_2 \in \mathbb{Z}^n}
T_{\sigma_{j,k, \nu}}
(f,g)
\bigg\|_{L^2}^2
=
\bigg\|
\sum_{\nu_2 \in \mathbb{Z}^n}
T_{\sigma_{j,k, \nu}}
(f,g_{j,\nu_2})
\bigg\|_{L^2}^2
\nonumber
\\
&
\lesssim
2^{kn}
\sum_{\nu_2 \in \mathbb{Z}^n}
\|
T_{\sigma_{j,k, \nu}}(f,g_{j,\nu_2})
\|_{L^2}^2
\nonumber
\\
&
\lesssim
2^{kn + 2(jm-kN)}
\sum_{\nu_2 \in \mathbb{Z}^n}
\int_{\mathbb{R}^n}
S_{2^{j\rho}}(f)(x)^2
S_{2^{j\rho}}(g_{j,\nu_2})(x)^2
\, dx.
\label{1111}
\end{align}
By Schwarz's inequality and
Lemma \ref{basic-estimate-0},
\begin{align*}
&
\sum_{\nu_2 \in \mathbb{Z}^n}
\int_{\mathbb{R}^n}
S_{2^{j\rho}}(f)(x)^2
S_{2^{j\rho}}(g_{j,\nu_2})(x)^2
\, dx
\\
&
\lesssim
\sum_{\nu_2 \in \mathbb{Z}^n}
\int_{\mathbb{R}^n}
S_{2^{j\rho}}(f^2)(x)
S_{2^{j\rho}}(g_{j,\nu_2}^2)(x)
\, dx
\\
&
\lesssim
\int_{\mathbb{R}^n}
S_{2^{j\rho}}(f^2)(x)
S_{2^{j\rho}}\big[S_{2^{j\rho}}(g^2)\big](x)
\, dx
\\
&
\lesssim
\|f\|_{L^2}^2
\|g\|_{L^{\infty}}^2.
\end{align*}
Since $N$ can be taken arbitrarily large,
we obtain the desired estimate.
Proof of (2).
By \eqref{replace-fjn-gjn}-\eqref{fjn-gjn},
Lemma \ref{basic-estimate-2}, and Schwarz's inequality,
we have
\begin{align*}
&
\bigg|
\sum_{\nu_1 + \nu_2=\mu}
T_{\sigma_{j,k,\nu}}
(f,g)(x)
\bigg|
\le
\sum_{\nu_1 + \nu_2=\mu}
|T_{\sigma_{j,k,\nu}}
(f_{j,\nu_1},g_{j,\nu_2})(x)|
\\
&
\lesssim
2^{jm-kN}
\sum_{\nu_1 + \nu_2=\mu}
S_{2^{j\rho}}(f_{j,\nu_1})(x)
S_{2^{j\rho}}(g_{j,\nu_2})(x)
\\
&
\le
2^{jm-kN}
\bigg(\sum_{\nu_1}
S_{2^{j\rho}}(f_{j,\nu_1})(x)^2
\bigg)^{1/2}
\bigg(\sum_{\nu_1}
S_{2^{j\rho}}(g_{j,\mu -\nu_1})(x)^2
\bigg)^{1/2}
\\
&
\lesssim
2^{jm-kN}
\bigg(\sum_{\nu_1}
S_{2^{j\rho}}(f_{j,\nu_1}^2)(x)
\bigg)^{1/2}
\bigg(\sum_{\nu_1}
S_{2^{j\rho}}(g_{j,\mu -\nu_1}^2)(x)
\bigg)^{1/2}.
\end{align*}
By Lemma \ref{basic-estimate-0},
we have
\begin{equation*}
\bigg(\sum_{\nu_1}
S_{2^{j\rho}}(f_{j,\nu_1}^2)(x)
\bigg)^{1/2}
\lesssim
S_{2^{j\rho}}\big[S_{2^{j\rho}}(f^2)\big]
(x)^{1/2}
\approx
S_{2^{j\rho}}(f^2)(x)^{1/2}.
\end{equation*}
Similarly,
\[
\bigg(\sum_{\nu_1}
S_{2^{j\rho}}(g_{j,\mu -\nu_1}^2)(x)
\bigg)^{1/2}
\lesssim
S_{2^{j\rho}}(g^2)(x)^{1/2}
\lesssim \|g\|_{L^{\infty}}.
\]
Thus we have the pointwise estimate
\[
\bigg|
\sum_{\nu_1 + \nu_2=\mu}
T_{\sigma_{j,k,\nu}}
(f,g)(x)
\bigg|
\lesssim
2^{jm-kN}
S_{2^{j\rho}}(f^2)(x)^{1/2}
\|g\|_{L^{\infty}},
\]
from which the desired $L^2$ inequality follows.
\end{proof}
\begin{lem}\label{sum-on-several-lines}
{\rm (1)} For each $N \in \mathbb{N}_0$, the estimate
\[
\bigg\|
\sum_{\nu_1 \in \Lambda}\sum_{\nu_2 \in \mathbb{Z}^n}
T_{\sigma_{j,k,\nu}}(f,g)\bigg\|_{L^2}
\lesssim
|\Lambda|^{1/2}
2^{jm-kN}\|f\|_{L^2}\|g\|_{L^\infty}
\]
holds for all $j,k \ge 0$ and all finite sets $\Lambda \subset \mathbb{Z}^n$.
\\
{\rm (2)} For each $N \in \mathbb{N}_0$, the estimate
\[
\bigg\|\sum_{\mu \in \Lambda}\sum_{\nu_1+\nu_2=\mu}
T_{\sigma_{j,k,\nu}}(f,g)\bigg\|_{L^2}
\lesssim
|\Lambda|^{1/2}
2^{jm-kN}\|f\|_{L^2}\|g\|_{L^\infty}
\]
holds for all $j,k \ge 0$ and all finite sets $\Lambda \subset \mathbb{Z}^n$.
\end{lem}
\begin{proof}
For the proof of (1) and (2),
we freeze $g \in L^{\infty}$
and consider the linear operator $T_{\sigma_{j,k,\nu}}(\cdot,g)$
defined by $[T_{\sigma_{j,k,\nu}}(\cdot,g)](f)=
T_{\sigma_{j,k,\nu}}(f,g)$
for $j,k \ge 0$ and $\nu \in \mathbb{Z}^n \times \mathbb{Z}^n$.
By \eqref{single-estimate},
$T_{\sigma_{j,k,\nu}}(\cdot,g)$ is a bounded linear operator in $L^2$.
Proof of (1).
Since
\[
\mathrm{supp}\, ast \bigg(\sum_{\nu_2\in Z^n} \sigma_{j,k,\nu}\bigg)
\subset
\mathrm{supp}\, \varphi (2^{-j\rho}\cdot -\nu_1) \times \mathbb{R}^n,
\]
we have
\[
\sum_{\nu_2 \in \mathbb{Z}^n}T_{\sigma_{j,k,\nu}}(f,g)
=\sum_{\nu_2 \in \mathbb{Z}^n}T_{\sigma_{j,k,\nu}}
(\mathbf{1}_{\mathrm{supp}\, \varphi (2^{-j\rho}\cdot -\nu_1)}(D)f,g).
\]
In terms of the linear operator, this can be written as
\[
\sum_{\nu_2 \in \mathbb{Z}^n}
T_{\sigma_{j,k,\nu}}(\cdot,g)=
\bigg(\sum_{\nu_2 \in \mathbb{Z}^n}
T_{\sigma_{j,k,\nu}}(\cdot,g)\bigg)
\mathbf{1}_{\mathrm{supp}\, \varphi (2^{-j\rho}\cdot -\nu_1)}(D).
\]
Since the interaction of the family
$\{\mathrm{supp}\, \varphi (2^{-j\rho}\cdot -\nu_1) \}_{\nu_{1}}$
is $\lesssim 1$,
we see that
\[
\bigg|\bigg\{\widetilde{\nu_1} \in \mathbb{Z}^n \,:\,
\bigg(\sum_{\nu_2 \in \mathbb{Z}^n}
T_{\sigma_{j,k,(\widetilde{\nu_1},\nu_2)}}(\cdot,g)\bigg)
\bigg(\sum_{\nu_2 \in \mathbb{Z}^n}
T_{\sigma_{j,k,(\nu_1,\nu_2)}}(\cdot,g)\bigg)^{\ast} \neq 0
\bigg\}\bigg|
\lesssim 1
\]
for all $\nu_1 \in \mathbb{Z}^n$.
Thus Lemma \ref{orthogonality} (3) yields
\[
\bigg\|
\sum_{\nu_1 \in \Lambda}
\sum_{\nu_2 \in \mathbb{Z}^n}
T_{\sigma_{j,k,\nu}}(\cdot ,g)
\bigg\|_{L^2\to L^2}^2
\lesssim
\sum_{\nu_1 \in \Lambda}
\bigg\|
\sum_{\nu_2 \in \mathbb{Z}^n}
T_{\sigma_{j,k,\nu}}(\cdot ,g)
\bigg\|_{L^2\to L^2}^2.
\]
By Lemma \ref{sum-on-line} (1), the
right-hand side of the above is $\lesssim 2^{2(jm-kN)}
|\Lambda|\|g\|_{L^{\infty}}^2$,
which implies the desired estimate.
Proof of (2).
As in \eqref{Fouriersupp-n1n2},
the formula \eqref{FouriersuppTsjkn} implies
\[
\mathrm{supp}\, \mathcal{F} \bigg[\sum_{\nu_1+\nu_2=\mu }
T_{\sigma_{j,k,\nu}}(f,g)
\bigg]
\subset
2^{j\rho}\mu
+ 2^{j\rho+ k+2}Q,
\]
which gives
\[
\sum_{\nu_1+\nu_2=\mu}
T_{\sigma_{j,k,\nu}}(f,g)
=\mathbf{1}_{2^{j\rho}\mu
+ 2^{j\rho+ k+2}Q}(D)
\bigg(\sum_{\nu_1+\nu_2=\mu}
T_{\sigma_{j,k,\nu}}(f,g)\bigg).
\]
Thus, in terms of the linear operator,
\[
\sum_{\nu_1+\nu_2=\mu}
T_{\sigma_{j,k,\nu}}(\cdot,g)
=\mathbf{1}_{2^{j\rho}\mu
+ 2^{j\rho+ k+2}Q}(D)
\bigg(\sum_{\nu_1+\nu_2=\mu}
T_{\sigma_{j,k,\nu}}(\cdot,g)\bigg).
\]
Since the interaction of the family
$\{2^{j\rho}\mu
+ 2^{j\rho+ k+2}Q\}_{\mu \in \mathbb{Z}^n}$
is $\lesssim 2^{kn}$,
we see that
\[
\bigg|\bigg\{\widetilde{\mu} \in \mathbb{Z}^n \,:\,
\bigg(\sum_{\nu_1+\nu_2=\widetilde{\mu}}
T_{\sigma_{j,k,\nu}}(\cdot,g)\bigg)^{\ast}
\bigg(\sum_{\nu_1+\nu_2=\mu}
T_{\sigma_{j,k,\nu}}(\cdot,g)\bigg)
\neq 0
\bigg\}\bigg|
\lesssim 2^{kn}
\]
for all $\mu \in \mathbb{Z}^n$.
Hence Lemma \ref{orthogonality} (2) yields
\[
\bigg\|
\sum_{\mu \in \Lambda}
\sum_{\nu_1 + \nu_2 =\mu}
T_{\sigma_{j,k,\nu}}(\cdot ,g)
\bigg\|_{L^2\to L^2}^2
\lesssim
2^{kn}
\sum_{\mu \in \Lambda}
\bigg\|
\sum_{\nu_1 + \nu_2 =\mu}
T_{\sigma_{j,k,\nu}}(\cdot ,g)
\bigg\|_{L^2\to L^2}^2.
\]
By Lemma \ref{sum-on-line} (2), the
right-hand side of the above is
$\lesssim 2^{kn+ 2(jm-kN)}|\Lambda|\|g\|_{L^{\infty}}^2$.
Since $N$ can be taken arbitrarily large, we obtain the desired estimate.
\end{proof}
Notice that
$\sigma_{j,k,\nu}\neq 0$ only for $|\nu_{1}|\lesssim 2^{j(1-\rho)}$
and $|\nu_2|\lesssim 2^{j(1-\rho)}$ and hence
$\sum_{\nu \in \mathbb{Z}^n \times \mathbb{Z}^n}\, T_{\sigma_{j,k,\nu}}$
can be written as the
sum of Lemma \ref{sum-on-several-lines} (1) or (2)
with $|\Lambda|\approx 2^{j(1-\rho)n}$.
Hence the following lemma
directly follows from Lemma \ref{sum-on-several-lines}.
\begin{lem}\label{sum-on-annulus}
If $m=-(1-\rho)n/2$ and $0\le \rho\le 1$,
then for each $N \in \mathbb{N}_0$ the estimate
\[
\bigg\|\sum_{\nu \in \mathbb{Z}^n \times \mathbb{Z}^n}\, T_{\sigma_{j,k,\nu}}
(f,g)\bigg\|_{L^2}
\lesssim
2^{-kN}\|f\|_{L^2}\|g\|_{L^\infty}
\]
holds for $j,k \ge 0$.
\end{lem}
\subsection{Proof of Theorem \ref{main-thm-1}}\label{subsection-proof}
Throughout this subsection, we assume $m$, $\rho$, and $\sigma$
satisfy the conditions of Theorem \ref{main-thm-1}, namely,
$0\le \rho <1$, $m=-(1-\rho)n/2$,
and $\sigma \in BS^{m}_{\rho,\rho}(\mathbb{R}^n)$.
Before proceeding to the main argument,
we shall see that it is sufficient to consider
the case where $\mathrm{supp}\, ast \sigma$
is included in a cone minus a ball centered at the origin.
To see this, take a function
$\mathbb{T}heta \in C_{0}^{\infty}(\mathbb{R}^n \times \mathbb{R}^n)$
such that
$\mathbb{T}heta (\xi, \eta)=1$ on
$\{(|\xi|^2+ |\eta|^2)^{1/2}\le 2\}$
and $\mathrm{supp}\, \mathbb{T}heta
\subset \{(|\xi|^2+ |\eta|^2)^{1/2}\le 4\}$,
and write $\sigma$ as
\[
\sigma(x, \xi,\eta)=\sigma(x, \xi,\eta)\mathbb{T}heta(\xi,\eta)
+\sigma(x, \xi,\eta)(1-\mathbb{T}heta(\xi,\eta)).
\]
By simply summing the estimate
of Lemma \ref{sum-on-annulus} over $k\ge 0$ and $0\le j\le 2$,
we obtain
\[
\|T_{\sigma\mathbb{T}heta}(f,g)\|_{L^2}
\lesssim \|f\|_{L^2}\|g\|_{L^{\infty}}.
\]
Hence
it is sufficient to treat only $T_{\sigma(1-\mathbb{T}heta)}$.
Next, if $(\xi,\eta)$ belongs to the unit
sphere $\Sigma$ of $\mathbb{R}^n \times \mathbb{R}^n$,
then either $\xi+\eta \neq 0$ or $\xi \neq 0$.
By the compactness of $\Sigma$,
this implies that there exists a constant $c>0$ such that
$\Sigma$ is covered by the two open sets
\begin{equation*}
V_1=\{(\xi,\eta) \in \Sigma \,:\,
|\xi+\eta|>c\},
\quad
V_2=\{(\xi,\eta) \in \Sigma \,:\,
|\xi|>c\}.
\end{equation*}
Taking a smooth partition of unity $\Phi_i$, $i=1,2$,
on $\Sigma$ such that $\mathrm{supp}\, \Phi_i \subset V_i$,
we decompose $\sigma (1-\mathbb{T}heta)$ as
\[
\sigma(x,\zeta)(1-\mathbb{T}heta(\zeta))
=\sum_{i=1}^{2}\sigma(x,\zeta)(1-\mathbb{T}heta(\zeta))
\Phi_i(\zeta/|\zeta|)
=\sum_{i=1}^{2}\sigma^{(i)}(x,\zeta),
\quad \zeta=(\xi,\eta).
\]
It is sufficient to prove the estimate
for each $T_{\sigma^{(i)}}$, $i=1,2$.
Obviously $\sigma^{(i)} \in BS^m_{\rho,\rho}(\mathbb{R}^n)$.
To sum up, by writing $\sigma^{(i)}$ simply as $\sigma$,
we may assume that $\sigma$ satisfies the additional condition
\[
\mathrm{supp}\, ast \sigma
\subset \Gamma(V_i)=\{\zeta \in \mathbb{R}^{2n}
\,:\, \zeta/|\zeta| \in V_i, \ |\zeta| \ge 2\}
\quad
\text{for $i=1$ or $2$}.
\]
For such $\sigma$, we have
$\sigma_{j,k,\nu}=0$ for $j=0$
and thus the decomposition \eqref{symbol-dec-0}
takes the form
\begin{equation}\label{symbol-dec-1}
\sigma =
\sum_{j\ge 1}\,
\sum_{\nu =(\nu _1, \nu_2)\in \mathbb{Z}^n \times \mathbb{Z}^n}\,
\sigma_{j,\nu}
=
\sum_{j\ge 1}\sum_{k\ge 0}\,
\sum_{\nu =(\nu _1, \nu_2)\in \mathbb{Z}^n \times \mathbb{Z}^n}\,
\sigma_{j,k,\nu}.
\end{equation}
In the rest of the proof, we shall consider the two cases
\[
\mathrm{supp}\, ast \sigma \subset \Gamma(V_1),
\quad
\mathrm{supp}\, ast \sigma \subset \Gamma(V_2)
\]
separately.
We shall prove the following
estimate for the trilinear form:
\[
|\langle T_{\sigma}(f,g), h \rangle |
\lesssim
\|f\|_{L^2}
\|g\|_{L^{\infty}}
\|h\|_{L^2},
\]
which is equivalent to the desired estimate
for the operator $T_{\sigma}$.
\noindent {\bf The case $\mathrm{supp}\, ast \sigma \subset \Gamma(V_1)$}.
In this case, all $(\xi, \eta) \in \mathrm{supp}\, ast \sigma$ satisfy
$|\xi+ \eta|\approx (|\xi|^2 + |\eta|^2)^{1/2}$
(but $|\xi|$ may be small compared with $(|\xi|^2 + |\eta|^2)^{1/2}$).
We take a positive integer $a$ such that
\begin{equation}\label{suppsigma-xi+eta}
(\xi, \eta)\in \mathrm{supp}\, ast (\sigma_{j,k,\nu})
\, \mathbb{R}ightarrow \,
2^{j-a}\le |\xi + \eta|\le 2^{j+a}.
\end{equation}
Using this $a$, we write \eqref{symbol-dec-1} as
\begin{equation*}
\sigma= \sum_{j \ge 1}\sum_{k \ge 0}\sum_{\nu}
\sigma_{j,k,\nu}
=
\sum_{\substack{
j \ge 1, \, k \ge 0
\\
k \le j(1-\rho)-a-2}}
\sum_{\nu}
\sigma_{j,k,\nu}
+\sum_{\substack{
j \ge 1, \, k \ge 0
\\
k > j(1-\rho)-a-2}}
\sum_{\nu}
\sigma_{j,k,\nu}.
\end{equation*}
According to this decomposition of
$\sigma$, we write
the trilinear form as
\begin{align*}
&\langle T_{\sigma}(f,g), h \rangle
\\
&=
\sum_{\substack{
j \ge 1, \, k \ge 0
\\
k \le j(1-\rho)-a-2}}
\sum_{\nu}
\langle T_{\sigma_{j,k,\nu}}(f,g), h \rangle
+\sum_{\substack{
j \ge 1, \, k \ge 0
\\
k > j(1-\rho)-a-2}}
\sum_{\nu}
\langle T_{\sigma_{j,k,\nu}}(f,g), h \rangle
\\
&=X_1 + X_2,
\quad
\text{say}.
\end{align*}
The estimate for the second term $X_2$ is easy.
In fact, Lemma \ref{sum-on-annulus} gives
\begin{align*}
&|X_2|
\le
\sum_{\substack{
j \ge 1, \, k \ge 0
\\
k > j(1-\rho)-a-2}}
\bigg\|
\sum_{\nu}
T_{\sigma_{j,k,\nu}}(f,g)
\bigg\|_{L^2}
\|h\|_{L^2}
\\
&
\lesssim
\sum_{\substack{
j \ge 1, \, k \ge 0
\\
k > j(1-\rho)-a-2}}
2^{-kN}
\|f\|_{L^2}
\|g\|_{L^{\infty}}
\|h\|_{L^2}
\approx
\|f\|_{L^2}
\|g\|_{L^{\infty}}
\|h\|_{L^2},
\end{align*}
where we used the assumption $\rho<1$ in the last $\approx$.
In order to estimate $X_1$, we use the decomposition
\[
f=\sum_{\ell } f_{\ell},
\quad
f_{\ell}=\psi_{\ell}(D)f,
\]
and write
\[
X_{1}
=
\sum_{\substack{
j \ge 1, \, k \ge 0
\\
k \le j(1-\rho)-a-2}}
\sum_{\ell \ge 0}
\sum_{\nu}
\langle T_{\sigma_{j,k,\nu}}(f_{\ell},g), h \rangle.
\]
Here we make simple observations.
First, if $k\le j(1-\rho) -a-2$, then
from \eqref{FouriersuppTsjkn} and \eqref{suppsigma-xi+eta}
we see that
\begin{align*}
\mathrm{supp}\, \mathcal{F} [T_{\sigma_{j,k,\nu}}(f,g)]
&\subset
\bigcup_{2^{j-a} \le |\xi+\eta|\le 2^{j+a}}
\{\zeta \in \mathbb{R}^n
\,:\,
|\zeta - \xi -\eta|\le 2^{j\rho + k +1}
\}
\\
&\subset
\{\zeta \in \mathbb{R}^n
\,:\,
2^{j-a-1}\le |\zeta|\le 2^{j+a+1}
\}.
\end{align*}
Hence, by \eqref{replace-h1},
$\langle T_{\sigma_{j,k,\nu}}(f_{\ell},g), h \rangle $
in $X_1$ can be written as
\[
\langle T_{\sigma_{j,k,\nu}}(f_{\ell},g), h \rangle
=\langle T_{\sigma_{j,k,\nu}}(f_{\ell},g), h_{j} \rangle,
\quad
h_j =\theta (2^{-j} D)h,
\]
where
$\theta$ is an appropriate
function supported in an annulus.
Secondly,
since $\mathrm{supp}\, \widehat{f}_{\ell}
\subset \{2^{\ell -1}\le |\xi|\le 2^{\ell +1}\}$
for $\ell >0$ and since
$\mathrm{supp}\, ast (\sigma_{j,k,\nu}) \subset \{|\xi|\le 2^{j+1}\}$,
it follows that
$T_{\sigma_{j,k,\nu}}(f_{\ell},g)=0$ for
$\ell > j+1$.
Thirdly, since
$\mathrm{supp}\, ast (\sigma_{j,k,\nu})
\subset
\mathrm{supp}\, \varphi (2^{-j \rho} \cdot - \nu_{1})
\times \mathbb{R}^n$
and $\mathrm{supp}\, \widehat{f}_{\ell} \subset \mathrm{supp}\, \psi_{\ell}$,
we have
\[
T_{\sigma_{j,k,\nu}}(f_{\ell},g)\neq 0
\; \mathbb{R}ightarrow \;
\mathrm{supp}\, \varphi (2^{- j\rho}\cdot - \nu_1)
\cap \mathrm{supp}\, \psi_{\ell} \neq
\emptyset .
\]
Combining these observations, we see that $X_1$ can be written as
\begin{equation}\label{X1}
X_1
=
\sum_{\substack{
j \ge 1, \, k \ge 0
\\
k \le j(1-\rho)-a-2}}
\,
\sum_{\ell =0}^{j+1}
\,
\sum_{\nu_{1}\in \Lambda_{j,\ell}}
\,
\sum_{\nu_2 \in \mathbb{Z}^n}
\,
\langle T_{\sigma_{j,k,\nu}}(f_{\ell},g), h_j \rangle,
\end{equation}
where
\[
\Lambda_{j,\ell}
=\{\nu_1 \in \mathbb{Z}^n
\,:\,
\mathrm{supp}\, \varphi (2^{-j\rho}\cdot - \nu_1)
\cap \mathrm{supp}\, \psi_{\ell}
\neq \emptyset
\}.
\]
The number of elements of $\Lambda_{j,\ell}$ satisfies
\[
|\Lambda_{j,\ell}|
\lesssim
(\max \{1, 2^{\ell - j\rho}\})^{n}.
\]
Thus Lemma \ref{sum-on-several-lines} (1) gives
\begin{align*}
&\bigg|
\sum_{\nu_{1}\in \Lambda_{j,\ell}}
\sum_{\nu_2 \in \mathbb{Z}^n}
\langle T_{\sigma_{j,k,\nu}}(f_{\ell},g), h_j \rangle
\bigg|
\\
&
\le
\bigg\|
\sum_{\nu_{1}\in \Lambda_{j,\ell}}
\sum_{\nu_2 \in \mathbb{Z}^n}
T_{\sigma_{j,k,\nu}}(f_{\ell},g)
\bigg\|_{L^2}
\|h_j \|_{L^2}
\\
&
\lesssim
\max \{1, 2^{(\ell -j\rho)n/2}\}
2^{jm -kN}
\|f_{\ell}\|_{L^2}
\|g\|_{L^{\infty}}
\|h_j \|_{L^2}.
\end{align*}
Hence
\begin{align}
\label{case1-final}
|X_1 |
&\lesssim
\sum_{\substack{
j \ge 1, \, k \ge 0
\\
k \le j(1-\rho)-a-2}}
\sum_{\ell =0}^{j+1}
\max \{1, 2^{(\ell -j\rho)n/2}\}
2^{jm -kN}
\|f_{\ell}\|_{L^2}
\|g\|_{L^{\infty}}
\|h_j \|_{L^2}
\nonumber
\\
&\le
\sum_{k\ge 0}\,
\sum_{\substack{
j\ge 1, \, \ell \ge 0
\\
\ell \le j+1
}}
\max \{1, 2^{(\ell -j\rho)n/2}\}
2^{jm -kN}
\|f_{\ell}\|_{L^2}
\|g\|_{L^{\infty}}
\|h_j \|_{L^2}.
\end{align}
Under our assumption
$m=-(1-\rho)n/2<0$, it holds that
\begin{equation}\label{Schur1}
\sum_{j \ge 1}
\mathbf{1} \{\ell \le j+1\}
\max\{1,2^{(\ell-j\rho)n/2}\}\, 2^{jm}
\approx 1
\quad \text{for all}\quad
\ell \ge 0
\end{equation}
and
\begin{equation}\label{Schur2}
\sum_{\ell \ge 0}
\mathbf{1} \{\ell \le j+1\}
\max\{1,2^{(\ell-j\rho)n/2}\}\, 2^{jm}
\approx 1
\quad \text{for all}\quad
j\ge 1.
\end{equation}
Hence, by Schur's lemma,
\eqref{case1-final} is bounded by
\[
\sum_{k\ge 0}
2^{-kN}
\bigg(
\sum_{\ell \ge 0} \|f_{\ell}\|_{L^2}^2
\bigg)^{1/2}
\bigg(
\sum_{j\ge 1} \|h_{j}\|_{L^2}^2
\bigg)^{1/2}
\|g\|_{L^{\infty}}
\lesssim
\|f\|_{L^2}
\|h\|_{L^2}
\|g\|_{L^{\infty}}.
\]
This completes the proof for the first case.
\noindent {\bf The case $\mathrm{supp}\, ast \sigma
\subset \Gamma(V_2)$}.
In this case,
all $(\xi, \eta) \in \mathrm{supp}\, ast \sigma$ satisfy
$|\xi| \approx (|\xi|^2 + |\eta|^2)^{1/2}$
(but $|\xi + \eta|$ may be small compared
with $(|\xi|^2 + |\eta|^2)^{1/2}$).
We divide the sum over $(j,k)$ in \eqref{symbol-dec-1} into
two parts $k \le j(1-\rho)$ and $k > j(1-\rho)$ and write
the trilinear form $\langle T_{\sigma}(f,g), h \rangle$ as
\begin{align*}
\langle T_{\sigma}(f,g), h \rangle
&=
\sum_{\substack{
j \ge 1, \, k \ge 0
\\
k \le j(1-\rho)}}
\,
\sum_{\nu}
\,
\langle T_{\sigma_{j,k,\nu}}(f,g), h \rangle
+
\sum_{\substack{
j \ge 1, \, k \ge 0
\\
k > j(1-\rho)}}
\,
\sum_{\nu}
\,
\langle T_{\sigma_{j,k,\nu}}(f,g), h \rangle
\\
&=Y_1 + Y_2,
\quad
\text{say}.
\end{align*}
As in the first case,
the estimate for the second term $Y_2$ is easy.
In fact, Lemma \ref{sum-on-annulus} gives
\begin{align*}
&|Y_2|
\le
\sum_{\substack{
j \ge 1, \, k \ge 0
\\
k > j(1-\rho)}}
\bigg\|
\sum_{\nu}
T_{\sigma_{j,k,\nu}}(f,g)
\bigg\|_{L^2}
\|h\|_{L^2}
\\
&
\lesssim
\sum_{\substack{
j \ge 1, \, k \ge 0
\\
k > j(1-\rho)}}
2^{-kN}
\|f\|_{L^2}
\|g\|_{L^{\infty}}
\|h\|_{L^2}
\approx
\|f\|_{L^2}
\|g\|_{L^{\infty}}
\|h\|_{L^2},
\end{align*}
where the last $\approx$ holds because $1-\rho>0$.
In order to estimate $Y_1$, we use the decomposition
\[
h=\sum_{\ell } h_{\ell},
\quad
h_{\ell}=\psi_{\ell}(D)h,
\]
and write
\[
Y_{1}
=
\sum_{\substack{
j \ge 1, \, k \ge 0
\\
k \le j(1-\rho)}}\,
\sum_{\ell \ge 0}\,
\sum_{\nu = (\nu_1, \nu_2)\in \mathbb{Z}^n \times \mathbb{Z}^n}\,
\langle T_{\sigma_{j,k,\nu}}(f,g), h_{\ell} \rangle .
\]
Here observe the following.
Firstly, since
$|\xi|\approx (|\xi|^2 + |\eta|^2)^{1/2}$
for $(\xi, \eta) \in \mathrm{supp}\, ast \sigma$,
there exists a positive integer $b$ such that
$\mathrm{supp}\, ast (\sigma_{j,k,\nu}) \subset \{(\xi, \eta)
\,:\,
2^{j-b}\le |\xi| \le 2^{j+b}\}$.
Hence, by \eqref{replace-f1-g1},
\[
T_{\sigma_{j,k,\nu}}(f,g)
= T_{\sigma_{j,k,\nu}}(f_j,g),
\quad
f_j =\theta (2^{-j} D)f,
\]
where
$\theta$ is an appropriate
function supported in an annulus.
Secondly, if $k\le j(1-\rho)$, then
\eqref{FouriersuppTsjkn} yields
\begin{align*}
\mathrm{supp}\, \mathcal{F} [T_{\sigma_{j,k,\nu}}(f_j,g)]
&\subset
\bigcup_{(|\xi|^2+|\eta|^2)^{1/2}\le 2^{j+1}}
\{\zeta \in \mathbb{R}^n
\,:\,
|\zeta - \xi -\eta|\le 2^{j\rho + k +1}
\}
\\
&\subset
\{\zeta \in \mathbb{R}^n
\,:\,
|\zeta|\le 2^{j+3}
\},
\end{align*}
which together with the fact
$\mathrm{supp}\, \widehat{h}_{\ell}\subset \mathrm{supp}\, \psi_{\ell}$
implies
$\langle T_{\sigma_{j,k,\nu}}(f_j,g), h_{\ell}\rangle =0$
for
$\ell > j+3$.
Thirdly, as we have already seen,
\eqref{Fouriersupp-n1n2} holds, and hence,
by \eqref{replace-h1},
\begin{equation*}
\langle T_{\sigma_{j,k,\nu}}(f_j,g), h_{\ell} \rangle \ne 0
\;\mathbb{R}ightarrow \;
(2^{j\rho} (\nu_1 +\nu_2) + 2^{j\rho + k+2} Q )
\cap \mathrm{supp}\, \psi_{\ell} \neq
\emptyset.
\]
Combining these observations, we see that $Y_1$ can be written as
\begin{equation}\label{Y1}
Y_1
=
\sum_{\substack{
j \ge 1, \, k \ge 0
\\
k \le j(1-\rho)}}
\,
\sum_{\ell =0}^{j+3}\,
\sum_{\mu \in \Lambda_{j,k,\ell}}\,
\sum_{\nu_{1}+ \nu_2 = \mu}\,
\langle T_{\sigma_{j,k,\nu}}(f_j,g), h_{\ell} \rangle,
\end{equation}
where
\[
\Lambda_{j,k,\ell}
=\{\mu \in \mathbb{Z}^n
\,:\,
(2^{j\rho} \mu + 2^{j\rho + k+2} Q )
\cap \mathrm{supp}\, \psi_{\ell} \neq
\emptyset
\}.
\]
The number of elements of $\Lambda_{j,k,\ell}$ is
estimated by
\begin{equation*}
|\Lambda_{j,k,\ell}|
\lesssim
(\max \{2^{k}, 2^{\ell -j \rho}\})^{n}
\lesssim
2^{kn}
\max \{1, 2^{(\ell -j \rho)n}\}.
\end{equation*}
Thus Lemma \ref{sum-on-several-lines} (2) gives
\begin{align*}
&\bigg|
\sum_{\mu \in \Lambda_{j,k,\ell}}
\sum_{\nu_1+ \nu_2 =\mu }
\langle T_{\sigma_{j,k,\nu}}(f_{\ell},g), h_j \rangle
\bigg|
\\
&
\le
\bigg\|
\sum_{\mu \in \Lambda_{j,k,\ell}}
\sum_{\nu_1+ \nu_2 =\mu }
T_{\sigma_{j,k,\nu}}(f_{\ell},g)
\bigg\|_{L^2}
\|h_j \|_{L^2}
\\
&
\lesssim
\max \{1, 2^{(\ell -j\rho)n/2}\}
2^{jm -k(N-n/2)}
\|f_{\ell}\|_{L^2}
\|g\|_{L^{\infty}}
\|h_j \|_{L^2}.
\end{align*}
Hence
\begin{align}
\label{case2-final}
|Y_1 |
&\lesssim
\sum_{\substack{
j \ge 1, \, k \ge 0
\\
k \le j(1-\rho)}}
\sum_{\ell =0}^{j+3}
\max \{1, 2^{(\ell -j\rho)n/2}\}
2^{jm -k(N-n/2)}
\|f_{\ell}\|_{L^2}
\|g\|_{L^{\infty}}
\|h_j \|_{L^2}
\nonumber
\\
&\le
\sum_{k\ge 0}\,
\sum_{\substack{
j\ge 1, \, \ell \ge 0
\\
\ell \le j+3
}}
\max \{1, 2^{(\ell -j\rho)n/2}\}
2^{jm -k(N-n/2)}
\|f_{\ell}\|_{L^2}
\|g\|_{L^{\infty}}
\|h_j \|_{L^2}.
\end{align}
Since \eqref{Schur1} and \eqref{Schur2} hold
if $\ell \le j+1$ is replaced by $\ell \le j+3$,
by Schur's lemma,
\eqref{case2-final} is bounded by
\[
\sum_{k\ge 0}
2^{-k(N-n/2)}
\bigg(
\sum_{\ell \ge 0} \|f_{\ell}\|_{L^2}^2
\bigg)^{1/2}
\bigg(
\sum_{j\ge 1} \|h_{j}\|_{L^2}^2
\bigg)^{1/2}
\|g\|_{L^{\infty}}
\lesssim
\|f\|_{L^2}
\|h\|_{L^2}
\|g\|_{L^{\infty}},
\]
which gives the desired estimate for $Y_1$.
This completes the proof of Theorem \ref{main-thm-1}.
\section{Proof of Theorem \ref{main-thm-2}}\label{section4}
In this section,
we shall prove Theorem \ref{main-thm-2}.
The main scheme of the arguments is the same as
that of Naibo \cite{Naibo}.
In the last step, we introduce a new idea
of using weak type estimates.
Since the theorem is already proved in the case $\rho=0$
(see \cite{Miyachi-Tomita}),
for the rest of this section, we assume
$0<\rho<1$, $m=-(1-\rho)n$, and
$\sigma \in BS^{m}_{\rho, \rho}(\mathbb{R}^n)$.
Using the function $\Psi_j$ of Subsection \ref{subsection-decomposition},
we decompose $\sigma$ as
\begin{align}
&
\sigma (x,\xi,\eta)
=
\sum_{j=0}^{\infty}
\sigma_{j} (x,\xi,\eta),
\label{sigma-sum-sigmaj}
\\
&
\sigma_{j} (x,\xi,\eta)
=\sigma (x,\xi,\eta)
\Psi_{j}(\xi, \eta).
\label{def-sigmaj}
\end{align}
We write the inverse Fourier transform of $\sigma_{j}$
with respect to $(\xi, \eta)$ as
\begin{equation*}
K_{j} (x,y,z)=\frac{1}{(2\pi )^{2n}}
\int_{\mathbb{R}^n \times \mathbb{R}^n}
e^{i(y \cdot \xi + z \cdot \eta)}
\sigma_{j} (x,\xi,\eta)\, d\xi d\eta.
\end{equation*}
First, we shall prove that $K_j$ satisfy the following estimates:
\begin{align}
&
\|(1+2^{j\rho}|y|)^{N_1}
(1+2^{j\rho}|z|)^{N_2}
K_{j}(x,y,z)
\|_{L^2_{y,z}}
\lesssim
2^{j (m+n)},
\label{weightKj-0}
\\
&
\|(1+2^{j\rho}|y|)^{N_1}
(1+2^{j\rho}|z|)^{N_2}
\nabla_{x}
K_{j}(x,y,z)
\|_{L^2_{y,z}}
\lesssim
2^{j (\rho + m+n)},
\label{weightKj-nabla-x}
\\
&
\|(1+2^{j\rho}|y|)^{N_1}
(1+2^{j\rho}|z|)^{N_2}
\nabla_{y}
K_{j}(x,y,z)
\|_{L^2_{y,z}}
\lesssim
2^{j (1+m+n)},
\label{weightKj-nabla-y}
\\
&
\|(1+2^{j\rho}|y|)^{N_1}
(1+2^{j\rho}|z|)^{N_2}
\nabla_{z}
K_{j}(x,y,z)
\|_{L^2_{y,z}}
\lesssim
2^{j (1+m+n)},
\label{weightKj-nabla-z}
\end{align}
where $\nabla_{x}, \nabla_{y}, \nabla_{z}$ denote the gradient
operator with respect to $x,y,z$ respectively,
and $N_1$ and $N_2$ can be arbitrary nonnegative real numbers.
To prove \eqref{weightKj-0}, observe that
$1+|\xi|+|\eta| \approx 2^{j}$ for
all $(\xi,\eta)\in \mathrm{supp}\, ast (\sigma_j)$
and $\sigma_j$ satisfies the estimate
\[
|\partial_{\xi}^{\beta}
\partial_{\eta}^{\gamma}
\sigma_{j} (x,\xi,\eta)|
\lesssim
(2^j)^{m-\rho |\beta|- \rho |\gamma|}
\mathbf{1} \{1+|\xi|+|\eta|\approx 2^j\}.
\]
Taking inverse Fourier transform with respect to $(\xi, \eta)$
and using Plancherel's theorem,
we obtain
\[
\|
(2^{j\rho}y)^{\beta}
(2^{j\rho}z)^{\gamma}
K_{j} (x,y,z)\|_{L^{2}_{y,z}}
\lesssim
(2^j)^{m+ n},
\]
from which \eqref{weightKj-0} follows.
The estimates \eqref{weightKj-nabla-x},
\eqref{weightKj-nabla-y}, and \eqref{weightKj-nabla-z}
can be derived from the estimates
\begin{align*}
&
|\partial_{\xi}^{\beta}
\partial_{\eta}^{\gamma}
\nabla_{x}\sigma_{j} (x,\xi,\eta)|
\lesssim
(2^j)^{m+\rho-\rho |\beta|- \rho |\gamma|}
\mathbf{1} \{1+|\xi|+|\eta|\approx 2^j\},
\\
&
|\partial_{\xi}^{\beta}
\partial_{\eta}^{\gamma}
\{\xi \sigma_{j} (x,\xi,\eta)\}|
\lesssim
(2^j)^{m+1 -\rho |\beta|- \rho |\gamma|}
\mathbf{1} \{1+|\xi|+|\eta|\approx 2^j\},
\\
&
|\partial_{\xi}^{\beta}
\partial_{\eta}^{\gamma}
\{\eta \sigma_{j} (x,\xi,\eta)\}|
\lesssim
(2^j)^{m+1 -\rho |\beta|- \rho |\gamma|}
\mathbf{1} \{1+|\xi|+|\eta|\approx 2^j\}
\end{align*}
in the same way.
Now we proceed to the proof of the $L^{\infty} \times L^{\infty} \to BMO$
boundedness of $T_{\sigma}$.
Let $f,g$ be functions satisfying
$\|f\|_{L^{\infty}}=\|g\|_{L^{\infty}}=1$
and let $Q$ be a cube in $\mathbb{R}^n$.
We denote by $\ell(Q)$ the side length of $Q$,
and by $x_Q$ the center of $Q$.
It is sufficient to prove that there exists a complex number
$C_{Q}$ such that
\begin{equation*}
\frac{1}{|Q|}
\int_{Q}
|T_{\sigma}(f,g)(x) - C_{Q}|\, dx \lesssim 1.
\end{equation*}
We write $h=\ell (Q)$ and take
the cube $\widetilde{Q}$ with the same center
as $Q$ and with the sidelength
\[
\ell (\widetilde{Q})
=
\left\{
\begin{array}{ll}
{2 h^{\rho}} & {\qquad\text{if $h\le 1$,}\quad }\\
{2h } & {\qquad\text{if $h> 1$.}\quad }
\end{array}
\right.
\]
We divide $f$ and $g$ as
\begin{align*}
&
f= f \mathbf{1}_{\widetilde{Q}} + f \mathbf{1}_{\widetilde{Q}^{c}} = f^{(0)}+ f^{(1)},
\\
&
g=g \mathbf{1}_{\widetilde{Q}} + g \mathbf{1}_{\widetilde{Q}^{c}} = g^{(0)}+ g^{(1)},
\end{align*}
and divide $T_{\sigma}(f,g)$ into four parts
\begin{align*}
T_{\sigma}(f,g)&=
T_{\sigma}(f^{(0)}, g^{(0)})
+
T_{\sigma}(f^{(0)}, g^{(1)})
+T_{\sigma}(f^{(1)}, g^{(0)})
+T_{\sigma}(f^{(1)}, g^{(1)})
\\
&=
F^{(1)}+F^{(2)}+F^{(3)}+F^{(4)},
\quad \text{say}.
\end{align*}
For each $i=1,2,3,4$, we shall show that there exists
a complex number $C^{(i)}_{Q}$ such that
\begin{equation}\label{BMOFi}
\frac{1}{|Q|}
\int_{Q}
|F^{(i)}(x) - C^{(i)}_{Q}|\, dx \lesssim 1.
\end{equation}
We divide the argument into two cases,
$h> 1$ and $h\le 1$.
\noindent {\bf The case $h=\ell (Q)>1$.}
In this case, we shall prove
\eqref{BMOFi} with $C^{(i)}_{Q}=0$ for all $i$.
{\it Estimate for $F^{(4)}$.\/}
We have
\begin{equation*}
F^{(4)}(x)
=T_{\sigma}(f^{(1)}, g^{(1)})(x)
=\sum_{j=0}^{\infty}T_{\sigma_j}(f^{(1)}, g^{(1)})(x).
\end{equation*}
Using the kernel $K_j$ and using
Schwarz's inequality,
we have
\begin{align*}
&
|T_{\sigma_{j}}(f^{(1)}, g^{(1)})(x)|
=\bigg|
\int_{\substack{
y\in \widetilde{Q}^{c}
\\
z\in \widetilde{Q}^{c}
}}
K_{j}(x,x-y, x-z)
f(y)g(z)\, dydz
\bigg|
\\
&\le
\bigg\|
h^{n}
\bigg(\frac{|x-y|}{h}\bigg)^{N_{1}}
\bigg(\frac{|x-z|}{h}\bigg)^{N_{2}}
K_{j}(x,x-y, x-z)
\bigg\|_{L^2( y\in \widetilde{Q}^{c},\, z\in \widetilde{Q}^c)}
\\
&\quad
\times \bigg\|
h^{-n}
\bigg(\frac{|x-y|}{h}\bigg)^{-N_1}
\bigg(\frac{|x-z|}{h}\bigg)^{-N_2}
f(y) g(z)
\bigg\|_{L^2( y\in \widetilde{Q}^{c},\, z\in \widetilde{Q}^c)},
\end{align*}
where $N_1, N_2 \ge 0$ can be taken arbitrarily.
The first $L^2$-norm above is estimated by \eqref{weightKj-0} as
\begin{align*}
&\bigg\|
h^{n}
\bigg(\frac{|x-y|}{h}\bigg)^{N_1}
\bigg(\frac{|x-z|}{h}\bigg)^{N_2}
K_{j}(x,x-y, x-z)
\bigg\|_{L^2( y\in \widetilde{Q}^{c},\, z\in \widetilde{Q}^{c})}
\\
&\le
h^{n}
(2^{j\rho} h)^{-N_1}
(2^{j\rho} h)^{-N_2}
\|(2^{j\rho}|y|)^{N_1}(2^{j\rho}|z|)^{N_2}
K_{j}(x,y,z)
\|_{L^2_{y,z}}
\\
&\lesssim
h^{n}
(2^{j\rho} h)^{-N_1}
(2^{j\rho} h)^{-N_2}
2^{j (m+n)}
=(2^{j\rho}h )^{-N_1 -N_2+n},
\end{align*}
where the last equality holds because of our assumption $m=-(1-\rho)n$.
If we take $N_1, N_2>n/2$, then, for $x\in Q$,
the second $L^2$-norm is
estimated as
\begin{align*}
&
\bigg\|
h^{-n}
\bigg(\frac{|x-z|}{h}\bigg)^{-N_1}
\bigg(\frac{|x-z|}{h}\bigg)^{-N_2}
f(y) g(z)
\bigg\|_{L^2( y\in \widetilde{Q}^{c},\, z\in \widetilde{Q}^{c})}
\\
&\le
\bigg\|
h^{-n}
\bigg(\frac{|x-y|}{h}\bigg)^{-N_1}
\bigg(\frac{|x-z|}{h}\bigg)^{-N_2}
\bigg\|_{L^2( y\in \widetilde{Q}^{c},\, z\in \widetilde{Q}^{c})}
\approx 1.
\end{align*}
Thus, by taking $N_1, N_2 >n/2$, we obtain
the pointwise estimate
\begin{align*}
&
|F^{(4)}(x)|
\le
\sum_{j=0}^{\infty}
|T_{\sigma_j}(f^{(1)}, g^{(1)})(x)|
\lesssim
\sum_{j=0}^{\infty}
(2^{j\rho}h )^{-N_1-N_2+n}
\approx h^{-N_1-N_2+n}\le 1
\end{align*}
for all $x\in Q$.
This certainly implies \eqref{BMOFi} for $i=4$ with $C^{(4)}_{Q}=0$.
{\it Estimate for $F^{(2)}$ and $F^{(3)}$.\/}
By symmetry, we consider only $F^{(2)}$.
We write
\begin{equation*}
F^{(2)}(x)
=T_{\sigma}(f^{(0)}, g^{(1)})(x)
=\sum_{j=0}^{\infty}T_{\sigma_j}(f^{(0)}, g^{(1)})(x).
\end{equation*}
By Schwarz's inequality,
we have
\begin{align*}
&|T_{\sigma_{j}}(f^{(0)}, g^{(1)})(x)|=
\bigg|
\int_{\substack{
y\in \widetilde{Q}
\\
z\in \widetilde{Q}^{c}
}}
K_{j}(x,x-y, x-z)
f(y)g(z)\, dydz
\bigg|
\\
&\le
\bigg\|
h^{n} \bigg(\frac{|x-z|}{h}\bigg)^{N_{2}}
K_{j}(x,x-y, x-z)
\bigg\|_{L^2( y\in \widetilde{Q},\, z\in \widetilde{Q}^c)}
\\
&\quad
\times \bigg\|
h^{-n}
\bigg(\frac{|x-z|}{h}\bigg)^{-N_2}
f(y) g(z)
\bigg\|_{L^2( y\in \widetilde{Q},\, z\in \widetilde{Q}^c)},
\end{align*}
where $N_2 \ge 0$ can be taken arbitrarily.
The first $L^2$-norm above is estimated by \eqref{weightKj-0} as
\begin{align*}
&\bigg\|
h^{n}
\bigg(\frac{|x-z|}{h}\bigg)^{N_2}
K_{j}(x,x-y, x-z)
\bigg\|_{L^2( y\in \widetilde{Q},\, z\in \widetilde{Q}^{c})}
\\
&\le
h^{n}
(2^{j\rho} h)^{-N_2}
\|(2^{j\rho}|z|)^{N_2}
K_{j}(x,y,z)
\|_{L^2_{y,z}}
\\
&\lesssim
h^{n}
(2^{j\rho} h)^{-N_2}
2^{j (m+n)}
=(2^{j\rho}h )^{-N_2+n}.
\end{align*}
If we take $N_2>n/2$, then, for $x\in Q$,
the second $L^2$-norm is
estimated as
\begin{align*}
&
\bigg\|
h^{-n}
\bigg(\frac{|x-z|}{h}\bigg)^{-N_2}
f(y) g(z)
\bigg\|_{L^2( y\in \widetilde{Q},\, z\in \widetilde{Q}^{c})}
\\
&\le
\bigg\|
h^{-n}
\bigg(\frac{|x-z|}{h}\bigg)^{-N_2}
\bigg\|_{L^2( y\in \widetilde{Q},\, z\in \widetilde{Q}^{c})}
\approx 1.
\end{align*}
Thus, by taking $N_2 >n$, we obtain
\begin{align*}
&
|F^{(2)}(x)|
\le
\sum_{j=0}^{\infty}
|T_{\sigma_j}(f^{(0)}, g^{(1)})(x)|
\lesssim
\sum_{j=0}^{\infty}
(2^{j\rho}h )^{-N_2+n}
\approx h^{-N_2+n}\le 1
\end{align*}
for all $x\in Q$.
This implies \eqref{BMOFi} for $i=2$ with $C^{(2)}_{Q}=0$.
{\it Estimate for $F^{(1)}$.\/}
Since $m=-(1-\rho)n<-(1-\rho)n/2=m_{\rho}(2,2)$,
the operator $T_{\sigma}$ is bounded in $L^{2}\times L^{2} \to L^{1}$
(see Proposition \ref{critical-order}).
Hence
\[
\frac{1}{|Q|}
\int_{Q}
|F^{(1)}(x)|\, dx
\le |Q|^{-1}
\|T_{\sigma}(f^{(0)}, g^{(0)})\|_{L^1}
\lesssim
|Q|^{-1}\|f^{(0)}\|_{L^2}
\|g^{(0)}\|_{L^2}
\lesssim
1.
\]
\noindent {\bf The case $h=\ell (Q)\le 1$.}
{\it Estimate for $F^{(4)}$.\/}
We shall prove the estimate \eqref{BMOFi} for $i=4$ with
$C^{(4)}_{Q}=F^{(4)}(x_Q)$.
In the following, $x$ always denotes arbitrary point in $Q$.
To estimate
$F^{(4)}(x)-F^{(4)}(x_Q)$,
we write
\begin{align*}
&F^{(4)}(x)-F^{(4)}(x_Q)
=
\sum_{j=0}^{\infty}
(T_{\sigma_j}(f^{(1)}, g^{(1)})(x)
-
T_{\sigma_j}(f^{(1)}, g^{(1)})(x_{Q}))
\\
&
=\sum_{j=0}^{\infty}
\int_{
\substack{
y\in \widetilde{Q}^{c}
\\
z\in \widetilde{Q}^{c}
}}
H_{j,Q}(x,y, z)
f(y)g(z)\, dydz,
\end{align*}
where
\begin{equation}\label{def-HjQ}
H_{j,Q}(x,y,z)
=K_{j}(x,x-y, x-z)-
K_{j}(x_{Q},x_{Q}-y, x_{Q}-z).
\end{equation}
By Schwarz's inequality,
\begin{equation}\label{F4jx-F4jxQ-Schwarz}
\begin{split}
&
|T_{\sigma_j}(f^{(1)}, g^{(1)})(x) - T_{\sigma_j}(f^{(1)}, g^{(1)})(x_{Q})|
\\
&
\le
\bigg\|
h^{\rho n}
\bigg(
\frac{|x-y|}{h^\rho }
\bigg)^{N_1}
\bigg(
\frac{|x-z|}{h^\rho }
\bigg)^{N_2}
H_{j,Q}(x,y,z)
\bigg\|_{L^2 (y\in \widetilde{Q}^{c},\,z\in \widetilde{Q}^{c})}
\\
&\quad
\times
\bigg\|
h^{-\rho n}
\bigg(
\frac{|x-y|}{h^\rho }
\bigg)^{-N_1}
\bigg(
\frac{|x-z|}{h^\rho }
\bigg)^{-N_2}
f(y) g(z)
\bigg\|_{L^2 (y\in \widetilde{Q}^{c},\,z\in \widetilde{Q}^{c})}.
\end{split}
\end{equation}
Since $\|f\|_{\infty}=\|g\|_{\infty}=1$,
if we take $N_1, N_2 >n/2$,
the latter $L^2$-norm of \eqref{F4jx-F4jxQ-Schwarz}
is $\lesssim 1$.
In order to estimate the former
$L^2$-norm of \eqref{F4jx-F4jxQ-Schwarz},
we write
\begin{equation}\label{def-HjQ-int}
H_{j,Q}(x,y,z)
=\int_{0}^{1}
\nabla K_{j}(x(t),x(t)-y, x(t)-z)
\cdot (x-x_{Q}, x-x_{Q}, x-x_{Q})
\, dt,
\end{equation}
where we used the notation $x(t)=x_{Q}+t (x-x_Q)$ and
\begin{align*}
&\nabla K_{j}(x,y, z)
\cdot (u, v, w)
\\
&=
\nabla_{x} K_{j}(x,y, z)
\cdot u
+ \nabla_{y} K_{j}(x,y, z)
\cdot v
+
\nabla_{z} K_{j}(x,y, z)
\cdot w.
\end{align*}
Notice that $|x-y|\approx |x(t)-y|$ and
$|x-z|\approx |x(t)-z|$ for all
$y,z \in \widetilde{Q}^{c}$ and
$0<t<1$.
Hence, by \eqref{def-HjQ-int} and by
\eqref{weightKj-nabla-x},
\eqref{weightKj-nabla-y}, \eqref{weightKj-nabla-z},
we can estimate the former
$L^2$-norm of \eqref{F4jx-F4jxQ-Schwarz}
as follows:
(here $\|\cdots \|_{L^2(\ast)}$ means
$\|\cdots \|_{L^2 (y\in \widetilde{Q}^{c},\,z\in \widetilde{Q}^{c})}$)
\begin{align*}
&
\bigg\|
h^{\rho n}
\bigg(
\frac{|x-y|}{h^\rho }
\bigg)^{N_1}
\bigg(
\frac{|x-z|}{h^\rho }
\bigg)^{N_2}
H_{j,Q}(x,y,z)
\bigg\|_{L^2 (\ast)}
\\
&\lesssim
\bigg\|
h^{1+\rho n}
\bigg(
\frac{|x-y|}{h^\rho }
\bigg)^{N_1}
\bigg(
\frac{|x-z|}{h^\rho }
\bigg)^{N_2}
\int_{0}^{1}
|\nabla K_{j}(x(t),x(t)-y, x(t)-z)|\, dt
\bigg\|_{L^2 (\ast)}
\\
&\approx
\bigg\|
h^{1+\rho n}
\int_{0}^{1}
\bigg(
\frac{|x(t)-y|}{h^\rho }
\bigg)^{N_1}
\bigg(
\frac{|x(t)-z|}{h^\rho }
\bigg)^{N_2}
|\nabla K_{j}(x(t),x(t)-y, x(t)-z)|
\, dt
\bigg\|_{L^2 (\ast)}
\\
&
\le
h^{1+\rho n} \int_{0}^{1}
\bigg\|
\bigg(
\frac{|x(t)-y|}{h^\rho }
\bigg)^{N_1}
\bigg(
\frac{|x(t)-z|}{h^\rho }
\bigg)^{N_2}
\nabla K_{j}(x(t), x(t)-y, x(t)-z)
\bigg\|_{L^2 (\ast)}
\, dt
\\
&
\lesssim
h^{1+\rho n}
(2^{j \rho}h^{\rho})^{-N_1}
(2^{j \rho}h^{\rho})^{-N_2}
2^{j(1+m+n)}
=(2^{j \rho}h^{\rho})^{-N_1 -N_2 +n +1/\rho},
\end{align*}
where we used the assumption $m=-(1-\rho)n$ to obtain the last equality.
On the other hand, if we use \eqref{weightKj-0}, then we can estimate
the former $L^2$-norm of \eqref{F4jx-F4jxQ-Schwarz} as
follows: (the notation $\|\cdots \|_{L^2(\ast)}$ is the same as above)
\begin{align*}
&\bigg\|
h^{\rho n}
\bigg(
\frac{|x-y|}{h^\rho }
\bigg)^{N_1}
\bigg(
\frac{|x-z|}{h^\rho }
\bigg)^{N_2}
H_{j,Q}(x,y,z)
\bigg\|_{L^2 (\ast)}
\\
&
\le
\bigg\|
h^{\rho n}
\bigg(
\frac{|x-y|}{h^\rho }
\bigg)^{N_1}
\bigg(
\frac{|x-z|}{h^\rho }
\bigg)^{N_2}
K_{j}(x,x-y, x-z)
\bigg\|_{L^2 (\ast)}
\\
&\quad
+ \bigg\|
h^{\rho n}
\bigg(
\frac{|x-y|}{h^\rho }
\bigg)^{N_1}
\bigg(
\frac{|x-z|}{h^\rho }
\bigg)^{N_2}
K_{j}(x_{Q},x_{Q}-y, x_{Q}-z)
\bigg\|_{L^2 (\ast)}
\\
&\approx
\bigg\|
h^{\rho n}
\bigg(
\frac{|x-y|}{h^\rho }
\bigg)^{N_1}
\bigg(
\frac{|x-z|}{h^\rho }
\bigg)^{N_2}
K_{j}(x,x-y, x-z)
\bigg\|_{L^2 (\ast)}
\\
&\quad
+
\bigg\|
h^{\rho n}
\bigg(
\frac{|x_{Q}-y|}{h^\rho }
\bigg)^{N_1}
\bigg(
\frac{|x_{Q}-z|}{h^\rho }
\bigg)^{N_2}
K_{j}(x_{Q},x_{Q}-y, x_{Q}-z)
\bigg\|_{L^2 (\ast)}
\\
&
\lesssim
h^{\rho n}
(2^{j \rho}h^{\rho})^{-N_1}
(2^{j \rho}h^{\rho})^{-N_2}
2^{j(m+n)}
=(2^{j \rho}h^{\rho})^{-N_1 -N_2 +n}.
\end{align*}
Combining the above estimates,
we have the following estimates for arbitrary
$N_1, N_2 >n/2$:
\[
|T_{\sigma_j}(f^{(1)}, g^{(1)})(x) - T_{\sigma_j}(f^{(1)}, g^{(1)})(x_{Q})|
\lesssim
\min \{(2^{j \rho}h^{\rho})^{-N_1 -N_2 +n +1/\rho},\,
(2^{j \rho}h^{\rho})^{-N_1 -N_2 +n}\}.
\]
Now we take $N_{1}=N_{2}=N$ such that
$-2N + n +1/\rho >0 > -2N +n$.
Then taking the sum of
the above estimates over $j\ge 0$, we obtain
\[
|F^{(4)}(x)- F^{(4)}(x_Q)|
\le
\sum_{j=0}^{\infty}
|T_{\sigma_j}(f^{(1)}, g^{(1)})(x) - T_{\sigma_j}(f^{(1)}, g^{(1)})(x_{Q})|
\lesssim 1,
\quad x\in Q,
\]
which a fortiori implies
\eqref{BMOFi} for $i=4$ with $C^{(4)}_{Q}=F^{(4)}(x_Q)$.
{\it Estimate for $F^{(2)}$ and $F^{(3)}$.\/}
By symmetry, we consider only $F^{(2)}$.
We shall prove the estimate \eqref{BMOFi} for $i=2$ with
$C^{(2)}_{Q}=F^{(2)}(x_Q)$.
In the following, $x$ always denotes arbitrary point in $Q$.
We write
\begin{align*}
&F^{(2)}(x)-F^{(2)}(x_Q)
=
\sum_{j=0}^{\infty}
(T_{\sigma_j}(f^{(0)}, g^{(1)})(x)
-
T_{\sigma_j}(f^{(0)}, g^{(1)})(x_{Q}))
\\
&
=
\sum_{j=0}^{\infty}
\int_{
\substack{
y\in \widetilde{Q}
\\
z\in \widetilde{Q}^{c}
}}
H_{j,Q}(x,y, z)
f(y)g(z)\, dydz
\end{align*}
with $H_{j,Q}$ given by \eqref{def-HjQ}.
By Schwarz's inequality,
we have
\begin{equation}\label{F2jx-F2jxQ-Schwarz}
\begin{split}
&
|T_{\sigma_j}(f^{(0)}, g^{(1)})(x) - T_{\sigma_j}(f^{(0)}, g^{(1)})(x_{Q})|
\\
&
\le
\bigg\|
h^{\rho n}
\bigg(
\frac{|x-z|}{h^\rho }
\bigg)^{N_2}
H_{j,Q}(x,y,z)
\bigg\|_{L^2 (y\in \widetilde{Q},\,z\in \widetilde{Q}^{c})}
\\
&\quad
\times
\bigg\|
h^{-\rho n}
\bigg(
\frac{|x-z|}{h^\rho }
\bigg)^{-N_2}
f(y) g(z)
\bigg\|_{L^2 (y\in \widetilde{Q},\,z\in \widetilde{Q}^{c})}.
\end{split}
\end{equation}
Since $\|f\|_{\infty}=\|g\|_{\infty}=1$,
if we take $N_2 >n/2$, then
the latter $L^2$-norm of \eqref{F2jx-F2jxQ-Schwarz}
is $\lesssim 1$.
By \eqref{def-HjQ-int} and by
\eqref{weightKj-nabla-x}, \eqref{weightKj-nabla-y}, and
\eqref{weightKj-nabla-z},
we can estimate the former
$L^2$-norm of \eqref{F2jx-F2jxQ-Schwarz}
as
\begin{align*}
&
\bigg\|
h^{\rho n}
\bigg(
\frac{|x-z|}{h^\rho }
\bigg)^{N_2}
H_{j,Q}(x,y,z)
\bigg\|_{L^2 (y\in \widetilde{Q},\,z\in \widetilde{Q}^{c})}
\\
&\lesssim
\bigg\|
h^{1+\rho n}
\bigg(
\frac{|x-z|}{h^\rho }
\bigg)^{N_2}
\int_{0}^{1}
|\nabla K_{j}(x(t),x(t)-y, x(t)-z)|\, dt
\bigg\|_{L^2 (y\in \widetilde{Q},\,z\in \widetilde{Q}^{c})}
\\
&\approx
\bigg\|
h^{1+\rho n}
\int_{0}^{1}
\bigg(
\frac{|x(t)-z|}{h^\rho }
\bigg)^{N_2}
|\nabla K_{j}(x(t),x(t)-y, x(t)-z)|
\, dt
\bigg\|_{L^2 (y\in \widetilde{Q},\,z\in \widetilde{Q}^{c})}
\\
&
\le
h^{1+\rho n} \int_{0}^{1}
\bigg\|
\bigg(
\frac{|x(t)-z|}{h^\rho }
\bigg)^{N_2}
\nabla K_{j}(x(t), x(t)-y, x(t)-z)
\bigg\|_{L^2 (y\in \widetilde{Q},\,z\in \widetilde{Q}^{c})}
\, dt
\\
&
\lesssim
h^{1+\rho n}
(2^{j \rho}h^{\rho})^{-N_2}
2^{j(1+m+n)}
=(2^{j \rho}h^{\rho})^{ -N_2 +n +1/\rho}.
\end{align*}
On the other hand, using \eqref{weightKj-0}, we can estimate
the former $L^2$-norm of \eqref{F2jx-F2jxQ-Schwarz} as
\begin{align*}
&\bigg\|
h^{\rho n}
\bigg(
\frac{|x-z|}{h^\rho }
\bigg)^{N_2}
H_{j,Q}(x,y,z)
\bigg\|_{L^2 (y\in \widetilde{Q},\,z\in \widetilde{Q}^{c})}
\\
&
\le
\bigg\|
h^{\rho n}
\bigg(
\frac{|x-z|}{h^\rho }
\bigg)^{N_2}
K_{j}(x,x-y, x-z)
\bigg\|_{L^2 (y\in \widetilde{Q},\,z\in \widetilde{Q}^{c})}
\\
&\quad
+ \bigg\|
h^{\rho n}
\bigg(
\frac{|x-z|}{h^\rho }
\bigg)^{N_2}
K_{j}(x_{Q},x_{Q}-y, x_{Q}-z)
\bigg\|_{L^2 (y\in \widetilde{Q},\,z\in \widetilde{Q}^{c})}
\\
&\approx
\bigg\|
h^{\rho n}
\bigg(
\frac{|x-z|}{h^\rho }
\bigg)^{N_2}
K_{j}(x,x-y, x-z)
\bigg\|_{L^2 (y\in \widetilde{Q},\,z\in \widetilde{Q}^{c})}
\\
&\quad
+
\bigg\|
h^{\rho n}
\bigg(
\frac{|x_{Q}-z|}{h^\rho }
\bigg)^{N_2}
K_{j}(x_{Q},x_{Q}-y, x_{Q}-z)
\bigg\|_{L^2 (y\in \widetilde{Q},\,z\in \widetilde{Q}^{c})}
\\
&
\lesssim
h^{\rho n}
(2^{j \rho}h^{\rho})^{-N_2}
2^{j(m+n)}
=(2^{j \rho}h^{\rho})^{-N_2 +n}.
\end{align*}
Combining the above estimates,
we have the estimates
\[
|T_{\sigma_j}(f^{(0)}, g^{(1)})(x) - T_{\sigma_j}(f^{(0)}, g^{(1)})(x_{Q})|
\lesssim
\min \{(2^{j \rho}h^{\rho})^{-N_2 +n +1/\rho},\,
(2^{j \rho}h^{\rho})^{-N_2 +n}\}
\]
for arbitrary $N_2 >n/2$.
Now we take $N_{2}$ such that
$-N_{2} + n +1/\rho >0 > -N_{2} +n$ and take the sum of
the above estimates over $j\ge 0$ to obtain
\[
|F^{(2)}(x)- F^{(2)}(x_Q)|
\le
\sum_{j=0}^{\infty}
|T_{\sigma_j}(f^{(0)}, g^{(1)})(x) - T_{\sigma_j}(f^{(0)}, g^{(1)})(x_{Q})|
\lesssim 1,
\quad x\in Q,
\]
which a fortiori implies
\eqref{BMOFi} for $i=2$ with $C^{(2)}_{Q}=F^{(2)}(x_Q)$.
{\it Estimate for $F^{(1)}$.\/}
We first prove an $L^2$ estimate of $T_{\sigma_j}(f^{(0)}, g^{(0)})$.
Let $\widetilde{\sigma}_{j}$ be the symbol
\begin{equation}\label{change-symbol}
\widetilde{\sigma}_{j}(x,\xi,\eta)
=
\sigma_{j}(2^{-j\rho}x,2^{j\rho}\xi,2^{j\rho}\eta).
\end{equation}
Then a simple change of variables gives
\begin{equation}\label{change-pdo}
T_{\sigma}(a,b)(2^{-j\rho} x)
=
T_{\widetilde{\sigma}}(a(2^{-j\rho}\cdot ),b(2^{-j\rho}\cdot ))
(x),
\end{equation}
which implies
\[
\|T_{\sigma_{j}}\|_{L^2 \times L^{\infty} \to L^2}
=
\|T_{\widetilde{\sigma}_{j}}\|_{L^2 \times L^{\infty} \to L^2}.
\]
Since $1+|\xi|+|\eta|\approx 2^{j}$ for all
$(\xi, \eta)\in \mathrm{supp}\, ast (\sigma_{j})$,
we see that $\widetilde{\sigma}_{j}$ satisfies
\begin{equation}\label{change-est}
\begin{split}
|\partial_{x}^{\alpha}
\partial_{\xi}^{\beta}
\partial_{\eta}^{\gamma}
\widetilde{\sigma}_{j} (x,\xi, \eta)|
&\lesssim
2^{jm}\mathbf{1} \{1+|\xi|+|\eta|\approx 2^{j(1-\rho)}\}
\\
&\lesssim
2^{-j(1-\rho)n/2}
(1+|\xi|+|\eta|)^{-n/2}.
\end{split}
\end{equation}
Hence the theorem of \cite{Miyachi-Tomita} or the case
$\rho =0$ of Theorem \ref{main-thm-1} yields
\[
\|T_{\widetilde{\sigma}_{j}}\|_{L^2 \times L^{\infty} \to L^2}\lesssim
2^{-j(1-\rho)n/2}.
\]
Thus we obtain
\begin{equation}\label{L2-estimate}
\begin{split}
&
\|T_{\sigma_{j}} (f^{(0)}, g^{(0)})\|_{L^2}
\lesssim
2^{-j(1-\rho)n/2}
\|f^{(0)}\|_{L^2}\|g^{(0)}\|_{L^{\infty}}
\\
&\le
2^{-j(1-\rho)n/2}
|\widetilde{Q}|^{1/2}
\approx
2^{-j(1-\rho)n/2}
|Q|^{\rho /2}.
\end{split}
\end{equation}
Next we prove an $L^{\infty}$ estimate of $T_{\sigma_j}(f^{(0)}, g^{(0)})$.
From the formula
\[
T_{\sigma_j}(a,b)(x)
=\int_{\mathbb{R}^n}
K_{j}(x,x-y,x-z) a(y) b(z)\, dydz
\]
and from \eqref{weightKj-0}, we have
\begin{align*}
&|T_{\sigma_j}(a,b)(x)|
\le
\|K_{j}(x,x-y,x-z) \|_{L^2_{y,z}}
\|a(y)b(z)\|_{L^2_{y,z}}
\\
&\lesssim
2^{j (m+n)}
\|a\|_{L^2}
\|b\|_{L^2}
=2^{j\rho n}
\|a\|_{L^2}
\|b\|_{L^2}.
\end{align*}
Hence
\begin{equation}\label{Linfty-estimate}
\begin{split}
&
\|T_{\sigma_{j}} (f^{(0)}, g^{(0)})\|_{L^{\infty}}
\lesssim
2^{j\rho n}
\|f^{(0)}\|_{2}\|g^{(0)}\|_{2}
\\
&\le
2^{j \rho n}
|\widetilde{Q}|
\approx
2^{j \rho n}
|Q|^{\rho}.
\end{split}
\end{equation}
Now by a characterization of weak $L^p$ functions
(see Lemma \ref{weakLp} to be given below),
the estimates \eqref{L2-estimate} and \eqref{Linfty-estimate}
imply the following weak type estimate
for $F^{(1)}=\sum_{j=0}^{\infty}T_{\sigma_{j}} (f^{(0)}, g^{(0)})$:
\[
|\{x \in \mathbb{R}^{n}
\,:\,
|F^{(1)}(x)| > \lambda
\}|
\lesssim
|Q| \lambda^{-1-1/\rho}, \quad \lambda >0.
\]
From this we obtain
\begin{align*}
\frac{1}{|Q|}\int_{Q}
|F^{(1)}(x)|\,
dx
&=
\int_{0}^{\infty}
|Q|^{-1}
|\{x \in Q
\,:\,
|F^{(1)}(x)| > \lambda
\}|\, d\lambda
\\
&\le
\int_{0}^{\infty}
\min \{
1,
\lambda^{-1-1/\rho}\}\, d\lambda
\approx 1,
\end{align*}
which is the estimate \eqref{BMOFi} for $i=1$ and $C^{(1)}_Q=0$.
This completes the proof of Theorem \ref{main-thm-2}.
Finally we shall give a proof of the fact that was used
at the last part of the above argument.
Here we shall give a slightly general lemma.
This lemma is equivalent to the fact that the space
$L^{(p,\infty)}$ is equal to the real interpolation space
$[L^{\infty}, L^{r}]_{\theta, \infty}$, $1/p=\theta/r$,
combined with the characterization of the latter space
by the $J$-method.
Although this may be known to many people, we shall give a proof
for reader's convenience.
\begin{lem}\label{weakLp}
Let $0<r<p<\infty$, $\alpha, \beta \in (0,\infty)$, and $0<\theta<1$
satisfy $1/p=\theta/r$ and $\alpha/ (\alpha+ \beta)=\theta$.
Then for nonnegative measurable functions $f$ on a measure space
the following two conditions are equivalent:
\begin{itemize}
\item[(i)]
there exists constants $A,B\in (0,\infty)$ and
a sequence of nonnegative measurable functions $\{f_j\}_{j\in \mathbb{Z}}$
such that
$\|f_j\|_{L^{\infty}} \le A 2^{j\alpha}$,
$\|f_j\|_{L^{r}} \le B 2^{-j\beta}$, and $f=\sum_{j\in \mathbb{Z}} f_j$.
\item[{\rm (ii)}]
$f\in L^{(p,\infty)}$, i.e., there exists a constant $C\in (0,\infty)$
such that
$|\{x \, :\, f(x)>\lambda\}| \le (C\lambda^{-1})^{p}$
for all $\lambda >0$.
\end{itemize}
To be precise, if {\rm (i)} holds
then {\rm (ii)} holds with $C=c(p,r,\alpha, \beta)A^{1-\theta}B^{\theta}$,
and, conversely,
if {\rm (ii)} holds then {\rm (i)} holds with $A, B\in (0, \infty)$
such that $A^{1-\theta}B^{\theta}=c(p,r,\alpha, \beta) C$.
\end{lem}
\begin{proof}
${\rm (i)} \mathbb{R}ightarrow{\rm (ii)}$.
Suppose {\rm (i)} holds and
write $\gamma =\alpha + \beta$.
Take an integer $j_0$ such that $A2^{j_0 \alpha}\approx B2^{-j_0 \beta}$
and set $C=A2^{j_0 \alpha}$.
Then $C\approx A^{1-\theta} B^{\theta}$,
$\|f_{j+j_0}\|_{L^{\infty}} \lesssim C 2^{j\gamma \theta}$, and
$\|f_{j+j_0}\|_{L^{r}} \lesssim C 2^{-j\gamma (1-\theta)}$.
For $\lambda \in (0,\infty)$ given, take an integer $j_1$ such that
$C 2^{j_{1}\gamma \theta}\approx \lambda$ and decompose $f$ as
\[
f=
\sum_{j\le j_1} f_{j+j_0}
+
\sum_{j> j_1} f_{j+j_0}
=f^{(0)}+ f^{(1)}.
\]
Then
$\|f^{(0)}\|_{L^{\infty}}\lesssim C 2^{j_{1}\gamma \theta} \approx \lambda$
and
$\|f^{(1)}\|_{L^{r}}\lesssim C 2^{-j_{1}\gamma (1-\theta)} \approx
C^{1/\theta} \lambda^{1-1/\theta}$.
Hence, if we take a sufficiently large constant $c_{0}$, which depends only
on $p,r,\alpha,\beta$, then we have
\begin{align*}
&
|\{x \, :\, f(x)>c_{0}\lambda\}|
\le
|\{x \, :\, f^{(1)}(x)>\lambda\}|
\\
&\le
\|f^{(1)}\|_{L^{r}}^{r}
\lambda^{-r}
\lesssim
(C^{1/\theta} \lambda^{1-1/\theta})^{r} \lambda^{-r}
=(C \lambda^{-1})^{p}.
\end{align*}
${\rm (ii)} \mathbb{R}ightarrow {\rm (i)}$.
Suppose {\rm (ii)} holds.
Take an $A\in (0, \infty)$ and decompose $f$ as
\[
f(x) = \sum_{j\in \mathbb{Z}} f_j (x),
\quad
f_j (x) = f(x) \mathbf{1} \{A2^{(j-1)\alpha} < f(x) \le A2^{j\alpha} \}.
\]
Then $\|f_j\|_{L^{\infty}}\le A2^{j\alpha}$
and
\[
\|f_j\|_{L^{r}}
\le
A2^{j\alpha}
|\{x \, :\, f(x) > A2^{(j-1)\alpha} \}|^{1/r}
\le
A2^{j\alpha}
(C A^{-1}2^{-(j-1)\alpha} )^{p/r}
= B 2^{-j \beta}
\]
with
$B\approx C^{1/\theta} A^{1-1/\theta} $.
The relations between the constants $A,B$, and $C$ are obvious from
the above arguments.
\end{proof}
\section{Proof of Corollary \ref{main-cor}}\label{section5}
It is known that there exist bijective mappings
$\sigma \mapsto \sigma^{\ast 1}$ and
$\sigma \mapsto \sigma^{\ast 2}$
of $BS^m_{\rho,\rho}$, $0 \le \rho <1$,
onto itself such that
\begin{equation}\label{eq310}
\int T_{\sigma}(f,g)(x)h(x)dx
=\int T_{\sigma^{\ast 1}}(h,g)(x)f(x)dx
=\int T_{\sigma^{\ast 2}}(f,h)(x)g(x)dx
\end{equation}
for all $f,g,h \in \mathcal{S}$ (see \cite[Theorem 2.1]{BMNT}).
By duality,
\begin{equation*}
\|T_{\sigma}\|_{L^2\times L^{\infty} \to L^2}
=
\|T_{\sigma^{\ast 2}}\|_{L^2 \times L^2 \to L^1}
=
\|T_{\left(\sigma^{\ast 2}\right)^{\ast1}}\|_{L^{\infty} \times L^2 \to L^2}.
\end{equation*}
In particular, if one of the above is finite, then
the other two are also finite.
Thus the desired result for $(p,q)=(2,\infty), (2,2), (\infty,2)$
follows from Theorem \ref{main-thm-1}.
Similarly, by the duality between $H^1$ and $BMO$,
\begin{equation*}
\|T_{\sigma}\|_{L^{\infty}\times L^{\infty} \to BMO}
\approx
\|T_{\sigma^{\ast 1}}\|_{H^1 \times L^{\infty} \to L^1}
\approx
\|T_{\sigma^{\ast 2}}\|_{L^{\infty} \times H^1 \to L^1}.
\end{equation*}
Hence the desired result for
$(p,q)=(\infty,\infty), (1,\infty), (\infty,1)$
follows from Theorem \ref{main-thm-2}.
Other cases can be obtained from interpolation.
As for the interpolation argument, see for example
\cite[Proof of Theorem 2.2]{BBMNT}.
\appendix\section{}\label{appendix}
In this appendix,
we shall prove Proposition \ref{critical-order}.
Let $0<p,q,r \le \infty$ and $1/p+1/q=1/r$.
We write
$m_{0}=m_{0}(p,q)$.
Recall that $m_{\rho}(p,q)=(1-\rho)m_0$.
For simplicity of notation, we only consider the case $r<\infty$,
but the argument below works in the case $r=\infty$ as well.
In fact, in the case $r=\infty$, all we need is to rewrite
$L^{r}$ by $BMO$.
In \cite[Theorem A.2]{Miyachi-Tomita},
it is already proved that
if $T_{\sigma}: H^p \times H^q \to L^r$
for all $m \in BS^{m}_{\rho,\rho}$ then
$m \le (1-\rho)m_0$.
Hence, in order to complete the proof of Proposition \ref{critical-order},
it is sufficient to show that
if $m<(1-\rho)m_{0}$ then $T_{\sigma}: H^p \times H^q \to L^r $
for all $\sigma \in BS^m_{\rho,\rho}$.
As we mentioned in Introduction, this has been proved
in \cite{MRS} and \cite{BBMNT}
in the range $1/p+1/q \le 1$.
Here we shall give a proof that is valid for all $0<p,q \le \infty$.
We use the fact that the case $\rho=0$ is already known.
To be precise, it is known that
$T_{\sigma}: H^p \times H^q \to L^r $
for all $\sigma \in BS^{m_0}_{0,0}$ (see \cite[Theorem 1.1]{Miyachi-Tomita}).
By virtue of the closed graph theorem,
this boundedness is equivalent to the claim that there exists
a positive integer $N$ and a constant $c$ such that
\begin{equation}\label{equiv-est}
\|T_{\sigma}\|_{H^p \times H^q \to L^r}
\le c \max_{|\alpha|, |\beta|, |\gamma| \le N}
\left(\sup_{x,\xi,\eta \in \mathbb{R}^n}
(1+|\xi|+|\eta|)^{-m_0}
|\partial^{\alpha}_x\partial^{\beta}_{\xi}
\partial^{\gamma}_{\eta}\sigma(x,\xi,\eta)|\right)
\end{equation}
for all $\sigma \in BS^{m_0}_{0,0}$
(see \cite[Lemma 2.6]{BBMNT}).
Now assume that $0<\rho<1$ and
$\sigma \in BS^m_{\rho,\rho}$ with $m<(1-\rho)m_0$.
In the same way as in Section \ref{section4},
we write $\sigma = \sum_{j=0}^{\infty}\sigma_j$
as in \eqref{sigma-sum-sigmaj} and \eqref{def-sigmaj},
and define $\widetilde{\sigma}_j$ by \eqref{change-symbol}.
Then \eqref{change-pdo} holds and this,
together with the relation $1/p + 1/q =1/r$, implies
\begin{equation}\label{s-tildes}
\|T_{\sigma_j}\|_{H^p \times H^q \to L^r}
=
\|T_{\widetilde{\sigma}_j}\|_{H^p \times H^q \to L^r}.
\end{equation}
Also, from the same argument as in \eqref{change-est},
we see that $\widetilde{\sigma}_j$ satisfies the
estimate
\[
|\partial^{\alpha}_x\partial^{\beta}_{\xi}
\partial^{\gamma}_{\eta}\widetilde{\sigma}_j(x,\xi,\eta)|
\le C_{\alpha,\beta,\gamma}\,
2^{j(m-(1-\rho)m_{0})}
(1+|\xi|+|\eta|)^{m_0}.
\]
Combining this with \eqref{s-tildes} and \eqref{equiv-est},
we have
\[
\|T_{{\sigma}_j}\|_{H^p \times H^q \to L^r}
=
\|T_{\widetilde{\sigma}_j}\|_{H^p \times H^q \to L^r}
\lesssim
2^{j (m- (1-\rho)m_0)}.
\]
Since $m<(1-\rho)m_0$, the above inequality implies
that $T_{\sigma}=\sum_{j=0}^{\infty}T_{\sigma_j}$
is bounded from $H^p \times H^q \to L^r$.
This completes the proof of Proposition \ref{critical-order}.
\end{document} | math | 78,740 |
\begin{document}
\preprint{APS/123-QED}
\title{Oscillating bound states in non-Markovian photonic lattices}
\author{Kian Hwee Lim}
\email{kianhwee\[email protected]}
\thanks{Equal contribution}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}
\author{Wai-Keong Mok}
\email{[email protected]}
\thanks{Equal contribution}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}
\affiliation{California Institute of Technology, Pasadena, CA 91125, USA}
\author{Leong-Chuan Kwek}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}
\affiliation{MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654, Singapore}
\affiliation{National Institute of Education,
Nanyang Technological University, 1 Nanyang Walk, Singapore 637616}
\affiliation{School of Electrical and Electronic Engineering
Block S2.1, 50 Nanyang Avenue,
Singapore 639798 }
\begin{abstract}
It is known that the superposition of two bound states in the continuum
(BIC) leads to the phenomenon of an oscillating bound state, where
excitations mediated by the continuum modes oscillate persistently. We
perform exact calculations for the oscillating BICs in a 1D photonic
lattice coupled to a ``giant atom" at multiple points. Our work is
significantly distinct from previous proposals of oscillating BICs in
continuous waveguide systems due to the presence of a finite energy band
contributing band-edge effects. In particular, we show that the bound
states outside the energy band are detrimental to the oscillating BIC
phenomenon, and can be suppressed by increasing either the number of
coupling points or the separation between each coupling point.
Crucially, non-Markovianity is necessary for the existence of oscillating
BIC, and the oscillation amplitude increases with the characteristic
delay time of the giant atom interactions. We also propose a novel
initialization scheme in the BIC subspace. Our work be experimentally
implemented on current photonic waveguide array platforms and opens up
new prospects in utilizing reservoir engineering for the storage of
quantum information in photonic lattices.
\end{abstract}
\maketitle
\section{Introduction}
The study of interactions between atoms and photons traces its history all
the way back to the inception of quantum mechanics itself. Since then, we
have acquired a better understanding of atom-photon interactions which underpins the foundation of many quantum technologies such as atomic clocks~\cite{ludlow2015optical} and trapped-ion quantum computers and simulators~\cite{bruzewics2019trapped,duan2010quantum,monroe2021programmable}, which harness the interaction of atoms
with lasers. In many studies of atom-photon interactions, one often makes the dipole approximation~\cite{tannoudji1992atom}, which assumes that the size of the atom is
much smaller than the wavelength of the light. This is especially valid in optical regimes where the length scale of the atom $(\approx 10^{-10} \, \text{m})$ is orders of magnitude smaller than the wavelength of light $(\approx 10^{-7} \, \text{m})$. Under the dipole approximation,
the time taken for the light to pass through a single atom is neglected, thus simplifying the interaction model. In
the field of waveguide quantum electrodynamics (QED)~\cite{liao2016photon,roy2017strongly}, which studies the interactions of atoms with a continuum of bosonic modes in a waveguide, the dipole approximation corresponds to
modelling the atoms as coupled to individual points along the waveguide~\cite{shen_coherent_2005_prl,shen_coherent_2005}. These
atoms could either be actual atoms~\cite{bajcsy_efficient_2009}, or
artificial atoms like quantum dots~\cite{akimov_generation_2007,alexander_coupling_2016,arcari_near-unity_2014}
and superconducting
qubits~\cite{astafiev_resonance_2010,astafiev_ultimate_2010,abdumalikov_electromagnetically_2010}.
A more complete overview of works done in this vein can be found in
\cite{chang_colloqium_2018,gu_microwave_2017,sheremet_waveguide_2021}.
However, this paradigm of dipole approximation in waveguide QED was recently broken with the discovery of the so-called ``giant atoms"~\cite{kockum_designing_2014,kockum_quantum_2021}, by coupling each atom to two or more points on the waveguide. This was originally achieved by coupling the superconducting artificial atom (working in the microwave regime) to surface acoustic waves (SAW). Due to the low SAW velocity, for a given frequency the wavelength of sound is no longer assumed to be large compared to the size of the superconducting artificial atom. An alternative method to engineer giant atom coupling is by meandering the transmission line such that the atom interacts with the waveguide at multiple locations~\cite{kockum_decoherence-free_2018,kannan_waveguide_2020}. In these setups, we can
no longer ignore the phase acquired by the light propagating in the 1D
waveguide during the interaction with the giant atoms. Remarkably, by tuning the acquired phase \cite{kockum_designing_2014}, one obtains fascinating phenomena such as prolonged coherence time of a giant atom~\cite{kockum_designing_2014, kannan_waveguide_2020}, decoherence-free interactions between two giant atoms
\cite{kockum_decoherence-free_2018,kannan_waveguide_2020} and the non-exponential
decay of a giant atom \cite{guo_giant_2017,andersson_non-exponential_2019}, which have also been experimentally demonstrated in recent years.
Another novel feature of giant atoms in waveguide QED is the existence of
oscillating bound states in the continuum (BIC), which is a genuine
non-Markovian effect due to the significant time delay for information to
propagate between the various coupling points of a giant
atom~\cite{guo_giant_2017}. The non-Markovianity manifests as a persistent
oscillation of energy in the waveguide trapped between the coupling
points of a giant atom, which behaves akin to a cavity. This is in stark
contrast to the irreversible loss of energy from the giant atom to the
waveguide in the Markovian regime. Thus, these oscillating BICs can
potentially be harnessed to preserve quantum information in a non-Markovian
bath by stabilizing the photonic quantum state, which we will show in this
paper.
In the continuous waveguide case, it is usual to linearize the dispersion relation about the atom's energy, since the coupling to the waveguide is weak~\cite{roy2017strongly}. The waveguide can then be regarded as having a linear dispersion with an infinite bandwidth. Instead of using a continuous waveguide such as a transmission line, we
consider a 1D photonic lattice which acts as the reservoir for the giant
atom. The key difference between the 1D photonic lattice that we consider here, and
the continuous waveguide proposed in \cite{guo_oscillating_2020}, is the
presence of a finite energy band where the band edge becomes significant,
which restricts the allowed BICs. Experimentally, this can be achieved using a photonic waveguide array
where each waveguide is side-coupled to each other via the evanescent field
produced by the photon propagating inside the
waveguides~\cite{jones_coupling_1965,somekh_channel_1973}. This has been
proposed to simulate the non-exponential decay of a photonic giant
atom~\cite{longhi_photonic_2020} which is simultaneously coupled to multiple
lattice sites. We also note that while oscillating BICs have been reported in a discrete lattice system with two giant atoms~\cite{longhi_rabi_2021}, manifesting as an effective Rabi oscillation between the atoms, the novelty of our work lies in only requiring a single giant atom to produce oscillating BICs.
As we will see, having a finite band gives rise to new conditions for the
oscillating BIC phenomenon, which are distinct from those derived for the
continuous waveguide. Moreover, we now need to consider the effect of bound
states outside the energy band. These bound states outside the energy band
are out of the continuum of allowed propagating modes and will henceforth be
called bound states outside the continuum (BOC). As will be explained in
more detail later, these BOCs are detrimental to quantum information storage
as they are states with an exponentially-decaying wave function around the
coupling points of the giant atom to the 1D photonic lattice. Hence, even
though it is possible to observe oscillatory behavior in the emitter
excitation probability with BOCs~\cite{ramos_nonMarkovian_2016}, we
distinguish the oscillating BICs which we study here, which allow for perfect
quantum information storage, from oscillations induced by BOCs which do not.
We also show that the oscillating BIC is a consequence of the time-delayed
interactions mediated by the 1D photonic lattice, and that a longer time
delay generally results in a higher amplitude for the BIC which reduces the
information leakage. A longer time delay also suppresses the unwanted
contributions from the BOCs which hinder the ability of the GA to store and
retrieve quantum information. This allows us to find the optimal conditions
for oscillating BICs.
This paper is organized as follows: firstly, we introduce the model
Hamiltonian and the theory behind BICs in Sec.~\ref{sec:theory}. Our main theoretical results are presented in Sec.~\ref{sec: oscillating bound states} where we derive the new conditions for oscillating BICs in our system as well as optimal conditions to
minimize the detrimental impact of BOCs.
Thereafter, we present some numerical results in Sec.~\ref{sec:numerics} which support our analytical
calculations. To demonstrate the feasibility of our theoretical results, we propose an experimental implementation of our
work achievable on state-of-the art photonic hardware in Sec.~\ref{sec:experiments}. Finally, we conclude in Sec.~\ref{sec:conclusion} and provide several directions for future research.
\section{Theory}
\label{sec:theory}
\subsection{Model Hamiltonian}
The Hamiltonian for the combined atom-lattice system can be written as $H =
H_a + H_{\text{wg}} + H_{\text{int}}$, given in \eqref{eqn: real space hamiltonian} as (setting $\hbar = 1$)
\begin{subequations}
\label{eqn: real space hamiltonian}
\begin{align}
H_{a} &= \omega_a a^\dag a + U a^{\dag 2} a^2 \\
\label{eqn: waveguide chain hamiltonian}
H_{\text{wg}} &= J\sum_{n=1}^{N-1}(b_n^\dagger b_{n+1} + \text{H.c}) \\
H_{\text{int}} &= \sum_{j=1}^{M} \rho_{j}(a^\dag b_{n_j} + \text{H.c})
\end{align}
\end{subequations}
where $a$ is the annihilation operator for the giant atom satisfying the
bosonic commutation relation $[a,a^\dag]=1$, and $b_n$ are the annihilation
operators for the 1D photonic lattice with $[b_n, b_m^\dag] = \delta_{mn}$.
$\omega_a$ is the detuning between the giant atom and the photonic lattice.
The $N$ lattice sites are coupled to each other via a tight-binding
Hamiltonian with interaction strength $J$. The giant atom is coupled to $M$
arbitrary lattice sites $\{n_1, \ldots, n_M\}$ with strength $\rho_j$,
$j=1,\ldots,M$. Here, $N$ is chosen to be a large number such that we can
treat the lattice as an infinite 1D chain in both the left and right
directions. The giant atom has an anharmonicity $U$, which we will take $U
\to \infty$ such that it is equivalent to treating the giant atom as a
two-level system. An illustration can be found in \figref{fig: hamiltonian
description}.
\begin{figure}\label{fig: hamiltonian description}
\end{figure}
As is shown in Appendix~\ref{Appendix: k-space hamiltonian derivation}, the
Hamiltonian described by \eqref{eqn: real space hamiltonian} can also be
written in $k$-space in the first Brillouin zone as
\begin{subequations}
\begin{align}
H_a &= \omega_a a^\dag a \\
H_{\text{wg}} &= \int_{-\pi}^\pi dk\, \omega(k) c^\dagger(k)c(k) \\
H_{\text{int}} &= \int_{-\pi}^\pi dk \left\{G(k)a^\dagger c(k) + \text{H.c}\right\}
\end{align}
\end{subequations}
by defining the $k$-space annihilation operators $c(k)$ through the discrete Fourier transform
\begin{equation}
\label{eqn: k-space annihilation operators}
c(k) = \frac{1}{\sqrt{2\pi}}\sum_{n=1}^N b_n e^{-ikn}, \quad
b_n = \frac{1}{\sqrt{2\pi}}\int_{-\pi}^\pi e^{ikn}c(k)\, dk.
\end{equation}
The operators for the lattice $c(k)$ obey the bosonic commutation relations
$\left[c(k),c^\dagger(k^\prime)\right] = \delta(k-k^\prime)$ with the dispersion relation $\omega(k) = 2J \cos(k)$ and the spectral
coupling function $G(k) = \frac{1}{\sqrt{2\pi}} \sum_{j=1}^{M} \rho_j e^{ikn_j}$. In general, $G(k)$ depends on the specific
geometry of our system, such as the number of coupling points $M$ between the
giant atom and the 1D photonic lattice and also the locations of the
coupling points $n_1, n_2, \dots n_M$. In previous works~\cite{guo_giant_2017,guo_oscillating_2020}, the dispersion relation is linearized and the energy band formed by the waveguide modes is approximated to be infinite such that the band-edge effects become negligible. As we will see later, by confining the allowed energies to be in $[-2J,2J]$, we obtain new conditions for oscillating BICs. Before that, it is helpful to first review the essential physics of BIC in this system.
\subsection{Bound states in the continuum}
A system is said to have a bound state in the continuum (BIC) if there is an
energy eigenstate with energy $\Omega$, where $\Omega$ lies in the band of
allowed energies of the system. BICs are theoretically very interesting
because conventionally, we would not expect a bound state to exist within a
continuum of propagating states that would carry the energy of the bound
state away, and yet these BICs truly exist and have been investigated both
theoretically and experimentally~\cite{stillinger_bound_1975,plotnik_experimental_2011,hsu_bound_2016,longhi_bound_2007}. Specifically, for the
setup that we are considering, the system has a BIC with energy $\Omega$ if
$\Omega \in [-2J, 2J]$ which is
defined by the tight-binding dispersion relation $\omega(k) = 2J\cos(k)$.
Furthermore, for a BIC at energy $\Omega$ to exist, either the density of
modes vanishes at $\omega=\Omega$ so that there is no mode in the continuum
for the bound state to decay into, or the coupling to the continuum vanishes
at $\omega=\Omega$. Lastly, since the BIC is a bound state by definition, we
also require the energy eigenstate at frequency $\Omega$ to have a finite
norm. The above conditions can be stated in more mathematically precise terms~\cite{longhi_bound_2007}.
Defining the density of modes $\rho(\omega) \equiv \frac{\partial k}{\partial
\omega}$, the density of modes vanishing at $\omega=\Omega$ means that we
require $\frac{\partial k}{\partial \omega}\bigr|_{\Omega} = 0$, which is not possible for the tight-binding dispersion relation. Thus, by designing the giant atom coupling, we enforce the
condition for the coupling to the continuum to vanish at $\Omega$
\begin{equation}
\label{eqn: coupling to continuum vanishes}
|G(k(\Omega))|^2 = 0 .
\end{equation}
If we restrict ourselves to the one-excitation subspace, a general
time-dependent state of the system can be written as
\begin{equation}
\label{eqn: one excitation subspace ansatz}
\ket{\psi(t)} = \psi_a(t) \ket{1_a} + \int_{-\pi}^\pi dk \,
\psi(k,t)\ket{1_k}
\end{equation}
where $\ket{1_a} = a^\dagger\ket{0}$, $\ket{1_k}=c^\dagger(k)\ket{0}$. By
considering an energy eigenstate $\ket{E}$ also in the one-excitation
subspace, we obtain
\begin{equation}
\label{eqn: Omega must be in band}
\Omega - \omega_a = \int_{-\pi}^{\pi}dk
\frac{|G(k)|^2}{\Omega - \omega(k)}
\end{equation}
by comparing the coefficients of $\ket{1_a}$ and $\ket{1_k}$ in the energy
eigenvalue equation $H\ket{E}= \Omega\ket{E}$. Hence, the requirement that
we have an energy eigenstate with energy $\Omega$ within the band implies that the solution of \eqref{eqn: Omega must be in band} for
$\Omega$ lies in the range $[-2J,
2J]$. The preceding calculation also gives us an expression for the
coefficient of $\ket{1_k}$, from which we can deduce that the finite norm requirement of the energy eigenstate is equivalent to $\rho(\omega)|G(k(\omega))|^2$ vanishing in the limit $\omega \to \Omega$ at
least as fast as $\sim(\Omega-\omega)^2$.
The integral in \eqref{eqn: Omega must be in band} can be evaluated by first
evaluating the self-energy $\Sigma(s)$ defined by
\begin{equation}
\label{eqn: Sigma(s) equation}
\Sigma(s) = \int_{-\pi}^\pi dk\,\frac{|G(k)|^2}{is - \omega(k)}.
\end{equation}
from which we get
\begin{equation}
\label{eqn: detuning in terms of Sigma(s)}
\Omega - \omega_a = \text{Re}[\Sigma(s=-i\Omega \pm 0^+)].
\end{equation}
A detailed derivation of the above equations is presented in
Appendix~\ref{Appendix: Omega equation derivation}.
\subsection{Decay dynamics}
In order to probe the decay dynamics of the giant atom into the 1D photonic lattice, we initialize the system with one
excitation in the giant atom and the lattice in the vacuum state. Mathematically, with reference to \eqref{eqn: one excitation subspace
ansatz}, we have $\psi_a(0)=1$ and $\psi(k,0)=0 \, \forall k\in[-\pi,\pi]$. From the Schr\"odinger equation $i\partial_t \ket{\psi(t)} = H\ket{\psi(t)}$ with these
initial conditions, it can be shown (see Appendix~\ref{Appendix: decay dynamics derivation}) that
$\Sigma(s)$ controls the time-dependent probability amplitude $\psi_a(t)$
through the equation
\begin{equation}
\label{eqn: decay dynamics}
\psi_a(t) = \sum_{\text{All residues}} \frac{ie^{st}}{is-\omega_a - \Sigma(s)}.
\end{equation}
From \eqref{eqn: decay dynamics}, we see that poles on the right-hand-side of
the equation with a non-zero real component will lead to a decay in
$\psi_a(t)$. On the other hand, for the poles on the right-hand-side of
the equation that lie on the imaginary axis, i.e if $s=-i\Omega$, the
exponential factor in the numerator will be $e^{-i\Omega t}$, which is
non-decaying and physically represents a BIC arising from the giant atom
decay. We note that (see Appendix~\ref{Appendix: Omega equation derivation})
when there exists an $\Omega$ that fulfils \eqref{eqn: detuning in terms of
Sigma(s)} as well as $|G(k(\Omega))|^2=0$, then $\Omega$ will be a pole on
the imaginary axis, which means that we will have a BIC at the frequency $\Omega$.
These BICs arising from giant atom decay are very interesting because by
construction they are immune to decay into the 1D lattice, and hence can be
used in a manner analogous to the so-called ``dark states'' for purposes like
storing quantum information
etc~\cite{johnsson_storing_2004,yang_quantum_2004}. Denoting the BIC
energies as $\Omega_j$, which satisfy both \eqref{eqn: detuning in terms of
Sigma(s)} and \eqref{eqn: coupling to continuum vanishes}, with simple poles
at $s=-i\Omega_j$, we have
\begin{align}
\psi_a(t) &= \sum_j
\lim_{s\to-i\Omega_j}\left[\frac{ie^{st}}{is-\omega_a-\Sigma(s)}(s+i\Omega)\right]
\nonumber \\
\label{eqn: residue in terms of sigma prime}
&= \sum_j \frac{e^{-i\Omega_j t}}{1+i\Sigma^\prime(-i\Omega_j)}
\end{align}
where we used L'Hopital's rule and also defined $\Sigma^\prime = \partial_s
\Sigma$ to get to the second line. Moreover, by noting that
\begin{equation}
\label{eqn: atom amplitude evolution equation}
\psi_a(t) = \sum_{E} e^{-iEt}|\braket{1_a}{E}|^2
\end{equation}
we obtain the emitter contribution of each BIC as
\begin{equation}
\label{eqn: emitter contribution BIC}
|\phi_a^{(j)}|^2 \equiv |\braket{1_a}{\Omega_j}|^2 = \frac{1}{1+i\Sigma^\prime(-i\Omega_j)}
\end{equation}
The usefulness of each of these BICs
can be quantified by the magnitude of $|\phi_a^{(j)}|^2$, since a large $|\phi_a^{(j)}|^2$
implies that the giant atom has a high probability of
being excited despite the existence of decay channels in the continuum for it to decay into.
\section{Oscillating bound states}
\label{sec: oscillating bound states}
Consider the case of giant atom decay in the one-excitation subspace again.
From \eqref{eqn: decay dynamics}, if there exists two BICs at frequency
$\Omega_\alpha$ and $\Omega_\beta$ that have relative large residues compared to the
other BICs, we have $\psi_a(t) \approx A e^{-i\Omega_\alpha t} + B e^{-i\Omega_\beta t}$
for some complex numbers $A$ and $B$. This means that the emitter probability
$|\psi_a(t)|^2$ oscillates sinusoidally with frequency $|\Omega_\alpha - \Omega_\beta|/2\pi$. We
can also infer the same fact by looking at \eqref{eqn: atom amplitude
evolution equation}. In this scenario, we say that our
system exhibits an oscillating BIC. Interestingly, we will show that these
oscillating BICs inherently require non-Markovianity in the system, resulting
in a bath-induced stabilization of a single-photon quantum state which can be
used both as a photon trapped in a cavity as well as a storage for quantum
information. Ideally, we would want $|A| = |B|$ so that at some time $t$, we
have $|\psi_a(t)|^2=0$ which means that by turning off the giant atom
couplings to the 1D lattice chain at that time, we can release the stored
photon into the 1D chain.
Let us now calculate the conditions in which the setup shown in
\figref{fig: hamiltonian description} exhibits an oscillating BIC. Consider
the case where the giant atom has $M$ coupling points equally spaced apart by $n_0$
sites on the photonic lattice. For $N$ lattice sites, let the giant atom be coupled to sites $0$, $n_0$, $2n_0$, $\dots$, $(M-1)n_0$ with a uniform coupling strength $\rho_0$. For this particular setup, we have
\begin{equation}
G(k) = \frac{\rho_0}{\sqrt{2\pi}} \sum_{j=0}^{M-1} e^{ij kn_0}
\end{equation}
As shown in Appendix~\ref{Appendix: oscillating BIC at n=2 CMI}, it is not
possible for an oscillating BIC to exist when $M=2$, consistent with the
results in a continuous linear waveguide~\cite{guo_oscillating_2020}. Hence,
we consider the case when $M\geq 3$ for which oscillating BICs exist.
Detailed calculations can be found in Appendix~\ref{Appendix: oscillating BIC calculations}. We will
summarize some of the key results here. We first calculate $|G(k)|^2$ to be
\begin{equation}
\label{eqn: |G(k)|^2 for M legs}
|G(k)|^2 = \frac{\rho_0^2}{2\pi}\left(M+2\sum_{r=1}^{M-1} (M-r)\cos(kn_0 r)\right)
\end{equation}
which means that when we enforce \eqref{eqn: coupling to continuum vanishes} for the coupling to the continuum to vanish, we have
\begin{equation}
\label{eqn: conditions for k for M legs oscillating BS}
k = \frac{2\pi}{n_0}\left(m\pm \frac{1}{M}\right)
\end{equation}
where $m\in \mathbb{Z}$. Furthermore, \eqref{eqn: Sigma(s) equation} and
\eqref{eqn: |G(k)|^2 for M legs} together give us
\begin{align}
\label{eqn: sigma(s) for M legs oscillating BS}
\Sigma(s) &= \frac{\mp i \rho_0^2}{\sqrt{s^2+4J^2}}\left[M +
2\sum_{r=1}^{M-1}(M-r) \alpha^{r n0}\right] \\
\label{eqn: big term in sigma(s) expression}
\alpha &= \left(\frac{\mp i\sqrt{s^2+4J^2}+is}{2J}\right)
\end{align}
where we have the negative sign in both $\alpha$ and $\Sigma(s)$ above when
$\text{Re}(s)>0$ and the positive sign when $\text{Re}(s)<0$. Since the
subsequent results are the same regardless of whether we consider
$\text{Re}(s)>0$ or $\text{Re}(s)<0$, we will restrict ourselves to the
$\text{Re}(s)>0$ case. Thereafter, from \eqref{eqn: detuning in terms of
Sigma(s)}, \eqref{eqn: sigma(s) for M legs oscillating BS} and \eqref{eqn:
big term in sigma(s) expression} we have
\begin{equation}
\label{eqn: delta(Omega) for M legs oscillating BS}
\Omega - \omega_a = \frac{-i \rho_0^2}{\sqrt{4J^2-\Omega^2}} \left[M +
2\sum_{r=1}^{M-1}(M-r) (e^{i n_0\theta})^r\right]
\end{equation}
where $\theta = \arctan(-\sqrt{4J^2-\Omega^2}/\Omega)$. Hence, to obtain an
oscillating BIC, we need to solve \eqref{eqn: delta(Omega) for M legs
oscillating BS} together with \eqref{eqn: conditions for k for M legs
oscillating BS} to obtain two eigenenergies $\Omega_1$ and $\Omega_2$ such
that the coupling to the continuum vanishes at these two energies.
At this juncture, we consider the case where $\omega_a=0$, such that the
giant atom energy is positioned at the band center. This is done so that the
BIC energies $\Omega_1, \Omega_2$ are symmetric about the band center, i.e
$\Omega_1 = - \Omega_2$, which is necessary to obtain perfectly sinusoidal
oscillations of the giant atom emitter probability. From \eqref{eqn: conditions for k for M legs oscillating BS}, we see that $n_0$ being an odd number will not give us BIC energies that are symmetric about the band centre. Hence, $n_0$ is
restricted to be an even number which give us two possibilities, $n_0 =
2(2l)$ or $n_0 = 2(2l+1)$ where $l\in \mathbb{Z}^+$. Since the giant atom
energy is positioned at the band center, we would expect the BIC energies
that are closer to the band center to correspond to states that have a larger
emitter probability. Using that criteria, we find that the optimal condition
for oscillating BIC occurs for the $n_0 = 4l$ case (see Appendix~\ref{Appendix: oscillating BIC calculations}) from which we get the BIC energies $\pm \Omega_{\text{BIC}}$ given
by
\begin{equation}
\label{eqn: BIC energies M legs n0=4l}
\Omega_{\text{BIC}} = 2J \sin\left(\frac{2\pi}{Mn_0}\right).
\end{equation}
We can then obtain the corresponding $\rho_0$ for $\pm \Omega_{\text{BIC}}$
by substituting \eqref{eqn: BIC energies M legs n0=4l} into \eqref{eqn:
delta(Omega) for M legs oscillating BS} together with $\omega_a=0$ to obtain
\begin{equation}
\label{eqn: BIC rho0 M legs n0=4l}
\left(\frac{\rho_0}{J}\right)^2 = \frac{2}{M}\tan\left(\frac{\pi}{M}\right)\sin\left(\frac{4\pi}{M n_0}\right)
\end{equation}
resulting in an oscillating BIC at the frequency $\Omega_\text{BIC}/ \pi$.
Now we can obtain the emitter probabilities of each BIC by substituting
\eqref{eqn: BIC energies M legs n0=4l} and \eqref{eqn: BIC rho0 M legs n0=4l}
into \eqref{eqn: emitter contribution BIC} to obtain
\begin{widetext}
\begin{equation}
\label{eqn: BIC emitter probs M legs n0=4l}
|\phi_a^{(\text{BIC})}|^2 \equiv \braket{1_a}{\pm \Omega_{\text{BIC}}}|^2 =
\frac{i e^{-\frac{4 i \pi }{M}} \left(1+e^{\frac{2 i \pi }{M}}\right) \left(-1+\left(i e^{\frac{2 i \pi }{M
n_0}}\right)^{n_0}\right)^3 \csc ^2\left(\frac{\pi }{M}\right) \cos ^2\left(\frac{2 \pi }{M n_0}\right)}{4 \left(2 n_0
\sin \left(\frac{4 \pi }{M n_0}\right)+\sin \left(\frac{2 \pi (n_0-2)}{M n_0}\right)+\sin \left(\frac{2 \pi
(n_0+2)}{M n_0}\right)\right)}
\end{equation}
\end{widetext}
From the emitter probabilities obtained above, using \eqref{eqn: decay
dynamics} and \eqref{eqn: residue in terms of sigma prime}, we see that in
the long time limit after all the non-BIC states have propagated away from
the giant atom to the left and right ends of the lattice chain, we would
expect oscillations in the emitter probability of the giant atom with
amplitude $(2|\phi_a^{(\text{BIC})}|^2)^2$. From \eqref{eqn: BIC emitter
probs M legs n0=4l}, we note that for all values of $n_0 = 4l, l\in
\mathbb{Z^+}$, $|\phi_a^{(\text{BIC})}|^2$ increases monotonically with $M$,
eventually saturating at the limiting value $|\phi_a^{(\text{BIC})}|^2 =
1/3$. Consequently, for any value of $n_0$, having a larger number of
coupling points $M$ leads to higher-amplitude oscillating BICs.
It is also helpful to use \eqref{eqn: BIC emitter probs M legs n0=4l} to
compute the asymptotic behavior of $|\phi_a^{(\text{BIC})}|^2$ as $n_0 \to
\infty$, which we can write as
\begin{align}
|\phi_a^{(\text{BIC})}|^2
\label{eqn: asymptotic emitter prob as n0=4l, n0 tends to infty}
&= \frac{1}{1+\frac{4\pi}{M}\csc\left(\frac{2\pi}{M}\right)} + \frac{A}{n_0^2} + \mathcal{O}\left(\frac{1}{n_0^4}\right) \\
\text{where }A&= \frac{4\pi^2\sin\left(\frac{2 \pi }{M}\right) \left(3 M
\sin \left(\frac{2 \pi }{M}\right)-4\pi\right)}{3 M \left(M \sin
\left(\frac{2 \pi }{M}\right)+4 \pi \right)^2}. \nonumber
\end{align}
By defining $\tau=(Mn_0)/\nu_g$ as the time taken for the photon to propagate
between the first coupling point and the $M$th coupling point (i.e, the size
of the giant atom), where $\nu_g=2J$ is the group velocity at the band
centre, and $\Gamma^{-1} = (M^2 \rho_0^2/J)^{-1}$ as the characteristic
timescale for the giant atom decay, we can quantify the amount of
non-Markovianity in our system through the quantity $\tau/\Gamma^{-1}$ which
can be written as
\begin{align}
\label{eqn: nonMarkovianity measure}
\frac{\tau}{\Gamma^{-1}} &= M^2 n_0 \sin\left(\frac{4\pi}{M n_0}\right)\tan\left(\frac{\pi}{M}\right) \\
\label{eqn: nonMarkovianity asymptotic measure}
&= 4M\pi\tan \left(\frac{\pi }{M}\right)-\frac{32 \left(\pi ^3 \tan
\left(\frac{\pi }{M}\right)\right)}{3 M
}\frac{1}{n_0^2}+\mathcal{O}\left(\frac{1}{n_0^4}\right)
\end{align}
where to get from the first line to the second line, we computed the
asymptotic behavior as $n_0 \to \infty$. From \eqref{eqn: nonMarkovianity
asymptotic measure}, we see that at large $n_0$ the non-Markovianity in our
system, which has the same $1/n_0^2$ scaling as the expressions for the BIC
emitter probabilities in \eqref{eqn: asymptotic emitter prob as n0=4l, n0
tends to infty}. This allows us to conclude that a stronger non-Markovianity
in our system arising from the time delay for information to propagate
between the giant atom coupling points results in better oscillating BICs,
though the amount of non-Markovianity quantified by $\tau/\Gamma^{-1}$
eventually reaches a plateau. The presence of the plateau means that even
though $\tau$ increases as $n_0$ increases, which leads to a greater
non-Markovianity in the system, this effect is quickly balanced by an
increase in the giant atom lifetime $\Gamma^{-1}$ which is a result of a
decreased coupling strength $\rho_0$. In practice, $n_0$ should of course
not be too large since the oscillation period scales as $\sim n_0$ which
might lead to more decoherence. Fortunately, the fast convergence
$\mathcal{O}(1/n_0^2)$ of the emitter probabilities means that a moderate
$n_0$ is already sufficient to observe good oscillating BICs.
Finally, we note that for all values of
$n_0 = 4l, l\in \mathbb{Z}^+$, as $M\to \infty$, $\tau/\Gamma^{-1}$ monotonically
decreases to a limiting value of $4\pi^2$. This implies that our system with an
oscillating BIC is inherently non-Markovian in nature, since there is a
non-negligible lower bound to $\tau/\Gamma^{-1}$.
\subsection{Role of imperfections: Bound states outside the continuum}
Bound states outside the continuum (BOCs) are energy eigenstates of the
Hamiltonian that have energy out of the range $[-2J,2J]$. For these states,
the wave number $k$ is complex~\cite{munro_optical_2017}, which means that
these states are unable to propagate in the 1D lattice chain and hence they
have a significant probability amplitude in $\ket{1_a}$, with an exponentially-decaying wavefunction around the coupling points of the giant atom. These states are imperfections to our
oscillating BIC for two reasons. Firstly, for an oscillating BIC produced by
giant atom decay, we want the emitter probability to be high for the two BICs
involved at $\pm \Omega_\text{BIC}$, and low for all the other energy eigenstates.
Yet these BOCs have a large atomic component and hence they act as
imperfections to our sinusoidal oscillation as per \eqref{eqn: decay
dynamics}. Secondly, these states leak energy outside the giant-atom
coupling points due to the exponential decay of the photon amplitude from the
coupling points.
To characterize the effect of BOCs on oscillating BIC produced by giant
atom decay, we first use \eqref{eqn: BIC rho0 M legs n0=4l} to obtain
$\rho_0$ corresponding to the oscillating BIC condition. Then, we solve for the BOC energies
$\Omega_{\text{BOC}}$ where $|\Omega_{\text{BOC}}|>2J$ in \eqref{eqn: Omega must be in band} with
$\omega_a=0$. In the limit of $n_0 \to \infty$, the two BOC energies can be
found as
\begin{equation}
\label{eqn: asymptotic BOC energies}
\Omega_{\text{BOC}} \approx \pm 2J \left(1 + \frac{2\pi^2 \tan(\pi/M)^2}{M^2n_0^2} \right)
\end{equation}
with the emitter probability
\begin{equation}
\label{eqn: asymptotic BOC probability}
\left|\phi_a^{(\text{BOC})}\right|^2 \sim \frac{4\pi^2 \tan(\pi/M)^2}{M^2}\frac{1}{n_0^2}.
\end{equation}
This means that for a given value of $M$, at large $n_0$, the contributions
from the BOCs to the oscillations are suppressed by a factor of $1/n_0^2$.
By comparing \eqref{eqn: asymptotic BOC probability} and \eqref{eqn:
nonMarkovianity asymptotic measure}, we see that a larger non-Markovianity in
our system characterized by a larger $\tau/\Gamma^{-1}$ leads to reduced
imperfections from the BOC. We also note that having a larger number of
coupling points $M$ leads to a diminished effect of the BOCs on our
oscillating bound states, which can be explained by how a larger value of $M$
leads to a smaller coupling $\rho_0$ between the giant atom and the lattice
chain.
\subsection{Initialization in the BIC subspace}
Up till now, we have considered the case of giant atom decay into the 1D
lattice chain. If instead we are given the ability to initialize the state
of the lattice sites in the chain, which is possible for some experimental
platforms such as a side-coupled waveguide array through pulse shaping
techniques, we can eliminate the effects of the BOCs even at low $n_0$ and
also obtain perfect storage of quantum information within the legs of the
giant atom. This is especially important for small values of $M$ like $M=3$,
since from \eqref{eqn: asymptotic BOC probability} we see that the emitter probability of the BOC states increases as $M$ decreases.
Hence, we shall temporarily restrict ourselves to $M=3$ here, though it
should be clear that the method below generalizes for any positive integer
values of $M$.
We first write the states corresponding to the BICs at $\pm
\Omega_{\text{BIC}}$ as
\begin{equation}
\ket{\pm} = \phi_{a,\pm} \ket{1_a} + \sum_{n} \phi_{n,\pm} \ket{n}.
\end{equation}
Without loss of generality, we can set $\phi_{a,\pm} =
|\phi_a^{(\text{BIC})}|$ since
eigenstates are defined up to a global phase. Here $\phi_{n,\pm}$ are the photon amplitudes in real
space, which for $M=3$ we can calculate $\phi_{n,\pm}$ to be given by
\begin{equation}
\label{eqn: M=3 bic probability dist}
\phi_{n,+} = C \times \begin{cases} \begin{aligned} &(-1)^n \exp{\left( i\frac{2\pi n}{3 n_0}\right)} \\&- \exp{\left( -i\frac{2\pi n}{3 n_0}\right)}, 0\leq n \leq n_0 \\ &(-1)^{n+1} \exp{\left( i\frac{2\pi}{3}\left(\frac{n}{n_0}-2\right)\right)} \\&+ \exp{\left( -i\frac{2\pi}{3}\left(\frac{n}{n_0}-2\right)\right)}, n_0\leq n \leq 2n_0 \\& 0, \quad \text{else} \end{aligned} \end{cases}
\end{equation}
where
\begin{equation}
C = \frac{-i^{n+1} \phi_a \rho_0}{2J \cos\left( \frac{2\pi}{3 n_0} \right)}
\end{equation}
and $\phi_{n,-} = (-1)^{n+1} \phi_{n,+}^*$. The above calculation also means that the state
\begin{align}
\label{eqn: BIC subspace state p}
\ket{p} &\equiv \frac{1}{\sqrt{2}}(\ket{+}-\ket{-}) \nonumber \\
&= \frac{1}{\sqrt{2}} \sum_n \left(\phi_{n,+}-\phi_{n,-}\right)\ket{n}
\end{align}
is a state with no probability amplitude in $\ket{1_a}$ and with photon
amplitudes in real space only within the $M$ coupling points of the giant
atom. Thus if we initialize the state in $\ket{p}$, then there will be zero excitation leakage outside of the giant atom and the lattice
sites within the $M$ coupling points. Furthermore, since the states
$\ket{\pm}$ are orthogonal to the BOCs, the imperfections in the oscillations
due to the BOCs are eliminated by construction. Lastly, we note that for $M=3$
the photon amplitudes are real and only differ in phase by $0$ or $\pi$, which makes it more feasible for practical implementation.
\section{Numerical results}
\label{sec:numerics}
\begin{figure*}\label{fig: M=3GAdecaySimulationResults}
\end{figure*}
Here we first present in \figref{fig: M=3GAdecaySimulationResults} some
results for the $M=3$ giant atom decay coupled to a 1D photonic lattice with
various values of $n_0$. For $M=3$, starting with a single excitation in the
giant atom, in the absence of imperfections due to the BOCs, we should expect
sinusoidal oscillations in the excitation probability of the giant atom
lattice site with oscillation amplitude $(2|\phi_a^{(\text{BIC})}|^2)^2
\approx 0.117411$, where we have used \eqref{eqn: asymptotic emitter prob as
n0=4l, n0 tends to infty} to obtain $|\phi_a^{(\text{BIC})}|^2 \approx 0.171$
for $M=3$. However, for the $M=3$ case, as is seen in \figref{fig:
nonMarkovianityProbPlot}, the BOC emitter probabilities are actually quite
substantial, especially at small values of $n_0$. Hence, this leads us to
consider a strategy for the $M=3$ case where instead of considering giant
atom decay, we initialize the lattice sites in the initial state \eqref{eqn:
BIC subspace state p} to eliminate the effects of the BOCs resulting in
complete storage of quantum information within the legs of the giant atoms,
and perfectly sinusoidal oscillations in the excitation probability of the
giant atom lattice site with oscillation amplitude
$2|\phi_a^{(\text{BIC})}|^2 \approx 0.33$. An example for the $n_0=4$ case
is shown in \figref{fig: M=3andn0=4BICsubspaceResults}. The amplitude and
phase of the initial photon excitation at each of the lattice sites can be
found using \eqref{eqn: BIC subspace state p}, where examples for various
values of $n_0$ are shown in \figref{fig: BICsubspaceinitialization}.
Finally, to show the effect of increasing $M$ on the quality of the giant
atom oscillating BIC, we plot the case of giant atom decay for $M=50$ and
$n_0=4$ case in \figref{fig: M=50n0=4giantatomdecay}. We note that for this
value of $M$, we should expect the excitation probability in the giant atom
lattice site to oscillate with an amplitude of
$(2|\phi_a^{(\text{BIC})}|^2)^2 \approx 4/9$ as $|\phi_a^{(\text{BIC})}|^2$
approaches the asymptotic value of $1/3$ for increasing values of $M$.
\begin{figure}\label{fig: nonMarkovianityProbPlot}
\end{figure}
\begin{figure*}\label{fig: M=3andn0=4BICsubspaceResults}
\end{figure*}
\begin{figure*}\label{fig: BICsubspaceinitialization}
\end{figure*}
\begin{figure}\label{fig: M=50n0=4giantatomdecay}
\end{figure}
\section{Experimental implementation}
\label{sec:experiments}
The Hamiltonian in \eqref{eqn: real space hamiltonian} can be simulated on a
variety of platforms, such as coupled cavity
arrays~\cite{hartmann_quantum_2008,majumdar_design_2012} and photonic
waveguide arrays~\cite{christodoulides_discretizing_2003,longhi_quantum_2009,
szameit_discrete_2010, aspuru-guzik_photonic_2010,garanovich_light_2012}. In
the case of a photonic waveguide array, we would have one photonic waveguide,
which we call the giant atom waveguide, coupled to $M$ different photonic
waveguides that are already coupled to each other to form a linear chain of
$N$ waveguides, where the coupling is due to the evanescent field produced by
the photon propagating within the waveguide. As the photon propagates in the
waveguide, we have the relation $z = ct$, where $c$ is the group velocity of the
photon in the waveguide and $z$ is the distance along the waveguide that the
photon has propagated for.
\begin{figure}\label{fig: proposed setup for oscillating BS BIC subspace}
\end{figure}
Following our formalism above, the nearest-neighbour coupling of the photonic
waveguides in the linear chain with coupling strength $J$ gives us the tight
binding Hamiltonian $H_{\text{wg}}$, whereas the coupling between the giant
atom waveguide and the linear chain of waveguides at $M$ different points,
each spaced $n_0$ apart, with coupling strength $\rho_0$ gives us the
interaction Hamiltonian $H_{\text{int}}$. Taking the constraints of current
experimental capabilities in mind, we propose an experimental setup for the
case where $M=3$ and $n_0 = 4$ using the BIC subspace initialization in
\figref{fig: proposed setup for oscillating BS BIC subspace}. For this
photonic waveguide array system, the BIC initialization according to
\figref{fig: proposed setup for oscillating BS BIC subspace} can be achieved
deterministically with a spatial light modulator that modulates a single
photon source \cite{tentrup_transmitting_2017}. Alternatively, one can also
prepare the oscillating BIC probabilistically by initializing an excitation
only in the giant atom waveguide and perform photodetection on the sites
outside of the giant atom coupling points, and postselect on the no-detection
events.
In \figref{fig: proposed setup for oscillating BS BIC subspace} we have also
denoted the next-nearest neighbor coupling between the giant atom waveguide
and the sites $0 \pm 1$, $n_0 \pm 1$ and $2n_0 \pm 1$ with $\rho_1$, and also
the next-nearest neighbor hopping between the lattice sites $0 \pm 1$, $n_0
\pm 1$ and $2n_0 \pm 1$ with $J^\prime$. In general, the presence of
$J^\prime$ and $\rho_1$ are unwanted imperfections, yet we note that by
choosing the geometry and the distances accordingly as per the inset in
\figref{fig: proposed setup for oscillating BS BIC subspace}, we can minimize
the contributions from $\rho_1$ as well as $J^\prime$. To do so, we first
use \eqref{eqn: BIC energies M legs n0=4l} to calculate the emitter energies
for $n_0=4$, which would give us the oscillation period $T = \pi$ for the
oscillating BIC. This means that to see an appreciable number of
oscillations, we could simulate up to $Jz = 5T \approx 15$. Now, suppose
that experimentally, we can only have photonic waveguides with length
$z_{\text{max}}$. This means that we require $J = 5T/z_{\text{max}}$.
Henceforth, we shall assume $z_{\text{max}} = 100$ mm which has been done
experimentally before~\cite{tang_generating_2022}. From \eqref{eqn: BIC rho0
M legs n0=4l}, we obtain $\rho_0/J=1$ which tells us to set $\rho_0=J$. It
is known that the evanescent coupling strength between waveguides decay
exponentially with the distance between
them~\cite{jiao_two-dimensional_2021}. Using the experimental values
obtained in~\cite{jiao_two-dimensional_2021} for the aforementioned
exponential relationship between coupling strength and distance, together
with the geometry of the proposed setup in the inset of \figref{fig: proposed
setup for oscillating BS BIC subspace}, we obtain $\rho_0=J=0.15 \,
\text{mm}^{-1}$, $\rho_1 = 0.0286 \rho_0$ and $J^\prime = 0.0286J$ which is
nearly negligible. Thus, our proposed oscillating BICs are experimentally
feasible using state-of-the-art photonic waveguide arrays.
Simulation results for $M=3$ and $n_0=4$ when $\rho_1=0$ and $J^\prime=0$ can
be found in \figref{fig: M=3andn0=4BICsubspaceResults}. The corresponding
results when $\rho_1 = 0.0286 \rho_0$ and $J^\prime = 0.0286J$ can be found in \figref{fig: experimental implementation simulation}.
\begin{figure*}\label{fig: experimental implementation simulation}
\end{figure*}
\section{Conclusion}
\label{sec:conclusion}
In this paper, we study the phenomenon of oscillating BIC in a discrete 1D photonic lattice using a single emitter coupled to multiple
lattice sites, which can be considered as the
discrete analog of a giant atom coupled to a continuous waveguide. The key difference between our work and the oscillating BICs found in continuous waveguide systems~\cite{guo_oscillating_2020} is the presence of a finite energy band, which contributes band-edge effects to the giant atom dynamics. This gives us new conditions for the existence of oscillating BICs which lead to persistent oscillations of energy between the coupling points of the giant atom to the 1D lattice. The presence of bound states outside the continuum (BOC) hinders the trapping of excitation between the giant atom coupling points and is detrimental to the sinusoidal oscillations in the giant atom probability. Crucially, we find that these unwanted BOCs can be suppressed drastically by increasing either the number of coupling points $M$ or the number of lattice sites $n_0$ between each coupling point, with the BOC contribution scaling as $1/M^2$ and $1/n_0^2$. With this, we can summarize our key results for the conditions to produce optimal oscillating BICs to be: (1) $n_0 = 4l, l \in \mathbb{Z}^+$ sites between each coupling point, (2) Large $n_0$ and (3) Large M. In practice however, we find that a moderate $M$ and $n_0$ suffice to achieve good oscillating BICs with significant giant atom probability. Alternatively, by initializing the lattice sites in the BIC subspace which we have calculated, the BOC contributes can be completely eliminated, resulting in perfect oscillating BICs even for small $M$ and $n_0$. We
stress that the oscillating BIC in our system is inherently a non-Markovian phenomenon due to the significant propagation time between the giant atom coupling points compared to the relaxation timescale of the giant atom. Moreover, we show that as the
non-Markovianity in our system increases, the oscillation amplitude of the BICs increases, improving the storage of quantum information within the coupling points. To illustrate the feasibility of our theoretical model, we propose an experimental implementation of our
system on photonic waveguide arrays and show that our oscillating BICs can be practically achieved even with current experimental limitations.
Our work provides a firm theoretical basis for oscillating BICs in discrete systems. In particular, oscillating BICs in discrete systems
offer new possibilities that cannot be replicated in continuous systems,
such as the ability to initialize the system in the BIC subspace by simply controlling the amplitude and phase of
the excitation at particular lattice sites. This allows us to achieve long-time storage of quantum information within the confines of the giant atom coupling points, limited only by the intrinsic coherence time of the photonic lattice. Our setup can also be regarded as an effective cavity, serving as a physical implementation of non-Markovian cavity-QED setups~\cite{crowder2020quantum,regidor2021cavitylike,guimond2016rabi,cernotik2019cavity,du2021single}. While we have
considered the tight-binding dispersion relation in this work, the phenomenon of oscillating BICs can be generally observed in discrete systems with other dispersion relations, which can be considered for future work. Another promising direction is to study the oscillating BIC phenomenon in higher-dimensional lattices~\cite{gonzalez2019engineering} or in synthetic dimensions~\cite{du2022giant,xiao2022bound}.
\begin{thebibliography}{56}
\makeatletter
\providecommand \@ifxundefined [1]{
\@ifx{#1\undefined}
}
\providecommand \@ifnum [1]{
\ifnum #1\expandafter \@firstoftwo
\else \expandafter \@secondoftwo
\fi
}
\providecommand \@ifx [1]{
\ifx #1\expandafter \@firstoftwo
\else \expandafter \@secondoftwo
\fi
}
\providecommand \natexlab [1]{#1}
\providecommand \enquote [1]{``#1''}
\providecommand \bibnamefont [1]{#1}
\providecommand \bibfnamefont [1]{#1}
\providecommand \citenamefont [1]{#1}
\providecommand \href@noop [0]{\@secondoftwo}
\providecommand \href [0]{\begingroup \@sanitize@url \@href}
\providecommand \@href[1]{\@@startlink{#1}\@@href}
\providecommand \@@href[1]{\endgroup#1\@@endlink}
\providecommand \@sanitize@url [0]{\catcode `\\12\catcode `\$12\catcode
`\&12\catcode `\#12\catcode `\^12\catcode `\_12\catcode `\%12\relax}
\providecommand \@@startlink[1]{}
\providecommand \@@endlink[0]{}
\providecommand \url [0]{\begingroup\@sanitize@url \@url }
\providecommand \@url [1]{\endgroup\@href {#1}{\urlprefix }}
\providecommand \urlprefix [0]{URL }
\providecommand \Eprint [0]{\href }
\providecommand \doibase [0]{http://dx.doi.org/}
\providecommand \selectlanguage [0]{\@gobble}
\providecommand \bibinfo [0]{\@secondoftwo}
\providecommand \bibfield [0]{\@secondoftwo}
\providecommand \translation [1]{[#1]}
\providecommand \BibitemOpen [0]{}
\providecommand \bibitemStop [0]{}
\providecommand \bibitemNoStop [0]{.\EOS\space}
\providecommand \EOS [0]{\spacefactor3000\relax}
\providecommand \BibitemShut [1]{\csname bibitem#1\endcsname}
\let\auto@bib@innerbib\@empty
\bibitem [{\citenamefont {Ludlow}\ \emph {et~al.}(2015)\citenamefont {Ludlow},
\citenamefont {Boyd}, \citenamefont {Ye}, \citenamefont {Peik},\ and\
\citenamefont {Schmidt}}]{ludlow2015optical}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Andrew~D.}\
\bibnamefont {Ludlow}}, \bibinfo {author} {\bibfnamefont {Martin~M.}\
\bibnamefont {Boyd}}, \bibinfo {author} {\bibfnamefont {Jun}\ \bibnamefont
{Ye}}, \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Peik}}, \ and\
\bibinfo {author} {\bibfnamefont {P.~O.}\ \bibnamefont {Schmidt}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Optical atomic clocks},}\
}\href {\doibase 10.1103/RevModPhys.87.637} {\bibfield {journal} {\bibinfo
{journal} {Rev. Mod. Phys.}\ }\textbf {\bibinfo {volume} {87}},\ \bibinfo
{pages} {637--701} (\bibinfo {year} {2015})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Bruzewicz}\ \emph {et~al.}(2019)\citenamefont
{Bruzewicz}, \citenamefont {Chiaverini}, \citenamefont {McConnell},\ and\
\citenamefont {Sage}}]{bruzewics2019trapped}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Colin~D.}\
\bibnamefont {Bruzewicz}}, \bibinfo {author} {\bibfnamefont {John}\
\bibnamefont {Chiaverini}}, \bibinfo {author} {\bibfnamefont {Robert}\
\bibnamefont {McConnell}}, \ and\ \bibinfo {author} {\bibfnamefont
{Jeremy~M.}\ \bibnamefont {Sage}},\ }\bibfield {title} {\enquote {\bibinfo
{title} {Trapped-ion quantum computing: Progress and challenges},}\ }\href
{\doibase 10.1063/1.5088164} {\bibfield {journal} {\bibinfo {journal}
{Applied Physics Reviews}\ }\textbf {\bibinfo {volume} {6}},\ \bibinfo
{pages} {021314} (\bibinfo {year} {2019})},\ \Eprint
{http://arxiv.org/abs/https://doi.org/10.1063/1.5088164}
{https://doi.org/10.1063/1.5088164} \BibitemShut {NoStop}
\bibitem [{\citenamefont {Duan}\ and\ \citenamefont
{Monroe}(2010)}]{duan2010quantum}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {L.-M.}\ \bibnamefont
{Duan}}\ and\ \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Monroe}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Colloquium: Quantum networks
with trapped ions},}\ }\href {\doibase 10.1103/RevModPhys.82.1209} {\bibfield
{journal} {\bibinfo {journal} {Rev. Mod. Phys.}\ }\textbf {\bibinfo
{volume} {82}},\ \bibinfo {pages} {1209--1224} (\bibinfo {year}
{2010})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Monroe}\ \emph {et~al.}(2021)\citenamefont {Monroe},
\citenamefont {Campbell}, \citenamefont {Duan}, \citenamefont {Gong},
\citenamefont {Gorshkov}, \citenamefont {Hess}, \citenamefont {Islam},
\citenamefont {Kim}, \citenamefont {Linke}, \citenamefont {Pagano},
\citenamefont {Richerme}, \citenamefont {Senko},\ and\ \citenamefont
{Yao}}]{monroe2021programmable}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
{Monroe}}, \bibinfo {author} {\bibfnamefont {W.~C.}\ \bibnamefont
{Campbell}}, \bibinfo {author} {\bibfnamefont {L.-M.}\ \bibnamefont {Duan}},
\bibinfo {author} {\bibfnamefont {Z.-X.}\ \bibnamefont {Gong}}, \bibinfo
{author} {\bibfnamefont {A.~V.}\ \bibnamefont {Gorshkov}}, \bibinfo {author}
{\bibfnamefont {P.~W.}\ \bibnamefont {Hess}}, \bibinfo {author}
{\bibfnamefont {R.}~\bibnamefont {Islam}}, \bibinfo {author} {\bibfnamefont
{K.}~\bibnamefont {Kim}}, \bibinfo {author} {\bibfnamefont {N.~M.}\
\bibnamefont {Linke}}, \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
{Pagano}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Richerme}},
\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Senko}}, \ and\ \bibinfo
{author} {\bibfnamefont {N.~Y.}\ \bibnamefont {Yao}},\ }\bibfield {title}
{\enquote {\bibinfo {title} {Programmable quantum simulations of spin systems
with trapped ions},}\ }\href {\doibase 10.1103/RevModPhys.93.025001}
{\bibfield {journal} {\bibinfo {journal} {Rev. Mod. Phys.}\ }\textbf
{\bibinfo {volume} {93}},\ \bibinfo {pages} {025001} (\bibinfo {year}
{2021})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Tannoudji}\ \emph {et~al.}(1992)\citenamefont
{Tannoudji}, \citenamefont {Grynberg},\ and\ \citenamefont
{Dupont-Roe}}]{tannoudji1992atom}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Claude~Cohen}\
\bibnamefont {Tannoudji}}, \bibinfo {author} {\bibfnamefont {Gilbert}\
\bibnamefont {Grynberg}}, \ and\ \bibinfo {author} {\bibfnamefont
{J}~\bibnamefont {Dupont-Roe}},\ }\bibfield {title} {\enquote {\bibinfo
{title} {Atom-photon interactions},}\ }\href
{https://www.osti.gov/biblio/5354747} {\ (\bibinfo {year}
{1992})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Liao}\ \emph {et~al.}(2016)\citenamefont {Liao},
\citenamefont {Zeng}, \citenamefont {Nha},\ and\ \citenamefont
{Zubairy}}]{liao2016photon}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Zeyang}\ \bibnamefont
{Liao}}, \bibinfo {author} {\bibfnamefont {Xiaodong}\ \bibnamefont {Zeng}},
\bibinfo {author} {\bibfnamefont {Hyunchul}\ \bibnamefont {Nha}}, \ and\
\bibinfo {author} {\bibfnamefont {M~Suhail}\ \bibnamefont {Zubairy}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Photon transport in a
one-dimensional nanophotonic waveguide {QED} system},}\ }\href {\doibase
10.1088/0031-8949/91/6/063004} {\bibfield {journal} {\bibinfo {journal}
{Physica Scripta}\ }\textbf {\bibinfo {volume} {91}},\ \bibinfo {pages}
{063004} (\bibinfo {year} {2016})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Roy}\ \emph {et~al.}(2017)\citenamefont {Roy},
\citenamefont {Wilson},\ and\ \citenamefont {Firstenberg}}]{roy2017strongly}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Dibyendu}\
\bibnamefont {Roy}}, \bibinfo {author} {\bibfnamefont {C.~M.}\ \bibnamefont
{Wilson}}, \ and\ \bibinfo {author} {\bibfnamefont {Ofer}\ \bibnamefont
{Firstenberg}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Colloquium:
Strongly interacting photons in one-dimensional continuum},}\ }\href
{\doibase 10.1103/RevModPhys.89.021001} {\bibfield {journal} {\bibinfo
{journal} {Rev. Mod. Phys.}\ }\textbf {\bibinfo {volume} {89}},\ \bibinfo
{pages} {021001} (\bibinfo {year} {2017})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Shen}\ and\ \citenamefont
{Fan}(2005{\natexlab{a}})}]{shen_coherent_2005_prl}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Jung-Tsung}\
\bibnamefont {Shen}}\ and\ \bibinfo {author} {\bibfnamefont {Shanhui}\
\bibnamefont {Fan}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Coherent single photon transport in a one-dimensional waveguide coupled with
superconducting quantum bits},}\ }\href {\doibase
10.1103/PhysRevLett.95.213001} {\bibfield {journal} {\bibinfo {journal}
{Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {95}},\ \bibinfo {pages}
{213001} (\bibinfo {year} {2005}{\natexlab{a}})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Shen}\ and\ \citenamefont
{Fan}(2005{\natexlab{b}})}]{shen_coherent_2005}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~T.}\ \bibnamefont
{Shen}}\ and\ \bibinfo {author} {\bibfnamefont {Shanhui}\ \bibnamefont
{Fan}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Coherent photon
transport from spontaneous emission in one-dimensional waveguides},}\ }\href
{\doibase 10.1364/OL.30.002001} {\bibfield {journal} {\bibinfo {journal}
{Opt. Lett.}\ }\textbf {\bibinfo {volume} {30}},\ \bibinfo {pages}
{2001--2003} (\bibinfo {year} {2005}{\natexlab{b}})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Bajcsy}\ \emph {et~al.}(2009)\citenamefont {Bajcsy},
\citenamefont {Hofferberth}, \citenamefont {Balic}, \citenamefont {Peyronel},
\citenamefont {Hafezi}, \citenamefont {Zibrov}, \citenamefont {Vuletic},\
and\ \citenamefont {Lukin}}]{bajcsy_efficient_2009}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Bajcsy}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Hofferberth}},
\bibinfo {author} {\bibfnamefont {V.}~\bibnamefont {Balic}}, \bibinfo
{author} {\bibfnamefont {T.}~\bibnamefont {Peyronel}}, \bibinfo {author}
{\bibfnamefont {M.}~\bibnamefont {Hafezi}}, \bibinfo {author} {\bibfnamefont
{A.~S.}\ \bibnamefont {Zibrov}}, \bibinfo {author} {\bibfnamefont
{V.}~\bibnamefont {Vuletic}}, \ and\ \bibinfo {author} {\bibfnamefont
{M.~D.}\ \bibnamefont {Lukin}},\ }\bibfield {title} {\enquote {\bibinfo
{title} {Efficient all-optical switching using slow light within a hollow
fiber},}\ }\href {\doibase 10.1103/PhysRevLett.102.203902} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
{volume} {102}},\ \bibinfo {pages} {203902} (\bibinfo {year}
{2009})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Akimov}\ \emph {et~al.}(2007)\citenamefont {Akimov},
\citenamefont {Mukherjee}, \citenamefont {Yu}, \citenamefont {Chang},
\citenamefont {Zibrov}, \citenamefont {Hemmer}, \citenamefont {Park},\ and\
\citenamefont {Lukin}}]{akimov_generation_2007}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Alexey}\ \bibnamefont
{Akimov}}, \bibinfo {author} {\bibfnamefont {A}~\bibnamefont {Mukherjee}},
\bibinfo {author} {\bibfnamefont {C}~\bibnamefont {Yu}}, \bibinfo {author}
{\bibfnamefont {D}~\bibnamefont {Chang}}, \bibinfo {author} {\bibfnamefont
{A.}~\bibnamefont {Zibrov}}, \bibinfo {author} {\bibfnamefont
{P}~\bibnamefont {Hemmer}}, \bibinfo {author} {\bibfnamefont {H}~\bibnamefont
{Park}}, \ and\ \bibinfo {author} {\bibfnamefont {M}~\bibnamefont {Lukin}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Generation of single optical
plasmons in metallic nanowires coupled to quantum dots},}\ }\href {\doibase
10.1038/nature06230} {\bibfield {journal} {\bibinfo {journal} {Nature}\
}\textbf {\bibinfo {volume} {450}},\ \bibinfo {pages} {402--6} (\bibinfo
{year} {2007})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Huck}\ and\ \citenamefont
{Andersen}(2016)}]{alexander_coupling_2016}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Alexander}\
\bibnamefont {Huck}}\ and\ \bibinfo {author} {\bibfnamefont {Ulrik~L.}\
\bibnamefont {Andersen}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Coupling single emitters to quantum plasmonic circuits},}\ }\href {\doibase
doi:10.1515/nanoph-2015-0153} {\bibfield {journal} {\bibinfo {journal}
{Nanophotonics}\ }\textbf {\bibinfo {volume} {5}},\ \bibinfo {pages}
{483--495} (\bibinfo {year} {2016})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Arcari}\ \emph {et~al.}(2014)\citenamefont {Arcari},
\citenamefont {S\"ollner}, \citenamefont {Javadi}, \citenamefont
{Lindskov~Hansen}, \citenamefont {Mahmoodian}, \citenamefont {Liu},
\citenamefont {Thyrrestrup}, \citenamefont {Lee}, \citenamefont {Song},
\citenamefont {Stobbe},\ and\ \citenamefont
{Lodahl}}]{arcari_near-unity_2014}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Arcari}}, \bibinfo {author} {\bibfnamefont {I.}~\bibnamefont {S\"ollner}},
\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Javadi}}, \bibinfo
{author} {\bibfnamefont {S.}~\bibnamefont {Lindskov~Hansen}}, \bibinfo
{author} {\bibfnamefont {S.}~\bibnamefont {Mahmoodian}}, \bibinfo {author}
{\bibfnamefont {J.}~\bibnamefont {Liu}}, \bibinfo {author} {\bibfnamefont
{H.}~\bibnamefont {Thyrrestrup}}, \bibinfo {author} {\bibfnamefont {E.~H.}\
\bibnamefont {Lee}}, \bibinfo {author} {\bibfnamefont {J.~D.}\ \bibnamefont
{Song}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Stobbe}}, \ and\
\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Lodahl}},\ }\bibfield
{title} {\enquote {\bibinfo {title} {Near-unity coupling efficiency of a
quantum emitter to a photonic crystal waveguide},}\ }\href {\doibase
10.1103/PhysRevLett.113.093603} {\bibfield {journal} {\bibinfo {journal}
{Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {113}},\ \bibinfo {pages}
{093603} (\bibinfo {year} {2014})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Astafiev}\ \emph
{et~al.}(2010{\natexlab{a}})\citenamefont {Astafiev}, \citenamefont
{Zagoskin}, \citenamefont {Abdumalikov}, \citenamefont {Pashkin},
\citenamefont {Yamamoto}, \citenamefont {Inomata}, \citenamefont {Nakamura},\
and\ \citenamefont {Tsai}}]{astafiev_resonance_2010}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {O.}~\bibnamefont
{Astafiev}}, \bibinfo {author} {\bibfnamefont {A.~M.}\ \bibnamefont
{Zagoskin}}, \bibinfo {author} {\bibfnamefont {A.~A.}\ \bibnamefont
{Abdumalikov}}, \bibinfo {author} {\bibfnamefont {Yu.~A.}\ \bibnamefont
{Pashkin}}, \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Yamamoto}},
\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Inomata}}, \bibinfo
{author} {\bibfnamefont {Y.}~\bibnamefont {Nakamura}}, \ and\ \bibinfo
{author} {\bibfnamefont {J.~S.}\ \bibnamefont {Tsai}},\ }\bibfield {title}
{\enquote {\bibinfo {title} {Resonance fluorescence of a single artificial
atom},}\ }\href {\doibase 10.1126/science.1181918} {\bibfield {journal}
{\bibinfo {journal} {Science}\ }\textbf {\bibinfo {volume} {327}},\ \bibinfo
{pages} {840--843} (\bibinfo {year} {2010}{\natexlab{a}})},\ \Eprint
{http://arxiv.org/abs/https://www.science.org/doi/pdf/10.1126/science.1181918}
{https://www.science.org/doi/pdf/10.1126/science.1181918} \BibitemShut
{NoStop}
\bibitem [{\citenamefont {Astafiev}\ \emph
{et~al.}(2010{\natexlab{b}})\citenamefont {Astafiev}, \citenamefont
{Abdumalikov}, \citenamefont {Zagoskin}, \citenamefont {Pashkin},
\citenamefont {Nakamura},\ and\ \citenamefont
{Tsai}}]{astafiev_ultimate_2010}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {O.~V.}\ \bibnamefont
{Astafiev}}, \bibinfo {author} {\bibfnamefont {A.~A.}\ \bibnamefont
{Abdumalikov}}, \bibinfo {author} {\bibfnamefont {A.~M.}\ \bibnamefont
{Zagoskin}}, \bibinfo {author} {\bibfnamefont {Yu.~A.}\ \bibnamefont
{Pashkin}}, \bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont {Nakamura}}, \
and\ \bibinfo {author} {\bibfnamefont {J.~S.}\ \bibnamefont {Tsai}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Ultimate on-chip quantum
amplifier},}\ }\href {\doibase 10.1103/PhysRevLett.104.183603} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
{volume} {104}},\ \bibinfo {pages} {183603} (\bibinfo {year}
{2010}{\natexlab{b}})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Abdumalikov}\ \emph {et~al.}(2010)\citenamefont
{Abdumalikov}, \citenamefont {Astafiev}, \citenamefont {Zagoskin},
\citenamefont {Pashkin}, \citenamefont {Nakamura},\ and\ \citenamefont
{Tsai}}]{abdumalikov_electromagnetically_2010}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~A.}\ \bibnamefont
{Abdumalikov}}, \bibinfo {author} {\bibfnamefont {O.}~\bibnamefont
{Astafiev}}, \bibinfo {author} {\bibfnamefont {A.~M.}\ \bibnamefont
{Zagoskin}}, \bibinfo {author} {\bibfnamefont {Yu.~A.}\ \bibnamefont
{Pashkin}}, \bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont {Nakamura}}, \
and\ \bibinfo {author} {\bibfnamefont {J.~S.}\ \bibnamefont {Tsai}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Electromagnetically induced
transparency on a single artificial atom},}\ }\href {\doibase
10.1103/PhysRevLett.104.193601} {\bibfield {journal} {\bibinfo {journal}
{Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {104}},\ \bibinfo {pages}
{193601} (\bibinfo {year} {2010})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Chang}\ \emph {et~al.}(2018)\citenamefont {Chang},
\citenamefont {Douglas}, \citenamefont {Gonz\'alez-Tudela}, \citenamefont
{Hung},\ and\ \citenamefont {Kimble}}]{chang_colloqium_2018}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~E.}\ \bibnamefont
{Chang}}, \bibinfo {author} {\bibfnamefont {J.~S.}\ \bibnamefont {Douglas}},
\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Gonz\'alez-Tudela}},
\bibinfo {author} {\bibfnamefont {C.-L.}\ \bibnamefont {Hung}}, \ and\
\bibinfo {author} {\bibfnamefont {H.~J.}\ \bibnamefont {Kimble}},\ }\bibfield
{title} {\enquote {\bibinfo {title} {Colloquium: Quantum matter built from
nanoscopic lattices of atoms and photons},}\ }\href {\doibase
10.1103/RevModPhys.90.031002} {\bibfield {journal} {\bibinfo {journal}
{Rev. Mod. Phys.}\ }\textbf {\bibinfo {volume} {90}},\ \bibinfo {pages}
{031002} (\bibinfo {year} {2018})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Gu}\ \emph {et~al.}(2017)\citenamefont {Gu},
\citenamefont {Kockum}, \citenamefont {Miranowicz}, \citenamefont {xi~Liu},\
and\ \citenamefont {Nori}}]{gu_microwave_2017}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Xiu}\ \bibnamefont
{Gu}}, \bibinfo {author} {\bibfnamefont {Anton~Frisk}\ \bibnamefont
{Kockum}}, \bibinfo {author} {\bibfnamefont {Adam}\ \bibnamefont
{Miranowicz}}, \bibinfo {author} {\bibfnamefont {Yu}~\bibnamefont {xi~Liu}},
\ and\ \bibinfo {author} {\bibfnamefont {Franco}\ \bibnamefont {Nori}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Microwave photonics with
superconducting quantum circuits},}\ }\href {\doibase
https://doi.org/10.1016/j.physrep.2017.10.002} {\bibfield {journal}
{\bibinfo {journal} {Physics Reports}\ }\textbf {\bibinfo {volume}
{718-719}},\ \bibinfo {pages} {1--102} (\bibinfo {year} {2017})},\ \bibinfo
{note} {microwave photonics with superconducting quantum
circuits}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Sheremet}\ \emph {et~al.}(2021)\citenamefont
{Sheremet}, \citenamefont {Petrov}, \citenamefont {Iorsh}, \citenamefont
{Poshakinskiy},\ and\ \citenamefont {Poddubny}}]{sheremet_waveguide_2021}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Alexandra~S.}\
\bibnamefont {Sheremet}}, \bibinfo {author} {\bibfnamefont {Mihail~I.}\
\bibnamefont {Petrov}}, \bibinfo {author} {\bibfnamefont {Ivan~V.}\
\bibnamefont {Iorsh}}, \bibinfo {author} {\bibfnamefont {Alexander~V.}\
\bibnamefont {Poshakinskiy}}, \ and\ \bibinfo {author} {\bibfnamefont
{Alexander~N.}\ \bibnamefont {Poddubny}},\ }\href {\doibase
10.48550/ARXIV.2103.06824} {\enquote {\bibinfo {title} {Waveguide quantum
electrodynamics: collective radiance and photon-photon correlations},}\ }
(\bibinfo {year} {2021})\BibitemShut {NoStop}
\bibitem [{\citenamefont {Frisk~Kockum}\ \emph {et~al.}(2014)\citenamefont
{Frisk~Kockum}, \citenamefont {Delsing},\ and\ \citenamefont
{Johansson}}]{kockum_designing_2014}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Anton}\ \bibnamefont
{Frisk~Kockum}}, \bibinfo {author} {\bibfnamefont {Per}\ \bibnamefont
{Delsing}}, \ and\ \bibinfo {author} {\bibfnamefont {G\"oran}\ \bibnamefont
{Johansson}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Designing
frequency-dependent relaxation rates and lamb shifts for a giant artificial
atom},}\ }\href {\doibase 10.1103/PhysRevA.90.013837} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {90}},\
\bibinfo {pages} {013837} (\bibinfo {year} {2014})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Frisk~Kockum}(2021)}]{kockum_quantum_2021}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Anton}\ \bibnamefont
{Frisk~Kockum}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum
optics with giant atoms---the first five years},}\ }in\ \href@noop {} {\emph
{\bibinfo {booktitle} {International Symposium on Mathematics, Quantum
Theory, and Cryptography}}},\ \bibinfo {editor} {edited by\ \bibinfo {editor}
{\bibfnamefont {Tsuyoshi}\ \bibnamefont {Takagi}}, \bibinfo {editor}
{\bibfnamefont {Masato}\ \bibnamefont {Wakayama}}, \bibinfo {editor}
{\bibfnamefont {Keisuke}\ \bibnamefont {Tanaka}}, \bibinfo {editor}
{\bibfnamefont {Noboru}\ \bibnamefont {Kunihiro}}, \bibinfo {editor}
{\bibfnamefont {Kazufumi}\ \bibnamefont {Kimoto}}, \ and\ \bibinfo {editor}
{\bibfnamefont {Yasuhiko}\ \bibnamefont {Ikematsu}}}\ (\bibinfo {publisher}
{Springer Singapore},\ \bibinfo {address} {Singapore},\ \bibinfo {year}
{2021})\ pp.\ \bibinfo {pages} {125--146}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Kockum}\ \emph {et~al.}(2018)\citenamefont {Kockum},
\citenamefont {Johansson},\ and\ \citenamefont
{Nori}}]{kockum_decoherence-free_2018}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Anton~Frisk}\
\bibnamefont {Kockum}}, \bibinfo {author} {\bibfnamefont {G\"oran}\
\bibnamefont {Johansson}}, \ and\ \bibinfo {author} {\bibfnamefont {Franco}\
\bibnamefont {Nori}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Decoherence-free interaction between giant atoms in waveguide quantum
electrodynamics},}\ }\href {\doibase 10.1103/PhysRevLett.120.140404}
{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf
{\bibinfo {volume} {120}},\ \bibinfo {pages} {140404} (\bibinfo {year}
{2018})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Kannan}\ \emph {et~al.}(2020)\citenamefont {Kannan},
\citenamefont {Ruckriegel}, \citenamefont {Campbell}, \citenamefont
{Frisk~Kockum}, \citenamefont {Braumüller}, \citenamefont {Kim},
\citenamefont {Kjaergaard}, \citenamefont {Krantz}, \citenamefont {Melville},
\citenamefont {Niedzielski}, \citenamefont {Vepsäläinen}, \citenamefont
{Winik}, \citenamefont {Yoder}, \citenamefont {Nori}, \citenamefont
{Orlando}, \citenamefont {Gustavsson},\ and\ \citenamefont
{Oliver}}]{kannan_waveguide_2020}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Bharath}\ \bibnamefont
{Kannan}}, \bibinfo {author} {\bibfnamefont {Max}\ \bibnamefont
{Ruckriegel}}, \bibinfo {author} {\bibfnamefont {Daniel}\ \bibnamefont
{Campbell}}, \bibinfo {author} {\bibfnamefont {Anton}\ \bibnamefont
{Frisk~Kockum}}, \bibinfo {author} {\bibfnamefont {Jochen}\ \bibnamefont
{Braumüller}}, \bibinfo {author} {\bibfnamefont {David}\ \bibnamefont
{Kim}}, \bibinfo {author} {\bibfnamefont {Morten}\ \bibnamefont
{Kjaergaard}}, \bibinfo {author} {\bibfnamefont {Philip}\ \bibnamefont
{Krantz}}, \bibinfo {author} {\bibfnamefont {Alexander}\ \bibnamefont
{Melville}}, \bibinfo {author} {\bibfnamefont {Bethany}\ \bibnamefont
{Niedzielski}}, \bibinfo {author} {\bibfnamefont {Antti}\ \bibnamefont
{Vepsäläinen}}, \bibinfo {author} {\bibfnamefont {Roni}\ \bibnamefont
{Winik}}, \bibinfo {author} {\bibfnamefont {Jonilyn}\ \bibnamefont {Yoder}},
\bibinfo {author} {\bibfnamefont {Franco}\ \bibnamefont {Nori}}, \bibinfo
{author} {\bibfnamefont {Terry}\ \bibnamefont {Orlando}}, \bibinfo {author}
{\bibfnamefont {Simon}\ \bibnamefont {Gustavsson}}, \ and\ \bibinfo {author}
{\bibfnamefont {William}\ \bibnamefont {Oliver}},\ }\bibfield {title}
{\enquote {\bibinfo {title} {Waveguide quantum electrodynamics with
superconducting artificial giant atoms},}\ }\href {\doibase
10.1038/s41586-020-2529-9} {\bibfield {journal} {\bibinfo {journal}
{Nature}\ }\textbf {\bibinfo {volume} {583}},\ \bibinfo {pages} {775--779}
(\bibinfo {year} {2020})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Guo}\ \emph {et~al.}(2017)\citenamefont {Guo},
\citenamefont {Grimsmo}, \citenamefont {Kockum}, \citenamefont {Pletyukhov},\
and\ \citenamefont {Johansson}}]{guo_giant_2017}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Lingzhen}\
\bibnamefont {Guo}}, \bibinfo {author} {\bibfnamefont {Arne}\ \bibnamefont
{Grimsmo}}, \bibinfo {author} {\bibfnamefont {Anton~Frisk}\ \bibnamefont
{Kockum}}, \bibinfo {author} {\bibfnamefont {Mikhail}\ \bibnamefont
{Pletyukhov}}, \ and\ \bibinfo {author} {\bibfnamefont {G\"oran}\
\bibnamefont {Johansson}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Giant acoustic atom: A single quantum system with a deterministic time
delay},}\ }\href {\doibase 10.1103/PhysRevA.95.053821} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {95}},\
\bibinfo {pages} {053821} (\bibinfo {year} {2017})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Andersson}\ \emph {et~al.}(2019)\citenamefont
{Andersson}, \citenamefont {Suri}, \citenamefont {Guo}, \citenamefont
{Aref},\ and\ \citenamefont {Delsing}}]{andersson_non-exponential_2019}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Gustav}\ \bibnamefont
{Andersson}}, \bibinfo {author} {\bibfnamefont {Baladitya}\ \bibnamefont
{Suri}}, \bibinfo {author} {\bibfnamefont {Lingzhen}\ \bibnamefont {Guo}},
\bibinfo {author} {\bibfnamefont {Thomas}\ \bibnamefont {Aref}}, \ and\
\bibinfo {author} {\bibfnamefont {Per}\ \bibnamefont {Delsing}},\ }\bibfield
{title} {\enquote {\bibinfo {title} {Non-exponential decay of a giant
artificial atom},}\ }\href {\doibase 10.1038/s41567-019-0605-6} {\bibfield
{journal} {\bibinfo {journal} {Nature Physics}\ }\textbf {\bibinfo {volume}
{15}} (\bibinfo {year} {2019}),\ 10.1038/s41567-019-0605-6}\BibitemShut
{NoStop}
\bibitem [{\citenamefont {Guo}\ \emph {et~al.}(2020)\citenamefont {Guo},
\citenamefont {Kockum}, \citenamefont {Marquardt},\ and\ \citenamefont
{Johansson}}]{guo_oscillating_2020}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Lingzhen}\
\bibnamefont {Guo}}, \bibinfo {author} {\bibfnamefont {Anton~Frisk}\
\bibnamefont {Kockum}}, \bibinfo {author} {\bibfnamefont {Florian}\
\bibnamefont {Marquardt}}, \ and\ \bibinfo {author} {\bibfnamefont {G\"oran}\
\bibnamefont {Johansson}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Oscillating bound states for a giant atom},}\ }\href {\doibase
10.1103/PhysRevResearch.2.043014} {\bibfield {journal} {\bibinfo {journal}
{Phys. Rev. Research}\ }\textbf {\bibinfo {volume} {2}},\ \bibinfo {pages}
{043014} (\bibinfo {year} {2020})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Jones}(1965)}]{jones_coupling_1965}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Alan~L.}\ \bibnamefont
{Jones}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Coupling of
optical fibers and scattering in fibers$\ast$},}\ }\href {\doibase
10.1364/JOSA.55.000261} {\bibfield {journal} {\bibinfo {journal} {J. Opt.
Soc. Am.}\ }\textbf {\bibinfo {volume} {55}},\ \bibinfo {pages} {261--271}
(\bibinfo {year} {1965})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Somekh}\ \emph {et~al.}(1973)\citenamefont {Somekh},
\citenamefont {Garmire}, \citenamefont {Yariv}, \citenamefont {Garvin},\ and\
\citenamefont {Hunsperger}}]{somekh_channel_1973}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
{Somekh}}, \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Garmire}},
\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Yariv}}, \bibinfo
{author} {\bibfnamefont {H.L.}\ \bibnamefont {Garvin}}, \ and\ \bibinfo
{author} {\bibfnamefont {R.G.}\ \bibnamefont {Hunsperger}},\ }\bibfield
{title} {\enquote {\bibinfo {title} {Channel optical waveguide directional
couplers},}\ }\href {\doibase 10.1063/1.1654468} {\bibfield {journal}
{\bibinfo {journal} {Applied Physics Letters}\ }\textbf {\bibinfo {volume}
{22}},\ \bibinfo {pages} {46--47} (\bibinfo {year} {1973})},\ \Eprint
{http://arxiv.org/abs/https://doi.org/10.1063/1.1654468}
{https://doi.org/10.1063/1.1654468} \BibitemShut {NoStop}
\bibitem [{\citenamefont {Longhi}(2020)}]{longhi_photonic_2020}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Stefano}\ \bibnamefont
{Longhi}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Photonic
simulation of giant atom decay},}\ }\href {\doibase 10.1364/OL.393578}
{\bibfield {journal} {\bibinfo {journal} {Optics Letters}\ }\textbf
{\bibinfo {volume} {45}},\ \bibinfo {pages} {3017} (\bibinfo {year}
{2020})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Longhi}(2021)}]{longhi_rabi_2021}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Stefano}\ \bibnamefont
{Longhi}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Rabi
oscillations of bound states in the continuum},}\ }\href@noop {} {\bibfield
{journal} {\bibinfo {journal} {Optics Letters}\ }\textbf {\bibinfo {volume}
{46}},\ \bibinfo {pages} {2091--2094} (\bibinfo {year} {2021})}\BibitemShut
{NoStop}
\bibitem [{\citenamefont {Ramos}\ \emph {et~al.}(2016)\citenamefont {Ramos},
\citenamefont {Vermersch}, \citenamefont {Hauke}, \citenamefont {Pichler},\
and\ \citenamefont {Zoller}}]{ramos_nonMarkovian_2016}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Tom\'as}\ \bibnamefont
{Ramos}}, \bibinfo {author} {\bibfnamefont {Beno\^{\i}t}\ \bibnamefont
{Vermersch}}, \bibinfo {author} {\bibfnamefont {Philipp}\ \bibnamefont
{Hauke}}, \bibinfo {author} {\bibfnamefont {Hannes}\ \bibnamefont {Pichler}},
\ and\ \bibinfo {author} {\bibfnamefont {Peter}\ \bibnamefont {Zoller}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Non-markovian dynamics in
chiral quantum networks with spins and photons},}\ }\href {\doibase
10.1103/PhysRevA.93.062104} {\bibfield {journal} {\bibinfo {journal} {Phys.
Rev. A}\ }\textbf {\bibinfo {volume} {93}},\ \bibinfo {pages} {062104}
(\bibinfo {year} {2016})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Stillinger}\ and\ \citenamefont
{Herrick}(1975)}]{stillinger_bound_1975}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Frank~H}\ \bibnamefont
{Stillinger}}\ and\ \bibinfo {author} {\bibfnamefont {David~R}\ \bibnamefont
{Herrick}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Bound states in
the continuum},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
{Physical Review A}\ }\textbf {\bibinfo {volume} {11}},\ \bibinfo {pages}
{446} (\bibinfo {year} {1975})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Plotnik}\ \emph {et~al.}(2011)\citenamefont
{Plotnik}, \citenamefont {Peleg}, \citenamefont {Dreisow}, \citenamefont
{Heinrich}, \citenamefont {Nolte}, \citenamefont {Szameit},\ and\
\citenamefont {Segev}}]{plotnik_experimental_2011}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Yonatan}\ \bibnamefont
{Plotnik}}, \bibinfo {author} {\bibfnamefont {Or}~\bibnamefont {Peleg}},
\bibinfo {author} {\bibfnamefont {Felix}\ \bibnamefont {Dreisow}}, \bibinfo
{author} {\bibfnamefont {Matthias}\ \bibnamefont {Heinrich}}, \bibinfo
{author} {\bibfnamefont {Stefan}\ \bibnamefont {Nolte}}, \bibinfo {author}
{\bibfnamefont {Alexander}\ \bibnamefont {Szameit}}, \ and\ \bibinfo {author}
{\bibfnamefont {Mordechai}\ \bibnamefont {Segev}},\ }\bibfield {title}
{\enquote {\bibinfo {title} {Experimental observation of optical bound states
in the continuum},}\ }\href@noop {} {\bibfield {journal} {\bibinfo
{journal} {Physical review letters}\ }\textbf {\bibinfo {volume} {107}},\
\bibinfo {pages} {183901} (\bibinfo {year} {2011})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Hsu}\ \emph {et~al.}(2016)\citenamefont {Hsu},
\citenamefont {Zhen}, \citenamefont {Stone}, \citenamefont {Joannopoulos},\
and\ \citenamefont {Solja{\v{c}}i{\'c}}}]{hsu_bound_2016}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Chia~Wei}\
\bibnamefont {Hsu}}, \bibinfo {author} {\bibfnamefont {Bo}~\bibnamefont
{Zhen}}, \bibinfo {author} {\bibfnamefont {A~Douglas}\ \bibnamefont {Stone}},
\bibinfo {author} {\bibfnamefont {John~D}\ \bibnamefont {Joannopoulos}}, \
and\ \bibinfo {author} {\bibfnamefont {Marin}\ \bibnamefont
{Solja{\v{c}}i{\'c}}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Bound states in the continuum},}\ }\href@noop {} {\bibfield {journal}
{\bibinfo {journal} {Nature Reviews Materials}\ }\textbf {\bibinfo {volume}
{1}},\ \bibinfo {pages} {1--13} (\bibinfo {year} {2016})}\BibitemShut
{NoStop}
\bibitem [{\citenamefont {Longhi}(2007)}]{longhi_bound_2007}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
{Longhi}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Bound states in
the continuum in a single-level {Fano}-{Anderson} model},}\ }\href {\doibase
10.1140/epjb/e2007-00143-2} {\bibfield {journal} {\bibinfo {journal} {The
European Physical Journal B}\ }\textbf {\bibinfo {volume} {57}},\ \bibinfo
{pages} {45--51} (\bibinfo {year} {2007})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Johnsson}\ and\ \citenamefont
{M{\o}lmer}(2004)}]{johnsson_storing_2004}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Mattias}\ \bibnamefont
{Johnsson}}\ and\ \bibinfo {author} {\bibfnamefont {Klaus}\ \bibnamefont
{M{\o}lmer}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Storing
quantum information in a solid using dark-state polaritons},}\ }\href@noop {}
{\bibfield {journal} {\bibinfo {journal} {Physical Review A}\ }\textbf
{\bibinfo {volume} {70}},\ \bibinfo {pages} {032320} (\bibinfo {year}
{2004})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Yang}\ \emph {et~al.}(2004)\citenamefont {Yang},
\citenamefont {Chu},\ and\ \citenamefont {Han}}]{yang_quantum_2004}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Chui-Ping}\
\bibnamefont {Yang}}, \bibinfo {author} {\bibfnamefont {Shih-I}\ \bibnamefont
{Chu}}, \ and\ \bibinfo {author} {\bibfnamefont {Siyuan}\ \bibnamefont
{Han}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum information
transfer and entanglement with squid qubits in cavity qed: A dark-state
scheme with tolerance for nonuniform device parameter},}\ }\href {\doibase
10.1103/PhysRevLett.92.117902} {\bibfield {journal} {\bibinfo {journal}
{Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {92}},\ \bibinfo {pages}
{117902} (\bibinfo {year} {2004})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Munro}\ \emph {et~al.}(2017)\citenamefont {Munro},
\citenamefont {Kwek},\ and\ \citenamefont {Chang}}]{munro_optical_2017}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Ewan}\ \bibnamefont
{Munro}}, \bibinfo {author} {\bibfnamefont {Leong~Chuan}\ \bibnamefont
{Kwek}}, \ and\ \bibinfo {author} {\bibfnamefont {Darrick~E}\ \bibnamefont
{Chang}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Optical
properties of an atomic ensemble coupled to a band edge of a photonic crystal
waveguide},}\ }\href {\doibase 10.1088/1367-2630/aa7fad} {\bibfield
{journal} {\bibinfo {journal} {New Journal of Physics}\ }\textbf {\bibinfo
{volume} {19}},\ \bibinfo {pages} {083018} (\bibinfo {year}
{2017})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Hartmann}\ \emph {et~al.}(2008)\citenamefont
{Hartmann}, \citenamefont {Brandao},\ and\ \citenamefont
{Plenio}}]{hartmann_quantum_2008}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Michael~J}\
\bibnamefont {Hartmann}}, \bibinfo {author} {\bibfnamefont {Fernando~GSL}\
\bibnamefont {Brandao}}, \ and\ \bibinfo {author} {\bibfnamefont {Martin~B}\
\bibnamefont {Plenio}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Quantum many-body phenomena in coupled cavity arrays},}\ }\href@noop {}
{\bibfield {journal} {\bibinfo {journal} {Laser \& Photonics Reviews}\
}\textbf {\bibinfo {volume} {2}},\ \bibinfo {pages} {527--556} (\bibinfo
{year} {2008})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Majumdar}\ \emph {et~al.}(2012)\citenamefont
{Majumdar}, \citenamefont {Rundquist}, \citenamefont {Bajcsy}, \citenamefont
{Dasika}, \citenamefont {Bank},\ and\ \citenamefont
{Vu{\v{c}}kovi{\'c}}}]{majumdar_design_2012}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Arka}\ \bibnamefont
{Majumdar}}, \bibinfo {author} {\bibfnamefont {Armand}\ \bibnamefont
{Rundquist}}, \bibinfo {author} {\bibfnamefont {Michal}\ \bibnamefont
{Bajcsy}}, \bibinfo {author} {\bibfnamefont {Vaishno~D}\ \bibnamefont
{Dasika}}, \bibinfo {author} {\bibfnamefont {Seth~R}\ \bibnamefont {Bank}}, \
and\ \bibinfo {author} {\bibfnamefont {Jelena}\ \bibnamefont
{Vu{\v{c}}kovi{\'c}}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Design and analysis of photonic crystal coupled cavity arrays for quantum
simulation},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
{Physical Review B}\ }\textbf {\bibinfo {volume} {86}},\ \bibinfo {pages}
{195312} (\bibinfo {year} {2012})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Christodoulides}\ \emph {et~al.}(2003)\citenamefont
{Christodoulides}, \citenamefont {Lederer},\ and\ \citenamefont
{Silberberg}}]{christodoulides_discretizing_2003}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Demetrios~N}\
\bibnamefont {Christodoulides}}, \bibinfo {author} {\bibfnamefont {Falk}\
\bibnamefont {Lederer}}, \ and\ \bibinfo {author} {\bibfnamefont {Yaron}\
\bibnamefont {Silberberg}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Discretizing light behaviour in linear and nonlinear waveguide lattices},}\
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Nature}\ }\textbf
{\bibinfo {volume} {424}},\ \bibinfo {pages} {817--823} (\bibinfo {year}
{2003})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Longhi}(2009)}]{longhi_quantum_2009}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
{Longhi}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum-optical
analogies using photonic structures},}\ }\href {\doibase
https://doi.org/10.1002/lpor.200810055} {\bibfield {journal} {\bibinfo
{journal} {Laser \& Photonics Reviews}\ }\textbf {\bibinfo {volume} {3}},\
\bibinfo {pages} {243--261} (\bibinfo {year} {2009})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Szameit}\ and\ \citenamefont
{Nolte}(2010)}]{szameit_discrete_2010}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Alexander}\
\bibnamefont {Szameit}}\ and\ \bibinfo {author} {\bibfnamefont {Stefan}\
\bibnamefont {Nolte}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Discrete optics in femtosecond-laser-written photonic structures},}\ }\href
{\doibase 10.1088/0953-4075/43/16/163001} {\bibfield {journal} {\bibinfo
{journal} {Journal of Physics B: Atomic, Molecular and Optical Physics}\
}\textbf {\bibinfo {volume} {43}},\ \bibinfo {pages} {163001} (\bibinfo
{year} {2010})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Aspuru-Guzik}\ and\ \citenamefont
{Walther}(2012)}]{aspuru-guzik_photonic_2010}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Al{\'a}n}\
\bibnamefont {Aspuru-Guzik}}\ and\ \bibinfo {author} {\bibfnamefont {Philip}\
\bibnamefont {Walther}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Photonic quantum simulators},}\ }\href@noop {} {\bibfield {journal}
{\bibinfo {journal} {Nature physics}\ }\textbf {\bibinfo {volume} {8}},\
\bibinfo {pages} {285--291} (\bibinfo {year} {2012})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Garanovich}\ \emph {et~al.}(2012)\citenamefont
{Garanovich}, \citenamefont {Longhi}, \citenamefont {Sukhorukov},\ and\
\citenamefont {Kivshar}}]{garanovich_light_2012}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Ivan~L.}\ \bibnamefont
{Garanovich}}, \bibinfo {author} {\bibfnamefont {Stefano}\ \bibnamefont
{Longhi}}, \bibinfo {author} {\bibfnamefont {Andrey~A.}\ \bibnamefont
{Sukhorukov}}, \ and\ \bibinfo {author} {\bibfnamefont {Yuri~S.}\
\bibnamefont {Kivshar}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Light propagation and localization in modulated photonic lattices and
waveguides},}\ }\href {\doibase
https://doi.org/10.1016/j.physrep.2012.03.005} {\bibfield {journal}
{\bibinfo {journal} {Physics Reports}\ }\textbf {\bibinfo {volume} {518}},\
\bibinfo {pages} {1--79} (\bibinfo {year} {2012})},\ \bibinfo {note} {light
propagation and localization in modulated photonic lattices and
waveguides}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Jiao}\ \emph {et~al.}(2021)\citenamefont {Jiao},
\citenamefont {Gao}, \citenamefont {Zhou}, \citenamefont {Wang},
\citenamefont {Ren}, \citenamefont {Xu}, \citenamefont {Qiao}, \citenamefont
{Wang},\ and\ \citenamefont {Jin}}]{jiao_two-dimensional_2021}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Zhi-Qiang}\
\bibnamefont {Jiao}}, \bibinfo {author} {\bibfnamefont {Jun}\ \bibnamefont
{Gao}}, \bibinfo {author} {\bibfnamefont {Wen-Hao}\ \bibnamefont {Zhou}},
\bibinfo {author} {\bibfnamefont {Xiao-Wei}\ \bibnamefont {Wang}}, \bibinfo
{author} {\bibfnamefont {Ruo-Jing}\ \bibnamefont {Ren}}, \bibinfo {author}
{\bibfnamefont {Xiao-Yun}\ \bibnamefont {Xu}}, \bibinfo {author}
{\bibfnamefont {Lu-Feng}\ \bibnamefont {Qiao}}, \bibinfo {author}
{\bibfnamefont {Yao}\ \bibnamefont {Wang}}, \ and\ \bibinfo {author}
{\bibfnamefont {Xian-Min}\ \bibnamefont {Jin}},\ }\bibfield {title}
{\enquote {\bibinfo {title} {Two-dimensional quantum walks of correlated
photons},}\ }\href {\doibase 10.1364/OPTICA.425879} {\bibfield {journal}
{\bibinfo {journal} {Optica}\ }\textbf {\bibinfo {volume} {8}},\ \bibinfo
{pages} {1129} (\bibinfo {year} {2021})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Tentrup}\ \emph {et~al.}(2017)\citenamefont
{Tentrup}, \citenamefont {Hummel}, \citenamefont {Wolterink}, \citenamefont
{Uppu}, \citenamefont {Mosk},\ and\ \citenamefont
{Pinkse}}]{tentrup_transmitting_2017}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Tristan B.~H.}\
\bibnamefont {Tentrup}}, \bibinfo {author} {\bibfnamefont {Thomas}\
\bibnamefont {Hummel}}, \bibinfo {author} {\bibfnamefont {Tom A.~W.}\
\bibnamefont {Wolterink}}, \bibinfo {author} {\bibfnamefont {Ravitej}\
\bibnamefont {Uppu}}, \bibinfo {author} {\bibfnamefont {Allard~P.}\
\bibnamefont {Mosk}}, \ and\ \bibinfo {author} {\bibfnamefont {Pepijn W.~H.}\
\bibnamefont {Pinkse}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Transmitting more than 10 bit with a single photon},}\ }\href {\doibase
10.1364/OE.25.002826} {\bibfield {journal} {\bibinfo {journal} {Opt.
Express}\ }\textbf {\bibinfo {volume} {25}},\ \bibinfo {pages} {2826--2833}
(\bibinfo {year} {2017})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Tang}\ \emph {et~al.}(2022)\citenamefont {Tang},
\citenamefont {Banchi}, \citenamefont {Wang}, \citenamefont {Shang},
\citenamefont {Tan}, \citenamefont {Zhou}, \citenamefont {Feng},
\citenamefont {Pal}, \citenamefont {Li}, \citenamefont {Hu}, \citenamefont
{Kim},\ and\ \citenamefont {Jin}}]{tang_generating_2022}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Hao}\ \bibnamefont
{Tang}}, \bibinfo {author} {\bibfnamefont {Leonardo}\ \bibnamefont {Banchi}},
\bibinfo {author} {\bibfnamefont {Tian-Yu}\ \bibnamefont {Wang}}, \bibinfo
{author} {\bibfnamefont {Xiao-Wen}\ \bibnamefont {Shang}}, \bibinfo {author}
{\bibfnamefont {Xi}~\bibnamefont {Tan}}, \bibinfo {author} {\bibfnamefont
{Wen-Hao}\ \bibnamefont {Zhou}}, \bibinfo {author} {\bibfnamefont {Zhen}\
\bibnamefont {Feng}}, \bibinfo {author} {\bibfnamefont {Anurag}\ \bibnamefont
{Pal}}, \bibinfo {author} {\bibfnamefont {Hang}\ \bibnamefont {Li}}, \bibinfo
{author} {\bibfnamefont {Cheng-Qiu}\ \bibnamefont {Hu}}, \bibinfo {author}
{\bibfnamefont {M.~S.}\ \bibnamefont {Kim}}, \ and\ \bibinfo {author}
{\bibfnamefont {Xian-Min}\ \bibnamefont {Jin}},\ }\bibfield {title}
{\enquote {\bibinfo {title} {Generating haar-uniform randomness using
stochastic quantum walks on a photonic chip},}\ }\href {\doibase
10.1103/PhysRevLett.128.050503} {\bibfield {journal} {\bibinfo {journal}
{Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {128}},\ \bibinfo {pages}
{050503} (\bibinfo {year} {2022})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Crowder}\ \emph {et~al.}(2020)\citenamefont
{Crowder}, \citenamefont {Carmichael},\ and\ \citenamefont
{Hughes}}]{crowder2020quantum}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Gavin}\ \bibnamefont
{Crowder}}, \bibinfo {author} {\bibfnamefont {Howard}\ \bibnamefont
{Carmichael}}, \ and\ \bibinfo {author} {\bibfnamefont {Stephen}\
\bibnamefont {Hughes}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Quantum trajectory theory of few-photon cavity-qed systems with a
time-delayed coherent feedback},}\ }\href {\doibase
10.1103/PhysRevA.101.023807} {\bibfield {journal} {\bibinfo {journal}
{Phys. Rev. A}\ }\textbf {\bibinfo {volume} {101}},\ \bibinfo {pages}
{023807} (\bibinfo {year} {2020})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Arranz~Regidor}\ and\ \citenamefont
{Hughes}(2021)}]{regidor2021cavitylike}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Sofia}\ \bibnamefont
{Arranz~Regidor}}\ and\ \bibinfo {author} {\bibfnamefont {Stephen}\
\bibnamefont {Hughes}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Cavitylike strong coupling in macroscopic waveguide qed using three coupled
qubits in the deep non-markovian regime},}\ }\href {\doibase
10.1103/PhysRevA.104.L031701} {\bibfield {journal} {\bibinfo {journal}
{Phys. Rev. A}\ }\textbf {\bibinfo {volume} {104}},\ \bibinfo {pages}
{L031701} (\bibinfo {year} {2021})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Guimond}\ \emph {et~al.}(2016)\citenamefont
{Guimond}, \citenamefont {Roulet}, \citenamefont {Le},\ and\ \citenamefont
{Scarani}}]{guimond2016rabi}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Pierre-Olivier}\
\bibnamefont {Guimond}}, \bibinfo {author} {\bibfnamefont {Alexandre}\
\bibnamefont {Roulet}}, \bibinfo {author} {\bibfnamefont {Huy~Nguyen}\
\bibnamefont {Le}}, \ and\ \bibinfo {author} {\bibfnamefont {Valerio}\
\bibnamefont {Scarani}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Rabi oscillation in a quantum cavity: Markovian and non-markovian
dynamics},}\ }\href {\doibase 10.1103/PhysRevA.93.023808} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume}
{93}},\ \bibinfo {pages} {023808} (\bibinfo {year} {2016})}\BibitemShut
{NoStop}
\bibitem [{\citenamefont {\ifmmode~\check{C}\else \v{C}\fi{}ernot\'{\i}k}\
\emph {et~al.}(2019)\citenamefont {\ifmmode~\check{C}\else
\v{C}\fi{}ernot\'{\i}k}, \citenamefont {Dantan},\ and\ \citenamefont
{Genes}}]{cernotik2019cavity}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Ond\ifmmode
\check{r}\else~\v{r}\fi{}ej}\ \bibnamefont {\ifmmode~\check{C}\else
\v{C}\fi{}ernot\'{\i}k}}, \bibinfo {author} {\bibfnamefont {Aur\'elien}\
\bibnamefont {Dantan}}, \ and\ \bibinfo {author} {\bibfnamefont {Claudiu}\
\bibnamefont {Genes}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Cavity quantum electrodynamics with frequency-dependent reflectors},}\
}\href {\doibase 10.1103/PhysRevLett.122.243601} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {122}},\
\bibinfo {pages} {243601} (\bibinfo {year} {2019})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Du}\ \emph {et~al.}(2021)\citenamefont {Du},
\citenamefont {Cai}, \citenamefont {Wu}, \citenamefont {Wang},\ and\
\citenamefont {Li}}]{du2021single}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Lei}\ \bibnamefont
{Du}}, \bibinfo {author} {\bibfnamefont {Mao-Rui}\ \bibnamefont {Cai}},
\bibinfo {author} {\bibfnamefont {Jin-Hui}\ \bibnamefont {Wu}}, \bibinfo
{author} {\bibfnamefont {Zhihai}\ \bibnamefont {Wang}}, \ and\ \bibinfo
{author} {\bibfnamefont {Yong}\ \bibnamefont {Li}},\ }\bibfield {title}
{\enquote {\bibinfo {title} {Single-photon nonreciprocal excitation transfer
with non-markovian retarded effects},}\ }\href {\doibase
10.1103/PhysRevA.103.053701} {\bibfield {journal} {\bibinfo {journal}
{Phys. Rev. A}\ }\textbf {\bibinfo {volume} {103}},\ \bibinfo {pages}
{053701} (\bibinfo {year} {2021})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Gonz\'alez-Tudela}\ \emph {et~al.}(2019)\citenamefont
{Gonz\'alez-Tudela}, \citenamefont {Mu\~noz},\ and\ \citenamefont
{Cirac}}]{gonzalez2019engineering}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Gonz\'alez-Tudela}}, \bibinfo {author} {\bibfnamefont {C.~S\'anchez}\
\bibnamefont {Mu\~noz}}, \ and\ \bibinfo {author} {\bibfnamefont {J.~I.}\
\bibnamefont {Cirac}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Engineering and harnessing giant atoms in high-dimensional baths: A proposal
for implementation with cold atoms},}\ }\href {\doibase
10.1103/PhysRevLett.122.203603} {\bibfield {journal} {\bibinfo {journal}
{Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {122}},\ \bibinfo {pages}
{203603} (\bibinfo {year} {2019})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Du}\ \emph {et~al.}(2022)\citenamefont {Du},
\citenamefont {Zhang}, \citenamefont {Wu}, \citenamefont {Kockum},\ and\
\citenamefont {Li}}]{du2022giant}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Lei}\ \bibnamefont
{Du}}, \bibinfo {author} {\bibfnamefont {Yan}\ \bibnamefont {Zhang}},
\bibinfo {author} {\bibfnamefont {Jin-Hui}\ \bibnamefont {Wu}}, \bibinfo
{author} {\bibfnamefont {Anton~Frisk}\ \bibnamefont {Kockum}}, \ and\
\bibinfo {author} {\bibfnamefont {Yong}\ \bibnamefont {Li}},\ }\bibfield
{title} {\enquote {\bibinfo {title} {Giant atoms in a synthetic frequency
dimension},}\ }\href {\doibase 10.1103/PhysRevLett.128.223602} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
{volume} {128}},\ \bibinfo {pages} {223602} (\bibinfo {year}
{2022})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Xiao}\ \emph {et~al.}(2022)\citenamefont {Xiao},
\citenamefont {Wang}, \citenamefont {Li}, \citenamefont {Chen},\ and\
\citenamefont {Yuan}}]{xiao2022bound}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Han}\ \bibnamefont
{Xiao}}, \bibinfo {author} {\bibfnamefont {Luojia}\ \bibnamefont {Wang}},
\bibinfo {author} {\bibfnamefont {Zheng-Hong}\ \bibnamefont {Li}}, \bibinfo
{author} {\bibfnamefont {Xianfeng}\ \bibnamefont {Chen}}, \ and\ \bibinfo
{author} {\bibfnamefont {Luqi}\ \bibnamefont {Yuan}},\ }\bibfield {title}
{\enquote {\bibinfo {title} {Bound state in a giant atom-modulated resonators
system},}\ }\href {\doibase 10.1038/s41534-022-00591-7} {\bibfield {journal}
{\bibinfo {journal} {npj Quantum Information}\ }\textbf {\bibinfo {volume}
{8}},\ \bibinfo {pages} {80} (\bibinfo {year} {2022})}\BibitemShut {NoStop}
\end{thebibliography}
\onecolumngrid
\appendix
\section{Writing \eqref{eqn: real space hamiltonian} in
\texorpdfstring{$k$}{k}-space}
\label{Appendix: k-space hamiltonian derivation}
From \eqref{eqn: k-space annihilation operators}, we can express
$H_{\text{wg}}$ as
\begin{align}
H_{\text{wg}}= & J\sum_{n}b_{n}^{\dagger}b_{n+1}+\text{H.c}\nonumber \\
= & \frac{J}{2\pi}\iint
dk\,dk^{\prime}\sum_{n}e^{i(k^{\prime}-k)n}e^{ik^{\prime}}c^{\dagger}(k)c(k^{\prime})+\frac{J}{2\pi}\iint
dk\,dk^{\prime}\sum_{n}e^{-i(k^{\prime}-k)n}e^{-ik^{\prime}}c^{\dagger}(k^{\prime})c(k)\nonumber
\\
= & \frac{J}{2\pi}\iint
dk\,dk^{\prime}\sum_{n}e^{i(k^{\prime}-k)n}e^{ik^{\prime}}c^{\dagger}(k)c(k^{\prime})+\frac{J}{2\pi}\iint
dk^{\prime}\,dk\sum_{n}e^{-i(k-k^{\prime})n}e^{-ik}c^{\dagger}(k)c(k^{\prime})\nonumber
\\
= & J\iint
dk\,dk^{\prime}\left(\frac{1}{2\pi}\sum_{n}e^{i(k^{\prime}-k)n}\right)\left(e^{ik^{\prime}}+e^{-ik}\right)c^{\dagger}(k)c(k^{\prime})\nonumber
\\
= & J\iint
dk\,dk^{\prime}\delta(k^{\prime}-k)\left(e^{ik^{\prime}}+e^{-ik}\right)c^{\dagger}(k)c(k^{\prime})\nonumber
\\
= & J\int dk\,\left(e^{ik}+e^{-ik}\right)c^{\dagger}(k)c(k)\nonumber \\
= & \int dk\,2J\cos(k)c^{\dagger}(k)c(k)\nonumber \\
= & \int dk\,\omega(k)c^{\dagger}(k)c(k),
\end{align}
where we have defined $\omega(k)\equiv2J\cos(k)$. Here, we have used the
fact that for $(k^\prime - k) \in [0,2\pi)$,
$\frac{1}{2\pi}\sum_{n}e^{i(k^{\prime}-k)n} = \delta(k^\prime - k)$.
Similarly, we can also express $H_{\text{int}}$ as
\begin{align}
H_{\text{int}} &= \sum_{j=1}^{k} \rho_{j}(a^\dag b_{n_j} + \text{H.c}) \nonumber \\
&= \sum_{j=1}^{k} \rho_{j}\left\{a^\dag
\left(\frac{1}{\sqrt{2\pi}}\int_{-\pi}^\pi dk\, e^{ikn_j}c(k)\right) +
\text{H.c}\right\} \nonumber \\
&= \int_{-\pi}^\pi dk\,\left\{\left(\frac{1}{\sqrt{2\pi}}\sum_{j=1}^k \rho_j
e^{ikn_j}\right)a^\dagger c(k) + \text{H.c} \right\} \nonumber \\
&\equiv \int_{-\pi}^\pi dk\,\left\{G(k)a^\dagger c(k) + \text{H.c} \right\},
\end{align}
where we have defined $G(k) = \left(\frac{1}{\sqrt{2\pi}}\sum_{j=1}^{K} \rho_j
e^{ikn_j}\right)$.
\section{Derivation of conditions for BIC}
\label{Appendix: Omega equation derivation}
First, to derive \eqref{eqn: Omega must be in band} and the condition for the
BIC to have a finite norm , we shall first write our energy eigenstate in the
one-excitation subspace as
\begin{equation}
\ket{E} = \phi_a \ket{1_a} + \int dk \,e(k) \ket{1_k}
\end{equation}
where $\phi_a = \braket{1_a}{E}$ and $e(k) = \braket{1_k}{E}$. Thereafter, we
consider the energy eigenvalue equation $H\ket{E} = \Omega \ket{E}$. To
evaluate $H\ket{E}$, we use the commutation relations $[a,a^\dagger] = 1$ and
$[c(k),c^\dagger(k^\prime)] = \delta(k-k^\prime)$, and finally we arrive at
\begin{equation}
H\ket{E} = \left(\omega_a \phi_a + \int dk\,G(k)e(k)\right)\ket{1_a} + \int
dk\,\left(\omega(k)e(k) + G^*(k)\phi_a\right)\ket{1_k}.
\end{equation}
Comparing the above equation with $\Omega\ket{E} = \Omega (\phi_a \ket{1_a} + \int dk \,e(k) \ket{1_k})$, we arrive at two simultaneous equations
\begin{subequations}
\label{eqn: one excitation subspace energy eigval eqn simultaneous eqn}
\begin{align}
\Omega \phi_a &= \omega_a \phi_a + \int dk\, G(k)e(k) \\
\label{eqn: finite norm condition}
\Omega e(k) &= \omega(k)e(k) + G^*(k)\phi_a
\end{align}
\end{subequations}
which we can then solve to obtain \eqref{eqn: Omega must be in band}. Next,
we note that in the one-excitation subspace, we have
\begin{align}
\braket{E}{E} &= |\phi_a|^2 + \int dk \, |e(k)|^2 \nonumber \\
&= |\phi_a|^2 \left(1 + \int dk\, \frac{|G(k)|^2}{(\Omega-\omega(k))^2}\right)
\nonumber \\
&= |\phi_a|^2 \left(1 + \int d\omega\, \frac{\rho(\omega)|G(k(\omega))|^2}{(\Omega-\omega)^2}\right)
\end{align}
where in the second line, we used \eqref{eqn: finite norm condition} and in
the third line, we used $\rho(\omega) \equiv \frac{\partial k}{\partial
\omega}$. Hence we see that the requirement that $\ket{E}$ has a finite norm
corresponds to $\rho(\omega)|G(k(\omega))|^2$ vanishing at least as fast as
$\sim(\Omega-\omega)^2$ as $\omega \to \Omega$.
Next, to derive \eqref{eqn: detuning in terms of Sigma(s)}, we note that
\begin{align}
\Sigma(s=-i\Omega \pm 0^+) &= \int_{-\pi}^\pi dk \frac{|G(k)|^2}{\Omega -
\omega(k)} \nonumber \\
&= \int_{-2J}^{2J} d\omega \frac{\rho(\omega)|G(k(\omega))|^2}{\Omega -
\omega} \nonumber \\
&= \lim_{\epsilon \to
0}\left(\int_{-2J}^{\Omega-\epsilon}d\omega\,\frac{\rho(\omega)|G(k(\omega))|^2}{\Omega
- \omega} + \int_{\Omega +
\epsilon}^{2J}d\omega\,\frac{\rho(\omega)|G(k(\omega))|^2}{\Omega - \omega}
\right) + \int_C d\omega\,\frac{\rho(\omega)|G(k(\omega))|^2}{\Omega - \omega} \nonumber \\
&= \mathcal{P}\left(\int_{-2J}^{2J} d\omega \frac{\rho(\omega)|G(k(\omega))|^2}{\Omega -
\omega}\right) \mp i \pi \rho(\Omega)|G(k(\Omega))|^2
\end{align}
where in the second last line, we note that the contour $C$ is a
semi-circular contour either in the top half or bottom half of the complex
plane, depending on the sign of $\pm 0^+$, and in the last line we used the
residue theorem for the special case of a semi-circular contour.
$\mathcal{P}()$ here denotes the Cauchy Principal value of the integral
enclosed in the parentheses. Taking the real part of the last line,
\eqref{eqn: detuning in terms of Sigma(s)} follows immediately. We note here
that if $\Omega$ also fulfils \eqref{eqn: coupling to continuum vanishes},
then the second term vanishes, which gives us
\begin{equation}
\Sigma(s=-i\Omega \pm 0^+) = \text{Re}[\Sigma(s=-i\Omega \pm 0^+)].
\end{equation}
\section{Decay dynamics}
\label{Appendix: decay dynamics derivation}
Here, we derive \eqref{eqn: decay dynamics}. By writing down the
Schrodinger equation $H\ket{\psi(t)} = i \frac{\partial}{\partial
t}\ket{\psi(t)}$ in the one-excitation subspace as per \eqref{eqn: one
excitation subspace ansatz}, we arrive at two coupled equations
\begin{subequations}
\begin{align}
\label{eqn: giant atom eqn from schrodinger eqn}
i\frac{\partial \psi_a(t)}{\partial t} &= \omega_a \psi_a(t) + \int_{-\pi}^\pi G(k)\psi(k,t)dk \\
\label{eqn: waveguide chain eqn from schrodinger eqn}
i\frac{\partial \psi(k,t)}{\partial t} &= \omega(k)\psi(k,t) + G^*(k)\psi_a(t).
\end{align}
\end{subequations}
To solve the two above equations for $\psi_a(t)$, we follow the standard
procedure of first integrating the ``bath'' equation, which is \eqref{eqn:
waveguide chain eqn from schrodinger eqn} in our case, to get
\begin{equation}
\psi(k,t) = -i G^*(k) e^{-i\omega(k)t}\int_0^t dt^\prime e^{i\omega(k)t^\prime}\psi_a(t^\prime) + e^{-i\omega(k)t}\psi(k,0).
\end{equation}
Then, we substitute the above equation into \eqref{eqn: giant atom eqn from
schrodinger eqn} to get
\begin{equation}
\frac{\partial \psi_a(t)}{\partial t} = -i\omega_a \psi_a(t) - \int_{-\pi}^\pi dk\,
|G(k)|^2 e^{-i\omega(k)t}\int_0^t
dt^\prime\,e^{i\omega(k)t^\prime}\psi_a(t^\prime)-iA(t)
\end{equation}
where $A(t) \equiv \int_{-\pi}^\pi dk\, G(k) e^{-i\omega(k)t}\psi(k,0)$.
Next, we take the Laplace transform on both sides of the above equation by
defining $\tilde{\psi}_a(s) = \int_0^\infty dt\,e^{-st}\psi_a(t)$. By also
using the fact that $\int_0^\infty dt\, e^{-st}\frac{\partial
\psi_a(t)}{\partial t} = s\tilde{\psi}_a(s) - \psi_a(0)$, and defining
$\tilde{A}(s)$ as the Laplace transfrom of $A(t)$, we arrive at
\begin{equation}
s\tilde{\psi}_a(s) - \psi_a(0) = -i\omega_a \tilde{\psi}_a(s) -i
\tilde{A}(s)-\int_{-\pi}^\pi dk\,|G(k)|^2\int_0^\infty dt\,
e^{-(s+i\omega(k))t} F(t)
\end{equation}
where $F(t) \equiv \int_0^t dt^\prime
e^{i\omega(k)t^\prime}\psi_a(t^\prime)$. Realising that $\frac{d}{dt}F(t) =
e^{i\omega(k)t}\psi_a(t)$, we perform integration by parts on $\int_0^\infty
dt\, e^{-(s+i\omega(k))t} F(t)$ and after some algebra finally arrive at
\begin{equation}
\left(s+i\omega_a + i\Sigma(s)\right)\tilde{\psi}_a(s) = -i\tilde{A}(s) + \psi_a(0)
\end{equation}
where $\Sigma(s)$ as per defined in \eqref{eqn: Sigma(s) equation}. By
inserting into the above equation the initial conditions corresponding to the
case of giant atom decay, $\psi_a(0) = 1$ and $\psi(k,0) = 0$ for all $k$, we
finally arrive at
\begin{equation}
\tilde{\psi}_a(s) = \frac{1}{s + i\omega_a + i\Sigma(s)}
\end{equation}
which we can invert via the Bromwich integral to give us $\psi_a(t)$ in the following manner,
\begin{align*}
\psi_a(t) &= \frac{1}{2\pi i} \int_{\lambda -i\infty}^{\lambda + i \infty}
\frac{e^{st}}{s+i\omega_a + i\Sigma(s)} \nonumber \\
&= \sum_{\text{All residues}}
\frac{e^{st}}{s+i\omega_a + i\Sigma(s)}
\end{align*}
where to go from the first line to the second line, we pick $\lambda$
sufficiently large so that all the poles of the integrand lie on the left of
the line $\lambda + it$, $t\in (-\infty,\infty)$. The second line is
\eqref{eqn: decay dynamics} in the main text. We note that in going from the
first line to the second line, we have ignored the contributions from the
integration over any paths induced by possible branch cuts. This is because
we will mainly use the above equation to study the behavior of BICs, which
are poles on the imaginary axis and hence lead to long-term, non-decaying
behavior of $\psi_a(t)$. The integration over any paths induced by possible
branch cuts leads to transient decay behavior which we are not interested in.
\section{Detailed calculations for section~\ref{sec: oscillating bound states} in the main text}
\label{Appendix: oscillating BIC calculations}
\subsection{Derivation of \eqref{eqn: sigma(s) for M legs oscillating BS}}
First of all, we note the important integral result
\begin{align}
\label{eqn: In(s) integral result}
I_n(s) &= \int_{-\pi}^{\pi} dk\,\frac{e^{ikn}}{is - \cos(k)} \nonumber \\
&= \frac{-2\pi i}{\sqrt{s^2+1}} \left(is \mp i \sqrt{s^2+1}\right)^{|n|}
\end{align}
where we have the minus sign when $\text{Re}(s)>0$ and the positive sign when
$\text{Re}(s)<0$. The result can be derived by making the subtitution
$z=e^{ik}$ and thereafter computing the resultant complex integral
\begin{align}
I_n(s) &= \oint_C dz \,\frac{z^n}{is - (1/2)(z+z^{-1})} \frac{1}{iz} \nonumber \\
&=2i \oint_C dz\, \frac{z^n}{z^2 - 2isz +1} \nonumber \\
&=2i \oint_C dz\, \frac{z^n}{(z-z_1)(z-z_2)}
\end{align}
where the contour $C$ is the unit circle in the complex plane and $z_1 =
i(s+\sqrt{s^2+1})$, $z_2 = i(s-\sqrt{s^2+1})$. The above integral can then
be easily evaluated by the Cauchy Residue Theorem. Now, we can show that
$\text{Re}(s)>0$ implies that $|z_1|>1$, $|z_2|<1$ which means that $z_2$ is
a simple pole in $C$. On the other hand, $\text{Re}(s)<0$ implies that
$|z_1|<1$, $|z_2|>1$ which means that $z_1$ is a simple pole in $C$.
Moreover, if $n <0$ then we also have a $n$th order pole at z=0 in $C$.
Putting all of these together with the fact that $z_1 z_2 = 1$ which means
that $z_1^{-|n|} = z_2^{|n|}$, we can arrive at \eqref{eqn: In(s) integral
result} after some algebra.
Thereafter, using the above result we can easily derive the following
integral, with $n$ being a non-negative integer
\begin{align}
\label{eqn: important cosine integral equation}
\int_{-\pi}^{\pi} dk \frac{\cos(nk)}{is-2J\cos(k)} = \frac{-2\pi i}{\sqrt{s^2+4J^2}} \left(\frac{is \mp i\sqrt{s^2 + 4J^2}}{2J}\right)^n
\end{align}
where again we have the minus sign when $\text{Re}(s)>0$ and the positive
sign when $\text{Re}(s) < 0$. Next, we can derive
\begin{align}
|G(k)|^2 &= \frac{\rho^2}{2\pi} \sum_{j=0}^{M-1}\sum_{l=0}^{M-1}e^{i k n_0
(j-l)} \nonumber \\
&= \frac{\rho^2}{2\pi}\left(M + 2\sum_{r=1}^{M-1}(M-r)\cos(k n_0 r)\right)
\end{align}
where in going from the first line to the next, we realise that there are $M$
terms of $e^{i k (0) n_0}$, $(M-1)$ terms of $e^{ik(1)n_0}$ and its complex
conjugate, $(M-2)$ terms of $e^{ik(2)n_0}$ and its complex conjugate, and so
on, until at last we have $1$ term of $e^{ik(M-1)n_0}$ and its complex
conjugate. With the form of $|G(k)|^2$ above and \eqref{eqn: important
cosine integral equation}, we can get \eqref{eqn: sigma(s) for M legs
oscillating BS} from \eqref{eqn: Sigma(s) equation}.
\subsection{Derivation of oscillating BIC conditions}
Since we are working with $\omega_a = 0$, we want to look for two BICs as
near the band centre as possible with energies $\pm \Omega_{\text{BIC}}$,
where $\Omega_{\text{BIC}}$ is a positive value to be determined. Hence, we
want to minimise the quantity
\begin{equation}
\label{eqn: quantity to be minimised BIC omega condition derivation}
\left|\frac{\Omega}{2J}\right| = \cos\left(\frac{2\pi}{n_0}\left(m \pm
\frac{1}{M}\right)\right)
\end{equation}
over all integer values of $m$. Note that in the above expression, we have
already substituted \eqref{eqn: conditions for k for M legs oscillating BS}
into the dispersion relation $\omega(k) = 2J\cos(k)$. One way to perform
this minimization is to first solve $m$ in the equation $|\Omega/2J|=0$ and
then rounding the value of $m$ to the closest integer. When we solve
$|\Omega/2J|=0$, we obtain two cases
\begin{subequations}
\begin{align}
\frac{2\pi}{n_0}\left(m \pm \frac{1}{m}\right) = \frac{\pi}{2} \implies m
= \frac{n_0}{4} \mp \frac{1}{m} \\
\frac{2\pi}{n_0}\left(m \pm \frac{1}{m}\right) = \frac{3\pi}{2} \implies
m = \frac{3n_0}{4} \mp \frac{1}{m}.
\end{align}
\end{subequations}
At this juncture, before we round the value of $m$ obtained above to the
closest integer, we note that there are two cases we need to consider: either
$n_0 = 2(2l)$ or $n_0 = 2(2l+1)$ where $l \in \mathbb{Z}$. In the former
case where $n_0 = 4l$, the closest integer value of $m$ would be $m = n_0/4$ or $m = 3n_0/4$,
which would give us the BIC frequencies in \eqref{eqn: BIC energies M legs
n0=4l} when we substitute those values of $m$ into \eqref{eqn: quantity to be
minimised BIC omega condition derivation}. Thereafter, we can obtain $\rho_0$ by substituting those frequencies into \eqref{eqn: delta(Omega) for M legs oscillating BS} to obtain \eqref{eqn: BIC rho0 M legs n0=4l}.
In the latter case where $n_0 = 4l+2$, the closest integer values of $m$
would be $m = n_0/4 \pm 1/2$ or $m = 3n_0/4 \pm 1/2$, corresponding to two
possible BIC frequencies,
\begin{equation*}
\Omega_{\text{BIC}} = \sin\left(\frac{\pi}{n_0}\left(1\pm \frac{2}{M}\right)\right).
\end{equation*}
However, when we substitute $\Omega_{\text{BIC}} =\sin((\pi/n_0)(1-2/M))$
into \eqref{eqn: delta(Omega) for M legs oscillating BS}, for the case of
$\omega_a=0$, we end up with $\rho_0^2<0$, which means that we are only left
with $\Omega_{\text{BIC}} =\sin((\pi/n_0)(1+2/M))$ for the $n_0 = 4l+2$ case.
However, since this is further from the band center as compared to the $n_0 =
4l$ case, the resultant bound states would have lower emitter probability. Hence, we can conclude that $n_0 = 4l$ would give optimal oscillating BICs.
As an example, for the case where $M=3$ and $n_0 = 4l+2, n_0>2$, we have the BIC
frequency $|\Omega| = 2 J \sin \left( \frac{5\pi}{3n_0} \right)$ which gives us
\begin{equation}
\rho_0^2 = \frac{2J^2}{\sqrt{3}} \sin \left( \frac{10\pi}{3 n_0} \right)
\end{equation}
Using \eqref{eqn: emitter contribution BIC}, we find that the emitter
probability
\begin{equation}
|\phi_a^{\text{BIC}}|^2 \to \frac{9}{9+20\sqrt{3}\pi} \approx 0.0764
\end{equation}
as $n_0 \to \infty$, which is considerably smaller than the optimal value
$\approx 0.171$ when $n_0$ is an integer multiple of $4$ for $M=3$.
\subsection{Derivation of \eqref{eqn: asymptotic BOC energies} and \eqref{eqn: asymptotic BOC probability}}
To derive \eqref{eqn: asymptotic BOC energies}, the method is to solve
\eqref{eqn: sigma(s) for M legs oscillating BS} using both \eqref{eqn: big
term in sigma(s) expression} and \eqref{eqn: detuning in terms of Sigma(s)}
for the case where $\omega_a =0$ and $|\Omega| > 2J$. The $|\Omega| > 2J$
condition is because we are trying to solve for $\Omega$ outside the energy
band. We note that there are two cases for us to consider, namely
$\text{Re}(s) > 0$, corresponding to $s = -i \Omega + 0^+$ and $\text{Re}(s)
< 0$, corresponding to $s = -i\Omega - 0^+$. As we will see, the
$\text{Re}(s) >0$ case gives us the BOC energy for $\Omega < -2J$, and the
$\text{Re}(s) < 0$ case gives use the BOC energy for $\Omega > 2J$. First,
we consider the $\text{Re}(s) > 0$ case. In this case, we have the equation
\begin{equation}
\left(\frac{\Omega}{2J}\right)^2 = -\frac{1}{4}\left(\frac{\rho_0}{J}\right)^2 \frac{1}{\sqrt{\left(\frac{\Omega}{2J}\right)^2-1}}\left(M + 2\sum_{r=1}^{M-1} (M-r) \left(\frac{\Omega}{2J} + \sqrt{\left(\frac{\Omega}{2J}\right)^2-1}\right)^{n_0 r}\right)
\end{equation}
where we have used $\sqrt{4J^2 - \Omega^2} = i\sqrt{\Omega^2 - 4J^2}$, since
$|\Omega|>2J$. Now, we substitute $\Omega^\prime = \Omega/2J$ as well as \eqref{eqn: BIC rho0 M legs n0=4l} for $\rho_0/J$ to get
\begin{equation}
\Omega^\prime \sqrt{{\Omega^\prime}^2 - 1} = -\frac{1}{4}\frac{2}{M}\tan\left(\frac{\pi}{M}\right)\sin\left(\frac{4\pi}{M n_0}\right) \left(M + 2\sum_{r=1}^{M-1} (M-r)\left(\Omega^\prime + \sqrt{{\Omega^\prime}^2 - 1}\right)^{n_0 r}\right)
\end{equation}
In the $n_0 \to \infty$ limit, for the above equation to have a solution, we
must have $\Omega^\prime + \sqrt{{\Omega^\prime}^2 - 1} < 1$, which means $\Omega < -2J$. Hence, taking the $n_0 \to \infty$ limit, we have
\begin{equation}
\Omega^\prime \sqrt{{\Omega^\prime}^2 - 1} = -\frac{2}{M}\tan\left(\frac{\pi}{M}\right)\left(\frac{\pi}{Mn_0}\right)M
\end{equation}
Solving for $\Omega^\prime$ in the above equation subject to the condition that $\Omega < -2J$, we have
\begin{align}
\Omega^\prime &= -\sqrt{\frac{1}{2} \sqrt{\frac{16 \pi ^2 \tan
^2\left(\frac{\pi }{M}\right)}{M^2 \text{n0}^2}+1}+\frac{1}{2}} \nonumber \\
&\approx -1 - \frac{2 \pi^2 \tan\left(\frac{\pi}{M}\right)^2}{M^2 n_0^2}.
\end{align}
The last line in the above equation is \eqref{eqn: asymptotic BOC energies}
for the case where $\Omega < -2J$. The derivation for the $\Omega > 2J$ case
can be done by choosing $\text{Re}(s) < 0$ and following the same steps above
to arrive at
\begin{align}
\Omega^\prime &= \sqrt{\frac{1}{2} \sqrt{\frac{16 \pi ^2 \tan
^2\left(\frac{\pi }{M}\right)}{M^2 \text{n0}^2}+1}+\frac{1}{2}} \nonumber \\
&\approx 1 + \frac{2 \pi^2 \tan\left(\frac{\pi}{M}\right)^2}{M^2 n_0^2}
\end{align}
where the last line in the above equation is \eqref{eqn: asymptotic BOC
energies} for the case where $\Omega > 2J$. To derive \eqref{eqn: asymptotic
BOC probability}, the method is to use \eqref{eqn: emitter contribution BIC}
together with \eqref{eqn: sigma(s) for M legs oscillating BS} and \eqref{eqn:
detuning in terms of Sigma(s)}. We will also use \eqref{eqn: asymptotic BOC
energies} derived above, and also \eqref{eqn: BIC rho0 M legs n0=4l} for
$\rho_0/J$. Thereafter, doing an asymptotic expansion about $\frac{1}{n_0}
\to 0$ and keeping the lowest order, we obtain \eqref{eqn: asymptotic BOC
probability}.
\subsection{Derivation of \eqref{eqn: M=3 bic probability dist}}
Firstly, we write our BICs $\ket{\pm}$ as
\begin{align}
\ket{\pm} &= \phi_a^{(\pm)} \ket{1_a} + \int_{-\pi}^\pi dk \, e(k)^{\pm}
\ket{1_k} \nonumber \\
&= \phi_a^{(\pm)} \ket{1_a} + \phi_a^{(\pm)} \int_{-\pi}^\pi dk \, \frac{G^*(k)}{\Omega_{\pm} - 2J \cos(k)} \ket{1_k}
\end{align}
where $\phi_a^{(\pm)} = \braket{1_a}{\pm}$ and $e(k)^{\pm} =
\braket{1_k}{\pm}$, and $\Omega_{\pm}$ is the energy of the $\ket{\pm}$ state
respectively. In going from the first line to the second line, we used
\eqref{eqn: one excitation subspace energy eigval eqn simultaneous eqn}. In
the second line above, we see that we can clearly factor out a global phase
corresponding to the phase of $\braket{1_a}{\pm}$. Hence, without loss of
generality, we can write $\phi_a \equiv \phi_a^{(\pm)} = \braket{1_a}{\pm}$.
Then, we have
\begin{align}
\ket{\pm}
&= \phi_a \ket{1_a} + \phi_a \int_{-\pi}^\pi dk \,
\frac{G^*(k)}{\Omega_{\pm} - 2J \cos(k)} \ket{1_k} \nonumber\\
&= \phi_a \ket{1_a} + \phi_a \int_{-\pi}^\pi dk \,
\frac{G^*(k)}{\Omega_{\pm} - 2J \cos(k)} c(k)^\dagger\ket{0} \nonumber\\
&= \phi_a \ket{1_a} + \phi_a \int_{-\pi}^\pi dk \,
\frac{G^*(k)}{\Omega_{\pm} - 2J \cos(k)} \sum_n e^{i k
n}\frac{1}{\sqrt{2\pi}}b_n^\dagger\ket{0} \nonumber\\
&= \phi_a \ket{1_a} + \phi_a \int_{-\pi}^\pi dk \,
\frac{(\rho_0/2\sqrt{\pi})\sum_{l=0}^{M-1} e^{-i k n_l}}{\Omega_{\pm} - 2J \cos(k)} \sum_n e^{i k
n}\frac{1}{\sqrt{2\pi}}b_n^\dagger\ket{0} \nonumber\\
&= \phi_a \ket{1_a} + \sum_n
\underbrace{\frac{\rho_0\phi_a}{2\pi}\sum_{l=0}^{M-1}\int_{-\pi}^\pi dk \, \frac{ e^{i k
(n-n_l)}}{\Omega_{\pm} - 2J \cos(k)}}_{\equiv \phi_{n, \pm}} b_n^\dagger\ket{0}
\end{align}
Thereafter, to evaluate $\phi_{n,\pm}$, we note that
\begin{align}
\phi_{n,\pm} &= \frac{\rho_0\phi_a}{2\pi}\sum_{l=0}^{M-1}\int_{-\pi}^\pi dk
\, \frac{ e^{i k (n-n_l)}}{\Omega_{\pm} - 2J \cos(k)} \nonumber \\
&= \frac{\rho_0\phi_a}{2\pi}\sum_{l=0}^{M-1} I_{n-n_l}(s=-i\Omega_\pm+0^+)
\end{align}
where $I_{n-n_l}(s)$ is given by \eqref{eqn: In(s) integral result}.
\section{Proof that there is no oscillating BIC with
\texorpdfstring{$M=2$}{M=2}}
\label{Appendix: oscillating BIC at n=2 CMI}
For $n=2$, we have
\begin{equation}
|G(k)|^2 = \frac{\rho_0^2}{\pi}\left(1 + \cos(kn_0)\right)
\end{equation}
which gives us
\begin{equation}
\Sigma(s) = \frac{-2i
\rho_0^2}{\sqrt{s^2+4J}}\left(1+\left(\frac{-i\sqrt{s^2+4J^2}+is}{2J}\right)^{n_0}\right).
\end{equation}
Thus, we have
\begin{align}
\Sigma(s=-i\Omega) &= \frac{-2i
\rho_0^2}{\sqrt{4J-\Omega^2}}\left(1+\left(\frac{-i\sqrt{4J^2-\Omega^2}+\Omega}{2J}\right)^{n_0}\right)
\nonumber \\
&= \frac{-2i \rho_0^2}{\sqrt{4J-\Omega^2}}\left(1+\cos(n_0
\theta)+i\sin(n_0\theta)\right)
\end{align}
where $\theta = \arctan\left(\frac{-\sqrt{4J^2-\Omega^2}}{\Omega}\right)$ and
$-2J<\Omega < 2J$.
Using $\Delta(\Omega) = \text{Re}(\Sigma(s=-i\Omega + 0^+))$, we have
\begin{equation}
\label{eqn: oscillating bs n=2 detuning}
\Omega - \omega_a = \frac{2\rho_0^2}{\sqrt{4J-\Omega^2}}\sin(n_0 \theta)
\end{equation}
Furthermore, enforcing $|G(k)|^2=0$ gives us
\begin{equation}
k = (2l+1)\frac{\pi}{n_0}
\end{equation}
where $l$ is an integer. Hence, we have
\begin{equation}
\Omega = 2J\cos\left(\frac{\pi}{n_0}(2l+1)\right)
\end{equation}
which we can substitute into $\theta = \arctan\left(\frac{-\sqrt{4J^2-\Omega^2}}{\Omega}\right)$ to get
\begin{equation}
\theta = -\frac{\pi}{n_0}(2l+1)
\end{equation}
Subtituting the above expression into \eqref{eqn: oscillating bs n=2 detuning}, we have
\begin{equation}
\Omega = \omega_a.
\end{equation}
Hence, when we have $K=2$ legs in our giant atom, there is only one BIC at
the frequency $\Omega = \omega_a$, provided that
\begin{equation}
\omega_a = \Omega = 2J\cos\left(\frac{\pi}{n_0}(2l+1)\right).
\end{equation}
Otherwise, there is no BIC for $K=2$ legs. Moreover, since we only have one
BIC, it is impossible to get an oscillating BIC, which requires at least two
BICs at two different frequencies in the band.
\section{Comparison of the oscillating BIC conditions with continuous waveguide}
For $M$ coupling points, and setting $\omega_a = 0$, the oscillating BIC found for a continuous (linearized) waveguide in Ref.~\cite{guo_oscillating_2020} are formed from the superposition of two BICs with energies
\begin{equation}
\Omega_{c} = \pm \frac{1}{2} M \gamma \cot\left( \frac{n\pi}{M} \right), \quad n=1,2,\ldots,\left\lfloor M/2 \right\rfloor
\end{equation}
where $\gamma$ is the giant atom decay rate into the waveguide. In our case, $\gamma$ corresponds to $\rho_0^2/J$. From~\eqref{eqn: BIC energies M legs n0=4l} and~\eqref{eqn: BIC rho0 M legs n0=4l}, we have, for our case,
\begin{equation}
\Omega_{\text{BIC}} = \pm 2J \sin\left( \frac{2\pi}{Mn_0} \right) = \pm 2 \gamma \frac{J^2}{\rho_0^2} \sin\left( \frac{2\pi}{Mn_0} \right) = \pm \frac{1}{2} M\gamma \cot\left( \frac{\pi}{M} \right) \sec\left( \frac{2\pi}{Mn_0} \right)
\end{equation}
which looks similar to the continuous-waveguide result for $n=1$. In fact, in the regime where $(M n_0)$ is large such that $\sec(2\pi/(Mn_0))\approx 1$,
the oscillating BIC energies are approximately the same as $\Omega_c$.
\end{document} | math | 123,725 |
\begin{document}
\begin{abstract}
The calculations of the Tibetan calendar are described, using modern
mathematical notations instead of the traditional methods.
\end{abstract}
\title{Tibetan calendar mathematics}
\section{Introduction}\label{Sintro}
The Tibetan calendar is derived from the Indian calendar tradition; it
has the same general structure as Indian calendars,
but the details differ significantly.
The basis for the Tibetan calendar is the \KT, which was translated
from Sanskrit into Tibetan in the 11th century.
(Traditional date of the translation is 1027
when the first 60 year cycle starts.)
It is based on Indian astronomy, but much modified.
The calendar became the standard in Tibet in the second half of the
thirteenth century.
As in Indian calendars
(see \citet{CC}),
months are lunar (from new moon to new moon) but numbered according to the
corresponding solar months,
\ie{} the position of the sun,
\xfootnote{Solar months are used only in the astronomical theory behind the
calendar, and not in the calendar itself.}
while days are numbered by the corresponding
lunar days. Since these correspondences are not perfect, there are occasionally
two months with the same number, in which case the first of them is
regarded as a leap month, and occasionally a skipped date or two days
with the same date
(then the first of them is regarded as a
leap day).
Unlike modern Indian calendars, there are no skipped months.
Various improvements of the calendar calculations have been suggested over the
centuries, see \cite{Schuh}, \cite[Chapter VI]{Henning},
\cite{tibetenc}, \cite{kalacakra},
and different traditions follow different rules for the details of the
calculation.
There are two main versions (\emph{\PH} and \emph{\TS})
of the Tibetan calendar in use today by
different groups inside and outside Tibet;
moreover, Bhutan and Mongolia also use versions of the Tibetan calendar.
See further \refApp{Aversions}.
The different versions frequently differ by a day or a month, see
\refApp{Acomp}.
The description below refers to the \PH{} version,
introduced 1447,
which is the most common version;
it is the version followed by \eg{} the Dalai Lama and
it can be regarded as the standard version of the Tibetan calendar.
The differences
in the \TS{} version and other versions are discussed in \refApp{Aversions}.
\begin{table}[t]
\begin{tabular}{r r l | r r l }
4/3 1927 & 28/2 1987 & Fire--Rabbit & 2/3 1957 & 27/2 2017 & Fire--Bird\\
22/2 1928 & 18/2 1988 & Earth--Dragon & 19/2 1958 & 16/2 2018 & Earth--Dog\\
10/2 1929 & 7/2 1989 & Earth--Snake & 8/2 1959 & 5/2 2019 & Earth--Pig\\
1/3 1930 & 26/2 1990 & Iron--Horse & 27/2 1960 & 24/2 2020 & Iron--Mouse\\
18/2 1931 & 15/2 1991 & Iron--Sheep & 16/2 1961 & 12/2 2021 & Iron--Ox\\
7/2 1932 & 5/3 1992 & Water--Monkey & 5/2 1962 & 3/3 2022 & Water--Tiger\\
25/2 1933 & 22/2 1993 & Water--Bird & 24/2 1963 & 21/2 2023 & Water--Rabbit\\
14/2 1934 & 11/2 1994 & Wood--Dog & 14/2 1964 & 10/2 2024 & Wood--Dragon\\
4/2 1935 & 2/3 1995 & Wood--Pig & 4/3 1965 & 28/2 2025 & Wood--Snake\\
23/2 1936 & 19/2 1996 & Fire--Mouse & 21/2 1966 & 18/2 2026 & Fire--Horse\\
12/2 1937 & 8/2 1997 & Fire--Ox & 10/2 1967 & 7/2 2027 & Fire--Sheep\\
3/3 1938 & 27/2 1998 & Earth--Tiger & 29/2 1968 & 26/2 2028 & Earth--Monkey\\
20/2 1939 & 17/2 1999 & Earth--Rabbit & 17/2 1969 & 14/2 2029 & Earth--Bird\\
9/2 1940 & 6/2 2000 & Iron--Dragon & 7/2 1970 & 5/3 2030 & Iron--Dog\\
26/2 1941 & 24/2 2001 & Iron--Snake & 26/2 1971 & 22/2 2031 & Iron--Pig\\
16/2 1942 & 13/2 2002 & Water--Horse & 15/2 1972 & 12/2 2032 & Water--Mouse\\
5/2 1943 & 3/3 2003 & Water--Sheep & 5/3 1973 & 2/3 2033 & Water--Ox\\
24/2 1944 & 21/2 2004 & Wood--Monkey & 22/2 1974 & 19/2 2034 & Wood--Tiger\\
13/2 1945 & 9/2 2005 & Wood--Bird & 11/2 1975 & 9/2 2035 & Wood--Rabbit\\
4/3 1946 & 28/2 2006 & Fire--Dog & 1/3 1976 & 27/2 2036 & Fire--Dragon\\
21/2 1947 & 18/2 2007 & Fire--Pig & 19/2 1977 & 15/2 2037 & Fire--Snake\\
10/2 1948 & 7/2 2008 & Earth--Mouse & 8/2 1978 & 6/3 2038 & Earth--Horse\\
28/2 1949 & 25/2 2009 & Earth--Ox & 27/2 1979 & 23/2 2039 & Earth--Sheep\\
17/2 1950 & 14/2 2010 & Iron--Tiger & 17/2 1980 & 13/2 2040 & Iron--Monkey\\
7/2 1951 & 5/3 2011 & Iron--Rabbit & 5/2 1981 & 3/3 2041 & Iron--Bird\\
26/2 1952 & 22/2 2012 & Water--Dragon & 24/2 1982 & 21/2 2042 & Water--Dog\\
14/2 1953 & 11/2 2013 & Water--Snake & 13/2 1983 & 10/2 2043 & Water--Pig\\
4/2 1954 & 2/3 2014 & Wood--Horse & 3/3 1984 & 29/2 2044 & Wood--Mouse\\
23/2 1955 & 19/2 2015 & Wood--Sheep & 20/2 1985 & 17/2 2045 & Wood--Ox\\
12/2 1956 & 9/2 2016 & Fire--Monkey & 9/2 1986 & 6/2 2046 & Fire--Tiger\\
\end{tabular}
\caption{Tibetan New Year, \tibx{Losar},
(\PH{} version) and year names for the last and
current 60 year cycles.}
\label{Tlosar}
\end{table}
\refT{Tlosar} gives the Gregorian dates of the Tibetan New Year
\tib{Losar}
for
the years in the last and current 60 year cycles (see \refS{Syear}).
See also \refT{T4Losar} in \refApp{Aversions}.
The table shows that New Year at present
occurs in February or the first week of March; the extreme dates during the
20th century are 4 February (\eg{} 1954) and 5 March (\eg{} 1992), and
during the 21st century 5 February (2019) and 7 March (2095). (The dates get
slowly later in the Gregorian calendar, see \refS{Smean}.)
\xfootnote{
The Tibetan New Year thus frequently coincides with the Chinese New Year,
which also is at new moon and always in the range 21 January -- 21 February
\cite[Section 17.6]{CC}, although often the New Years differ by a day
because of the different calculation methods; however, in about each second
year (at present), the Tibetan New Year is a month after the Chinese.
}
The purpose of this paper is to describe the mathematics of the Tibetan
calendar, using modern
mathematical notations instead of the traditional methods.
After some preliminaries (Sections \ref{Snot}--\ref{Sdef})
and a description of the naming of years (\refS{Syear} and \refApp{A60}),
the calculations of the Tibetan calendar are presented in two main
parts. In the first part (\refS{Smonths} and \refApp{Aleap}),
the months are regarded as units and I
discuss how they are numbered, which implies the partitioning of them
into years and also shows which months that are leap months.
In the second part (Sections \ref{Sdays}--\ref{Sweek}),
I discuss the coupling between months
and days, including finding the actual days when a month begins and
ends and the numbering of the days.
Finally, some further calculations are described
(Sections \ref{Sfurther}--\ref{Sholidays}) and
some mathematical consequences are given (Sections
\ref{Smean}--\ref{Speriod}).
Calculations for the planets and some other astrological
calculations are
described in Appendices \ref{Aplanet}--\ref{Aastro}.
The description is based mainly on the books by \citet{Schuh} and
\citet{Henning}, but the
analysis and mathematical formulations are often my own.
(Unfortunately I do not read Tibetan, so I have to use secondary
sources instead of Tibetan texts. This is of course is a serious drawback,
although I have been able to check the calculations against published
almanacs such as \cite{tib2013}.)
For further study I recommend in particular the detailed recent book by
\citet{Henning},
which contains much more material than this paper;
see also his web site \cite{kalacakra} with further information.
I (mostly) describe only the contemporary versions and ignore the historical
development; for the history of the calendar, see \citet{Schuh} and
\cite{tibetenc}.
See also \citet{CC} for a related description
and computer implementation of the Tibetan calendar.
Open source computer programs can be obtained from
\citet{kalacakra}.
Computer generated tables and calendars covering almost 1000 years are
given in \citet{Schuh} and by
\citet[Traditional Tibetan calendar archive]{kalacakra}.
Some further related references are
\citet[pp.~403--409]{Ginzel} (a classic, but dated and incomplete),
\citet{Petri} (on Tibetan astronomy),
\citet{Tseng1,Tseng2} (on astrology
\xfootnote{\label{f:astro}
\citet{Tseng1,Tseng2} and \citet{Schuh-review} make a distinction between
astrology
(using planets etc.)\
and divination (\tibx{nag-rtsis} or \tibx{'byung-rtsis}, using elements etc.).
I use ``astrology'' in a wider sense, encompassing both.
}
),
and further articles in the collection
\cite{Schuh-contributions}.
See also the review by \citet{Schuh-review} of the first (2007)
version of the present paper (together with the books \cite{CC} and
\cite{Henning}), and the reply by \citet{Henning-comments}.
\xfootnote{
The review by \citet{Schuh-review} is quite critical of many details.
I am grateful for some of his comments, and I have tried
to make some improvements in the present version.
(In other cases I agree that improvements of formulations might be
desirable, but I find the present version acceptable and leave it for
various reasons.
I also agree that several of the internet references are unreliable and not
scholarly; however, I have nevertheless included them, either to illustrate
actual usage among Buddhist groups or to show interesting claims that I have
been unable to verify or disprove; the reader is advised to regard these
with caution.)
I disagree with some other comments, of which I will mention a few here.
First, the present paper is primarily written for people who, like me, do not
know Tibetan, and I therefore find it useful to use more or less standard
Anglicized
forms of names rather than the scientific transliterations (Wylie) used by
Tibetologists (who hopefully will understand what I mean in any case),
although I sometimes give the latter as well (in italics).
(For example, I write
``\TS'',
used \eg{} by the \TS{} monastery itself on its English web pages
and on the almanacs from Rumtek shown in
\cite[Open source Tsurphu calendar software]{kalacakra}.)
Similarly, I have chosen to follow \cite{Henning} and call the main version of
the Tibetan
calendar the ``Phugpa version'', for convenience and without
going into the detailed historical background (see \cite{Schuh}).
Furthermore, I usually use the English terminology of \cite{Henning} for
Tibetan terms. Schuh does not like these translations; I cannot argue with
his expert linguistic remarks, but I am not convinced that
they are relevant for a description of the mathematical content rather than
a linguistic and cultural study of the calendar tradition
(see also \cite{Henning-comments});
moreover,
it seems better to use an existing English terminology \cite{Henning} rather
than making my own second translation from a German
translation \cite{Schuh} of the terms.
}
\begin{acks}
This study was made possible by the help of
Nachum Dershowitz, Edward Henning, Edward Reingold,
Olli Salmi
and Dieter Schuh,
who have provided me with essential references and given me helpful
comments;
in particular, Edward Henning has been very helpful by answering numerous
questions on many details.
\end{acks}
\section{Notation}\label{Snot}
\subsection*{Mixed radix notation}
Traditional Tibetan calculations are made
expressing the various quantities as sequences (written as columns) of
integers, which
should be interpreted as rational numbers
in a positional system with mixed radices;
\xfootnote{
In the same way as we denote time in days, hours, minutes and
seconds, with radices 24, 60, 60.
}
the radices are fixed but
different sequences of radices are used for different quantities.
I usually use standard notation for
rational numbers, but when quoting the traditional
expressions, I use notations of the type (of varying length)
\xfootnote{
This notation is taken from \citet{Henning}, although he usually omits all
or most of the radices since they are given by the context. \citet{Schuh},
\cite{tibetenc}
uses the similar notation
$[a_1,a_2,\dots,a_n]/(b_1,b_2,\dots,b_n)$ meaning either
$0;a_1,a_2,\dots,a_n \rr{b_1,b_2,\dots,b_n}$
or
$a_1;a_2,\dots,a_n \rr{b_2,\dots,b_n}$ depending on the context.
}
\begin{equation*}
a_0;a_1,a_2,a_3 \rr{b_1,b_2,b_3}
\qquad\text{meaning}\qquad
a_0+\frac{a_1+(a_2+(a_3/b_3))/b_2}{b_1}
.
\end{equation*}
Formally, we have the inductive definition
\begin{equation*}
a_0;a_1,a_2,\dots,a_n \rr{b_1,b_2,\dots,b_n}
=a_0+\frac{ a_1;a_2,\dots,a_n \rr{b_2,\dots,b_n}}{b_1}
\end{equation*}
for $n\ge2$, and $a_0;a_1\rr{b_1}=a_0+a_1/b_1$.
We will usually omit a leading 0 (and the semicolon) and write
just \eg{} $a_1,a_2,a_3 \rr{b_1,b_2,b_3}$ for numbers between 0 and 1.
For explanations and examples of the way the traditional hand calculations
are performed
\xfootnote{Traditionally in sand rather than on paper.},
see \citet{Henning}, \citet{Schuh} and
\cite[Kalenderrechnung, Sandabakus]{tibetenc}.
\subsection*{Angular units}
It will be convenient, although somewhat unconventional, to express
longitudes and other angular measurements in units of full circles
in our formulas.
To obtain values in degrees, the numbers should thus be multiplied by
360. (A Tibetan would probably prefer multiplying by 27 to obtain
lunar mansions (= lunar station = \tibx{naksatra}) and fractions thereof;
this is the unit usually used for longitudes.
A Western astrologer might prefer multiplying by 12 to obtain values
in signs. A mathematician might prefer multiplying by $2\pi$ (radians).)
For angular measurements, full circles are often to be ignored (but
see \refApp{Aleap}); this means with our convention that the numbers are
considered modulo 1, \ie{}, that only the fractional part matters.
\subsection*{Boolean variables}
For a Boolean variable $\ell$, \ie{} a variable taking one of the
values \true{} and \false, we use $\ell=\Bool{\cP}$ to denote that
$\ell=\true$ if and only if the condition $\cP$ holds; we further
let $\boolx{\ell}$ be the number defined by
$\boolx{\ell}=1$ when $\ell=\true$ and
$\boolx{\ell}=0$ when $\ell=\false$.
\subsection*{Julian day number}
The \emph{Julian day number} (which we abbreviate by JD) for a given day is
the number of days that have elapsed since the epoch 1 January 4713 BC
(Julian); for days before the epoch (which hardly concern the Tibetan
calendar), negative numbers are used. The Julian day numbers thus form
a continuous numbering of all days by $\dots,-1,0,1,2,\dots$. Such a
numbering is very convenient for many purposes, including conversions
between calendars. The choice of epoch for the day numbers is
arbitrary and for most purposes unimportant.
The conventional date 1 January 4713 BC ($-4712$ with astronomical numbering
of years) was
originally chosen by Scalinger in 1583 as the origin of the Julian
period, a (cyclic) numbering of years;
this was developed by 19th-century astronomers into a
numbering of days. See further
\cite[Section 12.7]{AA}, \cite[Section 15.1.10]{AA3}.
\xfootnote{
Dershowitz and Reingold \cite{CC} use another day number, denoted by RD,
with another epoch: RD 1 is 1 January 1 (Gregorian) which is JD 1721426.
Consequently, the two day numbers
are related by $\JD=\RD+1721425$.
}
A closely related version of this idea is the \emph{Julian date},
which is a real number that defines the time of a particular instant,
meaured (in days and fractions of days) from the same epoch.
The fractional part of the Julian date shows the fraction of a day
that has elapsed since noon GMT (UT); thus, if $n$ is an integer, then
the Julian date is $n.0$ (\ie, exactly $n$) at noon GMT on the
day with Julian day number $n$.
It is important to distinguish between the Julian
day number and the Julian date, even if they are closely related.
Both are extremely useful, but for different purposes, and much unnecessary
confusion has been caused by confusing and mixing them.
(We follow \cite{AA} in using different names for them, but
that is not always done by other authors.
\xfootnote{
Moreover,
astronomers simply define Julian day number as the integer part
of the Julian date \cite[Section 3.7]{AA3},
which is \emph{not} the version
described here
\cite[Section 12.7]{AA}, \cite[Section 15.1.10]{AA3},
used in historical chronology.
})
For this study, and most other work on calendars, the Julian day
number is the important concept.
(The Julian date is essential for exact astronomical calculations, but
no such calculations are used in the traditional Tibetan calendar.)
Note that the Julian day number is an integer, while the Julian date
is a real number. (A computer scientist would say that they have
different types.) Moreover, the Julian day number numbers the days
regardless of when they begin and end, while the Julian date depends
on the time of day, at Greenwich.
Hence, to convert a Julian date to a Julian day number, we need in practice
to know both the local time the day begins and the time zone, while these
are irrelevant for calculations with the Julian day number.
For example, 1 January 2007 has JD 2454102, everywhere. Thus the Julian date
2454102.0 is 1 January 2007, noon GMT (UT), and the new year began at
Julian date 2454101.5 in Britain, but at other Julian dates in other
time zones. The Tibetan day begins at dawn, about 5 am local mean solar time
(see \refR{Rdate} below),
but we do not have to find the Julian date of that instant.
\subsection*{Other notations}
We let $\mod$ denote the binary operation defined by $m \mod n =x$ if
$x\equiv m \pmod n$ and $0\le x<n$ (we only consider $n>0$, but $m$
may be of any sign; care has to be taken with this in computer
implementations).
Similarly (following \cite{CC}), we let
$\amod$ denote the binary operation defined by
$m\amod n =x$ if $x\equiv m \pmod n$ and $0< x\le n$.
This means that $m\amod n= m\mod n$ except when $m$ is a multiple of
$n$; then $m\mod n=0$ and $m\amod n=n$.
For integers $m$ and $n$ (the usual case, and the only one used in this paper),
$m\amod n=1+(m-1)\mod n$.
We use the standard notations $\floor{x}$ and $\ceil{x}$ for the
integers satisfying $x-1<\floor{x}\le x$ and
$x\le\ceil{x}< x+1$, \ie{} $x$ rounded down and up to an integer,
and $\FRAC(x)=x-\floor{x}=x\mod 1$ for the fractional part.
(Again, care has to be taken for $x<0$ in computer implementations.)
\section{Some concepts}\label{Sdef}
The calendar is based on considering several different types of days, months
and years, and we give a list of the most important ones here.
The calendar is in principle astronomical and based on the positions of the moon
and sun in the sky, more precisely their \emph{longitudes}
\xfootnote{\label{fequinox}
The longitude of a celestial object
is measured eastward along the ecliptic, with 0 at the first point of Aries,
both in Tibetan and Western astronomy.
(This is the vernal equinox, where the sun crosses the celestial equator
northwards every year; however, in Tibetan astronomy, the vernal equinox is
erroneously believed to be much earlier, see
\refF{fPHpoints} and \cite[p.~328]{Henning}.)
The sun is always on the ecliptic (with small
perturbations); the moon is up to $5.15\grad$ away from the ecliptic due to
the inclination of its orbit, but
only its longitude is relevant for the calendar.
\cite[\S1.3, Table 15.4, p.~731]{AA}
}
and in particular
the \emph{elongation} of the moon, \ie, the difference between the
longitudes of the moon and the sun.
The Tibetan calendar uses two different formulas to calculate each of these:
a simple (linear) formula giving the \emph{mean longitude} (assuming uniform
motions of the sun and moon) and a more complicated formula for the
\emph{true longitude}; thus the ``lunar'' and ``solar'' concepts defined below
have two versions,
one ``mean'' and one ``true''.
Note that the calendar always uses these theoretical values and not the
actual astronomical positions; do not confuse the ``true'' longitude with
the exact astronomical one.
(Nowadays, the ``true longitudes'' actually have large errors, see
\refS{Smean}.)
\begin{description}
\item [calendar day \rm(\tibx{gza'} or \tibx{nyin-zhag})]
Solar day or natural day; in Tibet the calendar day is from
dawn to dawn. The length is thus constant, 24 hours.
The calendar days are numbered by the number of the corresponding lunar day,
but since the correspondence is not perfect, sometimes a number is skipped
or repeated, see \refS{Sdays}.
Each calendar day has also a day of week, in the same way as in Western
calendars, see \refS{Sweek}.
\item [lunar day \rm(\tibx{tshes-zhag, \textup{Sanskrit} tithi})]
$1/30$ of a lunar month; more precisely the time during which the
elongation of the moon increases by $1/30$ ($=12\grad$).
(Since the moon does not travel at uniform speed; the lunar days have
different lengths, varying between about 21.5 and 25.7 hours, see
\refR{Rlunarday}.)
The lunar days are numbered by 1--30 in each lunar month,
with day 1 beginning at new moon. Thus lunar day $i$ is when the elongation
is between $(i-1)/30$ and $i/30$.
\item[calendar month \rm\tib{zla-ba}]
A period of 29 or 30 calendar days, approximating a lunar month.
(The calendar month gets the same number as the lunar month, and they are
often regarded as the same, but strictly speaking
the calendar month begins at the beginning of a calendar day, at dawn,
while the lunar month begins at the instant of new moon.)
\item[lunar month \rm\tib{tshes-zla}]
The period from the instant of a new moon to the next new moon.
(Synodic month in astronomic terminology.)
The lunar months are numbered 1--12, but sometimes there is a leap month and
a number is repeated, see \refS{Smonths}.
\item[solar month \rm\tib{khyim-zla}]
The period during which the sun travels through one sign.
(Each sign is 1/12 of the ecliptic, \ie, $30\grad$.)
This is thus 1/12 of a solar year, although the lengths of the true solar
months vary somewhat.
\item[calendar year]
12 or 13 calendar months, according to the rules for inserting leap months.
The length of the calendar year is 354, 355, 383, 384 or 385 days,
see \refR{Rlunarday}. The average length is close to the length of the solar
year, see \refS{Smean};
in principle, the calendar year is on the average fixed
with respect to the seasons, although this is in reality not exact.
\item[solar year]
The period during which the sun travels a full revolution around the ecliptic.
(Tropical year in astronomical terminology.)
\end{description}
\section{Numbering and naming of years}\label{Syear}
Several ways of numbering or naming Tibetan years are used,
see \eg{} \citet[pp.~142--145]{Schuh}, \cite[Kalender]{tibetenc}.
One common method, especially among Westeners, is to simply
number a Tibetan year by the Gregorian (or
Julian) year in which it starts (and where most of it falls).
For convenience, we will use this numbering below.
Another method is to number the years from an epoch 127 BC
(the traditional ascent of the first Tibetan king); the
Tibetan year starting in Gregorian year $Y$ will then be numbered
$Y+127$.
\xfootnote{This method has been used from the second half of the 20th century.
Epochs 255 and 1027 have also been used. See \cite[Kalender]{tibetenc}.
}
Both methods are used by Tibetans;
for example, the Tibetan calendar
\cite{tib2003} has titles in both Tibetan and English; with 2003 in
the English title and 2130 in the Tibetan.
Moreover, and more importantly, each year is named according
to a 60 year cycle. Actually, there are two different 60 year cycles
of names, one Indian and one Chinese. Of course, since the cycles have
the same length, there is a 1--1 correspondence between the names.
When naming years, the Chinese cycle is almost always used, and
sometimes the Indian and Chinese cycle names are used together.
(For example, the Chinese cycle names are used in the titles of the calendars
\cite{tib2003} and \cite{nitarthacal}.)
The Indian cycle is a list of 60 different names, in Sanskrit or
Tibetan, see \refApp{A60}. The cycle is named after its first year as
Prabhava \tib{rab byung}. The cycles are numbered, with the first
cycle beginning in AD 1027, which means that each year can be
unambiguously identified by its name in the cycle and the number of
the cycle; this method of naming years has sometimes been used
\cite[Sechzigjahreszyklen]{tibetenc}.
(See \refF{fmongol} and \cite{Terbish-GenghisKhan}
for modern Mongolian examples.)
Year $n$ in cycle $m$ (with $1\le n\le 60$ and, presumably, $m\ge1$) thus
corresponds to Gregorian or Julian year $Y$ given by
\begin{equation}\label{indian1}
Y = 1027 + (m-1)60+(n-1)
= 60m+n+966.
\end{equation}
Conversely,
\begin{align}
n&=(Y-1026)\amod60
=(Y-6)\amod60,
\label{indian60}
\\
m&=\Ceil{\frac{Y-1026}{60}}. \label{indian3}
\end{align}
For example, AD 2007 is the 21st year in the 17th Prabhava cycle,
which began in 1987.
The Chinese cycle is identical to the one used in the Chinese
calendar \cite{CC}.
The cycles start 3 years before the Indian ones, so the
first year (Prabhava, \tibx{rab byung}) in the Indian cycle is the
fourth year (Fire--female--Rabbit) in the Chinese cycle.
The full correspondence is given in \refApp{A60}.
The last cycle thus started 1984.
Hence, or by \eqref{indian60}, year $Y$ is number
\begin{equation}\label{chinese60}
(Y-3)\amod60
\end{equation}
in the Chinese cycle.
The Chinese cycles are not numbered in the Tibetan calendar.
The Chinese 60 year cycle is a combination of two cycles, of 10 and 12
years respectively. (Note that the least common multiple of 10 and 12
is 60; since 10 and 12 have greatest common divisor 2, only half of
the $10\times12$ combinations are possible.)
\xfootnote{
These cycles have been used in the Chinese calendar since at least 1400 BC,
first for naming days and from the Zhou dynasty (c.~1000 BC) for naming years.
\cite[p.~163]{Richards}
}
The 10 year cycle consists in China of 10 different names (proper
names with no other English translation) called celestial stems.
Each celestial stem is associated with an element (wood, fire, earth,
iron, water) and a gender (female or male), see \refT{T10}.
(In Chinese, the two genders are the well-known \emph{yin} and
\emph{yang}.)
Note that the 2 genders are alternating and thus are given by the year
mod 2, while the 5 elements are repeated 2 years each in the 10
year cycle. As a consequence, each celestial stem corresponds to a
unique (element, gender) pair, and in the Tibetan calendar, the
element and gender are used to name the year; the Chinese celestial stems are
usually not used \cite[p.~145]{Henning}.
It follows from \eqref{chinese60} that,
using the numberings in Tables \ref{T10} and \ref{T5}, year $Y$ is
$z=(Y-3)\amod 10$ in the Chinese 10 year cycle, and has element $\ceil{z/2}$.
The 12 year cycle is the well-known cycle of animals found in the Chinese
and many other Asian calendars, see \refT{T12}.
(The English translations are taken from \cite{Henning}; several easily
recognized variants exist.
The Tibetan names are from \cite{CC}.)
It follows from \eqref{chinese60} that
year $Y$ is
$z=(Y-3)\amod 12$ in the 12 year animal cycle.
The Tibetan name for a year according to the Chinese cycle is thus
given as Element--gender--Animal.
Note that the gender, being the year mod 2, also is determined by
the animal (since 2 divides 12), as shown in \refT{T12}. Indeed, the
gender is often omitted and only Element--Animal is used as the name
of the year.
For example, AD 2007 is the 24th year in the Chinese cycle (21st in
the Indian) and is thus Fire--female--Pig or simply Fire--Pig.
\begin{table}[htpb]
\begin{tabular}{r r l l l l}
\rlap{year}\phantom{0}& & element & gender &Tibetan
& celestial stem (Chinese) \\
\hline
1 &8& wood & male &\tibx{shing-pho} & ji\v{a} \\
2 &9& wood & female &\tibx{shing-mo} & y\v{\i} \\
3 &10& fire & male &\tibx{me-pho} & b\v{\i}ng \\
4 &1& fire & female &\tibx{me-mo} & d\={\i}ng \\
5 &2& earth & male &\tibx{sa-pho} & w\`u \\
6 &3& earth & female &\tibx{sa-mo} & j\v{\i} \\
7 &4& iron & male &\tibx{lcags-pho} & g\={e}ng \\
8 &5& iron & female &\tibx{lcags-mo} & x\={\i}n \\
9 &6& water & male &\tibx{chu-pho} & r\'{e}n \\
10 &7& water & female &\tibx{chu-mo} & gu\v{\i} \\
\end{tabular}
\caption{The 10 year cycle.
The first number on each line shows the year mod 10 counted from the
start of a Chinese cycle; the second shows the year mod 10 counted from the
start of a Prabhava cycle.}
\label{T10}
\end{table}
\begin{table}[htpb]
\begin{tabular}{r r l l l}
\rlap{year}\phantom{0} && animal & gender & Tibetan\\
\hline
1 &10& Mouse & male &\tibx{byi ba}\\
2 &11& Ox & female &\tibx{glang}\\
3 &12& Tiger & male &\tibx{stag} \\
4 &1& Rabbit & female & \tibx{yos}\\
5 &2& Dragon & male & \tibx{'brug}\\
6 &3& Snake & female &\tibx{sbrul} \\
7 &4& Horse & male &\tibx{rta} \\
8 &5& Sheep & female &\tibx{lug} \\
9 &6& Monkey & male &\tibx{spre'u} \\
10 &7& Bird & female &\tibx{bya}\\
11 &8& Dog & male &\tibx{khyi}\\
12 &9& Pig & female &\tibx{phag} \\
\end{tabular}
\caption{The 12 year cycle.
The first number on each line shows the year mod 12 counted from the
start of a Chinese cycle; the second shows the year mod 12 counted from the
start of a Prabhava cycle.}
\label{T12}
\end{table}
The civil year starts, unsurprisingly, with month 1; note that in
case there is a leap month 1, the year begins with the leap month,
which precedes the regular month 1.
(This happened in 2000. See also \refR{Rlosar}.)
The first day of the year is also celebrated as a major holiday
(\tibx{Losar}; the Tibetan New Year).
\begin{remark}\label{Rcaitra}
Traditional Tibetan almanacs (such as \cite{tib2013})
start, however, with month 3; this month is
identified with the \Kc{} month
\tibx{nag pa} (Tibetan)
or \Caitra{} (Sanskrit),
which was
considered the beginning of the \Kc{} year \cite[p.~194]{Henning}
and is the starting point in Tibetan astronomy.
(\Caitra{}
is the first month of the year
in most Indian calendars \cite{CC}.)
The standard numbering system is known as
\emph{Mongolian month} \tib{hor-zla}
(or, in recent almanacs, \emph{Tibetan month} \tib{bod-zla}),
and was introduced in the 13th century when the Tibetan
New Year was moved to be the same as the Mongolian
\xfootnote{
By the Tibetan spiritual and political leader
Chogyel Phagspa Lodro Gyeltsen
\tib{chos-rgyal 'Phags-pa Blo-gros rgyal-mtshan}
(1235--1280),
who was advisor of Kublai Khan and
Preceptor of Tibet, then part of the Mongol Empire.
\cite[Pakpa Lodro Gyeltsen]{treasury}
}
\cite[p.~145]{Henning},
\cite[Kalendar, Zeitma{\ss}e]{tibetenc}.
The year is nevertheless considered to consist of months 1--12 in the usual
way,
as said above,
so a traditional almanac contains the 10 last
months of the year and 2 months of the next (perhaps plus a leap month).
\end{remark}
\section{Numbering of months and leap months}\label{Smonths}
The Tibetan calendar months are, as said in \refS{Sdef}, lunar months, that
begin and end at new moon.
There are usually 12 months in a year,
but there is also
sometimes an extra \emph{leap} (or \emph{intercalary}) month, so that the
year then has 13 months (a \emph{leap year}),
in order to keep the calendar year roughly aligned with the (tropical) solar
year.
Non-leap months are called \emph{regular}. There are thus always 12 regular
months in a year.
The regular months in each year are numbered
1--12.
A leap month takes the same number (and belongs to the same year)
as the following month (as in Indian calendars \cite{CC}).
\xfootnote{
The original system in the \KT{} seems to have been the opposite
(although this is not stated explicitly),
with a leap month taking the number of the preceding month
(as in the Chinese calendar \cite{CC}),
see \cite[Published calendar explanation]{kalacakra}.
That system is also used in the Bhutanese version of the
calendar, see \refS{ABhutan}.
}
In a leap year there are thus two months having the same number and, by the
rule just given, the first of these is the leap month.
\subsection{Month names}\label{SSmonths}
Usually, the Tibetan months are just numbered, but month names exist too;
they are printed in almanacs
and are sometimes used, in particular in literature about the calendar.
Several different naming systems exist as follows, see \refT{Tmonths}.
(In all systems, a leap month gets the same name as the corresponding regular
month.)
See further
\citet[p.~145]{Schuh},
\cite{Schuh-review} and \cite[Zeitma{\ss}e]{tibetenc},
and \citet[pp.~147--149 and 194--196]{Henning} and
\cite[Early epochs]{kalacakra}.
\begin{romenumerate}[-15pt]
\item \label{month-mansion}
The Indian system of naming months by
twelve of the names of the
lunar mansions, chosen to be approximatively where the full moon occurs that
month (and thus opposite to the sign that the sun visits).
(Cf.\ \cite[Section 9.3]{CC}. See also \citet[pp.~358--359]{Henning} and
\citet[p.~92]{Petri}.)
See \refT{Tmonths}, where both the Tibetan names and the Sanskrit
names are given.
This system was introduced with the \Kc{} calendar in the 11th
century.
\item \label{month-animal}
Animal names, by the same 12 animal cycle as for years, see \refT{T12}.
This can be extended to Element + Animal, or Element + Gender + Animal,
as for years, see \refApp{ASattributes-months} for details.
(This is an old Chinese system, used in Tibet from the 12th century.)
Note that \PH{} almanacs set month 11 = Tiger (shown in \refT{Tmonths}),
while \TS{} almanacs (\refApp{ATS}) set month 1 = Tiger (as in
the Chinese calendar), see \refT{T12months}.
This method is still important for astrological purposes.
\item
Seasonal names, naming the 12 months
as beginning, middle and end of each of the four
seasons spring, summer, autumn, winter.
This is the oldest system, and was used at least from the 7th century.
\cite[Zeitma{\ss}e]{tibetenc}
Unfortunately, the seasonal months have been identified with the \Kc{}
months in several different ways, so there are several different versions.
As far as I know, none of them is really used today, but
they are mentioned \eg{} in almanacs, which often give two or several different
versions of them, see \eg{} \cite[pp.~195--196]{Henning}.
At least the following versions exist today.
(Including versions used by the \TS{} tradition, see \refApp{ATS}.)
\begin{enumerate}
\item \label{month-season-KT}
Month 1 = \tibx{mchu} = late-winter, etc.;
thus Month 3 = \nag{} = mid-spring.
This is the original \Kc{} identification, now used by the \TS{} tradition.
\item \label{month-season-PH}
Month 1 = \tibx{mchu} = early-spring, etc.;
thus Month 3 = \nag{} = late-spring.
See \refT{Tmonths}.
This is the \PH{} version of the \Kc{} system.
(Introduced c.~1200 by Drakpa Gyaltsen, see \cite[Early epochs]{kalacakra}.)
\item \label{month-season-animal}
Identifying the animal names in \ref{month-animal} by Tiger = early-spring,
etc.\ (as in the Chinese calendar).
Note that this gives different seasonal names for the \PH{} and \TS{}
versions.
The \PH{} version is given in \refT{Tmonths}.
\item
A different system with 6 seasons with two months each.
(This is an Indian system, see \eg{}
\cite[\S 75 and pp.~345--346]{Ginzel} and \cite[p.~195]{Richards}.)
Month 1 = \tibx{mchu} = early-winter;
Month 3 = \nag{} = early-spring;
see \cite[pp.~358--359]{Henning} for the complete list.
\end{enumerate}
\end{romenumerate}
The present numbering of months was, as said above, introduced in the 13th
century.
\xfootnote{In the \KT, \nag{} (\Caitra) is evidently the first month, as in
Indian calendars, although the months are not explicitly numbered.
}
\begin{table}[htpb]
\begin{tabular}{r l l l l l}
& Tibetan & Sanskrit & seasonal I & animal &seasonal II\\
\hline
1 & mchu & M\=agha & early-spring & dragon & late-spring\\
2 & dbo & Ph\=alguna & mid-spring & snake & early-summer\\
3 & \tibxx{nag pa} & \tibxx{Caitra} & late-spring & horse & mid-summer\\
4 & sa ga & Vai\'s\=akha & early-summer & sheep & late-summer\\
5 & snron & Jye{\d s}{\d t}ha & mid-summer & monkey &early-autumn\\
6 & chu stod & \=A\d{s}\=a\d{d}ha & late-summer & bird & mid-autumn\\
7 & gro bzhin & \'Sr\=ava\d{n}a & early-autumn & dog & late-autumn\\
8 & khrums & Bh\=adrapada & mid-autumn & pig & early-winter\\
9 & tha skar & \=A\'svina & late-autumn & mouse & mid-winter \\
10 & smin drug & K\=artikka & early-winter & ox & late-winter\\
11 & mgo & M\=arga\'s\=ir\d{s}a & mid-winter & tiger& early-spring \\
12 & rgyal & Pau\d{s}a & late-winter & rabbit &mid-spring
\end{tabular}
\caption{Month names (\PH{} system), see \refS{SSmonths}:
number;
Tibetan and Sanskrit lunar mansion names, \ref{month-mansion};
seasonal names, \ref{month-season-PH};
animal names, \ref{month-animal};
seasonal names, \ref{month-season-animal}.
}
\label{Tmonths}
\end{table}
\subsection{Leap months}
The Tibetan calendar is based on the relation
\begin{equation}\label{6765}
67 \text{ lunar months} = 65 \text{ solar months},
\end{equation}
which is regarded as exact.
(See \refS{SSratio} for the astronomical reality.)
This is a fundamental relation in the Tibetan calendar,
which distinguishes it from other calendars such as the Indian ones.
\xfootnote{
This relation derives, as all basic features of the calendar, from the \KT,
but there it is actually given as an approximation and not as an exact
relation, see \refApp{AKT}.
}
The leap months are regularly spaced in accordance with this relation, \ie,
with 2 leap months for each 65 solar months. In other words, there are 2
leap months for 65 regular months, and these are regularly spaced with
alternating 32 and 33 regular months between the leap months.
(Thus the distance between successive leap months alternates between 33 and
34 months.)
\xfootnote{The average length between intercalations is thus $32\frac12$ regular
months, or $33\frac12$ months including the leap month. Tibetan authors have
often interpreted this as meaning that each second intercalation really
should come in the middle of a month, although it is moved to the beginning of
the month, see \cite{Yamaguchi}, \cite[On intercalary months]{kalacakra}.
As far as I know, this view has only been used in theoretical discussions and
no Tibetan calendar has ever been produced with the leap month in the middle
of a month.
(However, leap months are inserted in this way in some Indian calendars
\cite[p.~277]{CC}.)
}
\begin{remark}\label{R6567}
It follows that the leap months repeat in a cycle of 65 years.
In 65 years, there are $65\cdot12=780$ regular months and
therefore
$2\cdot 12=24$ leap months, for a total of $804$ months.
Since 12 and 65 are relatively prime,
the leap month can occur at any place in the year; in 65 years, each
leap month 1--12 occurs exactly twice.
\end{remark}
We describe here the basic algorithmic calculations to determine leap months
and thus the numbering of all months. The astronomical theory behind the
rules is explained in \refApp{Aleap}.
The months can be described by year, number (from 1 to 12) and,
possibly, the label ``leap''. We can thus formally think of each month as
labelled by a triple $(Y,M,\ell)$, where $Y\in\bbZ$,
$m\in\set{1,\dots,12}$ and $\ell\in\set{\true,\false}$ is a Boolean variable.
(As said in \refS{Syear}, we number the Tibetan years by the Gregorian (or
Julian) year in which they start.)
The Tibetan calendar calculations also use a
consecutive numbering of all months
(regular or leap)
starting with 0 at some epoch.
(This is thus a linear numbering, ignoring the division into years, unlike
the standard cyclic numbering that starts again with each new year.)
The epoch could in principle be the beginning of
an arbitrarily chosen month; we assume in our formulas that the
epoch is the beginning of month
$M_0$ year $Y_0$.
Any epoch will give the same calendar (provided the correct initial data are
used), and in calculations only a single epoch is chosen.
Nevertheless, to illustrate the calendar mathematics,
we give (and compare) data for three different epochs.
The three epochs we use are
month 3 year 806
(the traditional epoch from \KT{} although it is
several centuries before the calendar came to Tibet,
used \eg{} by \citet{Schuh}),
month 3 year 1927
(used \eg{} by \citet{Henning})
and
month 3 year
1987
(used \eg{} by \citet{LaiDolma} and the almanac
\cite{tib2013}).
We denote these epochs by E806, E1927, E1987.
\xfootnote{
I have chosen these three epochs just as examples.
Another, historically important, traditional epoch is 1687
\cite[p.~331]{Henning}, \cite[Epoch data]{kalacakra}.
}
See further \refR{Repoch} below.
\begin{remark}\label{Repoch0}
By tradition, the epoch is always at the beginning of
month 3 (\tibx{nag pa}, \cf{} \refR{Rcaitra}),
so $M_0=3$.
(But see \refR{Repoch} for the correct interpretation.)
The year $Y_0$ is by tradition usually (but not always) chosen to be
the first year \tib{rab byung} of a Prabhava cycle; it is common to use the
first year of the present cycle (so for example in the almanac
\cite{tib2013} where the epoch is 1987).
(This is convenient for hand calculations
because it gives smaller numbers than older epochs.)
\end{remark}
The number of a month in the linear numbering from a given epoch
is called the
\emph{true month} \tib{zla-dag}, and is calculated as follows
for month $M$ year $Y$:
First, solar months are counted starting after the epoch (month $M_0$, year
$Y_0$); for month $M$ year $Y$,
the ``number of solar months'' is
\begin{equation}\label{MM}
\MM =12(Y-Y_0)+M-M_0.
\end{equation}
Next, one calculates a preliminary version of
the {true month}
as the rational number
\begin{align}\label{tm0}
& \frac{67}{65}\MM+\frac{\gbx}{65}
= \frac{67\bigpar{12(Y-Y_0)+(M-M_0)}+\gbx}{65},
\end{align}
where $\gbx$ is a constant depending on the epoch.
For our three example epochs we have
\begin{align}
\label{gbxS}
\gbx&=61
\qquad(\eS);
\\
\gbx&=55
\qquad(\eH);
\label{gbxH}
\\
\gbx&=0\phantom0
\qquad(\eX).
\label{gbxX}
\end{align}
We write the
fractional part of the true month \eqref{tm0}
as $ix/65$, where the integer $ix$ is called
the \emph{intercalation index}.
\xfootnote{
We use this name
from \cite{Henning};
the Tibetan name \tibx{zla-bshol rtsis-'phro} simply means
``remainder for the calculation of leap-month''
\cite{Schuh-review}, \cite[Kalenderrechnung]{tibetenc}.
}
Thus, the (preliminary version of the) intercalation index is
\begin{align}
\label{ix}
ix&=(67 \MM+\gbx) \bmod{65}
=(2\MM+\gbx) \bmod{65},
\end{align}
with $\MM$ given by \eqref{MM}.
Note that $\gbx$ is the initial value of $ix$ at the epoch.
The traditional \PH{} leap month rule is:
\begin{equation}\label{leaprule-P}
\hskip-1em
\vbox {\narrower\narrower\narrower\noindent\em
A leap month is inserted when the intercalation index\\
$ix = 48\text{ or\/ }49$.
}
\hskip-3em
\end{equation}
For the rest of the calendar calculations, the true month is rounded to an
integer by the following rule.
(We follow \cite{Henning} and use the same name ``true month'' for both the
rational version \eqref{tm0}
and the rounded integer version,
but in order to avoid confusion, we will often call the latter
``\tmc''.)
\begin{equation}\label{mcrule-P}
\hskip-1em
\vbox {\narrower\narrower\narrower\noindent\em
The true month \eqref{tm0} is rounded down to the nearest integer if
$ix<48$ and rounded up if $ix\ge48$, except that for a leap month (when $ix=48$
or $49$), always round down.
}
\hskip-3em
\raisetag\baselineskip
\end{equation}
When there is a leap month, there are two months with the same number $M$
the same year.
The rule just given means that the true month is rounded down for the first
of these (the leap month) and rounded up for the second (the regular month);
thus the \tmc{} will increase by 1 for each new month also immediately
before and after the leap month. Similarly, it is easily checked that the
\tmc{}
increases by 1 also when the intercalation index \eqref{ix} passes
65 and drops to 0 or 1 again. Hence the \tmc{}
calculated by \eqref{mcrule-P} is really a
continuous count of months.
\begin{remark}\label{Rix}
The formulation in \eqref{mcrule-P}
differs somewhat from the traditional formulation,
described \eg{} by \citet{Henning}.
Traditionally, the true month is calculated by \eqref{tm0} as above, but
if this yields an
intercalation index $>49$, one notes that there has been an earlier leap
month and therefore the numbering of months has to be corrected, so the
true month just calculated for month $M$ really applies to month $M-1$;
hence the true month for such a
month $M$
(and for a regular month immediately following a leap month)
is obtained by doing the calculation for
$M+1$ (which means adding
$1;2\rr{1,65}=1\frac{2}{65}$ to the true month calculated for
$M$).
\xfootnote{\label{f6566}
This leaves a gap when the intercalation index reaches 0 or 1:
If month $M$ has true month $n;0$ or $n;1\rr{1,65}$, then the
calculation for $M-1$ yields true month $(n-2);63$ or $(n-2);64$, which thus
applies to $M-2$.
Hence no starting month yields a calculation that applies to $M-1$;
the missing month $M-1$ is usually given
true month $(n-1);0$ or $(n-1);1$, respectively. Note that the integer part
is correct, and that the intercalation index 0 or 1 is repeated.
(Sometimes 65 and 66 are used instead [Henning, personal communication].)
These complications do not appear in my version \eqref{mcrule-P}.
}
The \tmc{} then is always the integer part of the true month
calculated in this way;
\ie, the true month is always rounded down.
This evidently yields the same \tmc{} as
\eqref{mcrule-P}.
However, the fractional part
will differ by $\frac{2}{65}$ for these months,
\ie{} the intercalation index will differ by 2.
(The intercalation index is not used further for the main calendar
calculations, but
almanacs publish for each month the true month with its fractional part,
\ie, the intercalation index,
see \refS{Sfurther}, and it has at least one use, see \refS{Stent}.)
Hence, in order to get the correct intercalation index,
\eqref{mcrule-P} should be extended by the rule that when the true month is
rounded up, the intercalation index is increased by 2.
(When the intercalation index from \eqref{ix} is 63 or 64,
the new value can be given as either 65 or 66, or as 0 or 1, see \refF{f6566}.)
\end{remark}
Since
$48 =65-17$,
\eqref{mcrule-P} can be written, using \eqref{tm0} and \eqref{MM}, and denoting
the \tmc{} by $n$,
\begin{equation}\label{c2}
n
=\Floor{\frac{67\MM+\gbx+17}{65}}-\boolx{\ell}.
\end{equation}
\begin{remark}\label{Repoch}
As said in \refR{Repoch0} above,
the epoch is by tradition
always at the beginning of
month 3 \tib{nag pa}.
However, because of the leap year rules, this has to be interpreted as
follows:
The calculation of the true month uses the number of solar months after
the epoch, calculated by \eqref{MM}.
For month 3 at the epoch, this is 0, so the true month is by \eqref{tm0}
$\gbx/65$; thus the
intercalation index is $\gbx$. According to the rounding rule \eqref{mcrule-P},
this month thus has \tmc{} 0 if $\gbx<48$ but \tmc{} 1 if
$\gbx\ge48$.
(If $\gbx=48$ or 49, there is a leap month 3 and the two months 3 have
\tmc{s} 0 and 1.)
Hence, it is only when $\gbx\le49$ that the epoch really is at the beginning
of calendar month 3; otherwise it is at the beginning of
the preceding month (calendar month 2),
and the nominal epoch month 3 really has \tmc{} 1.
(This means that for an epoch with $\gbx<48$, the \tmc{} is the number of
elapsed (lunar) months
\emph{after} month 3 in the epoch year, while if $\gbx\ge48$, the \tmc{} is
the number of elapsed
months \emph{starting with} month 3.)
For our three example epochs, the values of $\gbx$ in
\eqref{gbxS}--\eqref{gbxX} show that
for \eS{} and \eH{}, the epoch month with \tmc{} 0 is month 2,
while for \eX, it is month 3.
(See also the discussion in \cite{Schuh-review} and \cite{Henning-comments}
on the epoch of \eH.)
Note further that the epoch is not only a year and month; it is a specific
day, see
\refR{Repoch2} in \refS{Sastro}.
\end{remark}
\subsection{The inverse problem}
Although not needed for the traditional construction of a yearly calendar,
let us also consider the inverse problem:
to find the number $M$ and the year $Y$
of the month with \tmc{} $n$,
together with the indicator $\ell$ telling whether
the month is a leap month.
We begin by writing \eqref{c2} as
\begin{equation}\label{c3}
n
=\frac{67\MM+\gbx+17-r}{65},
\end{equation}
where $r$ is chosen such that the result is an integer;
for a regular month, $0 \le r \le 64$
($r$ is the remainder $(67\MM+\gbx+17)\bmod{65}$),
and for a leap month, $r=65$ or $66$
(the remainder is 0 or 1, but $[\ell]=1$ in \eqref{c2}).
Rearranging, we get
\begin{equation}
\MM = \frac{65n-\gbx-17+r}{67}
\end{equation}
with $0\le r \le 66$, and thus
\begin{equation}
\MM = \Ceil{\frac{65n-\gbx-17}{67}}.
\end{equation}
Recalling \eqref{MM}, and assuming $M_0=3$, this can be written
\begin{equation}
\label{jb1}
12(Y-Y_0)+M=\MM+3
=\Ceil{\frac{65n+184-\gbx}{67}}
=\Ceil{\frac{65n+\gb}{67}},
\end{equation}
where we define
\begin{equation}\label{gbgbx}
\gb=184-\gbx.
\end{equation}
For our three example epochs, this yields,
by \eqref{gbxS}--\eqref{gbxX},
\begin{align}
\label{gb1S0}
\gb&=123
\qquad(\eS),
\\
\gb&=129
\qquad(\eH),
\label{gb1H0}
\\
\gb&=184
\qquad(\eX).
\label{gb1X0}
\end{align}
Consequently, we can calculate $(Y,M)$ from $n$ by
\begin{align}
x&=\Ceil{\frac{65n+\gb}{67}},
\label{bx3}\\
M&= x \amod 12,
\label{bx4}\\
Y&=\frac{x-M}{12}+Y_0
= \Ceil{\frac{x}{12}}-1+Y_0.
\label{bx5}
\end{align}
To complete the calculations of $(Y,M,\ell)$ from $n$, we
note that a month is leap if and only if it gets
the same number as the following one. Thus,
\begin{equation}
\label{bx2}
\ell=\Bool{\Ceil{\frac{65n+\gb}{67}}=\Ceil{\frac{65(n+1)+\gb}{67}}}
=\bigbool{(65n+\gb)\bmod67 = 1\text{ or $2$}}.
\end{equation}
Note that $n=0$ in \eqref{bx3}--\eqref{bx4}
yields $M=\ceil{\gb/67}$, so $\ceil{\gb/67}$ is
the number of the epoch month. (This is 2 or 3,
see \refR{Repoch} above.)
\subsection{A general rule}
By \eqref{ix}, \eqref{MM}, \eqref{leaprule-P} and $M_0=3$,
there is a leap month $M$
year $Y$ if and only if
\begin{equation}\label{cx0}
{24(Y-Y_0)+2M-6+\gbx}
\equiv 48\text{ or }{49}\pmod{65}.
\end{equation}
Using \eqref{gbgbx}, $48+6-\gbx=\gb-130\equiv\gb\pmod{65}$ and thus
\eqref{cx0} can be written
\begin{Rule}
There is a leap month $M$ in year $Y$ if and only if
\begin{equation}\label{leaprule-gb0}
24(Y-Y_0)+2M
\equiv \gb\text{ or\/ }\gb+1\pmod{65}.
\end{equation}
\end{Rule}
We will see later, in Appendices \ref{Aversions} and \ref{Aleap},
that the rule \eqref{leaprule-gb0} holds
also for other versions of the Tibetan calendar, with appropriate $\gb$.
By combining \eqref{ix}, \eqref{MM} and $M_0=3$,
\begin{equation}\label{ixq}
ix = \bigpar{24(Y-Y_0)+2M+\gbx-6}\bxmod{65}
\end{equation}
(if necessary modified as in \refR{Rix})
and hence the rule \eqref{leaprule-gb0} says
that a
leap month has intercalation index
\begin{equation}\label{jeppe}
(\gb+\gbx-6)\bmod65
\qquad \text{or}\qquad
(\gb+\gbx-5)\bmod65 ,
\end{equation}
which is the general form of the relation between the \PH{} formulas
\eqref{gbgbx} and \eqref{leaprule-P}.
\subsection{Leap years}\label{SSleapyear}
We give some further formulas for
leap years and leap months, inspired by \cite{Salmi}, that follow from the
formulas above.
(These formulas are not traditional, and are not needed to construct
the calendar.)
We begin by stating them in a general form, valid also for other versions of
the calendar.
Since $M$ may be any number $1,\dots,12$, it follows
from \eqref{leaprule-gb0}
that $Y$ is a leap year
if and only if
\begin{equation}\label{cu}
24(Y-Y_0)-\gb \equiv 41,\,42,\dots,\text{ or }64\pmod{65},
\end{equation}
which also can be written as
\begin{equation}\label{cv}
\bigpar{ 24(Y-Y_0)-\gb} \bxmod 65 \ge 41.
\end{equation}
Since $24\cdot19=456\equiv 1\pmod{65}$, this can be further rewritten as
\begin{equation}\label{cw}
24(Y-Y_0-19\gb) \bxmod 65 \ge 41.
\end{equation}
Hence, if we define
\begin{equation}\label{gam}
\gamma= (-Y_0-19\gb)\bxmod{65},
\end{equation}
we can state the rule as:
\begin{Rule}
$Y$ is a leap year if and only if
\begin{equation}\label{lygam}
24(Y+\gamma) \bxmod 65 \ge 41.
\end{equation}
\end{Rule}
Equivalently, if we define
\begin{equation}\label{gamx}
\gamx=24\gamma \bxmod 65
=(-24 Y_0-\gb)\bxmod 65,
\end{equation}
then:
\begin{Rule}
$Y$ is a leap year if and only if
\begin{equation}\label{lygamx}
(24\,Y+\gamx) \bxmod 65 \ge 41.
\end{equation}
\end{Rule}
Furthermore, it easily seen from \eqref{leaprule-gb0}, \cf{} \eqref{cu},
that if $Y$ is a leap year, then the leap month has number
\begin{equation}\label{mk}
M = 1+\Floor{\frac{64-\bigpar{24(Y-Y_0)-\gb} \bxmod 65}2 }
\end{equation}
which, using \eqref{gamx}, easily is transformed to
\begin{equation}\label{mq}
M
= \Floor{33-\frac{(24\,Y+\gamx) \mod 65}2 }
= \Floor{33-\frac{24(Y+\gamma) \mod 65}2 };
\end{equation}
moreover,
if $Y$ is not a leap year, then \eqref{mk}--\eqref{mq} yield an impossible
value $M\ge13$.
Finally, if $Y_1 \le Y_2$, the number of leap years (and thus the number of
leap months) in the period from $Y_1$ to $Y_2$ (inclusive) is
\begin{equation}\label{lys}
\Floor{\frac{24(Y_2+1)+\gamx}{65}} - \Floor{\frac{24Y_1+\gamx}{65}}.
\end{equation}
To see this it suffices to consider the case $Y_2 =Y_1$, which easily
follows from \eqref{lygamx}.
For the standard \PH{} version,
\eqref{gam} can be written, using \eqref{gbgbx},
\begin{equation}\label{gam-P}
\gamma= (-Y_0-19\gb)\bxmod{65}
=(-Y_0+19\gbx+14)\bxmod{65},
\end{equation}
and
the values of $\gbx$ in
\eqref{gbxS}--\eqref{gbxX} yield
\begin{align}
\gamma &=42
\intertext{and thus}
\gamx &=24\cdot 42\mod 65 = 33.
\end{align}
(The values of $\gamma$ and $\gamx$ are the same for any epoch, since
the leap years are the same. However, for other versions of the calendar, the
formulas hold with other values of $\gamma$ and $\gamx$,
see \refApp{Aversions}.)
Thus the rule is \cite{Salmi}:
\begin{Rule}
$Y$ is a leap year if and only if
\begin{equation}
24(Y+42) \mod 65 \ge 41,
\end{equation}
\end{Rule}
or, equivalently,
\begin{Rule}
$Y$ is a leap year if and only if
\begin{equation}
(24\,Y+33) \mod 65 \ge 41.
\end{equation}
\end{Rule}
For the \PH{} version, \eqref{mq} is
\begin{equation}
M = \Floor{33-\frac{(24\,Y+33) \mod 65}2 }.
\end{equation}
\section{Days}\label{Sdays}
As said in \refS{Sdef},
each lunar month (from the instant of new moon to the next new moon)
is (as in the Indian calendars \cite{CC}) divided into 30
lunar days;
these have varying length of between 21.5 and 25.7 hours, and
do not correspond exactly to the
calendar days of 24 hours each.
During each of these lunar days,
the elongation of the moon
(\ie, the difference between lunar and solar longitude)
increases by 1/30 ($=12\grad$).
The calendar computations, unlike the Indian ones, do not include a
function calculating the elongation at a given time; instead the
computations use the inverse function, giving directly the time when the
elongation has a given value. We denote this function,
described in detail in \refS{Sastro}, by
$true\_date(d,n)$, giving the date at the end of the lunar day $d$ in
\tmc{} $n$. The value of this function is a real (rational) number;
traditionally it is counted modulo 7, and the integer part yields the
day of week, but we will treat it as a real number so that the integer
part directly gives the Julian day number JD.
(A different constant $m_0$ below will give RD \cite{CC} instead.)
The fractional part shows the time the lunar day ends;
it is used to calculate some further astronomical (and astrological)
information, see \refS{Sfurther}, but can be ignored for the present purpose.
The basic rule is:
\begin{Rule}
A calendar day is labelled by the lunar day
that is current at the beginning of the calendar day.
\end{Rule}
In other words,
a lunar day gives its name (number and month) to the calendar day
where the lunar day
ends. (Thus the JD of the calendar day is the
integer part
of $true\_date$ at the end of the corresponding lunar day.)
There are two special
cases covered by the rule above: if two lunar days end the same
calendar day, then the calendar day gets the name of the first of them;
if no lunar day ends a given calendar day, then that day gets the same
name as the following day.
The first special case occurs when a lunar day is completely contained in one
calendar day; in that case no calendar day gets the number of this lunar
day, so this date is skipped. (In the sense that the number is skipped in the
numbering of days in the
calendar. The lunar day itself exists and can be used for astrological
purposes. The calendar days exist in real life and, of course, as such
cannot be skipped or repeated.)
The second case occurs when a lunar day completely contains a calendar
day; in that case this calendar day gets the same number of the next day, so
the date is repeated. (In the sense that the number is repeated.)
When a date is repeated, the first of the two days with the same number is
regarded as a leap day, and denoted ``Extra'' in the almanacs.
\cite{Schuh-review}
(But see \refS{Sholidays}.)
Recall also that each calendar day has a day of week, in the same way as in
Western calendars, see \refS{Sweek}. In particular,
when a day is repeated, the
two days with the same number are distinguished by different days of week.
\begin{remark}\label{Rdate}
We do not have to worry about when the day starts; the calendar day is
from dawn
to dawn, but the formulas take this into account (at least
theoretically),
and no further modification is done.
In particular, no calculation of sunrise is required (as it is in Indian
calendar calculations \cite{CC}).
Thus, the $true\_date$ should be regarded as a
kind of local Julian date that is offset from the standard
astronomical one which assumes integer values at noon UT (= GMT), so
that it instead assume integer values at local (mean) dawn.
(Henning \cite[pp.~10--11]{Henning} and \cite{Henning-comments}
specifies the start of the day as
mean daybreak = 5 am local mean solar time.
Since the time difference is about 6 hours (Lhasa has longitude $91\grad$
which corresponds to $6^h4^m$), this is about
$-1$ UT, i.e.\ 11 p.m.\ UT the preceding day.)
\end{remark}
\begin{remark}
By definition, new moon is (exactly) at the end of lunar day 30 and full
moon is at the end of lunar day 15; the rules above imply that unless the
day is skipped, (true) new moon falls in calendar day 30 and (true)
full moon in calendar day 15 in every month. The true elongation differs
from the correct astronomical value by only about $2\grad$, corresponding to 4
hours, see \refS{SSdrift},
so usually the same holds for the astronomcal new moon and full moon as well.
\end{remark}
\section{Astronomical functions}\label{Sastro}
The $true\_date$ is calculated by first calculating a simpler
$mean\_date$, corresponding to the linear mean motion of the moon,
and then adjusting it by the equations of the moon and
sun, which are determined by the anomalies of the moon and sun
together with tables. (The tables are really approximation to sine,
suitably scaled. A similar table is used for each planet; no general
sine table is used, another difference from Indian calendars \cite{CC}.)
The anomalies, in turn, are also calculated by linear functions.
Traditional hand calculations calculate the mean quantities first for the
beginning of the month, \ie{} the end of the preceding month. (This
corresponds to taking day $d=0$ below. Note that this usually gives a time
during the last calendar day of the preceding calendar month.)
Then the quantities are adjusted to give the values for a given day.
We will combine the two steps into one, giving the values for \tm{} $n$
and day $d$ directly.
\xfootnote{
\citet{LaiDolma} refer to tables rather than doing multiplications to obtain
the adjustments for $d$ for $mean\_date$ and (without explanation)
for $mean\_sun$, but the results are the same.
}
The \PH{} tradition uses the following functions. As said above,
these give the values at the end of lunar day $d$ in \tmc{} $n$.
I use below the epoch \eS, but give also the corresponding constants for \eH{}
and \eX. (Recall that the different epochs yield the same calendar.)
The mean date \tib{gza' bar pa}
is (for \eS)
\begin{equation}\label{meandate}
mean\_date(d,n)=n\cdot m_1+d\cdot m_2+m_0,
\end{equation}
where
\begin{align}
m_1&=
29;31,50,0,480\rr{60,60,6,707}=\frac{167025}{5656} \quad(\approx 29.530587),
\label{m1}
\\
m_2&=0;59,3,4,16 \rr{60,60,6,707}=\frac{11135}{11312}
\quad\Bigpar{=\frac{m_1}{30}},
\\
m_0&=0;50,44,2,38 \rr{60,60,6,707}+2015501 =2015501+\frac{4783}{5656}.
\label{m0}
\end{align}
\begin{remark}\label{R7}
The traditional reckoning counts days modulo 7 only, \ie{} day of week
(see \refS{Sweek});
this is of course enough to construct a calendar month by month.
My version gives the JD directly. To be precise, the traditional
result is obtained by adding 2 to the value in \eqref{meandate} before
taking the remainder modulo 7,
\cf{} \eqref{weekday}.
(Hence, the integer added to the traditional value of $m_0$ has to be
congruent to $-2$ modulo 7.
Indeed,
$2015501\equiv -2 \pmod7$, and similarly for the other epochs below and in
\refApp{Aversions}.)
The traditional value is therefore
$m_1=1;31,50,0,480 \rr{60,60,6,707}$,
subtracting 28 from the value used in this paper.
(It can also, and perhaps better, be regarded as
$1,31,50,0,480 \rr{7,60,60,6,707}$, with the denominator 7 meaning that
we regard the numbers as fractions of weeks, thus ignoring integer parts).
Similarly, the constant 2015501 in $m_0$ (to get the result in JD) is my
addition and not traditional.
(To get RD, use 294076 instead.)
\end{remark}
\begin{remark}
For \eH{} \cite{Henning},
a simple calculation using \eqref{tm0} shows that the \tmc{} $n$ differs by
13866 from the value for \eS; thus
$m_0$ is instead
\begin{align*}
m_0+13866\cdot m_1
&= 2424972+\frac{5457}{5656}
\\
&= 2015501+409471+\frac{5457}{5656}
\\
&
=2015501+409465+6;57,53,2,20 \rr{60,60,6,707},
\end{align*}
traditionally written as $6;57,53,2,20 \rr{60,60,6,707}$.
(Note that $409465=7\cdot58495$ a multiple of 7, \ie{} a
whole number of weeks; hence this gives the correct day of week.)
Similarly,
for \eX{} \cite{LaiDolma},
with the value of $n$ differing by 14609, $m_1$ and $m_2$ are the same
but $m_0$ is instead
\begin{align*}
m_0+14609\cdot m_1
&= 2446914+\frac{135}{707}
\\
&= 2015501+431413+\frac{135}{707}
\\
&
=2015501+431410+3;11,27,2,332 \rr{60,60,6,707},
\end{align*}
traditionally written as $3;11,27,2,332 \rr{60,60,6,707}$.
\end{remark}
Similarly, the mean longitude of the sun \tib{nyi ma bar pa}
is
\begin{equation}\label{meansun}
mean\_sun(d,n)=n\cdot s_1+d\cdot s_2+s_0,
\end{equation}
where,
\xfootnote{\label{f67-707}
Note that the radices used for longitude are not the same as the
radices used for time, see \eg{} \eqref{m1} and \eqref{s1};
the last radix is 67
instead of 707. This ought to cause some problems when converting between
times and longitudes, but the difference is only in the last term
and is usually ignored.
}
\begin{align}
s_1&=2,10,58,1,17 \rr{27,60,60,6,67}=\frac{65}{804}
\quad\Bigpar{=\frac{65}{12\cdot67}}, \label{s1}
\\
s_2&=0,4,21,5,43 \rr{27,60,60,6,67}=\frac{13}{4824}
\quad\Bigpar{=\frac{s_1}{30}}, \label{s2}
\\
s_0&=24,57,5,2,16 \rr{27,60,60,6,67} =\frac{743}{804}. \label{s0}
\end{align}
\begin{remark}
For \eH{} \cite{Henning},
with the base for $n$ differing by 13866, $s_1$ and $s_2$ are the same
but
$s_0$ is instead (modulo 1, since only the fractional part matters)
\begin{align}\label{s0E1927}
s_0+13866\cdot s_1
\equiv
25,9,10,4,32 \rr{27,60,60,6,67}
=\frac{749}{804}
.
\end{align}
For \eX{} \cite{LaiDolma},
with the base for $n$ differing by 14609,
$s_0$ is instead
\xfootnote{
This vanishing of the initial value, which recurs every 65th year
(804th month),
is called \tibx{nyi ma stong bzhugs}
``sun empty enter'' \cite{LaiDolma}.}
\begin{align}\label{s0E1987}
s_0+14609\cdot s_1 \equiv 0.
\end{align}
\end{remark}
Thirdly,
the anomaly of the moon
\tib{ril-po dang cha-shas}
is
\begin{equation}\label{anomalymoon}
anomaly\_moon(d,n)=n\cdot a_1+d\cdot a_2+a_0,
\end{equation}
where (but see \refR{R30} below for an alternative for $a_2$)
\begin{align}
a_1&=2,1 \rr{28,126}=\frac{253}{3528},
\\
a_2&=1,0 \rr{28,126}=\frac1{28}, \label{a2}
\\
a_0&=3,97 \rr{28,126}=\frac{475}{3528}.
\end{align}
\begin{remark}
For \eH{} \cite{Henning},
with the base for $n$ differing by 13866,
$a_0$ is instead (modulo 1, since only the fractional part matters)
\begin{align}
a_0+13866\cdot a_1
\equiv 13,103 \rr{28,126}=\frac{1741}{3528}.
\end{align}
For \eX{} \cite{LaiDolma},
with the base for $n$ differing by 14609,
$a_0$ is instead
\begin{align}
a_0+14609\cdot a_1 \equiv 21,90 \rr{28,126}=\frac{38}{49}.
\end{align}
\end{remark}
\begin{remark}
For comparisons with modern astronomical calculations, note that
the anomaly is measured from the Moon's apogee, while Western
astronomy measures it from the perigee, which makes the values
differ by a half-circle.
\end{remark}
The equation of the moon \tib{zla rkang} is calculated by
\begin{equation}\label{moonequ}
moon\_equ=moon\_tab(28\cdot anomaly\_moon)
\end{equation}
where $moon\_tab(i)$ is listed in the following table for $i=0,\dots,7$,
which extends by the symmetry rules
$moon\_tab(14-i)=moon\_tab(i)$,
$moon\_tab(14+i)=-moon\_tab(i)$, and thus
$moon\_tab(28+i)=moon\_tab(i)$;
linear interpolation is used beween integer arguments.
\begin{equation}\label{moontab}
\begin{tabular}{l r r r r r r r r}
$i$ &0 & 1 & 2 & 3 & 4 & 5 & 6 & 7\\
$moon\_tab(i)$ & 0 & 5 & 10 & 15 & 19 & 22 & 24 & 25
\end{tabular}
\end{equation}
To find the equation of the sun \tib{nyi rkang}, first calculate the anomaly by
\begin{equation}
\label{anosun}
anomaly\_sun=mean\_sun-1/4
\end{equation}
and then take
\begin{equation}\label{sunequ}
sun\_equ=sun\_tab(12\cdot anomaly\_sun)
\end{equation}
where $sun\_tab(i)$ is listed in the following table for $i=0,\dots,3$,
which extends by the symmetry rules
$sun\_tab(6-i)=sun\_tab(i)$,
$sun\_tab(6+i)=-sun\_tab(i)$, and thus
$sun\_tab(12+i)=sun\_tab(i)$;
linear interpolation is used beween integer arguments.
\begin{equation}\label{suntab}
\begin{tabular}{l r r r r r r r r}
$i$ &0 & 1 & 2 & 3\\
$sun\_tab(i)$ & 0 & 6 & 10 & 11
\end{tabular}
\end{equation}
The date at the end of the lunar day \tib{gza' dag}
is finally calculated as
\begin{equation}\label{truedate}
true\_date =mean\_date +moon\_equ/60-sun\_equ/60.
\end{equation}
(The half-corrected $mean\_date +moon\_equ/60$ is called
semi-true date \tib{gza' phyed dag pa}.)
Similarly, although not needed to calculate the calendar date, the
true solar longitude
\tib{nyi dag} is
\begin{equation}\label{truesun}
true\_sun=mean\_sun-sun\_equ/(27\cdot 60).
\end{equation}
\begin{remark}
You will not find the factors 1/60 and $ 1/(27\cdot 60)$ explicit in the
references; they are consequences of the positional system with mixed
radices.
Furthermore, $1/4$ in \eqref{anosun} is traditionally expressed as
$6,45\rr{27,60}$.
\end{remark}
\begin{remark}\label{R30}
We have $m_2=m_1/30$ and $s_2=s_1/30$, which is very natural, since
it means that the functions $mean\_date$ and $mean\_sun$ are linear
functions of the lunar day count $d+30\cdot n$, and thus increase by the
same amount every day without any jumps at the
beginning of a new month.
\xfootnote{\label{fm'}
Schuh \cite{Schuh}, \cite[Kalenderrechnung]{tibetenc}
gives several examples ($m=3,4,5,6,7$ in his notation) of earlier versions
of the calendar (with $m_1$ and $s_1$ slightly different from the values
above), where simpler, rounded, values of $m_2$ and $s_2$ were
used. Moreover, most of these versions used $m_1$ and $s_1$ only for
calculating for the first month each year, and simplified value $m_1'$ and
$s_1'$ for the increments for the successive months within the year.
Some of them ($m=3,6,7$) also used the simplified $a_1'=2,0 \rr{28,126}\equiv
30 a_2$ for
the monthly increments of the anomaly.
See \refApp{AKT} for an example.
I do not know any currently used versions of the calendar that use such
simplifications.
}
For the anomaly, however, the standard value $a_2=1/28$ does not
conform to this. Note that $a_1$ could be replaced by
$1+a_1=30,1 \rr{28,126}$ since we count modulo 1 here;
in fact, this is the ``real'' value, since the
astronomical anomaly increases by 1 full circle
in a little less that one month, see also
\eqref{meanano}.
Moreover, $a_2=1,0\rr{28,126}=1/28$ is a close approximation to
$(1+a_1)/30$; the
conclusion is that one usually for convenience uses the rounded value
$a_2=1/28$. \citet{Henning}, however, uses instead the exact value
\begin{equation}\label{a2lochen}
a_2=\frac{1+a_1}{30}=\frac{3781}{105840}=\frac{1}{28}+\frac1{105840}
=1,0,1 \rr{28,126,30};
\end{equation}
this is also used in his computed calendars
\cite[Traditional Tibetan calendar archive]{kalacakra}.
\xfootnote{
The value \eqref{a2lochen} was proposed by
Minling Lochen Dharmashri (1654--1717).
\Hp.
}
The value \eqref{a2lochen} is mathematically more natural than
\eqref{a2}, since the latter value yields a (small) jump in the anomaly
at the end of each month while \eqref{a2lochen} yields same increase every
day.
Nevertheless, the simpler \eqref{a2} is usually used when calculating
calendars, for example in the almanac \cite{tib2013}.
\xfootnote{\label{fa2}
This is witnessed by, for example, the values given for the true day of week
(including fractional part), calculated by \eqref{truedate}.
This is given in the almanac with 3 terms (1,60,60), and
the two calculations often differ (more often towards the end of the month),
although
typically only by 1 unit in the third term.
A computer calculation for the 360 lunar days in 2013 yields
134 days with no difference in the true date (truncated to three terms),
149 days with a difference $\pm1$ in the third term, 75 days with $\pm2$ and
2 days with $\pm3$.
}
The difference $1/105840$ between \eqref{a2} and \eqref{a2lochen}
is small, and the resulting difference in the anomaly
is at most $30/105840=1/3528$; the difference in the argument to
$moon\_tab$ is thus at most $28/3528=1/126$; since the increments in
$moon\_tab$ are at most 5, the difference in $moon\_equ$ is at most
$5/126$; finally, by \eqref{truedate}, the difference in the true
date is at most $(5/126)/60 =1/1512$
(see also \refF{fa2}),
so when rounding to an integer
(see \eqref{jd} below), we would expect to obtain the same result
except, on the average, at most once in 1512 days.
We would thus expect that the two different values of $a_2$ would give
calendars that differ for at most one day in 1512 on the average.
Moreover, since this was a maximum value, and the average ought to be
less by a factor of about 1/2, or more precisely $15.5/30=31/60$, since the
difference is proportional to the day $d$ which on the average is
$15.5$, and by another factor of
$5/7$ since the average derivative of $moon\_tab$ is $25/7$ while we
just used the maximum derivative 5.
(For a sine function, the average absolute value of the derivative is
$2/\pi$ times the maximum value, but the ratio is closer to one for
the approximation in $moon\_tab$.)
Hence we would expect the average (absolute) difference in
$true\_date$ to be about
\begin{equation*}
\frac{31}{60}\cdot\frac{5}{7}\cdot\frac1{1512}
\approx\frac1{4100}.
\end{equation*}
Consequently, we expect
that the two versions of $a_2$ will lead to different
Tibetan dates for, on the average, one day in 4100,
or a little less than one day in 10 years.
This is confirmed by a computer search finding 9 examples in 1900--1999 and 8
examples in 2000--2099.
\xfootnote{On the other hand, there are 16 examples in the first 65 year
cycle 1027--1091, so the occurences are irregular.}
Some recent examples are
10 February 2001 (JD 2451951) and
10 May 2006 (JD 2453866); the next example is
19 November 2025 (JD 2460999).
(It would be interesting to check these dates in published calendars.)
\end{remark}
\begin{remark}\label{Rlunarday}
From one day to the next, the anomaly of the moon increases by 1/28 (or
slightly more if \eqref{a2lochen} is used, and by the slightly larger
$1,1\rr{28,126}= 127/3528$ at each new month if \eqref{a2lochen} is not
used; we can ignore these differences);
this means that the arguments used in \eqref{moonequ} differ by 1,
and thus the resulting values of $moon\_equ$ differ by at most 5 (see
\eqref{moontab}). Similarly, the anomaly of the sun differs by slightly less
than $1/360$, so by \eqref{sunequ} and \eqref{suntab}, the values of
$sun\_equ$ differ by at most about $1/5=0.2$.
The mean date increases by $m_2 =0.98435$ each lunar day;
this is thus the length of the mean lunar day, measured in calendar days,
which equals $24m_2=23.6245$ hours.
By \eqref{truedate} and the calculations just made
we see that the increase of the true date from one lunar
day to the next, \ie, the length of the true lunar day,
differs from this by at most $\pm (5+0.2)/60=0.087$ days, or $2.1$ hours.
Consequently, the length of the (true) lunar day varies between, roughly,
21.5 and 25.7 hours.
A similar calculation shows that the length of a (true) lunar month varies
between 29.263 and 29.798 days, or about $29^d6^h$ and $29^d19^h$.
Similarly, a lunar year of 12 lunar months has length
between 354.00
and 354.74 days,
and a lunar year of 13 lunar months has length between
383.67 and 384.13 days.
The length of the calendar year is one of these lengths for a lunar year
rounded up or down to one of the nearest integers; the possibilities are
thus 354, 355, 383, 384, 385.
A computer calculation (for 10000 years) gave the frequencies:
$$
\begin{tabular}{rrrrr}
354&355&383&384&385\\\hline
42\%&21\%&3\%&33\%&1\%
\end{tabular}.
$$
\end{remark}
\begin{remark}\label{Repoch2}
The epoch is a specific day (or instant), \viz{} the mean new moon at
the beginning of the month with \tmc{} 0 (see \refR{Repoch} for this month).
By \eqref{meandate}, the mean date of the epoch is $m_0$ (since $n=d=0$);
hence the JD of the epoch is $\floor{m_0}$.
Our version thus encodes the epoch in $m_0$, and the epoch is not needed
explicitly. Traditionally, with $m_0$ given modulo 7, see \refR{R7}, $m_0$
gives only the day of week of the epoch (with $\floor{m_0}$ giving the
number according to \refT{T7}, see \eqref{weekday}), and the epoch date is
needed to construct the calendar.
The epoch is always close to the beginning of month 2 or 3, depending on
$\gbx$ (the initial value of the intercalation index), see \refR{Repoch}.
Since the calculation is for the beginning of lunar day 1
(which equals the end of the last lunar day of the preceding lunar month),
the epoch is usually day 30 of calendar month 1 or 2, respectively,
\cf{} the rule for labelling calendar days in \refS{Sdays};
however, since the true date differs from the mean date, see
\eqref{truedate}, it may be one of the adjacent calendar days
(29/1 or 1/2, or 29/2 or 1/3).
For our three example epochs, the Tibetan dates are
29/1 806,
29/1 1927
and
1/3 1987.
\end{remark}
\section{Calendrical functions}\label{Scal}
As explained in \refS{Sdays},
the Julian day number JD of a given Tibetan date can be calculated as the
JD of the
calendar day containing the end of the corresponding lunar day, i.e.\
\begin{equation}\label{jd}
\JD=\floor{true\_date},
\end{equation}
with $true\_date$ given by \eqref{truedate},
except that if the Tibetan date is repeated, this gives the JD
of the second day; for the first we thus have to subtract 1.
If we do the calculations for a day that is skipped, formula
\eqref{jd} will still give the JD of the calendar day that lunar day
ends, which by the rule in \refS{Sdays} is the day with name of the preceding
lunar day (since that lunar day ends the same calendar day).
The calendar is really given by the inverse of this mapping; a day is
given the number of the corresponding lunar day (\ie, the lunar day ending
during the calendar day).
To find the Tibetan date for a given JD, we thus compute approximate
\tmc{} and day (using the mean motion in \eqref{meandate}) and then
search the neighbouring lunar days for an exact match, if any, taking care
of the special cases when there are 0 or 2 such lunar days. For a
detailed implementation, see
Dershowitz and Reingold \cite{CC}.
\subsection*{Beginning and end of months}
To find the last day of a month, we can just compute the JD for lunar
day 30 of the month; this gives the correct result also when day 30
is repeated or skipped. To find the first day, however, requires a little
care since lunar day 1 may end during the second day of the month
(when day 1 is repeated) or during the last day of the preceding month
(when day 1 is skipped); the simplest way to find the JD of first day of a
month is to add 1 to the JD of the last day of the preceding month.
By the comment on skipped days above, this can be computed as 1 + the
JD of day 30 the preceding month, regardless of whether day 30 is
skipped or not.
(There seems to be errors in the tables in \cite{Schuh} due to this.)
\subsection*{Tibetan New Year}
The Tibetan New Year \tib{Losar}
is celebrated starting the first day of the year. Since the
first month may be a leap month 1
(which happens twice every 65 year cycle,
the last time in 2000) and the first day may be day 2 (when day 1 is
skipped), some care has to be taken to calculate the
date. The simplest is to add 1 to the JD of the last day the preceding
year, which thus is 1 + JD of the last day of (regular) month 12 the
preceding year (and can be calculated as 1 + JD of day 30 of (regular)
month 12 the preceding year).
\begin{remark}\label{Rlosar}
Holidays are otherwise usually not celebrated in leap months.
However, the Tibetan New Year is really the first day of the year
even in a year that begins with a leap month 1.
\xfootnote{
This is verified by
\cite{Salden}
(written by a representative of the personal monastery of the Dalai Lama),
according to which
\tibx{Losar} was
celebrated on Sunday, February 6th (2000), which was the first day of
leap month 1.
}
\end{remark}
\section{Day of week}\label{Sweek}
Each calendar day is, as explained in Sections \refand{Sdays}{Scal},
given a number in the range 1,\dots,30.
It also has a day of week, which as in the Gregorian and many other
calendars simply repeats with a period of 7.
The day of week thus corresponds uniquely to the Western day of week.
The days of week are numbered 0,\dots,6, and they also have names.
Each day of week corresponds to one of the seven ``planets'' (including sun and
moon), and the name of the day is more or less identical to the name of the
planet.
\xfootnote{\label{fplanets}
The correspondence between days of week and planets in Tibetan
is the same as in Latin and (incompletely) in many modern European languages
(e.g., Sunday, Monday and Saturday in English).
This correspondence goes back to Roman astrology, see
\cite[pp.~268--273 and 391--397]{Richards} and \cite[\S12.12]{AA},
and came to Tibet through India.
}
The correspondence with
the English names is given in \refT{T7}.
(For the last column, see \refApp{Aastro}.)
\begin{table}[!htpb]
\begin{tabular}{r l l l l}
& English & Tibetan & planet & element\\
\hline
0 & Saturday & spen ma & Saturn & earth \\
1 & Sunday & nyi ma & Sun & fire \\
2 & Monday & zla ba & Moon & water \\
3 & Tuesday & mig dmar & Mars & fire \\
4 & Wednesday & lhag pa & Mercury & water \\
5 & Thursday & phur bu & Jupiter & wind \\
6 & Friday & pa sangs & Venus & earth
\end{tabular}
\caption{Days of week.}
\label{T7}
\end{table}
The day of week is simply calculated from the Julian day number found
in \eqref{jd} by
\begin{equation}\label{weekday}
day\_of\_week=(\JD+2) \mod 7.
\end{equation}
The date (\ie, the number of the day) and the day of week are often
given together. This resolves the ambiguity when days are repeated.
(It also helps to resolve most ambiguities when different rules of
calculation may have been used.)
\section{Further calculations}\label{Sfurther}
Tibetan almanacs traditionally also contain further information,
mainly for astrological purposes, see \citet[Chapter IV]{Henning}.
(See also his extensive computer calculated
examples \cite[Traditional Tibetan calendar archive]{kalacakra}
and programs with explanations
\cite[Open source Tibetan calendar software]{kalacakra},
\cite[Open source Tsurphu calendar software]{kalacakra}.)
In fact, the calendar is known as ``the five components''
\tib{lnga-bsdus},
\xfootnote{
Just as in Indian calendars \tib{panchang},
see \cite[Chapter 18, p.~312]{CC}.
}
where the five components are: the day of week,
the lunar day, the lunar mansion, the yoga and the \karana{}
(for these, see below).
\xfootnote{
\cite[Kalenderrechnung]{tibetenc} has a slightly different interpretation,
including the longitude of the sun as one of the five
and omitting the day of week (which is seen as given).
}
Nevertheless,
a complete Tibetan almanac
typically contains not only these five but also some further data.
(The following description is based on \cite{Henning} and the almanac
\cite{tib2013}, see also the examples in \cite[pp.~202--203]{Henning},
\cite{Schuh-review} and \cite[Kalenderrechnung]{tibetenc}.
There are certainly minor variations between different almanacs.)
The daily data in an almanac include (typically) the following.
See further \refApp{Aastro}.
The numbers are usually truncated to 3 significant terms, so not all radices
given below are used.
\begin{romenumerate}[-10pt]
\item \label{q-gza-dag}
The true day of week \tib{gza' dag},
\ie{}
the day of week and fractional part of the day when the lunar day
ends. This is
$(true\_date+2)\mod 7$,
calculated by \eqref{truedate} and
written with the radices (60,60,6,707).
The integer part (\ie, the first digit) of the true day of week
is thus the day of week;
its name (given by \refT{T7})
is also given in letters, together with the (lunar) date.
\item\label{q-tshes-khyud}
The (true) longitude of the moon at the end of the lunar day
\tib{tshes 'khyud zla skar},
calculated for lunar day $d$, \tm{} $n$ by
\begin{equation}\label{tseskhyud}
moon\_lunar\_day(d,n)=true\_sun(d,n)+d/30
\end{equation}
and written in lunar mansions with the radices (27,60,60,6,67).
(The rationale for \eqref{tseskhyud} is that the moon's elongation, \ie{}
the difference between lunar and solar longitude, by definition is
$d/30$ at the end of lunar day $d$, see \refS{Sdef}.)
\item\label{q-zla-skar}
The (true) longitude of the moon at the beginning of the calendar day
\tib{res 'grogs zla skar}
calculated from the values in \eqref{tseskhyud} and \eqref{truedate}
by (recalling that $\FRAC(x)$ denotes the fractional part of $x$)
\begin{equation}\label{zla-skar}
moon\_calendar\_day=moon\_lunar\_day-\FRAC(true\_date)/27,
\end{equation}
and written in lunar mansions with the radices (27,60,60,6,67).
(The idea here is that $\FRAC(true\_date)$ is the time from the beginning of
the calendar day to the end of the lunar day, so this formula is really an
approximation assuming that the moon moves with constant speed and making a
full circle in 27 days.
\xfootnote{\label{f27}
This is of course not exact, but it is a rather good approximation
since the tropical
\xpar{or sidereal; the difference is negligible}
month is 27.322 days
\cite[Table 15.3]{AA}.
In the calculation, one may also ignore that
in the traditional notations, the last radix differs betwen the true date
and the longitudes, \cf{} \refF{f67-707}.
}
The division by 27 is convenient since it is
simply a shift of the terms in the mixed radix notation.)
\item\label{q-naksatra}
The name of the lunar mansion (Sanskrit \emph{naksatra}), which is
determined by the Moon's longitude \eqref{zla-skar}
with fractions of mansions ignored.
More precisely, numbering the lunar mansions from 0 to 26,
the number of the lunar mansion is
\begin{equation}\label{mansion}
\floor{27moon\_calendar\_day}.
\end{equation}
(In the traditional notation, this is the first term of the lunar longitude
\eqref{zla-skar}.)
The names of the mansions (in Tibetan and Sanskrit)
are listed in \cite[Appendix I]{Henning}.
\item \label{q-nyi-dag}
The (true) longitude of the sun \tib{nyi dag},
given by
$true\_sun$
and written in mansions with the radices (27,60,60,6,67).
(Note that $true\_sun$ really is computed for the end of the lunar
day, but is regarded as valid also for the calendar day;
the motion of the sun during the day is thus ignored
and no correction as in \eqref{zla-skar} is made.
\xfootnote{
The motion of the sun is much slower than the motion of the moon; it is
about $1/365\approx1\grad$ each day.
})
\item \label{q-yoga-long}
The \emph{yoga} ``longitude'' \tib{sbyor ba} is the sum of the
longitudes of the sun and the moon, calculated by
\xfootnote{
As noted by \cite{Henning},
this is inconsistent, since we add one longitude at the beginning of the
calendar day and one longitude at the end of the lunar day. On the other
hand, this addition has in any case no physical or astronomical meaning.
}
\begin{equation}
\mathit{yoga\_longitude}=moon\_calendar\_day+true\_sun
\pmod1,
\end{equation}
and written in mansions with the radices (27,60,60,6,67).
\item \label{q-yoga}
The name of the yoga, which is
determined by the yoga longitude with fractions of mansions ignored.
More precisely, numbering the yogas from 0 to 26,
the number of the yoga is
\begin{equation}
\floor{27\mathit{yoga\_longitude}}.
\end{equation}
The names of the yogas (in Tibetan and Sanskrit)
are listed in \cite[Appendix I]{Henning}.
(The names differ from the names of the lunar mansions.)
\item \label{q-karana}
The \emph{\karana} \tib{byed pa}
in effect at the start of the calendar day. Each
lunar day is divided into two halves, and each half-day is assigned
one of 11 different \karana{s}. There are 4 ``fixed'' \karana{s} that occur
once each every month: the first half of the first lunar day, the
second half of the 29th lunar day, and the two halves of the 30th day;
the remaining 7 \karana{s}, called ``changing'', repeat cyclically for
the remaining 56 half-days. In other words, lunar day $D$ consists of
half-days $2D-1$ and $2D$, and half-day $H$ has one of the fixed
\karana{s} if $H=1,58,59,60$, and otherwise it has the changing \karana{}
number $(H-1)\amod 7$.
The names of the \karana{s} (in Tibetan and Sanskrit)
are listed in \cite[Appendix I]{Henning}.
(The exact rule for determining the time in the middle of the lunar
day that divides it into two halves is not completely clear to
me. \citet{Henning} divides each lunar day into two halves of equal lengths, but
there might be other versions.)
\item \label{q-nyi-bar}
The mean longitude
of the sun
\tib{nyi bar}
in signs and degrees (and minutes),
\ie{} $mean\_sun$ \eqref{meansun} written with the radices (12,30,60).
\item
\label{q-zla-skar-K}
The (true) longitude of the moon at the beginning of the calendar day
\tib{zla skar} as in \ref{q-zla-skar}, but calculated according to the
\karana{}
\xfootnote{The (Sanskrit) word \karana{} has a different meaning here
than in \ref{q-karana}. In contrast, the standard calculations described
above are called \siddhanta. See further \refApp{AKT}.
}
calculations, see \refApp{AKT}.
\xfootnote{
This means that all calendar
calculations have to be done also for the \karana{} version, in order to
find this value.}
This is, usually at least, calculated for the correct calendar day,
regardless of whether its
date is the same in the \PH{} and \karana{} versions or not; so for example
in the almanac \cite{tib2013}.
(There are exceptions, possibly mistakes, for example in the page
from a 2003 calendar shown in \cite[p.~202]{Henning}.)
\item \label{q-greg}
The Gregorian date. In modern almanacs written in Western (European) numerals.
\end{romenumerate}
As an example, the almanac \cite{tib2013}
lists for each day,
in addition to the Tibetan (lunar) date
and the Gregorian date \ref{q-greg},
the six numbers
\ref{q-gza-dag},
\ref{q-zla-skar},
\ref{q-nyi-dag},
\ref{q-yoga-long},
\ref{q-zla-skar-K},
\ref{q-nyi-bar}
above
\xfootnote{These numbers are truncated (not rounded) to 3 terms,
with radices $(1,60,60)$ or (27,60,60).
};
there is further (astrological) information as text.
\xfootnote{The same daily values are given in the 2003 almanac
shown in \cite[p.~202]{Henning}, see further the discussion there.
}
Note that \ref{q-gza-dag}, \ref{q-tshes-khyud}, \ref{q-nyi-dag},
\ref{q-nyi-bar},
refer to (the end of) the lunar day
while the others
refer to (the beginning of) the calendar day.
When a day is skipped (\ie, there is a lunar day without corresponding
calendar day), the almanac usually still gives
\ref{q-gza-dag} and \ref{q-nyi-dag} for the skipped lunar day;
when a day is repeated, so there are two calendar days corresponding to the
same lunar day, the data are given for both days, with the data referring to
the lunar day thus repeated, but
with modifications for the first day: In \cite{tib2013},
\ref{q-gza-dag} (otherwise the end of the lunar day) for the first day
is given as the end of the calendar day, written as
$x;60,0$ where $x$ is the day of week;
\ref{q-zla-skar} is obtained as
$moon\_lunar\_day-1/27$,
\xfootnote{
This is a reasonable approximation, since the end of the lunar day is just a
little more than one calendar day later than the beginning of the first
calendar day, \cf{} \refF{f27}. However, the resulting longitude is often
\emph{smaller} than the longitude at the end of the preceding lunar day, a
short time earlier. (The latter longitude is not printed in the almanac, so
the contradiction is not visible without further calculations.)
}
\cf{} \eqref{zla-skar};
\ref{q-yoga-long} is calculated separately for both days;
\ref{q-nyi-bar} is given only for the second day.
Thus only \ref{q-nyi-dag} is identical for the two days.
There is also further information at the beginning of each month, including
the mean date \tib{gza' bar} \eqref{meandate} and the mean solar longitude
\tib{nyi bar} \eqref{meansun} (both given with all 5 terms),
and also the lunar anomaly \tib{ril cha} \eqref{anomalymoon}, all calculated
for the beginning of the lunar month (day $d=0$ in the formulas above);
furthermore, the \tm{} \tib{zla dag} is given, with its fractional part
(the intercalation index), see \refR{Rix}.
These monthly values (mean date, mean
solar longitude, lunar anomaly and \tm) are also given for the \karana{}
calculation (see \refApp{AKT}).
There is also data on
the position of the planets (see \refApp{Aplanet}).
\subsection{Some special days}\label{Stent}
Furthermore, the almanac gives extra information on some special days, for
example
the days when the mean solar longitude passes certain values, including
the three series (with $30\grad$ intervals)
\begingroup
\addtolength{\leftmargini}{-10pt}
\begin{itemize}
\item
$0\grad, 30\grad, 60\grad\dots$ (when the mean sun enters a new sign);
\item $8\grad, 38\grad, 68\grad\dots$
(the \dpp{s} \tib{sgang}, see \refApp{SSdp});
\item $23\grad, 53\grad, 83\grad\dots$
(the midpoints between the \dpp{s} \tib{dbugs}).
\end{itemize}
(There are also 4 further such days, with longitudes
$66\grad$,
$132\grad$,
$147\grad$,
$235\grad$
in the 2013 almanac \cite{tib2013}.)
In all cases, the almanac gives the longitude and the
mean date for the instant $mean\_sun$ reaches this value.
This is easily found from \eqref{meansun} and \eqref{meandate}, now
letting $d$ be an arbitrary rational number denoting the lunar day including
a fractional part to show the exact instant.
\xfootnote{This is the only case that I know when the calculations involve
fractions of a lunar day.}
If we assume that $m_2=m_1/30$ and $s_2=s_1/30$, as is the case in the \PH{}
version and in all other modern versions that I know of, see \refR{R30},
then \eqref{meansun} shows that the mean solar longitude equals a given
value $\gl$ in the year $Y$ at lunar date
\begin{equation}
d = \frac{\gl+Y-Y_0-s_0}{s_2} = 30\frac{\gl+Y-Y_0-s_0}{s_1}
\end{equation}
after the epoch, where $Y_0$ is the epoch year (so $Y-Y_0+\gl$ can be
regarded as the desired longitude of the sun measured on a linear scale from
the epoch); however, in order for this to give the correct year,
$s_0$ has to be chosen such that $s_0$ is close to 0;
for \eX{} we thus take $s_0=0$ by \eqref{s0E1987}, but the values
\eqref{s0} and \eqref{s0E1927} for \eS{} and \eH{}
have to be decreased by 1 (recall that the
integer part of $s_0$ was irrelevant earlier); moreover, the integer part of
$\gl$ should similarly be adjusted so that $0\le\gl<1$, except at the
beginning of the year (before longitude $0\grad$)
when one should subtract 1 so that $-1<\gl<0$.
(Cf.\ \refApp{Aleap} for similar considerations regarding the mean solar
longitude as a real number on a linear scale.)
The mean date is then obtained from \eqref{meandate} as
\begin{equation}\label{jw}
d\cdot m_2+m_0 =
\frac{m_2}{s_2}
\xpar{\gl+Y-Y_0-s_0}+m_0
=\frac{m_1}{s_1}
\xpar{\gl+Y-Y_0-s_0}+m_0.
\end{equation}
\endgroup
The traditional calculation is somewhat different
\Hp.
First the month is determined (or guessed, for possible later correction)
-- this is easy since the mean sun always passes the definition point
$((M-3)\cdot 30+8)\grad$ during month $M$, see \refApp{Aleap}.
The lunar date of a new sign ($k\cdot30\grad$, for integer $k$)
is given by $6ix/13$, where $ix$ is the
intercalation index of the month, see \eqref{ix} and \refR{Rix}.
\xfootnote{
Note that $0\le 6ix/13\le 6\cdot64/13=384/13<30$.
}
The \dpp{} $8\grad$ later comes $8;3,1\rr{13,5}=8\frac{16}{65}$ lunar days
later and
the midpoint $7\grad$ earlier comes $7;2,4\rr{13,5}=7\frac{14}{65}$ lunar
days earlier.
Finally, the mean dates for these lunar dates are calculated by a
multiplication equivalent to \eqref{meandate}; tables exist to assist with the
multiplication by fractional parts of a lunar day.
To verify that this method works,
we first note that $8\frac{16}{65}=8\cdot 1\frac{2}{65}$
and $7\frac{14}{65}=7\cdot 1\frac{2}{65}$, and that during
$1\frac{2}{65}$ lunar day, the mean sun moves, see \eqref{s1}--\eqref{s2},
\begin{equation}
1\frac{2}{65}\cdot s_2 =
\frac{67}{65}\cdot s_2 =
\frac{67}{65}\cdot \frac{s_1}{30}=
\frac{67}{65}\cdot \frac{65}{67\cdot12}\cdot\frac{1}{30}=
\frac{1}{360} = 1\grad;
\end{equation}
hence the mean sun indeed moves $8\grad$ and $7\grad$ during these periods.
To verify the entry into signs, it is convenient to use the epoch \eX, since
then
both $\gbx=0$ \eqref{gbxX} and $s_0=0$ \eqref{s0E1987}.
Consider month $M$, year $Y$, and let as in \eqref{MM}
$\MM=12(Y-Y_0)+(M-M_0)$ be the number of solar months since the epoch
(year $Y_0=1987$, month $M_0=3$).
If this month has \tmc{} $n$ and \ixx{} $ix$, then the rule above yields a
lunar date (recalling that each month has 30 lunar days)
\begin{equation}\label{mi}
x= 30n+\frac{6ix}{13} = 30\Bigpar{n+\frac{ix}{65}}
\end{equation}
lunar days after the epoch. Note that $n+ix/65$ is the true month.
If the \ixx{} $ix<48$, then $n+ix/65$ is thus given by \eqref{tm0}, and is
thus $\frac{65}{67}\MM$ (since $\gbx=0$), so the lunar date \eqref{mi}
is
\begin{equation}
x = 30\cdot\frac{65}{67}\MM
\end{equation}
and consequently
the mean solar longitude then is,
by \eqref{meansun} (with $s_0=0$) and \eqref{s1}--\eqref{s2},
\begin{equation}
x\cdot s_2+s_0 = 30 \cdot\frac{65}{67} \cdot\MM \cdot \frac{s_1}{30}+0
=s_1 \cdot\frac{65}{67} \cdot\MM
=\frac{1}{12} \cdot\MM
=\MM\cdot30\grad,
\end{equation}
showing that the mean sun enters the sign with longitude $\MM\cdot 30\grad
\equiv (M-3)\cdot 30\grad$ (since longitudes are measured modulo $360\grad$).
If $ix\ge48$, then the
\tm{} is increased by $1\frac{2}{65}$, see \refR{Rix}, which corresponds to
an increase in the mean solar longitude of
$\frac{65}{67}s_1=\frac{1}{12}=30\grad$,
so the mean sun instead enters the next sign, with longitude
$(M-2)\cdot30\grad$
and the entry into the sought sign is found (by the same rule) in the preceding
month.
\xfootnote{It is easily verified that
$\frac{6\cdot47}{13} + 8\frac{16}{65} < 30 < \frac{6\cdot48}{13} +
8\frac{16}{65}$, which implies that in any case, the \dpp{}
$((M-3)\cdot 30+8)\grad$ is reached during month $M$ as said above,
see further \refApp{Aleap}.
}
(If $ix=48$ or $49$, this means leap month $M$.)
\begin{remark}
A complication for the almanac maker is that the lunar day found in this
way sometimes corresponds to a calendar day that is one day earlier or
later than the calendar day given by (the integer part of the) mean date
found above, since this correspondence uses the true date and not the mean
date. In this case, the data are entered in the almanac straddling both
days, see \cite{Schuh-review} for details and examples.
\end{remark}
The days discussed here have been defined by the mean solar longitude,
including day when it is 0, \ie, the mean sun passes the first point of Aries.
The day when the true solar longitude is $0$
is also marked in the almanac; in this case the true date is
given.
\xfootnote{The latter (``true'') day is about 2 days before the first
(``mean''), because of the
equation of the sun, see \eqref{truesun}; both are over a
month later then the astronomical vernal equinox when the real sun has
longitude $0$, see \refS{SSdrift}.
}
I do not
know exactly how this is calculated, but I assume that it is by some more
complicated version of the rule above, including adjustments for the
equations of sun and moon.
\section{Holidays}\label{Sholidays}
A list of holidays, each occuring on a fixed Tibetan date every year, is
given in \cite[Appendix II]{Henning}.
If a holiday is fixed to a given date, and that date is skipped, the
holiday is on the preceding day.
If the date appears twice, the holiday is on the first of these
(\ie, on the leap day, see \refS{Sdays}).
(These rules are given by \cite{Berzin}, but I have not checked them
against published calendars.)
Holidays are usually not celebrated in leap months, but see \refR{Rlosar}.
\section{Mean lengths and astronomical accuracy}\label{Smean}
The mean length of the month is
\begin{equation}\label{meanmonth}
m_1=29;31,50,0,480\rr{60,60,6,707}
=\frac{167025}{5656} \approx 29.530587
\text{ days},
\end{equation}
which is essentially identical to the modern astronomical value of the
synodic month $29.5305889$ (increasing by about $2\cdot10^{-7}$ each century)
\cite[(12.11-2)]{AA}.
Consequently, the mean length of the lunar day is
\begin{equation}
\label{meanlunarday}
\frac{m_1}{30}
=\frac{167025}{30\cdot5656}
=\frac{11135}{11312}
\approx 0.98435
\text{ days}.
\end{equation}
The mean length of the year is, \cf{} \eqref{6765},
\begin{equation}
\label{meanyear}
\frac{1}{s_1}
\text{ months}
= \frac{m_1}{s_1}
=\frac{804}{65}m_1
=\frac{6714405}{18382}
\approx 365.270645
\text{ days},
\end{equation}
which is $0.02846$ days more than
the modern astronomical value of the
tropical year $365.24219$
\cite[(12.11-1) and Table 15.3]{AA},
and $0.02815$ days longer than the mean Gregorian year $365.2425$ days.
(It is also longer than the
sidereal year $365.25636$ days
\cite[Table 15.3]{AA}.)
Hence the Tibetan year lags behind and starts on the average
later, compared to the
seasons or the Gregorian year, by almost 3 days per century or almost a month
(more precisely, 28 days) per millennium.
The year has thus drifted considerably since it was introduced, and
even more since the \KT{} was written; the drift during the 1200
years since the epoch 806 is 34 days, see further \refS{SSdrift}.
The mean length of the anomalistic month is
\begin{equation}\label{meanano}
\frac{1}{1+a_1} \text{ months}
=\frac{3528}{3781} \cdot\frac{167025}{5656}
=\frac{10522575}{381881}
\approx 27.55459
\text{ days}.
\end{equation}
This agrees well with the modern astronomical value
$27.55455$ days \cite[Table 15.3]{AA}; the drift is about 1 day in 2000
years.
See also \citet{Petri}.
\subsection{Accuracy of longitudes}\label{SSdrift}
As noted above, the mean length of the month is essentially equal to the
exact astronomical value. Indeed, the times (true date)
for new moons as calculated by
the Tibetan calendar are close to the exact astronomical times.
(Recall that the new moons signify the end of lunar day 30 in each month and the
beginning of a new month; in the almanac the time is thus given as the true
date of day 30 in the month.)
If we (following \cite{Henning}) regard the Tibetan day as starting at
mean daybreak, ca.~5 a.m.\ local mean solar time,
and set this equal to $-1$ UT (GMT), see \refR{Rdate},
then
the true dates given by Tibetan calculations are about 4 hours earlier
than the exact astronomical values.
\xfootnote{A calculation for the 12 new moons in the Gregorian year 2013
and comparison with a Swedish almanac gave differences
between $3^h16^m$ and $4^h21^m$.
}
Since the elongation increases by about $12\grad$ each day,
see \refS{Sdays},
this corresponds
to an error in the elongation of about $2\grad$ too large.
As also noted above, the mean length of the year is less accurate, and the
year has drifted 34 days since the epoch 806.
Indeed, the solar longitude as
computed by $true\_sun$ above passes $0\grad$ during
the 16th day in the third Tibetan month 2013, which equals April 26,
37 days after the astronomical vernal equinox on March 20,
which agrees well with this drift.
Another way to see this is to note that the true sun calculated at the
(astronomical) vernal equinox is only about $324\grad$, which means that the
solar longitude is about $36\grad$ too small.
\xfootnote{This varies slightly
over the year, since the equation of the sun given by \eqref{sunequ} is not
astronomically exact, both because it is intrinsically an approximation
only, and because the anomaly is by \eqref{anosun} also about $36\grad$
wrong.
However, the error does not vary by more than 1--$2\grad$ over the year
and is always about 36--$37\grad$.
}
Since the elongation was seen above to be about $2\grad$ too large,
the Tibetan longitude of the moon is about $34\grad$
(= 2.5 mansion) too small.
\subsection{The ratio between solar and lunar months}\label{SSratio}
The ratio of the exact astronomical lengths of solar and lunar months is
(with values correctly rounded for both today and 1000 years ago)
\begin{equation}
\frac{365.2422}{12\cdot 29.53059} \approx 1.030689.
\end{equation}
The standard way to find good rational approximations of a real number is
to expand it as a continued fraction, see \eg{} \cite{HardyWright}:
\begin{equation}
1.030689=
1+\frac{1}{32+\frac1{1+\frac1{1+\frac1{2+\frac1{2+\dots}}}}}
\end{equation}
and then calculate the partial quotients obtained by truncating the
continued fraction.
In this case the first partial quotients are (after 1)
\begin{align}
1+\frac{1}{32}&=\frac{33}{32}= 1.03125
\\
1+\frac{1}{32+\frac1{1}}&=\frac{34}{33}\approx1.030303
\\
1+\frac{1}{32+\frac1{1+\frac1{1}}}&=\frac{67}{65}\approx1.030769
\\
1+\frac{1}{32+\frac1{1+\frac1{1+\frac1{2}}}}&=\frac{168}{163}\approx1.030675
\\
1+\frac{1}{32+\frac1{1+\frac1{1+\frac1{2+\frac1{2}}}}}&=\frac{403}{391}
\approx 1.030690.
\end{align}
The first two approximations 33/32 and 34/33 have errors of about
$5\cdot 10^{-4}$ and $4\cdot 10^{-4}$, or about 1 month in 200 years.
The next approximation is 67/65, used in the Tibetan calendar, which has an
error of about $8\cdot 10^{-5}$, or as said above about 1 month in 1000 years.
The next approximation is 168/163, with an error of only
$1\cdot 10^{-5}$, about 1 month in 6\,000 years, and 403/391 has an error
$2\cdot 10^{-6}$, about 1 month in 50\,000 years
\xfootnote{
The latter accuracy is only fictional; the error 50\,000 years
from now is much larger, $10^{-4}$, because of changes in the lengths of the
month and the year.}.
The Tibetan approximation $67/65$, see \eqref{6765}, is thus a good choice
from a mathematical point of view, if we want a good approximation with
small numbers. (168/163 or 403/391 would yield more accurate but more
complicated calendars. As far as I know, they are not used in any calendar.)
\begin{remark}
An approximation used in several other calendars (\eg{} the Jewish,
and for calculation of the Christian Easter),
but not in the Tibetan,
is the relation
235 lunar months $\approx$ 228 solar months (19 solar years), known as
Meton's cycle \cite{CC}. We have $235/228= 1.030702$, with an error of
$1\cdot 10^{-5}$, about 1 month in 6000 years.
This does not appear in the list of partial quotients above, and indeed,
168/163 with smaller numbers is about as accurate.
However, the Metonic cycle has the advantage that it comprises a whole
number of years. (It appears as one of the partial quotients for the continued
fraction expansion of the number
of lunar months in a solar year.)
\end{remark}
\section{Period}\label{Speriod}
The $mean\_date$ given by \eqref{meandate} repeats after 5656 months,
but $true\_date$ depends also on $anomaly\_moon$ and $true\_sun$, which
repeat after 3528 and 804 months, respectively. The least common
multiple of 5656, 3528 and 804 is
$p=23873976$; consequently, all astronomical functions above repeat
after $p$ months,
which equals $m_1p=705012525$ days or
$s_1p=1930110$ years (these are necessarily
integers).
Note further that this period contains an integral number of leap year
cycles (804 months = 65 years), see \refS{Smonths}, so the numbering of
the months repeats too.
In other words:
\begin{Rule}
The calendar (days and months) repeats after
\begin{equation*}
705\,012\,525\text{ days}
=
23\,873\,976\text{ months}
=
1\,930\,110 \text{ years}.
\end{equation*}
\end{Rule}
Moreover, the number of days is divisible by 7, so also
the day of week repeats after this period.
However, $1930110\equiv 6 \pmod{12}$, so to repeat the animal names of
the years in
the 12-year cycle, or the names in the 60-year cycle, we need two of
these periods, \ie{} 3\,860\,220 years.
Of course, the period is so long that the calendar will be completely
out of phase with the tropical year and thus the seasons long before,
and it will move through the seasons hundreds of times during one
period.
If we also consider the planets, see \refApp{Aplanet},
the period becomes a whooping
2\,796\,235\,115\,048\,502\,090\,600 years, see \citet[pp.~332--333]{Henning}
and \cite[Weltzeitalter]{tibetenc}.
\appendix
\section{Different versions}\label{Aversions}
Different traditions follow different rules for the details of the
calculation of the Tibetan calendar, and as said in \refS{Sintro},
there are two main versions in use today.
(Several other versions survive according to \cite[p.~9]{Henning},
but no details are given.)
Schuh \cite{Schuh} gives historical information,
including many versions of the constants in \refS{Sastro} above
used or proposed during the centuries,
but says nothing about the
different versions today.
See also \cite[Kalenderrechnung]{tibetenc}.
\citet{Henning} discusses the \PH{} and \TS{} versions in detail, and also
one recent attempt at reform.
He gives epoch data for several versions in
\cite[Epoch data]{kalacakra}.
(Another source on different versions today is \citet{Berzin};
he gives the
impression that several versions are in use.
However, since \cite{Berzin} discusses the calendar and astrology
together, it is possible that some of these versions actually use
the same calendar but differ in other, astrological, calculations or
interpretations.)
See also the web pages of Nitartha \cite{nitartha}.
\subsection{\PH} (Phukluk, \tibx{phug-lugs}.) \label{APH}
This is the most widespread version, and is regarded as the official
Tibetan calendar.
(Also by at least some followers of the rival \TS{} tradition
\cite{nitartha}.)
It was started in 1447
(the first year, \tibx{rab byung}, of the 8th Prabhava cycle)
by
Phugpa Lhundrub Gyatso
\tib{phug-pa lhun-grub rgya-mtsho},
and was used by the Tibetan government from at least 1696 to 1959
\cite[Phugpa-Schule]{tibetenc}, \cite[pp.~8 and 321--337]{Henning},
\cite[p.~139]{Schuh}, \cite{Schuh-review}.
The \PH{} version is used by the Gelug, Sakya, Nyingma and Shangpa Kagyu
traditions of Tibetan buddhism, including the Dalai Lama,
and is used \eg{} in the calendars published in Dharamsala in India, where
the Tibetan exile government resides, see \cite{tib2013,men-tsee-khang}.
The Bon calendar is the same as the \PH{} \cite{Berzin}.
The \PH{} calendar is described in detail in the main body of the present paper.
\subsection{\TS}(Tsurluk.)\label{ATS}
This version was also introduced in 1447, by
Jamyang Dondrub Wozer
\tib{mtshur-phu 'jam-dbyangs chen-po don-grub 'od-zer}
and
derives from 14th century commentaries to \KT{} by the
3rd Karmapa Rangjung Dorje of \TS{} monastery
(the main seat of the Karma Kagyu tradition of Tibetan
buddhism) \cite{Berzin},
\cite[pp.~9 and 337--342]{Henning},
\cite[Tshurphu-Schule]{tibetenc}.
This version is used by the Karma Kagyu tradition, and it is used
\eg{} in calendars published
by the Rumtek monastery in India, the main exile seat of
the Karmapa (the head of Karma Kaygyu)
\cite[Open source Tsurphu calendar software]{kalacakra}.
Also the calendar
published by
Nitartha in USA \cite{nitarthacal,nitartha} gives the \TS{}
version (from 2004 the \PH{} version too is given).
The \TS{} tradition uses the astronomical functions in \refS{Sastro}
with the same values as given there for the \PH{} version of the
constants $m_1$, $s_1$ and $a_1$ (and $m_2$, $s_2$ and $a_2$, see \refR{R30})
for mean motions, while the epoch values $m_0$, $s_0$ and $a_0$ are different.
See further \cite[pp.~337--342]{Henning}.
\begin{remark}
Traditional \TS{} calculations use, however, somewhat different sets
of radices that the \PH{} versions, with one more term
(and thus potentially higher numerical accuracy in the calculations).
The same constants are thus written differently:
\begin{align}
m_1&=
29;31,50,0,8,584\rr{60,60,6,13,707}
=\frac{167025}{5656},
\\
s_1&
= 2,10,58,1,3,20 \rr{27,60,60,6,13,67}
=\frac{65}{804},
\end{align}
although $m_1$ is traditionally given as
$1;31,50,0,8,584\rr{60,60,6,13,707}$,
as always calculating modulo 7, and I have added 28, see \refR{R7}.
\end{remark}
Two epochs given in classical text \cite[p.~340]{Henning} are,
with
$m_0$ modified to yield JD, see \refR{R7},
\begin{align}
\JD&=2353745 \quad \text{(Wednesday, 26 March 1732 (Greg.))} \notag
\\
m_0&=
4;14,6,2,2,666 \rr{60,60,6,13,707}+2353741
=2353745+\frac{1795153}{7635600},
\\
s_0&=
- \bigpar{1,29,17,5,6,1 \rr{27,60,60,6,13,67}}
=-\frac{5983}{108540},
\label{s0T1732}
\\
a_0&= 14,99 \rr{28,126}
=\frac{207}{392}.
\end{align}
and 1485 months later (equivalent and giving the same calendar)
\begin{align}
\JD&=2397598 \quad \text{(Monday, 19 April 1852)} \notag
\\
m_0&=
2; 9,24,2,5,417 \rr{60,60,6,13,707}+2397596
=2397598+\frac{1197103}{7635600},
\\
s_0&=
0,1,22,2,4,18 \rr{27,60,60,6,13,67}
=\frac{23}{27135}, \label{s0T1852}
\\
a_0&=0,72 \rr{28,126}
=\frac{1}{49}.
\end{align}
We denote these by E1732 and E1852.
(See also \cite[Epoch data]{kalacakra}, where also a third epoch 1824 is
given.)
To find the leap months, the true month is calculated by
\eqref{tm0} with a constant $\gbx =59$ (E1732) or $14$ (E1852)
(the epoch value of the intercalation index);
the reason being that counting backwards to the \Kc{}
epoch \nag{} (\Caitra) 806 yields the intercalation index 0,
as in the \KT. (But unlike the \PH{} version, see \eqref{gbxS}.)
Moreover, instead of the Phugpa rule \eqref{leaprule-P},
the \TS{} version uses the simpler rule:
\begin{equation}\label{leaprule-T}
\hskip-1em
\vbox {\narrower\narrower\narrower\noindent\em
A leap month is inserted when the intercalation index\\
$ix = 0\text{ or\/ }1$.
}
\hskip-3em
\end{equation}
This implies that for a regular month, the \tmc{} $n$ is obtained from
the true month \eqref{tm0} by rounding down to the nearest integer; for
a leap month we further subtract 1.
(The \TS{} rules are thus simpler and more natural than the \PH{} rules
\eqref{leaprule-P} and \eqref{mcrule-P}
in \refS{Smonths}. They also follow the original \KT, \cf{} \refApp{AKT}.
In particular, if we extend the calendar backwards, there was a leap
month at
the \Kc{} epoch \nag{} (\Caitra) 806, in agreement with the \KT.)
\begin{remark}
This means that for the \TS\ version,
the intercalation index calculated in this paper
always agree with the traditional definition without further correction;
cf.\ \refR{Rix} for the \PH{} version.
\end{remark}
By \eqref{jeppe},
\eqref{leaprule-T} agrees with
the general rule \eqref{leaprule-gb0}
if $\gb$ is defined such that
\begin{equation}\label{gbx-T}
\gb+ \gbx \equiv 6 \bmod{65}.
\end{equation}
As we will see in \refApp{ASTSdp},
the values are
$\gb=142$ (E1732) and 187 (E1852), in accordance with
\eqref{gbx-T} and the values for $\gbx$ given above.
By \eqref{gam} and \eqref{gamx},
the rules
\eqref{lygam}, \eqref{lygamx} and the formula \eqref{lys} hold with
$\gamma =55$ and $\gamx =20$.
\begin{remark}\label{RTSkarana}
\TS{} almanacs give essentially the same information as almanacs for the
\PH{} version, see \refS{Sfurther}, but one difference is that
they by tradition give the true solar longitude for each day calculated by the
\karana{} calculation, see \refApp{AKT}, instead of the calculation above.
(Cf.\ \ref{q-zla-skar-K} in \refS{Sfurther}.)
In some \TS{} almanacs, the solar equation
from the \karana{} solar longitude calculation has also been used to
calculate the (\siddhanta) $true\_date$ in \eqref{truedate}.
\xfootnote{
This seems only to have been a time saving device, as
usually the \siddhanta{} and \karana{} calculations are kept seperate.
}
In other words, the values \eqref{s1KT} and \eqref{s0K} are used in the
calculations instead of the values for $s_1$ and $s_0$ given above.
See \cite[Open source Tsurphu calendar software]{kalacakra}.
This will lead to a slightly different $true\_date$, and occasionally
a different repeated or skipped day.
\xfootnote{
Calculations similar to the ones in \refR{R30} suggest that this will happen
about 5 times per year, and that the New Year will differ by a day about 2
times per century.
One example is day 13, month 6, 2013, for which the two versions yield
20 and 21 July.
}
However, the traditional version seems to be to use the values above for the
calculation of the true date, but to also calculate separately
the \karana{} version of
$true\_sun$ and publish it
[Henning, personal communication].
\end{remark}
\subsection{Mongolia}\label{AMongo}
The Mongolian calendar is a version of the Tibetan.
It became the official calendar in Mongolia in 1911 when
Mongolia declared independence from China
after the Chinese revolution that overthrew the last Chinese emperor;
however, it was replaced in the 1920s (under Communist rule) by
the Gregorian calendar (officially in 1948)
and the authorities even tried to abolish the traditional celebrations of
the Mongolian New Year.
The calendar has had a revival together with other traditions
after the end of Communist rule in the 1990s, although
the Gregorian calendar remains the official calendar and is used for
everyday civil use;
\xfootnote{\label{fmongol}
The first democratic constitution came into force ``from the horse hour of
the auspicious yellow horse day of the black tiger first spring month of the
water monkey year of the seventeenth 60-year cycle''.
[noon 9/1 = 12 February 1992]
\cite[p.~240]{mongolian}
}
in particular, the Mongolian New Year (\emph{Tsagaan Sar})
is again celebrated as a major national holiday. See
\cite{mongolian}, \cite{MongoliansWelcome},
\cite[4.1.3]{legalinfo.mn}.
(I guess that otherwise the Mongolian calendar
is mainly used for religious purposes and astrology.)
In 2012, a second public holiday was declared that is calculated by the
Mongolian calendar (the other holidays have fixed dates in the Gregorian
calendar \cite{legalinfo.mn}), \viz{} Genghis Khan's birthday,
the first day in the first winter month (month 10)
\xfootnote{
His actual date of birth is not known, and there are different dates in
different sources; for example Terbish \cite{Terbish-GenghisKhan} believes in
day 16 in month 4, 1162. (However, this was 500 years before the present
vesion of the calendar was introduced, so the calculation of the
corresponding Julian date (1 May) in \cite{Terbish-GenghisKhan} seems
uncertain.)}
\cite{Q++-chinggis},
\cite{embassy-holidays},
\cite[4.1.8]{legalinfo.mn}.
According to \cite{Berzin},
the Buryats and Tuvinians of
Siberia (in Russia) follow the New Genden version
(\ie, the version described here),
while
the Kalmyk Mongols in Russia follow
the \PH{} version.
(Inner Mongolia, in China,
uses instead the Chinese ``yellow system'', see \refS{Ayellow}.)
The information below is mainly based on
\citet{Berzin}, \citet[Epoch data]{kalacakra} and \citet{Salmi}, but I have no
reliable Mongolian sources, and not even a printed calendar as an example.
An example of its use is in the daily horoscope at \cite{news.mn},
but as noted by \cite{Salmi}, this contains some obvious errors and is thus not
reliable.
\xfootnote{In, for example, May, June and September 2012, some Mongolian
dates are out of order: others are missing.
Nevertheless, most of the skipped or repeated dates in \cite{news.mn}
for March 2011 -- August 2013 agree with the rules below, as does the
the leap month 6 in 2011.
(Note that the skipped and repeated days are sensitive to also
small changes in the constants.
For example, they usually differ between the New Genden version and the \PH{}
version, see the examples in \refT{T4+-}.)
The discrepancies that exist may be due to further errors in their dates. It
is also possible, although less likely, that different versions of the
Mongolian calendar are used.
}
A list of Gregorian dates of the Mongolian New Years
1896--2008 is given in \cite{olloo.mn}, and the same list extended to 2013
in
\cite{mongolnews}.
\xfootnote{These dates seem to be calculated by L. Terbish, who seems to be
the present expert on the calendar in Mongolia (see \eg{} \cite{mongolnews}).
The dates all agree with the rules below.
Note also that the list of New Years immediately yields the leap years.
Since the leap months are regularly spaced, see \refS{Smonths},
the sequence of leap years for 65 consecutive years uniquely determines the
leap months for all years, and they agree with the leap year rule below.
}\,
\xfootnote{However, there are also other, contradictory lists, for example
in \cite{mongolian} and on various unreliable web sites.
When this is written, Mongolian Wikipedia \cite[Tsagaan Sar]{mn.wikipedia}
gives two lists, one 1989--2013 (partial) and one 2000--2099; of the 13
common years, only 4 agree. (The first list is attributed to Terbish, and
agrees with the formulas here, except for a possible typo. The second list
agrees with neither the New Genden version,
the \PH{} version, nor the Chinese calendar.)
}
See \cite{Salmi} for many further references.
The Mongolian calendar follows,
see \cite{Berzin}, \cite{Salmi}, \cite{MongoliansWelcome} and
\cite{mongolnews},
a version of the Tibetan calendar known as New Genden
(Mongolian \emph{T\"ogs buyant}, ``Very virtuous'' \cite{Salmi})
which was created by Sumpa Khenpo \YP{}
\xfootnote{
An prominent 18th century Tibetan monk of Mongolian origin.
\cite{treasury}
}
(\tibx{sum-pa mkhan-po ye-shes dpal-'byor};
Mongolian S\"umbe Khamba Ishbaljir)
in 1786
\xfootnote{
According to \cite{Berzin}, who comments that the starting point
is the 40th year of the 60 year cycle and
claims that "Because of this
difference, the Mongolian calendar works out to be unique."
However, as said in \refR{Repoch}, the choice of epoch in itself has no
importance. Furthermore,
\YP{} used the epoch 1747
(the beginning of the corresponding 60 year cycle)
\cite[Epoch data]{kalacakra}.
}.
The constants $m_1$, $s_1$ and $a_1$ for mean motions are the same
in the New Genden version
as in the
\PH{} version, see \refS{Sastro},
and thus so are $m_2$, $s_2$ and $a_2$, see \refR{R30},
while the epoch values $m_0$, $s_0$ and $a_0$ are different and given by,
see \cite[Epoch data]{kalacakra}:
\begin{align}
\JD&=2359237 \quad \text{(Sunday, 9 April 1747 (Greg.))}
\notag
\\
m_0
&
= 1;55,13,3,31,394 \rr{60,60,6,67,707}
+ 2359236
\notag
\\&
=2359237+\frac{2603}{2828},
\label{m0genden}
\\
s_0&
= 26,39,51,0,18 \rr{27,60,60,6,67} =\frac{397}{402}, \label{s0genden}
\\
a_0&
= 24,22\rr{28,126} = \frac{1523}{1764}.
\label{a0genden}
\end{align}
I have here, as in \eqref{m0}, modified $m_0$ to yield the JD, see \refR{R7}.
\begin{remark}
In traditional calculations, one uses
a different sequence of radices
than the standard one, but the mean values can be expressed
exactly also with the standard \PH{} radices;
the constant $m_0$ in \eqref{m0genden} above equals
$1;55,13,3,333 \rr{60,60,6,707}$.
Similarly, $m_1$ is given by \YP{} as
$1;31,50,0,45,345 \rr{60,60,6,67,707}$, which equals the standard value
$1;31,50,0,480 \rr{60,60,6,707}$; as always these traditional values are
interpreted modulo 7, see \refR{R7},
and we add 28 and use the value in \eqref{m1}.
\end{remark}
\begin{remark}
For comparison,
the constants for the \PH{} version for this epoch, which thus give the
standard \PH{} calendar, can be calculated to be
\begin{align}
m_0 &
=
1;52,41,2,524 \rr{60,60,6,707}
+ 2359236
=2359237+\frac{4967}{5656},
\\
s_0 &
=
26,9,37,3,45 \rr{27,60,60,6,67} =\frac{779}{804},
\\
a_0 &
=
24,19\rr{28,126} = \frac{3043}{3528}.
\end{align}
\end{remark}
To find the leap months, the true month is calculated by
\eqref{tm0} with a constant $\gbx =10$
(the epoch value of the intercalation index). However, the Phugpa rule
\eqref{leaprule-P} is modified to
\xfootnote{The reason seems to be that counting backwards to the \Kc{}
epoch \nag{} (\Caitra) 806 yields the intercalation index 46 and thus a leap
month,
agreeing with the \KT{} (as in the \TS{} version but not in \PH).
As a consequence, the Mongolian version has the same leap months as the
\TS{} version (although one may start a day before the other),
see \refT{T4Leap}.
}
\begin{equation}\label{leaprule-G}
\hskip-1em
\vbox {\narrower\narrower\narrower\noindent\em
A leap month is inserted when the intercalation index\\
$ix = 46\text{ or\/ }47$.
}
\hskip-3em
\end{equation}
(There is also a corresponding change of \eqref{mcrule-P}, with 48 replaced
by 46. Similarly, \refR{Rix} applies again, with 49 replaced by 47.)
By \eqref{jeppe},
\eqref{leaprule-G} agrees with
the general rule \eqref{leaprule-gb0}
if $\gb$ is defined such that
\begin{equation}\label{gbx-G}
\gb+ \gbx \equiv 52 \bmod{65}.
\end{equation}
For the epoch above, with $\gbx=10$,
we take $\gb=172\equiv42\pmod{65}$, see
\refApp{ASMdp}.
By \eqref{gam} and \eqref{gamx},
the rules
\eqref{lygam}, \eqref{lygamx} and the formula \eqref{lys} hold with
$\gamma =55$ and $\gamx =20$,
\cf{} \cite{Salmi}.
(The same constants as for the \TS{} version in \refApp{ATS},
since as said above, these two versions have the same leap
months.)
The years in the Mongolian calendar
are named by Element + Animal as described in \refS{Syear}, but
often the element is replaced by the corresponding colour according to the
correspondence in \refT{T5} \cite{mongolian}.
\xfootnote{
The Mongolians use the colours in
parenthesis in \refT{T5} \cite[p.~155]{mongolian}.
(I suspect that this is a difference in translation of the colours more
than an actual difference in colour.
}\,
\xfootnote{
According to \cite[p.~155]{mongolian},
it is common to use the element in male years and the colour in female
years,
which would give the cycle Wood, Blue, Fire, Red, Earth, Yellow, Iron,
White, Water, Black, see \refT{T5}.
However, I have not seen this confirmed in other sources.
For example, the lists (in Mongolian) of New Years in \cite{olloo.mn}
and \cite{mongolnews} use
Element + Animal for all years.
}
The Tibetan 60-year \tibx{rab byung} cycle,
see \refS{Syear},
is recognized in Mongolia too
(called \tibx{jaran} in Mongolian), and the cycles are numbered as in Tibet
(with the first starting in 1027),
see
\cite{MongoliansWelcome}, \cite{olloo.mn} and
\cite{mongolnews} (both with formulas similar to
\eqref{indian1}--\eqref{indian3}),
and the example in \refF{fmongol}.
The months are not numbered, but are named.
\xfootnote{
Numbers are used for Gregorian months.
See \cite[p.~106]{mongolian} and, for examples, \cite{legalinfo.mn}.
}
Two different systems are used.
(Both have been used earlier in Tibet, see \refS{SSmonths}.)
One method is by Animal, or
Colour (or perhaps Element?) + Animal,
by
the same rules as described for the \TS{} version
in \refApp{ASattributes-months} below
(which are the rules of the Chinese calendar); thus month 1 is Tiger, etc.
The months are also named as beginning, middle and end of each of the four
seasons spring, summer, autumn, winter, with month 1 beginning of spring
(as in the Chinese calendar, and in \refS{SSmonths}\ref{month-season-animal})
\cite[pp.~155, 241--242]{mongolian}.
See \refF{fmongol} for an example where both methods are used together.
The calendar days can also be named by Colour (or Element?) + Animal,
in a simple 60-day cycle, see \refApp{ASattributes-day} and the example
in \refF{fmongol} \cite[p.~242]{mongolian}.
\newcommand\Dzo{Dzongkha}
\subsection{Bhutan}\label{ABhutan}
Bhutan uses a version of the Tibetan calendar as an official calendar;
see \eg{} the government's web page \cite{Bhutan} for an example, where
a calendar with both Bhutanese and Gregorian dates is shown.
In official documents such as acts, both the Bhutanese and Gregorian dates
are given (in both the \Dzo{} and English versions), see many examples at
\cite[Acts]{BhutanNA}. Of the public holidays, some have fixed dates in
the Bhutanese calendar
(New Year and some dates connected to Buddha and Buddhism);
some have fixed dates in the Gregorian calendar (\eg{} the National Day
and the King's Birthday); finally, the Winter Solstice is a holiday, but it is
calculated according to the Bhutanese calendar (see below); it usually
occurs at the Gregorian date 2 January. (See
\cite[Open source Bhutanese calendar software]
{kalacakra} and \cite[Public holidays]{Bhutan} for a complete list.)
The version was described by Lhawang Lodro
\tib{lha-dbang blo-gros}
in the 18th century
\cite[Open source Bhutanese calendar software]
{kalacakra}, but is said
to be older.
An important difference from the other main versions of the Tibetan
calendar is that a leap month is given the number of the \emph{preceding} month
insted of the next month,
see \cite[Open source Bhutanese calendar software]{kalacakra}.
(This is the system in the Chinese calendar, but not in Indian calendars
\cite{CC}, see \refR{RChina}; it seems to have been the original \KT{}
system,
see \refApp{AKT}.)
A unique feature of the Bhutanese calendar is that its day of week differs by
one day from Tibet (and the rest of the world, see \refF{fplanets}).
The names of the days of week are the same as in Tibetan
\xfootnote{
\Dzo{} (the national language of Bhutan) is, depending on one's view,
a dialect of Tibetan or a language closely related to Tibetan;
it is written with Tibetan script.
},
but day 0 (Saturday) is \tibx{spen pa} in Tibet but \tibx{nyi ma}
(really meaning sun) in Bhutan,
day 1 (Sunday) is \tibx{nyi ma} in Tibet but \tibx{zla ba} (really meaning
moon) in Bhutan,
etc., see \refT{T7}.
(This difference is lost when writing dates in English, since of
course the correct English day of week is chosen in both cases, for example
in the
calendar at \cite{Bhutan}.)
See further \cite[Bhutan calendar problem]{kalacakra}.
The constants $m_1$, $s_1$ and $a_1$ for mean motions are the same
in the Bhutanese version
as in the \PH{} version (and in the \TS{} and Mongolian versions), see
\refS{Sastro},
and thus so are $m_2$, $s_2$ and $a_2$, see \refR{R30},
while the epoch values are different and given by,
see Henning \cite[Epoch data]{kalacakra}:
\begin{align}
\JD&=2361807 \quad \text{(Monday, 22 April 1754 (Greg.))}
\notag
\\
m_0
&
= 2;4,24,552 \rr{60,60,707}
+ 2361805
=2361807+\frac{52}{707},
\label{m0bh}
\\
s_0&
= 0,24,10,50 \rr{27,60,60,67} =\frac{1}{67}, \label{s0bh}
\\
a_0&
= 3,30\rr{28,126} = \frac{17}{147}.
\label{a0bh}
\end{align}
Again, $m_0$ is here modified to yield the JD, see \refR{R7}.
\begin{remark}
The traditional calculations use
different sequences of radices than the standard ones (with fewer radices),
see
\eqref{m0bh}--\eqref{s0bh}, but the standard values can be expressed
exactly also with these radices.
The constant $m_1$ is in this tradition given by
$1;31,50,80 \rr{60,60,707}$, which equals the standard value \eqref{m1}
after our usual addition of 28;
similarly, $s_1$ is given by
$2,10,58,14 \rr{27,60,60,67}=65/804$ which equals \eqref{s1}.
\cite[Epoch data]{kalacakra}.
\end{remark}
To find the leap months, the true month is calculated by
\eqref{tm0} with a constant $\gbx =2$
(the epoch value of the intercalation index)
\cite[Epoch data]{kalacakra}.
Furthermore,
recalling that in Bhutan
a leap month gets the number of the preceding month,
the Phugpa rule
\eqref{leaprule-P} is modified to
\begin{equation}\label{leaprule-B2}
\hskip-1em
\vbox {\narrower\narrower\narrower\noindent\em
A month is a leap month if and only if
its intercalation index
$ix = 59\text{ or\/ }60$.
}
\hskip-3em
\end{equation}
Here have to use the correct (traditional) intercalation index, see
\refR{Rix},
which is \eqref{ix} increased by 2 for the leap month (and all later months
until $ix$ passes $65$).
A computationally simpler, but
more clumsy, formulation where our simplified
definition \eqref{ix} can be used directly is:
\begin{equation}\label{leaprule-B}
\hskip-1em
\vbox {\narrower\narrower\narrower\noindent\em
Regular month $M$ in year $Y$ is followed by a leap month if and only if
its intercalation index
$ix = 57\text{ or\/ }58$.
}
\hskip-3em
\end{equation}
There is a corresponding change of \eqref{mcrule-P}, with 48 replaced
by 59, and the true month rounded up for a leap month.
Comparing \eqref{leaprule-B} with
\eqref{jeppe} (and taking into account that now a leap month $M$ comes after
the regular month $M$), or better by \eqref{leaprule-B2} and \eqref{jeppe+}
below,
\eqref{leaprule-B2} and
\eqref{leaprule-B} agree with
the general rule \eqref{leaprule-gb0}
if $\gb$ is defined such that
\begin{equation}\label{gbx-B}
\gb+ \gbx \equiv 63 \bmod{65}.
\end{equation}
For the epoch above, with $\gbx=2$,
we take $\gb=191\equiv61\pmod{65}$, see
\refApp{ASBdp}.
By \eqref{gam} and \eqref{gamx},
\eqref{lygam} and \eqref{lygamx} hold with
$\gamma =12$ and $\gamx =28$.
The winter solstice is, as said above, a holiday in Bhutan.
It is defined as the
day the mean solar longitude \eqref{meansun} reaches
$250\grad$,
which at present occurs
on 2 January,
see \cite[Open source Bhutanese calendar software]{kalacakra}.
\xfootnote{The correct astronomical definition is $270\grad$, for the
(astronomical) true longitude; the correct Gregorian date is
(at present)
21 or 22 December.
}
(The date drifts slowly, by almost 3 days per century, see \refS{Smean}.
It will be 3 January for the first time in 2020. It coincided with the
astronomical winter solstice about 400 years ago, in the early 17th century,
which was before the present version of the Bhutanese calendar was described.)
The months in Bhutan are numbered, as in Tibet.
(Names exist as in Tibet, see \refS{SSmonths}, but are usually not used.)
The years are
named by Element + Gender + Animal, see \refS{Syear}.
(See the examples in
\cite[Acts]{BhutanNA}, where the Bhutanese dates are given in this way,
while the Gregorian dates are given with the number of the year.)
\subsection{\KT\ (\Karana)}\label{AKT}
The original \KT{} calculations are explained by
\citet[Chapter 5]{Schuh} and
\citet[Chapter V]{Henning}, \cite{Henning-Karana}; they are
(as far as I know) not used to produce calendars today;
however,
complete Tibetan almanacs usually give (by tradition, and for no other
obvious reason)
some values calculated by this version in addition
to values calculated by one of the versions above, see \refS{Sfurther}.
The \KT{} version is called \emph{\karana} \tib{byed rtsis}, while the \PH{}
and other similar versions are called \emph{\siddhanta} \tib{grub rtsis}.
The \KT{} (\karana)
version uses different values of the mean motions $m_1$ and $s_1$
than the versions described above but the same $a_1$
(see also \cite[Epoch data]{kalacakra}):
\begin{align}
m_1 &
=
29;31,50 \rr{60,60}
=\frac{10631}{360}
=29+\frac{191}{360},
\label{m1KT}
\\
s_1 &
=
2,10,58,2,10 \rr{27,60,60,6,13}
=\frac{1277}{15795},
\label{s1KT}
\\
a_1 &
=
2,1\rr{28,126} = \frac{253}{3528};
\intertext{the epoch values are}
\JD&=2015531 \quad \text{(Monday, 23 March 806 (Julian))}
\\
m_0 &
=
2;30,0 \rr{60,60}
+ 2015529
=2015531+\frac{1}{2},
\\
s_0 &
=
26,58 \rr{27,60} =\frac{809}{810}, \label{s0K}
\\
a_0 &
=
5,112\rr{28,126} = \frac{53}{252}.
\intertext{
Modern \karana{} calculations (\eg{} in the almanac \cite{tib2013})
use (at least in the cases I know) the natural values
$m_1/30$ and $s_1/30$
for the daily increments $m_2$ and $s_2$, while
$a_2=1/28$ is given by \eqref{a2} as for the versions above.
However, the \KT{} gives instead the
following simplified (rounded) values for $m_2$ and $s_2$,
\cf{} \refR{R30}, which thus not are used today:}
m_2 &
=
0;59 \rr{60}
=\frac{59}{60},
\\
s_2 &
=
0,4,20 \rr{27,60,60} = \frac{13}{4860},
\intertext{
Furthermore,
the \KT{} gives also
another computational simplification (perhaps suggested as an alternative?),
also not used today
(as far as I know, and certainly not in the almanac \cite{tib2013}),
see
\cite[p.~232--233]{Henning},
\cite[Chapter 2, v.~30]{Henning-Karana},
where
the values $m_1,s_1,a_1$ above are used only for the first month of a
calendar calculation (i.e, month 3, \tibx{nag pa}), and for each following
month one
adds instead the simpler\footnotemark}
m_1' &
=
29;32,0 \rr{60,60}
=\frac{443}{15}
=29+\frac{8}{15},
\label{m1'KT}
\\
s_1' &
=
2,11 \rr{27,60}
=\frac{131}{1620},
\label{s1'KT}
\\
a_1' &
=
2,0\rr{28,126} = \frac{1}{14}.
\end{align}
\footnotetext{It seems surprising that anyone should bother about these
minor simplifications,
once each for each month, which are very minor compared to the mass of
calculations for every day.}
The \KT{} calculates the true month by \eqref{tm0} with an epoch value
$\gbx=0$. The true month is simply rounded down,
as in the later \TS{} version, see \refApp{ATS}.
This would correspond to the
intercalation rule \eqref{leaprule-T}
if a leap month is given the same
number as the following month, as in Indian and later Tibetan calendars.
However, it seems that the \Kc{} system instead
gave a leap month the same name as the month preceding it,
as in the Bhutanese calendar in \refS{ABhutan}
(and the Chinese calendar \cite{CC}),
see \cite[Published calendar explanation]{kalacakra}.
In this case, the leap month is inserted before the month with
intercalation index $0$ or $1$; in the traditional formulation,
these indices are repeated (see \refR{Rix}) and thus the leap month
gets the same intercalation index, so
the rule \eqref{leaprule-T} nevertheless holds.
Cf.\ the rules \eqref{leaprule-B2} and \eqref{leaprule-B} for the Bhutanese
version.
\xfootnote{
The \Karana{} calculations in modern almanacs do not give any names or
numbers, or years, for the months, as far as I know,
so the question of leap months is irrelevant to them.
However, in \eg{} \cite{tib2013},
the \tmc{} and intercalation index are given for each month, for both \PH{}
and \karana{} calculations.
This almanac includes a month with \karana{}
intercalation index given as 65,
in accordance with the rules presented here; this month would thus be
a leap month (a second \tibx{M\=agha} = \tibx{mChu}, or month 1, 2014 if
we anachronistically use the same numbering as for the other
versions, although this was introduced long after the \KT,
see \refS{SSmonths}).
}
More precisely, this means that there is a leap month $(Y,M)$ when the
regular month $(Y,M)$ has intercalation index \eqref{ix} 63 or 64,
\ie, its \tm{} is $n+63/65$ or $n+64/65$ where $n$ is the \tmc.
The month $(Y,M+1)$ then has a \tm{} that is
$\xfrac{67}{65}=1+2/65$ larger, \ie{} $n+2$ or $n+2+1/65$; hence it has
\tmc{} that is $n+2$ and intercalation index $0$ or $1$.
There is thus a gap, which is filled by a leap month which is given the
missing \tmc{} $n+1$, and (conventionally) intercalation index $65$ or 66;
this month is thus leap month $(Y,M)$.
We thus have the rule, \cf{} \eqref{leaprule-B},
\begin{equation}\label{leaprule-K}
\hskip-1em
\vbox {\narrower\narrower\narrower\noindent\em
Regular month $M$ in year $Y$ is followed by a leap month if and only if
its intercalation index
$ix = 63\text{ or\/ }64$.
}
\hskip-3em
\end{equation}
Furthermore, we can say, \cf{} \eqref{leaprule-B},
\begin{equation}\label{leaprule-K2}
\hskip-1em
\vbox {\narrower\narrower\narrower\noindent\em
A month is a leap month if and only if
its intercalation index
$ix = 65\text{ or\/ }66$.
}
\hskip-3em
\end{equation}
This, however, is a tautology since we use the exceptional values 65 and 66
only for leap months. We cannot replace them by 0 and 1 in \eqref{leaprule-K2},
since the following (regular) month has intercalation index 0 or 1.
Summarizing,
in all cases, the \tmc{} is given by \eqref{tm0} rounded down, and
increased by 1 in the case of a leap month; the intercalation index is given
by \eqref{ix}, increased by 2 (to 65 or 66) for a leap month.
(Cf.\ \refR{Rix}.)
For the epoch 806 above, with $\gbx=0$, it follows from \eqref{ix} and the
discussion above that \eqref{leaprule-K} is equivalent to
\eqref{leaprule-gb0} with $\gb\equiv 4\pmod{65}$, see also \eqref{jeppe+} below.
By \eqref{gam} and \eqref{gamx},
\eqref{lygam} and \eqref{lygamx} hold with
$\gamma =28$ and $\gamx =22$.
The value \eqref{m1KT} of $m_1$ (the mean length of the month) is
$\approx 29.530556$ days, about 2.7 seconds shorter than the mean
length \eqref{meanmonth} for the standard versions. The \Kc{} value
differs from the modern astronomical value by about $2.9$ seconds, and is thus
less exact than the value \eqref{meanmonth}, but the difference amounts to
less than 1 hour every century.
The mean length of the year is, \cf{} \eqref{meanyear},
\begin{equation}
\label{meanyearK}
\frac{m_1}{s_1}
=\frac{3731481}{10216}
\approx 365.258516
\text{ days},
\end{equation}
which is better than the value in \eqref{meanyear} for the later versions;
it is $0.01633$ days longer than
the astronomical value of the tropical year ($365.24219$ days)
and only $0.002156$ days longer than the
sidereal year ($365.25636$ days),
see \refS{Smean} and \cite[Table 15.3]{AA}.
(Nevertheless, the year is intended to be a tropical year
\cite[p.~298]{Henning}.)
\subsection{Further historical versions}
Several further historical versions of the Tibetan calendar are described by
\citet{Schuh}. (I do not know to which extent these were actually used.)
A proposed (but never adopted) version by Zhonnu Pal \tib{gzhun-nu dpal}
from 1443
is described by
\citet[pp.~307--321]{Henning} and
\cite[Error correction system]{kalacakra}.
\subsection{Sherab Ling}\label{ASL}
A modern attempt at a reformed Tibetan calendar developed by Kojo
Tsewang Mangyal (Tsenam) at the Sherab Ling monastery in Bir, India,
is described in \cite[pp.~342--345]{Henning},
see also \cite[A reformed Tibetan calendar]{kalacakra}
and \cite[Epoch data]{kalacakra}.
It uses the value of $m_1$ in \refS{Sastro}, but uses
\begin{equation}
s_1=2,10,58,2,564,5546
\rr{27,60,60,6,707,6811}
=\frac{3114525}{38523016}.
\end{equation}
This deviation from the traditional value,
and from the equivalent fundamental relation \eqref{6765},
means that leap months no
longer will be regularly spaced in the traditional pattern with 2 leap
months every 65 regular months.
(However, the difference is small; the average number of leap months for 65
regular months is $1.9978$, about $0.1\%$ less than in the standard
calendars, which means that on the average a leap month is delayed a month
after about 75 years.)
\subsection{Sarnath}
A version of the Sherab Ling calendar (see \ref{ASL})
is published by the Jyotish
Department of
the Central University of Tibetan Studies
at Sarnath, India,
see
\cite[p.~346]{Henning},
\cite[A reformed Tibetan calendar]{kalacakra}.
This version differs from all other Tibetan calendars that I have
heard of in that the months begin at full moon (as in Indian calendars
in northern India \cite{CC}).
\subsection{Sherpa}
The Sherpas (a minority in Nepal) have a Tibetan calendar
called the Khunu almanac.
(The official Nepalese calendar is of the Indian type.)
The Sherpa calendar for 2002 I found on the web \cite{sherpa}
is identical to the \PH{} version, and this seems to be the general rule.
\subsection{Drikung Kagyu}
According to \cite{Berzin}, the Drikung Kagyu tradition follows a system
that combines the \TS{} and \PH{} traditions.
\subsection{Yellow calculations}\label{Ayellow} (Chinese-style)
According to \cite{Berzin},
the yellow system is like the Chinese in that it has no repeated or
skipped days. Months have 29 or 30 days, numbered consecutively "and
determined according to several traditions of calculation". The way it
adds leap months is similar to, but not equivalent to, the
Chinese. Unlike the Chinese calendar, it uses the basic calculations
from \KT.
This seems to be a version of the Chinese calendar
(or astrology) rather than the Tibetan; it might also be the Chinese
calendar (without modification), but then some of the statements just made
are incorrect.
See also \cite{MongoliansWelcome} and \cite{whenMongolia}.
Inner Mongolia (in China) follows the yellow system \cite{Berzin}.
\subsection{An astronomical version}
Henning \cite{kalacakra} has constructed
a reformed Tibetan calendar combining the principles of the \KT{}
with modern astronomical calculations.
\subsection{A comparison} \label{Acomp}
We give some examples of the (small) differences between the
four different versions of the Tibetan calendar in
Appendices \ref{APH}--\ref{ABhutan}.
The epoch values given above for the different versions use different
years. To enable an easy comparison, \refT{T4epok} gives the epoch values
for the four versions calculated for the same epoch, here chosen as
the \KT{} epoch at the beginning of \nag{} (\Caitra) 806.
(In all four versions, the epoch, \ie, the mean new moon,
is at JD 2015531, Monday 23 March 806 (Julian), \cf{}
\refR{Repoch2}.)
\xfootnote{
This happens to be the last day of the preceding month in all four versions;
this is day 30 in month 2 for the \PH{} and Bhutanese versions, and day 30
in leap month 3 in the \TS{} and Mongolian versions.
}
Since all four versions have the same mean motions $m_1,s_1,a_1$, the
differences between them for
mean dates, solar longitudes and lunar anomalies will be constant. For
example, the \TS{} mean solar longitude is always
$0.018261-0.004975=0.013286=4.78\grad$ larger than the \PH{} value,
and the mean dates differ by 0.046, about $1/20$.
The differences in true date and true solar longitude will be varying because of
the corrections in \eqref{truedate} and \eqref{truesun}, but the average
difference is the same as for the mean values.
Thus the
\PH{} and \TS{} dates will differ for, on the average, about one day in 20,
\ie, typically one or two days in a month. (See \refT{T4+-} for some
examples, with 0--4 days differing each month.)
\begin{table}[!htpb]
\begin{tabular}{lllll}
& \PH & \TS & Mongolia & Bhutan\\
\hline
$m_0$ & $2; 22, 34, 2, 518$ & $2; 25, 20, 2, 352$ & $2; 25, 6, 3, 327$
& $2; 24, 37, 5, 431$ \\
& 2.376238 & 2.422338 & 2.418494 & 2.410537 \\
\hline
$s_0$ & $0, 8, 3, 3, 33$ & $0, 29, 34, 5, 37$ & $0, 38, 17, 0, 6$
& $0, 28, 12, 3, 15$ \\
& 0.004975 & 0.018261 & 0.023632 & 0.017413 \\
\hline
$a_0$ & $5, 98$ & $5, 112$ & $5, 101$ & $6, 22$ \\
& 0.206349 & 0.210317 & 0.207200 & 0.220522 \\
\hline
\end{tabular}
\caption{Epoch data for four versions of the Tibetan calendar
for JD 2015531 (23 March 806), given both in
Tibetan form with radices
$\xpar{60,60,6,707}$,
$\xpar{27, 60, 60, 6, 67}$,
$\xpar{28,126}$
and with 6 decimals.
We have here given $m_0$ in the traditional form modulo 7; to get the result
in JD as in this paper, add 2015529.
}
\label{T4epok}
\end{table}
\begin{remark}
The differences in the correction terms can be estimated as follows:
Consider two versions whose epoch values $m_0,s_0,a_0$ differ by
$\gD m_0$, $\gD s_0$, $\gD a_0$. By \eqref{moonequ}, the arguments used
in the table $moon\_tab$ differs by $28\gD a_0$; since the maximum
derivative (slope) in the table \eqref{moontab}
is 5, the difference in $moon\_equ$
satisfies $|\gD moon\_equ|\le 140|\gD a_0|$; this correction is divided by
60 in \eqref{truedate},
so the corrections to the true date differ by at most $\frac73|\gD a_0|$.
Similarly, by \eqref{anosun}, \eqref{sunequ} and \eqref{suntab},
the difference in $sun\_equ$ satisfies
$|\gD sun\_equ|\le 6\cdot12|\gD s_0|$; this correction is divided by
60 in \eqref{truedate} and by $27\cdot60$ in \eqref{truesun},
so the corrections to the true date and true solar longitude differ by at
most $1.2|\gD s_0|$ and $0.045|\gD s_0|$.
In particular, the difference between the true solar longitudes differ
always by between $0.95$ and $1.05$ times the mean difference $\gD s_0$; for
\PH{} and \TS{} the difference is thus between $4.57\grad$ and $5.00\grad$.
For the date, the mean difference $\gD m_0$
between \PH{} and \TS{} is as said above
$0.046$, and the maximum differences in the correction terms
$\frac1{60}\gD moon\_equ$ and $\frac1{60}\gD sun\_equ$ are
$\frac73\gD a_0=0.009$ and $1.2\gD s_0=0.016$, respectively;
hence the difference of the true
dates is $0.046\pm0.009\pm0.016$, \ie, between $0.021$ and $0.071$,
with \TS{} always larger.
\xfootnote{
The two correction terms are periodic, with periods the anomalistic month
\eqref{meanano} and the year \eqref{meanyear};
the maximum values will thus coincide several times each year.
}
As a consequence, the \PH{} version is slightly
behind the \TS, so when the \PH{} and \TS{} dates
of a calendar day differ, the \PH{} date is always larger (by 1).
(See the example of differences in \refT{T4+-}.)
Similarly, the difference in mean date
between the Mongolian and \TS{} versions is
only $\gD m_0=0.0038$; the differences in solar longitude and lunar anomaly
are
$\gD s_0=-0.0054$ and
$\gD a_0=0.0031$; hence the true date differs by
$0.0038\pm0.0073\pm0.0064$,
\ie, between $-0.010$ and $0.018$. (So we expect a difference of the
calendar date only a few
days each year, \cf{} \refT{T4+-}.)
\end{remark}
\refT{T4Leap} shows leap months for a range of (Gregorian) years.
Since all four versions have a simple periodic pattern with alternating 32
or 33 regular months between the leap months, the same pattern repeats for
ever.
Note that the \TS{} and Mongolian versions have the same leap months, as said
before.
We see also that leap months in Bhutan come 2 or 3 months after the \PH{}
leap months (3 or 4 months later if we take into account the different
numbering of Bhutanese leap months), and the \TS{} and Mongolian leap months
come an additional 4 (really 3) months later.
\refT{T4+-} shows all repeated and skipped days during 2012 for the four
versions. (This year is chosen since none of the versions has a leap month.
The versions also all start this year on the same day.)
It is seen that four versions are very similar; often the same day is
repeated or skipped, but it also frequently happens that different
versions differ by a day (meaning that some days get different dates).
We can see that this year each month has the same
length in all four versions.
\refT{T4Losar} shows the Gregorian dates of New Year
for a range of (Gregorian) years.
We see that most years, all four calendars coincide. However, some times the
\PH{} and Bhutanese versions, or just the \PH{} version, is a month later
(due to the difference in leap months); sometimes there is also a difference
of a day between two versions.
\xfootnote{
Bhutan actually had New Year 3/3 2003, with day 1 of month 1 repeated
3/3 and 4/3, according to the government web site calendar \cite{Bhutan}
[Henning, personal communication];
the calculations described here yield
(as do the ones by Henning
\cite[Open source Bhutanese calendar software]{kalacakra})
3/3 instead as a repeated day 30 in
the last (leap) month of 2002. I have no explanation for this discrepancy.
}
In particular, we do not see any differences between the \TS{} and Mongolian
New Year in \refT{T4Losar}; a computer search reveals that the last time
these versions had different New Year was 1900 (31/1 vs 1/2) and the next
will be 2161 (26/2 vs 25/2), and only four more differences were found for
the present millenium.
\begin{table}[!htpb]
\begin{tabular}{r r r r r}
& \PH & \TS & Mongolia & Bhutan\\
\hline
2000 & 1 & 8 & 8 & 4 \\
2001 & & & & \\
2002 & 10 & & & 12 \\
2003 & & 4 & 4 & \\
2004 & & & & \\
2005 & 6 & & & 9 \\
2006 & & 1 & 1 & \\
2007 & & & & \\
2008 & 3 & 9 & 9 & 5 \\
2009 & & & & \\
2010 & 11 & & & \\
2011 & & 6 & 6 & 2 \\
2012 & & & & \\
2013 & 8 & & & 10 \\
2014 & & 2 & 2 & \\
2015 & & & & \\
2016 & 4 & 11 & 11 & 7 \\
2017 & & & & \\
2018 & & & & \\
2019 & 1 & 7 & 7 & 3 \\
2020 & & & & \\
\end{tabular}
\caption{Leap months for four versions of the Tibetan calendar.}
\label{T4Leap}
\end{table}
\begin{table}[!htpb]
\begin{tabular}{r| l|l|l|l}
month & \PH & \TS & Mongolia & Bhutan\\
\hline
1 & 5, -19 & 4, -20 & 4, -20 & 4, -19 \\
2 & 9, -12, -25, 27 & 8, -13 & 8, -13 & 8, -13 \\
3 & -17 & -17 & -17 & -17 \\
4 & 3, -10 & 2, -11 & 2, -11 & 2, -10 \\
5 & -13, 29 & -14, 28 & -14, 28 & -13, 28 \\
6 & -6 & -6 & -6 & -6 \\
7 & -9, 25 & -9, 25 & -9, 25 & -9, 24 \\
8 & -1 & -2 & -2 & -1 \\
9 & -5, 20, -29 & -6, 19, -29 & -6, 20, -29 & -5, 19, -29 \\
10 & & & & \\
11 & -3, 13, -27 & -3, 12, -28 & -4, 12, -28 & -3, 12, -27 \\
12 & 17, -21 & 15, -22 & 15, -22 & 15, -21 \\
\end{tabular}
\caption{Repeated and skipped (marked with -) days in each month
2012 for four versions of the Tibetan
calendar.}
\label{T4+-}
\end{table}
\begin{table}[!htpb]
\newcommand\xx{\textbf}
\begin{tabular}{r r r r r}
& \PH & \TS & Mongolia & Bhutan\\
\hline
2000 & 6/2 & 6/2 & 6/2 & 6/2 \\
2001 & 24/2 & 24/2 & 24/2 & 24/2 \\
2002 & 13/2 & 13/2 & 13/2 & 13/2 \\
2003 & \xx{3/3} & 2/2 & 2/2 & \xx{4/3} \\
2004 & 21/2 & 21/2 & 21/2 & 21/2 \\
2005 & 9/2 & 9/2 & 9/2 & 9/2 \\
2006 & \xx{28/2} & \xx{30/1} & \xx{30/1} & \xx{28/2} \\
2007 & 18/2 & 18/2 & 18/2 & 18/2 \\
2008 & \xx{7/2} & 8/2 & 8/2 & 8/2 \\
2009 & 25/2 & 25/2 & 25/2 & 25/2 \\
2010 & 14/2 & 14/2 & 14/2 & 14/2 \\
2011 & \xx{5/3} & 3/2 & 3/2 & 3/2 \\
2012 & 22/2 & 22/2 & 22/2 & 22/2 \\
2013 & 11/2 & 11/2 & 11/2 & 11/2 \\
2014 & \xx{2/3} & \xx{31/1} & \xx{31/1} & \xx{2/3} \\
2015 & 19/2 & 19/2 & 19/2 & 19/2 \\
2016 & 9/2 & 9/2 & 9/2 & 9/2 \\
2017 & 27/2 & 27/2 & 27/2 & 27/2 \\
2018 & 16/2 & 16/2 & 16/2 & 16/2 \\
2019 & 5/2 & 5/2 & 5/2 & 5/2 \\
2020 & 24/2 & 24/2 & 24/2 & 24/2 \\
2021 & 12/2 & 12/2 & 12/2 & 12/2 \\
2022 & \xx{3/3} & \xx{2/2} & \xx{2/2} & \xx{3/3} \\
2023 & 21/2 & 21/2 & 21/2 & 21/2 \\
2024 & 10/2 & 10/2 & 10/2 & 10/2 \\
2025 & \xx{28/2} & \xx{1/3} & \xx{1/3} & \xx{28/2} \\
2026 & 18/2 & 18/2 & 18/2 & 18/2 \\
2027 & 7/2 & 7/2 & 7/2 & 7/2 \\
2028 & 26/2 & 26/2 & 26/2 & 26/2 \\
2029 & 14/2 & 14/2 & 14/2 & 14/2 \\
2030 & \xx{5/3} & 3/2 & 3/2 & 3/2 \\
\end{tabular}
\caption{
Gregorian dates for
New Year (\tibx{Losar}) for four versions of the Tibetan calendar.
Dates differing from the majority in boldface.}
\label{T4Losar}
\end{table}
\section{The 60 year cycle}\label{A60}
The Indian 60 year cycle is derived from an Indian cycle of 60 names for the
``Jovian years''.
\xfootnote{
It takes Jupiter almost 12 years to orbit the sun.
A 1/12 of the orbital period can be called a Jovian year,
and
the traditional Indian Jovian cycle gives each
Jovian year a name (\emph{samvatsara})
from a list of 60 names, so the names repeat with a cycle of 5
revolutions. The solar years are given the same names,
based on the calculated position of
Jupiter at the beginning of the year; hence sometimes (every 85 or 86
years) a name is skipped (expunged) from the list. In southern India
this has from the 9th century been simplified to a simple 60 year
cycle of names,
and the same is done in Tibet. \cite{CC}, \cite[p.~143f]{Henning}.
}
\refT{T60}
(taken from \cite{Henning})
gives the full list of names, in Tibetan and Sanskrit,
together with the corresponding
year in the Chinese 60 year cycle and the
Gregorian years in the last and current cycle.
(Somewhat different transliterations of the Sanskrit names are given in
\cite{CC}.)
See also \refT{Tlosar}, which gives the Gregorian dates of New Year for
the years in the last and current cycles.
\begin{longtable}
{r r l l l r r }
\rlap{year}\phantom{0} &&
element--animal & Tibetan & Sanskrit & & \\
\hline
1 & 4 & Fire--Rabbit & rab byung & prabhava & 1927 & 1987\\
2 & 5 & Earth--Dragon & rnam byung & vibhava & 1928 & 1988\\
3 & 6 & Earth--Snake & dkar po & suklata & 1929 & 1989\\
4 & 7 & Iron--Horse & rab myos & pramadi & 1930 & 1990\\
5 & 8 & Iron--Sheep & skyes bdag & prajapati & 1931 & 1991\\
6 & 9 & Water--Monkey & anggi ra & ankira & 1932 & 1992\\
7 & 10 & Water--Bird & dpal gdong & srimukha & 1933 & 1993\\
8 & 11 & Wood--Dog & dngos po & bhava & 1934 & 1994\\
9 & 12 & Wood--Pig & na tshod ldan & yuvika & 1935 & 1995\\
10 & 13 & Fire--Mouse & 'dzin byed & dhritu & 1936 & 1996\\
11 & 14 & Fire--Ox & dbang phyug & isvara & 1937 & 1997\\
12 & 15 & Earth--Tiger & 'bru mang po & vahudhvanya & 1938 & 1998\\
13 & 16 & Earth--Rabbit & myos ldan & pramadi & 1939 & 1999\\
14 & 17 & Iron--Dragon & rnam gnon & vikrama & 1940 & 2000\\
15 & 18 & Iron--Snake & khyu mchog & brisabha & 1941 & 2001\\
16 & 19 & Water--Horse & sna tshogs & citra & 1942 & 2002\\
17 & 20 & Water--Sheep & nyi ma & bhanu & 1943 & 2003\\
18 & 21 & Wood--Monkey & nyi sgrol byed & bhanutara & 1944 & 2004\\
19 & 22 & Wood--Bird & sa skyong & virthapa & 1945 & 2005\\
20 & 23 & Fire--Dog & mi zad & aksaya & 1946 & 2006\\
21 & 24 & Fire--Pig & thams cad 'dul & sarvajit & 1947 & 2007\\
22 & 25 & Earth--Mouse & kun 'dzin & sarvadhari & 1948 & 2008\\
23 & 26 & Earth--Ox & 'gal ba & virodhi & 1949 & 2009\\
24 & 27 & Iron--Tiger & rnam 'gyur & vikrita & 1950 & 2010\\
25 & 28 & Iron--Rabbit & bong bu & khara & 1951 & 2011\\
26 & 29 & Water--Dragon & dga' ba & nanda & 1952 & 2012\\
27 & 30 & Water--Snake & rnam rgyal & vijaya & 1953 & 2013\\
28 & 31 & Wood--Horse & rgyal ba & jaya & 1954 & 2014\\
29 & 32 & Wood--Sheep & myos byed & mada & 1955 & 2015\\
30 & 33 & Fire--Monkey & gdong ngan & durmukha & 1956 & 2016\\
31 & 34 & Fire--Bird & gser 'phyang & hemalambha & 1957 & 2017\\
32 & 35 & Earth--Dog & rnam 'phyang & vilambhi & 1958 & 2018\\
33 & 36 & Earth--Pig & sgyur byed & vikari & 1959 & 2019\\
34 & 37 & Iron--Mouse & kun ldan & sarvavati & 1960 & 2020\\
35 & 38 & Iron--Ox & 'phar ba & slava & 1961 & 2021\\
36 & 39 & Water--Tiger & dge byed & subhakrita & 1962 & 2022\\
37 & 40 & Water--Rabbit & mdzes byed & sobhana & 1963 & 2023\\
38 & 41 & Wood--Dragon & khro mo & krodhi & 1964 & 2024\\
39 & 42 & Wood--Snake & sna tshogs dbyig & visvabandhu & 1965 & 2025\\
40 & 43 & Fire--Horse & zil gnon & parabhava & 1966 & 2026\\
41 & 44 & Fire--Sheep & spre'u & pravamga & 1967 & 2027\\
42 & 45 & Earth--Monkey & phur bu & kilaka & 1968 & 2028\\
43 & 46 & Earth--Bird & zhi ba & saumya & 1969 & 2029\\
44 & 47 & Iron--Dog & thun mong & sadharana & 1970 & 2030\\
45 & 48 & Iron--Pig & 'gal byed & virobhakrita & 1971 & 2031\\
46 & 49 & Water--Mouse & yongs 'dzin & paradhari & 1972 & 2032\\
47 & 50 & Water--Ox & bag med & pramadi & 1973 & 2033\\
48 & 51 & Wood--Tiger & kun dga' & ananda & 1974 & 2034\\
49 & 52 & Wood--Rabbit & srin bu & raksasa & 1975 & 2035\\
50 & 53 & Fire--Dragon & me & anala & 1976 & 2036\\
51 & 54 & Fire--Snake & dmar ser can & vingala & 1977 & 2037\\
52 & 55 & Earth--Horse & dus kyi pho nya & kaladuti & 1978 & 2038\\
53 & 56 & Earth--Sheep & don grub & siddhartha & 1979 & 2039\\
54 & 57 & Iron--Monkey & drag po & rudra & 1980 & 2040\\
55 & 58 & Iron--Bird & blo ngan & durmati & 1981 & 2041\\
56 & 59 & Water--Dog & rnga chen & dundubhi & 1982 & 2042\\
57 & 60 & Water--Pig & khrag skyug & rudhirura & 1983 & 2043\\
58 & 1 & Wood--Mouse & mig dmar & raktaksi & 1984 & 2044\\
59 & 2 & Wood--Ox & khro bo & krodhana & 1985 & 2045\\
60 & 3 & Fire--Tiger & zad pa & ksayaka & 1986 & 2046\\
\caption{The Chinese and Indian 60 year cycles of names, with the
names from the Indian cycle both in Tibetan and in Sanskrit.
The first number on each line shows the number in the
Prabhava cycle; the second shows the number in the
Chinese cycle. The last two numbers show the Gregorian
years in the last and current cycles.}
\label{T60}
\end{longtable}
\section{Leap months and the mean sun}\label{Aleap}
We give here an explanation of the leap month rules in \refS{Smonths},
based on Tibetan astronomy.
(This appendix is based on the description in \citet{Henning};
see also \cite[On intercalary months]{kalacakra}.
For a detailed historical discussion of different leap month rules,
see \citet[pp.~107--117]{Schuh};
see further
\citet{Yamaguchi} (which also contains tables with actually observed
leap months from historical data) and
\citet[Early epochs]{kalacakra}.
\xfootnote{
In particular, according to
Schuh \cite[Kalenderrechnung]{tibetenc},
see also \cite{Schuh},
the \dpp{s} described here were introduced (for the \PH{} version)
in 1696.
However, some similar method to align the year seems to have been used earlier;
the principle to use the position of the mean sun goes back to early
Indian calendars and
the calculations described
in \refS{Smonths} are only minor modifications of the ones in
the \KT{} which thus seem to be based on considerations related
to the ones
given here, although possibly in different formulations.
See also the discussion in \refS{ASKdp}.
})
\subsection{General theory}\label{ASdp}
The key is the position of the mean sun, which is a fictitious version of the
sun that travels along the ecliptic with uniform speed.
Its longitude (the mean solar longitude)
at the beginning of
\tm{} $n$
was denoted by $\MSL(0,n)$ in \refS{Sastro};
we use here the simplified notation $\MSL(n)$.
It is by \eqref{meansun} given
by the linear formula
\begin{equation}
\label{msl}
\MSL(n)=s_1n+s_0,
\end{equation}
where $s_0=\MSL(0)$ is the mean solar longitude at the epoch.
The Tibetan constant $s_1$, the mean motion of the sun per (lunar) month,
is given by
\begin{equation}\label{s1q}
s_1= \frac{65}{804}=\frac{65}{67\cdot12},
\end{equation}
see \eqref{s1}.
(This can also be expressed as 65/67 signs, or $29\frac{7}{67}\grad$.)
Note that \eqref{s1q} says that the mean sun goes
around the
ecliptic exactly 65 times in 804 lunar months, \ie,
\begin{equation}
804 \text{ lunar months } = 65 \text{ years} .
\end{equation}
Since $804=67\cdot12$, this is equivalent to the relation \eqref{6765},
and it explains the leap year cycle of 65 years, see \refR{R6567}.
Moreover, the zodiac contains 12 evenly spaced \emph{\dpp s}
\xfootnote{We use this term from \citet{Henning}.}
\tib{sgang},
one in each sign. Let us denote these (and their
longitudes) by $p_1,\dots,p_{12}$.
The rule for naming months is, see \cite{Henning}:
\begin{equation}\label{rule-dp}
\hskip-1em
\vbox {\narrower\narrower\narrower\noindent\em
A month where the mean sun passes a
\dpp{} $p_M$ is given number $M$.
}
\hskip-3em
\end{equation}
Since $s_1<\frac1{12}$, the
spacing of the \dpp{s}, the mean sun can never pass two \dpp{s} in one
month, but sometimes it does not pass any of them; in that case the
month is designated as a leap month, and is given the number of the
next month. In both cases, the number of the month is thus given by
the first \dpp{} $p_M$ that comes after $\MSL(n)$.
(We do not have to worry about the exact definition in the ambiguous
case when $\MSL(n)$ exactly equals some $p_M$; the constants in the
\PH{} system are such that this never will happen, see
\refR{Rnotequal}
below. I have therefore
just chosen one version in the formulas below.)
The leap month rule is thus:
\begin{equation}\label{leaprule-dp}
\hskip-1em
\vbox {\narrower\narrower\narrower\noindent\em
A month where the mean sun does not pass any definition point
is a leap month.
}
\hskip-3em
\end{equation}
\begin{remark}\label{RChina}
The rule above for numbering the months is
the same as in many Indian calendars \cite[Chapters 9 and 17]{CC},
and almost the same as the rule in the Chinese calendar
\cite[Chapters 16]{CC};
however, in these calendars, the
\dpp{s} are beginnings of the zodiacal signs, \ie, multiples of $30\grad=1/12$
while in the Tibetan calendar, the \dpp{s} are shifted, see \refS{SSdp}.
(In the Chinese calendar, the \dpp{s} are
called \emph{(major) solar terms}.
In most (but not all) Indian calendars,
month 1 (New Year) is defined by the vernal equinox, \ie, $p_1=0$ in
our notation;
the exact rule in the Chinese calendar is that the
winter solstice ($270\grad=3/4$) occurs in month 11, which corresponds to
$p_{11}=3/4$, but there are some complications for other months.)
Note however, that the (present)
Chinese and Indian calendars use the
true motion of sun and moon, while the Tibetan uses the mean motion,
leading to regularly spaced leap months in the Tibetan calendar, but
not in the Chinese and Indian ones. (It also leads to skipped months
sometimes in the Indian calendars.)
Note further that the numbering of leap months differs in the Chinese
calendar, where a leap month is given the number of the preceding month.
\end{remark}
The longitude in \eqref{msl} is naturally taken modulo 1, \ie,
considering only the fractional part. But a moment's consideration
shows that the integer part shows the number of elapsed full circles
of the sun, \ie, the number of years; in this appendix we thus use $\MSL(n)$
for the
real number defined by \eqref{msl}.
Let, as in \refS{Smonths},
$Y$ and $M$ be the year and the number of the month with \tmc{} $n$,
and let the Boolean variable $\ell$ indicate whether the month is a leap month.
The rule \eqref{rule-dp} then yields the following relations determining
$(Y,M,\ell)$,
with $p_0=p_{12}-1$,
\begin{gather}
Y-Y_0+p_{M-1}
<
\MSL(n)
\le
Y-Y_0+p_{M},
\label{xa1}
\\
\ell =[\MSL(n+1)\le Y-Y_0+p_{M}].
\label{xa2}
\end{gather}
Furthermore, the points $p_M$ are evenly spaced, so $p_M=p_0+M/12$.
\begin{remark}\label{Rs0}
The initial longitude $s_0$ is, as any longitude, really defined modulo 1,
\ie, only the fractional part matters. However, when we regard $\MSL(n)$ as a
real number, we have to make the right choice of integer part of $s_0$.
Since the epoch is assumed to be in year $Y_0$, with a month $M$
satisfying $1\le M\le12$, taking $n=0$ in \eqref{xa1} shows that
we must have
\begin{equation}\label{s0krav}
p_0<s_0\le p_{12}=p_0+1.
\end{equation}
Equivalently, $\ga$ defined in \eqref{ga} below must satisfy
\begin{equation}\label{gakrav}
0<\ga\le12.
\end{equation}
(This follows also by taking $n=0$ and $Y=Y_0$ in \eqref{b3}--\eqref{b5}
below, see \refR{Repoch-ga}.)
Consequently,
for the formulas for month numbers and leap months below,
we have to assume that the integer part of $s_0$ is chosen
such that \eqref{s0krav}--\eqref{gakrav} hold;
this sometimes means adding 1 to the traditional value
(which does not affect the solar longitude seen as an angle, \ie{}
modulo 1).
\xfootnote{
For calculations one can use any $s_0$ and instead normalize $\ga$ in
\eqref{ga} modulo 12 so that \eqref{gakrav} holds.
}
\end{remark}
Let us for simplicity write $Y'=Y-Y_0$. Then \eqref{xa1} can be
rewritten
\begin{equation}
\label{b0}
Y'+\frac{M-1}{12}+p_0
<
\MSL(n)
\le
Y'+\frac{M}{12}+p_0
\end{equation}
or
\begin{equation} \label{b0x}
12Y'+M-1
<
12(\MSL(n)-p_0)
\le
12Y'+M.
\end{equation}
Hence, if we use \eqref{msl} and further define
\begin{equation}
\label{ga}
\ga=12(s_0-p_0),
\end{equation}
we have
\begin{equation}
\label{b1}
12Y'+M=\ceil{12(\MSL(n)-p_0)}
=\ceil{12s_1n+\ga}.
\end{equation}
Consequently, we can calculate $(Y,M)$ from $n$ by
\begin{align}
x&=\ceil{12s_1n+\ga},
\label{b3}\\
M&= x \amod 12,
\label{b4}\\
Y&=\frac{x-M}{12}+Y_0.
\label{b5}
\end{align}
To complete the calculations of $(Y,M,\ell)$ from $n$, we
find similarly from \eqref{xa2}, or simpler by \eqref{b1}
because a month is leap if and only if it gets
the same number as the following one,
\begin{equation}
\label{b2}
\ell=\bigbool{\ceil{12s_1(n+1)+\ga}=\ceil{12s_1n+\ga}}.
\end{equation}
For the Tibetan value of $s_1$ in \eqref{s1q}, we have
$12s_1=\frac{65}{67}$,
and thus \eqref{b3} can be written
\begin{equation}\label{bxx3}
x = \Ceil{\frac{65}{67}n+\ga}
=\Ceil{\frac{65 n + 67\ga}{67}}
=\Ceil{\frac{65 n + \gb}{67}},
\end{equation}
where we define the integer $\gb$ to be $67\ga$ rounded up, \ie,
\begin{equation}\label{gb}
\gb=\ceil{67\ga}.
\end{equation}
Note that \eqref{bxx3} and \eqref{b4}--\eqref{b5} are the same as
\eqref{bx3}--\eqref{bx5}, which shows that the algorithmic
calculations in \refS{Smonths}
yield the same correspondence between $(Y,M)$ and \tmc{} $n$ as
the rule \eqref{rule-dp} used in this appendix, provided the value of
$\gb$ is the same; in particular, the two methods yield the same leap months.
\begin{remark}\label{Repoch-ga}
By \eqref{b3}--\eqref{b5}, the epoch month with \tmc{} $n=0$ is month
$M=\ceil{\ga}$ in the epoch year.
(Recall that we have $0<\ga\le12$ by \eqref{gakrav}, so this yields a value
$1\le M\le 12$. Conversely, we see again that \eqref{gakrav} is necessary
for $Y_0$ to be the epoch year.)
By \refR{Repoch}, the traditional choices of epoch always yield $M=2$ or 3
for $n=0$. There are thus only two possibilities:
either $1<\ga\le2$ and the epoch month with \tmc{} 0 is month 2
(so the nominal epoch month 3 has \tmc{} 1), or $2<\ga\le3$ and
the \tmc{} is 0 for month 3.
\end{remark}
\begin{remark}\label{Rnotequal}
To verify the assertion above that $\MSL(n)$ never equals some \dpp, note
that the calculations above show that this would happen if and only
if $12s_1n+\ga$ would be an integer. Since $12s_1=\frac{65}{67}$,
this can happen only if $67\ga=804(s_0-p_0)$ is an integer (and in that case it
would happen for some $n$); we will see in \eqref{ga1S} below that
for the \PH{} version,
this is not the case ($804\,s_0$ is an integer by \eqref{s0}
but $804\, p_0$ is not).
Similarly, it will never happen for the other versions of the Tibetan
calendar with the \dpp{s} defined for them below.
\end{remark}
Conversely,
given $(Y,M,\ell)$, we can find
the \tmc{} $n$ by
the theory in this appendix
from \eqref{b0},
noting that if there are two possible values of $n$, then we should
choose the smaller one if $\ell=\true$ (a leap month)
and the larger one if
$\ell=\false$ (a regular month).
If $\ell=\false$, then \eqref{b0} and \eqref{msl} thus show that $n$ is the
largest integer such that
\begin{equation}\label{qu6}
s_1 n+s_0-p_0 \le Y'+\frac{M}{12}
\end{equation}
or, recalling \eqref{ga},
\begin{equation}\label{qu7}
s_1 n \le Y'+\frac{M-\ga}{12}
=\frac{12(Y-Y_0)+M-\ga}{12};
\end{equation}
if $\ell=\true$, this value of $n$ should be decreased by 1.
Hence, in all cases, $n$ can be computed from $(Y,M,\ell)$ by
\begin{equation}\label{b6}
n =\Floor{\frac{12(Y-Y_0)+M-\ga}{12s_1}}-\boolx{\ell}.
\end{equation}
(For the \PH{} version, it is easily verified directly
that this is equivalent to \eqref{c2}, using \eqref{MM}
and \eqref{gbxga} below.)
An alternative formula, which has the advantage that if there is no
leap month $M$ in year $Y$, then $(Y,M,\true)$ gives the same result
as $(Y,M,\false)$, is given by
\begin{equation}\label{b6x}
n =\Floor{\frac{12(Y-Y_0)+M-\ga-(1-12s_1)\boolx{\ell}}{12s_1}};
\end{equation}
to see this, note that if $\ell=\false$, then \eqref{b6x} and \eqref{b6}
give the same result, while if $\ell=\true$, then \eqref{b6x} gives 1
more than \eqref{b6} applied to $(Y,M-1,\false)$, which is the month
preceding leap month $M$ (also if $M=1$, when this really is
$(Y-1,12,\false)$).
Let us write $M'=12(Y-Y_0)+M$; this can be interpreted as a solar
month count from the beginning of year $Y_0$
(or we can interpret month $M$ year $Y$ as month $M'$ year
$Y_0$); note that $\MM$ in \refS{Smonths} is
given by, by \eqref{MM},
\begin{equation}
\label{MMM}
\MM=M'-M_0=M'-3.
\end{equation}
Since $12s_1=\xfrac{65}{67}$, we can write \eqref{b6} as
\begin{equation}\label{c0}
n
=\Floor{\frac{67}{65}(M'-\ga)}-\boolx{\ell}
=\Floor{\frac{67M'-67\ga}{65}}-\boolx{\ell},
\end{equation}
and since ${67}M'$ is an integer,
this value is not affected if $67\ga$ is replaced by the integer
$\gb=\ceil{67\ga}$, see \eqref{gb}.
We thus have
\begin{equation}\label{c1}
n
=\Floor{\frac{67M'-\gb}{65}}-\boolx{\ell}
=M'+\Floor{\frac{2M'-\gb}{65}}-\boolx{\ell}.
\end{equation}
There is a leap month $M$ in year $Y$ if and only if
the \tmc{} for $(Y,M,\false)$ jumps by 2 from the preceding regular month
$(Y,M-1,\false)$. By \eqref{c1}, this happens exactly when $2M'-\gb$
just has passed a multiple of 65, \ie, when
$2M'-\gb\equiv \text{0 or 1}\pmod{65}$.
Thus, the general leap month rule \eqref{leaprule-dp} is equivalent to:
\begin{Rule}
There is a leap month $M$ in year $Y$ if and only if
\begin{equation}\label{leaprule-gb}
2M'\equiv \gb\text{ or\/ }\gb+1\pmod{65},
\end{equation}
where $M'=12(Y-Y_0)+M$.
\end{Rule}
Since this is the same as \eqref{leaprule-gb0}, we see again that
\eqref{leaprule-dp} leads to the same results as the rules in \refS{Smonths}.
\subsection{\PH{} definition points}\label{SSdp}
In the \PH{} version,
the first \dpp{} $p_1$ is
$23;6 \rr{60}$ mansions, \ie{}
\begin{equation}
\label{p1}
p_1=23,6 \rr{27,60}
=\frac{77}{90}
\end{equation}
and thus
\begin{equation}
\label{p0}
p_0=p_1-\frac1{12}
=\frac{139}{180};
\end{equation}
in degrees, this is $p_0=278\grad$, $p_1=308\grad$, and so on, with
intervals of $30\grad$, \ie{} 8 degrees after the beginning of each sign, see
\cite{Henning},
\cite{Schuh-review}.
\xfootnote{\label{fPHpoints}
For other purposes, Phugpa astronomy
regards the winter solstice to be at solar longitude
$18,31,30\rr{27,60,60}=247/360=247\grad$, one degree earlier than
$p_{11}=248\grad$, and similarly for the summer solstice,
see
\cite[p.~322--328]{Henning},
\cite[On intercalary months]{kalacakra},
\cite[pp.~114--115]{Schuh} and
\cite[pp.~223, 225]{Schuh-review}.
I have no explanation for this
difference, but it does not affect the calendar which uses the values above.
}
Recall that in the formulas above, the integer part of $s_0$ has to be
chosen such that \eqref{s0krav} holds, \ie, $139/180 < s_0 \le 319/180$.
This is satisfied by the values \eqref{s0} and \eqref{s0E1927} for the
epochs \eS{} and \eH, but for \eX{} the value $s_0=0$ in \eqref{s0E1987} has
to be replaced by 1. (Recall that this does not matter in \refS{Sastro}.)
The constant $\ga$ defined by \eqref{ga} equals thus
\begin{align}
\label{ga1S}
\ga&=12(s_0-p_0)
=\frac{1832}{1005}
=1+\frac{827}{1005}
\qquad(\eS);
\intertext{or}
\ga&=12(s_0-p_0)
=\frac{1922}{1005}
=1+\frac{917}{1005}
\qquad(\eH);
\\
\ga&=12(s_0-p_0)
=\frac{41}{15}
\phantom{00}
=2+\frac{11}{15}
\phantom{00}
\qquad(\eX). \label{ga1X}
\end{align}
Hence, \eqref{gb} yields
\begin{align}
\label{gb1}
\gb&=123
\qquad(\eS),
\\
\gb&=129
\qquad(\eH),
\label{gb1H}
\\
\gb&=184
\qquad(\eX)
\label{gb1X},
\end{align}
in agreement with
\eqref{gb1S0}--\eqref{gb1X0}.
Consequently, we have verified that the
arithmetic calculations in \refS{Smonths} and the astronomical theory in
this appendix yield the same result.
Recall that in \refS{Smonths}, $\gb$ was defined by \eqref{gbgbx}, so we see,
using \eqref{ga} and \eqref{gb},
that the initial value $\gbx$ in the \tm{}
calculation \eqref{tm0} is related to the definition points by
\begin{equation}\label{gbxga}
\gbx
= 184 - \gb
=184 - \ceil{67\ga}
=184 + \floor{804(p_0-s_0)}.
\end{equation}
This yields directly
\begin{align}
\label{gbxS=}
\gbx&=61
\qquad(\eS),
\\
\gbx&=55
\qquad(\eH),
\label{gbxH=}
\\
\gbx&=0\phantom0
\qquad(\eX),
\label{gbxX=}
\end{align}
in agreement with
\eqref{gbxS}--\eqref{gbxX}.
Note also that by \refR{Repoch-ga},
the values of $\ga$ in \eqref{ga1S}--\eqref{ga1X}
show again that
for \eS{} and \eH{}, the epoch month with \tmc{} 0 is month 2,
while for \eX, it is month 3.
\subsection{\TS{} \dpp{s}}\label{ASTSdp}
The values for $\gb$ given in
\refApp{ATS} are consistent with, \cf{} \eqref{p1} for \PH,
\begin{equation}
\label{p1t}
p_1=23,1,30 \rr{27,60,60}
=\frac{307}{360}
=307\grad,
\end{equation}
which gives
\begin{equation}\label{p0t}
p_0=\frac{277}{360}=277\grad.
\end{equation}
To see this, note first that
the values of $s_0$ in \eqref{s0T1732} and \eqref{s0T1852} both have to be
increased by 1 in order to satisfy \eqref{s0krav}, see \refR{Rs0}.
Thus we now take, for E1732 and E1852, respectively,
\begin{align}
s_0&
=1-\frac{5983}{108540}
=\frac{102557}{108540},
\\
s_0&=
0,1,22,2,4,18 \rr{27,60,60,6,13,67}+1
=1+\frac{23}{27135},
\end{align}
which together with \eqref{p0t} yield
\begin{align}
\ga&=12(s_0-p_0)
=2+\frac{1903}{18090}
\qquad\text{(E1732)},
\\
\ga&=12(s_0-p_0)
=2+\frac{14053}{18090}
\qquad\text{(E1852)}.
\end{align}
By \eqref{gb} this yields
the values $\gb=142$ (E1732) and 187 (E1852),
as said in \refApp{ATS}.
Note that $67\ga$ is not an integer, so (as for \PH{})
$\MSL(n)$ never equals some \dpp, see \refR{Rnotequal}.
\begin{remark}
Any $p_1$ satisfying
\begin{equation}
\frac{92432}{27\cdot4020}
<
p_1
<
\frac{92567}{27\cdot4020}
\end{equation}
would give the same $\gb$ and thus by \eqref{leaprule-gb} the same calendar.
The value \eqref{p1t} is from
Henning [personal communication];
it is determined from calendars and not explicitly taken from any text.
\end{remark}
\subsection{Mongolian (New Genden) \dpp{s}}\label{ASMdp}
As far as I know, \YP{} does not discuss \dpp{s} explicitly for the New
Genden version, but the value
of $\gbx$ is consistent with
increasing the \dpp{} $p_1$ from \eqref{p1} to
\begin{equation}
\label{p1mongo}
p_1=23,9 \rr{27,60}
=\frac{463}{540}
=308\tfrac23\grad,
\end{equation}
and thus
\begin{equation}
\label{p0mongo}
p_0
=\frac{209}{270}
=278\tfrac23\grad,
\end{equation}
which together with \eqref{s0genden}
yields, by \eqref{ga} and \eqref{gb},
\begin{equation}
\ga =\frac{7724}{3015}
=2+\frac{1694}{3015}
\end{equation}
and $\beta=172\equiv 42\pmod{65}$,
as said in \refApp{AMongo}.
(Again, note that $67\ga$ is not an integer, \cf{} \refR{Rnotequal}.)
In fact, any value of $p_1$ with
\begin{equation}
\frac{689}{804} < p_1 < \frac{690}{804}
\end{equation}
gives the same $\gb$, and thus the same calendar.
The value \eqref{p1mongo} is just our choice within this range.
We do not know whether such definition points are used at all in the
Mongolian calendar, or whether the rule \eqref{leaprule-G} is used as it is
without further justification.
\subsection{When the leap month is the second}\label{ASdp+}
In the Bhutanese version of the calendar (\refApp{ABhutan}, see also
\refApp{AKT}),
a leap month takes the number of the \emph{preceding} month, so the leap
month is the second of the two months with the same number.
For such versions, the theory above has to be modified.
The basic rules \eqref{rule-dp} and
\eqref{leaprule-dp} remain unchanged, but they now imply
that the number of a month is the number of
the last \dpp{} before the end of the month.
Hence \eqref{xa1}--\eqref{xa2} are replaced by
\begin{gather}
Y-Y_0+p_{M}
<
\MSL(n+1)
\le
Y-Y_0+p_{M+1},
\label{xa1b}
\\
\ell =[\MSL(n)> Y-Y_0+p_{M}].
\label{xa2b}
\end{gather}
Note that \eqref{xa1b} is the same as \eqref{xa1} with $M$ and $n$ replaced
by $M+1$ and $n+1$;
\xfootnote{
This implies that \eqref{s0krav} and \eqref{gakrav} have to be slightly
modified to
\begin{equation}
p_1<s_0+s_1\le p_{13}=p_1+1
\end{equation}
and, using \eqref{gabh},
\begin{equation}
0<
\gax=
\ga-\frac{2}{67}\le12;
\end{equation}
this is no difference in practice since the epochs traditionally are chosen at
the beginning of
month 3 (\nag), so $\ga$ (or rather $\ga-2/67$ in this case)
is between 1 and 3, see \refR{Repoch}.
}
thus \eqref{b0}--\eqref{b1} hold with the same modification,
which yields,
recalling $12s_1=\frac{65}{67}$,
\begin{equation}
\label{b1b}
\begin{split}
12Y'+M &
=\Ceil{12s_1(n+1)+\ga}-1
=\Ceil{12s_1n+\ga-\tfrac2{67}}
.
\end{split}
\end{equation}
Hence we now define, in addition to \eqref{ga},
\begin{equation}
\label{gabh}
\gax =\ga-\tfrac2{67} =12(s_0-p_0)-\tfrac2{67}
\end{equation}
and have, instead of \eqref{b1},
\begin{equation}
\label{b1bh}
12Y'+M
=\ceil{12s_1n+\gax}.
\end{equation}
Consequently, \eqref{b3}--\eqref{b2} hold with $\ga$ replaced by $\gax$.
We now change the definition \eqref{gb} to
\begin{equation}\label{gbb}
\gb=\ceil{67\gax}=\ceil{67\ga}-2;
\end{equation}
then \eqref{b1bh}
can be written, \cf{} \eqref{bxx3},
\begin{equation}\label{bxx3bh}
12Y'+M = \Ceil{\frac{65}{67}n+\gax}
=\Ceil{\frac{65 n + 67\gax}{67}}
=\Ceil{\frac{65 n + \gb}{67}}.
\end{equation}
Hence, with our new definition of $\gb$, \eqref{bx3}--\eqref{bx5} are valid
and yield the same correspondence between $(Y,M)$ and true month $n$ as
before (but we have to remember that the leap month now is the later of two
months with the same number).
For the converse problem, to find the true month $n$ given $(Y,M, \ell)$,
note first that if the month is regular ($\ell=\false$), then the month
before has number $M-1$ (interpreted modulo 12) and \tmc{} $n-1$, and thus by
\eqref{xa1b}
\begin{equation}
\MSL(n)\le Y-Y_0+p_M<\MSL(n+1).
\end{equation}
Hence, $n$ is the largest integer such that \eqref{qu6}--\eqref{qu7} hold,
just as in \refApp{Aleap}.
For a leap month ($\ell=\true$), this value of $n$ now should be increased by 1.
Hence, in all cases, \eqref{b6} holds if $-[\ell]$ is changed to $+[\ell]$.
Similarly,
\eqref{b6x} is modified to
\begin{equation}\label{b6xb}
n =\Floor{\frac{12(Y-Y_0)+M-\ga+(1-12s_1)\boolx{\ell}}{12s_1}};
\end{equation}
as before this gives the same result for
$(Y,M,\true)$ and $(Y,M,\false)$ if there is no leap month $M$.
(If $\ell=\false$, \eqref{b6xb} agrees with \eqref{b6}; if $\ell=\true$ it
gives 1 less than \eqref{b6} applied to $(Y,M+1,\false)$.)
Also \eqref{c0} holds with if $-[\ell]$ is changed to $+[\ell]$.
There is now a leap month $M$ in year $Y$ if and only if
the \tmc{} jumps by 2 from the regular month $(Y,M,\false)$ to the next
$(Y,M+1,\false)$. It follows from (the modified)
\eqref{c0}, \cf{} \eqref{c1} and the argument after it, that
this happens exactly
when $2(M'+1)-\ceil{67\ga}\equiv \text{0 or 1}\pmod{65}$.
With our new definition \eqref{gbb} of $\gb$,
this means that the rules \eqref{leaprule-gb} and the equivalent
\eqref{leaprule-gb0} still hold.
If there is a leap month $(Y,M)$, then \eqref{ixq} yields the intercalation
index for the preceding regular month, so for the leap month we have to
increase the result by $2$, yielding
$(2M'+\gbx-4)\bmod{65}$.
By \eqref{leaprule-gb}, this shows that \eqref{jeppe} is modifed when the
leap month comes after the
regular month with the same number;
a leap month now has intercalation index
\begin{equation}\label{jeppe+}
(\gb+\gbx-4)\bmod65
\qquad \text{or}\qquad
(\gb+\gbx-3)\bmod65 .
\end{equation}
Finally, \eqref{cu}--\eqref{lys} were derived from \eqref{leaprule-gb0}, and
thus hold in the present case too.
\subsection{Bhutanese \dpp{s}}\label{ASBdp}
The value $\gb=191\equiv 61\pmod{65}$ for the Bhutanese version of the calendar
in \refApp{ABhutan} (and the epoch used there) is
consistent
with the \dpp{s} defined by, for example,
\xfootnote{
This is our own choice. Any $p_1$ with
$\frac{690}{804} < p_1 <\frac{691}{804}$ yields the same result.
}
\begin{equation}
\label{p1B}
p_1=23,10,30 \rr{27,60,60}
=\frac{103}{120}
=309\grad
\end{equation}
and thus
\begin{equation}
\label{p0B}
p_0=p_1-\frac{1}{12} = 20,55,30 \rr{27,60,60}
=\frac{31}{40}
=279\grad.
\end{equation}
Again, in accordance with \refR{Rs0} and \eqref{s0krav},
we have to add 1 to the value of $s_0$ given in \eqref{s0bh};
we thus now use
\begin{equation}
s_0
= 0,24,10,50 \rr{27,60,60,67}+1 =1+\frac{1}{67}, \label{s0bh+}
\end{equation}
which together with \eqref{p0B}
yields, by \eqref{ga},
\begin{equation}
\ga =\frac{1929}{670}
=2+\frac{589}{670}
\end{equation}
and
thus by \eqref{gbb}
$\beta=193-2=191$ as asserted above.
By \eqref{gbx-B}, this is consistent with the value $\gbx=2$ used in
Bhutan (for the epoch above).
However, this is \emph{not} consistent with the original text by
Lhawang Lodro and published calendars, which give the definition points
as [Henning, personal communication]
\begin{equation}
\label{p1Bx}
p_1=23,15 \rr{27,60}
=\frac{31}{36}
=310\grad
\end{equation}
and thus
\begin{equation}
\label{p0Bx}
p_0=p_1-\frac{1}{12} = 21,0 \rr{27,60}
=\frac{7}{9}
=280\grad,
\end{equation}
which together with \eqref{s0bh}
would yield, by \eqref{ga} and \eqref{gbb},
\begin{equation}
\ga =\frac{572}{201}
=2+\frac{170}{201}
\end{equation}
and $\beta=189\equiv 59\pmod{65}$, which does not agree with \eqref{gbx-B},
nor with actual leap months.
\xfootnote{For example,
month 5 was a leap month in 2008.
(This is shown by the Election act
\cite[National Assembly, Acts]{BhutanNA}
who is enacted the
``26th Day of the Second 5th Month of the Earth Male Rat Year
corresponding to the 28th Day of the 7th Month of the Year 2008''.)
A simple calculation shows
that at the beginning of the second (leap) month 5, the mean solar longitude
\eqref{meansun} was
$5,14,19,47 \rr{27,60,60,67}=13/67=69.851\grad$,
so if the definition point
$p_5=5,15\rr{27,60}=70\grad$, as implied by \eqref{p1Bx},
then the mean sun reaches $p_5$ just after the beginning of the month, which
therefore would not be a leap month.
However, with \eqref{p1B} we have $p_5=69\grad$, passed at the end of the
preceding month, and the mean sun does not pass any \dpp{} during the month;
hence the month is (correctly) a leap month.
(The next \dpp{} is $p_6=99\grad$ and the mean sun has longitude $98.955\grad$
at the end of the month.)
}
I have no explanation for this. One possibility is that at some time, either
the definition points or the epoch value for true month and intercalation
index was adjusted, ignoring the fact that they in theory are connected and
that one cannot be changed without changing the other.
(Cf.\ the similar problem in \refF{fPHpoints}.)
Note also that, as said in \refApp{ABhutan}, the winter solstice
is defined as when the mean solar longitude is
$250\grad$; this is the definition point $p_{11}$ if we use
\eqref{p1Bx}, but not if we use \eqref{p1B}.
\subsection{\Karana{} \dpp{s}}\label{ASKdp}
The leap month rule \eqref{leaprule-K} may seem to be in accordance
with the definition points given by
\begin{equation}
p_1=22,30 \rr{27,60} = \frac56
=300\grad.
\end{equation}
Thus the \dpp{s} are at the beginning of zodiacal signs.
In particular,
the \dpp{} for \nag{} (\Caitra) is
$p_3=0\grad$,
the first point of Aries.
\xfootnote{
This agrees with Indian calendars, see \refR{RChina}, and seems to be the most
natural choice.}
This would give
\begin{equation}
p_0= \frac9{12}=\frac34
=270\grad
\end{equation}
and \eqref{s0K}, \eqref{ga} and \eqref{gbb} would yield
\begin{align}
\ga =\frac{403}{135}=2+\frac{133}{135}
\end{align}
and
$\gb=199\equiv4\pmod{65}$, in accordance with \refApp{AKT}, see also
\eqref{jeppe+}.
However, the theory of \dpp{s} above
is based on the standard value \eqref{s1} for the mean
solar motion $s_1$, and is thus logically inconsistent with the
slightly larger value
\eqref{s1KT} used in the \KT; the mean sun will move a little faster, and if
the leap year rule \eqref{leaprule-K}
is used, the mean sun will advance relative
to the position indicated by the number of the month.
(But the difference is
small (0.003\%), and amounts to about 1 sign (= 1 month) in 2600 years.)
\xfootnote{
The relation \eqref{6765} is thus only an approximation in
the original \KT{} version, but it has been treated as exact in later
versions of the calendar.}
Conversely, if the leap month rule \eqref{leaprule-dp} is used with any fixed
\dpp{s} and $s_1$ is the \karana{} value \eqref{s1KT}, then the leap months
will not follow a strict 65 year cycle.
See further \cite[On intercalary months]{kalacakra}.
In order to use the formulas in Section \ref{ASdp},
or
rather their modifications described in \refS{ASdp+}
with leap months numbered by the preceding month,
we thus have to use the \siddhanta{} value
$s_1=65/804$ from \eqref{s1} and not the \karana{} value \eqref{s1KT},
which seems illogical.
As far as I know, \dpp{s} are not mentioned explicitly in the \KT.
Nevertheless, it is obvious that the idea was to insert leap months so
that the year on the average agrees with the tropical solar year, and thus
the sun has more or less the same longitude for the, say, first month any year.
The simple relation \eqref{6765} was at some stage chosen as an
approximation, perhaps believed to be exact, and the mean solar motion $s_1$
was at some stage chosen to be the \siddhanta{} value $65/804$; these
choices are logically connected, but I do not know whether that was
realized when these values were introduced in the calendar, and whether
these choices were made together or at different times (as the \karana{}
system suggests).
\section{Planets}
\label{Aplanet}
The positions of the planets are,
for astrological purposes,
calculated by the following procedure, also based on the \KT.
The calculations yield the longitudes at the end of the calendar
(=solar) day.
In modern terms, the (geocentric) longitude of a planet is found by
first calculating the heliocentric longitude and the longitude of the
sun, and then combining them (using trigonometry).
The Tibetan calculations do effectively this, although the theory
behind (which is not explicitly mentioned) rather is an old Indian
system using epicycles similar to the Ptolemaic system \cite[p.~57]{Henning}.
As in the calculations of $true\_date$ in \refS{Sastro}, the
calculations use special tables as substitutes for trigonometrical
calculations.
For a more detailed description of the traditional calculation,
including conversions between different radices,
se \citet[Chapter II]{Henning}, which also includes discussions of
the astronomical background and geometry, and of the accuracy of the formulas.
The constants given below are
(as the main text of the present paper)
for the \PH{} tradition;
the \TS{} and other versions use their own
slightly different epoch values, see \cite[p.~340]{Henning} and
\cite[Epoch data]{kalacakra}.
\subsection{General day}
The general day \tib{spyi zhag}
is a count of (solar) days since the epoch. It is thus simply given by
\begin{equation}
general\_day=\JD-epoch.
\end{equation}
For \eH{} (as in \cite{Henning}),
the epoch is JD 2424972 (Friday, 1 April 1927; RD 703547)
and thus
\begin{equation}
general\_day=\JD-2424972.
\end{equation}
\begin{remark}
\label{Rgeneral}
The traditional method to calculate the general day is to first find
the true month $n$ as
in Section \ref{Smonths},
see \eqref{tm0} and \eqref{mcrule-P} or \eqref{c2}, and
then use it to find the
number of elapsed lunar days
since the epoch, $ld$ say, by
$ld=30n+D$, where $D$ is the date.
Next, $ld$ is multiplied by the ratio $11135/11312$ in
\eqref{meanlunarday}
(the mean length of a lunar day). A small constant $gd_0$ is
subtracted; for \eH, $gd_0=2,178 \rr{64,707}=199/5656$.
This constant is the fraction of the solar day remaining at the end of
the mean lunar day at the epoch.
The difference $\frac{11135}{11312}ld-gd_0$ is rounded up to the
nearest integer, and one sets provisionally
\begin{equation}
general\_day=\Ceil{\frac{11135}{11312}ld-gd_0}.
\end{equation}
This gives an approximation of the general day, but may be off by a
day because only the mean motion of the moon is considered (this is
equivalent to ignoring the corrections in \eqref{truedate}).
Hence the day of week is calculated by the simple formula
\begin{equation}
\label{wd}
\bigpar{general\_day+wd_0} \mod 7
\end{equation}
where the constant $wd_0$ is the day of week at the epoch; for \eH, $wd_0=6$.
If the value in \eqref{wd} differs from the correct day of week, then
$general\_day$ is adjusted by $\pm1$ (the error cannot be larger) so
that \eqref{wd} becomes correct.
(Since this final check and correction has to be done, one could as
well ignore the subtraction of $gd_0$ above, but it is traditionally done.)
\end{remark}
\subsection{Mean heliocentric motion}
The mean heliocentric position of a planet is represented by an
integer for each plane called \emph{particular day} \tib{sgos zhag},
calculated as
\begin{equation}
particular\_day
=
\begin{cases}
(100\cdot general\_day+pd_0) \bmod R, & \text{Mercury}, \\
(10\cdot general\_day+pd_0) \bmod R, & \text{Venus}, \\
(general\_day+pd_0) \bmod R, & \hskip-12pt\text{Mars, Jupiter, Saturn},
\end{cases}
\end{equation}
where the modulus $R$ and the epoch value
$pd_0$ are given in \refT{Tplanet}. The periods of the planets are
thus, exactly,
87.97, 224.7, 687, 4332 and 10766 days, respectively.
(Modern astronomical values are 87.9684, 224.695, 686.93, 4330.6,
and 10746.9 \cite[Table 15.6]{AA}.)
The particular day is 0 at the first point of Aries (\ie, when the
longitude is 0), and thus the mean heliocentric longitude is
\begin{equation}
mean\_helio\_long
= \frac{particular\_day}{R} \pmod 1.
\end{equation}
This is traditionally expressed in the radices $\rr{27,60,60,6,r}$
with the final radix $r$ depending on the planet and given in
\refT{Tplanet}.
(Note that $r$ always is a divisor of $R$.)
\begin{table}[!htpb]
\begin{tabular}
{l l l l l l}
& Mercury & Venus & Mars & Jupiter & Saturn\\
\hline
$R$ & 8797 & 2247 & 687 & 4332 & 10766 \\
$pd_0$ (\eH) & 4639 & 301 & 157 & 3964 & 6286 \\
$r$ & 8797 & 749 & 229 & 361 & 5383 \\
$birth\_sign$ & 11/18 & 2/9 &19/54 & 4/9 & 2/3 \\
trad.\ \rr{27,60}& 16,30 & 6,0 & 9,30 & 12,0 & 18,0 \\
\end{tabular}
\caption{Constants for planets.}
\label{Tplanet}
\end{table}
\subsection{Mean longitude of the sun}
The mean longitude of the sun (at the end of the calendar day) is
calculated from scratch and not using the
calculation of $mean\_sun$ (at the end of the lunar day) in
\refS{Sastro}, but the results are consistent
\cite[pp.~87--88]{Henning}.
(The \TS{} tradition uses the calculation of $mean\_sun$,
and in one version $true\_sun$
\cite[p.~341]{Henning}.)
The formula used is
\begin{equation}
mean\_solar\_long=s_1'\cdot general\_day+s_0',
\end{equation}
where
\begin{equation}
s_1'=0,4,26,0,93156 \rr{27,60,60,6,149209}
=\frac{18382}{6714405}
\quad\Bigpar{=\frac{11312}{11135}s_2}
\end{equation}
and the epoch value is, for \eH,
\begin{equation}
s_0'=25,9,20,0,97440 \rr{27,60,60,6,149209}
=1-\frac{458772}{6714405}
.
\end{equation}
\begin{table}[!htpb]
\begin{tabular}
{l l l l l l}
\qquad\qquad& Mercury & Venus & Mars & Jupiter & Saturn\\
\hline
0 & 0 & 0 & 0 & 0 & 0 \\
1 & 10 & 5 & 25 & 11 & 22 \\
2 & 17 & 9 & 43 & 20 & 37 \\
3 & 20 & 10 & 50 & 23 & 43 \\
\end{tabular}
\caption{Equation for planets.}
\label{Tplanet-equ}
\end{table}
\begin{table}[!htpb]
\begin{tabular}
{l l l l l l}
\qquad\qquad& Mercury & Venus & Mars & Jupiter & Saturn\\
\hline
0 & 0 & 0 & 0 & 0 & 0 \\
1 & 16 & 25 & 24 & 10 & 6 \\
2 & 32 & 50 & 47 & 20 & 11 \\
3 & 47 & 75 & 70 & 29 & 16 \\
4 & 61 & 99 & 93 & 37 & 20 \\
5 & 74 & 123 & 114 & 43 & 24 \\
6 & 85 & 145 & 135 & 49 & 26 \\
7 & 92 & 167 & 153 & 51 & 28 \\
8 & 97 & 185 & 168 & 52 & 28 \\
9 & 97 & 200 & 179 & 49 & 26 \\
10 & 93 & 208 & 182 & 43 & 22 \\
11 & 82 & 202 & 171 & 34 & 17 \\
12 & 62 & 172 & 133 & 23 & 11 \\
13 & 34 & 83 & 53 & 7 & 3 \\
\end{tabular}
\caption{Final correction for planets.}
\label{Tplanet-corr}
\end{table}
\subsection{Slow longitude and step index}
The remaining calculations are based on the mean heliocentric
longitude and the mean solar longitude, but these quantities are
treated differently for the inner (or ``peaceful'')
planets Mercury and Venus
and for the outer (or ``wrathful'') planets Mars, Jupiter,
Saturn.
The reason is that the mean motion of an outer planet is given by the
mean heliocentric longitude, while the mean motion of an inner planet is
given by the mean longitude of the sun.
In both cases, this main term is called the \emph{mean slow longitude}
\tib{dal ba},
while the other quantity is called the \emph{step index} \tib{rkang 'dzin}.
In other words, for the inner planets
\begin{align*}
mean\_slow\_long&=mean\_solar\_long, \\
step\_index&=mean\_helio\_long
\end{align*}
and for the outer planets
\begin{align*}
mean\_slow\_long&=mean\_helio\_long, \\
step\_index&=mean\_solar\_long.
\end{align*}
Next, \cf{} the calculations for the moon and sun in \refS{Sastro},
the anomaly is calculated by
\begin{equation}
anomaly=mean\_slow\_long - birth\_sign \pmod1,
\end{equation}
where the ``birth-sign'' \tib{skyes khyim}
is given in \refT{Tplanet}, both as a
rational number and in the traditional form in mansions \rr{27,60}.
The anomaly is used to find the equation from
\begin{equation}
equ=planet\_equ\_tab(12\cdot anomaly),
\end{equation}
where $planet\_equ\_tab(i)$ is given in \refT{Tplanet-equ} for $i=0,\dots,3$,
which extends by the symmetry rules
$planet\_equ\_tab(6-i)=planet\_equ\_tab(i)$ and
$planet\_equ\_tab(6+i)=-planet\_equ\_tab(i)$;
linear interpolation is used beween integer arguments.
Finally, the true slow longitude \tib{dal dag} is given by
\begin{equation}
true\_slow\_long=mean\_slow\_long - equ/(27\cdot 60).
\end{equation}
\subsection{Geocentric longitude}
The final step is to combine the true slow longitude and the step
index. First, the difference of these is found:
\begin{equation}
\mathit{diff}=step\_index - true\_slow\_long.
\end{equation}
This is used to find a correction by another table look-up:
\begin{equation}
corr=planet\_corr\_tab(27\cdot \mathit{diff}) \pmod1,
\end{equation}
where $planet\_corr\_tab(i)$ is given in \refT{Tplanet-corr} for
$i=0,\dots,13$,
which extends by the symmetry rules
$planet\_corr\_tab(27-i)=-planet\_corr\_tab(i)$ and
$planet\_corr\_tab(27+i)=planet\_corr\_tab(i)$;
as always, linear interpolation is used beween integer arguments.
Finally the (geocentric, or \emph{fast}) longitude
\tib{myur ba} is given by
\begin{equation}
fast\_long=true\_slow\_long + corr/(27\cdot 60).
\end{equation}
\subsection{Rahu}
\emph{Rahu} is the name of the nodes of the lunar orbit, \ie, the
intersections of the orbit and the ecliptic. More precisely, the
ascending node is called the \emph{Head of Rahu} and the descending node is
called the \emph{Tail of Rahu}. In Tibetan (as in Indian) astrology, Rahu is
treated as a planet, or perhaps two planets.
Rahu is further essential for prediction of eclipses
\cite[Chapter III]{Henning}.
Rahu has a slow motion that is retrograde (\ie, to the west, with
decreasing longitude, unlike the real planets). In the Tibetan system,
the period is exactly
230 lunar months = 6900 lunar days.
\begin{remark}
In calendar (solar) days, this is, \cf{} \eqref{meanlunarday},
\begin{equation}
\frac{11135}{11312} \cdot 6900 = \frac{19207875}{2828}
\approx 6792.035
\text{ days}
\end{equation}
or, \cf{} \eqref{meanyear},
\begin{equation}
\frac{11135}{11312} \cdot
6900 s_2 =6900\cdot\frac{s_1}{30}=\frac{7475}{402}
\approx 18.5945
\text{ Tibetan years}.
\end{equation}
The modern astronomical value is
6798 days = 18.61 Gregorian years
\cite[Table 15.4]{AA}.
\end{remark}
To find the position of Rahu at day $D$, month $M$, year $Y$, one
first calculates the \tmc{}
$n$, for example by \eqref{MM} and \eqref{c2}. Next, the number $x$
of elapsed lunar days since the Head of Rahu had longitude 0 is
calculated by, \cf{} the calculation of $ld$ in \refR{Rgeneral},
\begin{equation}
x=30(n+rd_0)+D = 30n+D+30rd_0 = ld+30rd_0,
\end{equation}
where $rd_0$ is an epoch value; $rd_0=187$ for \eH.
(For \eX, $rd_0=10$.)
Finally, the longitudes of the head and tail of Rahu are given by
\begin{align}
rahu\_head\_long&=-\frac{x}{6900} \pmod 1, \\
rahu\_tail\_long&=rahu\_head\_long +\frac12 \pmod 1.
\end{align}
\begin{remark}
Since Rahu has a retrograde motion, these are decreasing. In
traditional calculations,
$-rahu\_head\_long=\xfrac{x}{6900} \pmod 1$ is called the longitude
of the \emph{Source of Rahu}.
The longitudes are traditionally expressed in $\rr{27,60,60,6,23}$, and
$1/6900$ is then written as $0,0,14,0,12$.
\end{remark}
\section{Further astrological calculations}\label{Aastro}
As explained in \refS{Syear}, each year has a name consisting of an
element and an animal.
For astrological
\xfootnote{See Footnote \ref{f:astro}.}
purposes (in the Chinese or elemental astrological
system),
there are many further associations, assigning a year, month, lunar
day or solar day one of the 12 animals in \refT{T12},
one of the 5 elements in \refT{T5},
one of the 8 trigrams in \refT{T8},
or one of the 9 numbers $1,\dots,9$ in \refT{T9};
as shown in Tables \ref{T12}, \ref{T5}, \ref{T8}, \ref{T9}, these have further
associations to, for example, numbers, colours and directions.
There are also further attributes in the Indian system.
This appendix is only a brief introduction, and
only some of the attributes and their connections are mentioned here.
See \citet{Tseng1,Tseng2} and
\citet{Henning} for further
details.
\begin{table}[!p]
\begin{tabular}{c l l}
number & element & colour\\
\hline
1 & wood & green (blue) \\
2 & fire & red \\
3 & earth & yellow \\
4 & iron & white \\
5 & water & dark blue (black) \\
\end{tabular}
\caption{The 5 elements and the associated numbers and colours.}
\label{T5}
\end{table}
\newlength{\yinwd}
\newlength{\yinht}
\newlength{\yindp}
\setlength{\yinwd}{6pt}
\setlength{\yinht}{0pt}
\setlength{\yindp}{1pt}
\newcommand\yina{\vrule height \yinht width \yinwd depth\yindp}
\newcommand\yin{\hbox{\yina\hskip\yinwd\yina}}
\newcommand\yang{\hbox{\vrule height \yinht width 3\yinwd depth\yindp}}
\newcommand\yy[1]{\vskip2pt\ifodd#1\yang \else \yin \fi}
\newcommand\tri[3]{\vbox{\yy{#3}\yy{#2}\yy{#1}}}
\begin{table}[!htpb]
\begin{tabular}{l c l l l l l }
& binary & trigram & Tibetan & Chinese & direction & element\\
\hline
1 & 5 & \tri101 & li & l\'i & S & fire \\
2 & 0 & \tri000 & khon & k\=un & SW & earth \\
3 & 6 & \tri110 & dwa & du\`i & W & iron \\
4 & 7 & \tri111 & khen & qi\'an & NW & sky \\
5 & 2 & \tri010 & kham & k\v{a}n & N & water \\
6 & 1 & \tri001 & gin & g\`en & NE & mountain \\
7 & 4 & \tri100 & zin & zh\`en & E & wood \\
8 & 3 & \tri011 & zon & x\`un & SE & wind
\end{tabular}
\caption{The 8 trigrams with some attributes.
The ordering is the standard (King Wen, Later Heaven) order; the
numbering is perhaps not traditional, and the binary coding (reading
the trigrams bottom-up) is mathematically natural but not traditionally used.}
\label{T8}
\end{table}
\begin{table}[!htpb]
\begin{tabular}{ c l l l }
\qquad\quad& colour & element & direction\\
\hline
1 & white & iron & N \\
2 & black & water & SW \\
3 & blue & water & E \\
4 & green & wood & SE \\
5 & yellow & earth & Centre \\
6 & white & iron & NW \\
7 & red & fire & W \\
8 & white & iron & NE \\
9 & red & fire & S
\end{tabular}
\caption{The 9 numbers and their attributes.}
\label{T9}
\end{table}
\begin{remark}
The order of the directions in \refT{T9} may seem jumbled,
yet there is method in it.
The 9 numbers are often arranged in a $3\times3$ square according to
the directions as in \refT{T3x3} (upside down the standard Western
orientation), and then the numbers form a magic square with all rows,
columns and diagonals summing to 15.
\end{remark}
\begin{table}[!htpb]
\begin{tabular}{l | l | l}
4 & 9 & 2 \\
\hline
3 & 5 & 7 \\
\hline
8 & 1 & 6 \\
\end{tabular}
\qquad
\begin{tabular}{l | l | l}
SE & S & SW\\
\hline
E & C & W\\
\hline
NE & N & NW\\
\end{tabular}
\caption{A magic square of numbers and their directions}
\label{T3x3}
\end{table}
In formulas below, $Y$ is the Gregorian number of the year, see
\refS{Syear}. By \refS{Syear}, year $Y$ has in the Chinese 60 year
cycle number
\begin{equation}
\label{yz}
(Y-3) \amod 60,
\end{equation}
and hence numbers $(Y-3) \amod 10$ and $(Y-3) \amod 12$ in the Chinese
10 and 12 year cycles.
\subsection{Attributes for years}
\subsubsection{Elements}
Each year
is given 4 or 5 elements:
the power element \tib{dbang thang},
life element \tib{srog},
body element \tib{lus},
fortune element \tib{klung rta},
and sometimes also the
spirit element \tib{bla}.
The \emph{power element} is the element associated to the celestial
stem, see \refT{T10}; it is thus repeated in a cycle of 10 years, with
each element repeated 2 consecutive years, in the standard
order wood, fire, earth, iron, water
(the order in \refT{T5}).
As said above, year $Y$ is $(Y-3) \amod 10$ in the Chinese 10 year
cycle, and by Tables \refand{T10}{T5}, its power element has number
\begin{equation}\label{power}
\Ceil{\frac{(Y-3)\amod10}{2}}=\Ceil{\frac{Y-3}2} \amod 5.
\end{equation}
The \emph{life element} is repeated in a cycle of 12 years, and is
thus determined by the animal name of the year. The list is given in
\refT{Tlife}. Note that each third year is earth (the years $\equiv
2 \bmod 3$); the remaining four elements come repeated 2 years each, in the
same (cyclic) order wood, fire, iron, water as the power element.
\begin{table}[!tpb]
\begin{tabular}{r r l l l}
\rlap{year}\phantom{0} && animal & life element & spirit element\\
\hline
1 &10& Mouse & water & iron\\
2 &11& Ox & earth & fire\\
3 &12& Tiger & wood & water\\
4 &1& Rabbit & wood & water\\
5 &2& Dragon & earth & fire\\
6 &3& Snake & fire & wood \\
7 &4& Horse & fire & wood\\
8 &5& Sheep & earth & fire\\
9 &6& Monkey & iron & earth\\
10 &7& Bird & iron & earth\\
11 &8& Dog & earth & fire\\
12 &9& Pig & water & iron\\
\end{tabular}
\caption{The 12 year cycle of life elements.
The first number on each line shows the year mod 12 counted from the
start of a Chinese cycle; the second shows the year mod 12 counted from the
start of a Prabhava cycle.}
\label{Tlife}
\end{table}
The \emph{fortune element} is repeated in a cycle of 4 years,
in the order wood, water, iron, fire.
(Earth is not used. Note that
the order of the four used elements is different from the order used
for the power and life elements.)
Since 4 is a divisor of 12, the fortune element is determined by the
animal name of the year.
See \refT{Tfortune}.
\begin{table}[!htpb]
\begin{tabular}{r r l l }
\rlap{year}\phantom{0} && animals & fortune element\\
\hline
1 &2& Mouse, Dragon, Monkey & wood \\
2 &3& Ox, Snake, Bird & water\\
3 &4& Tiger, Horse, Dog & iron \\
4 &1& Rabbit, Sheep, Pig & fire\\
\end{tabular}
\caption{The 4 year cycle of fortune elements.
The first number on each line shows the year mod 4 counted from the
start of a Chinese cycle; the second shows the year mod 4 counted from the
start of a Prabhava cycle.}
\label{Tfortune}
\end{table}
The \emph{body element} is calculated in two steps.
(See \citet{Henning} for traditional ways of doing the calculations.)
First, an element is determined by the animal name; I do not know any
name for this intermediate element so let us call it $x$. The element
$x$ is repeated in a cycle of 6 years, with the 3 elements wood,
water, iron repeated 2 years each, see \refT{T6}.
Then count the number of steps from $x$ to the power element $y$, or
equivalently, replacing the elements by the corresponding numbers in
\refT{T5}, calculate $y-x\bmod 5$.
Finally, this difference determines the body element by
\refT{Tbody}.
(Note that the order in this table is not the standard one.)
\begin{table}[!htpb]
\begin{tabular}{r r l l c}
\rlap{year}\phantom{0} && animals & element & number\\
\hline
1 &4& Mouse, Horse & wood & 1\\
2 &5& Ox, Sheep & wood & 1\\
3 &6& Tiger, Monkey & water & 5 \\
4 &1& Rabbit, Bird & water & 5 \\
5 &2& Dragon, Dog & iron & 4 \\
6 &3& Snake, Pig & iron & 4 \\
\end{tabular}
\caption{The 6 year cycle of the element $x$
and its number
in the body element calculation.
The first number on each line shows the year mod 6 counted from the
start of a Chinese cycle; the second shows the year mod 6 counted from the
start of a Prabhava cycle.}
\label{T6}
\end{table}
\begin{table}[!htpb]
\begin{tabular}{ c l }
$y-x \mod 5$ & body element\\
\hline
0 & iron \\
1 & water \\
2 & fire \\
3 & earth \\
4 & wood \\
\end{tabular}
\caption{The final step in the body element calculation.}
\label{Tbody}
\end{table}
Since both $x$ and the power element $y$ are repeated 2 consecutive years each,
the same is true for the body element. If we consider only the even
years, say, then $x$ is by \refT{T6} given by
1, 0, $-1$, 1, 0, $-1$, $\dots\pmod5$,
while $y$ by \refT{T10} is given by 1, 2, 3, 4, 5, 1, 2, 3, \dots.
Hence, $y-x\mod 5$ repeats in a cycle of length 15:
0, 2, 4, 3, 0, 2, 1, 3, 0, 4, 1, 3, 2, 4, 1
(with differences $+2,+2,-1,+2,+2,-1,\dots$).
Consequently, the body element repeats in a cycle of 30 years, with
each element repeated for 2 consecutive years.
The full cycle is given in \refT{T30}.
The \emph{spirit element} is the element preceding the life element in
the standard order, see \refT{Tlife}.
Thus it too is repeated in a cycle of 12 years, and is
determined by the animal name of the year. Each third year is fire
(the years $\equiv 2 \bmod 3$) and the remaining four elements come
repeated 2 years each, in the standard (cyclic) order.
Since all elements for the year
have periods dividing 60, they all repeat in the
same order in each 60 year cycle as is shown in Table \ref{T60e}.
\subsubsection{Numbers}
Each year
is also associated with a set of three numbers,
in the range
$1,\dots,9$, the \emph{central number},
the \emph{life number} and the
\emph{power number}.
(These numbers are used for persons born that year. The central number is also
called the \emph{body number} or
\emph{birth number}.)
The numbers are associated with elements and directions according to
\refT{T9}.
The numbers decrease by 1 $\pmod9$ for each new year, and thus repeat
in a cycle of 9 years; they are given simply by, for Gregorian year
$Y$:
\begin{alignat}2
& \text{central number}&& = (2-Y) \amod 9,
\\
& \text{life number}&&
=
(\text{central number}-3) \amod 9
=(8-Y) \amod 9,
\\
& \text{power number}&&
=
(\text{central number}+3) \amod 9
=(5-Y) \amod 9.
\end{alignat}
Since 9 does not divide 60, these numbers do not follow the 60 year
cycle. The period for repeating all elements and numbers is 180 years,
\ie, 3 cycles of 60 years.
\begin{table}[!phbt]
\begin{tabular}{r r l l}
\rlap{year}\phantom{0} && power & body element \\
\hline
1 & 28 & wood & iron\\
2 & 29 & wood & iron\\
3 & 30 & fire & fire\\
4 & 1 & fire & fire\\
5 & 2 & earth & wood\\
6 & 3 & earth & wood\\
7 & 4 & iron & earth\\
8 & 5 & iron & earth\\
9 & 6 & water & iron\\
10 & 7 & water & iron\\
11 & 8 & wood & fire\\
12 & 9 & wood & fire\\
13 & 10 & fire & water\\
14 & 11 & fire & water\\
15 & 12 & earth & earth\\
16 & 13 & earth & earth\\
17 & 14 & iron & iron\\
18 & 15 & iron & iron\\
19 & 16 & water & wood\\
20 & 17 & water & wood\\
21 & 18 & wood & water\\
22 & 19 & wood & water\\
23 & 20 & fire & earth\\
24 & 21 & fire & earth\\
25 & 22 & earth & fire\\
26 & 23 & earth & fire\\
27 & 24 & iron & wood\\
28 & 25 & iron & wood\\
29 & 26 & water & water\\
30 & 27 & water & water\\
\end{tabular}
\caption{The 30 year cycle of the body element.
The first number on each line shows the year mod 30 counted from the
start of a Chinese cycle; the second shows the year mod 30 counted from the
start of a Prabhava cycle.}
\label{T30}
\end{table}
\begin{longtable}
{r r l l l l l}
\rlap{year}\phantom{0} && name (power) & life & body & fortune & spirit\\
\hline
1 & 58 & Wood--Mouse & water & iron & wood & iron\\
2 & 59 & Wood--Ox & earth & iron & water & fire\\
3 & 60 & Fire--Tiger & wood & fire & iron & water\\
4 & 1 & Fire--Rabbit & wood & fire & fire & water\\
5 & 2 & Earth--Dragon & earth & wood & wood & fire\\
6 & 3 & Earth--Snake & fire & wood & water & wood\\
7 & 4 & Iron--Horse & fire & earth & iron & wood\\
8 & 5 & Iron--Sheep & earth & earth & fire & fire\\
9 & 6 & Water--Monkey & iron & iron & wood & earth\\
10 & 7 & Water--Bird & iron & iron & water & earth\\
11 & 8 & Wood--Dog & earth & fire & iron & fire\\
12 & 9 & Wood--Pig & water & fire & fire & iron\\
13 & 10 & Fire--Mouse & water & water & wood & iron\\
14 & 11 & Fire--Ox & earth & water & water & fire\\
15 & 12 & Earth--Tiger & wood & earth & iron & water\\
16 & 13 & Earth--Rabbit & wood & earth & fire & water\\
17 & 14 & Iron--Dragon & earth & iron & wood & fire\\
18 & 15 & Iron--Snake & fire & iron & water & wood\\
19 & 16 & Water--Horse & fire & wood & iron & wood\\
20 & 17 & Water--Sheep & earth & wood & fire & fire\\
21 & 18 & Wood--Monkey & iron & water & wood & earth\\
22 & 19 & Wood--Bird & iron & water & water & earth\\
23 & 20 & Fire--Dog & earth & earth & iron & fire\\
24 & 21 & Fire--Pig & water & earth & fire & iron\\
25 & 22 & Earth--Mouse & water & fire & wood & iron\\
26 & 23 & Earth--Ox & earth & fire & water & fire\\
27 & 24 & Iron--Tiger & wood & wood & iron & water\\
28 & 25 & Iron--Rabbit & wood & wood & fire & water\\
29 & 26 & Water--Dragon & earth & water & wood & fire\\
30 & 27 & Water--Snake & fire & water & water & wood\\
31 & 28 & Wood--Horse & fire & iron & iron & wood\\
32 & 29 & Wood--Sheep & earth & iron & fire & fire\\
33 & 30 & Fire--Monkey & iron & fire & wood & earth\\
34 & 31 & Fire--Bird & iron & fire & water & earth\\
35 & 32 & Earth--Dog & earth & wood & iron & fire\\
36 & 33 & Earth--Pig & water & wood & fire & iron\\
37 & 34 & Iron--Mouse & water & earth & wood & iron\\
38 & 35 & Iron--Ox & earth & earth & water & fire\\
39 & 36 & Water--Tiger & wood & iron & iron & water\\
40 & 37 & Water--Rabbit & wood & iron & fire & water\\
41 & 38 & Wood--Dragon & earth & fire & wood & fire\\
42 & 39 & Wood--Snake & fire & fire & water & wood\\
43 & 40 & Fire--Horse & fire & water & iron & wood\\
44 & 41 & Fire--Sheep & earth & water & fire & fire\\
45 & 42 & Earth--Monkey & iron & earth & wood & earth\\
46 & 43 & Earth--Bird & iron & earth & water & earth\\
47 & 44 & Iron--Dog & earth & iron & iron & fire\\
48 & 45 & Iron--Pig & water & iron & fire & iron\\
49 & 46 & Water--Mouse & water & wood & wood & iron\\
50 & 47 & Water--Ox & earth & wood & water & fire\\
51 & 48 & Wood--Tiger & wood & water & iron & water\\
52 & 49 & Wood--Rabbit & wood & water & fire & water\\
53 & 50 & Fire--Dragon & earth & earth & wood & fire\\
54 & 51 & Fire--Snake & fire & earth & water & wood\\
55 & 52 & Earth--Horse & fire & fire & iron & wood\\
56 & 53 & Earth--Sheep & earth & fire & fire & fire\\
57 & 54 & Iron--Monkey & iron & wood & wood & earth\\
58 & 55 & Iron--Bird & iron & wood & water & earth\\
59 & 56 & Water--Dog & earth & water & iron & fire\\
60 & 57 & Water--Pig & water & water & fire & iron\\
\caption{The 60 year cycle of combinations of different elements.
The first number on each line shows the year mod 60 counted from the
start of a Chinese cycle; the second shows the year mod 60 counted from the
start of a Prabhava cycle. The power element is the first part of the name.}
\label{T60e}
\end{longtable}
\subsection{Attributes for months}\label{ASattributes-months}
Each regular calendar month is given attributes as follows.
A leap month is given the same attributes as the regular
month with the same number.
\subsubsection{Animals}
The 12 months are assigned one each of the 12 animals, in standard
order but with different starting points in the \PH{} and \TS{}
traditions,
see \refS{Smonths}\ref{month-animal}.
The full lists are given in \refT{T12months}.
(The \TS{} system is the same as the Chinese, see \eg{}
\cite[\S 126]{Ginzel} and \cite{10000}.
\xfootnote{\label{f10000}
The Chinese astrological system in \cite{10000} assigns animals,
gender and elements to solar months, defined by the minor solar terms, and
not to the (lunar) calendar months. However, there are different
astrological traditions in China.
})
\subsubsection{Gender}
Each month is given the gender (male or female) associated to its
animal. As Tables \refand{T12}{T12months} show, this simply means
(both in the \PH{} and \TS{} traditions) that odd-numbered months are
male and even-numbered female.
(This is in accordance with the general Chinese principle that odd
numbers are male (\emph{yang}) and even numbers female (\emph{yin}).)
\subsubsection{Elements}
Each month is assigned one of the 5 elements in \refT{T5}.
In the \TS{} version, the months cycle continuously through
the cycle in \refT{T10}, year after year.
This combines with the 12 month cycle for animals
to a 60 month cycle, exactly as for years, see \refS{Syear} and
Tables \ref{Tlosar} and \ref{T60}.
Since each element is repeated 2 months in the 10 month cycle, the element
of the first month advances one step in the list \refT{T5} each year.
More precisely,
the first month (Tiger) Gregorian year $Y$ has
month element number $(Y-2) \amod 5$.
(This is exactly as in the Chinese calendar, see \eg{}
\cite[\S 126]{Ginzel} and \cite{10000}.)
In the \PH{} version, the Tiger month
(the first in the Chinese calendar),
which is month 11 \emph{the preceding year}
(see \refT{T12months} and
\refS{Smonths}\ref{month-animal}),
gets the element following the
element of the year given in \refT{T10}, using the standard element
order in \refT{T5}. (The element following another, $x$, in this cyclic
order is called the \emph{son} of $x$.)
By \eqref{power},
this is element $\ceil{(Y-1)/2} \amod 5$.
Having determined the element of the Tiger month, the elements for the
12 month period starting with it are assigned in the same pattern as
for years in \refT{T10}: each element is repeated 2 months, the
first male and the second female, and then followed by the next
element. (This is the same sequence as for \TS, but only for 12 month periods.)
We thus obtain the following formulas for the number of the element
for month $M$ year $Y$.
\begin{description}
\item[\PH]
\begin{equation}
\begin{cases}
\lrpar{ \Ceil{\frac{Y-1}2}+\Floor{\frac{M+1}2}} \amod 5, & 1\le M\le 10,\\
\\
\lrpar{ \Ceil{\frac{Y}2}+\Floor{\frac{M-11}2}} \amod 5, & 11\le M\le 12 .
\end{cases}
\end{equation}
\item[\TS]
\begin{equation}
\lrpar{ Y-2+\Floor{\frac{M-1}2}} \amod 5.
\end{equation}
\end{description}
\begin{table}[!htpb]
\begin{tabular}{r l l l}
month & \PH & \TS & gender\\
\hline
1 & Dragon & Tiger & male \\
2 & Snake & Rabbit & female \\
3 & Horse & Dragon & male \\
4 & Sheep & Snake & female \\
5 & Monkey & Horse & male \\
6 & Bird & Sheep & female \\
7 & Dog & Monkey & male \\
8 & Pig & Bird & female \\
9 & Mouse & Dog & male\\
10 & Ox & Pig & female\\
11 & Tiger & Mouse & male\\
12 & Rabbit & Ox & female
\end{tabular}
\caption{The animals for the months.}
\label{T12months}
\end{table}
\subsubsection{Numbers}
Only \TS{} calendars give one of the 9 numbers to each month.
The number decreases by $1 \pmod 9$ for each month (except leap months),
and thus by $12\equiv 3\pmod 9$ for each year.
Month $M$ year $Y$ has number
\begin{equation}
\bigpar{3-(12Y+M)}\amod 9.
\end{equation}
The \TS{} rules for animal, gender and element agree with the rules in the
Chinese calendar, see \cite{10000}. Mongolia uses also the same rules, see
\refApp{AMongo}.
\goodbreak
\subsection{Attributes for lunar days}
\subsubsection{Animal}
Each lunar day has an animal. These repeat in the usual cycle of 12,
see \refT{T12}, with
each odd-numbered month starting with Tiger (number 3 in \refT{T12}) and
each even-numbered month starting with Monkey (number 9 in \refT{T12});
since there are exactly $30\equiv6\pmod{12}$ lunar days in each month,
the animals thus repeat in a continuous cycle broken only at leap
months, where the animals are repeated in the same order in the leap
month and the following regular month and there is a discontinuity
between the two months.
The number (in \refT{T12}) of the animal for lunar day $D$ in month
$M$ is thus
\begin{equation}
(D+30M+8) \amod 12
=
(D+6M+8) \amod 12.
\end{equation}
\subsubsection{Element}
Each lunar day has an element; these cycle inside each month in a
cycle of 5 in the usual order given in \refT{T5} (without repetitions
as for years and months), beginning with the element following the
element of the month calculated above.
If the month has element $x$, then lunar day $D$ of the month thus has
element $(x+D)\amod5$.
(There is thus often a jump in the sequence at the beginning of a new month.)
\subsubsection{Trigram}
The trigrams for lunar days cycle in the usual cycle of 8 in
\refT{T8};
as for the animals, this is continuous across months except for leap
months.
A Tiger month begins with trigram number 1, Li.
Thus the trigram for lunar day $D$ in a month with animal number $A$
(in \refT{T12}) has number
\begin{equation}
(D+30(A-3))\amod 8
=
(D-2A-2)\amod 8
=
(D+6A+6)\amod 8.
\end{equation}
Hence months Tiger, Horse, Dog begin with Li;
Rabbit, Sheep, Pig begin with Zin;
Mouse, Dragon, Monkey begin with Kham;
Ox, Snake, Bird begin with Dwa.
\subsubsection{Number}
The nine numbers (with their colours) for lunar days cycle forward
in the usual cycle of 9 in \refT{T9};
as for the animals, this is continuous across months except for leap
months.
A Tiger month begins with number 1 (white).
Thus the number for lunar day $D$ in a month with animal number $A$
(in \refT{T12}) has number
\begin{equation}
(D+30(A-3))\amod 9
=
(D+3A)\amod 9.
\end{equation}
Hence, months Tiger, Snake, Monkey, Pig begin with 1 (white);
Mouse, Rabbit, Horse, Bird begin with 4 (green);
Ox, Dragon, Sheep, Dog begin with 7 (red).
\subsection{Attributes for calendar days}\label{ASattributes-day}
As explained in Sections \ref{Sdays}, \refand{Scal}{Sweek}, each
calendar (solar) day has a number (the date) and a day of week.
Each (calendar) day is also given an element (and its colour according to
\refT{T5}, gender, animal, trigram and number
from the
Chinese system; these are simple cyclic with periods 10, 2, 12, 8, 9,
respectively, with the elements repeated twice each as for years in
\refT{T10}.
At least the element, gender and animal are the same as in the Chinese
calendar for the same day, see \cite{10000}.
\subsubsection{Element}
The element corresponds to number $\JD \amod 10$ in the (Chinese) cycle in
\refT{T10}.
The number of the element in \refT{T5} is thus
\begin{equation}\label{dayelement}
\Ceil{\frac{\JD \amod 10}{2}}
=
\Ceil{\frac{\JD}{2}} \amod 5.
\end{equation}
\subsubsection{Gender}
The gender is male when $ \JD$ is odd, and female when $\JD$ is even.
\xfootnote{
This is in accordance with the Chinese rule mentioned above that odd numbers
are male and even numbers female, but this is just a coincidence since
Julian Day Numbers were not invented as part of Chinese astrology.
}
\subsubsection{Animal}
The animal has number $(\JD+2) \amod12$ in the (Chinese) cycle in
\refT{T12}.
\subsubsection{Trigram}
The trigram has number $(\JD+2) \amod8$ in \refT{T8}.
\subsubsection{Number}
The number in \refT{T9} is $(-\JD) \amod9$.
\begin{remark}
\citet[pp.~208--209]{Henning} describe these using different,
and presumably traditional,
numberings for the 10 day cycle of elements and the 12 day cycle of
animals; his numbers (for the same element and animal as given
above)
are $(\JD-2) \amod 10$ and $(\JD-2)\amod12$.
He further calculates the number as $10-((\JD+1)\amod9)$.
\end{remark}
The element calculated in \eqref{dayelement} is the power element of the day.
Exactly as for years, see above, further elements (life, fortune,
body, spirit) can be calculated from the animal--element pair; these
elements thus follow a cycle of 60 days, which is equal to the cycle
in \refT{T60e}. A day has the elements given in \refT{T60e} on line
$(\JD -10) \amod 60$.
\begin{remark}
\TS{} calendars use a different method to assign the nine numbers;
the first Wood--Mouse day after the (true astronomical) winter
solstice is 5 (yellow); then the numbers increase by $1 \pmod 9$ each
day, until
the first Wood--Mouse day after the (true astronomical) summer
solstice, which is 4 (green), and then the numbers decrease by $1 \pmod
9$ each day for half a year until the first
Wood--Mouse day after the next winter solstice.
This requires accurate astronomical calculations of the solstices,
which is a central part of the Chinese calendar system, but foreign
to the Tibetan calendar calculations.
\end{remark}
\subsubsection{Elemental yoga}
In the Indian system, each day of week has an associated element from
the set \{earth, fire, water, wind\},
see \refT{T7}.
Further, each lunar mansion is also associated to an element from the
same set, see \citet[Appendix I]{Henning} for a list. Each calendar
day is thus given a combination of two elements, for the day of week
and for the lunar mansion (calculated in \eqref{weekday} and \eqref{mansion});
the order of these two elements does not matter and the combination is
regarded as an unordered pair. There are thus 10 possible different
combinations (\emph{yogas}), each having a name, see \cite[p.~204]{Henning}.
\newcommand\jour{\emph}
\newcommand\book{\emph}
\newcommand\vol{\textbf}
\newcommand\no{\unskip:}
\def\no#1#2,{\unskip#2, no. #1,}
\end{document} | math | 223,180 |
\begin{document}
\newcommand{{\rm Frac}}{{\rm Frac}}
\title{The number of rational numbers determined by
large sets of integers}
\author{Javier Cilleruelo}
\address{Instituto de Ciencias Matem\'{a}ticas (CSIC-UAM-UC3M-UCM) and Departamento de
Matem\'{a}ticas,
Universidad Aut\'{o}noma de Madrid,
Madrid-28049, Spain}
\email{[email protected]}
\author{D.S. Ramana}
\address{ Harish-Chandra Research Institute, Jhunsi, Allahabad -211 019, India.}
\email{[email protected] }
\author{Olivier Ramar\'{e}}
\address{Laboratoire Paul Painlev\'{e},
Universit\'{e} Lille 1,
59655 Villeneuve d'Ascq Cedex, France}
\email{[email protected]}
\makeatletter{\renewcommand*{\@makefnmark}{}
\footnotetext{{\it 2000 Mathematics Subject Classification : 11B05}}
\footnotetext{Keywords : rational numbers, large subsets, gaps, product sequence.}
\makeatother}
\begin{abstract} When $A$ and $B$ are subsets of the integers in $[1,X]$ and $[1,Y]$ respectively,
with $|A| \geq \alpha X$ and $|B| \geq \beta X$, we show that the
number of rational numbers expressible as $a/b$ with $(a,b)$ in $A
\times B$ is $\gg (\alpha \beta)^{1+\epsilon}XY$ for any $\epsilon >
0$, where the implied constant depends on $\epsilon$ alone. We then
construct examples that show that this bound cannot in general be
improved to $\gg \alpha \beta XY$. We also resolve the natural
generalisation of our problem to arbitrary subsets $C$ of the
integer points in $[1,X] \times [1,Y]$. Finally,
we apply our results to answer a question of S\'ark\"ozy concerning
the differences of consecutive terms of the product sequence of a
given integer sequence.\end{abstract}
\maketitle
\newcounter{thn}
\renewenvironment{th}
{\addtocounter{thn}{1}
\em
\noindent
{\sc Theorem \arabic{scn}.\arabic{thn}. ---}}{}
\newcounter{len}
\renewenvironment{le}
{\addtocounter{len}{1}
\em
\noindent
{\sc Lemma \arabic{scn}.\arabic{len}. ---}}{}
\newcounter{prn}
\newenvironment{prop}
{\addtocounter{prn}{1}
\em
\noindent
{\sc Proposition \arabic{scn}.\arabic{prn}. ---}}{}
\newcounter{corn}
\newenvironment{cor}
{\em
\addtocounter{corn}{1}
\noindent
{\sc Corollary \arabic{scn}.\arabic{corn}. ---}}{
}
\newcounter{scn}
\newenvironment{nsc}[1]
{\addtocounter{scn}{1}
\begin{center}
{\sc { \arabic{scn}. {#1}}}
\end{center}
\setcounter{equation}{0}
\setcounter{prn}{0}
\setcounter{len}{0}
\setcounter{thn}{0}
\setcounter{corn}{0}
}{}
\begin{nsc}{Introduction}
When $A$ and $B$ are intervals in the integers in $[1,X]$ and
$[1,Y]$ respectively, satisfying $|A| \geq \alpha X$ and $|B| \geq
\beta Y$, where $X$, $Y$ real numbers $\geq 1$, $\alpha$, $\beta$ are real numbers in $(0,1]$, a standard application of the M\"{o}bius
inversion formula shows that the number of rational numbers $a/b$
with $(a,b)$ in $A \times B$ is $\gg \alpha \beta XY$.
\noindent Our purpose is to investigate what might be
deduced when in place of {\em intervals} we consider {\em arbitrary}
subsets $A$ and $B$ of the integers in $[1,X]$ and $[1,Y]$
respectively with $|A| \geq \alpha X$ and $|B| \geq \beta Y$. When
$A$ and $B$ are not intervals, it may happen that an abnormally
large number of elements of these sets are multiples of certain
integers, determining which in general is not easy.
Nevertheless, since the sets under consideration are
large, popular heuristics suggest that a
non-trivial conclusion should still be accessible. What is pleasing
is that we in fact have the following theorem, which is our principal conclusion. In the
statement of this theorem and thereafter we write $A/B$ to denote
the subset of ${\bf Q}$ consisting of all rational numbers
expressible as $a/b$ with $(a,b)$ in $A \times B$ for any $A$ and
$B$ subsets of the integers $\geq 1$.
\begin{th}\label{main}
Let $\alpha$ and $\beta$ be real numbers in $(0,1]$ and $X$ and $Y$
real numbers $\geq 1$. When $A$ and $B$ are subsets of the integers
in $[1,X]$ and $[1,Y]$ respectively, with $|A| \geq
\alpha X$ and $|B| \geq \beta Y$ we have $|A/B| \gg
(\alpha\beta)^{1+\epsilon}XY$ for any $\epsilon > 0$, where the
implied constant depends on $\epsilon$ alone.
\end{th}
\noindent Deferring the detailed proof of Theorem 1.1 to Section~ 2, let us summarize our argument with the aid of the following notation. For any integer $d \geq 1$, $A$ and $B$ subsets of the integers $\geq 1$, we write ${\mathcal
M}(A,B,d)$ to denote the subset of $A \times B$ consisting of all $(a,b)$ in $A \times B$ with $\gcd(a,b)=d$.
We show in Proposition 2.1 that for $A$ and $B$ as in Theorem 1.1 we have $\sup_{d \geq 1} |{\mathcal M}(A,B,d)| \geq {\rm Frac}ac{1}{8} (\alpha \beta)^{2} XY$. Starting from this initial bound we then obtain $\sup_{d \geq 1} |{\mathcal M}(A,B,d)| \gg (\alpha \beta)^{1 +\epsilon} XY$ by a bootstrapping argument. Theorem 1.1 follows immediately from this last bound, since for any integer $d \geq 1$ we have $a/b \neq a_1/b_1$ for any two points $(a,b)$ and $(a_1,b_1)$ of
${\mathcal M}(A,B,d)$, and therefore $|A/B|
\geq \sup_{d \geq 1} |{\mathcal M}(A,B,d)|$.
\noindent
We supplement Theorem 1.1 with the following result which shows that the bound provided by Theorem 1.1 cannot be replaced with $|A/B| \gg \alpha\beta XY$. This bound, as we have already remarked, holds when $A$ and $B$ are intervals.
\begin{th}
For any $\epsilon >0$, there exists $\alpha>0$ such that for all
sufficiently large $X$ there exists a subset $A$ of the integers in
$[1,X]$ satisfying $|A|\ge \alpha X$ and $|A/A|<\epsilon
\alpha^2X^2$.
\end{th}
\noindent
We prove Theorem 1.2 in Section 3.
Our method depends on the observation that
for any $\epsilon > 0$ and any set of prime numbers ${\mathcal P}$ with $|{\mathcal P}|$ sufficiently large, we have $|
S({\mathcal P})/ S({\mathcal P})| \leq
\epsilon |S({\mathcal P})|^2$
, where $S({\mathcal P})$ is the set of squarefree
integers $d$ formed from the primes in the subsets of ${\mathcal P}$ containing about half the primes in ${\mathcal P}$.
By means of this observation we deduce that, for a suitable ${\mathcal P}$, the set of multiples of the elements of $S({\mathcal P})$ in $[1,X]$, meets the conditions of Theorem 1.2.
\noindent
The questions answered by the above theorems may be viewed as particular cases of a more general problem namely, for $X$ and $Y$ real numbers $\geq 1$ and $\gamma$ in $(0,1]$, given a subset $C$ of the integer points in $[1,X] \times [1,Y]$ satisfying $|C| \geq \gamma XY$, to determine in terms of $\gamma$, $X$ and $Y$ an optimal lower bound for ${\rm Frac}(C)$, the number of rational numbers $a/b$ with $(a,b)$ in $C$. Plainly, the above theorems take up the special case when $C$ is of the form $A \times B$, that is, when $C$ is equal to the product of its projections onto the co-ordinate axes.
\noindent
It turns out, however, that aforementioned general problem is somewhat easily resolved. In effect, the method of Proposition 2.1 generalizes without additional effort to give the bound $|{\rm Frac}(C)| \geq {\rm Frac}ac{1}{8}\gamma^2 XY$ and, interestingly, this bound is in fact optimal upto the constant ${\rm Frac}ac{1}{8}$. More precisely, we have the following theorem.
\begin{th}
For any $\gamma$ in $(0,1]$ and all sufficiently large $X$ and $Y$ there exists a subset $C$ of the integer points in $[1,X] \times [1,Y]$ satisfying $|C| \geq {\rm Frac}ac{\gamma}{8} XY$ and $|{\rm Frac}(C)| \leq {\rm Frac}ac{\gamma^2}{2}XY$.
\end{th}
\noindent
We prove Theorem 1.3 at the end of Section 3 by explicitly describing sets $C$ that satisfy the conditions of this theorem. Such sets are in general far from being of the form $A \times B$, which is only natural on account of Theorem 1.1. Indeed, our bootstrapping argument for Theorem 1.1 depends crucially on the fact that this theorem is, from the more general point view, about sets $C$ which are of the form $A \times B$.
\noindent We conclude this note with Section 4 where we
apply Theorem 1.1 to obtain a near-optimal answer to
the following question of S\'ark\"ozy. When ${\mathcal A}$, ${\mathcal B}$ are
sequences of integers, let ${\mathcal A}.{\mathcal B}$ be the sequence whose
terms are the integers of the form $ab$, for some $a\in
{\mathcal A}$, $b\in {\mathcal B}$. Then S\'ark\"ozy \cite{sar} asks if it is true that for
any $\alpha
> 0$ and ${\mathcal A}$ such that the lower asymptotic density
$\underline{d}({\mathcal A}) > \alpha$ there is a $c(\alpha)$ such that there
are infinitely many pairs of consecutive terms of ${\mathcal A}.{\mathcal A}$ the difference
between which is bounded by $c(\alpha)$.
\noindent Berczi \cite{ber} responded to the
aforementioned question of S\'ark\"ozy by showing that the minimum of
the differences between consecutive terms of ${\mathcal A}.{\mathcal A}$ is $\ll
{\rm Frac}ac{1}{\alpha^4}$, where $\alpha = \underline{d}({\mathcal A})$. Sandor
\cite{san} subsequently improved this by showing that this minimum
is in fact $\ll {\rm Frac}ac{1}{\alpha^3}$, with $\alpha$ now the upper
asymptotic density $\overline{d}({\mathcal A})$ of ${\mathcal A}$. Cilleruelo and Le
\cite{ch} obtained the same bound when $\alpha$ is the upper Banach
density of ${\mathcal A}$ and showed that this is the best possible bound for
this density. The following result improves upon and
generalizes Sandor's conclusion.
\begin{th}
Let $\alpha$ and $\beta$ be real numbers in $(0,1]$ and let
$\epsilon$ be $> 0$. When ${\mathcal A}$ and ${\mathcal B}$ are infinite sequences of
integers with upper asymptotic densities $\alpha$ and $\beta$
respectively, there are infinitely many pairs of
consecutive terms of the product sequence ${\mathcal A}.{\mathcal B}$ the difference
between which is $\ll {\rm Frac}ac{1}{(\alpha \beta)^{1+\epsilon}}$, where
the implied constant depends on $\epsilon$ alone.
\end{th}
\noindent When ${\mathcal A}$ and ${\mathcal B}$ are the sequences of
multiples of the integers $h$ and $k$ respectively, the difference
between any two consecutive terms of the sequence ${\mathcal A}.{\mathcal B}$ is $\geq
hk$. Since we have $\overline{d}({\mathcal A})= {\rm Frac}ac{1}{h}$ and
$\overline{d}({\mathcal B}) = {\rm Frac}ac{1}{k}$, we see that the conclusion of
Theorem 1.4 is optimal up to a factor ${\rm Frac}ac
1{(\alpha \beta)^{\epsilon}}$.
\noindent
Throughout this note, $X$, $Y$ shall denote real numbers $\geq 1$ and $\alpha$, $\beta$, $\gamma$
real numbers in $(0,1]$. Also, the letter $p$ shall denote a prime number. When $I$ and $J$ are subsets of a given set, $I \setminus J$ shall denote the set of elements of $I$ that are not in $J$. In addition to
the notation introduced so far, we shall write $A_d$ to denote the subset of a set of
integers $A$ consisting of all multiples of $d$ in $A$ for any integer $d$. Finally, if $B = \{b\}$ with
$b \geq 1$, we simply write $A/b$ in place of $A/B$, by an abuse of notation.
\end{nsc}
\begin{nsc}{Proof of the Bound}
\noindent
Let $A$ and $B$ be finite subsets of the integers $\geq 1$.
Then the family of subsets ${\mathcal M}(A,B,d)$ of $A \times B$, with $d$
varying over the integers $\geq 1$, is a partition of $A \times B$. Consequently, we have
\begin{equation}
\label{f1}
|A \times B| \; = \; \sum_{d \geq 1} |{\mathcal M}(A,B,d)| \; .
\end{equation}
\noindent When $A$ and $B$ are contained in $[1,X]$ and $[1,Y]$
respectively, we have $|A_d| \leq X/d$ and $|B_d| \leq
Y/d$, for any $d \geq 1$. Since ${\mathcal M}(A,B,d)$ is contained
in $A_d \times B_d$, we then obtain $|{\mathcal M}(A,B,d)| \leq
|A_d||B_d| \leq {\rm Frac}ac{XY}{d^2}$, for all $d \geq 1$.
\begin{prop}
When $A$ and $B$ are subsets of the integers in the intervals $[1,X]$ and $[1,Y]$
respectively, with $|A| \geq \alpha X$ and $|B| \geq \beta Y$, we have $\sup_{d \geq 1} |{\mathcal M}(A,B,d)| \geq {\rm Frac}ac{(\alpha \beta)^2 XY}{8}$.
\end{prop}
\noindent
{\sc proof.---} We adapt an argument from \cite{ch}. From (\ref{f1}) we have for any integer $T \geq 1$ that
\begin{equation}
\label{f2}
|A \times B| \; = \;
\sum_{1 \leq d \leq T} |{\mathcal M}(A,B,d)| + \sum_{T < d} |{\mathcal M}(A,B,d)| \; \leq \;
\sum_{1 \leq d \leq T} |{\mathcal M}(A,B,d)| + {\rm Frac}ac{XY}{T} \; ,
\end{equation}
\noindent
where the last inequality follows from $\sum_{T < d} |{\mathcal M}(A,B,d)| \leq \sum_{T < d} {\rm Frac}ac{XY}{d^2} \leq {\rm Frac}ac{XY}{T}$.
Since $|A \times B| \geq \alpha \beta XY$ we conclude
from (\ref{f2}) that
\begin{equation}
\label{f3}
\sup_{d \geq 1} |{\mathcal M}(A,B,d)|
\; \geq \;
{\rm Frac}ac{1}{T} \sum_{1 \leq d \leq T} |{\mathcal M}(A,B,d)|
\; \geq \;
\left({\rm Frac}ac{\alpha\beta - {\rm Frac}ac{1}{T}}{T}\right)XY \;
\end{equation}
\noindent
for any integer $T \geq 1$. Since $2 > \alpha \beta$, the interval $[{\rm Frac}ac{2}{\alpha\beta}, {\rm Frac}ac{4}{\alpha\beta}]$
contains an integer $\geq 1$. The proposition now follows on setting $T$ in (\ref{f3}) to be any such integer.
\noindent {\sc Definition 2.1---} We call a real number
$\delta$ an {\em admissible exponent} if there exists a real number
$C >0$ such that for any $\alpha$, $\beta$ real numbers in $(0,1]$,
any $X$, $Y$ real numbers $\geq 1$ and any subsets $A$ and $B$ of
the integers in $[1,X]$ and $[1,Y]$ with $|A| \geq \alpha X$ and $|B| \geq \beta Y$, we have $\sup_{d \geq 1} |{\mathcal M}(A,B,d)| \geq
C(\alpha\beta)^{\delta} XY$. We call a $C$ satisfying
these conditions a {\em constant associated to} the admissible
exponent $\delta$.
\noindent
Proposition 2.1 says that $\delta=2$ is an
admissible exponent. Proposition 2.2 will allow us to
conclude that every $\delta > 1$ is an admissible exponent.
The following lemma prepares us for an application of H\"{o}lder's inequality within the proof of Proposition 2.2.
\noindent
For any integer $n\geq 1$ let $\tau(n)$ denote, as usual, the number of integers $\geq 1$ that divide $n$.
When $D$ is an integer $\geq 1$ we write $\tau_D(n)$ to denote the number of divisors $d$ of $n$ satisfying the condition $p|d \implies p \leq D$ for any prime number $p$.
\begin{le}
When $q$ is an integer $\geq 0$ there is a real number $c(q) >0$ such that for all real numbers $X \geq 1$ and integers $D\geq 1$ we have
\begin{equation}
\label{f4} \sum_{1 \leq n \leq X} \tau_{D}(n)^q \; \leq \;
c(q) DX \; ,
\end{equation}
\end{le}
\noindent
{\sc Proof.---} In effect, we have
\begin{equation}
\label{6}
\sum_{1 \leq n \leq X}\tau_{D}(n)^q \ll X(\log 2D)^{2^q}
\, \ll \, (2^q!)\,D X \; ,
\end{equation}
\noindent
where the implied constants are absolute. Plainly, the second inequality results from the elementary inequality $(\log t)^n \leq n!\, t$ for $t \geq 1$. We now
prove the first inequality in (\ref{6}). Let us write ${\mathcal D}$ for the set of integers
$m$ satisfying the condition $p|m \implies p \leq D$. For any integer $n \geq 1$, let $k(n)$
be the largest of the divisors of $n$ lying in ${\mathcal D}$. We then have that
\begin{equation}
\label{7}
\sum_{1 \leq n \leq X}\tau_{D}(n)^q = \sum_{\stackrel{1 \leq m \leq X,}{m \in {\mathcal D}}} \tau(m)^q \sum_{\stackrel{1 \leq n \leq X,}{k(n) = m}} 1
\leq X \sum_{m \in {\mathcal D}} {\rm Frac}ac{\tau(m)^q}{m} \;
\; ,
\end{equation}
\noindent
where we have used the upper bound $X/m$ for the number of integers $n$ in $[1,X]$ with $k(n)=m$. Let us write $S(q)$ for any integer $q \geq 0$ to denote the last sum in (\ref{7}). Since Merten's formula gives
$\prod_{1 \leq p \leq D} (1-{\rm Frac}ac{1}{p}) \sim {\rm Frac}ac{e^{-\gamma}}{\log D}$, with $\gamma$ here being Euler's constant, we have
\begin{equation}
\label{9}
S(0) = \sum_{ m \in D} {\rm Frac}ac{1}{m} \; = \;
\prod_{1 \leq p \leq D}\left(1 + {\rm Frac}ac{1}{p} + {\rm Frac}ac{1}{p^2} +\ldots\right)\; = \;
\prod_{1 \leq p \leq D} \left(1 - {\rm Frac}ac{1}{p}\right)^{-1} \; \ll \; \log2D \; ,
\end{equation}
\noindent
where the implied constant is absolute. On noting that every divisor of an integer in ${\mathcal D}$ is again in ${\mathcal D}$ and using $\tau(dk) \leq \tau(d)\tau(k)$, valid for any integers $d$ and $k \geq 1$, we obtain
\begin{equation}
\label{8}
\sum_{m \in D} {\rm Frac}ac{\tau(m)^q}{m} \;
\;= \;
\sum_{m \in D} {\rm Frac}ac{\tau(m)^{q-1}}{m} \sum_{d | m} 1 \;
= \;
\sum_{(d,k) \in D\times D} {\rm Frac}ac{\tau(dk)^{q-1}}{dk}
\; \leq \;
\left(\sum_{d \in D} {\rm Frac}ac{\tau (d)^{q-1}}{d}\right)^2 \; .
\end{equation}
\noindent
In other words, $S(q) \leq S(q-1)^2$, for any $q \geq 1$. An induction on $q$ then shows that for any integer $q \geq 0$ we have $S(q) \leq S(0)^{2^q} \ll (\log D)^{2^q}$, where the implied constant is absolute. On combining this bound with (\ref{7}) we obtain the first inequality in (\ref{6}).
\begin{prop}
If $\delta>1$ is an admissible exponent then so is
${\rm Frac}ac{3\delta(1+1/q)-2}{2\delta-1}$ for every integer $q \geq 1$.
\end{prop}
\noindent {\sc Proof.---} Let $q$ be a given integer
$\geq 1$ and, for the sake of conciseness, let us write
$\delta^{\prime}$ to denote ${\rm Frac}ac{3\delta(1+1/q)-2}{2\delta-1}$,
which is $> 1$ since $\delta > 1$.
\noindent
When $C$ is a constant associated to $\delta$, let us set $C^{\prime}$ to be the unique real number $> 0$ satisfying
\begin{equation}
\label{f7.5} {\rm Frac}ac{1}{8C^{\prime}} \; =\;
\left({\rm Frac}ac{C^{\prime}}{C}\right)^{{\rm Frac}ac{1}{2(\delta-1)}}
8^{{\rm Frac}ac{\delta}{\delta-1}}
(4c(q))^{{\rm Frac}ac{\delta}{q(\delta-1)}} \; ,
\end{equation}
\noindent where $c(q)$ is the implied constant in
(\ref{f4}) of Lemma 2.1. It is easily seen from (\ref{f7.5}) that by replacing
$C$ with a smaller constant associated to $\delta$ if necessary we
may assume that ${\rm Frac}ac{1}{4} \geq C^{\prime}$.
\noindent
We shall show that $\delta^{\prime}$ is an admissible exponent with $C^{\prime}$ a constant associated to $\delta^{\prime}$.
Thus let $\alpha$, $\beta$ be real numbers in $(0,1]$ and $X$, $Y$ real numbers $\geq 1$. Also, let $A$ and $B$
be any subsets of the integers in $[1,X]$ and $[1,Y]$ satisfying $|A| \geq \alpha X$ and $|B| \geq \beta Y$. We shall show that
\begin{equation}
\label{f8}
\sup_{d \geq 1} |{\mathcal M}(A,B,d)| \geq C^{\prime} {(\alpha \beta)}^{\delta^{\prime}}XY \; .
\end{equation}
\noindent
Replacing $\alpha$ and $\beta$ with $\alpha^{\prime}\geq \alpha$ and $\beta^{\prime}\geq \beta$ such that $ \alpha^{\prime}\leq |A| \leq 2\alpha^{\prime}$ and $\beta^{\prime}\leq |B| \leq 2\beta^{\prime}$ if necessary, we reduce to the case when $|A| \leq 2\alpha X$ and $|B| \leq 2\beta Y$.
\noindent
Let us first dispose of the possibility that an abnormally large number of the integers in $A$ and $B$ are multiples of a given integer.
Thus let $\alpha_d = |A_d|/X$ and $\beta_d = |B_d|/Y$, for any integer $d \geq 1$. Suppose that there exists an integer $d \geq 1$ such that
\begin{equation}
\label{f9}
\alpha_d \beta_d \geq \left({\rm Frac}ac{C^{\prime}}{C}\right)^{{\rm Frac}ac{1}{\delta}} (\alpha \beta)^{{\rm Frac}ac{\delta^{\prime}}{\delta}} d^{{\rm Frac}ac{2}{\delta}-2}\; .
\end{equation}
\noindent
Then $A_d$ and $B_d$ are both non-empty and therefore $X$ and $Y$ are both $\geq d$. Further,
the sets $A_d/d$ and $B_d/d$ are subsets of the integers in $[1,X/d]$ and $[1,Y/d]$.
Since $\delta$ is an admissible exponent, $C$ a constant associated to $\delta$, and
we have $|A_d/d|= (d\alpha_d)|X/d|$, $|B_d/d|= (d\beta_d)|X/d|$, there exists an integer $d^{\prime} \geq 1$ such that
\begin{equation}
\label{f10}
|{\mathcal M}(A_d/d,B_d/d,d^{\prime})|\; \geq \; C(d^2\alpha_d\beta_d)^{\delta}{\rm Frac}ac{XY}{d^2}\; \geq \; C^{\prime}(\alpha \beta)^{\delta^{\prime}} XY \; ,
\end{equation}
\noindent
where the last inequality follows from (\ref{f9}). Since
$|{\mathcal M}(A_d/d,B_d/d,d^{\prime})|$ does not exceed $|{\mathcal M}(A,B,dd^{\prime})|$,
we obtain (\ref{f8}) from (\ref{f10}). We may therefore verify (\ref{f8}) assuming that for every integer $d \geq 1$ we have
\begin{equation}
\label{f11} \alpha_d \beta_d <
\left({\rm Frac}ac{C^{\prime}}{C}\right)^{{\rm Frac}ac{1}{ \delta}} (\alpha
\beta)^{{\rm Frac}ac{\delta^{\prime}}{\delta}} d^{{\rm Frac}ac{ 2}{\delta}-2}\; .
\end{equation}
\noindent
With the aid of (\ref{f11}) we shall in fact obtain a more precise conclusion than (\ref{f8}). Let us set
$K = {\rm Frac}ac{(\alpha\beta)^{1-\delta^{\prime}}}{8C^{\prime}}$ and $L=1 + [K]$. We shall show that
\begin{equation}
\label{f22}
{\rm Frac}ac{1}{L}\sum_{1 \leq d \leq L} |{\mathcal M}(A,B,d)| \geq C^{\prime}(\alpha\beta)^{\delta^\prime} XY \; ,
\end{equation}
\noindent
so that we have $|{\mathcal M}(A,B,d)| \geq C^{\prime}(\alpha\beta)^{\delta^\prime} XY$ for some integer $d \leq L$
, which of course implies (\ref{f8}). Note that since $L$ is roughly about ${\rm Frac}ac{(\alpha\beta)^{1-\delta^{\prime}}}{C^{\prime}}$, (\ref{f22}) is what one might expect from (\ref{f1}).
\noindent Let $D$ be an integer in
$[{\rm Frac}ac{2}{\alpha\beta},{\rm Frac}ac{4}{\alpha\beta}]$. Thus
in particular $D > 1$. When $L \geq D$ we obtain (\ref{f22}) even
without (\ref{f11}). In effect, we then have $K \geq 1$ and hence
that $L < 2K$ or, what is the same thing, that $L < {\rm Frac}ac{(\alpha
\beta)^{1-\delta^{\prime}}}{4C^{\prime}}$ from which (\ref{f22})
follows on noting that for any integer $T \geq D$, and in particular
for $T =L$, we have from (\ref{f3}) that
\begin{equation}
\label{f12} {\rm Frac}ac{1}{T} \sum_{1 \leq d \leq T} |{\mathcal M}(A,B,d)| \; \geq \;
\left({\rm Frac}ac{\alpha\beta - {\rm Frac}ac{1}{T}}{T}\right)XY \;
\geq \; \left({\rm Frac}ac{\alpha\beta -
{\rm Frac}ac{1}{D}}{T}\right)XY \; \geq \; {\rm Frac}ac{\alpha \beta XY}{2T} \; .
\end{equation}
\noindent Suppose now that $1 \leq L < D$. Let us first verify that for any integer $T$ such that $1 \leq T < D$ we have the following inequality on account of (\ref{f11}).
\begin{equation}
\label{f14}
\sum_{T < d \leq D} |{\mathcal M}(A,B,d)| \; \leq \;
\left({\rm Frac}ac{C^{\prime}}{C}\right)^{{\rm Frac}ac{1}{
2\delta}} (\alpha \beta)^{{\rm Frac}ac{\delta^{\prime}}{2\delta}} T^{{\rm Frac}ac{
1}{\delta}-1}\; (XY)^{{\rm Frac}ac{1}{2}}
\left(\sum_{T < d \leq D} |A_d|\right)^{{\rm Frac}ac{1}{2}}
\left(\sum_{T < d \leq D} |B_d|\right)^{{\rm Frac}ac{1}{2}}
\;.
\end{equation}
\noindent Indeed, for any integer $d$ satisfying $T <
d \leq D$ we have that
\begin{equation}
\label{f14.5}
|A_d||B_d| = (\alpha_d X \beta_d Y)^{{\rm Frac}ac{1}{2}} |A_d|^{{\rm Frac}ac{1}{2}}
|B_d|^{{\rm Frac}ac{1}{2}}\; \leq \;
\left({\rm Frac}ac{C^{\prime}}{C}\right)^{{\rm Frac}ac{1}{
\delta}} (\alpha \beta)^{{\rm Frac}ac{\delta^{\prime}}{2\delta}} T^{{\rm Frac}ac{
1}{\delta}-1}\; (XY)^{{\rm Frac}ac{1}{2}} |A_d|^{{\rm Frac}ac{1}{2}}
|B_d|^{{\rm Frac}ac{1}{2}} \; ,
\end{equation}
\noindent
where the last inequality follows from (\ref{f11}) on noting that
$d^{{\rm Frac}ac{1}{\delta}-1} \leq T^{{\rm Frac}ac{1}{\delta}-1}$ for $d$ satisfying $T < d \leq D$,
since $\delta \geq 1$. On combining the bound $|{\mathcal M}(A,B,d)| \leq |A_d||B_d|$ with (\ref{f14.5}) and an application of the
Cauchy-Schwarz inequality we obtain (\ref{f14}).
\noindent
We now estimate the sums on the right hand side of (\ref{f14}). An
application of H\"{o}lder's inequality gives
\begin{equation}
\label{f15}
\sum_{T < d \leq D} |A_d| \; = \;
\sum_{T < d \leq D} \sum_{\stackrel{n \in A,}{d|n}} 1
\;\leq \; \sum_{n \in A} \tau_{D}(n) \; \leq \;
|A|^{1 -{\rm Frac}ac{1}{q}} \left(\sum_{1 \leq n \leq X } \tau_{D}(n)^{q}\right)^{{\rm Frac}ac{1}{q}} \;.
\end{equation}
\noindent From Lemma 2.1 we have the upper bound $c(q)DX$ for the
last sum in (\ref{f15}). Since $|A| \leq 2\alpha X$
and $D \leq {\rm Frac}ac{4}{\alpha \beta}$, we deduce from (\ref{f15}) that
\begin{equation}
\label{f15.1} \sum_{T < d \leq D} |A_d| \leq
(2\alpha)^{1-{\rm Frac}ac{2}{q}}\beta^{-{\rm Frac}ac{1}{q}}(4c(q))^{{\rm Frac}ac{1}{q}}X.
\end{equation}
\noindent
Arguing similarly, we obtain the bound
\begin{equation}
\label{f15.2} \sum_{T < d \leq D} |B_d| \leq
(2\beta)^{1-{\rm Frac}ac{2}{q}}\alpha^{-{\rm Frac}ac{1}{q}}(4c(q))^{{\rm Frac}ac{1}{q}}Y.
\end{equation}
\noindent
With these estimates we conclude from (\ref{f14}) that for any integer $T$ satisfying $1 \leq T < D$ we have
\begin{equation}
\label{f16} \sum_{T < d \leq D} |{\mathcal M}(A,B,d)| \; \leq \;
2\left({\rm Frac}ac{C^{\prime}}{C}\right)^{ {\rm Frac}ac{1}{2\delta}}
(\alpha \beta)^{ {\rm Frac}ac{\delta^{\prime}}{2\delta} + {\rm Frac}ac{1}{2}
-{\rm Frac}ac{3}{2q}} T^{{\rm Frac}ac{1}{\delta} -1} (4c(q))^{{\rm Frac}ac{1}{q}} XY \; ,
\end{equation}
\noindent We now reveal that our choices for $C^{\prime}$ and
$\delta^{\prime}$ were made so that $K$ satisfies the relation
\begin{equation}
\label{f17} 2\left({\rm Frac}ac{C^{\prime}}{C}\right)^{{\rm Frac}ac{1}{2\delta}}
(\alpha \beta)^{ {\rm Frac}ac{\delta^{\prime}}{2\delta} + {\rm Frac}ac{1}{2}
-{\rm Frac}ac{3}{2q}} K^{{\rm Frac}ac{1}{\delta} -1} (4c(q))^{{\rm Frac}ac{1}{q}} \; = \;
{\rm Frac}ac{\alpha\beta}{4} \; ,
\end{equation}
\noindent as may be confirmed by a modest calculation
using the expressions defining $C^{\prime}$ and $\delta^{\prime}$ in
terms of $C$ and $\delta$.
\noindent We see that $\sum_{L < d \leq D} |{\mathcal
M}(A,B,d)| \leq {\rm Frac}ac{\alpha \beta}{ 4}
XY$ using (\ref{f16}) for $T = L$ together with (\ref{f17}) and
noting that $K < L$. Since (\ref{f12}) applied with $T =D$ gives us
$\sum_{1 \leq d \leq D} |{\mathcal M}(A,B,d)| \geq
{\rm Frac}ac{\alpha \beta}{2} XY$, we conclude that when $1
\leq L < D$ we have
\begin{equation}
\label{f18}
{\rm Frac}ac{1}{L}\sum_{1 \leq d \leq L} |{\mathcal M}(A,B,d)| \, \geq \, {\rm Frac}ac{\alpha \beta}{4L} XY \; .
\end{equation}
\noindent
If $L =1$ we obtain (\ref{f22}) from (\ref{f18}) on noting that ${\rm Frac}ac{\alpha \beta}{4} \geq C^{\prime} (\alpha \beta)^{\delta^{\prime}}$, since ${\rm Frac}ac{1}{4} \geq C^{\prime}$ and $1 \leq \delta^{\prime} $. When $1 < L < D$ we have $1 \leq K$ and hence $L < {\rm Frac}ac{(\alpha \beta)^{1-\delta^{\prime}}}{4C^{\prime}}$ so that (\ref{f22}) results from (\ref{f18}) in this final case as well.
\begin{cor} Every $\delta>1$ is an admissible exponent. \end{cor}
\noindent
{\sc Proof. ---} Let $q$ be any integer $\geq 4$ and let $\{\delta_n (q)\}_{n \geq 1}$ the sequence
of real numbers determined by the relations $\delta_1 (q) = 2$ and
\begin{equation}
\label{fe0}
\delta_{n+1} (q) = {\rm Frac}ac{3\delta_n(q)\left (1+{\rm Frac}ac 1q\right
)-2}{2\delta_n(q)-1}
\end{equation}
\noindent
for $n \geq 1$. Then each $\delta_n(q)$ is
an admissible exponent by Propositions 2.1 and 2.2. It is easily verified that the sequence $\delta_n(q)$ is decreasing and has a limit $\delta(q)$ given by the relation
\begin{equation}
\label{fe1}
\delta(q)=1+{\rm Frac}ac{3}{4q}+{\rm Frac}ac 12\sqrt{{\rm Frac}ac
6q+{\rm Frac}ac{9}{4q^2}}.
\end{equation}
\noindent
Plainly, any $\delta>\delta(q)$ is an admissible exponent. The corollary now follows on taking $q$ arbitrarily large in (\ref{fe1}).
\noindent
Theorem 1.1 follows from the above corollary and the definition of admissible exponents on recalling that $|A/B| \geq \sup_{d \geq 1} |{\mathcal M}(A,B,d)|$.
\end{nsc}
\begin{nsc}{Counterexamples}
\noindent
Let us first prove Theorem 1.2. To this end, given an integer $m \geq 1$ let ${\mathcal P}$ denote
any set of $2m$ prime numbers and, for any subset $I$ of ${\mathcal P}$, let $d(I) = \prod_{p \in I} p$.
If $S({\mathcal P})$ denotes the set of all $d(I)$ with $|I| = m$, we have the following lemma.
\begin{le}
For any $\epsilon >0$, we have $|S({\mathcal P})/S({\mathcal P})| \leq \epsilon |S({\mathcal P})|^2$ for all sufficiently large $m$.
\end{le}
\noindent
{\sc Proof. ---} Plainly, we have $|S({\mathcal P})| = \binom{2m}{m}$. Let ${\mathcal Q}$ be the set of
ordered pairs of disjoint subsets of ${\mathcal P}$. Then, for any $I$ and $J$ subsets of ${\mathcal P}$, we have
\begin{equation}
\label{e1}
{\rm Frac}ac{d(I)}{d(J)} = {\rm Frac}ac{d(I\setminus J)}{d(J\setminus I)} \; ,
\end{equation}
\noindent
and since $I\setminus J$ and $J\setminus I$ are disjoint, $(I\setminus J,J \setminus I)$ is in ${\mathcal Q}$.
Thus $|S({\mathcal P})/S({\mathcal P})| \leq |{\mathcal Q}|$. Let us associate any $(U,V)$ in ${\mathcal Q}$ to the map from ${\mathcal P}$ to the three element set $\{1,2,3\}$ that takes $U$ to 1, $V$ to 2 and the complement of $U \cup V$ in ${\mathcal P}$ to 3. It is easily seen that this association in fact gives a bijection from ${\mathcal Q}$ onto the set of maps from ${\mathcal P}$ to $\{1,2,3\}$ and hence that $|{\mathcal Q}| = 3^{2m}$. In summary, we deduce that
\begin{equation}
\label{s31}
|S({\mathcal P})/S({\mathcal P})| \; \leq \; |{\mathcal Q}|\; =\; 3^{2m} \; = \; {\rm Frac}ac{3^{2m}}{{\binom{2m}{m}}^2}\,|S({\mathcal P})|^2 \; \leq \; (2m+1)^2 \left({\rm Frac}ac{3}{4}\right)^{2m}|S({\mathcal P})|^2 \; ,
\end{equation}
\noindent
where we have used the inequality $\binom{2m}{m} \geq {\rm Frac}ac{2^{2m}}{2m+1}$.
The lemma follows from (\ref{s31}) on noting that $(2m+1)^2 \left({\rm Frac}ac{3}{4}\right)^{2m} \rightarrow 0$ as $m \rightarrow +\infty$.
\noindent
{\sc Proof of Theorem 1.2. ---} Given an integer $m \geq
1$, it is easily deduced from the prime number theorem that the
interval $[T, T + T/m]$ contains at least $2m$ prime numbers when
$T$ is sufficiently large. For such a $T$, let ${\mathcal P}$ be a
subset of $2m$ prime numbers in $[T, T + T/m]$. If ${\mathcal
A}({\mathcal P})$ is the sequence of integers $\geq 1$ that are
divisible by at least one of the integers $d(I)$ in $S({\mathcal
P})$ then a simple application of the principle of inclusion and
exclusion implies that ${\mathcal A}({\mathcal P})$ has an
asymptotic density $\alpha({\mathcal P})$ that is given by the
relation
\begin{equation}
\label{asym}
\alpha({\mathcal P}) \; = \;
\sum_{1 \leq r \leq \binom{2m}{m}} (-1)^{r+1} \sum_{1\leq i_1 < i_2 \ldots < i_r \leq \binom{2m}{m}} {\rm Frac}ac{1}{d(I_{i_1} \cup I_{i_2} \ldots \cup I_{i_r})} \; ,
\end{equation}
\noindent
where $I_1,I_2, \ldots, I_{\binom{2m}{m}}$ are the subsets of cardinality $m$ in ${\mathcal P}$.
\noindent
For any $i$ we have $T^m \leq
d(I_i) \leq (1+{\rm Frac}ac 1m)^m T^{m}<eT^m$. Consequently, for the term
$r =1$ in (\ref{asym}) we obtain
\begin{equation}
\sum_{1\leq i \leq \binom{2m}{m}} {\rm Frac}ac{1}{d(I_i)}\; \geq \;
{\rm Frac}ac{\binom{2m}{m}}{eT^m}.
\end{equation}
\noindent When $r \geq 2$, we have that $d(I_{i_1} \cup I_{i_2}
\ldots \cup I_{i_r})$, for any distinct indices $i_1, i_2 \ldots,
i_r$, has at least $k+1$ prime factors in ${\mathcal
P}$ and hence is $\geq T^{m+1}$. It follows from (\ref{asym}) and
these bounds that we have
\begin{equation}
\label{e2.1} \alpha({\mathcal P})\ge
{\rm Frac}ac{\binom{2m}{m}}{eT^m}-{\rm Frac}ac{2^{\binom{2m}{m} }}{T^{m+1}}\ge
{\rm Frac}ac{\binom{2m}{m}}{3T^m}
\end{equation}
\noindent when $T$ is sufficiently large. In particular, on
recalling that $|S({\mathcal P})| = \binom{2m}{m}$, we obtain that
for any integer $m \geq 1$, we have
\begin{equation}
\label{e2} \alpha({\mathcal P}) \; \geq \; {\rm Frac}ac{|S({\mathcal
P})|}{3T^m} \;
\end{equation}
\noindent for all sufficiently large $T$ and ${\mathcal P}$ any set
of $2m$ prime numbers in $[T, T+T/m]$.
\noindent
Finally, for ${\mathcal P}$ as above and any $X \geq 1$, let us set
$A = {\mathcal A}({\mathcal P}) \cap [1,X]$.
Since $\alpha({\mathcal P})$ is the asymptotic density of ${\mathcal A}({\mathcal P})$, we
have from (\ref{e2}) that $|A| \geq {\rm Frac}ac{|S({\mathcal P})|}{4T^m}X$, for all large enough $X$ and $T$. Clearly, each integer in $A$ is of the form $d(I)n$, for some $d(I)$ in $S({\mathcal P})$ and an integer $n$, which must necessarily be $\leq {\rm Frac}ac{X}{T^m}$, since $A$ is in $[1,X]$ and $d(I) \geq T^m$. Consequently, we have we have $|A/A| \leq {\rm Frac}ac{|S({\mathcal P})/S({\mathcal P})|}{T^{2m}} X^2$, for all large enough $X$ and $T$. On comparing $|A|$ and $|A/A|$ by means of Lemma 3.1, we see that $A$ meets the conditions of Theorem 1.2 when $m$, $T$ and $X$ are all sufficiently large.
\noindent {\sc Proof of Theorem 1.3. ---}
The number of primitive integer points, that is, integer points with
coprime co-ordinates, in $[1,\gamma X] \times [1,\gamma Y]$ is $\sim
{\rm Frac}ac{6}{\pi^2} \gamma^2 XY$ as $X$, $Y \rightarrow \infty$. Thus
for any $\gamma$ in $(0,1]$ and all sufficiently large $X$ and $Y$,
there is a subset $S$ of the primitive integer points in $[1,\gamma
X] \times [1,\gamma Y]$ satisfying ${\rm Frac}ac{\gamma^2}{4} XY \leq |S|
\leq {\rm Frac}ac{\gamma^2}{2} XY$. Let us take for $C$ the union of the
sets $d.S$ with $d$ varying over the interval
$[1,{\rm Frac}ac{1}{\gamma}]$, where each $d.S$ is the set of $(da,db)$
with $(a,b)$ varying over $S$. Then $C$ is contained in $[1, X] \times [1,
Y]$. Moreover, the sets $d.S$ are disjoint but ${\rm Frac}(d.S) =
{\rm Frac}(S)$, for each $d$, and
$|{\rm Frac}(S)| = |S|$. We therefore have $|C| = [{\rm Frac}ac{1}{\gamma}]|S|
\geq {\rm Frac}ac{\gamma}{8} XY$ but $|{\rm Frac}(C)|= |{\rm Frac}(S)| =
|S|\leq {\rm Frac}ac{\gamma^2}{2} XY$.
\end{nsc}
\begin{nsc}{Gaps in Product Sequences}
\noindent
We now deduce Theorem 1.4 from Theorem 1.1. Let ${\mathcal A}$ and ${\mathcal B}$ be sequences with upper asymptotic densities $\alpha$ and $\beta$.
Then there exist infinitely many real numbers $X$ and $Y$ $\geq 1$ such that $|{\mathcal A} \cap ({\rm Frac}ac{X}{2},X]| \geq {\rm Frac}ac{\alpha X}{4}$ and
$|{\mathcal B} \cap ({\rm Frac}ac{Y}{2},Y]| \geq {\rm Frac}ac{\beta Y}{4}$. For such $X$ and $Y$ let us apply Theorem 1.1 to the sets $A = {\mathcal A} \cap ({\rm Frac}ac{X}{2},X]$
and $B = {\mathcal B} \cap ({\rm Frac}ac{Y}{2},Y]$. We then have that $|A/B| \gg (\alpha \beta)^{1+\epsilon} XY$, where the implied constant
depends on $\epsilon$ alone. Since $A/B$ is a subset of the interval $[{\rm Frac}ac{X}{2Y}, {\rm Frac}ac{2X}{Y}]$, which is of length ${\rm Frac}ac{X}{Y}$,
we deduce that there are distinct $a/b$ and $a^{\prime}/b^{\prime}$ in $A/B$ such that
\begin{equation}
\label{e5} 0< \left|{\rm Frac}ac{a}{b} -
{\rm Frac}ac{a^{\prime}}{b^{\prime}}\right| \ll {\rm Frac}ac{X/Y}{(\alpha
\beta)^{1+\epsilon}XY} = {\rm Frac}ac{1}{(\alpha \beta)^{1+\epsilon}Y^2}
\; .
\end{equation}
\noindent
Since $|bb^{\prime}| \leq Y^2$, it
follows from (\ref{e5}) that difference between the
distinct terms $ba^{\prime}$ and $b^{\prime}a$ of the product
sequence ${\mathcal A}.{\mathcal B}$ is $\ll {\rm Frac}ac{1}{(\alpha \beta)^{1+\epsilon}}$.
Since there are infinitely many distinct $X$ and $Y$ satisfying the required conditions,
there are infinitely many such pairs of terms in the product sequence ${\mathcal A}.{\mathcal B}$.
\end{nsc}
\noindent
{\bf Acknowledgments : }
We arrived at Theorem 1.3 in response to a question of Professor Adrian Ubis, whom we gladly thank. We also wish to thank
the CRM, Barcelona and HRI, Allahabad for
opportunities that supported discussions on the problems addressed here.
\noindent J. Cilleruelo was supported by
Grant CCG08-UAM/ESP-3906 and MTM2008-03880 of the MYCIT
, Spain during the course of preparation of this note. D.S. Ramana
is with the Harish-Chandra Research Institute which is a
constituent institution of the Homi Bhabha National Institute,
Mumbai, India. Olivier Ramar\'{e} is supported by the CNRS, France.
\end{document} | math | 35,440 |
\begin{document}
\title[Expanding Ricci solitons with pinched curvature]
{Expanding Ricci solitons with pinched Ricci curvature}
\author{Li Ma }
\address{Li Ma, Department of Mathematical Sciences, Tsinghua University,
Peking 100084, P. R. China}
\email{[email protected]}
\thanks{The research is partially supported by the National Natural Science
Foundation of China 10631020 and SRFDP 20090002110019}
\begin{abstract}
In this paper, we prove that expanding gradient Ricci solitons
with (positively) pinched Ricci curvature are trivial ones.
Namely, they are either compact or flat.
{ \textbf{Mathematics Subject Classification 2000}: 35J60, 53C21,
58J05}
{ \textbf{Keywords}: Expanding Ricci soliton, curvature pinching,
asymptotic flat}
\end{abstract}
\maketitle
\section{Introduction}
In this paper we consider Problem 9.62 in the famous book
\cite{Cho06}, which is also the unanswered question in \cite{MD}.
Namely, when $n\geq 3$, do there exist expanding gradient Ricci
solitons with (positively) pinched Ricci curvature? In dimension
three, we settle the problem completely. Here we recall that the
(positively) pinched Ricci curvature for the Riemannian manifold
$(M^n,g)$ is in the sense that
\begin{equation}\label{pinch}
Rc\geq \epsilonsilon Rg\geq 0, \end{equation} where $R$ and $Rc$ are
the scalar and Ricci curvatures of the metric $g$ respectively,
$\epsilonsilon>0$ is a small constant. This concept plays an important
role in the seminal work of R.Hamilton \cite{Ham82}. We remark
that the compact expanding gradient Ricci solitons are Einstein.
This result is known in G.Perelman \cite{P02}. Then we may assume
that the expanding gradient Ricci soliton $(M^n,g(t),\phi)$ under
consideration is complete, non-compact, and in canonical form that
$$
D^2\phi=Rc+\frac{1}{2t}g, \ \ t>0.
$$
We denote by $d_g(x,o)$ the distance between the points $x$ and
$o$ in $(M,g)$.
We show that there are only trivial ones.
\begin{Thm}\label{thm:1}
Expanding gradient Ricci solitons (M,g) with (positively) pinched
Ricci curvature and curvature decay at the order
$d_g(x,o)^{-2-\delta}$ for some $\delta>0$, are trivial ones.
Namely, they are either compact or $R^n$ with flat metric.
\end{Thm}
We remark that in dimension three, the curvature decay condition
is automatically true \cite{MD}. This result is used in \cite{M}.
In the dimension bigger than three, the same is true for locally
conformally flat expanding gradient Ricci solitons with pinched
Ricci curvature. In general, since we are studying the Ricci flow,
we should have the curvature decay order as that of the Ricci
curvature. This will be considered in the future.
This paper is organized as follows. In section \ref{sect2} we
recall some famous results, which will be in use in section
\ref{sect3}. Theorem \ref{thm:1} is proved in section \ref{sect3}.
We shall use $r$ denote various uniform positive constants.
\section{Preliminary}\label{sect2}
Before we prove our main result Theorem \ref{thm:1}, we cite the
following results, which will be in use in next section. The first
is
\begin{Pro}\label{pro:1}(Hamilton, Proposition 9.46 in
\cite{Cho06}).
If $(M,g(t))$, $t>0$, is a complete non-compact expanding gradient
Ricci soliton with $Rc>0$, then
$$
AVR(g(t)):=\lim_{r\to \infty}\frac{B_{g(t)}(o,r)}{r^n}>0,
$$
where the definition of $AVR(g(t))$ is independent of the base
point $o\in M$.
\end{Pro}
The second is
\begin{Pro}\label{pro:2} (\cite{MD}). If $(M,g(t))$, $t>0$, is a complete non-compact expanding gradient
Ricci soliton with (\ref{pinch}), then the scalar curvature is
quadratic exponential decay.
\end{Pro}
The third one is Theorem 1.1 in \cite{B}. Roughly speaking, the
result says that for the complete and non-compact $(M,g)$, if
$AVR(g)>0$ and the curvature decay suitably, then it is an
asymptotic manifold. We invite the readers to papers \cite{B} and
\cite{LP} for the definitions of asymptotic flat manifold
$M_{\tau}$ and coordinates $(z)=(z^i)$ at infinity (also called
the asymptotic coordinates).
The last one is Proposition 10.2 in \cite{LP}. Namely,
\begin{Pro}\label{pro:3} (\cite{LP}).If $(M,g)$ is asymptotic flat
with $g\in \mathbf{M_{\tau}}$, for some $\tau>\frac{n-2}{2}$, and
the Ricci curvature is non-negative. Then the mass
$$
m(g):=\lim_{r\to\infty}\int_{S_r}\mu\lrcorner dz
$$
is non-negative, with $m(g)=0$
if and only if $(M,g)$ is isometric to $R^n$ with its Euclidean
metric. Here $S_r:\{x\in M; d_g(o,x)=1\}$, $$
\mu=(\partial_ig_{ij}-\partial_jg_{ii})\partial_j, \ \
\partial_j=\frac{\partial}{\partial z^j},
$$
and $(z^j)$ are the asymptotic coordinates.
\end{Pro}
\section{Proof of Theorem \ref{thm:1}}\label{sect3}
We argue by contradiction. So we assume that $(M,g(t))$ is not
flat.
Using the strong maximum principle \cite{Shi89a} to the Ricci
soliton, we may assume that the scalar curvature is positive,
i.e., $R>0$. Hence we know that $Rc>0$. According to the arguments
in \cite{Cho06} and \cite{MD}, we know that $\phi$ is a proper
strict convex function, which implies by using the Morse theory
that $M^n$ is diffeomorphic to $R$. Using Proposition \ref{pro:1},
Proposition \ref{pro:2}, and Theorem 1.1 in \cite{B} we know that
$(M,g(t))$ is an asymptotic flat manifold. We also know that
$$
\phi(x)\approx d_g(x,o)^2, \ \ |\nabla \phi(x)|\approx d_g(x,o).
$$
Recall that in coordinates $(z^j)$ at infinity, we have the Ricci
soliton equation
$$
\phi_{ij}=R_{ij}+\frac{1}{2t}g_{ij}.
$$
For notation simple, we let $t=1/2$. Then we have
$g_{ij}=\phi_{ij}-R_{ij}$. By Ricci pinching condition we know
that $R_{ij}$ decay exponentially and $$\Delta
\phi=\phi_{ii}=R+n.$$ Recall the Ricci formula
$$
\phi_{iji}=\phi_{iij}+R_{ji}\phi_i.
$$
This also implies that
$$
\nabla R=-2Rc(\nabla \phi).
$$
Hence $|\nabla R|$ decays in the exponent rate.
We now
compute the mass. Using the Ricci formula, we have
\begin{eqnarray*}
m(g)&=&\lim_{r\to\infty}\int_{S_r}(\partial_ig_{ij}-\partial_jg_{ii})\partial_j\lrcorner
dz
\\
&=&\lim_{r\to\infty}\int_{S_r}(\partial_i\phi_{ij}-\partial_j\phi_{ii})\partial_j\lrcorner
dz\\
&=&\lim_{r\to\infty}\int_{S_r}\phi_{iji}\partial_j\lrcorner dz\\
&=&\lim_{r\to\infty}\int_{S_r}(\phi_{iij}+R_{ij}\phi_i)\partial_j\lrcorner dz\\
&=&\lim_{r\to\infty}\int_{S_r}R_j\partial_j\lrcorner dz\\
&=&0.
\end{eqnarray*}
Using Proposition \ref{pro:3} we know that $(M,g(1/2))$ is $R^n$
with the Euclidean metric. A contradiction.
This completes the
proof of Theorem \ref{thm:1}. q.e.d.
\end{document} | math | 6,485 |
\begin{document}
\title{{Bounds for the first several prime character nonresidues}
\begin{abstract}\noindent Let $\varepsilon > 0$. We prove that there are constants $m_0=m_0(\varepsilon)$ and $\kappa=\kappa(\varepsilon) > 0$ for which the following holds: For every integer $m > m_0$ and every nontrivial Dirichlet character modulo $m$, there are more than $m^{\kappa}$ primes $\ell \le m^{\frac{1}{4\sqrt{e}}+\varepsilon}$ with $\chi(\ell)\notin \{0,1\}$. The proof uses the fundamental lemma of the sieve, Norton's refinement of the Burgess bounds, and a result of Tenenbaum on the distribution of smooth numbers satisfying a coprimality condition. For quadratic characters, we demonstrate a somewhat weaker lower bound on the number of primes $\ell \le m^{\frac14+\epsilon}$ with $\chi(\ell)=1$.
\end{abstract}
\section{Introduction}
Let $\chi$ be a nonprincipal Dirichlet character. An integer $n$ is called a $\chi$-\emph{nonresidue} if $\chi(n) \notin \{0,1\}$. Problems about character nonresidues go back to the beginnings of modern number theory. Indeed, one can read out of Gauss's \emph{Disquisitiones} that for primes $p\equiv 1\pmod{8}$ and $\chi(\cdot) = \leg{p}{\cdot}$, the smallest $\chi$-nonresidue does not exceed $2\sqrt{p}+1$ \cite[Article 129]{gauss86}. This was an auxiliary result required for Gauss's first proof of the quadratic reciprocity law.
In the early 20th century, I.\,M. Vinogradov initiated the study of how the quadratic residues and nonresidues modulo a prime $p$ are distributed in the interval $[1,p-1]$. A particularly natural problem is to estimate the size of $n_p$, the smallest quadratic nonresidue modulo $p$. Vinogradov conjectured that $n_p \ll_{\varepsilon} p^{\varepsilon}$, for each $\varepsilon >0$. By means of a novel estimate for character sums (independently discovered by P\'olya), coupled with a clever sieving argument, he showed \cite{vinogradov18} that $n_p \ll_{\varepsilon} p^{\frac{1}{2\sqrt{e}} + \varepsilon}$. Burgess's character sum bounds \cite{burgess57}, in conjunction with Vinogradov's methods, yield the sharper estimate
\begin{equation}\label{eq:burgessquadratic} n_p \ll_{\varepsilon} p^{\frac{1}{4\sqrt{e}}+\varepsilon}. \end{equation}
Fifty years of subsequent research has not led to any improvement in the exponent $\frac{1}{4\sqrt{e}}$. But generalizing \eqref{eq:burgessquadratic}, Norton showed that if $\chi$ is any nontrivial character modulo $m$, then the least $\chi$-nonresidue is $O_{\varepsilon}(m^{1/4\sqrt{e} + \varepsilon})$. See \cite[Theorem 1.30]{norton98}.
Since $\chi$ is completely multiplicative, the smallest $\chi$-nonresidue is necessarily prime. In this note, we prove that there are actually many prime $\chi$-nonresidues satisfying the Burgess--Norton upper bound.
\begin{thm}\label{thm:main} For each $\varepsilon > 0$, there are numbers $m_0(\varepsilon)$ and $\kappa=\kappa(\varepsilon)> 0$ for which the following holds: For all $m > m_0$ and each nontrivial character $\chi$ mod $m$, there are more than $m^{\kappa}$ prime $\chi$-nonresidues not exceeding $m^{\frac{1}{4\sqrt{e}}+\varepsilon}$.
\end{thm}
The problem of obtaining an upper bound on the first several prime character nonresidues was considered already by Vinogradov. In \cite{vinogradov18}, he showed that for large $p$, there are at least $\frac{\log{p}}{7\log\log{p}}$ prime quadratic nonresidues modulo $p$ not exceeding \[ p^{\frac{1}{2}-\frac{1}{\log\log{p}}}. \]
For characters to prime moduli, a result resembling Theorem \ref{thm:main} was proved by Hudson in 1983 \cite{hudson83}. (See also Hudson's earlier investigations \cite{hudson73,hudson74,hudson74A}.) But even restricted to prime $m$, Theorem \ref{thm:main} improves on \cite{hudson83} in multiple respects. In \cite{hudson83}, the exponent on $p$ is $\frac{1}{4}+\varepsilon$ instead of $\frac{1}{4\sqrt{e}} +\varepsilon$, and the number of nonresidues produced is only $c_{\varepsilon} \frac{\log{p}}{\log\log{p}}$. Moreover, it is assumed in \cite{hudson83} that the order of $\chi$ is fixed. Stronger results than those of \cite{hudson83} were announced by Norton already in 1973 \cite{norton74}.\footnote{Norton claims in \cite{norton74}: \emph{Let $\varepsilon>0$ and $k_0 \ge 2$. If $m \ge 3$ and $[(\mathbf{Z}/m\mathbf{Z})^{\times}: {(\mathbf{Z}/m\mathbf{Z})^{\times}}^k] \ge k_0$, then each of the smallest $\lfloor \log{m}/\log\log{m}\rfloor$ primes not dividing $m$ that are $k$th power nonresidues modulo $m$ is $\ll_{\varepsilon,k_0}n^{1/4u_{k_0} + \varepsilon}$}. Here $u_{k_0}$ has the same meaning as in our introduction.} Unfortunately, a full account of Norton's work seems to have never appeared.
It becomes easier to produce small character nonresidues as the order of $\chi$ increases. This phenomenon was noticed by Vinogradov \cite{vinogradov27} and further investigated by Buchstab \cite{buchstab49} and Davenport and Erd\H{o}s \cite{DE52}. To explain their results requires us to first recall the rudiments of the theory of smooth numbers. For each positive integer $n$, let $P^{+}(n)$ denote the largest prime factor of $n$, with the convention that $P^{+}(1)=1$. A natural number $n$ is called \emph{$y$-smooth} (or \emph{$y$-friable}) if $P^{+}(n) \le y$. For $x \ge y \ge 2$, we let $\Psi(x,y)$ be the count of $y$-smooth numbers up to $x$. We let $\rho$ be Dickman's function, defined by
\[ \rho(u)=1\text{ for $0 \le u \le 1$}, \quad \text{and}\quad u \rho'(u) = -\rho(u-1) \quad\text{for $u > 1$}. \]
The functions $\Psi(x,y)$ and $\rho(u)$ are intimately connected; it is known that $\Psi(x,y) \sim x\rho(u)$, where $u:=\frac{\log{x}}{\log{y}}$, in a wide range of $x$ and $y$. In fact, Hildebrand \cite{hildebrand86} has shown that this asymptotic formula holds whenever $x\to\infty$, as long as
\[ y \ge \exp((\log\log{x})^{5/3+\lambda}) \]
for some fixed positive $\lambda$. For this estimate to be useful, one needs to understand the behavior of $\rho(u)$. It is not hard to show that $\rho$ is strictly decreasing for $u > 1$ and that $\rho(u) \le 1/\Gamma(u+1)$. So for any $k > 1$, there is a unique $u_k > 1$ with $\rho(u_k)=\frac{1}{k}$. Buchstab and, independently, Davenport and Erd\H{o}s (developing ideas implicit in \cite{vinogradov27}) showed that if $\chi$ mod $p$ has order $k \ge 2$, then the least $\chi$-nonresidue is $O_{\varepsilon,k}(p^{1/2u_k+\varepsilon})$. If in their argument Burgess's method (which was not available at the time) is used in place of the P\'olya--Vinogradov inequality, then $1/2u_k$ may be replaced by $1/4u_k$ \cite{wy64}. We prove the following:
\begin{thm}\label{thm:fixedprime} Let $\varepsilon >0$ and $k_0 \ge 2$. There are numbers $m_0(\varepsilon,k_0)$ and $\kappa = \kappa(\varepsilon,k_0) > 0$ for which the following holds: For all $m > m_0$ and each nontrivial character $\chi$ mod $m$ of order $k \ge k_0$, there are more than $m^{\kappa}$ prime $\chi$-nonresidues not exceeding $m^{\frac{1}{4u_{k_0}}+\varepsilon}$.\end{thm}
\begin{rmk}\mbox{ }
\begin{itemize}
\item It follows readily from the definition that $\rho(u) = 1-\log{u}$ for $1 \le u \le 2$, and so $u_2 = e^{1/2} = 1.6487\ldots$ and $u_3 = e^{2/3} = 1.9477\ldots$. For $k > 3$, it does not seem that $u_k$ has a simple closed form expression.
\item Theorem \ref{thm:main} is the special case $k_0=2$ of Theorem \ref{thm:fixedprime}.
\end{itemize}
\end{rmk}
One might compare Theorem \ref{thm:main} for the quadratic character modulo a prime $p$ with a result of Banks--Garaev--Heath-Brown--Shparlinski \cite{BGHBS08}. They show that for each fixed $\varepsilon > 0$, and each $N \ge p^{1/4\sqrt{e}+\varepsilon}$, the proportion of quadratic nonresidues modulo $p$ in $[1,N]$ is $\gg_{\varepsilon} 1$ for all primes $p > p_0(\varepsilon)$.
Our arguments use the ideas of Vinogradov and Davenport--Erd\H{o}s but take advantage of modern developments in sieve methods and the theory of smooth numbers. A variant of the Burgess bounds developed by Norton also plays an important role. We note that an application of the sieve that is similar in spirit to ours appears in work of Bourgain and Lindenstrauss \cite[Theorem 5.1]{BL03}.\footnote{A special case of their result: \emph{Given $\varepsilon >0$, there is an $\alpha>0$ such that $\sum_{\substack{p^{\alpha} \le \ell \le p^{1/4+\varepsilon} \\ \leg{\ell}{p}=-1}}\frac{1}{\ell} > \frac{1}{2}-\varepsilon$, for all $p > p_0(\varepsilon)$.}}
It is equally natural to ask for small prime character \emph{residues}, i.e., primes $\ell$ with $\chi(\ell)=1$. The most significant unconditional result in this direction is due to Linnik and A.\,I. Vinogradov \cite{VL66}. They showed that if $\chi$ is the quadratic character modulo a prime $p$, then the smallest prime $\ell$ with $\chi(\ell)=1$ satisfies $\ell \ll_{\varepsilon} p^{1/4+\varepsilon}$. More generally, Elliott \cite{elliott71} proved that when $\chi$ has order $k$, the least such $\ell$ is $O_{k,\varepsilon}(p^{\frac{k-1}{4}+\epsilon})$. As Elliott notes, this bound is only interesting for small values of $k$; otherwise, it is inferior to what follows from known forms of Linnik's theorem on primes in progressions. For extensions of the Linnik--Vinogradov method in a different direction, see \cite{pollack14B, pollack14}.
Our final result is a partial analogue of Theorem \ref{thm:main} for prime residues of quadratic characters. Regrettably, the number of primes produced falls short of a fixed power of $m$.
\begin{thm}\label{thm:smallresidue} Let $\varepsilon > 0$ and let $A >0$. There is an $m_0=m_0(\varepsilon,A)$ with the following property: If $m > m_0$, and $\chi$ is a quadratic character modulo $m$, then there are at least $(\log{m})^{A}$ primes $\ell \le m^{\frac{1}{4}+\varepsilon}$ with $\chi(\ell)=1$.
\end{thm}
Results of the sort proven here have direct consequences for prime splitting in cyclic extensions of $\mathbf{Q}$. For example, Theorem \ref{thm:main} (respectively Theorem \ref{thm:smallresidue}) implies that there are more than $|\Delta|^{\kappa}$ inert (respectively, more than $(\log|\Delta|)^{A}$ split) primes $p \le |\Delta|^{\frac{1}{4\sqrt{e}}+\varepsilon}$ (respectively, $p \le |\Delta|^{\frac{1}{4}+\varepsilon}$) in the quadratic field of discriminant $\Delta$, as soon as $|\Delta|$ is large enough in terms of $\varepsilon$ (and $A$).
\section{Small prime nonresidues: Proofs of Theorems \ref{thm:main} and \ref{thm:fixedprime}}
\subsection{Preparation} As might be expected, the Burgess bounds play the key role in our analysis. The following version is due to Norton (see \cite[Theorem 1.6]{norton98}).
\begin{prop}\label{prop:norton} Let $\chi$ be a nontrivial character modulo $m$ of order dividing $k$. Let $r$ be a positive integer,
and let $\epsilon > 0$. For all $x > 0$,
\[ \sum_{n \le x} \chi(n) \ll_{\epsilon,r} R_k(m)^{1/r} x^{1-\frac{1}{r}} m^{\frac{r+1}{4r^2}+\epsilon}. \]
Here
\[ R_k(m) = \min\left\{M(m)^{3/4},Q(k)^{9/8}\right\}, \]
where
\[ M(m) = \prod_{p^e \parallel m,~e\ge 3} p^e \qquad\text{and}\quad Q(k) = \prod_{p^e \parallel k,~e\ge 2} p^e. \]
The factor of $R_k(m)^{1/r}$ can be omitted if $r \le 3$.
\end{prop}
Another crucial tool is a theorem of Tenenbaum concerning the distribution of smooth numbers satisfying a coprimality condition. For $x\ge y\ge 2$, let \[ \Psi_q(x,y) = \#\{n\le x: \gcd(n,q)=1, P^{+}(q) \le y\}. \]
\begin{prop}\label{prop:FT} For positive integers $q$ and real numbers $x, y$ satisfying
\[ P^{+}(q) \le y \le x \quad\text{and}\quad \omega(q) \le y^{1/\log(1+u)}, \]
we have
\[ \Psi_q(x,y) = \frac{\varphi(q)}{q} \Psi(x,y) \left(1+O\left(\frac{\log(1+u) \log(1+\omega(q))}{\log{y}}\right)\right). \]
As before, $u$ denotes the ratio $\log{x}/\log{y}$.
\end{prop}
\begin{proof} This is the main result of \cite{tenenbaum93} in the case $A=1$.\end{proof}
\begin{remark} If $q'$ is the largest divisor of $q$ supported on the primes not exceeding $y$, then $\Psi_{q}(x,y) = \Psi_{q'}(x,y)$. So the assumption in Proposition \ref{prop:FT} that $P^{+}(q) \le y$ does not entail any loss of generality.
\end{remark}
Theorem \ref{thm:fixedprime} will be deduced from two variant results claiming weaker upper bounds.
\begin{thm}\label{thm:fixedprime0} Let $\varepsilon >0$ and $k_0 \ge 2$. There are numbers $m_0(\varepsilon,k_0)$ and $\kappa = \kappa(\varepsilon,k_0) > 0$ for which the following holds: For all $m > m_0$ and each nontrivial character $\chi$ mod $m$ of order $k \ge k_0$, there are more than $m^{\kappa}$ prime $\chi$-nonresidues not exceeding $m^{\frac{1}{3u_{k_0}}+\varepsilon}$.\end{thm}
\begin{thm}\label{thm:fixedprime1} Let $\varepsilon >0$ and $k_0 \ge 2$. There are numbers $m_0(\varepsilon,k_0)$ and $\kappa = \kappa(\varepsilon,k_0) > 0$ for which the following holds: For all $m > m_0$ and each nontrivial character $\chi$ mod $m$ of order $k \ge k_0$, there are more than $m^{\kappa}$ prime $\chi$-nonresidues not exceeding $R_k(m) m^{\frac{1}{4u_{k_0}}+\varepsilon}$. Here $R_k(m)$ is as defined in Proposition \ref{prop:norton}.
\end{thm}
The proof of Theorem \ref{thm:fixedprime1} is given in detail in the next section. We include only a brief remark about the proof of Theorem \ref{thm:fixedprime0}, which is almost entirely analogous (but slightly simpler). We then present the derivation of Theorem \ref{thm:fixedprime} from Theorems \ref{thm:fixedprime0} and \ref{thm:fixedprime1}. We remind the reader that Theorem \ref{thm:main} is the special case $k_0=2$ of Theorem \ref{thm:fixedprime}.
\subsection{Proof of Theorem \ref{thm:fixedprime1}}\label{sec:proofs} We let $\chi$ be a nontrivial character modulo $m$ of order $k \ge k_0$, where $k_0 \ge 2$ is fixed. With $\delta \in (0,\frac{1}{4})$, we set
\[ x= R_k(m) \cdot m^{\frac14 + \delta}, \quad y = x^{\frac{1}{u_{k_0}} + \delta}. \]
To prove Theorem \ref{thm:fixedprime1}, it suffices to show that for all large $m$ (depending only on $k_0$ and $\delta$), there are at least $x^{\kappa}$ prime $\chi$-nonresidues in $[1,y]$ for a certain constant $\kappa = \kappa(k_0,\delta) > 0$.
Let $q$ be the product of the prime $\chi$-nonresidues in $[1,y]$. Note that $\gcd(q,m)=1$, from the definition of a $\chi$-nonresidue.
Our strategy is to estimate
\begin{equation}\label{eq:different} \sum_{\substack{n \le x \\ \gcd(n,mq)=1}} (1+\chi(n) + \chi^2(n) + \dots + \chi^{k-1}(n)) \end{equation}
in two different ways.
We first derive a lower bound on \eqref{eq:different}, under the assumption that there are not so many prime $\chi$-nonresidues in $[1,y]$.
\begin{lem}\label{lem:lower} There are constants $\eta = \eta(\delta,k_0) > 0$, $\kappa =\kappa(\delta,k_0) > 0$, and $m_0 = m_0(\delta,k_0)$ with the following property: If $m > m_0$ and $\omega(q) \le x^{\kappa}$, then
\[ \sum_{\substack{n \le x \\ \gcd(n,mq)=1}} (1+\chi(n) + \dots +\chi(n)^{k-1}) \ge \left(1+\frac{2k}{3}\eta\right) \frac{\varphi(mq)}{mq} x. \]
\end{lem}
\begin{proof} Observe that
\[ \sum_{\substack{n \le x \\ \gcd(n,mq)=1}} (1+\chi(n) + \dots + \chi(n)^{k-1}) = k \sum_{\substack{n \le x \\ \gcd(n,q)=1,~\chi(n)=1}} 1 \ge k\sum_{\substack{n \le x\\ \gcd(n,mq)=1 \\ p \mid n \Rightarrow p \le y}} 1 = k \cdot \Psi_{mq}(x,y).\]
We estimate $\Psi_{mq}(x,y)$ using Proposition \ref{prop:FT} and the succeeding remark. We have $u \asymp_{k_0} 1$, or equivalently, $\log y\asymp_{k_0} \log{x}$. So if $\kappa$ is sufficiently small in terms of $k_0$, and $\omega(q) \le x^{\kappa}$, Proposition \ref{prop:FT} gives
\begin{align*} \Psi_{mq}(x,y) &= \bigg(\Psi(x,y)\prod_{\substack{p \mid mq \\ p \le y}}\left(1-\frac{1}{p}\right)\bigg) \left(1+O_{k_0}\left(\frac{\log(1+x^{\kappa})}{\log{x}}\right)\right)\\
&\ge \Psi(x,y) \frac{\varphi(mq)}{mq} \left(1+O_{k_0}\left(\frac{\log(1+x^{\kappa})}{\log{x}}\right)\right).\end{align*}
Now the result of Hildebrand quoted in the introduction (or a much more elementary theorem) shows that $\Psi(x,y) = \Psi(x, x^{\frac{1}{u_{k_0}}+\delta}) \ge (\frac{1}{k_0}+\eta) x$ for a certain $\eta= \eta(k_0,\delta) > 0$ and all large $x$. So if $\kappa$ is fixed sufficiently small, depending on $k_0$ and $\delta$, and $x$ is sufficiently large,
\[ \Psi_{mq}(x,y) > \left(\frac{1}{k_0} + \frac{2}{3}\eta\right) \frac{\varphi(mq)}{mq} x. \]
Hence,
\[ \sum_{\substack{n \le x \\ \gcd(n,q)=1}} (1+\chi(n) + \dots + \chi(n)^{k-1}) \ge \left(\frac{k}{k_0} + \frac{2k}{3}\eta\right)\frac{\varphi(mq)}{mq} x \ge \left(1+\frac{2k}{3}\eta\right) \frac{\varphi(mq)}{mq} x. \qedhere\]
\end{proof}
We turn next to an upper bound.
\begin{lem}\label{lem:upper} Let $\beta > 0$. There are numbers $\eta' = \eta'(\delta) > 0$, $\kappa' = \kappa'(\delta,\beta)>0$ and $m_0 = m_0(\delta,\beta)$ with the following property: If $m > m_0$ and $\omega(q) \le x^{\kappa'}$, then
\[ \sum_{\substack{n \le x \\ \gcd(n,mq)=1}} (1+\chi(n) + \chi(n)^2 + \dots + \chi(n)^{k-1}) \le (1+\beta) \frac{\varphi(mq)}{mq}x +O_{\delta}(k x^{1-\eta'}). \]
\end{lem}
\begin{proof} We let $\mathcal{A} = \{n \le x: \gcd(n,m)=1,~\chi(n)=1\}$ and observe that
\begin{equation}\label{eq:fundidentity} \sum_{\substack{n \le x \\ \gcd(n,mq)=1}} (1+\chi(n) + \chi(n)^2 + \dots + \chi(n)^{k-1}) = k \sum_{\substack{n \in \mathcal{A} \\ \gcd(n,q)=1}} 1. \end{equation}
We apply the fundamental lemma of the sieve to estimate the right-hand sum. (The precise form of the fundamental lemma is not so important, but we have in mind \cite[Theorem 4.1, p. 29]{diamond08}.) Let $d \in [1,x]$ be a squarefree integer dividing $q$. Then
\begin{align*} \sum_{\substack{n \in \mathcal{A} \\ d \mid n}} 1 &= \frac{1}{k} \sum_{\substack{n \le x \\\gcd(n,m)=1,~d\mid n}} (1+\chi(n) + \dots + \chi(n)^{k-1}).
\end{align*}
For each $j=0,1,2,\dots, k-1$,
\[ \sum_{\substack{n \le x \\ \gcd(n,m)=1,~d\mid n}}\chi^j(n) = \chi^j(d) \sum_{\substack{e \le x/d \\ \gcd(e,m)=1}} \chi^j(e). \]
When $j=0$, the right-hand side is $\frac{x}{d}\frac{\varphi(m)}{m} +O_{\epsilon}(m^{\epsilon})$, by a straightforward inclusion-exclusion. For $j \in \{1,2,\dots, k-1\}$, Proposition \ref{prop:norton} gives
\begin{align*} \sum_{\substack{e \le x/d \\ \gcd(e,m)=1}} \chi^j(e) = \sum_{e \le x/d} \chi^j(e) \sum_{\substack{f \mid e \\ f\mid m}} \mu(f) &= \sum_{f \mid m} \mu(f) \chi^j(f) \sum_{g \le x/df} \chi^j(g) \\
&\ll_{\epsilon,r} R_k(m)^{1/r} x^{1-\frac1r} d^{-1+\frac{1}{r}} m^{\frac{r+1}{4r^2}+\epsilon} \sum_{f \mid m} f^{-1+\frac{1}{r}} \\&\ll_{\epsilon} R_k(m)^{1/r} x^{1-\frac1r} d^{-1+\frac{1}{r}} m^{\frac{r+1}{4r^2}+2\epsilon};
\end{align*}
here $r \ge 2$ and $\epsilon > 0$ are parameters to be chosen. (We used in the last step that the sum on $f$ has only $O_{\epsilon}(m^{\epsilon})$ terms, each of which is $O(1)$.) Assembling the preceding estimates,
\[ \sum_{\substack{n \in \mathcal{A} \\ d \mid n}} 1 = \frac{x}{dk}\frac{\varphi(m)}{m} + r(d),
\quad\text{where}\quad r(d) \ll_{\epsilon,r} R_k(m)^{1/r} x^{1-\frac1r} d^{-1+\frac{1}{r}} m^{\frac{r+1}{4r^2}+2\epsilon}. \]
By the fundamental lemma, for any choices of real parameters $z\ge 2$ and $v\ge 1$ with $z^{2v} < x$,
\begin{multline*} \sum_{\substack{n \in \mathcal{A}\\\gcd(n,q)=1}} 1 \le \sum_{\substack{n \in \mathcal{A} \\ p \mid \gcd(n,q) \Rightarrow p \ge z}} 1 = \Bigg(\frac{x}{k}\frac{\varphi(m)}{m} \prod_{\substack{p \mid q \\ p< z}}\left(1-\frac{1}{p}\right)\Bigg)\left(1 + O(v^{-v})\right) \\ + O_{\epsilon,r}\Bigg(R_k(m)^{1/r} x^{1-\frac1r} m^{\frac{r+1}{4r^2}+2\epsilon} \sum_{\substack{d < z^{2v} \\ d\mid q}} \mu^2(d) 3^{\omega(d)} d^{-1+\frac1r}\Bigg).\end{multline*}
We now make a choice of parameters. Let $r = \lceil \frac{1}{2\delta}\rceil$ (so that $\delta \ge \frac{1}{2r}$). Since $x=R_k(m)\cdot m^{1/4+\delta}$, we have
\[ R_k(m)^{1/r} x^{1-\frac1r} m^{\frac{r+1}{4r^2}} = x \cdot m^{-\frac{1}{4r} -\delta/r} m^{\frac{r+1}{4r^2}} = x \cdot m^{\frac{1}{r}(\frac{1}{4r}-\delta)} \le x\cdot m^{-\frac{\delta}{4r^2}}. \]
We take $\epsilon = \frac{\delta}{16r^2}$, so that
\[ m^{2\epsilon} = m^{\frac{\delta}{8r^2}}. \]
Since $r\ge 2$ and $3^{\omega(d)} \ll d^{1/2}$, each term in the sum on $d$ is $O(1)$. Putting it all together, the $O$-term above is
\[ \ll_{\delta} x \cdot m^{-\frac{\delta}{4r^2}} \cdot m^{\frac{\delta}{8r^2}} \cdot z^{2v}. \]
Since $x = R_k(m) \cdot m^{1/4+\delta} \le m^{3/4} \cdot m^{1/4+\delta} < m^2$, this upper bound is $\ll_{\delta} x^{1-\frac{\delta}{16r^2}} z^{2v}$. Taking $z = x^{\frac{\delta}{64r^2 v}}$ gives a final upper bound on the $O$-term of
\[ \ll_{\delta} x^{1-\eta'},\quad\text{where}\quad \eta' = \frac{\delta}{32r^2}. \]
Turning attention to the main term, we fix $v$ large enough that the factor $1+O(v^{-v})$ is smaller than $1+\frac{1}{2}\beta$. Then our main term above does not exceed
\begin{align*} \frac{x}{k} \frac{\varphi(mq)}{mq} \left(1+\frac{1}{2}\beta\right) \prod_{\substack{p \mid q\\ p \ge z}} \left(1-\frac{1}{p}\right)^{-1} &\le \frac{x}{k} \frac{\varphi(mq)}{mq} \left(1+\frac{1}{2}\beta\right) \exp\bigg(2\sum_{\substack{p \mid q \\ p \ge z}}\frac{1}{p}\bigg) \\&\le \frac{x}{k} \frac{\varphi(mq)}{mq} \left(1+\frac{1}{2}\beta\right) \exp(2\omega(q) z^{-1}). \end{align*}
Take $\kappa' = \frac{\delta}{128r^2 v}$. Under the assumption that $\omega(q) \le x^{\kappa'}$, we have $2 \omega(q) z^{-1} \le 2x^{-\delta/128r^2 v}$, and $\exp(2\omega(q) z^{-1}) = 1 + O(x^{-\delta/128r^2 v})$. So once $x$ (or equivalently, $m$) is large enough, our main term is smaller than $\frac{x}{k}\frac{\varphi(mq)}{mq}(1+\beta)$. So we have shown that for large $m$,
\[ \sum_{\substack{n \in \mathcal{A} \\ \gcd(n,q)=1}} 1 \le \frac{x}{k}\frac{\varphi(mq)}{mq}(1+\beta) + O_{\delta}(x^{1-\eta'}). \]
Recalling \eqref{eq:fundidentity} finishes the proof.
\end{proof}
\begin{proof}[Completion of the proof of Theorem \ref{thm:fixedprime}] We keep the notation from earlier in this section. Let $\eta$, $\kappa$ be as specified in Lemma \ref{lem:lower}. With $\beta = \eta/2$, choose $\eta'$ and $\kappa'$ as in Lemma \ref{lem:upper}. If $m$ is large and we assume that
\[ \omega(q) \le x^{\kappa''}, \quad\text{where} \quad\kappa'' = \min\{\kappa,\kappa'\}, \]
then these lemmas imply that
\[ \bigg(1+\frac{2k}{3}\eta\bigg) \frac{\varphi(mq)}{mq} x \le \bigg(1+\frac{1}{2}\eta\bigg) \frac{\varphi(mq)}{mq} x + O_{\delta}(kx^{1-\eta'}). \]
Rearranging,
\[ k \eta \frac{\varphi(mq)}{mq} x \ll \frac{4k-3}{6} \eta \cdot \frac{\varphi(mq)}{mq}x \ll_{\delta} k x^{1-\eta'}, \]
and so
\[ \frac{mq}{\varphi(mq)} \gg_{k_0,\delta} x^{\eta'}. \]
Noting that $m < x^{4}$ and $q \le y^{\omega(q)}\le x^{\omega(q)}$, we see that for large $x$,
\[ \frac{mq}{\varphi(mq)} \ll \log\log(mq+2) \ll \log\log{x} + \log(\omega(q)+2) \ll \log{x}. \]
Comparing with the above lower bound, we see that $x$, and hence $m$, is bounded. Turning it around, for $m$ large enough, there are at least $x^{\kappa''}$ prime $\chi$-nonresidues in $[1,y]$.
\end{proof}
\begin{proof}[Sketch of the proof of Theorem \ref{thm:fixedprime0}] The proof of Theorem \ref{thm:fixedprime0} is quite similar, except that now we take $x = m^{1/3+\delta}$. With this choice of $x$, we can apply the Burgess bounds with $r=3$, which allows us to omit the factor of $R_k(m)$ in the resulting estimates.\end{proof}
\subsection{Deduction of Theorem \ref{thm:fixedprime}} Let $\varepsilon> 0$ and $k_0 \ge 2$ be fixed. Let $\chi$ be a nonprincipal character mod $m$ of order $k$, where $k \ge k_0$. We would like to show that as long as $m$ is large enough there must be at least $m^{\kappa}$ prime $\chi$-nonresidues not exceeding $x^{1/4u_{k_0}+\varepsilon}$, for a certain $\kappa = \kappa(\varepsilon,k_0) > 0$. Let $k_1$ be the smallest positive integer with $3u_{k_1} > 4u_{k_0}$. If $k \ge k_1$, apply Theorem \ref{thm:fixedprime0}: We find that for large $m$, there are at least $m^{\kappa_0}$ prime $\chi$-nonresidues
\[ \le m^{\frac{1}{3u_{k_1}} + \varepsilon} \le m^{\frac{1}{4u_{k_0}} + \varepsilon}, \]
where $\kappa_0 = \kappa(\varepsilon,k_1)$ in the notation of Theorem \ref{thm:fixedprime0}. Suppose instead that $k_0 \le k< k_1$. Then $R_k(m)$ is bounded in terms of $k_0$. Theorem \ref{thm:fixedprime1} thus shows that for large $m$, there are at least $m^{\kappa_1}$ prime $\chi$-nonresidues
\[ \le R_k(m) m^{\frac{1}{4u_{k_0}} + \varepsilon/2} \le m^{\frac{1}{4u_{k_0}}+\varepsilon}, \]
where $\kappa_1 = \kappa(\varepsilon/2,k_0)$ in the notation of Theorem \ref{thm:fixedprime1}. Theorem \ref{thm:fixedprime} follows with $\kappa = \min\{\kappa_0,\kappa_1\}$.
\begin{remark} By a minor modification of our proof, one can establish the following more general result. Theorem \ref{thm:fixedprime} corresponds to the case $H = \ker\chi$.
\begin{thm} Let $\varepsilon >0$ and $k_0 \ge 2$. There are numbers $m_0(\varepsilon,k_0)$ and $\kappa = \kappa(\varepsilon,k_0) > 0$ for which the following holds: For all $m > m_0$ and every proper subgroup $H$ of $G=(\mathbf{Z}/m\mathbf{Z})^{\times}$ of index $k \ge k_0$, there are more than $m^{\kappa}$ primes $\ell$ not exceeding $m^{\frac{1}{4u_{k_0}}+\varepsilon}$ with $\ell \nmid m$ and $\ell \bmod{m}\notin H$.\end{thm}
\noindent This strengthens \cite[Theorem 1.20]{norton98}, where the bound $O_{k_0,\epsilon}(m^{\frac{1}{4u_{k_0}}+\varepsilon})$ is established for the first such prime $\ell$.
The main idea in the proof of the generalization is to replace $1+\chi(n)+\dots + \chi(n)^{k-1}$ with $\sum_{\chi\in \widehat{G/H}} \chi(n)$, where $\widehat{G/H}$ denotes the group of characters $\chi$ mod $m$ with $\ker \chi \supset H$. We leave the remaining details to the reader.
\end{remark}
\section{Small prime residues of quadratic characters: Proof of Theorem \ref{thm:smallresidue}}
The next proposition is a variant of \cite[Theorem 2]{VL66}. Given a character $\chi$, we let $r_{\chi}(n) = \sum_{d \mid n} \chi(d)$. Since $\chi$ will be clear from context, we will suppress the subscript.
\begin{prop}\label{prop:LV} For each $\epsilon > 0$, there is a constant $\eta = \eta(\epsilon) >0$ for which the following holds: If $\chi$ is a quadratic character modulo $m$ and $x \ge m^{1/4+\epsilon}$, then
\[ \sum_{n \le x} r(n) = L(1,\chi) x + O_{\epsilon}(x^{1-\eta}). \] \end{prop}
\begin{proof} With $\upsilon = \frac{1/4+\epsilon/2}{1/4+\epsilon}$, put $y = x^{\upsilon}$, so that $y \ge m^{\frac{1}{4}+\frac{1}{2}\epsilon}$. Put $z=x/y$. By Dirichlet's hyperbola method,
\begin{equation}\label{eq:hyperbola} \sum_{n \le x} r(n) = \sum_{d \le y}\chi(d) \sum_{e \le x/d} 1 + \sum_{e \le z} \sum_{d \le x/e} \chi(d) - \sum_{d\le y}\chi(d)\sum_{e\le z}1. \end{equation}
By Proposition \ref{prop:norton} (with $k=2$, so that $R_k(m)^{1/r}=1$), there is an $\eta_0 =\eta_0(\epsilon) > 0$ with $\sum_{d \le T} \chi(d) \ll_{\epsilon} T^{1-\eta_0} \quad\text{for all}\quad T \ge y$. Thus, the second double sum on the right of \eqref{eq:hyperbola} is $\ll_{\delta} x^{1-\eta_0} \sum_{e \le z} e^{\eta_0-1} \ll_{\delta} x (z/x)^{\eta_0} = x y^{-\eta_0}$. Similarly, the third double sum is $\ll_{\epsilon} z y^{1-\eta_0} = x y^{-\eta_0}$. Finally,
\[ \sum_{d \le y}\chi(d) \sum_{e \le x/d} 1=\sum_{d\le y} \chi(d) \left(\frac{x}{d}+O(1)\right) = xL(1,\chi) - x\sum_{d > y} \frac{\chi(d)}{d} + O(y) = x L(1,\chi) + O_{\epsilon}(xy^{-\eta_0}) + O(y). \]
(Here the sum on $d>y$ has been handled by partial summation.)
Collecting our estimates and keeping in mind that $y=x^\upsilon$, we obtain the theorem with $\eta$ defined by $1-\eta = \max\{\upsilon,1-v\eta_0\}$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:smallresidue}] Let $\varepsilon \in (0, \frac14)$ and let $\chi$ be a quadratic character modulo $m$. Let
\[ x = m^{\frac{1}{4}+\varepsilon}, \]
and let $q$ be the product of the primes $\ell \le x$ with $\chi(\ell)=1$. We suppose that $\omega(q) \le (\log{m})^{A}$, and we show this implies that $m$ is bounded by a constant depending on $\varepsilon$ and $A$. Throughout this proof, we suppress any dependence on $\varepsilon$ and $A$ in our $O$-notation.
By Proposition \ref{prop:LV},
\begin{equation} \sum_{n \le x} r(n) = L(1,\chi) \cdot x + O(x^{1-\eta}). \label{eq:ldub}\end{equation}
We can estimate the sum in a second way. Observe that
\begin{equation}\label{eq:rnexpression} r(n) = \prod_{\ell^e \parallel n} \left(1+\chi(\ell) + \dots + \chi(\ell^e)\right) \ge 0. \end{equation}
Hence, if the subset $\mathcal{S}$ of $[1,x]$ is chosen to contain the support of $r(n)$ on $[1,x]$, then
\[ 0 \le \sum_{n \le x} r(n) \le \#\mathcal{S} \cdot \left(\max_{n \in \mathcal{S}} r(n)\right). \]
Examining the expression in \eqref{eq:rnexpression} for $r(n)$, we see $\mathcal{S}$ can be chosen as the set of $n\le x$ where every prime that appears to the first power in the factorization of $n$ divides $mq$. For each $n \in \mathcal{S}$, we can write $n=n_1 n_2$, where $n_1$ is a squarefree divisor of $mq$ and $n_2$ is squarefull. The number of elements of $\mathcal{S}$ with $n_2 > x^{1/2}$ is $O(x^{3/4})$. For the remaining elements of $\mathcal{S}$, we have $n_1 \le x/n_2$ and $n_1$ is a squarefree product of primes dividing $mq$. There is a bijection
\[ \iota\colon \{\text{squarefree divisors of $mq$}\} \to \{\text{squarefrees composed of the first $\omega(mq)$ primes}\}
\]
with $\iota(r) \le r$ for all $r$. Hence, given $n_2$, the number of choices for $n_1$ is at most the number of integers in $[1,x/n_2]$ supported on the product of the first $\omega(mq)$ primes. By our assumption on $\omega(q)$, those primes all belong to the interval $[1, (\log{x})^{A+1}]$, once $x$ is large. Hence, given $n_2$, the number of possible values of $n_1$ is at most
\[ \Psi(x/n_2, (\log{x})^{A+1}). \]
For fixed $\theta \ge 1$, a classical theorem of de Bruijn \cite{dB66} asserts that $\Psi(X,(\log{X})^{\theta}) = X^{1-\frac{1}{\theta}+o(1)}$, as $X\to\infty$. Since $x/n_2 \ge x^{1/2}$, we deduce that
\[ \Psi(x/n_2, (\log{x})^{A+1}) \le (x/n_2)^{1-\frac{1}{A+2}} \]
if $x$ is large. Summing on squarefull $n_2 \le x^{1/4}$, we see that the number of elements of $\mathcal{S}$ arising in this way is $O(x^{1-\frac{1}{A+2}})$. Hence,
\[ \#\mathcal{S} \ll x^{3/4} + x^{1-\frac{1}{A+2}} \ll x^{1-\eta'}, \quad\text{where}\quad \eta'=\min\left\{\frac14,\frac{1}{A+2}\right\}. \]
Since $r(n) \le \tau(n) \ll x^{\eta'/2}$ for $n \le x$,
\begin{equation}\label{eq:udub} \sum_{n\le x} r(n) \ll \#\mathcal{S}\cdot x^{\eta'/2} \ll x^{1-\eta'/2}. \end{equation}
Comparing \eqref{eq:ldub} and \eqref{eq:udub} gives
\[ L(1,\chi) \ll x^{-\min\{\eta'/2,\eta\}}. \]
But for large $x$, this contradicts Siegel's theorem \cite[Theorem 11.14, p. 372]{MV07}.
\end{proof}
\begin{remark} Any improvement on Siegel's lower bound for $L(1,\chi)$ would boost the number of $\ell$ produced in Theorem \ref{thm:smallresidue}. Substantial improvements of this kind would have other closely related implications. For example, a simple modification of an argument of Wolke \cite{wolke69} shows that for any quadratic character $\chi$ mod $m$,
\[ \sum_{\substack{\ell \le m \\ \chi(\ell)= 1}}\frac{1}{\ell} \ge \frac{1}{2} \log\left(\frac{\varphi(m)}{m} L(1,\chi) \log{m}\right) + O(1), \]
where the $O(1)$ constant is absolute. {(Here is the short proof: By Proposition \ref{prop:LV}, $\frac{1}{m}\sum_{n \le m}r(n) \gg L(1,\chi)$. On the other hand, \cite[Theorem 5, p. 308]{tenenbaum95} yields $\frac{1}{m}\sum_{n \le m} r(n) \ll \frac{1}{\log{m}} \sum_{n \le m} \frac{r(n)}{n} \ll \frac{1}{\log{m}} \cdot \frac{m}{\varphi(m)} \cdot \exp\left(2 \sum_{\ell \le m,~\chi(\ell)=1}\frac{1}{\ell}\right)$.)}
\end{remark}
\section*{Acknowledgments} This work was motivated in part by observations made on \texttt{mathoverflow} by ``GH from MO'' \cite{52393}. The author is also grateful to ``Lucia'' for pointing out there the work of Bourgain--Lindenstrauss. He thanks Enrique Trevi\~no for useful feedback on an early draft. This research was supported by NSF award DMS-1402268.
{\small
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
}
\end{document} | math | 32,114 |
\begin{document}
\title{The Entropic Approach to Causal Correlations}
\author{Nikolai Miklin}
\thanks{These authors contributed equally to this work.}
\affiliation{Naturwissenschaftlich-Technische Fakult\"{a}t, Universit\"{a}t Siegen, Walter-Flex-Strasse 3, 57068 Siegen, Germany}
\author{Alastair A. Abbott}
\thanks{These authors contributed equally to this work.}
\affiliation{Institut N\'{e}el, CNRS and Universit\'{e} Grenoble Alpes, 38042 Grenoble Cedex 9, France}
\author{Cyril Branciard}
\affiliation{Institut N\'{e}el, CNRS and Universit\'{e} Grenoble Alpes, 38042 Grenoble Cedex 9, France}
\author{Rafael Chaves}
\affiliation{International Institute of Physics, Federal University of Rio Grande do Norte, 59070-405 Natal, Brazil}
\author{Costantino Budroni}
\affiliation{Institute for Quantum Optics and Quantum Information (IQOQI), Boltzmanngasse 3 1090 Vienna, Austria}
\date{September 19, 2017}
\begin{abstract}
The existence of a global causal order between events places constraints on the correlations that parties may share.
Such ``causal correlations'' have been the focus of recent attention, driven by the realization that some extensions of quantum mechanics may violate so-called causal inequalities.
In this paper we study causal correlations from an entropic perspective, and we show how to use this framework to derive entropic causal inequalities.
We consider two different ways to derive such inequalities.
Firstly, we consider a method based on the causal Bayesian networks describing the causal relations between the parties.
In contrast to the Bell-nonlocality scenario, where this method has previously been shown to be ineffective, we show that it leads to several interesting entropic causal inequalities.
Secondly, we consider an alternative method based on counterfactual variables that has previously been used to derive entropic Bell inequalities.
We compare the inequalities obtained via these two methods and discuss their violation by noncausal correlations.
As an application of our approach, we derive bounds on the quantity of information -- which is more naturally expressed in the entropic framework -- that parties can communicate when operating in a definite causal order.
\end{abstract}
\maketitle
\section{Introduction}
When describing most physical phenomena it seems natural to assume that physical events take place in a well-defined causal structure.
For instance, earlier events can influence later ones but not the opposite, or, if two events are distant enough (typically, space-like separated) from each other, any correlation between them can only be due to some common cause in their past. This intuition is formalized in Reichenbach's principle~\cite{Reichenbachbook} and generalized by the mathematical theory of causal models~\cite{Pearlbook} that form the basis for our current understanding of how to infer causation from empirically observed correlations.
Not surprisingly, it has found a wide range of applications~\cite{Pearlbook,Angrist1996,Friedman2004}.
Yet, quantum phenomena defy such an intuitive notion of cause and effect.
As shown by Bell's Theorem~\cite{Bell1964}, quantum correlations obtained by measurements on distant entangled parties are incompatible with Reichenbach's principle~\cite{Cavalcanti2014,Wood2015} or, more generally, with classical theories of causality, forcing us to generalize the notion of causal models~\cite{Leifer2013,Henson2014,Chaves2015,Ried2015,Costa2016,Allen2016}.
In a scenario where different experimenters interact only once with a given system that is exchanged between them, one could expect that no simultaneous causal influences between them should be possible but rather only one-way influences.
However, it has been realized that physical theories do not necessarily have to comply with the idea of a definite causal order~\cite{oreshkov12,Portmann2017}.
One can also imagine theories where the causal order itself is in a sort of ``quantum superposition''~\cite{oreshkov12,Chiribella2013}, which can be verified using so-called causal witnesses~\cite{Araujo2015,Branciard2016}.
As for entanglement witnesses~\cite{horodecki96,Horodecki2009}, the use of causal witnesses assumes that we have a precise description of the measurement apparatus, that is, they are relevant in a device-dependent framework. Nevertheless, by allowing physical theories that are locally equivalent to quantum mechanics but relax the assumption of a fixed global causal structure, it is possible to verify causal indefiniteness also in a device-independent manner. With the aim of providing a general framework to such scenarios, the process matrix formalism~\cite{oreshkov12} has been introduced and shown to allow for the violation of so-called causal inequalities~\cite{oreshkov12,baumeler13,baumeler14,oreshkov15,Branciard:2016aa,Abbott:2016aa}, which are device-independent constraints that play a similar role to that of Bell inequalities~\cite{Bell1964}. However, whether violations of causal inequalities can be experimentally observed is still an important open question.
Our goal in this paper is to introduce a new framework for the derivation of causal inequalities and the study of their potential violations: the entropic approach to causal correlations. The idea of using entropies to understand sets of correlations has its origins in the context of Bell inequalities~\cite{Braunstein1988,Chaves:2012aa,Fritz:2013aa,Chaves2014} but since then has also found various other applications in quantum contextuality~\cite{Kurzynski2012,Chaves:2013aa,Raesi2015}, device-independent applications~\cite{Chaves2015DI,Zhu2016}, causal inference~\cite{Chaves2014b,Henson2014,Pienaar2016} and in the characterization of nonsignaling correlations~\cite{Chaves:2016aa}. As for these previous applications, the interest in characterizing the entropies compatible with causal correlations stems not only from practical and technical issues, but also from a more fundamental point.
To begin with, causal inequalities expressed in terms of probabilities are constructed for a fixed number of inputs and outputs, and their systematic derivation becomes harder as this number increases~\cite{Branciard:2016aa,Abbott:2016aa}.
In contrast, we will derive entropic causal inequalities that are valid for arbitrary finite alphabets either for the input and output variables, or just for the output variables. Furthermore, entropic inequalities can be easily combined with extra assumptions, such as conditional independence relations or information theoretic constraints (e.g., bounds on the amount of communication), which would be hard to treat in the probabilistic framework~\cite{Chaves2014b,Chaves:2016aa,Chaves2016Po}.
More fundamentally, given that entropies are a core concept in classical and quantum information theory, it is of clear relevance to have a framework that focuses on these quantities rather than on probabilities, and it may help connect causal inequalities with principles such as information causality~\cite{Pawlowski2009}.
The paper is organized as follows.
In Sec.~\ref{sec:preliminary}, we will introduce the basic notions relevant for our investigation, namely causal correlations and the entropic approach to causal structures, and elaborate two complementary ways in which the approach can be applied.
In Sec.~\ref{sec:bipartite} we will show how to derive entropic causal inequalities for the bipartite scenario, and discuss
their violation.
In Sec.~\ref{sec:multipartite}, we will explain how this approach can be generalized to multipartite scenarios.
Finally, as an application, in Sec.~\ref{sec:infoBounds} we use this approach to derive bounds on mutual informations in causal games.
\section{Preliminaries}\label{sec:preliminary}
\subsection{Causal correlations}
\label{sec:causalcorr}
Causal correlations are most easily introduced in the bipartite case, where we consider two parties, Alice (${\rm A}$) and Bob (${\rm B}$), who together conduct a joint experiment while each having control over a separate closed laboratory.
During each round of the experiment, Alice and Bob each receive, operate on, and send out a single physical system, which is the only means by which they may communicate.
In addition, they each receive some (external) classical inputs $X$ and $Y$, for Alice and Bob respectively, and produce some classical outputs $A$ and $B$, respectively.
Throughout the paper we use upper-case letters (e.g., $X$) to denote random variables, and corresponding lower-case letters (e.g., $x$) to denote the specific values they take.
Their probability distributions will generically be denoted by $P$; we will also use the shorthand notations $P(x)$ for $P(X=x)$, $P(x_{\text{\tiny (}},_{\text{\tiny )\!\!}}y)$ for $P(X=x,Y=y)$, $P(a|x)$ for $P(A=a|X=x)$, etc.
The joint conditional probability distributions $P(ab|xy)$ that can be produced in such an experiment depend on the causal relation between Alice and Bob.
If Bob cannot signal to Alice their correlations should obey $P(a|xy)=P(a|xy')$ for all $x,y,y',a$, where $P(a|xy)=\sum_bP(ab|xy)$.
We denote this situation by $\mathrm{A}\prec\mathrm{B}$, and write $P = P^{\mathrm{A}\prec \mathrm{B}}$ in this case.
Note that this does not necessarily imply that Alice is in the causal past of Bob since the events could be space-like separated, but merely that the correlation is compatible with such a causal order.
Similarly, if the correlation is compatible with Bob being in the causal past of Alice we write $\mathrm{B}\prec \mathrm{A}$ and we have $P^{\mathrm{B}\prec \mathrm{A}}(b|xy)=P^{\mathrm{B}\prec \mathrm{A}}(b|x'y)$ for all $x,x',y,b$.
The correlations that satisfy both these conditions (and are thus consistent both with $\mathrm{A}\prec \mathrm{B}$ and $\mathrm{B}\prec \mathrm{A}$) are precisely the nonsignaling correlations~\cite{Popescu1994}.
More generally, we are interested in the correlations achievable under the assumption of a definite causal order in each round of the experiment, even if the causal relation between Alice and Bob may be different (e.g., chosen randomly) for each individual round. We thus say that a correlation $P(ab|xy)$ is \emph{causal} if it can be written as
\begin{equation}\label{eq:causalCorreltn}
P(ab|xy)=q_0\,P^{\mathrm{A}\prec \mathrm{B}}(ab|xy) + q_1\,P^{\mathrm{B}\prec \mathrm{A}}(ab|xy),
\end{equation}
with $q_0,q_1\in [0,1]$ and $q_0+q_1=1$, where $P^{\mathrm{A}\prec \mathrm{B}}(ab|xy)$ and $P^{\mathrm{B}\prec \mathrm{A}}(ab|xy)$ satisfy the respective (one-way) no-signaling conditions defined above~\cite{oreshkov12}.
It was shown in Ref.~\cite{Branciard:2016aa} that the set of bipartite causal correlations forms a convex polytope, whose vertices are simply the deterministic causal correlations (i.e., causal correlations for which the outputs $A,B$ are deterministic functions of the inputs $X,Y$).
The facets of this polytope specify \emph{causal inequalities}, analogous to Bell inequalities for local correlations, that any causal correlation must satisfy~\cite{oreshkov12}.
The situation with binary input and output variables was characterized completely in~\cite{Branciard:2016aa}, where it was shown that there are only two nonequivalent causal inequalities (up to symmetries).
The simplest of these is perhaps the ``guess your neighbor's input'' (GYNI) inequality, which has a simple interpretation as a game (up to a relabeling of the inputs and outputs) in which the inputs $X,Y$ are chosen uniformly at random and the goal is for each party to output the other party's input.
One such form of this inequality can be written~\cite{Branciard:2016aa}
\begin{equation}\label{eq:GYNI}
\frac{1}{4}\sum_{x,y,a,b}\delta_{a,y}\,\delta_{b,x}\,P(ab|xy)\le \frac{1}{2},
\end{equation}
where $\delta$ is the Kronecker delta function.
The notion of causal correlations can be generalized to more parties, although one has to take into account the fact that, in a given round of the experiment, the causal order of some parties may depend on the inputs and outputs of previous parties~\cite{oreshkov15,Abbott:2016aa}.
In this paper we will primarily, in Sec.~\ref{sec:bipartite}, focus on applying the entropic approach to bipartite causal correlations, before returning to the multipartite case in Sec.~\ref{sec:multipartite}.
\subsection{The entropic approach and marginal problems}
\label{sec:entropymarginal}
Below we introduce the basic notions concerning entropy cones and marginal scenarios.
We then review the entropic characterization of marginal scenarios~\cite{Fritz:2013aa} using two complementary methods, the first considering the entropies of the variables composing a given causal model, and the second based on the counterfactual approach to correlations.
To illustrate concretely and contrast these two methods, we apply them to the well-known Bell scenario.
Readers well-familiarized with the entropic approach may prefer to skip these expository examples.
\subsubsection{Entropy and Shannon cones}
Let $S=\{X_1, \dots, X_n\}$ be a set of $n$ random variables taking values $x_1, \dots, x_n$, whose joint distribution $P(x_1, \dots, x_n)$ we wish to characterize entropically.
For every nonempty subset $T\subset S$ we shall denote by $\bm{X}_T = (X_i)_{X_i \in T}$ the joint random variable that involves all variables in $T$, taking values $\bm{x}_T = (x_i)_{X_i \in T}$.
We can then compute the marginal \emph{Shannon entropies} $H(\bm{X}_T)=H(T)$ from the marginal probability distributions $P(\bm{X}_T=\bm{x}_T)=P(\bm{x}_T)$ as
\begin{equation}
\label{ShannonE}
H(T)\coloneqq -\sum_{\bm{x}_T}P(\bm{x}_T)\log_2{P(\bm{x}_T)}.
\end{equation}
Together with $H(\emptyset)\coloneqq 0$, every global probability distribution $P(x_1,\ldots,x_n)$ thus specifies $2^n$ real numbers in the entropic description, which can be expressed as the components of a $(2^n)$-dimensional vector ${\bm{h}=(H(\emptyset),H(X_1),\ldots,H(X_{1}X_{2}),\ldots, H(X_1 \dots X_n))} = (H(T))_{T \subset S}$ in $\mathds{R}^{2^n}$.
A fundamental problem in information theory is to decide whether a given vector is an \emph{entropy vector}, that is, if it is obtainable from some probability distribution.
The (closure of the) region of valid entropy vectors
\begin{equation}\label{eq:entropyCone}
\Gamma_{S}^* \coloneqq \overline{\left\{ \bm{h} \in \mathds{R}^{2^n} \,|\, \bm{h} = (H(T))_{T\subset S} \right\}},
\end{equation}
is known to be a convex cone, called the \emph{entropy cone} (see Ref.~\cite{Yeung2008} for a comprehensive discussion of entropy cones).
There is no known explicit description of $\Gamma_{S}^*$, so one generally has to rely on an approximation of it.
A well-known and very useful outer approximation of $\Gamma_{S}^*$ is the so-called \emph{Shannon cone} $\Gamma_{S}$, defined by the elemental inequalities
\begin{align}
\label{shannonineqs_basic}
H(S\setminus\{X_i\}) &\leq H(S), \\
H(T) + H(T\cup\{X_i,X_j\}) &\leq H(T\cup \{X_i\}) + H(T\cup \{X_j\}),\notag
\end{align}
for all $1\le i, j\le n$, $i \neq j$, and $T \subset S \setminus\{X_i,X_j\}$.
That is, the Shannon cone $\Gamma_{S}$ is described by a finite system of $m=n+2^{n-2}\binom{n}{2}$ linear inequalities, which one can write in the form $I\bm{h}\leq \bm{0}$, where $I$ is an $m\times 2^n$ real matrix and $\bm{0}$ a vector with null entries.
The inequalities in Eq.~\eqref{shannonineqs_basic} are the minimal set of inequalities implying the monotonicity of entropy, i.e., $H(U|T) \coloneqq H(TU) - H(T) \geq 0$, and the submodularity (or strong subadditivity), i.e., $I(U:V|T)\coloneqq H(TU)+H(TV)-H(TUV)-H(T)\geq 0$, for any subsets $T,U,V\subset S$.
These inequalities and any combination thereof are known as \emph{Shannon-type inequalities}.
It is known that for $n\le 3$ variables every inequality delimiting the entropy cone $\Gamma_{S}^*$ is of the Shannon type; however, this is not the case for $n> 3$~\cite{Yeung2008}.
The inequalities characterizing the Shannon cone simply arise from demanding that the function $P(\bm{x}_T)$ appearing in~\eqref{ShannonE} should be identified with a valid probability distribution (i.e., it should be nonnegative and normalized).
However, one often wishes to consider (and characterize the entropy vectors for) situations where additional constraints on the random variables are known.
For example, $X_i$ and $X_j$ might be known to be independent, which implies that $P(x_i,x_j)=P(x_i)P(x_j)$.
Such independence constraints, which are nonlinear in terms of probabilities, define simple linear constraints in terms of entropies, e.g., $P(x_i,x_j)=P(x_i)P(x_j) \rightarrow H(X_iX_j)=H(X_i)+H(X_j)$.
These extra constraints can be easily incorporated into the entropic framework since they define a linear subspace, which we denote $\textrm{L}_{\mathcal{C}}$, characterized by linear equalities.
When combined with the elemental inequalities one obtains a new finite system of inequalities $I^{\prime}\bm{h}\leq \bm{0}$ characterizing the ``constrained Shannon cone'' $\Gamma_{S}\bigcap \textrm{L}_{\mathcal{C}}$.
In some cases, one may also wish to add linear inequality constraints which, in general, may give rise to more general polyhedra described by inhomogeneous systems of linear inequalities $I^{\prime}\bm{h}\leq \bm{\beta}$~\cite{Schrijver_book}.
In such cases we will again denote the set of vectors $\bm{h}$ satisfying these additional constraints as $\textrm{L}_{\mathcal{C}}$; we will return to this point in more detail in Sec.~\ref{sec:bipartite}.
\subsubsection{Marginal scenarios}
\label{subsec:marginal}
Consider again a set of random variables $\{X_1,\dots,X_n\}$ with a joint probability distribution $P(x_1,\ldots,x_n)$.
We often encounter situations where not all variables, or combinations thereof, are empirically accessible.
For example, our system of interest could be composed of three random variables $X_1,X_2,X_3$ but, for some reason, we can access at most two of them at a time, thus implying that we cannot know their joint entropy $H(X_1X_2X_3)$.
Alternatively, there might be variables that represent latent factors~\cite{Pearlbook} and that, for this reason, are unobservable.
In such cases, we face a \emph{marginal problem}: decide whether some given information on the marginals is compatible with a global description fulfilling certain constraints (for example the elemental entropy inequalities).
In the example with three variables, it is easy to see that the elemental inequalities imply that
\begin{equation}
H(X_1) + H(X_2X_3) \leq H(X_1X_2)+H(X_1X_3).
\end{equation}
That is, the global structure of entropy vectors implies nontrivial constraints (which are not elemental inequalities~\eqref{shannonineqs_basic}) that should be respected by any marginal information compatible with it.
More formally, given a set of random variables $S=\{X_1,\ldots,X_n\}$, a \emph{marginal scenario} is a collection of subsets $\mathcal{M}=\{ M_1,\ldots,M_{|\mathcal{M}|}\}$, $M_j\subset S$ representing those variables for which we have access to the probability distribution $P(\bm{x}_{M_j})$ (and thus to $H(M_j)$).
Clearly, $M_j\in \mathcal{M}$ and $M_j'\subset M_j$ implies $M_j'\in \mathcal{M}$, that is, given some probability distribution we also have access to any marginal of it.
In a slight abuse of notation we will therefore write $\mathcal{M}$ only in terms of its maximal subsets, since these are sufficient to specify the entire marginal scenario; the complete representation of $\mathcal{M}$, which explicitly includes all (not necessarily maximal) subsets $T$ for which the marginal distribution $P(\bm{x}_{T})$ is accessible, will be denoted $\mathcal{M}^{\rm c} = \{T \mid T\subset M_j, M_j \in \mathcal{M}\}$.
In the example above the marginal scenario would then be represented as $\mathcal{M}=\big\{ \{X_1,X_2\},\{X_1,X_3\},\{X_2,X_3\}\big\}$, or $\mathcal{M}^{\rm c}=\big\{ \emptyset, \{X_1\}, \{X_2\}, \{X_3\}, \{X_1,X_2\},\{X_1,X_3\},\{X_2,X_3\}\big\}$.
In general we are interested in characterizing the entropy cone $\Gamma^*_{\mathcal{M}}$ associated with a marginal scenario $\mathcal{M}$, thus obtaining constraints implied by the global entropy cone on the marginal subspace of interest.
Geometrically, this corresponds to the projection of the original entropy cone onto the subspace of entropy vectors $\bm{h}=(H(T))_{T \in \mathcal{M}^{\rm c}} \in \mathds{R}^{|\mathcal{M}^{\rm c}|}$, corresponding to the variables in $\mathcal{M}^{\rm c}$.
Since, in practice, we work with the Shannon cone $\Gamma_{S}$ -- possibly constrained by some further linear constraints specifying a subset of entropy vectors $\textrm{L}_{\mathcal{C}}$, as described previously -- which is characterized by a finite system of inequalities, this projection corresponds to a simple variable elimination of all the terms not contained in $\mathcal{M}^{\rm c}$~\cite{Williams1986,budroni2012bell2,Fritz:2013aa}.
After removing redundant inequalities, the remaining inequalities are facets (i.e., the boundaries) of the Shannon cone, or more generally polyhedron, in the observable marginal subspace.
Formally, the marginal Shannon polyhedron $\Gamma_{\mathcal{M}}$ is defined as
\begin{equation}
\label{eq:ProjMargCone}
\Gamma_{\mathcal{M}}=\Pi_\mathcal{M}\left(\Gamma_{S} \bigcap \textrm{L}_{\mathcal{C}} \right),
\end{equation}
where $\Pi_\mathcal{M}$ denotes the projection onto the coordinates associated with the marginal scenario $\mathcal{M}$ -- i.e., onto the coordinates $H(T)$ with $T \in \mathcal{M}^{\rm c}$.
\subsubsection{Probability structures}
\label{sec:probStructures}
The characterization of entropy cones (or polyhedra) and marginal problems outlined above can be easily extended to the case where we no longer assume that there is a well-defined global probability distribution over all the variables in the set $S$.
Instead, we may assume that only certain subsets of variables have such a joint distribution, and that only the marginals of certain subsets of these subsets are empirically accessible.
This type of restriction may be imposed by assumptions about the underlying physical theory being described, as will be clear in the example we discuss in Sec.~\ref{sec:counterfactuals}.
We will denote the collection of subsets of $S$ for which we assume joint probability distributions exist by $\mathcal{S} = \{S_1,\ldots,S_{|\mathcal{S}|}\}$, with each $S_i \subset S$ such that $\cup_i S_i = S$; we call $\mathcal{S}$ the \emph{probability structure}.\footnote{Of course the probability assignment should be consistent. That is, for two subsets $S_i$ and $S_i'$ of $\mathcal{S}$, the corresponding probability distributions $P_i$ and $P_i'$ should coincide on $S_i \cap S_i'$, so that one must have $P_i(\bm{x}_{T}) = P_i'(\bm{x}_{T})$ for all $T \subset S_i \cap S_i'$. This allows one to define $H(T)$ for all $T \in \mathcal{S}^{\rm c}$ unambiguously.}
As for the marginal scenario, we will represent $\mathcal{S}$ by just its maximal subsets in a slight abuse of notation; the complete representation of $\mathcal{S}$, that explicitly includes all subsets for which a joint probability distribution exists, will similarly be denoted $\mathcal{S}^{\rm c}$.
In such a situation the entropies $H(T)$ cannot be defined for all subsets $T \subset S$, but only for the subsets in $\mathcal{S}^{\rm c}$. The entropy vectors we shall consider will thus be defined here as $\bm{h}=(H(T))_{T \in \mathcal{S}^{\rm c}} \in \mathds{R}^{|\mathcal{S}^{\rm c}|}$.
Again, no explicit characterization is known for the set of valid entropy vectors; we will instead rely on its outer approximation characterized via the Shannon constraints, now restricted to each subset $S_i \in \mathcal{S}$. Namely, the Shannon cone of interest is now
\begin{equation}\label{eq:defconeS}
\Gamma^{\mathcal{S}}=\bigcap_{S_i\in \mathcal{S}} \Gamma_{S_i},
\end{equation}
where $\Gamma_{S_i}\subset \mathds{R}^{|\mathcal{S}^{\rm c}|}$ is the cone defined by the Shannon inequalities on the variables in $S_i$, which, in particular, leave the other variables in $S \setminus S_i$ unconstrained.
In the extremal case where we do assume a global joint probability distribution for all variables we have $\mathcal{S}=\{S\}$, $\mathcal{S}^{\rm c}=2^S$, and we recover $\Gamma^{\mathcal{S}}=\Gamma_{S}$.
One can similarly consider marginal scenarios under a given probability structure $\mathcal{S}$, with the constraint that marginals must arise from existing probability distributions, i.e., for all $M_j\in \mathcal{M}$ there must exist an $S_i\in \mathcal{S}$ such that $M_j\subset S_i$.
One can also add linear constraints to the entropy vectors under consideration, as before, represented by some subset of entropy vectors $\textrm{L}_{\mathcal{C}}$.
We can thus define the marginal Shannon polyhedron associated with $\mathcal{S},\mathcal{M}$, and $\textrm{L}_{\mathcal{C}}$ as
\begin{equation}
\label{eq:ProjMargConeS}
\Gamma_{\mathcal{M}}^{\mathcal{S}}=\Pi_\mathcal{M}\left(\Gamma^{\mathcal{S}} \bigcap \textrm{L}_{\mathcal{C}} \right).
\end{equation}
The choice of probability structure can generally be considered on a case-by-case basis depending on the scenario being modeled.
Unless otherwise stated we will take $\mathcal{S}=\{S\}$ but, as we will discuss, this will not always be the most pertinent choice.
\subsubsection{The entropic characterization of causal Bayesian networks}
\label{sec:causal_str}
In order to describe the causal relations between random variables, we will first use the framework of causal Bayesian networks\footnote{Note that although notions of causal correlations and causal Bayesian networks both share the ``causal'' qualifier, they are distinct concepts: a causal correlation is \emph{not} simply one that can be obtained from any particular causal Bayesian network.}~\cite{Pearlbook}.
Such networks can be conveniently represented as \emph{directed acyclic graphs} (DAGs), in which each node represents a variable and directed edges (arrows) encode the causal relations between them.
A set of variables $S=\{X_1,\dots,X_n\}$ forms a Bayesian network with respect to a given DAG if and only if the variables admit a global probability distribution $P(x_1,\dots,x_n)$, i.e., $\mathcal{S}=\{ S\}$, that factorizes according to
\begin{equation}
P(x_1,\dots,x_n)= \prod_{i=1}^{n} P(x_i\vert \mathrm{Pa}_i),
\end{equation}
where $\mathrm{Pa}_i$ stands for the graph-theoretical parents of variable $X_i$, that is, all those variables $X_j$ that have an outgoing edge pointing to $X_i$ in the DAG under consideration.
The decomposition above implies a set of conditional independences (CIs), which are either independence relations of the type $P(x_i,x_j)=P(x_i)P(x_j)$ (in which case we write $X_i {\perp\!\!\!\perp} X_j$) or conditional independence relations such as $P(x_i,x_j | x_k)=P(x_i\vert x_k)P(x_j \vert x_k)$ (denoted $X_i{\perp\!\!\!\perp} X_j \mid X_k$).\footnote{For CIs between more than two variables, we use the natural extension of this notion. For example, if $P(x_i,x_j,x_k)=P(x_i)P(x_j)P(x_k)$ we write $X_i {\perp\!\!\!\perp} X_j {\perp\!\!\!\perp} X_k$.}
Given a DAG, a complete list of CIs can be obtained via the $d$-separation criterion~\cite{Pearlbook}.
If the arrows in the DAG representation of a Bayesian network describe the direct causal relations between the variables in question, then we call it a \emph{causal Bayesian network}.
Entropically, these CIs correspond to simple linear relations: $X_i {\perp\!\!\!\perp} X_j \rightarrow H(X_iX_j)=H(X_i)+H(X_j)$ and $X_i{\perp\!\!\!\perp} X_k \mid X_k \rightarrow H(X_iX_j\vert X_k)=H(X_i\vert X_k)+H(X_j \vert X_k)$.
As a result, the set of entropy vectors compatible with a given DAG is the intersection of the entropy cone $\Gamma_{S}^*$ with the linear subspace $\textrm{L}_{\mathrm{CI}}$ defined by the set of linear constraints that characterize the CIs associated with the DAG~\cite{Chaves2014,Chaves2014b}.
In practice, we again rely on the outer approximation given by the intersection of the Shannon cone $\Gamma_{S}$ with $\textrm{L}_{\mathrm{CI}}$.
If all the variables in a DAG are observable, in order to check the compatibility of a given entropy vector with the DAG it suffices to check whether all the entropic CIs are satisfied.
However, we are often interested in DAGs containing latent, non-observable, variables.
Splitting the $n$ variables making up the DAG into $j$ observable variables $O_1,\dots,O_j$ and $n-j$ latent variables $\Lambda_1,\dots,\Lambda_{n-j}$ we thus need to compute the marginal Shannon cone $\Pi_\mathcal{M}\left(\Gamma_{S} \bigcap \textrm{L}_{\mathrm{CI}} \right)$ where $\mathcal{M}=\big\{ \{O_1,\dots,O_j\}\big\}$.
\begin{figure}
\caption{DAG showing the causal structure of a local hidden variable model for the Bell scenario. \label{fig:bell}
\label{fig:bell}
\end{figure}
As an illustration, consider the paradigmatic causal Bayesian network for a local hidden variable model satisfying Bell's assumption of local causality~\cite{Bell1964,Wood2015}.
The relevant DAG, shown on Fig.~\ref{fig:bell}, has five variables, four of which are observable while the hidden variable $\Lambda$ is not: in the context of Bell's Theorem the ``hidden variables'' indeed refer to the latent factors introduced above.
This DAG represents the physical scenario where two distant observers receive physical systems produced by a common source (the hidden variable $\Lambda$) and make different measurements (choices of which are labelled by $X$ and $Y$), obtaining measurement outcomes (represented by the variables $A$ and $B$).
That is, the probability structure is $\mathcal{S}=\{S\}$ with $S=\{X,Y,A,B,\Lambda\}$, and the marginal scenario is $\mathcal{M}=\big\{ \{X,Y,A,B\}\big\}$.
Some of the conditional independences implied by this DAG are given by $P(xy\lambda)=P(x)P(y)P(\lambda)$ (the measurement independence assumption), $P(a\vert xyb\lambda)=P(a\vert x\lambda)$ and $P(b\vert xya\lambda)=P(b\vert y\lambda)$ (the locality assumption) that in turn imply (after eliminating the hidden variable $\Lambda$) Bell inequalities for the observed variables~\cite{Bell1964,Wood2015}.
These constraints also imply the no-signaling constraints $P(a\vert xy)=P(a\vert x)$ and $P(b\vert xy)=P(b\vert y)$.
This example shows that, in general, DAGs with latent variables imply CIs both on the level of observable and unobservable variables.
The CIs involving latent variables are not directly testable but imply further constraints (Bell inequalities, in the example above) that can be tested to check whether the observable behavior is compatible with the proposed underlying DAG.
If, instead of characterizing the allowed probability distributions, we consider the entropic description of the Bell scenario, i.e., the Shannon cone together with the linear constraints arising from the DAG's CIs, then after eliminating the latent variable $\Lambda$ one obtains no further constraints other than the elemental inequalities (which are trivial since they are respected by all probability distributions) and the observable CIs implied by the DAG: $H(XY)=H(X)+H(Y)$, $H(A\vert XY)=H(A\vert X)$ and $H(B\vert XY)=H(B\vert Y)$~\cite{Weilenmann:2016aa}.
The first CI relation represents the independence of the two measurement choices, while the two latter ones are no-signaling conditions.
Thus, for this particular causal Bayesian network, when the entropic approach is applied to the variables making up the DAG one does not obtain any nontrivial constraints (i.e., entropic Bell inequalities)~\cite{Weilenmann:2016aa}.
However, there are many examples of Bayesian networks for which one does obtain such nontrivial constraints~\cite{Chaves2014,Chaves2014b,Henson2014}.
In fact, as we will see in Sec.~\ref{sec:bipartite}, a slight modification of this method also leads to nontrivial constraints on causal correlations.
\subsubsection{The entropic characterization of counterfactuals}
\label{sec:counterfactuals}
While the DAG method fails to provide nontrivial constraints for the Bell scenario (a result that can be extended to a larger class of ``line-like'' Bayesian networks~\cite{Weilenmann:2016aa}), it has been known for some time that entropic Bell inequalities can be derived using different methods~\cite{Braunstein1988}.
Interestingly, these inequalities can even be turned into necessary and sufficient conditions for a given probability distribution to satisfy Bell's local causality assumption~\cite{Chaves:2013aa}.
The method that allows such inequalities to be derived is motivated by the realization that the entropic approach can be applied to any marginal scenario for a relevant set of random variables~\cite{Fritz:2013aa}, and not only those arising from causal Bayesian networks.
In particular, when we are interested in constraints on conditional distributions of the form $P(ab|xy)$, where we have distinct sets of input and output variables, we may consider the output variables conditioned on certain relevant input variables (e.g.\ $A_{xy}$ and $B_{xy}$, where the notation $A_{xy}$ denotes the random variable $A|(X=x,Y=y)$).\footnote{We focus here on the bipartite case for concreteness, but the method readily generalizes to multipartite scenarios.}
The choice of relevant input variables to condition on, as well as the appropriate probability structure, will depend on the physical situation being considered.
In general, a global probability distribution may not exist on such ``counterfactual'' variables even if one does exist on the unconditioned variables.
Let us illustrate how this method may be applied by considering again its application to the Bell scenario.
Instead of considering all the input and output variables as in the DAG approach (e.g.\ $X,Y,A,B$), one can consider copies of the output variables conditioned on the corresponding party's input, i.e., $A_{x},B_{y}$, where $A_{x}$ denotes the random variable $A{|(X=x)}$.
Indeed, due to the no-signaling constraints, the output variables can only depend on the corresponding local input.
Furthermore, from Fine's Theorem~\cite{Fine1982} we know that Bell's local causality assumption is equivalent to the existence of a well defined (although empirically inaccessible) joint probability distribution $P(a_1,\dots, a_{\vert \cal X \vert},b_1,\dots, b_{\vert \cal Y \vert})$ (where ${\cal X} = \{1,\ldots,|{\cal X}|\}$ and ${\cal Y} = \{1,\ldots,|{\cal Y}|\}$ denote the alphabets of Alice and Bob's inputs) on these variables\footnote{In particular, by invoking Fine's Theorem we do not need to explicitly include the hidden variable $\Lambda$ in this method, contrary to the DAG method outlined previously.} that marginalizes to the observable one given by $P(ab \vert xy)=P(a_x,b_y)$.
Hence, the appropriate probability structure for local correlations in the Bell scenario is $\mathcal{S}=\{S \}$ with $S=\{A_1\dots,A_{|\mathcal{X}|},B_1,\dots,B_{|\mathcal{Y}|}\}$, and we consider the Shannon cone $\Gamma_{S}=\Gamma^{\mathcal{S}}$ that contains all $2^{|\mathcal{X}|+|\mathcal{Y}|}$-dimensional entropy vectors $\bm{h}=\big(H(\emptyset), H(A_1),\dots, H(B_1),\dots, H(A_1\dots A_{\vert \cal X \vert}B_1\dots B_{\vert \cal Y \vert}) \big)$.
The marginal scenario in this case is simply $\mathcal{M}=\big\{ \{A_x,B_y\} \big\}_{x,y}$ and local correlations are then characterized by the cone $\Pi_\mathcal{M}\left(\Gamma_S\right)$.
In contrast to the characterization based directly on the DAG variables, this approach leads to nontrivial entropic inequalities (i.e., not obtainable from the elemental inequalities in Eqs.~\eqref{shannonineqs_basic}) in the Bell scenario.
For example, for two measurement settings per party, which we label in this case $x,y=0,1$, one obtains the Braunstein-Caves inequality~\cite{Braunstein1988} together with its symmetries obtained by relabeling the inputs, namely,
\begin{align}\label{eq:entropicBellIneq}
I(A_0:B_0)&+I(A_0:B_1) +I(A_1:B_0) \notag\\
& -I(A_1:B_1) -H(A_0)-H(B_0) \leq 0 ,
\end{align}
where $I(A_x:B_y)\coloneqq H(A_x)+H(B_y)-H(A_xB_y)$ is the mutual information between the variables $A_x$ and $B_y$.
This inequality can be understood as the entropic counterpart of the paradigmatic CHSH inequality~\cite{Clauser1969}.
Although the choice of probability structure above corresponds, via Fine's theorem, to the assumption of a local hidden variable theory, one can also consider other possibilities.
For instance, taking $\mathcal{S}=\mathcal{M}$ amounts to assuming a nonsignaling theory~\cite{Popescu1994}.
In this case, the entropy cone is characterized only by the Shannon inequalities and one can obtain a characterization of the extremal rays of the cone, corresponding to the entropic analogue of Popescu-Rohrlich (PR) boxes~\cite{Chaves:2016aa}.
In general (i.e., beyond the simplest Bell scenario), both methods based on the variables in a causal Bayesian network and on counterfactual variables can lead to nontrivial constraints~\cite{Fritz2012,Chaves2014,Chaves2014b,Henson2014,Steudel2015,Chaves:2016aa}.
To conclude this section, let us nonetheless highlight an important difference between the two methods: while the former is valid for arbitrary input alphabets, the latter fixes the number of inputs to which the inequalities apply.
\section{Bipartite entropic causal inequalities}
\label{sec:bipartite}
With the entropic approach to characterizing sets of correlations outlined, we can now proceed to apply this approach to causal correlations, so as to derive \emph{entropic causal inequalities}.
We consider in this section the bipartite case. We first show how the method based on causal Bayesian networks can be adapted to characterize causal correlations, before considering also the method based on counterfactual variables.
\subsection{Characterization based on causal Bayesian networks}
\label{sec:causal_str_bi}
\subsubsection{Conditional DAGs for bipartite causal correlations}
The ability to apply the entropic approach to DAGs, as outlined in Sec.~\ref{sec:causal_str}, is a powerful tool for characterizing the correlations obtainable within arbitrary causal networks.
However, the notion of causal correlations defined in Eq.~\eqref{eq:causalCorreltn} is somewhat more general and cannot be directly expressed within the framework of causal Bayesian networks.
In order to see why this is the case, let us first note that the random variables of interest are $X,Y,A,B$, representing the inputs $X,Y$ and outputs $A,B$ for Alice and Bob.
Note that since we consider signaling scenarios here, unlike in the Bell scenario, we do not need to include any latent variable $\Lambda$ in our description to account for shared randomness, since this can be established via local randomness and communication.
\begin{figure}
\caption{DAGs for bipartite causal correlations. The latent ``switch'' variable $Q$ determines which DAG, corresponding to the fixed causal order $\mathrm{A}
\label{fig:dag_bi}
\end{figure}
If Alice and Bob share a correlation compatible with a fixed causal order (i.e.\ either $\mathrm{A}\prec \mathrm{B}$ or $\mathrm{B}\prec \mathrm{A}$), then the functional dependences between $X,Y,A,B$ can indeed be expressed as a DAG -- specifically, the two DAGs containing these variables in Fig.~\ref{fig:dag_bi}.
However, a causal correlation may in general not be compatible with any fixed causal order, but may require a mixture thereof.
This has some similarities with the situation in the Svetlichny definition of genuine multipartite nonlocality~\cite{Svetlichny1987,Chaves2016causal} where a convex mixture of different DAGs has to be considered.
To tackle this problem it is necessary to find a way to take into account the constraints arising separately from each of the two fixed causal orders, and then to combine them to obtain those satisfied by causal correlations.
In order to do this, we exploit the fact that any mixture of fixed-order causal correlations can be seen as arising from a latent variable that determines the causal order for each individual experiment~\cite{oreshkov15}. We thus introduce a new random variable $Q$ which we call a ``switch'', and which determines univocally the appropriate causal Bayesian network for each trial.
The resulting causal model is shown in Fig.~\ref{fig:dag_bi}, where the DAG with $\mathrm{A}\prec\mathrm{B}$ is used for $Q=0$, and the one with $\mathrm{B}\prec\mathrm{A}$ for $Q=1$. By identifying $q_0,q_1$ in Eq.~\eqref{eq:causalCorreltn} as $q_0=P(Q=0)$, and $q_1=P(Q=1)$, one can readily see that this description is equivalent to the definition of causal correlations in Eq.~\eqref{eq:causalCorreltn}.
Both DAGs imply the independence of the inputs, $X{\perp\!\!\!\perp} Y$. The DAG for $Q=0$ (i.e., for $\mathrm{A}\prec\mathrm{B}$) also implies the CI $A{\perp\!\!\!\perp} Y\mid X$ (i.e.\ that there is no signaling from $\mathrm{B}$ to $\mathrm{A}$), while the DAG for $Q=1$ implies $B{\perp\!\!\!\perp} X\mid Y$ instead.
In addition, the switch variable $Q$ should be independent of Alice and Bob's inputs $X$ and $Y$, so that we have $XY{\perp\!\!\!\perp} Q$, which, together with $X{\perp\!\!\!\perp} Y$, implies that $X{\perp\!\!\!\perp} Y{\perp\!\!\!\perp} Q$.
\subsubsection{Shannon polyhedra of causal correlations}\label{sec:Sh_pol_dag}
In order to use the ``conditional'' causal Bayesian network in Fig.~\ref{fig:dag_bi} to characterize the set of entropy vectors obtainable from causal correlations, we first note that we can directly use the techniques of Sec.~\ref{sec:causal_str} to construct the Shannon cones for each of the two DAGs appearing in the figure conditioned on $Q$ (i.e., for fixed-order correlations with $\mathrm{A}\prec\mathrm{B}$ or $\mathrm{B}\prec\mathrm{A}$).
Denoting these cones $\Gamma^{\mathrm{A}\prec\mathrm{B}}$ and $\Gamma^{\mathrm{B}\prec\mathrm{A}}$, we have
\begin{equation}\label{eq:coneAB}
\Gamma^{\mathrm{A}\prec\mathrm{B}}=\Gamma_{S}\cap\textrm{L}_{\mathcal{C}}^{\mathrm{A}\prec\mathrm{B}}
\end{equation}
and
\begin{equation}\label{eq:coneBA}
\Gamma^{\mathrm{B}\prec\mathrm{A}}=\Gamma_{S}\cap\textrm{L}_{\mathcal{C}}^{\mathrm{B}\prec\mathrm{A}},
\end{equation}
where $\Gamma_{S}$ is the Shannon cone for the four variables in $S=\{X,Y,A,B\}$, the probability structure is simply $\mathcal{S}=\{S\}$, and $\textrm{L}_{\mathcal{C}}^{\mathrm{A}\prec\mathrm{B}}$ denotes the linear subspace defined by the CI constraints for the case ${\mathrm{A}\prec\mathrm{B}}$, namely, the equations $H(XY)=H(X)+H(Y)$ and $H(YA| X)= H(Y| X) + H(A|X)$, and similarly for $\textrm{L}_{\mathcal{C}}^{\mathrm{B}\prec\mathrm{A}}$.
These cones are characterized by the systems of inequalities $I_0\bm{h}\le \bm{0}$ and $I_1\bm{h}\le \bm{0}$, where $\bm{h}=(H(T))_{T\subset S}$.
Recall that in the probabilistic case the polytope of causal correlations is simply the convex hull of the polytopes of correlations for $\mathrm{A}\prec\mathrm{B}$ and $\mathrm{B}\prec\mathrm{A}$~\cite{Branciard:2016aa}, and with the new variable $Q$ the definition in Eq.~\eqref{eq:causalCorreltn} can be rewritten as
\begin{align}\label{eq:causalCorreltn_Q}
P(ab|xy)=& \ P(Q=0)P^{\mathrm{A}\prec\mathrm{B}}(ab|xy,Q=0) \notag \\
&+P(Q=1)P^{\mathrm{B}\prec\mathrm{A}}(ab|xy,Q=1).
\end{align}
In contrast, the convex hull of the cones $\Gamma^{\mathrm{A}\prec\mathrm{B}}$ and $\Gamma^{\mathrm{B}\prec\mathrm{A}}$ does not contain all entropy vectors of causal correlations due to the concavity of the Shannon entropy.
Indeed, in Appendix~\ref{apndx:counterex} we provide an explicit example of a causal correlation whose entropy vector is not contained in the convex hull $\conv(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}})$.
To see more precisely why this is the case, and how to give a correct entropic characterization of causal correlations, observe that, when taking a convex mixture of two causal correlations with different causal orders, the ``conditional entropy vectors'' $\bm{h}_0=(H(T|Q=0))_{T\subset S}$ and $\bm{h}_1=(H(T|Q=1))_{T\subset S}$ must be contained in $\Gamma^{\mathrm{A}\prec\mathrm{B}}$ and $\Gamma^{\mathrm{B}\prec\mathrm{A}}$, respectively, and thus satisfy $I_0\bm{h}_0\le\bm{0}$ and $I_1\bm{h}_1\le\bm{0}$.
For any causal correlation, the convex mixture
\begin{equation}\label{eq:convHvec}
\bm{h}_{\textrm{conv}} = P(Q=0)\bm{h}_0 + P(Q=1)\bm{h}_1
\end{equation}
is thus contained in $\conv(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}})$.
Observe now that, in contrast to the convex sum Eq.~\eqref{eq:causalCorreltn_Q} defining causal correlations, $\bm{h}_{\textrm{conv}}$ thus defined is equal to $(H(T|Q))_{T\subset S}$, rather than just $(H(T))_{T\subset S}$, and hence the convex hull of the fixed-order cones characterizes the conditional entropies (conditioned on the switch variable $Q$) obtainable with causal correlations, rather than the entropy vectors of causal correlations directly.
With the appropriate transformation, the inequalities $I\bm{h}\le\bm{0}$ characterizing\footnote{In practice these can be obtained by taking the union of the extremal rays of the two cones $\Gamma^{\mathrm{A}\prec\mathrm{B}}$ and $\Gamma^{\mathrm{B}\prec\mathrm{A}}$ and solving the facet enumeration problem to obtain the inequality representation of $\conv(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}})$ using standard software for convex polyhedra such as PANDA~\cite{PANDA}.}
$\conv(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}})$ can be transformed into inequalities satisfied by the standard (i.e., non-conditional) entropy vector $\tilde{\bm{h}}=(H(T))_{T\subset \tilde{S}}$ for the variables now in $\tilde{S}=S\cup\{Q\}$ (and the probability structure is consequently extended to $\widetilde{\mathcal{S}}=\{\tilde{S}\}$).
Specifically, each row $\bm{I}$ of the matrix $I$ (defining each individual inequality $\bm{I}\cdot\bm{h}\le 0$) must undergo the linear transformation $\mathcal{T}_{Q}: \mathds{R}^{2^{|S|}}\rightarrow \mathds{R}^{2^{|S|+1}}$ mapping $\bm{I}\mapsto\tilde{\bm{I}}\coloneqq \mathcal{T}_{Q}(\bm{I})$ with the components of $\tilde{\bm{I}}$ given by\footnote{Note that $(\tilde{\bm{I}})_{\emptyset}$ multiplies $H(\emptyset) = 0$ in the scalar product $\tilde{\bm{I}}\tilde{\bm{h}}$, so its value is irrelevant.}
\begin{equation}\label{eq:lin_tran}
(\tilde{\bm{I}})_{T \cup \{Q\}} = (\bm{I})_T, \ (\tilde{\bm{I}})_{\{Q\}} = -\sum_{T \neq \emptyset} (\bm{I})_T, \text{ and } (\tilde{\bm{I}})_T=0
\end{equation}
for all nonempty subsets $T\subset S$.
We will denote by $\conv_Q(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}})$ the cone of vectors $\tilde{\bm{h}}$ satisfying the resulting inequalities $\tilde{I}\tilde{\bm{h}}\le\bm{0}$.
To complete the characterization of entropy vectors for causal correlations, we recall that, in addition to the fact that any distribution on $\tilde{S}$ must give an entropy vector in the Shannon cone $\Gamma_{\tilde{S}}$, the conditional DAG in Fig.~\ref{fig:dag_bi} gives us the CI constraints $X{\perp\!\!\!\perp} Y{\perp\!\!\!\perp} Q$. Moreover, since $Q$ is a binary variable (as there are only two orders to switch between) we have $H(Q)\le 1$.
A consequence of this final inequality constraint is that the set of entropy vectors under consideration will be characterized by an inhomogeneous system of inequalities of the form $\tilde{I}\tilde{\bm{h}}\le \tilde{\bm{\beta}}$ for some $\tilde{\bm{\beta}}\in\mathbb{R}^{2^{|S|+1}}$ and is thus no longer a cone but a polyhedron.
The polyhedron characterizing entropy vectors associated with the conditional DAG (when still including $Q$) is thus given by
\begin{align}\label{eq:bip_cau_coneQ}
\widetilde\Gamma^{\mathrm{causal}}_{\rm AB} =& \Gamma_{\tilde{S}} \cap \conv_Q(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}}) \notag\\
& \cap \textrm{L}_{\mathcal{C}}\big(\{(X{\perp\!\!\!\perp} Y{\perp\!\!\!\perp} Q), H(Q)\leq 1\}\big),
\end{align}
where the notation $\textrm{L}_{\mathcal{C}}(\cdot)$ denotes the subset (here, a polyhedron) in the entropy vector space defined by the corresponding linear constraints.
Finally, following the general approach presented in Sec.~\ref{sec:entropymarginal}, it remains just to eliminate the terms containing the (unobservable) switch variable $Q$ in order to obtain the inequalities characterizing bipartite causal correlations.
This is done by projecting $\widetilde\Gamma^{\mathrm{causal}}_{\rm AB}$ onto the marginal scenario $\mathcal{M}= \{S\} = \big\{ \{X,Y,A,B\}\big\}$.
We thus finally obtain the polyhedron
\begin{align}\label{eq:bip_cau_cone}
&\Gamma^{\mathrm{causal}}_{\rm AB} = \Pi_{\mathcal{M}} \big( \widetilde\Gamma^{\mathrm{causal}}_{\rm AB} \big),
\end{align}
which we shall refer to as the \emph{causal Shannon polyhedron} or simply the \emph{causal polyhedron} and is again characterized by an inhomogeneous system of inequalities $I'\bm{h}\le \bm{\beta}$ for some $\bm{\beta}\in\mathbb{R}^{2^{|S|}}$.
We emphasize that the construction given above is in fact not at all restricted to the description of causal correlations, and can be used to characterize arbitrary convex mixtures of different Bayesian networks. Furthermore, as we will see in Sec.~\ref{sec:multipartite}, this method can be generalized to convex combinations of more distributions, in our case corresponding to more than two causal orders in multipartite scenarios (and even correlations with ``dynamical causal order''~\cite{hardy2005probability,oreshkov15,Abbott:2016aa}).
\subsubsection{Entropic causal inequalities and their violation}\label{sec:DAGviolations}
The constructive description of the causal polyhedron $\Gamma^{\textrm{causal}}_{\rm AB}$ from Eqs.~\eqref{eq:bip_cau_coneQ} and~\eqref{eq:bip_cau_cone} also makes it clear how we can characterize it, in practice, as a system of linear inequalities.
A description of $\widetilde\Gamma^{\mathrm{causal}}_{\rm AB}$ in terms of its facets is straightforwardly obtained by taking the union of the inequalities describing the individual cones and polyhedron appearing in Eq.~\eqref{eq:bip_cau_coneQ} and eliminating redundant ones.
The inequalities characterizing $\Gamma^{\textrm{causal}}_{\rm AB}$ can then be found by eliminating the terms not contained in the marginal scenario $\mathcal{M}= \{S\}$, either by Fourier-Motzkin elimination~\cite{Williams1986} or by finding its extremal rays and projecting out the unwanted coordinates.
The resulting system of inequalities is thus satisfied by any bipartite causal correlation.
However, many of these inequalities are either elemental inequalities (as in Eq.~\eqref{shannonineqs_basic}) or can be obtained from these by using the independence constraint $X{\perp\!\!\!\perp} Y$, and thus represent trivial constraints.
After characterizing the polyhedron in Eq.~\eqref{eq:bip_cau_cone} and eliminating all trivial inequalities, i.e., those satisfied by any distribution $P(xyab)$ with $X{\perp\!\!\!\perp} Y$, we find 35 novel entropic causal inequalities.
Several of these inequalities are equivalent under the exchange of parties (i.e., exchanging $(X,A)\leftrightarrow(Y,B)$), and under this symmetry there are in fact 20 equivalence classes of entropic causal inequalities, the full list of which is given in Appendix~\ref{apndx:DAGineqs}.
Of these, 10 have bounds of $0$ (i.e., are of the form $\bm{I}\cdot\bm{h}\le 0$), while the remaining 10 have nonzero bounds (resulting from a nontrivial dependence on $H(Q)$ before this variable was eliminated; see Appendix~\ref{apndx:DAGineqs}).
Simple interpretations of the entropic causal inequalities seem to be less forthcoming than for the bipartite causal inequalities in terms of probabilities~\cite{Branciard:2016aa} (for binary inputs and outputs -- recall that the entropic inequalities given here are, in contrast, valid for any number of possible inputs and outputs).
One of the simpler examples, which is symmetric under the exchange of parties, is
\begin{equation}\label{eq:dagineq1}
I(X:YA) + I(Y:XB) - H(AB) \leq 0.
\end{equation}
Note that the fact that we find nontrivial inequalities is in stark contrast to the situation for Bell-type inequalities (and line-like causal Bayesian networks), where the DAG-based entropic method only leads to trivial inequalities obtainable from the elemental inequalities and no-signaling conditions~\cite{Weilenmann:2016aa}.
While these entropic inequalities are obeyed by any bipartite causal correlation, we note that \emph{a priori} they need not be tight.
Indeed, recall that the Shannon cone is only an outer approximation to the true entropy cone, so it is thus interesting to study the tightness and violation of these inequalities more carefully.
Although one generally would not expect every point on the boundary of $\Gamma^{\mathrm{causal}}_{\rm AB}$ to be obtainable by a causal correlation, it is nonetheless desirable to be able to saturate each inequality by some causal probability distribution for appropriate distributions for $X$ and $Y$.
By looking at deterministic causal distributions with binary inputs and outputs, which can easily be enumerated, we readily verified that all 10 families of inequalities that are bounded by $0$ (given in Eq.~\eqref{eq:dag_v1}) can indeed be saturated when taking uniformly distributed inputs.
However, we were unable to find causal distributions, either by mixing binary ones or by considering more outputs, that saturate the remaining inequalities, and their tightness remains an open question.
To understand now the violation by noncausal distributions of the entropic inequalities, we consider the extremal rays of the constrained Shannon cone
\begin{equation}\label{eq:constrainedShannonCone}
\Gamma_S\cap\textrm{L}_{\mathcal{C}}\big( \{X{\perp\!\!\!\perp} Y\} \big)
\end{equation}
which violate the inequalities.\footnote{Note that the nontriviality of the inequalities implies that such extremal rays indeed exist.}
A crucial question is whether or not these extremal rays actually correspond to valid probability distributions (i.e., whether they support entropy vectors), and if not, whether the inequalities can nonetheless be violated.
In order to look at this, it is instructive to first restrict our attention to distributions satisfying $H(X)\le 1$, $H(Y)\le 1$, $H(A)\le 1$ and $H(B)\le 1$.
These constraints are satisfied by all distributions with binary inputs and outputs, and this therefore also allows us to compare the violation of the entropic causal inequalities to the violation of standard causal inequalities that are understood well in this scenario~\cite{Branciard:2016aa}.
Imposing these constraints on the cone in Eq.~\eqref{eq:constrainedShannonCone}, one obtains a polytope with extremal points corresponding to the extremal rays of the cone scaled to satisfy these constraints (together with the null vertex $\bm{0}$).
Under these we found that the 10 inequalities in Eq.~\eqref{eq:dag_v1} and the two inequalities in Eq.~\eqref{eq:dag_v2} could be violated, although the latter are (in that case) weaker than, and implied by, the former and are thus redundant.
The remaining 8 inequalities in Eqs.~\eqref{eq:dag_v3} and~\eqref{eq:dag_v4} cannot be violated.
All in all, the set of binary causal correlations is entropically characterized by the 10 inequalities in Eq.~\eqref{eq:dag_v1} that are bounded by $0$.
Amongst the extremal points violating each of these inequalities, those that give the maximal violation all satisfy $H(X)=H(Y)=1$ and $H(XY)=H(XYAB)$ and thus, if realizable, correspond to deterministic conditional distributions taken with uniformly distributed inputs $X$ and $Y$.
\footnote{Note that there are nonetheless extremal points that are only realizable by distributions with non-uniformly distributed inputs $X$ and $Y$, and which violate some of the inequalities. However, such distributions never yield the maximal violation obtainable.}
In fact, all but one of these 10 inequalities are maximally violated (by which we henceforth mean with respect to the Shannon cone augmented with the independence constraint $X {\perp\!\!\!\perp} Y$) by one of the three following deterministic distributions taken with uniform inputs:
\begin{align}\label{eq:GYNIdists}
P(ab|xy) &= \delta_{a,y}\,\delta_{b\oplus y,x} \notag\\
P(ab|xy) &= \delta_{a\oplus x, y}\,\delta_{b,x}\\
P(ab|xy) &= \delta_{a\oplus x, y}\,\delta_{b\oplus y,x}, \notag
\end{align}
where $x,y,a,b$ take the binary values $0,1$, and $\oplus$ denotes addition modulo $2$.
For example, Eq.~\eqref{eq:dagineq1} is violated by the third distribution above with a value for the left-hand side of $1$.
The one exception not violated by the distributions in Eq.~\eqref{eq:GYNIdists} is the second inequality in~\eqref{eq:dag_v1},
\begin{align}
I(A:B) &- I(A:B|X) - I(A:B|Y) - 2H(AB|XY) \leq 0,
\label{eq:dagineq2}
\end{align}
which, in turn, is violated by the deterministic distribution (again taken with uniform inputs)
\begin{equation}\label{eq:otherdist}
P(ab|xy) = \delta_{a\oplus x,xy}\,\delta_{b\oplus y,xy}.
\end{equation}
However, unlike for the other inequalities, this distribution does not give the maximal possible violation of inequality~\eqref{eq:dagineq2} (which is $1/2$),
as the corresponding extremal point $\bm{h}_{\mathrm{ext}}$ that does maximally violate it is not reachable by a valid probability distribution with binary inputs and outputs.
This is easily verified by making use of the previous observation that this extremal point must correspond to a deterministic distribution taken with uniform inputs, the set of which can easily be enumerated for binary inputs and outputs.
Amongst such distributions, the one in Eq.~\eqref{eq:otherdist} gives the best violation of $1-\frac{3}{2}\log_2\frac{3}{2}\approx 0.123 > 0$.
The distributions in Eq.~\eqref{eq:GYNIdists} are particularly interesting, as they all violate maximally some symmetries of the GYNI inequality~\eqref{eq:GYNI} (under relabeling of the inputs and outputs), but not Eq.~\eqref{eq:GYNI} itself.
Interestingly, it turns out that \emph{all} binary deterministic noncausal distributions, when taken with uniform inputs, violate at least one of our entropic inequalities \emph{except} the distribution $P^{\mathrm{GYNI}}(ab|xy) = \delta_{a,y}\delta_{b,x}$ (which violates maximally Eq.~\eqref{eq:GYNI}) and its four symmetries under input-independent relabeling of outputs only.
Note, however, that if Alice and Bob have a noncausal resource producing the distribution $P^{\mathrm{GYNI}}$, they can produce any of the distributions in Eq.~\eqref{eq:GYNIdists} by appropriately XORing their input with their output, and thus still obtain an operational violation of an entropic causal inequality.\footnote{This illustrates an important difference between the probabilistic and entropic frameworks: while all symmetries of a correlation obtained by flipping inputs and outputs (possibly conditioned on the local inputs for the latter) are equivalent in the probabilistic case (in the sense that if one violates a causal inequality, then all other ones violate a symmetry of that inequality) this is not the case in the entropic approach. The entropy vectors of two different symmetries of a correlation may be inequivalent, with one violating an entropic causal inequality while the other does not.}
It is interesting to observe that distributions maximally violating GYNI-type inequalities have such a crucial role in violating the entropic causal inequalities given that the entropic inequalities superficially bear little resemblance to these, and are valid for arbitrary numbers of inputs and outputs.
Returning to the more general situation with no upper bound imposed on $H(X), H(Y), H(A)$ and $H(B)$, we see that all the remaining entropic causal inequalities can be violated by entropy vectors that are parallel to the realizable entropy vectors giving violations in the restricted scenario -- more precisely, those obtained from the distributions Eq.~\eqref{eq:GYNIdists} (for all but one of the remaining inequalities) and Eq.~\eqref{eq:otherdist} (for the remaining one).
This shows that, given large enough alphabets for the input and output variables, all the entropic causal inequalities we obtained can indeed be violated by noncausal probability distributions, since if the distribution $P(xyab)$ has entropy vector $\bm{h}$ then the distribution
\begin{equation}\label{eq:mult_cp_nc}
P(\bm{x}\bm{y}\bm{a}\bm{b})=P(x_1y_1a_1b_1)\times \cdots\times P(x_ny_na_nb_n),
\end{equation}
where $\bm{x}=(x_1,\dots,x_n)$ and similarly for $\bm{y}$, $\bm{a}$ and $\bm{b}$, has entropy vector $n\cdot\bm{h}$.
\footnote{Using this approach (and the distributions in Eqs.~\eqref{eq:GYNIdists} and~\eqref{eq:otherdist}), one must take $n=17$ (and thus $2^{17}$ inputs) in order to violate all the remaining inequalities (although $n=2$ is sufficient for all but one of these inequalities). However, we expect that more intelligent approaches may allow them to be violated using less inputs.}
One should be careful, however, to note that the operation of sharing multiple independent correlations among the same parties is not a free operation either in the framework of causal correlations (since, for example, two independent copies of a causal distribution may give rise to a noncausal one), or in the process matrix framework (where two independent copies of a process matrix does not, in general, produce a valid process matrix).
Nevertheless, $P(\bm{a}\bm{b}|\bm{x}\bm{y}) = P(\bm{x}\bm{y}\bm{a}\bm{b})/P(\bm{x}\bm{y})$ obtained from Eq.~\eqref{eq:mult_cp_nc} still represents a valid (possibly noncausal) distribution.
It is interesting also to ask how sensitive the entropic causal inequalities are for detecting noncausality.
Since it does not appear possible to saturate the inequalities~\eqref{eq:dag_v2}--\eqref{eq:dag_v4} with non-zero bounds using causal distributions, these inequalities are not tight and, consequentially, unable to detect noncausal correlations that are very close to being causal.
For the other inequalities in Eq.~\eqref{eq:dag_v1} this is nonetheless a pertinent question.
More precisely, one may ask whether there exists a distribution $P^\varepsilon$ of the form
\begin{equation}
\label{eq:mix_distr}
P^\varepsilon(ab|xy)= \varepsilon P^{\textrm{NC}}(ab|xy) + (1-\varepsilon)P^{\textrm{C}}(ab|xy),
\end{equation}
where $P^{\textrm{NC}}$ is a noncausal distribution and $P^{\textrm{C}}$ is causal, that violates any of these entropic inequalities for arbitrarily small $\varepsilon>0$.
We looked in detail at this question for the case of binary inputs and outputs, where the inequalities in Eq.~\eqref{eq:dag_v1} can all both be saturated by causal distributions, and violated by noncausal ones.
By trying exhaustively all deterministic distributions $P^{\textrm{NC}}$ and $P^{\textrm{C}}$, we found that such behaviour was exhibited (for such distributions) only by the two inequalities
\begin{equation}\label{eq:goodmixingineq1}
I(A:B|X) - I(Y:B) - 2H(B|XY) \leq 0
\end{equation}
and
\begin{equation}\label{eq:goodmixingineq2}
I(XA:Y) + I(YB:X) - H(X|YA) - H(A) \leq 0.
\end{equation}
Equation~\eqref{eq:goodmixingineq1}, for example, is violated by $P^\varepsilon$ for all $\varepsilon>0$ when taking $P^{\textrm{NC}}(ab|xy)=\delta_{a\oplus x,y}\,\delta_{b\oplus y,x}$ and $P^{\textrm{C}}(ab|xy)=\delta_{a,0}\,\delta_{b\oplus y,x}$ along with uniformly distributed inputs $X$ and $Y$, which also gives a violation of the GYNI-type causal inequality
\begin{equation}
\frac{1}{4}\sum_{x,y,a,b} \delta_{a\oplus x,y}\, \delta_{b\oplus y,x}\, P(ab|xy)\le\frac{1}{2}
\end{equation}
with a left-hand side value of $\frac{1+\varepsilon}{2} > \frac{1}{2}$.
For the remaining inequalities, such mixtures that violate a standard causal inequality for arbitrarily small $\varepsilon$ only violate an entropic causal inequality when $\varepsilon>\varepsilon_0$ for some $\varepsilon_0$ bounded away from $0$.
We observed identical behavior when we extended our consideration also to various non-deterministic distributions $P^{\textrm{NC}}$ and $P^{\textrm{C}}$, and it thus seems that only Eqs.~\eqref{eq:goodmixingineq1} and~\eqref{eq:goodmixingineq2} exhibit this ability to detect the noncausality of distributions that are arbitrarily close to being causal.
A final point worth discussing relates to the physical interpretation of the distributions violating entropic causal inequalities.
One of the motivations in introducing the notion of causal correlations was whether nature permits more general causal structures that might allow such correlations to be realized, for example in quantum gravity.
In particular, the authors of Ref.~\cite{oreshkov12} introduced the so-called process matrix formalism, in which quantum mechanics is assumed to hold locally for each party, while no global order is assumed between the parties.
They showed that causal inequalities can be violated within this framework, and this helped motivate further studies of causal and noncausal correlations, where it has been shown that the violation of causal inequalities is ubiquitous within this framework~\cite{Abbott:2016aa,baumeler13,baumeler14,Baumelerspace2016,Branciard:2016aa,feix16}.
It is thus interesting to see whether entropic causal inequalities share this property and can also be violated within the process matrix framework.
To look for such violations, we used the optimization techniques of Refs.~\cite{Branciard:2016aa,Abbott:2016aa} with qubit systems to try and optimize the violation of the GYNI-type inequalities that the distributions in Eq.~\eqref{eq:GYNIdists} violate maximally.
We also tried minimizing the distance to other deterministic noncausal correlations such as Eq.~\eqref{eq:otherdist}, as well as optimizing in random directions in probability space.
Unfortunately, we were unable to find any process matrices operating on qubits that violate our entropic causal inequalities with such techniques.
We additionally attempted to reproduce (as closely as possible) distributions of the form~\eqref{eq:mix_distr} for small $\varepsilon$ in order to violate inequalities~\eqref{eq:goodmixingineq1} and~\eqref{eq:goodmixingineq2}, but similarly found no violation.
Finally, we looked at correlations obtained by mixing noncausal correlations realizable by process matrices with causal correlations.
An analogous mixing procedure was shown to enable all nonlocal distributions to violate the entropic Bell inequalities described in Sec.~\ref{sec:counterfactuals}~\cite{Chaves:2013aa}, but we were unable to find violations of any entropic causal inequalities with this approach.
This lack of violation is perhaps unsurprising given the general lack of sensitivity of the entropic inequalities to nearly-causal distributions, and the fact that the best-known violations of causal inequalities for this scenario with process matrices are relatively small~\cite{Branciard:2016aa}.
Nonetheless, it remains possible that violations can be found with higher-dimensional systems or more inputs and outputs; we leave this as an open question.
\subsection{Characterization based on counterfactual variables}
\label{sec:counterfactuals_causal}
In this section we will consider counterfactual variables as outlined in Sec.~\ref{sec:counterfactuals}.
Rather than considering the inputs as random variables $X$ and $Y$, we take copies of each output variable for all input combinations, i.e.~$A_{xy}$ and $B_{xy}$.
In contrast to the method based on causal Bayesian networks, this method fixes the number of inputs that the inequalities apply to but may lead to novel constraints, as is the case in the Bell scenario.
\subsubsection{Counterfactual variables for bipartite causal correlations}
To keep the discussion simple, we will consider only the case of binary inputs, but the generalization to arbitrary inputs is straightforward.
We thus consider the variables in
\begin{equation}
S = \{A_{00},A_{01},A_{10},A_{11},B_{00},B_{01},B_{10},B_{11}\}.
\end{equation}
Note that, in contrast to the example of Bell inequalities discussed in Sec.~\ref{sec:counterfactuals}, we need to consider copies of each variable for each input pair $(x,y)$.
This is a consequence of the fact that the correlations which we want to characterize may be signaling, e.g., for the causal order $\mathrm{A}\prec\mathrm{B}$, $B_{00}$ and $B_{10}$ will in general be different.
Since $A_{xy}$ and $B_{x'y'}$ are jointly observable only if $x=x'$ and $y=y'$, the marginal scenario in this case is
\begin{equation}\label{eq:counterfactualM}
\mathcal{M}=\Big\{\{A_{00},B_{00}\},\{A_{01},B_{01}\},\{A_{10},B_{10}\},\{A_{11},B_{11}\}\Big\}.
\end{equation}
In contrast to the DAG-based method, several choices of probability structure $\mathcal{S}$ compatible with $\mathcal{M}$ are possible, and the particular choice must be motivated on the basis of physical assumptions.
One natural possibility would be to take $\mathcal{S}=\mathcal{M}$, as one may have no \emph{a priori} reason to think that the variables $A_{xy}$ and $A_{x'y'}$ have simultaneous physical meaning for $(x,y)\neq (x',y')$, and hence may not have a well-defined joint probability distribution.
On the other hand, in some cases one may imagine that such inputs correspond to the choice of measurements of some physical properties that are simultaneously well-defined, as in a classical theory; hence, one may alternatively take $\mathcal{S}= \{ \cup_{M_j\in \mathcal{M}} M_j\}=\{S\}$.
In the following, we will adopt the former approach and take $\mathcal{S}=\mathcal{M}$, since this constitutes the minimum assumptions compatible with the marginal scenario.
The Shannon cone for $\mathcal{S}$ is thus
\begin{equation}\label{eq:shannonCone_counterfact}
\Gamma^{\mathcal{S}} = \Gamma_{\{A_{00},B_{00}\}}\cap\Gamma_{\{A_{01}B_{01}\}}\cap\Gamma_{\{A_{10}B_{10}\}}\cap\Gamma_{\{A_{11}B_{11}\}},
\end{equation}
as in Eq.~\eqref{eq:defconeS}.
We note however that this physically motivated choice for $\mathcal{S}$ implies, for this particular scenario, that a global probability distribution does in fact exist.\footnote{
This is the result of the more general fact that different choices of $\mathcal{S}$ may provide equivalent descriptions of marginal probabilities~\cite{Vorob1962} and entropies~\cite{BMC2016}.}
Taking $\mathcal{S}=\{S\}$ would thus provide an equivalent entropic characterization, and moreover, this equivalence also holds at the level of Shannon (rather than entropy) cones (see Appendix~\ref{sec:A_gp} for a more detailed discussion).
We follow a method analogous to that used in Sec.~\ref{sec:causal_str_bi}.
First, we characterize the cones $\Gamma^{\mathrm{A}\prec\mathrm{B}}$ and $\Gamma^{\mathrm{B}\prec\mathrm{A}}$ of entropy vectors for fixed-order causal correlations, then, we characterize the convex mixtures of such correlations.
To do this, we note that the no-signaling conditions obeyed by fixed-order correlations (see Sec.~\ref{sec:causalcorr}) impose constraints on the counterfactual variables.
For example, correlations consistent with the order $\mathrm{A}\prec\mathrm{B}$ obey $P(a|xy)=P(a|xy')$ for all $x,y,y',a$, which implies $A_{xy}=A_{xy'}$ and thus $H(A_{xy})=H(A_{xy'})$ also.
Similarly, for $\mathrm{B}\prec\mathrm{A}$, we have $H(B_{xy})=H(B_{x'y})$ for all $x,x',y$.
The cones $\Gamma^{\mathrm{A}\prec\mathrm{B}}$ and $\Gamma^{\mathrm{B}\prec\mathrm{A}}$ are thus given by
\begin{equation}\label{eq:coneABcoutnerfact}
\Gamma^{\mathrm{A}\prec\mathrm{B}} = \Gamma^\mathcal{S}\cap\textrm{L}_{\mathcal{C}}\big(\{A_{00}=A_{01},\; A_{10}=A_{11}\}\big)
\end{equation}
and
\begin{equation}\label{eq:coneBAcoutnerfact}
\Gamma^{\mathrm{B}\prec\mathrm{A}} = \Gamma^\mathcal{S}\cap\textrm{L}_{\mathcal{C}}\big(\{B_{00}=B_{10},\; B_{01}=B_{11}\}\big),
\end{equation}
where $\textrm{L}_{\mathcal{C}}(\cdot)$ again denotes the linear subspace defined by the corresponding constraints.
As in Sec.~\ref{sec:causal_str_bi}, we introduce the latent switch variable $Q$, denote the augmented set of random variables $\tilde{S}=S\cup\{Q\}$, and extend the probability structure as
\begin{equation}\label{eq:tildeS}
\widetilde{\mathcal{S}}=\Big\{ \{A_{xy},B_{xy},Q\} \mid x,y\in\{0,1\} \Big\}
\end{equation}
(in Appendix~\ref{sec:A_gp} we discuss further the implications of different choices of probability structures).
With this extra variable we note again that the convex hull $\conv(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}})$ contains the vectors $\bm{h}_{\textrm{conv}} = (H(T|Q))_{T\in\mathcal{S}^{\rm c}}$ for causal correlations. The system of inequalities $I\bm{h}\le\bm{0}$ characterizing $\conv(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}})$ can then again be transformed in a similar way to Eq.~\eqref{eq:lin_tran} into a new system $\tilde{I}\tilde{\bm{h}}\le\bm{0}$ defining the cone of corresponding entropy vectors $\tilde{\bm{h}}=(H(T))_{T\in\widetilde{\mathcal{S}}^{\rm c}}$, which we again denote by $\conv_Q(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}})$.
In contrast to the DAG-based method, the only constraint on $Q$ is, now, $H(Q)\le 1$, since $Q$ need not be independent of the counterfactual output variables $A_{xy},B_{xy}$.
Finally, we need to project onto the marginal scenario $\mathcal{M}$ in Eq.~\eqref{eq:counterfactualM}.
The causal polyhedron is thus given, in analogy to Eqs.~\eqref{eq:bip_cau_coneQ} and~\eqref{eq:bip_cau_cone}, by
\begin{align}
\label{eq:cone_counter_bi}
\Gamma^{\mathrm{causal}}_{\rm AB} = \Pi_\mathcal{M} \Big[&\Gamma^{\widetilde{\mathcal{S}}}\cap \conv_Q(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}}) \notag \\[-1mm]
& \hspace{14mm} \cap\textrm{L}_{\mathcal{C}}\big(\{H(Q)\leq 1\}\big) \Big],
\end{align}
where we have $\Gamma^{\widetilde{\mathcal{S}}}=\bigcap_{x,y\in\{0,1\}}\Gamma_{\{A_{xy},B_{xy},Q\}}$.
\subsubsection{Entropic causal inequalities for counterfactual variables and their violation}
As in Sec.~\ref{sec:causal_str_bi}, the construction above allows one to obtain the full list of entropic inequalities characterizing $\Gamma^{\mathrm{causal}}_{\rm AB}$.
After removing the trivial inequalities directly implied by Shannon constraints on $\mathcal{M}$, we find that there are 6 nontrivial entropic causal inequalities, which can be grouped into two equivalence classes of inequalities under the relabeling of inputs:
\begin{equation}\label{eq:marg_ineq1}
I(A_{00}:B_{00}) - H(A_{01}) - H(B_{10})\leq 1
\end{equation}
and
\begin{align}\label{eq:marg_ineq2}
& I(A_{00}:B_{00}) + I(A_{11}:B_{11})\notag\\
& \quad - H(A_{01}B_{01}) - H(A_{10}B_{10})\leq 2.
\end{align}
The fact that these inequalities have nontrivial bounds is, as for the DAG-based method, a result of the constraint $H(Q)\le 1$ which means $\Gamma^{\mathrm{causal}}_{\rm AB}$ is a polyhedron characterized by a set of inhomogeneous inequalities.
Indeed, if one chooses not to eliminate $Q$ from the entropic description, one obtains a convex cone characterized by the above equations, except that the right-hand sides are multiplied by $H(Q)$ (see the discussion in Appendix~\ref{apndx:DAGineqs}).
In contrast to the case for the DAG-based approach, where violation of the causal inequalities we obtained was possible even with deterministic distributions, it is clear that such distributions provide no interesting behavior in the counterfactual approach since any such distribution will have a null entropy vector.
By looking at equal mixtures of deterministic causal distributions $P^{\mathrm{A}\prec \mathrm{B}}$ and $P^{\mathrm{B}\prec \mathrm{A}}$, however, we were able to verify that the inequalities in Eqs.~\eqref{eq:marg_ineq1}--\eqref{eq:marg_ineq2} can indeed be saturated by such (causal) distributions and are thus tight.
In order to study the potential violation of these entropic inequalities, we again need to look at nondeterministic distributions.
One can easily see, however, that Eqs.~\eqref{eq:marg_ineq1}--\eqref{eq:marg_ineq2} cannot be violated when restricted to distributions satisfying $H(A_{xy})\le 1$ and $H(B_{xy})\le 1$ for all $x,y\in\{0,1\}$, as this also implies that $I(A_{xy}:B_{xy})\le 1$.
This means that the inequalities for counterfactual variables are unable to detect noncausality when both parties are restricted to binary outputs.
To find violations we again look at the extremal rays of the Shannon cone $\Gamma^{\mathcal{S}}$ of Eq.~\eqref{eq:shannonCone_counterfact} which violate one of the inequalities, and examine whether these rays can be reached by any probability distribution.
Considering bounds on $H(A_{xy})$ and $H(B_{xy})$ strictly larger than $1$, we find that violations are possible for any such bound.
Moreover, the entropy vectors giving maximal violation of Eqs.~\eqref{eq:marg_ineq1} and~\eqref{eq:marg_ineq2} are generally realizable with equal mixtures of causal and noncausal distributions.
For example, given the constraints $H(A_{xy})\le \log_2 k$ and $H(B_{xy})\le \log_2 k$ for some integer $k \ge 2$, the distribution
\begin{equation}\label{eq:margScenViolatingDist}
P_k(ab|xy)=\delta_{x,y}\ \frac{1}{k}\delta_{a,b}+(1{-}\delta_{x,y})\ \delta_{a,0}\delta_{b,0},
\end{equation}
where $a,b\in\{0,\dots,k-1\}$, realizes such an extremal point for all $k \ge 2$, and provides a violation of both Eqs.~\eqref{eq:marg_ineq1} and~\eqref{eq:marg_ineq2} for $k > 2$.
For $k=2$ (binary outputs), this distribution can be written as the convex combination
\begin{equation}
P_2(ab|xy)=\frac{1}{2} P^{\textrm{NC}}(ab|xy) + \frac{1}{2} P^{\textrm{C}}(ab|xy),
\end{equation}
where $P^{\textrm{NC}}(ab|xy)=\delta_{a\oplus x \oplus 1,y}\delta_{b\oplus y\oplus 1,x}$ maximally violates a GYNI-type inequality (it is simply a symmetry of the third distribution in Eq.~\eqref{eq:GYNIdists}, obtained by flipping all outputs), and $P^{\textrm{C}}(ab|xy)=\delta_{a,0}\delta_{b,0}$ is causal.
Even though it does not violate Eq.~\eqref{eq:marg_ineq1} or~\eqref{eq:marg_ineq2}, $P_2$ is noncausal.
The distribution $P_k$ can be seen as a possible generalization of a GYNI-violating distribution.
This link to the GYNI-type inequalities and correlations can be made more explicit by considering the related distribution
\begin{equation}\label{eq:GYNIdistGen}
P_k'(ab|xy)=\delta_{x,y}\ \frac{1}{k{-}1}\delta_{a,b}(1{-}\delta_{a,0}\delta_{b,0}) + (1{-}\delta_{x,y})\ \delta_{a,0}\delta_{b,0},
\end{equation}
with again $a,b\in\{0,\dots,k-1\}$.
We have $P_2'=P^{\textrm{NC}}$, and, for $k \ge 3$, $P_k'$ has the same entropy vector as $P_{k-1}$.
$P_k'$ can be clearly simulated from $P_2'=P^{\textrm{NC}}$ by making use of shared randomness and by letting both parties replace the output $1$ obtained from $P_2'$ by a shared random value $a=b \in \{1,\dots,k-1\}$.
It is interesting to see, then, that the GYNI-maximally-violating distributions also provide the best behavior entropically when augmented with shared randomness, even though they fail to violate the inequalities when the parties have only binary outputs.
As for the DAG-based method, it is also interesting to look at the sensitivity of the inequalities with respect to the detection of noncausality.
To do so, we again looked at distributions of the form given in Eq.~\eqref{eq:mix_distr}, but where $P^{\mathrm{NC}}$ and $P^{\mathrm{C}}$ are now equal mixtures of 3-outcome deterministic noncausal and causal distributions, respectively.
The number of such distributions makes an exhaustive search difficult, but by sampling randomly we were nonetheless able to verify the existence of distributions $P^{\varepsilon}(ab|xy)$ which violate the entropic inequalities~\eqref{eq:marg_ineq1} and~\eqref{eq:marg_ineq2} for arbitrary small $\varepsilon$.
Although the examples we found are not particularly informative or simple (and thus we refrain from giving them explicitly), they nevertheless show that the entropic inequalities~\eqref{eq:marg_ineq1} and~\eqref{eq:marg_ineq2} exhibit the desired sensitivity.
Finally, one may again ask whether one can violate any of the entropic inequalities for counterfactuals within the process matrix formalism, or whether any noncausal correlation can be mixed with a causal one to violate an entropic inequality, as is the case for entropic Bell inequalities obtained from the counterfactual approach~\cite{Chaves:2013aa}.
We leave this as an open question, but note only that we were not able to find a way to do so: for example, we were unable to find a violation (with or without the use of shared randomness) for noncausal distributions realizable within the process matrix framework.
\section{Multipartite entropic causal inequalities}
\label{sec:multipartite}
The notion of causal correlations can be extended to more than two parties in a recursive manner~\cite{Abbott:2016aa,oreshkov15}.
Consider $N$ parties ${\rm A}_1,\dots,{\rm A}_N$, with inputs $\bm{x}=(x_1,\dots,x_N)$ and outputs $\bm{a}=(a_1,\dots,a_N)$.
In any given run, one party, say ${\rm A}_k$, must act first, and none of the other parties can signal to them, which implies $P(a_k|\bm{x})=P(a_k|x_k)$.
The correlations shared by the remaining $N-1$ parties, conditioned on the input and output of the first, must also in turn be causal.
However, note that the causal order itself (and not only the response functions) of the remaining parties may depend on the input and output of the first, a phenomenon called dynamical causal order~\cite{hardy2005probability,oreshkov15,Abbott:2016aa}, and which goes beyond the standard model of fixed causal Bayesian networks.
An $N$-partite correlation $P(\bm{a}|\bm{x})$ is thus called causal if it can be decomposed in the following way~\cite{Abbott:2016aa,oreshkov15}:
\begin{equation}\label{eq:causal_corr}
P(\bm{a}|\bm{x})=\sum_{k=1}^N q_k\, P_k(a_k|x_k) \, P_{k,x_k,a_k}(\bm{a}_{\backslash k}|\bm{x}_{\backslash k}),
\end{equation}
where $\bm{x}_{\backslash k}=(x_1,\dots,x_{k-1},x_{k+1},\dots,x_N)$ and $\bm{a}_{\backslash k}=(a_1,\dots,a_{k-1},a_{k+1},\dots,a_N)$, with $q_k \ge 0$, $\sum_k q_k = 1$, and where for each $k, x_k, a_k$, $P_{k,x_k,a_k}(\bm{a}_{\backslash k}|\bm{x}_{\backslash k})$ is a causal $(N{-}1)$-partite correlation (down to the lowest level of this recursive definition, where any $1$-partite correlation is considered to be causal).
Note that, for $N=2$ this reduces to Eq.~\eqref{eq:causalCorreltn}.
The entropic approach can be generalized to the multipartite scenario using a similar recursive method.
\subsection{Causal Bayesian network method}
\begin{figure}
\caption{DAGs for tripartite causal correlations. The latent ``switch'' variable $Q$ determines which DAG is ``activated''. Correlations among variables from shaded rectangles are causal conditionally on the input and output of the party acting first. \label{fig:dag_tri}
\label{fig:dag_tri}
\end{figure}
It is instructive to first look into the details of the tripartite case -- in which case we shall denote the parties Alice (${\rm A}$), Bob (${\rm B}$) and Charlie (${\rm C}$), as is standard -- before generalizing the method to more parties.
The general method follows that used for the bipartite case in Sec.~\ref{sec:causal_str_bi}, and the relevant conditional DAG is shown in Fig.~\ref{fig:dag_tri}.
The set of observable variables to be considered here is $S = \{X,Y,Z,A,B,C\}$, the marginal scenario and the probability structure are ${\mathcal M} = {\mathcal S} = \{S\}$.
The polytope of tripartite causal correlations (i.e., of the form Eq.~\eqref{eq:causal_corr}) can be written as
\begin{equation}
\mathcal{P}_{\rm ABC}^{\rm causal} = \conv(\mathcal{P}^{\rm A},\mathcal{P}^{\rm B},\mathcal{P}^{\rm C}),
\end{equation}
where $\mathcal{P}^{\rm A}$ is the polytope of causal distributions consistent with Alice acting first and such that the remaining conditional correlation shared by Bob and Charlie is causal, and analogously for $\mathcal{P}^{\rm B}$ and $\mathcal{P}^{\rm C}$.
As a consequence, in order to define the polyhedron characterizing entropically tripartite causal correlations, which we denote $\Gamma^{\rm causal}_{\rm ABC}$, we first need to define the corresponding polyhedra, namely $\Gamma^{\rm A},\Gamma^{\rm B},\Gamma^{\rm C}$, associated with each party acting first.
Let us thus consider $\Gamma^{\rm A}$.
According to the recursive definition given in Eq.~\eqref{eq:causal_corr}, for any $x,a$, the conditional entropy vector $\bm{h}^{xa}_{\rm BC}=(H(T|X=x,A=a))_{T\subset \{Y,Z,B,C\}}$ for a correlation in $\mathcal{P}^{\rm A}$ must be contained in the bipartite causal polyhedron $\Gamma^{\rm causal}_{\rm BC}$, defined for Bob and Charlie as in Eqs.~\eqref{eq:bip_cau_coneQ}--\eqref{eq:bip_cau_cone}.
By convexity this also implies that $\bm{h}_{\rm BC}=(H(T|XA))_{T\subset \{Y,Z,B,C\}}= \sum_{x,a} P(x,a) \bm{h}^{xa}_{\rm BC}$ is in $\Gamma^{\rm causal}_{\rm BC}$.
We can then use a similar transformation to Eq.~\eqref{eq:lin_tran} to obtain constraints on $\Gamma^{\rm A}$: if entropy vectors $\bm{h}_{\rm BC}$ in $\Gamma^{\rm causal}_{\rm BC}$ satisfy the inequalities $\bm{I}\cdot\bm{h}_{\rm BC}\le \beta$, then the corresponding (unconditional) entropy vector $\bm{h}=(H(T))_{T\subset S}$ must satisfy the inequalities $\mathcal{T}_{XA}(\bm{I})\cdot\bm{h}\le\beta$.
Writing $\mathcal{T}^*_{XA}$ for the dual transformation on the space of entropy vectors, we thus have that $\bm{h}\in\mathcal{T}^*_{XA}(\Gamma_{\rm BC}^{\textrm{causal}})$.
Together with the facts that $\bm{h}$ must lie in the Shannon cone $\Gamma_S$ for the relevant variables, that all the inputs must be independent from each other, and that Alice's output must be independent from Bob and Charlie's inputs (conditioned on her input), we obtain the characterization
\begin{align}\label{eq:con_A_first}
\Gamma^{\rm A} =\,& \Gamma_S \cap \mathcal{T}^*_{XA}(\Gamma_{\rm BC}^{\rm causal}) \notag\\
& \quad \cap \textrm{L}_{\mathcal{C}}(\{ X {\perp\!\!\!\perp} Y {\perp\!\!\!\perp} Z, A {\perp\!\!\!\perp} YZ | X \}),
\end{align}
with similar expressions for $\Gamma^{\rm B}$ and $\Gamma^{\rm C}$.
Following the same approach as in Sec.~\ref{sec:causal_str_bi}, we introduce a (now three-valued) switch variable $Q$ (see Fig.~\ref{fig:dag_tri}).
Similarly to what we observed in the bipartite case, the convex hull $\conv(\Gamma^{\rm A},\Gamma^{\rm B},\Gamma^{\rm C})$ contains the conditional entropy vectors $(H(T|Q))_{T\subset S}$ for tripartite causal correlations.
The inequalities characterizing $\conv(\Gamma^{\rm A},\Gamma^{\rm B},\Gamma^{\rm C})$ can again be transformed into inequalities satisfied by the entropy vector $\tilde{\bm{h}}= (H(T))_{T\subset \tilde S}$, for variables in $\tilde S = S\cup\{Q\}$, by introducing a transformation $\mathcal{T}_Q$ as in Eq.~\eqref{eq:lin_tran}, thus defining the polyhedron ${\rm conv}_{Q}(\Gamma^{\rm A},\Gamma^{\rm B},\Gamma^{\rm C} )$ as before.
Taking into account the Shannon constraints for all variables in $\tilde S$, the independence constraints ${\rm CI}_{Q} = (X{\perp\!\!\!\perp} Y {\perp\!\!\!\perp} Z {\perp\!\!\!\perp} Q)$ and the bound $H(Q)\leq \log_2 3$, and finally projecting onto the observable variables in $S$, we see that the entropy vectors for tripartite causal correlations belong to the polyhedron
\begin{align}
\hspace{-2mm} (\Gamma^{\mathrm{causal}}_{\rm ABC})_0 = & \ \Pi_S \left[ \Gamma_{\tilde S} \cap {\rm conv}_{Q}(\Gamma^{\rm A},\Gamma^{\rm B},\Gamma^{\rm C} ) \right.\notag\\[-1mm]
& \qquad \quad \cap \textrm{L}_{\mathcal{C}}(\{{\rm CI}_{Q}, \, H(Q)\leq \log_2 3\}\big)\Big].
\end{align}
While this characterization is certainly valid, some subtleties arising from the differences between the probabilistic and entropic descriptions allow one to actually make it tighter.
Specifically, certain conditions implied by the definition~\eqref{eq:causal_corr} need not be implied by the corresponding entropic definition outlined above.
For example, if $P(abc|xyz)$ is a causal correlation, then the bipartite marginal distributions $P_{x}(bc|yz)=\sum_{a}P(abc|xyz)$ and $P(bc|yz)=\sum_{x}P(x)P_{x}(bc|yz)$ are both causal (as are the corresponding marginals for each other pair of parties)~\cite{Abbott:2016aa}.
This implies that the entropy vectors $(H(T|X))_{T\subset\{Y,Z,B,C\}}$ and $(H(T))_{T\subset\{Y,Z,B,C\}}$ corresponding to a tripartite causal correlation must also satisfy all the inequalities characterizing the bipartite causal polyhedron $\Gamma^{\rm causal}_{\rm BC}$ -- which may not necessarily be implied by the characterization of $(\Gamma^{\mathrm{causal}}_{\rm ABC})_0$ above.
We can thus tighten the previous characterization, and define the tripartite causal polyhedron as\footnote{In Eq.~\eqref{eq:tri_causal_polyhedron} we abuse the notation slightly and denote by $\Gamma_{\rm BC}^{\rm causal}$ the set of entropy vectors $(H(T))_{T\subset S}$ -- instead of $(H(T))_{T\subset \{Y,Z,B,C\}}$ -- which satisfy the constraints characterizing $\Gamma_{\rm BC}^{\rm causal}$ as defined in Eqs.~\eqref{eq:bip_cau_coneQ}--\eqref{eq:bip_cau_cone}. The transformation $\mathcal{T}_{X}$, of which $\mathcal{T}^*_{X}$ is the dual, is again defined in a similar way as in Eq.~\eqref{eq:lin_tran}.}
\begin{equation} \label{eq:tri_causal_polyhedron}
\Gamma_{\rm ABC}^{\rm causal} = (\Gamma^{\mathrm{causal}}_{\rm ABC})_0 \cap \Gamma_{\rm BC}^{\rm causal} \cap \mathcal{T}^*_{X}(\Gamma_{\rm BC}^{\rm causal}) \cap \text{[perms.]},
\end{equation}
where $\text{[perms.]}$ denotes the permutations of the preceding two terms for the other parties.
Note that such extra constraints do not need to be imposed in the bipartite case since the causality of all one-party marginals is equivalent to them being valid probability distributions, which is already assured by the elemental inequalities.
To extend the above idea to the general multipartite case of Eq.~\eqref{eq:causal_corr}, we simply define recursively (here the notation should be self-evident)
\begin{align}
\Gamma^{{\rm A}_k} = \Gamma_{\{\bm{X},\bm{A}\}}\cap \mathcal{T}^*_{X_k A_k}\left(\Gamma_{\bm{\mathrm{A}}_{\backslash k}}^{\rm causal}\right) \cap \textrm{L}_{\mathcal{C}}({\rm CI}_{{\rm A}_k}),
\end{align}
where ${\rm CI}_{{\rm A}_k}$ denotes the set of independence constraints resulting from the assumption that all parties' inputs are independent, i.e.~$X_1 {\perp\!\!\!\perp} \dots {\perp\!\!\!\perp} X_N$, and that party $k$ acts first, which implies $A_k {\perp\!\!\!\perp} \bm{X}_{\backslash k}|X_k$.
The causal polyhedron is then defined as
\begin{align}\label{eq:con_cau_mult}
\Gamma^{\rm causal}_{\bm{\mathrm{A}}}= & \ \Pi_{\bm{X},\bm{A}} \left[ \Gamma_{\{\bm{X},\bm{A},Q\}} \cap {\rm conv}_{Q}(\{\Gamma^{{\rm A}_k}\}_k ) \right. \notag\\[-1mm]
& \hspace{11mm} \cap \textrm{L}_{\mathcal{C}}(\{{\rm CI}_{Q}, \, H(Q)\leq \log_2 N\}\big)\Big]\notag\\
& \ \bigcap_k \left[ \Gamma_{\bm{\mathrm{A}}_{\backslash k}}^{\rm causal} \cap \mathcal{T}^*_{X_k}\left(\Gamma_{\bm{\mathrm{A}}_{\backslash k}}^{\rm causal}\right) \right],
\end{align}
where ${\rm CI}_{Q}$ denotes the independence relation between all inputs and $Q$, i.e.~$X_1 {\perp\!\!\!\perp} \cdots {\perp\!\!\!\perp} X_N {\perp\!\!\!\perp} Q$.
\subsection{Counterfactual variable method}\label{sec:count_mult}
A similar generalization is possible also for the counterfactual method.
Again, it is instructive to look first at the tripartite case, where the set of variables to be considered is $S = \{A_{xyz},B_{xyz},C_{xyz}\}_{x,y,z}$, the marginal scenario is ${\mathcal M} = \{\{A_{xyz},B_{xyz},C_{xyz}\}\}_{x,y,z}$ and we take the probability structure to be ${\mathcal S} = {\mathcal M}$.
We start by defining the polyhedron for the case in which Alice acts first,
\begin{align}
\Gamma^{\rm A} = \bigcap_{xyz} \Big[& \Gamma_{\{A_{xyz},B_{xyz},C_{xyz}\}} \cap \mathcal{T}^*_{A_{xyz}}(\Gamma_{\rm BC}^{\rm causal}) \notag\\[-3mm]
& \qquad \cap \textrm{L}_{\mathcal{C}}\big( \{A_{xyz} = A_{xy'z'} \}_{y'z'}\big)\Big],
\end{align}
which is the analogue, for the counterfactual method, of the polyhedron in Eq.~\eqref{eq:con_A_first}. Similar definitions hold for $\Gamma^{\rm B}$ and $\Gamma^{\rm C}$.
The tripartite polyhedron of causal counterfactual inequalities can then be defined, following a similar reasoning to the previous case, as
\begin{align}\label{eq:count_trip_pol}
&\Gamma_{\rm ABC}^{\rm causal} = \Pi_{\mathcal{M}}\! \Big[ \Gamma^{\widetilde{\mathcal{S}}}\cap {\rm conv}_{Q}(\Gamma^{\rm A},\Gamma^{\rm B},\Gamma^{\rm C} ) \notag\\[-2mm]
& \hspace{30mm} \cap \textrm{L}_{\mathcal{C}}\!\big(\!\{\!H(Q)\!\leq\! \log_2 \!3\}\!\big)\!\Big]\notag\\
& \hspace{15mm} \bigcap_x \Gamma_{{\rm BC}|x}^{\rm causal} \ \bigcap_y \Gamma_{{\rm AC}|y}^{\rm causal} \ \bigcap_z \Gamma_{{\rm AB}|z}^{\rm causal},
\end{align}
where $\widetilde{\mathcal{S}}=\{ \{A_{xyz},B_{xyz},C_{xyz},Q\}\}_{x,y,z}$ and
$\Gamma_{{\rm BC}|x}^{\rm causal}$ is defined by imposing the constraints characterizing $\Gamma_{\rm BC}$ (\emph{a priori} defined for some variables $B_{yz}, C_{yz}$) to the variables $B_{xyz}, C_{xyz}$, and with similar definitions for $\Gamma_{{\rm AC}|y}^{\rm causal}$ and $\Gamma_{{\rm AB}|z}^{\rm causal}$.
As for the case based on causal Bayesian networks, the construction in Eq.~\eqref{eq:count_trip_pol} can then be generalized to an arbitrary number of parties in a recursive way.
\section{Information bounds in causal games}\label{sec:infoBounds}
One of the advantages of the entropic approach is that it allows information theoretic constraints to be naturally imposed, derived, and interpreted~\cite{Pawlowski2009,Chaves2015}.
As an illustration, we consider a simple application of our approach to understanding the role of bounded communication in causal games.
Consider the generalization of the GYNI game described in Sec.~\ref{sec:causalcorr} to arbitrary numbers of inputs and outputs, in which two parties try to maximize the winning probability $p_{\mathrm{succ}}=P(a=y,b=x)$.
If the parties operate causally, then in any given round of the game only one-way communication may occur.
One may be interested in the effect of limiting the amount communication that can occur in any such round.
In the entropic framework, this can easily be taken into account by adding an additional constraint of the form $I(X:B)\leq I_{\mathrm{max}}$ to $\Gamma^{\mathrm{A}\prec\mathrm{B}}$ in order to restrict $B$'s dependency on $X$, and similarly imposing $I(Y:A)\leq I_{\mathrm{max}}$ to $\Gamma^{\mathrm{B}\prec\mathrm{A}}$, where the quantity $I_{\mathrm{max}}$ represents the maximum allowable entropy of the classical message communicated by the parties.
For example, if the parties are permitted, in each round, to exchange a classical $d$-dimensional system, then $I_{\mathrm{max}}=\log_2 d$.
In general, the amount of one-way communication $I_{\mathrm{max}}$ does not need to be specified in advance, it will appear as a parameter in our inequalities.
By applying the approach of Sec.~\ref{sec:causal_str_bi} to this scenario one finds that causal correlations must then obey the inequality
\begin{equation}\label{eq:inf_caus_simpler}
I(X:B) + I(Y:A) \leq I_{\mathrm{max}},
\end{equation}
i.e., the sum of the two mutual informations is similarly bounded by $I_{\mathrm{max}}$.
Although this is perhaps not unexpected, it shows the ease with which such bounds can be derived in the entropic framework.
A more subtle variant is obtained by considering a slight generalization of the causal game proposed by Oreshkov, Costa, and Brukner (OCB) in Ref.~\cite{oreshkov12}.
In this game, the goal is also for one party to guess the other party's input; in contrast to the GYNI game, however, an additional input random bit $Y'$ is given,\footnote{In the original OCB game, only Bob receives the input $Y'$, whereas in the variant we consider here, both parties have access to it.} which determines whether it is Bob who should guess Alice's input (if $Y'=0$) or vice versa (if $Y'=1$).
The parties thus now attempt to maximize the winning probability
\begin{equation}
p_{\mathrm{succ}} = \frac{1}{2}\Big(P(b=x\,|\,Y'=0) + P(a=y\,|\,Y'=1)\Big).
\end{equation}
An analogous entropic inequality can be obtained via a combination of the methods discussed in Sec.~\ref{sec:bipartite}.
Since the relevant direction of communication in each round of this game depends on the additional input $Y'$, we will combine the DAG-based method for the variables $X,Y,A,B$ with the counterfactual approach to condition on $Y'$.
More precisely, one may take $\mathcal{S}=\mathcal{M} = \big\{ \{X,Y,A_{y'},B_{y'}\} \big\}_{y'}$ and $\widetilde{\mathcal{S}} = \big\{ \{X,Y,A_{y'},B_{y'},Q\} \big\}_{y'}$;
the relevant causal constraints for the cones $\Gamma^{\mathrm{A}\prec\mathrm{B}}$ and $\Gamma^{\mathrm{B}\prec\mathrm{A}}$ and the polyhedron $\widetilde{\Gamma}^{\text{causal}}_{\rm AB}$ are the same as those imposed on $X,Y,A,B,Q$ in the DAG-based method, except that now they are applied to each copy of the conditional variables $A_{y'}$ and $B_{y'}$, and the communication bounds $I(X:B_{y'})\leq I_{\mathrm{max}}$ and $I(Y:A_{y'})\leq I_{\mathrm{max}}$ are imposed on the corresponding cones.
Notice that, in this way, we are assuming that $X {\perp\!\!\!\perp} Y {\perp\!\!\!\perp} Y' {\perp\!\!\!\perp} Q$.
Combining the above constraints with the analysis in Sec.~\ref{sec:bipartite}, one finds that causal correlations must obey
\begin{equation}
\label{eq:inf_caus}
I(X:B\,|\,Y'=0) + I(Y:A\,|\,Y'=1) \leq I_{\mathrm{max}}.
\end{equation}
This inequality, for the special case of binary inputs and outputs and with $I_{\mathrm{max}}=1$, was proposed in
Ref.~\cite{Ibnouhsein2015} as a potential principle to bound the set of correlations obtainable within the process matrix formalism,\footnote{Ref.~\cite{Ibnouhsein2015} proposed this inequality in the framework of the original OCB game. However, one can easily see that our derivation of Eq.~\eqref{eq:inf_caus} in the more general scenario implies that it must hold in that framework too. Indeed, if only Bob receives $Y'$, then this implies the additional constraint $H(A_{0})=H(A_1)$ when $\mathrm{A}\prec\mathrm{B}$. The set of correlations obtainable is thus a subset of those obtainable in the more general version of the game, and thus Eq.~\eqref{eq:inf_caus} must again hold true.} in analogy with the celebrated information causality principle~\cite{Pawlowski2009} that provides bounds on the strength of bipartite quantum correlations.
Our approach allowed us to show that Eq.~\eqref{eq:inf_caus} indeed holds for causal processes, but it remains to be seen whether such a constraint on mutual informations for causal correlations can be violated within the process matrix framework.
This example, however, highlights the potential of the entropic approach to causal correlations for studying information-theoretic principles.
\section{Discussion}
Since Bell first formulated his eponymous theorem, understanding the role of causality within quantum mechanics has been a central yet thorny goal.
Complicating matters further, the very idea of a definite causal order itself has begun to be questioned.
While sophisticated frameworks have been introduced in an effort to free physical theories from the shackles of a rigid causal framework, the issue of whether nature permits violations of causal inequalities remains an elusive question.
Against this backdrop, our aim in this paper was to introduce an entropic approach to studying causal correlations, and to this end we presented two complementary methods: the first based on the consideration of the entropies of the variables appearing in the causal Bayesian networks describing causal scenarios, and the second based on a counterfactual description of the outcome variables appearing in such networks.
Focusing on bipartite causal scenarios, we described in detail the successful application of both methods to derive nontrivial entropic causal inequalities, before showing how the characterizations can be generalized to multipartite scenarios.
In contrast to the usual approach to causal correlations based on probability distributions, the entropic causal inequalities we derived using both methods are valid for any finite number of possible outcomes, as well as for any number of inputs for the first method based on causal Bayesian networks, and thus provide a very concise description of causal correlations.
We discussed the ability for the derived entropic causal inequalities to witness the noncausality of several classes of interesting noncausal correlations, but were nonetheless unable to find violations of the inequalities by correlations obtainable within the process matrix formalism~\cite{oreshkov12} using qubit systems.
In light of the coarse-grained description provided by entropic inequalities and the fact that the known violations of standard causal inequalities are in general rather small~\cite{Branciard:2016aa}, this is arguably an unsurprising negative result.
The question of whether entropic causal inequalities can be violated within the process matrix formalism and (more importantly) by quantum correlations thus remains open.
More generally, our construction can be used to characterize arbitrary convex combinations of different causal Bayesian networks, and thus provides, for example, a natural tool to investigate stronger notions of multipartite Bell nonlocality~\cite{Svetlichny1987,Gallego2012,Bancal2013,Chaves2016causal} from the entropic perspective.
In view of this new framework for the study of causal correlations we believe that several other directions of research can naturally be pursued.
Here we focused on using the Shannon entropies of the relevant variables, but it is known that, at least in particular scenarios, the same approach can be used to derive constraints using certain generalized entropies~\cite{Rastegin2014,Wajs2015} and even with non-statistical information measures such as the Kolmogorov complexity~\cite{Chaves2014b}.
Can our framework be extended to these other information measures, and if so, are they more sensitive to violations of causality?
Similarly, one may wonder whether the addition of non-Shannon-type inequalities to the entropic descriptions of causal correlations considered might lead to tighter constraints~\cite{zhang98,Fritz:2013aa,weilenmann16}.
Moreover, in the multipartite characterization of causal correlations in Sec.~\ref{sec:multipartite} we also saw that causality imposes additional entropic constraints on marginal and conditional distributions that allow us to tighten our characterization in Eq.~\eqref{eq:tri_causal_polyhedron}.
It remains an open question whether additional such constraints can be found that tighten even further the characterization.
Another important direction to consider would be the ability to formulate, and perhaps violate, information-theoretical principles~\cite{Chaves2015} of causality.
We provided, as a simple application, an idea for one possible approach, showing how simple bounds on mutual informations can be derived for causal games where communication is limited in each direction.
It would be interesting to see, in particular, whether such principles could be violated within the process matrix formalism and, if so, the connection to the violation of causal inequalities.
For example, does the violation of causal inequalities imply the violation of some principle implied by quantum mechanics?
We expect our results to motivate these and many more future investigations.
\section*{Acknowledgments}
We acknowledge fruitful discussions with Philippe Allard
Gu\'erin, Flavio Baccari, and \v{C}aslav Brukner.
This work was funded by the DAAD, the ``Retour Post-Doctorants'' program (ANR-13-PDOC-0026) of the French National Research Agency, the Brazilian ministries MEC and MCTIC, and the FWF (Project: M 2107 Meitner-Programm).
\onecolumngrid
\appendix
\section{Causal correlations not contained in $\conv(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}})$}
\label{apndx:counterex}
Starting with the systems of inequalities $I_0\bm{h}\le\bm{0}$ and $I_1\bm{h}\le\bm{0}$ characterizing the cones $\Gamma^{\mathrm{A}\prec\mathrm{B}}$ and $\Gamma^{\mathrm{B}\prec\mathrm{A}}$ defined in Eqs.~\eqref{eq:coneAB} and~\eqref{eq:coneBA}, the characterization $I\bm{h}\le\bm{0}$ of $\conv(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}})$ can be found by first solving the extremal ray enumeration problem for the extremal rays of $\Gamma^{\mathrm{A}\prec\mathrm{B}}$ and $\Gamma^{\mathrm{B}\prec\mathrm{A}}$, taking the union of these rays and finally solving the facet enumeration problem for the inequalities characterizing $\conv(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}})$.
We find that there are six nontrivial inequalities (i.e., non Shannon-type inequalities) for $\conv(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}})$, which correspond to four equivalence classes of inequalities under exchange of parties:\footnote{For compactness we generically write entropic inequalities not just in terms of Shannon entropies (as defined in Eq.~\eqref{ShannonE}), but also in terms of conditional entropies (of the form $H(A|B) \coloneqq H(AB) - H(B)$), of mutual information ($I(A:B) \coloneqq H(A)+H(B) - H(AB)$) and of conditional mutual information ($I(A:B|C)\coloneqq H(AC)+H(BC)-H(ABC)-H(C)$). The expressions given for the inequalities are of course not unique.}
\begin{align}
\label{eq:dag_conv_hull}
& I(X:YA) + I(Y:XB) - I(XY:AB) \leq 0\notag \\
& I(A:B) - I(A:B|X) - I(A:B|Y) \leq 0\notag\\
& I(X:A|B) - I(XB:A|Y) \leq 0\\
& I(A:B|X) - I(A:B|XY) - I(Y:B) \leq 0.\notag
\end{align}
In order to see that there are causal bipartite correlations that have entropy vectors not contained in $\conv(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}})$, consider the following counterexample.
Take $P^{\mathrm{A}\prec\mathrm{B}}(ab|xy)=\delta_{a,x}\delta_{b,x}$ and $P^{\mathrm{B}\prec\mathrm{A}}(ab|xy)=\delta_{a,y}\delta_{b,y}$ and consider the inputs $x,y$ to be uniformly distributed so that $P^{\mathrm{A}\prec\mathrm{B}}(xyab)=\frac{1}{4}P^{\mathrm{A}\prec\mathrm{B}}(ab|xy)$ and $P^{\mathrm{B}\prec\mathrm{A}}(xyab)=\frac{1}{4}P^{\mathrm{B}\prec\mathrm{A}}(ab|xy)$.
The distribution $P(xyab)=\frac{1}{2}P^{\mathrm{A}\prec\mathrm{B}}(xyab)+\frac{1}{2}P^{\mathrm{B}\prec\mathrm{A}}(xyab)$ thus also defines a causal correlation $P(ab|xy)$, but one can verify that the entropy vector for $P(xyab)$ violates the first and last inequalities in~\eqref{eq:dag_conv_hull} with a value for the left-hand sides of $1-\frac{3}{2}\log_2\frac{3}{2}\approx 0.123 > 0$.
A similar conclusion can also be reached for the method based on counterfactual variables: starting from the definitions of $\Gamma^{\mathrm{A}\prec\mathrm{B}}$ and $\Gamma^{\mathrm{B}\prec\mathrm{A}}$ in Eqs.~\eqref{eq:coneABcoutnerfact} and~\eqref{eq:coneBAcoutnerfact} one finds that the inequalities characterizing $\conv(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}})$ are precisely the same as the causal inequalities in Eqs.~\eqref{eq:marg_ineq1} and Eq.~\eqref{eq:marg_ineq2} except with bounds on the right-hand side of $0$.
One can easily verify that Eqs.~\eqref{eq:marg_ineq1} and Eq.~\eqref{eq:marg_ineq2} can be saturated by causal correlations (for some equal mixtures of correlations $P^{\mathrm{A}\prec \mathrm{B}}$ and $P^{\mathrm{B}\prec \mathrm{A}}$), thus providing such a counterexample.
\section{Bipartite entropic causal inequalities from the DAG method}
\label{apndx:DAGineqs}
The following is the full list of (equivalence classes of) entropic causal inequalities obtained from the DAG method, up to their symmetries under the exchange of parties.
Ten (of the twenty) families of inequalities have bounds of $0$ and can be violated by binary distributions:
\begin{align}
\label{eq:dag_v1}
& I(X:YA) + I(Y:XB) - H(AB) \leq 0\notag \\
& I(A:B) - I(A:B|X) - I(A:B|Y) -2H(AB|XY) \leq 0\notag\\
& I(X:YA) +I(Y:AB) -H(B|X) -H(A) \leq 0\notag\\
& I(A:B|X) - I(Y:B) -2H(B|XY) \leq 0\notag \\
& I(A:B|X) - I(A:B) - H(A|YB)-2H(B|XY) \leq 0 \notag\\
& I(XA:Y) + I(YB:X) - H(X|YA) - H(A) \leq 0\\
& I(XA:Y) + I(YB:X) - H(B|YA) - H(A) \leq 0 \notag \\
& I(XA:Y) + I(YB:X) - I(A:B) -H(B|YA)-H(YA|X) \leq 0\notag\\
& I(XA:Y) + I(YB:X) - I(A:B) -H(B|YA) -H(AB|X) \leq 0 \notag \\
& I(XA:Y) + I(YB:X)- I(A:B) +I(X:A|Y) -H(XAB) \leq 0.\notag
\end{align}
Two more have non-zero bounds but, under the constraints that $H(A)\le 1$, $H(B)\le 1$, $H(X)\le 1$, $H(Y)\le 1$, turn out to be implied by the previous inequalities in Eq.~\eqref{eq:dag_v1}:
\begin{align}
\label{eq:dag_v2}
& H(X|B)+H(Y|A) - I(A:B|XY) -2H(X|YB) -2H(Y|XA) \le 1\notag\\
& I(XB:A) -3I(X:A) -3I(Y:B) - 4I(A:B|XY)-2H(XB|YA)+2H(B) \le 2.
\end{align}
Four correspond to ``corrected'' versions of the inequalities~\eqref{eq:dag_conv_hull} characterizing $\conv(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}})$, and cannot be violated by binary distributions:
\begin{align}
\label{eq:dag_v3}
& I(X:YA) + I(Y:XB) - I(XY:AB) \le 1\notag \\
& I(A:B) - I(A:B|X) - I(A:B|Y) \le 2\notag\\
& I(X:A|B) - I(XB:A|Y) \le 1\\
& I(A:B|X) - I(A:B|XY) - I(Y:B) \le 1,\notag
\end{align}
while a further four can also not be violated by binary distributions:
\begin{align}
\label{eq:dag_v4}
& I(A:B|X) - I(A:B) - I(X:A|YB) - H(B|XY) \le 1\notag \\
& I(A:B|X) - I(A:B) - I(A:B|XY) - H(X|YB) \le 1\notag \\
& I(A:B|X) - I(A:B) - I(A:B|XY) - H(A|YB) \le 1 \\
& I(A:B|X) - I(A:B) - I(A:B|XY) + I(X:YA)+H(B|Y)-H(XB) \le 1\notag.
\end{align}
We note that, instead of projecting $\widetilde\Gamma^{\mathrm{causal}}_{\rm AB}$ (as defined in Eq.~\eqref{eq:bip_cau_coneQ}) onto the marginal scenario $\mathcal{M}=\big\{\{X,Y,A,B\}\big\}$ to obtain these entropic causal inequalities, one could start by projecting it onto the marginal scenario $\mathcal{M}'=\big\{\{X,Y,A,B\big\},\{Q\}\}$ which would amount to eliminating all entropies $H(T \cup \{Q\})$ for all nonempty subsets $T \subset \{X,Y,A,B\}$ from the description while keeping $H(Q)$.
By doing so, one obtains the same inequalities given in Eqs.~\eqref{eq:dag_v1} to~\eqref{eq:dag_v4}, except with all the right-hand sides multiplied by $H(Q)$.
The inequalities in Eq.~\eqref{eq:dag_v1} thus have no dependence on $H(Q)$ (i.e., on the exent to which correlations of different causal orders are mixed), while the remaining inequalities have a nontrivial dependence on it.
By eliminating $H(Q)$ using the constraint $H(Q)\le 1$ one then recovers the entropic causal inequalities above.\footnote{A similar procedure can also be followed for the approach with counterfactual variables, in which case one obtains upper-bounds of $H(Q)$ and $2H(Q)$ in Eqs.~\eqref{eq:marg_ineq1} and~\eqref{eq:marg_ineq2} (or $0$ for fixed-order correlations when $H(Q)=0$), before eliminating $H(Q)$ and obtaining Eqs.~\eqref{eq:marg_ineq1}--\eqref{eq:marg_ineq2} again.}
The inequalities containing $H(Q)$ may be of interest if, for some reason, one puts a nontrivial bound on $H(Q)$ (e.g., if one knows that one fixed causal order is more probable than the other), as they give novel constraints in such situations.
In the extreme case, if we know that $H(Q) = 0$, then the inequalities we obtain (namely Eqs.~\eqref{eq:dag_v1}--\eqref{eq:dag_v4}, with all upper bounds replaced by $0$) are valid for fixed-order causal correlations.
All of the inequalities in Eqs.~\eqref{eq:dag_v3}--\eqref{eq:dag_v4} with upper-bounds multiplied by $H(Q)$, except the second one in~\eqref{eq:dag_v3}, can be violated by binary noncausal correlations for any $H(Q)<1$; for the second inequality in Eq.~\eqref{eq:dag_v3} we were only able to find a violation for $H(Q)<\frac{1}{2}(1+\frac{3}{2}\log_2\frac{3}{2})\approx 0.939$.
\section{Relations between different probability structures}\label{sec:A_gp}
In the application of the counterfactual method to causal correlations discussed in Sec.~\ref{sec:counterfactuals_causal}, as a result of the structure of the marginal scenario one can prove that different choices of probability structure $\mathcal{S}$ give rise to the same observed marginal distributions.
This is due to the fact that since all the marginals $M_j\in\mathcal{M}$ are disjoint, they are always consistent with the global product probability distribution
\begin{equation}\label{eq:prodext}
P(a_{00},\ldots,b_{11}) = \prod_{xy} P(a_{xy},b_{xy}).
\end{equation}
Hence, whichever probability structure $\mathcal{S}$ we choose (consistent with $\mathcal{M}$), the observed marginal probabilities can always be interpreted as arising from a global probability distribution.
Similarly, the choice of extended probability structure $\widetilde{\mathcal{S}}$ including the switch variable $Q$ in Eq.~\eqref{eq:tildeS} implies also the existence of a global probability distribution
\begin{equation}\label{eq:prodcondext}
P(a_{00},\ldots,b_{11},q) = P(q)\prod_{xy} P(a_{xy},b_{xy}|q).
\end{equation}
(Such a construction is also possible in some other types of scenarios; see Ref.~\cite{Vorob1962} for more general results.)
It thus follows that the probability structures $\widetilde{\mathcal{S}}$ that we chose and $\widetilde{\mathcal{S}}'=\{\tilde{S}\}$ again give rise to the same marginal distributions on $\mathcal{M}$.
A similar analysis can also be applied to the recursive method presented for the multipartite case in Sec.~\ref{sec:count_mult}.
At the level of entropic inequalities, however, the fact that we are considering Shannon inequalities that provide only an outer approximation of the entropy cone means that one may \emph{a priori} obtain different constraints depending on which of these equivalent probability structures one assumes.
For the specific case of a marginal scenario with disjoint elements, i.e., $M_i\cap M_j = \emptyset$ for all $M_i,M_j\in \mathcal{M}$, a result by Mat\'u\v{s} (see Remark 1 in Ref.~\cite{matus2007}) implies, nevertheless, that choosing $\mathcal{S}=\mathcal{M}$ or $\mathcal{S}'=\{S\}$, with $S=\cup_{M_i\in \mathcal{M}} M_i$, also provides an equivalent description for the Shannon cone. More precisely, we have
\begin{equation}\label{eq:adh_disj}
\Pi_{\mathcal{M}}( \Gamma^{\mathcal{S}'} ) = \Pi_{\mathcal{M}}( \Gamma_S ) = \Pi_{\mathcal{M}}\big( \Gamma_S \cap \textrm{L}_{\mathcal{C}}(\{\bm{X}_{M_i} {\perp\!\!\!\perp} \bm{X}_{M_j}\}_{M_i,M_j\in \mathcal{M},i\neq j})\big) = \Pi_{\mathcal{M}} ( \bigcap_{M_i\in \mathcal{M}} \Gamma_{M_i} ) = \Pi_{\mathcal{M}}( \Gamma^{\mathcal{S}} ),
\end{equation}
where $\bm{X}_{M_i}$ denotes the joint random variable associated with the subset of variables $M_i\in\mathcal{M}$. The linear constraints in Eqs.~\eqref{eq:coneABcoutnerfact} and~\eqref{eq:coneBAcoutnerfact} can then be imposed after the projection. Hence, the use of $\mathcal{S}=\mathcal{M}$ or $\mathcal{S}'=\{S\}$ is equivalent, in this case, even at the level of the Shannon cone description of $\Gamma^{\mathrm{A}\prec\mathrm{B}}$ and $\Gamma^{\mathrm{B}\prec\mathrm{A}}$.
One may hope that a similar analysis can be applied to show the equivalence of the probability structures $\widetilde{\mathcal{S}}$ and $\widetilde{\mathcal{S}}'=\{\tilde{S}\}$, where $\tilde{S}_i\cap \tilde{S}_j=\{Q\}$ for all distinct $\tilde{S}_i,\tilde{S}_j\in \widetilde{\mathcal{S}}$.
However, even though the marginal scenario of interest remains the same as above, one no longer has $\widetilde{\mathcal{S}}=\mathcal{M}$ and, moreover, Eq.~\eqref{eq:cone_counter_bi} involves extra constraints given by $\conv_Q(\Gamma^{\mathrm{A}\prec\mathrm{B}},\Gamma^{\mathrm{B}\prec\mathrm{A}})$ and $H(Q)\leq 1$.
As a result, the previous approach does not allow us to show the equivalence of choice between $\widetilde{\mathcal{S}}$ and $\widetilde{\mathcal{S}}'$ in this situation, which we were indeed unable to prove.
Nonetheless, we stress that any possible differences in tightness between the entropic inequalities arising here from one particular probability structure or another are not due to stricter physical assumptions (i.e., the existence of joint probability distributions), but are rather due to different outer approximations of the true entropy cone (or polyhedron) via Shannon inequalities.
We remark, however, that the choice of a minimal probability structure is computationally easier to handle due to the much lower number of variables; for example, compare the case $\mathcal{S}=\mathcal{M}$ in Eq.~\eqref{eq:shannonCone_counterfact}, where $|\mathcal{S}^{\rm c}| = 13$ and thus $\Gamma^{\mathcal{S}} \subset \mathbb{R}^{13}$ (and where the entropy vectors to be considered are effectively $12$-dimensional, since $H(\emptyset)$ is fixed to be $0$), with the corresponding case for $\mathcal{S}'=\{S\}$, where $\Gamma_S \subset \mathbb{R}^{2^8}=\mathbb{R}^{256}$ (with effectively $255$-dimensional entropy vectors).
For an extensive discussion of the role of such constraints in the computation of tighter approximations to the entropy cone we refer the reader to Ref.~\cite{BMC2016}.
\end{document} | math | 117,304 |
\begin{document}
\FXRegisterAuthor{sm}{asm}{SM}
\FXRegisterAuthor{ja}{aja}{JA}
\FXRegisterAuthor{ls}{als}{LS}
\FXRegisterAuthor{tw}{twm}{TW}
\title{On Finitary Functors}
\begin{abstract}
A simple criterion for a functor to be finitary is presented: we call $F$
finitely bounded if for all objects $X$ every finitely generated subobject of
$FX$ factorizes through the $F$-image of a finitely generated subobject of
$X$. This is equivalent to $F$ being finitary for all functors between
`reasonable' locally finitely presentable categories, provided that $F$
preserves monomorphisms. We also discuss the question when that last
assumption can be dropped.
The answer is affirmative for functors between categories such as Set,
K-Vec (vector spaces), boolean algebras, and actions of any finite group
either on Set or on K-Vec for fields K of characteristic 0.
All this generalizes to locally $\lambda$-presentable categories,
$\lambda$-accessible functors and $\lambda$-presentable algebras. As
an application we obtain an easy proof that the Hausdorff functor on
the category of complete metric spaces is $\aleph_1$-accessible.
\end{abstract}
\section{Introduction}
\label{sec:intro}
In a number of applications of categorical algebra, {\em finitary
functors}, i.e.~functors preserving filtered colimits, play an
important role. For example, the classical varieties are precisely the
categories of algebras for finitary monads over $\Set$. How does one
recognize that a functor $F$ is finitary? For endofunctors of $\Set$
there is a simple necessary and sufficient condition: given a set $X$,
every finite subset of $FX$ factorizes through the image by $F$ of a
finite subset of $X$. This condition can be formulated for general
functors $F\colon \A\to \B$:
given an object $X$ of $\A$, every finitely generated subobject of $FX$ in $\B$ is required to factorize through the image by $F$ of a finitely generated subobject of $X$ in $\A$. We call such functors {\em finitely bounded}. For functors between locally finitely presentable categories which preserve monomorphisms we prove
$$\text{finitary} \iff \text{finitely bounded}$$
whenever finitely generated objects are finitely presentable. (The last condition is, in fact, not only sufficient but also necessary for the above equivalence.)
What about general functors, not necessarily preserving monomorphisms? We prove
the above equivalence whenever $\A$ is a strictly locally finitely presentable
category, see Definition \ref{D:strict}. Examples of such categories are sets,
vector spaces, group actions of finite groups, and $S$-sorted sets with $S$
finite. Conversely, if the above equivalence is true for all functors from $\A$
to $\Set$, we prove that a weaker form of
strictness holds for $\A$.
All of the above results can be also formulated for locally
$\lambda$-presentable categories and $\lambda$-accessible functors.
We use this to provide a
simple proof that the Hausdorff functor on the category of complete metric
spaces is countably accessible.
\paragraph{Acknowledgement.}
We are very grateful to the anonymous referee: he/she
found a substantial simplification of the main definition (strictly and semi-strictly lfp
category) and pointed us to atomic toposes (see
Example \ref{E:strict}\ref{e:topos}).
We are also grateful for discussions about pure subobjects with John Bourke, Ivan
Di Liberti, and Ji\v{r}\'\i~Rosick\'y.
\section{Preliminaries}\label{sec:prel}
In this section we present properties of finitely presentable and finitely generated objects which will be useful in the subsequent sections.
Recall that an
object $A$ in a category $\A$ is called \emph{finitely presentable} if its
hom-functor $\A(A,-)$ preserves filtered colimits, and $A$ is called
\emph{finitely generated} if $\A(A,-)$ preserves filtered colimits of
monomorphisms -- more precisely, colimits of filtered diagrams $D\colon \D
\to \A$ for which $Dh$ is a monomorphism in $\A$ for every morphism
$h$ of $\D$.
\begin{notation}
For a category $\A$ we denote by
\[
\Afp \qquad\text{and}\qquad\Afg
\]
full subcategories of $\A$ representing (up to isomorphism) all finitely
presentable and finitely generated objects, respectively.
Subobjects $m\colon M \monoto A$ with $M$ finitely generated are called
\emph{finitely generated subobjects}.
\end{notation}
Recall that $\A$ is a \emph{locally finitely presentable} category, shortly
\emph{lfp} category, if it is cocomplete, $\Afp$ is small, and every
object is a colimit of a filtered diagram in $\Afp$.
We now recall a number of standard facts about
lfp categories~\cite{AdamekR94}.
\begin{remark}\label{R:prelim}
Let $\A$ be an lfp category.
\begin{enumerate}
\item \label{I:factorization} By~\cite[Proposition~1.61]{AdamekR94}, $\A$ has
(strong epi, mono)-factorizations of morphisms.
\item \label{I:canColim} By~\cite[Proposition~1.57]{AdamekR94}, every object
$A$ of $\A$ is the colimit of its \emph{canonical filtered diagram}
\[
D_A\colon \Afp/A \to \A \qquad (P \xrightarrow{p} A) \mapsto P,
\]
with colimit injections given by the $p$'s.
\item By~\cite[Theorem~2.26]{AdamekR94}, $\A$ is a free completion
of $\Afp$ under filtered colimits. That is, for every functor
$H\colon \Afp \to \B$, where $\B$ has filtered colimits, there is an
(essentially unique) extension of $H$ to a finitary functor
$\bar{H}\colon\A\to\B $. Moreover, this extensions can be formed as
follows: for every object $A\in \A$ put
\[
\bar{H} A = \colim H\cdot D_A.
\]
\item \label{I:colimono} By~\cite[Proposition~1.62]{AdamekR94}, a colimit of a filtered
diagram of monomorphisms has monomorphisms as colimit injections.
Moreover, for every compatible cocone formed by monomorphisms, the
unique induced morphism from the colimit is a monomorphism too.
\item \label{I:fingen} By~\cite[Proposition~1.69]{AdamekR94}, an object $A$ is
finitely generated iff it is a strong quotient of a finitely
presentable object, i.e.~there exists a finitely presentable
object $A_0$ and a strong epimorphism $e\colon A_0 \epito A$.
\item It is easy to verify that every split quotient of a finitely
presentable object is finitely presentable again.
\end{enumerate}
\end{remark}
\begin{lemma}\label{L:union} Let $\A$ be an lfp category. A cocone of monomorphisms
$c_i\colon Di\monoto C\; (i\in I)$ of a filtered diagram $D$ of
monomorphisms is a colimit of $D$ iff it is a {\em union}; that is,
iff $\id_C$ is the supremum of the subobjects $c_i\colon Di\monoto C$.
\end{lemma}
\begin{proof} The `only if' direction is clear. For the `if' direction
suppose that $c_i\colon Di \monoto C$ have the union $C$, and let
$\ell_i\colon Di \to L$ be the colimit of $D$. Then, since $c_i$ is a
cocone of $D$, we get a unique morphism $m\colon L \to C$ with
$m \cdot \ell_i = c_i$ for every $i$. By
Remark~\ref{R:prelim}\ref{I:colimono}, all the $\ell_i$ and $m$ are
monomorphisms, hence $m$ is a subobject of $C$. Moreover, we have
that $c_i \leq m$, for every $i$. Consequently, since $C$ is the
union of all $c_i$, $L$ must be isomorphic to $C$ via $m$, because
$id_C$ is the largest subobject of $C$. Thus, the original cocone
$c_i$ is a colimit cocone.
\end{proof}
\begin{remark} \label{R:refl}
Colimits of filtered diagrams $D\colon \D\to \Set$ are precisely those cocones
$c_i\colon D_i\to C$ ($i\in \obj \D$) of $D$ that have the following
properties:
\begin{enumerate}
\item $(c_i)$ is jointly surjective, i.e.~$C=\bigcup c_i[D_i]$, and
\item given $i$ and elements $x,y\in D_i$ merged by $c_i$, then they are also
merged by a connecting morphism $D_i\to D_j$ of $D$.
\end{enumerate}
This is easy to see: for every cocone $c_i'\colon D_i\to C'$ of $D$ define
$f\colon C\to
C'$ by choosing for every $x\in C$ some $y\in D_i$ with $x=c_i(y)$ and putting
$f(x) = c_i'(y)$. By the two properties above, this is well defined and is unique
with $f\cdot c_i = c_i'$ for all $i$.
\end{remark}
\begin{lemma}[Finitely presentable objects collectively reflect
filtered colimits.]\label{L:refl}\\
Let $\A$ be an lfp category and $D\colon \D\to \A$ a filtered
diagram with objects $D_i$ ($i\in I$). A cocone $c_i\colon D_i\to C$
of $D$ is a colimit iff for every $A\in \Afp$ the cocone
\[
c_i\cdot (-)\colon \A(A,D_i) \longrightarrow \A(A,C)
\]
is a colimit of the diagram $\A(A,D-)$ in $\Set$.
\end{lemma}
Explicitly, the above property of the cocone $(c_i)$ states that for every
morphism $f\colon A\to C$ where $A\in \A_\fp$
\begin{enumerate}
\item a factorization through some $c_i$ exists, and
\item given two factorizations $f=c_i\cdot q_k$ for $k=1,2$, then
$q_1,q_2\colon A\to D_i$ are merged by a connecting morphism of $\D$. The
proof that this describes $\colim \A(A,D-)$ follows from Remark \ref{R:refl}.
\end{enumerate}
\begin{proof}
If $(c_i)$ is a colimit, then since $\A(A,-)$ preserves filtered colimits,
the cocone of all $\A(A,c_i)=c_i\cdot (-)$ is a colimit in $\Set$.
Conversely, assume that, for every $A\in \A_\fp$, the colimit cocone
of the functor $\A(A,D-)$ is $\big(\A(A,c_i)\big)_{i\in \D}$. For
every cocone $g_i\colon D_i\to G$ it is our task to prove that there
exists a unique $g\colon C\to G$ with $g_i = g\cdot c_i$ for all
$i$. We first prove uniqueness of $g$. If $g\cdot c_i=g'\cdot c_i$
for all $i$, then $\A(A,g)\cdot \A(A,c_i)=\A(A,g')\cdot
\A(A,c_i)$. Since the $\A(A,c_i)$ are jointly surjective, we obtain
$\A(A,g)=\A(A,g')$. Since this holds for all $A\in \A_{\fp}$, and
$\A_\fp$ is a generator, we have $g=g'$.
Now $\big(\A(A,g_i)\big)_{i\in \D}$ forms a cocone of the functor
$\A(A,-)\cdot D$. Consequently, there is a unique map
$\phi_A\colon \A(A,C)\to \A(A,G)$ with $\phi_A\cdot \A(A,c_i)=\A(A,g_i)$
for all $i\in \D$.
For every morphism $h\colon A_1\to A_2$ between objects of $\A_\fp$
the square on the right of the following diagram is commutative:
\[
\xymatrix{
\A(A_1,D_i)\ar[rrr]^{\A(A_1,c_i)}
&&&
\A(A_1,C)\ar[rr]^{\phi_{A_1}}
&&
\A(A_1,G)
\ar@{<-} `u[l] `[lllll]_-{\A(A_1,g_i)} [lllll]
\\
\A(A_2,D_i)\ar[u]^{\A(h,D_{i})}\ar[rrr]^{\A(A_2,c_i)}
&&&
\A(A_2,C)\ar[rr]^{\phi_{A_2}}\ar[u]_{\A(h,C)}
&&
\A(A_2,G)\ar[u]_{\A(h,G)}
\ar@{<-} `d[l] `[lllll]^-{\A(A_2,g_i)} [lllll]}
\]
This follows from the commutativity of the left-hand square and
the outside one combined with the fact that
$\big(\A(A_2,c_i)\big)_{i\in \D}$, being a colimit cocone, is
jointly epic.
As a consequence, the morphisms
\[
A\xrightarrow{\phi_A(a)}C\qquad \text{with $a\colon A\to C$ in $\A_\fp/C $,}
\]
form a cocone for the canonical filtered diagram $D_C\colon \A_\fp/C \to \A$, of
which $C$ is the colimit. Indeed, given a morphism $h$ in $\A_\fp/C$
\[
\xymatrix{
A_1
\ar[rd]_{a_1}
\ar@{->}[rr]^-{h}
&&
A_2
\ar[ld]^{a_2}
\\
&
C
&
}
\]
we have
$$
\phi_{A_1}(a_1)=\phi_{A_1}(a_2\cdot h)=\phi_{A_1}\cdot\A(h,C)(a_2)=\A(h,G)\cdot
\phi_{A_2}(a_2)=\phi_{A_2}(a_2)\cdot h.
$$
Thus there is a unique morphism $g\colon C\to G$ making for each
$a\colon A\to C$ in $\A_\fp/C$ the following triangle commute:
\[
\xymatrix{
& A
\ar[dl]_{a}
\ar[dr]^{\phi_A(a)}
\\
C
\ar@{->}[rr]^{g}
&&
G
}
\]
It satisfies $g\cdot c_i=g_i$ for all $i\in \D$. Indeed, fix $i$;
for every $A\in \A_{\fp}$ and $b\colon A\to D_i$, we have
$g_ib=\A(A,g_i)(b)=\phi_A\cdot
\A(A,c_i)(b)=\phi_A(c_ib)=gc_ib$. And the morphisms
$b\in \A_\fp/D_i$ are jointly epimorphic, thus $g_i=g\cdot c_i$.
Thus $g$ is the desired factorization morphism.
\end{proof}
\begin{lemma}[Finitely generated objects collectively reflect filtered
colimits of monomorphisms.]\label{L:refl2}
Let $\A$ be an lfp category and $D\colon \D\to \A$ a filtered
diagram of monomorphisms with ojects $D_i\, (i\in I)$. A cocone
$c_i\colon D_i\to C$ of $D$ is a colimit iff for every $A\in \A_\fg$
the cocone
\[
c_i\cdot (-)\colon \A(A,D_i) \longrightarrow \A(A,C) \qquad (i\in I)
\]
is a colimit of the diagram $\A(A,D-)$ in $\Set$.
\end{lemma}
\begin{proof}
If $(c_i)$ is a colimit, then since $\A(A,-)$ preserves filtered colimits of monomorphisms, the
cocone $c_i\cdot(-)\colon \A(A,D_i)\to \A(A,C)$ is a colimit in $\Set$.
Conversely, if for every $A\in \A_\fg$, the cocone $c_i\cdot (-)\colon \A(A,D_i)\to
\A(A,C)$, $i\in I$, is a colimit of the
diagram $\A(A,D-)$, then we have for every $A\in \A_\fp$ that the cocone $c_i\cdot
(-), i\in I$, is a colimit of the diagram $\A(A,D-)$.
Hence by Lemma~\ref{L:refl}, the cocone $(c_i)$ is a colimit.
\end{proof}
\begin{corollary} \label{C:refl}
A functor $F\colon \A\to \B$ between lfp categories is finitary iff it
preserves the canonical colimits: $FA = \colim FD_A$ for every object $A$ of $\A$.
\end{corollary}
\begin{proof}
Indeed, in the notation of Lemma~\ref{L:refl} we are to verify that
$Fc_i\colon FD_i\to FC$ ($i\in I$) is a colimit of $FD$. For this,
taking into account that lemma and Remark \ref{R:refl}, we take any
$B\in \B_\fp$ and prove that every morphism $b\colon B\to FC$
factorizes essentially uniquely through $Fc_i$ for some $i\in
\D$. Since $FC=\colim FD_C$ we have a factorization
\[
\vcenter{\xymatrix{
& FA
\ar[d]^{Fa}
\\
B
\ar@{-->}[ur]^{b_0}
\ar[r]_{b}
& FC
}}
\qquad\quad (A\in \A_\fp)
\]
By Lemma~\ref{L:refl} there is some $i\in \D$ and $a_0\in \A(A,D_i)$ with
$a=c_i\cdot a_0$ and hence $b=Fc_i\cdot (Fa_0\cdot b_0)$.
The essential uniqueness is clear.
\end{proof}
\begin{notation} \label{N:image} Throughout the paper, given a morphism $f\colon
X
\to Y$ we denote by $\Im f$ the {\em image of $f$}, that is, any choice of the
intermediate object defined by taking the (strong epi, mono)-factorization of
$f$:
\[
f = (\xymatrix@1{X \ar@{->>}[r]^-e & \Im f \ar@{ >->}[r]^-m & Y}).
\]
\end{notation}
We will make use of the next lemma in the proof of Proposition~\ref{P:finmono}.
\begin{lemma}\label{L:im}
In an lfp category, images of filtered colimits are directed unions
of images.
\end{lemma}
More precisely, suppose we have a filtered diagram $D\colon \D \to \A$ with objects $D_i\, (i\in I)$ and
a colimit cocone $(c_i\colon D_i \to C)_{i\in I}$. Given a morphism
$f\colon C \to B$, take the factorizations of $f$ and all $f\cdot c_i$ as follows:
\begin{equation}\label{E:unionofimages}
\vcenter{
\xymatrix{
D_i
\ar[d]_{c_i} \ar@{->>}[rr]^-{e_i}
&&
\Im(f \cdot c_i) \ar@{ >->}[d]^{m_i} \ar@{-->}[ld]_-{d_i}
\\
C
\ar@{->>}[r]^-e
&
\Im f \ar@{ >->}[r]^-{m}
&
B \ar@{<-} `d[l] `[ll]^-{f}
}}\qquad \quad (i\in I)
\end{equation}
Then the subobject
$m$ is the union of the subobjects $m_i$.
\begin{proof}
We have the
commutative diagram~\eqref{E:unionofimages}, where $d_i$ is the
diagonal fill-in. Since $m\cdot d_i = m_i$, we see that $d_i$
is monic. Furthermore, for every connecting morphism $Dg\colon D_i\to D_j$ we
get a monomorphism $\bar g\colon \Im (f\cdot c_i)\monoto \Im (f\cdot c_j)$ as a diagonal fill-in
in the diagram below:
\[
\xymatrix{
D_i\ar@{->>}[r]^-{e_i}
\ar[d]_{Dg}
& \Im (f\cdot c_i)
\ar@{>->}[dr]^-{d_i}
\ar@{>-->}[d]_{\bar g}
\\
D_j
\ar@{->>}[r]^-{e_j}
&
\Im (f\cdot c_j)
\ar@{>->}[r]_-{d_j}
&
\Im f
}
\]
Since $D$ is a filtered diagram, we see that the objects $\Im (f\cdot c_i)$ form a
filtered diagram of monomorphisms; in fact, since $d_i$ and $d_j$ are monic
there is at most one connecting morphism $\Im (f\cdot c_i) \to \Im (f\cdot c_j)$.
In order to see that $m$ is the union of the subobjects $m_i$, let
$d^{\prime}_i\colon \Im (f\cdot c_i)\monoto N$ and $n\colon N \monoto \Im f$ be
monomorphisms such that $n\cdot d^{\prime}_i=d_i$ for every $i\in I$.
\[
\xymatrix{
Di\ar@{->>}[r]^-{e_i}
\ar[d]_{c_i}
& \Im (f\cdot c_i)
\ar@{>->}[d]^-{d^{\prime}_i}
\ar@{>->}[r]^-{d_i}
&
\Im f
\\
C
\ar@{-->}[r]^-{t}
&
N
\ar@{>->}[ru]_-{n}
&
}
\]
Since $n$ is monic, the morphisms $d^{\prime}_i\cdot e_i$ clearly form a
cocone of $D$, and this induces a unique morphism $t\colon C\to N$ such that $t\cdot
c_i=d^{\prime}_i\cdot e_i$. Then $n\cdot t\cdot
c_i=e\cdot c_i$; hence, $n\cdot t=e$. Since $n$ is monic,
it follows that it is an isomorphism, i.e.~the subobjects $\id_{\Im f}$ and
$n$ are isomorphic. This shows that $m$ is the desired union.
\end{proof}
\section{Finitary and Finitely Bounded Functors}\label{sec:fin}
In this section we introduce the notion of a finitely bounded functor on a
locally presentable category, and investigate when these functors are precisely
the finitary ones.
\begin{definition}\label{D:bounded}
A functor $F\colon \A \to \B$ is called \emph{finitely bounded} provided
that, given an object $A$ of $\A$, every finitely generated
subobject of $FA$ in $\B$ factorizes through the $F$-image of a finitely generated
subobject of $A$ in $\A$.
\end{definition}
\noindent
In more detail, given a monomorphism $m_0\colon M_0 \monoto FA$ with $M_0
\in \B_{\mathsf{fg}}$ there exists a monomorphism $m\colon M \monoto A$ with
$M \in \Afg$ and a factorization as follows:
\[
\xymatrix{
& FM\ar[d]^{Fm}\\
M_0\ar@{-->}[ru]\ar@{ >->}[r]_{m_0} & FA
}
\]
\begin{example}\label{E:bounded}
\begin{enumerate}
\item\label{E:bounded:1} If $\B$ is the category of $S$-sorted sets,
then $F$ is finitely bounded iff for every object $A$ of $\A$ and
every element $x \in FA$ there exists a finitely generated
subobject $m\colon X \monoto A$ such that $x \in Fm[FX]$.
\item\label{E:bounded:2} Let $\A$ be a category with (strong
epi, mono)-factorizations. An object of $\A$ is finitely generated iff its
hom-functor is finitely bounded. Indeed, by applying~\ref{E:bounded:1} we
see that $\A(A,-)$ is finitely bounded iff for every morphism $f\colon A
\to B$ there exists a factorization $f = m \cdot g$, where $m\colon A'
\monoto B$ is monic and $A'$ is finitely generated. This implies that $A$ is
finitely generated: for $f=\id_A$ we see that $m$ is invertible. Conversely,
if $A$ is finitely generated, then we can take the (strong epi,
mono)-factorization of $f$ and use that finitely generated objects are
closed under strong quotients~\cite{AdamekR94}.
\end{enumerate}
\end{example}
\begin{proposition}\label{P:finmono}
Let $F$ be a functor between lfp categories preserving
monomorphisms. Then $F$ is finitely bounded iff it preserves
filtered colimits of monomorphisms.
\end{proposition}
\begin{proof}
We are given lfp categories $\A$ and $\B$ and a functor $F\colon \A\to \B$
preserving monomorphisms.
\begin{enumerate}
\item Let $F$ preserve filtered colimits of monomorphisms. Then, for
every object $A$ we express it as a canonical filtered colimit of
all $p\colon P \to A$ in $\Afp/A$ (see Remark~\ref{R:prelim}\ref{I:canColim}). By
Lemma~\ref{L:im} applied to $f = \id_A$ we see that $A$ is the
colimit of its subobjects $\Im p$ where $p$ ranges over $\Afp/A$.
Hence, $F$ preserves this colimit,
\[
FA = \colim_{p \in \Afp/A} F(\Im p),
\]
and it is a colimit of monomorphisms since $F$ preserves monomorphisms. Given a finitely generated subobject $m_0\colon M_0\monoto FA$, we thus obtain
some $p$ in $\Afp/A$ such that $m_0$ factorizes through the
$F$-image of $\Im(p) \monoto A$. Hence $F$ is finitely bounded.
\item Let $F$ be finitely bounded. Let $D\colon \D \to \A$ be a filtered
diagram of monomorphisms with a colimit
cocone:
\[
c_i\colon D_i\monoto C\qquad (i\in I).
\]
In order to prove that $Fc_i\colon FD_i\to FC$, $i\in I$, is a colimit cocone, we
show that its image under $\B(B,-)$ is a colimit cocone for every finitely
generated object $B$ in $\B$ (cf.~Lemma~\ref{L:refl2}). In other words,
given $f\colon B\to FC$ with $B\in \B_\mathsf{fg}$ then
\begin{enumerate}
\item $f$ factorizes through $Fc_i$ for some $i$ in $I$, and
\item the factorization is unique.
\end{enumerate}
We do not need to take care of (b): since every $c_i$ is monic by
Remark~\ref{R:prelim}(4), so is every $Fc_i$. In order to prove (a),
factorize $f\colon B\to FC$ as a strong epimorphism $q\colon B \epito M_0$
followed by a monomorphism $m_0\colon M_0\monoto FC$. Then $M_0$ is
finitely generated by Remark~\ref{R:prelim}(5). Thus, there exists a
finitely generated subobject $m\colon M\monoto C$ with $m_0 = Fm \cdot u$ for
some $u\colon M_0 \to FM$. Furthermore, since $\A(M,-)$ preserves the
colimit of $D$, $m$ factorizes as $m = c_i\cdot \overline{m}$ for some
$i\in I$. Thus $F\overline{m}\cdot u\cdot q$ is the desired
factorization:
\[
f
= m_0 \cdot q
= Fm \cdot u \cdot q
= Fc_i \cdot F\overline{m} \cdot u \cdot q.
\tag*{\endproofbox}
\]
\end{enumerate}
\makeatletter
\def\endproof{}
\makeatother
\end{proof}
In the following theorem we work with an lfp category whose finitely generated
objects are finitely presentable. This holds e.g.~for the categories of sets,
many-sorted sets, posets, graphs, vector spaces, unary algebras on one operation
and nominal sets. Further examples are the categories of commutative monoids
(this is known as Redei's theorem~\cite{Redei65}, see Freyd~\cite{Freyd68} for a
rather short proof), positive convex algebras (i.e.~the Eilenberg-Moore algebras
for the (sub-)distribution monad on sets~\cite{SokolovaW15}), semimodules for
Noetherian semirings (see e.g.~\cite{bms13} for a proof). The category of
finitary endofunctors of sets also has this property as we verify in
Corollary~\ref{cor:setFunNotSStrict}.
On the other hand, the categories of groups, lattices or monoids do
not have that property.\smnote{Why lattices? Can we give a citation? TW: so we
now know that the free lattice on 3 generators is infinite, but we could not
find in the literature any statement about fp vs fg.}
A particularly simple counter-example is the slice category
$\Nat/\Set$; equivalently, this is the category of algebras with a set
of constants indexed by $\Nat$. Hence, an object $a\colon \Nat\to A$ is
finitely generated iff $A$ has a finite set of generators,
i.e.~$A\setminus a[\Nat]$ is a finite set. It is finitely presentable
iff, moreover, $A$ is presented by finitely many relations, i.e.~the
kernel of~$a$ is a finite subset of $\Nat\times \Nat$.
\begin{theorem}\label{T:finbound}
Let $\A$ be an lfp category in which every finitely generated object is
finitely presentable ($\Afp=\Afg$). Then for all functors preserving
monomorphisms from $\A$ to lfp categories we have the equivalence
\[
\text{finitary} \iff \text{finitely bounded}.
\]
\end{theorem}
\begin{proof} Let $F\colon \A\to \B$ be a finitely bounded functor preserving
monomorphisms, where $\B$ is lfp. We prove that $F$ is finitary. The
converse follows from Proposition~\ref{P:finmono}.
According to Corollary~\ref{C:refl} it suffices to prove that $F$ preserves
the colimit of the canonical filtered diagram of every object $A$. The proof
that $FD_A$ has the colimit cocone given by $Fp$ for all $p\colon P\to A$ in $\Afp /
A$ uses the fact that this is a filtered diagram in the lfp category $\B$. By
Remark~\ref{R:refl}, it is therefore sufficient to prove that for every object
$C\in \B_\mathsf{fp}$ and every morphism $c\colon C\to FA$ we have the following two
properties:
\begin{enumerate}
\item\label{T:finbound:1}
$c$ factorizes through some of the colimit maps
\[
\vcenter{
\xymatrix{
& FP\ar[d]^{Fp}\\
C\ar@{-->}[ur]^{u}\ar[r]_-{c}
& FA
}
}
\qquad
(P\in\Afp),
\]
\item\label{T:finbound:2} given another such factorization, $c=Fp\cdot v$, then $u$ and
$v$ are merged by some connecting morphism; i.e.~we have a
commutative triangle
\[
\vcenter{
\xymatrix{
P\ar[rr]^-{h}\ar[rd]_{p}
&&
P'\ar[ld]^{p'}\\
&A&
}}
\qquad
(P,P'\in\Afp)
\]
with $Fh\cdot u = Fh\cdot v$.
\end{enumerate}
Indeed, by applying Lemma~\ref{L:im} to
$f = \id_A$, we see that the monomorphisms $m_p\colon \Im p \monoto A$ for $p \in
\A_\fp/A$ form a colimit cocone of a diagram of monomorphisms. By
Proposition~\ref{P:finmono}, $F$ preserves this colimit,
therefore any $c\colon C\to FA$ factorizes through some
$Fm_p\colon F(\Im p)\to FA$. Observe that, since $\Afg=\Afp$, we know by
Remark~\ref{R:prelim}(5) that every $\Im p$ is finitely presentable,
hence the morphisms $m_p$ are colimit injections and all $e_p\colon P\epito \Im p$ are connecting
morphisms of $D_A$. Consequently, (1) is clearly satisfied.
Moreover, given $u,v\colon C\to FP$ with $Fp \cdot u= Fp \cdot v$, we have
that $Fe_p\cdot u=Fe_p\cdot v$, since $Fm_p$ is monic, thus (2) is
satisfied, too.
\end{proof}
\begin{remark}
Conversely, if every functor from $\A$ to an lfp category fulfils
the equivalence in the above theorem, then $\Afp=\Afg$. Indeed, for
every finitely generated object $A$, since $F=\A(A,-)$ preserves
monomorphisms, we can apply Proposition~\ref{P:finmono} and conclude
that $F$ is finitary, i.e.~$A\in \Afp$.
\end{remark}
\begin{example}\label{E:unary}
For $\Un$, the category of algebras with one unary operation, we
present a finitely bounded endofunctor that is not finitary. Since
in $\Un$ finitely generated algebras are finitely presentable, this
shows that the condition of preservation of monomorphisms cannot be
removed from Theorem~\ref{T:finbound}.
Let $C_p$ denote the algebra on $p$ elements whose operation forms a
cycle. Define $F\colon\Un\to \Un$ on objects by
\[
FX =
\begin{cases}
C_1 + X &
\text{if $\Un(C_p,X)=\emptyset$ for some prime $p$,}
\\
C_1 & \text{else.}
\end{cases}
\]
Given a homomorphism $f\colon X \to Y$ with $FY=C_1 + Y$, then also
$FX=C_1 + X$; indeed, in case $FX = C_1$ we would have
$\Un(C_p, X) \neq \emptyset$ for all prime numbers $p$, and then the
same would hold for $Y$, a contradiction. Thus we can put
$Ff=\id_{C_1} + f$. Otherwise $Ff$ is the unique homomorphism to
$C_1$.
\begin{enumerate}
\item We now prove that $F$ is finitely bounded. Suppose we
are given a finitely generated subalgebra $m_0\colon M_0\monoto FX$. If
$FX = C_1$ then take $M = \emptyset$ and $m\colon \emptyset \monoto X$
the unique homomorphism. Otherwise we have $FX = C_1 + X$, and we
take the preimages of the coproduct injections under $Ff$ to see that $m_0 = u
+ m$, where $u$ is the unique homomorphism into the terminal algebra
$C_1$ as shown below:
\[
\xymatrix@C+2pc{
M' \ar@{ >->}[d] \ar[r]^-u & C_1 \ar@{ >->}[d] \\
M_0 \ar[r]^-{m_0} & C_1 + X \\
M \ar@{ >->}[u] \ar[r]_-m & X \ar@{ >->}[u]
}
\]
Then we obtain the desired factorization of $m_0$:
\[
\xymatrix{
& C_1 + M = FM\ar[d]^{\id_{C_1} + m = Fm}
\\
M_0 = M' + M
\ar[r]_-{u + m}
\ar[ru]^-{u + M}
&
C_1 + X = FX
}
\]
\item However, $F$ is not finitary; indeed, it does not preserve the
colimit of the following chain of inclusions
\[
C_2\hookrightarrow C_2+C_3\hookrightarrow
C_2+C_3+C_5\hookrightarrow \cdots
\]
since every object $A$ in this chain is mapped by $F$ to $C_1+A$ while its
colimit $X = \coprod_{i\text{ prime}} C_i$ is mapped to $C_1$.
\end{enumerate}
\end{example}
We now turn to the question for which lfp categories $\A$ the equivalence
\[
\text{finitary} \iff \text{finitely bounded}
\]
holds for \emph{all} functors with domain $\A$.
In the following we call a morphism $u\colon X\to Y$ \emph{finitary}
if it factorizes through a finitely presentable object:
\begin{equation}\label{eq:finitary}
\vcenter{
\xymatrix{
& C \mathrlap{~\in \A_\fp}
\ar[d]^{w}
\\
X\ar[ru]^v
\ar[r]_{u}
& Y
}}
\end{equation}
\begin{example}\label{E:loop}
In the category of graphs consider the following graph on $\Nat$:
\[
\xymatrix{
0
\ar@(ul,dl)[]
& 1
\ar[r]
& 2
\ar[r]
& 3
\ar[r]
& \cdots
}
\]
The constant self-map of value $0$ is finitary, but no other endomorphism on this
graph is finitary.
\end{example}
\begin{remark}
\begin{enumerate}
\item If a morphism in an lfp category has a finitely presentable image
(see Notation~\ref{N:image}), then it is of course finitary.
\item The converse, namely that every finitary morphism has a
finitely presentable image, holds whenever $\A_\fp$ is closed under subobjects and
$\A_\fp= \A_\fg$. Indeed, given a finitary morphism $u\colon X\to Y$, let
$w\cdot v$ be a factorization through a finitely presentable object $C$.
Take a (strong epi, mono)-factorization $v = v_2\cdot v_1$ of
$v$:
\[
\xymatrix{
& C_1
\ar@{>->}[r]^{v_2}
\ar[d]^{d}
& C
\ar[d]^{w}
\\
X
\ar@{->>}[r]
\ar@{->>}[ur]^{v_1}
& \Im(u)
\ar@{>->}[r]
& Y
}
\]
Then $C_1$ is finitely presentable and the diagonal fill-in $d$ is strongly
epic thus, $\Im(u)$ is finitely presentable. This holds e.g.~for sets,
graphs, posets, vector spaces and semilattices.
\end{enumerate}
\end{remark}
\begin{definition} \label{D:strict}
An lfp category is called
\begin{enumerate}
\item \emph{semi-strictly lfp} if every object has a finitary endomorphism;
\item \emph{strictly lfp} if every object has, for each finitely generated
subobject $m$, a finitary endomorphism $u$ fixing that subobject
(i.e.~$u\cdot m = m$).
\end{enumerate}
\end{definition}
\begin{remark} \label{R:strict}
\begin{enumerate}
\item \label{R:strict:2semi} `strictly' implies `semi-strictly' due to $0\in
\A_\fp$: use the image of the unique $b\colon 0\to A$.
\item \label{R:strict:str} An lfp category is strictly lfp iff for every morphism
$b\colon B\to A$ with $B\in \A_\fp$ there exist morphisms $b'\colon B'\to A$
and $f\colon A\to B'$ with $B'\in \A_\fp$ such that the square below
commutes.
\[
\xymatrix{
B
\ar[r]^b
\ar[d]_b
& A
\\
A
\ar[r]_f
& B'
\ar[u]_{b'}
}
\]
Indeed, this condition is necessary: choose, for the image $m$ of $b$,
a finitary $u\colon A\to A$ with $m=u\cdot m$, thus $b=u\cdot b$.
We have a
factorization $u=b'\cdot f$ where $b'\colon B'\to A$ has a finitely
presentable domain.
The condition is also sufficient: given a square as above, the morphism
$u=b'\cdot f$ is finitary and $b = u\cdot b$.
\item \label{R:strict:semi} An lfp category is semi-strictly lfp iff for every
morphism $b\colon B\to A$ with $B\in \A_\fp$ there exists a factorization of
$b$ through a morphism $b'\colon B'\to A$ with $B'\in \A_\fp$ such that
$\A(B',A) \neq \emptyset$.
\[
\xymatrix{B\ar[rd]_b\ar@{-->}[rr]&&B'\ar@<2pt>[ld]^{b'}\\
&A\ar@<2pt>[ur]^f&}
\]
Indeed, this condition is necessary: given a finitary morphism
$u\colon A\to A$ we have $u=w\cdot v$ as
in~\eqref{eq:finitary}. Moreover, $B'=B+C$ is finitely presentable
since both $B$ and $C$ are. Put $b'=[b,w]\colon B'\to A$ and
\[
f = \big(
A\xrightarrow{~v~}
C\xrightarrow{~\inr~}
B+C
\big),
\]
where $\inr$ is the right-hand coproduct injection.
Then $b$ factorizes through $b'$ via the left-hand coproduct
injection $\inl\colon B\to B+C$.
The condition is also sufficient: consider $b\colon 0 \to A$ and put $a=b'\cdot f$.
\item \label{R:strict:fpfg} In every strictly lfp category we have
$\Afg=\Afp$. Indeed, given $A\in \Afg$ express it as a strong quotient
$b\colon B\epito A$ of some $B\in \Afp$, see Remark \ref{R:prelim}(5). Then the
equality $b=b'\cdot f\cdot b$ in \ref{R:strict:str} above implies $b'\cdot f=\id$. Thus,
$A$ is a split quotient of a finitely presentable object $B'$, hence, $A$ is
finitely presentable by Remark \ref{R:prelim}(6).
\end{enumerate}
\end{remark}
\begin{examples}\label{E:set}
\begin{enumerate}
\item $\Set$ is strictly lfp: given $b\colon B\to A$ with $B\not=\emptyset$
factorize it as $e\colon B\epito \Im b$ followed by a split monomorphism
$b'\colon \Im
b\to A$. Given a splitting, $f\cdot b'=\id$, we have $b=b'\cdot f\cdot b$.
The case $B=\emptyset$ is trivial: for $A\not=\emptyset$, $b'$ may be any
map from a singleton set to $A$.
\item Vector spaces (over a given field) form a strictly lfp
category. This can be seen directly quite easily, we show this in
Example \ref{E:strict}\ref{e:Vec} as a consequence of
Proposition~\ref{P:semi-simple}.
\item For every finite group $G$ the category $G$-$\Set$ of sets
with an action of $G$ is strictly lfp. This category is equivalent
to that of presheaves on $G^{\text{op}}$, see
Lemma~\ref{L:gpd}.
\item \label{R:strict:zeroobject} Every lfp category with a zero
object $0\cong 1$ is semi-strictly lfp. This follows from the fact
that $0$ is finitely presentable and every object $A$ has the
finitary endomorphism
$\big( A \to 1 \cong 0 \to A \big)$. Examples
include the categories of monoids and groups, which are not
strictly lfp because in both cases the classes of finitely
presentable and finitely generated objects differ.
A bit more generally: let an lfp category $\A$ have a finitely
presentable terminal object from which morphisms exist to all
objects outside of $\A_{\fp}$. Then it is semi-strictly lfp. For
example, the category of posets is semi-strictly lfp.
\item \label{E:strictNotSemi} An example of an lfp category $\A$ which fulfils
$\A_\fp = \A_\fg$ but is not semi-strictly lfp is the category of graphs.
The subgraph of the graph of Example~\ref{E:loop} on $\Nat\setminus\{0\}$ has no
finitary endomorphism. Another such example is the category of nominal sets
which is discussed in Example~\ref{ex:Nom}.
\end{enumerate}
\end{examples}
We will see other examples (and non-examples) below. The following
figure shows the relationships between the different properties:
\[
\begin{tikzpicture}[
every node/.append style={
align=center,
anchor=center,
inner sep=2pt,
minimum height=2em,
}
]
\node (strict) at (90:25mm) {strictly\\ lfp};
\node (semi) at (-150:37mm) {semi-\\ strictly\\ lfp};
\node (fpfg) at (-30:37mm){$\Afg = \Afp$};
\begin{scope}[
every edge/.append style={-implies,double equal sign distance,draw,bend left=10},
every node/.style={sloped,above},
]
\path (strict) edge[bend right] node {\ref{R:strict}\ref{R:strict:2semi}} (semi);
\path (semi) edge[bend right] node[anchor=center] {/}
node[below,outer sep=1mm] {\ref{E:set}\ref{R:strict:zeroobject}}
(strict);
\path (strict) edge node {\ref{R:strict}\ref{R:strict:fpfg}} (fpfg);
\path (fpfg) edge node[anchor=center] {/}
node[below,outer sep=1mm] {\ref{E:set}\ref{E:strictNotSemi}, \ref{ex:Nom}}
(strict);
\path (fpfg) edge
node[anchor=center] {/}
node[below,outer sep=1mm] {\ref{E:set}\ref{E:strictNotSemi}, \ref{ex:Nom}}
(semi);
\path (semi) edge
node[anchor=center] {/}
node[outer sep=1mm] {\ref{E:set}\ref{R:strict:zeroobject}}
(fpfg);
\end{scope}
\end{tikzpicture}
\]
\begin{theorem}\label{T:boundstrict}
Let $\A$ be a strictly lfp category, and $\B$ an lfp category
with $\B_{\mathsf{fg}} = \B_{\mathsf{fp}}$. Then for all functors from
$\A$ to $\B$ we have the equivalence
\[
\text{finitary} \iff \text{finitely bounded}.
\]
\end{theorem}
\begin{proof}
($\Longrightarrow$) Let $F\colon \A \to \B$ be finitary. By
Remark~\ref{R:strict}\ref{R:strict:fpfg} we know that $\Afp = \Afg$. Given a
finitely generated subobject $m\colon M \monoto FA$, write $A$ as the
directed colimit of all of its finitely generated subobjects
$m_i\colon A_i \monoto A$. Since $F$ is finitary, it preserves this
colimit, and since $M$ is finitely generated, whence finitely
presentable, we obtain some $i$ and some $f\colon M \to FA_i$ such that
$Fm_i \cdot f = m$ as desired.
($\Longleftarrow$) Suppose that $F\colon \A \to \B$ is finitely bounded. We verify the
two properties~\ref{T:finbound:1} and~\ref{T:finbound:2} in the proof of
Theorem~\ref{T:finbound}. In order to verify~\ref{T:finbound:1}, let $c\colon C \to
FA$ be a morphism with $C$ finitely presentable. Then we have the finitely
generated subobject $\Im c \monoto FA$, and this factorizes through $Fm\colon FM
\to FA$ for some finitely generated subobject $m\colon M \monoto A$ since $F$ is
finitely bounded. Then $c$ factorizes through $Fm$, too, and we are done since
$M$ is finitely presentable by Remark~\ref{R:strict}(4).
To verify~(2), suppose that we have $u,v\colon C \to FB$ and
$b\colon B \to A$ in $\Afp/A$ such that $Fb \cdot u = Fb \cdot v$. Now
choose $f\colon A \to B'$ with $b = b' \cdot (b \cdot f)$ (see
Remark~\ref{R:strict}\ref{R:strict:str}). Put $h = f\cdot b$ to get $b = b'\cdot h$
as required. Since $Fb\cdot u = Fb\cdot v$, we conclude
$Fh\cdot u = Ff \cdot Fb \cdot u = Ff \cdot Fb \cdot u = Fh\cdot v$.
\end{proof}
\begin{corollary}
A functor between strictly lfp categories is finitary iff it is
finitely bounded.
\end{corollary}
\begin{remark}
Consequently, a set functor $F$ is finitary if and only if it is
finitely bounded. The latter means precisely that every element of
$FX$ is contained in $Fm[FM]$ for some finite subset $m\colon M \subto
X$.
This result was formulated already in~\cite{AdamekP04}, but the
proof there is unfortunately incorrect.
\end{remark}
\begin{openproblem}
Is the above implication an equivalence? That is, given an lfp category $\A$
such that every finitely bounded functor into lfp categories is finitary, does
this imply that $\A$ is strictly lfp?
\end{openproblem}
\begin{theorem}
\label{T:equiv2semstrict}
Let $\A$ be an lfp category such that for functors
$F\colon \A \to \Set$ we have the equivalence
\[
\text{finitary} \iff \text{finitely bounded}.
\]
Then $\A$ is semi-strictly lfp and $\Afg = \Afp$.
\end{theorem}
\proof
The second statement easily follows from Example~\ref{E:bounded}\ref{E:bounded:2}.
Suppose that $\A$ is an lfp category such that the above equivalence
holds for all functors from $\A$ to $\Set$. Then the same
equivalence holds for all functors $F\colon \A \to \Set^S$, for $S$ a set of
sorts. To see this, denote by $C\colon \Set^S \to \Set$ the functor forming
the coproduct of all sorts. It is easy to see that $C$ creates
filtered colimits. Thus, a functor $F\colon \A \to \Set^S$ is finitary iff
$C\cdot F\colon \A \to \Set$ is. Moreover, $F$ is finitely bounded iff
$C\cdot F$ is; indeed, this follows immediately from Example~\ref{E:bounded}\ref{E:bounded:1}.
We proceed to prove the semi-strictness of $\A$. Put
$S= \Afp$. Given a
morphism
\[
b\colon B\to A\qquad\text{with $B\in \Afp$}
\]
we present $b'$ and $f$ as required in Remark~\ref{R:strict}\ref{R:strict:str}. Define a functor
$F\colon \A\to Set^S$ on objects $Z$ of $\A$ by
\[
FZ=\begin{cases}
\mathds{1}+(\A(s,Z))_{s\in S}& \text{if $\A(A,Z)=\emptyset$}\\
\mathds{1}&\text{else,}
\end{cases}
\]
where $ \mathds{1}$ denotes the terminal $S$-sorted set. Given a
morphism $f\colon Z \to Z'$ we need to specify $Ff$ in the case where
$\A(A,Z')= \emptyset$: this implies $\A(A,Z) =\emptyset$ and we put
\[
Ff = \id_\mathds{1} + (\A(s,f))_{s \in S}.
\]
Here $\A(s,f)\colon \A(s, Z) \to \A(s, Z')$ is given by $u \mapsto f
\cdot u$, as usual. It is easy to verify that $F$ is a well-defined
functor.
\begin{enumerate}
\item Let us prove that $F$ is finitely bounded. The category
$\Set^S$ is lfp with finitely generated objects $(X)_{s\in S}$
precisely those for which the set $\coprod_{s \in S} X_s$ is
finite. Let $m_0\colon M_0 \monoto FZ$ be a finitely generated
subobject. We present a finitely generated subobject $m\colon M \monoto
Z$ such that $m_0$ factorizes through $Fm$. This is trivial in the
case where $\A(A,Z) \neq \emptyset$: choose any finitely generated
subobject $m\colon M \monoto Z$ (e.g.~the image of the unique morphism
from the initial object to $Z$: cf.~Remark~\ref{R:prelim}(5)). Then
$Fm$ is either $\id_\mathds{1}$ or a split epimorphism, since $FZ =
\mathds{1}$ and in $FM$ each sort is non-empty. Thus, we have $t$ with $Fm\cdot t=\id$ and $m_0$
factorizes through $Fm$:
\[
\xymatrix{
& FM \ar@<-2pt>@{->>}[d]_{Fm} \\
M_0\ar[ru]^-{t \cdot m_0} \ar@{ >->}[r]_-{m_0}
&
FZ = \mathds{1}
\ar@<-2pt>@{>->}[u]_t
}
\]
In the case where $\A(A,Z) = \emptyset$ we have $m_0 = m_1 + m_2$
for subobjects
\[
m_1 \colon M_1 \monoto \mathds{1}
\qquad
\text{and}
\qquad
m_2\colon M_2 \monoto (\A(s,Z))_{s \in S}.
\]
For notational convenience, assume $(M_2)_s \subseteq \A(s, Z)$ and $(m_2)_s$
is the inclusion map for every $s \in S$. Since $M_0$ is finitely generated,
$M_2$ contains only finitely many elements $u_i\colon s_i \to Z$, $i = 1, \ldots,
n$. Factorize $[u_1, \ldots, u_n]$ as a strong epimorphism $e$ followed by a
monomorphism $m$ in $\A$ (see Remark~\ref{R:prelim}\ref{I:factorization}):
\[
\xymatrix{
\coprod_{i=1}^n s_i \ar@{->>}[r]^-{e}
&
M \ar@{ >->}[r]^-m
&
Z
}.
\]
Then $\A(A,M) = \emptyset$, therefore $Fm = \id_\mathds{1} +
(\A(s,m))_{s\in S}$. Since every element $u_i\colon s_i \to Z$ of $M_2$
factorizes through $m$ in $\A$, we have
\[
u_i = m \cdot u_i'
\quad
\text{for $u_i'\colon s_i \to M$ with $[u_1', \ldots, u_n'] = e$.}
\]
Let $v:M_2 \to \A(s,M)$ be the $S$-sorted map taking each $u_i$ to $u'_i$.
Then the inclusion map $m_2\colon M_2 \to (\A(s,Z))_{s \in S}$ has the
following form
\[
m_2 = \left(M_2 \xrightarrow{v} (\A(s,M))_{s \in S}
\xrightarrow{(\A(s,m))_{s\in S}} (\A(s,Z))_{s\in S}\right).
\]
The desired factorization of $m_0 = m_1 + m_2$ through $Fm =
\id_\mathds{1} + (\A(s,m))_{s\in S}$ is as follows:
\[
\xymatrix@C+2pc{
&
\mathds{1} + (\A(s,M))_{s\in S}
\ar[d]^{\id + (\A(s,m))_{s\in S}}
\\
M_0 = M_1 + M_2
\ar@{ >->}[r]_-{m_0 = m_1 + m_2}
\ar[ru]^-{m_1 + v}
&
\mathds{1} + (\A(s,Z))_{s \in S}
}
\]
\item We thus know that $F$ is finitary, and we will use this to prove that $\A$
is semi-strictly lfp. That is, as in Remark~\ref{R:strict}\ref{R:strict:semi}
we find $b'\colon B' \to A$ in $\Afp/A$ through which $b$ factorizes and which
fulfils $\A(A,B') \neq \emptyset$. Recall from Remark~\ref{R:prelim}(2) that
$A = \colim D_A$. Our morphism $b$ is an object of the diagram scheme $\Afp/A$
of $D_A$. Let $D_A'$ be the full subdiagram of $D_A$ on all objects $b'$ such
that $b$ factorizes through $b'$ in $\A$ (that is, such that a connecting
morphism $b \to b'$ exists in $\Afp/A$). Then $D_A'$ is also a filtered
diagram and has the same colimit, i.e.~$A = \colim D_A'$. Since $F$ preserves
this colimit and $FA = \mathds{1}$, we get
\[
\mathds{1} \cong \colim FD_A'.
\]
Assuming that $\A(A,B') = \emptyset$ for all $b'\colon B'\to A$ in
$D_A'$, we obtain a contradiction: the objects of $FD_A'$ are
$\mathds{1} + (\A(s,B'))_{s\in S}$, and since for every $s \in S$
the functor $\A(s, -)$ is finitary, the colimit of all $\A(s,B')$ is
$\A(s,A)$. We thus obtain an isomorphism
\[
\mathds{1} \cong \mathds{1} + (\A(s,A))_{s\in S}.
\]
This means $\A(s,A) = \emptyset$ for all $s \in S$, in particular
$\A(B,A) = \emptyset$, in contradiction to the existence of the given
morphism $b\colon B \to A$.
Therefore, there exists $b'\colon B' \to A$ in $D_A'$, i.e.~$b'$ through
which $b$ factorizes with $\A(A,B') \neq \emptyset$, as required.
\endproof
\end{enumerate}
We now present examples of strictly lfp categories. All of them happen
to be either atomic toposes or semi-simple (aka atomic) abelian
categories. Recall that an object $A$ is called \textit{simple}, or an
\emph{atom}, if it has no nontrivial subobject. That is, every
subobject of $A$ is either invertible or has the initial object as a
domain.
\begin{definition}\label{D:semi-simple}
A category is called {\em semi-simple} or {\em atomic} if every
object is a coproduct of simple objects.
\end{definition}
\begin{proposition}\label{P:semi-simple}
Let a semi-simple, cocomplete category have only finitely many
simple objects (up to isomorphism), all of them finitely
presentable. Then it is strictly lfp.
\end{proposition}
\begin{proof}
\begin{enumerate}
\item The given category $\A$ is lfp. Indeed, it is cocomplete and every finite coproduct of simple objects is finitely presentable. Moreover, every object $\coprod_{i\in I}A_i$, $A_i$ simple, is a filtered colimit of finite subcoproducts. Conversely, every finitely presentable object is, obviously, a split quotient of a finite coproduct of simple objects. Thus, for the countable set $M$ representing all these finite coproducts we see that $\Afp$ consists of split quotients of objects in $M$. Therefore $\Afp$ is essentially a
set: split quotients of any object $X$ correspond bijectively to idempotent endomorphisms of $X$, and thus form a set. Hence, $\A$ is lfp.
\item\label{P:semi-simple:2} Let $b\colon B\to A=\coprod_{i\in I}A_i$ be a morphism with all
$A_i$ simple and $B$ finitely presentable. Then $b$ factorizes
through a finite subcoproduct $a_J\colon \coprod_{i\in J}A_i\to A$
($J\subseteq I$ finite), say, $b=a_J\cdot b'$. Since $\A$ has
essentially only a finite set of simple objects, $J$ can be chosen
so that each $A_i$ is isomorphic to some $A_j$, $j\in
J$. Consequently, there exists a morphism
$g\colon \coprod_{i\in I\setminus J}A_i \to \coprod_{j\in
J}A_j$. The following composite $u\colon A\to A$
$$
A = \big(\coprod_{j\in J}A_j+\coprod_{i\in I\setminus J}A_i\big)
\xrightarrow{[\id,g]}
\coprod_{j\in J}A_j\xrightarrow{a_J} A
$$
is finitary and fulfils, since $[\id,g]\cdot a_J=\id$, the desired equation
$$u\cdot b=a_J\cdot [\id,g]\cdot a_J\cdot b'=a_J\cdot b'=b.$$
\end{enumerate}
\end{proof}
\begin{examples}\label{E:strict}
\begin{enumerate}
\item\label{e:SetS}$\Set^S$ is strictly lfp iff $S$ is
finite. Indeed, the sufficiency is a clear consequence of Proposition \ref{P:semi-simple}.
Conversely, if $S$ is
infinite then the identity on the terminal object, which is its
unique endomorphism, is not finitary, whence $\Set^S$ is not
semi-strictly lfp.
\item\label{e:Vec} For every field $K$ the category
$K$-$\mathsf{Vec}$ of vector spaces is strictly lfp. Indeed, the
simple spaces are those of dimension $0$ or $1$, and every space is a
coproduct of copies of $K$.
\item\label{e:Mod} We recall that a ring $R$ is called \emph{semi-simple}
if the category $R$-$\mathsf{Mod}$ of left modules is
semi-simple. For example, the matrix ring $K^{(n)}$ for every field
$K$ and every finite $n$ is semi-simple.
The category $R$-$\mathsf{Mod}$ is strictly lfp for every finite
semi-simple ring $R$. Indeed, every simple module $A$ is a
quotient of the module $R$: in case $A\not=\mathbf{0}$, choose
$a\in A\setminus \{0\}$. Since $Ra$ is a submodule of $A$, we conclude
$$
A=Ra \cong R/\mathord{\sim}
$$
where $\sim$ is the congruence defined by $x\sim y$ iff $Rx=Ry$.
Each quotient module $R/\mathord{\sim}$ is finitely presentable. Indeed, let
$a_i\colon A_i\to A$, $i\in I$, be a filtered colimit and
$f\colon R/\mathord{\sim} \to A$ a homomorphism. Since $R/\mathord{\sim}$ is
finite, $f$ factorizes in $\Set$ through $a_j$ for some $j\in J$:
$f=a_j\cdot f'$. It remains to choose $j$ so that
$f'\colon R/\mathord{\sim} \to A_j$ is a homomorphism. Given $r, s\in R$ we
know that $rf([s])=f([rs])$, thus $a_j$ merges $rf'([s])$ and
$f'([rs])$. Our colimit is filtered, hence for the given pair we
can assume, without loss of generality, that
$rf'([s])=f'([rs])$. Moreover, since $R\times R$ is finite, this
assumption can be made for all pairs $(r,s)$ at once. That is, by
a suitable choice of $j$ we achieve that $f'$ preserves scalar
multiplication. A completely analogous argument shows that $j$ can
be chosen so that, moreover, $f'$ preserves addition. Thus, it is
a homomorphism.
\item\label{e:topos} A Grothendieck topos is called \textit{atomic},
see \cite{BarrDiaconescu}, if it is semi-simple. For example, the
presheaf topos $\Set^{{\cal{C}}^\text{op}}$ is atomic iff
$\cal{C}$ is a groupoid, i.e.~its morphisms are all invertible,
see Sect.~7(2) in op.~cit. It follows from the Proposition~\ref{P:semi-simple}
that every atomic Grothendieck topos with a finite set of finitely
presentable atoms (up to isomorphism) is strictly lfp.
More atomic toposes can be found in
\cite[Example~3.5.9]{JohnstoneElephant2vol}. Not all atomic Grothendieck
toposes are semi-strictly lfp. See Example~\ref{ex:Nom} below: in the
category of nominal sets (aka the Schanuel topos), the set of atoms
is infinite. Next we provide a class of examples of strict lfp toposes, see
also Example~\ref{E:gpd-lambda} below.
\end{enumerate}
\end{examples}
\begin{lemma}\label{L:gpd}
The category of presheaves on a finite groupoid is strictly lfp.
\end{lemma}
\begin{proof} In view of Example \ref{E:strict}(4) all we need proving is that
for every finite groupoid $\G$
the category $\Set^{\G^{\text{op}}}$ has, up to isomorphism, a finite set of finitely presentable
atoms.
(1)~Put $S=\obj \G$. Then the category
$\Set^{\G^{\text{op}}}$ can be considered as a
variety of $S$-sorted unary algebras. The signature is given by
the set of all morphisms of $\G^{\text{op}}$: every morphism $f\colon X
\to Y$ of $\G^{\text{op}}$ corresponds to an operation symbol of
arity $X \to Y$ (i.e.~variables are of sort $X$ and results of sort
$Y$). This variety is presented by the equations corresponding to the composition in
$\G^{\text{op}}$: represent $g\cdot f=h\colon X\to Y$ in
$\G^{\text{op}}$ by $g(f(x))=h(x)$ for a variable $x$ of sort
$X$. Moreover, for every
object $X$, add the equation $\id_X(x) = x$ with $x$ of sort $X$.
For every algebra $A$ and every element $x\in A$ of sort $X$ the
subalgebra which $x$ generates is denoted by $A^x$. Denote by $\sim_A$ the
equivalence on the set of all elements of $A$ defined by $x\sim_Ay$
iff $A^x=A^y$. If $I(A)$ is a choice class of this equivalence, then
we obtain a representation of $A$ as the following coproduct:
\[
A=\coprod_{x\in I(A)}A^x.
\]
This follows from $\G$ being a groupoid: whenever
$A^x\cap A^y\not=\emptyset$, then $x\sim_A y$.
Moreover, for every homomorphism $h\colon A\to B$ there exists a function
$h_0\colon I(A) \to I(B)$ such that on each $A^x$, $x\in I(A)$, $h$
restricts to a homomorphism $h_0\colon A^x\to B^{h(x)}$. Indeed,
define $h_0(x)$ as the representative of $\sim_B$ with
$B^{h(x)}=B^{h_0(x)}$.
\noindent (2)~Given $x\in A$ of sort $X$, the algebra $A^x$ is a
quotient of the representable algebra $\G(-,X)$. Indeed, the Yoneda
transformation corresponding to $x$, an element of $A^x_X$ of sort $X$, has
surjective components (by the definition of $A^x$).
Observe that every representable algebra has only finitely many
quotients. This follows from the fact that $\G(-,X)$ has finitely
many elements, hence, finitely many equivalence relations exist
on the set of all elements.
\noindent (3)~We conclude that the finite set $\B$
of all algebras representing quotients of representable algebras
$\G(-,X)$ consists of finitely presentable algebras. Moreover, every
algebra is a coproduct of algebras from $\B$.
\end{proof}
\begin{remark}
Recall from \cite[Proposition 2.30]{AdamekR94} that \emph{pure subobjects}
$b\colon B\monoto A$ in an lfp category $\A$ are precisely the filtered
colimits of split subobjects of $A$ in the slice category $\A/A$.
\end{remark}
\begin{proposition}
Let $\A$ be an lfp category in which all subobjects are pure. If $\A_{\fp} =
\A_{\fg}$, then $\A$ is strictly lfp.
\end{proposition}
\begin{proof}
Let $b\colon B\monoto A$ be a finitely generated subobject. Express it as a
filtered colimit of split subobjects $b_i\colon B_i\monoto A$ (with $e_i\cdot
b_i = \id_{B_i}$ for $e_i\colon A\to B_i$), $i\in I$, with the following colimit
cocone in $\A/A$:
\[
\xymatrix{B_i\ar@<-2pt>@{>->}[rd]_{b_i} \ar[rr]^{c_i}&&B\ar@{>->}[ld]^{b}\\
&A\ar@<-2pt>@{-->}[ul]_{e_i}
&
}
\]
Then in $\A$ we have expressed $B$ as a filtered colimit of the objects $B_i$
with the cocone $(c_i)_{i\in I}$. It follows from our assumptions that $B$ is
also finitely presentable, and therefore $\A(B,-)$ preserves that colimit.
Hence, some $c_i$ is invertible (being both monic, due to $b_i=b\cdot c_i$,
and split epic). Consequently, $B_i$ is finitely presentable. The finitary
endomorphism $f=b_i\cdot e_i$ fixes the subobject $b$, as desired:
\[
f\cdot b = (b_i\cdot e_i) \cdot (b_i\cdot c_i^{-1})
= b_i\cdot c_i^{-1}= b.
\tag*{\endproofbox}
\]
\def\endproof{}
\end{proof}
\begin{example}
The following categories are strictly lfp because they satisfy all the
assumptions
of the above proposition. By a variety we mean an equational class of finitary
(one-sorted) algebras.
\begin{enumerate}
\item\label{i:varietyFpFg} A variety $\A$ of algebras with $\A_{\fp} =
\A_{\fg}$ in which every finitely generated subobject of a finitely
generated object splits. By \cite[Theorem~2.1]{borceuxRosicky} all monomorphisms are
pure.
An example of such a variety are boolean algebras. Here $\A_{\fg} =
\A_{\fp}$ are precisely the finite algebras. Since every epimorphism in
$\Set_{\fp}$ splits, by Stone's Duality every monomorphism between finite
boolean algebras splits.
\item\label{i:RModNoether} \rmod{} for all regular, left-Noetherian rings $R$.
Recall
that $R$ is left-Noetherian if every left ideal $I\subseteq R$ is
finitely generated; this implies that finitely generated left modules are
finitely presentable \cite[Example 3.8.28]{Rowen88}.
Recall further that regularity (in von Neumann's sense) means that for every $a\in
R$ there exists $\bar a\in R$ with $a=a\cdot \bar a \cdot a$. For
left-Noetherian rings, this condition is
equivalent to \rmod{} having all monomorphisms pure, see \cite[Proposition 2.11.20]{Rowen88}.
Regular rings are a wider class than semi-simple rings, so in the realm of
left-Noetherian rings we have a simplification of the argument of
Example~\ref{E:strict}\ref{e:Mod}.
\item A special case of \ref{i:varietyFpFg}, which is the `non-abelian
generalization' of \ref{i:RModNoether}, are varieties $\A$ with
$\A_\fp=\A_\fg$ such that for every morphism $a\colon X\to Y$ of $\A_{\fp}$
there exists $\bar a\colon Y\to X$ with $a=a\cdot \bar a\cdot a$, See
\cite[Proposition 3.4]{borceuxRosicky}.
\item $G$-modules over a field $K$, i.e.~the functor category
\[
(\kvec)^G,
\]
for a finite group $G$ and a field of characteristic $0$. (More generally:
every field whose characteristic does not divide $|G|$.)
By the classical Maschke's Theorem \cite[Theorem XIII.1.1]{Lang65} for every
subobject $b\colon B\monoto A$ there exists a coproduct $A=B+C$ with $b$ as
the left injection. Thus $b$ splits: consider $[\id_B,0]\colon A\to B$.
Hence all monomorphisms are pure.
The forgetful functor to $\kvec$ preserves colimits (computed object-wise).
The free $G$-module $\actualphi n$ on $n$ generators thus has finite dimension
(of the underling vector space). Indeed, $\actualphi 1$ has dimension $|G|$
because its underlying space is spanned by $G$, see XIII, Section 1 of
\cite{Lang65}.
Hence $\actualphi n = \actualphi 1 + \cdots + \actualphi 1$ has dimension
$n\cdot |G|$.
It follows that every finitely generated $G$-module is finitely presentable.
Indeed, it is a quotient of $\actualphi n$ for some $n$, thus, it is
finite-dimensional. And every finite-dimensional $G$-module $A$ is finitely
presentable in $(\kvec)^G$. This follows easily from $A$ being finitely
presentable in $\kvec$, since the group action $G\times A\to A$ is
determined by its domain restriction to the finite set $G\times X$, where
$X$ is a base of $A$.
\end{enumerate}
\end{example}
\begin{examples}\label{E:nonsemistrict} Here we present
lfp categories $\A$ which are not semi-strictly lfp.
For that it would be sufficient to exhibit an object $A$ such that no
endomorphism is finitary. However, we also provide something stronger:
In each case we present a non-finitary \emph{endofunctor} that is finitely
bounded.
\begin{enumerate}
\item The category $\Un$. In Example~\ref{E:unary} we have already shown
the promised endofunctor. Thus
$\Un$ is not semi-strictly lfp. For the algebra $A = \coprod_{p} C_p$, where $p$
ranges over all prime numbers, there exists no finitary endomorphism.
\item The category $\Int$-$\Set$ (of actions of the integers on sets). Since
this category is equivalent to that of unary algebras with one invertible
operation, the argument is as in~(1).
\item The category $\Gra$ of graphs and their homomorphisms is not
semi-strictly lfp (see
Example~\ref{E:set}\ref{E:strictNotSemi}).
Analogously to Example~\ref{E:unary} define an endofunctor
$F$ on $\Gra$ by
\[
FX =
\begin{cases}
\mathds{1} + X & \text{if $X$ contains no cycle and no infinite path}\\
\mathds{1} & \text{else},
\end{cases}
\]
where $\mathds{1}$ is the terminal object, and $Ff = \id_\mathds{1} + f$ if the
codomain $X$ of $f$ fulfils $FX = \mathds{1} + X$. This functor is clearly
finitely bounded, but for the graph $A$ consisting of a single
infinite path, it does not preserve the
colimit $A = \colim D_A$ of Remark~\ref{R:prelim}(2).
\item $\Set^\Nat$. If $\mathds{1}$ is the terminal object, then
$\Set^\Nat(\mathds{1}, B') = \emptyset$ for all finitely presentable
objects $B$. We define $F$ on $\Set^\Nat$ by $FX = \mathds{1} + X$ if $X$
has only finitely many non-empty components, and $FX = \mathds{1}$ else.
\end{enumerate}
\end{examples}
\begin{openproblem}
Is the category $\Pos$ of posets strictly lfp? Is every finitely bounded
endofunctor on $\Pos$ finitary?
\end{openproblem}
We next present two examples of rather important categories for which we prove
that they are not semi-strictly lfp either.
\begin{example}
\label{ex:Nom}
Nominal sets are not semi-strictly lfp. Let us first recall the definition of
the category $\Nom$ of nominal sets (see e.g.~\cite{Pitts13}). We fix a
countably infinite set $\V$ of \emph{atomic names}. Let $\perms(\V)$ denote
the group of all finite permutations on $\V$ (generated by all
transpositions). Consider a set $X$ with an action of this group, denoted by
$\pi \cdot x$ for a finite permutation $\pi$ and $x \in X$. A subset $A
\subseteq \V$ is called a \emph{support} of an element $x \in X$ provided that
every permutation $\pi \in \perms(\V)$ that fixes all elements of $A$ also
fixes $x$:
\[
\text{$\pi(a) = a$ for all $a \in A \implies \pi \cdot x = x$}.
\]
A \emph{nominal set} is a set with an action of the group
$\perms(\V)$ where every element has a finite support. The category
$\Nom$ is formed by nominal sets and \emph{equivariant maps},
i.e.~maps preserving the given group action. $\Nom$ is a
Grothendieck topos, it is an lfp category (see
e.g.~Pitts~\cite[Remark~5.17]{Pitts13}), and, as shown by
Petri\c{s}an~\cite[Proposition~2.3.7]{petrisanphd}, the finitely
presentable nominal sets are precisely those with finitely many
orbits (where an orbit of $x$ is the set of all $\pi \cdot x$).
It is a standard result that every element $x$ of a nominal set has the least
support, denoted by $\supp(x)$. In fact, $\supp\colon X\to \powf(\V)$ is itself an
equivariant map, where $\powf(\V)$ is the set of all finite subsets of $\V$ with the action given by $\pi \cdot Y = \{ \pi(v) \mid v \in
Y\}$.
Consequently, any two elements of the same orbit $x_1$ and $x_2 = \pi\cdot x_1$ have a
support of the same size. In addition, if $f\colon X \to Y$ is an equivariant map, it is clear that
\begin{equation}\label{eq-supp}{\supp (f(x)) \subseteq \supp(x),} \quad\text{for every
$x \in X$}.
\end{equation}
Now we present a non-finitary endofunctor on $\Nom$ which is finitely bounded.
Consider for every natural number $n$ the nominal set $P_n = \{ Y \subseteq \V
\mid |Y| = n\}$ with the nominal structure given
element-wise, as for $\powf(\V)$ above. Clearly, $\supp(Y) = Y$ for every $Y \in P_n$.
For $A = \coprod_{0 < n < \omega} P_n$ the existence of a finitary
endomorphism leads to a contradiction. In fact, let the
corresponding pair of morphisms
$\xymatrix@1{A\ar@<2pt>[r]^f&X\ar@<2pt>[l]^{g}}$ with $X$
orbit-finite be given. It is clear that, for every
$x\in X$, $\supp(x)\not=\emptyset$, otherwise, by \eqref{eq-supp},
we would have $\supp(g(x))=\emptyset$, which contradicts the fact
that $\supp(Y)=Y\not=\emptyset$ for all $Y\in A$. We show below that for
every $Y\in A$, $\supp(f(Y))=\supp(Y)=Y$, thus $X$ admits infinitely
many cardinalities for $\supp(x)$ with $x\in X$, contradicting the
orbit-finiteness of $X$.
By~\eqref{eq-supp}, it remains to prove that
$\supp(Y) \subseteq \supp(f(Y))$. To see this, fix an element $v$ of
$\supp(f(Y))$, which is already known to be nonempty. Now for any given
element $w$ of $\supp(Y) = Y$, the equivariance of $f$ applied to
the transposition $\pi$ of $v$ and $w$ implies that
\[
w \in \pi \cdot \supp(f(Y)) = \supp(\pi \cdot f(Y)) = \supp(f(\pi \cdot Y)) = \supp(f(Y)).
\]
This proves that $\Nom$ is not semi-strictly lfp.
Analogously to Example~\ref{E:unary} we define an endofunctor $F$ on
$\Nom$ by
\[
FX =
\begin{cases}
\mathds{1} + X & \text{if $\Nom(P_n,X) = \emptyset$ for some $n <
\omega$}
\\
\mathds{1} & \text{else.}
\end{cases}
\]
For an equivariant map $f\colon X \to Y$, if $FY = \mathds{1} + Y$, then also $FX =
\mathds{1} + X$: given $\Nom(P_n, Y) = \emptyset$ for some $n$, then also
$\Nom(P_n, X) = \emptyset$ holds. In that case put
$Ff=\id_{ \mathds{1}}+f$ and else $Ff$ is the unique equivariant map to $FY =
\mathds{1}$. A very similar argument as in Example~\ref{E:unary} shows that
$F$ is finitely bounded. However, $F$ is not finitary, as it does not preserve
the colimit $\coprod_{n<\omega}P_n$ of the chain $P_1 \subto P_1 + P_2 \subto
P_1 + P_2 + P_3 \subto \cdots$.
\end{example}
We prove next that in the category $[\Set,\Set]_\fin$
of finitary set functors (known to be lfp \cite[Theorem 1.46]{AdamekR94}) finitely generated objects
coincide with the finitely presentable ones, yet this category fails to be
semi-strictly lfp.
\begin{remark} \label{R:quot}
Recall that a quotient of an object $F$ of $[\Set,\Set]_\fin$ is represented
by a natural transformation $\varepsilon\colon F\to G$ with epic components.
Equivalently, $G$ is isomorphic to $F$ modulo a \emph{congruence} $\sim$. This
is a collection of equivalence relations $\sim_X$ on $FX$ ($X\in \Set$) such
that for every function $f\colon X\to Y$ given $p_1\sim_X p_2$ in $FX$, it
follows that $Ff(p_1) \sim_Y Ff(p_2)$.
\end{remark}
We are going to characterize finitely presentable objects of $[\Set,\Set]_\fin$
as the super-finitary functors introduced in \cite{Adamek:1990:AAC:575450}:
\begin{definition} \label{D:fin}
A set functor $F$ is called \emph{super-finitary} if there exists a natural
number $n$ such that $Fn$ is finite and for every set $X$, the maps
$Ff$ for $f\colon n\to X$ are jointly surjective, i.e.~they fulfil
$FX=\bigcup_{f\colon n\to X} Ff[Fn]$.
\end{definition}
\begin{examples}\label{E:supfin}
\begin{enumerate}
\item \label{supfin:automata}
The functors $A\times \Id^n$ are super-finitary for all finite sets $A$ and
all $n \in \Nat$.
\item \label{supfin:sig}
More generally, let $\Sigma$ be a finitary signature, i.e.~a set of
operation symbols $\sigma$ of finite arities $|\sigma|$. The corresponding
\emph{polynomial set functor}
\[
H_\Sigma X= \coprod_{\sigma\in \Sigma} X^{|\sigma|}
\]
is super-finitary iff the signature has only finitely many symbols.
We call such signatures \emph{super-finitary}.
\item \label{superfin:sub}
Every subfunctor $F$ of $\Set(n,-)$, $n\in \Nat$, is super-finitary.
Indeed, assuming $FX\subseteq \Set(n,X)$ for all $X$, we are to find, for
each $p\colon n\to X$ in $FX$, a member $q\colon n\to n$ of $Fn$ with
$p=Ff(q)$ for some $f\colon n\to X$. That is, with $p=f\cdot q$.
Choose a function $g\colon X\to n$ monic on $p[n]$. Then there exists
$f\colon n\to X$ with $p=f\cdot g\cdot p$. From $p\in FX$ we deduce
$Fg(p)\in Fn$, that is, $g\cdot p\in Fn$. Thus $q=g\cdot p$ is the desired
element: we have $p=f\cdot q=Ff(q)$.
\item \label{supfin:quotients} Every quotient $\varepsilon\colon F\twoheadrightarrow G$ of a
super-finitary functor $F$ is super-finitary. Indeed, given $p\in GX$,
find $p'\in FX$ with $p=\varepsilon_X(p')$. There exists $q'\in
Fn$ with $p'=Ff(q')$ for some $f\colon n\to X$. We conclude
that $q=\varepsilon_n(q')$ fulfils $p=Gf(q)$ from the naturality of $\varepsilon$.
\end{enumerate}
\end{examples}
\begin{lemma} \label{L:fin}
The following conditions are equivalent for every set functor $F$:
\begin{enumerate}
\item $F$ is super-finitary \label{L:fin:supfin}
\item $F$ is a quotient of the polynomial functor $H_\Sigma$ for a
super-finitary signature $\Sigma$, and \label{L:fin:poly}
\item $F$ is a quotient of a functor $A\times \Id^n$ for $A$ finite and $n\in \Nat$. \label{L:fin:quot}
\end{enumerate}
\end{lemma}
\begin{proof}
\ref{L:fin:quot}$\implies$\ref{L:fin:poly} is clear and for \ref{L:fin:poly}$\implies$\ref{L:fin:supfin}
see the Examples \ref{supfin:sig} and \ref{supfin:quotients} above. To
prove \ref{L:fin:supfin}$\implies$\ref{L:fin:quot}, let $F$ be super-finitary and
put $A=Fn$ in the above definition. Apply Yoneda Lemma to $\Id^n\cong
\Set(n,-)$ and use that $[\Set,\Set]_\fin$ is cartesian closed:
\[
\frac{
Fn \xrightarrow{~~\cong~~} [\Set,\Set]_\fin(\Set(n,-), F)
}{
\varepsilon\colon Fn \times \Set(n,-)\xrightarrow{~~\phantom{\cong}~~} F
}
\]
The definition of super-finitary shows that $\varepsilon_X$ is surjective for
every $X$.
\end{proof}
\begin{proposition} \label{P:fin}
Super-finitary functors are closed in $[\Set,\Set]_\fin$ under finite products,
finite coproducts, subfunctors, and hence under finite limits.
\end{proposition}
\begin{proof}
\begin{enumerate}
\item Finite products and coproducts are clear: given quotients
$\varepsilon_{i}\colon A_i\times \Id^{n_i}\twoheadrightarrow F_i$, $i\in \{1,2\}$, then
$F_1\times F_2$ is super-finitary due to the quotient
\[
\varepsilon_1 \times \varepsilon_2\colon (A_1\times A_2)\times \Id^{n_1+n_2}
\to F_1\times F_2.
\]
Suppose $n_1\ge n_2$, then we can choose a quotient $\varphi\colon
A_2\times \Id^{n_1} \twoheadrightarrow A_2\times \Id^{n_2}$.
This proves that $F_1+F_2$ is super-finitary due to the quotient
\[
\varepsilon_1+(\varepsilon_2\cdot \varphi)\colon
(A_1+A_2)\times \Id^{n_1}
\cong A_1\times \Id^{n_1}+A_2\times \Id^{n_1}
\to F_1+F_2.
\]
\item Let $\mu\colon G\rightarrowtail F$ be a subfunctor of a super-finitary
functor $F$ with a quotient $\varepsilon\colon A\times
\Id^n\twoheadrightarrow F$. Form a pullback (object-wise in $\Set$) of
$\varepsilon$ and $\mu$:
\[
\begin{tikzcd}
H
\arrow[>->]{r}{\bar \mu}
\arrow[->>]{d}[swap]{\bar \varepsilon}
\pullbackangle{-45}
&
A \times \Id^n
\arrow[->>]{d}{\varepsilon}
\\
G
\arrow[>->]{r}{\mu} & F
\end{tikzcd}
\]
For each $a\in A$, the preimage $H_a$ of $\{a\}\times \Id^n\cong
\Set(n,-)$ under $\bar\mu$ is super-finitary by Example \ref{superfin:sub}
above. Since $A\times \Id^n = \coprod_{a\in A} \{a\}\times \Id^n$ and
preimages under $\bar \mu$ preserve coproducts, we have $H=\coprod_{a\in
A} H_a$ and so $G$ is a quotient of the super-finitary functor~$H$.
\end{enumerate}
\end{proof}
\begin{lemma} \label{L:kernel2fpfg} Let $\C$ be an lfp
category with finitely generated objects closed under kernel pairs and in
which strong epimorphisms are regular. Then finitely presentable
and finitely generated objects coincide.
\end{lemma}
\begin{proof} We apply Remark \ref{R:prelim}\ref{I:fingen}:
Consider a strong epimorphism $c\colon X\twoheadrightarrow Y$ with $X$
finitely presentable. We are to show that $Y$ is finitely presentable. Let
$p,q\colon K\rightrightarrows X$ be the kernel pair of $c$, then $K$ is
finitely generated. Hence there is some finitely presentable object $K'$ and a
strong epimorphism $e\colon K'\twoheadrightarrow K$:
\[
\xymatrix{
K'
\ar@{->>}[r]^-{e}
& K
\ar@<2pt>[r]^-{p}
\ar@<-2pt>[r]_-{q}
& X
\ar@{->>}[r]^-{c}
& Y
}
\]
Since the strong epimorphism $c$ is also regular, it is the coequalizer of its
kernel pair $(p,q)$; furthermore $e$ is epic, thus $c$ is also the coequalizer of
$p\cdot e$ and $q\cdot e$. This means that $Y$ is a finite colimit of finitely
presentable objects and thus it is finitely presentable.
\end{proof}
\begin{corollary}
\label{cor:setFunNotSStrict}
$[\Set,\Set]_\fin$ is not semi-strictly lfp.
\end{corollary}
\begin{proof}
We use the subfunctors
\[
\bar \pow \subseteq \pow_0 \subseteq \pow
\]
of the power-set functor $\pow$ given by $\pow_0X=\pow X\setminus\{\emptyset\}$ and $\bar \pow X
=\{M\in \pow_0 X\mid M\text{ finite}\}$. Then $\bar\pow$ is an object of
$[\Set,\Set]_\fin$ which is clearly not super-finitary. The only
endomorphism of $\bar \pow$ is $\id_{\bar \pow}$. Indeed for $\pow_0$ this has
been proven in \cite[Proposition 5.4]{adamekSousa}; the same proof applies to
$\bar \pow$.
And $\id_{\bar \pow}$ is not finitary: otherwise $\bar \pow$ would be a
quotient of a finitely presentable object, thus, it would be super-finitary
(due to Lemma~\ref{L:fin}).
\end{proof}
\begin{corollary} \label{cor:supfinfgfp}
For a finitary set functor, as an object of $[\Set,\Set]_\fin$, the following
conditions are equivalent:
\begin{enumerate}
\item \label{supfin:fp} finitely presentable,
\item \label{supfin:fg} finitely generated, and
\item \label{supfin:supfin} super-finitary.
\end{enumerate}
\end{corollary}
\begin{proof}
To verify
\ref{supfin:fg}$\implies$\ref{supfin:supfin}, let $F$ be finitely generated.
For every finite subset $A\subseteq Fn$, $n\in \Nat$, we have a subfunctor
$F_{n,A}\subseteq F$ given by
\[
F_{n,A} X = \,\bigcup_{\mathclap{f\colon n\to X}}\, Ff[A].
\]
Since $F$ is finitary, it is a directed union of all these
subfunctors. This implies $F\cong
F_{n,A}$ for some $n$ and $A$, and $F_{n,A}$ is clearly super-finitary.
For \ref{supfin:supfin}$\implies$\ref{supfin:fg}, combine Lemma \ref{L:fin} and Example \ref{E:supfin}\ref{supfin:automata}.
\ref{supfin:fp}$\iff$\ref{supfin:fg} follows by Lemma~\ref{L:kernel2fpfg}.
\end{proof}
\section{\texorpdfstring{$\lambda$}{Lambda}-Accessible Functors}
Almost everything we have proved above generalizes to locally
$\lambda$-presentable categories for every infinite regular cardinal
$\lambda$. Recall that an object $A$ of a category $\A$ is
\emph{$\lambda$-presentable} (\emph{$\lambda$-generated}) if its
hom-functor $\A(A,-)$ preserves $\lambda$-filtered colimits (of
monomorphisms). A category $\A$ is \emph{locally
$\lambda$-presentable} if it is cocomplete and has a set of
$\lambda$-presentable objects whose closure under $\lambda$-filtered
colimits is all of $\A$. Functors preserving $\lambda$-filtered
colimits are called \emph{$\lambda$-accessible}. We denote by $\A_\lp$ and
$\A_\lg$ full subcategories representing (up to isomorphism) all $\lambda$-presentable and $\lambda$-generated
objects, respectively.
All of Remark~\ref{R:prelim} holds for $\lambda$ in lieu of
$\aleph_0$, with the same references in~\cite{AdamekR94}.
If $\lambda = \aleph_1$ we speak about {\em locally countably presentable
categories}, {\em countably presentable objects}, etc.
\begin{examples}
\begin{enumerate}
\item Complete metric spaces. We denote by
\[
\CMS
\]
the category of complete metric spaces of diameter $\leq 1$ and
non-expanding functions, i.e.~functions $f\colon X \to Y$ such that for
all $x,y \in X$ we have $d_Y(f(x),f(y)) \leq d_X(x,y)$. This
category is locally countably presentable. The classes of
countably presentable and countably generated objects coincide and
these are precisely the compact spaces.
Indeed, every compact (= separable) complete metric space is countably
presentable, see \cite[Corollaries~2.9]{ammu15}. And every countably
generated space $A$ in $\CMS$ is separable: consider the countably filtered
diagram of all spaces $\bar X\subseteq A$ where $X$ ranges over countable
subsets of $A$ and $\bar X$ is the closure in $A$. Since $A$ is the colimit
of this diagram, $\id_A$ factorizes through one of the embeddings $\bar
X\hookrightarrow A$, i.e.~$A=\bar X$ is separable.
\item Complete partial orders. Denote by
\[
\CPO
\]
the category of $\omega$-cpos, i.e.~of posets with joins of
$\omega$-chains and monotone functions preserving joins of
$\omega$-chains. This is also a locally countably presentable
category. An $\omega$-cpo is countably presentable (equivalently,
countably generated) iff it has a countable subset which is dense
w.r.t.~joins of $\omega$-chains.
\end{enumerate}
\end{examples}
Following our convention in Section~\ref{sec:fin} we speak about a
\emph{$\lambda$-generated subobject} $m\colon M \monoto A$ of $A$ if $M$ is
a $\lambda$-generated object of $\A$. This leads to a generalization
of the notion of finitely bounded functors to $\lambda$-bounded
ones. The latter terminology stems from Kawahara and
Mori~\cite{KawaharaM00}, where endofunctors on sets were
considered. Our terminology is slightly different in that
$\lambda$-generated subobjects in $\Set$ have cardinality less than
$\lambda$, whereas subsets of cardinality less than or equal to $\lambda$
were considered in loc.~cit.
\begin{definition}
A functor $F\colon \A \to \B$ is called \emph{$\lambda$-bounded} provided
that given an object $A$ of $\A$, every $\lambda$-generated
subobject $m_0\colon M_0 \monoto FA$ in $\B$ factorizes through the $F$-image of a
$\lambda$-generated subobject $m\colon M \monoto A$ in $\A$:
\[
\xymatrix{
& FM\ar[d]^{Fm}\\
M_0\ar@{-->}[ru]\ar@{ >->}[r]_{m_0} & FA
}
\]
\end{definition}
\begin{theorem}\label{T:bound}
Let $\A$ be a locally $\lambda$-presentable category in which every
$\lambda$-generated object is $\lambda$-presentable. Then for all
functors from $\A$ to locally $\lambda$-presentable
categories preserving monomorphisms we have the equivalence
\[
\text{$\lambda$-accessible} \iff \text{$\lambda$-bounded}.
\]
\end{theorem}
The proof is completely analogous to that of Theorem~\ref{T:finbound}.
\begin{example}
The Hausdorff endofunctor $\H$ on $\CMS$ was proved to be accessible (for some
$\lambda$) by van Breugel et al.~\cite{vanBreugelEA07}. Later it was shown to
be even finitary~\cite{ammu15}. However, these proofs are a bit involved.
Using Theorem~\ref{T:bound} we provide an easy argument why the Hausdorff
functor is countably accessible. (Which, since $\CMS$ is not lfp but is
locally countably presentable, seems to be the `natural' property.)
Recall that for a given metric space $(X,d)$ the distance of a point $x\in X$
to a subset $M \subseteq X$ is defined by $d(x, M) = \inf_{y \in M} d(x,y)$.
The \emph{Hausdorff distance} of subsets $M, N \subseteq X$ is defined as the
maximum of $\sup_{x \in M} d(x,N)$ and $\sup_{y\in N} d(y, M)$. The
\emph{Hausdorff functor} assigns to every complete metric space $X$ the space
$\H X$ of all non-empty compact subsets of $X$ equipped with the Hausdorff
metric. It is defined on non-expanding maps by taking the direct images.
We now easily see that $\H$ is countably accessible:
\begin{enumerate}
\item $\H$ preserves monomorphisms. Indeed, given $f\colon X \monoto Y$
monic, then $f[M] \neq f[N]$ for every pair $M, N$ of distinct elements of
$\H X$, thus $\H f$ is monic, too.
\item $\H$ is countably bounded. In order to see this, let $m_0\colon M_0
\subto \H X$ be a subspace with $M_0$ compact, and choose a
countable dense subset $S \subseteq M_0$. For every element $s \in
S$ the set $m_0(s) \subseteq X$ is compact, hence, separable; choose
a countable dense set $T_s \subseteq m_0(s)$. For the countable set
$T = \bigcup_{s \in S} T_s$ form the closure in $X$ and denote it
by $m\colon M \subto X$. Then $M$ is countably generated, and $M_0
\subseteq \H m[\H M]$; indeed, for every $x \in M_0$ we have $m_0(x)
\subseteq M$ because $M$ is closed, and this holds whenever $x \in
S$ (due to $m_0(x) = \overline{T_x}$).
\end{enumerate}
\end{example}
In the following definition a morphism is called \emph{$\lambda$-ary} if it factorizes
through a $\lambda$-presentable object.
\takeout{
\begin{definition}
A locally $\lambda$-presentable category $\A$ is called {\em strictly} or {\em
semi-strictly} locally $\lambda$-presentable provided that every morphism
$b\colon B\to A$ in $\A_{\lambda}/A$ factorizes through a morphism $b'\colon B'\to A$ in
$\A_{\lambda}/A$ for which some $f\colon A\to B'$ exists and, in the case of strict
locally $\lambda$-presentable, $f\cdot b$ is such a factor,
i.e.~$b=b'\cdot(f\cdot b)$.
\[
\begin{array}{c}
\xymatrix{B\ar[rd]_b\ar@{-->}[rr]&&B'\ar@<2pt>[ld]^{b'}\\
&A\ar@<2pt>[ur]^f&}
\\[5pt]
\text{\em semi-strictly locally $\lambda$-presentable}
\end{array}
\qquad \qquad
\begin{array}{c}
\xymatrix{
B\ar[rd]_b\ar[rr]^{f\cdot b}
&&
B'\ar@<2pt>[ld]^{b'}\\
&
A\ar@<2pt>[ur]^f&}
\\[5pt]
\text{\em strictly locally $\lambda$-presentable}
\end{array}
\]
\end{definition}
}
\begin{definition}
A locally $\lambda$-presentable category is called
\begin{enumerate}
\item \emph{semi-strictly locally $\lambda$-presentable} if every object has a $\lambda$-ary endomorphism;
\item \emph{strictly locally $\lambda$-presentable} if every object has, for each $\lambda$-generated
subobject $m$, a finitary endomorphism $u$ fixing that subobject
(i.e.~$u\cdot m = m$).
\end{enumerate}
\end{definition}
Observe that Remark \ref{R:strict} immediately generalizes to an
arbitrary $\lambda$.
\begin{examples}
\begin{enumerate}
\item $\Set^S$ is strictly locally $\lambda$-presentable iff $\card S
< \lambda$. This is analogous to Example~\ref{E:strict}(1).
\item The category $\mathsf{Grp}$ of groups is semi-strictly locally
$\lambda$-presentable by the same argument as in
Example~\ref{E:set}\ref{R:strict:zeroobject}. However, $\mathsf{Grp}$ is not
strictly locally $\lambda$-presentable for any infinite
cardinal $\lambda$.
To see this, let $A$ be a simple group of cardinality at least $\lambda^\lambda$.
(Recall that for every set $X$ of cardinality $\geq 5$ the group of even
permutations on $X$ is simple.) Since $\mathsf{Grp}$ is an lfp category, there exists
a non-zero homomorphism $b\colon B \to A$ with $B$ finitely presentable. Given a
commutative diagram
\[
\vcenter{
\xymatrix{
B \ar[rr]^-{f \cdot b} \ar[rd]_b && B' \ar[ld]^{b'}\\
& A
}
}
\qquad
\text{for some $f\colon A \to B'$}
\]
we show that $B'$ is not $\lambda$-presentable. Indeed, since $b$ is non-zero,
we see that so is $f\colon A \to B'$. Since $A$ is simple, $f$ is monic, hence
$\card B' \geq \lambda^\lambda$. However, every $\lambda$-presentable group
has cardinality at most $\lambda$. Thus, by an argument analogous to
Remark~\ref{R:strict}\ref{R:strict:str}, $\mathsf{Grp}$ is not strictly
locally $\lambda$-presentable.
\takeout{
The category of groups is not semi-strictly\smnote{I believe
this should be `semi-strictly'.} \lsnote{I agree with Stefan: the category of groups should be semi-strictly locally
$\lambda$-presentable, because it is easy to see that the initial object is always $\lambda$-presentable, for any $\lambda$, thus we may prove semi-strictness exactly as in \ref{E:set}\ref{R:strict:zeroobject}.}locally
$\lambda$-presentable for any infinite cardinal $\lambda$.
Indeed, let $A$ be a simple group of cardinality at least $\lambda^\lambda$.
(Recall that for every set $X$ of cardinality $\geq 5$ the group of even
permutations on $X$ is simple.)
Then every endomorphism of $A$ is monic, hence its image has cardinality at
least $\lambda^\lambda$, too. However,
every $\lambda$-presentable group has cardinality at most $\lambda$.
Thus, $A$ has no $\lambda$-ary endomorphism.}
\item The category $\Nom$ of nominal sets is strictly locally countably
presentable. In order to prove this, we first verify that countably
presentable objects are precisely the countable nominal
sets.
\begin{enumerate}
\item
Let $X$ be a countably presentable nominal set. Then every countable
choice of orbits of $X$ yields a countable subobject of $X$ in
$\Nom$. Thus $X$ is a countably directed union of countable
subobjects. Since $X$ is countably presentable, it follows that $X$
is isomorphic to one of these subobjects. Thus, $X$ is countable.
\item
Conversely, every countable nominal set is countably presentable
since countably filtered colimits of nominal sets are formed on the
level of sets (i.e.~these colimits are preserved and reflected by
the forgetful functor $\Nom \to \Set$).
\end{enumerate}
Now let $b\colon B \to A$ be a morphism in $\Nom$ with $B$
countable. We have $A = \Im(b) + C$ for some subobject $C$
of $A$. Indeed, every nominal set is a coproduct of its orbits, and the
equivariance of $b$ implies that $\Im(b)$ is a coproduct of some of
the orbits of $A$. Furthermore, let $m\colon C_1 \monoto C$ be a
subobject obtained by choosing one orbit from each isomorphism class
of orbits of $C$. We obtain a surjective equivariant map
$e\colon C \epito C_1$ by choosing, for every orbit in
$C \setminus C_1$, a concrete isomorphism to an orbit of $C_1$ and
for every $x \in C_1 \subseteq C$ putting $e(x) = x$. Then we have
$e \cdot m = \id_{C_1}$, i.e.~$m$ is a split monomorphism of
$\Nom$. In the appendix we prove that there are (up to isomorphism)
only countably many single-orbit nominal sets. Hence, $C_1$ is
countable, and thus so is $B' = \Im(b) + C_1$. Moreover, the
morphisms $b' = \id + m\colon B' \to A$ and
$f\colon \id + e\colon A \to B'$ clearly satisfy the desired
property $b = b'\cdot f \cdot b$, see Remark~\ref{R:strict}\ref{R:strict:str}.
\end{enumerate}
\end{examples}
\begin{proposition}
Every semi-simple locally presentable category is strictly locally
$\lambda$-presentable for some $\lambda$.
\end{proposition}
\begin{proof} Let $\A$ be a locally $\kappa$-presentable category that is semi-simple.
\begin{enumerate}
\item $\A$ has only a set of simple objects up to
isomorphism. Indeed, we have a set $\A_{\kappa}$ representing all
$\kappa$-presentable objects. Given a simple object $A$, express
it as a colimit of a $\kappa$-filtered diagram in $\A_{\kappa}$
with a colimit cocone $c_i\colon C_i\to A$, $i\in I$. Since $\A$
is locally presentable, it has (strong epi,
mono)-factorizations~\cite[Proposition~1.61]{AdamekR94}. Then,
since $A$ is simple, either it is a strong quotient of some $C_i$
or it is an initial object. Thus, every simple object
is a strong quotient of a $\kappa$-presentable one. The desired
statement follows since every locally
presentable category is cowellpowered~\cite[Theorem~1.58]{AdamekR94}.
\item Let $\lambda\geq \kappa$ be a regular cardinal such that every
semi-simple object is $\lambda$-presentable. Then $\A$ is locally
$\lambda$-presentable, and the rest of the proof is completely
analogous to point \ref{P:semi-simple:2} in the proof of
Proposition~\ref{P:semi-simple}.
\end{enumerate}
\end{proof}
\begin{corollary}
For every semi-simple ring $R$ the category
$R$-$\mathsf{Mod}$ is strictly locally $\lambda$-presentable
provided that $\lambda > 2^{|R\times R|}$.
\end{corollary}
\noindent
Indeed, the module $R$ has less than $\lambda$ quotient modules. As in
Example~\ref{E:strict}\ref{e:Mod} each quotient is
$\lambda$-presentable in $R$-$\mathsf{Mod}$, and the rest is as in
that example.
\begin{corollary}
Every atomic Grothendieck topos with a set of atoms
(up to isomorphism) is strictly locally $\lambda$-presentable for
some $\lambda$.
\end{corollary}
\noindent
Being a Grothendieck topos, our category is locally
$\lambda$-presentable for some $\lambda$. We can choose $\lambda$ to
be (a) larger than the number of atoms up to isomorphism and (b) such
that every atom is $\lambda$-presentable. Then our topos is strictly
locally $\lambda$-presentable.
\begin{example}\label{E:gpd-lambda}
The category of presheaves on a small groupoid is strictly locally
$\lambda$-presentable. Indeed, the proof that there is, up to
isomorphism, only a set of atomic presheaves is analogous to
Lemma~\ref{L:gpd}.
\end{example}
\begin{theorem}
Let $\A$ be a locally $\lambda$-presentable category.
\begin{enumerate}
\item If $\A$ is strictly locally $\lambda$-presentable, then
for all functors from $\A$ to a locally $\lambda$-presentable
category $\B$ with $\B_\lp= \B_\lg$ we have
\[
\text{$\lambda$-accessible} \iff \text{$\lambda$-bounded}.
\]
\item Conversely, if this equivalence holds for all functors to $\Set$, then
$\A$ is semi-strictly locally $\lambda$-presentable and $\A_\lp = \A_\lg$.
\end{enumerate}
\end{theorem}
The proofs are completely analogous to those of Theorems~\ref{T:boundstrict} and \ref{T:equiv2semstrict}.
\begin{remark} Assume that we work in a set theory distinguishing between sets and classes (e.g. Zermelo-Fraenkel theory) or distinguishing universes, so that by `a class' we take a member of the next higher universe of that of all small sets.
Then we form a super-large category
$$\Class$$
of classes and class functions. It plays a central role in the paper of Aczel and Mendler \cite{AczelMen89} on terminal coalgebras. An endofunctor $F$ of $\Class$ in that paper is called {\em set-based}
if for every class $X$ and every element $x\in FX$ there exists a subset
$i\colon Y\monoto X$ such that $x$ lies in $Fi[FX]$. This corresponds to $\infty$-bounded where $\infty$ stands for `being large'. The corresponding concept of $\infty$-accessibility is evident:
\end{remark}
\begin{definition} A diagram $D\colon \D\to \Class$, with $\D$ not necessarily small, is called {\em $\infty$-filtered} if every small subcategory of $\D$
has a cocone in $\D$. An endofunctor of $\Class$ is called {\em $\infty$-accessible} if it preserves colimits of $\infty$-filtered diagrams.
\end{definition}
\begin{proposition} An endofunctor of $\Class$ is set-based iff it is $\infty$-accessible.
\end{proposition}
\begin{proof}(1) For every morphism $b\colon B\to A$ in $\Class$ with $B$ small
factorizes in $\Set/A$ through a morphism $b'\colon B'\to A$ in $\Set/A$ where the factorization $f$ fulfils $b=b'\cdot (f\cdot b)$.
(Shortly: $\Class$ is strictly locally $\infty$-presentable.) The proof is the same as that of Example \ref{E:set}(2).
(2) The rest is completely analogous to part (1) of the proof of Theorem \ref{T:boundstrict}
\end{proof}
\begin{remark} Assuming, moreover, that all proper classes are mutually bijective, it follows that {\em every} endofunctor on $\Class$ is $\infty$-accessible, see \cite{AMV04}.
\end{remark}
\begin{appendix}
\section{Details on Single-Orbit Nominal Sets}
In this appendix we prove that in the category $\Nom$ of nominal
sets there are (up to isomorphism) only countably many nominal sets
having only one orbit. To this end we consider the nominal sets
$\V^{\#n}$ of injective maps from $n=\{0,1,\dots,\, n-1\}$ to $\V$.
The group action on $\V^{\#n}$ is component-wise, in other words, it
is given by postcomposition: for $t\colon n \monoto \V$ and
$\pi \in \perms(\V)$ (which is a bijective map $\pi\colon \V \to \V$)
the group action is the composed map
$\pi \cdot t\colon n \monoto \V$. Thus, for every $t:n\monoto \V$
of $\V^{\#n}$, $\supp(t)=\{t(i)\mid i<n\}$.
\begin{lemma}
Up to isomorphism, there are only countably many single-orbit nominal sets.
\end{lemma}
\proof
Every single-orbit nominal set $Q$ whose elements have supports of
cardinality $n$ is a quotient of the (single-orbit) nominal set
$\V^{\# n}$ (see~\cite[Exercise~5.1]{Pitts13}). Indeed, if
$Q=\{ \pi\cdot x \mid \pi \in \perms(\V)\}$ with
$\supp(x)=\{a_0,\ldots,a_{n-1}\}$, let $t\colon n \monoto \V$ be
the element of $\V^{\# n}$ with $t(i)=a_i$ and define
$q\colon \V^{\# n} \epito Q$ as follows: for every
$u\in \V^{\# n}$ it is clear that there is some
$\pi \in \perms(\V)$ with $u=\pi \cdot t$; put $q(u)=\pi \cdot
x$. This way, $q$ is well-defined (since
$\supp(x)=\{t(i)\mid i<n\}$) and equivariant.
For every $n\in \Nat$, the quotients of $\V^{\#n}$ are given by equivariant
equivalence relations on $\V^{\#n}$. We prove that we have a bijective
correspondence between the set of all quotients with $|\supp([t]_{\sim})| =
n$ for all $t \in \V^{\#n}$ and the set of all subgroups of $\perms(n)$.
\begin{enumerate}
\item Given an equivariant equivalence $\sim$ on $\V^{\# n}$ put
\[
S = \{ \sigma \in \perms(n) \mid \forall (t\colon n\monoto
\V)\colon t\cdot \sigma \sim t\}.
\]
Note that since $\sim$ is equivariant (and composition of
maps is associative), $\forall$ can equivalently be replaced by
$\exists$:
\[
S = \{ \sigma \in \perms(n) \mid \exists (t\colon n\monoto
\V)\colon t\cdot \sigma \sim t\}.
\]
It is easy to verify that $S$ is a subgroup of
$\perms(n)$. Moreover, we have that, for every $t,u\in \V^{\#n}$,
\begin{equation}\label{Aa}
t\sim u \quad\iff\quad u=t\cdot \sigma\quad \text{for some $\sigma \in S$}.
\end{equation}
Indeed, ``$\Longleftarrow$'' is obvious. For ``$\Longrightarrow$'' suppose that
$t\sim u$. Since $|\supp([t]_{\sim})|=n$, we have that
$\supp(t)=\supp([t]_{\sim})=\supp([u]_{\sim})=\supp(u)$; thus,
there is some $\sigma \in \perms(n)$ such that $u=t\cdot
\sigma$. Consequently, $t \sim t\cdot \sigma$,
showing that $\sigma \in S$.
\enlargethispage{1pt}
\item For every subgroup $S$ of $\perms(n)$, it is clear that the
relation $\sim$ defined by \eqref{Aa} is an equivariant
equivalence. We show that, moreover, $|\supp([t]_{\sim})|=n$ for
every $t\in \V^{\#n}$. We have $|\supp([t]_\sim)| \le n$ because
the canonical quotient map $[-]_\sim$ is equivariant. In order
to see that $|\supp([t]_\sim)|$ is not smaller than $n$, assume
$a \in \supp(t)\setminus \supp([t]_\sim)$ and take any element
$b\not\in \supp(t)$. Then $(a\,b)\cdot [t]_\sim = [t]_\sim$,
i.e.~there is some $\sigma\in \perms(n)$ with
$(a\,b)\cdot t\cdot \sigma = t$, which is a contradiction to
$b\not\in\supp(t) = \supp(t \cdot \sigma) = \{t(i) \mid i <
n\}$.
\item It remains to show that, given two subgroups $S$ and $S'$
which determine the same equivariant equivalence relations
$\sim$ via \eqref{Aa}, then $S=S'$. Indeed, given $\sigma\in S$,
we have $t = (t \cdot \sigma) \cdot \sigma^{-1}$ and therefore
$t\cdot \sigma \sim t$ for every $t\in \V^{\#n}$. By \eqref{Aa}
applied to $S'$, this implies that
$t =t\cdot\sigma \cdot \sigma'$ for some $\sigma' \in S'$. Since
$t$ is monic, we obtain $\sigma \cdot \sigma'=\id_n$,
i.e.~$\sigma=(\sigma')^{-1}\in S'$. This proves $S\subseteq S'$,
and the reverse inclusion holds by symmetry. \endproof
\end{enumerate}
\end{appendix}
\end{document}
\begin{appendix}
\section{Details on Single-Orbit Nominal Sets}
In this appendix we prove that in the category $\Nom$ of nominal
sets there are (up to isomorphism) only countably many nominal sets
having only one orbit. To this end we consider the nominal sets
$\V^{\#n}$ of distinct $n$-tuples:
\[
\V^{\#n} := \{ (a_1,\ldots,a_n)\in \A^n \mid |\{a_1,\ldots,a_n\}| = n\}.
\]
The group action on those sets is component-wise and thus
$\supp(a_1,\ldots,a_n)=\{a_1,\ldots,a_n\}$. Note that an element
$t\in \V^{\#n}$ is, equivalently, an injective map
$t\colon n\monoto \V$, and the group action is given by
postcomposition: for $t\colon n \monoto \V$ and $\pi \in
\perms(\V)$ (i.e.~a bijective map $\pi\colon \V \to \V$) the group
action is the composed map $\pi \cdot t\colon n \monoto \V$.
Recall from Pitts' book~\cite[Exercise~5.1]{Pitts13} that every
one-orbit nominal set is a quotient of one of the nominal sets
$\V^{\# n}$.
\begin{lemma}
\label{lemSubgroupQuot}
For every $n\in \Nat$, we have a bijective correspondence between
the set of quotients $q\colon \V^{\#n}\epito Q$ such that
$|\supp(x)| = n$ for all $x \in Q$ and the set of subgroups $S
\leq \perms(n)$.
\end{lemma}
\proof
The direction from quotients to subgroups is defined by:
\begin{align*}
G(q) &= \{ \sigma \in \perms(n) \mid \forall t\colon n\monoto \V\colon q(t\cdot \sigma) = q(t)\}
\\
&= \{ \sigma \in \perms(n) \mid \exists t\colon n\monoto \V\colon q(t\cdot \sigma) = q(t)\}.
\end{align*}
Note that since $q$ is equivariant (and composition of maps is associative),
$\forall$ and $\exists$ is equivalent here.\footnote{This is an instance of
the freshness quantifier, see \cite[Chapter~3]{Pitts13}.}
The converse direction is defined by
\[
F(S) = [-]_\sim\colon \V^{\#n }\twoheadrightarrow \V^{\#n }/\mathord{\sim}
\quad\text{where }\mathord{\sim} = \{ (t,t\cdot \sigma)\mid t\colon
n\monoto \V, \sigma \in S\}.
\]
\begin{itemize}
\item Let us check that $G$ is well-defined. Given a quotient $q\colon
\V^{\# n}\twoheadrightarrow Q$ in $\Nom$, we need to show that $G(q)$ is
indeed a subgroup of $\perms(n)$. Obviously, the neutral element $\id_n$
lies in $G(q)$. If $\sigma_1,\sigma_2\in G(q)$, then we have $\sigma_1\cdot
\sigma_2\in G(q)$:
\[
q(t\cdot (\sigma_1\cdot\sigma_2))
= q((t\cdot \sigma_1)\cdot\sigma_2)
\overset{\sigma_2\in G(q)}{=} q(t\cdot \sigma_1)
\overset{\sigma_1\in G(q)}{=} q(t)
\qquad\text{ for all }t\colon n\monoto \V.
\]
If $\sigma \in G(q)$, then $\sigma^{-1}\in G(q)$, because
\[
q(t\cdot \sigma^{-1})
\overset{\sigma\in G(q)}{=} q((t\cdot \sigma^{-1}) \cdot\sigma)
= q(t\cdot (\sigma^{-1} \cdot\sigma))
= q(t)
\qquad\text{ for all }t\colon n\monoto \V.
\]
\item Let us check that $F$ is well-defined. For a subgroup $S\le \perms(n)$, the relation $\sim$ is an equivalence
relation. It is reflexive because $\id_n\in S$, transitive because
$S$ is closed under composition, and symmetric because $S$ is closed
under inverses. Furthermore, the subset $\mathord{\sim}\subseteq
\V^{\# n}\times \V^{\# n}$ is equivariant, because composition is
associative. Hence, the quotient $A^{\# n}/\mathord{\sim}$ is
formed in $\Set$ and is a well-defined nominal set. The
equivalence classes are:
\[
[t]_\sim = \{ t\cdot \sigma \mid \sigma \in S\}.
\]
For every $t\in A^{\# n}$, we have $|\supp([t]_\sim)| \le n$ because the
canonical quotient is equivariant. To see that $|\supp([t]_\sim)|$ is not
smaller than $n$, assume $a \in \supp(t)\setminus \supp([t]_\sim)$ and
take any fresh
$b\not\in \supp(t)$. Then $(a\,b)\cdot [t]_\sim = [t]_\sim$, i.e.~there is
some $\sigma\in \perms(n)$ with $(a\,b)\cdot t\cdot \sigma = t$, which is
a contradiction to $b\not\in\supp(t) = \supp(t \cdot \sigma) =
\{t(i) \mid i < n\}$.
\end{itemize}
It remains to show that $G$ and $F$ are mutually inverse.
\begin{itemize}
\item For $S = G(F(S))$ we compute as follows:
\begin{equation*}
\begin{array}{cl}
&\sigma \in G(F(S)) \\
\Leftrightarrow& \exists t\in (n\monoto \V): [t]_\sim = [t\cdot \sigma]_\sim \\
\Leftrightarrow& \exists t\in (n\monoto \V), \exists \sigma'\in S: t = t\cdot \sigma\cdot \sigma' \\
\overset{\text{$t$ mono}}\Leftrightarrow& \exists t\in (n\monoto \V), \exists \sigma'\in S: \id_n = \sigma\cdot \sigma' \\
\Leftrightarrow& \exists \sigma'\in S: \id_n = \sigma\cdot \sigma' \\
\Leftrightarrow& \exists \sigma'\in S: \sigma = \sigma'^{-1} \\
\Leftrightarrow& \sigma \in S \\
\end{array}
\end{equation*}
\item Finally, we prove $q\cong FG(q)$. Given $q\colon
\V^{\#n}\twoheadrightarrow Q$ in $\Nom$ with $|\supp(x)| =n $ for all
$x\in Q$. Hence, for every $t\in \V^{\#n}$ we have that $\supp(t) =
\supp(q(t))$. We need to show that $t\sim t'$ iff $q(t) = q(t')$, where
$\sim$ is given by $F(G(q))$:
\begin{enumerate}
\item[($\Leftarrow$)]
Assume $q(t) = q(t')$. Hence, $\supp(t) = \supp(q(t)) = \supp(q(t')) =
\supp(t')$, and hence there is some $\sigma \in \perms(n)$ with $t' =
t\cdot \sigma$. Hence, $q(t) = q(t') = q(t\cdot \sigma)$ and so $\sigma
\in G(q)$ by the definition of $G(q)$. Consequently $t\sim t'$, because
$t' = t\cdot \sigma$ and $\sigma \in G(q)$
\item[($\Rightarrow$)]
Given $t\sim t'$, then $t' = t\cdot \sigma$ for some $\sigma \in G(q)$
and hence $q(t') = q(t\cdot \sigma) = q(t)$ by the definition
of $G(q)$.\endproof
\end{enumerate}
\end{itemize}
\begin{corollary}
Up to isomorphism, there are only countably many single-orbit sets.
\end{corollary}
\begin{proof}
The desired result follows from Pitts~\cite[Exercise~5.1]{Pitts13}
and the fact that for every natural number $n$ there are only
finitely many quotients $q\colon \V^{\#n}\twoheadrightarrow Q$,
which we now prove by induction on $n$. By the equivariance of
$q$, $Q$ must consist of a single orbit, and every element
$x \in Q$ has at most $n$ atoms in its support:
$|\supp(q(x))| \le |\supp(x)| = n$.
For $n=0$, $\V^{\#0} = 1$ has only one quotient, namely itself. In
the induction step, we consider $\V^{\#n+1}$. By
Lemma~\ref{lemSubgroupQuot}, there are only finitely many
quotients $q\colon \V^{\#n+1}\twoheadrightarrow Q$ with
$|\supp(x)| = n+1$ for all $x\in Q$. Furthermore, there are only
finitely many quotients $q\colon \V^{\#n+1}\twoheadrightarrow Q$
with $|\supp(x)| \le n$ for all $x\in Q$, because every such
quotient must factorize through one of the $n+1$ equivariant
projection maps $p\colon \V^{\#n+1} \twoheadrightarrow \V^{\#n}$,
and $\V^{\#n}$ has only finitely many quotients by the induction
hypothesis. In total, $\V^{\#n+1}$ has only finitely many
quotients.
\end{proof}
\end{appendix}
\end{document}
\begin{appendix}
\section{Is Nom strictly locally $\lambda$-presentable?}
\begin{definition}
Let $\A^{\#n}$ be the set of distinct $n$-tuples:
\[
\A^{\#n} := \{ (a_1,\ldots,a_n)\in \A^n \mid |\{a_1,\ldots,a_n\}| = n\}.
\]
where the nominal structure is component-wise and thus $\supp(a_1,\ldots,a_n)=\{a_1,\ldots,a_n\}$.
Equivalently, $A^{\#n}\cong (n\monoto \V)$, i.e.~every $t\in \V^{\#n}$ is nothing but an injective
map $t\colon n\monoto \V$, where the nominal structure $\pi \cdot t$, $\pi
\in \perms(\V)\subseteq (\V\monoto \V)$ is
defined to be the composition $\pi\cdot t\colon n\monoto \V$. As usual, the
set $n$ here is defined to be $n := \{m\in \Nat \mid m \lneqq n\}$.
\end{definition}
\begin{lemma}
\label{lemSubgroupQuot}
For every $n\in \Nat$, we have a bijections between the sets
\twnote{I think it's even an isomorphism between posets}
\[
\{q\colon \V^{\#n}\twoheadrightarrow Q\text{ in }\Nom\mid \text{for all
}x\in Q: |\supp(x)| =n\}
\]
and
\[
\{S\mid \text{subgroup }S\text{ of }\perms(n)\}.
\]
\end{lemma}
\begin{proof}
The direction from quotients to subgroups is defined by:
\begin{align*}
G(q) &= \{ \sigma \in \perms(n) \mid \forall t\in (n\monoto \V)\colon q(t\cdot \sigma) = q(t)\}
\\
&= \{ \sigma \in \perms(n) \mid \exists t\in (n\monoto \V)\colon q(t\cdot \sigma) = q(t)\}.
\end{align*}
Note that since $q$ is equivariant (and composition of maps is associative),
$\forall$ and $\exists$ is equivalent here.\footnote{This is an instance of
the freshness quantifier, see \cite[Chapter~3]{Pitts13}.}
The converse direction is defined by
\[
F(S) = [-]_\sim\colon \V^{\#n }\twoheadrightarrow \V^{\#n }/\mathord{\sim}
\quad\text{where }\mathord{\sim} = \{ (t,t\cdot \sigma)\mid t\colon
n\monoto \V, \sigma \in S\}.
\]
\begin{itemize}
\item Let us check that $G$ is well-defined. Given a quotient $q\colon
\V^{\# n}\twoheadrightarrow Q$ in $\Nom$, we need to show that $G(q)$ is
indeed a subgroup of $\perms(n)$. Obviously, the neutral element $\id_n$
lies in $G(q)$. If $\sigma_1,\sigma_2\in G(q)$, then we have $\sigma_1\cdot
\sigma_2\in G(q)$:
\[
q(t\cdot (\sigma_1\cdot\sigma_2))
= q((t\cdot \sigma_1)\cdot\sigma_2)
\overset{\sigma_2\in G(q)}{=} q(t\cdot \sigma_1)
\overset{\sigma_1\in G(q)}{=} q(t)
\qquad\text{ for all }t\colon n\monoto \V.
\]
If $\sigma \in G(q)$, then $\sigma^{-1}\in G(q)$, because
\[
q(t\cdot \sigma^{-1})
\overset{\sigma\in G(q)}{=} q((t\cdot \sigma^{-1}) \cdot\sigma)
= q(t\cdot (\sigma^{-1} \cdot\sigma))
= q(t)
\qquad\text{ for all }t\colon n\monoto \V.
\]
\item Let us check that $F$ is well-defined. For a subgroup $S\le \perms(n)$, the relation $\sim$ is an equivalence
relation. It is reflexive because $\id_n\in S$, transitive because
$S$ is closed under composition, and symmetric because $S$ is closed
under inverses. Furthermore, the subset $\mathord{\sim}\subseteq (n\monoto
\V)\times (n\monoto \V)$ is equivariant, because composition is
associative. Hence, the quotient $A^{\# n}/\mathord{\sim}$ is
formed in $\Set$ and is a well-defined nominal set. The
equivalence classes are:
\[
[t]_\sim = \{ t\cdot \sigma \mid \sigma \in S\}.
\]
For every $t\in A^{\# n}$, we have $|\supp([t]_\sim)| \le n$ because the
canonical quotient is equivariant. To see that $|\supp([t]_\sim)|$ is not
smaller than $n$, assume $a \in \supp(t)\setminus \supp([t]_\sim)$ and
take any fresh
$b\not\in \supp(t)$. Then $(a\,b)\cdot [t]_\sim = [t]_\sim$, i.e.~there is
some $\sigma\in \perms(n)$ with $(a\,b)\cdot t\cdot \sigma = t$, which is
a contradiction to $b\not\in\supp(t) = \supp(t \cdot \sigma) =
\{t(i) \mid i < n\}$.
\end{itemize}
It remains to show that $G$ and $F$ are mutually inverse.
\begin{itemize}
\item For $S = G(F(S))$ we compute as follows:
\begin{equation*}
\begin{array}{cl}
&\sigma \in G(F(S)) \\
\Leftrightarrow& \exists t\in (n\monoto \V): [t]_\sim = [t\cdot \sigma]_\sim \\
\Leftrightarrow& \exists t\in (n\monoto \V), \exists \sigma'\in S: t = t\cdot \sigma\cdot \sigma' \\
\overset{\text{$t$ mono}}\Leftrightarrow& \exists t\in (n\monoto \V), \exists \sigma'\in S: \id_n = \sigma\cdot \sigma' \\
\Leftrightarrow& \exists \sigma'\in S: \id_n = \sigma\cdot \sigma' \\
\Leftrightarrow& \exists \sigma'\in S: \sigma = \sigma'^{-1} \\
\Leftrightarrow& \sigma \in S \\
\end{array}
\end{equation*}
\item Finally, we prove $q\cong FG(q)$. Given $q\colon
\V^{\#n}\twoheadrightarrow Q$ in $\Nom$ with $|\supp(x)| =n $ for all
$x\in Q$. Hence, for every $t\in \V^{\#n}$ we have that $\supp(t) =
\supp(q(t))$. We need to show that $t\sim t'$ iff $q(t) = q(t')$, where
$\sim$ is given by $F(G(q))$:
\begin{enumerate}
\item[($\Leftarrow$)]
Assume $q(t) = q(t')$. Hence, $\supp(t) = \supp(q(t)) = \supp(q(t')) =
\supp(t')$, and hence there is some $\sigma \in \perms(n)$ with $t' =
t\cdot \sigma$. Hence, $q(t) = q(t') = q(t\cdot \sigma)$ and so $\sigma
\in G(q)$ by the definition of $G(q)$. Consequently $t\sim t'$, because
$t' = t\cdot \sigma$ and $\sigma \in G(q)$
\item[($\Rightarrow$)]
Given $t\sim t'$, then $t' = t\cdot \sigma$ for some $\sigma \in G(q)$
and hence $q(t') = q(t\cdot \sigma) = q(t)$ by the definition of $G(q)$.
\end{enumerate}
\end{itemize}
\end{proof}
\begin{corollary}
Up to isomorphism, there are only countably many orbit-finite sets.
\end{corollary}
\begin{proof}
We first prove by induction on $n$ that there are only finitely many
quotients $q\colon \V^{\#n}\twoheadrightarrow Q$. Recall that $Q$ must
consist of a single orbit, and every element $x \in Q$ has at most $n$ atoms
in its support: $|\supp(q(x))| \le n$.
For $n=0$, $\V^{\#0} = 1$ has only one quotient, namely itself. In the step,
consider $\V^{\#n+1}$. By Lemma~\ref{lemSubgroupQuot}, there are only
finitely many quotients $q\colon \V^{\#n+1}\twoheadrightarrow Q$ with
$|\supp(x)| = n+1$ for all $x\in Q$. Furthermore, there are only finitely
many quotients $q\colon \V^{\#n+1}\twoheadrightarrow Q$ with $|\supp(x)| \le
n$ for all $x\in Q$, because every such quotient factors through one of the
$n+1$ equivariant maps $p\colon \V^{\#n+1} \twoheadrightarrow \V^{\#n}$, and
$\V^{\#n}$ has only finitely many quotients by the induction hypothesis. In
total, $\V^{\#n+1}$ has only finitely many quotients.
Consequently, for every $n\in \Nat$, there are (up to isomorphism) only
finitely many orbit-finite sets $X$ with $|\supp(x)| \le n$ for all $x\in
X$.
\end{proof}
Let $\mathcal{O}(X)$ the set of orbits of a nominal set $X$:
\[
\mathcal{O}(X) = \big\{\,\{\pi\cdot x\mid \pi \in \perms(\V)\} \subseteq X \mid
x\in X\,
\big\}
\]
Note that one has the equivalence
\[
X\text{ is finitely presentable}
\quad\Longleftrightarrow\quad
X\text{ has finitely many orbits}
\quad\Longleftrightarrow\quad
|\mathcal{O}(X)| \in \omega
\]
\paragraph{Conjecture.}
We conjecture the following equivalence:
\[
X\text{ is $\lambda$-presentable}
\quad\Longleftrightarrow\quad
X\text{ has less than $\lambda$-many orbits}
\quad\Longleftrightarrow\quad
|\mathcal{O}(X)| \in \lambda
\]
\begin{proposition}
Assuming the above conjecture, \Nom is strictly locally $\lambda$-presentable.
\end{proposition}
\begin{proof}
Given a nominal set $X$, let $\mathord{\sim} \subseteq \mathcal{O}(X)\times
\mathcal{O}(X)$ be the equivalence relation defined by
\[
O_1 \sim O_2
\quad\Leftrightarrow\quad
\text{ there exists an isomorphism }\phi\colon O_1\to O_2\text{ (in $\Nom$)}
\]
For each $\sim$-isomorphism class, choose one orbit and for each orbit choose
one witnessing isomorphism. This means we have some choice function $c\colon
\mathcal{O}(X)/\mathord{\sim} \hookrightarrow \mathcal{O}(X)$ and for every
$O\in \mathcal{O}(X)$ an isomorphism $\phi_O\colon O\to c([O]_\sim)$. This
choice of orbits defines a nominal set as follows. Let the nominal set
$m\colon X'\hookrightarrow X$ consist of only the chosen orbits:
\[
X' := \coprod_{\substack{O\in \mathcal{O}(X)\\ c([O]_\sim)\,=\,O}} O
\]
Note that there are only countably many isomorphism classes of orbits, and so
$X'$ has only countably many orbits.
We can now check that $\Nom$ is strictly locally $\lambda$-presentable, for
$\omega < \lambda$. Assume some $b\colon B\to X$ with $B\in \Nom_\lambda$.
Define
\[
f\colon X\to X'+\Im(b)
\qquad
f(x) = \begin{cases}
\inr(x)
& \text{if }x\in \Im(b)
\\
\phi_O(x)
& \text{otherwise, where }x\in O\in \mathcal{O}(X)
\end{cases}
\]
Since $B$ is $\lambda$-presentable, so is $\Im(b)$ and thus also $X'+\Im(b)$
-- recall that $X'$ has at most $\omega$-many orbits so is
$\lambda$-presentable as well. With $[m,m']\colon X'+\Im(b) \to X$, where
$m'\colon \Im(b) \hookrightarrow X$ is the inclusion of the image, we have \(
[m,m']\cdot f\cdot b = b \), as desired.
\end{proof}
\end{appendix}
\end{document} | math | 111,281 |
\begin{document}
\begin{center}
{{\large\bf Phase transition of disordered random networks on quasi-transitive graphs}\footnote{The project is supported partially by CNNSF (No.~11671216) and by
Hu Xiang Gao Ceng Ci Ren Cai Ju Jiao Gong Cheng-Chuang Xin Ren Cai (No. 2019RS1057).}}
{\mathrm{e}}nd{center}
\begin{center}
Liu Yuelin$^{a}$\ \& \ Xiang Kainan$^b$
\vskip 1mm
\footnotesize{$^a$Department of Mathematics, Tianjin University of Finance and Economics\\
Tianjin City 300222, P. R. China}\\
\footnotesize{$^b$ Hunan Key Laboratory for Computation and Simulation in Science and Engineering \&\\
Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education \&\\
School of Mathematics and Computational Science, Xiangtan University\\
Xiangtan City 411105, Hunan Province, P. R. China}\\
\footnotesize{Emails: \texttt{[email protected]} (Liu)\ \ \ \ \ \texttt{[email protected]} (Xiang)}
{\mathrm{e}}nd{center}
\begin{center}
{\mathrm{e}}mph{This paper is dedicated to the memory of Vladas Sidoravicius.}
{\mathrm{e}}nd{center}
\begin{abstract}
Given a quasi-transitive infinite graph $G$ with volume growth rate ${\rm gr}(G),$ a transient biased electric network $(G,\,\mathbf{c}_1)$ with bias $\lambda_1\in (0,\,{\rm gr}(G))$ and a recurrent biased one $(G,\,\mathbf{c}_2)$ with bias $\lambda_2\in ({\rm gr}(G),\infty).$ Write $G(p)$ for the Bernoulli-$p$ bond percolation on $G$, and define percolation process $(G(p))_{p\in [0,\, 1]}$ by the grand coupling. Let $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ be the following biased disordered random network: Open edges $e$ in $G(p)$ take the conductance $\mathbf{c}_1(e)$, and closed edges $g$ in $G(p)$ take the conductance $\mathbf{c}_2(g)$. We mainly study recurrence/transience phase transition for $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ when $p$ varies from $0$ to $1$, and our main results are as follows:
\begin{enumerate}[{\bf (i)}]
\item On connected quasi-transitive infinite graph $G$ with percolation threshold $p_c\in (0,\, 1),$ the biased disordered random network $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ has a non-trivial recurrence/transience phase transition such that the threshold $p_{c}^{*}\in (0,\, 1)$ is deterministic, and almost surely $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ is recurrent for any $p<p_c^*$ and transient for any $p>p_c^*.$
On any Cayley graph $G$ of any group which is virtually ${\mathbb{Z}}$, there is no non-trivial recurrence/transience phase transition for $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$, i.e. $p_{c}^{*}=p_c= 1.$
Note $p_c<1$ for an infinite finitely generated group if and only if it is not virtually ${\mathbb{Z}}.$ Thus there is a non-trivial recurrence/transience phase transition for $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ with $G$ being a Cayley graph if and only if the corresponding group is not virtually ${\mathbb{Z}}$.
\item On ${\mathbb{Z}}^d$ for any $d\geq 1,$ $p_c^{*}= p_c$ (note $p_c=1$ if and only if $d=1$). And on $d$-regular trees $\mathbb{T}^d$ with $d\geq 3$, $p_c^{*}=(\lambda_1\vee 1) p_c$, and thus $p_c^{*}>p_c$ for any $\lambda_1\in (1,\,{\rm gr}(\mathbb{T}^d)).$ Critical $\left({\mathbb{Z}}^d,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,p_c\right)$ with $0<\lambda_1<1<\lambda_2$ and $d=2$ or $0<\lambda_1\leq 1<\lambda_2$ and $d\geq 11$ is recurrent almost surely, and so is critical $\left(\mathbb{T}^d,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,\frac{1}{d-1}\right)$ with $0<\lambda_1\leq 1<d-1<\lambda_2$ and $d\geq 3.$
Generally, we propose a conjecture characterizing the $p_c^*.$
{\mathrm{e}}nd{enumerate}
As a contrast, we also consider phase transition of having unique currents or not for $({\mathbb{Z}}^d,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ with $d\geq 2$ when $p$ varies from $0$ to $1$ (the case $d=1$ is trivial due to $p_c^*=1$), and prove that
almost surely $({\mathbb{Z}}^2,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ with $\lambda_1<1\leq\lambda_2$ has unique currents for any $p\in [0,1]$ (and thus has no current uniqueness/non-uniqueness phase transition), and conjecture that the same conclusion holds for $d\geq 3.$
\vskip 3mm
\noindent{\bf AMS 2020 subject classifications}. 60K35, 60K37, 60J10, 82B43, 05C80, 05C81.
\vskip 3mm
\noindent{\bf Key words and phrases.} Phase transition, disordered random network, recurrence/transience, percolation, biased random walk.
{\mathrm{e}}nd{abstract}
\section{Introduction}
\setcounter{equation}{0}
\noindent
Let $G=(V,\, E)$ be a locally finite infinite connected graph with vertex set $V$ and edge set $E$, and fixed root $o\in V.$ When two vertices $x$ and $y$ of $G$ are adjacent, write $x\sim y$ and denote by $\{x,y\}$ (resp. $xy$) the corresponding undirected edge (resp. directed edge from $x$ to $y$). Write ${\mathbb{Z}}$ (resp. ${\mathbb{N}}$) for the set of all integers (resp. natural numbers). Let each $B_n(o)$ be the closed ball in $G$ centered at $o$ with radius $n,$ and $\vert A\vert$ the cardinality of a set $A.$ Define the lower (volume) growth rate and the (volume) growth rate of graph $G$ respectively by
\begin{eqnarray}\label{eq-growth-rate}
\underline{{\rm gr}}(G)=\liminf\limits_{n\rightarrow\infty}\sqrt[n]{\vert B_n(o)\vert}\ \mbox{and}\
{\rm gr}(G)=\lim\limits_{n\rightarrow\infty}\sqrt[n]{\vert B_n(o)\vert}\ (\mbox{if exists}).
{\mathrm{e}}nd{eqnarray}
Call an edge weighted graph $(G,\,\mathbf{c})$ is a network (or an electrical network), and $\mathbf{c}:\ E\rightarrow{\mathbb{R}}_{+}=[0,\infty)$ the conductance function and its reciprocal $\mathbf{r}=1/\mathbf{c}$ the resistance function. Recall the random walk associated to a network $(G,\,\mathbf{c})$ is a random walk $(X_n)_{n\geq 0}$ on graph $G=(V,\,E)$ with transition probability $\mathbf{p}(\cdot, \cdot)$ such that
\[
\mathbf{p}(x,y):= \frac{c(\{x,y\})}{\sum\limits_{x\in e} \mathbf{c}(e)},\ x,y\in V,\ x\sim y.
\]
Say network $(G,\,\mathbf{c})$ is transient (resp. recurrent) if so is its associated random walk $(X_n)_{n\geq 0}$. For any $\lambda\in (0,\,\infty),$ let
$$\mathbf{C}_\lambda(e)=\lambda^{-\vert e\vert},\ e=\{x,y\}\in E,$$
where with ${\rm dist}(\cdot,\cdot)$ being the graph distance on $G$,
$$\vert e\vert={\rm dist}(x,o)\wedge {\rm dist}(y,o)={\rm dist}(e,o).$$
Say $(G,\,\mathbf{C}_\lambda)$ is a biased network with bias $\lambda$ and its associated random walk ${\rm RW}_\lambda$ on $G$ a biased random walk with bias $\lambda$.
Suppose $(G,\, \mathbf{c}_1)$ and $(G,\, \mathbf{c}_2)$ are two electrical networks. Introduce Bernoulli bond percolation process $\omega=(\omega_p)_{p\in [0,\,1]}:=(G(p))_{p\in [0,\,1]}$ on $G$ by the grand coupling: Let $(U_e)_{e\in E}$ be an i.i.d.\,family of the uniform distribution on $[0,\,1].$ An edge $e$ is open in Bernoulli-$p$ bond percolation $G(p)$ if $U_e\leq p$ and closed in $G(p)$ otherwise. Namely, for any $p\in [0,\,1],$ $\omega_p(e)=I_{\{U_e\leq p\}},\ e\in E.$ Write $\mathbb{P}_p$ for the law of $\omega_p$. Define the following disordered random network process $\left((G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,p)\right)_{p\in [0,1]}$ on $G$: For any $p\in [0,\,1],$ $(G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,p)$
is a random network such that each open edge $e$ in $G(p)$ takes the conductance $\mathbf{c}_1(e)$, while each closed edge $g$ in $G(p)$
takes the conductance $\mathbf{c}_2(g)$. Specially for any $0<\lambda_1<\lambda_2<\infty,$ each $(G,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,p)$ is called a biased disordered random network (with biases $\lambda_1$ and $\lambda_2$). When $(G,\, \mathbf{c}_1)$ is transient and $(G,\, \mathbf{c}_2)$ is recurrent, each $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ with $p\in (0,1)$ is called a competing disordered random network in the sense that $(G,\, \mathbf{c}_1)$ wins $(G,\, \mathbf{c}_2)$ if $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ is transient, and otherwise $(G,\, \mathbf{c}_2)$ wins $(G,\, \mathbf{c}_1).$
This paper mainly studies recurrence/transience phase transition for competing disordered random networks $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ when $p$ varies from $0$ to $1$. To state our main results, recall the following preliminaries:
\begin{enumerate}[{\bf (i)}]
\item
Let ${\rm Aut}(G)$ be the group consisting of all automorphisms of graph $G$. $G$ is quasi-transitive (resp. transitive) if
there are only finitely many orbits (resp. is only one orbit) under group action of ${\rm Aut}(G)$. When $G$ is quasi-transitive, $\underline{{\rm gr}}(G)={\rm gr}(G),$
and the critical parameter $\lambda_c(G)$ such that ${\rm RW}_\lambda$ is transient for $\lambda<\lambda_c(G)$ and recurrent for $\lambda>\lambda_c(G)$ is just ${\rm gr}(G)$ (which can be proved similarly to \cite[Theorem 1.1]{RL1995}).
\item A group $\Gamma$ is an extension of a group $H$ by $Q$ if there is a short exact sequence
\[
1 \longrightarrow Q \stackrel{f}\longrightarrow \Gamma\stackrel{g}\longrightarrow H\longrightarrow 1
\]
such that $f,g$ are group homomorphisms and ${\rm Im}(f)={\rm Ker}(g)$, equivalently $Q$ is a normal subgroup of $\Gamma$ and $H$ is isomorphic to quotient group $\Gamma/Q$. If $Q$ is a finite group, $\Gamma$ is called a finite extension of $H$ or is virtually $H.$ In other words, $\Gamma$ is a finite extension of $H$ if $H$ is a subgroup of $\Gamma$ with a finite index $[\Gamma:\,H].$
{\mathrm{e}}nd{enumerate}
Then our main results, Theorems {\mathrm{Re}}f{generalgraph01}, {\mathrm{Re}}f{recurthm}, {\mathrm{Re}}f{generald}-{\mathrm{Re}}f{thm-current-unique}, are summarized as follows:
\begin{enumerate}[{\bf (i)}]
\item On connected quasi-transitive infinite graph $G$ with percolation threshold $p_c=p_c(G)\in (0,\, 1),$ for any $0<\lambda_1<\lambda_c(G)<\lambda_2,$ $(G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$ has a non-trivial recurrence/transience phase transition such that the threshold $p_{c}^{*}\in (0,\, 1)$ is deterministic, and almost surely $(G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$ is recurrent for any $p<p_c^*$ and transient for any $p>p_c^*.$ On any Cayley graph $G$ of any group which is virtually ${\mathbb{Z}}$, there is no non-trivial recurrence/transience phase transition for $(G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$, i.e. $p_{c}^{*}=p_c= 1.$
Note $p_c<1$ for an infinite finitely generated group if and only if it is not virtually ${\mathbb{Z}}.$ Thus there is a non-trivial recurrence/transience phase transition for $(G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$ with $G$ being a Cayley graph if and only if the corresponding group is not virtually ${\mathbb{Z}}$.
\item On ${\mathbb{Z}}^d$ for any $d\geq 1,$ $p_c^{*}= p_c$ (note $p_c=1$ if and only if $d=1$). And on $d$-regular trees $\mathbb{T}^d$ with $d\geq 3$, $p_c^{*}=(\lambda_1\vee 1) p_c$, and thus $p_c^{*}>p_c$ for any $\lambda_1\in (1,\,\lambda_c(\mathbb{T}^d)).$ Generally, we propose Conjecture {\mathrm{Re}}f{conj-p_c^*} to characterize the $p_c^*$. Critical $\left({\mathbb{Z}}^d,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,p_c\right)$ with $0<\lambda_1<1<\lambda_2$ and $d=2$ or $0<\lambda_1\leq 1<\lambda_2$ and $d\geq 11$ is recurrent almost surely, and so is critical $\left(\mathbb{T}^d,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,\frac{1}{d-1}\right)$ with $0<\lambda_1\leq 1<d-1<\lambda_2$ and $d\geq 3$; and moreover we have Conjectures
{\mathrm{Re}}f{conj-critical-recurrent/transient} and {\mathrm{Re}}f{conj-critical-recurrent-tree}.
\item As a contrast, for $({\mathbb{Z}}^d,\,\mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$ with $d\geq 2$ (the case $d=1$ is trivial due to $p_c^*=1$), we also consider phase transition of having unique currents or not when $p$ varies from $0$ to $1$, and prove that almost surely $({\mathbb{Z}}^2,\,\mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$ with $\lambda_1<1\leq\lambda_2$ has unique currents for any $p\in [0,1]$ (no current uniqueness/non-uniqueness phase transition!), and think that the same conclusion holds for $d\geq 3$ (Conjecture {\mathrm{Re}}f{conj-current-unique}).
{\mathrm{e}}nd{enumerate}
Now we are in the position to describe backgrounds, motivations and interests on disordered random networks and our main results.
Disordered random network is one of the most important models in discrete probability theory and is also widely applied in physics and biology. The natural physical background of disordered random network is to study the effective conductance in the doped semiconductors where each edge has different resistance which decays along temperature (\cite{MA1960, AHL1971, POF2013}). Percolation theory plays an important role in analysis of above models. In biology, disordered random network can be seen in several statistical biology models such as DNA-unzipping experiments or DNA-polymerase phenomenon (\cite{BBCEM2006, BBCEM2007, HFR2009, KSJW2002}). Recall from \cite[pp.\,6-7, pp.\,380-382]{GG1999}, disordered random network $(G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,p)$ on finite (and infinite) graphs $G$ is a mathematical modelling of a disordered mixture of two conductor materials $A$ and $B;$ and effective resistance $\mathscr{R}_i$ of
disordered random network $\left(\{0,\,1,\,\ldots,\,i\}^d,\, 1,\, 0,\, p\right)$ between the bottom and top sides of $\{0,\,1,\,\ldots,\,i\}^d$ satisfies that for a constant $p_c(d)\in (0,\, 1),$
$$
\mathscr{R}_i= \infty\ \mbox{a.s.\,for all large $i$}\ \mbox{if}\ p< p_c(d)\ \mbox{and}\ \mbox{a.s.}\,
\lim\limits_{i\rightarrow \infty}\frac{\mathscr{R}_{i}}{i^{2-d}}\in (0,\,\infty)\ \mbox{exists if}\ p>p_c(d)\ (\cite{RK1983,VJ1994}).
$$
Theoretically, Chernov \cite{AAC1967} introduced the idea of random walk generated in a random environment in 1967 as a mathematical model to study the transport in a random media in biology. Random walk in random environment (RWRE) has become one of the most popular probability models in recent decades (\cite{AAC1967, FS1975, YS1982, RY1995, SS2004, AS2004, OZ2006}). Typical RWRE on Euclidean lattice ${\mathbb{Z}}^d$ can be defined as follows: Suppose $\omega=\{p_x\}_{x\in{\mathbb{Z}}^d}$ is an i.i.d.\,family of random probability measures on $\mathscr{S}_d=\{\pm e_i,\, 1\leq i\leq d\},$
where each $e_i$ is the $i$th standard unit vector in ${\mathbb{Z}}^d.$ Given random environment $\omega,$ define a nearest-neighbour random walk $(X_n)_{n=0}^{\infty}$ on ${\mathbb{Z}}^d$ by
$${\mathbb{P}}_\omega[X_{n+1}=x+y\,\vert\, X_n=x]=p_x(y),\ y\in\mathscr{S}_d,\ n\in{\mathbb{Z}}_{+}:=\{0,\,1,\,2,\,\ldots\}.$$
Such a model was firstly defined on ${\mathbb{Z}}$ by Solomon 1975 \cite{FS1975} (in this case it is also known as Sinai's simple random walk in random environment \cite{YS1982}). The above definition can be extended to more general (random) graphs with random environment being ergodic (e.g. translation invariant independent random environment on transitive graphs). See \cite{AS2004, OZ2006} and \cite[pp.\,56-57]{LP2017}.
There are two layers of randomness for RWRE which makes the model very interesting: the first is the random environment; the second is the random walk in a given random environment. As written in \cite[p.\,56]{LP2017}, ``The topic of RWRE with any i.i.d.\,transition probabilities, is quite natural and extensive but only partially understood, except on trees." Secondly, there is a class of interesting random walks in inhomogeneous random environment such that the law of the random environment is stationary and ergodic with respect to space-time shifts (see \cite{MVK1985}, \cite{BR2017} and \cite{OZ2006}). Thirdly, a special class of RWRE models, which are reversible Markov chain in the environment, is given by the class of nearest-neighbour random conductance models (RCMs). In RCM, random environment is generally translation-invariant and usually random conductance function on edges is an i.i.d.\,family. See \cite{MTB2004, SS2004, BB2007, BP2007, MP2007, BD2010, ABDH2013} and particularly survey \cite{MB2011}.
For fixed $p\in (0,\,1),$ disordered random network $(G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,p)$ is a random electrical network (hence a random conductance model and a random resistance one) and can be viewed naturally as a RWRE; and different with usual RWRE, RCM models, we remove the assumption of stationarity, ergodicity and translation-invariance of random environments (i.e. random conductances) which calls for new techniques and more precise estimation on percolation structure to study a disordered random network. In a certain sense, $(G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,p)$ is a new type of RWRE models. When $p$ evolves from $0$ to $1$, disordered random networks $(G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,p)$ is an interesting interpolation of two deterministic networks $(G,\,\mathbf{c}_1)$ and $(G,\,\mathbf{c}_2);$ and to understand typical probability behaviours varying in $p$ of the interpolation between two networks (or their associated random walks), is an original motivation to study disordered random network process $((G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,p))_{p\in [0,\,1]}.$ As said before, $((G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,p))_{p\in [0,\,1]}$ is also
a competing stochastic process (in fact, a new competing stochastic process) when $(G,\, \mathbf{c}_1)$ is transient and $(G,\, \mathbf{c}_2)$ is recurrent in the sense that $(G,\, \mathbf{c}_1)$ wins $(G,\, \mathbf{c}_2)$ if $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ is transient, and otherwise $(G,\, \mathbf{c}_2)$ wins $(G,\, \mathbf{c}_1).$ The interpolation and competition lead to a natural featured topic of disordered random networks: recurrence/transience phase transitions for $((G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,p))_{p\in [0,\,1]}$ with $p$ varying from $0$ to $1.$
For other phase transitions related to disordered random networks, see Problem {\mathrm{Re}}f{prob-PhaseTransition}. We hope that the percolation theory can lead to a sequence of profound and interesting results for disordered random networks, and conversely disordered random networks can provide new interesting topics (insights) to the percolation theory.
Note interpolation and competition are interesting topics for stochastic processes. Recall that \cite{WLE2018} introduced a $p$-rotor walk on ${\mathbb{Z}}$ which is an interpolation between simple random walk and deterministic rotor walk, and proved an invariance principle such that the limiting process is a doubly perturbed Brownian motion multiplying constant $\sqrt{\frac{1-p}{p}}$. Here the interpolation is to choose random transition probability through site percolation (refer to Subsection {\mathrm{Re}}f{sec-disorderedRW} for such a similar interpolation). Additionally, there are some models studying the competing behaviour such as competing frogs model \cite{MTF2019}, and competing first passage percolation (Richardson model) \cite{DR1973, HP1998, HP2000} and so on.
Our aforementioned main results show that for disordered random networks, recurrence vs transience phase transition and current uniqueness vs non-uniqueness one may present different phase transition vs no phase transition phenomena, and there are interesting universal properties; and disordered random networks can provide new interesting topics to the percolation theory (for this viewpoint see also Section {\mathrm{Re}}f{sec-concluding}). To study systematically disordered random networks is our future goal.
Finally we need to explain the reason for choosing biased conductances $\mathbf{C}_{\lambda}$ to study disordered random networks.
Recall an original motivation for introducing ${\rm RW}_\lambda$ on graphs $G$ by Berretti and Sokal \cite{BA-SA-1985} in 1985 is to design a new Monte Carlo algorithm for self-avoiding walks, see \cite{LG-SA1988,SA-JM1989, RD1994} for refinements of this idea. And ${\rm RW}_\lambda$ has received much attention recently, see \cite{BG-FA2016}, \cite{LP2017} and references therein.
When $G$ is a locally finite quasi-transitive infinite graph, ${\rm RW}_\lambda$s capture geometric information on $G$: notably critical parameter $\lambda_c(G)$, such that ${\rm RW}_\lambda$ is transient for $\lambda<\lambda_c(G)$ and recurrent for $\lambda>\lambda_c(G)$, is just the volume growth rate ${\rm gr}(G)$ for $G.$ While growth of groups is an important area for group theory (\cite{BE2014, HH2015}). Secondly when $G$ is a random graph (e.g. Galton-Watson tree), ${\rm RW}_\lambda$ has close relation with trapping phenomenon of RWREs (\cite{BG-FA2016}). Thirdly, networks $(G,\,\mathbf{C}_\lambda)$ ($\lambda\not=1$) are not transitive, and may provide a very useful setting to check some properties for probability models in a non-ergodic situation. All these facts will make geometry of percolation play an important role in studying recurrence/transience phase transition of disordered random network process $((G,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,p))_{p\in [0,1]}$
and can lead to some interesting results of the mentioned phase transition.\\
\noindent{\bf Notations.} For any graph $G$, use $V=V(G)$ and $E=E(G)$ to denote its vertex and edge sets respectively, and let $\overrightarrow{E}$ be the set of all directed edges of graph $G.$ For any $e\in \overrightarrow{E},$ write $e_{-}$ and $e_{+}$ for its tail and head respectively. Note Bernoulli bond percolation process $\omega=(\omega_p)_{p\in [0,\,1]}:=(G(p))_{p\in [0,\,1]}$ on $G$ is defined by the grand coupling,
and $p_c=p_c(G)$ is the corresponding percolation threshold for infinite $G$. And when $G$ is infinite and quasi-transitive, $\lambda_c(G)={\rm gr}(G).$
For two nonnegative functions $f$ and $g$ defined on a set, denote
$f\asymp g$ if for two positive constants $c_1$ and $c_2$, $c_1g\leq f\leq c_2g$; and denote $f(x)\asymp g(x)$ as $x\rightarrow x_0$ if
for two positive constants $c_1$ and $c_2,$ $c_1g(x)\leq f(x)\leq c_2g(x)$ for $x$ sufficiently close to $x_0.$ Recall ${\mathbb{Z}}$ (resp. ${\mathbb{N}}$) is the set of all integers (resp. natural numbers),
and ${\mathbb{Z}}_+=\{0,\,1,\,2,\ldots\}.$
\section{Main results}
\setcounter{equation}{0}
\noindent
\begin{thm}\label{generalgraph01}
On connected quasi-transitive locally finite infinite graph $G$ with percolation threshold $p_c\in (0,\, 1),$ for any $0<\lambda_1<\lambda_c(G)<\lambda_2,$ $((G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p))_{p\in [0,\,1]}$ has a non-trivial recurrence/transience phase transition such that the threshold $p_{c}^{*}\in (0,\, 1)$ is deterministic, and almost surely $(G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$ is recurrent for any $p<p_c^*$ and transient for any $p>p_c^*.$ On any Cayley graph $G$ of any group which is virtually ${\mathbb{Z}}$, there is no non-trivial recurrence/transience phase transition for $((G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p))_{p\in [0,\,1]}$, i.e. $p_{c}^{*}=p_c= 1.$
{\mathrm{e}}nd{thm}
\vskip 2mm
\begin{remark}\label{remark-PT}
{\bf (i)} Note $p_c<1$ holds for an infinite finitely generated group if and only if it is not virtually ${\mathbb{Z}}$ (\cite[Theorem~1.3]{DGRSY2018}). Thus there is a non-trivial recurrence/transience phase transition for $((G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p))_{p\in [0,\,1]}$ with $G$ being a Cayley graph if and only if the corresponding group is not virtually ${\mathbb{Z}}$.
Additionally, when $\lambda_1,\lambda_2< \lambda_c(G)$ (resp. $\lambda_1,\lambda_2>\lambda_c(G)$), from the Rayleigh's monotonicity principle, almost surely every $(G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$ is transient (resp. recurrent).
{\bf (ii)} There are two thresholds, $p_c^*$ and $\widehat{p}_c^*,$ for recurrence/transient phase transition of
disordered random networks $((G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p))_{p\in [0,\,1]}$ on quasi-transitive infinite graph $G:$
\begin{eqnarray*}
&&p_c^*=\sup\left\{p\in [0,\,1]:\ \mbox{almost surely},\ (G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, q)\ \mbox{is recurrent
for all}\ q\in [0,\,p)\right\},\\
&&\widehat{p}_c^*=\sup\left\{p\in [0,\,1]:\ (G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)\ \mbox{is almost surely recurrent}\right\}.
{\mathrm{e}}nd{eqnarray*}
Due to ergodicity of Bernoulli bond percolation on $G$ (\cite[Proposition~7.3]{LP2017}) and the Rayleigh's monotonicity principle,
it is easy to check that $p_c^*=\widehat{p}_c^*.$ See {\bf (i)} in proving Theorem {\mathrm{Re}}f{generalgraph01}.
{\bf (iii)} To prove recurrence/transience of random networks, a known approach is to take the average network (see \cite[Exercises~2.96-2.97]{LP2017}): Suppose $R$ (resp. $C$) is a random resistance (resp. conductance) function on graph $G$ such that for any $e\in E,$
$$r(e)=\mathbb{E}[R(e)]\in [0,\infty)\ (\mbox{resp.}~c(e)=\mathbb{E}[C(e)]\in [0,\infty)).$$
If $(G,\,r)$ is transient (resp. $(G,\,c)$ is recurrent), then $(G,\,R)$ is a.s. transient (resp. $(G,\,C)$ is a.s. recurrent).
The method lose effect for competing disordered random networks $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ with $p\in (0,\,1):$ When taking average for random conductance function $C(\cdot)$, we have $(G,\,c)$ is a transient network; while when taking average for random resistance function $R(\cdot)$, we see $(G,\, r)$ is a recurrent network.
To get the phase transition of recurrence/transience and value of critical parameter $p_c^{*}$, we need to use one or more ingredients such as the Nash-Williams criterion, the Rayleigh's monotonicity principle, the energy transience/recurrence criterion, geodesic spanning tree, rough embeddings, and more delicate properties with respect to structure of percolations in the cases of Theorems {\mathrm{Re}}f{generalgraph01}, {\mathrm{Re}}f{recurthm}, {\mathrm{Re}}f{generald} and {\mathrm{Re}}f{regulartree1}.
{\mathrm{e}}nd{remark}
\vskip 2mm
\begin{thm}\label{recurthm}
{\bf (i)} Given any two networks $({\mathbb{Z}},\,\mathbf{c}_1)$ and $({\mathbb{Z}},\,\mathbf{c}_2)$ such that $({\mathbb{Z}},\,\mathbf{c}_2)$ is recurrent. Then almost surely, all disordered random networks $({\mathbb{Z}},\,\mathbf{c}_1,\,\mathbf{c}_2,\, p)$ with $p\in [0,\,1)$ are recurrent.
{\bf (ii)} Let $G$ be a Cayley graph of ${\mathbb{Z}}$, and $(G,\,\mathbf{c}_1)$ and $(G,\,\mathbf{c}_2)$ two networks with
$c=\sup\limits_{e\in E}\{\mathbf{c}_2(e)\}<\infty.$ Then almost surely, all $(G,\,\mathbf{c}_1,\,\mathbf{c}_2,\, p)$ with $p\in [0,\,1)$ are recurrent.
{\bf (iii)} Consider graph ${\mathbb{Z}}\times G$ with $G$ being a finite connected graph, two connected networks $({\mathbb{Z}}\times G,\, \mathbf{c}_1)$ and $({\mathbb{Z}}\times G,\, \mathbf{c}_2)$ such that $\mathbf{c}_1$ and $\mathbf{c}_2$ are positive functions, and network on ${\mathbb{Z}}$ with the conductance function
$$\mathbf{c}_2^\prime(\{x,x+1\})=\max\limits_{y\in G}\left\{\mathbf{c}_2\left(\{(x,y), (x+1,y)\}\right)\right\},\ x\in{\mathbb{Z}}$$
is recurrent. Then almost surely, for any $p\in [0,\, 1),$ $({\mathbb{Z}}\times G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,p)$ is recurrent.
{\bf (iv)} Let $\Gamma$ be a finite extension of group ${\mathbb{Z}}$ and $G$ a Cayley graph of $\Gamma.$ Assume $(G,\,\mathbf{c}_1)$ and $(G,\,\mathbf{c}_2)$ are two connected networks such that $\mathbf{c}_1$ and $\mathbf{c}_2$ are positive functions, and $c=\sup\limits_{e\in E}\{\mathbf{c}_2(e)\}<\infty.$ Then almost surely, all $(G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,p)$ with $p\in [0,1)$ are recurrent.
{\mathrm{e}}nd{thm}
\vskip 2mm
\begin{prob}\label{prob-Z-Cayley-Graph}
Do there exist a Cayley graph $\widehat{{\mathbb{Z}}}$ of ${\mathbb{Z}}$, a transient network $\left(\widehat{{\mathbb{Z}}},\,\mathbf{c}_1\right)$ and a recurrent one $\left(\widehat{{\mathbb{Z}}},\,\mathbf{c}_2\right)$ such that $\left(\left(\widehat{{\mathbb{Z}}},\,\mathbf{c}_{1},\,\mathbf{c}_{2},\, p\right)\right)_{p\in [0,\,1]}$ has a nontrivial recurrence/transience phase transition?
{\mathrm{e}}nd{prob}
\vskip 2mm
When $p_c^*\in (0,1),$ does $p_c^*=p_c$ hold? We will see in Theorem {\mathrm{Re}}f{generald}\ (i) and Theorem {\mathrm{Re}}f{regulartree1}\,(i) that both $p_c^*=p_c$ (on ${\mathbb{Z}}^d,\ d\geq 2$) and $p_c^*>p_c$ (on $\mathbb{T}^d,\ d\geq 3$) may be true; and generally we propose Conjecture {\mathrm{Re}}f{conj-p_c^*} to characterize $p_c^*.$ Additionally, whether critical $(G,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,p_c^*)$ on ${\mathbb{Z}}^d$ and $\mathbb{T}^d$ is recurrent almost surely or not, we have Theorem {\mathrm{Re}}f{generald}\ (ii) and Theorem {\mathrm{Re}}f{regulartree1}\,(ii), and Conjectures {\mathrm{Re}}f{conj-critical-recurrent/transient} and {\mathrm{Re}}f{conj-critical-recurrent-tree}.
\begin{thm}\label{generald}
\begin{enumerate}[{\bf (i)}]
\item For $\left(\left({\mathbb{Z}}^d,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,p\right)\right)_{p\in [0,\,1]}$ with $0<\lambda_1\leq 1<\lambda_2$ and $d\geq 3$ or $0<\lambda_1<1\leq \lambda_2$ and $d=2$, $p_c^*$ is just $p_c.$
\item
Critical $\left({\mathbb{Z}}^d,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,p_c\right)$ with $0<\lambda_1<1<\lambda_2$ and $d=2$ or $0<\lambda_1\leq 1<\lambda_2$ and $d\geq 11$ is recurrent almost surely.
{\mathrm{e}}nd{enumerate}
{\mathrm{e}}nd{thm}
\vskip 2mm
For $\mathbb{T}^d$ with $d\geq 3,$ $p_c=\frac{1}{d-1},$ $\lambda_c(\mathbb{T}^d)={\rm gr}(\mathbb{T}^d)= d-1$.
\begin{thm}\label{regulartree1}
\begin{enumerate}[{\bf (i)}]
\item
For $\left(\left(\mathbb{T}^d,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\, p\right)\right)_{p\in [0,\,1]}$ with $0<\lambda_1<d-1\leq \lambda_2$ and $d\geq 3,$
$$p_c^*=\frac{(\lambda_1\vee 1)}{d-1}=(\lambda_1\vee 1)p_c.$$
\item
Critical $\left(\mathbb{T}^d,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,\frac{1}{d-1}\right)$ with $0<\lambda_1\leq 1<d-1<\lambda_2$ and $d\geq 3$ is recurrent almost surely.
{\mathrm{e}}nd{enumerate}
{\mathrm{e}}nd{thm}
\vskip 2mm
For any current $i$ on network $(G,\,\mathbf{c})$, define
\[
{\rm d}^*i(x)= \sum\limits_{y\sim x} i(xy), \ x\in V.
\]
Say currents are unique on $(G,\,\mathbf{c})$ if for any currents $i, i'$ satisfying $d^*i= d^*i'$, we have $i= i'$.
As a contrast to recurrence/transience phase transition, we also consider phase transition of having unique currents or not for $({\mathbb{Z}}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$ with $d\geq 2$ when $p$ varies from $0$ to $1$ (the case $d=1$ is trivial due to $p_c^*=1$), and prove that
almost surely $({\mathbb{Z}}^2,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$ with $\lambda_1<1\leq\lambda_2$ has no current uniqueness/non-uniqueness phase transition, and think that the same conclusion holds for $d\geq 3$ (Conjecture {\mathrm{Re}}f{conj-current-unique}).
Note disordered random walk in Subsection {\mathrm{Re}}f{sec-disorderedRW} can have different features from those of random networks $(G,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,p)$ (e.g. Theorem {\mathrm{Re}}f{thm-sz1}).
\begin{thm}\label{thm-current-unique}
Almost surely, for any $p\in [0,\,1],$ $({\mathbb{Z}}^2,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$ with $0<\lambda_1<\lambda_2<\infty$ has unique currents.
{\mathrm{e}}nd{thm}
\section{Proofs of main results}
\setcounter{equation}{0}
\noindent
In this section, we firstly introduce some necessary preliminaries in Subsection {\mathrm{Re}}f{pre}, then prove Theorem {\mathrm{Re}}f{generalgraph01}, Theorems {\mathrm{Re}}f{recurthm}, {\mathrm{Re}}f{generald} and {\mathrm{Re}}f{thm-current-unique},
and Theorem {\mathrm{Re}}f{regulartree1} in respectively Subsection {\mathrm{Re}}f{quasi}, Subsection {\mathrm{Re}}f{rzd}, and Subsection {\mathrm{Re}}f{rrt}.
\subsection{Preliminaries}\label{pre}
\noindent
Given two subsets $A$ and $Z$ of $V,$ call $v:\ V\rightarrow\mathbb{R}$ is a voltage function if it is harmonic at any $x\notin A\cup Z$.
Call a function $\theta:\ \overrightarrow{E}\rightarrow\mathbb{R}$ is a flow between $A$ and $Z$ if
$$\theta(xy)=-\theta(yx),\ \forall\,x,y\in V,\ x\sim y,\ \mbox{and}\ \sum_{w\sim z}\theta(zw)= 0,\ \forall z\notin A\cup Z.$$
For an antisymmetric function $\theta$ on $\overrightarrow{E}$, define its energy to be $\mathscr{E}(\theta)=\frac{1}{2}\sum_{xy\in \overrightarrow{E}} \theta^2(xy)\mathbf{r}(\{x,y\})$.
Call a flow $i(\cdot)$ is a current on network $(G,\,\mathbf{c})$ between $A$ and $Z$ if there is a voltage function $v$ satisfying the Ohm's law:
$$\mbox{for any}\ x\sim y,\ v(x)- v(y) = i(xy)/\mathbf{c}(\{x,y\}):=i(xy)\mathbf{r}(\{x,y\}).$$
Since $\sum_{x\sim a} i(ax)$ is the total amount of current flowing into the circuit at vertex $a$, one can regard the entire circuit between $a$ and $Z$ as a single conductor with effective conductance
\[C_{\rm eff}:= \sum\limits_{x\sim a} \mathbf{c}(\{a, x\}){\mathbb{P}}\left[a\rightarrow Z\right] =: \mathscr{C}(a\leftrightarrow Z)= \mathscr{C}_{\mathbf{c}}(a\leftrightarrow Z)= \mathscr{C}_{\mathbf{c}}(a\leftrightarrow Z;\,G),\]
where ${\mathbb{P}}(a\rightarrow Z)$ is the probability that $(X_n)_{n\geq 0}$, the random walk associated to network $(G,\,\mathbf{c})$ starting at $a$, hits $Z$ before visiting $a$ again. Define the effective resistance between $a$ and $Z$ as
$$\mathscr{R}(a\leftrightarrow Z)=\mathscr{R}(a\leftrightarrow Z;\,G)=\frac{1}{\mathscr{C}(a\leftrightarrow Z)}.$$
When $A$ is not a singleton, define $\mathscr{C}(A\leftrightarrow Z)$ to be $\mathscr{C}(a\leftrightarrow Z)$ by identifying $A$ to a single vertex $a,$ and $\mathscr{R}(A\leftrightarrow Z)=\frac{1}{\mathscr{C}(A\leftrightarrow Z)}.$
To define $\mathscr{C}(a\leftrightarrow \infty)$, take a sequence $(G_n)_n$ of finite subgraphs of $G$ exhausting $G,$ i.e., $G_n\subseteq G_{n+1}$ and $G=\bigcup\limits_{n}G_n.$ Let $Z_n$ be the vertex set of $G\setminus G_n$ and $G_n^W$ the graph obtained from $G$ by identifying $Z_n$ to a single vertex $z_n$ and removing loops (but keeping multiple edges). Call
$$\mathscr{C}(a\leftrightarrow \infty)=\mathscr{C}_{\mathbf{c}}(a\leftrightarrow \infty):=\lim\limits_{n\rightarrow\infty}\mathscr{C}_{\mathbf{c}}\left(a\leftrightarrow z_n;\,G_n^W\right)$$
the effective conductance from $a$ to $\infty$ in $G,$ and its reciprocal $\mathscr{R}(a\leftrightarrow\infty)=\frac{1}{\mathscr{C}(a\leftrightarrow \infty)}$ the effective resistance.
Recall that on connected network $(G,\,\mathbf{c}),$
$$(X_n)_{n\geq 0}\ \mbox{is transient (resp.\,recurrent)}\ {\mathbb{L}}ongleftrightarrow \mathscr{C}(x\leftrightarrow \infty)>0\ (\mbox{resp}.\,=0)\ \mbox{for any vertex}\ x.$$
\begin{lem}[Rayleigh's monotonicity principle]\label{lem-ray}
Let $G$ be a connected graph with two conductances $\mathbf{c}$ and $\mathbf{c}'$ such that $\mathbf{c}(e)\leq \mathbf{c}'(e),\,e\in E$.
\begin{enumerate}[{\bf (i)}]
\item For finite $G$ and any its two disjoint vertex subsets $A$ and $Z$,
\[\mathscr{C}_{\mathbf{c}}(A\leftrightarrow Z) \leq \mathscr{C}_{\mathbf{c}'}(A\leftrightarrow Z).\]
\item For infinite $G$ and any its vertex $a,$
\[\mathscr{C}_{\mathbf{c}}(a\leftrightarrow \infty) \leq \mathscr{C}_{\mathbf{c}'}(a\leftrightarrow \infty).\]
In particular, $(G,\,\mathbf{c})$ is transient implies so is $(G,\,\mathbf{c}^\prime)$ (equivalently, the recurrence of $(G,\,\mathbf{c}')$ implies that of $(G,\,\mathbf{c})$).
{\mathrm{e}}nd{enumerate}
{\mathrm{e}}nd{lem}
\vskip 2mm
\begin{lem}[The Nash-Williams inequality and recurrence criterion]\label{lem-Nash}
For any distinct vertices $a$ and $z$ separated by pairwise disjoint cutsets ${\mathbb{P}}i_1,\, \cdots,\, {\mathbb{P}}i_{n}$ in a finite network,
\[
\mathscr{R}(a\leftrightarrow z)\geq \sum\limits_{k=1}^{n} \left(\sum\limits_{e\in {\mathbb{P}}i_k}\mathbf{c}(e)\right)^{-1}.
\]
For any sequence $\left\{{\mathbb{P}}i_n\right\}_n$ of pairwise disjoint finite cutsets in an infinite locally finite network $G$ such that
each ${\mathbb{P}}i_n$ separates $a$ from $\infty,$
\[
\mathscr{R}(a\leftrightarrow \infty)\geq \sum\limits_{n}^{\infty} \left(\sum\limits_{e\in {\mathbb{P}}i_n}\mathbf{c}(e)\right)^{-1};
\]
and particularly $G$ is recurrent when the right-hand side is $\infty.$
{\mathrm{e}}nd{lem}
\vskip 2mm
\begin{lem}[Energy transience criterion \mbox{\cite[Theorem 2.11]{LP2017}}]\label{lem-energy}
Connected infinite network $(G,\, \mathbf{c})$ is transient if and only if there exists an unit flow from some (every) vertex to $\infty$ with finite energy.
{\mathrm{e}}nd{lem}
\vskip 2mm
Given two networks $G=((V,\,E),\, \mathbf{c})$ and $G'=((V',\,E'),\,\mathbf{c}')$. Say $G$ can be roughly embedded into $G'$ if there exists a map $\phi:\,V\longmapsto V'$ such that there are constants $\alpha,\beta<\infty$ and
a map ${\mathbb{P}}hi$ mapping oriented edges $xy$ in $G$ to a non-empty simple oriented path ${\mathbb{P}}hi(xy)$ in $G'$ from $\phi(x)$ to $\phi(y)$ such that
\begin{eqnarray*}
&&\mbox{$\sum\limits_{e'\in {\mathbb{P}}hi(xy)} \mathbf{r}'(e')\leq \alpha \mathbf{r}(\{x,y\})$ and ${\mathbb{P}}hi(yx)$ is the reverse of ${\mathbb{P}}hi(xy)$; and for any $e'\in E'$,}\\
&&\mbox{there are no more than $\beta$ edges in $G$ whose image under ${\mathbb{P}}hi$ contains $e'$.}
{\mathrm{e}}nd{eqnarray*}
Call $G$ and $G'$ are roughly equivalent if $G$ and $G'$ can be roughly embedded into each other.
\begin{lem}[Rough embeddings and transience \mbox{\cite[Theorem 2.17]{LP2017}}] For two roughly equivalent connected networks $G$ and $G'$, $G$ is transient iff so is $G'$. In fact, if there is a rough embedding from $G$ to $G'$, then $G$ is transient implies so is $G'$ (equivalently, the recurrence of $G'$ implies the recurrence of $G$).
{\mathrm{e}}nd{lem}
\subsection{Proof of Theorem {\mathrm{Re}}f{generalgraph01}}\label{quasi}
\noindent
To begin, define a geodesic spanning tree $\mathcal{T}$ on a quasi-transitive graph $G$ as follows:
{\bf (i)} Define an order for oriented edges adjacent to each vertex in $G$. Under group action of automorphism group ${\rm Aut}(G)$ of $G$, there are only finitely many orbits $\{\mathcal{O}_i \}_{i=1}^k$. Let $\mathcal{O}_1=\{y_1,\, y_2,\,\ldots\}$. Choose an order `$<$' for all oriented edges $y_1\cdot$ starting at $y_1$, and a sequence $\{\phi_i\}_{i=2}^{\infty}\subseteq {\rm Aut}(G)$ such that $\phi_i(y_1)= y_i\in \mathcal{O}_1$ for any $i\geq 2.$ Then there is a natural way to define an order of oriented edges starting at $\phi_i(y_1)$, namely $\phi_i(y_1)\phi_i(u)< \phi_i(y_1)\phi_i(v)$ if and only if $y_1u<y_1v$. Here $y_1\sim u$ and $y_1\sim v$. Then define similarly `$<$' for
oriented edges starting at vertices in other orbits. Finally, at each vertex of $G$, all oriented edges starting at this vertex have a well-defined order `$<$'.
{\bf (ii)} Notice $o$ is the root of $G$. Then each $x\in V$ can be uniquely denoted by lexicographically minimal finite words of vertices in $G$ as $x= \gamma_0\gamma_1\gamma_2 \cdots\gamma_{|x|}$. Here $\gamma_0= o$ and $\gamma_i\in V,\, 1\leq i\leq |x|$, and $\vert x\vert$ is the graph distance between $x$ and $o.$ {\it In fact}, for any finite words $x= \beta_0\beta_1\beta_2 \cdots \beta_{|x|}$ such that
$$\beta_0=o,\ \{\beta_0,\, \beta_1,\, \beta_2,\, \ldots,\, \beta_{|x|}\}\neq \{\gamma_0,\, \gamma_1,\, \gamma_2,\, \ldots,\, \gamma_{|x|}\},$$
there must be some $0< s\leq |x|$ such that $\gamma_i= \beta_i$ for $0\leq i \leq s-1$ and $\gamma_s\neq \beta_s$. Lexicographical
minimality implies that $\gamma_{s-1}\gamma_s<\beta_{s-1}\beta_{s}.$ Denote lexicographically minimal finite words representation of $x$ by $w_x= w_x(0)w_x(1)\cdots w_x(\vert x\vert).$
{\bf (iii)} A geodesic spanning tree $\mathcal{T}$ of $G$ is a subgraph of $G$ with no loop and contains all vertices in $G$ such that there is an edge between any two vertices $x$ and $y$ of $\mathcal{T}$ iff $|x|= |y|+1$ and $w_y(j)=w_x(j)$ for $0\leq j\leq |y|$ or $|y|= |x|+1$ and $w_x(j)=w_y(j)$ for $0\leq j\leq |x|.$ Now the construction of $\mathcal{T}$ is done.\\
Given a locally finite infinite tree $T$ with root $o.$ Recall branching number of $T$ is defined as
$${\rm br}(T)=\sup\limits\left\{\lambda\geq 0:\ {\mathrm{e}}xists\ \mbox{a nonzero flow}\ \theta\ \mbox{on}\ T\ \mbox{such that}\ \vert\theta\vert (e)\leq \lambda^{-\vert e\vert},\ \forall\, \mbox{directed edge}\ e\right\};$$
and by the max-flow min-cut theorem,
$${\rm br}(T)=\sup\left\{\lambda\geq 0:\ \inf\limits_{{\mathbb{P}}i}\sum\limits_{e\in{\mathbb{P}}i}\lambda^{-\vert e\vert}>0\right\},$$
where the $\inf$ is over all cutsets ${\mathbb{P}}i$ separating $o$ from $\infty.$ By \cite[Theorem 3.5]{LP2017}, ${\rm RW}_{\lambda}$ on $T$ is transient if $\lambda<{\rm br}(T)$ and recurrent if $\lambda>{\rm br}(T);$ and by \cite[Theorem 5.15]{LP2017},
\begin{eqnarray}\label{eq-p_c-br-tree}
p_c(T)=\frac{1}{{\rm br}(T)}.
{\mathrm{e}}nd{eqnarray}
\begin{lem}\label{brgr}
For a locally finite quasi-transitive infinite connected graph $G$ with a geodesic spanning tree $\mathcal{T}$,
\begin{equation}\label{quasi-relation}
\lambda_c(G)={\rm br}(\mathcal{T})= {\rm gr}(\mathcal{T})={\rm gr}(G).
{\mathrm{e}}nd{equation}
{\mathrm{e}}nd{lem}
\noindent {\it{Proof.}\hskip 2pt}
By quasi-transitivity, the geodesic spanning tree $\mathcal{T}$ of $G$ is a sub-periodic tree. Then \cite[Theorem 3.8]{LP2017} implies that growth rate ${\rm gr}(\mathcal{T})$ exists and ${\rm gr}(\mathcal{T})={\rm br}(\mathcal{T})$. Note that the graph distances between $o$ and any vertex are the same in tree $\mathcal{T}$ and in original graph $G$. Thus ${\rm gr}(\mathcal{T})={\rm gr}(G)$. Recall $\lambda_c(G)={\rm gr}(G)$. We obtain the lemma immediately.
\rule{4pt}{7pt}
\vskip 2mm
\begin{lem}[\mbox{\cite[Proposition 6.1]{RL1990}}]\label{gpercolation}
Assume $G$ is a locally finite connected infinite graph and $G({\omega}_p)$ the open subgraph of $G$ in Bernoulli-$p$ bond percolation $\omega_p$ with $p\in [0,\,1].$ Given $\omega_p,$ let
$$p_c\left(G({\omega}_p)\right)=\sup\left\{q:\ \mathbb{P}\left[\mbox{Bernoulli-$q$ bond percolation on}\ G(\omega_p)\ \mbox{has an infinite cluster}\right]=0\right\}.$$
Then
\[
p_c\left(G({\omega}_p)\right)=\left(p_c(G)/p\right)\wedge 1 \ \mbox{a.s.}
\]
{\mathrm{e}}nd{lem}
\vskip 2mm
For a tree $\Gamma$ with root $o$, let $\Gamma^{\sigma}= \{\pi\in \Gamma: \sigma\leq \pi\}$ denote the subtree of $\Gamma$ with $\sigma$ and all its descendents. Recall from \cite[Corollary 6.3]{RL1990} that if $K_{\sigma}(\omega_p)$ denotes the cluster of $\sigma$ in $\omega_p$, then when $p> ({\rm br}(\Gamma))^{-1},$
\begin{equation}\label{ppercolation}
\sup\limits_{\sigma\in \Gamma}{\rm br}(K_{\sigma}(\omega_p))= \sup\limits_{\sigma\in \Gamma} {\rm br}(K^{\sigma}(\omega_p))= p \cdot {\rm br}(\Gamma) \ \ a.s.,
{\mathrm{e}}nd{equation}
where $K^{\sigma}(\omega_p)=\Gamma^{\sigma}\cap K_{\sigma}(\omega_p)$ and the branching number of a finite tree is regarded as zero. Therefore,
$${\rm ess} \sup {\rm br}(K_{o}(\omega_p))= p\cdot {\rm br}(\Gamma).$$
\vskip 2mm
\noindent{\bf Proof of Theorem {\mathrm{Re}}f{generalgraph01}.} {\bf (i)} Fix $p\in [0,\,1].$ Define an event on percolation configuration space:
\[
A_p=\{(G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)\ \mbox{is recurrent}\}.
\]
Then $A_p$ is an invariant event under group action of ${\rm Aut}(G)$, and thus $\mathbb{P}(A_p)\in\{0,1\}$ by the ergodicity of Bernoulli percolation $\omega_p$ (\cite[Proposition~7.3]{LP2017}),
namely either $(G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$ is a.s.\,recurrent or a.s.\,transient.
{\it In fact}, let $\mathbf{C}_{\omega_p}$ be the conductance function of $(G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p),$ then for any $\gamma\in {\rm Aut}(G),$ when $e$ is open,
\begin{eqnarray*}
\mathbf{C}_{\gamma\omega_p}(\gamma e)=\lambda_1^{-\vert\gamma e\vert}\in\lambda_1^{-\vert e\vert}\left[(\lambda_1\wedge 1)^{\vert\gamma o\vert}\wedge(\lambda_1\vee 1)^{-\vert\gamma o\vert} ,\, (\lambda_1\vee 1)^{\vert\gamma o\vert}\vee(\lambda_1\wedge 1)^{-\vert\gamma o\vert}\right],
{\mathrm{e}}nd{eqnarray*}
and when $e$ is closed,
\begin{eqnarray*}
\mathbf{C}_{\gamma\omega_p}(\gamma e)=\lambda_2^{-\vert\gamma e\vert}\in\lambda_2^{-\vert e\vert}\left[\lambda_2^{-\vert\gamma o\vert},\, \lambda_2^{\vert\gamma o\vert}\right].
{\mathrm{e}}nd{eqnarray*}
Write $\mathbf{C}^\prime_{\gamma\omega_p}(e)=\mathbf{C}_{\gamma\omega_p}(\gamma e),\ e\in E.$ Then
$$\mathbf{C}^\prime_{\gamma\omega_p}(e)\asymp \mathbf{C}_{\omega_p}(e),\ e\in E,$$
namely networks $(G,\,\mathbf{C}^{\prime}_{\gamma\omega_p})$ and $(G,\,\mathbf{C}_{\omega_p})$ are equivalent.
Notice $(G,\,\mathbf{C}^{\prime}_{\gamma\omega_p})$ is the image network of $(G,\,\mathbf{C}_{\gamma\omega_p})$ under the automorphism $\gamma.$
Therefore, $(G,\,\mathbf{C}_{\gamma\omega_p})$ is recurrent iff so is $(G,\,\mathbf{C}_{\omega_p}),$ and further $A_p$ is an ${\rm Aut}(G)$-invariant event.
Note that for any $0\leq p\leq q\leq 1,$
$$\omega_p(e)\leq\omega_q(e),\ \mathbf{C}_{\omega_p}(e)\leq\mathbf{C}_{\omega_q}(e),\ \forall\,e\in E;$$
and by the Rayleigh's monotonicity principle (Lemma {\mathrm{Re}}f{lem-ray}), $(G,\,\mathbf{C}_{\omega_p})$ is recurrent if so is $(G,\,\mathbf{C}_{\omega_q}).$
On one hand, remembering $(G,\,\mathbf{C}_{\omega_0})=(G,\,\mathbf{C}_{\lambda_2})$ is recurrent, we have that
\begin{eqnarray*}
p_c^*=\sup\{p\geq 0:\ \mathbb{P}[(G,\,\mathbf{C}_{\omega_p})\ \mbox{is recurrent}]=1\}
{\mathrm{e}}nd{eqnarray*}
satisfies that for any $p<p_c^*$, $(G,\,\mathbf{C}_{\omega_p})$ is a.s. recurrent; and for any $p>p_c^*,$ $(G,\,\mathbf{C}_{\omega_p})$ is a.s. transient.
On the other hand, again by the Rayleigh's monotonicity principle (Lemma {\mathrm{Re}}f{lem-ray}), for any percolation environment $\omega,$
there exists $p_c^{*}(\omega)\in[0,\, 1]$ such that $(G,\, \mathbf{C}_{\omega_p})$ is recurrent for any $p< p_c^{*}(\omega)$ and is transient for any $p> p_c^{*}(\omega)$. So
\begin{eqnarray*}
&&\mbox{almost surely, $p_c^*=p_c(\omega),$ and $(G,\, \mathbf{C}_{\omega_p})$ is recurrent for any $p< p_c^{*}$}\\
&&\mbox{and is transient for any $p> p_c^{*}$.}
{\mathrm{e}}nd{eqnarray*}
{\it Indeed}, for any rational number $p\in [0,\,p_c^*)$ (if exists), almost surely $(G,\,\mathbf{C}_{\omega_p})$ is recurrent; and hence almost surely $p_c^*(\omega)\geq p.$ Let $p\uparrow p_c^*,$ we get that $p_c^*(\omega)
\geq p_c^*$ almost surely. Additionally, for any rational number $q\in (p_c^*,\,1]$ (if exists), almost surely $(G,\,\mathbf{C}_{\omega_q})$ is transient; and thus almost surely $p_c^*(\omega)\leq q.$ Let $q\downarrow p_c^*,$
we have that $p_c^*(\omega)\leq p_c^*$ almost surely. Therefore, almost surely, $p_c^*(\omega)=p_c^*.$
\vskip 2mm
{\bf (ii)} Assume $p_c=p_c(G)\in (0,1).$ To prove $p_c^*\in (0,1).$
Let $\mathcal{T}$ be a geodesic spanning tree of quasi-transitive graph $G$. Then by ({\mathrm{Re}}f{quasi-relation}),
\begin{equation*}
\lambda_c(G)={\rm br}(\mathcal{T})={\rm gr}(\mathcal{T})={\rm gr}(G).
{\mathrm{e}}nd{equation*}
Assume firstly $0<\lambda_1<\lambda_c(G)=1.$ Then for any $p>p_c$, almost surely, there is an infinite open cluster $K$ in $\omega_p.$ By the energy transience criterion (Lemma {\mathrm{Re}}f{lem-energy}),
$(K,\,\mathbf{C}_{\omega_p})=(K,\,\mathbf{C}_{\lambda_1})$ is transient. Thus from the Rayleigh's monotonicity principle (Lemma {\mathrm{Re}}f{lem-ray}),
$(G,\,\mathbf{C}_{\omega_p})$ is transient; and further
$$p_c^*\leq p_c<1.$$
Suppose $0<\lambda_1<\lambda_c(G)$ and $\lambda_c(G)>1.$ Take $p>\frac{\lambda_1\vee 1}{\lambda_c(G)}$ and $\varepsilon\in (0,1)$ such that
$$(p-\varepsilon)\lambda_c(G)>\lambda_1\vee 1.$$
Then by Lemma {\mathrm{Re}}f{gpercolation} and {\mathrm{e}}qref{ppercolation}, almost surely, there exists an infinite open cluster $K_{\sigma}(\omega_{p,\mathcal{T}})$ of some $\sigma\in\mathcal{T}$ in percolation $\omega_{p,\mathcal{T}}$, which is the restriction of $\omega_p$ to $\mathcal{T},$ such that
$${\rm br}\left(K_{\sigma}(\omega_{p,\mathcal{T}})\right)>(p-\varepsilon){\rm br}(\mathcal{T})=(p-\varepsilon)\lambda_c(G)>\lambda_1\vee 1.$$
Fix such an $\omega_{p}$. Note $K_{\sigma}(\omega_{p,\mathcal{T}})$ is a tree. Hence $\left(K_{\sigma}(\omega_{p,\mathcal{T}}),\,\mathbf{C}_{\lambda_1}\right)=\left(K_{\sigma}(\omega_{p,\mathcal{T}}),\,\mathbf{C}_{\omega_p}\right)$ is transient.
Then by the Rayleigh's monotonicity principle (Lemma {\mathrm{Re}}f{lem-ray}), $(G,\,\mathbf{C}_{\omega_p})$ is transient; and further
$$p_c^*\leq\frac{\lambda_1\vee 1}{\lambda_c(G)}<1.$$
Now we are in the position to prove $p_c^*>0.$ Fix any $0<\varepsilon <\lambda_2- \lambda_c(G)$ and let
$$L_{k}:= \{x\in G:\ |x|= k\},\ k\in\mathbb{N}.$$
Use $A\stackrel{\omega_p}{\longleftrightarrow}B$ to denote vertex sets $A$ and $B$ are connected to each other in $\omega_p.$ Given any vertex $x\in G.$ Let $a_n(x)$ be the number of self-avoiding walks (paths) on $G$ with length $n$ starting at $x.$ Then due to $G$ is quasi-transitive,
$$\mu=\lim\limits_{n\rightarrow\infty}\sqrt[n]{a_n(x)}\in [1,\,\infty)\ \mbox{exists and independ of}\ x;$$
and call $\mu$ the connective constant of $G.$ Choose constant $C\in (0,\,\infty)$ such that
$$\vert B_n(o)\vert\leq C(\lambda_c(G)+\varepsilon)^n,\ a_n(x)\leq C(\mu+\varepsilon)^n,\ n\in\mathbb{N},\ x\in G.$$
Then for any $\alpha\in (1,\,\infty)$ with $\lambda_2^{1/\alpha}>\lambda_c(G)+\varepsilon,$ when $p\in \left[0,\,(\mu+\varepsilon)^{-1}(\lambda_c(G)+\varepsilon)^{-\frac{\alpha}{\alpha-1}}\right),$
\begin{equation*}
\begin{split}
\sum\limits_{n= n_0}^{\infty} {\mathbb{P}}\left(L_{\alpha^n}\stackrel{\omega_p}{\longleftrightarrow}L_{\alpha^{n+1}}\right)&\leq \sum\limits_{n=n_0}^{\infty} \sum\limits_{x\in L_{\alpha^{n+1}}} {\mathbb{P}}\left(x
\stackrel{\omega_p}{\longleftrightarrow}L_{\alpha^n}\right)\\
&\leq \sum\limits_{n=n_0}^{\infty}C(\lambda_c(G)+\varepsilon)^{\alpha^{n+1}}\sum\limits_{j\geq \alpha^{n+1}-\alpha^n-1}p^jC(\mu+\varepsilon)^j \\
&=\frac{C^2}{1-p(\mu+\varepsilon)}\sum\limits_{n= n_0}^{\infty}\left[(p(\mu+\varepsilon))^{1-\alpha^{-1}-\alpha^{-(n+1)}}(\lambda_c(G)+\varepsilon)\right]^{\alpha^{n+1}}\\
&<\infty,
{\mathrm{e}}nd{split}
{\mathrm{e}}nd{equation*}
where $\mathbb{N}\ni n_0>\left(-\frac{\log(\alpha-1)}{\log\alpha}\right)\vee 0,$ each $L_{\alpha^n}$ is viewed as $L_{\lfloor\alpha^n\rfloor}$ with $\lfloor\alpha^n\rfloor$ being the integer part of
$\alpha^n.$ By the Borel-Cantelli lemma, almost surely, we can find a sequence $\{{\mathbb{P}}i_n\}_{n=n_1}^{\infty}$ of minimum closed cutsets such that each ${\mathbb{P}}i_n$ is between $L_{\alpha^n}$ and $L_{\alpha^{n+1}}$,
where $n_1$ is a large random natural number. Fix such a percolation configuration. By the Nash-Williams recurrence criterion (Lemma {\mathrm{Re}}f{lem-Nash}),
\begin{equation*}
\begin{split}
\mathscr{R}(o\leftrightarrow \infty)&\geq \sum\limits_{n=n_1}^{\infty} \left(\sum\limits_{e\in {\mathbb{P}}i_n} \mathbf{C}_{\lambda_2}(e)\right)^{-1}\geq \sum\limits_{n=n_1}^{\infty}\left(\lambda_2^{-\lfloor\alpha^n\rfloor} d C(\lambda_c(G)+\varepsilon)^{\alpha^{n+1}}\right)^{-1}\\
&\geq \sum\limits_{n=n_1}^{\infty}\left(\lambda_2^{-\alpha^n+1} d C(\lambda_c(G)+\varepsilon)^{\alpha^{n+1}}\right)^{-1}\\
&= \frac{1}{\lambda_2 dC}\sum\limits_{n=n_1}^{\infty}\left(\frac{\lambda_2^{1/\alpha}}{\lambda_c(G)+\varepsilon}\right)^{\alpha^{n+1}}= \infty,
{\mathrm{e}}nd{split}
{\mathrm{e}}nd{equation*}
and $(G,\,\mathbf{C}_{\omega_p})$ is recurrent. Here we have used that $\vert{\mathbb{P}}i_n\vert$ is no more than the number of edges in $B_{\lfloor\alpha^{n+1}\rfloor}(o)\setminus B_{\lfloor\alpha^n\rfloor}(o)$,
and clearly the latter is at most $d\left\vert B_{\lfloor\alpha^{n+1}\rfloor}(o)\right\vert$ with $d$ being the maximum of vertex degrees of $G.$
Therefore, $p_c^*>0.$
\vskip 2mm
{\bf (iii)} On any Cayley graph $G$ of any group which is virtually ${\mathbb{Z}}$, by Theorem {\mathrm{Re}}f{recurthm} (iv), there is no non-trivial recurrence/transience phase transition for $((G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p))_{p\in [0,\,1]}$, i.e. $p_{c}^{*}=p_c= 1.$
\rule{4pt}{7pt}
\vskip 2mm
\subsection{Proofs of Theorems {\mathrm{Re}}f{recurthm}, {\mathrm{Re}}f{generald} and {\mathrm{Re}}f{thm-current-unique}}\label{rzd}
\noindent\textbf{Proof of Theorem {\mathrm{Re}}f{recurthm}\,(i)-(ii).}
{\bf (i)} Write $e_{i}^{+}$ and $e_{i}^{-}$ for the directed edges from $i$ to $i+1$ and $i-1$
respectively for any $i\in{\mathbb{Z}}.$ Note that any unit flow $\theta$ from $i_0$ to infinity on ${\mathbb{Z}}$ must have the following form:
\begin{equation}\label{eq-flow-structure}
\begin{split}
&\mbox{For some constant $a\in\mathbb{R}$,}\
\theta(e_{i_0+i}^{+})=a,\ \theta(e_{i_0-i}^{-})=1-a,\ i\in \mathbb{Z}_+;\ \mbox{and $\theta$ is}\\
&\mbox{the only unit flow on $i_0+{\mathbb{Z}}_{+}$ (resp. $i_0-{\mathbb{Z}}_{+}$) if $a=1$ (resp. $a=0$).}
{\mathrm{e}}nd{split}
{\mathrm{e}}nd{equation}
For any unit flow $\theta$ on ${\mathbb{Z}}$ and any $p\in [0,\,1],$ the energy $\mathscr{E}_p(\theta)$ of $\theta$ on $({\mathbb{Z}},\,\mathbf{c}_1,\,\mathbf{c}_2,\,p)$ satisfies that
\begin{eqnarray*}
\mathscr{E}_p(\theta)= \sum\limits_{e\in E({\mathbb{Z}})} \frac{\theta^2(e)}{\mathbf{c}_1(e)}I_{\{U_e\leq p\}}+\sum\limits_{e\in E({\mathbb{Z}})} \frac{\theta^2(e)}{\mathbf{c}_2(e)}I_{\{U_e>p\}}
\geq \sum\limits_{e\in E({\mathbb{Z}})} \frac{\theta^2(e)}{\mathbf{c}_2(e)}I_{\{U_e>p\}}.
{\mathrm{e}}nd{eqnarray*}
And $\sum\limits_{e\in E({\mathbb{Z}})} \frac{\theta^2(e)}{\mathbf{c}_2(e)}I_{\{U_e>p\}}$ is a decreasing function in $p.$ So to prove that
\begin{eqnarray}\label{eq-flow-energy-Z}
\mbox{almost surely},\ \mbox{for any unit flow}\ \theta\ \mbox{on}\ {\mathbb{Z}}\ \mbox{and any}\ p\in [0,1),\ \mathscr{E}_p(\theta)=\infty,
{\mathrm{e}}nd{eqnarray}
it suffices to prove that for any fixed $p\in [0,\,1),$ almost surely,
\begin{eqnarray}\label{eq-energy-Z}
\sum\limits_{i\in i_0+{\mathbb{Z}}_+} \frac{1}{\mathbf{c}_2(\{i,i+1\})}I_{\{U_{\{i,i+1\}}>p\}}=\sum\limits_{i\in i_0-{\mathbb{Z}}_+} \frac{1}{\mathbf{c}_2(\{i,i-1\})}I_{\{U_{\{i,i-1\}}>p\}}=\infty,\ i_0\in{\mathbb{Z}}.
{\mathrm{e}}nd{eqnarray}
Once {\mathrm{e}}qref{eq-energy-Z} is true, then {\mathrm{e}}qref{eq-flow-energy-Z} holds; and let $\mathbf{c}_p$ be the conductance function of $({\mathbb{Z}},\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ and
${\mathbb{Z}}(\mathbf{c}_p)$ the graph on ${\mathbb{Z}}$ induced by $\{e\in E({\mathbb{Z}}):\ \mathbf{c}_p(e)>0\}$ for any $p\in [0,1);$ and fix a percolation environment satisfying {\mathrm{e}}qref{eq-flow-energy-Z}.
Note a connected component of network $({\mathbb{Z}},\,\mathbf{c}_p)$ means a connected component of ${\mathbb{Z}}(\mathbf{c}_p).$ Trivially the associated random walk on every finite connected component of network $({\mathbb{Z}},\,\mathbf{c}_p)$ is recurrent. While for the associated random walk on every infinite connected component of network $({\mathbb{Z}},\,\mathbf{c}_p),$ note {\mathrm{e}}qref{eq-flow-structure} and {\mathrm{e}}qref{eq-flow-energy-Z}, by Lemma {\mathrm{Re}}f{lem-energy},
it is recurrent. Namely, assuming {\mathrm{e}}qref{eq-energy-Z}, almost surely, for any $p\in [0,\,1),$ $({\mathbb{Z}},\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ is recurrent. We prove {\mathrm{e}}qref{eq-energy-Z} in two steps.
{\bf (i.1)} We prove firstly
\begin{eqnarray}\label{eq-energy-Z-2}
\sum\limits_{i\in i_0+{\mathbb{Z}}_+} \frac{1}{\mathbf{c}_2(\{i,i+1\})}=\sum\limits_{i\in i_0-{\mathbb{Z}}_+} \frac{1}{\mathbf{c}_2(\{i,i-1\})}=\infty,\ i_0\in{\mathbb{Z}}.
{\mathrm{e}}nd{eqnarray}
Let ${\mathbb{Z}}({\mathbf{c}_2})$ be the graph on ${\mathbb{Z}}$ induced by $\{e\in E({\mathbb{Z}}):\ \mathbf{c}_2(e)>0\}.$ When $({\mathbb{Z}},\,\mathbf{c}_2)$ is connected (i.e., ${\mathbb{Z}}(\mathbf{c}_2)$ is the graph $({\mathbb{Z}},\,E({\mathbb{Z}}))$),
by {\mathrm{e}}qref{eq-flow-structure}, recurrence of connected network $({\mathbb{Z}},\,\mathbf{c}_2)$ and Lemma {\mathrm{Re}}f{lem-energy}, {\mathrm{e}}qref{eq-energy-Z-2} is true.
Clearly {\mathrm{e}}qref{eq-energy-Z-2} holds if all connected components of ${\mathbb{Z}}(\mathbf{c}_2)$ are finite. If ${\mathbb{Z}}(\mathbf{c}_2)$ is not connected and has only one infinite connected component, say $i_1+{\mathbb{Z}}_+,$
then trivially
\begin{eqnarray*}
\sum\limits_{i\in i_0-{\mathbb{Z}}_+} \frac{1}{\mathbf{c}_2(\{i,i-1\})}=\infty,\ i_0\in{\mathbb{Z}};
{\mathrm{e}}nd{eqnarray*}
and by recurrence of connected network $(i_1+{\mathbb{Z}}_+,\,\mathbf{c}_2)$, Lemma {\mathrm{Re}}f{lem-energy} and {\mathrm{e}}qref{eq-flow-structure},
\begin{eqnarray}\label{eq-energy-Z-3}
\sum\limits_{i\in i_1+{\mathbb{Z}}_+} \frac{1}{\mathbf{c}_2(\{i,i+1\})}=\infty,\ \mbox{and further}\ \sum\limits_{i\in i_0+{\mathbb{Z}}_+} \frac{1}{\mathbf{c}_2(\{i,i+1\})}=\infty,\ i_0\in {\mathbb{Z}}.
{\mathrm{e}}nd{eqnarray}
If ${\mathbb{Z}}(\mathbf{c}_2)$ is not connected and has two infinite connected components, $i_1+{\mathbb{Z}}_+$ and $i_2-{\mathbb{Z}}_{+}$ with $i_2<i_1,$ similarly to {\mathrm{e}}qref{eq-energy-Z-3}, one can prove
{\mathrm{e}}qref{eq-energy-Z-2}.
{\bf (i.2)} By {\mathrm{e}}qref{eq-energy-Z-2}, for any $i_0\in {\mathbb{Z}},$
\begin{eqnarray*}
\sum\limits_{i\in i_0+{\mathbb{Z}}_+}\left(1-e^{-1/\mathbf{c}_2(\{i,i+1\})}\right)=\infty
=\sum\limits_{i\in i_0-{\mathbb{Z}}_+}\left(1-e^{-1/\mathbf{c}_2(\{i,i-1\})}\right),
{\mathrm{e}}nd{eqnarray*}
and further for any $p\in [0,\,1),$
\begin{eqnarray*}
\prod\limits_{i\in i_0+{\mathbb{Z}}_+}\left\{(1-p)\left(e^{-1/\mathbf{c}_2(\{i,i+1\})}-1\right)+1\right\}=0
=\prod\limits_{i\in i_0-{\mathbb{Z}}_+}\left\{(1-p)\left(e^{-1/\mathbf{c}_2(\{i,i-1\})}-1\right)+1\right\}.
{\mathrm{e}}nd{eqnarray*}
Namely for any $p\in [0,\,1),$
\begin{eqnarray*}
\mathbb{E}\left[\prod\limits_{i\in i_0+{\mathbb{Z}}_+}e^{-\frac{1}{\mathbf{c}_2(\{i,i+1\})}I_{\left\{U_{\{i,i+1\}}>p\right\}}}\right]=0=
\mathbb{E}\left[\prod\limits_{i\in i_0-{\mathbb{Z}}_+}e^{-\frac{1}{\mathbf{c}_2(\{i,i-1\})}I_{\left\{U_{\{i,i-1\}}>p\right\}}}\right],
{\mathrm{e}}nd{eqnarray*}
where we have used that
\begin{eqnarray*}
&&\mathbb{E}\left[\prod\limits_{i\in i_0+{\mathbb{Z}}_+}e^{-\frac{1}{\mathbf{c}_2(\{i,i+1\})}I_{\left\{U_{\{i,i+1\}}>p\right\}}}\right]=
\prod\limits_{i\in i_0+{\mathbb{Z}}_+}\left\{(1-p)\left(e^{-1/\mathbf{c}_2(\{i,i+1\})}-1\right)+1\right\},\\
&&\mathbb{E}\left[\prod\limits_{i\in i_0-{\mathbb{Z}}_+}e^{-\frac{1}{\mathbf{c}_2(\{i,i-1\})}I_{\left\{U_{\{i,i-1\}}>p\right\}}}\right]
=\prod\limits_{i\in i_0-{\mathbb{Z}}_+}\left\{(1-p)\left(e^{-1/\mathbf{c}_2(\{i,i-1\})}-1\right)+1\right\}.
{\mathrm{e}}nd{eqnarray*}
Therefore, {\mathrm{e}}qref{eq-energy-Z} is true.
\vskip 2mm
{\bf (ii)} Since ${\mathbb{Z}}$ is Abelian, so any its Cayley graph corresponds to a certain symmetric generating set. Assume the generating set of $G$ is
$$S=\{\pm a_i:\ 1\leq i\leq{\mathrm{e}}ll,\ 0\leq a_1<a_2<\ldots<a_{\mathrm{e}}ll\}.$$
\begin{figure}[!htp]
\centering
\includegraphics[width=9cm, height= 1.5cm]{cayleyz.pdf}
\caption{A Cayley graph on ${\mathbb{Z}}$ with generating set $\{\pm 2, \pm 3\}$.}
\label{gcayleyz}
{\mathrm{e}}nd{figure}
Given any $p\in (0,\,1).$ For any $x\in{\mathbb{Z}}_+,$ let
$$A_x(p)=\left\{\forall\,y\in (x,\,x+a_{{\mathrm{e}}ll}],\ \{y,y+a_i\}\ \mbox{is closed in}\ \omega_p\ \mbox{for any}\ 1\leq i\leq{\mathrm{e}}ll\right\};$$
and for any negative $x\in{\mathbb{Z}},$ let
$$A_x(p)=\left\{\forall\,y\in [x-a_{{\mathrm{e}}ll},\,x),\ \{y,y-a_i\}\ \mbox{is closed in}\ \omega_p\ \mbox{for any}\ 1\leq i\leq{\mathrm{e}}ll\right\}.$$
Clearly, $\{A_{ka_{{\mathrm{e}}ll}}(p)\}_{k\in{\mathbb{Z}}}$ is an i.i.d.\,sequence of events and $\mathbb{P}[A_0(p)]>0.$ By the law of large numbers, almost surely,
$$\frac{1}{n+1}\sum\limits_{k=0}^n I_{A_{ka_{{\mathrm{e}}ll}}(p)}\rightarrow \mathbb{P}[A_0(p)]>0,$$
and thus infinitely many $A_{ka_{{\mathrm{e}}ll}}(p)$s, say $A_{k_ja_{{\mathrm{e}}ll}}(p),\ j=1,2,\ldots,$ occur. Similarly, almost surely, infinitely many $A_{-ka_{{\mathrm{e}}ll}}(p)$s ($k>0$), say $A_{-s_ja_{{\mathrm{e}}ll}}(p),\ j=1,2,\ldots,$ occur. Fix a percolation environment $\omega=(\omega_q)_{q\in [0,\,1]}$ such that these events hold. Then
\begin{eqnarray*}
&&{\mathbb{P}}i_n=\left\{\{y,y+a_i\}:\ 1\leq i\leq {\mathrm{e}}ll,\ y\in (k_na_{{\mathrm{e}}ll},\,(k_n+1)a_{{\mathrm{e}}ll}]\right\}\\
&&\ \ \ \ \ \ \ \ \ \ \cup \left\{\{y,y-a_i\}:\ 1\leq i\leq {\mathrm{e}}ll,\ y\in [-(s_n+1)a_{{\mathrm{e}}ll},\,-s_na_{{\mathrm{e}}ll})\right\}
{\mathrm{e}}nd{eqnarray*}
is a set of closed edges in $\omega_p$ (and all $\omega_q$ with $q\leq p$) and a cutset of $G$ separating $0$ from infinity. Then effective resistance $\mathscr{R}_q(0\leftrightarrow\infty)$ between $0$ and infinity of $(G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,q)$ with $q\leq p$ satisfies that
\begin{eqnarray*}
\mathscr{R}_q(0\leftrightarrow\infty)\geq \sum\limits_{n=1}^{\infty}\left(\sum\limits_{e\in{\mathbb{P}}i_n}\mathbf{c}_2(e)\right)^{-1}
\geq\sum\limits_{n=1}^{\infty}\left(\sum\limits_{e\in{\mathbb{P}}i_n}c\right)^{-1}
=\frac{1}{c}\sum\limits_{n=1}^{\infty}\frac{1}{\vert{\mathbb{P}}i_n\vert}=\frac{1}{c}\sum\limits_{n=1}^{\infty}\frac{1}{\vert{\mathbb{P}}i_1\vert}=\infty.
{\mathrm{e}}nd{eqnarray*}
By the Nash-Williams criterion, for the percolation environment $\omega,$ all networks $(G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,q)$ with $q\leq p$ are recurrent. Due to $p$ is arbitrary, we are done.
\rule{4pt}{7pt}
\vskip 2mm
\begin{figure}
\centering
\subfigure{
\includegraphics[width=7cm, height= 3cm]{ladder.pdf}}
\hspace{0cm}
\subfigure{
\includegraphics[width=3cm]{finitegraph.pdf}}
\caption{An example of a ladder graph ${\mathbb{Z}} \times G$; $G$ is isomorphic to a square.}
\label{graphladder}
{\mathrm{e}}nd{figure}
\noindent\textbf{Proof of Theorem {\mathrm{Re}}f{recurthm}\,(iii).}
{\bf Step 1.} Consider an infinite connected multi-graph $H=(V(H),\,E(H))$ with loop edges. Let $(H,\,\mathbf{c})$ and $(H,\,\mathbf{c}^*)$ be two networks such that $\mathbf{c}^*(e)=\mathbf{c}(e)$ for any non-loop edge $e$ and otherwise $\mathbf{c}^*(e)=0,$ and
$\sum\limits_{u\in e}\mathbf{c}(e)I_{\{e\ \mbox{is not a loop}\}}>0$ for any vertex $u\in H.$ Then $(H,\,\mathbf{c})$ is recurrent iff so is $(H,\,\mathbf{c}^*).$
{\it Indeed}, let $(X_n)_{n=0}^{\infty}$ be the random walk associated to $(H,\,\mathbf{c})$ starting at $o\in V(H).$ Define
$$\tau_0=0,\ \tau_{k+1}=\inf\{n>\tau_k:\ X_n\not=X_{\tau_{k}}\},\ k\in{\mathbb{Z}}_+.$$
Note that for any $k\in{\mathbb{Z}}_{+},$ given $X_{\tau_k}=u,$ $\tau_{k+1}-\tau_k$ obeys a geometric distribution with parameter
$$q_u=\frac{\sum\limits_{u\in e}\mathbf{c}(e)I_{\{e\ \mbox{is not a loop}\}}}{\sum\limits_{u\in e}\mathbf{c}(e)}>0.$$
This implies that each $\tau_k$ is finite a.s.. It is easy to see that $(X_{\tau_k})_{k=0}^{\infty}$ is just a random walk associated to $(H,\,\mathbf{c}^*)$ starting at $o$. Thus $(X_n)_{n=0}^{\infty}$ is recurrent iff so is $(X_{\tau_k})_{k=0}^{\infty},$ the claim holds.
\vskip 2mm
{\bf Step 2.} Write $\mathbf{c}_p$ for the conductance function of $({\mathbb{Z}}\times G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,p).$
For any $x\in {\mathbb{Z}}$, identify vertices $(x,y),\, y\in G$ as one point $x$. Then we obtain naturally a multi-graph $\widehat{{\mathbb{Z}}}$ on ${\mathbb{Z}}$ with loop edges such that there are $|G|$ parallel edges between any two adjacent points in ${\mathbb{Z}},$ and a disordered random network $\left(\widehat{{\mathbb{Z}}},\,\mathbf{\hat{c}}_1,\,\mathbf{\hat{c}}_2,\,p\right)=:\left(\widehat{{\mathbb{Z}}},\,\mathbf{\hat{c}}_p\right)$ for any $p\in [0,\,1].$ Here $\mathbf{\hat{c}}_1,\,\mathbf{\hat{c}}_2$ and $\mathbf{\hat{c}}_p$ inherit naturally from $\mathbf{c}_1,\,\mathbf{c}_2$ and $\mathbf{c}_p$ respectively. Let each $\mathbf{\hat{c}}_p^*$ be the restriction of $\mathbf{\hat{c}}_p$ to all non-loop edges. Then
by Step 1,
\begin{eqnarray}\label{eq-recurrent-equiv-1}
\mbox{each}\ \left(\widehat{{\mathbb{Z}}},\,\mathbf{\hat{c}}_p\right)\ \mbox{is recurrent iff so is each}\ \left(\widehat{{\mathbb{Z}}},\,\mathbf{\hat{c}}_p^*\right).
{\mathrm{e}}nd{eqnarray}
Denote by $\widetilde{{\mathbb{Z}}}$ the multi-graph obtained from $\widehat{{\mathbb{Z}}}$ by deleting all loop edges of $\widehat{{\mathbb{Z}}}.$ Clearly, each $\left(\widehat{{\mathbb{Z}}},\,\mathbf{\hat{c}}_p^*\right)$ is just the network $\left(\widetilde{{\mathbb{Z}}},\,\mathbf{\hat{c}}_p^*\right).$ Since $\mathbf{c}_1$ and $\mathbf{c}_2$ are positive functions, we have that $\mathbf{c}_p$ and $\mathbf{\hat{c}}_p$ are also positive functions, both
$({\mathbb{Z}}\times G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,p)=({\mathbb{Z}}\times G,\,\mathbf{c}_p)$ and $\left(\widehat{{\mathbb{Z}}},\,\mathbf{\hat{c}}_1,\,\mathbf{\hat{c}}_2,\,p\right)=\left(\widehat{{\mathbb{Z}}},\,\mathbf{\hat{c}}_p\right)$ are connected
networks. Notice
$$({\mathbb{Z}}\times G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,p)=({\mathbb{Z}}\times G,\,\mathbf{c}_p)\longmapsto \left(\widehat{{\mathbb{Z}}},\,\mathbf{\hat{c}}_1,\,\mathbf{\hat{c}}_2,\,p\right)=\left(\widehat{{\mathbb{Z}}},\,\mathbf{\hat{c}}_p\right)$$
is a rough embedding (see \cite{LP2017} p.\,44), by {\mathrm{e}}qref{eq-recurrent-equiv-1} and the version of \cite[Theorem 2.17]{LP2017} on networks with multiple edges and loop ones (which can be proved identically to \cite[Theorem 2.17]{LP2017}), if
\begin{eqnarray}\label{eq-recurrent-equiv-2}
\mbox{almost surely},\ \left(\widehat{{\mathbb{Z}}},\,\mathbf{\hat{c}}_p^*\right)=\left(\widetilde{{\mathbb{Z}}},\,\mathbf{\hat{c}}_p^*\right)\ \mbox{is recurrent for all}\ p\in [0,1),
{\mathrm{e}}nd{eqnarray}
then
\begin{eqnarray*}
\mbox{almost surely},\ \mbox{for any}\ p\in [0,\,1),\ \left({\mathbb{Z}}\times G,\,\mathbf{c}_p\right)\ \mbox{is recurrent}.
{\mathrm{e}}nd{eqnarray*}
Therefore, it suffices to prove {\mathrm{e}}qref{eq-recurrent-equiv-2}.
\vskip 2mm
{\bf Step 3.} This step devotes to prove {\mathrm{e}}qref{eq-recurrent-equiv-2}.
For any $x\in{\mathbb{Z}}$ and $y\in G,$ write $\widetilde{e}_{x,y}$ for the parallel edge of $\widetilde{{\mathbb{Z}}}$ between $x$ and $x+1$, which comes from the edge $e_{x,y}=\{(x,y),(x+1,y)\}$ of ${\mathbb{Z}}\times G.$ Clearly, for any $p\in [0,\,1),$
$\left(\widetilde{{\mathbb{Z}}},\,\mathbf{\hat{c}}_p^*\right)$ is recurrent iff so is $({\mathbb{Z}},\,\mathbf{\tilde{c}}_p)$, where
$U_{x,y}:=U_{e_{x,y}},$ and $B_{x}(p)=\sum\limits_{y\in G}I_{\{U_{x,y}\leq p\}},$
\begin{eqnarray*}
\mathbf{\tilde{c}}_p(\{x,x+1\})&:=&\sum\limits_{y\in G}\mathbf{\hat{c}}_p^*(\widetilde{e}_{x,y})=
\sum\limits_{y\in G}\left\{\mathbf{c}_1(e_{x,y})I_{\{U_{x,y}\leq p\}}+\mathbf{c}_2(e_{x,y})I_{\{U_{x,y}>p\}}\right\}\\
&\leq & \sum\limits_{y\in G}\mathbf{c}_1(e_{x,y})I_{\{U_{x,y}\leq p\}}+\mathbf{c}_2^\prime(\{x,x+1\})(\vert G\vert-B_x(p)),\
x\in{\mathbb{Z}}.
{\mathrm{e}}nd{eqnarray*}
Recall the proof of Theorem {\mathrm{Re}}f{recurthm}\,(i). For any unit flow $\theta$ from $0$ to infinity on ${\mathbb{Z}}$ and any $p\in [0,\,1],$ the energy $\widetilde{\mathscr{E}}_p(\theta)$ of $\theta$ on $({\mathbb{Z}},\,\mathbf{\tilde{c}}_p)$ satisfies that
\begin{eqnarray*}
\widetilde{\mathscr{E}}_p(\theta)= \sum\limits_{i\in {\mathbb{Z}}} \frac{\theta^2(\{i,i+1\})}{\mathbf{\tilde{c}}_p(\{i,i+1\})}
\geq \sum\limits_{i\in {\mathbb{Z}}} \frac{\theta^2(\{i,i+1\})}{\vert G\vert \mathbf{c}_2^\prime(\{i,i+1\})}I_{\{B_i(p)=0\}}.
{\mathrm{e}}nd{eqnarray*}
Note network $({\mathbb{Z}},\,\mathbf{\tilde{c}}_p)$ is connected for any $p\in [0,1).$ Similarly to prove Theorem {\mathrm{Re}}f{recurthm}\,(i), to prove that
$$\mbox{almost surely},\ \mbox{for all}\ p\in [0,1),\ ({\mathbb{Z}},\,\mathbf{\tilde{c}}_p)\ \mbox{is recurrent},$$
it only needs to prove that for any fixed $p\in [0,\,1),$ almost surely,
\begin{eqnarray}\label{eq-energy-Z-1}
\sum\limits_{i\in {\mathbb{Z}}_+} \frac{1}{\mathbf{c}_2^\prime(\{i,i+1\})}I_{\{B_{i}(p)=0\}}=\sum\limits_{i\in {\mathbb{Z}}_+} \frac{1}{\mathbf{c}_2^\prime(\{-i,-i-1\})}I_{\{B_{-i-1}(p)=0\}}=\infty.
{\mathrm{e}}nd{eqnarray}
Note that connected network $({\mathbb{Z}},\,\mathbf{c}_2^\prime)$ is recurrent by the assumption, and
$${\mathbb{P}}[B_i(p)=0]=(1-p)^{\vert G\vert}\in (0,1],\ i\in{\mathbb{Z}}.$$
Similarly to {\mathrm{e}}qref{eq-energy-Z}, one can verify {\mathrm{e}}qref{eq-energy-Z-1}, which implies {\mathrm{e}}qref{eq-recurrent-equiv-2}.
\rule{4pt}{7pt}
\vskip 2mm
\noindent{\bf Proof of Theorem {\mathrm{Re}}f{recurthm}\,(iv).}
{\bf Step 1.} By the assumption, there is a finite normal subgroup $Q$ of $\Gamma$ such that
$$\Gamma/Q\cong{\mathbb{Z}}\ \mbox{and}\ \Gamma=Q\ltimes \Gamma/Q\cong Q\ltimes {\mathbb{Z}}. $$
Assume $T=\{t_1,\,\ldots,\,t_k\}$ is the generating set corresponding to $G.$ Then multiple set
$$\langle T\rangle=\{t_1Q,\,\ldots,\,t_kQ\}$$
is a generating set of $\Gamma/Q\cong{\mathbb{Z}}.$ For convenience, write
$$\langle T\rangle=\{a_1,\,\ldots,\,a_k\}\subset{\mathbb{Z}}\ \mbox{with}\ a_1\leq a_2\leq\ldots\leq a_k.$$
Due to ${\mathbb{Z}}$ is Abelian, the Cayley graphs on ${\mathbb{Z}}$ for $\langle T\rangle$ and $\langle T\rangle_{\pm}=\{\pm a_1,\,\ldots,\,\pm a_k\}$ are identical. Therefore, the image graph $\langle G\rangle$ of $G$ under the map
$$\pi:\ \gamma\in\Gamma\longmapsto \gamma Q\in \Gamma/Q\cong{\mathbb{Z}}$$
is a multi-graph on ${\mathbb{Z}}$ possibly with loop edges such that for any $x,y\in{\mathbb{Z}},$ $x\sim y$ iff $x-y\in\langle T\rangle_{\pm}.$ With abusing notations, let $\left(\langle G\rangle,\,\mathbf{c}_1\right)$ and $\left(\langle G\rangle,\,\mathbf{c}_2\right)$ be the image networks of $(G,\,\mathbf{c}_1)$ and $(G,\,\mathbf{c}_2)$ respectively under $\pi,$ and
$$(\langle G\rangle,\,\mathbf{c}_1,\,\mathbf{c}_2,\, p)\ \mbox{the image network of}\ (G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,p)\ \mbox{for any}\ p\in [0,1]\ \mbox{under}\ \pi.
$$
Write $\langle G\rangle_*$ for the multi-graph obtained from $\langle G\rangle$ by deleting all loop edges, $\mathbf{c}_1^*$ and $\mathbf{c}_2^*$ respectively for the restrictions of $\mathbf{c}_1$ and $\mathbf{c}_2$ to all non-loop edges of $\langle G\rangle.$
Then each $(\langle G\rangle_*,\,\mathbf{c}_1^*,\,\mathbf{c}_2^*,\, p)$ is just the restriction of $(\langle G\rangle,\,\mathbf{c}_1,\,\mathbf{c}_2,\, p)$ to $\langle G\rangle_*.$ And by Step 1 in proving Theorem {\mathrm{Re}}f{recurthm}\,(iii), for any $p\in [0,\,1),$
\begin{eqnarray}\label{eq-recurrent-equiv-3}
(\langle G\rangle_*,\,\mathbf{c}_1^*,\,\mathbf{c}_2^*,\, p)=:(\langle G\rangle_*,\,\mathbf{c}_p^*)\ \mbox{is recurrent iff so is}\
(\langle G\rangle,\,\mathbf{c}_1,\,\mathbf{c}_2,\, p)=:(\langle G\rangle,\,\mathbf{c}_p).
{\mathrm{e}}nd{eqnarray}
Notice that
$$(G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,p)=:(G,\,\mathbf{c}_p)\longmapsto (\langle G\rangle,\,\mathbf{c}_1,\,\mathbf{c}_2,\, p)
=\left(\langle G\rangle,\,\mathbf{c}_p\right)\ \mbox{is a rough embedding}.
$$
Similarly to Step 2 in proving Theorem {\mathrm{Re}}f{recurthm}\,(iii), we have that if
\begin{eqnarray}\label{eq-recurrent-equiv-4}
\mbox{almost surely},\
\left(\langle G\rangle_*,\,\mathbf{c}_p^*\right)\ \mbox{is recurrent for all}\ p\in [0,1),
{\mathrm{e}}nd{eqnarray}
then
\begin{eqnarray*}
\mbox{almost surely},\ \mbox{for any}\ p\in [0,\,1),\ \left(G,\,\mathbf{c}_p\right)\ \mbox{is recurrent}.
{\mathrm{e}}nd{eqnarray*}
Therefore, it suffices to prove {\mathrm{e}}qref{eq-recurrent-equiv-4}.
\vskip 2mm
{\bf Step 2.} Clearly the generating set of $\langle G\rangle_*$ is the multiple set
$S=\{\pm a_i:\ a_i\not=0,\ 1\leq i\leq k\}.$ Removing multiplicities of elements in $S$, we get a set
$\widetilde{S}=\{\pm b_j:\ 0<b_1<\ldots<b_{{\mathrm{e}}ll}\}.$
For any $x\in \widetilde{S},$ let
$$\mathbf{\tilde{c}}_p(\{0,x\})=\sum\limits_{e\in\pi^{-1}(\{0,x\})}\mathbf{c}_p(e),\ p\in [0,1),$$
and identify all parallel edges in $\pi^{-1}(\{0,x\})$ to the edge $\{0,x\}.$ Then we get the Cayley graph $\langle G\rangle_{\sim}$
on ${\mathbb{Z}}$ corresponding to $\widetilde{S},$ and a network process $\left((\langle G\rangle_{\sim},\,\mathbf{\tilde{c}}_p)\right)_{p\in [0,\,1)}.$
Clearly
\begin{eqnarray*}
\mbox{each}\ (\langle G\rangle_{\sim},\,\mathbf{\tilde{c}}_p)\ \mbox{is recurrent iff so is}\ \left(\langle
G\rangle_*,\,\mathbf{c}_p^*\right).
{\mathrm{e}}nd{eqnarray*}
So {\mathrm{e}}qref{eq-recurrent-equiv-4} boils down to that
\begin{eqnarray}\label{eq-recurrent-equiv-5}
\mbox{almost surely},\ (\langle G\rangle_{\sim},\,\mathbf{\tilde{c}}_p)
\ \mbox{is recurrent for all}\ p\in [0,1).
{\mathrm{e}}nd{eqnarray}
\vskip 2mm
{\bf Step 3.} This step is to prove {\mathrm{e}}qref{eq-recurrent-equiv-5}.
Given any $p\in (0,\,1).$ For any $x\in{\mathbb{Z}}_+,$ let
$$A_x(p)=\left\{\forall\,y\in (x,\,x+b_{{\mathrm{e}}ll}],\ \pi^{-1}(\{y,y+b_i\})\ \mbox{is a closed edge set in}\ \omega_p\ \mbox{for any}\ 1\leq i\leq{\mathrm{e}}ll\right\};$$
and for any negative $x\in{\mathbb{Z}},$ let
$$A_x(p)=\left\{\forall\,y\in [x-b_{{\mathrm{e}}ll},\,x),\ \pi^{-1}(\{y,y-b_i\})\ \mbox{is a closed edge set in}\ \omega_p\ \mbox{for any}\ 1\leq i\leq{\mathrm{e}}ll\right\}.$$
Clearly, $\{A_{kb_{{\mathrm{e}}ll}}(p)\}_{k\in{\mathbb{Z}}}$ is an i.i.d.\,sequence of events and $\mathbb{P}[A_0(p)]>0.$ By the law of large numbers, almost surely,
$$\frac{1}{n+1}\sum\limits_{k=0}^n I_{A_{kb_{{\mathrm{e}}ll}}(p)}\rightarrow \mathbb{P}[A_0(p)]>0,$$
and thus infinitely many $A_{kb_{{\mathrm{e}}ll}}(p)$s, say $A_{k_jb_{{\mathrm{e}}ll}}(p),\ j=1,\,2,\,\ldots,$ occur. Similarly, almost surely, infinitely many $A_{-kb_{{\mathrm{e}}ll}}(p)$s ($k>0$), say $A_{-s_jb_{{\mathrm{e}}ll}}(p),\ j=1,\,2,\,\ldots,$ occur. Fix a percolation environment $\omega=(\omega_q)_{q\in [0,\,1]}$ such that these events hold. Then
\begin{eqnarray*}
&&{\mathbb{P}}i_n=\left\{\{y,y+b_i\}:\ 1\leq i\leq {\mathrm{e}}ll,\ y\in (k_nb_{{\mathrm{e}}ll},\,(k_n+1)b_{{\mathrm{e}}ll}]\right\}\\
&&\ \ \ \ \ \ \ \ \ \ \cup \left\{\{y,y-b_i\}:\ 1\leq i\leq {\mathrm{e}}ll,\ y\in [-(s_n+1)b_{{\mathrm{e}}ll},\,-s_nb_{{\mathrm{e}}ll})\right\}
{\mathrm{e}}nd{eqnarray*}
is a cutset of $\langle G\rangle_{\sim}$ separating $0$ from infinity such that $\pi^{-1}(e)$ is a closed edge set in $\omega_p$ (and all $\omega_q$ with $q\leq p$) for any $e\in{\mathbb{P}}i_n.$ Notice that for any edge $e$ in ${\mathbb{P}}i_n$ and $q\leq p,$
\begin{eqnarray*}
\mathbf{\tilde{c}}_q(e)&=&\sum\limits_{e^\prime\in\pi^{-1}(e)}\mathbf{c}_q(e^\prime)=\sum\limits_{e^\prime\in\pi^{-1}(e)}\left\{
\mathbf{c}_1(e^\prime)I_{\{U_{e^\prime}\leq q\}}+ \mathbf{c}_2(e^\prime)I_{\{U_{e^\prime}>q\}}\right\}\\
&=&\sum\limits_{e^\prime\in\pi^{-1}(e)}\mathbf{c}_2(e^\prime)\leq c\left\vert\pi^{-1}(e)\right\vert
\leq c\max\limits_{e\in{\mathbb{P}}i_n}\left\{\left\vert\pi^{-1}(e)\right\vert\right\}
=c\max\limits_{e\in{\mathbb{P}}i_1}\left\{\left\vert\pi^{-1}(e)\right\vert\right\}:=C.
{\mathrm{e}}nd{eqnarray*}
Thus effective resistance $\mathscr{R}_q(0\leftrightarrow\infty)$ between $0$ and infinity of
$(\langle G\rangle_{\sim},\,\mathbf{\tilde{c}}_q)$ with $q\leq p$ satisfies that
\begin{eqnarray*}
\mathscr{R}_q(0\leftrightarrow\infty)\geq \sum\limits_{n=1}^{\infty}\left(\sum\limits_{e\in{\mathbb{P}}i_n}\mathbf{\tilde{c}}_q(e)\right)^{-1}
\geq\sum\limits_{n=1}^{\infty}\left(\sum\limits_{e\in{\mathbb{P}}i_n}C\right)^{-1}
=\frac{1}{C}\sum\limits_{n=1}^{\infty}\frac{1}{\vert{\mathbb{P}}i_n\vert}=\frac{1}{C}\sum\limits_{n=1}^{\infty}\frac{1}{\vert{\mathbb{P}}i_1\vert}=\infty.
{\mathrm{e}}nd{eqnarray*}
By the Nash-Williams criterion, for the percolation environment $\omega,$ all networks $(\langle G\rangle_{\sim},\,\mathbf{\tilde{c}}_q)$
with $q\leq p$ are recurrent. Due to $p$ is arbitrary, we are done.
\rule{4pt}{7pt}
\vskip 2mm
The following Lemmas {\mathrm{Re}}f{clustersize} and {\mathrm{Re}}f{supdense} are necessary preliminaries for proving Theorem {\mathrm{Re}}f{generald}. To state them, for any $n\in {\mathbb{N}}$, let
\[
R_n:= \{x\in {\mathbb{Z}}^d:\ \vert x\vert \leq n\},\
\partial R_n:= \{x\in {\mathbb{Z}}^d:\ \vert x\vert= n\}.
\]
Here $\vert \cdot \vert$ denotes ${\mathrm{e}}ll_{1}$ norm (graph distance) on ${\mathbb{Z}}^d$. Recall $A\stackrel{\omega_p}{\longleftrightarrow}B$ denotes vertex sets $A$ and $B$ are connected to each other in $\omega_p.$
When $p$ is fixed, write $A{\longleftrightarrow}B$ for $A\stackrel{\omega_p}{\longleftrightarrow}B.$
\begin{lem}\label{clustersize}
{\bf (i)} {\rm \cite[Theorem 5.4]{GG1999}.} For Bernoulli-$p$ bond percolation $\omega_p$ on ${\mathbb{Z}}^d$ with $p< p_{c}$, there is a constant $\psi(p,d)>0$ such that for any $n,$
\[
{\mathbb{P}}_p\left(\mathbf{0}\longleftrightarrow \partial R_n\right)\leq e^{-n\psi(p,d)}.
\]
{\bf (ii)} {\rm \cite[Theorem 11.89]{GG1999}, \cite[Corollary 4.7]{HD2018}.} For critical Bernoulli bond percolation $\omega_{1/2}$ on ${\mathbb{Z}}^2$, there is a constant $\alpha>0$ such that
\[
\frac{1}{2n}\leq {\mathbb{P}}_{1/2}\left(\mathbf{0}\longleftrightarrow \partial R_n\right)\leq \frac{1}{n^{\alpha}},\ \forall\,n\geq 1.
\]
{\bf (iii)} {\rm \cite[Theorem 11.5]{HH2016}.} On ${\mathbb{Z}}^d$ with $d\geq 11,$ for critical Bernoulli bond percolation $\omega_{p_c},$
\[
{\mathbb{P}}_{p_c}\left(\mathbf{0}\longleftrightarrow \partial R_n\right)\asymp n^{-2}.
\]
{\mathrm{e}}nd{lem}
\vskip 2mm
The next lemma is a slight improvement of estimation in \cite[Appendix~\uppercase{\mathrm{e}}xpandafter{\romannumeral1}, Theorem~5]{FDTT2007}. Let $C_m$ be a box with size $m\times \kappa\log m$ whose boundaries are all edges in ${\mathbb{Z}}^2$,
where $\kappa\log m$ is understood as its integer part $\lfloor \kappa\log m\rfloor.$ Now consider Bernoulli bond percolation on ${\mathbb{Z}}^2$. A horizontal open crossing of $C_m$ is a simple path within $C_{m}$ connecting its left boundary with its right boundary.
\begin{lem}\label{supdense}
Consider Bernoulli bond percolation on ${\mathbb{Z}}^2.$ For any $p>1/2$ and $\kappa>\frac{2}{\psi\left(3/4-p/2\right)}$, there exists a $\delta(\kappa,p)> 0$ such that almost surely, the number of edge-disjoint horizontal open crossings of $C_m$ is no less than $\delta(\kappa,p)\log m$ when $m$ is large enough. Here $\psi(\cdot)$ is given in Lemma {\mathrm{Re}}f{clustersize}\,(i).
{\mathrm{e}}nd{lem}
\noindent {\it{Proof.}\hskip 2pt} Let ${\rm LR}(B_{m,n})$ be the event that there exists an open crossing from left boundary to right boundary of a box $B_{m,n}$ with size $m\times n$ in Bernoulli-$p$ bond percolation $\omega_p$ on ${\mathbb{Z}}^2.$ Define similarly the vertical crossing event ${\rm UD}(B_{m,n})$. Note that $C_m= B_{m,\kappa\log m}$. Write $B_n= B_{n,\, n}$. Recall a natural coupling of two bond percolations on ${\mathbb{Z}}^2$ and its dual lattice $({\mathbb{Z}}^2)^{*}$: For any edge $e$ of ${\mathbb{Z}}^2$, write $e_*$ for its dual edge. Define the dual percolation $\omega_p^*$ on $({\mathbb{Z}}^2)^{*}$ by
$$\omega_p^*(e_*)=1-\omega_p(e),\ \forall\,e\in E({\mathbb{Z}}^2).$$
Clearly $\omega_p^*$ is a Bernoulli-$(1-p)$ bond percolation on $({\mathbb{Z}}^2)^{*}$. Then the event that there is an open horizontal crossing in box $B_{m,n}$ in ${\mathbb{Z}}^2$ is equivalent to the event that there is no open vertical crossing in $B^{*}_{m,n}$ in $({\mathbb{Z}}^2)^*$.
{\it In fact}, define the vertices in $B_{m,n}$ which is within or can be connected to left boundary by an open path to be `black vertices' and the other vertices to be `white vertices'. When ${\rm LR}(B_{m,n})$ does not occur, in $B^*_{m,n}$, we can find a unique interface which is a path separating `black vertices' from `white vertices' such that each type of vertices distribute in the same side of the interface, and the interface is an open vertical crossing in $B^*_{m,n}$, namely ${\rm UD}(B^{*}_{m,n})$ occurs. And clearly when ${\rm LR}(B_{m, n})$ occurs, ${\rm UD}(B^{*}_{m,n})$ in $({\mathbb{Z}}^2)^*$ does not occur.
\begin{figure}[!htp]
\centering
\includegraphics[width=6.5cm, height= 6.5cm]{dualgraph.pdf}
\caption{$B_{n}$ is in the real line and filled dot; and $B_{n}^{*}$ is in the dashed line and hollow dot.}
\label{graphduallattice}
{\mathrm{e}}nd{figure}
Suppose $1/2< p_1< p$. By Lemma {\mathrm{Re}}f{clustersize},
\begin{eqnarray*}
{\mathbb{P}}_{p_1}\left({\rm LR}(B_{m,\kappa\log m})\right)& = &1- {\mathbb{P}}_{1-p_1}\left({\rm UD}(B^*_{m,\kappa\log m})\right)\\
&\geq & 1-(m+1){\mathbb{P}}_{1-p_1}\left(\mathbf{0}\longleftrightarrow \partial R_{\lfloor\kappa\log m\rfloor}\right)\\
&\geq & 1-(m+1) e^{-\psi(1-p_1)\lfloor\kappa\log m\rfloor }.
{\mathrm{e}}nd{eqnarray*}
On ${\mathbb{Z}}^2$, for any $\omega\in\{0,1\}^{E({\mathbb{Z}}^2)},$ define
\[
E_r(\omega):=\left\{\omega'\in \{0,1\}^{E({\mathbb{Z}}^2)}:\ \mbox{there are at most}\ r\ \mbox{edges}\ e\ \mbox{such that}\ \omega'(e)\neq \omega(e)\right\},
\]
and for any measurable set $A$, let
\[
I_r(A):= \left\{\omega\in \{0,1\}^{E({\mathbb{Z}}^2)}:\ E_r(\omega)\subseteq A\right\}.
\]
Then by the max-flow min-cut theorem and the Menger's theorem, $I_{r}\left({\rm LR}(B_{m,\kappa\log m})\right)$ is the event that there are at least $r+1$ edge-disjoint open horizontal crossings from left boundary to right boundary of box $B_{m,\kappa\log m}$.
By \cite[Theorem~2.45]{GG1999}, for any $r\in{\mathbb{N}},$
\[
1- {\mathbb{P}}_p\left(I_r\left({\rm LR}(B_{m,\kappa\log m})\right)\right)\leq \left(\frac{p}{p-p_1}\right)^r \left(1- {\mathbb{P}}_{p_1}\left({\rm LR}(B_{m,\kappa\log m})\right)\right).
\]
Let $p_1= p/2+ 1/4$. Notice $\kappa >\frac{2}{\psi\left(3/4- p/2\right)}$. We can choose $\delta=\delta(\kappa,p)>0$ small enough such that
\begin{eqnarray*}
\delta\log \frac{p}{p-p_1}+1-\kappa\psi(1- p_1)< -1.
{\mathrm{e}}nd{eqnarray*}
Therefore, as $m\rightarrow\infty,$
\begin{eqnarray*}
&&{\mathbb{P}}_{p}(\mbox{Number of edge-disjoint open horizontal crossings in}\ B_{m,\kappa \log m}\ \mbox{is at most}\ \delta\log m)\\
&&\leq\left(\frac{p}{p-p_1}\right)^{\lfloor\delta\log m\rfloor} (m+1) e^{-\psi(1-p_1)\lfloor\kappa\log m\rfloor }\\
&&\sim m^{\delta\log\frac{p}{p-p_1}+1-\kappa\psi(1-p_1)}.
{\mathrm{e}}nd{eqnarray*}
By the Borel-Cantelli lemma, we obtain the lemma immediately.
\rule{4pt}{7pt}
\vskip 2mm
\vskip 2mm
\noindent\textbf{Proof of Theorem {\mathrm{Re}}f{generald}.} {\bf (a)} Verify $p_c^{*}= p_c$ for $0<\lambda_1<1<\lambda_2$ and $d\geq 2.$ We only prove $p_c^{*}= p_c$ for $d=2$ due to similarity for the case $d\geq 3.$
{\bf (a.1)} When $p>p_c=1/2$, almost surely, there is an infinite open cluster $\mathcal{C}$ on which all edges $e$ have conductances $\mathbf{C}_{\lambda_1}(e)$. Fix such a percolation configuration and choose a simple infinite path $\gamma=x_0x_1x_2\cdots$ in $\mathcal{C}.$
Define a unit flow $\theta$ on $\gamma$ such that
$$\theta(x_ix_{i+1})=1,\ \theta(x_{i+1}x_i)=-1,\ i\in\mathbb{Z}_{+}.$$
Then the energy $\mathscr{E}(\theta)$ of $\theta$ on network $(\gamma,\,\mathbf{C}_{\lambda_1})$ satisfies that
\begin{eqnarray*}
\mathscr{E}(\theta)=\sum\limits_{i\in{\mathbb{Z}}_{+}}\vert\theta(x_ix_{i+1})\vert^2\frac{1}{\mathbf{C}_{\lambda_1}(\{x_i,x_{i+1}\})}=\sum\limits_{i\in{\mathbb{Z}}_+}\lambda_1^{\vert x_i\vert\wedge\vert x_{i+1}\vert}
<\sum\limits_{e\in E({\mathbb{Z}}^2)}\lambda_1^{\vert e\vert}<\infty.
{\mathrm{e}}nd{eqnarray*}
By Lemma {\mathrm{Re}}f{lem-energy}, $(\gamma,\,\mathbf{C}_{\lambda_1})$ is transient. Then by the Rayleigh's monotonicity principle (Lemma {\mathrm{Re}}f{lem-ray}), both $(\mathcal{C},\mathbf{C}_{\lambda_1})$ and $({\mathbb{Z}}^2,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,p)$ are transient.
{\bf (a.2)} When $p< 1/2$, let $A_{n}$ be the event that $\partial R_{n^2}$ and $\partial R_{(n+1)^2}$ are connected by an open path for any $n\in {\mathbb{N}}$. Then with probability $1$, $\{A_n\}_n$ occurs for only finite times. {\it In fact}, by Lemma {\mathrm{Re}}f{clustersize}, there exists $\psi(p)>0$ such that for a positive constant $C,$
\begin{eqnarray*}
{\mathbb{P}}_p\left(A_n\right)&\leq & \sum\limits_{x\in \partial R_{n^2}} {\mathbb{P}}_p\left(x\longleftrightarrow \partial R_{(n+1)^2}\right)\\
&\leq & \sum\limits_{x\in \partial R_{n^2}} {\mathbb{P}}_p\left(\mathbf{0} \longleftrightarrow \partial R_{(n+1)^2- n^2}\right)\leq C n^2 e^{-\left(2n+1\right)\psi(p)},
{\mathrm{e}}nd{eqnarray*}
which verifies the claim by the Borel-Cantelli lemma. Thus, almost surely, there exists a random natural number $N$ such that for any $n\geq N$, there is a closed cutset ${\mathbb{P}}i_n$ in $R_{(n+1)^2}\setminus R_{n^2-1}$ separating $\bf{0}$ and $\infty$; and further for some positive constant $C',$
\begin{eqnarray*}
\mathscr{R}(\bf{0}\leftrightarrow \infty)&\geq &\sum\limits_{n=N}^{\infty}\left(\sum\limits_{e\in {\mathbb{P}}i_{n}} \mathbf{C}_{\lambda_2}(e)\right)^{-1}\geq C'\sum\limits_{n= N}^{\infty} \left( (n+1)^4 \lambda_2^{-n^2}\right)^{-1}\\
&=&C'\sum\limits_{n=N}^{\infty} \frac{\lambda_2^{n^2}}{(n+1)^4}= \infty.
{\mathrm{e}}nd{eqnarray*}
By the Nash-Williams recurrence criterion (Lemma {\mathrm{Re}}f{lem-Nash}), $({\mathbb{Z}}^2,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,p)$ is recurrent almost surely.
\vskip 2mm
{\bf (b)} Prove $p_c^*=p_c$ for $d\geq 3$ and $\lambda_1=1<\lambda_2.$
The recurrence for $p< p_c$ can be proved similarly to {\bf (a.2)}. For the supercritical case $p>p_c$, note that simple random walk on the infinite open cluster is transient almost surely (\cite[Theorem 1]{GHY1993}). Thus the transience for $p> p_c$ holds directly from this conclusion by the Rayleigh's monotonicity principle.
\vskip 2mm
{\bf (c)} Show that critical $\left({\mathbb{Z}}^d,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,p_c\right)$ with $0<\lambda_1<1<\lambda_2$ and $d=2$ or $0<\lambda_1\leq 1<\lambda_2$ and $d\geq 11$ is recurrent almost surely.
{\bf (c.1)} Assume $d=2$. Then $p_c=1/2$, by Lemma {\mathrm{Re}}f{clustersize}, for a positive constant $\alpha,$
\[
\frac{1}{2n}\leq {\mathbb{P}}_{1/2}(\mathbf{0}\longleftrightarrow \partial R_n)\leq \frac{1}{n^{\alpha}}.
\]
For any $n\in {\mathbb{N}}$, we take $A_n$ to be the event that there is no open path connecting $\partial R_{K^{l^n}}$ with $\partial R_{K^{l^{n+1}}}$ with $1<K\in{\mathbb{N}}$ and $1/\alpha< l\in{\mathbb{N}}$. Then
for some positive constant $C_1$,
\begin{eqnarray*}
{\mathbb{P}}_{1/2}\left(A_n\right)&\leq & \sum\limits_{x\in \partial R_{K^{l^n}}} {\mathbb{P}}_{1/2}\left(x\longleftrightarrow \partial R_{K^{l^{n+1}}}\right)\leq \sum\limits_{x\in \partial R_{K^{l^n}}} {\mathbb{P}}_{1/2}\left(\mathbf{0} \longleftrightarrow \partial R_{K^{l^{n+1}}- K^{l^{n}}}\right)\\
&\leq & C_1 K^{l^n} \left(K^{l^{n+1}}- K^{l^{n}}\right)^{-\alpha}\leq C_1 \left(\frac{K}{K-1}\right)^{\alpha} K^{l^n(1-\alpha l)}.
{\mathrm{e}}nd{eqnarray*}
Then similarly to {\bf (a.2)}, almost surely, there is a sequence $\{{\mathbb{P}}i_n\}_{n=N}^{\infty}$ of disjoint closed cutsets with each ${\mathbb{P}}i_n$ being in $R\left(K^{l^{n+1}}\right)\setminus R\left(K^{l^{n}}-1\right)$ and separating
$\bf{0}$ from $\infty;$ and for a positive constant $C_2,$
\begin{eqnarray*}
\mathscr{R}(\bf{0}\leftrightarrow\infty)&\geq &\sum\limits_{n=N}^{\infty}\left(\sum\limits_{e\in {\mathbb{P}}i_{n}}\mathbf{C}_{\lambda_2}(e)\right)^{-1}\geq C_2 \sum\limits_{n=N}^{\infty} \left(K^{2 l^{n+1}} \lambda_2^{-K^{l^{n}}}\right)^{-1}\\
&=&C_2 \sum\limits_{n=N}^{\infty} \frac{\lambda_2^{K^{l^{n}}}}{K^{2 l^{n+1}}} = \infty,
{\mathrm{e}}nd{eqnarray*}
which shows $({\mathbb{Z}}^2,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, 1/2)$ is recurrent almost surely.
{\bf (c.2)} Similarly to {\bf (c.1)}, one can prove that critical $\left({\mathbb{Z}}^d,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,p_c\right)$ with $0<\lambda_1\leq 1<\lambda_2$ and $d\geq 11$ is recurrent almost surely.
\vskip 2mm
{\bf (d)} Prove $p_c^*=p_c=1/2$ for $d=2$ and $0< \lambda_1< 1=\lambda_2.$
The transience when $p>1/2$ can be proved similarly as in {\bf (a.1)}.
Assume $p\in [0,\, 1/2)$. we claim $({\mathbb{Z}}^2,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_1,\, p)$ is recurrent almost surely. Simple random walk (SRW) on supercritical infinite open cluster in ${\mathbb{Z}}^2$ is recurrent by the Rayleigh's monotonicity principle.
Compared with SRW on supercritical infinite open cluster, to prove the recurrence of $({\mathbb{Z}}^2,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_1,\, p)$ here is more interesting.
We prove this in two steps.
\vskip 2mm
\begin{figure}[!htp]
\centering
\includegraphics[width=7.5cm, height= 7cm]{parellelcircle.pdf}
\caption{Open circuits in annulus of box $S_{r_n}$}
\label{graphopencircuits}
{\mathrm{e}}nd{figure}
\textbf{Step (d.1). Preparation.} Let $S_{n}= [-n,\, n]\times [-n,\, n]$ be a square in ${\mathbb{Z}}^2$ for any $n\in (0,\,\infty).$ Fix $p_0\in (1/2,\,1)$ and $\kappa>2/\psi(3/4- p_0/2)$.
We aim to prove that there exists a constant $\delta=\delta(\kappa,p_0)>0$ such that when $n$ is large enough, number of edge-disjoint open circuits in the annuli $S_{r_{n+1}}\backslash S_{r_n}$ (c.f. Figure {\mathrm{Re}}f{graphopencircuits}) with $r_n= 2\kappa\log{n!}$ is no less than $\delta \log(4 \kappa\log{(n+1)!})$ almost surely.
{\it In fact}, $S_{r_{n+1}}\backslash S_{r_{n}}$ is the union of
\begin{eqnarray*}
&&A_n^u:=[-2\kappa \log{(n+1)!},\, 2\kappa \log{(n+1)!}]\times [2\kappa \log{n!},\, 2\kappa \log{(n+1)!}],\\
&&A_n^d:=[-2\kappa \log{(n+1)!},\, 2\kappa \log{(n+1)!}]\times [-2\kappa \log{(n+1)!},\, -2\kappa \log{n!}],\\
&&A_n^l:=[-2\kappa \log{(n+1)!},\, -2\kappa \log{n!}]\times [-2\kappa \log{(n+1)!},\, 2\kappa \log{(n+1)!}],\\
&& A_n^r:=[2\kappa \log{n!},\, 2\kappa \log{(n+1)!}]\times [-2\kappa \log{(n+1)!},\, 2\kappa \log{(n+1)!}].
{\mathrm{e}}nd{eqnarray*}
Define
\begin{eqnarray*}
&&C_n^{u}=\{\mbox{There are at least}\ \delta \log(4 \kappa\log{(n+1)!})\ \mbox{edge-disjoint open horizontal crossings of}\ A_n^{u}\},\\
&&C_n^{d}=\{\mbox{There are at least}\ \delta \log(4 \kappa\log{(n+1)!})\ \mbox{edge-disjoint open horizontal crossings of}\ A_n^{d}\};\\
&&C_n^{l}=\{\mbox{There are at least}\ \delta \log(4 \kappa\log{(n+1)!})\ \mbox{edge-disjoint open vertical crossings of}\ A_n^{l}\},\\
&&C_n^{r}=\{\mbox{There are at least}\ \delta \log(4 \kappa\log{(n+1)!})\ \mbox{edge-disjoint open vertical crossings of}\ A_n^{r}\}.
{\mathrm{e}}nd{eqnarray*}
Note that as $n\rightarrow \infty$,
\[
2\kappa\log (n+1)!- 2\kappa\log n!> \kappa \log(4\kappa\log (n+1)!).
\]
Then similarly to Lemma {\mathrm{Re}}f{supdense}, we can prove that for some constant $\delta=\delta(\kappa,p_0)>0,$ almost surely, $C_n^u$, $C_n^d$, $C_n^l$
and $C_n^r$ occur when $n$ is large enough.
Thus, by taking intersections of vertical crossings in $A_n^l$ and $A_n^r$ with horizontal crossings in $A_n^u$ and $A_n^d$, we have that almost surely,
number of edge-disjoint open circuits in the annuli $S_{r_{n+1}}\backslash S_{r_n}$ is no less than $\delta \log(4 \kappa\log{(n+1)!})$ for large enough $n$.
\vskip 2mm
\textbf{Step (d.2). Completing proof.} On $({\mathbb{Z}}^2)^*$, each edge is open independently with probability $1-p$ by the coupling introduced in proof of Lemma {\mathrm{Re}}f{supdense}.
Recall an edge $e_*$ in $({\mathbb{Z}}^2)^*$ is open iff its dual edge $e$ in ${\mathbb{Z}}^2$ is closed. Note that the set of dual edges of an open circuit in $({\mathbb{Z}}^2)^*$ is a closed cutset separating $\bf{0}$ from $\infty$ in ${\mathbb{Z}}^2$, and
edges $e$ in this cutset are of conductances $\mathbf{C}_{2}(e)=1$.
Choose $n_0\in{\mathbb{N}}$ such that
$$\delta\log(4\kappa\log(n_0+1)!)\geq 2.$$
For any $n_0\leq n\in {\mathbb{N}},$ on $({\mathbb{Z}}^2)^*$, let $c_n^1,\, c_n^2,\, \cdots,\, c_n^{m(n)}$ be edge-disjoint open concentric circuits in $S_{r_{n+1}}\backslash S_{r_n}$ with $c_n^i$ being in the interior of $c_n^{i+1},\, 1\leq i\leq m(n)-1$. And $\pi_n^1,\, \pi_n^2,\, \cdots,\, \pi_n^{m(n)}$ denote dual edge sets of $c_n^1,\ c_n^2,\, \cdots,\, c_n^{m(n)}$ which is a sequence of disjoint cutsets on ${\mathbb{Z}}^2$. By the Nash-Williams inequality (Lemma {\mathrm{Re}}f{lem-Nash}) on ${\mathbb{Z}}^2$ and Step {\bf (d.1)}, almost surely,
\begin{eqnarray*}
\mathscr{R}(\bf{0}\leftrightarrow\infty)&\geq &\sum\limits_{n=n_0}^{\infty} \sum\limits_{i=1}^{m(n)-1} |\pi_n^{i}|^{-1}=\sum\limits_{n=n_0}^{\infty} \sum\limits_{i=1}^{m(n)-1} |c_n^{i}|^{-1}\\
&\geq &\sum\limits_{n=n_0}^{\infty} \sum\limits_{i=1}^{\lfloor\delta \log(4\kappa\log{(n+1)!})\rfloor- 1} |c_n^{i}|^{-1}.
{\mathrm{e}}nd{eqnarray*}
Here $|c_n^i|$ denotes the number of edges in $c_n^i$. Note that
\[
\frac{(2\kappa\log (n+1)!)^2- (2\kappa\log n!)^2}{\{\delta \log(4\kappa\log{(n+1)!})\}^2}\asymp n,
\]
and for some positive constant $C_3,$
\[|c_n^1|+ |c_n^2|+ |c_n^3|+ \cdots+ |c_n^{m(n)}| \leq C_3\left\{(2\kappa\log (n+1)!)^2- (2\kappa\log n!)^2\right\},
\]
and by the Jensen's inequality, with $a_n=\lfloor\delta \log(4\kappa\log{(n+1)!})\rfloor- 1\leq m(n)-1,$
\begin{eqnarray*}
\sum\limits_{i=1}^{a_n} |c_n^{i}|^{-1}\geq a_n\left\{\frac{1}{a_n}\sum\limits_{i=1}^{a_n} |c_n^{i}|\right\}^{-1}\geq a_n^2\left\{\sum\limits_{i=1}^{m(n)} |c_n^{i}|\right\}^{-1}.
{\mathrm{e}}nd{eqnarray*}
Thus, for some positive constant $C_4,$ almost surely,
\begin{eqnarray*}
\mathscr{R}(\bf{0}\leftrightarrow \infty)&\geq & \sum\limits_{n=n_0}^{\infty}\frac{\{\lfloor\delta \log(4\kappa\log{(n+1)!})\rfloor-1\}^2}{C_3\left\{(2\kappa\log (n+1)!)^2- (2\kappa\log n!)^2\right\}}\\
&\geq & C_4\sum\limits_{n=n_0}^{\infty} \frac{1}{n}=\infty,
{\mathrm{e}}nd{eqnarray*}
which implies that $({\mathbb{Z}}^2,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_1,\, p)$ is recurrent almost surely.
\rule{4pt}{7pt}\\
\noindent\textbf{Proof of Theorem {\mathrm{Re}}f{thm-current-unique}.} {\bf Step 1.} Some preliminaries. Any recurrent infinite network $(G,\,\mathbf{c})$ has unique currents. So by the Rayleigh's monotonicity principle, Theorem {\mathrm{Re}}f{thm-current-unique} is trivial for $1\leq \lambda_1<\lambda_2.$ Recall the following two criteria on current uniqueness from \cite[Chapter 9]{LP2017}.
Consider infinite network $(G,\, \mathbf{c})$. Note $\mathbf{r}=1/\mathbf{c}.$ Let ${\mathrm{e}}ll^2\left(\overrightarrow{E},\,\mathbf{r}\right)$ be the space of anti-symmetric functions $\theta$ satisfying $\sum\limits_{e\in \overrightarrow{E}} \theta^2(e)\mathbf{r}(e)<\infty$. For any function $f$ on $V$, define its gradient as follows:
\[
\nabla f(e)=\mathbf{c}(\{x,y\}) {\rm d}f (e)=\mathbf{c}(\{x,y\}) \left(f(x)- f(y)\right),\ e=xy\in \overrightarrow{E}.
\]
Call ${\bf D}=\left\{f:\ \nabla f\in {\mathrm{e}}ll^2\left(\overrightarrow{E},\,\mathbf{r}\right)\right\}$ the Dirichlet space.
Write {\bf HD} for the harmonic Dirichlet space consisting of harmonic functions in {\bf D}. Then $(G,\,\mathbf{c})$ is current unique iff
${\bf HD}=\mathbb{R}.$
For any finite subgraph $A=(V(A),\,E(A))$ of $G$, define
\[
{\rm RD}(A)= \sup\left\{\mathscr{R}\left(x\leftrightarrow y;\ A\right): \ x,\, y\in V(A)\right\},
\]
where $\mathscr{R}(x\leftrightarrow y;\ A)$ is the effective resistance between $x$ and $y$ in finite network $(A,\,\mathbf{c})$.
Then $(G,\,\mathbf{c})$ has unique currents if there is a sequence $\{W_n\}_{n=1}^{\infty}$ of pairwise edge-disjoint finite subnetworks of $(G,\,\mathbf{c})$ such that each (equivalently, some) vertex is separated from $\infty$ by all but finitely many $W_n$ and
\begin{equation}\label{currentu}
\sum\limits_{n=1}^{\infty}\frac{1}{{\rm RD}(W_n)}= \infty.
{\mathrm{e}}nd{equation}
\vskip 2mm
\noindent{\bf Step 2.} Assume $0< \lambda_1<\lambda_2\leq 1$. For any $n\geq 1,$ let $W_n$ be the set of edges $e$ of ${\mathbb{Z}}^2$ with $\vert e\vert=n.$ View naturally each $W_n$ as a subnetwork $W_n(p)$ of any given $({\mathbb{Z}}^2,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,p).$
Let $\mathbf{c}_p$ (resp.\,$\mathbf{r}_p$) be the conductance (resp. resistance) function of $({\mathbb{Z}}^2,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,p).$ Denote by $\mathcal{P}_n(x,y)$ any edge-disjoint path in $W_n$ starting from $x$ and ending at $y.$ Clearly the length of $\mathcal{P}_n(x,y)$ is at most $4(n+1).$ Then
\begin{eqnarray*}
{\rm RD}(W_n(p))&=&\sup\left\{\mathscr{R}(x\leftrightarrow y;\ W_n(p)):\ x,y\in V(W_n)\right\}\\
&\leq &\sup\left\{\sum\limits_{e\in \mathcal{P}_n(x,y)}\mathbf{r}_p(e):\ x,y\in V(W_n)\right\}
\leq 4(n+1)(\lambda_1 \vee \lambda_2)^{n}\leq 4(n+1),
{\mathrm{e}}nd{eqnarray*}
and thus
\begin{equation*}
\sum\limits_{n=1}^{\infty}\frac{1}{{\rm RD}(W_n(p))}= \infty,\ p\in [0,\,1].
{\mathrm{e}}nd{equation*}
Therefore, almost surely, for all $p\in [0,\,1],$ $({\mathbb{Z}}^2,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,p)$ is current unique.
\vskip 2mm
\noindent{\bf Step 3.} Suppose $0<\lambda_1<1<\lambda_2.$ Note that almost surely,
for any $p\in [0,\,1/2],$ $({\mathbb{Z}}^2,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,p)$ is recurrent.
It suffices to prove the theorem for all $p>1/2$ by proving that
$$\mbox{almost surely},\ \mathbf{HD}(p)= {\mathbb{R}},\ \forall\, p\in (1/2,1].$$
Here $\mathbf{HD}(p)$ is the harmonic Dirichlet space of $({\mathbb{Z}}^2,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,p).$
{\it In fact}, for any fixed $p>1/2,$ almost surely, there exist infinitely many open circuits $\{C_n\}_{n=1}^{\infty}$ in $\omega_p$ and thus in all $\omega_q$ with $q\geq p$ such that each $[-n,\, n]^2$ is within the interior of $C_n,$
and $C_n$ is within the interior of $C_{n+1}.$ Fix such a percolation environment $\omega.$ Clearly on $C_n$,
$$\mathbf{c}_q(e)=\lambda_1^{-|e|},\ e\in C_n,\ q\geq p. $$
For any $f\in \mathbf{HD}(q)$ with $q\geq p$, suppose $x_1^n, x_2^n$ are respectively the maximal and minimal points of $f$ in the area \[B(C_n)=\{x\in {\mathbb{Z}}^2:\ x\ \mbox{is on or within}\ C_n\}\]
whose boundary is $C_n$. Namely,
\begin{eqnarray*}
f(x_1^n)= \max\{f(y):\ y\in B(C_n)\},\ f(x_2^n)= \min\{f(y):\ y\in B(C_n)\}.
{\mathrm{e}}nd{eqnarray*}
Note $x_1^n,x_2^n\in C_n$ since $f$ is harmonic on $B(C_n)$. Let $\mathcal{P}_n(x_1^n,x_2^n)$ be the path in $C_n$ connecting $x_1^n$ and $x_2^n.$ Then as $n\rightarrow\infty,$
\begin{eqnarray*}
\left|f(x_1^n)- f(x_2^n)\right|&\leq &\sum\limits_{\{u,v\}\in \mathcal{P}_n(x_1^n,x_2^n)} |f(u)- f(v)|\\
&\leq & \sum\limits_{\{u,v\}\in \mathcal{P}_n(x_1^n,x_2^n)}\mathbf{r}_q(\{u,v\})
\sum\limits_{\{u,v\}\in \mathcal{P}_n(x_1^n,x_2^n)} \mathbf{c}_q(\{u,v\}) |f(u)- f(v)|^2\\
&\leq & \sum\limits_{\{u,v\}\in \mathcal{P}_n(x_1^n,x_2^n)}\lambda_1^{\vert u\vert\wedge\vert v\vert}
\left(\sum\limits_{\vert z-w\vert=1}\mathbf{c}_q(\{z,w\})\vert f(z)-f(w)\vert^2\right)\\
&\leq & \sum\limits_{\stackrel{\vert u-v\vert=1}{\vert u\vert,\vert v\vert\geq n}}\lambda_1^{\vert u\vert\wedge\vert v\vert}
\left(\sum\limits_{\vert z-w\vert=1}\mathbf{c}_q(\{z,w\})\vert f(z)-f(w)\vert^2\right)\\
&\rightarrow & 0.
{\mathrm{e}}nd{eqnarray*}
Namely as $n\rightarrow\infty,$
$$\max\limits_{x,y\in B(C_n)}\vert f(x)-f(y)\vert\rightarrow 0.$$
Therefore, $f$ is constant, and further ${\bf HD}(q)={\mathbb{R}}.$ So far we have proved that for any fixed $p>1/2,$ almost surely,
for all $q\geq p,\ {\bf HD}(q)={\mathbb{R}};$
which completes proving the theorem.
\rule{4pt}{7pt}
\vskip 2mm
\subsection{Proof of Theorem {\mathrm{Re}}f{regulartree1}}\label{rrt}
\noindent
\begin{lem}[Heathcote, Seneta and Vere-Jones (1967) \cite{HSV1967}]\label{heightgw}
For a Galton-Watson tree $T$, let $T_n$ be the number of vertices in $n$-th generation from the origin $o$ and $L$ the random number of offsprings of $o$. Assume $m:= {\mathbb{E}}[L]\in (0,\,\infty)$. Then $\{{\mathbb{P}}(T_n> 0)/ m^{n}\}$ is a decreasing sequence. When $m< 1$, the following are equivalent:
\begin{eqnarray*}
{\bf (i)}\ \lim\limits_{n\rightarrow \infty} {\mathbb{P}}[T_n> 0]/ m^{n} >0;\ {\bf (ii)}\ \sup {\mathbb{E}}[\left. T_n\ \right\vert\ T_n> 0]< \infty;\ {\bf (iii)}\ {\mathbb{E}}[L \log^{+} L]< \infty.
{\mathrm{e}}nd{eqnarray*}
{\mathrm{e}}nd{lem}
Before proving Theorem {\mathrm{Re}}f{regulartree1}, we calculate the growth rate of infinite open cluster of supercritical percolation on $\mathbb{T}^d$ with $d\geq 3$ and root $o.$
Let $\mathcal{T}_{\omega_p}$ be the open cluster of root $o$ in Bernoulli-$p$ bond percolation $\omega_p$ on $\mathbb{T}^d$. Write $B(n,p)$ for the binomial distribution with trial number $n\in {\mathbb{N}}$ and success probability $p\in [0,\,1].$
Then $\mathcal{T}_{\omega_p}$ is a Galton-Watson tree with a slight modification, namely offspring number of root $o$ obeys $B(d,p)$ and the other vertices in $\mathcal{T}_{\omega_p}$ have an i.i.d.\,$B(d-1,p)$ family of offsprings. So many limit properties for Galton-Watson trees are still applicable for $\mathcal{T}_{\omega_p}$.
Let $|T_n|$ be the number of edges at distance $n$ from $o$ in $\mathcal{T}_{\omega_p}$. Similarly to \cite[Proposition~5.5]{LP2017}, one can prove that
$(|T_n|/[d(d-1)^{n-1}p^n])_{n\geq 1}$ is a non-negative martingale, and has a finite limit $\mathscr{W}$ a.s. by the martingale convergence theorem. Similarly to the Kesten-Stigum Theorem (\cite[Section 12.2]{LP2017}), one
can show that when $(d-1)p>1$ namely $p>\frac{1}{d-1}$,
\[
{\mathbb{P}}(\mathscr{W}=0)={\mathbb{P}}[\vert \mathcal{T}_{\omega_p}\vert <\infty]=:q.
\]
Thus for $p>\frac{1}{d-1}$, on the event $\left\{\vert \mathcal{T}_{\omega_p}\vert =\infty\right\}$, almost surely,
\begin{eqnarray*}
\lim\limits_{n\rightarrow \infty} \frac{|T_n|}{d(d-1)^{n-1}p^n} =\mathscr{W}> 0\ \mbox{and}\
{\rm gr}(\mathcal{T}_{\omega_p})=\lim\limits_{n\rightarrow \infty} {\sqrt[n]{|T_n|}} = (d-1)p.
{\mathrm{e}}nd{eqnarray*}
Note the percolation process $(\omega_p)_{p\in [0,\,1]}$ on $\mathbb{T}^d$ is constructed by the grand coupling. By the indistinguishability of infinite open clusters of Bernoulli percolation on $\mathbb{T}^d$ (see Subsection {\mathrm{Re}}f{sec-discussion-growth}), almost surely, as $\frac{1}{d-1}<p\uparrow 1$, each infinite open cluster of $\omega_p$ has growth rate $(d-1)p\uparrow d-1={\rm gr}(\mathbb{T}^d)$.\\
\noindent{\bf Proof of Theorem {\mathrm{Re}}f{regulartree1}.} {\bf (i)} When $0<\lambda_1\leq 1<d-1\leq\lambda_2$ and $d\geq 3,$ $p_c^*=p_c=\frac{1}{d-1}.$
{\bf (i.1)} For any $p>\frac{1}{d-1}$, by {\mathrm{e}}qref{eq-p_c-br-tree} and Lemma {\mathrm{Re}}f{gpercolation}, condition on $\left\{\vert \mathcal{T}_{\omega_p}\vert =\infty\right\},$ almost surely,
\[({\rm br}(\mathcal{T}_{\omega_p}))^{-1}= p_{c}(\mathcal{T}_{\omega_p})=\frac{1}{(d-1)p},\ \mbox{and thus}\ {\rm br}(T_{\omega_p})= (d-1)p={\rm gr}(\mathcal{T}_{\omega_p}).\]
By \cite[Theorem~3.5]{LP2017}, when $p>\frac{\lambda_1\vee 1}{d-1}=\frac{1}{d-1}$, condition on $\left\{\vert \mathcal{T}_{\omega_p}\vert =\infty\right\},$ almost surely, ${\mathbb{R}}W_{\lambda_1}$ constrained on $\mathcal{T}_{\omega_p}$ is transient. So by the Rayleigh's monotonicity principle and $0$-$1$ law for recurrence/transience (see {\bf (i)} in proving Theorem {\mathrm{Re}}f{generalgraph01}), for any $p>\frac{1}{d-1},$
$(\mathbb{T}^d,\,\mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\,p)$ is transient almost surely.
{\bf (i.2)} Assume $p\leq\frac{1}{d-1}$. For any $n\in{\mathbb{N}},$ let $L_n$ be the edge set in which all edges are at distance $n$ from the root $o$ on $\mathbb{T}^d$. Note $p_c=\frac{1}{d-1}$ and critical percolation $\omega_{p_c}$ on $\mathbb{T}^d$ does not percolate almost surely (\cite[Theorem 8.21]{LP2017}).
Define a random sequence $\{X_n\}_{n=0}^{\infty}$ as follows: Let $X_0= 0$ and $L_{X_0}= \{o\}$. For any $i\in{\mathbb{N}},$ define $X_i$ to be the smallest natural number $n> X_{i-1}$ such that $L_{X_{i-1}}$ cannot connect to $L_{n}$ in $\omega_p.$ Here $L_{i}$ and $L_{j}$ $(j> i)$ are connected if there exists an open path in $\omega_p$ with its two endpoints belonging to $L_{i}$ and $L_{j}$ respectively.
Clearly, almost surely, $\{X_n\}_{n=0}^{\infty}\subseteq{\mathbb{Z}}_{+}$ is a strictly increasing sequence.
For any subtree $T\subset \mathbb{T}^d$, its outer boundary is defined by
\[
\partial T:= \left\{e\in E(\mathbb{T}^d):\ e=\{x,y\},\ x\in T,\ y\notin T,\ \vert y\vert>\vert x\vert\right\}.
\]
For any $x\in\mathbb{T}^d,$ let $\mathcal{T}(x)$ be the open subtree of $\mathbb{T}^d$ consisting of $x$ and its offsprings connected to $x$ by an open path in $\omega_p$.
Clearly, $\mathcal{T}(x)$ is a Galton-Watson tree with root $x$ and offspring distribution $B(d-1,p).$ Let
$$W(x):=\sum_{e\in \partial \mathcal{T}(x)} \mathbf{C}_{\lambda_2}(e).$$
And code the individuals of $n$-th generation in $\mathbb{T}^d$ by
\[x^{n}_1,\, x^{n}_2,\, \cdots,\, x^{n}_{d(d-1)^{n-1}}.\]
Then we claim that when $p<\frac{1}{d-1}$,
\begin{eqnarray}\label{eq-claim}
{\mathbb{P}}\left[\sum\limits_{n=1}^{\infty} \left(W\left(x^{X_{n}}_1\right)+ W\left(x^{X_{n}}_2\right)+\cdots +W\left(x^{X_{n}}_{d(d-1)^{X_{n}-1}}\right)\right)^{-1}=\infty\right]=1.
{\mathrm{e}}nd{eqnarray}
\begin{figure}[!htp]
\centering
\includegraphics[width=13cm, height= 9cm]{regulartree.pdf}
\caption{Bernoulli percolation ($p= 0.4$) on binary tree: open edges are in black line; closed edges are in grey line; $\partial \mathcal{T}(o)$ are in dash line.}
\label{graphregulartree}
{\mathrm{e}}nd{figure}
To see this, for $j\in{\mathbb{N}},$ let $R_j= d(d-1)^{X_j- 1},$ and $Y_{i}^{j}= \lambda_2^{X_j} W\left(x_{i}^{X_j}\right),\ 1\leq i\leq R_j,$ and
$$S_j:=\frac{\sum\limits_{i= 1}^{R_j} \left[Y_{i}^{j}- {\mathbb{E}}\left(Y_{1}^{1}\right)\right]}{R_j}.$$
By Lemma {\mathrm{Re}}f{heightgw}, we have that when $p<\frac{1}{d-1}$,
\begin{eqnarray}\label{eq-2nd-moment}
{\mathbb{E}}\left[(Y_1^1)^2\right]= \lambda_2^2 {\mathbb{E}}\left[W^2\left(x_1^{1}\right)\right]\leq \lambda_2^2\sum\limits_{k=0}^{\infty} p_k \left(\sum\limits_{i=0}^{k} \lambda_2^{-i} d (d-1)^{i}\right)^2=:C < \infty,
{\mathrm{e}}nd{eqnarray}
where $p_k= {\mathbb{P}}\left[\mbox{The height of}\ \mathcal{T}\left(x_1^{1}\right)\ \mbox{is}\ k\right]$. Then when $p<\frac{1}{d-1}$, for any $j\in{\mathbb{N}},$
\begin{eqnarray*}
{\mathbb{E}}\left[S_j^2\right]= {\mathbb{E}}\left[{\mathbb{E}}\left[\left. S_j^2\ \right\vert\ R_j\right]\right]= {\mathbb{E}}\left[\frac{{\mathbb{E}}[(Y_{i}^{j}- {\mathbb{E}}(Y_{1}^{1}))^2]}{R_j}\right]\leq \frac{C}{d(d-1)^{j-1}},
{\mathrm{e}}nd{eqnarray*}
and by the Chebyshev inequality, for any ${\mathrm{e}}psilon>0,$
\begin{eqnarray*}
{\mathbb{P}}[|S_j|> {\mathrm{e}}psilon]\leq \frac{{\mathbb{E}}\left[S_j^2\right]}{{\mathrm{e}}psilon^2}\leq \frac{C}{{\mathrm{e}}psilon^2 d(d-1)^{j-1}};
{\mathrm{e}}nd{eqnarray*}
which leads to
\[
\lim\limits_{j\rightarrow \infty} S_j= 0 \ \ a.s.
\]
by the Borel-Cantelli lemma. Namely, when $p<\frac{1}{d-1}$,
\[
\frac{W\left(x^{X_n}_1\right)+ W\left(x^{X_n}_2\right)+\cdots + W\left(x^{X_n}_{d(d-1)^{X_n-1}}\right)}{d(d-1)^{X_n-1} \lambda_2^{- X_n}}\xrightarrow[n\rightarrow \infty]{a.s.} \lambda_2 {\mathbb{E}}\left[W\left(x^{1}_1\right)\right]
\in (0,\,\infty).
\]
Since ${\mathbb{P}}\left[\sum\limits_{n=1}^{\infty} d(d-1)^{-(X_{n}-1)} \lambda_2^{X_{n}}=\infty\right]=1$, we obtain {\mathrm{e}}qref{eq-claim}.
Therefore, when $p< \frac{1}{d-1}$, by the Nash-Williams inequality and recurrence criterion (Lemma {\mathrm{Re}}f{lem-Nash}), almost surely,
\begin{align*}
\mathscr{R}\left(o\leftrightarrow \infty\right)\geq \sum\limits_{j=1}^{\infty} \left(W\left(x^{X_{n_j}}_1\right)+ W\left(x^{X_{n_j}}_2\right)+\cdots +W\left(x^{X_{n_j}}_{d(d-1)^{X_{n_j}-1}}\right)\right)^{-1}
=\infty,
{\mathrm{e}}nd{align*}
and $(\mathbb{T}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$ is recurrent almost surely.
\vskip 2mm
{\bf (ii)} Critical $\left(\mathbb{T}^d,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,\frac{1}{d-1}\right)$ with $0<\lambda_1\leq 1<d-1<\lambda_2$ and $d\geq 3$ is recurrent almost surely.
{\it In fact}, note that {\mathrm{e}}qref{eq-2nd-moment} still holds in this case due to $\lambda_2>d-1.$ Similarly to {\bf (i.2)}, we can prove that almost surely $\left(\mathbb{T}^d,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\,\frac{1}{d-1}\right)$ is recurrent.
\vskip 2mm
{\bf (iii)} Suppose $1<\lambda_1<d-1\leq\lambda_2$ and $d\geq 3.$ Then $p_c^*=\lambda_1p_c=\frac{\lambda_1}{d-1}.$
{\bf (iii.1)} For $p>\frac{\lambda_1}{d-1},$ almost sure transience of $\left(\mathbb{T}^d,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\, p\right)$ can be proved similarly to {\bf (i.1)}.
{\bf (iii.2)} When $p< \frac{1}{d-1}$, similarly to {\bf (i.2)}, one can verify that $\left(\mathbb{T}^d,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\, p\right)$ is recurrent almost surely.
To prove that when $\frac{1}{d-1}< p< \frac{\lambda_1}{d-1}$, $\left(\mathbb{T}^d,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\, p\right)$ is recurrent almost surely. Note by the Rayleigh's monotonicity principle,
this implies that $\left(\mathbb{T}^d,\,\mathbf{C}_{\lambda_1},\,\mathbf{C}_{\lambda_2},\, q\right)$ is a.s.\,recurrent for any $q<\frac{\lambda_1}{d-1}.$
{\bf Assume firstly $\lambda_2> d-1$.} Define a sequence $\{a_n\}_{n=0}^{\infty}\subset{\mathbb{Z}}_+$ by letting
$$a_0= 0,\ a_1= d-1,\ \mbox{and}\ a_{i+1}=(d-1)^{a_i},\ i\in\mathbb{N}.$$
Then define a sequence of cutsets $\{{\mathbb{P}}i_i\}_{i=1}^{\infty}$ in $\mathbb{T}^d$ as follows. On tree $\mathbb{T}^d,$ call an edge $e'$ in the geodesic path connecting root $o$ and edge $e$ an ancestor edge of $e$.
Recall $L_n=\{e\in E(\mathbb{T}^d):\ \vert e\vert =n\},\ n\in{\mathbb{N}}.$ For any $i\in {\mathbb{N}},$ let
\begin{equation*}
\begin{aligned}
&{\mathbb{P}}i^1_i= \left\{e^*:\ e\in E(\mathbb{T}^d),\ |e|= a_{i+1}-1,\ e\ \mbox{is not connected to}\ L_{a_i}\ \mbox{in}\ \omega_p,\ e^*\ \mbox{is a closed ancestor}\right.\\
&\hskip 1.75cm \left.\mbox{of}\ e\ \mbox{with}\ \vert e^*\vert=\min\left\{\vert e'\vert:\ \vert e'\vert\geq a_i,\ e'\ \mbox{is a closed ancestor of}\ e\right\}\right\},\\
&{\mathbb{P}}i^2_i= \left\{e\in E(\mathbb{T}^d):\ e\ \mbox{is open and connected to}\ L_{a_i}\ \mbox{in}\ \omega_p,\ |e|=a_{i+1}-1\right\},\\
&{\mathbb{P}}i_i= {\mathbb{P}}i_i^1\cup {\mathbb{P}}i_i^2.
{\mathrm{e}}nd{aligned}
{\mathrm{e}}nd{equation*}
Then all ${\mathbb{P}}i_i$s are edge-disjoint cutsets separating $o$ from $\infty$ in $\mathbb{T}^d$. Clearly
\begin{eqnarray}\label{eq-conductance-sum}
C_1:=\sum\limits_{i=1}^{\infty} \sum\limits_{e\in{\mathbb{P}}i_i^1} \mathbf{C}_{\lambda_2}(e)\leq \sum\limits_{i=1}^{\infty} d(d-1)^{i} \lambda_2^{-i}<\infty.
{\mathrm{e}}nd{eqnarray}
To continue, recall the following useful facts. For the Galton-Watson tree $T$ with offspring distribution $B(d-1,p)$, write
$$m=(d-1)p\ \mbox{and}\ \sigma^2=(d-1)p(1-p).$$
Let $Z_n(T)$ be the number of individuals in the $n$th generation of $T.$ It is easy to check that
\[
{\mathbb{E}}\left[Z^2_{n+1}(T)\right]= m^2 {\mathbb{E}}\left[Z_n^2(T)\right]+ \sigma^2 m^n.
\]
Note $m>1.$ Then
$$\lim\limits_{n\rightarrow\infty}{\mathbb{E}}\left[\left(\frac{Z_n(T)}{m^n}\right)^2\right]=1+\sum\limits_{k=0}^{\infty}\frac{\sigma^2}{m^{k+2}}\in (0,\infty).$$
It is well-known that almost surely, $\left\{\frac{Z_n(T)}{m^n}\right\}$ converges to a random variable $W\in [0,\,\infty)$ with mean $1.$
Thus by the Doob's maximum inequality and the dominated convergence theorem,
\begin{eqnarray}\label{eq-uniform-2nd-moment}
{\mathbb{E}}\left[W^2\right]=\lim\limits_{n\rightarrow\infty}{\mathbb{E}}\left[\left(\frac{Z_n(T)}{m^n}\right)^2\right]=\sup\limits_{n} {\mathbb{E}}\left[\left(\frac{Z_n(T)}{m^n}\right)^2\right]<\infty.
{\mathrm{e}}nd{eqnarray}
Notice that from {\bf (i.2)}, $x^i_{k}$ is the $k$-th $(1\leq k\leq d(d-1)^{i-1})$ individual in the $i$-th generation in $\mathbb{T}^d$; and $\mathcal{T}(x^i_{k})$ denotes random open descendant subtree rooted at $x^i_{k}$,
a Galton-Watson tree with offspring distribution $B(d-1,p).$ Write $Z_m\left(x^i_{k}\right):=Z_m\left(\mathcal{T}(x^i_{k})\right)$ for the number of individuals of the $m$-th generation of random tree $\mathcal{T}(x^i_{k})$. Then for any $i\in{\mathbb{N}},$
\begin{equation*}
\begin{aligned}
\left|{\mathbb{P}}i_i^2\right|=\sum\limits_{k=1}^{d(d-1)^{a_i-1}} Z_{a_{i+1}-a_{i}}(x^{a_i}_{k}).
{\mathrm{e}}nd{aligned}
{\mathrm{e}}nd{equation*}
For any $j\in{\mathbb{N}},$ let $\mathcal{R}_j= d(d-1)^{a_j- 1}$ and
$$\mathcal{Y}_{i}^{j}=\frac{Z_{a_{j+1}-a_{j}}(x^{a_j}_{i})}{m^{a_{j+1}-a_{j}}},\, 1\leq i\leq\mathcal{R}_j.$$
By the standard theory of Galton-Watson branching processes, almost surely, for any $j\in{\mathbb{N}}$ and $1\leq i\leq\mathcal{R}_j,$
$$W_i^j=\lim\limits_{k\rightarrow\infty}\frac{Z_k(x_i^{a_j})}{m^k}\in [0,\,\infty)\ \mbox{exists};$$
and all $W_i^j$s have a common distribution as that of $W$. Let
$$\mathcal{S}_j:=\frac{\sum\limits_{i= 1}^{\mathcal{R}_j} \left(\mathcal{Y}_{i}^{j}- 1\right)}{\mathcal{R}_j}.$$
By {\mathrm{e}}qref{eq-uniform-2nd-moment},
\begin{eqnarray*}
\sup\limits_{j\in{\mathbb{N}},\,1\leq i\leq \mathcal{R}_j}{\mathbb{E}}\left[\left(\mathcal{Y}_i^j\right)^2\right] < +\infty.
{\mathrm{e}}nd{eqnarray*}
Since for any $j\in{\mathbb{N}},$ $\{\mathcal{Y}_i^j-1\}_{1\leq i\leq\mathcal{R}_j}$ is an i.i.d.\,family with ${\mathbb{E}}\left[\mathcal{Y}_i^j-1\right]=0,$ then for some constant $C_2\in (0,\,\infty),$
\begin{eqnarray*}
{\mathbb{E}}\left[\mathcal{S}_j^2\right]={\mathbb{E}}\left[\frac{{\mathbb{E}}\left[\left(\mathcal{Y}_{1}^{j}- 1\right)^2\right]}{\mathcal{R}_j}\right]\leq \frac{C_2}{d(d-1)^{a_j-1}}.
{\mathrm{e}}nd{eqnarray*}
By the Chebyshev inequality, for any ${\mathrm{e}}psilon\in (0,\,\infty),$
\begin{eqnarray*}
{\mathbb{P}}[|\mathcal{S}_j|>{\mathrm{e}}psilon]\leq \frac{{\mathbb{E}}\left[\mathcal{S}_j^2\right]}{{\mathrm{e}}psilon^2}\leq \frac{C_2}{{\mathrm{e}}psilon^2 d(d-1)^{a_j-1}}
{\mathrm{e}}nd{eqnarray*}
which leads to
\[
\lim\limits_{j\rightarrow \infty} \mathcal{S}_j= 0\ a.s.
\]
by the Borel-Cantelli lemma. Namely, almost surely,
\[
\lim\limits_{j\rightarrow \infty} \frac{\sum\limits_{i=1}^{\mathcal{R}_j} Z_{a_{j+1}-a_{j}}(x^{a_j}_{i})}{m^{a_{j+1}-a_{j}}d(d-1)^{a_j-1}}=1.
\]
Since
\[
\sum\limits_{j=1}^{\infty} \lambda_1^{-a_{j+1}+1} m^{a_{j+1}-a_{j}}d(d-1)^{a_j-1}=\sum\limits_{j=1}^{\infty}\frac{d\lambda_1}{d-1}\left(\frac{(d-1)p}{\lambda_1}\right)^{a_{j+1}}
\left(\frac{1}{p}\right)^{a_j}<\infty,
\]
we have that almost surely
\begin{equation}\label{eq-conductance-sum-2}
\begin{aligned}
C_3:=\sum\limits_{j=1}^{\infty} \sum\limits_{e\in {\mathbb{P}}i_j^2} \mathbf{C}_{\lambda_1}(e)&= \sum\limits_{j=1}^{\infty} \lambda_1^{-a_{j+1}+1} |{\mathbb{P}}i_{j}^2|\leq \sum\limits_{j=1}^{\infty} \lambda_1^{-a_{j+1}+1} \sum\limits_{i=1}^{\mathcal{R}_j} Z_{a_{j+1}-a_{j}}(x^{a_j}_{i})< \infty.
{\mathrm{e}}nd{aligned}
{\mathrm{e}}nd{equation}
By {\mathrm{e}}qref{eq-conductance-sum} and {\mathrm{e}}qref{eq-conductance-sum-2}, and the Nash-Williams inequality and recurrence criterion (Lemma {\mathrm{Re}}f{lem-Nash}), when $\frac{1}{d-1}< p< \frac{\lambda_1}{d-1}$, almost surely,
\begin{align*}
\mathscr{R}\left(o\leftrightarrow \infty\right)&\geq \sum\limits_{j=1}^{\infty} \left(\sum\limits_{e\in {\mathbb{P}}i_{j}^1}\mathbf{C}_{\lambda_2}(e)+\sum\limits_{e\in{\mathbb{P}}i_j^2}\mathbf{C}_{\lambda_1}(e)\right)^{-1}
\geq \sum\limits_{j=1}^{\infty}(C_1+C_3)^{-1}=\infty,
{\mathrm{e}}nd{align*}
and $(\mathbb{T}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$ is recurrent.
\vskip 2mm
{\bf Assume secondly $\lambda_2=d-1$.} Recall the following facts. Let $T$ be a finite tree with root $o$ and leaf set $A$ such that
\begin{eqnarray}\label{eq-degree-condition}
\deg_T(o)=d-1,\ \deg_T(x)=d,\ x\in T\setminus (A\cup\{o\}).
{\mathrm{e}}nd{eqnarray}
Here $\deg_T(\cdot)$ is the degree function of vertices. For any $x\in T\setminus\{o\},$ let $x_*$ be the parent vertex of $x$:
$$x_*\sim x,\ \vert x_*\vert=\vert x\vert-1.$$
Define a unit flow $\theta_T$ on $T$ from $o$ to $A$: For any $x\in T\setminus\{o\},$
$$\theta_T(x_*x)=(d-1)^{-(\vert x_*\vert+1)},\ \theta_T(xx_*)=-(d-1)^{-(\vert x_*\vert+1)}.$$
By the flow conservation property (\cite[Lemma 2.8]{LP2017}),
$$\sum\limits_{x\in A}\theta_T(x_*x)=\sum\limits_{x\sim o}\theta_T(ox)=1.$$
Let $T'$ be a subtree of $T$ with the root $o$ and leaf set $A'$ that is a subset of $A.$ Then
\begin{eqnarray*}
\sum\limits_{x\in A'}\theta_{T}(x_*x)\leq \sum\limits_{x\in A}\theta_T(x_*x)=1,
\ \mbox{namely}\ \sum\limits_{x\in A'}(d-1)^{-\vert x\vert}\leq 1.
{\mathrm{e}}nd{eqnarray*}
Note any finite tree $T'$ with root $o$ and leaf set $A'$ satisfying
\begin{eqnarray}\label{eq-degree-condition-1}
\deg_{T'}(o)\leq d-1,\ \deg_{T'}(x)\leq d,\ x\in T'\setminus (A'\cup\{o\}),
{\mathrm{e}}nd{eqnarray}
can be embedded into a finite tree $T$ with root $o$ and leaf set $A$ such that both $A'\subseteq A$ and {\mathrm{e}}qref{eq-degree-condition} are true.
Therefore, for any finite tree $T'$ with root $o$ and leaf set $A'$ satisfying {\mathrm{e}}qref{eq-degree-condition-1},
\begin{eqnarray}\label{eq-leaf-mass}
\sum\limits_{x\in A'}(d-1)^{-\vert x\vert}\leq 1.
{\mathrm{e}}nd{eqnarray}
For any $i\in {\mathbb{N}}$ and $x\in L_{a_i},$ let $\mathcal{T}_i(x)$ be the open subtree of $\mathbb{T}^d$ consisting of $x$ and its offsprings $y$ connected to $x$ by an open path in $\omega_p$
such that $\vert y\vert \in [a_i+1,a_{i+1}-1].$ Write $T_i(x)=\mathcal{T}_i(x)\cup\partial\mathcal{T}_i(x).$
Let $A_i(x)$ be the leaf set of finite tree $T_i(x).$ Then
\begin{eqnarray*}
\sum\limits_{e\in{\mathbb{P}}i_i^1}\mathbf{C}_{d-1}(e)&\leq &\sum\limits_{x\in L_{a_i}}\sum\limits_{e\in\partial\mathcal{T}_i(x)}\mathbf{C}_{d-1}(e)=\sum\limits_{x\in L_{a_i}}\sum\limits_{e\in\partial\mathcal{T}_i(x)}(d-1)^{-\vert
e\vert}\\
&=& \sum\limits_{x\in L_{a_i}}\sum\limits_{y\in A_i(x)}(d-1)^{-(\vert y\vert-1)}\\
&=&(d-1)^{-a_i+1}\sum\limits_{x\in L_{a_i}}\sum\limits_{y\in A_i(x)}(d-1)^{-(\vert y\vert-a_i)},
{\mathrm{e}}nd{eqnarray*}
and by {\mathrm{e}}qref{eq-leaf-mass},
\begin{eqnarray}\label{eq-sum-closed}
\sum\limits_{e\in{\mathbb{P}}i_i^1}\mathbf{C}_{d-1}(e)\leq (d-1)^{-a_i+1}\sum\limits_{x\in L_{a_i}}1=(d-1)^{-a_i+1}d(d-1)^{a_i-1}=d.
{\mathrm{e}}nd{eqnarray}
By {\mathrm{e}}qref{eq-conductance-sum-2} and {\mathrm{e}}qref{eq-sum-closed}, and the Nash-Williams inequality and recurrence criterion, when $\frac{1}{d-1}< p< \frac{\lambda_1}{d-1}$, almost surely,
\begin{align*}
\mathscr{R}\left(o\leftrightarrow \infty\right)&\geq \sum\limits_{j=1}^{\infty} \left(\sum\limits_{e\in {\mathbb{P}}i_{j}^1}\mathbf{C}_{d-1}(e)+\sum\limits_{e\in{\mathbb{P}}i_j^2}\mathbf{C}_{\lambda_1}(e)\right)^{-1}
\geq \sum\limits_{j=1}^{\infty}(d+C_3)^{-1}=\infty,
{\mathrm{e}}nd{align*}
and $(\mathbb{T}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{d-1},\, p)$ is recurrent.
\rule{4pt}{7pt}
\vskip 2mm
\section{Concluding remarks and problems}\label{sec-concluding}
\setcounter{equation}{0}
\subsection{Biased disordered random networks $(G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$}
\noindent
Recall Theorem {\mathrm{Re}}f{generald} and $p_c\in (0,\,1/2)$ on ${\mathbb{Z}}^d$ with $d\geq 3.$ Note that on ${\mathbb{Z}}^d$ with $d\geq 2,$ the non-existence of infinite cluster at critical percolation is a well-known conjecture, which holds for $d=2$ and $d\geq 11.$ While $p_c=1/2$ for ${\mathbb{Z}}^2,$ the competing behavior of biased $({\mathbb{Z}}^2,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, 1/2)$ with $0< \lambda_1<\lambda_2=1$ is more subtle; and in this case transience seems to dominate recurrence.
These lead to the following
\begin{conj}\label{conj-critical-recurrent/transient}
When $3\leq d\leq 10$ and $0<\lambda_1\leq 1<\lambda_2,$ almost surely $({\mathbb{Z}}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p_c)$ is recurrent. And biased $({\mathbb{Z}}^2,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, 1/2)$ with $0< \lambda_1<\lambda_2=1$ is transient almost surely.
{\mathrm{e}}nd{conj}
\vskip 2mm
The following conjecture on having unique currents arises naturally.
\begin{conj}\label{conj-current-unique}
For $d\geq 3$ and $0<\lambda_1\leq 1<\lambda_2,$ almost surely, all transient biased $({\mathbb{Z}}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$ are current unique.
{\mathrm{e}}nd{conj}
\vskip 2mm
For $d$-regular tree $\mathbb{T}^d$ with $d\geq 3,$ $p_c=\frac{1}{d-1}$ and $\lambda_c=d-1.$
\begin{conj}\label{conj-critical-recurrent-tree}
For any $d\geq 3,$ biased $(\mathbb{T}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, \frac{\lambda_1\vee 1}{d-1})$ with $1<\lambda_1<d-1\leq \lambda_2$ or
$0<\lambda_1\leq 1<d-1= \lambda_2$ is recurrent almost surely.
{\mathrm{e}}nd{conj}
\vskip 2mm
Recall for any $x=(x_1,\,\ldots,\, x_d)\in{\mathbb{Z}}^d,$ $\vert x\vert=\sum\limits_{i=1}^d\vert x_i\vert.$
Note the invariance principle and the large deviation for ${\mathbb{R}}W_\lambda$ with $\lambda\in (0,1)$ on ${\mathbb{Z}}^d$ was proved in \cite{LS2020}. And from \cite{SSSWX2018}, on ${\mathbb{Z}}^d$,
${\mathbb{R}}W_\lambda$ $(X_n)_{n=0}^{\infty}$ with $\lambda\in (0,1)$ almost surely has positive speed, i.e., $\lim\limits_{n\rightarrow\infty}\frac{\vert X_n\vert}{n}=\frac{1-\lambda}{1+\lambda};$
and the heat kernel of $(X_n)_{n=0}^{\infty}$ decays exponentially.
\begin{prob}\label{plimit}
Let $(X_n)_{n=0}^{\infty}$ be the random walk associated to any biased $({\mathbb{Z}}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$ with $0<\lambda_1<\lambda_2<\infty.$
{\bf (i)} For transient biased $({\mathbb{Z}}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$, does almost surely the quenched speed $\lim\limits_{n\rightarrow\infty}\frac{\vert X_n\vert}{n}$ exist? If yes, is it positive and constant almost surely?
{\bf (ii)} Prove the quenched invariance principle for random walk $(X_n)_{n=0}^{\infty}$.
{\bf (iii)} Assume $0<\lambda_1<1<\lambda_2.$ Almost surely, is there an exponential vs polynomial decay in time
phase transition for the quenched heat kernel of $(X_n)_{n=0}^{\infty}$ when $p$ varies from $0$ to $1$?
{\mathrm{e}}nd{prob}
\vskip 2mm
Let $K_{o}(\omega_p)$ denote the open cluster of $o$ in Bernoulli-$p$ bond percolation $\omega_p$ on graph $G.$ Define
$$p_u=\inf\{p\in [0,\,1]:\ \omega_p\ \mbox{has a unique infinite open cluster with positive probability}\}.$$
To characterize $p_c^*$ for biased disordered random networks $(G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$ with $G$ being a Cayley graph, we recall the following Lalley's conjecture.
\begin{conj}[Lalley \mbox{\cite[Conjecture 17]{SL2006}}]\label{conj-Lalley}
Given a transitive non-amenable graph $G$ with $p_c<p_u$ and fixed vertex $o$. If there exists a unique cluster a.s. at $p= p_u$,
then condition on $K_{o}(\omega_{p_u})$ is infinite,
\[\lim\limits_{p\uparrow p_u}\underline{\rm gr}\left(K_{o}(\omega_p)\right)= {\rm gr}(G)\ \mbox{almost surely}.\]
{\mathrm{e}}nd{conj}
\vskip 2mm
\begin{conj}\label{conj-p_c^*}
Let $G$ be an infinite Cayley graph with fixed vertex $o.$ Then the following hold.
\begin{enumerate}[{\bf (i)}]
\item If $G$ is amenable with $\lambda_c(G)=1,$ and $(G,\,\mathbf{C}_1)$ is transient (resp. recurrent), then $p_c^*=p_c$ for any $((G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p))_{p\in [0,\,1]}$ with $0<\lambda_1\leq 1<\lambda_2$ (resp. $0<\lambda_1< 1\leq \lambda_2 $).
\item When $G$ is amenable with $\lambda_c(G)>1,$ and $\left(G,\,\mathbf{C}_{\lambda_c(G)}\right)$ is transient (resp. recurrent),
the threshold $p_c^*$ of any $((G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p))_{p\in [0,\,1]}$ with $0<\lambda_1\leq\lambda_c(G)<\lambda_2$ (resp. $0<\lambda_1<\lambda_c(G)\leq\lambda_2$) satisfies that
\begin{eqnarray*}
p_c^*=p_c^*(\lambda_1)=\inf\left\{p\in (p_c,1]:\ \mathbb{P}\left[\left. \underline{\rm gr}(K_o(\omega_p))>(\lambda_1\vee 1)\,\right\vert \vert K_o(\omega_p)\vert=\infty\right]>0\right\}
{\mathrm{e}}nd{eqnarray*}
with convention $\inf{\mathrm{e}}mptyset=p_c,$ and $p_c^*(\lambda_1)\in [p_c,\,1)$ is continuous in $\lambda_1\in(0,\,\lambda_c(G))$ and strictly increasing in $\lambda_1\in [1,\,\lambda_c(G)),$ and
$\lim\limits_{\lambda_1\uparrow\lambda_c(G)}p_c^*(\lambda_1)=1.$
\item For any non-amenable $G$ with $p_c<p_u$ such that $\omega_{p_u}$ has a.s.\,a unique infinite cluster and $\left(G,\,\mathbf{C}_{\lambda_c(G)}\right)$ is transient (resp. recurrent),
any $((G,\,\mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p))_{p\in [0,\,1]}$
with $0<\lambda_1\leq\lambda_c(G)<\lambda_2$ (resp. $0<\lambda_1<\lambda_c(G)\leq\lambda_2$) has the threshold
\begin{eqnarray*}
p_c^*=p_c^*(\lambda_1)=\inf\left\{p\in (p_c,1]:\ \mathbb{P}\left[\left. \underline{\rm gr}(K_o(\omega_p))>(\lambda_1\vee 1)\,\right\vert \vert K_o(\omega_p)\vert=\infty\right]>0\right\}\in [p_c,\,p_u),
{\mathrm{e}}nd{eqnarray*}
and $p_c^*(\lambda_1)$ is continuous in $\lambda_1\in(0,\,\lambda_c(G))$ and strictly increasing in $\lambda_1\in [1,\,\lambda_c(G)),$ and
$\lim\limits_{\lambda_1\uparrow\lambda_c(G)}p_c^*(\lambda_1)=p_u.$
\item Assume $G$ is non-amenable such that $p_c<p_u$ and $\omega_{p_u}$ has a.s.\, infinitely many infinite clusters, and $\left(G,\,\mathbf{C}_{\lambda_c(G)}\right)$ is transient (resp. recurrent).
Then for any $((G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p))_{p\in [0,\,1]}$ with $0<\lambda_1\leq \lambda_c(G)<\lambda_2$ (resp. $0<\lambda_1<\lambda_c(G)\leq\lambda_2$),
\begin{eqnarray*}
p_c^*=p_c^*(\lambda_1)=\inf\left\{p\in (p_c,1]:\ \mathbb{P}\left[\left. \underline{\rm gr}(K_o(\omega_p))>(\lambda_1\vee 1)\,\right\vert \vert K_o(\omega_p)\vert=\infty\right]>0\right\}\in [p_c,\,p_u],
{\mathrm{e}}nd{eqnarray*}
and $p_c^*(\lambda_1)$ is continuous in $\lambda_1\in (0,\,\lambda_c(G))$, and there is a $\lambda_1^*\in (1,\lambda_c(G))$ such that $p_c^*(\lambda_1)=p_u$ for $\lambda_1\in [\lambda_1^*,\,\lambda_c(G))$
and $p_c^*(\lambda_1)$ is strictly increasing in $\lambda_1\in [1,\,\lambda_1^*].$
{\mathrm{e}}nd{enumerate}
{\mathrm{e}}nd{conj}
\vskip 2mm
For $(G,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\, p)$, let $\theta(p)$ be the probability of event $\{X_0=o,\, X_n\not=o,\,\forall n\geq 1\}$ given $\omega_p.$ Note $\theta(p)$ is a measurable function of $\omega_p$ and thus of whole process $(\omega_q)_{q\in [0,1]}.$ So if $G$ is quasi-transitive, then by the ergodic property of $(\omega_q)_{q\in [0,1]},$ almost surely, $\theta(p)$ is a constant for any $p\in [0,1].$
\begin{prob}\label{prob-exit-probability}
Assume $G$ is quasi-transitive with $p_c\in (0,1).$ Is $\theta(p_c^*)$ zero? Is $\theta(p)$ continuous in $p\in [p_c^*,1]?$ Is $\theta(p)$ strictly increasing in $p\in [p_c^*,1]?$ And if $\theta(p)$ is right continuous at $p_c^*\in (0,1),$ is there a critical exponent $\alpha>0$ such that
$\theta(p)-\theta(p_c^*)\asymp \vert p-p_c^*\vert ^{\alpha+o(1)}$ as $p\downarrow p_c^*?$
{\mathrm{e}}nd{prob}
\subsection{Other disordered random networks}
\noindent
\begin{prob}\label{pphasetransition}
Given a quasi-transitive infinite graph $G$, and any two conductance functions $\mathbf{c}_1$ and $\mathbf{c}_2$ such that $\mathbf{c}_1(e)\geq \mathbf{c}_2(e)$ for any $e$ of $G$, $(G,\, \mathbf{c}_1)$ is transient and $(G,\, \mathbf{c}_2)$ is recurrent.
Almost surely, when is there a nontrivial recurrence/transience phase transition for $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ as $p$ varies from $0$ to $1$? Does there exist a class of graphs $G$ such that almost surely there is always a nontrivial recurrence/transience phase transition of $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ for any fixed $\mathbf{c}_1$ and $\mathbf{c}_2$?
{\mathrm{e}}nd{prob}
\vskip 2mm
\begin{prob}\label{prob-PhaseTransition}
Let $G$ be a quasi-transitive infinite graph. {\bf (i)} Assume $(G,\,\mathbf{c}_1)$ is not current unique while $(G,\,\mathbf{c}_2)$ is. Study current uniqueness/nonuniqueness phase transition for $(G,\,\mathbf{c}_1,\,\mathbf{c}_2,\,p)$ when $p$ varies from $0$ to $1.$
{\bf (ii)} Suppose the free (wired) Ising model on $G=(V,\,E)$ with coupling constants $(\mathbf{c}_1(e))_{e\in E}$ (resp. $(\mathbf{c}_2(e))_{e\in E}$) and inverse temperature $1$ is in a ferromagnet (resp. paramagnet) regime. Investigate paramagnet/ferromagnet
phase transition for the free (wired) Ising model on $G$ with coupling constants $\left(\mathbf{c}_1(e)I_{\{e\in\omega_p\}}+\mathbf{c}_2(e)I_{\{e\notin\omega_p\}}\right)_{e\in E}$ and inverse temperature $1$ when $p$ varies from $0$ to $1.$
{\mathrm{e}}nd{prob}
\vskip 2mm
\begin{prob}\label{prob-synchronize}
Given a quasi-transitive infinite graph $G$. {\bf (i)} When $(G,\, \mathbf{c}_1)$ and $(G,\, \mathbf{c}_2)$ are both recurrent (resp. transient) networks, is $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ recurrent (resp. transient) almost surely for $p\in [0,\, 1]$? {\bf (ii)} If both $(G,\, \mathbf{c}_1)$ and $(G,\, \mathbf{c}_2)$ have current uniqueness, does $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ have current uniqueness almost surely for $p\in [0,\, 1]$?
{\mathrm{e}}nd{prob}
\vskip 2mm
\begin{remark}[Discussion on Problem {\mathrm{Re}}f{prob-synchronize} (i)] Assume $(G,\, \mathbf{c}_1)$ and $(G,\, \mathbf{c}_2)$ are both recurrent, and one of the following conditions is true: {\bf (a)} $(G,\, \mathbf{c}_1\wedge \mathbf{c}_2)$ is recurrent or $\mathbf{c}_1\asymp \mathbf{c}_2$; {\bf (b)} $(G,\, \mathbf{c}_1)$ and $(G,\, \mathbf{c}_2)$ are both positive recurrent. Then $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ is recurrent almost surely for $p\in [0,\, 1]$. It is unknown whether this is true generally.
There are examples that $(G,\, \mathbf{c}_1)$ and $(G,\, \mathbf{c}_2)$ are both transient, but $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ is recurrent almost surely for $p\in (0,\, 1)$.
On ${\mathbb{Z}}$, define for $0< \lambda< 1$,
$$\mathbf{c}_1(\{i,i-1\}) =\lambda^{i}I_{\{i\leq 0\}}+I_{\{i>0\}},\ \mathbf{c}_2(\{i,i+1\}) =\lambda^{-i}I_{\{i\geq 0\}}+I_{\{i<0\}},\ i\in{\mathbb{Z}}.$$
Then $({\mathbb{Z}},\, \mathbf{c}_{1})$ and $({\mathbb{Z}},\, \mathbf{c}_{2})$ are transient; and by the Nash-Williams criterion, $({\mathbb{Z}},\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$ is a.s. recurrent for $p\in(0,\, 1).$
To obtain an example on ${\mathbb{Z}}^2$, define for $0< \lambda< 1$, $\mathbf{c}_1(e)=\lambda^{-\vert e\vert}$ when $e$ is in the negative $x$-axis and $1$ otherwise; and similarly $\mathbf{c}_2(e)=\lambda^{-\vert e\vert}$ when $e$ is in the positive $x$-axis and $1$ otherwise. Then $({\mathbb{Z}}^2,\, \mathbf{c}_{1})$ and $({\mathbb{Z}}^2,\, \mathbf{c}_{2})$ are transient. However, $({\mathbb{Z}}^2,\,\mathbf{c}_1,\, \mathbf{c}_2,\, p)$ is a.s. recurrent for $p\in (0,\, 1)$,
which can be proved by the Nash-Williams criterion,
\[
\mathscr{R}(0\leftrightarrow \infty)\geq \sum\limits_{k=1}^{\infty}\left(\sum\limits_{e\in {\mathbb{P}}i_{k}} c(e)\right)^{-1}= \infty,
\]
where each ${\mathbb{P}}i_{k}$ is taken as follows: Let $\{e_{k,+}\}_{k=1}^{\infty}$ (resp. $\{e_{k,-}\}_{k=1}^{\infty}$) be all open (resp. closed) edges in the positive (resp. negative) axis such that $\vert e_{k,+}\vert$ (resp.
$\vert e_{k,-}\vert$) is strictly increasing in $k$. Let ${\mathbb{P}}i_k$ be the set of all edges in $[-\vert e_{k,-}\vert-1,\vert e_{k,+}\vert+1]^2$ with exactly one endpoint in $[-\vert e_{k,-}\vert,\vert e_{k,+}\vert]^2.$
To apply the Nash-Williams criterion, one needs to note that almost surely,
$$\limsup\limits_{k\rightarrow\infty}\frac{\vert e_{k,+}\vert}{k}<\infty\ \mbox{and}\ \limsup\limits_{k\rightarrow\infty}\frac{\vert e_{k,-}\vert}{k}<\infty.$$
{\mathrm{e}}nd{remark}
\vskip 2mm
Note for $(G,\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)$, the random environment is given by Bernoulli-$p$ bond percolation. Instead of Bernoulli bond percolation on graph $G=(V,\,E),$ one can define disordered random networks with random environments described by point processes on $E$ such as Poisson point process (PPP), determinant point process (DPP) and random cluster model through an obvious ways. To study phase transition, one needs to let the edge density in these environments vary from $0$ to $1$ via some natural means.
Additionally, one can also define disordered random network in random environment provided respectively by PPP, DPP and Bernoulli-$p$ site percolation on $V$ as follows: Let random subset $V_1$ of $V$ follow one of the distributions of PPP, DPP and Bernoulli-$p$ site percolation on $V.$ Write $E(V_1)$ (resp. $E_o(V_1)$) for the set of edges of $G$ whose one endpoint intersects (resp. two endpoints intersect) $V_1$. By letting edges in $E(V_1)$ (resp. $E_o(V_1)$) take conductance $\mathbf{c}_1$ and other edges take conductance $\mathbf{c}_2,$ one gets the desired disordered random network.
Finally, let $\mathbf{c}_{V,1}$ and $\mathbf{c}_{V,2}$ be two nonnegative weight functions on $V$ such that $(G,\,\mathbf{c}_1)$ is transient and $(G,\,\mathbf{c}_2)$ is recurrent, where
$$\mathbf{c}_1(\{x,y\})=\mathbf{c}_{V,1}(x)\mathbf{c}_{V,1}(y)\ \mbox{and}\ \mathbf{c}_2(\{x,y\})=\mathbf{c}_{V,2}(x)\mathbf{c}_{V,2}(y)\
\mbox{for any edge}\ \{x,y\}\ \mbox{of}\ G.$$
For the just mentioned random subset $V_1$, let $V_2=V\setminus V_1$ and
$$\mathbf{c}_{1,2}(\{x,y\})=\mathbf{c}_{V,i}(x)\mathbf{c}_{V,j}(y)\ \mbox{if}\ x\in V_i,\, y\in V_j\ \mbox{and}\ \{x,y\}\in E.$$
Thus one obtain a disordered random network $(G,\,\mathbf{c}_{1,2}).$
It is very interesting to study various typical properties for the above disordered random networks. Recall from \cite{MA1960, FM2019} that there is an interesting random resistor network built in a different way in a homogeneous Poisson point process environment, which is called Miller-Abrahams random resistor network.
\subsection{Disordered random walks}\label{sec-disorderedRW}
\noindent Let $\mathbf{p}_1=\mathbf{p}_1(\cdot,\,\cdot)$ and $\mathbf{p}_2=\mathbf{p}_2(\cdot,\,\cdot)$ be respectively two 1-step transition probabilities of two Markov chains on graph $G=(V,\,E).$ Let $V_1=V_1(p)$ be the set of open vertices in Bernoulli-$p$ site percolation on $G$ and $V_2=V\setminus V_1.$ Use $(G,\,\mathbf{p}_1,\,\mathbf{p}_2,\,p)$ to denote the Markov chains on $G$ with $1$-step transition probability given by
$$\mathbf{p}(x,\,y)=\mathbf{p}_1(x,\,y)I_{\{x\in V_1\}}+\mathbf{p}_2(x,\,y)I_{\{y\in V_2\}},\ x,y\in V.$$
Like the disordered random network $(G,\,\mathbf{c}_1,\,\mathbf{c}_2,\, p)$, we also define Bernoulli site percolation process $(V_1(p))_{p\in [0,\,1]}$ on $G$ by the grand coupling, and also study recurrence/transience phase transition (and other ones) for
disordered random walk $(G,\,\mathbf{p}_1,\,\mathbf{p}_2,\,p)$ as $p$ varies from $0$ to $1.$
Notice that recurrence/transience phase transition for biased $(G,\,\mathbf{p}_1,\,\mathbf{p}_2,\,p)$ is more subtle than that of biased $(G,\,\mathbf{c}_1,\,\mathbf{c}_2,\, p)$ due to that unlike the latter, there is no the Rayleigh's monotonicity principle for the former. For example, similarly to Remark {\mathrm{Re}}f{remark-PT}\,(ii), one can also define $p_c^*$ and $\widehat{p}_c^*$ for biased $(G,\,\mathbf{p}_1,\,\mathbf{p}_2,\,p)$, but $p_c^*=\widehat{p}_c^*$ may not hold and transient regime may not be $(p_c^*,1]$ or $(\widehat{p}_c^*,1].$ The disordered random walks $(G,\, \mathbf{p}_1,\, \mathbf{p}_2,\, p)$ are a new kind of RWREs. Replacing Bernoulli site percolation by a PPP or DPP on $V$, one gets another disordered random walk.
\begin{prob}
Suppose $G$ is a quasi-transitive infinite graph, $\mathbf{p}_1$-random walk is transient and $\mathbf{p}_2$-random walk is recurrent. Characterize the recurrent and transient regimes for the disordered random walk family $\left((G,\, \mathbf{p}_1,\, \mathbf{p}_2,\, p)\right)_{p\in [0,1]}$. When are these regimes almost surely a single subinterval of $[0,\,1]$ such that almost surely, there is a threshold $p_c^*$ (may be random) satisfying $(G,\, \mathbf{p}_1,\, \mathbf{p}_2,\, p)$ is recurrent for any $p<p_c^*$ and transient for any $p>p_c^*?$ Study various typical properties for the disordered random walks (particularly the biased ones).
{\mathrm{e}}nd{prob}
\vskip 2mm
\begin{thm}\label{thm-sz1}
Let $0<\lambda_1<\lambda_2<\infty$ and each $\mathbf{p}_i$-random walk be ${{\mathbb{R}}W}_{\lambda_i}$ on ${\mathbb{Z}},$ and
$$p_c^*=\left(\frac{\log \lambda_2}{\log \lambda_2- \log \lambda_1}\vee 0\right) \wedge 1.$$
Then almost surely, $({\mathbb{Z}},\, \mathbf{p}_1,\, \mathbf{p}_2,\, p)$ is recurrent for any $p\in [0,p_c^*]$ and transient for any
$p\in (p_c^*,1].$
{\mathrm{e}}nd{thm}
\vskip 2mm
For the recurrence/transience phase transition of biased $\left(({\mathbb{Z}},\, \mathbf{p}_1,\, \mathbf{p}_2,\, p)\right)_{p\in [0,1]}$ with each $\mathbf{p}_i$-random walk being ${{\mathbb{R}}W}_{\lambda_i}$ and $0<\lambda_1<1\leq\lambda_2<\infty,$ $p_c^*=\frac{\log \lambda_2}{\log \lambda_2- \log \lambda_1}$, which is not related to the phase transition of the Bernoulli site percolation on ${\mathbb{Z}}.$ And hence it has a novel nature
different from that of biased random networks $\left(({\mathbb{Z}},\, \mathbf{c}_1,\, \mathbf{c}_2,\, p)\right)_{p\in [0,1]}.$
Note that on regular trees $\mathbb{T}^d$ with $d\geq 3,$ the biased disordered random walk has i.i.d.\,random transition probabilities on all vertices but the root. Recall from \cite{LP1992} and \cite{RY1995}, Lyons, Pemantle and Peres gave complete recurrent/transience criteria for RWREs with some special i.i.d.\,random environments on trees in 1990s. By these criteria and some monotonicity similarly to {\mathrm{e}}qref{stoc-monotone}, we have the following clear picture of phase transition for the biased disordered random walks on $\mathbb{T}^d$:
\begin{enumerate}
\item[] On $\mathbb{T}^d$ with $d\geq 3$, let each $\mathbf{p}_i$-random walk be ${{\mathbb{R}}W}_{\lambda_i}$ with $0<\lambda_1<\lambda_2<\infty.$
Then when $\lambda_1\geq d-1$ (resp. $\lambda_2<d-1$), almost surely, $(\mathbb{T}^d,\, \mathbf{p}_1,\, \mathbf{p}_2,\, p)$ is recurrent (resp. transient) for all $p\in [0,\, 1];$ when $\lambda_2=d-1,$ almost surely, $(\mathbb{T}^d,\, \mathbf{p}_1,\, \mathbf{p}_2,\, p)$ is transient for all $p\in (0,\, 1]$ (note $(\mathbb{T}^d,\, \mathbf{p}_1,\, \mathbf{p}_2,\, 0)$ is recurrent). And when $\lambda_1<d-1<\lambda_2$, there is a non-trivial recurrence/transience phase transition for the biased disordered random walks $(\mathbb{T}^d,\, \mathbf{p}_1,\, \mathbf{p}_2,\, p)$ such that for a constant $p^{*}_c\in (0,\, 1)$, almost surely, $(\mathbb{T}^d,\, \mathbf{p}_1,\, \mathbf{p}_2,\, p)$ is recurrent for any $p\leq p_c^{*}$ and transient for $p> p_c^{*};$ where $p_c^*$ is the unique solution to
$$f(p)=\min\limits_{0\leq x\leq 1}\left\{\left(\frac{1}{\lambda_1}\right)^xp+\left(\frac{1}{\lambda_2}\right)^x(1-p)\right\}=\frac{1}{d-1},\ p\in [0,1];$$
and explicitly $p_c^*=\frac{\frac{1}{d-1}-\frac{1}{\lambda_2}}{\frac{1}{\lambda_1}-\frac{1}{\lambda_2}}$ when $1\leq\lambda_1<d-1<\lambda_2.$
{\mathrm{e}}nd{enumerate}
\noindent{\bf Proof of Theorem {\mathrm{Re}}f{thm-sz1}.}
{\bf Step 1.} Recall some preliminaries on random walks.
\begin{lem}[\cite{FS1975}]\label{Cprop}
Let $\{X_n\}_{n=0}^{\infty}$ be a sequence of i.i.d.\,non-degenerate finite-value random variables and $S_n= \sum\limits_{i=1}^{n} X_i,\, n\in {\mathbb{N}}$. Then {\bf (i)} $\sum\limits_{n=1}^{\infty} n^{-1}{\mathbb{P}}\left(S_n >0\right)< \infty {\mathbb{L}}ongleftrightarrow\lim\limits_{n\rightarrow \infty} S_n= -\infty \ \mbox{a.s.},$ and under this condition $\sum\limits_{n=1}^{\infty} e^{S_n}< \infty$ a.s.; and {\bf (ii)}
$\sum\limits_{n=1}^{\infty} n^{-1}{\mathbb{P}}\left(S_n >0\right)= \infty=\sum\limits_{n=1}^{\infty} n^{-1}{\mathbb{P}}\left(S_n <0\right)$
is equivalent to
\[
-\infty= \liminf\limits_{n\rightarrow \infty} S_n< \limsup\limits_{n\rightarrow \infty} S_n= \infty\ a.s.,
\]
and in this case $\sum\limits_{n=1}^{\infty} e^{S_n}= \infty= \sum_{n=1}^{\infty} e^{-S_n}$ a.s..
{\mathrm{e}}nd{lem}
For any countably infinite set $\Sigma$, say a sequence $(x_n)_{n\geq 1}\subseteq \Sigma$ converges to $\infty$ if for any finite subset $A$ of $\Sigma,$ $x_n\notin A$ for large enough $n.$
\begin{lem}[Lyapunov recurrence/transience criterion \mbox{(\cite[Chapter~2.5]{MSA2015}, \cite{FMS1998})}]\label{Lcriterion} Suppose $\{X_n\}_n$ is an irreducible Markov chain on a countably infinite state space $\Sigma$. Fix $o\in\Sigma.$
\begin{enumerate}[{\bf (i)}]
\item $\{X_n\}_n$ is recurrent if and only if there are a function $f:\, \Sigma \rightarrow {\mathbb{R}}_{+}$ and a finite non-empty set $A$ of $\Sigma$ such that $f(x)\rightarrow \infty$ as $x\rightarrow \infty$, and
\[
{\mathbb{E}}[f(X_{n+1})- f(X_n) \, \vert\, X_n= x]\leq 0\ \mbox{for all}\ x\in \Sigma \backslash A\ \mbox{and}\ n\geq 0.
\]
$\{X_n\}_n$ is transient if and only if there exist a function $f:\, \Sigma \rightarrow {\mathbb{R}}_{+}$ and a non-empty set $A\subset\Sigma$
satisfying $f(y)<\inf\limits_{x\in A} f(x)$ for at least one $y\in \Sigma\backslash A,$ and
\[
{\mathbb{E}}[f(X_{n+1})- f(X_n) \ \vert\ X_n= x]\leq 0\ \mbox{for all}\ x\in \Sigma \backslash A \ \mbox{and}\ n\geq 0.
\]
\item $\{X_n\}_n$ is recurrent if there is a function $f:\, \Sigma\rightarrow {\mathbb{R}}$ with that $f(x)\rightarrow\infty$ as $x\rightarrow\infty$ and
\[{\mathbb{E}}\left[f(X_{n+1})- f(X_n) \, \vert\, X_n=x\right]\leq 0,\ x\not=o,\ n\geq 0;\]
and is transient if there is a bounded and non-constant function $f:\, \Sigma \rightarrow \mathbb{R}$ such that
\[{\mathbb{E}}\left[f(X_{n+1})- f(X_n) \, \vert\, X_n=x\right]= 0,\ x\not=o,\ n\geq 0.\]
{\mathrm{e}}nd{enumerate}
{\mathrm{e}}nd{lem}
\noindent{{\bf Step 2. Turn to prove Theorem {\mathrm{Re}}f{thm-sz1}.}
The proof is routine. Let $\omega_p={\mathbb{Z}}(p)$ be the Bernoulli-$p$ site percolation on ${\mathbb{Z}}$, and $\omega=(\omega_p)_{0\leq p\leq 1}$ the Bernoulli site percolation process constructed by the grand coupling. Fix a random environment $\omega$, for $x>0$, let
\begin{eqnarray*}
&&p_x(p)=\mathbf{p}_1(x,x-1)I_{\{\omega_p(x)=1\}}+\mathbf{p}_2(x,x-1)I_{\{\omega_p(x)=0\}}
=\frac{\lambda_1}{1+\lambda_1}I_{\{\omega_p(x)=1\}}+\frac{\lambda_2}{1+\lambda_2}I_{\{\omega_p(x)=0\}},\\
&&q_x(p)=\mathbf{p}_1(x,x+1)I_{\{\omega_p(x)=1\}}+\mathbf{p}_2(x,x+1)I_{\{\omega_p(x)=0\}}
=\frac{1}{1+\lambda_1}I_{\{\omega_p(x)=1\}}+\frac{1}{1+\lambda_2}I_{\{\omega_p(x)=0\}};
{\mathrm{e}}nd{eqnarray*}
and for $x<0,$ let
\begin{eqnarray*}
&&p_x(p)=\mathbf{p}_1(x,x+1)I_{\{\omega_p(x)=1\}}+\mathbf{p}_2(x,x+1)I_{\{\omega_p(x)=0\}}
=\frac{\lambda_1}{1+\lambda_1}I_{\{\omega_p(x)=1\}}+\frac{\lambda_2}{1+\lambda_2}I_{\{\omega_p(x)=0\}},\\
&&q_x(p)=\mathbf{p}_1(x,x-1)I_{\{\omega_p(x)=1\}}+\mathbf{p}_2(x,x-1)I_{\{\omega_p(x)=0\}}
=\frac{1}{1+\lambda_1}I_{\{\omega_p(x)=1\}}+\frac{1}{1+\lambda_2}I_{\{\omega_p(x)=0\}}.
{\mathrm{e}}nd{eqnarray*}
To construct a nonconstant bounded function $f=f_{\omega,p}:\ {\mathbb{Z}}\rightarrow{\mathbb{R}}$ which is harmonic on ${\mathbb{Z}}\backslash\{0\}$ for $({\mathbb{Z}},\,\mathbf{p}_1,\,\mathbf{p}_2,\,p).$ Define $f(0)= 0$ and $f(1)= f(-1)= 1$. Since for any $x> 0$,
$$\left(f(x-1)- f(x)\right)p_x(p)+ \left(f(x+1)- f(x)\right)q_x(p)= 0,$$
we have that
\[
\frac{f(x+1)- f(x)}{f(x)- f(x-1)}= \frac{p_x(p)}{q_x(p)}\ \mbox{and}\
f(x+1)= 1+ \sum\limits_{i=1}^{x} e^{\sum_{k=1}^i Y_k.}
\]
Here $\{Y_k\}_{k\in {\mathbb{Z}}\setminus\{0\}}$ is a random sequence with
$$Y_k:=Y_k(p)=\log(p_k(p)/q_k(p))=\log\left(\lambda_1I_{\{\omega_p(k)=1\}}+\lambda_2I_{\{\omega_p(k)=0\}}\right).$$
Similarly for any $x< 0$,
$$f(x-1)= 1+ \sum_{i=1}^{x} e^{\sum_{k=1}^i Y_{-k}}.$$
Clearly $\{p_x(p)/q_x(p)\}_{x\in{\mathbb{Z}}\setminus\{0\}}$ is a sequence of i.i.d.\,random variables.
By Lemma {\mathrm{Re}}f{Cprop}\,(i),
$f$ is almost surely a non-constant bounded function on ${\mathbb{Z}}$ if
\[{\mathbb{E}}\left(Y_x\right)=p\cdot \log \lambda_1+ (1-p) \cdot \log \lambda_2 < 0,\ x\in{\mathbb{Z}}\setminus\{0\},\ \mbox{namely}\ p>p_c^*.\]
Note that
\begin{eqnarray}\label{stoc-monotone}
\mbox{each}\ Y_k(p)\ \mbox{is a decreasing function in}\ p\in [0,\,1]\ \mbox{for any environment}\ \omega,\ \mbox{so is every}\ f(k).
{\mathrm{e}}nd{eqnarray}
Then by {\mathrm{e}}qref{stoc-monotone} and Lemma {\mathrm{Re}}f{Lcriterion}\,(ii), almost surely, $({\mathbb{Z}},\, \mathbf{p}_1,\, \mathbf{p}_2,\, p)$ is transient for all $p\in \left(p_c^*,\, 1\right].$ When each ${\mathbb{E}}\left(Y_x \right)>0$, by the law of large numbers, $\lim\limits_{x\rightarrow\infty}f(x)=\lim\limits_{x\rightarrow -\infty}f(x)=\infty$ a.s.. Thus by {\mathrm{e}}qref{stoc-monotone} and Lemma {\mathrm{Re}}f{Lcriterion}\,(ii) again, almost surely, $({\mathbb{Z}},\, \mathbf{p}_1,\, \mathbf{p}_2,\, p)$ is recurrent for all $p\in \left[0,\,p_c^*\right).$
When $p_c^*= \frac{\log \lambda_2}{\log \lambda_2- \log \lambda_1}\in (0,\, 1)$, for any $n\in\mathbb{N},$ define
$$Z_n=\frac{Y_n(p_c^*)- \log \lambda_1}{\log \lambda_2- \log \lambda_1},\ S_n=\sum_{i=1}^{n}Y_i(p_c^*),\ S'_n=\sum_{i=1}^{n} Z_i.$$
Then $S'_n$ has binomial distribution $B(n,\, 1-p_c^*)$, and
\begin{equation*}
\begin{split}
\sum\limits_{n=1}^{\infty} n^{-1} {\mathbb{P}}\left(S_n > 0\right)&=\sum\limits_{n=1}^{\infty} n^{-1} {\mathbb{P}}\left(S'_n > \frac{-n \log \lambda_1}{\log \lambda_2- \log \lambda_1}\right)= \sum\limits_{n=1}^{\infty} n^{-1} {\mathbb{P}}\left(S'_n > n(1-p_c^*)\right)=\infty,\\
\sum\limits_{n=1}^{\infty} n^{-1} {\mathbb{P}}\left(S_n < 0\right)&= \sum\limits_{n=1}^{\infty} n^{-1} {\mathbb{P}}\left(S'_n< n(1-p_c^*)\right)=\infty.
{\mathrm{e}}nd{split}
{\mathrm{e}}nd{equation*}
Here we have used that by the central limit theorem,
$$\lim\limits_{n\rightarrow\infty}{\mathbb{P}}\left(S'_n > n(1-p_c^*)\right)=\lim\limits_{n\rightarrow\infty}{\mathbb{P}}\left(S'_n < n(1-p_c^*)\right)=\frac{1}{2}.$$
By Lemma {\mathrm{Re}}f{Cprop}\,(ii), almost surely, $f_{\omega,p_c^*}(x+1)= 1+ \sum_{i=1}^{x} e^{S_i}\rightarrow \infty,\ x\rightarrow\infty.$
Similarly,
$$\mbox{almost surely},\ f_{\omega,p_c^*}(x-1)\rightarrow \infty,\ x\rightarrow-\infty.$$
Thus $\left({\mathbb{Z}},\, \mathbf{p}_1,\, \mathbf{p}_2,\, p_c^*\right)$ is recurrent almost surely by Lemma {\mathrm{Re}}f{Lcriterion}\,(ii). The proof is done.
\rule{4pt}{7pt}
\subsection{Discussion on volume growth rate of percolation clusters}\label{sec-discussion-growth}
\noindent
On a connected transitive graph $G$, the number of infinite open clusters is constant almost surely which can only be $0$, $1$, or $\infty$ (\cite{NS1981}). And the constant cannot be $\infty$ when $G$ is amenable (\cite{BK1989, GKN1992}). On non-amenable graphs, there might exists two phase transitions, namely there exist $0<p_c<p_u\leq 1$ such that there are infinitely many infinite open clusters when $p\in (p_c, p_u)$. Benjamini and Schramm \cite{BS1996} conjectured that $p_u< 1$ when $G$ is quasi-transitive with one end and $p_c< p_u$ when $G$ is quasi-transitive and non-amenable. For recent progresses on `$p_c< p_u$'-conjecture, see Hutchcroft \cite{HT2019, HT2020}.
\vskip 2mm
Here we can verify Conjecture {\mathrm{Re}}f{conj-Lalley} on transitive non-amenable graphs $G$ with $0< p_c< p_u=1$.
Let $\mathcal{T}$ be the geodesic spanning tree of $G$ in Lemma {\mathrm{Re}}f{brgr}. Note the basic property of trees indicates that the lower growth rate is no less than its branching number. Then by {\mathrm{e}}qref{ppercolation} and \cite[Theorem~5.15]{LP2017}, almost surely, when $p_c< p\rightarrow p_u=1$,
\[
\sup\limits_{\sigma \in \mathcal{T}} \underline{\rm gr}\left(K_{\sigma}(\omega_{p,\mathcal{T}})\right)\geq \sup\limits_{\sigma \in \mathcal{T}} {\rm br}\left(K_{\sigma}(\omega_{p,\mathcal{T}})\right)= p\cdot {\rm br}(\mathcal{T})\rightarrow
{\rm br}(\mathcal{T})={\rm gr}(G),
\]
\[\sup\limits_{\sigma \in G} \underline{\rm gr}\left(K_{\sigma}(\omega_p)\right) \geq \sup\limits_{\sigma \in G} \underline{\rm gr}\left(K_{\sigma}(\omega_{p,\mathcal{T}})\right)\rightarrow K \geq {\rm br}(\mathcal{T});\]
where $\omega_{p,\mathcal{T}}$ is the restriction of Bernoulli percolation $\omega_p$ to $\mathcal{T}.$ Together with $\sup\limits_{\sigma \in G} \underline{\rm gr}\left(K_{\sigma}(\omega_p)\right)\leq {\rm gr}(G)$, we obtain that almost surely, as $p_c< p\rightarrow 1$,
\begin{eqnarray}\label{rate-limit}
\sup\limits_{\sigma \in G} \underline{\rm gr}\left(K_{\sigma}(\omega_{p})\right)\rightarrow {\rm br}(\mathcal{T})={\rm gr}(G).
{\mathrm{e}}nd{eqnarray}
Recall that a set of subgraphs of $G$ is automorphism-invariant if and only if it is invariant under action of any element of ${\rm Aut}(G)$; and a percolation measure ${\mathbb{P}}$ has indistinguishable infinite clusters if and only if for any automorphism-invariant property $\mathcal{A}$ of $subgraphs$, ${\mathbb{P}}$ a.s. either all infinite clusters satisfy $\mathcal{A}$, or all infinite clusters don't satisfy $\mathcal{A}$. Note that Lyons and Schramm \cite{RO1999}
proved that on any quasi-transitive unimodular graph, an insertion-tolerant invariant bond percolation has indistinguishable infinite clusters.
Since the (lower) volume growth rate is an automorphism-invariant property and Cayley graph is a class of transitive and unimodular graphs, we have that ${\mathbb{P}}$ almost surely, all open infinite clusters have the same lower volume growth rate. This together with {\mathrm{e}}qref{rate-limit} imply that Conjecture {\mathrm{Re}}f{conj-Lalley} holds in the mentioned case.
\begin{thebibliography}{999}
\bibitem{AB1987}
M. Aizenman, D. J. Barsky. (1987). Sharpness of the phase transition in percolation models. {\it Comm. Math. Phys.} {\bf 108}, 489-526.
\bibitem{AE2008}
E. A\"{\i}d\'ekon. (2008). Transient random walks in random environment on a Galton-Watson tree. {\it Probab. Th. Rel. Fields.} {\bf 142(3-4)}, 525-559.
\bibitem{AKN1987}
M. Aizenman, H. Kesten, C. M. Newman. (1987). Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. {\it Comm. Math. Phys.} {\bf 111}, 505-531.
\bibitem{AHL1971}
V. Ambegoakar, B. I. Halperin and J. S. Langer. (1971). Hopping conductivity in disordered systems. {\it Phys. Rev. B.} {\bf 4}, 2612-2620.
\bibitem{ABDH2013}
S. Andres, M. T. Barlow, J.-D. Deuschel, B. M. Hambly. (2013). Invariance principle for the random conductance model. {\it Probab. Th. Rel. Fields.} {\bf 156}, 535-580.
\bibitem{BBCEM2006}
V. Baldazzi, S. Bradde, S. Cocco, E. Marinari and R. Manasson. (2006). Inference of DNA sequences from mechanical unzipping: an ideal-case study. {\it Phys. Rev. Lett.} {\bf 96(12)}, 128102.
\bibitem{BBCEM2007}
V. Baldazzi, S. Bradde, S. Cocco, E. Marinari and R. Manasson. (2007). Inferring DNA sequences from mechanical unzipping data: the large-bandwidth case. {\it Phys. Rev. E.} {\bf 75}, 011904.
\bibitem{MTB2004}
M. T. Barlow. (2004). Random walks on supercritical percolation clusters. {\it Ann. Probab.} {\bf 32(4)}, 3024-3084.
\bibitem{BD2010}
M. T. Barlow, J. D. Deuschel (2010). Invariance principle for the random conductance model with unbounded conductances. {\it Ann. Probab.} {\bf 38(1)}, 234-276.
\bibitem{BG-FA2016}
G. Ben Arous, A. Fribergh. (2016). Biased random walks on random graphs. {\it Probability and statistical physics in St. Petersburg}, 99-153.
Proc. Sympos. Pure Math., {\bf 91}, Amer. Math. Soc., Providence, RI.
\bibitem{BS1996}
I. Benjamini, O. Schramm. (1996). Percolation beyond ${\mathbb{Z}}^d$, many questions and a few answers. {\it Electron. Comm. Probab.} {\bf 1(8)}, 71-82.
\bibitem{BB2007}
N. Berger, M. Biskup. (2007). Quenched invariance principle for simple random walk on percolation clusters. {\it Probab. Th. Rel. Fields.} {\bf 137(1-2)}, 83-120.
\bibitem{BA-SA-1985}
A. Berretti, A. D. Sokal. (1985). New Monte Carlo method for the self-avoiding walk. {\it J. Stat. Phys.} {\bf 40(3)}, 483-531.
\bibitem{MB2011}
M. Biskup. (2011). Recent progress on the random conductance model. {\it Probab. Surv.} {\bf 8}, 294-373.
\bibitem{BP2007}
M. Biskup, T. M. Prescott. (2007). Functional CLT for random walk among bounded conductances. {\it Electron. J. Probab.} {\bf 12(49)}, 1323-1348.
\bibitem{BR2017}
M. Biskup and P. F. Rodriguez. (2018). Limit theory for random walks in degenerate time-dependent random environments. {\it J. Funct. Anal.} {\bf 274(4)}, 985-1046.
\bibitem{BE2014}
E. Breuillard. (2014). Diophantine geometry and uniform growth of finite and infinite groups. {\it Proc. ICM2014-Seoul.} {\bf III}, 27-50.
Kyung Moon Sa Co. Ltd.
\bibitem{BH1957}
S. R. Broadbent, J. M. Hammersley. (1957). Percolation processes I. Crystals and mazes. {\it Proc. Cambridge Philos. Soc.} {\bf 53}, 629-641.
\bibitem{BK1989}
R. M. Burton, M. Keane. (1989). Density and uniqueness in percolation. {\it Comm. Math. Phys.} {\bf 121}, 501-505.
\bibitem{AAC1967}
A. A. Chernov. (1967). Replication of a multicomponent chain by the ``lightning" mechanism. {\it Biophysics.} {\bf 12(2)}, 336-341.
\bibitem{FMS1998}
F. Comets, M. Menshikov, S. Popov. (1998). Lyapunov functions for random walks and strings in random environment. {\it Ann. Probab.} {\bf 26(4)}, 1433-1445.
\bibitem{MTF2019}
M. Deijfen, T. Hiescher, F. Lopes. (2019). Competing frogs on ${\mathbb{Z}}^d$ (\url{https://arxiv.org/abs/1902.01849}). arXiv:1902.01849 [math.PR].
\bibitem{DFGW1989}
A. De Masi, P. A. Ferrari, S. Goldstein, and W. D. Wick. (1989). An invariance principle for reversible Markov processes - Applications to random motions in random environments. {\it J. Stat. Phys.} {\bf 55}, 787-855.
\bibitem{HD2017}
H. Duminil-Copin. (2018). Sixty years of percolation. {\it Proc. ICM2018. Rio de Janeiro.} Vol. {\bf 3}, 2817-2847. SBM \& World Scientific. \url{https://eta.impa.br/dl/086.pdf}
\bibitem{HD2018}
H. Duminil-Copin. (2018). Introduction of Bernoulli percolation. (\url{http://www.ihes.fr/~duminil/publi/2017percolation.pdf}).
\bibitem{DGRSY2018}
H. Duminil-Copin, S. Goswami, A. Raoufi, F. Severo and A. Yadin. (2018). Existence of phase transition for percolation using the Gaussian Free Field (\url{https://arxiv.org/abs/1806.07733}). arXiv: 1806.07733. [math. PR].
\bibitem{FM2019}
A. Faggionato, H. A. Mimun. (2019). Left-right crossings in the Miller-Abrahams random resistor network on a poisson point process (\url{https://arxiv.org/abs/1912.07482}). arXiv:1912.07482v1 [math.PR].
\bibitem{GF2011}
G. Faraud. (2011). A central limit theorem for random walk in random environment on marked Galton-Watson trees. {\it Electron. J. Probab.} {\bf 16}, 174-215.
\bibitem{FDTT2007}
M. Franceschetti, O. Dousse, D. N. C. Tse, P. Thiran. (2007). Closing the gap in the capacity of wireless networks via percolation theory. {\it IEEE Trans. Inform. Theory.} {\bf 53(3)}, 1009-1018.
\bibitem{HF1945}
H. Freudenthal. (1945). \"{U}ber die Enden diskreter R\"{a}ume und Gruppen. {\it Comment. Math. Helv.} {\bf 17}, 1-38.
\bibitem{GKN1992}
A. Gandolfi, M. S. Keane, C. M. Newman. (1992). Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. {\it Probab. Th. Rel. Fields.} {\bf 92(4)}, 511-527.
\bibitem{GG1999}
G. R. Grimmett. (1999). {\it Percolation} (2nd edition). Springer.
\bibitem{GG2014}
G. R. Grimmett. (2014). Criticality, universality, and isoradiality. {\it Proc. ICM2014-Seoul.} Vol. {\bf IV}, 25-47. Kyung Moon Sa, Seoul.
\bibitem{GHY1993}
G. R. Grimmett, H. Kesten, Y. Zhang. (1993). Random walk on the infinite cluster of the percolation model. {\it Probab. Th. Rel. Fields.} {\bf 96}, 33-44.
\bibitem{MG1981}
M. Gromov. (1981). Groups of polynomial growth and expanding maps. {\it Inst. Hautes \'{E}tudes Sci. Publ. Math.} {\bf 53}, 53-73.
\bibitem{AH2002}
A. Hatcher. (2002). {\it Algebraic topology.} Cambridge University Press.
\bibitem{HP1998}
O. H\"aggstr\"om, R. Pemantle. (1998). First passage percoaltion and a model for competing spatial growth. {\it J. Appl. Probab.} {\bf 35}, 683-692.
\bibitem{HP2000}
O. H\"aggstr\"om, R. Pemantle. (2000). Absence of mutual unbounded growth for almost all parameter values in the two-type Richardson model. {\it Stoc. Proc. Appl.} {\bf 20}, 207-222.
\bibitem{TH1960}
T. E. Harris. (1960). A lower bound for the critical probability in a certain percolation process. {\it Proc. Cambridge
Philos. Soc.} {\bf 56}, 13-20.
\bibitem{HS1990}
T. Hara, G. Slade. (1990). Mean-field critical behaviour for percolation in high dimensions. {\it Comm. Math.
Phys.} {\bf 128}, 333-391.
\bibitem{HSV1967}
C. R. Heathcote, E. Seneta, D. Vere-Jones. (1967). A refinement of two theorems in the theory of branching processes. {\it Theory Probab. Appl.} {\bf 12}, 297-301.
\bibitem{HH2015}
H. A. Helfgott. (2015). Growth in groups: ideas and perspectives. {\it Bull. Amer. Math. Soc.} {\bf 52(3)}, 357-413.
\bibitem{HH2016}
M. Heydenreich, R. van der Hofstad. (2017). {\it Progress in high-dimensional percolation and random graphs.} CRM Short Courses. Springer, Cham; Centre de Recherches Math\'{e}matiques, Montreal, QC.
\bibitem{MT2014}
M. Holmes, T. S. Salisbury. (2014). Random walks in degenerate random environments. {\it Canad. J. Math.} {\bf 66(5)}, 1050-1077.
\bibitem{MT2017}
M. Holmes, T. S. Salisbury. (2017). Conditions for ballisticity and invariance principle for random walk in non-elliptic random environment. {\it Electron. J. Probab.} {\bf 22(81)}, {1-18}.
\bibitem{HFR2009}
J. M. Huguet, N. Forns and F. Ritort. (2009). Statistical properties of metastable intermediates in DNA unzipping. {\it Phys. Rev. Lett.} {\bf 103(24)}, 248106.
\bibitem{WLE2018}
W. Huss, L. Levine, E. S. Huss. (2018). Interpolating between random walk and rotor walk. {\it Random Struct. Algorithms.} {\bf 52}, 263-282.
\bibitem{HT2019}
T. Hutchcroft. (2019). Percolation on hyperbolic graphs. {\it Geom. Funct. Anal.} {\bf 29(3)}, 766-810.
\bibitem{HT2020}
T. Hutchcroft. (2020). Non-uniqueness and mean-field criticality for percolation on nonunimodular transitive graphs. {\it J. Amer. Math. Soc.} (to appear).
arXiv:1711.02590v3 [math.PR] (\url{https://arxiv.org/pdf/1711.02590.pdf}).
\bibitem{VJ1994}
V. Jikov, S. Kozlov, O. Oleinik. (1994). {\it Homogenization of differential operators and integral functionals.} Springer.
\bibitem{HK1980}
H. Kesten. (1980). The critical probability of bond percolation on the square lattice equals $1/2$. {\it Comm. Math. Phys.} {\bf 74}, 41-59.
\bibitem{MVK1985}
M. V. Kozlov. (1985). The method of averaging and walks in inhomogeneous environments. {\it Russ. Math. Surv.} {\bf 40}, 73-145.
\bibitem{KSJW2002}
S. Koch, A. Shundrovsky, B. C. Jantzen and M. D. Wang, (2002). Probing protein-DNA interactions by unzipping a single DNA double helix. {\it Biophysical J.} {\bf 83 (2)}, 1098-1105.
\bibitem{RK1983}
R. K\"unnemann. (1983). The diffusion limit for reversible jump processes on ${\mathbb{Z}}^d$ with ergodic random bond conductivities. {\it Comm. Math. Phys.} {\bf 90(1)}, 27-68.
\bibitem{GL1982}
G. F. Lawler. (1982). Weak convergence of a random walk in a random environment. {\it Comm. Math. Phys.} {\bf 87}, 81-87.
\bibitem{SL2006}
S. P. Lalley. (2006). The weak/strong survival transition on trees and nonamenable graphs. {\it Proc. ICM2006.} Vol. {\bf III}, 637-647. Eur. Math. Soc., Z\"{u}rich.
\bibitem{JL1960}
J. Lamperti. (1960). Criterion for recurrence or transience of stochastic process. {\it J. Math. Anal. Appl.} {\bf 1}, 314-330.
\bibitem{LG-SA1988}
G. F. Lawler, A. D. Sokal. (1988). Bounds on the $L^2$ spectrum for Markov chains and Markov processes: a generalization of Cheeger's inequality. {\it Tran. Amer. Math. Soc.} {\bf 309(2)}, 557-580.
\bibitem{JG2006}
J.-F. Le Gall. (2006). {\it Int\'egration, probabilit\'es et processus al\'eatoires.} Cambridge University Press.
\bibitem{LS2020}
Y. Liu, V. Sidoravicius, L. Wang, K. Xiang. (2020). The invariance principle and the large deviation for the biased random walk on ${\mathbb{Z}}^d$. {\it J. Appl. Prob.} {\bf 57}, 295-313.
\bibitem{RL1990}
R. Lyons. (1990). Random walks and percolation on trees. {\it Ann. Probab.} {\bf 18(3)}, 931-958.
\bibitem{RL1992}
R. Lyons. (1992). Random walk, capacity and percolation on trees. {\it Ann. Probab.} {\bf 20}, 2043-2088.
\bibitem{RL1995}
R. Lyons. (1995). Random walks and the growth of groups. {\it C. R. Acad. Sci. Paris S\'{e}r. I Math.} {\bf 320(11)}, 1361-1366.
\bibitem{LP1992}
R. Lyons, R. Pemantle. (1992). Random walk in random environment and first-passage percolation on trees. {\it Ann. Probab.} {\bf 20(1)}, 125-136.
\bibitem{RO1999}
R. Lyons, O. Schramm. (1999). Indistinguishability of percolation clusters. {\it Ann. Probab.} {\bf 27(4)}, 1809-1836.
\bibitem{LP2017}
R. Lyons, Y. Peres. (2016). {\it Probability on trees and networks.} Cambridge University Press.
\bibitem{MP2007}
P. Mathieu, A. L. Piatnitski. (2007). Quenched invariance principles for random walks on percolation clusters. {\it Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.} {\bf 463}, 2287-2307.
\bibitem{MA1960}
A. Miller and E. Abrahams. (1960). Impurity conduction at low concentrations. {\it Phys. Rev.} {\bf 120}, 745-755.
\bibitem{MV1986}
M. V. Menshikov. (1986). Coincidence of critical points in percolation problems. {\it Dokl. Akad. Nauk SSSR}. {\bf 288}, 1308-1311.
\bibitem{MSA2015}
M. Menshikov, S. Popov, A. Wade. (2015). {\it Non-homogeneous random walks. Lyapunov function methods for near-critical stochastic systems.} Cambridge University Press.
\bibitem{NS1981}
C. M. Newman, L. S. Schulman. (1981). Infinite clusters in percolation models. {\it J. Stat. Phys.} {\bf 26(3)}, 613-628.
\bibitem{RY1995}
R. Pemantle, Y. Peres. (1995). Critical random walk in radom environment on trees. {\it Ann. Probab.} {\bf 23(1)}, 105-140.
\bibitem{POF2013}
M. Pollak, M. Ortu${\rm \tilde{n}}$o and A. Frydman. (2013). {\it The electron glass.} Cambridge University Press.
\bibitem{RD1994}
D. Randall. (1994). {\it Counting in lattices: combinatorial problems for statistical mechanics.} PhD thesis, University of California, Berkeley.
\bibitem{DR1973}
D. Richardson. (1973). Random growth in a tessellation. {\it Proc. Cambridge Phil. Soc.} {\bf 74,} 515-528.
\bibitem{RH2017}
F. Robert, R. van der Hofstad. (2017). Mean-field behavior for nearest-neighbor percolation in $d>10$. {\it Electron. J. Probab.} {\bf 22}, 43-65.
\bibitem{SSSWX2018}
Z. Shi, V. Sidoravicius, H. Song, L. Wang, K. Xiang. (2020). Uniform spanning forests on biased Euclidean lattices. {\it Ann. Inst. H. Poincar\'{e} Probab. Statist.}
Minor revision required. (arXiv version: (2018). Uniform spanning forests associated with biased random walks on euclidean lattices. \url{https://arxiv.org/abs/1805.01615})
\bibitem{CS2004}
C. Sabot. (2004). Ballistic random walks in random environment at low disorder. {\it Ann. Probab.} {\bf 32(4)}, 2996-3023.
\bibitem{SS2004}
V. Sidoravicius, A. S. Sznitman. (2004). Quenched invariance principles for walks on clusters of percolation or among random conductances. {\it Probab. Th. Rel. Fields.} {\bf 129(2)}, 219-244.
\bibitem{YS1982}
Y. G. Sinai. (1982). The limit behavior of a one-dimensional random walk in a random environment. {\it Teor. Veroyatn. Primen.} {\bf 27(2)}, 247-258.
\bibitem{SA-JM1989}
A. J. Sinclair, M. R. Jerrum. (1989). Approximate counting, uniform generation and rapidly mixing Markov chains. {\it Information and Computation.} {\bf 82}, 93-133.
\bibitem{AS2004}
A. S. Sznitman. (2004). Topics in random walks in random environment. In G. F. Lawler, editor, {\it School and Conference on Probability Theory}, ICTP Lect. Notes. {\bf XVII}, 203-266. Abdus Salam Int. Cent. Theoret. Phys., Trieste. Expanded lecture notes from the school and conference held in Trieste, May 2002.
\bibitem{SZ1999}
A. S. Sznitman, M. Zerner. (1999). A law of large numbers for random walks in random environment. {\it Ann. Probab.} {\bf 27(4)}, 1851-1869.
\bibitem{FS1975}
F. Solomon. (1975). Random walks in a random environment. {\it Ann. Probab.} {\bf 3(1)}, 1-31.
\bibitem{CW1967}
C. T. C. Wall. (1967). Poincar\'{e} complexes: I. {\it Ann. Math.} {\bf 86(2)}, 213-245.
\bibitem{OZ2006}
O. Zeitounni. (2006). Random walks in random envirionment. {\it Lectures on probability theory and statistics.} {\bf 1837}, 189-312. Springer.
{\mathrm{e}}nd{thebibliography}
{\mathrm{e}}nd{document} | math | 163,643 |
\begin{document}
\title{Polarization Entangled {\it W} State using Parametric Down-Conversion}
\author{Takashi Yamamoto}
\email{[email protected]}
\author{Kiyoshi Tamaki}
\author{Masato Koashi}
\author{Nobuyuki Imoto}
\homepage{http://www.soken.ac.jp/quantum/index.html}
\address{CREST Research Team for Interacting Carrier Electronics, School of Advanced Sciences,\\ The Graduate University for
Advanced Studies (SOKENDAI), Hayama, Kanagawa, 240-0193, Japan
}
\date{\today}
\begin{abstract}
An experimental scheme for preparing a polarization entangled {\it W} states from four photons emitted by parametric down-conversion is proposed. We consider two different configurations and a method of improving the yield by using single photon sources. In the proposed scheme, one uses only linear optical elements and photon detectors, so that this scheme is feasible by current technologies.
\pacs{03.67.-a, 42.50.-p}
\end{abstract}
\maketitle
In the quantum information processing including many quantum protocols and quantum
computation \cite{N-H}, quantum entanglement plays a crucial rule.
Most of the quantum protocols concern with the bipartite
system, mainly because the nature of
multipartite entanglement has not been clarified yet.
Recently, however, the nature of multipartite entanglement, especially,
that of tripartite entanglement begins to be clarified.
In \cite{dur00}, D\"ur {\it et al.} have classified the tripartite
pure states based on the equivalence under stochastic LOCC (local operations and classical communication).
They showed that there are two different kinds of genuine
tripartite entanglement. One is Greenberger-Horne-Zeilinger (GHZ) states
\cite{ghz89}, which is represented, for example, as
\begin{eqnarray}
\ket{{\rm{GHZ}}}=\frac{1}{\sqrt{2}}\left(\ket{000}+\ket{111}\right)\,,
\label{eq-ghz}
\end{eqnarray}
where $\{\ket{0}, \ket{1}\}$ is the orthonormal basis for a qubit.
The other is {\it W} states, which is represented, for example, by
\begin{eqnarray}
\ket{W}=\frac{1}{\sqrt{3}}\left(\ket{001}+\ket{010}+\ket{010}\right)\,.
\label{eq-w}
\end{eqnarray}
These two states cannot be converted to each other by LOCC with nonzero success probability.
These states show different behavior when one of the qubits is discarded.
For three qubits in GHZ states, the remaining two qubits are completely unentangled. But, for {\it W} states, the remaining two qubits are still entangled.
Indeed it was shown that {\it W} states are optimal in the amount of such pairwise entanglement\cite{Koashi00}.
Many works have been devoted to the study of GHZ states in connection with Bell's theorem \cite{Mer90-1,GHSZ90,Mer90-2,Roy91}, and
violation of Bell inequalities are demonstrated experimentally \cite{zei00}.
Besides the fundamental studies of GHZ states, several applications of
these states such as the quantum teleportation \cite{Karlsson01},
the quantum secret sharing \cite{hillery99,Cleve99} and the
quantum key distribution protocol \cite{Kempe99,Dukin01} have been
proposed.
On the other hand, the study of {\it W} states has not been done until recently. For application, the quantum key distribution (QKD) with {\it W} states is proposed \cite{Jaewoo02}, and a {\it W}-class state is used for the optimal universal quantum cloning machine \cite{Buzek96,Buzek97,Gisin97,Buzek97-2,Bruss98,Murao99}.
In fundamental aspects, Cabello \cite{Cabello02} has illustrated some differences between the violation of local realism exhibited by {\it W} states and that by GHZ states.
The {\it W} states have a clearer prescription for selecting a pair of qubits to be subjected to a Bell's theorem test than the GHZ states have. Thus, not only for the purpose of the realization of some applications, but also for the fundamental interests, it is important to prepare {\it W} states experimentally.
\begin{figure}\label{fig:setup1}
\end{figure}
So far, several schemes for preparation of {\it W} states have been proposed. Zeilinger, Horne, and Greenberger proposed a scheme using third order nonlinearity for path entangled photons \cite{Zei97}. Guo and Zhang proposed a scheme for three entangled atoms via cavity quantum electrodynamics \cite{Guo02}.
In this paper, we propose an experimentally feasible scheme for preparing a polarization entangled {\it W} states. The scheme is composed of parametric down-conversion (PDC), linear optical elements, and photon detectors, so that our scheme is feasible by current technologies. In our proposal, there is no interference
between the photons passing through different paths, which makes it easy for us to align the optical elements and makes our system insensitive to fluctuations of optical path lengths.
In our scheme, we utilize four photons emitted collinearly from type-II PDC, which are in the following state,
\begin{eqnarray}
\ket{2}_{0{\rm H}}\ket{2}_{0{\rm V}},
\label{eq:1}
\end{eqnarray}
where $\ket{n}$ is the normalized $n$-photon number state.
The subscript numbers label the spatial modes, and ${\rm H}$ and ${\rm V}$ represent horizontal and vertical polarization modes, respectively. As shown in Fig.~\ref{fig:setup1}, these photons are split into four spatial modes (1, 2, 3, and 3$^\prime$) by beam splitters (BS$_k$, $k=1, 2, 3$), whose reflectivity and transmissivity are independent of polarization. The transformation by BS$_k$ is expressed by
\begin{eqnarray}
\ket{1}_{{\rm H}}&\to &r_k\ket{1}_{k{\rm H}}+t_k\ket{1}_{k^\prime {\rm H}},
\nonumber \\
\ket{1}_{{\rm V}}&\to &r_ke^{i\phi_{k}}\ket{1}_{k{\rm V}}+t_ke^{i\psi_{k}}\ket{1}_{k^\prime {\rm V}},
\nonumber \\
\ket{2}_{{\rm H}}&\to &r^2_k\ket{2}_{k{\rm H}}+t^2_k\ket{2}_{k^\prime {\rm H}}+2r_kt_k\ket{1}_{k{\rm H}}\ket{1}_{k^\prime {\rm H}},
\end{eqnarray}
and
\begin{eqnarray}
\ket{2}_{{\rm V}}&\to &r^2_ke^{2i\phi_{k}}\ket{2}_{k{\rm V}}+t^2_ke^{2i\psi_{k}}\ket{2}_{k^\prime {\rm V}} \nonumber \\
& &+2r_kt_ke^{i(\phi_{k}+\psi_{k})}\ket{1}_{k{\rm V}}\ket{1}_{k^\prime {\rm V}},
\end{eqnarray}
where $r_k$ and $t_k$ are the reflection and transmission coefficients of BS$_k$, respectively, which satisfy $|r_{k}|^2+|t_{k}|^2=1$. We assume that $r_k$ and $t_k$ are real, without loss of generality. Here $\phi_{k}$ and $\psi_{k}$ are the phase differences between mode ${\rm H}$ and ${\rm V}$ for reflected and transmitted photons, respectively. For simplicity, we omit the modes in the vacuum, using abbreviations such as $\ket{1}_{k{\rm V}}\ket{0}_{k^\prime{\rm V}} \to \ket{1}_{k{\rm V}}$. After these transformations, the phase offsets for the photons in mode 2 and 3 are compensated by birefringent phase shifters (BPS$_k$, $k=2,3$). The amount of compensation is chosen as
\begin{eqnarray}
\ket{1}_{{2\rm V}}\to e^{i(-\phi_{2}+\psi_{2}+\psi_{3})}\ket{1}_{{2\rm V}}
\end{eqnarray}
for BPS$_2$, and
\begin{eqnarray}
\ket{1}_{{3\rm V}}\to e^{i(-\phi_{3}+\psi_{3})}\ket{1}_{{3\rm V}}
\end{eqnarray}
for BPS$_3$.
After compensating these phase differences, we are only interested in the case where there is a single photon in each spatial mode ($1$, $2$, $3$, and $3^\prime$). If such a case is successfully selected, these photons are in the following state,
\begin{eqnarray}
\frac{1}{\sqrt{2}}(e^{i\phi_{1}}\ket{1}_{1{\rm V}}\ket{W_{{\rm V}}}_{233^\prime}+e^{i(\psi_{1}+\psi_{2}+\psi_{3})}\ket{1}_{1{\rm H}}\ket{W_{{\rm H}}}_{233^\prime})
\label{eq:2}
\end{eqnarray}
where $\ket{W_{{\rm V}}}_{233^\prime}$ and $\ket{W_{{\rm H}}}_{233^\prime}$ are the {\it W} states which can be written as
\begin{eqnarray}
\ket{W_{{\rm V}}}_{233^\prime}&\equiv &\frac{1}{\sqrt{3}}(\ket{1}_{2{\rm H}}\ket{1}_{3H}\ket{1}_{3^\prime {\rm V}}+\ket{1}_{2{\rm H}}\ket{1}_{3{\rm V}}\ket{1}_{3^\prime {\rm H}} \nonumber \\
& &+\ket{1}_{2{\rm V}}\ket{1}_{3{\rm H}}\ket{1}_{3^\prime {\rm H}}) \nonumber
\end{eqnarray}
and
\begin{eqnarray}
\ket{W_{{\rm H}}}_{233^\prime}&\equiv &\frac{1}{\sqrt{3}}(\ket{1}_{2{\rm V}}\ket{1}_{3{\rm V}}\ket{1}_{3^\prime {\rm H}}+\ket{1}_{2{\rm V}}\ket{1}_{3{\rm H}}\ket{1}_{3^\prime {\rm V}} \nonumber \\
& &+\ket{1}_{2{\rm H}}\ket{1}_{3{\rm V}}\ket{1}_{3^\prime {\rm V}}). \nonumber
\end{eqnarray}
The probability of obtaining the photons in the state of Eq.~(\ref{eq:2}) is $(2\sqrt{6}r_{1}t^3_{1}r_{2}t^2_{2}r_{3}t_{3})^2$. If we detect a single photon at the photon detector D$_{1{\rm V}}$ and the state is projected to $\ket{1}_{1{\rm V}}\ket{W_{{\rm V}}}_{233^\prime}$, we obtain three photons in the state $\ket{W_{{\rm V}}}_{233^\prime}$. Even if we detect a single photon at the photon detector D$_{1{\rm H}}$ and the state is projected to $\ket{1}_{1{\rm H}}\ket{W_{{\rm H}}}_{233^\prime}$, we can also obtain the state $\ket{W_{{\rm V}}}_{233^\prime}$ after rotating the polarization by $90^{\circ}$ in mode 2, 3, and 3$^\prime$. In this case, the maximum probability of obtaining the photons in the state $\ket{W_{{\rm V}}}_{233^\prime}$ is $3/32$ when we set $r_{1}^2=1/4$, $r_{2}^2=1/3$, and $r_{3}^2=1/2$.
\begin{figure}\label{fig:setup2}
\end{figure}
Although it is difficult to select the single photon in each spatial mode without destroying the photons, we can discard the photocounts caused by the non-{\it W} states if we are allowed to perform the postselection where we select the events of the photocounts in mode $2$, $3$, and $3^\prime$.
In practice, to implement our scheme experimentally, we have to pay attention to the errors and the efficiency of generating the photons in {\it W} states. The
errors in the selected state are mainly caused by generation of three photon pairs at PDC and the dark counts of photon detectors. In PDC, the photon pair generation rate per pulse $\gamma$ is approximately $10^{-4}$ in typical multi-photon experiments \cite{zei99,zei00,Zeilinger01,Zeilinger02}. The three-pair generation rate $O(\gamma^3)$ is approximately $10^{-4}$ lower than two-pair generation rate $O(\gamma^2)$. The dark counts of current photon detectors is quite low for multi-photon coincidence measurement, so that these errors are negligible.
(See also \cite{TYamamoto01} about this kind of errors.) To see whether the efficiency of generating three photons in the {\it W} state is acceptable, we compare the yield of the {\it W} state with that of GHZ states in \cite{zei99,zei00,Zeilinger01,Zeilinger02} where type-II PDC is also used for generating three photons. In the GHZ experiment, the probability of obtaining the photons in the GHZ state after generating two photon pairs is $3/8$. Compared with this probability, the yield of {\it W} states in our scheme is smaller by a factor $1/4$ . However, using stimulated PDC \cite{Bouwmeester01}, the four-photon generation rate can be 16 times higher than spontaneous PDC, which suggest that our proposal is experimentally feasible.
We can also consider another setup (scheme II) as shown
in Fig.~\ref{fig:setup2}.
In this scheme, after compensations similar to scheme I expressed by
\begin{eqnarray}
\ket{1}_{{2\rm V}}\to e^{i(-\phi_{1}-\phi_{2}+\psi_{1}+\psi_{3})}\ket{1}_{{2\rm V}}
\end{eqnarray}
and
\begin{eqnarray}
\ket{1}_{{3\rm V}}\to e^{i(-\phi_{3}+\psi_{3})}\ket{1}_{{3\rm V}},
\end{eqnarray}
we obtain the photons in the following state,
\begin{eqnarray}
\frac{1}{\sqrt{2}}(e^{i\phi_{1}}\ket{1}_{1{\rm V}}\ket{W_{{\rm V}}}_{233^\prime}+e^{i(\psi_{1}-\psi_{2}+\psi_{3})}\ket{1}_{1{\rm H}}\ket{W_{{\rm H}}}_{233^\prime})
\end{eqnarray}
with the probability $(2\sqrt{6}r^2_{1}t^2_{1}r_{2}t_{2}r_{3}t_{3})^2$. The maximum probability of obtaining these photons in the state {\it W} is $3/32$, which is the same as scheme I, when we set $r_{1}^2=r_{2}^2=r_{3}^2=1/2$. This scheme has an advantage that the maximum probability can be obtained by using only symmetric beam splitters.
\begin{figure}\label{fig:setup3}
\end{figure}
So far, we have assumed that the reflectivity and transmissivity of BS$_k$ are independent of polarization. If these depend on the polarization, the fidelity of the final state to the desired {\it W} state becomes lower.
In this case, scheme I and scheme II show slightly different behavior. Here, we represent the polarization-dependent reflection and transmission coefficient of BS$_k$ as $r_{kL}$ and $t_{kL}$, respectively, which satisfy $r_{kL}^2+t_{kL}^2=1$ where $L={\rm H},{\rm V}$. We also introduce the error factor $\delta _{k{\rm L}}$ defined by $\delta _{k{\rm L}}=r_{kL}^2-(r^{opt}_{k})^2$ where $r^{opt}_{k}$ is the optimal reflectivity,
namely
$(r^{opt}_{1})^2=1/4$, $(r^{opt}_{2})^2=1/3$, and $(r^{opt}_{3})^2=1/2$ in scheme I and $(r^{opt}_{1})^2=(r^{opt}_{2})^2=(r^{opt}_{3})^2=1/2$ in scheme II.
When $\delta _{k{\rm L}}$ are small, the fidelity ${\rm F_{I}}$ in scheme I and ${\rm F_{II}}$ in scheme II are given by
\begin{eqnarray}
{\rm F_{I}}&\approx &1-\frac{1}{24}(27\delta^2_{2}+16\delta^2_{3})+O(\delta _{k}^{3})
\end{eqnarray}
and
\begin{eqnarray}
{\rm F_{II}}&\approx &1-\frac{2}{9}[(2\delta_{1}+\delta_{2})^2 +3\delta^2_{3}]+O(\delta _{k}^{3}).
\end{eqnarray}
where $\delta _{k}=\delta _{k{\rm H}}- \delta _{k{\rm V}}$.
In scheme I, $\delta _{1}$ merely changes the amplitudes of $\ket{1}_{1{\rm V}}\ket{W_{{\rm V}}}_{233^\prime}$ and $\ket{1}_{1{\rm H}}\ket{W_{{\rm H}}}_{233^\prime}$ in Eq.~(\ref{eq:2}), so that this does not affect the fidelity unlike scheme II.
The use of a single photon source (SPS), which is currently being developed \cite{YYamamoto01,YYamamoto02}, will improve the rate of generating the photons in {\it W} states. An ideal SPS emits a single photon in a single mode at a desired time. In this case, we can start from only three photons in the state $\ket{2}_{0{\rm H}}\ket{1}_{0{\rm V}}$ (or $\ket{2}_{0{\rm V}}\ket{1}_{0{\rm H}}$ ). To prepare this initial state, three SPSs and a symmetric beam splitter (BS$_1$) are arranged as shown in Fig~\ref{fig:setup3} (SPS1 and SPS2 emit a photon in mode H and SPS3 emits a photon in mode V). The state at the output ports of BS$_1$ is $(\ket{2}\ket{0}+\ket{0}\ket{2})/\sqrt{2}$, so that we obtain three photons in the state $\ket{2}_{0{\rm H}}\ket{1}_{0{\rm V}}$ with probability $1/2$ under the condition that each SPS has emitted a photon.
After we transform these photons by BS$_2$ and BS$_3$, and in the case where there is one photon in each spatial mode, we can obtain the photons in the state $\ket{W_{{\rm V}}}_{233^\prime}$. The probability of obtaining the photons in the state $\ket{W_{{\rm V}}}_{233^\prime}$ after generating a photon from each SPS is $1/2(\sqrt{6}r_{2}t^2_{2}r_{3}t_{3})^2$ and the maximum of this probability is $3/32$, which is the same as above schemes, at $r_{2}^2=r_{3}^2=1/2$.
The generation rate of one photon from SPS is approximately $0.4$ per pulse \cite{YYamamoto02} so that three-photon generation rate is about $0.064$ per pulse, which is significantly larger than $\sim 10^{-8}$ per pulse for PDC \cite{zei99}. Since SPS and PDC currently achieve almost the same repetition rate, using SPS improves the rate of preparing the state {\it W}.
In our scheme, one can also prepare non-equally weighted states belonging to {\it W}-class. An example is the state used for the optimal universal quantum cloning machine via teleportation by three distant parties \cite{Bruss98},
\begin{eqnarray}
& &\sqrt{\frac{2}{3}}\ket{1}_{2{\rm H}}\ket{1}_{3H}\ket{1}_{3^\prime {\rm V}}
-\frac{1}{\sqrt{6}}\ket{1}_{2{\rm H}}\ket{1}_{3{\rm V}}\ket{1}_{3^\prime {\rm H}} \nonumber \\
& &-\frac{1}{\sqrt{6}}\ket{1}_{2{\rm V}}\ket{1}_{3{\rm H}}\ket{1}_{3^\prime {\rm H}}.
\end{eqnarray}
To prepare such states, one can generally include additional polarization dependent losses in mode $2$, $3$, and $3^\prime$ and adjust BPS$_k$ properly.
In summary, we have proposed simple schemes for preparing the the photons in {\it W} states by using parametric down-conversion, linear optical elements, and photon detectors.
The schemes are easy to implement and feasible by current technologies. Our schemes can be improved by using single photon sources to obtain a higher rate of generating the photons.
We thank K. Nagata, J. Shimamura, and S. K. \"Ozdemir for helpful discussions.
\end{document} | math | 16,091 |
\begin{document}
\title[Noncoercive quasilinear equations]{Noncoercive quasilinear elliptic operators with singular lower order terms}
\author[Farroni, Greco, Moscariello, Zecca]{Fernando Farroni, Luigi Greco,\\Gioconda Moscariello and Gabriella Zecca}
\address{Dipartimento di Ingegneria Elettrica e delle Tecnologie dell'Informazione, Universit\`a degli Studi di Napoli ``Federico~II'', Via Claudio~21, 80125 Napoli, Italy}\email{[email protected]}
\address{Dipartimento di Matematica e Applicazioni ``R. Caccioppoli'', Universit\`a degli Studi di Napoli ``Federico~II'', Via Cintia, 80126 Napoli, Italy}\email{[email protected]} \email{[email protected]} \email{[email protected]}
\thanks{The authors are members of Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni (GNAMPA) of INdAM. The research of G.M. has been partially supported by the National Research Project PRIN \lq\lq Gradient flows, Optimal Transport and Metric Measure Structures'', code~2017TEXA3H}
\keywords{Dirichlet problems, Noncoercive elliptic operators, Obstacle problems}
\subjclass[2020]{35J25, 35J87}
\maketitle
\begin{abstract}
We consider a family of quasilinear second order elliptic differential operators which are not coercive and are defined by functions in Marcinkiewicz spaces. We prove the existence of a solution to the corresponding Dirichlet problem. The associated
obstacle problem is also solved. Finally, we show higher integrability of a solution to the Dirichlet problem when the datum is more regular.
\end{abstract}
\section{Introduction}
Given a bounded domain
$\Omegaega$ of $\mathbb R^N$, $N\ge 2$, we consider
\[A\colon (x,u,\xi)\in \Omegaega\times {\mathbb R}\times \mathbb R^N\quad\mappaa\quad \mathbb R^N\]
a Carath\'eodory vector field (i.e.~measurable in $x\in \Omegaega$ and continuous in $(u,\xi)\in {\mathbb R}\times \mathbb R^N$)
satisfying for a.e.\ $x\in\Omega$ and for every $u\in{\mathbb R}$ and $\xi,\eta\in\reale^N$ the following structural conditions:
\begin{equation}\label{coercivita}
\langle A(x,u,\xi),\xi\rangle\geqslant \alpha|\xi|^p-\big(b(x)\,|u|\big)^p-\varphi(x)^{p}
\end{equation}
\begin{equation}\label{limitatezza}
|A(x,u,\xi)|\leqslant \beta|\xi|^{p-1}+\big(b(x)\,|u|\big)^{p-1}+\varphi(x)^{p-1}
\end{equation}
\begin{equation}\label{monotonia}
\langle A(x,u,\xi)- A (x,u,\eta ), \xi -\eta \rangle>0\qquad \text{for }\xi\ne\eta
\end{equation}
where $0< \alpha < \beta $ are positive constants, $1<p<N$, and $b$ and $\varphi$ are positive functions verifying
$b\in L^{N,\infty}(\Omegaega)$ and $\varphi\in L^{p}(\Omegaega)$. In view of Sobolev embedding theorem in Lorentz spaces \cite{O,A,GM}, by \eqref{limitatezza} and the assumptions on $b$ and $\varphi$, for each $u\in\SobpO$ we have
\[A(x,u,\nabla u)\in L^{p'}(\Omegaega,\mathbb R^N)\]
Hence, we can define the quasilinear elliptic distributional operator
\begin{equation}\label{loperatore}
-\mathop{\mathrm{div}} A(x,u,\nabla u)
\end{equation}
setting for any $w\in \SobpO$
\begin{equation}
\langle -\mathop{\mathrm{div}} A(x,u,\nabla u),w\rangle =\int_\Omegaega \langle A(x,u,\nabla u),\nabla w\rangle \,\mathrm d x\,.
\end{equation}
Given $\Phi\in\Sobdual{\Omegaega}$, we study the Dirichlet problem
\begin{equation}\label{Dirichlet}
\left\{\begin{array}{c}
-\mathop{\mathrm{div}} A(x,u,\nabla u)= \Phi
\\
u\in \SobpO
\end{array}
\right.
\end{equation}
By a solution to Problem \eqref{Dirichlet} we mean a function $u \in W_0^{1,p}(\Omegaega)$ such that
\begin{equation}\label{soluzint}
\int_\Omegaega A(x,u,\nabla u)\nabla w \,\mathrm d x
=
\langle
\Phi , w
\rangle\,,
\qquad \forall w \in C^\infty_0(\Omegaega)\,,
\end{equation}
where $\langle\cdot,\cdot \rangle$ denotes the duality product of
$ W ^{-1,p^\prime}(\Omegaega)$
and
$ W_0^{1,p}(\Omegaega)$. Clearly, \eqref{soluzint} extends to all $w \in W_0^{1,p}(\Omegaega)$.\par
Our structural conditions allow for a singular lower order term depending on $u$. As an example, we consider the following operator
\begin{equation}\label{modello}
A(x,u,\xi):=\langle \mathcal H(x)\xi , \xi \rangle ^{\frac {p-2} 2} \mathcal H(x) \xi + B(x) |u|^{p-2}u
\end{equation}
with $1<p<N$. Here $\mathcal H=\mathcal H(x)\colon \Omegaega \rightarrow \mathbb R^{N \times N}$ is a symmetric, bounded matrix field of $\mathbb R^{N \times N}$ such that
\[
\langle \mathcal H(x)\xi , \xi \rangle \geq \alpha |\xi|^2 \quad \text{for a.e. $x \in \Omegaega$ and for all }\xi \in \mathbb R^N\,,
\]
for a given $\alpha>0$.
The vector field $B\colon \Omegaega \rightarrow \mathbb R^{N}$ is a measurable function satisfying $|B(x)|\leqslant (b(x))^{p-1}$ for a.e.\ $x \in \Omegaega$ and for some $b \in L^{N, \infty }(\Omegaega)$.
The feature of Problem \eqref{Dirichlet} is the lack of coercivity for the operator \eqref{loperatore}
and the singularity in the lower order term due to property of $b$.
It is well known that, if the operator in (1.4)-(1.5) is coercive, then a solution to problem (1.6) exists. It can for instance be shown by monotone operator theory \cite{LL,HB B,Br,BBM}.
On the other hand,
the existence of a bounded solution
can be expected when $\Phi$ and $b$ are sufficiently smooth. For example, in the model case and
even for the corresponding linear case,
a solution to
Problem \eqref{Dirichlet} is bounded whenever $\Phi$ and $b$ are in $W^{-1,p^\prime }(\Omegaega)$ and
$L^{p^\prime }(\Omegaega,\mathbb R^N)$, respectively, with $p^\prime > N/(p-1)$ (see \cite{S,GMZ}).\par
The space $L^{N,\infty}(\Omegaega)$ is slightly larger than
$L^{N}(\Omegaega)$. Nevertheless, there are essential differences between the case $b \in L^{N}(\Omegaega)$
(\cite{BoBumi,Droniou}), or even $b \in L^{N,q}(\Omegaega)$ (\cite{M1})
with $ N \leqslant q < \infty$, and the case $b\in L^{N,\infty}(\Omegaega)$.
In $ L^{N,\infty}(\Omegaega)$ bounded functions are not dense. Furthermore, in $ L^{N,\infty}(\Omegaega)$
the norm is not absolutely continuous, namely a function can have large norm even if restricted to a set with small measure.
Our first result is the following
\begin{theorem}\label{main}
Let $\Phi\in\Sobdual{\Omegaega}$. Under the assumptions
\eqref{coercivita},
\eqref{limitatezza} and
\eqref{monotonia},
if
\begin{equation}\label{lndeboleinfty}
\mathrm{dist}_{L^{N,\infty}} (b,L^\infty (\Omegaega)) < \frac {\alpha^\frac1p}{S_{N,p}}
\end{equation}
then Problem~\eqref{Dirichlet} admits a solution.
\end{theorem}
Here $S_{N,p}$ denotes the Sobolev constant of Theorem
\ref{lorentz} below. As an illustration, we state the following immediate consequence. We denote by $\chiusura N(\Omegaega)$ the closure of $L^\infty(\Omegaega)$ in $L^{N,\infty}(\Omegaega)$.
\begin{corollary}\label{corollary}
Assume \eqref{coercivita},
\eqref{limitatezza} and
\eqref{monotonia}, with $b\in \chiusura N(\Omegaega)$.
Then Problem~\eqref{Dirichlet} admits a solution, for every $\Phi\in\Sobdual{\Omegaega}$.
\end{corollary}
The closure $\chiusura N(\Omegaega)$ contains for example all Lorentz spaces $L^{N,q}(\Omegaega)$, for $1<q<+\infty$, see Subsection~\ref{funct sp}.
\par
It is not clear if the bound in \eqref{lndeboleinfty} is sharp.
However, existence of a solution could fail if
no restriction on the distance is imposed, even in the liner case,
as the Example
\ref{Ex1}
in Subsection \ref{example sec} shows
(see also \cite{GMZ3}).
Notice that condition \eqref{lndeboleinfty} does not imply the smallness of the norm of $b$ in
$L^{N,\infty}(\Omegaega)$ (see \cite{GMZ3}).
In the case $p=2$ existence results analogous to that of Theorem \ref{main} have been proved in
\cite{GGM,GMZ3,Z} and in \cite{DiG Z,RZ,Z2} when the principal part has a coefficient bound in BMO (i.e. the space of functions of bounded mean oscillation).
We explicitly remark that in this
context the operator \eqref{loperatore} has the same integrability properties of the principal part (see also~\cite{KK}).
The evolutionary counterpart of Problem~\eqref{Dirichlet}
has been studied in~\cite{FarMos}.
Other related results can be found in
\cite{AlfMo,
KTAIPENG,MV}.
An additional difficulty in proving Theorem~\ref{main} lies in the lack of compactness that the operator
\begin{equation}\label{lack}
u\in \SobpO\quad \quad\mappaa\quad \quad \big(b(x)\,|u|\big)^{p-1}\in L^{p'}(\Omegaega)
\end{equation}
exhibits in the case $b\in L^{N,\infty}(\Omegaega)$,
in contrast with the case $b\in L^{N}(\Omegaega)$
(see Example
\ref{Ex2}
in Subsection \ref{example sec}).
In order to overcome this issue, first we consider the case in which $b\in L^\infty(\Omegaega)$. Under this assumption, we deduce the existence of a solution to Problem~\eqref{Dirichlet} by means of Leray--Schauder fixed point theorem. The a priori estimate required follows from a lemma that could be interesting in itself (see
Lemma \ref{lemma weak compact}
below).
In order to reduce the general case $b\in L^{N,\infty}(\Omegaega)$ to
the previous one, we consider a sequence of approximating problems, defined essentially by truncating the vector field $A=A(x,u,\xi)$ in the $u$--variable.
A bound on the sequence of the solutions is achieved due to the assumption
\eqref{lndeboleinfty}.
We emphasize
that our result is also new when $b\in L^{N}(\Omegaega)$, in the sense that our approach allows us to treat the general nonlinear operator in
\eqref{Dirichlet}.
Finally, by testing the problems with a suitable admissible test functions, we show that the sequence of solutions to the approximating problems is compact and its limit is a solution to the original problem \eqref{Dirichlet}.
In Section~\ref{pb ostacolo}, we show that our approach is robust enough to handle also the corresponding obstacle problem.
We prove an existence result in the same spirit of \cite{GMZ2} (where the case $p=2$ is taken into account).
In Section~\ref{reg sol} we present a regularity result.
When
$b \in L^N(\Omegaega)$,
the study of the higher integrability of a solution to \eqref{Dirichlet}
has been developed in \cite{GG,Giustibook}
by using the theory of quasiminima. Local summability properties have been recently considered in
\cite{CDLP,KangKim}
in the linear case.
Here, following
\cite{GMZ},
we prove higher summability of a solution $u$ to \eqref{Dirichlet} according to that of the data $\Phi$ and $\varphi$.
\section{Preliminaries and examples}\label{section2}
\subsection{Notation and function spaces}\label{funct sp}
Let $\Omega$ be a bounded domain in $\mathbb R^N$.
Given $1<p<\infty$ and $1\leqslant q<\infty$, the Lorentz space $L^{p,q}(\Omega)$ consists of all measurable functions $f$ defined on $\Omegaega$ for which the quantity
\begin{equation}\label{norma Lorentz}
\|f\|_{p,q}^q=p\int_{0}^{+\infty} |\Omega_t|^{\frac{q}{p}}t^{q-1}\,\mathrm d t
\end{equation}
is finite, where $\Omega_t= \left\{ x\in \Omega: |f(x)|>t \right\}$ and $|\Omega_t|$ is the Lebesgue measure of $\Omega_t$, that is, $\lambda_f(t)=|\Omega_t|$ is the distribution function of $f$. Note that $\|\cdot\|_{p,q}$ is equivalent to a norm and $L^{p,q}$ becomes a Banach space when endowed with it (see \cite{O,bs,g}). For $p=q$, the Lorentz space
$L^{p,p}(\Omegaega)$ reduces to the Lebesgue space $L^p(\Omegaega)$. For $q=\infty$, the class $L^{p,\infty}(\Omegaega)$ consists of all measurable functions $f$ defined on $\Omegaega$ such that
\begin{equation*}\label{2.1}
\|f\|^p_{p,\infty}=\sup_{t>0}t^{ p}|\Omega_t|<+\infty
\end{equation*}
and it coincides with the Marcinkiewicz class, weak-$L^p(\Omegaega)$.
For Lorentz spaces the following inclusions hold
\[
L^r (\Omega)\subset L^{p,q}(\Omega)\subset L^{p,r} (\Omega) \subset L^{p,\infty}(\Omega)\subset L^q(\Omega),
\]
whenever $1\leqslant q<p<r\leqslant \infty.$ Moreover, for $1<p<\infty$, $1\leqslant q\leqslant \infty$ and $\frac 1 p+\frac 1 {p'}=1$, $\frac 1 q+\frac 1 {q'}=1$, if $f\in L^{p,q}(\Omegaega)$, $g\in L^{p',q'}(\Omegaega)$ we have the H\"{o}lder--type inequality
\begin{equation*}\label{holder}
\int_{\Omegaega}|f(x)g(x)|\,\mathrm d x \leqslant \|f\|_{p,q}\|g\|_{p',q'}.
\end{equation*}
As it is well known, $L^\infty(\Omega)$ is not dense in $L^{p,\infty}(\Omega)$.
For a function $f \in
L^{p,\infty} (\Omega)$ we define
\begin{equation}\label{dist}
\mathrm{dist}_{L^{p,\infty} (\Omega) } (f,L^\infty (\Omega)) =\inf_{g\in L^\infty(\Omega)} \|f-g\|_{L^{p,\infty}(\Omega)}.
\end{equation}
In order to characterize the distance in \eqref{dist},
we introduce for all $k>0$ the truncation operator
\begin{equation*}\label{201206272}
T_k(y):=\frac{y}{|y|}\min\{|y|, k\}\,.
\end{equation*}
It is easy to verify that
\begin{equation}\label{distlim}
\lim_{k\to\infty}\|f-T_kf\|_{p,\infty}
=
\mathrm{dist}_{L^{p,\infty} (\Omega) } (f,L^\infty (\Omega))\,.
\end{equation}
We denote by $\chiusura p(\Omegaega)$ the closure of $L^\infty(\Omegaega)$. We have (see \cite[Lemma~2.3]{GZ})
\begin{equation}\label{caratterizzazione chiusura}
f\in\chiusura p(\Omegaega)\iff \lim_{t\to+\infty}t\,[\lambda_f(t)]^{1/p}=0\,.
\end{equation}
Clearly, for $1\leqslant q<\infty$ we have $L^{p,q}(\Omegaega)\subset \chiusura p(\Omegaega)$, that is, any function in $L^{p,q}(\Omegaega)$ has vanishing distance zero to $L^\infty(\Omegaega)$. Indeed, $L^\infty(\Omegaega)$ is dense in $L^{p,q}(\Omegaega)$, the latter being continuously embedded into $L^{p,\infty}(\Omegaega)$. Actually, the inclusion also follows from \eqref{caratterizzazione chiusura}, since $\lambda_f(t)=|\Omegaega_t|$ is decreasing and hence the convergence of the integral at \eqref{norma Lorentz} implies the condition on the right of \eqref{caratterizzazione chiusura}.\par
Assuming the origin $0\in\Omegaega$, a typical element of $L^{N,\infty}(\Omega)$ is $b(x)=B/|x|$, with $B$ a positive constant. An elementary calculation shows that
\begin{equation}\label{esempio-dist}
\mathrm{dist}_{L^{N,\infty} (\Omega) } (b,L^\infty (\Omega))=B\,\omegaega_N^{1/N}
\end{equation}
where $\omegaega_N$ stands for the Lebesgue measure of the unit ball of $\reale^N$.\par\medbreak
The Sobolev embedding theorem in Lorentz spaces reads as
\begin{theorem}[\cite{O,A,GM}]\label{lorentz}
Let us assume that $1<p<N$, $1\leqslant q\leqslant p$, then every function $g\in W_0^{1,1}(\Omegaega)$ verifying $|\nabla g|\in L^{p,q}(\Omega)$ actually belongs to $L^{p^*,q}(\Omega)$, where $p^*=\frac{Np}{N-p}$ and
$$
\|g\|_{p^*,q}\leqslant S_{N,p}\|\nabla g\|_{p,q}
$$
where $S_{N,p}$ is the Sobolev constant.
\end{theorem}
\subsection{A version of the Leray--Schauder fixed point theorem}
We shall use the well known Leray--Schauder fixed point theorem in the following form (see \cite[Theorem 11.3 pg. 280]{gt}). A continuous mapping between two Banach spaces is called compact if the images of bounded sets are precompact.
\begin{theorem}\label{LerSch2}
Let $\mathcal F$ be a compact mapping of a Banach space $X$ into itself, and suppose there exists a constant $M$ such that $\|x\|_{X}<M$ for all $x\in X$ and $t\in [0,1]$ satisfying $x=t\mathcal F(x).$ Then, $\mathcal F$ has a fixed point.
\end{theorem}
\subsection{Critical examples}\label{example sec}
Our first example shows that the only assumption that $b \in L^{N,\infty} (\Omegaega)$
does not guarantee the existence of a solution to Problem \eqref{Dirichlet}.
\begin{example}\label{Ex1}
Let $\Omegaega$ be the unit ball. For $\frac N2<\gamma+1\leqslant N$, the problem
\begin{equation}\label{controesempio}
\left\{
\begin{array}{cl}
\displaystyle -\mathbb{D}elta u-\mathop{\mathrm{div}}\left(\gamma\,u\,\frac x{|x|^2}\right)=-\mathop{\mathrm{div}}\left(\frac x{|x|^{N-\gamma}}\right)&\text{ in }\Omegaega\\
\displaystyle u=0&\text{ on }\partial\Omegaega
\end{array}
\right.
\end{equation}
does not admit a solution.
Assume to the contrary that $u$ is a solution of \eqref{controesempio}.
In the right hand side of the equation we recognize that
\[\frac x{|x|^{N-\gamma}}=\nabla v(x)\,,\]
where $v\in W^{1,2}_0(\Omegaega)$
is given by
\[v(x)=\left\{
\begin{array}{ll}
\displaystyle \frac1{2-N+\gamma}\,(|x|^{2-N+\gamma}-1)&\text{ for }\gamma\not=N-2\\
\displaystyle \vrule width 0pt height 1.2em
\log|x|&\text{ for }\gamma=N-2
\end{array}
\right.\]
Moreover, $v$ solves the adjoint problem
\[\left\{
\begin{array}{cl}
\displaystyle -\mathbb{D}elta v+\gamma\,\frac x{|x|^2}\cdot \nabla v=0&\text{ in }\Omegaega\\
\displaystyle v=0&\text{ on }\partial\Omegaega
\end{array}
\right.\]
Testing the equation in \eqref{controesempio} by $v$ we have
\[\int_\Omegaega |\nabla v|^2\,\mathrm d x=0\,.\]
which readly implies $v \equiv 0$ in $\Omegaega$, which is clearly not the case. \qed
\end{example}
Next example shows that for the complete operator
\[\mathop{\mathrm{div}} A(x,u,\nabla u)+B(x,u,\nabla u)\]
in general we do not have existence, even in the linear case.
\begin{example}\label{operatore completo}
Let $\lambda$ be an eigenvalue of Laplace operator and $w$ a corresponding eigenfunction
\[\left\{
\begin{array}{ll}
\displaystyle -\mathbb{D}elta w=\lambda\, w\\
\displaystyle w\in\Sob2\Omegaega\setminus\{0\}
\end{array}
\right.\]
Then the equation
\[-\mathbb{D}elta u-\lambda\,u=w\]
has no solution of class $\Sob2\Omegaega$.
\end{example}
Our final example shows that
compactness of the operator
\eqref{lack} in the Introduction could fail.
\begin{example}\label{Ex2}
Assume $N\geqslant 2$ and $1<p<N$. Let $\Omegaega$ be the ball of $\mathbb R^N$ centered at the origin of radius $3$.
Our aim is to construct a sequence of functions
$\{
u_n
\}_{n\in\mathbb N}$
in $W^{1,p}_0(\Omegaega)$ and a function $b\in L^{N,\infty}(\Omegaega)$
such that $\{
\nabla u_n
\}_{n\in\mathbb N}$ is bounded in $L^p(\Omegaega,\mathbb R^N)$, however it is not possible to extract from $\{(b|u_n|)^{p-1} \}_{n\in\mathbb N}$ any subsequence strongly converging
$L^{p^\prime}(\Omegaega)$.
To this aim, let $$ b(x):=\frac 1 {|x|} $$ and
\[
\gamma : = 1 - \frac N p\,.
\]
We define a sequence
$\{
u_n
\}_{n\in\mathbb N}$
setting for $x \in \Omegaega$
\begin{equation}\label{controesempio2}
\begin{split}
u_1(x) & :=
\left\{
\begin{array}{cl}
1-2^\gamma &\text{if $|x|<1$}
\\
|x|^\gamma -2^\gamma &\text{if $1 \leqslant |x| < 2$}
\\
0 &\text{if $|x|\geqslant 2$}
\end{array}
\right.
\\
u_n (x)& : = n^{-\gamma} u_1 (nx) \qquad \text{for $n\geqslant 2$}
\end{split}
\end{equation}
Observe that
$u_n \in W^{1,p}_0(\Omegaega)$ since
\begin{equation}\label{contr2.1}
\begin{split}
| \nabla u_n(x) | =
\left\{
\begin{array}{cl}
|\gamma |
|x|^\gamma &\text{if $\frac 1 n \leqslant |x| < \frac 2 n$}
\\
0 &\text{otherwise}
\end{array}
\right.
\end{split}
\end{equation}
and
\begin{equation}
\| \nabla u_n\|^p_{L^p(\Omegaega)} = |\gamma|^p N \omegaega_N \log 2\,,
\end{equation}
where $\omegaega_N$ denotes the measure of the unit ball of $\mathbb R^N$.
In particular, $\| \nabla u_n\|^p_{L^p(\Omegaega)}$ is independent of $n$.
On the other hand, a direct calculation shows that
\begin{equation}
\begin{split}
\left\|
\left(
b|u_n|
\right)^{p-1}
\right\|^{p^\prime} _{L^{p^\prime} (\Omegaega)}
& =
\int_{|x|<\frac 3 n} (b |u_n|)^p \,\mathrm d x
\\
&
=\int_{|x|<\frac 1 n} (b |u_n|)^p \, dx + \int_{\frac 1 n \leqslant |x|<\frac 2 n} (b |u_n|)^p \,\mathrm d x
\\
&
=
N\omegaega_N
\left[
\frac{ (1-2^\gamma)^p}{N-p} + \int_1^2 r^{N-p} \left(r^\gamma-2^\gamma\right)^p \, \frac{\mathrm d r} r
\right]
\end{split}
\end{equation}
Hence, we see that the norm of $(b|u_n|)^{p-1}$ in $L^{p^\prime}(\Omegaega)$
is independent of $n$ as well
and strictly positive.
On the other hand, $(b |u_n|)^{p-1}\rightarrow 0$ pointwise in $\Omegaega$
and this readily implies that
there is no subsequence of
$\{ (b|u_n|)^{p-1} \}_{n\in \mathbb N}$ strongly converging in $L^{p^\prime}(\Omegaega)$.
\qed
\end{example}
\subsection{An elementary lemma}
\begin{lemma}\label{lemma elementare}
Assume $f_n\to f$ a.e. Moreover, let $g_n$, $n\in {\mathbb N}$, and $g$ in $L^q$, $1\leqslant q<+\infty$, verify $g_n\to g$ a.e.,
$|f_n|\leqslant g_n$ a.e., $\forall n\in{\mathbb N}$, and
\[\int_\Omegaega g_n^q\,\mathrm d x\to \int_\Omegaega g^q\,\mathrm d x\,.\]
Then $f_n,f\in L^q$ and
\[f_n\to f\text{ in }L^q\,.\]
\end{lemma}
It suffices to apply Fatou lemma to the sequence of nonnegative functions
\[F_n=2^{q-1}(g_n^q+g^q)-|f_n-f|^q\,.\]
\section{A weak compactness result}
The aim of this section is to establish a weak compactness criterion in the space $W^{1,p}_0(\Omegaega)$
that has an interest by itself.
\begin{lemma}\label{lemma weak compact}
Let $\mathcal B$ be a nonempty subset of $W^{1,p}_0(\Omegaega)$. Assume that there exists a constant $C>0$ such that
\begin{equation}\label{stima vkn3 bis}
\|\nabla u \|_{L^p(\Omegaega\setminus \Omegaega_\sigma)}^p\leqslant
C\left(
1+\| u \|_{L^p(\Omegaega\setminus \Omegaega_\sigma)}^p
\right)
\end{equation}
for any $\sigma>0$ and $u \in \mathcal B$, where $\Omegaega_\sigma:=\{x\in \Omegaega\colon\,|u(x)| \geqslant \sigma\}$. Then, there exists a constant $M>0$ such that
\begin{equation}\label{3.23quater}
\| u \|_{W^{1,p}(\Omegaega)}\leqslant M\,
\end{equation}
for any
$u \in \mathcal B$.
\end{lemma}
\begin{proof}
We argue by contradiction and assume $\mathcal B$ unbounded. Then we construct a sequence
$\{u_k\}_k$ in $\mathcal B$ such that
\[\| u_k \|:=\| \nabla u_k \|_{p}\rightarrow \infty\]
as $k\rightarrow \infty$.
By \eqref{stima vkn3 bis} we get, for any $k \in \mathbb N$ and $\varepsilon >0$
\begin{equation}\label{3.3ter}
\int_\Omegaega |\nabla T_{\varepsilon\|u_k\|}u_k|^p
\,\mathrm d x
\leqslant
C\left(
1+
\int_{\Omegaega} |u_k|^p\chi_{\{|u_k|<\varepsilon\|u_k\|\}} \,\mathrm d x
\right)
\end{equation}
We set
\[v_k=\frac{u_k}{\|u_k\|}\,.\]
Hence, there exists $v \in W^{1,p}_0(\Omegaega)$ such that (up to a subsequence)
$v_k\rightharpoonup v$ weakly in $\Sob p{}$, $v_k\to v$ strongly in $L^p$ and $v_k(x)\to v(x)$ for a.e.~$x\in \Omegaega$.
Notice that
\[\frac {T_{\varepsilon \|u_k\| } u_k } {\|u_k\|}= T_{\varepsilon} v_k\,,\]
thus $\nabla {T_{\varepsilon \|u_k\| } u_k }=0$ on the set $\{x\in\Omegaega:|v_k(x)|\geqslant \varepsilon\}$.
Dividing
\eqref{3.3ter}
by $\|u_k\|^{p}$ we have
\begin{equation}\label{stima vk ter}
\int_\Omegaega |\nabla T_{\varepsilon}v_k|^p \,\mathrm d x \leqslant
C\left(
\|u_k\|^{ -p} +
\int_{\Omegaega} |v_k|^p\chi_{\{|v_k|<\varepsilon \}} \,\mathrm d x
\right)
\end{equation}
Now, we let $k\to+\infty$. To this end, we note that $T_\varepsilon v_k\rightharpoonup T_\varepsilon v$ weakly in $\Sob p{\Omegaega}$ and $T_\varepsilon v_k\to T_\varepsilon v$ strongly in $L^p(\Omegaega)$. In the left hand side of \eqref{stima vk ter}, we use semicontinuity of the norm with respect to weak convergence, while in the right hand side we observe that $\|u_k\|^{-1}\to0$. Moreover, if
\begin{equation}\label{e eccezionale ter}
|\{x\in\Omegaega:|v(x)|=\varepsilon\}|=0\,,
\end{equation}
then we have $\chi_{\{|v_k|<\varepsilon\}}\to \chi_{\{|v|<\varepsilon\}}$ a.e.\ in $\Omegaega$ and hence
\[v_k\,\chi_{\{|v_k|<\varepsilon\}}\to v\,\chi_{\{|v|<\varepsilon\}}\]
strongly in $L^p$. Note that the set of values $\varepsilon>0$ for which \eqref{e eccezionale ter} fails is at most countable.
Thus, we end up with the following estimate
\begin{equation}\label{stima v ter}
\int_\Omegaega |\nabla T_{\varepsilon}v|^p \,\mathrm d x \leqslant C \int_{\Omegaega}|v|^p\chi_{\{|v|<\varepsilon\}} \,\mathrm d x
\end{equation}
Using Poincar\'e inequality in the left hand side, this yields
\[\varepsilon^p\,|\{x:|v|\geqslant\varepsilon\}|\leqslant C\,\varepsilon^p\,|\{x:0<|v|<\varepsilon\}|\,.\]
Passing to the limit as $\varepsilon\downarrow 0$ (assuming \eqref{e eccezionale ter}), we deduce
\[|\{x:|v|>0\}|=0\,,\]
that is, $v(x)=0$ a.e. Once we know that $v_k\rightharpoonup 0$ weakly in $\SobpO$, the above argument (formally with $\varepsilon=+\infty$, i.e.~without truncating $v_k$) actually shows that $v_k\to0$ strongly in $\SobpO$, compare with \eqref{stima v ter}, and this is not possible, as $\|v_k\|=1$, for all $k$.
\end{proof}
\section{Proof of Theorem \ref{main}}\label{Section3}
\subsection{The case of bounded coefficient}\label{coefficienti limitati}
In this subsection we assume $b\in L^\infty(\Omegaega)$. For a given function $v\in L^p(\Omegaega)$, we define the vector field on $\Omegaega\times\reale^N$
\begin{equation}\label{campoAv}
A_v(x,\xi):=A(x,v(x),\xi)
\end{equation}
which satisfies similar conditions as $A$, namely
\begin{equation}\label{coercivita-v}
\langle A_v(x,\xi),\xi\rangle\geqslant \alpha|\xi|^p-\big(b(x)\,|v|\big)^p-\varphi(x)^{p}
\end{equation}
\begin{equation}\label{limitatezza-v}
|A_v(x,\xi)|\leqslant \beta|\xi|^{p-1}+\big(b(x)\,|v|\big)^{p-1}+\varphi(x)^{p-1}
\end{equation}
\begin{equation}\label{monotonia-v}
\langle A_v(x,\xi)- A_v(x,\eta ), \xi -\eta \rangle>0\qquad \text{for }\xi\ne\eta
\end{equation}
Hence, we can consider a quasilinear elliptic
operator similar to \eqref{loperatore}
\begin{equation}\label{loperatore-v}
u\in \SobpO\quad\mappaa\quad-\mathop{\mathrm{div}} A_v(x,\nabla u)\in \Sobdual{\Omegaega}
\end{equation}
defined by the rule
\begin{equation}
\langle -\mathop{\mathrm{div}} A_v(x,\nabla u),w\rangle =\int_\Omegaega \langle A(x,v,\nabla u),\nabla w\rangle \,\mathrm d x
\end{equation}
for any $w\in \SobpO$. The operator at \eqref{loperatore-v} is invertible. Indeed,
\begin{proposition}\label{la proposizione}
For every $\Phi\in\Sobdual{\Omegaega}$, there exists a unique $u\in \SobpO$ such that
\begin{equation}\label{equazione-v}
-\mathop{\mathrm{div}} A_v(x,\nabla u)=\Phi
\end{equation}
Moreover, the mapping
\begin{equation}\label{mappa continua}
(v,\Phi)\in L^p(\Omegaega)\times \Sobdual{\Omegaega}\quad\mappaa\quad u\in \SobpO
\end{equation}
is continuous.
\end{proposition}
\begin{proof}
Existence of a solution is classical, see e.g.~\cite{LL}, \cite[pg.~27]{Br}, or \cite[Th\'eor\`eme~2.8, pg.~183]{L}. Uniqueness trivially holds by monotonicity.\par
For the sake of completeness, we prove continuity of the map \eqref{mappa continua}.
Given $v_n\to v$ in $L^p(\Omegaega)$ and $\Phi_n\to\Phi$ in $\Sobdual{\Omegaega}$,
let $u_n\in \SobpO$ solve
\begin{equation}\label{202005251}
-\mathop{\mathrm{div}} A(x,v_n,\nabla u_n)=\Phi_n\,.
\end{equation}
The sequence $\{u_n\}_n$ is clearly bounded, hence we may assume $u_n \rightharpoonup u$ weakly in $\SobpO$. Moreover, testing equation \eqref{202005251} with $u_n-u$, we have
\begin{equation}\label{202005022}
\lim_{n\rightarrow \infty}
\int_\Omegaega A(x,v_n,\nabla u_n) (\nabla u_n-\nabla u)\,\mathrm d x=\lim_{n\rightarrow \infty}\langle \Phi_n,u_n-u\rangle= 0\,.
\end{equation}
On the other hand, we easily see that $A(x,v_n,\nabla u)\to A(x,v,\nabla u)$ strongly in $L^{p'}(\Omegaega,{\mathbb R}^N)$ and thus \eqref{202005022} implies
\begin{equation}\label{202005023}
\lim_{n\rightarrow \infty}
\int_\Omegaega \left[ A(x,v_n,\nabla u_n) - A(x,v_n,\nabla u) \right]\nabla (u_n-u)\,\mathrm d x= 0\,.
\end{equation}
The integrands in \eqref{202005023} are nonnegative by monotonicity. Hence, arguing as in the proof of \cite[Lemma~3.3]{LL}, we also get $\nabla u_n(x)\to\nabla u(x)$ a.e.\ in $\Omegaega$, and
\[A (x,v_n,\nabla u_n)\rightharpoonup A (x,v ,\nabla u )\]
weakly in $L^{p'} (\Omegaega,\mathbb R^N)$. Combining this with \eqref{202005022} yields
\begin{equation}\label{202005033}
\lim_{n\rightarrow \infty}
\int_\Omegaega A(x,v_n,\nabla u_n) \nabla u_n\,\mathrm d x=\int_\Omegaega A(x,v,\nabla u) \nabla u\,\mathrm d x\,.
\end{equation}
By coercivity condition \eqref{coercivita}, we deduce
\[\alpha |\nabla u_n|^p\leqslant A(x,v_n,\nabla u_n) \nabla u_n+(b|v_n|)^p+\varphi^{p}\]
Trivially $\int_\Omegaega(b|v_n|)^p\,\mathrm d x$ converges to $\int_\Omegaega(b|v|)^p\,\mathrm d x$. In view of \eqref{202005033}, by Lemma~\ref{lemma elementare} we get $u_n \to u$ strongly in $\SobpO$, and $u$ solves the equation
\[-\mathop{\mathrm{div}} A(x,v,\nabla u)=\Phi\,.\]
\end{proof}
In view of Rellich Theorem, we have
\begin{corollary}\label{corollary compattezza}
For fixed $\Phi\in\Sobdual{\Omegaega}$, the mapping
\begin{equation}\label{risolvente}
\mathcal F\colon v\in \SobpO\quad\mappaa\quad u\in \SobpO
\end{equation}
which takes $v$ to the unique solution $u$ of equation~\eqref{equazione-v} is compact.
\end{corollary}
Now we state an existence result to Problem \eqref{Dirichlet}
when $b\in L^\infty(\Omegaega)$.
\begin{proposition}\label{b-limitata}
Let
\eqref{coercivita}, \eqref{limitatezza} and \eqref{monotonia}
be in charge
with $b\in L^\infty(\Omegaega)$.
Then Problem \eqref{Dirichlet} has a solution $u \in W_0^{1,p} (\Omegaega)$.
\end{proposition}
\begin{proof}
If $\mathcal{F}$ is the operator defined in Corollary \ref{corollary compattezza}, clearly
a fixed point of $\mathcal F$ is a solution to Problem~\eqref{Dirichlet}. To apply Leray-Schauder theorem, we need an a~priori estimate on the solution $u\in \SobpO$ of the equation
\[u=t\mathcal F[u]\]
that is
\begin{equation}\label{equazione-k ter}
-\mathop{\mathrm{div}} A\left(x,u ,\frac1{t }\,\nabla u \right)=\Phi\,,
\end{equation}
as $t\in{}]0,1]$ varies.
By using $T_\sigma u$ with $\sigma >0$ as a test function in \eqref{equazione-k ter} we get
\begin{equation}
\int _\Omegaega
\left \langle
A\left(x,u ,\frac1{t }\,\nabla u \right)
,
\nabla T_\sigma u
\right \rangle\,\mathrm d x
=
\left \langle
\Phi
,
T_\sigma u
\right \rangle
\end{equation}
Therefore, using the point-wise condition \eqref{coercivita-v} we get
\begin{equation}\label{stima vk0 ter}
\alpha\, t ^{1-p}\int_\Omegaega |\nabla T_{\sigma} u |^p \,\mathrm d x
\leqslant \|\Phi\|\,\|\nabla T_\sigma u_k\|_p+\int_{\Omegaega}\Big[b(x)^p|u |^p\chi_{\{|u |< \sigma \} }+\varphi(x)^{p}\Big]\,\mathrm d x
\end{equation}
As $0<t\leqslant 1$, by Young inequality \eqref{stima vk0 ter} yields
\begin{equation}\label{final proposition ter}
\frac \alpha 2
\int_\Omegaega |\nabla T_{\sigma} u |^p \,\mathrm d x
\leqslant
\|\Phi\|^{p^\prime}
+
\| b \|_\infty ^p
\int_{\Omegaega}
|u|^p \chi_{ \left\{ |u|\leqslant \sigma \right\} }
\,\mathrm d x
+
\|\varphi\|_p^{p}
\end{equation}
The conclusion follows by Lemma \ref{lemma weak compact}.
\end{proof}
\subsection{The approximating problems}\label{approssimazione}
For each $n\in {\mathbb N}$, we set
\begin{equation}\label{theta}
\vartheta_n(x)=\frac{T_nb(x)}{b(x)},\qquad \mbox{ a.e. }x\in \Omega\,,
\end{equation}
and define the vector field
\begin{equation}\label{A n}
A_n\colon (x,u,\xi)\in \Omegaega\times {\mathbb R}\times \reale^N\quad\mappaa\quad \reale^N
\end{equation}
letting
\begin{equation}\label{theta2}
A_n(x,u,\xi)=A(x,\vartheta_n u,\xi)\,
\end{equation}
The vector field $A_n$ has similar properties as $A$, with $b$ replaced by $T_nb$. More precisely,
\begin{equation}\label{coercivita n}
\langle A_n(x,u,\xi),\xi\rangle\geqslant \alpha|\xi|^p-\big(T_nb(x)\,|u|\big)^p-\varphi(x)^{p}
\end{equation}
\begin{equation}\label{limitatezza n}
|A_n(x,u,\xi)|\leqslant \beta|\xi|^{p-1}+\big(T_nb(x)\,|u|\big)^{p-1}+\varphi(x)^{p-1}
\end{equation}
\begin{equation}\label{monotonia n}
\langle A_n(x,u,\xi)- A_n(x,u,\eta ), \xi -\eta \rangle>0\qquad \text{for }\xi\ne\eta
\end{equation}
Applying Proposition~\ref{b-limitata} with $A_n$ in place of $A$, fixed $\Phi\in \Sobdual\Omegaega$, we find $u_n\in \SobpO$ such that
\begin{equation}\label{equazione n}
-\mathop{\mathrm{div}} A_n(x,u_n,\nabla u_n)= \Phi\,.
\end{equation}
Notice that we have, for $\sigma>0$
\begin{equation}\label{stima vkn}
\alpha\, \int_\Omegaega |\nabla T_\sigma u_n|^p
\,\mathrm d x
\leqslant \|\Phi\|\,\|\nabla T_\sigma u_n\|_p+\int_{\Omegaega}\Big[ (T_n b) ^p\,|u_n|^p
\chi_{\{|u_n|<\sigma\}}
+\varphi^{p}\Big]\,\mathrm d x
\end{equation}
which implies
\begin{equation}\label{stima vkn norme}
\alpha^{\frac1p}\|\nabla T_\sigma u_n\|_p
\leqslant (\|\Phi\|\,\|\nabla T_\sigma u_n\|_p)^{\frac1p}+\|(T_n b)\,u_n
\chi_{\{|u_n|<\sigma\}}\|_p
+\|\varphi\|_p
\end{equation}
Our next step consists in showing that the sequence $\{u_n\}_n$
is bounded in $\SobpO$. Let $m$ be a positive integer to be chosen later. For $n\geqslant m$ we have
\[T_n b\leqslant T_mb+(b-T_mb)\]
and hence
\begin{equation}\label{migliore costante}
\|(T_n b)\,u_n \chi_{\{|u_n|<\sigma\}}\|_p\leqslant \|(T_m b)\,u_n \chi_{\{|u_n|<\sigma\}}\|_p+\|(b-T_m b)\,u_n \chi_{\{|u_n|<\sigma\}}\|_p
\end{equation}
Using H\"older and Sobolev inequalities we get
\[\|(b-T_mb)\,u_n
\chi_{\{|u_n|<\sigma\}}
\|_p\leqslant \|b-T_mb\|_{N,\infty}\|T_\sigma u_n\|_{p^*,p}\le
\,S_{N,p}\,
\|b-T_mb\|_{N,\infty}\|\nabla T_\sigma u_n\|_p\]
Then \eqref{stima vkn norme} and \eqref{migliore costante} give
\begin{equation}\label{stima vkn3bis}
\begin{split}
\alpha^{\frac1p}\|\nabla T_\sigma u_n\|_p\leqslant
(\|\Phi\|\,\|\nabla T_\sigma u_n\|_p)^{\frac1p}&+\|(T_m b)\,u_n \chi_{\{|u_n|<\sigma\}}\|_p
+\|\varphi\|_p
\\
&
+S_{N,p} \|b-T_m b\|_{L^{N,\infty} (\Omegaega) } \|
\nabla T_\sigma u_n\|_{L^{p} (\Omegaega) }
\end{split}
\end{equation}
By our assumption \eqref{lndeboleinfty}, the level $m$ can be chosen large enough so that
\[
S_{N,p} \|b-T_m b\|_{L^{N,\infty} (\Omegaega) }<\alpha^{\frac1p}
\]
Then, by absorbing
in
\eqref{stima vkn3bis} the latter term of the right hand side in the left hand side, we get
\begin{equation}\label{stima vkn3}
C\int_\Omegaega |\nabla T_\sigma u_n|^p\,\mathrm d x\leqslant \|\Phi\|\,\|\nabla T_\sigma u_n\|_p+\int_{\Omegaega}\Big[(T_m b)^p\,|T_\sigma u_n|^p
\chi_{\{|u_n|<\sigma\}}
+\varphi^{p}\Big]\,\mathrm d x
\end{equation}
for a positive constant $C$ which is independent of $n$. Now, it is clear that \eqref{stima vkn3}, via Young inequality, allows us to apply Lemma~\ref{lemma weak compact}, then
\begin{equation}\label{3.23bis}
\|u_n\|\leqslant M\,
\end{equation}
for a constant $M$ independent of $n$.
In the model case \eqref{modello}, it is easy to show that the operator $\mathcal F$ defined in \eqref{risolvente} is compact, also for $b\in L^N(\Omegaega)$ (see Remark \ref{202005252} below). In the general case, in which
$b\in L^{N,\infty}(\Omegaega)$ we need more work.
\subsection{Passing to the limit}\label{pass lim}
Now, we are in a position to conclude the proof of Theorem~\ref{main}.
Taking into account estimate \eqref{3.23bis}
we may assume
\begin{equation}\label{22}
\begin{split}
u_n \rightharpoonup u & \qquad \text{in $\SobpO$ weakly} \\
u_n \rightarrow u & \qquad \text{in $L^q(\Omegaega)$ strongly for any $q<p^*$, and also a.e.\ in $\Omegaega$}
\end{split}
\end{equation}
for some $u \in W^{1,p}_0(\Omegaega)$.
We shall conclude our proof showing that $u$ solves Problem~\eqref{Dirichlet}.
In the rest of our argument, we let for simplicity $\gamma(t):=\arctan t$. Obviously, $\gamma \in C^1(\mathbb R)$, $|\gamma (t)|\leqslant |t|$ and $0 \leqslant \gamma^\prime(t) \leqslant 1$ for all $t \in \mathbb R$. In particular,
$\gamma$ is Lipschitz continuous in the whole of $\mathbb R$ and therefore
\[
u_n,u\in \SobpO \quad \Longrightarrow \quad \gamma(u_n-u)\in \SobpO\,.
\]
Moreover, since $\gamma(0)=0$ we have
\begin{equation}\label{23}
\gamma(u_n-u) \rightharpoonup 0 \qquad \text{in $\SobpO$ weakly}\,.
\end{equation}
Testing equation \eqref{equazione n} with the function $\gamma(u_n-u)$ we get
\[
\int_\Omegaega A_n(x,u_n,\nabla u_n) \nabla \gamma(u_n-u)\,\mathrm d x= \left\langle \Phi , \gamma(u_n-u) \right \rangle
\]
where $\nabla \gamma(u_n-u) = \gamma^\prime(u_n-u) (\nabla u_n - \nabla u)$.
In view of \eqref{23} we necessarily have
\begin{equation}\label{24}
\lim_{n\rightarrow \infty}
\int_\Omegaega A_n(x,u_n,\nabla u_n) \nabla \gamma(u_n-u)\,\mathrm d x= 0\,.
\end{equation}
We claim that
\begin{equation}\label{26}
\lim_{n\rightarrow \infty} \int_\Omegaega A_n(x,u_n,\nabla u) \nabla \gamma(u_n-u)\,\mathrm d x= 0\,.
\end{equation}
In order to prove \eqref{26}, since $\nabla u_n - \nabla u\rightharpoonup 0$, it suffices to show that
\begin{equation}\label{compattezza}
\gamma'(u_n-u)\,A_n(x,u_n,\nabla u)= \frac { A_n(x,u_n,\nabla u) }{1+|u_n-u|^2}\qquad \text{is compact in }L^{p'}\,.
\end{equation}
Preliminarily, we observe that combining \eqref{22} with the property that $\vartheta_n\rightarrow 1$ as $n\rightarrow \infty$, we have
\[
\frac { A_n(x,u_n,\nabla u) }{1+|u_n-u|^2} \rightarrow
A (x,u ,\nabla u) \qquad \text{a.e.\ in }\Omegaega\,.
\]
We are going to use Lemma~\ref{lemma elementare}. To this end, by \eqref{limitatezza n} we deduce that
\[
\left|\frac { A_n(x,u_n,\nabla u) }{1+|u_n-u|^2}\right|^{p'}\leqslant C
\left[|\nabla u|^p+\varphi^{p} + (b|u|)^{p}+\frac {(b |u_n-u|)^{p} }{1+|u_n-u|^2}\right]
\]
for a positive constant $C=C(p,\beta)$. Hence, we can pass to the limit if $1<p\leqslant 2$. For $p>2$ we choose $s$ satisfying
\[
\frac {p^\ast}{p} < s < \frac {p^\ast}{p-2}\,,
\]
so that $ps^\prime<N$, and we conclude also in this case, further estimating with the aid of Young inequality
\[
\frac {(b |u_n-u|)^{p} }{1+|u_n-u|^2}\leqslant b^{ps'}+ |u_n-u|^{(p-2)s}\,.
\]
Now, from \eqref{24} and \eqref{26} we get
\begin{equation}\label{27}
\lim_{n\rightarrow \infty}
\int_\Omegaega \left[ A_n(x,u_n,\nabla u_n) - A_n(x,u_n,\nabla u) \right]\nabla \gamma(u_n-u)\,\mathrm d x= 0\,.
\end{equation}
As the integrand is nonnegative, we have (up to a subsequence)
\[\left[ A_n(x,u_n,\nabla u_n) - A_n(x,u_n,\nabla u) \right]\nabla \gamma(u_n-u)\to 0\]
a.e.\ in $\Omegaega$. Moreover, since $\gamma'(u_n-u)\to1$ a.e.\ in $\Omegaega$, the above in turn implies
\begin{equation}\label{202005021}
\left[ A_n(x,u_n,\nabla u_n) - A_n(x,u_n,\nabla u) \right]\,(\nabla u_n-\nabla u)\to 0
\end{equation}
Arguing as in the proof of \cite[Lemma~3.3]{LL}, we see that
\begin{equation}\label{29.4}
\nabla u_n \rightarrow \nabla u \qquad \text{a.e.\ in $\Omegaega$}
\end{equation}
and
\begin{equation}\label{29.6}
A_n (x,u_n,\nabla u_n)
\rightharpoonup
A (x,u ,\nabla u )
\qquad \text{in
$L^{p^\prime} (\Omegaega,\mathbb R^N)$ weakly}
\end{equation}
and we conclude that $u$ is a solution to the original problem~\eqref{Dirichlet}.
\begin{remark}\label{202005252}
We discuss briefly the particular case in which the operator has the form
\[A(x,v,\xi)=A'(x,\xi)+A''(x,v)\,,\]
with
\[|A''(x,v)|\leqslant (b(x)\,|v|)^{p-1}+\varphi(x)^{p-1}\,.\]
and $b\in L^N(\Omegaega)$ (see also \cite{BoBumi}).
We can easily show that the operator $\mathcal F$ defined in \eqref{risolvente} is compact, also for $b\in L^N(\Omegaega)$. Indeed, equation \eqref{equazione-v} in this case becomes
\begin{equation}\label{equazione-v particolare}
-\mathop{\mathrm{div}} A'(x,\nabla u)=\Phi+\mathop{\mathrm{div}} A''(x,v)\,.
\end{equation}
Defined $\vartheta_n$ as in \eqref{theta}, each mapping
\[v\in \SobpO\quad\mappaa\quad A''(x,\vartheta_n\,v)\in L^{p'}(\Omegaega,{\mathbb R}^N)\]
is clearly compact. Moreover,
\begin{equation}\label{202005253}
|A''(x,v)-A''(x,\vartheta_n\,v)|\leqslant 2[(b\,|v|)^{p-1}+\varphi^{p-1}]\,\chi_{E_n}\,,
\end{equation}
where
\[E_n=\{x\in \Omegaega:|b(x)|>n\}\,.\]
Therefore, as $n\to+\infty$ we have
\[A''(x,\vartheta_n\,v)\to A''(x,v)\qquad \text{strongly in }L^{p'}(\Omegaega,{\mathbb R}^N)\,,\]
the convergence being uniform when $v$ varies in a bounded subset of $\SobpO$, and compactness is preserved for the limit mapping.\par
An a~priori bound for solutions of equation
\[u=t\,\mathcal F[u]\]
can be easily obtained as above, splitting $b\in L^N(\Omegaega)$ as
\[b=T_m b+(b-T_mb)\]
for a sufficiently large $m$. Therefore, in this particular case the existence result of Theorem~\ref{main} follows simply applying Leray--Schauder fixed point theorem.\par
\end{remark}
\section{The obstacle problem}\label{pb ostacolo}
This section is devoted to the obstacle problem naturally related with problem
\eqref{Dirichlet} (see \cite{KinStam} for a comprehensive treatment of the topic).
We again assume that
\eqref{coercivita}, \eqref{limitatezza} and \eqref{monotonia}
are in charge
and we let $\Phi \in W^{-1,p}(\Omegaega)$.
Given a measurable function $\psi\colon \Omegaega \rightarrow \overline {\mathbb R}$,
where $\overline {\mathbb R}:=[-\infty,\infty]$, we consider the convex subset
of $\mathcal K_\psi(\Omegaega)$ of
$
W_0^{1,p}(\Omegaega)
$
given by
\begin{equation}
\mathcal K_\psi (\Omegaega):=
\left\{
w \in W_0^{1,p}(\Omegaega) \colon
\,
w \geqslant \psi \text{ a.e.\ in }\Omegaega
\right\}.
\end{equation}
We will assume that $\mathcal K_\psi(\Omegaega)$ is nonempty.
An element $u \in \mathcal K_\psi (\Omegaega)$ is a solution to the obstacle problem associated with \eqref{Dirichlet} if the following variational inequality holds
\begin{equation}\label{obst}
\begin{split}
\int_{\Omegaega }
\langle
A(x,u,\nabla u) , \nabla (w-u) \rangle
\,\mathrm d x
\geqslant
\langle \Phi,
w-u
\rangle\,,
\qquad \forall w \in \mathcal K_\psi(\Omegaega)\,.
\end{split}
\end{equation}
As
$\mathcal K_\psi(\Omegaega)
\neq \emptyset
$,
we may assume without loss of generality that
\begin{equation}\label{nonneg}
\text{$\psi\leqslant 0$ a.e. in $\Omegaega$.}
\end{equation}
In fact, if $g \in \mathcal K _\psi(\Omegaega)$, then one can consider the operator defined by the vector field
\[
\tilde A(x,u,\xi):=A(x,u+g(x),\xi+\nabla g(x))\,,
\]
satisfying conditions similar to
\eqref{coercivita},
\eqref{limitatezza} and
\eqref{monotonia}.
Now it is clear that, if function
$\tilde u \in {\mathcal K} _{\psi-g}(\Omegaega)$ satisfies
the following
variational inequality
\begin{equation}\label{obst tilde}
\begin{split}
\int_{\Omegaega }
\langle \tilde A(x,\tilde u,\nabla \tilde u) , \nabla (w-\tilde u) \rangle
\,\mathrm d x
\geqslant
\langle \Phi,
w-\tilde u
\rangle
\qquad \forall w \in {\mathcal K} _{\psi-g}(\Omegaega)
\end{split}
\end{equation}
correspondingly
$u = \tilde u + g$ is a solution to \eqref{obst}. Notice that the obstacle function for problem \eqref{obst tilde}
is nonpositive, as we are assuming for the original problem.
\begin{theorem}\label{ob theorem}
Let $\Phi\in\Sobdual{\Omegaega}$
and $\psi:\Omegaega\rightarrow [-\infty,0]$ be a measurable function. Under the assumption
\eqref{coercivita},
\eqref{limitatezza} and
\eqref{monotonia},
if \eqref{lndeboleinfty} holds,
then the ostacle problem~\eqref{obst} admits a solution.
\end{theorem}
\begin{proof}
We follow closely the arguments of
Section \ref{Section3}.
For each $n\in {\mathbb N}$, we consider the function $\vartheta_n$ as in \eqref{theta}
and define the vector fields
$
A_n=A_n (x,u,\xi)
$
as in \eqref{theta2}.
We consider a sequence of obstacle problems provided by
\begin{equation}\label{obst n}
\begin{split}
\int_{\Omegaega }
\langle A_n(x,u_n,\nabla u_n) , \nabla (w-u_n) \rangle
\,\mathrm d x
\geqslant
\langle \Phi,
w-u_n
\rangle\,,
\qquad \forall w \in \mathcal K_\psi(\Omegaega)\,.
\end{split}
\end{equation}
The existence of a
solution $u_n \in \mathcal K_\psi(\Omegaega)$ to \eqref{obst n} is proven applying \cite[Th\'eor\`eme~8.2, pg.~247]{L} to the operator
\[-\mathop{\mathrm{div}} A_n(x,v,\nabla u)\,,\]
for a fixed $v\in \SobpO$, and then using Leray--Schauder Theorem, arguing as in Subsection~\ref{coefficienti limitati}. Due to \eqref{nonneg},
for every
$ k >0$
the function
\[
w:=u_n - T_k u_n \in \mathcal K_\psi(\Omegaega)
\]
is a test function for
\eqref{obst n}.
Arguing
as in Section \ref{approssimazione} we obtain
\[
\|
u_n
\|\leqslant M
\]
with $M$ independent of $n$ (as in \eqref{3.23bis}). Therefore
\eqref{22} holds
for some $u \in W^{1,p}_0(\Omegaega)$. It is clear from \eqref{22} itself that
\begin{equation}\label{u in K}
u \in \mathcal K_\psi(\Omegaega)
\end{equation}
As for Theorem \ref{main}, we shall prove that $u$ is a solution to the original problem \eqref{obst}. We proceed as follows. We use
\begin{equation}\label{arct ob}
w:=u_n - \gamma(u_n-v)
\end{equation}
in \eqref{obst n}, where $\gamma (s)=\lambda\arctan(s/\lambda)$, for $\lambda>0$, and $v\in \mathcal K_\psi(\Omegaega)$ is arbitrary. Note that this is a legitimate test function, that is $w\in \mathcal K_\psi(\Omegaega)$. Indeed, on the set where $u_n\geqslant v$ we have
$\gamma(u_n-v)\leqslant u_n-v$ and so $w \geqslant v$; on the other hand, on the set where $u_n\leqslant v$ we have
$ \gamma (u_n-v)\leqslant 0$ and so $w\geqslant u_n$. Therefore, from \eqref{obst n} we get
\begin{equation}\label{var in n}
\int_\Omegaega A_n(x,u_n,\nabla u_n) \nabla \gamma(u_n-v)\,\mathrm d x \leqslant \left\langle \Phi , \gamma(u_n-v) \right \rangle\,.
\end{equation}
Following the lines of the proof of Theorem \ref{main} (where $\lambda=1$), we get in turn \eqref{27}, \eqref{29.4} and finally \eqref{29.6}. To pass to the limit for fixed general $\lambda>0$ in \eqref{var in n}, we rewrite it as follows:
\begin{equation}\label{var in n 2}
\begin{split}
\int_\Omegaega [A_n(x,u_n,\nabla u_n)&-A_n(x,u_n,\nabla v)]\,\nabla \gamma(u_n-v)\,\mathrm d x \\
&\leqslant \left\langle \Phi , \gamma(u_n-v) \right \rangle-\int_\Omegaega A_n(x,u_n,\nabla v)\nabla \gamma(u_n-v)\,\mathrm d x \,.
\end{split}
\end{equation}
In the left hand side we use Fatou lemma, as by monotonicity condition \eqref{monotonia} the integrand is nonnegative. In the right hand side, we note that $A_n(x,u_n,\nabla v)\,\gamma'(u_n-v)$ converges to $A (x,u ,\nabla v)\,\gamma'(u-v)$ in $L^{p'}$, compare with \eqref{compattezza} where we did not use that $u_n\to u$. Hence, we deduce from \eqref{var in n 2}
\[\begin{split}
\int_\Omegaega [A(x,u,\nabla u)&-A(x,u,\nabla v)]\,\nabla \gamma(u-v)\,\mathrm d x \\
&\leqslant \left\langle \Phi , \gamma(u-v) \right \rangle-\int_\Omegaega A(x,u,\nabla v)\nabla \gamma(u-v)\,\mathrm d x \,,
\end{split}
\]
that is
\begin{equation}\label{var in n 3}
\int_\Omegaega A(x,u,\nabla u)\,\nabla \gamma(u-v)\,\mathrm d x\leqslant \left\langle \Phi , \gamma(u-v) \right \rangle\,.
\end{equation}
Now we let $\lambda\to\infty$ in \eqref{var in n 3}, noting that $\gamma(u-v)\to u-v$ strongly in $\SobpO$. Therefore, we get
\[\int_\Omegaega A(x,u,\nabla u)\,\nabla (u-v)\,\mathrm d x \leqslant \left\langle \Phi , u-v \right \rangle\,,\]
for all $v\in K_\psi(\Omegaega)$, which means exactly that $u$ is a solution to our obstacle problem.
\end{proof}
\begin{remark}
Clearly, Theorem \ref{ob theorem} is more general than Theorem \ref{main} since we are allowed to choose $\psi \equiv -\infty$.
Indeed, in this case, the obstacle problem \eqref{obst} reduces to \eqref{Dirichlet}.
\end{remark}
\section{Regularity of the solution}\label{reg sol}
In this Section, following \cite{GMZ} we study regularity of the problem \eqref{Dirichlet}.
\begin{theorem}
Let $1 < p<r<N$ and
$\Phi \in W^{-1, \frac {r}{p-1} } (\Omegaega)$.
Assume that
\eqref{coercivita},
\eqref{limitatezza} and
\eqref{monotonia} hold with
$\varphi \in L^r(\Omegaega)$. Under these hypotheses, there exists $\partiallta=\partiallta(\alpha, N,p,r)>0$ such that
if
\begin{equation}\label{lndeboleinfty r}
\mathrm{dist}_{L^{N,\infty}} (b,L^\infty (\Omegaega))<\partiallta,
\end{equation}
then any solution $u \in W^{1,p}_0(\Omegaega)$ of \eqref{Dirichlet}
satisfies
\begin{equation}\label{6.2}
|u|^{r^\ast/p^\ast} \in W^{1,p}_0(\Omegaega)
\end{equation}
In particular $u \in L^{r^\ast} (\Omegaega)$.
\end{theorem}
\begin{proof}
Let $u \in W^{1,p}_0(\Omegaega)$ be a solution
of \eqref{Dirichlet}. We may write $\Phi \in W^{-1,\frac {r}{p-1}} (\Omegaega)$ as
\[\Phi=\mathop{\mathrm{div}}(|F|^{p-2} F)\]
for a suitable $F \in L^{r} (\Omegaega,\mathbb R^N)$.\par
For fixed $k>0$, we use $v:=u-T_k u$ as a test function in \eqref{soluzint} to get
\begin{equation}\label{202005221}
\alpha\int_{\Omegaega_k}|\nabla u|^p\,\mathrm d x\leqslant \int_{\Omegaega_k}|F|^{p-1}\,|\nabla u|\,\mathrm d x+\int_{\Omegaega_k}(b^p|u|^p+\varphi^p)\,\mathrm d x
\end{equation}
where $\Omegaega_k$ denotes the superlevel set $\{|u|>k\}$. For $0<\varepsilon<\alpha$, by Young inequality we get
\begin{equation}\label{202005226}
(\alpha-\varepsilon)\int_{\Omegaega_k}|\nabla u|^p\,\mathrm d x\leqslant \int_{\Omegaega_k}(C\,|F|^p+b^p|u|^p+\varphi^p)\,\mathrm d x
\end{equation}
with $C=C(p,\varepsilon)>0$. We let
\begin{equation}\label{202005227}
\lambda=\frac{r^*}{p^*}-1
\end{equation}
and multiply both sides of \eqref{202005226} by $k^{p\lambda-1}$ and integrate w.r.t.\ $k$ over the interval $[0,K]$, for $K>0$ fixed. By Fubini theorem we have
\begin{equation}\label{202005222}
(\alpha-\varepsilon)\int_{\Omegaega}|\nabla u|^p\,|T_Ku|^{p\lambda}\,\mathrm d x\leqslant \int_{\Omegaega}(C\,|F|^p+b^p|u|^p+\varphi^p)\,|T_Ku|^{p\lambda}\,\mathrm d x
\end{equation}
which implies
\begin{equation}\label{202005234}
(\alpha-\varepsilon)^{\frac1p}\|\nabla u\,|T_Ku|^{\lambda}\|_p\leqslant C\|F\,|T_Ku|^{\lambda}\|_p+\|b\,u\,|T_Ku|^{\lambda}\|_p+\|\varphi\,|T_Ku|^{\lambda}\|_p
\end{equation}
For $M>0$ we write
\begin{equation}\label{202005223}
\|b\,u\,|T_Ku|^\lambda\|_p\leqslant \|(b-T_Mb)\,u\,|T_Ku|^\lambda\|_p+M\|u\,|T_Ku|^\lambda\|_p
\end{equation}
By H\"older inequality and Sobolev embedding Theorem \ref{lorentz}
\begin{equation}\label{202005224}
\begin{array}{rl}
\displaystyle \|(b-T_Mb)\,u\,|T_Ku|^\lambda\|_p\kern-.7em &\displaystyle \leqslant \|b-T_M b\|_{N,\infty}\,\|u\,|T_Ku|^\lambda\|_{p^*,p}
\\
&\displaystyle \leqslant \|b-T_M b\|_{N,\infty}\,S_{N,p}\,\|\nabla(u\,|T_Ku|^\lambda)\|_p
\end{array}
\end{equation}
Moreover,
\begin{equation}\label{202005225}
|\nabla(u\,|T_Ku|^\lambda)|\leqslant (1+\lambda)\,|\nabla u|\,|T_Ku|^\lambda
\end{equation}
Therefore
\begin{equation}\label{202005228}
\|(b-T_Mb)\,u\,|T_Ku|^\lambda\|_{L^p(\Omegaega)}\leqslant \|b-T_M b\|_{N,\infty}\,S_{N,p}\,(1+\lambda)\,\|\nabla u\,|T_Ku|^\lambda\|_p
\end{equation}
Under the assumption
\begin{equation}\label{202005229}
\|b-T_M b\|_{N,\infty}\,S_{N,p}\,(1+\lambda)<\alpha^{\frac1p}
\end{equation}
choosing $\varepsilon$ small enough we get from \eqref{202005234}
\begin{equation}\label{2020052210}
\|\nabla u\,|T_Ku|^{\lambda}\|_p\leqslant C\,\|G\,|T_Ku|^{\lambda}\|_p
\end{equation}
with $C=C(p,r,M,\alpha)>0$, where we set
\begin{equation}\label{202005231}
G^p=|F|^p+|u|^p+\varphi^p\,.
\end{equation}\par
We first show the claim under the additional assumption $u\in L^r(\Omegaega)$, so that $G\in L^r(\Omegaega)$. By H\"{o}lder inequality we have
\begin{equation}\label{I1}
\|G\,|T_Ku|^{\lambda}\|_p\leqslant
\|G\|_r\,\|T_K u\|^{\lambda}_{\lambda\,\frac {rp}{r-p}}
\end{equation}
From \eqref{202005227} we get
\begin{equation}\label{beta2}
\lambda\,\frac {rp}{r-p}=r^*\,.
\end{equation}
Hence, by Sobolev embedding theorem we have
\begin{equation}\label{I1bis}
\begin{array}{rl}
\displaystyle \|T_K u\|^{\lambda}_{\lambda\,\frac {rp}{r-p}}\kern -.7em &\displaystyle=\|T_K u\|^{\lambda}_{r^*}=\||T_K u|^{\frac{r^*}{p^*}}\|^{\lambda\frac{p^*}{r^*}}_{p^*}\leqslant C\,\|\nabla |T_K u|^{\frac{r^*}{p^*}}\|^{\lambda\frac{p^*}{r^*}}_{p}
\\
&\displaystyle\leqslant C\,\|\nabla u\,|T_Ku|^{\lambda}\|_p^{\frac\lambda{\lambda+1}}
\end{array}
\end{equation}
Then, combining \eqref{2020052210}, \eqref{I1} and \eqref{I1bis}, we get
\begin{equation}\label{5.7bis}
\|\nabla u\,|T_Ku|^{\lambda}\|_p^{\frac{p^*}{r^*}}\leqslant C\,\|G\|_r
\end{equation}
Passing to the limit as $K\rightarrow + \infty$ and recalling \eqref{202005231},
we have
\begin{equation}\label{202005232}
\|\nabla u\,|u|^{\lambda}\|_p^{\frac{p^*}{r^*}}\leqslant C\,(\|F\|_r+\|\varphi\|_r+\|u\|_r)
\end{equation}
that is
\begin{equation}\label{202005233}
\|\nabla |u|^{\frac{r^*}{p^*}}\|_p\leqslant C\,(\|F\|_r+\|\varphi\|_r+\|u\|_r)^{\frac{r^*}{p^*}}
\end{equation}
Hence, \eqref{6.2} holds as long as $u \in L^r(\Omegaega)$. At this point we observe that if
$r\leqslant p^\ast$, using the
Sobolev embedding theorem, $u\in L^{p^*}(\Omega)$ and the proof is concluded.
In the complementary case $r > p^\ast$, we use a bootstrap approach. Precisely, we repeat the previous
argument replacing $r$ with $p^\ast$ to get $u \in L^{p^{\ast\ast}}(\Omegaega)$. Using this information, if $r\leqslant p^{\ast\ast} $, there is nothing left to prove. Otherwise we repeat previous argument again. In a finite number of similar steps we can conclude our proof.
\end{proof}
\begin{remark}
In view of \eqref{202005229}, we may take
\[\partiallta=\frac {\alpha^\frac1p}{S_{N,p}}\,\frac{p^*}{r^*}\]
in \eqref{lndeboleinfty r}. Since $r\mappaa r^*$ is increasing, a similar condition to~\eqref{202005229} will hold in all intermediate steps, in case we need the bootstrap argument. Note that $\partiallta$ reduces to the distance in \eqref{lndeboleinfty}, for $r=p$.
\end{remark}
\end{document} | math | 52,431 |
\begin{document}
\title{Regularity and uniqueness for a class of solutions to the hydrodynamic flow of nematic liquid crystals}
\author{Tao Huang \\
Department of Mathematics, The Pennsylvania State University\\
University Park, PA 16802, USA\\
[email protected]}
\date{}
\maketitle
\begin{abstract} In this paper, we establish an $\epsilon$-regularity criterion for any weak solution
$(u,d)$ to the nematic liquid crystal flow (\ref{lce}) such that
$(u,\nabla d)\in L^p_tL^q_x$ for some $p\ge 2$ and $q\ge n$ satisfying the condition
(\ref{serrin-condition}). As consequences, we prove the interior smoothness of any such a solution
when $p>2$ and $q>n$. We also show that uniqueness holds for the class of weak solutions $(u,d)$
the Cauchy problem of the nematic liquid crystal flow (\ref{lce}) that satisfy $(u,\nabla d)\in L^p_tL^q_x$
for some $p>2$ and $q>n$ satisfying (\ref{serrin-condition}).
\end{abstract}
\section {Introduction}
\setcounter{equation}{0}
\setcounter{theorem}{0}
For any $n\geq 3$,
the hydrodynamic flow of nematic liquid crystals in $\mathbb R^n\times[0,T]$, for some $0<T<+\infty$, is given by
\begin{equation}\label{lce}
\begin{cases}
u_t+u\cdot\nabla u-\widetilde{D}}\def\del{\widetilde{\Delta}}\def\na{\widetilde{\nabla}elta u+\nabla P=-\nabla\cdot(\nabla d\otimes\nabla d-\frac12{|\nabla d|^2}\mathbb{I}_n)
& \ {\rm{in}}\ \mathbb R^n\times (0,T)\\
\nabla\cdot u=0 & \ {\rm{in}}\ \mathbb R^n\times (0,T)\\
d_t+u\cdot\nabla d=\widetilde{D}}\def\del{\widetilde{\Delta}}\def\na{\widetilde{\nabla}elta d+|\nabla d|^2d & \ {\rm{in}}\ \mathbb R^n\times (0,T)\\
(u,d)=(u_0,d_0)& \ {\rm{on}}\ \mathbb R^n\times\{0\}
\end{cases}
\end{equation}
where $u:\mathbb R^n\times[0,T]\rightarrow\R^n$ is the velocity field of underlying incompressible fluid,
$d:\mathbb R^n\times[0,T]\rightarrow S^2$ is the director field of nematic liquid crystal molecules,
$P:\mathbb R^n\times[0,T]\rightarrow\R$ is the pressure function,
$\nabla\cdot$ denotes the divergence operator on $\mathbb R^n$, $\nabla
d\otimes\nabla d =\left(\frac{\partialrtial d}{\partialrtial
x_i}\cdot\frac{\partialrtial d}{\partialrtial x_j}\right)_{1\leq i,j\leq n}$
is the stress tensor induced by the director field $d$,
$\mathbb{I}_n$ is the identity matrix of order $n$, $u_0:\mathbb R^n\to\mathbb R^n$ is the initial
velocity field with $\nabla\cdot u_0=0$, and $d_0:\mathbb R^n\to S^2$ is the initial
director field.
The system (\ref{lce}) is a simplified version of the
Ericksen-Leslie system modeling the hydrodynamics of liquid crystal
materials, proposed by Ericksen \cite{ericksen} and Leslie
\cite{leslie} in 1960's. It is a macroscopic continuum description
of the time evolution of the material under the influence of both
the flow field and the macroscopic description of the microscopic
orientation configurations of rod-like liquid crystals.
The interested readers can refer to
\cite{ericksen}, \cite{leslie}, \cite{lin}, and \cite{lin-liu} for
more detail. Mathematically, the system (\ref{lce}) is strongly
coupling the Naiver-Stokes equations and the
(transported) heat flow of harmonic maps into $S^2$.
For $n=2$, Lin-Lin-Wang \cite{lin-lin-wang} have proved the existence of global Leray-Hopf type weak solutions to (\ref{lce}) with initial and boundary conditions, which is smooth away from finitely many possible singular times
(see Hong \cite{hong} and Xu-Zhang \cite{xu-zhang} for related works). Lin-Wang \cite{lin-wang} proved the uniqueness for such weak solutions. It remains a very challenge open problem to prove the global existence of Leray-Hopf type weak solutions
and partial regularity of suitable weak solutions to (\ref{lce}) in higher dimensions. A BKM type blow-up criterion was obtained
for the local strong solution to (\ref{lce}) for $n=3$ by \cite{huang-wang2}, i.e., if $0<T_*<+\infty$ is the maximum time
interval of the strong solution to (\ref{lce}), then
$$\int_0^{T_*}\left(\|\nabla\times u\|_{L^{\infty}}+\|\nabla d\|^2_{L^{\infty}}\right)\,dt=+\infty.$$
Recently, the local well-posedness of (\ref{lce}) was obtained for initial data $(u_0,d_0)$ with $(u_0,\nabla d_0)\in L^3_{\rm{uloc}}(\mathbb R^3)$, the space of uniformly locally $L^3$-integrable functions, of small norm for $n=3$
by \cite{hineman-wang}. While the global well-posedness of (\ref{lce}) was obtained by \cite{wang1}
for $(u_0,d_0)\in$ BMO$\times$ BMO$^{-1}$ of small norm for $n\ge 3$.
The existence of global Leray-Hopf weak
solutions to the Naiver-Stokes equations has long been established by Leray \cite{leray} and
Hopf \cite{hopf}. However the uniqueness (regularity) of Leray-Hopf solutions
in dimension three remains largely open. In
\cite{serrin}, Serrin proved the so called `weak-strong' uniqueness,
i.e., the uniqueness holds for Leray-Hopf solutions $u,v$ with the same initial data, if
$u\in L^{p}_tL^q_x(\mathbb R^n\times[0,T])$, where $p\geq 2$ and $q\ge n$ satisfy
\begin{equation}\label{serrin-condition}
\frac{2}{p}+\frac{n}{q}=1.
\end{equation}
The smoothness of such solutions was established by Ladyzhenskaya in \cite{ladyzhen} for $p>2$ and $q>n$.
In the fundamental work \cite{ESS}, Escauriaza-Seregin-$\check{\mbox{S}}$ver$\acute{\mbox{a}}$k have proved the smoothness of Serrin's solutions for the endpoint case $(p,q)=(+\infty,n)$ when $n=3$ (see also \cite{dong-du} for $n\ge 4$).
Wang \cite{wang} proved smoothness of weak solutions $u$ to the heat flow of harmonic maps
such that $\nabla u\in L^{p}_tL^q_x(\mathbb R^n\times[0,T])$ with $\frac{2}{p}+\frac{n}q=1$
for $n\geq 4$ (or $q\geq 4$ for $2\leq n<4$, see \cite{huang-wang1} for the case $2<q<4$ when $2\le n<4$).
In \cite{huang-wang1}, the uniqueness of Serrin's solutions to the heat flow of harmonic maps is also established
when $p> 2$ and $q>n$. These results motivate us
to investigate the regularity and uniqueness of Serrin's ($p,q$)-solutions to the system (\ref{lce}) of
nematic liquid crystal flows.
Before stating our main theorems, we need to introduce some notations.
\noindent{\bf Notations}: For $1\le p, q\le +\infty$, $0<T\le \infty$, define the Sobolev space
$$H^{1}(\mathbb R^n\times[0,T], \mathbb R^n)
=\Big\{f\in L^2([0,T], H^{1}(\mathbb R^n, \mathbb R^n)): \ \partialrtial_t
f\in L^2([0,T], L^2(\mathbb R^n,\mathbb R^n))\Big\},$$
$$\mathbb E^p(\mathbb R^n\times[0,T], \mathbb R^n)=\{f\in L^p(\mathbb R^n\times[0,T], \mathbb R^n)
\ | \ \nabla\cdot f=0\},$$
and the Morrey space $M^{p,\lambda}(U)$ for $0\le\lambda\le n+2$
and $U=U_1\times [t_1,t_2]\subset\mathbb R^{n}\times\mathbb R$:
$${M}^{p,\lambda}(U)=\left\{f\in L^{p}_{\mbox{loc}}(U):
\Big\|f\Big\|_{{M}^{p,\lambda}(U)}<+\infty\right\},$$
where
$$\Big\|f\Big\|_{{M}^{p,\lambda}(U)}=\Big(\sup\limits_{(x,t)\in U}\sup\limits_{0<r<\min\delta((x,t),\partialrtial_p U)}\
r^{\lambda-n-2}\int_{P_r(x,t)} |f|^p\Big)^{\frac{1}{p}},$$
$$B_r(x)=\{y\in\mathbb R^n: \,|y-x|\leq r\},
\ P_r(x,t)=B_r(x)\times[t-r^2,t], \partialrtial_p U=(\partialrtial U_1\times [t_1,t_2])\cup (U_1\times\{t_1\}),$$
and
$$\delta((x,t),\partialrtial_p U)=\inf_{(y,s)\in\partialrtial_p U}\delta\left((x,t), (y,s)\right),
\ {\rm{and}}\ \ \delta((x,t), (y,s))=\min\left\{|x-y|, \sqrt{|t-s|}\right\}.$$
Denote $B_r$ (or $P_r$) for
$B_r(0)$ (or $P_r(0)$ respectively). We also
recall the weak Morrey space, $M_*^{p,\lambda}(U)$,
that is the set of functions $f$ on $U$ such that
$$\|f\|^p_{M_*^{p,\lambda}(U)}=\sup\limits_{r>0,(x,t)\in U }\,\Big\{r^{\lambda-(n+2)}
\|f\|_{L^{p,*}(P_r(x,t)\cap U)}^p \Big\}<+\infty,$$
where $L^{p,*}(P_r(x,t)\cap U)$ is the weak $L^p$-space, that is the collection of functions $v$ on $P_r(x,t)\cap U$
such that
$$
\|v\|^p_{L^{p,*}(P_r(x,t)\cap U)}=\sup\limits_{a>0}\, \Big\{\ a^p\left|\{z\in P_r(x,t)\cap U\ : \, |v(z)|>a\}\right|\Big\}<+\infty.
$$
Recall that $(u,d)\in H^{1}(\mathbb R^n\times[0,T],\mathbb R^n\times S^2)$ is a weak solution to
(\ref{lce}) if $(u,d)$ satisfies
(\ref{lce})$_1$-(\ref{lce})$_3$ in sense of distribution and (\ref{lce})$_4$ in sense of trace.
A weak solution $(u,d)\in H^{1}(\mathbb R^n\times[0,T],\mathbb R^n\times S^2)$ of (\ref{lce}) if called
a Serrin's ($p,q$)-solution, if $(u, \nabla d)\in L^p_tL^q_x(\mathbb R^n\times [0,T])$ for some
($p,q$) satisfying (\ref{serrin-condition}).
Our first result concerns an $\epsilon_0$-regularity criterion for Serrin's ($p,q$)-solutions to (\ref{lce}).
\btm\label{loc-reg-th-lcf}{\it There exists $\epsilon_0>0$ such that
if a weak solution $(u,d)\in H^1(P_r(x_0,t_0), \mathbb R^n\times S^2)$ to (\ref{lce}) satisfies
\beq\label{u-lcf2.1}
\|u\|_{L^{p}_t L^{q}_x(P_r(x_0,t_0))}+\|\nabla
d\|_{L^{p}_t L^{q}_x(P_r(x_0,t_0))}\leq\epsilon_0,
\eeq where $p\geq 2$ and
$q\geq n$ satisfy (\ref{serrin-condition}),
then $(u,d)\in
C^{\infty}(P_{\frac{r}{16}}(x_0,t_0))$, and
\beq\label{u-lcf2.2}
r\|u\|_{L^{\infty}(P_{\frac{r}{16}}(x_0,t_0))}+r\|\nabla
d\|_{L^{\infty}(P_{\frac{r}{16}}(x_0,t_0))}
\leq C\left(\|u\|_{L^{p}_t L^{q}_x(P_r(x_0,t_0))}+\|\nabla
d\|_{L^{p}_t L^{q}_x(P_r(x_0,t_0))}\right).
\eeq
}
\etm
A direct corollary of Theorem \ref{loc-reg-th-lcf} is the following regularity theorem
for Serrin's ($p,q$)-solutions to (\ref{lce}).
\begin{corollary}\label{reg-th-lcf}{\it For some $0<T< +\infty$,
suppose that $(u,d)\in H^1(\mathbb R^n\times[0,T],\mathbb R^n\times S^2)$ is a weak solution to (\ref{lce}) with $(u,\nabla d)\in L^p_tL^q_x(\mathbb R^n\times[0,T])$, for some $p>2$ and
$q>n$ satisfying (\ref{serrin-condition}). Then $(u,d)\in
C^{\infty}(\mathbb R^n\times(0,T],\mathbb R^n\times S^2)$.
}
\end{corollary}
\begin{remark}{\it (i) For the heat flow of harmonic maps and the Navier-Stokes equations, Corollary \ref{reg-th-lcf} is valid for the end point case $(p,q)=(+\infty,n)$. It is an interesting open question to investigate the regularity of Serrin's solutions
to (\ref{lce})
in this end point case. \\
(ii) If $(u_0,\nabla d_0)\in L^{\gamma}(\mathbb R^n)$ for some $\gamma>n$, then the local existence of Serrin's solutions in $L^p_tL^q_x$ for some $p>2$ and $q>n$ can be obtained by the fixed point argument (see e.g., \cite{FJR} $\S 4$).\\
}
\end{remark}
As a corollary of Theorem \ref{loc-reg-th-lcf} and Corollary \ref{reg-th-lcf}, we can prove the uniqueness of
Serrin's ($p,q$)-solutions to (\ref{lce}).
\btm\label{unique3}{\it For $n\geq 2$, $0<T< +\infty$, and
$i=1,2$, if $(u_i, d_i):\mathbb R^n\times[0,T]\to\mathbb R^n\times S^2$ are two
weak solutions to (\ref{lce}) with the same initial
data $(u_0,d_0):\mathbb R^n\to \mathbb R^n\times S^2$. Suppose, in
additions, there exists $p>2$ and $q>n$ satisfying
(\ref{serrin-condition}) such that $(u_1,\nabla d_1), (u_2,\nabla
d_2)\in L^{p}_tL^{q}_x(\mathbb R^n\times[0,T])$. Then $(u_1,d_1)\equiv (u_2,d_2)$ on
$\mathbb R^n\times[0,T]$.
}\etm
\begin{remark} {\rm For $n=2$, Lin-Wang \cite{lin-wang} have proved the uniqueness of (\ref{lce})
for $p=q=4$. More precisely, if, for $i=1,2$,
$$
\begin{cases}
u_i\in L^{\infty}([0,T], L^{2}(\R^2,\R^2))
\cap L^{2}([0,T], H^{1}(\R^2,\R^2));\\
\nabla d_i\in L^{\infty}([0,T], L^2(\R^2))\cap L^{2}([0,T], {H}^{1}(\R^2))
\end{cases}
$$
are weak solutions to (\ref{lce}) under the same initial condition,
then $(u_1,d_1)\equiv (u_2,d_2)$ on $\R^2\times[0,T]$. For $n\geq 3$, Lin-Wang \cite{lin-wang}
proved the uniqueness,
provided that $u_i\in C([0,T], L^{n}(\R^n))$ and $\nabla d_i\in C([0,T], {L}^{n}(\R^n))$ for $i=1,2$.}
\end{remark}
\section{Proof of Theorem \ref{loc-reg-th-lcf} and Corollary \ref{reg-th-lcf}}
\setcounter{equation}{0}
\setcounter{theorem}{0}
In this section, we will prove Theorem
\ref{loc-reg-th-lcf} and Corollary \ref{reg-th-lcf} for nematic liquid crystal flows (\ref{lce}). The crucial step
is to establish an $\epsilon_0$-regularity criterion.
\blm\label{unqlc-lemma3.1}{\it There exists $\epsilon_0>0$ such that
if $(u,\nabla d)\in L^p_tL^q_x(P_1(0,1))$, for some $p\geq 2$ and
$q\geq n$ satisfying (\ref{serrin-condition}), is a weak
solution to (\ref{lce}) that satisfies
\beq\label{unqlc3.3}
\|u\|_{L^{p}_t L^{q}_x(P_1(0,1))}+\|\nabla
d\|_{L^{p}_t L^{q}_x(P_1(0,1)))}\leq\epsilon_0,
\eeq
then $(u,d)\in
C^{\infty}(P_{\frac{1}{16}}(0,1))$, and
\beq\label{unqlc3.4}
\|u\|_{L^{\infty}(P_{\frac{1}{16}}(0,1))}+\|\nabla
d\|_{L^{\infty}(P_{\frac{1}{16}}(0,1))}\leq C\epsilon_0.
\eeq
}\elm
Before proving this lemma, we need the following inequality, due to Serrin \cite{serrin}.
\blm\label{unq-serrin-lemma}{\it For any open set $U\subset\mathbb
R^n$ and any open interval $I\subset\mathbb R$, let $f$, $g$, $h\in
L^2_tH^1_x(U\times I)$ and $f\in L^p_tL^q_x(U\times I)$ with $3\le
n\le q\le +\infty$ and $2\le p\le +\infty$ satisfying
(\ref{serrin-condition}). Then
\beq\label{unq2.5}
\int_{U\times I}|f||g||\nabla h| \leq C\|\nabla h\|_{L^2(U\times I)}
\|g\|_{L^2_tH^1_x(U\times I)}^{\frac{n}{q}}
\left\{\int_I\|f\|_{L^q(\mathbb R^n)}^{p}\|g\|_{L^2(\mathbb R^n)}^2\,dt\right\}^{\frac{1}{p}},
\eeq where $C>0$ depends only on $n$. }\elm
\noindent{\bf Proof of Lemma \ref{unqlc-lemma3.1}}.
For any $(x,t)\in
P_{\frac{1}{2}}(0,1)$ and $0<r<\frac{1}{2}$, we have, by
(\ref{unqlc3.3}),
\beq\label{unqlc3.5}
\|u\|_{L^{p}_tL^q_x(P_r(x,t))}+\|\nabla
d\|_{L^{p}_tL^q_x(P_r(x,t))}\leq\epsilon_0.
\eeq
We will divide the proof into two claims.
\noindent{\bf Claim 1}. $\nabla d\in L^{\gamma}(P_{\frac{1}{2}}(0,1))$ for any $1<\gamma<\infty$, and
\begin{equation}
\|\nabla d\|_{L^{\gamma}(P_{\frac{1}{4}}(0,1))}\leq C(\gamma)\|\nabla d\|_{L^p_tL^q_x(P_1(0,1))}.
\end{equation}
To show it, let $d_1: P_r(x,t)\rightarrow \R^3$ solve
\beq\label{unqlc3.6}
\left\{
\begin{split}
\partialrtial_t d_1-\widetilde{D}}\def\del{\widetilde{\Delta}}\def\na{\widetilde{\nabla}elta d_1&=0, \quad\mbox{in } P_r(x,t)\\
d_1&=d, \quad\mbox{on }\partialrtial_pP_r(x,t).
\end{split}\right.
\eeq Set $d_2=d-d_1$. Multiplying (\ref{lce})$_3$ and
(\ref{unqlc3.6}) by $d_2$, subtracting the resulting equations and
integrating over $P_r(x,t)$, we obtain
\beq\label{unqlc3.7}
\begin{split}
&\sup\limits_{t-r^2\leq \tau\leq t}\int_{B_r(x)}|d_2|^2(\cdot,\tau)+2\int_{P_r(x,t)}|\nabla d_2|^2\\
\leq& C\int_{P_r(x,t)}(|u||d_2||\nabla d|+|\nabla d||d_2||\nabla d|)=J_1+J_2.
\end{split}
\eeq By (\ref{unq2.5}), the Poincar$\acute{\mbox{e}}$ inequality and
the Young inequality, we have
\begin{eqnarray*}
|J_1|
&\lesssim&\begin{cases}
\|\nabla d\|_{L^2(P_r(x,t))} \|\nabla d_2\|_{L^2(P_r(x,t))}^{\frac{n}{q}}
\left\{\int_{t-r^2}^t\|u\|_{L^{q}(B_{r}(x))}^{p}\|d_2\|_{L^2(B_r(x))}^2\,d\tau\right\}^{\frac{1}{p}},\ & p<+\infty\\
\|\nabla d\|_{L^2(P_r(x,t))}\|\nabla d_2\|_{L^2(P_r(x,t))}
\|u\|_{L^{\infty}_tL^n_x(P_r(x,t))},\ & p=+\infty,
\end{cases}\\
&\leq& \begin{cases}\frac{1}{2}\|\nabla
d_2\|_{L^2(P_r(x,t))}^{2}+C\epsilon_0\|\nabla
d\|_{L^2(P_r(x,t))}^2
+C\epsilon_0^{\frac{p}2}\|d_2\|_{L^\infty_tL^2_x(P_r(x,t))}^2,\ & p<+\infty\\
\frac{1}{2}\|\nabla
d_2\|_{L^2(P_r(x,t))}^{2}+C\epsilon_0\|\nabla
d\|_{L^2(P_r(x,t))}^2, \ & p=+\infty.
\end{cases}
\end{eqnarray*}
Similarly, for $J_2$, we have
$$
|J_2| \leq \begin{cases}\frac{1}{2}\|\nabla
d_2\|_{L^2(P_r(x,t))}^{2}+C\epsilon_0\|\nabla
d\|_{L^2(P_r(x,t))}^2
+C\epsilon_0^{\frac{p}2}\|d_2\|_{L^\infty_tL^2_x(P_r(x,t))}^2,\ & p<+\infty\\
\frac{1}{2}\|\nabla
d_2\|_{L^2(P_r(x,t))}^{2}+C\epsilon_0\|\nabla
d\|_{L^2(P_r(x,t))}^2, \ & p=+\infty.
\end{cases}
$$
Putting these estimates into (\ref{unqlc3.7}), applying (\ref{unqlc3.5}), and choosing sufficiently small $\epsilon_0$,
we have
\beq\label{unqlc3.12}
\int_{P_r(x,t)}|\nabla d_2|^2
\leq C \epsilon_0\|\nabla d\|_{L^2(P_r(x,t))}^2.
\eeq
This, combined with the standard estimate on $d_1$, implies that for any $\theta\in (0,1)$,
\beq\label{unqlc3.14}
(\theta r)^{-n}\int_{P_{\theta r}(x,t)}|\nabla d|^2\le
C\Big(\theta^2+\theta^{-n}\epsilon_0\Big) r^{-n}\int_{P_r(x,t)}|\nabla d|^2.
\eeq
By iterations, we obtain for any $(x,t)\in P_{\frac{1}{2}}(0,1)$,
$0<r\leq\frac{1}{2}$ and $0<\alpha<1$,
\beq\label{unqlc3.15}
\begin{split}
r^{-n}\int_{P_{r}(x,t)}|\nabla d|^2
\leq Cr^{2\alpha}\int_{P_1(0,1)}|\nabla d|^2.
\end{split}
\eeq
Hence $\nabla d\in\mathcal{M}^{2,2-2\alpha}(P_\frac12(0,1))$ and
\begin{equation}\label{Dd_morrey_estimate}
\|\nabla d\|_{\mathcal M^{2,2-2\alpha}(P_\frac12(0,1))}\le
C\|\nabla d\|_{L^p_tL^q_x(P_1(0,1))}.
\end{equation}
Now Claim 1 follows
by the same estimate of Riesz potentials between parabolic Morrey spaces
as in \cite{huang-wang} (Theorem 1.5) and \cite{lin-wang} (Lemma 2.1).
\noindent{\bf Claim 2}. $u\in L^{\gamma}(P_{\frac{1}{4}}(0,1))$ for
any $1<\gamma<\infty$, and
\begin{equation} \label{integral_estimate_u}
\|u\|_{L^\gamma(P_\frac14(0,1))}\le C(\gamma)\|u\|_{L^p_tL^q_x(P_1(0,1))}.
\end{equation}
Let $\mathbb{E}^\gamma$ be the closure in
$L^{\gamma}(\R^n,\R^n)$ of all divergence-free vector fields with
compact supports. Let $\mathbb{P}:L^{2}(\R^n,\R^n)\rightarrow
\mathbb{E}^2$ be the Leray projection operator. It is well-known
that $\mathbb{P}$ can be extended to a bounded linear operator from
$L^\gamma(\R^n,\R^n)$ to $\mathbb{E}^{\gamma}$ for all
$1<\gamma<+\infty$. Let $\mathbb{A}=\mathbb{P}\widetilde{D}}\def\del{\widetilde{\Delta}}\def\na{\widetilde{\nabla}elta$ denote the Stokes
operator.
For any $(x,t)\in P_{\frac{1}{4}}(0,1)$ and
$0<r\leq \frac{1}{4}$, let $\eta\in C_0^\infty(P_{2r}(x,t))$ be such that
$0\le\eta\le 1$, $\eta\equiv 1$ on $P_r(x,t)$, $|\nabla\eta|\le 4r^{-1}$,
and $|\partialrtial_t\eta|\le 16r^{-2}$. Let $(v,P^1):\mathbb R^n\times (0,1)\to\mathbb R^n\times \mathbb R$ solve
\begin{equation}\label{stokes}
\begin{cases}
\partialrtial_t v-\widetilde{D}}\def\del{\widetilde{\Delta}}\def\na{\widetilde{\nabla}elta v+\nabla P^1=-\nabla\cdot\Big(\eta^2 (u\otimes u+\nabla d\otimes\nabla d-\frac12{|\nabla d|^2}\mathbb{I}_n)\Big) &\ {\rm{in}}\ \mathbb R^n\times (0,1)\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nabla\cdot v= 0 &\ {\rm{in}}\ \mathbb R^n\times (0,1)\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ v=0 &\ {\rm{on}}\ \mathbb R^n\times \{0\}.
\end{cases}
\end{equation}
Define $w: P_r(x,t)\rightarrow \R^n$ by $w=u-v$. Then $w$ solves
the Stokes equation in $P_r(x,t)$:
\beq\label{unqlc3.17} \left\{
\begin{split}
\partialrtial_t w-\widetilde{D}}\def\del{\widetilde{\Delta}}\def\na{\widetilde{\nabla}elta w+\nabla Q^1&=0 \quad\mbox{in } P_r(x,t)\\
\nabla\cdot w&=0\quad\mbox{in } P_r(x,t).
\end{split}\right.
\eeq
By the standard theory of linear Stokes' equations, we have that
$w\in C^\infty(P_r(x,t))$ and,
for any $\theta\in (0, 1)$,
\begin{equation}\label{stokes_est}
\|w\|_{L^p_tL^q_x(P_{\theta r}(x,t))}
\le C\theta\ \|w\|_{L^p_tL^q_x(P_r(x,t))}.
\end{equation}
To estimate $v$, we apply $\mathbb P$ to both sides of the equation (\ref{stokes})$_1$ to obtain
$$\partialrtial_t v-\mathbb A v=-\mathbb P\nabla\cdot\Big(\eta^2 (u\otimes u+\nabla d\otimes\nabla d-\frac12{|\nabla d|^2}\mathbb{I}_n)\Big)
\ {\rm{in}}\ \mathbb R^n\times (0,1); \
v=0 \ {\rm{on}}\ \mathbb R^n\times \{0\}.$$
By the Duhamel formula, we have
\begin{equation}\label{duhamel}
v(t)=-\int_0^te^{-(t-\tau)\mathbb{A}}\mathbb{P}\nabla\cdot\Big(\eta^2(u\otimes u+\nabla d\otimes\nabla d-\frac12{|\nabla d|^2}\mathbb{I}_n)\Big)\,d\tau,
\ 0<t\le 1.
\end{equation}
Now we can apply Fabes-Jones-Riviere \cite{FJR} Theorem 3.1 (see also Kato \cite{kato} page 474, ($2.3'$)) to conclude
that $v\in L^p_tL^q_x(\mathbb R^n\times [0,1])$ and
\begin{eqnarray}\label{v-estimate}
\|v\|_{L^p_tL^q_x(\mathbb R^n\times [0,1])}
&\le& C(\|\eta u\|_{L^p_tL^q_x(\mathbb R^n\times [0,1])}^2+\|\eta \nabla d\|_{L^p_tL^q_x(\mathbb R^n\times [0,1])}^2)\nonumber\\
&\le& C\epsilon_0 (\|u\|_{L^p_tL^q_x(P_{2r}(x,t)))}+\|\nabla d\|_{L^p_tL^q_x(P_{2r}(x,t))}).
\end{eqnarray}
Putting (\ref{stokes_est}) and (\ref{v-estimate}) together, we have that for any $\theta\in (0, 1)$,
\begin{equation}\label{decay_est1}
\|u\|_{L^p_tL^q_x(P_{\theta r}(x,t))}
\le C(\theta+\epsilon_0)\|u\|_{L^p_tL^q_x(P_{2r}(x,t))}
+C\epsilon_0\|\nabla d\|_{L^p_tL^q_x(P_{2r}(x,t))}.
\end{equation}
By Claim 1, we have that
for any $\alpha\in (0,1)$, there exists $\epsilon_0>0$ depending on $\alpha$ such that
\begin{equation}\label{Dd_estimate}
\|\nabla d\|_{L^p_tL^q_x(P_{2r}(x,t))}\le Cr^{\alpha}\|\nabla d\|_{L^p_tL^q_x(P_1(0,1))}.
\end{equation}
Substituting (\ref{Dd_estimate}) into (\ref{decay_est1}) yields
\begin{equation}\label{decay_est2}
\|u\|_{L^p_tL^q_x(P_{\theta r}(x,t))}
\le C(\theta+\epsilon_0)\|u\|_{L^p_tL^q_x(P_{2r}(x,t))}+Cr^\alpha\|\nabla d\|_{L^p_tL^q_x(P_{1}(0,1))}.
\end{equation}
It is standard that by choosing $\theta=\theta_0(\alpha)>0$ and
iterating (\ref{decay_est2}) finitely many times, we conclude that
for any $(x,t)\in P_{\frac{1}{4}}$,
$0<r\leq\frac{1}{4}$ and $0<\alpha<1$,
\begin{equation}\label{decay_est3}
\|u\|_{L^p_tL^q_x(P_{r}(x,t))}
\le C\Big(\|u\|_{L^p_tL^q_x(P_1(0,1))}+\|\nabla d\|_{L^p_tL^q_x(P_{1}(0,1))}\Big)r^\alpha.
\end{equation}
By H\"older's inequality, (\ref{decay_est3}) implies that
$u\in \mathcal M^{2,2-2\alpha}(P_\frac38(0,1))$, and
\begin{equation}\label{u_morrey_estimate}
\|u\|_{\mathcal M^{2,2-2\alpha}(P_\frac38(0,1))}
\le C\Big[\|u\|_{L^p_tL^q_x(P_1(0,1))}+\|\nabla d\|_{L^p_tL^q_x(P_1(0,1))}\Big].
\end{equation}
The higher integrability estimate of $u$ on $P_\frac14(0,1)$ can be done by the parabolic Riesz potential estimate
in parabolic Morrey spaces. Here we will sketch it.
Let $\phi\in C_{0}^{\infty}(P_{\frac{3}{8}}(0,1))$ such that $0\leq\phi\leq 1$, $\phi\equiv1$ on
$P_{\frac{5}{16}}(0,1)$, and
$$|\partialrtial_t\phi|+|\nabla \phi|+|\nabla^2\phi|\leq C.$$
Define $\widetilde{u}:\mathbb R^n\times [0,1]\to\mathbb R^n$ by
\begin{equation}\label{duhamel1}
\widetilde{u}(t)=-\int_0^te^{-(t-\tau)\mathbb{A}}\mathbb{P}\nabla\cdot\Big(\phi^2(u\otimes u+\nabla d\otimes\nabla d-\frac12{|\nabla d|^2}\mathbb{I}_n)\Big)\,d\tau,
\ 0<t\le 1.
\end{equation}
Then, as in the proof of Theorem 3.1 (i) of \cite{FJR}, we have that for any $(x,t)\in\mathbb R^n\times (0,1]$,
\begin{equation}\label{duhamel2}
|\widetilde{u}(x,t)|
\le C\int_0^t\int_{\mathbb R^n}\frac{1}{\delta^{n+1}((x,t), (y,s))}
(|\phi u|^2+|\phi\nabla d|^2)(y,s)\,dyds.
\end{equation}
Recall the parabolic Riesz potential of
order $1$, $I_1(\cdot)$, is defined by
$$I_{1}(f)(z):=\int_{\R^{n+1}}\frac{|f(w)|}{\delta^{n+1}(z,w)}\, dw, \ f\in L^1(\mathbb R^{n+1}).$$
Then we have
\begin{equation}\label{para_morrey}
|\widetilde{u}(x,t)|
\le CI_1({F})(x,t), \ (x,t)\in\mathbb R^n\times (0,1],
\end{equation}
where
$${F}=\phi^2(|u|^2+|\nabla d|^2).$$
By H\"older's inequality, (\ref{Dd_morrey_estimate}), and (\ref{u_morrey_estimate}), we have
that ${F}\in \mathcal{M}^{1,2-2\alpha}(\R^{n+1})$ and
\begin{equation}\label{f_estimate}
\|{F}\|_{\mathcal M^{1,2-2\alpha}(\mathbb R^{n+1})}
\le C\Big(\|\nabla d\|_{L^p_tL^q_x(P_1(0,1))}^2+\|u\|_{L^p_tL^q_x(P_1(0,1))}^2\Big).
\end{equation}
Hence, by \cite{huang-wang} Theorem 3.1 (ii), we conclude that
$\widetilde{u}\in \mathcal M_*^{\frac{2-2\alpha}{1-2\alpha},2-2\alpha}(\R^{n}\times [0,1])$,
and
\begin{eqnarray}\label{morrey_estimate_u}
\|\widetilde{u}\|_{\mathcal M_*^{\frac{2-2\alpha}{1-2\alpha},2-2\alpha}(\R^{n}\times [0,1])}
&\le& C\|{F}\|_{\mathcal{M}^{1,2-2\alpha}(\R^{n+1})}\nonumber\\
&\le& C\left(\|\nabla d\|_{L^p_tL^q_x(P_1(0,1))}^2+\|u\|_{L^p_tL^q_x(P_1(0,1))}^2\right).
\end{eqnarray}
As $\lim\limits_{\alpha\uparrow \frac{1}{2}}\frac{2-2\alpha}{1-2\alpha}=+\infty$,
we have that $\widetilde{u}\in L^{\gamma}(P_{\frac{5}{16}}(0,1))$ for any
$1<\gamma<+\infty$, and
\begin{equation}\label{integral_estimate}
\|\widetilde u\|_{L^\gamma(P_\frac{5}{16})}\le C(\gamma)\Big(\|\nabla d\|_{L^p_tL^q_x(P_1(0,1))}^2+\|u\|_{L^p_tL^q_x(P_1(0,1))}^2\Big).
\end{equation}
Set $\widetilde w=u-\widetilde u$ on $P_\frac{5}{16}(0.1)$. Then it follows from (\ref{lce}) and (\ref{duhamel1})
that
$$\partialrtial_t \widetilde w-\widetilde{D}}\def\del{\widetilde{\Delta}}\def\na{\widetilde{\nabla}elta\widetilde w+\nabla \widetilde{Q}=0;
\ \nabla\cdot\widetilde w=0\ \ \ {\rm{in}} \ \ P_{\frac{5}{16}}(0,1).$$
By the standard theory of linear Stokes' equations, we have that
$\widetilde w\in L^\infty(P_\frac14(0,1))$, and
\begin{eqnarray}\label{integral_estimate1}
\|\widetilde w\|_{L^\infty(P_\frac14(0,1))}&\le&
C\|\widetilde w\|_{L^1(P_\frac5{16}(0,1))}
\le C\Big(\|u\|_{L^1(P_\frac5{16}(0,1))}+\|\widetilde u\|_{L^1(P_\frac5{16}(0,1))}\Big)\nonumber\\
&\le& C\Big(\|\nabla d\|_{L^p_tL^q_x(P_1(0,1))}+\|u\|_{L^p_tL^q_x(P_1(0,1))}\Big).
\end{eqnarray}
It is clear that (\ref{integral_estimate_u}) follows from (\ref{integral_estimate}) and (\ref{integral_estimate1}).
This completes the proof of Claim 2.
Finally, it is not hard to see that by the $W^{2,1}_\gamma$-theory for the heat equation and the linear Stokes equation,
and the Sobolev embedding theorem, we have that $(u,\nabla d)\in L^\infty(P_\frac18(0,1))$. Then the Schauder's theory
and the bootstrap argument can imply that $(u,d)\in C^\infty(P_\frac1{16}(0,1))$.
Furthermore, the estimate (\ref{unqlc3.4}) holds.
This completes the proof. \endpf
\noindent{\bf Proof of Corollary \ref{reg-th-lcf}}: It is easy to see that when $p> 2$, $q>n$, for any $(x,t)\in\mathbb R^n\times(0,T]$, we can find $R_0>0$ such that
\beq\label{u-lcf-3}
\|u\|_{L^p_t L^q_x(P_{R_0}(x,t))}
+\|\nabla d\|_{L^p_t L^q_x(P_{R_0}(x,t))}\leq \epsilon_0,
\eeq
where $\epsilon_0$ is given in Lemma \ref{unqlc-lemma3.1}. By Theorem \ref{loc-reg-th-lcf}, we conclude that
$(u,d)\in C^{\infty}(P_{\frac{R_0}{16}}(x,t))$. This completes the proof of Theorem \ref{reg-th-lcf}
\endpf
\section{Proof of Theorem \ref{unique3}}
\setcounter{equation}{0}
\setcounter{theorem}{0}
In this section, we will prove Theorem \ref{unique3}.
To do this, we need the
following estimate.
\blm\label{unqlc-lemma3.2}{For $T>0$, suppose that $(u,d)$ is a
weak solution to (\ref{lce}) in $\mathbb R^n\times (0,T]$, which
satisfies the assumption of Theorem \ref{unique3}. Then $(u,d)\in
C^{\infty}(\R^n\times(0,T],\mathbb R^n\times S^2)$, and there exists $t_0>0$ such that
for $0<t\le t_0$, it holds
\beq\label{unqlc3.32} \sup\limits_{0<\tau\leq
t}\sqrt{\tau}\Big(\|u(\tau)\|_{L^{\infty}(\R^n)}+\|\nabla
d(\tau)\|_{L^{\infty}(\R^n)}\Big)\leq C\Big(\|u\|_{L^p_t L^q_x(\R^n\times [0,t])}
+\|\nabla d\|_{L^p_t L^q_x(\R^n\times [0,t])}\Big).
\eeq
In particular, we have
\beq\label{unqlc3.36}
\lim\limits_{t\downarrow
0^+}\sqrt{t}\Big(\|u\|_{L^{\infty}(\R^n)}+\|\nabla
d\|_{L^{\infty}(\R^n)}\Big)=0. \eeq}
\elm
\noindent{\bf Proof.\quad}}\def\endpf{
$\Box$ Let $\epsilon_0$ be given by Lemma \ref{unqlc-lemma3.1}.
Since $p>2$ and $q>n$ satisfy (\ref{serrin-condition}),
for any $0<\epsilon\leq\epsilon_0$ we can find $t_0>0$ such that for any $0<\tau\leq \sqrt{t_0}$
\beq\label{u-lcf-3}
\|u\|_{L^p_t L^q_x(\mathbb R^n\times[0,\tau^2])}
+\|\nabla d\|_{L^p_t L^q_x(\mathbb R^n\times[0,\tau^2])}\leq \epsilon.
\eeq
For any $x_0\in \mathbb R^n$, define
\beq\notag
\begin{split}
&\bar{u}(y,s)=\tau u(x_0+y\tau ,s\tau ^2)\\
&\bar{P}(y,s)=\tau^2P(x_0+y\tau ,s\tau ^2)\\
&\bar{d}(y,s)=d(x_0+y\tau ,s\tau ^2).
\end{split}
\eeq
Then $(\bar{u},\bar{P},\bar{d})$ is a weak solution to
(\ref{lce}) on $P_1(0,1)$, and by (\ref{u-lcf-3}),
\beq\label{u-lcf-4} \|\bar u\|_{L^p_t
L^q_x(P_1(0,1))} +\|\nabla \bar d\|_{L^p_t L^q_x(P_1(0,1))}\leq
\epsilon.
\eeq
By Lemma \ref{unqlc-lemma3.1}, we conclude that
\beq\label{u-lcf-5}
|\bar u(0,1)|+|\nabla \bar d(0,1)|\leq C
\left(\|\bar u\|_{L^p_t L^q_x(P_1(0,1))} +\|\nabla \bar d\|_{L^p_t
L^q_x(P_1(0,1))}\right).
\eeq
By rescaling, this implies
\beq\label{u-lcf-6}
\tau\left(| u(x_0,\tau^2)|+|\nabla d(x_0,\tau^2)|\right)\leq C
\left(\|u\|_{L^p_t L^q_x(\mathbb R^n\times[0,\tau^2])}
+\|\nabla d\|_{L^p_t L^q_x(\mathbb R^n\times[0,\tau^2])}\right)\leq C\epsilon.
\eeq
Taking supremum over all $x_0\in\mathbb R^n$ completes the proof.
\endpf
\noindent{\bf Proof of Theorem \ref{unique3}}: By (\ref{unqlc3.36}), we have that for any $\epsilon>0$,
there exists $t_0=t_0(\epsilon)>0$ such that
\begin{eqnarray}\label{max_bound}
\mathcal {A}(t_0)&=&\sum_{i=1}^2\Big[\sup_{0\le t\le t_0}\sqrt{t}(\|u_i(t)\|_{L^{\infty}(\R^n)}+\|\nabla
d_i(t)\|_{L^{\infty}(\R^n)})\nonumber\\
&&+(\|u_i\|_{L^p_t L^{q}_x(\R^n\times [0,t_0]))}+\|\nabla
d_i\|_{L^p_t L^{q}_x(\R^n\times [0,t_0]))})\Big]\le \epsilon.
\end{eqnarray}
It suffices to show $(u_1,d_1)=(u_2,d_2)$ on $\mathbb R^n\times [0,t_0]$. To do so,
let $u=u_1-u_2$ and $d=d_1-d_2$.
Applying $\mathbb{P}$ to both (\ref{lce})$_1$ for $u_1$ and $u_2$ and taking the difference of
resulting equations, we have that
\begin{equation}
\begin{cases}
u_t-\mathbb{A} u=-\mathbb{P}\nabla\cdot\left(u\otimes u_1+u_2\otimes u
+\nabla d\otimes\nabla d_1+\nabla d_2\otimes\nabla d+(|\nabla d_1|+|\nabla d_2|)|\nabla d|\mathbb I_n\right), \\
\nabla\cdot u=0, \\
d_t-\widetilde{D}}\def\del{\widetilde{\Delta}}\def\na{\widetilde{\nabla}elta d=[(\nabla d_1+\nabla d_2)\cdot\nabla d\ d_2+|\nabla d_1|^2d]
-[u\cdot\nabla d_1+u_2\cdot\nabla d], \\
(u,d)\Big|_{t=0}=(0,0).
\end{cases}
\end{equation}
By the Duhamel formula, we have that for any $0<t\le t_0$,
\beq\notag
\begin{split}
u(t)=-\int_0^te^{-(t-\tau)\mathbb{A}}\mathbb{P}\nabla\cdot\Big(u\otimes u_1+u_2\otimes u
+\nabla d\otimes\nabla d_1+\nabla d_2\otimes\nabla d+(|\nabla d_1|+|\nabla d_2|)|\nabla d|\mathbb I_n\Big)\,d\tau,
\end{split}
\eeq
\beq\label{unqlc4.6}
\begin{split}
d(t)=\int_0^te^{-(t-\tau)\widetilde{D}}\def\del{\widetilde{\Delta}}\def\na{\widetilde{\nabla}elta}\Big((\nabla d_1+\nabla d_2)\cdot\nabla
d \ d_2+|\nabla d_1|^2d -u\cdot\nabla d_1-u_2\cdot\nabla
d\Big)\,d\tau.
\end{split}
\eeq
For $0<t\le t_0$, set
$$\Phi(t)=\|u\|_{L^p_tL^{q}_x(\R^n\times [0,t]))}+\|\nabla
d\|_{L^p_tL^{q}_x(\R^n\times [0,t]))}+\sup\limits_{0\leq\tau\leq t}\|d(\cdot,\tau)\|_{L^{\infty}(\R^n)}.$$
By (\ref{unqlc4.6}) and the standard estimate on the heat kernel, we obtain that
\beq\label{unqlc4.9}
\begin{split}
\Big\|\nabla
d(t)\Big\|_{L^{q}(\R^n)}\leq &C\Big[\sum_{i=1}^2\int_0^{t}(t-\tau)^{\frac{1}{p}-1}
\|\nabla d_i\|_{L^{q}(\R^n)}\|\nabla
d\|_{L^{q}(\R^n)}\,d\tau\\
&+\|d\|_{L^{\infty}(\R^n)}\int_0^{t}(t-\tau)^{\frac{1}{p}-1}
\|\nabla d_1\|_{L^{q}(\R^n)}^2\,d\tau\\
&+\int_0^{t}(t-\tau)^{\frac{1}{p}-1}
\|\nabla d_1\|_{L^{q}(\R^n)}\|u\|_{L^{q}(\R^n)}\,d\tau\\
&+\int_0^{t}(t-\tau)^{\frac{1}{p}-1}
\|u_2\|_{L^{q}(\R^n)}\|\nabla d\|_{L^{q}(\R^n)}\,d\tau\Big].
\end{split}
\eeq
By the standard Riesz potential estimate in $L^p$-spaces (see \cite{FJR} Theorem 3.0), we see that
$\nabla d\in L^p_tL^q_x(\mathbb R^n\times [0,t_0])$, and
\beq\label{unqlc4.10}
\begin{split}
\Big\|\nabla d\Big\|_{L^p_tL^q_x(\R^n\times [0,t_0])}\leq&C\Big[\sum_{i=1}^2\|\nabla
d_i\|_{L^p_tL^{q}_x(\R^n\times [0,t_0]))}\|\nabla
d\|_{L^p_tL^{q}_x(\R^n\times [0,t_0])}\\
&+\|d\|_{L^{\infty}(\R^n\times[0,t_0])}\|\nabla
d_1\|^2_{L^p_tL^{q}_x(\R^n\times [0,t_0])}\\
&+\|\nabla
d_1\|_{L^p_tL^{q}_x(\R^n\times [0,t_0])}\|u\|_{L^p_tL^{q}_x(\R^n\times [0,t_0])}\\
&+\|u_2\|_{L^p_tL^{q}_x(\R^n\times [0,t_0])}\|\nabla
d\|_{L^p_tL^{q}_x(\R^n\times [0,t_0])}\Big]\\
\leq&C\mathcal A(t_0)\Phi(t_0).
\end{split}
\eeq
Similarly, by using the estimate of Theorem 3.1 (i) of \cite{FJR}, we have that
$u\in L^p_tL^q_x(\mathbb R^n\times [0,t_0])$, and
\beq\label{unqlc4.11}
\begin{split}
\|u\|_{L^p_tL^{q}_x(\R^n\times [0,t_0])} \leq& C\mathcal A(t_0)\Phi(t_0).
\end{split}
\eeq
Now we need to estimate
$\sup\limits_{0\leq\tau\leq t_0}\|d(\cdot,\tau)\|_{L^{\infty}(\R^n)}$. We claim
\begin{equation}\label{max_estimate_d}
\|d\|_{L^\infty(\mathbb R^n\times [0,t_0])}\le C\mathcal A(t_0)\Phi(t_0).
\end{equation}
To show (\ref{max_estimate_d}), let
$\displaystyle H(x,t)$
be the heat kernel of $\R^n$.
By (\ref{unqlc4.6}), we have
\beq\label{unqlc3.35}
\begin{split}|d(x,t)|=&\Big|\int_0^t\int_{\R^n}H(x-y,t-\tau)\left((\nabla d_1+\nabla d_2)\cdot\nabla d\ d_2+|\nabla
d_1|^2d\right)(y,\tau)\,dyd\tau\\
&-\int_0^t\int_{\R^n}H(x-y,t-\tau)\left(u\cdot\nabla d_1+u_2\cdot\nabla d\right)(y,\tau)\,dyd\tau\Big|\\
\leq& C\Big[\int_0^t\int_{\mathbb R^n} H(x-y,t-\tau)K(y,\tau)\,dyd\tau\\
&\quad +\int_0^t\int_{\R^n} H(x-y,t-\tau)|\nabla d_1|^2(y,\tau)\,dyd\tau
\cdot\sup\limits_{0\leq\tau\leq
t}\|d(\cdot,\tau)\|_{L^{\infty}(\R^n)}\Big], \\
\end{split}\eeq
where
$$K(y,\tau):=\sum\limits_{i=1}^2(|u_i|+|\nabla d_i|)(|u|+|\nabla d|)(y,\tau).$$
By (\ref{max_bound}), we have that for any $0<t\le t_0$,
\beq\label{unqlc3.37}
\begin{split}
&\int_0^t\int_{\mathbb R^n} H(x-y,t-\tau)K(y,\tau)\,dyd\tau\\
\leq&\mathcal{A}(t_0)\int_0^{t}(t-\tau)^{-\frac{n}2}\tau^{-\frac12}\int_{\mathbb R^n}(|u|+|\nabla d|)\exp\Big(-\frac{|x-y|^2}{4(t-\tau)}\Big)
\,dyd\tau\\
\leq&\mathcal{A}(t_0)\Big\|(t-\tau)^{-\frac{n}{2q}}\tau^{-\frac12}\Big\|_{L^{\frac{p}{p-1}}([0,t])}
\Big\||u|+|\nabla d|\Big\|_{L^p_tL^q_x(\mathbb R^n\times [0,t])}\\
\leq& C\mathcal {A}(t_0)\Phi(t_0),
\end{split}\eeq
where we have used H\"older inequality and
\begin{eqnarray*}
\Big\|(t-\tau)^{-\frac{n}{2q}}\tau^{-\frac12}\Big\|_{L^{\frac{p}{p-1}}([0,t])}^{\frac{p}{p-1}}
&=&t^{(\frac12-(\frac{n}{2q}+\frac{1}{p}))\frac{p}{p-1}}
\int_0^1 (1-\tau)^{-\frac{np}{2(p-1)q}}\tau^{-\frac{p}{2(p-1)}}\,d\tau\\
&=&\int_0^1 (1-\tau)^{-\frac{p-2}{2(p-1)}}\tau^{-\frac{p}{2(p-1)}}\,d\tau<+\infty,
\end{eqnarray*}
as (i) $\frac{n}{2q}+\frac{1}{p}=\frac12$,
and (ii) $2<p<+\infty$ yields $\frac{p}{2(p-1)}<1$ and $\frac{p-2}{2(p-1)}<1$.
Similarly, we can obtain that for $0\le t\le t_0$,
\begin{equation}\label{unqlc3.38}
\int_0^t\int_{\R^n} H(x-y,t-\tau)|\nabla d_1|^2(y,\tau)\,dyd\tau
\le C\mathcal {A}^2(t_0).
\end{equation}
Putting (\ref{unqlc3.37}) and (\ref{unqlc3.38}) into (\ref{unqlc3.35}) and taking supremum over $(x,t)\in\mathbb R^n\times [0,t_0]$, we have
\begin{equation}\label{max_bound1}
\sup_{0\le t\le t_0}\|d\|_{L^\infty(\mathbb R^n)}
\le C\mathcal{A}(t_0)\Phi(t_0)+C\mathcal {A}^2(t_0)\sup_{0\le t\le t_0}\|d\|_{L^\infty(\mathbb R^n)}.
\end{equation}
Therefore, if we choose $\epsilon\le \sqrt{\frac{1}{2C}}$ so that
$C\mathcal{A}^2(t_0)\le C\epsilon^2\le \frac12$, then we obtain (\ref{max_estimate_d}).
Putting (\ref{unqlc4.10}), (\ref{unqlc4.11}) and (\ref{max_estimate_d}) together, and choosing $\epsilon\le\frac{1}{2C}$,
we obtain
$$\Phi(t_0)\leq C\mathcal{A}(t_0)\Phi(t_0)\le \frac12\Phi(t_0).$$
This implies that $\Phi(t_0)=0$ and hence $(u_1,d_1)\equiv (u_2,d_2)$ on $\mathbb R^n\times [0,t_0]$.
If $t_0<T$, then we can repeat the argument for $t\in [t_0,T]$ and eventually show that $(u_1,d_1)\equiv (u_2,d_2)$ on $\mathbb R^n\times [0,T]$.
This completes the proof. \endpf
\noindent{\bf Acknowledgements}. The paper is part of my Ph.D. thesis in University of Kentucky. I would like to thank my advisor Professor Changyou Wang for his helpful discussion and constant encouragement.
\end{document} | math | 34,477 |
\begin{document}
\title{Mixed Hodge modules and mixed twistor modules}
\author[M. Saito]{Morihiko Saito}
\begin{abstract} We explain some fundamental differences between the theories of mixed Hodge modules and mixed twistor modules (including the difference in weight system on the nearby cycle functor) which do not seem to be clarified explicitly in the literature.
\end{abstract}
\maketitle
\centerline{\bf Introduction}
\par
n
In the introduction of the first version of \cite{Sab4}, it is stated that there is a {\it fully faithful\,} functor from the category of mixed Hodge modules ${\rm MHM}(X)$ (see \cite{mhm}) to the category of mixed twistor modules ${\rm MTM}(X)$ (see \cite{Mo2}).
If $X$ is a point, then this implies that there is a fully faithful functor from the category of mixed (complex) Hodge structures to the category of mixed twistor structures in the sense of Simpson \cite{Si} (see also \cite[2.1]{Sab2}).
The ``full faithfulness" implies that the {\it Hodge numbers} can be recovered from the image of a mixed Hodge structure under this functor.
However, it is shown by Simpson \cite{Si} that this can be done only by using a ${\mathbb C}^*$-action.
This ``full faithfulness" does not seem to be attained even by replacing ${\rm MTM}(X)$ with the category of {\it integrable} mixed twistor modules ${\rm MTM}^{\rm int}(X)$ as in the second version of \cite{Sab4}, if the integrability condition is taken in a weak sense as in \cite[7.1]{Sab2}, \cite[2.8--9]{Sab4}. Indeed, it does not seem enough to assume that a mixed twistor module {\it admits} an action of $z^2\partial_z$ (or $\lambdambda^2\partial_{\lambdambda}$ in \cite{Mo2}) satisfying certain properties as in \cite{Sab2}, \cite{Sab4}, but we would have to {\it fix} such an action in order to capture the Hodge filtration $F$ as in \cite{Si}. In fact, without it, ${\rm MTM}^{\rm int}(X)$ is still a {\it full subcategory} of ${\rm MTM}(X)$, and ${\rm MTM}^{\rm int}(X)$ would not be called naturally a {\it subcategory} of ${\rm MTM}(X)$ as in the introduction of \cite{Sab4} unless one can {\it choose} naturally a good action of $z^2\partial_z$ for each object in this subcategory, although this does not seem quite easy (since the situation is entirely different from the ``integrability" of a connection), see Remark~(1.6) below for more detailed explanations. Anyway it seems rather difficult to say that the natural functor from ${\rm MTM}^{\rm int}(X)$ to ${\rm MTM}(X)$ is a {\it forgetful functor} forgetting the action of $z^2\partial_z$.
\par
As an example, consider pure Hodge structures of rank $2$ and type
$$\{(p,-p),(-p,p)\}\quad\hbox{for}\,\,\,\,p\in{\mathbb Z}_{>0},
\leqno(1)$$
(see \cite{th2}), which are denoted by $H_{(p)}$. It is quite unclear how one can find a difference between their images for $p\in{\mathbb Z}_{>0}$ in the category of twistor structures of weight 0 without using a ${\mathbb C}^*$-action (which is essentially equivalent to a $z\partial_z$-action) as in \cite{Si}.
Recall that the category of twistor structures of weight $k$ consists of vector bundles $E$ on ${\mathbb P}^1$ which are (non-canonically) isomorphic to direct sums of copies of ${\mathcal O}_{{\mathbb P}^1}(k)$, see \cite{Si}, \cite[2.1]{Sab2}.
\par
Here it is unclear whether a polarization of a pure Hodge structure using the Weil operator is {\it correctly} compared in \cite[Lemma~3.46]{Mo1} with a polarization of the corresponding pure twistor structure endowed with a ${\mathbb C}^*$-action or a $z\partial_z$-action.
Notice that there are two ways of {\it sign convention} for polarizations of Hodge structures used by Deligne and Griffiths, where the difference comes from the place of the Weil operator $C$, and produces a {\it certain difference} in the sign of polarizations of Hodge structures depending on the {\it dimensions of strict supports} (more precisely, see \cite[1.4.7]{ypg}).
Indeed, Deligne's convention is used in mixed Hodge modules (see for instance \cite[Remark before Theorem~3.20]{mhm}), although Griffiths' one is mainly used in Hodge theory recently (including papers of Kashiwara and Kawai), see for instance \cite[Section 5.5]{St}.
It does not seem very clear which convention is used in \cite{Mo1}, \cite{Mo2}, \cite{Sab2}, since the notation is rather complicated.
\par
It is also usually unnoticed that there is a fundamental difference in the {\it weight system} on the nearby cycle functor for Hodge modules and twistor modules, and the weight of a pure twistor module does not not change under non-characteristic restriction, see Section~2 below.
\par
I would like to thank T.~Mochizuki for very important information \cite{Mo3}, \cite{Mo4} giving some explanations about the correspondence between Hodge and twistor modules in \cite[13.5]{Mo2} and also about \cite[Lemma 3.7.9]{Sab2}.
\par
\par
\centerline{\bf 1. ``Integrable ${\mathcal R}$-triples" or ``${\mathcal R}R$-triples"}
\par
n
{\bf 1.1.~${\mathcal R}R$-modules.} Let $X$ be a complex manifold. Set ${\mathcal X}=X\times{\mathbb C}$. By definition ${\mathcal R}_{{\mathcal X}}$ is the subalgebra of ${\mathcal D}_{{\mathcal X}}$ generated by $z\,p_1^*\xi$ for $\xi\in\Theta_X$ over ${\mathcal O}_{{\mathcal X}}$, where $z$ is the coordinate of ${\mathbb C}$, $p_1:{\mathcal X}\to X$ is the first projection, and $\Theta_X$ is the sheaf of vector fields. This ${\mathcal R}_{{\mathcal X}}$ is contained in the subalgebra ${\mathcal R}R_{{\mathcal X}}$ of ${\mathcal D}_{{\mathcal X}}$ generated by $z^2\partial_z$ over ${\mathcal R}_{{\mathcal X}}$.
\par
An ${\mathcal R}$-triple consists of $({\mathcal M}',{\mathcal M}'',C)$ where the ${\mathcal M}',{\mathcal M}''$ are ${\mathcal R}_{{\mathcal X}}$-modules and $C$ is a certain pairing between ${\mathcal M}'|_{X\times S}$ and $\sigma^*{\mathcal M}''|_{X\times S}$ where $S:=\{z\in{\mathbb C}\mid|z|=1\}$ and $\sigma$ is induced by the involution of ${\mathbb C}^*$ defined by $z\mapsto-1/\overline{z}$.
It is called {\it integrable} if the ${\mathcal R}_{{\mathcal X}}$-module structure of ${\mathcal M}',{\mathcal M}''$ is {\it liftable} to an ${\mathcal R}R_{{\mathcal X}}$-module structure so that the pairing $C$ satisfies a certain compatibility condition for the action of $z^2\partial_z$ (see \cite[7.1]{Sab2}).
\par
It seems much better to use the terminology ``${\mathcal R}R$-triple", rather than ``integrable ${\mathcal R}$-triple" here, since the ${\mathcal R}R$-module structure is not uniquely determined by the underlying ${\mathcal R}$-module structure as is shown in (1.2) below. This seems rather misleading for non-specialists.
\par
n
{\bf 1.2.~Case $X=pt$.} The above problem becomes clearer in the case $X=pt$. This case was studied by C.~Simpson \cite{Si}, who showed that ${\mathbb C}^*$-action is needed to capture the Hodge filtration. Here we have
$${\mathcal R}_{{\mathcal X}}={\mathcal O}_{{\mathbb C}},\quad{\mathcal R}R_{{\mathcal X}}={\mathcal O}_{{\mathbb C}}\lambdangle z^2\partial_z\rangle,$$
with $z$ the coordinate of ${\mathbb C}$.
For simplicity, assume furthermore that ${\mathcal M}',{\mathcal M}''$ are finite free ${\mathcal O}_{{\mathbb C}}$-modules and the action of $z^2\partial_z$ comes from that of $z\partial_z$, that is, the corresponding connection has a logarithmic pole.
(Note that a ${\mathbb C}^*$-action is essentially equivalent to a $z\partial_z$-action.)
The eigenvalues of the {\it residue} of the logarithmic connection should be closely related to the complex numbers $p$, $q$ with
$${\rm Gr}_F^pi_1^*{\mathcal M}'\ne 0,\,\,\,{\rm Gr}_F^qi_1^*{\mathcal M}''\ne 0\quad\hbox{(counted with multiplicities)},$$
where $i_1:\{1\}\hookrightarrow{\mathbb C}$ is the inclusion, see \cite{Sab2}.
(Note that these are closely related to the ambiguities of the integrable structure in \cite[Remark~2.9]{Sab4}.)
\par
Indeed, the twistor structure associated with a complex Hodge structure $(H;F',F'')$ can be defined by using
$$\h{$\bigoplus$}_{p\in{\mathbb Z}}\,\,\overline{\!F'^pH}\otimes z^{-p},\quad\h{$\bigoplus$}_{q\in{\mathbb Z}}\,F''^qH\otimes z^{-q}.
\leqno(1.2.1)$$
More precisely, we have to use the {\it dual} of the first term, since the {\it first} term is {\it contravariant} in the case of twistor modules, and a {\it polarization} of complex Hodge structure is {\it not} used here as is noted in an earlier version of this note, see \cite[2.1]{Sab2}, and also (A.2--4) below. Indeed, if we do not use the dual, then we would get a twistor structure of weight 0, since the dual Hodge stricture $H^{\vee}$ is isomorphic to the conjugate Hodge structure $\,\overline{\!H}$ up to the Tate twist $(w)$ with $w$ the weight of the Hodge structure.
\par
The above argument shows that there is a {\it canonical representative} of ${\mathcal R}$-modules (up to a non-canonical isomorphism) using the action of $z^2\partial_z$ in the case of twistor modules coming from variation of complex Hodge structures.
\par
n
{\bf Remark~1.3.} The graded ${\mathbb C}[z]$ modules in (1.2.1) are quite similar to Brieskorn lattices of Gauss-Manin systems associated with {\it weighted homogeneous} polynomials having isolated singularities, see \cite{Sab3}, \cite{bl}, \cite{ScSt}, etc.) In the general ``integrable" case where $z^2\partial_z$-actions are simply assumed, one may have a situation similar to the Brieskorn lattices of certain {\it non-weighted-homogeneous} polynomials.
Here one has to use the graded-quotients of the $V$-filtration along $z=0$. This may be related to the rescaling limit argument in \cite{Sab4} in the case $X=pt$, since the latter seems to be related to the theory of asymptotic Hodge structures by Varchenko \cite{Va} where one takes the {\it leading terms} of asymptotic integrals, see also \cite{ScSt}.
(Note that the irregular case with respect to $\partial_z$ is also possible if we consider $z^2\partial_z$-actions.)
\par
n
{\bf 1.4.~Description of the Hodge filtration $F$.} The argument in (1.2) implies that the Hodge filtration $F$ {\it cannot be captured} unless one {\it fixes} an ${\mathcal R}R$-module structure for ${\mathcal M}',{\mathcal M}''$. Indeed, the eigenvalues of the residues depend heavily on the lifting as an ${\mathcal R}R$-module, see also the example in the introduction which gives {\it various liftings as ${\mathcal R}R$-module structures} for various $p\in{\mathbb Z}_{>0}$. (This may be related with \cite[2.8--9]{Sab4}.) So it would be better to use ``${\mathcal R}R$-triple" rather than ``integrable ${\mathcal R}$-triple" in order to avoid any possible confusions.
(Indeed, ``integrable ${\mathcal R}$-triple" may strongly suggest that an ${\mathcal R}R$-module structure is not fixed although it admits such a structure, see also Remark~(1.6) below.)
\par
Related to the above problem, it does not seem very clear how to interpret, for instance, an ``integrable morphism of the underlying filtered ${\mathcal R}$-triples" in \cite[p.~188, $\ell$.~5]{Mo2}. It is noted in \cite[7.1.c]{Sab2} that a morphism of integrable triples is called ``integrable" if it commutes with ``some representatives of the $\bar{\partial}_z$-actions" (with $\bar{\partial}_z:=z^2\partial_z$).
However, it does not seem quite clear whether some ``equivalence class" is fixed in \cite[p.~188, $\ell$.~5]{Mo2} or not.
(Two actions of $\bar{\partial}_z$ on $({\mathcal M}',{\mathcal M}'')$ are called ``equivalent" in \cite[7.1.c]{Sab2}, if their difference is given by $\lambdambda z$ on ${\mathcal M}'$ and $\bar{\lambdambda}z$ on ${\mathcal M}''$ for some $\lambdambda\in{\mathbb C}$.)
This is closely related to \cite[2.8-9]{Sab4} in the {\it irreducible} (or {\it simple}) case.
Here it does not seem very clear whether {\it direct sums} of integrable triples can be well-defined under the above definition of integrable morphisms, since the ambiguity $\lambdambda$ might depend on direct factors.
This problem seems to be closely related to the example in the introduction, since the Hodge structures there are direct sums of two {\it complex Hodge structures}, see \cite{Si}, \cite[2.1.d]{Sab2}.
\par
As a conclusion, it does not seem quite easy to show that the integrability condition in the weak sense explained above together with the real structure is sufficient to get a fully faithful functor from the category of mixed Hodge modules.
Note that the full faithfulness implies that, for an object in the essential image, the corresponding mixed Hodge module could be determined up to a canonical isomorphism; in particular, the Hodge numbers could be determined uniquely in the case $X=pt$.
\par
n
{\bf 1.5.~Relation with Kashiwara's conjecture.}
It seems {\it highly desirable} to construct a certain category of twistor modules on any variety in such a way that this category contains the category of Hodge modules {\it as a full subcategory} (or at least as a {\it subcategory}) and moreover the underlying ${\mathcal D}$-modules of its objets contain {\it any} irreducible holonomic ${\mathcal D}$-modules. However, one cannot cover all the irreducible local systems on smooth complex varieties if he imposes the above integrability condition (in the weak sense as explained above) on twistor deformations.
In the case of rank 1 local systems on smooth projective varieties, for instance, the integrability condition implies that the twistor deformation (which is slightly different from Simpson's ${\mathbb C}^*$-action) is constant so that the Higgs field vanishes, and hence the local system is {\it unitary}, see \cite{Sab2} and also \cite{tdef} (here it is not very clear whether one can get a correct definition of twistor deformations of local systems in the non-compact case without assuming the extendability to a compactification, see \cite[Remark~2.5(iii)]{tdef}).
So it would be better to avoid the same notation MTM for the one with integrability condition imposed (as in someone's talk in Kyoto).
Perhaps ``(irregular) Hodge-twistor modules" might be more suitable for these objects, since they seem to be quite close to ``irregular Hodge modules", see \cite{Sab4}.
\par
n
{\bf Remark~1.6.} In the introduction of (the first version of) \cite{Sab4}, it is stated as follows: {\it ``The subcategory ${\rm MTM}^{\rm int}(X)$ of integrable objects and morphisms plays an important intermediate role in what follows. The ${\mathcal R}_{{\mathcal X}}$-modules underlying the objects in this subcategory are equipped with a compatible action of $z^2\partial_z$ and the pairing is supposed to be compatible with it. The morphisms in this subcategory are those morphisms in ${\rm MTM}(X)$ which are compatible with the $z^2\partial_z$."}
Here the most nontrivial point is the following question: Of which category is ${\rm MTM}^{\rm int}(X)$ considered as a ``{\it subcategory}"? In view of the above expression, this seems to be regarded as a ``subcategory" of ${\rm MTM}(X)$, and there does not seem to be no other choices. Indeed, one does not seem to be talking about the category whose objects are objects of ${\rm MTM}(X)$ endowed with a good action of $z^2\partial_z$, that is, the category of $W$-filtered ${\mathcal R}R_{{\mathcal X}}$-triples satisfying certain good conditions (the latter category will be denoted by ${\rm MTM}^{\sim}(X)$ in these notes). In fact, ``this subcategory" would be replaced by ``this category" in such a case. Moreover the last sentence of the quoted phrases seems to describe the way in which the author {\it shrinks} the {\it groups of morphisms} of ``this subcategory".
However, there seems to be {\it some non-trivial difference} between the category ${\rm MTM}^{\sim}(X)$ and the situation mentioned in the above quoted sentences. It does not seem quite easy to {\it realize} the above situation without solving some {\it set-theoretic problem} as is explained below.
\par
Indeed, in order to realize the above sentence, one would have to ``give", or rather ``choose", an action of $z^2\partial_z$ (among all the possible choices) for each object in this ``subcategory" of ${\rm MTM}(X)$.
This seems to be equivalent to choosing an {\it objectwise section} (forgetting about morphisms) on the image of the forgetful functor ${\rm MTM}^{\sim}(X)\to{\rm MTM}(X)$. The image of the latter coincides with the {\it objects} of ${\rm MTM}^{\rm int}(X)$. However, this forgetful functor is never injective except for certain special cases, since there is an {\it ambiguity} of choice of the action of $z^2\partial_z$. This means the the same object of ${\rm MTM}(X)$ may underlie many different objects of ${\rm MTM}^{\sim}(X)$. It implies that the natural functor from ${\rm MTM}^{\sim}(X)$ to ${\rm MTM}(X)$ {\it never factors through} ${\rm MTM}^{\rm int}(X)$, and the {\it groups of morphisms of $\,{\rm MTM}^{\rm int}(X)$ do depend} on the choice of the objectwise section, that is, on the choice of the action of $z^2\partial_z$. In particular, ${\rm MTM}^{\rm int}(X)$ would be called rather ``{\it a} subcategory" instead of ``{\it the} subcategory" unless the choice of the $z^2\partial_z$-action is explicitly given.
(In order to understand the situation, the reader may restrict to the case $X=pt$ and consider only pure objects of weight 0 as in the example in the introduction.
It may be also helpful to compare the situation to the case where one identifies a {\it certain full subcategory} of the category ${\bf M}(k[x])$ of $k[x]$-modules with a {\it ``subcategory"} of the category ${\bf M}(k)$ of $k$-vector spaces by {\it choosing} an element of ${\rm End}_k(V)$ as an action of $x$ for each $V\in{\bf M}(k)$, where $k$ is a field and $k[x]$ is the polynomial ring in one variable $x$. Here one gets only a {\it full subcategory} of ${\bf M}(k[x])$ since for each $k$-vector space $V$, there is only {\it one object} in this full subcategory such that its underlying $k$-vector space is $V$. Note that this full subcategory of ${\bf M}(k[x])$ is not necessarily equivalent to the whole category ${\bf M}(k[x])$ if the above ``subcategory" of ${\bf M}(k)$ is badly given.)
\par
It does not seem very clear, however, how the above choice is made in \cite{Sab4}.
This seems to be rather nontrivial even in the case of mixed twistor structures where $X=pt$.
There does not seem to be a {\it canonical} way to do it in general.
It does not seem even clear whether there is a {\it good} way to do it, although it seems relatively easy to choose a {\it very bad} action of $z^2\partial_z$ in such a way that for any two objects in this subcategory which are isomorphic to each other in ${\rm MTM}(X)$, the actions of $z^2\partial_z$ on these two objects are compatible with {\it some} isomorphism between these in ${\rm MTM}(X)$. (Indeed, choose an object in each isomorphism class of objects of ${\rm MTM}(X)$, choose one isomorphism between the chosen object and any other object in the isomorphism class, and then choose an action of $z^2\partial_z$ for the chosen object in the isomorphism class.
In the case of twistor structures of weight $0$, this would correspond to assigning, for instance, a trivial Hodge structure of type $(0,0)$ to {\it any} twistor structure of weight $0$.)
It does not seem very clear in general how these bad choices are excluded in the above quoted situation, for instance, if we choose the action of $z^2\partial_z$ {\it arbitrarily} among possible choices for each object of ``a subcategory" ${\rm MTM}^{\rm int}(X)$.
\par
Actually it may be possible to give a {\it very artificial choice} of the action of $z^2\partial_z$ by dividing each isomorphism class of objects of ${\rm MTM}(X)$ into a disjoint union indexed by all the possible actions of $z^2\partial_z$ on the chosen object in this isomorphism class (provided that the associated set-theoretical problem can be resolved).
In this case, however, it seems more appropriate to say that ``a subcategory" ${\rm MTM}^{\rm int}(X)$ of ${\rm MTM}(X)$ is identified with a certain ``full subcategory" of ${\rm MTM}^{\sim}(X)$ which is equivalent to ${\rm MTM}^{\sim}(X)$, by using this ``very artificial choice".
Here the condition: ``which is equivalent to ${\rm MTM}^{\sim}(X)$" disappears if we take a bad choice of the $z^2\partial_z$-action as above.
Anyway, how to choose an action of $z^2\partial_z$ on each object of ``a subcategory" ${\rm MTM}^{\rm int}(X)$ does not seem to be a trivial matter which can be left without giving any details.
\par
n
{\bf Remark~1.7.} In the third version of \cite{Sab4}, it is stated as follows: {\it ``The category ${\rm MTM}^{\rm int}(X)$ of integrable objects and morphisms plays an important intermediate role in what follows. The ${\mathcal R}_{{\mathcal X}}$-modules underlying the objects in this category are equipped with a compatible action of $z^2\partial_z$ and the pairing is supposed to be compatible with it. The morphisms in this category are defined as are the morphisms in ${\rm MTM}(X)$, with the supplementary condition that they are compatible with the $z^2\partial_z$-action."} Here it does not seem very clear whether the problem explained in Remark~1.6 above is completely solved by this change. The main problem is that the {\it same object} of ${\rm MTM}(X)$ {\it can} underlie {\it many different objects} of ${\rm MTM}^{\rm int}(X)$, or ${\rm MTM}^{\sim}(X)$ following the notation in Remark~(1.6) above, as is repeatedly explained there. In view of this phenomenon, the expression: ``The ${\mathcal R}_{{\mathcal X}}$-modules underlying the objects in this category are equipped with a compatible action of $z^2\partial_z$" could be rather confusing, since it is not quite clear whether the $z^2\partial_z$-action on the underlying ${\mathcal R}_{{\mathcal X}}$-modules {\it can really depend on each object of ${\rm MTM}^{\rm int}(X)$} (or ${\rm MTM}^{\sim}(X)$ in the notation of Remark~(1.6)). Indeed, it seems quite possible for the reader to understand that the author is simply considering ``the ${\mathcal R}_{{\mathcal X}}$-modules underlying the objects in this category" without caring so much about the object of this category that each ${\mathcal R}_{{\mathcal X}}$-module underlies. It may be better to note, for instance, as follows: ``The ${\mathcal R}_{{\mathcal X}}$-modules underlying each object in this category are equipped with a compatible action of $z^2\partial_z$ depending on each object, and not only on the underlying ${\mathcal R}_{{\mathcal X}}$-modules". (Here note that a pair of ${\mathcal R}_{{\mathcal X}}$-modules is needed for each object of ${\rm MTM}(X)$.)
\par
The above problem may have some relation to the difference between an ``inverse functor" and a ``quasi-inverse functor" for a functor of categories $F:{\mathcal C}\to {\mathcal C}'$. Indeed, the former is a functor $G:{\mathcal C}'\to{\mathcal C}$ such that $G\,\raise.15ex\h{${\scriptstyle\circ}$}\, F$ and $F\,\raise.15ex\h{${\scriptstyle\circ}$}\, G$ are the {\it identity} functors on ${\mathcal C}$ and ${\mathcal C}'$. This is, however, rather difficult to construct in practice. What we can usually construct is the latter which satisfies the following conditions: There are functorial isomorphisms $\psi_A:G(F(A))\cong A$ and $\phi_B:F(G(B))\cong B$ for $A\in{\mathcal C}$, $B\in{\mathcal C}'$.
\par
\par
\centerline{\bf 2. Weight system of mixed twistor modules}
\par
n
{\bf 2.1.~Weights on nearby cycles.} Let $X$ be a smooth variety.
The weight of the structure sheaf ${\mathcal O}_X$ is always 0 {\it independently} of the dimension of $X$, according to the authors of \cite{Sab2} and \cite{Mo2}. Here we consider the twistor module corresponding to the {\it constant} twistor deformation of ${\mathcal O}_X$ over ${\mathbb P}^1$ or a constant variation of twistor structure of rank 1 and weight 0, see \cite{Sab2}, \cite{Si}, and also Remark~(2.8) below.
The above assertion follows from the definition of twistor modules using the nearby cycle functors along holomorphic functions (see \cite[4.1]{Sab2} and also \cite[p.~9, $\ell$.~10]{Mo2}), which we apply to local coordinates of $X$ inductively. Indeed, according to these, the weight filtration $W$ on the nearby cycle functor $\psi_f$ of a pure twistor module ${\mathcal M}$ of weight $w$ is given by the monodromy filtration shifted by $w$, instead of $w-1$ as in the Hodge module case. Note that the latter shift of weight by $-1$ in the mixed Hodge module case implies that the weight of a pure Hodge module with a strict support is the sum of the dimension of the support and the pointwise weight at a generic point of the support as in the $\ell$-adic case (see also Kashiwara's remark explained below).
\par
n
{\bf 2.2.~Kashiwara's remark.} If we define the weight filtration $W$ on the vanishing cycle functor $\varphi_{f,1}$ in the same way as $\psi_f$ (that is, the monodromy filtration $W$ is shifted by the weight $w$), then we get the {\it vanishing} of
$${\rm can}:{\rm Gr}^W_k\psi_{f,1}{\mathcal M}\to{\rm Gr}^W_k\varphi_{f,1}{\mathcal M},\quad{\rm var}:{\rm Gr}^W_k\varphi_{f,1}{\mathcal M}\to({\rm Gr}^W_k\psi_{f,1}{\mathcal M})(-1),
\leqno(2.2.1)$$
see Kashiwara's remark before \cite[Theorem 3.21 on p.~303]{mhm}. This shift for $\varphi_{f,1}$ is quite reasonable if one considers the case where the support of ${\mathcal M}$ is contained in $f^{-1}(0)$.
\par
n
{\bf 2.3.~Weights on direct images.} It turns out that the weights of the direct images of twistor modules under projective morphisms would be given (as is noted in \cite[Theorem 6.1.1]{Sab2}) as follows:
\par
n
(A)\quad If $f:X\to Y$ is a projective morphism and ${\mathcal M}$ is pure of weight $w$, then ${\mathcal H}^jf_+{\mathcal M}$ is pure of weight $w+j$.
\par
This seems to hold, for instance, in the case where
$${\mathcal M}={\mathcal O}_X\,\,\,\hbox{with}\,\,\,w=0,\,\,\,Y=pt,\,\,\,\hbox{and}\,\,\,{\mathcal H}^0f_+{\mathcal O}_X={\mathcal H}^0(a_X)_+{\mathcal O}_X\,\,\,\hbox{has weight}\,\,\,0.
\leqno(2.3.1)$$
Here $a_X:X\to pt$ denotes the structure morphism, and we consider the constant twistor deformation of ${\mathcal O}_X$ over ${\mathbb P}^1$ as in (2.1). Recall that the direct image functor of twistor modules for the projection $a_X:X\to pt$ is defined by using the relative de Rham complex which is locally the Koszul complex of the $z\partial_{x_i}$ using the bases $z^{-p}\,\partiald x_{i_1}\wedge\cdots\wedge\partiald x_{i_p}$, where the division by $z^p$ corresponds to the shift of the Hodge filtration $F$. We have, for instance, the induced pairing between $H^0(X,\Omega_X^{d_X})\otimes z^{-d_X}$ and its complex conjugate, where $d_X:=\dim X$. This seems to define a twistor structure of weight $0$, and not $d_X$, as is explained a remark after (1.2.1). Indeed, there is a canonical pairing between $H^0(X,\Omega_X^{d_X})\otimes z^0$ and its complex conjugate, which is {\it constant} on $S^1:=\{z\in{\mathbb C}\mid|z|=1\}$. This would imply the assertion for $H^0(X,\Omega_X^{d_X})\otimes z^{-d_X}$ and its complex conjugate, since $z\bar z=1$ on $S^1$.
\par
More generally, $(p,q)$-forms have a perfect pairing with $(d_X-p,d_X-q)$ forms, and the conjugates of the former are $(q,p)$ forms. This would imply that $H^j(a_X)_+{\mathcal O}_X$ has weight $j$. Indeed, $H^j(a_X)_*{\mathcal O}_X$ should be defined by
$$\bigl(H^{d_X-j}(X,{\mathbb C}),H^{d_X+j}(X,{\mathbb C}),C\bigr),$$
(see \cite[1.6.13]{Sab2}), and we have
$$q-(d_X-p)=j\quad\hbox{if}\quad p+q=d_X+j.$$
(This fundamental example does not seem to be explained in the literature.)
\par
In the ``integrable" case (more precisely, in the ``${\mathcal R}R$-triple case"), the above argument should give the ``Hodge numbers" of the direct image.
\par
The assertion (A) implies that the weights of twistor modules {\it do not} change under the direct images by closed embeddings. (This is {\it different} from the earlier version of this note.)
\par
n
{\bf Remark~2.4.} There is a Tate twist $(-1/2)$ (in the notation of \cite{Sab2}) in the isomorphism between the two nearby cycle functors in \cite[Proposition 4.3.1]{Mo2}, where the usual one is as defined in \cite{Sab2}, and is denoted simply by $\psi$ in this note, and the other one, which is denoted by $\psi^{(1)}$, is defined in \cite{Mo2} by using Beilinson's construction. It may be more natural to use the latter for the inductive definition of twistor modules from the beginning in \cite{Sab2}.
\par
n
{\bf Remark~2.5.} In the calculation of the nearby cycles for a special case which was used in the proof of the decomposition theorem (see \cite[Lemma 3.7.9]{Sab2}), we would have
$${\rm Gr}_k^W\psi_f{\mathcal M}=\begin{cases}(i_Z)_+{\mathcal M}|_Z(1/2)&\hbox{if}\,\,\,k=w-1,\\(i_{Y_1})_+{\mathcal M}|_{Y_1}\oplus(i_{Y_2})_+{\mathcal M}|_{Y_2}&\hbox{if}\,\,\,k=w,\\(i_Z)_+{\mathcal M}|_Z(-1/2)&\hbox{if}\,\,\,k=w+1,\end{cases}
\leqno(2.5.1)$$
Here $w$ is the weight of a pure twistor module ${\mathcal M}$, $f^{-1}(0)$ is a union of normally crossing two smooth hypersurfaces $Y_1,Y_2$ in $X$, and $Y_1,Y_2,Z:=Y_1\cap Y_2$ are transversal to a Whitney stratification of ${\mathcal M}$.
\par
The above Tate twists are not stated in \cite{Sab2} (by the reason that the weights of twistor modules can be changed arbitrarily since there are Tate twists with half-integer values, that is, with any integer weights, {\it according to the author of} \cite{Sab2}).
It turns out that {\it the above Tate twists do not appear at all on the level of ${\mathcal R}$-modules, and they are reflected only on the pairings $C$} (in the manner noted at the end of \cite[Lemma 3.7.9]{Sab2}) according to \cite{Mo4}.
\par
In the ``integrable" case (or more precisely, in the ${\mathcal R}R$-triple case), however, the above assertion may sound rather strange, since there would be a {\it canonical} representative of ${\mathcal R}R$-modules using the action of $z^2\partial_z$ at least in the variation of complex Hodge structure case as is explained in (1.2). In this case, some more arguments concerning the action of $z^2\partial_z$ may be needed for the proof of \cite[Lemma 3.7.9]{Sab2}. (For instance, the Tate twists in (2.5.1) must be {\it more precise} in the ``integrable" case, see (2.9) below.)
\par
Even in the non-integrable case, the argument in \cite{Sab2} (and \cite{Mo4}) may be a little bit confusing because of the definition of $P\psi$ in \cite[3.6.14]{Sab2} since the relation with ${\mathcal M}|_Z$, the usual restriction of ${\mathcal M}$ to $Z$, is not stated there (indeed, the assertion for the underlying ${\mathcal R}$-modules is proved for the first time in the first part of \cite[Lemma 3.7.9]{Sab2}). So the last part of \cite[Lemma 3.7.9]{Sab2} may be misstated. Actually ${\rm Gr}_1^W\psi_fC$ seems to be compared with the pairing of ${\mathcal M}|_Z$ by using {\it two-variable} Mellin transform in the {\it proof} of \cite[Lemma 3.7.9]{Sab2}. However, the argument is rather difficult to follow for non-experts, since the assertion of \cite[Lemma 3.7.8]{Sab2} states only the information about the orders of poles and some assertion from the proof of \cite[Proposition 3.8.1]{Sab2} (that is, (3.8.2) which is shown only in the constant variation case) is quoted.
\par
n
{\bf Remark~2.6.} Let $f:X\to S$ be a projective morphism of smooth varieties with $\dim S=1$. Assume $D:=f^{-1}(0)$ is a reduced divisor with simple normal crossings for $0\in S$. Let $D_i$ be the irreducible components of $D$. Set
$$D_I:=\h{$\bigcap$}_{i\in I}\,D_i\quad\hbox{with $\,\,i_I:D_I\hookrightarrow X\,\,$ the natural inclusion.}$$ Let $t$ be a local coordinate of $(S,0)$. Then the {\it co-primitive} part of the graded quotient of weight $-k$ of the weight filtration on $\psi_f{\mathcal O}_X:=\psi_t(i_f)_+{\mathcal O}_X$ is described as
$$\bigl({\rm Gr}^W_{-k}\psi_f{\mathcal O}_X\bigr){}_{\rm copr}=\h{$\bigoplus$}_{|I|=k+1}\,(i_I)_+{\mathcal O}_{D_I}(k/2)\quad(k\geqslant 0),
\leqno(2.6.1)$$
and the {\it primitive} part of the graded quotient of weight $k$ as
$$\bigl({\rm Gr}^W_{k}\psi_f{\mathcal O}_X\bigr){}_{\rm prim}=\h{$\bigoplus$}_{|I|=k+1}\,(i_I)_+{\mathcal O}_{D_I}(-k/2)\quad(k\geqslant 0),
\leqno(2.6.2)$$
where the latter is related to the former by $N^k$, see \cite[Proposition 3.8.1]{Sab2}. (Note, however, that $i_J^*$ used there is the usual restriction, and is quite different from the pull-back functor in (2.6.4) below, see also \cite[14.3.3]{Mo2}.)
\par
Note that the co-primitive part of ${\rm Gr}^W_{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\psi_f{\mathcal O}_X$ should be related closely to the ${\mathcal D}$-module corresponding to ${\mathbb C}_Y$ by the {\it local invariant cycle theorem} asserting that
$$\alphaigned{\mathbb Q}_Y[\dim Y]=\,&{\rm Ker}\bigl(N:\psi_{f,1}{\mathbb Q}_X\to\psi_{f,1}{\mathbb Q}_X(-1)\bigr)\\\bigl(=\,&{\rm Ker}({\rm can}:\psi_{f,1}{\mathbb Q}_X\to\varphi_{f,1}{\mathbb Q}_X)\bigr).\endaligned
\leqno(2.6.3)$$
These suggest that the native restriction morphism does not work in twistor theory. More precisely, for a closed embedding $i:X\hookrightarrow Y$ of smooth complex manifolds of codimension $r$, and for a twistor module ${\mathcal M}$ on $Y$ which is {\it non-characteristic} to $X$ (for instance, $X$ is transversal to a Whitney stratification of ${\mathcal M}$), the pull-back functor $i^*$ in the derived category (see \cite[14.3.3]{Mo2}) should satisfy
$$i^*{\mathcal M}\cong{\mathcal M}|_X(r/2)[r],
\leqno(2.6.4)$$
where $M|_X$ denotes the usual restriction as in \cite{Sab2}.
(This may be explained somewhere in the literature, see \cite[Lemma 14.3.26]{Mo2} for the Tate twisted constant variation case.) It is closely related to Remark~(2.7) below.
\par
Note that (2.6.4) suggests that, for a smooth morphism $f:X\to Y$ of relative dimension $r$, we would have
$$f^*{\mathcal M}\cong f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}{\mathcal M}(-r/2)[-r],
\leqno(2.6.5)$$
where $f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}{\mathcal M}$ denotes the usual smooth pull-back as in \cite{Sab2}. In the case $Y=pt$, the Tate twist in (2.6.5) seems to be compatible with \cite[13.5]{Mo2}, see (2.8.1) below.
\par
n
{\bf Remark~2.7.} Let $X$ be a smooth projective curve, and $P$ be a point of $X$ with $i_P:\{P\}\hookrightarrow X$ the inclusion. In this note, we denote $a_X^*{\mathbb Q}\in D^b{\rm MHM}(X)$ by ${\mathbb Q}_{h,X}$ with $a_X:X\to pt$ the structure morphism (and similarly for ${\mathbb Q}_{h,P}$). There is a canonical morphism (coming from adjunction morphisms)
$$({\mathbb Q}_{h,X}[1])\to({\mathbb Q}_{h,P})[1],
\leqno(2.7.1)$$
in the derived category of mixed Hodge modules on $X$, where $(M)$ indicates that $M$ is a Hodge module for $M={\mathbb Q}_{h,X}[1]$ or ${\mathbb Q}_{h,P}$ (and $(i_P)_*$ before ${\mathbb Q}_{h,P}$ is omitted to simplify the notation). More precisely, the distinguished triangle associated to this morphism is expressed by the following short exact sequence in the abelian category of mixed Hodge modules ${\rm MHM}(X)$:
$$0\to{\mathbb Q}_{h,P}\to(j_U)_!{\mathbb Q}_{h,U}[1]\to{\mathbb Q}_{h,X}[1]\to 0,
\leqno(2.7.2)$$
where $U:=X\setminus\{P\}$ with inclusion $j_U:U\hookrightarrow X$.
\par
Taking the direct image of (2.7.1) by $a_X:X\to pt$, we then get the canonical isomorphism
$$H^{-1}(a_X)_*({\mathbb Q}_{h,X}[1])=H^0(X,{\mathbb Q})\buildrel\sim\over\longrightarrow H^{-1}(a_X)_*({\mathbb Q}_{h,P}[1])={\mathbb Q}.
\leqno(2.7.3)$$
\par
n
Here one problem is as follows:
\par
n
\rlap{(Q)}\quad\quad Is there a morphism corresponding to (2.7.1) in the bounded derived category\par\par\noindent\quad\quad
of mixed twistor modules?
\par
It does not seem that there is a corresponding morphism in twistor theory if one considers ${\mathcal O}_X$ (with constant twistor deformation over ${\mathbb P}^1$) and its usual restriction to $P\in X$. Here one would have to use Beilinson's nearby cycle functor $\psi^{(1)}$ to define the restriction to $P$. This functor is different from the usual one in \cite{Sab2} (that is, $\psi$ in this note) by the Tate twist $(1/2)$ as is explained in Remark~(2.4). These are closely related to (2.6.4).
\par
Note also that the twistor module ${\mathcal O}_X(-d_X/2)$ in the notation of \cite{Sab2} is used in \cite[13.5]{Mo2} as the twistor module corresponding to the Hodge module ${\mathbb Q}_{h,X}[d_X]$. Here the notation is rather complicated since the dual filtered ${\mathcal D}$-modules are used, see also Remark~(2.8) below. (These follow by interpreting Mochizuki's comments \cite{Mo3} given {\it very recently}.)
If one uses $\widetilde{{\mathcal O}}_X:={\mathcal O}_X(-d_X/2)$ instead of ${\mathcal O}_X$ in Remark~(2.6), then one would have to use Beilinson's nearby cycle functor instead of the usual one to get a good formula.
\par
n
{\bf Remark~2.8.} If $({\mathcal O}_{{\mathcal X}},{\mathcal O}_{{\mathcal X}},C)$ denotes the twistor module of weight 0 corresponding to the constant twistor deformation of ${\mathcal O}_X$ over ${\mathbb P}^1$ as in (2.1) (where ${\mathcal X}:=X\times{\mathbb C}$), then ${\mathcal O}_X(-d_X/2)$ in Remark~(2.7) above is isomorphic to
$$(z^{d_X}{\mathcal O}_{{\mathcal X}},{\mathcal O}_{{\mathcal X}},C)\cong({\mathcal O}_{{\mathcal X}},z^{-d_X}{\mathcal O}_{{\mathcal X}},C)\bigr)\,\bigl(\cong({\mathcal O}_{{\mathcal X}},{\mathcal O}_{{\mathcal X}},C)(-d_X/2)\bigr)
\leqno(2.8.1)$$
where $z^{d_X}$ in the first term corresponds to the shift of the Hodge filtration $F$ (that is, the Tate twist). Note that the first isomorphism in (2.8.1) would not hold unconditionally in the ``integrable" case (that is, in the ${\mathcal R}R$-triple case), since this would change the Hodge numbers if the representative is taken canonically by using the action of $z^2\partial_z$ as in the last remark in (1.2).
\par
The above shift of the Hodge filtration comes from the self-duality isomorphism
$${\mathcal D}D({\mathcal O}_X,F)=({\mathcal O}_X,F[d_X]).
\leqno(2.8.2)$$
A similar isomorphism holds for any pure Hodge modules with $d_X$ replaced by the weight $w$ in general, and this is used in \cite[13.5]{Mo2} in an essential way. This may work at least if one uses {\it right} ${\mathcal D}$-modules where the filtration $F$ on $\Omega_X^{d_X}$ is shifted by $-d_X$. This shift may imply some shift of the filtration $F$ in \cite[13.5]{Mo2}.
If we use {\it left} ${\mathcal D}$-modules instead, then the filtration $F$ on the dualizing sheaf is shifted by $2d_X$, and this may induce a shift of the filtration $F$ in the duality isomorphism for the direct images by projective morphisms, which is used in an essential way in \cite[13.5]{Mo2}. (Note also that there is a shift of filtration $F$ in the transformation between left and right ${\mathcal D}$-modules, and this shift depends on the ambient dimension. This will also induces a shift of filtration $F$ in the duality isomorphism for the direct images by proper morphisms.)
It is rather complicated for the author to follow the arguments in \cite[13.1.1--2]{Mo2} (see also \cite{Mo4}).
\par
n
{\bf 2.9.~Tate twists.} In the ``integrable" case (that is, in the ${\mathcal R}R$-triple case), {\it Tate twists} should be indexed by {\it two} integers $p,q$. For instance, the ``integrable" twistor structure associated to a complex Hodge structure of rank 1 as in (1.2) should have some type $(p,q)$ coming from the Hodge-type of the Hodge structure, and the Tate twists should contain the information about the change of types. These {\it refined Tate twists} may be denoted, for instance, by $(\!(a,b)\!)$, where $-a,\,-b$ give the {\it shift} of type $(p,q)$ (that is, $H(\!(a,b)\!)$ has type $(p-a,q-b)$ if $H$ is an ``integrable" twistor structure of type $(p,q)$). For ${\mathcal M}=({\mathcal M}',{\mathcal M}'',C)$, these can be defined by
$${\mathcal M}(\!(a,b)\!):=(z^{-a}{\mathcal M}',z^b{\mathcal M}'',C).
\leqno(2.9.1)$$
They coincide with the Tate twists ${\mathcal U}(-a,b)$ in \cite[2.1.8.1]{Mo2}, and give $((a+b)/2)$ in \cite{Sab2} {\it forgetting} the action of $z^2\partial_z$. In the ``integrable" case, it may be possible to {\it define} that the Tate twist $(m)$ means $(\!(m,m)\!)$ for $m\in{\mathbb Z}$, but this is {\it unclear} for $m\in\tfrac{1}{2}\,{\mathbb Z}\setminus{\mathbb Z}$. (Note that the Tate twist $(m)$ changes the weights by $-2m$.)
\par
Note that, by considering the argument in (2.3), it may be possible to use also
$${\mathcal M}(\!(a,b)\!)^t:=(z^{-b}{\mathcal M}',z^a{\mathcal M}'',C),
\leqno(2.9.2)$$
where ${\mathcal M}(\!(a,b)\!)^t={\mathcal M}(\!(b,a)\!)$. (In this case, (1.2.1) might be modified appropriately.)
\par
Anyway the Tate twists in (2.5.1), (2.6.1--2), (2.6.4--5) should be replaced by the above {\it refined} Tate twists in the ``integrable" case.
\par
n
{\bf Remark~2.10.} As for the definition of twistor modules, it seems necessary to {\it assume} that the pairing $C$ of twistor modules depends {\it holomorphically} on $z\in S^1:=\{z\in{\mathbb C}^*\mid\,|z|=1\}$ in some sense; more precisely, it should depend {\it real analytically} by using an automorphism of ${\mathbb P}^1$ moving $S^1$ to the real axis ${\mathcal R}RR\cup\{\infty\}\subset{\mathbb P}^1$. Indeed, we would have to glue {\it holomorphic vector bundles}, and it does not seem easy to replace these by $C^{\infty}$ bundles. As for \cite[Lemma 1.5.3]{Sab2}, it does not seem very clear how ``Grothendieck's Dolbeault lemma" is used for its proof if the definition of ${\mathcal C}_{{\mathcal X}|S}^{\infty,{\rm an}}$ is really as in \cite[Example 0.5.2]{Sab2}; for instance, in the case $X=pt$.
\par
As to the $E_2$-degeneration argument of the weight spectral sequence, the same remark as in (A.6) below may apply.
\par
\par
\centerline{\bf Appendix. Some remarks on complex Hodge modules}
\par
n
Recently the theory of {\it complex} Hodge modules is studied by some people. These modules are between real Hodge modules and twistor modules, and the theory seems to be based on some ideas of Kashiwara (see \cite{Ka}, \cite{Sab1}, \cite[(5.1)]{ScVi}). We note here some remarks related to them.
\par
n
{\bf A.1.~Complex conjugation.}
Let $X$ be a complex manifold, and ${\mathcal O}_X$ be the sheaf of holomorphic functions. We denote respectively by ${\mathcal O}_{{\mathcal X}X}$, ${\mathcal O}_{X_{{\mathcal R}RR}}$ the sheaf of anti-holomorphic functions and complex-valued real analytic functions (or $C^{\infty}$ functions if one prefers) on the underlying topological space of $X$.
\par
Let $L$ be a ${\mathbb C}$-local system on $X$. Its complex conjugate is denoted by $\LL$. This is defined by the sheaf $L$ with action of ${\mathbb C}$ given via the {\it complex conjugation}
$$c\mapsto\overline{c}\quad(c\in{\mathbb C}).
\leqno({\rm A}.1.1)$$
If $L$ has rank 1 and a local monodromy of $L$ is given by the multiplication by $\lambda\in{\mathbb C}^*$, then the corresponding local monodromy of $\LL$ is the multiplication by $\overline{\lambda}$.
\par
Set
$$M:={\mathcal O}_X\otimes_{{\mathbb C}}L.$$
This is a locally free ${\mathcal O}_X$-module with holomorphic connection $\nabla$. Set
$${\mathcal M}M:={\mathcal O}_{{\mathcal X}X}\otimes_{{\mathbb C}}\LL,$$
which is the complex conjugate of $M$. This is a locally free ${\mathcal O}_{{\mathcal X}X}$-module with anti-holomorphic connection $\overline{\nabla}$, and is defined by the sheaf $M$ with action of ${\mathcal O}_{{\mathcal X}X}$ given via the {\it complex conjugation}
$$g\mapsto\overline{g}\quad(g\in{\mathcal O}_{{\mathcal X}X}).
\leqno({\rm A}.1.2)$$
\par
n
{\bf A.2.~Variations of complex Hodge structure.}
With the above notation, assume the local system $L$ underlies a polarizable variation of {\it complex} Hodge structure. Here the Hodge filtration $F$ is defined on $M$. We have the {\it opposite Hodge filtration} $F^c$ which, together with the Hodge filtration $F$, gives the {\it Hodge decomposition} of $L_x$ at each $x\in X$. This $F^c$ is defined on
$${\mathcal M}Mc:={\mathcal O}_{{\mathcal X}X}\otimes_{{\mathbb C}}L,$$
inducing a filtration of ${\mathcal O}_{X_{{\mathcal R}RR}}\otimes L$ by scalar extension, {\it but not on} ${\mathcal M}M={\mathcal O}_{{\mathcal X}X}\otimes_{{\mathbb C}}\LL$ in general (unless $L$ is defined over ${\mathcal R}RR$). Its complex conjugate $\overline{F^c}$ is defined on
$$M^c:={\mathcal O}_X\otimes_{{\mathbb C}}\LL,$$
which is the complex conjugate of ${\mathcal M}Mc$, and $\overline{F^c}$ can be described as in (A.3) below by using a {\it polarization} of variation of complex Hodge structure
$$L\otimes_{{\mathbb C}}\LL\to{\mathbb C}.
\leqno({\rm A}.2.1)$$
Note that the latter is equivalent to a ${\mathcal D}_X{\otimes}_{{\mathbb C}}{\mathcal D}_{{\mathcal X}X}$-linear pairing:
$$M\otimes_{{\mathbb C}}{\mathcal M}M\to{\mathcal O}_{X_{{\mathcal R}RR}}\,(\subset{\mathcal D}b_X),
\leqno({\rm A}.2.2)$$
with ${\mathcal D}b_X$ is the sheaf of distributions, see also \cite[(5.1)]{ScVi}.
Here we use ${\mathcal D}_X{\otimes}_{{\mathbb C}}{\mathcal D}_{{\mathcal X}X}$-linear pairings as above. This should be distinguished from the ``sesquilinear pairing":
$$M\times M\to{\mathcal O}_{X_{{\mathcal R}RR}}\,(\subset{\mathcal D}b_X),
\leqno({\rm A}.2.3)$$
which uses the {\it twist} of the action of ${\mathcal O}_X$ by the complex conjugation as in (A.1.2) {\it for one factor}. We will not use sesquilinear pairings except for the case $X$ is a {\it point}, since the use of the complex conjugation in this way could be {\it very confusing} when {\it monodromies} are considered.
\par
n
{\bf A.3.~Description of the opposite filtration.} We can verify that the Hodge filtration $F$ on $M={\mathcal O}_X\otimes_{{\mathbb C}}L$ induces the associated {\it dual} filtration $F^{\vee}$ on
$$M^c={\mathcal O}_X\otimes_{{\mathbb C}}\LL,$$
with respect to the polarization (A.2.1), and $F^{\vee}$ coincides, up to a shift of filtration, with the complex conjugate $\overline{F^c}$ of the opposite filtration $F^c$ on
$${\mathcal M}Mc={\mathcal O}_{{\mathcal X}X}\otimes_{{\mathbb C}}L.$$
\par
Indeed, using the {\it point-wise dual filtration}, we see that the filtration $F$ on $M={\mathcal O}_X\otimes_{{\mathbb C}}L$ induces the filtration $F^{\vee}$ on $M^c={\mathcal O}_X\otimes_{{\mathbb C}}\LL$, or equivalently, $\overline{F^{\vee}}$ on ${\mathcal M}Mc={\mathcal O}_{{\mathcal X}X}\otimes_{{\mathbb C}}L$ (by taking the complex conjugation as in (A.1.2)), and the latter coincides with the opposite filtration $F^c$ up to a shift of the filtration.
In fact, the non-degenerate pairing (A.2.1) induces an isomorphism at each $x\in X$
$$L_x\cong\LL^{\vee}_x,
\leqno({\rm A}.3.1)$$
with left-hand side having the filtration $F$, and this induces the dual filtration $F^{\vee}$ on $\LL_x$, which coincides with the opposite filtration $F^c$ up to a shift of filtration. Here it seems necessary to view the polarization as a {\it sesquilinear pairing} as in (A.2.3) after restricting it over $x\in X$, in order to get the {\it Hodge decomposition} of $L_x$ at each $x\in X$, where $L_x$ and $\LL_x$ are identified with each other up to the action of ${\mathbb C}$.
\par
n
{\bf Remark~A.4.} It has been informed from T.~Mochizuki that a pairing corresponding to the following is used in the twistor theory (where the first factor behaves contravariantly):
$${\mathcal D}D M\otimes{\mathcal M}Mc\to{\mathcal O}_{X_{{\mathcal R}RR}}\,(\subset{\mathcal D}b_X),
\leqno({\rm A}.4.1)$$
which is induced by the canonical morphism $L^{\vee}\otimes L\to{\mathbb C}$.
Here ${\mathcal D}D M$ is the dual as a filtered ${\mathcal D}$-module, and this is different from the dual as a filtered ${\mathcal O}$-module by a shift of filtration $F$, see (A.5.2) below. Note that this shift is very important in the theory, see \cite[13.5]{Mo2}.
\par
n
{\bf Remark~A.5.} In the case of variations of complex Hodge structure, it may be more natural to use the following ${\mathcal D}_X{\otimes}_{{\mathbb C}}{\mathcal D}_{{\mathcal X}X}$-linear pairing:
$$M\otimes_{{\mathbb C}}\overline{M'}\to{\mathcal O}_{X_{{\mathcal R}RR}}\,(\subset{\mathcal D}b_X),
\leqno({\rm A}.5.1)$$
where
$$(M',F):={\mathcal D}DD(M^c,\FFc[-d_X])={\mathcal D}DD_{{\mathcal O}}(M^c,\FFc)\,\bigl(:={\mathcal H} om_{{\mathcal O}_X}\bigl((M^c,\FFc),({\mathcal O}_X,F)\bigr)\bigr),
\leqno({\rm A}.5.2)$$
with ${\rm Gr}_F^p{\mathcal O}_X=0$ ($p\ne 0$). Indeed, the argument in (A.3) implies
$${\mathcal D}DD_{{\mathcal O}}(M^c,\FFc)=(M,F[w]).$$
Here ${\mathcal D}DD$ and ${\mathcal D}DD_{{\mathcal O}}$ denote respectively the dual functors for filtered holonomic ${\mathcal D}$-modules and for filtered locally free sheaves.
\par
Since the second factor of the pairing must behave {\it contravariantly} in the case of complex Hodge modules, the above formula (using the dual functor ${\mathcal D}DD$) may be useful especially in the {\it mixed} case where we have the weight filtration $W$. (Indeed, we would have to use a weight ``co-filtration" for the second factor otherwise.)
\par
Here it seems useful to introduce the {\it Hermitian dual}\,:
$${\mathcal D}DD^H(M):=\overline{{\mathcal H} om_{{\mathcal O}_X}(M,{\mathcal D}b_X)},$$
as well as the {\it Hermitian conjugate}\,:
$$M^{HC}:={\mathcal D}DD({\mathcal D}DD^H(M)).$$
These are isomorphic to ${\mathcal D}DD(M^c)$ and $M^c$ respectively in the case of variations of complex Hodge structure by the above argument. Setting $M''={\mathcal D}DD(M')$, we could use the isomorphism
$$M^{HC}\cong M'',\quad\hbox{or equivalently,}\,\,\,{\mathcal D}DD^H(M)\cong{\mathcal D}DD(M''),$$
(instead of a pairing) for the definition of complex Hodge modules, where $M''={\mathcal D}DD(M')$ has the filtration $F$ (corresponding to the {\it opposite filtration} $F^c$ by the complex conjugation) so that the above isomorphism induces complex Hodge structures at any $x\in X$ in the case of polarizable variations of complex Hodge structure.
\par
n
{\bf Remark~A.6.} The formulation in Remark~A.5 above seems to be useful for the proof of the $E_2$-{\it degeneration} of the weight spectral sequence, where one has to show the {\it compatibility} of the above structure with the {\it differentials} $d_r$ of the spectral sequences for any $r\geqslant 2$. More precisely, the {\it compatibility} of each differential $d_r$ of the two spectral sequences with the induced pairing between the $E_2$-terms (which are identified with the $E_r$-terms) must be proved {\it before showing their vanishing} for each $r\geqslant 2$. This problem does not seem quite easy to solve by using a {\it pairing argument} unless we adopt the above formulation together with the standard argument of spectral sequences associated with filtered complexes. Note that it is enough to prove a {\it comparison isomorphism} between two spectral sequences in the latter case. Here one would have to use the vanishing of ${\mathcal E}xt^i_{{\mathcal D}_X}(M,{\mathcal D}b_X)$ for $i\ne 0$ (see \cite{Ka}) in order to apply the derived functor argument.
\par
As for \cite{Ka} and \cite{Sab1}, these seem to be closely related to {\it ideas} behind the theories of twistor modules and complex Hodge modules, although they do not seem to be quoted explicitly in \cite{Sab2}. (This may be rather strange, see also a sentence before \cite[(5.1)]{ScVi} or p.~19, \raise1pt\hbox{$\uparrow\,$}l.~6 in arXiv:1206.5547.)
\par
n
{\bf Remark~A.7.} In the case of {\it real\,} Hodge modules, the above ``weight spectral sequence" is {\it defined} (or constructed) in the category $MF_{rh}({\mathcal D}_X,{\mathcal R}RR)$ consisting of filtered regular holonomic ${\mathcal D}$-modules endowed with a real structure. However, the ``corresponding category" together with the ``corresponding argument" for {\it complex} Hodge modules does not seem to be very clear.
\par
If we use the above Hermitian conjugate $M^{HC}$, then this category can be given by the category whose objects are pairs of filtered holonomic ${\mathcal D}$-modules $(M,F)$, $(M'',F)$ endowed with an isomorphism $\alpha:M^{HC}\cong M''$ as in Remark~A.5 above, and the argument is relatively easy as is explained in Remark~A.6 above.
\par
If the Hermitian conjugate is not used, then one would have to use a {\it pairing argument} between two ``{\it spectral objects}" in the sense of Verdier \cite{Ve} in order to replace the above {\it comparison isomorphism} argument between two spectral sequences, where the argument would be much more complicated. (This pairing argument between two spectral objects seems to be useful also in the twistor case.)
\par
n
{\bf Remark~A.8.} It would be highly desirable to write down {\it complete proofs} of main theorems in the theory of {\it complex} Hodge modules before giving courses about these. Otherwise, it would be rather irresponsible toward the audience.
For instance, generalizations of results of Schmid and Zucker to the {\it non-quasi-unipotent} monodromy case seem to be far from trivial.
\par
There does not seem to be any example where rational or real Hodge modules are not enough, and complex Hodge modules are really needed in algebraic geometry, except for certain cases related to representation theory as in \cite{ScVi}.
\end{document} | math | 52,992 |
\begin{document}
\title{Private Quantum Subsystems}
\author{Tomas Jochym-O'Connor}
\affiliation{Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada}
\affiliation{Department of Physics \& Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada}
\author{David W. Kribs}
\affiliation{Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada}
\affiliation{Department of Mathematics \& Statistics, University of Guelph, Guelph, Ontario, N1G 2W1, Canada}
\author{Raymond Laflamme}
\affiliation{Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada}
\affiliation{Department of Physics \& Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada}
\affiliation{Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, Canada}
\author{Sarah Plosker}
\affiliation{Department of Mathematics \& Statistics, University of Guelph, Guelph, Ontario, N1G 2W1, Canada}
\date{\today}
\begin{abstract}
We investigate the most general notion of a private quantum code, which involves the encoding of qubits into quantum subsystems and subspaces. We contribute to the structure theory for private quantum codes by deriving testable conditions for private quantum subsystems in terms of Kraus operators for channels; establishing an analogue of the Knill-Laflamme conditions in this setting. For a large class of naturally arising quantum channels, we show that private subsystems can exist even in the absence of private subspaces. In doing so, we also discover the first examples of private subsystems that are not complemented by operator quantum error correcting codes; implying that the complementarity of private codes and quantum error correcting codes fails for the general notion of private quantum subsystem.
\end{abstract}
\pacs{03.67.Hk; 03.67.Pp}
\mathcal aketitle
\section{Introduction \& Background}
The most essential primitive for private communication between two parties, Alice and Bob, in classical computation is the one-time pad. In such a scheme, the two parties share a secret key that is unknown to an external observer Eve; this key enables reliable communication by the parties as the message appears to be a random mixture of input bits from Eve's viewpoint without the key.
Private quantum codes were initially introduced as the quantum analogue of the classical one-time pad. The basic setting for a ``private quantum channel'' \cite{AMTW00,BR03} is as follows: Alice and Bob share a private classical key that Alice uses to inform Bob which of a set of unitary operators $\{ U_i \}$ she has used to encode her quantum state: $\rho \mathcal apsto U_i \rho U_i^\dagger$. With this information in hand, Bob can decode and recover the state $\rho$ without disturbing it. The set of unitaries $\{U_i\}$ and the probability distribution $\{p_i\}$ that makes up the random key which determines the encoding unitary are shared publicly. Thus, without further information, Eve's description of the system is given by the random unitary channel $\Phi(\rho) = \sum_i p_i U_i \rho U_i^\dagger$. By selecting certain sets of unitary operators with appropriate coefficients, the random unitary channel will provide Eve with no information about the input state.
The body of work on private quantum codes now includes a variety of other applications, with realizations both as subspaces and subsystems of Hilbert space. Private shared reference frames exploit private subspaces and subsystems that also arise from the ignorance associated with an eavesdropper's description of a system \cite{BRS04,BHS05}. The notion of using mixed state ancilla qubits to encode information, which can be viewed as subsystem encodings, has also been studied in the context of quantum secret sharing~\cite{CGL99,CGS02}. There, the goal is to encode information into a globally mixed state of $n$~qubits such that to recover the quantum information one would need access to $k$ qubits of the global state, where any fewer would yield no information regarding the initial state. Using mixed states allows for the increase of $k$ for a fixed $n$, thus solidifying the idea that mixed state encodings increase privacy. There are also bridges between these works and quantum error correction, formalized by the complementarity results of \cite{KKS08}. Connections between the study of private quantum codes and the theory of operator algebras have recently been found as well \cite{CKPP12}.
In this Letter we consider the most general notion of a private quantum code \cite{AMTW00,BR03,BRS04}, which involves the encoding of quantum bits into subsystems. Private quantum channels, private subspaces, and what we refer to as ``operator'' private subsystems---are captured as special cases of this general phenomena. We consider a class of phase damping channels throughout the presentation that highlights the main differences between mappings on subsystems and subspaces. Most surprisingly, we show that certain classes of channels can only be private in the subtle subsystem sense; thus establishing that private subsystems can exist in the absence of private subspaces.
We also make the first significant move toward a structure theory for private quantum codes; specifically we set out algebraic conditions that characterize privacy of a code in terms of the Kraus operators for a given quantum channel. This can be viewed as an analogue of the set of Knill-Laflamme conditions \cite{KL97} from quantum error correction to this setting, and indeed we discuss further connections with error correction. In particular we show that complementarity of private and error-correcting codes fails at the most general level, and we point out a potentially new type of quantum error-correcting code.
We now describe our notation and nomenclature. Given a quantum system $S$, with (finite-dimensional and complex) Hilbert space also denoted by $S$, we will use customary notation such as $\rho$, $\sigma$ for density operators. The set of linear operators on $S$ is denoted by $\mathcal athcal{L}(S)$. Linear maps on $\mathcal athcal{L}(S)$ can be viewed as operators acting on the operator space $\mathcal athcal{L}(S)$. We use the term \emph{(quantum) channel} to refer to a completely positive and trace-preserving linear map $\Phi:\mathcal athcal{L}(S)\rightarrow \mathcal athcal{L}(S)$. Such maps describe (discrete) time evolution of open quantum systems in the Schr\"{o}dinger picture, and can always be written in the Choi-Kraus operator-sum form $\Phi(\rho) = \sum_i V_i \rho V_i^\dagger$ for some operators $V_i$ in $\mathcal athcal{L}(S)$ satisfying $\sum_iV_i^\dagger V_i=I$. The composition of two maps will be denoted by $\Phi\circ \Psi(\sigma )=\Phi(
\Psi(\sigma ))$.
A (linearly closed) subspace $C$ of $S$ is said to be a \emph{private subspace} for $\Phi$ if
there is a density operator $\rho_0$ on $S$ such that
$\Phi (\ket{\psi}\bra{\psi}) = \rho_0$ for all pure states $\ket{\psi}$ in $C$. By linearity, $\Phi (\rho)
= \rho_0$ for all $\rho$ in $\mathcal athcal L(C)$. We could also consider a collection of private states not associated with a subspace of the Hilbert space, but, as in quantum error correction, we wish to allow for arbitrary superpositions of our code states and this demands the set of states considered are linearly closed.
A quantum system $B$ is a \textit{subsystem} of $S$ if we can write
$S=(A\otimes B)\oplus (A\otimes B)^{\perp }$.
This definition is symmetric in that $A$ is also considered a subsystem of $S$. The sub\emph{spaces} of $S$ can
be viewed as subsystems $B$ for which $A$ is one-dimensional. A subscript such as $\sigma_B$ means the operator belongs to $\mathcal athcal{L}(B)$. A subsystem $B$ is a \emph{private subsystem} for $\Phi$ if there is a $\rho_0 \in \mathcal athcal{L}(S)$ and $\sigma_A \in \mathcal athcal{L}(A)$ such that
\begin{eqnarray}\label{private_definition}
\Phi (\sigma_A \otimes \sigma_B) = \rho_0 \quad \forall \sigma_B\in \mathcal athcal{L}(B).
\end{eqnarray}
The case of random unitary channels $\Phi$ in
Eq.~(\ref{private_definition}) was first considered in
\cite{AMTW00,BR03} where the terminology \emph{private quantum channels} was used, and the case of general channels $\Phi$ was
formalized in \cite{BRS04} where private subsystems were given the extra prefix ``completely'' that we have dropped. If Eq.~(\ref{private_definition}) holds for all $\sigma_A$, as opposed to a single state
$\sigma_A$, then we shall refer to $B$ as an \emph{operator private subsystem} (since these are precisely the private subsystems that are complementary to operator quantum error-correcting subsystems discussed below).
\section{Private Subsystems In The Absence Of Private Subspaces}
An operator private subsystem is one in which the private channel
splits into a product of maps on the individual subsystems $A$ and $B$ when
the channel is restricted to the combined product subspace $A \otimes B$. Such private subsystems cannot exist
without the existence of private subspaces; indeed, if Eq.~(\ref{private_definition})
holds for all states on $A$, it follows that every sub\emph{space}
$\ket{\psi}\otimes B$ is private for $\Phi$ for any fixed pure state $\ket{\psi}$ on $A$.
Even though the definition given by
Eq.~(\ref{private_definition}) allows for the possibility of
examples of private subsystems that do not extend to private subspaces, the private subsystems
exhibited in the literature
\cite{AMTW00,BR03,BRS04,BHS05} thus far have either been of operator type,
or are already subspaces. Here we present the first examples of private
subsystems for which there are no private subspaces that exist; in particular these are private subsystems that are
not of operator type. Our motivating class of channels is
built upon a simple phase damping model. We begin the discussion
by recalling the most basic private quantum channel and asking
some basic questions on quantum privacy.
The completely depolarizing channel ($\Phi(\rho) = \frac{1}{\dim
S}I$ for all $\rho$) is an easy to describe example of a
quantum channel that is private. In this case the entire Hilbert
space acts as a private code for the channel, and so in order to
implement such a private channel a full set of Pauli rotations
must be available. This leads to a very basic question in the
study of private quantum codes: Do there exist channels with fewer
physical operations such that we can still encode qubits for
privacy?
Perhaps the simplest class of channels one could imagine would
be the family of phase damping channels that can be applied to any
qubit of a larger Hilbert space~$\mathcal {S}$ of $n$~qubits,
\begin{eqnarray}
\Lambda_i (\rho) = \frac{1}{2}(\rho + Z_i \rho Z_i ), \qquad \forall \rho \in \mathcal {L}(S).
\end{eqnarray}
A single qubit phase damping channel is not private. Yet we can
ask: can composing the phase damping channel on multiple qubits
yield a private subspace~$C \subseteq\mathcal {S}$? Such a question is
analogous to the sort of questions that have been asked in quantum
error correction for some time; for example, given a set of errors
that are uncorrectable on a single qubit, does there exist a
larger Hilbert space such that the action of the error on the
encoded Hilbert space is correctable? The answer to such a
question in quantum error correction is yes, as demonstrated by
the five-qubit code which corrects for arbitrary single-qubit
errors, an error that would be uncorrectable if one did not have
access to a larger Hilbert space to encode the quantum information
into a quantum code.
We shall define the map $\Lambda$ as the composition of the maps $\Lambda_i$ on each of the $n$~qubits of the state $\rho \in S$,
\begin{eqnarray}
\Lambda (\rho) &= \Lambda_n \circ \Lambda_{n-1} \circ \dots \circ \Lambda_1 (\rho).
\end{eqnarray}
Equivalently one could consider the $n$-product map
$\Lambda_1^{\otimes n}$ of the single qubit channel $\Lambda_1$.
For any input state $\rho$, this channel will decohere all
off-diagonal terms in the computational basis; as such, the
resulting output density matrix will be diagonal.
Consider the case when $n=2$. Every output state of $\Lambda$ has the form
\begin{eqnarray}
\label{eq:outputstate}
\rho_0 = \frac{1}{4}\Big( II + \alpha IZ + \beta ZI + \gamma ZZ \Big),
\end{eqnarray}
where $I$ and $Z$ are the one-qubit identity and Pauli $Z$
matrices. The goal is to find a subspace $C$ of dimension~2 and a
state $\rho_0 \in \mathcal {L}(S)$ such that $\Lambda(\rho) = \rho_0$
$\forall \rho \in \mathcal {L}(C)$. This would show that $\Lambda$ has a
private qubit subspace, defined by a pair of orthogonal logical states
$\ket{0_L}$, $\ket{1_L}$ in $C$.
However, one can show that such a subspace \emph{does not} exist.
In fact we can prove the following more general result, which
applies to the channels $\Lambda$ directly, and can be extended to channels
with commuting normal Kraus operators as well. We leave the proof for \cite{JKLP12}.
\begin{lemma}
Let $\Phi$ be a random unitary channel with mutually commuting Kraus operators.
Then $\Phi$ has no private subspaces.
\label{lem:RandomUnitaryChannel}
\end{lemma}
Is this the end of the story? This result is intuitive---at first
glance it certainly does not ``feel'' as though we should be able
to find private codes for channels such as the phase damping maps
$\Lambda = \Lambda_n \circ \cdots \circ \Lambda_2 \circ
\Lambda_1$ due to the preservation of information stored in
the diagonal elements of the initial density matrix.
Moreover, the experience with operator private
subsystems, which demand the existence of private subspaces, also
suggests we can go no further with these channels. Somewhat
surprisingly, we do find private subsystems for these channels,
and necessarily they are not of the type exhibited before.
Indeed, consider the following logically encoded qubits in two-qubit
Hilbert space:
\begin{eqnarray}\label{subsystem_encoding}
\rho_L=\frac14(II+\alpha XX + \beta YI+\gamma ZX).
\end{eqnarray}
This describes a single qubit encoding, as
Eq.~(\ref{subsystem_encoding}) describes the coordinates for a
logical Bloch sphere in two-qubit Hilbert space with logical Pauli
operators given by $X_L=XX, Y_L=YI, Z_L=ZX$. Now, observe that the dephasing map $\Lambda = \Lambda_2 \circ
\Lambda_1$ acting on each density operator $\rho_L$ produces an
output state that is maximally mixed; that is, $\Lambda(\rho_L) =
\frac14 \, II$ for all $\rho_L$. Thus, we see that
Eq.~(\ref{subsystem_encoding}) yields a private two-qubit code for the
dephasing map $\Lambda$. However, we know from
Lemma~\ref{lem:RandomUnitaryChannel} that the input space cannot
be a subspace, and we have already noted this implies it also
cannot be an operator subsystem. It is however still a private
subsystem in the sense of Eq.~(\ref{private_definition}). Let us discuss the encoding in more detail.
\begin{figure}
\caption{The gates in the red box implement the encoding of an arbitrary two-qubit state belonging to the $I \otimes \mathcal athbb{C}
\label{fig:PrivateChannel}
\end{figure}
The logical encoding of a single qubit into a two-qubit subsystem is shown by the unitary operation given by the red boxed region in Figure~\ref{fig:PrivateChannel}. The mapping, given by a pair of CNOT gates and the $T = \frac1{\sqrt{2}} (\ket{0}(\bra{0} + \bra{1}) + i \ket{1}(\bra{0} - \bra{1}))$, shows a unitary equivalency between the two-qubit operator algebra~$I_2 \otimes \mathcal athbb{C}^{2\times 2}$ and the encoded logical qubit through the following transformation of the basis elements of $\mathcal athbb{C}^{2\times 2}$,
\begin{eqnarray*}
IX \longmapsto & IX &\longmapsto XX \longmapsto ZX\\
IY \longmapsto & ZY &\longmapsto YX \longmapsto XX\\
IZ \longmapsto & ZZ &\longmapsto ZI \longmapsto IY.
\end{eqnarray*}
More generally, a logical qubit encoding into a subsystem of a $n$-qubit Hilbert space can be constructed to privatize the $n$-qubit phase damping channel $\Lambda = \Lambda_n \circ \cdots \circ \Lambda_2 \circ \Lambda_1$, which by Lemma~\ref{lem:RandomUnitaryChannel} cannot have a private subspace.
\begin{theorem}
For any $n$-qubit Hilbert space $\mathcal athcal{H}$, there exist quantum channels $\Phi$ for which a private quantum subsystem $B$ of $\mathcal athcal{H}$ can be constructed in the absence of the existence of any private quantum subspace $C \subseteq \mathcal athcal{H}$.
\label{thm:RandomUnitaryChannel}
\end{theorem}
\section{Testable Conditions For Private Quantum Codes}
If we are given a quantum channel $\Phi(\rho) = \sum_i V_i \rho V_i^\dagger$ and a subsystem $B$, we can ask if it is possible to decide whether $B$ is private for $\Phi$; and more to the point, we can ask if this can be answered in terms of the Kraus operators $V_i$ for the channel. The analogous question in quantum error correction is answered by the fundamental Knill-Laflamme conditions \cite{KL97}, which provide an explicit set of algebraic constraints in terms of the Kraus operators and the code, and allow one to test whether a given code is correctable for a channel. The generalization of these conditions to the case of operator error-correcting subsystems was established in \cite{KLP05,KLPL05,NiPo07}.
The following result answers this question for private quantum subsystems. In addition to Kraus operators, we would expect the algebra to include the fixed $A$ state $\sigma_A$ and output state $\rho_0$---observe that this information is indeed included in the conditions.
\begin{theorem}
A subsystem $B$ is private for a channel
$\Phi(\rho) = \sum_i V_i \rho V_i^\dagger$ with fixed $A$
state $\sigma_A$ and output state $\rho_0$ if and only if
there are complex scalars $\lambda_{ijkl}$ forming an isometry matrix $\lambda = (\lambda_{ijkl})$ such that
$\sqrt{p_k }V_j\ket{\psi_{A,k}} = \sum_{i,l}\lambda_{ijkl} \sqrt{q_l}\ket{\phi_{l}}\bra{\psi_{B,i}}$,
where $\ket{\psi_{A,k}}$ ($p_k$) and $\ket{\phi_{l}}$ ($q_l$) are eigenstates (eigenvalues) of
$\sigma_A$ and $\rho_0$ respectively, $\ket{\psi_{B,i}}$ is an orthonormal basis
for $B$, and where $\ket{\psi_{A,k}}$ is viewed as a channel from $B$ into $S$.
\end{theorem}
The key observation in establishing this result is that the left and right hand sides of Eq.~(\ref{private_definition}) each define channels from $B$ to $S$ which are in fact the same. One can then use basic results from the theory of completely positive maps to obtain the equations spelled out in the theorem. More details on the theory will be presented in \cite{JKLP12}.
It is important to note that this result is new even for private sub\emph{spaces}. In the notation of the theorem for that case, $A$ is one-dimensional and $B$ is the subspace. If we let $P_B$ be the projector of $S$ onto $B$, then we see that the characterization of privacy is given by the conditions: $V_j P_B = \sum_{i,l}\lambda_{ijl} \sqrt{q_l}\ket{\phi_{l}}\bra{\psi_{B,i}}$ for all $j$. As a simple illustration, in the case of the completely depolarizing channel on $N$-dimensional Hilbert space, $P_B$ is the identity operator and these conditions reduce to the Kraus operators satisfying $\sqrt{N}\, V_j = \sum_{i_1,i_2} \lambda_{i_1i_2 j} \ket{i_1}\bra{i_2}$, for some choice of orthonormal bases $\ket{i_1}$ and $\ket{i_2}$ and unitary matrix $(\lambda_{i_1i_2 j})_{i_1,i_2}$. One can also phrase the private subspace conditions neatly via the Heisenberg picture in terms of the dual map $\Phi^\dagger$, which has Kraus operators $V_j^\dagger$, as follows~\cite{KP12}: $P_B \Phi^\dagger (M) P_B = \tr(M\rho_0) P_B$ for all (arbitrary) observables $M$.
Of course the theorem applies to general private subsystems as well. Here we point out how the 2-qubit phase damping channel $\Lambda$ can be assembled from this result. In that case both $A$ and $B$ are spanned by $\{\ket{0}$, $\ket{1}\}$. The eigenstates of $\rho_0=\frac14 I_4$ are $\{\ket{00}, \ket{01}, \ket{10}, \ket{11}\}$, each having eigenvalue $\frac14$. The eigenstates of $\sigma_A=\frac12 I_2$ are $\{\ket{\psi_{A,k}}\}=\{\ket{0}, \ket{1}\}$, with corresponding eigenvalues $\frac12$. For brevity, we omit the matrix calculations for the operators $V_j\ket{\psi_{A,k}}$ here; we simply note that they are $4\times 2$ matrices formed with $2\times 2$ Pauli operators and zero blocks. The scalar-valued matrix $\lambda=(\lambda_{ijkl})$ is indeed an isometry. Furthermore, because the number of operators $V_j\ket{\psi_{A,k}}$ agrees with the number of operators $\ket{\phi_{l}}\bra{\psi_{B,i}}$ (namely, 8), the matrix $\lambda$ is in fact unitary.
\subsection{Complementarity and Quantum Error Correction} Several links have been made between quantum error correction and quantum privacy. In the case of operator private subsystems and operator error-correcting subsystems, the complementarity theorem of \cite{KKS08} discussed below established an algebraic bridge between the two subjects. This firmly links the operator quantum error correction theory to that of operator private subsystems---results in one field can immediately be exported to the other. Thus, it is natural to ask whether such a result holds in this more general setting. To answer this we need the concept of complementary channels.
As a consequence of the Stinespring dilation theorem, every channel $\Phi$ may be seen to arise from an environment Hilbert space $E$, a pure state $|\psi \rangle $ on the environment, and a unitary operator $U$ on the composite $
S\otimes E $ in the following sense: $\Phi(\rho )=\tr_{E}\big(
U(\rho \otimes \ket{\psi}\bra{\psi})U^\dagger\big).$ Tracing out the system instead yields a complementary channel: $\Phi^{\sharp}(\rho )=\tr_{S}\big(U(\rho \otimes
\ket{\psi}\bra{\psi})U^\dagger\big).$ The uniqueness built into the theorem yields a certain uniqueness for such a pair of channels, so that we talk of ``the'' complementary channel $\Phi^\sharp$ for a given channel $\Phi$ \cite{Hol06,KMNR07}.
We have already discussed operator private subsystems---the essential difference being that instead of a single state on $A$, it is demanded that Eq.~(\ref{private_definition}) holds for all states on $A$. Similarly, an operator quantum error-correcting subsystem $B$ for a channel $\mathcal athcal E$ \cite{KLP05,KLPL05} requires the existence of a correction operation $\mathcal athcal R$ such that: $\forall \sigma_A$ $\forall \sigma_B$, $\exists \tau_A$ for which $\mathcal athcal R \circ \mathcal athcal E (\sigma_A \otimes \sigma_B) = \tau_A \otimes \sigma_B$. The main result of \cite{KKS08} shows that $B$ is private for $\Phi$ if and only if it is error-correcting for $\Phi^\sharp$.
Does this result extend to the more general setting? The Kraus operators of the complementary map~$\Lambda^{\sharp}$ are four orthogonal rank-one projectors in two-qubit Hilbert space, and in particular the map determines a von Neumann measurement. No error-correcting subsystem can be extracted in such a setting; moreover, when the input space is restricted to be that of our example, the complementary map is private. Thus, not only does the complementarity result fail, it fails dramatically. We save these calculations and further analysis for \cite{JKLP12}.
This discussion motivates the following observation: The notion of an operator quantum error-correcting subsystem can be expanded to mimic the general definition of a private quantum subsystem. Indeed, a revised definition analogous to that of Eq.~(\ref{private_definition}) could be proposed as follows: $B$ is \emph{correctable} for $\mathcal athcal E$ if there exists an operation $\mathcal athcal R$ such that for all $\sigma_B$ and some \emph{fixed} states $\sigma_A,\tau_A$, we have $\mathcal athcal R \circ \mathcal athcal E (\sigma_A \otimes \sigma_B) = \tau_A \otimes \sigma_B.$ This is a potentially new notion of quantum error-correcting code.
\section{Conclusion \& Outlook}
We have studied the most general notion of private quantum subsystems. Taking motivation from quantum error correction, specifically the Knill-Laflamme conditions, we presented algebraic conditions that characterize when a code is private for a given channel. This result is new even for private subspaces, and opens up questions such as: is there an analogue of the stabilizer formalism from quantum error correction for quantum privacy? We analyzed the development of private subsystems for the special case given by the composition of phase damping channels on many qubit Hilbert~spaces. While each individual channel of this form is not private, the composition of such channels were shown to contain a private single qubit subsystem. Yet, for such channels, and for a wide class of more general channels, no private subspace or operator private subsystem exists. Moreover, we discussed how the channel fails to have the corresponding complementary error-correctable pair as in the case of operator subsystems. Nevertheless, this analysis naturally led us to define a potentially new form of quantum error-correcting code and warrants, along with the topic of channel complementarity, further investigation. Finally, preliminary discussions also suggest there may be yet unexplored connections between the study of private quantum subsystems and privacy in classical communication. We will continue and expand on the work initiated here elsewhere.
\section{Acknowledgments}
We are grateful to Robert Spekkens for an interesting discussion, and we thank the referees for helpful comments on our initial submission. T.~J.-O.\ was supported by an Ontario Graduate Scholarship. D.W.K.\ was supported by NSERC. R.L.\ was supported by NSERC, CIFAR, and Industry Canada. S.P.\ was supported by an NSERC Graduate Scholarship.
\end{document} | math | 25,813 |
\begin{document}
\title[A sufficient criterion]{A sufficient criterion for control of generalised error rates in multiple testing}
\date{\today}
\begin{abstract}
Based on the work of \citet{RomanoShaikh2006,RomanoShaikh2006AOS,LehmannRomano2005} we give a sufficient criterion for controlling generalised error rates for arbitrarily dependent $p$-values. This criterion is formulated in terms of matrices associated with the corresponding error rates and thus it is possible to view the corresponding critical constants as solutions of sets of certain linear inequalities. This property can in some cases be used to improve the power of existing procedures by finding optimal solutions to an associated linear programming problem.
\end{abstract}
\maketitle
\section{Introduction}
Consider the problem of testing $n$ hypotheses $H_1, \ldots, H_n$ simultaneously. A classical approach to dealing with the multiplicity problem is to control the familywise error rate (FWER) i.e. the probability of one or more false rejections. However, when the number $n$ of hypotheses is large, the ability to reject false hypotheses is small. Therefore, alternative type I error rates have been proposed that relax control of FWER in order to reject more false hypotheses (for a survey, see e.g. \citet{DudoitLaan2007}).
One such generalised error rate is the \textnormal{k-FWER}, i.e. the probability of $k$ or more false rejections for some integer $k\ge 1$, $\textnormal{k-FWER}\le \alpha$ considered by \citet{HommHoff87} and \citet{LehmannRomano2005}. For $k=1$ the usual FWER is obtained. Alternatively, instead of controlling the absolute number of false rejections it may be desirable to control the proportion of false rejections amongst all rejected hypotheses. This ratio is called the false discovery proportion (FDP). More specifically, if $R$ denotes the number of rejected hypotheses and $V$ the number of falsely rejected hypotheses, then $\textnormal{FDP}=V/R$ (and equal to $0$ if there are no rejections). For the $\textnormal{FDP}$, mainly two types of control have been considered in the literature. One aim might be to control the tail probability $P(\textnormal{FDP}>\gamma)\le \alpha$ for some user-specified value $\gamma \in [0,1)$. This error measure has been termed $\gamma-\textnormal{FDP}$ by \citet{LehmannRomano2005} and tail probability for the proportion of false positives ($TPPFP(\gamma)$) in \citet{DudoitLaan2007}. Instead of controlling a specific tail probability, the false discovery rate (FDR) requires that $\textnormal{FDR}=\textnormal{E}(\textnormal{FDP})\le \gamma$, i.e. control in the mean. As \citet{RomanoWolf2010} point out, probabilistic control of the FDP allows one to make useful statements about the realized FDP in applications, whereas this is not possible when controlling the FDR.
Recently, a number of methods have been proposed that control these generalised error rates under various assumptions. In this paper we focus on multiple testing procedures that are based on marginal $p$-values and are valid for finite sample sizes under no assumptions on the type of dependency of these $p$ values. For $\textnormal{k-FWER}$ and $\gamma-\textnormal{FDP}$, step-up and step-down methods have been obtained in \citet{RomanoShaikh2006,RomanoShaikh2006AOS,LehmannRomano2005}. For $\textnormal{FDR}$, \citet{BenjaminiYekutieli01} have shown that a rescaled version of the original step-up procedure of \citet{BenjaminiHochberg95} controls the $\textnormal{FDR}$ under arbitrary dependencies. \citet{GuoRao2008} have extended these results and have also given corresponding upper bounds for step-down $\textnormal{FDR}$ procedures (see \citet{GuoRao2008} and the references cited therein for more details).
The aim of this paper is two-fold. First, we present a sufficient condition for control of $\textnormal{k-FWER}$ and $\gamma-\textnormal{FDP}$ based on matrices that are associated with a specific error-rate and direction of stepping. This result is mainly a rephrasing of results obtained by \citet{RomanoShaikh2006,RomanoShaikh2006AOS,LehmannRomano2005}. In the second step we show how the rescaled procedures introduced by \citet{RomanoShaikh2006,RomanoShaikh2006AOS,LehmannRomano2005} can in some cases be improved. In particular, we introduce a linear programming approach which uses the above-mentioned matrices.
The paper is organized as follows. First, we introduce some terminology and assumptions that will be used in what follows. In section three we will state the main theoretical results which will be used in the following section to define new modified FDP controlling procedures. Section \ref{sec:Proofs} contains the proof of the main theorem. In section \ref{sec:SimulationStudy} we investigate the power of the new modified procedures in a simulation setting and in section \ref{sec:EmpiricalApplications} we apply them to the analysis of empirical data. The paper concludes with a discussion.
\section{Notation, Definitions and assumptions} \label{sec:NotationAssumptions}
In this section we introduce some terminology and assumptions that will be used in the sequel. When testing hypotheses $H_1, \ldots,H_n$, we assume that corresponding $p$-values $PV_1, \ldots, PV_n$ are available. For any true hypothesis $i$ we assume that the distribution of the $p$-values $PV_i$ is stochastically larger than a uniform rv, i.e.
\begin{align*}
P(PV_i\le u)&\le u
\end{align*}
for all $u \in (0,1)$. Let $PV_{(1)}\le \cdots \le PV_{(n)}$ denote the ordered $p$-values and $H_{(1)}, \ldots, H_{(n)}$ the associated (null-) hypotheses. Let
\begin{align*}
\mathcal{C}&=\{c \in {\mathbb R}_{+}^n | c_1 \le \cdots \le c_n\}
\end{align*}
denote the set of non-decreasing non-negative critical constants. For $c \in \mathcal{C}$ the associated step-up procedure rejects hypotheses $H_{(1)}, \ldots, H_{(k)}$, where $k= \max \{i| PV_{(i)} \le c_i\}$. If no such $i$ exists, no hypothesis is rejected. For the corresponding step-down procedure, reject $H_{(1)}, \ldots, H_{(k)}$, where $k= \max \{i| PV_{(j)} \le c_j, \quad j=1, \ldots, i\}$. If $PV_{(1)}>c_1$, no hypothesis is rejected.
\mbox{s.\,u.}\xspacebsection{Generalized error rates} \label{ssec:GeneralizedErrorRates}
In the following definition we introduce the sets of $\textnormal{k-FWER}$- and $\textnormal{FDP}$-controlling procedures weconsider in this paper.
\begin{definition} \label{def:ControllingPorcedures}
Let $\alpha \in (0,1)$.
\begin{itemize}
\item[(a)] For $1 \le k \le n$ define the set of step-up procedures that (strongly) control the $\textnormal{k-FWER}$:
\begin{align*}
S^{\textnormal{k-FWER}SU}(\alpha,k)&= \{ c\in \mathcal{C} |\max_{1\le |I| \le n} \textnormal{k-FWER}SU (c) \le \alpha\},\\
S^{\textnormal{k-FWER}SD}(\alpha,k)&= \{ c\in \mathcal{C} |\max_{1\le |I| \le n} \textnormal{k-FWER}SD (c) \le \alpha\}
\end{align*}
\item[(b)] For $\gamma \in [0,1)$ define the set of step-up and step-down procedures that (strongly) control the $\textnormal{FDP}$:
\begin{align*}
S^{\textnormal{FDP}SU}(\alpha,\gamma)&= \{ c\in \mathcal{C} |\max_{1\le |I| \le n} P(\textnormal{FDP}SU(c)> \gamma) \le \alpha\},\\
S^{\textnormal{FDP}SD}(\alpha,\gamma)&= \{ c\in \mathcal{C} |\max_{1\le |I| \le n} P(\textnormal{FDP}SD(c)> \gamma) \le \alpha\}.
\end{align*}
\end{itemize}
\end{definition}
In order to formulate the main results of this paper we introduce subsets of $\mathcal{C}$ that are defined by
\begin{align*}
\mathcal{F}(A)&:= \{c \in \mathcal{C} | ||A\cdot c||_{\infty} \le 1\},
\end{align*}
where $A \in {\mathbb R}_{+}^{n \times n}$ and $||x||_{\infty} =\max_{1\le i \le n} |x|_i$ denotes the maximum norm. The elements of $\mathcal{F}(A)$ can be interpreted as the set of feasible points given by a set of linear constraints (inequalities). We will show in Theorem \ref{theorem:MainTheorem} that for each error rate $\textnormal{k-FWER}$ and $\textnormal{FDP}$ and direction of stepping we can define an associated matrix $A$, such that any procedure in $\alpha \cdot \mathcal{F}(A)$ controls the correponding error rate at level $\alpha$.
\mbox{s.\,u.}\xspacebsection{Associated matrices} \label{ssec:AssociatedMatrices}
In this section we introduce the matrices associated with the error rates mentioned above.
\begin{definition}[\textnormal{k-FWER}SU] \label{def:MatrixkFWERSU}
Let $k \in \{1,\ldots,n\}$. Define
\begin{align}
A^{\textnormal{k-FWER}SU}_{ij}(k)&=i\cdot
\begin{cases}
0 & \qquad i<k\\
0 & \qquad i\ge k, j< n+k-i\\
\left(\frac{1}{j-n+i} -\frac{1}{j-n+i+1}\right) & \qquad i\ge k, n+k-i \le j <n\\
\frac{1}{i} & \qquad i\ge k, j =n
\end{cases}
\end{align}
\end{definition}
\begin{definition}[\textnormal{k-FWER}SD] \label{def:MatrixkFWERSD}
Let $k \in \{1,\ldots,n\}$. Define
\begin{align}
A^{\textnormal{k-FWER}SD}_{ij}(k)&= i \cdot
\begin{cases}
\frac{1}{k} & \qquad i\ge k, j=n-i+k\\
0 & \qquad \text{else}
\end{cases}
\end{align}
\end{definition}
\begin{definition} \label{def:AuxFDPSU}
Let $\gamma \in [0,1)$ and define $m(j)=\lfloor \gamma j \rfloor +1$ where $\lfloor x \rfloor$ is the greatest integer $\le x$. For $i \in \{1, \ldots,n\}$ define
\begin{align*}
\widetilde{M}(i)&:=\widetilde{M}(\gamma,n,i)=\max\{l \in \{1,\ldots,n\}| m(l) \le i\}\\
g_i(l)&:= \max \{i-n+l,m(l)\}\\
M(i)&:=g_i(\widetilde{M}(i))\\
t_k (i)&:=\max g_i^{-1}(\{k\}), \text{ for } k \in \{1,\ldots,M(i)\}.
\end{align*}
\end{definition}
\begin{definition}[\textnormal{FDP}SU] \label{def:MatrixFDPSU}
Let $\gamma \in [0,1)$. Using the notation from definition \ref{def:AuxFDPSU}, define
\begin{align}
A^{\textnormal{FDP}SU}_{ij}(\gamma) &=i \cdot
\begin{cases}
\left(\frac{1}{k}- \frac{1}{k+1} \right) & \text{for $j=t_k(i)$, if $1\le k < M(i)$}\\
\frac{1}{M(i)} & \text{for $j=t_{M(i)}(i)$,}\\
0 & \text{else.}
\end{cases}
\end{align}
\end{definition}
\begin{definition} \label{def:AuxFDPSD}
Let $c \in\mathcal{C}$. For $i \in \{1, \ldots,n\}$ define
\begin{align*}
k_{i}(l)&= \min \{n,n+l-i,\left\lceil\frac{l}{\gamma}\right\rceil-1\}\qquad l=1, \ldots,\lfloor \gamma n \rfloor+1 \qquad \text{and}\\
N(i)&=N(\gamma,n,i)=\min \{ \lfloor \gamma n \rfloor +1,i, \left\lfloor \gamma \cdot \left( \frac{n-i}{1-\gamma}+1\right) \right\rfloor +1 \} ,\\
M(i)&=k_i(\{1, \ldots,\lfloor \gamma n \rfloor+1\}).
\end{align*}
\end{definition}
\begin{definition}[\textnormal{FDP}SD] \label{def:MatrixFDPSD}
Let $\gamma \in [0,1)$. Using the notation from definition \ref{def:AuxFDPSD} define
\begin{itemize}
\item[(a)] $\widetilde{A} \in {\mathbb R}^{n \times n}$ by
\begin{align*}
\widetilde{A}_{ij} &=i \cdot
\begin{cases}
\left(\frac{1}{j}- \frac{1}{j+1}\right) & \qquad \text{for $1\le j < N(i)$},\\
\frac{1}{N(i)} & \qquad \text{for $j=N(i)$},\\
0 & \qquad \text{else}.
\end{cases}
\end{align*}
\item[(b)] and
\begin{align*}
A^{\textnormal{FDP}SD}_{ij}(\gamma)&=
\begin{cases}
\mbox{s.\,u.}\xspacem_{l \in k_i^{-1}(\{j\})}\widetilde{A}_{il}& \qquad j \in M(i),\\
0 & \qquad \text{else.}
\end{cases}
\end{align*}
\end{itemize}
\end{definition}
\section{Main results} \label{sec:TheoreticalResults}
First we state the main results of this paper which will serve as the starting point for mofifying some existing MTPs. As the proof in section \ref{sec:Proofs} shows, it is actually a rephrasing of results of \citet{RomanoShaikh2006,RomanoShaikh2006AOS} in terms of the associated matrices introduced in section \ref{sec:NotationAssumptions}.
\begin{thm}\label{theorem:MainTheorem}
Let $\alpha \in (0,1)$.
\begin{itemize}
\item[(a)] For $1\le k \le n$ it holds $\alpha \cdot\mathcal{F}(A^{\textnormal{k-FWER}SU}(k)) \mbox{s.\,u.}\xspacebset S^{\textnormal{k-FWER}SU}(\alpha,k)$.
\item[(b)] For $1\le k \le n$ it holds $\alpha \cdot\mathcal{F}(A^{\textnormal{k-FWER}SD}(k)) \mbox{s.\,u.}\xspacebset S^{\textnormal{k-FWER}SD}(\alpha,k)$.
\item[(c)] For $\gamma \in [0,1)$ it holds $\alpha \cdot\mathcal{F}(A^{\textnormal{FDP}SU}(\gamma)) \mbox{s.\,u.}\xspacebset S^{\textnormal{FDP}SU}(\alpha,\gamma)$.
\item[(d)] For $\gamma \in [0,1)$ it holds $\alpha \cdot\mathcal{F}(A^{\textnormal{FDP}SD}(\gamma)) \mbox{s.\,u.}\xspacebset S^{\textnormal{FDP}SD}(\alpha,\gamma)$.
\end{itemize}
\end{thm}
The theorem provides generic sufficient conditions for control of generalised error rates, i.e. if $d \in \mathcal{F}(A)$ then $\alpha \cdot d$ controls the corresponding error rate at the desired level. Since the sets $\mathcal{F}(A)$ from the theorem are convex, it follows immediately that for any matrix $A$ from theorem \ref{theorem:MainTheorem} and level $\alpha \in (0,1)$ the set of procedures $\alpha \cdot \mathcal{F}(A)$ is also convex. \citet{GuoHeSarkar2012} have introduced the $\gamma-\textnormal{kFDP}=P(\textnormal{kFDP}> \gamma)$ where $\textnormal{kFDP}=V/R$ ($V$ and $R$ defined as in the introduction) if $V \ge k$ and $0$ else. Under the assumption that $PV_i \sim U(0,1)$ under any true hypothesis $i$ they obtain linear bounds for the $\gamma-\textnormal{kFDP}$ in the proofs of their Theorems 4.1 and 4.2. These bounds can again be used to define appropriate associated matrices and establish a result similar to the above theorem for the $\gamma-\textnormal{kFDP}$ but we do not pursue this any further here.
One immediate consequence of the theorem is the following corollary.
\begin{coro} \label{coro:RescalingProcedures}
Let $1\le k \le n$, $\gamma \in [0,1)$. For $c \in \mathcal{C}$ and $x \in \{\textnormal{k-FWER}SU, \textnormal{k-FWER}SD, \textnormal{FDP}SU,\textnormal{FDP}SD\}$ define
\begin{align*}
D^x(c) &= ||A^x \cdot c||_{\infty}.
\end{align*}
Then the rescaled procedure $\widetilde{c}:= \alpha \cdot c / D^x(c)$ yields control of the error rate $x$ at level $\alpha$.
\end{coro}
Thus we can always achieve control of generalised error rates by using the rescaling approach.
The proof of the above theorem relies on two key tools. The first is the following generalised Bonferroni inequality due to \citet{LehmannRomano2005}.
\begin{lemma}[\citet{LehmannRomano2005}] \label{lemma:BasicLemma}
Let $X_1,\ldots,X_t: \Omega \rightarrow (0,1]$ be $p$-values that satisfy the above distributional assumption i.e. $P(X_i\le u)\le u$ for all $i$ and $u \in (0,1)$. Denote their ordered values by $X_{(1)}\le \cdots \le X_{(t)}$ and let $0=c_0 \le c_1 \le \cdots \le c_m\le 1$ for some $m \le t$.
\begin{itemize}
\item[(i)] Then it holds
\begin{align}
P(\{X_{(1)}\le c_1 \} \cup \cdots \cup \{X_{(m)}\le c_m \} ) &\le t \cdot\mbox{s.\,u.}\xspacem_{i=1}^{m} \frac{c_i - c_{i-1}}{i}. \label{eq:GenBonf}
\end{align}
\item[(ii)] As long as the right-hand side of \eqref{eq:GenBonf} is $\le 1$, the bound is sharp in the sense that there exists a joint distribution for the $p$-values for which the inequality is an equality.
\end{itemize}
\end{lemma}
Related inequalities have been obtained previously by \citet{Hommel1983}, \citet{RoehmelStreitberg1987} and \citet{Falk1989}.
The second step uses the observation that the generalised error rates considered here can all be bounded by probabilities of the type
\begin{align}
P &\left( \bigcup_{i =1}^{M(|I|)} \{PV_{(i)} \le c_{t_i(|I|)}\} \right), \label{eq:GenProbBound}
\end{align}
where $|I|$ is the number of true hypotheses, $M(|I|)\in \{0,\ldots,n\}$ and $t_i(|I|) \in \{0,\ldots,n\}$ is an increasing sequence in $i$ (depending on $|I|$) and the $PV$ in \eqref{eq:GenProbBound} are taken under the null hypotheses. Then the probability in \eqref{eq:GenProbBound} can be bounded using lemma \ref{lemma:BasicLemma}. We call the resulting bound the LR-bound of the corresponding error rate.
For the procedures considered here, adjusted $p$-values can be defined in the generic way decribed in \citep[Procedures 1.3 and 1.4]{DudoitLaan2007}: For raw $p$-values $pv_1, \ldots, pv_n$ and $c=(c_1, \ldots,c_n) \in \mathcal{F}(A)$ define step-up $p$-values
\begin{align}
\widetilde{pv}_{(i)} &= \min_{j=i, \ldots,n} \left\{ \min \left( \frac{pv_{(j)}}{c_j},1\right) \right\}
\intertext{and step-down $p$-values}
\widetilde{pv}_{(i)} &= \max_{j=1, \ldots,i} \left\{ \min \left( \frac{pv_{(j)}}{c_j},1 \right) \right\} .
\end{align}
In what follows we will focus on $\textnormal{FDP}$ controlling procedures.
\section{Modified FDP-controlling procedures} \label{sec:NewFDPProcedures}
In addition to providing an easily verifiable condition for FDP controlling procedures, theorem \ref{theorem:MainTheorem} can be used to construct new or modify existing procedures. In this section we describe an approach based on linear programming. Our focus in this section is on improving classical procedures based on rescaled constants as considered in \citet{RomanoShaikh2006,RomanoShaikh2006AOS}. First we define new modified FDP procedures as the solutions of a linear programming problem.
\begin{definition}\label{def:NewGenericFDPProcedure}
Let $A \in {\mathbb R}_{+}^{n \times n}$ and $c \in \mathcal{F}(A)$. Define the modified procedure $\xi=\xi(c) $ as the solution to the following linear programming problem (P):
\begin{align}
\text{maximize} \qquad & F(\xi)= a \cdot \xi &&\notag\\
\text{subject to} \qquad & A_{i \cdot} \cdot \xi \le 1 &&\qquad i=1, \ldots, n \tag{P}\\
& -\xi_i+\xi_{i-1} \le 0 &&\qquad i=1, \ldots, n \notag\\
& -\xi_i+c_{i} \le 0 &&\qquad i=1, \ldots, n, \notag
\end{align}
where $\xi_0=0$ and $a_j=\mbox{s.\,u.}\xspacem_{i=1}^n A_{ij}$.
\end{definition}
Note that the third constraint in (P) implies that $\xi\ge c$ while the first and second constraints guarantee that $\xi \in \mathcal{F}(A)$. Note also that if $c=0$ then $\mathcal{F}(A)$ is identical with the feasible points of the optimisation problem, so that this approach could be used to find optimal solutions within the whole class $\mathcal{F}(A)$ instead of $\mathcal{F}(A) \cap \{\xi \ge c\}$. Since we are primarily interested in improving existing procedures we do not pursue this any further.
For problems like (P), standard numerical methods like the simplex algorithm \citep{Dantzig63} are available. From a statistical viewpoint, it would be desirable to optimise the power of the MTP (defined in a suitable sense, see also section \ref{sec:SimulationStudy}), subject to the given constraints. The rationale for using the objective function $F$ is the following: Let $b_i=\mbox{s.\,u.}\xspacem_{j=1}^n A_{ij} \cdot \xi_j$, so that by Theorem \ref{theorem:MainTheorem} under $|I|=i$ the error rate is bounded by $b_i$ and the sum $b_1+ \cdots +b_n=F(\xi)$ can thus be interpreted as the sum of the maximum significance levels of the procedure. Since we are aiming for a powerful procedure it seems plausible to optimise this objective function in the sense that the best we can do without violating the bounds from lemma \ref{lemma:BasicLemma} is $F(\xi)=n$. Thus $F(\xi)$ may be thought of as a surrogate-measure of power. It can also be interpreted in a Bayesian framework by observing that optimising it is equivalent to optimising the mean maximum level of significance if the number of true hypotheses $|I|$ is distributed uniformly on $\{1, \ldots,n\}$. Thus, if prior knowledge is available for the distribution of $|I|$, we could also use the weighted objective function $F(\xi,w)=\mbox{s.\,u.}\xspacem_{i=1}^n w_i \cdot b_i$ where $w_i=P(|I|=i)$.
Using theorem \ref{theorem:MainTheorem} we immediately obtain the following result.
\begin{coro}
Let $\gamma \in [0,1)$, $A \in \{ A^{\textnormal{FDP}SU}(\gamma),A^{\textnormal{FDP}SD}(\gamma) \}$ and $c \in \mathcal{F}(A)$. Let $\xi=\xi(c)$ as defined in definition \ref{def:NewGenericFDPProcedure}. Then $\xi \in \mathcal{F}(A)$ and therefore the procedure $\alpha \cdot \xi$ controls the FDP for any $\alpha\in (0,1)$. This procedure is at least as powerful as procedure $\alpha\cdot c$.
\end{coro}
Clearly, if $F(\xi) > F(c)$, then $\xi>c$. This means that this approach will always find a strict improvement over $c$ whenever one exists and we may thus expect a gain in power. Since, by construction, $\xi$ can not be improved uniformly within class $\mathcal{F}(A)$, $\alpha\cdot \xi$ can be seen as an optimal procedure within the subset $\alpha \cdot \mathcal{F}(A)$ of all $\alpha$-controlling FDP procedures.
We now consider two specific types of critical constants in more detail.
\begin{itemize}
\item[(a)] The Benjamini-Hochberg constants:
\begin{align*}
c_i^{BH}&=c_i^{BH}(n)=\frac{i}{n}
\end{align*}
\item[(b)] The Lehmann-Romano constants:
\begin{align*}
c_i^{LR}&=c_i^{LR}(\gamma,n)=\frac{\lfloor \gamma i \rfloor +1}{n + \lfloor \gamma i \rfloor +1 -i}
\end{align*}
\end{itemize}
In \citet{RomanoShaikh2006,RomanoShaikh2006AOS} normalising constants were introduced for $c^{BH}$ and $c^{RS}$ for step-up and step-down procedures. These constants were defined (in our notation) by
\begin{align*}
D^{BH-SU}(\gamma) &= ||A^{\textnormal{FDP}SU}(\gamma) \cdot c^{BH}||_{\infty},\\
D^{RS-SU}(\gamma) &= ||A^{\textnormal{FDP}SU}(\gamma) \cdot c^{RS}(\gamma)||_{\infty},\\
D^{BH-SD}(\gamma) &= ||A^{\textnormal{FDP}SD}(\gamma) \cdot c^{BH}||_{\infty},\\
D^{RS-SD}(\gamma) &= ||A^{\textnormal{FDP}SD}(\gamma) \cdot c^{RS}(\gamma)||_{\infty},
\end{align*}
and due to corollary \ref{coro:RescalingProcedures} the rescaled procedures $\alpha \cdot c/D(\gamma)$ all control the $\gamma$-FDP at level $\alpha$.
\mbox{s.\,u.}\xspacebsection*{Example} Figure \ref{fig:Explanationl150} illustrates the possible gains resulting from the optimisation approach for $n=50$ and $\gamma=0.05$.
\begin{sidewaysfigure}[htbp]
\centering
\includegraphics[width=1.00\textwidth]{Explanation50_05.pdf}
\caption{Illustration for $\textnormal{FDP}$ with $n=50$ and $\gamma=0.05$. Left panels: Ratios of modified critical constants $\xi$ to original rescaled constants $c$. Right panels: Values of $(A \cdot \xi)_{|I|}$ (dashed lines) and $(A \cdot c)_{|I|}$ (solid lines). The procedures BH-SU, BH-SD, RS-SU and RS-SD are sorted from top to bottom.}
\label{fig:Explanationl150}
\end{sidewaysfigure}
In all cases the modified procedures are strictly better than the rescaled procedures. To investigate where the gains come from, we consider the BH-SU procedure in more detail. For the rescaled procedure $c=c^{BH}/D^{BH-SU}(\gamma)$, $(A^{\textnormal{FDP}SU}(0.05)\cdot c)_{|I|}=1$ for $|I|=32$. The column entries for row 32 of matrix $A^{\textnormal{FDP}SU}(0.05)$ are strictly greater than zero for columns $19$ to $50$ and therefore the associated critical constants $c_{19}, \ldots,c_{50}$ can not be improved upon (any increase would violate the constraint $\max (A \cdot c)_{|I|}\le 1$). However, since $A_{32,1}=\cdots =A_{32,18}=0$ there is some potential for increasing the remaining critical constants $c_{1}, \ldots,c_{18}$. This is exactly what the optimisation in the linear program (P) accomplishes. Ideally, this would result in a new procedure $\xi$ with $(A \cdot \xi)_{|I|}= 1$ for all $|I|$, yielding a completely unimprovable procedure within class $\mathcal{F}(A)$. This happens e.g. for $A=A^{\textnormal{k-FWER}SD}$, when $\xi$ is the vector of Lehmann-Romano constants, see section \ref{ssec:ProofMainTheoremb}. However, due to the structure of the matrix $A$, this is usually impossible. In the case of BH-SD we obtain $A\cdot \xi_{32}= \cdots =A\cdot \xi_{50}=1$ (see uppermost right panel in figure \ref{fig:Explanationl150}).
Figure \ref{fig:Explanationl150} suggests that the gains derived from the modifications are considerably larger for the BH than for the RS procedures. This is also supported by the numerical values in table \ref{tab:CompareProcedures}. If we follow the arguments given above for justifying the choice of objective function we would expect the modified BH-SD procedure to be the most powerful procedure (indicated by the highest values of $F(\xi)$), followed closely by the modified RS-SD procedure. This is also consistent with the simulation results in section \ref{sec:SimulationStudy}.
\begin{sidewaystable}[htb]
\centering
\begin{tabular}{rrrrrrrrrrrrrrrrr}
\toprule
& \multicolumn{ 8}{c}{SU} & \multicolumn{ 8}{c}{SD} \\ \cmidrule(lr){2-9}\cmidrule(lr){10-17}
& \multicolumn{ 4}{c}{BH} & \multicolumn{ 4}{c}{RS} & \multicolumn{ 4}{c}{BH} & \multicolumn{ 4}{c}{RS} \\ \cmidrule(lr){2-5}\cmidrule(lr){6-9}\cmidrule(lr){10-13}\cmidrule(lr){14-17}
$n$ & $F(c)$ & $F(\xi)$ & $M_1$ & $M_2$ & $F(c)$ & $F(\xi)$ & $M_1$ & $M_2$ & $F(c)$ & $F(\xi)$ & $M_1$ & $M_2$ & $F(c)$ & $F(\xi)$ & $M_1$ & $M_2$ \\ \hline
10 & 7.75 & 8.16 & 2.61 & 1.34 & 8.76 & 8.76 & 1.00 & 1.00 & 7.33 & 10.00 & 3.00 & 3.00 & 10.00 & 10.00 & 1.00 & 1.00 \\
25 & 18.32 & 20.39 & 6.42 & 2.09 & 21.32 & 22.75 & 1.23 & 1.11 & 17.18 & 24.14 & 6.76 & 6.76 & 17.90 & 23.50 & 1.43 & 1.43 \\
50 & 32.78 & 37.90 & 12.36 & 3.14 & 41.75 & 43.39 & 1.56 & 1.23 & 31.55 & 48.17 & 12.4 & 12.4 & 38.69 & 44.94 & 1.50 & 1.50 \\
100 & 66.97 & 74.02 & 18.39 & 3.92 & 83.63 & 85.47 & 1.29 & 1.09 & 65.24 & 94.89 & 18.39 & 18.39 & 77.47 & 87.01 & 1.57 & 1.51 \\
250 & 165.51 & 173.72 & 19.00 & 3.59 & 207.72 & 209.11 & 1.11 & 1.03 & 164.27 & 230.50 & 24.41 & 19.00 & 196.77 & 219.11 & 1.82 & 1.71 \\
500 & 328.09 & 336.90 & 19.00 & 3.34 & 411.57 & 412.68 & 1.05 & 1.02 & 328.13 & 459.61 & 31.05 & 19.00 & 392.67 & 444.89 & 2.15 & 1.96 \\
1000 & 650.00 & 659.18 & 19.00 & 3.13 & 812.64 & 813.49 & 1.03 & 1.01 & 653.11 & 921.70 & 39.52 & 19.00 & 778.33 & 902.52 & 2.49 & 2.22 \\
\bottomrule
\end{tabular}
\caption{Values of $F$ for rescaled and modified critical constants and maximum ratios $M_1=\max \xi_i/c_i$, $M_2=\max (A \cdot \xi)_{|I|}/(A \cdot c)_{|I|}$ for $\textnormal{FDP}$ procedures with $\gamma=0.05$.}
\label{tab:CompareProcedures}
\end{sidewaystable}
\section{Proofs}\label{sec:Proofs}
In this section we prove the statements of the theorem. Actually, the main work is to rephrase the results of \citet{RomanoShaikh2006,RomanoShaikh2006AOS} in terms of the matrices introduced in section \ref{sec:NotationAssumptions}. The structure of the proofs is the same in all cases.
\mbox{s.\,u.}\xspacebsection{Proof of theorem \ref{theorem:MainTheorem}, part (a)}
\begin{proof}
Let $d \in \mathcal{F}(A^{\textnormal{k-FWER}SU}(k))$, define $c=\alpha \cdot d$ and let $I \mbox{s.\,u.}\xspacebset \{1,\ldots,n\}$ be the set of true hypotheses. By \citet[lemma 3.1]{RomanoShaikh2006AOS} we have
\begin{align*}
\textnormal{k-FWER}(c) &\le P\left( \bigcup_{k\le \ell\le |I|} \{PV_{(\ell)} \le c_{n-|I|+\ell}\} \right)
\end{align*}
where $PV_{(1)},\ldots,PV_{(|I|)}$ are the $p$-values corresponding to the null hypotheses. By lemma \ref{lemma:BasicLemma} with $t=|I|=m$ and $\tilde{c}_0=\ldots=\tilde{c}_{k-1}=0,\tilde{c}_k=c_{n-|I|+k},\ldots,\tilde{c}_{|I|}=c_n$ this probability can be bounded by
\begin{align}
&|I| \cdot \left\{\frac{c_{n-|I|+k}}{k} + \frac{c_{n-|I|+k+1} -c_{n-|I|+k}}{k+1} + \cdots + \frac{c_{n} -c_{n-1}}{|I|} \right\} \notag\\
&= |I| \cdot \left\{c_{n-|I|+k}\cdot \left(\frac{1}{k}-\frac{1}{k+1} \right) + \cdots + c_{n-1}\cdot \left(\frac{1}{|I|-1}-\frac{1}{|I|}\right) + c_n \cdot \frac{1}{|I|} \right\} \label{eq:kFWERMatrix1}\\
&= \mbox{s.\,u.}\xspacem_{j=1}^n A_{|I|j}\cdot c_j, \label{eq:kFWERMatrix2}\\
&= \alpha \cdot (A\cdot d)_{|I|}\notag
\end{align}
where $A=A^{\textnormal{k-FWER}SU}(k)$. Equality \eqref{eq:kFWERMatrix2} can be verified by considering the four cases in definition \ref{def:MatrixkFWERSU} separately:
\begin{itemize}
\item For $|I|<k$ definition \ref{def:MatrixkFWERSU} yields $A_{|I|1}=\cdots=A_{|I|n}=0$ so that the bound in \eqref{eq:kFWERMatrix2} is equal to 0 which is correct, since $\textnormal{k-FWER}(c)=0$ for $|I|<k$.
\item For $|I|\ge k$ and $j<n+k-|I|$ the coefficient of $c_j $ is easily seen to equal 0 from equation \eqref{eq:kFWERMatrix1}.
\item For $|I|\ge k$ note that the sum \eqref{eq:kFWERMatrix2} can be reexpressed as $\mbox{s.\,u.}\xspacem_{p=0}^{|I|-k} c_{n-|I|+k+p} \cdot A_{|I|,n-|I|+k+p}$. For $n+k-|I|\le j <n$ ($\Leftrightarrow$ $0\le p <|I|-k$) the coefficient of $c_{n-|I|+k+p}$ is $A_{|I|,n-|I|+k+p}=|I| \cdot (1/(k+p)-1/(k+p+1)$. Since $k+p=j+|I|-n$ the claim follows from the third part of the definition of $A$.
\item For $|I|\ge k$ and $l=n$ the coefficient equals 1 as seen from equation \eqref{eq:kFWERMatrix1}.
\end{itemize}
Since $d \in \mathcal{F}(A^{\textnormal{k-FWER}SU}(k))$, it follows
\begin{align*}
\max_{I \mbox{s.\,u.}\xspacebset \{1,\ldots,n\}} \textnormal{k-FWER}(c) &\le \alpha \cdot \max_{I \mbox{s.\,u.}\xspacebset \{1,\ldots,n\}} (A \cdot d)_{|I|} \\
&\le \alpha
\end{align*}
\end{proof}
\mbox{s.\,u.}\xspacebsection{Proof of theorem \ref{theorem:MainTheorem}, part (b)} \label{ssec:ProofMainTheoremb}
\begin{proof}
To prove that $\alpha \cdot \mathcal{F}(A^{\textnormal{k-FWER}SD}(k)) \mbox{s.\,u.}\xspacebset S^{\textnormal{k-FWER}SD}(\alpha,k)$ let $d \in \mathcal{F}(A^{\textnormal{k-FWER}SD}(k))$, define $c=\alpha \cdot d$ and let $I \mbox{s.\,u.}\xspacebset \{1,\ldots,n\}$ be the set of true hypotheses. From the proof of Theorem 2.2 in \citet{LehmannRomano2005} it follows that
\begin{align*}
\textnormal{k-FWER}(c) &\le P(PV_{(k)} \le c_{n-|I|+k})
\end{align*}
and by lemma \ref{lemma:BasicLemma} with $t=|I|$, $m=k$ and $0=\tilde{c}_0=\cdots=\tilde{c}_{m-1},\tilde{c}_m=c_{n-|I|+k}$ this probability can be bounded by
\begin{align}
\frac{|I|}{k}\cdot c_{n-|I|+k} &= \alpha \cdot (A\cdot d)_{|I|}\notag
\end{align}
where $A=A^{\textnormal{k-FWER}SD}(k)$.
To prove that $ S^{\textnormal{k-FWER}SD}(\alpha,k) \mbox{s.\,u.}\xspacebset\alpha \cdot \mathcal{F}(A^{\textnormal{k-FWER}SD}(k)$ we use the optimality property of the Lehmann-Romano procedure. Let $c \in S^{\textnormal{k-FWER}SD}(\alpha,k)$. By Theorem 2.3 (ii) in \citet{LehmannRomano2005} it follows that for $i\ge k$ $c_i \le \alpha \cdot d^{LR}_i$ where $d^{LR}_i=k/(n+k-i)$ are the Lehmann-Romano critical constants. Now let $|I|\in \{1, \ldots,n\}$. Then it follows
\begin{align*}
(A \cdot d^{LR})_{|I|} &= A_{|I|,n-|I|+k} \cdot d^{LR}_{n-|I|+k}= \frac{|I|}{k} \cdot \frac{k}{n+k-(n-|I|+k)}=1
\end{align*}
so that $d^{LR} \in \mathcal{F}(A^{\textnormal{k-FWER}SD}(k))$ and the claim is proved.
\end{proof}
\mbox{s.\,u.}\xspacebsection{Proof of theorem \ref{theorem:MainTheorem}, part (c)}The following lemma is a re-phrasing of Lemma 4.1 in \citet{RomanoShaikh2006AOS} and states that the event $\{\textnormal{FDR}> \gamma\}$ is a subset of the union of sets of the type $\{PV_{(j)} \le c_{i_j}\}$.
\begin{lemma}\label{lemma:FDPRepresentation}
Let the notation from definition \ref{def:AuxFDPSU} be given. Consider testing $n$ null hypotheses, with $|I|\ge 1$ of them true. Let $PV_{(1)},\ldots,PV_{(|I|)}$ denote the sorted $p$-values under the null hypotheses and let $\gamma \in [0,1)$. Then it holds for the step-up procedure based on the constants $c_1 \le \cdots \le c_n\le 1$
\begin{align*}
\{\textnormal{FDP}(c) > \gamma\} &\mbox{s.\,u.}\xspacebset \bigcup_{k =1}^{M(|I|)} \{PV_{(k)} \le c_{t_k (|I|)}\}
\end{align*}
\end{lemma}
\begin{proof}
We use the bound
\begin{align*}
\{\textnormal{FDP} > \gamma\} &\mbox{s.\,u.}\xspacebset \bigcup_{0 \le j \le n-1, |I|\ge m(n-j)} \{PV_{(\max[(|I|-j),m(n-j)])} \le c_{n-j}\}
\end{align*}
given at the bottom of p. 1861 in \citet{RomanoShaikh2006AOS}. With $\ell=n-j$ the index set is now $\ell=1, \ldots,n$ with $m(\ell)\le |I|$ and so we have
\begin{align*}
\{\textnormal{FDP} > \gamma\} &\mbox{s.\,u.}\xspacebset \bigcup_{\ell=1}^{\widetilde{M}(|I|)} \{PV_{(max[(|I|-n+\ell),m(\ell)])} \le c_{\ell}\}\\
&=\bigcup_{\ell=1}^{\widetilde{M}(|I|)} \{PV_{(g_{|I|}(\ell))} \le c_{\ell}\}
\end{align*}
where the last equality follows from the definition of $g_{|I|}$ (see definition \ref{def:AuxFDPSU}), defined on $\{1, \ldots, \widetilde{M}(|I|)\}$. Clearly, $g_{|I|}$ is non-decreasing. Since $g_{|I|}(\ell+1)-g_{|I|}(\ell) \le 1$ and $g_{|I|}(1)=1$ it follows that $g_{|I|}(\ell) \le \ell$ and from the definition of $M_{|I|}$ that $g_{|I|}(\{1, \ldots, \widetilde{M}(|I|)\})=\{1, \ldots, M(|I|)\}$.
For $k \in \{1, \ldots, M(|I|)\}$ we now claim that $\{PV_{(g_{|I|}(\ell))} \le c_{\ell}\} \mbox{s.\,u.}\xspacebset \{PV_{(k)} \le c_{t_k (|I|)} \}$ for any $\ell \in g_{|I|}^{-1}(\{k\})$. To see this, let $\ell \in g_{|I|}^{-1}(\{k\})$. By the definition of $t_k$ it follows $\ell \le t_k(|I|)$. We thus obtain
\begin{align*}
\{PV_{(g_{|I|}(\ell))} \le c_{\ell} \} &= \{PV_{(k)} \le c_{\ell}\} \qquad \text{(since $g_{|I|}(\ell)=k$)}\\
&\mbox{s.\,u.}\xspacebset \{ PV_{(k)} \le c_{t_k (|I|)} \},
\end{align*}
since $\ell \le t_k (|I|)$. Altogether this yields
\begin{align*}
\{\textnormal{FDP} > \gamma\} &\mbox{s.\,u.}\xspacebset \bigcup_{\ell=1}^{\widetilde{M}(|I|)} \{PV_{(g_{|I|}(\ell))} \le c_{\ell}\}\\
&\mbox{s.\,u.}\xspacebset \bigcup_{k=1}^{M(|I|)}\{PV_{(k)} \le c_{t_k (|I|)}\}.
\end{align*}
\end{proof}
\begin{proof}[Proof of theorem \ref{theorem:MainTheorem}, part (c)]
Let $d \in \mathcal{F}(A^{\textnormal{FDP}SU}(\gamma))$, define $c=\alpha \cdot d$ and let $I \mbox{s.\,u.}\xspacebset \{1,\ldots,n\}$ be the set of true hypotheses. Be lemma \ref{lemma:FDPRepresentation} we have
\begin{align*}
\{\textnormal{FDP}(c) > \gamma\} &\mbox{s.\,u.}\xspacebset \bigcup_{k =1}^{M(|I|)} \{PV_{(k)} \le c_{t_k (|I|)}\}
\end{align*}
and by lemma \ref{lemma:BasicLemma} this probability can be bounded by
\begin{align}
&|I| \cdot \left\{\frac{c_{t_1 (|I|)}}{1} + \frac{c_{t_2 (|I|)} -c_{t_1 (|I|)}}{2} + \cdots + \frac{c_{t_{M(|I|)} (|I|)}-c_{t_{M(|I|)-1} (|I|)}}{M(|I|)} \right\} \notag\\
&= |I| \cdot \left\{ c_{t_1 (|I|)}\cdot \left(1-\frac{1}{2} \right) + \cdots + c_{t_{M(|I|)-1}(|I|)}\cdot \left(\frac{1}{M(|I|)-1}-\frac{1}{M(|I|)}\right) + \frac{c_{t_{M(|I|)} (|I|)}}{M(|I|)} \right\} \label{eq:FDPSUMatrix1}\\
&= \mbox{s.\,u.}\xspacem_{j=1}^n A_{|I|j}\cdot c_j, \label{eq:FDPSUMatrix2}\\
&= \alpha \cdot (A\cdot d)_{|I|}\notag
\end{align}
where $A=A^{\textnormal{FDP}SU}(\gamma)$. Equality \eqref{eq:FDPSUMatrix2} can be verified by considering the following two cases:
\begin{itemize}
\item If $M(|I|)=1$, the above upper bound equals $|I|\cdot c_{t_1 (|I|)}$ which is identical with \eqref{eq:FDPSUMatrix2} due to the second case in definition \ref{def:MatrixFDPSU}.
\item If $M(|I|)>1$, the sum in \eqref{eq:FDPSUMatrix2} consists only of terms with $j=t_1 (|I|),t_2 (|I|), \ldots, t_{M(|I|)} (|I|)$ and the non-zero entries of row $|I|$ of $A$ are exactly the coefficents of $c_{t_1 (|I|)}, \ldots , c_{t_{M(|I|)} (|I|)}$ in \eqref{eq:FDPSUMatrix1}, corresponding to the first case in definition \ref{def:MatrixFDPSU}.
\end{itemize}
Since $d \in \mathcal{F}(A^{\textnormal{FDP}SU}(\gamma))$ it now follows
\begin{align*}
\max_{I \mbox{s.\,u.}\xspacebset \{1,\ldots,n\}} P(\{\textnormal{FDP}(c) > \gamma\}) &\le \alpha \cdot \max_{I \mbox{s.\,u.}\xspacebset \{1,\ldots,n\}} (A \cdot d)_{|I|} \\
&\le \alpha
\end{align*}
\end{proof}
\mbox{s.\,u.}\xspacebsection{Proof of theorem \ref{theorem:MainTheorem}, statement (d)}
The following is a rephrasing of results from \citet{RomanoShaikh2006}.
\begin{prop} \label{prop:FDP.SD.bound}
Let the notation from definition \ref{def:AuxFDPSD} be given and let $c \in\mathcal{C}$. For $1\le |I|\le n$ define
\begin{align*}
\beta_\ell &= \beta_\ell(|I|)=c_{k_{|I|}(l)}, \qquad \ell=1, \ldots,\lfloor \gamma n \rfloor+1.
\intertext{Then it holds}
P(\textnormal{FDP}(c)>\gamma) &\le |I| \cdot \mbox{s.\,u.}\xspacem_{i=1}^{N(|I|)} \frac{\beta_i(|I|)-\beta_{i-1}(|I|)}{i}.
\end{align*}
\end{prop}
\begin{proof} Note that $N(i)$ from definition \ref{def:AuxFDPSD} is identical to (3.11) in \citet{RomanoShaikh2006}, $k_i$ corresponds to $k(s,\gamma,m,|I|)$ on p. 42 there, and $\beta$ defined above agrees with $\beta$ in (3.15) in \citet{RomanoShaikh2006}. As noted by \citet{RomanoShaikh2006}, the arguments used in the proof of Theorem 3.4 do not depend on the specific form of the original constants. This implies, as in the proof of Theorem 3.4 (bottom of p. 40 and top of p. 41) that
\begin{align*}
P(\textnormal{FDP}(c)>\gamma) &\le P(\bigcup_{i =1}^{N(|I|)} \{PV_{(i)} \le \beta_i (|I|)\}) \le |I| \cdot \mbox{s.\,u.}\xspacem_{i=1}^{N(|I|)} \frac{\beta_i(|I|)-\beta_{i-1}(|I|)}{i}
\end{align*}
where the last bound is obtained by lemma \ref{lemma:BasicLemma}.
\end{proof}
\begin{coro}\label{coro:FDP.SD.bound.1}
Let $c \in\mathcal{C}$ and $\beta$ be defined as in proposition \ref{prop:FDP.SD.bound} and $\widetilde{A}$ as in definition \ref{def:MatrixFDPSD}. Denote by $\widetilde{A}_{i \cdot}$ the $i$-th row of $\widetilde{A}$ and for $I \mbox{s.\,u.}\xspacebset \{1,\ldots,n\}$ define $\beta(|I|)=(\beta_1(|I|),\ldots,\beta_{\lfloor \gamma n \rfloor+1}(|I|),0, \ldots,0) \in {\mathbb R}^n$. For $\alpha \in (0,1)$ it holds: If
\begin{align*}
\widetilde{A}_{1\cdot}\cdot \beta(1)^t &\le \alpha\\
\widetilde{A}_{2\cdot}\cdot \beta(2)^t &\le \alpha\\
&\vdots \\
\widetilde{A}_{n\cdot}\cdot \beta(n)^t &\le \alpha
\end{align*}
then $\max_{I \mbox{s.\,u.}\xspacebset \{1,\ldots,n\}} P(\{\textnormal{FDP}(c) > \gamma\})\le \alpha$.
\end{coro}
\begin{proof} For any set $I \mbox{s.\,u.}\xspacebset \{1,\ldots,n\}$ of true hypotheses by proposition \ref{prop:FDP.SD.bound} the probability $P(\textnormal{FDP}(c) > \gamma)$ is bounded by
\begin{align*}
& |I| \cdot \left\{\frac{\beta_1(|I|)}{1} + \frac{\beta_2(|I|) -\beta_1(|I|)}{2} + \cdots + \frac{\beta_{N(|I|)}(|I|) -\beta_{N(|I|)-1}(|I|)}{N(|I|)} \right\} \notag\\
&= |I| \cdot \left\{ \beta_1(|I|)\cdot \left(1-\frac{1}{2} \right) + \cdots + \beta_{N(|I|)-1}(|I|)\cdot \left(\frac{1}{N(|I|)-1}-\frac{1}{N(|I|)}\right) + \frac{\beta_{N(|I|)}(|I|)}{N(|I|)} \right\}\\
&=\widetilde{A}_{|I|\cdot}\cdot \beta(|I|)^t.
\end{align*}
\end{proof}
\begin{coro} Let $\gamma \in [0,1)$.
\begin{itemize}
\item[(a)] Let $1\le i\le n$. For any $\delta \in {\mathbb R}^n$ and $\beta_m(i):=\delta_{k_i(m)}$ it holds that $\widetilde{A}_{i \cdot} \cdot \beta(i)^t=(A^\textnormal{FDP}SD \cdot \delta^t)_i$.
\item[(b)] For $\alpha \in (0,1)$ and $c \in\mathcal{C}$ it holds: If $||A^\textnormal{FDP}SD\cdot c||_{\infty} \le \alpha$ then $P(\textnormal{FDP}(c)> \gamma)\le \alpha$.
\end{itemize}
\end{coro}
\begin{proof} For (a) we have
\begin{align*}
\widetilde{A}_{i \cdot} \cdot \beta(i)^t &= \mbox{s.\,u.}\xspacem_{\ell=1}^{\lfloor \gamma n \rfloor+1} \widetilde{A}_{i\ell} \cdot \beta_\ell (i) = \mbox{s.\,u.}\xspacem_{\ell=1}^{\lfloor \gamma n \rfloor+1} \widetilde{A}_{i\ell} \cdot \delta_{k_i(\ell)} \qquad \text{(by definition of $\beta$)}\\
&=\mbox{s.\,u.}\xspacem_{j=1}^n \delta_j \cdot \left( \mbox{s.\,u.}\xspacem_{\ell: k_i(\ell)=j} \widetilde{A}_{i\ell} \right) =\mbox{s.\,u.}\xspacem_{j=1}^n \delta_j \cdot \left( \mbox{s.\,u.}\xspacem_{\ell \in k_i^{-1}(\{j\})} \widetilde{A}_{i\ell} \right)\\
&=(A^{\textnormal{FDP-SD}}\cdot \delta^t)_i,
\end{align*}
where in the first equality of the second row the convention $\mbox{s.\,u.}\xspacem_{\textnormal{Var}nothing}\widetilde{A}_{i\ell}=0$ was used.
For part (b) note that if $||A^\textnormal{FDP}SD\cdot c||_{\infty} \le \alpha$ then this means by part (a) that $\max(\widetilde{A}_{1\cdot}\cdot \beta(1)^t, \ldots , \widetilde{A}_{n\cdot}\cdot \beta(n)^t)\le \alpha$ for $\beta_m(i):=c_{k_i(m)}$ and the claim then follows from corollary \ref{coro:FDP.SD.bound.1}.
\end{proof}
Thus theorem \ref{theorem:MainTheorem}, statement (d) is proved since for $d \in \mathcal{F}(A^{\textnormal{FDP}SD}(\gamma))$ and $c=\alpha \cdot d$ it now follows $||A^\textnormal{FDP}SD\cdot c||_{\infty}= \alpha \cdot||A^\textnormal{FDP}SD\cdot d||_{\infty}\le \alpha$ and part (b) from the above corollary yields the result.
\mbox{s.\,u.}\xspacebsection{Comments}
For the step-up $\textnormal{k-FWER}$ and $\textnormal{FDP}$ procedures, \citet{RomanoShaikh2006AOS} have proved that the choice $D=||A \cdot c||_{\infty}$ (with associated matrix $A$) is the smallest possible constant one can use for rescaled procedures of the form $c/D$ and still maintain control of the corresponding error rates. The key ingredient to their proof is part (ii) of lemma \ref{lemma:BasicLemma}.
For $\textnormal{k-FWER}$ step-down procedures \citet[Theorem 2.3 (ii)]{LehmannRomano2005} show that none of the Lehmann-Romano constants $c_i=\frac{k}{n+k-i}$ for $i>k$ can be improved without violating the $\textnormal{k-FWER}$. For $\textnormal{FDP}$ step-down procedures, \citet{RomanoShaikh2006} give an example that suggests that $D=||A \cdot c||_{\infty}$ is very nearly the smallest possible constant $d$ such that $c/d$ still controls $\textnormal{FDP}$, but no proof is given that this constant possesses the same optimality property as in the step-up case.
The modified FDP procedures introduced in section \ref{sec:NewFDPProcedures} by construction can not be improved without violating the LR bounds, i.e. without leading to $||A \cdot \xi||_{\infty}>1$. However, it is unclear whether this can also imply $P(\textnormal{FDP} > \gamma) > \alpha$. In the step-up case, the arguments given by \citet{RomanoShaikh2006} depend crucially on considering only linear modifications of the original procedures. Therefore these arguments do not seem applicable to investigating whether the modified procedures from section \ref{sec:NewFDPProcedures} can be improved any further.
\section{Simulation study} \label{sec:SimulationStudy}
In this section we investigate the power of the different FDP procedures in a simulation study. We consider the following procedures:
\begin{itemize}
\item FDP-BH-SU and its modified variant FDP-BH-SU (mod),
\item FDP-RS-SU and its modified variant FDP-RS-SU (mod),
\item FDP-BH-SD and its modified variant FDP-BH-SD (mod),
\item FDP-RS-SD and its modified variant FDP-RS-SD (mod).
\end{itemize}
The goals of the study are three-fold:
\begin{enumerate}
\item to compare the power of the modified procedures with their original counterparts,
\item to compare the power between the modified procedures,
\item to compare the best FDP procedure (if it exists) with FDR controlling procedures.
\end{enumerate}
To make the last comparison more consistent, we use for the step-up direction the \citet{BenjaminiYekutieli01} procedure FDR-BY-SU with critical constants
\begin{align*}
c_i^{BY}&=c_i^{BY}(n)=c_i^{BH}/D, \quad \text{where} \quad D=1+\frac{1}{2}+ \cdots + \frac{1}{n}
\end{align*}
which controls the FDR under arbitrary dependence. For the step-down direction we use the rescaled BH constants obtained by \citet{GuoRao2008}, i.e.
\begin{align*}
c_i^{GR}&=c_i^{GR}(n)=c_i^{BH}/D, \quad \text{where} \\
D& = \max_{i=1, \ldots,n} \frac{i}{n} \left\{\mbox{s.\,u.}\xspacem_{j=1}^{n-i+1} \frac{1}{j} + \frac{n-i}{n-i+1}- \frac{n-i}{n} \right\}.
\end{align*}
We denote this approach by FDR-GR-SD. Similarly to \citet{RomShaWoETh2008} we control the median FDP as an alternative to controlling the FDR. We do this at the $.05$-level, i.e. $P(\textnormal{FDP}>0.05)\le 0.5$, while the FDR procedures control the expectation $\textnormal{E}(\textnormal{FDP})\le 0.05$. As \citet{RomShaWoETh2008} point out, the median FDP is a less stringent measure than the FDR in the sense that the probability of the FDP exceeding $0.05$ can be much bigger when the median FDP is controlled than when the FDR is controlled.
For MTPs there are several ways to measure power, see e.g. \citep[Section 1.2.10]{DudoitLaan2007}. We use average power, i.e. the average proportion of rejected false hypotheses, for comparing procedures. We assume equicorrelated multivariate normal test statistics, i.e. $T=(T_1, \ldots, T_n) \sim \textbf{{N}}(\mu, \Sigma)$ with $\mu_i=0 $ for $i=1, \ldots, |I|$, $\mu_i=d$ for $i= |I|+1, \ldots,n$ and $\Sigma_{ij}=1/2$ for $i \neq j$ and $\Sigma_{ij}=1$ else. For the parameter $d$, three nonzero values were used: $d = 0.1, 1$ and $3$, reflecting small, moderate and large deviations from the null hypotheses. For each simulated vector of test-statistics $p$-values were calculated for the gaussian test of the null hypotheses $H^0_i: \mu_i = 0$ (two-sided). The number of tests performed was set to one of the values $10$, $50$, $100$ and $500$ reflecting small, medium and (moderately) large multiplicity of tests. We used $20000$ simulations in the simulation study which gives a uniform upper bound for the standard errors of $0.0035$.
Figure \ref{fig:FDPvsModifiedFDPAverageProportionRejections_20000} depicts the gains in average power of the modified FDP procedures over the original (rescaled) variants.
\begin{sidewaysfigure}[htbp]
\centering
\includegraphics[width=1.00\textwidth]{AverageProportionRejectionsDependentDifferenceFDPs_20000.pdf}
\caption{Difference of simulated average power for modified FDP procedures vs original (rescaled) FDP procedures (the $x$-axis is the number of true hypotheses). Shown are BH-SU (blue), RS-SU (violet), BH-SD (red), RS-SD (orange).}
\label{fig:FDPvsModifiedFDPAverageProportionRejections_20000}
\end{sidewaysfigure}
For most constellations, the gains in power are considerably larger for the BH-type procedures than for the RS-type procedures. Put differently, the RS procedures perform so well that in many situations none or only little improvement is possible.
\begin{sidewaysfigure}[htbp]
\centering
\includegraphics[width=1.00\textwidth]{AverageProportionRejectionsDependentModFDPs_20000.pdf}
\caption{Simulated average power for modified FDP procedures. Shown are BH-SU (blue), RS-SU (violet), BH-SD (red), RS-SD (orange).}
\label{fig:SelectedAverageProportionRejections_20000}
\end{sidewaysfigure}
Figure \ref{fig:SelectedAverageProportionRejections_20000} presents a comparison of the four modified FDP-controlling procedures. The FDP-BH-SD procedure usually performs best and is followed closely by FDP-RS-SD and FDP-BH-SU.
\begin{sidewaysfigure}[htbp]
\centering
\includegraphics[width=1.00\textwidth]{AverageProportionRejectionsDependentFDPvsFDR_20000.pdf}
\caption{Simulated average power for modified FDP-BH-SD (red), FDR-BY-SU (black) and FDR-GR-SD (grey).}
\label{fig:FDPvsFDRAverageProportionRejections_20000}
\end{sidewaysfigure}
Figure \ref{fig:FDPvsFDRAverageProportionRejections_20000} compares the modified procedure FDP-BH-SD (mod) with the FDR procedures BY-SU and GR-SD. The median FDP-BH-SD posesses the highest power for all constellations while FDR-GR-SD and FDR-BY-SU perform very similarly.
Altogether we conclude that
\begin{itemize}
\item modifying the rescaled FDP procedures resulted in increased power for all four procedures. The largest gains were achieved for the BH-type procedures,
\item for the constellations considered here, FDP-BH-SD (mod) performed best, with FDP-SR-SD (mod) or FDP-BH-SU (mod) usually coming in a close second,
\item the best modified median FDP procedure outperformed the FDR-controlling procedures that were rescaled in order to account for general dependence.
\end{itemize}
\section{Empirical applications}\label{sec:EmpiricalApplications}
In this section we compare the performance of the FDP and FDR approaches from the previous section for some empirical data.
\mbox{s.\,u.}\xspacebsection{Benjamini-Hochberg data} \label{ssec:BHData}
We revisit the data analysed in \cite{BenjaminiHochberg95}, consisting of 15 $p$-values from a study on myocardial infarctation. Table \ref{tab:BHData} gives the results of applying the median FDP and FDR procedures at levels $q=0.05$ (note that in this case $\gamma-\textnormal{FDP}=\textnormal{FWER}$) and $q=0.10$, i.e. $P(\textnormal{FDP}>q)\le 0.5$ and $\textnormal{E}(\textnormal{FDP})\le q$.
\begin{table}[htb]
\centering
\begin{tabular}{lcc}
& \multicolumn{ 2}{c}{Number of rejections} \\ \cmidrule(lr){2-3}
Method & $q=0.05$ & $q=0.10$ \\
\hline
\hline
FDP-BH-SU & 9 & 9 \\
FDP-BH-SU (mod) & 9 & 9 \\
FDP-RS-SU & 5 & 4 \\
FDP-RS-SU (mod) & 5 & 5 \\\hline
FDR-BY-SU & 3 & 3 \\\hline
FDP-BH-SD & 10 & 10 \\
FDP-BH-SD (mod) & 10 & 10 \\
FDP-RS-SD & 10 & 10 \\
FDP-RS-SD (mod) & 10 & 10 \\\hline
FDR-GR-SD & 3 & 4 \\
\hline
\end{tabular}
\caption{Number of rejected hypotheses for the Benjamini-Hochberg data. }
\label{tab:BHData}
\end{table}
For $q=0.05$, the step-down procedures performed best, followed by the step-up FDP-BH and FDP-RS methods. The FDR procedures rejected the fewest hypotheses. Note that the FDP-RS-SU procedure rejects fewer hypotheses at level $0.10$ than at level $0.05$. This behavior is due to the fact that both the original constants and the scaling constant $D$ depend on the parameter $\gamma$. In this special case it means that $c_i^{0.05}\le c_i^{0.10}$ only for $i \in \{10,\ldots,14\}$. For the FDP-BH procedures this can not happen, since the original constants do not depend on the parameter $\gamma$ and the scaling constants are increasing in $\gamma$.
\mbox{s.\,u.}\xspacebsection{Westfall-Young data} \citet{WestYoung93} use resampling methods to analyze data from a complex epidemiological survey designed to assess the mental health of urban and rural individuals living in central North Carolina. The data consists of 72 raw $p$-values (see \citet[table 7.42]{WestYoung93}), with 25 of them $<0.05$ and 9 of the adjusted $p$-values $<0.05$. Table \ref{tab:WYData} displays the number of rejections when using the median FDP and FDR controlling procedures introduced above.
\begin{table}[htb]
\centering
\begin{tabular}{lcc}
& \multicolumn{ 2}{c}{Number of rejections} \\
Method & $q=0.05$ & $q=0.10$ \\
\hline
\hline
FDP-BH-SU & 11 & 11 \\
FDP-BH-SU (mod) & 11 & 12 \\
FDP-RS-SU & 10 & 11 \\
FDP-RS-SU (mod) & 11 & 11 \\\hline
FDR-BY-SU & 10 & 10 \\\hline
FDP-BH-SD & 11 & 11 \\
FDP-BH-SD (mod) & 12 & 12 \\
FDP-RS-SD & 11 & 12 \\
FDP-RS-SD (mod) & 11 & 12 \\\hline
FDR-GR-SD & 10 & 11 \\
\hline
\end{tabular}
\caption{Number of rejected hypotheses for the Westfall-Young data. }
\label{tab:WYData}
\end{table}
All procedures reject at least one additional hypothesis. For level $q=0.05$, all median FDP procedures except RS-SU perform better than the FDR procedures; the modified BH-SD procedure is the only procedure that rejects three additional hypotheses. For $q=0.10$ the step-down FDP procedures seem to work best.
\mbox{s.\,u.}\xspacebsection{Hedenfalk data} \label{ssec:HedenfalkData}
The data come from the breast cancer cDNA microarray experiment of Hedenfalk et al. (2001). In the original experiment, comparison was made between 3,226 genes of two mutation types, BRCA1 (7 arrays) and BRCA2 (8 arrays). The data included here are $p$-values obtained from a two- sample t-test analysis on a subset of 3,170 genes, as described in \citet{Storey2003}. Table \ref{tab:HedenfalkData} gives the results of applying the median FDP and FDR procedures at levels $q=0.05$ and $q=0.10$, i.e. $P(\textnormal{FDP}>q)\le 0.5$ and $\textnormal{E}(FDP)\le q$.
\begin{table}[htb]
\centering
\begin{tabular}{lcc}
& \multicolumn{ 2}{c}{Number of rejections} \\
Method & $q=0.05$ & $q=0.10$ \\
\hline
\hline
FDP-BH-SU & 0 & 1\\
FDP-BH-SU (mod) & 6 & 10 \\
FDP-RS-SU & 3 & 3 \\
FDP-RS-SU (mod) & 3 & 3\\\hline
FDR-BY-SU & 0 & 1 \\\hline
FDP-BH-SD & 0 & 1 \\
FDP-BH-SD (mod) & 7 & 4 \\
FDP-RS-SD & 6 & 4 \\
FDP-RS-SD (mod) & 6 & 4 \\\hline
FDR-GR-SD & 0 & 1 \\
\hline
\end{tabular}
\caption{Number of rejected hypotheses for the Hedenfalk data. }
\label{tab:HedenfalkData}
\end{table}
\begin{figure}
\caption{$p$-values (solid triangles) and multiple testing procedures for the Hedenfalk data (left panel: $\gamma=0.05$, right panel: $\gamma=0.01$).}
\end{figure}
Again, the step-down FDP procedures perform better than their step-up counterparts, the modified median BH-SD procedure rejecting the most hypotheses. While all FDP procedrues except BU-SU reject more hypothese than both FDR approaches, we might hope for more powerful procedures. One alternative idea could be to use resampling methods in order to account for dependencies. However, as \citet{Pounds2006} points out, the power of these methods "will be severely limited, when the sample size is small". When the dependency between the $p$-values is assumed to be strong and extensive he tentatively recommends the FDR-BY-SU procedure.
\section{Discussion} \label{sec:Discussion}
In this paper we have used results from \citet{RomanoShaikh2006,RomanoShaikh2006AOS} to obtain sufficient criteria for generalised error rates under general dependence in terms of systems of linear inequalities. These systems of linear inequalities describe the set of feasible points of a suitable linear optimisation problem. This property can be used to obtain modified multiple testing procedures which can improve on the rescaled procedures introduced in \citet{RomanoShaikh2006,RomanoShaikh2006AOS}. In a simulation study we have observed that these modified procedures can posess considerably more power than the original procedures.
While the focus of this work was on developing more powerful multiple testing procedures, \citet{HomBretz2008} have formulated additional desirable properties for such pocedures. Even though all methods considered here satisfy the property of coherence they are not particularly simple to describe and to communicate to non-statisticians. Since the modified procedures are obtained from a computationally complex numerical optimisation technique, the resulting sequence of critical constants will generally not exhibit any aesthetic mathematical patterns like e.g. the Bonferroni-Holm procedures. Another potential drawback from an aesthetical perspective may be related to what \citet{HomBretz2008} describe as monotonicity properties of multiple testing procedures. While all procedures considered here yield monotonic decisions with respect to the corresponding type 1 error, it could be pointed out that additional monotonicity properties are conceivable that are not satisified by some of them. As a case in point, reconsider for $n=15$ the RS-procedures for $\gamma=0.05$ and $\gamma=0.10$ (see section \ref{ssec:BHData}). A numerical evaluation shows that $c^{0.05}_i<c^{0.10}_i$ holds true only for $i=10,\ldots,14$. Thus it may happen that more hypotheses are rejected for $\gamma=0.05$ than for $\gamma=0.1$ (at the same level of type 1 error) even though one would expect that the requirement $\textnormal{FDP} \le 0.05$ is more stringent than $\textnormal{FDP} \le 0.10$. The reason for this behavior is that both the original critical constants and the scaling constant $D$ depend on the parameter $\gamma$. For the FDP-BH procedures the original critical constants do not depend on $\gamma$. Numerical computations suggest that the scaling constants for FDP-BH are increasing in $\gamma$ and so this effect seems to be avoided by the FDP-BH procedures.
Another issue is the computational complexity of solving the linear programming problem which is needed to obtain the modified procedures. As a case in point, the calculation of the modified procedures used for analysing the Hedenfalk data with $n=3170$ in section \ref{ssec:HedenfalkData} took approximately nine hours on a Intel Xeon 5620 processor using the R-function \verb+Rglpk_solve_LP+. For multiple testing problems where the number of tests is significantly larger we thus expect run-time problems depending on the software and hardware available.
Finally, concerning other error rates like the FDR it seems natural to ask whether there are similar ways of modifying existing procedures under arbitrary dependence of the $p$-values. Recall that the key to modifying FDP procedures in section \ref{sec:NewFDPProcedures} was the observation that an improvement is possible whenever the $|I|^\star$-th row of matrix $A$ (with $|I|^\star= \arg\max_{|I|} (A \cdot c)_{|I|}$) contains at least one zero entry. \citet{GuoRao2008} have obtained bounds for the FDR that are similar to the bounds in theorem \ref{theorem:MainTheorem}, i.e. with $\textnormal{FDR}(c) \le \| A \cdot c\|_{\infty}$ for a suitable matrix $A$. However, as their Theorems 4.2 and 5.2 show, the corresponding step-up and step-down matrices do not contain any zero elements. Therefore, while the linear optimisation approach could still be used to define new procedures e.g. via an unconstrained linear program, it will not be possible to attain strict improvements along the lines of section \ref{sec:NewFDPProcedures}.
\label{bibliography}
\end{document} | math | 57,839 |
\begin{document}
\title{Causal Inference on Win Ratio for Observational Data with Dependent Subjects}
\begin{abstract}
Composite endpoints are commonly used with an anticipation that clinically relevant endpoints as a whole would yield meaningful treatment benefits. The win ratio is a rank-based statistic to summarize composite endpoints, allowing prioritizing the important components of the composite endpoints. Recent development in statistical inference for the win ratio statistic has been focusing on independent subjects without any potential confounding. When analyzing composite endpoints using observational data, one of the important challenges is confounding at baseline. Additionally, hierarchical observational data structures are commonly seen in practice, especially in multi-center studies with patients nesting within hospitals. Such hierarchical structure can introduce potential dependency or cluster effects among observations in the analysis. To address these two issues when using the win ratio statistic, we propose a weighted stratified causal win ratio estimator with calibrated weights. The calibrated weights create balanced patient-level covariates and cluster effect distributions between comparison groups. We conducted extensive simulation studies and showed promising performance of the proposed estimator in terms of bias, variance estimation, type I error and power analysis, regardless of the allocation of treatment assignments at baseline and intra-cluster correlations within clusters. Lastly, the proposed estimator was applied to an observational study among children with traumatic brain injury.
\end{abstract}
\noindent {\it Keywords:} Calibration; Causal Inference; Cluster; U-statistic; Weighting
\section{Introduction} \label{intro}
Composite endpoints are commonly used in clinical trials and observational studies. One of the main advantages of using the composite endpoints is a higher study power compared to a study design based on a single endpoint \cite{huque2011addressing}. When the composite endpoints imply different clinical importance, however, the analysis should take into account the magnitude of importance of each endpoint to reach a more sensible conclusion. One approach to account for the importance of composite endpoints is the win ratio statistic proposed by \cite{pocock2011win}. Without loss of generality, we consider composite endpoints of a non-terminal (e.g. hospitalization) and a terminal (e.g. death) event. To construct the win ratio statistic, time to the terminal events are first being compared between patients from treatment and control groups. If the “win” or “loss” decision cannot be made due to censoring, then the non-terminal events are compared subsequently to reach a decision. The verdict is a tie when at least one of the paired times is censored and one censoring time occurs prior to the other censoring time or event time. Eventually, the win ratio for the treatment group is the ratio of the number of wins to the number of losses in the treatment group.
\\ \\
Numerous studies have investigated the statistical properties of the win ratio statistic in clinical trial settings. \cite{luo2015alternative}, \cite{dong2016generalized} and \cite{oakes2016win} established the statistical framework for the win ratio. \cite{bebu2015large} formulated the win ratio through a bivariate $U$-statistics for independent data based on large sample inference. \cite{luo2017weighted} proposed a weighted win ratio statistic to improve efficiency. Several studies have also extended the use of the win ratio to non-time-to-event outcomes \citep{wang2016win}. \cite{dong2019win}, and \cite{finkelstein2019graphing} discussed the interpretation of the win ratio. \cite{dong2018stratified} and \cite{mao2019alternative} further generalized the use of the win ratio to multivariate and stratified settings.
\\ \\
The application of the win ratio statistic in observational studies raises additional challenges. The Approaches and Decisions for Acute Pediatric (ADAPT) Traumatic Brain Injury (TBI) trial is a multi-center observational study to investigate the impacts of various medical interventions on children under 18 years old with TBI. During the first 7 days of hospitalization after brain injuries, one of the study interests is to investigate the impact of the Cerebrospinal Fluid (CSF) drainage treatment compared to no CSF treatment on the composite endpoints of time to first most severe neurological complication and time to death. Patients and physicians value death as a more important event than complications, and the investigators want to reflect this information in the analysis. Total 26 potential baseline confounders were identified in the study, and the data were collected from 37 sites. The medical resources and physician training may be similar within the same site; however, they could vary substantially across sites. These challenges should be accounted for in the analysis to allow legitimate inference.
\\ \\
To adopt the win ratio statistic in the ADAPT observational study, we face two main challenges: potential confounders and cluster effects due to sites. The existing literature related to the inference of win ratio statistic has been mainly focusing on independent subjects without potential confounding. Mao proposed an efficient inverse probability weighted (IPW) estimator to make causal inference on the U-statistics \citep{mao2017causal}. In epidemiological literature, the approach of IPW type estimators in casual inference is also called propensity score (PS) analysis with weighting \citep{austin2015moving}. The propensity scores can be used in various ways to balance the baseline confounding covariates. Since the win ratio can be expressed as the rank-based U-statistic, in this paper we adopt Mao's efficient estimator for causal inference on the win ratio statistic to account for the baseline confounding. In terms of accounting for the cluster effects in observational studies, existing literature has been mainly focusing on the PS-based approaches. \cite{li2013propensity} and \cite{thoemmes2011use} have shown that one should account for the cluster effects both at the PS estimation stage and the outcome analysis stage. \cite{arpino2016propensity} proposed a PS matching algorithm with the PS estimation from fixed effects and random effects models. However, there are limitations based on both fixed effects and random effects models to estimate the PS \citep{li2013propensity} due to parametric assumptions. In addition, the cluster effects estimated from the random effects model can be distorted due to the estimation shrinkage towards the grand mean in the generalized mixed model. \cite{chan2016globally} introduced a general class of the calibration estimator for the weights to balance the baseline covariates for the independent subject case. This calibration estimation is attractive due to the non-parametric nature and its robustness against model misspecification. The calibration weighting technique is commonly used in survey sampling analysis \citep{chen2002using, kim2010calibration}, and recently, \cite{yang2018propensity} adapted this technique to estimate the weights for clustered continuous and binary outcomes.
\\ \\
We propose a weighted stratified win ratio estimator to analyze the composite endpoints in the presence of baseline confounding and cluster effects. We adopt the causal U-statistics by \cite{mao2017causal} to account for baseline confounding and the calibrated weights by \cite{yang2018propensity} to account for cluster effects in the treatment selection process. The proposed estimator was implemented in an R function available on GitHub. This article contributes to the methodological gap in the literature and in practice, regarding analyzing hierarchical observational data with subject-level confounding for composite endpoints, prioritizing important components of the composite endpoints. The rest of the article is organized as follows. Section \ref{wr_independent_no_confounding} reviews the win ratio statistic for independent subjects without confounding. Section \ref{wr_independent_confounding} presents the inference on the causal win ratio for independent subjects. Section \ref{weighted_estimator} introduces the proposed weighted stratified estimator for dependent subjects with confounding. Section \ref{weight_estimation} presents the estimation of the calibrated weights. Through simulation studies in Section \ref{calibrationWR_simulation}, we show that the proposed estimator performs well compared to the estimators with the weights estimated from the fixed effects or random effect models. An application to the TBI data is presented in Section \ref{calibrationWR_example}. Conclusions and limitations are discussed in Section \ref{calibrationWR_conclusion}.
\section{Review Win Ratio Statistic}
\label{wr_independent_no_confounding}
We first review the win ratio statistic for data with independent subjects without confounding \cite{pocock2011win}. Consider composite endpoints including a terminal event (e.g. death) and a non-terminal event (e.g. hospitalization). To decide if a treatment patient wins or loses compared to a control patient, times to the terminal events are first compared. If the control patient has an earlier terminal event, then the treatment patient wins, vice versa. If a decision cannot be reached using the terminal events due to censoring, then times to non-terminal events are compared between these two patients following the same logic. Similar procedures apply for all pairwise comparisons between patients from treatment and control groups. All possible scenarios of between-patient comparisons can be represented using the indicator functions as shown by \cite{luo2015alternative}. Let $N_w$ and $N_l$ be the numbers of wins and losses in the treatment group, respectively. The win ratio statistic for independent data is defined as $WR=E(\frac{N_w}{N_l})$, which can be approximated by $WR\approx \frac{E(N_w)}{E(N_l)}=\frac{\tau_1}{\tau_2}$ based on the first order Taylor series expansion.
\section{Win Ratio for Independent Subjects with Confounding}
\label{wr_independent_confounding}
Based on the formulation of independent win ratio by \cite{luo2015alternative} and \cite{bebu2015large}, we adopt the efficient estimator for the causal U-statistics by \cite{mao2017causal} to account for patient-level confounding. The win ratio statistic is essentially a ratio of two ordinal outcomes and they can be represented by the U-statistics. \cite{mao2017causal} established the causal inference for the U-statistics using inverse probability weights (IPWs) based on the semi-parametric theory. We define $n$ to be the total number of independent subjects. Let $Y_i$ and $Y_j$ be the observed composite endpoints for subject $i$ and $j$, respectively, where $i, j=1,...,n$. We define two kernels as, $i\ne j$,
\begin{equation}\label{kernel_causalWR1}
\phi_1(Y_i,Y_j)=\textbf{1}\{Y_i>Y_j\}=
\left\{
\begin{array}{@{}ll@{}}
1 & \text{if}\ Y_i \text{ wins}, \\
0 & \text{otherwise},
\end{array}\right.
\end{equation}
and
\begin{equation}\label{kernel_causalWR2}
\phi_2(Y_i,Y_j)=\textbf{1}\{Y_i<Y_j\}=
\left\{
\begin{array}{@{}ll@{}}
1 & \text{if}\ Y_j \text{ wins}, \\
0 & \text{otherwise}.
\end{array}\right.
\end{equation}
The within-pair comparisons to determine whether $Y_i$ or $Y_j$ wins follow the description in Section \ref{wr_independent_no_confounding}. An efficient IPW estimator \citep{mao2017causal} can be constructed as below, using the observed data $Y_i$ and $Y_j$ in the kernel function (\ref{kernel_causalWR1})
\begin{equation}
\label{ipw}
\begin{split}
\hat{\tau_1}=&{n \choose 2}^{-1} \sum_{i=1}^{n-1}\sum_{j>i}^{n}\{\frac{1}{2}\{\frac{Z_i(1-Z_j)}{\hat{\pi} (X_i; \hat{\alpha})(1-\hat{\pi}(X_j, \hat{\alpha}))}\phi_1(Y_i, Y_j)\\
&+\frac{Z_j(1-Z_i)}{\hat{\pi}(X_j; \hat{\alpha})(1-\hat{\pi}(X_i; \hat{\alpha}))}\phi_1(Y_j, Y_i)\}\},
\end{split}
\end{equation}
where $Z_i$ is the observed treatment status for subject $i$ and $\pi(X_i; \alpha)$ is the probability of receiving the treatment of interest given a vector of confounders $X_i$ for subject $i$. The only quantity that needs to be estimated in $\hat{\tau_1}$ is $\pi(\cdot)$ and it's usually done through some parametric modeling, such as a logistic regression with a model parameter vector $\alpha$. This average effect for the ordinal outcomes is defined as average superiority effect (ASE) \citep{mao2017causal}. Similar formulation can be applied for $\tau_2$. \cite{mao2017causal} showed that the IPW estimator for the average treatment effect in (\ref{ipw}) is consistent and asymptotically normally distributed under some suitable regularity conditions. Therefore, as $n \rightarrow \infty$, by the delta method the estimator of the causal win ratio, $\hat{\tau_1}/\hat{\tau_2}$, would also follow an asymptotic normal distribution as does $\log[\hat{\tau_1}/\hat{\tau_2}]$.
\section{Win Ratio for Dependent Subjects with Confounding}
\label{weighted_estimator}
We propose to use a weighted stratified causal win ratio estimator to account for potential baseline confounding and cluster effects. The stratification unit is a cluster. We re-define some notations in this section. Let $\tau_{1i}$ be the average effect of treatment in cluster $i$, and $\tau_{2i}$ be the average effect of control in cluster $i$ ($i=1, ..., m$). Define the cluster-specific win ratio for cluster $i$ be $\mu_i = \frac{\tau_{1i}}{\tau_{2i}}$. To estimate $\tau_{1i}$ accounting for the patient-level confounding, we adopt the efficient estimator for the causal U-statistic by \cite{mao2017causal}, shown in Section \ref{wr_independent_confounding}. The estimator $\hat{\tau}_{1i}$ for cluster $i$ can be defined as
\begin{equation}
\label{cluster_tao}
\hat{\tau}_{1i}={n_i \choose 2}^{-1} \sum_{j=1}^{n_i-1}\sum_{j'>j}^{n_i}\{\frac{1}{2}\{Z_{ij}(1-Z_{ij'})\hat{w}_{ij}\hat{w}_{ij'}\phi_1(Y_{ij}, Y_{ij'})+Z_{ij'}(1-Z_{ij})\hat{w}_{ij}\hat{w}_{ij'}\phi_1(Y_{ij'}, Y_{ij})\}\},
\end{equation}
where $n_i$ is the total number of independent subjects in cluster $i$, $Z_{ij}$ and $Z_{ij'}$ are treatment status for subject $j$ and $j'$ ($j, j'=1, ..., n_i$) in cluster $i$, respectively, and $\hat{w}_{ij}$ and $\hat{w}_{ij'}$ are the estimated weights for subject $j$ and $j'$. The weight is a function of patient-level confounders and cluster effects. Under the independent subjects setting, the weight corresponds to the inverse of probability of receiving the observed treatment, given patient-level confounders. As shown in Equation (\ref{ipw}), when subject $i$ receives treatment and subject $j$ receives control, $Z_i=1$ and $Z_j=0$. The weights correspond to $\frac{1}{\hat{\pi} (X_i; \hat{\alpha})(1-\hat{\pi}(X_j, \hat{\alpha}))}$. Under the dependent subjects setting, we incorporate the calibrated weights as proposed by \cite{yang2018propensity} to further account for cluster effects. More details of the calibrated weights are presented in Section \ref{weight_estimation}. Let $Y_{ij}$ and $Y_{ij'}$ be the composite endpoints for subjects $j$ and $j'$ in cluster $i$, respectively. The two kernels, where $j\ne j'$, become
\begin{equation}\label{kernel_causalWR3}
\phi_1(Y_{ij},Y_{ij'})=\textbf{1}\{Y_{ij}>Y_{ij'}\}=
\left\{
\begin{array}{@{}ll@{}}
1 & \text{if}\ Y_{ij} \text{ wins}, \\
0 & \text{otherwise},
\end{array}\right.
\end{equation}
and
\begin{equation}\label{kernel_causalWR4}
\phi_2(Y_{ij},Y_{ij'})=\textbf{1}\{Y_{ij}<Y_{ij'}\}=
\left\{
\begin{array}{@{}ll@{}}
1 & \text{if}\ Y_{ij'} \text{ wins}, \\
0 & \text{otherwise}.
\end{array}\right.
\end{equation}
Similar estimator can be derived for $\tau_{2i}$ with the kernel $\phi_2(Y_{ij},Y_{ij'})$.
\\ \\
Define $\hat{w}_{i\cdot}=\sum_{j=1}^{n_i-1}\sum_{j'>j}^{n_i}\{ \frac{1}{2} Z_{ij}(1-Z_{ij'})\hat{w}_{ij}\hat{w}_{ij'}\}$ as sum of all possible pairwise combinations of estimated weights multiplied for cluster $i$. Let $\hat{w}_{\cdot \cdot}=\sum_{i=1}^{m} \hat{w}_{i\cdot}$ be the overall estimated weights across all clusters. Therefore, the estimated average treatment effect of the causal win ratio for clustered data is
\begin{equation}
\label{equ7}
\hat{\mu} = \frac{\sum_{i=1}^{m} \hat{w}_{i\cdot}\hat{\mu}_i}{\hat{w}_{\cdot \cdot}}.
\end{equation}
Heuristically, assuming that $\hat{w}_{i \cdot}$ ($i=1,2,...,m$) and $\hat{w}_{\cdot \cdot}$ converge in probability to their limiting values $\omega_i$ ($i=1,2,...,n_i$) and $\omega$, as $m, n_i \rightarrow \infty$, $\hat{\mu}$ would follow an asymptotic normal distribution by the Slutsky theorem as $\hat{\mu}_i$ does. In practice, the variance of this estimator can be estimated using empirical bootstrap variance estimation by resampling clusters with replacement. We first draw samples of $m$ clusters independently with replacement. Then estimate the causal win ratio as shown in Equation (\ref{equ7}) in this bootstrap sample. Repeat this resampling and estimation procedure for $B$ times. The bootstrap variance is the sample variance of the causal win ratio estimates from the bootstrap samples. We show the empirical distribution of the proposed estimator follows a normal distribution in the Appendix, based on our simulation set-up in Section \ref{set-up}. As the number of clusters and the cluster sizes increase, the density of the distribution concentrates more around the mean.
\\ \\
Under the potential outcomes framework \citep{rubin1974estimating}, we assume that the consistency assumption holds true, which defines that the observed outcome corresponds to the potential outcome under the specific treatment. Furthermore, we make the stable unit and treatment version assumption (SUTVA), which implies that the potential outcomes for each unit are not affected by the treatments assigned to other units. Conditional on the observed patient-level confounders and cluster effects, the ignoreability assumption holds, where the potential outcomes and treatment variable are independent. Yang defines this assumption as latent ignorability assumption for cluster data \citep{yang2018propensity}. Lastly, we make positivity assumption that every unit has a positive probability to receive the treatment of interest. This is reflected in $0<\pi(X_{ij}; \alpha)<1$ for the initial weight estimation in Section \ref{weight_estimation}.
\section{Calibrated Weight Estimation}\label{weight_estimation}
We use calibrated weights, proposed by \cite{yang2018propensity}, to account for potential subject-level confounders and cluster effects imposed on treatment selections in observational studies. Following the notations in \cite{yang2018propensity}, there are two steps in the estimation procedure:
\begin{enumerate}
\item Estimate the initial weights $d_{ij}$ using some parametric working model, such as the logistic regression. Define $d_{ij}=\frac{Z_{ij}}{\pi(X_{ij}; \alpha)}+\frac{1-Z_{ij}}{1-\pi(X_{ij}; \alpha)}$ as the initial weights, where $\pi(X_{ij}; \alpha)$ is the probability of receiving treatment of interest conditional on confounders for subject $j$ in cluster $i$, and $\alpha$ is a vector of coefficient parameters for the parametric working model.
\item Estimate the calibration weights $w_{ij}$ using some distance function $D(w_{ij}, d_{ij})$ with constraints. Define $w_{ij}$ as the final weight.
\end{enumerate}
The constrains in step 2 of the procedure are used to ensure the balance of covariates and cluster effects between comparison groups so that valid causal inference can be made for observational clustered data. We refer to \cite{yang2018propensity} for more detailed discussion on the constrains. Different distance functions can lead to different estimation methods. Following \cite{yang2018propensity}, we adopted the Kullback-Leibler (KL) distance function $D(w_{ij}, d_{ij})=w_{ij}\ln \frac{w_{ij}}{d_{ij}}$. This distance function measures distance between probability distributions. The KL distance function provides exponential tilting estimation \citep{schennach2007point}, which is asymptotically equivalent to the regression estimator, but avoids extreme weights \citep{kim2010calibration, schennach2007point}. Specifically in this problem, the KL distance function can incorporate the constraints as a closed form, which is more computationally efficient.
\\ \\
The calibrated weights can be estimated by solving the distance function with constrains through iterative optimization. A detailed derivation of the calibrated weights following \cite{yang2018propensity} was provided in the Appendix. We implemented the proposed weighted stratified estimator with calibrated weights in a R function \emph{WR.causal} of package \verb|cWR| on GitHub at \url{https://github.com/dee1008/cWR}.
\section{Simulation Studies}
\label{calibrationWR_simulation}
\subsection{Set-up}
\label{set-up}
Several simulation studies were conducted to assess the behavior of the proposed weighted stratified causal win ratio estimator in clustered settings. The following estimators were compared, (1) the independent causal win ratio estimator without stratification and with IPW estimated from standard logistic regression (i.e. unadjusted estimator); (2) the weighted stratified estimator with IPW estimated from standard logistic regression (i.e. logistic estimator); (3) the weighted stratified estimator with IPW estimated from logistic regression with clusters as fixed effects (i.e. fixed estimator); (4) the weighted stratified estimator with IPW estimated from logistic regression with clusters as random intercepts (i.e. random estimator); (5) the weighted stratified estimator with calibrated weights (i.e. calibration estimator). In the calibrated weight estimation, we used a standard logistic regression as the working model to estimate the initial weights $d_{ij}$. Since the statistical framework we assumed requires both treatment and control groups within the same cluster, we excluded clusters that only contained one group in the simulation. Two covariates were considered, $X_1$ was a standard normal variable, and $X_2$ was normally distributed with mean 1 and variance 4. The same set of covariates were used to generate the outcomes and the treatments. Let $\gamma_1$ be the cluster effect on the outcomes and $\gamma_2$ be the cluster effect on the treatment selections. We assume $\gamma_1$ follows a gamma distribution that induces the intra-cluster correlation (ICC) of 0.2 or 0.067. We assume that $\gamma_2$ follows a normal distribution with the mean of 0 and the variance of 4 or a gamma distribution with the shape parameter of 2 and the rate parameter of 10. We assume $\gamma_1$ and $\gamma_2$ are correlated through a normal copula with a correlation parameter of 0.4. The cluster effects were generated using the R package \verb|Copula| \citep{kojadinovic2010modeling}.
\\ \\
Assuming a proportional hazards model, let $\lambda_{H_{z_{ij}}}=\gamma_{1i}\lambda_H\exp(-\eta_{H}z_{ij}+x_{ij}\beta_1)$ be the hazard function for time to non-terminal event given covariates for patient $j$ in cluster $i$, where $\lambda_H$ was the baseline hazard function of the distribution of time to non-terminal events, $\eta_H$ was the non-terminal event rate per time unit, and $\beta_1$ was the coefficients of the covariate vector $x_{ij}$ with dimension 2. The treatment status $z_{ij}$ can be 0 or 1, which corresponds to treatment or control group, respectively. Similarly, let $\lambda_{D_{z_{ij}}}=\gamma_{1i}\lambda_D\exp(-\eta_{D}z_{ij}+x_{ij}\beta_2)$ be the hazard function of the distribution of time to terminal events. The bivariate exponential distribution with Gumbel-Hougaard copula was used to generate the bivariate distribution of time to non-terminal events $T_{H_{z_{ij}}}$ and time to terminal events $T_{D_{z_{ij}}}$ with the joint survival function
\begin{equation}
P(T_{H_{z_{ij}}}>y_1, T_{D_{z_{ij}}}>y_2|z_{ij}, x_{ij}, \gamma_{1i})=\exp\{-[(\lambda_{H_{z_{ij}}}y_1)^\varphi+(\lambda_{D_{z_{ij}}}y_2)^\varphi]^\frac{1}{\varphi}\},
\end{equation}
where $\varphi\ge 1$ is an association parameter between $T_{H_{z_{ij}}}$ and $T_{D_{z_{ij}}}$. Specifically to structure semi-competing risks data with covariates, the R package \verb|Gumbel| \citep{caillat2009package} was modified, such that the bivariate uniform random variables were first generated for the marginal survival functions in the copula based on the stable distribution \citep{marshall1988families}. Through the probability integral transformation, the true bivariate event times were obtained, adjusted for event types, covariates, and treatment effect. We assumed that the censoring time $T_{C_{z_{ij}}}$ was independent from $T_{H_{z_{ij}}}$, $T_{D_{z_{ij}}}$ and covariates, and followed an exponential distribution with the hazard function of $\lambda_{C_{z_{ij}}}=\lambda_C\exp(-\eta_{C}z_{ij})$. Treatment indicators were generated from a logistic regression, where $p(z_{ij}=1|x_{ij}, \gamma_{2i})=\frac{1}{1+\exp(-x_{ij}\alpha+\gamma_{2i})}$.
\\ \\
The total sample size was 1000. We varied the number of clusters to be 20 or 50, and cluster sizes to be 50 or 20. Different percentage of treatment assignments were considered: 50\% (equal) and 30\% (unequal). Throughout the simulation study, the true parameter values were fixed as follows: $\lambda_H=0.1$, $\lambda_D=0.08$, $\lambda_C=0.09$, $\eta_C=0.1$, $\varphi=2$, $\beta_1=(0.1, 0.3)^T$, $\beta_2=(0.2, 0.4)^T$, $\alpha=(-0.2, 0.5, 0.5)^T$ for equal percentage of treatment assignment, and $\alpha=(-1.8, 0.5, 0.5)^T$ for unequal percentage of treatment assignment, assuming a normal distribution for the cluster effects on treatment selections. When the cluster effects on treatment selections follow the gamma distribution, $\alpha=(-0.6, 0.5, 0.5)^T$ for equal percentage of treatment assignment. Other parameters stay the same. To assess type I error probabilities, the data were simulated under null where $\log(\mu)=0$. For the power analysis, under the null hypothesis of $\log(\mu)=0$, the data were simulated from three different alternative values of $\log(\mu)=0.098$, 0.223, and 0.328, respectively. Total 3,000 Monte Carlo simulations were performed for each scenario. The impact of model misspecification was also explored for the proposed stratified estimator with calibrated weights. Instead of using a logit link in the working model, a cloglog link was used for the misspecified model.
\subsection{Results}
\subsubsection{Type I Error Study}
\label{typeIerror_causal2}
The results under the correct specification of the PS model are presented in Table \ref{causal_WR_cluster_typeIerror}. The unadjusted estimator ignoring the cluster effect produces large bias and inflated type I error rate, especially when the within-cluster correlation is high. In all the scenarios, the calibration estimator produces the smallest bias. The bias of the calibration estimator tends to be slightly larger under unbalance designs. However, the discrepancy decreases when the number of clusters increases. The random effects estimator shows the largest bias. This is not surprising due to the shrinkage estimation of the random effects in the generalized mixed model. The fixed effects estimator produces larger bias compared to the calibration estimator, and the bias increases when the number of clusters increases. When the number of clusters is large, the fixed effects model tends to be unstable due to the large number of parameters estimated in the model. The bootstrap variance estimation underestimates the variance of the fixed effects and random effects estimators, but provides the variances of the calibration estimator close to the empirical variances. In general, the calibration estimator adequately maintains the nominal type I error level. When the number of clusters is small, the calibration estimator has slightly inflated type I error rate. Overall, under the setting of correct model specification, the fixed and the random effects estimators perform poorly, while the calibration estimator performs well, in terms of bias, variance estimation and type I error controls.
\\ \\
We further investigated the impacts of different distributional assumptions on calibration estimator only (Table \ref{causal_WR_cluster_typeIerror_mis}). We considered different distributions of cluster effects on treatment selections and the misspecification of the error distribution of the PS model. Estimators with a misspecified link function of the PS model had similar performance with the one with the correctly specified link function. Specifically with the KL distance we chose for the calibration estimator, it produced small bias for both normal and gamma distributions of cluster effects on treatment selections. Type I errors for the gamma distribution were inflated, especially for the small number of clusters, but type I errors were closer to the nominal level when the number of clusters increased. Overall, the calibration estimator with KL distance performs better if one assumes normal distribution for the cluster effects on treatment selections. Nevertheless, the performance improves for other distributions if the number of clusters increases. Here, we argue that generally it's reasonable to assume normal distribution for cluster effects. Additionally, when the number of clusters is large, the distribution of the cluster effects will approach normal. Therefore, the proposed calibration estimator will have reasonable performance in settings where the number of clusters is moderate or large.
\begin{table}[!htbp]
\centering
\caption{Type I Error Study on the Causal Win Ratio for Clustered-Dependent Subjects}
\label{causal_WR_cluster_typeIerror}
\resizebox{0.7\textwidth}{!}{
\begin{threeparttable}
\begin{tabular}{l|cccccc}
\hline\hline
\multicolumn{1}{c|}{} & & Estimator & Bias & Empirical SE & Estimated SE & Type I Error \\ \hline
\multirow{24}{*}{$m=20$, $n_i=50$} & \multicolumn{6}{l}{\% Treatment Assignment: 50\%} \\
& & & & & & \\
& ICC=0.07 & Unadjusted & -0.108 & 0.111 & 0.105 & 0.187 \\
& & Logistic & -0.232 & 0.128 & 0.118 & 0.508 \\
& & Fixed & 0.085 & 0.330 & 0.201 & 0.101 \\
& & Random & 0.294 & 0.481 & 0.176 & 0.523 \\
& & Calibration & 0.002 & 0.141 & 0.136 & 0.064 \\ \cline{3-7}
& ICC=0.2 & Unadjusted & -0.184 & 0.145 & 0.106 & 0.426 \\
& & Logistic & -0.233 & 0.129 & 0.120 & 0.497 \\
& & Fixed & 0.085 & 0.334 & 0.204 & 0.098 \\
& & Random & 0.293 & 0.493 & 0.179 & 0.529 \\
& & Calibration & 0.000 & 0.143 & 0.139 & 0.065 \\ \cline{2-7}
& \multicolumn{6}{l}{\% Treatment Assignment: 30\%} \\
& & & & & & \\
& ICC=0.07 & Unadjusted & -0.103 & 0.111 & 0.114 & 0.157 \\
& & Logistic & -0.216 & 0.136 & 0.129 & 0.404 \\
& & Fixed & 0.118 & 0.338 & 0.215 & 0.096 \\
& & Random & 0.278 & 0.447 & 0.187 & 0.486 \\
& & Calibration & 0.008 & 0.151 & 0.159 & 0.048 \\ \cline{3-7}
& ICC=0.2 & Unadjusted & -0.179 & 0.143 & 0.114 & 0.379 \\
& & Logistic & -0.216 & 0.137 & 0.129 & 0.395 \\
& & Fixed & 0.119 & 0.338 & 0.216 & 0.094 \\
& & Random & 0.275 & 0.451 & 0.189 & 0.479 \\
& & Calibration & 0.010 & 0.153 & 0.161 & 0.047 \\ \hline
\multirow{24}{*}{$m=50$, $n_i=20$} & \multicolumn{6}{l}{\% Treatment Assignment: 50\%} \\
& & & & & & \\
& ICC=0.07 & Unadjusted & -0.109 & 0.096 & 0.104 & 0.165 \\
& & Logistic & -0.223 & 0.123 & 0.121 & 0.454 \\
& & Fixed & 0.202 & 0.373 & 0.247 & 0.108 \\
& & Random & 0.366 & 0.582 & 0.193 & 0.567 \\
& & Calibration & 0.001 & 0.129 & 0.133 & 0.044 \\ \cline{3-7}
& ICC=0.2 & Unadjusted & -0.189 & 0.114 & 0.105 & 0.451 \\
& & Logistic & -0.224 & 0.124 & 0.123 & 0.441 \\
& & Fixed & 0.199 & 0.375 & 0.250 & 0.101 \\
& & Random & 0.363 & 0.584 & 0.196 & 0.569 \\
& & Calibration & 0.000 & 0.133 & 0.135 & 0.050 \\ \cline{2-7}
& \multicolumn{6}{l}{\% Treatment Assignment: 30\%} \\
& & & & & & \\
& ICC=0.07 & Unadjusted & -0.109 & 0.095 & 0.113 & 0.120 \\
& & Logistic & -0.208 & 0.131 & 0.132 & 0.369 \\
& & Fixed & 0.236 & 0.412 & 0.269 & 0.137 \\
& & Random & 0.355 & 0.550 & 0.212 & 0.508 \\
& & Calibration & 0.000 & 0.148 & 0.150 & 0.044 \\ \cline{3-7}
& ICC=0.2 & Unadjusted & -0.186 & 0.111 & 0.113 & 0.380 \\
& & Logistic & -0.208 & 0.133 & 0.133 & 0.357 \\
& & Fixed & 0.236 & 0.407 & 0.269 & 0.148 \\
& & Random & 0.363 & 0.553 & 0.212 & 0.512 \\
& & Calibration & 0.002 & 0.150 & 0.152 & 0.040 \\ \hline
\end{tabular}
\begin{tablenotes}
\footnotesize
\item $m$ is the number of clusters and $n_i$ is the cluster size for the $i^{th}$ cluster, where $i=1,...,m$. Equal cluster sizes are assumed across all clusters, however, the number of treatment and control patients within the same cluster could be different. The Unadjusted estimator is the IPW causal win ratio estimator assuming subjects are independent; The Logistic, Fixed effects, Random effects, and the Calibration estimators use the weighted stratified estimator. The Logistic estimator is with the PS estimated from logistic regression without considering cluster effects; the Fixed effects estimator is with the PS estimated from logistic regression with cluster effects as fixed effects; the Random effects estimator is with the PS estimated from logistic regression with cluster effects as random effects; the Calibration estimator is with the PS estimated from calibration method using the correct working model (logit link).
\item Acronym: SE: standard error, ICC: intra-cluster correlation, PS: propensity score
\end{tablenotes}
\end{threeparttable}
}
\end{table}
\begin{table}[!htbp]
\centering
\caption{Type I Error Study on Calibration Estimator under Model Misspecification}
\label{causal_WR_cluster_typeIerror_mis}
\resizebox{0.95\textwidth}{!}{
\begin{threeparttable}
\begin{tabular}{l|cccccc}
\hline\hline
\multicolumn{1}{c|}{} & & Estimator & Bias & Empirical SE & Estimated SE & Type I Error \\ \hline
\multirow{20}{*}{$m=20, n_i=50$} & \multicolumn{6}{l}{\% Treatment Assignment: 50\%} \\
& & & & & & \\
& ICC=0.07 & True\_Normal PS+logit & 0.002 & 0.141 & 0.136 & 0.058 \\
& & Mis\_Normal PS+cloglog & 0.002 & 0.140 & 0.135 & 0.059 \\
& & True\_Gamma PS+logit & 0.004 & 0.095 & 0.091 & 0.077 \\
& & Mis\_Gamma PS+cloglog & 0.004 & 0.095 & 0.090 & 0.079 \\ \cline{3-7}
& ICC=0.2 & True\_Normal PS+logit & 0.000 & 0.143 & 0.139 & 0.061 \\
& & Mis\_Normal PS+cloglog & 0.000 & 0.142 & 0.138 & 0.064 \\
& & True\_Gamma PS+logit & 0.004 & 0.096 & 0.092 & 0.078 \\
& & Mis\_Gamma PS+cloglog & 0.004 & 0.095 & 0.092 & 0.077 \\ \cline{2-7}
& \multicolumn{6}{l}{\% Treatment Assignment: 30\%} \\
& & & & & & \\
& ICC=0.07 & True\_Normal PS+logit & 0.008 & 0.151 & 0.159 & 0.043 \\
& & Mis\_Normal PS+cloglog & 0.009 & 0.151 & 0.158 & 0.044 \\
& & True\_Gamma PS+logit & 0.001 & 0.114 & 0.108 & 0.069 \\
& & Mis\_Gamma PS+cloglog & 0.002 & 0.114 & 0.108 & 0.069 \\ \cline{3-7}
& ICC=0.2 & True\_Normal PS+logit & 0.010 & 0.153 & 0.161 & 0.041 \\
& & Mis\_Normal PS+cloglog & 0.011 & 0.153 & 0.161 & 0.041 \\
& & True\_Gamma PS+logit & 0.002 & 0.116 & 0.110 & 0.063 \\
& & Mis\_Gamma PS+cloglog & 0.003 & 0.115 & 0.110 & 0.061 \\ \hline
\multirow{20}{*}{$m=50, n_i=20$} & \multicolumn{6}{l}{\% Treatment Assignment: 50\%} \\
& & & & & & \\
& ICC=0.07 & True\_Normal PS+logit & 0.001 & 0.129 & 0.133 & 0.042 \\
& & Mis\_Normal PS+cloglog & 0.001 & 0.129 & 0.133 & 0.041 \\
& & True\_Gamma PS+logit & -0.002 & 0.100 & 0.098 & 0.057 \\
& & Mis\_Gamma PS+cloglog & -0.002 & 0.099 & 0.098 & 0.060 \\ \cline{3-7}
& ICC=0.2 & True\_Normal PS+logit & 0.000 & 0.133 & 0.136 & 0.049 \\
& & Mis\_Normal PS+cloglog & 0.000 & 0.132 & 0.135 & 0.047 \\
& & True\_Gamma PS+logit & -0.003 & 0.102 & 0.100 & 0.065 \\
& & Mis\_Gamma PS+cloglog & -0.003 & 0.101 & 0.099 & 0.069 \\ \cline{2-7}
& \multicolumn{6}{l}{\% Treatment Assignment: 30\%} \\
& & & & & & \\
& ICC=0.07 & True\_Normal PS+logit & 0.001 & 0.148 & 0.151 & 0.041 \\
& & Mis\_Normal PS+cloglog & 0.002 & 0.148 & 0.150 & 0.041 \\
& & True\_Gamma PS+logit & 0.004 & 0.118 & 0.122 & 0.043 \\
& & Mis\_Gamma PS+cloglog & 0.006 & 0.118 & 0.121 & 0.046 \\ \cline{3-7}
& ICC=0.2 & True\_Normal PS+logit & 0.003 & 0.150 & 0.153 & 0.039 \\
& & Mis\_Normal PS+cloglog & 0.004 & 0.150 & 0.153 & 0.041 \\
& & True\_Gamma PS+logit & 0.005 & 0.120 & 0.124 & 0.045 \\
& & Mis\_Gamma PS+cloglog & 0.007 & 0.119 & 0.123 & 0.049 \\ \hline
\end{tabular}
\begin{tablenotes}
\footnotesize
\item $m$ is the number of clusters and $n_i$ is the cluster size for the $i^{th}$ cluster, where $i=1,...,m$. Equal cluster sizes are assumed across all clusters, although the number of treatment and control patients within the same cluster could be different. ``Normal PS"/``Gamma PS" indicates that data are generated using normal/gamma distribution for the cluster effects on treatment selections. The treatment status are generated using logit link for all scenarios. ``logit"/``cloglog" indicates that the working PS model is fitted using logit/cloglog link. ``True\_"/``Mis\_" indicates that the working PS model is correctly/incorrectly specified.
\item Acronym: SE: standard error, ICC: intra-cluster correlation, PS: propensity score
\end{tablenotes}
\end{threeparttable}
}
\end{table}
\subsubsection{Power Analysis}
Power analysis was conducted under the alternative hypothesis $\log(\mu)=0.42$. The power and the 95\% coverage probability for the true $\log(\mu)$ for all estimators were evaluated using the same simulation settings (Figure \ref{power_depPS}). All estimators have similar or slightly less power in the large ICC setting compared to the small ICC setting. Generally, when there is equal percentage of treatment assignments, the power is higher than the one with unequal percentage of treatment assignments. The calibration estimators with and without correctly specified working model have similar powers in all scenarios, and both of the estimators have close to 95\% coverage of the true $\log(\mu)$. In cases where the fixed and random effects estimators have higher power than the calibration estimators, the coverage probability of fixed and random effects estimators are much less than 95\% due to the inflated type I errors shown in Table \ref{causal_WR_cluster_typeIerror}.
\begin{figure}
\caption{
Power Analysis of Causal Win Ratio for Dependent Subjects
}
\label{power_depPS}
\end{figure}
\section{Data Example}
\label{calibrationWR_example}
Traumatic brain injury (TBI) remains the leading cause of death and disability in children. Despite the preventative measures, thousands of children every year in the US and abroad are affected \citep{michaud1993traumatic, kurihara2000traumatic, mckinlay2008prevalence}. None of the 33 randomized controlled trials (RCTs) of various therapies so far has demonstrated the improved patients outcomes. The Approaches and Decisions for Acute Pediatric TBI (ADAPT) trial was an observational study to investigate the impacts of various medical interventions on children under 18 years old with TBI. For demonstrative purposes, we focus on the comparison between the Cerebrospinal Fluid (CSF) drainage intervention and no CSF treatment in the first 7 days of hospitalization after brain injuries. Total 880 patients from 37 sites were included in this analysis. Thirty-four percent (34\%) of the patients were treated with CSF and 66\% of them had no CSF treatment. The sites are clusters and we assume that patients in the same site share similar cluster level characteristics. For instance, medical facilities, resources and physician training may be different across sites, but similar within sites. All sites have patients from both groups. We are interested in the composite endpoints of time to first most severe neurological complication and time to death. Intuitively, we value death as a more important event than complications, and we want to reflect this information in the analysis. Total 26 confounders were considered, including patient demographics, TBI causes and several pre-hospitalization measures. Missing values of confounders were imputed using multiple imputation implemented in the R package \verb|mice| \citep{buuren2010mice} with default settings.
\\ \\
All patients were followed up daily during hospitalization. Total 13\% of the patients died and 29\% had neurological complications. Nine percent (9\%) of them experienced neurological complications before their death, and 4\% died without complications. About 20\% of the patients experienced neurological complications but survived the first 7 days. The Kaplan-Meier curves of the time to neurological complication and time to death between groups are presented in Figure \ref{time_neuro_death}. For both endpoints, the no drainage group tends to be more beneficial compared to the CSF drainage group. The difference between groups is bigger in time to complications than in time to death.
\\ \\
We first compared the causal win ratio approach to traditional survival analysis methods accounting for confounders, assuming subjects are independent (Table \ref{causal_timeFirst_WR}). The traditional methods include analyses on time-to-complications only, time-to-death only, and time-to-first-event analysis for the composite endpoints. We considered both Cox proportional hazards model with IPWs and additive hazards model with IPWs \citep{lin1994semiparametric}. \cite{aalen2015does} noted that the hazard ratio in a Cox model is not a natural causal quantity to consider, thereby we also provided analysis using additive hazards model for the reader to consider. The CSF drainage group was the treatment of interest and the group without CSF treatment was the reference group. The probability of receiving the CSF treatment conditional on pre-specified confounders (or PS) was calculated using a logistic regression model. The IPWs were calculated as the reciprocal of the PS for CSF drainage group and the reciprocal of one minus the PS for the reference group. Under the Cox model, the treatment effect estimate is the hazard ratio (HR) on a log scale, and a positive value indicates that the reference group is favored. Under the additive hazards model, the effect estimate is the absolute difference of the hazards, and a positive value indicates that the reference group is favored. The effect estimate from the causal win ratio is the win ratio (WR) on a log scale, and a negative value indicates that the reference group is beneficial. For single-event analyses, the hazard of neurological complications or death for CSF drainage patients is slightly higher than the one for the no drainage patients. Since the time-to-first-event analysis treats
both endpoints equally important and complications always happen before death, the neurological complications drive the results of the traditional time-to-first-event analysis. For the win ratio approach, the IPW estimator was implemented according to Equation (\ref{ipw}) in Section \ref{wr_independent_confounding}. We observe that being in the CSF group is 15\% less beneficial than in the no drainage group in delaying time-to-neurological-complications at least, if not time-to-death as the primary event, yet the difference is insignificant.
\begin{figure}
\caption{Kaplan-Meier Curves of Time to Events in First 7 Days of Hospitalization}
\label{time_neuro_death}
\end{figure}
\begin{table}[!htbp]
\caption{Causal Inference of Time-to-First-Event Analysis and Win Ratio for Independent Subjects}
\label{causal_timeFirst_WR}
\centering
\resizebox{\textwidth}{!}{
\begin{threeparttable}
\begin{tabular}{c|ccccccc}
\hline
\textbf{Method} & \textbf{Endpoint} & \textbf{$\beta$} & \textbf{$\exp(\beta)$} & \textbf{$se(\beta)$} & \textbf{95\% CI} & \textbf{Test Statistic} & \textbf{P-value} \\ \hline
Additive Hazards with IPW & Death Only & 0.002 & - & 0.004 & (-0.0058, 0.0099) & 0.513 & 0.608 \\
($\beta:$ difference in hazards) & Neurological Complications Only & 0.009 & - & 0.007 & (-0.0049, 0.0237) & 1.292 & 0.196 \\
& Composite (Time-to-First-Event) & 0.011 & - & 0.008 & (-0.0047, 0.0268) & 1.377 & 0.168 \\ \hline
Cox Regression with IPW & Death Only & 0.097 & 1.102 & 0.188 & (-0.27, 0.46) & 0.516 & 0.606 \\
($\beta: \log(HR)$) & Neurological Complications Only & 0.169 & 1.184 & 0.125 & (-0.08, 0.41) & 1.353 & 0.176 \\
& Composite (Time-to-First-Event) & 0.170 & 1.185 & 0.117 & (-0.06, 0.4) & 1.453 & 0.146 \\ \hline
Causal Win Ratio & \multirow{2}{*}{Composite} & \multirow{2}{*}{-0.158} & \multirow{2}{*}{0.854} & \multirow{2}{*}{0.125} & \multirow{2}{*}{(-0.4, 0.09)} & \multirow{2}{*}{-1.264} & \multirow{2}{*}{0.206} \\
($\beta: \log(WR)$) & & & & & & & \\ \hline
\end{tabular}
\begin{tablenotes}
\footnotesize
\item $\beta$ is $\log(HR)$ for Cox regression and $\log(WR)$ for win ratio approach. 95\% CI is for $\beta$.
\item Acronym: IPW: inverse probability weights, HR: hazard ratio, WR: win ratio, se: standard error, CI: confidence interval
\end{tablenotes}
\end{threeparttable}
}
\end{table}
Second, we compared the three weighted stratified causal WR estimators discussed in the simulation study, accounting for cluster effects. For the calibration esitmator, the initial weights were first calculated using IPWs, and the calibrated weights were estimated according to Section \ref{weight_estimation}. Then the cluster-specific weighted win ratios were estimated based on Equation (\ref{cluster_tao}), and the estimated average treatment effect of the causal win ratio for clustered data was estimated acccording to Equation (\ref{equ7}). The balance of covariates between groups in overall sample was calculated using the absolute mean difference criterion \citep{austin2015moving}. As shown in Figure 3, the distributions of covariates between groups achieve exact balance for the calibration estimator, while imbalance are observed for some covariates when using the weights calculated by the logistic regression, fixed or random effects models. Without accounting for cluster effects, the unadjusted estimator yields a larger effect estimate compared to other cluster-adjusted weighted estimators (Table \ref{data_multipleEst}). Although the logistic estimator does not account for cluster effects in the treatment selection process, i.e. estimation of weights, the effect estimate by the logistic estimator is close to the one from the calibration estimator. This implies that, on average, the treatment decision processes were similar by physicians across sites. However, by taking the ratio between the variances of the two $log(WR)$, the calibration estimator shows 37\% more variability of the treatment effects compared to logistic estimator. Comparing the logistic estimator with the unadjusted estimator, the logistic estimator has 52\% more variation in the treatment effect than the unadjusted estimator. The fixed effects and random effects estimators have the effect estimates with opposite signs compared to the other estimators, but all of their 95\% confidence intervals include 0, leading to the same conclusions in hypothesis testing. Recall the simulation results in Section \ref{typeIerror_causal2}, the fixed effects and random effects estimators have much greater variations compared to the calibration estimator, but the SEs are similar among these three estimators in this real example. This may be due to the large number of confounders included in the analysis. Finally, we interpret the result based on the calibration weighted estimator as follows. Accounting for the potential confounders and cluster effects, being in the CSF group is about 3\% less beneficial than in the no drainage group in delaying time-to-neurological-complications at least, if not time-to-death as the primary event, yet the difference is insignificant. There is no sufficient evidence to support any causal relationship between CSF drainage treatment and time-to-neurological-complication or time-to-death during the 7-day follow-up period.
\begin{figure}
\caption{Balance of Covariates between Groups in Overall Sample}
\label{balance}
\end{figure}
\begin{table}[!htbp]
\caption{Data Analysis using Cluster-Adjusted Weighted Estimators}
\label{data_multipleEst}
\centering
\resizebox{0.8\textwidth}{!}{
\begin{threeparttable}
\begin{tabular}{c|cccccc}
\hline
\textbf{Estimator} & \textbf{$\beta$} & \textbf{$\exp(\beta)$} & \textbf{$se(\beta)$} & \textbf{95\% CI} & \textbf{Test Statistic} & \textbf{P-value} \\ \hline
Unadjusted & -0.158 & 0.854 & 0.125 & (-0.4, 0.09) & -1.264 & 0.206 \\ \hline
Logistic & -0.024 & 0.977 & 0.154 & (-0.32, 0.28) & -0.154 & 0.878 \\ \hline
Fixed Effects & 0.062 & 1.064 & 0.211 & (-0.35, 0.48) & 0.295 & 0.768 \\ \hline
Random Effects & 0.006 & 1.006 & 0.176 & (-0.34, 0.35) & 0.036 & 0.971 \\ \hline
Calibration & -0.034 & 0.966 & 0.180 & (-0.39, 0.32) & -0.191 & 0.848 \\ \hline
\end{tabular}
\begin{tablenotes}
\footnotesize
\item $\beta$ is $\log(WR)$. 95\% CI is for $\beta$.
\item Acronym: WR: win ratio, se: standard error, CI: confidence interval
\end{tablenotes}
\end{threeparttable}
}
\end{table}
\section{Discussions}
\label{calibrationWR_conclusion}
Composite endpoints are commonly used in clinical trials and observational studies with an anticipation that clinically relevant endpoints as a whole would yield meaningful treatment benefits \citep{cordoba2010definition, capodanno2016computing, weintraub2016statistical}. For some diseases such as cardiovascular disease, the composite endpoints are often required to be the primary endpoint for clinical studies \citep{tong2012weighting}. When the composite endpoints imply different clinical importance, the analysis should take into account the magnitude of importance of each endpoint to reach a more sensible conclusion. For example, progression-free survival is a commonly used endpoint in cancer studies with death as fatal (or terminal) event and disease progression as non-fatal event. Death is more important than disease progression. However, the traditional time-to-first-event analysis emphasizes disease progression since it always happens before death. Therefore, accounting for the importance of endpoints in a hierarchical fashion would provide clinically valuable interpretation of the data.
\\
A calibration weighted stratified win ratio estimator was proposed for causal inference for the composite endpoints with cluster-dependent subjects. This estimator is able to prioritize the important endpoint, and account for confounders and correlations within clusters. The calibration weights create balanced covariates and cluster effects distributions between groups. Additionally, they are robust against the distribution assumptions of the treatment selection modeling. Compared to fixed or random effects estimator, the proposed estimator showed superior performances in terms of bias, variance, type I error and power, regardless of the percentage of treatment assignments and intra-cluster correlations. We focus on two-group comparison in our current study; however, the proposed approach can be easily extended to settings with multiple-treatment comparisons. A multinomial logistic regression can be used to calculate the initial weights for the calibration weight estimation and similar procedures can be applied to estimate the win ratio as treatment effect. Additionally, the proposed calibration weighted stratified win ratio estimator can be further extended to hierarchical structures that have more than two levels, which might merit further investigation.
\section*{Appendix 1: Estimation of calibrated weights proposed by Yang \cite{yang2018propensity}.}
\subsection*{Appendix 1.1: Balance of Covariates for Clustered Data}
Based on the features of the propensity score as a balancing score for independent subjects \citep{chan2016globally, imai2014covariate}, for each covariate $X_{ijl}$, we define the propensity score as a balancing score for clustered data as
\begin{equation}
\label{cluster_balance}
E[\frac{Z_{ij} X_{ijl}}{\pi(X_{ij}; \alpha, \gamma_i)}]=E[\frac{(1-Z_{ij})X_{ijl}}{1-\pi(X_{ij}; \alpha, \gamma_i)}]=E[X_{ijl}],
\end{equation}
where $l=1,...,p$, and $\pi(X_{ij}; \alpha, \gamma_i)$ be the propensity score with parameter $\alpha$, which is the probability of receiving treatment, given covariates and cluster effect. If the propensity score is correct, then the weighted covariates in the treatment group will have similar distributions as the weighted covariates in the control group.
\\ \\
The empirical version of Equation (\ref{cluster_balance}) is
\begin{equation}
\label{cluster_balance_emp1}
\sum_{i=1}^{m} \sum_{j=1}^{n_i} \frac{Z_{ij} X_{ijl}}{\hat{\pi}(X_{ij}; \hat{\alpha}, \hat{\gamma_i})}
=\sum_{i=1}^{m} \sum_{j=1}^{n_i} \frac{(1-Z_{ij}) X_{ijl}}{1-\hat{\pi}(X_{ij}; \hat{\alpha}, \hat{\gamma_i})}
=\sum_{i=1}^{m} \sum_{j=1}^{n_i} X_{ijl}.
\end{equation}
To ensure the balance of the cluster effects between treatment and control group, we want to satisfy additional conditions \citep{yang2018propensity}
\begin{equation}
\label{cluster_balance_gamma}
E[\frac{Z_{ij} \gamma_i}{\pi(X_{ij}; \alpha, \gamma_i)}]=E[\frac{(1-Z_{ij})\gamma_i}{1-\pi(X_{ij}; \alpha, \gamma_i)}]=E[\gamma_i].
\end{equation}
Therefore, given a specific cluster effect $\gamma_i$, Equation (\ref{cluster_balance_gamma}) becomes
\begin{equation}
\label{cluster_balance_condition}
E[\frac{Z_{ij}}{\pi(X_{ij}; \alpha, \gamma_i)}|\gamma_i]=E[\frac{(1-Z_{ij})}{1-\pi(X_{ij}; \alpha, \gamma_i)}|\gamma_i]=E[1|\gamma_i].
\end{equation}
The empirical version of the Equation (\ref{cluster_balance_condition}) for a specific cluster $i$ is
\begin{equation}
\label{cluster_balance_emp2}
\sum_{j=1}^{n_i} \frac{Z_{ij}}{\hat{\pi}(X_{ij}; \hat{\alpha}, \hat{\gamma_i})}
=\sum_{j=1}^{n_i} \frac{(1-Z_{ij})}{1-\hat{\pi}(X_{ij}; \hat{\alpha}, \hat{\gamma_i})}
=\sum_{j=1}^{n_i}1 = n_i.
\end{equation}
These two constrains (Equation (\ref{cluster_balance_emp1}) and (\ref{cluster_balance_emp2})) are used in the calibration estimation of the weights. They ensure the balance of covariates and cluster effects between groups so that valid causal inference can be made for observational clustered data.
\subsection*{Appendix 1.2: Estimation}
The procedure of estimating weights using calibration is
\begin{enumerate}
\item Estimate the initial weights $d_{ij}$ using some parametric working model, such as logistic regression. Define $d_{ij}=\frac{Z_{ij}}{\pi(X_{ij}; \alpha)}+\frac{1-Z_{ij}}{1-\pi(X_{ij}; \alpha)}$ as the initial weights.
\item Estimate the calibration weights $w_{ij}$ using some distance function $D(w_{ij}, d_{ij})$ with constraints. Define $w_{ij}$ as the final weights.
\end{enumerate}
After estimation of the initial weights $\hat{d_{ij}}$, we want to minimize the objective function with multiple constraints, respect to $w_{ij}$:
\begin{equation}
\label{app_objective_fn}
\begin{split}
O=& \sum_{i=1}^{m} \sum_{j=1}^{n_i} w_{ij}\ln \frac{w_{ij}}{d_{ij}} \\
&+{\lambda_1}^T\{\sum_{i=1}^{m} \sum_{j=1}^{n_i} [Z_{ij} w_{ij} X_{ij} - X_{ij}] \}\\
&+{\lambda_2}^T\{\sum_{i=1}^{m} \sum_{j=1}^{n_i} [(1-Z_{ij}) w_{ij} X_{ij} - X_{ij}] \}\\
=&\sum_{i=1}^{m} \sum_{j=1}^{n_i} w_{ij}\ln \frac{w_{ij}}{d_{ij}}+{\lambda_1}^T\{\sum_{i=1}^{m} \sum_{j=1}^{n_i}C_{ij1}\}+{\lambda_2}^T\{\sum_{i=1}^{m} \sum_{j=1}^{n_i}C_{ij2}\},
\end{split}
\end{equation}
with additional constraints within each cluster $i$,
\begin{equation}
\label{app_objective_fn_add}
\sum_{j=1}^{n_i} Z_{ij} w_{ij}=\sum_{j=1}^{n_i} (1-Z_{ij}) w_{ij}= n_i,
\end{equation}
Let the first derivative of the objective function equal to 0:
\begin{equation}
\begin{split}
&\frac{\partial g}{\partial w_{ij}}=\ln(\frac{w_{ij}}{d_{ij}})+1+{\lambda_1}^T Z_{ij} X_{ij} + {\lambda_2}^T (1-Z_{ij}) X_{ij} =0\\
\Longrightarrow &\hat{w}_{ij}=d_{ij}Z_{ij}\exp(-1-{\lambda_1}^T Z_{ij} X_{ij}) + d_{ij}(1-Z_{ij}) \exp(-1-{\lambda_2}^T (1-Z_{ij}) X_{ij}).
\end{split}
\end{equation}
Since when $Z_{ij}=1$, $Z_{ij}w_{ij}=d_{ij}Z_{ij}\exp(-1-{\lambda_1}^T Z_{ij} X_{ij})$. According to Equation \ref{app_objective_fn_add}, we have
\begin{equation}
\label{app_add_cond1}
\begin{split}
\sum_{j=1}^{n_i} Z_{ij} w_{ij}&= n_i\\
&=\sum_{j=1}^{n_i} d_{ij}Z_{ij}\exp(-1-{\lambda_1}^T Z_{ij} X_{ij}),\\
\Longrightarrow n_i &= \sum_{j=1}^{n_i} d_{ij}Z_{ij}\exp(-1-{\lambda_1}^T Z_{ij} X_{ij}).
\end{split}
\end{equation}
Same reasoning applies to when $1-Z_{ij}=1$,
\begin{equation}
\label{app_add_cond2}
n_i = \sum_{j=1}^{n_i} d_{ij}(1-Z_{ij})\exp(-1-{\lambda_2}^T (1-Z_{ij}) X_{ij}).
\end{equation}
Therefore, we can rewrite the estimator $\hat{w}_{ij}$ with conditions \ref{app_add_cond1} and \ref{app_add_cond2} as
\begin{equation}
\hat{w}_{ij}=\frac{n_i d_{ij}Z_{ij}\exp(-{\lambda_1}^T Z_{ij} X_{ij})}{\sum_{j=1}^{n_i} d_{ij}Z_{ij}\exp(-{\lambda_1}^T Z_{ij} X_{ij})}
+\frac{n_i d_{ij}(1-Z_{ij})\exp(-{\lambda_2}^T (1-Z_{ij}) X_{ij})}{\sum_{j=1}^{n_i} d_{ij}(1-Z_{ij})\exp(-{\lambda_2}^T (1-Z_{ij}) X_{ij})}
\end{equation}
Since $\hat{w}_{ij}$ is a function of $\hat{\lambda}_1$ and $\hat{\lambda}_2$, we can estimate $\lambda_1$ and $\lambda_2$ iteratively from $\sum_{i=1}^{m} \sum_{j=1}^{n_i}C_{ij1}=0$ and $\sum_{i=1}^{m} \sum_{j=1}^{n_i}C_{ij2}=0$. \\
Different from the initial weights calculated from the propensity score estimates, the calibration weight is calculated directly by mapping the initial weights to the final weights based on a distance function.
\section*{Appendix 2: Empirical Distribution of the Proposed Estimator}
We show the empirical distribution of the proposed weighted stratified causal win ratio estimator with calibrated weight, based on the simulation set-up (Section \ref{set-up}) for ICC=0.067, varying number of clusters and cluster sizes from 10, 20 to 50. The true underlying causal win ratio on log scale is 0. Similar patterns were observed for ICC=0.2.\\
\begin{figure}
\caption{Empirical Distribution of the Proposed Weighted Stratified Estimator with Calibrated Weight on Log Scale}
\label{balance}
\end{figure}
{}
\end{document} | math | 62,780 |
\begin{document}
\title{\mbox{Generalized Maneuvers in Route Planning}}
\author{Petr Hlin\v{e}n\'y ~\and~ Ondrej Mori\v{s}}
\institute{Faculty of Informatics, Masaryk University \\
Botanick\'a 68a, 602 00 Brno, Czech Republic \\
\email{[email protected], [email protected]}}
\maketitle
\begin{abstract}
We study an important practical aspect of the route planning problem in
real-world road networks -- \emph{maneuvers}. Informally, maneuvers
represent various irregularities of the road network graph such as
turn-prohibitions, traffic light delays, round-abouts, forbidden passages
and so on. We propose a generalized model which can handle arbitrarily complex
(and even negative) maneuvers, and outline how to enhance Dijkstra's
algorithm in order to solve route planning queries in this model without
prior adjustments of the underlying road network graph.
\end{abstract}
\section{Introduction}
Since mass introduction of GPS navigation devices, the \emph{route planning
problem}, has received considerable attention. This problem is in fact
an instance of the well-known single pair shortest path (SPSP) problem
in graphs representing real-world road networks. However, it involves
many challenging difficulties compared to ordinary SPSP. Firstly, classical
algorithms such as Dijkstra's \cite{Dijkstra1959}, A* \cite{Hart1972} or their
bidirectional variants \cite{Pohl1969} are not well suited for the route
planning despite their optimality in wide theoretical sense. It is mainly
because graphs representing real-world road networks are so huge that even
an algorithm with linear time and space complexity cannot be feasibly run
on typical mobile devices.
Secondly, these classical approaches disregard certain important aspects of
real-world road networks, namely route restrictions, traffic regulations, or
actual traffic info. Hence a route found by such algorithms might not be
optimal or not even feasible. Additional attributes are needed in this regard.
The first difficulty has been intensively studied in the past decade, and
complexity overheads of classical algorithms have been largely improved by
using various preprocessing approaches. For a brief overview, we refer the
readers to \cite{Cherkassky1994,Delling2009B,Schultes2008} or our
\cite{HM2011A}. In this paper we focus on the second mentioned difficulty
as it is still receiving significantly less attention.
\subsubsection{Related Work. }
The common way to model required additional attributes of road networks is
with so called {\em maneuvers}\/; Definition~\ref{def:maneuver}. Maneuvers
do not seem to be in the center of interest of route-planning research papers:
They are either assumed to be encoded into the underlying graph of a road
network, or they are addressed only partially with rather simple types of
restriction attributes such as turn-penalties and path prohibitions.
Basically, the research directions are represented either by modifications
of the underlying graph during preprocessing
\cite{Jiang2002,Pallottino1997,Ziliaskopoulos1996}, or by adjusting a query
algorithm \cite{Kirby1969,Villeneuve2005} in order to resolve simple types
of restrictions during queries.
The first, and seemingly the simplest, solution is commonly used as it makes
a~road network graph maneuver-free and so there is no need to adjust the
queries in any way. Unfortunately, it can significantly increase the size
of the graph \cite{Winter2002}; for instance, replacing a single
turn-prohibition can add up to eight new vertices in place of one
original~\cite{Gutierrez2008}. A solution like this one thus conflicts with
the aforementioned (graph-size) objectives. Another approach \cite{Anez1996}
uses so-called dual graph representation instead of the original one, where
allowed turns are modeled by dual edges.
To summarize, a sufficiently general approach for arbitrarily complex
maneuvers seems to be missing in the literature despite the fact that such
a solution could be really important. We would like to emphasize that all
the cited works suffer from the fact that they consider only ``simple'' types
of maneuvers.
\subsubsection{Our Contribution. } Firstly, we introduce a formal model of a
generic maneuver -- from a single vertex to a long self-intersecting walk
-- with either positive or negative effects (penalties); being enforced,
recommended, not recommended or even prohibited. Our model can capture
virtually any route restriction, most traffic regulations and even some
dynamic properties of real-world road networks.
Secondly, we integrate this model into Dijkstra's algorithm, rising its
worst-case time complexity only slightly (depending on a structure of
maneuvers). The underlying graph is not modified at all and no preprocessing
is needed. Even though our idea is fairly simple and relative easy to
understand, it is novel in the respect that no comparable solution has been
published to date. Furthermore, some important added benefits of our
algorithm are as follows:
\begin{itemize}
\parskip 2pt
\item It can be directly used bidirectionally with any alternation strategy
using an appropriate termination condition; it can be extended also to the
A* algorithm by applying a ``potential function to maneuver effects''.
\item Many route planning approaches use Dijkstra or A* in the core of their
query algorithms, and hence our solution can be incorporated into many of
them (for example, those based on a reach, landmarks or various types of
separators) quite naturally under additional assumptions.
\item Our algorithm tackles maneuvers ``on-line'' -- that is\ no maneuver
is processed before it is reached. And since the underlying graph of a
road network is not changed (no vertices or edges are removed or added),
it is possible to add or remove maneuvers dynamically even during queries
to some extent.
\end{itemize}
\section{Maneuvers: Basic Terms}
\label{sec:maneuvers}
A \emph{(directed) graph} $G=(V,E)$ is a pair of a finite set $V$ of vertices
and a finite multiset $E \subseteq V \times V$ of edges (self-loops and
parallel edges are allowed). The vertex set of $G$ is referred to as $V(G)$,
its edge multiset as $E(G)$. \emph{A~subgraph} $H$ of a graph $G$ is denoted
by $H \subseteq G$.
\emph{A walk} $P \in G$ is an alternating sequence of vertices and
edges $(u_0,e_1,u_1,\ldots,$ $e_k,u_k) \subseteq G$ such that $e_i =
(u_{i-1},u_i)$ for $i = 1,\ldots,k$, the multiset of all edges of a walk $P$
is denoted by $E(P)$. \emph{A concatenation} $P_1 .\,P_2$ of
walks $P_1=(u_0,e_1,u_1,\ldots,e_k,u_k)$ and $P_2=(u_k,e_{k+1},u_{k+1},\ldots,
e_l,u_l)$ is the walk $(u_0,e_1,u_1,$ $\ldots,e_k,u_k,e_{k+1},\ldots,e_l,u_l)$.
If $P_2=(u,f,v)$ represents a single edge, we write~$P_1.\,f$. If edges are
clear from the graph, then we write a walk simply as $(u_0,u_1,\dots,u_k)$.
A walk $Q$ is a {\em prefix} of another walk $P$ if $Q$ is a subwalk of
$P$ starting with the same index, and analogically with {\em suffix}.
The \emph{prefix set} of a walk $P=(u_0,e_1,\ldots,e_k,u_k)$ is $\mathit{Prefix}
(P) = \{(u_0,e_1,\ldots,e_i,u_i) |\> 0 \le i \le k\},$ and analogically
$\mathit{Suf\!fix}(P)=\{(u_i,e_{i+1},\ldots,e_k,u_k) |\> 0 \le i\le k\}$.
A prefix (suffix) of a walk $P$ thus is a member of $\mathit{Prefix}(P)$
($\mathit{Suf\!fix}(P)$), and it is {\em nontrivial} if $i\geq1$.
\emph{The weight} of a walk $P \subseteq G$ with respect to a weighting $w:
E(G) \mapsto \mathbb{R}$ of $G$ is defined as $\sum_{e \in E(P)} w(e)$ and
denoted by $|P|_w$. \emph{A distance} from $u$ to $v$ in $G$, $\delta_w(u,v)$,
is the minimum weight of a walk $P=(u, \ldots, v) \subseteq G$ over all such
walks and $P$ is then called \emph{optimal} (with respect to weighting $w$).
If there no such walk then $\delta_w(u,v) = \infty$. \emph{A~path} is a walk
without repeating vertices and edges.
Virtually any route restriction or traffic regulation in a road network,
such as turn-prohibitions, traffic lights delays, forbidden passages,
turn-out lanes, suggested directions or car accidents by contrast, can be
modeled by \emph{maneuvers} -- walks having extra (either positive or negative)
``cost effects''. Formally:
\begin{definition}[Maneuver]
\label{def:maneuver}
\emph{A maneuver} $M$ of $G$ is a walk in $G$ that is assigned a penalty
$\Delta(M)\! \in\! \mathbb{R}\!\cup\! \infty$. A set of all maneuvers of $G$ is
denoted by ${\cal{M}}$.
\end{definition}
\begin{remark}
A maneuver with a negative or positive penalty is called \emph{negative}
or \emph{positive}, respectively. Furthermore, there are two special kinds
of maneuvers the \emph{restricted} ones of penalty 0 and the \emph{prohibited}
ones of penalty~$\infty$.
\end{remark}
The cost effect of a maneuver is formalized next:
\begin{definition}[Penalized Weight]
\label{def:penalized_weight}
Let $G$ be a graph with a weighting $w$ and a set of maneuvers $\cal{M}$.
The \emph{penalized weight} of a walk $P \subseteq G$ containing the maneuvers
$M_1,\ldots,M_r \in {\cal{M}}$ as subwalks is defined as $|P|_w^{\cal{M}} =
|P|_w + \sum_{i=1}^{r} \Delta(M_i)$.
\end{definition}
Then, the intended meaning of maneuvers in route planning is as follows.
\begin{itemize}
\parskip2pt
\item If a driver enters a restricted maneuver, she must pass it completely
(cf.~Definition~\ref{def:valid_walk});
she must obey the given direction(s) regardless of the cost effect.
Examples are headings to be followed or specific round-abouts.
\item By contrast, if a driver enters a prohibited maneuver, she must
not pass it completely. She must get off it before reaching its end,
otherwise it makes her route infinitely bad. Examples are forbidden
passages or temporal closures.
\item Finally, if a driver enters a positive or negative maneuver, she is
not required to pass it completely;
but if she does, then this will increase or decrease the
cost of her route accordingly. Negative maneuvers make her route
better (more desirable) and positive ones make it worse.
Examples of positive maneuvers are, for instance, traffic lights delays, lane
changes, or left-turns.
Examples of negative ones are turn-out lanes, shortcuts, or implicit routes.
\end{itemize}
\begin{definition}[Valid Walk]
\label{def:valid_walk}
Let $G,w,\cal{M}$ be as in Definition~\ref{def:penalized_weight}. A walk
$P$ in $G$ is \emph{valid} if and only if $|P|_w^{\cal{M}} < \infty$ and, for any
restricted maneuver $M\in\cal{M}$, it holds that if a nontrivial prefix of
$M$ is a subwalk of $P$, then whole $M$ is a~subwalk of $P$ or a suffix of $P$
is contained in~$M$ (that is $P$ ends there).
\end{definition}
We finally get to the summarizing definition. A structure of a road network
is naturally represented by a graph $G$ such that the junctions are
represented by $V(G)$ and the roads by $E(G)$. The chosen cost function
(for example travel time, distance, expenses) is represented by a \emph{non-negative}
weighting $w: E(G) \mapsto\mathbb{R}^+_0$ assigned to $G$, and the additional
attributes such as traffic regulations are represented by maneuvers as above.
We say that two walks $Q_1,Q_2$ are {\em divergent} if, up to symmetry
between $Q_1,Q_2$, a nontrivial prefix of $Q_1$ is contained in $Q_2$ but
the whole $Q_1$ is not a subwalk of~$Q_2$. Moreover, we say that $Q_2$
{\em overhangs} $Q_1$ if a~nontrivial prefix of $Q_2$ is a suffix of~$Q_1$
(particularly, $E(Q_1)\cap E(Q_2)\neq \emptyset$).
\begin{definition}[Road Network]
\label{def:road_network}
Let $G$ be a graph with a non-negative weighting $w$ and a set of maneuvers
$\cal M$. \emph{A road network} is the triple $(G,w,{\cal M})$.
Furthermore, it is called \emph{proper} if:
\begin{itemize}
\parskip 2pt
\item[i.] no two restricted maneuvers in $\cal M$ are divergent,
\item[ii.] no two negative maneuvers in $\cal M$ overhang one another, and
\item[iii.] for all $N \in {\cal{M}}$, $\Delta(N) \geq -|N|_w^{{\cal{M}}
\setminus \{N\}}$ (that is, the penalized weight of every walk in $G$ is
non-negative).
\end{itemize}
\end{definition}
\begin{figure}
\caption{A road network containing maneuvers $M_1=(a,ab,b,bc,c)$
with $\Delta(M_1)=\infty$ (prohibited left turn) and $M_2=(a,ab,b,bf,f)$
with $\Delta(M_2) = 1$ (right turn traffic lights delay). All edges
have weight 1. The penalized weight of the walk $(a,ab,b,bc,c)$ is $2 +
\infty$, the penalized weight of the walk $(a,ab,b,bf,f,fe,e,ed,d,db,b,bd,
c)$ is $6 + 1$. Therefore the optimal walk (with respect to the
penalized weight) from $a$ to $c$ is $(a,ab,b,bd,d,de,e,ef,f,fb,b,bc,c)$
with the penalized weight $6 + 0$.}
\label{fig:maneuver}
\end{figure}
Within a road network, only valid walks (Definition~\ref{def:valid_walk})
are allowed further, and the distance from $u$ to $v$, $\delta_w^{\cal{M}}
(u,v)$, is the minimum penalized weight (Definition~\ref{def:penalized_weight})
of a valid walk $P=(u, \ldots, v) \subseteq G$; such a walk $P$ is then
called \emph{optimal with respect to the penalized weight}. If there is no
such walk, then $\delta_w^{\cal{M}}(u,v) = \infty$. See~Fig.~\ref{fig:maneuver}.
Motivation for the required properties i.--iii. in
Definition~\ref{def:road_network} is of both natural and practical character:
As for i., it simply says that no two restricted maneuvers are in a
conflict (that is no route planning deadlocks). Point ii. concerning only
negative maneuvers is needed for a fast query algorithm, and it is indeed
a~natural requirement (to certain extent, overhanging maneuvers can be modeled
without overhangs). We remark that other studies usually allow no negative
maneuvers at all. Finally, iii. states that no negative maneuvers can result
in a negative overall cost of any walk -- another very natural property.
In informal words, a negative penalty of a maneuver somehow ``cannot
influence'' suitability of a~route before entering and after exiting the
maneuver.
\subsection{Strongly Connected Road Network}
\label{sec:connectivity}
The traditional graph theoretical notion of strong connectivity also needs to
be refined, it must suit our road networks to dismiss possible route
planning traps now imposed by maneuvers.
First, we need to define a notion of a \emph{``context''} of a vertex $v$
in $G$ -- a~maximal walk in $G$ ending at $v$ such that it is a proper
prefix of a maneuver in $\cal M$, or $\emptyset$ otherwise. A set of all such
walks for $v$ is denoted by ${\cal{X}}_{\cal{M}}$. For example, on the road
network depicted on Fig.~\ref{fig:maneuver},
${\cal{X}}_{\cal M}(b)=\{(a,b),\emptyset\}$. More formally:
\begin{definition}
\label{def:context}
Let $\cal M$ be a set of maneuvers. We define
$$ {\cal{X}}_{{\cal{M}}}(v) \stackrel{\textrm{\tiny{def}}}{=}
\big\{ X \in \mathit{Prefix}^{<}({\cal M}) \, | (v) \in
\mathit{Suf\!fix}(X)\big\} \cup \{\emptyset\} $$
$$ \mathit{Prefix}^{<}(M) \stackrel{\textrm{\tiny{def}}}{=} \mathit{Prefix}(M)
\setminus \{M\}, \quad \mathit{Prefix}^{<}({\cal M})
\stackrel{\textrm{\tiny{def}}}{=} \bigcup\nolimits_{M\in
\cal M}\mathit{Prefix}^{<}(M). $$
This ${\cal{X}}_{{\cal{M}}}(v)$ is the {\em maneuver-prefix set} at $v$,
that is\ the set of all proper prefixes of walks from $\cal M$ that end
right at~$v$, including the mandatory empty walk.
An element of ${\cal{X}}_{{\cal{M}}}(v)$ is called a {\em context} of the
position $v$ within the road network.
\end{definition}
\emph{The reverse graph} $G^R$ of $G$ is a graph on the same set of vertices
with all of the edges reversed. Let $(G,w,{\cal{M}})$ be a road network,
a \emph{reverse road network} is defined as $(G^R,w^R,{\cal{M}}^R)$, where
$w^R: E(G^R) \mapsto \mathbb{R}_0^+$, $\forall (u,v) \in E(G^R):\, w^R(u,v)
= w(v,u)$ and ${\cal{M}}^R = \{ M^R | M \in {\cal{M}}\}$, $\forall M^R \in
{\cal{M}}^R:\, \Delta(M^R) = \Delta(M)$.
\begin{definition}
\label{def:Mconnectivity}
A road network $(G,w,{\cal M})$ is \emph{strongly connected} if, for every
pair of edges $e=(u',u),\>f=(v,v') \in E(G)$ and for each possible
context $X=X_1\cdot\,e\in{\cal X}_{\cal M}(u)$ of $u$ in $G$ and each one
of $v$ in $G^R$, that is \ $Y^R=Y_1^R.\,f^R \in
{\cal X}_{{\cal M}^R}(v)$, there exists a valid walk starting with $X$ and
ending with~$Y$.
\end{definition}
We remark that Definition~\ref{def:Mconnectivity} naturally corresponds to
strong connectivity in an~amplified road network modeling the maneuvers
within underlying graph.
\section{Route Planning Queries}
\label{sec:query}
At first, let us recall classical Dijkstra's algorithm \cite{Dijkstra1959}.
It solves SPSP\footnote{Given a graph and two vertices find a shortest path
from one to another.} problem a graph $G$ with a~non-negative weighting $w$
for a pair $s,t\in V(G)$ of vertices.
\begin{itemize}
\parskip 3pt
\item The algorithm maintains, for all $v \in V(G)$, a
{\em (temporary) distance estimate} of the shortest path from $s$ to $v$
found so far in $d[v]$, and a predecessor of $v$ on that path in $\pi[v]$.
\item The scanned vertices, that is those with $d[v] = \delta_w(s,v)$,
are stored in the set $T$; and the reached but not yet scanned vertices,
that is those with $\infty >d[v] \geq \delta_w(s,v)$, are stored in the set $Q$.
\item The algorithm work as follows: it iteratively picks a vertex $u \in Q$
with minimum value $d[u]$ and relaxes all the edges $(u,v)$ leaving $u$.
Then $u$ is removed from $Q$ and added to $T$. {\em Relaxing} an edge $(u,v)$
means to check if a shortest path estimate from $s$ to $v$ may be improved
via $u$; if so, then $d[v]$ and $\pi[v]$ are updated. Finally, $v$ is added
into $Q$ if is not there already.
\item The algorithm terminates when $t$ is scanned or when $Q$ is empty.
\end{itemize}
Time complexity depends on the implementation of $Q$; such as it is
${\cal{O}}(|E(G)| + |V(G)|\log|V(G)|)$ with the Fibonacci heap.
\subsection{$\cal{M}$-Dijkstra's Algorithm}
\label{sec:Mdijkstra}
In this section we will briefly sketch the core ideas of our natural extension
of Dijkstra's algorithm. We refer a reader to Algorithm \ref{alg:m-dijkstra}
for a full-scale pseudocode of this $\cal{M}$-Dijkstra's algorithm.
\begin{enumerate}
\parskip 3pt
\item Every vertex $v\in V(G)$ scanned during the algorithm is considered
together with its context $X \in {\cal X}_{\cal M}(v)$
(Definition~\ref{def:context}); that is\ as a pair $(v,X)$.
The intention is for $X$ to record how $v$ has been reached in the
algorithm, and same $v$ can obviously be reached and scanned more than
once, with different contexts.
For instance, $b$ can be reached with the empty or $(a,b)$ contexts
on the road network depicted on Fig.~\ref{fig:maneuver}.
\item Temporary distance estimates are stored in the algorithm as $d[v,X]$
for such vertex-context pairs $(v,X)$. At each step the
algorithm selects a next pair $(u,Y)$ such that it is minimal with respect
to the following partial order $\le_{\cal M}$.
\begin{remark}{Partial order $\le_{\cal M}$:}
\label{rmk:order}
$$
(v_1,X_1) \le_{\cal M} (v_2,X_2)
\stackrel{\textrm{\tiny{def}}}{\Longleftrightarrow}
\left.\begin{array}{ll}\big(d[v_1,X_1] < d[v_2,X_2] &\lor\\[2pt]
(d[v_1,X_1] = d[v_2,X_2] &\land \> X_1 \in \mathit{Suf\!fix}(X_2)\,)\big).
\end{array}\right.
$$
\end{remark}
\item Edge relaxation from a selected vertex-context pair $(u,Y)$ respects
all maneuvers related to the context $Y$ (there can be more such maneuvers).
If one of them is restricted, then only its unique (cf.\ Definition
\ref{def:road_network},\,i.) subsequent edge is taken, cf.\
Algorithm~\ref{alg:m-dijkstra}, \textsc{RestrictedDirection}.
Otherwise, every edge $f=(u,v)$ is relaxed such that the distance estimate
at $v$ -- together with its context as derived from the concatenation
$(Y.\,f)$ -- is (possibly) updated with the weight $w(f)$ plus the sum of
penalties of all the maneuvers in $(Y.\,f)$ ending at $v$,
cf.~Algorithm~\ref{alg:m-dijkstra}, \textsc{Relax}.
\item If an edge relaxed is the first one of a negative maneuver,
a specific process is executed before scanning the next vertex-context pair.
See below.
\end{enumerate}
\subsection{Processing Negative Maneuvers}
\label{sec:process_negative_maneuvers}
Note that the presence of a maneuver of negative penalty
{\em may violate} the basic assumption of ordinary Dijkstra's algorithm;
that relaxing an edge never decreases the nearest temporary distance
estimate in the graph. An example of such a violation can be seen in
Fig.~\ref{fig:negative-maneuver}, for instance, at vertex $v_5$ which would
not be processed in its correct place by ordinary Dijkstra's algorithm.
That is why a negative maneuver $M$ must be processed by $\cal{M}$-Dijkstra's
algorithm at once -- whenever its starting edge is relaxed,
cf.~Algorithm~\ref{alg:m-dijkstra}, \textsc{ProcessNegative}.
Suppose that an edge $f=(u,v)$ is relaxed from a selected vertex-context pair
$(u,X)$ and there is a negative maneuver $M = (v_0,f_1,v_1,\ldots,v_{n-1},f_n,
v_n)$, $u=v_0$, $v=v_1$ starting with $f$ (that is $f=f_1$), processing negative
maneuver $M$ works as follow:
\begin{figure}
\caption{A road network containing two negative maneuvers,
$M_1=(v_0,\ldots,v_5)$ and $M_3=(w_0,\ldots,,w_5)$,
a restricted maneuver $M_2=(v_2,v_3,v_4)$, and a prohibited
maneuver $M_4=(w_2,w_3,w_4)$.
When $u$ is being processed (with its implicit context),
$x_1,x_2$ and $v_1,w_1$ are relaxed normally. Furthermore,
negative maneuver processing is executed for both $M_1$ and $M_3$.
As a result, $v_5$ will be immediately reached and inserted to $Q$ with
distance estimate equal to that of $u$ which is less than those of $x_1,
x_2$ (5 from $u$) and of $v_1,w_1$ (1 from $u$). On the other hand,
$w_5$ will not be reached in the process because the distance estimate
of $w_4$ bounces to $\infty$ while handling $M_4$.}
\label{fig:negative-maneuver}
\end{figure}
\begin{enumerate}
\parskip 2pt
\item Vertex-context pairs $(v_i,X_i), 0 \le i \le n$ along $M$ are
scanned one by one towards the end of $M$. The other vertices leaving
these $v_i$ are ignored.
\item Scanned vertex-context pairs are added to $Q$ and their distance
estimates are updated, but none of them is added into $T$. They must be
properly scanned during the main loop of the algorithm.
\item This process terminates when the end $(v_n,X_n)$ is reached or the
distance estimate of some $(v_i,X_i)$ bounces to $\infty$ (that is there is a
prohibited maneuver ending at $v_i$) or when some restricted maneuver
forces us to get off $M$ (and thus $M$ cannot be completed).
\end{enumerate}
\begin{algorithm}[H]
\caption{~$\cal{M}$-Dijkstra's Algorithm}
\label{alg:m-dijkstra}
\begin{algorithmic}[1]
\REQUIRE A proper road network $(G,w,{\cal M})$ and vertices $s,t \in V(G)$.
\ENSURE A valid walk from $s$ to $t$ in $G$ optimal with respect to the
penalized weight.
\end{algorithmic}
\textsc{${\cal{M}}$-Dijkstra}$(G,w,{\cal{M}},s,t)$
\begin{algorithmic}[1]
\small
\FORALL[
/* Initialization. */]{$v \in V(G)$, $X \in
{\cal X}_{\cal M}(v)$}
\STATE $d[v,X] \leftarrow \infty;~ \pi[v,X] \leftarrow \bot$
\ENDFOR
\STATE $d[s,\emptyset] \leftarrow 0$;~ $Q \leftarrow \{(s,\emptyset)\}$;~ $T
\leftarrow \emptyset$
\OLIF{$(s) \in {\cal{M}}$}{
$d[s,\emptyset] \leftarrow d[s,\emptyset] + \Delta(s)$}
\vskip 0pt
\COMMENT{/* The main loop starts at $(s,\emptyset)$ and terminates
when either all reachable vertex-\\
context pairs have been scanned
or when $t$ is reached with some of its contexts. */}
\WHILE{$Q \neq \emptyset \land [\not\exists\, X \in {\cal{X}}_{\cal{M}}(t)$
s.t. $(t,X) \in T]$}
\STATE $(u,X) \leftarrow \min_{\le_{\cal M}}(Q)$;~$Q \leftarrow Q \setminus
\{(u,X)\}$
\COMMENT{
/* Recall $\le_{\cal{M}}$ (Remark \ref{rmk:order}) */}
\STATE $F \leftarrow \textsc{RestrictedDirection}(u,X)$
\COMMENT{
/* Possible restricted dir.\ from $u$. */}
\OLIF{$F = \emptyset$}{
$F \leftarrow \{ (u,v) \in E(G) \, | \, v \in V(G)\}$}
\FORALL{$ f=(u,v) \in F$}
\STATE \textsc{Relax}$(u,X,f,v)$
\FORALL{$M = (u,f,v,\ldots) \in {\cal{M}}$ s.t.
$\Delta(M)<0 \,\wedge\, |E(M)| > 1$}
\STATE \textsc{ProcessNegative}$\,(X,M)$
\ENDFOR
\COMMENT{
/* Negative man.\ starting with $f$ are processed
separately. */}
\ENDFOR
\STATE $T \leftarrow T \cup \{(u,X)\}$
\ENDWHILE
\STATE \textsc{ConstructWalk}$\,(G,d,\pi)$
\COMMENT{
/* Use ``access'' information stored in $\pi[v,X]$. */}
\end{algorithmic}
\end{algorithm}
\small{
\noindent
\textsc{LongestPrefix}$\,(P) :\> \textrm{a walk } P' \subseteq G$
\begin{algorithmic}[1]
\item[]
\COMMENT{
/* The longest (proper) prefix of some maneuver contained
as a suffix of $P$ */}\vskip 0pt
\STATE $\begin{array}{l} P' \leftarrow \max_{\subseteq}
\big[\,(\mathit{Suf\!fix}(P) \cap \mathit{Prefix}^<({\cal{M}}))
\cup \{\emptyset\}\big] \\ ~~~~~~~~~~~~~~~~~~~
\mbox{where }\mathit{Prefix}^{<}({\cal M})
\stackrel{\textrm{\tiny{def}}}{=} \bigcup\nolimits_{M\in
\cal M}\mathit{Prefix}(M)\setminus\{M\} \end{array}$
\RETURN $P'$
\end{algorithmic}
\noindent
\textsc{RestrictedDirection}$(u,X) :\> F\subseteq E(G)$
\begin{algorithmic}[1]
\item[]
\COMMENT{
/* Looking for edge $f$ leaving $u$ that follows
in a restricted man.\ in context $X$.*/}\vskip 0pt
\STATE $\begin{array}{l}F \leftarrow \{ f = (u,v) \in E(G) \, | \,
~\exists \textrm{ restricted } R \in {\cal{M}}: \\ ~~~~~~~~~~~~~~~~~~~
E(X)\cap E(R)\neq \emptyset \, \land
\mathit{Suf\!fix}(X.\, f) \cap \mathit{Prefix}(R)\neq \emptyset\}
\end{array}$
\RETURN $F$
\end{algorithmic}
\noindent
\textsc{Relax}$\,(u,X,f,v)$
\begin{algorithmic}[1]
\vspace*{-\baselineskip}
\item[]
\COMMENT{
/* Relaxing an edge $f$ from vertex $u$ with context $X$.
*/}\vskip 0pt
\STATE $\delta \leftarrow w(f) + \sum_{N \in {\cal{N}}} \Delta(N)$ ~where
${\cal{N}} = {\cal{M}} \cap \mathit{Suf\!fix}(X .\,f)$
\STATE $X' \leftarrow \textsc{LongestPrefix}(X . f)$
\IF{$d[u,X] + \delta < d[v,X']$}
\STATE $Q \leftarrow Q \cup \{(v,X')\}$;~
$d[v,X'] \leftarrow d[u,X] + \delta$;~
$\pi[v,X'] \leftarrow (u,X)$
\ENDIF
\end{algorithmic}
\noindent
\textsc{ProcessNegative}$(X,\, M = (v_0,e_1,\ldots,e_n,v_n)\,)$
\begin{algorithmic}[1]
\STATE $i \leftarrow 1; \, X_0 \leftarrow X; \, F \leftarrow \emptyset$
\COMMENT{
/* Relaxing sequentially all the edges of $M$. */}
\WHILE{$i \leq n \land d[v_{i-1},X_i] <\infty \land F = \emptyset$}
\STATE \textsc{Relax}$(v_{i-1},X_{i-1},e_i,v_i)$
\STATE $X_{i}\leftarrow \textsc{LongestPrefix}(X_{i-1} . e_i)$;
$F \leftarrow \textsc{RestrictedDirection}(v_i,X_i)
\setminus\{e_{i+1}\}$
\STATE $i \leftarrow i + 1$
\ENDWHILE
\end{algorithmic}
}
\subsection{Correctness and Complexity Analysis}
\label{sec:analysis}
Assuming validity of Definition \ref{def:road_network}\, ii. in a~proper road
network, correctness of above $\cal{M}$-Dijkstra's algorithm can be argued
analogously to a traditional proof of Dijkstra's algorithm. Hereafter, the
time complexity growth of the algorithm depends solely on the number of
vertex-context pairs.
\begin{theorem}
\label{thm:MDijkstra}
Let a proper road network $(G,w,\cal{M})$ and vertices $s,t \in
V(G)$ be given. ${\cal{M}}$-Dijk\-stra's algorithm (Algorithm~\ref{alg:m-dijkstra})
computes a valid walk from $s$ to $t$ in $G$ optimal with respect to the
penalized weight, in time ${\cal O} \big(c^2_{\cal M}|E(G)| + c_{\cal M}|V(G)|
\log(c_{\cal M}|V(G)|) \big)$ where $c_{\cal M} =\max_{v\in V(G)}|
\{M\in{\cal{M}}\, |\, v\in V(M)\}|$ is the maximum number of maneuvers per
vertex.
\end{theorem}
\begin{proof}
We follow a traditional proof of ordinary Dijkstra's algorithm with a~simple
modification -- instead of vertices we consider vertex-context pairs as in
Definition~\ref{def:Mconnectivity} and in Algorithm~\ref{alg:m-dijkstra}.
For a walk $P$ let $\chi(P)=\max_{\subseteq} \big[\,(\mathit{Suf\!fix}(P) \cap
\mathit{Prefix}^<({\cal{M}})) \cup \{\emptyset\}\big]$ denote the context of
the endvertex of $P$ with respect to maneuvers $\cal M$. Let $P_x$ stand for
the prefix of $P$ up to a vertex $x\in V(P)$. The following invariant holds at
every iteration of the algorithm:
\begin{itemize}
\item[i.] For every $(u,X)\in T$, the final distance estimate $d[u,X]$ equals
the smallest penalized weight of a valid walk $P$ from $s$ to $u$ such that
$X=\chi(P)$. Every vertex-context pair directly accessible from a member
of~$T$ belongs to $Q$.
\item[ii.] For every $(v,X')\in Q$, the temporary distance estimate
$d[v,X']$ equals the smallest penalized weight of a walk $R$ from
$s$ to $v$ such that $X'=\chi(R)$ and, moreover,
$(x,\chi(R_x))\in T$ for each internal vertex $x\in V(R)$
(except vertices reached during \textsc{ProcessNegative}, if any).
\end{itemize}
This invariant is trivially true after the initialization. By induction we
assume it is true at the beginning of the while loop on line~6, and line 7 is
now being executed -- selecting the pair $(u,X)\in Q$. Then, by minimality of
this selection, $(u,X)$ is such that the distance estimate $d[u,X]$ gives the
optimal penalized weight of a walk $P$ from $s$ to $u$ such that
$X=\chi(P)$. Hence the first part of the invariant (concerning $T$, line~16)
will be true also after finishing this iteration.
Concerning the second claim of the invariant, we have to examine the effect
of lines~8--15 of the algorithm. Consider an edge $f=(u,v) \in E(G)$ starting
in~$u$, and any walk $R$ from $s$ to $v$ such that $\chi(R_u)=X$. Since
$\chi(R)$ must be contained in $X.\,f$ by definition; it is, \textsc{Relax},
line 2, $\chi(R)=X'$. Furthermore, every maneuver contained in $R$ and not
in $R_u$ must be a suffix of $X.\,f$ by definition. So the penalized weight
increase $\delta$ is correctly computed in \textsc{Relax}, line 1. Therefore,
\textsc{Relax} correctly updates the temporary distance estimate $d[v,X']$
for every such~$f$. Finally, any negative maneuver starting from $u$ along
$f$ is correctly reached towards its end $w$ on line 13, its distance estimate
is updated by successive relaxation of its edges and, by
Definition~\ref{def:road_network}, ii. and iii., this distance estimate of $w$ and
its context is not smaller than $d[u,X]$; thus the second part of claimed
invariant remains true.
Validity of a walk is given by line 8 -- \textsc{RestrictedDirection}, that is
enforcing entered restricted maneuvers; and line 1 in \textsc{Relax} --
$\delta$ grows to infinity when completing prohibited maneuvers,
``if'' condition on line 3 in \textsc{Relax} is then false and therefore
prohibited maneuver cannot be contained in an optimal walk.
Lastly, we examine the worst-case time complexity of this algorithm. We
assume $G$ is efficiently implemented using neighborhood lists, the maneuvers
in $\cal M$ are directly indexed from all their vertices and their number is
polynomial in the graph size, and that $Q$ is implemented as Fibonacci heap.
\begin{itemize}
\item The maximal number of vertex-context pairs that may enter $Q$ is
$$ m= |V(G)|+\sum_{M\in\cal M}(|M| - 1)\leq c_{\cal{M}} \cdot |V(G)| \,,$$
and time complexity of the Fibonacci heap operations is $O(m\log m)$.
\item Every edge of $G$ starting in $u$ is relaxed at most those many times as
there are contexts in ${\cal X_M}(u)$ and edges of negative maneuvers
are relaxed one more time during \textsc{ProcessNegative}. Hence the
maximal overall number of relaxations is
$$ r= \sum_{u\in V(G)}|{\cal X_M}(u)|\cdot\mathit{out\mbox-deg}(u) + q
\leq (c_{\cal{M}} + 1) \cdot|E(G)|$$
where $q$ is the number of edges belonging to negative maneuvers.
\item The operations in \textsc{Relax} on line 1, \textsc{LongestPrefix}
as well as \textsc{RestrictedDirection} can be implemented in time
$O(c_{\cal{M}})$.
\end{itemize}
The claimed runtime bound follows.
\vskip 0pt
\qed
\end{proof}
Notice that, in real-world road networks, the number $c_{\cal M}$ of maneuvers
per vertex is usually quite small and independent of the road network size,
and thus it can be bounded by a reasonable minor constant. Although road
networks in practice may have huge maneuver sets, particular maneuvers do
not cross or interlap too much there. for example, $c_{\cal M}=5$ in the current
OpenStreetMaps of Prague.
\subsection{Route Planning Example}
\label{sec:example}
In this section we will demonstrate $\cal{M}$-Dijkstra's algorithm on
a road network containing maneuvers. Consider the road network depicted
below with a weighting representing travel times. There are five
maneuvers (their edges are depicted by dotted lines): a~traffic jam detour
($M_1$), a forbidden passage ($M_2$), a~traffic light left turn delay
($M_3$), a~traffic light delay ($M_4$) and a~direct to be followed ($M_5$).
\begin{tabular}{m{170pt}m{150pt}}
\epsfig{file=./alg-example-01.ps, scale=0.44}
&
\hskip 5pt Road network $(G,w,{\cal{M}})$, where
\begin{itemize}
\item ~$G$ is depicted on the left,
\vskip 2pt
\item ~$\forall e \in E(G): w(e)=1$,
\vskip 2pt
\item ~${\cal{M}}=\{M_1,M_2,M_3,M_4,M_5\}$
\vskip 5pt
\begin{tabular}{l@{~}l}
$M_1 = (b,c,d,e,f)$ & $\Delta(M_1)=-3$ \\[3pt]
$M_2 = (b,r,l)$ & $\Delta(M_2)=~\infty$ \\[3pt]
$M_3 = (g,h,s)$ & $\Delta(M_3)=~~5$ \\[3pt]
$M_4 = (s)$ & $\Delta(M_4) = ~~9$ \\[3pt]
$M_5 = (i,j,k,l)$ & $\Delta(M_5)=~~0$
\end{tabular}
\end{itemize}
\end{tabular}
The goal of out driver is to get from $a$ to $m$ as fast as possible. Classical
Dijkstra's algorithm finds \mbox{$P_1=(a,b,r,l,m)$} with $|P_1|_w = 4$,
unfortunately $|P_1|_w^{\cal{M}} = \infty$ and hence it is impossible for our
driver -- it contains a forbidden passage ($M_2$). On the other hand,
$\cal{M}$-Dijkstra's algorithm finds $P_2=(a,b,c,d,e,f,g,h,i,j,k,l,m)$ with
$|P_2|_w^{\cal{M}} = 9$ and $P_2$ is optimal w.r.t. the penalized weight.
Steps are outlined in Tab.~\ref{tab:example} and Fig.~\ref{fig:example}.
\begin{table}[H]
\label{tab:example}
\caption{State of selected data structures during the steps of
Alg.~\ref{alg:m-dijkstra}. Second column shows a vertex-context pair chosen
at the beginning of the while-loop, i.e. $ \min_{\le_{\cal{M}}}(Q)$. Third column
shows its final distance estimate, i.e. $d[u,X]=\delta_w^{\cal{M}}(a,u)$ and,
finally, the last column depicts elements of the queue $Q$ at the end of
the while-loop.}
\centering
\begin{tabular}{|@{~~}c@{~~}|@{~~}c@{~~}|@{~~}c@{~~}|@{~}c|}
\hline
\textit{Step} & $(u,X)$ \textit{(line 7)} & $d[u,X]$ & $Q$ \textit{(line 16)} \\[2pt]
\hline\hline
1 & $[a,\emptyset]$ & 0 & $[b,\emptyset]$ \\[2pt]
2 & $[b,\emptyset]$ & 1 & $[c,(b,c)]; [t,\emptyset],[r,(b,r)]$ \\[2pt]
3 & $[c,(b,c)]$ & 2 & $[t,\emptyset]; [r,(b,r)]; [d,(b,c,d)]; [e,(b,c,d,e)];
[f,\emptyset]$ \\[2pt]
4 & $[t,\emptyset]$ & 2 & $[(r,(b,r)]; [d,(b,c,d)]; [e,(b,c,d,e)]; [f,\emptyset]$
\\[2pt]
5 & $[r,(b,r)]$ & 2 & $[d,(b,c,d)]; [e,(b,c,d,e)]; [f,\emptyset]; [l,\emptyset]$
\\[2pt]
6 & $[f,\emptyset]$ & 2 & $[g,\emptyset]; [s,\emptyset]; [d,(b,c,d)];
[e,(b,c,d,e)]; [l,\emptyset]$ \\[2pt]
7 & $[d,(b,c,d)]$ & 3 & $[g,\emptyset]; [s,\emptyset]; [e,(b,c,d,e)];
[l,\emptyset]$ \\[2pt]
8 & $[g,\emptyset]$ & 3 & $[h,(g,h)]; [s,\emptyset]; [e,(b,c,d,e)]; [l,\emptyset]$
\\[2pt]
9 & $[e,(b,c,d,e)]$ & 4 & $[h,(g,h)]; [s,\emptyset]; [l,\emptyset]$ \\[2pt]
10 & $[h,(g,h)]$ & 4 & $[i,\emptyset]; [s,\emptyset]; [l,\emptyset]$ \\[2pt]
11 & $[i,\emptyset]$ & 5 & $[j,(i,j)]; [s,\emptyset]; [l,\emptyset]$ \\[2pt]
12 & $[j,(i,j)]$ & 6 & $[k,(i,j,k)]; [s,\emptyset]; [l,\emptyset]$ \\[2pt]
13 & $[k,(i,j,k)]$ & 7 & $[s,\emptyset]; [l,\emptyset]$ \\[2pt]
14 & $[l,\emptyset]$ & 8 & $[m,\emptyset]; [s,\emptyset]$ \\[2pt]
15 & $[m,\emptyset]$ & 9 & $[s,\emptyset]$ \\
\hline
\end{tabular}
\end{table}
\begin{figure}
\caption{A computation of an optimal walk w.r.t. the penalized weight from
$a$ to $m$ in $G$. Numbers represent the distance from the start $a$.
Black vertices are reached or scanned and black edges were relaxed.
Dotted edges represent maneuver edges.
Steps 6 and 7 are depicted in the same figure (they are equal),
analogously for steps 8 and 9.}
\label{fig:example}
\end{figure}
\section{Conclusion}
We have introduced a novel generic model of maneuvers that is able to capture
almost arbitrarily complex route restrictions, traffic regulations and even
some dynamic aspects of the route planning problem. It can model anything
from single vertices to long self-intersecting walks as restricted, negative,
positive or prohibited maneuvers. We have shown how to incorporate this model
into Dijkstra's algorithm so that no adjustment of the underlying road network
graph is needed. The running time of the proposed Algorithm~\ref{alg:m-dijkstra}
is only marginally larger than that of ordinary Dijkstra's algorithm
(Theorem~\ref{thm:MDijkstra}) in practical networks.
Our algorithm can be relatively straightforwardly extended to a bidirectional
algorithm by running it simultaneously from the start vertex in the original
network and from the target vertex in the reversed network. A termination
condition must reflect the fact that chained contexts of vertex-context pairs
scanned in both directions might contain maneuvers as subwalks.
Furthermore, since the A* algorithm is just an ordinary Dijkstra's algorithm
with edge weights adjusted by a potential function, our extension remains
correct for A* if the road network is proper (Definition~\ref{def:road_network},
namely iii.) even with respect to this potential function.
Finally, we would like to highlight that, under reasonable assumptions, our
model can be incorporated into many established route planning approaches.
\end{document} | math | 38,248 |
\begin{document}
\title{
The \ksmt calculus is a $\delta$-complete decision procedure for non-linear constraints\thanks{
This research was partially supported by an Intel research grant, the DFG grant WERA MU 1801/5-1 and the RFBR-JSPS 20-51-5000 grant.
}
\begin{abstract}
\ksmt is a CDCL-style calculus for solving non-linear constraints over real numbers involving polynomials and transcendental functions. In this paper we investigate properties of the \ksmt calculus and show that it is a $\delta$-complete decision procedure for bounded problems.
We also propose an extension with local linearisations,
which allow for more efficient treatment of non-linear constraints.
\end{abstract}
\section{Introduction}\label{sec:intro}
\strut\KK{Spell check final version!}
\todo{All references to appendices need to be replaced. a) Upcoming journal version, b) TR on arXiv, c) just drop \fb b \kk b \mk b
Remember to eliminate all words ``appendix''.}
Solving
non-linear constraints is important in many applications, including verification of
cyber-physical systems
,
software verification
,
proof assistants for mathematics
~\cite{DBLP:books/sp/Platzer18,DBLP:conf/formats/KuratkoR14,https://doi.org/10.1002/rnc.2914,DBLP:conf/fm/BardBD19,DBLP:journals/corr/HalesABDHHKMMNNNOPRSTTTUVZ15,DBLP:conf/fmcad/BrausseKK20}.
Hence there has been a number of approaches for solving non-linear constraints, involving symbolic methods~\cite{DBLP:journals/cca/JovanovicM12,DBLP:conf/cade/MouraP13,DBLP:journals/fmsd/TiwariL16,DBLP:conf/casc/KorovinKS14} as well as numerically inspired ones, in particular for dealing with transcendental functions~\cite{DBLP:conf/cade/GaoAC12,DBLP:conf/cade/TungKO16}, and
combinations of symbolic and numeric methods~\cite{DBLP:conf/frocos/BrausseKKM19,DBLP:journals/tocl/CimattiGIRS18,DBLP:conf/frocos/FontaineOSV17}.
In \cite{DBLP:conf/frocos/BrausseKKM19} we introduced the \ksmt calculus
for solving non-linear constraints over a large class of functions including polynomial, exponential and trigonometric functions.
The \ksmt calculus
\footnote{Implementation is available at \url{http://informatik.uni-trier.de/~brausse/ksmt/}} combines CDCL-style
reasoning~\cite{DBLP:conf/iccad/SilvaS96,DBLP:conf/vmcai/MouraJ13,DBLP:journals/jar/BonacinaGS20} over reals based on conflict resolution~\cite{CR:KTV09} with incremental linearisations of non-linear
functions using methods from computable analysis~\cite{DBLP:series/txtcs/Weihrauch00,DBLP:conf/cca/Muller00}.
Our approach is based on computable analysis and exact real arithmetic which avoids limitations of double precision computations caused by rounding errors and instabilities in numerical methods. In particular, satisfiable and unsatisfiable results returned by \ksmt are exact as required in many applications.
This approach also supports implicit representations of functions as solutions of ODEs and PDEs~\cite{DBLP:books/daglib/0087495}.
It is
well known that in the presence of transcendental functions the constraint satisfiability problem is undecidable~\cite{DBLP:journals/jsyml/Richardson68}.
However if we
only require solutions up to some specified precision $\delta$, then the problem can be solved algorithmically on bounded instances and that is the motivation behind $\delta$-completeness, which was introduced in~\cite{DBLP:conf/cade/GaoAC12}.
In essence a $\delta$-complete procedure decides if a formula is
unsatisfiable or a $\delta$ weakening of the formula is satisfiable.
In this paper we investigate theoretical properties of the \ksmt calculus, and its extension $\delta$-\ksmt for the $\delta$-SMT setting.
Our main results are as follows:
\begin{enumerate}
\item We introduced a notion of \emph{\epsfull{} linearisations} and prove that all \epsfull{} runs of \ksmt are terminating on bounded instances.
\item We extended the $\ksmt$ calculus to the $\delta$-satisfiability setting and proved that $\delta$-\ksmt is a
\emph{$\delta$-complete decision procedure} for bounded instances.
\item We introduced an algorithm for computing \epsfull{} \emph{local linearisations} and integrated it into $\delta$-\ksmt. Local linearisations can be used to considerably narrow the search space
by taking into account local behaviour of non-linear functions avoiding computationally expensive global analysis.
\end{enumerate}
In \Cref{sec:ksmt-overview}, we give an overview about the \ksmt calculus and
introduce the notion of \epsfull linearisation used throughout the rest of the
paper. We also present a completeness theorem. \Cref{sec:delta-sat} introduces
the notion of $\delta$-completeness and related concepts. In
\Cref{sec:delta-ksmt-cade}
we introduce the $\delta$-\ksmt adaptation, prove it is correct and
$\delta$-complete, and give concrete effective linearisations
based on a uniform modulus of continuity.
Finally in \Cref{sec:eps-from-M-cade}, we introduce local linearisations and show that
termination
is independent of computing uniform moduli of continuity,
before we conclude in \Cref{sec:concl}.
\section{Preliminaries}\label{sec:prelim}
\irrelcade{
\todo[inline]{NM, KK: Reframe notions to intervals which might be easier to digest for target audience. \fb n}
}
The following conventions are used throughout this paper.
By $\Vert\cdot\Vert$ we denote the maximum-norm $\Vert (x_1,x_2,\ldots,x_n)\Vert=\max\{|x_i|:1\leq i\leq n\}$.
When it helps clarity, we write finite and infinite sequences $\vec x=(x_1,\ldots,x_n)$ and $\vec y=(y_i)_i$ in bold typeface.
We are going to use open balls
$B(\vec c,\epsilon)=\{\vec x:\Vert\vec x-\vec c\Vert<\epsilon\}\subseteq\mathbb R^n$
for $\vec c\in\mathbb R^n$ and $\eps>0$ and $\widebar A$ to denote the closure of
the set $A\subseteq\mathbb R^n$ in the
standard topology induced by the norm.
By $\mathbb Q_{>0}$ we denote the set $\{q\in\mathbb Q:q>0\}$.
For sets $X,Y$, a (possibly partial) function from $X$ to $Y$ is written as $X\to Y$.
We use
the notion of compactness: a set $A$ is compact iff every open cover of $A$ has a finite subcover.
\todofb{Note: We use this for $\widebar{D_P}$ in the proof of \Cref{th:int:lin:alt-cade}.}
In Euclidean spaces this is equivalent to $A$ being bounded and closed~\cite{Willard70}.
\subsection*{Basic notions of Computable Analysis}
\label{sec:ca}
Let us recall the notion of computability of
functions over real numbers used throughout this paper.
A rational number $q$ is an {\it $n$-approximation} of a real number $x$ if $\Vert q-x\Vert\leq2^{-n}$.
Informally, a function $f$ is {\it computed} by a function-oracle Turing machine
$M_f^?$, where $^?$ is a placeholder for the oracle representing the argument of the function, in the following way.
The real argument $x$ is represented by an oracle function $\varphi:\mathbb N\to\mathbb Q$, for each $n$ returning
an $n$-approximation $\varphi_n$ of $x$.
For simplicity, we refer to $\varphi$ by the sequence $(\varphi_n)_n$.
When run with argument $p\in\mathbb N$, $M_f^\varphi(p)$ computes a rational
$p$-approximation
of $f(x)$ by querying its oracle $\varphi$ for approximations of $x$.
Let us note that the definition of the oracle machine does not depend on the concrete oracle, i.e., the oracle can be seen as a parameter. In case only the machine without a concrete oracle is of interest, we write $M_f^?$.
We refer to~\cite{DBLP:books/daglib/0067010} for a precise definition of the model of computation by function-oracle Turing machines which is standard in computable analysis.
\begin{definition}[\cite{DBLP:books/daglib/0067010}]\label{def:computable}
Consider $\vec x \in \mathbb R^n$. A \emph{name} for $\vec x$ is a rational sequence
$\vec\varphi=(\vec\varphi_k)_k$
such that $\forall k:\Vert\vec\varphi_k-\vec x\Vert\leq2^{-k}$.
A function $f:\mathbb R^n\to\mathbb R$ is \emph{computable} iff there is a
function-oracle Turing machine
$M_f^?$
such that for all $\vec x\in\dom f$ and names $\vec\varphi$ for $\vec x$,
$|M_f^{\vec\varphi}(p)-f(\vec x)|\leq2^{-p}$ holds for all $p\in\mathbb N$.
\end{definition}
This definition is closely related to interval arithmetic with unrestricted precision, but enhanced with the guarantee of convergence and it is equivalent to the notion of computability used in~\cite{DBLP:series/txtcs/Weihrauch00}.
The class of computable functions contains polynomials and transcendental functions like $\sin$, $\cos$, $\exp$, among others.
It is well known \cite{DBLP:books/daglib/0067010,DBLP:series/txtcs/Weihrauch00}
that this class is closed under composition and
that computable functions are continuous.
By continuity, a computable function $f:\mathbb R^n\to\mathbb R$ total on a compact $D\subset\mathbb R^n$
has a computable \emph{uniform modulus of continuity} $\mu_f:\mathbb N\to\mathbb N$ on $D$~\cite[Theorem 6.2.7]{DBLP:series/txtcs/Weihrauch00},
that is,
\begin{align}
\forall k\in\mathbb N\,\forall\vec y,\vec z\in D:
\Vert\vec y-\vec z\Vert\leq2^{-\mu(k)}\implies|f(\vec y)-f(\vec z)|\leq2^{-k}
\text.
\label{eq:mu}
\end{align}
A uniform modulus of continuity of $f$
expresses how changes in the value of $f$ depend on changes of the arguments
in a uniform way.
\section{The \ksmt calculus}\label{sec:ksmt-overview}
We first describe the \ksmt calculus
for solving non-linear constraints~\cite{DBLP:conf/frocos/BrausseKKM19} informally, and subsequently recall the main definitions which we use in this paper.
The \ksmt calculus consists of transition rules, which, for any formula in linear separated form, allow deriving lemmas consistent with the formula and, in case of termination,
produce a satisfying assignment for the formula or show that it is unsatisfiable.
A quantifier-free formula is in separated linear form $\mathcal L\cup\mathcal N$ if $\mathcal L$ is a set of clauses over linear constraints and $\mathcal N$ is a set of
non-linear atomic constraints; this notion is rigorously defined below.
In the \ksmt calculus there are four transition rules applied to its states: Assignment refinement $(A)$, Conflict resolution $(R)$, Backjumping $(B)$ and Linearisation $(L)$.
The final \ksmt states are \SAT and \UNSAT.
A non-final \ksmt state is
a triple $(\alpha,\mathcal L,\mathcal N)$ where $\alpha$ is a (partial) assignment of variables to rationals.
A \ksmt derivation
starts with an initial state where $\alpha$ is
empty and tries to extend this assignment to a solution of
$\mathcal L \cup \mathcal N$ by
repeatedly
applying the Assignment refinement rule.
When such assignment extension is not possible
we either obtain a linear conflict which is resolved using the conflict resolution rule, or a non-linear conflict which is resolved using the linearisation rule.
The main idea behind the linearisation rule is to approximate the non-linear constraints around the conflict using linear constraints in such a way that the conflict will be shifted into the linear part where it will be resolved using conflict resolution.
Applying either of these two rules results in a state containing a clause evaluating to \False under the current assignment. They either result in application
of the backjumping rule, which undoes assignments or in termination in case the formula is \UNSAT.
In this procedure, only the assignment and linear part of the state change and the non-linear part stays fixed.
\begin{figure}
\caption{Core of \ksmt calculus. Derivations terminate in
red nodes.}
\label{fig:white-box}
\end{figure}
\paragraph{Notations.}
Let
$\mathcal F_{\mathrm{lin}}$ consist of rational constants, addition and multiplication
by rational constants; $\mathcal F_{\mathrm{nl}}$ denotes an arbitrary collection of non-linear
computable
functions including transcendental functions and polynomials over the reals.
We consider the structure
$(\mathbb R,\langle\mathcal F_{\mathrm{lin}}\cup\mathcal F_{\mathrm{nl}},\mathcal P\rangle)$
where $\mathcal P=\{{<},{\leq},{>},{\geq},{=},{\neq}\}$ and a set of variables $V=\{x_1,x_2,\ldots,x_n,\ldots\}$.
We will use, possibly with indices, $x$ to denote variables and
$q,c,e$ for rational constants.
Define terms, predicates and formulas over $V$ in the standard way.
An \emph{atomic linear constraint} is a formula of the form:
$q+c_1x_1+\ldots+c_nx_n \diamond 0$ where $q,c_1,\ldots,c_n\in\mathbb Q$ and $\diamond\in\mathcal P$. Negations of atomic formulas can be eliminated by rewriting the predicate symbol $\diamond$ in the standard way, hence we assume that all literals are positive.
A \emph{linear constraint} is a disjunction of atomic linear constraints, also called \emph{(linear) clause}.
An \emph{atomic non-linear} constraint is a formula of the form
$f(\vec x)\diamond 0$, where $\diamond\in\mathcal P$
and $f$ is a composition of computable non-linear functions from $\mathcal F_{\mathrm{nl}}$ over variables $\vec x$.
Throughout this paper
for every computable real function $f$
we use $M_f^?$ to denote a function-oracle Turing machine computing $f$.
We assume quantifier-free formulas in \emph{separated linear
form}~\cite[Definition~1]{DBLP:conf/frocos/BrausseKKM19},
that is, $\mathcal L\cup\mathcal N$ where $\mathcal L$ is a set of linear constraints and $\mathcal N$ is a set of non-linear atomic constraints.
Arbitrary quantifier-free formulas can be transformed equi-satisfiably into separated linear form in polynomial time~\cite[Lemma~1]{DBLP:conf/frocos/BrausseKKM19}.
Since in separated linear form all non-linear constraints are atomic we will call them just \emph{non-linear constraints}.
Let $\alpha:V\to\mathbb Q$ be a partial variable assignment.
The interpretation $\interp{\vec x}^\alpha$ of a vector of variables $\vec x$ under $\alpha$ is defined in a standard way as component-wise application of $\alpha$.
Define the notation $\interp{t}^\alpha$ as
evaluation of term $t$ under assignment $\alpha$, that can be partial,
in which case
$\interp{t}^\alpha$ is treated symbolically.
We extend $\interp\cdot^\alpha$ to predicates, clauses and CNF in the
usual way and $\True,\False$ denote the constants of the Boolean domain.
The evaluation $\interp{t\diamond 0}^\alpha$ for a predicate $\diamond$ and a term $t$ results in $\True$ or $\False$ only if all variables in $t$ are assigned by $\alpha$.
In order to formally restate the calculus, the notions of linear resolvent and linearisation are essential.
A resolvent $R_{\alpha,\mathcal L,z}$ on a variable $z$ is a set of linear constraints
that do not contain $z$, are implied by the formula $\mathcal L$ and which evaluate to $\False$ under the current partial assignment $\alpha$; for more
details see~\cite{CR:KTV09,DBLP:conf/frocos/BrausseKKM19}.
\begin{definition}\label{def:linearisation}
Let $P$ be a non-linear constraint and let $\alpha$ be an
assignment with $\interp P^\alpha=\False$.
A \emph{linearisation of $P$ at $\alpha$} is a linear clause $C$
with the properties:
\begin{enumerate}
\item\label{def:linearisation:it1}
$\forall\beta:\interp P^\beta=\True\implies\interp C^\beta=\True$,
and
\item\label{def:linearisation:it2}
$\interp C^\alpha=\False$.
\end{enumerate}
\end{definition}
Wlog.\ we can assume that the variables of $C$ are a subset of the variables of $P$.
Let us note that any linear clause $C$ represents the complement of a rational polytope $R$ and we will use both
interchangeably. Thus for a rational polytope $R$, $\vec{x}\not \in R$ also stands for a linear clause.
In particular, any linearisation excludes a rational polytope containing the conflicting assignment from the search space.
\paragraph{Transition rules.}
For a formula $\mathcal L_0\cup\mathcal N$ in separated linear form,
the initial \ksmt state is $(\nil,\mathcal L_0,\mathcal N)$. The calculus consists of the following transition rules from a state $S=(\alpha,\mathcal L,\mathcal N)$ to $S'$:
\begin{description}
\item[$(A)$]\label{rule:A}
\emph{Assignment.}
$S'=(\alpha::z\mapsto q,\mathcal L,\mathcal N)$ iff $\interp{\mathcal L}^\alpha\neq\False$ and there is a variable $z$ unassigned in $\alpha$ and $q\in\mathbb Q$ with $\interp{\mathcal L}^{\alpha::z\mapsto q}\neq\False$.
\item[$(R)$]\label{rule:R'}
\emph{Resolution.}
$S'=(\alpha,\mathcal L\cup R_{\alpha,\mathcal L,z},\mathcal N)$ iff
$\interp{\mathcal L}^\alpha\neq\False$ and there is a variable $z$ unassigned in $\alpha$ with $\forall q\in\mathbb Q:\interp{\mathcal L}^{\alpha::z\mapsto q}=\False$
and $R_{\alpha,\mathcal L,z}$ is a resolvent.
\item[$(B)$]\label{rule:B}
\emph{Backjump.}
$S'=(\gamma,\mathcal L,\mathcal N)$ iff $\interp{\mathcal L}^\alpha=\False$ and there is a maximal prefix $\gamma$ of $\alpha$ such that $\interp{\mathcal L}^\gamma\neq\False$.
\item[$(L)$]\label{rule:L}
\emph{Linearisation.}
$S'=(\alpha,\mathcal L\cup \{L_{\alpha, P}\},\mathcal N)$ iff
$\interp{\mathcal L}^\alpha\neq\False$,
there is $P$ in $\mathcal N$ with $\interp P^\alpha=\False$ and
there is a linearisation
$L_{\alpha,P}$ of $P$ at $\alpha$.
\item[\Fsat]
\emph{Final \SAT.}
\todo{prefer $\SAT_\alpha$ \kk y \mk y}
$S'=\SAT$ if all variables are assigned in $\alpha$,
$\interp{\mathcal L}^\alpha=\True$ and
none of the rules $(A),(R),(B),(L)$ is applicable.
\item[\Funsat]
\emph{Final \UNSAT.}
$S'=\UNSAT$ if $\interp{\mathcal L}^\nil=\False$. In other words a trivial contradiction, e.g., $0>1$ is in $\mathcal L$.
\end{description}
A path (or a run) is a derivation in a \ksmt.
A procedure is an effective (possibly non-deterministic) way to construct a path.
\paragraph{Termination.}\label{par:termination}
If no transition rule is applicable, the derivation terminates.
For clarity, we added the explicit rules $\Fsat$ and $\Funsat$ which lead to the final states.
This calculus is sound \cite[Lemma~2]{DBLP:conf/frocos/BrausseKKM19}: if the final transition is $\Fsat$, then $\alpha$ is a solution to the original formula, or $\Funsat$, then a trivial contradiction $0>1$ was derived and the original formula is unsatisfiable. The calculus also makes progress by reducing the search space~\cite[Lemma~3]{DBLP:conf/frocos/BrausseKKM19}.
\begin{figure}
\caption{\texttt{unsat}
\label{fig:lin-exampe-unsat-run}
\end{figure}
An example run of the \ksmt calculus is presented in \Cref{fig:lin-exampe-unsat-run}. We start in a state with a non-linear part $\mathcal N=\{y\leq 1/x\}$, which defines the pink area and the linear part $\mathcal L=\{ (x/4+1\leq y), (y\leq4\cdot(x-1))\}$, shaded in green. Then we successively apply \ksmt rules excluding regions around candidate solutions by
linearisations,
until we derive linearisations
which separates the pink area from the green area thus deriving a contradiction.
\begin{remark}\label{rem:lin-infty-often}
In general a derivation may not terminate.
The only cause of non-termination is
the linearisation rule which adds new linear constraints and can
be applied infinitely many times.
To see this, observe that \ksmt with only
the rules $(A),(R),(B)$ corresponds to the conflict resolution
calculus which is known to be terminating~\cite{CR:KTV09,DBLP:conf/cade/KorovinV11}. Thus, in infinite \ksmt runs the linearisation rule $(L)$ is applied infinitely often. This argument is used in the proof of \Cref{th:bounded} below.
Let us note that during a run the \ksmt calculus neither conflicts nor lemmas can be generated more than once. In fact, any generated linearisation is not implied by the linear part,
prior to adding this linearisation.
\end{remark}
\subsection{Sufficient termination conditions}
In this section we will assume that $(\alpha,\mathcal L,\mathcal N)$ is a \ksmt state obtained by applying \ksmt inference rules to an initial state.
As in \cite{DBLP:conf/cade/GaoAC12} we only consider bounded instances. In many applications this is a natural assumption as variables usually range within some (possibly large) bounds.
\irrelcade{\todofb{remove sentence, implied by `bounded set'}}
We can assume that these bounds are made explicit as linear constraints in the system.
\begin{definition}
Let $F$ be the formula $\mathcal L_0\land\mathcal N$ in separated linear form over variables $x_1,\ldots,x_n$ and
let $B_i$ be the set defined by the conjunction of all clauses in $\mathcal L_0$ univariate in $x_i$,
for $i=1,\ldots,n$; in particular, if there are no univariate linear constraints over $x_i$ then $B_i=\mathbb R$.
We call $F$ a \emph{bounded instance} if:
\begin{itemize}
\item
$D_F\coloneqq\bigtimes_{i=1}^n B_i$ is bounded,
and
\item for each non-linear constraint $P:f(x_{i_1},\ldots,x_{i_k})\diamond 0$ in $\mathcal N$
with $i_j\in\{1,\ldots,n\}$ for $j\in\{1,\ldots,k\}$
it holds that
$\widebar{D_P}\subseteq\dom f$ where
$D_P\coloneqq\bigtimes_{j=1}^k B_{i_j}$.
\end{itemize}
\end{definition}
By this definition, already the linear part of bounded instances
explicitly defines a bounded set by univariate constraints.
Consequently, the set of solutions of $F$ is bounded as well.
In \Cref{th:bounded} we show that when we consider bounded instances and restrict linearisations to so-called $\epsilon$-full linearisations, then the procedure terminates. We use this to show that the \ksmt-based decision procedure we introduce in \Cref{sec:delta-ksmt-cade} is $\delta$-complete.
\begin{definition}\label{def:eps-full}
Let $\eps>0$, $P$ be a non-linear constraint over variables $\vec x$ and let
$\alpha$ be an assignment of $\vec x$.
A linearisation $C$ of $P$ at $\alpha$ is called \emph{\epsfull} iff
for all assignments $\beta$ of $\vec x$ with
$\interp{\vec x}^\beta\in B(\interp{\vec x}^\alpha,\eps)$, $\interp{C}^\beta=\False$.
A \ksmt
\todofb{`almost all' weakens \Cref{th:int:lin-epsfull-cade,th:int:lin:alt-cade}.}
run is called \epsfull for some $\eps>0$, if
all but finitely many linearisations in this run are $\epsilon$-full.
\end{definition}
\begin{comment}
As in \cite{DBLP:conf/cade/GaoAC12} we only consider bounded instances. In many applications this is a natural assumption as variables usually range within some (possibly large) bounds.
\todofb{remove sentence, implied by `bounded set'}
We can assume that these bounds are made explicit as linear constraints in the system.
\end{comment}
The next theorem provides a basis for termination of \ksmt-based decision procedures for satisfiability.
\begin{theorem}\label{th:bounded}
Let $\eps>0$.
On bounded instances,
\epsfull \ksmt runs
are terminating.
\end{theorem}
\begin{proof}
Let $F:\mathcal L_0 \wedge \mathcal N$ be a bounded instance and $\eps>0$.
Towards a contradiction assume
there is an infinite
\epsfull
derivation $(\alpha_0,\mathcal L_0,\mathcal N),\dots, (\alpha_n,\mathcal L_n,\mathcal N), \dots $ in the \ksmt calculus.
Then, by definition of the transition rules, $\mathcal L_k\subseteq\mathcal L_l$ for all $k,l$ with $0\leq k\leq l$.
According to \Cref{rem:lin-infty-often} in any infinite derivation the linearisation rule must be applied infinitely many times.
During any run of \ksmt the set of non-linear constraints $\mathcal N$ is fixed and therefore
there is a non-linear
constraint $P$ in $\mathcal N$ over variables $\vec x$ to which linearisation is applied infinitely often. Let
$(\alpha_{i_1},\mathcal L_{i_1},\mathcal N),\dots, (\alpha_{i_n},\mathcal L_{i_n},\mathcal N), \dots$ be a corresponding subsequence in the derivation such that
$C_{i_1}\in \mathcal L_{i_1+1},\ldots,C_{i_n}\in \mathcal L_{i_n+1},\ldots$ are $\epsilon$-full linearisations of $P$.
Consider
two different linearisation steps $k,\ell\in\{i_j:j\in\mathbb N\}$
in the derivation
where $k < \ell$.
By the precondition $\interp{\mathcal L_\ell}^{\alpha_\ell}\neq\False$ of rule $(L)$ applied in step $\ell$, in particular the linearisation $C_k\in\mathcal L_{k+1}\subseteq\mathcal L_\ell$ of $P$ constructed in step $k$ does not evaluate to \False under $\alpha_\ell$.
Since the set of variables in $C_k$ is a subset of those in $P$, $\interp{C_k}^{\alpha_\ell}\neq\False$ implies $\interp{C_k}^{\alpha_\ell}=\True$.
By assumption, the linearisation $C_k$ is \epsfull, thus
from Definition~\ref{def:eps-full} it follows that
$\interp{\vec x}^{\alpha_\ell}\notin B(\interp{\vec x}^{\alpha_k},\epsilon)$.
Therefore the distance between $\interp{\vec x}^{\alpha_k}$ and $\interp{\vec x}^{\alpha_\ell}$ is at least $\epsilon$.
However, every conflict satisfies the variable bounds defining $D_F$, so there could be only finitely many conflicts with pairwise distance at least $\epsilon$.
This contradicts the above.
\end{proof}
Concrete algorithms to compute \epsfull linearisations are presented in \Cref{sec:delta-ksmt-cade,sec:eps-from-M-cade}.
\section{$\delta$-decidability}\label{sec:delta-sat}
In the last section, we proved termination of the \ksmt calculus on bounded instances
when linearisations are \epsfull.
Let us now investigate how \epsfull linearisations of
constraints involving non-linear computable functions can be constructed.
To that end, we assume that all non-linear functions are defined on the closure of the bounded space $D_F$ defined by the bounded instance $F$.
So far we described an
approach which gives exact results but at the same time is necessarily incomplete due to undecidability of non-linear constraints in general.
On the other hand, non-linear constraints usually can be approximated using numerical methods allowing to obtain approximate solutions to the problem.
This gives rise to the
bounded $\delta$-SMT problem~\cite{DBLP:conf/cade/GaoAC12} which allows an overlap between the properties $\delta$-\SAT and \UNSAT of formulas
as illustrated by \Cref{fig:lin-mv-cade}.
It is precisely this overlap that enables $\delta$-decidability of bounded instances.
\irrelcade{
The notion of computability for functions over the reals is \todofb{cite}
closely related to effective continuity.
That way, the problem of deciding satisfiability of a formula can be lifted to a continuous domain where functions are no longer handled on a purely algebraic basis only as symbols, but also gain the quality of approximability.
In particular, computation of these functions is performed on continuous spaces of sequences of approximations of real numbers.
}
Let us recall the notion of $\delta$-decidability, adapted from~\cite{DBLP:conf/cade/GaoAC12}.
\begin{definition}\label{def:pred-approx}
Let $F$ be a formula in separated linear form and
let $\delta\in\mathbb Q_{>0}$.
We inductively define the $\delta$-weakening $F_\delta$ of $F$.
\begin{itemize}
\item
If $F$ is linear, let $F_\delta\coloneqq F$.
\item
If $F$ is a non-linear constraint $f(\vec x)\diamond 0$, let
\[
F_\delta\coloneqq\begin{cases}
f(\vec x)-\delta\diamond 0,&\text{if}~\diamond\in\{<,\leq\} \\
f(\vec x)+\delta\diamond 0,&\text{if}~\diamond\in\{>,\geq\} \\
|f(\vec x)|-\delta\leq 0,&\text{if}~\diamond\in\{=\} \\
(f(\vec x)<0\lor f(\vec x)>0)_\delta,&\text{if}~\diamond\in\{\neq\}\text.
\end{cases}
\]
\item
Otherwise, $F$ is $A\circ B$ with $\circ\in\{\land,\lor\}$. Let $F_\delta\coloneqq(A_\delta\circ B_\delta)$.
\end{itemize}
\noindent\emph{$\delta$-deciding} $F$ designates
computing
\[
\begin{cases}
\text{\UNSAT},&\text{if}~\interp F^\alpha=\False~\text{for all}~\alpha \\
\text{$\delta$-\SAT},&\text{if}~\interp{F_\delta}^\alpha=\True~\text{for some}~\alpha\text.
\end{cases} \]
In case both answers are valid, the algorithm may output any.
An assignment $\alpha$ with $\interp{F_\delta}^\alpha=\True$ we call a \emph{$\delta$-satisfying assignment} for $F$.
\end{definition}
\begin{figure}
\caption{
The overlapping cases in the $\delta$-SMT problem $f(x)\leq 0$.
}
\label{fig:lin-mv-cade}
\end{figure}
For non-linear constraints $P$ this definition of the $\delta$-weakening $P_\delta$ corresponds exactly to the notion of $\delta$-weakening $P^{-\delta}$ used in the introduction of $\delta$-decidability \cite[Definition 4.1]{DBLP:conf/lics/GaoAC12}.
\begin{remark}\label{rem:delta:eq-le-ge}
The $\delta$-weakening of a non-linear constraint $f(\vec x)\neq 0$ is a tautology.
\end{remark}
We now consider the problem of $\delta$-deciding quantifier-free formulas
in separated linear form.
The notion of $\delta$-decidability is slightly stronger than in \cite{DBLP:conf/cade/GaoAC12}
in the sense that we do not weaken linear constraints.
Consider a formula $F$ in separated linear form.
As before, we assume variables $\vec x$ to be bounded by linear constraints $\vec x\in D_F$.
We additionally assume that for all non-linear constraints
$P:f(\vec x)\diamond0$ in $\mathcal N$, $f$ is defined on
$\widebar{D_P}$ and,
in order to simplify the presentation, throughout the rest of paper we will assume only the predicates $\diamond\in\{{>},{\geq}\}$ are part of formulas, since the remaining ones ${<},{\leq},{=}$
can easily be expressed by the former using
simple arithmetic transformations,
and by \Cref{rem:delta:eq-le-ge} predicates $\neq$ are irrelevant for $\delta$-deciding formulas.
An algorithm is \emph{$\delta$-complete}, if it $\delta$-decides
bounded instances
~\cite{DBLP:conf/cade/GaoAC12}.
\section{$\delta$-\ksmt}\label{sec:delta-ksmt-cade}
Since $\delta$-decidability as introduced above
adapts the condition when a formula is considered to be satisfied to $\delta$-\SAT, this condition has to be reflected in the calculus, which we show solves the bounded $\delta$-SMT problem in this section.
Adding the following
rule $\Fsat[\delta]$ together with the new final state $\delta$-\SAT to \ksmt relaxes the
termination conditions
and turns it into the extended calculus we call $\delta$-\ksmt.
\begin{description}
\item[\ensuremath{\Fsat[\delta]}]
\emph{Final $\delta$-\SAT.}
If $(\alpha,\mathcal L,\mathcal N)$ is a $\delta$-\ksmt state
where $\alpha$ is a total assignment
and $\interp{\mathcal L\land\mathcal N_\delta}^\alpha=\True$, transition to the $\delta$-\SAT state.
\end{description}
The applicability conditions on the rules $(L)$ and $\Fsat[\delta]$ individually are not decidable~\cite{DBLP:journals/jsyml/Richardson68,Brattka2008},
however, when we compute them simultaneously, we can effectively apply one of these rules,
as we will show in \Cref{th:int:lin-correct-cade}.
In combination with \epsfull{}ness of the computed linearisations (\Cref{th:int:lin-epsfull-cade}),
this leads to \Cref{thm:delta-ksmt-complete-cade},
showing that $\delta$-\ksmt is a $\delta$-complete decision procedure.
Let us note that if we assume $\delta=0$ then $\delta$-\ksmt would just reduce to \ksmt
as $\Fsat$ and $\Fsat[\delta]$ become indistinguishable, but in the following we always assume $\delta>0$.
In the following sub-section, we prove that terminating derivations of the $\delta$-\ksmt calculus lead to correct results.
Then, in \Cref{sec:pf-snd-thm-cade},
we present a concrete algorithm
for applying rules $(L)$ and $\Fsat[\delta]$ and show its linearisations to be \epsfull,
which is sufficient to ensure termination, as shown in \Cref{th:bounded}.
These properties lead to a $\delta$-complete decision procedure.
In \Cref{sec:eps-from-M-cade} we develop a more practical algorithm for $\epsilon$-full linearisations that does not require computing a uniform modulus of continuity.
\subsection{Soundness}
In this section we show soundness of the $\delta$-\ksmt calculus, that is,
validity of its derivations.
In particular, this implies that
derivability of the final states $\UNSAT$, $\delta$-\SAT and \SAT directly corresponds to unsatisfiability, $\delta$-satisfiability and satisfiability of the original formula, respectively.
\begin{lemma}\label{lem:preserve_assign-cade}
For all $\delta$-\ksmt derivations of $S'=(\alpha',\mathcal L',\mathcal N)$ from a state $S=(\alpha,\mathcal L,\mathcal N)$
and for all total assignments $\beta$,
$\interp{\mathcal L\land\mathcal N}^\beta=
\interp{\mathcal L'\land\mathcal N}^\beta$.
\end{lemma}
\begin{proof}
Let $\beta$ be a total assignment of the variables in $\mathcal L\land\mathcal N$.
Since the set of variables remains unchanged by $\delta$-\ksmt derivations,
$\beta$ is a total assignment for $\mathcal L'\land\mathcal N$ as well.
Let $S'=(\alpha',\mathcal L',\mathcal N)$ be derived
from $S=(\alpha,\mathcal L,\mathcal N)$
by a single application of one of $\delta$-\ksmt rules.
By the structure of $S'$, its derivation was not caused by
neither $\Funsat,\Fsat$ or $\Fsat[\delta]$.
For rules $(A)$ and $(B)$ there is nothing to show since
$\mathcal L=\mathcal L'$.
If $(R)$ caused $S\mapsto S'$, the claim holds by
soundness of arithmetical
resolution. Otherwise $(L)$ caused
$S\mapsto S'$ in which
case
the direction $\Rightarrow$ follows from the definition of a linearisation (condition~\ref{def:linearisation:it1} in \Cref{def:linearisation})
while the other direction trivially holds since $\mathcal L\subseteq\mathcal L'$.
The condition on derivations of arbitrary lengths then follows by induction.
\end{proof}
\begin{lemma}\label{thm:delta_sound-cade''}
Let $\delta\in\mathbb Q_{>0}$.
Consider a formula $G=\mathcal L_0\land\mathcal N$ in separated linear form and
let $S=(\alpha,\mathcal L,\mathcal N)$ be a $\delta$-\ksmt state derivable from the initial state $S_0=(\nil,\mathcal L_0,\mathcal N)$.
The following hold.
\begin{itemize}
\item
If rule $\Funsat$ is applicable to $S$
then $G$ is unsatisfiable.
\item
If rule $\Fsat[\delta]$ is applicable to $S$
then $\alpha$ is a $\delta$-satisfying assignment for $G$, hence $G$ is $\delta$-satisfiable.
\item
If rule $\Fsat$ is applicable to $S$
then $\alpha$ is a satisfying assignment for $G$,
hence $G$ is satisfiable.
\end{itemize}
\end{lemma}
\begin{proof}
Let formula $G$ and states $S_0,S$ be as in the premise.
As $S$ is not final in $\delta$-\ksmt, only \ksmt rules have been applied in deriving it. The statements for rules $\Funsat$ and $\Fsat$ thus hold by soundness of \ksmt~\cite[Lemma~2]{DBLP:conf/frocos/BrausseKKM19}.
Assume $\Fsat[\delta]$ is applicable to $S$,
that is, $\interp{\mathcal L\land\mathcal N_\delta}^\alpha$ is \True.
Then, since $\mathcal L_0\subseteq\mathcal L$, we conclude that $\alpha$ satisfies $\mathcal L_0\land\mathcal N_\delta$ which, according to \Cref{def:pred-approx}, equals $G_\delta$. Therefore $\alpha$ is a $\delta$-satisfying assignment for $G$.
\end{proof}
Since the only way to derive one of the final states \UNSAT, $\delta$-\SAT and \SAT from the initial state in $\delta$-\ksmt is by application of the rule $\Funsat,\Fsat[\delta]$ and $\Fsat$,
respectively,
as corollary of \Cref{lem:preserve_assign-cade,thm:delta_sound-cade''} we obtain soundness.
\begin{theorem}[Soundness]\label{thm:delta_sound-cade}
Let $\delta\in\mathbb Q_{>0}$. The $\delta$-\ksmt calculus is sound.
\end{theorem}
\subsection{$\delta$-completeness}\label{sec:pf-snd-thm-cade}
We proceed by
introducing \Cref{alg:box-linearisation'-cade}
computing linearisations and
deciding which of the rules $\Fsat[\delta]$ and $(L)$ to apply.
These linearisations are then shown to be \epsfull for some $\eps>0$ depending on the bounded instance.
By \Cref{th:bounded}, this property implies termination, showing that $\delta$-\ksmt is a $\delta$-complete decision procedure.
Given a non-final $\delta$-\ksmt state,
the function \Call{nlinStep$_\delta$}{} in \Cref{alg:box-linearisation'-cade}
computes a $\delta$-\ksmt state derivable from it by application of $\Fsat[\delta]$ or $(L)$. This is done by evaluating the non-linear functions and adding a linearisation $\ell$ based on their uniform moduli of continuity as needed.
To simplify the algorithm, it assumes
total assignments as input.
It is possible to relax this requirement, e.g., by invoking rules $(A)$ or $(R)$ instead of returning $\delta$-\SAT for partial assignments.
\begin{algorithm}[tp]
\setlength{\columnsep}{0cm}
\begin{multicols}{2}
\begin{algorithmic}
\Function{linearise$_\delta$}{$f,\vec x,\diamond,\alpha$}
\State compute
$p\geq-\lfloor\log_2(\min\{1,\delta/4\})\rfloor$
\State $\varphi\gets(n\mapsto\interp{\vec x}^\alpha)$
\State $\eps\gets2^{-\mu_f(p)}$
\State $\tilde y\gets M_f^\varphi(p)$
\If{$\tilde y\diamond-\delta/2$}
\State\Return $\None$
\EndIf
\State\Return $(\vec x\notin B(\interp{\vec x}^\alpha, \eps))$
\EndFunction
\columnbreak
\Function{nlinStep$_\delta$}{$\alpha,\mathcal L,\mathcal N$}
\For{$P:(f(\vec x)\diamond 0)$ \textbf{in} $\mathcal N$}
\State $\ell\gets{}$\Call{linearise$_\delta$}{$f,\vec x,\diamond,\alpha$}
\If{$\ell\neq\None
$}
\State\Return $(\alpha,\mathcal L\cup\{\ell\},\mathcal N)$
\Comment
$(L)$
\EndIf
\EndFor
\State \Return $\delta$-\SAT
\Comment
$\Fsat[\delta]$
\EndFunction
\end{algorithmic}
\end{multicols}
\caption{
(\textsc{nlinStep$_\delta$})
Algorithm computing a $\delta$-\ksmt derivation
according to either rule $(L)$ or $\Fsat[\delta]$
from a state $(\alpha,\mathcal L,\mathcal N)$ where $\alpha$ is total.
The functions $f$ are assumed to be computed by machines $M_f^?$
and $\mu_f$ to be a computable uniform modulus of continuity of $f$.
}
\label{alg:box-linearisation'-cade}
\end{algorithm}
\begin{lemma}\label{th:int:lin-correct-cade}
Let $\delta\in\mathbb Q_{>0}$
and let $S=(\alpha,\mathcal L,\mathcal N)$ be a $\delta$-\ksmt state
where $\alpha$ is total and
$\interp{\mathcal L}^\alpha=\True$. Then
\Call{nlinStep$_\delta$}{$\alpha,\mathcal L,\mathcal N$}
computes
a state derivable by application of either $(L)$ or $\Fsat[\delta]$ to $S$.
\end{lemma}
\begin{proof}
In the proof we will use notions from computable analysis, as defined in \Cref{sec:ca}.
Let $(\alpha,\mathcal L,\mathcal N)$ be a state as in the premise
and let $P:f(\vec x)\diamond0$ be a non-linear constraint in $\mathcal N$.
Let $M_f^?$ compute $f$ as in \Cref{alg:box-linearisation'-cade}.
The algorithm computes a rational approximation $\tilde y=M_f^{(\interp{\vec x}^\alpha)_i}(p)$
of $f(\interp{\vec x}^\alpha)$
where $p\geq-\lfloor\log_2(\min\{1,\delta/4\})\rfloor\in\mathbb N$.
$\interp{\mathcal L}^\alpha=\True$ implies
$\interp{\vec x}^\alpha\in D_P\subseteq\dom f$, thus the computation of $\tilde y$ terminates.
Since $M_f^?$ computes $f$, $\tilde y$ is accurate up to $2^{-p}\leq\delta/4$, that is,
$\tilde y\in[f(\interp{\vec x}^\alpha)\pm\delta/4]$.
By assumption $\diamond\in\{{>},{\geq}\}$, thus
\begin{enumerate}
\item\label{th:int:lin:delta-sat} $\tilde y\mathrel\diamond-\delta/2$
implies $f(\interp{\vec x}^\alpha)\mathrel\diamond-\delta$, which is equivalent to $\interp{P_\delta}^\alpha=\True$, and
\item\label{th:int:lin:unsat} $\neg(\tilde y\mathrel\diamond-\delta/2)$
implies $\neg(f(\interp{\vec x}^\alpha)\mathrel\diamond-\delta/2+\delta/4)$, which in turn implies $\interp P^\alpha=\False$
and the applicability of rule $(L)$.
\end{enumerate}
For \cref{th:int:lin:delta-sat} no linearisation is necessary and indeed the algorithm does not linearise $P$.
Otherwise (\Cref{th:int:lin:unsat}),
it adds the linearisation $(\vec x\notin B(\interp{\vec x}^\alpha,\eps_P))$ to the linear clauses.
Since $\interp{\vec x}^\alpha\in D_P$
by \cref{eq:mu} we obtain that
$0\notin B(f(\vec z),\delta/4)$ holds,
implying $\neg(f(\vec z)\diamond 0)$,
for all $\vec z\in B(\interp{\vec x}^\alpha,\eps_P)\cap\widebar{D_P}$.
Hence,
$(\vec x\notin B(\interp{\vec x}^\alpha,\eps_P))$
is a linearisation of $P$ at $\alpha$.
In case \Call{nlinStep$_\delta$}{$\alpha,\mathcal L,\mathcal N$} returns
$\delta$-\SAT, the premise of \Cref{th:int:lin:delta-sat} holds for every non-linear constraint in $\mathcal N$, that is, $\interp{\mathcal N_\delta}^\alpha=\True$.
By assumption $\interp{\mathcal L}^\alpha=\True$, hence the application of the $\Fsat[\delta]$ rule deriving $\delta$-\SAT is possible in $\delta$-\ksmt.
\end{proof}
\begin{lemma}\label{th:int:lin-epsfull-cade}
For any bounded instance $\mathcal L_0\land\mathcal N$
there is a computable $\eps\in\mathbb Q_{>0}$ such that
any $\delta$-\ksmt
run starting in $(\nil,\mathcal L_0,\mathcal N)$, where applications of $(L)$ and $\Fsat[\delta]$ are performed
by \Call{nlinStep$_\delta$}{}, is \epsfull.
\end{lemma}
\begin{proof}
Let $P:f(\vec x)\diamond 0$ be a non-linear constraint in $\mathcal N$.
Since $\mathcal L_0\land\mathcal N$ is a bounded instance, $D_P\subseteq\mathbb R^n$ is also bounded.
Let $\eps_P\coloneqq2^{-\mu_f(p)}$
where $p\geq-\lfloor\log_2(\min\{1,\delta/4\})\rfloor\in\mathbb N$
as in \Cref{alg:box-linearisation'-cade}.
As $\mu_f$ is a uniform modulus of continuity,
the inequalities in the following construction hold on the whole domain
$\widebar{D_P}$ of $f$ and do not depend on the concrete assignment $\alpha$ where the linearisation is performed.
Since $\log_2$ and $\mu_f$ are computable, so are $p$ and $\eps_P$.
There are finitely many non-linear constraints $P$ in $\mathcal N$,
therefore the linearisations the algorithm \Call{nlinStep$_\delta$}{}
computes are \epsfull with $\epsilon=\min\{\eps_P:P~\text{in}~\mathcal N\}>0$.
\end{proof}
We call $\delta$-\ksmt derivations when linearisation are computed using Algorithm~\ref{alg:box-linearisation'-cade} $\delta$-\ksmt with full-box linearisations, or \emph{$\delta$-\ksmt-fb} for short.
As the runs computed by it are \epsfull for $\eps>0$, by \Cref{th:bounded} they terminate.
\begin{theorem}\label{thm:delta-ksmt-complete-cade}
$\delta$-\ksmt-fb is a $\delta$-complete decision procedure.
\end{theorem}
\begin{proof}
$\delta$-\ksmt-fb is sound (\Cref{thm:delta_sound-cade}) and
terminates on bounded instances (\Cref{th:bounded,th:int:lin-epsfull-cade}).
\end{proof}
\section{Local \epsfull linearisations} \label{sec:eps-from-M-cade}
In practice, when the algorithm computing \epsfull linearisations described in the previous section is going to be implemented, the question arises of how to get a good uniform modulus of continuity $\mu_f$ for a computable function $f$. Depending on how $f$ is given, there may be several ways of computing it.
Implementations of exact real arithmetic, e.g.,
iRRAM~
\cite{DBLP:conf/cca/Muller00} and
Ariadne~
\cite{https://doi.org/10.1002/rnc.2914}, are usually based on the formalism of function-oracle Turing machines (see \Cref{def:computable}) which allow to compute with representations of computable functions~\cite{DBLP:journals/corr/BrausseS17} including
implicit representations of functions as solutions of
ODEs/PDEs~\cite{DBLP:books/daglib/0087495,DBLP:conf/ershov/BrausseKM15}.
If $f$ is only available as a function-oracle Turing machine $M_f^?$ computing it,
a modulus $\mu_f$ valid on a compact domain can be computed, however,
in general this
is not possible without exploring the behaviour of the function on the whole domain, which in many cases is computationally expensive.
Moreover, since $\mu_f$ is uniform, $\mu_f(n)$ is constant throughout $D_F$, independent of the actual assignment $\alpha$ determining where $f$ is evaluated.
Yet, computable functions admit \emph{local} moduli of continuity that additionally depend on the concrete point in their domain.
In most cases these would provide linearisations with $\eps$ larger than that determined by $\mu_f$ leading to larger regions being excluded,
ultimately resulting in fewer linearisation steps and general speed-up.
Indeed, machines producing finite approximations of $f(x)$ from finite approximations of $x$ internally have to compute some form of local modulus
to guarantee correctness.
In this section, we explore
this approach of obtaining linearisations covering a larger part of the function's domain.
In order to guarantee a positive
bound on the local modulus of continuity extracted directly from
the run of the machine $M_f^?$ computing $f$, it is necessary to employ a restriction on the names of real numbers $M_f^?$ computes on.
The set of
names should in a very precise sense be ``small'', i.e., it has to be compact.
The very general notion of names used in \Cref{def:computable} is too broad to satisfy this criterion since
the space of rational approximations
is not even locally compact.
Here, we present an approach using practical names of real numbers as sequences of dyadic rationals of lengths restricted by accuracy.
For that purpose, we introduce another representation~\cite{DBLP:series/txtcs/Weihrauch00} of $\mathbb R$,
that is, the surjective mapping $\xi:\mathbb D_\omega\to\mathbb R$.
Here, $\mathbb D_\omega$ denotes the set of infinite sequences $\varphi$ of dyadic rationals with bounded length. If $\varphi$ has a limit (in $\mathbb R$), we write $\lim\varphi$.
\begin{definition}\label{def:Domega-xi-Cauchy}
\begin{itemize}
\item
For $k\in\omega$ let
$\mathbb D_k\coloneqq\mathbb Z\cdot2^{-(k+1)}=\{m/2^{k+1}:m\in\mathbb Z\}\subset\mathbb Q$
and let
$\mathbb D_\omega\coloneqq\bigtimes_{k\in\omega}\mathbb D_k$
be the set of all sequences $(\varphi_k)_k$ with $\varphi_k\in\mathbb D_k$ for all $k\in\omega$.
By default, $\mathbb D_\omega$ is endowed with the Baire space topology,
which corresponds to that induced by the metric
\[ d:(\varphi,\psi)\mapsto\begin{cases}
0&\text{if}~\varphi=\psi \\
1/{\min\{1+n:n\in\omega,\varphi_n\neq\psi_n\}}&\text{otherwise.}
\end{cases} \]
\item
Define $\xi:\mathbb D_\omega\to\mathbb R$ as the partial function mapping
$\varphi\in\mathbb D_\omega$ to
$\lim\varphi$ iff $\forall i,j:|\varphi_i-\varphi_{i+j}|\leq2^{-(i+1)}$.
Any $\varphi\in\xi^{-1}(x)$ is called a \emph{$\xi$-name} of $x\in\mathbb R$.
\item
The representation $\rho:(x_k)_k\mapsto x$ mapping names $(x_k)_k$ of $x\in\mathbb R$ to $x$ as per \Cref{def:computable} is called \emph{Cauchy representation}.
\end{itemize}
\end{definition}
Using a standard product construction we can easily generalise the notion of $\xi$-names to $\xi^n$-names of $\mathbb R^n$.
When clear from the context, we will drop $n$ and just write $\xi$ to denote the corresponding generalised representation $\mathbb D_\omega^n\to\mathbb R^n$.
Computable equivalence between two representations not only implies that there are continuous maps between them but also that names can computably be transformed~\cite{DBLP:series/txtcs/Weihrauch00}.
Since the Cauchy representation itself is continuous~\cite{DBLP:journals/tcs/BrattkaH02} we
derive continuity of $\xi$,
which is used below to show compactness of preimages $\xi^{-1}(X)$ of compact sets $X\subseteq\mathbb R$ under $\xi$.
\cade{
All proofs can be found in~\todo[inlinepar]{\cite{}}.
}{
All proofs can be found in the appendix.
}
\begin{lemma}\label{prop:cade}
The following properties hold for $\xi$.
\begin{enumerate}
\item $\xi$ is a representation of $\mathbb R^n$:
it is well-defined and surjective.
\item\label{it:xi2rho-cade}
Any $\xi$-name of $\vec x\in\mathbb R^n$ is a Cauchy-name of $\vec x$.
\item\label{it:xi-equiv-rho-cade}
$\xi$ is computably equivalent to the Cauchy representation.
\item $\xi$ is continuous.
\end{enumerate}
\end{lemma}
The converse of \cref{it:xi2rho-cade} does not hold. An example for a Cauchy-name of $0\in\mathbb R$ is the sequence $(x_n)_n$ with $x_n=(-2)^{-n}$ for all $n\in\omega$, which does not satisfy $\forall i,j:|x_i-x_{i+j}|\leq2^{-(i+1)}$.
However, given a name of a real number, we can compute a corresponding $\xi$-name, this is one direction of the property in \cref{it:xi-equiv-rho-cade}.
As a consequence of \cref{it:xi2rho-cade} a function-oracle machine $M^?$ computing $f:\mathbb R^n\to\mathbb R$ according to \Cref{def:computable} can be run on $\xi$-names of $\vec x\in\mathbb R^n$ leading to valid Cauchy-names of $f(\vec x)$.
Note that this proposition does not require $M_f^?$ to compute a $\xi$-name of $f(\vec x)$.
Any rational sequence rapidly converging to $f(\vec x)$ is a valid output.
This means, that the model of computation remains unchanged with respect to the
earlier parts of this paper.
It is the set of names the machines are operated on, which is restricted.
This is reflected in \Cref{alg:2}
by computing dyadic rational approximations $\tilde{\vec x}_k$ of $\interp{\vec x}^\alpha$ such that
$\tilde{\vec x}_k\in\mathbb D_k^n$ instead of keeping the name of $\interp{\vec x}^\alpha$ constant
as has been done in \Cref{alg:box-linearisation'-cade}.
\begin{algorithm}[tp]
\begin{algorithmic}
\Function{LineariseLocal$_\delta$}{$f,\vec x,\diamond,\alpha$}
\State $\varphi\gets(m\mapsto \Approx(\interp{\vec x}^\alpha,m))$
\Comment then $\varphi$ is a $\xi$-name of $\interp{\vec x}^\alpha$
\State compute $p\geq-\lfloor\log_2(\min\{1,\delta/4\})\rfloor$
\State run $M_f^\varphi(p+2)$, record its output $\tilde y$ and its maximum query $k\in\omega$ to $\varphi$
\If{$\tilde y\diamond-\delta/2$}
\State \Return $\None$
\Else
\State \Return $(\vec x\notin B(\interp{\vec x}^\alpha,2^{-k}))$
\EndIf
\EndFunction
\end{algorithmic}
\caption{\textbf{(Local linearisation)}
Algorithm $\delta$-deciding $P:f(\vec x)\diamond 0$ and -- in case \UNSAT{} -- computing a linearisation at $\alpha$ or returning ``\None'' and in this case $\alpha$ satisfies $P_\delta$.
The function $f$ is computed by machine $M_f^?$.
}
\label{alg:2}
\end{algorithm}
In particular, in \Cref{th:int:lin:alt-cade} we show that linearisations for the $(L_\delta)$ rule can be computed by \Cref{alg:2}, which -- in contrast to \Call{linearise$_\delta$}{} in \Cref{alg:box-linearisation'-cade} -- does not require access to a procedure computing an upper bound $\mu_f$ on the uniform modulus of continuity of the non-linear function $f\in\mathcal F_{\mathrm{nl}}$ valid on the entire bounded domain.
It not just runs the machine $M_f^?$, but also
observes the queries $M_f^\varphi$ poses to its oracle in order to
obtain a local modulus of continuity of $f$ at the point of evaluation.
The function $\Approx(\vec x,m)\coloneqq\round{\vec x\cdot 2^{m+1}}/2^{m+1}$ used to define \Cref{alg:2} computes a dyadic approximation of $\vec x$, with
$\round\cdot:\mathbb Q^n\to\mathbb Z^n$ denoting a rounding operation, that is, it satisfies
$\forall \vec q:\Vert\round{\vec q}-\vec q\Vert\leq\frac12$. On rationals (our use-case), $\round\cdot$ is computable by a classical Turing machine.
\begin{definition}[{\cite[Definition 6.2.6]{DBLP:series/txtcs/Weihrauch00}}]\label{def:local-mod}
Let $f:\mathbb R^n\to\mathbb R$ and $\vec x\in\dom f$.
A function $\gamma:\mathbb N\to\mathbb N$ is called \emph{a (local) modulus of continuity of $f$ at $\vec x$} if for all $p\in\mathbb N$ and $\vec y\in\dom f$,
$\Vert\vec x-\vec y\Vert\leq 2^{-\gamma(p)}\implies\vert f(\vec x)-f(\vec y)\vert\leq2^{-p}$
holds.
\end{definition}
We note that in most cases a local modulus of continuity of $f$ at $\vec x$
is smaller than the best uniform modulus of $f$ on its domain, since it only depends on the local behaviour of $f$ around $x$.
One way of computing a local modulus of $f$ at $\vec x$ is using the function-oracle machine $M_f^?$ as defined next.
\begin{definition}\label{def:local-mod-M}
Let $M^?_f$ compute $f:\mathbb R^n\to\mathbb R$ and let $\vec x\in\dom f$ have Cauchy-name $\varphi$.
The function
$\gamma_{M_f^?,\varphi}:p\mapsto\max\{0,k:k~\text{is queried by}~M_f^\varphi(p+2)\}$
is called
\emph{the effective local modulus of continuity induced by $M_f^?$ at $\varphi$}.
\end{definition}
The effective local modulus of continuity of $f$ at a name $\varphi$ of $\vec x\in\dom f$ indeed is a local modulus of continuity of $f$ at $\vec x$~\cite[Theorem~2.13]{DBLP:books/daglib/0067010}.
We prove that \Cref{alg:2} indeed computes linearisations
in \Cref{prf:local-lin''}.
\begin{lemma}\label{lem:local-lin''}
Let $P:f(\vec x)\diamond0$ be a non-linear constraint in $\mathcal N$ and
$\alpha$ be an assignment of $\vec x$ to rationals in $\dom f$.
Whenever $C={}$\Call{LineariseLocal$_\delta$}{$f,\vec x,\diamond,\alpha$}
and $C\neq\None$, $C$ is an \epsfull linearisation of $P$ at $\alpha$,
with $\epsilon$
corresponding to the effective local modulus of continuity
induced by $M_f^?$ at a $\xi$-name of $\interp{\vec x}^\alpha$.
\end{lemma}
Thus, the function \textsc{lineariseLocal$_\delta$}{} in \Cref{alg:2}
is a drop-in replacement for \textsc{linearise$_\delta$}{} in \Cref{alg:box-linearisation'-cade} since the condition on returning a linearisation of $P$ versus accepting $P_\delta$ is identical.
The linearisations however differ in the radius $\epsilon$, which now, according to \Cref{lem:local-lin''}, corresponds to the effective local modulus of continuity.
The resulting procedure we call \Call{nlinStepLocal$_\delta$}{}.
One of its advantages over \Call{nlinStep$_\delta$}{} is
running $M_f^?$ on $\xi$-names instead of Cauchy-names, is that they form a compact set for bounded instances, unlike the latter. This allows us
to bound $\eps>0$ for the computed \epsfull local linearisations of otherwise arbitrary $\delta$-\ksmt runs.
A proof of the following Lemma showing
compactness of preimages $\xi^{-1}(X)$ of compact sets $X\subseteq\mathbb R$ under $\xi$ is given in \Cref{prf:names-compact-cade}.
\begin{lemma}\label{prop:names-compact-cade}
Let $X\subset\mathbb R^n$ be compact.
Then the set $\xi^{-1}(X)\subset\mathbb D_\omega^n$ of
$\xi$-names of elements in $X$ is compact as well.
\end{lemma}
The proof involves showing $\xi^{-1}(X)$ to be closed and uses the fact that for each component $\varphi_k$ of names $(\varphi_k)_k$ of $\vec x\in X$ there are just finitely many choices from $\mathbb D_k$ due to the restriction of the length of the dyadics. This is not the case for the Cauchy representation used in \Cref{def:computable} and
\todorev[2]{`sceptical'. \emph{`We will [...] add clarifications'}}
it is the key for deriving existence of a strictly positive lower bound $\epsilon$ on the \epsfull{}ness of linearisations.
\begin{theorem}\label{th:int:lin:alt-cade}
Let $\delta\in\mathbb Q_{>0}$.
For any bounded instance $\mathcal L_0\land\mathcal N$
there is $\epsilon>0$ such that any $\delta$-\ksmt run starting in $(\nil,\mathcal L_0,\mathcal N)$, where applications of $(L)$ and $\Fsat[\delta]$ are performed according to \Call{nlinStepLocal$_\delta$}{}, is \epsfull.
\end{theorem}
\begin{proof}
Assume $\mathcal L_0\land\mathcal N$ is a bounded instance.
Set $\eps\coloneqq\min\{\eps_P:P\in\mathcal N\}$, where $\eps_P$ is defined as follows.
Let $P:f(\vec x)\diamond 0$ in $\mathcal N$.
Then the closure $\widebar{D_P}$ of the
bounded set $D_P$ is compact.
Let $E$ be the set of $\xi$-names of elements of $\widebar{D_P}\subseteq\dom f$ (see \Cref{def:Domega-xi-Cauchy})
and for any $\varphi\in E$
let $k_\varphi$ be the maximum index queried by $M_f^\varphi(p)$ where $p$ is computed from $\delta$ as in \Cref{alg:2}.
Therefore $\varphi\mapsto k_\varphi$ is continuous.
By \Cref{prop:names-compact-cade} $E$ is compact, thus,
there
\todofb{could also state $k_\psi=\max\{k_\varphi:\varphi\in E\}$ instead; which is clearer? \fb{keep} \mk{keep}}
is $\psi\in E$ such that $2^{-k_\psi}=
\inf\{2^{-k_\varphi}:\varphi\in E\}$.
Set $\eps_P\coloneqq2^{-k_\psi}$.
The claim then follows by \Cref{lem:local-lin''}.
\end{proof}
Thus we can conclude.
\begin{corollary}
$\delta$-\ksmt with local linearisations is a $\delta$-complete decision procedure.
\end{corollary}
\section{Conclusion}\label{sec:concl}
In this paper we extended the the \ksmt calculus to the $\delta$-satisfiability setting and proved that
the resulting $\delta$-\ksmt calculus is a $\delta$-complete decision procedure for solving non-linear constraints over computable functions which include polynomials, exponentials, logarithms, trigonometric and
many other functions used in applications.
We presented algorithms for constructing $\epsilon$-full linearisations
ensuring termination of $\delta$-\ksmt.
Based on methods from computable analysis we presented an algorithm for constructing local linearisations. Local linearisations exclude larger regions from the search space and can be used to avoid computationally expensive global analysis of non-linear functions.
\cade{}{
\appendix
\section{Proofs}
\subsection{Proof of \Cref{prop:cade}}
\subsubsection{Any $\xi$-name of $\vec x\in\mathbb R^n$ is a Cauchy-name of $\vec x$}
\begin{lemma}\label{lem:dist-cade}
For any $\vec x\in\mathbb R^n$ and $\xi$-name $\varphi$ of $\vec x$, $\forall k:|\vec x-\varphi_k|\leq 2^{-k}$ holds.
\end{lemma}
\begin{proof}
For simplicity we assume a dimension of $n=1$.
The general case can be proved similarly.
Let $x\in\mathbb R$ and $\varphi$ be a $\xi$-name of $x$
and let $k\in\omega$.
By construction $x=\lim\varphi$, hence there is
$n_0\in\omega$ such that for every $n\geq n_0$ the bound
$|\varphi_n-x|<2^{-k-1}$ holds.
If $n_0\leq k$, the previous bound already gives the required property.
Otherwise $n_0>k$, then
$|\varphi_k-x|\leq|\varphi_k-\varphi_{n_0}|+|\varphi_{n_0}-x|$
holds. Since $\varphi\in\dom\xi$, the first summand is bounded by $2^{-\min(k,n_0)-1}=2^{-(k+1)}$.
By the property above, so is the second.
Ergo $|\varphi_k-x|\leq2^{-k}$.
\end{proof}
The property that $\xi$-names are Cauchy-names follows directly from \Cref{lem:dist-cade}.
\subsubsection{$\xi$ is computably equivalent to the Cauchy representation}
\begin{proof}
For simplicity we assume a dimension of $1$. The general case can be proved similarly.
\begin{itemize}
\item[$\Rightarrow$)]
Let $\psi$ be a $\xi$-name of $x\in\mathbb R$.
By \Cref{it:xi2rho-cade} of \Cref{prop:cade}, $\psi$
is a
name of $x$.
\item[$\Leftarrow$)]
Given $\varphi\in\dom\rho$ and $n\in\omega$.
Compute $\psi_n\coloneqq\round{\varphi_{n+4}\cdot2^{n+1}}/2^{n+1}\in\mathbb D_n$
where $\round\cdot:\mathbb Q\to\mathbb Z$ is a computable rounding operation
with $|\round q-q|\leq1/2$.
Then with $x\coloneqq\lim\varphi$:
\begin{align*}
|\psi_n-x|&=|\round{\varphi_{n+4}\cdot2^{n+1}}/2^{n+1}-x| \\
&\leq|\round{\varphi_{n+4}\cdot2^{n+1}}-\varphi_{n+4}\cdot2^{n+1}|/2^{n+1}+
|\varphi_{n+4}-x| \\
&\leq2^{-(n+2)}+2^{-(n+4)}
\end{align*}
We show $\psi\coloneqq(\psi_n)_n$ is a $\xi$-name of $x$.
Let $n,k\in\omega$ with $k>0$.
\begin{align*}
|\psi_n-\psi_{n+k}|
&\leq|\psi_n-x|+|\psi_{n+k}-x| \\
&\leq2^{-(n+2)}+2^{-(n+4)}+2^{-(n+k+2)}+2^{-(n+k+4)} \\
&\leq2^{-(n+2)}+2^{-(n+4)}+2^{-(n+3)}+2^{-(n+5)} \\
&\leq2^{-(n+1)}
\end{align*}
Thus, $\psi\in\dom\xi$ and
therefore $\psi$ is a $\xi$-name of $\lim\psi=x$.
\end{itemize}
\end{proof}
\subsubsection{$\xi$ is continuous}
\begin{proof}
Computable equivalence between two representations implies there are continuous maps between them.
Since the Cauchy representation is continuous itself~\cite{DBLP:journals/tcs/BrattkaH02}, so is $\xi$.
\end{proof}
\subsection{Proof of \Cref{lem:local-lin''}}\label{prf:local-lin''}
\paragraph{\upshape\textbf{\Cref{lem:local-lin''}.}} \emph{
Let $P:f(\vec x)\diamond0$ be a non-linear constraint in $\mathcal N$ and
$\alpha$ be an assignment of $\vec x$ to rationals in $\dom f$.
Whenever $C={}$\Call{LineariseLocal$_\delta$}{$f,\vec x,\diamond,\alpha$}
and $C\neq\None$, $C$ is an \epsfull linearisation of $P$ at $\alpha$,
with $\epsilon$
corresponding to the effective local modulus of continuity
induced by $M_f^?$ at a $\xi$-name of $\interp{\vec x}^\alpha$.
}
\begin{proof}
Let $P:f(\vec x)\diamond 0$, $\alpha$ and $C\neq\None$ be as in the premise and
let $p,\tilde y$ and $\varphi$ as in \Cref{alg:2}.
Since $C\neq\None$ by construction
$C=(\vec x\notin B(\interp{\vec x}^\alpha,2^{-k}))$ where
$k=\gamma_{M^?_f,\varphi}(p)$ is the maximum query $M_f^\varphi(p+2)$ poses to its oracle.
Thus, by definition, $C$ is \epsfull with $\epsilon=2^{-k}$.
In order to show that $C$ indeed is a linearisation of $P$ at $\alpha$,
let $\vec z\in B(\interp{\vec x}^\alpha,2^{-k})\cap\widebar{D_P}\subseteq\dom f$.
As $\gamma_{M^?_f,\varphi}$ is a local modulus of continuity of $f$ at $\interp{\vec x}^\alpha$~\cite[Theorem~2.13]{DBLP:books/daglib/0067010},
$f(\vec z)$ is within distance $2^{-p}\leq\delta/4$ of $f(\interp{\vec x}^\alpha)$, which, by definition of $M_f^?$, is at most $2^{-(p+2)}<\delta/4$ away from $\tilde y$.
By construction of $C$ in \Cref{alg:2}
the property $\neg(\tilde y\diamond-\delta/2)$ holds.
As in case~\ref{th:int:lin:unsat} in the proof of \Cref{th:int:lin-correct-cade},
this property implies
$\neg(f(\vec z)\diamond 0)$.
Therefore, according to \Cref{def:eps-full}, $C$ is an \epsfull linearisation of $P$ at $\alpha$.
\end{proof}
\subsection{Proof of \Cref{prop:names-compact-cade}}\label{prf:names-compact-cade}
The proof of the following lemma
follows that of~\cite[Theorem 7.2.5.2]{DBLP:series/txtcs/Weihrauch00}.
\begin{lemma}\label{prop:names-closed-cade}
Let $X\subset\mathbb R^n$ be closed.
Then the set $\xi^{-1}(X)\subset\mathbb D_\omega^n$ of
$\xi$-names of elements in $X$ is closed as well.
\end{lemma}
\begin{proof}
Again, for sake of simplicity we assume $n=1$ while the
general case be proved in a similar manner.
We first introduce the notation $\mathbb D_*$ for the
set of finite prefixes of elements in $\mathbb D_\omega$
and for any prefix $u\in\mathbb D_*$ let
$\mathbb D_\omega[u]\coloneqq\{u\varphi\mid \varphi\in\omega^\omega,u\varphi\in\mathbb D_\omega\}$ denote `the
ball' around $u$ in $\mathbb D_\omega$.
Observe that $\mathbb D_\omega[u]$ is a basic open set
in $\mathbb D_\omega$ for any
prefix $u\in\mathbb D_*$.
In order to show $\xi^{-1}(X)$ is closed, we prove that
its complement is open.
Since $X$ is closed there is a
\KK{$\mathcal B$ not needed \fb y}
collection $\mathcal B$ of open
subsets of $\mathbb R$ such that $\bigcup\mathcal B=\mathbb R\setminus X$. Then
\[ C\coloneqq\{u\in\omega^*\mid\exists B\in\mathcal B~\text{s.t.}~
\xi(\mathbb D_\omega[u])\subseteq B\}
\]
is a subset of $\omega^*$. Define
\begin{align*}
U_1&\coloneqq\bigcup_{u\in C}\mathbb D_\omega[u] \\
U_2&\coloneqq\{(p_n)_n\in\mathbb D_\omega\mid
\exists i,j~\text{s.t.}~|p_i-p_{i+j}|>2^{-(i+1)}\}
\end{align*}
and $U\coloneqq U_1\cup U_2$.
By definition, $U_1$ is open in $\mathbb D_\omega$.
To see that $U_2$ is open in $\mathbb D_\omega$ as well,
observe that
\[ U_2=\mathbb D_\omega\setminus\dom\xi=\bigcup_{w\in C'}\mathbb D_\omega[w] \]
where $w\in C'\subseteq\mathbb D_*$ iff there are $i,j\in\omega$ such that $|w_i-w_{i+j}|>2^{-(i+1)}$.
We show $\xi^{-1}(X)=\mathbb D_\omega\setminus U$.
\begin{itemize}
\item Assume $p\in\xi^{-1}(X)$, that is, $\xi(p)\in X$.
\begin{itemize}
\item
Then $p=(p_k)_k$ with $p_k\cdot2^{k+1}\in\mathbb Z$ for every $k$
such that $\forall i,j:|p_i-p_{i+j}|\leq2^{-(i+1)}$, thus, $p\notin U_2$.
\item
Suppose $p\in U_1$.
Then there is $u\in C$ and $B\in\mathcal B$ such that
$p\in\mathbb D_\omega[u]$ and $\xi(\mathbb D_\omega[u])\subseteq B$ or
$\mathbb D_\omega[u]\cap\dom\xi=\varnothing$.
In both cases $p\notin\xi^{-1}(X)$, a contradiction.
\end{itemize}
From $p\notin U_1$ and $p\notin U_2$ we obtain $p\notin U$.
\item Assume $p\notin\xi^{-1}(X)$.
\begin{itemize}
\item First, consider $p\notin\dom\xi$. Then $p\in U_2$.
\item Finally, consider $p\in\dom\xi$.
Since $\xi(p)\notin X$, there are
\KK{elaborate: whole prefix will be outside \fb y}
some prefix $u$ of $p$ and
some $B\in\mathcal B$ such that $\xi(\mathbb D_\omega[u])\subseteq B$,
and so $p\in U_1$.
\end{itemize}
From $p\in U_1$ or $p\in U_2$ we obtain $p\in U$.
\end{itemize}
Therefore, $U$ is the open complement of $\xi^{-1}(X)$, which is a closed subset of $\mathbb D_\omega$.
\end{proof}
Now, the proof of \Cref{prop:names-compact-cade} follows from Tychonoff's theorem,
which states that arbitrary products of non-empty compact spaces again are compact~\cite{Tychonoff1930}.
\paragraph{\upshape\textbf{\Cref{prop:names-compact-cade}.}} \emph{
Let $X\subset\mathbb R^n$ be compact.
Then the set $\xi^{-1}(X)\subset\mathbb D_\omega^n$ of
$\xi$-names of elements in $X$ is compact as well.
}
\begin{proof}
By \Cref{lem:dist-cade},
$\xi^{-1}(X)$ is a subset of the product of the
finite and therefore compact spaces
\[ \{\vec y\in\mathbb D_k^n:\Vert\vec x-\vec y\Vert\leq2^{-k},\vec x\in X\} \] over $k\in\omega$.
As a closed (\Cref{prop:names-closed-cade}) subset of a compact space, $\xi^{-1}(X)$ is compact as well.
\end{proof}
}
\end{document} | math | 64,637 |
\begin{document}
\title{On Coherence of Assistance and Regularized Coherence of Assistance}
\author{Ming-Jing Zhao$^1$}
\author{Teng Ma$^2$ }
\author{Shao-Ming Fei$^{3,4}$ }
\affiliation{
$^1$School of Science,
Beijing Information Science and Technology University, Beijing, 100192, China\\
$^2$State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China\\
$^3$School of Mathematical Sciences, Capital Normal
University, Beijing
100048, China\\
$^4$Max-Planck-Institute for Mathematics in the Sciences, 04103
Leipzig, Germany}
\pacs{03.65.Ud, 03.67.-a}
\begin{abstract}
We study the relation between the coherence of assistance and the regularized coherence of assistance introduced in [Phys. Rev. Lett. {\bf 116}, 070402 (2016)]. The
necessary and sufficient conditions that these two quantities coincide are provided.
Detailed examples are analyzed and the optimal pure state decompositions such that the coherence of assistance equals to the regularized coherence of assistance are derived.
Moreover, we present the protocol for obtaining the maximal relative entropy coherence, assisted by another party under local measurement and one-way communication in one copy setting.
\end{abstract}
\maketitle
\section{Introduction}
Quantum coherence is an important feature in quantum physics \cite{A. Streltsov-rev}. It is also a powerful resource for quantum metrology \cite{V. Giovannetti}, entanglement creation \cite{J. K. Asboth}, and biological processes \cite{E. Collini,N. Lambert,J. Cai,E. J.OReilly}.
Due to the significant roles played in many novel quantum phenomena, it has attracted much attention recently. A rigorous framework for the quantification of coherence is introduced and some intuitive and
computable measures of coherence are identified, for example, the relative entropy coherence and $l_1$ norm coherence \cite{T. Baumgratz}. The relative entropy coherence of a state is defined as the difference of von Neumann entropy between the density matrix and the diagonal matrix given by its diagonal entries. The $l_1$ norm coherence depends on the magnitudes of off-diagonal entries of a density matrix.
Trace norm coherence is a coherence measure for qubits \cite{L. H. Shao}, but it is only
a coherence monotone for X states \cite{S. Rana}. Besides, the coherence can also be quantified via the convex roof construction \cite{X. Yuan}.
More than that, there are operational coherence measures such as distillable coherence and
coherence cost which characterize the optimal rate of performance for certain information processing tasks \cite{A. Winter}. In Ref. \cite{A. Winter}, they
reveal the appealing feature of distillable coherence being equal to the relative entropy coherence.
As the maximal average relative entropy coherence of a quantum state, the coherence of assistance $C_a$ is another coherence monotone \cite{E. Chitambar}.
This quantity $C_a$ has an operational interpretation.
Suppose Bob holds a state $\rho^B$. Alice holds another part of the purified state of $\rho^B$. With the help of Alice by performing local measurements and telling Bob her measurement outcomes by classical communication,
the relative entropy coherence of $\rho^B$ can be increased to $C_a(\rho^B)$ maximally.
In many copy setting, if Alice is allowed to make joint measurement across her many copies and telling Bob her measurement results by classical communication,
averagely,
the relative entropy coherence of $\rho^B$ can be increased to $C_a^{\infty}(\rho^B)$, which is called the regularized coherence of assistance \cite{E. Chitambar}.
For the process of increasing relative entropy coherence with the help of another party under the local measurement and one way classical communication, an interesting and meaningful question is when the coherence $C_a$ obtained in one copy setting equals to that $C_a^{\infty}$ in many copy setting.
Obviously, for quantum states $\rho$ such that $C_a(\rho)=C_a^{\infty}(\rho)$, one copy setting is enough, and many copy setting is redundant and wasteful. In this paper, we aim to answer this question and provide analytical results for the equivalence of the coherence of assistance and the regularized coherence of assistance.
First we present the necessary and sufficient conditions when the coherence of assistance attains the regularized coherence of assistance.
Detailed examples are analyzed for two dimensional, three dimensional and high dimensional systems.
In these examples, the optimal decompositions for the saturation of the coherence of assistance $C_a$ with the regularized coherence of assistance $C_a^{\infty}$ are provided. The optimal protocol of obtaining maximal relative entropy coherence assisted by an assistant using local measurement and one way communication in one copy setting is designed finally.
\section{Coherence of assistance}
Under fixed reference basis, the coherence of assistance of a state $\rho$ is characterized by the maximal average relative entropy coherence,
\begin{eqnarray}
C_a(\rho)=\max \sum_i p_i C_r(|\psi_i\rangle),
\end{eqnarray}
where the maximization is taken over all pure state decompositions of $\rho=\sum_i p_i |\psi_i\rangle\langle\psi_i|$, $C_r(\rho)=S(\Delta(\rho))-S(\rho)$ is the relative entropy of coherence,
$\Delta(\rho)$ denotes the state given by the diagonal entries of $\rho$, $S(\rho)$ is the von Neumann entropy \cite{E. Chitambar}.
Coherence of assistance can be interpreted operationally.
For given quantum state $\rho$, its initial relative entropy coherence is $C_r(\rho)$. Now suppose Bob holds a state $\rho^B$ and an assistant Alice holds another part of a purification of $\rho^B$. With the help of Alice by performing local measurement and telling Bob her measurement outcomes by classical communication, the quantum state in Bob will be in one pure state ensemble $\{ p_i,\ |\psi_i\rangle\}$ with relative entropy coherence $\sum_i p_i C_r(|\psi_i\rangle)$. The relative entropy coherence in Bob is increased as relative entropy coherence is monotonic under selective measurements on average. Maximally, the relative entropy coherence can be increased to $C_a(\rho^B)$ in this process.
Similarly,
the regularized coherence of assistance is introduced as the average coherence of assistance in many copy setting,
\begin{eqnarray}\label{def regularized ca}
C_a^{\infty}(\rho)=\lim_{n\to \infty} \frac{1}{n}C_a(\rho^{\otimes n}).
\end{eqnarray}
It is obvious that the coherence of assistance is bounded by the regularized coherence of assistance from above
\begin{eqnarray}\label{relation between ca and cainf}
C_a(\rho)\leq C_a^{\infty}(\rho).
\end{eqnarray}
Utilizing the relation between the regularized coherence of assistance and the regularized entanglement of assistance \cite{D. DiVincenzo, E. Rains}, the authors in \cite{E. Chitambar} have shown a closed form expression for the regularized coherence of assistance,
\begin{eqnarray}\label{exp cainf}
C_a^{\infty}(\rho)=S(\Delta(\rho)).
\end{eqnarray}
Based on this formula, we can get the first necessary and sufficient condition for the saturation of the coherence of assistance with the regularized coherence of assistance as follows.
\begin{theorem}\label{upper bound of ca}
For any quantum state $\rho$, $C_a(\rho)= C_a^{\infty}(\rho)$
if and only if there exists a pure state decomposition
$\rho=\sum_i p_i |\psi_i\rangle\langle\psi_i|$ such that all $\Delta(|\psi_i\rangle)=\Delta(\rho)$.
\end{theorem}
[{Proof}]. By definition, we have
$C_a(\rho)=\max \sum_i p_i C_r(|\psi_i\rangle)
=\max \sum_i p_i S(\Delta(|\psi_i\rangle))
\leq \max S(\sum_i p_i \Delta(|\psi_i\rangle))
=S(\Delta(\rho))=C_a^{\infty}(\rho)$,
where the second equation is due to $S(|\psi_i\rangle)=0$ for pure state $|\psi_i\rangle$, and the third inequality is from the concavity of the Von Neumann entropy. The third inequality becomes equality if and only if $\Delta(|\psi_i\rangle)$ are the same for all $i$. Hence, the coherence of assistance equals to the regularized coherence of assistance if and only if there exists a pure state decomposition $\rho=\sum_i p_i |\psi_i\rangle\langle\psi_i|$ such that all $\Delta(|\psi_i\rangle)=\Delta(\rho)$, that is all components in the pure state decomposition have the same diagonal entries as the density matrix. \qed
From theorem \ref{upper bound of ca} one can get another necessary and sufficient condition which is easy to prove.
\begin{corollary}\label{th 1'}
For any quantum state $\rho$, $C_a(\rho)= C_a^{\infty}(\rho)$
if and only if there exists a pure state decomposition
$\rho=\sum_i p_i |\psi_i\rangle\langle\psi_i|$ such that each pure state $|\psi_i\rangle$ has relative entropy coherence $S(\Delta(\rho))$.
\end{corollary}
Theorem \ref{upper bound of ca} and corollary \ref{th 1'} are both necessary and sufficient conditions for the coincidence of the coherence of assistance and the regularized coherence of assistance. The former gives more explicit form of the optimal pure state ensemble and the latter is more easy to understand.
$C_a$ is called additive theoretically if $C_a= C_a^{\infty}$. In
Ref. \cite{E. Chitambar} it has been shown that $C_a$ fails to be additive in general, with an example in 4 dimensional system showing the nonadditivity.
Nevertheless, $C_a$ is additive in two dimensional system. Furthermore, we can find one optimal decomposition for the balance of the coherence of assistance and the regularized coherence of assistance by theorem \ref{upper bound of ca}.
Consider two dimensional quantum states
\begin{equation}\label{2-dim state}
\rho=\sum_{i,j=1}^2 \rho_{ij} |i\rangle\langle j|.
\end{equation}
If the coefficient $\rho_{12}$ is real, we choose
\begin{equation}
\begin{array}{rcl}
|\psi_0\rangle&=&\sqrt{\rho_{11}}|1\rangle + \sqrt{\rho_{22}}|2\rangle,\\
|\psi_1\rangle&=&\sqrt{\rho_{11}}|1\rangle - \sqrt{\rho_{22}}|2\rangle,
\end{array}
\end{equation}
and $p_0=\frac{1}{2}(1+\rho_{12}/\sqrt{\rho_{11}\rho_{22}})$, $p_1=\frac{1}{2}(1-\rho_{12}/\sqrt{\rho_{11}\rho_{22}})$ for nonzero $\rho_{11}$ and $\rho_{22}$.
If the coefficient $\rho_{12}$ is complex, with $|\rho_{12}|$ the magnitude and $\arg(\rho_{12})$ the argument, we set
\begin{equation}
\begin{array}{rcl}
|\psi_0\rangle&=&\sqrt{\rho_{11}}|1\rangle + \sqrt{\rho_{22}}e^{-{\rm i}\arg(\rho_{12})}|2\rangle,\\
|\psi_1\rangle&=&\sqrt{\rho_{11}}|1\rangle + \sqrt{\rho_{22}}e^{-{\rm i}(\pi+\arg(\rho_{12}))}|2\rangle,
\end{array}
\end{equation}
and $p_0=\frac{1}{2}(1+|\rho_{12}|/\sqrt{\rho_{11}\rho_{22}})$, $p_1=\frac{1}{2}(1-|\rho_{12}|/\sqrt{\rho_{11}\rho_{22}})$ for nonzero $\rho_{11}$ and $\rho_{22}$.
Thus $\{p_i, |\psi_i\rangle\}$ is an optimal pure state decomposition of $\rho$ such that the coherence of assistance attains the regularized coherence of assistance.
In fact, there are infinitely many optimal decompositions as the choices of the relative phase in $|\psi_0\rangle$ are infinite.
However, once $|\psi_0\rangle$ is fixed, $|\psi_1\rangle$ and the corresponding probabilities $p_0$ and $p_1$ are determined.
Moreover, if one of the elements $\rho_{11}$ and $\rho_{22}$ is zero, the quantum state $\rho$ is pure, and
its coherence of assistance, regularized coherence of assistance and relative entropy coherence are the same.
Now we consider the equality $C_a(\rho)= C_a^{\infty}(\rho)$ in $n$-dimensional systems and investigate the requirement quantum states should satisfy.
For an $n$-dimensional quantum state $\rho=\sum_{ij} \rho_{ij} |i\rangle\langle j|$, we define the matrix equation,
\begin{equation}\label{equation for p}
AP=B.
\end{equation}
Here
\begin{equation}
A=\left(
\begin{array}{ccccccc}
e^{{\rm i}\theta^{(1)}_{{12}}} & e^{{\rm i}\theta^{(2)}_{{12}}} & \cdots & e^{{\rm i}\theta^{(T)}_{{12}}}\\
e^{{\rm i}\theta^{(1)}_{{13}}} & e^{{\rm i}\theta^{(2)}_{{13}}} & \cdots & e^{{\rm i}\theta^{(T)}_{{13}}}\\
\cdots & \cdots & \cdots & \cdots \\
e^{{\rm i}\theta^{(1)}_{{1n}}} & e^{{\rm i}\theta^{(2)}_{{1n}}} & \cdots & e^{{\rm i}\theta^{(T)}_{{1n}}}\\
e^{{\rm i}\theta^{(1)}_{{23}}} & e^{{\rm i}\theta^{(2)}_{{23}}} & \cdots & e^{{\rm i}\theta^{(T)}_{{23}}}\\
\cdots & \cdots & \cdots & \cdots \\
e^{{\rm i}\theta^{(1)}_{{2n}}} & e^{{\rm i}\theta^{(2)}_{{2n}}} & \cdots & e^{{\rm i}\theta^{(T)}_{{2n}}}\\
\cdots & \cdots & \cdots & \cdots \\
e^{{\rm i}\theta^{(1)}_{{n-1,n}}} & e^{{\rm i}\theta^{(2)}_{{n-1,n}}} & \cdots & e^{{\rm i}\theta^{(T)}_{{n-1,n}}}\\
1 & 1 & \cdots & 1
\end{array}
\right)_{(\frac{n(n-1)}{2}+1)\times T}
\end{equation}
with ${(n-1)(n-2)}/{2}$ constraints
\begin{equation}\label{theta constrains}
\left\{\begin{array}{rcl}
\theta^{(k)}_{1s}-\theta^{(k)}_{2s}&=&\theta^{(k)}_{12},\ \ s=3,\cdots,n,\\
\theta^{(k)}_{2s}-\theta^{(k)}_{3s}&=&\theta^{(k)}_{23},\ \ s=4,\cdots,n,\\
\cdots\\
\theta^{(k)}_{n-2,n}-\theta^{(k)}_{n-1,n}&=&\theta^{(k)}_{n-2,n-1},
\end{array}
\right.
\end{equation}
for all $k$. There are essentially $n-1$ independent variables $\theta^{(k)}_{ij}$ for each $k$, $k=1,2,\cdots,T$, which are all between 0 and $2\pi$. $P=(p_1,p_2,\cdots,p_T)^t$, $0\leq p_k\leq 1$ for $k=1,2...,T$.
\begin{small}
\begin{equation}
B=(\frac{\rho_{12}}{\sqrt{\rho_{11}\rho_{22}}}, \frac{\rho_{13}}{\sqrt{\rho_{11}\rho_{33}}},\cdots,\frac{\rho_{1n}}{\sqrt{\rho_{11}\rho_{nn}}},\frac{\rho_{23}}{\sqrt{\rho_{22}\rho_{33}}}, \cdots,
\frac{\rho_{2n}}{\sqrt{\rho_{22}\rho_{nn}}},\cdots,\frac{\rho_{n-1,n}}{\sqrt{\rho_{n-1,n-1}\rho_{nn}}},1)^t,
\end{equation}
\end{small}
with superscript $t$ denoting transpose.
For vector $B$, although its components are all fractions, if one denominator is zero, then the corresponding numerator must be zero because of the positivity of density matrix. Therefore, vector $B$ is a well defined $\frac{n(n-1)}{2}+1$ dimensional vector and is decided by the coefficients of density matrix. In Eq. (\ref{equation for p}), the vector $B$ is known and given by the density matrix, the
matrix $A$ and the vector $P$ are unknown.
\begin{theorem}\label{th n if and only if}
For $n$-dimensional quantum state $\rho$, $C_a(\rho)=C_a^{\infty}(\rho)$ if and only if the equation (\ref{equation for p}) has solutions for unknowns $P$ and $\theta^{(k)}_{{ij}}$ satisfying conditions (\ref{theta constrains}).
\end{theorem}
[{Proof}].
Let $\{p_k, |\psi_k\rangle\}$
be an optimal pure state ensemble such that $C_a(\rho)=\sum_{k=1}^T p_k C_r(|\psi_k\rangle\langle\psi_k|)$. If $C_a(\rho)$ attains its upper bound $C_a^{\infty}(\rho)$, then $C_r(|\psi_k\rangle\langle\psi_k|)=S(\Delta(\rho))$ by corollary \ref{th 1'} and
$|\psi_k\rangle\langle\psi_k|$ should be of the form
\begin{equation}\label{eq n pure}
\left(
\begin{array}{cccccc}
\rho_{11}& \sqrt{\rho_{11}\rho_{22}}e^{{\rm i}\theta^{(k)}_{{12}}} & \sqrt{\rho_{11}\rho_{33}}e^{{\rm i}\theta^{(k)}_{{13}}} & \cdots
& \sqrt{\rho_{11}\rho_{nn}}e^{{\rm i}\theta^{(k)}_{{1n}}}\\
\sqrt{\rho_{11}\rho_{22}}e^{{\rm -i}\theta^{(k)}_{{12}}} & \rho_{22} & \sqrt{\rho_{22}\rho_{33}}e^{{\rm i}\theta^{(k)}_{{23}}} & \cdots
& \sqrt{\rho_{22}\rho_{nn}}e^{{\rm i}\theta^{(k)}_{{2n}}}\\
\sqrt{\rho_{11}\rho_{33}}e^{{\rm -i}\theta^{(k)}_{{13}}} & \sqrt{\rho_{22}\rho_{33}}e^{{-\rm i}\theta^{(k)}_{{23}}} & \rho_{33}& \cdots
& \sqrt{\rho_{33}\rho_{n,n}}e^{{\rm i}\theta^{(k)}_{{3n}}}\\
\cdots &\cdots &\cdots &\cdots &\cdots\\
\sqrt{\rho_{11}\rho_{nn}}e^{{\rm -i}\theta^{(k)}_{{1n}}} & \sqrt{\rho_{22}\rho_{nn}}e^{{\rm -i}\theta^{(k)}_{{2n}}}
&\sqrt{\rho_{33}\rho_{nn}}e^{{\rm -i}\theta^{(k)}_{{3n}}} & \cdots
&\rho_{nn}
\end{array}
\right)
\end{equation}
by theorem \ref{upper bound of ca} for all $k$.
The ${(n-1)(n-2)}/{2}$ constraints in Eqs. (\ref{theta constrains})
guarantee that the rank of $|\psi_k\rangle\langle\psi_k|$ in Eq. (\ref{eq n pure}) is one.
$\rho=\sum_{k=1}^T p_k |\psi_k\rangle\langle\psi_k|$ demands
\begin{equation}
\sum_{k=1}^T p_k \sqrt{\rho_{ii}\rho_{jj}}e^{{\rm i}\theta^{(k)}_{{ij}}}=\rho_{ij},
\end{equation}
or
\begin{equation}
\sum_{k=1}^T p_k e^{{\rm i}\theta^{(k)}_{{ij}}}=\rho_{ij}/\sqrt{\rho_{ii}\rho_{jj}},
\end{equation}
for $1\leq i<j\leq n$, which gives rise to the equation (\ref{equation for p}). Thus $C_a(\rho)=C_a^{\infty}(\rho)$ if and only if the equation (\ref{equation for p}) has solutions for $P$ and $\theta^{(k)}_{{ij}}$ satisfying conditions (\ref{theta constrains}).
\qed
In Theorem 1 and Corollary 1, the necessary and sufficient conditions are provided for the saturation of the coherence of assistance $C_a(\rho)$ with the regularized coherence of assistance $C_a^{\infty}(\rho)$.
In theorem 2, we present the way to find the optimal pure state ensemble for this saturation. The solution $P$ in matrix equation (8)
is just the probabilities $\{p_k\}$ in the optimal decomposition $\{p_k, |\psi_k\rangle\}$. The solution $\theta^{(k)}_{{ij}}$ in $A$ in (8) is the argument of the entries in the $i$-th row and the
$j$-th column with magnitude $\sqrt{\rho_{ii}\rho_{jj}}$ for the component $|\psi_k\rangle\langle\psi_k|$ in the optimal decomposition $\{p_k, |\psi_k\rangle\}$. The problem of theorem 2
is that the matrix $A$ and $P$ scale quadratically with respect to the dimension of the density matrix, which implies
more unknowns $P$ and arguments $\theta$ in $A$ are involved when the dimension increases. In solving the matrix equation,
one can select proper independent arguments first, then subsequently the matrix $A$.
The vector $P$ is then determined by $A$ and the previous vector $B$.
If $P=(p_1,p_2,\cdots,p_T)^t$ is the solution satisfying $0\leq p_k\leq 1$ for $k=1,2...,T$, then the solution is obtained
and the coherence of assistance $C_a(\rho)$ equals to regularized coherence of assistance $C_a^{\infty}(\rho)$.
Otherwise, one chooses different independent arguments.
{\it Example 1}. Consider the following three dimensional state,
\begin{equation}\label{3-dim state}
\rho=\sum_{i,j=1}^3 \rho_{ij} |i\rangle\langle j|.
\end{equation}
According to Theorem 2, $C_a(\rho)=C_a^{\infty}(\rho)$ if and only if matrix equation
\begin{equation}\label{eq dim 3}
\left(
\begin{array}{ccccccc}
e^{{\rm i}\theta^{(0)}_{{12}}} & e^{{\rm i}\theta^{(1)}_{{12}}} & \cdots & e^{{\rm i}\theta^{(T-1)}_{{12}}}\\
e^{{\rm i}\theta^{(0)}_{{23}}} & e^{{\rm i}\theta^{(1)}_{{23}}} & \cdots & e^{{\rm i}\theta^{(T-1)}_{{23}}}\\
e^{{\rm i}\theta^{(0)}_{{13}}} & e^{{\rm i}\theta^{(1)}_{{13}}} & \cdots & e^{{\rm i}\theta^{(T-1)}_{{13}}}\\
1 & 1 & \cdots & 1
\end{array}
\right)
\left(
\begin{array}{ccccccc}
p_0\\p_1\\\cdots\\p_{T-1}
\end{array}
\right)
=\left(
\begin{array}{ccccccc}
\rho_{12}/\sqrt{\rho_{11}\rho_{22}}\\
\rho_{23}/\sqrt{\rho_{22}\rho_{33}}\\
\rho_{13}/\sqrt{\rho_{11}\rho_{33}}\\
1
\end{array}
\right)
\end{equation}
with $\theta^{(k)}_{{12}}+\theta^{(k)}_{{23}}=\theta^{(k)}_{{13}}$,
have solutions for $P$ satisfying $0\leq p_k\leq 1$ and free arguments $\theta^{(k)}_{{12}}$ and $\theta^{(k)}_{{23}}$.
For simplicity, suppose $\rho_{12}$, $\rho_{23}$ and $\rho_{13}$ are all non-zero real numbers.
Denote $\rho_{12}/\sqrt{\rho_{11}\rho_{22}}=r_1$, $\rho_{23}/\sqrt{\rho_{22}\rho_{33}}=r_2$ and $\rho_{13}/\sqrt{\rho_{11}\rho_{33}}=r_3$.
First, set $T=4$ and $\theta^{(0)}_{{12}}=\theta^{(0)}_{{23}}=0$, $\theta^{(1)}_{{12}}=\pi$, $\theta^{(1)}_{{23}}=0$, $\theta^{(2)}_{{12}}=\theta^{(2)}_{{23}}=\pi$, $\theta^{(3)}_{{12}}=0$, $\theta^{(3)}_{{23}}=\pi$.
Then the matrix equation (\ref{eq dim 3}) becomes
\begin{equation}
\left(
\begin{array}{ccccccc}
1 & -1 & -1 & 1\\
1 & 1 & -1 & -1\\
1& -1 & 1 & -1\\
1 & 1 & 1 & 1
\end{array}
\right)
\left(
\begin{array}{ccccccc}
p_0\\p_1\\p_2\\p_3
\end{array}
\right)
=\left(
\begin{array}{ccccccc}
r_1\\
r_2\\
r_3\\
1
\end{array}
\right).
\end{equation}
The unique solution of the matrix equation above is $p_0=\frac{1}{4}(r_1+r_2+r_3+1)$, $p_1=\frac{1}{4}(r_2-r_1-r_3+1)$, $p_2=\frac{1}{4}(r_3-r_1-r_2+1)$, $p_3=\frac{1}{4}(r_1-r_2-r_3+1)$.
Obviously, $p_0,p_1,p_2,p_3\leq 1$. Therefore, if $r_1+r_2+r_3+1\geq 0$, $r_1-r_2-r_3+1\geq 0$, $r_2-r_1-r_3+1\geq 0$ and $r_3-r_1-r_2+1\geq 0$, then $\{p_i\}$ and $\{\theta_{ij}^{(k)}\}$ are one set of solutions of Eq. (\ref{eq dim 3}) for $C_a(\rho)=C_a^{\infty}(\rho)$. Therefore the probabilities $\{p_i\}$ with pure states
\begin{equation}\label{3-dim optimal dec}
\begin{array}{rcl}
|\psi_0\rangle&=&\sqrt{\rho_{11}}|1\rangle+\sqrt{\rho_{22}}|2\rangle+\sqrt{\rho_{33}}|3\rangle, \\ |\psi_1\rangle&=&-\sqrt{\rho_{11}}|1\rangle+\sqrt{\rho_{22}}|2\rangle+\sqrt{\rho_{33}}|3\rangle,\\
|\psi_2\rangle&=&\sqrt{\rho_{11}}|1\rangle-\sqrt{\rho_{22}}|2\rangle+\sqrt{\rho_{33}}|3\rangle,\\
|\psi_3\rangle&=&\sqrt{\rho_{11}}|1\rangle+\sqrt{\rho_{22}}|2\rangle-\sqrt{\rho_{33}}|3\rangle.
\end{array}
\end{equation}
constitute the optimal decomposition of $\rho$ in Eq. (\ref{3-dim state}) giving $C_a(\rho)=C_a^{\infty}(\rho)$.
Such quantum states all belongs to the polyhedron in Fig. 1.
\begin{center}
\begin{figure}
\caption{(Color online) Quantum states in this polyhedron satisfy four inequalities: $r_1+r_2+r_3+1\geq 0$, $r_1-r_2-r_3+1\geq 0$, $r_2-r_1-r_3+1\geq 0$ and $r_3-r_1-r_2+1\geq 0$. The coherence of assistance
attains the regularized coherence of assistance for these quantum states.}
\label{fig}
\end{figure}
\end{center}
{\it Example 2}. Consider an $n$-dimensional state $\rho=\sum_{i,j=1}^n \rho_{ij} |i\rangle \langle j|$
such that
\begin{eqnarray}\label{eq n-dim}
\sum_{k=1}^{n-1} p_k f(k)+p_0=\rho_{ij}/\sqrt{\rho_{ii}\rho_{jj}}, \ \ i<j,
\end{eqnarray}
holds for some probabilities $p_k$, where $f(k)=1$ for $i\leq k<j$, and $f(k)=-1$ otherwise, $0\leq p_k\leq 1$ for $k=0,1,\cdots,n-1$.
Eq. (\ref{eq n-dim}) is derived by inserting Eq. (\ref{equation for p}) with
$\theta_{1j}^{(1)}=\pi$, $j=2,\cdots, n$; $\theta_{1j}^{(2)}=\theta_{2j}^{(2)}=\pi$, $j=3,\cdots, n$;
$\cdots$;
$\theta_{1n}^{(n-1)}=\cdots=\theta_{nn}^{(n-1)}=\pi$; and other arguments 0.
Therefore if $n$ dimensional quantum state $\rho$ satisfies Eq. (\ref{eq n-dim}) for some probabilities $p_k$, then it allows solution for Eq. (\ref{equation for p}) for some probabilities $p_k$ and $\theta$ defined above. Such quantum state $\rho$ satisfying Eq. (\ref{eq n-dim}) makes $C_a(\rho)=C_a^{\infty}(\rho)$.
For the given arguments
$\theta_{ij}^{(k)}$, we find the corresponding pure states are
\begin{equation}\label{n-dim optimal dec}
\begin{array}{rcl}
|\psi_0\rangle&=&\sqrt{\rho_{11}}|1\rangle+\sqrt{\rho_{22}}|2\rangle+\cdots+\sqrt{\rho_{nn}}|n\rangle,\\ |\psi_1\rangle&=&-\sqrt{\rho_{11}}|1\rangle+\sqrt{\rho_{22}}|2\rangle+\cdots+\sqrt{\rho_{nn}}|n\rangle, \\ |\psi_2\rangle&=&-\sqrt{\rho_{11}}|1\rangle-\sqrt{\rho_{22}}|2\rangle+\cdots+\sqrt{\rho_{nn}}|n\rangle,\\ &\cdots&\\
|\psi_{n-1}\rangle&=&-\sqrt{\rho_{11}}|1\rangle-\sqrt{\rho_{22}}|2\rangle+\cdots-\sqrt{\rho_{n-1,n-1}}|n-1\rangle+\sqrt{\rho_{nn}}|n\rangle.
\end{array}
\end{equation}
Then $\{p_k, |\psi_k\rangle\}$ constitutes an optimal decomposition of $\rho$ with $P=(p_0,p_1,\cdots,p_n)^t$ the solution of Eq. (\ref{eq n-dim}) and $\{|\psi_k\rangle\}$ in Eqs. (\ref{n-dim optimal dec}).
As coherence of assistance $C_a(\rho)$ is the maximal relative entropy coherence obtained with the help of another party making local measurement and one way classical communication in one copy setting. It can be increased more generally in many copy setting. For quantum state $\rho$, the equality $C_a(\rho)=C_a^{\infty}(\rho)$ means to increase the relative entropy coherence in one copy setting is equivalent to the result in many copy setting. Therefore, many copy setting and joint measurement of assistant is redundant.
By theorem 2 we have presented some classes of quantum states whose coherence of assistance $C_a(\rho)$ reaches regularized coherence of assistance $C_a^{\infty}(\rho)$, together with the corresponding optimal pure state decompositions for each class of quantum states. Based on these results, the protocol of obtaining the maximal relative entropy coherence with the help of assistant using local measurement and one way communication can be schemed explicitly.
As an example let us consider the three dimensional quantum state given by Eq. (\ref{3-dim state}), denoted as $\rho_B$, which is held by Bob. As a purification
we first prepare a pure entangled state $|\psi\rangle_{AB}=\sum_{i=0}^3 |i\rangle_A|\psi_i\rangle_B$, with $\{|\psi_i\rangle\}_{i=0}^3$ given in Eqs. (\ref{3-dim optimal dec}). Then Alice performs optimal von Neumann measurements on the basis $\{|i\rangle_A\}$.
If Alice's part is projected to state $|i\rangle_A$, the state of Bob will be collapsed to $|\psi_i\rangle_B$, with relative entropy coherence $S(\Delta(\rho_B))$.
After receiving Alice's measurement outcomes via
classical communication channel, Bob can obtain his state in a four-state ensemble that each state has the same relative entropy coherence $S(\Delta(\rho_B))$. Therefore the final relative entropy coherence for Bob is $S(\Delta(\rho_B))$, which is the maximal relative entropy coherence he can get in this one way assisted protocol.
\section{Conclusions}
To summarize, we have investigated the saturation of the coherence of assistance $C_a(\rho)$ with its upper bound regularized coherence of assistance $C_a^{\infty}(\rho)$. Necessary and sufficient conditions have been provided. Especially, for some special quantum states in two dimensional, three dimensional and general high dimensional systems, the optimal decompositions for the coincidence of $C_a(\rho)$ and $C_a^{\infty}(\rho)$ have been
presented. And the corresponding optimal protocol of obtaining the maximal relative entropy coherence with the help of assistant using local measurement and one way communication has been schemed.
These results are of significant implications in two folds. Firstly, the equality $C_a(\rho)=C_a^{\infty}(\rho)$ implies the additivity of coherence of assistance $C_a(\rho)$. We have investigated which kind of quantum states allow the coherence of assistance additive mathematically. Secondly,
the equality $C_a(\rho)=C_a^{\infty}(\rho)$ shows the equivalence of the maximal relative entropy coherence in one way assisted protocol in one copy setting and that in many copy setting. Here we have revealed the conditions for which kind of quantum states
the maximal relative entropy coherence obtained in one way assisted protocol
with one copy setting is enough.
Note that coherence of assistance $C_a(\rho)$ is the maximal relative entropy coherence attained with the help of another part by local measurements and one way
communication in one copy setting, while the relative entropy coherence is in fact the distillable coherence. Therefore, coherence of assistance $C_a$ quantifies
the one way coherence distillation rate with the help of another part in one copy setting.
In many copy setting, higher one way coherence distillation rate can be obtained. In average
$C_a^{\infty}(\rho)$ characterizes the one way coherence distillation rate in infinite copy setting.
The equality $C_a(\rho)=C_a^{\infty}(\rho)$ shows the equivalence of one way distillation rate in one copy setting and the one way distillation rate in many copy setting assisted by another party.
In Ref. \cite{K. D. Wu}, an experimental realization in linear optical system for obtaining the maximal relative entropy coherence for two dimensional quantum states in assisted distillation
protocol has been presented. Their results are based on one copy setting as the optimal distillable rate of two dimensional quantum states can be reached with one copy scenario. Our research may help for assisted distillation of coherence in high dimensional systems experimentally.
\noindent{\bf Acknowledgments}\, This work is supported by the NSF of China under
Grant Nos. 11401032 and 11675113.
\end{document} | math | 27,670 |
\begin{document}
\title[Green functions and weights]{Green functions and weights
of polynomial skew products on $\mathbb{C}^2$}
\author[K. Ueno]{Kohei Ueno}
\address{Toba National College of Maritime Technology,
Mie 517-8501, Japan}
\curraddr{}
\email{[email protected]}
\urladdr{}
\dedicatory{}
\date{}
\thanks{}
\translator{}
\keywords{Complex dynamics, skew product, Green function}
\subjclass[2010]{32H50 (primary), 30D05 (secondary)}
\begin{abstract}
We study the dynamics of polynomial skew products on $\mathbb{C}^2$.
By using suitable weights,
we prove the existence of several types of Green functions.
Largely,
continuity and plurisubharmonicity follow.
Moreover,
it relates to the dynamics of the rational extensions
to weighted projective spaces.
\end{abstract}
\maketitle
\section{Introduction}
The Green function $G_f$ of $f$ is a powerful tool
for the study of the dynamics of a polynomial map $f$
from $\mathbb{C}^{2}$ to $\mathbb{C}^{2}$,
which is defined as
\[
G_f(z,w) = \lim_{n \to \infty} \frac{1}{{\lambda}^n} \log^{+} |f^n(z,w)|,
\]
where $f^n$ denotes the $n$-th iterate of $f$,
$\lambda$ is the dynamical degree of $f$,
and $|(z,w)| = \max \{ |z|, |w| \}$.
In one dimension, it is well known
that the Green function $G_p$ of a polynomial $p$ is defined,
continuous and subharmonic on $\mathbb{C}$ and
that it coincides with the Green function of $K_p$ with pole at infinity,
where $K_p$ is the set of points whose orbits under $p$ are bounded.
However, in two dimension, it is not known
whether the limit $G_f$ is defined on $\mathbb{C}^{2}$.
One strategy to study the dynamics in two dimension is
to assume that $f$ has a good property that induces the existence of $G_f$.
For example,
it is well known that if $f$ is regular, that is,
if $f$ extends to a holomorphic map on the projective space $\mathbb{P}^{2}$,
then $G_f$ is defined,
continuous and plurisubharmonic on $\mathbb{C}^2$.
Moreover, it coincides with the pluricomplex Green function of $K_f$,
where $K_f$ is the set of points whose orbits under $f$ are bounded
(see e.g. \cite{bj}).
To relax this regularity condition,
many authors have paid attention to
another condition called algebraically stability
(e.g. \cite{fs}, \cite{g}, \cite{dg}, \cite{ddg1}, \cite{ddg2} and \cite{ddg3}).
Recently,
Favre and Jonsson~\cite{fj} proved that any polynomial map has an extension
that is algebraically stable in some weak sense,
to a projective compactification of $\mathbb{C}^{2}$
with at worst quotient singularities.
Their proof is based on the valuative techniques developed in \cite{fj-ev},
with which they proved that if $f$ is not conjugate to a skew product,
then there is a plurisubharmonic function $u$,
which is close to $\log^{+} |(z,w)|$,
such that $\lambda^{-n} u \circ f^n$
decreases to a plurisubharmonic function, which is not identically zero.
Here $\circ$ denotes the composition.
The distinctiveness of polynomial skew products
was pointed out at numerous instances in their papers.
In this paper
we study the dynamics of polynomial skew products on $\mathbb{C}^2$.
A polynomial skew product is a polynomial map of the form
$f(z,w) = (p(z),q(z,w))$.
We assume that $\delta = \deg p \geq 2$ and $d = \deg_w q \geq 2$.
Let $b(z)$ be the coefficient of $w^d$ and $\gamma$ its degree.
Many authors have studied the dynamics of regular polynomial skew products
(e.g. \cite{h}, \cite{h2}, \cite{j} and \cite{dh});
$f$ is regular if and only if $\delta = d = \deg q$,
which implies $\gamma = 0$.
We have studied the dynamics of
nondegenerate polynomial skew products in \cite{u-weight};
we say that $f$ is nondegenerate if $\gamma = 0$.
Besides giving examples of polynomial skew products
whose Green functions are not defined on some curves,
we introduced the weighted Green function $G_f^{\alpha}$ of $f$ in \cite{u-weight},
\[
G_f^{\alpha} (z,w) = \lim_{n \to \infty}
\frac{1}{{\lambda}^n} \log^{+} |f^n(z,w)|_{\alpha},
\]
where $|(z,w)|_{\alpha} = \max \{ |z|^{\alpha},|w| \}$,
which is defined, continuous and plurisubharmonic on $\mathbb{C}^2$
for a suitable rational number $\alpha \geq 0$.
Moreover, $f$ extends to an algebraically stable map
on a weighted projective space,
whose dynamics relates to $G_f^{\alpha}$.
Although the dynamics becomes much more difficult
without the nondegeneracy condition,
the idea of imposing suitable weights is still effective.
The dynamics of $f$ consists of
the dynamics of $p$ on the base space and the dynamics on fibers:
$f^n(z,w) = (p^n(z),Q_z^n(w))$, where
$Q_z^n = q_{p^{n-1}(z)} \circ \cdots \circ q_{p(z)} \circ q_{z}$
and $q_z(w) = q(z,w)$.
As the Green function $G_p$ of $p$,
we define the fiberwise Green function of $f$ as
\[
G_z (w) = \lim_{n \to \infty} \frac{1}{d^n} \log^{+} | Q_z^n(w) |.
\]
Since $G_p$ exists on $\mathbb{C}$,
the existence of $G_z$ implies the existence of $G_f$ and $G_f^{\alpha}$.
Favre and Guedj proved the existence of $G_z$ on $K_p \times \mathbb{C}$
in \cite[Theorem 6.1]{fg},
which is continuous and plurisubharmonic if $b^{-1}(0) \cap K_p = \emptyset$,
and gave examples
whose fiberwise Green functions are discontinuous over $J_p = \partial K_p$
in \cite[Proposition 6.5]{fg}.
In this paper, we investigate the existence,
continuity and plurisubharmonicity of $G_z$ on $A_p \times \mathbb{C}$,
where $A_p$ denotes the set of points whose orbits under $p$ tend to infinity.
A summary of our results is as follows.
We replace the definition of $|(z,w)|_{\alpha}$
by $\max \{ |z|^{max \{ \alpha, 0 \}},|w| \}$
or $\max \{ (|z|+1)^{\alpha},|w| \}$.
In the case $\delta \neq d$,
the weighted Green function $G_f^{\alpha}$ is defined on $\mathbb{C}^2$
and continuous and plurisubharmonic on $A_p \times \mathbb{C}$
for a suitable rational number $\alpha$,
which can be negative if $\delta < d$.
If $\delta > d$, then $\alpha$ is nonnegative,
$\limsup_{n \to \infty} \delta^{-n} \log^{+} |Q_z^n| \leq \alpha G_p$
and so $G_f^{\alpha} = \alpha G_p$ on $\mathbb{C}^2$.
Moreover,
if $\alpha = \gamma /(\delta - d)$,
then the limit
\[
G_z^{\alpha} (w) = \lim_{n \to \infty} \frac{1}{d^n} \log^{+}
\left| \dfrac{Q_z^n(w)}{p^n(z)^{\alpha}} \right|
\]
is defined, continuous and plurisubharmonic on $A_p \times \mathbb{C}$.
If $\delta < d$,
then $G_z$ coincides with $G_f$ and $G_f^{\alpha}$ on $\mathbb{C}^2$,
and with $G_z^{\alpha}$ on $A_p \times \mathbb{C}$.
Moreover,
if $\delta \neq d$ and $\alpha = \gamma /(\delta - d)$,
then we obtain certain uniform convergence to $G_z^{\alpha}$
on $A_p \times \mathbb{C}$
and the asymptotics of $G_z^{\alpha}$ near infinity.
In the case $\delta = d$, the dynamics differs
depending on whether $f$ is nondegenerate.
If $\gamma \neq 0$, then $G_f^{\alpha}$ is defined on $\mathbb{C}^2$
if we admit plus infinity
for a suitable rational number $\alpha$, which can be negative.
Moreover, the limit
$\lim_{n \to \infty} (n \gamma d^{n-1})^{-1} \log^{+} |Q_z^n(w)|$
is defined and plurisubharmonic on $\mathbb{C}^2$.
It coincides with $G_p$ on a region in $A_p \times \mathbb{C}$
and with $0$ on the complement of the region in $A_p \times \mathbb{C}$.
Furthermore, the limit
\[
G(z,w) = \lim_{n \to \infty} \frac{1}{d^n} \log
\left| \dfrac{Q_z^n(w)}{p^n(z)^{n \frac{\gamma}{d}}} \right|
\]
is defined and plurisubharmonic on $A_p \times \mathbb{C}$
if we admit minus infinity.
It is continuous on the region in $A_p \times \mathbb{C}$.
Throughout the paper
we make use of the extensions of $f$
to rational maps on weighted projective spaces,
which can be algebraically stable
if and only if $\gamma = 0$ or $\delta > d$.
Some results on the Green functions
can be illustrated with these extensions.
In particular,
if $\delta > d$ and $\alpha = \gamma /(\delta - d)$
or if $\delta \leq d$,
then $f \sim (z^{\delta}, z^{\gamma} w^d)$ on a region
in the attracting basin of the indeterminacy point $[0:1:0]$,
which induces the results on the Green functions.
Here the notation $f \sim g$ means that
the ratios of the first and second components
of the maps tend to $1$, respectively.
Note that
Guedj pointed out in \cite[Example 3.2]{g}
that $f$ extends to algebraically stable maps
on Hirzebruch surfaces if $\delta \leq d$.
Under an additional condition,
he derived the existence of a weighted Green function of $f$
in \cite[Theorem 4.1]{g},
which is continuous and plurisubharmonic on $\mathbb{C}^2$.
This paper is organized as follows.
First,
we briefly recall the dynamics of polynomial skew products
and give the definition of rational extensions on weighted projective spaces
in Section 2.
In Section 3 we recall the results for the nondegenerate case.
The generalized results
in the cases $\delta > d$, $\delta < d$ and $\delta = d$
are presented in Sections 4, 5 and 6, respectively.
In these sections we offer two examples:
monomial maps, and skew products that are semiconjugate to polynomial products.
The weight $\alpha$ contributes also to the degree growth of $f$;
in Section 7 we provide a result on the degree growth
and a corollary on a weighted Green function that relates to the degree growth.
\section{Preliminaries}
Let $f(z,w)=(p(z),q(z,w))$ be
a polynomial skew product such that
$p(z) = a z^{\delta} + O( z^{\delta - 1} )$ and
$q(z,w) = b(z) w^{d} + O_z( w^{d - 1} )$,
and let $\gamma = \deg b$.
We assume that $\delta \geq 2$ and $d \geq 2$.
Then we may assume that polynomials $p$ and $b$ are monic
by taking an affine conjugate;
$p(z) = z^{\delta} + O( z^{\delta - 1} )$ and
$b(z) = z^{\gamma} + O( z^{\gamma - 1} )$.
Let $\lambda = \max \{ \delta, d \}$ and
$\deg f$ be the algebraic degree of $f$,
i.e. $\max \{ \deg p, \deg q \}$.
In general, $\deg f$ may be greater than $\lambda$.
However, the dynamical degree of $f$,
$\lim_{n \to \infty} \sqrt[n]{\deg (f^n)}$,
is equal to $\lambda$.
Let us briefly recall
the dynamics of polynomial skew products.
Roughly speaking,
the dynamics of $f$ consists of
the dynamics on the base space and the dynamics on the fibers.
The first component $p$ defines
the dynamics on the base space $\mathbb{C}$.
Note that $f$ preserves the set of vertical lines in $\mathbb{C}^{2}$.
For this reason,
we often use the notation $q_z (w)$ instead of $q(z,w)$.
The restriction of $f^n$ to the vertical line $\{ z \} \times \mathbb{C}$
can be viewed as the composition of $n$ polynomials on $\mathbb{C}$,
$q_{p^{n-1}(z)} \circ \cdots \circ q_{p(z)} \circ q_{z}$.
A useful tool in the study of the dynamics of $p$ on the base space
is the Green function $G_p$ of $p$,
\[
G_p(z) = \lim_{n \to \infty} \frac{1}{\delta^n} \log^{+} |p^n(z)|.
\]
It is well known that $G_p$ is defined,
continuous and subharmonic on $\mathbb{C}$.
More precisely,
$G_p$ is harmonic and positive on $A_p$ and zero on $K_p$,
and $G_p(z)= \log |z| + o(1)$ as $z \to \infty$.
Here $A_p = \{ z : p^n (z) \to \infty \}$,
$K_p = \{ z : \{ p^n (z) \}_{n \geq 1} \ \text{bounded} \}$
and $A_p \sqcup K_p = \mathbb{C}$.
By definition, $G_p(p(z))= \delta G_p(z)$.
In a similar fashion,
we consider the fiberwise Green function of $f$,
\[
G_z(w) = \lim_{n \to \infty} \frac{1}{d^n} \log^{+} |Q_z^n(w)|
\text{ or }
G_z^{\lambda} (w) = \lim_{n \to \infty} \frac{1}{{\lambda}^n} \log^{+} | Q_z^n(w) |,
\]
where
$Q_z^n = q_{p^{n-1}(z)} \circ \cdots \circ q_{p(z)} \circ q_{z}$.
By definition, $G_{p(z)}(q_z(w)) = d G_z(w)$ if it exists.
Since the limit $G_p$ exists on $\mathbb{C}$,
the existence of $G_z$ or $G_z^{\lambda}$ implies those of $G_f$ and $G_f^{\alpha}$.
Since the existence of $G_z$
on $K_p \times \mathbb{C}$ is proved in \cite{fg},
the remaining problem lies in investigating
the existence on $A_p \times \mathbb{C}$.
We note that it is unclear even if $f$ is regular.
For the nondegenerate case,
using an argument in the proof of \cite[Theorem 6.1]{fg},
the author showed the existence of $G_z$
on an open subset of $A_p \times \mathbb{C}$ in \cite[Lemma 2.3]{u-sym}
with the assumption $\delta \leq d$,
which was improved in \cite{u-weight}.
In this paper we generalize these results for the case $\gamma \neq 0$.
It is useful to consider the dynamics of the rational extensions of $f$.
Let $r$ and $s$ be any two positive integers.
The weighted projective space $\mathbb{P} (r,s,1)$ is
a quotient space of $\mathbb{C}^{3} - \{ O \}$,
\[
\mathbb{P} (r,s,1) = \mathbb{C}^{3} - \{ O \} \ \big/ \sim,
\]
where $(z, w, t) \sim (c^r z, c^s w, c t)$
for any $c$ in $\mathbb{C} - \{ 0 \}$.
Thus $\mathbb{P} (r,s,1) = \mathbb{C}^{2} \sqcup L_{\infty}$,
where $L_{\infty}$ denotes the line at infinity $\{ t = 0 \}$.
We denote a point in $\mathbb{P} (r,s,1)$
by weighted homogeneous coordinates $[ z : w : t ]$.
It follows that
$f$ extends to a rational map $\tilde{f}$ on $\mathbb{P} (r,s,1)$,
\[
\tilde{f} [ z : w : t ]
= \left[ p \left( \frac{z}{t^r} \right) t^{\lambda r}
: q \left( \frac{z}{t^r}, \frac{w}{t^s} \right) t^{\lambda s}
: t^{\lambda} \right].
\]
We say that $\tilde{f}$ is algebraically stable
if there is no integer $n$ and no hypersurface $V$
such that ${\tilde{f}}^n (V - I_{{\tilde{f}}^n}) \subset I_{\tilde{f}}$,
where $I_{\tilde{f}}$ denotes the indeterminacy set of $\tilde{f}$.
It is well known that if $\tilde{f}$ is algebraically stable on $\mathbb{P}^2$,
then $\deg (f^n) = (\deg f)^n$.
In the final section
we present a similar claim on $\mathbb{P} (r,s,1)$.
We define the Fatou set $F_{\tilde{f}}$ of $\tilde{f}$ as
the maximal open set of $\mathbb{P} (r,s,1)$
where the family of iterates $\{ \tilde{f}^n \}_{n \geq 0}$ is normal.
The Julia set $J_{\tilde{f}}$ of $\tilde{f}$ is defined as
the complement of the Fatou set of $\tilde{f}$.
\section{Nondegenerate case}
To recall the main results in \cite{u-weight},
we assume that $f$ is nondegenerate throughout this section.
We defined the rational number $\alpha$ as
\[
\min
\left\{ l \in \mathbb{Q} \ \Big|
\begin{array}{lcr}
l \lambda \geq n_j + l m_j
\text{ for any integers $n_j$ and $m_j$} \\
\text{s.t. } c_j z^{n_j} w^{m_j} \text{ is a term in } q
\text{ for some } c_j \neq 0
\end{array}
\right\}
\]
if $\deg_z q > 0$
and as $0$ if $\deg_z q = 0$.
Since $q$ has only finitely many terms, one can take the minimum.
Indeed, $\alpha$ is equal to
\[
\max \left\{ \frac{n_j}{\lambda - m_j} \ \Big|
\begin{array}{lcr}
\ c_j z^{n_j} w^{m_j} \text{ is a term in } q \\
\text{ with $c_j \neq 0$ and } m_j < \lambda
\end{array}
\right\}.
\]
Clearly, $\alpha \geq 0$,
and $\alpha = 0$ if and only if $f$ is a polynomial product.
By definition,
$\alpha \leq \deg_z q$ and $\alpha < \deg q$.
Moreover,
$\lambda^n \leq \deg (f^n) \leq
\max \{ 1, \alpha \} \lambda^n$ for any positive integer $n$.
We define the weight of a monomial $z^n w^m$ mainly as $n + \alpha m$.
The weight of $q$ is defined as the maximum of the weights of all terms in $q$.
Then the weight of $q$ is $\alpha \lambda$
and the weight of $Q_z^n(w)$ is $\alpha \lambda^n$.
Let $h$ be the weighted homogeneous part of $q$ of highest weight $\alpha \lambda$.
In the case $\delta \leq d$,
it follows from definition that $h$ contains $w^d$.
On the other hand,
if we replace $\alpha$
in the definition of the weight of a monomial
by a positive number which is larger than $\alpha$,
then $h$ coincides with $w^d$,
and if we replace $\alpha$ by a positive number which is smaller than $\alpha$,
then $h$ does not contain $w^d$.
We explain the importance of $\alpha$ in terms of
the rational extensions of $f$,
assuming that $f$ is not a polynomial product.
We saw that $f$ extends to a rational map $\tilde{f}$
on $\mathbb{P} (r,s,1)$ for any two positive integers $r$ and $s$.
Moreover, it extends to
a weighted homogeneous polynomial on $\mathbb{C}^{3}$
if and only if $s/r \geq \alpha$.
Similarly,
$\tilde{f}$ is algebraically stable
if and only if $s/r \geq \alpha$;
if $s/r < \alpha$
then $\tilde{f}$ contracts $L_{\infty} - I_{\tilde{f}}$
to the indeterminacy point $p_{\infty} = [0:1:0]$.
The map $\tilde{f}$ is holomorphic
if and only if $\delta = d$ and $s/r \geq \alpha$.
Note that the holomorphic extensions of polynomial maps
to weighted projective spaces are also mentioned in \cite[Section 5.3]{fj}.
Let us list some more details of $\tilde{f}$
on $\mathbb{P} (r,s,1)$ for $s/r \geq \alpha$,
which deffer depending on the magnitude relation of $\delta$ and $d$.
If $\delta > d$, then $I_{\tilde{f}} = \{ p_{\infty} \}$.
Moreover,
if $s/r = \alpha$
then the dynamics on $L_{\infty} - \{ p_{\infty} \}$ is induced by the polynomial $h$,
and if $s/r > \alpha$
then $\tilde{f}$ contracts $L_{\infty} - \{ p_{\infty} \}$
to the attracting fixed point $[1:0:0]$.
On the other hand,
$p_{\infty}$ becomes an attracting fixed point if $\delta \leq d$.
If $\delta < d$,
then $\tilde{f}$ contracts $L_{\infty} - I_{\tilde{f}}$ to $p_{\infty}$.
If $\delta = d$,
then the dynamics on $L_{\infty}$ is determined by $h$.
Here $h$ is the weighted homogeneous part of $q$ of highest weight $sd/r$,
which coincides with $w^d$ if $s/r > \alpha$.
Via $\alpha$,
we obtain the results on the Green functions of $f$.
In the case $\delta > d$,
the dynamics of $\tilde{f}$ described above implies the following upper estimate.
\begin{theorem}[\cite{u-weight}] \label{}
Let $\gamma = 0$.
If $\delta > d$, then
$\tilde{G}_{z}^{\lambda} \leq \alpha G_p$
on $\mathbb{C}^{2}$, where
$\tilde{G}_{z}^{\lambda} = \limsup_{n \to \infty}$ $\delta^{-n} \log^{+}$ $|Q_z^n|$.
In particular, if $\alpha \leq 1$ then $G_f = G_p$ on $\mathbb{C}^{2}$.
\end{theorem}
Hence if $\delta > d$, then $G_f^{\alpha} = \alpha G_p$ on $\mathbb{C}^{2}$.
In the case $\delta \leq d$,
it follows from definition that $f \sim (z^{\delta}, w^d)$ on $W_R$
and that $f(W_R) \subset W_R$ for large $R > 0$,
where $W_R = \{ |w| > R|z|^{\alpha}, |w| > R^{\alpha + 1} \}$.
This implies that $G_z$ is defined, continuous and pluriharmonic on $W_R$.
Although we use the same notation $W_R$ in the sections below,
the definition of $W_R$ differs depending on whether $f$ is nondegenerate.
Let $A_f = \cup_{n \geq 0} f^{-n} (W_R)$.
\begin{theorem}[\cite{u-weight}]\label{}
Let $\gamma = 0$.
If $\delta \leq d$,
then $G_z$ is defined, continuous and pluriharmonic on $A_f$.
Moreover,
$G_z (w)$ tends
to $0$ if $\delta < d$ and
to $\alpha G_p (z)$ if $\delta = d$
as $(z,w)$ in $A_f$ tends to $\partial A_f$.
\end{theorem}
Hence if $\delta \leq d$,
then $G_f$ is defined, continuous and pluriharmonic on $A_f$.
Note that $A_f$ is the restriction of
the attracting basin of $p_{\infty}$ to $\mathbb{C}^{2}$.
If $\delta < d$,
then $G_z$ is defined, continuous and plurisubharmonic on $\mathbb{C}^{2}$
and it coincides with both $G_f$ and $G_f^{\alpha}$.
If $\delta = d$,
then $G_f^{\alpha}$ is defined, continuous and plurisubharmonic on $\mathbb{C}^{2}$;
roughly speaking, it is the maximum of $\alpha G_p$ and $G_z$.
Moreover,
$G_f^{\alpha}$ determines the Fatou and Julia sets of
the holomorphic map $\tilde{f}$ on $\mathbb{P} (r,s,1)$,
where $s/r \geq \alpha$.
Since two theorems above hold with suitable modifications for any $l \geq \alpha$,
we arrive at the following corollary.
\begin{cor}[\cite{u-weight}]
Let $\gamma = 0$.
For any $l \geq \alpha$, the limit $G_f^{l}$ is defined,
continuous and plurisubharmonic on $\mathbb{C}^{2}$.
\end{cor}
The $\alpha$ is optimal in the sense that
in this theorem $\alpha$ can not be replaced by any smaller number.
See \cite[Remark 2, Examples 5.2 and 5.3]{u-weight} for details.
\section{$\delta > d$}
We first study the case $\delta > d$ toward a generalization of
the results for nondegenerate polynomial skew products.
Assume that $f$ satisfies the condition $\delta > d$
throughout this section.
Besides showing the same upper estimate of $G_z$ as the nondegenerate case,
we also analyze the new phenomena that does not appear in the nondegenerate case.
The definition of $\alpha$ is the same,
which is positive unless $f$ is a polynomial product.
The map $f$ extends to the rational map $\tilde{f}$ on $\mathbb{P} (r,s,1)$,
which is algebraically stable if $s/r \geq \alpha$.
The dynamics of $\tilde{f}$ implies
the upper estimate of $G_z$ on $\mathbb{C}^2$,
which induces $G_f^{\alpha} = \alpha G_p$ on $\mathbb{C}^2$.
Moreover,
if $\alpha = \gamma / (\delta - d)$,
then the dynamics of $f$ is much more understandable;
we show the existence of $G_z^{\alpha}$ on $A_p \times \mathbb{C}$.
This function relates to the dynamics of $\tilde{f}$,
and induces the existence of $G_z$ and $G_f$
on some region in $A_p \times \mathbb{C}$.
The organization of this section is as follows.
In Section 4.1,
we state the definition of $\alpha$,
whose importance is illustrated
with the weighted homogeneous part of $q$ or $Q_z^n$,
and with the rational map $\tilde{f}$ on $\mathbb{P} (r,s,1)$.
Then the upper estimate of $G_z$ is described without a complete proof.
In addition,
we present two types of examples of polynomial skew products
whose dynamics is rather understandable.
The first type is monomial maps,
whose Green functions are completely described.
The second type is polynomial skew products
that are semiconjugate to polynomial products,
whose Green functions are well understood.
With the assumption $\alpha = \gamma /(\delta - d)$,
we prove the existence of $G_z^{\alpha}$ in Section 4.2
and the uniform convergence to $G_z^{\alpha}$ in Section 4.3,
which implies the asymptotics of $G_z^{\alpha}$ near infinity.
\subsection{Weights}
The definition of $\alpha$ is the same as in the nondegenerate case:
\[
\min \left\{ l \in \mathbb{Q} \ \Big|
\begin{array}{lcr}
l \delta \geq n_j + l m_j \text{ for any integers $n_j$ and $m_j$ s.t.} \\
z^{n_j} w^{m_j} \text{ is a term in } q \text{ with nonzero coefficient}
\end{array}
\right\},
\]
which is equal to
\[
\max \left\{ \dfrac{n_j}{\delta - m_j} \ \Big|
\begin{array}{lcr}
z^{n_j} w^{m_j} \text{ is a term in } q \\
\text{with nonzero coefficient}
\end{array}
\right\}.
\]
Clearly, $\alpha \geq 0$,
and $\alpha = 0$ if and only if $f$ is a polynomial product.
By definition,
$\alpha \leq \deg_z q$ and $\alpha < \deg q$.
Moreover,
$\delta^n \leq \deg (f^n) \leq \max \{ 1, \alpha \} \delta^n$
for any positive integer $n$.
We define the weight of a monomial $z^n w^m$ mainly as $n + \alpha m$.
As same as the nondegenerate case,
the weight of $q$ is $\alpha \delta$
and the weight of $Q_z^n(w)$ is $\alpha \delta^n$.
Let $h$ be the weighted homogeneous part of $q$
of highest weight $\alpha \delta$,
which contains $z^{\gamma} w^d$ if $\alpha = \gamma /(\delta - d)$
and does not if $\alpha > \gamma /(\delta - d)$.
To begin with,
let us consider the case where the rational number $\alpha$ is an integer.
Put $w = cz^{\alpha}$, then
$h(z, cz^{\alpha}) = h(1,c) z^{\alpha \delta}$.
Fix $c$ so that $q(z,cz^{\alpha})$ can be regarded as a polynomial in $z$.
By letting $h(c) = h(1,c)$, it follows from definition that
$h(c) z^{\alpha \delta}$ is the homogeneous part of
$q(z,cz^{\alpha})$ of degree $\alpha \delta$.
Moreover,
$h^n(c) z^{\alpha {\delta}^n}$ is the homogeneous part of
$Q_z^n (cz^{\alpha})$ of degree $\alpha {\delta}^n$.
Therefore,
$z^{\alpha {\delta}^n} h^n(z^{- \alpha} w)$ is the weighted homogeneous part
of $Q_z^n (w)$ of weight $\alpha \delta^n$.
If $\alpha$ is not an integer, then
$z^{\alpha}$ is not well defined
and $c$ is not uniquely determined by $z$ and $w$.
However, in that case,
the polynomial $h$ has some symmetries
related to the denominator of $\alpha$,
and these notations are still helpful.
The dynamics of $\tilde{f}$ on $\mathbb{P} (r,s,1)$ is also the same as the nondegenerate case.
The details are as follows.
If $s/r < \alpha$,
then $\tilde{f}$ contracts $L_{\infty} - I_{\tilde{f}}$
to the indeterminacy point $p_{\infty} = [0:1:0]$;
thus it is not algebraically stable.
On the other hand,
if $s/r \geq \alpha$, then $\tilde{f}$ is algebraically stable.
If $s/r = \alpha$,
then $\tilde{f} [z:w:t] = [z^{\delta} + tu(z,t): h(z,w) + tv(z,w,t): t^{\delta}]$,
where $u$ and $v$ are polynomials.
Since $h$ is divisible by $z$,
it follows that $I_{\tilde{f}} = \{ p_{\infty} \}$.
Since $\tilde{f} [z:w:0] = [z^{\delta}: h(z,w): 0]$,
the dynamics of $\tilde{f}$ on $L_{\infty} - \{ p_{\infty} \}$ is induced
by the dynamics of $h(1,w)$.
If $s/r > \alpha$,
then $\tilde{f} [z:w:t] = [z^{\delta} + tu(z,t): tv(z,w,t): t^{\delta}]$.
Hence $I_{\tilde{f}} = \{ p_{\infty} \}$ and
$\tilde{f}$ contracts $L_{\infty} - I_{\tilde{f}}$ to the attracting fixed point $[1:0:0]$.
Because the fixed point $[1:0:0]$ is attracting in a very strong sense as above,
we obtain the upper estimate
$\tilde{G}_z^{\lambda} \leq l G_p$ on $A_p \times \mathbb{C}$
for any $l = s/r > \alpha$, where $\tilde{G}_z^{\lambda}
= \limsup_{n \to \infty} \delta^{-n} \log^{+} |Q_z^n|$.
Therefore,
\begin{pro}
If $\delta > d$, then
$\tilde{G}_z^{\lambda} \leq \alpha G_p$ on $A_p \times \mathbb{C}$.
\end{pro}
One can also prove this proposition along the same line
as the proof of \cite[Theorem 3.2]{u-weight}.
Hence if $\delta > d$, then
$G_p \leq \tilde{G}_f \leq \max \{ \alpha, 1 \} G_p$
on $A_p \times \mathbb{C}$,
where $\tilde{G}_f
= \limsup_{n \to \infty} \delta^{-n} \log^{+} |f^n|$.
In particular, if $\alpha \leq 1$,
then $G_f = G_p$ on $\mathbb{C}^2$.
In addition,
the existence of $G_z$ on $K_p \times \mathbb{C}$ implies that
$G_f^{\alpha} = G_f = G_z^{\lambda} = 0$ on $K_p \times \mathbb{C}$,
since $\lambda = \delta > d$.
Consequently,
the proposition above implies the following corollary.
\begin{cor}\label{delta > d; main cor}
If $\delta > d$, then
$G_f^{\alpha} = \alpha G_p$ on $\mathbb{C}^2$.
\end{cor}
We end this subsection with two examples of polynomial skew products
whose dynamics are well understood:
monomial maps, and skew products that are semiconjugate to polynomial products.
\begin{ex}[monomial maps]
Let $f = (z^{\delta}, z^{\gamma} w^d)$, $\delta > d$ and $\gamma \neq 0$.
Then $\alpha = \gamma /(\delta - d) > 0$ and
$f^n = (z^{\delta^n}, z^{\gamma_n} w^{d^n})$,
where
\[
\gamma_n = (\delta^{n - 1} + \delta^{n - 2} d + \cdots + d^{n - 1}) \gamma
= \alpha {\delta}^n \left\{ 1 - \left( \dfrac{d}{\delta} \right)^n \right\}.
\]
Hence $G_z$ is $\infty$ on $\{ |z| > 1, w \neq 0 \}$,
$\log^{+} |w|$ on $\{ |z| = 1 \}$, and
$0$ on $\{ |z| < 1 \} \cup \{ w = 0 \}$.
Since $G_p = \log^{+} |z|$,
\[
G_z^{\lambda} =
\begin{cases}
\alpha \log^{+} |z| & \text{ on } \{ w \neq 0 \} \\
0 & \text{ on } \{ w = 0 \},
\end{cases}
\]
which is not continuous on $\{ |z| \geq 1, w = 0 \}$, and
\[
G_f =
\begin{cases}
\max \left\{ \alpha, 1 \right\} \log^{+} |z| & \text{ on } \{ w \neq 0 \} \\
\log^{+} |z| & \text{ on } \{ w = 0 \},
\end{cases}
\]
which is continuous on $\mathbb{C}^2$ if $\alpha \leq 1$
and is not on $\{ |z| \geq 1, w = 0 \}$ if $\alpha > 1$.
Therefore, $G_f^{\alpha} = \alpha \log^{+} |z|$ on $\mathbb{C}^2$,
which is continuous on $\mathbb{C}^2$.
The limits $G_z^{\lambda}$, $G_f$ and $G_f^{\alpha}$
are all plurisubharmonic on $\mathbb{C}^2$.
\end{ex}
\begin{ex}\label{delta > d; ex2}
Let $f = (z^{\delta}, q(z,w))$ be a polynomial skew product,
where $q = z^{\gamma} w^d + O_z(w^{d-1})$,
$\gamma \neq 0$ and $\delta > d$,
that is semiconjugate to $f_0 = (z^{\delta}, h(w))$
by $\pi = (z^r,z^s w)$ for some positive integers $r$ and $s$;
$f \pi = \pi f_0$.
Note that $h(w) = q(1,w)$ and so the degree of $h$ is $d$.
The identity $q(z^r,z^s w) = z^{s \delta} q(1,w)$ implies
that $\alpha = s/r > 0$ and
$q(z,w) = z^{\alpha \delta} h(w/z^{\alpha})$.
Moreover,
$f^n = (z^{\delta^n}, z^{\alpha \delta^n} h^n (w/z^{\alpha}))$.
Hence
$w_n/z_n^{\alpha} = h^n (w/z^{\alpha})$ and so
\[
G_z^{\alpha} (w) =
\lim_{n \to \infty} \frac{1}{d^n}
\log^{+} \left| \dfrac{w_n}{z_n^{\alpha}} \right|
= G_h \left( \dfrac{w}{z^{\alpha}} \right)
\text{ on } \mathbb{C}^{2} - \{ z = 0 \},
\]
where $(z_n,w_n) = f^n(z,w)$.
Define
\[
E_f = \bigcup_{|z| > 1} \{ z \} \times z^{\alpha} E_h
\text{ and }
E_h = \bigcap_{l \geq 0} \overline{ \bigcup_{n \geq l} h^{-n} (0) }
\]
as in \cite{u-semiconj}.
Then
$G_z^{\lambda} = \alpha \log^{+} |z|$ on $\mathbb{C}^{2} - E_f$,
which implies that
$G_f = \max \{ \alpha, 1 \} \log^{+} |z|$ on $\mathbb{C}^{2} - E_f$
and that
$G_f^{\alpha} = \alpha \log^{+} |z|$ on $\mathbb{C}^{2}$.
If $0 \not\in E_h$ then the equalities above extend to $\mathbb{C}^{2}$,
and if $\alpha \leq 1$ then $G_f = \log^{+} |z|$ on $\mathbb{C}^{2}$.
\end{ex}
The claims in this example follow from the same line as in \cite{u-semiconj}.
These maps can be characterized by the symmetries of the Julia sets,
see \cite[Theorems 5.2 and 5.5]{u-sym2} for details.
\subsection{Existence of Green functions: $\alpha = \gamma/(\delta - d)$}
We saw that,
for the special map $f$ in Example \ref{delta > d; ex2},
the limit $G_z^{\alpha}$ is defined,
continuous and plurisubharmonic on $\mathbb{C}^{2} - \{ z = 0 \}$.
In this subsection,
assuming that $\alpha = \gamma/(\delta - d)$,
we derive the existence, continuity and plurisubharmonicity
of $G_z^{\alpha}$ on $A_p \times \mathbb{C}$.
We also assume that $f$ is not a polynomial product
for simplicity,
which implies that $\alpha > 0$ and $\gamma \neq 0$.
Let $W_R = \{ |z| > R, |w| > R|z|^{\alpha} \}$ for large $R > 0$.
Note that the definition of $W_R$ differs from the nondegenerate case.
We often use the new variety $c = z^{- \alpha} w$.
Although $c$ depends on the choice of the branch of $z^{- \alpha}$,
$|c|$ does not and $W_R = \{ |z| > R, |c| > R \}$.
The following important lemma follows from
the definition of $\alpha$.
\begin{lem}\label{delta > d: main lem}
If $\delta > d$ and $\alpha = \gamma/(\delta - d)$, then
\[
\left| \dfrac{q(z,w)}{p(z)^{\alpha}} \right|
\sim \left| \dfrac{w}{z^{\alpha}} \right|^d
\text{ on } W_R
\]
and $f$ preserves $W_R$; that is, $f(W_R) \subset W_R$.
\end{lem}
\begin{proof}
We explain this claim by using the notation $|c| = |z^{- \alpha} w|$.
Let $z^{n_j} w^{m_j}$ be a term of $q$ with nonzero coefficient.
If $z^{n_j} w^{m_j} \neq z^{\gamma} w^d$, then
\[
\left| \dfrac{z^{n_j} w^{m_j}}{c^d z^{\alpha \delta}} \right|
= \left| \dfrac{c^{m_j} z^{n_j + \alpha m_j}}{c^d z^{\alpha \delta}} \right|
\to 0 \text{ as $z$ and $c$} \to \infty
\]
since at least one of the inequalities
$d > m_j$ and $\alpha \delta > n_j + \alpha m_j$ holds.
Thus
\[
\left| \dfrac{q(z,w)}{z^{\alpha \delta}} \right|
= \left| \dfrac{q(z,cz^{\alpha})}{z^{\alpha \delta}} \right|
= |c|^d \{ 1 + o(1) \}
= \left| \dfrac{w}{z^{\alpha}} \right|^d \{ 1 + o(1) \}.
\]
Hence there exist positive constants $r_1 < 1 < r_2$ such that
\begin{equation}\label{delta > d: eq1}
r_1 \left| \dfrac{w}{z^{\alpha}} \right|^d
< \left| \dfrac{q(z,w)}{p(z)^{\alpha}} \right|
< r_2 \left| \dfrac{w}{z^{\alpha}} \right|^d
\end{equation}
on $W_R$ for large $R$,
since $p(z) \sim z^{\delta}$ as $z \to \infty$.
Let us show that $f$ preserves $W_R$.
It is clear that if $R$ is large enough,
then $|p(z)| > R$
since $p(z) \sim z^{\delta}$ as $z \to \infty$.
Hence it is enough to show that
$|q(z,w)| > R|p(z)|^{\alpha}$ on $W_R$ for large $R$,
which follows from inequality $(\ref{delta > d: eq1})$
since $d \geq 2$.
\end{proof}
\begin{rem}
We can also show the following
as Lemmas \ref{delta < d: main lem} and \ref{delta = d: main lem}
with a slight change of the proof:
if $\delta > d$ and $\alpha = \gamma/(\delta - d)$, then
$q(z,w) \sim z^{\gamma} w^d$ on $W_R$ for large $R > 0$.
\end{rem}
Let $\tilde{f}$ be the rational extension to $\mathbb{P} (r,s,1)$,
where $s/r = \alpha$.
Recall that the dynamics of $\tilde{f}$
restricted to $L_{\infty} - \{ p_{\infty} \}$
is induced by the polynomial $h(1,w)$ of degree $d$.
Hence $p_{\infty}$ attracts most nearby points in $A_p \times \mathbb{C}$,
and $W_R$ is included in the attracting basin of $p_{\infty}$.
Therefore,
the inclusion $f(W_R) \subset W_R$ is natural.
Let $A_f = \cup_{n \geq 0} f^{-n} (W_R)$,
which is the restriction of the attracting basin of $p_{\infty}$
to $A_p \times \mathbb{C}$.
\begin{theorem}\label{delta > d: main thm}
If $\delta > d$ and $\alpha = \gamma /(\delta - d)$, then
the limit $G_z^{\alpha}$ is defined,
continuous and pluriharmonic on $A_f$.
Moreover,
$G_z^{\alpha} = \log |z^{-\alpha} w| + o(1)$ on $W_R$,
$G_z^{\alpha} \sim \log |w|$ as $w \to \infty$ for fixed $z$ in $A_p$,
and $G_z^{\alpha}$ tends to $0$ as $(z,w)$ in $A_f$
tends to $\partial A_f - J_p \times \mathbb{C}$.
\end{theorem}
\begin{proof}
First,
we prove the uniform convergence of $G_n$ to $G_z^{\alpha}$ on $W_R$,
where $G_n(z,w) = d^{-n} \log | z_n^{- \alpha} w_n |$
and $(z_n,w_n) = f^n (z,w)$.
It follows from inequality $(\ref{delta > d: eq1})$
in the proof of Lemma {\rmfamily \ref{delta > d: main lem}} that,
for any $(z,w)$ in $W_R$,
\[
r_1 \left| \dfrac{w_n}{z_n^{\alpha}} \right|^d
< \left| \dfrac{w_{n+1}}{z_{n+1}^{\alpha}} \right|
< r_2 \left| \dfrac{w_n}{z_n^{\alpha}} \right|^d.
\]
Hence, for any $(z,w)$ in $W_R$ and for any positive integer $n$,
\[
\left| G_{n+1} (z,w) - G_n (z,w) \right|
= \left| \dfrac{1}{d^{n+1}} \log \left| \dfrac{w_{n+1}}{z_{n+1}^{\alpha}} \right|
\cdot \left| \dfrac{w_{n}}{z_{n}^{\alpha}} \right|^{-d} \right|
< \dfrac{\log r}{d^{n+1}},
\]
where $\log r = \max \{ - \log r_1, \log r_2 \}$.
Therefore,
$G_n$ converges uniformly to $G_z^{\alpha}$ on $W_R$.
Since $G_n$ is continuous and pluriharmonic on $W_R$,
the limit $G_z^{\alpha}$ is also continuous and pluriharmonic on $W_R$.
By the inequality above, for any $(z,w)$ in $W_R$,
\begin{eqnarray}\label{delta > d: eq2}
\begin{split}
\left| G_z^{\alpha}(w) - \log \left| \dfrac{w}{z^{\alpha}} \right| \right|
\leq & \sum_{n=0}^{\infty} \left| G_{n+1} (z,w) - G_n (z,w) \right| \\
< & \sum_{n=0}^{\infty} \dfrac{\log r}{d^{n+1}} = \dfrac{\log r}{d-1} =: C_R.
\end{split}
\end{eqnarray}
In particular,
$G_z^{\alpha} \sim \log |w|$ as $w \to \infty$ for fixed $z$ in $W_R$.
We can extend the domain of $G_z^{\alpha}$ from $W_R$ to $A_f$.
Indeed, for any $(z,w)$ in $A_f$,
there exists a positive integer $n$
such that $f^n (z,w)$ belongs to $W_R$.
Then we define $G_z^{\alpha}(w)$ as $d^{-n} G_{p^n(z)}^{\alpha}(Q_z^n(w))$.
Clearly, $G_z^{\alpha}$ is continuous and pluriharmonic on $A_f$,
and $G_z^{\alpha} \sim \log |w|$ as $w \to \infty$ for fixed $z$ in $A_p$.
Next, we use inequality $(\ref{delta > d: eq2})$
to calculate the asymptotic value of $G_z^{\alpha}(w)$ as $(z,w)$ in $A_f$
tends to $\partial A_f - J_p \times \mathbb{C}$.
Let $E = \{ |w| = R|z|^{\alpha}, |z| > R \} \subset \partial W_R$.
It then follows from inequality $(\ref{delta > d: eq2})$ that
\[
\left| \ G_{p^n(z)}^{\alpha}(Q_z^n(w)) - \log R \ \right|
\leq C_R \ \text{ on } f^{-n} (E),
\]
since $f^n (z,w)$ belongs to $E$
for any $(z,w)$ in $f^{-n} (E)$.
Thus it follows from equation
$G_{p^n(z)}^{\alpha}(Q_z^n(w)) = d^n G_z^{\alpha}(w)$
that
\begin{equation}\label{delta > d: eq3}
\left| \ G_z^{\alpha}(w) - d^{-n} \log R \ \right| \leq d^{-n} C_R
\ \text{ on } f^{-n} (E).
\end{equation}
Since $f^{-n} (E)$ converges to
$\partial A_f - J_p \times \mathbb{C}$ as $n$ tends to infinity,
$G_z^{\alpha}(w)$ converges to $0$
as $(z,w)$ in $A_f$ tends to $\partial A_f - J_p \times \mathbb{C}$.
\end{proof}
We remark that the set $\partial A_f - J_p \times \mathbb{C}$
in Theorem {\rmfamily \ref{delta > d: main thm}}
can be replaced by its closure.
Let $B_f = A_p \times \mathbb{C} - A_f$.
Since $G_z^{\alpha} = 0$ on $B_f$,
we get the following corollary.
\begin{cor}
If $\delta > d$ and $\alpha = \gamma /(\delta - d)$, then
the limit $G_z^{\alpha}$ is defined,
continuous and plurisubharmonic on $A_p \times \mathbb{C}$.
Moreover,
it is pluriharmonic on $A_f$ and int$B_f$.
\end{cor}
Here are some properties of $A_f$ and $B_f$.
First, $A_f$ and $B_f$ are invariant under $f$;
that is, $f(A_f) \subset A_f = f^{-1} (A_f)$
and $f(B_f) \subset B_f = f^{-1} (B_f)$.
Theorem {\rmfamily \ref{delta > d: main thm}}
guarantees that the set
$\{ \{ z \} \times \mathbb{C} : \deg_w q_z = 0 \}$ of degenerate fibers
does not intersect with $A_f$.
Hence $B_f$ is not empty; more precisely,
$B_f \cap (\{ z \} \times \mathbb{C}) \neq \emptyset$
for any $z$ in $A_p$.
The existence of $G_z^{\alpha}$ implies
the following two corollaries on the existence of other Green functions.
\begin{cor}
If $\delta > d$ and $\alpha = \gamma /(\delta - d)$, then
$G_z = \infty$ on $A_f$.
\end{cor}
We note that there exist polynomial skew products such that
$G_z = \infty$ on $A_p \times \mathbb{C}$.
Indeed,
let $f = (z^{\delta}, q(z,w))$ be a polynomial skew product
that is semiconjugate to $(z^{\delta}, h(w))$
by $\pi = (z^r, z^s w)$.
If $0 \in A_h$,
then $G_z = \infty$ on $\{ |z| > 1 \}$.
\begin{cor}
If $\delta > d$ and $\alpha = \gamma /(\delta - d)$, then
$G_z^{\lambda} = \alpha G_p$ and
$G_f = \max \{ \alpha, 1 \} G_p$ on $\mathbb{C}^2 - B_f$.
\end{cor}
In particular,
if $\delta > d$ and $\alpha = \gamma /(\delta - d)$,
then $G_z^{\lambda}$ exists on $A_f$.
We can insist on the optimality of $\alpha$ and $A_f$
as in \cite[Remark 2, Examples 5.2 and 5.3]{u-weight},
using polynomial skew products
that are semiconjugate to polynomial products.
We end this subsection with a description of the dynamics of
$\tilde{f}$ on $\mathbb{P} (r,s,1)$,
where $s/r = \alpha$.
Let $U = \text{int} \overline{A_p \times \mathbb{C}}$,
where the interior and closure are taken in $\mathbb{P} (r,s,1)$.
Then $U = (A_p \times \mathbb{C}) \cup (L_{\infty} - \{ p_{\infty} \})$.
Let $A_{\tilde{f}}$ be the restriction of
the attracting basin of $p_{\infty}$ to $U$.
It is equal to the union of preimages
$\tilde{f}^{-n} (\text{int} \overline{W_R})$ and hence open.
Moreover,
$A_{\tilde{f}} = A_f \cup A_h$,
where $A_h$ denotes the set of points in $L_{\infty} - \{ p_{\infty} \}$
whose orbits converge to $p_{\infty}$.
Since $A_{\tilde{f}}$ is included in the attracting basin of $p_{\infty}$,
it follows that $A_{\tilde{f}} \subset F_{\tilde{f}}$.
Let $B_{\tilde{f}} = U - A_{\tilde{f}}$.
Then $B_{\tilde{f}} = B_f \cup K_h$,
where $K_h$ denotes the set of points in $L_{\infty} - \{ p_{\infty} \}$
whose orbits do not converge to $p_{\infty}$.
Since int$B_{\tilde{f}} \cap \text{int} \overline{\{ |z| > R \}}$ is Kobayashi hyperbolic
and preserved by $\tilde{f}$,
it follows int$B_{\tilde{f}} \subset F_{\tilde{f}}$
(see \cite{u-weight} for details).
\begin{pro}\label{delta > d: Fatou and Julia}
Let $\delta > d$ and $\alpha = \gamma/(\delta - d)$.
The restriction of $F_{\tilde{f}}$ to $U$
consists of $A_{\tilde{f}}$ and int$B_{\tilde{f}}$.
The restriction of $J_{\tilde{f}}$ to $U$
is equal to the restriction of $\partial A_{\tilde{f}}$ to $U$,
and to the restriction of $\partial B_{\tilde{f}}$ to $U$.
Moreover, it coincides with the restriction of the closure of
$\{ (z,w) \in A_p \times \mathbb{C} :
G_z^{\alpha} \text{ is not pluriharmonic} \}$ to $U$.
\end{pro}
The dynamics of $\tilde{f}$ on $B_{\tilde{f}}$ is as follows.
Any point in $B_{\tilde{f}}$ is attracted to $L_{\infty}$ under iteration.
Eventually, the dynamics on $L_{\infty}$,
which is induced by $h$,
should determine the dynamics of $\tilde{f}$ on $B_{\tilde{f}}$.
It follows that $h(c) := h(1,c)$ can be written as $c^{l} H(c^r)$
for some integer $l \geq 0$ and some polynomial $H$,
which is semiconjugate to $c^{l} H(c)^r$ by $c^r$.
The restriction of $\tilde{f}$ to $L_{\infty}$ is conjugate to $c^{l} H(c)^r$.
\subsection{Uniformly convergence and Asymptotics: $\alpha = \gamma/(\delta - d)$}
Continuing with the assumption $\alpha = \gamma/(\delta - d)$,
we show that the convergence to $G_z^{\alpha}$ is uniform
on some region in $A_p \times \mathbb{C}$,
which induces the asymptotic of $G_z^{\alpha}$ near infinity.
\begin{pro}\label{delta > d: unif}
If $\delta > d$ and $\alpha = \gamma /(\delta - d)$, then
the convergence to $G_z^{\alpha}$ is uniform on $V \times \mathbb{C}$,
where $\overline{V} \subset A_p$.
\end{pro}
\begin{proof}
We show that, for any $\epsilon > 0$ and
for any subset $V$ such that $\overline{V} \subset A_p$,
there exists $N$ such that
$|G_n - G_z^{\alpha}| < \epsilon$ on $V \times \mathbb{C}$ for any $n \geq N$,
where $G_n(z,w) = d^{-n} \log^{+} | z_n^{- \alpha} w_n |$
and $(z_n,w_n) = f^n(z,w)$.
Let $U = \{ z \in V, G_z^{\alpha} \geq \epsilon /3 \}$.
It is enough to show the uniform convergence of $G_n$ to $G_z^{\alpha}$ on $U$.
Indeed, if this holds,
then there exists $N$ such that
$|G_n - G_z^{\alpha}| < \epsilon /3$ on $U$ for any $n \geq N$.
Hence $G_n < 2 \epsilon /3$ on $\partial U$ for any $n \geq N$.
By Maximum Principle for subharmonic functions on vertical lines,
$G_n < 2 \epsilon /3$
and so $| G_n - G_z^{\alpha} | < \epsilon$ on $U^c$ for any $n \geq N$,
which completes the proof.
Now we show the uniform convergence on $U$.
The equation $G_n \circ f = d G_{n+1}$ extends
the uniform convergent region from $W_R$ to $f^{-n} (W_R)$.
Hence it is enough to show that $U \subset f^{-n} (W_R)$ for large $n$.
From equation $(\ref{delta > d: eq3})$ in the proof of
Theorem {\rmfamily \ref{delta > d: main thm}},
it follows that $G_z^{\alpha} < \epsilon /3$ on $f^{-n} (E)$ for large $n$,
which implies that $U \subset f^{-n} (W_R)$.
\end{proof}
We saw that, for fixed $c = z^{- \alpha} w$,
the polynomial
$h^n(c) z^{\alpha d^n}$ is the homogeneous part of
$Q_z^n (cz^{\alpha})$ of degree $\alpha \delta^n$.
Although $c$ is not well defined if $\alpha$ is not an integer,
the polynomial $h$ and the Green function $G_h$
have some symmetries related to the denominator $r$ of $\alpha$
in that case:
$h(c)$ can be written as $c^{l} H(c^r)$,
the Julia set $J_h$ is preserved by the rotation $\rho c$,
where $\rho$ is a $r$-th root of $1$,
and $G_h (c) = G_h (z^{- \alpha} w)$ is a well defined function in $z$ and $w$.
Hence we get the following asymptotics of $G_z^{\alpha}$ near infinity.
\begin{lem}\label{delta > d: asy lem}
If $\delta > d$ and $\alpha = \gamma /(\delta - d)$, then
$G_z^{\alpha}(cz^{\alpha}) = G_h(c) + o(1)$ as $z \to \infty$
for fixed $c$.
\end{lem}
\begin{proof}
We prove that for any $\epsilon > 0$ there exists $R > 0$
such that
\begin{eqnarray}\label{delta > d: eq4}
|G_z^{\alpha}(cz^{\alpha}) - G_h(c)| < \epsilon
\end{eqnarray}
on $\{ |z| > R \}$.
Assume the rational number $\alpha$ is an integer.
Since $Q_z^n(cz^{\alpha}) = h^n(c)z^{\alpha {\delta}^n} + o(z^{\alpha {\delta}^n})$,
it follows that
\[
\frac{1}{d^n} \log^{+} \left| \dfrac{Q_z^n(cz^{\alpha})}{z^{\alpha {\delta}^n}} \right|
= \frac{1}{d^n} \log^{+} |h^n(c)| + o(1).
\]
Since $p(z) \sim z^{\delta}$ as $z \to \infty$,
\begin{eqnarray}\label{delta > d: eq5}
\frac{1}{d^n} \log^{+}
\left| \dfrac{Q_z^n(cz^{\alpha})}{p^n(z)^{\alpha}} \right|
= \frac{1}{d^n} \log^{+} |h^n(c)| + o(1).
\end{eqnarray}
By Proposition {\rmfamily \ref{delta > d: unif}},
there exist an integer $N_1$ and a number $R > 0$ such that
$|G_z^{\alpha}(w) - G_n(z,w)| < \epsilon /3$ on $\{ |z| > R \}$
for any $n \geq N_1$.
Since the convergence of $G_h^n = d^{-n} \log^{+} |h^n|$
to $G_h$ is uniform on $\mathbb{C}$,
there exists $N_2$ such that
$|G_h(c) - G_h^n(c)| < \epsilon /3$ for any $n \geq N_2$.
From equation $(\ref{delta > d: eq5})$
it follows that if $R$ is large enough, then
$|G_N(z,w) - G_h^N(c)| < \epsilon /3$ on $\{ |z| > R \}$,
where $N = \max \{ N_1, N_2 \}$.
Consequently, inequality $(\ref{delta > d: eq4})$ holds on $\{ |z| > R \}$.
Even if $\alpha$ is not an integer, it follows that
$|Q_z^n(cz^{\alpha})| =
|h^n(c)||z|^{\alpha {\delta}^n} + o(|z|^{\alpha {\delta}^n})$.
A similar argument as above implies inequality $(\ref{delta > d: eq4})$.
\end{proof}
\begin{pro}\label{delta > d: asy pro}
If $\delta > d$ and $\alpha = \gamma /(\delta - d)$, then
$G_z^{\alpha}(w) = G_h \left( z^{- \alpha} w \right) + o(1)$
as $z \to \infty$.
\end{pro}
\begin{proof}
We prove that for any $\epsilon > 0$ there exists $R>0$
such that
\begin{eqnarray}\label{delta > d: eq6}
| G_z^{\alpha}(w) - G_h(z^{- \alpha} w) | < \epsilon
\end{eqnarray}
on $\{ |z| > R \}$.
Separating the region $\{ |z| > R \}$
into two open sets whose union contains the region,
we get this inequality.
First,
we show that this inequality holds on $W_R$ for large $R$,
where $W_R = \{ |z| > R, |w| > R |z|^{\alpha} \}$.
Inequality $(\ref{delta > d: eq2})$ in the proof of
Theorem \ref{delta > d: main thm} implies that
$|G_z^{\alpha}(w) - \log|z^{- \alpha} w|| < \epsilon /2$
on $W_R$ for large $R$.
Since
$| \log|z^{- \alpha} w| - G_h(z^{- \alpha} w) | < \epsilon /2$
on $W_R$ for large $R$,
there exists $R_1 > 0$ such that
inequality $(\ref{delta > d: eq6})$ holds on $W_{R_1}$.
Next,
let $V_R = \{ |z| > R, |w| < 2 R_1 |z|^{\alpha} \}$.
From a similar argument as the proof of
Lemma \ref{delta > d: asy lem},
it follows that
$G_z^{\alpha}(w) = G_h (z^{- \alpha} w) + o(1) \text{ on } V_R$,
since the projection of $V_R$ to $\mathbb{C}$
by the multi-valued function $z^{- \alpha} w$
is a relatively compact subset of $\mathbb{C}$.
Hence there exists $R_2 \geq R_1$ such that
inequality $(\ref{delta > d: eq6})$ holds on $V_{R_2}$.
Since $W_{R_2}$ and $V_{R_2}$ cover
the region $\{ |z| > R_2 \}$,
inequality $(\ref{delta > d: eq6})$ holds on
$\{ |z| > R_2 \}$.
\end{proof}
\section{$\delta < d$}
Next we study the case $\delta < d$,
assuming that $\gamma \neq 0$.
We generalize the definition of $\alpha$,
and prove the existence
of $G_z$ on $\mathbb{C}^2$ as the nondegenerate case,
which is continuous and plurisubharmonic on $A_p \times \mathbb{C}$.
Moreover,
$G_z = G_f = G_f^{\alpha}$ on $\mathbb{C}^2$
and $G_z = G_z^{\alpha}$ on $A_p \times \mathbb{C}$.
Unlike the nondegenerate case,
$\alpha$ can be negative and
$\tilde{f}$ is not algebraically stable unless $f$ is nondegenerate.
However, the indeterminacy point $p_{\infty}$
is still attracting in some sense,
and it follows that $f \sim (z^{\delta}, z^{\gamma} w^d)$
on a region in the attracting basin of $p_{\infty}$,
which induces the results above.
To obtain uniform convergence to $G_z$,
we have to consider unusual plurisubharmonic functions
that converge to $G_z$.
In particular,
if $\alpha = \gamma /(\delta - d)$,
then the convergence to $G_z^{\alpha}$ is uniform
on a region in $A_p \times \mathbb{C}$,
and we get the asymptotics of $G_z^{\alpha}$ near infinity.
This section is divided into three subsections:
the definition and the properties of $\alpha$,
the existence of $G_z$ and other Green functions,
and the uniform convergence to $G_z$.
We remark that many claims hold even if $\gamma = 0$.
\subsection{Weights}
We generalize the definition of $\alpha$ as
\[
\min \left\{ l \in \mathbb{Q} \ \Big|
\begin{array}{lcr}
\gamma + ld \geq l \delta \text{ and } \gamma + ld \geq n_j + l m_j \text{ for} \\
\text{any integers $n_j$ and $m_j$ s.t. } z^{n_j} w^{m_j} \text{ is} \\
\text{a term in $q$ with nonzero coefficient}
\end{array}
\right\},
\]
which is equal to
\[
\max \left\{ \dfrac{-\gamma}{d - \delta},
\dfrac{n_j - \gamma}{d - m_j} \ \Big|
\begin{array}{lcr}
z^{n_j} w^{m_j} \text{ is a term in } q \text{ with} \\
\text{nonzero coefficient s.t. } m_j < d
\end{array}
\right\}.
\]
By definition,
$- \gamma \leq \alpha \leq \deg_z q - \gamma$
and $\alpha < \deg q - \gamma$.
Moreover, $\alpha$ induces inequalities for the degree growth of $f$,
given in Section 7.
Let the weight of $z^n w^m$ be $n + \alpha m$.
Then the weight of $q$ is $\gamma + \alpha d$ and
the weight of $Q_z^n(w)$ is $\gamma_n + \alpha d^n$,
where $\gamma_n =(\delta^{n - 1} + \delta^{n - 2} d + \cdots + d^{n - 1}) \gamma$,
which coincides with $\alpha \delta^n$ if $\alpha = \gamma /(\delta - d)$.
Let $h$ be the weighted homogeneous part of $q$
of highest weight $\gamma + \alpha d$,
which contains $z^{\delta} w^d$.
If $\alpha = \gamma /(\delta - d)$, then
$z^{\alpha {\delta}^n} h^n(z^{- \alpha} w)$ is the weighted homogeneous part of
$Q_z^n (w)$ of weight $\alpha {\delta}^n$.
If $\alpha > \gamma /(\delta - d)$, then
$z^{\gamma_n} (z^{- \gamma} h(z,w))^{d^{n-1}}$ is the weighted homogeneous part of
$Q_z^n (w)$ of weight $\gamma_n + \alpha d^n$.
The rational map $\tilde{f}$ on $\mathbb{P} (r,s,1)$ is not algebraically stable
for any positive integers $r$ and $s$;
it contracts $L_{\infty} - I_{\tilde{f}}$
to the indeterminacy point $p_{\infty}$.
However,
$p_{\infty}$ attracts most points in $A_p \times \mathbb{C}$;
more precisely, it attracts all points
which converge to $(L_{\infty} - I_{\tilde{f}}) \cup \{ p_{\infty} \}$.
As in Section 4.1,
we end this subsection with two examples:
monomial maps, and polynomial skew products
that are semiconjugate to polynomial products.
\begin{ex}[monomial maps]
Let $f = (z^{\delta}, z^{\gamma} w^d)$ and $\delta < d$.
Then $\alpha = \gamma /(\delta - d) \leq 0$
and $f^n = (z^{\delta^n}, z^{\gamma_n} w^{d^n})$,
where
\[
\gamma_n = (\delta^{n - 1} + \delta^{n - 2} d + \cdots + d^{n - 1}) \gamma
= - \alpha d^n \left\{ 1 - \left( \dfrac{\delta}{d} \right)^n \right\}
\]
Hence $G_z = G_z^{\alpha} = G_f = G_f^{\alpha} = \log^{+} |z^{- \alpha} w|$,
which is continuous and plurisubharmonic on $\mathbb{C}^{2}$.
\end{ex}
\begin{ex}
Let $f = (z^{\delta}, q(z,w))$ be a polynomial skew product,
where $q = z^{\gamma} w^d + O_z(w^{d-1})$, $\gamma \neq 0$ and $\delta < d$,
that is semiconjugate to a polynomial product $f_0 = (z^{\delta}, h(w))$
by $\pi = (z^r, z^{-s} w)$ for some positive integers $r$ and $s$;
$f \pi = \pi f_0$.
Note that $h(w) = q(1,w)$,
which is divisible by $w$.
It follows from the identity $q(z^r, z^{-s} w) = z^{-s \delta} q(1,w)$
that $\alpha = - s/r < 0$ and
$q(z,w) = z^{- \alpha \delta} h(z^{- \alpha} w)$.
Moreover,
$f^n = ( z^{\delta^n}, z^{- \alpha \delta^n} h^n (z^{- \alpha} w))$.
Hence $z_n^{- \alpha} w_n = h^n (z^{- \alpha} w)$ and so
\[
G_z^{\alpha}(w) =
\lim_{n \to \infty} \frac{1}{d^n}
\log^{+} \left| z_n^{- \alpha} w_n \right|
= G_h \left( z^{- \alpha} w \right)
\text{ on } \mathbb{C}^{2},
\]
where $(z_n, w_n) = f^n (z, w)$.
Moreover,
$G_z = G_z^{\alpha} = G_f = G_f^{\alpha}$,
which is continuous and plurisubharmonic on $\mathbb{C}^{2}$.
\end{ex}
\subsection{Existence of Green functions}
We defined $W_R$
as $\{ |z| > R, |w| > R|z|^{\alpha} \}$ if $\gamma \neq 0$.
The following important lemma follows from
the definition of $\alpha$.
\begin{lem}\label{delta < d: main lem}
If $\delta < d$, then
$q(z,w) \sim z^{\gamma} w^d$ on $W_R$ for large $R > 0$,
and $f$ preserves $W_R$; that is, $f(W_R) \subset W_R$.
\end{lem}
\begin{proof}
We explain this claim by using the notation $|c| = |z^{- \alpha} w|$.
Let $z^{n_j} w^{m_j}$ be a term of $q$ with nonzero coefficient.
If $z^{n_j} w^{m_j} \neq z^{\gamma} w^d$, then
\[
\left| \dfrac{z^{n_j} w^{m_j}}{z^{\gamma} w^d} \right|
= \left| \dfrac{c^{m_j} z^{n_j + \alpha m_j}}{c^d z^{\gamma + \alpha d}} \right|
\to 0 \text{ as $z$ and $c$} \to \infty
\]
since at least one of the inequalities
$d > m_j$ and $\gamma + \alpha d > n_j + \alpha m_j$ holds.
Therefore, $q(z,w) \sim z^{\gamma} w^d$ on $W_R$.
In other word,
\begin{equation}\label{delta < d: eq1}
r_1 < \left| \dfrac{q(z,w)}{z^{\gamma} w^d} \right| < r_2
\end{equation}
on $W_R$ for some positive constants $r_1 < 1 < r_2$.
Let us show that $f$ preserves $W_R$.
It is enough to show that
$|q(z,w)| > R|p(z)|^{\alpha}$ on $W_R$ for large $R$.
Since $p(z) \sim z^{\delta}$,
\[
\left| \dfrac{q(z,w)}{p(z)^{\alpha}} \right|
= \left| \dfrac{z^{\delta}}{p(z)} \right|^{\alpha}
\cdot \left| \dfrac{z^{\gamma + \alpha d}}{z^{\alpha \delta}} \right|
\cdot \left| \dfrac{q(z,w)}{z^{\gamma + \alpha d}} \right|
\sim \left| \dfrac{z^{\gamma + \alpha d}}{z^{\alpha \delta}} \right| \cdot |c|^d.
\]
This is larger than $R$ if it is large enough
since $\gamma + \alpha d \geq \alpha \delta$ and $d \geq 2$.
\end{proof}
This lemma implies that
$W_R$ is included in the attracting basin of $p_{\infty}$.
Let $A_f$ be the union of preimages $f^{-n} (W_R)$.
\begin{theorem}\label{delta < d: main thm}
If $\delta < d$, then
the limit $G_z$ is defined, continuous and pluriharmonic on $A_f$.
Moreover,
$G_z \sim \log |w|$ as $w \to \infty$ for fixed $z$ in $A_p$,
and $G_z$ tends to $0$ as $(z,w)$ in $A_f$
tends to $\partial A_f - J_p \times \mathbb{C}$.
\end{theorem}
\begin{proof}
The proof is similar to that of Theorem {\rmfamily \ref{delta > d: main thm}}.
We first prove the locally uniform convergence of $G_n$ to $G_z$ on $W_R$,
where $G_n = d^{-n} \log |Q_z^n|$.
It follows from inequality $(\ref{delta < d: eq1})$
in the proof of Lemma {\rmfamily \ref{delta < d: main lem}} that,
for any $(z,w)$ in $W_R$,
\[
r_1 |(p^n(z))^{\gamma}|
< \left| \dfrac{Q_z^{n+1} (w)}{Q_z^n(w)^{d}} \right|
< r_2 |(p^n(z))^{\gamma}|.
\]
Hence, for any $(z,w)$ in $W_R$ and for any positive integer $n$,
\[
\left| G_{n+1} - G_n \right|
= \left| \dfrac{1}{d^{n+1}} \log \dfrac{|Q_z^{n+1}(w)|}{|Q_z^n(w)|^d} \right|
< \dfrac{1}{d^{n+1}} \log \left( r|p^n(z)|^{\gamma} \right),
\]
where $\log r = \max \{ - \log r_1, \log r_2 \}$.
Therefore, $G_n$ converges locally uniformly to $G_z$ on $W_R$,
which is continuous and pluriharmonic.
By the inequality above, for any $(z,w)$ in $W_R$,
\[
| G_z(w) - \log |w| | < \sum_{n=0}^{\infty}
\dfrac{\gamma}{d^{n+1}} \log |p^n(z)| + \dfrac{\log r}{d - 1}.
\]
Although this infinite sum converges since $\deg p = \delta < d$,
we can restate this inequality to a simpler form.
Since $p(z) \sim z^{\delta}$,
there exists a constant $r_0 > 1$ such that
$|p(z)| < r_0 |z|^{\delta}$ and so $|p^n(z)| < (r_0 |z|)^{\delta^n}$
if $|z| > R$.
Hence
\begin{equation}\label{delta < d: eq2}
\left| G_z(w) - \log |w| \right| < \dfrac{\gamma}{d - \delta} \log |z| + C_R
\end{equation}
for any $(z,w)$ in $W_R$,
where
\[
C_R = \dfrac{\gamma}{d - \delta} \log r_0 + \dfrac{\log r}{d - 1}.
\]
In particular,
$G_z \sim \log |w|$ as $w \to \infty$ for fixed $z$ in $W_R$.
We can extend the domain of $G_z$ from $W_R$ to $A_f$
by the equation $G_z(w) = d^{-n} G_{p^n(z)} (Q_z^n(w))$.
Next, we
calculate the asymptotic value of $G_z(w)$ as $(z,w)$ in $A_f$
tends to $\partial A_f - J_p \times \mathbb{C}$.
By inequality $(\ref{delta < d: eq2})$,
\[
\left| G_{p^n(z)} (Q_z^n(w)) - \log R |p^n(z)|^{\alpha} \right|
\leq \frac{\gamma}{d - \delta} \log |p^n(z)| + C_R
\]
for any $(z,w)$ in $f^{-n} (E)$,
where $E = \{ |w| = R|z|^{\alpha}, |z| > R \} \subset \partial W_R$.
Thus it follows from the equation $G_z(w) = d^{-n} G_{p^n(z)} (Q_z^n(w))$
that
\begin{equation}\label{delta < d: eq3}
\left| G_{z} (w) - \dfrac{1}{d^{n}} \log R |p^n(z)|^{\alpha} \right|
\leq \dfrac{1}{d^{n}}
\left( \frac{\gamma}{d - \delta} \log |p^n(z)| + C_R \right)
\end{equation}
on $f^{-n} (E)$.
Since $\deg p = \delta < d$
and since $f^{-n} (E)$ converges to
$\partial A_f - J_p \times \mathbb{C}$ as $n$ tends to infinity,
$G_z(w)$ converges to $0$
as $(z,w)$ in $A_f$ tends to $\partial A_f - J_p \times \mathbb{C}$.
\end{proof}
Moreover, it follows that
$G_z = \log |z^{\gamma /(d - \delta)} w| + o(1)$ on $W_R$;
see Remark {\rmfamily \ref{delta < d: asy of G on W_R} in the next subsection.
Since $G_z = 0$ on $B_f$,
the theorem above implies the following.
\begin{cor}\label{delta < d; main cor}
If $\delta < d$, then
$G_z$ is defined on $\mathbb{C}^2$,
which is continuous and plurisubharmonic on $A_p \times \mathbb{C}$.
Moreover,
$G_z = G_f = G_f^{\alpha}$ on $\mathbb{C}^2$,
and $G_z = G_z^{\alpha}$ on $A_p \times \mathbb{C}$.
\end{cor}
We end this subsection with a description of
the dynamics of $\tilde{f}$ on $\mathbb{P} (r,s,1)$
as in Section 4.2.
By definition,
\[
\tilde{f} [z:w:t] =
\left[ t^{r(d - \delta)} \{ z^{\delta} + tu(z,t) \} :
\dfrac{h(z,w) + tv(z,w,t)}{t^{r \gamma}}: t^d \right]
\]
\[
= \left[ z^{\delta} + tu(z,t) :
\dfrac{h(z,w) + tv(z,w,t)}{t^{r \gamma + sd - s \delta}} : t^{\delta} \right]
\]
\[
= [ t^{\frac{r}{s}A} \{ z^{\delta} + tu(z,t) \} :
h(z,w) + tv(z,w,t) : t^{\frac{1}{s}A + \delta} ],
\]
where $u$ and $v$ are polynomials,
$h$ is the weighted homogeneous part of $q$ of weight $sd/r$,
and $A = r \gamma + sd - s \delta > 0$.
The polynomial $h$ contains $z^{\gamma} w^d$
if and only if $s/r \geq \alpha$,
which coincides with $z^{\gamma} w^d$ if $s/r > \alpha$.
Since $I_{\tilde{f}} = \{ [z:w:0] : h(z,w) = 0 \}$
and $h$ is divisible by $z$,
the point $p_{\infty} = [0:1:0]$ is always an indeterminacy point.
Note that $\tilde{f}$ has other indeterminacy points
than $p_{\infty}$ if $s/r \geq \alpha$.
In particular,
$I_{\tilde{f}} = \{ p_{\infty}, [1:0:0] \}$ if $s/r > \alpha$.
Recall that $p_{\infty}$ attracts all points
which converge to $(L_{\infty} - I_{\tilde{f}}) \cup \{ p_{\infty} \}$.
For the sets $A_{\tilde{f}}$ and $B_{\tilde{f}}$ as before,
it follows that
$A_{\tilde{f}} = A_f \cup (L_{\infty} - I_{\tilde{f}})$
and $B_{\tilde{f}} = B_f \cup (I_{\tilde{f}} - \{ p_{\infty} \})$.
Since $A_{\tilde{f}}$ is included in the attracting basin of $p_{\infty}$
and since $B_{\tilde{f}}$ is the attracting basin of $I_{\tilde{f}} - \{ p_{\infty} \}$,
it follows that $A_{\tilde{f}} \cup \text{int} B_{\tilde{f}} \subset F_{\tilde{f}}$.
Hence the same claim as
Proposition {\rmfamily \ref{delta > d: Fatou and Julia}} holds.
\subsection{Uniformly convergence and Asymptotics}
It seems impossible to prove that
the convergence of $G_n$ to $G_z$ is uniform,
where $G_n = d^{-n} \log^{+} |Q_z^n|$.
To overcome this problem,
we define
\[
\tilde{G}_n(z,w) = G_n(z,w)
+ \sum_{j = n}^{\infty} \dfrac{\gamma}{d^{j+1}} \log^{+} |p^j (z)|,
\]
which converges to $G_z$ on $\mathbb{C}^2$.
\begin{pro}\label{delta < d: unif}
If $\delta < d$, then
the convergence of $\tilde{G}_n$ to $G_z$ is
uniform on $V \times \mathbb{C}$,
where $\overline{V} \subset A_p$.
\end{pro}
\begin{proof}
We first show that
$\tilde{G}_n$ converges uniformly to $G_z$ on $W_R$.
By
Lemma {\rmfamily \ref{delta < d: main lem}},
there is a constant $r > 1$ such that,
for any $(z,w)$ in $W_R$,
\[
\left| \tilde{G}_{n+1} - \tilde{G}_n \right|
= \left| G_{n+1} - G_n - \dfrac{\gamma}{d^{n+1}} \log |p^n(z)| \right|
\]
\[
= \dfrac{1}{d^{n+1}} \log \left| \dfrac{Q_z^{n+1}(w)}{(p^n(z))^{\gamma} (Q_z^n(w))^d} \right|
< \dfrac{\log r}{d^{n+1}}.
\]
Hence $\tilde{G}_n$ converges uniformly to $G_z$ on $W_R$.
The left part of the proof is the same as the proof of
Proposition {\rmfamily \ref{delta > d: unif}}.
The equation $\tilde{G}_{n} \circ f = d \tilde{G}_{n+1}$
extends the uniform convergent region from $W_R$ to $f^{-n} (W_R)$.
It follows from equation $(\ref{delta < d: eq3})$
in the proof of Theorem {\rmfamily \ref{delta < d: main thm}}
that $U = \{ z \in V, G_z \geq \epsilon /3 \} \subset f^{-n} (W_R)$
for large $n$,
which induces the required uniform convergence.
\end{proof}
This proposition holds even if we replace $\log^{+} |p^j|$
in the definition of $\tilde{G}_n$ by $\log |p^j|$,
since $|p^j| > 1$ on $V$ if $j$ is large enough.
Moreover,
in Proposition {\rmfamily \ref{delta < d: unif}}
we can replace $\tilde{G}_n$ by
\[
\hat{G}_n
= G_n + \sum_{j = n}^{\infty} \dfrac{\gamma}{d^{j+1}} \log |z^{\delta^j}|
= G_n + \dfrac{\gamma}{d - \delta} \left( \dfrac{\delta}{d} \right)^n \log |z|
\]
with a slight modification of the convergent region.
However,
the proof is not the same as before, because
\[
\hat{G}_n \circ f - d \hat{G}_{n+1}
= \dfrac{\gamma}{d - \delta} \left( \dfrac{\delta}{d} \right)^n
\log \left| \dfrac{p(z)}{z^{\delta}} \right|,
\]
which is not zero in general.
\begin{pro}\label{delta < d: unif2}
If $\delta < d$, then
the convergence of $\hat{G}_n$ to $G_z$ is uniform on $V \times \mathbb{C}$,
where $\overline{V} \subset A_p \cap \{ |z| > 1 \}$.
\end{pro}
\begin{proof}
We first show that
$\hat{G}_n$ converges uniformly to $G_z$ on $W_R$.
Since $p(z) \sim z^{\delta}$,
there exists constants $0 < r_1 < 1 < r_2$ such that
$r_1 |z^{\delta}| < |p(z)| < r_2 |z^{\delta}|$ and so
$(r_1 |z|)^{\delta^n} < |p^n(z)| < (r_2 |z|)^{\delta^n}$ if $|z| > R$.
Hence if $|z| > R$, then
\[
\left| \log \left| \dfrac{p^n(z)}{z^{\delta^n}} \right| \right|
< \log r_0^{\delta^n} = \delta^n \log r_0,
\]
where $\log r_0 = \max \{ - \log r_1, \log r_2 \}$.
By Lemma {\rmfamily \ref{delta < d: main lem}},
there is a constant $r > 1$ such that,
for any $(z,w)$ in $W_R$,
\[
\left| \log \left| \dfrac{Q_z^{n+1}(w)}{(p^n(z))^{\gamma} (Q_z^n(w))^d} \right| \right|
< \log r.
\]
With these inequalities, the equation
\[
\left| \dfrac{Q_z^{n+1}(w)}{(z^{\delta^n})^{\gamma} (Q_z^n(w))^d} \right|
= \left| \dfrac{p^n(z)}{z^{\delta^n}} \right|^{\gamma}
\cdot \left| \dfrac{Q_z^{n+1}(w)}{(p^n(z))^{\gamma} (Q_z^n(w))^d} \right|
\]
implies that, for any $(z,w)$ in $W_R$,
\begin{eqnarray}\label{delta < d: eq8}
\begin{split}
\left| \hat{G}_{n+1} - \hat{G}_n \right|
&= \left| G_{n+1} - G_n - \dfrac{\gamma}{d^{n+1}} \log |z^{\delta^n}| \right| \\
&= \left| \dfrac{1}{d^{n+1}} \log \left|
\dfrac{Q_z^{n+1}(w)}{(z^{\delta^n})^{\gamma} (Q_z^n(w))^d} \right| \right| \\
&< \dfrac{\gamma}{d} \left( \dfrac{\delta}{d} \right)^n \log r_0
+ \dfrac{1}{d^{n+1}} \log r.
\end{split}
\end{eqnarray}
Hence $\hat{G}_n$ converges uniformly to $G_z$ on $W_R$.
Next, we show the uniform convergence
on $f^{-N} (W_R) \cap ( p_0^{-N} (U) \times \mathbb{C} )$
for any $N$,
where $p_0 (z) = z^{\delta}$ and $U = \{ |z| > R \}$.
By definition,
there exist constants $0 < m_N < 1 < M_N$ such that
$m_N |z^{\delta^N}| < |p^N (z)| < M_N |z^{\delta^N}|$
on $U_N = p^{-N} (U) \cap p_0^{-N} (U)$.
Since
\[
\left| \dfrac{p^n(z)}{z^{\delta^n}} \right|
= \left| \dfrac{p^{n-N}(p^N(z))}{p_0^{n-N}(p^N(z))}
\cdot \dfrac{p_0^{n-N}(p^N(z))}{p_0^{n-N}(p_0^N(z))} \right|,
\]
it follows that,
for any $z$ in $U_N$ and for any $n \geq N$,
\[
(r_1 m_N)^{\delta^{n-N}}
< \left| \dfrac{p^n(z)}{z^{\delta^n}} \right|
< (r_2 M_N)^{\delta^{n-N}}.
\]
Applying a similar argument as above for $n \geq N$,
we get the uniform convergence
on the required region,
which converges to $A_f \cap ( \{ |z| > 1 \} \times \mathbb{C} )$.
The left part is the same as the proof of
Proposition {\rmfamily \ref{delta > d: unif}}.
\end{proof}
\begin{rem}\label{delta < d: asy of G on W_R}
It follows from inequality $(\ref{delta < d: eq8})$ that
\[
\left| G_z(w) - \log |z^{\gamma /(d - \delta)} w| \right| < C_R
\text{ on } W_R,
\]
where $C_R$ converges to $0$ as $R$ tends to $\infty$.
This inequality is better than
inequality $(\ref{delta < d: eq2})$
in the proof of Theorem {\rmfamily \ref{delta < d: main thm}}
and resembles inequality $(\ref{delta > d: eq2})$
in the proof of Theorem {\rmfamily \ref{delta > d: main thm}}.
\end{rem}
Moreover, if $\alpha = \gamma /(\delta - d) < 0$, then
we can show that the convergence to $G_z^{\alpha}$ is uniform.
Let $\bar{G}_n = d^{-n} \log^{+} |z_n^{- \alpha} w_n|$,
where $(z_n,w_n) = f^n(z,w)$.
If $\alpha = \gamma /(\delta - d) < 0$, then
$\hat{G}_n = d^{-n} \log |z^{- \alpha \delta^n} w_n|$ and so
\[
\bar{G}_n
= \hat{G}_n + \frac{- \alpha}{d^n} \log \left| \dfrac{p^n(z)}{z^{\delta^n}} \right|
\text{ on } W_R.
\]
Hence we get the following convergence theorem.
\begin{cor}
If $\delta < d$ and $\alpha = \gamma /(\delta - d) < 0$, then
the convergence of $\bar{G}_n$ to $G_z^{\alpha}$ is uniform on $V \times \mathbb{C}$,
where $\overline{V} \subset A_p$.
\end{cor}
\begin{proof}
Since $\hat{G}_n$ converges uniformly to $G_z^{\alpha}$ on $W_R$,
so does $\bar{G}_n$.
The equation $\bar{G}_n \circ f = d \bar{G}_{n+1}$
extends the uniform convergent region from $W_R$ to $f^{-n} (W_R)$.
The left part is the same as the proof of
Proposition {\rmfamily \ref{delta > d: unif}}.
\end{proof}
This uniform convergence induces
the following asymptotics of $G_z^{\alpha}$ near infinity.
The proofs are similar to those of
Lemma {\rmfamily \ref{delta > d: asy lem}}
and Proposition {\rmfamily \ref{delta > d: asy pro}}.
\begin{lem}
If $\delta < d$ and $\alpha = \gamma /(\delta - d)$,
then $G_z^{\alpha}(cz^{\alpha}) = G_h (c) + o(1)$
as $z \to \infty$ for fixed $c$.
\end{lem}
\begin{proof}
Because
$Q_z^n (cz^{\alpha}) = z^{\alpha \delta^n} h^n(c) \{ 1 + o(1) \}$
as $z \to \infty$,
\[
\hat{G}_n (z, cz^{\alpha})
= d^{-n} \log^{+} |h^n(c)| + o(1).
\]
Since $\hat{G}_n$ and $d^{-n} \log^{+} |h^n|$ converge uniformly
to $G_z^{\alpha}$ and $G_h$ on suitable sets respectively,
we get the required asymptotics.
\end{proof}
In this proof we can replace $\hat{G}_n$ by $\bar{G}_n$.
\begin{pro}\label{delta < d: asy}
If $\delta < d$ and $\alpha = \gamma /(\delta - d)$,
then $G_z^{\alpha}(w) = G_h (z^{- \alpha} w) + o(1)$
as $z \to \infty$.
\end{pro}
\section{$\delta = d$}
In this section we deal with the last case $\delta = d$.
The results for the Green functions of $f$
and the dynamics of $\tilde{f}$ are different
depending on whether $f$ is nondegenerate.
However, it is common that $p_{\infty}$ is attracting in some sense
and that $f \sim (z^{d}, z^{\gamma} w^d)$
on a region in the attracting basin of $p_{\infty}$.
In Section 6.1,
we give the definition of $\alpha$ and an example of monomial maps.
In Section 6.2,
we prove the existence of three types of Green functions
under the assumption $\gamma \neq 0$.
First,
we show that $G_z$ is defined on $A_f$
and $G_f^{\alpha}$ is defined on $\mathbb{C}^2$
if we admit plus infinity.
Next,
we show the existence of
$\lim_{n \to \infty} (n \gamma d^{n-1})^{-1} \log^{+} |Q_z^n(w)|$
on $\mathbb{C}^2$,
which is continuous on $\mathbb{C}^2 - \partial A_f \cap \partial B_f$
and
plurisubharmonic on $\mathbb{C}^2$.
Finally,
we show that the limit $G$ is defined and plurisubharmonic
on $A_p \times \mathbb{C}$ if we admit minus infinity.
It is continuous and pluriharmonic on $A_f$.
\subsection{Weights}
We generalize the definition of $\alpha$ as
\[
\inf \left\{ l \in \mathbb{Q} \ \Big|
\begin{array}{lcr}
\gamma + ld \geq n_j + l m_j \text{ for any integers $n_j$ and $m_j$ s.t.} \\
z^{n_j} w^{m_j} \text{ is a term in } q \text{ with nonzero coefficient}
\end{array}
\right\}.
\]
This definition is similar to the previous case $\delta < d$,
since the inequality $\gamma + ld \geq l \delta$ is trivial if $\delta = d$.
If $q(z,w) \neq b(z) w^d$, then
we can replace the infimum in the definition of $\alpha $ by the minimum,
which is equal to
\[
\max \left\{ \dfrac{n_j - \gamma}{d - m_j} \ \Big|
\begin{array}{lcr}
z^{n_j} w^{m_j} \text{ is a term in } q \text{ with} \\
\text{nonzero coefficient s.t. } m_j < d
\end{array}
\right\}.
\]
For this case,
$- \gamma \leq \alpha \leq \deg_z q - \gamma$
and $\alpha < \deg q - \gamma$.
If $q(z,w) = b(z) w^d$, then $\alpha = - \infty$.
Thus $\alpha = - \infty$ even if $q = w^d$,
which differs with the definition we used for the nondegenerate case.
See Section 7 for a claim regarding the degree growth of $f$.
Let the weight of $z^n w^m$ be $n + \alpha m$.
Then the weight of $q$ is $\gamma + \alpha d$ and
the weight of $Q_z^n(w)$ is $n \gamma d^{n - 1} + \alpha d^n$.
Let $h$ be the weighted homogeneous part of $q$
of highest weight $\gamma + \alpha d$,
which contains $z^{\delta} w^d$.
If $\gamma = 0$, then
$z^{\alpha {\delta}^n} h^n(z^{- \alpha} w)$ is the weighted homogeneous part of
$Q_z^n (w)$ of weight $\alpha d^n$.
If $\gamma \neq 0$, then
$z^{n \gamma d^{n - 1}} (z^{- \gamma} h(z,w))^{d^{n-1}}$ is the weighted homogeneous part of
$Q_z^n (w)$ of weight $n \gamma d^{n - 1} + \alpha d^n$.
The dynamics of $\tilde{f}$ on $\mathbb{P} (r,s,1)$ is
the same as in the case $\delta < d$
if $\gamma \neq 0$ for any positive integers $r$ and $s$.
Because it contracts $L_{\infty} - I_{\tilde{f}}$ to the indeterminacy point $p_{\infty}$,
the point $p_{\infty}$ attracts most nearby points in $A_p \times \mathbb{C}$.
In this subsection we give only one example,
i.e. monomial maps.
If $\gamma = 0$,
then there are many polynomial skew products
that are semiconjugate to polynomial products;
such maps are studied in \cite[Theorem 3.7 and Proposition 3.9]{u-sym},
\cite[Examples 5.2 and 5.3]{u-weight} and \cite{u-semiconj}.
However,
we have no such maps if $\gamma \neq 0$.
\begin{ex}[monomial maps]
Let $f = (z^{d}, z^{\gamma} w^d)$ and $\gamma \neq 0$.
Then $\alpha = - \infty$
and $f^n = (z^{d^n}, z^{\gamma_n} w^{d^n})$,
where $\gamma_n = n \gamma d^{n-1} + d^n$.
Hence
\[
G_f = G_z =
\begin{cases}
\infty & \text{ on } \{ |z| > 1, w \neq 0 \} \\
\log^{+} |w| & \text{ on } \{ |z| = 1 \} \\
0 & \text{ on } \{ |z| < 1 \} \cup \{ w = 0 \}.
\end{cases}
\]
Moreover,
\[
\lim_{n \to \infty} \frac{1}{\deg (f^n)} \log^{+} |f^n(z,w)|
= \lim_{n \to \infty} \frac{1}{n \gamma d^{n-1} + d^n} \log^{+} |Q_z^n(w)|
\]
\[
=
\begin{cases}
\log^{+} |z| & \text{ on } \{ w \neq 0 \} \\
0 & \text{ on } \{ w = 0 \},
\end{cases}
\]
which is plurisubharmonic on $\mathbb{C}^2$
but not continuous on $\{ |z| \geq 1, w = 0 \}$,
and $G(z,w) = \log |w|$ on $\{ z \neq 0 \}$,
which is continuous and pluriharmonic on $\{ zw \neq 0 \}$
and plurisubharmonic on $\{ z \neq 0 \}$.
\end{ex}
\subsection{Existence of Green functions}
We defined $W_R$
as $\{ |z| > R, |w| > R|z|^{\alpha} \}$ if $\gamma \neq 0$.
If $\alpha = - \infty$,
then $W_R = \{ |z| > R, |w| > 0 \}$
since we may assume that $R > 1$.
As same as the case $\delta < d$, we have the following lemma.
\begin{lem}\label{delta = d: main lem}
If $\delta = d$,
then $q(z,w) \sim z^{\gamma} w^d$ on $W_R$ for large $R > 0$,
and $f$ preserves $W_R$; that is, $f(W_R) \subset W_R$.
\end{lem}
\begin{proof}
If $q(z,w) \neq b(z)w^d$, then $\alpha > - \infty$
and the proof is the same as the case $\delta < d$,
the proof of Lemma {\rmfamily \ref{delta < d: main lem}}.
If $q(z,w) = b(z)w^d$,
then $\alpha = - \infty$ and this claim is trivial.
Moreover,
$q(z,w) \sim z^{\gamma} w^d$ on $\{ |z| > R \} \times \mathbb{C}$.
\end{proof}
From now on, we deal with only the case $\gamma \neq 0$.
The lemma above implies the following two theorems.
\begin{theorem}\label{delta = d: main thm1}
Let $\delta = d$ and $\gamma \neq 0$.
If $\alpha > 0$, then
$G_z = \infty$ on $A_f$
and $\tilde{G}_z \leq \alpha G_p$ on $B_f$.
\end{theorem}
\begin{proof}
By Lemma {\rmfamily \ref{delta = d: main lem}},
there exists a positive constant $r < 1$ such that
$| q(z,w) | > r |z^{\gamma} w^d|$ on $W_R$.
Since $p(z) \sim z^{d}$,
there exists a positive constant $r_0 < 1$ such that
$|p(z)| > r_0 |z|^d$
and so $|p^n(z)| > |r_0 z|^{d^n}$ if $|z| > R$.
Using these inequalities inductively, we get
\[
|Q_z^n(w)| > r^{1 + d + \cdots + d^{n-1}} |(r_0 z)^{n \gamma d^{n-1}} w^{d^n}|
\]
and so
\[
\dfrac{1}{d^n} \log |Q_z^n(w)|
> \log r + \dfrac{n \gamma}{d} \log |r_0 z| + \log |w|,
\]
which tends to $\infty$ as $n \to \infty$.
Therefore, $G_z = \infty$ on $W_R$, which extends to $A_f$.
Let $(z,w)$ be a point in $B_f$.
Then $f^n(z,w)$ never belong to $W_R$
and so $|Q_z^n(w)| < R |p^n(z)|^{\alpha}$.
Hence $\tilde{G}_z (w) \leq \alpha G_p(z)$ on $B_f$.
\end{proof}
The existence of $G_z$ on $B_f$ is still unclear.
We exhibit three examples that relate to this problem.
\begin{ex}
For any positive integer $s$,
let $f = f_s = (z^2, z(w^2 - z^s) + z^{2s})$,
which is conjugate to $f_1 = (z^2, zw^2)$ by $\pi = (z, w + z^s)$.
Then $\alpha = s$ and
$G_z = \alpha \log |z|$ on $B_f = \{ |z| > 1, w = z^s \}$.
\end{ex}
\begin{ex}
Let $f = (z^r, z^{\gamma} (w^r - z^s) + z^s)$.
Then $\alpha = s/r$ and $G_z = \alpha \log |z|$
on $\{ |z| > 1, w^r = z^s \} \subset B_f$.
\end{ex}
\begin{ex}
Let $f = (z^2, zw^2 + z^2w)$.
Then $\alpha = 1$ and
$G_z = 0$ on $\{ w = 0 \} \subset B_f$.
\end{ex}
Let us return to the statement on the Green function
in the remaining case $\alpha \leq 0$.
\begin{theorem}
Let $\delta = d$ and $\gamma \neq 0$.
If $\alpha \leq 0$, then
$G_z$ is $\infty$ on $A_f$ and $0$ on $B_f$.
\end{theorem}
\begin{proof}
If $q(z,w) \neq b(z)w^d$, then $\alpha > - \infty$.
The proof of the claim $G_z = \infty$ on $A_f$ is the same as
the proof of Theorem {\rmfamily \ref{delta = d: main thm1}}.
The claim $G_z = 0$ on $B_f$ follows from
the definition of $B_f$ and the assumption $\alpha \leq 0$.
If $q(z,w) = b(z)w^d$, then $\alpha = - \infty$
and this claim follows from the direct calculation;
see the proof of
Proposition {\rmfamily \ref{delta = d: q = b(z) w^d}} below
for detail.
\end{proof}
More precisely,
$B_f$ consists of infinitely many lines if $q = b(z)w^d$.
\begin{pro}\label{delta = d: q = b(z) w^d}
Let $\delta = d$ and $\gamma \neq 0$.
If $q(z,w) = b(z) w^d$, then
$G_z$ is $\infty$ on $A_f$ and $0$ on $B_f$.
Moreover,
$B_f$ coincides with the union of
the preimages of $\{ z \in A_p, w = 0 \}$ under $f$,
which is equal to
\[
\bigcup_{n \geq 0} p^{-n}(b^{-1}(0) \cap A_p) \times \mathbb{C}
\cup \{ z \in A_p, w = 0 \}.
\]
\end{pro}
\begin{proof}
Let $f(z,w) = (p(z), b(z)w^d)$.
Then $f^n(z,w) = (p^n(z), B_n (z) w^{d^n})$,
where $B_n (z) = b(p^{n-1}(z)) \cdots b(p(z))^{d^{n-2}} b(z)^{d^{n-1}}$.
Hence
\[
\dfrac{1}{d^n} \log |Q_z^n(w)|
= \log |w| + \sum_{j = 0}^{n - 1} \dfrac{1}{d^{j+1}} \log |b(p^j(z))|.
\]
This finite sum tends to $\infty$ as $n \to \infty$
unless $b(p^j(z)) = 0$ for some $j \geq 0$,
since $\deg p = d$.
\end{proof}
Combining two theorems above, we get the following corollary.
\begin{cor}
If $\delta = d$ and $\gamma \neq 0$, then
\[
G_f^{\alpha} (z,w) =
\begin{cases}
\infty & \text{ on } A_f \\
\max \{ \alpha, 0 \} G_p(z) & \text{ on } B_f.
\end{cases}
\]
\end{cor}
The dynamics of $\tilde{f}$ is similar to
that mentioned in the previous section.
The set $A_{\tilde{f}}$ is included in the attracting basin of $p_{\infty}$,
and $B_{\tilde{f}}$ is the attracting basin of $I_{\tilde{f}} - \{ p_{\infty} \}$.
Hence the same claim as
Proposition {\rmfamily \ref{delta > d: Fatou and Julia}} holds
except the description of $J_{\tilde{f}}$ in terms of a Green function,
for which we use
the Green function in Corollary {\rmfamily \ref{delta = d; cor2}} below
instead of $G_z^{\alpha}$.
Now we show the existence of
other Green functions
that are locally bounded on $\mathbb{C}^{2}$.
\begin{theorem}
If $\delta = d$ and $\gamma \neq 0$, then
\[
\lim_{n \to \infty} \frac{1}{n \gamma d^{n-1}} \log |Q_z^n(w)| = G_p(z)
\text{ on } A_f.
\]
\end{theorem}
\begin{proof}
By Lemma {\rmfamily \ref{delta = d: main lem}},
there exist constants $r_1 < 1 < r_2$ such that
$r_1 |z^{\gamma} w^d| < |q(z,w)| < r_2 |z^{\gamma} w^d|$ on $W_R$.
By using $|q(z,w)| < r_2 |z^{\gamma} w^d|$ inductively,
we get the upper estimate
\[
|Q_z^n(w)| < r_2^{1 + d + \cdots + d^{n-1}} |p^{n-1} (z)|^{\gamma}
|p^{n-2} (z)|^{\gamma d} \cdots |z|^{\gamma d^{n-1}} |w|^{d^n}
\]
and so
\[
\frac{1}{n \gamma d^{n-1}} \log |Q_z^n(w)|
< \dfrac{d}{n \gamma} \left( \log r_2 + \log |w| \right)
\]
\[
+ \dfrac{1}{n} \left\{ \log |z| + \dfrac{1}{d} \log |p(z)|
+ \cdots + \dfrac{1}{d^{n-1}} \log |p^{n-1}(z)| \right\}.
\]
Since $d^{-n} \log|p^{n}|$ converges to $G_p$ on $A_p$,
the right hand side converges to $G_p$ as $n \to \infty$.
Thus the inequality
\[
\limsup_{n \to \infty} \frac{1}{n \gamma d^{n-1}} \log |Q_z^n(w)| \leq G_p(z)
\]
holds on $W_R$.
By the same argument in terms of $r_1 |z^{\gamma} w^d| < |q(z,w)|$,
we get the inverse inequality.
Therefore, we get the required equation on $W_R$.
A similar argument as above induces the required equation on $A_f$,
because if $f^N (z,w)$ belongs to $W_R$
then $Q_z^n(w) = Q_{p^N(z)}^{n - N} (Q_z^N(w))$ approximates to
$p^{n-1} (z)^{\gamma} p^{n-2} (z)^{\gamma d} \cdots p^N (z)^{\gamma d^{n-N-1}} Q_z^N(w)^{d^{n-N}}$
for $n \geq N$.
\end{proof}
It follows from this theorem that
$\lim_{n \to \infty} (n \gamma d^{n-1})^{-1} \log^{+} |Q_z^n| = 0$ on $B_f$.
Therefore,
\begin{cor}\label{delta = d; cor2}
If $\delta = d$ and $\gamma \neq 0$, then
\[
\lim_{n \to \infty} \frac{1}{n \gamma d^{n-1}} \log^{+} |f^n(z,w)| =
\lim_{n \to \infty} \frac{1}{n \gamma d^{n-1}} \log^{+} |Q_z^n(w)|
\]
\[
=
\begin{cases}
G_p(z) & \text{ on } \mathbb{C}^{2} - B_f \\
0 & \text{ on } B_f.
\end{cases}
\]
\end{cor}
Finally, we prove the existence of $G$ using
Lemma {\rmfamily \ref{delta = d: main lem}}.
By definition,
$G(f^n(z,w)) = d^n G(z,w) + n \gamma d^{n-1} G_p(z)$
if it exists.
\begin{theorem}
The limit $G$ is defined, continuous and pluriharmonic on $A_f$.
Moreover,
$G = \log|w| + o(1)$ on $W_R$,
$G \sim \log |w|$ as $w \to \infty$ for fixed $z$ in $A_p$,
and $G$ tends to $- \infty$ as $(z,w)$ in $A_f$ tends to
any point in $\partial A_f - J_p \times \mathbb{C}$.
\end{theorem}
\begin{proof}
The proof is similar to those of Theorems
{\rmfamily \ref{delta > d: main thm}} and
{\rmfamily \ref{delta < d: main thm}}.
We first show the uniform convergence to $G$ on $W_R$.
Let $G_n = d^{-n} \log |z_n^{- n \gamma /d} w_n|$,
where $(z_n,w_n) = f^n(z,w)$.
By Lemma {\rmfamily \ref{delta = d: main lem}},
there are constants $r > 1$ and $r_0 > 1$ such that
$|w_{n+1}| < r|z_n^{\gamma} w_n^d|$ and $|z_n^d| < r_0 |z_{n+1}|$ on $W_R$.
Hence
\[
G_{n+1} - G_n =
\dfrac{1}{d^{n+1}} \log \left| \dfrac{w_{n+1}}{w_n^d} \cdot
\dfrac{z_n^{n \gamma}}{z_{n+1}^{(n+1) \frac{\gamma}{d}}} \right|
< \dfrac{1}{d^{n+1}} \log \left| r \left( \dfrac{z_n^d}{z_{n+1}} \right)^{(n+1) \frac{\gamma}{d}} \right|
\]
\[
< \dfrac{1}{d^{n+1}} \log \left| r {r_0}^{(n+1) \frac{\gamma}{d}} \right|
= \dfrac{1}{d^{n+1}} \log r + \dfrac{n+1}{d^{n+1}} \cdot \dfrac{\gamma}{d} \log r_0.
\]
Therefore, $G_n$ converges uniformly to $G$ on $W_R$,
which is continuous and pluriharmonic.
By the inequality above,
\begin{equation}\label{delta = d: eq}
|G(z,w) - \log |w|| < C_R \text{ on } W_R,
\end{equation}
where the constant $C_R > 0$ converges to $0$ as $R \to \infty$.
The equation $G = d^{-n} G \circ f^n - n \gamma d^{-1} G_p$
extends the domain of $G$ from $W_R$ to $A_f$.
Next, we show the last statement.
By inequality (\ref{delta = d: eq}),
\[
\left |G(f^n(z,w)) - \log R|p^n(z)|^{\alpha} \right| < C_R \text{ on } f^{-n}(E),
\]
where $E = \{ |w| = R|z|^{\alpha}, |z| > R \} \subset \partial W_R$.
Thus, by the identity $G \circ f^n = d^n G + n \gamma d^{n-1} G_p$,
\[
\left |G(z,w) + n \dfrac{\gamma}{d} G_p(z) - \dfrac{\log R|p^n(z)|^{\alpha}}{d^n} \right|
< \dfrac{C_R}{d^n} \text{ on } f^{-n}(E).
\]
Therefore, the values of $G$ on $f^{-n}(E)$ tend to $- \infty$ as $n \to \infty$.
\end{proof}
This theorem guarantees that
$B_f \cap ( \{ z \} \times \mathbb{C} ) \neq \emptyset$
for any $z$ in $A_p$.
Since $G = - \infty$ on $B_f$,
it follows that $G$ is defined and plurisubharmonic on $A_p \times \mathbb{C}$,
and $A_f = \{ (z,w) \in A_p \times \mathbb{C} : G > - \infty \}$.
The identity $G \circ f^n = d^n G + n \gamma d^{n-1} G_p$ induces that
$G(z,w)$ converges to $0$ as $(z,w)$ in $A_f$ tends to the intersection of
the closure of $\partial A_f - J_p \times \mathbb{C}$
and $J_p \times \mathbb{C}$.
Hence the description of the last statement is different
from those in the case $\delta \neq d$.
\section{Degree growth}
In this section we give
a remark on the relationship between algebraically stability and weight growth,
the list of inequalities of degree growth in terms of $\alpha$,
and a corollary on the existence of a weighted Green function
that is normalized by $\deg (f^n)$.
\subsection{Algebraically stability and Weight growth}
Assume that $f$ is not a polynomial product.
We saw that if $\gamma \neq 0$ or $\delta > d$,
then $\alpha$ is a positive rational number,
$f$ extends to an algebraically stable map on $\mathbb{P} (r,s,1)$,
where $s/r = \alpha$,
and the weight of $Q_z^n$ is $\alpha \lambda^n$.
Hence the weight of $f^n$ is equal to $\lambda^n$
if we define it as the maximum of
the degree of $p^n$ and the weight of $Q_z^n$ times $\alpha^{-1}$;
namely,
$\text{weight} (f^n) = ( \text{weight} f )^n$.
This is an analogue of the well known fact:
if $\tilde{f}$ is algebraically stable on $\mathbb{P}^2$,
then $\deg (f^n) = (\deg f)^n$.
\subsection{Degree growth}
Next, we give the list of inequalities on the degree growth of $f$,
which follows from the definition of $\alpha$.
\begin{enumerate}
\item If $\delta > d$, then
$\delta^n \leq \deg (f^n) \leq \max \{ \alpha, 1 \} \delta^n$.
\item If $\delta < d$ and $\alpha > 0$, then
\[
\left[ 1 + \dfrac{\gamma}{d - \delta}
\left\{ 1 - \left( \dfrac{\delta}{d} \right)^n \right\} \right] d^n
\leq \deg (f^n)
\]
\[
\leq \left[ \max \{ \alpha, 1 \} + \max \left\{ \dfrac{1}{\alpha}, 1 \right\}
\dfrac{\gamma}{d - \delta} \left\{ 1 - \left( \dfrac{\delta}{d} \right)^n \right\} \right] d^n.
\]
If $\delta < d$ and $\alpha \leq 0$, then
\[
\deg (f^n) = \left[ 1 + \dfrac{\gamma}{d - \delta}
\left\{ 1 - \left( \dfrac{\delta}{d} \right)^n \right\} \right] d^n.
\]
\item If $\delta = d$ and $\alpha > 0$, then
\[
\left( \dfrac{\gamma}{d} n + 1 \right) d^n
\leq \deg (f^n) \leq
\left[ \max \left\{ \dfrac{1}{\alpha}, 1 \right\} \dfrac{\gamma}{d} n + \max \{ \alpha, 1 \} \right] d^n.
\]
If $\delta = d$ and $\alpha \leq 0$, then
\[
\deg (f^n) = \left( \dfrac{\gamma}{d} n + 1 \right) d^n.
\]
\end{enumerate}
Note that it follows from the equalities above that,
if $\gamma = 0$, then
$\lambda^n \leq \deg (f^n) \leq \max \{ \alpha, 1 \} \lambda^n$,
which was already stated in \cite{u-weight}.
In general,
it is proved in \cite{fj-ev} that,
for any polynomial map $F$
which is not conjugate to a skew product
such that $\delta = d$ and $\gamma \neq 0$,
if the dynamical degree $\lambda$ is larger than $1$,
then there is $D \geq 1$
such that $\lambda^n \leq \deg (F^n) \leq D \lambda^n$.
\subsection{Another Green function}
In the final subsection
we consider the existence of the limit
\[
\lim_{n \to \infty} \frac{1}{\deg (f^n)} \log^{+} |f^n(z,w)|_{\alpha}.
\]
If $\delta \neq d$,
then the topological degree $\delta d$ is smaller
than $\lambda^2$.
In this case, the existence of the limit follows from
Corollaries {\rmfamily \ref{delta > d; main cor}}
and {\rmfamily \ref{delta < d; main cor}}
and the main result in \cite{bfj},
which can be restated as follows:
for a polynomial map $F$
whose topological degree is smaller
than the square $\lambda^2$ of the dynamical degree,
the ratio of $\deg (F^n)$ and $\lambda^n$
converges to some positive number.
For the case $\delta = d$,
the existence follows if $\alpha \leq 0$
because of Corollary {\rmfamily \ref{delta = d; cor2}}
and the result above on the degree growth.
Consequently,
\begin{cor}
If $\delta \neq d$ or $\alpha \leq 0$, then
the limit of $(\deg (f^n))^{-1} \log^{+}$ $|f^n(z,w)|_{\alpha}$
is defined on $\mathbb{C}^2$.
Moreover, it is continuous on $\mathbb{C}^2$ if $\delta > d$,
on $A_p \times \mathbb{C}$ if $\delta < d$,
and on $\mathbb{C}^2 - B_f$ if $\delta = d$ and $\alpha \leq 0$.
\end{cor}
\end{document} | math | 82,530 |
\begin{document}
\begin{frontmatter}
\title{The competition numbers of ternary Hamming graphs}
\author[label1]{{\sc Boram PARK} \corref{cor1}\fnref{label3}}
\author[label2]{{\sc Yoshio SANO} \fnref{label4} }
\address[label1]{Department of Mathematics Education,
Seoul National University, Seoul 151-742, Korea}
\address[label2]{Pohang Mathematics Institute, POSTECH, Pohang 790-784, Korea}
\fntext[label3]{
{\it E-mail addresses:}
{\tt [email protected]}; {\tt [email protected]} \\
This research was supported by Basic Science Research Program
through the National Research Foundation of Korea (NRF) funded by
the Ministry of Education, Science and Technology (700-20100058).}
\fntext[label4]{
{\it E-mail addresses:}
{\tt [email protected]}; {\tt [email protected]} \\
This work was supported by Priority Research Centers Program
through the National Research Foundation of Korea (NRF) funded by
the Ministry of Education, Science and Technology (MEST) (No. 2010-0029638).}
\cortext[cor1]{Corresponding author}
\begin{abstract}
It is known to be a hard problem to compute the competition number $k(G)$
of a graph $G$ in general.
Park and Sano \cite{PS1} gave the exact values
of the competition numbers of Hamming graphs $H(n,q)$
if $1 \leq n \leq 3$ or $1 \leq q \leq 2$.
In this paper, we give an explicit formula of the competition numbers
of ternary Hamming graphs.
\end{abstract}
\begin{keyword}
competition graph; competition number;
edge clique cover; Hamming graph
\MSC[2010] 05C69, 05C20
\end{keyword}
\end{frontmatter}
\section{Introduction}
The notion of a competition graph was introduced by Cohen \cite{Cohen1}
as a means of determining the smallest dimension of ecological phase space.
The {\it competition graph} $C(D)$ of a digraph $D$ is
a (simple undirected) graph
which has the same vertex set
as $D$ and an edge between vertices $u$ and $v$ if and only if
there is a vertex $x$ in $D$ such that $(u,x)$ and $(v,x)$ are arcs of $D$.
For any graph $G$, $G$ together
with sufficiently many isolated vertices is the competition graph of an
acyclic digraph.
Roberts \cite{MR0504054} defined the {\em competition number} $k(G)$ of
a graph $G$ to be the smallest number $k$ such that $G$ together with
$k$ isolated vertices is the competition graph of an acyclic
digraph.
Opsut \cite{MR679638} showed that the computation of the
competition number of a graph is an NP-hard problem
(see \cite{kimsu} for graphs whose competition numbers are known).
It has been one of important research problems in the study of
competition graphs to compute the competition numbers for
various graph classes
(see \cite{KLS}, \cite{KPS-Johnson}, \cite{KimSano},
\cite{LC}, \cite{LiChang}, \cite{PLK},
\cite{PKS1}, \cite{PKS2}, \cite{Sano}, \cite{ZhaoChang2009},
\cite{ZhaoChang2010}, for recent research).
For some special graph families, we have explicit formulas
for computing competition numbers.
For example, if $G$ is a chordal graph without isolated vertices
then $k(G)=1$,
and if $G$ is a nontrivial triangle-free connected graph
then $k(G)=|E(G)|-|V(G)|+2$ (see \cite{MR0504054}).
In this paper, we study the competition numbers of Hamming graphs.
For a positive integer $q$,
we denote a $q$-set $\{1,2 \ldots, q \}$ by $[q]$.
Also we denote the set of $n$-tuple over $[q]$ by $[q]^n$.
For positive integers $n$ and $q$,
the {\it Hamming graph} $H(n,q)$
is the graph which has the vertex set ${[q]^n}$ and
in which
two vertices $x$ and $y$ are adjacent if $d_H(x,y)=1$,
where $d_H:[q]^n \times [q]^n \to \ensuremath{\mathbb{Z}}$ is
the {\it Hamming distance} defined by
$d_H(x,y):=|\{i \in [n] \mid x_i \neq y_i \}|$.
Note that the diameter of the Hamming graph
$H(n,q)$ is equal to $n$ if $q \geq 2$
and that the number of edges of
the Hamming graph $H(n,q)$ is equal to $\frac{1}{2} n(q-1)q^n$.
Park and Sano \cite{PS1} gave the exact values
of the competition numbers of Hamming graphs $H(n,q)$
if $1 \leq n \leq 3$ or $1 \leq q \leq 2$ as follows.
For $n \geq 1$, $H(n,1)$ has no edge and so $k(H(n,1))=0$.
For $n \geq 1$, $H(n,2)$ is a $n$-cube and $k(H(n,2))=(n-2)2^{n-1} +2$.
For $q \geq 2$, $H(1,q)$ is complete graph and so $k(H(1,q))=1$.
\begin{Thm}[\cite{PS1}] \label{thm:H(2,q)}
For $q \geq 2$, we have $k(H(2,q)) =2$.
\end{Thm}
\begin{Thm}[\cite{PS1}]\label{thm:H(3,q)}
For $q \geq 3$, we have $k(H(3,q)) =6$.
\end{Thm}
In this paper, we give the exact values of
the competition numbers of ternary Hamming graphs $H(n,3)$.
Our main result is the following:
\begin{Thm}\label{Mainthm:H(n,3)}
For $n \geq 3$, we have
\[
k(H(n,3)) =(n-3)3^{n-1}+6.
\]
\end{Thm}
\section{Preliminaries}
We use the following notation and terminology in this paper.
For a digraph $D$,
a sequence $v_1, v_2, \ldots, v_n$ of the vertices of $D$
is called an \emph{acyclic ordering} of $D$
if $(v_i,v_j) \in A(D)$ implies $i<j$.
It is well-known that a digraph $D$ is acyclic
if and only if there exists an acyclic ordering of $D$.
For a digraph $D$ and a vertex $v$ of $D$,
we define
the {\it in-neighborhood} $N^{-}_D(v)$
of $v$ in $D$ to be the set $\{w \in V(D) \mid (w,v)\in A(D)\}$,
and a vertex in $N^{-}_D(v)$ is called an {\it in-neighbor} of $v$ in $D$.
For a graph $G$ and a nonnegative integer $k$,
we denote by $G\cup I_k$ the graph such that
$V(G\cup I_k)=V(G)\cup I_k$ and
$E(G\cup I_k)=E(G)$, where $I_k$ is a set of $k$ isolated vertices.
For a clique $S$ of a graph $G$
and an edge $e$ of $G$,
we say {\it $e$ is covered by $S$}
if both of the endpoints of $e$ are contained in $S$.
An \textit{edge clique cover} of a graph $G$
is a family of cliques of $G$ such that
each edge of $G$ is covered by some clique in the family.
The {\it edge clique cover number} $\theta_E(G)$ of a graph $G$
is the minimum size of an edge clique cover of $G$.
An edge clique cover of $G$
is called a {\it minimum edge clique cover} of $G$
if its size is equal to $\theta_E(G)$.
Let $\pi_j:[q]^n \to [q]^{n-1}$ be a map defined by
$
(x_1, ..., x_{j-1}, x_j, x_{j+1}, ..., x_n) \mapsto
(x_1, ..., x_{j-1}, x_{j+1}, ..., x_n).
$
For $j \in [n]$ and ${ p} \in [q]^{n-1}$,
we let
\begin{equation}\label{eq:Sjp}
S_j(p) := \pi^{-1}_j(p) = \{x \in [q]^n \mid \pi_j(x)= { p} \}.
\end{equation}
Note that $S_j(p)$ is a clique of $H(n,q)$ with size $q$. Let
\begin{equation}\label{eq:Fnq}
\ensuremath{\mathcal{F}}(n,q):=\{S_j(p) \mid j \in [n], p \in [q]^{n-1} \}.
\end{equation}
Then $\ensuremath{\mathcal{F}}(n,q)$ is the family of maximal cliques of $H(n,q)$.
Park and Sano \cite{PS1} showed the following:
\begin{Lem}[\cite{PS1}]\label{lem:LB2}
The following hold:
\begin{itemize}
\item{}
Let $n\ge 2$ and $q \ge 2$,
and let $K$ be a clique of $H(n,q)$ with size at least $2$.
Then there exists a unique maximal clique $S$ of $H(n,q)$
containing $K$.
\item{}
The edge clique cover number of $H(n,q)$ is equal to $nq^{n-1}$.
\item{}
Any minimum edge clique cover of $H(n,q)$ consists of
edge disjoint maximum cliques.
\item{}
The family $\ensuremath{\mathcal{F}}(n,q)$ defined by (\ref{eq:Fnq})
is a minimum edge clique cover of $H(n,q)$.
\end{itemize}
\end{Lem}
Now we present the following lemma:
\begin{Lem} \label{lem:ExistD_nq}
Let $G$ be a graph. Suppose that any edge of $G$
is contained in exactly one maximal clique of $G$.
Let $\ensuremath{\mathcal{F}}$ be the family of all maximal cliques of size
at least two in $G$,
and let $k$ be an integer with $k\ge k(G)$.
Then there exists an acyclic digraph $D$
satisfying the following:
\begin{itemize}
\item[{\rm (a)}]
The competition graph of $D$ is $G \cup I_{k}$.
\item[{\rm (b)}]
For any vertex $v$ in $D$,
$N^-_D(v) \in \ensuremath{\mathcal{F}} \cup \{\emptyset\}$.
\item[{\rm (c)}]
The number of vertices which have no in-neighbor in $D$ is
equal to
$k + |V(G)| - |\ensuremath{\mathcal{F}}|$.
\end{itemize}
\end{Lem}
\begin{proof}
Let $k$ be an integer such that $k \ge k(G)$.
By the definition of the competition number of a graph,
there exists an acyclic digraph satisfying (a).
Suppose that any acyclic digraphs satisfying (a) does not satisfies (b).
Let $D$ be an acyclic digraph which maximizes
$|\{ v \in V(D) \mid N^-_D(v)=\emptyset\}|$
among all acyclic digraphs satisfying (a) but not (b).
Since $D$ does not satisfy (b), there exists a vertex $v^*$ in $D$
such that $N^-_D(v^*) \not\in \ensuremath{\mathcal{F}} \cup \{ \emptyset \}$.
Since
$D$ maximizes $|\{ v \in V(D) \mid N^-_D(v)=\emptyset\}|$,
we may assume that
$|N^-_D(v)| \neq 1$ for any $v \in V(D)$.
By the assumption
that any edge of $G$
is contained in exactly one maximal clique of $G$,
any clique of size at least two is contained in
a unique maximal clique in $G$.
Since $N^-_D(v^*)$ is a clique of size at least two,
there exists a unique maximal clique $S \in \ensuremath{\mathcal{F}}$ containing $N^-_D(v^*)$.
Let $\sigma:=(v_1,v_2,\ldots,v_{|V(D)|})$ be an acyclic ordering of $D$,
i.e., if $(v_i,v_j) \in A(D)$ then $i<j$.
Let $v_i$ and $v_j$ be two vertices of $S$
whose indices are largest in the acyclic ordering $\sigma$.
Since $v_i$ and $v_j$ are adjacent in $G$,
there is a vertex $v_l \in V(D)$ such that
$(v_i,v_l)$ and $(v_j,v_l)$ are arcs of $D$.
Then we define a digraph $D_1$ by $V(D_1) = V(D)$ and
\begin{eqnarray*}
A(D_1) &=& (A(D) \setminus \{ (x, v^*) \mid x \in N^-_D(v^*) \} )
\cup \{ ( x, v_l) \mid x \in S \}.
\end{eqnarray*}
Then the digraph $D_1$ is acyclic,
since the index $l$ of $v_l$ is larger
than the index of any vertex in $S$.
By the assumption
that any edge of $G$
is contained in exactly one maximal clique of $G$,
we have $N^-_D(v_l) \subseteq S$, which implies that $C(D_1) = G \cup I_k$.
Since
$|\{ v \mid N^-_{D_1}(v) = \emptyset \}|
> |\{ v \mid N^-_{D}(v) = \emptyset \}|$,
we reach a contradiction to the choice of $D$.
Thus, there exists an acyclic digraph satisfying both (a) and (b).
Let $D_2$ be an acyclic digraph satisfying both (a) and (b).
If the digraph $D_2$ has two vertices $u,w$
such that $N^-_{D_2}(u)=N^-_{D_2}(w) \neq \emptyset$,
then we can obtain an acyclic digraph $D_3$
such that all the nonempty in-neighborhoods of vertices in $D_3$
are distinct by deleting arcs in
$\{(x,u) \mid x \in N^-_{D_2}(u) \}$ from $D_2$.
Therefore,
we may assume that all the
nonempty in-neighborhoods $N^-_{D_3}(v)$ are distinct.
Then the number of vertices $v$ such that
$N^-_{D_3}(v) \neq \emptyset$ is exactly equal to $|\ensuremath{\mathcal{F}}|$.
Therefore,
$
|\{ v \in V(D_3) \mid N^-_{D_3}(v) = \emptyset\}|
=|V(D_3)|- |\ensuremath{\mathcal{F}}|= k+|V(G)|-|\ensuremath{\mathcal{F}}|.
$
Thus the digraph $D_3$ satisfies (a), (b), and (c).
Hence the lemma holds.
\end{proof}
By Lemmas \ref{lem:LB2} and \ref{lem:ExistD_nq},
the following holds.
\begin{Cor}\label{cor:Dabc}
Let $k$ be an integer with $k\ge k(H(n,q))$.
Then there exists an acyclic digraph $D$
satisfying the following:
\begin{itemize}
\item[{\rm (a)}]
$C(D)=H(n,q) \cup I_{k}$.
\item[{\rm (b)}]
$N^-_D(v) \in \ensuremath{\mathcal{F}}(n,q) \cup \{\emptyset\}$
for any $v \in V(D)$.
\item[{\rm (c)}]
$|\{ v \in V(D) \mid N^-_D(v) = \emptyset\}| = k - (n-q)q^{n-1}$.
\end{itemize}
\end{Cor}
\section{Proof of Theorem \ref{Mainthm:H(n,3)}}
In this section, we present the proof of the main result.
The following theorem gives an upper bound for
the competition numbers of Hamming graphs $H(n,q)$ where $2 \leq q \leq n$.
\begin{Thm} \label{thm:UH_nq}
For $2 \leq q \leq n$, we have
\[
k(H(n,q)) \leq (n-q)q^{n-1} + k(H(q,q)).
\]
\end{Thm}
\begin{proof}
We prove the theorem by induction on $n$.
If $n=2$, then $n=q=2$ and the theorem trivially holds.
For simplicity, for any $2 \leq q \leq n$,
let
$
\alpha(n,q) := (n-q)q^{n-1} + k(H(q,q)).
$
Let $n\ge 3$ and we assume that
$k(H(n-1,q)) \leq \alpha(n-1,q)$ holds for any $q$
such that $2 \leq q \leq n-1$.
Now consider a Hamming graph $H(n,q)$ where $q$ is an integer
satisfying $2 \leq q \leq n$.
If $n=q$, then the theorem clearly holds.
Suppose that $2\leq q \leq n-1$.
For $i \in [q]$,
let $H^{(i)}$ be the subgraph of $H(n,q)$ induced
by a vertex set
$\{ (x_1, x_2, \ldots, x_n) \in [q]^n \mid x_n=i\}$.
Then each $H^{(i)}$ is isomorphic to the Hamming graph $H(n-1,q)$.
We denote the minimum edge clique cover of $H^{(i)}$ by $\ensuremath{\mathcal{F}}^{(i)}(n-1,q)$.
By the induction hypothesis,
it holds that $k(H^{(i)})= k(H(n-1,q)) \leq \alpha(n-1,q)$.
By Corollary \ref{cor:Dabc},
there exists an acyclic digraph $D^{(i)}$
for each $i\in [q]$ such that
\begin{itemize}
\item[(a)] $C(D^{(i)})=H^{(i)} \cup I^{(i)}_{\alpha(n-1,q)}$,
\item[(b)] $N^-_{D^{(i)}}(v) \in \ensuremath{\mathcal{F}}^{(i)}(n-1,q) \cup \{ \emptyset \}$
for any $v \in V(D^{(i)})$,
\item[(c)] $|\{v \in V(D^{(i)}) \mid N^-_{D^{(i)}}(v) = \emptyset \}|
=\alpha(n-1,q) - (n-1-q)q^{n-2} = k(H(q,q))$.
\end{itemize}
where $I^{(i)}_{\alpha(n-1,q)}$ is a set of ${\alpha(n-1,q)}$ isolated vertices.
Then by (c), we may let, for each $i\in [q]$,
\[
W^{(i)} :=\{v \in V(D^{(i)}) \mid N^-_{D^{(i)}}(v) = \emptyset \}
= \{w^{(i)}_1, \ldots, w^{(i)}_{k(H(q,q))} \}.
\]
Since $q \leq n-1$, it holds that
$k(H(q,q))=\alpha(n-1,q)-((n-1)-q)q^{n-2} \leq \alpha(n-1,q)$.
By (a), there are at least $k(H(q,q))$ isolated vertices
in $I^{(i)}_{\alpha(n-1,q)}$ of $C(D^{(i)})$.
Let $J^{(i)}:= \{j^{(i)}_1, \ldots, j^{(i)}_{k(H(q,q))} \}$
be a set of $k(H(q,q))$ vertices which belong to $I^{(i)}_{\alpha(n-1,q)}$
in $C(D^{(i)})$. Let $D$ be the digraph defined by
\begin{eqnarray*}
V(D) &:=&
V(D^{(1)}) \cup
\bigcup_{i=2}^{q} (V(D^{(i)}) \setminus J^{(i)}), \\
A(D) &:=& A(D^{(1)}) \cup \bigcup_{i=2}^{q}
(A(D^{(i)}) \setminus
\{ (x,j^{(i)}_l) \mid 1 \leq l \leq k(H(q,q)),
x \in N_{D^{(i)}}^{-}(j^{(i)}_l) \} ) \\
&& \qquad \quad \,\,\cup
\bigcup_{i=2}^{q}
\{ (x,w^{(i-1)}_l) \mid 1 \leq l \leq k(H(q,q)),
x \in N_{D^{(i)}}^{-}(j^{(i)}_l) \}.
\end{eqnarray*}
Since each digraph $D^{(i)}$ is acyclic, the digraph $D$ is also acyclic.
In addition, $C(D)= (\bigcup_{i=1}^{q} H^{(i)}) \cup I_{k}$ where
$k := q \cdot \alpha(n-1,q) - (q-1) \cdot k(H(q,q))$.
Note that by the definition of $H^{(i)}$,
the graph $\bigcup_{i=1}^{q} H^{(i)}$ is a spanning subgraph of $H(n,q)$
except the edges of cliques in $\{S_n(x)\mid x \in [q]^{n-1} \}$.
Therefore, by letting$D^*$ be the digraph defined by
\begin{eqnarray*}
V(D^*) &:=& V(D) \cup I_{q^{n-1}}
\ = \ V(D) \cup \{z_{x} \mid x \in [q]^{n-1} \}, \\
A(D^*) &:=& A(D) \cup \bigcup_{x\in [q]^{n-1}} \{(y,z_x) \mid y\in S_n(x)\},
\end{eqnarray*}
we obtain an acyclic digraph $D^*$ such that $C(D^*) = H(n,q) \cup I_{k^*}$,
where
\begin{eqnarray*}
k^* &:=& k + q^{n-1}\\
&= & q \cdot \alpha(n-1,q) - (q-1) \cdot k(H(q,q)) + q^{n-1} \\
&=& q \cdot ((n-1-q)q^{n-2} + k(H(q,q)))
- (q-1) \cdot k(H(q,q)) +q^{n-1} \\
&=& (n-q)q^{n-1} + k(H(q,q))
\ = \ \alpha(n,q).
\end{eqnarray*}
Therefore, $k(H(n,q)) \leq \alpha(n,q)$.
Hence, the theorem holds.
\end{proof}
The following corollary follows from Theorem \ref{thm:UH_nq}.
\begin{Cor} \label{cor:UH_n3}
For $n\ge 3$, we have
$k(H(n,3)) \leq (n-3)3^{n-1}+6$.
\end{Cor}
In the following, we show a lower bound for the competition numbers
of ternary Hamming graphs $H(n,3)$ where $n\ge 3$.
\begin{Lem} \label{lem2}
Let $n \ge 3$, and let $G$ be a subgraph
of the ternary Hamming graph $H(n,3)$
with $10$ vertices.
Then the number of triangles in $G$
is at most $6$.
Moreover, it is exactly equal to $6$ if and only if $G$ is
isomorphic to either $H_1 \cup I_1$ or $H_2$ in Figure~\ref{fig:H1H2H3}.
\end{Lem}
\begin{proof}
We denote by $t_G$ the number of triangles in a graph $G$.
For a vertex $v$ in a graph $G$,
we denote by $t_G(v)$ the number of triangles in $G$
containing the vertex $v$.
For any graph $G$, it holds that
\begin{equation}\label{nG}
\sum_{v\in V(G)} t_G(v) = 3\times t_G.
\end{equation}
Let $G$ be a subgraph of $H(n,3)$ with $10$ vertices
such that $t_G$ is the maximum
among all the subgraphs of $H(n,3)$ with $10$ vertices.
Since the graph $H_2$ drawn in Figure
\ref{fig:H1H2H3}
is a subgraph of $H(n,3)$
with $10$ vertices and $6$ triangles,
we have $t_G \ge 6$.
Let $v_1, v_2, v_3, \ldots, v_9, v_{10}$
be the vertices of $G$.
\begin{figure}
\caption{Induced subgraphs of $H(n,3)$}
\label{fig:H1H2H3}
\end{figure}
Suppose that there exists a vertex $v_i$
such that $t_G(v_i) \ge 3$. Then we will reach a contradiction.
Without loss of generality, we may assume that $t_G(v_1) \ge 3$
and that $\{v_1,v_2,v_3\}$, $\{v_1,v_4,v_5\}$, $\{v_1,v_6,v_7\}$
are cliques which contain $v_1$.
Since each edge is contained exactly one triangle by Lemma \ref{lem:LB2},
then the subgraph of $H(n,3)$ induced by $\{v_1, v_2, \ldots, v_7\}$
is isomorphic to
the graph $H_4$ drawn in Figure~\ref{fig;H4H5}.
\begin{figure}
\caption{Subgraphs of $H(n,3)$}
\label{fig;H4H5}
\end{figure}
Since $G$ has at least $6$ triangles,
there exist at least three triangles $T_1,T_2,T_3$ in $G$
which are not in the induced subgraph $H_4$.
For each $1 \leq i\leq 3$,
the triangle $T_i$ has at least two vertices in $\{v_8,v_9,v_{10}\}$
since each edge of $G$ is contained in exactly one triangle.
We may assume that $T_1 \supseteq \{v_8,v_9\}$,
$T_2 \supseteq \{v_8,v_{10}\} $, and $T_3 \supseteq \{v_9,v_{10}\}$.
Then $\{v_8, v_9, v_{10}\}$ forms a triangle,
which contradicts the fact that each edge of $G$
is contained exactly one triangle.
Therefore, $t_G(v_i) \leq 2$ for all $v_i \in V(G)$.
Since $t_G(v_i) \leq 2$ for all $v_i \in V(G)$,
it holds that $\sum_{i=1}^{10} t_G(v_i) \le 20$.
By (\ref{nG}),
$\sum_{i=1}^{10} t_G(v_i)$ must be a multiple of $3$,
$\sum_{i=1}^{10} t_G(v_i) \le 18$.
Since $t_G \ge 6$, we also have
$\sum_{i=1}^{10} t_G(v_i) \ge 18$ by (\ref{nG}).
Thus, we have $\sum_{i=1}^{10} t_G(v_i) = 18$ and $t_G=6$.
Since $\sum_{i=1}^{10} t_G(v_i) = 18$ and $t_G(v_i) \leq 2$
for all $v_i \in V(G)$,
without loss of generality,
we may assume that $t_{G}(v_{10}) \le 1$.
Let $G - v_{10}$ be the graph obtained from $G$ by deleting $v_{10}$.
To show the ``moreover'' part,
it is sufficient to show that $G - v_{10}$ is isomorphic to the graph
$H_1 (\cong H(2,3))$ drawn in Figure \ref{fig:H1H2H3}.
Note that $15 \le \sum_{i=1}^{9} t_{G-v_{10}} (v_i) \le 18$,
and $t_{G - v_{10}} (v) \le 2$ for all $v\in V(G - v_{10})$.
Then $G - v_{10}$ has at least $6$ vertices $v$
such that $t_{G - v_{10}}(v) = 2$, and so
$G - v_{10}$ has a triangle $T_0=\{v_1,v_2,v_3\}$
such that $t_{G - v_{10}} (v_i)=2$ for any $i=1,2,3$.
For $i=1,2,3$, let $T_i$ be the triangle containing
$v_i$ and $T_i \neq T_0 $.
Then the triangles $T_1$, $T_2$, and $T_3$ are mutually vertex disjoint,
and we may assume that
$T_1=\{v_1, v_4, v_5\}$, $T_2=\{v_2, v_6, v_7\}$,
and $T_3=\{v_3, v_8, v_9\}$
(see the graph $H_5$ drawn
in Figure~\ref{fig;H4H5}).
Since $G - v_{10}$ has at least $6$ vertices $v$
such that $t_{G - v_{10}}(v) = 2$,
we may assume that $t_{G - v_{10}}(v_4) = 2$.
Then the triangle containing $v_4$ in $G - v_{10}$, which is not $T_1$,
are containing one of $\{v_6,v_7\}$ and one of $\{v_8,v_9\}$.
Without loss of generality, we may assume that $\{v_4,v_6,v_8\}$
is a triangle.
Since $\{v_1, v_2, v_6, v_4\}$ forms a cycle of length four,
without loss of generality, we may let
\[
\begin{array}{llll}
v_1 := (1,1,1, \ldots, 1), & v_2 := (2,1,1, \ldots, 1),
& v_4 := (1,2,1, \ldots, 1), & v_6 := (2,2,1, \ldots, 1).
\end{array}
\]
Since $\{v_1,v_2,v_3\}$ is a triangle,
we have $v_3=(3,1,1, \ldots,1)$.
Since $\{v_1,v_4,v_5\}$ is a triangle,
we have $v_5=(1,3,1, \ldots,1)$.
Since $\{v_2,v_6,v_7\}$ is a triangle,
we have $v_7=(2,3,1, \ldots,1)$.
Since $\{v_4,v_6,v_8\}$ is a triangle,
we have $v_8=(3,2,1, \ldots,1)$.
Since $\{v_3,v_8,v_9\}$ is a triangle,
we have $v_9=(3,3,1, \ldots,1)$.
Then $\{v_5,v_7,v_9\}$ forms a triangle,
and hence $G - v_{10}$ is isomorphic to $H_1$
in Figure \ref{fig:H1H2H3}.
We complete the proof.
\end{proof}
\begin{Thm} \label{thm:LH_n3}
For $n\ge 3$, we have
$k(H(n,3)) \geq (n-3)3^{n-1} +6$.
\end{Thm}
\begin{proof}
Suppose that $k(H(n,3)) \leq (n-3)3^{n-1}+5$.
Then, by Corollary \ref{cor:Dabc},
there exists an acyclic digraph $D$
such that $C(D)=H(n,3)\cup I_{(n-3)3^{n-1}+5}$,
$N^-_D(v) \in \ensuremath{\mathcal{F}}(n,3) \cup \{\emptyset\}$ for any $v \in V(D)$,
and
$|\{ v \in V(D) \mid N^-_D(v) = \emptyset\}| =5$.
Let $v_1, v_2, \ldots, v_{(n-3)3^{n-1} + 5}$ be an acyclic ordering of $D$.
We may assume that the vertices $v_1, v_2, \ldots, v_5$
have no in-neighbors in $D$.
For $6 \le i \le 12$, let
$\ensuremath{\mathcal{F}}_i:= \{ N^-_{D}(x) \mid x \in \{ v_1, v_2, \ldots, v_{i}, v_{i+1} \} \}$
and
let $G_i$ be the subgraph of $H(n,3)$ induced
by $\{ v_1,v_2,\ldots, v_{i} \}$.
Note that $\ensuremath{\mathcal{F}}_i$ contains $i-4$ triangles
whose vertices are in $\{ v_1,v_2,\ldots, v_{i} \}$ and that
$G_i$ contains all the triangles in $\ensuremath{\mathcal{F}}_i$.
Since $G_{10}$ is a subgraph of $H(n,3)$
with $10$ vertices containing $6$ triangles, by Lemma \ref{lem2},
$G_{10}$ is isomorphic to $H_1 \cup I_1$ or $H_2$,
where $H_1$ and $H_2$ are the graphs drawn in
Figure \ref{fig:H1H2H3}.
Since $G_{10}$ is a subgraph of $G_{11}$
and $G_{11}$ has $7$ triangles,
$G_{10}$ must be isomorphic to $H_2$
and $G_{11}$ must be isomorphic to $H_3$ in Figure \ref{fig:H1H2H3}.
Then we can observe that any subgraph of $H(n,3)$ with $12$ vertices
containing $H_3$ cannot have $8$ triangles.
However, $G_{12}$ is a subgraph of $H(n,3)$
with $12$ vertices and $8$ triangles,
which is a contradiction.
Hence $k(H(n,3)) \geq (n-3)3^{n-1}+6$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{Mainthm:H(n,3)}]
Theorem \ref{Mainthm:H(n,3)} follows from
Corollary \ref{cor:UH_n3} and
Theorem \ref{thm:LH_n3}.
\end{proof}
\section{Concluding Remarks}
In this paper, we gave the exact values of
the competition numbers of ternary Hamming graphs.
Note that the bound given in Theorem \ref{thm:UH_nq} is tight
when $q=2$ or $3$, that is,
$k(H(n,q))=(n-q)q^{n-1} + k(H(q,q))$ holds for $n \geq q$ and $q \in\{2,3\}$.
We left a question for a further research
whether or not the bound in Theorem \ref{thm:UH_nq}
is tight for any $2 \le q \le n$.
\end{document} | math | 21,982 |
\begin{document}
\title{there exists\;xtbf{A comparison of the Almgren-Pitts and the Allen-Cahn min-max theory}}
\author{Akashdeep Dey \thanks{Email: [email protected], [email protected]}}
\date{}
\maketitle
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{pro}[thm]{Proposition}
\newtheorem{clm}[thm]{Claim}
\newtheorem*{thm*}{Theorem}
\newtheorem*{lem*}{Lemma}
\newtheorem*{clm*}{Claim}
\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{ex}[thm]{Example}
\theoremstyle{remark}
\newtheorem{rmk}[thm]{Remark}
\numberwithin{equation}{section}
\newcommand{manifold\;}{manifold\;}
\newcommand{varifold\;}{varifold\;}
\newcommand{hypersurface\;}{hypersurface\;}
\newcommand{Riemannian\;}{Riemannian\;}
\newcommand{constant\;}{constant\;}
\newcommand{metric\;}{metric\;}
\newcommand{such that\;}{such that\;}
\newcommand{Theorem\;}{Theorem\;}
\newcommand{Lemma\;}{Lemma\;}
\newcommand{Proposition\;}{Proposition\;}
\newcommand{Equation\;}{Equation\;}
\newcommand{equation}{equation}
\newcommand{there exists\;}{there exists\;}
\newcommand{Therefore, \;}{Therefore, \;}
\newcommand{with respect to\;}{with respect to\;}
\newcommand{\mathbb{R}}{\mathbb{R}}
\newcommand{\mathbb{N}}{\mathbb{N}}
\newcommand{\mathbb{Z}}{\mathbb{Z}}
\newcommand{\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}}{\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}}
\newcommand{\rightarrow}{\rightarrow}
\newcommand{\longrightarrow}{\longrightarrow}
\newcommand{function\;}{function\;}
\newcommand{Suppose\;}{Suppose\;}
\newcommand{\partial}{\partial}
\newcommand{sequence\;}{sequence\;}
\newcommand{continuous\;}{continuous\;}
\newcommand{\mathbf{F}}{\mathbf{F}}
\newcommand{\mathbf{M}}{\mathbf{M}}
\newcommand{\mathbf{L}}{\mathbf{L}}
\newcommand{\mathcal{C}(M)}{\mathcal{C}(M)}
\newcommand{\mathcal{Z}_n(M^{n+1}; \mathbb{Z}_2)}{\mathcal{Z}_n(M^{n+1}; \mathbb{Z}_2)}
\newcommand{\mathbf{B}}{\mathbf{B}}
\newcommand{\tilde{X}}{\tilde{X}}
\newcommand{\tilde{X}}{\tilde{X}}
\newcommand{\tilde{X}k}{\tilde{X}[K]}
\newcommand{\tilde{\Phi}}{\tilde{\Phi}}
\newcommand{\tilde{\Phi}}{\tilde{\Phi}}
\newcommand{\partialst}{\partial^{*}}
\newcommand{\tilde{\Psi}}{\tilde{\Psi}}
\newcommand{\mathcal{Z}}{\mathcal{Z}}
\newcommand{\al}{\alpha}\newcommand{\be}{\beta}\newcommand{\ga}{\gamma}\newcommand{\de}{\partialta}\newcommand{\ve}{\varepsilon}\newcommand{\et}{\eta}\newcommand{\ph}{\phi}\newcommand{\vp}{\varphi}\newcommand{\ps}{\psi}\newcommand{\ka}{\kappa}\newcommand{\la}{\lambda}\newcommand{\om}{\omega}\newcommand{\rh}{\rho}\newcommand{\si}{\sigma}\newcommand{\tht}{\theta}\newcommand{\ta}{\tau}\newcommand{\ch}{\chi}\newcommand{\ze}{\zeta}\newcommand{\Ga}{\Gamma}\newcommand{\De}{\Delta}\newcommand{\Ph}{\Phi}\newcommand{\Ps}{\Psi}\newcommand{\La}{\Lambda}\newcommand{\Om}{\Omega}\newcommand{\Si}{\Sigma}\newcommand{\Th}{\Theta}
\newcommand{\norm}[1]{\left\|#1\right\|}\newcommand{\md}[1]{\left|#1\right|}\newcommand{\Md}[1]{\Big|#1\Big|}
\newcommand{\db}[1]{[\![#1]\!]}
\newcommand{\ov}[1]{\overline{#1}}
\newcommand{\operatorname{Vol}}{\operatorname{Vol}}
\newcommand{\operatorname{Ind}}{\operatorname{Ind}}
\newcommand{\vto}{
}\newcommand{\vth}{
}\newcommand{varifold\;o}{
}
\begin{abstract}
\noindent
Min-max theory for the Allen-Cahn equation was developed by Guaraco \cite{Guaraco} and Gaspar-Guaraco \cite{GG1}. They showed that the Allen-Cahn widths are greater than or equal to the Almgren-Pitts widths. In this article we will prove that the reverse inequalities also hold i.e. the Allen-Cahn widths are less than or equal to the Almgren-Pitts widths. Hence, the Almgren-Pitts widths and the Allen-Cahn widths coincide. We will also show that all the closed minimal hypersurfaces (with optimal regularity) which are obtained from the Allen-Cahn min-max theory are also produced by the Almgren-Pitts min-max theory. As a consequence, we will point out that the index upper bound in the Almgren-Pitts setting, proved by Marques-Neves \cite{MN_index} and Li \cite{Li_index}, can also be obtained from the index upper bound in the Allen-Cahn setting, proved by Gaspar \cite{Gaspar} and Hiesmayr \cite{H}.
\end{abstract}
\section{Introduction}
Minimal submanifolds are defined by the condition that they are the critical points of the area functional. In \cite{alm_article,Alm}, Almgren studied the topology of the space of cycles and developed a min-max theory for the area functional. He proved that any closed, Riemannian manifold \((M^{n+1},g)\) contains a minimal variety of dimension \(l\) for every \(1\leq l \leq n\). The regularity theory in the co-dimension $1$ case was further developed by Pitts \cite{Pitts} and Schoen-Simon \cite{SS}. They proved that in a closed, Riemannian manifold \((M^{n+1},g)\), \(n+1 \geq 3\), there exists a closed, minimal hypersurface which is smooth and embedded outside a singular set of Hausdorff dimension \(\leq n-7.\)
In recent years, there have been a lot of research activities in the Almgren-Pitts min-max theory. By the work of Marques-Neves \cite{MN_ricci_positive} and Song \cite{Song}, every closed Riemannian manifold \((M^{n+1},g)\), \(3\leq n+1\leq 7\), contains infinitely many closed, minimal hypersurfaces. This was conjectured by Yau \cite{yau}. In \cite{IMN}, Irie, Marques and Neves proved that for a generic metric\; \(g\) on \(M\), the union of all closed, minimal hypersurfaces is dense in \((M,g)\). This theorem was later quantified by Marques, Neves and Song in \cite{MNS} where they proved that for a generic metric there exists an equidistributed sequence of closed, minimal hypersurfaces in \((M,g)\). In higher dimensions, Li \cite{Li} has proved that every closed Riemannian manifold equipped with a generic metric\; contains infinitely many closed minimal hypersurfaces with optimal regularity. The Weyl law for the volume spectrum $\{\om_k\}_{k=1}^{\infty}$, proved by Liokumovich, Marques and Neves \cite{LMN} played a major role in the arguments of \cite{IMN,MNS,Li}.
The Morse index of the minimal hypersurfaces produced by the Almgren-Pitts min-max theory has been obtained by Marques and Neves when the ambient dimension $3\leq n+1\leq 7$. In \cite{MN_index}, Marques and Neves showed that the index of the min-max minimal hypersurface is bounded from above by the dimension of the parameter space. Zhou \cite{Zhou2} has proved that for a generic (bumpy) metric, the min-max minimal hypersurfaces have multiplicity one which was conjectured by Marques and Neves. Using the Morse index upper bound \cite{MN_index} and multiplicity one theorem \cite{Zhou2}, Marques and Neves \cite{MN_index_2} have proved the following theorem. For a generic (bumpy) metric\; there exists a sequence of closed, embedded, two-sided minimal hypersurfaces \(\{\Si_k\}_{k=1}^{\infty}\) in \((M^{n+1},g)\) such that\; \(\Ind(\Si_k)=k\) and \(\cH^n(\Si_k)=\om_k\sim k^{\frac{1}{n+1}}\). This theorem has been generalized by Marques, Montezuma and Neves in \cite{mmn} where they have proved the strong Morse inequalities for the area functional. In higher dimensions, Morse index upper bound has been proved by Li \cite{Li_index}.
In \cite{Guaraco}, Guaraco introduced a new approach for the min-max construction of minimal hypersurfaces which was further developed by Gaspar and Guaraco in \cite{GG1}. This approach is based on the study of the limiting behaviour of solutions to the Allen-Cahn equation. The Allen-Cahn equation (with parameter \(\ve >0\)) is the following semi-linear, elliptic PDE
\[AC_{\ve}(u):=\ve\De u-\ve^{-1} W'(u)=0\]
where \(W:\mathbb{R} \rightarrow \mathbb{R}\) is a double well potential e.g. \(W(t)=\frac{1}{4}(1-t^2)^2.\) The solutions of this equation are precisely the critical points of the energy functional
\[E_{\ve}(u)=\int_M\ve\frac{|\nabla u|^2}{2}+\frac{W(u)}{\ve}.\]
Building on the work of Hutchinson-Tonegawa \cite{HT}, Tonegawa \cite{t} and Tonegawa-Wickramasekera \cite{TW}, Guaraco \cite{Guaraco} proved that if \(\{u_i\}_{i=1}^{\infty}\) is a sequence of solutions to the Allen-Cahn equation \(AC_{\ve_i}(u_i)=0\), \(\ve_i \rightarrow 0\) with \(E_{\ve_i}(u_i)\) and \(\Ind(u_i)\) are uniformly bounded, then, possibly after passing to a subsequence, the level sets of \(u_i\) accumulate around a closed, minimal hypersurface with optimal regularity. (Such a minimal hypersurface is called a there exists\;xtit{limit-interface}.) Moreover, by a mountain-pass argument he proved the existence of critical points of \(E_{\ve}\) with uniformly bounded energy and Morse index. In this way he obtained a new proof of the previously mentioned theorem of Almgren-Pitts-Schoen-Simon. The index of the limit-interface is bounded by the index of the solutions. This was proved by Hiesmayr \cite{H} assuming the limit-interface is two-sided and by Gaspar \cite{Gaspar} in the general case.
In \cite{GG1,GG}, Gaspar and Guaraco studied the phase transition spectrum which is the Allen-Cahn analogue of the volume spectrum. They proved that the phase transition spectrum satisfies a Weyl law similar to the volume spectrum and gave alternative proofs of the density \cite{IMN} and the equidistribution \cite{MNS} theorems. In \cite{CM}, Chodosh and Mantoulidis proved the multiplicity one conjecture in the Allen-Cahn setting in dimension \(3\) and the upper semicontinuity of the Morse index when the limit-interface has multiplicity one. As a consequence, they proved that for a generic (bumpy) metric \(g\) on a closed manifold \(M^3\), there exists a sequence of closed, embedded, two-sided minimal surfaces \(\{\Si_k\}_{k=1}^{\infty}\) in \((M^3,g)\) such that\; \(\Ind(\Si_k)=k\) and \(there exists\;xt{area}(\Si_k)\sim k^{1/3}\).
If \(\Si\) is a non-degenerate, separating, closed, embedded minimal hypersurface in a closed Riemannian manifold, Pacard and Ritor\'{e} \cite{pr} constructed solutions of the Allen-Cahn equation \(AC_{\ve}(u)=0\) for sufficiently small \(\ve>0\) whose level sets converge to \(\Si\). The uniqueness of these solutions has been proved by Guaraco, Marques and Neves \cite{gmn}. The construction of Pacard and Ritor\'{e} has been extended by Caju and Gaspar \cite{cg} in the case when all the Jacobi fields of \(\Si\) are induced by the ambient isometries.
In the present article we will be interested in the question to what extent the Almgren-Pitts min-max theory and the Allen-Cahn min-max theory agree. Part of this question has been answered by Guaraco \cite{Guaraco} and Gaspar-Guaraco \cite{GG1}; they proved that the Almgren-Pitts widths are less than or equal to the Allen-Cahn widths. The aim of this article is to prove the reverse inequality i.e the Allen-Cahn widths are less than or equal to the Almgren-Pitts widths.
To precisely state our main result, we need some facts about the universal \(G\)-principal bundle. We will follow the book by Dieck \cite{Dieck}*{Chapter 14.4} and the paper by Gaspar and Guaraco \cite{GG1}*{Appendix B} where further details can be found. Let \(G\) be a topological group and \(p_{G}:EG \rightarrow BG\) be a universal \(G\)-principal bundle (which is unique upto isomorphism). Given a topological space \(B\), there exists a one-to-one correspondence between the set of homotopy classes of maps \(B \rightarrow BG\) and the set of isomorphism classes of numerable \(G\)-principal bundles over \(B\). If \(f_1,f_2:B \rightarrow BG\) are homopotic, \(f_1^{*}EG\) and \(f_2^{*}EG\) are isomorphic numerable principal \(G\)-bundles over \(B\). Conversely, if \(E\) is a numerable free \(G\)-space, there exists a \(G\)-map from \(E\) to \(EG\) which is unique upto \(G\)-homotopy. Denoting \(B=E/G\), if \(F_1,F_2:E\rightarrow EG\) are \(G\)-maps, they descend to homotopic maps \(f_1,\;f_2 :B \rightarrow BG\). We also note the following facts: a numerable \(G\)-principal bundle \(\mathfrak{p}:\cE \rightarrow \cB\) is universal if \(\cE\) is a contractible topological space \cite{Dieck}*{14.4.12}; each open covering of a paracompact space is numerable \cite{Dieck}*{13.1.3}.
We refer to Section 2 for the definitions and notations used in the rest of this section. Let \((M^{n+1},g)\) be a closed Riemannian manifold, \(n+1\geq 3\). Let \(X\) be a cubical complex and we fix a double cover \(\pi:\tilde{X} \rightarrow X\). Since the space \(\mathbf{I}_{n+1}(M^{n+1};\mathbf{F};\mathbb{Z}_2)\) is contractible \cite{MN_index_2} and every metric space is paracompact, \(\partial:\mathbf{I}_{n+1}(M^{n+1};\mathbf{F};\mathbb{Z}_2) \rightarrow \cZ_n(M^{n+1};\mathbf{F};\mathbb{Z}_2)\) is a universal \(\mathbb{Z}_2\)-principal bundle. We denote by \(\Pi\) the homotopty class of maps \(X \rightarrow \cZ_n(M^{n+1};\mathbf{F};\mathbb{Z}_2)\) corresponding to the double cover \(\pi:\tilde{X} \rightarrow X\). More concretely, \(\Pi\) is the set of all maps \(\Ph: X \rightarrow \cZ_n(M^{n+1};\mathbf{F};\mathbb{Z}_2)\) such that\; \(\ker(\Ph_{*})=there exists\;xt{im}(\pi_{*})\) where
\[\Ph_{*}:\pi_1(X)\rightarrow \pi_1\left(\cZ_n(M^{n+1};\mathbf{F};\mathbb{Z}_2)\right)(=\mathbb{Z}_2),\quad \pi_{*}:\pi_1(\tilde{X})\rightarrow \pi_1(X)\]
are the maps induced by \(\Ph,\;\pi\).
Similarly, \(H^1(M)\setminus\{0\}\) is contractible and there is a free \(\mathbb{Z}_2\) action on this space given by \(u \mapsto -u\). Therefore, \(H^1(M)\setminus\{0\}\) (equipped with the \(\mathbb{Z}_2\) action) is the total space of a universal \(\mathbb{Z}_2\)-principal bundle. Let \(\tilde{\Pi}\) denote the set of all \(\mathbb{Z}_2\)-equivariant maps \(h:\tilde{X} \rightarrow H^1(M)\setminus\{0\}\) i.e. if \(T:\tilde{X} \rightarrow \tilde{X}\) is the deck transformation, \(h(T(x))=-h(x)\) for all \(x \in \tilde{X}\).
The following theorem follows from the work of Guaraco \cite{Guaraco} and Gaspar-Guaraco \cite{GG1}.
\begin{thm}[\cite{Guaraco,GG1}]\label{thm GG}
Let \(\mathbf{L}_{AP}(\Pi)\) be the Almgren-Pitts width of \(\Pi\) (equationref{2 def AP width}) and \(\mathbf{L}_{\ve}(\tilde{\Pi})\) be the \(\ve\)-Allen-Cahn width of \(\tilde{\Pi}\) (equationref{2 def AC width}). Then the following inequality holds.
\begin{equation}
\mathbf{L}_{AP}(\Pi)\leq \frac{1}{2\si}\liminf_{\ve\rightarrow 0^{+}} \mathbf{L}_{\ve}(\tilde{\Pi}).\label{width ineq of GG}
\end{equation}
As a consequence, the following inequality holds between the volume spectrum and the phase transition spectrum.
\begin{equation}
\om_p \leq \frac{1}{2\si}\liminf_{\ve\rightarrow 0^{+}}c_{\ve}(p) \; \forall p\in \mathbb{N}.\label{spec ineq of GG}
\end{equation}
\end{thm}
In the present article we will show that the reverse inequality also holds. More precisely, we will prove the following theorem.
\begin{thm}\label{thm main thm}
We have the following inequality between the Almgren-Pitts width and the \(\ve\)-Allen-Cahn width.
\begin{equation}
\frac{1}{2\si}\limsup_{\ve \rightarrow 0^{+}} \mathbf{L}_{\ve}(\tilde{\Pi}) \leq \mathbf{L}_{AP}(\Pi).\label{width ineq}
\end{equation}
As a consequence we have,
\begin{equation}
\frac{1}{2\si}\limsup_{\ve \rightarrow 0^{+}}c_{\ve}(p)\leq \om_p\;\forall p\in \mathbb{N}.\label{spec ineq}
\end{equation}
\end{thm}
Hence, combining equationref{width ineq of GG} and equationref{width ineq} we conclude that \(\frac{1}{2\si}\lim_{\ve\rightarrow 0^{+}}\mathbf{L}_{\ve}(\tilde{\Pi})\) exists and is equal to \(\mathbf{L}_{AP}(\Pi)\). Similarly, equationref{spec ineq of GG} and equationref{spec ineq} together imply that \(\frac{1}{2\si}\lim_{\ve\rightarrow 0^{+}}c_{\ve}(p)\) exists and is equal to \(\om_p\) for all \(p \in \mathbb{N}\). When the ambient dimension \(3\leq n+1 \leq 7\), it was proved by Gaspar and Guaraco \cite{GG} that \(\lim_{\ve\rightarrow 0^{+}}c_{\ve}(p)\) exists.
The next theorem essentially follows from the work of Hutchinson-Tonegawa \cite{HT}, Guaraco \cite{Guaraco} and Gaspar-Guaraco \cite{GG1}. Informally speaking, it says that all the minimal hypersurfaces obtained from the Allen-Cahn min-max theory are also produced by the Almgren-Pitts min-max theory.
\begin{thm}\label{thm critical set}
Let \(\mathbf{C}_{AC}(\tilde{\Pi})\) be as defined at the end of Section \ref{section 2.5} and \(\mathbf{C}_{AP}(\Pi)\) be as defined at the end of Section \ref{section 2.3}. If \(V \in \mathbf{C}_{AC}(\tilde{\Pi})\), then \(V \in \mathbf{C}_{AP}(\Pi)\) as well.
\end{thm}
Combining the index estimate of Gaspar \cite{Gaspar} (Theorem \ref{thm index}) and the above Theorem \ref{thm critical set}, one can obtain an alternative proof of the following Morse index upper bound in the Almgren-Pitts min-max theory proved by Marques-Neves \cite{MN_index} and Li \cite{Li_index}.
\begin{thm}[\cite{MN_index,Li_index}]
Let $\dim(X)=\dim(\tilde{X})=k$. There exists \(V \in \mathbf{C}_{AP}(\Pi)\) such that\; \(\Ind(\spt(V))\) is less than or equal to \(k\).
\end{thm}\vto
Indeed, by the min-max theory for the Allen-Cahn functional (see Section 2.4), for all sufficiently small \(\ve>0\) there exists a min-max critical point \(u_{\ve}\) of \(E_{\ve}\) (corresponding to the homotopy class \(\tilde{\Pi}\)) such that\; \(\Ind(u_{\ve})\leq k\). Hence, by Theorem \ref{thm interface}, \ref{thm index} and \ref{thm critical set}, there exists
\[V\in\mathbf{C}_{AC}(\tilde{\Pi})\subset \mathbf{C}_{AP}(\Pi)\]
such that\; \(\Ind(\spt(V))\leq k.\)
there exists\;xtbf{Acknowledgements.} I am very grateful to my advisor Prof. Fernando Cod√° Marques for many helpful discussions and for his support and guidance. The author is partially supported by NSF grant DMS-1811840.
\section{Notations and Preliminaries}
\subsection{Notations}
Here we summarize the notations which will be frequently used later.
\begin{tabular}{ll}
\([m]\) & the set \(\{1,2,\dots,m\}\)\\
$\mathcal{H}^s$ & the Hausdorff measure of dimension $s$ \\
\(A\;\dot\cup\; B\) & the disjoint union of \(A\) and \(B\)\\
$int(A),\ov{A}$ & the interior of \(A\), the closure of \(A\) (in a topological space)\\
\(\mathcal{C}(M)\) & the space of Caccippoli sets in \(M\)\\
\(\partial A\) & the topological boundary of \(A\) (in a topological space) = \(\ov{A}\setminus int(A)\); \(\partial\) will also \\ & denote the boundary of a current or the boundary of a cell in a cell-complex.\\
\(\partialst E\) & the reduced boundary of a Caccioppoli set \(E\)\\
\(\db{S}\) & the current associated to the rectifiable set \(S\)\\
\(\md{\Si}\) & the varifold associated to the rectifiable set \(\Si\)\\
\(\norm{V}\) & the Radon measure associated to the varifold \(V\) \\
\(B^c\) & the complement of \(B\) in \(M\) i.e. \(M\setminus B\)\\
\(B(p,r)\) & the geodesic ball centered at \(p\) with radius \(r\)\\
\(A(p,r,R)\) & the annulus centered at \(p\) with radii \(r<R\)\\
\(d(-,S) \) & distance from a set \(S \subset (M,g)\) \\
\(\cN_{\rh}(S)\) & the set of points \(x \in (M,g)\) such that\; \(d(x,S)\leq \rh\) \\
\(\cT_{\rh}(S)\) & the set of points \(x \in (M,g)\) such that\; \(d(x,S)=\rh\) \\
\(H^1(M)\) & the Sobolev space \(\{f \in L^2(M): there exists\;xt{distributional derivative } \nabla f \in L^2(M) \} \)
\end{tabular}
\subsection{Varifolds}
Here we will briefly discuss the notion of varifold; further details can be found in Simon's book \cite{Sim}. Given a Riemannian manifold\; $(M^{n+1},g)$, let $G_nM$ denote the Grassmanian bundle of $n$-dimensional hyperplanes over $M$. An $n$there exists\;xtit{-varifold} in $M$ is a positive Radon measure on $G_nM$. If $V$ is an \(n\)-varifold and $\mathbf{p}:G_nM \rightarrow M$ is the canonical projection map, $\|V\| = \mathbf{p}_{*}V$ is a Radon measure on $M$; $\|V\|(A)=V(\mathbf{p}^{-1}(A))$. The topology on the space of $n$-varifolds is given by the weak* topology i.e. a net $\{V_i\}_{i \in \cI}$ converges to $V$ if and only if
$$\int _{G_nM} f(x, \omega)dV_i(x, \omega) \rightarrow \int _{G_nM} f(x, \omega)dV(x, \omega)$$
for all $f \in C_c(G_nM)$. If \(\Si \subset M\) is \(n\)-rectifiable and \(\tht : \Si \rightarrow [0,\infty)\) is in \(L^1_{there exists\;xt{loc}}(\Si,\cH^n)\), the \(n\)-varifold \(\mathbf{v}(\Si,\tht)\) is defined by
\[\mathbf{v}(\Si,\tht)(f)=\int_{\Si}f(x,T_x\Si)\tht(x)\; d\cH^n(x)\]
where \(T_x\Si\) is the approximate tangent space of \(\Si\) at $x$ which exists $\cH^n \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \Si$ a.e. Such a varifold is called a there exists\;xtit{rectifiable $n$-varifold}. When $\tht$ is the constant function $1$, $\mathbf{v}(\Si,\tht)$ is denoted by $\md{\Si}$; it is called the there exists\;xtit{varifold associated to $\Si$.}
If $\varphi : M \rightarrow M$ is a \(C^1\) map and $ V$ is an \(n\)-varifold in \(M\), the push-forward varifold $\varphi_{\#}V$ is defined as follows.
$$(\varphi_{\#}V)(f) = \int _{G_n^{+}M}f\left(\varphi(x),D\varphi|_x(\omega)\right)J\varphi(x, \omega)dV(x, \omega)$$
where
$$J\varphi(x, \omega) = \left(\det\left(\left(D \varphi(x) \big |_{\omega}\right)^t \circ \left(D\varphi(x) \big |_{\omega}\right)\right)\right)^{1/2}$$
is the Jacobian factor and \[G_n^{+}M=\{(x,\om)\in G_nM: J\vp(x,\om)\neq 0\}.\] If \(V=\mathbf{v}(\Si,\tht)\) is a rectifiable \(n\)-varifold, \(\vp_{\#}V=\mathbf{v}(\vp(\Si),\tilde{\tht})\); \(\tilde{\tht}:\vp(\Si)\rightarrow \mathbb{R}\) is defined by \[\tilde{\tht}(y)=\sum_{x \in \vp^{-1}(y)\cap \Si}\tht(x).\]
We denote by \(\cV_n(M)\) the closure of the space of rectifiable \(n\)-varifolds in \(M\) with respect to\; the above varifold weak topology. The \(\mathbf{F}\) metric on \(\cV_n(M)\) is defined as follows \cite{Pitts}*{page 66}.
\[\mathbf{F}(V,W)=\sup \{V(f)-W(f): f\in C_c^{0,1}(G_nM), \md{f}\leq 1, there exists\;xt{Lip}(f)\leq 1\}.\]
For every \(a>0\), the \(\mathbf{F}\)-metric topology and the varifold weak topology coincide on the set \(\{V\in \cV_n(M):\norm{V}(M)\leq a\}\).
\subsection{Almgren-Pitts min-max theory}
In this subsection, we will recall some of the definitions in the Almgren-Pitts min-max theory; we refer to the papers by Marques and Neves \cite{MN_Willmore,MN_ricci_positive,MN_index,MN_index_2}, Schoen and Simon \cite{SS} and the book by Pitts \cite{Pitts} for more details. To discuss the Almgren-Pitts min-max theory we need to introduce the following spaces of currents. Let \((M^{n+1},g)\) be a closed Riemannian manifold. \(\mathbf{I}_l(M^{n+1};\mathbb{Z}_2)\) is the space of \(l\)-dimensional mod \(2\) flat chains in \(M\); we only need to consider \(l=n,n+1\). \(\mathcal{Z}_n(M^{n+1}; \mathbb{Z}_2)\) denotes the space of flat chains \(T \in \mathbf{I}_n(M;\mathbb{Z}_2)\) such that\; \(T=\partial U\) for some \(U \in \mathbf{I}_{n+1}(M;\mathbb{Z}_2)\). For \(T \in \cZ_n(M;\mathbb{Z}_2)\), \(\md{T}\) stands for the varifold associated to \(T\) and \(\norm{T}\) is the radon measure associated to \(\md{T}\). \(\cF\) and \(\mathbf{M}\) denote the flat and mass norm on \(\mathbf{I}_l(M;\mathbb{Z}_2)\). When \(l=n+1\), these two norms coincide. The \(\mathbf{F}\) metric on the space of currents is defined as follows.
\begin{align*}
&\mathbf{F}(U_1,U_2)=\cF(U_1,U_2)+\mathbf{F}(\md{\partial U_1},\md{\partial U_2}) there exists\;xt{ if } U_1, U_2 \in \mathbf{I}_{n+1}(M;\mathbb{Z}_2);\\
&\mathbf{F}(T_1,T_2)=\cF(T_1,T_2)+\mathbf{F}(\md{T_1},\md{ T_2}) there exists\;xt{ if } T_1, T_2 \in \mathbf{I}_{n}(M;\mathbb{Z}_2).
\end{align*}
It is proved in \cite{MN_index_2} that the space \(\mathbf{I}_{n+1}(M;\mathbf{F};\mathbb{Z}_2)\) is contractible and the boundary map \(\partial:\mathbf{I}_{n+1}(M;\mathbf{F};\mathbb{Z}_2)\rightarrow \cZ_n(M;\mathbf{F};\mathbb{Z}_2)\) is a two sheeted covering map. By the constancy theorem, if \(U_1,U_2 \in \mathbf{I}_{n+1}(M;\mathbb{Z}_2)\) such that\; \(\partial U_1=\partial U_2\), either \(U_1=U_2\) or \(U_1+U_2=\db{M}\).
Let \(X,\; \Pi\) be as in Section 1. The there exists\;xtit{Almgren-Pitts width} of the homotopy class \(\Pi\) is defined by
\begin{equation}
\mathbf{L}_{AP}(\Pi)=\inf_{\Ph \in \Pi}\sup_{x\in X}\left\{\mathbf{M}(\Ph(x))\right\}.\label{2 def AP width}
\end{equation}
A sequence of maps \(\Ph_i:X \rightarrow \cZ_n(M;\mathbf{F};\mathbb{Z}_2)\) in \(\Pi\) is called a there exists\;xtit{minimizing sequence} if
\begin{equation*}
\limsup_{i \rightarrow \infty}\sup_{x\in X}\left\{\mathbf{M}(\Ph_i(x))\right\}=\mathbf{L}_{AP}(\Pi).
\end{equation*}
The there exists\;xtit{critical set} of a minimizing sequence\; \(\{\Ph_i\}\), denoted by \(\mathbf{C}\left(\{\Ph_i\}\right)\), is the set of all varifolds \(V \in \cV_n(M)\) such that\; \(\norm{V}(M)=\mathbf{L}_{AP}(\Pi)\) and there exist sequences \(\{i_j\}\subset \{i\}\) and \(\{x_j\} \subset X\) such that\;
\[\lim_{j \rightarrow \infty}\mathbf{F}\left(V,\md{\Ph_{i_j}(x_{j})}\right)=0.\]
We define \(\mathbf{C}_{AP}(\Pi)\) to be the set of all varifolds \(V\in \cV_n(M)\) such that\; \(V \in \mathbf{C}\left(\{\Ph_i\}\right)\) for some minimizing sequence \(\{\Ph_i\}\subset \Pi\), \(V\) is a stationary, integral varifold and \(\spt(V)\) is a closed, minimal hypersurface with optimal regularity (i.e. smooth and embedded outside a singular set of Hausdorff dimension \(\leq n-7\)). The theorem of Almgren-Pitts-Schoen-Simon guarantees that \(\mathbf{C}_{AP}(\Pi)\) is non-empty.\label{section 2.3}
\subsection{Allen-Cahn min-max theory}
We now briefly discuss the min-max theory for the Allen-Cahn functional following the papers by Guaraco \cite{Guaraco} and Gaspar-Guaraco \cite{GG1} where further details can be found. Let \(W:\mathbb{R} \rightarrow \mathbb{R}\) be a smooth, symmetric, double well potential. More precisely, \(W\) has the following properties. \(W\geq 0\); \(W(-t)=W(t)\) for all \(t \in \mathbb{R}\); \(W\) has exactly three critical points \(0,\pm 1\); \(W(\pm 1)=0\) and \(W''(\pm 1)>0\) i.e. \(\pm 1\) are non-degenerate minima; \(0\) is a local maximum. The Allen-Cahn energy (with parameter \(\ve>0\)) is given by
\[E_{\ve}(u)=\int_M\ve\frac{|\nabla u|^2}{2}+\frac{W(u)}{\ve}.\]
As mentioned earlier,
\[AC_{\ve}(u):=\ve\De u-\ve^{-1} W'(u)=0\]
if and only if \(u\) is a critical point of \(E_{\ve}\).
Let \(\tilde{X},\; \tilde{\Pi}\) be as in Section 1. The there exists\;xtit{\(\ve\)-Allen-Cahn width} of the homotopy class \(\tilde{\Pi}\) is defined by
\begin{equation}
\mathbf{L}_{\ve}(\tilde{\Pi})=\inf_{h \in \tilde{\Pi}} \sup_{x \in \tilde{X}} E_{\ve}(h(x)). \label{2 def AC width}
\end{equation}
A sequence of maps \(h_i:\tilde{X} \rightarrow H^1(M)\setminus \{0\}\) in \(\tilde{\Pi}\) is called a there exists\;xtit{minimizing sequence} for \(E_{\ve}\) if
\[\limsup_{i \rightarrow \infty}\sup_{x \in \tilde{X}}E_{\ve}(h_i(x))=\mathbf{L}_{\ve}(\tilde{\Pi})\]
\(u\) is called a there exists\;xtit{min-max critical point} of \(E_{\ve}\) (corresponding to the homotopy class \(\tilde{\Pi}\)) if \(u\) is a critical point of \(E_{\ve}\) with \(E_{\ve}(u)=\mathbf{L}_{\ve}(\tilde{\Pi})\) and
\[\lim_{i \rightarrow \infty} d_{H^1(M)}\left( u, h_i(\tilde{X})\right) = 0 \]
where \(\{h_i\}\) is a minimizing sequence for \(E_{\ve}\) in \(\tilde{\Pi}\).
As \(W\) is an even function, \(E_{\ve}\) is invariant under the \(\mathbb{Z}_2\) action on \(H^1(M)\) given by \(u \mapsto -u\) i.e. \(E_{\ve}(u)=E_{\ve}(-u)\). Moreover, as proved in \cite{Guaraco}*{Proposition 4.4}, \(E_{\ve}\) satisfies the Palais-Smale condition for bounded sequences. Hence, as explained in \cite{GG1}, if \(\ve>0\) satisfies
\begin{equation}\label{condition}
\mathbf{L}_{\ve}(\tilde{\Pi})<E_{\ve}(0)=\frac{W(0)}{\ve}\operatorname{Vol}(M,g)
\end{equation}
(which holds for \(\ve\) sufficiently small by equationref{width ineq}), one can apply Corollary 10.5 of \cite{gh} to the \(\mathbb{Z}_2\)-homotopic family \(\tilde{\Pi}\) to conclude that there exists a min-max critical point \(u_{\ve}\) of \(E_{\ve}\) (corresponding to the homotopy class \(\tilde{\Pi}\)) such that\; \(\Ind(u_{\ve})\leq k\). (Here \(k\) is the dimension of the parameter space \(\tilde{X}.\)) The restriction on \(\ve\) given by equationref{condition} is due to the fact that the space \(H^1(M)\setminus\{0\}\) is not complete; equationref{condition} ensures that a minimizing sequence for \(E_{\ve}\) is bounded away from \(0\).
\subsection{Convergence of the phase interfaces}
Let us define \(F:\mathbb{R} \rightarrow \mathbb{R}\) and the energy constant \(\si\) as follows.
\begin{equation}
F(a)=\int_{0}^{a}\sqrt{W(s)/2}\; ds;\quad \quad \si=\int_{-1}^{1}\sqrt{W(s)/2}\;ds\quad there exists\;xt{so that}\quad F(\pm 1)=\pm \frac{\si}{2}.\label{2 defn F}
\end{equation}
Let \(u \in C^1(M)\), \(w=F \circ u\). The \(n\)-varifold associated to \(u\) is defined by
\[V[u](A)=\frac{1}{\si}\int_{-\infty}^{\infty}\md{\{w=s\}}(A)\; ds\]
for every Borel set \(A \subset G_nM\). On a closed manifold, if \(AC_{\ve}(u)=0\) and \(u\) is not identically equal to \(\pm 1\), \(\md{u}< 1\) \cite{GG1}*{Lemma 2.2}; in that case, in the definition of \(V[u]\) the integral can be taken over the interval \((-\si/2,\si/2).\)
Building on the work of Hutchinson-Tonegawa \cite{HT}, Tonegawa \cite{t} and Tonegawa-Wickramasekera \cite{TW}, Guaraco \cite{Guaraco} has proved the following theorem.
\begin{thm}[\cite{HT,t,TW,Guaraco}]\label{thm interface}
Let \(\{u_{i}:M\rightarrow (-1,1)\}_{i=1}^{\infty}\) be a sequence of smooth functions such that\;
\begin{itemize}
\item[(i)] \(AC_{\ve_i}(u_i)=0\) with \(\ve_i\rightarrow 0\) as \(i \rightarrow \infty\);
\item[(ii)] There exists \(E_0>0\) and \(I_0\in \mathbb{N}_0\) such that\; \(E_{\ve_i}(u_i)\leq E_0\) and \(\Ind(u_i)\leq I_0\) for all \(i \in \mathbb{N}\).
\end{itemize}
Then, there exists a stationary, integral varifold \(V\) such that\; possibly after passing to a subsequence, \(V[u_i]\rightarrow V\) in the sense of varifolds. Moreover,
\[\norm{V}(M)=\frac{1}{2\si}\lim_{i \rightarrow \infty} E_{\ve_i}(u_i)\]
and \(\spt(V)\) is a closed, minimal hypersurface with optimal regularity.
\end{thm}\vto
The proof of the regularity of the limit-interface depends on the regularity theory of stable, minimal hypersurfaces developed by Wickramasekera \cite{w}.
The upper bound for the Morse index of the limit-interface in the above theorem was proved by Gaspar \cite{Gaspar} and Hiesmayr \cite{H} (when the limit-interface is two sided).
\begin{thm}[\cite{Gaspar,H}]\label{thm index}
The Morse index of the limit-interface \(\spt(V)\) in the above Theorem \ref{thm interface} is less than or equal to \(I_0\).
\end{thm}\vth
Lastly, we introduce the following definition. \(\mathbf{C}_{AC}(\tilde{\Pi})\) is the set of all stationary, integral \(n\)-varifolds \(V\) such that\; \(\spt(V)\) is a closed, minimal hypersurface with optimal regularity and \(V\) is the varifold limit of \(V[u_i]\) for some sequence\; \(\{u_i\}_{i=1}^{\infty}\) such that\; \(u_i\) is a min-max critical point of \(E_{\ve_i}\) (with \(\ve_i\rightarrow 0\)) corresponding to the homotopy class \(\tilde{\Pi}\). By the discussion of Section 2.4 and Theorem \ref{thm interface}, \(\mathbf{C}_{AC}(\tilde{\Pi})\) is non-empty.\label{section 2.5}
\section{Proof of the width inequality}
In this section we will prove our main Theorem \ref{thm main thm}. Let us fix \(\et >0\). Let \(L=\mathbf{L}_{AP}(\Pi)\). By the interpolation theorems of Pitts and Marques-Neves \cite{MN_ricci_positive,MN_index_2} there exists $\Ph:X \rightarrow \mathcal{Z}_n(M^{n+1}; \mathbf{M};\mathbb{Z}_2)$ such that
\begin{equation}
\sup_{x \in X}\{\mathbf{M}(\Ph(x))\}<L+\et. \label{eq mass of Phi is bounded}
\end{equation}
We choose \(\tilde{\Phi}:\tilde{X} \rightarrow \mathcal{C}(M)\) which is a lift of \(\Ph\) i.e. for all \(x \in \tilde{X}\),
\[\db{\partial^{*}\tilde{\Phi}(x)}=\partial \db{\tilde{\Phi}(x)}=\Ph(\pi(x)).\]
\(\tilde{\Phi}\) is \(\mathbb{Z}_2\)-equivariant i.e. if \(T:\tilde{X} \rightarrow \tilde{X}\) is the deck transformation, \(\db{\tilde{\Phi}(x)}+\db{\tilde{\Phi}(T(x))}=\db{M}\) for all \(x \in \tilde{X}\).
\subsection{Approximation of a Caccippoli set by open sets with smooth boundary}
In this subsection, following the book by Giusti \cite{giu} and the paper by Miranda-Pallara-Paronetto-Preunkert \cite{MPPP}, we briefly discuss the fact that a Caccioppoli set can be approximated by open sets with smooth boundary. We begin with the following theorem.
\begin{thm}[\cite{giu}*{Theorem 1.17}, \cite{MPPP}*{Proposition 1.4}]\label{thm 3.2}
Let \(E\in \mathcal{C}(M)\). There exists a sequence of smooth functions \(\{f_j:M \rightarrow \mathbb{R}\}_{j=1}^{\infty}\) such that\; \(0\leq f_j \leq 1\) for all \(j\) and
\[\lim_{j \rightarrow \infty}\int_M|f_j -\ch_E|\;d\cH^{n+1} =0 \quad there exists\;xt{ and } \quad \int_M|D\ch_E|=\lim_{j\rightarrow \infty}\int_M|Df_j|.\]
\end{thm}
Following \cite{giu}*{Proof of Theorem 1.24}, for \(t\in(0,1)\), let us define \(E_{j,t}=\{f_j>t\}\). Then,
\begin{equation}
|f_j-\ch_E|> t there exists\;xt{ on } E_{j,t}\setminus E \quad there exists\;xt{ and } \quad |f_j-\ch_E|\geq 1-t there exists\;xt{ on } E \setminus E_{j,t} \label{eq 3.2.1}
\end{equation}
which implies
\begin{equation}
\int_M|f_j-\ch_{E}|\;d\cH^{n+1}\geq \min \{t, 1-t\}\int_M |\ch_{E_{j,t}}-\ch_E|\;d\cH^{n+1}. \label{eq 3.2.2}
\end{equation}
Hence, for all \(t\in (0,1)\),
\begin{align}
&\lim_{j\rightarrow \infty} \int_M |\ch_{E_{j,t}}-\ch_E|\;d\cH^{n+1} =0 \label{eq: convergence in L1 }\\
&\Longrightarrow \int_M|D\ch_E| \leq \liminf_{j \rightarrow \infty}\int_M |D\ch_{E_{j,t}}|.\label{eq 3.2.3}
\end{align}
Therefore, \; using Theorem \ref{thm 3.2}, the co-area formula for the BV function and equationref{eq 3.2.3} we obtain the following inequalities.
\[\int_M |D\ch_E|=\lim_{j\rightarrow \infty}\int_M|Df_j|\geq \int_{0}^{1}\left(\liminf_{j \rightarrow \infty}\int_M |D\ch_{E_{j,t}}|\right)dt\geq \int_M|D\ch_E|.\]
This implies
\begin{equation}
\liminf_{j \rightarrow \infty}\int_M |D\ch_{E_{j,t}}|= \int_M|D\ch_E| there exists\;xt{ for a.e. } t\in (0,1). \label{eq: convergence of liminf}
\end{equation}
We choose \(t_0\in(0,1)\) such that\; \(t_0\) is a regular value of \(f_j\) for all \(j\) and equationref{eq: convergence of liminf} holds for \(t=t_0\). Further, possibly after passing to a subsequence, we can assume that
\begin{equation}
\lim_{j \rightarrow \infty}\int_M |D\ch_{E_{j,t_0}}|= \int_M|D\ch_E|. \label{eq: convergence of lim}
\end{equation}
Let us define \(E_j=\ov{E}_{j,t_0}\). Since
\[E_{j,t_0} \subset \ov{E}_{j,t_0} \subset E_{j,t_0} \cup \{f_j=t_0\}, \]
we have \(\cH^{n+1}\left(\ov{E}_{j,t_0} \setminus E_{j,t_0} \right).\) From equationref{eq: convergence in L1 } and equationref{eq: convergence of lim} we conclude that
\begin{equation}
\ch_{E_j} \rightarrow \ch_E there exists\;xt{ in } L^1(M) there exists\;xt{ and } \lim_{j \rightarrow \infty}\int_M |D\ch_{E_{j}}|= \int_M|D\ch_E|. \label{eq: convergence in F}
\end{equation}
By \cite{Pitts}*{2.1(18)(f), page-63}, equationref{eq: convergence in F} implies that \([\![\partial^{*} E_j]\!]\) converges to \([\![\partial^{*}E]\!]\) in \(\mathbf{F}\).
Let us fix \(p \in M\) and \(R>0\). Using equationref{eq: convergence in F},
\[\lim_{j\rightarrow \infty}\int_{0}^{R}\bigg(\int_{\partial B(p,t)}|\ch_{E_j}-\ch_E| \; d\cH^n\bigg)dt = \lim_{j \rightarrow \infty}\int_{B(p,R)}|\ch_{E_j}-\ch_E| \; d\cH^{n+1}=0.\]
Hence, there exists a subsequence \(\{\ch_{E_{j_s}}\} \subset \{\ch_{E_j}\}\) such that\;
\[\lim_{s \rightarrow \infty}\int_{\partial B(p,t)}|\ch_{E_{j_s}}-\ch_E| \; d\cH^n =0 there exists\;xt{ for a.e. } t \in (0,R).\]
Next, we define \(F_j=\ov{\{f_j< t_0\}}\) and \(F=M\setminus E\). Then, \(E_j \cap F_j\subset \{f_j=t_0\} \) and \(E_j \cup F_j=M\). Therefore, \([\![E_j]\!]+[\![F_j]\!]=[\![M]\!]\), \(\partial^{*}E_j=\partial^{*}F_j\) and \(\ch_{F_j} \rightarrow \ch_F\) in \(L^1(M)\). We note that \(\partial E_j,\; \partial F_j \subset \{f_j=t_0\}\). As the reduced boundary is a subset of the topological boundary, we also have \(\partial^{*}E_j=\partial^{*}F_j\subset \{f_j=t_0\}.\) Since \(\{f_j=t_0\}\) is a smooth, closed hypersurface\; in \(M\), for each \(a \in \{f_j=t_0\}\) there exist \(\rh>0\) and co-ordinates \(\{x_1,x_2,\dots , x_{n+1}\}\) on \(B(a,\rh)\) such that\; \(x_i(a)=0\) for all \(i\) and
\[\left(B(a,\rh)\cap \{f_j=t_0\}\right)=\{x \in B(a,\rh):x_{n+1}=0\}.\]
Let \(G_1=\{x \in B(a,\rh):x_{n+1}<0\}\) and \(G_2=\{x \in B(a,\rh):x_{n+1}>0\}\) so that
\[\left(B(a,\rh)\setminus \{f_j=t_0\}\right)=G_1\; \dot\cup \; G_2 .\]
\(G_1\) and \(G_2\) are connected open sets. We have the following three mutually exclusive cases.
\begin{itemize}
\item[(1)] \(f_j >t_0\) on both \(G_1\) and \(G_2\); this implies \(a \in int(E_j)\setminus F_j\);
\item[(2)] \(f_j <t_0\) on both \(G_1\) and \(G_2\); this implies \(a \in int(F_j) \setminus E_j\);
\item[(3)] \(f_j >t_0\) on one of \(G_1\) and \(G_2\), and \(f_j <t_0\) on the other; this implies \(a \in E_j \cap F_j\).
\end{itemize}
From the above three cases we can conclude that \(\partial^{*}E_j=\partial^{*}F_j= E_j \cap F_j= \partial E_j=\partial F_j\) which is a smooth, closed, embedded hypersurface\; in \(M\). Indeed, the set of points \(a \in \{f_j=t_0\}\) for which the above item (3) holds is both open and closed in \(\{f_j=t_0\}\); hence it is the union of certain connected components of \(\{f_j=t_0\}.\)
From the above discussion we arrive at the following proposition.
\begin{pro}\label{pro: modified smooth approximation}
Let \(E \in \mathcal{C}(M)\) and \(F=M \setminus E\). Then, for each \(j \in \mathbb{N}\) there exist closed sets \(E_j,\; F_j \in \mathcal{C}(M)\) such that\; the followings hold.
\begin{itemize}
\item[(i)]\([\![E_j]\!]+[\![F_j]\!]=[\![M]\!]\) and \(M=E_j \cup F_j\).
\item[(ii)] \(\ch_{E_{j}} \rightarrow \ch_{E}\) and \(\ch_{F_{j}} \rightarrow \ch_{F}\) in \(L^1(M)\).
\item[(iii)] \(\partial^{*}E_j=\partial^{*}F_j= E_j \cap F_j= \partial E_j=\partial F_j\) is a smooth, closed, embedded hypersurface\; in \(M\).
\item[(iv)] \([\![\partial^{*} E_j]\!]=[\![\partial^{*} F_j]\!]\) converges to \([\![\partial^{*}E]\!]=[\![\partial^{*}F]\!]\) in \(\mathbf{F}\).
\item[(v)] For every \(p \in M\) and \(R>0\) there exist subsequences \(\{\ch_{E_{j_s}}\} \subset \{\ch_{E_j}\}\) and \(\{\ch_{F_{j_s}}\} \subset \{\ch_{F_j}\}\) such that\;
\[\lim_{s \rightarrow \infty}\int_{\partial B(p,t)}|\ch_{E_{j_s}}-\ch_E| \; d\cH^n =0 \; there exists\;xt{ and }\; \lim_{s \rightarrow \infty}\int_{\partial B(p,t)}|\ch_{F_{j_s}}-\ch_F| \; d\cH^n =0\]
for a.e. \(t \in (0,R).\)
\end{itemize}
\end{pro}
\subsection{Preliminary constructions}
Let \(D\) be a countable, dense subset of \(M\) and \(there exists\;xt{inj}(M)\) be the injectivity radius of \((M,g)\). We consider
\[\mathscr{B}=\{B(p,t): p\in D, t \in (0, there exists\;xt{inj}(M))\cap \mathbb{Q}\}\]
which is also a countable set. Let us assume that \(M\) is isometrically embedded in some Euclidean space \(\mathbb{R}^m\). We have the following theorem which is a consequence of Sard's theorem.
\begin{thm}(\cite{Nic}*{Corollary 1.25})
Let \(\Si\) be a closed submanifold of \(\mathbb{R}^m\). For \(v \in \mathbb{R}^m\) we define \(f_v: \mathbb{R}^m \rightarrow \mathbb{R}\) by \(f_v(x)=\langle x, v\rightarrowngle\). Then, there exists a generic set \(V \subset \mathbb{R}^m\) (depending on \(\Si\)) such that\; for all \(v \in V\), \(f_v\vert_{\Si}\) is a Morse function on \(\Si\).
\end{thm}
By the above theorem, there exists \(\om \in \mathbb{R}^m\) such that\; \(f_{\om}\vert_{M}\) and \(f_{\om}\vert_{\partial B}\) are Morse functions for all \(B \in \mathscr{B}\). By composing scaling and translation with \(f_{\om}\), we can assume that \(f_{\om}(M)=[1/3,2/3]\). From now on, whenever we will consider \(f_{\om}\), it will be assumed that \(f_{\om}: M \rightarrow [1/3,2/3]\). Let us choose \(r_0\in (0, there exists\;xt{inj}(M))\) such that\;
\begin{itemize}
\item \(\cH^n\left(f_{\om}^{-1}(t)\cap \overline{B}(p,r_0)\right)<\et/2\) for all \(t \in [1/3,2/3],\; p\in M\);
\item \(\mathbf{M} \left(\Ph(x)\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \overline{B}(p,r_0)\right)<\et\) for all \(x \in X,\; p \in M\);
\end{itemize}
where \(\Ph\) is as chosen at the beginning of Section 3. Such a choice of \(r_0\) is possible because of the `no concentration of mass property'(\cite{MN_ricci_positive}). We choose \(p_i \in D\) such that\;
\begin{equation}
M=\bigcup_{i=1}^{I}B\left(p_i,r_0/4\right) ; \qquad \mathbf{B}_i^0 :=B(p_i,r_0). \label{def: bbb0}
\end{equation}
Hence, in particular, we have
\begin{align}
&\cH^n\left(f_{\om}^{-1}(t)\cap \overline{\mathbf{B}_i^0}\right)<\et/2 \;\;\forall t \in [1/3,2/3], i \in [I];\nonumber\\
&\mathbf{M} \left(\Ph(x)\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \overline{\mathbf{B}_i^0}\right)<\et\;\; \forall x \in X, i \in [I]. \label{eq mass of Phi in Biz}
\end{align}
\begin{lem}
There exists \(r_1 \in (r_0/2, 3r_0/4)\cap \mathbb{Q}\), \( \partialta \in (0, r_0/8)\) such that\;
\[\mathbf{M}\left( \Ph(x)\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \ov{A}(p_i, r_1-2\partialta, r_1+\partialta)\right)<\frac{\et}{I}\]
for all \(x \in X\) and \(i \in [I].\)\label{lem: mass of phi in annulus}
\end{lem}
\begin{proof}
By the compactness of \(X\), there exists\; \(\{x_j\}_{j=1}^J \subset X\) such that\;
\begin{equation}
\forall x \in X, \; \exists j \in [J] there exists\;xt{ such that } \mathbf{M} \left( \Ph (x) - \Ph(x_j)\right) < \frac{\et}{2I}.\label{eqn: xj's}
\end{equation}
For \(i \in [I], j\in [J]\) we define
\[\cR_{ij}=\left\{ r' \in \left(r_0/2,3r_0/4\right):\mathbf{M}\left(\Ph(x_j)\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \partial B(p_i, r')\right)<\frac{\et}{2I}\right\}.\]
We note that
\[\left(r_0/2,3r_0/4\right)\setminus \cR_{ij} there exists\;xt{ is finite } \forall\; i,j.\]
Hence, we can choose
\begin{equation}
r_1 \in \Big(\bigcap_{i\in [I], j\in [J]}\cR_{ij}\Big)\cap\left(r_0/2,3r_0/4\right)\cap \mathbb{Q}. \label{eqn: choosing r1}
\end{equation}
Therefore, \; we have
\[\mathbf{M}\left(\Ph(x_j)\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \partial B(p_i,r_1)\right)< \frac{\et}{2I} \quad \forall \; i,j.\]
Hence, we can choose \(\partialta \in (0,r_0/8)\) such that\;
\[\mathbf{M}\left(\Ph(x_j)\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \ov{A}(p_i,r_1-2\partialta,r_1+\partialta)\right)<\frac{\et}{2I} \quad \forall \; i,j;\]
which implies (by equationref{eqn: xj's})
\[\mathbf{M}\left( \Ph(x)\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \ov{A}(p_i, r_1-2\partialta, r_1+\partialta)\right)<\frac{\et}{I} \quad \forall \; x \in X \; i \in [I].\]
\end{proof}
\begin{lem}
There exists \(\partialta\in (0,r_0/8)\) such that
\[\cH^n\left(f_{\om}^{-1}(t)\cap \ov{A}(p_i, r_1-2\de, r_1+\partialta)\right)<\frac{\et}{2I}\]
for all \(t \in [1/3,2/3]\), \(i \in [I]\). \label{lem: mass of initial morse fn level set}
\end{lem}
\begin{proof}
We assume by contradiction that there exist \(\{t_j\}_{j=1}^{\infty}\subset [1/3,2/3]\) and \(\{d_j\}_{j=1}^{\infty}\subset \mathbb{R}^{+}\) such that\; \(d_j \rightarrow 0\) and for some \(i \in [I]\)
\[\cH^n\left(f_{\om}^{-1}(t_j)\cap \ov{A}(p_i, r_1-2d_j, r_1+d_j)\right)\geq\frac{\et}{2I}\]
holds for all \(j \in \mathbb{N}\). Without loss of generality we can assume that \(t_j \rightarrow t\). Denoting
\[\mu_j=\cH^n\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} f_{\om}^{-1}(t_j) \qquad there exists\;xt{ and } \qquad \mu=\cH^n\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} f_{\om}^{-1}(t),\]
we have that \(\mu_j\) weakly converges to \(\mu\) (in the sense of radon measure). Let us fix \(l \in \mathbb{N}\). There exists \(j_0 \in \mathbb{N}\) such that\; \(d_j \leq l^{-1}\) for all \(j \geq j_0\). Hence,
\[\mu_j(\ov{A}(p_i, r_1-2l^{-1}, r_1+l^{-1}))\geq \frac{\et}{2I}\]
for all \(j \geq j_0\). Therefore,
\[\mu(\ov{A}(p_i, r_1-2l^{-1}, r_1+l^{-1}))\geq \limsup_{j \rightarrow \infty} \mu_j(\ov{A}(p_i,r_1-2l^{-1}, r_1+l^{-1} ))\geq \frac{\et}{2I}.\]
This holds for all \(l \in \mathbb{N}\). Denoting \(A_l = \ov{A}(p_i, r_1-2l^{-1}, r_1+l^{-1})\) we have
\[A_{l+1}\subset A_l \quad there exists\;xt{ and } \quad \bigcap_{l=1}^{\infty}A_l= \partial B(p_i, r_1).\]
This implies
\[\cH^n\left(f_{\om}^{-1}(t)\cap \partial B(p_i, r_1)\right)=\mu(\partial B(p_i, r_1))=\lim_{l \rightarrow \infty} \mu(A_l) \geq \frac{\et}{2I}.\]
However, this is not possible. Indeed, \(r_1 \in \mathbb{Q}\) (Lemma \ref{lem: mass of phi in annulus}) and by the choice of \(\om\), \(f_{\om}\vert_{\partial B(p_i, r_1)}\) is a Morse function on $\partial B(p_i, r_1)$; hence \( \cH^n\left(f_{\om}^{-1}(t)\cap \partial B(p_i, r_1)\right)\) must be \(0\).
\end{proof}
We can assume that the two \(\partialta\)'s appearing in Lemma \ref{lem: mass of phi in annulus} and Lemma \ref{lem: mass of initial morse fn level set} are the same. Next, we modify \(f_{\om}\) near the points where it achieves local maxima or local minima (which are not global maxima or minima) to get another Morse function \(f:M \rightarrow [1/3, 2/3]\) such that\; \(f\) has no non-global local maxima or local minima and for all \(t \in [1/3,2/3]\), \(i \in [I]\)
\begin{equation}
\label{eq: mass of modified morse fn level set}
\cH^n\left(f^{-1}(t)\cap \overline{\mathbf{B}^0_i}\right)<\et \quad there exists\;xt{ and } \quad \cH^n\left(f^{-1}(t)\cap \ov{A}(p_i, r_1-2\de, r_1+\partialta)\right)<\frac{\et}{I}.
\end{equation}
Let us introduce the following notation which will be used later.
\begin{align}
&\mathbf{B}_i^1 = B(p_i,r_1); \qquad \cA_1=\bigcup_{i=1}^IA(p_i,r_1-2\de, r_1+\de); \nonumber \\
&\cA_2=\bigcup_{i=1}^IA\left(p_i,r_1-3\de/2, r_1+\de/2\right) ; \qquad \cA=\bigcup_{i=1}^IA(p_i,r_1-\de, r_1). \label{eq: notation for ball and annulus}
\end{align}
\subsection{Cell complex structure on the parameter space}
For \(l \in \mathbb{N}\), \(\mathscr{I}[l]\) is the cell complex on \(\mathscr{I}=[0,1]\) whose \(0\)-cells are
\[[0],\; [l^{-1}], \dots , [1-{l^{-1}}],\; [1]\]
and \(1\)-cells are
\[[0,l^{-1}],\;[l^{-1},2l^{-1}],\dots, [1-l^{-1},1].\]
\(\mathscr{I}^m[l]\) denotes the cell complex on \(\mathscr{I}^m=[0,1]^m\) whose cells are \(\al_1 \otimes \al_2 \otimes \dots \otimes \al_m\) where each \(\al_j \in \mathscr{I}[l]\). We note that \(\mathscr{I}^m[1]\) is the standard cell complex on \(\mathscr{I}^m\). By abuse of notation, we will identify a cell \(\al_1 \otimes \al_2 \otimes \dots \otimes \al_m\) with its support \(\al_1 \times \al_2 \times \dots \times \al_m \subset \mathscr{I}^m\). Similarly, if \(Y\) is a subcomplex of \(\mathscr{I}^m[1]\), \(Y[l]\) is the union of all the cells in \(\mathscr{I}^m[l]\) whose support is contained in \(Y\). If \(\cY\) is a cell complex, \(\cY_p\) will denote the set of all \(p\)-cells in \(\cY\); if \(\al,\be \in \cY\) such that\; \(\be\) is a face of \(\al\) (in the definition of the face, we do not insist that \(\dim(\be)<\dim(\al)\) or \(\dim(\be)=\dim(\al)-1\)), we use the notation \(\be \prec \al\).
\newcommand{\mathscr{I}}{\mathscr{I}}
If \(\la=[il^{-1},(i+1)l^{-1}] \in \il_1\), there exists a canonical map
\begin{equation}
\De_{\la}:\mathscr{I} \rightarrow \mathscr{I};\qquad \De_{\la}(t)=(i+t)l^{-1}\label{eq defining delta lambda}
\end{equation}
such that \(\De_{\la}:\mathscr{I} \rightarrow \la\) is a homeomorphism. Similarly, if \(\al = \al_1 \otimes \al_2 \otimes \dots \otimes \al_m \in \iml_p\), there exists a canonical map \(\De_{\al}:\mathscr{I}^p \rightarrow \sci^m\) defined as follows. There exist precisely \(p\) indices
\[j_1<j_2<\dots <j_p \;there exists\;xt{ such that } \dim(\al_{j_s})=1 \; \forall s=1,2,\dots,p.\]
We define
\begin{equation}
\left(\De_{\al}(t_1,t_2, \dots, t_p)\right)_j=
\begin{cases}
\De_{\al_{j_s}}(t_s) & there exists\;xt{if } j =j_s,\\
\al_j & there exists\;xt{if } j \notin \{j_1,j_2,\dots,j_p\} \Leftrightarrow \dim(\al_j)=0.
\end{cases}
\end{equation}
\(\De_{\al}:\mathscr{I}^p \rightarrow \al\) is a homeomorphism. Let \(D_{\al}:\al \rightarrow \sci^p\) be the inverse of \(\De_{\al}\).
If \(\be \prec \al\) and \(D_{\al}(\be)=\varrho \prec \sci^p\), then the following compatibility relation holds.
\begin{equation}
\De_{\al}\circ \De_{\varrho}=\De_{\be}. \label{eq compatibility}
\end{equation}
We recall from Section 1 that \(X\) is a subcomplex of \(\sci^N[1]\) for some \(N\) and \( \pi: \tilde{X} \rightarrow X\) is a double cover. For \(l\in\mathbb{N}\), \(X[l]\) is the cell complex on \(X\) as defined above. \(\tilde{X}[l]\) denotes the cell complex on \(\tilde{X}\) whose cells are pre-images of the cells of \(X[l]\) via the map \(\pi\). We choose \( K \in \mathbb{N}\) such that\; the followings hold. (Here \(\Ph\) and \(\tilde{\Ph}\) are as chosen at the beginning of Section 3.)
\begin{align}
&there exists\;xt{If } x_1,x_2 \in X[K]_0 there exists\;xt{ belong to a common cell in } X[K],\; \mathbf{M}\left(\Ph(x_1)-\Ph(x_2)\right)<\et. \label{cond: estimating mass norm}\\
&there exists\;xt{If } y_1,y_2 \in \tilde{X}[K]_0 there exists\;xt{ belong to a common cell in } \tilde{X}[K],\; \cH^{n+1}(\tilde{\Ph}(y_1)\De\tilde{\Ph}(y_2))<\frac{\et\de}{I}; \label{cond: estimating flat norm}
\end{align}
where \(\de\) is as in Section 3.2 (Lemma \ref{lem: mass of phi in annulus}, equationref{eq: mass of modified morse fn level set}).
Let \(\{c_q:q \in [Q]\}\) be all the cells of \(X[K]\) indexed in such a way that \(\dim(c_{q_1})\leq \dim(c_{q_2})\) if \(q_1 \leq q_2\). Let \(\{e_q,f_q:q \in [Q]\}\) be the cells of \(\tilde{X}k\) so that \(\pi^{-1}(c_q)=e_q \;\dot\cup\; f_q\). Since \(c_q\) is contractible, \(\pi|_{e_q}:e_q \rightarrow c_q\) and \(\pi|_{f_q}:f_q \rightarrow c_q\) are homeomorphisms. Let us introduce the following notation (\(d=\dim(c_q)\)).
\begin{align}
\De_{c_q}=\De_{q}:\sci^{d} \rightarrow c_q \quad&;\quad D_{c_q}=D_{q}:c_q \rightarrow \sci^{d}; \nonumber\\
\left(\pi|_{e_q}\right)^{-1}\circ \De_q=\De^1_q:\sci^{d} \rightarrow e_q \quad &; \quad D_q \circ \left(\pi|_{e_q}\right)= D^1_q:e_q \rightarrow \sci^{d}; \label{eq canonical maps}\\
\left(\pi|_{f_q}\right)^{-1}\circ \De_q=\De^2_q:\sci^{d} \rightarrow f_q \quad &; \quad D_q \circ \left(\pi|_{f_q}\right)=D^2_q:f_q \rightarrow \sci^{d}. \nonumber
\end{align}
\subsection{Construction of an almost smooth sweepout}
The next proposition follows from Proposition \ref{pro: modified smooth approximation}.
\begin{pro}
\label{pro: seq of good approximations}
There exists a sequence\; \(\{\tilde{\Phi}_j : \tilde{X}[K]_0 \rightarrow \mathcal{C}(M)\}_{j=1}^{\infty}\) such that\; the followings hold for all \(j \in \mathbb{N}\) and \(x \in \tilde{X}[K]_0\).
\begin{itemize}
\item[(i)] \(\tilde{\Phi}_j(x)\) is a closed subset of \(M\).
\item[(ii)] \(\db{\tilde{\Phi}_j(x)}+\db{\tilde{\Phi}_j(T(x))}=\db{M}\) and \(M= \tilde{\Phi}_j(x) \cup \tilde{\Phi}_j(T(x))\).
\item[(iii)] As \(j \rightarrow \infty\), \(\ch_{\tilde{\Phi}_j(x)} \rightarrow \ch_{\tilde{\Phi}(x)}\) in \(L^1(M)\).
\item[(iv)] \(\partial\tilde{\Phi}_j(x)=\partial\tilde{\Phi}_j(T(x))=\tilde{\Phi}_j(x)\cap \tilde{\Phi}_j(T(x))=\partialst\tilde{\Phi}_j(x)=\partialst\tilde{\Phi}_j(T(x))\) is a smooth, closed hypersurface in \(M\).
\item[(v)] Defining \(\Ph_j(x)= \partialst\tilde{\Phi}_j(x)\), as \(j \rightarrow \infty\), \(\db{\Ph_j(x)} \rightarrow \Ph(\pi(x))\) in \(\mathbf{F}\).
\item[(vi)] For all \(i \in [I]\),
\[\lim_{j \rightarrow \infty}\int_{\partial B(p_i,t)}|\ch_{\tilde{\Phi}_j(x)}-\ch_{\tilde{\Phi}(x)}|\;d\cH^n=0\]
for a.e. \(t \in (0,r_1)\).
\end{itemize}
\end{pro}
We will approximate \(\tilde{\Phi}\) by a discrete and `almost smooth' sweepout \(\tilde{\Psi}:\tilde{X}[KI]_0 \rightarrow \mathcal{C}(M)\) which will be constructed using the \(\tilde{\Phi}_j\)'s. The construction is motivated by the interpolation theorems of Almgren \cite{alm_article}, Pitts \cite{Pitts}*{4.5}, Marques-Neves \cite{MN_Willmore}*{Theorem 14.1} and Chambers-Liokumovich \cite{CL}*{Lemma 6.2}. The construction is divided into three parts.
there exists\;xtbf{Part 1.} We recall that \(\{e_q,f_q:q \in [Q]\}\) are the cells of \(\tilde{X}k\). For \(q\in [Q]\) we define the following collection of balls
\begin{equation}
\label{eq: defining the colloection of balls}
\mathscr{B}(q)=\{B(p_i,r_i(q)):i \in [I]\}
\end{equation}
where \(r_i(q)\in (r_1-\de,r_1)\) (\(\de\) is as in Section 3.2) and \(r_i(q)\) is chosen inductively so that the following conditions are satisfied.
\begin{itemize}
\item[(i)] \(\norm{\Ph(\pi(x))}(\partial B(p_i,r_i(q)))=0 \;\;\forall\; x \in (e_q)_0.\)
\item[(ii)] \(\partial B(p_i,r_i(q))\) is transverse to \(\Ph_j(x)\) for all \(x \in (e_q)_0\) and \(j \in \mathbb{N}.\)
\item[(iii)] \(\partial B(p_i,r_i(q))\) is transverse to \(\partial B(p_s,r_s(q))\) for all \(s<i\).
\item[(iv)] \(\partial B(p_i,r_i(q))\) is transverse to \(\partial B(p_j,r_j(q'))\) for all \(q' < q\) and \(j \in [I]\).
\item[(v)] \(\ch_{\tilde{\Phi}_j(x)}\) converges to \(\ch_{\tilde{\Phi}(x)}\) in \(L^1(M, \cH^n\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \partial B(p_i,r_i(q)))\) for all \(x \in (e_q)_0\cup (f_q)_0\)\footnote{For this item we need to use item (vi) of Proposition \ref{pro: seq of good approximations}.}.
\item[(vi)] If \(m=\dim(e_q)=\dim(f_q)\),
\[\int_{\partial B(p_i, r_i(q))}|\ch_{\tilde{\Phi}(x)}-\ch_{\tilde{\Phi}(x')}| \; d\cH^{n}< \frac{2^{2m}\et}{I}\]
for all \(x,x' \in (e_q)_0\) and for all \(x,x' \in (f_q)_0\)\footnote{For this item we need to use equationref{cond: estimating flat norm}.}.
\end{itemize}
Next we choose
\begin{equation*}
r_1>r>\max\{r_i(q):i \in [I], q \in [Q]\}
\end{equation*}
such that\; the followings hold.
\begin{align}
&\norm{\Ph(\pi(x))}(\partial B(p_i,r))=0\; \forall \; x \in \tilde{X}k_0,\; i \in [I];\label{eq defining r}\\
&\norm{\Ph_j(x)}(\partial B(p_i,r))=0\; \forall \; j \in \mathbb{N},\; x \in \tilde{X}k_0,\; i \in [I].\label{eq defining r bis}
\end{align}
We introduce the notation
\begin{equation}
B_i(q)=B(p_i,r_i(q)) \quad;\quad \mathbf{B}_i=B(p_i,r). \label{eq defining biq and bbbi}
\end{equation}
We note that by equationref{def: bbb0}, for all \(q \in [Q]\),
\[\bigcup_{i \in [I]}B_i(q)=M = \bigcup_{i \in [I]}\mathbf{B}_i.\]
there exists\;xtbf{Part 2.} Let us introduce the following definitions.
\begin{equation*}
\mathscr{R}_1=\{\mathbf{B}_i:i \in [I]\}\cup \{M \setminus\ov{\mathbf{B}}_i: i \in [I]\},
\quad
\mathscr{R}_2=\{U_1\cap U_2 \cap \dots\cap U_s:s\in \mathbb{N} there exists\;xt{ and each } U_j \in \mathscr{R}_1\},
\end{equation*}
\begin{equation*}
\mathscr{R}=\{V_1\cup V_2\cup \dots \cup V_t: t \in \mathbb{N} there exists\;xt{ and each }V_j \in \mathscr{R}_2\}
\end{equation*}
\(\mathscr{R}_1, \mathscr{R}_2\) and \(\mathscr{R}\) are finite sets. In a topological space for any two sets \(A\) and \(B\),
\begin{equation}
\partial (A \cap B), \;\partial(A \cup B) \subset \partial A \cup \partial B.\label{eq boundary}
\end{equation}
Hence, for any \(R \in \mathscr{R}\),
\[\partial R \subset \bigcup_{i \in [I]} \partial \mathbf{B}_i \Longrightarrow \norm{\Ph(\pi(x))}(\partial R)=0\; \forall \; x \in \tilde{X}k_0.\]
Moreover, \(R\) is an open subset of \(M\). Therefore, \; by Proposition \ref{pro: seq of good approximations} item (v), as \(j \rightarrow \infty\)
\begin{equation}
\norm{\Ph_j(x)}(R) \longrightarrow \norm{\Ph(\pi(x))}(R)\; \forall \; x \in \tilde{X}k_0. \label{eq phi-j-R converges to phi-R }
\end{equation}
We also note that \(M \in \mathscr{R}\) as \(\cup_{i \in [I]}\mathbf{B}_i=M.\)
\begin{pro}\label{pro existence of alpha}
There exists \(\ga\in \mathbb{N}\) such that\; the followings hold.
\begin{itemize}
\item[(i)] $\big|\norm{\Ph_{\ga}(x)}(R)-\norm{\Ph(\pi(x))}(R)\big|<\et$ for all \(x \in \tilde{X}k_0\) and \(R \in \mathscr{R}\).
\item[(ii)] \(\cH^n(\Ph_{\ga}(x)) < L+2\et \; \forall x \in \tilde{X}k_0.\)
\item[(iii)] \(\cH^n(\Ph_{\ga}\cap \ov{\mathbf{B}_i^0})<\et \; \forall x \in \tilde{X}k_0\).
\item[(iv)] \(\cH^n(\Ph_{\ga}\cap \ov{\cA_1})<\et \; \forall x \in \tilde{X}k_0\).
\item[(v)] For \(q \in [Q]\) and \(i \in [I]\), if \(\dim(e_q)=\dim(f_q)=m\),
\begin{equation}
\int_{\partial B_i(q)}|\ch_{\tilde{\Phi}_{\ga}(x)}-\ch_{\tilde{\Phi}_{\ga}(x')}| \; d\cH^{n}< \frac{2^{2m}\et}{I} \label{eq boundary sphere has small mass}
\end{equation}
if \(x,x' \in (e_q)_0\) or if \(x,x' \in (f_q)_0\).
\end{itemize}
\end{pro}
\begin{proof}
(i) follows from equationref{eq phi-j-R converges to phi-R }. Since, \(M \in \mathscr{R}\), (ii) follows from equationref{eq mass of Phi is bounded} and (i). To obtain (iii) we note that Proposition \ref{pro: seq of good approximations} item (v) and equationref{eq mass of Phi in Biz} imply for each \(x \in \tilde{X}k_0\),
\[\limsup_{j \rightarrow \infty}\norm{\Ph_j(x)}(\ov{\mathbf{B}_i^0})\leq \norm{\Ph(\pi(x))}(\ov{\mathbf{B}_i^0})< \et. \]
Similarly, (iv) follows from Proposition \ref{pro: seq of good approximations} item (v) and Lemma \ref{lem: mass of phi in annulus}. Finally, item (v) follows from items (v) and (vi) of the definition of \(r_i(q)\).
\end{proof}
Next we define
\[\mathscr{S}=\left\{\Ph_{\ga}(x) \cap \partial B_i(q): x \in \tilde{X}k_0, i \in [I], q \in [Q]\right\} \bigcup \Big\{\partial B_i(q) \cap \partial B_j(q'): i,j \in [I]; q,q' \in [Q]\Big\}\]
where \(\ga\) is as in Proposition \ref{pro existence of alpha}. Let \(\cS\) be the closed subset of \(M\) which is the union of all the elements of \(\mathscr{S}\). By the transversality assumptions in the definition of \(r_i(q)\), each non-empty element of \(\mathscr{S}\) is a smooth, closed, co-dimension \(2\) submanifold of \(M\). Hence, for all \(\Si \in \mathscr{S}\)
\[\cH^n(\cT_{\rh}(\Si))=O(\rh)\]
where the constants in \(O(\rh)\) depend on \(\Si\). Therefore, \; there exist constants \(C,\; \rh_0\) depending on the submanifolds contained in the set \(\mathscr{S}\) such that\;
\begin{equation}
\cH^n(\cT_{\rh}(\cS)) \leq C\rh \quad \forall \rh \leq \rh_0. \label{eq area of equidistant hypersurface}
\end{equation}
there exists\;xtbf{Part 3.} \newcommand{\xit}{\tilde{\Xi}}\newcommand{\tilde{X}ki}{\tilde{X}[KI]_0}\newcommand{equationiz}{e_q[I]_0}\newcommand{\fqiz}{f_q[I]_0}\newcommand{equationz}{(e_q)_0}\newcommand{\fqz}{(f_q)_0}
Let \(\cK(M)\) be the set of all closed subsets of \(M\). We are now going to prove that there exists a discrete, `almost smooth' sweepout \(\tilde{\Psi} :\tilde{X}[KI]_0 \rightarrow \cK(M)\) which approximates \(\tilde{\Phi}: \tilde{X} \rightarrow \mathcal{C}(M).\) With some additional work it is possible to ensure that for all \(v \in \tilde{X}ki\), \(\tilde{\Psi}(v) \in \mathcal{C}(M)\); however we do not need this fact to prove Theorem \ref{thm main thm}. Let us introduce the following notation. Let \(\mathbb{K},\; \mathbb{K}'\) be closed subsets of \(M\). \(B\) is a normal geodesic ball and \(A=M\setminus \overline{B}.\) We define
\begin{equation}
\Om(\mathbb{K},\mathbb{K}';B) = (\mathbb{K} \cap \ov{A} ) \cup (\mathbb{K}' \cap \ov{B}) \label{eq defining Omega operation}
\end{equation}
which is also a closed subset of \(M\). This definition is motivated by the construction of ``nested sweepouts" by Chambers and Liokumovich \cite{CL}*{Section 6}. In the following proposition and in its proof, we will use various notations which were introduced previously.
\begin{pro}\label{pro big}
There exists a map \(\tilde{\Psi}:\tilde{X}[KI]_0 \rightarrow \cK(M)\) such that\; \(\tilde{\Psi}\) has the property
\begin{itemize}
\item[\((P)_0\)] \(\tilde{\Psi}(x)=\tilde{\Phi}_{\ga}(x)\) for all \(x \in \tilde{X}k_0\).
\end{itemize}
Moreover, denoting
\[\Ps(v)=\Ps(T(v))=\tilde{\Psi}(v)\cap \tilde{\Psi}(T(v))there exists\;xt{ for } v \in \tilde{X}ki\; ;\]
\[\tilde{\Psi}\circ\De^1_q =\xit^1_q,\quad \quad \tilde{\Psi}\circ\De^2_q =\xit^2_q ,\]
for \(q \in [Q]\) and \(m =\dim(e_q)=\dim(f_q)\), \(\tilde{\Psi}|_{e_q[I]_0}\), \(\tilde{\Psi}|_{f_q[I]_0}\) have the following properties.
\begin{itemize}
\item[$(P0)_{m,q}$] For \(s=1,2\),
\[\xit^s_q(\xi',iI^{-1})=\Om\left(\xit^s_q(\xi',(i-1)I^{-1}),\; \xit^s_q(\xi',1);\;B_i(q)\right)\] for all \(\xi' \in \sci^{m-1}[I]_0 \setminus \left(\partial \sci^{m-1}\right)[I]_0\) and \(i \in [I]\).
\item[\((P1)_{m,q}\)] For all \(v \in equationiz\), \(M=\tilde{\Psi}(v) \cup \tilde{\Psi}(T(v))\).
\item[\((P2)_{m,q}\)] For all \(v \in e_q[I]_0\), \[\tilde{\Psi}(v) \subset \bigcup_{x \in (e_q)_0}\tilde{\Phi}_{\ga}(x).\]
Similarly, for all \(v' \in f_q[I]_0\),
\[\tilde{\Psi}(v') \subset \bigcup_{x' \in (f_q)_0}\tilde{\Phi}_{\ga}(x').\]
\item[\((P3)_{m,q}\)] Let \[\mathscr{F}_q=\{s \in [Q]: e_s \prec e_q there exists\;xt{ or } f_s \prec e_q\}.\]
For all \(v \in e_q[I]_0\),
\[\partial\tilde{\Psi}(v),\partial \tilde{\Psi}(T(v)) \subset \bigg\{\bigcup_{x \in equationz} \Ph_{\ga}(x) \bigg\} \bigcup \bigg\{\bigcup_{i \in [I],\; s \in \mathscr{F}_q } \partial B_i(s) \bigg\}.\]
\item[\((P4)_{m,q}\)] For all \(v \in equationiz\), \(\Ps(v)\setminus \cS=\mathbb{S}_1\;\dot\cup\;\mathbb{S}_2\;\dot\cup\;\dots\;\dot\cup\;\mathbb{S}_{l}\) where each \(\mathbb{S}_j\) is an open subset of a hypersurface belonging the set
\[\Big\{\Ph_{\ga}(x):x \in equationz \Big\} \bigcup \Big\{\partial B_i(s):i \in [I],\; s \in \mathscr{F}_q \Big\}.\]
Here \(\mathscr{F}_q\) is as in \((P3)_{m,q}\).
\item[\((P5)_{m,q}\)] Suppose\; \(v,v'\in equationiz\) or \(v,v'\in \fqiz\); \(e\in e_q[I]_1\cup f_q[I]_1\) such that\; \(v,v'\prec e \). Then, there exists \(\cB=\mathbf{B}_{i(e)}\) for some \(i(e)\in [I]\) such that\;
\begin{align*}
\Ps(v)\cap \left(M\setminus(\cA \cup \cB)\right) & = \Ps(v')\cap \left(M\setminus(\cA \cup \cB)\right);\\
\tilde{\Psi}(v)\cap \left(M\setminus(\cA \cup \cB)\right) & = \tilde{\Psi}(v')\cap \left(M\setminus(\cA \cup \cB)\right);\\
\tilde{\Psi}(T(v))\cap \left(M\setminus(\cA \cup \cB)\right) & = \tilde{\Psi}(T(v')\cap \left(M\setminus(\cA \cup \cB)\right).
\end{align*}
Moreover, \(\cB\) can be characterized as follows. Without loss of generality, let \(v,v'\in equationiz\). If \(D^1_q(v)=(\xi_1,\xi_2,\dots, \xi_m)\) and \(D^1_q(v')=(\xi'_1,\xi'_2,\dots, \xi'_m)\), there exists a unique index \(j \in [m]\) such that\; \(\xi_j \neq \xi'_j\). Let \(i \in [I]\) be such that \(\{\xi_j,\xi'_j\}=\{(i-1)I^{-1},iI^{-1}\}\). Then \(\cB=\mathbf{B}_i.\)
\item[\((P6)_{m,q}\)] For all \(i \in [I]\) and \(v \in equationiz\),
\[\cH^n\left(\Ps(v)\cap \ov{\mathbf{B}^0_i}\right)<2^{4m+2}\et.\]
\item[\((P7)_{m,q}\)] For all \(v \in equationiz\),
\[\cH^n\left(\Ps(v)\cap \ov{\cA}_1\right)<2^{4m+2}\et.\]
\item[\((P8)_{m,q}\)] For all \(v \in equationiz\), \(x \in (e_q)_0\) and \(R \in \mathscr{R}\),
\[\Big|\cH^n\left(\Ps(v) \cap R\right)-\cH^n\left(\Ph_{\ga}(x) \cap R\right)\Big|<2^{4m+2}\et.\]
In particular,
\begin{equation}
\cH^n(\Ps(v))<L+(2^{4m+2}+2) \et.\label{H^n of Psi(v)}
\end{equation}
\end{itemize}
\end{pro}
\begin{proof}
We will inductively construct \(\tilde{\Psi}:equationiz, \fqiz \rightarrow \cK(M)\). If \(\dim(e_q)=\dim(f_q)=0\), then \(e_q[I]=e_q\) and \(f_q[I]=f_q\). In that case we define \[\tilde{\Psi}(e_q)=\tilde{\Phi}_{\ga}(e_q) \quad there exists\;xt{ and } \quad \tilde{\Psi}(f_q)= \tilde{\Phi}_{\ga}(f_q)\] which is precisely the property \((P)_0\). Moreover, for this definition of \(\tilde{\Psi}\) on the \(0\)-cells \(e_q\),\(f_q\), properties \((P0)_{0,q}-(P8)_{0,q}\) are also satisfied. (We need to use Proposition \ref{pro existence of alpha}.)
Let us assume that \(d\geq 1\) and \(\tilde{\Psi}\) is defined on
\[\bigcup_{\dim(e_q)\leq d-1}\Big(equationiz \cup \fqiz\Big)\]
and if \(\dim(e_s)=\dim(f_s)=(d-1)\), \(\tilde{\Psi}|_{e_s[I]_0}\), \(\tilde{\Psi}|_{f_s[I]_0}\) have the properties \((P0)_{(d-1),s}-(P8)_{(d-1),s}.\)
\newcommand{\eniz}{e_{\nu}[I]_0}\newcommand{function\;iz}{f_{\nu}[I]_0}
Let \(\nu \in [Q]\) be such that\; \(\dim(e_{\nu})=\dim(f_{\nu})=d\). We define \(\xit^1_{\nu}, \;\xit^2_{\nu}\) on \(\sci^d[I]_0\) and \(\tilde{\Psi}\) on \(e_{\nu}[I]_0,\; f_{\nu}[I]_0\) as follows. By our assumption, \(\tilde{\Psi}\) is defined on \((\partial e_{\nu})[I]_0, \; (\partial f_{\nu})[I]_0\). For \(z \in (\partial \sci^d)[I]_0,\; \De^1_{\nu}(z)\in (\partial e_{\nu})[I]_0\) and \(\De^2_{\nu}(z) \in (\partial f_{\nu})[I]_0\); for \(s=1,2\) we define
\begin{equation}
\xit^s_{\nu}(z)=\left(\tilde{\Psi} \circ \De^s_{\nu}\right)(z). \label{eq defining xit-s on boundary}
\end{equation}
If \(z \in \sci ^d[I]_0 \setminus \left(\partial \sci^d\right)[I]_0\), we write \(z=(z',iI^{-1})\) with \(z' \in \sci^{d-1}[I]_0\) and \(1 \leq i \leq 1-I^{-1}\); for \(s=1,2\), we define
\begin{equation}
\xit^s_{\nu}(z', iI^{-1}) = \Om \left(\xit^s_{\nu}(z',(i-1)I^{-1}),\; \xit^s_{\nu}(z',1);\;B_i(\nu) \right). \label{eq defining xit-s on the whole cube}
\end{equation}
For \(v \in e_{\nu}[I]_0 \cup f_{\nu}[I]_0\), we define
\begin{equation}
\tilde{\Psi}(v)=
\begin{cases}
\left(\xit^1_{\nu}\circ D^1_{\nu}\right)(v) & there exists\;xt{if } v \in e_{\nu}[I]_0; \\
\left(\xit^2_{\nu}\circ D^2_{\nu}\right)(v) & there exists\;xt{if } v \in f_{\nu}[I]_0.
\end{cases}
\label{eq defining psi-tilde on eq, fq}
\end{equation}
We note that by equationref{eq defining xit-s on boundary}, \(\tilde{\Psi}\) is well-defined on \((\partial e_{\nu})[I]_0\), \((\partial f_{\nu})[I]_0\). Let us also set
\begin{equation}
\Xi_{\nu}(z)=\xit^1_{\nu}(z) \cap \xit^2_{\nu}(z), \; z \in \sci^d[I]_0 \label{eq defining xi}
\end{equation}
so that
\begin{equation}
\Ps(v)=
\begin{cases}
\left(\Xi_{\nu}\circ D^1_{\nu}\right)(v) & there exists\;xt{if } v \in e_{\nu}[I]_0; \\
\left(\Xi_{\nu}\circ D^2_{\nu}\right)(v) & there exists\;xt{if } v \in f_{\nu}[I]_0.
\end{cases}\label{eq defining psi on eq, fq}
\end{equation}
We need to show that \(\tilde{\Psi}\) defined in this way on \(\eniz,function\;iz\) have the properties \((P0)_{d,\nu}-(P8)_{d, \nu}\).
Before we start, for convenience let us introduce the following notation.
\[B_i=B_i(\nu),\quad A_i=M\setminus\ov{B}_i, \quad S_i=\partial B_i = \partial A_i = \ov{A}_i \cap \ov{B}_i.\]
We fix \(z' \in \sci^{d-1}[I]_0\setminus \left(\partial \sci^{d-1}\right)[I]_0\) and define
\[K_i = \xit^1_{\nu}(z', iI^{-1}), \quad L_i=\xit^2_{\nu}(z', iI^{-1}),\quad \Si_i=\Xi_{\nu}(z',iI^{-1})=K_i\cap L_i.\]
From equationref{eq defining xit-s on the whole cube}, for \(i=1,2,\dots, (I-1)\) we obtain,
\begin{align}
& K_i = (K_{i-1}\cap \ov{A}_i) \cup (K_I \cap \ov{B}_i);\nonumber\\
&L_i = (L_{i-1}\cap \ov{A}_i) \cup (L_I \cap \ov{B}_i); \label{eq inductive defn of K_i, L_i, Si_i} \\
& \Si_i=(\Si_{i-1}\cap \ov{A}_i)\cup (\Si_I \cap \ov{B}_i) \cup (K_{i-1}\cap L_I\cap S_i) \cup (K_I\cap L_{i-1}\cap S_i). \nonumber
\end{align}
Using induction one can prove the following.
\begin{align}
& K_i=\Big[K_0\cap \left(\cup_{s=1}^i B_s\right)^c\Big] \bigcup \Big[K_I \cap \left(\cup_{s=1}^i \ov{B}_s\right)\Big]; \nonumber\\
& L_i=\Big[L_0\cap \left(\cup_{s=1}^i B_s\right)^c\Big] \bigcup \Big[L_I \cap \left(\cup_{s=1}^i \ov{B}_s\right)\Big]; \label{eq explicit defn of K_i, L_i}\\
& \Si_i=\Big[\Si_0\cap \left(\cup_{s=1}^i B_s\right)^c\Big] \bigcup \Big[\Si_I \cap \left(\cup_{s=1}^i \ov{B}_s\right)\Big] \bigcup \La_i \quad there exists\;xt{where} \quad \La_i \subset \bigcup_{t=1}^i S_t.\nonumber
\end{align}
For \(i>1\) we have,
\begin{equation}
\La_i=(\La_{i-1}\cap \ov{A}_i) \cup (K_{i-1}\cap L_I \cap S_i) \cup (K_I\cap L_{i-1} \cap S_i). \label{eq reccurence for La_i}
\end{equation}
From the equations in equationref{eq explicit defn of K_i, L_i}, we conclude that the equations in equationref{eq inductive defn of K_i, L_i, Si_i} hold for \(i=I\) as well. We want to denote the top and bottom faces of \(e_{\nu}\) and \(f_{\nu}\) by the indices \(\tau\) and \(\be \in [Q]\) i.e.
\begin{align*}
& \Big\{\De^1_{\nu}(\sci^{d-1} \times \{1\}),\;\De^2_{\nu}(\sci^{d-1} \times \{1\})\Big\}=\{e_{\ta}, f_{\ta}\}; \\
& \Big\{\De^1_{\nu}(\sci^{d-1} \times \{0\}),\;\De^2_{\nu}(\sci^{d-1} \times \{0\})\Big\}=\{e_{\be}, f_{\be}\}.
\end{align*}
We note that the top face of \(e_{\nu}\) could be either \(e_{\ta}\) (in that case the top face of \(f_{\nu}\) is $f_{\tau}$) or \(f_{\tau}\) (in that case the top face of \(f_{\nu}\) is $e_{\tau}$); similar remark applies for the bottom face. For the ease of the presentation, we will assume that \(e_{\ta}\) is the top face of \(e_{\nu}\) and \(e_{\be}\) is the bottom face of \(e_{\nu}\). The other cases will be entirely analogous.
We now proceed in steps to check that \(\tilde{\Psi}\vert_{\eniz}\), \(\tilde{\Psi}\vert_{function\;iz}\) have the properties \((P0)_{d,\nu}-(P8)_{d,\nu}\).
there exists\;xtbf{Step 0.} \((P0)_{d, \nu}\) is satisfied because of our definition of \(\xit_{\nu}^1\) and \(\xit_{\nu}^2\) i.e. equation equationref{eq defining xit-s on the whole cube}.
there exists\;xtbf{Step 1.} By the induction hypothesis, \(\tilde{\Psi}\) restricted to \(e_{\be}[I]_0, \;f_{\be}[I]_0\) (and \(e_{\ta}[I]_0,\; f_{\ta}[I]_0\)) satisfy \((P1)_{(d-1), \be}\) (and \((P1)_{(d-1), \ta}\)). Hence,
\[K_0 \cup L_0=M=K_I \cup L_I\]
which together with equationref{eq explicit defn of K_i, L_i} imply
\[M=K_i \cup L_i.\]
there exists\;xtbf{Step 2.} By our assumption, \(\tilde{\Psi}\) restricted to \(e_{\be}[I]_0, \;f_{\be}[I]_0\) (and \(e_{\ta}[I]_0,\; f_{\ta}[I]_0\)) satisfy \((P2)_{(d-1), \be}\) (and \((P2)_{(d-1), \ta}\)). Hence,
\[K_0 \subset \bigcup_{x \in (e_{\be})_0}\tilde{\Phi}_{\ga}(x); \quad K_{I}\subset \bigcup_{x \in (e_{\ta})_0} \tilde{\Phi}_{\ga}(x).\]
Therefore, \; using equationref{eq explicit defn of K_i, L_i},
\[K_i \subset \bigcup_{x \in (e_{\nu})_0}\tilde{\Phi}_{\ga}(x)\quad \forall i \in [I].\]
Similarly, we can show that
\[L_i \subset \bigcup_{x \in (f_{\nu})_0}\tilde{\Phi}_{\ga}(x)\quad \forall i \in [I].\]
there exists\;xtbf{Step 3.} By the induction hypothesis, \newcommand{(e_{\nu})_0}{(e_{\nu})_0}
\begin{align}
& \partial K_0 \subset \bigg\{\bigcup_{x \in \ebz} \Ph_{\ga}(x) \bigg\} \bigcup \bigg\{\bigcup_{i \in [I],\; s \in \mathscr{F}_{\be} } \partial B_i(s) \bigg\}; \label{eq del K_0}\\
& \partial K_I \subset\bigg\{\bigcup_{x \in \etz} \Ph_{\ga}(x) \bigg\} \bigcup \bigg\{\bigcup_{i \in [I],\; s \in \mathscr{F}_{\ta} } \partial B_i(s) \bigg\}. \label{eq del K_I}
\end{align}
Moreover, by equationref{eq boundary} and equationref{eq inductive defn of K_i, L_i, Si_i}
\begin{align}
\partial K_i \subset \partial K_{i-1} \cup \partial K_{I} \cup S_i \Longrightarrow \partial K_i \subset \partial K_0 \cup \partial K_I \cup \left(\cup_{j=1}^i S_j\right). \label{eq del K_i}
\end{align}
Combining equationref{eq del K_0}, equationref{eq del K_I} and equationref{eq del K_i}, we obtain
\begin{equation}
\partial K_i \subset\bigg\{\bigcup_{x \in \enz} \Ph_{\ga}(x) \bigg\} \bigcup \bigg\{\bigcup_{i \in [I],\; s \in \mathscr{F}_{\be} \cup \mathscr{F}_{\ta} } \partial B_i(s) \bigg\}\bigcup \bigg\{\bigcup_{t=1}^{i} \partial B_t\bigg\}.\label{eq del K_i final}
\end{equation}
We can arrive at a similar conclusion for \(\partial L_i\) as well.
there exists\;xtbf{Step 4.} For \(j = 0,1,\dots,I\), let \(\hat{\Si}_j:=\Si_j \setminus \cS\). By the induction hypothesis \[\hat{\Si}_0=\mathbb{S}^0_1\;\dot\cup\;\mathbb{S}^0_2\;\dot\cup\;\dots\;\dot\cup\;\mathbb{S}^0_{l_0}\]
where each \(\mathbb{S}^0_j\) is an open subset of a hypersurface belonging to the set
\begin{equation*}
\Big\{\Ph_{\ga}(x):x \in \ebz \Big\} \bigcup \Big\{\partial B_i(s):i \in [I],\; s \in \mathscr{F}_{\be}\Big\};
\end{equation*}
and
\[\hat{\Si}_I=\mathbb{S}^I_1\;\dot\cup\;\mathbb{S}^I_2\;\dot\cup\;\dots\;\dot\cup\;\mathbb{S}^I_{l_I}\]
where each \(\mathbb{S}^I_j\) is an open subset of a hypersurface belonging to the set
\begin{equation}\label{eq induction hyp on Si_I-hat}
\Big\{\Ph_{\ga}(x):x \in \etz \Big\} \bigcup \Big\{\partial B_i(s):i \in [I],\; s \in \mathscr{F}_{\ta} \Big\}.
\end{equation}
By induction let us assume that
\[\hat{\Si}_{i-1}=\mathbb{S}^{i-1}_1\;\dot\cup\;\mathbb{S}^{i-1}_2\;\dot\cup\;\dots\;\dot\cup\;\mathbb{S}^{i-1}_{l_{i-1}}\]
where each \(\mathbb{S}^{i-1}_j\) is an open subset of a hypersurface belonging to the set
\begin{equation}
\Big\{\Ph_{\ga}(x):x \in \enz \Big\} \bigcup \Big\{\partial B_i(s):i \in [I],\; s \in \mathscr{F}_{\be} \cup \mathscr{F}_{\ta} \Big\}\bigcup \Big\{\partial B_t: t\in [i-1]\Big\}.\label{eq induction hypothesis for Si-hat-i}
\end{equation}
Then,
\begin{equation}
(\Si_{i-1} \cap \ov{A}_i)\setminus \cS = (\hat{\Si}_{i-1} \cap \ov{A}_i)\setminus \cS = (\hat{\Si}_{i-1} \cap A_i)\setminus \cS \label{eq Si-hat cap A_i}
\end{equation}
as by equationref{eq induction hypothesis for Si-hat-i} \(\hat{\Si}_{i-1}\cap \partial B_i \subset \cS.\) Further, \((\hat{\Si}_{i-1} \cap A_i)\setminus \cS\) is an open subset of \(\hat{\Si}_{i-1}\) as \(A_i \subset M\) is open and \(\cS \subset M\) is closed. Similarly, using equationref{eq induction hyp on Si_I-hat},
\begin{equation}
(\Si_{I} \cap \ov{B}_i)\setminus \cS = (\hat{\Si}_{I} \cap B_i)\setminus \cS \label{eq Si_I cap A_i}
\end{equation}
which is an open subset of \(\hat{\Si}_{I}\).
\begin{equation}
(K_{i-1} \cap L_I \cap S_i)\setminus \cS = (int(K_{i-1} \cap L_I )\cap S_i)\setminus \cS \label{eq S_i term one}
\end{equation}
as by equationref{eq boundary}, \(\partial(K_{i-1} \cap L_I) \subset \partial K_{i-1} \cup \partial L_I\); by equationref{eq del K_i final} and by the induction hypothesis \((P3)_{(d-1), \ta}\),
\[(\partial K_{i-1} \cup \partial L_I)\cap S_i \subset \cS.\]
Similarly,
\begin{equation}
(K_{I} \cap L_{i-1} \cap S_i)\setminus \cS = (int(K_{I} \cap L_{i-1} )\cap S_i)\setminus \cS. \label{eq S_i term two}
\end{equation}
Moreover,
\[M = B_i \;\dot\cup\; S_i\; \dot\cup \;A_i.\]
Hence, using equationref{eq inductive defn of K_i, L_i, Si_i} and equations equationref{eq induction hyp on Si_I-hat} -- equationref{eq S_i term two}, we obtain \[\hat{\Si}_{i}=\mathbb{S}^{i}_1\;\dot\cup\;\mathbb{S}^{i}_2\;\dot\cup\;\dots\;\dot\cup\;\mathbb{S}^{i}_{l_{i}}\]
where each \(\mathbb{S}^{i}_j\) is an open subset of a hypersurface belonging to the set
\begin{equation}
\Big\{\Ph_{\ga}(x):x \in \enz \Big\} \bigcup \Big\{\partial B_i(s):i \in [I],\; s \in \mathscr{F}_{\be} \cup \mathscr{F}_{\ta} \Big\}\bigcup \Big\{\partial B_t: t\in [i]\Big\}.
\end{equation}
there exists\;xtbf{Step 5.}
Suppose\; \(v, v'\in (\partial e_{\nu})[I]_0\). Let \(\rh \in [Q]\) such that\; \(\la_{\rh} \prec e_{\nu}\) (\(\la_{\rh}\) could be \(e_{\rh}\) or \(f_{\rh}\)) with \(\dim(\la_{\rh})=d-1\) and \(v, v' \in \la_{\rh}[I]_0\). Then, by the inductive hypothesis \((P5)_{(d-1),\rh}\) and by the compatibility relation equationref{eq compatibility}, we get \((P5)_{d, \nu}\) for this case.
Now, we assume that \(v \notin (\partial e_{\nu})[I]_0\); \(D^1_q(v)=(z_1,z_2,\dots ,z_d)=(z',z_d)\), \(D^1_q(v')=(z'_1,z'_2,\dots ,z'_d)=(z'',z'_d)\). If \(z_d \neq z'_d\), using equationref{eq inductive defn of K_i, L_i, Si_i} we get \((P5)_{d, \nu}\). Otherwise, \(z'_d=z_d\) and there are two possibilities: either \(v' \notin (\partial e_{\nu})[I]_0\) or \(v' \in (\partial e_{\nu})[I]_0\). Let us assume the second possibility (the other case is similar), i.e. there exists \(\rh \in [Q]\) such that\; \(\la_{\rh} \prec e_{\nu}\) (\(\la_{\rh}=e_{\rh} there exists\;xt{ or } f_{\rh}\)) with \(\dim(\la_{\rh})=d-1\) and \(v' \in \la_{\rh}[I]_0\). We define,
\[B'_i=B_i(\rh),\quad A'_i=M\setminus\ov{B'_i}, \quad S'_i=\partial B'_i = \partial A'_i ;\]
\[K'_i = \xit^1_{\nu}(z'', iI^{-1}), \quad L'_i=\xit^2_{\nu}(z'', iI^{-1}),\quad \Si'_i=\Xi_{\nu}(z'',iI^{-1})=K'_i\cap L'_i.\]
Using \((P0)_{(d-1),\rh}\) and the compatibility relation equationref{eq compatibility}, we can deduce the following relations similar to those in equationref{eq explicit defn of K_i, L_i}.
\begin{align}
& K'_i=\Big[K'_0\cap \left(\cup_{s=1}^i B'_s\right)^c\Big] \bigcup \Big[K'_I \cap \left(\cup_{s=1}^i \ov{B'_s}\right)\Big]; \nonumber\\
& L'_i=\Big[L'_0\cap \left(\cup_{s=1}^i B'_s\right)^c\Big] \bigcup \Big[L'_I \cap \left(\cup_{s=1}^i \ov{B'_s}\right)\Big]; \label{eq explicit defn of K'_i, L'_i}\\
& \Si'_i=\Big[\Si'_0\cap \left(\cup_{s=1}^i B'_s\right)^c\Big] \bigcup \Big[\Si'_I \cap \left(\cup_{s=1}^i \ov{B'_s}\right)\Big] \bigcup \La'_i \quad there exists\;xt{where} \quad \La'_i \subset \bigcup_{t=1}^i S'_t.\nonumber
\end{align}
\((P5)_{(d-1),\be}\) and \((P5)_{(d-1),\ta}\) imply
\begin{align}
& \Si_0\cap \left(M\setminus(\cA \cup \cB)\right) = \Si'_0\cap \left(M\setminus(\cA \cup \cB)\right); \nonumber\\
& \Si_I\cap \left(M\setminus(\cA \cup \cB)\right) = \Si_I'\cap \left(M\setminus(\cA \cup \cB)\right); \label{P5 Si_I}
\end{align}
where \(\cB=\mathbf{B}_j\); \(j\) is such that if \(z_l \neq z'_l\), \(\{z_l,z'_l\}=\{(j-1)I^{-1}, jI^{-1}\}\). We note that for all \(s \in [I]\),
\begin{equation*}
B_s \;\De\; B'_s,\; \partial B_s,\; \partial B'_s,\; \subset \cA.
\end{equation*}
Hence,
\begin{align}
\left(\cup_{s=1}^i B_s\right)^c \De \left(\cup_{s=1}^i B'_s\right)^c \subset \cA \implies &\left(\cup_{s=1}^i B_s\right)^c \cap (\cA \cup \cB)^c = \left(\cup_{s=1}^i B'_s\right)^c \cap (\cA \cup \cB)^c;\nonumber \\
\left(\cup_{s=1}^i \ov{B_s}\right) \De \left(\cup_{s=1}^i \ov{B'_s}\right) \subset \cA \implies & \left(\cup_{s=1}^i \ov{B_s}\right)\cap (\cA \cup \cB)^c = \left(\cup_{s=1}^i \ov{B'_s}\right)\cap (\cA \cup \cB)^c;\nonumber\\
\La_i \cap (\cA \cup \cB)^c &= \emptyset = \La_i'\cap (\cA \cup \cB)^c \label{P5 intersection w. A^c}
\end{align}
Using the equations equationref{eq explicit defn of K_i, L_i}, equationref{eq explicit defn of K'_i, L'_i}, equationref{P5 Si_I} and equationref{P5 intersection w. A^c}, we obtain
\[\Si_i\cap \left(M\setminus(\cA \cup \cB)\right) = \Si'_i\cap \left(M\setminus(\cA \cup \cB)\right).\]
for all \(i \in [I]\). By a similar argument, one can also show that
\[K_i\cap \left(M\setminus(\cA \cup \cB)\right) = K'_i\cap \left(M\setminus(\cA \cup \cB)\right);\]
and
\[L_i\cap \left(M\setminus(\cA \cup \cB)\right) = L'_i\cap \left(M\setminus(\cA \cup \cB)\right).\]
there exists\;xtbf{Step 6.} Using equationref{eq reccurence for La_i}, for \(i>1\),
\begin{equation}
\cH^n(\La_i) \leq \cH^n(\La_{i-1})+\cH^n(K_{i-1}\cap L_{I} \cap S_i) + \cH^n(K_I\cap L_{i-1} \cap S_i). \label{eq H^n of La_i}
\end{equation}
By \((P2)_{d,\nu}\),
\begin{equation*}
K_{i-1} \subset \bigcup_{x \in \enz} \tilde{\Phi}_{\ga}(x), \quad L_I \subset \bigcup_{x \in (f_{\nu})_0} \tilde{\Phi}_{\ga}(x).
\end{equation*}
Hence,\newcommand{\tilde{\Phi}g}{\tilde{\Phi}_{\ga}}\newcommand{function\;z}{(f_{\nu})_0}
\begin{equation}
\cH^n(S_i \cap K_{i-1} \cap L_I) \leq \sum_{\substack{x_1 \in \enz,\\ x_2 \in (f_{\nu})_0}} \cH^n\left(S_i \cap \tilde{\Phi}_{\ga}(x_1) \cap \tilde{\Phi}g(x_2)\right). \label{eq H^n Si cap dots 1}
\end{equation}
For \(x_1 \in \enz,\; x_2 \in function\;z,\)
\begin{align}
\cH^n\left(S_i \cap \tilde{\Phi}_{\ga}(x_1) \cap \tilde{\Phi}g(x_2)\right) &= \cH^n\left(S_i \cap \tilde{\Phi}_{\ga}(x_1) \cap \tilde{\Phi}g(x_2) \cap \tilde{\Phi}g(T(x_2))\right) \nonumber\\
& + \cH^n\left(S_i \cap \tilde{\Phi}_{\ga}(x_1) \cap \tilde{\Phi}g(x_2) \cap \left(M -\tilde{\Phi}g(T(x_2))\right)\right) \nonumber \\
& = 0 + \cH^n\left(S_i \cap \left(\tilde{\Phi}_{\ga}(x_1)-\tilde{\Phi}g(T(x_2)\right)\right) \nonumber \\
& <\frac{2^{2d}\et}{I}. \label{eq H^n Si cap dots 2}
\end{align}
In the second equality we have used the fact that \(S_i\) is transverse to \(\Ph_{\ga}(x_2)=\tilde{\Phi}g(x_2) \cap \tilde{\Phi}g(T(x_2))\) (which was assumed in the definition of \(r_i(q)\)) and \((P1)_{d,\nu}\). In the last inequality we have used Proposition \ref{pro existence of alpha}, item (v). Therefore, \; combining equationref{eq H^n of La_i} -- equationref{eq H^n Si cap dots 2}, we obtain
\begin{equation*}
\cH^n(\La_i) < \cH^n(\La_{i-1})+\frac{2^{4d+1}\et}{I}\; there exists\;xt{ for } i>1.
\end{equation*}
Moreover, by the above argument,
\begin{equation*}
\cH^n(\La_1) < \frac{2^{4d+1}\et}{I}.
\end{equation*}
Hence, for all \(i \in [I],\)
\begin{equation}
\cH^n(\La_i) < \frac{2^{4d+1}\et i}{I}\leq 2^{4d+1}\et. \label{eq H^n La_i final}
\end{equation}
Using this inequality along with equationref{eq explicit defn of K_i, L_i} and \((P6)_{(d-1),\be},\;(P6)_{(d-1),\ta}\), we conclude that for all \(j \in [I],\)
\begin{align}
\cH^n\left(\Si_i \cap \ov{\mathbf{B}_j^0}\right) & \leq \cH^n\left(\Si_0 \cap \ov{\mathbf{B}_j^0}\right) + \cH^n\left(\Si_I \cap \ov{\mathbf{B}_j^0}\right) + \cH^n(\La_i) \nonumber \\
& <(2^{4d-2}+2^{4d-2}+2^{4d+1})\et\nonumber\\
& < 2^{4d+2}\et. \label{eq proving P6}
\end{align}
there exists\;xtbf{Step 7.} Using equationref{eq H^n La_i final}, equationref{eq explicit defn of K_i, L_i} and \((P7)_{(d-1),\be},\;(P7)_{(d-1),\ta}\), we obtain as in Step 6,
\begin{align}
\cH^n\left(\Si_i \cap \ov{\cA}_1\right) & \leq \cH^n\left(\Si_0 \cap \ov{\cA}_1\right) + \cH^n\left(\Si_I \cap \ov{\cA}_1\right) + \cH^n(\La_i) \nonumber \\
& <(2^{4d-2}+2^{4d-2}+2^{4d+1})\et \nonumber\\
& < 2^{4d+2}\et. \label{eq proving P7}
\end{align}
there exists\;xtbf{Step 8.} We note that for all \(s \in [I]\),
\[\ov{\mathbf{B}}_s^c \subset B_s^c \subset \ov{\mathbf{B}}_s^c \cup \ov{\cA}; \qquad \ov{B}_s \subset \mathbf{B}_s \subset \ov{B}_s \cup \ov{\cA}.\]
Therefore, \; for all \(i \in [I]\),
\begin{align*}
& \Big(\bigcup_{s=1}^i\ov{\mathbf{B}}_s\Big)^c \subset \Big(\bigcup_{s=1}^iB_s\Big)^c \subset \Big(\bigcup_{s=1}^i\ov{\mathbf{B}}_s\Big)^c \cup \ov{\cA};\\
& \bigcup_{s=1}^i\ov{B}_s \subset \bigcup_{s=1}^i\mathbf{B}_s \subset \Big(\bigcup_{s=1}^i\ov{B}_s\Big) \cup \ov{\cA}.
\end{align*}
Hence, using equationref{eq explicit defn of K_i, L_i}, we deduce the following inequalities.
\begin{align}
&\cH^n(\Si_i \cap R) \leq \cH^n\left(\Si_0\cap \left(\cup_{s=1}^i\ov{\mathbf{B}}_s\right)^c\cap R\right)+\cH^n\left(\Si_0 \cap \ov{\cA}\right)+\cH^n\left(\Si_I\cap \left(\cup_{s=1}^i\mathbf{B}_s\right)\cap R\right) + \cH^n(\La_i);\label{inequ 1}\\
& \cH^n(\Si_i \cap R) \geq \cH^n\left(\Si_0\cap \left(\cup_{s=1}^i\ov{\mathbf{B}}_s\right)^c\cap R\right)+\cH^n\left(\Si_I\cap \left(\cup_{s=1}^i\mathbf{B}_s\right)\cap R\right)-\cH^n\left(\Si_I \cap \ov{\cA}\right).\label{inequ 2}
\end{align}
Denoting
\begin{equation}
R_1 = \left(\cup_{s=1}^i\ov{\mathbf{B}}_s\right)^c\cap R, \qquad R_2 = \left(\cup_{s=1}^i\mathbf{B}_s\right)\cap R, \label{eq defn of R1, R2}
\end{equation}
using equationref{inequ 1}, equationref{inequ 2}, \((P7)_{(d-1),\be}\), \((P7)_{(d-1),\ta}\) and equationref{eq H^n La_i final} we obtain,
\begin{align}
&\left|\cH^n(\Si_i \cap R)-\cH^n(\Si_0 \cap R_1)-\cH^n(\Si_I \cap R_2)\right|\nonumber\\
& \leq \cH^n\left(\Si_0 \cap \ov{\cA}\right) + \cH^n\left(\Si_I \cap \ov{\cA}\right) + \cH^n(\La_i) \nonumber \\
& \leq (2.2^{4d-2}+2^{4d+1})\et.\label{inequ 3}
\end{align}
Let \(x_0 \in \enz\); without loss of generality we can assume that \(x_0 \in \ebz\) i.e. \(D^1_{\nu}(x_0)=(\xi, 0)\) for some \(\xi \in \sci^{d-1}\). Let \(x_1\in\etz \) such that\; \(x_1=\De^1_{\nu}(\xi,1).\)
We note that \(R \in \mathscr{R}\) implies \(R_1, R_2 \in \mathscr{R}\) as well. Further, by the definition of \(\mathbf{B}_s\), \(\norm{\Ph_{\ga}(x_0)}(\partial \mathbf{B}_s)=0\); hence,
\[\cH^n\left(\Ph_{\ga}(x_0)\cap R\right)=\cH^n\left(\Ph_{\ga}(x_0)\cap R_1\right) + \cH^n\left(\Ph_{\ga}(x_0)\cap R_2\right).\]
Therefore, \; using \(equationref{inequ 3}\), \((P8)_{(d-1),\be}\), \((P8)_{(d-1), \ta}\), Proposition \ref{pro existence of alpha} (i) and equationref{cond: estimating mass norm} we get the following estimate.
\begin{align}
&\left|\cH^n(\Si_i \cap R)-\cH^n\left(\Ph_{\ga}(x_0) \cap R\right)\right| \nonumber\\
& \leq \left|\cH^n(\Si_i \cap R)-\cH^n(\Si_0 \cap R_1)-\cH^n(\Si_I \cap R_2)\right| + \left|\cH^n(\Si_0 \cap R_1)-\cH^n\left(\Ph_{\ga}(x_0) \cap R_1\right)\right| \nonumber\\
& + \left|\cH^n(\Si_I \cap R_2)-\cH^n\left(\Ph_{\ga}(x_1) \cap R_2\right)\right| + \left|\cH^n\left(\Ph_{\ga}(x_1) \cap R_2\right)-\cH^n\left(\Ph_{\ga}(x_0) \cap R_2\right)\right|\nonumber\\
& <(2^{4d-1}+2^{4d+1}+2^{4d-2}+2^{4d-2}+3)\et\nonumber\\
& <2^{4d+2}\et. \label{eq proving P8}
\end{align}
Since \(M \in \mathscr{R}\), equationref{eq proving P8} and Proposition \ref{pro existence of alpha}, (ii) imply
\[\cH^n(\Si_i)\leq L+(2^{4d+2}+2)\et.\]
\end{proof}
\subsection{Approximate solution of the Allen-Cahn equation}
Here we will briefly discuss an approximate solution of the Allen-Cahn equation whose energy is concentrated in a tubular neighbourhood of a closed, two-sided hypersurface (with mild singularities). We will follow the paper by Guaraco \cite{Guaraco}; further details can be found there.
Let \(h: \mathbb{R} \rightarrow \mathbb{R}\) be the unique solution of the following ODE.
\[\dot{\vp}(t)= \sqrt{2W(\vp(t))};\quad \vp(0)=0.\]
For all \(t \in \mathbb{R}\), \(-1<h(t)<1\) and as \(t \rightarrow \pm \infty\), \((h(t)\mp 1)\) converges to zero exponentially fast.
\(h_{\ve}(t)=h(t/\ve)\) is a solution of the one dimensional Allen-Cahn equation
\[\ve^2\ddot{\vp}(t)=W'(\vp(t))\]
with finite total energy:
\[\int_{-\infty}^{\infty}\left(\frac{\ve}{2}\dot{h}_{\ve}(t)^2+W(h_{\ve}(t))\right) dt=2\si.\]
For $\ve >0$, we define Lipschitz continuous function \newcommand{u_{\ve}}{u_{\ve}}
\begin{equation*}
g_{\ve}(t)=
\begin{cases}
h_{\ve}(t) & there exists\;xt{if } |t|\leq \sqrt{\ve};\\
h_{\ve}(\se)+\left(\frac{t}{\se}-1\right)(1-\he(\se)) &there exists\;xt{if } \se \leq t \leq 2 \se;\\
1 &there exists\;xt{if } t\geq 2\se;\\
\he(-\se)+ \left(\frac{t}{\se}+1\right)(1+\he(-\se)) &there exists\;xt{if } -2\se \leq t \leq -\se;\\
-1 &there exists\;xt{if } t\leq -2\se.
\end{cases}
\end{equation*}
Suppose \(d:M \rightarrow \mathbb{R}\) is a Lipschitz continuous function such that\; \(\norm{\nabla d}=1\) a.e. Let \(\ue=g_{\ve}\circ d\), \(U \subset M\). Using the notation
\[E_{\ve}(u,U)=\int_U\ve \frac{|\nabla u|^2}{2}+\frac{W(u)}{\ve},\]
we compute
\begin{align}
E_{\ve}(u_{\ve},U) & = \int_{U \cap \{|d|\leq 2\se\}} \left[\frac{\ve}{2}\dot{g}_{\ve}(d(y))^2+\frac{1}{\ve}W\Big(g_{\ve}(d(y))\Big)\right] \; d\cH^{n+1}(y) \nonumber \\
& = \int_{|\ta|\leq 2\se} \left[\frac{\ve}{2}\dot{g}_{\ve}(\ta)^2+\frac{1}{\ve}W(g_{\ve}(\ta))\right] \cH^n \left(U \cap \{d=\ta\}\right) d\ta \nonumber \\
& \leq \int_{|\ta|\leq \se} \left[\frac{\ve}{2}\dot{h}_{\ve}(\ta)^2+\frac{1}{\ve}W(h_{\ve}(\ta))\right] \cH^n \left(U \cap \{d=\ta\}\right) d\ta \label{estimate energy of u_ep1} \\
& + \left(\frac{1}{2}(1-\he(\se))^2+\frac{1}{\ve}W(\he(\se))\right)\operatorname{Vol}(M,g) \label{estimate energy of u_ep2}
\end{align}
Let
\[\La_{\ve} = \sup_{|\ta|\leq \se} \cH^n\left(U \cap \{d = \ta\}\right).\]
Then, the integral in equationref{estimate energy of u_ep1} is bounded by \(2 \si \La_{\ve}\). Further, there exists \(\ve_0=\ve_0(W,g,\et)\) such that\; for all \(\ve \leq \ve_0\) the expression in equationref{estimate energy of u_ep2} is bounded by \(2\si \et\). Hence,
\begin{equation}
\label{estimate energy of u_ep final}
E_{\ve}(u_{\ve},U) \leq 2\si(\La_{\ve} + \et)\; \forall \ve \leq \ve_0.
\end{equation}
For \(v \in \tilde{X}ki\), we define \(d_v:M \rightarrow \mathbb{R}\) as follows.
\begin{equation}
d_v(p)=
\begin{cases}
-d(p,\Ps(v)) &there exists\;xt{if } p \in \tilde{\Psi}(v);\\
d(p,\Ps(v)) &there exists\;xt{if } p \in \tilde{\Psi}(T(v)).
\end{cases}
\label{def d_v}
\end{equation}
By the definition of \(\Ps(v)\) and \((P1)\) of Proposition \ref{pro big}, \(d_v\) is a well-defined continuous function on \(M\). Moreover, by \cite{Guaraco}*{Proposition 9.1}, \(d_v\) is Lipschitz continuous and \(\norm{\nabla d_v}=1\) a.e.
Let \(p \in U \cap \{d_v = \tau\}\). Then, either \(d(p,\cS)=\md{\ta}\) or there exists \(z \in (\Ps(v)\setminus \cS)\) with \(d(p,z)=\md{\ta}\) such that\; \(p = \exp_z(\ta \mathbf{n}(z))\) where \(\mathbf{n}(z)\) is the unit normal to \(\Ps(v)\setminus \cS\) at \(z\) pointing inside \(\tilde{\Psi}(T(v))\). We must have
\[z \in \cN_{\md{\ta}}(U)\cap \left(\Ps(v)\setminus \cS\right).\]
Hence, we can write
\begin{equation}
\cH^n\left(U\cap \{d_v=\ta\}\right) \leq \cH^n(\cT_{\md{\ta}}(\cS))+\int_{\cN_{\md{\ta}}(U)\cap (\Ps(v)\setminus \cS)}\left|J \exp_{z}(\ta \mathbf{n}(z))\right|\; d\cH^n(z)
\label{inequ H^n of d_v=tau}
\end{equation}
where \(J \exp_{z}(\ta \mathbf{n}(z))\) is the Jacobian factor of the map \(z \mapsto \exp_{z}(\ta \mathbf{n}(z))\) which can be estimated as follows.
Let
\[\mathscr{S}'=\big\{\Ph_{\ga}(x): x \in \tilde{X}[K]_0\big\} \bigcup \left\{\partial B_{i}(q):i \in [I], q \in [Q]\right\}.\]
Using \((P4)\) of Proposition \ref{pro big} and \cite{warner}*{Corollary 4.2, Theorem 4.3}, \cite{Guaraco}*{Proposition 9.4}, one can deduce the following estimate. Let \(\la>0\) be such that\;
\[ I\!I_{\Si}(v,v)\leq \la \langle v,v \rightarrowngle \; \forall\; \Si \in \mathscr{S}', \; v \in T\Si.\]
Here \(I\!I_{\Si}\) denotes the second fundamental form of \(\Si.\) Then, there exist \(\ta_0\) and \(C_1\) depending only on \(\la\), the ambient dimension \(n+1\) and \(g\) such that\;
\[\left|J \exp_{z}(\ta \mathbf{n}(z))\right|\leq (1+C_1|\ta|) \; \forall z \in \Ps(v)\setminus \cS.\]
Hence, using equationref{eq area of equidistant hypersurface} and equationref{inequ H^n of d_v=tau}, there exists \(\ta_1=\ta_1(\mathscr{S},\mathscr{S}',n,g)\) such that\; for all \(\md{\ta}\leq \ta_1\)
\[\cH^n\left(U\cap \{d_v=\ta\}\right) \leq C|\ta|+(1+C_1|\ta|)\cH^n\left(\cN_{\md{\ta}}(U) \cap \Ps(v) \right).\]
This estimate along with \((P6)\) and \((P7)\) of Proposition \ref{pro big}, equationref{H^n of Psi(v)} and equationref{estimate energy of u_ep final} gives the following proposition. We recall that \(k\) is the dimension of the parameter spaces \(X\) and \(\tilde{X}\).
\begin{pro}\label{pro est energy of g_ep circ d_v}
There exists \(\ve_1=\ve_1(\et,\ta_1,\ve_0,\de,r_0-r_1)\) such that\; for all \(v \in \tilde{X}ki\), \(\ve \leq \ve_1\) and \(i \in [I]\), denoting \(\vartheta_{\ve}^v=g_{\ve}\circ d_v\), we have
\[E_{\ve}\left(\vartheta_{\ve}^v,\mathbf{B}_i^1\right) \leq 2\si (2^{4k+2}+2)\et; \; E_{\ve}\left(\vartheta_{\ve}^v,\cA_2\right) \leq 2\si(2^{4k+2}+2)\et; \; E_{\ve}\left(\vartheta_{\ve}^v,M\right) \leq 2\si(L+ (2^{4k+2}+4)\et).\]
\end{pro}
We recall from Section 3.2 that \(f:M \rightarrow [1/3,2/3]\) is a Morse function with no non-global local maxima or minima; hence the map \(t \mapsto f^{-1}(t)\) is continuous in the Hausdorff topology. Following \cite{Guaraco}*{Section 7}, we define the following functions. For \(t \in [1/3,2/3]\), let
\[d^{(t)}(p)=
\begin{cases}
-d\left(p, f^{-1}(t)\right) & there exists\;xt{if } f(p)\leq t;\\
d\left(p,f^{-1}(t)\right) & there exists\;xt{if } f(p) \geq t.
\end{cases}\]
We define \(w_{\ve}:[0,1] \rightarrow H^1(M)\) as follows.
\begin{equation}w_{\ve}(t)=
\begin{cases}
g_{\ve}\circ d^{(t)} & there exists\;xt{if } \frac{1}{3}\leq t \leq \frac{2}{3}; \\
1-3t(1-w_{\ve}(1/3)) & there exists\;xt{if } 0 \leq t \leq \frac{1}{3};\\
-1+3(1-t)(1+w_{\ve}(2/3)) & there exists\;xt{if } \frac{2}{3}\leq t\leq 1.
\end{cases}\label{w_ep}
\end{equation}
Since \(f^{-1}(t)\) varies continuously in the Hausdorff topology, \(w_{\ve}:[0,1] \rightarrow H^1(M)\) is continuous \cite{Guaraco}*{Proposition 9.2}. The following proposition follows from \cite{Guaraco}*{Section 9.6} and the estimates equationref{eq: mass of modified morse fn level set}, equationref{estimate energy of u_ep final}.
\begin{pro}\label{pro est energy of w_t}
There exists \(\ve_2=\ve_2(\et,f,\ve_0,\de,r_0-r_1)\) such that\; for all \(t \in [0,1]\), \(\ve \leq \ve_2\) and \(i \in [I]\), we have
\[E_{\ve}\left(w_{\ve}(t),\mathbf{B}_i^1\right) \leq 6\si\et; \quad E_{\ve}\left(w_{\ve}(t),\cA_2\right) \leq 6\si\et.\]
\end{pro}
\subsection{Construction of sweepout in \(H^1(M)\setminus\{0\}\) and proof of Theorem \ref{thm main thm}}
The next proposition essentially follows from the argument in \cite{gt}*{Proof of Lemma 7.5}.
\begin{pro}\label{mod u}
\(u \mapsto |u|\) is a continuous map from \(H^1(M)\) to itself.
\end{pro}
\begin{proof}
We define \( \ka:\mathbb{R} \rightarrow \mathbb{R}\) as follows
\[\ka(t)=
\begin{cases}
1 &there exists\;xt{if } t>0;\\
0 &there exists\;xt{if } t=0;\\
-1&there exists\;xt{if } t<0.
\end{cases}\]
By \cite{gt}*{Lemma 7.6} \(v \in H^1(M)\) implies \(|v|\in H^1(M)\) with \(D|v|=\ka(v)Dv\).
We need to show that if \(u_n \rightarrow u\) in \(H^1(M)\), \(|u_n| \rightarrow |u|\) in \(H^1(M)\). We will use the following fact. Let \(\{x_n\}\) be a sequence in a topological space \(\cX\) and \(x \in \cX\) such that\; every subsequence \(\{x_{n_k}\}\) of \(\{x_n\}\) has a further subsequence \(\{x_{n_{k_i}}\}\) which converges to \(x\). Then \(\{x_n\}\) converges to \(x\). Therefore, without loss of generality we can assume that \(u_n\) converges to \(u\) pointwise a.e.
\(|u_n|\rightarrow |u|\) in \(L^2(M)\) because of the inequality \(||u_n|-|u||\leq |u_n-u|\). Using the fact that \(Du=0\) a.e. on \(\{u=0\}\), we compute
\begin{align}
\int_M |\ka(u_n)Du_{n}-\ka(u)Du|^2 & \leq \int_M 2|\ka(u_n)|^2|Du_n-Du|^2+2|\ka(u_n)-\ka(u)|^2|Du|^2 \nonumber\\
& \leq 2\int_M|Du_n-Du|^2 +2\int_{\{u \neq 0\}} |\ka(u_n)-\ka(u)|^2|Du|^2. \nonumber
\end{align}
The first integral converges to \(0\). If \(u(y) \neq 0\) and \(u_n(y)\) converges to \(u(y)\), \(\ka(u_n(y))\) converges to \(\ka(u(y))\) as well. Hence, by the dominated convergence theorem, the second integral also converges to \(0\).
\end{proof}
For \(x \in \mathbb{R}\), let \(x^{+}=\max\{x,0\}\); \(x^{-}=\min\{x,0\}\). We define the maps \(\ph,\ps,\tht :H^1(M)\times H^1(M)\times H^1(M) \rightarrow H^1(M)\) as follows.
\begin{align}
&\ph(u_0,u_1,w)=\min\{\max\{u_0,-w\},\max\{u_1,w\}\}; \nonumber\\
&\ps(u_0,u_1,w)=\max\{\min\{u_0,w\},\min\{u_1,-w\}\}; \label{def of phi, psi}\\
&\tht(u_0,u_1,w)=\ph(u_0,u_1,w)^{+}+\ps(u_0,u_1,w)^{-}.\nonumber
\end{align}
\begin{pro}
The map \(\tht\) defined above has the following properties.
\begin{itemize}
\label{pro tht}
\item[(i)] \(\tht\) is a continuous map.
\item[(ii)] \(\tht(-u_0,-u_1,w)=-\tht(u_0,u_1,w)\).
\item[(iii)] If \(|u_0|,|u_1| \leq 1\), then \(\tht(u_0,u_1,\mathbf{1})=u_0\), \(\tht(u_0,u_1,-\mathbf{1})=u_1\) where \(\mathbf{1}\) denotes the constant function \(1\).
\item[(iv)] If \(u_0(p)=u_1(p)\), then \(\tht(u_0,u_1,w)(p)=u_0(p)=u_1(p)\).
\item[(v)] For every \(U \subset M\), \(E_{\ve}(\tht(u_0,u_1,w),U)\leq E_{\ve}(u_0,U)+E_{\ve}(u_1,U)+E_{\ve}(w,U)\).
\item[(vi)] \(\tht(u_0,u_1,w)(p)=0\) implies \(p\in \{u_0=0\}\cup \{u_1=0\}\cup\{w=0\}\).
\end{itemize}
\end{pro}
\begin{proof}
We have the following identities.
\[\max\{a,b\}=\frac{a+b+|a-b|}{2}; \quad \min\{a,b\}=\frac{a+b-|a-b|}{2}.\]
Hence, the continuity of \(\ph,\;\ps\) and \(\tht\) follows from Proposition \ref{mod u}. We note that \(\ph(-u_0,-u_1,w)=-\ps(u_0,u_1,w)\) which implies (ii). (iii) also follows from direct computation. Indeed, if \(|u_0|,|u_1| \leq 1\), \(\ph(u_0,u_1,\mathbf{1})=u_0=\ps(u_0,u_1,\mathbf{1})\) and \(\ph(u_0,u_1,-\mathbf{1})=u_1=\ps(u_0,u_1,-\mathbf{1})\). To prove (iv), we need the following identity. If \(a\geq 0\),
\begin{equation}
\min\{\max\{a,-b\},\max\{a,b\}\}=a. \label{id}
\end{equation}
Indeed, the identity holds if \(\max\{a,b\}=a=\max\{a,-b\}\). It also holds if \(\max\{a,-b\}=a\) and \(\max\{a,b\}=b>a\). It is not possible for both \(\max\{a,-b\}\) and \(\max\{a,b\}\) to be different from \(a\) as this will imply \(b>a,\; -b>a\; \implies 0>a\). Moreover, if \(a\geq 0\), either \(\min\{a,b\}\geq 0\) (if \(b \geq 0\)) or \(\min\{a,-b\}\geq 0\) (if \(b \leq 0\)); hence
\begin{equation}
\max\{\min\{a,b\},\min\{a,-b\}\}\geq 0. \label{id'}
\end{equation}
equationref{id} and equationref{id'} give (iv) if \(u_0(p)=u_1(p)\geq 0\). The other case is similar.
For \(i=0,1\), let \(U_i=\{u_i<0\}\) and \(V_i=\{u_i>0\}\). We also set \(G=\{w<0\}\), \(H=\{w>0\}\), \(Z=\{w=0\}\). Then we have (suppressing the dependence of \(\ph,\;\ps,\;\tht\) on \(u_0,\;u_1,\;w\)),
\begin{align}
&\ph(y) <0 \Leftrightarrow y \in (U_0\cap H)\dot\cup(U_1\cap G);\; \ph(y)>0 \Leftrightarrow y \in (V_0\cap H)\dot\cup (V_1\cap G)\dot\cup(V_0\cap V_1 \cap Z);\nonumber\\
&\ps(y)<0 \Leftrightarrow y \in (U_0\cap H)\dot\cup (U_1\cap G)\dot\cup(U_0\cap U_1 \cap Z); \; \ps(y)>0 \Leftrightarrow y \in (V_0\cap H)\dot\cup (V_1\cap G). \label{U_i}
\end{align}
Further,
\begin{equation}
\tht(y)=
\begin{cases}
\ph(y) &there exists\;xt{if } y \in \{\ph\geq 0\}\cap \{\ps\geq 0\};\\
\ps(y) &there exists\;xt{if } y \in \{\ph\leq 0\}\cap \{\ps\leq 0\}.
\end{cases}\label{theta}
\end{equation}
It follows from equationref{U_i} that
\[M= \big(\{\ph\geq 0\}\cap \{\ps\geq 0\}\big) \cup \big(\{\ph\leq 0\}\cap \{\ps\leq 0\}\big).\]
Hence, using the definition of \(\ph\) and \(\ps\), equationref{theta} implies that
\begin{equation}
\tht(y) \in \{u_0(y),u_1(y),w(y),-w(y)\};\quad \nabla\tht(y) \in \{\nabla u_0(y),\nabla u_1(y), \nabla w(y), -\nabla w(y)\}. \label{pt w}
\end{equation}
Therefore, \; we have the following point-wise estimates.
\[|\nabla \tht (y)|^2 \leq |\nabla u_0 (y)|^2+|\nabla u_1 (y)|^2+|\nabla w (y)|^2;\; W(\tht(y))\leq W(u_0(y))+W(u_1(y))+W(w(y))\]
which give item (v) of the proposition. equationref{pt w} also implies item (vi).
\end{proof}\newcommand{\tilde{X}kin}{\tilde{X}[KI]}\newcommand{\vr}{\varrho}
Let \(\al \in \tilde{X}kin_j\) so that \(\mu=\pi(\al)\in X[KI]_j\). We define the following maps.
\begin{equation*}
\left(\pi|_{\al}\right)^{-1}\circ \De_{\mu}=\De^{\bullet}_{\al}:\sci^j\rightarrow \al;\quad D_{\mu}\circ\left(\pi|_{\al}\right)=D^{\bullet}_{\al}:\al \rightarrow \sci^j.
\end{equation*}\newcommand{\hat{\ze}_{\ve}}{\hat{\ze}_{\ve}}
\begin{pro}\label{pro zeta_ep}
Let \(\ve^*=\min\{\ve_1,\ve_2,(\de/4)^2, 4^{-1}(r_1-r)^2\}\). There exists a continuous, \(\mathbb{Z}_2\)-equivariant map \(\ze_{\ve}:\tilde{X} \rightarrow H^1(M)\setminus \{0\}\) with the property
\begin{itemize}
\item[\((\cP)_0\)] For all \(v \in \tilde{X}ki\), \(\zee(v)=\vartheta_{\ve}^v\);
\end{itemize}
where \(\vartheta_{\ve}^v\) is as in Proposition \ref{pro est energy of g_ep circ d_v}. Moreover, if \(\al \in \tilde{X}kin_j\), \(\zee|_{\al}\) has the following properties.
\begin{itemize}
\item[\((\cP0)_{j,\al}\)] Let \(\zeh^{\al}=\zee\circ\De^{\bullet}_{\al}\). Then, for all \(z=(z',z_j)\in \sci^j\) (with \(z'\in \sci^{j-1}\)),
\[\zeh^{\al}(z',z_j)=\tht\left(\zeh^{\al}(z',0),\zeh^{\al}(z',1),w_{\ve}(z_j)\right);\]
where \(w_{\ve}\) is as defined in equationref{w_ep}.
\item[\((\cP1)_{j,\al}\)] We define \(\cI_{\al}=\{i(e):e \in \al_1\}\) where \(i(e)\) is as in Proposition \ref{pro big}, \((P5)\). For all \(\ve\leq \ve^*\), \(t \in \al\) and \(v \in \al_0\),
\[y \notin \cA_2\bigcup\Big(\bigcup_{i \in \cI_{\al}}\mathbf{B}^1_{i}\Big) \implies \zee(t)(y)=\zee(v)(y).\]
\item[\((\cP2)_{j,\al}\)] For all \(\ve\leq \ve^*\), \(i\in [I]\) and \(t \in \al\),
\[\frac{1}{2\si }E_{\ve}\left(\zee(t),\mathbf{B}^1_i\right)\leq 3^j(2^{4k+2}+2)\et;\quad \frac{1}{2\si }E_{\ve}\left(\zee(t),\cA_2\right)\leq 3^j(2^{4k+2}+2)\et.\]
As a consequence, for all \(\ve\leq \ve^*\) and \(t \in \al\),
\begin{equation}
\frac{1}{2\si }E_{\ve}\left(\zee(t),M\right)\leq L+(2^{4k+2}+4)\et +(j+1)3^j(2^{4k+2}+2)\et. \label{eqn of pro 3.13}
\end{equation}
\end{itemize}
\end{pro}
\begin{proof}
The map \(\zee\) will be defined inductively on the cells of \(\tilde{X}kin\). The equivariance of \(\zee\) and the fact that \(\zee(x)\) is not identically \(0\) for all \(x \in \tilde{X}\) will be proved at the end (after proving the other properties of \(\zee\)). If \(v \in \tilde{X}ki\), we define \(\zee(v)=\vartheta_{\ve}^v\) which is precisely the property \((\cP)_0\). The properties \((\cP0)_{0,v}\) and \((\cP1)_{0,v}\) are vacuous; \((\cP2)_{0,v}\) is satisfied because of Proposition \ref{pro est energy of g_ep circ d_v}. Let us assume that for some \(p\geq 1\), \(\zee\) has been defined on
\[\bigcup_{j\leq p-1}\tilde{X}kin_j\]
and if \(\vr \in \tilde{X}kin_{p-1}\), \(\zee\vert_{\vr}\) has the properties \((\cP0)_{(p-1),\vr}-(\cP2)_{(p-1),\vr}.\)
Let \(\la \in \tilde{X}kin_p.\) By our assumption, \(\zee\) is already defined on \(\partial \la.\) For \(z \in \partial \sci^p\), we define
\[\zeh^{\la}(z)=\left(\zee \circ \De^{\bullet}_{\la}\right)(z).\]
For arbitrary \(z=(z',z_p) \in \sci^p\) with \(z'\in \sci^{p-1}\), we define
\begin{equation}
\zeh^{\la}(z',z_p)=\tht\left(\zeh^{\la}(z',0),\zeh^{\la}(z',1), w_{\ve}(z_p)\right) \label{defn zeh}
\end{equation}
where \(w_{\ve}\) is as defined in equationref{w_ep}. By Proposition \ref{pro tht} (iii), this equation indeed holds for \(z_p=0,1\). Further, equationref{defn zeh} is also valid if \(z \in \partial \sci^p\) because of our induction hypothesis that \(\zee\) restricted \(\partial \la\) satisfies \((\cP0)\) and the compatibility relation equationref{eq compatibility}. For \(t \in \la\), \(\zee(t)\) is defined by \(\zee(t)=\left(\zeh^{\la} \circ D^{\bullet}_{\la}\right)(t)\). \(\zee\) is continuous on \(\la\) as \(\zee|_{\partial \la}\) is continuous\; and the maps \(\tht,\;w_{\ve}\) are continuous.
It remains to verify that \(\zee|_{\la}\) satisfies the properties \((\cP0)_{p,\la}-(\cP2)_{p,\la}\). \((\cP0)_{p,\la}\) holds by the definition equationref{defn zeh}. To prove \((\cP1)_{p,\la}\), we first show that if \(v,v' \in \tilde{X}ki\) and \(e \in \tilde{X}kin_1\) such that\; \(v,v' \prec e\),
\begin{equation}
y \notin \cA_2 \cup \mathbf{B}_{i(e)}^1 \implies \zee(v)(y)=\zee(v')(y). \label{cP2 1st}
\end{equation}
We set \(\cB^1=\mathbf{B}_{i(e)}^1\) and \(\cB=\mathbf{B}_{i(e)}\). Either \(y \in \tilde{\Psi}(v)\) or \(y \in \tilde{\Psi}(T(v))\). We assume that \(y \in \tilde{\Psi}(T(v))\) (the other case is analogous); hence, (using Proposition \ref{pro big} \((P5)\))
\[y \in \tilde{\Psi}(T(v))\cap \left(\cA_2 \cup \cB^1\right)^c = \tilde{\Psi}(T(v'))\cap \left(\cA_2 \cup \cB^1\right)^c \]
Thus, \(d_v(y)=d(y,\Ps(v))\), \(d_{v'}(y)=d(y,\Ps(v'))\). If both \(d_v(y)\) and \(d_{v'}(y)\) are \(\geq 2\sqrt{\ve}\), \(\zee(v)(y)=1=\zee(v')(y)\).
Otherwise, let us say \(d_v(y)=d(y,\Ps(v))<2\sqrt{\ve}\). We will show that \(d_v(y)=d_{v'}(y).\) Let \(y'\in \Ps(v)\) such that\; \(d(y,y')= d(y,\Ps(v))\). Therefore, \;
\begin{equation}
d(y,y')= d(y,\Ps(v)) <2\sqrt{\ve}\leq \min\{\de/2, r_1-r\}. \label{d(y,y')}
\end{equation}
Hence, \(y \notin \cA_2 \cup \cB^1\) implies \(y' \notin \cA \cup \cB\). Thus, (again using Proposition \ref{pro big} \((P5)\))
\[y'\in \Ps(v)\cap \left(\cA \cup \cB\right)^c = \Ps(v')\cap \left(\cA \cup \cB\right)^c.\]
This gives
\[d(y, \Ps(v))=d\left(y,\Ps(v)\cap \left(\cA \cup \cB\right)^c\right);\]
and \(d(y, \Ps(v'))\leq d(y,y')< 2\sqrt{\ve}\) which also implies (by the above reasoning)
\[d(y, \Ps(v'))=d\left(y,\Ps(v')\cap \left(\cA \cup \cB\right)^c\right).\]
Since \(\Ps(v)\cap \left(\cA \cup \cB\right)^c = \Ps(v')\cap \left(\cA \cup \cB\right)^c\), we get \(d_v(y)=d_{v'}(y).\)
Thus we have proved equationref{cP2 1st} which together with equationref{defn zeh}, induction hypothesis \((\cP1)_{(p-1),\partial \la}\) and Proposition \ref{pro tht} (iv) gives \((\cP1)_{p,\la}\).
Finally we prove \((\cP2)_{p,\la}\). Using equationref{defn zeh} together with Proposition \ref{pro tht} (v), the induction hypothesis \((\cP2)_{(p-1),\partial \la}\) and Proposition \ref{pro est energy of w_t} we obtain the following estimate (\(\cU\) stands for \(\mathbf{B}_i^1\) for some \(i \in [I]\) or \(\cA_2\).)
\begin{align*}
\frac{1}{2\si}E_{\ve}\left(\zeh^{\la}(z',z_p), \cU\right)& \leq \frac{1}{2\si}E_{\ve}\left(\zeh^{\la}(z',0),\cU\right)+ \frac{1}{2\si}E_{\ve}\left(\zeh^{\la}(z',1), \cU\right) + \frac{1}{2\si}E_{\ve}\left(w_{\ve}(z_p), \cU\right)\\
&\leq 2.3^{p-1}(2^{4k+2}+2)\et + 3\et\\
&\leq 3^{p}(2^{4k+2}+2)\et.
\end{align*}
This estimate along with \((\cP1)_{p,\la}\), \((\cP)_0\) and Proposition \ref{pro est energy of g_ep circ d_v} gives for all \(\ve\leq \ve^*\), \(t \in \al\) and \(v \in \al_0\),
\begin{align*}
\frac{1}{2\si}E_{\ve}\left(\ze_{\ve}(t),M\right) & \leq \frac{1}{2\si}E_{\ve}\left(\ze_{\ve}(v),M\right)+\sum_{i \in \cI_{\la}}\frac{1}{2\si}E_{\ve}\left(\ze_{\ve}(t),\mathbf{B}_i^1\right) +\frac{1}{2\si}E_{\ve}\left(\ze_{\ve}(t),\cA_2\right)\\
&\leq L+(2^{4k+2}+4)\et +(p+1)3^p(2^{4k+2}+2)\et.
\end{align*}
In the last line we have used the fact that \(\md{\cI_{\la}}\leq p\) which follows from the characterization of \(i(e)\) in Proposition \ref{pro big} \((P5).\)
Now we show that the map \(\zee\) constructed above is \(\mathbb{Z}_2\)-equivariant. From the definition of \(d_v\) (equationref{def d_v}), it follows that \(d_{T(v)}=-d_v\) for all \(v \in \tilde{X}ki.\) Hence, by \((\cP)_0\), \(\zee(T(v))=-\zee(v)\) for all \(v \in \tilde{X}ki.\) Now we can use induction along with \((\cP0)\) and Proposition \ref{pro tht} (ii) to conclude that \(\zee(T(x))=-\zee(x)\) for all \(x \in \tilde{X}.\)
Lastly, we prove that for all \(x \in \tilde{X}\), \(\cH^{n+1}(\{\zee(x)=0\})=0\). This will in particular imply that for all \(x \in \tilde{X}\), \(\zee(x)\) is not identically equal to \(0\). If \(v \in \tilde{X}ki\), by equationref{def d_v} and \((\cP)_0\), \(\{\zee(v)=0\}=\Ps(v)\) and \(\cH^{n+1}(\Ps(v))=0\) by Proposition \ref{pro big} $(P4)$. Hence, \(\cH^{n+1}(\{\zee(v)=0\})=0\) for all \(v \in \tilde{X}ki\). Now, as before we can use induction along with \((\cP0)\) and Proposition \ref{pro tht} (vi) to conclude that \(\cH^{n+1}(\{\zee(x)=0\})=0\) for all \(x \in \tilde{X}.\)
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm main thm}]
By Proposition \ref{pro zeta_ep}, for every \(\et>0\) there exists \(\ve^*>0\) such that\; for all \(\ve\leq \ve^*\) there exists \(\ze_{\ve}\in \tilde{\Pi}\) such that\;
\[\frac{1}{2\si}\sup_{x \in \tilde{X}}E_{\ve}(\ze_{\ve}(x)) \leq L+\al(k)\et \;(there exists\;xt{by } equationref{eqn of pro 3.13})\]
where \(\al(k)\) is a constant which depends only on \(k=\dim (\tilde{X})\). Hence,
\[\frac{1}{2\si}\mathbf{L}_{\ve}(\tilde{\Pi})\leq L+\al(k)\et \;\; \forall\ve \leq \ve^* \implies\frac{1}{2\si}\limsup_{\ve \rightarrow 0^{+}} \mathbf{L}_{\ve}(\tilde{\Pi})\leq L+\al(k)\et.\]
Since \(\et>0\) is arbitrary, this implies equationref{width ineq}.
The following facts follow from \cite{GG1}*{Section 6}. \(\Ind_{\mathbb{Z}_2}(\tilde{X})\geq p+1\) if and only if each map \(\Ph \in \Pi \) is a \(p\)-sweepout. Moreover,
\[c_{\ve}(p)=\inf_{\tilde{\Pi}} \mathbf{L}_{\ve}(\tilde{\Pi})\]
where the infimum is taken over all \(\tilde{\Pi}\) such that\; \(\Ind_{\mathbb{Z}_2}(\tilde{X})\geq p+1\). Similarly,
\[\om_p = \inf_{\Pi}\mathbf{L}_{AP}(\Pi)\]
where the infimum is taken over all \(\Pi\) such that\; each map in the homotopy class \(\Pi\) is a \(p\)-sweepout.
We fix \(p\in \mathbb{N}\). For each \(j \in \mathbb{N}\), there exist double cover \(\tilde{X}_j \rightarrow X_j\) and the corresponding homotopy classes \(\Pi_j\), \(\tilde{\Pi}_j\) (as discussed in Section 1) such that\; \(\Ind_{\mathbb{Z}_2}(\tilde{X}_j)\geq p+1\) and
\[\mathbf{L}_{AP}(\Pi_j)<\om_p+\frac{1}{j}.\]
By equationref{width ineq},
\[\frac{1}{2\si}\limsup_{\ve \rightarrow 0^{+}}\mathbf{L}_{\ve}(\tilde{\Pi}_j)\leq \mathbf{L}_{AP}(\Pi_j)<\om_p+\frac{1}{j}.\]
Hence, there exists \(\tilde{\ve}>0\) such that\; for all \(\ve\leq \tilde{\ve}\)
\[\frac{1}{2\si}c_{\ve}(p)\leq \frac{1}{2\si}\mathbf{L}_{\ve}(\tilde{\Pi}_j)<\om_p+\frac{1}{j}\]
which implies
\[\frac{1}{2\si}\limsup_{\ve \rightarrow 0^{+}}c_{\ve}(p)\leq \om_p+\frac{1}{j}.\]
Since this holds for all \(j \in \mathbb{N}\), we obtain equationref{spec ineq}.
\end{proof}
\section{Proof of Theorem \ref{thm critical set}}
In this section we will discuss how Theorem \ref{thm critical set} can be proved using the results contained in the papers \cite{HT,Guaraco,GG1}. We recall the function \(F\) defined in equationref{2 defn F}. For \(\be\in (0,1)\), we set \(\si_{\be}=F^{-1}(1-\be).\)
\begin{pro}\label{pro 4.1}
Let \(\{u_{i}:M\rightarrow (-1,1)\}_{i=1}^{\infty}\) be a sequence smooth functions such that\; the followings hold.
\begin{itemize}
\item[(i)] \(AC_{\ve_i}(u_i)=0\) with \(\ve_i\rightarrow 0\) as \(i \rightarrow \infty\).
\item[(ii)] There exists \(E_0\) such that\; \(E_{\ve_i}(u_i)\leq E_0\) for all \(i \in \mathbb{N}\).
\item[(iii)] \(V\) is a stationary, integral varifold such that\; \(V_i=V[u_i]\rightarrow V\) and \(\spt(V)\) has optimal regularity.
\end{itemize}
Then, for all \(s>0\) there exists \(b_0\in (0,1/2]\) such that\; the following holds. Denoting \(w_i=F \circ u_i\), for all \(b \in (0,b_0]\) there exists \(i_0 \in \mathbb{N}\) depending on \(s\) and \(b\) such that\; for all \(i \geq i_0\) there exists \(t_i \in [-\si_b/2,\si_b/2]\) for which \(\{w_i>t_i\},\{w_i<t_i\}\in \mathcal{C}(M)\), \(\db{\{w_i>t_i\}}+\db{\{w_i<t_i\}}=\db{M}\), \(\partial\db{\{w_i>t_i\}}=\db{\{w_i=t_i\}} \) and \(\mathbf{F}(V,\md{\{w_i=t_i\}})\leq s.\)
\end{pro}
\begin{proof}
Throughout the proof, we will use the notation of \cite{HT}. Let \(p\in (M,g)\) and \(0<r<there exists\;xt{inj}(M)/4\). Identifying \(\mathbb{R}^{n+1}\) with \(T_pM\), let us define \(f_{p,r}:B^{n+1}(\mathbf{0},4)\rightarrow M\),
\[f_{p,r}(v)=\exp_{p}(rv)\quad there exists\;xt{ and }\quad g_{p,r}=r^{-2}f_{p,r}^{*}g.\]
As explained in \cite{Guaraco}, there exists \(0<r_0<there exists\;xt{inj}(M)/4\), depending on \((M^{n+1},g)\), such that\; if \(p \in M\), \(0<r\leq r_0\) and \(\{u_i\}_{i=1}^{\infty}\) are solutions of \(AC_{\ve_i}(u_i)=0\) on \((B^{n+1}(\mathbf{0},4),g_{p,r})\) with \(\ve_i\rightarrow 0\), all the results of \cite{HT} continue to hold.
For simplicity, let us assume that \(V=N|\Si|\) where \(\Si\) is a closed, minimal hypersurface with optimal regularity. The general case will be similar. Let
\begin{equation}
s_0=2\si(N+4+2\si N)^{-1}. \label{4 defn of s_0}
\end{equation}
and we fix \(0<s\leq s_0\). (The assumption \(s\leq s_0\) will be useful later.) By \cite{HT}*{Proposition 5.1} there exists \(b_0 \in (0,1/2]\) such that\;
\begin{equation*}
\limsup_{i \rightarrow \infty}\int_{\{|u_i|\geq 1-b_0\}}\frac{W(u_i)}{\ve_i} \leq s.
\end{equation*}\newcommand{\subset\joinrel\subset}{\subset\joinrel\subset}
Let us fix \(b \in (0,b_0]\). Using the above equation and \cite{HT}*{Proposition 4.3},
\begin{equation}
\label{4 small energy}
\limsup_{i \rightarrow \infty}\int_{\{|u_i|\geq 1-b\}}|\nabla w_i| \leq s.
\end{equation}
There exists an open set \(\Om\) containing \(reg(\Si)\) such that\; \(d_{\Si}=d(-,\Si)\) is smooth on \(\Om\) and the nearest point projection map \(P:\Om \rightarrow \Si\) is also smooth on \(\Om\). By \cite{MN_JDG_survey}*{Proof of Theorem 5.2}, \(im(P)=reg(\Si)\). We choose \(\cU_1 \subset\joinrel\subset \cU_2 \subset\joinrel\subset reg(\Si)\) such that\;
\begin{equation}
N\cH^n(\Si \setminus\cU_1)<s.\label{4 U_0^c has small area}
\end{equation}
For \(\cU \subset\joinrel\subset reg(\Si)\), let
\[\mathfrak{N}_{r}\cU= \{v: v\in T^{\perp}_p\Si there exists\;xt{ with } \norm{v}<r, p \in \cU\}.\]
There exists \(\rh>0\) such that\; \(\exp:\mathfrak{N}_{2\rh}\cU_2 \rightarrow \Om\) is a diffeomorphism onto its image. For \(j=1,2\), let \(U_j=\exp(\mathfrak{N}_{\rh}\cU_j)\) so that
\[P(\ov{U}_j)=\ov{\cU}_j=\Si \cap \ov{U}_j \quad there exists\;xt{ and }\quad U_1 \subset\joinrel\subset U_2 \subset\joinrel\subset \Om.\]
We can assume that (choosing a smaller \(\rh\) if necessary), the Jacobian factor
\begin{equation}
JP(y,S)\leq 2 \; \forall\; (y,S)\in G_n\ov{U}_2.\label{4 JP is bdd}
\end{equation}
On \(\Om\), following \cite{HT}*{Section 5}, we define
\begin{equation*}
v_i=\begin{cases}
\frac{\langle\nabla u_i,\nabla d_{\Si}\rightarrowngle}{|\nabla u_i|} & there exists\;xt{if } |\nabla u_i|\neq 0;\\
0 & there exists\;xt{if } |\nabla u_i|=0.
\end{cases}
\end{equation*}
Let \(\ta_i=(1-v_i^2)\ve_i|\nabla u_i|^2\). We choose a compactly supported function \(0\leq \ch_1\leq 1\) such that\; \(there exists\;xt{spt}(\ch_1)\subset \Om\) and \(\ch_1 equationuiv 1\) on \(\ov{U}_2.\) Since, \(V_i\rightarrow V\) is the varifold sense, using \cite{HT}*{Proposition 4.3} we obtain (\cite{HT}*{(5.7)})
\begin{equation}
\lim_{i\rightarrow \infty}\int_{\Om}\ch_1\ta_i=\lim_{i\rightarrow \infty}\int_{\Om}\ch_1(1-v_i^2)|\nabla w_i|=0\implies \lim_{i\rightarrow \infty}\int_{\ov{U}_2}\ta_i=0 .\label{4 tilt converges to 0}
\end{equation}
Let us use the notation
\[\xi_i=\frac{\ve_i|\nabla u_i|^2}{2}-\frac{W(u_i)}{\ve_i}.\]
Since the level sets \(u_i^{-1}(t)\) converge to \(\Si\) in the Hausdorff sense, we can choose a sequence \(r_i\) such that\;
\begin{itemize}
\item \(r_i \rightarrow 0\);
\item \(\{|u_i|\leq 1-b\}\subset \cN_{r_i}(\Si);\)
\item \(\ve_i/r_i \rightarrow 0.\)
\end{itemize}
By \cite{HT}*{Proposition 4.3} and equationref{4 tilt converges to 0},
\[\lim_{i \rightarrow \infty}\int_{\ov{U}_2}|\xi_i|+\ta_i =0.\]
Let us choose a sequence \(\et_i \rightarrow 0\) such that\;
\[\lim_{i \rightarrow \infty}\et_i^{-1}\int_{\ov{U}_2}|\xi_i|+\ta_i =0.\]
There exists \(i_1 \in \mathbb{N}\) such that\; for all \(i \geq i_1\) we have
\begin{itemize}\vth
\item \(r_i \leq r_0 \) where \(r_0\) is as defined at the beginning of the proof;
\item \(r_i<d(\ov{U}_1,\partial U_2)\).
\end{itemize}\vth\newcommand{\mathscr{G}_i}{\mathscr{G}_i}
As we discussed at the beginning of the proof, for any \(p \in M\) and \(0<r\leq r_0\), if \(u_{\ve}\) is a solution of \(AC_{\ve}(u_{\ve})=0\) on \((B^{n+1}(\mathbf{0},4), g_{p,r})\), Proposition 5.5 and 5.6 of \cite{HT} hold (with the constants depending additionally on \(r_0\)). For our fixed choice of \(s\) and \(b\), we choose \(L\) via Proposition 5.5 and 5.6. For \(i \geq i_1\), let
\[G_i=\ov{U}_1\cap \{y:\int_{B(y,r)}|\xi_i|+\ta_i \leq \et_ir^n there exists\;xt{ if }4\ve_iL\leq r \leq r_i\}.\]
\(G_i\) may not be \(\cH^n\)-measurable; for our later purpose we choose an \(\cH^n\)-measurable set \(\mathscr{G}_i \subset G_i\) as follows. By the Besicovitch covering theorem, there exist \(\{\mathscr{B}_j\}_{j=1}^{\ell}\) such that\; each \(\mathscr{B}_j\) is a collection of at-most countably many mutually disjoint open balls and
\[\ov{U}_1\setminus G_i \subset \bigcup_{j=1}^{\ell}\bigcup_{B \in \mathscr{B}_j} B.\]
Let
\[\mathscr{G}_i = \left( \bigcup_{j=1}^{\ell}\bigcup_{B \in \mathscr{B}_j}B \right)^c \cap \ov{U}_1\subset G_i \]
which is a compact set. Using the monotonicity formula for the scaled energy one can show the following (\cite{HT}*{(5.9)}).
\begin{equation}
\norm{V_i}\left(\ov{U}_1\setminus \mathscr{G}_i\right) + \cH^{n}\left(P(\ov{U}_1\setminus \mathscr{G}_i)\right)\leq c(s,W,g)\et_i^{-1}\int_{\ov{U}_2}|\xi_i|+\ta_i \label{4 G_i}
\end{equation}
which converges to \(0\) as \(i \rightarrow \infty\) by our choice of the sequence \(\{\et_i\}\). \newcommand{\mathscr{G}_i}{\mathscr{G}_i}
We define
\[S_i^t=\{w_i=t\}.\]
For a.e. $t\in [-\si_b/2,\si_b/2]$, \(\{w_i>t\},\{w_i<t\}\in \mathcal{C}(M)\); \(\db{\{w_i>t\}}+\db{\{w_i<t\}}=\db{M}\); and \(\partial\db{\{w_i>t\}}=\db{\{w_i=t\}} \). For such \(t\in [-\si_b/2,\si_b/2]\), \(i\geq i_1\) and Lipschitz continuous function \(\vp:G_nM^{n+1}\rightarrow \mathbb{R}\) with \(|\vp|\leq 1\), \(there exists\;xt{Lip}(\vp)\leq 1\), we obtain the following estimates.\newcommand{S^t_i}{S^t_i}
\begin{align}
&\Md{|S^t_i|(\vp)-N\md{\Si}(\vp)}\leq \Md{\md{S^t_i \cap \ov{U}_1}(\vp)-N\md{\Si \cap \ov{U}_1}(\vp)} + \norm{S^t_i}(U_1^c) + N\norm{\Si}(U_1^c); \label{4.1}
\end{align}varifold\;o
\begin{align}
\Md{\md{S^t_i \cap \ov{U}_1}(\vp)-N\md{\Si \cap \ov{U}_1}(\vp)} \leq \Md{&\md{S^t_i \cap \mathscr{G}_i}(\vp)-N\md{P(\mathscr{G}_i)}(\vp)}\nonumber\\
&+\norm{S^t_i}(\ov{U}_1\setminus \mathscr{G}_i)+ N \cH^n(P(\ov{U}_1\setminus\mathscr{G}_i)); \label{4.2}
\end{align}varifold\;o
\begin{align}
\Md{\md{S^t_i \cap \mathscr{G}_i}(\vp)-N\md{P(\mathscr{G}_i)}(\vp)}& \leq \Md{\md{S^t_i \cap \mathscr{G}_i}(\vp)-P_{\#}\md{S^t_i \cap \mathscr{G}_i}(\vp)}\nonumber\\
& +\Md{P_{\#}\md{S^t_i \cap \mathscr{G}_i}(\vp)-N\md{P(\mathscr{G}_i)}(\vp)}; \label{4.3}
\end{align}varifold\;o
\begin{align}
& \Md{\md{S^t_i \cap \mathscr{G}_i}(\vp)-P_{\#}\md{S^t_i \cap \mathscr{G}_i}(\vp)}\nonumber\\
& \leq \int_{S_i^t\cap\; \mathscr{G}_i}\Md{\vp(y,T_yS^t_i)-\vp\left(P(y),DP|_y(T_yS^t_i)\right)JP(y,T_yS^t_i)}\; d\cH^n(y).\label{4.4}
\end{align}
Using \cite{HT}*{Proposition 5.6} and a scaling argument as in the Proof of Theorem 1 of \cite{HT}, as \(i \rightarrow \infty\)
\begin{equation*}
d_{G_nM}\left(\left(y,T_yS^t_i\right),\left(P(y), DP|_y(T_yS^t_i)\right)\right) \rightarrow 0 there exists\;xt{ and }JP(y,T_yS^t_i) \rightarrow 1
\end{equation*}
uniformly for \(|t|\leq \si_b/2\) and \(y\in S_i^t\cap \mathscr{G}_i\). Here \(d_{G_nM}\) denotes the distance in \(G_nM\). Hence, there exists a sequence of positive real numbers \(\{\tht_i\}_{i=1}^{\infty}\) (which does not depend on \(t\)) such that\;
\begin{equation}
\lim_{i \rightarrow \infty} \tht_i = 0\; there exists\;xt{ and }\; \sup_{\vp}\Md{\md{S^t_i \cap \mathscr{G}_i}(\vp)-P_{\#}\md{S^t_i \cap \mathscr{G}_i}(\vp)} \leq \tht_i \cH^n(S^t_i) \label{4.5}
\end{equation}
for all \(t\in [-\si_b/2,\si_b/2].\)
Let us choose a cut-off function \(0\leq \ch'_2\leq 1\) with \(\spt(\ch'_2)\subset U_2\) and \(\ch'_2 equationuiv 1\) on \(\ov{U}_1\). If \(\mathbf{p}:G_nM\rightarrow M\) is the canonical projection map, \(\ch_2=\ch'_2\circ \mathbf{p}.\)
For \(y \in P(\mathscr{G}_i)\), let \(m_i^t(y)\) be the cardinality of the set $P^{-1}(y)\capS^t_i\cap\mathscr{G}_i$. Using \cite{HT}*{Proposition 5.5, 5.6} and a scaling argument as in the Proof of Theorem 1 of \cite{HT}, there exists \(i_2\in \mathbb{N}\) such that\; for all \(i \geq i_2\), \(y \in P(\mathscr{G}_i)\) and \(|t|\leq \si_b/2\), \(m_i^t(y)\leq N\). (Here we need to use \(s\leq s_0\).) Hence,
\begin{align}
& \Md{P_{\#}\md{S^t_i \cap \mathscr{G}_i}(\vp)-N\md{P(\mathscr{G}_i)}(\vp)}\nonumber\\
&\leq \int_{P(\mathscr{G}_i)}(N-m_i^t(y))\; d\cH^n(y)\nonumber\\
&= N\md{P(\mathscr{G}_i)}(\ch_2)-P_{\#}\md{S^t_i \cap \mathscr{G}_i}(\ch_2)\nonumber\\
&\leq N\md{\Si}(\ch_2)-P_{\#}\md{S^t_i}(\ch_2)+2\norm{S^t_i}(\ov{U}_1\setminus\mathscr{G}_i)+2\norm{S^t_i}(U_1^c)\;(there exists\;xt{by }equationref{4 JP is bdd}).\label{4.6}
\end{align}
We will now integrate the various error terms obtained above with respect to \(t\) in the interval \([-\si_b/2,\si_b/2]\) and let \(i \rightarrow \infty\).
\begin{align}
&\limsup_{i \rightarrow \infty}\int_{-\si_b/2}^{\si_b/2}\norm{S^t_i}(U_1^c)\;dt\leq \limsup_{i \rightarrow \infty}\si\norm{V_i}(U_1^c)\leq \si s \;(there exists\;xt{by } equationref{4 U_0^c has small area});\\
&\int_{-\si_b/2}^{\si_b/2}N\norm{\Si}(U_1^c)\; dt\leq \si s \;(there exists\;xt{by } equationref{4 U_0^c has small area});\\
&\lim_{i \rightarrow \infty}\int_{-\si_b/2}^{\si_b/2}\norm{S^t_i}(\ov{U}_1\setminus \mathscr{G}_i)+ N \cH^n(P(\ov{U}_1\setminus\mathscr{G}_i))\;dt =0 \; (there exists\;xt{by equationref{4 G_i}});\\
&\limsup_{i \rightarrow \infty}\int_{-\si_b/2}^{\si_b/2}\norm{S^t_i}(M) \;dt \leq \limsup_{i \rightarrow \infty}\si\norm{V_i}(M)=\si N \cH^n(\Si).
\end{align}
We note that \(V_i \rightarrow N|\Si|\) in the varifold sense implies \(P_{\#}V_i(\ch_2)\rightarrow N|\Si|(\ch_2).\) (\(P_{\#}V_i(\ch_2)\) is well-defined as \(\spt(\ch_2)\subset U_2.\)) Hence, using equationref{4 JP is bdd} and equationref{4 small energy},
\begin{align}
&\limsup_{i \rightarrow \infty}\int_{-\si_b/2}^{\si_b/2}\left(N\md{\Si}(\ch_2)-P_{\#}\md{S^t_i}(\ch_2)\right) dt \nonumber\\
&\leq \lim_{i \rightarrow \infty}\si \left(N\md{\Si}(\ch_2)-P_{\#}V_i(\ch_2)\right)+\limsup_{i \rightarrow \infty}2\int_{\{|u_i|\geq 1-b\}}|\nabla w_i|\nonumber\\
&\leq 2s. \label{4.7}
\end{align}
Hence, using the equations equationref{4.1}-equationref{4.7}, we conclude that for all \(i \geq i_2\), there exists a measurable function \(\Th_i:[-\si_b/2,\si_b/2]\rightarrow \mathbb{R}\) such that\;
\[\mathbf{F}(N|\Si|,\md{S^t_i})\leq \Th_i(t)\;\forall t\in [-\si_b/2, \si_b/2] \; there exists\;xt{ and }\; \limsup_{i \rightarrow \infty}\int_{-\si_b/2}^{\si_b/2}\Th_i(t) \; dt\leq (2+4\si)s.\]
Hence, there exists \(i_3 \in \mathbb{N}\) such that\; for all \(i\geq i_3\),
\[\int_{-\si_b/2}^{\si_b/2}\Th_i(t) \; dt\leq (3+4\si)s.\]
Therefore, for \(i \geq i_3\), there exists \(t_i \in [-\si_b/2, \si_b/2] \) such that\;
\[\mathbf{F}\left(N|\Si|,\md{S_i^{t_i}}\right)\leq \Th_i(t_i)\leq (3+4\si)\si_b^{-1}s.\]
Since \(s\in (0,s_0]\) is arbitrary and \(b\leq b_0\leq 1/2\), this finishes the proof of the proposition.
\end{proof}
\begin{pro}\label{pro 4.2}
Let \(\{u_{i}:M \rightarrow (-1,1) \}_{i=1}^{\infty}\) be a sequence of smooth functions such that\; items (i) -- (iii) of Proposition \ref{pro 4.1} are satisfied. Additionally, we assume that \(u_i\) is a min-max critical point of \(E_{\ve_i}\) corresponding to the homotopy class \(\tilde{\Pi}\). Let us set \(L_{\ve_i}=\mathbf{L}_{\ve_i}(\tilde{\Pi})\) so that
\[L=\mathbf{L}_{AP}(\Pi)=\frac{1}{2\si} \lim_{i \rightarrow \infty}L_{\ve_i}=\norm{V}(M).\]
Then, (using the notation from Proposition \ref{pro 4.1}) for every \(s>0\) and \(b \in (0,b_0(s)]\), there exists \(i_0^{*}\geq i_0\) such that\; the following holds. For all \(i \geq i_0^{*}\), there exists \(\Ph_i:X \rightarrow \cZ_n(M^{n+1};\mathbf{M};\mathbb{Z}_2)\), \(x_i^{*}\in X\) and \(\de_i > 0\) such that\; \(\Ph_i \in \Pi\), \(\de_i \rightarrow 0\),
\begin{equation}
\sup_{x \in X}\mathbf{M}\left(\Ph_i(x)\right)\leq \max \left\{\frac{1}{2\si_b}(L_{\ve_i}+\ve_i),L+s\right\} + \de_i \quad there exists\;xt{ and } \quad \mathbf{F}\left(V,\md{\Ph_i(x_i^{*})}\right)\leq s. \label{eqn of prop 4.2}
\end{equation}
\end{pro}varifold\;o
\begin{proof}
Since \(u_i\) is a min-max critical point of \(E_{\ve_i}\) corresponding to the homotopy class \(\tilde{\Pi}\), for each \(i\), there exists a sequence of continuous, \(\mathbb{Z}_2\)-equivariant maps \(\{h^i_j:\tilde{X} \rightarrow H^1(M)\setminus\{0\}\}_{j=1}^{\infty}\) such that\;
\[\sup_{x \in \tilde{X},\; j \in \mathbb{N}}E_{\ve_i}\left(h^i_j(x)\right)\leq L_{\ve_i}+\ve_i\quad there exists\;xt{ and }\quad \lim_{j\rightarrow \infty}d_{H^1(M)}\left(u_i,h^i_j(\tilde{X})\right)=0. \]
The next Lemma is a restatement of Lemma 8.10 and 8.11 of \cite{Guaraco}.
\begin{lem}[\cite{Guaraco}*{Lemma 8.10, 8.11}]
Let \(h_1,h_2\in H^1(M)\). For \(\de\in (0,1)\), we set \(C_{\de}=W(1-\de)>0.\) Then, for all \(\ve>0\), \[\cH^{n+1}\left(\{\md{h_1}\leq 1-\de\}\right)\leq \ve C_{\de}^{-1}E_{\ve}(h_1).\]
Let \(\al_0\in (-1+\de,1-\de)\) be such that\; for \(j=1,2\), \(\Om_j=\{h_j > \al_0\}\in \mathcal{C}(M).\) Then, for all \(\ve>0\),
\[\cH^{n+1}\left(\Om_1 \setminus\Om_2\right)\leq \ve C_{\de}^{-1}E_{\ve}(h_2)+(\al_0+1-\de)^2\norm{h_1-h_2}^2_{H^1(M)}.\]
As a consequence, for \(j=1,2\), if \(\la_j\in (-1+\de,1-\de)\) such that \(T_j=\partial \db{\{h_j>\la_j\}}\in \cZ_n(M^{n+1};\mathbb{Z}_2)\),
\[\cF(T_1,T_2)\leq 2\ve C_{\de}^{-1}(E_{\ve}(h_1)+E_{\ve}(h_2))+2(\al_0+1-\de)^2\norm{h_1-h_2}^2_{H^1(M)}\; \forall\ve>0.\]
\label{lem 4.3}
\end{lem}varifold\;o
We recall that \(X\) is a subcomplex of \(\sci^N[1]\) for some \(N \in \mathbb{N}\). There exists \(\al \in (-1+b, 1-b)\) such that\; for all \(i,j\in \mathbb{N}\) and \(x \in \pi^{-1}(X\cap \mathbb{Q}^N)\), \(\{h^i_j(x)> \al\}\in \mathcal{C}(M)\). Let us fix \(i\geq i_0\) (\(i_0\) is as in Proposition \ref{pro 4.1}); let \(j_0\in \mathbb{N}\) such that\;
\[d_{H^1(M)}\left(u_i,h^i_{j_0}(\tilde{X})\right)\leq \ve_i/2.\]
For simplicity, let us denote the map \(h^i_{j_0}\) by \(h\). We choose \(l_i\in \mathbb{N}\) such that\; if \(x,x'\) belong to a common cell in \(\tilde{X}[3^{-l_{i}}]\), \(\norm{h(x)-h(x')}_{H^1(M)}\leq \ve_i/2.\) Let \(x_i^*\in \tilde{X}[3^{-l_{i}}]_0\) such that\;
\[d_{H^1(M)}\left(u_i,h(x_i^*)\right)=d_{H^1(M)}\left( u_i ,h\left(\tilde{X}[3^{-l_{i}}]_0\right) \right)\]
which is bounded by \(\ve_i\) by our choice of \(j_0\) and \(l_{i}\). Following the argument in \cite{Guaraco,GG1}, there exists a function \(\la:\tilde{X} \rightarrow (-1+b, 1+b)\) such that\; for all \(x \in \tilde{X}\) the following conditions are satisfied.
\begin{itemize}\vth
\item $\la(T(x))=-\la(x)$;
\item \(\{h(x)>\la(x)\}, \{h(x)<\la(x)\}\in \mathcal{C}(M)\) with \(\db{\{h(x)>\la(x)\}}+\db{\{h(x)<\la(x)\}}=\db{M}\);
\item Denoting \(\tilde{h}=F\circ h\) and \(\tilde{\la}=F\circ \la\); \(2\si_b\mathbf{M}\big(\partial\db{\{\tilde{h}(x)>\tilde{\la}(x)\}}\big) \leq E_{\ve_i}(h(x)). \)
\end{itemize}\vth
One can define a discrete, \(\mathbb{Z}_2\)-equivariant map \(\tilde{\vp}_i:\tilde{X}[3^{-l_{i}}]_0 \rightarrow \mathbf{I}_{n+1}(M^{n+1};\mathbb{Z}_2)\) which is fine in the flat norm as follows (\(w_i\), \(t_i\) are as in Proposition \ref{pro 4.1}).
\begin{equation*}
\tilde{\vp}_i(x)=
\begin{cases}
\db{\{\tilde{h}(x)>\tilde{\la}(x)\}} & there exists\;xt{if } x \notin \{x_i^*,T(x_i^*)\};\\
\db{\{w_i>t_i\}} & there exists\;xt{if } x = x_i^*;\\
\db{\{w_i<t_i\}} & there exists\;xt{if } x = T(x_i^*).
\end{cases}
\end{equation*}
If \(\vp_i=\partial \circ \tilde{\vp}_i\), using Lemma \ref{lem 4.3}, fineness of \(\vp_i\) with respect to\; the flat norm
\[\mathbf{f}^{\cF}(\vp_i)\leq 4 \ve_i C^{-1}_b(L_{\ve_i}+\ve_i)+2(1-b+\al)^{-2}\ve_i^2\]
which converges to \(0\) as \(i \rightarrow \infty\). Moreover, \(\mathbf{F}(V,\md{\{w_i=t_i\}})\leq s\) implies that \(\mathbf{M}\left(\db{\{w_i=t_i\}}\right)\leq L+s\). Hence,
\[\sup_{x\in X[3^{-l_{i}}]_0}\mathbf{M}(\vp_i(x))\leq \max \left\{\frac{1}{2\si_b}(L_{\ve_i}+\ve_i),L+s\right\}.\]
As argued in \cite{Guaraco,GG1}, one can apply the interpolation theorem of Zhou \cite{Zhou}*{Proposition 5.8} to produce a sequence of discrete maps whose fineness with respect to\; the mass norm converges to \(0\) and then, using the interpolation theorem of Marques and Neves \cite{MN_Willmore}*{Theorem 14.1}, one can find a sequence of maps continuous in the mass norm. More precisely, there exists \(i_0^*\geq i_0\) such that\; for all \(i\geq i_0^*\) there exist \(\Ph_i:X \rightarrow \cZ_n(M^{n+1};\mathbf{M};\mathbb{Z}_2)\) and \(\de_i > 0\) such that\; \(\Ph_i \in \Pi\), \(\de_i \rightarrow 0\) and
\begin{equation*}
\sup_{x \in X}\mathbf{M}\left(\Ph_i(x)\right)\leq \max \left\{\frac{1}{2\si_b}(L_{\ve_i}+\ve_i),L+s\right\}+\de_i.
\end{equation*}
Moreover, \(\Ph_i(x)=\vp_i(x)\) for all $x\in X[3^{-l_{i}}]_0 $. In particular, \(\Ph_i(x_i^{*})=\vp_i(x_i^*)=\db{\{w_i=t_i\}}\); hence,
\[\mathbf{F}\left(V,\md{\Ph_i(x_i^{*})}\right)\leq s.\]
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm critical set}]
By letting \(s\rightarrow 0\), \(b \rightarrow 0\) and \(i \rightarrow \infty\) in the above Proposition \ref{pro 4.2}, we obtain Theorem \ref{thm critical set}. More precisely, let \(\{s_m\}_{m=1}^{\infty}\) be a sequence such that\; \(s_m \rightarrow 0\). We choose \(b_m\in (0, b_0(s_m)]\) such that\; \(b_m \rightarrow 0\). By Proposition \ref{pro 4.2}, for every \(m \in \mathbb{N}\), there exist \(\Ps_m:X \rightarrow \cZ_n(M^{n+1};\mathbf{M};\mathbb{Z}_2)\), \(i(m)\in \mathbb{N}\) and \(x_m^{\bullet}\in X\) such that\; \(\Ps_m \in \Pi\), \(i(m)>i(m-1)\),
\begin{equation*}
\sup_{x \in X}\mathbf{M}\left(\Ps_m(x)\right)\leq \max \left\{\frac{1}{2\si_{b_m}}(L_{\ve_{i(m)}}+\ve_{i(m)}),L+s_m\right\} + s_m \; there exists\;xt{ and } \; \mathbf{F}\left(V,\md{\Ps_m(x_m^{\bullet})}\right)\leq s_m. \label{eqn of prop 4.2 bis}
\end{equation*}
This implies \(\{\Ps_m\}_{m=1}^{\infty}\) is a minimizing sequence in \(\Pi\) and \(V \in \mathbf{C}\left(\{\Ps_m\}\right)\).
\end{proof}
\nocite{*}
\end{document} | math | 127,594 |
\begin{document}
\title[A hierarchy on non-archimedean CLI Polish groups]{A hierarchy on non-archimedean Polish groups admitting a compatible complete left-invariant metric}
\author{Longyun Ding}
\address{School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, P.R.China}
{\rm e}mail{[email protected]}
\thanks{Research is partially supported by the National Natural Science Foundation of China (Grant No. 11725103).}
\author{Xu Wang}
{\rm e}mail{[email protected]}
\subjclass[2010]{Primary 03E15, 22A05}
\keywords{hierarchy, non-archimedean Polish group, tree}
\begin{abstract}
In this article, a hierarchy named $\alpha$-CLI for $\alpha<\omega_1$ is defined on non-archimedean Polish groups admitting a compatible complete left-invariant metric. Then (1) $G$ is $0$-CLI iff $G=\{1_G\}$; and (2) $G$ is $1$-CLI iff $G$ admit a compatible complete two sided invariant metric.
This notion forms a proper hierarchy because, for any $\alpha<\omega_1$, there exist non-archimedean CLI Polish groups $G_\alpha$ and $H_\alpha$ such that $H_\alpha$ is $\alpha$-CLI but not contains any open subgroup which is $\beta$-CLI for $\beta<\alpha$; and $G_\alpha$ is not $\alpha$-CLI but contains an open subgroup which is $\alpha$-CLI, and hence $G_\alpha$ is $(\alpha+1)$-CLI.
{\rm e}nd{abstract}
\maketitle
\section{Introduction}
A Polish group is {{\rm i}t non-archimedean} if it has a neighborhood base of its identity element consisting of open subgroups. By a theorem of Becker and Kechris (cf.~\cite[Theorem 1.5.1]{BK}), a Polish group is non-archimedean iff it is homeomorphic to a closed subgroup of $S_{\rm i}nfty$, the group of all permutations of ${\mathbb{N}}$ equipped with the pointwise convergence topology. A metric $d$ on a group $G$ is {{\rm i}t left-invariant} if $d(gh,gk)=d(h,k)$ for all $g,h,k{\rm i}n G$. A Polish group is {{\rm i}t CLI} if it admits a compatible complete left-invariant metric.
Malicki \cite{malicki11} defined a notion of orbit tree $T_G$ for each closed subgroup $G$ of $S_{\rm i}nfty$, and showed that $G$ is CLI iff $T_G$ is well-founded. Moreover, he proved that the heights of orbit trees of all CLI closed subgroups of $S_{\rm i}nfty$ are cofinal in $\omega_1$. Granting this cofinality, Malicki proved that the family of all CLI groups is coanalytic non-Borel. After that, Xuan defined a deferent kind of orbit trees and showed that, a closed subgroup of $S_{\rm i}nfty$ is locally compact iff its orbit tree has finite height (cf. \cite[Theorem 3.7]{xuan}). It is worth noting that both kinds of orbit trees defined by Malicki and Xuan are all defined on closed subgroups of $S_{\rm i}nfty$ rather than directly on non-archimedean Polish groups. As a result, two mutual topological isomorphic closed subgroups of $S_{\rm i}nfty$ can have completely different orbit trees, and even the rank of their orbit trees can be different. Therefore, we cannot directly use the ranks of their orbit trees as a hierarchy on these groups.
In this article, for a given non-archimedean CLI Polish group $G$, instead of using a closed subgroup of $S_{\rm i}nfty$ that is topologically isomorphic to it, we use a neighborhood base of the identity element $1_G$ to define a new kind of orbit trees. More specifically, let $\mathcal G=(G_n)$ be a decreasing sequence of open subgroups of $G$ with $G_0=G$ such that $(G_n)$ forms a neighborhood base of $1_G$, we will define a well-founded tree $T^{X(\mathcal G)}_\mathcal G$, whose rank denoted by $\rho(\mathcal G)$. We will show that (a) the biggest countable ordinal $\beta$ with $\rho(\mathcal G)\ge\omega\cdot\beta$, denoted by ${\rm rank}(G)$, is independent to the choice of $\mathcal G$; and (b) whether $\rho(\mathcal G)=\omega\cdot\beta$ is also independent to the choice of $\mathcal G$. This makes the following definitions form a well-defined hierarchy on non-archimedean CLI Polish groups: given an ordinal $\alpha<\omega_1$,
\begin{enumerate}
{\rm i}tem[(1)] if $\rho(\mathcal G)\le\omega\cdot\alpha$, we say $G$ is $\alpha$-CLI;
{\rm i}tem[(2)] if ${\rm rank}(G)\le\alpha$, we say $G$ is L-$\alpha$-CLI.
{\rm e}nd{enumerate}
It is clear that, if $G$ is L-$\alpha$-CLI, it is also $(\alpha+1)$-CLI.
The following results show that this hierarchy can well classify non-archimedean CLI Polish groups:
\begin{theorem}
Let $G$ be a non-archimedean CLI Polish group, $\alpha<\omega_1$. Then we have
\begin{enumerate}
{\rm i}tem[(1)] $G$ is $0$-CLI iff $G=\{1_G\}$;
{\rm i}tem[(2)] $G$ is L-$0$-CLI iff $G$ is discrete;
{\rm i}tem[(3)] $G$ is $1$-CLI iff $G$ is TSI, i.e., $G$ admit a compatible complete two sided invariant metirc;
{\rm i}tem[(4)] $G$ is L-$\alpha$-CLI iff $G$ is locally $\alpha$-CLI, i.e., $G$ has an open subgroup which is $\alpha$-CLI.
{\rm e}nd{enumerate}
{\rm e}nd{theorem}
It is well know that all compact Polish groups are TSI (c.f.~\cite[Theorem 2.1.5]{gaobook}), and all locally compact Polish groups are CLI (c.f.~\cite[Theorem 2.2.5]{gaobook}). Now we know that all compact non-archimedean Polish groups are $1$-CLI, and all locally compact non-archimedean Polish groups are L-$1$-CLI.
\begin{theorem}
Let $G$ be a non-archimedean CLI Polish group, $H$ a closed subgroup of $G$, $N$ a closed normal subgroup of $G$, and $\alpha<\omega_1$. If $G$ is $\alpha$-CLI (or L-$\alpha$-CLI), so are $H$ and $G/N$. In particular, we have ${\rm rank}(H)\le{\rm rank}(G)$ and ${\rm rank}(G/N)\le{\rm rank}(G)$.
{\rm e}nd{theorem}
\begin{theorem}
Let $(G^i)$ be a sequence of non-archimedean CLI Polish groups, $\alpha<\omega_1$, and let $G=\prod_iG^i$. Then we have
\begin{enumerate}
{\rm i}tem[(1)] $G$ is $\alpha$-CLI iff all $G^i$ are $\alpha$-CLI;
{\rm i}tem[(2)] $G$ is L-$\alpha$-CLI iff all $G^i$ are L-$\alpha$-CLI and for all but finitely many $i$, $G^i$ is $\alpha$-CLI.
{\rm e}nd{enumerate}
{\rm e}nd{theorem}
In the end, we show that this hierarchy is proper by prove the following:
\begin{theorem}
For any $\alpha<\omega_1$, there exist non-archimedean CLI Polish groups $G_\alpha$ and $H_\alpha$ with ${\rm rank}(G_\alpha)={\rm rank}(H_\alpha)=\alpha$ such that $H_\alpha$ is $\alpha$-CLI and $G_\alpha$ is L-$\alpha$-CLI but not $\alpha$-CLI.
{\rm e}nd{theorem}
\section{Preliminaries}
We denote ${\rm Ord}$ the class of all ordinal. For $\alpha{\rm i}n{\rm Ord}$, we denote
$$\omega(\alpha)=\max\{0,\lambda:\lambda\le\alpha\mbox{ is a limit ordinal}\}.$$
Then $\alpha=\omega(\alpha)+m$ for some $m<\omega$.
Let $E$ be an equivalence relation on a set $X$, $x{\rm i}n X$, and $A\subseteq X$. The $E$-equivalence class of $x$ is $[x]_E=\{y{\rm i}n X:xEy\}$. Similarly, the $E$-saturation of $A$ is $[A]_E=\{y{\rm i}n X:{\rm e}xists z{\rm i}n A\,(yEz)\}$.
The identity element of a group $G$ is denoted as $1_G$. Let $H$ be a subgroup of $G$, we denote $G/H=\{gH:g{\rm i}n G\}$ the set of all left-cosets of $H$.
A topological space is {{\rm i}t Polish} if it is separable and completely metrizable. A topological group is {{\rm i}t Polish} if its underlying topology is Polish. Let $G$ be a Polish group and $X$ a Polish space, an action of $G$ on $X$, denoted by $G\curvearrowright X$, is a map $a:G\times X\to X$ satisfies that $a(1_G,x)=x$ and $a(gh,x)=a(g,a(h,x))$ for $g,h{\rm i}n G$ and $x{\rm i}n X$. The pair $(X,a)$ is called a {{\rm i}t Polish $G$-space} if $a$ is continuous. For brevity, we write $g\cdot x$ in place of $a(g,x)$. The {{\rm i}t orbit equivalence relation} $E_G^X$ defined as, $xE_G^Xy{\rm i}ff{\rm e}xists g{\rm i}n G\,(g\cdot x=y)$. Note that the $E_G^X$-equivalence class of $x$ is $G\cdot x=\{g\cdot x:g{\rm i}n G\}$, which is also called the {{\rm i}t $G$-orbit} of $x$. Similarly, for $A\subseteq X$, the $E_G^X$-saturation of $A$ is $G\cdot A=\{g\cdot x:g{\rm i}n G\wedge x{\rm i}n A\}$.
Let $<$ be a binary relation on a set $T$. We say $(T,<)$ is a {{\rm i}t tree} if
\begin{enumerate}
{\rm i}tem[(1)] $\forall s{\rm i}n T\,(s\not<s)$,
{\rm i}tem[(2)] $\forall s,t,u{\rm i}n T\,((s<t\wedge t<u)\Rightarrow s<u)$,
{\rm i}tem[(3)] $\forall s{\rm i}n T\,(|\{t{\rm i}n T:t<s\}|<\omega\wedge\forall t,u<s\,(t=u\vee t<u\vee u<t))$.
{\rm e}nd{enumerate}
For $s{\rm i}n T$, we denote ${\rm lh}(s)=|\{t{\rm i}n T:t<s\}|$, which is called the length of $s$. It is clear that $s<t$ implies ${\rm lh}(s)<{\rm lh}(t)$. For $n<\omega$, we denote the $n$-th level of $T$ by
$$L_n(T)=\{s{\rm i}n T:{\rm lh}(s)=n\}.$$
Each element in $L_0(T)$ is called a {{\rm i}t root} of $T$.
Let $(T,<)$ be a tree. We say $T$ is {{\rm i}t well-founded} if any non-empty subset of $T$ contains a maximal element, or equivalently (under AC), there is no infinite strictly increasing sequence in $T$. Let $T$ be a well-founded tree, we define $\rho_T:T\to{\rm Ord}$ by transfinite induction as
$$\rho_T(s)=\sup\{\rho_T(t)+1:s<t\wedge t{\rm i}n T\}.$$
If $\rho_T(s)=0$, we say $s$ is a {{\rm i}t terminal} of $T$. Then we denote
$$\rho(T)=\sup\{\rho_T(s)+1:s{\rm i}n T\}.$$
So $\rho(T)=0$ iff $T={\rm e}mptyset$. It is clear that $\rho(T)=\sup\{\rho_T(s)+1:s{\rm i}n L_0(T)\}$. If $L_0(T)=\{s_0\}$ is a singleton, then $\rho(T)=\rho_T(s_0)+1$ is a successor ordinal.
For $s{\rm i}n T$, we denote
$$T_s=\{t{\rm i}n T:s=t\vee s<t\}.$$
Since $L_0(T_s)=\{s\}$, we have $\rho(T_s)=\rho_T(s)+1$. For the convenience of discussion, while $s\notin T$, we also denote $T_s={\rm e}mptyset$, and thus $\rho(T_s)=0$. Therefore, $\rho(T_s)$ is always a non-limit ordinal no matter $s$ in $T$ or not.
\begin{proposition}\label{tree}
Let $T$ be a well-founded tree, then
$$\rho(T)=\sup\{\rho(T_s):s{\rm i}n T\}=\sup\{\rho(T_s):s{\rm i}n T\wedge s{\rm i}n L_0(T)\},$$
and for all $s{\rm i}n T$,
$$\begin{array}{ll}\rho(T_s) &=\sup\{\rho(T_t):s<t\wedge t{\rm i}n T\}+1\cr
&=\sup\{\rho(T_t):s<t\wedge t{\rm i}n T\wedge{\rm lh}(t)={\rm lh}(s)+1\}+1.{\rm e}nd{array}$$
{\rm e}nd{proposition}
\begin{proposition}\label{L_k(T)}
Let $(T,<)$ be a well-founded tree, $k<\omega$. Then we have
\begin{enumerate}
{\rm i}tem[(1)] $\sup\{\rho(T_s):s{\rm i}n L_k(T)\}\le\rho(T)\le\sup\{\rho(T_s):s{\rm i}n L_k(T)\}+k$;
{\rm i}tem[(2)] $\omega(\rho(T))=\omega(\sup\{\rho(T_s):s{\rm i}n L_k(T)\})$;
{\rm i}tem[(3)] if $\rho(T_s)\ge\alpha$ for some $s{\rm i}n L_k(T)$, then $\rho(T)\ge\alpha+k$.
{\rm e}nd{enumerate}
{\rm e}nd{proposition}
\begin{proof}
It is routine to prove clause (1) by induction on $k$. Clause (2) is an easy corollary of (1). And clause (3) is trivial.
{\rm e}nd{proof}
Let $(S,<)$ and $(T,<)$ be two trees. A map $\phi:S\to T$ is said to be an {{\rm i}t order preserving map} if
$$\forall s,t{\rm i}n S\,(s<t\Rightarrow\phi(s)<\phi(t)).$$
It is said be an {{\rm i}t order preserving embedding (isomorphism)} if it is injective (bijective) and
$$\forall s,t{\rm i}n S\,(s<t{\rm i}ff\phi(s)<\phi(t)).$$
In particular, an order preserving map $\phi$ is said to be {{\rm i}t Lipschitz} if ${\rm lh}(\phi(s))={\rm lh}(s)$ for all $s{\rm i}n S$.
\begin{proposition}\label{subtree}
Let $(S,<)$ be a tree, $(T,<)$ a well-founded tree. If there exists an order preserving map $\phi:S\to T$, then $(S,<)$ is well-founded too, and we have $\rho(S)\le\rho(T)$.
{\rm e}nd{proposition}
\begin{proof}
It $S$ contains an infinite strictly increasing sequence $(s_n)$, then $(\phi(s_n))$ is an infinite strictly increasing sequence in $T$, contradicting that $(T,<)$ is well-founded.
We prove by induction that $\rho_S(s)\le\rho_T(\phi(s))$ for $s{\rm i}n S$ as follows:
$$\begin{array}{ll}\rho_S(s) &=\sup\{\rho_S(u)+1:s<u\wedge u{\rm i}n S\}\cr
&\le\sup\{\rho_T(\phi(u))+1:s<u\wedge u{\rm i}n S\}\cr
&\le\sup\{\rho_T(t)+1:\phi(s)<t\wedge t{\rm i}n T\}=\rho_T(\phi(s)).{\rm e}nd{array}$$
And hence $\rho(S)\le\rho(T)$.
{\rm e}nd{proof}
\section{Definition of the hierarchy}
\begin{definition}
Let $X$ be a set, $\mathcal E=(E_n)$ a decreasing sequence of equivalence relations on $X$, i.e., $E_n\supseteq E_{n+1}$ for each $n<\omega$. We denote
$$T_\mathcal E^X=\{(n,C):{\rm e}xists x{\rm i}n X\,(C=[x]_{E_n}\ne\{x\})\}.$$
For $(n,C),(m,D){\rm i}n T_\mathcal E^X$, we define
$$(n,C)<(m,D){\rm i}ff n<m\wedge C\supseteq D.$$
{\rm e}nd{definition}
It is straightforward to check that $(T_\mathcal E^X,<)$ is a tree. Note that $(n,C){\rm i}n T_\mathcal E^X$ iff $C$ is a non-singleton equivalence class of $E_n$.
\begin{definition}
Let $G$ be a non-archimedean Polish group, we denote by ${\rm dgnb}(G)$ the set of all decreasing sequences $\mathcal G=(G_n)$ of open subgroups of $G$ with $G_0=G$ such that $(G_n)$ forms a neighborhood base of $1_G$.
Let $X$ be a countable discrete Polish $G$-space, $\mathcal G=(G_n){\rm i}n{\rm dgnb}(G)$. We denote $E_n=E_{G_n}^X$, i.e., $xE_ny{\rm i}ff{\rm e}xists g{\rm i}n G_n\,(g\cdot x=y)$, and hence $[x]_{E_n}=G_n\cdot x$. Then we write $\mathcal E=(E_n)$ and
$$T_\mathcal G^X=T_\mathcal E^X.$$
{\rm e}nd{definition}
Therefore, $(n,C){\rm i}n T_\mathcal G^X$ iff $C$ is a non-singleton $G_n$-orbit. In contrast, the definitions of both orbit trees from Malicki and Xuan are based on infinite orbits.
\begin{lemma}\label{CLItoWF}
Let $G$ be a non-archimedean CLI Polish group, $X$ a countable discrete Polish $G$-space, and let $\mathcal G=(G_n){\rm i}n{\rm dgnb}(G)$. Then $T_\mathcal G^X$ is well-founded.
{\rm e}nd{lemma}
\begin{proof}
Assume for contradiction that $T_\mathcal G^X$ is ill-founded, then there exists an infinite sequence $(n,C_n),\,n{\rm i}n\omega$ in $T_\mathcal G^X$ with $C_n\supseteq C_{n+1}$ for each $n$.
Let $d$ be a compatible complete left-invariant metric on $G$.
Fix an $x_0{\rm i}n C_0$. Then we have $G_0\cdot x_0=C_0\supseteq C_1$, so we can find a $g_0{\rm i}n G_0$ such that $g_0\cdot x_0{\rm i}n C_1$. Inductively, we can find a $g_n{\rm i}n G_n$ for each $n$ such that $g_ng_{n-1}\cdots g_0\cdot x_0{\rm i}n C_{n+1}$. Put $h_n=g_0^{-1}\cdots g_n^{-1}$ for $n{\rm i}n\omega$. For any $n,p{\rm i}n\omega$, we have $d(h_{n+p},h_n)=d(h_n^{-1}h_{n+p},1_G)=d(g_{n+1}^{-1}\cdots g_{n+p}^{-1},1_G)\le{\rm diam}(G_{n+1})\to 0$ as $n\to{\rm i}nfty$. It follows that $(h_n)$ is a $d$-Cauchy sequence in $G$, so converges to some $h{\rm i}n G$.
Denote $x_{\rm i}nfty=h^{-1}\cdot x_0$. By $h_n\to h$, we have $h_n^{-1}\to h^{-1}$, and hence $h_n^{-1}\cdot x_0\to h^{-1}\cdot x_0=x_{\rm i}nfty$. Note that $X$ is discrete, there exists $N$ such that $h_n^{-1}\cdot x_0=x_{\rm i}nfty$ for any $n>N$, so $x_{\rm i}nfty=g_ng_{n-1}\cdots g_0\cdot x_0{\rm i}n C_{n+1}$. This implies that $x_{\rm i}nfty{\rm i}n\bigcap_nC_n$ and $G_n\cdot x_{\rm i}nfty=C_n$ for each $n$.
In the end, we denote $G_{x_{\rm i}nfty}=\{g{\rm i}n G:g\cdot x_{\rm i}nfty=x_{\rm i}nfty\}$ and put $f:G\to X$ as $f(g)=g\cdot x_{\rm i}nfty$. Since $f$ is continuous and $\{x_{\rm i}nfty\}$ is clopen in $X$, it follows that $G_{x_{\rm i}nfty}=f^{-1}(x_{\rm i}nfty)$ is a clopen subgroup of $G$. So there exists an $m$ such that $G_m\subseteq G_{x_{\rm i}nfty}$. Then we have $C_m=G_m\cdot x_{\rm i}nfty=\{x_{\rm i}nfty\}$, contradicting that $(m,C_m){\rm i}n T_\mathcal G^X$.
{\rm e}nd{proof}
Given two sets $X$ and $Y$. Let $\mathcal E=(E_n)$ and $\mathcal F=(F_n)$ be two decreasing sequences of equivalence relations on $X$ and $Y$ respectively. Let $\theta:X\to Y$ be an injection such that $\theta$ is a reduction of $E_n$ to $F_n$ for each $n<\omega$, i.e.,
$$\forall n<\omega\,\forall x,x'{\rm i}n X\,(xE_nx'{\rm i}ff\theta(x)F_n\theta(x')).$$
For $(n,C){\rm i}n T_\mathcal E^X$, put $\phi(n,C)=(n,[\theta(C)]_{F_n})$.
\begin{proposition}\label{embedding}
$\phi$ is a Lipschitz embedding from $T_\mathcal E^X$ to $T_\mathcal F^Y$. In particular, if $\theta$ is a bijection, then $\phi$ is an order preserving isomorphism.
{\rm e}nd{proposition}
\begin{proof}
Note that $\theta$ is injective. For $(n,C){\rm i}n T_\mathcal E^X$, $C$ is not a singleton, so neither is $[\theta(C)]_{F_n}$, and thus we have $\phi(n,C){\rm i}n T_\mathcal F^Y$. The rest of the proof is trivial.
{\rm e}nd{proof}
\begin{definition}
Let $(T,<)$ be a tree, $(n_i)$ a strictly increasing sequence of natural numbers. We define
$$T|(n_i)=\bigcup_iL_{n_i}(T).$$
It is trivial to see that $(T|(n_i),<)$ is a tree too, and $L_j(T|(n_i))=L_{n_j}(T)$ for each $j<\omega$. We call $T|(n_i)$ a {{\rm i}t level-subtree} of $T$.
{\rm e}nd{definition}
\begin{lemma}
Let $(T,<)$ be a well-founded tree, $(n_i)$ a strictly increasing sequence of natural numbers. Then we have
$$\omega(\rho(T))\le\rho(T|(n_i))\le\rho(T).$$
In particular, if $\rho(T)$ is a limit ordinal, then $\rho(T|(n_i))=\rho(T)$.
{\rm e}nd{lemma}
\begin{proof}
$\rho(T|(n_i))\le\rho(T)$ follows from Proposition~\ref{subtree}. We prove $\omega(\rho(T))\le\rho(T|(n_i))$ by induction on $\rho(T)$.
First, if $\rho(T)<\omega$, then $\rho(T)=\min\{n:L_n(T)={\rm e}mptyset\}$, and hence $\rho(T|(n_i))=\min\{i:L_{n_i}(T)={\rm e}mptyset\}$. So we have $\omega(\rho(T))=0\le\rho(T|(n_i))$.
For $t{\rm i}n T|(n_i)$, note that $(T|(n_i))_t=\{u{\rm i}n T|(n_i):t=u\vee t<u\}$ is a level-subtree of $T_t$ as well.
{\sl Case 1:} If $\rho(T)$ is a limit ordinal, then $\omega(\rho(T))=\rho(T)$. Proposition~\ref{tree} gives $\rho(T)=\sup\{\rho(T_t):t{\rm i}n T\}$.
Since $\rho(T)$ is a limit ordinal and $\rho(T_t)$ is a successor ordinal, we have $\rho(T_t)<\rho(T)$ for $t{\rm i}n T$.
{\sl Subcase 1.1:} If there is no maximum in $\{\omega(\rho(T_t)):t{\rm i}n T\}$, we have
$$\rho(T)=\sup\{\rho(T_t):t{\rm i}n T\}=\sup\{\omega(\rho(T_t)):t{\rm i}n T\}.$$
By induction hypothesis, we have $\omega(\rho(T_t))\le\rho((T|(n_i))_t)$. Proposition~\ref{subtree} gives $\rho((T|(n_i))_t)\le\rho(T|(n_i))$ for each $t{\rm i}n T$, so we have $\rho(T|(n_i))=\rho(T)$.
{\sl Subcase 1.2:} Otherwise, let $\alpha=\max\{\omega(\rho(T_t)):t{\rm i}n T\}$. Since $\rho(T_t)<\rho(T)$ for $t{\rm i}n T$, we have $\rho(T)=\alpha+\omega$. We can find a sequence of $t_m,\,m<\omega$ in $L_0$ such that $\rho(T_{t_m})=\alpha+k_m$ with $\sup\{k_m:m<\omega\}=\omega$. By Proposition~\ref{tree}, for each $m<\omega$ and $n<k_m$ we can find $t_m^n{\rm i}n L_n(T)$ such that $t_m=t_m^0<t_m^1<\dots<t_m^{k_m-1}$ and $\rho(T_{t_m^n})=\alpha+(k_m-n)$. For $k_m>n_0$, let $i_m$ be the biggest $i$ such that $n_i<k_m$, then $t_m^{n_j}{\rm i}n L_j(T|(n_i))=L_{n_j}(T)$ for $j\le i_m$. By induction hypothesis, $\alpha=\omega(\rho(T_{t_m^{n_j}}))\le\rho((T|(n_i))_{t_m^{n_j}})$. Since $\rho((T|(n_i))_{t_m^{n_j}})\ge\rho((T|(n_i))_{t_m^{n_{j+1}}})+1$ for each $j<i_m$, we have $\rho((T|(n_i))_{t_m^{n_0}})\ge\alpha+i_m$. By the definition of $i_m$, we have $\sup\{i_m:m<\omega\}=\omega$. This gives $\rho(T|(n_i))=\alpha+\omega=\rho(T)$.
{\sl Case 2:} If $\rho(T)=\omega(\rho(T))+n$ with $1\le n<\omega$, then there exists some $t_0{\rm i}n L_0(T)$ such that $\rho(T)=\rho_T(t_0)+1$. Since $\{u{\rm i}n T|(n_i):t_0<u\}$ is a level-subtree of $\{u{\rm i}n T:t_0<u\}$ and $\rho(\{u{\rm i}n T:t_0<u\})=\rho_T(t_0)<\rho(T)$, by induction hypothesis and Proposition~\ref{subtree}, we have
$$\omega(\rho_T(t_0))=\omega(\rho(\{u{\rm i}n T:t_0<u\}))\le\rho(\{u{\rm i}n T|(n_i):t_0<u\})\le\rho(T|(n_i)).$$
It follows that $\omega(\rho(T))=\omega(\rho_T(t_0))\le\rho(T|(n_i))$.
{\rm e}nd{proof}
In general, the tree $T_\mathcal G^X$ and the ordinal $\rho(T_\mathcal G^X)$ depends on $\mathcal G$, not only on the action $G\curvearrowright X$. The following key lemma shows that, $\omega(\rho(T_\mathcal G^X))$ is independent to the choice of $\mathcal G$.
\begin{lemma}\label{two dgnb}
Let $G$ be a non-archimedean CLI Polish group, $X$ a countable discrete Polish $G$-space, and let $\mathcal G=(G_n),\mathcal G'=(G_n'){\rm i}n{\rm dgnb}(G)$. Then we have $$\omega(\rho(T_\mathcal G^X))=\omega(\rho(T_{\mathcal G'}^X)).$$
{\rm e}nd{lemma}
\begin{proof}
(1) First, we consider the case that $(G_n')$ is a subsequence of $(G_n)$, i.e., there is a strictly increasing sequence $(n_i)$ of natural numbers such that $G_i'=G_{n_i}$ for each $i<\omega$.
We define $\psi:T_{\mathcal G'}^X\to T_\mathcal G^X$ as $\psi(i,C)=(n_i,C)$. It is clear that $\psi$ is an order preserving isomorphism from $T_{\mathcal G'}^X$ onto $T_\mathcal G^X|(n_i)$. It follows that
$$\omega(\rho(T_\mathcal G^X))\le\rho(T_{\mathcal G'}^X)=\rho(T_\mathcal G^X|(n_i))\le\rho(T_\mathcal G^X).$$
So we have $\omega(\rho(T_\mathcal G^X))=\omega(\rho(T_{\mathcal G'}^X))$.
(2) Since $(G_n),(G_n'){\rm i}n{\rm dgnb}(G)$, we can find two strictly increasing natural numbers $(n_i)$ and $(m_j)$ such that $n_0=0,m_0=0$, and
$$G_0\supseteq G_{m_0}'\supseteq G_{n_1}\supseteq G_{m_1}'\supseteq G_{n_2}\supseteq\cdots.$$
Denote $H_{2i}=G_{n_i}$ and $H_{2i+1}=G_{m_i}'$ for each $i<\omega$. Then $(H_k){\rm i}n{\rm dgnb}(G)$. Put $\mathcal H=(H_k),\mathcal K=(G_{n_i})$, and $\mathcal K'=(G_{m_i}')$.
Note that $(G_{n_i})$ is a subsequence of $(G_n)$ and also a subsequence of $(H_k)$. From (1), we have
$$\omega(\rho(T_\mathcal G^X))=\omega(\rho(T_\mathcal K^X))=\omega(\rho(T_\mathcal H^X)).$$
Similarly, we have
$$\omega(\rho(T_{\mathcal G'}^X))=\omega(\rho(T_{\mathcal K'}^X))=\omega(\rho(T_\mathcal H^X)).$$
So we have $\omega(\rho(T_\mathcal G^X))=\omega(\rho(T_{\mathcal G'}^X))$.
{\rm e}nd{proof}
Now we are going to find a special $G$-space $X(\mathcal G)$ such that $\omega(\rho(T_\mathcal G^{X(\mathcal G)}))$ reaches the maximum value among all $\omega(\rho(T_\mathcal G^X))$. This leads to that the value of $\omega(\rho(T_\mathcal G^{X(\mathcal G)}))$ is determined by $G$ itself.
\begin{lemma}\label{surjection-tree}
Given two sets $X$ and $Y$. Let $\mathcal E=(E_n)$ and $\mathcal F=(F_n)$ be two decreasing sequences of equivalence relations on $X$ and $Y$ respectively. Let $\theta:X\to Y$ be a surjection such that
$$\forall n<\omega\,\forall x{\rm i}n X\,(\theta([x]_{E_n})=[\theta(x)]_{F_n}).$$
Then there exists a Lipschitz embedding $\psi:T_\mathcal F^Y\to T_\mathcal E^X$. In particular, if $T_\mathcal E^X$ is well-founded, so is $T_\mathcal F^Y$, and then $\rho(T_\mathcal F^Y)\le\rho(T_\mathcal E^X)$.
{\rm e}nd{lemma}
\begin{proof}
For any $(n,C){\rm i}n T_\mathcal F^Y$, we construct $\psi(n,C)$ by induction on $n$ such that $\psi(n,C)=(n,[x]_{E_n})$ for some $x{\rm i}n X$ with $[\theta(x)]_{F_n}=C$.
If $n=0$, since $\theta$ is a surjection, we can find an $x{\rm i}n X$ with $\theta(x){\rm i}n C$. Then we put $\psi(0,C)=(0,[x]_{E_0})$.
If $n>0$, since $C$ is an $F_n$-equivalence class, there exists an unique $F_{n-1}$-equivalence class $D\supseteq C$. By induction hypothesis, we can find an $x'{\rm i}n X$ such that $\psi(n-1,D)=(n-1,[x']_{E_{n-1}})$ with $[\theta(x')]_{F_{n-1}}=D$. Since $\theta([x']_{E_{n-1}})=[\theta(x')]_{F_{n-1}}=D\supseteq C$, we can find $x{\rm i}n[x']_{E_{n-1}}$ such that $\theta(x){\rm i}n C$. Then we put $\psi(n,C)=(n,[x]_{E_n})$.
Since $C$ is not a singleton, by $\theta([x]_{E_n})=[\theta(x)]_{F_n}=C$, we can see that $[x]_{E_n}$ is not a singleton. So $\psi(n,C){\rm i}n T_\mathcal E^X$. From the construction, it is routine to check that $\psi:T_\mathcal F^Y\to T_\mathcal E^X$ is a Lipschitz embedding.
In the end, if $T_\mathcal E^X$ is well-founded, by Proposition~\ref{subtree}, $T_\mathcal F^Y$ is well-founded too, and then $\rho(T_\mathcal F^Y)\le\rho(T_\mathcal E^X)$.
{\rm e}nd{proof}
\begin{definition}
Let $G$ be a non-archimedean CLI Polish group, and let $\mathcal G=(G_n){\rm i}n{\rm dgnb}(G)$. For $k<\omega$, we define an action $G\curvearrowright G/G_k$ as, $g\cdot hG_k=ghG_k$ for $g,h{\rm i}n G$. We denote $\rho^k(\mathcal G)=\rho(T^{G/G_k}_\mathcal G)$. Furthermore, letting $X(\mathcal G)=\bigcup_kG/G_k$, we denote $\rho(\mathcal G)=\rho(T^{X(\mathcal G)}_\mathcal G)$.
{\rm e}nd{definition}
Note that $G\cdot gG_k=\{hgG_k:h{\rm i}n G\}=G/G_k$ for any $g{\rm i}n G$.
It is clear that $\rho^0(\mathcal G)=0$, since $G=G_0$.
\begin{lemma}\label{rho}
\begin{enumerate}
{\rm i}tem[(1)] $\rho(\mathcal G)=\sup\{\rho^k(\mathcal G):k<\omega\}$.
{\rm i}tem[(2)] $(\rho^k(\mathcal G))$ is an increasing sequence of countable non-limit ordinals.
{\rm e}nd{enumerate}
{\rm e}nd{lemma}
\begin{proof}
(1) Note that $L_0(T^{X(\mathcal G)}_\mathcal G)=\{(0,G/G_k):G\ne G_k\}$. By
$$(T^{X(\mathcal G)}_\mathcal G)_{(0,G/G_k)}\cong T^{G/G_k}_\mathcal G$$
for $G\ne G_k$, and $(T^{X(\mathcal G)}_\mathcal G)_{(0,G/G_k)}=T^{G/G_k}_\mathcal G={\rm e}mptyset$ for $G=G_k$, so
$$\begin{array}{ll}\rho(\mathcal G)&=\rho(T^{X(\mathcal G)}_\mathcal G)=\sup\{\rho((T^{X(\mathcal G)}_\mathcal G)_{(0,G/G_k)}):k<\omega\}\cr
&=\sup\{\rho(T^{G/G_k}_\mathcal G):k<\omega\}=\sup\{\rho^k(\mathcal G):k<\omega\}.{\rm e}nd{array}$$
(2) Given $k<\omega$, we define $\theta:G/G_{k+1}\to G/G_k$ as $\theta(gG_{k+1})=gG_k$ for $g{\rm i}n G$. It is clear that $\theta$ is well defined and is a surjection. Furthermore, for $n<\omega$ and $g{\rm i}n G$, we have
$$\begin{array}{ll}\theta(G_n\cdot gG_{k+1})&=\{\theta(hgG_{k+1}):h{\rm i}n G_n\}\cr
&=\{hgG_k:h{\rm i}n G_n\}=G_n\cdot gG_k=G_n\cdot\theta(gG_{k+1}).{\rm e}nd{array}$$
From Lemma~\ref{surjection-tree}, we have
$$\rho(T_\mathcal G^{G/G_k})\le\rho(T_\mathcal G^{G/G_{k+1}}),$$
i.e., $(\rho^k(\mathcal G))$ is increasing.
For each $k<\omega$, since $T_\mathcal G^{G/G_k}$ is countable, $\rho^k(\mathcal G)$ is countable too. If $G=G_k$, then $T^{G/G_k}_\mathcal G={\rm e}mptyset$; else if $G\ne G_k$, then $L_0(T^{G/G_k}_\mathcal G)=\{(0,G/G_k)\}$ is a singleton. So $\rho^k(\mathcal G)=\rho(T^{G/G_k}_\mathcal G)$ is either $0$ or a successor ordinal.
{\rm e}nd{proof}
Recall that a $G$-space $X$ is said to be transitive if $X$ itself is an orbit.
\begin{lemma}\label{transitive}
Let $G$ be a non-archimedean CLI Polish group, $X$ a countable discrete transitive Polish $G$-space, and let $\mathcal G=(G_n){\rm i}n{\rm dgnb}(G)$. Then we can find some $k<\omega$ such that $\rho(T_\mathcal G^X)\le\rho^k(\mathcal G)$.
{\rm e}nd{lemma}
\begin{proof}
Fix an $x{\rm i}n X$. Since $\{x\}$ is clopen in $X$, by the continuity of the group action of $G$ on $X$, we have $G_x$ is a clopen subgroup of $G$. So there is some $k<\omega$ such that $G_k\subseteq G_x$. Then we can define $\theta:G/G_k\to X$ as $\theta(gG_k)=g\cdot x$ for $g{\rm i}n G$. Since $X$ is transitive $G$-space, $\theta$ is surjective and $\theta(G_n\cdot gG_k)=G_n\cdot\theta(gG_k)$ for each $n<\omega$ and $g{\rm i}n G$. From Lemma~\ref{surjection-tree}, we have $\rho(T_\mathcal G^X)\le\rho^k(\mathcal G)$.
{\rm e}nd{proof}
\begin{corollary}\label{non-transitive}
Let $G$ be a non-archimedean CLI Polish group, $X$ a countable discrete Polish $G$-space, and let $\mathcal G=(G_n){\rm i}n{\rm dgnb}(G)$. Then we have $$\rho(T_\mathcal G^X)\le\rho(\mathcal G).$$
{\rm e}nd{corollary}
\begin{proof}
Note that $L_0(T_\mathcal G^X)=\{(0,G\cdot x):x{\rm i}n X\wedge G\cdot x\ne\{x\}\}$ and $T_\mathcal G^{G\cdot x}\cong(T_\mathcal G^X)_{(0,G\cdot x)}$ for $G\cdot x\ne\{x\}$, so we have $$\rho(T_\mathcal G^X)=\sup\{\rho((T_\mathcal G^X)_{(0,G\cdot x)}):x{\rm i}n X\}=\sup\{\rho(T_\mathcal G^{G\cdot x}):x{\rm i}n X\}.$$
Then by Lemma~\ref{transitive}, we have
$$\rho(T_\mathcal G^X)\le\sup\{\rho^k(\mathcal G):k<\omega\}=\rho(\mathcal G).$$
{\rm e}nd{proof}
\begin{corollary}
Let $G$ be a non-archimedean Polish group, $\mathcal G=(G_n){\rm i}n{\rm dgnb}(G)$. Then $G$ is CLI iff $T_\mathcal G^{G/G_k}$ is well-founded for any $k<\omega$.
{\rm e}nd{corollary}
\begin{proof}
$(\Rightarrow)$ part follows from Lemma~\ref{CLItoWF}.
$(\Leftarrow)$. Given a countable Polish $G$-space $X$. Following the arguments in the proof of Lemma~\ref{transitive}, we can see that, for any $x{\rm i}n X$, there is a $k<\omega$ and a Lipschitz embedding from $T^{G\cdot x}_\mathcal G$ to $T_\mathcal G^{G/G_k}$. Since $T_\mathcal G^{G/G_k}$ is well-founded, so is $T^{G\cdot x}_\mathcal G$. By the arbitrary of $x{\rm i}n X$, we have $T^X_\mathcal G$ is also well-founded. Then \cite[Theorem 6]{malicki11} gives that $G$ is CLI.
{\rm e}nd{proof}
\begin{theorem}\label{omega=omega}
Let $G$ be a non-archimedean CLI Polish group, $\mathcal G=(G_n),\mathcal G'=(G_n'){\rm i}n{\rm dgnb}(G)$. Then we have $\omega(\rho(\mathcal G))=\omega(\rho(\mathcal G'))$.
{\rm e}nd{theorem}
\begin{proof}
By Corollary~\ref{non-transitive}, we have $\rho(T^{X(\mathcal G)}_{\mathcal G'})\le\rho(\mathcal G')$. Then Lemma~\ref{two dgnb} gives
$$\omega(\rho(\mathcal G))=\omega(\rho(T^{X(\mathcal G)}_\mathcal G))=\omega(\rho(T^{X(\mathcal G)}_{\mathcal G'}))\le\omega(\rho(\mathcal G')),$$
and vice verse.
{\rm e}nd{proof}
By the preceding theorem, there is an unique ordinal $\beta<\omega_1$ which is independent to the choice of $\mathcal G=(G_n){\rm i}n{\rm dgnb}(G)$ with
$$\omega(\rho(\mathcal G))=\omega\cdot\beta,$$
denoted by $\beta={\rm rank}(G)$.
\begin{lemma}\label{rank}
\begin{enumerate}
{\rm i}tem[(1)] If $\rho(\mathcal G)=\omega\cdot{\rm rank}(G)$, then either ${\rm rank}(G)=0$ or $\rho^k(\mathcal G)<\omega\cdot{\rm rank}(G)$ for any $k<\omega$.
{\rm i}tem[(2)] If $\rho(\mathcal G)>\omega\cdot{\rm rank}(G)$, then there exists an $m>0$ such that $\rho^k(\mathcal G)=\omega\cdot{\rm rank}(G)+m$ for large enough $k<\omega$.
{\rm e}nd{enumerate}
{\rm e}nd{lemma}
\begin{proof}
(1) If ${\rm rank}(G)>0$, then $\omega\cdot{\rm rank}(G)$ is a limit ordinal. By Lemma~\ref{rho}, $\rho^k(\mathcal G)$ is either $0$ or a successor ordinal, so $\rho^k(\mathcal G)<\omega\cdot{\rm rank}(G)$ for any $k<\omega$.
(2) Clearly, $\rho(\mathcal G)=\omega\cdot{\rm rank}(G)+m$ for some $0<m<\omega$. Again by Lemma~\ref{rho}, $(\rho^k(\mathcal G))$ is increasing, so $\rho^k(\mathcal G)=\omega\cdot{\rm rank}(G)+m$ for large enough $k<\omega$.
{\rm e}nd{proof}
\begin{theorem}\label{equi-omega}
Let $G$ be a non-archimedean CLI Polish group, $\mathcal G=(G_n),\mathcal G'=(G_n'){\rm i}n{\rm dgnb}(G)$. Then $\rho(\mathcal G)=\omega\cdot{\rm rank}(G)$ iff $\rho(\mathcal G')=\omega\cdot{\rm rank}(G)$.
{\rm e}nd{theorem}
\begin{proof}
If ${\rm rank}(G)=0$, then $\rho(\mathcal G)=\omega\cdot{\rm rank}(G)$ implies that $\rho^k(\mathcal G)=0$ for any $k<\omega$. So $T_\mathcal G^{G/G_k}={\rm e}mptyset$, and hence $G_0\cdot G_k\notin T_\mathcal G^{G/G_k}$. It follows that $G_0\cdot G_k=\{G_k\}$, i.e., $G=G_0=G_k$ for any $k<\omega$. This gives $G=\{1_G\}$. Then we can easily see that $T_{\mathcal G'}^{G/G_k'}={\rm e}mptyset$ for $k<\omega$. So $\rho(\mathcal G')=\omega\cdot{\rm rank}(G)$ holds. And vice verse.
If ${\rm rank}(G)>0$, assume for contradiction that $\rho(\mathcal G)=\omega\cdot{\rm rank}(G)$, but $\rho(\mathcal G')>\omega\cdot{\rm rank}(G)$. From Lemma~\ref{rank}, we have $\rho^k(\mathcal G)<\omega\cdot{\rm rank}(G)$ for any $k<\omega$, but $\rho^l(\mathcal G')=\omega\cdot{\rm rank}(G)+m$ for some $0<m<\omega$ and large enough $l<\omega$. From lemmas~\ref{two dgnb} and~\ref{transitive}, for any $l<\omega$, there is some $k<\omega$ with
$$\omega(\rho^l(\mathcal G'))=\omega(\rho(T_{\mathcal G'}^{G/G_l'}))=\omega(\rho(T_\mathcal G^{G/G_l'}))\le\rho(T_\mathcal G^{G/G_l'})\le\rho^k(\mathcal G).$$
A contradiction!
{\rm e}nd{proof}
Now we are ready to define a hierarchy on non-archimedean CLI Polish groups.
\begin{definition}
Let $G$ be a non-archimedean CLI Polish group, $\mathcal G=(G_n){\rm i}n{\rm dgnb}(G)$, and let $\alpha<\omega_1$ be an ordinal.
\begin{enumerate}
{\rm i}tem[(1)] If $\rho(\mathcal G)\le\omega\cdot\alpha$, we say $G$ is {{\rm i}t $\alpha$-CLI};
{\rm i}tem[(2)] if $\omega(\rho(\mathcal G))\le\omega\cdot\alpha$, i.e., ${\rm rank}(G)\le\alpha$, we say $G$ is {{\rm i}t L-$\alpha$-CLI}.
{\rm e}nd{enumerate}
It is clear that, if $G$ is L-$\alpha$-CLI, it is also $(\alpha+1)$-CLI.
From theorems~\ref{omega=omega} and~\ref{equi-omega}, we see that the definitions $\alpha$-CLI and L-$\alpha$-CLI are independent to the choice of $\mathcal G{\rm i}n{\rm dgnb}(G)$.
{\rm e}nd{definition}
Recall that a metric $d$ on a group $G$ is {{\rm i}t two sided invariant} if $d(gh,gk)=d(h,k)=d(hg,kg)$ for all $g,h,k{\rm i}n G$. A Polish group is {{\rm i}t TSI} if it admits a compatible complete two sided invariant metric.
\begin{theorem}\label{0-1}
Let $G$ be a non-archimedean CLI Polish group. Then we have
\begin{enumerate}
{\rm i}tem[(1)] $G$ is $0$-CLI iff $G=\{1_G\}$;
{\rm i}tem[(2)] $G$ is L-$0$-CLI iff $G$ is discrete;
{\rm i}tem[(3)] $G$ is $1$-CLI iff $G$ is TSI.
{\rm e}nd{enumerate}
{\rm e}nd{theorem}
\begin{proof}
Fix a sequence $\mathcal G=(G_n){\rm i}n{\rm dgnb}(G)$.
(1) It follows from the first paragraph of the proof of Theorem~\ref{equi-omega}.
(2) If $G$ is L-$0$-CLI, then we have ${\rm rank}(G)=0$. So there is an $m<\omega$ such that $\rho(T_\mathcal G^{G/G_k})=\rho^k(\mathcal G)=m$ for large enough $k<\omega$. Then we have $L_m(T_\mathcal G^{G/G_k})={\rm e}mptyset$. This implies that $G_m\cdot G_k=\{G_k\}$. So $G_m\subseteq G_k$ for large enough $k<\omega$, and thus $G_m=\{1_G\}$. It follows that $G$ is discrete.
On the other hand, if $G$ is discrete, then there is an $m<\omega$ such that $G_m=\{1_G\}$. Therefore, for any $k<\omega$, we have $L_m(T_\mathcal G^{G/G_k})={\rm e}mptyset$, and hence $\rho^k(\mathcal G)\le m$. This gives ${\rm rank}(G)=0$, i.e., $G$ is L-$0$-CLI.
(3) If $G$ is $1$-CLI, then Lemma~\ref{rank} implies that $\rho^k(\mathcal G)<\omega$ for any $k<\omega$. So there is $m_k<\omega$ such that $L_{m_k}(T_\mathcal G^{G/G_k})={\rm e}mptyset$. Then it follows that, for $g{\rm i}n G$, $G_{m_k}\cdot gG_k=\{gG_k\}$, so $g^{-1}G_{m_k}g\subseteq G_k$. Put $U_k=\bigcup\{g^{-1}G_{m_k}g:g{\rm i}n G\}\subseteq G_k$. Then $(U_k)$ is a neighborhood base of $1_G$ with $g^{-1}U_kg=U_k$ for all $g{\rm i}n G$. By Klee's theorem (c.f.~\cite{klee} or~\cite[Exercise 2.1.4]{gaobook}), $G$ is TSI.
On the other hand, if $G$ is TSI, again by Klee's theorem, we can find a neighborhood base $(U_m)$ of $1_G$ with $g^{-1}U_mg=U_m$ for all $g{\rm i}n G$. For any $n<\omega$, there is an $m_n<\omega$ such that $U_{m_n}\subseteq G_n$. Let $V_n=U_{m_n}\cap U_{m_n}^{-1}$ and $G_n'=\bigcup_iV_n^i$. Then $G_n'$ is an open normal subgroup of $G$ with $G_n'\subseteq G_n$. So $(G_n'){\rm i}n{\rm dgnb}(G)$. Put $\mathcal G'=(G_n')$. Then $G_k'\cdot gG_k'=\{gG_k'\}$ for all $g{\rm i}n G$ and $k<\omega$, thus $L_k(T_{\mathcal G'}^{G/G_k'})={\rm e}mptyset$. So $\rho^k(\mathcal G')\le k<\omega$, and hence $G$ is $1$-CLI.
{\rm e}nd{proof}
Clause (2) in the preceding theorem can be generalize to all $\alpha<\omega_1$.
\begin{definition}
Let $G$ be a non-archimedean CLI Polish group, $\alpha<\omega_1$. We say $G$ is {{\rm i}t locally $\alpha$-CLI} if $G$ has an open subgroup which is $\alpha$-CLI.
{\rm e}nd{definition}
\begin{theorem}\label{locally}
Let $G$ be a non-archimedean CLI Polish group, $\alpha<\omega_1$. Then $G$ is L-$\alpha$-CLI iff $G$ is locally $\alpha$-CLI.
{\rm e}nd{theorem}
\begin{proof}
$(\Rightarrow)$. If $G$ is L-$\alpha$-CLI, without loss of generality, we may assume that $G$ is not $\alpha$-CLI. Fix a sequence $\mathcal G=(G_n){\rm i}n{\rm dgnb}(G)$. There exists an $m\ge 1$ such that $\rho(\mathcal G)=\omega\cdot\alpha+m$, and thus we can pick a $k_0>m$ such that $\rho^k(\mathcal G)=\omega\cdot\alpha+m$ for any $k\ge k_0$. We will show that $G_{k_0}$ is $\alpha$-CLI.
Put $H=G_{k_0}$ and $H_n=G_{n+k_0}$ for $n<\omega$. Then $(H_n){\rm i}n{\rm dgnb}(H)$. Put $\mathcal H=(H_n)$. Given $k<\omega$, define $\phi:T^{H/H_k}_\mathcal H\to(T^{G/G_{k+k_0}}_\mathcal G)_{(k_0,G_{k_0}/G_{k+k_0})}$ as $\phi(n,C)=(n+k_0,C)$. It is trivial to see that $\phi$ is an order preserving isomorphism. From Proposition~\ref{L_k(T)}, since $k_0>m$, we have
$$\rho^k(\mathcal H)=\rho(T^{H/H_k}_\mathcal H)=\rho((T^{G/G_{k+k_0}}_\mathcal G)_{(k_0,G_{k_0}/G_{k+k_0})})\le\omega\cdot\alpha.$$
So $\rho(\mathcal H)\le\omega\cdot\alpha$, and hence $H=G_{k_0}$ is $\alpha$-CLI.
$(\Leftarrow)$. If $G$ is locally $\alpha$-CLI, let $H$ be an open subgroup of $G$ which is $\alpha$-CLI, and let $\mathcal H=(H_n){\rm i}n{\rm dgnb}(H)$. Then $\rho^k(\mathcal H)\le\omega\cdot\alpha$ for $k<\omega$.
Put $G_0=G$ and $G_n=H_{n-1}$ for $n\ge 1$. Then $(G_n){\rm i}n{\rm dgnb}(G)$. Put $\mathcal G=(G_n)$. Given $g{\rm i}n G$ and $k<\omega$, by the similar arguments in $(\Rightarrow)$ part, we have $T^{H\cdot gH_k}_\mathcal H\cong(T^{G/G_{k+1}}_\mathcal G)_{(1,G_1\cdot gG_{k+1})}$. By Lemma~\ref{transitive}, there exists an $l<\omega$ such that $\rho(T^{H\cdot gH_k}_\mathcal H)\le\rho^l(\mathcal H)$. Therefore, by Proposition~\ref{tree},
$$\begin{array}{ll}\rho^{k+1}(\mathcal G)&=\rho(T^{G/G_{k+1}}_\mathcal G)\cr
&\le\sup\{\rho((T^{G/G_{k+1}}_\mathcal G)_{(1,G_1\cdot gG_{k+1})}):g{\rm i}n G\}+1\cr
&=\sup\{\rho(T^{H\cdot gH_k}_\mathcal H):g{\rm i}n G\}+1\cr
&\le\sup\{\rho^l(\mathcal H):l<\omega\}+1\cr
&\le\omega\cdot\alpha+1.{\rm e}nd{array}$$
So $\rho(\mathcal G)\le\omega\cdot\alpha+1$, and hence $G$ is L-$\alpha$-CLI.
{\rm e}nd{proof}
\section{Properties of the hierarchy}
\begin{theorem}\label{subgroup}
Let $G$ be a non-archimedean CLI Polish group, $H$ a closed subgroup of $G$, and $\alpha<\omega_1$. If $G$ is $\alpha$-CLI (or L-$\alpha$-CLI), so is $H$. In particular, we have ${\rm rank}(H)\le{\rm rank}(G)$.
{\rm e}nd{theorem}
\begin{proof}
Let $\mathcal G=(G_n){\rm i}n{\rm dgnb}(G)$, and put $H_n=H\cap G_n$ for $n<\omega$. It is clear that $(H_n){\rm i}n{\rm dgnb}(H)$. Put $\mathcal H=(H_n)$. We only need to show that $\rho(\mathcal H)\le\rho(\mathcal G)$.
Given $k<\omega$, define $\theta:H/H_k\to G/G_k$ as $\theta(hH_k)=hG_k$ for $h{\rm i}n H$. By Proposition~\ref{embedding}, there is a Lipschitz embedding from $T^{H/H_k}_\mathcal H$ to $T^{G/G_k}_\mathcal G$. So
$$\rho^k(\mathcal H)=\rho(T^{H/H_k}_\mathcal H)\le\rho(T^{G/G_k}_\mathcal G)=\rho^k(\mathcal G).$$
Then we have $\rho(\mathcal H)\le\rho(\mathcal G)$ as desired.
{\rm e}nd{proof}
\begin{theorem}
Let $G$ be a non-archimedean CLI Polish group, $N$ a closed normal subgroup of $G$, and $\alpha<\omega_1$. If $G$ is $\alpha$-CLI (or L-$\alpha$-CLI), so is $G/N$. In particular, we have ${\rm rank}(G/N)\le{\rm rank}(G)$.
{\rm e}nd{theorem}
\begin{proof}
Let $\mathcal G=(G_n){\rm i}n{\rm dgnb}(G)$, and put $H_n=G_n\cdot N=\{\hat gN:\hat g{\rm i}n G_n\}$ for $n<\omega$. It is clear that $H_0=G/N$ and $(H_n){\rm i}n{\rm dgnb}(G/N)$. Put $\mathcal H=(H_n)$. We only need to show that $\rho(\mathcal H)\le\rho(\mathcal G)$.
Given $k<\omega$, define $\theta:G/G_k\to(G/N)/H_k$ as $\theta(gG_k)=(gN)H_k$ for $g{\rm i}n G$. Note that for $n<\omega$,
$$\begin{array}{ll}\theta(G_n\cdot gG_k)&=\theta(\{\hat ggG_k:\hat g{\rm i}n G_n\})\cr
&=\{(\hat ggN)H_k:\hat g{\rm i}n G_n\}\cr
&=\{(\hat gN)(gN)H_k:\hat g{\rm i}n G_n\}\cr
&=\{(\hat gN)\theta(gG_k):\hat g{\rm i}n G_n\}\cr
&=\{\hat gN:\hat g{\rm i}n G_n\}\theta(gG_k)=H_n\cdot\theta(gG_k).{\rm e}nd{array}$$
By Lemma~\ref{surjection-tree}, we have
$$\rho^k(\mathcal H)=\rho(T^{(G/N)/H_k}_\mathcal H)\le\rho(T^{G/G_k}_\mathcal G)=\rho^k(\mathcal G).$$
Then we have $\rho(\mathcal H)\le\rho(\mathcal G)$ as desired.
{\rm e}nd{proof}
The above two theorems involve closed subgroups and quotient groups. Now we turn to discuss product groups, which are more complicated. We discuss finite product groups first.
\begin{lemma}\label{XtimesY}
Let $X,Y$ be two sets, $\mathcal E=(E_n)$ and $\mathcal F=(F_n)$ two decreasing sequence of equivalence relations on $X$ and $Y$ respectively. Denote $\mathcal{E\times F}=(E_n\times F_n)$.
\begin{enumerate}
{\rm i}tem[(1)] $T_\mathcal{E\times F}^{X\times Y}$ is well-founded iff $T_\mathcal E^X$ and $T_\mathcal F^Y$ are well-founded.
{\rm i}tem[(2)] If $T_\mathcal{E\times F}^{X\times Y}$ is well-founded, then we have
$$\rho(T_\mathcal{E\times F}^{X\times Y})=\max\{\rho(T_\mathcal E^X),\rho(T_\mathcal F^Y)\}.$$
{\rm e}nd{enumerate}
{\rm e}nd{lemma}
\begin{proof}
First, note that $[(x,y)]_{E_n\times F_n}=[x]_{E_n}\times[y]_{F_n}$ for all $(x,y){\rm i}n X\times Y$ and $n<\omega$.
(1) For any sequences $(x_n),(y_n)$ in $X,Y$ respectively, $((n,[(x_n,y_n)]_{E_n\times F_n}))$ is an infinite branch of $T_\mathcal{E\times F}^{X\times Y}$ iff either $((n,[x_n]_{E_n}))$ or $((n,[y_n]_{F_n}))$ is an infinite branch of $T_\mathcal E^X$ or $T_\mathcal F^Y$ respectively.
(2) If $T_\mathcal{E\times F}^{X\times Y}$ is well-founded, by (1), $T_\mathcal E^X$ and $T_\mathcal F^Y$ are also well-founded. For all $(x,y){\rm i}n X\times Y$ and $n<\omega$, note that
$$\begin{array}{ll} & (n,[(x,y)]_{E_n\times F_n}){\rm i}n T^{X\times Y}_\mathcal{E\times F}\cr
{\rm i}ff & [(x,y)]_{E_n\times F_n}\ne\{(x,y)\}\cr
{\rm i}ff & [x]_{E_n}\ne\{x\}\vee[y]_{F_n}\ne\{y\}\cr
{\rm i}ff & (n,[x]_{E_n}){\rm i}n T_\mathcal E^X\vee(n,[y]_{F_n}){\rm i}n T_\mathcal F^Y.{\rm e}nd{array}$$
By Proposition~\ref{tree}, it is routine to prove
$$\rho((T_\mathcal{E\times F}^{X\times Y})_{(n,[(x,y)]_{E_n\times F_n})})=\max\{\rho((T_\mathcal E^X)_{(n,[x]_{E_n})}),\rho((T_\mathcal F^Y)_{(n,[y]_{F_n})})\}$$
by induction on $\rho((T_\mathcal{E\times F}^{X\times Y})_{(n,[(x,y)]_{E_n\times F_n})})$. Taking supremum on both sides of the above formula, we get
$$\rho(T_\mathcal{E\times F}^{X\times Y})=\max\{\rho(T_\mathcal E^X),\rho(T_\mathcal F^Y)\}.$$
{\rm e}nd{proof}
\begin{corollary}\label{GtimesH}
Let $G$ and $H$ be two non-archimedean CLI Polish groups, $\mathcal G=(G_n){\rm i}n{\rm dgnb}(G)$ and $\mathcal H=(H_n){\rm i}n{\rm dgnb}(H)$. Then we have $\mathcal{G\times H}=(G_n\times H_n){\rm i}n{\rm dgnb}(G\times H)$ and
$$\rho^k(\mathcal{G\times H})=\max\{\rho^k(\mathcal G),\rho^k(\mathcal H)\}\quad(\forall k<\omega),$$
$$\rho(\mathcal{G\times H})=\max\{\rho(\mathcal G),\rho(\mathcal H)\},$$
$${\rm rank}(G\times H)=\max\{{\rm rank}(G),{\rm rank}(H)\}.$$
{\rm e}nd{corollary}
\begin{proof}
For any $k<\omega$, put $X=G/G_k,Y=H/H_k$, and define $E_n,F_n$ on $X$ and $Y$ for each $n<\omega$ respectively by
$$(\hat gG_k,\tilde gG_k){\rm i}n E_n{\rm i}ff{\rm e}xists g{\rm i}n G_n\,(g\hat gG_k=\tilde gG_k)\quad(\forall\hat g,\tilde g{\rm i}n G),$$
$$(\hat hH_k,\tilde hH_k){\rm i}n F_n{\rm i}ff{\rm e}xists h{\rm i}n H_n\,(h\hat hH_k=\tilde hH_k)\quad(\forall\hat h,\tilde h{\rm i}n H).$$
Then Lemma~\ref{XtimesY} gives $\rho^k(\mathcal{G\times H})=\max\{\rho^k(\mathcal G),\rho^k(\mathcal H)\}$. The rest follows trivially.
{\rm e}nd{proof}
Now we are ready to discuss countably infinite product groups.
\begin{lemma}\label{times}
Let $(G^i)$ be a sequence of non-archimedean CLI Polish groups and $\mathcal G^i=(G^i_n){\rm i}n{\rm dgnb}(G^i)$ for each $i<\omega$. We denote $G=\prod_iG^i$ and
$$G_n=\prod_{i<n}G^i_n\times\prod_{i\ge n}G^i\quad(\forall n<\omega).$$
Then $\mathcal G=(G_n){\rm i}n{\rm dgnb}(G)$ and for $k<\omega$, we have
$$\rho^k(\mathcal G)\le\max\{\rho^k(\mathcal G^i):i<k\}+k.$$
{\rm e}nd{lemma}
\begin{proof}
Given $k<\omega$, we denote $Y=G^0/G^0_k\times\dots\times G^{k-1}/G^{k-1}_k$ and define $\theta:G/G_k\to Y$ as, for $(g_i){\rm i}n G=\prod_iG^i$,
$$\theta((g_i)G_k)=(g_0G^0_k,\dots,g_{k-1}G^{k-1}_k).$$
By the definition of $G_k$, it is trivial to see that $\theta$ is a bijection. Moveover, we put
$$H_n=\left\{\begin{array}{ll}\prod_{i<n}G^i_n\times\prod_{n\le i<k}G^i, & n<k,\cr
\prod_{i<k}G^i_n, & n\ge k,{\rm e}nd{array}\right.$$
then $\theta$ is a reduction of $E^{G/G_k}_{G_n}$ to $E^Y_{H_n}$ for each $n<\omega$. Put $\mathcal H=(H_n)$. By Proposition~\ref{embedding}, $(n,G_n\cdot (g_i)G_k)\mapsto(n,H_n\cdot\theta((g_i)G_k))$ is an order preserving isomorphism from $T^{G/G_k}_\mathcal G$ to $T^Y_\mathcal H$. So $\rho^k(\mathcal G)=\rho(T^{G/G_k}_\mathcal G)=\rho(T^Y_\mathcal H)$.
For $n\ge k$ and $(g_i){\rm i}n G$, we have
$$H_n\cdot\theta((g_i)G_k)=(G^0_n\cdot g_0G^0_k)\times\dots\times(G^{k-1}_n\cdot g_{k-1}G^{k-1}_k).$$
Lemma~\ref{XtimesY} gives
$$\rho((T^Y_\mathcal H)_{(k,H_k\cdot\theta((g_i)G_k))})=\max\{\rho((T^{G^i/G^i_k}_{\mathcal G^i})_{(k,G^i_k\cdot g_iG^i_k)}):i<k\}.$$
Hence by Proposition~\ref{L_k(T)}.(1),
$$\begin{array}{ll}\rho^k(\mathcal G)&=\rho(T^{G/G_k}_\mathcal G)=\rho(T^Y_\mathcal H)\cr
&\le\sup\{\rho((T^Y_\mathcal H)_{(k,H_k\cdot\theta((g_i)G_k))}):(g_i){\rm i}n G\}+k\cr
&=\sup\{\max\{\rho((T^{G^i/G^i_k}_{\mathcal G^i})_{(k,G^i_k\cdot g_iG^i_k)}):i<k\}:(g_i){\rm i}n G\}+k\cr
&=\max\{\sup\{\rho((T^{G^i/G^i_k}_{\mathcal G^i})_{(k,G^i_k\cdot gG^i_k)}):g{\rm i}n G^i\}:i<k\}+k\cr
&\le\max\{\rho(T^{G^i/G^i_k}_{\mathcal G^i}):i<k\}+k\cr
&=\max\{\rho^k(\mathcal G^i):i<k\}+k.{\rm e}nd{array}$$
{\rm e}nd{proof}
\begin{theorem}\label{prod}
Let $(G^i)$ be a sequence of non-archimedean CLI Polish groups, $\alpha<\omega_1$, and let $G=\prod_iG^i$. Then we have
\begin{enumerate}
{\rm i}tem[(1)] $G$ is $\alpha$-CLI iff all $G^i$ are $\alpha$-CLI;
{\rm i}tem[(2)] $G$ is L-$\alpha$-CLI iff all $G^i$ are L-$\alpha$-CLI and for all but finitely many $i$, $G^i$ is $\alpha$-CLI.
{\rm e}nd{enumerate}
{\rm e}nd{theorem}
\begin{proof}
Fix a $\mathcal G^i=(G^i_n){\rm i}n{\rm dgnb}(G^i)$ for each $i<\omega$. Put
$$G_n=\prod_{i<n}G^i_n\times\prod_{i\ge n}G^i\quad(\forall n<\omega).$$
(1) If $G$ is $\alpha$-CLI, since each $G^i$ is topologically isomorphic to a closed subgroup of $G$, by Theorem~\ref{subgroup}, $G^i$ is $\alpha$-CLI too.
On the other hand, if all $G^i$ are $\alpha$-CLI, by Lemma~\ref{rank}.(1), $\rho^k(\mathcal G^i)<\omega\cdot\alpha$ for all $i,k<\omega$. Then by Lemma~\ref{times},
$$\rho^k(\mathcal G)\le\max\{\rho^k(\mathcal G^i):i<k\}+k<\omega\cdot\alpha$$
for each $k<\omega$. Therefore, $G$ is $\alpha$-CLI.
(2) If $G$ is L-$\alpha$-CLI, since each $G^i$ is topologically isomorphic to a closed subgroup of $G$, we have $G^i$ is L-$\alpha$-CLI too. Moreover, by Theorem~\ref{locally}, there is an open subgroup $H$ of $G$ which is $\alpha$-CLI. And hence there is some $n<\omega$ such that $G_n$ is a clopen subgroup of $H$, thus is $\alpha$-CLI too. By (1), for all $i\ge n$, $G^i$ is $\alpha$-CLI.
On the other hand, if all $G^i$ are L-$\alpha$-CLI and there is an $m<\omega$ such that $G^i$ is $\alpha$-CLI for $i\ge m$, then (1) implies that $\prod_{i\ge m}G^i$ is $\alpha$-CLI. Note that $G=G^0\times\dots\times G^{m-1}\times\prod_{i\ge m}G^i$. By Corollary~\ref{GtimesH}, we have
$${\rm rank}(G)=\max\{{\rm rank}(G^0),\dots,{\rm rank}(G^{m-1}),{\rm rank}(\prod_{i\ge m}G^i)\}\le\alpha,$$
i.e., $G$ is L-$\alpha$-CLI.
{\rm e}nd{proof}
In the rest of this article, we will show that the notions of $\alpha$-CLI, L-$\alpha$-CLI, together with ${\rm rank}(G)$, forms a proper hierarchy on non-archimedean CLI Polish groups. To to so, we will construct groups which are $\alpha$-CLI but not L-$\beta$-CLI for all $\beta<\alpha$ and groups which are L-$\alpha$-CLI but not $\alpha$-CLI inductively for each $\alpha<\omega_1$. We consider the case concerning successor ordinals first.
\begin{corollary}\label{+1}
Let $(G^i)$ be a sequence of non-archimedean CLI Polish groups, $\alpha<\omega_1$, and let $G=\prod_iG^i$. If all $G^i$ are L-$\alpha$-CLI but not $\alpha$-CLI, then $G$ is $(\alpha+1)$-CLI but not L-$\alpha$-CLI.
{\rm e}nd{corollary}
\begin{proof}
Note that all $G^i$ are $(\alpha+1)$-CLI but not $\alpha$-CLI.
{\rm e}nd{proof}
From Theorem~\ref{0-1} and \cite[Theorem 1.1]{GX}, a non-archimedean CLI Polish group $G$ is $1$-CLI iff $G$ is isomorphic to a closed subgroup of a product $\prod_iG^i$, where each $G^i$ is L-$0$-CLI. Comparing the preceding corollary, we may ask the following question:
\begin{question}
Let $0<\alpha<\omega_1$, and let $G$ be an $(\alpha+1)$-CLI group. Can we find a sequence of L-$\alpha$-CLI groups $G^i$ such that $G$ is isomorphic to a closed subgroup of $\prod_iG^i$?
{\rm e}nd{question}
Let $G$ and $\Lambda$ be two groups. Recall that the {{\rm i}t wreath product} $\Lambda\wr G$ is the set $\Lambda\times G^\Lambda$ equipped with group operation as, for $(\hat\lambda,\hat\chi),(\tilde\lambda,\tilde\chi){\rm i}n\Lambda\times G^\Lambda$,
$$(\hat\lambda,\hat\chi)(\tilde\lambda,\tilde\chi)=(\hat\lambda\tilde\lambda,\chi)$$
with $\chi(\lambda)=\hat\chi(\lambda)\tilde\chi(\hat\lambda^{-1}\lambda)$ for $\lambda{\rm i}n\Lambda$. If $\Lambda$ is countable discrete and $G$ is Polish, $\Lambda\wr G$ equipped the product topology of $\Lambda\times G^\Lambda$ is also a Polish group.
\begin{theorem}\label{wr}
Let $G$ be a non-archimedean CLI Polish group, $\Lambda$ an infinite countable discrete group, $\alpha<\omega_1$. If $G$ is $(\alpha+1)$-CLI but not $\alpha$-CLI, then $\Lambda\wr G$ is L-$(\alpha+1)$-CLI but not $(\alpha+1)$-CLI.
{\rm e}nd{theorem}
\begin{proof}
For the sake of brevity, we denote $H=\Lambda\wr G$. Let $\lambda_i,\,i<\omega$ be a non-repeat enumeration of $\Lambda$. Note that the underlying space of $\Lambda\wr G$ is $\Lambda\times G^\Lambda$. For $(\lambda,\chi){\rm i}n\Lambda\wr G$, put $\pi_\Lambda(\lambda,\chi)=\lambda$ and $\pi_G^i(\lambda,\chi)=\chi(\lambda_i)$.
Let $\mathcal G=(G_n){\rm i}n{\rm dgnb}(G)$. Since $G$ is not $\alpha$-CLI, $G\ne\{1_G\}$. Without loss of generality, we can assume that $G\ne G_1$. Put $H_0=H$ and for $n<\omega$,
$$H_{n+1}=\{(1_\Lambda,\chi):\chi{\rm i}n G^\Lambda\wedge\forall i<n\,(\chi(\lambda_i){\rm i}n G_n)\}.$$
Then $\mathcal H=(H_n){\rm i}n{\rm dgnb}(H)$. Note that the open subgroup $H_1=\{1_\Lambda\}\times G^\Lambda$ is topologically isomorphic to $G^\omega$. By Theorem~\ref{prod}, $H_1$ is $(\alpha+1)$-CLI, so $H$ is L-$(\alpha+1)$-CLI.
Since $G$ is $(\alpha+1)$-CLI but not $\alpha$-CLI, $\omega\cdot\alpha<\rho(\mathcal G)\le\omega\cdot(\alpha+1)$. By Lemma~\ref{rank}, there exist $1\le m,k<\omega$ such that $\rho^k(\mathcal G)=\omega\cdot\alpha+m$. To see that $H$ is not $(\alpha+1)$-CLI, we show that $\rho^{k+1}(\mathcal H)>\omega\cdot(\alpha+1)$ as follows.
For any $(\lambda_l,\hat\chi){\rm i}n H$ and $(1_\Lambda,\tilde\chi){\rm i}n H_{k+1}$, note that $(\lambda_l,\hat\chi)(1_\Lambda,\tilde\chi)=(\lambda_l,\hat\chi\tilde\chi_l)$, where $\tilde\chi_l(\lambda)=\tilde\chi(\lambda_l^{-1}\lambda)$ for $\lambda{\rm i}n\Lambda$. It follows that
$$(\lambda_l,\hat\chi)H_{k+1}=\{(\lambda_l,\chi):\chi{\rm i}n G^\Lambda\wedge\forall i<k\,(\chi(\lambda_l\lambda_i){\rm i}n\hat\chi(\lambda_l\lambda_i)G_k)\}.$$
There is an unique $l_i<\omega$ with $\lambda_{l_i}=\lambda_l\lambda_i$ for $i<k$. It is clear that $\pi_\Lambda((\lambda_l,\hat\chi)H_{k+1})=\{\lambda_l\}$ and for $j<\omega$,
$$\pi_G^j((\lambda_l,\hat\chi)H_{k+1})=\left\{\begin{array}{ll}\hat\chi(\lambda_{l_i})G_k, & j=l_i,i<k,\cr G, & \mbox{otherwise.}{\rm e}nd{array}\right.$$
Denote $m_l=\max\{l_i:i<k\}$, then it is clear that $m_l\ge k-1$ for $l<\omega$ and $\sup\{m_l:l<\omega\}=\omega$. Note that
$\pi_G^{m_l}(H_{m_l+1}\cdot(\lambda_l,\hat\chi)H_{k+1})=G/G_k$ is not a singleton, so $(m_l+1,H_{m_l+1}\cdot(\lambda_l,\hat\chi)H_{k+1}){\rm i}n T^{H/H_{k+1}}_\mathcal H$. Also note that
$$(T^{H/H_{k+1}}_\mathcal H)_{(m_l+2,H_{m_l+2}\cdot(\lambda_l,\hat\chi)H_{k+1})}\cong T^{H_{m_l+2}\cdot(\lambda_l,\hat\chi)H_{k+1}}_{\mathcal H'},$$
$$(T^{G/G_k}_\mathcal G)_{(m_l+1,G_{m_l+1}\cdot gG_k)}\cong T^{G_{m_l+1}\cdot gG_k}_{\mathcal G'},$$
where $\mathcal H'=(H_{n+m_l+2})$ and $\mathcal G'=(G_{n+m_l+1})$ for $n<\omega$. Define $\theta:H/H_{k+1}\to G/G_k$ as $\theta(C)=\pi_G^{m_l}(C)$. Applying Lemma~\ref{surjection-tree} on the restriction of $\theta$ from $H_{m_l+2}\cdot(\lambda_l,\hat\chi)H_{k+1}$ to $G_{m_l+1}\cdot\hat\chi(\lambda_{m_l})G_k$, and also applying Proposition~\ref{L_k(T)}, we get
$$\begin{array}{ll}&\rho((T^{H/H_{k+1}}_\mathcal H)_{(m_l+1,H_{m_l+1}\cdot(\lambda_l,\bar\chi)H_{k+1})})\cr
\ge &\sup\{\rho((T^{H/H_{k+1}}_\mathcal H)_{(m_l+2,H_{m_l+2}\cdot(\lambda_l,\hat\chi)H_{k+1})}):\forall i<m_l\,(\bar\chi(\lambda_i)=\hat\chi(\lambda_i))\}+1\cr
\ge &\sup\{\rho((T^{G/G_k}_\mathcal G)_{(m_l+1,G_{m_l+1}\cdot gG_k)}):g{\rm i}n G\}+1\cr
\ge &\omega(\rho((T^{G/G_k}_\mathcal G))+1\cr
= &\omega\cdot\alpha+1.{\rm e}nd{array}$$
So $\rho(T^{H/H_{k+1}}_\mathcal H)\ge\omega\cdot\alpha+m_l+2$ for all $l<\omega$, and hence
$$\rho^{k+1}(\mathcal H)=\rho(T^{H/H_{k+1}}_\mathcal H)\ge\omega\cdot\alpha+\omega=\omega\cdot(\alpha+1).$$
Since $\rho^{k+1}(\mathcal H)$ is a successor ordinal, we have $\rho^{k+1}(\mathcal H)>\omega\cdot(\alpha+1)$.
{\rm e}nd{proof}
Now we turn to consider the case concerning limit ordinals. To do this, we need to prepare two lemmas first.
\begin{lemma}
Let $(G^i)$ be a sequence of Polish groups, and let $H^i$ be an open subgroup of $G^i$ for each $i<\omega$. Suppose $H=\prod_iH^i$ and
$$G=\{(g_i){\rm i}n\prod_iG^i:\forall^{\rm i}nfty i\,(g_i{\rm i}n H^i)\}$$
equipped with the topology $\tau$ generated by the sets of the form $(g_i)U$ for $(g_i){\rm i}n G$ and $U$ open in $H$. Then $(G,\tau)$ is a Polish group and $\tau$ is the unique group topology on $G$ such that $H$ is an open subgroup of $G$.
{\rm e}nd{lemma}
\begin{proof}
For each $(g_i){\rm i}n G$, the subspace $(g_i)H$ of $(G,\tau)$ is homeomorphic to $H$, so is Polish. Let $D^i\subseteq G^i$ meets every coset of $H^i$ at exactly one point. Then $D^i$ is countable for $i<\omega$. We denote
$$D=\{(g_i){\rm i}n\prod_iD^i:\forall^{\rm i}nfty i\,(g_i=1_{G^i})\}.$$
It is clear that $G/H=\{(g_i)H:(g_i){\rm i}n D\}$ is countable. So $(G,\tau)$ is a sum of countably many Polish spaces, thus is a Polish space.
For $(g_i),(h_i){\rm i}n G$ and $U$ open in $H$ with $1_H{\rm i}n H$, there exists an $m<\omega$ such that $g_i,h_i{\rm i}n H^i$ for $i>m$ and $U^0\times\dots\times U^m\times\prod_{i>m}H^i\subseteq U$, where $U^i$ is an open subset of $H^i$ with $1_{H^i}{\rm i}n U^i$ for each $i\le m$. We can find open neighborhoods $V^i$ and $W^i$ of $1_{H^i}$ with $(g_iV^i)(h_iW^i)^{-1}\subseteq g_ih_i^{-1}U^i$ for $i\le m$. We denote $V=V^0\times\dots\times V^m\times\prod_{i\ge m}H^i$ and $W=W^0\times\dots\times W^m\times\prod_{i\ge m}H^i$. Then $V$ and $W$ are open neighborhoods of $1_H$, and $((g_i)V)((h_i)W)^{-1}\subseteq(g_ih_i^{-1})U$. So $(G,\tau)$ is Polish group.
In the end, suppose $\tau'$ is another group topology on $G$ such that $H$ is an open subgroup of $G$. Then for each $(g_i){\rm i}n G$, the subspace $(g_i)H$ of $(G,\tau')$ is homeomorphic to $H$, so the restrictions of $\tau$ and $\tau'$ on $(g_i)H$ are the same. Hence $\tau=\tau'$.
{\rm e}nd{proof}
\begin{lemma}\label{limit}
Let $(G^i)$ be a sequence of non-archimedean CLI Polish groups, $\mathcal G^i=(G^i_n){\rm i}n{\rm dgnb}(G^i)$ for each $i<\omega$, and let $0<\alpha<\omega_1$. Suppose
$$\sup\{\rho^1(\mathcal G^i):i<\omega\}=\omega\cdot\alpha,$$
$$G=\{(g_i){\rm i}n\prod_iG^i:\forall^{\rm i}nfty i\,(g_i{\rm i}n G^i_1)\}$$
equipped with the unique group topology so that $\prod_iG^i_1$ is an open subgroup of $G$. Then $G$ is not $\alpha$-CLI.
{\rm e}nd{lemma}
\begin{proof}
Put $G_0=G$ and for $n\ge 1$,
$$G_n=\prod_{i<n-1}G^i_n\times\prod_{i\ge n-1}G^i_1.$$
It is clear that $G_1=\prod_iG^i_1$ and $\mathcal G=(G_n){\rm i}n{\rm dgnb}(G)$.
Given $j<\omega$, define $\theta:G/G_1\to G^j/G^j_1$ as $\theta((g_i)G_1)=g_jG^j_1$ for $(g_i){\rm i}n G$. Applying Lemma~\ref{surjection-tree} on the restriction of $\theta$ as in the proof of Theorem~\ref{wr}, we have
$$\begin{array}{ll}\rho(T^{G/G_1}_\mathcal G) &=\sup\{\rho((T^{G/G_1}_\mathcal G)_{(1,G_1\cdot(g_i)G_1)}):(g_i){\rm i}n G\}+1\cr
&\ge\sup\{\rho((T^{G^j/G^j_1}_{\mathcal G^j})_{(1,G^j_1\cdot gG^j_1)}):g{\rm i}n G^j\}+1\cr
&=\rho(T^{G^j/G^j_1}_{\mathcal G^j})=\rho^1(\mathcal G^j).{\rm e}nd{array}$$
Therefore,
$$\rho^1(\mathcal G)=\rho(T^{G/G_1}_\mathcal G)\ge\sup\{\rho^1(\mathcal G^j):j<\omega\}=\omega\cdot\alpha.$$
Since $\rho^1(\mathcal G)$ is a non-limit ordinal and $\alpha>0$, we have $\rho^1(\mathcal G)>\omega\cdot\alpha$, so $G$ is not $\alpha$-CLI.
{\rm e}nd{proof}
Finally, we complete all the construction in the following theorem.
\begin{theorem}
For any $\alpha<\omega_1$, there exist non-archimedean CLI Polish groups $G_\alpha$ and $H_\alpha$ with ${\rm rank}(G_\alpha)={\rm rank}(H_\alpha)=\alpha$ such that $H_\alpha$ is $\alpha$-CLI and $G_\alpha$ is L-$\alpha$-CLI but not $\alpha$-CLI.
{\rm e}nd{theorem}
\begin{proof}
We construct $G_\alpha$ and $H_\alpha$ by induction on $\alpha$. From Corollary~\ref{+1} and Theorem~\ref{wr}, we only need to consider the case that $\alpha$ is a limit ordinal.
Let $(\alpha_i)$ be a sequence of ordinals less than $\alpha$ with $\sup\{\alpha_i:i<\omega\}=\alpha$. By induction hypothesis, we can find a non-archimedean CLI Polish group $G^i$ for each $i<\omega$ such that $G^i$ is L-$\alpha_i$-CLI but not $\alpha_i$-CLI. It is clear that ${\rm rank}(G^i)=\alpha_i<\alpha$.
Put $H_\alpha=\prod_iG^i$. Theorem~\ref{prod}.(1) implies that $H$ is $\alpha$-CLI. By Theorem~\ref{subgroup}, ${\rm rank}(H_\alpha)\ge{\rm rank}(G^i)$ for each $i<\omega$. So ${\rm rank}(H_\alpha)=\alpha$.
For $i<\omega$, let $\mathcal G^i=(G^i_n){\rm i}n{\rm dgnb}(G^i)$. By Lemma~\ref{rank}, there exist $0<k_i<\omega$ such that $\omega(\rho^{k_i}(\mathcal G^i))=\omega\cdot\alpha_i$. Put $H^i_0=G^i$ and $H^i_{n+1}=G^i_{n+k_i}$ for $n<\omega$. Then $\mathcal H^i=(H^i_n){\rm i}n{\rm dgnb}(G^i)$ and $H^i_1=G^i_{k_i}$. By Lemma~\ref{two dgnb}, we have
$$\omega(\rho^1(\mathcal H^i))=\omega(\rho(T^{G^i/H^i_1}_{\mathcal H^i}))=\omega(\rho(T^{G^i/G^i_{k_i}}_{\mathcal G^i}))=\omega(\rho^{k_i}(\mathcal G^i))=\omega\cdot\alpha_i.$$
So $\sup\{\rho^1(\mathcal H^i):i<\omega\}=\omega\cdot\alpha$. Now we put
$$G_\alpha=\{(g_i){\rm i}n\prod_iG^i:\forall^{\rm i}nfty i\,(g_i{\rm i}n G^i_{k_i})\}$$
equipped with the unique group topology so that $\prod_iG^i_{k_i}$ is an open subgroup of $G_\alpha$. By Lemma~\ref{limit}, $G_\alpha$ is not $\alpha$-CLI. It is clear that the open subgroup $\prod_iG^i_{k_i}$ is $\alpha$-CLI, so $G_\alpha$ is L-$\alpha$-CLI, and hence ${\rm rank}(G_\alpha)=\alpha$.
{\rm e}nd{proof}
\begin{thebibliography}{99}
\bibitem{BK} H. Becker, A.S. Kechris, The Descriptive Set Theory of Polish Group Actions, Lond. Math. Soc. Lect. Note Ser., vol. 232, Cambridge University Press, 1996.
\bibitem{gaobook} S. Gao, Invariant Descriptive Set Theory, Monographs and Textbooks in Pure and Applied Mathematics, vol. 293, CRC Press, 2009.
\bibitem{GX} S. Gao, M. Xuan, \textit{On non-Archimedean Polish groups with two-sided invariant metrics}, Topol. Appl. 161 (2014) 343--353.
\bibitem{klee} V.L. Klee, Invariant metrics in groups (solution of a problem of Banach), Proc. Amer. Math. Soc. 3 (1952) 484--487.
\bibitem{malicki11} M. Malicki, \textit{On Polish groups admitting a compatible complete left-invariant metric}, J. Symb. Logic 76 (2011) 437--447.
\bibitem{xuan} M. Xuan, On steinhaus sets, orbit trees and universal properties of various subgroups in the permutation group of natural numbers, Ph.D. thesis, University of North Texas, 2012.
{\rm e}nd{thebibliography}
{\rm e}nd{document} | math | 58,453 |
\begin{document}
\maketitle
\begin{abstract} In this paper we face the study of the representations of the exceptional Lie superalgebra $E(5,10)$.
We recall the construction of generalized Verma modules and give a combinatorial description of the restriction to $\mathfrak{sl}_5$ of the Verma module induced
by the trivial representation. We use this description to classify morphisms between Verma modules of degree one, two and three
proving in these cases a conjecture given by Rudakov \cite{R}. A key tool is the notion of dual morphism between Verma modules.
\end{abstract}
\section{Introduction}
Infinite dimensional linearly compact simple Lie superalgebras over the complex numbers were classified by Victor Kac in 1998 \cite{K}. A complete list, up to isomorphisms, consists of ten
infinite series and five exceptions, denoted by $E(1,6)$, $E(3,6)$, $E(3,8)$, $E(5,10)$ and $E(4,4)$. See also \cite{CK, S1, S2, S3} for the genesis of these superalgebras.
Some years later Kac and Rudakov
initiated the study of the representations of these algebras \cite{KR1, KR2, KR3, KR} developing a general theory of Verma modules that we briefly recall.
Let $L=\oplus_{j\in\mathbb{Z}}L_j$ be a $\mathbb{Z}$-graded Lie superalgebra, let $L_-=\oplus_{j<0}L_j$, $L_+=\oplus_{j>0}L_j$ and $L_{\geq 0}=L_0\oplus L_+$.
We denote by $U(L)$ the universal enveloping algebra of $L$.
If $F$ is an irreducible $L_0$-module we define
$$M(F)=U(L)\otimes_{U(L_{\geq 0})} F$$
where we extend the action of $L_0$ to $L_{\geq 0}$ by letting $L_+$ act trivially on $F$.
We call $M(F)$ a minimal generalized Verma module associated to $F$. If $M(F)$ is not irreducible we say that it is degenerate.
In \cite{KR1, KR2, KR3, KR}, a complete description of the degenerate Verma modules for $E(3,6)$ and
$E(3,8)$ is given, as well as of their unique irreducible quotients.
In \cite{KR} some basic ideas and constructions are settled also for $E(5,10)$. In this case Kac and Rudakov conjecture a complete list of $L_0$-modules which
give rise to the degenerate
Verma modules (see Conjecture \ref{conjecture}).
In 2010 Rudakov tackled the proof of the conjecture through the study of morphisms between Verma modules. The existence of a degenerate Verma module
is indeed strictly related to the existence of such morphisms of positive degree (see Proposition \ref{morphism}). In \cite{R} Rudakov classified
morphisms of degree one and gave some examples of morphisms of degree at most 5. He also conjectured that there exists no morphism of higher degree and
that his list exhausts all the examples. A more general family of modules,
possibly induced from infinite-dimensional $\mathfrak{sl}_5$-modules,
had been studied in \cite{GLS}, where some of Rudakov's examples in degree one and two had been obtained through the use of the
computer.
In this paper we study morphisms between generalized Verma modules and to this aim we analyze the structure of the universal enveloping algebra $U_-=
U(L_-)$
as an $L_0$-module.
This analysis has its own interest and
provides an explicit combinatorial description of the action of $L_0$. This description is the main ingredient in our study of morphisms,
together with a systematic use of the dominance order of the weights of the $L_0$-modules.
Our main result is the proof of Rudakov's conjecture in degree two and three (see Theorems \ref{teorema2}, \ref{teorema3}).
A useful observation that we made is that if there exists a morphism $\varphi: M(V) \rightarrow M(W)$
between generalized Verma modules of degree $d$, then there exists a dual morphism $\psi: M(W^*) \rightarrow M(V^*)$ of the same degree.
This duality is here proved in low degree for the purpose of this work but it holds in a much wider context as a consequence of the fact that
the conformal dual of a Verma module is itself a Verma module. This will be shown in a forthcoming paper.
The paper is organized as follows: in Section \ref{S1} we recall the basic definitions and fix the notation. Section \ref{S3} is dedicated to Verma modules. Here we
characterize degenerate Verma modules in terms of singular vectors and morphisms. In Section \ref{S4}, following \cite{R}, we give examples of morphisms of degree one, two and three.
Section \ref{S5} contains our first main result on the structure of $U_-$ as an $L_0$-module: we construct an explicit basis of $U_-$ and describe
its combinatorial properties. Section \ref{S6} is dedicated to the analysis of the dominance order of the weights of the basis elements of $U_-$.
In Section \ref{six} we develop the idea of dual morphism between generalized Verma modules and establish sufficient conditions for the existence of such a morphism
(see Remark \ref{lemdual2}). Finally, Sections \ref{S7}, \ref{eight} and \ref{nine} contain the classification of morphisms of degree one, two and three, respectively.
We thank Victor Kac for useful discussions.
\section{Preliminaries}\label{S1}
We let $\mathbb{N}=\{0,1,2,3,\dots\}$ be the set of non-negative integers and for $n\in\mathbb{N}$ we set $[n]=\{i\in\mathbb{N} ~|~ 1\leq i\leq n\}$.
If $P$ is a proposition we let $\chi_{P}=1$ if $P$ is true and $\chi_{P}=0$ if $P$ is false.
We consider the simple, linearly compact Lie superalgebra of exceptional type $L=E(5,10)$ whose even and odd parts are
as follows: $L_{\bar{0}}$ consists of zero-divergence vector fields in five (even) indeterminates $x_1,\ldots,x_5$, i.e.,
\[L_{\bar{0}}=S_5=\{X=\sum_{i=1}^5f_i\partial_i ~|~ f_i\in\mathbb{C}[[x_1,\dots,x_5]], \textrm{div}(X)=0\},\]
where $\partial_i=\partial_{x_i}$,
and $L_{\bar{1}}=\Omega^2_{cl}$ consists of closed two-forms in the five indeterminates $x_1,\ldots,x_5$.
The bracket between a vector field and a form is given by the Lie derivative and for $f,g\in \mathbb{C}[[x_1,\dots,x_5]]$ we have
$$[fdx_i\wedge dx_j,g dx_k\wedge dx_l]=\varepsilon_{ijkl}fg\partial_{t_{ijkl}}$$
where, for $i,j,k,l\in [5]$, $\varepsilon_{ijkl}$ and $t_{ijkl}$ are defined as follows: if $|\{i,j,k,l\}|=4$ we let $t_{ijkl}\in [5]$ be such that $|\{i,j,k,l,t_{ijkl}\}|=5$ and $\varepsilon_{ijkl}$ be the sign of the permutation
$(i,j,k,l,t_{ijkl})$. If $|\{i,j,k,l\}|<4$ we let $t_{ijkl}=1$ (this choice will be irrelevant) and $\varepsilon_{ijkl}=0$.
From now on we shall denote $dx_i\wedge dx_j$ simply by $d_{ij}$.
The Lie superalgebra $L$ has a consistent, irreducible, transitive $\mathbb{Z}$-grading of depth 2 where,
for $k\in\mathbb{N}$,
\begin{align*}
L_{2k-2}&=\langle f\partial_i ~|~i=1,\dots,5, f\in\mathbb{C}[[x_1,\dots, x_5]]_{k}\rangle\cap S_5\\
L_{2k-1}&=\langle fd_{ij} ~|~ i,j=1,\dots,5, f\in\mathbb{C}[[x_1,\dots, x_5]]_{k}\rangle\cap\Omega^2_{cl}
\end{align*}
where
by $\mathbb{C}[[x_1,\dots, x_5]]_{k}$ we denote the homogeneous component of $\mathbb{C}[[x_1,\dots, x_5]]$ of degree $k$.
Note that $L_0\cong \mathfrak{sl}_5$, $L_{-2}\cong (\mathbb{C}^5)^*$, $L_{-1}\cong \inlinewedge^2\mathbb{C}^5$ as $L_0$-modules (where $\mathbb{C}^5$ denotes the
standard $\mathfrak{sl}_5$-module).
We set $L_{-}=L_{-2}\oplus L_{-1}$, $L_{+}=\oplus_{j>0}L_j$ and $L_{\geq 0}=L_0\oplus L_+$.
We denote by $U$ (resp.\ $U_{-}$) the universal enveloping algebra of $L$ (resp.\ $L_-$). Note that $U_-$ is
an $L_0$-module with respect to the adjoint action: for $x\in L_0$ and $u\in U_-$,
$$x.u=[x,u]=xu-ux.$$
We also point out that the $\mathbb{Z}$-grading of $L$ induces a $\mathbb{Z}$-grading on the enveloping algebra $U_-$.
It is customary, though, to invert the sign of the degrees hence getting a grading over $\mathbb{N}$. Note that the homogeneous component
$(U_-)_d$ of degree $d$
of $U_-$ under this grading is an $L_0$-submodule. Section \ref{S3} will be dedicated to the study of
these homogeneous components.
We fix the Borel subalgebra $\langle x_i\partial_j, h_{ij}=x_i\partial_i-x_j\partial_j ~|~ i<j\rangle$ of $L_0$ and we consider the usual base of the corresponding root system given by $\{\alpha_{12},\ldots,\alpha_{45}\}$. We let $\Lambda$ be the weight lattice of $\frak{sl}_5$ and we express all weights of $\frak{sl}_5$ using their coordinates with respect to the fundamental weights $\omega_{12},\omega_{23},\omega_{34},\omega_{45}$, i.e., for $\lambda\in \Lambda$ we write $\lambda=(\lambda_{12},\ldots,\lambda_{45})$ for some $\lambda_{i\,i+1}\in \mathbb Z$ to mean $\lambda=\lambda_{12}\omega_{12}+\cdots+\lambda_{45}\omega_{45}$.
For $i<j$ we denote as usual
\[
\alpha_{ij}=\sum_{k=i}^{j-1}\alpha_{k\,k+1}
\]
and $\alpha_{ji}=-\alpha_{ij}$. For notational convenience we also let $\alpha_{ii}=0$.
Viewed as elements in the weight lattice we have
\[
\alpha_{12}=(2,-1,0,0),\,\alpha_{23}=(-1,2,-1,0),\,\alpha_{34}=(0,-1,2,-1),\, \alpha_{45}=(0,0,-1,2).
\]
If $\lambda\in \Lambda$ is a weight, we use the following convention: for all $1\leq i<j\leq 5$ we let
\[
\lambda_{ij}=\sum_{k=i}^{j-1}\lambda_{k\,k+1}.
\]
If $V$ is a $\frak {sl}_5$-module and $v\in V$ is a weight vector we denote by $\lambda(v)$ the weight of $v$ and by $\lambda_{ij}(v)=(\lambda(v))_{ij}$.
If $\lambda=(a,b,c,d)\in \Lambda$ is a dominant weight, i.e. $a,b,c,d\geq 0$, let us denote by $F(\lambda)=F(a,b,c,d)$ the irreducible $\mathfrak{sl}_5$-module of highest
weight $\lambda$. In this paper we always think of $F(a,b,c,d)$ as the irreducible submodule of \[\Sym^a(\mathbb{C}^5)\otimes \Sym^b(\displaywedge^2(\mathbb{C}^5))\otimes
\Sym^c(\displaywedge^2(\mathbb{C}^5)^*)\otimes \Sym^d((\mathbb{C}^5)^*)\] generated by the highest weight vector $x_1^ax_{12}^b{x_{45}^*}^c{x_5^*}^d$
where $\{x_1,\dots, x_5\}$ denotes the standard basis of $\mathbb{C}^5$, $x_{ij}=x_i\wedge x_j$, and $x_i^*$ and $x_{ij}^*$ are the corresponding dual
basis elements. Besides, for a weight $\lambda=(a,b,c,d)$ we let $\lambda^*=(d,c,b,a)$, so that $F(\lambda)^*\cong F(\lambda^*)$.
Notice that $L_1\cong F(1,1,0,0)$ and that $x_5d_{45}$ is a lowest weight vector in $L_1$. Moreover,
for $j\geq 1$, we have $L_j=L_1^j$.
\section{Generalized Verma modules and morphisms}\label{S3}
We recall the definition of generalized Verma modules introduced in \cite{KR1}. For the reader's convenience we also sketch some proofs of basic results.
Given an $L_0$-module $V$ we extend it to
an $L_{\geq 0}$-module by letting $L_+$ act trivially, and define
$$M(V)=U\otimes_{U(L_{\geq 0})}V.$$
Note that $M(V)$ has a $L$-module structure by multiplication on the left, and is called the (generalized) Verma module associated to $V$. We also observe that $M(V)\cong U_{-}\otimes_{\mathbb{C}}V$ as $\mathbb{C}$-vector spaces.
If $V$ is finite-dimensional and irreducible, then $M(V)$ is called a minimal Verma module. We denote by $M(\lambda)$ the minimal Verma module $M(F(\lambda))$. A minimal Verma module is said to be non-degenerate if it is irreducible and degenerate if it is not irreducible.
\begin{definition}
We say that an element $w\in M(V)$ is homogeneous of degree $d$ if $w\in (U_-)_d\otimes V$.
\end{definition}
\begin{definition} A vector $w\in M(V)$ is called a singular vector if it satisfies the following conditions:
\begin{itemize}
\item[(i)] $x_i\partial_{i+1}w=0$ for every $i=1,\dots,4$;
\item[(ii)] $zw=0$ for every $z\in L_1$;
\item[(iii)]
$w$ does not lie in $V$.
\end{itemize}
\end{definition}
We observe that the homogeneous components of positive degree of a singular vector are singular vectors.
The same holds for its weight components. From now on we will thus assume that a singular vector is a homogeneous weight
vector unless otherwise specified.
Notice that if condition (i) is satisfied then condition (ii) holds if $x_5d_{45}w=0$ since $x_5d_{45}$ is a lowest weight vector in $L_1$.
\begin{proposition} \label{dege=sing}A minimal Verma module $M(V)$ is degenerate if and only if it contains a singular vector.
\end{proposition}
\begin{proof}
Let $w\in M(V)$ be a singular vector. We may assume that $w$ is homogeneous of degree $d>0$. Hence the singular vector $w$ generates a submodule
of $M(V)$ which is proper since it is contained in $\oplus_{k\geq d} (U_-)_k\otimes V$.
On the other hand, if $M(V)$ is degenerate let us consider a proper non-zero submodule $W$ of $M(V)$. Let
$z\in W$ be a non-zero vector. By repeatedly applying $L_1$ to $z$ if necessary we can find a non-zero element $w\in W$
such that $L_1w=0$, since the action of $L_1$ lowers the degree of the homogeneous components of $z$ by 1.
We observe that $L_1$ vanishes on the $L_0$-module generated by $w$. Any highest weight vector in
such a module is a singular vector.
\end{proof}
Degenerate Verma modules can also be described in terms of morphisms.
A linear map $\varphi: M(V)\rightarrow M(W)$ can always be associated to an element $\Phi\in U_{-}\otimes \Hom(V,W)$ as follows: for $u\in U_-$ and $v\in V$ we let
$$\varphi(u\otimes v)=u\Phi(v)$$
where, if $\Phi=\sum_iu_i\otimes \theta_i$ with $u_i\in U_-$,
$\theta_i\in \Hom(V,W),$ we let $\Phi(v)=\sum_iu_i\otimes \theta_i(v)$.
We will say that $\varphi$ (or $\Phi$) is a morphism of degree $d$ if $u_i\in (U_-)_d$ for every $i$.
The following proposition characterizes morphisms between Verma modules.
\begin{proposition}\cite{R}\label{morphisms}
Let $\varphi: M(V)\rightarrow M(W)$ be the linear map associated with the element $\Phi\in U_{-}\otimes \Hom(V,W)$.
Then $\varphi$ is a morphism of $L$-modules if and only if the following conditions
hold:
\begin{itemize}
\item[(a)] $L_0.\Phi=0$;
\item[(b)] $t\Phi(v)=0$ for every $t\in L_1$ and for every $v\in V$.
\end{itemize}
\end{proposition}
We observe that if $M(V)$ is a minimal Verma module and condition $(a)$ holds it is enough to verify condition $(b)$ for an element $t$ generating
$L_1$ as an $L_0$-module and for $v$ a highest weight vector in $V$.
\begin{proposition}\label{morphism} Let $M(\mu)$ be a minimal Verma module. Then the following are equivalent:
\begin{itemize}
\item[(a)] $M(\mu)$ is degenerate;
\item[(b)] $M(\mu)$ contains a singular vector;
\item[(c)] there exists a minimal Verma module $M(\lambda)$ and a morphism $\varphi:M(\lambda)\rightarrow M(\mu)$ of positive degree.
\end{itemize}
\end{proposition}
\begin{proof}
We already know that condition (a) is equivalent to condition (b) by Proposition \ref{dege=sing}.
Assume condition (c) holds: if $s\in F(\lambda)$ is a highest weight vector, then $\varphi(1\otimes s)$ is a singular vector in $M(\mu)$.
On the other hand, if $w$ is a singular vector in $M(\mu)$, we can define
$\varphi: M(\lambda(w))\rightarrow M(\mu)$ as the unique morphism of $L$-modules such that $\varphi(1\otimes s)=w$, $s$ being a highest weight vector in $M(\lambda(w))$.
\end{proof}
\begin{remark}\label{dual}
Let $\varphi: M(V)\rightarrow M(W)$ be a linear map of degree $d$ associated to an element $\Phi\in U_-\otimes \Hom(V,W)$ that satisfies condition $(a)$ of Proposition \ref{morphisms}. Then there exists an $L_0$-morphism $\psi: (U_-)_d^*\rightarrow \Hom(V,W)$ such that
$\Phi= \sum_i u_i\otimes \psi(u_i^*)$ where $\{u_i, i\in I\}$ is any basis of $(U_-)_d$ and $\{u_i^*, i\in I\}$ is the corresponding dual basis.
\end{remark}
\begin{definition} Let $M(\mu)$ be a minimal Verma module and let $\pi: M(\mu)\rightarrow U_-\otimes F(\mu)_{\mu}$ be the natural projection,
$F(\mu)_{\mu}$ being the weight space of $F(\mu)$ of weight $\mu$. Given a singular vector $w\in M(\mu)$ we call $\pi(w)$ the leading term of $w$.
\end{definition}
\begin{proposition}\label{leading} If $w$ is a singular vector in $M(\mu)$ then:
\begin{itemize}
\item[(i)] $\pi(w)\neq 0$;
\item[(ii)] if two singular vectors in $M(\mu)$ have the same leading term then they coincide.
\end{itemize}
\end{proposition}
\begin{proof} If $w$ is a weight vector homogeneous of degree $d$ then we can write $w=\sum_iu_i\otimes v_i$ for some basis $\{u_i\}$ of $(U_-)_d$ consisting of weight vectors
and $v_i\in F(\mu)_{\lambda_i}$ for some weight $\lambda_i$.
Let $\lambda_{i_0}$ be maximal in the dominance order such that $v_{i_0}\neq 0$. Then $v_{i_0}$ is a highest weight vector in $F(\mu)$. Indeed, for
$r<s$ we have:
\[0=x_r\partial_s w=\sum_i[x_r\partial_s,u_i]\otimes v_i+\sum_i u_i\otimes x_r\partial_s.v_i. \]
By the maximality of $\lambda_{i_0}$ it follows that $x_r\partial_s.v_{i_0}=0$.
$(ii)$ follows from $(i)$.
\end{proof}
\section{Examples}\label{S4}
In this section we give some examples of singular vectors and the corresponding morphisms of Verma modules. These were described in \cite{R}. We will need the following technical result.
\begin{lemma}\label{esempi}
Let $\varphi: M(\lambda)\rightarrow M(W)$ be a morphism of Verma modules of degree one associated to
$\Phi=\sum_{i<j} d_{ij}\otimes \theta_{ij}$ and let $s$ be a highest weight vector in $F(\lambda)$.
Let $\tilde W$ be an $L_0$-module containing $W$ and let $\tilde{\theta}_{ij}\in \Hom(F(\lambda),\tilde W)$ be such that
the map $(U_-)_1^*\rightarrow \Hom(F(\lambda),\tilde W)$ given by $d_{ij}^*\mapsto \tilde{\theta}_{ij}$ is well defined and $L_0$-equivariant. Then
$\tilde{\theta}_{ij}(s)={\theta}_{ij}(s)$ implies
$\tilde{\theta}_{ij}(v)={\theta}_{ij}(v)$ for all $v\in F(\lambda)$.
\end{lemma}
\begin{proof} It is enough to show that if $\tilde{\theta}_{ij}(v)={\theta}_{ij}(v)$
for some $v\in F(\lambda)$ and all $i\neq j$, then
$\theta_{ij}(x_h\partial_k.v)= \tilde{\theta}_{ij}(x_h\partial_k.v)$ for all $i\neq j$ and $h\neq k$. We have:
\begin{align*}
\tilde{\theta}_{ij}(x_h\partial_k.v)& =x_h\partial_k(\tilde{\theta}_{ij}(v))-(x_h\partial_k.\tilde{\theta}_{ij})(v)=
x_h\partial_k(\tilde{\theta}_{ij}(v))+\delta_{hi}\tilde{\theta}_{kj}(v)+\delta_{hj}\tilde{\theta}_{ik}(v)\\
&=x_h\partial_k({\theta}_{ij}(v))+\delta_{hi}{\theta}_{kj}(v)+\delta_{hj}{\theta}_{ik}(v)={\theta}_{ij}(x_h\partial_k.v)
\end{align*}
where we used Remark \ref{dual} in order to write the action of $L_0$ on the $\theta_{ij}$'s.
Namely, we have:
$$x_h\partial_k.\theta_{ij}=-\delta_{hi}\theta_{kj}-\delta_{hj}\theta_{ik}$$
where if $r>s$, $\theta_{rs}=-\theta_{sr}$.
\end{proof}
\begin{example}\label{nablaA}
Let us consider the Verma module $M(m,n,0,0)$. We first observe that $d_{12}\otimes x_1^mx_{12}^n$ is a singular vector
in $M(m,n,0,0)$. Indeed, for $i=1,\dots, 4$,
$$x_i\partial_{i+1}d_{12}\otimes x_1^mx_{12}^n=0;$$
besides,
$$x_5d_{45}d_{12}\otimes x_1^mx_{12}^n=x_5\partial_3\otimes x_1^mx_{12}^n=0.$$
By Proposition \ref{morphism} we can define a morphism of Verma modules $\nabla_A: M(m,n+1,0,0) \rightarrow M(m,n,0,0)$
by setting $\nabla_A(1\otimes s)=d_{12}\otimes x_1^mx_{12}^n$. By Lemma \ref{esempi} used with $\tilde W=\Sym^m(\mathbb{C}^5)\otimes \Sym^n(\inlinewedge^2\mathbb{C}^5)$ we have that $\nabla_A$ is associated to:
$$\sum_{i<j}d_{ij}\otimes \frac{\partial}{\partial x_{ij}}\in U_-\otimes \Hom(F(m,n+1,0,0),F(m,n,0,0)).$$
\end{example}
\begin{example}\label{nablaB}
Let us consider the Verma module $M(m,0,0,n+1)$. One can check that $\sum_{j=2}^5d_{1j}\otimes x_1^mx_j^*(x_{5}^*)^n$ is a singular vector
in $M(m,0,0,n+1)$, with leading term $d_{15}\otimes x_1^m(x_{5}^*)^{n+1}$. By Remark \ref{morphism} we can define a morphism of Verma modules $\nabla_B: M(m+1,0,0,n) \rightarrow M(m,0,0,n+1)$
by setting $\nabla_B(1\otimes s)=\sum_{j=2}^5d_{1j}\otimes x_1^mx_j^*(x_{5}^*)^n$. By Lemma \ref{esempi}, we have that $\nabla_B$ is associated to
$$\sum_{i<j}d_{ij}\otimes (x_i^*\partial_j-x_j^*\partial_i).$$
\end{example}
\begin{example} \label{nablaC} We shall now exhibit a singular vector in $M(0,0,m+1,n)$. To this aim it is convenient to think
of $F(0,0,m+1,n)$ as the dual $L_0$-module $F(n,m+1,0,0)^*$. We shall later investigate the role of duality between Verma modules in Section \ref{six}, where we will show, in particular, that the morphism we are going to construct can be seen in a certain sense as the dual of the morphism $\nabla_A$ defined in Example \ref{nablaA}.
Let us observe that the vector $\sum_{i<j}d_{ij}\otimes x_{ij}^*(x_{45}^*)^m(x_{5}^*)^n$ is a singular vector
in $M(F(n,m+1,0,0)^*)$ (with leading term $d_{45}\otimes (x_{45}^*)^{m+1}(x_{5}^*)^n$). Indeed, one immediately checks that $x_k\partial_{k+1}(\sum_{i<j}d_{ij}\otimes x_{ij}^*(x_{45}^*)^m(x_{5}^*)^n)=0$
for every $k=1,\dots, 4$. Besides, we have:
\begin{align*} x_5d_{45}&(\sum_{i<j}d_{ij}\otimes x_{ij}^*(x_{45}^*)^m(x_{5}^*)^n)\\
&=
x_5\partial_3x_{12}^*(x_{45}^*)^m(x_5^*)^n-x_5\partial_2x_{13}^*(x_{45}^*)^m(x_5^*)^n+x_5\partial_1x_{23}^*(x_{45}^*)^m(x_5^*)^n\\
&=m(x_{45}^*)^{m-1}(x_5^*)^n(x_{12}^*x_{34}^*+x_{13}^*x_{42}^*+x_{14}^*x_{23}^*)
-n(x_{45}^*)^{m}(x_5^*)^{n-1}(x_{12}^*x_{3}^*+x_{23}^*x_{1}^*+x_{31}^*x_{2}^*)
&=0.
\end{align*}
Notice that, in fact,
\[ x_{ab}^*x_{cd}^*+x_{ac}^*x_{db}^*+x_{ad}^*x_{bc}^*=0\]
and
\[ x_{ab}^*x_{c}^*+x_{bc}^*x_{a}^*+x_{ca}^*x_{b}^*=0\] in $F(n,m+1,0,0)^*$
for all $a,b,c,d \in[5]$,
as one can check by applying these elements to the highest weight vector $x_1^nx_{12}^{m+1}$ in $F(n,m+1,0,0)$
and using the $L_0$-action.
By Remark \ref{morphism} we can thus define a morphism of Verma modules $\nabla_C: M(0,0,m,n) \rightarrow M(F(n,m+1,0,0)^*)$
by setting $\nabla_C(1\otimes s)=\sum_{i<j}d_{ij}\otimes x_{ij}^*(x_{45}^*)^m(x_{5}^*)^n$. Once again, Lemma \ref{esempi} implies that the morphism $\nabla_C$ is associated to
$$\sum_{i<j}d_{ij}\otimes x_{ij}^*.$$
\end{example}
Examples \ref{nablaA}, \ref{nablaB} and \ref{nablaC} imply the following result.
\begin{proposition}
Let $m,n\geq 0$. Then $M(m,n,0,0)$, $M(m,0,0,n)$ and $M(0,0,m,n)$ are degenerate Verma modules.
\end{proposition}
Kac and Rudakov proposed the following conjecture \cite{KR}:
\begin{conjecture}\label{conjecture}
Let $a,b,c,d\geq 0$ be such that $M(a,b,c,d)$ is a degenerate Verma module. Then $a=b=0$ or $b=c=0$ or $c=d=0$.
\end{conjecture}
By Proposition \ref{morphism} a possible strategy to prove Conjecture \ref{conjecture} is to construct all possible morphisms between minimal Verma modules. One of
the main results of this paper is a complete classification of such morphisms of degree at most 3.
\begin{example}\label{exampledeg2}
The following are nonzero morphisms of degree 2:
\begin{itemize}
\item $\nabla_B\nabla_A:M(m,1,0,0)\rightarrow M(m-1,0,0,1)$;
\item $\nabla_C\nabla_B:M(1,0,0,n)\rightarrow M(0,0,1,n+1)$;
\item $\nabla_C\nabla_A:M(0,1,0,0)\rightarrow M(0,0,1,0)$;
\end{itemize}
Indeed,
\[
\nabla_B\nabla_A(1\otimes x_1^mx_{12})=\nabla_B(d_{12}\otimes x_1^m)=-m\sum_{j>1}d_{12}d_{1j}\otimes x_1^{m-1}x_j^*\neq 0
\]
\[
\nabla_C\nabla_B(1\otimes x_1(x_5^*)^n)=\sum_{j>1}\sum_{h<k} d_{1j}d_{hk}\otimes x_{hk}^*x_j^*(x_5^*)^n\neq 0
\]
\[
\nabla_C\nabla_A (1\otimes x_{12})=\sum_{i<j}d_{12}d_{ij}\otimes x_{ij}^*\neq 0.
\]
We observe that the leading terms of these singular vectors are $d_{12}d_{15}\otimes x_1^{m-1}x_5^*$, $d_{15}d_{45}\otimes x_{45}^*(x_5^*)^{n+1}$
and $d_{12}d_{45}\otimes x_{45}^*$, respectively.
(We also observe that the other compositions $\nabla_A\nabla_B$, $\nabla_A\nabla_C$, $\nabla_B\nabla_C$ are not defined). Moreover, one can also verify that $\nabla_A^2=\nabla_B^2=\nabla_C^2=0$ whenever they are defined: this will also be a consequence of the general treatment of morphisms of degree 2 in Section \ref{eight}.
\end{example}
\begin{example}
\[
\nabla_C\nabla_B\nabla_A:M(1,1,0,0)\rightarrow M(0,0,1,1)
\]
is a nonzero morphism of degree 3. We have that $\nabla_C\nabla_B\nabla_A(x_1x_{12})=\sum_{j>1, k<l} d_{12}d_{1j}d_{kl}\otimes x_{j}^*x_{kl}^*$
is a singular vector in $M(0,0,1,1)$ with leading term $d_{12}d_{15}d_{45}\otimes x_{45}^*x_5^*$ .
\end{example}
We will prove that the morphisms described in this section are all possible morphisms between minimal Verma modules of degree at most 3.
\section{Structure of $U_-$}\label{S5}
In order to classify morphisms between generalized Verma modules of a given degree we need to better understand the structure of $U_-$ as an $L_0$-module. The main result of this section is the construction of an explicit linear basis of $U_-$ which realizes its structure of $L_0$-module in a combinatorial way.
We recall that $(U_-)_d$ denotes the homogeneous component of $U_-$ of degree $d$.
We let
\[\mathcal I_d=\{ I=(I_1,\ldots,I_d):\, I_l=(i_l,j_l) \textrm{ with $ 1\leq i_l,j_l\leq 5$ for every $l=1,\ldots,d$}\}.
\]
If $I=(I_1,\ldots,I_d)\in \mathcal I_d$ we let $d_I=d_{I_1}\cdots d_{I_d}\in (U_-)_d$, with $d_{I_l}=d_{i_l j_l}$.
We set
$[5]^k=\{(t_1,\dots, t_k)~|~ t_i\in [5]\}$ and
for $T=(t_1,\dots, t_k)\in [5]^k$ we let
$\partial_T=\partial_{t_1}\dots\partial_{t_k}$.
We have that $(U_-)_d$ is spanned by all elements of the form $d_I$ as $I$ varies in $\mathcal I_d$. One can also consider the following filtration of subspaces of $(U_-)_d$: for all $k\leq d/2$ we let
\[
(U_-)_{d,k}=\textrm{Span}\{\partial_Td_I:\, T\in \mathbb [5]^k,\, I\in \mathcal I_{d-2k}\}.
\]
We have the following chain of inclusions
\[
(U_-)_d=(U_-)_{d,0}\supseteq (U_-)_{d,1}\supseteq (U_-)_{d,2}\supseteq\cdots.
\]
We observe that for all $k\leq d/2$ the subspace $(U_-)_{d,k}$ is also an $L_0$-submodule of $(U_-)_d$ and so we have the following isomorphism of $L_0$-modules
\[
(U_-)_d\cong \bigoplus_{k\leq d/2} (U_-)_{d,k}/(U_-)_{d,k+1},
\]
where we let $(U_-)_{d,k}=0$ if $k>d/2$. For example, we have
\[
(U_-)_5\cong \frac{(U_-)_{5,0}}{(U_-)_{5,1}}\oplus \frac{(U_-)_{5,1}}{(U_-)_{5,2}}\oplus (U_-)_{5,2}.
\]
Moreover, one can check that there is an isomorphism of $L_0$-modules $\psi:(U_-)_{d,k}/(U_-)_{d,k+1} \rightarrow \Sym^k({\mathbb C^5}^*)\otimes {\inlinewedge}^{d-2k}(\inlinewedge^2\mathbb C^5)$: this isomorphism is simply given by extending multiplicatively the following formulas
\[
\psi(\partial_i)=x_i^*,\,\,\psi(d_{ij})=x_{ij}.
\]
and so we have that
\[
(U_-)_d\cong\bigoplus_{k< d/2} \Sym^k({\mathbb C^5}^*)\otimes \displaywedge^{d-2k}({\displaywedge}^2\mathbb C^5)
\]
as $L_0$-modules. The main goal of this section is to explicitly construct such isomorphism.
We need some further technical notation. If $1\leq i,j\leq 5$ we let $\overline{(i,j)}=(j,i)$. There is a natural action of $B_d$, the Weyl group of type $B$ and rank $d$, on $\mathcal I_d$ that can be described in the following way.
If $w=(\eta_1\sigma_1,\ldots,\eta_d \sigma_d)\in B_d$, where $\sigma=(\sigma_1,\ldots,\sigma_d)$ is a permutation of $[d]$ and $\eta_j=\pm 1$ for all $j\in [d]$, we let
\[
w(I)=J
\]
where
\[
J_j=\begin{cases}
I_{\sigma_j}& \textrm{if $\eta_j=1$}\\ \overline{I_{\sigma_j}}& \textrm{if $\eta_j=-1$}.
\end{cases}
\]
The fact that this is a $B_d$-action is an easy verification and is left to the reader.
We let $\mathcal S_d$ be the set of subsets of $[d]$ of cardinality 2, so that $|\mathcal S_d|=\binom{d}{2}$.
Note that elements in $\mathcal I_d$ are ordered tuples of ordered pairs, while elements in $\mathcal S_d$ are unordered tuples of unordered pairs.
If $\{k,l\}\in \mathcal S_d$ and $I\in \mathcal I_d$ we let $t_{I_k,I_l}=t_{i_k,j_k, i_l,j_l}$ and $
\varepsilon_{I_k,I_l}=\varepsilon_{i_k,j_k, i_l,j_l}
$ (see Section \ref{S1}).
Note that the definitions of $t_{I_k,I_l}$ and $ \varepsilon_{I_k,I_l}$ do not depend on the order of $k$ and $l$ but only on the set $\{k,l\}$. We also let
\[
D_{\{k,l\}}(I)=\frac{1}{2} (-1)^{l+k} \varepsilon_{I_k,I_l}\partial_{t_{I_k,I_l}}\in (U_-)_2. \]
For example, if $I=((1,2),(2,3),(3,5))\in \mathcal I_3$ then $D_{\{1,3\}}(I)=\frac{1}{2} (-1)^4 \varepsilon_{12354}\partial_4=-\frac{1}{2}\partial_4$.
\begin{definition}
A subset $S$ of $\mathcal S_d$ is \emph{self-intersection free} if its elements are pairwise disjoint.\end{definition}
For example $S=\{\{1,3\},\{2,5\}, \{4,7\}\}$ is self-intersection free while $\{\{1,3\},\{2,5\}, \{3,7\}\}$ is not. We denote by $\textrm{SIF}_d$ the set of self-intersection free subsets of $\mathcal S_d$.
\begin{definition}
Let $\{k,l\}, \{h,m\}\in \mathcal S_d$ be disjoint. We say that $\{k,l\}$ and $\{h,m\}$ \emph{cross} if exactly one element in $\{k,l\}$ is between $h$ and $m$. If $S\in \textrm{SIF}_d$ we let the crossing number $c(S)$ of $S$ be the number of pairs of elements in $S$ that cross.
\end{definition}
For example, if $S=\{\{1,3\},\{2,5\}, \{4,7\}\}$ then $\{1,3\}$ and $\{2,5\}$ cross, $\{1,3\}$ and $\{4,7\}$ do not cross, and $\{2,5\}$ and $\{4,7\}$ cross, so the crossing number of $S$ is $c(S)=2$ (see Figure \ref{figcross} for a graphical interpretation).
\begin{figure}
\caption{A graphical interpretation of the crossing number}
\label{figcross}
\end{figure}
\begin{definition}
Let $S=\{S_1,\ldots,S_r\}\in \textrm{SIF}_d$. We let
\[
D_S(I)=\prod_{j=1}^rD_{S_j}(I)\in (U_-)_{2r}
\]
if $r\geq 2$ and $D_{\emptyset}(I)=1$ (note that the order of multiplication is irrelevant as the elements $D_{S_j}(I)$ commute among themselves).
\end{definition}
\begin{definition}
For $I=(I_1,\ldots,I_d)\in \mathcal I_d$ and $S=\{S_1,\ldots,S_r\}\in \textrm{SIF}_d$ we let $C_S(I)\in \mathcal I_{d-2r}$ be obtained from $I$ by removing all $I_j$ such that $j\in S_k$ for some $k\in [r]$.
\end{definition}
For example, if $d=7$ and $S=\{\{1,4\},\{2,7\}\}$ then $C_S(I)=(I_3,I_5,I_6)$.
We are now ready to give the main definition of this section.
\begin{definition}\label{defomega}
For all $I\in \mathcal I_d$ we let
\[
\omega_I=\sum_{S\in \textrm{SIF}_d}(-1)^{c(S)}D_S(I)\,d_{C_S(I)}\in (U_-)_d.
\]
\end{definition}
For example, if $I=(21,13,45,25)\in \mathcal I_4$ we have
\begin{itemize}
\item $D_{\emptyset}(I)=1$;
\item $D_{\{1,3\}}(I)=-\frac{1}{2}\partial_3$;
\item $D_{\{2,3\}}(I)= +\frac{1}{2}\partial_2$;
\item $D_{\{2,4\}}(I)=+\frac{1}{2}\partial_4$;
\item $D_{\{1,3\},\{2,4\}}(I)=D_{\{1,3\}}(I)D_{\{2,4\}}(I)=-\frac{1}{4}\partial_3\partial_4$
\end{itemize}
and all other $D_S(I)$ vanish. We also have, $c(\{\{1,3\},\{2,4\}\})=1$ so
\[
\omega_I=d_I-\frac{1}{2}\partial_3d_{13}d_{25}+\frac{1}{2}\partial_2d_{21}d_{25}+\frac{1}{2}\partial_4 d_{21}d_{45}+\frac{1}{4}\partial_3\partial_4.
\]
\begin{proposition}
For all $I\in \mathcal I_d$ and all $g\in B_d$ we have
\[
\omega_{g(I)}=(-1)^{\ell(g)}\omega_I,
\]
where $\ell(g)$ is the length of $g$ with respect to the Coxeter generators $\{s_0,s_1,s_2,\ldots,s_{d-1}\}$, with $s_0=(-1,2,3,\ldots,d)$ and $s_1,\ldots,s_{d-1}$ the usual simple transpositions.
\end{proposition}
\begin{proof}
It is enough to verify the statement for $g\in \{s_0,\ldots,s_{d-1}\}$. If $g=s_0$ we have, for all $k,l$, $1\leq k,l\leq d$:
\begin{itemize}
\item $\varepsilon_{s_0(I)_k,s_0(I)_l}=(-1)^{\chi_{1\in \{k,l\}}}\varepsilon_{I_k,I_l}$;
\item $t_{s_0(I)_k,s_0(I)_l}=t_{I_k,I_l}$;
\end{itemize}
hence $D_S(s_0(I))=(-1)^{\chi_{1\in S}}D_S(I)$ while $d_{\mathcal C_{S}(s_0(I))}=(-1)^{\chi_{1\notin S}}d_{\mathcal C_{S}(I)}$, and
therefore we have
\begin{align*} \omega_{s_0(I)}&= \sum_{S\in SIF_d}(-1)^{c(S)}D_S(s_0(I))d_{C_S(s_0(I))}\\
&= \sum_{S\in SIF_d}(-1)^{c(S)} (-1)^{\chi_{1\in S}}D_S(I) (-1)^{\chi_{1\notin S}}d_{\mathcal C_{S}}(I)\\
&=-\omega_I.
\end{align*}
Now let $h\in \{1,\ldots,d-1\}$ and, for notational convenience, let $\sigma=s_h$. We have:
\begin{itemize}
\item $\varepsilon_{{\sigma}(I)_k,\sigma(I)_l}=\varepsilon_{I_{\sigma(k)},I_{\sigma(l)}}$;
\item $t_{{\sigma}(I)_k,\sigma(I)_l}=t_{I_{\sigma(k)},I_{\sigma(l)}}$;
\item $(-1)^{k+l}=(-1)^{{\sigma}(k)+{\sigma}(l)+\chi_{h\in \{k,l\}}+\chi_{h+1\in \{k,l\}}}$
\end{itemize}
hence $D_S({\sigma}(I))=(-1)^{\chi_{h\in S}+\chi_{h+1\in S}}D_{{\sigma}(S)}(I)$, where ``$h\in S$'' means that $h$ belongs to some element of $S$. We also observe that
\[
(-1)^{c(S)}=(-1)^{c({\sigma}(S))}(-1)^{\chi_{h\in S,\,h+1\in S,\,\{h,h+1\}\notin S}}
\]
i.e. the parity of the crossing number of $S$ is opposite to the parity of the crossing number of ${\sigma}(S)$ precisely if $h$ and $h+1$ belong to two distinct elements of $S$.
Moreover we observe that $d_{C_S({\sigma}(I))}=d_{C_{{\sigma}(S)}(I)}$ if $h$ or $h+1$ belong to $S$. If $h,h+1$ do not belong to $S$ we have
\[
d_{C_S({\sigma}(I))}=-d_{C_{{\sigma}(S)}(I)}-2D_{\{h,h+1\}}(I)d_{C_{{\sigma}(\tilde S)}(I)}
\]
where $\tilde S$ is obtained from $S$ by adding the pair $\{h,h+1\}$. We are now ready to compute $\omega_{{\sigma}(I)}$. We have
\begin{align*}
\omega_{{\sigma}(I)}&=\sum_{S\in SIF_d}(-1)^{c(S)}D_S({\sigma}(I))d_{C_S({\sigma}(I))}\\
&= \sum_{S\ni h\,or\, S\ni h+1 \,but\, S\not \ni \{h,h+1\}} (-1)^{c(S)}D_S({\sigma}(I))d_{C_S({\sigma}(I))}\\
&\hspace{3mm}+\sum_{S \not \ni h,h+1}\Big((-1)^{c(S)}D_S({\sigma}(I))d_{C_S(s_0(I))}+(-1)^{c(\tilde S)}D_{\tilde S}({\sigma}(I))d_{C_ {\tilde S}(\sigma(I))}\Big)\\
&= \sum_{S\ni h\,or\, S\ni h+1 \,but\, S\not \ni \{h,h+1\}} (-1)^{\chi_{h\in S,\,h+1 \in S}}(-1)^{c({\sigma}(S))}(-1)^{\chi_{h\in S}+\chi_{h+1\in S}} D_{{\sigma}(S)}(I)d_{C_{{\sigma}(S)}(I)}\\
&\hspace{3mm}+ \sum_{S \not \ni h,h+1}\Big( (-1)^{c({\sigma}(S))}D_{{\sigma}(S)}(I)(-d_{C_{{\sigma}(S)}(I)}-2D_{\{h,h+1\}}(I)d_{C_{{\sigma}(\tilde S)}(I)})+(-1)^{c({\sigma}(\tilde S))}D_{{\sigma}(\tilde S)}(I)d_{C_{\sigma(\tilde S)}(I)}\Big)\\
&=-\sum_{S\ni h\,or\, S\ni h+1 \,but\, S\not \ni \{h,h+1\}} (-1)^{c({\sigma}(S))} D_{{\sigma}(S)}(I)d_{C_{{\sigma}(S)}(I)}-\sum_{S \not \ni h,h+1}(-1)^{c({\sigma}(S))}D_{{\sigma}(S)}(I)d_{C_{{\sigma}(S)}(I)}\\
&\hspace{3mm}+\sum _{S \not \ni h,h+1}\Big( (-2)(-1)^{c({\sigma}(S))}D_{{\sigma}(S)}(I)D_{\{h,h+1\}}(I)d_{C_{{\sigma}(\tilde S)}(I)})+(-1)^{c({\sigma}(\tilde S))}D_{{\sigma}(\tilde S)}(I)d_{C_{\sigma(\tilde S)}(I)}\Big)\\
&=-\sum_{S\in SIF_d}(-1)^{c(\sigma(S))}D_{\sigma(S)}(I)\, d_{C_{\sigma(S)}(I)}\\
&=-\omega_I
\end{align*}
where we used that $D_{{\sigma}(S)}(I)D_{h,h+1}(I)=D_{\sigma(\tilde S)}(I)$ and $(-1)^{c({\sigma}(S))}=(-1)^{c({\sigma}(\tilde S))}$.
\end{proof}
\begin{corollary}
If $I=(I_1,\ldots,I_d)$ is such that $I_j= I_k$ for some $j<k$, then $\omega_I=0$; if $I_j=\overline I_k$ for some $j\leq k$, then $\omega_I=0$.
\end{corollary}
Now we want to study the action of $L_0$ on the elements $\omega_I$.
If $I=(I_1,\ldots,I_d)$ and $r$ appears once in $I_b$ for some $b$ we let $I^{b,s,r}$ be the sequence obtained from $I$ by substituting the letter $r$ in $I_b$ by $s$. We want to prove the following
\begin{theorem}\label{act} Let $I\in \mathcal I_d$ and $r,s\in [5]$, $r\neq s$.
Assume that the letter $r$ appears in $I_1,\ldots,I_c$, once in each pair, and does not appear in $I_{c+1},\ldots,I_{d}$. Then
\[
x_s\partial _r. \omega_I=\sum_{b=1}^c \omega_{I^{b,s,r}}.\]
\end{theorem}
\begin{proof} For notational convenience, since $r$ and $s$ are fixed in this proof, we simply let $I^b=I^{b,s,r}$ for all $1\leq b\leq c$.
We start by calculating the left-hand side. We have
\begin{align*}
x_s\partial _r. \omega_I&= x_s \partial _r. \sum_S(-1)^{c(S)} D_S(I) d_{C_S(I)}.
\end{align*}
Now we observe that $x_s\partial_r. D_{\{k,l\}}(I)$ is non zero if and only if $I_k$ and $I_l$ have the four indices distinct from $s$, hence $k$ and $l$ cannot be both less than or equal to $c$
or both strictly greater than $c$. We then assume that $k\leq c$ and $l>c$; in this case we have
\[
x_s\partial_r. D_{\{k,l\}}(S)=x_s \partial _r. \Big(\frac{1}{2}(-1)^{k+l}\varepsilon_{I_k,I_l}\partial_{t_{I_k,I_l}}\Big)=\frac{1}{2}(-1)^{k+l+1}\varepsilon_{I_k,I_l}\partial_r.
\]
So we have
\begin{align*}
x_s\partial _r \omega_I=& \sum_{k\leq c<l,\,s\notin I_k,\,s\notin I_l}\frac{1}{2}(-1)^{k+l+1}\varepsilon_{I_k,I_l}\partial_r\sum_{S\not \ni k,l}(-1)^{c(S)}D_S(I)d_{C_{S\cup \{k,l\}}(I)}+\\
&+\sum_S (-1)^{c(S)}D_S(I) \sum_{b\leq c,\,b\notin S}d_{C_S(I^b)}.
\end{align*}
Now we compute the right-hand side:
\begin{align*}
\sum_{b\leq c} \omega_{I^{b}}=\sum_{b\leq c} \sum_{S}(-1)^{c(S)}D_S(I^b)d_{C_S(I^b)}.
\end{align*}
Now we observe that if $b\notin S$ we have $D_S(I^b)=D_S(I)$ and so we reduce to prove the following:
\[
\sum_{k\leq c<l,\,s\notin I_k,\,s\notin I_l}\frac{1}{2}(-1)^{k+l+1}\varepsilon_{I_k,I_l}\partial_r\sum_{S\not \ni k,l}(-1)^{c(S)}D_S(I)d_{C_{S\cup \{k,l\}}(I)}=\sum_{S,b: b\leq c,\,b\in S} (-1)^{c(S)}D_S(I^b)d_{C_S(I^b)}
\]
We notice that if $\{b,b'\}\in S$ with both $b,b'\leq c$ then $D_S(I^b)=-D_S(I^{b'})$ hence we reduce to prove that
\begin{align*}
\sum_{k\leq c<l,\,s\notin I_k,\,s\notin I_l}\frac{1}{2}(-1)^{k+l+1}\varepsilon_{I_k,I_l}\partial_r\sum_{S\not \ni k,l}&(-1)^{c(S)}D_S(I)d_{C_{S\cup \{k,l\}}(I)}
=\\
&=\sum_{b\leq c<l}\sum_ {S:\, S\not\ni b,l} (-1)^{c(S)}D_{\{b,l\}}(I^b)D_S(I^b)d_{C_{S\cup\{b,l\}}(I^b)}.
\end{align*}
Finally, in order to prove this last equation we observe that if $b\leq c<l$ then $D_{\{b,l\}}(I^b)$ is nonzero only if $s\notin I_b,I_l$, that in this case $\varepsilon_{(I^b)_b,(I^b)_l}=-\varepsilon_{I_b,I_l}$, that $D_{\{b,l\}}(I^b)= -\frac{1}{2}(-1)^{b+l}\varepsilon_{I_b,I_l}\partial _r$ and that $d_{\mathcal C_{S\cup \{b,l\}}(I^{b})}=d_{\mathcal C_{S\cup \{b,l\}}(I)}$.
The proof is complete.
\end{proof}
If $I=(I_1,\ldots,I_d)$ with $I_k=(i_k,j_k)$ we let
\[
D_{s\rightarrow r}(\omega _I)=\delta_{r,i_{1}}\omega_{((s,j_{1}),I_2,\ldots,I_d)}+\delta_{r,j_{1}}\omega_{((i_{1},s),I_2,\ldots,I_d)}+\delta_{r,i_{2}}\omega_{(I_1,(s,j_{2}),I_3,\ldots,I_d)}+\cdots+\delta_{r,j_{d}}\omega_{(I_1,\ldots,I_{d-1},(i_{d},s))}.
\]
\begin{corollary}\label{omegaction}
Let $I=(I_1,\ldots,I_d)$ be arbitrary. Then
\[
x_s\partial_r \omega_I= D_{s\rightarrow r}(\omega _I).
\]
\end{corollary}
\begin{proof}
If there exists $k$ such that $i_{k}=j_{k}$ then $\omega_I=0$ and clearly also $D_{s\rightarrow r}(\omega _I)=0$ since all summands in the definition above vanish except possibly two of them which cancel out.
If such $k$ does not exist let $w\in B_d$ be such that $J=w(I)$ satisfies the following property: there exists $0\leq c\leq d$ such that $r$ appears in $J_1,\ldots,J_c$ and does not appear in $J_{c+1},\ldots,J_d$. By Theorem \ref{act} we know that the result holds for $J$ hence the result follows since $D_{s\rightarrow r}$ commutes with the action of $B_d$ (we leave this to the reader).
\end{proof}
\begin{corollary}\label{basi}
The map
\[
\varphi:\bigoplus_{k}\Sym^k({\mathbb C^5}^*)\otimes \displaywedge^{d-2k}(\displaywedge^2 \mathbb C^5)\rightarrow (U_-)_d
\]
given by
\[
\varphi(x_{t_1}^*\cdots x_{t_k}^*\otimes x_{i_1 j_1}\wedge \cdots\wedge x_{i_{d-2k} j_{d-2k}})=\partial_{t_1}\cdots \partial_{t_k} \omega_{(i_1,j_1),\ldots,(i_{d-2k},j_{d-2k})}
\]
for all $k\leq d/2$ and $t_1,\dots,t_k, i_1, j_1,\dots, i_{d-2k}, j_{d-2k}\in [5]$ is an isomorphism of $L_0$-modules, hence the set
$$\bigcup_{k\leq d/2}\{\partial_T\omega_I~|~ T=(t_1,\dots,t_k)\in[5]^k, t_1<\dots<t_k, I\in{\mathcal{I}}_{d-2k}/B_{d-2k}\}$$
is a basis of $(U_-)_d$.
\end{corollary}
\section{Properties of the dominance order}\label{S6}
In this section we establish simple combinatorial criteria to determine whether the weights of vectors in $U_-$ and $(U_-)^*$ are comparable.
\begin{remark}\label{thetaction} If $\varphi:M(V)\rightarrow M(W)$ is a linear map of degree $d$ which satisfies condition (a) of Proposition \ref{morphisms} let $\psi:(U_-)_d^*\rightarrow \Hom(V,W)$ be as in Remark \ref{dual}. By Corollary \ref{basi} we can identify $(U_-)_d^*$ with $\bigoplus_{k}\Sym^k({\mathbb \mathbb{C}^5})\otimes \inlinewedge^{d-2k}(\inlinewedge^2 \mathbb (\mathbb{C}^5)^*)$ and we let for all $T=(t_1,\dots,t_k)\in [5]^k$ and
$I=(I_1,\dots, I_{d-2k})\in \mathcal{I}_{d-2k}$ with $I_h=(i_h,j_h)$,
\[\theta_I^T=\psi(x_{t_1}\cdots x_{t_k}\otimes x_{i_1 j_1}^*\wedge \cdots\wedge x_{i_{d-2k} j_{d-2k}}^*).
\]
We observe that $\theta_{g(I)}^T=(-1)^{\ell(g)}\theta_I^T$ for every $g\in B_{d-2k}$ hence $\partial_T\omega_I\otimes \theta_I^T$ is invariant
with respect to the action of $B_{d-2k}$ on $I$. We can thus write
$$\Phi=\sum_{\substack{T=(t_1,\ldots,t_k):\\1\leq t_1\leq \cdots \leq t_k\leq 5}}\sum_{I\in {\mathcal I}_{d-2k}/B_{d-2k}}\partial_T\omega_I\otimes\theta_{I}^T.$$
Moreover, we have:
\[x_s\partial_r.\theta_I^T=x_s\partial_r.\theta_{I_1,\dots,I_{d-2k}}^{t_1,\dots,t_k}=\sum_{h=1}^k\Delta_{s\rightarrow r}^{h}\theta_{I}^T
-\sum_{l=1}^{d-2k}D_{r\rightarrow s}^{l}\theta_I^T\]
where $\Delta_{s\rightarrow r}^h(\theta_I^T)=\delta_{r,t_h}\theta_I^{t_1,\dots,t_{h-1},s,t_{h+1},\dots, t_k}$ and
\[D_{r\rightarrow s}^l(\theta_I^T)=\delta_{i_l,s}\theta^T_{I_1,\dots,I_{l-1},(r,j_l),I_{l+1},\dots, I_{d-2k}}+\delta_{s,j_l}\theta^T_{I_1,\dots,I_{l-1},(i_l,r),I_{l+1},\dots, I_{d-2k}}.\]
\end{remark}
We now study the dominance order on the weights of the elements $d_I$, $\omega_I$ and $\theta_I^T$. This will turn out to play a fundamental role in the study of morphisms of
Verma modules.
We observe that $d_{kl}$ is a weight vector for $L_0$. Indeed we have:
\[[h_{ij},d_{kl}]=(\delta_{i,k}+\delta_{i,l}-\delta_{j,k}-\delta_{j,l})d_{k,l}
\]
and so $\lambda_{ij}(d_{kl})$ is the number of occurrences of $i$ minus the number of occurrences of $j$ in $\{k,l\}$.
If $I=(i_1,\ldots,i_d)$ is a sequence of integers and we let
\[
m_k(I)=|\{s\in [d]:\, i_s=k\}|\
\]
be the multiplicity of $k$ in $I$,
we have
\[
\lambda_{ij}(d_{kl})=m_i(k,l)-m_j(k,l).
\]
More generally, if $I=\{i_{1},j_{1},\ldots,i_{d},j_{d}\}$ and $d_I=d_{i_{1}j_{1}}\cdots d_{i_{d}j_{d}}$ we have
\[
\lambda_{ij}(d_I)=m_i(I)-m_j(I).
\]
In order to understand when the weights of $d_I$ and $d_K$ are comparable in the dominance order, we first observe that the weight of $d_I$ does not depend on the order of its entries.
If $I=(i_1,\ldots,i_{2d})$ we let $I_o=(i'_1,\ldots,i'_{2d})$ be the non decreasing reordering of $I$. We write $I\leq K$ if $i'_1\leq k'_1,\,\ldots, i'_{2d}\leq k'_{2d}$ and $I<K$ if $I\leq K$ and at least one of the previous inequalities is strict (notice that this is different that requiring $I\neq K$).
\begin{proposition}\label{dominance}For all $I,K\in \mathcal I_d$ we have $\lambda(d_I)\geq \lambda (d_K)$ if and only if $I\leq K$.
\end{proposition}
\begin{proof}
We can assume that $I=(i_1,\ldots,i_{2d})$ and $K=(k_1,\ldots,k_{2d})$ are such that $I=I_o$ and $K=K_o$.\
We express the difference of the weights as a linear combination of roots. First assume that all entries of $I$ and $K$ coincide except in position $r$ and that $i_r=h$ and $k_r={h+1}$.
We have $m_l(I)=m_l(K)$ for all $l\neq h,h+1$, $m_h(I)=m_h(K)+1$ and $m_{h+1}(I)=m_{h+1}(K)-1$. Therefore $\lambda_{l,l+1}(d_I)=\lambda_{l,l+1}(d_K)$ for all $l\neq h-1,h,h+1$, $\lambda_{h-1,h}(d_I)=\lambda_{h-1,h}(d_K)+1$, (if $h\neq 1$), $\lambda_{h,h+1}(d_I)=\lambda_{h,h+1}(d_K)-2$ and $\lambda_{h+1,h+2}(d_I)=\lambda_{h+1,h+2}(d_K)+1$ (if $h\neq 4$). Therefore
\[
\lambda(d_I)-\lambda(d_K)=\alpha_{h,h+1}.
\]
From this we can deduce that
\[
\lambda(d_I)-\lambda(d_K)=\alpha_{i_1,k_1}+\alpha_{i_2,k_2}+\cdots+\alpha_{i_{2d},k_{2d}}.
\]
In particular, if $i_1\leq k_1,\ldots,i_{2d}\leq k_{2d}$ then $\lambda(d_I)\geq \lambda(d_K)$. Now we assume that the inequalities $i_1\leq k_1,\ldots,i_{2d}\leq k_{2d}$ are not all satisfied and we let $r$ be minimum such that $i_r>k_r$. If we express $\lambda(d_I)-\lambda(d_K)$ as a linear combination of the simple roots then $\alpha_{k_r,k_{r+1}}$ necessarily appears with a negative coefficient and we are done.
\end{proof}
\begin{corollary}\label{dominancetheta} For all $I,K\in \mathcal I_d$ and all $T,R\in [5]^k$ we have:
\begin{itemize}
\item[(i)] $\lambda(\theta^T_I)\leq \lambda (\theta_K^T)$ if and only if $I\leq K$;
\item[(ii)] $\lambda(\theta^T_I)\geq \lambda (\theta_I^R)$ if and only if $T\leq R$.
\end{itemize}
\end{corollary}
\begin{proof} In order to prove $(i)$ it is sufficient to notice that $\lambda(\theta_I^T)=-(\lambda(\partial_T\omega_I))
=-\lambda(\partial_T)-\lambda(d_I)$ and then use Proposition \ref{dominance}.
In order to prove $(ii)$ it is convenient to introduce the following notation. For $t\in[5]$ we let $t^{(1)}<t^{(2)}< t^{(3)} <t^{(4)}$
such that $\{t, t^{(1)}, t^{(2)}, t^{(3)} ,t^{(4)}\}=[5]$ and,
for
$T=(t_1,\dots, t_k)\in[5]^k$, $T^c=(t_1^{(1)} t_1^{(2)}, t_1^{(3)} t_1^{(4)}, \dots, t_k^{(1)} t_k^{(2)}, t_k^{(3)} t_k^{(4)})\in\mathcal{I}_{2k}$.
Then it is enough to notice that $\lambda(\partial_T)=\lambda(d_{T^c})$ and that $T\leq R$ if and only if $T^c\geq R^c$. Then one can use $(i)$.
\end{proof}
\section{Duality}\label{six}
Consider a morphism $\varphi:M(V)\rightarrow M(W)$ of generalized Verma modules of degree $d$ associated to an element $\Phi\in (U_-)_d\otimes \Hom(V,W)$. We ask the natural question: does it exist a ``related'' morphism $\psi:M(W^*)\rightarrow M(V^*)$ of the same degree $d$? The first natural candidate to look at is the following: if $\Phi=\sum_i u_i\otimes \theta_i$, where $\{u_i~|~i\in I\}$ is any basis of $(U_-)_d$ and $\theta_i\in \Hom(V,W)$ then we can consider the linear map $\psi:M(W^*)\rightarrow M(V^*)$ associated to $\Psi=\sum_i u_i \otimes \theta_i^*$, where, for all $\theta\in \Hom(V,W)$ we denote by $\theta^*\in \Hom(W^*,V^*)$ the pull-back of $\theta$ given by $\theta^*(f)=f\circ \theta$ for all $f\in W^*$. One can easily check that the map $\psi$ does not depend on the chosen basis $\{u_i~|~i\in I\}$ of $(U_-)_d$. It turns out that for $d=1$ the map $\psi$ is also a morphism of $L$-modules, but this is not the case in general if the degree $d$ is at least 2.
In this section we develop some tools which will allow us to construct a morphism of $L$-modules $\psi:M(W^*)\rightarrow M(V^*)$ starting from a morphism $\varphi:M(V)\rightarrow M(W)$ of degree at most 3 and we conjecture that our construction provides such morphism in all degrees.
The main result that we will need is the following.
\begin{proposition}\label{lemdual}
Let $\theta_1,\ldots,\theta_r, \sigma_1,\ldots, \sigma_s\in \Hom(V,W)$ for some $L_0$-modules $V$, $W$, and let $z_1,\ldots,z_t\in L_0$. Let $a_i, b_{j,k}\in \mathbb C$ be such that
\[
\sum_i a_i \theta_i(v)+\sum_{j,k}b_{j,k}\big(z_k.(\sigma_j(v))+\sigma_j(z_k.v)\big)=0\in W
\]
for all $v\in V$. Then
\[
\sum_i a_i \theta^*_i(f)+\sum_{j,k}b_{j,k}\big(z_k.((-\sigma_j^*)(f))+(-\sigma_j^*)(z_k.f)\big)=0\in V^*
\]
for all $f\in W^*$.
\end{proposition}
\begin{proof}
For all $v\in V$ we have
\begin{align*}
\Big( \sum_i a_i &\theta^*_i(f)+\sum_{j,k}b_{j,k}\big(z_k.((-\sigma_j^*)(f))+(-\sigma_j^*)(z_k.f)\big)\Big)(v)\\
&=\sum_i a_i f(\theta_i(v))+\sum_{j,k}b_{j,k}\big(\sigma_j^*(f))(z_k.v)+(z_k.f)(-\sigma_j(v))\big)\\
&=\sum_i a_i f(\theta_i(v))+\sum_{j,k}b_{j,k}\big(f(\sigma_j(z_k.v))+f(z_k.(\sigma_j(v))\big)\\
&=f\Big( \sum_i a_i \theta_i(v)+\sum_{j,k}b_{j,k}\big(\sigma_j(z_k.v)+z_k.(\sigma_j(v)\big) \Big)\\
&=0.
\end{align*}
\end{proof}
\begin{remark}\label{lemdual2}
We will use Proposition \ref{lemdual} also in the following equivalent formulation: let $\theta_1,\ldots,\theta_r$, $\sigma_1,\ldots, \sigma_s\in \Hom(V,W)$ for some $L_0$-modules $V$, $W$ and $z_1,\ldots,z_t\in L_0$. Let $a_i, b_{j,k}\in \mathbb C$ be such that
\[
\sum_i a_i \theta_i(v)+\sum_{j,k}b_{j,k}\big(2z_k.(\sigma_j(v))-(z_k.\sigma_j)(v)\big)=0\in W
\]
for all $v\in V$. Then
\[
\sum_i a_i \theta^*_i(f)+\sum_{j,k}b_{j,k}\big(2z_k.((-\sigma_j^*)(f))-(z_k.(-\sigma_j^*))(f)\big)=0\in V^*
\]
for all $f\in W^*$.
\end{remark}
\begin{conjecture}\label{conjdual}
Let $\varphi:M(V)\rightarrow M(W)$ be a morphism of degree $d$ associated to $\Phi:=\sum_{T,I}{\partial_T\omega_I}\otimes \theta^T_I$ for some $\theta^T_I\in \Hom(V,W)$. Then the linear map $\psi:M(W^*)\rightarrow M(V^*)$ associated to $\Psi:=\sum_{T,I}\partial_T\omega_I\otimes (-1)^{\ell(T)}(\theta^T_I)^*$ is also a morphism of Verma modules, where
if $T\in[5]^k$, we let $\ell(T)=k$.
\end{conjecture}
In the following sections we will verify Conjecture \ref{conjdual} for morphisms of degree at most 3 as a straightforward application of Proposition \ref{lemdual}.
\begin{definition}
Let $\varphi:M(\lambda)\rightarrow M(\mu)$ be a morphism of Verma modules. The weight $\mu-\lambda$ is called the leading weight of $\varphi$.
\end{definition}
The reason of the terminology in the previous definition is motivated by the following observation.
\begin{remark}
Let $\varphi:M(\lambda)\rightarrow M(\mu)$ be a morphism of Verma modules of leading weight $\nu$. If $\varphi$ is associated to $\Phi=\sum_{i}u_i\otimes \theta_i$, where $\{u_i ~|~ i\in I\}$ is a basis of $(U_-)_d$ consisting of weight vectors, let $\theta_{i_0}$ be of maximal weight such that $\theta_{i_0}(s)\neq 0$ for a highest weight vector $s\in F(\lambda)$. Then $\theta_{i_0}(s)$ is a highest weight vector in $F(\mu)$ and so the weight of $\theta_{i_0}$ is the leading weight of $\varphi$. Therefore if $\varphi$ has leading weight $\nu$ the leading
term of the singular vector $\varphi(1\otimes s)$ is
$$\sum_{i: \lambda(\theta_i)=\nu}u_i\otimes \theta_i(s).$$
We also say that $\theta\in \Hom(V,W)$ has the leading weight of $\varphi$ if $\theta(s)\neq 0$ and the weight of $\theta$ is $\nu$. A general strategy to study a morphism $\varphi:M(V)\rightarrow M(W)$ is to understand elements $\theta\in \Hom(V,W)$ which have the leading weight of $\varphi$; in particular we will often show that there is no such morphism by showing that there is no $\theta\in \Hom(V,W)$ that may possibly have the leading weight of a morphism.
\end{remark}
Whenever Conjecture \ref{conjdual} holds the next result allows us to simplify the classification of morphisms.
\begin{remark}
Let $\varphi:M(V)\rightarrow M(W)$ and $\psi:M(W^*)\rightarrow M(V^*)$ be morphisms of Verma modules and let $\nu=(a,b,c,d)$ be the leading weight of $\varphi$. Then the leading weight of $\psi$ is $-\nu^*=-(d,c,b,a)$.
\end{remark}
\section{Morphisms of degree one}\label{S7}
In this section we classify morphisms of degree one between generalized Verma modules, slightly simplifying Rudakov's argument \cite{R}.
We let $C(a,b,c)$ be the set of cyclic permutations of $a,b,c$, i.e., $C(a,b,c)=\{(a,b,c), (b,c,a)$, $(c,a,b)\}$.
\begin{theorem}
Let $\varphi:M(V)\rightarrow M(W)$ be a linear map of degree one associated to
$$\Phi=\sum_{I\in{\mathcal I}_1/B_1}\omega_I\otimes \theta_I$$
such that $L_0.\Phi=0$. Then $\varphi$ is a morphism of Verma modules if and only if
for all distinct $a,b,c,p\in [5]$ and for all $v\in V$ we have
\begin{equation}
\sum_{(\alpha,\beta,\gamma)\in C(a,b,c)}
x_p\partial_\gamma. (\theta_{\alpha\beta}(v))=0.
\label{equazione1}
\end{equation}
\end{theorem}
\begin{proof}
By Proposition \ref{morphisms} it is enough to check when $x_pd_{pq}\Phi(v)=0$ for all $p,q\in [5]$.
For notational convenience we let $Q=(p,q)$ and $\{a,b,c,p,q\}=[5]$.
We have:
\begin{align*}
x_pd_Q\Phi(v)&=x_pd_Q\sum_{I\in{\mathcal I}_1/B_1}\omega_I\otimes \theta_I(v)=x_pd_Q\sum_{I\in{\mathcal I}_1/B_1}d_I\otimes \theta_I(v)\\
& =\sum_{I\in{\mathcal I}_1/B_1}\varepsilon_{Q,I}x_p\partial_{t_{Q,I}}.(\theta_I(v))=
\varepsilon_{pqabc}\sum_{(\alpha,\beta,\gamma)\in C(a,b,c)}
x_p\partial_\gamma. (\theta_{\alpha\beta}(v)).
\end{align*}
\end{proof}
\begin{remark} We point out that Equation (\ref{equazione1}) satisfies the hypotheses of Proposition \ref{lemdual} since in this case
\[x_p\partial_{\gamma}.(\theta_{\alpha\beta}(v))=\theta_{\alpha\beta}(x_p\partial_{\gamma}.v)\]
hence we can write
\[x_p\partial_{\gamma}.(\theta_{\alpha\beta}(v))=\frac{1}{2}\big(x_p\partial_{\gamma}.(\theta_{\alpha\beta}(v))+\theta_{\alpha\beta}(x_p\partial_{\gamma}.v)\big).\]
Conjecture \ref{conjdual} then holds in degree one. This will be also confirmed by Theorem \ref{gradouno}.
\end{remark}
\begin{proposition}\label{pesigrado1} Let $\varphi: M(\lambda)\rightarrow M(\mu)$ be a morphism of Verma modules of degree one and let
$\theta_{hk}$ have the leading weight of $\varphi$. Then if $i<j$ are distinct from $h,k$ we have
\[
\mu_{ij}=-\chi_{i<h<j}-\chi_{i<k<j}.
\]
\end{proposition}
\begin{proof}
Consider Equation (\ref{equazione1}) with $p=j$, $c=i$, $a=h$, $b=k$ and $v=s$ a highest weight vector in $F(\lambda)$:
\[
x_j\partial _i.(\theta_{hk}(s))+x_j\partial_k.(\theta_{ih}(s))+x_j\partial_h.(\theta_{ki}(s))=0.
\]
Now we apply $x_i\partial_j$ to this equation. We have
\begin{align*}
h_{ij}.(\theta_{hk}(s))-\chi_{i<k<j}\theta_{kh}(s)-\chi_{i<h<j}\theta_{kh}(s)=0
\end{align*}
and the result follows.
\end{proof}
\begin{theorem}\label{gradouno}
Let $\varphi: M(\lambda)\rightarrow M(\mu)$ be a morphism of Verma modules of degree one.
Then one of the following occurs:
\begin{itemize}
\item
$\lambda=(m,n+1,0,0)$, $\mu=(m,n,0,0)$ for some $m,n\geq 0$ and, up to a scalar, $\varphi=\nabla_A$.
\item
$\lambda=(m+1,0,0,n)$, $\mu=(m,0,0,n+1)$ for some $m,n\geq 0$ and, up to a scalar, $\varphi=\nabla_B$.
\item
$\lambda=(0,0,m,n)$, $\mu=(0,0,m+1,n)$ for some $m,n\geq 0$ and, up to a scalar, $\varphi=\nabla_C$.
\end{itemize}
\end{theorem}
\begin{proof} Let $\theta_{hk}$ have the leading weight of $\varphi$.
By Proposition \ref{pesigrado1} we have that if $(h,k)\neq (1,2),(1,5),(4,5)$ we can find $i,j$ such that $\mu_{i,j}<0$, a contradiction.
Proposition \ref{pesigrado1} also provides
\begin{itemize}
\item $\mu_{3,5}=0$ if $(h,k)=(1,2)$;
\item $\mu_{2,4}=0$ if $(h,k)=(1,5)$;
\item $\mu_{1,3}=0$ if $(h,k)=(4,5)$,
\end{itemize}
and the rest follows using Lemma \ref{esempi} and Proposition \ref{leading} recalling that $\lambda(\theta_{hk})=-\lambda(d_{hk})$.
\end{proof}
\section{Morphisms of degree 2}\label{eight}
In this section we provide a complete classification of morphisms between Verma modules of degree 2. We will make use of the following preliminary result which holds in a much wider generality.
Here and in what follows we denote by $(p,q,a,b,c)$ any permutation of $[5]$ and we set $Q=(p,q)$.
\begin{lemma} \label{diquaedila}
Suppose that $\Phi=\sum_{T,I}\partial_T\omega_I\otimes\theta^T_I$ defines a morphism of Verma modules $\varphi: M(V)\rightarrow M(W)$. Then for all $t_1\dots t_h\in [5]$,
$I_1,\ldots,I_k\in \mathcal I_1$ and $v\in V$ we have
\begin{align*}
&\sum_{I,J_1,\ldots,J_r\in\mathcal{I}_1}\varepsilon_{Q,I}x_p\partial_{t_{Q,I}}d_{J_1}\cdots d_{J_r}\otimes \theta^{t_1,\ldots,t_h}_{I_1,\ldots,I_k,I,J_1,\ldots,J_r}(v)
=2\sum_{\substack{(\alpha,\beta, \gamma)\in C(a,b,c) \\ H_1,\ldots,H_r\in\mathcal{I}_1}}\varepsilon_{pqabc}d_{H_1} \cdots d_{H_r}\otimes\\
& \Big(\theta^{t_1,\ldots,t_h}_{I_1,\ldots,I_k,\alpha\beta,H_1,\ldots,H_r}(x_p\partial_\gamma.v)+\sum_{s=1}^h \Delta_{p\rightarrow\gamma}^s\theta^{t_1,\ldots,t_h}_{I_1,\ldots,I_k,\alpha\beta,H_1,\ldots,H_r}(v)-\sum_{s=1}^kD_{\gamma\rightarrow p}^s\theta^{t_1,\ldots,t_h}_{I_1,\ldots,I_k,\alpha\beta,H_1,\ldots,H_r}(v)\Big)
\end{align*}
\end{lemma}
\begin{proof}
Using the definitions of $D_{a\rightarrow b}^h$, of $\Delta_{a\rightarrow b}^h,$ of $\theta^{t_1,\ldots,t_h}_{I_1,\ldots,I_k}$ and of the action of $L_0$ on the latter elements, we have
\begin{align*}
&\sum_{I,J_1,\ldots,J_r}\varepsilon_{Q,I}x_p\partial_{t_{Q,I}}d_{J_1}\cdots d_{J_r}\otimes \theta^{t_1,\ldots,t_h}_{I_1,\ldots,I_k,I,J_1,\ldots,J_r}(v)= -2\sum_{\substack{(\alpha,\beta, \gamma)
\in C(a,b,c) \\ H_1,\ldots,H_r\in\mathcal{I}_1}}\varepsilon_{pqabc}d_{H_1} \cdots d_{H_r}\otimes\\
&\Big((x_p \partial_\gamma. \theta^{t_1,\ldots,t_h}_{I_1,\ldots,I_k,\alpha\beta,H_1,\ldots,H_r})(v)-\sum_{s=1}^h \Delta_{p\rightarrow\gamma}^s\theta^{t_1,\ldots,t_h}_{I_1,\ldots,I_k,\alpha\beta,H_1,\ldots,H_r}(v)+\sum_{s=1}^kD^s_{\gamma\rightarrow p}\theta^{t_1,\ldots,t_h}_{I_1,\ldots,I_k,\alpha\beta,H_1,\ldots,H_r}(v)\\
&- x_p\partial_\gamma (\theta^{t_1,\ldots,t_h}_{I_1,\ldots,I_k,\alpha\beta,H_1,\ldots,H_r}(v))\Big)\\
\end{align*}
from which the thesis follows.
\end{proof}
We are now ready to state the following characterization result.
\begin{theorem}\label{equationdeg2}
Let $\varphi:M(V)\rightarrow M(W)$ be a linear map of degree 2 associated to
\[\Phi=\sum_{(I,J)\in \mathcal I_2/B_2}\omega_{I,J}\otimes \theta_{I,J}+\sum_{t=1}^5\partial_t\otimes \theta^t\]
such that $x.\Phi=0$ for all $x\in L_0$. Then $\varphi$ is a morphism of Verma modules if and only if for all $K\in \mathcal I_1$ and all $v\in V$ we have
\[
-\chi_{(K\in B_1 Q)}\theta^p(v)+\frac{1}{2}\varepsilon_{pqabc}\sum_{(\alpha \beta \gamma) \in C(a,b,c)}\Big(-\big((x_p\partial_\gamma).\theta_{\alpha\beta , K}\big)(v)+2x_p\partial_\gamma .(\theta_{\alpha \beta,K}(v))\Big)=0
\]
\end{theorem}
\begin{proof}
By Proposition \ref{morphisms} we have that $\varphi$ is a morphism of Verma modules if and only if
\[
x_pd_Q \Big(\sum_{(I,J)\in \mathcal I_2/B_2}\omega_{I,J}\otimes \theta_{I,J}(v)+\sum_{t}\partial_t\otimes \theta^t(v)\Big)=0
\]
for all $v\in V$. It is convenient for us to consider the first sum running over all $(I,J)\in \mathcal I_2$ and so we have
\begin{align}\label{aryu}
\nonumber x_pd_{Q}\Big(\frac{1}{8}\sum_{(I,J)\in\mathcal{I}_2}&\omega_{I,J}\otimes \theta_{I,J}(v)+\sum_t \partial _t\otimes \theta^t(v) \Big)\\
&=x_pd_{Q}\Big(\frac{1}{8}\sum_{I,J}(d_Id_J-\frac{1}{2}\varepsilon_{I,J}\partial_{t_{I,J}})\otimes \theta_{I,J}(v)+\sum_t \partial _t\otimes \theta^t(v) \Big).
\end{align}
We split Equation \eqref{aryu} into three parts:
In the first part of Equation \eqref{aryu} we have, using Lemma \ref{diquaedila},
\begin{align*}
x_pd_{Q}\sum_{I,J}d_Id_J\otimes \theta_{I,J}(v)&=\sum_{I,J}\Big(\varepsilon_{Q,I}(x_p \partial _{t_{Q,I}})d_J-\varepsilon_{Q,J}d_I (x_p \partial_{t_{Q,J}})\Big)\otimes \theta_{I,J}(v)\\
&=2\sum_{H}d_H \otimes \varepsilon_{pqabc} \sum_{\alpha\beta\gamma}\theta_{\alpha\beta,H}(x_p\partial_\gamma.v)\\
&-2\sum_{I}d_I\otimes \varepsilon_{pqabc}\sum_{\alpha \beta\gamma}(\theta_{I,\alpha\beta}(x_p\partial_\gamma.v)-D^1_{\gamma\rightarrow p}\theta_{I,\alpha\beta}(v))\\
&=4\sum_Hd_H\otimes \varepsilon_{pqabc} \sum_{\alpha \beta\gamma}\big(\theta_{\alpha\beta,H}(x_p\partial_\gamma.v)+\frac{1}{2}(x_p\partial_\gamma.\theta_{\alpha\beta,H})(v)\big)\\
&=4\sum_Hd_H\otimes \varepsilon_{pqabc} \sum_{\alpha \beta\gamma}\big(x_p\partial_\gamma.(\theta_{\alpha\beta,H}(v))-\frac{1}{2}(x_p\partial_\gamma.\theta_{\alpha\beta,H})(v)\big)
\end{align*}
where the sums run over $I,J,H\in\mathcal{I}_1$ and $(\alpha, \beta,\gamma)\in C(a,b,c)$.
In the second part of Equation \eqref{aryu} we have
\begin{align*}
\sum_{I,J}\frac{1}{2}\varepsilon_{I,J}\partial_{t_{I,J}}\otimes \theta_{I,J}(v)=0
\end{align*}
since the term indexed by $(I,J)$ cancels the term indexed by $(J,I)$.
In the third part of Equation \eqref{aryu} we have:
\begin{align*}
\sum_t x_pd_Q \partial _t\otimes \theta^t(v)=- d_Q\otimes \theta^p(v).
\end{align*}
Putting the three parts together Equation \eqref{aryu} becomes
\begin{align*}
& x_pd_Q\Big(\frac{1}{8}\sum_{I,J\in\mathcal{I}_1}\omega_{I,J}\otimes \theta_{I,J}(v)+\sum_t \partial _t\otimes \theta^t(v) \Big)\\
&=\sum_{K\in \mathcal I_1/B_1}d_K\otimes \Big(-\chi_{(K\in B_1Q)}\theta^p(v)+\varepsilon_{pqabc}\sum_{(\alpha \beta \gamma) \in C(a,b,c)}-\frac{1}{2}\big(x_p\partial_\gamma.\theta_{\alpha\beta , K}\big)(v)+x_p\partial_\gamma .(\theta_{\alpha \beta,K}(v))\Big)
\end{align*}
and the result follows.
\end{proof}
We deduce that Conjecture \ref{conjdual} holds for morphisms of degree 2 and in particular we have the following duality result for degree 2 morphisms.
\begin{corollary}\label{degree2dual}
Let $\varphi:M(V)\rightarrow M(W)$ be a morphism of Verma modules of degree 2 associated to
\[\Phi=\sum_{(I,J)\in \mathcal I_2/B_2}\omega_{I,J}\otimes \theta_{I,J}+\sum_t \partial_t \otimes \theta^t.\]
Then the linear map $\psi:M(W^*)\rightarrow M(V^*)$ associated to
\[
\Psi=\sum_{(I,J)\in \mathcal I_2/B_2}\omega_{I,J}\otimes \theta^*_{I,J}+\sum_t \partial_t \otimes (-\theta^t)^*
\]
is also a morphism of Verma modules.
\end{corollary}
\begin{proof}
This is an immediate consequence of Remark \ref{lemdual2} and Theorem \ref{equationdeg2}.
\end{proof}
\begin{corollary}\label{equationdegree2highest}
Let $\varphi:M(\lambda)\rightarrow M(\mu)$ be a morphism of Verma modules and $s\in F(\lambda)$ a highest weight vector. Then for all $K\in \mathcal I_1$ we have
\[
2 \chi_{K\in B_1Q}\varepsilon_{pqabc} \theta^p(s)+ \sum_{(\alpha\beta\gamma)\in C(abc)}\Big((-1)^{\chi_{p>\gamma}}(x_p\partial_{\gamma}.\theta_{\alpha\beta,K})(s)+2\chi_{p>\gamma}x_p\partial_\gamma. (\theta_{\alpha\beta,K}(s))\Big)=0
\]
\end{corollary}
\begin{proof}
This result immediately follows from Theorem \ref{equationdeg2} by observing that if $p<\gamma$ then $x_p\partial_\gamma.s=0$.
\end{proof}
In the following results we fix a morphism $\varphi:M(\lambda)\rightarrow M(\mu)$ of Verma modules of degree 2 associated to $\Phi=\sum \omega_{I,J}\otimes \theta_{I,J}+\sum \partial_t\otimes \theta^t$ and we exploit Corollary \ref{equationdegree2highest} to obtain some constraints on the weights $\lambda$ and $\mu$. The next result is analogous to Proposition \ref{pesigrado1}.
\begin{proposition}\label{proppesi1}Let $h,k,l,m \in [5]$ be such that $\theta_{hk,lm}$ has the leading weight of $\varphi$. Let $1\leq i<j\leq 5$ be such that $j\neq h,k,l,m $ and $i\neq h,k$. Then
\[
\mu_{ij}=-\chi_{i<h<j}-\chi_{i<k <j}.
\]
\end{proposition}
\begin{proof}By Corollary \ref{equationdegree2highest} used with $a=i$, $b=h$, $c=k$, $p=j$ and $K=(l,m)$, observing that $x_j\partial_\gamma.\theta_{\alpha\beta,K}=0$ for all $(\alpha,\beta,\gamma)\in C(i,h,k)$, we obtain the following relation
\[
x_j\partial_i.(\theta_{hk,lm}(s))+\chi_{h<j} x_j\partial_h .(\theta_{ k i ,lm }(s))+\chi_{k<j}x_j\partial_k.( \theta_{i h, kl }(s))=0.
\]
Applying $x_i\partial_j$ to this equation we have
\begin{align*}
h_{ij}.(\theta_{hk,lm}(s))&+\chi_{h<j}\big(x_i\partial_h.(\theta_{k i,lm }(s))-x_j\partial_h.(\theta_{k j,lm}(s))\big)\\
&+\chi_{k <j}\big(x_i\partial_k.(\theta_{ih,lm }(s))-x_j\partial_k.(\theta_{jh,lm }(s))\big)=0
\end{align*}
Since $\theta_{hk,lm}$ has the leading weight of $\varphi$, if $h<j$ we necessarily have $\theta_{k j,lm }(s)=0$, by Corollary \ref{dominancetheta}. Similarly, if $k<j$, we have $\theta_{jh,lm }(s)=0$. Therefore the previous equation becomes
\[
h_{ij}.(\theta_{hk,lm }(s))+\chi_{h<j}x_i\partial_h.(\theta_{k i,lm }(s))
+\chi_{k <j}x_i\partial_k.( \theta_{ih,lm }(s))=0
\]
Again, if $i>h$, we have $\theta_{k i,lm }(s)=0$ and otherwise we have $x_i\partial_h.(\theta_{k i,lm }(s))=-\theta_{kh,lm}(s)$ and similarly for the other term, and so we have
\[
h_{ij}.(\theta_{hk,lm}(s))-\chi_{h<j}\chi_{i<h}\theta_{kh,lm}(s)
-\chi_{k <j}\chi_{i<k }\theta_{kh,lm}(s)=0
\]
i.e.,
\[
h_{ij}.(\theta_{hk,lm}(s))=-(\chi_{i<h<j}+\chi_{i<k <j})\theta_{hk,lm}(s).
\]
\end{proof}
\begin{proposition}\label{proppesi3}
Let $i,h,k,l,m\in [5]$, with $i,h,k,m$ distinct and $i<m$, be such that $\theta_{hk,lm}$ has the leading weight of $\varphi$. Then
\begin{align*}
&h_{im}.( \theta_{hk,lm}(s))=\\
&\Big(\frac{1}{2}-\chi_{i<h<m}-\chi_{i<k<m}\Big)\theta_{hk,lm}(s)-\varepsilon_{mlhki}\theta^{i}(s)-\frac{1}{2}\Big((-1)^{\chi_{h<m}}\theta_{hl,km}(s)+(-1)^{\chi_{k<m}}\theta_{hm,kl}\Big).
\end{align*}
\end{proposition}
\begin{proof}
We consider Corollary \ref{equationdegree2highest} with $a=h$, $b=k$, $c=i$, $p=m$ and $K=(l,m)$. We observe that \[\varepsilon_{pqabc}\chi_{K\in B_1Q}=\varepsilon_{mqhki}\chi_{l=q}=\varepsilon_{mlhki}\] and so we obtain
\begin{align*}
\varepsilon_{mlhki}\theta^m(s)
&+\frac{1}{2}\Big((-1)^{\chi_{h<m}}\theta_{k i ,hl}(s)
+(-1)^{\chi_{k<m}}\theta_{i h,kl}(s)
-\theta_{hk,il}(s)\Big)\\
&+\chi_{h<m}x_m \partial_h.( \theta_{k i ,lm}(s))
+\chi_{k<m}x_m \partial_k. (\theta_{i h ,lm}(s))
+x_m \partial_i. (\theta_{hk,lm}(s))=0
\end{align*}
We apply $x_i\partial_m$ to this equation and we obtain
\begin{align*}
\varepsilon_{mlhki}\theta^i(s)
&-\frac{1}{2}\Big((-1)^{\chi_{h<m }}\theta_{km,hl}(s)
+(-1)^{\chi_{k<m}}\theta_{mh,kl}(s)
+\theta_{hk,ml}(s)\Big)\\
&-\chi_{i<h <m }\theta_{kh,lm}(s)
-\chi_{i<k <m } \theta_{kh ,lm}(s)
+h_{im}.(\theta_{hk,lm}(s))=0
\end{align*}
and the result follows.
\end{proof}
\begin{proposition}\label{proppesi2}
Let $h,k,m,i\in [5]$ be distinct, $i<m$, be such that $\theta_{hk,hm}$ has the leading weight of $\varphi:M(\lambda)\rightarrow M(\mu)$. Then
\[
\mu_{i,m}=\chi_{k<m}-\chi_{i<h<m}-\chi_{i<k<m}
\]
and
\[
\lambda_{i,m}=\chi_{k<m}-\chi_{i<h<m}-\chi_{i<k<m}-1.
\]
\end{proposition}
\begin{proof}
We use Proposition \ref{proppesi3} with $l=h$ and deduce
\begin{align*}
h_{im}.(\theta_{hk,hm}(s))&=\Big(\frac{1}{2}-\chi_{i<h<m}-\chi_{i<k<m}\Big)\theta_{hk,hm}(s)-\frac{1}{2}(-1)^{\chi_{k<m}}\theta_{hm,kh}\\
&=\Big(\frac{1}{2}-\frac{1}{2}(-1)^{\chi_{k<m}}-\chi_{i<h<m}-\chi_{i<k<m}\Big)\theta_{hk,hm}(s)\\
&=(\chi_{k<m}-\chi_{i<h<m}-\chi_{i<k<m}\Big)\theta_{hk,hm}(s).
\end{align*}
and the first part of the statement follows. The second part is an easy consequence since
\[
\lambda_{i,m}(\theta_{hk,hm})=1.
\]
\end{proof}
\begin{theorem}\label{teorema2} Let $\varphi:M(\lambda)\rightarrow M(\mu)$ be a morphism of degree 2.
Then one of the following occurs:
\begin{enumerate}
\item
$\lambda=(1,0,0,n)$, $\mu=(0,0,1,n+1)$ for some $n\geq 0$ and, up to a scalar, $\varphi=\nabla_C\nabla_B$;
\item
$\lambda=(n+1,1,0,0)$, $\mu=(n,0,0,1)$ for some $n\geq 0$ and, up to a scalar, $\varphi=\nabla_B\nabla_A$;
\item
$\lambda=(0,1,0,0)$, $\mu=(0,0,1,0)$, and, up to a scalar, $\varphi=\nabla_C\nabla_A$.
\end{enumerate}
\end{theorem}
\begin{proof}
We first make the following observation that will allow us to simplify several arguments. If $\nu\in \Lambda$ is any weight, by Corollary \ref{degree2dual}, if the statement holds for all morphisms of leading weight $\nu$ then it holds also for all morphisms of leading weight $-\nu^*$.
We let $s$ be a highest weight vector of $F(\lambda)$ and we suppose that $\theta_{hk,lm}$ has the leading weight of $\varphi$. Let us first assume $|\{h,k,l,m\}|=3$ i.e., without loss of generality, $h=l$.
By Corollary \ref{equationdegree2highest} with $K=(p,a)$
we have:
\begin{align}
\nonumber-((-1)^{\chi_{b<p}}&+(-1)^{\chi_{c<p}})\theta_{ab,ca}(s)\\+
&2\chi_{a<p}x_p\partial_a.(\theta_{bc,pa}(s))
+2\chi_{b<p}x_p\partial_b.(\theta_{ca,pa}(s))+2\chi_{c<p}x_p\partial_c.(\theta_{ab,pa}(s))=0.
\label{equation2m=a}
\end{align}
Using this equation with $a=h$, $b=k$, $c=m$, since $\theta_{hk,hm}$ has the leading weight of $\varphi$, we immediately obtain
\[
((-1)^{\chi_{k<p}}+(-1)^{\chi_{m<p}})\theta_{hk,hm}(s)=0.
\]
In particular, if we can choose $p$ such that $p>k,m$ or $p<k,m$ we have $\theta_{hk,hm}(s)=0$, a contradiction.
So we reduce to study the following cases: (a) $k=1, m=5$; (b) $k=2, m=5, h=1$; (c) $k=1, m=4, h=5.$
\begin{enumerate}
\item [(a)] By duality, since $\lambda(\theta_{21,25})=-(\lambda(\theta_{41,45}))^*$, it is enough to consider only the cases $h=2,3$; we have, by Proposition \ref{proppesi1},
\[
\mu_{14}=-\chi_{1<h<4}-\chi_{1<5<4}=-1,
\]
a contradiction.
\item [(b)] In this case we have, by Proposition \ref{proppesi1}
\[
\mu_{23}=-\chi_{2<1<3}-\chi_{2<5<3}=0
\]
and by Proposition \ref{proppesi2} we have
\[
\mu_{35}=\chi_{2<5}-\chi_{3<1<5}-\chi_{3<2<5}=1.
\]
Since the leading weight of $\varphi$ is $\lambda(\theta_{12,15})=(-1,-1,0,1)$ we conclude that $\mu=(n,0,0,1)$ for some $n\geq 0$ and so $\lambda=(n+1,1,0,0)$.
The leading term of the singular vector $\varphi(1\otimes s)$ is $\omega_{12,15}\otimes \theta_{12,15}(s)=d_{12}d_{15}\otimes \theta_{12,15}(s)$ hence, up to a scalar,
$\varphi=\nabla_B \nabla_A$ by Proposition \ref{leading}.
\item[(c)] Since $\lambda(\theta_{51,54})=-\lambda(\theta_{12,15})^*$ this follows from case (b) and we obtain in this case the morphism $\nabla_C\nabla_B$.
\end{enumerate}
This concludes the study of all possible $\theta_{hk,lm}$ having the leading weight of $\varphi$ with $h,k,l,m$ not distinct.
In order to deal with the case where $h,k,l,m$ are distinct we let $p$ be different from $h,k,l,m$.
If $p=4,5$ we apply Proposition \ref{proppesi1} with $i=1$ and $j=p$ and we get that $\mu_{1 p}<0$ hence $\theta_{hk,lm}$ does not have the leading
weight of $\varphi$. By Corollary \ref{equationdegree2highest} we also have $\theta^p(s)=0$ and so also $\theta^p$ can not have the leading weight of $\varphi$.
For $p=1$ we have $\lambda(\theta^1)=-\lambda(\theta^5)^*$ and if $p=2$ we have $\lambda(\theta^2)=-\lambda(\theta^4)^*$ and so these cases follows from the previous discussion
by Corollary \ref{degree2dual}.
For $p=3$ Proposition \ref{proppesi1} with $i=1$, $j=3$ shows that $\theta_{14,25}$ and $\theta_{15,24}$ cannot have the leading weight of $\varphi$,
i.e.\ $\theta_{14,25}(s)=\theta_{15,24}(s)=0$, and
that if $\theta_{12,45}$ has leading weight then $\mu_{1,3}=0$. Besides, by Corollary \ref{equationdegree2highest},
$\theta_{12,45}(s)=2\theta^3(s)$.
By Proposition \ref{proppesi3} we immediately get
\[
h_{35}.(\theta_{12,45}(s))=\theta_{12,45}(s)
\]
and so $\mu_{3,5}=1$.
Since the leading weight is $\lambda(\theta_{12,45})=(0,-1,1,0)$ we conclude that $\mu=(0,0,1,0)$ and so $\lambda=(0,1,0,0)$.
The leading term of $\varphi(1\otimes s)$ is
$$\omega_{12,45}\otimes \theta_{12,45}(s)+\partial_3\otimes\theta^3(s)=d_{12}d_{45}\otimes \theta^3(s)$$
hence, up to a scalar, $\varphi=\nabla_{C}\nabla_A$ by Proposition \ref{leading}.
\end{proof}
\section{Morphisms of degree 3}\label{nine}
This section is dedicated to the study of morphisms of Verma modules of degree three. We consider a linear map $\varphi: M(\lambda)\rightarrow M(\mu)$
of degree three associated to \[\Phi=\sum_{I\in{\mathcal I}_3/B_3}\omega_I\otimes \theta_I+\sum_{t\in[5], I\in{\mathcal I}_1/B_1}\partial_t\omega_I\otimes \theta^t_I.\]
As in the case of morphisms of degree one and two, our goal is to establish necessary and sufficient conditions to ensure that $\varphi$ is a morphism of Verma modules.
\begin{lemma}\label{fybj} If $x.\Phi=0$ for every $x\in L_0$, then
the following relation holds for every $v\in F(\lambda)$:
\[
\sum_{I\in \mathcal I_3}\omega_I\otimes \theta_I(v)=\sum_{I\in \mathcal I_3}d_I\otimes \theta_I(v).
\]
\end{lemma}
\begin{proof}
Indeed we have
\begin{align*}
\sum_{I\in \mathcal I_3}\omega_I\otimes \theta_I(v)&=\sum_{I\in \mathcal I_3}d_I\otimes \theta_I(v)\\
& +\sum_{I_1,I_2,I_3}(-\frac{1}{2}
\varepsilon_{I_1,I_2}\partial_{t_{I_1,I_2}}d_{I_3}+\frac{1}{2}
\varepsilon_{I_1,I_3}\partial_{t_{I_1,I_3}}d_{I_2}- \frac{1}{2}
\varepsilon_{I_2,I_3}\partial_{t_{I_2,I_3}}d_{I_1})\otimes \theta_{I_1,I_2,I_3}(v) \end{align*}
and the last sum vanishes since the coefficients of $\theta_{I_1,I_2,I_3}(v)$ and $\theta_{I_3,I_2,I_1}(v)$ coincide.
\end{proof}
\begin{theorem}\label{grado3} Let us assume that $x.\Phi=0$ for every $x\in L_0$. Then $\varphi$ is a morphism of Verma modules if and only if for every $H,L\in \mathcal{I}_1$, every permutation $(p,q,a,b,c)$ of $[5]$ and every $v\in F(\lambda)$, the following equations hold:
\begin{align}
\label{deg3n1}& \chi_{L\in B_1Q}\theta^p_{H}(v)+\frac{1}{2}\varepsilon_{pqabc}\sum_{(\alpha,\beta,\gamma)\in C(a,b,c)}
\big(-(x_p\partial_{\gamma}.\theta_{\alpha\beta,H,L})(v)+2 x_p\partial_\gamma. (\theta_{\alpha \beta, H,L}(v))\big)=0\\
\label{deg3n2}&\frac{1}{4}\theta_{ab,bc,cq}(v)+\frac{1}{4}\theta_{ac,cb,bq}(v)+\frac{1}{2}\varepsilon_{pqabc}\sum_{(\alpha,\beta,\gamma)\in C(a,b,c)}\big(-(x_p\partial_\gamma.\theta^a_{\alpha\beta})(v)+2x_p\partial_\gamma.(\theta^a_{\alpha\beta}(v))\big)=0\\
\label{deg3n3}& \sum_{(\alpha, \beta, \gamma)\in C(a,b,c)}x_p\partial_{\gamma}.(\theta^p_{\alpha \beta}(v))=0\\
\label{deg3n4}& \varepsilon_{pqabc}\sum_{(\alpha, \beta, \gamma)\in C(a,b,c)}x_p\partial_{\gamma}.(\theta^q_{\alpha \beta}(v))-\frac{1}{2}\theta_{ab,bc,ca}(v)=0.
\end{align}
\end{theorem}
\begin{proof} By Proposition \ref{morphisms} we need to compute $x_pd_Q\Phi(v)$ for $v\in F(\lambda)$. We compute the different summands separately.
Using Lemma \ref{fybj} and Lemma \ref{diquaedila} we have
\begin{align*} x_pd_Q &\sum_{(I,J,K)\in\mathcal{I}_3/B_3}\omega_{I,J,K}\otimes \theta_{I,J,K}(v)=
\frac{1}{48}\sum_{I,J,K\in\mathcal{I}_3}\omega_{I,J,K}\otimes \theta_{I,J,K}(v)\\
&=\frac{1}{48} x_pd_Q \sum_{I,J,K}d_I d_J d_K\otimes \theta_{I,J,K}(v)\\
&=\frac{1}{48}\sum_{I,J,K}(\varepsilon_{Q,I}x_p\partial _{t_{Q,I}}d_Jd_K-d_I \varepsilon_{Q,J}\,x_p\partial_{t_{Q,J}}d_K+d_Id_J\varepsilon_{Q,K}x_p\partial_{t_{Q,K}})\otimes {\theta_{I,J,K}(v)}\\
&=\frac{1}{48} \sum_{H,L}d_Hd_L\otimes 2\varepsilon_{pqabc}\sum_{\alpha\beta\gamma}\Big(D^2_{\gamma \rightarrow p}\theta_{\alpha\beta,H,L}(v)+
2D^3_{\gamma \rightarrow p}\theta_{\alpha \beta, H,L}(v)+3x_p\partial_\gamma.(\theta_{\alpha \beta, H,L}(v))\Big),
\end{align*}
where the sums run over $I,J,K\in \mathcal I_1$ and $(\alpha,\beta,\gamma)\in C(a,b,c)$.
Recalling that $d_Hd_L=\omega_{H,L}+\frac{1}{2}\varepsilon_{H,L}\partial_{t_{H,L}}$ we have:
\begin{align*}
x_pd_Q &\sum_{(I,J,K)\in \mathcal I_3/B_3}\omega_{I,J,K}\otimes \theta_{I,J,K}(v)\\
&= \frac{1}{48}\sum_{H,L}\omega_{H,L}\otimes 2\varepsilon_{pqabc}\sum_{\alpha\beta\gamma }\Big(D^2_{\gamma\rightarrow p}\theta_{\alpha\beta,H,L}(v)+
2D^3_{\gamma \rightarrow p}\theta_{\alpha \beta, H,L}(v)+3x_p\partial_\gamma.( \theta_{\alpha \beta, H,L}(v))\Big)\\
&\,\,\,+\frac{1}{48}\sum_{H,L}\partial_{t_{H,L}}\otimes \varepsilon_{H,L}\varepsilon_{pqabc}\sum_{\alpha\beta\gamma }\Big(D^2_{\gamma\rightarrow p}\theta_{\alpha\beta,H,L}(v)
+2D^3_{\gamma\rightarrow p}\theta_{\alpha \beta, H,L}(v)+3x_p\partial_\gamma.(\theta_{\alpha \beta, H,L}(v))\Big)\\
&=\frac{1}{48}\sum_{(H,L)\in\mathcal{I}_2/B_2}\omega_{H,L}\otimes 2\varepsilon_{pqabc}\sum_{\alpha\beta\gamma }\Big(12D^2_{\gamma\rightarrow p}\theta_{\alpha\beta,H,L}(v)+
12D^3_{\gamma\rightarrow p}\theta_{\alpha \beta, H,L}(v)+24x_p\partial_\gamma.( \theta_{\alpha \beta, H,L}(v))\Big)\\
&\,\,\,+\frac{1}{48}\sum_{(H,L)\in\mathcal{I}_2/B_2}\partial_{t_{H,L}}\otimes \varepsilon_{H,L}\varepsilon_{pqabc}\sum_{\alpha\beta\gamma }\Big(-4D^2_{\gamma\rightarrow p}\theta_{\alpha\beta,H,L}(v)+4 D^3_{\gamma\rightarrow p}\theta_{\alpha \beta, H,L}(v)\Big)\\
&=\sum_{(H,L)\in\mathcal{I}_2/B_2}\omega_{H,L}\otimes \frac{1}{2}\varepsilon_{pqabc}\sum_{\alpha\beta\gamma }
\Big(-(x_p\partial_{\gamma}.\theta_{\alpha\beta,H,L})(v)+
2x_p\partial_\gamma.( \theta_{\alpha \beta, H,L}(v))\Big)\\
&\,\,\,+\partial_q\otimes- \frac{1}{2} \theta_{ab,bc,ca}(v) +\sum_{\alpha\beta\gamma}\partial_\alpha\otimes \frac{1}{4} (\theta_{\alpha\beta,\beta\gamma,\gamma q}(v)+\theta_{\alpha\gamma,\gamma \beta,\beta q}(v)).
\end{align*}
We also need the following computation
\begin{align*}
x_pd_Q \sum_{t\in [5]}\sum_{I\in \mathcal I_1/B_1} &\partial_t\omega_I\otimes \theta^t_I(v)=-\frac{1}{2}\sum_{I\in \mathcal I_1}d_Qd_I\otimes \theta^p_I(v)+\frac{1}{2}\sum_{t}\partial_t x_pd_Q\sum_I d_I\otimes \theta^t_I(v)\\
&=-\frac{1}{2}\sum_Id_Qd_I\otimes \theta^p_I(v)+\sum_t\partial_t\otimes \varepsilon_{pqabc}\sum_{\alpha\beta\gamma}x_p\partial_\gamma.( \theta^t_{\alpha\beta}(v))\\
&=\frac{1}{2}\sum_I (\omega_{I,Q}-\frac{1}{2}\varepsilon_{Q,I}\partial_{t_{Q,I}})\otimes \theta^p_I(v)+\sum_t\partial_t\otimes \varepsilon_{pqabc}\sum_{\alpha\beta\gamma}x_p\partial_\gamma.( \theta^t_{\alpha\beta}(v))\\
&= \sum_{I\in \mathcal I_1/B_1} (\omega_{I,Q}\otimes \theta^p_I(v)-\partial_{t_{Q,I}}\otimes\frac{1}{2}\varepsilon_{Q,I}\theta^p_I(v))+\sum_t\partial_t\otimes \varepsilon_{pqabc}\sum_{\alpha\beta\gamma}x_p\partial_\gamma.( \theta^t_{\alpha\beta}(v)).
\end{align*}
Now we can use these two relations and compute
\begin{align*}
x_p&d_Q\Phi(v)=x_pd_Q\Big(\sum_{(I,J,K)\in \mathcal I_3/B_3}\omega_{I,J,K}\otimes \theta_{I,J,K}(v)+\sum_{t\in[5]}\sum_{I\in \mathcal I_1/B_1}\partial_t\omega_I\otimes \theta^t_{I}(v)\Big)\\
&=\sum_{(H,L)\in\mathcal{I}_2/B_2}\omega_{H,L}\otimes \frac{1}{2}\varepsilon_{pqabc}\sum_{\alpha\beta\gamma }
\Big(-(x_p\partial_{\gamma}.\theta_{\alpha\beta,H,L})(v)+
2x_p\partial_\gamma.( \theta_{\alpha \beta, H,L}(v))\Big)\\
&\,\,\,+\partial_q\otimes- \frac{1}{2} \theta_{ab,bc,ca}(v) +\sum_{\alpha\beta\gamma}\partial_\alpha\otimes \frac{1}{4} (\theta_{\alpha\beta,\beta\gamma,\gamma q}(v)+\theta_{\alpha\gamma,\gamma \beta,\beta q}(v))\\
&+\sum_{I\in \mathcal I_1/B_1} (\omega_{I,Q}\otimes \theta^p_I(v)-\partial_{t_{Q,I}}\otimes\frac{1}{2}\varepsilon_{Q,I}\theta^p_I(v))+\sum_t\partial_t\otimes \varepsilon_{pqabc}\sum_{\alpha\beta\gamma}x_p\partial_\gamma.( \theta^t_{\alpha\beta}(v))\\
&=\sum_{(H,L)\in\mathcal{I}_2/B_2}\omega_{H,L}\otimes \big(\chi_{L\in B_1Q}\theta^p_{H}(v)+\varepsilon_{pqabc}\sum_{\alpha\beta\gamma}
\big(-\frac{1}{2}(x_p\partial_{\gamma}.\theta_{\alpha\beta,H,L})(v)+
x_p\partial_\gamma.(\theta_{\alpha \beta, H,L}(v))\big)\\
&\hspace{5mm}+\partial_p\otimes\varepsilon_{pqabc}\sum_{\alpha\beta\gamma}x_p\partial_\gamma.( \theta^p_{\alpha\beta}(v))+\partial_q\otimes \big(\varepsilon_{pqabc}\sum_{\alpha\beta\gamma}x_p\partial_\gamma.( \theta^q_{\alpha\beta}(v))-\frac{1}{2}\theta_{ab,bc,ca}(v)\big)\\
&\hspace{5mm}+\sum_{\alpha\beta\gamma}\partial_\alpha \otimes\Big(\frac{1}{4}\theta_{\alpha\beta,\beta\gamma,\gamma q}(v)+\frac{1}{4}\theta_{\alpha\gamma,\gamma\beta,\beta q}(v)+\varepsilon_{pqabc}\big(-\frac{1}{2}\theta^p_{\beta \gamma}(v)+x_p\partial_c.( \theta^\alpha_{ab}(v))\\&\hspace{30mm}+x_p \partial_b .(\theta^\alpha_{ca}(v))+x_p\partial_a .(\theta^\alpha_{bc}(v))\big)\Big).
\end{align*}
This completes the proof of Equations \eqref{deg3n1}, \eqref{deg3n3} and \eqref{deg3n4}. In order to deduce Equation \eqref{deg3n2} we consider the coefficient of $\partial_a$ in the previous equation (the coefficients of $\partial_b$ and $\partial_c$ provide equivalent conditions) and we have
\begin{align*}
\frac{1}{4}&\theta_{ab,bc,cq}(v)+\frac{1}{4}\theta_{ac,cb,bq}(v)+\varepsilon_{pqabc}\big(-\frac{1}{2}\theta^p_{bc}(v)+x_p\partial_c .(\theta^a_{ab}(v))+x_p \partial_b.( \theta^a_{ca}(v))+x_p\partial_a. (\theta^a_{bc}(v))\big)\\
&=\frac{1}{4}\theta_{ab,bc,c q}(v)+\frac{1}{4}\theta_{ac,cb,bq}(v)+\varepsilon_{pqabc}\Big(-\frac{1}{2}\big((x_p\partial_a.\theta^a_{bc})(v)+(x_p\partial_b. \theta^a_{ca})(v)+(x_p\partial_c. \theta^a_{ab})(v)\big)\\ &\hspace{5mm}+x_p\partial_c .(\theta^a_{ab}(v))+x_p \partial_b.( \theta^a_{ca}(v))+x_p\partial_a. (\theta^a_{bc}(v))\Big)\\
&=\frac{1}{4}\theta_{ab,bc,cq}(v)+\frac{1}{4}\theta_{ac,cb,bq}(v)+\frac{1}{2}\varepsilon_{pqabc}\sum_{\alpha\beta\gamma}\big(-(x_p\partial_\gamma.\theta^a_{\alpha\beta})(v)+2x_p\partial_\gamma.(\theta^a_{\alpha\beta}(v))\big).
\end{align*}
\end{proof}
\begin{corollary}\label{degree3dual}
Let $\varphi:M(\lambda)\rightarrow M(\mu)$ be a morphism of Verma modules of degree 3 associated to
\[\Phi=\sum_{I\in \mathcal I_3/B_3}\omega_I\otimes \theta_I+\sum_{t\in [5]}\sum_{I\in \mathcal I_1/B_1} \partial_t\omega_I \otimes \theta^t_I.\]
Then the linear map $\psi:M(\mu^*)\rightarrow M(\lambda^*)$ associated to
\[
\Psi=\sum_{I\in \mathcal I_3/B_3}\omega_I\otimes \theta_I^*+\sum_{t\in [5]}\sum_{I\in \mathcal I_1/B_1} \partial_t\omega_I \otimes (-\theta^t_I)^*
\]
is also a morphism of Verma modules.
\end{corollary}
\begin{proof}
This is an immediate consequence of Remark \ref{lemdual2} and Theorem \ref{grado3}.
\end{proof}
If we consider Equation \eqref{deg3n1} on a highest weight vector $s\in F(\lambda)$ (and we multiply it by $2\varepsilon_{pqabc}$) we obtain the following equation:
\begin{equation}\label{maxdeg3}
2\varepsilon_{pqabc}\chi_{L\in B_1Q}\theta^p_{H}(s)+\sum_{\alpha\beta\gamma}
\big((-1)^{\chi_{p>\gamma}}(x_p\partial_{\gamma}.\theta_{\alpha\beta,H,L})(s)+2\chi_{p>\gamma}x_p\partial_\gamma.(\theta_{\alpha\beta,H,L}(s))\big)=0.
\end{equation}
\begin{remark}\label{asdf}
If $x_p\partial_c.\theta_{ab,H,L}$ has the leading weight of $\varphi$ then $\chi_{p>\gamma}x_p\partial_\gamma.(\theta_{\alpha\beta,H,L}(s))=0$ for all $(\alpha,\beta,\gamma)\in C(a,b,c)$ and so we obtain the following
\begin{equation}
2\varepsilon_{pqabc}\chi_{L\in B_1Q}\theta^p_{H}(s)+\sum_{\alpha\beta\gamma}
(-1)^{\chi_{p>\gamma}}(x_p\partial_{\gamma}.\theta_{\alpha\beta,H,L})(s)=0.
\end{equation}
\end{remark}
This equation has several immediate consequences.
\begin{lemma} \label{a,b,c,d}If $a,b,c,d\in [5]$ are distinct then $\theta_{ab,ac,ad}$ does not have the leading weight of $\varphi$.
\end{lemma}
\begin{proof}
Without loss of generality we can assume that the fifth element $p$ is either bigger than both $b$ and $c$ or smaller than both $b$ and $c$. Otherwise we can rename $b,c,d$ accordingly. Remark \ref{asdf} applies with $H=(a,p)$, $q=d$ and $L=(a,d)$ so we have
\[
(-1)^{\chi_{p>c}}\theta_{ab,ac,ad}(s)+(-1)^{\chi_{p>b}}\theta_{ab,ac,ad}(s)=0.
\]
\end{proof}
\begin{lemma}\label{a,b,c}
If $a,b,c\in [5]$ are distinct then $\theta_{ab,bc,ca}$ does not have the leading weight of $\varphi$.
\end{lemma}
\begin{proof}
Without loss of generality we can choose $p$ such that $p$ is either bigger than both $a$ and $c$ or smaller than both $a$ and $c$. Remark \ref{asdf} applies with $H=(b,p)$ and $L=(c,a)$ so we have
\[
(-1)^{\chi_{p>c}}\theta_{ab,bc,ca}(s)+(-1)^{\chi_{p>a}}\theta_{ab,bc,ca}(s)=0.
\]
\end{proof}
\begin{lemma}\label{12e45}
If $x,y,z,w\in[5]$ are distinct and $\theta_{xy,zw,xw}$ has the leading weight of $\varphi$, then $\theta_{xy,zw,xw}=\theta_{12,45,kl}$ for some $k,l\in\{1,2,4,5\}$.
\end{lemma}
\begin{proof}
Let us first assume that $\{x,y,z,w\}\neq \{1,2,4,5\}$.
This assumption ensures that we can assume that the fifth element $p$ is either bigger or smaller than both $y$ and $w$ (otherwise exchange the roles of $x,z$ and $y,w$).
Use Remark \ref{asdf} with $a=x$, $b=y$, $c=w$, $q=z$, $H=(x,p)$, $L=(z,w)$. Then we have:
\[
(-1)^{\chi_{p>y}}\theta_{xy,xw,zw}(s)+(-1)^{\chi_{p>w}}\theta_{xy,xw,zw}(s)=0.
\]
Now let $\{x,y,z,w\}= \{1,2,4,5\}$. If either
$\{y,w\}=\{1,2\}$ or $\{y,w\}=\{4,5\}$ then we can use the same argument as above.
Now let $\{y,z\}=\{4,5\}$ so that $\theta_{1y,2z,12}$ has the leading weight of $\varphi$. Equation \eqref{maxdeg3} with $a=1$, $b=2$, $q=3$, $c=y$ $p=z$, $H=(2,p)$ and $L=(1,2)$ gives
$$x_z\partial_2.(\theta_{1y,2z,12}(s))=0$$
hence if we apply $x_2\partial_z$ we get $h_{2z}.(\theta_{1y,2z,12}(s))=0$ which implies in particular that
$\mu_{34}=0$. Since $\lambda_{34}(\theta_{1y,2z,12})=1$ this contradicts the dominance of $\lambda$.
The thesis follows.
\end{proof}
\begin{lemma} \label{1245}The elements $\theta_{12,45,14}$, $\theta_{12,45,25}$ and $\theta_{12,45,24}$ do not have the leading weight of $\varphi$.
\end{lemma}
\begin{proof}
Use Equation \eqref{maxdeg3} with $a=1$ $b=2$ $c=4$, $q=3$ and $p=5$, $H=(4,5)$ and $L=(1,2)$. We obtain
\begin{equation}\label{czwp}
\theta_{24,41,12}(s)+\theta_{41,42,12}(s)+2x_5\partial_1. (\theta_{24,45,12}(s))+2x_5\partial_2. (\theta_{41,45,12}(s))=0.
\end{equation}
Assume $\theta_{12,45,14}$ has the leading weight of $\varphi$. Then $\theta_{24,45,12}(s)=0$ and we apply $x_2\partial_5$ to Equation \eqref{czwp} to obtain
\[
-\theta_{54,41,12}(s)-\theta_{24,41,15}(s)-\theta_{41,45,12}(s)-\theta_{41,42,15}(s)+2h_{25}.(\theta_{41,45,12}(s))=0.
\]
But by Lemma \ref{12e45} we have $\theta_{24,41,15}(s)=0$ and so we have
\[
-2\theta_{41,45,12}(s)+2h_{25}.(\theta_{41,45,12}(s))=0.
\]
It follows that $\lambda_{25}(\theta_{41,45,12}(s))=1$ and so $\lambda_{34}(\theta_{41,45,12}(s))\leq 1$ and, since $\lambda_{34}(\theta_{41,45,12})=2$ this would imply $\lambda_{34}(s)\leq -1$, a contradiction.
By Corollary \ref{degree3dual} the element $\theta_{12,45,25}$ does not have the leading weight of $\varphi$ since $\lambda(\theta_{12,45,25})=-\lambda(\theta_{12,45,14})^*$.
Now we assume that $\theta_{12,45,24}$ has the leading weight of $\varphi$. We apply $x_1\partial_5$ to Equation \eqref{czwp} to obtain
\[
-\theta_{24,45,12}(s)-\theta_{24,41,52}(s)-\theta_{45,42,12}-\theta_{41,42,52}+2h_{15}.(\theta_{24,45,12}(s))+2x_1\partial_2.(\theta_{41,45,12}(s))=0.
\]
Lemma \ref{12e45} ensures $\theta_{24,41,52}(s)=0$ and so we obtain
\[
-2\theta_{24,45,12}(s)+2h_{15}.(\theta_{24,45,12}(s)-2\theta_{42,45,12}(s)=0
\]
and we conclude
\[
h_{15}.(\theta_{24,45,12}(s))=0.
\]
We obtain a contradiction with the same argument used in the other case.
\end{proof}
\begin{lemma}\label{morfismogrado3}
Assume that $\theta_{12,15,45}$ has the leading weight of $\varphi$. Then $\lambda=(1,1,0,0)$, $\mu=(0,0,1,1)$ and $\varphi=\nabla_C\nabla_B\nabla_A$ (up to a scalar).
\end{lemma}
\begin{proof}
Use Equation \eqref{maxdeg3} with $a=1$, $b=2$, $c=4$, $q=3$, $p=5$, $H=(1,5)$ and $L=(4,5)$. We obtain
\[
\theta_{12,14,45}(s)+\theta_{24,15,41}(s)+\theta_{41,12,45}(s)+\theta_{41,15,42}(s)+2x_5\partial_4.(\theta_{12,15,45}(s))=0
\]
since
$\theta_{24,14,45}(s)=\theta_{41,15,45}(s)=0$.
Applying $x_4\partial_5$ we get
\[
-\theta_{12,15,45}(s)-\theta_{25,15,41}(s)-\theta_{51,12,45}(s)-\theta_{41,15,52}(s)+2h_{45}.(\theta_{12,15,45}(s))=0.
\]
By Lemma \ref{12e45} we have $\theta_{25,15,41}(s)=0$ and so we obtain
\[
-2\theta_{12,15,45}(s)+2h_{45}.(\theta_{12,15,45}(s))=0
\]
and so
\[
\lambda_{45}(\theta_{12,15,45}(s))=1.
\]
Now we consider Equation \eqref{maxdeg3} with $a=1$, $b=3$, $c=5$, $q=2$, $p=4$, $H=(1,2)$ and $L=(4,5)$. We obtain
\[
\theta_{35,12,15}(s)+\theta_{51,12,35}(s)+2x_4\partial_3.(\theta_{51,12,45}(s))=0.
\]
Applying $x_3\partial_4$ to this equation we have
\[
-\theta_{45,12,15}(s)-\theta_{51,12,45}(s)+2h_{34}.(\theta_{51,12,45}(s))=0
\]
and from this we get $\lambda_{34}(\theta_{12,15,45}(s))=1$.
Finally, we use again Equation \eqref{maxdeg3} with $a=1$, $b=4$, $c=5$, $q=2$, $p=3$, $H=(1,2)$, $L=(1,5)$ which gives $2x_3\partial_1.(\theta_{45,12,15}(s))=0$, hence
\[
\lambda_{13}(\theta_{12,15,45}(s))=0
\]
proving that $\mu=(0,0,1,1)$. It follows that $\lambda=(1,1,0,0)$ since $\lambda(\theta_{12,15,45})=(-1,-1,1,1)$.
By Remark \ref{asdf} we have $-2\theta^3_{15}(s)-\theta_{12,15,45}(s)=0$ hence the leading term of the singular vector $\varphi(1\otimes s)$ is
$\omega_{12,15,45}\otimes \theta_{12,15,45}(s)+\partial_3d_{15}\otimes\theta^3_{15}(s)= d_{12}d_{15}d_{45}\otimes \theta_{12,15,45}(s)$.
It follows that $\varphi=\nabla_C\nabla_B\nabla_A$ due to Proposition \ref{leading}.
\end{proof}
In the next result, for notational convenience, for all $a,b\in [5]$ we let $(-1)^{a<b}=(-1)^{\chi_{a<b}}$.
\begin{proposition}\label{12equazioni}
Let $\{x,y,z,w,t\}=[5]$ and let $s$ be a highest weight vector in $F(\lambda)$. Assume that $\theta_{xy,xz,wt}$ has the leading weight of $\varphi$. Then
the following equations hold:
\begin{align}
\label{eq1} -2\varepsilon_{xyzwt}\theta^y_{xy}(s)+(-1)^{y<t}\theta_{xz,xt,yw}(s)&+(-1)^{y<t}\theta_{xz,xy,tw}(s)+ (-1)^{y<z}\theta_{xz,xt,yw}(s)\\
\nonumber & +(-1)^{y<z}\theta_{xy,xt,zw}(s)+(-1)^{y<x}\theta_{xy,xw,zt}(s)=0
\end{align}
\begin{align}
\label{eq2} 2\varepsilon_{xyzwt}\theta^y_{xy}(s)+(-1)^{y<t}\theta_{xw,xt,yz}(s)&+(-1)^{y<t}\theta_{xw,xy,tz}(s)+ (-1)^{y<w}\theta_{xw,xt,yz}(s)\\
\nonumber &+(-1)^{y<w}\theta_{xy,xt,wz}(s)+(-1)^{y<x}\theta_{xy,xz,wt}(s)=0
\end{align}
\begin{align}
\label{eq3} -2\varepsilon_{xyzwt}\theta^y_{xy}(s)+(-1)^{y<z}\theta_{xw,xz,yt}(s)&+(-1)^{y<z}\theta_{xw,xy,zt}(s)+ (-1)^{y<w}\theta_{xw,xz,yt}(s)\\
\nonumber &+(-1)^{y<w}\theta_{xy,xz,wt}(s)+(-1)^{y<x}\theta_{xy,xt,wz}(s)=0
\end{align}
\begin{align}
\label{eq4} 2\varepsilon_{xyzwt}\theta^z_{xz}(s)+(-1)^{z<t}\theta_{xy,xt,zw}(s)&+(-1)^{z<t}\theta_{xy,xz,tw}(s)+ (-1)^{z<y}\theta_{xy,xt,zw}(s)\\
\nonumber &+(-1)^{z<y}\theta_{xz,xt,yw}(s)+(-1)^{z<x}\theta_{xz,xw,yt}(s)=0
\end{align}
\begin{align}
\label{eq5} -2\varepsilon_{xyzwt}\theta^z_{xz}(s)+(-1)^{z<t}\theta_{xw,xt,zy}(s)&+(-1)^{z<t}\theta_{xw,xz,ty}(s)+ (-1)^{z<w}\theta_{xw,xt,zy}(s)\\
\nonumber &+(-1)^{z<w}\theta_{xz,xt,wy}(s)+(-1)^{z<x}\theta_{xz,xy,wt}(s)=0
\end{align}
\begin{align}
\label{eq6} 2\varepsilon_{xyzwt}\theta^z_{xz}(s)+(-1)^{z<y}\theta_{xw,xy,zt}(s)&+(-1)^{z<y}\theta_{xw,xz,yt}(s)+ (-1)^{z<w}\theta_{xw,xy,zt}(s)\\
\nonumber &+(-1)^{z<w}\theta_{xz,xy,wt}(s)+(-1)^{z<x}\theta_{xz,xt,wy}(s)=0
\end{align}
\begin{align}
\label{eq7} 2\varepsilon_{xyzwt}\theta^t_{xt}(s)+(-1)^{t<y}\theta_{xz,xy,tw}(s)&+(-1)^{t<y}\theta_{xz,xt,yw}(s)+ (-1)^{t<z}\theta_{xz,xy,tw}(s)\\
\nonumber &+(-1)^{t<z}\theta_{xt,xy,zw}(s)+(-1)^{t<x}\theta_{xt,xw,zy}(s)=0
\end{align}
\begin{align}
\label{eq8} -2\varepsilon_{xyzwt}\theta^t_{xt}(s)+(-1)^{t<w}\theta_{xz,xw,ty}(s)&+(-1)^{t<w}\theta_{xz,xt,wy}(s)+ (-1)^{t<z}\theta_{xz,xw,ty}(s)\\
\nonumber &+(-1)^{t<z}\theta_{xt,xw,zy}(s)+(-1)^{t<x}\theta_{xt,xy,zw}(s)=0
\end{align}
\begin{align}
\label{eq9} 2\varepsilon_{xyzwt}\theta^t_{xt}(s)+(-1)^{t<w}\theta_{xy,xw,tz}(s)&+(-1)^{t<w}\theta_{xy,xt,wz}(s)+ (-1)^{t<y}\theta_{xy,xw,tz}(s)\\
\nonumber &+(-1)^{t<y}\theta_{xt,xw,yz}(s)+(-1)^{t<x}\theta_{xt,xz,yw}(s)=0
\end{align}
\begin{align}
\label{eq10} -2\varepsilon_{xyzwt}\theta^w_{xw}(s)+(-1)^{w<t}\theta_{xy,xt,wz}(s)&+(-1)^{w<t}\theta_{xy,xw,tz}(s)+ (-1)^{w<y}\theta_{xy,xt,wz}(s)\\
\nonumber &+(-1)^{w<y}\theta_{xw,xt,yz}(s)+(-1)^{w<x}\theta_{xw,xz,yt}(s)=0
\end{align}
\begin{align}
\label{eq11} 2\varepsilon_{xyzwt}\theta^w_{xw}(s)+(-1)^{w<t}\theta_{xz,xt,wy}(s)&+(-1)^{w<t}\theta_{xz,xw,ty}(s)+ (-1)^{w<z}\theta_{xz,xt,wy}(s)\\
\nonumber &+(-1)^{w<z}\theta_{xw,xt,zy}(s)+(-1)^{w<x}\theta_{xw,xy,zt}(s)=0
\end{align}
\begin{align}
\label{eq12} -2\varepsilon_{xyzwt}\theta^w_{xw}(s)+(-1)^{w<y}\theta_{xz,xy,wt}(s)&+(-1)^{w<y}\theta_{xz,xw,yt}(s)+ (-1)^{w<z}\theta_{xz,xy,wt}(s)\\
\nonumber &+(-1)^{w<z}\theta_{xw,xy,zt}(s)+(-1)^{w<x}\theta_{xw,xt,zy}(s)=0
\end{align}
\end{proposition}
\begin{proof} We use Remark \ref{asdf} twelve times with $L=Q=(p,q)$ any ordered pair in $\{y,z,w,t\}$ and $H=(x,p)$ to obtain the stated equations.
More precisely we get Equation \eqref{eq1} with $p=y$, $q=w$;
Equation \eqref{eq2} with $p=y$, $q=z$;
Equation \eqref{eq3} with $p=y$, $q=t$;
Equation \eqref{eq4} with $p=z$, $q=w$;
Equation \eqref{eq5} with $p=z$, $q=y$;
Equation \eqref{eq6} with $p=z$, $q=t$;
Equation \eqref{eq7} with $p=t$, $q=w$;
Equation \eqref{eq8} with $p=t$, $q=y$;
Equation \eqref{eq9} with $p=t$, $q=z$;
Equation \eqref{eq10} with $p=w$, $q=z$;
Equation \eqref{eq11} with $p=w$, $q=y$;
Equation \eqref{eq12} with $p=w$, $q=t$.
\end{proof}
Proposition \ref{12equazioni} provides 12 linear equations in the ten unknown
$\theta_{xy,xz,wt}(s)=f_{wt}$, $\theta_{xy,xw,zt}(s)=f_{zt}$, $\theta_{xy,xt,zw}(s)=f_{zw}$, $\theta_{xz,xw,yt}(s)=f_{yt}$, $\theta_{xz,xt,yw}(s)=f_{yw}$, $\theta_{xw,xt,yz}(s)=f_{yz}$,
$\varepsilon_{xyzwt}\theta^y_{xy}(s)=b_y$, $\varepsilon_{xyzwt}\theta^{z}_{xz}(s)=b_z$, $\varepsilon_{xyzwt}\theta^w_{xw}(s)=b_w$, $\varepsilon_{xyzwt}\theta^t_{xt}(s)=b_t$.
We are now interested in the study of the weights $\lambda_{i,j}(\theta_{xy,xz,wt}(s))$.
\begin{proposition}\label{aaa} Let $\{p,q,a,b,c\}=[5]$ with $c<p$, let $s$ be a highest weight vector in $F(\lambda)$, $H,L\in\mathcal{I}_1$
and assume that $\theta_{ab,H,L}$ has the leading weight of $\varphi$. Then we have
\begin{align*}
&2h_{cp}.(\theta_{ab,H,L}(s))=\\
&-2\varepsilon_{pqabc}\chi_{L\in B_1Q}(x_c\partial_p.\theta^p_{H})(s)+(x_c\partial_p.(x_p\partial_c \theta_{ab,H,L}))(s)+(-1)^{\chi_{p<b}}(x_c\partial_p.(x_p\partial_{b}.\theta_{ca,H,L}))(s)\\
&+(-1)^{\chi_{p<a}}(x_c\partial_p.(x_p\partial_{a}.\theta_{bc,H,L}))(s)-2\chi_{c<b<p}(x_c\partial_b.\theta_{ca,H,L})(s)-2\chi_{c<a<p}(x_c\partial_a.\theta_{bc,H,L})(s)
\end{align*}
\end{proposition}
\begin{proof}
Equation \eqref{maxdeg3} is equivalent to the following
\begin{align*}
2\varepsilon_{pqabc}\chi_{L\in B_1Q}\theta^p_{H}(s)&-(x_p\partial_{c}.\theta_{ab,H,L})(s)+(-1)^{\chi_{p>b}}(x_p\partial_{b}.\theta_{ca,H,L})(s)+(-1)^{\chi_{p>a}}(x_p\partial_{a}.\theta_{bc,H,L})(s)\\
&+ 2x_p\partial_c.(\theta_{ab,H,L}(s))+2\chi_{p>b}x_p\partial_b.(\theta_{ca,H,L}(s))+2\chi_{p>a}x_p\partial_a.(\theta_{bc,H,L}(s))=0.
\end{align*}
We apply $x_c\partial_p$ to this equation and we obtain
\begin{align*}
2&\varepsilon_{pqabc}\chi_{L\in B_1Q}x_c\partial_p.(\theta^p_{H}(s))-(x_c\partial_p.(x_p\partial_c.\theta_{ab,H,L}))(s)+(-1)^{\chi_{p>b}}(x_c\partial_p.(x_p\partial_{b}.\theta_{ca,H,L}))(s)\\
&+(-1)^{\chi_{p>a}}(x_c\partial_p.(x_p\partial_{a}.\theta_{bc,H,L}))(s)+ 2h_{cp}.(\theta_{ab,H,L}(s))+2\chi_{c<b<p}(x_c\partial_b.\theta_{ca,H,L})(s)\\
&+2\chi_{c<a<p}(x_c\partial_a.\theta_{bc,H,L})(s)=0.
\end{align*}
The result follows.
\end{proof}
\begin{corollary} Let $\{x,y,z,w,t\}=[5]$ and assume that $\theta_{xy,xz,wt}$ has the leading weight of $\varphi$. Then we have
\noindent
if $z<w$,
\begin{align}
\label{9141} 2h_{zw}.f_{wt}=2(b_w-b_z)+f_{zt}+f_{wt}+(-1)^{\chi_{w<x}}(f_{yw}+f_{yz})-2\chi_{z<x<w}f_{wt};
\end{align}
if $y<z$,
\begin{align}
\label{9142} 2h_{yz}.f_{wt}&=2(b_y-b_z)+(-1)^{\chi_{t<z}}(-f_{zw}-f_{yw})+(-1)^{\chi_{w<z}}(f_{zt}+f_{yt})\\ &
\nonumber -2\chi_{y<t<z}(f_{wt}+f_{yw})-2\chi_{y<w<z}(f_{wt}-f_{yt})
\end{align}
if $w<t$,
\begin{align}
\label{9143}2h_{wt}.f_{wt}&=(-1)^{\chi_{y<t}}(f_{yw}+f_{yt})+(-1)^{\chi_{x<t}}(f_{yt}+f_{yw})\\
\nonumber &-2\chi_{w<y<t}(f_{wt}-f_{yt})-2\chi_{w<x<t}(f_{wt}-f_{yw}).
\end{align}
if $w<z$,
\begin{align}
\label{9144} 2h_{wz}.f_{wt}&=f_{wt}+f_{zt}+(-1)^{\chi_{z<y}}(f_{wt}+f_{zt})+2\chi_{w<y<z}(-f_{wt}+f_{yt})+
2\chi_{w<x<z}(-f_{wt}+f_{yw})
\end{align}
if $x<y$,
\begin{align}
\label{9145} 2h_{xy}.f_{wt}&=((-1)^{\chi_{y<t}}+(-1)^{\chi_{y<w}})(-f_{yw}+f_{yt}+f_{yz})\\
\nonumber &-2\chi_{x<t<y}(f_{wt}+f_{yt}-f_{zt})
-2\chi_{x<w<y}(f_{wt}+f_{yw}-f_{yz})
\end{align}
\end{corollary}
\begin{proof}
The statement follows from Proposition \ref{aaa} with the following choices:
\begin{enumerate}
\item $a=x$ $b=y$, $c=z$, $p=w$, $q=t$, $H=(x,z)$, $L=(w,t)$.
\item $c=y$, $p=z$, $a=w$, $b=t$, $q=x$, $H=(x,y)$, $L=(z,x)$.
\item $c=w$, $p=t$, $a=x$, $b=y$, $q=z$, $H=(x,z)$, $L=(w,t)$.
\item $c=w$, $p=z$, $a=x$, $b=y$, $q=t$, $H=(x,z)$, $L=(w,t)$.
\item $c=x$, $p=y$, $a=w$ $b=t$, $q=z$, $H=(x,y)$, $L=(x,z)$.
\end{enumerate}
\end{proof}
\begin{proposition}\label{trequarti}
Let $s$ be a highest weight vector in $F(\lambda)$. For $c<p$ we have
\begin{align*}
4h_{cp}.(\theta^a_{ab}(s))&=(-4\chi_{c<b<p}-4\chi_{c<a<p})\theta^a_{ab}(s)+(2-4\chi_{c<a})\theta^c_{bc}(s)+(-2+4\chi_{p<a})\theta^p_{bp}(s)\\
&\hspace{5mm}+\varepsilon_{pqabc}\big(\theta_{ab,bp,cq}(s)+\theta_{ab,bc,pq}(s)+\theta_{ap,cb,bq}(s)+\theta_{ac,pb,bq}(s)\big)
\end{align*}
\end{proposition}
\begin{proof}
We start from Equation \eqref{deg3n2}:
\begin{equation}\label{coefparta}
\theta_{ab,bc,cq}(s)+\theta_{ac,cb,bq}(s)+\varepsilon_{pqabc}\big(-2\theta^p_{bc}(s)+4x_p\partial_c.(\theta^a_{ab}(s))+4x_p\partial_b.(\theta^a_{ca}(s))+4x_p\partial_a. (\theta^a_{bc}(s))\big)=0.
\end{equation}
We want to apply $x_c\partial_p$ to this equation and so we do the following two preliminary calculations:
\begin{align*}
x_c\partial_p.(x_p\partial_b.(\theta^a_{ca}(s)))&=\chi_{c<b}x_c\partial_p.(x_p\partial_b.(\theta^a_{ca}(s)))\\
&=\chi_{c<b}x_c\partial_b.(\theta^a_{ca}(s))+\chi_{c<b}x_p\partial_b.(x_c\partial_p.(\theta^a_{ca}(s)))\\
&=-\chi_{c<b}\theta^a_{ba}(s)-\chi_{c<b}x_p\partial_b.(\theta^a_{pa}(s))\\
&= \chi_{c<b}\theta^a_{ab}(s)+\chi_{c<b}\chi_{p<b}\theta^a_{ba}(s)\\
&=\chi_{c<b}(1-\chi_{p<b})\theta^a_{ab}(s)\\
&=\chi_{c<b<p}\theta^a_{ab}(s)
\end{align*}
\begin{align*}
x_c\partial_p.(x_p\partial_a.(\theta^a_{bc}(s)))&=\chi_{c<a}x_c\partial_p.(x_p\partial_a.(\theta^a_{bc}(s)))\\
&=\chi_{c<a}x_c\partial_a.(\theta^a_{bc}(s))+\chi_{c<a}x_p\partial_a.(x_c\partial_p.(\theta^a_{bc}(s)))\\
&=\chi_{c<a}(\theta^c_{bc}(s)-\theta^a_{ba}(s))-\chi_{c<a}x_p\partial_a.(\theta^a_{bp}(s))\\
&=\chi_{c<a}\theta^c_{bc}(s)+\chi_{c<a}\theta^a_{ab}(s)-\chi_{c<a}\chi_{p<a}(\theta^p_{bp}(s)-\theta^a_{ba}(s))\\
&=\chi_{c<a}\theta^c_{bc}(s)-\chi_{p<a}\theta^p_{bp}(s)+\chi_{c<a<p}\theta^a_{ab}(s)
\end{align*}
Therefore, if we apply $x_c\partial_p$ to Equation \eqref{coefparta}, using the previous computations, we obtain
\begin{align*}
-&\theta_{ab,bp,cq}(s)-\theta_{ab,bc,pq}(s)-\theta_{ap,cb,bq}(s)-\theta_{ac,pb,bq}(s)+\varepsilon_{pqabc}\big(-2\theta^c_{bc}(s)+2\theta^p_{bp}(s)\\&+4h_{cp}.(\theta^a_{ab}(s))+4\chi_{c<b<p}\theta^a_{ab}(s)+4\chi_{c<a}\theta^c_{bc}(s)-4\chi_{p<a}\theta^p_{bp}(s)+4\chi_{c<a<p}\theta^a_{ab}(s)\big)=0
\end{align*}
hence we get the statement.
\end{proof}
\begin{proposition} \label{xyxzwt}Let $\{h,k,l,m,n\}=[5]$. Then $\theta_{hk,hl,mn}$ and $\theta^k_{hk}$ do not have the leading weight of $\varphi$.
\end{proposition}
\begin{proof}
We first assume $h=1$ and we let $x=1$, $y=2$, $z=3$, $w=4$, $t=5$. We use notation introduced after the proof of Proposition \ref{12equazioni}
and we observe that, up to a sign, $\theta_{1k,1l,mn}(s)\in \{f_{23}, f_{24}, f_{25}, f_{34}, f_{35}, f_{45}\}$ and $\theta^k_{1k}(s)\in\{b_2,b_3,b_4,b_5\}$.
We solve the linear system provided by Proposition \ref{12equazioni} and
we have:
\begin{itemize}
\item $f_{35}=-f_{45}=-f_{34}$
\item $f_{24}=-f_{25}=-f_{23}$
\item $2b_2=-3f_{34}+2f_{23}$
\item $2b_3=2b_5=2b_4=-f_{34}$.
\end{itemize}
We use Proposition \ref{trequarti} with $a=4$, $b=1$, $c=2$, $p=3$, $q=5$ and we obtain
\begin{align*}
h_{23}.b_4= &\frac{1}{2}b_2-\frac{1}{2}b_3+\frac{1}{4}(f_{25}+f_{35}+f_{34}+f_{24})\\
&=\frac{1}{4}(-3f_{34}+2f_{23})+\frac{1}{4}f_{34}+\frac{1}{4}(f_{23}-f_{34}+f_{34}-f_{23})\\
&=-\frac{1}{2}f_{34}+\frac{1}{2}f_{23}
\end{align*}
therefore
\[
h_{23}.f_{34}=f_{34}-f_{23}.
\]
Now we use Equation \eqref{9142}:
\[
2h_{23}.f_{45}=2(b_2-b_3)-f_{34}-f_{24}+(f_{35}+f_{25})
\]
i.e.
\[
2h_{23}.f_{34}=-3f_{34}+2f_{23}+f_{34}-f_{34}+f_{23}-f_{34}+f_{23}=-4f_{34}+4f_{23}
\]
or
\[
h_{23}.f_{34}=-2f_{34}+2f_{23}.
\]
Comparing this with the previous equation we obtain $f_{34}=f_{23}$.
Now we use Equation \eqref{9145}:
\[
2h_{12}.f_{45}=2f_{24}-2f_{25}-2f_{23}
\]
i.e.
\[
2h_{12}.f_{34}=-2f_{23}-2f_{23}-2f_{23}=-6f_{34}
\]
This implies that $f_{34}=f_{23}=0$. It follows that $\theta_{1k,1l,mn}(s)=0$ and $\theta^k_{1k}(s)=0$.
Now let $h=2$ and $x=2$, $y=1$, $z=3$, $w=4$, $t=5$. Similarly as above
we have, up to a sign, $\theta_{2k,2l,mn}(s)\in \{f_{13}, f_{14}, f_{15}, f_{34}, f_{35}, f_{45}\}$ and $\theta^k_{2k}(s)\in\{b_1,b_3,b_4,b_5\}$.
We solve the linear system provided by Proposition \ref{12equazioni} and
we have:
\begin{itemize}
\item $f_{35}=-f_{45}=-f_{34}$
\item $f_{14}=-f_{15}=-f_{13}$
\item $2b_1=-f_{34}+2f_{13}$
\item $2b_3=2b_4=2b_5=-f_{34}$
\end{itemize}
We use Proposition \ref{trequarti} with $a=4$, $b=2$, $c=1$, $p=5$, $q=3$ and we obtain:
\[
h_{15}.b_4=\frac{1}{2}f_{34}+\frac{1}{2}f_{13}
\]
i.e.,
\[
h_{15}.f_{34}=-f_{34}-f_{13}
\]
Now we use Equations \eqref{9141}, \eqref{9142}, \eqref{9143} and we obtain:
\[
h_{15}.f_{34}=2f_{13}-f_{34}\]
It follows that:
\[
2f_{13}-f_{34}=-f_{34}-f_{13}\]
i.e., $f_{13}=0$, hence $h_{15}.f_{34}=-f_{34}$ which implies $f_{34}=0$. It follows that $\theta_{2k,2l,mn}(s)=0$ and $\theta^k_{2k}=0$.
Now let $h=3$ and $x=3$, $y=1$, $z=2$, $w=4$, $t=5$. Similarly as above
we have, up to a sign, $\theta_{3k,3l,mn}(s)\in \{f_{12}, f_{14}, f_{15}, f_{24}, f_{25}, f_{45}\}$ and $\theta^k_{3k}(s)\in\{b_1,b_2,b_4,b_5\}$.
We solve the linear system provided by Proposition \ref{12equazioni} and
we have:
\begin{itemize}
\item $f_{15}=f_{24}=-f_{25}=-f_{14}$
\item $f_{45}=-2b_4=-2b_5=-2f_{14}-f_{12}$
\item $2b_1=2b_2=f_{12}$
\end{itemize}
We use Proposition \ref{trequarti} with $a=2$, $b=3$, $c=1$, $p=5$, $q=4$ and we obtain:
\[
-h_{15}(b_2)=\frac{1}{2}f_{12}-\frac{1}{2}f_{14}
\]
i.e.,
\[
h_{15}(f_{12})=f_{14}-f_{12}
\]
Now we use Equations \eqref{9141}, \eqref{9142}, \eqref{9143} and we obtain:
\[h_{15}(f_{45})=3f_{14}+f_{12}.\]
It follows that:
\[h_{15}.f_{14}=-\frac{1}{2}h_{15}.(f_{45}+f_{12})=-2f_{14}\]
hence $f_{14}=0$ and $h_{15}.f_{12}=-f_{12}$ from which it follows that $f_{12}=0$.
We conclude that $\theta_{3k,3l,mn}(s)=0$ and $\theta^k_{3k}(s)=0$.
If $h=4,5$ the result follows from Corollary \ref{degree3dual}.
\end{proof}
Now we can summarize the classification of morphisms of degree 3 in the next result.
\begin{theorem}\label{teorema3}
Let $\varphi:M(\lambda)\rightarrow M(\mu)$ be a morphism of degree 3. Then $\lambda=(1,1,0,0)$, $\mu=(0,0,1,1)$ and up to a scalar $\varphi=\nabla_C\nabla_B\nabla_C$.
\end{theorem}
\begin{proof}
This follows from Lemmas \ref{a,b,c,d}, \ref{a,b,c}, \ref{12e45}, \ref{1245}, \ref{morfismogrado3} and Proposition \ref{xyxzwt}.
\end{proof}
\end{document}
\begin{theorem} Morphisms $\nabla_{AB} : M(m,1,0,0)\rightarrow M(m-1,0,0,1)$, $\nabla_{BC} : M(1,0,0,n)\rightarrow M(0,0,1,n+1)$,
$\nabla_{AC}: M(0,1,0,0)\rightarrow M(0,0,1,0)$ are the only morphisms of Verma modules of degree two.
\end{theorem}
\begin{proof}
\end{proof}
It follows that if the complement $\overline{\{b,c,k,l\}}$ of the set $\{b,c,k,l\}$
does not consist of two subsequent numbers, then the weight of $\theta_{bc,kl}(v)$
is not dominant. Therefore, if $\theta_{ij,rs}(v)=0$ for every $(ijrs)>(bckl)$
then $\theta_{bc,kl}(v)=0$.
Now set $m=a$ in Equation (\ref{equation2}), hence getting the following equation:
\begin{equation}
\frac{1}{2}((-1)^{\chi_{b<p}}+(-1)^{\chi_{c<p}})\theta_{ab,ac}(v)+
\chi_{a<p}x_p\partial_a\theta_{bc,pa}(v)
-\chi_{b<p}x_p\partial_b\theta_{ac,pa}(v)+\chi_{c<p}x_p\partial_c\theta_{ab,pa}(v)=0
\label{equation2m=a}
\end{equation}
It follows that if $\theta_{ij,rs}(v)=0$ for every $(ijrs)>(abac)$ and
$5\notin\{a,b,c\}$ then $\theta_{abac}(v)=0$. It is indeed sufficient to set $p=5$ in Equation
\eqref{equation2m=a}. Moreover, if $1\notin\{a,b,c\}$ we can use Equation \eqref{equation2m=a} with $p=1$ to deduce that $\theta_{ab,ac}(v)=0$
It also follows that if we can choose $p>b,c$ or $p<b,c$ we have $\theta_{abac}(s)=0$ (by assuming that $\theta_{ijrs}(v)=0$
for every $(ijrs)>(abac)$. In particular $\theta_{1525}(s)=0$ and $\theta_{1415}(s)=0$.
Consider Equation \eqref{equation1} with $a=k=1$ and $l=b=2$ and $c=5$ and $p=3$:
\[
x_3\partial_1 \theta_{2512}(s)-x_3\partial_2 \theta_{1512}(s)=0
\]
and applying $x_1\partial_3$ we get
\[
h_{13}\theta_{1225}(s)=-\theta_{1225}(s).
\]
Since the weight of a highest weight vector is dominant we conclude that $\theta_{1225}(s)=0$.
If we take Equation \eqref{equation2} with $a=3$, $b=1$, $c=m=4$ and $p=5$ we obtain
\[
\theta_{1434}(s)-x_5\partial_3 \theta_{1445}(s)+x_5\partial_4\theta_{1345}(s)=0
\]
and applying $x_3\partial_5$ we getting
\[
h_{35}(\theta_{1445}(s))=0.
\]
Since $h_{35}(\theta_{1445})=1$ by Remark... and the weight of $s$ is dominant we conclude that $\theta_{1445}(s)=0$.
Now we compute the weight of $\theta_{1215}(s)$. By Equation \eqref{equation1} with $a=l=2,\,p=3,\,b=k=1,\,c=5$ we have
\[
x_3\partial_2 \theta_{1512}(s)=0
\]
and so $h_{23}\theta_{1215}(s)=0$.
If we take \eqref{equation2} with $a=3$, $b=m=1$, $c=2$, $p=5$ we obtain
\[
\theta_{1231}(s)-x_5\partial_3\theta_{1215}(s)=0;
\]
if we apply $x_3\partial_5$ we obtain
\[
h_{35}\theta_{1215}(s)=\theta_{1215}(s).
\]
Since the weight of $\theta_{1215}$ is $(-1,-1,0,1)$ the weight of $\theta_{1215}(s)$ is necessarily $(m-1,0,0,1)$ and so the weight of $s$ is $(m,1,0,0)$ and so $\varphi=\nabla_A \nabla_B$.
Now we compute the weight of $\theta_{1545}(s)$. By Equation \eqref{equation1} with $a=k=1$, $p=3$, $b=4$ and $c=l=5$ we immediately get
\[
h_{13}\theta_{1545}(s)=0.
\]
Using Equation \eqref{equation2} with $a=3$, $b=1$, $c=5=m$ and $p=4$ implies
\[
h_{34}(\theta_{1545}(s)=\theta_{1545}(s).
\]
Since the weight of $\theta_{1545}$ is $(-1,0,1,1)$ the weight of $\theta_{1545}(s)$ is necessarily $(0,0,1,n+1)$ and the weight of $s$ is $(1,0,0,n)$.
This concludes the study of $\theta_{abac}$.
In order to deal with weight vectors $\theta_{abcm}(s)$ with distinct $a,b,c,m$ we consider the system of equations that we get using Equation \eqref{equation2} fixing $p$ and choosing $(ab)<(cm)$ such that $\{a,b,c,m,p\}=[5]$. We obtaing in this way for every $p$ four homogeneous linear equations in four indeterminates $\sigma_p(s)$, $\theta_{abcm}(s)$, $\theta_{bcam}(s)$ and $\theta_{acbm}(s)$ (we always use induction hypothesis on higher weight vectors).
For $p=1,5$ one easily checks that the system has a unique (trivial) solution.
For $p=2$ the system is equivalent to $\sigma_2(s)=0$ and $\theta_{1435}(s)=\theta_{1345}(s)+\theta_{1534}(s)$.
Using Equation \eqref{equation2} with $p=5$ $m=1$ and $a=2,b=3,c=4$ we have
\[
-\sigma_5(s)-\frac{1}{2}(-\theta_{3421}(s)+\theta_{2431}(s)-\theta_{2341}(s))+x_5\partial_2\theta_{3451}(s)-x_5\partial_3\theta_{2451}(s)+x_5\partial_4 \theta_{2351}(s)=0
\]
We apply $x_2\partial _5$ and we obtain
\[
h_{25}(\theta_{1534}(s)=-2\theta_{1534}(s)
\]
which implies $\theta_{1534}(s)=0$.
Using Equation \eqref{equation2} with $p=4$ and $m=5$ and $a=1,b=2,c=3$ and we obtain
\[
-\sigma_4-\frac{1}{2}(-\theta_{2315}(s)+\theta_{1325}(s)-\theta_{1235}(s))-x_4\partial_2 \theta_{1345}(s)+x_4\partial_3 \theta_{1245}(s)=0.
\]
Applyins $x_2\partial_4$ and using $\theta_{1534}(s)=0$ we have
\[
h_{24}\theta_{1345}(s)=-\theta_{1345}(s)
\]
hence $\theta_{1345}(s)=0$ and also $\theta_{1345}(s)=0$.
For $p=4$ the system is equivalent to $\sigma_4(s)=0$ and $\theta_{1235}(s)=\theta_{1325}(s)+\theta_{1523}(s)$.
Using Equation \eqref{equation2} with $p=5$ $m=1$ and $a=2,b=3,c=4$ and applying $x_4\partial _5$ and we obtain
\[
h_{45}(\theta_{1523}(s)=0;
\]
but $h_{45}(\theta_{1523})=\theta_{1523}$ and so $\theta_{1523}(s)=0$.
Finally, using Equation \eqref{equation2} with $p=5$ $m=2$ and $a=1,b=3,c=4$ and applying $x_4\partial_5$ we have
\[
h_{45}(\theta_{1325}(s)=0;
\]
as above, this implies $\theta_{1325}(s)=0$ and so also $\theta_{1235}(s)=0$.
For $p=3$ the system is equivalent to $\theta_{1425}(s)=\theta_{1524}(s)=0$ and $\theta_{1245}(s)=2\sigma_3(s)$.
We now compute the weight of $\theta_{1245}(s)$; we use Equation \eqref{equation2} with $a=1$, $b=2$, $c=3$, $m=4$, $p=5$ and we apply $x_3\partial _5$ we obtain
\[
h_{35}(\theta_{1245}(s)=\theta_{1245}(s);
\]
Similarly, using Equation \eqref{equation2} with $a=1$, $b=4$, $c=5$, $m=1$, $p=2$ and applying $x_1\partial_2$ we obtain
\[
h_{12}(\theta_{1245}(s)=0;
\]
moreover Equation 1 with $a=2$, $b=4$, $c=5$ and $k=1$, $l=2$ $p=3$ and applying $x_2\partial_3$ we obtain
\[
h_{23}(\theta_{1245}(s)=0;
\]
since the weight of $\theta_{1245}$ is $(0,-1,1,0)$ we conclude that the weight of $\theta_{1245}(s)$ is $(0,0,1,0)$ and so the weight of $s$ is $(0,1,0,0)$ and therefore $\varphi=\nabla_{AC}$.
Now suppose $b<p$. If we apply $x_b\partial_p$ to Equation (\ref{equation2m=a}) the we obtain
the following equation:
\[
\frac{1}{2}(-1+(-1)^{\chi_{c<p}})(-\theta_{ap,ac}(v))+
\chi_{a<p}(x_b\partial_a\theta_{bc,pa}(v)-x_p\partial_a\theta_{pc,pa}(v))
-h_{bp}\theta_{ac,pa}(v)+\chi_{c<p}(x_b\partial_c\theta_{ab,pa}(v))=0
\]
Now suppose that $\theta_{ij,rs}(v)=0$ for every $(ijrs)>(acpa)$. Then
\[
h_{bp}\theta_{ac,pa}(v)=
(\chi_{c<p}-\chi_{b<a<p}-\chi_{b<c<p})\theta_{ac,pa}(v)
\]
We also know that
\[
x_5d_{45}\varphi(v)=0
\]
for all $v\in F(m,n,p,q)$. We have
\[
x_5d_{45} \varphi(v)=\sum_{(i,j)<(k,l)}(x_5d_{45}. \omega_{ijkl})\otimes\theta_{ijkl}(v)-d_{45}\otimes \sigma_5(v)
\]
(I do not like this notation as there is no action of $L$ on $U_-$...)
We compute summands separately (always assuming $(i,j)<(k,l)$: we have
\[
(x_5d_{45})\omega_{1234}\otimes v=(x_5d_{45})(d_{12}d_{34}-\frac{1}{2}\partial_5)\otimes v=-\frac{1}{2}d_{45}\otimes v+d_{34}x_5\partial_3\otimes v
\]
\[
(x_5d_{45})\omega_{1324}\otimes v=(x_5d_{45})(d_{13}d_{24}+\frac{1}{2}\partial_5)\otimes v=\frac{1}{2}d_{45}\otimes v-d_{24}x_5\partial_2\otimes v
\]
\[
(x_5d_{45})\omega_{1423}\otimes v=(x_5d_{45})(d_{14}d_{23}-\frac{1}{2}\partial_5)\otimes v=-d_{14}\otimes x_5\partial_1.v+\frac{1}{2}d_{45}\otimes v
\]
If $t\neq 5$ we have $x_5d_{45} .\partial_t=0$ and so the term $\partial_t$ can be ignored.
If $\{i,4,k,l\}\neq \{1,2,3,4\}$ we have
\[
x_5d_{45}.\omega_{i4kl}=0.
\]
If $\{k,l\}\neq \{2,3\}$
\[
x_5d_{45}.\omega_{15kl}=0
\]
and \[
x_5d_{45}.\omega_{1523}=-d_{15}x_5\partial_1
\]
We have
\[x_5d_{45}\omega_{12kl}\otimes v=\begin{cases}
d_{kl}\otimes x_5\partial_3.v&\textrm{if }(k,l)\neq (1,3),(2,3), (3,4)\\
d_{15}\otimes v+d_{12}\otimes x_5\partial_2. v+d_{13}\otimes x_5\partial_ 3 .v&\textrm{if }(k,l)=(1,3)\\
d_{25}\otimes v-d_{12}\otimes x_5\partial_1.v+d_{23} \otimes x_5\partial_ 3.v &\textrm{if }(k,l)=(2,3)\\
\end{cases}
\]
\[
x_5d_{45}.\omega_{13kl}=\begin{cases}
-d_{kl}\otimes x_5\partial_2.v&\textrm{if }(k,l)\neq (2,3), (2,4)\\
d_{35}\otimes v-d_{13}\otimes x_5\partial _1\otimes v-d_{23}\otimes x_5\partial_2.v& \textrm{if }(i,j)=(2,3)
\end{cases}
\]
\[
x_5d_{45}.\omega_{23kl}=d_{kl}\otimes x_5\partial_1.v
\]
So the condition $x_5d_{45}\varphi(v)=0$ is thus equivalent to the following system
\[
\begin{cases}
-\frac{1}{2}\theta_{1234}(v)+\frac{1}{2}\theta_{1324}(v)+\frac{1}{2}\theta_{1423}(v)-\sigma_5(v)+x_5\partial_3.(\theta_{1245}(v))-x_5\partial_2.(\theta_{1345}(v))+x_5\partial_1.(\theta_{2345}(v))=0\\
-x_5\partial_1. \theta_{1423}(v)+x_5\partial_3.(\theta_{1214}(v))-x_5\partial_2.(\theta_{1314}(v))=0\\
-x_5\partial_1.( \theta_{1523}(v))+x_5\partial_3.(\theta_{1215}(v))+\theta_{1213}(v)-x_5\partial_2.(\theta_{1315}(v))=0\\
x_5\partial_3. (\theta_{1213}(v))-x_5\partial_1.(\theta_{1323}(v))=0\\
x_5\partial_3.(\theta_{1223}(v))-x_5\partial_2.(\theta_{1323}(v))=0\\
x_5\partial_2.(\theta_{1213}(v))-x_5\partial_1.(\theta_{1223}(v))=0\\
x_5\partial_3.(\theta_{1234}(v))-x_5\partial_2.(\theta_{1334}(v))+x_5\partial_1.(\theta_{2334}(v))=0\\
-x_5\partial_2.(\theta_{1324}(v))+x_5\partial_3.(\theta_{1224}(v))+x_5\partial_1.(\theta_{2324}(v))=0\\
x_5\partial_3. \theta_{1235}(v)-x_5\partial_2.(\theta_{1335}(v))+x_5\partial_1.(\theta_{2335}(v))+\theta_{1323}(v)=0\\
x_5\partial_3. \theta_{1225}(v)-x_5\partial_2.(\theta_{1325}(v))+x_5\partial_1.(\theta_{2325}(v))+\theta_{1223}(v)=0\\
\end{cases}
\]
In particular, if $a$ is a lowest weight vector in $F(m,n,p,q)$ this system implies
$\theta_{1213}(a)=\theta_{1223}(a)=\theta_{1323}(a)=0$ and
\[
\frac{1}{2}\theta_{1234}(a)-\frac{1}{2}\theta_{1324}(a)-\frac{1}{2}\theta_{1423}(a)-\sigma_5(a)=0
\]
\dots
\end{document} | math | 112,263 |
\begin{document}
\title[Complexity of some classes of Banach spaces.]{On the complexity of some inevitable classes of separable Banach spaces.}
\subjclass[2010]{Primary: 46B20}
\keywords{ Minimal Banach spaces, continuously tight Banach spaces, HI spaces, descriptive set theory, trees, Banach spaces dichotomies.}
\author{ B. M. Braga.}
\address{Department of Mathematics, Statistics, and Computer Science (M/C 249)\\
University of Illinois at Chicago\\
851 S. Morgan St.\\
Chicago, IL 60607-7045\\
USA}\email{[email protected]}
\date{}
\date{}
\begin{abstract}
In this paper, we study the descriptive complexity of some inevitable classes of Banach spaces. Precisely, as shown in \cite{Go}, every Banach space either contains a hereditarily indecomposable subspace or an unconditional basis, and, as shown in \cite{FR}, every Banach space either contains a minimal subspace or a continuously tight subspace. In these notes, we study the complexity of those inevitable classes as well as the complexity of containing a subspace in any of those classes.
\end{abstract}
\maketitle
\section{Introduction.}\label{intro}
Let $\text{SB}=\{X\subset C(\Delta)\mid X\text{ is linear and closed}\}$ be our coding for the separable Banach spaces, and endow $\text{SB}$ with the Effros-Borel structure (for definitions, see Section \ref{definition}). Given a separable Banach space $X\in\text{SB}$, it is natural to ask about the complexity of the isomorphism class $\langle X\rangle=\{Y\in\text{SB}\mid Y\cong X\}$. Is $\langle X\rangle$ Borel, analytic, coanalytic? This question has only been solved for some very specific Banach spaces. For example, S. Kwapien showed (see \cite{Kw}, Proposition $3.1$) that a Banach space $X$ is isomorphic to a Hilbert space if, and only if, $X$ has both type and cotype equal $2$. As $\ell_2$ is the only separable Hilbert space (up to isomorphism), this gives us a characterization of separable Banach spaces which are isomorphic to $\ell_2$ in terms of type and cotype. Using this characterization, it is not hard to show that $\langle \ell_2\rangle$ is Borel (see \cite{B}, page $130$). In fact, $\langle \ell_2\rangle$ is the only known example of a Borel isomorphism class. We recall the following problem (see \cite{B}, Problem 2.9).
\begin{problem}
Let $X\in\text{SB}$ be a separable Banach space whose isomorphism class $\langle X\rangle$ is Borel. Is $X$ isomorphic to $\ell_2$?
\end{problem}
For some classical spaces, such as $L_p[0,1]$, with $p\neq 2$, Pelczynski's universal space $U$, and $C(\Delta)$, it had been shown that their isomorphism classes are analytic non Borel (see \cite{B}, page $130$ and Theorem $2.3$, and \cite{K}, Theorem $33.24$, respectively).
The problem of classifying Banach spaces up to isomorphism is extremely complex. Indeed, considering the isomorphism relation $\cong$ as a subset of $\text{SB}\times\text{SB}$, we have that $\cong$ is a complete analytic equivalence relation (see \cite{FLR}, Theorem $31$). In a more precise sense, separable Banach spaces up to isomorphism may serve as a complete invariant for any other reasonable class of mathematical objects. Therefore, it makes sense to study easier problems such as classifying Banach spaces up to its subspaces. We have then another natural problem: given a Banach space $X\in\text{SB}$, what can we say about the complexity of $\text{C}_X=\{Y\in\text{SB}\mid X\emb Y\}$? This question had been solved by B. Bossard (see \cite{B}, Corollary $3.3$).
\begin{thm}\textbf{(Bossard)}\label{nonborel}
Let $X\in\text{SB}$. If $X$ is finite dimensional, then $\text{C}_X$ is Borel. If $X$ is infinite dimensional, then $\text{C}_X$ is complete analytic.
\end{thm}
In \cite{Go2}, W. T. Gowers had started with a new classification theory for Banach spaces. Indeed, after Gowers and Maurey had solved the unconditional basis problem, and proved the existence of hereditarily indecomposable Banach spaces (see \cite{GM}), Gowers proved that every Banach space either contains a subspace with an unconditional basis, or a hereditarily indecomposable subspace (see \cite{Go}). Later, in \cite{Go2}, Gowers used Ramsey methods in order to refine his dichotomy. Many other dichotomies had been proved by V. Ferenczi and C. Rosendal in \cite{FR}, for example, their main dichotomy says that every Banach space either contains a minimal subspace or a continuously tight subspace (see \cite{FR}, Theorem 3.13).
In these notes, we study the descriptive complexity of the inevitable classes above as well as the complexity of containing a subspace in any of those inevitable classes. The table below summarizes the lower and upper bounds known for the complexity of those classes. In the table below, an asterisk before the complexity indicates that this bound was already known before this paper. All the other bounds are either computed in this paper or are given by
trivial computations.
\begin{center}
\begin{tabular}{ lll }
\hline
Classes of Banach spaces& Lower bound&Upper bound\\
\hline
Minimal spaces&&$\ \ \ \ \ \Delta^1_2$ \\
Containing a minimal subspace\ \ \ \ \ &$\ \ \ \ \ \Sigma^1_1$-hard &$(*)\ \Sigma^1_2$ \\
Continuously tight spaces&$\ \ \ \ \ \Pi^1_1$-hard& $\ \ \ \ \ \Sigma^1_2$\\
Containing a continuously tight subspace&$\ \ \ \ \ \Sigma^1_1$-hard&$\ \ \ \ \ \Sigma^1_2$ \\
HI spaces&$\ \ \ \ \ \Pi^1_1$-hard \ \ \ &$\ \ \ \ \ \Pi^1_1$\\
Containing a HI subspace&$\ \ \ \ \ \Sigma^1_1$-hard&$\ \ \ \ \ \Sigma^1_2$\\
Spaces with unconditional basis & &$\ \ \ \ \ \Sigma^1_1$\\
Containing unconditional basis&$\ \ \ \ \ \Sigma^1_1$-hard&$\ \ \ \ \ \Sigma^1_1$\\
\hline
\end{tabular}
\end{center}
In particular, we completely compute the descriptive complexity of the class of hereditarily indecomposable spaces, and the class of spaces containing an unconditional basis, by showing those classes are complete coanalytic and complete analytic, respectively.
The properties of being a continuously tight space and of having an unconditional basis are more properties about basis then about the spaces themselves. Therefore, it is more natural to ask what is the complexity of the set of continuously tight basis, and the complexity of the set of unconditional basis. We have the following.
\begin{center}
\begin{tabular}{ lll }
\hline
Classes of basis& Lower bound&Upper bound\\
\hline
Continuously tight basis& $\ \ \ \ \ \Pi^1_1$-hard\ \ \ \ &$\ \ \ \ \ \Sigma^1_2$ \\
Unconditional basis\ \ \ \ \ \ & &\ \ \ \ \ Borel\\
\hline
\end{tabular}
\end{center}
This paper is organized in the following way. In Section \ref{definition}, we give all the basic background on both descriptive set theory and the theory of Banach spaces necessary for these notes. In Section \ref{sectionlema}, we prove a general lemma for Banach spaces with an unconditional basis $(x_{s})_{s\in\theta}$ indexed by a tree $\theta\in\text{WF}$. In Section \ref{sectionlp}, we work with $\ell_p$-Baire sums of basic sequences as a tool to compute lower bounds for the complexity of classes of Banach spaces. Then, Theorem \ref{lema} gives a sufficient condition for the set of separable Banach spaces containing a subspace in a given class to be $\Sigma^1_1$-hard (in particular, non Borel).
Section \ref{newnew} is dedicated to the computation of the complexities in the tables above. We apply Theorem \ref{lema} to the class of spaces containing a hereditarily indecomposable subspace (Subsection \ref{masha}, Corollary \ref{hhii}), and to the class of spaces containing a continuously tight subspace (Subsection \ref{sectiontight}, Corollary \ref{cu}), obtaining a lower bound for the complexity of those classes.
In order to obtain the results above, we rely on the method of $\ell_p$-Baire sums of separable Banach spaces (see Lemma \ref{arroto}). However, this method does not allow us to obtain any information about the complexity of the class of spaces (i) containing an unconditional basis, (ii) containing a minimal subspace, or (iii) containing a continuously tight subspace. Hence, in Subsection \ref{sectionminimal}, we define a parameterized version of Tsirelson space in order to show that the class of spaces containing a minimal subspace is $\Sigma^1_1$-hard. Such parametrization will show that it is impossible to Borel separate the set of separable Banach spaces containing $c_0$ from the set of Tsirelson-saturated Banach spaces (see Theorem \ref{lemaminimal}). In Subsection \ref{sectiontightmesmo}, we use the methods of Subsection \ref{sectionminimal} in order to show that the class of continuously tight spaces is $\Pi^1_1$-hard (Theorem \ref{tighttighthu}).
At last, in Subsection \ref{hiGM}, we follow Argyros presentation of how to construct HI extensions of ground norms (\cite{AT}, Chapters II and III) in order to show that the set of hereditarily indecomposable separable Banach spaces is complete coanalytic (Theorem \ref{hhhhiiii}). We also obtain that the set of hereditary indecomposable Banach spaces cannot be Borel separated from the set of Banach spaces containing $\ell_1$. By the same methods, in Subsection \ref{ubub}, we show that the class of Banach spaces containing an unconditional basis is complete analytic (Theorem \ref{ubca}).
\begin{problem}
For all the classes in the tables above, we have only completely computed the descriptive complexity of the set of HI spaces, and the set of spaces containing an unconditional basis. What are the exact complexities of the remaining classes?
\end{problem}
At last, we would like to make a small remark. In \cite{FR}, the authors actually present Theorem 1.1 as their main dichotomy, which says that every Banach space contains either a minimal subspace or a tight subspace. This dichotomy does not explicitly say anything about continuously tight spaces. However, their proof for Theorem 1.1 shows that any Banach space $X$ must contain either a minimal Banach subspace or a continuously tight Banach subspace (see \cite{FR}, Theorem 3.13).
We have two reasons for choosing to deal with continuously tight spaces instead of tight spaces, (i) we easily get better estimates for the upper bound of continuously tight spaces and spaces containing a continuously tight subspace, (ii) Rosendal had shown (see \cite{R}, Appendix) that, for minimal Banach spaces not containing $c_0$, there exists a notion of being continuously minimal, and this notion coincides with being minimal (see Subsection \ref{outrodia}).
\section{Background.}\label{definition}
A separable metric space is said to be a \emph{Polish space} if there exists an equivalent metric in which it is complete. A continuous image of a Polish space into another Polish
space is called an \emph{analytic} set and a set whose complement is analytic is called \emph{coanalytic}. A measure space $(X,\mathcal{A})$, where $X$ is a set and $\mathcal{A}$ is a
$\sigma$-algebra of subsets of $X$, is called a \emph{standard Borel space} if there exists a Polish topology on this set whose Borel $\sigma$-algebra coincides with $\mathcal{A}$. We
define Borel, analytic and coanalytic sets in standard Borel spaces by saying that these are the sets that, by considering a compatible Polish topology, are Borel, analytic, and
coanalytic, respectively. Observe that this is well defined, i.e., this definition does not depend on the Polish topology itself but on its Borel structure. A function between two
standard Borel spaces is called \emph{Borel measurable} if the inverse image of each Borel subset of its codomain is Borel in its domain. We usually refer to Borel measurable functions
just as Borel functions. Notice that, if you consider a Borel function between two standard Borel spaces, its inverse image of analytic sets (resp. coanalytic) is analytic (resp.
coanalytic) (see \cite{S}, Proposition $1.3$, page $50$).
Given a Polish space $X$, the set of analytic (resp. coanalytic) subsets of $X$ is denoted by $\Sigma^1_1(X)$ (resp. $\Pi^1_1(X)$). We usually omit $X$, and simply write $\Sigma^1_1$, or $\Pi^1_1$. Hence, the terminology $\Sigma^1_1$-set (resp. $\Pi^1_1$-set) is used to refer to analytic sets (resp. coanalytic sets). We define the \emph{projective hierarchy} by induction of $n$. We say a subset of a standard Borel space is $\Pi^1_n$ if it is the complement of a $\Sigma^1_n$-set, and we say that a subset is $\Sigma^1_{n+1}$ if it is a Borel image of a $\Pi^1_n$-set (see \cite{D}, Chapter 1). For each $n\in\mathbb{N}$, we denote by $\Delta^1_n$ the subsets of a standard Borel space which are both $\Sigma^1_n$ and $\Pi^1_n$, i.e., $\Delta^1_n=\Sigma^1_n\cap \Pi^1_n$.
Let $X$ be a standard Borel space. An analytic (resp. coanalytic) subset $A\subset X$ is said to be \emph{complete analytic} (resp. \emph{complete coanalytic}) if for every standard Borel space $Y$ and every analytic subset $B\subset Y$ (resp. coanalytic), there exists a Borel function $f:Y\to X$ such that $f^{-1}(A)=B$. This function is called a \emph{Borel reduction} from $B$ to $A$, and $B$ is said to be \emph{Borel reducible} to $A$.
Let $X$ be a standard Borel space. We call a subset $A\subset X$ \emph{$\Sigma^1_n$-hard} (resp. \emph{$\Pi^1_n$-hard}) if for all standard Borel space $Y$ and all $\Sigma^1_n$-set $B\subset Y$ (resp. $\Pi^1_n$-set) there exists a Borel reduction from $B$ to $A$. Therefore, saying that a set $A\subset X$ is $\Sigma^1_n$-hard (resp. $\Pi^1_n$-hard) means that $A$ is at least as complex as any $\Sigma^1_n$-set (resp. $\Pi^1_n$-set) in the projective hierarchy. With this terminology we have that $A\subset X$ is $\Sigma^1_n$-complete (resp. $\Pi^1_n$-complete) if, and only if, $A$ is $\Sigma^1_n$-hard (resp. $\Pi^1_n$-hard) and $\Sigma^1_n$ (resp. $\Pi^1_n$).
As there exist analytic non Borel (resp. coanalytic non Borel) sets we have that $\Sigma^1_1$-hard (resp. $\Pi^1_1$-hard) sets are non Borel. Also, if $X$ is a standard Borel space, $A\subset X$, and there exists a Borel reduction from a $\Sigma^1_1$-hard (resp. $\Pi^1_1$-hard) subset $B$ of a standard Borel space $Y$ to $A$, then $A$ is $\Sigma^1_1$-hard (resp. $\Pi^1_1$-hard). If $A$ is also analytic (resp. coanalytic), then $A$ is $\Sigma^1_1$-complete (resp. $\Pi^1_1$-complete).
Consider a Polish space $X$ and let $\mathcal{F}(X)$ be the set of all its non empty closed sets. We endow $\mathcal{F}(X)$ with the \emph{Effros-Borel structure}, i.e., the $\sigma$-algebra
generated by
$$\{F\subset X \mid F\cap U\neq \emptyset\},$$\\
\noindent where $U$ varies among the open sets of $X$. It can be shown that $\mathcal{F}(X)$, endowed with the Effros-Borel structure, is a standard Borel space (see \cite{K}, Theorem $12.6$). The following well known lemma (see
\cite{K}, Theorem $12.13$) will be crucial in some of our proofs.
\begin{lemma}\textbf{(Kuratowski and Ryll-Nardzewski selection principle)}\label{lll}
Let $X$ be a Polish space. There exists a sequence of Borel functions $(S_n)_{n\in\mathbb{N}}:\mathcal{F}(X)\to X$ such that $\{S_n(F)\}_{n\in\mathbb{N}}$ is dense in $F$, for all closed $F\subset X$.
\end{lemma}
In these notes we will only work with separable Banach spaces. We denote the closed unit ball of a Banach space $X$ by $B_X$. It is well
known that every separable Banach space is isometrically isomorphic to a closed linear subspace of $C(\Delta)$ (see \cite{K}, page $79$), where $\Delta$ denotes the Cantor set. Therefore,
$C(\Delta)$ is called \emph{universal} for the class of separable Banach spaces and we can code the class of separable Banach spaces, denoting it by $\text{SB}$, by
$\text{SB}=\{X\subset C(\Delta)\mid X \text{ is a closed }\allowbreak\text{linear subspace of }\allowbreak C(\Delta)\}$. As $C(\Delta)$ is clearly a Polish space we can endow
$\mathcal{F}(C(\Delta))$ with the Effros-Borel structure. It can be shown that $\text{SB}$ is a Borel set in $\mathcal{F}(C(\Delta))$ and hence it is also a standard Borel space (see \cite{D}, Theorem $2.2$). It now makes
sense to wonder if specific classes of separable Banach spaces are Borel or not (in our coding $\text{SB}$).
Throughout these notes we will denote by $\{S_n\}_{n\in\mathbb{N}}$ the sequence of Borel functions $S_n:\text{SB}\to C(\Delta)$ given by \text{Lemma \ref{lll}}, with $X=C(\Delta)$ (more precisely, the restriction of those functions to $\text{SB}$).
Consider the standard Borel space $C(\Delta)^\mathbb{N}$, and let
$$\mathcal{B}=\{(x_j)_{j\in\mathbb{N}}\in C(\Delta)^\mathbb{N}\mid (x_j)_{j\in\mathbb{N}}\text{ is a basic sequence}\}.$$\\
\noindent It is easy to see that $\mathcal{B}$ is a Borel set. Therefore, we have that $\mathcal{B}$ is a standard Borel space, and we can code all the basic sequences as elements of $\mathcal{B}$. We can now wonder about the descriptive complexity of some specific classes of basis.
Let $(e_n)_{n\in\mathbb{N}}$ be a basic sequence in a Banach space $X\in\text{SB}$. By a standard Skolem hull construction, there is a countable subfield $\mathbb{F}$ of $\mathbb{R}$ containing the rationals such that for any finite linear combination
$$\lambda_0 e_0+...+\lambda_n e_n,$$\\
\noindent with $n\in\mathbb{N}$, and $\lambda_1,...,\lambda_n\in\mathbb{F}$, we have $\|\lambda_0 e_0+...+\lambda_n e_n\|\in\mathbb{F}$. By working with $\mathbb{F}$ instead of $\Q$, we guarantee that any $\mathbb{F}$-linear combination of $(e_n)_{n\in\mathbb{N}}$ can be normalized and remain a $\mathbb{F}$-linear combination of $(e_n)_{n\in\mathbb{N}}$. Clearly, the $\mathbb{F}$-span of $(e_n)_{n\in\mathbb{N}}$ is dense in $\overline{\text{span}}\{e_n\}$.
Let $D$ be the set of normalized blocks of $(e_n)_{n\in\mathbb{N}}$, i.e., the set of all $\lambda_0 e_0+...+\lambda_n e_n,$ with $n\in\mathbb{N}$, and $\lambda_1,...,\lambda_n\in\mathbb{F}$. Clearly, $D$ is countable. A \emph{block sequence} of $(e_n)_{n\in\mathbb{N}}$ is a sequence $(y_n)_{n\in\mathbb{N}}$ such that, (i) there exist increasing sequences of natural numbers $(p_n)_{n\in\mathbb{N}}$ and $(q_n)_{n\in\mathbb{N}}$ such that $p_i\leq q_i<p_{i+1}$, for all $i\in\mathbb{N}$, and (ii) for all $n\in\mathbb{N}$
$$y_n=\sum_{j=p_n}^{q_n}a_je_j,$$\\
\noindent for some sequence $(a_n)_{n\in\mathbb{N}}$ of real numbers. We define a \emph{finite block sequence} analogously. Denote by $bb(e_n)$ the set of normalized block sequences with the coefficients $(a_n)_{n\in\mathbb{N}}$ in $\mathbb{F}$. Endowing $D$ with the discrete topology, we can see $bb(e_n)$ as a closed subset of $D^\mathbb{N}$, so $bb(e_n)$ is a standard Borel space. Denote by $fbb(e_n)$ the set of normalized finite block sequences with the coefficients $(a_n)_{n\in\mathbb{N}}$ in $\mathbb{F}$. Letting $D^{<\mathbb{N}}=\cup_nD^n$, we can see $fbb(e_n)$ as a closed subset of $D^{<\mathbb{N}}$. Hence, $fbb(e_n)$ is a standard Borel space.
Let $X$ and $Y$ be Banach spaces. We write $X\hookrightarrow Y$ if $X$ can be linearly embedded into $Y$. If $K>0$, we write $X\hookrightarrow_K Y$ if $X$ can be $K$-embedded in $Y$, i.e., if there exists an embedding $T:X\to Y$ such that $\|T\|\|T^{-1}\|\leq K$. If $(x_n)_{n\in\mathbb{N}}$ and $(y_n)_{n\in\mathbb{N}}$ are two sequences in Banach spaces, we write $(x_n)_{n\in\mathbb{N}}\approx (y_n)_{n\in\mathbb{N}}$ if $(x_n)_{n\in\mathbb{N}}$ is equivalent to $(y_n)_{n\in\mathbb{N}}$, i.e., if the map $x_n\mapsto y_n$ induces a linear isomorphism between $\overline{\text{span}}\{x_n\}$ and $\overline{\text{span}}\{y_n\}$. Similarly, if $K>0$, we write $(x_n)_{n\in\mathbb{N}}\approx_K (y_n)_{n\in\mathbb{N}}$ if the induced isomorphism if a $K$-isomorphism.
Denote by $\mathbb{N}N$ the set of all finite tuples of natural numbers plus the empty set. Given $s=(s_0,...,s_{n-1}),\allowbreak t=(t_0,...,t_{m-1})\in\mathbb{N}N$ we say that the length of $s$ is
$|s|=n$, $s_{|i}=(s_0,...,s_{i-1})$, for all $i\in\{1,...,n\}$, $s_0=\{\emptyset\}$, $s\preceq t$ \emph{iff} $n\leqslant m$ and $s_i=t_i$, for all $i\in\{0,...,n-1\}$, i.e., if $t$ is an
extension of $s$. We define $s<t$ analogously. Define the concatenation of $s$ and $t$ as $s^\smallfrown t=(s_0,...,s_{n-1},t_0,...,t_{m-1})$. A subset $T$ of $\mathbb{N}N$ is called a
\emph{tree} if $t\in T$ implies $t_{|i}\in T$, for all $i\in\{0,...,|t|\}$. We denote the set of trees on $\mathbb{N}$ by $\text{Tr}$. A subset $I$ of a tree $T$ is called a segment if $I$ is
completely ordered and if $s,t\in I$ with $s\preceq t$, then $l\in I$, for all $l\in T$ such that $s\preceq l\preceq t$. Two segments $I_1,\ I_2$ are called completely incomparable if neither
$s\preceq t$ nor $t\preceq s$ hold if $s\in I_1$ and $t\in I_2$.
As $\mathbb{N}N$ is countable, $2^\mathbb{N}N$ (the power set of $\mathbb{N}N$) is Polish with its standard product topology. If we think about $\text{Tr}$ as a subset of $2^\mathbb{N}N$, it is easy to see that $\text{Tr}$ is a $G_\delta$ set in $2^\mathbb{N}N$. Thus, it is also
Borel in $2^\mathbb{N}N$. As $\text{Tr}$ is Borel in the Polish space $2^{\mathbb{N}N}$, we have that $\text{Tr}$ is a standard Borel space. A $\beta\in\mathbb{N}^\mathbb{N}$ is called a \emph{branch} of
a tree $T$ if $\beta_{|i}\in T$, for all $i\in\mathbb{N}$, where $\beta_{|i}$ is defined analogously as above. We call a tree $T$ \emph{well-founded} if $T$ has no branches and
\emph{ill-founded} otherwise, we denote the set of well-founded and ill-founded trees by $\text{WF}$ and $\text{IF}$, respectively. It is well known that $\text{WF}$ is a complete coanalytic set of
$\text{Tr}$, hence $\text{IF}$ is complete analytic (see \cite{K}, Theorem $27.1$).
\section{A Lemma.}\label{sectionlema}
In this section, we prove a basic lemma that will be essential in many of the main results of this paper. Fix a compatible enumeration of $\mathbb{N}N$, i.e., a sequence $(s_n)_{n\in\mathbb{N}}$ in $\mathbb{N}N$ such that $s_n\preceq s_m$ implies $n\leqslant m$ and for all $s\in\mathbb{N}N$ there exists $n\in\mathbb{N}$ such that $s_n=s$. With this enumeration in mind, if $\theta\in\text{Tr}$, we say that a sequence $(x_s)_{s\in\theta}$ is a basis for a given Banach space $X$, if $(x_{s_n})_{n\in N}$ is a basis for $X$, where $N=\{n\in\mathbb{N}\mid s_n\in \theta\}$. We now show the following.
\begin{lemma}\label{lemageral}
Let $\theta\in\text{WF}$, and let $X$ be a Banach space with an unconditional basis $(e_s)_{s\in\theta}$. Let $Y$ be an infinite dimensional subspace of $X$. Then $Y$ contains a basic sequence $(y_k)_{k\in\mathbb{N}}$ equivalent to a semi-normalized block sequence $(x_k)_{k\in\mathbb{N}}$ of $(e_s)_{s\in\theta}$ with completely incomparable supports.
\end{lemma}
Before we prove this lemma, let's show a simple lemma that will be important in our proof. We say that an operator $T:X\to Y$ is \emph{strictly singular} if for all infinite dimensional subspace $Z\subset X$, $T_{|Z}:Z\to Y$ is not an embedding.
\begin{lemma}\label{u8u}
Let $(X_1,\|\cdot\|_1),...,(X_n,\|\cdot\|_n)$ be Banach spaces, and let $Y\subset \oplus_{i=1}^n X_i$ be an infinite dimensional subspace. Consider the standard projections $P_j:\oplus_{i=1}^nX_i\to X_j$, for all $j\in\{1,...,n\}$. Then, there exists $j\in\{1,...,n\}$ such that $P_j:Y\to X_j$ is not strictly singular.
\end{lemma}
\begin{proof}
Let $X=\oplus_{i=1}^nX_i$. As this is a finite sum, we can assume $X=(\oplus_{j=1}^nX_j)_{\ell_1}$, i.e., if $(x_1,...,x_n)\in X$, then $\|x\|_X=\sum_j\|x_j\|_{j}$. Assume towards a contradiction that $P_j$ is strictly singular, for all $j\in\{1,...,n\}$. By a classic property of strictly singular operators (see \cite{D}, Proposition B.5), we know that for all $\varepsilon>0$ there exists an infinite dimensional subspace $A\subset Y$ such that $\|P_{j|A}\|<\varepsilon$, for all $j\in\{1,...,n\}$. Pick $x\in A$ of norm one. Then, as $x=(P_1(x),...,P_n(x))$, we have $\|x\|_X\leqslant n\varepsilon$. By choosing $\varepsilon<1/n$ we get a contradiction.
\end{proof}
\noindent\emph{Proof of Lemma \ref{lemageral}.}
For each $s\in\theta$, let $\Lambda_s=\{\lambda\in\mathbb{N}\mid s^\smallfrown(\lambda)\in \theta\}$, and enumerate each $\Lambda_s$, say $\Lambda_s=\{\lambda^i_s\mid i\in\mathbb{N}\}$. For each $s\in\theta$, let $\theta_{s}=\{\tau\in\theta\mid
s\preceq\tau\}$.
For each $s\in\theta$, let $P_s:X\to X$ be the projection of $X$ onto
$$\overline{\text{span}} \{e_\tau\mid s\preceq \tau\}.$$\\
\noindent For each $s\in\theta$, and each $n\in\mathbb{N}$, consider the projections
\begin{align*}
Q_{s,n}:\ \ \ X\ \ &\to \oplus_{i=1}^nP_{s^\smallfrown{(\lambda^i_s)}}(X)\\
(a_\tau)_{\tau \in \theta}&\to (a_\tau)_{\tau\in\cup_{i=1}^n \theta_{s^\smallfrown{(\lambda^i_s)} } }.
\end{align*}\\
\indent\textbf{Claim:} There exists $s\in\theta$ such that $P_s:Y\to X$ is not strictly singular, but $Q_{s,n}:Y\to \oplus_{i=1}^nP_{s^\smallfrown{(\lambda^i_s)}}(X)$ is strictly singular, for all $n\in\mathbb{N}$.\\
Let's assume the claim is true and finish the proof of the lemma. Indeed, if $P_s:Y\to X$ is not strictly singular, we can substitute $Y$ by an infinite dimensional $Z\subset Y$ such that $P_s:Z\to X$ is an isomorphism onto its image. Let $E=P_s(Z)$, and notice that
$$E\subset \overline{\text{span}} \{e_\tau\mid s\preceq \tau\}.$$\\
\noindent By Lemma \ref{u8u}, we can actually assume that $E\subset \overline{\text{span}} \{e_\tau\mid s\prec \tau\}$. Hence, for all $x\in E$, we have that $x=\lim_n Q_{s,n}(x)$.\\
\textbf{Claim:} There exists a normalized sequence $(y_j)_{j\in\mathbb{N}}$ in $E$ such that $Q_{s,n}(y_j)\to 0$, as $j\to \infty$, $\forall n\in\mathbb{N}$.\\
Indeed, for all $n\in\mathbb{N}$, there exists a normalized sequence $(y^n_j)_{j\in\mathbb{N}}$ in $E$ such that
$$\|Q_{s,n}(y^n_j)\|<1/j , \text{ \ for all\ } j\in\mathbb{N}.$$\\
\indent Let $(y_j)_{j\in\mathbb{N}}$ be the diagonal sequence of the sequences $(y^n_j)_{j\in\mathbb{N}}$, i.e., $y_j=y^j_j$, for all $j\in\mathbb{N}$. Say $M$ is the unconditional constant of $(e_s)_{s\in\theta}$. Then, $m\leqslant n$
implies $\|Q_{s,m}(x)\|\leqslant M \|Q_{s,n}(x)\|$, for all $x\in E$. Hence, $(y_j)_{j\in\mathbb{N}}$ has the required property.
Say $(\varepsilon_i)_{i\in\mathbb{N}}$ is a sequence of positive real numbers converging to zero. As $Q_{s,n}(x)\to x$, as $n\to\mathbb{N}$, for all
$x\in E$, we can pick increasing sequences of natural numbers $(n_k)_{k\in\mathbb{N}}$, and $(l_k)_{k\in\mathbb{N}}$ such that
\begin{enumerate}[(i)]
\item $\|Q_{s,{l_k}}(y_{n_k})-y_{n_k}\|_\theta<\varepsilon_k$, for all $k\in\mathbb{N}$, and
\item $\|Q_{s,{l_k}}(y_{n_{k+1}})\|_\theta<\varepsilon_k$, for all $k\in\mathbb{N}$.
\end{enumerate}
For each $k\in \mathbb{N}$, let
$$x_k=Q_{s,{l_k}}(y_{n_k})-Q_{s,{l_{k-1}}}(y_{n_k}).$$ \\
\noindent Choosing $(\varepsilon_k)_{k\in\mathbb{N}}$ converging to zero fast enough, we have that $(x_k)_{k\in\mathbb{N}}$ is semi-normalized and, by the principle of small perturbations, that $(y_{n_k})_{k\in\mathbb{N}}$ is equivalent to $(x_{k})_{k\in\mathbb{N}}$ (see \cite{AK}, \text{Theorem 1.3.9}). Clearly, $(x_k)_{k\in\mathbb{N}}$ has completely incomparable supports. As $Y$ contains a sequence equivalent to $(x_k)_{k\in\mathbb{N}}$, the proof is complete.
We now prove our first claim. Suppose the claim is false, i.e., suppose that for all $s\in\theta$ such that $P_s:Y\to X$ is not strictly singular, there exists $n\in\mathbb{N}$ such that $Q_{s,n}:Y\to \oplus_{i=1}^nP_{s^\smallfrown{(\lambda^i_s)}}(X)$ is not strictly singular. By Lemma \ref{u8u}, if $Q_{s,n}:Y\to \oplus_{i=1}^nP_{s^\smallfrown{(\lambda^i_s)}}(X)$ is not strictly singular, there exists $m\leqslant n$ such that
$$P_{s^\smallfrown \lambda^m_s}=P_{s^\smallfrown \lambda^m_s}\circ Q_{s,n}:Y\to X$$\\
\noindent is not strictly singular. Therefore, for all $s\in \theta$ such that $P_s:Y\to X$ is not strictly singular, there exists $s'\succ s$ such that $P_{s'}:Y\to X$ is not strictly singular.
Now notice that $P_\emptyset:Y\to X$ is not strictly singular, indeed, $P_\emptyset=Id$. Therefore, by applying the last paragraph $\omega$ times, we get a sequence $(s_n)_{n\in\mathbb{N}}$ such that $P_{s_n}:Y\to X$ is not strictly singular, and $s_n\prec s_{n+1}$, for all $n\in\mathbb{N}$. In particular, $s_n\in\theta$, for all $n$, absurd, as $\theta$ is well-founded.\qed
\section{$\ell_p$-Baire sums.}\label{sectionlp}
We now deal with $\ell_p$-Baire sums of basic sequences, this tool will be crucial in many of our results in these notes. Fix a basic sequence $\mathcal{E}=(e_n)_{n\in\mathbb{N}}$, and $p\in[1,\infty)$. Let us define a Borel function $\varphi:\text{Tr}\to\text{SB}$ in the following manner. For each $\theta\in\text{Tr}$ and $x=(x(s))_{s\in\theta}\in c_{00}(\theta)$ we define
\begin{align*}
\left\|x\right\|_{\mathcal{E},p,\theta}=\sup\Big\{\Big(\sum_{i=1}^{n}\big\|\sum_{s\in I_i} x(s)e_{|s|}\big\|^p_{\mathcal{E}}\Big)^{\frac{1}{p}}| \ n\in\mathbb{N},\ I_1, ...,\ I_n &\text{ incomparable}\\ &\ \ \ \text{segments of
}\theta\Big\},
\end{align*}\\
\noindent where $\|.\|_{\mathcal{E}}$ is the norm of $\overline{\text{span}}\{\mathcal{E}\}$. Define $\varphi_{\mathcal{E},p}(\theta)$ as the completion of $c_{00}(\theta)$ under the norm $\left\|.\right\|_{\mathcal{E},p,\theta}$. The space $\varphi_{\mathcal{E},p}(\theta)$ is known as the \emph{$\ell_p$-Baire sum} of $\overline{\text{span}}\{\mathcal{E}\}$ (indexed by $\theta$).
Similarly, we define $\|.\|_{\mathcal{E},0,\theta}$ as
$$\left\|x\right\|_{\mathcal{E},0,\theta}=\sup\Big\{\big\|\sum_{s\in I} x(s)e_{|s|}\big\|_{\mathcal{E}}\mid I \text{ segment of }\theta\Big\},$$\\
\noindent and let $\varphi_{\mathcal{E},0}(\theta)$ be the completion of $(c_{00}(\theta),\|.\|_{\mathcal{E},0,\theta})$. We denote by $(e_s)_{s\in\theta}$ the sequence in $c_{00}(\theta)$ such that, for each $\tau\in\theta$, the coordinate $e_s(\tau)$ equals $1$ if $s=\tau$ and zero otherwise. Considering a compatible enumeration of $\mathbb{N}^{<\mathbb{N}}$, as in Section \ref{sectionlema}, the sequence $(e_s)_{s\in\theta}$ is clearly a basis for $\varphi(\theta)$.
Pick $Y\subset C(\Delta)$ such that $\varphi_{\mathcal{E},p}(\mathbb{N}N)$ is isometrically isomorphic to $Y$. If we consider the natural isometries of $\varphi_{\mathcal{E},p}(\theta)$ into $\varphi_{\mathcal{E},p}(\mathbb{N}N)$, we can see $\varphi_{\mathcal{E},p}$ as a Borel function from $\text{Tr}$ into $\text{SB}$. This gives us the following (see \cite{S}, Proposition 3.1, page 79).
\begin{prop}
Let $p\in[1,\infty)$ or $p=0$. Then, the map $\varphi_{\mathcal{E},p}:\text{Tr}\to\text{SB}$ defined above is Borel.
\end{prop}
The following lemma summarizes the main properties of the $\ell_p$-Baire sum that we will need later in these notes.
\begin{lemma}\label{arroto}
Let $\mathcal{E}$ be a basic sequence. The Borel function $\varphi_{\mathcal{E},p}:\text{Tr}\to\text{SB}$ defined above has the following properties:
\begin{enumerate}[(i)]
\item If $\theta\in\text{IF}$, then $\varphi_{\mathcal{E},p}(\theta)$ contains $\overline{\text{span}}\{\mathcal{E}\}$.
\item If $\theta\in\text{WF}$, then $\varphi_{\mathcal{E},p}(\theta)$ is $\ell_p$-saturated, i.e., every infinite dimensional subspace of $\varphi_{\mathcal{E},p}(\theta)$ contains an isomorphic copy of $\ell_p$.
\end{enumerate}
\noindent The analogous is true for $\varphi_{\mathcal{E},0}:\text{Tr}\to\text{SB}$, i.e.,
\begin{enumerate}[(i)]
\item If $\theta\in\text{IF}$, then $\varphi_{\mathcal{E},0}(\theta)$ contains $\overline{\text{span}}\{\mathcal{E}\}$.
\item If $\theta\in\text{WF}$, then $\varphi_{\mathcal{E},0}(\theta)$ is $c_0$-saturated, i.e., every infinite dimensional subspace of $\varphi_{\mathcal{E},0}(\theta)$ contains an isomorphic copy of $c_0$.
\end{enumerate}
\end{lemma}
\begin{proof} If $\theta\in\text{IF}$, clearly $\varphi_{\mathcal{E},p}(\theta)$ contains $\overline{\text{span}}\{\mathcal{E}\}$. Indeed, let $\beta$ be a branch of $\theta$, then $\overline{\text{span}}\{\mathcal{E}\}\cong\varphi_{\mathcal{E},p}(\beta)\emb\varphi_{\mathcal{E},p}(\theta)$, where by $\varphi_{\mathcal{E},p}(\beta)$ we mean $\varphi_{\mathcal{E},p}$ applied to the tree $\{s\in\mathbb{N}N\mid s<\beta\}$.
Say $\theta\in\text{WF}$, and let $E$ be an infinite dimensional subspace of $\varphi_{\mathcal{E},p}(\theta)$. By Lemma \ref{lemageral}, $E$ has a basic sequence equivalent to a semi-normalized block sequence $(x_k)_{k\in\mathbb{N}}$ with completely incomparable supports. It is trivial to check that a semi-normalized block sequence with completely incomparable supports is equivalent to the $\ell_p$-basis (resp. $c_0$-basis). So we are done.
\end{proof}
Let $\mathcal{P}\subset \text{SB}$. We say that $\mathcal{P}$ is a \emph{class} of Banach spaces if $\mathcal{P}$ is closed under isomorphism, i.e., for all $X,Y\in\text{SB}$, $X\in\mathcal{P}$ and $Y\cong X$ imply $Y\in \mathcal{P}$. We say that a class of Banach spaces $\mathcal{P}\subset \text{SB}$ is \emph{pure} if $X\in\mathcal{P}$ implies $Y\in \mathcal{P}$, for all subspace $Y\subset X$. We say a class $\mathcal{P}\subset\text{SB}$ is \emph{almost-pure} if for all $X\in \mathcal{P}$ and all infinite dimensional $Y\subset X$ there exists an infinite dimensional subspace $Z\subset Y$ such that $Z\in\mathcal{P}$.
\begin{thm}\label{lema}
Say $\mathcal{P}\subset \text{SB}$ is almost-pure and that $\ell_p$ (reps. $c_0$) does not embed in any $Y\in \mathcal{P}$, for some $p\in[1,\infty)$. Then $\text{C}_\mathcal{P}=\{Y\in\text{SB}\mid \exists Z\in \mathcal{P},\ Z\emb Y\}$ is $\Sigma^1_1$-hard. In particular, the same is true if $\mathcal{P}$ is pure and does not contain $\ell_p$ (reps. $c_0$), for some $p\in[1,\infty)$.
\end{thm}
\begin{proof}
This is a simple application of Lemma \ref{arroto}. Indeed, let $\mathcal{E}$ be a basis for $C(\Delta)$, and consider the restriction of $\varphi_{\mathcal{E},p}$ to the set of infinite trees, say $\text{ITr}$. It is easy to see that $\text{ITr}$ is Borel (see \cite{S}, Proposition 1.6, page 72), so ${\varphi_{\mathcal{E},p}}_{|\text{ITr}}$ is a Borel function. By Lemma \ref{arroto}, ${\varphi_{\mathcal{E},p}}_{|\text{ITr}}:\text{ITr}\to \text{SB}$ is a Borel reduction from $\text{IF}$ to $\text{C}_\mathcal{P}$. Therefore, as $\text{IF}$ is $\Sigma^1_1$-hard, $\text{C}_\mathcal{P}$ is $\Sigma^1_1$-hard.
\end{proof}
\section{Descriptive complexity of the inevitable classes.}\label{newnew}
\subsection{Spaces containing a hereditarily indecomposable subspace.}\label{masha}
In 1991 W. T Gowers and B. Maurey independently solved the unconditional basic sequence problem, i.e., they constructed a Banach space with no unconditional basic sequence (see \cite{GM}). It was noticed by W. B. Johnson that the space constructed by Gowers and B. Maurey not only had no unconditional basic sequence but was also hereditarily indecomposable.
We say that an infinite dimensional Banach space $X$ is \emph{hereditatily indecomposable} if none of $X$ subspaces can be decomposed as a direct sum of two infinite dimensional subspaces. Clearly, the class $\text{HI}=\{X\in\text{SB}\mid X\text{ is hereditarily} \allowbreak\text{indecomposable}\}$ is a pure class and it contains no $\ell_p$. Hence, Theorem \ref{lema} gives us the following.
\begin{cor}\label{hhii}
$\text{C}_{\text{HI}}$ is $\Sigma^1_1$-hard.
\end{cor}
We come back to the class of hereditarily indecomposable spaces in Subsection \ref{hiGM}, where we show that the set \text{HI} is complete coanalytic, and that $\text{C}_{\text{HI}}$ is at most $\Sigma^1_2$.
\subsection{Spaces containing a continuously tight subspace.}\label{sectiontight}
In \cite{FR}, Ferenczi and Rosendal defined a new class of Banach spaces, the class of continuously tight spaces, and proved many interesting properties about this class. In Theorem 3.13 of \cite{FR}, for example, Ferenczi and Rosendal have shown that every Banach space must contain either a minimal subspace or a continuously tight subspace, giving us another dichotomy for Banach spaces.
Denote by $[\mathbb{N}]$ the set of increasing sequences of natural numbers. We can see $[\mathbb{N}]$ as a Borel subset of $\mathbb{N}^\mathbb{N}$. A basic sequence $(e_n)_{n\in\mathbb{N}}$ is called \emph{continuously tight} if there exists a continuous function $f:bb(e_n)\to [\mathbb{N}]$ such that, for all block basis $\bar{y}=(y_n)_{n\in\mathbb{N}}\in bb(e_n)$, if we set $I_j=\{m\in\mathbb{N}\mid f(\bar{y})_{2j}\leq m\leq f(\bar{y})_{2j+1}\}$, then for all infinite set $A\subset \mathbb{N}$,
$$\overline{\text{span}}\{\bar{y}\}\not\hookrightarrow\overline{\text{span}}\{e_n\mid n\not\in \cup_{j\in A} I_j\},$$\\
\noindent i.e., $\overline{\text{span}}\{\bar{y}\}$ does not embed into $\overline{\text{span}}\{e_n\}$ avoiding an infinite number of the intervals $I_j$. A space with a continuously tight basis is called \emph{continuously tight}. The Tsirelson space is an example of a continuously tight Banach space (see \cite{FR}, Corollary 4.3).
For a detailed study of continuously tight spaces and other related properties (e.g., tight spaces, tight with constants, tight by range, etc) see \cite{FR}. Let $\text{CT}=\{X\in\text{SB}\mid X\text{ is continuoulsy tight}\}$ be our coding for the class of continuously tight separable Banach spaces.
\begin{cor}\label{cu}
$\text{C}_\text{CT}$ is $\Sigma^1_1$-hard.
\end{cor}
\begin{proof}
\text{Proposition 3.3} of \cite{FR} says that continuously tight spaces contain no minimal subspaces, and \text{Theorem 3.13} of \cite{FR} says that a space with no minimal subspaces contains a continuously tight subspace. Therefore, $\text{CT}$ is an almost-pure class. Also, again by Proposition 3.3 of \cite{FR}, we have that no elements of $\text{CT}$ contain an isomorphic copy of $\ell_p$. Hence, Theorem \ref{lema} gives us that $\text{C}_\text{CT}$ is $\Sigma^1_1$-hard.
\end{proof}
We cannot obtain any lower bound for the complexity of the set of continuously tight Banach spaces with the method of $\ell_p$-Baire sums. We come back to the class of continuously tight spaces in Subsection \ref{sectiontightmesmo}, where we show that the set \text{CT} is $\Pi^1_1$-hard by using a different method.
\subsection{Spaces containing a minimal subspace.}\label{sectionminimal}
A Banach space $X$ is called \emph{minimal} if every infinite dimensional subspace of $X$ contains an isomorphic copy of $X$. We now turn our attention to the following: Although $\text{M}=\{X\in\text{SB}\mid X\text{ is}\allowbreak \text{ minimal}\}$ is clearly a pure class, $\text{M}$ contains $c_0$ and $\ell_p$, for all $p\in[1,\infty)$. Therefore, \text{Theorem \ref{lema}} does not say anything about the complexity of $\text{C}_\text{M}$. However, we can use the construction of Tsirelson space by T. Figiel and W. B. Johnson (see \cite{FJ}, Section $2$) in order to construct a $\varphi:\text{Tr}\to\text{SB}$ that will solve our problem.
Fix a compatible enumeration of $\mathbb{N}N$, i.e., $(s_n)_{n\in\mathbb{N}}$ such that $s_n\preceq s_m$ implies $n\leqslant m$ and for all $s\in\mathbb{N}N$ there exists $n\in\mathbb{N}$ such that $s_n=s$. This enumeration give us an order on $\mathbb{N}^{<\mathbb{N}}$. With this ordering in mind, we say that $E_0<E_1$ if $\max E_0<\min E_1$, for all finite sets $E_0, E_1\subset \mathbb{N}N$. We write $k<E$ if $\{s_k\}<E$.
Given $\theta\in\text{Tr}$, $E\subset\theta$, and $x=\sum_{n\in\mathbb{N}}a_{s_n}e_{s_n}\in c_{00}(\theta)$ (for some $(a_{s_n})_{n\in\mathbb{N}}\in\mathbb{R}^\mathbb{N}$), we let $Ex=\sum_{s_n\in E}a_{s_n}e_{s_n}$.
We define a Borel function $\varphi:\text{Tr}\to\text{SB}$ as, for each $\theta\in\text{Tr}$ and each $x=(x(s))_{s\in\theta}\in c_{00}(\theta)$, let $(\|.\|_{\theta,m})_{m\in\mathbb{N}}$ be inductively defined by
\begin{align*}
\|x\|_{\theta,0}&=\|x\|_0,\text{ and }\\
\|x\|_{\theta,m+1}&=\max\{\|x\|_0,\displaystyle{\frac{1}{2}}\max\sum_{i=1}^k\|E_ix\|_{\theta,m}\},
\end{align*}
\noindent for all $m\in\mathbb{N}$, where the ``inner$"$ maximum above is taken over all $k\in\mathbb{N}$ and all completely incomparable finite sets $(E_i)_{i=1}^k$ ($E_i\subset \theta$, for all $i\in\{1,...,k\}$) such that $k\leqslant E_1<...<E_k$ (for the definition of completely incomparable sets of a tree, see Section \ref{definition}). Exactly as we have for the standard Tsirelson space, we can define a norm $\|.\|_\theta$ as
$$\|x\|_\theta=\underset{m\to\infty}{\lim}\|x\|_{\theta,m},$$\\
\noindent for all $x\in c_{00}(\theta)$. We define $\varphi(\theta)$ to be the completion of $c_{00}(\theta)$ under this norm. This norm can be implicitly defined as
$$\left\|x\right\|_\theta=\max\{\|x\|_0,\displaystyle{\frac{1}{2}}\max\sum_{i=1}^k\|E_ix\|_\theta\},$$\\
\noindent where the ``inner$"$ maximum above is taken over all $k\in\mathbb{N}$ and all completely incomparable finite sets $(E_i)_{i=1}^k$ ($E_i\subset \theta$, for all $i\in\{1,...,k\}$) such that $k\leqslant E_1<...<E_k$.
By the universality of $C(\Delta)$ for separable Banach spaces, we can identify $\varphi(\mathbb{N}N)$ with an isometric copy inside of $C(\Delta)$. As we can identify each $\varphi(\theta)$ with a subspace of $\varphi(\mathbb{N}N)$ in a natural fashion, we can see $\varphi$ as a Borel function from $\text{Tr}$ to $\text{SB}$ (see \cite{S}, Proposition 3.1, page 79, for similar arguments).
\begin{thm}\label{lemaminimal}
Let $\varphi:\text{Tr}\to\text{SB}$ be defined as above. Then $\varphi$ is a Borel function with the following properties
\begin{enumerate}[(i)]
\item $c_0\emb\varphi(\theta)$, for all $\theta\in\text{IF}$, and
\item $\varphi(\theta)$ has the following property if $\theta\in\text{WF}$: for every infinite dimensional subspace $E\subset\varphi(\theta)$, there exists a further subspace $F\subset E$ isomorphic to an infinite dimensional subspace of Tsirelson space, i.e., $\varphi(\theta)$ is Tsirelson-saturated.
\end{enumerate}
\noindent In particular, $\text{C}_\text{M}$ is $\Sigma^1_1$-hard, and $\text{C}_{c_0}$ cannot be Borel separated from the set of Tsirelson-saturated Banach spaces.
\end{thm}
Before proving this theorem, notice the following trivial consequence of Lemma \ref{u8u}.
\begin{lemma}\label{u8uu}
A finite sum of spaces satisfying property $\text{(ii)}$ of Theorem \ref{lemaminimal} still has property $\text{(ii)}$.
\end{lemma}
\noindent\emph{Proof of Theorem \ref{lemaminimal}.}
If $\theta\in\text{IF}$, it is clear that $c_0\emb\varphi(\theta)$. Say $\theta\in\text{WF}$, let us show that every infinite dimensional subspace of $\varphi(\theta)$ contains a subspace isomorphic to an infinite dimensional subspace of Tsirelson's space. Say $E\subset \varphi(\theta)$ is an infinite dimensional subspace.
As $\theta\in\text{WF}$, Lemma \ref{lemageral} gives us that $E$ contains a sequence equivalent to a block sequence $(y_n)_{n\in\mathbb{N}}$ of $\varphi(\theta)$ with completely incomparable supports. We will be done once we prove the following claim.\\
\textbf{Claim:} $(y_n)_{n\in\mathbb{N}}$ is equivalent to a subsequence of the standard basis of Tsirelson space.\\
Let $\|\cdot \|_{T,\theta}$ be the standard Tsirelson norm on $c_{00}(\theta)$, i.e.,
$$\left\|x\right\|_{T,\theta}=\max\{\|x\|_0,\displaystyle{\frac{1}{2}}\max\sum_{i=1}^k\|E_ix\|_{T,\theta}\},$$\\
\noindent where the ``inner$"$ maximum above is taken over all $k\in\mathbb{N}$ and all finite sets $(E_i)_{i=1}^k$ ($E_i\subset \theta$, for all $i\in\{1,...,k\}$) such that $k\leqslant E_1<...<E_k$. The only difference between $\|\cdot\|_\theta$ and $\|\cdot \|_{T,\theta}$ is that in $\|\cdot\|_{T,\theta}$ we do not have the restriction of $(E_i)_{i=1}^k$ being completely incomparable.
It is clear that the the basis $(e_s)_{s\in\mathbb{N}N}$ of the completion of $(c_{00}(\theta),\|\cdot \|_{T,\theta})$ is equivalent to the standard basis of Tsirelson space. Moreover, the basis $(e_s)_{s\in\theta}$ of the completion of $(c_{00}(\theta),\|\cdot \|_{T,\theta})$ is equivalent to a subsequence of the standard basis of Tsirelson space, if $\theta$ is infinite (this because this norm has no dependance on the structure of the tree $T$). Also, we clearly have
\begin{align}\label{uperb}
\|\sum_{i=1}^ka_iy_i\|_\theta\leqslant\|\sum_{i=1}^ka_iy_i\|_{T,\theta},
\end{align}\\
\noindent for all $a_1,...,a_k\in\mathbb{R}$. Mimicking the proof of \text{Lemma II.1} of \cite{CS} we have the following lemma.
\begin{lemma}\label{leamm}
Let $(p_n)_{n\in\mathbb{N}}$ be a increasing sequence of natural numbers. Let $(e_{s_n})_{n\in\mathbb{N}}$ be the standard basis of $\varphi(\theta)$. Let $y_n=\sum_{i=p_n+1}^{p_{n+1}}b_ie_{s_i}$ (for all $n\in\mathbb{N}$) be a normalized block sequence of $(e_{s_n})_{n\in\mathbb{N}}$ and assume $(y_n)_{n\in\mathbb{N}}$ has completely incomparable supports. Then
$$\|\sum_{n\in\mathbb{N}}a_ne_{s_{p_n+1}}\|_\theta\leqslant\|\sum_{n\in\mathbb{N}}a_ny_n\|_\theta,$$\\
\noindent for any sequence of scalars $(a_n)_{n\in\mathbb{N}}$.
\end{lemma}
\begin{proof}
It is enough to show that, for any sequence $(a_n)_{n\in\mathbb{N}}$, we have
$$\|\sum_{n\in\mathbb{N}}a_ne_{s_{p_n+1}}\|_{\theta,m}\leqslant\|\sum_{n\in\mathbb{N}}a_ny_n\|_\theta,$$\\
\noindent for all $m\in\mathbb{N}$. Let us proceed by induction on $m\in\mathbb{N}$. For $m=0$ the result is clear. Assume the equation above holds for a fixed $m\in\mathbb{N}$. Let $x=\sum_{n\in\mathbb{N}}a_ne_{p_n+1}$ and $y=\sum_{n\in\mathbb{N}}a_ny_n$. Fix $k\in\mathbb{N}$, and completely incomparable finite sets $(E_n)_{n=1}^k$ such that $k\leqslant E_1<...<E_k$. Consider the sum
$$\displaystyle{\frac{1}{2}}\sum_{i=1}^k\|E_ix\|_{\theta,m}.$$\\
\noindent Since the support of $x$ is contained in $\{p_n+1\mid n\in\mathbb{N}\}$, we may assume that
$$E_i\subset \{p_n+1\mid n\in\mathbb{N}\},$$\\
\noindent for all $i\in\mathbb{N}$. Applying the inductive hypothesis, we have
$$\displaystyle{\frac{1}{2}}\sum_{i=1}^k\|E_ix\|_{\theta,m}\leqslant \displaystyle{\frac{1}{2}}\sum_{i=1}^k\|\sum_{\substack{n\in\mathbb{N}\\ p_n+1\in E_i}}a_ny_n\|_{\theta}$$\\
\noindent As $p_n+1\in E_i$ implies $k\leqslant p_n+1$, and as $(y_n)_{n\in\mathbb{N}}$ has completely incomparable supports, the sum on the right hand side of the equation above is allowed as an ``inner$"$ sum in the definition of the norm $\|.\|_{\theta}$. Therefore, we have
$$\displaystyle{\frac{1}{2}}\sum_{i=1}^k\|E_ix\|_{\theta,m}\leqslant\|y\|_{\theta},$$\\
\noindent for all $k\in\mathbb{N}$, and all completely incomparable finite sets $(E_n)_{n=1}^k$ such that $k\leqslant E_1<...<E_k$. Hence, for any sequence of scalars $(a_n)_{n\in\mathbb{N}}$, we have
$$\|\sum_{n\in\mathbb{N}}a_ne_{s_{p_n+1}}\|_{\theta,m+1}\leqslant\|\sum_{n\in\mathbb{N}}a_ny_n\|_\theta,$$\\
\noindent and we are done.
\end{proof}
Let $(b_n)_{n\in\mathbb{N}}$ be the sequence of scalars such that our block sequence $(y_n)_{n\in\mathbb{N}}$ can be written as $y_n=\sum_{i=p_n+1}^{p_{n+1}}b_ie_{s_i}$. Then, \text{Lemma \ref{leamm}} gives us that
\begin{align}\label{eq2}
\|\sum_{n\in\mathbb{N}}a_ne_{s_{p_n+1}}\|_\theta\leqslant\|\sum_{n\in\mathbb{N}}a_ny_n\|_\theta,
\end{align}\\
\noindent for any sequence of scalars $(a_n)_{n\in\mathbb{N}}$. As the supports of $(y_n)_{n\in\mathbb{N}}$ are completely incomparable we have that
\begin{align}\label{eq3}
\|\sum_{n\in\mathbb{N}}a_ne_{s_{p_n+1}}\|_\theta=\|\sum_{n\in\mathbb{N}}a_ne_{s_{p_n+1}}\|_{T,\theta},
\end{align}\\
\noindent for any sequence of scalars $(a_n)_{n\in\mathbb{N}}$. By \text{Proposition II.4} of \cite{CS}, we have
\begin{align}\label{lowerb}
\displaystyle{\frac{1}{18}}\|\sum_{n\in\mathbb{N}}a_ny_n\|_{T,\theta}\leqslant\|\sum_{n\in\mathbb{N}}a_ne_{s_{p_n+1}}\|_{T,\theta},
\end{align}\\
\noindent for all sequence of scalars $(a_n)_{n\in\mathbb{N}}$. Therefore, putting Equation \ref{uperb}, Equation \ref{eq2}, Equation \ref{eq3}, and Equation \ref{lowerb} together, we have that
$$\|\sum_{i=1}^ka_ne_{s_{p_n+1}}\|_{T,\theta}\leqslant\|\sum_{i=1}^ka_ny_n\|_\theta\leqslant18\|\sum_{i=1}^ka_ne_{s_{p_n+1}}\|_{T,\theta},$$\\
\noindent for all $a_1,...,a_k\in\mathbb{R}$. Hence, the sequence $(y_n)_{n\in\mathbb{N}}$ as a sequence in $\varphi(\theta)$ is equivalent to the sequence $(e_{s_{p_n+1}})_{n\in\mathbb{N}}$ as a sequence in the Tsirelson space (the completion of $(c_{00}(\theta), \|\cdot\|_{T,\theta})$).
To conclude that $\varphi(\theta)$ contains no minimal subspaces if $\theta\in\text{WF}$, recall that Tsirelson space contains no minimal subspaces (see \cite{CS}, Corollary VI.b.6), so we are done.\qed\\
It is easy to see, by simply counting quantifiers, that the set $\text{C}_\text{M}$ is at most $\Sigma^1_3$, i.e., the Borel image of a set that can be written as the complement of the Borel image of a coanalytic set. However, by using Gowers' theorem, Rosendal was able to find a better upper bound for $\text{C}_\text{M}$ (see \cite{R}, Appendix).
\begin{thm}\textbf{(C. Rosendal)}
$\text{C}_\text{M}$ is $\Sigma^1_2$.
\end{thm}
\begin{problem}
What is the exact complexity of $\text{C}_\text{M}$? Is it $\Sigma^1_2$-complete?
\end{problem}
In Subsection \ref{outrodia}, we talk a little bit about Rosendal's proof for $\text{C}_\text{M}$ being $ \Sigma^1_2$. We will notice that Rosendal's proof also gives us that \text{M} is at most $\Delta^1_2$.
\subsection{Continuously tight Banach spaces.}\label{sectiontightmesmo}
We are now capable of giving a lower bound for the complexity of CT, the set of continuously tight spaces.
Recall, a Banach space $X$ with a basis $(x_k)_{k\in\mathbb{N}}$ is said to be \emph{strongly asymptotic $\ell_p$} if there exists a function $f:\mathbb{N}\to\mathbb{N}$ and a constant $C>0$ such that any set of $m$ unit vectors in $\overline{\text{span}}\{x_k\mid k\geq f(m)\}$ with disjoint supports is $C$-equivalent to the basis of $\ell_p^m$. The Tsirelson space with its standard basis is an example of an strongly asymptotic $\ell_1$ space (see \cite{CS}, Chapter V). Also, a Banach space $X$ is said to be \emph{crudely finitely representable} in a Banach space $Y$ if there exists $M>0$ such that every finite subspace of $X$ $M$-embeds into $Y$.
The proof below is an adaptation of the proof of Proposition 4.2, in \cite{FR}.
\begin{thm}\label{tighttighthu}
The set of continuously tight spaces \text{CT} is $\Pi^1_1$-hard. Moreover, the set of continuously tight basis, say $\mathcal{CT}$, is $\Pi^1_1$-hard.
\end{thm}
\begin{proof}
Let $\varphi:\text{Tr}\to \text{SB}$ be the map in Lemma \ref{lemaminimal}. As $c_0$ embeds into $\varphi(\theta)$ if $\theta\in\text{IF}$, we only need to show that $\varphi(\theta)$ is continuously tight if $\theta\in\text{WF}$. Indeed, as the map $\theta\in\text{Tr}\mapsto (e_{s})_{s\in\theta}\in\mathcal{B}$ is a Borel map, this is enough to prove both assertions of the theorem.
A Banach space $X$ with a basis is said to be \emph{tight with constants} if no Banach space embeds uniformly into its tail subspaces (see \cite{FR}, Proposition 4.1). Let's show that $\varphi(\theta)$ is tight with constants, for all $\theta\in\text{WF}$. As spaces which are tight with constants are also continuously tight (see Proposition \ref{versa} below) we will be done.
Assume towards a contradiction that, for some $K>0$, there exists a space $Y$ which $K$-embeds into all tail subspaces of $\varphi(\theta)$. By Theorem \ref{lemaminimal}, we can assume, by taking a subspace, that $Y$ is generated by a sequence $(y_k)_{k\in\mathbb{N}}$ which is equivalent to a subsequence of the basis of Tsirelson space. Therefore, $(y_k)_{k\in\mathbb{N}}$ is unconditional and strongly asymptotic $\ell_1$. Let $C>0$ and $f:\mathbb{N}\to \mathbb{N}$ be as in the definition of strongly asymptotic $\ell_1$ spaces.
By Proposition 1 of \cite{Jo}, we have that for all $m\in\mathbb{N}$, there exists $N(m)\in\mathbb{N}$ such that $(y_1,...,y_m)$ is $2K$-equivalent to a sequence of vectors in the linear span of $N(m)$ disjointly supported unit vectors in any tail of $Y$. In particular, in the tail $\overline{\text{span}}\{y_k\mid k\geq f(N(m))\}$. Therefore, as $Y$ is strongly asymptotic $\ell_1$, we have that $(y_1,...,y_m)$ $2KC$-embeds into $\ell_1$, for all $m\in\mathbb{N}$. So $Y$ is crudely finitely representable in $\ell_1$, and therefore $Y$ embeds into $
L_1$ (see \cite{AK}, Theorem 11.1.8). Hence, as $(y_k)_{k\in\mathbb{N}}$ is unconditional asymptotic $\ell_1$, we have that $Y$ contains $\ell_1$ (see \cite{DFKO}, Proposition 5), absurd, because $Y$ is a subspace of Tsirelson space.
\end{proof}
\begin{prop}\label{versa}
Let $X$ be a Banach space with basis $(e_n)_{n\in\mathbb{N}}$. Say $(e_n)_{n\in\mathbb{N}}$ is tight with constants (see definition in the proof above). Then $(e_n)_{n\in\mathbb{N}}$ is continuously tight.
\end{prop}
\begin{proof}
For this, we will use details of the proof of Proposition 4.1 of \cite{FR}. Precisely, let $X\in\text{SB}$ be a Banach space with a basis $(e_n)_{n\in\mathbb{N}}$ which is tight with constants. For each $L\in\mathbb{N}$, let $c(L)>0$ be a constant such that if two block sequences of $(e_n)_{n\in\mathbb{N}}$ differ from at most $L$ terms, then they are $c(L)$-equivalent. For each $(y_n)_{n\in\mathbb{N}}\in bb(e_n)$, let us define a sequence $(I_j)_{j\in\mathbb{N}}$ of finite intervals of natural numbers.
By Proposition 4.1 of \cite{FR}, for each $K,m\in\mathbb{N}$, there exists an $l>m$ such that
\begin{align}\label{bedbug}
\text{span}\{y_n\mid m\leq n\leq l\}\not\hookrightarrow_K\overline{\text{span}}\{e_n\mid n\geq l\}.
\end{align}\\
\noindent Let $l_1\in\mathbb{N}$ be the minimal $l\in\mathbb{N}$ as above, for $m=1$, and $K=c(1)$. Let $I_1=[1,l_1]$, where if $a\leq b\in\mathbb{N}$, $[a,b]=\{n\in\mathbb{N}\mid a\leq n\leq b\}$. Assume we had already defined finite intervals $I_1<...<I_{j-1}\subset \mathbb{N}$ and numbers $l_1<...<l_{j-1}\in\mathbb{N}$. Define $l_{j}$ as the minimal $l\in\mathbb{N}$ as in (\ref{bedbug}) above, for $m=\max\{I_{j-1}\}+1$, and $K=j\cdot c(\max\{I_{j-1}\}+1)$. Let $I_j=[\max\{I_{j-1}\}+1,l_j]$.
By the proof of Proposition 4.1 of \cite{FR}, we have that, for all $K\in\mathbb{N}$,
$$\text{span}\{y_n\mid n\in I_K\}\not\hookrightarrow_K \overline{\text{span}}\{e_n\mid n\not\in I_K\}.$$\\
\noindent In particular, for all infinite set $A\subset \mathbb{N}$,
$$\overline{\text{span}}\{y_n\}\not\hookrightarrow\overline{\text{span}}\{e_n\mid n\not\in \cup_{j\in A}I_j\}.$$\\
\noindent Hence, we had defined a map $\bar{y}=(y_n)\mapsto (I_j)_{j\in\mathbb{N}}$, and we will be done if this assignment is continuous, i.e., if there exists a continuous function $f:bb(e_n)\to [\mathbb{N}]$ such that $I_j=[f(\bar{y})_{2j},f(\bar{y})_{2j+1}]$. For this, we only need to notice that in order to obtain a finite chunk of the sequence of intervals $(I_j)_{j\in\mathbb{N}}$, say $I_1,...,I_K$, we only need to know a finite chunk of $(y_n)_{n\in\mathbb{N}}$, precisely, $y_1,...,y_{\max\{I_K\}}$. So we are done.
\end{proof}
\begin{prop}
$\text{CT}$, $\text{C}_\text{CT}$ and $\mathcal{CT}$ are at most $\Sigma^1_2$.
\end{prop}
\begin{proof}
This is a simple matter of counting quantifiers and the fact that we only quantify over standard Borel spaces in the definition of those three classes. Indeed, for $\mathcal{CT}$, for example, we have
\begin{align*}
(e_n)_{n\in\mathbb{N}}\in \mathcal{CT}\Leftrightarrow\ &\exists \text{ continuous }\ f:bb(e_n)\to[\mathbb{N}],\ \forall (y_n)_{n\in\mathbb{N}}\in bb(e_n),\\ &\forall\text{ infinite }\ A\subset \mathbb{N},\ \forall (x_n)_{n\in\mathbb{N}}\in \text{span}\{e_n\mid n\not\in \cup_{j\in A} I_j\}^\mathbb{N},\\ &\forall K\in\mathbb{N},\ (x_n)_{n\in\mathbb{N}}\not\approx_K (y_n)_{n\in\mathbb{N}}.
\end{align*}\\
\indent The only quantifier that demands some explanation is $``\exists \text{ continuous }\ f:bb(e_n)\to[\mathbb{N}]"$. For this, let $\mathbb{N}^{[<N]}$ be the set of finite increasing sequence of natural numbers. Then it is easy to see that a continuous function $f:bb(e_n)\to[\mathbb{N}]$ gives us a function $g:fbb(e_n)\to\mathbb{N}^{[<N]}$ such that $g(\bar{y})\preceq g(\bar{x})$, if $\bar{y}\preceq\bar{x}$, and vice versa, where $fbb(e_n)$ is the set of finite block sequences of $(e_n)_{n\in\mathbb{N}}$ (see Section \ref{definition}). So, as $fbb(e_n)$ is countable, the space of functions $fbb(e_n)\to\mathbb{N}^{[<N]}$ is a standard Borel space, so we are done. The same arguments work for $\text{CT}$ and $\text{C}_\text{CT}$.
\noindent
\end{proof}
\subsection{Mininal spaces.}\label{outrodia}
It follows straight forward from the definition of minimal Banach spaces that $\text{M}=\{X\in\text{SB}\mid X\text{ is minimal}\}$ is $\Pi^1_2$. In this subsection we show that $\text{M}$ is also $\Sigma^1_2$. Hence, $\text{M}$ is at most $\Delta^1_2$.
Using Gowers' theorem (see \cite{Go2}), and a corollary of the solution of the distortio problem (see \cite{OS}, and \cite{OS2}), Rosendal had shown (see \cite{R}, Appendix) that if a Banach space $X$ not containing $c_0$ contains a minimal subspace then there exists a basic sequence $(e_n)_{n\in\mathbb{N}}$ in $X$, a block subsequence $(y_n)_{n\in\mathbb{N}}\in bb(e_n)$, and a continuous function $f: bb(y_n)\to \overline{\text{span}}\{e_n\}^\mathbb{N}$ such that, for all $\bar{z}=(z_n)_{n\in\mathbb{N}}\in bb(y_n)$, we have
$$\overline{\text{span}}\{f(\bar{z})\}\subset \overline{\text{span}}\{z_n\}\ \ \text{ and }\ \ (e_n)_{n\in\mathbb{N}}\approx f(\bar{z}).$$\\
\indent By counting quantifiers, the set of Banach spaces satisfying the property above is at most $\Sigma^1_2$. Clearly, if a Banach space satisfies the property above, then it contains a minimal subspace, indeed, $\overline{\text{span}}\{y_n\}$ is minimal. As the set of Banach spaces containing $c_0$ is $\Sigma^1_1$ ($\Sigma^1_1$-complete actually), this gives us that $\text{C}_\text{M}$ is at most $\Sigma^1_2$.
We now notice that this also gives us an equivalent characterization of minimality. Indeed, Rosendal's result clearly implies that if $X$ is a minimal Banach space not containing $c_0$ then there exists a basic sequence $(e_n)_{n\in\mathbb{N}}$ in $X$, a block subsequence $(y_n)_{n\in\mathbb{N}}\in bb(e_n)$, such that
$$X\hookrightarrow\overline{\text{span}}\{y_n\},$$\\
\noindent and there exists a continuous function $f: bb(y_n)\to \overline{\text{span}}\{e_n\}^\mathbb{N}$ such that, for all $\bar{z}=(z_n)_{n\in\mathbb{N}}\in bb(y_n)$, we have
$$\overline{\text{span}}\{f(\bar{z})\}\subset \overline{\text{span}}\{z_n\}\ \ \text{ and }\ \ (e_n)_{n\in\mathbb{N}}\approx f(\bar{z}).$$\\
\indent By counting quantifiers, the set of Banach spaces satisfying the property above is at most $\Sigma^1_2$. Notice that if a Banach space satisfies the property above, then it is minimal. Therefore, as the set of minimal Banach spaces containing $c_0$ is the set of spaces that embed into $c_0$, and as this set is easily seen to be analytic, we have that $\text{M}$ is at most $\Sigma ^1_2$. As M is also $\Pi^1_2$, we have the following.
\begin{prop}
The class of minimal Banach spaces M is at most $\Delta^1_2.$
\end{prop}
\subsection{Hereditarily indecomposable spaces.}\label{hiGM}
Let $\text{HI}=\{X\in\text{SB}\mid X\text{ is hereditarily }\allowbreak\text{indecomposable}\}$. In Subsection \ref{masha}, we proved, using the method of $\ell_p$-Baire sums (Corollary \ref{hhii}), that $\text{C}_\text{HI}$ is $\Sigma^1_1$-hard. However, this method does not allow us to obtain any information about the complexity of $\text{HI}$. In this subsection, we will use a more complex construction in order to compute the complexity of \text{HI}.
Precisely, we follow Argyros presentation (see \cite{AT}, Chapters II and III) of how to construct HI extensions of ground norms in order to define a Borel function $\varphi:\text{Tr}\to\text{SB}$ such that $\varphi^{-1}(\text{HI})=\text{WF}_\infty$, where $\text{WF}_\infty$ denotes the subset of infinite well-founded trees. This will show that $\text{HI}$ is $\Pi^1_1$-hard. As a Banach space $X$ is hereditarily indecomposable if, and only if, for all subspaces $Z,W\subset X$ and all $\varepsilon>0$, there exist $z\in Z$ and $w\in W$ such that $\|z-w\|<\varepsilon\|z+w\|$, we can easily show that $\text{HI}$ is coanalytic. Therefore, we will show that $\text{HI}$ is complete coanalytic.
\begin{thm}\label{hhhiii}
\text{HI} is coanalytic.
\end{thm}
\begin{proof}
This is a simple consequence of the fact that $X\in\text{HI}$ if, and only if, for all subspaces $Z,W\subset X$ and all $\varepsilon>0$, there exists $z\in Z$ and $w\in W$ such that $\|z-w\|<\varepsilon\|z+w\|$. Indeed, notice that
\begin{align*}
X\in\text{HI}\Leftrightarrow\ &\forall Z,W \subset X,\ \forall \varepsilon\in\Q_+,\ \exists n,m\in\mathbb{N} \\
&\|S_{n}(Z)-S_{m}(W)\|<\varepsilon\|S_{n}(Z)+S_{m}(W)\|,
\end{align*}
\noindent where ``$\forall Z,W \subset X"$ means ``for all subspaces $Z,W\subset X"$. As $\{(Z,W,X)\in\text{SB}^3|Z,W\subset X\}$ is well known to be Borel (\cite{S}, see Lemma 1.9, page 73), we are done.
\end{proof}
This trivially gives us the following upper bound for the complexity of $\text{C}_\text{HI}$.
\begin{cor}
$\text{C}_\text{HI}$ is $\Sigma^1_2$.
\end{cor}
The only thing left to show is that $\text{HI}$ is $\Pi^1_1$-hard. For this, let us define a special Borel map $\varphi:\text{Tr}\to \text{SB}$ such that $\varphi^{-1}(\text{HI})=\text{WF}_\infty$. As the construction of such $\varphi$ will heavily rely on the construction of HI extensions of a ground norm, we tried to be consistent with Argyros notation (\cite{AT}, Chapters II and III). We believe this will make the presentation more clear for the reader which is familiar with HI spaces. Therefore, the notation used to denote some spaces in this subsection will be slightly different from the notation chosen in the rest of these notes. We will make sure to point out the differences as they appear though. To start with, we will denote by $\mathfrak{X}(D_{G}{(\theta)})$ the resulting space $\varphi(\theta)$.
First, we fix a compatible enumeration for $\mathbb{N}N$, say $(s_i)_{i\in\mathbb{N}}$. Let $(e_{s_i})_{i\in\mathbb{N}}$ be the standard unit basis of $c_{00}(\mathbb{N}^{<\mathbb{N}})$.
Let $x=(x(s))_{s\in\mathbb{N}^{<\mathbb{N}}}\in c_{00}(\mathbb{N}^{<\mathbb{N}})$, and $x^*=(x^*(s))_{s\in\mathbb{N}^{<\mathbb{N}}}\in c_{00}(\mathbb{N}^{<\mathbb{N}})$, we define
$$x^*(x)=\sum_{s\in\mathbb{N}^{<\mathbb{N}}}x(s)x^*(s),$$\\
\noindent i.e., the notation $ ``^*"$ means that we will consider $x^*$ as a functional on $c_{00}(\mathbb{N}^{<\mathbb{N}})$. For $x^*=(x^*(s))_{s\in\mathbb{N}^{<\mathbb{N}}}\in c_{00}(\mathbb{N}^{<\mathbb{N}})$, we let $\text{supp}(x^*)=\{s\in\mathbb{N}^{<\mathbb{N}}\mid x^*(s)\neq 0\}$.
Let
$$G=\big\{\sum_{i=1}^n a_ie_{s_{j_i}}\mid n\in\mathbb{N}, s_{j_i}\in\mathbb{N}^{<\mathbb{N}},s_{j_1}\prec...\prec s_{j_n},|a_i|= 1\big\}.$$\\
\indent The set $G$ is called a \emph{ground set} (for a definition of ground sets see \cite{AT}, Definition II.1, page 21). We define $Y_{G}$ as the completion of $c_{00}(\mathbb{N}^{<\mathbb{N}})$ under the norm
$$\|x\|_{G} =\sup\{g(x)\mid g\in G\},$$\\
\noindent for all $x\in c_{00}(\mathbb{N}^{<\mathbb{N}})$. For each $\theta\in \text{Tr}$, we let $Y_{G}(\theta)$ denote the subspace of $Y_G$ generated by $\{e_s\mid s\in\theta\}$. So $Y_G(\mathbb{N}^{<\mathbb{N}})=Y_G$. We will define $\mathfrak{X}(D_{G}{(\theta)})$ as a ground norm extension of $Y_{G}{(\theta)}$.
\textbf{Remark:} Notice that, according to the notation of \text{Lemma} \ref{arroto}, it is clear that $Y_{G}{(\theta)}\cong \varphi_{\mathcal{E},0}(\theta)$, if $\mathcal{E}$ is the standard $\ell_1$-basis.
For each $j\in \mathbb{N}$, let $\mathcal{A}_j=\{F\subset \mathbb{N}\mid |F|\leqslant j\}$, where $|F|$ is the cardinality of $F\subset \mathbb{N}$. As in Subsection \ref{sectionminimal}, for finite sets $E_1,E_2\subset \mathbb{N}$, we write $E_1<E_2$ if $\max E_1<\min E_2$. Let $g_1,...,g_n\in c_{00}(\mathbb{N}^{<\mathbb{N}})$. We write $g_1<...<g_n$ if $\text{supp} (g_1)<...<\text{supp} (g_n)$. Let $(g_l)_{l=1}^{n}$ be a finite sequence in $c_{00}(\mathbb{N}^{<\mathbb{N}})$ such that $g_1<...<g_n$, and $m\in\mathbb{N}$, we define the \emph{$(\mathcal{A}_{n}, \frac{1}{m})$-operation on $(g_l)_{l=1}^{n}$} as the functional $g=\frac{1}{m}(g_1+...+g_{n})$.
Fix two sequences of natural numbers $(m_j)_{j\in\mathbb{N}}$ and $(n_j)_{j\in\mathbb{N}}$ such that $m_1=2$, $m_{j+1}=m_j^5$, $n_1=4$, and $n_{j+1}=(5n_j)^s$, where $s_j=\log_2 m_{j+1}^3$.
We now define a norming set $D_{G}\subset c_{00}(\mathbb{N}^{<\mathbb{N}})$ that will give us the norm we will use to define $\mathfrak{X}(D_{G}{(\theta)})$, i.e., $\varphi(\theta)$. In the definition below we use the term \emph{``$n_{2j-1}$-special sequence$"$}. As this definition will play no role in our proof of the theorem and as it is a really technical definition, we chose to omit it here. The interested reader can find the precise definition of an $n_{2j-1}$-special sequence in \cite{AT}, Chapter III, page $40$.
Let $E\subset \mathbb{N}^{<\mathbb{N}}$, and $x^*=(x^*(s))_{s\in \mathbb{N}^{<\mathbb{N}}}\in c_{00}(\mathbb{N}^{<\mathbb{N}})$. We define $Ex^*=(x^*(s))_{s\in E}$.
\begin{defn}
We define $D_{G}$ as the minimal subset of $c_{00}(\mathbb{N}^{<\mathbb{N}})$ satisfying
\begin{enumerate}[(i)]
\item $G\subset D_{G}$.
\item $D_{G}$ is symmetric, i.e., $g\in D_{G}$ implies $-g\in D_{G}$.
\item $D_{G}$ is closed under the restriction of its elements to intervals of $\mathbb{N}^{<\mathbb{N}}$, i.e., if $E\subset\mathbb{N}^{<\mathbb{N}}$ is an interval and $g\in D_{G}$, then $Eg\in D_{G}$.
\item $D_{G}$ is closed under the $(\mathcal{A}_{n_{2j}}, \frac{1}{m_{2j}})$-operations, i.e., if $(g_l)_{l=1}^{n_{2j}}$ is a sequence in $D_{G}$ such that $g_1<...<g_{n_{2j}}$, then $g=\frac{1}{m_{2j}}(g_1+...+g_{n_{2j}})$ belongs to $D_{G}$.
\item $D_{G}$ is closed under $(\mathcal{A}_{n_{2j-1}}, \frac{1}{m_{2j-1}})$-operations on special sequences, i.e., for every $n_{2j-1}$-special sequence $(g_1,...,g_{n_{2j-1}})$ in $D_{G}$, the functional $g=\frac{1}{m_{2j-1}}(g_1+...+g_{n_{2j-1}})$ belongs to $D_{G}$.
\item $D_{G}$ is rationally convex.
\end{enumerate}
\end{defn}
We define a norm on $c_{00}(\mathbb{N}^{<\mathbb{N}})$, as
$$\|x\|_{D_G}=\sup\{g(x)\mid g\in D_{G}\},$$\\
\noindent for all $x\in c_{00}(\mathbb{N}^{<\mathbb{N}})$. Let $\mathfrak{X}(D_{G})$ be the completion of $c_{00}(\mathbb{N}^{<\mathbb{N}})$ under this norm. For each $\theta\in\text{Tr}$, let $\mathfrak{X}(D_{G}(\theta))$ be the subspace of $\mathfrak{X}(D_{G})$ generated by $\{e_s\mid s\in\theta\}$. Therefore, for each $\theta\in\text{Tr}$, we assign a space $\varphi(\theta)=\mathfrak{X}(D_{G}{(\theta)})$.
Identify $\mathfrak{X}(D_{G}(\mathbb{N}^{<\mathbb{N}}))=\mathfrak{X}(D_{G})$ with one of its isometric copies in $\text{SB}$. It is clear that the map $\varphi:\text{Tr}\to \text{SB}$ such that $\varphi(\theta)=\mathfrak{X}(D_{G}(\theta))$, is Borel (see \cite{S}, Proposition 3.1, page 79, for similar arguments).
\begin{thm}\label{ultimoo}
Let $\varphi:\text{Tr}\to\text{SB}$ be the function defined above. Then
\begin{enumerate}[(i)]
\item $\varphi(\theta)=\mathfrak{X}(D_{G}{(\theta)})$ contains $\ell_1$, for all $\theta\in\text{IF}$.
\item $\varphi(\theta)=\mathfrak{X}(D_{G}{(\theta)})$ is hereditarily indecomposable, for all $\theta\in\text{WF}_\infty$.
\end{enumerate}
\noindent In particular, $\text{HI}$ is $\Pi^1_1$-hard, and $\text{C}_{\ell_1}$ cannot be Borel separated from $\text{HI}$.
\end{thm}
\begin{proof} First, notice that if $\theta\in\text{IF}$ then $\ell_1\emb\varphi(\theta)$. Indeed, on segments $I \subset\theta$, the $\ell_1$-norm given by the ground set $G$ is greater than the norm given by $D_{G}$. So, if $\theta$ has a branch, say $\beta$, we have $\mathfrak{X}(D_{G}{(\beta)})\cong\ell_1$.
In order to show the second part of the theorem consider the ``identity$"$ map $Id:\mathfrak{X}(D_{G}{(\theta)})\to Y_{G}{(\theta)}$. We will show that $Id:\mathfrak{X}(D_{G}{(\theta)})\to Y_{G}{(\theta)}$ is strictly singular, for all $\theta\in\text{WF}_\infty$. Once we do that, we will be done by \text{Theorem III.7} of \cite{AT} (page $42$). Indeed, it is clear from the proof of \text{Theorem III.7} of \cite{AT}, that we have the following.
\begin{thm}
Let $G$ be a ground set in $c_{00}(\mathbb{N}^{<\mathbb{N}})$, and let $Y_G$, and $\mathfrak{X}(D_{G})$ be the spaces obtained as above. If $Z\subset \mathfrak{X}(D_{G})$ is an infinite dimensional subspace, and the restriction $Id_{|Z}:Z\to Y_G$ is strictly singular, then $Z$ is hereditarily indecomposable.
\end{thm}
\begin{prop}\label{ss}
Let $\theta\in\text{WF}_\infty$. Then $Id:\mathfrak{X}(D_{G}{(\theta)})\to Y_{G}{(\theta)}$ is strictly singular.
\end{prop}
\begin{proof}
Suppose not. Then there exists an infinite dimensional subspace $Y\subset \mathfrak{X}(D_{G}{(\theta)})$ such that $Id_{|Y}$ is an isomorphism with its image. We now look at $Y=Id(Y)\subset Y_{G}{(\theta)}$. By \text{Lemma} \ref{arroto} and the remark following the definition of $Y_{G}{(\theta)}$, there is a normalized sequence $(y_i)_{i\in\mathbb{N}}$ in $Y\subset Y_{D_{G}{(\theta)}}$ which is equivalent to the standard basis of $c_0$. In particular, there exists $C>0$ such that
$$\big\|\sum_{i=1}^ny_i\big\|_{G}<C,\ \text{ for all }n\in \mathbb{N}.$$\\
\indent Moreover, we can assume, by Lemma \ref{porque} below, that there exists a sequence $(g_i)_{i\in\mathbb{N}}$ in $G$ such that $g_i(y_i)>\frac{1}{2}$ and $g_i<g_{i+1}$, for all $i\in\mathbb{N}$, and $|g_i(y_k)|<2^{-(k+2)}$, for all $i\neq k$. Therefore, by the definition of the norm of $\mathfrak{X}(D_{G}{(\theta)})$, we have that
\begin{align*}\big\|\sum_{i=1}^{n_{2j}}y_i\big\|_{D_G}
&\geq\frac{1}{m_{2j}}\sum_{i=1}^{n_{2j}}g_i(y_i)+\frac{1}{m_{2j}}\sum_{\substack{i,k=1\\ i\neq k}}^{n_{2j}}g_i(y_k)\\
&\geq\frac{n_{2j}}{2m_{2j}}-\frac{n_{2j}}{m_{2j}}\sum_{k=1}^\infty 2^{-(k+2)}\\
&=\frac{n_{2j}}{4m_{2j}}
\end{align*}\\
As $\frac{n_{2j}}{m_{2j}}\to\infty$, as $j\to\infty$, and as $Id_{|Y}$ is an isomorphism, we get a contradiction.
\end{proof}
The proof of Theorem \ref{ultimoo} is now done.
\end{proof}
\begin{thm}\label{hhhhiiii}
\text{HI} is complete coanalytic.
\end{thm}
In order to prove the result above we made use of the following lemma.
\begin{lemma}\label{porque}
Let $\varphi_{\mathcal{E},0}:\text{Tr}\to\text{SB}$ be as in Lemma \ref{arroto}. If $\theta\in\text{WF}$, and $Y\subset \varphi_{\mathcal{E},0}(\theta)$ is infinite dimensional, then there exists a normalized sequence $(y_i)_{i\in\mathbb{N}}$ in $Y$ equivalent to the $c_0$-basis. Moreover, there exists a sequence $(g_i)_{i\in\mathbb{N}}$ in $G(\theta)$ such that $g_i<g_{i+1}$, for all $i\in\mathbb{N}$, $g_i(y_i)>\frac{1}{2}$, for all $i\in\mathbb{N}$, and $|g_i(y_k)|<2^{-(k+2)}$, for all $i\neq k$.
\end{lemma}
\begin{proof}
This can be obtained by a simple modification in the proof of Lemma \ref{lemageral}. For completeness, we write the modifications here. Assume all the notation in the proof of Lemma \ref{lemageral}. So we have $X=\varphi_{\varepsilon,0}(\theta)$, and $s\in\theta$ is such that $P_s:Y\to X$ is not strictly singular, but $Q_{s,n}:Y\to X$ is strictly singular, for all $n\in\mathbb{N}$. Let $E=P_s(Z)$, where $Z\subset Y$ is a subspace such that $P_s:Z\to X$ is an isomorphism with its image. As in Lemma \ref{lemageral}, we can assume that
$$E\subset \overline{\text{span}} \{e_\tau \mid s\prec\tau\}.$$\\
\indent For each $m\in\mathbb{N}$, let $I_m:X\to X$ be the standard projection over the first $m$ coordinates of the basis $(e_s)_{s\in\theta}$, i.e., $I_m(\sum_{s\in\theta}a_se_s)=\sum_{i=1}^ma_{s_i}e_{s_i}$. Let $(y_k)_{k\in\mathbb{N}}$ be a normalized sequence in $E$ such that $Q_{s,n}(y_k)\to 0$, as $k\to\infty$, for all $n\in\mathbb{N}$ (we showed such sequence exists in the proof of Lemma \ref{lemageral}). As $Q_{s,n}(x)\to x$, as $n\to\mathbb{N}$, for all
$x\in E$, we have that given a sequence $(\varepsilon_k)_{k\in\mathbb{N}}$ of positive real numbers, we can pick increasing sequences of natural numbers $(n_k)_{k\in\mathbb{N}}$, $(m_k)_{k\in\mathbb{N}}$, and $(l_k)_{k\in\mathbb{N}}$ such that
\begin{enumerate}[(i)]
\item $\|Q_{s,{l_k}}(y_{n_k})-y_{n_k}\|_\theta<\varepsilon_k$, for all $k\in\mathbb{N}$, and
\item $\|Q_{s,{l_k}}(y_{n_{k+1}})\|_\theta<\varepsilon_k$, for all $k\in\mathbb{N}$.
\item$\|I_{m_k}\big(Q_{s,{l_k}}(y_{n_k})-Q_{s,{l_{k-1}}}(y_{n_k})\big)-\big(Q_{s,{l_k}}(y_{n_k})-Q_{s,{l_{k-1}}}(y_{n_k})\big)\|_\theta<\varepsilon_k$, for all $k\in\mathbb{N}$.
\end{enumerate}
For each $k\in \mathbb{N}$, let
$$x_k=I_{m_k}\big(Q_{s,{l_k}}(y_{n_k})-Q_{s,{l_{k-1}}}(y_{n_k})\big).$$ \\
\noindent Choosing $(\varepsilon_k)_{k\in\mathbb{N}}$ converging to $0$ sufficiently fast, we have that $(x_k)_{k\in\mathbb{N}}$ is equivalent to $(y_{n_k})_{k\in\mathbb{N}}$, and, as $(x_k)_{k\in\mathbb{N}}$ has completely incomparable supports, it is easy to see that $(x_k)_{k\in\mathbb{N}}$ is also equivalent to the $c_0$-basis (as in Lemma \ref{arroto}). Also, by taking a subsequence if necessary, we can assume that $(x_k)_{k\in\mathbb{N}}$ is a block sequence. Hence, as we can assume $\|x_i\|_\theta>\frac{1}{2}$, for all $i\in\mathbb{N}$, there exists a sequence $(g_i)_{i\in\mathbb{N}}$ in $G(\theta)$ such that $g_i<g_{i+1}$, for all $i\in\mathbb{N}$, and $g_i(x_i)>\frac{1}{2}$, for all $i\in\mathbb{N}$, and $g_i(x_k)=0$, for all $i\neq k$.
Clearly, we can assume $\text{supp}(g_i)\subset \text{supp}(x_i)$, for all $i\in\mathbb{N}$. Hence, if $(y'_i)_{i\in\mathbb{N}}$ is a sequence in $Z$ such that $P_s(y'_i)=y_{n_i}$, for all $i\in\mathbb{N}$, we have that $g_i(y'_i)=g_i(y_{n_i})=g_i(x_i)$, for all $i\in\mathbb{N}$. Therefore, $g_i(y'_i)>\frac{1}{2}$, for all $i\in\mathbb{N}$.
On the other hand, it is easy to see that $g_i(y'_k)=g_i(y_{n_k}-x_k)$, for all $i\neq k$. Hence, as $\|y_{n_k}-x_k\|_\theta<2\varepsilon_k+\varepsilon_{k-1}$, we can assume that $|g_i(y'_k)|<2^{-(k+2)}$, for all $i\neq k$. Although $(y'_k)_{k\in\mathbb{N}}$ is only semi-normalized, it is clear from the proof, that we can assume $(y'_k)_{k\in\mathbb{N}}$ is normalized, so we are done.
\end{proof}
\subsection{Unconditional basis.}\label{ubub} A basic sequence $(x_j)_{j\in\mathbb{N}}$ in a Banach space $X$ is unconditional if, and only if, there exists $M>0$ such that, for all $n\in\mathbb{N}$, for all $a_1,...,a_n\in\Q$, and all $b_1,...,b_n\in\Q$ such that $|a_j|\leqslant |b_j|$, for all $j\in\{1,...,n\}$, we have
$$\|\sum_{j=1}^ka_jx_j\|\leqslant \|\sum_{j=1}^kb_jx_j\|.$$\\
\noindent Therefore, it is clear that the set $\mathcal{UB}\subset \mathcal{B}$ of unconditional basis is Borel, where $\mathcal{B}$ is our coding for the basic sequences (see Section \ref{definition}).
We now consider instead of the set of unconditional basis, the set of Banach spaces with an unconditional basis, say $\text{UB}$, and the set of Banach spaces containing an unconditional basis, say $\text{C}_\text{UB}$. As
\begin{align*}
X\in\text{UB} \Leftrightarrow\ &\exists (x_j)_{j\in\mathbb{N}}\in C(\Delta)^\mathbb{N} \text{ such that }\\
&(x_j)_{j\in\mathbb{N}}\text{ is an unconditional basis for }X,
\end{align*}\\
\noindent and the condition $``(x_j)_{j\in\mathbb{N}}\text{ is a basis for X}"$ is easily seen to be Borel, we have that $\text{UB}$ is analytic. Analogously, $\text{C}_\text{UB}$ is also analytic.
Let us now give a lower bound for $\text{C}_\text{UB}$. As hereditarily indecomposable spaces cannot contain an unconditional basis, Theorem \ref{ultimoo} gives us the following.
\begin{thm}\label{ubca}
The set of separable Banach spaces containing an unconditional basis $\text{C}_{\text{UB}}$ is $\Sigma^1_1$-hard. Moreover, $\text{C}_{\text{UB}}$ is $\Sigma^1_1$-complete.
\end{thm}
\begin{problem}
Is \text{UB} Borel? Is \text{UB} complete analytic? What about $S=\{X\in\text{SB}\mid X\text{ has a Schauder basis}\}$? Similarly, as we have for UB, we can easily see that S is analytic. Is S Borel? Is S complete analytic?
\end{problem}
\textbf{Acknowledgements:} The author would like to thank his adviser C. Rosendal
for all the help and attention he gave to this paper. Also, the author would like to thank Joe Diestel for reading this manuscript, and Spyros Argyros for his help in the HI part of these notes.
\end{document} | math | 76,379 |
\begin{document}
\title{\bf A Connection between the Riemann Hypothesis and Uniqueness of the Riemann zeta function }
\author{ Pei-Chu Hu and Bao Qin Li}
\date{}
\maketitle
Department of Mathematics, Shandong University, Jinan 250100,
Shandong, P. R. China E-mail: [email protected]
Department of Mathematics and Statistics, Florida International
University, Miami, FL 33199 USA E-mail: [email protected]
\begin{abstract}
In this paper, we give a connection between the Riemann hypothesis and uniqueness of the Riemann zeta function and an analogue for L-functions.
\end{abstract}
\pagenumbering{arabic}
\section{Introduction}
The Riemann $\zeta$ function is defined by the Dirichlet series
\begin{equation}\label{zeta-Dirichletser}
\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^{s}},\quad s=\sigma+it
\end{equation}
for ${\rm Re}(s)>1$, which is absolutely convergent, and admits an analytical continuation as a meromorphic function in the complex plane $\mathbb{C}$ of order $1$, which has only a simple pole at $s=1$ with residue equal to $1$. It satisfies the following Riemann functional equation:
\begin{equation}\label{r-equation}
\zeta(1-s)=2(2\pi)^{-s}\cos\left(\frac{\pi
s}{2}\right)\Gamma(s)\zeta(s),
\end{equation}
where $\Gamma$ is the Euler gamma function
$$
\Gamma(z) =\int_0^{\infty}t^{z-1}e^{-t}dt, \ \ \ \ \mbox{Re} z
>0,
$$
analytically continued as a meromorphic function in $\mathbb{C}$ of order $1$ without any zeros and with simple poles at $s=0$, $-1$, $-2$, $\cdots$.
The allied function
\begin{equation}\label{xi}
\xi(s)=\frac{s}{2}(s-1)\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\zeta(s)
\end{equation}
is an entire function of order equal to $1$ satisfying the functional equation
\begin{equation}\label{xi-func-eq}
\xi(1-s)=\xi(s)
\end{equation}
(see e.g. \cite{T2}, p.16 and p.29).
It is easy to see that $\zeta(s)$ has no zeros for ${\rm Re}(s)>1$ and, by the functional equation, the only zeros of $\zeta(s)$ in the domain ${\rm Re}(s)<0$ are the poles of $\Gamma(s/2)$. These are called the
{\it trivial zeros}\index{trivial zero} of $\zeta(s)$. Other zeros, called {\it nontrivial zeros}, lie in the {\it critical strip}\index{critical strip} $0\leq{\rm Re}(s)\leq 1$ (actually lie in the open strip $0<{\rm Re}(s)<1$). It is a well-known theorem of G. H. Hardy that there are an infinity of
zeros on ${\rm Re}(s)=\frac{1}{2}$. The famous, as yet unproven, Riemann hypothesis states as follows:
\begin{conjecture}[Riemann Hypothesis]
The nontrivial zeros of $\zeta(s)$ lie on the line ${\rm
Re}(s)=\frac{1}{2}$.
\end{conjecture}
The uniqueness problem for the Riemann zeta function (more generally, for L-functions, see below) is to study how the Riemann zeta function $\zeta$ (or an L-function) is uniquely determined by its zeros or by its $a$-values, i.e., the zeros of $\zeta(s)-a$, where $a$ is a complex value. Uniqueness problems have extensively been studied in the value distribution theory of meromorphic functions in terms of shared values (see e.g. the monographs \cite{HLY} and \cite{YY}), in which two meromorphic functions $f$ and $g$ are called to share a value $a$ if $Z(f-a)=Z(g-a)$, where $Z(F)$ denotes the zero set of $F$ (counting or not counting multiplicities, depending on the questions under consideration). The problem for the Riemann zeta function and L-functions has recently been studied in various settings (see e.g. \cite{CY}, \cite{GHK}, \cite{Ki}, \cite{KL}, \cite{Li1}, \cite{Li2}, \cite{St}, to list a few). In particular, in \cite{KL} and \cite{Li1}, the problem was considered by relaxing the set equality $Z(f-a)=Z(g-a)$ to the set inclusion $Z(f-a)\subseteq Z(g-a)$ for uniqueness of L-functions, which will be seen to be crucial in \S 2. Roughly speaking, two L-functions satisfying the same functional equation are identically equal if they have sufficiently many common zeros (see \cite{KL} and \cite{Li1} for the details and related results as well as references), which gives a uniqueness theorem for solutions of the Riemann functional equation or, more generally, Riemann type functional equations (cf. \S 2 and see [1], [4], [9], etc. for studies of solutions of the Riemann functional equation).
In the present paper, we will discover a connection (an equivalence) between the Riemann Hypothesis and the above mentioned uniqueness problem and then an analogue for L-functions, which, as a consequence, also implies a simply stated necessary and sufficient condition for the Riemann Hypothesis to hold in terms of the limit of an allied function as $\sigma\to +\infty$. This connection does not seem to have been observed before. The results it has brought out in this paper are of a neat and best possible form. Given the fact that uniqueness problems have been studied extensively for meromorphic functions and various techniques have been developed over the years, it would be profitable to further explore this approach with the connection in mind.
\section{Results}
Let $\rho_n$ be the nontrivial zeros of $\zeta$ in the critical strip
$0\leq{\rm Re}(s)\leq 1$. It follows that
\begin{equation*}
\zeta(\rho_n)=\zeta(1-\rho_n)=\zeta(\bar{\rho}_n)
=\zeta(1-\bar{\rho}_n)=0
\end{equation*}
from the functional equation and the identity
$
\zeta(\bar{s})=\overline{\zeta(s)},
$
that is, $\bar{\rho}_n$, $1-\rho_n$, $1-\bar{\rho}_n$ are zeros of
$\zeta(s)$, too. In other words, nontrivial zeros of $\zeta(s)$
are distributed symmetrically with respect to the real axis and to
the critical line ${\rm Re}(s)=\frac{1}{2}$.
Now, let $s_\nu$ be the zeros of $\zeta$ on the half-line ${\rm Re}(s)=\frac{1}{2}$, ${\rm
Im}(s)>0$. Assume that $\rho_n$, $s_\nu$ are ordered with respect
to increasing absolute values of their imaginary parts.
We would like to define a meromorphic function that captures the key features of the Riemann zeta function $\zeta$ with, however, all its zeros in the critical strip being located exactly at those nontrivial zeros of $\zeta$ on the critical line ${\rm Re}(s)=\frac{1}{2}$. If this function is defined ``ideally" and turns out to be identically equal to $\zeta$ (It is here where the uniqueness problem arises, cf. below), then the Riemann hypothesis must follow by the distribution of the zeros of the constructed function.
To realize this goal, we first construct an entire function which plays the role of $\xi$ with, however, zeros at $s_\nu$, which are, as mentioned above, distributed symmetrically with respect to the critical line. We define
\begin{equation}\label{h-symmetry}
h(s)=\frac{1}{2}\prod_{\nu=1}^\infty\left(1-\frac{s-s^2}{|s_\nu|^2}\right).
\end{equation}
This function $h$ possesses the following properties, which are important in serving our purposes:
(a) $h$ is an entire function of order $\le 1$;
(b) The general factor $1-\frac{s-s^2}{|s_\nu|^2}$ in the infinite product has exactly the zeros at $s_v$ and $\overline{s_\nu}$ and thus the zeros of the function $h$ are exactly $s_v$ and $\overline{s_\nu}$ (symmetrically with respect to the critical line), $v=1, 2, \cdots$;
(c) $\lim\limits_{s\to 1}h(s)=\frac{1}{2}$;
(d) $h$ satisfies the same equation (\ref{xi-func-eq}) as $\xi$ does, i.e.,
\begin{equation}\label{h(s)}
h(1-s)=h(s).
\end{equation}
To see (a), it is clear from the definition of $\xi$ in (\ref{xi}) that all the zeros of $\xi$ lie in the critical strip and they are zeros of $\zeta$, i.e., $\rho_n$. Recall that $\xi$ is of order $1$. It follows from Jensen's formula that
\begin{equation*}
n(r, \{ s_{\nu} \})\le n(r, \{\rho_n\})\le Kr^{1+\epsilon}
\end{equation*}
for any $\epsilon>0$, where $K>0$ is a constant and $n(r, \{\rho_n\})$ (resp. $n(r, \{ s_{\nu} \})$) denotes the number of the points $\rho_n, n=1, 2,\cdots$ (resp. $s_\nu, \nu=1, 2, \cdots$) lying in the disc $|s|\le r$ (see e.g. \cite{T1}, p.249). Thus for $|s|\ge 1$,
\begin{eqnarray*}
& &\log |h(s)|\le \sum_{\nu=1}^{\infty}\log\left(1+|\frac{2s^2}{s_{\nu}^2}|\right)-\log 2 \nonumber \\
& &=\int_0^{\infty}\log\left(1+\frac{2|s^2|}{r^2}\right)dn(r, \{ s_{\nu} \})-\log 2 \nonumber\\
& &=4|s|^2\int_0^{\infty}\frac{n(r, \{ s_{\nu} \})}{ r(r^2+2|s|^2)}dr -\log 2 \nonumber\\
& &\le 4|s|^2\big(\int_0^{|s|}\frac{Kr^{1+\epsilon}}{ 2r|s|^2}dr+\int_{|s|}^{\infty}\frac{Kr^{1+\epsilon}}{ r^3}dr\bigr) -\log 2 \nonumber\\
& &=\frac{2K}{1+\epsilon}|s|^{1+\epsilon}+\frac{4K}{1-\epsilon}|s|^{1+\epsilon}-\log 2,
\end{eqnarray*}
which implies that $h$ is an entire function of order $\le 1$. This shows (a). The property (b) is immediate by the fact that $(s-s_v)(s-\overline{s_\nu})=s^2-s+|s^2_{\nu}|$ since ${\rm Re}(s_\nu)=\frac{1}{2}$. The properties (c) and (d) are also immediate, directly from the expression of $h$ in (\ref{h-symmetry}).
Further, we define a meromorphic function $\eta(s)$ using the same expression as that for $\zeta$ in (\ref{xi}) (with the role of $\xi$ there being replaced by $h$),
\begin{equation}\label{eta}
h(s)=\frac{s}{2}(s-1)\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\eta(s).
\end{equation}
Replacing $s$ by $1-s$ yields that
\begin{equation*}
h(1-s)=\frac{s}{2}(s-1)\pi^{-\frac{1-s}{2}}\Gamma\left(\frac{1-s}{2}\right)\eta(1-s),
\end{equation*}
which implies, in view of (\ref{h(s)}) and (\ref{eta}), that
\begin{eqnarray*}
& &\eta(1-s)=\frac{h(1-s)}{ \frac{s}{2}(s-1)\pi^{-\frac{1-s}{2}}\Gamma\left(\frac{1-s}{2}\right)}\\
& &=\frac{\frac{s}{2}(s-1)\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\eta(s)}{ \frac{s}{2}(s-1)\pi^{-\frac{1-s}{2}}\Gamma\left(\frac{1-s}{2}\right)}\\
& &=\frac{\pi^{-s+\frac{1}{2}}\Gamma(\frac{s}{2})}{\Gamma(\frac{1-s}{2})}\eta(s)\\
& &=2(2\pi)^{-s}\cos\left(\frac{\pi s}{2}\right)\Gamma(s)\eta(s),
\end{eqnarray*}
by virtue of the identity $\frac{\pi^{-s+\frac{1}{2}}\Gamma(\frac{s}{2})}{\Gamma(\frac{1-s}{2})}=2(2\pi)^{-s}\cos\left(\frac{\pi s}{2}\right)\Gamma(s)$ (see e.g. \cite{T2}, p.16). That is, the function $\eta$ also satisfies the Riemann functional equation (\ref{r-equation}) as $\zeta$ does:
\begin{equation}\label{eta1}
\eta(1-s)=2(2\pi)^{-s}\cos\left(\frac{\pi
s}{2}\right)\Gamma(s)\eta(s).
\end{equation}
We see from (\ref{eta}) that only zeros of $\eta(s)$ in the domain ${\rm Re}(s)<0$ are the poles of $\Gamma(s/2)$, which are the trivial zeros of $\zeta$. Other zeros of $\eta$ lie on the line ${\rm Re}(s)=\frac{1}{2}$ in view of the construction of $h(s)$ and $\eta(s)$ (see Property (b) of $h$). The point $s=1$ is the only pole of $\eta(s)$, which is a simple pole with residue
\begin{equation}\label{residue}
{\rm Res}_{s=1}\eta(s)=\lim_{s\to 1}(s-1)\eta(s)=\lim_{s\to 1}\frac{2\pi^{\frac{s}{2}}h(s)}{s\Gamma\left(\frac{s}{2}\right)}=\frac{\sqrt{\pi}}{\Gamma\left(\frac{1}{2}\right)}=1,
\end{equation}
using (\ref{eta}) and Property (c) of $h$. It also follows from (\ref{eta}) that $(s-1)\eta$ is an entire function of order $\le 1$ in view of Property (a) of $h$.
The function $\eta$ possesses the characteristics we desire, as described above; in fact, we can now establish the following
\begin{theorem}\label{RH-eqthm}
The Riemann hypothesis is true if and only if $\zeta(s)\equiv \eta(s)$.
\end{theorem}
\begin{proof}
The sufficiency is clear since all the zeros of $\eta$ on ${\rm Re}(s)\ge 0$ lie on the line ${\rm Re}(s)=\frac{1}{2}$ by the construction of $h(s)$ and $\eta(s)$.
For the necessity, if the Riemann hypothesis holds, then $\zeta$ and $\eta$ have the same zeros in the entire complex plan; thus we have that
$\eta(s)=e^{as+b}\zeta(s)$ for some complex numbers $a, b$, in view of the fact that $\zeta$ is of order $1$ and $\eta$ is of order $\le 1$. We deduce, by applying (\ref{eta}), (\ref{xi}), (\ref{h(s)}) and (\ref{xi-func-eq}), that
\begin{eqnarray*}
& &e^{as+b}=\frac{\eta(s)}{\zeta(s)}=\frac{h(s)}{\xi(s)}\\
& &=\frac{h(1-s)}{\xi(1-s)}=\frac{\eta(1-s)}{\zeta(1-s)}=e^{a(1-s)+b}.
\end{eqnarray*}
Thus, $e^{as+b}=e^{a(1-s)+b},$ which implies that $a=0$. Then $\eta(s)=e^{b}\zeta(s)$ and then $(s-1)\eta(s)=e^{b}(s-1)\zeta(s)$. Taking the limit $s\to 1$ and by (\ref{residue}) and the fact that $\zeta$ has residue $1$ at $s=1$ also, we obtain that $e^b=1$. This proves that $\eta(s)\equiv \zeta(s).$
\end{proof}
From Theorem 2.1, to prove the Riemann hypothesis we now only need to prove that $\zeta(s)\equiv\eta(s)$, from which the uniqueness problem arises. Note that the function $\eta$ is a meromorphic function in $\mathbb{C}$ of order $\le 1$ that satisfies the following important properties:
(i) $\eta$ and $\zeta$ satisfy the same functional equation;
(ii) the zero set of $\eta$ is a subset of the zero set of $\zeta$ (counting multiplicities), i.e., $Z(\eta)\subseteq Z(\zeta)$, where $Z(f)$ denotes the set of the zeros of $f$ with counting multiplicities.
The property (i) means that $\eta$ is a solution of the Riemann functional equation, which is known to have different solutions with certain relations (see \cite{BC}, \cite{Ha}, \cite{Kn}, etc. for studies of solutions of the Riemann functional equation). Clearly we are seeking the conditions that force the solutions to become the unique one - the Riemann zeta function. This leads to the following uniqueness problem:
\begin{problem}[Uniqueness problem] Let $f$ be a meromorphic function (of order $\le 1$) in $\mathbb{C}$ such that
(i) $f$ and $\zeta$ satisfy the same functional equation;
(ii) $Z(f)\subseteq Z(\zeta)$.
Under what conditions are $f$ and $\zeta$ identically equal?
\end{problem}
This is the uniqueness problem considered in \cite{Li1} and then in \cite{KL} for two L-functions; but to serve our purpose here we now need to consider the uniqueness problem when one of the functions is a meromorphic function $f$ satisfying the above two conditions (i) and (ii) in Problem 2.2.
It is clear that if $f$ satisfies the above two conditions (i) and (ii), then for any nonzero constant $c$, $cf$ also satisfies the these two conditions. An obvious property of the Riemann zeta function (simply from its Dirichlet series) is that $\zeta$ tends to $1$ as $\sigma\to +\infty$. In order to have the uniqueness of $f$ and $\zeta$, $f$ must necessarily tend to $1$ as $\sigma\to +\infty$. Thus, this naturally becomes the condition we use, as given in the theorem below.
\begin{theorem}\label{zeta-thm}
Let $f(s)$ be a nonconstant meromorphic function in $\mathbb{C}$ of order $\le 1$ with $\lim\limits_{\sigma\to +\infty}f(s)=1$. Then $f\equiv \zeta$ if and only if $f$ satisfies the Riemann functional equation and $Z(f)\subseteq Z(\zeta)$.
\end{theorem}
The theorem will be proved later and treated as a consequence of a more general result for L-functions (see Theorem 2.5 below). Since the function $\eta$ satisfies the conditions (i) and (ii) in Problem 2.2, Theorem 2.3 and Theorem 2.1 yield immediately the following theorem for the Riemann hypothesis, which is of a particularly neat and simple statement:
\begin{theorem}
The Riemann hypothesis is true if and only if $\lim\limits_{\sigma\to +\infty}\eta(s)=1$.
\end{theorem}
In fact, if $\lim\limits_{\sigma\to +\infty}\eta(s)=1$, then $\eta$ satisfies all the conditions of Theorem 2.3 and thus, $\eta(s)\equiv \zeta(s)$, which implies that the Riemann hypothesis is true by the sufficient condition of Theorem 2.1. Conversely, if the Riemann hypothesis holds, then by the necessary condition of Theorem 2.1, $\zeta(s)\equiv\eta(s)$, which then implies that $\lim\limits_{\sigma\to +\infty}\eta(s)=\lim\limits_{\sigma\to +\infty}\zeta(s)=1$.
We are going to generalize Theorem 2.3 so that one of the functions in the theorem is a meromorphic function $f$ as described above and the other is a Dirichlet series in the extended Selberg class, which takes the Riemann zeta function as a special case, so the above approach can then be pushed over to L-functions (see Theorem 2.5 below). The result we present is more than what we need, which is inspired by and based on our earlier work \cite{Li1} and \cite{KL}, and which, as a uniqueness theorem, is of its own independent interest. The observation that one of the functions is not necessarily assumed to be an L-function is essential for the purpose of the connection as analyzed above.
Recall that the Selberg class of $L$-functions is the set of all Dirichlet series $L(s)=\sum_{n=1}^{\infty} {a(n)\over n^s}$ with $a(1)=1$, satisfying the following axioms (see \cite{Se}):
\noindent (i) (Dirichlet series) For $\sigma >1 $, $L(s)$ is an absolutely convergent Dirichlet series;
\hfil\break (ii) (Analytic
continuation) There is a non-negative integer $k$ such that
$(s-1)^kL(s)$ is an entire function of finite order; \hfil\break
(iii) (Functional equation) $L$ satisfies a functional equation of
type
$$\Lambda_L(s)=\omega\overline{\Lambda_L(1-{\bar
s})},$$ where $\Lambda_L(s)=L(s)Q^s\prod_{j=1}^K\Gamma(\lambda_j
s+\mu_j)$ with positive real numbers $Q, \lambda_j$, and complex
numbers $\mu_j, \omega$ with $\hbox{Re}\mu_j\geq 0$ and
$|\omega|=1$; \hfil\break (iv) (Ramanujan hypothesis) $a(n)\ll
n^{\varepsilon}$ for every $\varepsilon>0$;
\hfil\break(v) (Euler product) $\log L(s)=\sum_{n=1}^{\infty}{b(n)\over n^s}$, where
$b(n)=0$ unless $n$ is a positive power of a prime and $b(n)\ll
n^{\theta}$ for some $\theta<{1\over 2}$.
The Selberg class includes the Riemann zeta-function $\zeta$ and
essentially all Dirichlet series where one might expect the
analogue of the Riemann hypothesis. In the uniqueness theorem given below, all $L$-functions are assumed to be in the extended Selberg class, i.e., Dirichlet series
$L(s)=\sum_{n=1}^{\infty} {a(n)\over n^s}$ with $a(1)=1$ satisfying only the axioms (i)-(iii). Thus, the result below particularly applies to $L$-functions in the Selberg class.
\begin{theorem}\label{M-thm}
Let $f(s)$ be a nonconstant meromorphic function in $\mathbb{C}$ of order $\le 1$ with $\lim\limits_{\sigma\to +\infty}f(s)=1$ and $L$ an L-function. Then $f\equiv L$ if and only if $f$ and $L$ satisfy the same functional equation and
$Z^+(f)\setminus G\subseteq Z^+(L)$ for a set $G$ (counted with multiplicity) satisfying
that
\begin{equation}\label{G-notation}
\limsup\limits_{r\rightarrow \infty}\frac{n(r, G)}{r}<{\log 4\over \pi}.
\end{equation}
Furthermore, the inequality (\ref{G-notation}) is best possible.
\end{theorem}
On the above, $n(r, G)$ denotes the number of points of $G$ (counting multiplicities) lying in the disc $|s|\leq r$. And, $Z^+(L)$ denotes the set of nontrivial zeros of $L$ counted with multiplicity. As usual, the trivial zeros of $L$ are those coming from the poles of the $\Gamma$ factors in the functional equation of the axiom (iii), and the other zeros are called nontrivial zeros. The set $Z^+(f)$ is defined in the same way using the same functional equation.
In addition to the sharpness of (\ref{G-notation}), the conditions in Theorem 2.5 (and thus in Theorem 2.3) are tight and the result is best possible in the sense that the theorem breaks down if any of the conditions is dropped, as shown by the counterexamples in the Remark after the proof.
\begin{proof} By the assumption on the set $G$, it is easy to check that the infinite product $\sum\limits_{\rho \in
G}\log\left(1-\frac{s^2}{\rho^2}\right)$ converges to an entire function in the complex plane (cf. (\ref{product}) below). Since $L$ satisfies the analytic continuation axiom (ii), $L$ has at most one pole at $s=1$. We can thus properly choose integers $m, n$ such that the auxiliary function
\begin{equation}\label{function}
F(s):=\left(s^2-1\right)^ms^n{L(s)-f(s)\over f(s)}{L(-s)-f(-s)\over f(-s)}\prod_{\rho\in
G}\left(1-{s^2\over \rho^2}\right)
\end{equation}
does not have a pole at $s=\pm 1$ and that $s=0$ is a zero of $F$ (we may then assume that $s=0$ is not in $G$). Since $f$ and $L$ satisfy the same functional equation, the function $f-L$ must satisfy the same functional equation. Thus, $f$ and $f-L$ have the same trivial zeros that are located at the poles of the $\Gamma$ factors in the functional equation of the axiom (iii). These zeros do not produce any poles of $F$ due to cancelation. Other zeros of $f$ are canceled by those of $L-f$ in (\ref{function}). Any pole of $f$ clearly does not produce a pole of $F$ by the construction of $F$. Hence, $F$ is an entire function.
Choose
$0<D_1<D_2<1$ with $\limsup\limits_{r\rightarrow \infty}\frac{n(r, G)}{r}<D_1{\log 4\over \pi}.$
Then, there is a positive number $r_0>0$ such that ${n(r, G)\over r}<D_1{\log 4\over \pi}$ for $r\geq r_0$. We deduce that for large $|s|$,
\begin{eqnarray}\label{product}
& &\log\left|\prod_{\rho\in
G}\left(1-\frac{s^2}{\rho^2}\right)\right|\leq\sum_{\rho \in
G}\log\left(1+|\frac{s^2}{\rho^2}|\right) \nonumber \\
& &=\int_0^{\infty}\log\left(1+\frac{|s^2|}{r^2}\right)dn(r, G) \nonumber\\
& &=2|s|^2\int_0^{\infty}\frac{n(r, G)}{ r(r^2+|s|^2)}dr \nonumber\\
& &\leq 2|s|^2\left\{\frac{1}{ |s|^2}\int_0^{r_0}\frac{n(r, G)}{ r}dr+ D_1\frac{\log 4}{\pi}\int_{r_0}^{\infty}\frac{1}{ r^2+|s|^2}dr \right\} \nonumber \\
& &= 2|s|^2\left\{\frac{1}{ |s|^2}\int_0^{r_0}\frac{n(r, G)}{ r}dr+ D_1\frac{\log 4}{\pi}\frac{1}{|s|}(\frac{\pi}{2}-\arctan\frac{r_0}{|s|})\right\} \nonumber \\
& &\le 2\int_0^{r_0}\frac{n(r, G)}{r}dr+D_1|s|\log 4\leq D_2|s|\log 4.
\end{eqnarray}
Recall that $L(s)=\sum_{n=1}^{\infty}{a(n)\over n^s}$ with
$a(1)=1$ and the series converges absolutely as $\sigma>1$. It is elementary to check that $(\frac{n}{2})^{\sigma}\geq n^2$ for $n\ge 4$ and $\sigma\ge 4$. Thus as $\sigma\ge 4$, we have that
\begin{eqnarray*}
& &\sum_{n=4}^{\infty}|\frac{a(n)}{n^s}|\le \frac{1}{2^{\sigma}}\sum_{n=4}^{\infty}|\frac{a(n)}{(\frac{n}{2})^{\sigma}}| \\
& &\le \frac{1}{2^{\sigma}}\sum_{n=4}^{\infty}|\frac{a(n)}{n^2}|=\frac{C}{2^{\sigma}},
\end{eqnarray*}
where $C=\sum\limits_{n=4}^{\infty}|\frac{a(n)}{n^2}|<+\infty$. Hence,
$$|f(s)-L(s)|=|f(s)-1-\sum\limits_{n=2}^{\infty}\frac{a_n}{n^s}|\le |f(s)-1|+\frac{1}{2^{\sigma}}O(1)$$
and then for a fixed $\epsilon>0$ (to be specified later),
$$|\frac{f(s)-L(s)}{f(s)}|=(\epsilon+\frac{1}{2^\sigma})O(1)$$
for large $\sigma$, in view of the assumption that $f(s)\to 1$ as $\sigma\to +\infty$.
Dividing the functional equation of $f-L$ by the same functional equation satisfied by $f$ and $L$, we obtain that
\begin{equation*}
{L(s)-f(s)\over f(s)}={\overline{L(1-\overline{s})}-\overline{f(1-\overline{s})}\over \overline{f(1-\overline{s})}}.
\end{equation*}
We thus obtain that
$$\left|{L(s)-f(s)\over f(s)}\cdot{L(-s)-f(-s)\over f(-s)}\right|=(\epsilon+\frac{1}{2^{|\sigma|}})^2O(1)$$
as $\sigma\rightarrow \pm \infty$. By applying this estimate and the estimate (\ref{product}) to (\ref{function}), we have
that for a number $D_3$ with $D_2<D_3<1$,
\begin{eqnarray*}
& &\log |F(s)|\leq D_3|s|\log 4+2\log (\epsilon+\frac{1}{2^{|\sigma|}})\\
& &=D_3|s|\log 4-|\sigma|\log 4+2\log (1+\epsilon 2^{|\sigma|})
\end{eqnarray*}
as $\sigma\rightarrow\pm \infty.$
Define $g(\epsilon)=\log (1+\epsilon 2^{|\sigma|})-\epsilon\log 2^{|\sigma|}$ for $\epsilon>0$. Then $g(0)=0$ and it is easy to check that $g'(\epsilon)<0$ for sufficiently large $|\sigma|$. Thus, as $\sigma\rightarrow\pm \infty,$ we have that
$$\log (1+\epsilon 2^{|\sigma|})\le \epsilon\log 2^{|\sigma|}.$$
We can now take $\epsilon$ such that $D:=D_3+\epsilon<1$. Then
\begin{eqnarray*}
& &\log |F(s)|\leq D_3|s|\log 4-|\sigma|\log 4+\epsilon |\sigma|\log 4\\
& & \le (D|s|-|\sigma|)\log 4=|s|(D-\frac{|\sigma|}{|s|})\log 4.
\end{eqnarray*}
It is then easy to see that $F$ is bounded on the rays
$\hbox{arg}(s)=\theta, \pi-\theta, \pi+\theta, 2\pi-\theta$, where
$0<\theta<\pi/2$ with $\cos \theta=D$, since on these rays, $|\cos\theta|={|\sigma|\over |s|}=D$. Note that $f$ is of order $\le 1$ by the assumption,
a nonconstant $L$-function is of order $1$ (see e.g. \cite{Se} and \cite{St}), and the infinite product
in (\ref{function}) is also of order $\le 1$, which follows from (\ref{product}). Thus, $F$ must be of order at most $1$. We
then have that $F(s)=O\left(e^{|s|^{1+\epsilon}}\right)$ for any
$\epsilon>0$. Recall the Phragm\'en-Lindel\"of theorem (see e.g. \cite{T1}, p.177): Let $f$ be holomorphic in a sector between two straight lines making an angle of $\pi/\alpha$ at the origin and continuous on the boundary. If $|f(s)|\le M$ on the boundary and $f(s)=O(e^{r^{\beta}})$ as $r\to\infty$ uniformly in the sector, where $\beta<\alpha$, then $|f(s)|\le M$ in the entire sector. We see that $F$ satisfies the conditions of the theorem in each of the sectors bounded by the
above rays and thus $f$ is bounded in each of the sectors and thus in the entire complex plane. Therefore the entire function $F$ must
be a constant. But, $F$ has a zero at $s=0$ (see the choice of $n$). Thus $F$ and then $f-L$ must
be identically zero.
Next, we prove that the inequality in (\ref{G-notation}) is best possible. We will present a counterexample, in which $f$ is even not a Dirichlet series (and thus the theorem fails badly). To this end, consider
$$ L(s)=1+{2\over 4^s}, \quad f(s)=(1+\frac{1}{s(1-s)})L(s).$$
Then it is easy to verify that
$$2^sL(s)=2^{1-s}\overline{L(1-{\overline s})},$$
which also clearly implies that
$$2^sf(s)=2^{1-s}\overline{f(1-{\overline s})}.$$
That is, both $f$ and $L$ satisfy the same functional equation. The zeros of $L(s)$ are $\frac{1}{\ln 4}(\ln 2+\pi i+2k\pi i)$, where $k$ is an integer, which readily implies that
$$\lim\limits_{r\to\infty}{n(r, Z(L))\over r}={\log 4 \over \pi}$$
and also that
$$\lim\limits_{r\to\infty}{n(r, Z(f))\over r}={\log 4 \over \pi}.$$
Now, take the exceptional set $G$ to be the entire set $Z(f)$. Then, $Z^+(f)\setminus G=\emptyset\subset Z^+(L)$. However, $f\not=L$. This proves the theorem.
\end{proof}
\noindent{\bf Remark} {\bf (i)} It would be tempting to try to drop the condition of the order $\le 1$ for $f$ in Theorem 2.5. But, it is not the case. Consider
$$ L(s)=1+\frac{2}{4^s}, \quad f(s)=\frac{1}{1+e^{s(1-s)}}L(s).$$
Then it is easy to check that $f$ is of order equal to $2$ with $\lim\limits_{\sigma\to +\infty}f(s)=1$. From the proof of Theorem 2.5, we see that both $f$ and $L$ satisfy the same functional equation. Note that $L$ and $f$ have the same zeros. Take the exceptional set $G$ to be the empty set. Then, $Z^+(f)\setminus G=Z^+(L)$. However, $f\not=L$.
\noindent
{\bf (ii)} The condition that $\lim\limits_{\sigma\to +\infty}f(s)=1$ in the theorem cannot be dropped either. To see this, use the same function $ L(s)=1+{2\over 4^s}$ as in (i) but set $f(s)=\frac{1}{s(1-s)}L(s).$ Then $L$ and $f$ satisfy all the conditions of Theorem 2.5 with $G$ being the empty set, except that $\lim\limits_{\sigma\to +\infty}f(s)=0$. But, $f\not=L$.
The above ideas may be carried over to L-functions. To demonstrate, we will do this specifically for the Dedekind zeta function of an algebraic number field, which encodes important arithmetic information of the field and has extensively been studied in number theory (see e.g. the monographs \cite{Ne} and \cite{HY}). Let $\kappa$ be a number field. Its Dedekind zeta function is defined by the Dirichlet series
$$\zeta_{\kappa}(s) =\sum\limits_{\mathfrak{a}}\frac{1}{\mathcal{N}(\mathfrak{a})^{s}}$$
for $\sigma>1$, where $\mathfrak{a}$ runs over the non-zero ideals of the ring $\kappa$ of integers of $\kappa$ and $\mathcal{N}(\mathfrak{a})$ denotes the absolute norm of $\mathfrak{a}$. It becomes the Riemann zeta function when the field is the rational numbers $\mathbb{Q}$.
The Dirichlet series converges absolutely for $\sigma>1$, it has an analytic continuation to a meromorphic function in $\mathbb{C}$ of order equal to $1$ with only a simple pole at $s = 1$. By the well-known Analytic Class Number Formula (see e.g. \cite{Ne}, p.467), the residue of $\zeta_{\kappa}$ at $s=1$ is given by
\begin{equation}\label{residue-formula}
\lim\limits_{s\to 1}(s-1)\zeta_{\kappa}=\frac{2^{r_1}(2\pi)^{r_2}c_{\kappa}R_{\kappa}}{w_{\kappa}{\sqrt{|D_{\kappa/\mathbb{Q}}|}}},
\end{equation}
where $r_1$ (resp. $r_2$) is the number of real (resp. complex) places of $\kappa$, $c_{\kappa}$ is the class number of $\kappa$, $R_{\kappa}$ is the regulator of $\kappa$, $D_{\kappa/\mathbb{Q}}$ is the discriminant of the field $\kappa$, and $w_{\kappa}$ denotes the number of roots of unity in $\kappa$. The function $\zeta_\kappa$ satisfies the
following functional equation (see e.g. \cite{Ne}, p.467)
\begin{equation}\label{kappa}
\zeta_\kappa(1-s)= |D_{\kappa/\mathbb{Q}}|^{s-\frac{1}{2}}
\left(\cos\frac{\pi s}{2}\right)^{r_1+r_2} \left(\sin\frac{\pi
s}{2}\right)^{r_2} \Gamma_{\mathbb{C}}(s)^n \zeta_\kappa(s),
\end{equation}
where $n=[\kappa:\mathbb{Q}]=r_1+2r_2$ and
$
\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s).
$
Let
$\Gamma_{\mathbb{R}}(s)=\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right).$ Then, the function
\begin{equation}\label{xi-a}
\xi_\kappa(s)=\frac{s}{2}(s-1)|D_{\kappa/\mathbb{Q}}|^{s/2}
\Gamma_{\mathbb{R}}(s)^{r_1}\Gamma_{\mathbb{C}}(s)^{r_2}\zeta_\kappa(s),
\end{equation}
is an entire function of order $1$ and satisfies the functional equation (see e.g. \cite{Ne}, p.467)
\begin{equation}\label{xi-k}
\xi_\kappa(s)=\xi_\kappa(1-s).
\end{equation}
\begin{conjecture}[Grand Riemann Hypothesis for the Dedekind zeta function]
If $\zeta_\kappa(s)= 0$ and $0 \le {\rm Re}(s) \le 1$, then ${\rm Re}(s) = 1/2.$
\end{conjecture}
As before and for convenience of comparison, we still use $\rho_n$ to denote the zeros of $\zeta_{\kappa}$ satisfying $0\leq {\rm Re}(\rho_n)\leq 1$ and $s_\nu$ the zeros of $\zeta_{\kappa}$ on the half-line ${\rm Re}(s)=\frac{1}{2}$, ${\rm Im}(s)>0$, ordered with respect to increasing absolute values of their imaginary parts.
We define
\begin{equation}\label{h-product}
h_\kappa(s)=\frac{2^{r_1+r_2-1}}{w_{\kappa}}c_{\kappa}R_{\kappa}\prod_{\nu=1}^\infty\left(1-\frac{s-s^2}{|s_\nu|^2}\right).
\end{equation}
Then with the same arguments for $h(s)$, we can show that the function $h_\kappa(s)$ possesses the following properties:
(a) $h_\kappa$ is an entire function of order $\le 1$;
(b) The general factor $1-\frac{s-s^2}{|s_\nu|^2}$ in the infinite product has exactly the zeros at $s_v$ and $\overline{s_\nu}$ and thus the zeros of the function $h_\kappa$ are exactly $s_v$ and $\overline{s_\nu}$ (symmetrically with respect to the critical line), $v=1, 2, \cdots$;
(c) $\lim\limits_{s\to 1}h_\kappa(s)=\frac{2^{r_1+r_2-1}}{w_{\kappa}}c_{\kappa}R_{\kappa}$;
(d) $h_\kappa$ satisfies the same equation (\ref{xi-k}) as $\xi_\kappa$ does, i.e.,
\begin{equation}\label{h-kappa}
h_\kappa(1-s)=h_\kappa(s).
\end{equation}
Further, we define $\eta_\kappa(s)$ by
\begin{equation}\label{eta-k}
h_\kappa(s)=\frac{s}{2}(s-1)|D_{\kappa/\mathbb{Q}}|^{s/2}
\Gamma_{\mathbb{R}}(s)^{r_1}\Gamma_{\mathbb{C}}(s)^{r_2}\eta_\kappa(s).
\end{equation}
Replacing $s$ by $1-s$, we then have that
\begin{equation*}
h_\kappa(1-s)=\frac{s}{2}(s-1)|D_{\kappa/\mathbb{Q}}|^{\frac{1-s}{2}}
\Gamma_{\mathbb{R}}(1-s)^{r_1}\Gamma_{\mathbb{C}}(1-s)^{r_2}\eta_\kappa(1-s),
\end{equation*}
which implies, in view of (\ref{eta-k}), that
\begin{eqnarray*}
& &\eta_{\kappa}(1-s)=\frac{h_{\kappa}(1-s)}{ \frac{s}{2}(s-1)|D_{\kappa/\mathbb{Q}}|^{ \frac{1-s}{2} }
\Gamma_{\mathbb{R}}(1-s)^{r_1}\Gamma_{\mathbb{C}}(1-s)^{r_2}\eta_\kappa(s) }\\
& &=|D_{\kappa/\mathbb{Q}}|^{s-\frac{1}{2}}\frac{\Gamma_{\mathbb{R}}(s)^{r_1}\Gamma_{\mathbb{C}}(s)^{r_2}}{\Gamma_{\mathbb{R}}
(1-s)^{r_1}\Gamma_{\mathbb{C}}(1-s)^{r_2}}\eta_{\kappa}(s) \\
& &=|D_{\kappa/\mathbb{Q}}|^{s-\frac{1}{2}}\left(\cos\frac{\pi s}{2}\right)^{r_1+r_2} \left(\sin\frac{\pi
s}{2}\right)^{r_2}\Gamma_{\mathbb{C}}(s)^n \eta_\kappa(s)
\end{eqnarray*}
by virtue of the identity
\begin{eqnarray*}
& &\frac{\Gamma_{\mathbb{R}}(s)^{r_1}\Gamma_{\mathbb{C}}(s)^{r_2}}{\Gamma_{\mathbb{R}}
(1-s)^{r_1}\Gamma_{\mathbb{C}}(1-s)^{r_2}} \\
& &=\left(\cos\frac{\pi s}{2}\right)^{r_1+r_2} \left(\sin\frac{\pi
s}{2}\right)^{r_2} \Gamma_{\mathbb{C}}(s)^n,
\end{eqnarray*}
which can be directly verified or obtained from (\ref{xi-k}), (\ref{xi-a}) and (\ref{kappa}). Hence, $\eta_\kappa$ satisfies the same functional equation (\ref{kappa}) as $\zeta_{\kappa}$ does:
\begin{equation*}
\eta_\kappa(1-s)= |D_{\kappa/\mathbb{Q}}|^{s-\frac{1}{2}}
\left(\cos\frac{\pi s}{2}\right)^{r_1+r_2} \left(\sin\frac{\pi
s}{2}\right)^{r_2} \Gamma_{\mathbb{C}}(s)^n \eta_\kappa(s).
\end{equation*}
\begin{theorem}\label{GRH-eqthm}
The Grand Riemann hypothesis is true if and only if $\zeta_{\kappa}(s)=\eta_{\kappa}(s)$.
\end{theorem}
\begin{proof}
The sufficiency is clear since all the zeros of $\eta_{\kappa}$ on ${\rm Re}(s)\ge 0$ lie on the line ${\rm Re}(s)=\frac{1}{2}$ by the construction of $\eta_{\kappa}(s)$ and $h_{\kappa}(s)$ (cf. Property (b) of $h_{\kappa}$).
For the necessity, if the Grand Riemann Hypothesis holds, then $\zeta_{\kappa}(s)$ and $\eta_{\kappa}(s)$ have the same zeros in the entire complex plan; thus we have that
$\eta_{\kappa}(s)=e^{as+b}\zeta_{\kappa}(s)$ for some complex numbers $a, b$, in view of the fact that $\zeta_{\kappa}$ is of order $1$ and $\eta_{\kappa}$ is of order $\le 1$ (cf. Property (a) of $h_{\kappa}$). By applying (\ref{eta-k}), (\ref{h-kappa}), (\ref{xi-k}), and (\ref{xi-a}), we deduce that
\begin{eqnarray*}
& &e^{as+b}=\frac{\eta_{\kappa}(s)}{\zeta_{\kappa}(s)}=\frac{h_{\kappa}(s)}{\xi_{\kappa}(s)}\\
& &=\frac{h_{\kappa}(1-s)}{\xi_{\kappa}(1-s)}=\frac{\eta_{\kappa}(1-s)}{\zeta_{\kappa}(1-s)}=e^{a(1-s)+b},
\end{eqnarray*}
which implies that $a=0$. Then $\eta_{\kappa}(s)=e^{b}\zeta_{\kappa}(s)$ and thus \begin{equation}\label{limit}
(s-1)\eta_{\kappa}(s)=e^{b}(s-1)\zeta_{\kappa}(s).
\end{equation}
Next, by the Analytic Class Number Formula (\ref{residue-formula}), the residue of $\zeta_{\kappa}$ at $s=1$ is
$$\lim\limits_{s\to 1}(s-1)\zeta_{\kappa}=\frac{2^{r_1}(2\pi)^{r_2}c_{\kappa}R_{\kappa}}{w_{\kappa}{\sqrt{|D_{\kappa/\mathbb{Q}}|}}}.$$
On the other hand, by (\ref{eta-k}) and Property (c) of $h_{\kappa}$ we have that
\begin{eqnarray*}
& &\lim\limits_{s\to 1}(s-1)\eta_{\kappa}=\lim\limits_{s\to 1}\frac{(s-1)
{h_\kappa(s)}}{{\frac{s}{2}(s-1)|D_{\kappa/\mathbb{Q}}|^{s/2}
\Gamma_{\mathbb{R}}(s)^{r_1}\Gamma_{\mathbb{C}}(s)^{r_2}}}\\
& &=\frac{2^{r_1+r_2-1}c_{\kappa}R_{\kappa}}{\frac{1}{2}\pi^{-r_2}w_{\kappa}\sqrt{|D_{\kappa/\mathbb{Q}}|}}\\
& &=\frac{2^{r_1}(2\pi)^{r_2}c_{\kappa}R_{\kappa}}{w_{\kappa}{\sqrt{|D_{\kappa/\mathbb{Q}}|}}}.
\end{eqnarray*}
Taking the limit $s\to 1$ in (\ref{limit}), we deduce that $1=e^b$. This proves that $\eta_{\kappa}=\zeta_{\kappa}.$
\end{proof}
To conclude the paper, we give the analogue of Theorem 2.4 for the Dedekind zeta function. We can write $\zeta_{\kappa}$ as a normal Dirichlet series,
$\zeta_{\kappa}=\sum\limits_{n=1}^{\infty}\frac{a(n)}{n^s}$, where the coefficients $a(n)$ now represent the number of ideals of norm $n$ and $a(1)=1$, corresponding to the ideal $\mathcal{O}_{\kappa}$. If the Grand Riemann Hypothesis is true, then by the necessary condition of Theorem 2.7, $\zeta_{\kappa}=\eta_{\kappa}$. Thus, $\lim\limits_{\sigma\to +\infty}\eta_\kappa(s)=\lim\limits_{\sigma\to +\infty}\zeta_\kappa(s)=1.$ Conversely, if $\lim\limits_{\sigma\to +\infty}\eta_\kappa(s)=1$, then by Theorem 2.5 and in view of the fact that all the conditions there are satisfied with $f=\eta_{\kappa}$ and $L=\zeta_{\kappa}$, we have the uniqueness that $\zeta_{\kappa}=\eta_{\kappa}$ and thus the Grand Riemann Hypothesis holds by the sufficient condition of Theorem 2.7. This shows that we have the following
\begin{theorem}
The Grand Riemann Hypothesis is true if and only if
$$
\lim_{\sigma\to +\infty}\eta_\kappa(s)=1.
$$
\end{theorem}
\end{document} | math | 35,464 |
\betagin{document}
\title{\bf Capture zones of the family of functions $\lambda z^m \exp(z)$. }
\centerline{\bf Abstract}
\betagin{quotation}
{\noindent \small We consider the family of entire transcendental maps
given by $F_{\lambdambda,m} (z ) \, = \, \lambdambda z^m \exp(z) $ where $ m \ge 2$.
All functions $F_{\lambdambda,m}$ have a superattracting
fixed point at $z=0$, and a critical point at $z=-m$. In the dynamical plane
we study the topology of the basin of attraction of $z=0$. In the parameter plane we focus
on the capture behaviour, i.e., $\lambdambda$ values such that the critical point
belongs to the basin of attraction of $z=0$. In particular, we find a capture zone for
which this basin has a unique connected component, whose boundary is then non-locally connected.
However, there are parameter values for which the boundary of the immediate basin of $z=0$ is a
quasicircle.}
\end{quotation}
\section{Introduction}
Our goal in this paper is to study some dynamical aspects of the families of
entire transcendental maps
\[
F_{\lambdambda,m} (z ) \, = \, \lambdambda
z^m \exp(z), \quad m \ge 2.
\]
Observe that $m=0$ corresponds to the exponential family $E_{\lambdambda} (z) \,
= \, \lambdambda \exp(z)$, the simplest example of an entire transcendental
map with a unique asymptotic value, $z=0$, in analogy with the well known quadratic family of
polynomials $z \to z^2 +c$. The exponential map has been thoroughly studied by many authors (see
for example, \cite{DK}, \cite{DT}).
The case $m=1$ corresponds to $G_{\lambdambda}(z) \, = \, \lambdambda z \exp(z)$ which appeared for the first time in \cite{Ba1} as an example of an entire transcendental map whose Julia set is the whole plane
(for an appropriate value of $\lambda$). Later on this family was studied in \cite{F} and \cite{G}. The asymptotic value $z=0$ of $G_{\lambdambda}(z)$ is fixed and its multiplier is $G'_{\lambdambda}(0) \, = \, \lambdambda$. Hence its dynamical character depends on the parameter $\lambda$. Besides this point, the
dynamical behavior of $G_{\lambdambda}$ is determined by the orbit of the critical point $z=-1$.
Some functions in the family $F_{\lambdambda,m}=\lambdambda z^m \exp(z)$ for $m \ge 2$ have been used in the literature
as examples of certain dynamical phenomena (see for example \cite{B}, for a Baker domain
at a positive distance from any singular orbit). But, to our knowledge, no systematic study
has been made before this work.
All functions of the form $F_{\lambdambda,m}$, with $m \ge 2$, have a superattracting fixed point
at $z=0$ of multiplicity $m$, which is also an asymptotic value. The only other
critical point is $z=-m$. The coexistence of a superattracting fixed point and a free critical
point makes this family very much analogous to the family of cubic polinomials $C_a(z) \, =
\, z^3 \, - \, 3a^2z + 2a^3+a $ which is described in \cite{M}.
Let $f$ be a transcendental entire function. It is known that the dynamical plane can be decomposed
into two invariant sets. The first one is an open set, namely the Fatou set or stable set, denoted by
${\cal F}(f)$, and it is formed by points, $z_0 \in {\mathbb C}$ whose iterates
$\{f^{\circ n}\}$ form a normal family, in the sense of Montel, in some neihbourhood of $z_0$. The
second one, namely the Julia or chaotic set, denoted by ${\cal J} (f)$, is defined as the
complement of the Fatou set, that is ${\cal J} (f) = {\mathbb C} - {\cal F}(f)$. Fatou (\cite{Fa}) showed
that the Julia set of an entire transcendental function is a completely invariant, closed, nonempty
and perfect set. As in the polynomial case, it may also be defined as the closure of the set of repelling periodic
points.
We denote by $A(0)\,=\,A_{\lambdambda,m}(0)$ the basin of attraction of $z=0$, i.e., the set of all $z$ such that $F_{\lambdambda,m}^{\circ n}(z) \to 0 $ as $n$ tends to $+ \infty$. We also denote by $A^*(0)\,=\,A_{\lambdambda,m}^*(0)$ the
connected component of $A(0)$ containing $z=0$.
In Sec. 2 we concentrate on the dynamical plane and study the basin of attraction
$A(0)$ (see Fig. \ref{a0}). The skeleton of the main components of $A(0)$ is needed to study later the parameter
planes. We summarize the main results with regard to $A(0)$ in the following theorem.
\betagin{thmA}
$\,$ \par
\betagin{enumerate}
\item There exists $\epsilon_0 = \epsilon_0 (|\lambdambda|,m) \, > \, 0$, defined as the unique positive solution of $x^{m-1
} e^x = 1/|\lambdambda| $, such that $A^*(0)$ contains the
disk $D_{\epsilon_{0}} \, = \, \{ z \in {\mathbb C} \, ; \, |z| \, < \, \epsilon_0 \} $.
\item There exist $x_0\,=\,x_0(|\lambdambda|,m) <0$ and a function $\, C(x) \ge 0$ such that the open set
$$ H_{ |\lambdambda|,m} \, = \, \left\{ z \, =
\, x \, + \, y i \, \left| \betagin{array}{ll}
x \,& \in \, (-\infty,x_0) \\
y \,& \in (-C(x),C(x)) \end{array}\right. \right\} $$
satisfies $F_{\lambdambda,m} \, (H_{|\lambdambda|,m}) \, \subset \, D_{\epsilon_0}$.
\item There exist infinitely many strips, denoted by $S_{\lambdambda,m}^k$, which are preimages of $H_{ |\lambdambda|,m}$. These
horizontal strips extend to $+\infty$, and they have asymptotic width equal to $\pi$.
\end{enumerate}
\end{thmA}
\betagin{figure}[hbt]
\psfrag{d}[][]{\small $D_{\epsilon_0}$}
\psfrag{s0}[][]{\small $S^0_{\lambdambda,m}$}
\psfrag{s1}[][]{\small $S^1_{\lambdambda,m}$}
\psfrag{s2}[][]{\small $S^2_{\lambdambda,m}$}
\psfrag{sm1}[][]{\small $S^{-1}_{\lambdambda,m}$}
\psfrag{sm2}[][]{\small $S^{-2}_{\lambdambda,m}$}
\psfrag{H}[][]{\small $H_{|\lambdambda|,m}$}
\centerline{\includegraphics[width=0.4 \textwidth]{fg1.eps}}
\caption{\small{Sketch of the basin of attraction of $z=0$, satisfying that $D_{\epsilon_0} \subset
A^*(0)\, , \, F_{\lambdambda,m} (H_{|\lambdambda|,m}) \subset D_{\epsilon_0} \, \hbox{and} \,
F_{\lambdambda,m} \, (S_{\lambdambda,m}^k) \subset H_{|\lambdambda|,m}$.}}
\lambdabel{a0}
\end{figure}
Section 3 is dedicated to parameter planes. For some parameter values, the free critical point
$z=-m$ belongs to the basin of attraction of $z=0$, A(0), in which case we say that it has
been ``captured''. The connected components of parameter space for which this phenomenon occurs are
called {\it capture zones}, and they clearly do not exist for members of the family with $m<2$ such as
the exponential. Hence it is natural to study the set of parameters for which the orbit of
$z=-m$ is bounded. That is, we define the sets
$$B_m \, = \, \{ \lambdambda \in {\mathbb C} \, | \, F_{\lambdambda,m}^{\circ n} (-m) \nrightarrow
\infty \}.$$
In each of these sets, we may also distinguish between two different behaviours. Those parameter
values for which $-m \in A(0)$ and those for which this does not occur. Let $\stackrel{\circ}{B_m}$ denote the
interior of $B_m$. We will study the sets
\[
C_m^n = \{ \lambdambda \in \stackrel{\circ}{B_m} | F_{\lambdambda,m}^n (-m) \in A^*(0) \hbox{ and $n$ is the smallest number with this property} \}
\]
Although each $B_m$ contains infinitely many different capture zones, there is one which is
dynamically very different from all others.
We define the {\it main capture zone} $C_m^0$ as the set of parameter values $\lambdambda$ for which the
critical point $\,z=-m\,$ belongs to the immediate basin of $0$. That is,
\[
C_m^0 = \, \{ \lambdambda \in \stackrel{\circ}{B_m} \, | \, -m \in A^*(0) \}.
\]
As we shall see, this is a quite special component of $B_m$ since its boundary separates the parameter
values for which $\partial A^*(0)$ is a Cantor bouquet from those for which it is a Jordan curve (also,
this boundary separates the parameter values for which ${\cal F}(F_{\lambdambda,m})$ has one or infinitely
many components). The detailed study of this boundary will be the object of a future paper. Our goal
in Sec. 3 is to describe the main features of the parameter planes of the functions
$F_{\lambdambda,m}$ and, in particular, the structure of the capture zones (Fig. \ref{zona}). We
summarize some of these facts in the following theorem.
\betagin{thmB}
$\,$ \par
\betagin{enumerate}
\item The critical point $-m$ belongs to $A^*(0)$ if and only if the critical value $F_{\lambdambda,m}(-m)$ belongs
to $A^*(0)$. Hence $C_m^1 \, = \, \o$.
\item The main capture zone $C_m^0$ is bounded.
\item The set $C_m^0$ contains the disk $\{ \lambdambda \in {\mathbb C} \, ;\, |\lambdambda| \, < \,
min(\frac{1}{e},(\frac{e}{m})^m ) \}$.
\item If $\lambdambda \in C_m^0$ then $A(0)=A^*(0)$, i.e., the basin of attraction of $z=0$ has a unique connected
component and hence it is totally invariant. Moreover, the boundary of $A^*(0)$ (which equals the Julia set) is a
Cantor bouquet and hence it is disconnected and non-locally connected.
\item If $\lambdambda \notin C_m^0$ then $A(0)$ has infinitely many components. Moreover, if $|\lambdambda| > (\frac{e}{m-1})^{m-1} $, the boundary of $A^*(0)$ is a quasi-circle.
\end{enumerate}
\end{thmB}
We also summarize some properties of the most obvious capture zones $C_m^2$ and $C_m^3$.
\betagin{thmC}
$\,$ \par
\betagin{enumerate}
\item The set $C_m^2 $ contains an unbounded set to the left or to the right depending on the oddity of $m$. More
precisely, there exists a real constant $D_0(m) > 0$, and a function $\alpha=\alpha(|\lambdambda|,m) \in (\pi/2,\pi)$, such that
\[
\betagin{array}{ll}
\bullet \hbox{ for $m$ even, the set $C_m^2$ contains the open set} & \left\{ \lambdambda \in {\mathbb C} \, \left|
\betagin{array}{ll}
|\lambdambda| & \, > \, D_0 \\
|Arg(\lambdambda)| & \, > \, \alpha \end{array}\right. \right\} \\
\bullet \hbox{ for $m$ odd, the set $C_m^2$ contains the open set} & \left\{ \lambdambda \in {\mathbb C} \, \left|
\betagin{array}{ll}
|\lambdambda| & \, > \, D_0 \\
|Arg(\lambdambda)| & \, < \, \pi- \alpha \end{array}\right. \right\}
\end{array}
\]
\item There exists infinitely many strips in $C_m^3$. If $m$ is even (resp. odd) then these horizontal strips
extend to $+\infty$ (resp. $-\infty$) and they have an asymptotic width equal to $(\frac{e}{m})^m \pi$.
\end{enumerate}
\end{thmC}
\betagin{figure}[hbt]
\psfrag{ti1}[][]{\small $m \,$ even}
\psfrag{ti2}[][]{\small $m \,$ odd}
\psfrag{H}[][]{\small $C_m^2$}
\psfrag{D}[][]{\small $C_m^0$}
\psfrag{S}[][]{\small $C_m^3$}
\centerline{\includegraphics[width=0.8 \textwidth]{fg2.eps}}
\caption{\small{Sketch of capture zones contained in $\stackrel{\circ}{B_m}$. On the left hand side when $m$ is even and on the right hand side when $m$ is odd.}}
\lambdabel{zona}
\end{figure}
\section{The Dynamical Planes}
Our goal in this Sec. is to describe the dynamical plane of the families of maps
$F_{\lambdambda,m}$ given by the equation $F_{\lambdambda,m} (z ) \, = \, \lambdambda
z^m \exp(z), \quad \hbox{where } m \ge 2$.
The function $F_{\lambdambda,m}$ is a critically finite entire function, that is, it has a finite number of
asymptotic values ($z=0$), and critical values ($z=0$ and $z=(-1)^m \lambdambda (\frac{m}{e})^m$). For this kind of
functions there exists a characterization of the Julia set (\cite{DT}), namely as the closure of the
set of points whose orbits tend to $\infty$.
Using the characterization above we can plot an
approximation of ${\cal J}(F_{\lambdambda,m})$. Generally, orbits tend to $\infty$ in specific directions. In
our case, if $\lim_{n \to \infty} |F^{\circ n}_{\lambdambda,m} (z) | \, = \, +\infty \,$, then we have
$ \, \lim_{n \to \infty} Re(F^{\circ n}_{\lambdambda,m} (z) ) \, = \, +\infty$. Thus, an approximation
of the Julia set is given by the set of points whose orbit containts a point with real part greater
than, say, $50$.
In Figs. \ref{jul2}-\ref{julm}, we display the Julia set of $F_{\lambdambda,m}$ for differents values of
$\lambdambda$ and $m$. The basin of attraction of $z=0$ is shown in red, while the components
of the Fatou set different from A(0) are shown in blue. Points in the Julia set are shown in black.
\betagin{figure}[hbt]
\centering
\subfigure[\scriptsize{$\lambdambda = -2.1 $} ]{
\includegraphics[width=0.3 \textwidth]{fg3.eps}}
\hspace{0.1in}
\subfigure[\scriptsize{$\lambdambda = -8$} ]{
\includegraphics[width=0.3 \textwidth]{fg4.eps}}
\hspace{0.1in}
\subfigure[\scriptsize{$\lambdambda = 6.9 $} ]{
\includegraphics[width=0.3 \textwidth]{fg5.eps}}
\caption{\small{The Julia set for $F_{\lambdambda,2}$. Range $(-10,10) \times (-10,10)$.}}
\lambdabel{jul2}
\end{figure}
In Fig. \ref{jul2} we show the dynamical plane of function $F_{\lambdambda,2} \, =
\, \lambdambda z^2 \exp(z) \,$, for three different values of $\lambdambda$. Apparently, the basin of $0$
contains an infinite number of horizontal strips, that extend to $+\infty$ as their real parts
tend to $+\infty$. Between these strips we find the well known structures, named Cantor Bouquets which
are invariant sets of curves governed by some symbolic dynamics. The existence of this kind of structures
in the Julia set are typical for critically finite entire transcendental functions (\cite{DT}).
As we change the parameter $\lambdambda$ we observe that the relative position of these bands
also changes, but not their width. Also, we can see the existence
of an unbounded region that extends to $-\infty$ contained in $A(0)$.
In the next figure (Fig. \ref{julm}) we show a mosaic of different dynamical planes for some values of $m$,
specifically for $m=2, \, 3, \, 4,\,
\hbox{and } 5$. We choose $\lambdambda$ such that all these dynamical planes exhibit the same dynamical
behaviour, or more precisely, so that the critical point $z \, = \, -m$, is a superattracting
fixed point. A simple computation gives $\lambdambda \, = (-1)^{m-1} m (\frac{e}{m})^m$. In these
dynamical planes we see similar structures as in Fig. \ref{jul2}, even though the values of $m$ are
different.
\betagin{figure}[hbt]
\centering
\subfigure[\scriptsize{ $m=2$} ]{
\includegraphics[width=0.3 \textwidth]{fg6.eps}}
\hspace{0.5in}
\subfigure[\scriptsize{ $m=3$} ]{
\includegraphics[width=0.3 \textwidth]{fg7.eps}}
\hspace{0.5in}
\subfigure[\scriptsize{ $m=4$} ]{
\includegraphics[width=0.3 \textwidth]{fg8.eps}}
\hspace{0.5in}
\subfigure[\scriptsize{ $m=5$} ]{
\includegraphics[width=0.3 \textwidth]{fg9.eps}}
\caption{\small{The dynamical plane of $F_{\lambdambda,m}$ for different values of $m$.
In every case $\lambdambda=(-1)^{m-1} m (\frac{e}{m})^m$. Range $(-10,10) \times (-10,10)$.}}
\lambdabel{julm}
\end{figure}
We start with the following general result regarding $A(0)$.
\betagin{lemma} \lambdabel{pa00}
$A(0)$ has either one or infinitely many connected components. Moreover, connected components different from $A^*(0)$ are
unbounded.
\end{lemma}
\betagin{proof}
Using Sullivan's theorem (\cite{S}) we have that $A^*(0)$ is the unique fixed connected component of $A(0)$. For all other
connected components of $A(0)$ there exists a number $ i>0$ such that $F_{\lambdambda,m}^i \,(U)\, = A^*(0)$, and $i$ is the smallest
number with this property. Suppose that there exist a finite number of connected components, and let $U_0 = A^*(0) \,
, \, U_1 \, , \, U_2 \, , \, ... \, , \, U_N$ the connected components of $A(0)$. We may choose the index $i$
in the natural way so that $F_{\lambdambda,m}^i \,(U_i)\, = A^*(0)$. Let $z \in U_N$ such that is not exceptional; then,
points in $F_{\lambdambda,m}^{-1}\, (z) \,$ belong to $A(0)$, but not to $ U_0 \cup U_1 \cup ...
\cup U_N$, wich is a contradiction.
Now suppose that $U$ is a connected component of $A(0)$ different from $A^*(0)$, and let $i>0$ be the smallest number such
that $F_{\lambdambda,m}^i \,(U)\, = A^*(0)$. Let $z \in U$, and denote by $\gamma$ a simple path in $A^*(0)$ that
joins $F_{\lambdambda,m}^i(z)$ and 0. The preimage of $\gamma$ in $U$ must include a path $\gamma_1$ that joins $z$ and
$\infty$, since $0$ is an asymptotical value with no other finite preimage than itself. Thus we conclude that $U$ is unbounded.
\end{proof}
\subsection{Proof of Theorem A}
In this Sec. we describe the basin of attraction of the superattracting fixed point $z=0$.
Since $z\, =\,0$ is a superattracting fixed point, there exists $\epsilon_0 \, > \, 0$ such that the
open disk $D_{\epsilon_{0}} \, = \, \{ z \in {\mathbb C} \, ; \,|z| \, < \, \epsilon_0 \} $ is contained
in the immediate basin of attraction of $z=0$. First, we give an estimate of the size of the immediate basin of
attraction, $A^*(0)$ (Proposition \ref{p21}), which will prove the first part of theorem A. Secondly, we find
the first preimage of $D_{\epsilon_0}$
(Proposition \ref{hoho} and Proposition \ref{th}), proving the second part of theorem A. Finally, we find
the second preimage of $D_{\epsilon_0}$ (Proposition \ref{thbandas}), and prove the third part of
theorem A.
Before proving Proposition \ref{p21} we first look at some properties of the real funtion $h(x) \, = \, x^m e^x \, $
where $\, m \ge 1$. In Fig. \ref{hx} we show the graph of this function. It has a relative extremum
at $x=-m$, it is a monotone function on $(-\infty,-m)$ and it satisfies that
$|h(x)| \le |h(-m)| \, $ for all $\, x \le 0$. Also, it is an increasing function in $(0,+\infty)$.
\betagin{figure}[hbt]
\psfrag{m}[][]{\scriptsize $-m$}
\psfrag{fm}[][]{\scriptsize $h(-m)$}
\centerline{\includegraphics[width=0.8 \textwidth]{fg10.eps}}
\caption{\small{Graph of $h(x)=x^m \exp(x)$. The left hand side corresponds to $m$ even and
the right hand side to $m$
odd.}}
\lambdabel{hx}
\end{figure}
Using these properties it is easy to prove the following auxiliary result.
\betagin{lemma} \lambdabel{lem2}
Given $r \in (0,|h(-m)|]\,=\,(0,(\frac{m}{e})^m]$, the equation $ |h(x)| = r $ has a unique solution in $(-\infty,-m]$.
Moreover, given $s > 0$, the equation $h(x)=s$ has a unique solution in $(0,+\infty)$.
\end{lemma}
We now turn to estimate the function $\epsilon_0(|\lambdambda|,m)$, and find its dependency on $\lambdambda \in
{\mathbb C}-\{0\} \, \hbox{ and } m \ge 2$.
\betagin{proposition}\lambdabel{p21}
If we define $\epsilon_0$ as the unique positive solution of the real
equation \par $x^{m-1}e^x = \frac{1}{|\lambdambda|}$, then $D_{\epsilon_0} \, = \, \{ z \in {\mathbb C} \, ;
\, |z| \, < \, \epsilon_0 \} \subset A^*(0)$. Moreover, if $\lambdambda \in {\mathbb R}^{+}$ we have that $\epsilon_0$ lies
in $\partial A^*(0)$.
\end{proposition}
\betagin{proof}
In order to prove that $D_{\epsilon_0}$ is contained in $A^*(0)$, we use Schwartz's lemma. That is, it suffices to prove that
if $ |z| \, \le \, \epsilon_0 \,$ then $\, |F_{\lambdambda,m} (z)| \, \le \, \epsilon_0$.
Suppose $|z| \le \epsilon_0$, we have
\[
|F_{\lambdambda,m} \, (z) | \, = \, | \lambdambda \, z^m \, e^z | \, = \,
| \lambdambda | \, |z|^m e^{Re(z)} \, \le \, |z|\, |\lambdambda| \, |z|^{m-1} \, e^{|z|} .
\]
Since $|z|<\epsilon_0$ it follows that $|z|^{m-1} e^{|z|} < \epsilon_0^{m-1} e^{\epsilon_0}$, and using that
$\epsilon_0^{m-1} e^{\epsilon_0} = 1/|\lambdambda|$, we conclude
\[
|F_{\lambdambda,m} \, (z) | \, \le \,
|z| \, |\lambdambda| |z|^{m-1} e^{|z|} \, \le \, |z| \le \epsilon_0 .
\]
If $\lambdambda \in {\mathbb R}^{+}$, we have that $\lambdambda \epsilon_0^m \exp(\epsilon_0) \, = \epsilon_0$, i.e., $\epsilon_0$ is a
fixed point. The multiplier of this fixed point is $\epsilon_0 + m >1$, and hence $\epsilon_0$ lies in the Julia set. By
definition we have that $D_{\epsilon_0} \subset A^*(0)$, then $\epsilon_0$ lies in the boundary of $A^*(0)$.
\end{proof}
In the following auxiliary result we find a lower bound for $\epsilon_0$, which will be used in
the next Sec..
\betagin{lemma}\lambdabel{lem_e0}
The value of $\epsilon_0 $ is always larger or equal than $ \min\{1,(\frac{1}{|\lambdambda|e})^{\frac{1}{m-1}}\}$
\end{lemma}
\betagin{proof}
Suppose $|\lambdambda| \, \ge \, 1/e$. This condition is equivalent to
$\frac{1}{|\lambdambda|e} \,\le \, 1$, hence we must prove that $\epsilon_0 \, \ge \,
(\frac{1}{|\lambdambda| \, e })^{1/(m-1)}$. Using that $x^{m-1}e^x$ is an increasing function on
$(0,+\infty)$, this condition is equivalent to
\[
\epsilon_0^{m-1} e^{\epsilon_0} \, \ge \, \frac{1}{|\lambdambda|e}\quad e^{(\frac{1}{|\lambdambda|e})^{\frac{1}{m-1}}}.
\]
By definition we have that $\epsilon_0^{m-1} e^{\epsilon_0} \, = \, \frac{1}{|\lambdambda|}$, then
\[
\frac{1}{|\lambdambda|} \, \ge \,\frac{1}{|\lambdambda|e} \, e^{(\frac{1}{|\lambdambda|e})^{\frac{1}{m-1}}}
\]
or equivalently
\[
e \, \ge \,e^{(\frac{1}{|\lambdambda|e})^{\frac{1}{m-1}}}
\]
\noindent and this follows if $|\lambdambda| \, \ge \,1/e $.
If $|\lambdambda| \, \le \, 1/e$, we must prove that $\epsilon_0 \, \ge \,1$.
Using the same argument, i.e., that $x^{m-1} e^x$ is an increasing function, it follows that this condition
is equivalent to
\[
\epsilon_0^{m-1} e^{\epsilon_0} \, \ge e
\]
and this follows if $|\lambdambda| \, \le \,1/e $.
\end{proof}
Next we want to find an open set $H_{|\lambdambda|,m} \, \subset \, {\mathbb C} $
such that $ F_{\lambdambda,m} (H_{|\lambdambda|,m} ) \, \subset \, D_{\epsilon_0}\,$ (Fig. \ref{hll}). To that end, we first
obtain a value in ${\mathbb R}^{-}$, namely $x_0=x_0(|\lambdambda|,m)$, such that for all $
x \in {\mathbb R}^{-} \, \hbox{ with } \, x \le x_0 \,$ we have that $\, F_{\lambdambda,m}
(x) \in D_{\epsilon_0}$ (Proposition \ref{hoho}). After finding $x_0$, we will look for an upper bound $C (x) \, \ge 0$,
such that if $z= x+yi, \, \hbox{with }\, x<x_0 \hbox{ and } |y|\le C(x) \, $, then $\, F_{\lambdambda,m} (z) \, \in \,
D_{\epsilon_0}$ (Proposition \ref{th}).
\betagin{figure}[hbt]
\psfrag{HH}[][]{\small $H_{|\lambdambda|,m}$}
\psfrag{D}[][]{\small $D_{\epsilon_0}$}
\psfrag{x0}[][]{\small $x_0$}
\psfrag{yc}[][]{\small $y = C(x)$}
\psfrag{ymc}[][]{\small $y=-C(x)$}
\psfrag{x}[][]{\small $x$}
\centerline{\includegraphics[width=0.4 \textwidth]{fg11.eps}}
\caption{\small{Sketch of $H_{|\lambdambda|,m} \,.\, \hbox{ satisfying } F_{\lambdambda,m} \, (H_{|\lambdambda|,m}) \,
\subset \, D_{\epsilon_0} \subset A^*(0)$.}}
\lambdabel{hll}
\end{figure}
\betagin{proposition}\lambdabel{hoho}
For all $\lambdambda \in {\mathbb C} \hbox{ and } m \ge 2 \,$,
there exists $x_0 \in (-\infty,-m]$, such that for all $ x \le x_0$, we have $\, F_{\lambdambda,m}(x) \in D_{\epsilon_0}$.
\end{proposition}
\betagin{proof}
We suppose that $z=x+0i$ and we impose $|F_{\lambdambda,m}(z)| = \epsilon_0$, that is
\[
|F_{\lambdambda,m}(z)| \, = \, |\lambdambda| |x|^m e^x \, = \, \epsilon_0 \,
\]
\noindent or equivalently
\[
|h(x)|\,=\,|x|^m e^x \, = \, \frac{\epsilon_0}{|\lambdambda|},
\]
\noindent where $h(x)$ is the auxiliary function defined above.
If $|h(-m)| \le \frac{\epsilon_0}{|\lambdambda|}$, then we take $x_0 = -m$, and for all $ x \in (-\infty,x_0) \,$,
we have $\, |h(x)| \le |h(-m)| \le \frac{\epsilon_0}{|\lambdambda|}$.
On the other hand, if $|h(-m)| > \frac{\epsilon_0}{|\lambdambda|}>0$,
we define $x_0$ as the unique solution of equation $|h(x)|=\frac{\epsilon_0}{|\lambdambda|}$ in the interval $(-\infty,-m)$.
Since $|h(x)|$ is an increasing function in $(-\infty,-m)$, it follows that for all
$ x \in (-\infty,x_0) \,$, then $\, |h(x)| \le |h(x_0)| = \frac{\epsilon_0}
{|\lambdambda|}$.
\end{proof}
\betagin{proposition} \lambdabel{th}
Let $x_0=x_0(|\lambda|,m)\,$ be as in Proposition \ref{hoho}. There exists $\, C(x) \ge 0$ such that the
open set (Fig. \ref{hll})
$$ H_{ |\lambdambda|,m} \, = \, \left\{ z \, =
\, x \, + \, y i \, \left| \betagin{array}{ll}
x \,& \in \, (-\infty,x_0) \\
y \,& \in (-C(x),C(x)) \end{array}\right. \right\} $$
satisfies $F_{\lambdambda,m} \, (H_{|\lambdambda|,m}) \, \subset \, D_{\epsilon_0}$.
\end{proposition}
\betagin{proof}
We want to calculate an upper bound $C(x) \ge 0$, such that if $z \, = \, x +
yi \, $, with $x \in (-\infty,x_0) \, \hbox{ and } \, |y| \le C(x) \, $, then $\,
F_{\lambdambda,m}(z) \in D_{\epsilon_0}$. If we require that $F_{\lambdambda,m}(z) \, \in \, D_{\epsilon_0}$,
we obtain the definition of $C(x)$. Let $z \, = \, x \pm y i $, with $x \in (-\infty,x_0)$. Then
\[
|F_{\lambdambda,m} (z)| \, = \, |\lambdambda| |z|^m \exp(Re(z)) \, = \,|\lambdambda| \left[ \sqrt{x^2 + y^2} \right]^m \exp(x) \, = \, |\lambdambda| (x^2 \, + \, y^2)^{m/2} \exp(x).
\]
\noindent Using the expression above, and requiring $|F_{\lambdambda,m}(z)| \, \le \, \epsilon_0 $, we obtain
\[
|y| \, \le \,
+ \sqrt{ \,\left[ \frac{\epsilon_0}{|\lambdambda|} \right]^{2/m} \exp(-x\frac{2}{m})\, -
\, x^2 }.
\]
\noindent Thus, the right hand side of this inequality gives an analytic expression for the function $C(x)$.
\end{proof}
The next lemma gives a simpler condition to assure that a point
$z \in {\mathbb C}$ lies in $H_{|\lambdambda|,m}$.
See Fig. \ref{esq}.
\betagin{lemma} \lambdabel{pro}
The point $z= x + y i \, $ lies in $ H_{|\lambdambda|,m}$ if there exists $ \, k \ge 1 \, \hbox{ and } \, A \ge 0$, such
that $|y| \, \le \, A\,|x|^k$ and $|x| \,$ is large enough.
\end{lemma}
\betagin{proof}
We will prove that
$A|x|^k \, \le \, C(x) \hbox{ as } x \to -\infty$. Using the definition of $C(x)$, this is equivalent to showing
\[
A|x|^k \, < \, \sqrt{ \,\left[ \frac{\epsilon_0}{|\lambdambda|} \right]^{m/2}
\exp(-x\frac{m}{2})\, - \, x^2 },
\]
\noindent or
\[
\left[A^2|x|^{2k} \, + \, x^2 \right]\exp(x\frac{m}{2}) \, < \,
\left[ \frac{\epsilon_0}{|\lambdambda|}\right]^{m/2}.
\]
\noindent The left hand side of this inequality is a function that tends to zero as $x \, $ tends to $\, -\infty$, whereas
the right hand side is positive.
\end{proof}
\betagin{figure}[hbt]
\psfrag{a}[][]{\scriptsize $y=A|x|$}
\psfrag{b}[][]{\scriptsize $y=A|x|^2$}
\psfrag{c}[][]{\scriptsize $H_{|\lambdambda|,m}$}
\centerline{\includegraphics[width=0.4 \textwidth]{fg12.eps}}
\caption{\small{Relation between $ H_{|\lambdambda|,m}\, \hbox{ and } \, y=A|x|^k \, \hbox{ for } \, k=1,2$}}
\lambdabel{esq}
\end{figure}
We proceed now to the third iterate, by proving the existence of some strips in dynamical plane, such that
the image of these open sets under $F_{\lambda,m}$ is contained in $H_{|\lambdambda|,m}$ (see Fig. \ref{a0}).
Before calculating the preimage of the set $H_{|\lambdambda|,m}$, we first find the preimage of
the negative real axis under the function $F_{\lambdambda,m}$. Hereafter, we denote by $Arg(.) \in (-\pi,\pi]$ the
argument. Using the definition of $F_{\lambdambda,m}$ it is easy to see that
\[
Arg(F_{\lambdambda,m}(z)) = Arg(\lambdambda) + m Arg(z) + Im(z) \qquad (\hbox{ mod } 2 \pi).
\]
Finding the preimages of ${\mathbb R}^-$ is equivalent to solving
\[
Arg(F_{\lambdambda,m}(z)) = \pi.
\]
We denote $r=|z|$ and $\alpha = Arg(z)$. Then the equation above is equivalent to
\[
Arg(\lambdambda) + m \alpha + r sin(\alpha) = (2k+1)\pi \qquad k \in {\mathbb Z}.
\]
Hence, we obtain
\[
r = \rho(\alpha) = \frac{(2k+1)\pi - m \alpha - Arg(\lambdambda)}{sin(\alpha)} \qquad \alpha \in (-\pi,\pi).
\]
We denote each of these curves by $\sigma_k\, =\,\sigma_k(\lambdambda,m)$, where the possible values of the argument depend on $k$. More precisely,
if $m=2j$ for $j \in {\mathbb Z}$
\[
\sigma_k = \rho(\alpha) e^{i\alpha} \hbox{ where } \left\{ \betagin{array}{llll} 0< \alpha < \pi \qquad & \hbox{ if }\quad k \ge j \\
0< \alpha <\frac{(2k+1)\pi-Arg(\lambdambda)}{m} \qquad & \hbox{ if }\quad 0 \le k \le j-1 \\ \frac{(2k+1)\pi-Arg(\lambdambda)}{m}< \alpha <0 \qquad & \hbox{ if }\quad -j \le k \le 0 \\ -\pi< \alpha <0 \qquad & \hbox{ if }\quad k \le -(j+1) \\ \end{array}\right. ;
\]
if $m=2j+1$ for $j \in {\mathbb Z}$
\[
\sigma_k = \rho(\alpha) e^{i\alpha} \hbox{ where } \left\{ \betagin{array}{llll} 0< \alpha < \pi \qquad & \hbox{ if }\quad k \ge j+1 \\
0< \alpha <\frac{(2k+1)\pi-Arg(\lambdambda)}{m} \qquad & \hbox{ if }\quad 0 \le k \le j \\
\frac{(2k+1)\pi-Arg(\lambdambda)}{m}< \alpha <0 \qquad & \hbox{ if }\quad -j \le k \le 0 \\
-\pi< \alpha <0 \qquad & \hbox{ if }\quad k \le -(j+1) \\ \end{array}\right.
\]
In Fig. \ref{par2} we show some of these curves for $m=5$. As their real parts tend to $+\infty$, the
$\sigma_k$'s are asymptotic to the lines
$Im(z) = (2k+1)\pi-Arg(\lambdambda)$. There are $m$ of these curves, that start at the origin
and tend to $+\infty$. The others are asymptotic to the lines
$Im(z) = (2k+1)\pi-Arg(\lambdambda)\,$ when $k<0$, or $Im(z) = 2k\pi-Arg(\lambdambda)\,$ when $k>0$.
\betagin{figure}[hbt]
\centering
\subfigure[\scriptsize{Graph of $\sigma_k$ for $\lambda = 0.45 + 0.35i$ and $m=5$.} ]{
\psfrag{a}[][]{\scriptsize $\sigma_0$}
\psfrag{b}[][]{\scriptsize $\sigma_1$}
\psfrag{c}[][]{\scriptsize $\sigma_2$}
\psfrag{d}[][]{\scriptsize $\sigma_{-1}$}
\psfrag{e}[][]{\scriptsize $\sigma_{-2}$}
\psfrag{m}[][]{\scriptsize $\sigma_{-3}$}
\psfrag{f}[][]{\scriptsize $\sigma_{-4}$}
\psfrag{g}[][]{\scriptsize $\sigma_{-5}$}
\psfrag{h}[][]{\scriptsize $\sigma_{-6}$}
\psfrag{i}[][]{\scriptsize $\sigma_3$}
\psfrag{j}[][]{\scriptsize $\sigma_4$}
\psfrag{k}[][]{\scriptsize $\sigma_5$}
\psfrag{l}[][]{\scriptsize $\sigma_6$}
\fbox{\includegraphics[width=0.4 \textwidth]{fg13.eps}}}
\hspace{0.5in}
\subfigure[\scriptsize{The julia set of $F_{0.45+0.35i,5}$} ]{
\includegraphics[width=0.4 \textwidth]{fg14.eps}}
\caption{\small{Strips in the dynamical plane.}}
\lambdabel{par2}
\end{figure}
Now we may find preimages of the open set $H_{|\lambdambda|,m}$. First we find preimages of the interval
$(-\infty,x_0)$. There exists one preimage of this interval on each curve $\sigma_k$. Moreover, the preimage of a real
number tending to $-\infty$, is a complex number on $\sigma_k$ whose real part tends to $+\infty$. Hence, the preimages of the set $H_{|\lambdambda|,m}$ contain some strips, namely $S_{\lambdambda,m}^k$, around $\sigma_k$. (See Fig. \ref{par2}).
Let $z \in \sigma_k$. If we evaluate $F_{\lambda,m}(z)$, we have
\[
F_{\lambda,m}(z)= |\lambda| |z|^m e^{Re(z)} \, e^{(Arg(\lambda)+m Arg(z) + Im(z))i} = - |\lambda| |z|^m e^{Re(z)}
\]
\noindent since each $\sigma_k$ is a preimage of the negative real axis. The expression above shows that
if we keep $Re(z)$ constant, and we increase the index $k$ of the curve $\sigma_k$, we obtain values tending to
$-\infty$, since $|z|$ increases. Hence, if we denote by $q_0(k)$ the preimage of $x_0$ on $\sigma_k$,
its real part decreases as $|k|$ increases (at least after a certain point). This fact explains
the apparent arangement of the strips in dynamical plane (Fig. \ref{par2}).
We now prove that these strips have an asymptotic
width equal to $\pi$. We fix a value $k \in {\mathbb Z}$, and we recall that $\sigma_k$ tends asymptotically to the line
$Im(z) = (2k+1) \pi - Arg( \lambdambda)$, as its real part tends to $+\infty$.
\betagin{proposition} \lambdabel{thbandas}
Given any $\, y \in ( (2k+1) \pi - Arg( \lambdambda) - \frac{\pi}{2} \, , \, (2k+1) \pi - Arg( \lambdambda) + \frac{\pi}{2}) , \, $ there exists
a real number $\, x_*$ such that for all $\,x \ge x_* $, $F_{\lambdambda,m} (x+yi) \, \in \, H_{|\lambdambda|,m}$.
\end{proposition}
\betagin{proof}
Let $z = x + i y $, where $ y \in ( (2k+1) \pi - Arg( \lambdambda) - \frac{\pi}{2} \, , \, (2k+1) \pi - Arg( \lambdambda) +
\frac{\pi}{2}) $. If we write the parameter $\lambda$ in polar
coordinates, $\lambdambda \, = \, se^{i\betata} $, then
\[
\betagin{array}{ll}
F_{\lambdambda,m} (z) \, = \, & \lambdambda \, z^m \, \exp(z) \, = \, s e^{i\betata} \, (x \, + y i)^m
e^x \, e^{i y} \, = \, s (x \, + y i)^m \, e^x \, e^{i(y \, + \, \betata)} \, = \, \\
& \{P(x) \, + Q(x)i \} \, e^x \, \{\cos(y \, + \, \betata) \, + \, \sin(y \, + \,
\betata)i \}
\end{array}
\]
\noindent where
\[
\left\{ \betagin{array}{ll} P(x) \, = & Re(s(x \, + \,y i)^m) \, = \,
sx^m \, + \, {\cal O} (x^{m-2}) \\
Q(x) \, = & Im(s(x \, + \,y i)^m) \, = \,
msy x^{m-1} \, + \, {\cal O} (x^{m-3}) . \end{array}\right.
\]
\noindent Using the expressions above in $F_{\lambdambda,m} (z)\,$, we obtain
\betagin{equation}
\betagin{array}{llllll}
F_{\lambdambda,m} (z) & = \{P(x)\cos(y + \betata) - Q(x)\sin(y + \betata) \} e^x + \\
& \quad \{ P(x)\sin(y + \betata) + Q(x)\cos(y + \betata) \}e^x i \\
& = \{ P(x)\cos(y \, + \, \betata) \, + {\cal O}(x^{m-1}) \} e^x \, + \, \\ \lambdabel{eqnum}
& \quad \{ P(x)\sin(y \, + \, \betata) \, + {\cal O}(x^{m-1}) \} \, e^x \, i \, \\
& = \{sx^m\cos(y\, + \, \betata) \, + \, {\cal O}(x^{m-1}) \} e^x \, + \, \\
& \quad \{sx^m\sin(y \, + \, \betata) \, + \, {\cal O}(x^{m-1}) \} e^x \, i.
\end{array}
\end{equation}
\noindent We recall that $ \, y \in (\frac{\pi}{2} + 2k\pi - \betata \, ,
\, \frac{3\pi}{2} + 2k\pi - \betata ) \, $, or equivalently $ \, y \, + \, \betata \in (\frac{\pi}{2} \, + \, 2k\pi , \frac{3\pi}{2} \, + \, 2k\pi) \,$ which implies that $ -1 \le cos(y \, + \, \betata ) < 0$
Since $Re(F_{\lambdambda,m} (z)) \, = \, \{sx^m cos(y \, + \, \betata ) \, + \, {\cal O}
(x^{m-1}) \}e^x$, it follows that
\[
\lim_{x \, \to \, +\infty} Re(F_{\lambdambda,m} (z)) \, = \,
\lim_{x \, \to \, +\infty} \{ cos(y \, + \, \betata ) x^m s \, + \, {\cal O} (x^{m-1}) \}e^x \, = \,
-\infty.
\]
Hence, there exists $x_*$ large enough, such that $Re(F_{\lambdambda,m} (z)) \, < \, x_{0} $ for $x \ge x_*$.
Finally, we use lemma \ref{pro} to conclude the proof. From equation (\ref{eqnum}) we have
\[
\lim_{x \to +\infty} \frac{Re(F_{\lambdambda,m}(z))}{Im(F_{\lambdambda,m}(z))} \, = \,
\lim_{x \to +\infty} \frac{sx^m \cos(y \,+\betata) \, + \, {\cal O} (x^{m-1})}
{sx^m \sin(y \,+\betata) \, + \, {\cal O} (x^{m-1})} \, = \, \frac{\cos(y \,+\betata)}
{\sin(y \,+\betata)}.
\]
Hence, if $\sin(y \, + \, \betata) \neq 0 \,$, it follows that $\, Im(F_{\lambdambda,m}(z)) \, \approx \,
\frac {\sin(y \,+\betata)}{\cos(y + \betata)} Re(F_{\lambdambda,m}(z)) $, and we can apply
lemma \ref{pro} ( with $A \, = \, |\frac{\sin(\alpha \,+\betata)}{\cos(\alpha + \betata)}|
\, \hbox{ and } \, k=1$). Using this lemma, we conclude that $ F_{\lambdambda,m}(z)\, $ lies
in $\, H_{|\lambdambda|,m}$, if $x$ is large enough.
On the other hand, if $\sin(y \, + \, \betata) = 0 \,,$ then $\, cos(y \, + \, \betata) = -1$, and we obtain
\[
\betagin{array}{ll}
F_{\lambdambda,m} (z) \, & = \, \{-P(x) -Q(x)i\}e^x \, \\
& = \{-sx^m + {\cal O} (x^{m-1})\}e^x \, + \,
\{m s y x^{m-1}+ {\cal O} (x^{m-2})\}ie^x.
\end{array}
\]
\noindent Hence
\[
\lim_{x \to +\infty} \frac{Re(F_{\lambdambda,m}(z))}{Im(F_{\lambdambda,m}(z))} \, = \,
\lim_{x \to +\infty} \frac{-sx^m + {\cal O} (x^{m-1})}{-msyx^{m-1}+ {\cal O} (x^{m-2})} \, = \,
+\infty .
\]
If $x$ is large enough, there exists $K>0$ such that
\[
\left| \frac{Re(F_{\lambdambda,m}(z))}{Im(F_{\lambdambda,m}(z))} \right|
\, > K \,
\]
\noindent that is, $ | Im(F_{\lambdambda,m}(z)) | \, \le \, \frac{1}{K} |Re(F_{\lambdambda,m}(z))|$. Hence, using lemma \ref{pro}
($A = \frac{1}{K}$ , $k=1$), we also obtain that $ F_{\lambdambda,m}(z)\,$ lies in $\,H_{|\lambdambda|,m}.$
\end{proof}
\section{The Parameter Planes}
The orbit of the free critical point $z=-m$, determines in large measure the dynamics of $F_{\lambda,m}$.
Indeed, the functions $F_{\lambda,m}(z) = \lambda z^m \exp(z) $ are entire maps with a finite
number of critical and asymptotic values. These kind of functions do not have wandering domains
nor Baker domains. By the Sullivan classification, we know that
if the orbit of $z=-m$ tends to $\infty$ then the Fatou set must coincide with the basin of $0$, i.e.,
${\cal F} (F_{\lambdambda,m}) \, = \, A(0)$, since no other Fatou components can exist besides those that belong to
$A(0)$. The set $B_m$ is defined as before as
$$B_m \, = \, \{ \lambdambda \in {\mathbb C} \, | \, F_{\lambdambda,m}^{\circ n} (-m) \nrightarrow
\infty \}.$$
\betagin{figure}[hbt]
\centering
\subfigure[\scriptsize{Range $(-25,25) \times (-25,25)$} ]{
\includegraphics[width=0.45 \textwidth]{fg15.eps}}
\hspace{0.5in}
\subfigure[\scriptsize{Range $(-15,5) \times (-8,8)$} ]{
\includegraphics[width=0.45 \textwidth]{fg16.eps}}
\caption{\small{Parameter plane for $F_{\lambdambda,2}$. Color codes are explained in the text.}}
\lambdabel{pp2}
\end{figure}
In each of these sets, we may also distinguish between two different behaviours: those parameters values for which
$-m \in A(0)$ and those for which this does not occur. Let $\stackrel{\circ}{B_m}$ denote the interior of $B_m$.
\betagin{definition} Let $ U $ be a connected component of $\stackrel{\circ}{B_m}$. We say that $U$ is a {\it capture zone} if
for all $\, \lambdambda $ in $\, U$ it is true that $\,\lim_{n \to +\infty} F_{\lambdambda,m}^{\circ n}(-m) = 0$, or in other words,
$ -m \in A(0)$. We then say that the orbit of the critical
point is captured by the basin of attraction of the superattracting fixed point $z=0$.
\end{definition}
In Figs. \ref{pp2}- \ref{ppm}, we show a numerical approximation of the set $B_m$ for different values of $m$.
The capture zones are shown in red, while other components of $B_m$ are
shown in blue. The parameter values for which the orbit of the free critical point tends to $\infty$ are shown
in black. In these sets we can see a countable quantity of horizontal strips. If $m$ is even these strips
extend to $+\infty$ as
the real part of $\lambda$ tends to $+\infty$, whereas if $m$ is odd these strips
extend to $-\infty$ as the real part of $\lambda$ tends to $-\infty$. Notsurpringly, the distribution of these capture zones
in the parameter plane (Fig. \ref{zona}) appears to be similar to the distribution of $A(0)$ in the dynamical plane (Fig. \ref{a0}).
We start with the following simple facts.
\betagin{proposition}
Let $U$ be a capture zone of $B_m$ and let $ \lambdambda \in U\,$. Then $ \,{\cal F} (F_{\lambdambda,m}) \, =
\, A(0)\,$ and $\,{\cal J} (F_{\lambdambda,m}) \, = \, \partial A(0)$.
\end{proposition}
\betagin{proof}
As in the case of the critical value tending to $\infty$, since the only free critical point of $F_{\lambda,m}$ lies
in the basin of $0$, no other components different from those in $A(0)$ can exist in ${\cal F} (F_{\lambdambda,m})$.
Let ${\cal D}$ be the union of all the components of the Fatou set, then ${\cal J} \, (F_{\lambdambda,m}) \, = \, \partial
{\cal D}$ (\cite{CG}). In our case, if $\lambdambda \in U \, $ and $\, U$ is a capture zone, then ${\cal D} \,
= \, A(0).$
\end{proof}
\betagin{figure}[hbt]
\centering
\subfigure[\scriptsize{$B_3$. Range $(-12,12) \times (-12,12)$} ]{
\includegraphics[width=0.36 \textwidth]{fg17.eps}}
\hspace{0.5in}
\subfigure[\scriptsize{$B_4$. Range $(-4,2) \times (-3,3)$} ]{
\includegraphics[width=0.36 \textwidth]{fg18.eps}}
\hspace{0.5in}
\subfigure[\scriptsize{$B_5$. Range $(-0.8,0.8) \times (-0.8,0.8)$} ]{
\includegraphics[width=0.36 \textwidth]{fg19.eps}}
\hspace{0.5in}
\subfigure[\scriptsize{$B_6$. Range $(-0.15,0.15) \times (-0.15,0.15)$} ]{
\includegraphics[width=0.36 \textwidth]{fg20.eps}}
\caption{\small{Parameter plane for $F_{\lambdambda,m}$, for differents values of $m$.}}
\lambdabel{ppm}
\end{figure}
The main objective of this Sec. is to describe the most obvious capture zones contained in
$B_m$, as well as to describe the dynamical plane for parameter values that belong to such components.
\subsection{Proof of Theorem B}
In this Sec. we describe the main capture zone $C_m^0$. We recall that
\[
C_m^n = \{ \lambdambda \in \stackrel{\circ}{B_m} | F_{\lambdambda,m}^n (-m) \in A^*(0) \hbox{ and $n$ is the smallest number with this property} \}
\]
We prove each statement of theorem B in a different proposition.
\betagin{proposition}\lambdabel{ant}
The critical point $-m$ belongs to $A^*(0)$ if and only if the critical value $F_{\lambdambda,m}(-m)$ belongs
to $A^*(0)$. Hence $C_m^1 \, = \, \o$.
\end{proposition}
\betagin{proof}
Supose that $F_{\lambdambda,m}(-m) \in A^*(0)$. Let $\gamma$ be a simple path in $A^*(0)$ that joins $F_{\lambdambda,m}(-m)$ and $0$. The
set of preimages of $\gamma$ must include a path $\gamma_1$ that joins $-\infty$ with $-m$, and also a path $\gamma_2$ that
joins $-m$ and $0$ (since $-m$ is a critical point and $0$ is a fixed point and asymptotic value). Hence $\gamma_1 \cup
\gamma_2 \subset A^*(0)$ and so does $-m$. Conversely, if $-m \in A^*(0)$ we have that $F_{\lambdambda,m}(-m) \in A^*(0)$.
\end{proof}
\betagin{proposition} \lambdabel{pepe}
The set $C_m^0$ contains the disk $\{ \lambdambda \in {\mathbb C} \, ;\, |\lambdambda| \, < \,
\min(\frac{1}{e},(\frac{e}{m})^m ) \}$.
\end{proposition}
\betagin{proof}
We denote $D_m$ the open disk $ \{ \lambdambda \in {\mathbb C} \, ;\, |\lambdambda| \, < \,
\min(\frac{1}{e},(\frac{e}{m})^m ) \} \,$. Let $\lambdambda \in D_m$, we will prove that $F_{\lambdambda,m}(-m) $
lies in $D_{\epsilon_0}$ which we know belongs to $A^*(0)$. In order to do so, we use that
$\epsilon_0 \ge \min(1,(\frac{1}{|\lambdambda| e})^{1/(m-1)})$ (lemma \ref{lem_e0}).
We choose $\lambdambda \in D_m$. Then $\, |\lambdambda| < \frac{1}{e} \,$, and hence
$\, \epsilon_0 \ge 1$. The condition $\lambda \in D_m$ also implies that
$|\lambdambda| < (\frac{e}{m})^m$. Hence
\[
|F_{\lambdambda,m}(-m)| \, = \, |\lambdambda| |(-m)^m e^{-m}| \,= \,
|\lambdambda| \left(\frac{m}{e}\right)^m \, < \, 1
\le \epsilon_0,
\]
\noindent and $F_{\lambdambda,m}(-m) \, $ lies in $ \,A^*(0)$.
\end{proof}
\betagin{proposition}\lambdabel{prim}
The set $C_m^0$ is bounded. In fact it is contained in the closed
disk $\{ \lambdambda \in {\mathbb C} \, ;\, |\lambdambda|
\le (\frac{e}{m-1})^{m-1} \}$.
\end{proposition}
\betagin{proof}
We will prove that $-m \notin A^*(0)$ for all $\lambdambda \in {\mathbb C} $ such that $|\lambdambda| > (\frac{e}{m-1})^{m-1}$.
Let $D$ the disk centered at $0$ of radius $m-1$. If we calculate the image of its
boundary, $\{|z|=m-1\}$, we obtain
\[
|F_{\lambdambda,m}(z)|= |\lambdambda| |z|^m e^{Re(z)} \ge |\lambdambda| (m-1)^m e^{-(m-1)} > m-1
\]
\noindent where the inequality is obtained using $|\lambdambda| > (\frac{e}{m-1})^{m-1}$. This shows that $D \subset F_{\lambdambda,m}(D)$, and
hence $A^*(0) \subset D$. Since $-m \notin D$, the proposition follows.
\end{proof}
\betagin{proposition}
If $\lambdambda \in C_m^0$ then $A(0)=A^*(0)$, i.e., the basin of attraction of $z=0$ has a unique connected
component and hence it is totally invariant. Moreover, the boundary of $A^*(0)$ (which equals the Julia set) is a Cantor
bouquet and hence it is disconnected and non-locally connected.
\end{proposition}
\betagin{proof}
Let $\lambdambda \in C_m^0$. As in proposition \ref{ant}, let $\gamma$ be a simple path in $A^*(0)$ that
joins $F_{\lambdambda,m}(-m)$ and $0$. The
preimage of $\gamma$ must include a path $\tilde{\gamma}$ contained in $A^*(0)$ that joins $-\infty$ with $0$ passing through
$-m$ ($\tilde{\gamma}$ maps 2-1 to $\gamma$). Since $H_{|\lambdambda|,m}$ intersects $\tilde{\gamma}$ so it follows that $H_{|\lambdambda|,m} \subset A^*(0)$. All preimages of $\tilde{\gamma}$,
are contained in $A^*(0)$ as well, since they all intersect $H_{|\lambdambda|,m}$. In fact, we have
that $A(0)=A^*(0)$ since any preimage of $D_{\epsilon_0}$ must contain points of $H_{|\lambdambda|,m}$. Hence $A(0)$ has a unique
connected component. In fact, from \cite{DG},\cite{BD}, it follows that the Julia set has an uncountable number of connected
components
and it is not locally connected at any point.
Using \cite{DT} one can show that the Julia set contains a
Cantor Bouquet tending to $\infty$ in the direction of the positive real axis. To see this, it is sufficient to construct
a hyperbolic exponential tract on which $F_{\lambdambda,m}$ has asymptotic direction $\theta^*$. Let $B_r$ an open disk
containing $F_{\lambdambda,m}(-m)$, the preimage of this set is an open set similar to $H_{|\lambdambda|,m}$. Let $D$ the complement
of this set. We have that $F_{\lambdambda,m}$ maps $D$ onto the exterior of $B_r$, then $D$ is an exponential tract
for $F_{\lambdambda,m}$. We may choose the negative real axis to define the fundamental domains in $D$. Since the
curves $\sigma_k$ for $k \in {\mathbb Z}$ are mapped by $F_{\lambdambda,m}$ onto this axis, it follows that $D$ has asymptotic
direction $\theta^* = 0$. Furthermore, since $F_{\lambdambda,m}(z) \, = \, \lambdambda z^m \exp(z)$, one may check readily that
$D$ is a hyperbolic exponential tract.
\end{proof}
\betagin{proposition}
If $\lambdambda \notin C_m^0$ then $A(0)$ has infinitely many components. Moreover, if $|\lambdambda| > (\frac{e}{m-1})^{m-1} $, the boundary of $A^*(0)$ is a quasi-circle.
\end{proposition}
\betagin{proof}
Using lemma \ref{pa00} we have that $A(0)$ has either one or infinitely many connected components. If we suppose
that $A(0)$ has only one connected component, then $A(0)$ is a completely invariant component of the Fatou set. We have
that all the critical values of $F_{\lambdambda,m}$ are in $A(0)$ (see \cite{Ba2}), and hence we conclude
that $-m$ belongs to $A(0)$. However, is imposible if $\lambdambda \notin C_m^0$.
Let $\lambdambda \notin C_m^0$ such that $|\lambdambda| > (\frac{e}{m-1})^{m-1} $. The main idea of this proof is the same as that
used by Bergweiler in (\cite{B}). We will show that $F_{\lambda,m}$ is a polynomial-like of degree $m$ in a
neighbourhood of $0$, which includes the whole immediate basin. From the proof of proposition \ref{prim}, the disc
$D$ centered at $0$ of radius $m-1$ satisfies $\overline{D} \subset F_{\lambdambda,m}(D)$,
and hence $A^*(0) \subset D$.
Let $W$ be the component of $F^{-1}_{\lambdambda,m}(D)$ that contains the origin. It is clear that $\overline{W} \subset D$ and
$\overline{A^*(0)} \subset W$. Moreover, $F_{\lambdambda,m}$ is a proper function of degree $m$ from $W$ onto $D$, (see Fig. \ref{quasi}).
In the terminology of polynomial-like mappings, developed by Douady and Hubbard (\cite{DH}),
the triple $(F_{\lambdambda,m};W,D)$ is a polynomial-like mapping of degree $m$. By the Straightening theorem,
there exists a quasiconformal mapping, $\phi$, that conjugates $F_{\lambdambda,m}$ to a polynomial $P$
of degree $m$, on the set $W$. That is $(\phi^{-1} \circ F_{\lambdambda,m} \circ \phi )(z) = P(z) $ for all $z \in W$.
Since $z=0$ is superattracting for $ F_{\lambdambda,m}$ and $\phi$ is a conjugacy, we have that $z=0$ is
superattracting for $P$. Hence, after perhaps a holomorphic change of variables, we may assume that $P(z)=z^m$.
Hence, $\partial A^*(0) = \phi ({\mathbb T})$, and the theorem follows.
\end{proof}
\betagin{figure}[hbt]
\psfrag{a}[][]{\small $A^*(0)$}
\psfrag{b}[][]{\small $W$}
\psfrag{c}[][]{\small $D$}
\psfrag{d}[][]{\tiny $m \, - \,1 $}
\psfrag{de}[][]{\small ${\mathbb D}$}
\psfrag{fi}[][]{\small $\phi$}
\psfrag{fu}[][]{\small $F_{\lambda,m}$}
\psfrag{pol}[][]{\small $z \to z^m$}
\centerline{\includegraphics[width=0.7 \textwidth]{fg21.eps}}
\caption{\small{ $F_{\lambdambda,m}$ is a polynomial-like mapping of degree $m$ near the origin.}}
\lambdabel{quasi}
\end{figure}
\betagin{remark}
The reason to ask for $|\lambda| > (\frac{e}{m-1})^{m-1}$ as a condition is as follows. We want to find
a value $K>0$ such that if $|z|=K$ then $|F_{\lambdambda,m}(z)| > K$. This condition is equivalent to
\[
|F_{\lambdambda,m}(z)| \ge |\lambdambda| |z|^m e^{-|z|} = |\lambdambda| (K)^m e^{-K} > K
\]
\noindent or equivalently
\[
|\lambdambda| > K^{1-m}e^K.
\]
We want to use this argument for the largest possible region of values of $\lambdambda$. Hence,
we choose $K>0$, such that $K^{1-m}e^K$ is minimum. This minimum value is reached exactly at $K=m-1$.
\end{remark}
\subsection{Proof of Theorem C}
In this Sec. we describe the capture zones $C_m^2$ (Proposition \ref{par_real} and
Proposition \ref{thi}) and $C_m^3$ (Proposition \ref{banpp}).
We will construct the open set $C_m^2$ in two steps. In the first (Proposition \ref{par_real}) we obtain an
unbounded interval of real numbers $I$, such that for all $\lambdambda \in I, \, F^2_{\lambdambda,m}(-m) \,$
lies in $\, A^*(0)$. In the second (Proposition \ref{thi}), we will extend this construction to $\lambdambda \, $ in
$\, {\mathbb C}$. We denote $\lambdambda = \lambdambda_1 + \lambdambda_2 i$, where $\lambdambda_1 \hbox{ and } \lambdambda_2$, are
the real and imaginary parts of $\lambdambda$.
\betagin{proposition}\lambdabel{par_real}
There exists an unbounded interval,$\,I$, such that for all real numbers $\lambdambda_1 \in I$, we have that
$F_{\lambdambda_1,m}^2(-m) \in D_{\epsilon_0} \subset A^*(0)$.
\end{proposition}
\betagin{proof}
Hereafter we denote $r_m = (\frac{e}{m})^m$. We take $ \lambdambda_1 \in {\mathbb R} $, and we impose that
$F_{\lambdambda_1,m}(-m) \in H_{|\lambdambda_1|,m}$.
If we calculate $F_{\lambdambda_1,m}(-m)$ we obtain
\[
F_{\lambdambda_1,m}(-m) \, = \, \lambdambda_1 (-m)^m \exp(-m) \, = \, (-1)^m \frac{\lambdambda_1}{r_m}.
\]
\noindent This real value lies in $H_{|\lambdambda_1|,m}$, if and only if
\[
|h(F_{\lambdambda_1,m}(-m))| \, < \, \frac{\epsilon_0}{|\lambdambda_1|}.
\]
\noindent Recall that $h(x) = x^m \exp(x)$, and $\epsilon_0$ only depends on $|\lambdambda_1|$ and $m$. This condition
is equivalent to
\[
\frac{|\lambdambda_1|^{m+1}}{r_m^m} \exp((-1)^m \frac{\lambdambda_1}{r_m}) \, < \, \epsilon_0.
\]
Using lemma \ref{lem_e0} we have that $\epsilon_0 \, \ge \, \min\{1,(\frac{1}{|\lambdambda_1|e})^{1/(m-1)}\}$. If we use this
explicit lower bound we may impose
\[
\frac{|\lambdambda_1|^{m+1}}{r_m^m} \exp((-1)^m \frac{\lambdambda_1}{r_m}) \, < \,\min\{1,(\frac{1}{|\lambdambda_1|e})^{1/(m-1)}\}
\]
We define the auxiliary function
\[
l(\lambdambda_1) \, = \, \left\{
\betagin{array}{ll}
|\lambdambda_1|^{m+1+\frac{1}{m-1}} \, \exp((-1)^m \, \frac{\lambdambda_1}{r_m}) \exp(1/(m-1)) & \, \hbox{if } \, |\lambdambda_1| > 1/e \\
|\lambdambda_1|^{m+1} \exp((-1)^m \, \frac{\lambdambda_1}{r_m}) & \, \hbox{if } \, |\lambdambda_1| \le 1/e
\end{array}\right.
\]
\noindent and the above inequality is transformed into $l(\lambdambda_1)< r_m^m$.
Using some elementary properties of function $l(\lambdambda_1)$, one can see that
\[
\left\{
\betagin{array}{ll}
lim_{\lambdambda_1 \to -\infty} l(\lambdambda_1) \, = \, 0 & \, \hbox{if } $m$ \hbox { is even}\\
lim_{\lambdambda_1 \to +\infty} l(\lambdambda_1) \, = \, 0 & \, \hbox{if } $m$ \hbox { is odd}.
\end{array}\right.
\]
Since $l(\lambdambda_1)$ is continous and positive and it has a finite number of relative maxima and minima,
we can find an unbounded interval of real numbers such that $ l(\lambdambda_1)< r_m^m $.
If $m$ is even, we define $I=(-\infty,-D_0)$, where $-D_0=-D_0(m) \le 0$, is the smallest of the values such that $l(\lambdambda_1)=r_m^m$. If $m$
is odd, we choose $I=(D_0,+\infty)$, with $D_0 =D_0(m)\ge 0$, such that $D_0$ is the largest of the values
for which $l(\lambdambda_1)=r_m^m$.
\end{proof}
\betagin{proposition}\lambdabel{thi}
Let $D_0(m) > 0$ be as in Proposition \ref{par_real}.There exists a function $\alpha=\alpha(|\lambdambda|,m) \in
(\pi/2,\pi)$, such that
\[
\betagin{array}{ll}
\bullet \hbox{ for $m$ even, the set $C_m^2$ contains the open set} & \left\{ \lambdambda \in {\mathbb C} \, \left|
\betagin{array}{ll}
|\lambdambda| & \, > \, D_0 \\
|Arg(\lambdambda)| & \, > \, \alpha \end{array}\right. \right\} \\
\bullet \hbox{ for $m$ odd, the set $C_m^2$ contains the open set} & \left\{ \lambdambda \in {\mathbb C} \, \left|
\betagin{array}{ll}
|\lambdambda| & \, > \, D_0 \\
|Arg(\lambdambda)| & \, < \, \pi - \alpha \end{array}\right. \right\}
\end{array}
\]
\end{proposition}
\betagin{proof}
Given $\lambdambda_1^* \in I$, we denote by $S$ the circle of radius $|\lambdambda_1^*|$ and centered at the origin. We will
find all complex
numbers $\lambdambda \, $ in $\, S$, such that $F_{\lambdambda,m}(-m) \in H_{|\lambdambda|,m}$.
All complex numbers $\lambda \in S$ have the same $H_{|\lambdambda|,m}$ set, since this set only depends on $|\lambdambda|$ and $m$. We denote it by $H_S$.
When $\lambdambda $ belongs to $S$, the image of the critical point, $F_{\lambdambda,m}(-m)=(-1)^m \frac{\lambdambda}{r_m}$,
belongs to another circle, namely $\widetilde{S}$, and its argument verifies
\[
Arg(F_{\lambdambda,m}(-m)) \, = \,
\left\{
\betagin{array}{ll}
Arg(\lambda) & \, \hbox{ if } m \hbox{ is even} \\
Arg(\lambda)+ \pi & \, \hbox{ if } m \hbox{ is odd} \\
\end{array}\right.
\]
This circle is concentric with respect to $S$ and its radius is
equal to $\frac{|\lambdambda_1^*|}{r_m}$ (see Fig. \ref{para}), which is larger
than the radius of $S$ if $m=2$, and smaller if $m \ge 3$.
We take $\lambdambda_1^* \,$ on $\, I$. Using the construction above of the interval I, we obtain that $F_{\lambdambda_1^*,m}(-m) \in H_{|\lambdambda_1^*|,m}
=H_S$. This fact assures a non-empty intersection of $\partial H_S$ with $\widetilde{S}$.
Using the analytic definition of $H_S$ (proof of Proposition \ref{th}), we can calculate $\partial H_S \cap \widetilde{S}$.
We find this intersection by solving:
\[
\left\{
\betagin{array}{ll}
\sqrt{ \lambda_1^2 \, + \, \lambda_2^2} \, = \, \frac{|\lambdambda_1^*|}{r_m} & \, \lambda_1+i \lambda_2 \in \widetilde{S} \\
\lambda_2 = +C(\lambda_1) = \sqrt{ \left[ \frac{\epsilon_0}
{|\lambda_1^*|} \right]^{2/m} \exp(-\lambda_1 \frac{2}{m}) - \lambda_1^2 } & \, \lambda_1 +i \lambda_2 \in \partial H_S
\end{array}\right.
\]
It is not difficult to show that this system has two conjugate solutions namely $\zeta$ and $\bar{\zeta}$. If
we write $\zeta=\widetilde{\lambda_1} + i \widetilde{\lambda_2}$, then
\[
\widetilde{\lambda_1} = \ln{\frac{\epsilon_0 r_m^m}{|\lambdambda_1^*|^{m+1}}} \qquad \widetilde{\lambda_2} = +C(\widetilde{\lambda_1})
\]
Let $\alpha=\alpha(|\lambda_1^*|,m) \in (\pi/2,\pi)\,$ be the argument of $\zeta$ (see Fig. \ref{para}).
If $m$ is even then for all complex numbers with modulus equal to $|\lambda_1^*|$, where $ \lambda_1^* \,$
lies in $\, I$, and argument greater than $\alpha$ in absolute value, it is verified that $F_{\lambdambda,m}(-m)
\in H_{|\lambdambda|,m}$. If $m$ is odd, the same is true for all complex numbers with modulus equal
to $|\lambda_1^*|$, where $\lambda_1^* \, $ lies in $\, I$, and argument, in absolute value, smaller than $\pi-\alpha$.
\end{proof}
\betagin{figure}[hbt]
\lambdabel{}
\psfrag{x}[][]{ $\lambda_1$}
\psfrag{y}[][]{$\lambda_2$}
\psfrag{s}[][]{ $S$}
\psfrag{s1}[][]{$\widetilde{S}$}
\psfrag{h}[][]{ $H_S$}
\psfrag{a}[][]{ $\alpha$}
\psfrag{z}[][]{ $\zeta$}
\psfrag{i}[][]{ $\lambda_1^*$}
\psfrag{z1}[][]{ $\bar{\zeta}$}
\centerline{\includegraphics[width=0.5 \textwidth]{fg22.eps}}
\caption{\small{Sketch of construction of the set $H_m, \, m=2.$ }}
\lambdabel{para}
\end{figure}
Parallel to the construction in dynamical plane we will now show the existence of a countable
number of horizontal bands, all of which are also capture zones. See Fig. \ref{pp2}- \ref{ppm}. Apparently, when $m$ is even, these
strips extend to $+\infty$, while if $m$ is odd they extend to
$-\infty$. It also seems that their width decreases as $m$ increases.
In the Sec. above we constructed similar strips around the curves $\sigma_k$. These curves were preimages of the negative
real axis. In this case, we define
\[
\Gamma_k = \{ \lambda \in {\mathbb C} \hbox{ such that } F_{\lambda,m}(-m) \in \sigma_k \}.
\]
To find an expression for the curves $\Gamma_k$ we first write
\[
F_{\lambda,m}(-m) = \lambda (-m)^m \exp(-m) = (-1)^m \frac{\lambda}{r_m}
\]
where $r_m = (\frac{e}{m})^m$. Recall that $z \in \sigma_k$ when
\[
Arg(\lambdambda) + m Arg(z) + Im(z) = (2k+1)\pi.
\]
Hence we need that
\[
Arg(\lambda) + m Arg(F_{\lambda,m}(-m)) + Im(F_{\lambda,m}(-m)) = (2k+1)\pi.
\]
If $m$ is even then $Arg(F_{\lambda,m}(-m)) = Arg(\lambda)$, while if $m$ is odd then $Arg(F_{\lambda,m}(-m)) = Arg(\lambda) + \pi$;
thus we obtain the condition for $F_{\lambda,m}(-m) \in \sigma_k$
\[
\left\{
\betagin{array}{ll}
Arg(\lambda) + m Arg(\lambda) + \frac{|\lambda|}{r_m} \sin(Arg(\lambda)) = (2k+1)\pi &\, \hbox{if } $m$ \hbox { is even}\\
Arg(\lambda) + m (Arg(\lambda)+\pi) + \frac{|\lambda|}{r_m} (-1)\sin(Arg(\lambda)) = (2k+1)\pi & \, \hbox{if } $m$ \hbox { is odd}
\end{array}\right.
\]
Solving for $|\lambda|$, we obtain a function of $Arg(\lambda)$, which we denote by $\phi$. Explicitly, the curve $\Gamma_k$ can
be written as
\[
\left\{
\betagin{array}{ll}
|\lambda| = \phi(Arg(\lambda)) = r_m \frac{(2k+1)\pi - (m+1) Arg(\lambda)} {\sin(Arg(\lambda))} \qquad -\pi \le Arg(\lambda) \le \pi & \hbox{if } $m$ \hbox{ is even} \\
|\lambda| = \phi(Arg(\lambda)) = r_m \frac{(2k+1-m)\pi - (m+1) Arg(\lambda)} {-\sin(Arg(\lambda))} \qquad -\pi \le Arg(\lambda) \le \pi & \hbox{if } $m$ \hbox{ is odd}
\end{array}\right.
\]
As in the Sec. above, we need to impose $\phi(Arg(\lambda)) \ge 0$. If we denote $\theta = Arg(\lambda)$, we have:
\noindent if $m=2j \hbox{ for } j \in {\mathbb Z}$
\[
\Gamma_k = \phi(\theta) e^{i\theta} \left\{ \betagin{array}{llll} 0< \theta < \pi \qquad & \hbox{ if }\quad
k \ge j+1 \\
0< \theta <\frac{(2k+1)\pi}{m+1} \qquad & \hbox{ if }\quad 0 \le k \le j \\
\frac{(2k+1)\pi}{m+1}< \theta <0 \qquad & \hbox{ if }\quad -(j+1) \le k \le 0 \\
-\pi< \theta <0 \qquad & \hbox{ if }\quad k \le -(j+2) \\ \end{array}\right. ;
\]
\noindent if $m=2j+1 \hbox { for } j \in {\mathbb Z}$
\[
\Gamma_k = \phi(\theta) e^{i\theta} \left\{ \betagin{array}{lllll} 0< \theta < \pi \qquad & \hbox{ si }\quad k \ge m \\
-\pi < \theta < 0 \, \cup \, \frac{2k+1-m}{m+1} < \theta < \pi \qquad & \hbox{ if }\quad j+1 \le k \le m-1 \\
-\pi< \theta < \pi \qquad & \hbox{ if }\quad k = j \\
-\pi< \theta < \frac{2k+1-m}{m+1} \, \cup \, 0 < \theta < \pi \qquad & \hbox{ if }\quad j-1 \le k \le 0 \\
-\pi< \theta <0 \qquad & \hbox{ if }\quad k \le -1 \\ \end{array}\right.
\]
In Fig. \ref{gamma} we show some of these curves for some values of $m$. If we suppose that $m=2j$ is even, then
each $\Gamma_k$ tends asympotically to the line $Im(z) = (2k+1)\pi \, r_m$ as its real part tends to $+\infty$. We can
classify these curves in three types. The first one is formed by curves whose real part runs from $-\infty$ to $+\infty$. There
are two curves of the second kind, $\Gamma_j \hbox { and } \Gamma_{-(j+1)}$, with real part in $[-(m+1)r_m,\infty]$. The
third group is formed by $m$ curves, starting at the origin and tending to $+\infty$. These $m$ curves
have indexes between $j-1 \hbox { and } -j$.
\betagin{figure}[hbt]
\centering
\subfigure[\scriptsize{Graph of $\Gamma_k$ for $m=3$} ]{
\psfrag{a}[][]{\scriptsize $\Gamma_4$}
\psfrag{b}[][]{\scriptsize $\Gamma_3$}
\psfrag{c}[][]{\scriptsize $\Gamma_2$}
\psfrag{d}[][]{\scriptsize $\Gamma_0$}
\psfrag{e}[][]{\scriptsize $\Gamma_{-1}$}
\psfrag{m}[][]{\scriptsize $\Gamma_{-}$}
\psfrag{f}[][]{\scriptsize $\Gamma_{-2}$}
\psfrag{g}[][]{\scriptsize $\Gamma_{-3}$}
\psfrag{h}[][]{\scriptsize $\Gamma_1$}
\psfrag{i}[][]{\scriptsize $\Gamma_0$}
\psfrag{j}[][]{\scriptsize $\Gamma_2$}
\psfrag{k}[][]{\scriptsize $\Gamma_1$}
\fbox{ \includegraphics[width=0.3 \textwidth]{fg23.eps}}}
\hspace{0.5in}
\subfigure[\scriptsize{The parameter plane of $F_{\lambda,3}$} ]{
\includegraphics[width=0.3 \textwidth]{fg24.eps}}
\hspace{0.5in}
\subfigure[\scriptsize{ Graph of $\Gamma_k$ for $m=4$} ]{
\psfrag{a}[][]{\scriptsize $\Gamma_{-3}$}
\psfrag{b}[][]{\scriptsize $\Gamma_{-2}$}
\psfrag{c}[][]{\scriptsize $\Gamma_{-1}$}
\psfrag{d}[][]{\scriptsize $\Gamma_0$}
\psfrag{e}[][]{\scriptsize $\Gamma_1$}
\psfrag{f}[][]{\scriptsize $\Gamma_2$}
\psfrag{p}[][]{\scriptsize $\Gamma_{-6}$}
\psfrag{g}[][]{\scriptsize $\Gamma_{-5}$}
\psfrag{h}[][]{\scriptsize $\Gamma_{-4}$}
\psfrag{i}[][]{\scriptsize $\Gamma_3$}
\psfrag{j}[][]{\scriptsize $\Gamma_4$}
\psfrag{k}[][]{\scriptsize $\Gamma_5$}
\fbox{ \includegraphics[width=0.3 \textwidth]{fg25.eps}}}
\hspace{0.5in}
\subfigure[\scriptsize{Parameter plane of $F_{\lambda,4}$} ]{
\includegraphics[width=0.3 \textwidth]{fg26.eps}}
\caption{\small{Strips in the parameter plane.}}
\lambdabel{gamma}
\end{figure}
If we take $m$ an odd index ($m=2j+1$), these curves tend asymptotically to the lines $Im(z) = 2(k-m)\pi \, r_m$ as their real part
tend to $-\infty$. As above, we can classify these curves in three types. The first one is formed by curves that extend from $-\infty$ to $+\infty$. The
second one is formed by the curve $\Gamma_j $, has a horseshoe shape, and cuts the real axis at the point
$(m+1)r_m$. The third one is formed by $m-1$ curves, starting at the origin and tending to $-\infty$. These $m$ curves
have indexes between $0 \hbox { and } m-1$, except for $\Gamma_j$.
Hence we have obtained some curves $\Gamma_k$ such that if $\lambda \in \Gamma_k$, then
$F_{\lambda,m}(-m) \in \sigma_k $. Hence, choosing $\lambda \in \Gamma_k$ with $Re(F_{\lambda,m}(-m))$ large enough, we obtain that
$F_{\lambda,m}(F_{\lambda,m}(-m)) \in H_{|\lambda|,m}$. Since $Re(F_{\lambda,m}(-m))= Re(\lambda)(-1)^m r_m$, this corresponds to taking $Re(\lambda)$ or $-Re(\lambda)$ large enough depending on $m$
being even or odd.
By construction, the half curves we just defined belong each to $C_m^3$. We will
now show that a neighbourhood of $\Gamma_k$ of asymptotic width equal to $r_m \pi$
is also part of $C_m^3$. We fix a value $k \in {\mathbb Z}$ and we suppose that
$\lambda = \lambda_1 + i\lambda_2$, where $\lambda_1 \hbox{ and } \lambda_2$ are real numbers. We will prove the following result.
\betagin{proposition} \lambdabel{banpp}
If $m$ is even, for all $ \lambda_2 \in (r_m (\frac{\pi}{2}+2k\pi) \, ,
\, r_m(\frac{3\pi}{2} + 2k\pi)) $
there exists $\lambda_1^* \,$ such that, for all $ \lambda_1 > \lambda_1^* \,$ then $ F_{\lambdambda,m}^3(-m) \in A^*(0)$.
If $m$ is odd, for all $ \lambda_2 \in (r_m(-\frac{\pi}{2} + 2k\pi) \, , \, r_m(\frac{\pi}{2} + 2k\pi)) $
there exists $ \lambda_1^* \,$ such that, for all $ \lambda_1 < \lambda_1^* \,$ then $F_{\lambdambda,m}^3 (-m) \in A^*(0)$.
\end{proposition}
\betagin{proof}
Assume that $m$ is even (the odd case is completely symetric), and let $\lambda_2 \in (r_m (\frac{\pi}{2}+2k\pi) \, , \, r_m(\frac{3\pi}{2} +
2k\pi)) $. We recall
that proposition \ref{thbandas} assures that, for all $\, y \in ( (2k+1) \pi - Arg( \lambdambda) - \frac{\pi}{2} \, , \, (2k+1) \pi -
Arg( \lambdambda) + \frac{\pi}{2}) , \, $ there exists $\, x_* \in {\mathbb R} $ such that, for all $\,x \ge x_* $, the point
$F_{\lambdambda,m} (x+yi) \in H_{|\lambdambda|,m}$.
Using that $F_{\lambdambda,m} (-m)\, = \, \frac{\lambda_1}{r_m} \, + \, \frac{\lambda_2}{r_m} i $, it suffices to prove that
\[
Im(F_{\lambda,m}(-m)) = \frac{\lambda_2}{r_m} \in (\frac{\pi}{2} + 2k\pi - Arg(\lambdambda) \, , \, \frac{3\pi}{2} + 2k\pi - Arg(\lambdambda) ).
\]
\noindent Choosing
\[
Re(F_{\lambda,m}(-m)) = \frac{\lambda_1}{r_m} > x^*.
\]
\noindent The first condition is equivalent to $Arg(\lambdambda) \in (\alpha_1 , \alpha_2)$,
\betagin{figure}[hbt]
\psfrag{a1}[][]{ \scriptsize{$r_m(\frac{3\pi}{2}+2k\pi) i$}}
\psfrag{a2}[][]{\scriptsize{$r_m(\frac{\pi}{2}+2k\pi) i$}}
\psfrag{yr}[][]{ \scriptsize{$\lambda_2 / r_m i$}}
\psfrag{j}[][]{$\alpha_2$}
\psfrag{l}[][]{ $\alpha_1$}
\psfrag{al}[][]{ $Arg(\lambdambda)$}
\psfrag{rp}[][]{ $\scriptsize{\widetilde{\lambda_1}}$}
\psfrag{c1}[][]{\small{$\alpha_2 > 0$}}
\psfrag{c2}[][]{$\alpha_1 < 0$}
\psfrag{r}[][]{$r$}
\psfrag{s}[][]{$s$}
\centerline{\includegraphics[width=0.4 \textwidth]{fg27.eps}}
\caption{\small{Construction of the value $\widetilde{\lambda_1}$ }}
\lambdabel{ban}
\end{figure}
\noindent where $\alpha_1 = \frac{\pi}{2} + 2k\pi - \frac{\lambda_2}{r_m} < 0 \,\hbox{ and } \, \alpha_2 = \frac{3\pi}{2} + 2k\pi - \frac{\lambda_2}{m} >0$ (Fig. \ref{ban}).
Suppose that $k>0$. We denote by $r$ the line through the origin with slope $tan(\alpha_2)$, and let $s$ be the horizontal
line through $\frac{\lambda_2}{r_m} i$. We also denote by $\widetilde{\lambda_1}$ the abscissa of the intersection
point between the lines $r \, \hbox{ and } \, s $ (Fig. \ref{ban}). All values of $\lambda$ on $s$, with abscissa greater than $\widetilde{\lambda_1}$ verify that $0< arg(\lambda) < \alpha_2$. Finally,
we define $\lambda_1^* = max\{r_m x_* ,r_m \widetilde{\lambda_1}\}$, and for this value both conditions are verified.
Therefore, $F^2_{\lambda,m} (-m) \in H_{|\lambda|,m}$, and it follows that $F^3_{\lambda,m} (-m) \in D_{\epsilon_0}
\subset A^*(0)$. If $k \le 0$, we replace the line of slope $tan(\alpha_2)$, by a line with slope
$tan(\alpha_1)$.
\end{proof}
\noindent{\bf Acknowledgments}
We wish to thank Robert Devaney and Xavier Jarque for very helpful
discussions.
\betagin{thebibliography}{}
\bibitem[Baker, 1970]{Ba1} Baker I. N. [1970] ``Limit functions and sets of non-normality in iteration theory''. {\it Ann. Acad.
Sci. Fenn. Ser. A I Math}. {\bf 467} 1-11.
\bibitem[Baker, 1984]{Ba2} Baker I. N. [1984] ``Wandering Domains in the Iteration of Entire Functions''. {\it Proc. London Math. Soc.}
{\bf 49} 563-576.
\bibitem[Baker \& Dominguez, 2000]{BD} Baker I. N. \& Dominguez P. [2000] ``Some connectedness properties of Julia sets''. {\it Complex variables Theory Appl}. {\bf 41} 371-389.
\bibitem[Bergweiler, 1995]{B} Bergweiler W. [1995] ``Invariant domains and singularities''. {\it Math. Proc. Camb. Phil. Soc. }
{\bf 117} 525-532.
\bibitem[Carleson \& Gamelin, 1993]{CG} Carleson L. \& Gamelin Th. [1993] {\it Complex Dynamics}. Springer.
\bibitem[Devaney \& Goldberg, 1987]{DG} Devaney R. L. \& Goldberg L. R. [1987] ``Uniformation of attracting basins for
exponential maps''. {\it Duke Math. Journal}. {\bf 55} 253-266.
\bibitem[Devaney \& Krych, 1984]{DK} Devaney R. L. \& Krych M. [1984] ``Dynamics of $\exp(z)$''.{\it Ergodic Theory Dynam.
Systems}. {\bf 4} 35-52.
\bibitem[Devaney \& Tangerman, 1986]{DT} Devaney R. L. \& Tangerman F. [1986] ``Dynamics of entire functions near the
essential singularity''. {\it Ergodic Theory Dynam. Systems.} {\bf 6} 489-503.
\bibitem[Douady \& Hubbard, 1985]{DH} Douady A. \& Hubbard J. H. [1985] ``On the dynamics of Polynomial-like Mappings''. {\it Ann.
Scient. Ec. norm. Sup.} {\bf 18}, 287-343.
\bibitem[Fagella, 1995]{F} Fagella N. [1995] ``Limiting dynamics for the complex standard family''.
{\it International Journal of Bifurcation and Chaos (3)}. {\bf 5} 673-699.
\bibitem[Fatou, 1926]{Fa} Fatou P. [1926] ``Sur l'iter\'ation des fonctions transcendentes
enti\`eres''. {\it Acta Math.} {\bf 47} 337-370.
\bibitem[Geyer, 2001]{G} Geyer L. [2001] ``Siegel discs, Herman rings and the Arnold Family''.
{\it Trans. Amer. Math. Soc.} {\bf 353} 3661-3683.
\bibitem[Milnor, 1991]{M} Milnor J. [1991] ``On cubic polynomials with periodic critical point''.
{\it Stony Brook Institute for Mathematical Sciences. http://www.math.sunysb.edu/dynamics/surveys.html}.
\bibitem[Sullivan, 1985]{S} Sullivan D. [1985] ``Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia
problem on wandering domains''. {\it Ann. of Math. (2)} {\bf 122} 401-418.
\end{thebibliography}
\end{document} | math | 67,058 |
\begin{document}
{\Large \bf From Incompleteness Towards \\ \\ Completeness} \\
{\bf Elem\'{e}r E Rosinger} \\
Department of Mathematics \\
and Applied Mathematics \\
University of Pretoria \\
Pretoria \\
0002 South Africa \\
[email protected] \\ \\
{\bf Abstract} \\
It is argued that G\"{o}del's incompleteness theorem should be seen as self-evident, rather
than unexpected or surprising. \\ \\
{\bf A Surprising Shock and its widespread disregard} \\
Recently has been celebrated the centenary of the birth of Kurt G\"{o}del, [N]. That occasion
also recalls the 75 years since the publication of his famous paper on incompleteness. At the
time, in the 1930s, there were still strong reverberations in Mathematics following the
paradoxes in Set Theory discovered some decades earlier around the turn of the century. Three
main directions of thought had emerged in this regard. They were the so called "logicism",
following Frege, Russell and Whitehead, "formalism" supported by Hilbert, and the somewhat
less main line but rather radical "intuitionism" strongly promoted by Brouwer. \\
Logicism did not prove to be highly popular among mathematicians, since not many were ready to
accept the idea that Logic contained all of Mathematics, thus there was nothing more in
Mathematics than could be encompassed by Logic. Intuitionism proved even less popular, due to
the most severe restrictions it imposed even on Logic, let alone on Mathematics. \\
In this way, to some extent by default, Hilbert formalism came to prominence, only to be
severely questioned by G\"{o}del's 1931 incompleteness result which, in everyday terms, states
that "Every sufficiently strong axiomatic theory which is consistent must be incomplete",
[F]. \\
At the time, and to a good extent ever since, this incompleteness theorem has been considered
by so many among mathematicians who encounter it for the first time as being most surprisingly
unexpected, and in fact, rather unsettling. As it happens, however, most of the usual
mathematicians consider that, regardless of the mentioned features of the incompleteness
theorem, they can quite safely disregard it within their everyday professional pursuit. \\
Needless to say, Hilbert himself tried assiduously to diminish, if not completely dismiss, its
obvious impact on his program of formalism. \\ \\
{\bf What is self-evident should in fact not at all surprise ...} \\
And yet, what may appear to be no less surprising, this incompleteness theorem should at a
somewhat better and deeper consideration be {\it self-evident}, instead of being seen so
surprising ... \\
However, what prevents it from being self-evident is a long time and most powerfully
entrenched, even if not quite perceived "cut" in the way thinking in Mathematics, as well as
in so many other realms, is being practiced. \\
And by its very nature, Mathematics, or for that matter, Mathematical Logic, manifests this
"cut" in a most trenchant manner. \\ \\
{\bf The ever ongoing "cut" ...} \\
So then, what is this "cut" all about ? \\
For convenience, let us consider it in the case of Mathematics. Typically, and in fact, rather
without any exception, what is considered to be Mathematics, as for instance in its form
publishable in professional journals, is one thing, while mathematical thinking which leads to
such Mathematics is seen to be quite another. There may be just about one not particularly
central exception, namely, with conjectures or open problems. However, they can hardly be
published as such and all alone in professional journals. Instead, they are supposed to be
supported by a convincing argument along the lines of what is considered to be Mathematics. \\
And what is considered to be Mathematics has to be a rigorous, logically developed system
based on earlier definitions and proven results, plus possibly new definitions, and of course,
the required new results. As for axioms, they are supposed to be clear in the given context,
or occasionally, new ones may be put forward. \\
Indeed, from this point of view what is to be Mathematics is at least as clear as it may ever
happen with anything else in science. \\
But then, most obviously, hardly any mathematician does ever do Mathematics in this way. And
to make things easier to understand, let us use the analogy of what is going on even in the
very best restaurants. \\
The dishes are prepared in the kitchen, and then taken to the eating hall where the customers
sit at the neat tables, and where the dishes are served up in front of them in a spectacularly
appetizing manner. \\
Of course, the customers are not supposed to have even a look, let alone go, into the kitchen
where things are quite inevitably really messy ... \\
A similar "cut" between the "kitchen" and the "eating hall" happens in Mathematics as well,
and does so quite inevitably. \\
And needless to say, just like with restaurants, the creative mathematical activity happens in
the "kitchen", which can be the thinking of one single mathematician, or of a group who
collaborate with one another. \\ \\
{\bf The other side of the "cut" ...} \\
Strangely enough, however, what goes on in such "kitchens" is seldom brought to the attention
of the profession. And even less so of what has gone wrong in such "kitchens". Yet that
process of "cooking" new Mathematics - or in general, science - is not only essential, but
often it can also be {\it essentially different} from what is at the end seen in the "eating
halls". And in this regard we can mention several aspects, the following three of which can be
most important, no matter how much they may happen to be overlooked. \\
{\it Self-referentiality.} In present day Mathematics, or for that matter, Mathematical Logic,
so called "circular definitions" are not quite welcome, in spite of recent research in this
direction, [B-M]. However, within mathematical thinking, that is, inside the "kitchen",
nothing stops one from indulging in self-referential concepts. After all, the famous 1903
Russell paradox is but the expression of such a thinking in terms of Set Theory. \\
{\it Instantaneity.} Here, turning to Physics may better help in understanding, [R]. It is a
basic principle of both Special and General Relativity that no physical action whatever,
including information transmission can occur faster than the speed of light, [C-K]. And yet,
in the thinking of anybody, physicists or not, one can simultaneously conceive of arbitrarily
faraway objects which may possibly act upon one another. As for Quantum Mechanics, one can
conceive of two entangled quantum particles arbitrarily far from one another, yet with a
correlation of their states, which means that in one's thinking one can instantaneously
register the fact that, given the state of one of the particles, the state of the other one is
determined. \\
{\it Unattributable sources.} A good deal of original scientific ideas appear in one's mind as
if from nowhere. Furthermore, often one cannot trace precisely their origin and process of
emergence into a constituted idea even with hindsight. \\ \\
{\bf Should Incompleteness still be such a surprise ?} \\
Let us consider G\"{o}del's incompleteness theorem as relating to the Peano axioms of natural
numbers. It implies that there are statements $S$ which, together with their negation non-$S$,
do not follow as theorems in that theory. In this way, the respective theory of natural
numbers {\it branches}\, at $S$ and non-$S$, in the sense that one can add to the Peano axioms
$S$ and obtain one theory of natural numbers, or alternatively, one can add non-$S$ and,
clearly, obtain a rather different theory. \\
The decision which of such two branches to take does of course happen in the "kitchen". And
there, any of the mentioned three phenomena can manifest itself, not mention other ones not
considered above. And clearly, the third of the above phenomena - namely, the possible
presence of unattributable sources of our ideas - can be a prime source of {\it
incompleteness}. \\
Consequently, the moment we no longer consider Mathematics as being all of it contained only
on the "eating hall" side of that long ongoing and disregarded "cut", that very moment we can
only be surprised how and why we did not hit prior to G\"{o}del upon a variety of
incompleteness results. \\ \\
{\bf Not many "cuts" outside of hard science ...} \\
It may be important to enquire why such a "cut" exists, persists, and fails to be noted, let
alone, taken into account in various sciences. \\
One immediate reason can be the following. Outside of sciences - and here one means "hard
sciences" - that "cut" hardly exists. In other words, the respective "restaurants" do not have
the sharp division of "kitchen" and "eating hall". \\
Indeed, in non-intellectual realms, such as sport for instance, such a division would in fact
be highly counterproductive. \\
As for a large variety of intellectual realms, including philosophy or "soft sciences", even
if such a "cut" may be claimed to exist, that claim is questionable. And even if that "cut"
may to some extent be present, it happens - even if often regrettably so - not to be
particularly important in fact. \\
In this way, since the practice of modern "hard science" is relatively new in human history,
it may be less surprising that the mentioned "cut" has passed insufficiently noticed, let
alone considered in its implications. \\
As for the fact that Mathematics is a couple of millennia older than modern "hard science",
and nevertheless it still does not give enough attention to that "cut", one can only speculate
about the reason possibly being in the singular - and thus, insufficiently consequential -
character of Mathematics within human experience at large, as well as within the life
experience of individual mathematicians. \\ \\
\end{document} | math | 9,861 |
\betagin{document}
\maketitle
\betagin{abstract}
Under the Riemann Hypothesis, we prove for any natural number $r$ there exist infinitely many natural numbers $n$ such that $(\gammammama_{n+r}-\gammammama_n)/(2\pi r /\log \gammammama_n) > 1 + \Theta/\sqrt{r}$
and $(\gammammama_{n+r}-\gammammama_n)/(2\pi r /\log \gammammama_n) < 1 - \vartheta/\sqrt{r}$ for explicit absolute positive constants $\Theta$ and $\vartheta$, where $\gammammama$ denotes an ordinate of a zero of the Riemann zeta-function on the critical line. Selberg published announcements of this result several times without proof.
\end{abstract}
\section{Introduction}
Let $\zeta(s)$ denote the Riemann zeta-function, and let $\rho = \betata + i\gammammama$ denote a nontrivial zero of $\zeta(s)$. Consider the sequence of ordinates of zeros in the upper half-plane
\[
0 < \gammammama_1 \le \gammammama_2 \le \ldots \le \gammammama_n \le \gammammama_{n+1} \le \ldots.
\]
It is well known that
\[
N(T) := \sum_{0<\gammammama\le T}1 \sim \frac{T}{2\pi}\log T,
\]
from which it follows that the average gap between consecutive zeros is $2\pi/\log \gammammama_n$. Assuming the Riemann Hypothesis, $\betata = 1/2$ and $\gammammama \in \mathbb{R}$. The result of this note is a proof of the following theorem.
\betagin{theorem*}\label{Selberg}
Assuming the Riemann Hypothesis, for any natural number $r$ there exist infinitely many $n$ such that
\[
\frac{\gammammama_{n+r}-\gammammama_n}{2\pi r /\log \gammammama_n} > 1 + \frac{\Theta}{\sqrt{r}} \qquad \text{ and } \qquad
\frac{\gammammama_{n+r}-\gammammama_n}{2\pi r /\log \gammammama_n} < 1 - \frac{\vartheta}{\sqrt{r}}
\]
for the absolute positive constants $\Theta=0.574271$ and $\vartheta= 0.299856$. Moreover, for $r$ sufficiently large, we may take $\Theta= \vartheta = 0.9065$.
\end{theorem*}
There are discrepancies in the literature regarding the correct statement of this result, which we hope to now clarify. In \cite[p. 199]{Sel47}, Selberg announced, without proof, that there exists an absolute positive constant $\theta$ such that for all positive integers $r$
\[
\limsup_{n\to \infty}\frac{\gammammama_{n+r}-\gammammama_n}{2\pi r / \log \gammammama_n} > 1+ \theta \qquad \text{ and } \qquad \liminf_{n\to \infty}\frac{\gammammama_{n+r}-\gammammama_n}{2\pi r/ \log \gammammama_n} < 1- \theta .
\]
This statement was later updated in the Acknowledgements section of \cite{Mue82}, with the $\theta$ appearing above replaced with $\theta/\sqrt{r}$. Finally, in the errata of Volume 1 of his collected papers \cite[p. 355]{SelCollected}, Selberg clarified the correct statement of his result.
\betagin{Selberg2*}
There exist an absolute positive constant $\theta$ such that for all positive integers $r$
\[
\limsup_{n\to \infty}\frac{\gammammama_{n+r}-\gammammama_n}{2\pi r / \log \gammammama_n} > 1+ \theta r^{-\alphapha}\qquad \text{ and } \qquad \liminf_{n\to \infty}\frac{\gammammama_{n+r}-\gammammama_n}{2\pi r / \log \gammammama_n} < 1- \theta r^{-\alphapha},
\]
where $\alphapha$ may be taken as 2/3, and if one assumes the Riemann Hypothesis as 1/2.
\end{Selberg2*}
Selberg did not give an indication of a proof for either statement, however Heath-Brown in \cite[p. 246-249]{Titchmarsh} provides an unconditional proof of Selberg's result in the case $r=1$ using the work of Fujii \cite{Fujii1} concerning the mean value of $S(t)$ in short intervals. (Note that $\pi S(t)$ is the argument of $\zeta(s)$ at the point $s = 1/2 + it.$) We remark that Heath-Brown's proof for $r=1$ shows that the result holds for a positive proportion of integers $n$.
The goal of this note is to give a proof of Selberg's conditional result for all $r\ge1$ with explicit constants. To prove our theorem, we adapt a method developed by Conrey, Ghosh, and Gonek \cite{CGG84} on gaps between consecutive nontrivial zeros of $\zeta(s)$ in the interval $[0,T]$ for $T$ large. The method is conditional on the Riemann Hypothesis. To our knowledge, our proof is the first to appear in the literature for $r>1$.
For a fixed, positive integer $r$, let
\betagin{equation}\label{mulambda}
\lambdabda_r := \limsup_{n\to \infty} \frac{\gammammama_{n+r}-\gammammama_n}{2\pi /\log \gammammama_n} \qquad \text{and} \qquad \mu_r := \liminf_{n\to \infty} \frac{\gammammama_{n+r}-\gammammama_n}{2\pi /\log \gammammama_n}.
\end{equation}
By definition $\lambdabda_r \ge r$ and similarly $\mu_r \le r$, however random matrix theory predicts that $\lambdabda_r = \infty$ and $\mu_r =0$. Following \cite{CGG84}, we compare averages of a well-chosen polynomial of the form
\betagin{equation}\label{polynomial}
A(t) := \sum_{n\le X}\frac{a^\pm(n)}{n^{it}},
\end{equation}
where $X=T^{1-\deltata}$ for some small $\deltata>0$. To adapt for $r$-gaps, we set
\betagin{equation*}
M_1 := \int_{T}^{2T}\left|A(t)\right|^2\,dt
\end{equation*}
and
\betagin{equation*}
M_{2}(c_r) := \int_{-\pi c_r/\log T}^{\pi c_r/\log T}\sum_{T\le \gammammama \le 2T}\left|A(\gammammama+\alphapha)\right|^2\,d\alphapha,
\end{equation*}
where $c_r$ is some nonzero real number. We see that $M_{2}(c_r)$ is monotonically increasing and
\[
M_{2}(\mu_r) \le rM_1 \le M_{2}(\lambdabda_r).
\]
Therefore, if $M_{2}(c_r)<rM_1$ for some choice of $a^+(n)$ and $c_r$ then $\lambdabda_r > c_r$. Similarly, if $M_2(c_r)>rM_1$ for some choice of $a^-(n)$ and $c_r$ then $\mu_r < c_r$.
Connecting their work to a previous result of Montgomery and Odlyzko \cite{MO}, Conrey, Ghosh, and Gonek show
\betagin{equation*}
\frac{M_{2}(c_r)}{M_1} = h^\pm(c_r) + o(1),
\end{equation*}
where $h(c_r)$ is defined by
\betagin{equation}\label{hfunction}
h^{\pm}(c_r):=c_r\mp\frac{\mathbb{R}e\left( \displaystyle\sum_{kn\le X}\frac{a^\pm(n)\overline{a^\pm(kn)}g_{c_r}(k)\Lambdabda(k)}{kn}\right)}{\displaystyle\sum_{n\le X}\frac{|a^\pm(n)|^2}{n}}
\end{equation}
and
\betagin{equation*}
g_{c_r}(k)=\frac{2\sin\left(\pi c_r\frac{\log k}{\log{T}}\right)}{\pi\log k}
\end{equation*}
so that $|g_{c_r}(k)| \le 2c_r/\log T$. The function $h^\pm(c_r)$ was introduced by Montgomery and Odlyzko to study gaps between consecutive zeros of $\zeta(s)$. In particular, they show that if one is able to find $c_r$ such that $h^+(c_r) < r$ then $\lambdabda_r > c_r$ and such that if $h^-(c_r) > r$ then $\mu_r < c_r$.
Letting $r=1$ in \eqref{mulambda}, it follows from our theorem that $\lambdabda_1 >1$ and $\mu_1 <1$. Quantitative bounds on $\lambdabda_1$ and $\mu_1$ have been obtained using the above approach, with different choices of $a(n)$ leading to improved results. See \cite{BMN10} and subsequently \cite{FengWu} for discussions of these choices. The best current quantitative bounds concerning gaps between consecutive zeros of the Riemann zeta function (under the assumption of the Riemann Hypothesis) are $\lambdabda_1 > 3.18$, due to Bui and Milinovich \cite{BM17} , and $\mu_1 <0.515396$, due to Preobrazhenskii \cite{Preo}. We note that the method employed in \cite{BM17}, which is based on the work of Hall \cite{Hall} and different from the method discussed above, is unconditional if one restricts the analysis to critical zeros.
\section{Proof of the theorem for fixed $r\ge 1$}\label{proof}
For large gaps for any fixed $r\ge 1$, we choose $a^+(n) = d_{\ell}(n)$, where $d_\ell$ is multiplicative and defined on prime powers by
\[
d_\ell(p^m)= \frac{\Gammamma(m+\ell)}{\Gammamma(\ell)m!}.
\]
Fix $\ell\ge 1$. (In the proof, we will ultimately set $\ell$ to be an explicit value depending on $r$.) Similarly, for small gaps for any fixed $r\ge 1$, we choose $a^-(n) = \lambdabda(n)d_{\ell}(n)$, where $\lambdabda(n)$ denotes the Liouville function.\\
We now prove the result for large gaps for any fixed $r \ge 1$. Take $a^+(n) = d_\ell(n)$ for $\ell \ge 1$ an integer to be determined later. In this case the relevant mean-value to compute is well known:
\betagin{equation*}
\sum_{n \le x} \frac{d_\ell(n)^2}{n} = C_\ell(\log x)^{\ell^2}+O((\log T)^{\ell^2-1})
\end{equation*}
for fixed $\ell \ge 1$, uniformly for $x \le T$, where $C_\ell$ is a constant which will not have an effect in our application. It is shown in \cite[p.422]{CGG84} that for this choice of $a^+(n)$, the equation $M_2(c_r) / M_1 = h^+(c_r)+o(1)$ reduces to
\betagin{equation}\label{large}
h^+(c_r) = c_r -2\ell \int_{0}^{1}\frac{\sin(\pi c_r v(1-\deltata))}{\pi v}(1-v)^{\ell^2}\,dv + O(1/\log T)
\end{equation}
where $\deltata>0$ is as in \eqref{polynomial} and will be taken to be sufficiently small. To detect large gaps, we must show that $h^+(c_r) < r$ for fixed $r\ge 1$. By the previous discussion, this will imply $\lambdabda_r > c_r$. For example, using \eqref{large} we can compute the following table of values.
\betagin{table}[h]
\centering
\betagin{tabular}{ |c|c|c|c| }
{\bf h}line
$r$ & $\ell$ & $c_r$ & $h^+(c_r)$\\
{\bf h}line
1 & 2.2 & 2.337 & 0.99965\\
2 & 2.8 & 3.708 & 1.99937\\
3 & 3.3 & 4.994 & 2.99975\\
4 & 3.7 & 6.235 & 3.99950\\
5 & 4.0 & 7.448 & 4.99978\\
{\bf h}line
\end{tabular}
\caption{For fixed $r$, the table gives values of $\ell,c_r$ for which $h^+(c_r) < r$, implying $\lambdabda_r > c_r$.}
\label{figure: table example}
\end{table}
In general, to prove large gaps of the desired shape, we show that $h^+(c_r) < r$ for fixed $r\ge 1$ and $c_r = r + \Theta\sqrt{r}$ with $\Theta>0$. We estimate the integral appearing in \eqref{large} as follows. Let
\[
\int_{0}^{1}\frac{\sin(\pi c_r(1-\deltata) v)}{\pi v}(1-v)^{\ell^2}\,dv = I_1 + I_2,
\]
where
\[I_1:= \int_{0}^{1/c_r}\frac{\sin(\pi c_r(1-\deltata) v)}{\pi v}(1-v)^{\ell^2}\,dv
\qquad \text{ and } \qquad
I_2:=\int_{1/c_r}^{1}\frac{\sin(\pi c_r(1-\deltata) v)}{\pi v}(1-v)^{\ell^2}\,dv.
\]
For $I_1$, we first observe that the integrand is positive in the range of integration and write
\betagin{equation}\label{I1}
I_1 \ge I_{1,a} + I_{1,b},
\end{equation}
say, where
\[I_{1,a}:= \int_{0}^{1/4c_r}\frac{\sin(\pi c_r(1-\deltata) v)}{\pi v}(1-v)^{\ell^2}\,dv,
\]
\[ I_{1,b}:= \int_{1/4c_r}^{1/2c_r}\frac{\sin(\pi c_r(1-\deltata) v)}{\pi v}(1-v)^{\ell^2}\,dv,
\]
and we have discarded the portion of the integral from $1/(2c_r)$ to $1/c_r$. Now we estimate $I_{1,a}$ and $I_{1,b}$.
For $I_{1,a}$, we compare $\sin(\pi c_r(1-\deltata) v)$ to $2\sqrt{2}c_r(1-\deltata)v$ and find
\betagin{equation}\label{I1a}
I_{1,a} \ge \int_{0}^{1/4c_r}\frac{2\sqrt{2}c_r(1-\deltata)v}{\pi v}(1-v)^{\ell^2}\,dv= \frac{2\sqrt{2}c_r(1-\deltata)}{\pi(\ell^2+1)}\left(1- \left(1-\frac{1}{4c_r}\right)^{\ell^2+1}\right).
\end{equation}
Similarly for $I_{1,b}$, we compare $\sin(\pi c_r(1-\deltata) v)$ to $(4-2\sqrt{2})c_r(1-\deltata)v$ and find
\betagin{equation}\label{I1b}
I_{1,b} \ge \frac{(4-2\sqrt{2})c_r(1-\deltata)}{\pi(\ell^2+1)}\left(\left(1-\frac{1}{2c_r}\right)^{\ell^2+1}- \left(1-\frac{1}{4c_r}\right)^{\ell^2+1}\right).
\end{equation}
Thus by \eqref{I1}, \eqref{I1a}, and \eqref{I1b}, we have
\[
I_1 \ge \frac{2c_r(1-\deltata)}{\pi(\ell^2+1)}\left(\sqrt{2}- (2\sqrt{2}-2)\left(1-\frac{1}{4c_r}\right)^{\ell^2+1} - (2-\sqrt{2})\left(1-\frac{1}{2c_r}\right)^{\ell^2+1}\right).
\]
Furthermore, since $\exp(-x)\ge 1-x$ for $x\ge0$, it follows that
\betagin{equation}\label{I1estimate}
I_1 \ge \frac{2c_r(1-\deltata)}{\pi(\ell^2+1)}\left\{\sqrt{2}- (2\sqrt{2}-2)\exp\left(\frac{-(\ell^2+1)}{4c_r} \right) - (2-\sqrt{2})\exp\left(\frac{-(\ell^2+1)}{2c_r} \right)\right\}.
\end{equation}
We now estimate the second integral $I_2$. Since $v\ge0$, we have
\[
|I_2| \le \frac{1}{\pi}\int_{1/c_r}^{1}\frac{(1-v)^{\ell^2}}{v}\,dv \le \frac{1}{\pi}\int_{1/c_r}^{1}\frac{\exp(-\ell^2v)}{v}\,dv.
\]
Thus, by the change of variable $u=\ell^2v$, we find
\betagin{equation}\label{I2estimate}
I_2 \ge \frac{-1}{\pi}\int_{\ell^2/c_r}^{\infty}\frac{\exp(-u)}{u}\,du.
\end{equation}
Combining the estimates in \eqref{I1estimate} and \eqref{I2estimate}, we have
\betagin{equation*}\label{Hestimate}
\betagin{split}
h^+(c_r) \le c_r&-2\ell\biggl\{\frac{2c_r(1-\deltata)}{\pi(\ell^2+1)}\biggl(\sqrt{2}- (2\sqrt{2}-2)\exp\left(\frac{-(\ell^2+1)}{4c_r} \right)\\ &-(2-\sqrt{2})\exp\left(\frac{-(\ell^2+1)}{2c_r} \right)\biggr)- \frac{1}{\pi}\int_{\ell^2/c_r}^{\infty}\frac{\exp(-u)}{u}\,du\biggr\} +O\left(1/\log T\right).
\end{split}
\end{equation*}
In this case, $c_r = r+\Theta\sqrt{r}$ where $\Theta>0$, and thus $c_r >1$ for any $r\ge 1$. Thus, letting $\ell= \sqrt{bc_r-1}$, where $b>1$ is a real number that will be chosen later, we have
\[
\ell = \sqrt{bc_r-1} \ge \sqrt{br}\sqrt{1-\frac{1}{b}}
\]
for any $r\ge 1$. Furthermore, since we always have $c_r > 1$, for any $r\ge 1$ it follows that
\[
\frac{\ell^2}{c_r}> b-1,
\]
and thus we may again increase the length of integration in $I_2$ to write
\[
\int_{\ell^2/c_r}^{\infty}\frac{\exp(-u)}{u}\,du < \int_{b-1}^{\infty}\frac{\exp(-u)}{u}\,du.
\]
Combining these estimates, we find
\betagin{equation*}\betagin{split}
h^+(c_r) < r+ \Theta\sqrt{r} &- 2\sqrt{br}\sqrt{1-\frac{1}{b}}\biggl\{\frac{2(1-\deltata)}{\pi b}\biggl(\sqrt{2}-(2\sqrt{2}-2)\exp\left(\frac{-b}{4}\right)\\
&-(2-\sqrt{2})\exp\left(\frac{-b}{2}\right)\biggr)-\frac{1}{\pi}\int_{b-1}^{\infty}\frac{\exp(-u)}{u}\,du\biggr\} +O\left(\frac{1}{\log T}\right).
\end{split}\end{equation*}
To show $h^+(c_r)< r$ and prove the theorem, we set
\betagin{equation*}\betagin{split}
\Theta = \max_{b}\biggl\{2\sqrt{b}\sqrt{1-\frac{1}{b}}&\biggl(\frac{2}{\pi b}\biggl(\sqrt{2}-(2\sqrt{2}-2)\exp\left(\frac{-b}{4}\right)\\
&-(2-\sqrt{2})\exp\left(\frac{-b}{2}\right)\biggr)-\frac{1}{\pi}\int_{b-1}^{\infty}\frac{\exp(-u)}{u}\,du\biggr)\biggr\}.
\end{split}\end{equation*}
The choice $b=5.0107$ yields $\Theta=0.574271$. With $\deltata$ sufficiently small and $T$ sufficiently large, these choices guarantee that $h^+(c_r)<r$, as desired.
We now prove the result for small gaps for any fixed $r\ge1$. The proof for small gaps is similar to the proof for large gaps, so we indicate the necessary changes. Take $a^-(n) = \lambdabda(n)d_\ell(n)$ for $\ell \ge 1$ fixed. It is given in \cite[p.422]{CGG84} that this choice of $a^-(n)$, yields
\betagin{equation}\label{small}
h^-(c_r) = c_r +2\ell \int_{0}^{1}\frac{\sin(\pi c_r v(1-\deltata))}{\pi v}(1-v)^{\ell^2}\,dv +O\left(1/\log T\right).\\
\end{equation}
To detect small gaps, we must show that $h^-(c_r) > r$ for fixed $r\ge 1$. By the previous discussion, this will imply $\mu_r < c_r$. For example, using \eqref{small} we can compute the following table of values.
\betagin{table}[h]
\centering
\betagin{tabular}{ |c|c|c|c| }
{\bf h}line
$r$ & $\ell$ & $c_r$ & $h^-(c_r)$\\
{\bf h}line
1 & 1.1 & 0.5172 & 1.00012\\
2 & 1.4 & 1.126 & 2.00118\\
3 & 1.9 & 1.831 & 3.00072\\
4 & 2.3 & 2.588 & 4.00099\\
5 & 2.7 & 3.375 & 5.00116 \\
{\bf h}line
\end{tabular}
\caption{For fixed $r$, the table gives values of $\ell,c_r$ for which $h^-(c_r) > r$, implying $\mu_r < c_r$.}
\label{figure: table example}
\end{table}
In general, to prove small gaps of the desired shape, we show that $h^-(c_r) < r$ for fixed $r\ge 1$ and $c_r = r - \Theta\sqrt{r}$ with $\Theta>0$. We estimate the integral appearing in \eqref{small} as before, however for brevity we will perform the calculation without writing $I_1$ as the sum of two integrals of equal length.\footnote{To see how these choices affect the size of $\vartheta$ here and in the large gaps setting, please refer to the remark following the proof.} We find
\betagin{equation*}\betagin{split}
h^-(c_r) &\ge c_r+2\ell\biggl\{ \frac{2c_r(1-\deltata)}{\pi(\ell^2+1)}\biggl(1-\exp\left(\frac{-(\ell^2+1)}{c_r} \right)- \frac{1}{\pi}\int_{\ell^2/c_r}^{\infty}\frac{\exp(-u)}{u}\,du\biggr\} +O\left(\frac{1}{\log T}\right).
\end{split}\end{equation*}
Let $\ell= \sqrt{bc_r-1}$ and $c_r = r - \vartheta\sqrt{r}$, with $\vartheta>0$. In this case, we do not always have $c_r >1$. Indeed, since $\vartheta>0$, if $r=1$ then $0< c_r <1$. However, if we require that $\vartheta \le 0.5$, the estimate
\[
\ell = \sqrt{bc_r-1} > \sqrt{br}\sqrt{\frac{1}{2}-\frac{1}{b}}
\]
holds for any $r\ge 1$. The requirement that $\vartheta \le 0.5$ also implies
\[
\frac{\ell^2}{c_r}\ge b-2
\]
for any $r\ge 1$, and we may increase the length of integration in $I_2$ to write
\[
\int_{\ell^2/c_r}^{\infty}\frac{\exp(-u)}{u}\,du \le \int_{b-2}^{\infty}\frac{\exp(-u)}{u}\,du.
\]
Thus, requiring that $\vartheta\le0.5$, we may put these estimates together to write
\betagin{equation*}\betagin{split}
h^-(c_r)&> r - \vartheta\sqrt{r}+ 2\sqrt{br}\sqrt{\frac{1}{2}-\frac{1}{b}}\bigg\{\frac{2(1-\deltata)}{\pi b}\biggl(1-\exp\left(-b\right) \biggr)-\frac{1}{\pi}\int_{b-2}^{\infty}\frac{\exp(-u)}{u}\,du\biggr\} + O\left(\frac{1}{\log T}\right).\end{split}\end{equation*}
To show $h^-(c_r)> r$ and thus prove the theorem, we set
\betagin{equation*}\betagin{split}
\vartheta = \max_{b}\biggl\{2\sqrt{b}\sqrt{\frac{1}{2}-\frac{1}{b}}&\biggl(\frac{2}{\pi b}(1-\exp\left(-b\right)\biggr)-\frac{1}{\pi}\int_{b-2}^{\infty}\frac{\exp{-u}}{u}\,du\biggr)\biggr\}.
\end{split}\end{equation*}
The choice $b=5.17305$ yields $\vartheta=0.299856$. (We note that the condition $\vartheta <0.5$ is satisfied.) With $\deltata$ sufficiently small and $T$ sufficiently large, these choices guarantee that $h^+(c_r)<r$, as desired.
\betagin{remark*}In the argument above for large gaps, if we had not divided the remaining portion of $I_1$ into two smaller integrals and instead compared $\sin(\pi c_r(1-\deltata)v)$ to $2c_r(1-\deltata)v$ over the interval $[0, 1/2c_r]$, we would have ultimately found that one can take $\Theta = 0.447$. Instead, by carrying out the analysis on $I_1 > I_{1,a} + I_{1,b}$ (see \eqref{I1}) and estimating $I_{1,a}$ and $I_{1,b}$ separately, we were able to provide the stronger constant $\Theta = 0.570717$. One could thus slightly improve the absolute constant $\Theta$ by breaking up $I_1$ into smaller pieces over the interval $[0, 1/2c_r]$, and estimating each piece accordingly. For example, writing $I_1 > I_{1,a'} + I_{1,b'}+I_{1,c'}+ I_{1,d'}$ where each integral has equal length of integration over the interval $[0, 1/2c_r]$, one can obtain $\Theta = 0.593234$, and comparing $I_1$ to the sum of sixteen such smaller integrals $\Theta = 0.599648$. Similarly, for small gaps, comparing $I_1$ to the sum of two smaller integrals of equal length over the interval $[0, 1/2c_r]$ yields $\vartheta = 0.359222$; using sixteen smaller integrals of equal length of integration over the interval $[0, 1/2c_r]$ yields $\vartheta = 0.379674$.
\end{remark*}
\section{Proof of the theorem for $r$ sufficiently large} We can improve the constants $\Theta$ and $\vartheta$ appearing in the theorem if we take $r$ to be large. In fact, we will see that in this setting, we may take $\Theta = \vartheta = 0.9065.$
We first consider large gaps for sufficiently large $r$. Starting with \eqref{large}, to detect large gaps of the desired size, we must show that $h^+(c_r) < r$ for sufficiently large $r$ and $c_r = r + \Theta\sqrt{r}$ with $\Theta>0$. Choosing $\ell = B\sqrt{r}$, we have
\[
h^+(c_r) < c_r -2B\sqrt{r} \int_{0}^{1}\frac{\sin(\pi r v(1-\deltata))}{\pi v}(1-v)^{B^2r}\,dv + O(1/\log T)
\]
for sufficiently large $r$. Making the change of variable $rv = w$, the above inequality becomes
\betagin{equation}\label{argument}\betagin{split}
h^+(c_r) &< c_r -2B\sqrt{r} \int_{0}^{r}\frac{\sin(\pi w(1-\deltata))}{\pi w}\left(1-\frac{w}{r}\right)^{B^2r}\,dw + O(1/\log T)\\
&< c_r -2B\sqrt{r} \int_{0}^{r}\frac{\sin(\pi w(1-\deltata))}{\pi w}\exp\left(-B^2w\right)\,dw + O(1/\log T)\\
&=c_r -2B\sqrt{r} \int_{0}^{\infty}\frac{\sin(\pi w(1-\deltata))}{\pi w}\exp\left(-B^2w\right)\,dw -2B\sqrt{r}E(r)+ O(1/\log T), \\
\end{split}\end{equation}
where
\[
E(r) =\int_{r}^{\infty}\frac{\sin(\pi w(1-\deltata))}{\pi w}\exp\left(-B^2w\right)\,dw.
\]
Note that as $r\to \infty$, $\sqrt{r}E(r) \to 0$, so for sufficiently large $r$ this term is negligible. Thus we set
\betagin{equation*}\betagin{split}
\Theta = \max_{B}\left\{2B\int_{0}^{\infty}\frac{\sin(\pi w)}{\pi w}\exp\left(-B^2w\right)\,dw\right\} = \max_{B}\left\{\frac{2B}{\pi}\arctan\left(\frac{\pi}{B^2} \right)\right\}. \\
\end{split}\end{equation*}
The choice $B=1.502243.$ yields $\Theta = 0.9065$. With $\deltata$ sufficiently small, $T$ and $r$ sufficiently large, these choices guarantee that $h^+(c_r) < r$.
We now consider small gaps for $r$ sufficiently large. We begin with \eqref{small} and let $\ell = B\sqrt{r-\sqrt{r}}$. If we assume $\vartheta <1$, then $r-\vartheta \sqrt{r} > r - \sqrt{r}$ for all $r$, and we have
\[
h^-(c_r) > c_r +2B\sqrt{r-\sqrt{r}} \int_{0}^{1}\frac{\sin(\pi (r-\sqrt{r})v(1-\deltata))}{\pi v}(1-v)^{B^2(r-\sqrt{r})}\,dv + O(1/\log T).
\]
Using the change of variable $(r-\sqrt{r})v=w$, we follow an analogous argument as in the previous subsection and ultimately set
\betagin{equation*}\betagin{split}
\vartheta = \max_{B}\left\{2B\int_{0}^{\infty}\frac{\sin(\pi w)}{\pi w}\exp\left(-B^2w\right)\,dw\right\}= \max_{B}\left\{\frac{2B}{\pi}\arctan\left(\frac{\pi}{B^2} \right)\right\}. \\
\end{split}\end{equation*}
As before, the choice $B=1.502243$ yields $\vartheta = 0.9065$. With $\deltata$ sufficiently small, $T$ and $r$ sufficiently large, these choices guarantee that $h^-(c_r) > r$. \\
\noindent{\bf Acknowledgements.} Turnage-Butterbaugh was supported by the National Science Foundation Grant DMS-1440140 while in residence at the Mathematical Sciences Research Institute during the Spring 2017 semester. The authors thank D.A. Goldston and M.B. Milinovich for helpful comments on an earlier version of the article. We also thank the anonymous referee for a suggestion that led to improved constants in the main theorem.
\betagin{thebibliography}{99}
\bibitem{BM17}
{H.~M. Bui \and M.~Milinovich}, Gaps between zeros of the {R}iemann zeta-function, {\em Quart. J. Math. Oxford, to appear}, 2017.
\bibitem{BMN10}
{H.~M. Bui, M.~B. Milinovich, and N.~C. Ng}, A note on the gaps between consecutive zeros of the {R}iemann
zeta-function, {\em Proc. Amer. Math. Soc.}, 138(12):4167--4175, 2010.
\bibitem{CGG84}
{J.~B. Conrey, A.~Ghosh, and S.~M. Gonek}, A note on gaps between zeros of the zeta function,
\newblock {\em Bull. London Math. Soc.}, 16(4):421--424, 1984.
\bibitem{Fujii2}
{A.~Fujii}, On the difference between {$r$} consecutive ordinates of the zeros of
the {R}iemann zeta function,
\newblock {\em Proc. Japan Acad.}, 51(10):741--743, 1975.
\bibitem{Fujii1}
{A.~Fujii}, On the distribution of the zeros of the {R}iemann zeta function in
short intervals,
\newblock {\em Bull. Amer. Math. Soc.}, 81:139--142, 1975.
\bibitem{FengWu}
{S.~Feng and X.~Wu}, On gaps between zeros of the {R}iemann zeta-function,
\newblock {\em J. Number Theory}, 132(7):1385--1397, 2012.
\bibitem{Hall}
{R.~R. Hall}, The behaviour of the {R}iemann zeta-function on the critical line,
\newblock {\em Mathematika}, 46(2):281--313, 1999.
\bibitem{MO}
{H.~L. Montgomery and A.~M. Odlyzko}, Gaps between zeros of the zeta function,
\newblock In {\em Topics in classical number theory, {V}ol. {I}, {II}
({B}udapest, 1981)}, volume~34 of {\em Colloq. Math. Soc. J\'anos Bolyai},
pages 1079--1106. North-Holland, Amsterdam, 1984.
\bibitem{Mue82}
{J.~Mueller}, On the difference between consecutive zeros of the {R}iemann zeta
function,
\newblock {\em J. Number Theory}, 14(3):327--331, 1982.
\bibitem{Preo}
{S.~Preobrazhenski\u\i}, A small improvement in the gaps between consecutive zeros of the
{R}iemann zeta-function,
\newblock {\em Res. Number Theory}, 2:Art. 28, 11, 2016.
\bibitem{Sel47}
{A.~Selberg}, The zeta-function and the {R}iemann hypothesis,
\newblock In {\em C. {R}. {D}ixi\`eme {C}ongr\`es {M}ath. {S}candinaves 1946},
pages 187--200. Jul. Gjellerups Forlag, Copenhagen, 1947.
\bibitem{SelCollected}
{A.~Selberg}, {\em Collected papers. {V}ol. {I}},
\newblock Springer-Verlag, Berlin, 1989.
\newblock With a foreword by K. Chandrasekharan.
\bibitem{Titchmarsh}
{E.~C. Titchmarsh}, {\em The theory of the {R}iemann zeta-function},
\newblock The Clarendon Press, Oxford University Press, New York, second
edition, 1986.
\newblock Edited and with a preface by D. R. Heath-Brown.
\end{thebibliography}
\end{document} | math | 24,097 |
\begin{document}
\begin{abstract}
We show that a pseudoeffective ${\mathbb R}$-divisor has numerical dimension 0 if it is numerically trivial on a subvariety with ample normal bundle. This implies that the cycle class of a curve with ample normal bundle is big, which gives an affirmative answer to a conjecture of Peternell. We also give other positivity properties of such subvarieties.
{\mathbf e}nd{abstract}
\title{On subvarieties with ample normal bundle}
\thispagestyle{empty}
A well-established principle in algebraic geometry is that geometric properties of an algebraic variety is reflected in the subvarieties which are in various senses `positively embedded' in it. The primary example is the hyperplane section in a projective embedding of the variety, which gives rise to the notion of an ample divisor. However, in higher codimension it is less clear what it should mean in general for a subvariety to be `ample'. In his book \cite{Har70}, Hartshorne considers several approaches, including the condition that the normal bundle of the subvariety should be an ample vector bundle. Even for divisors this condition is weaker than ampleness, in the sense that it is a condition that concerns a vector bundle on the subvariety itself, rather than a global condition on the ambient variety. Still the condition guarantees certain good properties of the subvariety, e.g., that its cycle class is nef. For a more global definition of ampleness, see \cite{Ott12}.
Our first result is the following theorem, which gives a positive answer to a question of Peternell \cite{Pet11}. The theorem essentially says that a pseudoeffective divisor which is numerically trivial on a subvariety with ample normal bundle is very far from being big.
\begin{theorem}\label{interiortheorem}
Let $X$ be a smooth projective variety over an algebraically closed field of characteristic 0 and let $Y$ be a smooth subvariety of dimension $>0$ with ample normal bundle. If $D$ is a pseudoeffective ${\mathbb R}$-divisor such that $D|_Y{\mathbf e}quiv 0$, then its numerical dimension $\nu(D)$ is 0. In particular, if $D$ is nef, then $D{\mathbf e}quiv 0$.
{\mathbf e}nd{theorem}
See Definition |ef{numericaldimension} for the precise definition of numerical dimension of a divisor. In particular, this implies that the Iitaka dimension $\kappa(D)$ is non-positive.
Combining this result with the duality theorem of Boucksom--Demailly--Paun--Peternell \cite{BDPP}, we prove the following result about curves with ample normal bundle. The first part of the theorem was also conjectured by Peternell \cite{OP04,Pet08,Pet11}.
\begin{theorem}\label{interiortheorem2}
Let $X$ be a smooth projective variety over ${\mathbb C}$, let $C$ be a smooth curve with ample normal bundle. Then the cycle class of $C$ is big, i.e., it lies in the interior of the cone of curves, $\overline{{ N}E}(X)$.
If in addition $C$ is strictly nef (i.e., $C\operatorname{cd}ot D>0$ for any irreducible divisor $D$), then the cycle class of $C$ lies in the interior of the cone of movable curves, $\overline{{\mathscr M}E}(X)$. In particular, $C$ is numerically equivalent to a ${\mathbb Q}$-linear combination of strongly movable curves.
{\mathbf e}nd{theorem}
Interestingly, Voisin \cite{Voi08} showed that the corresponding result is false for subvarieties of higher dimensions. More precisely, she gives an examples of smooth projective varieties in any dimension $\ge 4$, containing a codimension 2 subvariety with ample normal bundle, but whose class is in the boundary of the pseudoeffective cone. In these examples, the subvariety deforms in a family covering the ambient variety and the normal bundle is even globally generated.
The strictly nef assumption in the second part of the theorem is necessary. Indeed, take any smooth projective variety with a curve with ample normal bundle and blow up a point outside it. Then on the blow-up, the preimage of the curve, $C$, has ample normal bundle, but the exceptional divisor satisfies $E\operatorname{cd}ot C=0$, so $C$ lies in the boundary of the cone of movable curves.
On the other hand, the following theorem says that there can be at most finitely many prime divisors disjoint from $C$.
\begin{theorem}\label{finitelymany}Let $X$ be a smooth projective variety over ${\mathbb C}$ and let $Y\subseteqset X$ be a smooth subvariety of dimension at least one with ample normal bundle. Then $Y$ intersects all but finitely many prime divisors on $X$. In fact, the number of such divisors is less than the Picard number of $X$.
{\mathbf e}nd{theorem}
All of the above results remain valid if $Y$ is assumed to be locally complete intersection instead of smooth.
Thanks to Fr\'ed\'eric Campana and Burt Totaro for comments and useful discussions.
\section{Curves with positivity properties}
Let $X$ be a smooth projective variety. An ${\mathbb R}$-divisor is a finite sum $D=\sum \lambda_i D_i$ where $\lambda_i^{-1}n {\mathbb R}$ and each $D_i$ is an irreducible divisor in $X$. Write $N^1(X)=\operatorname{Pic}(X)\otimes {\mathbb R}/{\mathbf e}quiv$ for the N\'eron-Severi group of $X$, i.e., the ${\mathbb R}$-vector space of divisors modulo numerical equivalence. In $N^1(X)$ we define the effective cone ${\mathbf e}ff(X)$ as the cone spanned by effective divisors and the nef cone $\operatorname{Nef}(X)$ the cone of nef divisors, i.e., ${\mathbb R}$-divisors such that $D\operatorname{cd}ot C\ge 0$ for every curve $C$ on $X$. An ${\mathbb R}$-divisor is {\mathbf e}mph{pseudoeffective} if its class lies in the closure $\overline{{\mathbf e}ff}(X)$ of the effective cone.
We let $N_1(X)$ denote the vector space of 1-cycles modulo numerical equivalence, and ${ N}E(X)$ the cone spanned by curves on $X$. We call a cycle $\alpha^{-1}n N_1(X)$ {\mathbf e}mph{big} if it lies in the interior of ${ N}E(X)$. Inside ${ N}E(X)$, there is the subcone ${\mathscr M}E(X)$ spanned by curves that are {\mathbf e}mph{movable}. Here a curve $C$ is called movable if it is a member of a family of curves that dominates $X$. By definition, the cones $\operatorname{Nef}(X)$ and $\overline{{ N}E}(X)$ are dual with respect to the intersection pairing. A fundamental result of Boucksom--Demailly--Paun--Peternell\cite{BDPP}, states that for a smooth variety over ${\mathbb C}$, also the cones $\overline{{\mathscr M}E}(X)$ and $\overline{{\mathbf e}ff}(X)$ are dual, i.e., a divisor $D$ is pseudoeffective if and only if $D\operatorname{cd}ot C\ge 0$ for all movable curves $C$. Moreover, they show that $\overline{{\mathscr M}E}(X)$ coincides with the closure of the cone spanned by curves which are strongly movable, that is, 1-cycles of the form $f_*(H_1\cap \operatorname{cd}ots \cap H_{n-1})$, where the $H_i$ are very ample divisors on $X'$ and $f:X'\to X$ is birational.
\subseteqsection{Subvarieties with ample normal bundle}
Recall that a vector bundle ${\mathscr E}$ on a variety is {\mathbf e}mph{ample} if the line bundle ${\mathscr O}(1)$ is ample on ${\mathbb P}({\mathscr E})$. Here and throughout the paper we use the Grothendieck notation for projectivized bundles, i.e., ${\mathbb P}({\mathscr E})$ is the variety of hyperplanes in ${\mathscr E}$. If ${\mathscr E}$ is a vector bundle on a curve, ${\mathscr E}$ is ample if and only if every quotient line bundle of ${\mathscr E}$ has positive degree \cite{Harcurves}. We will mainly consider the case when ${\mathscr E}$ is the normal bundle $N_Y=(\mathcal I/\mathcal I^2)^*$, of a subvariety $Y\subseteqset X$, which is a vector bundle of rank equal to the codimension when $Y$ is smooth (or more generally locally complete intersection.)
Subvarieties with ample normal bundle share many interesting geometric properties with ample divisors (see e.g., \cite{Har70} or \cite{Laz04}). For example, for every coherent sheaf ${\mathscr F}$, the cohomology groups $H^i(X-Y,{\mathscr F})$ are finite-dimensional vector spaces for $i<\dim Y$ \cite{Har70}. Also, if $\dim Y\ge 1$, a result of Napier and Ramachandran \cite{NR98} says $^{-1}m(\pi_1(Y)\to \pi_1(X))$ has finite index in $\pi_1(X)$. A property which will be important for our purposes is the following:
\begin{proposition}\cite[Corollary 8.4.3]{Laz04}
Let $Y\subseteqset X$ be a subvariety with ample normal bundle, then $Y$ is nef, i.e., $Y\operatorname{cd}ot Z\ge 0$ for any subvariety with $\dim Y+\dim Z=\dim X$.
{\mathbf e}nd{proposition}
In his book \cite{Har70}, Hartshorne presented two of his influential conjectures about such subvarieties:
\begin{conja}[Hartshorne]\label{Hartshorne2}
Let $Y\subseteqset X$ be a smooth subvariety of $X$ such that the normal bundle $N_{Y}$ is
an ample vector bundle. Is it true that some multiple of $Y$ deforms (as a cycle) in a family covering
$X$?
{\mathbf e}nd{conja}
\begin{conjb}[Hartshorne]\label{Hartshorne1}
Let $Y,Z$ be smooth subvarieties of $X$ with such that the normal bundles $N_Y,N_Z$ are ample vector bundles. If $\dim Y+\dim Z\ge \dim X$, then $Y\cap Z\neq {\mathbf e}mptyset$.
{\mathbf e}nd{conjb}
It is known by results of \cite{FL81} that Conjecture A implies Conjecture B. Unfortunately, Fulton and Lazarsfeld also showed that Conjecture A is false in general when $\dim Y\ge 2$. Their counterexample is based on constructing a certain ample rank two vector
bundle on $Y={\mathbb P}^2$, so that no multiple of the zero-section moves in the total space of the bundle.
Given this, one asks whether it is still true that a {\mathbf e}mph{curve} $C$ with ample normal bundle
has a multiple that moves in $X$. This question is open in general, but is known when $X$ is a surface or when $g(C)\le 1$. Further evidence for this is given by the result of Campana and Flenner \cite{CF90}, which states that some multiple of the zero-section moves in the normal bundle. In particular, this implies that the example of Fulton and Lazarsfeld cannot be modified to the dimension 1 case.
Theorem |ef{interiortheorem2} can be viewed as a weak form of Hartshorne's Conjecture A. Indeed, since the cycle class of $C\subseteqset X$ is big, a multiple of it can be written as $h+e$ where $h$ is the class of a complete intersection of $n-1$ sufficiently ample divisors and $e$ is an effective 1-cycle. In particular, for any finite set of points in $X$, there is an effective cycle numerically equivalent to $mC$ which passes through them. Note also that if $C$ is in addition assumed to be strictly nef, Theorem |ef{interiortheorem2} implies that some integral multiple of $C$ is even equivalent to a sum of strongly movable curves in $X$.
\subseteqsection{Examples}
(i) If $D$ is a divisor with ample normal bundle, then $D$ is nef and big \cite{Voi08}. In particular, when $X$ is a surface, Theorems |ef{interiortheorem} and |ef{finitelymany} follow directly from the Hodge index theorem.
(ii) If $Y$ is a complete intersection, or more generally, a transverse intersection of subvarieties with ample normal bundle, then $N_Y$ is ample.
(iii) Any smooth subvariety of projective space has ample normal bundle, since it is the quotient of the tangent bundle $T_{{\mathbb P}^n}$ which is ample \cite{Laz04}.
(iv) If the normal bundle $N_C$ is sufficiently ample in the sense that $h^0(N_C)\ge \dim X-1$ and $h^1(N_C)=0$, then the curve $C$ itself moves in a family covering $X$. In this case it is known that $C$ is big by \cite[Theorem 4.11]{Pet11}. In particular, this holds when $C$ is rational or elliptic. In fact, a variety is rationally connected if and only if it contains a rational curve with ample normal bundle. In \cite{OP04}, Oguiso and Peternell give an analogous geometric characterization when $C$ is an elliptic curve in a threefold.
(v) If $C$ has genus $g\ge 2$, we can consider the embedding of $C$ in its Jacobian $\operatorname{Jac}(C)$. Here the normal bundle of $C$ is ample \cite{Laz04}. In this example, it is classically known that the cycle-class of $C$ is in the interior of the cone of curves of $\operatorname{Jac}(C)$. In fact, Poincare's formula gives that $C{\mathbf e}quiv \Gammarac1{(g-1)!}{\mathscr T}heta^{g-1},$ where ${\mathscr T}heta$ is the theta divisor of $\operatorname{Jac}(C)$, which is ample.
(vi) If $X$ is a homogenous manifold, then the ampleness of the normal bundle of a subvariety can often be interpreted geometrically. For example, $Y$ is non-degenerate. If $X$ is an abelian variety and $C$ is a curve, then $N_C$ is ample if and only if a translate of $C$ generates $X$ as a group \cite{Laz04}. If $X$ is a quadric, then by \cite[Theorem 1]{Ballico}, the normal bundle $N_{C}$ is ample if and only if $C$ is not a line. In general, a line in a homogeneous manifold has ample bundle if and only if $X={\mathbb P}^n$.
(vii) Bigness of the cycle class of $C$ has however no implications for the positivity of the normal bundle. Indeed, take any 3-fold with Picard number one containing a $(-1,-1)$ curve: then $N_C={\mathscr O}(-1)\oplus {\mathscr O}(-1)$, and $C$ is big because $\overline{{ N}E}(X)$ is 1-dimensional.
\section{Proof of Theorem |ef{interiortheorem} and |ef{interiortheorem2}}
\subseteqsection{Divisorial Zariski decomposition}
We briefly recall the divisorial Zariski decomposition introduced by Boucksom \cite{Bou04} and Nakayama \cite{Nak04}. Let $X$ be a smooth projective variety and let $D$ be a pseudoeffective ${\mathbb R}$-divisor. We define the {\mathbf e}mph{diminished base locus} of $D$ by $$\mathbf B_-(D)=\bigcup_{A} \mathbf B_{\mathbb R}(D+A),$$ where $A$ runs over all ample divisors and $\mathbf B_{\mathbb R}(D)=\bigcap\{\operatorname{Supp}(D') | D'\ge 0\mbox{ and } D'\sim_{\mathbb R} D\}$. By \cite[Theorem V.1.3]{Nak04}, $\mathbf B_-(D)$ is a countable union of closed subsets. Let $H$ be an ample line bundle on $X$. For each prime divisor ${\mathscr G}amma$ on $X$ define the coefficient
$$\sigma_{\mathscr G}amma(D)=\lim_{{\mathbf e}psilon\to 0^+}^{-1}nf\{\mbox{mult}_{\mathscr G}amma(D') | D'\sim_{\mathbb R} D+{\mathbf e}psilon H \mbox{ and }D'\ge 0\}
$$It was shown by Nakayama \cite[III.1.5]{Nak04} that these numbers do not depend on the choice of $H$ and that there are only finitely many prime divisors ${\mathscr G}amma$ such that $\sigma_{{\mathscr G}amma}(D)>0$. Following \cite{Nak04} we then define $N_\sigma(D)=\sum_{\mathscr G}amma \sigma_{\mathscr G}amma(D){\mathscr G}amma$ and $P_\sigma(D)=D-N_\sigma(D)$, and call $D=N_\sigma(D)+P_\sigma(D)$ the {\mathbf e}mph{divisorial Zariski decomposition} of $D$.
The main properties of this decomposition is captured by the following
\begin{proposition}\cite[III.1.4, III.1.9, V.1.3]{Nak04} Let $D$ be a pseudoeffective ${\mathbb R}$-divisor.
\begin{enumerate}[(i)]
^{-1}tem $N_\sigma(D)$ is effective and $\operatorname{Supp}(N_
\sigma(D))$ coincides with the divisorial part of $\mathbf B_-(D)$.
^{-1}tem $N_\sigma(D)=0$ when $D$ is nef.
^{-1}tem For all $m\ge 0$, $H^0(X,{\mathscr O}_X(\Gammaloor{mP_\sigma(D)}))\simeq H^0(X,{\mathscr O}_X(\Gammaloor{mD}))$
{\mathbf e}nd{enumerate}
{\mathbf e}nd{proposition}
\begin{definition}\label{numericaldimension}
Let $D$ be a pseudoeffective ${\mathbb R}$-divisor. For an ample divisor $H$ define $\nu(D,H)$ as the maximal non-negative integer $k$ such that
$$
\limsup_{m\to^{-1}nfty}\Gammarac{h^0(X,{\mathscr O}_X(\Gammaloor{mD}+H)}{m^k}>0
$$We define the {\mathbf e}mph{numerical dimension} $\nu(D)$ has the maximal value of $\nu(D,H)$ when $H$ varies over all ample divisors on $X$. (Although the paper \cite{BDPP} uses a different definition of $\nu(D)$, it is equivalent to ours by the main theorem in \cite{Leh11}.)
{\mathbf e}nd{definition}
\begin{lemma}\label{slowgrowth}\cite[Proposition V.2.7]{Nak04}
Let $X$ be a smooth projective variety and let $D$ be a pseudoeffective ${\mathbb R}$-divisor. Then $\nu(D)=0$ if and only if $D{\mathbf e}quiv N_\sigma(D)$.
{\mathbf e}nd{lemma}
Since this result is vital in the proof of Theorem 1, we give a proof in the case $D$ is a nef divisor. In fact, this special case is enough to prove the first part of Theorem 2. We will prove the following statement: If $H$ is a smooth very ample divisor, then $D{\mathbf e}quiv 0$ if and only if for all sufficiently large $k$, $\nu(D,kH)=0$. We'll use the observation that $D{\mathbf e}quiv 0$ if and only if $D|_H{\mathbf e}quiv 0$ (which comes from the fact that $D{\mathbf e}quiv 0$ if and only if $D\operatorname{cd}ot H^{n-1}=0$). By Fujiita's vanishing theorem, there is a $k_0$ such that $H^1(X,{\mathscr O}_X(mD+(k-1)H))=0$ for all $m\ge0$ and $k\ge k_0$. Consider now the restriction map
$$H^0(X,{\mathscr O}_X(mD+kH))\to H^0(H,{\mathscr O}_H(mD+kH)).$$By construction, this map is surjective for every $m\ge 0,k\ge k_0$, so in particular also $\nu(D|_H,kH|_H)=0$ for all $k\ge k_0$. By induction on the dimension, $D|_H{\mathbf e}quiv 0$ and hence also $D{\mathbf e}quiv 0$.
When $D$ is only pseudoeffective, essentially the same idea can be used, but a different vanishing theorem is required (cf. \cite{Nak04}).
\begin{lemma}\label{h0vanishing}
Let ${\mathscr E}$ be an ample vector bundle on a curve $C$ and let $d$ be an integer. Then there is an integer
$m_0=m_0(d)>0$ so that
$$
H^0(C,\operatorname{Sym}^m {\mathscr E}^* \otimes {\mathscr O}_C(L))=0
$$for all $m\ge m_0$, and all line bundles $L$ of degree $d$.
{\mathbf e}nd{lemma}
\begin{proof}
Let ${\mathbb P}({\mathscr E})$ denote the variety of hyperplanes in ${\mathscr E}$ with projection $\pi:Y\to C$. By the ampleness of ${\mathscr E}$, the line bundle ${\mathscr O}_{{\mathbb P}({\mathscr E})}(1)$ is ample on ${\mathbb P}({\mathscr E})$. Hence by Serre duality and the Leray spectral sequence,
\begin{eqnarray*}
H^0(C,\operatorname{Sym}^m {\mathscr E}^* \otimes {\mathscr O}_C(L))&=&H^1(C,{\mathscr O}_C(K_C-L)\otimes \operatorname{Sym}^m({\mathscr E}))\\
&=&H^1({{\mathbb P}({\mathscr E})},\pi^*(K_C-L)\otimes {\mathscr O}(m))=0{\mathbf e}nd{eqnarray*}
The last cohomology group vanishes for all $m\ge m_0$, where $m_0$ depends only on $d$ (e.g., by Fujita's vanishing theorem \cite{Laz04}).
{\mathbf e}nd{proof}
Note that proof uses the characteristic $0$ assumption in the isomorphism $(\operatorname{Sym}^m {\mathscr E}^*)^*=\operatorname{Sym} {\mathscr E}$.
\begin{lemma}\label{h0bounded}
Let $C\subseteqset X$ be a smooth curve with ample normal bundle and let $D$ be a pseudoeffective ${\mathbb R}$-divisor on $X$ such that $D\operatorname{cd}ot C=0$. Then for any ample divisor $H$, the function $h(t)= h^0(X,{\mathscr O}_X(\Gammaloor{tD}+H))$ is bounded.
{\mathbf e}nd{lemma}
\begin{proof}Let $I$ be the ideal sheaf of $C$ in $X$. Since $C$ is locally complete intersection, we have $I^{k}/I^{k+1}=\operatorname{Sym}^k N_C^*$. By taking global sections of the exact sequences
$$
0\to I^{k+1} (\Gammaloor{tD}+H)\to I^{k}(\Gammaloor{tD}+H) \to \operatorname{Sym}^k N_C^*\otimes {\mathscr O}_C(\Gammaloor{tD}+H)\to 0
$$for $k=0,1,\ldots$, we deduce that
\begin{equation*}\label{h0sum}
h^0(X,{\mathscr O}_X(\Gammaloor{tD}+H))\le \sum_{k=0}^^{-1}nfty h^0(C, \operatorname{Sym}^k N_C^* \otimes {\mathscr O}_C(\Gammaloor{tD}+H))
{\mathbf e}nd{equation*}Note that we have $\Gammaloor{tD}\operatorname{cd}ot C\le tD\operatorname{cd}ot C=0$. So in particular, $\deg {\mathscr O}_C(\Gammaloor{tD}+H)$ is bounded above by some constant $K>0$ depending only on $D$ and $H$.
By Lemma |ef{h0vanishing}, there is a $k_0\ge 1$ so that the cohomology groups on the right-hand side of {\mathbf e}qref{h0sum} vanish for $k\ge k_0$ and all $t$. In particular,
\begin{equation*}\label{h0sum2}
h^0(X,{\mathscr O}_X(\Gammaloor{tD}+H))\le \sum_{k=0}^{k_0} h^0(C, \operatorname{Sym}^k N_C^* \otimes {\mathscr O}_C(\Gammaloor{tD}+H))
{\mathbf e}nd{equation*}Moreover, as each of the terms on the right-and side are bounded above by a constant independent of $t$, we see that the same holds for $h^0(X,{\mathscr O}_X(\Gammaloor{tD}+H))$.
{\mathbf e}nd{proof}
With these results, we are now in position to prove Theorem 1 and 2.
\begin{proof}[Proof of Theorem |ef{interiortheorem}]It suffices to prove the theorem when $Y$ is a curve. Indeed, if $\dim Y\ge 2$ and $A_1,\dots,A_{\dim Y-1}$ are sufficiently general, smooth, ample divisors, then $C=Y\cap A_1\cap \operatorname{cd}ots \cap A_{\dim Y-1}$ will be a smooth curve and $D|_Y{\mathbf e}quiv 0$ if and only if $D\operatorname{cd}ot C=0$. Moreover, the normal bundle of $C$ is ample, because it is an extension of the ample vector bundles $N_{Y|X}|_C$ and $N_{C|Y}$ (see e.g., \cite[III.\S 1]{Har70}).
So suppose that $Y=C$ is a curve with ample normal bundle and let $D$ be a pseudoeffective ${\mathbb R}$-divisor such that $D\operatorname{cd}ot C=0$ and let $H$ be any ample divisor. By Lemma |ef{h0bounded}, we have that the dimensions of the cohomology groups $H^0(X,{\mathscr O}_X(\Gammaloor{tD}+H))$ are bounded above, so in particular $\nu(D)=0$. Moreover, if $D$ is nef, from the definition, $N_\sigma(D)=0$, so in particular $D{\mathbf e}quiv 0$.{\mathbf e}nd{proof}
\begin{proof}[Proof of Theorem |ef{interiortheorem2}]
Let $C$ be a curve with ample normal bundle. By definition, the cone of curves $\overline{{ N}E}(X)\subseteqset N_1(X)$ is dual to $\operatorname{Nef}(X)\subseteqset N^1(X)$. Hence, to show that the class of $C$ is in the interior of the cone of curves it suffices to show that if $D$ is a nef ${\mathbb R}$-divisor such that $D\operatorname{cd}ot C=0$, then $D{\mathbf e}quiv 0$. But this is exactly the first part of Theorem |ef{interiortheorem}.
Suppose now that $C$ is strictly nef (i.e., $C\operatorname{cd}ot D>0$ for all effective divisors $D$), we need to show that the class of $C$ is in the interior of the cone of {\mathbf e}mph{movable curves}, $\overline{{\mathscr M}E}(X)\subseteqset N_1(X)$. By \cite{BDPP}, the movable cone is dual to the pseudoeffective cone, so we need only check that $C\operatorname{cd}ot D>0$ for every pseudoeffective ${\mathbb R}$-divisor which is not numerically trivial. Let $D$ be a pseudoeffective ${\mathbb R}$-divisor such that $C\operatorname{cd}ot D=0$. By Lemma |ef{slowgrowth} and Lemma |ef{h0bounded}, we have that $D{\mathbf e}quiv N_\sigma(D)=\sum \sigma_{\mathscr G}amma {\mathscr G}amma$, so in particular also $N_\sigma(D)\operatorname{cd}ot C=0$, contradicting the strictly nefness of $C$.{\mathbf e}nd{proof}
\def\operatorname{mob}{\operatorname{mob}}
\begin{remark} The paper \cite{Ott12} presents a definition of ampleness for subschemes of arbitrary codimension, generalizing the usual notion for divisors. In short, a subscheme is defined to be {\mathbf e}mph{ample} if the exceptional divisor on the blow-up along the subscheme satisfies a certain partial positivity condition, namely that its asymptotic cohomology groups vanish in certain degrees (it is `$q$-ample' in the sense of \cite{Tot10}, with $q=\operatorname{codim} Y-1$). When $Y$ is smooth, or locally complete intersection, it is known that this condition implies that the normal bundle of $Y$ is ample and $Y$ is strictly nef \cite[Corollary 5.6]{Ott12}.
{\mathbf e}nd{remark}
\section{Proof of Theorem |ef{finitelymany}}Let $X$ be a smooth complex variety over ${\mathbb C}$ and let $Y$ be a smooth subvariety with ample normal bundle and let $D\subseteqset X$ be any effective divisor (reducible or non-reduced) such that $Y\cap D={\mathbf e}mptyset$. By Theorem |ef{interiortheorem}, $D$ must have numerical dimension 0, so in particular its Iitaka dimension $\kappa(D)$ is also 0. From this we have
\begin{lemma}\label{uniquelin}
Let $D$ be an effective divisor disjoint from $Y$. Then $H^0(X,{\mathscr O}_X(D))={\mathbb C}$, i.e., $D$ is the unique effective divisor in its linear equivalence class.
{\mathbf e}nd{lemma}
The following lemma is the essential ingredient in the proof of Theorem |ef{finitelymany}. The idea of using the Albanese variety was inspired by an argument used by Totaro \cite{Tot00}.
\begin{lemma}
Let $Y\subseteqset X$ be a smooth subvariety with ample normal bundle. Then the restriction map \begin{equation}\label{restrH1}
H^1(X,{\mathbb Q})\to H^1(Y,{\mathbb Q})
{\mathbf e}nd{equation}
is injective.{\mathbf e}nd{lemma}
\begin{proof}This essentially follows since a subvariety with ample normal bundle can not be contracted to a point by a non-constant morphism. Fix a base-point on $Y$ and consider the map of Albanese varieties
\begin{equation*}
\alpha: {\mathscr A}lb(Y)\to {\mathscr A}lb(X).
{\mathbf e}nd{equation*}If {\mathbf e}qref{restrH1} is not injective, then $\alpha$ is not surjective, i.e., the quotient abelian variety $B={\mathscr A}lb(X)/\alpha({\mathscr A}lb(Y))$ has positive dimension. Note that the composition$$Y\to X\to {\mathscr A}lb(X)\to B$$sends $Y$ to a point $b^{-1}n B$. Let $f$ be a non-constant holomorphic function in a neighbourhood of $b$, which vanishes at $b$. Note that the above composition pulls the function $f$ back to a global section of $H^0(Y,I_Y^m/I_Y^{m+1})$ for some $m>0$. But $I_Y^m/I_Y^{m+1}=\operatorname{Sym}^m N_C^*$ cannot have global sections if the normal bundle of $Y$ is ample.
{\mathbf e}nd{proof}
In particular, this implies that the map of abelian varieties
\begin{equation}\label{finiteker}
\operatorname{Pic}^0(X)\to \operatorname{Pic}^0(Y)
{\mathbf e}nd{equation}has finite kernel.
\begin{lemma}\label{uniquenum}
Suppose $D_1,D_2$ are numerically equivalent effective divisors whose supports are disjoint from $Y$. Then $D_1$ and $D_2$ are equal as divisors.{\mathbf e}nd{lemma}
\begin{proof}Suppose first that $D_1$ and $D_2$ are algebraically equivalent. By definition, the element $D_1-D_2$ defines an element of $\operatorname{Pic}^0(X)$. Note that $D_1-D_2$ restricts to $0$ in $\operatorname{Pic}^0(Y)$ (since both $D_1$ or $D_2$ are disjoint from $Y$). Since the kernel of {\mathbf e}qref{finiteker} is finite, this means that there is a positive integer $m>0$ such that $m(D_1-D_2)=0$ in $\operatorname{Pic}^0(X)$ and hence $mD_1$ and $mD_2$ are linearly equivalent. By Lemma |ef{uniquelin}, we have $mD_1=mD_2$, and also $D_1=D_2$.
If $D_1$ and $D_2$ are numerically equivalent, then by Matsusaka's theorem, there is an integer $m>0$ such that $mD_1$ and $mD_2$ are algebraically equivalent. Using the same argument again, we find that $D_1=D_2$.
{\mathbf e}nd{proof}
With this we can complete the proof of Theorem |ef{finitelymany}:
\begin{proof}[Proof of Theorem |ef{finitelymany}]After replacing $Y$ with an appropriate linear section in some projective embedding, we may assume that $Y$ is a smooth curve. We may also suppose that the Picard number $|ho$ is greater than 1, otherwise there is nothing to prove.
Now take any distinct $|ho$ prime divisors $D_1,\ldots,D_|ho$ disjoint from $Y$. Since $D_i\operatorname{cd}ot Y=0$ for $i=0,\ldots, |ho$, we see that the $D_i$ lie in a rational hyperplane in $N^1(X)$. Hence after re-ordering the $D_i$, there is a relation of the form
$$
m_1D_1+\operatorname{cd}ots+m_s D_s {\mathbf e}quiv m_{s+1} D_{s+1}+\operatorname{cd}ots+m_|ho D_|ho
$$where $m_i$ are non-negative integers. Now let $E$ (resp. $F$) denote the divisor on the left hand side (resp. right hand side) of this equation. Note that the supports of $E$ and $F$ are disjoint from $Y$, so by Lemma |ef{uniquenum} the divisors $E,F$ are equal. This contradicts the assumption that the components $D_1,\ldots,D_|ho$ are different.
{\mathbf e}nd{proof}
\begin{thebibliography}{-9}
\bibitem{Ballico}
E.~Ballico.
\newblock Normal bundle to curves in quadrics.
\newblock {{\mathbf e}m Bull. Soc. Math. France}, 109, (1981), 227--235.
\bibitem{Bou04}
S. Boucksom, Divisorial Zariski Decompositions on Compact Complex Manifolds. Ann. Scient. \'Ec. Norm. Sup., $4^e$ s\'erie, 37 (2004) 45--76.
\bibitem{BDPP}S. Boucksom, J.-P. Demailly, M. Paun, T. Peternell, The pseudo-effective cone of a compact K\"ahler manifold and varieties of negative Kodaira dimension. {{\mathbf e}m J. Algebraic Geom. 22} (2013), 201--248.
\bibitem{CF90}
F. Campana and H. Flenner, A characterization of ample vector bundles on a curve, {{\mathbf e}m Math. Ann.} 287 (1990), 571--575.
\bibitem{FL81}
W. Fulton, R. Lazarsfeld.
\newblock Positivity and excess intersection, in {{\mathbf e}m Enumerative and classical algebraic
geometry, Nice, 1981,
\newblock Progress in Math. 24, Birkhauser} (1982), 97--105.
\bibitem{Har70}
R.~Hartshorne.
\newblock {{\mathbf e}m Ample subvarieties of algebraic varieties}.
\newblock Lecture Notes in Mathematics, Vol. 156. Springer-Verlag, Berlin,
(1970).
\bibitem{Harcurves}
R.~Hartshorne.
\newblock {Ample vector bundles on curves}.
\newblock {{\mathbf e}m Nagaoya Math. J.} 43 (1971), 73--89.
\bibitem{Laz04}
R.~Lazarsfeld.
\newblock {{\mathbf e}m Positivity in algebraic geometry {I and II}}.
\newblock Springer-Verlag, Berlin, 2004.
\bibitem{Leh11}
B. Lehmann. Comparing numerical dimensions. To appear in J. of Alg. and Num. Theory.
\bibitem{Nak04} N. Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004.
\bibitem{NR98}
T.~Napier and M.~Ramachandran.
\newblock The {$L^2\ \overline \partial$}-method, weak {L}efschetz theorems,
and the topology of {K}\"ahler manifolds.
\newblock {{\mathbf e}m J. Amer. Math. Soc.}, 11 (1998) 375--396.
\bibitem{OP04}K. Oguiso, T. Peternell. The dual K¨ahler cone of compact K¨ahler threefolds. Comm.
Anal. Geom. 12 (2004) 1131--1154.
\bibitem{Ott12}J. C. Ottem. {Ample subvarieties and $q$-ample divisors}. {{\mathbf e}m Advances
in Mathematics} {229} (2012) 2868--2887.
\bibitem{Pet08}T. Peternell, Submanifolds with ample normal bundles and a conjecture of Hartshorne. {\mathbf e}mph{Interactions of classical and numerical algebraic geometry}, 317--330, Contemp. Math., 496, Amer. Math. Soc., Providence, RI, 2009.
\bibitem{Pet11} T. Peternell, Compact subvarieties with ample normal bundles, algebraicity, and cones of cycles. {{\mathbf e}m Michigan Math. J.} 61 (2012), no. 4, 875--889.
\bibitem{Tot10}
B.~Totaro.
\newblock Line bundles with partially vanishing cohomology.
\newblock {{\mathbf e}m J. Eur. Math. Soc. 15} (2013), 731-754.
\bibitem{Tot00}
B.~Totaro. The topology of smooth divisors and the arithmetic of abelian varieties. {{\mathbf e}m Michigan Math. J.} Volume 48, Issue 1 (2000), 611-624.
\bibitem{Voi08}C. Voisin. Coniveau 2 complete intersections and effective cones. {{\mathbf e}m Geom. Funct. Anal.} Vol. 19 (2010) 1494--1513
{\mathbf e}nd{thebibliography}
{\mathbf e}nd{document} | math | 30,464 |
\begin{document}
\title{Flips in Graphs}
\begin{abstract}
We study a problem motivated by a question related to quantum-error-correcting codes.
Combinatorially, it involves the following graph parameter:
$$f(G)=\min\set{|A|+|\{x\in V\setminus A : d_A(x)\text{ is odd}\}| : A\neq\varepsilonmptyset},$$
where $V$ is the vertex set of $G$ and $d_A(x)$ is the number of neighbors of~$x$ in~$A$.
We give asymptotically tight estimates of~$f$ for the random graph $G_{n,p}$ when $p$ is constant. Also,
if
$$f(n)=\max\set{f(G):\;|V(G)|=n}$$
then we show that $f(n)\lambdaeq (0.382+o(1))n$.
\varepsilonnd{abstract}
\section{Introduction}
In this paper we consider a problem which is motivated by a question from quantum-error-correcting codes. To see how to use graphs to construct quantum-error-correcting codes see, \textit{e.g.}, \cite{HDERNB, LYGG, YCO}.
Given a graph $G$ with $\pm1$ signs on vertices, each vertex can perform at most
one of the following three operations: $O_1$ (flip all of its neighbors, \textit{i.e.}, change
their signs), $O_2$ (flip itself), and $O_3$ (flip itself and all of its
neighbors). We want to start with all $+1$'s, execute some non-zero number of operations and return to all $+1$'s.
The \varepsilonmph{diagonal distance} $f(G)$ is the minimum
number of operations needed (with each vertex doing at most one operation).
Trivially,
\begin{equation}\lambdaabel{trivial}
f(G)\lambdae \delta(G)+1
\varepsilonnd{equation}
holds, where $\delta(G)$ denotes the minimum degree. Indeed, a vertex with the minimum degree applies $O_1$ and then its neighbors fix
themselves applying $O_2$.
Let $$f(n) = \max f(G),$$ where the maximum is taken over all non-empty graphs of order~$n$.
Shiang Yong Looi (personal communication) asked for a good approximation on $f(n)$.
In this paper we asymptotically determine the diagonal distance of the random graph $G_{n,p}$ for any $p\in(0,1)$.
We denote the \textit{symmetric difference} of two sets~$A$ and~$B$ by $A\bigtriangleup B$ and the \textit{logarithmic function} with base~e as~$\lambdaog$.
\begin{thm}\lambdaabel{thm:1}
There are absolute constants $\lambdaambda_0\alphapprox 0.189$ and $p_0\alphapprox 0.894$, see~\varepsilonqref{const:lambda} and~\varepsilonqref{const:p}, such that for
$G=G_{n,p}$ asymptotically almost surely:
\begin{enumerate}[(i)]
\item $f(G) = \delta(G)+1$ for $0<p<\lambdaambda_0$ or $p=o(1)$,
\item $|f(G) - \lambdaambda_0 n|=\tilde{O}(n^{1/2})$ for $\lambdaambda_0\lambdae p\lambdae p_0$,
\item $f(G)=2+\min_{x,y\in V(G)} \lambdaeft|\lambdaeft(N(x)\bigtriangleup N(y)\rhoight)\setminus\{x,y\}\rhoight|$ for $p_0<p<1$ or $p=1-o(1)$.
\varepsilonnd{enumerate}
\varepsilonnd{thm}
\noindent
(Here $\tilde{O}(n^{1/2})$ hides a polylog factor).
Figure~\rhoef{fig:p} visualizes the behavior of the diagonal distance of~$G_{n,p}$.
In addition to Theorem~\rhoef{thm:1} we find the following upper bound on~$f(n)$.
\begin{thm}\lambdaabel{thm:2}
$f(n)\lambdae (0.382+o(1))n$.
\varepsilonnd{thm}
\begin{figure}
\centering
\begin{picture}(0,0)
\includegraphics{f.pdf}
\varepsilonnd{picture}
\setlength{\unitlength}{4144sp}
\begingroup\makeatletter\ifx\SetFigFont\undefined
\gdef\SetFigFont#1#2#3#4#5{
\rhoeset@font\fontsize{#1}{#2pt}
\fontfamily{#3}\fontseries{#4}\fontshape{#5}
\selectfont}
\fi\varepsilonndgroup
\begin{picture}(3921,1411)(4816,-5690)
\put(8461,-5641){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\rhomdefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$p$}
}}}}
\put(8056,-5641){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\rhomdefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$1$}
}}}}
\put(7561,-5641){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\rhomdefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$p_0$}
}}}}
\put(5671,-5641){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\rhomdefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$\lambdaambda_0$}
}}}}
\put(5041,-5641){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\rhomdefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$0$}
}}}}
\put(4996,-4876){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\rhomdefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$\lambdaambda_0$}
}}}}
\put(4816,-4426){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\rhomdefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$\hat{f}(p)$}
}}}}
\varepsilonnd{picture}
\caption{The behavior of $\hat{f}(p)=\lambdaim_{n\to\infty}f(G_{n,p})/n$ as a function of~$p$.}\lambdaabel{fig:p}
\varepsilonnd{figure}
In the remainder of the paper we will use a more convenient restatement of~$f(G)$.
Observe that the order of execution of operations does not affect the final outcome.
For any $A\subset V=V(G)$,
let $B$ consist of those vertices in $V\setminus A$ that have odd number of
neighbors in $A$. Let $a=|A|$ and $b=|B|$. We want to minimize $a+b$ over all
non-empty $A\subset V(G)$.
The vertices of $A$ do an $O_1/O_3$ operation, depending on the even/odd parity of their neighborhood in~$A$.
The vertices in $B$ then do an $O_2$-operation to change back to $+1$.
\section{Random Graphs for $p=1/2$}\lambdaabel{sec:random}
Here we prove a special case of Theorem~\rhoef{thm:1} when $p=1/2$. This case is somewhat easier to handle.
Let $G=G_{n,1/2}$ be a binomial random graph. First we find a lower bound on $f(G)$. If we choose a non-empty
$A\subset V$ and then generate $G$, then the distribution of $b$ is binomial with parameters $n-a$ and $1/2$, which we denote here by
$Bin(n-a,1/2)$.
Hence, if $l$ is such that
\begin{equation}\lambdaabel{eq:a}
\sum_{a=1}^{l-1} \binom{n}{a}\Pr\lambdaeft(Bin(n-a,1/2)\lambdae l-1-a\rhoight) =o(1),
\varepsilonnd{equation}
then asymptotically almost surely the diagonal distance of $G$ is at least $l$.
Let $\lambdaambda=l/n$ and $\alphalpha=a/n$. We can approximate the summand in~\varepsilonqref{eq:a}
by
\begin{equation}\lambdaabel{eq:summand}
2^{n\lambdaeft(H(\alphalpha)+(1-\alphalpha)\lambdaeft(H\lambdaeft(\frac{\lambdaambda-\alphalpha}{1-\alphalpha}\rhoight)-1\rhoight) +O(\lambdaog n/n)\rhoight)},
\varepsilonnd{equation}
where $H$ is the binary entropy function defined as $H(p)=-p\lambdaog_2{p}-(1-p)\lambdaog_2(1-p)$. For more information about the entropy function and
its
properties see, \textit{e.g.},~\cite{AS}.
Let
\begin{equation}\lambdaabel{defg}
g_\lambda(\alphalpha) = H(\alphalpha)+(1-\alphalpha)\lambdaeft(H\lambdaeft(\frac{\lambdaambda-\alphalpha}{1-\alphalpha}\rhoight)-1\rhoight).
\varepsilonnd{equation}
The maximum of $g_\lambda(\alphalpha)$ is attained exactly for $\alphalpha=2\lambdaambda/3$, since
$$g_\lambda'(\alphalpha)=\lambdaog_2 \frac{2(\lambdaambda-\alphalpha)}{\alphalpha}.$$
Now the function
\begin{equation}\lambdaabel{defh}
h(\lambdaambda)=g_\lambda(2\lambda/3)
\varepsilonnd{equation}
is concave on $\lambdaambda\in[0,1]$ since
$$h''(\lambdaambda)=\frac{1}{(\lambdaambda-1)\lambdaambda\lambdaog 2}<0.$$
Moreover,
observe that $h(0)=-1$ and $h(1)=H(2/3)-1/3>0$. Thus the equation $h(\lambda)=0$ has a unique solution $\lambda_0$ and one can compute that
\begin{equation}\lambdaabel{const:lambda}
\lambdaambda_0 = 0.1892896249152306\lambdadots
\varepsilonnd{equation}
Therefore, if $\lambdaambda=\lambdaambda_0-K\lambdaog n/n$ for large enough $K>0$, then the left hand side of~\varepsilonqref{eq:a} goes to zero and similarly for
$\lambdaambda=\lambdaambda_0+K\lambdaog n/n$ it goes to infinity. In particular, $f(G)> (\lambdaambda_0-o(1))n$ asymptotically almost surely.
Let us show that this constant $\lambdaambda_0$ is best possible, \textit{i.e.}, asymptotically almost surely $f(G)<(\lambdaambda_0+K\lambdaog n/n)n$.
Let $\lambdaambda=\lambdaambda_0+K\lambdaog n/n$, $n$ be large, and $l=\lambdaambda n$. Let
$\alphalpha=2\lambdaambda/3$ and $a=\lambdafloor \alphalpha n\rhofloor$. We pick a random $a$-set $A\subset V$
and compute $b$. Let $X_A$ be an indicator random variable so that $X_A=1$ if and only if $b=b(A)\lambdae l-a$. Let $X=\sum_{|A|=a} X_A$.
We succeed if $X>0$.
The expectation $E(X)=\binom{n}{a} \Pr\lambdaeft(Bin(n-a,1/2)\lambdae l-a\rhoight)$ tends to infinity, by our choice of $\lambda$.
We now show that $X>0$ asymptotically almost surely by using the Chebyshev inequality.
First note that for $A\cap C \neq \varepsilonmptyset$ we have
$$Cov(X_A,X_C)=\Pr(X_A=X_C=1)-\Pr(X_A=1)\Pr(X_C=1)=0.$$
Indeed, if $x\in V\setminus (A\cup C)$,
then $\Pr(x\in B(A)|X_C=1)=1/2$, since $A\setminus C\neq\varepsilonmptyset$ and no adjacency between~$x$ and all vertices in~$A\setminus C$ is exposed
by
the event~$X_C=1$.
Similarly, if $x\in C\setminus A$, then $A\cap C\neq\varepsilonmptyset$ and an adjacency between~$x$ and $A\cap C$ is independent of the
occurrence of $X_C=1$. This implies that
$\Pr(x\in B(A)\mid X_C=1)=1/2$ as well. Thus
$\Pr(X_A=1|X_C=1)=\Pr\lambdaeft(Bin(n-a,1/2)\lambdae l-a\rhoight) = \Pr(X_A=1)$, and consequently, $Cov(X_A,X_C)=0$.
Now consider the case when $A\cap C=\varepsilonmptyset$. Let $s$ be a vertex in~$A$. Define a new indicator random variable $Y$ which takes the value~$1$
if
and only if $|B(C)\setminus\{s\}|\lambdae l-a$. Observe that
\begin{align*}
\Pr(Y=1)&=\Pr\lambdaeft(Bin(n-a-1,1/2)\lambdae l-a\rhoight)\\
&\lambdae 2\Pr\lambdaeft(Bin(n-a,1/2)\lambdae l-a\rhoight) = 2\Pr(X_A=1).
\varepsilonnd{align*}
Moreover,
$$\Pr(X_A=1|Y=1) = \Pr\lambdaeft(Bin(n-a,1/2)\lambdae l-a\rhoight) = \Pr(X_A=1),$$
since for every $x\in V\setminus A$ the adjacency between~$x$ and~$s$ is not influenced by~$Y=1$.
Finally note that $X_C\lambdae Y$. Thus,
\begin{align*}
Cov(X_A,X_C)&\lambdae \Pr(X_A=X_C=1)\lambdae \Pr(X_A=Y=1)\\
&= \Pr(Y=1)\Pr(X_A=1|Y=1)\lambdae 2\lambdaeft(\Pr(X_A=1)\rhoight)^2.
\varepsilonnd{align*}
Consequently,
\begin{align*}
Var(X) &= E(X) + \sum_{A\cap C\neq\varepsilonmptyset, A\neq C} Cov(X_A,X_C) + \sum_{A\cap C = \varepsilonmptyset} Cov(X_A,X_C)\\
&\lambdae E(X) + 2\sum_{A\cap C = \varepsilonmptyset} \lambdaeft(\Pr(X_A=1)\rhoight)^2\\
&= E(X) + 2\binom{n}{a}\binom{n-a}{a} \lambdaeft(\Pr(X_A=1)\rhoight)^2 = o\lambdaeft(E(X)^2\rhoight),
\varepsilonnd{align*}
as $E(X) = \binom{n}{a} \Pr(X_A=1)$ tends to infinity and $\binom{n-a}{a}=o\lambdaeft(\binom{n}{a}\rhoight)$. Hence, Chebyshev's inequality yields that
$X>0$ asymptotically almost surely.
\begin{rem}
A version of the well-known Gilbert-Varshamov bound (see, \textit{e.g.},~\cite{Lint}) states that if
\begin{equation}\lambdaabel{eq:gv}
2^{-n}\sum_{i=1}^{l-1} \binom{n}{i}3^i <1,
\varepsilonnd{equation}
then $f(n)\ge l$. Observe that this is consistent with bound~\varepsilonqref{eq:a}.
Let $\lambdaambda=l/n$. We can approximate the left hand side of~\varepsilonqref{eq:gv} by
$$
2^{n\lambdaeft( H(\lambdaambda) + \lambdaambda\lambdaog_2{3} -1 +o(1) \rhoight)}.
$$
One can check after some computation that
$$
H(\lambdaambda) + \lambdaambda\lambdaog_2{3} -1 = g_{\lambdaambda}(2\lambdaambda/3).
$$
Therefore, \varepsilonqref{eq:a} and~\varepsilonqref{eq:gv} give asymptotically the same lower bound on~$f(n)$.
\varepsilonnd{rem}
\section{Random Graphs for Arbitrary $p$}
Let $G=G_{n,p}$ be a random graph with constant $p\in (0,1)$.
Observe that for a fixed set~$A\subset V$, $|A|=a$, the probability that a vertex from $V\setminus A$ belongs to~$B(A)$ is
$$
p(a) = \sum_{0\lambdae i<\frac{a}{2}} \binom{a}{2i+1} p^{2i+1} (1-p)^{a-(2i+1)}
= \frac{1-(1-2p)^a}{2}.
$$
(If this is unfamiliar, expand $(1-2p)^n$ as $((1-p)-p)^n$ and compare).
\subsection{$0<p< \lambdaambda_0$}\lambdaabel{sec:case1}
For $p<\lambdaambda_0$ we begin with the upper bound $f(G)\lambdae \delta(G)+1$, see \varepsilonqref{trivial}. For the lower bound it is enough to show that
\begin{equation}\lambdaabel{small}
\sum_{2\lambdae a\lambdae pn} \binom{n}{a}\Pr\lambdaeft(Bin(n-a,p(a))\lambdae pn-a\rhoight) =o(1),
\varepsilonnd{equation}
since $\delta(G)+1 \lambdaeq np$
asymptotically almost surely. (We may assume that $p=\Omega\lambdaeft(\frac{\lambdaog n}{n}\rhoight)$; for otherwise $\delta(G)=0$ with high probability and the theorem is trivially true.) This implies that if $|A|+|B| \lambdae pn$, then $|A|=1$.
\subsubsection{$p$ Constant}
We split this sum into two sums for $2\lambdae a\lambdae \sqrt{n}$ and $\sqrt{n}<a\lambdae pn$, respectively.
Let $X=Bin(n-a,p(a))$ and
\begin{equation}\lambdaabel{defeps}
\varepsilonps =1-\frac{pn-a}{(n-a)p(a)} \ge 1-\frac{p}{p(2)} = 1-\frac{1}{2-2p}>0.
\varepsilonnd{equation}
Thus, by Chernoff's bound,
\begin{equation}\lambdaabel{Chernoff}
\Pr(Bin(N,\rho)\lambdaeq (1-\theta)N\rho)\lambdaeq e^{-\theta^2N\rho/2}
\varepsilonnd{equation}
we see that
\begin{align*}
\Pr\lambdaeft(Bin(n-a,p(a))\lambdae pn-a\rhoight) &= \Pr\lambdaeft( X\lambdae (1-\varepsilonps)E(X) \rhoight)\\
&\lambdae \varepsilonxp\{-\varepsilonps^2E(X)/2\}\\
&= \varepsilonxp\{ -\Theta(n)\},
\varepsilonnd{align*}
and consequently,
\begin{align*}
\sum_{2\lambdae a < \sqrt{n}} \binom{n}{a}\Pr\lambdaeft(Bin(n-a,p(a))\lambdae pn-a\rhoight) &\lambdae \sqrt{n} \binom{n}{\sqrt{n}} \varepsilonxp\{ -\Theta(n)\}\\
&\lambdae \varepsilonxp \{ O(\sqrt{n}\lambdaog n)\} \varepsilonxp\{-\Theta(n)\}\\
& =o(1).
\varepsilonnd{align*}
Now we bound the second sum corresponding to $\sqrt{n}<a\lambdae pn$. Note that
\begin{align*}\sum_{\sqrt{n}\lambdae a \lambdae pn} \binom{n}{a}&\Pr\lambdaeft(Bin(n-a,p(a))\lambdae pn-a\rhoight)\\
&=\sum_{\sqrt{n}\lambdae a \lambdae pn} \binom{n}{a}\Pr\lambdaeft(Bin\lambdaeft(n-a,\frac12+O(e^{-\Omega(n^{1/2})})\rhoight)\lambdae pn-a\rhoight)\\
&\lambdaeq n2^{nh(p)+o(1)}=o(1).
\varepsilonnd{align*}
Here $h$ is defined in \varepsilonqref{defh} and the right hand limit is zero since $p < \lambdaambda_0$.
\subsubsection{$p=o(1)$}\lambdaabel{sec:p_little_1}
We follow basically the same strategy as above and show that \varepsilonqref{small} holds for large $a$
and something similar when $a$ is small. Suppose then that $p=1/\omega$ where $\omega=\omega(n)\to\infty$.
First consider those $a$ for which $ap\geq 1/\omega^{1/2}$. In this case $p(a)\geq (1-e^{-2ap})/2$.
Thus,
\begin{multline*}
\sum_{\substack{ap\geq 1/\omega^{1/2}\\alpha\lambdaeq np}}\binom{n}{a} \Pr\lambdaeft(Bin(n-a,p(a))\lambdae pn-a\rhoight)\\
=\sum_{\substack{ap\geq 1/\omega^{1/2}\\alpha\lambdaeq np}}e^{O(n\lambdaog\omega/\omega)} e^{-\Omega(n/\omega^{1/2})}=o(1).
\varepsilonnd{multline*}
If $ap\lambdaeq 1/\omega^{1/2}$ then $p(a)= ap(1+O(ap))$.
Then
\begin{multline}\lambdaabel{a}
\sum_{\substack{ap<1/\omega^{1/2}\\2\lambdaeq a\lambdaeq np}}\binom{n}{a} \Pr\lambdaeft(Bin(n-a,p(a))\lambdae pn-a\rhoight)\\
\lambdaeq \sum_{\substack{ap<1/\omega^{1/2}\\2\lambdaeq a\lambdaeq np}}\brac{\frac{ne}{a}e^{-np/10}}^a=o(1)
\varepsilonnd{multline}
provided $np\geq 11\lambdaog n$.
If $np\lambdaeq \lambdaog n-\lambdaog\lambdaog n$ then $G=G_{n,p}$ has isolated vertices asymptotically almost surely
and then $f(G)=1$. So we are left with the case where $\lambdaog n-\lambdaog\lambdaog n\lambdaeq np\lambdaeq 11\lambdaog n$.
We next observe that if there is a set $A$ for which
$2\lambdaeq |A|$ and $|A|+|B(A)|\lambdaeq np$ then there is a minimal size such set. Let $H_A=(A,E_A)$ be a graph
with vertex set $A$ and an edge $(v,w)\in E_A$ if and only if $v,w$ have a common neighbor in $G$. $H_A$ must be
connected, else $A$ is not minimal. So we can find $t\lambdaeq a-1$ vertices $T$ such that $A\cup T$ spans
at least $t+a-1$ edges between~$A$ and~$T$. Thus we can replace the estimate \varepsilonqref{a} by
\begin{align*}
\sum_{\substack{ap<1/\omega^{1/2}\\2\lambdaeq a\lambdaeq np}}\sum_{t=1}^{a-1}&\binom{n}{a}\binom{n}{t}\binom{ta}{t+a-1}p^{t+a-1}
\Pr\lambdaeft(Bin(n-a-t,p(a))\lambdae pn-a\rhoight)\\
&\lambdaeq\sum_{\substack{ap<1/\omega^{1/2}\\2\lambdaeq a\lambdaeq np}}\sum_{t=1}^{a-1}\bfrac{ne}{a}^a\bfrac{ne}{t}^t\bfrac{taep}{t+a-1}^{t+a-1}e^{-anp/10}\\
&\lambdaeq \frac{1}{e^2np}\sum_{\substack{ap<1/\omega^{1/2}\\2\lambdaeq a\lambdaeq np}}a\brac{(e^2np)^2e^{-np/10}}^a=o(1).
\varepsilonnd{align*}
\subsection{$p_0 < p<1$}
First let us define the constant $p_0$. Let
\begin{equation}\lambdaabel{const:p}
p_0\alphapprox 0.8941512242051071\lambdadots
\varepsilonnd{equation}
be a root of $2p-2p^2=\lambdaambda_0$.
For the upper bound let $A=\{x,y\}$, where $x$ and $y$ satisfy $|N(x)\bigtriangleup N(y)| \lambdae |N(x')\bigtriangleup N(y')|$ for any $x',y'\in
V(G)$. Then $B=B(A) = N(x)\bigtriangleup N(y)$, and thus, asymptotically almost surely $|B|\lambdaeq (2p-2p^2)n$ plus a negligible error term~$o(n)$. (We may assume that $1-p=\Omega\lambdaeft(\frac{\lambdaog n}{n}\rhoight)$; for otherwise we have two vertices of degree~$n-1$ with high probability, and hence, $f(G)$=2.)
To show the lower bound it is enough to prove that
\begin{equation*}
\sum_{3\lambdae a\lambdae (2p-2p^2)n} \binom{n}{a}\Pr\lambdaeft(Bin(n-a,p(a))\lambdae (2p-2p^2)n-a\rhoight)=o(1).
\varepsilonnd{equation*}
Indeed, this implies that if $|A|+|B| \lambdae (2p-2p^2)n$, then $|A|=1$ or~$2$. But if $|A|=1$, then in a typical graph $|B| = (p+o(1))n >
(2p-2p^2)n$ since $p>1/2$.
\subsubsection{$p$ Constant}
As in the previous section we split the sum into two sums for $3\lambdae a\lambdae \sqrt{n}$ and $\sqrt{n}<a\lambdae pn$, respectively. Let
$$\varepsilonps=1-\frac{(2p-2p^2)n-a}{(n-a)p(a)}\geq 1-\frac{2p-2p^2}{p(a)} >0.$$
To confirm the second inequality we have to consider two cases. The first one is for~$a$ odd and at least~$3$. Here,
$$1-\frac{2p-2p^2}{p(a)} > 1-\frac{2p-2p^2}{1/2} = (2p-1)^2 > 0.$$
The second case, for~$a$ even and at least~$4$, gives
$$1-\frac{2p-2p^2}{p(a)} > 1-\frac{2p-2p^2}{p(2)} = 0.$$
Now one can apply Chernoff bounds with the given~$\varepsilonps$ to show that
$$
\sum_{3\lambdae a < \sqrt{n}} \binom{n}{a}\Pr\lambdaeft(Bin(n-a,p(a))\lambdae (2p-2p^2)n-a\rhoight) =o(1).
$$
Now we bound the second sum corresponding to $\sqrt{n}<a\lambdae (2p-2p^2)n$. Note that
\begin{align*}&\sum_{\sqrt{n}\lambdae a \lambdae (2p-2p^2)n} \binom{n}{a}\Pr\lambdaeft(Bin(n-a,p(a))\lambdae (2p-2p^2)n-a\rhoight)\\
&=\sum_{\sqrt{n}\lambdae a \lambdae (2p-2p^2)n} \binom{n}{a}\Pr\lambdaeft(Bin\lambdaeft(n-a,\frac12+O(e^{-\Omega(n^{1/2})})\rhoight)\lambdae (2p-2p^2)n-a\rhoight)\\
&\lambdaeq n2^{nh(2p-2p^2)+o(1)}=o(1)
\varepsilonnd{align*}
since $p>p_0$ implies that $2p-2p^2<\lambda_0$.
\subsubsection{$p=1-o(1)$}
One can check it by following the same strategy as above and in Section~\rhoef{sec:p_little_1}.
\subsection{$\lambdaambda_0 \lambdae p \lambdae p_0$}
Let $\alphalpha =2\lambdaambda_0/3$, $a=\lambdafloor \alphalpha n\rhofloor$. Fix an $a$-set $A\subset V$ and generate
our random graph and determine $B=B(A)$ with $b=|B|$. Let $\varepsilon=(\lambdaog n)^4/\sqrt{n}$ and let $X_A$ be the indicator
random variable for $a+b\lambdae (\lambdaambda_0+\varepsilon)n$ and $X=\sum_A X_A$.
Then
$$p(a)=\frac12+e^{-\Omega(n)}$$
and with $g_\lambda(\alphalpha)$ as defined in \varepsilonqref{defg},
\begin{equation}\lambdaabel{EX}
E(X)=\varepsilonxp\{(g_{\lambda_0+\varepsilonps}(2\lambda_0/3)+o(1))n\lambdaog 2\}.
\varepsilonnd{equation}
Now
\begin{eqnarray*}
g_{\lambda+\varepsilonps}(\alphalpha)&=&g_{\lambda}(\alphalpha)+(1-\alphalpha)\brac{H\bfrac{\lambda+\varepsilonps-\alpha}{1-\alpha}-H\bfrac{\lambda-\alpha}{1-\alpha}}\\
&=&g_{\lambda}(\alphalpha)+\varepsilonps \lambdaog_2\bfrac{1-\lambda}{\lambda-\alpha}+O(\varepsilon^2).
\varepsilonnd{eqnarray*}
Plugging this into \varepsilonqref{EX} with $\lambda=\lambda_0$ and $\alpha=2\lambda_0/3$ we see that
\begin{equation}\lambdaabel{EX2}
E(X)=\varepsilonxp\set{\brac{\varepsilonps\lambdaog_2\bfrac{1-\lambda_0}{\lambda_0/3}+O(\varepsilon^2)}n\lambdaog 2}=e^{\Omega((\lambdaog n)^4n^{1/2})}.
\varepsilonnd{equation}
Next, we estimate the variance of $X$. We will argue that for $A,C\in \binom{V}{a}$ either
$|A\bigtriangleup C|$ is small (but the number of such pairs is small) or
$|A\bigtriangleup C|$ is large (but then the covariance $Cov(X_A,X_C)$ is very
small since if we fix the adjacency of some vertex $x$ to $C$, then the parity of
$|N(x)\cap (A\setminus C)|$ is almost a fair coin flip). Formally,
$$
\begin{array}{rcl}
Var(X) &= E(X) &+ \quad\sum_{A\neq C} Cov(X_A, X_C) \\
&\lambdae E(X) &+ \quad\sum_{|A\bigtriangleup C|<2\sqrt{n}} \Pr(X_A=X_C=1)\\
& &+ \quad\sum_{|A\bigtriangleup C|\ge 2\sqrt{n}, |A\cap C|\ge \sqrt{n}} Cov(X_A, X_C)\\
& &+ \quad\sum_{|A\cap C|< \sqrt{n}} \Pr(X_A=X_C=1).
\varepsilonnd{array}
$$
Since $E(X)$ goes to infinity, clearly $E(X)=o(E(X)^2)$. We show in Claims~\rhoef{clm:1}, \rhoef{clm:2} and~\rhoef{clm:3} that the remaining part is also bounded by $o(E(X)^2)$.
Then
Chebyshev's inequality will imply that $X>0$ asymptotically almost surely.
\begin{clm}\lambdaabel{clm:1}
$\sum_{|A\bigtriangleup C|<2\sqrt{n}} \Pr(X_A=X_C=1)=o(E(X)^2)$
\varepsilonnd{clm}
\begin{proof}
We estimate trivially $\Pr(X_A=X_C=1)\lambdae \Pr(X_A=1)$. Then,
\begin{align*}
\sum_{|A\bigtriangleup C|<2\sqrt{n}}\Pr(X_A=1) &= \binom{n}{a} \sum_{0\lambdae i< \sqrt{n}} \binom{n-a}{i} \binom{a}{a-i} \Pr(X_A=1)\\
&= E(X) \sum_{0\lambdae i< \sqrt{n}} \binom{n-a}{i} \binom{a}{a-i} \\
&\lambdae E(X)\ 2^{O(\sqrt{n}\lambdaog n)}.
\varepsilonnd{align*}
Thus, \varepsilonqref{EX2} yields that $\sum_{|A\bigtriangleup C|<2\sqrt{n}} \Pr(X_A=X_C=1)=o(E(X)^2)$.
\varepsilonnd{proof}
\begin{clm}\lambdaabel{clm:2}
$\sum_{|A\bigtriangleup C|\ge 2\sqrt{n}, |A\cap C|\ge \sqrt{n}} Cov(X_A, X_C) = o(E(X)^2)$
\varepsilonnd{clm}
\begin{proof}
If $x\in V\setminus (A\cup C)$, then $\Pr(x\in B(A)|X_C=1)=2^{-1+o(1/n)}$,
since we can always find at least $\sqrt{n}$ vertices in~$A\setminus C$
with no adjacency with~$x$ determined by the event~$X_C=1$. Similarly, if $x\in C\setminus A$, then there are at
least $\sqrt{n}-1$ vertices in $A\cap C$ such that their adjacency with $x$ is independent of the occurrence of $X_C=1$. This implies that
$$\Pr(X_A=1|X_C=1)= \sum_{0\lambdae i\lambdae l-a} \binom{n-a}{i} 2^{-(n-a)+o(1)} = 2^{o(1)}\Pr(X_A=1),$$
and consequently, $Cov(X_A,X_C) = o\lambdaeft(\Pr(X_A=1)^2\rhoight)$. Hence,
\begin{align*}
\sum_{|A\bigtriangleup C|\ge 2\sqrt{n}, |A\cap C|\ge \sqrt{n}} Cov(X_A, X_C) \lambdae \binom{n}{a}^2 o\lambdaeft(\Pr(X_A=1)^2\rhoight) = o(E(X)^2).
\varepsilonnd{align*}
\varepsilonnd{proof}
\begin{clm}\lambdaabel{clm:3}
$\sum_{|A\cap C|< \sqrt{n}} \Pr(X_A=X_C=1) = o(E(X)^2)$
\varepsilonnd{clm}
\begin{proof}
First let us estimate the number of ordered pairs $(A,C)$ for which $|A\cap C|<\sqrt{n}$. Note,
\begin{align}\lambdaabel{eq:clm3:1}
\sum_{|A\cap C|<\sqrt{n}} {1} &= \binom{n}{a} \sum_{0\lambdae i<\sqrt{n}} \binom{n-a}{a-i}\binom{a}{i}\notag\\
&\lambdae \sqrt{n}\binom{n}{a}\binom{n-a}{a}\binom{a}{\sqrt{n}}\notag\\
&= 2^{n\lambdaeft(H(\alphalpha) + H\lambdaeft( \frac{\alphalpha}{1-\alphalpha} \rhoight)(1-\alphalpha) +o(1)\rhoight)}.
\varepsilonnd{align}
Now we will bound $\Pr(X_A=X_C=1)$ for fixed $a$-sets $A$ and~$C$. Let $S\subset A\setminus C$ be a set of size $s=|S|=\lambdafloor \sqrt{n} \rhofloor$.
Define a new indicator random variable $Y$ which takes the value $1$ if and only if $|B(C)\setminus S|\lambdae (\lambdaambda_0+\varepsilon)n-a$. Clearly,
$X_C\lambdaeq Y$ and
\begin{align*}
\Pr(Y=1) &= \Pr\lambdaeft(Bin(n-a-s, p(a))\lambdae (\lambdaambda_0+\varepsilon)n-a\rhoight)\\
&\lambdae 2^{s+o(1)} \sum_{0\lambdae i\lambdae (\lambdaambda_0+\varepsilon)n-a} \binom{n-a}{i}2^{-(n-a)}\\
&= 2^{s+o(1)} \Pr(X_A=1),
\varepsilonnd{align*}
Now if we condition on the existence or otherwise of all edges $F'$ between $C$ and $V\setminus S$
then if $x\in V \setminus A$
$$\Pr(x\in B(A)\mid F' \mbox{ and } F'') \in \lambdaeft[\frac{1-(1-2p)^{s}}{2}, \frac{1+(1-2p)^{s}}{2}\rhoight],$$
where $F''$ is the set of edges between~$x$ and $A\setminus S$.
This implies that
\begin{align*}
\Pr(X_A=1|Y=1) &= \sum_{0\lambdae i\lambdae (\lambdaambda_0+\varepsilon)n-a} \binom{n-a}{i}2^{-(n-a)+o(1)}\\
&= 2^{o(1)} \Pr(X_A=1),
\varepsilonnd{align*}
Consequently,
$$\Pr(X_A=X_C=1)\lambdae \Pr(X_A=Y=1)\lambdae 2^{\sqrt{n}+o(1)}\Pr(X_A=1)^2.$$
Hence, \varepsilonqref{eq:clm3:1} implies
$$
\sum_{|A\cap C|< \sqrt{n}} \Pr(X_A=X_C=1) \lambdae 2^{n\lambdaeft(H(\alphalpha) + H\lambdaeft( \frac{\alphalpha}{1-\alphalpha} \rhoight)(1-\alphalpha) +o(1)\rhoight)}
\Pr(X_A=1)^2.
$$
To complete the proof it is enough to note that
$$E(X)^2=2^{n\lambdaeft(2H(\alphalpha)+o(1)\rhoight)} \Pr(X_A=1)^2$$
and
$$2H(\alphalpha) > H(\alphalpha) + H\lambdaeft( \frac{\alphalpha}{1-\alphalpha}\rhoight)(1-\alphalpha).$$
Indeed, the last inequality follows from the strict concavity of the entropy
function, since then
$(1-\alphalpha)H\lambdaeft( \frac{\alphalpha}{1-\alphalpha}\rhoight) + \alphalpha H(0) \lambdae H(\alphalpha)$ with the equality for $\alphalpha=0$ only.
\varepsilonnd{proof}
Now we show that $f(G_{n,p})\ge (\lambdaambda_0-\varepsilon)n$.
We show that
$$
\sum_{1\lambdae a\lambdae (\lambdaambda_0-\varepsilon)n} \binom{n}{a} \Pr\lambdaeft(Bin(n-a,p(a))\lambdae (\lambdaambda_0-\varepsilon)n-a\rhoight)=o(1).
$$
As in previous sections we split this sum into two sums but this time we make the break into $1\lambdae a\lambdae (\lambdaog n)^2$ and $(\lambdaog n)^2<a\lambdae
(\lambdaambda_0-\varepsilon)n$, respectively.
In order to estimate the first sum we use the Chernoff bounds with deviation $1-\theta$ from the mean where
$$\theta=1-\frac{(\lambdaambda_0-\varepsilon)n-a}{(n-a)p(a)}\ge 1-\frac{\lambdaambda_0-\varepsilon}{p(a)} \ge 1-\frac{\lambdaambda_0-\varepsilon}{\lambdaambda_0} =\frac{\varepsilon}{\lambda_0}.$$
Consequently,
\begin{align*}
\sum_{2\lambdae a < (\lambdaog n)^2} \binom{n}{a}&\Pr\lambdaeft(Bin(n-a,p(a))\lambdae (\lambdaambda_0-\varepsilon)n-a\rhoight)\\
&\lambdae (\lambdaog n)^2 \binom{n}{(\lambdaog n)^2} \varepsilonxp\{ -\Omega(\lambdaog n)^4\}\\
&\lambdae \varepsilonxp \{ -\Omega(-(\lambdaog n)^4)\}=o(1).
\varepsilonnd{align*}
Now we bound the second sum corresponding to $\lambdaog\lambdaog n<a\lambdae (\lambdaambda_0-\varepsilon)n$.
\begin{multline*}
\sum_{\lambdaog\lambdaog n\lambdae a \lambdae (\lambdaambda_0-\varepsilon)n} \binom{n}{a}\Pr\lambdaeft(Bin(n-a,p(a))\lambdae (\lambdaambda_0-\varepsilon)n-a\rhoight)\\
=2^{n(h(\lambda_0-\varepsilonps)+o(1/n))}=o(1).
\varepsilonnd{multline*}
\section{General Graphs}\lambdaabel{GG}
Here we present the proof of Theorem~\rhoef{thm:2}. First, we prove a weaker result $f(n)\lambdae (0.440\lambdadots+o(1)) n$.
Suppose we aim at showing that $f(n)\lambdae \lambdaambda n$. We fix some $\alphalpha$ and
$\rhoho$ and let $a=\alphalpha n$ and $r=\rhoho n$. For each $a$-set $A$ let $R(A)$
consist of all sets that have Hamming distance at most~$r$ from~$B(A)$. If
\begin{equation}\lambdaabel{eq:vol}
\binom{n}{a}\sum_{i=0}^{r}\binom{n}{i}=2^{n(H(\alpha)+H(\rho)+o(1))}>2^n,
\varepsilonnd{equation}
then there are $A,A'$ such that
$R(A)\cap R(A')\ni C$ is non-empty. This means that $C$ is within Hamming distance $r$ from
both $B=B(A)$ and $B'=B(A')$. Thus $|B\bigtriangleup B'|\lambdae 2r$.
Let all vertices in $A''=A\bigtriangleup A'$ flip their neighbors, \textit{i.e.}, execute operation $O_1$.
The only vertices outside of~$A''$ that
can have an odd number of neighbors in $A''$ are restricted to
$(B\bigtriangleup B')\cup (A\cap A')$. Thus
\begin{equation}\lambdaabel{eq:h}
f(G)\lambdae |A\bigtriangleup A'|+|(B\bigtriangleup B')\cup (A\cap A')|\lambdae 2a+2r = 2n(\alphalpha+\rhoho).
\varepsilonnd{equation}
Consequently, we try to minimise $\alphalpha+\rhoho$ subject to $H(\alphalpha)+H(\rhoho)> 1$. Since
the entropy function is strictly concave, the optimum satisfies $\alphalpha=\rhoho$, otherwise
replacing each of $\alphalpha,\rhoho$ by $(\alphalpha+\rhoho)/2$ we strictly increase
$H(\alphalpha)+H(\rhoho)$
without changing the sum.
Hence, the optimum choice is
\begin{equation*}
\alphalpha=\rhoho\alphapprox 0.11002786443835959\lambdadots
\varepsilonnd{equation*}
the smaller root of
$H(x)=1/2$, proving that $f(n)\lambdae (0.440\lambdadots+o(1)) n$.
In order to obtain a better constant we modify the approach taken in~\varepsilonqref{eq:vol}.
Let us take $\delta=0.275$,
$\alphalpha=0.0535$, $a=\lambdafloor\alphalpha n\rhofloor$, $d=\lambdafloor \delta n\rhofloor$. Look
at the collection of sets $B(A)$, $A\in {[n]\choose a}$. This gives ${n\choose
a}= 2^{n(H(\alphalpha)+o(1))}$ binary $n$-vectors.
We claim that some two of these vectors are at distance at most $d$. If not,
then inequality $(5.4.1)$ in~\cite{Lint} says that
$$
H(\alphalpha)+o(1)\lambdae \min\{1+g(u^2)-g(u^2+2\delta u + 2\delta) : 0\lambdae u\lambdae
1-2\delta\},
$$
where $g(x)=H((1-\sqrt{1-x})/2)$.
In particular, if we take $u=1-2\delta=0.45$, we get $0.30108+o(1)\lambdae
0.30103$, a contradiction.
Thus, we can find two different $a$-sets $A$ and $A'$ such that
$|B(A)\bigtriangleup B(A')|\lambdae d$. As in~\varepsilonqref{eq:h}, we can conclude
that $f(G)\lambdae 2a+d\lambdae (0.382+o(1))n$.
\section{Acknowledgment}
The authors would like to thank Shiang Yong Looi for suggesting this problem.
\begin{thebibliography}{1}
\bibitem{AS} N. Alon and J. Spencer,
{\varepsilonm The Probabilistic Method},
third ed., Wiley, New York,~2008.
\bibitem{HDERNB} M.~Hein, W.~D{\"u}r, J.~Eisert, R.~Raussendorf, M.~van den~Nest, H.~J.~Briegel,
{\varepsilonm Entanglement in graph states and its applications},
E-print \texttt{arXiv:quant-ph/0602096}, Version 1, 2006.
\bibitem{Lint} J.~H.~van~Lint,
{\varepsilonm Introduction to Coding Theory},
third ed., Springer-Verlag,~1999.
\bibitem{LYGG} S.~Y. Looi, L.~Yu, V.~Gheorghiu, and R.~B. Griffiths,
{\varepsilonm Quantum error-correcting codes using qudit graph states},
E-print \texttt{arXiv.org:0712.1979}, Version 4, 2008.
\bibitem{YCO} S.~Yu, Q.~Chen, C.~H.~Oh,
{\varepsilonm Graphical quantum error-correcting codes},
E-print \texttt{arXiv:0709.1780v1}, Version 1, 2007.
\varepsilonnd{thebibliography}
\varepsilonnd{document} | math | 30,152 |
\begin{document}
\begin{abstract}
We prove the existence of a minimal action nodal solution for the quadratic Choquard equation
\begin{equation*}
-\Delta u + u = \bigl(I_\alpha \ast \abs{u}^2\bigr)u \quad\text{in \({\mathbb R}^N\)},
\end{equation*}
where $I_\alpha$ is the Riesz potential of order $\alpha\in(0,N)$.
The solution is constructed as the limit of minimal action nodal solutions for the
nonlinear Choquard equations
\begin{equation*}
-\Delta u + u = \bigl(I_\alpha \ast \abs{u}^p\bigr)|u|^{p-2}u \quad\text{in \({\mathbb R}^N\)}
\end{equation*}
when $p\searrow 2$. The existence of minimal action nodal solutions for $p>2$ can be proved
using a variational minimax procedure over Nehari nodal set.
No minimal action nodal solutions exist when $p<2$.
\end{abstract}
\maketitle
\section{Introduction}
We study least action nodal solutions of the quadratic Choquard equation
\begin{equation}
\label{eqChoquard}
\tag{$\mathcal{C}_2$}
-\Delta u + u = \bigl(I_\alpha \ast \abs{u}^2\bigr) u \quad\text{in \({\mathbb R}^N\)},
\end{equation}
for \(N \in {\mathbb N}\) and \(\alpha \in (0, N)\). Here \(I_\alpha : {\mathbb R}^N \to {\mathbb R}\) is the Riesz potential
defined for each \(x \in {\mathbb R}^N \setminus \{0\}\) by
\begin{align*}
I_\alpha (x) &= \frac{A_\alpha}{\abs{x}^{N - \alpha}}, &
&\text{where }
&A_\alpha = \frac{\Gamma(\tfrac{N-\alpha}{2})}
{\Gamma(\tfrac{\alpha}{2})\pi^{N/2}2^{\alpha} }.
\end{align*}
For \(N = 3\) and \(\alpha = 2\) equation \eqref{eqChoquard} is the \emph{Choquard--Pekar equation}
which goes back to the 1954's work by S.\thinspace I.\thinspace Pekar on quantum theory of a polaron at rest \citelist{\cite{Pekar1954}\cite{DevreeseAlexandrov2009}*{Section 2.1}}
and to 1976's model of P.\thinspace Choquard of an electron trapped in its own hole, in an approximation to Hartree-Fock theory of one-component plasma \cite{Lieb1977}. In the 1990's the same equation reemerged as a model of self-gravitating matter \citelist{\cite{KRWJones1995newtonian}\cite{Moroz-Penrose-Tod-1998}}
and is known in that context as the \emph{Schr\"odinger--Newton equation}.
Equation \eqref{eqChoquard} is also studied as a nonrelativistic model of boson stars \citelist{\cite{Frohlich-Lenzmann}\cite{Lenzmann-2009}}.
Mathematically, the existence and some qualitative properties of solutions of Choquard equation \eqref{eqChoquard}
have been studied by variational methods in the early 1980s,
see \citelist{\cite{Lieb1977}\cite{Lions1980}\cite{Menzala1980}\cite{Lions1984-1}*{Chapter III}} for earlier work on the problem.
Recently, nonlinear Choquard equation
\begin{equation}
\label{eqChoquard-p}
\tag{$\mathcal{C}_p$}
-\Delta u + u = \bigl(I_\alpha \ast \abs{u}^p\bigr) \abs{u}^{p - 2} u \quad\text{in \({\mathbb R}^N\)},
\end{equation}
with a parameter $p>1$ attracted interest of mathematicians, see
\citelist{\cite{Ma-Zhao-2010}\cite{CingolaniClappSecchi2012}\cite{CingolaniSecchi}\cite{ClappSalazar}\cite{MorozVanSchaftingen13}\cite{Ghimenti-soliton}} and further references therein; see also \citelist{\cite{CingolaniSecchi-2015}\cite{fractionalChoquard}} for modifications of \eqref{eqChoquard-p} involving fractional Laplacian.
Most of the recent works on Choquard type equations \eqref{eqChoquard-p} so far were dedicated to the study of positive solutions.
Nodal solutions were studied in \citelist{\cite{CingolaniClappSecchi2012}\cite{CingolaniSecchi}\cite{ClappSalazar}\cite{GhimentiVanSchaftingen}\cite{Weth2001}*{theorem 9.5}}.
Equation \eqref{eqChoquard-p} is the Euler equation of the Choquard action functional \(\mathcal{A}_p\)
which is defined for each function \(u\) in the Sobolev space
\(H^1 ({\mathbb R}^N)\) by
\[
\mathcal{A}_p (u) = \frac{1}{2} \int_{{\mathbb R}^N} \abs{\nabla u}^2 + \abs{u}^2 - \frac{1}{2 p} \int_{{\mathbb R}^N} \bigl(I_\alpha \ast \abs{u}^p\bigr) \abs{u}^p.
\]
By the Hardy--Littlewood--Sobolev inequality, if \(s
\in (1, \frac{N}{\alpha})\) then for
every \(v \in L^s ({\mathbb R}^N)\), \(I_\alpha \ast v\in L^\frac{N s}{N - \alpha s}
({\mathbb R}^N)\) and
\begin{equation}
\label{eqHLS}
\int_{{\mathbb R}^N} \abs{I_\alpha \ast v}^\frac{N s}{N - \alpha s} \le C
\Bigl(\int_{{\mathbb R}^N} \abs{v}^s \Bigr)^\frac{N}{N - \alpha s},
\end{equation} (see for example \cite{LiebLoss2001}*{theorem 4.3}).
In view of the classical Sobolev embedding, the action functional \(\mathcal{A}_p\) is
well-defined and continuously differentiable if and only if
\[
\frac{N - 2}{N + \alpha} \le \frac{1}{p} \le \frac{N}{N + \alpha}.
\]
By testing equation~\eqref{eqChoquard-p} against \(u\), the natural
\emph{Nehari constraint}
$\dualprod{\mathcal{A}' (u)}{u} = 0$ appears.
Then positive solutions of \eqref{eqChoquard-p} can be obtained by studying the infimum
\[
c_{0, p} = \inf\, \bigl\{\mathcal{A}_p (u) \;:\; u \in \mathcal{N}_{0, p}\bigr\}
\]
over the \emph{Nehari manifold}
\[
\mathcal{N}_{0, p} = \bigl\{u \in H^1 ({\mathbb R}^N) \setminus \{0\} \;:\; \dualprod{\mathcal{A}_p' (u)}{u} = 0\}.
\]
It turns out that the infimum $c_{0, p}$ is achieved when
\[
\frac{N - 2}{N + \alpha} < \frac{1}{p} < \frac{N}{N + \alpha},
\]
and these assumptions are optimal \citelist{\cite{Lieb1977}\cite{Lions1980}\cite{MorozVanSchaftingen13}}.
Besides positive solutions minimising $c_{0, p}$, which are known as {\em groundstates} or {\em least action solutions},
additional solutions can be constructed by several variational construction.
In particular, one can consider \emph{least action nodal solutions}, the sign-changing counterpart of least actions solutions.
One way to search for such solutions is to consider the infimum
\[
c_{\mathrm{nod}, p} = \inf\, \bigl\{\mathcal{A}_p (u) \;:\; u \in \mathcal{N}_{0, p}\bigr\}
\]
over the \emph{Nehari nodal set}
\begin{multline*}
\mathcal{N}_{\mathrm{nod}, p}
=\bigl\{ u \in H^1 ({\mathbb R}^N) \;:\; u^+ \ne 0 \ne u^-,\,\\
\dualprod{\mathcal{A}_p'(u)}{u^+} = 0
\text{ and } \dualprod{\mathcal{A}_p'(u)}{u^-} = 0\bigr\},
\end{multline*}
where \(u = u^+ - u^-\).
Such construction has been performed for local elliptic problems on bounded domains of \({\mathbb R}^N\),
see \citelist{\cite{CeramiSoliminiStruwe1986}\cite{CastroCossioNeuberger1997}\cite{CastorCossioNeuberger1998}},
whereas the approach fails for the nonlinear Schr\"odinger equation
\begin{equation}\label{e-NLS}
-\Delta u+u=\abs{u}^{2p - 2} u \quad\text{in \({\mathbb R}^N\)},
\end{equation}
which has no least action nodal solutions \citelist{\cite{AckermannWeth2005}*{lemma 2.4}\cite{Weth2006}\cite{GhimentiVanSchaftingen}}.
Moreover, the least action energy on the Nehari nodal set is not approximated by nodal solutions of \eqref{e-NLS},
see \cite{Weth2006}*{theorem 1.5}.
Surprisingly, it has been proved that unlike its local counterpart \eqref{e-NLS},
the nonlocal Choquard equation \eqref{eqChoquard-p} admits least action nodal solutions when
\[
\frac{N - 2}{N + \alpha} < \frac{1}{p} < \frac{1}{2}
\]
\cite{GhimentiVanSchaftingen}*{theorem 2}; while the infimum $c_{\mathrm{nod}, p}$ is not achieved when
\[
\frac{1}{2} < \frac{1}{p} < \frac{N}{N + \alpha},
\]
because \(c_{\mathrm{nod}, p} = c_{0, p}\) \cite{GhimentiVanSchaftingen}*{theorem 3}.
The borderline quadratic case \(p = 2\) was not covered by either existence or non-existence proofs in \cite{GhimentiVanSchaftingen}, because of the possible degeneracy of the minimax reformulation of the problem that was introduced (see \cite{GhimentiVanSchaftingen}*{eq.~(3.3)})
and of difficulties in controlling the norms of the positive and negative parts of Palais--Smale sequences.
The goal of the present work is to study the existence of least action nodal solutions for Choquard equation \eqref{eqChoquard-p}
in the physically most relevant quadratic case \(p = 2\). Because the minimax procedure for capturing $c_{\mathrm{nod}, p}$ introduced in \cite{GhimentiVanSchaftingen} apparently fails for $p=2$, a different approach is needed.
Instead of directly minimizing $c_{\mathrm{nod}, 2}$, our strategy will be to employ Choquard equations \eqref{eqChoquard-p} with $p>2$ as a regularisation family for the quadratic Choquard equation \eqref{eqChoquard} and to pass to the limit when $p\searrow 2$.
Our main result is the following.
\begin{theorem}
\label{theoremMain}
If \(N \in \mathbb N\) and \(\alpha \in ((N - 4)^+, N)\),
then there exists a weak solution \(u \in H^1 ({\mathbb R}^N)\) of Choquard equation \eqref{eqChoquard}
such that
\(u^+ \ne 0 \ne u^-\) and \(\mathcal{A}_2 (u) = c_{\mathrm{nod},2}\).
\end{theorem}
The constructed nodal solution $u$ is regular, that is, $u\in L^1({\mathbb R}^N)\cap C^2({\mathbb R}^N)$, see \cite{MorozVanSchaftingen13}*{proposition 4.1}.
The conditions of the theorem are optimal, as for \(\alpha\not\in ((N - 4)^+, N)\) no sufficiently regular solutions to \eqref{eqChoquard} exist in $H^1 ({\mathbb R}^N)$ by the Poho\v zaev identity \cite{MorozVanSchaftingen13}*{theorem 2}.
In order to prove theorem \eqref{theoremMain}, we will approximate a least action nodal solution of quadratic
Choquard equation \eqref{eqChoquard} by renormalised least action nodal solution of \eqref{eqChoquard-p} with $p\searrow 2$.
To do this, in section~\ref{section2} we first establish continuity of the energy level \(c_{0, p}\) with respect to $p$.
Then in section~\ref{section3} we prove theorem \ref{theoremMain} by showing that as \(p \searrow 2\), positive and negative parts
of the renormalised least action nodal solutions of \eqref{eqChoquard-p} do not vanish and do not diverge apart from each other.
\section{Continuity of the critical levels}
\label{section2}
In the course of the proof of theorem~\ref{theoremMain}, we will need the following strict inequality on critical levels.
\begin{proposition}
\label{propositionStrictInequality}
If \(\frac{N - 2}{N + \alpha} < \frac{1}{p} < \frac{N}{N + \alpha}\), then
\[
c_{\mathrm{nod}, p} < 2 c_{0, p}.
\]
\end{proposition}
This follows directly from \cite{GhimentiVanSchaftingen}*{propositions 2.4 and 3.7}.
The construction in the latter reference is done by taking translated pa ositive and a negative copy of the groundstate
of \eqref{eqChoquard-p} and by estimating carefully the balance between the truncation and the effect of the Riesz potential interaction. When \(p < 2\), it is known that \(c_{\mathrm{nod}, p} = c_{0, p}\) \cite{GhimentiVanSchaftingen}*{theorem 3} and proposition~\ref{propositionStrictInequality} loses its interest.
Because we shall approximate the quadratic case \(p = 2\) by \(p > 2\), it will be useful to have some information about the continuity of \(c_{0, p}\).
\begin{proposition}
\label{continuityGroundstate}
The function
\(
p \in (\frac{N + \alpha}{N}, \frac{N + \alpha}{(N - 2)_+}) \mapsto c_{0, p} \in {\mathbb R}
\)
is continuous.
\end{proposition}
\begin{proof}
It can be observed that
\[
c_{0, p} = \inf \Biggl\{ \Bigl(\frac{1}{2} - \frac{1}{2 p}\Bigr) \Biggl(\frac{\Bigl(\displaystyle \int_{{\mathbb R}^N} \abs{\nabla u}^2 + \abs{u}^2\Bigr)^\frac{p}{p - 1}}{\Bigl(\displaystyle \int_{{\mathbb R}^N} \bigl(I_\alpha \ast \abs{u}^p\bigr) \abs{u}^p\Bigr)^\frac{1}{p - 1}} \Biggr) \;:\; u \in H^1 ({\mathbb R}^N) \setminus \{0\} \Biggr\}.
\]
Since for every \(u \in H^1 ({\mathbb R}^N)\), the function
\[
p \in (\tfrac{N + \alpha}{N}, \tfrac{N + \alpha}{(N - 2)_+}) \mapsto \int_{{\mathbb R}^N} \bigl(I_\alpha \ast \abs{u}^p\bigr) \abs{u}^p
\]
is continuous, the function \(p \mapsto c_{0, p}\) is then upper semicontinuous as an infimum of continuous functions.
We now consider the more delicate question of the lower semicontinuity. There exists a family of functions \(u_p \in H^1 ({\mathbb R}^N)\) such that \eqref{eqChoquard-p} holds
and \(\mathcal{A}_p (u_p) = c_{0, p}\).
By the upper semicontinuity, it follows that the function \(p \in (\frac{N + \alpha}{N}, \frac{N + \alpha}{(N - 2)_+}) \mapsto u_p\in H^1 ({\mathbb R}^N)\) is locally bounded.
In view of the equation \eqref{eqChoquard-p} and of the Hardy--Littlewood--Sobolev inequality,
we have
\[
\int_{{\mathbb R}^N} \abs{\nabla u_p}^2 + \abs{u_p}^2
= \int_{{\mathbb R}^N} \bigl(I_\alpha \ast \abs{u_p}^p\bigr) \abs{u_p}^p
\le \refstepcounter{cte} C_{\thecte} \Bigl(\int_{{\mathbb R}^N} \abs{u_p}^\frac{2 N p}{N + \alpha}\Bigr)^\frac{N + \alpha}{N},
\]
where constant $C_1$ could be chosen uniformly bounded when \(p\) remains in a compact subset of \((\frac{N + \alpha}{N}, \frac{N + \alpha}{(N - 2)_+})\).
This implies that \citelist{\cite{Lions1984CC2}*{lemma I.1}\cite{Willem1996}*{lemma
1.21}\cite{MorozVanSchaftingen13}*{lemma 2.3}\cite{VanSchaftingen2014}*{(2.4)}}
\begin{multline*}
\int_{{\mathbb R}^N} \abs{\nabla u_p}^2 + \abs{u_p}^2\\
\le \refstepcounter{cte} C_{\thecte} \Bigl(\int_{{\mathbb R}^N} \abs{\nabla u_p}^2 + \abs{u_p}^2 \Bigr)^\frac{N + \alpha}{N} \Bigl( \sup_{a \in {\mathbb R}^N} \int_{B_1 (a)} \abs{u_p}^\frac{2 N p}{N + \alpha} \Bigr)^{(N + \alpha)\bigl(\frac{1}{N} - \frac{N + \alpha}{p} \bigr)},
\end{multline*}
where \(C_{\thecte}\) can be also chosen uniformly bounded when \(p\) is in a compact subset of \((\frac{N + \alpha}{N}, \frac{N + \alpha}{(N - 2)_+})\).
Up to a translation in \({\mathbb R}^N\), we can thus assume that the function
\[
p \in (\tfrac{N + \alpha}{N}, \tfrac{N + \alpha}{(N - 2)_+}) \mapsto \int_{B_1} \abs{u}^\frac{2 N p}{N + \alpha}
\]
is locally bounded away from \(0\).
We assume now \((p_n)_{n \in {\mathbb N}}\) to be a sequence in the interval \((\tfrac{N + \alpha}{N}, \tfrac{N + \alpha}{(N - 2)_+})\) that converges to \(p_* \in (\tfrac{N + \alpha}{N}, \tfrac{N + \alpha}{(N - 2)_+})\). Since the sequence \((u_{p_n})_{n \in {\mathbb N}}\) is bounded in the space \(H^1 ({\mathbb R}^N)\), there exists a sequence \((n_k)_{k \in {\mathbb N}}\) diverging to infinity and \(u_* \in H^1 ({\mathbb R}^N)\) such that the subsequence \((u_{p_{n_k}})_{k \in {\mathbb N}}\) converges weakly in \(H^1 ({\mathbb R}^N)\) to \(u_*\).
Moreover, we have by the Rellich--Kondrachov compactness theorem
\[
\int_{B_1} \abs{u_*}^\frac{2 N p_*}{N + \alpha}
= \lim_{k \to \infty} \int_{B_1} \abs{u_{p_{n_k}}}^\frac{2 N p}{N + \alpha} > 0.
\]
Thus \(u_* \ne 0\) and \(u_*\) satisfies
\[
-\Delta u_* + u_* = \bigl(I_\alpha \ast \abs{u}^{p_*}\bigr) \abs{u}^{p_* - 2} u.
\]
We have thus
\[
\begin{split}
c_{0, p_*} \le \mathcal{A}_{p_*} (u_*)
&= \Bigl(\frac{1}{2} - \frac{1}{2 p_*}\Bigr)
\int_{{\mathbb R}^N} \abs{\nabla u_*}^2 + \abs{u_*}^2\\
&\le \liminf_{k \to \infty} \Bigl(\frac{1}{2} - \frac{1}{2 p_{n_k}}\Bigr)
\int_{{\mathbb R}^N} \abs{\nabla u_{p_{n_k}}}^2 + \abs{u_{p_{n_k}}}^2\\
&= \liminf_{k \to \infty} \mathcal{A}_{p_{n_k}} (u_{p_{n_k}})
= \liminf_{k \to \infty} c_{0, p_{n_k}}.
\end{split}
\]
Since the sequence \((p_n)_{n \in {\mathbb N}}\) is arbitrary, this proves the lower semicontinuity.
\end{proof}
\section{Proof of the main theorem}
\label{section3}
\begin{proof}
[Proof of theorem~\ref{theoremMain}]
\resetclaim
We shall successively construct a family of solutions of \eqref{eqChoquard}, prove that neither the positive nor the negative part of this family goes to \(0\), and show that the negative part and the positive part cannot diverge from each other as \(p \to 2\). The theorem will then follow from a classical weak convergence and local compactness argument.
\begin{claim}
\label{claimConstructionSequence}
There exists a family \((u_p)_{p \in (2, \frac{N + \alpha}{N - 2})}\) in \(H^1 ({\mathbb R}^N)\) such that for each \(p \in (2, \frac{N + \alpha}{N - 2})\), the function \(u_p\) changes sign and
\[
-\Delta u_p + u_p = \bigl(I_\alpha \ast \abs{u_p}^p\bigr) \abs{u_p}^{p - 2} u_p.
\]
Moreover
\[
\limsup_{p \to 2} \mathcal{A}_{p} (u_p) \le c_{\mathrm{nod},2}
\]
and
\[
\limsup_{p \to 2} \int_{{\mathbb R}^N} \abs{\nabla u_p}^2 + \abs{u_p}^2 \le
4\, c_{\mathrm{nod},2}.
\]
\end{claim}
The claim implies the uniform boundedness in \(H^1 ({\mathbb R}^N)\) of the solutions \(u_p\) since \(c_{\mathrm{nod}, 2} \le c_{\mathrm{odd}, 2} < 2 c_{0,2} < \infty\) \cite{GhimentiVanSchaftingen}.
\begin{proofclaim}
The existence of a function \(u_p \in H^1 ({\mathbb R}^N)\) that changes sign and satisfies Choquard equation \eqref{eqChoquard-p}
has been proved in \cite{GhimentiVanSchaftingen}*{theorem 2}. Moreover,
\[
\Bigl(\frac{1}{2} - \frac{1}{2p}\Bigr) \int_{{\mathbb R}^N} \abs{\nabla u_p}^2 + \abs{u_p}^2 = \mathcal{A}_p (u_p) = c_{\mathrm{nod}, p}.
\]
It remains to obtain some upper asymptotics on \(c_{\mathrm{nod}, p}\) as \(p \searrow 2\).
Let \(w \in \mathcal{N}_{2, \mathrm{nod}}\) and define
\(w_p = t_{+, p}^{{1}/{p}} w^+ - t_{-, p}^{{1}/{p}} w^-\), where \((t_{+, p}, t_{-, p}) \in [0, \infty)^2\) is the unique maximizer of the concave function
\begin{multline*}
(t_+, t_-) \in [0, \infty)^2
\mapsto E_p (t_+, t_-) = \mathcal{A}_p (t_+^\frac{1}{p} w^+ - t_-^\frac{1}{p} w^-)\\
= \frac{t_+^\frac{2}{p}}{2} \int_{{\mathbb R}^N} \abs{\nabla w^+}^2+\abs{w^+}^2
+ \frac{t_-^\frac{2}{p}}{2} \int_{{\mathbb R}^N} \abs{\nabla w^-}^2+\abs{w^-}^2\\
- \frac{1}{2 p} \int_{{\mathbb R}^N} \bigabs{I_{\alpha/2} \ast \bigl(t_+ \abs{w^+}^p + t_- \abs{w^-}^p)}^2.
\end{multline*}
Since \((t_{+, p}, t_{-, p}) \in (0, \infty)\), we have \(w_p \in \mathcal{N}_{\mathrm{nod}, p}\) and
\[
c_{\mathrm{nod}, p} \le \mathcal{A}_p (w_p).
\]
Since \(E_p (t_+, t_-) \to -\infty\) as \((t_+, t_-) \to \infty\) uniformly in \(p\) in bounded sets and since \(E_p \to E_2\) as \(p \to 2\) uniformly over compact subsets of \([0, \infty)^2\), we have \(t_{\pm, p} \to 1\) as \(p \to 2\). Therefore
\[
\lim_{p \to 2} \mathcal{A}_p (w_p) = \mathcal{A}_2 (w).
\]
Since the function \(w \in \mathcal{N}_{\mathrm{nod}, 2}\) is arbitrary, we deduce that
\[
\limsup_{p \to 2} c_{\mathrm{nod}, p} \le c_{\mathrm{nod}, 2},
\]
from which the upper bounds of the claim follow.
\end{proofclaim}
\begin{claim}
\label{claimNonzero}
\[
\liminf_{p \to 2} \int_{{\mathbb R}^N} \abs{\nabla u_p^\pm}^2 + \abs{u_p^\pm}^2
= \liminf_{p \to 2} \int_{{\mathbb R}^N} \bigl(I_\alpha \ast \abs{u_p}^p\bigr) \abs{u_p^\pm}^p > 0.
\]
\end{claim}
\begin{proofclaim}
We first compute by the Hardy--Littlewood--Sobolev inequality
\[
\begin{split}
\int_{{\mathbb R}^N} \abs{\nabla u_p}^2 + \abs{u_p}^2
= \int_{{\mathbb R}^N} \bigl(I_\alpha \ast \abs{u_p}^p\bigr) \abs{u_p}^p
&\le \refstepcounter{cte} C_{\thecte} \Bigl(\int_{{\mathbb R}^N} \abs{u_p}^{\frac{2 N p}{N + \alpha}} \Bigr)^{1 + \frac{\alpha}{N}}\\
&\le \refstepcounter{cte} C_{\thecte} \Bigl(\int_{{\mathbb R}^N} \abs{\nabla u_p}^2 + \abs{u_p}^2 \Bigr)^p,
\end{split}
\]
where the constant \(C_{\thecte}\) can be taken independently of \(p \in (2, \frac{N + \alpha}{N - 2})\) once \(p\) remains bounded.
It follows then that
\[
\liminf_{p \to 2} C_{\thecte} \Bigl(\int_{{\mathbb R}^N} \abs{\nabla u_p}^2 + \abs{u_p}^2\Bigr)^{p - 1} \ge 1,
\]
and thus,
\[
\liminf_{p \to 2} \int_{{\mathbb R}^N} \abs{\nabla u_p}^2 + \abs{u_p}^2 > 0.
\]
We assume now that there is a sequence \((p_n)_{n \in {\mathbb N}}\) such that
\begin{equation}
\label{eqPositivePartContradictionAssumption}
\lim_{n \to \infty} \int_{{\mathbb R}^N} \abs{\nabla u_{p_n}^-}^2 + \abs{u_{p_n}^-}^2 = 0.
\end{equation}
For \(p \in (2, \frac{N + \alpha}{(N - 2)_+})\) we define the renormalised negative part
\[
v_{p} = \frac{u_{p}^-}{\norm{u_{p}^-}_{H^1 ({\mathbb R}^N)}}.
\]
We first observe that
\begin{equation}
\label{eqRenormalizedInteractionLowerBound}
\int_{{\mathbb R}^N} \bigl(I_{\alpha} \ast \abs{u_{p}}^{p}\bigr) \abs{v_p}^p
= 1.
\end{equation}
By \cite{GhimentiVanSchaftingen}*{lemma~3.6},
for every \(\beta \in \bigl(\alpha, N\bigr)\) there exist \(C_5,C_6
> 0\) such that
\begin{multline*}
\int_{{\mathbb R}^N} \bigl(I_\alpha \ast \abs{u_p}^p\bigr) \abs{v_p}^p
\le \refstepcounter{cte} C_{\thecte} \Bigl(\int_{{\mathbb R}^N} \abs{\nabla
u_p}^2 + \abs{u_p}^2 \int_{{\mathbb R}^N} \abs{\nabla
v_p}^2 + \abs{v_p}^2 \Bigr)^\frac{1}{2}\\
\shoveright{\times \Bigl(\sup_{a \in {\mathbb R}^N} \int_{B_R
(a)} \abs{u_p}^\frac{2 N p}{N + \alpha}\int_{B_R (a)}
\abs{v_p}^\frac{2 N p}{N + \alpha}\Bigr)^{\frac{N + \alpha}{2
N}(1 - \frac{1}{p})}}\\
+ \frac{\refstepcounter{cte} C_{\thecte}}{R^{\beta - \alpha}} \Bigl(\int_{{\mathbb R}^N} \abs{\nabla
u_p}^2 + \abs{u_p}^2 \int_{{\mathbb R}^N} \abs{\nabla
v_p}^2 + \abs{v_p}^2 \Bigr)^\frac{p}{2}
\end{multline*}
The constants come from the Hardy--Littlewood--Sobolev inequality and from the Sobolev inequality;
they can thus be taken to be uniform as \(p \to 2\).
Since \(u_p\) and \(v_p\) remain bounded in \(H^1 ({\mathbb R}^N)\) as \(p \to 2\), we have
\begin{multline*}
\int_{{\mathbb R}^N} \bigl(I_\alpha \ast \abs{u_p}^p\bigr) \abs{v_p}^p\\
\le \refstepcounter{cte} C_{\thecte} \Bigl(\sup_{a \in {\mathbb R}^N} \int_{B_R
(a)} \abs{u_p}^\frac{2 N p}{N + \alpha}\int_{B_R (a)}
\abs{v_p}^\frac{2 N p}{N + \alpha}\Bigr)^{\frac{N + \alpha}{2
N}(1 - \frac{1}{p})} + \frac{\refstepcounter{cte} C_{\thecte}}{R^{\beta - \alpha}}.
\end{multline*}
In view of \eqref{eqRenormalizedInteractionLowerBound}, there exists \(R > 0\) such that
\[
\liminf_{p \to 2} \sup_{a \in {\mathbb R}^N} \Bigl(\int_{B_R
(a)} \abs{u_p}^\frac{2 N p}{N + \alpha}\int_{B_R (a)}
\abs{v_p}^\frac{2 N p}{N + \alpha} \Bigr) > 0.
\]
In particular, there exists a sequence of vectors \((a_n)_{n \in {\mathbb N}}\) in \({\mathbb R}^N\) and a sequence of real numbers $(p_n)_{n \in {\mathbb N}}$ in \((2, \frac{N + \alpha}{N - 2}) \) converging to $2$
such that
\begin{equation}
\label{ineqLiminfuv}
\liminf_{n \to \infty} \Bigl(\int_{B_R
(a_n)} \abs{u_{p_n}}^\frac{2 N {p_n}}{N + \alpha}\int_{B_R (a_n)}
\abs{v_{p_n}}^\frac{2 N {p_n}}{N + \alpha} \Bigr) > 0.
\end{equation}
There exists thus a subsequence \((n_k)_{k \in {\mathbb N}}\) such that
the sequences of functions \((u_{p_{n_k}} (\cdot - a_{n_k}))_{k \in {\mathbb N}}\) and \((v_{p_{n_k}} (\cdot - a_{n_k}))_{k \in {\mathbb N}}\) both converge weakly in the space \(H^1 ({\mathbb R}^N)\) to some functions $u$ and $v\in H^1({\mathbb R}^N)$.
By our contradiction assumption \eqref{eqPositivePartContradictionAssumption} and the Rellich--Kondrachov compactness theorem, we have \(u \ge 0\).
By the classical Rellich--Kondrachov compactness theorem, it follows from \eqref{ineqLiminfuv} that
\begin{align}
\label{eqUPositivity}
\int_{B_R} \abs{u}^\frac{2 N p}{N + \alpha}& > 0&
&\text{ and }&
\int_{B_R}
\abs{v}^\frac{2 N p}{N + \alpha} & > 0.
\end{align}
We also observe that by definition of \(v_p\),
\[
\{x \in {\mathbb R}^N \;:\; v_p (x) < 0\} \subseteq \{ x \in {\mathbb R}^N \;:\; u_p (x) \le 0\},
\]
so that by the Rellich--Kondrachov theorem, we have
\begin{equation}
\label{eqUNegativity}
\{ x \in {\mathbb R}^N \;:\; v (x) < 0\}
\subseteq \{ x \in {\mathbb R}^N \;:\; u (x) \le 0\}.
\end{equation}
Since by the classical Rellich--Kondrachov compactness theorem, the sequence \((\abs{u_{p_{n_k}} (\cdot - a_{n_k})}^{p_n})_{k \in {\mathbb N}}\) converges locally in measure to \(\abs{u}^2\) and is bounded in \(L^{2 N/(N + \alpha)} ({\mathbb R}^N)\),
it converges weakly to \(\abs{u}^2\) in the space \(L^{2N/(N + \alpha)} ({\mathbb R}^N)\)
\citelist{\cite{Bogachev2007}*{proposition~4.7.12}\cite{Willem2013}*{
proposition 5.4.7}}.
In view of the Hardy--Littlewood--Sobolev inequality \eqref{eqHLS} and the continuity of bounded linear operators for the weak topology, the sequence \((I_\alpha \ast \abs{u_{p_{n_k}} (\cdot - a_{n_k})}^{p_{n_k}})_{k \in {\mathbb N}}\) converges weakly to \(I_\alpha \ast \abs{u}^2\)
in \(L^{2N/(N - \alpha)} ({\mathbb R}^N)\).
Since \(((\abs{u_{p_{n_k}}}^{p_{n_k} - 2} u_{p_{n_k}})(\cdot - a_{n_k}))_{k \in {\mathbb N}}\) converges to \(u\) in \(L^2_{\mathrm{loc}} ({\mathbb R}^N)\),
we conclude that
\[
\bigl(I_\alpha \ast \abs{u_{p_{n_k}} (\cdot - a_{n_k})}^{p_{n_k}}\bigr)\bigl(\abs{u_{p_{n_k}}}^{p_ {n_k} - 2} u_{p_{n_k}}\bigr)\,(\cdot - a_{n_k}) \to \bigl(I_\alpha \ast \abs{u}^2\bigr)u
\]
in \(L^{2N/(2 N - \alpha)} ({\mathbb R}^N)\), as \(k \to \infty\).
By construction of the function \(u_p\), we deduce from that the function \(u \in H^1 ({\mathbb R}^N)\) is a weak solution of the problem
\[
-\Delta u + u = \bigl(I_{\alpha} \ast \abs{u}^2\bigr) u.
\]
By the classical bootstrap method for subcritical semilinear elliptic problems applied to the Choquard equation (see for example \citelist{\cite{MorozVanSchaftingen13}*{proposition 4.1}\cite{CingolaniClappSecchi2012}*{lemma A.1}}), \(u\) is smooth. Since \(u \ge 0\), by the strong maximum principle we have either \(u=0\) or \(u > 0\), in contradiction with \eqref{eqUPositivity} and \eqref{eqUNegativity}.
The claim is thus proved by contradiction.
\end{proofclaim}
\begin{claim}
There exists \(R > 0\) such that
\[
\limsup_{p \to 2} \sup_{a \in {\mathbb R}^N} \int_{B_R (a)} \abs{u_p^+}^\frac{2
N p}{N + \alpha} \int_{B_R (a)} \abs{u_p^-}^\frac{2 N
p}{N + \alpha} > 0.
\]
\end{claim}
\begin{proofclaim}
We assume by contradiction that for every \(R > 0\),
\begin{equation}
\label{eqRepulsionContradiction}
\lim_{p \to 2} \sup_{a \in {\mathbb R}^N} \int_{B_R (a)} \abs{u_p^+}^\frac{2
N p}{N + \alpha} \int_{B_R (a)} \abs{u_p^-}^\frac{2 N
p}{N + \alpha} = 0.
\end{equation}
In view of \cite{GhimentiVanSchaftingen}*{lemma~3.6} and since the sequences \((u_n^+)_{n
\in
{\mathbb N}}\) and \((u_n^-)_{n \in {\mathbb N}}\) are both bounded in \(H^1 ({\mathbb R}^N)\), we have, as in the proof of claim~\ref{claimNonzero}, for every \(\beta \in (\alpha, N)\) and \(R > 0\),
\begin{multline*}
\int_{{\mathbb R}^N} (I_\alpha \ast \abs{u_p^+}^p) \abs{u_p^-}^p\\
\le \refstepcounter{cte} C_{\thecte} \Bigl(\sup_{a \in {\mathbb R}^N} \int_{B_R
(a)} \abs{u_p^+}^\frac{2 N p}{N + \alpha}\int_{B_R (a)}
\abs{u_p^-}^\frac{2 N p}{N + \alpha}\Bigr)^{\frac{N + \alpha}{2
N}(1 - \frac{1}{p})} + \frac{\refstepcounter{cte} C_{\thecte}}{R^{\beta - \alpha}}.
\end{multline*}
By our assumption \eqref{eqRepulsionContradiction},
we have thus
\begin{equation}
\label{eqNoInteraction}
\lim_{p \to 2} \int_{{\mathbb R}^N} \bigl(I_\alpha \ast \abs{u_p^+}^p\bigr)
\abs{u_p^-}^p = 0.
\end{equation}
We define now the pair \((t_{p, +}, t_{p, -}) \in (0, \infty)^2\) by the condition that \(t_{p, \pm} u_p^\pm \in
\mathcal{N}_{0, p}\), or equivalently,
\[
t_{p, \pm}^{2 p - 2} =
\frac{\displaystyle \int_{{\mathbb R}^N}\abs{\nabla u_p^\pm}^2 + \abs{u_p^\pm}^2}{
\displaystyle \int_{{\mathbb R}^N} \bigl(I_\alpha
\ast \abs{u_p^\pm}^p\bigr) \abs{u_p^\pm}^p}
=\frac{\displaystyle \int_{{\mathbb R}^N} \bigl(I_\alpha
\ast \abs{u_p}^p\bigr) \abs{u_p^\pm}^p}{
\displaystyle \int_{{\mathbb R}^N} \bigl(I_\alpha
\ast \abs{u_p^\pm}^p\bigr) \abs{u_p^\pm}^p}
=1 + o (1)
\]
as \(p \to 2\), in view of claim~\ref{claimNonzero} and of \eqref{eqNoInteraction}.
Since the family \(u_p\) remains bounded in \(H^1 ({\mathbb R}^N)\), we have
\[
\lim_{p \to 2} \mathcal{A}_p (t_{p, +} u_p^+ + t_{p, -} u_p^-) - \mathcal{A}_p (u_p)
= 0.
\]
In view of the identity
\begin{multline*}
\mathcal{A}_p (t_{p, +} u_p^+ + t_{p, -} u_p^-)\\
= \mathcal{A}_p (t_{p, +} u_p^+) + \mathcal{A}_p(t_{p, -} u_p^-)
-\frac{t_{p, +}^p t_{p, -}^p}{p} \int_{{\mathbb R}^N} \bigl(I_\alpha \ast
\abs{u_p^+}^p\bigr) \abs{u_p^-}^p
\end{multline*}
and by \eqref{eqNoInteraction}, we conclude that
\[
\liminf_{p \to 2} \mathcal{A}_p (u_p) \ge 2 \liminf_{p \to 2} c_{0, p}.
\]
By claim~\ref{claimConstructionSequence} and by proposition~\ref{continuityGroundstate}, this implies that
\[
c_{\mathrm{nod}, 2} \ge 2 c_{0, 2},
\]
in contradiction with proposition~\ref{propositionStrictInequality}.
\end{proofclaim}
We are now in a position to conclude the proof.
Up to a translation, there exist $R>0$ and a sequence \((p_n)_{n \in {\mathbb N}}\) in \((2, \frac{N + \alpha}{N - 2})\) such that \(p_n \searrow 2\) as \(n \to \infty\),
\[
\liminf_{n \to \infty} \int_{B_R} \abs{u_{p_n}^\pm}^\frac{2
N p}{N + \alpha} \ge 0
\]
and the sequence \((u_{p_n})_{n \in {\mathbb N}}\) converges weakly in \(H^1 ({\mathbb R}^N)\) to some function \(u \in H^1
({\mathbb R}^N)\).
As in the proof of claim~\ref{claimNonzero}, by the
weak convergence and by the classical Rellich--Kondrachov compactness
theorem, we have \(\mathcal{A}'_2 (u) = 0\) and \(u^\pm \ne 0\), whence \(u \in
\mathcal{N}_{2, \mathrm{nod}}\).
We also have by the weak lower semicontinuity of the norm,
\[
\begin{split}
\lim_{n \to \infty} \mathcal{A}_{p_n} (u_{p_n}) &= \lim_{n \to \infty} \Bigl(\frac{1}{2} - \frac{1}{2 p_n} \Bigr)\int_{{\mathbb R}^N}
\abs{\nabla u_{p_n}}^2 + \abs{u_{p_n}}^2\\
&\ge \frac{1}{4} \int_{{\mathbb R}^N} \abs{\nabla u}^2 +
\abs{u}^2 =\mathcal{A}_2 (u),
\end{split}
\]
and thus \(\mathcal{A}_2 (u) = c_{\mathrm{nod}, 2}\).
\end{proof}
In claim~\ref{claimNonzero}, the study of the renormalised negative part to prevent vanishing is reminiscent of the idea of taking the renormalised approximate solution to bypass the Ambrosetti--Rabinowitz superlinearity condition \citelist{\cite{LiuWang2004}\cite{VanSchaftigenWillem2008}}.
\begin{bibdiv}
\begin{biblist}
\bib{AckermannWeth2005}{article}{
author={Ackermann, Nils},
author={Weth, Tobias},
title={Multibump solutions of nonlinear periodic Schr\"odinger equations
in a degenerate setting},
journal={Commun. Contemp. Math.},
volume={7},
date={2005},
number={3},
pages={269--298},
issn={0219-1997},
}
\bib{Bogachev2007}{book}{
author={Bogachev, V. I.},
title={Measure theory},
publisher={Springer},
place={Berlin},
date={2007},
isbn={978-3-540-34513-8},
isbn={3-540-34513-2},
}
\bib{Ghimenti-soliton}{article}{
author={Bonanno, Claudio},
author={d'Avenia, Pietro},
author={Ghimenti, Marco},
author={Squassina, Marco},
title={Soliton dynamics for the generalized Choquard equation},
journal={J. Math. Anal. Appl.},
volume={417},
date={2014},
number={1},
pages={180--199},
issn={0022-247X},
}
\bib{CastroCossioNeuberger1997}{article}{
author={Castro, Alfonso},
author={Cossio, Jorge},
author={Neuberger, John M.},
title={A sign-changing solution for a superlinear Dirichlet problem},
journal={Rocky Mountain J. Math.},
volume={27},
date={1997},
number={4},
pages={1041--1053},
issn={0035-7596},
}
\bib{CastorCossioNeuberger1998}{article}{
author={Castro, Alfonso},
author={Cossio, Jorge},
author={Neuberger, John M.},
title={A minmax principle, index of the critical point, and existence of
sign-changing solutions to elliptic boundary value problems},
journal={Electron. J. Differential Equations},
volume={1998},
date={1998},
number={2},
pages={18},
issn={1072-6691},
}
\bib{CeramiSoliminiStruwe1986}{article}{
author={Cerami, G.},
author={Solimini, S.},
author={Struwe, M.},
title={Some existence results for superlinear elliptic boundary value
problems involving critical exponents},
journal={J. Funct. Anal.},
volume={69},
date={1986},
number={3},
pages={289--306},
issn={0022-1236},
}
\bib{CingolaniClappSecchi2012}{article}{
author={Cingolani, Silvia},
author={Clapp, M{\'o}nica},
author={Secchi, Simone},
title={Multiple solutions to a magnetic nonlinear Choquard equation},
journal={Z. Angew. Math. Phys.},
volume={63},
date={2012},
number={2},
pages={233--248},
issn={0044-2275},
}
\bib{CingolaniSecchi}{article}{
author={Cingolani, Silvia},
author={Secchi, Simone},
title={Multiple \(\mathbb{S}^{1}\)-orbits for the Schr\"odinger-Newton system},
journal={Differential and Integral Equations},
volume={26},
number={9/10},
pages={867--884},
date={2013},
}
\bib{CingolaniSecchi-2015}{article}{
author={Cingolani, Silvia},
author={Secchi, Simone},
title={Ground states for the pseudo-relativistic Hartree equation with
external potential},
journal={Proc. Roy. Soc. Edinburgh Sect. A},
volume={145},
date={2015},
number={1},
pages={73--90},
issn={0308-2105},
}
\bib{ClappSalazar}{article}{
author={Clapp, M{\'o}nica},
author={Salazar, Dora},
title={Positive and sign changing solutions to a nonlinear Choquard
equation},
journal={J. Math. Anal. Appl.},
volume={407},
date={2013},
number={1},
pages={1--15},
issn={0022-247X},
}
\bib{fractionalChoquard}{article}{
author={d'Avenia, Pietro},
author={Siciliano, Gaetano},
author={Squassina, Marco},
title={On fractional Choquard equations},
journal={Math. Models Methods Appl. Sci.},
volume={25},
date={2015},
number={8},
pages={1447--1476},
issn={0218-2025},
}
\bib{DevreeseAlexandrov2009}{book}{
author={Devreese, Jozef T.},
author={Alexandrov, Alexandre S. },
title={Advances in polaron physics},
year={2010},
publisher={Springer},
volume={159},
series={Springer Series in Solid-State Sciences},
pages={ix+167},
}
\bib{Frohlich-Lenzmann}{article}{
author={Fr{\"o}hlich, J{\"u}rg},
author={Lenzmann, Enno},
title={Mean-field limit of quantum Bose gases and nonlinear Hartree
equation},
conference={
title={S\'eminaire: \'Equations aux D\'eriv\'ees Partielles.
2003--2004},
},
book={
series={S\'emin. \'Equ. D\'eriv. Partielles},
publisher={\'Ecole Polytech., Palaiseau},
},
date={2004},
pages={Exp. No. XIX, 26},
}
\bib{GhimentiVanSchaftingen}{article}{
author={Ghimenti, Marco},
author={Van Schaftingen, Jean},
title={Nodal solutions for the Choquard equation},
eprint={arXiv:1503.06031},
}
\bib{KRWJones1995newtonian}{article}{
title={Newtonian Quantum Gravity},
author={Jones, K. R. W.},
journal={Australian Journal of Physics},
volume={48},
number={6},
pages={1055-1081},
year={1995},
}
\bib{Lenzmann-2009}{article}{
author={Lenzmann, Enno},
title={Uniqueness of ground states for pseudorelativistic Hartree
equations},
journal={Anal. PDE},
volume={2},
date={2009},
number={1},
pages={1--27},
issn={1948-206X},
}
\bib{Lieb1977}{article}{
author={Lieb, Elliott H.},
title={Existence and uniqueness of the minimizing solution of Choquard's
nonlinear equation},
journal={Studies in Appl. Math.},
volume={57},
date={1976/77},
number={2},
pages={93--105},
}
\bib{LiebLoss2001}{book}{
author={Lieb, Elliott H.},
author={Loss, Michael},
title={Analysis},
series={Graduate Studies in Mathematics},
volume={14},
edition={2},
publisher={American Mathematical Society},
place={Providence, RI},
date={2001},
pages={xxii+346},
isbn={0-8218-2783-9},
}
\bib{Lions1980}{article}{
author={Lions, P.-L.},
title={The Choquard equation and related questions},
journal={Nonlinear Anal.},
volume={4},
date={1980},
number={6},
pages={1063--1072},
issn={0362-546X},
}
\bib{Lions1984-1}{article}{
author={Lions, P.-L.},
title={The concentration-compactness principle in the calculus of
variations. The locally compact case.},
part = {I},
journal={Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire},
volume={1},
date={1984},
number={2},
pages={109--145},
issn={0294-1449},
}
\bib{Lions1984CC2}{article}{
author={Lions, Pierre-Louis},
title={The concentration-compactness principle in the calculus of
variations. The locally compact case. II},
journal={Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire},
volume={1},
date={1984},
number={4},
pages={223--283},
issn={0294-1449},
}
\bib{LiuWang2004}{article}{
author={Liu, Zhaoli},
author={Wang, Zhi-Qiang},
title={On the Ambrosetti-Rabinowitz superlinear condition},
journal={Adv. Nonlinear Stud.},
volume={4},
date={2004},
number={4},
pages={563--574},
issn={1536-1365},
}
\bib{Ma-Zhao-2010}{article}{
author={Ma,Li},
author={Zhao, Lin},
title={Classification of positive solitary solutions of the nonlinear
Choquard equation},
journal={Arch. Ration. Mech. Anal.},
volume={195},
date={2010},
number={2},
pages={455--467},
issn={0003-9527},
}
\bib{Menzala1980}{article}{
author={Menzala, Gustavo Perla},
title={On regular solutions of a nonlinear equation of Choquard's type},
journal={Proc. Roy. Soc. Edinburgh Sect. A},
volume={86},
date={1980},
number={3-4},
pages={291--301},
issn={0308-2105},
}
\bib{Moroz-Penrose-Tod-1998}{article}{
author={Moroz, Irene M.},
author={Penrose, Roger},
author={Tod, Paul},
title={Spherically-symmetric solutions of the Schr\"odinger-Newton
equations},
journal={Classical Quantum Gravity},
volume={15},
date={1998},
number={9},
pages={2733--2742},
issn={0264-9381},
}
\bib{MorozVanSchaftingen13}{article}{
author={Moroz, Vitaly},
author={Van Schaftingen, Jean},
title={Groundstates of nonlinear Choquard equations: existence, qualitative
properties and decay asymptotics},
journal={J. Funct. Anal.},
volume={265},
date={2013},
pages={153--184},
}
\bib{Pekar1954}{book}{
author={Pekar, S.},
title={Untersuchung {\"u}ber die Elektronentheorie der Kristalle},
publisher={Akademie Verlag},
place={Berlin},
date={1954},
pages={184},
}
\bib{VanSchaftingen2014}{article}{
author={Van Schaftingen, Jean},
title={Interpolation inequalities between Sobolev and Morrey--Campanato
spaces: A common gateway to concentration-compactness and
Gagliardo--Nirenberg interpolation inequalities},
journal={Port. Math.},
volume={71},
date={2014},
number={3-4},
pages={159--175},
issn={0032-5155},
}
\bib{VanSchaftigenWillem2008}{article}{
author={Van Schaftingen, Jean},
author={Willem, Michel},
title={Symmetry of solutions of semilinear elliptic problems},
journal={J. Eur. Math. Soc. (JEMS)},
volume={10},
date={2008},
number={2},
pages={439--456},
issn={1435-9855},
}
\bib{Weth2001}{thesis}{
author={Weth, Tobias},
title={Spectral and variational characterizations
of solutions to semilinear eigenvalue problems},
date={2001},
organization={Johannes Gutenberg-Universit\"at},
address={Mainz},
}
\bib{Weth2006}{article}{
author={Weth, Tobias},
title={Energy bounds for entire nodal solutions of autonomous superlinear
equations},
journal={Calc. Var. Partial Differential Equations},
volume={27},
date={2006},
number={4},
pages={421--437},
issn={0944-2669},
}
\bib{Willem1996}{book}{
author={Willem, Michel},
title={Minimax theorems},
series={Progress in Nonlinear Differential Equations and their
Applications, 24},
publisher={Birkh\"auser},
address={Boston, Mass.},
date={1996},
pages={x+162},
isbn={0-8176-3913-6},
}
\bib{Willem2013}{book}{
author = {Willem, Michel},
title = {Functional analysis},
subtitle = {Fundamentals and Applications},
series={Cornerstones},
publisher = {Birkh\"auser},
place = {Basel},
volume = {XIV},
pages = {213},
date={2013},
}
\end{biblist}
\end{bibdiv}
\end{document} | math | 39,446 |
\begin{document}
\title{Bipyramid decompositions of multicrossing link complements}
\begin{abstract}
Generalizing previous constructions, we present a dual pair of decompositions of the complement of a link $L$ into bipyramids, given any multicrossing projection of $L$. When $L$ is hyperbolic, this gives new upper bounds on the volume of $L$ given its multicrossing projection. These bounds are realized by three closely related infinite tiling weaves.
\end{abstract}
\author[Colin Adams]{Colin Adams}
\address{Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267}
\email{[email protected]}
\author{Gregory Kehne}
\address{Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213}
\email{[email protected]}
\date{\today}
\maketitle
\section{Introduction} The standard planar projections of links embedded in $S^3$ are \textit{2-crossing projections}, which means that strands only cross one another two at a time in the projection---or equivalently, that the vertices of the projection graph $G$ all have degree $4$. Recently, 2-crossing projections have been generalized to \textit{$n$-crossing projections}, which are projections of a link in which all strands cross one another $n$ at a time; equivalently, $G$ is $2n$-regular.
Every link has a projection consisting solely of $n$-crossings for every $n \ge 2$, and many of the ideas that apply to 2-crossing projections can be generalized to $n$-crossing projections. See for instance \cite{Adams2012}, \cite{SMALL2014}, and \cite{Adams2013}. More generally, \textit{multicrossing projections} are projections $P$ in which crossings of varying multiplicities are permitted, and the vertices of $G$ need only be of even degree.
A particular case of interest is the \textit{\"ubercrossing projections}, which are the link projections consisting of a lone multicrossing. When the multiplicity of the lone multicrossing is odd and each strand is connected to both of its neighboring strands in the multicrossing, it is a \textit{petal projection} of a knot. Petal knot projections and \"ubercrossing projections were shown to exist for any knot and link (respectively) in \cite{SMALL2012}, and further studied in \cite{SMALL2013} and \cite{Hass2014}.
When referencing an $n$-crossing $c$ in a projection of a link, it will be useful to refer in a standard way to the strands $s_1,\dots,s_n$ and the levels $l_1,\dots,l_n$ at which each of the strands passes through the $n$-crossing. These labels are shown in Figure \ref{schematic}. Throughout we refer to the levels of adjacent strands $l_i, l_{i+1}$ and the strands corresponding to adjacent levels $s_i, s_{i+1}$; the `wraparound cases' $l_n, l_1$ and $s_n, s_1$ are implied. We keep track of a particular multicrossing by noting the permutation $l_1, \ldots l_n$ obtained by considering the levels of its strands clockwise from above, starting on the top level of the crossing.
\begin{figure}
\caption{A general multicrossing $c$ of size $n$, with strands and associated strand levels labelled.}
\label{diagram}
\caption{The specific multicrossing 13524, which appears in an \"ubercrossing projection of the figure-eight knot.}
\label{41diagram}
\caption{multicrossings are represented by their strands and the levels at which the strands enter the crossing. The levels of the strands give a permutation on $n$ elements that encodes the multicrossing.}
\label{schematic}
\end{figure}
By work of W. Thurston \cite{ThurstonNotes}, the complements of many knots and links admit a hyperbolic metric, and when this is the case, the Mostow/Prasad Rigidity Theorem says the metric is uniquely determined. This implies that geometric invariants of a link complement derived from its hyperbolic metric are topological invariants of the link. The \textit{hyperbolic volume} of a link, defined to be $vol(L)=vol(S^3\setminus L)$, is particularly discerning invariant for distinguishing among hyperbolic knots and links. Additionally, it offers an interesting measure of the complexity of a link.
As hyperbolic 3-manifolds, the complements of links produce polyhedral fundamental domains in the universal covering space $\mathbb{H}^3$, where the components of the link correspond to collections of \textit{ideal points} on $\partial\mathbb{H}^3$ ``at infinity''. The corresponding decompositions of the manifold $S^3\setminus L$ into hyperbolic polyhedra contain ideal vertices and edges that extend to them. Decompositions can also contain \textit{finite vertices}, which correspond to points in the interior of the link complement. For any such combinatorial polyhedron $P$, there is an upper bound on the hyperbolic volume $vol(P)$ in $\mathbb{H}^3$ which holds across all embeddings of $P$ in $\mathbb{H}^3$ with geodesic faces and edges. We make special use of the maximal hyperbolic volume across all octahedra, $v_{oct}\approx 3.6639$, which is realized by the ideal regular octahedron.
Since each polyhedron has a maximum achievable volume when embedded in $\mathbb{H}^3$, a decomposition of a link complement into combinatorial polyhedra provides an upper bound on the volume of the link.
D. Thurston showed that given a 2-crossing projection $P$ of a link $L$, the complement $S^3\setminus L$ can be constructed by placing an octahedron at each crossing, where each octahedron's top and bottom vertices are ideal points in the cusps of the upper and lower strands of its crossing, as in Figure \ref{octa}(A) (see \cite{DThurston}). These octahedra are \textit{crossing-centered bipyramids}, which are bipyramids in the complement of a link with finite equatorial vertices and top and bottom vertices that are ideal points at the cusps of adjacent-level strands in a multi-crossing. The equatorial vertices of each octahedron are pulled up and down to finite vertices $U$ and $D$, which sit above and below the projection plane in the complement of the link. The edges extending from the ideal top and bottom vertices of the octahedra become ``vertical'' semi-finite edges from $U$ or $D$ to the cusps after gluing, and the finite equatorial edges of the octahedra become ``vertical'' edges between $U$ and $D$ that pass through each face in the projection. The edge labellings in Figure \ref{octa} depict these operations. The faces of the crossing-centered octahedra glue to the faces of the octahedra at adjacent crossings on the strand, and they together form a decomposition of the link complement into octahedra. This gives an upper bound of
$$vol(L)\leq c(L)v_{oct},$$ where $c(L)$ is the crossing number.
This decomposition and associated bounds have been applied and modified, as in \cite{SMALL2015a}, \cite{Gar} and \cite{Murakami}. These and other bounds are described in \cite{Adams2015}, where the octahedra are each cut into four tetrahedra as in Figure \ref{octa}(B), then recombined about the finite ``vertical'' edges that pass perpendicularly through the faces of $G$ in the projection plane with endpoints $U$ and $D$. The tetrahedra glue together about each of these edges to form a bipyramid that we call a \textit{face-centered bipyramid}, with finite top and bottom vertices at $U$ and $D$ and ideal equatorial vertices corresponding to the cusps. For a given 2-crossing projection the face-centered bipyramid decomposition offers a more stringent upper bound on the volume of a link than the octahedral decomposition. This is because the volume of a maximum-size hyperbolic bipyramid grows logarithmically in $|B|$, where $|B|$ is the number of equatorial edges of $B$, and is referred to as the {\it size} of $B$. The volume bound derived from this face-centered bipyramid decomposition is referred to as the FCB bound.
\begin{figure}
\caption{The crossing-centered octahedron.}
\label{octahedronpic}
\caption{The octahedron is split into tetrahedra.}
\label{octahedroncutpic}
\caption{Decomposing the 2-crossing-centered octahedra into tetrahedra to form the face-centered bipyramid decomposition.}
\label{octa}
\end{figure}
In Section 2, given any multicrossing projection of $L$, we develop a dual pair of bipyramid decompositions of the complement of $L$. These are the multicrossing generalizations of the 2-crossing projection decompositions into octahedra at the crossings and into face-centered bipyramids described above.
In Section 3, we consider upper bounds on hyperbolic volume that these decompositions yield in the case when $L$ is hyperbolic. In a subsequent paper, a more in depth analysis of the resulting volume bounds will appear.
In Section 4, we apply these upper bounds, together with properties of the infinite square weave, in order to establish two new planar tiling weaves that contain multicrossings. These are the \textit{triple weave}, consisting of 3-crossings, and the \textit{right triangle weave}, consisting of 2- and 4-crossings. Like the square weave, the triple weave and the right triangle weave realize the maximum possible volume per crossing in any link with these types of crossings. Somewhat surprisingly, the complement manifolds of the minimal finite representations of these three weaves in a thickened torus are homeomorphic to one another.
We would like to thank the referees, who substantially helped to improve the readability of this paper, especially with regard to the proof of Theorem 4.
\section{The Construction}
We first develop the \textit{face-centered bipyramid decomposition}, which holds for all multicrossing projections of all links. The face-centered bipyramids in this decomposition have finite top and bottom vertices and ideal equatorial vertices. We then demonstrate how, as in the 2-crossing case, these face-centered bipyramids can be cut into tetrahedra and reglued into a dual decomposition of the complement into crossing-centered bipyramids.
\subsection{Face-Centered Bipyramid Construction}
\begin{figure}
\caption{A face of the projection graph.}
\label{step1face}
\caption{The equatorial edges of a face-centered bipyramid.}
\label{step1}
\caption{The face-centered bipyramid.}
\label{step1bip}
\caption{The construction of a face-centered bipyramid.}
\label{facecentered}
\end{figure}
We begin with a particular fixed multi-crossing projection of $L$ and associated projection graph $G$. In each face $F$ of $G$, such as in Figure \ref{facecentered}(A), we produce a cycle of edges, shown dashed, bounding a polygon $P_F$, as in Figure \ref{facecentered}(B). Each strand of the link around the boundary of $F$ contributes a vertex for $P_F$ and at each crossing on the boundary of $F$, each strand that is between the heights of the two edges of the face at that crossing also contributes a vertex of $P_F$. Then we add a finite vertex $U$ above the projection plane and a finite vertex $D$ below the projection plane, and cone each polygon up to $U$ and down to $D$. The result is a bipyramid $B_F$ corresponding to each face of the projection, including the outermost face, as in Figure \ref{facecentered}(C). We call $P_F$ the equatorial polygon of $B_F$.
We now describe how to glue the faces of these bipyramids together to fill the complement of the link. We depict the construction in detail for a triple-crossing, but the general case is similar. In Figure \ref{triplecentered}(A), we see the top view of a triple crossing with part of each equatorial polygon corresponding to the adjacent face-centered bipyramids. We have also added in the top edges of the bipyramids, which meet at $U$. In Figure \ref{triplecentered}(B), we see a side view of the same crossing.
\begin{figure}
\caption{The view from above of how the face-centered bipyramids around a 3-crossing fit together.}
\label{step2}
\caption{The view from the side of how the face-centered bipyramids around a 3-crossing fit together.}
\label{step3}
\caption{The face-centered bipyramid decomposition.}
\label{triplecentered}
\end{figure}
Here we can see how the various faces of the top half of the bipyramids glue together. For instance, representing faces by the edge classes on their boundaries, we see face $bxd$ of $I$ will glue to face $dxb$ of $III$. And face $dye$ of $I$ glues to face $eyd$ of $II$. Note that a vertical edge can slide along its link strand if there is no obstruction from another strand to doing so-this is why the two vertical edges coming out of strand 1 are both labelled with $b$.
In general, consider any face-centered bipyramid $B_F$ adjacent to a multicrossing $c$, with $F$ bounded by strands that enter $c$ at levels $j$ and $k$, where the faces and strands about the multicrossing are considered in clockwise order. Consider an upper face $pqr$ of $B_F$, where $p$ extends from $U$ down to the strand in $c$ at level $i+1$, $q$ lies between the strand at level $i+1$ and the strand at level $i$, and $r$ extends from the strand at level $i$ back up to $U$, where $j\leq i<i+1\leq k$. This face will glue to its partner face $rqp$ of $B_F'$, where $B_{F'}$ is the next face-centered bipyramid encountered such that $F'$ is bounded by strands entering $c$ at levels $k'$ and $j'$, where $j' \leq i < i+1 \leq k'$. All upper faces will either be one such $pqr$ or the $rqp$ partner of some $pqr$, and so all upper faces will be paired and glued to fill space above the link.
Finally, the faces of the bottom halves of the face-centered bipyramids glue together in a similar fashion, filling the entire complement of the link.
The discussion above is summarized in the following theorem:
\begin{thm}\label{facebip}
Given any link $L$ and any multicrossing projection of $L$ with projection graph $G$, the complement of $L$ can be decomposed into a collection of bipyramids $\mathcal{B}_f = \{B_F :~ F \text{ a face of $G$}\}$. Furthermore, the size of each $B_F \in \mathcal{B}_F$ is given by
\begin{equation}\label{facesize}
|B_F|=\sum_{c_i \in \partial F} \left| l(s_i,c_i)-l(s_{i+1}, c_i) \right|,
\end{equation}
where $\partial F=s_1, c_1,\dots, s_m,c_m$ is the boundary of $F$ and $l(s,c)$ is the level at which strand $s$ enters crossing $c$.
\end{thm}
Note that in general a multi-crossing face-centered bipyramid may consist of more tetrahedra than there are edges bounding the face, whereas in the 2-crossing case $|B_F|$ necessarily equals the number of edges of $F$.
\subsection{Crossing-Centered Bipyramid Construction}
\begin{figure}
\caption{The 3-crossing face-centered decomposition of a link complement.}
\label{3crossingf}
\caption{The 3-crossing crossing-centered decomposition of a link complement.}
\label{3crossingc}
\caption{The crossing-centered and face-centered bipyramid decompositions.}
\label{3crossingbips}
\end{figure}
From the face-centered decomposition we derive the \textit{crossing-centered bipyramid decomposition} by first cutting each face-centered bipyramid $B_F$ into its constituent $|B_F|$ tetrahedra. These tetrahedra share an edge from $U$ to $D$ in the center of $B_F$, and the opposite edge of each tetrahedron lies between two adjacent-level strands in a multicrossing in $\partial F$. For a given $c$, consider the tetrahedra from all face-centered bipyramids neighboring $c$ which have an edge passing between adjacent-level strands of $c$. The 3-crossing case is shown in Figure \ref{3crossingbips}(A) and (B). We regroup these tetrahedra according to which of the two adjacent-level strands in $c$ at levels $i$ and $i+1$ that they touch, and so for each $i$, the edge between the level $i$ and $i+1$ strands is shared by all of the tetrahedra in this group. We can glue each of these groups of tetrahedra together about this shared edge to form bipyramid with top vertex at the level $i$ strand and bottom vertex at the level $i+1$ strand. The size of this crossing-centered bipyramid between levels $i$ and $i+1$ of $c$ is determined by the number of faces neighboring $c$ that contribute tetrahedra to it, since each such face contributes exactly one. This is captured by the following theorem:
\begin{thm}\label{crossingbipsize}
Given any link $L$ and any projection $P$ of $L$ with multicrossings $C$, the complement of $L$ can be decomposed into a collection of bipyramids $\mathcal{B}_C = \{B_{c, i} :~ c \in C, i\in \{1, \dots , |c| -1 \}\}$. Furthermore, the sizes of these bipyramids are given by
\begin{equation}\label{crossingbipsizeeqn}
|B_{c,i}|=2\left| \left\{j \in \{1, \dots, |c|\}: \min\{l_j, l_{j+1}\} < i + 1/2 < \max\{l_j,l_{j+1}\} \right\} \right|,
\end{equation}
where $c$ is a multicrossing composed of the strands $s_1, \dots, s_j, \dots, s_n$ at crossing levels $l_1, \dots l_j, \dots, l_n$.
\end{thm}
\begin{proof}
The crossing-centered bipyramid $B_{c,i}$ can be cut into a collection of $|B_{c,i}|$ tetrahedra that share the bipyramid's central edge and glue face-to-face around it. Each of these tetrahedra comes from exactly one face-centered bipyramid $B_F$, where $c$ is in $\partial F$. By the construction of the face-centered bipyramids above, if the boundary of $F$ is of the form $\partial F = \dots , s_{j+1}, c, s_j, \dots$, then $B_F$ contributes a tetrahedron to $B_{c,i}$ exactly when either $l_{j+1} < i + 1/2 < l_j$ or $l_{j+1} > i + 1/2 > l_j$. Therefore as the adjacent pairs of strands at levels $i$ and $i+1$ of $c$ are considered in turn, for the two face-centered bipyramids in the two faces bounded by $s_j$ and $s_{j+1}$ and opposite $c$ from one another, either both face-centered bipyramids contribute a tetrahedron to $B_{c, i}$ or neither does.
This shows that these collections of tetrahedra are of the stated size; it remains to show that they glue together to form bipyramids. But the gluings that merge these tetrahedra into bipyramids are exactly the gluings that describe how the face-centered bipyramids glue up to fill the complement of $L$. The pairs of triangular faces that meet around the central edge of each face-centered bipyramid are alternating pairs of the partnered upper faces and lower faces of the face-centered bipyramids surrounding $c$. In the construction of the crossing-centered bipyramids from the face-centered bipyramids, the equatorial edges of each become the central edges of the other.
\end{proof}
\begin{figure}
\caption{$1234$}
\label{1234}
\caption{$1243$}
\label{1243}
\caption{$1324$}
\label{1324}
\caption{The six 4-crossing configurations, identified up to reflection by their permutations, and their crossing-centered bipyramid decompositions, with edges labelled by their edge classes.}
\label{bips}
\end{figure}
For convenience, the criterion for whether a given face-centered bipyramid $B_F$ with face boundary $\partial F = \dots, s_j, c, s_{j+1}, \dots$ contributes a tetrahedron to the crossing-centered bipyramid $B_{c, i}$ can be reframed in terms of interval containment in the following way. The interval $[i, i+1]$ represents the position of $B_{c, i}$ in the crossing, and $[l_j, l_{j+1}]$ represents the range of levels of $c$ that are spanned by $B_F$. Then $B_F$ contributes a tetrahedron to $B_{c,i}$ if and only if $[i, i+1] \subseteq [l_j, l_{j+1}]$.
\begin{coro}\label{boundedbipgrowth}
For any multicrossing $c$, $|B_{c, 1}| = |B_{c, |c|-1}| = 4$ and the sizes of adjacent crossing-centered bipyramids must satisfy
$$\left||B_{c, i}| - |B_{c,i+1}|\right| = 0 \hspace{.1in} \text{or} \hspace{.1in} 4$$
\end{coro}
\begin{proof}
To see that $|B_{c,1}| = |B_{c, |c|-1}| = 4$, first consider $B_{c,1}$. If the top strand in $c$ is $s_i$ with level $l_i=1$, then $[l_{i-1}, l_i]$ and $[l_i, l_{i+1}]$ are the only adjacent-strand level intervals containing $[1,2]$. Therefore by Theorem \ref{crossingbipsize}, $B_{c,1}$ is composed of 4 tetrahedra glued face-to-face and sharing a common edge, and it is therefore an octahedron. The bottom bipyramid $B_{c, |c|-1}$ is an octahedron for the same reason.
Within $c$, the sizes of two neighboring crossing-centered bipyramids $B_{c,i}$ and $B_{c,i+1}$ correspond to the frequency with which the intervals $[i,i+1]$ and $[i+1, i+2]$ are contained in the strand level intervals $[l_j, l_{j+1}]$. If $s_k$ is the strand with $l_k=i+1$ that passes between these two bipyramids, then the intervals $[l_j, l_{j+1}]$ will contain either both $[i,i+1]$ and $[i+1, i+2]$ or neither, unless $s_j=s_k$ or $s_{j+1} = s_k$. Therefore, the difference between $|B_{c,i}|$ and $|B_{c,i+1}|$ is determined by how the tetrahedra are allocated from the four face-centered bipyramids around $c$ that are bordered by $s_k$ and correspond to the strand level intervals $[l_{k-1}, i+1]$ and $[i+1, l_{k+1}]$. If $l_{k-1} < l_k$ and $l_{k+1} < l_k$, then these four face-centered bipyramids will contribute four tetrahedra to $B_{c,i}$ and none to $B_{c,i+1}$, and $|B_{c,i}| = |B_{c,i+1}|+4$. If $l_{k-1} > l_k$ and $l_{k+1} > l_k$, then these four tetrahedra will be allocated to $B_{c,i+1}$, and $|B_{c,i}| = |B_{c,i+1}|-4$. And if $l_{j-1} < l_j < l_{j+1}$ or $l_{j-1} > l_j > l_{j+1}$, then the contributions to the two bipyramids will be the same, so $|B_{c,i}|=|B_{c,i+1}|$.
\end{proof}
In Figure \ref{bips}, we see the crossing-centered bipyramid decompositions for all six 4-crossing configurations (identified up to reflection). This includes the first instance of a non-octahedral crossing-centered bipyramid, shown in Figure \ref{bips}(C). In light of the constraints on crossing-centered bipyramid sizes given by Corollary \ref{boundedbipgrowth}, it is natural to ask which sequences of crossing-centered bipyramid sizes are realizable. It turns out that these conditions constitute a classification of the realizable crossing-centered bipyramid size sequences:
\begin{thm}\label{xingalg}
Every sequence $m_1, m_2, \dots,m_{n-1}$ of positive integers such that \\ $m_1=m_{n-1}=4$ and $|m_i- m_{i+1}| = 0 \hspace{.1in}\text{or} \hspace{.1in}4$ is realized as the signature of crossing-centered bipyramid sizes for some $n$-crossing.
\end{thm}
In order to prove this theorem, we first prove two lemmas.
\begin{lem}\label{add4} If $m_1, m_2, \dots,m_{n-1}$ is realized, so is $4, m_1+4, m_2+4, \dots,m_{n-1}+4, 4$.
\end{lem}
\begin{proof} Let $c$ be the $n$-crossing that realizes $m_1, m_2, \dots,m_{n-1}$ with level sequence $l_1,l_2,\dots,l_n$. We create an $(n+2)$-crossing $c'$ by adding a strand above and below $c$ in the following manner. Add the new overstrand clockwise from the understrand of $c$ and the new understrand just clockwise from the new overstrand. The contributions of the intervals $[l_j, l_{j+1}]$ to the sizes of the bipyramids between strands remain unchanged, except for in three cases. Moving clockwise from the old understrand of $c$, the interval between the old understrand of $c$ and the new overstrand of $c'$ will contribute to every bipyramid except the bipyramid above the very bottom strand of $c'$. The interval between the new overstrand and the new understrand will contribute to every bipyramid between strands in $c'$. And the interval between the new understrand of $c'$ and the strand that was clockwise from the understrand in $c$ will contribute to the same set of bipyramids it did before, as well as to the bottom bipyramid above the new understrand. Since each such contribution is doubled when we consider the intervals on the opposite side of the crossing, this means that the sequence of bipyramid sizes for $c'$ is $4, m_1+4, m_2+4, \dots,m_{n-1}+4, 4$.
\end{proof}
\begin{lem}\label{concatenate} If sequences $p_1, p_2, \dots,p_{u-1}$ and $q_1, q_2, \dots,q_{v-1}$ are realized, then so is
$p_1, p_2, \dots,p_{u-1}, q_2, \dots, q_{v-1}$.
\end{lem}
\begin{proof} Note that both realized sequences begin and end with 4's, so in the concluding sequence, the last 4 of the first sequence given by $p_{u-1}$ is identified with the beginning 4 of the second sequence, given by $q_1$. Let $c_1$ and $c_2$ be multicrossings realizing the two given sequences. Construct a new $(n+q-2)$-crossing $c_3$ by starting with the first crossing and then placing directly beneath it the second crossing, such that from above, the entire second crossing appears in the two opposite regions just clockwise from the bottom strand in $c_1$. Moreover, do so such that the topmost strand of $c_2$ is clockwise from the bottom strand of $c_1$. (See Figure \ref{concatenation}.) Now remove the bottom strand of $c_1$ and the top strand of $c_2$ to obtain our new crossing $c_3$.
\begin{figure}
\caption{Creating crossing $c_3$ from $c_1$ and $c_2$ to``concatenate" bipyramid size seqences.}
\label{concatenation}
\end{figure}
Let $j$ and $k$ be the heights of the strands counterclockwise and clockwise from the bottom strand in $c_1$. Let $r$ and $s$ be the heights of the strands counterclockwise and clockwise from the top strand in $c_2$. In $c_3$, the only intervals originally from $c_1$ with their contributions to bipyramids between strands affected are $[j,n]$ and $[k,n]$. Similarly, the only intervals originally from $c_2$ with their contributions to bipyramids between strands affected are $[1,r]$ and $[1,s]$. In $c_3$, we also have the new intervals $[j, s+n-2]$ and $[k, r+ n-2]$. Then the contributions to the sizes of bipyramids by $[j,n]$ and $[1,s]$ in $c_1$ and $c_2$ are exactly replaced by the contributions from $[j, s+n-2]$, with the exception that there is a single intermediate bipyramid that is contributed to rather than separate bipyramids at the bottom of $c_1$ and the top of $c_2$.
The same holds for replacing the contributions of intervals $[k,n]$ and $[1,r]$ by \\$[k, r+n-2]$. Hence $c_3$ realizes the desired sequence.
\end{proof}
\begin{proof}[Proof of Theorem \ref{xingalg}]
We induct on the sum of the bipyramid size sequence $\sum_{i = 1}^{n-1} m_i$ in a sequence $\{m_i\}$. We can realize the single integer sequence $\{4\}$ with a 2-crossing. Suppose that we can realize all sequences $\{m_i\}$ such that
\begin{enumerate}
\item $m_1 = m_{n-1} = 4 \label{4end}$,
\item $|m_i - m_{i+1}| \leq 4$ for all $i$ \label{smallgap}, and
\item $\sum_{i = 1}^{n-1} m_i \leq 4t$ \label{smallsum}.
\end{enumerate}
Then given a sequence $\{m_i\}$ satisfying (\ref{4end}) and (\ref{smallgap}) and for which $\sum_{i = 1}^{n-1} m_i = 4(t+1)$, either $\{m_i\}$ contains a 4 that is not at the beginning or end, or it does not. If it does, then $\{m_i\}$ is of the form $p_1, \dots , p_{k-1}, q_2, \dots, q_{l-1}$ for two sequences $\{p_i\}$ and $\{q_i\}$ that both satisfy (\ref{4end}), (\ref{smallgap}), and (\ref{smallsum}). These sequences are therefore realizable, and so $\{m_i\}$ is realizable by Lemma \ref{concatenate}.
If $\{m_i\}$ does not contain a 4 in its interior, then it is of the form $4, p_1, 4+p_2, \dots 4+p_{n-1}, 4$ for some sequence $\{p_i\}$ satisfying (\ref{4end}), (\ref{smallgap}), and (\ref{smallsum}). Therefore $\{p_i\}$ is realizable, and so by Lemma \ref{add4} $\{m_i\}$ is realizable as well.
\end{proof}
This crossing-centered bipyramid decomposition agrees with the construction used in \cite{Adams2012} Theorem 5.2, which shows that for a link $L$ in a 3-crossing projection, the complement can be decomposed into pairs of octahedra positioned between the strands of each 3-crossing, as in Figure \ref{3crossingbips}(B).
\section{Hyperbolic Volume Bounds}
In hyperbolic space $\mathbb{H}^3$, for each fixed $n$ there is a maximum $n$-bipyramid volume. Therefore, given a decomposition of a hyperbolic link complement $S^3\setminus L$ into bipyramids, the volume
of the entire manifold $S^3\setminus L$ is bounded above by the sum of the maximum possible volumes of each of its constituent bipyramids. We pursue this strategy in order to develop upper bounds for volumes of hyperbolic link complements, given their multicrossing projections and the corresponding bipyramid decompositions developed in Section 2. To begin, we know from \cite{Adams2015} Theorem 2.2 that the volumes of these maximal size-$n$ bipyramids, here denoted $B_n$, grow logarithmically in $n$:
\begin{thm} \label{logbound}
$vol(B_n)<2\pi\log(n/2)$ for $n\geq 3$ and $vol(B_n)$ grows asymptotically like $2\pi\log(n/2)$:
$$\lim_{n\rightarrow \infty} \frac{vol(B_n)}{2\pi\log(n/2)}=1.$$
\end{thm}
In a subsequent paper, we will explore the volume bounds to which these decompositions give rise and consider techniques for improving upon these bounds.
Here we note that the multicrossing-centered bipyramid decomposition for a hyperbolic link $L$ in a multi-crossing projection, the derived crossing-centered bipyramids $\mathcal{B}_c$ give an upper bound on the volume of $L$. This bound is
\begin{equation}\label{MCCB}
vol(L) < \sum_{B \in \mathcal{B}_C} vol(B_{|B|}),
\end{equation}
where for $B \in \mathcal{B}_C$, $|B|$ is given by Theorem\ref{crossingbipsize}. This multicrossing crossing-centered bipyramid bound will be referred to as the MCCB bound on volume.
Similarly, the multicrossing face-centered bipyramid decomposition also gives us an upper bound on volume. For a given link $L$ in a multi-crossing projection with derived face-centered bipyramids $\mathcal{B}_F$, this bound is
\begin{equation} \label{MFCB}
vol(L)< \sum_{B \in \mathcal{B}_F} vol(B_{|B|}),
\end{equation}
where for $B \in \mathcal{B}_C$, $|B|$ is given by Theorem \ref{facebip}. This multicrossing face-centered bipyramid bound will be referred to at the MFCB bound on volume.
Note that for both the face-centered and crossing-centered bipyramid decompositions, the specific configurations of the $n$-crossings affect the sizes of the bipyramids, and as a result the MCCB and MFCB bounds depend on the crossing configurations. Certain crossing configurations yield larger volumes and volume upper bounds than others. This variation will be investigated further in the subsequent paper. Note also that if we apply the Thurston 2-crossing octahedral upper bound on volume to a multi-crossing projection, then for each $n$-crossing it gives an upper bound of ${n \choose 2} v_{oct}$. This is because each $n$-crossing must be perturbed into ${n \choose 2}$ 2-crossings. On the other hand, the MCCB bound applied to an $n$-crossing gives an upper bound of $(n-1) v_{oct}$ in the best case, and an upper bound that is $O(n \log n)$ in the worst case. Table \ref{boundcomparison} compares these bounds for some values of $n$.
\begin{table}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
n & Best-case MCCB bound & Worst-case MCCB bound & Octahedral bound \\ \hline
3 & 7.32772 & 7.32772 & 10.9916 \\ \hline
4 & 10.9916 & 15.1827 & 21.9832 \\ \hline
5 & 14.6554 & 23.0377 & 36.6386 \\ \hline
10 & 32.9747 & 81.6887 & 164.874 \\ \hline
100 & 362.722 & 2,183.09 & 18,136.1 \\
\hline
\end{tabular}
\caption{Contribution to best-case, worst-case, and octahedral upper bounds on volume from $n$-crossings.} \label{boundcomparison}
\end{table}
\section{Maximal Weaves}
In \cite{CKP} Champanerkar, Kofman, and Purcell introduced the \textit{infinite weave} $\mathcal{W}$, which is the unique infinite alternating link embedded in $\mathbb{R}^3$ with the $(4^4)$ regular tiling of the Euclidean plane as its projection graph, as in Figure \ref{weave}(A). They also study the \textit{volume density} of hyperbolic links, which is defined for a link $L$ to be
\begin{equation}
\mathcal{D}_{vol}(L)=\frac{vol(L)}{c(L)},
\end{equation}
and they considered $\mathcal{W}$ as the limit of an infinite sequence of finite links that contain increasingly large patches of the square weave.
In this manner, they showed that $\mathcal{W}$ is \textit{geometrically maximal}, meaning that in the limit it attains the maximal value of $\mathcal{D}_{vol}(\mathcal{W})=v_{oct}$, which realizes D. Thurston's octahedral upper bound on volume. Since each face of $\mathcal{W}$ has 4 sides, it also realizes the face-centered bipyramid upper bound of \cite{Adams2015}.
\begin{figure}
\caption{Square weave.}
\label{squareweave}
\caption{Triple weave.}
\label{tripleweave}
\caption{Right triangle weave.}
\label{rightweave}
\caption{Three weaves.}
\label{weave}
\end{figure}
We now apply these new decompositions and corresponding volume upper bounds to the \textit{triple weave} $\mathcal{W}_T$ corresponding to the $(3^6)$ regular tiling, shown in Figure \ref{weave}(B), and to the \textit{isoceles right triangle weave} $\mathcal{W}_R$, corresponding to the $[4.8^2]$ Laves tiling, shown in Figure \ref{weave}(C). These are periodic infinite links embedded in $\mathbb{R}^3$. The triple weave has triple crossings of type 123 and 132 in alternate rows. The right triangle weave has an equal ratio of 2-crossings and 4-crossings, with 4-crossings given by the permutation 1243, where the top strand in the 4-crossing passes through the 2-crossing as an understrand and the bottom strand in the 4-crossing passes through the 2-crossing as an overstrand.
For all three weaves we can take the quotient of $\mathbb{R}^3$ by $\mathbb{Z}^2$, its discrete subgroup of translational isometries, to obtain a link in a thickened torus $T \times (0,1)$. Equivalently, we can view these as links in $S^3$, where we have added two components, each of which is a core curve of one of the solid tori to either side of the projection torus, so the complement of these two components is $T \times (0,1)$.
For the square weave, we denote this six-component link complement in $S^3$ by $\mathcal{W}'$. There are four 2-crossings on the projection torus and the projection of the four components coming from the square weave is alternating on the torus, which is apparent from Figure \ref{tripleconstruction}. The core curves of the solid torus are shown in pink and light blue. The four link components of $\mathcal{W}'$ each bound a twice-punctured disk in the complement in $S^3$, and two of these twice-punctured disks are shown here shaded.
For the triple weave, we denote the link complement by $\mathcal{W}_T'$. There are two 3-crossings on the torus, as in Figure \ref{torus}(B). And for the isoceles right triangle weave, the corresponding link complement is denoted $\mathcal{W}_R'$ and there is a single 2-crossing and a single 4-crossing on the torus, as in Figure \ref{torus}(D).
\begin{figure}
\caption{The link complement $\mathcal{W}
\caption{The link $\mathcal{W}
\caption{The link complement $\mathcal{W}
\caption{The link $\mathcal{W}
\caption{On the left, two pairs of twice-punctured disks in the complement of $\mathcal{W}
\label{triplefigs}
\label{tripleconstruction}
\label{tripletorus}
\label{rightconstruction}
\label{righttriangletorus}
\label{torus}
\end{figure}
\begin{thm}\label{WT}
$\mathcal{W}', \mathcal{W}_T'$ and $\mathcal{W}_R'$ are all isometric with volume equal to $4v_{oct}$.
\end{thm}
\begin{proof}
By cutting $\mathcal{W}'$ open along the two thrice-punctured spheres highlighted in Figure \ref{torus}(A), introducing a full twist on each, and regluing them, we obtain $\mathcal{W}_T'$, as shown in Figure \ref{torus}(B). The resulting manifold is isometric to the original and therefore has the same volume.
Similarly, introducing a full twist to each of the thrice-punctured spheres highlighted in Figure \ref{torus}(C) yields $\mathcal{W}_R'$ as shown in Figure \ref{torus}(D). Therefore all three are isometric. Using either results of \cite{CKP} as applied to $\mathcal{W}'$ or the decomposition of any of these link complements into four ideal octahedra meeting four along each edge yields a volume, via the Mostow/Prasad Rigidity Theorem, of $4 v_{oct}$. \end{proof}
For all three weaves, the volume of this shared manifold achieves both the MFCB bound and MCCB bound. In the case of $\mathcal{W}'$, there is one ideal regular octahedron corresponding to each of the crossings of the link in $T \times (0,1)$. In the case of $\mathcal{W}_T'$, there are two regular ideal octahedra at each of the two triple crossings in $T \times (0,1)$. In the case of $\mathcal{W}_R'$, there is one ideal regular octahedron at the 2-crossing and three octahedra at the single 1243-crossing in $T \times (0,1)$. For the MFCB bound, we consider the bipyramids coming from the faces of the projection onto the torus, and we obtain one regular ideal octahedron per face for the four faces, in each of these three cases, again realizing the upper bound on volume. Note that we must remove additional link components in order to be in $T \times (0,1)$ and make all vertices ideal on the octahedra, which is necessary in order to realize the upper bounds on volume.
\begin{coro}
$\mathcal{W}_T$ is geometrically maximal among all 3-crossing links in $T\times (0,1)$.
\end{coro}
\begin{proof}
From the preceding theorem, $vol(\mathcal{W}_T)=4v_{oct}$. Since it contains two 3-crossings on the torus, its triple-crossing number is $c_3(\mathcal{W}_T)= 2$. This implies that the \textit{triple volume density} of $\mathcal{W}_T$ is
\begin{equation}
\mathcal{D}^3_{vol}(\mathcal{W}_T)=\frac{vol(\mathcal{W}_T)}{c_3(\mathcal{W}_T)}=2v_{oct}
\end{equation}
This also realizes the MCCB bound for 3-crossings in general, where the region surrounding each 3-crossing can be decomposed into two octahedra. From the MCCB bound it follows that no link embedded in $T\times (0,1)$ can have a higher 3-crossing volume density.
\end{proof}
For finite links with 3-crossing planar projections in $S^3$, the MCCB bound of $2v_{oct}$ per crossing given by \cite{Adams2012} (and the equivalent decomposition above) certainly holds. However volume bound improvements can be made by collapsing the finite $U$ and $D$ vertices to the cusp, so equality is unattainable. In \cite{CKP} Champanerkar, Kofman, and Purcell were able to show that certain sequences of finite links that contain ever-increasing patches of the square weave also approach the infinite square weave in volume density. Their argument used lower bounds on volume attained by guts, which were derived from the essentiality of the checkerboard surfaces that came from the alternating projections that they considered. We expect analogous sequences of finite links containing increasing patches of the triple weave as in Figure \ref{patch} to similarly approach $\mathcal{D}^3_{vol}(\mathcal{W}_T)=2v_{oct}$ in triple volume density, but the corresponding theory for links in triple-crossing projections is not yet developed enough to permit a similar argument.
\begin{figure}
\caption{Links containing ever larger patches of the triple weave should have volume density approaching $2v_{oct}
\label{patch}
\end{figure}
\begin{conj}
For links $L$ in a 3-crossing projection on the plane, the triple-crossing volume density bound
\begin{equation*}
\mathcal{D}^3_{vol}(L)< 2v_{oct}
\end{equation*}
is sharp, and is realized by a sequence of links as in Figure \ref{patch}.
\end{conj}
\end{document} | math | 38,861 |
\begin{document}
\title[The Swing Lemma and $\E C_1$-diagrams]
{Using the Swing Lemma and $\E C_1$-diagrams
for congruences of planar semimodular lattices}
\author[G.\ Gr\"atzer]{George Gr\"atzer}
\email{[email protected]}
\urladdr{http://server.maths.umanitoba.ca/homepages/gratzer/}
\address{University of Manitoba}
\date{June 6, 2021}
\begin{abstract}
A planar semimodular lattice $K$ is \emph{slim}
if $\SM{3}$ is not a sublattice of~$K$.
In a recent paper, G. Cz\'edli found four new properties
of congruence lattices of slim, planar, semimodular lattices,
including the \emph{No Child Property}:
\emph{Let~$\mathcal{P}$ be the ordered set of join-irreducible congruences of $K$.
Let $x,y,z \in \mathcal{P}$ and let $z$ be a~maximal element of $\mathcal{P}$.
If $x \neq y$ and $x, y \prec z$ in $\mathcal{P}$,
then there is no element $u$ of $\mathcal{P}$ such that $u \prec x, y$ in $\mathcal{P}$.}
We are applying my Swing Lemma, 2015,
and a type of standardized diagrams of Cz\'edli's, to verify his four properties.
\end{abstract}
\subjclass[2000]{06C10}
\keywords{Rectangular lattice, slim planar semimodular lattice, congruence lattice}
\maketitle
\section{Introduction}\label{S:Introduction}
Let $K$ be a planar semimodular lattice.
We call the lattice $K$ \emph{slim} if $\SM{3}$ is not a~sublattice of~$K$.
In the paper \cite[Theorem 1.5]{gG14a}, I found a property of congruences of slim, planar, semimodular lattices.
In the same paper (see also Problem 24.1 in G. Gr\"atzer~\cite{CFL2}),
I~proposed the following:
\tbf{Problem.} Characterize the congruence lattices of slim planar semimodular lattices.
G. Cz\'edli ~\cite[Corollaries 3.4, 3.5, Theorem 4.3]{gCa} found four new properties
of congruence lattices of slim, planar, semimodular lattices.
\begin{theoremn}\label{T:main}
Let $K$ be a slim, planar, semimodular lattice
with at least three elements and let~$\E P$ be
the ordered set of join-irreducible congruences of $K$.
\begin{enumeratei}
\item \emph{Partition Property:}
The set of maximal elements of $\E P$ can be divided into the disjoint union
of two nonempty subsets such that no two distinct elements
in the same subset have a common lower cover.\label{E:LC}
\item \emph{Maximal Cover Property:}
If $v \in \E P$ is covered by a maximal element $u$ of $\E P$,
then $u$ is not the only cover of $v$.
\item \emph{No Child Property:}
Let $x \neq y \in \E P$ and let $u$ be a maximal element of $\E P$.
If $x,y \prec u$ in $\E P$,
then there is no element $z \in \E P$ such that $z \prec x, y$ in $\E P$.
\item \emph{Four-Crown Two-pendant Property:}
There is no cover-preserving embedding of the ordered set $\E R$ in Figure~\ref{F:notation}
into $\E P$ satisfying the property\tup{:} any maximal element of~$\E R$
is a maximal element of $\E P$.
\end{enumeratei}
\end{theoremn}
In this paper, we will provide a short and direct proof of this theorem
using only the Swing Lemma and $\E C_1$-diagrams, see Section~\ref{S:Tools}.
\begin{figure}
\caption{The Four-crown Two-pendant ordered set $\E R$ with notation;
the covering $\SfS 7$ sublattice with edge and element notation}
\label{F:notation}
\end{figure}
\subsection*{Outline}
Section~\ref{S:Motivation} provides the motivation for Cz\'edli's Theorem.
Section~\ref{S:Tools} provides the tools we need:
the Swing Lemma, $\E C_1$-diagrams, and forks.
Section~\ref{S:partition} proves the Partition Property,
Section~\ref{S:Maximal} does the Maximal Cover Property,
while Section~\ref{S:Child} verifies the No Child Property.
Finally, The Four-Crown Two-pendant Property is proved in Section~\ref{S:Crown}.
\section{Motivation}\label{S:Motivation}
In my paper \cite{GLS98a} with H. Lakser and E.\,T. Schmidt,
we proved that every finite distributive lattice $D$ can be represented
as the congruence lattice of a semimodular lattice $L$.
To our surprise, the semimodular lattice $K$ we constructed was \emph{planar}.
G.~Gr\"atzer and E.~Knapp~\cite{GKn07}--\cite{GK10}
started the study of planar semimodular lattices.
I continued it with my ``Notes on planar semimodular lattices'' series (started with Knapp):
\cite{gG13}, \cite{GW10} (with T. Wares), \cite{CG12} (with G. Cz\'edli),
\cite{gG19}, \cite{gG21b}.
See also G. Cz\'edli and E.\,T. Schmidt \cite{CS13} and G. Cz\'edli \cite{gC14}--\cite{gCb}.
A major subchapter of the theory of planar semimodular lattices
started with the observation that in the construction of the lattice $K$,
as in the first paragraph of this section,
$\SM{3}$ sublattices play a crucial role.
It was natural to raise the question
what can be said about congruence lattices of slim, planar, semimodular (SPS) lattices
(see [CFL2, Problem~24.1], originally raised in G. Gr\"atzer~\cite{gG14a}).
In~\cite{gG14a}, I~found the first necessary condition and
G. Cz\'edli \cite{gC14a} proved that this condition is not sufficient
(see also my related papers \cite{gG15a} and \cite{gG19}).
A number of papers developed tools to tackle this problem:
the Swing Lemma (G. Gr\"atzer~\cite{gG15}), trajectory coloring (G. Cz\'edli \cite{gC14}),
special diagrams (G. Cz\'edli \cite{gC17}), lamps (G. Cz\'edli \cite{gCa}).
Some of these results require long proofs.
The proof of the trajectory coloring theorem is just shy of 20 pages,
while the basic theory of lamps and its application to Theorem~\ref{T:main} is 23 pages.
There are a number of surveys of this field,
see the book chapters G.~Cz\'edli and G.~Gr\"atzer~\cite{CG14}
and G.~Gr\"atzer~\cite{gG13b}
in G.~Gr\"atzer and F.~Wehrung, eds.\,~\cite{LTS1}.
My~presentation \cite{gG21a} provides a gentle review
for the background of this topic.
\section{The tools we need}\label{S:Tools}
Most basic concepts and notation not defined in this paper
are available in Part~I of the book \cite{CFL2}, see
\verb+https://www.researchgate.net/publication/299594715+\\
\indent {\tt arXiv:2104.06539}
\noindent It is available to the reader.
We will reference it, for instance, as [CFL2, page 52].
In particular, we use the notation $C \persp D$, $C \perspup D$,
and $C \perspdn D$ for perspectivity,
up-perspectivity, and down-perspectivity, respectively.
As usual, for planar lattices, a prime interval (or covering interval) is called an \emph{edge}.
For a finite lattice $K$ and a~finite ordered set $R$,
a \emph{cover-preserving} embedding $\ge \colon R \to K$ is an embedding~$\ge$
mapping edges of $R$ to edges of $K$.
We define a \emph{cover-preserving} sublattice similarly.
For the lattice $\SfS 7$ of Figure~\ref{F:notation},
we need a variant: an $\SfS 7$ sublattice $\SfS{}$
(a sublattice isomorphic to $\SfS 7$) is a \emph{peak sublattice}
if the three top edges ($L$, $M$, and $R$ in Figure~\ref{F:notation}) are edges in $K$.
By G. Gr\"atzer and E. Knapp \cite{GKn09},
every slim, planar, semimodular lattice $K$ has a congruence-preserving
extension (see [CFL2, page 43]) $\hat K$ to a slim rectangular lattice.
Any of the properties (i)--(iv) holds for $K$ if{}f it holds for $\hat K$.
Therefore, in~the rest of this paper, we can assume that $K$ is a slim rectangular lattice,
simplifying the discussion.
\subsection{Swing Lemma}\label{S:Swing}
For an edge $E$ of an SPS lattice $K$, let $E = [0_E, 1_E]$
and define $\Col{E}$, the \emph{color of}~$E$, as $\con E$,
the (join-irreducible) congruence generated by collapsing $E$
(see [CFL2, Section 3.2]).
We write $\E P$ for $\Ji {\Con K}$,
the ordered set of join-irreducible congruences of $K$.
As in my paper~\cite{gG15}, for the edges $U, V$
of an SPS lattice $K$, we define a binary relation:
$U$~\emph{swings} to $V$, in formula, $U \swing V$, if $1_U = 1_V$,
the element $1_U = 1_V$ of~$K$ covers at least three elements,
and $0_V$ is neither the left-most
nor the right-most element covered by $1_U = 1_V$;
if also $0_U$ is such, then the swing is \emph{interior},
otherwise, it is \emph{exterior}, denoted by $U \inswing V$ and $U \exswing V$, respectively.
\begin{named}{Swing Lemma [G. Gr\"atzer~\cite{gG15}]}
Let $K$ be an SPS lattice and let $U$ and $V$ be edges in $K$.
Then $\Col V \leq\Col U$ if{}f there exists an edge $R$
such that $U$ is up-perspective to $R$
and there exists a sequence of edges and a~sequence of binary relations
\begin{equation*}\label{E:sequence}
R = R_0 \mathbin{\gr}_1 R_1 \mathbin{\gr}_2 \dots \mathbin{\gr}_n R_n = V,
\end{equation*}
where each relation $\mathbin{\gr}_i$ is $\perspdn$ \pr{down-perspective} or $\swing$ \pr{swing}.
In~addition, this sequence also satisfies
\begin{equation*}\label{E:geq}
1_{R_0} \geq 1_{R_1} \geq \dots \geq 1_{R_n}.
\end{equation*}
\end{named}
The following statements are immediate consequences of the Swing Lemma,
see my papers~\cite{gG15} and \cite{gG14e}.
\begin{corollary}\label{C:equal}
We use the assumptions of the Swing Lemma.
\begin{enumeratei}
\item The equality $\Col U = \Col V$ holds in $\E P$ if{}f there exist edges $S$ and $T$ in $K$,
such that
\begin{equation*}\label{E:xx}
U \perspup S,\ S \inswing T,\ T \perspdn V.
\end{equation*}
\item Let us further assume that the element $0_U$ is meet-irreducible.
Then the equality $\Col U = \Col V$ holds in $\E P$
if{}f there exists an edge $T$ such that $U \inswing T \perspdn V$.
\item If the lattice $K$ is rectangular and $U$ is on the upper boundary of $K$,
then the equality $\Col U = \Col V$ holds in $\E P$ if{}f $U \perspdn V$.
\end{enumeratei}
\end{corollary}
Note that in (i) the edges $S, T, U, V$ need not be distinct,
so we have as special cases $U = V$, $U \persp V$, $S = T$, and others.
\begin{corollary}\label{C:cov}
We use the assumptions of the Swing Lemma.
\begin{enumeratei}
\item The covering $\Col V \prec \Col U$ holds in $\E P$
if{}f there exist edges $R_1, \dots, R_4$ in~$K$, such that
\begin{equation*}
U \perspup R_1,\ R_1 \inswing R_2,\ R_2 \perspdn R_3,\
R_3 \exswing R_4,\ R_4 \perspdn V.
\end{equation*}
\item If the element $0_U$ is meet-irreducible,
then the covering $\Col V \prec \Col U$ holds in $\E P$
if{}f there exist edges $S, T$ in $K$, so that
\begin{equation*}
U \perspdn S \exswing T \perspdn V.
\end{equation*}
\end{enumeratei}
\end{corollary}
\begin{corollary}\label{C:covnew}
Let $K$ be a slim rectangular lattice, let $U$ and $V$ be edges in $K$,
and let $U$ be in the upper-left boundary of $K$.
\begin{enumeratei}
\item The covering $\Col V \prec \Col U$ holds in $\E P$
if{}f there exist edges $S, T$ in $K$, such that
\begin{equation}\label{E:seq5}
U \perspdn S \exswing T \perspdn V.
\end{equation}
\item Define the element $t = 1_S = 1_T \in K$
and let $S = E_1, E_2, \dots, E_n = W$ enumerate, from left to right,
all the edges $E$ of $K$ with $1_E = t$.
Then
\begin{align}
\col{S} &\neq \col{W},\label{E:1}\\%\eqref{E:1}
\col{E_2} = \cdots &= \col{E_{n-1}} = \col{T},\label{E:2}\\%\eqref{E:2}
\col{T} &\prec \col{S}, \col{W}.\label{E:3}
\end{align}
\end{enumeratei}
\end{corollary}
\begin{corollary}\label{C:max}
Let the edge $U$ be on the upper edge of the rectangular lattice $K$.
Then $\Col U$ is a maximal element of $\E P$.
\end{corollary}
The converse of this statement is stated in Corollary~\ref{C:max1}.
\subsection{$\E C_1$-diagrams}\label{S:diagrams}
In the diagram of a planar lattice $K$,
a \emph{normal edge} (\emph{line}) has a~slope of $45\degree$ or $135\degree$.
If it is the first, we call it a~\emph{normal-up edge} (\emph{line}),
otherwise, a \emph{normal-down edge} (\emph{line}).
Any edge of slope strictly between $45\degree$ and $135\degree$ is \emph{steep}.
\begin{definition}\label{D:well}
A diagram of an rectangular lattice $K$ is a
\emph{$\E C_1$-diagram} if the middle edge of any covering $\SfS 7$ is steep
and all other edges are normal.
\end{definition}
This concept was introduced in G.~Cz\'edli~\cite[Definition 5.3(B)] {gC17},
see also G.~Cz\'edli \cite[Definition 2.1]{gCa}
and G. Cz\'edli and G.~Gr\"atzer~\cite[Definition 3.1]{CG21}.
The following is the existence theorem of $\E C_1$-diagrams
in G. Cz\'edli \cite[Theorem 5.5]{gC17}.
\begin{theorem}\label{T:well}
Every rectangular lattice lattice $K$ has a $\E C_1$-diagram.
\end{theorem}
See the illustrations in this paper for examples of $\E C_1$-diagrams.
For a short and direct proof for the existence of $\E C_1$-diagrams,
see my paper~\cite{gG21b}.
\emph{In this paper, $K$ denotes a slim rectangular lattice with a fixed $\E C_1$-diagram
and~$\E P$ is the ordered set of join-irreducible congruences of $K$}.
Let $C$ and $D$ be maximal chains in an interval $[a,b]$ of $K$
such that $C \ii D = \set{a,b}$.
If there is no element of~$K$ between $C$ and $D$,
then we call $C \uu D$ a~\emph{cell}.
A~four-element cell is a \text{\emph{$4$-cell}}.
Opposite edges of a $4$-cell are called \emph{adjacent}.
Planar semimodular lattices are $4$-cell lattices,
that is, all of its cells are $4$-cells,
see G.~Gr\"atzer and E. Knapp \cite[Lemmas 4, 5]{GKn07}
and [CFL2,~Section 4.1] for more detail.
The following statement illustrates the use of $\E C_1$-diagrams.
\begin{lemma}\label{L:application}
Let $K$ be a slim rectangular lattice $K$ with a fixed $\E C_1$-diagram
and let~$X$ be a normal-up edge of $K$.
Then $X$ is up-perspective either to an edge in the upper-left boundary of $K$
or to a steep edge.
\end{lemma}
\begin{proof}
If $X$ is not steep nor it is in the upper-left boundary of $K$,
then there is a~$4$-cell $C$ whose lower-right edge is $X$.
If the upper-left edge is steep or it is in the upper-left boundary, then we are done.
Otherwise, we proceed the same way until we reach a~steep edge
or an edge the upper-left boundary.
\end{proof}
\begin{corollary}\label{C:max1}
Let the edge $U$ be on the upper edge of $K$.
Then $\Col U$ is a maximal element of $\E P$.
Conversely, if $u$ is a maximal element of $\E P$,
then there is an edge $U$ on the upper edge of $K$ so that $\Col U = u$.
\end{corollary}
\subsection{Trajectories}\label{S:Trajectories}
G. Cz\'edli and E.\,T. Schmidt \cite{CS11} introduced a \emph{trajectory} in $K$
as a maximal sequence of consecutive edges, see also [CFL2, Section~4.1].
The \emph{top edge}~$T$ of a trajectory
is either in the upper boundary of $K$ or it is steep by Lemma~\ref{L:application}.
For such an edge~$T$, we denote by $\traj T$ the trajectory with top edge~$T$.
By G.~Gr\"atzer and E. Knapp \cite[Lemma 8]{GKn07},
an element $a$ in an SPS lattice~$K$ has at most two covers.
Therefore, a trajectory has at most one top edge and at most one steep edge.
So we conclude the following statement.
\begin{lemma}\label{L:disj}
Let $K$ be a slim rectangular lattice $K$ with a fixed $\E C_1$-diagram.
Let $X$ and $Y$ be distinct steep edges of $K$.
Then $\traj X$ and $\traj Y$ are disjoint.
\end{lemma}
\section{The Partition Property}\label{S:partition}
First, we verify the Partition Property for the slim rectangular lattice $K$
and with a fixed $\E C_1$-diagram.
We start with a lemma.
\begin{lemma}\label{L:disjoint}
Let $X $ and $Y$ be distinct edges on the upper-left boundary of $K$.
Then there is no edge $Z$ of $K$ such that $\col Z \prec \col X, \col Y$.
\end{lemma}
\begin{proof}
By way of contradiction, let $Z$ be an edge such that $\col Z \prec \col X, \col Y$.
Since $X$ and $Y$ are on the upper-left boundary,
Corollary~\ref{C:covnew}(i) applies.
Therefore, there exist normal-up edges $S_X, S_Y$ and steep edges $T_X, T_Y$ such that
\[
X \perspdn S_X \exswing T_X,\q Y \perspdn S_Y \exswing T_Y,\q
Z \in \traj {T_X} \ii \traj {T_Y}.
\]
By Lemma~\ref{L:disj}, the third formula implies that $T_X = T_Y$
and xo $X = Y$, contrary to the assumption.
\end{proof}
By Corollary~\ref{C:max1}, the set of maximal elements of $\E P$ is the same
as the set of colors of edges in the upper boundaries.
We can partition the set of edges in the upper boundaries
into the set of edges~$\E L$ in the upper-left boundary
and the set of edges~$\E R$ in the upper-right boundary.
If $X $ and $Y$ are distinct edges in $\E L$,
then there is no edge $Z$ of $K$ such that $\col Z \prec \col X, \col Y$
by Lemma~\ref{L:disjoint}.
By symmetry, this verifies the Partition Property.
\section{The Maximal Cover Property}\label{S:Maximal}
Next, we verify the Maximal Cover Property for the slim rectangular lattice~$K$
and with a fixed $\E C_1$-diagram.
Let $x \in \E P$ be covered by a maximal element $u$ of $\E P$ in $K$.
By Corollary~\ref{C:max1}, we can choose an edge $U$ of color $u$ on the upper boundary of $K$,
by symmetry, on the upper-left boundary of $K$.
By Corollary~\ref{C:covnew}(ii), we can choose the edges $S, T$ in $K$
so that $U \perspdn S \exswing T$, $\col S = u$, and $\col T = x$.
By Corollary~\ref{C:covnew}(ii), specifically, by equations \eqref{E:1} and \eqref{E:3},
we have $x \prec u, \col{W}$ and $u \neq \col{W}$, verifying the Maximal Cover Property.
\section{The No Child Property}\label{S:Child}
In this section, we verify the No Child Property for the slim rectangular lattice~$K$
and with a fixed $\E C_1$-diagram.
Let $x,y,z,u \in \E P$ with $x \neq y \in \E P$,
let $u$ be a maximal element of $\E P$,
and let $x, y \prec u$ in $\E P$.
By way of contradiction,
let us assume that there is an element $z \in \E P$
such that $z \prec x,y$ in $\E P$.
By Corollary~\ref{C:max1}, the element~$u$ colors an edge~$U$ on the upper boundary of~$K$,
say, in the upper-left boundary.
By Corollary~\ref{C:cov}(i), for $z \prec x \in \E P$,
we get a peak sublattice $\SfS 7$
in which the middle edge $Z$ is colored by $z$ and upper-left edge $X$
is colored by $x$, or symmetrically.
The upper-right edge $Y$ must have color~$y$.
Now we apply Corollary~\ref{C:covnew}(ii)
to the edge $U$ and middle edge $Z$ of the peak sublattice $\SfS 7$,
obtaining that $U \perspdn Y \swing Z$, in particular, $U \perspdn Y$.
This is a contradiction, since $U$ is normal-up and $Y$ is normal-down.
\section{The Four-Crown Two-pendant Property}\label{S:Crown}
Finally, we verify the Four-Crown Two-pendant Property for the slim rectangular lattice $K$
and with a fixed $\E C_1$-diagram.
By way of contradiction, assume that the ordered set $\E R$ of Figure~\ref{F:notation}
is a cover-preserving ordered subset of $\E P$,
where $a,b,c,d$ are maximal elements of $\E P$.
By~Corollary~\ref{C:max1}, there are edges $A,B,C,D$ on the upper boundary of $K$,
so that $\col A = a$, $\col B = b$, $\col C=c$, $\col D = d$.
By left-right symmetry, we can assume that the edge $A$
is on the upper-left boundary of $K$. Since $p \prec a, b$ in $\E P$,
it follows from Lemma~\ref{L:disjoint} that the edge $B$
is on the upper-right boundary of $K$, and so is $D$.
Similarly, $C$ is on the upper-left boundary of $K$.
There are four cases, (i) $C$ is below $A$ and $B$ is below $D$;
(ii)~$C$~is below $A$ and $D$ is below $B$; and so on.
The first two are illustrated in Figure~\ref{F:CABDx}.
\begin{figure}
\caption{Illustrating the proof of The Four-Crown Two-pendant Property}
\label{F:CABDx}
\end{figure}
We consider the first case.
By Corollary~\ref{C:cov}(ii),
there is a peak sublattice $\SfS 7$ with middle edge $P$
(as in the first diagram of Figure~\ref{F:CABDx})
so that $A$ and $B$ are down-perspective
to the upper-left edge and the upper-right edge of this peak sublattice, respectively.
We define, similarly, the edge $Q$ for $C$ and $B$,
the edge $S$ for $A$ and~$D$, the edge $R$ for $C$ and $D$,
and the edge $U$ for $R$ and $P$.
The ordered set $\E R$ is a cover-preserving subset of $\E P$,
so we get, similarly, the peak sublattice~$\SfS 7$ with middle edge $U$.
Finally, $v \prec q, s$ in $\E R$,
therefore, there is a peak sublattice~$\SfS 7$ with middle edge $V$
with upper-left edge $V_l$ and the upper-right edge~$V_r$
so that $S \perspdn V_l$ and $S \perspdn V_r$, or symmetrically.
This concludes the proof of the Four-Crown Two-pendant Property
and of Cz\'edli's Theorem.
Of course, the diagrams in Figure~\ref{F:CABDx} are only illustrations.
The grid could be much larger, the edges $A, C$ and $B, D$ may not be adjacent,
and there maybe lots of other elements in $K$.
However, our argument does not utilize the special circumstances in the diagrams.
The second case is similar, except that we get the edge $V$
and cannot get the edge $U$.
The third and fourth cases follow the same way.
\appendix
\section{Two more illustrations for Section~\ref{S:Crown}}\label{S:appendix}
\begin{figure}
\caption{Two more illustrations for Section~\ref{S:Crown}
\label{F:CABDx2}
\end{figure}
\end{document} | math | 20,603 |
\begin{document}
\baselineskip=15.5pt
\title[Moduli of unstables]{Moduli of unstable bundles of HN-length two with fixed algebra of endomorphisms}
\author{L. Brambila-Paz}
\address{CIMAT, Apdo. Postal 402,C.P. 36240, Guanajuato, Mexico}
\email{[email protected]}
\author{Roc\'io R\'ios Sierra}
\address{CIMAT, Apdo. Postal 402,C.P. 36240, Guanajuato, Mexico}
\email{[email protected]}
\subjclass[2010]{14H60, 14J60}
\date{\today}
\thanks{Both authors acknowledges the support of CONACYT, in particular of CONACYT grant 251938.}
\begin{abstract}
Let $X$ be a smooth irreducible complex projective curve of genus $g \geq 2$ and $U_{{\mu _1}}(n,d)$ the moduli scheme of indecomposable vector bundles over $X$ with fixed Harder-Narasimhan type $\sigma=(\mu _1, \mu_2)$. In this paper, we give necessary and sufficient conditions for a vector bundle $E\in U_{{\mu _1}}(n,d)$ to have $\mathbb{C}[x_1,\dots , x_k]/(x_1,\dots , x_k)^2$ as its algebra of endomorphisms. Fixing the dimension of the algebra of endomorphisms we obtain a stratification of $ U_ {\mu_1} (n, d) $ such that each stratum $ U_ {\mu_1} (n, d, k) $
is an algebraic variety, moreover, a coarse moduli space. A particular case of interest is when the unstable bundles are simple. In that case the moduli space is fine. Topological properties of $ U_ {\mu_1} (n, d, k) $ will depend on the generality of the curve $X$. Such results differ from the corresponding results for the moduli space of stable bundles, where non-emptiness, dimension etc. are independent of the curve.
\end{abstract}
\maketitle
\section{Introduction}\label{intro}
Let $X$ be a complex smooth projective variety. It is well known that the moduli space of semistable bundles with fixed invariants exists, and that sometimes is fine, i.e. there exists a universal family. Some moduli spaces of unstable bundles have been constructed by adding some extra information: the moduli spaces of unstable vector bundles of rank $2$ and $3$ were constructed in \cite{bm} and in \cite{rocio}, respectively, when $\dim X=1$ and the algebra of endomorphisms is fixed; in \cite{str} when $X$ is the projective plane and in \cite{banc1} when $X=\mathbb{P}^3(\mathbb{C})$, in both cases they consider the degree of instability. J. M. Drezet studied in \cite{drezet} the case of very unstable bundles when $\dim X >2$. In \cite{hk} the authors construct the moduli spaces of pure sheaves with fixed Harder-Narasimhan type which have some additional data called a $m$-rigidification. The moduli spaces of unstable sheaves via non-reductive GIT has been studied by V. Hoskins, G. Berczi, J. Jackson and F. Kirwan in \cite{frances}.
Unless otherwise stated we assume now that $X$ is a smooth irreducible complex projective curve of genus $g \geq 2.$ The aim of this paper is to study unstable bundles over $X$ using their algebra of endomorphisms.
The advantage of using algebra of endomorphisms lies in the fact that it allow us to construct coarse moduli spaces, and even fine moduli spaces.
In order to state our results let us recall that any vector bundle over $X$ has a unique filtration (called Harder-Narasimhan filtration)
$$ 0 = E_0\subset E_1 \subset \cdots \subset E_m = E$$
such that for $0 \leq i \leq m-1, \ E_{i+1}/E_{i}$ is semistable and
\begin{equation}\label{eq01}
\mu (E_1) > \mu (E_2/E_1) > \cdots > \mu (E_m/E_{m-1}).
\end{equation}
The sequence of slopes $\sigma=(\mu (E_1) ,\mu (E_2/E_1), \cdots , \mu (E_m/E_{m-1}))$ is called the {\it Harder-Narasimhan type (HN-type for short)} of $E$. The moduli space of vector bundles of HN- type $\sigma =(\mu (E_1))$ is the moduli space of semistable bundles. It was constructed by Mumford \cite{mun} in 1960’s using Geometric Invariant Theory and by M.S. Narasimhan and S. Seshadri \cite{ns} using representation theory. For a treatment of a more general case we refer the reader to \cite{sim}, \cite{geiseker}, \cite{maruyama1} and \cite{maruyama2}. The moduli space of some non-simple semistable vector bundles with a fixed algebra of endomorphisms was constructed in \cite{yo2}, \cite{yo3} and \cite{yo4}.
Denote by $U_{{\mu _1}}(n,d)$ the set of indecomposable vector bundles of rank $n$ and degree $d$ of coprime-type $\sigma =(\mu _1, \mu _2)$ (see Definition \ref{definitions}). If $\mu _1-\mu _2 > 2g-1$, $U_{{\mu _1}}(n,d)= \emptyset $ (see Proposition \ref{propfm}). For $0<\mu _1-\mu _2 \leq 2g-1$, $U_{{\mu _1}}(n,d)$ has a projective scheme structure that makes it an moduli scheme (see Theorem \ref{teo2}).\footnote{At the time of writing this article, some results on vector bundles of type $\sigma =(\mu _1, \mu _2)$ were obtained independently, and by different methods, in \cite{vickyjosua} and \cite{josua}}
Our purpose is to use one more numerical invariant to describe a stratification of $U_{{\mu _1}}(n,d)$ such that each stratum is an algebraic variety, and a coarse moduli space. Such invariant is the dimension, as a $\mathbb{C}$-vector space, of the algebra of global endomorphisms.
First we determine the structure of the algebra of endomorphism of vector bundles in $U_{{\mu _1}}(n,d)$. If $\mu _1=\frac{d_1}{n_1}$, write $d_2=d-d_1$ and $n_2=d-d_1.$ We prove that (see Corollary \ref{cor2})
$$ \mathbbox{if} \ \ E\in U_{{\mu _1}}(n,d)\ \ \mathbbox{then} \ \ End(E)= \mathbb{C}[x_1,\dots , x_k]/(x_1,\dots , x_k)^2$$
where $0\leq k \leq \frac{d_1n_2-d_2n_1}{2}+n_1n_2.$
Set $A_k:=\mathbb{C}[x_1,\dots , x_k]/(x_1,\dots , x_k)^2$ and $A_0:= \mathbb{C}$. For any integer $0\leq k \leq \frac{d_1n_2-d_2n_1}{2}+n_1n_2$ we will denote by $U_{{\mu _1}}(n,d,k)$ the set
$$U_{{\mu _1}}(n,d,k):=\{ E\in U_{{\mu _1}}(n,d): End(E)\cong A_k\}.$$
Fixing $\dim End(-)$ as invariant we prove that a flattening stratification of an $\it{Ext}^1$-sheaf over a convenient variety $Y$ (see Theorems \ref{teo1}) gives the existence of a schematic stratification of the variety $Y$ with a universal property. We use such
sub schemes and the ideas of twisted Brill-Noether theory (see \cite{hitching}) to give an algebraic structure to $U_{{\mu _1}}(n,d,k)$ and prove the following theorems. By $B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)$ we mean the twisted Brill-Noether locus of product of two stable bundles with at least $k$ section and by $\mathcal{Y}_k\subset B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)$ those with exactly $k$ sections (see (\ref{support}) and Section \ref{moduli} for the definition of the twisted Brill-Noether locus $B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)$ and of the number $h^1$).
\begin{theorem} (Theorems \ref{teo2} and Corollary \ref{corprin0}) \ $\mathcal{U}_{{\mu _1}}(n,d,k)$ is a coarse moduli space and $\mathcal{U}_{{\mu _1}}(n,d,0)$ is a fine moduli space. Moreover, if $\mathcal{Y}_k$ is irreducible and smooth of dimension $\rho$, then $U_{{\mu _1}}(n,d,k)$ is irreducible and smooth of dimension $\rho + h^1-1.$ In particular, if $U_{{\mu _1}}(n,d,k)$ is non-empty, $B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)$ is non-empty.
\end{theorem}
\begin{corollary} (Corollary \ref{corprin}) If $\mathcal{Y}_k$ is irreducible and smooth then $H^i(\mathcal{U}_{{\mu _1}}(n,d,k),\mathbb{C})\cong H^i(\mathcal{Y}_k, \mathbb{C})$ for $i\geq 0$.
\end{corollary}
Non-emptiness and topological properties of $\mathcal{U}_{{\mu _1}}(n,d,k)$ are given in the following theorems. Of particular interest are the vector bundles of HN-type $\sigma =(\frac{d-a}{n-1},a)$, where $a$ is an integer. That is, indecomposable unstable bundles that are extensions of a line bundle by a semistable bundle. In this case the results are a reformulation of the known results of the Brill-Noether theory
in terms of the moduli of unstable bundles. The expected dimension $\beta (g,n_1,d_1,n_2,d_2,k)$ of $U_{{\mu _1}}(n,d,k)$ is given in Section \ref{moduli}. Recall that the {\it Brill-Noether loci} are defined as
$$B(n,d,k):=\{G\in M(n,d): h^0(G)\geq k \},$$
where $M(n,d)$ is the moduli space of stable vector bundles of degree $d$ and rank $n$ over $X$.
\begin{theorem} (Theorem \ref{teop3}) Assume $0<d-an<2(n-1)$ and $(n-1,d-an,k)\ne (n-1,n-1,n-1)$. Then for $\mu _1=\frac{d-a}{n-1}$, $U_{{\mu _1}}(n,d,k)$ is non-empty if and only if $k\leq n-1+\frac{d-n(a+1)+1}{g}$. Moreover, if $U_{{\mu _1}}(n,d,k)$ is non-empty then it is irreducible and smooth of the expected dimension.
\end{theorem}
There are special results for general and Petri curves;
\begin{theorem} (Theorem \ref{teop4})
Let $(g,n-1,d-na,k)$ be integers that satisfies the conditions given in Theorem \ref{bn},(5),(6) and (7).
For general curve, $U_{{\mu _1}}(n,d,k)$ is non-empty and has an irreducible component of the expected dimension. Moreover, if $X$ is a Petri curve of genus $g\geq 3, n\geq 5$ and $g\geq 2n-4$ then $U_{{\mu _1}}(n,d,k)$ is non-empty.
\end{theorem}
As in the Brill-Noether theory for vector bundles, it is possible that for
special curves the above conclusion not holds, and where the Brill-Noether locus, and hence the moduli space, is not even reduced.
Thus, the above results differ from the corresponding results for the moduli space of
stable bundles, where non-emptiness, dimension etc. are independent of the curve.
To our best knowledge the following theorems give also new results in the Brill-Noether and twisted Bril-Noether theory (see \cite{tbn}).
\begin{theorem}(Theorem \ref{teop05})
Assume that $B(n_1,d_1,n_1+a)$ is non-empty with $a>0$. If $2n_1<d_1<a(g+1)$ and $d_2> 2gm$ then for any $0\leq k\leq (d_2+m(1-g))(n_1+a) -(d_2n_1+d_1m +mn_1(1-g))$, $\mathcal{Y}_k\subset B^k(U_1,U_2^*)$ is non-empty. Moreover, if $\mu_1 = \frac{d_1}{n_1}$ then $U_{{\mu_1}}({n},{d},k)$ is non-empty, where $n=n_1+n_2$ and $d=d_1+d_2$.
\end{theorem}
For Perti curves we prove
\begin{theorem}(Theorem \ref{teopetrif}) Let $X$ be a Petri curve of genus $g\geq 3$ and $(\mathcal{O}(D), V)$ a general generated linear system of degree $d_2\geq g+1$ and $\dim V =n_2+1$ with $n_2\leq 4 $ or if $n_2\geq 5$ then $g \geq 2n_2-4$. Assume that $B(n_1,d_1,t)$ is non-empty and $\frac{d_2}{n_2}<\frac{d_1}{n_1}$. For any $0\leq k \leq n_2t-n_1d_2$, $ \mathcal{Y}_k \subset B^k(U_1,U_2^*)$ is non-empty and if $n=n_2+n_1$, $d=d_2+d_1$ and $\mu _1=\frac{d_1}{n_1}$, $U_{{\mu_1}}(n,d,k)$, is non-empty.
\end{theorem}
In the remainder of the last section we give necessary and sufficient conditions for $U_{{\mu _1}}(n,d,k)$ be smooth on $E \in U_{{\mu _1}}(n,d,k)$.
Let $0\subset E_1\subset E$ be the HN-filtration of $E$ in $U_{{\mu _1}}(n,d,k)$. Write $E/E_1=F_1$. We use the following diagram
$$
\begin{array}{ccc}
H^1(End(E_1))\oplus H^1(End(F_1))&\stackrel{d\Phi}{\longrightarrow } & H^1(End(E_1\otimes F^*_1))\\
&\eta _E\searrow & \beta\downarrow \\
&&H^0(E_1\otimes F^*_1)^*\otimes H^1(E_1\otimes F^*_1)
\end{array}
$$
to define $\eta _E$ as $\eta _E:=\beta \circ d\Phi $ and prove
\begin{theorem}( Theorem \ref{teop8}) $U_{{\mu _1}}(n,d,k)$ is smooth at $E$ and of the expected dimension
if and only if $\eta _E $ is surjective.
\end{theorem}
In Section \ref{unstables} we review some of the standard
facts on unstable bundles. Section \ref{algebra} will be concerned with the algebra of endomorphisms of unstable bundles. In Sections \ref{moduli} and \ref{nonempty} our main results are stated and proved.
{\bf Acknowledgments:} The first author gratefully acknowledges the many helpful suggestions of Peter E. Newstead
during the preparation of the paper. Her thanks are also to Alfonso Zamora Saiz for drawing the author’s attention to some unclear points.
The first author is a member of the international research group VBAC (Vector Bundles on Algebraic Curves).
{\bf Notation} The rank and degree of a vector bundle $E$ are denoted by $rk(E)$ and $d(E)$ respectively and the slope as the rational number $\mu (E):=\frac{d(E)}{rk(E)}$. We will write the projection in the i-factor as $p_i$ and by $<E\stackrel{f}{\to} F>$ the linear space generated by the function $f:E\to F$. The cohomology groups $H^i(X,E)$ as $H^i(E)$ and its dimension as $h^i(E)$. Given an exact sequences
$$
\rho: 0\to G \to E \to F \to 0,
$$
\begin{itemize}
\item by $\rho ^*$ we mean the dual sequence $\rho ^*: 0\to F^* \to E^* \to G^* \to 0,$
\item by $(M\otimes \rho )$ the sequence $\rho $ tensor by the vector bundle $M$ i.e.
$$
(M\otimes \rho): 0\to M\otimes G \to M\otimes E \to M\otimes F \to 0.
$$
\item By $H^* (\rho)$ we mean the cohomology sequence of $\rho $
$$ 0\to H^0(X,G) \to H^0(X,E) \to H^0(X,F) \stackrel{\delta}{\to} H^1(X,G) \to H^1(X,E) \to \dots .$$
\item To shorten notation, sometimes we write $E\in H^1(X,F^*\otimes G)$ instead of $(\rho: 0\to G \to E \to F \to 0) \in H^1(X,F^*\otimes G)$
\end{itemize}
\section{Unstable bundles of HN-lenght 2}\label{unstables}
From now on, $X$ will be a smooth irreducible complex projective curve of genus $g \geq 2$. Recall that a vector bundle $E$ over $X$ is semistable if for all proper subbundle $F\subset E$ the slopes satisfy the following inequality
$$\mu(F)\leq \mu (E).$$
The vector bundle $E$ is stable if the inequality is strict and unstable if it is not semistable.
In \cite{hn} it was proved that any vector bundle over $X$ has a unique filtration, called {\it Harder-Narasimhan filtration},
\begin{equation}
0 = E_0\subset E_1 \subset \cdots \subset E_m = E
\end{equation}
such that for $1 \leq i \leq m,$
\begin{itemize}
\item $E_i/E_{i-1}$ is semistable and
\item \begin{equation}\label{desigualdadmu} \mu (E_1) > \mu (E_2/E_1) > \cdots > \mu (E_m/E_{m-1}).
\end{equation}
\end{itemize}
To shorten notation, we write HN-filtration instead of Harder-Narasimhan filtration and for abbreviation, we write $F_i$ instead of the quotient $E_{i+1}/E_i$, when no confusion can arise. Note that $F_0=E_1$.
The HN-max and HN-min of $E$ are defined as $$\mu _{max}(E):=\mu (E_1)\ \ \mathbbox{and} \ \ \mu _{min}(E):=\mu (F_{m-1}). $$ The vector bundle $E_1$ is called {\it the maximal destabilizing subbundle}.
The following definitions will be used throughout all the paper.
\begin{definition}\begin{em}\label{definitions}
Let $ 0 = E_0\subset E_1 \subset \cdots \subset E_m = E$ be the Harder-Narasimhan filtration of $E$.
\begin{itemize}
\item The HN-type is the sequence of slopes $\sigma=(\mu (E_1), \dots ,\mu( E_m/E_{m-1})).$
\item The number $m$ is called {\it the HN-length} of the HN-filtration.
\item The HN-filtration is called of {\it simple type} if each $E_i$ and $F_{i}$ are simple for $i=1,\dots ,m-1$.
\item The HN-filtration is called of {\it coprime type} if the numbers in each slope $\mu (F_{i})$ and $\mu (E_i)$ are coprime for $i=1,\dots ,m-1$.
\item The HN-filtration is called {\it HN-indecomposable} if each $E_i$ and $F_i$ is indecomposable $i=1,\dots ,m-1$.
\item $E$ is called {\it $HN$-general} (respectively {\it HN-special}) if all the quotient $F_i$ are general (respectively special) in the Brill-Noether theory.
\item The extension $\rho _i: 0\to E_{i} \to E_{i+1} \to F_{i} \to 0$ is called the HN(i)-extension and the sequence of extensions $\rho=(\rho _1, \cdots, \rho _{m-1})$ is called the {\it HN-sequences of $E$}.
\end{itemize}
\end{em}
\end{definition}
\begin{remark}\begin{em}\label{remigual}
Let $ 0 = E_0\subset E_1 \subset \cdots \subset E_m = E$ be the HN-filtration of $E$. Note that:
\begin{enumerate}
\item the condition to being of simple/coprime type or HN-indecomposable is for $i=1,\dots ,m-1$. Therefore, $E_m=E$ is not necessarily simple or indecomposable.
\item Simple type $\Longrightarrow$ HN-indecomposable type. However, HN-indecomposable $\nRightarrow$ simple type.
\item If $i\ne 0$, $E_i$ is unstable and $ 0 = E_0\subset E_1 \subset \cdots \subset E_i$ is the HN-filtration of $E_i$. Moreover, if the filtration of $E$ is of coprime, simple, HN-indecomposable, or general type, the same holds for the HN-filtration of any $E_i.$ This will allow us to make induction on the HN-length.
\item For each $E_i$ we have an exact sequence
\begin{equation}\label{extension}
\rho _i: 0\to E_{i-1} \to E_i \to F_{i-1} \to 0.
\end{equation}
Thus, $E$ is constructed by a successive sequence of extensions.
\end{enumerate}
\end{em}\end{remark}
In this section we will restrict our attention to the case of vector bundles of HN-type $\sigma =(\mu _1, \mu _2)$. Note that the value of $\mu _1$ fix the value of $\mu _2$ and viceversa. Indeed, if $\mu _1=\frac{d_1}{n_1} $ then $\mu _2=\frac{d(E)-d_1}{rk(E)-n_1}$. Our aim is to describe some properties and parameterize vector bundles of HN-type $\sigma =(\mu _1, \mu _2)$.
Let
\begin{equation}\label{extension tipo2}
(\rho _1: 0\to E_{1} \stackrel{i}{\to} E_2 \stackrel{p}{\to} F_1 \to 0) \in Ext^1(F_1,E_1)\cong H^1(X,F_1^*\otimes E_1).
\end{equation} be an extension of two semistable bundles with $\mu_1:=\mu(E_1)>\mu (F_1)$.
It follows that $E_2$ is unstable and $0\subset E_1\subset E_2$ is its HN-filtration, with $E_2/E_1=F_1$.
\begin{remark}\begin{em}\label{rem1} Note that given two extensions $(\rho _1: 0\to E_{1} \stackrel{i}{\to} E_2 \stackrel{p}{\to} F_1 \to 0)$ and $(\rho '_1: 0\to E_{1} \stackrel{i}{\to} E_2' \stackrel{p}{\to} F_1 \to 0)$ in $Ext^1(F_1,E_1)$, $E_2\cong E_2'$ if and only if $ \rho _1\sim \rho _1 '$ in $Ext^1(F_1,E_1)/(Aut(E_1)\times Aut(F_1))$. Moreover, if $E_1$ and $F_1$ are simple, $E_2\cong E_2'$ iff $\rho =\lambda \rho '$ with $\lambda \in \mathbb{C}^*$.
\end{em}\end{remark}
To prove our first results we use the following lemmas.
\begin{lemma}\label{lema01}\label{dim1} Let
$\rho _1: 0\to E_{1} \stackrel{i}{\to} E_2 \stackrel{p}{\to} F_1 \to 0$
be an extension of two semistable bundles with $\mu_1:=\mu(E_1)>\mu (F_1)$. If $\mu (E_1)-\mu(F_1)> 2g-2$, $\rho _1=0$.
Moreover, if $\rho _1$ is non-trivial then
\begin{enumerate}
\item $H^0(X,E^*_1\otimes E_1)=H^0(X,E^*_1\otimes E_2 )$. Moreover, if $E_1$ is simple, $h^0(E^*_1\otimes E_2)=1$ and $H^0(E^*_1\otimes E_2)=<E_{1} \stackrel{i}{\to} E_2>$.
\item $H^0(F_1^*\otimes F_1)= H^0(E_2^*\otimes F_1)$. Moreover, if $F_1$ is simple, $h^0(E_2^*\otimes F_1)=1 $ and $H^0(E_2^*\otimes F_1)=<E_2 \stackrel{p}{\to} F_1>$.
\item If $F_1$ is simple then $H^0(F_1^*\otimes E_1)= H^0(F_1^* \otimes E_2 )$.
\end{enumerate}
\end{lemma}
\begin{proof}
The inequality $\mu (E_1)-\mu(F_1)> 2g-2$ implies that $\mu (F_1^*\otimes E_1)>2g-2 $. Hence, since $F_1^*\otimes E_1$ is semistable, $H^1(X,F_1^*\otimes E_1)=0$.
Since $H^0(X,E_1^*\otimes F_1)=0$, $(1)$ and $(2)$ follow from the cohomology sequences
\begin{equation}\label{eq1}
H^*(E_1^*\otimes (\rho _1 )): 0\to H^0(X, E_1^*\otimes E_1)\to H^0(X,E_1^*\otimes E_2)\to H^0(X, E_1^*\otimes F_1)\stackrel{\delta _0}{\to} \cdots
\end{equation}
and
\begin{equation}\label{eq3}
H^*((\rho _1 )^*\otimes F_1): 0\to H^0(X,F_1^*\otimes F_1)\to H^0(X,E_2^*\otimes F_1)\to H^0(X, E_{1}^*\otimes F_1)\stackrel{\delta _1}{\to} \cdots
\end{equation}
$(3).-$ From the cohomology sequence $H^*(F_1^*\otimes (\rho _1 ))$
\begin{equation}\label{eq2}
0\to H^0(X,F_1^*\otimes E_1)\to H^0(X,F_1^* \otimes E_2 )\to H^0(X, F_{1}^*\otimes F_1)\stackrel{\delta _2}{\to} H^1(X,F_1^*\otimes E_1) \cdots
\end{equation}
$H^0(X,F_1^*\otimes E_1)= H^0(X,F_1^* \otimes E_2 )$, since $F_1$ is simple and $\rho _1\ne 0$.
\end{proof}
\begin{lemma}\label{indecom} Assume $E_1$ and $F_1$ are semistable with $E_1$ simple and $F_1$ indecomposable. Then $E_2$ is indecomposable if and only if $\rho _1 \ne 0$.
\end{lemma}
\begin{proof} One direction is trivial. Assume $\rho _1\ne 0.$
Suppose, contrary to our claim, that $E_2=G_1\oplus G_2$ and the extension
$$0= \rho : 0\to G_1\stackrel{j}{\to} E_2\stackrel{q}{\to} G_2\to 0$$ splits. Denote by $\gamma :G_2\to E_2$ the splitting morphism, i.e. $q\circ \gamma = id_{G_2}$. We can certainly assume that $\mu (G_2) \leq \mu (E_2)$, for if not, we replace $G_2$ by $G_1$.
The inclusion $i:E_1\to E_2$ in the extension $\rho _1$ induces the following diagram
$$
\begin{array}{ccccccccc}
&& 0 && 0&& 0&& \\
&& \downarrow& & \downarrow&& \downarrow&& \\
0&\to& M& {\to}& E_1&\stackrel{\sigma}{\to}& H&\to & 0\\
&& \downarrow& & \downarrow i&& \downarrow \nu&& \\
0&\to &G_1&\stackrel{j}{\to} & E_2 &\stackrel{q}{\rightleftarrows}_{\gamma} & G_2& \to& 0
\end{array}
$$
We have different cases.
Case (1)-. $H\ne 0$ and $M\ne 0.$
In this case there is a contradiction to Lemma \ref{lema01},(1) since the morphism $\gamma\circ \nu \circ \sigma :E_1\to E_2$ is not a scalar multiple of the inclusion $i:E_1\to E_2$.
Case (2)-. $H=E_1$.
The diagram
$$
\begin{array}{ccccccccc}
&& && 0&& 0&& \\
&& & & \downarrow&& \downarrow&& \\
&& & & E_1&=& E_1& & \\
&&& & \downarrow i&& \downarrow && \\
0&\to &G_1&\stackrel{j}{\to} & E_2 &\stackrel{q}{\rightleftarrows}_{\gamma} & G_2& \to& 0
\end{array}
$$
induces the following diagram
$$
\begin{array}{ccccccccc}
&& && 0&& 0&& \\
&& & & \downarrow&& \downarrow&& \\
&& & & E_1&=& E_1& & \\
&&& & \downarrow i&& \downarrow && \\
0&\to &G_1&\stackrel{<}{\to} & E_2 &\stackrel{<}{\rightleftarrows}_{\gamma} & G_2& \to& 0 \\
&& \|& & \downarrow \vee && \downarrow&& \\
0&\to& G_1& \stackrel{\leq }{\to}& F_1&\stackrel{\sigma}{\to}& T&\to & 0\\
&& \downarrow& & \downarrow && \downarrow && \\
&& 0 && 0&& 0&&.
\end{array}
$$
The semistability of $F_1$ implies that $\mu (G_1) \leq \mu(F_1)< \mu (E_2)$, hence that $\mu (E_2)< \mu (G_2)$, which is a contradiction to our assumption.
Case (3)-. $H=0$.
In this case we have the following diagram
$$
\begin{array}{ccccccccc}
&& 0&& 0&& && \\
& & \downarrow&& \downarrow&& && \\
& & E_1&=& E_1& & && \\
& & \downarrow i&& \downarrow i && && \\
0&\to &G_1&\stackrel{j}{\to} & E_2 &\stackrel{q}{\rightleftarrows}_{\gamma} & G_2& \to& 0 \\
&& \downarrow& & \downarrow p && \|&& \\
0&\to& T_1& \stackrel{\jmath }{\to}& F_1&\stackrel{\ell}{\to}& G_2&\to & 0\\
&& \downarrow& & \downarrow && \downarrow && \\
&& 0 && 0&& 0&&.
\end{array}
$$
If $0\ne \eta := p\circ \gamma : G_2\to F_1$, $$\ell\circ \eta = \ell\circ p\circ \gamma =q\circ \gamma =id _{{G_2}}$$ and hence $F_1$ is decomposable which is a contradiction, since $F_1$ is indecomposable.
If $\eta =0$ then $G_2\subset E_2\subset G_1$, which is also a contradiction since $E_2=G_1\oplus G_2$.
Therefore, $E_2$ is indecomposable, as claimed.
\end{proof}
Let $T=Ext^1(F_1,E_1)=H^1(X,F_1^*\otimes E_1)$. It is well known (see \cite{nr}) that $\mathbb{P}(T)$ parameterize non-trivial extensions $\rho _1: 0\to E_{1} \stackrel{i}{\to} E \stackrel{p}{\to} F_1 \to 0$ and that if $\mathbb{H}$ is the hyperplane bundle over $\mathbb{P}(T)$ the extension
$$ 0\to p_1^*( E_1)\otimes p_2^*(\mathbb{H}) \to \mathcal{E} \to p_1^*( F_1)\to 0 $$
that corresponds to the identity under the isomorphism $$ End(T)\cong H^1(X\times \mathbb{P}(T), p_1^*(Hom(F_1,E_1))\otimes p_2^*(\mathbb{H}))$$ has universal properties. Thus, Lemma \ref{indecom} leads us to the following result.
\begin{proposition}\label{propfm} If $E_1$ and $F_1$ are simple semistable vector bundles with $\mu (E_1)>\mu (F_1)$ then the pair $(\mathbb{P}(T), \mathcal{E})$ is a fine moduli space for
indecomposable unstable bundles $E$ with $E_1$ as the maximal destabilizing subbundle of $E$ and $F_1=E/E_1$. Moreover, if $\mu (E_1)-\mu(F_1)> 2g-2$, $\mathbb{P}(T)= \emptyset$.
\end{proposition}
From now on we tacitly assume that $0<\mu (E_1)-\mu(F_1)\leq 2g-2.$ By Riemann-Roch
\begin{equation}\label{dimension}
\dim \mathbb{P}(T)=h^0(F_1^*\otimes E_1)-\tilde{d}+\tilde{r}(g-1)-1 ,
\end{equation}
where $\tilde{d}:= deg(E_1)rk(F_1) -deg(F_1)rk(E_1)$ and $\tilde{r}:=rk(E_1)rk(F_1)$.
The dimension of $\mathbb{P}(T)$ depends on $h^0(F_1^*\otimes E_1)$. Actually, $\dim \mathbb{P}(T)>0$ if and only if $h^0(F_1^*\otimes E_1)>\tilde{d}+\tilde{r}(1-g)+1$. Since $F_1^*\otimes E_1$ is semistable and $0<\mu (E_1)-\mu(F_1)\leq 2g-2 $, the study of the dimension $h^0(F_1^*\otimes E_1)$ belongs to the Brill-Noether Theory for vector bundles.
We will touch only a few aspects of the theory. Recall that a vector bundle $G$ is special if $h^0(G)\cdot h^1(G)\ne 0$, otherwise is called general. The {\it Brill-Noether loci} are defined as
$$B(n,d,k):=\{G\in M(n,d): h^0(G)\geq k \},$$
where $M(n,d)$ is the moduli space of stable vector bundles of degree $d$ and rank $n$ over $X$. Moreover, the Brill-Noether loci induce the following filtration:
$$M(n,d)\supseteq B(n,d, 1)\supseteq \cdots \supseteq B(n,d, k) \supseteq B(n,d,k+1) \supseteq \cdots. $$
The number $$\rho (g,n,d,k):= n^2(g-1)+1 -k(k-d+n(g-1))$$ is called {\it the Brill-Noether number} and is the expected dimension of $B(n,d,k)$. The Brill-Noether loci are also defined on the moduli space of $S$-equivalent semistable bundles (see \cite{bgn}).
\begin{remark}\begin{em}\label{remigualk}
Denote by $\operatorname{ch}^0i $ the number $\operatorname{ch}^0i :=d+n(1-g)$.
Note that if $k\leq \operatorname{ch}^0i$, $B(n,d,k)= M(n,d)$. Thus, to have proper Brill-Noether loci we will assume that $k> \operatorname{ch}^0i$.
\end{em}\end{remark}
For the convenience of the reader we summarize the main results of these concepts and the relevant material for us
without proofs, thus making our exposition self-contained. See \cite{im} for a survey of the main results on the Brill-Noether Theory. Recall that $X$ is a {\it Petri curve} if the multiplication map $\mu : H^0(X,L)\otimes H^0(X, K_X\otimes L^*) \to H^0(X, K_X)$ is injective for every line bundle $L$ on $X$ (see \cite{arb}).
\begin{theorem}\label{bn} Let $G$ be a semistable vector bundle of rank $n$ and degree $d$ on $X$.
\begin{enumerate}
\item If $G$ is general,
$$
h^0(G) =
\left\{ \begin{array}{ll}
0 & \text{if $ 0<\mu(G) <g-1$,}\\
{d}+{r}(1-g) & \text{if $g-1 \leq \mu(G) <2(g-1)$.}
\end{array}\right.
$$
\item If $B(n,d,k)$ is proper, the general point $E$ of every proper component of $B(n,d,k)$ has $h^0(E)=k$.
\item (Clifford's Theorem for vector bundles \cite[Theorem 2.1]{bgn}) If $G$ is special, $$h^0(G)\leq \frac{d}{2} +n.$$
\item (\cite{bgn} and \cite{mer}) For any curve $X$, if $0<\mu(G)<2$,
$$B(n,d,k)\ne \emptyset \ \mathbbox{ if and only if} \ n<d+(n-k)g \ \mathbbox{ and} \ (n,d,k)\ne (n,n,n).$$
Moreover, if $B(n,d,k)$ is not empty, then it is irreducible, of dimension $\rho (g,n,d,k)$ and $Sing(B(n,d,k))= B(n,d,k+1)$.
\item (\cite{bmno}) For general curve $B(n,d,k)$ is not empty if $d = nd'+ d''$ and $ n\leq d'' +(n-k)g$ and the following conditions are satisfied $$0 < d'' < 2n, \ d' \geq \frac{(s-1)(s+g)}{s}, \ \mathbbox{with} \ 1 < s < g, \ \ (d'', k)\ne (n,n).$$
\item (\cite{im}) If $X$ is generic and $d = nd_1+d_2, \ k =nk_1+k_2, \ d_2 < n, \ k_2 < n$ for some positive integers $d_i$ and $k_i$ then $B(n,d,k)\ne \emptyset $ and has one component of the expected dimension if one of the following conditions is satisfied:
$$
\begin{array}{ll}
g-(k_1+1)(g-d_1+k_1-1)\geq 1, & 0=d_2 \geq k_2 \\
g-k_1(g-d_1+k_1+1) > 1, & d_2=k_2=0 \\
g-(k_1+1)(g-d_1+k_1) \geq 1, & d_2 < k_2.
\end{array}
$$
\item (\cite{bbn2}) Let $X$ be a Petri curve of genus $g\geq 3, n\geq 5$ and $g\geq 2n-4$. Then $B(n,d,n+1)$ is non-empty.
\end{enumerate}
\end{theorem}
\begin{remark}\begin{em} The non-emptiness of the locus $B(n,d,n+1)$ has been proved with different relation among $n,d$ and $g$ (see \cite{bbn2}, \cite{bbn1} or \cite{b} and the bibliography there). For simplicity we just quote one of the results in Theorem \ref{bn},(7).
\end{em}\end{remark}
According to the above theorem, in some cases the non emptiness and dimension of the moduli space $\mathbb{P}(T)$
differ from general or special curves.
Theorem \ref{bn} will be relevant in Section \ref{nonempty} when we consider families of extensions of semistable bundles.
The remainder of this section will be devoted to the general properties of unstable bundles.
It is well know that $H^0(E)=0$ for semistable vector bundles $E$ of negative degree. However, unstable bundles of negative degree can have sections. Indeed, let $F$ be a bundle of positive degree with $H^1(F)\ne 0$. Any extension $\rho: 0\to \mathcal{O} \to G \to F^* \to 0 \in H^1(F)$ defines an unstable bundle $G$ with at least one section. In the following proposition we proof for unstable bundles some of the well known properties of semistable bundles. They follow directly from the properties of the HN-filtration. Maybe some are already known but we prefer to include them in one proposition.
\begin{proposition}\label{propunstable} Let $E$ be an indecomposable unstable vector bundle of rank $n$ and degree $d$ and $ 0 = E_0\subset E_1 \subset \cdots \subset E_m = E$ its Harder-Narasimhan filtration.
\begin{enumerate}
\item If $\mu _{min}(E)>2g-1$ then $H^1(X,E)=0$ and $E$ is (globally) generated. Moreover, indecomposable unstable bundles with fix HN-filtration are bounded.
\item If $E$ is HN-special, $h^0(E)\leq \frac{d}{2}+n$.
\item If $E$ is HN-general and $\mu _{min}(E)>g+1$, $H^1(X,E)=0$ and $h^0(E)= d(E)+rk(E)(1-g)$. Moreover, if $0<\mu _{max}(E) <g, \ h^0(E)=0.$
\end{enumerate}
\end{proposition}
\begin{proof}
Let $\rho _i: 0\to E_{i-1} \to E_i \to F_{i-1} \to 0$ be the HN(i)-extension of $E$ and
\begin{equation}\label{cohom0}
0\to H^0(X,E_{i-1}) \to H^0(X,E_i) \to H^0(X,F_{i-1}) \stackrel{\delta}{\to}
\end{equation}
\begin{equation}\label{cohom1}
\stackrel{\delta}{\to} H^1(X,E_{i-1}) \to H^1(X,E_i) \to H^1(X,F_{i-1}) \to 0
\end{equation}
its cohomology sequence. Recall that we have the inequalities $$\mu _{max}(E)=\mu (E_1) > \mu (F_1) > \cdots > \mu (F_{m-1})= \mu _{min}(E)>2g-1.$$
The proofs are by induction on the HN-length of $E$. We give the proof for the case $m=2$ using the HN(2)-extension.
$(1)$.- Since $\mu (E_1) > \mu (F_1)= \mu _{min}(E_2)>2g-1$, (\ref{cohom1}) shows that $H^1(E_2)=0$, by the semistability of $E_1$ and $F_1$. As $E_1$ and $F_1$ are generated we have that $E_2$ is generated. That are bounded follows from \cite{maruyama1} (see also \cite{hk} ).
$(2)$.- We conclude from (\ref{cohom0}) that $h^0(E_2)\leq h^0(E_1) +h^0(F_1), $ hence that $$ h^0(E_1) +h^0(F_1)\leq \frac{d(E_1)}{2}+rk(E_1)+\frac{d(F_1)}{2}+rk(F_1)$$ (see Theorem \ref{bn} (2)) and finally that $$h^0(E_2)\leq \frac{d(E_2)}{2}+rk(E_2).$$
$(3)$.- Follows also from (\ref{cohom0}) and (\ref{cohom1}), Riemann-Roch and the assumption that HN-filtration is HN-general.
From Remark \ref{remigual} we can apply induction. Analysis similar to that in the proof of $m=2$ (now using the HN(i)-extension), shows that $(1),(2)$ and $(3)$ are satisfied for any $m>2$. The details are left to the reader.
\end{proof}
It is known (see \cite[Lemma 1.3.3]{danle}) that if $E$ and $E'$ are two unstable vector bundles with $\mu_{min}(E)>\mu_{max}(E')$ then $Hom(E,E')=0$. Our aim is to study $Hom(E,E)=End(E)$ for unstable bundles.
\section{Algebra of endomorphisms}\label{algebra}
In this section, we determine the structure of the algebra of endomorphisms $End(-)$ of a HN-indecomposable vector bundle of rank $n$, degree $d$ and of HN-length $2$. Using the HN-filtration, we give also an upper bound for $\dim End(-)$ when the HN-length $>2$.
Recall from \cite{atiyah}, that if $E$ is indecomposable vector bundle then
\begin{equation}\label{nil}
End(E)\cong (Id_E)\oplus Nil(E),
\end{equation}
where $Nil(E)$ is the subset $Nil(E)\subset End(E)$ consisting of all nilpotent global endomorphisms of $E$. In \cite{yo4}, the following upper bound
\begin{equation}\label{dimalgs}
\dim End(E)\leq 1+\frac{n(n-1)}{2}
\end{equation}
was given for indecomposable semistable bundles of rank $n$. The bound for indecomposable bundles of HN-length $2$ is established by our next proposition.
\begin{proposition}\label{propalg} Let $E_2$ be a indecomposable unstable bundle and $ 0 = E_0\subset E_1 \subset E_2$ be its HN-filtration. Then, if $E_1$ and $F_1=E_2/E_1$ are indecomposable then
\begin{equation}\label{dimalg2}
\dim End(E_2)\leq 1+\frac{n(n-1)}{2} + h^0(F_1^*\otimes E_1),
\end{equation}
where $F_1=E_2/E_1$.
\end{proposition}
\begin{proof} Let $\rho _1 : 0\to E_1 \stackrel{\iota}{\to} E_2\stackrel{p}{\to} F_1 \to 0$ be the HN(2)-sequence of $E_2$. Assume $n_1:=rk(E_1)$ and $n_2:=rk(F_1)$.
We will use the following sequences
\begin{equation}\label{eqalg1}
0\to H^0(X, E_1\otimes E_2^*)\to H^0(X, E_2\otimes E_2^*)\to H^0(X, F_1\otimes E_2^*)\stackrel{\delta _0}{\to} H^1(X, E_1\otimes E_2^*) \cdots
\end{equation}
\begin{equation}\label{eqalg2}
0\to H^0(X, F_1^*\otimes F_1)\to H^0(X, E_2^*\otimes F_1)\to H^0(X, E_{1}^*\otimes F_1)\stackrel{\delta _1}{\to} H^1(X, F_1^*\otimes F_1) \cdots
\end{equation}
\begin{equation}\label{eqalg3}
0\to H^0(X,F_1^*\otimes E_1)\to H^0(X,E_2^* \otimes E_1 )\to H^0(X, E_{1}^*\otimes E_1)\stackrel{\delta _2}{\to} H^1(X,F_1^*\otimes E_1)
\end{equation}
that are part of the cohomology sequences of $H^*((\rho _1 )\otimes E_2^*)$, $H^*((\rho _1 )^*\otimes F_1)$ and of $H^*((\rho _1 )^*\otimes E_1)$ , respectively.
From (\ref{eqalg1})
\begin{equation}\label{alg2}
\dim End(E_2)\leq h^0(E_1\otimes E_2^*) + h^0( F_1\otimes E_2^*).
\end{equation}
The semistability of the $F_1$ and $E_1$ and the inequalities (\ref{desigualdadmu}) imply that $H^0(X, E_{1}^*\otimes F_1)=0$, hence from (\ref{eqalg2})
\begin{equation}\label{eqigualdad} H^0(X, F_1\otimes E_2^*)= H^0(X,F_1^*\otimes F_1)=End(F_1).
\end{equation}
From (\ref{eqalg3})
$$h^0( E_1 \otimes E_2^* )\leq h^0(F_1^*\otimes E_1) + h^0( E_{1}^*\otimes E_1) $$
since the extension $\delta_2(Id)$ is the element classifying the extension $(\rho_1)^*$ and $\rho ^*_1$ is non-trivial.
Therefore, since $E_1$ and $F_1$ are indecomposables
$$\begin{array}{ccl} \label{alg2f}
\dim End(E_2)&\leq &h^0(E_1\otimes E_2^*) + h^0( F_1\otimes E_2^*)\\
&=& h^0(E_1\otimes E_2^*) + \dim End(F_1)\\
&\leq & h^0(F_1^*\otimes E_1) + \dim End(E_1) + \dim End(F_1)\\
&\leq & h^0(F_1^*\otimes E_1) +\frac{n_1(n_1-1)}{2} +1+\frac{n_2(n_2-1)}{2}+1 \ \ \ \ (\mathbbox{see} \ \ (\ref{dimalgs})) \\
& \leq& h^0(F_1^*\otimes E_1)+ 1+\frac{(n_1^2+n_2^2)}{2} -\frac{n_1+n_2}{2} +1\\
& \leq& h^0(F_1^*\otimes E_1)+ 1+\frac{n(n-1)}{2} +1 -n_1n_2\\
&\leq & h^0(F_1^*\otimes E_1)+ 1+\frac{n(n-1)}{2}
\end{array}
$$
as claimed.
\end{proof}
\begin{corollary}\label{cor2} Let $E_2$ be a indecomposable bundle of rank $n$ of HN-length $2$.
If $E_1$ and $F_1$ are simple then $\dim End(E_2)= 1+ h^0(F_1^*\otimes E_1)$ and $$End(E_2)\cong \mathbb{C}[x_1,\dots , x_k]/(x_1,\dots , x_k)^2,$$
where $k=h^0(F_1^*\otimes E_1)$. Moreover, $E_2$ is simple if and only if $h^0(F_1^*\otimes E_1)=0$.
\end{corollary}
\begin{proof}
The vector bundle $E_2$ is indecomposable of simple type, hence Lemma \ref{dim1}, (2) shows that $h^0(F_1\otimes E_2^*)=1$ and from (\ref{eqalg1}) we conclude that $\delta _0(\lambda p)= 0$. Thus,
$$\dim End(E_2)=h^0(E_1\otimes E_2^*) + h^0( F_1\otimes F_1^*)= h^0(E_1\otimes E_2^*) + 1.$$
It follows from (\ref{eqalg3}) that $\delta _2(Id_{E_1})\ne 0$, hence that $h^0(F_1^*\otimes E_1)=h^0( E_1 \otimes E_2^*)$, and finally that $$\dim End(E_2)= 1+ h^0(F_1^*\otimes E_1).$$
From (\ref{nil}) we deduce that $\dim Nil(E)=h^0(F_1^*\otimes E_1)$.
The map $$\varphi :H^0(F_1^*\otimes E_1) \to Nil(E)$$ defined as $\sigma \mapsto \iota \circ\sigma \circ p$ is a well defined injective homomorphism. Hence, $H^0(F_1^*\otimes E_1)\cong Nil(E)$. Moreover, $(\varphi (\sigma ))^2=0$ and
$$(\varphi (\sigma _1 ))\circ (\varphi (\sigma _2))=(\varphi (\sigma _2))\circ(\varphi (\sigma _1))=0.$$
We thus get $End(E)\cong \mathbb{C}[x_1,\dots , x_k]/(x_1,\dots , x_k)^2$ where $k=h^0(F_1^*\otimes E_1)$. Moreover, $E_2$ is simple if and only if $h^0(F_1^*\otimes E_1)=0$ which is our claim.
\end{proof}
\begin{remark}\begin{em}\label{decomposable} Note that if $F_1^*\otimes E_1$ is special $h^0(F_1^*\otimes E_1)\cdot h^1(F_1^*\otimes E_1)\ne 0$ and if $0\ne \rho:0\to E_1\to E_2\to F_1\to 0\in H^1(X,F_1^*\otimes E_1)$, $E_2$ is indecomposable and $\dim End(E_2) >1$. Moreover, if $F_1^*\otimes E_1$ is no special. $h^0(F_1^*\otimes E_1)\cdot h^1(F_1^*\otimes E_1)= 0$, consequently, either $E_2$ is decomposable or is simple.
\end{em}\end{remark}
The next propositions is a fairly straightforward generalization for unstable bundles of HN-length $m>2$.
\begin{proposition}\label{dimensionr} Let $E$ be an indecomposable unstable vector bundle of rank $n$. If the HN-filtration $ 0 = E_0\subset E_1 \subset \cdots \subset E_m = E$ is HN-indecomposable
then
\begin{equation}\label{dimalgr}
\dim End(E)\leq 1+\frac{n(n-1)}{2} + \sum ^{m-1} h^0(F_i^*\otimes E_{i})
\end{equation}
\end{proposition}
For the proof we will use the following lemma.
\begin{lemma}\label{lema1}\label{lemfin} For all $0<i< j\leq m $
\begin{enumerate}
\item $H^0(X,F_i^*\otimes F_j)=0$ and
\item for all $0<i\leq j\leq m $, $H^0(X, E_i^*\otimes F_j)=0.$
\end{enumerate}
\end{lemma}
\begin{proof} $(1)$ follows from the semistability of the quotients $F_i$'s and the inequalities (\ref{desigualdadmu}).
$(2)$ The semistability of $E_1$ and $F_1$ and the inequalities (\ref{desigualdadmu}) imply that
\[
H^0(X,F_1^*\otimes F_j)=H^0(X,E_1^*\otimes F_j)=0,
\]
Hence, from the cohomology sequence
$$H^*((\rho _1)^*\otimes F_j): 0\to H^0(X,F_1^*\otimes F_j) \to H^0(X,E_2^*\otimes F_j) \to H^0(X,E_1^*\otimes F_j) \to \cdots $$
we conclude that $H^0(X,E_2^*\otimes F_j)=0$.
In the same manner we can see that $H^0(X,E_i^*\otimes F_j)=0$ for all $i\leq j\leq m.$ The detailed verification being left to the reader.
\end{proof}
{\it Proof of Proposition \ref{dimensionr}}
The proof is by induction on the HN-length. The first step of induction is Proposition \ref{propalg}. From the cohomology sequence $H^*(\rho _{m-1} \otimes E_m^*)$
\begin{equation}\label{algm}
\dim End(E_m)\leq h^0(E_{m-1}\otimes E_m^*) + h^0( F_{m-1}\otimes E_m^*).
\end{equation}
We can now proceed analogously to the proof of Proposition \ref{propalg}. From the cohomology sequence $H^*((\rho _{m-1})^*\otimes F_{m-1})$ and Lemma \ref{lema1},(2) it follows that
\[
H^0( F_{m-1}\otimes E_m^*)=H^0(F_{m-1}^*\otimes F_{m-1}).
\]
From the cohomology sequence $H^*((\rho _{m-1})^*\otimes E_{m-1})$
we have the inequality
\[
h^0(E_{m-1}\otimes E_m^*)\leq h^0(End(E_{m-1}))+ h^0(F_{m-1}^*\otimes E_{m-1}).
\]
Recall that $E_m$ is HN-indecomposable.
Now (\ref{algm} ) becomes
$$
\begin{array}{cll}
\dim End(E_m) &\leq & h^0(End(E_{m-1}))+ h^0(F_{m-1}^*\otimes E_{m-1}) + h^0(F_{m-1}^*\otimes F_{m-1})\\
&&\\
&\leq &\frac{rk(E_{m-1})(rk(E_{m-1})-1)}{2} +\frac{rk(F_{m-1})(rk(F_{m-1})-1)}{2} +2+\\
&& \sum ^{m-2} h^0(F_i^*\otimes E_{i}) + h^0(F_{m-1}^*\otimes E_{m-1})\\
&&\\
&\leq & 1+\frac{rk(E_{m})(rk(E_{m})-1)}{2} + \sum ^{m-1} h^0(F_i^*\otimes E_{i}),
\end{array}
$$
which is the desired conclusion.
$
{\Box}$
The following result may be proved in much the same way as Corollary \ref{cor2}.
\begin{corollary} Let $E$ be an indecomposable unstable vector bundle of rank $n$ of simple type. If $ 0 = E_0\subset E_1 \subset \cdots \subset E_m = E$ is its HN-filtration
then
\begin{equation}\label{coro3}
\dim End(E)= 1+ h^0(F_{m-1}^*\otimes E_{m-1}).
\end{equation}
Moreover, $h^0(F_{m-1}^*\otimes E_{m-1})\leq \sum _0 ^{m-2} h^0(F_{m-1}^*\otimes F_{i}).$
\end{corollary}
\begin{proof} We give only the main ideas of the proof. We can proceed analogously to the proof of Corollary \ref{cor2}. The equality $\dim End(E)= 1+ h^0(F_{m-1}^*\otimes E_{m-1})$ follows from Lemma \ref{lemfin} and the cohomology of the exact sequences
$$0\to E_{m-1}\otimes E^*\to E\otimes E^* \to F_{m-1}\otimes E^* \to 0,$$
$$0\to F_{m-1}^*\otimes F_{m-1}\to E^*\otimes F_{m-1} \to E_{m-1}^*\otimes F_{m-1} \to 0,$$
$$0\to F_{m-1}^*\otimes E_{m-1}\to E^*\otimes E_{m-1} \to E_{m-1}^*\otimes E_{m-1} \to 0,$$
since $E$ is of simple type, and hence HN-indecomposable.
The basic idea of the proof of the bound for $h^0(F_{m-1}^*\otimes E_{m-1})$ is to take the cohomology of the sequences
$$0\to E_{i-1}\otimes F_{m-1}^*\to E_i\otimes F_{m-1}^* \to F_{i-1}\otimes F_{m-1}^* \to 0,$$
and proceed by induction.
The details are left to the reader.
\end{proof}
\section{Moduli spaces} \label{moduli}
In this section Proposition \ref{propfm} is given in a more general setting. Our aim is to consider families of semistable bundles. Recall that if the vector bundles $E_1$ and $F_1$ have automorphisms the $\mathbb{C}^*$-action does not identify all the isomorphic classes of the vector bundles in $H^1(X,E_1\otimes F_1^*)$. The simple vector bundles over X with fixed rank and degree possess a
coarse moduli space (see \cite[Corollary 6.5]{KO}), but there is no universal family. It is possibly that it is non-separated and by the work of M. Artin is an algebraic space. Therefore, to construct a moduli scheme of indecomposable vector bundles of HN-length $2$ we will consider those of coprime type $\sigma=(\mu _1, \mu _2)$. Note that if $\mu _1$ is given by $\mu _1=\frac{d_1}{n_1}$ then $\mu _2=\frac{d-d_1}{n-n_1}:=\frac{d_2}{n_2}.$ From now on we tacitly assume that $\mu _1=\frac{d_1}{n_1} >\frac{d_2}{n_2}=\mu _2 $ and denote by $U_{{\mu _1}}(n,d)$ the set of such bundles. Let
$$U_{{\mu _1}}(n,d,k):=\{E\in U_{{\mu _1}}(n,d): \dim End(E)=1+k\}.$$
The moduli space $M(n_i,d_i)$ of semistable vector bundles of rank $n_i$ and degree $d_i$ for $1=1,2$ with $gcd(n_i,d_i)=1$
carries a universal bundle
$\mathcal{U}_i \to X\times {M}(n_i,d_i)$. In general, $\mathcal{U}_i$ is determined up to tensoring by a line bundle lifted from ${M}(n_i,d_i)$. In this paper it will be fixed, unless
otherwise stated.
Denote by $p_{ij}$ the projection of $X\times {M}(n_1,d_1) \times {M}(n_2,d_2)$ in the $ij$-factors. We will denote by ${\mathcal{R}}_{{\mu _1}}$ the $1$st- direct image sheaf
$${\mathcal{R}}_{{\mu _1}}:=\mathcal{R}^1_{{p_{23}}}(p_{12}^*{\mathcal{U}_{{1}}}\otimes p_{13}^*{\mathcal{U}^*_{{2}}})$$ over ${M}(n_1,d_1) \times {M}(n_2,d_2)$.
With the notation $M_i:=M(n_i,d_i)$, the following diagram summarise the notation.
\begin{equation}\label{diag1}
\xymatrix@1{ & p_{12}^*\mathcal{U}_{{1}}\otimes p_{13}^*\mathcal{U}^*_{{2}}\ar[d] &
\mathcal{R}_{{\mu _1}}:=\mathcal{R}^1_{{p_{23}}}(p_{12}^*\mathcal{U}_{{1}}\otimes
p_{13}^*\mathcal{U}^*_{{2}})\ar[d] \\
\mathcal{U}_{{1}}\ar[d] & X\times M_1 \times M_2\ar[dl]^{p_{12}}\ar[r]^{p_{23}}\ar[dr]^{p_{13}} &M_1 \times M_2\\
X\times M_1& & X\times M_2.\\}
\end{equation}
The following theorem follows from \cite{grot}, \cite{lange}, Proposition \ref{indecom} and Lemma \ref{lema1}.
\begin{theorem}\label{teo1} If $\mu _1-\frac{d_2}{n_2}\geq 2g-1$, $U_{{\mu _1}}(n,d)=\emptyset$ and if $\mu _1-\frac{d_2}{n_2}< 2g-1$, $U_{{\mu _1}}(n,d)$ has a projective scheme structure
that makes it an moduli scheme. Moreover, there is a natural isomorphism between $U_{{\mu _1}}(n,d)$ and $Proj(\mathcal{R}_{{\mu _1}})$.
\end{theorem}
\begin{proof}
Let $E\in U_{{\mu _1}}(n,d)$. Note that being of coprime type implies that $E$ is an extension of two stable bundles, that is
$$\rho _1: 0\to E_1 \to E \to F_1 \to 0.$$
Since ${\it Ext}^2(p_{13}^*{\mathcal{U}_{{2}}}, p_{12}^*{\mathcal{U}_{{1}}})=0$, it follows that $\mathcal{R}^1_{{p_{23}}}(p_{12}^*{\mathcal{U}_{{1}}}\otimes p_{13}^*{\mathcal{U}^*_{{2}}})_{{|_{(E_1,F_1)}}} \to
H^1(X, E_1\otimes F_1^*)$ is an isomorphism. From \cite{grot}, \cite{lange}, the coherent sheaf ${\mathcal{R}}_{{\mu _1}}:=\mathcal{R}^1_{{p_{23}}}(p_{12}^*{\mathcal{U}_{{1}}}\otimes p_{13}^*{\mathcal{U}^*_{{2}}})$ parameterizes the classes of extensions of two stable bundles $(E_1,F_1)\in M_1\times M_2$. From Lemma \ref{decomposable},$(1)$ $E$ is indecomposable if and only if $\rho _2\ne 0$. Therefore, the theorem follows from Remark \ref{rem1}.
\end{proof}
\begin{remark}\begin{em} Note that the coherent sheaf ${\mathcal{R}}_{{\mu _1}}$ could be defined even if $\gcd(n_0,d_0)\ne1$. Indeed, if $\gcd(n_i,d_i)\ne1$, there exists an \'etale covering $\widetilde{M_i}$ of $M_i$ such that a universal bundle $\widetilde{\mathcal{U}_i}$ exists on $X\times \widetilde{M_i}$ (see \cite[Proposition 2.4]{nr}).
Thus, the coherent sheaf $\widetilde{{\mathcal{R}}_{{\mu _1}}}:=\mathcal{R}^1_{{p_{23}}}(p_{12}^*{\widetilde{\mathcal{U}_{{1}}}}\otimes p_{13}^*{\widetilde{\mathcal{U}^*_{{2}}}})$ will parameterize extensions of two semistable bundles $(E_1,F_1)\in \widetilde{M_1}\times \widetilde{M_2}$. However, in this case we can not apply Lemma \ref{decomposable},$(1)$ (see Remark \ref{rem1}) since semistable bundles can be non-simple.
\end{em}
\end{remark}
In general, the coherent sheaf ${\mathcal{R}}_{{\mu _1}}=\mathcal{R}^1_{{p_{23}}}(p_{12}^*{\mathcal{U}_{{1}}}\otimes p_{13}^*{\mathcal{U}^*_{{2}}})$ is not locally free. We will give a flattening stratification of $M_1 \times M_2$ for ${\mathcal{R}}_{{\mu _1}}$. To give the schematic semi-continuity stratification of $M_1 \times M_2$ we will use the algebra of endomorphisms and the twisted Brill-Noether theory. For a recent account of the theory we refer the reader to \cite{hitching}. We follow \cite{arb} in the construction of the determinantal varieties that we need.
Let $D$ be an effective divisor on $X$ of degree $d_0>>0$ such that for any $(E_1,F_1)\in M_1 \times M_2$, $H^1(X,E_1\otimes F_1^*\otimes \mathcal{O}(D))=0.$ Let
$$0\to H^0(X,E_1\otimes F_1^*) \to H^0(X,E_1\otimes F_1^*\otimes \mathcal{O}(D)) \stackrel{\phi}{\to} H^0(X,(E_1\otimes F_1^*)_{{|_D}}) \to H^1(X,E_1\otimes F_1^*)\to 0$$
be the cohomology sequence of
$$0\to E_1\otimes F_1^* \to E_1\otimes F_1^*\otimes \mathcal{O}(D) \to (E_1\otimes F_1^*)_{{|_D}} \to 0.$$
Thus, $h^0(E_1\otimes F_1^*) \geq k$ if and only if $rk(\phi) < h^0(E_1\otimes F_1^*\otimes \mathcal{O}(D))-k$.
We want to vary $(E_1, F_1)$ in $ M_1\times M_2.$
Let $\Gamma :=D\times M_1 \times M_2$ be the product divisor in $X\times M_1 \times M_2$. We have the following sequence
$$ 0\to p_{12}^*{\mathcal{U}_{{1}}}\otimes p_{13}^*{\mathcal{U}^*_{{2}}}\to p_{12}^*{\mathcal{U}_{{1}}}\otimes p_{13}^*{\mathcal{U}^*_{{2}}}\otimes \mathcal{O}(\Gamma)\to \mathcal{N} \to 0, $$
over $X\times M_1 \times M_2$, where $\mathcal{N}$ is the quotient $p_{12}^*{\mathcal{U}_{{1}}}\otimes p_{13}^*{\mathcal{U}^*_{{2}}}\otimes \mathcal{O}(\Gamma)/ p_{12}^*{\mathcal{U}_{{1}}}\otimes p_{13}^*{\mathcal{U}^*_{{2}}}.$
Since $H^1(E_1\otimes F_1^*\otimes \mathcal{O}(D))=0$, the direct image induces the complex $\phi :\mathcal{K}_0\to \mathcal{K}_1$ of locally free sheaves over $M_1 \times M_2$ where $\mathcal{K}^0:=\mathcal{R}^0_{{p_{23}}}(p_{12}^*\mathcal{U}_{{1}}\otimes p_{13}^*\mathcal{U}^*_{{2}}\otimes \mathcal{O}(\Gamma))$ is of rank $d(E_1\otimes F_1^*\otimes \mathcal{O}(D) ) +n_1n_2(1-g) $ and $\mathcal{K}^1:=\mathcal{R}^0_{{p_{23}}}(\mathcal{N})$. We follow the notation of \cite{hitching} and denote by $B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)$ the $k$th-determinantal variety of the complex $\phi :\mathcal{K}_0\to \mathcal{K}_1$. That is,
\begin{equation}\label{support}
Supp(B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)):=\{(E_1,F_1)\in M_1 \times M_2: h^0( E_1\otimes F_1^*)\geq k\}.
\end{equation}
\begin{remark}\begin{em}\label{remnotation}When no confusion can arise and to simplify notation, from now on we use the following notation:
\begin{itemize}
\item $n=n_1+n_2$,
\item $d=d_1+d_2$,
\item $n_0=n_1n_2$,
\item $d_0=n_2d_1-n_1d_2$,
\item $h^0=k$ and
\item $h^1=k-d_0 +n_0(g-1)$.
\end{itemize}
\end{em}\end{remark}
Hence, as $k$th-determinantal variety, the {\it expected dimension} of $B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)$ is the number
$$
\begin{array}{cll}
\rho (g,n_1,d_1,n_2,d_2,k)&:=&\dim (M_1 \times M_2)-h^0\cdot h^1\\
&=&(n_1^2+n_2^2)(g-1) +2-k(k-d_0 +n_0(g-1)).\\
\end{array}
$$
and $B^{k+1}(\mathcal{U}_1,\mathcal{U}^*_2)\subseteq Sing B^{k}(\mathcal{U}_1,\mathcal{U}^*_2).$ Moreover, the filtration
$$M_1 \times M_2\supset B^{1}(\mathcal{U}_1,\mathcal{U}^*_2)\supset \cdots \cdots \supset B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)\supset B^{k+1}(\mathcal{U}_1,\mathcal{U}^*_2) \supset \cdots \cdots $$
define a stratification by closed subsets.
We will denote by $\mathcal{Y}_k$ the stratum
$$\mathcal{Y}_k:=B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)-B^{k+1}(\mathcal{U}_1,\mathcal{U}^*_2).$$
That is, $$\mathcal{Y}_k=\{(E_1,F_1)\in M_1 \times M_2: h^0(E_1\otimes F_1^*)=k\}. $$ For any $(E_1,F_1)\in \mathcal{Y}_k$,
$$\dim H^1(X,E_1\otimes F_1^*)=k-d(E_1\otimes F_1^*)+n_1n_2(g-1)=k-d_0+n_0(g-1).$$ Therefore, {\it the restriction of} ${\mathcal{R}}_{{\mu _1}}$ to $\mathcal{Y}_k$, denoted by ${\mathcal{R}}_{{\mu _1}}(\mathcal{Y}_k)\to \mathcal{Y}_k$, is locally free of rank $k-d_0 +n_0(g-1)$. Let
$$\mathbb{P}({\mathcal{R}}_{{\mu _1}}(\mathcal{Y}_k))\to \mathcal{Y}_k $$
be the projective bundle associated to ${\mathcal{R}}_{{\mu _1}}(\mathcal{Y}_k)\to \mathcal{Y}_k$.
{\it The expected dimension} of $\mathbb{P}({\mathcal{R}}_{{\mu _1}}(\mathcal{Y}_k))$ is
$$
\begin{array}{cll}
\beta(g,n_1,d_1,n_2,d_2,k)&:=&\dim (M_1 \times M_2)-h^0h^1 +h^1-1\\
&=&\dim (M_1 \times M_2)-h^1(h^0 -1) -1\\
&=&(n_1^2+n_2^2)(g-1) +1 -(k-1)(k-d_0 +n_0(g-1)).
\end{array}
$$
\begin{remark}\begin{em}\label{remy0} $\mathcal{Y}_0$ is the locus where $ h^0(E_1\otimes F_1^*)=0$, and hence, the indecomposable bundles $E\in H^1(X,E_1\otimes F_1^*)$ are simple (see Corollary \ref{cor2}).
Under the assumption that $M_1 \times M_2\ne B^{1}(\mathcal{U}_1,\mathcal{U}^*_2)$,
$$\mathcal{Y}_0=M_1 \times M_2- B^{1}(\mathcal{U}_1,\mathcal{U}^*_2)$$
is an open set and, in consequence,
$$\dim \mathbb{P}({\mathcal{R}}_{{\mu _1}}(\mathcal{Y}_0))=\dim M_1+\dim M_2 +h^1-1.$$
\end{em}\end{remark}
\begin{remark}\begin{em} \begin{enumerate}
\item If $p_2:B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)\subset M_1\times M_2 \to M_2$ is the projection then for $F_1\in p_2(B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)) $ the inverse image $p_2^{-1}(F_1)$ is the locus
$$B^{k}(\mathcal{U}_1,F^*_1):=\{E_1\in M_1:h^0(E_1\otimes F_1^*)\geq k \}.$$
\item In particular, if $M_2=Pic^0(X)$ and $F_1=\mathcal{O}_X$ then $B^{k}(\mathcal{U}_1,\mathcal{O}_X)$ is the Brill-Noether locus $B(n_1,d_1,k)$. In this case we denote $\mathcal{Y}_k$ as $Y_k$. That is,
$$Y_k=B(n_1,d_1,k)-B(n_1,d_1,k+1).$$
\item In general, the product of two stable bundles is semistable. However, from \cite[Lemma 3.5]{bbnpic} if on of the bundles is general then $E\otimes F^* $ is stable, or if $(n_0,d_0)=1$. Thus, in this case if
$ B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)\ne \emptyset$ then $B(n_0,d_0,k)\ne \emptyset $.
\item If $M_1 \times M_2\ne B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)$ and $B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)\ne \emptyset$ then $\mathcal{Y}_k\ne \emptyset$.
\end{enumerate}
\end{em}\end{remark}
Using the notation of Remark \ref{remnotation} we can now formulate one of our main results.
For any $0\leq k\leq \frac{d_0}{2}+n_0,$ let $U_{{\mu _1}}(n,d,k)$ be the set
$$U_{{\mu _1}}(n,d,k):=\{E\in U_{{\mu _1}}(n,d): \dim End(E)=1+k\}.$$
\begin{theorem}\label{teo2} $U_{{\mu _1}}(n,d,k)=\mathbb{P}({\mathcal{R}}_{{\mu _1}}(\mathcal{Y}_k))$ is coarse moduli space and $U_{{\mu _1}}(n,d,0)$ is a fine moduli space.
Moreover, if $\mathcal{Y}_k$ is irreducible and smooth of dimension $\rho$, then $U_{{\mu _1}}(n,d,k)$ is irreducible and smooth of dimension $\rho + h^1-1. $
\end{theorem}
\begin{proof} From what has already
been proved and Corollary \ref{cor2} it follows that $$U_{{\mu _1}}(n,d,k)=\mathbb{P}({\mathcal{R}}_{{\mu _1}}({\mathcal{Y}_k})).$$
If $E\in U_{{\mu _1}}(n,d,0)$, E is simple and hence from Corollary \ref{cor2} $h^0(E_1\otimes F_1^*)=0$. Thus, $(\mathcal{R}^0_{{p_{23}}}(p_{12}^*{\mathcal{U}_{{1}}}\otimes p_{13}^*{\mathcal{U}^*_{{2}}})){{|_{{\mathcal{Y}_0}}}}=0$. Hence, from
\cite[Corollary 4.5]{lange}, $\mathbb{P}({\mathcal{R}}_{{\mu _1}}(\mathcal{Y}_0))$ parameterize a universal extension
$$0\to q^*(p_{12}^*(\mathcal{U}_1))\otimes p_2^*\mathbb{H}\to \mathcal{E}\to q^*p_{13}^*(\mathcal{U}_2)\to 0,$$
where $q:X\times \mathbb{P}({\mathcal{R}}_{{\mu _1}}(\mathcal{Y}_0)) \to X\times \mathcal{Y}_0$ is the induced map, $p_2:X\times \mathbb{P}({\mathcal{R}}_{{\mu _1}}(\mathcal{Y}_0))\to \mathbb{P}({\mathcal{R}}_{{\mu _1}}(\mathcal{Y}_0))$ is the projection and $\mathbb{H}$ the hyperplane bundle over $\mathbb{P}({\mathcal{R}}_{{\mu _1}}(\mathcal{Y}_0))$.
The universal properties of the family $\mathcal{E}$ imply that the pair $(\mathbb{P}({\mathcal{R}}_{{\mu _1}}(\mathcal{Y}_0)),\mathcal{E}) $ is the fine moduli space for
$U_{{\mu _1}}(n,d,0)$.
If $\mathcal{Y}_k$ is irreducible and smooth of dimension $\rho$, a straightforward computation shows $U_{{\mu _1}}(n,d,k)$ is irreducible and smooth of dimension $\rho + h^1-1$, since $\mathbb{P}({\mathcal{R}}_{{\mu _1}}(\mathcal{Y}_k)) $ is a projective bundle with fibre $\mathbb{P}(H^1(X, E_1\otimes F_1^*))$ at $(E_1,F_1)\in \mathcal{Y}_k$.
\end{proof}
\begin{corollary}\label{corprin0}
If $U_{{\mu _1}}(n,d,k)$ is non-empty, $B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)$ is non-empty.
\end{corollary}
\begin{corollary}\label{corprin} If $\mathcal{Y}_k$ is irreducible and smooth then $H^i(\mathcal{U}_{{\mu _1}}(n,d,k), \mathbb{C})\cong H^i(\mathcal{Y}_k, \mathbb{C})$ for $i\geq 0$.
\end{corollary}
\begin{remark}\begin{em} The moduli space of simple bundles of type $\sigma=(\mu (E_1), \dots ,\mu( E_m/E_{m-1}))$, for $m>2$, are considered in \cite{buns}.
\end{em}\end{remark}
\begin{remark}\begin{em} Theorem \ref{teo2} expresses the equivalence of the existence and topology of the $U_{{\mu _1}}(n,d,k)$ and that of twisted Brill-Noether loci. For some values of $(g, n,d,k)$, non emptiness, dimension, and irreducibility of $B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)$, and of $\mathcal{Y}_k$, are known for general curve (see \cite{hitching}). Thus, as in the Brill-Noether and twisted Brill-Noether theory for vector bundles, it is possible that for special curves the moduli space $U_{{\mu _1}}(n,d,k)$ is even reduced. Thus, the above results differ from the corresponding results for the moduli space of
stable bundles, where non-emptiness, dimension etc. are independent of the curve.
\end{em}\end{remark}
\section{Non-emptiness of $U_{{\mu _1}}(n,d,k)$ }\label{nonempty}
In this section we will prove non emptiness of $U_{{\mu _1}}(n,d,k)$ for some values of $(g,n,d,k)$. First we consider the case of bundles of type $\sigma =(\mu _1, \mu _2)$ where $\mu _1=\frac{d-a}{n-1}$ and $\mu _2=a$ is an integer. That is, the quotient of the maximal destabilizing subbundle is a line bundle of degree $a$. In this case $\mathcal{Y}_k $ is a subset of the Brill-Noether locus $B(n-1,d-an,k)$, and will be denoted as $Y_k$.
The next theorems are applications the known results on $B(n-1,d-an,k)$ and Theorem \ref{teo2}.
\begin{theorem}\label{teop3} Let $n,d,a $ and $k$ be positive integers. Assume $0<d-an<2(n-1)$ and $(n-1,d-an,k)\ne (n-1,n-1,n-1)$. Then for $\mu _1=\frac{d-a}{n-1}$, $U_{{\mu _1}}(n,d,k)$ is non-empty if and only if $k\leq n-1+\frac{d-n(a+1)+1}{g}$.
Moreover, if $U_{{\mu _1}}(n,d,k)$ is non-empty then it is irreducible and smooth of the expected dimension.
\end{theorem}
\begin{proof} The theorem follows from Theorem \ref{teo2}, since from \cite[Theorem A]{bgn} and \cite[A-1 Th\'{e}or\`{e}me]{mer}, $Y_k\subset B(n-1,d-an,k)$ has the required properties.
\end{proof}
\begin{remark}\begin{em} Note that under the assumptions of Theorem \ref{teop3}, the positivity of the expected dimension $\rho (n-1,d-an,1,a,k)$ does not imply non emptiness of $U_{{\mu _1}}(n,d,k) .$
\end{em}\end{remark}
For $g=2$ we have a complete description.
\begin{corollary} Assume $g=2$. If $0<d-an<2(n-1)$ and $0\leq k\leq \frac{d-an}{2}+\frac{n-1}{2}$ then $U_{{\mu _1}}(n,d,k)$ is irreducible and smooth of the expected dimension. If $d-an\geq 2(n-1)$, $U_{{\mu _1}}(n,d,k)= \emptyset .$
\end{corollary}
We now rephrase Theorem \ref{bn},(5),(6) and (7) as follows.
\begin{theorem}\label{teop4}
Let $(g,n-1,d-na,k)$ be integers that satisfies the conditions given in Theorem \ref{bn},(5),(6) and (7).
For general curve, $U_{{\mu _1}}(n,d,k)$ is non-empty and has an irreducible component of the expected dimension and if $X$ is a Petri curve of genus $g\geq 3, n\geq 5$ and $g\geq 2n-4$ then $U_{{\mu _1}}(n,d,n)$ is non-empty.
\end{theorem}
In the above case, to the best of our knowledge, it is know that there exists an irreducible component of the expected dimension, however, in general, it is not known that $B(n,d,k)$ is irreducible.
Let us now consider the general case. We will use the notation given in Remark \ref{remnotation}. Recall that $B^{k}(\mathcal{U}_1,\mathcal{U}^*_2) \subset M_1\times M_2$.
Denote by $M_{12}$ the moduli space of stable bundles ${M}(n_0,d_0 )$. From what has already been proved, we conclude that if $B(n_0,d_0,k)=\emptyset $ then $U_{{\mu _1}}(n,d,k)=\emptyset .$ However, if $B(n_0,d_0,k)\ne \emptyset $ does not imply that $U_{{\mu _1}}(n,d,n)\ne \emptyset .$
Let $\Phi :M_1\times M_2 \to M_{12}$ be the morphism defined as $(E, F)\mapsto E\otimes F^*$. This gives
$$\Phi (B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)) \subset B(n_0,d_0,k).$$ We want to describe $\Phi (B^{k}(\mathcal{U}_1,\mathcal{U}^*_2))$ in some cases.
\begin{remark}\begin{em}\label{remmulti} Let $G$ be a vector bundle of rank $m$ and degree $d_2$ generated by linear subspace $V\subset H^0(X,E)$ of dimension $k_2=m+n_2$. Let $D_{E,V}^*$ be the kernel of the evaluation map (or the syzygy bundle). Let us introduce the temporary notation $F_1^*$ for $D_{E,V}^*$. That is, $F_1^*$ has rank $rk (F_1)=n_2$, degree $d(F_1^*)=-d_2$ and fits into the following exact sequence
\begin{equation}\label{eqdualm}
0\to F_1^*\to V\otimes \mathcal{O}\to G\to 0.
\end{equation}
Tensor (\ref{eqdualm}) with a vector bundle $E_1$ of rank $n_1$ and degree $d_1$ with $h^0(E_1)\geq k_1$. The injectivity of the multiplication map $\mu _{{V,E_1}}: V\otimes H^0(E_1)\to H^0(G\otimes E_1)$ is measure by $h^0(F_1^*\otimes E_1)$. Indeed, if $H^0(F_1^*\otimes E_1)=0$, $\mu _{{V,E_1}}$ is injective. From the cohomology sequence
$$0\to H^0(F_1^*\otimes E_1)\to V\otimes H^0(E_1)\to H^0(G\otimes E_1)\to$$
we obtain the inequality
\begin{equation}\label{eqdesigk}
\dim V\cdot h^0(F)- h^0(G\otimes E_1)\leq h^0(F_1^*\otimes E_1).
\end{equation}
Thus, if
\begin{equation}\label{eqdesigk1}
k\leq k_1k_2-h^0(G\otimes E_1)
\end{equation}
then $k\leq h^0(F_1^*\otimes E_1).$
\end{em}\end{remark}
With the above notation, $n_0=rk(F_1^*\otimes E_1)=n_1n_2$ and $d_0=d(F_1^*\otimes E_1)=d_1n_2-d_2n_1.$ Let $\tilde{n}:= rk(G\otimes E_1)=mn_1$ and $\tilde{d}:=d(G\otimes E_1)=d_2n_1+d_1m$.
Our interest is to apply Remark \ref{remmulti} to situations in which $E_1$ is general in $M(n_1,d_1)=M_1$, $F_1\in M(n_2,d_2)=M_2 $ and $k\geq 0$, in particular when $\mu _{{V,E_1}}$ is not injective. In that case, $(E_1,F_1^*) \in B^k(U_1,U_2^*)$ and $E_1\otimes F_1^*\in B(n_0,d_0,k)$. The following theorems give existence of some Brill-Noether and twisted Brill-Noether loci, and therefore the non emptiness of $U_{{\mu_1}}({n},{d},k)$. The proofs are different of those of \cite{hitching} and some values, to our best knowledge, are not been include in \cite{hitching} nor in Theorem \ref{bn} (see \cite{tbn}).
\begin{theorem}\label{teop05}
Assume that $B(n_1,d_1,n_1+a)$ is non-empty with $a>0$. If $2n_1<d_1<a(g+1)$ and $d_2> 2gm$ then for any $0\leq k\leq (d_2+m(1-g))(n_1+a) -(d_2n_1+d_1m +mn_1(1-g))$, $\mathcal{Y}_k\subset B^k(U_1,U_2^*)$ is non-empty. Moreover, if $\mu_1 = \frac{d_1}{n_1}$ then $U_{{\mu_1}}({n},{d},k)$ is non-empty, where $n=n_1+n_2$ and $d=d_1+d_2$.
\end{theorem}
\begin{proof} Let $G\in M(m,d_2)$ and $E_1\in B(n_1,d_1,n_1+a)$. We want to prove that $0\leq k_1k_2-h^0(G\otimes E_1)$, where $k_1=n_1+a$ and $k_2=d_2+m(1-g)$.
If $d_2> 2gm$, $G$ is generated and from \cite[Theorem 1.2]{but} we have the exact sequence
\begin{equation}\label{eqdual01}
0\to F_1^*\to H^0(G)\otimes \mathcal{O}\to G\to 0
\end{equation}
with $F_1\in M(n_2, d_2) $ where $n_2:= h^0(G)-m=d_2-mg.$
From Remark \ref{remmulti} and (\ref{eqdesigk1})
\begin{equation}\label{eqdesigk2}
h^0(G)\cdot (n_1+a) \leq h^0(G)\cdot h^0(E_1)- h^0(G\otimes E_1)\leq h^0(E_1\otimes F_1^*).
\end{equation}
If $d_1<a(g+1)$ and $d_2> 2gm$ then $\frac{d_1}{a}+g-1<2g<\frac{d_2}{m}$. Hence,
\begin{equation}\label{eqdesig}
d_1m+am(g-1)<ad_2.
\end{equation}
Now add in both sides of \ref{eqdesig} $d_2n_1+mn_1(g-1)$ to obtain
$$
d_2n_1+d_1m +(n_1+a)m(g-1)<d_2(n_1+a)+n_1m(g-1).
$$
Hence,
\begin{equation}\label{eqdesigg}
0<(n_1+a)(d_2+m(1-g))-(d_2n_1+d_1m + mn_1(1-g))=k_1k_2-h^0(G\otimes E_1).
\end{equation}
This gives $0\leq k\leq h^0(E_1\otimes F_1^*)$, for any $0\leq k\leq (d_2+m(1-g))(n_1+a) -(d_2n_1+d_1m +mn_1(1-g))$. It follows that $B^k(U_1,U_2^*)$ is non-empty and, in consequence, $U_{{\mu_1}}(n,d,k)$ is non-empty.
\end{proof}
\begin{theorem}\label{teopetrif} Let $X$ be a Petri curve of genus $g\geq 3$ and $(\mathcal{O}(D), V)$ a general generated linear system of degree $d_2\geq g+1$ and $\dim V =n_2+1$ with $n_2\leq 4 $ or if $n_2\geq 5$ then $g \geq 2n_2-4$. Assume that $B(n_1,d_1,t)$ is non-empty and $\frac{d_2}{n_2}<\frac{d_1}{n_1}$. For any $0\leq k \leq n_2t-n_1d_2$, $ \mathcal{Y}_k \subset B^k(U_1,U_2^*)$ is non-empty and if $n=n_2+n_1$, $d=d_2+d_1$ and $\mu _1=\frac{d_1}{n_1}$, $U_{{\mu_1}}(n,d,k)$, is non-empty.
\end{theorem}
\begin{proof} Under the above conditions there exist an exact sequence
\begin{equation}\label{eqdual02}
0\to F_1^*\to V\otimes \mathcal{O}\to \mathcal{O}(D)\to 0
\end{equation}
such that $F_1\in M(n_2,d_2) $ (see \cite{b}, \cite{bbn1} and \cite{bbn2}).
Let $E_1\in B(n_1,d_1,n_1+a)$ such that $h^0(E_1)=t\geq n_1+a$. From Remark \ref{remmulti} and (\ref{eqdesigk1})
\begin{equation}\label{eqdesigk02}
(n_2+1)\cdot t- h^0(\mathcal{O}(D)\otimes E_1)\leq h^0(F_1^*\otimes E_1).
\end{equation}
From the assumtion $\frac{d_2}{n_2}<\frac{d_1}{n_1}$ and the exact sequence
\begin{equation}
0\to E_1\to \mathcal{O}(D)\otimes E_1\to (E_1)_D\to 0
\end{equation}
we deduce that
\begin{equation}\label{eqdesigk002}
tn_2-d_2n_1 < (n_2+1)\cdot t- h^0(\mathcal{O}(D)\otimes E_1)\leq h^0(F_1^*\otimes E_1).
\end{equation}
Therefore, for any $0\leq k\leq tn_2-d_2n_1$, $\mathcal{Y}_k \subset B^k(U_1,U_2^*)$ is non-empty and if $n=n_2+n_1$, $d=d_2+d_1$ and $\mu _1=\frac{d_1}{n_1}$, $U_{{\mu_1}}(n,d,k)$, is non-empty, as claimed.
\end{proof}
\begin{remark}\begin{em} Under the hypothesis of Theorems \ref{teop05} and \ref{teopetrif} we can choose the $d_2,n_2,n_1$ and $d_1$ such that $(n_0,d_0)=1$, and hence $B(n_0,d_0,k) $ is non-empty.
\end{em}\end{remark}
We conclude now with a description of a smooth point of $U_{{\mu _1}}(n,d,k)$. Recall that $U_{{\mu _1}}(n,d,k)$ is a projective bundle over $\mathcal{Y}_k$.
Let $0\subset E_1\subset E$ be the HN-filtration of $E\in U_{{\mu _1}}(n,d,k)$. Write $E/E_1=F_1$.
Let us describe the tangent bundle of $B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)$ at a point $(E_1,F_1^*)$. We abbreviate $B^{k}(\mathcal{U}_1,\mathcal{U}^*_2)$ to $\mathcal{B}_{12}$. Since $\mathcal{B}_{12}\subset M_1\times M_2$, for any $z:=(E_1,F_1)\in \mathcal{B}_{12}$ we have the sequence
\begin{equation}\label{tan1}
0\to T_{{z}} \mathcal{B}_{12} \to T_{{E_1}}M_1\oplus T_{{F_1}}M_2\to N\to 0
\end{equation}
where $N$ is the normal bundle. Moreover, from the morphism $\Phi :M_1\times M_2 \to M_{12}$ we get
\begin{equation}\label{tan2}
T_{{E_1}}M_1\oplus T_{{F_1}}M_2\stackrel{d\Phi}{\to } T_{{z}}M_{1,2}
\end{equation}
That is,
$$H^1(X,End(E_1))\oplus H^1(X,End(F_1))\stackrel{d\Phi}{\to } H^1(X,End(E_1\otimes F_1^*)). $$
Moreover, from the dual of the multiplication map
$$ H^0(X,E_1\otimes F_1^*)\otimes H^0(X,(E_1\otimes F_1^*)^*\otimes K)\to H^0(X,End (E_1\otimes F_1^*)\otimes K)$$
we have the morphism
$$ H^1(X,End(E_1\otimes F_1^*))\stackrel{\beta}{\to } H^0(X,E_1\otimes F_1^*)^*\otimes H^1(X,E_1\otimes F_1^*). $$
Thus,
$$
\begin{array}{ccc}
H^1(X,End(E_1))\oplus H^1(X,End(F_1))&\stackrel{d\Phi}{\to } & H^1(X,End(E_1\otimes F_1^*))\\
&\eta\searrow & \beta\downarrow \\
&&H^0(X,E_1\otimes F_1^*)^*\otimes H^1(X,E_1\otimes F_1^*)
\end{array}
$$
where $\eta=\beta \circ d\Phi $ and the image of $\eta $ is precisely the normal bundle. Hence from \cite[Proposition 3.5]{hitching} and Theorem \ref{teo2} we conclude
\begin{theorem}\label{teop8} $U_{{\mu _1}}(n,d,k)$ is smooth at $E$ and of expected dimension
$$\beta (g,n_1,d_1,n_2,d_2,k)\ \mathbbox{if and only if} \ \eta \ \mathbbox{is surjective}. \hspace{6cm}$$
\end{theorem}
\end{document}
\end{document} | math | 64,372 |
\begin{document}
\title{The Emergence of 4-Cycles in Polynomial Maps over the Extended Integers}
\begin{abstract}
Let $f(x) \in \ensuremath{\mathbb{Z}}[x]$; for each integer $\alpha$ it is interesting to consider the number of iterates $n_{\alpha}$, if possible, needed to satisfy $f^{n_{\alpha}}(\alpha) = \alpha$. The sets $\{\alpha, f(\alpha), \ldots, f^{n_{\alpha} - 1}(\alpha), \alpha\}$ generated by the iterates of $f$ are called cycles. For $\ensuremath{\mathbb{Z}}[x]$ it is known that cycles of length 1 and 2 occur, and no others. While much is known for extensions to number fields, we concentrate on extending $\ensuremath{\mathbb{Z}}$ by adjoining reciprocals of primes. Let $\ensuremath{\mathbb{Z}}[1/p_1, \ldots, 1/p_n]$ denote $\ensuremath{\mathbb{Z}}$ extended by adding in the reciprocals of the $n$ primes $p_1, \ldots, p_n$ and all their products and powers with each other and the elements of $\ensuremath{\mathbb{Z}}$.
Interestingly, cycles of length 4, called 4-cycles, emerge for polynomials in $\ensuremath{\mathbb{Z}}\left[1/p_1, \ldots, 1/p_n\right][x]$ under the appropriate conditions. The problem of finding criteria under which 4-cycles emerge is equivalent to determining how often a sum of four terms is zero, where the terms are $\pm 1$ times a product of elements from the list of $n$ primes. We investigate conditions on sets of primes under which 4-cycles emerge. We characterize when 4-cycles emerge if the set has one or two primes, and (assuming a generalization of the ABC conjecture) find conditions on sets of primes guaranteed not to cause 4-cycles to emerge.
\end{abstract}
\title{The Emergence of 4-Cycles in Polynomial Maps over the Extended Integers}
\tableofcontents
\section{Introduction}
\subsection{Background and Motivation}
Let $R$ be a ring and $f(x) \in R[x]$ a polynomial over $R$. For any fixed $\alpha \in R$ we define the cycle starting at $\alpha$ to be the sequence $(\alpha, f(\alpha)$, $f(f(\alpha)) := f^2(\alpha)$, $\dots)$. When this cycle consists of only finitely many distinct elements of $R$, then $\alpha$ is said to be pre-periodic. We study the cases where $\alpha$ is periodic; that is, where $f^{n}(\alpha) = \alpha$ for some integer $n$.
\begin{defn}
Given a ring $R$ and a polynomial $f$ in $R[x]$, an \textbf{$n$-cycle} (or a \textbf{cycle of length $n$}) is a sequence of $n$ distinct elements of the ring, $(x_1, \dots, x_n)$, such that \begin{equation} f(x_1) \ = \ x_2,\ \ \ f(x_2) \ = \ x_3,\ \ \ \dots, \ \ \ f(x_n) \ = \ x_1.\end{equation} \end{defn}
It is well known that when $R = \ensuremath{\mathbb{Z}}$ the only possible cycle lengths are 1 and 2, both of which occur. For one proof, see \cite[Lemma 28]{Zieve}. In more generality, the possible cycle lengths for a polynomial in a number field has been related to the unit group of the ring of integers, see \cite{Lenstra}. In his thesis Zieve \cite{Zieve} showed that if $R = \ensuremath{\mathbb{Z}}_{(2)}$, the localization\widehat{\eta}ootnote{That is, $\ensuremath{\mathbb{Z}}_{(2)}$ consists of all fractions where the numerator is an integer and the denominator is not divisible by 2.} of $\ensuremath{\mathbb{Z}}$ at the ideal $(2)$, then the only possible cycle lengths are 1, 2, and 4. It is thus natural to consider rings properly contained between $\ensuremath{\mathbb{Z}}$ and $\ensuremath{\mathbb{Z}}_{(2)}$. In particular, we are interested in the rings $\ensuremath{\mathbb{Z}} \left[1/p_1, \dots, 1/p_n \right]$ which are formed by adjoining the reciprocals of $n$ odd primes $\{p_1, \ldots, p_n\}$ along with all their products and powers with each other and the elements of $\ensuremath{\mathbb{Z}}$. We call $\{p_1,\ldots, p_n\}$ the \emph{\textbf{inversion set}} associated to $\ensuremath{\mathbb{Z}} \left[1/p_1, \ldots, 1/p_n \right]$. Because $\ensuremath{\mathbb{Z}} \subset \ensuremath{\mathbb{Z}} \left[1/p_1, \ldots, 1/p_n \right]$, these intermediary rings of course have cycles of length 1 and 2.
While it is not known which rings $\ensuremath{\mathbb{Z}} \left[1/p_1, \dots, 1/p_n \right]$ have polynomials that exhibit 4-cycles, there is an elegant connection between the existence of 4-cycles in a ring $\ensuremath{\mathbb{Z}} \left[1/p_1\right.$, $\dots$, $\left.1/p_n \right]$ and the solvability of special equations involving products of the primes in its inversion set.
\begin{lem} If there is a polynomial in $\ensuremath{\mathbb{Z}} \left[1/p_1, \ldots, 1/p_n \right]$ that exhibits a 4-cycle, then we can write
\be u_1 \ + \ u_2 \ + \ u_3 \ + \ u_4\ = \ 0, \ee
where $u_i = \pm p_1^{a_{i1}} \cdots p_n^{a_{in}}$ and each $a_{ij}$ is a nonnegative integer. In order to discard pathological examples like $1 - 1 + 1 - 1 = 0$ or $p - p + 1 - 1 = 0$, we also insist that the $u_i$'s have no proper subsum equal to 0.
\end{lem}
This fact is a consequence of Corollary 20 in \cite{Zieve}, and we now provide a paraphrasing of an explanation from his thesis.
To show the necessity of the existence of such a linear relation
suppose that $f \in R[x]$ has the 4-cycle $(x_1, x_2, x_3, x_4)$. As the polynomial $x-y$ divides $f(x) - f(y)$ in $R[x,y]$, we find
\begin{align}
x_{i} \ - \ x_{i-1} \ | \ f(x_{i}) \ - \ f(x_{i-1}) \ \ = \ \ x_{i+1} - x_{i}, \hspace{15 mm} 1 \leq i \leq 4.
\end{align}
\noindent From this, we obtain the chain of divisors
\begin{align}
x_2 \ - \ x_1 \ | \ x_3 \ - \ x_2 \ | \ x_4 \ - \ x_3 \ | \ x_1 \ - \ x_4 \ | \ x_2 \ - \ x_1.
\end{align}
This shows that the pairwise ratios of $u_1 = x_2 - x_1, \ \ u_2 = x_3 - x_2, \ \ u_3 = x_4 - x_3,$ and $u_4 = x_1 - x_4$ are units in $R$. Therefore, if an orbit exists, we are guaranteed a sum of units equaling zero.
Note that this condition is not sufficient for there to be a 4-cycle; see Lemma \ref{lem:cor27}.
\begin{defn} \label{defn:AdmitCycle}
We say that a set of primes $\{p_1, \dots, p_n\}$ \tbf{admits a 4-cycle} if we can write $\epsilon_1 u_1 \ + \ \epsilon_2 u_2 \ + \ \epsilon_3 u_3 \ + \ \epsilon_4 u_4 = 0$ with $\epsilon_i \in \{-1, 1\}$ and $u_i = p_1^{a_{i1}} \cdots p_n^{a_{in}}$ where each $a_{ij}$ is a nonnegative integer; we further require that the $u_i$'s have no zero proper subsum. If a set of primes does not admit a 4-cycle, we say it \tbf{avoids a 4-cycle}. Moreover, we say that this set \tbf{linearly admits (or avoids) a 4-cycle} if each $a_{ij} \in \{0, 1\}$, and in general, we say that this set \tbf{admits (or avoids) a 4-cycle with $n$-powers} if each $a_{ij} \in \{0,1,\ldots,n\}$.
\end{defn}
We have justified in a natural way the requirements of Definition \ref{defn:AdmitCycle} in this section. With only a minor abuse of notation, we apply the same terminology for sets of primes to the ring $R$.
\subsection{Summary of Main Results}
We first attempt to classify inversion sets of low cardinality by whether they admit 4-cycles. Theorems \ref{prop:singleton} and \ref{thm:doubletonclass} also appear in Narkiewicz \cite{Nar}[Theorem 1 \& Theorem 2] with similar proofs.
\begin{theorem}\label{prop:singleton}
$\ensuremath{\mathbb{Z}}[1/p]$ admits a 4-cycle if and only if $p = 2$ or $3$.
\end{theorem}
\begin{theorem}\label{thm:doubletonclass} Fix a positive integer $n$. An inversion set with two elements admits a 4-cycle if any of the following hold:
\begin{enumerate}
\item it is of the form $\{p,p+2\}$, with $p$ and $p+2$ both prime,
\item it is of the form $\{p,p^n-2\}$, with $p$ and $p^n-2$ both prime,
\item it is of the form $\{p,2p+1\}$, with $p$ and $2p+1$ both prime.
\end{enumerate}
\end{theorem}
We prove related results for infinite sets, such as Corollary \ref{cor:upperdensity4cycle} (which states that any inversion set with positive upper density not only admits a 4-cycle, but does so linearly).
We then turn to the much harder problem of constructing inversion sets that are proven to avoid 4-cycles. Our main result assumes a generalized ABC conjecture. If we do not assume this conjecture we can prove that certain sets avoid 4-cycles with $n$ powers (i.e., no prime occurs to a power greater than $n$); see \S\ref{subsec:avoiding4cycles} for detailed constructions.
\begin{theorem}\label{t:main}
If Conjecture \ref{c:BrowBrz} is true, then there exist infinitely many pairs of distinct primes $p_1$ and $p_2$ such that $\ensuremath{\mathbb{Z}}\left[\widehat{\eta}rac{1}{p_1}, \widehat{\eta}rac{1}{p_2}\right]$ does not have a 4-cycle.
\end{theorem}
After proving some useful auxiliary results, we prove the above theorems in \S\ref{sec:main}, and give conditions on the two primes in Theorem \ref{t:main} that, under Conjecture \ref{c:BrowBrz} holding, ensure there is no 4-cycle. We conclude with a discussion of some future research problems in \S\ref{sec:futurework} and some examples in the appendices.
\section{Proofs of Main Results}\label{sec:main}
We begin with a result of Zieve that will be useful throughout the paper.
\begin{lemma}\label{lem:cor27}(Corollary 27, \cite{Zieve}) Let $R$ be an integral domain. There exists a polynomial in $R[x]$ having a 4-cycle in $R$ if and only if there exist units $u$ and $v$ for which $u+v$ and $u+1$ are associates, and for which $1+u+v$ is a unit.
\end{lemma}
This leads us to a partial reformulation of the problem of characterizing sets of primes that admit a 4-cycle.
\begin{proposition}
If the set of primes $\{p_1, \dots, p_n\}$ does not admit a 4-cycle, then the ring $\ensuremath{\mathbb{Z}}\left[1/p_1\right.$, $\ldots$, $\left.1/p_n \right]$ has no polynomial with a 4-cycle.
\end{proposition}
\begin{proof}
By means of contraposition, assume that $R = \ensuremath{\mathbb{Z}}\left[1/p_1, \ldots, 1/p_n \right]$ has a polynomial with a 4-cycle. Then by Lemma \ref{lem:cor27} we know that there exist units $u, v, w \in \ensuremath{\mathbb{R}}$ such that $1 + u + v = w$. Units in $R$ are of the form $p_1^{a_1} \cdots p_n^{a_n}$ with $a_1, \dots, a_n \in \ensuremath{\mathbb{Z}}$. We multiply through to eliminate negative exponents on the primes, which yields an equation of the form
\begin{equation}\epsilon_1t_1 \ + \ \epsilon_2 t_2 \ + \ \epsilon_3 t_3 \ + \ \epsilon_4 t_4 \\ = \ \ 0, \end{equation}
where $\epsilon_i \ \in \ \{-1, 1\}$ and $t_i \ = \ p_1^{a_1} \cdots p_n^{a_n}$ with each $a_i$ a positive integer. Therefore, we see that $\{p_1, \dots, p_n\}$ admits a 4-cycle.
\end{proof}
\subsection{Singleton Inversion Sets}
We turn to the proof of Theorem \ref{prop:singleton}, which states a singleton inversion set $\ensuremath{\mathbb{Z}}[1/p]$ admits a 4-cycle if and only if $p = 2$ or $3$.
\begin{proof}[Proof of Theorem \ref{prop:singleton}]
First, consider the case when $p = 2$. Let $R = \ensuremath{\mathbb{Z}} \left[ 1/2 \right], u = 2, v = 1$. Then, by Lemma \ref{lem:cor27}, $\ensuremath{\mathbb{Z}}\left[ 1/2 \right]$ admits a 4-cycle.
Next, consider the case when $p = 3$. Letting $R = \ensuremath{\mathbb{Z}} \left[ 1/3 \right]$ and $u = v = 1$, by Lemma \ref{lem:cor27} we now have that $\ensuremath{\mathbb{Z}}\left[ 1/3 \right] $ admits a 4-cycle.
Otherwise, let $p > 3$ be a prime. We know that $\ensuremath{\mathbb{Z}} \left[1/p \right]$ admits a 4-cycle if and only if there exist values of $a_i$ such that
\begin{equation}\pm p^{a_1} \pm p^{a_2} \pm p^{a_3} \pm p^{a_4}\ = \ 0\end{equation}
and this equation admits no zero proper subsum.
By multiplying by the appropriate power of $p$, namely $p^{- \min a_i}$, we rewrite the equation as
\begin{equation}1 \pm p^{b_1} \pm p^{b_2} \pm p^{b_3}\ = \ 0, \end{equation}
where $b_1, b_2, b_3 \geq 0$ and at least one sign is negative. Note that, disregarding solutions to this equation that admit a zero proper subsum, looking at this equation $\bmod \, p$ we have either
\begin{equation}1 \equiv 0, \; 2 \equiv 0, \text{ or } 3 \equiv 0 \bmod p,\end{equation}
depending on the number of $b_i = 0$. However, since $p > 3$, this is a contradiction. Therefore $\ensuremath{\mathbb{Z}} \left[1/p \right]$ admits a 4-cycle if and only if $p = 2$ or $3$.
\end{proof}
\begin{example}
The polynomial $f(x) = -\widehat{\eta}rac{2}{3}x^3 + 4x^2 - \widehat{\eta}rac{19}{3} x + 5$ has the 4-cycle $(1,2,3,4)$ in $\ensuremath{\mathbb{Z}} \left[ 1/3 \right]$.
\end{example}
The proposition above answers for almost all rings $\ensuremath{\mathbb{Z}}\left[1/p \right]$ (except $p = 2$) the question of which cycle lengths are allowed. The case of $p = 2$ is handled completely by Narkiewicz \cite{Nar}[Lemmas 5 \& 6]. In addition, Narkiewicz provides every example of a polynomial in $\ensuremath{\mathbb{Z}}[\widehat{\eta}rac{1}{2}]$ with a $4$-cycle.
\subsection{Other Inversion Sets Admitting 4-Cycles} \label{sec:doubletonsadmit4cycles}
As soon as we consider slightly larger inversion sets, say of cardinality 2, the picture turns murky. Using our reformulation of the problem in the introduction, it is often a matter of algebra to find inversion sets with special structure that admit 4-cycles. We give three examples in Theorem \ref{thm:doubletonclass} (restated below for convenience), with full explanation for the first, and suggest others.\\ \
\noindent \textbf{Theorem \ref{thm:doubletonclass}} (Partial Classification of Doubleton Inversion Sets) \emph{Fix a positive integer $n$. An inversion set with two elements admits a 4-cycle if any of the following hold:
\begin{enumerate}
\item it is of the form $\{p,p+2\}$, with $p$ and $p+2$ both prime,
\item it is of the form $\{p,p^n-2\}$, with $p$ and $p^n-2$ both prime,
\item it is of the form $\{p,2p+1\}$, with $p$ and $2p+1$ both prime.
\end{enumerate}
}
\begin{proof}[Proof of (1)] Using our reformulation, $\{p,p+2\}$ admits a 4-cycle if we can write
\begin{equation} u_1 \ + \ u_2 + \ u_3 \ + \ u_4 \ \ = \ \ 0 \end{equation}
with $u_i = \pm p^{a_{i1}}(p+2)^{a_{i2}}$, each $a_{ij}$ a nonnegative integer, and the set of $u_i$'s has no zero proper subsum.
Write
\begin{align}
u_1 \ & = \ p+2 \nonumber \\
u_2 \ & = \ -1 \nonumber \\
u_3 \ & = \ -1 \nonumber \\
u_4 \ & = \ -p.
\end{align}
The result follows.
\end{proof}
Similar proofs for the other cases are given in Appendix \ref{app:doubletonclass}.
To actually construct a polynomial $f(x)$ in $\ensuremath{\mathbb{Z}}[1/p,1/(p+2)][x]$ that admits a 4-cycle, simply choose a 4-cycle $(x_1,x_2,x_3,x_4)$ with appropriate step sizes $u_i$. Using Lagrange interpolation with $f(x_1) \ = \ x_2$, and so on, one can construct $f$.
\begin{example}[Example of (1)] \label{4cyclearbtriv} Consider the polynomial
\begin{equation} f(x) \ \ = \ \ -\widehat{\eta}rac{2}{35}x^3 \ - \ \widehat{\eta}rac{4}{35}x^2 \ + \ \widehat{\eta}rac{221}{35}x \ + \ \widehat{\eta}rac{101}{7}\ \in\ \ensuremath{\mathbb{Z}}\left[\widehat{\eta}rac 15, \widehat{\eta}rac 17 \right][x]. \end{equation}
It is easy to verify that $f$ has the 4-cycle $(-10,-3,-4,-9)$. This example also shows that the step sizes $u_i$ may be reordered.
\end{example}
Other interesting inversion sets of size 2 that admit 4-cycles exist. There are also larger inversion sets that admit 4-cycles. In fact, it is trivial to find inversion sets of arbitrary size that admit 4-cycles: Simply add in primes to an inversion set of size 2 that already admits 4-cycles. For example, to get an inversion set of size 3 that admits 4-cycles, you might consider the polynomial $f$ in Example \ref{4cyclearbtriv} viewed as an element of the ring $\ensuremath{\mathbb{Z}} [1/5,1/7,1/37][x]$. Thus, the bulk of our paper is dedicated to investigating inversion sets that \textbf{avoid} 4-cycles, as this problem is more interesting.
\subsection{Separations and Avoiding 4-Cycles}\label{subsec:avoiding4cycles}
The following theorem from Green is the starting point of our investigations of inversion sets avoiding 4-cycles.
\begin{theorem}[Theorem 1.4, \cite{Green}] \label{thm:Green}
Write $\mathcal{P}$ for the set of primes. Every subset of $\mathcal{P}$ of positive upper density contains a 3-term arithmetic progression.
\end{theorem}
Green's result immediately implies the following useful characterization.
\begin{prop}
If a set of primes does not linearly admit a 4-cycle, then it has density 0 in the primes.
\end{prop}
\begin{proof}
Suppose we have a set of primes with a 3-term arithmetic progression; that is, we have distinct primes $p_1, \ p_2, \ p_3$ such that $p_2 = p_1 + a$ and $p_3 = p_2 + 2a$ where $a \in \ensuremath{\mathbb{Z}}$. Then
\begin{align}\label{eq:arithprog}
p_3 - p_2 - p_2 + p_1 \ = \ p_1 + 2 a - (p_1 + a) - (p_1 + a) + p_1 \ = \ 0,
\end{align} and this set of primes linearly admits a 4-cycle. Thus we have that if a set of primes does not linearly admit a 4-cycle, then it contains no arithmetic progressions. Then, by Theorem \ref{thm:Green}, we have that any set of primes that does not linearly admit a 4-cycle has density 0 in the primes.
\end{proof}
Note that this does not guarantee that $u+v$ and $u+1$ will be associates, to satisfy the conditions of Lemma \ref{lem:cor27}.
\begin{cor}\label{cor:upperdensity4cycle}
Every subset of $\mathcal{P}$ of positive upper density must linearly admit a 4-cycle.
\end{cor}
We now construct sets of primes that linearly avoid 4-cycles. We consider equations of the form
\begin{align} \label{eq:FourTerms}
\epsilon_1t_1 + \epsilon_2 t_2 + \epsilon_3 t_3 + \epsilon_4 t_4 & \ = \ 0, \ \ \ \epsilon_i \in \{-1, 1\},\ \ \ t_i \ = \ p_1^{a_{i1}} \cdots p_n^{a_{in}}, \ \ \ a_{ij} \in \{0, 1\}
\end{align}
and discount trivial solutions, that is, instances where the set $\{\epsilon_i t_i\}$ contains a proper subsum equal to 0, as discussed in Definition \ref{defn:AdmitCycle}.
Without loss of generality, let the term $t_1$ be maximal in the set of terms $\{t_1, t_2, t_3, t_4\}$. If $t_1 > t_2 + t_3 + t_4$ then it is easy to see that $\epsilon_1t_1 + \epsilon_2 t_2 + \epsilon_3 t_3 + \epsilon_4 t_4 \neq 0$ for all choices of $\epsilon_i$.
\begin{lemma}[Separation Lemma]\label{sep_lemma} Suppose we have an ordering of positive terms $\{x_1,x_2,\ldots,x_n\}$ such that $x_{i-1} < x_i$ holds for each $2 \leq i \leq n$. Also suppose that given an $x_i$, for any three distinct terms $x_{j_1}, x_{j_2}, x_{j_3} < x_i$ we have $x_i > x_{j_1} + x_{j_2} + x_{j_3}$. Then there is no set of $t_i$'s that non-trivially satisfy
\[ \epsilon_1t_1 + \epsilon_2 t_2 + \epsilon_3 t_3 + \epsilon_4 t_4 \ = \ 0,
\]
where $t_i \in \{x_i\}$ and $\epsilon_i = \{-1,1\}$.
\end{lemma}
\begin{proof} It suffices to show the result holds for the ordering of terms $\{x_1, x_2, x_3, x_4\}$ such that $x_{i-1} < x_{i}$ for each $2 \leq i \leq 4$ and $x_4 > x_1 + x_2 + x_3$. In this case, let $t_i = x_i$ for $1 \leq i \leq 4$. Then $t_1 + t_2 + t_3 - t_4 < 0$. It is now easy to see that $\epsilon_i t_i + \epsilon_i t_i + \epsilon_i t_i + \epsilon_i t_i \neq 0$ for all choices of $\epsilon_i$.
\end{proof}
With Lemma $\ref{sep_lemma}$, we characterize a set of primes which linearly avoids a 4-cycle in the next result. In particular, we note that the conditions of the lemma are satisfied if $x_i > 3 x_{i-1}$.
\begin{prop}
Fix a positive integer $k$, and let $p_1 > 3, \; p_j > 3\prod_{i=1}^{j-1}p_i$ for $2 \le j \le k$ be prime. Then the inversion set $\{p_1,p_2,\ldots,p_k\}$ linearly avoids a 4-cycle.
\end{prop}
\begin{proof}
We establish that for any two terms $s,t$ drawn from the set of products $\{ p_1^{\alpha_1}\cdots p_{n}^{\alpha_{n}} : \alpha_i \in \{0,1\} \}$, if $s < t$ then $3s < t$, giving enough separation to linearly avoid a 4-cycle.
We proceed by induction on the number of primes in the inversion set. If we only have one prime $p > 3$, then $\{p\}$ linearly (and in fact generally by Proposition \ref{prop:singleton}) avoids a 4-cycle. Now suppose that a set of $k$ primes $\{p_1, \dots, p_k\}$ with the separation properties above linearly avoids a 4-cycle. Consider adding in another prime $p_{k+1}$, satisfying $p_{k+1} > 3p_1 \cdots p_k$. Let
\[ S \ := \ \{p_1^{\alpha_1}\cdots p_{k}^{\alpha_{k}} : \alpha_i \in \{0,1\} \}
\]
denote the set of products for which the induction hypothesis applies, and let
\[ S^* \ := \ \{p_1^{\alpha_1}\cdots p_{k+1}^{\alpha_{k+1}} : \alpha_i \in \{0,1\} \}
\]
denote the set of products extended with the possibility of a $p_{k+1}$ factor.
\begin{cla}
For all $s, t \in S^*$, if $s < t$ then $3s < t$.
\end{cla}
To see this, choose $s = p_1^{\alpha_1}\cdots p_{k+1}^{\alpha_{k+1}}$ and $t = p_1^{\beta_1}\cdots p_{k+1}^{\beta_{k+1}}$ in $S^*$ such that $s < t$.
First, if $\alpha_{k+1} = 0$ and $\beta_{n+1} = 0$, then $s, t \in S$, so that $3s < t$ by hypothesis.
Next, if instead we have $\alpha_{k+1} = 1$ and $\beta_{k+1} = 1$, then $s/p_{k+1},t/p_{k+1} \in S$ with $s/p_{k+1} < t/p_{k+1}$, so that by hypothesis we have $3s/p_{k+1} < t/p_{k+1}$, so that $3s < t$.
Further, suppose $\alpha_{k+1} = 1$ and $\beta_{k+1} = 0$. Then $s \geq p_{k+1} > 3p_1\cdots p_k > t$, which contradicts $s < t$.
Finally, suppose that $\alpha_{k+1} = 0$ and $\beta_{k+1} = 1$. Then we have that $3s \leq 3p_1\cdots p_k < p_{k+1} \leq t$, so that $3s < t$. The claim follows.
Therefore, following the discussion above, for any $k$, provided the separation conditions are met, the inversion set $\{p_1,p_2,\ldots,p_k\}$ linearly avoids a 4-cycle.
\end{proof}
We have discussed the case in which the powers of primes are restricted to first powers (linear avoidance). We now consider the case where we allow powers of primes up to some integer $n$. The ideas behind the proof are very similar.
\begin{prop}
Fix positive integers $n$ and $k$, and let $p_1 > 3, \; p_j > 3\prod_{i=1}^{j-1}p_i^n$ for $2\le j \le k$ be prime. Then the inversion set $\{p_1, p_2, \ldots, p_k\}$ avoids a 4-cycle with $n$-powers.
\end{prop}
\begin{proof}
We establish that for any two terms $s,t$ drawn from the set of products $\{ p_1^{\alpha_1}\cdots p_{n}^{\alpha_{n}} : \alpha_i \in \{0,1,\ldots,n\} \}$, if $s < t$ then $3s < t$, giving enough separation to avoid a 4-cycle with $n$-powers.
We proceed by induction on the number of primes in the inversion set. If we only have one prime $p > 3$, then $\{p\}$ avoids a 4-cycle by Proposition \ref{prop:singleton}). Now suppose that a set of $k$ primes $\{p_1, \dots, p_k\}$ with the separation properties above avoids a 4-cycle with $n$-powers. Consider adding in another prime $p_{k+1}$, satisfying $p_{k+1} > 3p_1^n \cdots p_k^n$. Let
\[ S \ := \ \{p_1^{\alpha_1}\cdots p_{k}^{\alpha_{k}} : \alpha_i \in \{0,1,\ldots,n\} \}
\]
denote the set of products for which the induction hypothesis applies, and let
\[ S^* \ := \ \{p_1^{\alpha_1}\cdots p_{k+1}^{\alpha_{k+1}} : \alpha_i \in \{0,1,\ldots,n\} \}
\]
denote the set of products extended with the possibility of a $p_{k+1}$ factor.
\begin{cla}
For all $s, t \in S^*$, if $s < t$ then $3s < t$.
\end{cla}
To see this, choose $s = p_1^{\alpha_1}\cdots p_{k+1}^{\alpha_{k+1}}$ and $t = p_1^{\beta_1}\cdots p_{k+1}^{\beta_{k_1}}$ in $S^*$ such that $s < t$.
First, if $\alpha_{k+1} = 0$ and $\beta_{k+1} = 0$, then $s,t \in S$, so that $3s < t$ by hypothesis.
Next, if instead we have $\alpha_{k+1} = \beta_{n+1} \neq 0$, then $s/p_{k+1}^{\alpha_{k+1}}, t/p_{k+1}^{\alpha_{k+1}} \in S$ with $s/p_{k+1}^{\alpha_{k+1}} < t/p_{k+1}^{\alpha_{k+1}}$, so that by hypothesis we have $3s/p_{k+1}^{\alpha_{k+1}} < t/p_{k+1}^{\alpha_{k+1}}$, so that $3s < t$.
Further, suppose $\alpha_{k+1} > \beta_{k+1}$. Then we have
\[ s\ \geq\ p_{k+1}^{\alpha_{k+1}}\ >\ 3p_1^n\cdots p_k^n p_{k+1}^{\alpha_{k+1} - 1}\ \geq\ 3p_1^n\cdots p_k^n p_{k+1}^{\beta_{k+1}}\ >\ p_1^n\cdots p_k^n p_{k+1}^{\beta_{k+1}}\ \geq\ t,
\]
which contradicts $s < t$.
Finally, suppose that $\alpha_{k+1} < \beta_{k+1}$. Then we have
\[ 3s\ \leq\ 3p_1^n\cdots p_k^n p_{k+1}^{\alpha_{k+1}}\ <\ p_{k+1}^{\alpha_{k+1}+1} \ \leq \ p_{k+1}^{\beta_{k+1}} \ \leq \ t,
\]
so that $3s < t$.
Therefore, following the discussion above, for any $k$ and $n$, provided the separation conditions are met, the inversion set $\{p_1,p_2,\ldots,p_k\}$ avoids a 4-cycle with $n$-powers.
\end{proof}
Up until now, we have restricted the powers we allow on primes when constructing sets that avoid 4-cycles to some degree. We remove this restriction of possible powers after first proving the following helpful lemma.
\begin{lemma}\label{lem:primeordering}
Let $m > 7$ be an integer.
Fix primes $p_1$ and $p_2$ such that for all non-negative integers $k$ and all integers $\ell$ with $0\le \ell < m$ it holds that \[p_1^{\ell + k}\ <\ p_1^k p_2^\ell\ <\ \widehat{\eta}rac{1}{3} p_1^{\ell + \widehat{\eta}rac{\ell}{m} + k}.\] Then for any non-negative integers $\ell, k, s,$ and $t$ satisfying \begin{enumerate}\item $0 \le t <m$;\item $0 \le \ell < m$;\item $t(1 + \widehat{\eta}rac{1}{m}) + s > \ell(1 + \widehat{\eta}rac{1}{m}) + k$,\end{enumerate} it holds that $p_1^s p_2^t > p_1^kp_2^{\ell}$.
\end{lemma}
\begin{proof}
The lemma is clear whenever $t + s > \ell\left(1 + \widehat{\eta}rac{1}{m}\right) + k$, as $p_1^s p_2^t > p_1^{t + s} > p_1^{\ell\left(1 + \widehat{\eta}rac{1}{m}\right) + k} > p_1^kp_2^\ell$.
Otherwise, we must have that $t(1 + \widehat{\eta}rac{1}{m}) + s > \ell(1+\widehat{\eta}rac{1}{m}) + k \ge t+s$. Since both $\widehat{\eta}rac{t}{m}$ and $\widehat{\eta}rac{\ell}{m}$ are less than $1$, it must be that $t + s = k + \ell.$ In this case, since $t(1 + \widehat{\eta}rac{1}{m}) + s > \ell(1 + \widehat{\eta}rac{1}{m}) + k$, we have that $t/m > \ell/m$ and so $t > \ell$. Therefore, $k = s + t- \ell$. Thus,
\begin{equation}p_1^sp_2^t \ = \ p_1^sp_2^{t-\ell}p_2^\ell > p_1^s p_1^{t-\ell}p_2^\ell \ = \ p_1^k p_2^\ell.\end{equation}
\end{proof}
For our main theorems, we need to assume the following generalization of the ABC conjecture.
\begin{conjecture}\label{c:BrowBrz}[Browkin-Brzezinski]
Given an integer $n > 2$ and an $\epsilon> 0$, there exists a constant $C_{n, \epsilon}$, such that for all integers $a_1, \dots, a_n $ with $a_1+\cdots+ a_n=0$ (and no proper subset having a zero sum), and $\gcd( a_1, \dots, a_n)=1$ we have
\begin{equation}\label{eq:generalabc}
\max(|a_1|,\dots,|a_n|) \ \leq \ C_{n,\epsilon} ({\rm rad}(a_1 \times \cdots \times a_n))^{2n-5+\epsilon},
\end{equation}
where ${\rm rad}(n)$ is the product of the distinct prime factors of $n$.
\end{conjecture}
\begin{proof}[Proof of Theorem \ref{t:main}]
We may assume without loss of generality that $p_2 > p_1$, and that $a_1 + a_2 + a_3 + a_4 = 0$ is a sum of powers of $p_1$ and $p_2$ which vanish, and no sum with fewer terms is zero. Our proof deals with two cases: the first has at least one summand divisible by a large power of $p_2$, and the second assumes that all four summands are only divisible by a small number of multiples of $p_2$.
In applying the conjecture, we are concerned about the case when $n = 4$, and we fix $\epsilon = 1$ for simplicity. Note that each $a_i$ is a product of powers of $p_1$ and $p_2$ and so ${\rm rad}(a_1 \times \cdots \times a_n) = p_1 p_2$. We then get
\begin{equation}\label{eq:ourgeneralabc}
\max(|a_1|,|a_2|,|a_3|,|a_4|)\ \leq\ C_{4, 1} (p_1 p_2)^{4}.
\end{equation}
Choose $m \in \ensuremath{\mathbb{Z}}$ such that $m > \max\{8, \log_3(C_{4,1})\}$ and choose primes $p_1$ and $p_2$ such that the following conditions are satisfied:
\begin{enumerate}
\item $p_1 \geq 18^m$;
\item $p_2 \in \left(3p_1, \widehat{\eta}rac{1}{3}p_1^{1+\widehat{\eta}rac{1}{m}}\right)$.
\end{enumerate}
We first show that we can always find a prime satisfying Condition 2. From Condition 1 we have that $p_1 \geq 18^m$, which we can obviously satisfy for any choice of $m$. We then get
\bea\label{eq:prime1conditions}
p_1\ \geq\ 18^m\ \ \ \
\ensuremath{\mathbb{R}}ightarrow \ \ \ \ \widehat{\eta}rac{1}{3} p_1^{1 + \widehat{\eta}rac{1}{m}}\ \geq\ 2 \cdot 3p_1.
\eea
By Bertrand's postulate (see for example \cite{Da}) there is always a prime in $(x, 2x)$ for all $x > 1$. Thus we see that we can find a prime $p_2 \in (3p_1, 2\cdot 3p_1) \subseteq \left(3p_1, \widehat{\eta}rac{1}{3}p_1^{1+\widehat{\eta}rac{1}{m}}\right)$.
We first deal with the case where at least one summand is divisible by a large power of $p_2$; in particular, we define large to be greater than $m$.
To show that any term divisible by $p_2^k$ where $k \geq m$ cannot be a part of a 4-term sum, we only need to show that $p_2^{m} > C_{4,1}(p_1 p_2)^{4}$.
We required that $m > \max\{8, \log_3(C_{4,1})\}$. This gives us that $C_{4,1} < 3^m$,
because $m > \log_3(C_{4,1})$;
it also gives that $\widehat{\eta}rac{8m + 4}{m^2} < 1$, because $m$ is at least $9$.
Combining these inequalities, we see that
\begin{equation} C_{4,1}^\widehat{\eta}rac{1}{m} p_1^{\widehat{\eta}rac{8m + 4}{m^2}}\ <\ 3p_1.\end{equation}
Finally, since $3p_1 < p_2$, we get that
\begin{equation} p_2\ >\ C_{4,1}^\widehat{\eta}rac{1}{m} p_1^{\widehat{\eta}rac{8m + 4}{m^2}}\end{equation}
and so
\bea
p_2^m & \ \ge \ & C_{4,1} p_1^{\left(8 + \widehat{\eta}rac{4}{m}\right)} \nonumber \\
&>& C_{4,1} p_1^4 p_1^{4\left({1 + \widehat{\eta}rac{1}{m}}\right)} \nonumber \\
&>& C_{4,1} (p_1 p_2)^4,
\eea
the last substitution coming from the fact that $p_2 < p_1^{1+\widehat{\eta}rac{1}{m}}.$
Thus if a 4-term sum exists, it cannot have any term divisible by $p_2^m$.
Next, we invoke Lemma \ref{lem:primeordering} to show that when all the terms are divisible only by powers of $p_2$ that are less than $m$, we have enough separation between possible products of powers of primes to make the sum impossible.
To see this, we need to show that letting $S = \{ p_1^k p_2^\ell | k \in \ensuremath{\mathbb{Z}}_{\geq 0}, 0 \leq \ell < m\}$ that for all $a, b \in S$ if $a < b$ then $3a < b$.
Our lemma gives us an ordering on the elements -- now that we have this ordering, we only need to verify the two following cases.
\begin{enumerate}
\item $3p_1^kp_2^\ell < p_1^{k-1}p_2^{\ell+1}$
\item $3p_2^\ell < p_1^{\ell + 1}$.
\end{enumerate}
Case 1 corresponds to the case in which $a = p_1^k p_2^\ell$ and $b =a \widehat{\eta}rac{p_2}{p_1}$. By our original conditions, we have that $\widehat{\eta}rac{p_2}{p_1} > 3$ and therefore $3a < b$.
In Case 2 we have $a = p_2^\ell$ and $b = p_1^{\ell+1}$, and by our initial conditions, since $\ell \leq m$, we see that
\begin{equation} p_2^\ell\ <\ \widehat{\eta}rac{1}{3}p_1^{\ell + \widehat{\eta}rac{\ell}{m}} \end{equation}
so
\begin{equation} 3p_2^\ell\ <\ p_1^{\ell + 1}.\end{equation}
Therefore, for all $a, b \in S$, if $a < b$ then $3a < b$.
With all of these conditions in place, we then see that $\ensuremath{\mathbb{Z}}\left[1/p_1, 1/p_2\right]$ does not have a 4-cycle.
\end{proof}
\subsection{Numerics}
We end by examining patterns that occur when counting the number of cycles for a given prime list. Based on our observations and the formulation of our results, we conjecture that the number of cycles that occur when considering a specific list of primes correlates with the spacing between the primes. Intuitively, if the primes are spaced far apart, the likelihood of them ``interacting'' in a way that gives a cycle -- that is, finding some combination of four products of the primes that sums to zero -- is small. We examine this conjecture through computation and find the pattern to hold.
Figure \ref{fig:mingap} gives a plot of the number of cycles based on the minimum gap between primes in the inversion set. The points are based on lists of five of the first 50 primes. For example, the inversion set $\{37, 73, 83, 127, 157\}$ admits two cycles. The minimum gap associated to this list is 10. In the plots, the size of the point at any given position represents the number of lists associated with that gap and number of cycles.
\begin{center}
\begin{figure}
\caption{\label{fig:mingap}
\label{fig:mingap}
\end{figure}
\end{center}
\section{Future Work}\label{sec:futurework}
In terms of the main result, there are at least two directions in which to proceed. First, we would like to extend Theorem \ref{t:main} to sets of $n$ primes. The main difficulty with extending our method of proof is constructing a set of primes so that both methods in the proof still apply. In particular, a generalization of Lemma \ref{lem:primeordering} would be needed. Second, we would like to eliminate the dependence of Theorem \ref{t:main} on the generalized ABC conjecture given as Conjecture \ref{c:BrowBrz}.
In Section \ref{sec:doubletonsadmit4cycles} we considered particular shapes of doubleton inversion sets that admit 4-cycles. Related questions of the following flavor suggest themselves: Given an inversion set consisting of a particular odd prime $p$, what is the minimal number of primes we need to add to the set to ensure that it admits a 4-cycle? Per the results in that subsection, this answer might usually be one (if $p$ happens to be a twin prime, for example), but occasionally it might be two. Or, if we have an inversion set with two primes that is known to avoid 4-cycles, how ``easy'' is it to introduce 4-cycles by adding primes to it? (Maybe such sets happen to be difficult to disrupt in this way, and require essentially grafting an entire inversion set that is known to work, such as a pair of twin primes, or maybe not.) These and other questions would be interesting to consider further.
\appendix
\section{Cycle Lengths in $\ensuremath{\mathbb{Z}}[1/2]$}\label{app:p=2case}
A few results from \cite{Zieve} will be helpful. The \tbf{Lenstra constant} of a ring $R$ was defined in \cite{Lenstra} to be
\begin{align}
L(R) =& \sup \{k : \text{there exist $x_1, \ldots, x_k \in \ensuremath{\mathbb{R}}$ such that $x_i - x_j \in R^{*}$} \\
& \text{for all $i, j$ for which $1 \leq i < j \leq k$ } \nonumber \}
\end{align}
\begin{example} $L(\ensuremath{\mathbb{Z}}) = 2$. To show that $L(\ensuremath{\mathbb{Z}}) \geq 2$, just consider the set $\{0,1\}$. To see that the Lenstra constant cannot exceed 2, without loss of generality we can shift all our elements so the first is 0. As the units are $\pm 1$, without loss of generality $x_2 = 1$, and there is no choice for $x_3$ such that $x_3 - 0$ and $x_3 - 1$ are both units.
\end{example}
\begin{lem}[Lemma 22, \cite{Zieve}] \label{lem:Lem22Zieve}
If a polynomial over $R$ has a $p$-cycle in $R$, where $p$ is prime, then $p \leq L(R)$.
\end{lem}
First we prove a helpful lemma.
\begin{lemma}\label{lem:Z2nobigpcycles}
$\ensuremath{\mathbb{Z}} \left[ 1/2 \right]$ admits no cycles of prime length $p > 3$.
\end{lemma}
\begin{proof}
We first compute the Lenstra constant $L\left(\ensuremath{\mathbb{Z}}\left[ 1/2 \right]\right)$. Note that $f(x) = -(3/2) x^{2} + (11/2)x - 2$ has the 3-cycle $(1,2,3)$, so by Lemma \ref{lem:Lem22Zieve}, $L\left(\ensuremath{\mathbb{Z}} \left[ 1/2 \right]\right) \geq 3$.
Then, assume to the contrary that $L\left(\ensuremath{\mathbb{Z}} \left[ 1/2 \right]\right) \geq 4$; that is, assume there exist $x_1, \dots , x_4 \in \ensuremath{\mathbb{Z}} \left[ 1/2 \right]$ such that $x_i - x_j \in \ensuremath{\mathbb{Z}} \left[ 1/2 \right]^*$ for all $i,j$ for which $1 \leq i <j \leq 4$. Then, for some $k_1, k_2, k_3 \in \ensuremath{\mathbb{Z}}$ we have the following:
\be x_1 - x_2\ =\ 2^{k_1} \ee
\be x_2 - x_3 \ =\ 2^{k_2} \ee
\be x_3 - x_4 \ =\ 2^{k_3} \ee
\begin{equation} \label{eqn:ktrick12}
x_1 - x_3 = 2^{k_1}+2^{k_2} \ =\ 2^{k_1}(2^{k_2-k_1}+1)
\end{equation}
\begin{equation} \label{eqn:ktrick23}
x_2 - x_4 = 2^{k_2}+2^{k_3} \ =\ 2^{k_2}(2^{k_3-k_2}+1). \end{equation}
Equation \eqref{eqn:ktrick12} implies that $k_2 - k_1 = 0$, for otherwise, $x_1 - x_3$ would not be a unit. Similarly, Equation \eqref{eqn:ktrick23} implies that $k_3 - k_2 = 0$, so that $k_1 = k_2 = k_3$. Then we have that
\begin{equation}
x_1 - x_4 = 2^{k_1}+2^{k_2}+2^{k_3} = 3 \cdot 2^{k_1},
\end{equation}
so that $x_1 - x_4$ is not a unit, which is the desired contradiction. Thus, $L\left(\ensuremath{\mathbb{Z}}\left[ 1/2 \right]\right) < 4$, so that $L\left(\ensuremath{\mathbb{Z}}\left[ 1/2 \right]\right) = 3$.
Finally, by Corollary 24 in \cite{Zieve}, the only cycles of prime length that are admitted are of length 2 or 3. The result follows.
\end{proof}
We can obtain a slightly stronger result by considering the 3-smooth numbers.
\begin{defn} Let $B$ be a fixed integer. An integer $n$ is said to be \emph{\textbf{$B$-smooth}} if none of its prime factors are larger than $B$. That is, if $p$ is prime and $p \ | \ n$, then $p \leq B$.
\end{defn}
\begin{corollary}
If $\ensuremath{\mathbb{Z}} \left[ 1/2 \right]$ admits a cycle of length $k$, then $k$ is 3-smooth.
\end{corollary}
\begin{proof}
Assume to the contrary that $f(x) \in \ensuremath{\mathbb{Z}}\left[ 1/2 \right][x]$ has a cycle $(a_1,a_2,\ldots,a_k)$ of length $k$, with $k$ not 3-smooth. Then there exists a prime $p > 3$ such that $p \ | \ k$. But then $f^\widehat{\eta}rac{k}{p}(x) \in \ensuremath{\mathbb{Z}}\left[ 1/2 \right][x]$ has a cycle of length $p$, namely $(a_{k/p},a_{2k/p},\ldots,a_k)$, which contradicts Lemma \ref{lem:Z2nobigpcycles}.
\end{proof}
\section{Proofs and Examples of Theorem \ref{thm:doubletonclass} } \label{app:doubletonclass}
\begin{proof}[Proof of Theorem \ref{thm:doubletonclass} (2)] Using our reformulation, we write
\begin{align}
u_1 \ & = \ p^n-2 \nonumber \\
u_2 \ & = \ 1 \nonumber \\
u_3 \ & = \ 1 \nonumber \\
u_4 \ & = \ -p^n.
\end{align}
It is clear that $u_1 \ + \ u_2 \ + \ u_3 \ + \ u_4 = 0$ and there are no zero proper subsums of $u_i$'s, so that $\{p,p^n-2\}$ admits a 4-cycle.
\end{proof}
We give an example of this case when $n=2$ and $p=5$.
\begin{example}[Example of (2)] Consider the polynomial
\begin{equation}
g(x) \ \ = \ \ -\widehat{\eta}rac{2}{575}x^3 \ + \ \widehat{\eta}rac{112}{115}x^2 \ + \ \widehat{\eta}rac{3127}{575}x \ - \ \widehat{\eta}rac{16019}{115} \ \in \ensuremath{\mathbb{Z}}\left[\widehat{\eta}rac 15, \widehat{\eta}rac{1}{23}\right][x].
\end{equation}
It is easy to verify that $g$ has the 4-cycle $(-14,-15,10,9)$. This example also shows that the step sizes $u_i$ can appear with all polarities reversed, too.
\end{example}
\begin{proof}[Proof of Theorem \ref{thm:doubletonclass} (3)] Using our reformulation, we write
\begin{align}
u_1 \ & = \ 2p+1 \nonumber \\
u_2 \ & = \ -p \nonumber \\
u_3 \ & = \ -p \nonumber \\
u_4 \ & = \ -1.
\end{align}
It is clear that $u_1 \ + \ u_2 \ + \ u_3 \ + \ u_4 = 0$ and the set of $u_i$'s has no zero proper subsum, so that $\{p,2p+1\}$ admits a 4-cycle.
\end{proof}
\begin{example}[Example of (3)] Consider the polynomial
\begin{equation}
h(x) \ \ = \ \ -\widehat{\eta}rac{2}{11}x^3 \ - \ \widehat{\eta}rac{146}{55}x^2 \ - \ \widehat{\eta}rac{39}{5}x \ + \ 7/11 \ \in \ensuremath{\mathbb{Z}}\left[\widehat{\eta}rac 15, \widehat{\eta}rac {1}{11}\right][x].
\end{equation}
It is easy to verify that $h$ has the 4-cycle $(-10,-5,-4,1)$.
\end{example}
\end{document} | math | 38,883 |
\begin{document}
\begin{center}
{\large \bf Representations of the Drazin inverse involving idempotents in a ring}\\
{\small \bf Huihui Zhu, Jianlong Chen\footnote{Corresponding author.
Department of Mathematics, Southeast University, Nanjing 210096, China.
Email: [email protected]}}
\end{center}
{ \bf Abstract:} \leftskip0truemm\rightskip0truemm We present some formulae for the Drazin inverse of difference and product of idempotents in a ring. A number of results of bounded linear operators in Banach spaces are extended to the ring case.
\\{ {\textup e}xtbf{Keywords:}} Idempotent, Drazin inverse, Spectral idempotent, involution
\\\noindent { {\textup e}xtbf{2010 Mathematics Subject Classification:}} 15A09, 16U99
\section { \bf Introduction}
Let $R$ be an associative ring with unity $1\neq 0$. The symbols $R^{-1}$, $R^D$ and $R^{\rm nil}$ denote the sets of invertible, Drazin invertible and nilpotent elements of $R$, respectively. The commutant of an element $a\in R$ is defined as ${\rm comm}(a)=\{x\in R:xa=ax \}$.
An element $a\in R$ is said to have a Drazin inverse [6] if there exists $b\in R$ such that
\begin{center}
$b\in {\rm comm}(a)$, $bab=b$, $a-a^2b\in R^{\rm nil}$.
\end{center}
The element $b\in R$ above is unique and denoted by $a^D$. The nilpotency index of $a-a^2b$ is called the Drazin index of $a$, denoted by ${\rm ind}(a)$. If ${\rm ind}(a)=1$, then $a$ is group invertible and the group inverse of $a$ is denoted by $a^\#$. By $a^\pi=1-aa^D$ we mean the spectral idempotent of $a$. It is well known that $a\in R^D$ implies that $a^2\in R^D$ and $(a^2)^D=(a^D)^2$.
Gro{\ss} and Trenkler [7] considered the nonsingularity of $p-q$ for general matrix projectors $p$ and $q$. Koliha and Rako\v{c}evi\'{c} [10] studied the invertibility of the sum $f+g$ when $f$ and $g$ are idempotents in a ring or bounded linear operators in Hilbert or Banach spaces, and proved that $f+g$ is invertible if and only if $f-g$ is invertible. Koliha and Rako\v{c}evi\'{c} [11] obtained the equivalent conditions for the invertibility of $f-g$ in a ring. They also gave applications to bounded linear operators in Banach and Hilbert spaces. Koliha, Rako\v{c}evi\'{c} and Stra\v{s}kraba [12] presented new results on the invertibility of the sum of projectors and obtained the formulae of invertibility of $p-q$ and $p+q$. The problems of Drazin inverse of difference and product of idempotents were studied by many researchers, such as[3,4,5,9,13].
Deng and Wei [5] presented the formulae for the Drazin inverse of difference and product of idempotent bounded linear operators in Banach spaces. In this paper, we give a algebraic proof of the Drazin inverse of sum, difference and product of idempotents in a ring. Moreover, the formulae of the Drazin inverse involving idempotents are established. Hence, we extend the results in [4,5] to the ring case.
\section{\bf Some lemmas }
In what follows, $p$, $q$ always mean any two idempotents in a ring $R$. We state several known results in the form of lemmas without proofs.
\begin{lemma} $[2, {\rm Proposition}~3.1, {\rm Theorem}~3.3]$ Let $\sum=\{ p-q, 1-pq, p-pq, p-qp, p-pqp, 1-qp, q-pq, q-qp, p+q-pq\}$. If one of the elements in the $\sum$ is Drazin invertible, then all other elements in $\sum$ are Drazin invertible.
\end{lemma}
\begin{lemma}$[2, {\rm Theorem}~3.4]$ The following statements are equivalent:\\
$(1)$ $pq\in R^D,$\\
$(2)$ $1-p-q\in R^D$,\\
$(3)$ $(1-p)(1-q)\in R^D$.
\end{lemma}
\begin{lemma} Let $a, b\in R^D$. Then $(ba)^D=b((ab)^D)^2a$. If $ab=ba$, then $(ab)^D=b^Da^D=a^Db^D$.
\end{lemma}
\begin{lemma}$[1,{\rm Theorem~3.6}]$ Let $a, b\in R$. If $1-ab\in R^D$ with ${\rm ind}(1-ab)=k$, then $1-ba\in R^D$ with ${\rm ind}(1-ba)=k$ and
\begin{center}
$(1-ba)^D=1+b((1-ab)^D-(1-ab)^\pi r)a$,
\end{center}
where $r=\displaystyle{\sum_{i=0}^{k-1}(1-ab)^i}$.
\end{lemma}
\section{\bf Main results}
In this section, we present some formulae on the Drazin inverse of difference and product of idempotents of ring $R$.
\begin{definition} Let $p-q\in R^D$. Define $F$, $G$ and $H$ as
\begin{center}
$F=p(p-q)^D$, $G=(p-q)^D p$, $H=(p-q)^D(p-q)$.
\end{center}
\end{definition}
\begin{theorem} Let $p-q\in R^D$. Then $F$, $G$ and $H$ above are idempotents and\\
$(1)$ $F=(p-q)^D (1-q),$\\
$(2)$ $G=(1-q)(p-q)^D$.
\end{theorem}
\begin{proof} Since $p$, $q$ are idempotents, we get
\begin{center}
$p(p-q)^2=(p-q)^2p=p-pqp$.
\end{center}
Note that $a\in R^D$ and $ab=ba$ imply $a^Db=ba^D$ by [6, Corollary 2]. It follows that $p\in {\rm comm}((p-q)^D)^2$. Hence, we have
\begin{eqnarray*}
F &=& p(p-q)^D=p((p-q)^D)^2(p-q)\\
&=&((p-q)^D)^2p(p-q)=((p-q)^D)^2(p-q)(1-q)\\
&=&(p-q)^D (1-q).
\end{eqnarray*}
Next, we prove that $F$ is idempotent. From
\begin{center}
$p(p-q)^D=(p-q)^D (1-q)$,
\end{center}
we have
\begin{eqnarray*}
F^2 &=& (p-q)^D(1-q)p(p-q)^D = (p-q)^D(1-q)(p-q)(p-q)^D \\
&=& p(p-q)^D(p-q)(p-q)^D = p(p-q)^D\\
&=&F.
\end{eqnarray*}
Similarly, $G^2=G=(1-q)(p-q)^D$. It is obvious that $H$ is idempotent and $H=(p-q)(p-q)^D=(p-q)^D(p-q)$.
\end{proof}
We replace $p$ by $q$ in Theorem 3.2 to obtain more relations among $F$, $G$ and $H$.
\begin{corollary} Let $p-q\in R^D$. Then \\
$(1)$ $q(p-q)^D=(p-q)^D(1-p),$\\
$(2)$ $(p-q)^Dq=(1-p)(p-q)^D,$\\
$(3)$ $qH=Hq,$\\
$(4)$ $G(1-q)=(1-q)F.$
\end{corollary}
\begin{proof} (1) We can get (1) and (2) in a similar way of Theorem 3.2.
(3) Since $H=(p-q)^D(p-q)$, we have
\begin{eqnarray*}
qH &=& q(p-q)^D(p-q)=(p-q)^D(1-p)(p-q) \\
&=& (p-q)^D(p-q)q\\
&=&Hq.
\end{eqnarray*}
(4) By Theorem 3.2, we have
\begin{eqnarray*}
G(1-q) &=& (p-q)^Dp(p-q)=(1-q)(p-q)^D(p-q) \\
&=& (1-q)(1-q-1+p)(p-q)^D=(1-q)p(p-q)^D\\
&=&(1-q)F.
\end{eqnarray*}
The proof is complete.
\end{proof}
\begin{theorem} Let $p-q\in R^D$. Then \\
$(1)$ $Fp=pG=pH=Hp,$\\
$(2)$ $qHq=qH=Hq=HqH.$
\end{theorem}
\begin{proof} (1) It is obvious $Fp=pG$, we only need to show $pG=pH$ and $pH=Hp$.
\begin{eqnarray*}
pG &=& p(p-q)^D p = (p-q)^D(1-q)p \\
&=& (p-q)^D (p-q)p\\
&=& Hp.
\end{eqnarray*}
According to Theorem 3.2, we get
\begin{eqnarray*}
pH &=& p(p-q)^D (p-q)=(p-q)^D(1-q)(p-q) \\
&=& (p-q)^D (p-q)p\\
&=& Hp.
\end{eqnarray*}
Hence, (1) holds.
(2) Note that $qH=Hq$ in Corollary 3.3(3). We obtain that $qHq=(Hq)q=Hq$. Since $H$ is idempotent, $HqH=H^2q=Hq$.
Thus, $qHq=qH=Hq=HqH$.
\end{proof}
The following theorems, the main result of this paper, give the formulae of the Drazin inverses of product and difference of idempotents in a ring $R$.
\begin{theorem} Let $p-q\in R^D$. Then \\
$(1)$ $ (1-pqp)^D=[(p-q)^D]^2p+1-p,$\\
$(2)$ $ (p-pqp)^D= [(p-q)^D]^2p=p[(p-q)^D]^2,$\\
$(3)$ $ (p-pq)^D=p[(p-q)^D]^3,$\\
$(4)$ $ (p-qp)^D=[(p-q)^D]^3p,$\\
$(5)$ If ${\rm ind}(p-q)=k$, then
$$(1-pq)^D=1-p+[(p-q)^D]^2[p+pq(1-p)]+[\sum_{i=0}^{k-1}(p-q)^\pi(p-q)^{2i}]pq(p-1).$$
\end{theorem}
\begin{proof}
(1) Note that $1-pqp=(p-q)^2p+1-p$ and $((p-q)^2)^D=((p-q)^D)^2$. Since $(p-q)^2p(1-p)=(1-p)(p-q)^2p=0$, $(1-pqp)^D= [(p-q)^D]^2p+1-p$ by [6, Corollary 1].
(2) Observing that $p-pqp=p(p-q)^2=(p-q)^2p$, we get $(p-pqp)^D=[(p-q)^D]^2p=p[(p-q)^D]^2$ from Lemma 2.3.
(3) Let $x=p[(p-q)^D]^3$. We prove that $x$ is the Drazin inverse of $p-pq$ by showing the following conditions hold.
(a) From $p(p-q)^2=(p-q)^2p=(p-pq)p$, it follows that
\begin{eqnarray*}
~~~ (p-pq)x &=& (p-pq)p[(p-q)^D]^3 = p(p-q)^2[(p-q)^D]^3 \\
&=& p(p-q)^D
\end{eqnarray*}
and
\begin{eqnarray*}
x(p-pq) &=& p[(p-q)^D]^3(p-pq)= p(p-q)^D[(p-q)^D ]^2p(p-q)\\
&=& p(p-q)^D p(p-q)^D= p(p-q)^D\\
&=&(p-pq)x.
\end{eqnarray*}
(b) Note that $(p-pq)x=p(p-q)^D$. We have
\begin{eqnarray*}
x(p-pq)x &=& p[(p-q)^D]^3p(p-q)^D= [(p-q)^D]^2p(p-q)^Dp(p-q)^D \\
&=& [(p-q)^D]^2p(p-q)^D=p[(p-q)^D]^3 \\
&=& x.
\end{eqnarray*}
(c) Since $x(p-pq)^D=p(p-q)^D$, we obtain that
\begin{eqnarray*}
(p-pq)-(p-pq)^2x &=& (p-pq)-(p-pq)p(p-q)^D \\
&=& p(p-q)-p(p-q)^2(p-q)^D \\
&=& p(p-q)(p-q)^\pi.
\end{eqnarray*}
According to $pH=Hp$ and $qH=Hq$, it follows that $p(p-q)(p-q)^\pi=(p-q)^\pi p(p-q)$. By induction, $(p(p-q))^m=p(p-q)^{2m-1}$. Take $m \gammaeqslant {\rm ind}(p-q)$, then
\begin{center}
$[(p(p-q)(p-q)^\pi)]^m=p(p-q)^{2m-1}(p-q)^\pi=0$.
\end{center}
Hence, $(p-pq)-(p-pq)^2x$ is nilpotent.
Therefore, $(p-pq)^D=p[(p-q)^D]^3$.
(4) Use a similar proof of (3).
(5) Since $p-q\in R^D$, $1-pq \in R^D$ according to Lemma 2.1. By Lemma 2.4, we obtain
\begin{center}
$(1-pq)^D=1+p[(1-pqp)^D-(1-pqp)^\pi r]pq$,
\end{center}
where $r=\displaystyle{\sum_{i=0}^{k-1}}(1-pqp)^i$. Note that (1). We have
$$(1-pq)^D=1-p+[(p-q)^D]^2[p+pq(1-p)]+[\sum_{i=0}^{k-1}(p-q)^\pi(p-q)^{2i}]pq(p-1).$$
\end{proof}
\begin{theorem} Let $1-p-q\in R^D$. Then\\
$(1)$ $(pqp)^D=[(1-p-q)^D]^2p=p[(1-p-q)^D]^2,$\\
$(2)$ $(pq)^D=[(1-p-q)^D]^4pq.$
\end{theorem}
\begin{proof} (1) By $pqp=p(1-p-q)^2=(1-p-q)^2p$ and Lemma 2.2, it follows that $(pqp)^D=[(1-p-q)^D]^2p=p[(1-p-q)^D]^2$.
(2) From $pq=ppq$ and Lemma 2.3, we have
\begin{center}
$(pq)^D=p[(pqp)^D]^2pq=[(pqp)^D]^2pq$.
\end{center}
According to (1), we obtain
\begin{center}
$(pq)^D=[(pqp)^D]^2pq=[(1-p-q)^D]^4pq$.
\end{center}
\end{proof}
\begin{theorem} Let $pq\in R^D$. Then\\
$(1)$ $(pq)^D=qp$ if and only if $pq=qp,$\\
$(2)$ $(pq)^D=(pqp)^D-p((1-q)(1-p))^D,$\\
$(3)$ $(pq)^Dpq=(pqp)^Dpq.$
\end{theorem}
\begin{proof}
(1) If $pq=qp$, we can get $(pq)^D=qp$ by a direct calculation.
Conversely, since $(pq)^D=qp$, we obtain $pqp=qpq$ and $qp=qpqp$. On the other hand, we also get $pqpq=pq$. Hence,
\begin{center}
$pq=pqpq=ppqp=pqp=qpq=qpqp=qp$.
\end{center}
(2) By Theorem 3.5(4), we have $(p-qp)^D=((p-q)^D)^3p$ and $$(q-pq)^D=((q-p)^D)^3q=-((p-q)^D)^3q.$$
Hence,
$$(q-pq)^D+(p-qp)^D=((p-q)^D)^3(-q)+((p-q)^D)^3p=((p-q)^D)^2. \eqno(3.1)$$
We replace $p$, $q$ by $1-p$ and $q$ in the equality (3.1) to get
$$(pq)^D+((1-q)(1-p))^D=((1-p-q)^D)^2. \eqno(3.2) $$
Multiplying the equality (3.2) by $p$ on the left yields
$$p(pq)^D+p((1-q)(1-p))^D=p((1-p-q)^D)^2. \eqno(3.3) $$
Note that $p(pq)^D=p(pq)(pq)^D(pq)^D=(pq)^D$ and Theorem 3.6. We have $$(pq)^D=(pqp)^D-p((1-q)(1-p))^D.$$
(3) By Lemma 2.3, we have
\begin{center}
$(pqp)^Dpq=pq((pq)^D)^2pq=(pq)^Dpq$.
\end{center}
Completing the proof.
\end{proof}
\begin{theorem} Let $1-pq\in R^D$. Then $p-q\in R^D$ and
\begin{center}
$(p-q)^D=(1-pq)^D(p-pq)+(p+q-pq)^D(pq-q)$.
\end{center}
\end{theorem}
\begin{proof}
By Theorem 3.5(5), we have
$$(1-pq)^D=1-p+[(p-q)^D]^2[p+pq(1-p)]+[\sum_{i=0}^{k-1}(p-q)^\pi(p-q)^{2i}]pq(p-1).\eqno(3.4)$$
we replace $p$ and $q$ by $1-p$ and $1-q$ respectively in the equality (3.4) to obtain
$$(p+q-pq)^D=p+[(p-q)^D]^2[1-p+(1-p)(1-q)p]+[\sum_{i=0}^{k-1}(p-q)^\pi(p-q)^{2i}](1-p)(1-q)p.\eqno(3.5)$$
Multiplying the equality (3.4) by $p-pq$ on the right yields $$(1-pq)^D(p-pq)=p(p-q)^D=(p-q)^D(1-q).\eqno(3.6)$$
Multiplying the equality (3.5) by $pq-p$ on the right yields
$$(p+q-pq)^D(pq-q)=(p-q)^Dq. \eqno(3.7)$$
Notice that (3.6) and (3.7). One has
\begin{eqnarray*}
(1-pq)^D(p-pq)+(p+q-pq)^D(pq-q) &=& (p-q)^D(1-q)+(p-q)^Dq\\
&=&(p-q)^D.
\end{eqnarray*}
The proof is complete.
\end{proof}
An involution $x\mapsto x^*$ in a ring $R$ is an anti-isomorphism of degree 2, that is,
\begin{center}
$(a^*)^*=a$, $(a+b)^*=a^*+b^*$, $(ab)^*=b^*a^*$
\end{center}
for all $a, b \in R$. Idempotent element $p$ is a \emph{projector} if $p=p^*$. An element $a\in R$ is \emph{$\ast$-cancellable} if
\begin{center}
$a^*ax=0\Rightarrow ax=0$ and $xaa^*=0\Rightarrow xa=0$.
\end{center}
A ring $R$ is \emph{$\ast$-reducing} if all elements in $R$ are $\ast$-cancellable. This is equivalent to $a^*a=0\Rightarrow a=0$ for all $a\in R$.
\begin{theorem} Let $R$ be a $\ast$-reducing ring and $p$, $q$ be two projectors. Then\\
$(1)$ $(p-q)^D=p-q$ if and only if $pq=qp$,\\
$(2)$ If $6\in R^{-1}$, then $(p+q)^D=p+q$ if and only if $pq=qp=0$.
\end{theorem}
\begin{proof} (1) If $pq=qp$, it is easy to check that $(p-q)^D=p-q$.
Conversely, $(p-q)^D=p-q$ implies that $(p-q)^3=p-q$, that is $pqp=qpq$. Since $R$ is a $\ast$-reducing ring and $(pq-qp)^*(pq-qp)=0$, it follows that $pq=qp$.
(2) If $pq=qp=0$, by [6, Corollary 1], $(p+q)^D=p+q$.
Conversely, $(p+q)^D=p+q$ implies $(p+q)^3=p+q$, i.e.,
$$2pq+2qp+pqp+qpq=0. \eqno(3.8)$$
Multiplying the equality (3.8) by $p$ on the left yields
$$2pq+3pqp+pqpq = 0.\eqno(3.9)$$
Multiplying the equality $(3.8)$ by $q$ on the right yields
$$2pq+3qpq+pqpq=0. \eqno(3.10)$$
Combining the equalities $(3.9)$ and $(3.10)$, we obtain that $pqp=qpq$. By the proof of (1), we get $pq=qp$. Hence, equality $(3.8)$ can be simplified to $6pq=0$. Therefore, $pq=qp=0$.
\end{proof}
Let $p$, $q$ be two idempotents in a Banach algebra. Then, $p+q$ is Drazin invertible if and only if $p-q$ is Drazin invertible [9]. However, in general, this need not be true in a ring. For example, let $R=\mathbb{Z}$ and $p=q=1$. Then $p-q=0$ is Drazin invertible, but $p+q=2$ is not Drazin invertible. Next, we consider what conditions $p$ and $q$ satisfy, $p-q\in R^D$ implies that $p+q\in R^D$. Deng and Wei [5] proved the following results for bounded linear operators in Banach spaces. Now we present it in a ring without proofs.
\begin{theorem} Let $p-q\in R^D$. If $F$, $G$ and $H$ are given by Definition $3.1$ and $(p+q)(p-q)^\pi \in R^{\rm nil}$, then\\
$(1)$ $(p+q)^D=(p-q)^D (p+q)(p-q)^D,$\\
$(2)$ $(p-q)^D=(p+q)^D (p-q)(p+q)^D,$\\
$(3)$ $(p-q)^\pi=(p+q)^\pi,$\\
$(4)$ $(p-q)^D= F+G-H,$\\
$(5)$ $(p+q)^D=(2G-H)(F+G-H).$
\end{theorem}
\begin{theorem} Let $p-qp\in R^D$. Then \\
$(1)$ $(p-q)^D=(p-q)^2((p-qp)^D-(q-qp)^D),$\\
$(2)$ If $(p+q)(p-q)^\pi\in R^D$, we have $p((p+q)^D-(p-q)^D)(p-q)^2=0$.
\end{theorem}
\begin{proof} Since $(p-qp)^D=((p-q)^D)^3p$ and $(q-qp)^D=q((q-p)^D)^3$, we get
\begin{eqnarray*}
(p-q)^2((p-qp)^D-(q-qp)^D) &=& (p-q)^2(((p-q)^D)^3p-q((q-p)^D)^3) \\
&=& (p-q)^Dp+q(p-q)^D\\
&=& (p-q)^Dp+(p-q)^D(1-p)\\
&=& (p-q)^D.
\end{eqnarray*}
(2) Note that $(p+q)^D=(p-q)^D(p+q)(p-q)^D$. We have
\begin{eqnarray*}
p((p+q)^D-(p-q)^D)(p-q)^2 &=& p(p-q)^D(p+q)(p-q)-p(p-q) \\
&=& p(1-q+1-p)(p-q)^D(p-q)-p(p-q)^2(p-q)^D\\
&=& p(p-q)^2(p-q)^D-p(p-q)^2(p-q)^D\\
&=& 0.
\end{eqnarray*}
\end{proof}
In Theorem 3.10, we know that if $p-q\in R^D$ and $(p+q)(p-q)^\pi$ is nilpotent, then $p+q\in R^D$. Naturally, does the reverse statement above hold? Next, we give an example to illustrate it is not true.
Take $R= \mathbb{Z}_7$, \begin{center}
$p=\left(
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}
\right)
\in M_2(R)$ and $q=\left(
\begin{array}{cc}
1 & 0 \\
0 & 0 \\
\end{array}
\right)
\in M_2(R)$,
\end{center}
$p$ and $q$ are obvious two idempotents. Moreover, \begin{center}
$p+q=\left(
\begin{array}{cc}
2 & 0 \\
0 & 1 \\
\end{array}
\right)
$, $p-q=\left(
\begin{array}{cc}
0 & 0 \\
0 & 1 \\
\end{array}
\right)
$.
\end{center}
Hence, $p+q$ and $p-q$ are Drazin invertible. However, $(p+q)(p-q)^\pi$ is not nilpotent.
Koliha and Rako\v{c}evi\'{c} [10] proved that $p-q\in R^{-1}$ implies that $p+q\in R^{-1}$ for idempotents $p$ and $q$ in a ring $R$. Moreover, if $2\in R^{-1}$, then $p-q\in R^{-1}$ is equivalent to $p+q \in R^{-1}$ and $1-pq\in R^{-1}$. Hence, we have the following result.
\begin{corollary} $[12, {\rm Theorem~2.2}]$ Let $p-q\in R^{-1}$. If $F=p(p-q)^{-1}$ and $G=(p-q)^{-1}p$, then\\
$(1)$ $(p+q)^{-1}=(p-q)^{-1} (p+q)(p-q)^{-1},$\\
$(2)$ $(p-q)^{-1}=(p+q)^{-1}(p-q)(p+q)^{-1},$\\
$(3)$ $(p-q)^{-1}= F+G-1,$\\
$(4)$ $(p+q)^{-1}=(2G-1)(F+G-1)$.
\end{corollary}
\centerline {\bf ACKNOWLEDGMENTS} This research is supported by the National Natural Science Foundation of China (10971024),
the Specialized Research Fund for the Doctoral Program of Higher Education (20120092110020),
and the Natural Science Foundation of Jiangsu Province (BK2010393) and the Foundation of Graduate
Innovation Program of Jiangsu Province(CXLX13-072).
\end{document} | math | 16,625 |
\begin{document}
\title{Wigner coefficient for Lie algebras of series $B,C,D$ and a base of Gelfand-Tsetlin type. hanks
{The work was supported by grants NSch-5998.2012.1, RFFI-12-01-31414, MK-4594.2013.1}
\renewcommand{\abstractname}{}
\begin{abstract} For the Lie algebras $g_n= \mathfrak{o}_{2n+1},\mathfrak{sp}_{2n},\mathfrak{o}_{2n}$
a simple construction of a base in an irreducible representation is
given. The construction of this base uses the method of
$Z$-invariants of Zhelobenko and the technique of Wigner
coefficients, which was applied by Biedenharn and Baird to the
construction of a Gelfand-Tsetlin base in the case
$\mathfrak{gl}_n$. A relation between matrix elements and Wigner
coefficients for
$g_n$ and analogous objects for $\mathfrak{gl}_{n+1}$ is established.
\end{abstract}
\section{Introduction}
One can construct explicitly a representation of a simple Lie
algebra of the type $A$ using the technique of Wigner coefficients.
In the case $\mathfrak{sl}_2$ such a construction was obtained by
Wigner and Racah, and in the case
$\mathfrak{sl}_n$ it was obtained by Biedenharn, Baird, Louck and others. The aim
of the present paper is a generalization of this technique to the
case of simple Lie algebras $g_n $ of types $B,C,D$. Such
constructions
can be applied in the nuclear physics, for example in the theory of nuclear shells.
The construction given in the present paper establishes a remarkable relation between Gelfand-Tsetlin bases for the algebras of series
$A$ and of series $B,C,D$. Moreover we express matrix elements for generators in the base and Wigner coefficients for algebras of series $B,C,D$ through the analogous objects of series $A$.
The construction in the paper is inductive. For the algebras
$g_n=\mathfrak{o}_{2n+1},\mathfrak{sp}_{2n},\mathfrak{o}_{2n}$ the
constructions are different in the case $n=1$ and also in the case
$n=2$. But the consideration for $n>2$ are similar for all algebras
and also they are similar to analogous constructions for the
algebra $\mathfrak{gl}_{n+1}$. This fact plays a key role in all
constructions. It is a direct corollary of the fact that for the
Dynkin diagrams $A_{n+1},B_{n},C_{n},D_{n}$ transform in a similar
way when we change $2$ to $n$.
The construction uses a well-known inductive procedure of
Gelfand and Tsetlin. The main step in this construction is an
investigation of a branching of an irreducible representation of
$g_n$ when one restricts the algebra $g_{n}\downarrow g_{n-1}$. For this
investigation we use the method of $Z$-invariants of Zhelobenko.
To obtain formulas for the action of generators of the algebra in
the base we use the technique of Wigner coefficients. More precise, the matrix
elements are expressed through the simplest Wigner coefficients that
correspond to a decomposition of a tensor product of an arbitrary
representation with a standard representation of the algebra.
Earlier analogous results were obtained by Biedenharn and Baird.
They expressed matrix elements for operators $E_{i,i-1}$ in the
Gelfant-Tsetlin base for the algebras of series $A$ through Wigner coefficients \cite{1963}.
I.M. Gelfand in the late 80-th several times said about the desire
to generalize the works of
Biedenharn and Louck. We have understood this as a problem of a generalization of the construction
of Biedenharn and Louck for the Lie algebras of series $A$ to the case of algebras of series
$B,C,D$. This is done in the present paper.
The problem of construction of base for orthogonal algebras
algebras was investigated in the works of Gelfand
and Tsetlin, but it's construction is based on restrictions $\mathfrak{o}_N\downarrow
\mathfrak{o}_{N-1})$ \footnote{Thus this base is not a base for a series $B$ or $D$ since the algebras of both series are involved in it's construction}.
In \cite{zh} using the method of $Z$-invariants of
Zhelobenko there was constructed a Gelfanf-Tsetlin type base for
orthogonal algebras (based on and for symplectic algebras
(see also
\cite{Sch1}, \cite{Sch2}, \cite{Sch3}). But in these papers only an indexation is constructed, the formulas for the action of generators are not obtained.
Nevertheless there exists a construction of a Gelfand-Tsetlin type
base for the algebras of series $B,C,D$ which belongs to Molev
\cite{M}. But it uses a difficult technique of
Mickelsson-Zhelobenko algebras and the action of Jangians on the
multiplicity spaces\footnote{ An indexation of base vectors in the base that is constructed in the present paper is the same as
in the Gelfand-Tsetlin-Molev \cite{M}. But we have not managed to construct an exlicit isomorphism betweeen
our base and the Gelfand-Tsetlin-Molev base}.
The construction given in the present paper is
much simpler. Is's main advantage is that it establishes a remarkable relation between Gelfant-Tsetlin bases for
series $A,B,C,D$. Also as a by-product of our construction we obtain
explicit formulas for Wigner coefficients for algebras of series
$B,C,D$. The indexation of base vectors is the same as Molev's. But
also our base is not orthonormal as Molev's.
The Gelfand-Tsetlin type bases for Lie $g_n $ of
types $B_n,C_n,D_n$ based on restrictions $g_n\downarrow g_{n-1}$
are important in nuclear physics. Such a base is used in
problems where the algebra of five-dimensional quasi-spin (isomorphic to
$\mathfrak{o}_5=\mathfrak{sp}_4$) is involved, for example in the
theory of nuclear shells, \cite{He}, in the Bohr-Mottelson model
\cite{BM}, in the model of interacting bosons \cite{IBM}, in the
models of high-temperature superconductivity \cite{16}, \cite{17},
\cite{GoLi}.
The typical application of Wigner coefficients is the following. If one
identify an irreducible representation with a (quasi)particle, then Wigner
coefficients that define a decomposition of a tensor product of two representation
describe a spectrum of (quasi)particles that appear after their interaction
(см. \cite{Lip}).
The Wigner coefficients that are calculated in the present paper correspond to the adding of one (quasi)particle. Thus in the case of a nuclear shell model the multiplication to the standard representation corresponds to the adding to the system of one nucleon that is a proton or a neutron. The explicit formulas for Wigner coefficients allow to write the selection rulers for the quantum numbers that define the states of quasi-spin and calculate the probabilities of transitions into these states.
There exist also completely different applications. Thus the
high-temperature decomposition in the classical $N$-vector model is
described using Wigner coefficients of the orthogonal algebra
$\mathfrak{o}_N$ \cite{joyce}.
In all these problems mostly the Wigner coefficients that define the
decomposition of two symmetric representations are used. Such
coefficients were explicitly obtained in many particular cases in
\cite{Gav}, \cite{Ki}, \cite{Junkr}, \cite{Al1}, \cite{Al2}. But mention
the authors of these papers used the Gelfand-Tsetlin base whose
construction is based on restrictions $\mathfrak{o}_N\downarrow\mathfrak{o}_{N-1}$
($\mathfrak{o}_5\downarrow\mathfrak{o}_4\downarrow\mathfrak{o}_3$ in the case $\mathfrak{o}_5$).
This base is not natural form the physical point of view. In the
present paper we use the base for a representation of
$\mathfrak{o}_5$, whose construction is based on restrictions within
the series $B$.
\subsection{The structure of the paper}
The structure of the paper is the following. In Section \ref{s2}
definitions and notations are introduced.
The technical details, the discussion of definitions is placed in
Appendix \ref{appendi}.
In Section \ref{oldvector} the method of $Z$-invariants is
explained, it allows to solve effectively the problem of description
of a branching of a representation when one restricts an algebra.
In Section \ref{shema} we give the scheme of a solution of the problem
of restriction $g_n\downarrow g_{n-1}$\footnote{When we write "the
problem of restriction $g_n\downarrow g_{n-1}$" we mean the problem
of an explicit description of a base in the space of
$g_{n-1}$-highest vectors in a
$g_{n}$-representation.}. We explain how to construct a Gelfand-Tselin type
base, obtain coefficients and restricted Wigner coefficients and
matrix elements of the generators.
The main construction is given in Sections \ref{step1}, \ref{step2},
\ref{step3}. In these Section the Gelfand-Tsetlin type base for
series $B,C,D$ and explicit formulas for the action of generators
are constructed for $n=1$, $n=2$, $n>2$ respectively. Note that
Wigner coefficients for $g_{n-1}$ are calculated when the algebra
$g_n$ is considered.
The cases $n=1$, $n=2$ are considered in Sections
\ref{step1}, \ref{step2} separately for each series $B,C,D$.
In Section \ref{step3} further steps are discussed, they are similar for all series.
In all cases all values are expressed through the analogous values
for the algebras $\mathfrak{gl}_N$, which were obtained in an explicit
form by Biedenharn and Baird in \cite{1963} and also by Zhelobenko
in \cite{zh}.
In Appendix the facts from the representation theory that are not
well-known are given. Mostly that can be found in \cite{1968}. The
reader can find them here of in Appendix \ref{appendi}.
\section{Basic definitions}
\label{s2}
In the present Section the basic definitions and notations are introduced. See also Appendix \ref{appendi}.
In the paper we use the Lie algebras $\mathfrak{sp}_{2n}$ and $\mathfrak{o}_{N}$
in split realization. These algebras act in the space with coordinates $x_{-n},...,x_{-1},x_{1},...,x_{n}$, in the cases $\mathfrak{sp}_{2n}$ and $\mathfrak{o}_{2n}$, and in the space
with coordinates $x_{-n},...,x_{-1}, x_{0},x_{1},...,x_{n}$ in the case $\mathfrak{o}_{2n+1}$.
The generators of the symplectic algebra are the matrices $$F_{i,j}=E_{i,j}-sign(i)sing(j)E_{-j,-i},$$
and the generators of the orthogonal algebra are the matrices $$F_{i,j}=E_{i,j}-E_{-j,-i}.$$
\subsection{Gelfand-Tsetlin tableaux}
\label{baspred}
In the presen paper a Gelfand-Tsetlin base in an irreducible representation of
$g_n$ is constructed. Base vectors are indexed by Gelfand-Tsetlin type tableaux, these tableaux have similar structure, let us describe it here.
Let us be given an irreducible representation $V$ of the algebra $g_n$ with the highest weight $[m_{-n,n},...,m_{-1,n}]$. Then the base vectors are indexed by tableaux $(m)$ of type
\begin{align}\label{gcmspo}
(m)=&\begin{pmatrix} [m]_{n}\\
[m']_{n}\\
[m]_{n-1}\\
[m']_{n-1}\\
...\\
[m]_1\\
[m']_{1}\\
\end{pmatrix}.
\end{align}
The row $[m]_{n}$ is the highest weight of the considered representation
$V$, the row $[m]_{n-1}$ is the highest weight of an irreducible
$g_{n-1}$-representation that contains the vector $(m)$. The row $[m']_{n-1}$ is a base element in the space of $g_{n-1}$-highest vectors with highest weight
$[m]_{n-1}$ and so on. The structure of the rows depends on the series $B_n$, $C_n$ or $D_n$.
The weight of the vector $(m)$ is denoted as
$\cal Delta(m)$.
Denote the tableau in which all indices take the maximum possible values as $(m)_{max}$.
The vector $(m)_{max}$ is a highest vector.
\subsection{Wigner coefficients and reduced Wigner coefficients}\label{coefvigner}
Let us define Wigner coefficients, reduced Wigner coefficients and let us give a solution of the multiplicity problem. See also Appendix \ref{appendi}.
Denote an irreducible representation with highest weight $[m]_{n}$ as
$V^{[m]_{n}}$.
The tableau $(m)$ from which the first row is removed is denoted as $(m)_{n-1}$.
The Wigner coefficients are matrix elements of an interwinnig operator
$\Phi: V^{[\bar{m}]_{n}}\rightarrow V^{[M]_{n}}\otimes V^{[m]_{n}}$.
All such operators are indexed by tableaux
such that
$$\Phi((m)_{max})=(\Gamma)\otimes (m)_{max}+l.o.t.,\,\,\, (\Gamma)\in V^{[M]_{n}}$$ where $l.o.t.$ (lower order terms)
denotes a sum of tensor products of weight vectors where the second vector has a weight lower than $[m]_n$.
The corresponding Wigner coefficient is denoted as
\begin{equation}\label{viggenn}
<\begin{pmatrix} [\bar{m}]_n \\ (\bar{m})_{n-1} \end{pmatrix}
\begin{pmatrix} (\Gamma)_{n-1} \\ [ M]_n\\ (M)_{n-1}\end{pmatrix}
\begin{pmatrix}[m]_n\\ (m)_{n-1}\end{pmatrix}>.
\end{equation}
This coefficient can be non-zero only if $[\bar{m}]_n=\cal Delta(\Gamma)+[m]_n$.
The corresponding Wigner coefficient is denoted as
\begin{align}\label{redviggenn}
<\begin{pmatrix} [\bar{m}]_n \\ [\bar{m}']_{n} \\
[\bar{m}]_{n-1}
\end{pmatrix} \begin{vmatrix} (\Gamma)_{n-1} \\ [ M]_n\\ (\gamma)_{n-1}
\end{vmatrix} \begin{pmatrix}[m]_n\\ [m']_{n} \\ [m]_{n-1}
\end{pmatrix}>
\end{align}
\subsection{ Fundamental Wigner coefficients}
In the present paper only the Wigner coefficients are considered for which $[M]_n=[1,0,...,0]=[1 \dot{0}]_{n}$, that is when the tensor factor
$V^{[M]_n}$ is a standard representation. Such Wigner coefficients are called fundamental.
Note that weight vectors $(m)$ of the standard representations are completely defined by their weights $\cal Delta(m)=[0,...,\pm 1,...,0]$, where $\pm 1$ occurs at the place $i$. If $i=0$ then only $1$ is allowed. Thus the Wigner coefficient is of type can be denoted as
\begin{equation}
\begin{pmatrix} i \\ [1 \dot{0}]_{n}\\j \end{pmatrix}
\end{equation}
\section{The method of $Z$-invariants}
\label{oldvector}
Let us explain the method of $Z$-invariants, that allows to describe effectively the space of $g_{n-1}$-highest vectors in a $g_n$-representation.
\subsection{ Realization of a representation on the space of functions on uppertriangular matrices.}
Let $g$ be a classical Lie algebra, denote as $n$ its rank, and let $G$ be the corresponding Lie group.
Let $$G=Z_{-}DZ_{+}$$be the Gauss decomposition. Denote the group $Z_{+}$ of upper triangular matrices with units on the diagonal shortly as $Z$.
Let us construct a realization of a representation with the highest weight $[m_{-n,n},...,m_{-1,n}]$
on the space of polynomial functions on $Z$ .
The elements of the matrix $z\in Z_{}$ are denoted as $z_{ij}$.
Thus the functions are polynomials in variables $z_{ij}$.
Define the function $\alpha$ on the space of diagonal matrices as follows
$$\alpha(\delta)=\delta_{-n}^{m_{-n,n}}...\delta_{-1}^{m_{-1,n}},$$ where
$\delta\in D$ and $\delta_{-n}$,...,$\delta_{-1}$ are diagonal elements of $\delta$.
Define the action $T_g$ of the element $g\in G$ on a function $f(z)$, $z\in
Z$ as follows
\begin{equation}\label{deist}(T_{g}f)(z)=\alpha(\tilde{\delta})f(\tilde{z}),\end{equation} where
$\tilde{\delta}$ and $\tilde{z}$ througn the Gauss decomposition of $zg$
$$zg=\tilde{\zeta}\tilde{\delta}\tilde{z}, \,\,\,\tilde{\zeta}\in
Z_{-},\,\,\,\, \tilde{\delta}\in D,\,\,\, \tilde{z}\in Z.$$
In the space of function on $Z$ the subspace of functions $f$,
that form an irreducible representation with the highest weight $m=[m_{-n,n},...,m_{-1,n}]$
is defined as follows.
Let $\mathcal{O}_i$ be an operator on the space of functions on $Z$, which is a left infinitesimal shift on the $i$-root element. Then the space of functions we are looking for is the solution space of the following system of differential equations which is called the indicator system
\begin{equation}\label{indic}\mathcal{O}_i^{r_i+1}f=0,\,\,\,r_i=\frac{2(m,\omega_i)}{(\omega_i,\omega_i)},\,\,\,i=1,...,n,\end{equation}
where $\omega_i$ are fundamental weights for $g_n$. We call $r_i$ the exponents of the system \eqref{indic}.
\subsection{The highest vectors for the subalgebra $g_{n-1}\subset
g_n$, that preseves the coordinates $x_{-n},x_{n}$.}\label{ve}
Let $G_n=Sp_{2n}$, $O_{2n+1}$, $O_{2n}$.
Identify $G_{n-1}\subset G_n$ with a subgroup in $G_n$, that preserves coordinates $x_{-1}$, $x_{1}$. The subgroup $Z$ in
$G_n$ is denoted as $Z_{n}$. Obviously $Z_{n-1}=Z_{n}\cap
G_{n-1}$. Then $G_{n-1}$-highest vectors correspond to polynomials that are invariant under the action of the subgroup $Z_{n-1}$ and satisfy the indicator system.
Let us find polynomials invariant under the action of $Z_{n-1}$. In the case $Sp_{2n}$ it is done in \cite{zh}, in the case $O_{2n+1}$,
$O_{2n}$ this can be done in a similar way. The answer is the following.
The polynomials invariant under the action of $Z_{n-1}$ are polynomials in variables $z_{-1,2},...,z_{-1,n}$,
$z_{1,2},...,z_{1,n}$ in the case of orthogonal groups and in variables
$z_{-1,1},z_{-1,2}...,z_{-1,-2}$, $z_{1,2},...,z_{1n}$ in the case of symplectic groups.
\section{The sketch of the construction}
\label{shema}
Let us give a sketch the construction of the Gelfand-Tsetlin type base, the calculation of Wigner coefficients and reduced Wigner coefficients and the derivation of formulas for the action of generators of the algebra.
The induction in $n$ is used. The subalgebra
$g_{n-1}\subset g_n$ is identified with the subalgebra that preserves the coordinates $x_{-1},x_{1}$.
At first the case $n=1$ is considered, this is the case of algebras
$\mathfrak{o}_3$, $\mathfrak{sp}_2$, $\mathfrak{o}_2$, then the case $n=2$ is considered,
this is the case of algebras $\mathfrak{o}_5$, $\mathfrak{sp}_4$,
$\mathfrak{o}_4$.
Finally the the passage from $g_{2}$ to $g_n$ is considered. This passage is done simultaneously for all algebras $B$, $C$, $D$.
In this step the Gelfand-Tsetlin base for $g_n$ is constructed,
Wigner coefficients for $g_{n-1}$ corresponding to tensor multiplication on the standard representation are obtained.
As before the Wigner coefficients in the case $g_2$ are calculated separately for the three series of algebras and the Wigner coefficients for $g_n$, $n>2$ are calculated in a similar way.
Finally the explicit formulas for the action of generator $g_n$ in the base are derived.
Note that the Wigner coefficients for $g_{n-1}$ are calculated when the algebra $g_n$ is considered.
Let us stress that the passage from $g_{2}$ to $g_{n}$ is very close to the passage from
$\mathfrak{gl}_{3}$ to $\mathfrak{gl}_{n+1}$. This relation plays a key role in the construction.
Is is not necessary to calculate the formulas for the action of all generators. Indeed, it is enough to obtain the formulas for the action of elements $e_{\pm\alpha}$, where $\alpha
$ are simple roots and also formulas for the action of Cartan elements.
Also let us note that the matrices that define the action the element $e_{-\alpha}$ can be obtained from the matrix that correspond to $e_{-\alpha}$ by conjugation, hence it is enough to consider only one of the roots $\pm \alpha$.
It is well-know that for the Gelfand-Tsetlin type base the following statement is true. The subalgebra $g_k\subset g_n$ changes only the part of the Gelfans-Tsetlin tableau that corresponds to $g_k$, and this action does not depend on the upper rows.
Thus we obtain that it is enough to calculate the weight of the base vector and also the action of the operators $F_{-1,0}$ for $n=1$, and $F_{-1,-2}$, $F_{-2,1}$ for $n=2$
and $F_{-1,-2}$ for $n>2$.
\subsection{Notations}
For the rows $[m]_n=[m_{-n,n},...,m_{-1,n}]$ and
$[m]_{n-1}=[m_{-n,n-1},...,m_{-2,n-1}]$ introduce as in
\cite{1968} the following symbols \footnote{In \cite{1968} it is suggested that $m_{-1,n}=0$,
that is why the formula \eqref{redsp2} is slightly different from
\cite{1968}}.
Let
\begin{align}
\begin{split}\label{redsp2}&\begin{vmatrix}n \\
i_1: n-1\end{vmatrix}^{[m]_n,[m]_{n-1}}=\big(
(m_{-2,n-1}-m_{-1,n})\frac{\Pi_{j=-1}^{-n}(m_{j,n}-m_{i_1,n-1}-j+i_1+1)}{\Pi_{j=-2,j\neq
i}^{-n}(m_{j,n-1}-m_{i_1,n-1}-j+i_1+1)}\big)^{\frac{1}{2}},\end{split}\end{align}
be the reduced matrix element for the
$\mathfrak{gl}_{n-1}$-tensor operator $E_{n,i}$.
Let
\begin{align}\label{tredsp22}\begin{split}&\begin{vmatrix}i_1: n \\ i_2 : n-1\end{vmatrix}^{[m]_n,[m]_{n-1}}=S(i_2-i_1)\big(
\frac{\Pi_{j=-2,j\neq
i_1}^{-n}(m_{j,n-1}-m_{i_1,n}-j+i_1)}{\Pi_{j=-1,j\neq
i_1}^{-n}(m_{j,n}-m_{i_1,n}-j+i_1)}\cdot \\&\cdot
\frac{\Pi_{j=-1,j\neq
i_2}^{-n}(m_{j,n}-m_{i_2,n-1}-j+i_2+1)}{\Pi_{j=-2,j\neq
i_2}^{-n}(m_{j,n-1}-m_{i_2,n-1}-j+i_2+1)}\big)^{\frac{1}{2}}
,\end{split},
\end{align}
be a $\mathfrak{gl}_n$ reduced Wigner coefficient. Here $S(x)=sign(x)$, $S(0)=1$.
Let
\begin{align}
\begin{split}
\label{wigsp222}
&\begin{vmatrix} i: n \\ n-1\end{vmatrix}^{[m]_n,[m]_{n-1}}=\big ( \frac{\Pi_{j=-2}^{-n} (m_{j,n-1}-m_{i,n}-j+i )
}{ \Pi_{j=-1,j\neq i}^{-n} ( m_{i,n}-m_{i,n}-j+i ) }
\big)^{\frac{1}{2}}.
\end{split}
\end{align}
be $\mathfrak{gl}_{n}$ Wigner coefficient.
\section{$n=1$: the construction for the algebras $\mathfrak{o}_3$, $\mathfrak{sp}_2$, $\mathfrak{o}_2$.}
\label{step1}
Let us construct the Gelfand-Tsetlin base and derive formulas for the action of generators of these algebras.
For the algebras $\mathfrak{sp}_2$, $\mathfrak{o}_2$ this is a trivial task, since these
algebras are one-dimensional and their irreducible representations are also one-dimensional.
They are defined by the highest weight which is just one number $m_{-1}$. Consider the nontrivial case $\mathfrak{o}_3$.
\subsection{The case $\mathfrak{o}_3$}
The Gelfand-Tsetlin
base coincides with the standard weight base in a
$\mathfrak{sl}_2$-representation.
But an indexation that we construct differs from the standard
one.
The nonstandard indexation is useful in further investigations since it allows to establish
a relation for Gelfand-Tsetlin bases for the algebras of series $B$ and $A$ for $n=2$.
Let us use the realization of a representations on the space of
functions on $Z$. The indicator system in the case of the highest
weight $m_{-1,1}$ (which is an inter or a half-integer number) is
\begin{equation}
(\frac{\partial }{\partial z_{-1,0}})^{2m_{-1,1}}f=0.
\end{equation}
Thus the base vectors in the representation in this realization are
monomials $z_{-1,0}^k$, $k=0,...,2m_{-1,1}$. In \cite{zh} it is
shown that the vector corresponding to the monomial $z_{-1,0}^k$,
has a weight $m_{-1,1}-k$.
However when the Wigner coefficient for $\mathfrak{sl}_2$ are calculated the orthonomal
base is used, that is the base
$$e_{\mu}=\frac{z_{-1,0}^{m_{-1,1}-\mu}}{\sqrt{(m_{-1,1}-\mu)!(m_{-1,1}+\mu)!}},$$
$\mu=-m_{-1,1},...,m_{-1,1}$ is the weight of the vector. Below we use this base.
Let us construct two numbers. Put $m'_{-1,1}=[\frac{k}{2}]$, where
$[ \,.\,]$ is an integer part and let $\sigma_{-1}=0,1$ be the
residue of the division of $k$ by 2. A base vector can be encoded by
a tableau
\begin{align}\begin{split}\label{gco3}
& m_{-1,1}\\
& \sigma_{-1}\,\,m'_{-1,1},\end{split}
\end{align}
The following inequality holds $m_{-1,1}\geq m'_{-1,1}\geq 0$. Also
if $m_{-1,1}=m'_{-1,1}$ and $m_{-1,1}$ is an integer then
$\sigma_{-1}=0$.
The weight of the vector encoded by a tableau can be calculated by
the formula $m_{-1,1}-2m'_{-1,1}-\sigma_1$.
Let us find the action of the operator $F_{-1,0}$. Put
$g=exp(tF_{-1,0} )\in Z$, then $T_gf(z)=f(zg)$. Under the action of
$F_{-1,0}$ the vector $z_{-1,0}^k$
is mapped to $kz_{-1,0}^{k-1}$. One can easily write how the numbers $m'_{-1,1}$ and $\sigma_{-1}$ change under the action of $F_{-1,0}$.
This gives us a necessary condition for the matrix element to be non-zero. In this case the matrix element equals to the reduced
matrix element of a $\mathfrak{o}_1$-tensor operator $F_{-1,0}$.
Thus the following theorem takes place.
Put $[m]_{1}=[m_{-1,1},0]$, $[m']_{1}=[m'_{-1,1}]$.
\begin{thm}
Under the action of $F_{-1,0}$ the tableau $(m)$ changes in the
following way.
\begin{enumerate}
\item If $\sigma_{-1}=0$, then $\sigma_{-1}$ diminishes by $1$,
$m'_{-1,1}$ remain unchanged, the resulting tableau is multiplied
to
$\begin{vmatrix}2 \\
-2: 1\end{vmatrix}^{[m]_{1},[m']_{1}}$.
\item If $\sigma_{-1}=1$, then $m'_{-1,1}$ diminishes by $1$,
$\sigma_{-1}$ turns to $0$, the resulting tableau is multiplied to
$\begin{vmatrix}2 \\
-2: 1\end{vmatrix}^{[m]_{1},[m']_{1}}$.
\end{enumerate}
\end{thm}
\section{$n=2$: the construction for the algebras $\mathfrak{o}_5$, $\mathfrak{sp}_4$, $\mathfrak{o}_4$.}
\label{step2}
In the present Section the restrictions $\mathfrak{o}_5\downarrow
\mathfrak{o}_3$, $\mathfrak{sp}_{4}\downarrow\mathfrak{sp}_2$,
$\mathfrak{o}_4\downarrow \mathfrak{o}_2$ are investigated. Then the
Gelfand-Tsetlin base is constructed and the action of the
generators in this base is investigated.
The construction in the three cases $\mathfrak{o}_5$,
$\mathfrak{sp}_4$, $\mathfrak{o}_4$ are different. For
$\mathfrak{o}_5$ and $\mathfrak{sp}_4$ the corresponding problem of
restriction is solved by establishing a relation with the analogous
problem of restriction $\mathfrak{gl}_3\downarrow \mathfrak{gl}_1$.
Note that the Wigner coefficients for
$\mathfrak{o}_3=\mathfrak{sl}_2$
are known thus for $n=2$ it is not necessary to calculate the Wigner coefficients.
\subsubsection{The auxiliary problem of restriction $\mathfrak{gl}_3\downarrow \mathfrak{gl}_1$}
Identify the algebra $\mathfrak{gl}_3$ with the algebra of
endomorphisms of the space with coordinates $x_{-2},x_{-1},x_1$, and
$\mathfrak{gl}_1$ is the subalgebra that preserves the last two
coordinates. The problem of restriction $\mathfrak{gl}_3\downarrow
\mathfrak{gl}_1$ is equivalent to the problem of construction of
weight base in a $\mathfrak{gl}_3$-representation.
We use the realization of a representation with the highest weight
$[\lambda_{-2},\lambda_{-1},\lambda_{1}]$ on the space of polynomials
on the group $Z$, that is on the space of polynomials in variables
$z_{-2,-1},z_{-2,1},z_{-1,1}$. The indicator system is of type
\begin{align}\begin{split}\label{indgl3}
& (\frac{\partial}{\partial z_{-2,-1}}+z_{-1,1}\frac{\partial}{\partial z_{-2,1}})^{r_{-2}+1}f=0,\\&
(\frac{\partial}{\partial z_{-1,1}})^{r_{-1}+1}f=0, \end{split}
\end{align}
where $r_{-2}=\lambda_{-2}-\lambda_{-1}$, $r_{-1}=\lambda_{-1}$.
The solution space of this system is spanned by polynomials of type
\begin{equation}\label{solgl3}
f=f_0(z_{-2,-1},z_{-2,1},z_{-1,1})z_{-1,1}^{p},\end{equation} where
$f_0$ is a polynomial solution of the first equation, that is not divisible by
$z_{-1,1}$. The following inequality must hold
$$deg_{z_{-1,1}}f_0+p\leq r_{-1},$$
where $deg_{z_{-1,1}}f_0$ is the degree of $f_0$ as a polynomial in $z_{-1,1}$.
There exist a base in the representation whose vectors are encoded by Gelfand-Tsetlin tableaux
\begin{align}\begin{split}\label{gcgl3}
& \lambda_{-2} \,\,\,\, \lambda_{-1} \,\,\,\, \lambda_{1}\\
& \,\,\,\lambda_{-2,2}\,\,\lambda_{-1,2}\\
&\,\,\,\,\,\,\,\,\lambda_{-2,1}.\end{split}
\end{align}
\subsection{The construction in the case $\mathfrak{o}_5$}
Consider a representation of $\mathfrak{o}_5$ with the highest weight $[m_{-2,2},m_{-1,2}]$.
\subsubsection{The problem of restriction $\mathfrak{o}_5 \downarrow \mathfrak{o}_3$}\label{indb}
Using the method of $Z$-invariants (see Section \ref{ve}) let us investigate how an $\mathfrak{o}_5$-representation branches when we restrict the algebra
$\mathfrak{o}_5 \downarrow \mathfrak{o}_3$. The polynomials on
$Z_{\mathfrak{o}_5}$, that are invariant under the action of
$Z_{\mathfrak{o}_3}$, are polynomials in variables $z_{-2,-1},
z_{-2,1}, z_{-1,0}$.
The indicator system is
\begin{align}\begin{split}\label{indo5}&
(\frac{\partial}{\partial z_{-2,-1}}+z_{-1,1}\frac{\partial}{\partial z_{-2,1}})^{r_{-2}+1}f=0,\\&
(\frac{\partial}{\partial z_{-1,0}})^{r_{-1}+1}f=0,\end{split}
\end{align}
where $r_{-2}=m_{-2,2}-m_{-1,2}$, $r_{-1}=2m_{-1,2}$. Note that $z_{-1,1}=-\frac{z_{-1,0}^2}{2}.$
The solution space of this system is spanned by polynomials of type
\begin{equation}\label{solo5}
f=f_0(z_{-2,-1},z_{-2,1},z_{-1,1})z_{-1,0}^{p},\end{equation} where
$f_0$ is a polynomial solution of the first equation that is not divisible by
$z_{-1,1}$. The following inequality must take place
$$2deg_{z_{-1,1}}f_0+p\leq r_{-1}.$$
Let us establish a correspondence between the problems of restriction $\mathfrak{o}_5 \downarrow
\mathfrak{o}_3$ and $\mathfrak{gl}_3\downarrow \mathfrak{gl}_1$.
Consider the cases when $p=2p'$ is even and when $p=2p'+1$ is odd.
\begin{enumerate}
\item In the case $p=2p'$ to the solution \eqref{solo5} with the parameter $p$ of the system \eqref{indo5} with exponents $r_{-2},r_{-1}$
there correponds a solution \eqref{solgl3} with the parameter $p'$ of the system \eqref{indgl3} with exponents $r_{-2},\frac{r_{-1}}{2}$
$$f_0(z_{-2,-1},z_{-2,1},z_{-1,1})z_{-1,0}^{p}\mapsto f_0(z_{-2,-1},z_{-2,1},z_{-1,1})z_{-1,1}^{[\frac{p}{2}]}. $$
This correspondence is a bijection between the solution space of \eqref{indo5} with an even parameter $p$ and the solution space of
\eqref{indgl3}.
\item In the case $p=2p'+1$ to the solution \eqref{solo5} with the parameter $p$ of the system \eqref{indo5} with exponents $r_{-2},r_{-1}$
there correponds a solution \eqref{solgl3} with the parameter $p'$ of the system \eqref{indgl3} with exponents $r_{-2},\frac{r_{-1}}{2}$
$$f_0(z_{-2,-1},z_{-2,1},z_{-1,1})z_{-1,0}^{p}\mapsto f_0(z_{-2,-1},z_{-2,1},z_{-1,1})z_{-1,1}^{[\frac{p}{2}]}. $$
This correspondence is a bijection between the solution space of \eqref{indo5} with an odd parameter $p$ and the solution space of
\eqref{indgl3}.
\end{enumerate}
\subsubsection{The Gelfand-Tsetlin base for $\mathfrak{o}_5$}\label{gcbaseo5}
In the case $\mathfrak{gl}_{3}$
the indicator system with exponents
$r_{-2},\frac{r_{-1}}{2}$ corresponds to the highest weight $[m_{-2,2},m_{-1,2},0]$.
Base vectors of a representation of $\mathfrak{gl}_{3}$ are encoded by tableaux \eqref{gcgl3}.
Denote as $\sigma_{-1}$ the residue of the division of $p$ by $2$.
Using the correspondence that was established in the previous subsection one obtains that $\mathfrak{o}_3$-highest vectors in a
$\mathfrak{o}_5$-representation are encoded by
$\mathfrak{gl}_{3}$-tableaux (from which the zero in the upper row is removed) with an additional number $\sigma_{-2}$, that is by tableaux of type
\begin{align}\begin{split}\label{gco5}
& \,\,\,m_{-2,2} \,\,\,\, m_{-1,2} \,\,\,\, \\
& \sigma_{-2} \,\,\,m'_{-2,2}\,\,m'_{-1,2}\\
&\,\,\,\,\,\,\,\,\,\,\,m_{-2,1}.\end{split}
\end{align}
The following inequalities must take place
\begin{align}
& m_{-2,2}\geq m'_{-2,2}\geq m_{-1,2}\geq m'_{-1,2}\geq 0\\& m'_{-2,2}\geq
m_{-2,1}\geq m'_{-1,2}.
\end{align}
Also if $m'_{-2,2}=0$, then $\sigma_{-2}=0$.
Indeed, consider the inequality $2deg_{z_{-1,1}}f_0+p\leq r_{-1},$ and divide it by two. One obtaines
$deg_{z_{-1,1}}f_0+p'+\sigma_{-2}\leq \frac{r_{-1}}{2}.$
From here we conclude that if $r_{-1}$ is even (that is when the highest weight
$[m_{-2,2},m_{-1,2}]$ is integer) and $p'$ takes he maximm value then $\sigma_{-2}=0$. The fact that $p'$ takes the maximum value means that
$m'_{-2,2}=0$ (see explicit formula for the polynomial on $Z$,
corresponding to a $\mathfrak{gl}_3$-tableau in \cite{zh}).
To be able to use this indexation of
$\mathfrak{o}_3$-highest vectors for the construction of the Gelfand-Tsetlin type base we must prove the following Proposition.
\begin{prop}\label{stroka05} The number $m_{-2,1}$ in the tableau \eqref{gco5} is the
$\mathfrak{o}_3$-weight of the vector that is encoded by the tableau.
\end{prop}
The proof is elementary, it can be found in Appendix in Section \ref{sootv}.
Using the inductive procedure of the construction of the Gelfand-Tsetlin type base we obtain that the vectors of the representation are encoded by a tableau $(m)$, that is obtained by adding to the tableau
\eqref{gco5} the tableau \eqref{gco3}, that is by a tableau $(m)$ of type
\begin{align}\begin{split}\label{gco55}
& \,\,\,m_{-2,2} \,\,\,\, m_{-1,2} \,\,\,\, \\
& \sigma_{-2} \,\,\,m'_{-2,2}\,\,m'_{-1,2}\\
&\,\,\,\,\,\,\,\,\,\,\,m_{-2,1}\\
&\,\,\,\,\,\,\,\,\,\,\,\sigma_{-1} \,\,\,m'_{-2,1}.\end{split}
\end{align}
The elements of this tableau must satisfy the inequalities corresponding to the tableaux
\eqref{gco5}, \eqref{gco3}.
Let us give formulas for the weight
$[\cal Delta(m)_{-2},\cal Delta(m)_{-1}]$ of the vector encoded by tableau. It is enough to give a formula for
$\cal Delta(m)_{-1}$.
\begin{prop}
\label{weightofo5}
$\cal Delta(m)_{-1}=-2\sum_i m'_{i,2}+\sum_i
m_{i,2}+m_{-2,1}-\sigma_{-2}$
\end{prop}
The proof can be found in Appendix in Section
\ref{sootv}.
\subsubsection{Reduced matrix elements for $\mathfrak{o}_5$}
\label{reduct05}
To calculate matrix elements of generators in the base that we have constructed we need explicit formulas for reduced matrix elements of operators
$F_{-1,-2}$ and
$F_{1,-2}$, viewed as $\mathfrak{o}_3$-tensor operators acting between $\mathfrak{o}_3$-representations into which an $\mathfrak{o}_5$-representation splits
Let $(\bar{m})_{red}$, $(m)_{red}$ be two tableaux of type
\eqref{gco5} that define $\mathfrak{o}_3$-highest vectors in a
$\mathfrak{o}_5$ representation. Let $(\bar{m})^{*}_{red}$ and
$(m)^{*}_{red}$ be two $\mathfrak{gl}_3$-tableux, that are obtained from $(\bar{m})_{red}$, $(m)_{red}$ by removing the number $\sigma_1$ and adding $0$ to the upper row in the right.
The reduced matrix element of a $\mathfrak{o}_3$-tensor operator $F_{\pm 1,-2}$ depends on tableaux $(\bar{m})_{red}$, $(m)_{red}$. Denote it $<(\bar{m})_{red}|F_{\pm 1, -2}|(m)_{red}>_{red}$.
Denote as $<(\bar{m})^{*}_{red}|Е_{\pm 1, -2}|(m)^{*}_{red}>$ a $\mathfrak{gl}_3$-matrix element.
The following lemma takes place.
\begin{lem}\label{gl3o5} The matrix elements for $\mathfrak{o}_5$ and
$\mathfrak{gl}_3$ are equal
\begin{equation} <(\bar{m})_{red}|F_{\pm 1, -2}|(m)_{red}>_{red}=<(\bar{m})^{*}_{red}|Е_{\pm 1, -2}|(m)^{*}_{red}>
\end{equation}
\end{lem}
\proof
The proof uses the following fact. Let
$(m)_{max}$ be a $\mathfrak{o}_{5}$-tableau that is obtained from
$(m)_{red}$ by adding maximal rows below.
\begin{prop}\label{pp}
The following equality takes place
\begin{align}
<(\bar{m})_{max} \mid F_{\pm 1,-2} \mid (m)_{max}
>=<(\bar{m})_{red} \mid F_{\pm 1,-2} \mid (m)_{red}>_{red}
\end{align}
\end{prop}
The proof can be found in Appendix in Section \ref{redel}.
Let us return to the proof of Lemma. It enough to prove that
$$<(\bar{m})_{max} \mid F_{\pm 1,-2} \mid (m)_{max}
>=<(\bar{m})^{*}_{red}|Е_{\pm 1, -2}|(m)^{*}_{red}>.$$
Let $f=z_{-1,0}^pf_0(z_{-2,-1},z_{-2,1},z_{-1,1})$ be a polynomial corresponding to $(m)_{max}$. One can easily check that under the action of $e^{tF_{-2,-1}}$ it is transformed into the polynomial
$z_{-1,0}^pf_0(z_{-2,-1}+t,z_{-2,1},z_{-1,1})$.
To the vector $(m)^{*}_{red}$ there corresponds a polynomial
$f^*=z_{-1,1}^{[\frac{p}{2}]}f_0(z_{-2,-1},z_{-2,1},z_{-1,1})$. One can easily check that under the action of $e^{tE_{-2,-1}}$ it is transformed into the polynomial
$z_{-1,1}^{[\frac{p}{2}]}f_0(z_{-2,-1}+t,z_{-2,1},z_{-1,1})$.
Thus the actions of $F_{-2,-1}$ and $E_{-2,-1}$ on the highest $\mathfrak{o}_3$ and $\mathfrak{gl}_1$ vectors are agreed with the correspondence that was constructed in Section \ref{indb}. Thus the matrix elements of these operators are equal. Hence the matrix elements of the operators $F_{-1,-2}$ and $E_{-1,-2}$ are equal. Using Proposition \ref{pp} we prove the lemma for the operators $F_{-1,-2}$ and $E_{-1,-2}$.
Analogously under the action of the operator $e^{tF_{-2,1}}$, the polynomial $f$ is transformed into the polynomial
$z_{-1,0}^pf_0(z_{-2,-1},z_{-2,1}+t,z_{-1,1})$. The polynomial $f^*=z_{-1,1}^{[\frac{p}{2}]}f_0(z_{-2,1},z_{-2,1},z_{-1,1})$ corresponding to $(m)^{*}_{red}$ under the action f the operator $e^{tE_{-2,1}}$ is transformed into the polynomial
$z_{-1,1}^{[\frac{p}{2}]}f_0(z_{-2,-1},z_{-2,1}+t,z_{-1,1})$.
Form this fact we obtain the statement of Lemma for operators $F_{1,-2}$ and $E_{1,-2}$.
\endproof
Let us find reduced matrix elements for operators $F_{-1,2}$ and $F_{1,2}$ using the lemma.
Put $[m]_{2}=[m_{-2,2},m_{-1,2},0]$, $[m']_{2}=[m'_{-2,2},m'_{-1,2}]$, $[m]_{1}=[m_{-2,1}]$.
Let us prove theorems
\begin{thm}
\label{t1o5}
The reduced matrix element $<(\bar{m})_{red}
\mid F_{1,-2} \mid (m)_{red}>_{red}$ can be nonzero only if the following condition holds. There exist a unique index $i_1=-2$ or $-1$ such that
$\bar{m}'_{i_1,2}=m'_{i_1,2}-1$, and for the other index $i$ one has
$\bar{m}'_{i,2}=m'_{i,2}$. Also $\bar{m}_{-2,1}=m_{-2,1}-1$.
If this condition holds one has
\begin{align}<(\bar{m})_{red} \mid F_{1,-2} \mid (m)_{red}>_{red}=
\begin{vmatrix}3 \\ i_1 :
2\end{vmatrix}^{[m]_2,[m']_{2}}\begin{vmatrix}i_1: 2 \\ -2 :
1\end{vmatrix}^{[m']_2,[m]_{1}},
\end{align}
\end{thm}
\proof
By Lemma \ref{gl3o5} it is enough to find matrix elements for the operator $E_{1,-2}$.
In \cite{1963} in the case $\mathfrak{gl}_{n}$ the following formula is proved
\begin{align}
<(\bar{m}) \mid E_{1,-2} \mid (m)>=\begin{vmatrix}n \\
i_1 : n-1\end{vmatrix}^{[m]_n,[m]_{n-1}}\begin{vmatrix}i_1: n-1 \\ i_2 :
n-2\end{vmatrix}^{[m]_{n-1},[m]_{n-2}}\begin{vmatrix}i_2 : n-2 \\ n-3\end{vmatrix}^{[m]_{n-2},[m]_{n-3}},
\end{align}
where $[m]_k$ is the $k$-th row in the Gelfand-Tsetlin tableau $(m)$.
It is suggested that one has $\bar{m}_{i_j,j}=m_{i_j,j}-1$, and for $k\neq i_j$
one has $\bar{m}_{i,j}=m_{i,j}$. In all other cases
$<(\bar{m}) \mid E_{1i} \mid (m)>$ vanishes.
If one considered $E_{1,-2}$ as a $\mathfrak{gl}_{n-2}$-tensor operator then the first two factors are a reduced matrix element and the last is a Wigner coefficient.
If one puts $n=3$ then (since $i_2=-2$)
one gets
\begin{align}
<(\bar{m})_{red} \mid E_{1,-2} \mid (m)_{red}>_{red}=\begin{vmatrix}3 \\
i_1 : 2\end{vmatrix}^{[m]_3,[m]_2}\begin{vmatrix}i_1: 2 \\ -2 :
1\end{vmatrix}^{[m]_2,[m]_1}.
\end{align}
Using now Lemma \ref{gl3o5} we prove the Theorem.
\endproof
Let us find the formula for the reduced matrix element of the operator
$F_{-1,-2}$.
\begin{thm}
\label{t2o5}
The reduced matrix element $<(\bar{m})_{red}
\mid F_{-1,-2} \mid (m)_{red}>_{red}$ can be nonzero only if the following condition holds. One has $\bar{m}_{-2,1}=m_{-2,1}-1$ and all other entries of $(\bar{m})$ and $(m)$ are equal.
If this condition holds one has
\begin{align}<(\bar{m})_{red} \mid F_{-1,-2} \mid (m)_{red}>_{red}=
\begin{vmatrix}2 \\ -2 : 1\end{vmatrix}^{[m']_{2},[m]_{1}},
\end{align}
\end{thm}
\proof By Lemma \ref{gl3o5} it is enough to find matrix elements $E_{-1,-2}$. One has the formula
\begin{align}
<(\bar{m})_{red} \mid E_{-1-2} \mid (m)_{red}>=\begin{vmatrix}n \\
i_1 : n-1\end{vmatrix}^{[m]_{n},[m]_{n-1}}\begin{vmatrix}i_1: n-1 \\ n-2\end{vmatrix}^{[m]_{n-1},[m]_{n-2}},
\end{align}
where $[m]_k$ is the $k$-th row of the Gelfand-Tsetlin tableau $(m)$.
It is suggested that $\bar{m}_{i_1,n}=m_{i_1,n}-1$ and
$\bar{m}_{j,n}=m_{j,n}$ for $j\neq i_1$. The first factor is the reduced matrix element. Put $n=2$, using Lemma \ref{gl3o5} we obtain the statement of the Theorem
\endproof
\subsubsection{Matrix elements}
Using calculation that were done in the previous sections let us derive formulas for matrix elements of generators of $\mathfrak{o}_{5}$ in the base that we have constructed.
It is enough to give formulas for the matrix elements of operators.
$F_{-1,-2}$ и $F_{1,-2}$.
The following theorem takes place.
\begin{thm}
\label{t3o5}
The matrix element $<(\bar{m})_{red}
\mid F_{1,-2} \mid (m)_{red}>_{red}$ can be nonzero only if the following condition holds. There exist a unique index $i_1=-2$ or $-1$ such that
$\bar{m}'_{i_1,2}=m'_{i_1,2}-1$, and for the other index $i$ one has
$\bar{m}'_{i,2}=m'_{i,2}$. Also $\bar{m}_{-2,1}=m_{-2,1}-1$.
If this condition holds one has
\begin{align}\begin{split}&
<(\bar{m})| F_{1,-2} |(m)>=
\begin{vmatrix}3
\\ i_1 : 2\end{vmatrix}^{[m]_2,[m']_2}\begin{vmatrix}i_1: 2 \\ -2 :
1\end{vmatrix}^{[m']_2,[m]_1}\begin{vmatrix}-2 : 2
\\ 1 \end{vmatrix}^{[m]_1,[m']_1}
.\end{split}
\end{align}
The matrix element $<(\bar{m}) \mid
F_{-1,-2} \mid (m)>$ can be nonzero only if the following condition holds. One has
$\bar{m}_{-2,1}=m_{-2,1}-1$ and all other entries of $(\bar{m})$ and $(m)$ are equal.
If this condition holds one has
\begin{align}\begin{split}&
<(\bar{m})| F_{-1,-2} |(m)>=
\begin{vmatrix}2 \\
-2 : 1\end{vmatrix}^{[m']_2,[m]_1}\begin{vmatrix}-2: 2 \\ 1 \end{vmatrix}^{[m]_1,[m']_1},
\end{split}
\end{align}
\end{thm}
\proof The Theorem is immediately proved using the Wigner-Ekcart theorem.
\endproof
\subsection{The construction in the case $\mathfrak{sp}_4$}
Consider an irreducible representation of $\mathfrak{sp}_4$ with the highest weight $[m_{-2,2},m_{-1,2}]$. Since the algebra $\mathfrak{sp}_2$
is one dimensional the problem of restriction $\mathfrak{sp}_4\downarrow
\mathfrak{sp}_2$ is equivalent to the problem of construction of a base in a
$\mathfrak{sp}_4$-representation.
\subsubsection{The problem of restriction $\mathfrak{sp}_4 \downarrow \mathfrak{sp}_2$}\label{indc}
Let us investigate the problem of restriction $\mathfrak{sp}_4 \downarrow
\mathfrak{sp}_2$ using the method of $Z$-invariants. The polynomials $Z_{\mathfrak{sp}_4}$ that are invariant under $Z_{\mathfrak{sp}_2}$ are polynomials in variables
$z_{-2,-1}, z_{-2,1}, z_{-1,1}$.
The indicator system is
\begin{align}\begin{split}\label{indsp4}&
(\frac{\partial}{\partial z_{-2,-1}}+z_{-1,1}\frac{\partial}{\partial z_{-2,1}})^{r_{-2}+1}f=0\\&
(\frac{\partial}{\partial z_{-1,1}})^{r_{-1}+1}f=0,\end{split}
\end{align}
where $r_{-2}=m_{-2,2}-m_{-1,2}$, $r_{-1}=m_{-1,2}$.
The system \eqref{indsp4} is exatly the system \eqref{indgl3}, that appears in the problem of restricton $\mathfrak{gl}_3\downarrow \mathfrak{gl}_1$.
\subsubsection{The Gelfand-Tsetln type base for $\mathfrak{sp}_4$}\label{gcbasesp4}
In the case $\mathfrak{gl}_{3}$ the indicator system with exponents
$r_{-2},r_{-1}$ corresponds to the highest weight $[m_{-2,2},m_{-1,2},0]$.
Base vectors of a $\mathfrak{gl}_{3}$-representation are encoded by tableaux \eqref{gcgl3}. Hence the base vectors of a
$\mathfrak{sp}_4$-representation can be encoded by a $\mathfrak{gl}_3$-tableau (form which we remove zero in the upper row), that is by tableaux of type
\begin{align}\begin{split}\label{gcsp4}
& \,\,\,m_{-2,2} \,\,\,\, m_{-1,2} \,\,\,\, \\
&\,\,\,\, \,\,\,m'_{-2,2}\,\,m'_{-1,2}\\
&\,\,\,\,\,\,\,\,\,\,\,m_{-2,1}.\end{split}
\end{align}
The entries of this tableau must satisfy the inequalities
\begin{align}
& m_{-2,2}\geq m'_{-2,2}\geq m_{-1,2}\geq m'_{-1,2}\geq 0\\& m'_{-2,2}\geq
m_{-2,1}\geq m'_{-1,2}.
\end{align}
The following Proposition takes place
\begin{prop}\label{strokasp4} The number $m_{-2,1}$ in the tableau \eqref{gcsp4} is the
$\mathfrak{sp}_4$-weight of the corresponding vector
\end{prop}
The proof is analogous to the proof of the Proposition
\ref{stroka05}.
Let us give the formula for the weight
$[\cal Delta(m)_{-2},\cal Delta(m)_{-1}]$ of the vector encoded by the tableau $(m)$. It is enough to consider
$\cal Delta(m)_{-1}$.
\begin{prop}
\label{weightofsp4}
$\cal Delta(m)_{-1}=-2\sum_i m'_{i,2}+\sum_i m_{i,2}+m_{-2,1}$
\end{prop}
The proof of this Proposition is analogous to the proof of Proposition \ref{weightofsp4}.
\subsubsection{Matrix elements for $\mathfrak{sp}_4$}
Matrix elements for
$F_{-1,-2}$ and $F_{1,-2}$ can be calculeted if we consider these opearators as
$\mathfrak{sp}_2$-tensor operators acting between
$\mathfrak{sp}_2$-representations into which a
$\mathfrak{sp}_4$-representation splits.
Let us we write the Wigner-Eckart theorem for $F_{-1,-2}$
and $F_{1,-2}$. Since the algebra $\mathfrak{sp}_2$is one dimensional, the corresponding Wigner coefficient is zero. Hence the matrix elements of $F_{-1,-2}$ and $F_{1,-2}$ equal to the reduced matrix elements.
As in Section \ref{reduct05} let us find the relation of these reduced matrix elements with the analogous matrix elements for $\mathfrak{gl}_3$.
Let $(\bar{m})$, $(m)$ be two tableaux of type \eqref{gcsp4}, and let
$(\bar{m})^{*}$, $(m)^{*}$ be two tableaux for
$\mathfrak{gl}_3$,
that are obtained from $(\bar{m})$, $(m)$ by adding zero $0$ to the upper row in the right.
The following Lemma takes place.
\begin{lem}\label{gl3sp4} Matrix elements for $\mathfrak{sp}_4$ and
$\mathfrak{gl}_3$ are equal
\begin{equation} <(\bar{m})|F_{\pm 1, 2}|(m)>=<(\bar{m})^*|Е_{\pm 1, 2}|(m)^*>
\end{equation}
\end{lem}
The proof is analogous to the proof of Lemma \ref{gl3o5}.
Introduce notations $[m]_2=[m_{-2,2},m_{-1,2},0]$, $[m']_2=[m'_{-2,2},m'_{-1,2}]$, $[m]_{-1}=[m_{-2,1}]$.
Using Lemma \ref{gl3sp4},let us calculate matrix elements for $F_{-1,2}$ and
$F_{1,2}$. They are given by the follolwing theorem.
\begin{thm}
The matrix element $<(\bar{m}) \mid F_{1,-2} \mid
(m)>$ can be nonzero only if the following condition holds. For one index $i_1=-2$
or $-1$ one has $\bar{m}'_{i_1,2}=m'_{i_1,2}-1$, and for another one has
$\bar{m}_{ij}=m_{ij}$. Also $\bar{m}_{2,1}=m_{2,1}-1$. If this condition holds one has
\label{t1sp4}
\begin{align}<(\bar{m})\mid F_{1,-2} \mid (m)>=
\begin{vmatrix}3 \\ i_1 :
2\end{vmatrix}^{[m]_2,[m']_2}\begin{vmatrix}i_1: -2 \\ -2 : 1\end{vmatrix}^{[m']_2,[m]_1},
\end{align}
The matrix element $<(\bar{m}) \mid F_{-1,-2} \mid
(m)>$ can be nonzero only if the following condition holds. One has
$\bar{m}_{-2,1}=m_{-2,1}-1$ and all othe entries of $(\bar{m})$ and $(m)$ are equal. If this condition holds one has
\begin{align}<(\bar{m}) \mid F_{-1,-2} \mid (m)>=
\begin{vmatrix}2 \\ -2 : 1\end{vmatrix}^{[m']_2,[m]_1},
\end{align}
\end{thm}
The proof of this Theorem is base on Lemma \ref{gl3sp4} and is analogous to the proof of Theorem \ref{t1o5}.
\subsection{The construction in the case $\mathfrak{o}_4$}
Let us be given a $\mathfrak{o}_4$-represention with the highest weight $[m_{-2,2},m_{-1,2}]$. Since the algebra $\mathfrak{o}_2$
is one dimensional the problem of restriction $\mathfrak{o}_4\downarrow
\mathfrak{o}_2$ is equivalent to the problem of construction of a base in a
$\mathfrak{o}_4$-representation.
\subsubsection{The problem of restriction $\mathfrak{o}_4 \downarrow \mathfrak{o}_2$}\label{indd}
Consider the problem of restriction
$\mathfrak{o}_4 \downarrow \mathfrak{o}_2$. The polynomials on
$Z_{\mathfrak{o}_4}$ that are invariant under
$Z_{\mathfrak{o}_2}$ are the polynomials in variables $z_{-2,-1},
z_{-2,1}$.
The indicator system is
\begin{align}\begin{split}\label{indo4}
(\frac{\partial}{\partial z_{-2,-1}})^{r_{-2}+1}f=0,\\
(\frac{\partial}{\partial z_{-2,1}})^{r_{-1}+1}f=0,\end{split}
\end{align}
where $r_{-2}=m_{-2,2}-m_{-1,2}$, $r_{-1}=m_{-2,2}+m_{-1,2}$.
All polynomial solutions are linear combination of the following polynomials (we use the standard normalization for
$\mathfrak{sl}_2$)
\begin{align}\begin{split}\label{solo4}
& f=\frac{z_{-2,-1}^{p}}{\sqrt{p!(2r_{-1}-p)!}}\frac{ z_{-2,1}^{q}}{\sqrt{q!(2r_{-2}-q)!}},\\
& 0\leq p\leq r_{-2},\\
& 0\leq q \leq r_{-1}.\end{split}
\end{align}
\subsubsection{The Gelfand-Tsetlin type base for $\mathfrak{o}_4$}\label{gcbaseo4}
We see that there exist a base whose vectors are defined by two numbers $p$, $q$. Following Molev (see \cite{M})
let us define another indexation. Put
\begin{equation}
m'_{-2,2}=m_{-2,2}-min\{p,q\},\,\,\,
m_{-2,1}=m_{-1,2}-p+q.
\end{equation}
\begin{prop}\label{pq}
The numbers $p$ and $q$ are reconstructed as follows
\begin{enumerate}
\item Если $m'_{-2,1}-m_{-1,2}>0$, то $p=m_{-2,2}-m'_{-2,2}$, $q=m_{-2,2}-m'_{-2,2}+m'_{-2,1}-m_{-1,2}$.
\item Если $m'_{-2,1}-m_{-1,2}\leq 0$, то $q=m_{-2,2}-m'_{-2,2}$, $p=m_{-2,2}-m'_{-2,2}-m'_{-2,1}+m_{-1,2}$.
\end{enumerate}
\end{prop}
Let us construct the Gelfand-Tsetlin type tableau in the following way
\begin{align}\begin{split}\label{gco4}
& \,\,\,m_{-2,2} \,\,\,\, m_{-1,2} \,\,\,\, \\
&\,\,\,\,\,\, \,\,\,m'_{-2,2}\\
&\,\,\,\,\,\,\,\,\,\,\,m_{-2,1}\end{split}
\end{align}
The equalities \eqref{solo4} are satisfied if and only if (see \cite{M}), when the following inequalities take place
\begin{align}
& m_{-2,2}\geq m'_{-2,2}\geq |m_{-1,2}|\\
& m'_{-2,2}\geq |m_{-2,1}|
\end{align}
One has
\begin{prop}\label{stroka04} The number $m_{-2,1}$ in the tableau \eqref{gco4} is the
$\mathfrak{o}_2$-weight of the corresponding vector.
\end{prop}
The proof is analogous to the proof of Proposition \ref{stroka05}.
Let us give formulas for the weight
$[\cal Delta(m)_{-2},\cal Delta(m)_{-1}]$ of the vector encoded by the tableau $(m)$. It is enough to consider the component
$\cal Delta(m)_{-1}$.
\begin{prop}
\label{weightofo4}
$\cal Delta(m)_{-1}=-2\sum_i m'_{i,2}+\sum_i m_{i,2}+m_{-2,1}$
\end{prop}
The proof is analogous to the proof of Proposition
\ref{weightofo5}.
\subsubsection{Matrix elements}
Since the construction of the Gelfand-Tsetlin type base for
$\mathfrak{o}_4$ has another character than the construction for the algebras
$\mathfrak{o}_5$ and $\mathfrak{sp}_4$ the calculation of matrix elements does not use the Wigner-Eckart theorem. It is based on the isomorphism $\mathfrak{o}_4\simeq
\mathfrak{sl}_2\oplus\mathfrak{sl}_2$.
The operator $F_{-1,-2}$ increases the number $p$ by $1$ and multiplies the vector on the reduced matrix element
$\sqrt{p_{}!(2r_{-2}-p_{})!}$.
The operator $F_{-1,2}$ increases the number $q$ by $1$ and multiplies the vector on the reduced matrix element
$\sqrt{q_{}!(2r_{-1}-q_{})!}$.
When we pass to the Gelfand-Tsetlin type tableaux then using Proposition \ref{pq}, we obtain the theorem
\begin{thm}
The action of $F_{-1,-2}$ on the tableau \eqref{gco4} is described as follows. If $m'_{2,1}-m_{-1,2}+1\leq 0$, then $F_{-1,-2}$ diminishes $m_{-2,1}$ by $1$. If $m'_{2,1}-m_{-1,2}+1> 0$ then also $m'_{-2,2}$ diminishes by $1$.
In both cases the vector is multiplied onto
$\sqrt{p_{}!(2r_{-2}-p_{})!}$ (see Proposition \ref{pq}).
The action of $F_{-1,2}$ on the tableau \eqref{gco4} is described as follows. If $m'_{2,1}-m_{-1,2}-1\geq 0$, then $F_{-1,-2}$ increases $m_{-2,1}$ by $1$. If $m'_{2,1}-m_{-1,2}+1< 0$ then also $m'_{-2,2}$ diminishes by $1$.
In both cases the vector is multiplied by
$\sqrt{q_{}!(2r_{-1}-q_{})!}$ (see Proposition \ref{pq}).
\end{thm}
\section{The construction in the case $g_n$}
\label{step3} In the present section in the case $n>2$ we construct a Gelfand-Tsetlin type base in a $g_{n}$-representation, the construction is similar for all algebras. Then we calculate the Wigner coefficients for $g_{n-1}$, the calculations for $g_2$ are different for series $B,C,D$ but the calculations for $g_n$, $n>2$ are similar for all algebras. Then we derive explicit formulas for the action of generators in the Gelfand-Tsetlin type base.
Consider a $g_n$-representation with the highest weight
$[m_{-n,n},...,m_{-1,n}]$.
\subsection{The indicator system in the problem of restriction $g_n\downarrow g_{n-1}$}\label{indn}
The indicator system $I_n$ can be presented as a union of two systems of equation: $I'_n$ and $I_2$. Нere $I_2$ is an indicator system that appears in the problem of restriction $g_2\downarrow g_1$, it is different for series $B$, $C$, $D$ (see Sections \ref{indb},
\ref{indc}, \ref{indd}), and the system $I'_n$ is the same for all algebras, this is the following system
\begin{align}\begin{split}
&(z_{-n+1,-1}\frac{\partial }{\partial z_{-n,-1}}+z_{-n+1,1}\frac{\partial }{\partial z_{-n,1}})^{r_{-n}+1}f=0,\\
&...,\\
&(z_{-2,-1}\frac{\partial }{\partial z_{-3,-1}}+z_{-2,1}\frac{\partial }{\partial z_{-3,1}})^{r_{-3}+1}f=0.\end{split}
\end{align}
The exponents $r_{i}$ are defined by formulas
\begin{equation}
r_{-n}=m_{-n,n}-m_{-n+1,n},...,r_{-3}=m_{-3,n}-m_{-2,n}.
\end{equation}
The indicator system has this type not only in the problem of restriction $g_n\downarrow g_{n-1}$
for all algebras $g_n=\mathfrak{o}_{2n+1},\mathfrak{sp}_{2n},\mathfrak{o}_{2n}$,
but also in the problem of restriction $\mathfrak{gl}_{n+1}\downarrow \mathfrak{gl}_{n-1}$.
Using this remark let us construct a base in the solution space of $I_{n}$, starting from a base in the solution space of
$I_2$. Firstly let us show how this base is constructed in the case
$\mathfrak{gl}_{n+1}$,
and then let us show that this construction is valid also in the case $g_n$.
Consider the algebra $\mathfrak{gl}_{n+1}$. Let this algebra act in the space with coordinates
$x_{-n},x_{-n+1},...,x_{-1},x_{1}$. Then $\mathfrak{gl}_{n-1}$-highest vector are indexed by tableaux
\begin{align}\begin{split}\label{dcgln1}
&m_{-n,n},\,\,\,\,m_{-n+1,n},\,\,\, m_{-n+2,n},,....\,\,\,\,\,\,\,\,\,m_{-3,n},\,\,\,\, m_{-2,n},\,\,\,\, m_{-1,n},\,\,\,\, m_{1,n}\\
&\,\,\,\,m_{-n,n-1},\,\,\, m_{-n+1,n-1},,....\,\,\,\,\,\,\,\,\,\,\,\,\, m_{-3,n-1},\,\,\,\, m_{-2,n-1},\, m_{-1,n-1},\,\,\,\, \\
&\,\,\,\,\,\,\,\,m_{-n,n-2},\,\,\, m_{-n+1,n-2},,....\,\,\,\,\,\,\,\,\,\,\,\,\, m_{-3,n-2},\,\,\,\, \,\,\,\,\,m_{-2,n-2},\,\,\,\,\,\,\,\,
\end{split}\end{align}
the entries of this tableau must satisfy the betweeness conditions.
The procedure of construction of solutions consists of two steps.
{\bf Step 1.} Let us start with a solution $f(z_{-2,-1},z_{-2,1},z_{-1,1})$ of the system $I_2$,
corresponding to a $\mathfrak{gl}_3$-tableau
\begin{align}\begin{split}\label{gl3f}
& m_{-2,n},\, m_{-1,n},\, m_{1,n}\\
& \,\,\,\,\,\,\,m_{-2,n-1},\, m_{-1,n-1}\,, \\
& \,\,\,\,\,\, \,\,\,\,\,\, m_{-2,n-2},
\end{split}
\end{align}
Also $f$ is a solution of the sysytem $I_n$, which is encoded by a tableau
\begin{align}
\begin{split}
&m_{-n,n},\,\,\,\,m_{-n+1,n},\,\,\, m_{-n+2,n},,....\,\,\,\,\,\,\,\,\,m_{-3,n},\,\,\,\, m_{-2,n},\,\,\,\, m_{-1,n},\,\,\,\, m_{1,n}\\
&\,\,\,\,m_{-n,n},\,\,\, m_{-n+1,n},,....\,\,\,\,\,\,\,\,\,\,\,\,\, m_{-3,n},\,\,\,\, m_{-2,n-1},\, m_{-1,n-1},\,\,\,\, \\
&\,\,\,\,\,\, \,\,\,\,\,\,\,\,m_{-n,n},\,\,\, m_{-n+1,n},,....\,\,\,\,\,\,\,\,\,\,\,\,\, m_{-3,n},\,\,\,\, \,\,\,\,\,m_{-2,n-2},\,\,\,\,\,\,\,\,
\end{split}
\end{align}
{\bf Step 2.} Let us transform the solution $f$ of the system $I_n$ to the solution that corresponds to the tableau \eqref{dcgln1}. In \cite{zh} it is shown that the polynomial corresponding to the tableau \eqref{dcgln1} is of type
\begin{equation}
\prod_{i=-n}^{-2}
\nabla_{-1,i}^{m_{i,n-1}-m_{i,n-2}}\prod_{i=-n}^{-1} z_{i,1}^{m_{i,n}-m_{i,n-1}}
\end{equation}
where
\begin{align}
\begin{split}\label{om2}
\nabla_{k,i}=\sum_{j_1<...< j_s} c_{j_1...j_s}
E_{k,j_1}E_{j_1,j_2}...E_{j_s,j},\\
c_{j_1,....,j_s}=\prod_{k>j>i, j\neq j_k}(E_{i,i}-E_{j,j}+j-i).
\end{split}
\end{align}
One can easily check that the operators $\nabla_{-1,i}$ and the operator of multiplication on $z_{j,1}$ commute for $i<j$, thus the polynomial corresponding to \eqref{dcgln1} can be written as
\begin{equation}\label{om3}
(\prod_{i=-n}^{-3}
\nabla_{-1,i}^{m_{i,n-1}-m_{i,n-2}}
\prod_{i=-n}^{-3}z_{i,1}^{m_{i,n}-m_{i,n-1}})(\nabla_{-1,-2}^{m_{-2,n-1}-m_{-2,n-2}}z_{-2,1}^{m_{-2,n}-m_{-2,n-1}}z_{-1,1}^{m_{-1,n}-m_{-1,n-1}}).
\end{equation}
The second expression in \eqref{om3} is the polynomial $f$ up to multiplication on a constant. Introduce a notation for the first factor.
\begin{equation}\label{om1}
\Omega^{\mathfrak{gl}_{n+1}}=\prod_{i=-n}^{-3}
\nabla_{-1,i}^{m_{i,n-1}-m_{i,n-2}}
\prod_{i=-n}^{-3}z_{i,1}^{m_{i,n}-m_{i,n-1}}.
\end{equation}
Thus to the tableau \eqref{dcgln1} there corresponds the vector
\begin{equation}\label{f52}
\Omega^{\mathfrak{gl}_{n+1}}f.
\end{equation}
Let us fomulate the statement that we have proved. For this let us give a definition.
\begin{defn}
Let $f$ be a polynomial corresponding to the tableau \eqref{gl3f}. A set of operators of type \eqref{om1} is called admissible, if after application of operators from this set to $f$ one obtaines all solutions of
$I'_n$ in the space of polinomials
$F(z_{-n,-1},z_{-n,1},....,z_{-3,1},z_{-2,-1},z_{-2,1})$ with coeffisients in
$\mathbb{C}$ with the initial condition
$F(0,0,....,0,z_{-2,-1},z_{-2,1})=f(z_{-2,-1},z_{-2,1})$. And also $\Omega_1 f\neq \Omega_2 f$, where
$\Omega_1\neq \Omega_2$ are admissible operators.
\end{defn}
The previous discussion shows that the set of admissible operators depend on the elements $m_{-2,n},m_{-2,n-1},m_{-2,n-2}$ of the tableau \eqref{gl3f} only. An operator $\Omega^{\mathfrak{gl}_{n+1}}$ is admissible if and only if the nubmers $m_{i,n},m_{i,n-1},m_{i,n-2}$ satisfy
\begin{align}\begin{split}\label{ineq1}
& m_{-n,n}\geq m_{-n,n-1}\geq m_{-n+1,n}\geq....\geq m_{-3,n}\geq m_{-3,n-1}\geq m_{-2,n}\\
& m_{-n,n-1}\geq m_{-n,n-2}\geq m_{-n+1,n-1}\geq .... \geq m_{-3,n-1} \geq m_{-3,n-2} \geq m_{-2,n-1},
\end{split}
\end{align}
(
Using explicit formula for the polynomial $f$ (see \cite{zh}) corresponding to the tableau \eqref{gl3f}, one can see that the numbers $m_{-2,n},m_{-2,n-1},m_{-2,n-2}$ are defined by the degrees of variables $z_{-2,-1},z_{-2,1}$ in the polynomial $f$.
Put $z_{-1,1}=1$, one gets the proposition.
\begin{prop}\label{p3r}
In the case $\mathfrak{gl}_{n+1}$ all solution of the system
$I'_n$ in the space of polynomias of type
$F(z_{-n,-1},z_{-n,1},....,z_{-3,1},z_{-2,-1},z_{-2,1})$ with coefficients in
$\mathbb{C}$ with the initial condition
$F(0,0,....,0,z_{-2,-1},z_{-2,1})=f(z_{-2,-1},z_{-2,1})$.
can be written as $\Omega^{\mathfrak{gl}_{n+1}}f$, where the set
of admissible operators $\Omega^{\mathfrak{gl}_{n+1}}$ is defined by the monomials of $f$ or by inequalities
\eqref{ineq1}.
\end{prop}
Let us give an analogous construction in the case $g_n$.
Put $\mathfrak{k}=\mathbb{C}(z_{-1,0})$ in the case
$\mathfrak{o}_5$, put $\mathfrak{k}=\mathbb{C}(z_{-1,1})$ in the case
$\mathfrak{sp}_4$, put $\mathfrak{k}=\mathbb{C}$ in the case
$\mathfrak{o}_4$.
Note that in the case $g_{n}$ the operator $F_{ij}$ acts on a polynomial in variables $z_{-2,-1},z_{-2,1},....,z_{-n,-1},z_{-n,1}$ exactly in the same way as the operator $E_{ij}$ acts on this polynomial.
\begin{defn}
Define a operator $\Omega^{g_{n}}$ by formulas \eqref{om1}, \eqref{om2}, where $E_{ij}$ is replaced to
$F_{ij}$.
\end{defn}
Now let us construct the set of admissible operators $\Omega^{g_{n}}$.
Let us be given a $g_2$-tableau $D$ and denote the corresponding polynomial as $f$.
Our next pupose is to describe the set of admissible operators $\Omega^{g_{n}}$, such that applying them to
$f$ we obtain all solutions of $I'_n$ with the initial consdition $f$ as in Proposition \ref{p3r}.
In the cases $\mathfrak{o}_{2n+1}$, $\mathfrak{sp}_{2n}$ in Sections \ref{indb}, \ref{indc} a correspondence between problems of restriction $\mathfrak{gl}_3\downarrow\mathfrak{gl}_1$ and $\mathfrak{o}_5\downarrow\mathfrak{o}_3$, $\mathfrak{sp}_4\downarrow\mathfrak{sp}_2$
was established. This correpondence is generated by a corresponce between solution spaces of indicatir systems. The later correspondence preserves the degrees of variables $z_{-2,-1},z_{-2,1}$.
The system $I'_n$ is the same in the case $g_n$ and in the case $\mathfrak{gl}_{n+1}$. Thus one gets that the condition providing that an operator is admissible in the cases $\Omega^{\mathfrak{o}_{2n+1}}$, $\Omega^{\mathfrak{sp}_{2n}}$ and in the case $\Omega^{\mathfrak{gl}_{n+1}}$ are the same. This condition is the set of inequalities \eqref{ineq1}.
Now let us find the admissible operators in the case $\mathfrak{o}_{2n}$. In the case $\mathfrak{gl}_3$ the element $m_{-2,n-1}$ of the Gelfand-Tsetlin tableau can be charateried as follows. We act on the vector corresponding to the tableau by the raising operator $E_{-1,-2}^k$ for the maximum possible $k$. We obtain a $\mathfrak{gl}_2$-highest vector. Then $m_{-2,n-1}$ is a $(-2)$-component of it's highest weight.
Consider a vector $\mathfrak{gl}_3$-representation defind by a polynomial $z_{-2,-1}^pz_{-2,1}^q$, it is a linear combination of tableaux. Let us give a formula for the biggest element $m_{-2,n-1}$ of these tableaux. Under the action
$E_{-1,-2}^k$ this polynomial is tranformed into $constz_{-2,-1}^{p-k}z_{-2,1}^{q-k}$. Thus the maximum value of $k$ is $min(p,q)$. Hence
\begin{equation} \label{mpro4} m_{-2,n-1}=m_{-2,n}-min(p,q).\end{equation}
In the case $\mathfrak{o}_4$ to a Gelfand-Tsetlin type tableax there corresponds a polynomial
$const z_{-2,-1}^pz_{-2,1}^q$. The formula for the element $m'_{-2,2}$ of this tableau is just \eqref{mpro4}.
Also in the case of the problem of restriction $\mathfrak{gl}_3\downarrow \mathfrak{gl}_1$ the element $m_{-2,n-2}$ is the weight of the corresponding vector relatively the subalgebra $\mathfrak{gl}_1$. In the case $g_{n}=\mathfrak{o}_4$ and the problem of the restriction $g_n\downarrow g_{n-1}$ the element $m_{-2,1}$ is the weight relatively $g_{n-1}=\mathfrak{o}_2$.
Thus in the case $\Omega^{\mathfrak{o}_{2n}}$ and in the case $\Omega^{\mathfrak{gl}_{n+1}}$ are the same. This condition is the set of inequalities \eqref{ineq1}.
The following analog of the Proposition \ref{p3r} takes place.
\begin{prop}\label{p33r}
In the case $g_{n}$ all solutions of the system $I'_n$ in the space of polynomials $F(z_{-n,-1},z_{-n,1},....,z_{-3,1},z_{-2,-1},z_{-2,1})$
with coefficients in $\mathfrak{k}$ with the initial condition
$F(0,0,....,0,z_{-2,-1},z_{-2,1})=f(z_{-2,-1},z_{-2,1})$.
can be writtem as $\Omega^{g_{n}}f$, where only the admissible
operators $\Omega^{g_{n}}$ are taken.
\end{prop}
Note that if $f$ is a polynomial over $\mathbb{C}$ then
$\Omega^{g_{n}}f$ is a polynomial over $\mathbb{C}$.
Using an explicit description of admissible operators one obtains a proposition
\begin{prop}\label{p4rg}
In the case $g_n$ all polynomial soltions $I_n$ can be written as
$\Omega^{g_{n}}f$, they are encoded by tableaux of type
\begin{align}\begin{split}\label{dcgln2}
&m_{-n,n},\,\,\,\,m_{-n+1,n},\,\,\, m_{-n+2,n},,....\,\,\,\,\,\,\,\,\,m_{-3,n},\,\,\,\,\\
&\,\,\,\,m'_{-n,n},\,\,\, m'_{-n+1,n},,....\,\,\,\,\,\,\,\,\,\,\,\,\, m'_{-3,n},\,\,\,\, D \\
&\,\,\,\,\,\,\,\,m_{-n,n-1},\,\,\, m_{-n+1,n-1},,....\,\,\,\,\,\,\,\,\,\,\,\,\, m_{-3,n-1},\,\,\,\, \,\,\,\, ,
\end{split}\end{align}
where $D$ is a Gelfand-Tsetlin type tableaux for a $g_2$-representation with the highest weight $[m_{-2,n},m_{-1,n}]$. The following inequalities must take place
\begin{align}\begin{split}\label{ineq}
& m_{-n,n}\geq m'_{-n,n}\geq m_{-n+1,n}\geq....\geq m_{-3,n}\geq m'_{-3,n}\geq m_{-2,n} \geq...\\
& m'_{-n,n}\geq m_{-n,n-1}\geq m'_{-n+1,n}\geq .... \geq m'_{-3,n} \geq m_{-3,n-1} \geq m'_{-2,n}\geq ...
,\end{split}
\end{align}
supplied with the inequalities corresponding to the $g_2$-tableau $D$.
The polynomial $f$ is defined by $D$, and $\Omega^{g_{n}}$ is defined by the rest part of the tableau \eqref{dcgln2}.
\end{prop}
\subsection{The Gelfand-Tsetlin type base for $g_{n}$}
To use this indexation of
$g_{n-1}$-highest obtain in Proposition \ref{p4rg} vectors for the construction of the Gelfand-Tsetlin type base we must prove the following Proposition.
\begin{prop}
The lower row in \eqref{dcgln2} is the $g_{n-1}$-weight of the corresponding
$g_{n-1}$-highest vector.
\end{prop}
\proof The proof is based on the following fact. Let us be given a polynomial from
$\mathfrak{k}[z_{-3,-1},z_{-3,1},...,z_{-n,-1},z_{-n,1}]$,
corresponding to the tableau \eqref{dcgln2}. The action on it of the operators $E_{i,i}$ $i=-n,...,-3$, in the case $\mathfrak{gl}_{n+1}$,
and of the operators $F_{i,i}$, in the case $g_n$, coincide.
For the operators $E_{i,i}$ this polynomial is an eigenvector with the eigenvalue $m_{i,n-2}$. Thus for the components of the weight with indices $i=-n,...,-3$ the statement is proved. For $i=-2$ the statement follows form the analogous statement for $g_2$.
\endproof
Applying the standard procedure of construction of the Gelfand-Tsetlin base we obtain the Theorem.
\begin{thm} In a represention of $g_{n}$ there exist a base called the Gelfand-Tsetlin type base. Its vector are encoded by tableaux \eqref{gcmspo}. The rows $[m]_{k}, [m']_{k},[m]_{k-1}$ of these tableaux are of type \eqref{dcgln2}, where $D$ is a Gelfand-Tsetlin tableau for $g_2$.
The elements of these tableaux satisfy the inequalities \eqref{ineq},
supplied with inequalities corresponding to the $g_2$-tableau $D$.
\end{thm}
Let us give the formulas for the weight of the vector encoded by a tableau $(m)$. Denote it as
\begin{equation}
\cal Delta(m)=[\cal Delta(m)_{-n},...,\cal Delta(m)_{-1}],
\end{equation}
\begin{prop}\label{strokan}
\begin{align*}
\cal Delta(m)_{-k+n-1}=-2\sum_{i}m'_{i,k}+\sum_{i}m_{i,k}+\sum_{i}m_{i,k-1} \text { in the cases } \mathfrak{sp}_{2n},\mathfrak{o}_{2n},\\
\cal Delta(m)_{-k+n-1}=-2\sum_{i}m'_{i,k}+\sum_{i}m_{i,k}+\sum_{i}m_{i,k-1}-\sigma_{-k}\text { in the cases } \mathfrak{o}_{2n+1}\\
\end{align*}
\end{prop}
The proof can be found in Appendix in Section
\ref{sootvn}.
\subsection{Reduced matrix elements}
\label{reductgn}
To calculate Wigner coefficients and matrix elements of generators we need reduced matrix elements of the operator $F_{-1,-2}$,
viewed as $g_{n-1}$-tensor operator that acts between
$g_{n-1}$-representations into which a
$g_n$-representation splits.
Let $(\bar{m})$, $(m)$be two tableaux for the algebra $g_n$.
In the calculation of the reduced matrix elements we can suggest that these tableaux are maximal with respect to the subalgebra $g_{n-1}$.
\begin{defn}
Denote as $(m)_{red}$ the part of the tableau $(m)$, formed by three upper rows (that is the tableau \eqref{dcgln2}) from which the
$g_2$-tableau $D$ is removed. Thus the three upper rows of $(m)$ (on which the reduced matrix element depend) can be written as $(m)_{red}D$.
\end{defn}
Note that $(m)_{red}$ is also a part of a
$\mathfrak{gl}_{n+1}$-tableau. Its three upper rows can be written as $(m)_{red}D$, where $D$ is a $\mathfrak{gl}_3$-tableau. The action of $E_{-1,-2}$ on the tableau
$(m)_{red}D$ is know (see \cite{zh}). The result is a linear combination of tableaux, the $i$-th tableau in this combination is obtained by subtracting $1$ from the $i$-th element of the third row.
Thus one can write
\begin{equation}\label{expr1} E_{-1,-2}((m)_{red}D)=(E_{-1,-2}(m)_{red})D+(m)_{red}(E_{-1,-2}D).
\end{equation}
To obtain an analogous equality for $F_{-1,-2}$,
let us use the formula for the polynomial
$f$, corresponding to $(m)_{red}D$ that was obtained in \ref{indn}. One has
\begin{equation}\label{expr2} f=\Omega^{g_n}c,
\end{equation}
where $c$is a polynomial, corresponding to $D$.
The action of $E_{-1,-2}$ on $f$ is described as follows. We multiply the expression \eqref{expr2} onto $E_{-1,-2}$ in the left and then move $E_{-1,-2}$ to $f_0$. From the commutation relations new summands appear. They correspond to the member
$(E_{-1,-2}(m)_{red})D$ in the expression \eqref{expr1}.
The member corresponding to the action of $E_{-1,-2}$ on $f_0$, corresponds to the member $(m)_{red}(E_{-1,-2}D)$ in the expression \eqref{expr1}.
Let us formulate the ruler for calculation of $<(\bar{m})_{red}|F_{- 1, -2}|(m)_{red}>_{red}$.
\begin{lem}\label{redgn}
\begin{equation} F_{-1,-2}((m)_{red}D)=(E_{-1,-2}(m)_{red})D+(m)_{red}(F_{-1,-2}D).
\end{equation}
\end{lem}
\proof
The proof is an immediate consequence of the following fact. The correspondence: $E_{i,-2}\mapsto F_{i,-2}$, $i=-n,...,-1$ is agreed with commutators.
\endproof
Let us give an explicit formula for the reduced matrix elements.
Note that when we apply the operator
\begin{equation}
\Omega^{\mathfrak{gl}_{n+1}}=\prod_{i=-n}^{-3}
\nabla_{-1,i}^{m'_{i,n}-m_{i,n-1}}
\prod_{i=-n}^{-3}z_{i,1}^{m_{i,n}-m'_{i,n}},
\end{equation}
to the function, which is identically equal to one, we get a vector corresponding to the diagram
\begin{align}\begin{split}
&m_{-n,n},\,\,\,\,m_{-n+1,n},\,\,\, m_{-n+2,n},,....\,\,\,\,\,\,\,\,\,m_{-3,n},\,\,\,\, m_{-2,n},\,\,\,\, m_{-1,n},\,\,\,\, m_{1,n}\\
&\,\,\,\,m'_{-n,n},\,\,\, m'_{-n+1,n},,....\,\,\,\,\,\,\,\,\,\,\,\,\, m'_{-3,n},\,\,\,\,m_{-2,n},\, m_{-1,n},\,\,\,\, \\
&\,\,\,\,\,\,\,\,m_{-n,n-2},\,\,\,
m_{-n+1,n-1},,....\,\,\,\,\,\,\,\,\,\,\,\,\, m_{-3,n-1},\,\,\,\,
\,\,\,\,\,m_{-2,n},\,\,\,\,\,\,\,\,
\end{split}\end{align}
where the $\mathfrak{gl}_3$-tableau on the right is maximal. Denote this $\mathfrak{gl}_3$-tableau as $D_{max}$.
Define the symbol $<(\bar{m})_{red} \mid E_{-1,-2} \mid
(m)_{red}>_{red}$ as follows. If these exists $i_1=-n,...,-3$,
such that $\bar{m}_{i_1,n-1}=m_{i_1,n-1}-1$ and all other elements of rows $(\bar{m})_{red}$, $(m)_{red}$ coincide that it equals the $\mathfrak{gl}_{n+1}$-matrix element $<(\bar{m})_{red}D_{max}\mid E_{-1,-2}\mid (m)_{red}D_{max}>$. Otherwise it is zero.
An explicit formula for the matrix element $<(\bar{m})_{red}D_{max}\mid
E_{-1,-2}\mid (m)_{red}D_{max}>$ is obtained in \cite{zh}. It equals
\begin{equation}
\frac{\prod_{i=-n}^{-1} (l'_{i,n}-l_{i_1,n-1})}{ \prod_{i=-n,i\neq
i_1}^{-2} (l_{i,n-1}-l_{i_1,n-1}) },
\end{equation}
where
\begin{equation}
l'_{i,n}=m'_{i,n}+i,\,\,\,\, l_{i,n-1}=m_{i,n-1}+i.
\end{equation}
The following theorem takes place. It is a direct corollary of Lemma
\ref{redgn}.
\begin{thm}
\label{t2gn}
\begin{align}\begin{split}&
<(\bar{m})_{red} \bar{D} \mid F_{-1,-2} \mid
(m)_{red}D>_{red}=\delta_{D,\hat{D}}<(\bar{m})_{red} \mid E_{-1,-2}
\mid (m)_{red}>_{red}+\\&+\delta_{(\bar{m})_{red},(m)_{red}}<\bar{D}
\mid F_{-1,-2} \mid D>_{red}.
\end{split}
\end{align}
It is suggested that
$\bar{m}_{i_1,n-1}=m_{i_1,n-1}-1$ and $\bar{m}_{j,n-1}=m_{j,n-1}$ for
$j\neq i$.
\end{thm}
\subsection{Wigner coefficients}
\label{coew}
Let us obtain formulas for the Wigner coefficients for $g_{n-1}$.
Following \cite{1963}, let us first obtain formulas for the coefficients
\begin{align}
\label{vig} <(\bar{m}) \begin{pmatrix} j \\ [1\dot{0}]_{n-1} \\ -2
\end{pmatrix} (m)>.
\end{align}
We use the following fact.
Let us be given a diagram $(m)$ that define a vector in s $g_{n-1}$-representation
with the highest weight $[m_{-n,n-1},...,m_{-2,n-1}]$.
A polynomial on the group $Z_{n-1}$ corresponds to this vector. Consider it as a polynomial on a bigger group $Z_n$.
\begin{prop}\label{predl}
The vector that corresponds to this polynomial belong to a $g_{n}$-representation with the highest weight $[m_{-n,n-1},...,m_{-2,n-1},0]$. The corresponding tableau is of type $\begin{pmatrix}max\\ m\end{pmatrix}$, this is a $g_{n}$-tableau that is obtained form
$(m)$ by adding two maximum row.
\end{prop}
The proof can be found in Appendix in Section
\ref{docpredl}.
Now return to the calculation of the Wigner coefficient
\eqref{vig}. Let us be given two $g_{n-1}$-tableaux $(\bar{m})$ and $(m)$. We can suggest that they are $g_{n-2}$-maximal. The three upper rows of these tableau are of type $(\bar{m})_{red}\bar{D}$ and $(m)_{red}D$. As in Proposition \ref{predl}
add to them two maximal
$g_n$-rows, denote the five rows that we obtain as $\widetilde{(\bar{m})_{red}\bar{D}}$ and $\widetilde{(m)_{red}D}$.
Let us prove the equality.
\begin{prop}\label{pr12}
\begin{equation}
<(\bar{m}) \begin{pmatrix} j \\ [1\dot{0}]_{n-1} \\ -2
\end{pmatrix} (m)>=<\widetilde{((\bar{m})_{red}\bar{D})} \mid F_{-1,-2} \mid \widetilde{((m)_{red}D)}>.
\end{equation}
\end{prop}
\proof
Let us apply the Wigner-Eckart theorem to the matrix element on the right. It equals to the product of a reduced matrix element
and a Wigner coefficient that occurs on the left side of the equality. Since the upper two rows of tableaux $(\bar{m})_{red}\bar{D})$, $\widetilde{((m)_{red}D)}$ are maximal the reduced matrix element equals to $1$. This proves the equality.
\endproof
Let us calculate the matrix element that occurs on the right in the equality \ref{pr12}.
Let us be given a tableau that defines a
$g_{n-2}$-highest vector in a $g_n$-representation. This is a tableau of type
\begin{align}
\begin{split}\label{diagc}
& m_{-n,n},\,\,\,....\,\,\,m_{-4,n}\,\,\, \\
&\,\,\,m'_{-n,n},\,\,\, ....\,\,\,m'_{-4,n}\,\,\,\,\\
&\,\,\,\,\,\,m_{-n,n-1},\,\,\, ....\,\,\,m_{-4,n-1}\,\,\,\, C\\
&\,\,\,\,\,\,\,\,\,m'_{-n,n-1},\,\,\, ....\,\,\,m'_{-4,n-1}\,\,\,\,\\
&\,\,\,\,\,\,\,\,\,\,\,\,m_{-n,n-2},\,\,\, ....\,\,\,m_{-4,n-2}\,\,\,\,,\\
\end{split}
\end{align}
where $C$ is a $g_3$-tableau. Denote the tableau \eqref{diagc} shortly as
$(k)C$.
Using the technique of raising operators $\nabla_{ij}$ (see \cite{zh}), and applying the arguments that were used in the poorf of the formula \eqref{f52}, one obtains that the polynomial that corresponds to this tableau
can be written as
$\Omega f,$ where $f$ is a polynomial that corresponds to a
$g_3$-tableau $C$ and the operator $\Omega $ is defined as follows
\begin{equation}
\Omega =\prod_{i=-n}^{-3} \nabla_{-3,i}^{m'_{i,n-1}-m_{i,n-2}}
\prod_{i=-n}^{-3}
\nabla_{-2,i}^{m_{i,n-1}-m'_{i,n-1}}\prod_{i=-n}^{-3}
\nabla_{-1,i}^{m'_{i,n}-m_{i,n-1}}
\prod_{i=-n}^{-3}z_{i,1}^{m_{i,n}-m'_{i,n}}
\end{equation}
The considered tableau $\widetilde{((m)_{red}D)}$ is of type
\eqref{diagc}. One has $(n)=\widetilde{(m)_{red}}$ and
$C=\widetilde{D}$. By analogy with the proof of Lemma \ref{redgn} one concludes that
\begin{equation}
F_{-1,-2}\widetilde{((m)_{red}D)}=\widetilde{(E_{-1,-2}(m)_{red})}\widetilde{D}+\widetilde{(m)_{red}}F_{-1,-2}\widetilde{D}.
\end{equation}
When one passes to matrix elements, one gets
\begin{align}\begin{split}&
<\widetilde{((\bar{m})_{red}\bar{D})} \mid F_{-1,-2} \mid \widetilde{((m)_{red}D)}>=
\delta_{\bar{D},D}<\widetilde{((\bar{m})_{red})} \mid E_{-1,-2} \mid \widetilde{((m)_{red}}>+\\
&+\delta_{(\bar{m})_{red},(m)_{red}}< \widetilde{\bar{D}} \mid F_{-1,-2} \mid \widetilde{D}>.
\end{split}
\end{align}
Thus we have proved a Theorem
\begin{thm}
\label{t3gn}
\begin{align}
\begin{split}
\label{vign}& <(\bar{m}) \begin{pmatrix} j \\ [1\dot{0}]_{n-1} \\ -2
\end{pmatrix} (m)>=\\&=
\delta_{\bar{D},D}<(\bar{m})_{red} \mid E_{-1,-2} \mid (m)_{red}>+\delta_{(\bar{m})_{red},(m)_{red}}<\bar{D} \mid F_{-1,-2} \mid D>.
\end{split}
\end{align}
\end{thm}
Let us give rulers for calculation of summands that occur on the right hand side in Theorem \ref{t3gn}.
\subsubsection{The matrix element $<\widetilde{((\bar{m})_{red})} \mid E_{-1,-2} \mid \widetilde{((m)_{red}}>$}
As in previous section this matrix element can expressed through a matrix element of the algebra $\mathfrak{gl}_{n+1}$.
\begin{thm} If there exists
$i_1=-n,...,-3$,
such that $\bar{m}_{i_1,n-1}=m_{i_1,n-1}-1$, and all other elements of rows $(\bar{m})_{red}$, $(m)_{red}$ coincide than the considered matrix
elements equals
\begin{equation}
\frac{\prod_{i=-n}^{-1} (l_{i,n-1}-l_{i_1,n-2})}{ \prod_{i=-n,i\neq
i_1}^{-2} (l_{i,n-2}-l_{i_1,n-2}) },
\end{equation}
where
\begin{equation}
l_{i,n-1}=m_{i,n-1}+i,\,\,\,l_{i,n-2}=m'_{i,n-1}+i.
\end{equation}
If thexe exist no such index than the considered matrix element equals zero
\end{thm}
\subsubsection{The matrix element $< \widetilde{\bar{D}} \mid F_{-1,-2} \mid \widetilde{D}>$. The case $\mathfrak{o}_5$}
This matrix element equals to a $\mathfrak{o}_5$-Wigner coefficiens
\begin{equation}
< \widetilde{\bar{D}} \mid F_{-1,-2} \mid \widetilde{D}>=<\bar{D} \begin{pmatrix} j \\ [10]_{\mathfrak{o}_5} \\ -2
\end{pmatrix} D>.
\end{equation}
Let us calculate it directly. One can suggest that
$\mathfrak{o}_5$-tableaux $\bar{D}$ and $D$ are
$\mathfrak{o}_3$-maximal.
To the tableau $D$ there corresponds a polynomial $f$ on $Z_{\mathfrak{o}_5}$.
ТSince $D$ is $\mathfrak{o}_3$-maximal, then
$f=f(z_{-3,-2},z_{-3,2})$. By Proposition \ref{predl},
to the tableau $\widetilde{D}$ there corresponds the same polynomial, but considered as a polynomial on $Z_{\mathfrak{o}_5}$.
Let find the action of the operator $e^{tF_{-1,-2}}$ on the polynomial $f$.
The explicit calculation gives that
\begin{equation} \label{o5wig}
f(z_{-3,-2},z_{-3,2})\mapsto
(1+tz_{-2,-1})^{m_{-2,2}-m_{-1,2}}(z_{-3,-2}+tz_{-3,-1},z_{-3,2})
\end{equation}
When we were defining the Gelfand-Tsetlin type base for
$\mathfrak{o}_5$ in Section \ref{indb} for the polynomial $f$ we have constructed a polynomial $f^*$ on the group $Z_{\mathfrak{gl}_4}$.
By explicit calculations it can be shown that on the polynomial $f^*$ the operator $e^{tE_{-1,-2}}$ acts by the same formula \eqref{o5wig}. Thus the correspondence conjugates the actions of $e^{tF_{-1,-2}}$ and $e^{tE_{-1,-2}}$.
In Section \ref{gcbaseo5}
using this correspondence the Gelfand-Tsetlin type base for $\mathfrak{o}_5$ was constructed. To a $\mathfrak{o}_5$-tableau $D$, that defines a $\mathfrak{o}_3$-highest vector (and a polynomial $f$), there corresponds a $\mathfrak{gl}_3$-tabelau $D^*$, which is obtained from $D$ by removing the zero from the upper row and $\sigma_{-2}$ (to the tableau $D^*$ there corresponds the polynomial $f^*$).
Thus we have
\begin{equation}\label{o5gl3wig}
< \widetilde{\bar{D}} \mid F_{-1,-2} \mid \widetilde{D}>=< \widetilde{\bar{D^*}} \mid E_{-1,-2} \mid \widetilde{D^*}>
\end{equation}
The matrix element on the right in \eqref{o5gl3wig} is a $\mathfrak{gl}_3$-Wigner coefficient.
Thus we have proved the theorem
\begin{thm}
The Wigner coefficient for $\mathfrak{o}_5$ and $\mathfrak{gl}_3$ are equal
\begin{equation}
<\bar{D} \begin{pmatrix} j \\ [10]_{\mathfrak{o}_5} \\ -2
\end{pmatrix} D>=<\bar{D^*} \begin{pmatrix} j \\ [100]_{\mathfrak{gl}_3} \\ -2
\end{pmatrix} D^*>=\begin{vmatrix} j: 3 \\ 2\end{vmatrix}^{[m]_2,[m']_{2}},
\end{equation}
where $[m]_2=[m_{-2,2},m_{-1,2},0]$ и $[m']_2=[m'_{-2,2},m'_{-1,2}]$.
\end{thm}
\begin{cor}
\begin{equation}
< \widetilde{\bar{D}} \mid F_{-1,-2} \mid \widetilde{D}>=\begin{vmatrix} j: 3 \\ 2\end{vmatrix}^{[m]_2,[m']_{2}},
\end{equation}
where $[m]_2=[m_{-2,2},m_{-1,2},0]$ и $[m']_2=[m'_{-2,2},m'_{-1,2}]$.
\end{cor}
\subsubsection{The matrix element $< \widetilde{\bar{D}} \mid F_{-1,-2} \mid \widetilde{D}>$. The case $\mathfrak{sp}_4$}
In this case the matrix element equals to a $\mathfrak{sp}_4$-Wigner coefficient
\begin{equation}
< \widetilde{\bar{D}} \mid F_{-1,-2} \mid \widetilde{D}>=<\bar{D} \begin{pmatrix} j \\ [10]_{\mathfrak{sp}_4} \\ -2
\end{pmatrix} D>.
\end{equation}
In section \ref{gcbasesp4} the Gelfand-Tsetlin type base for $\mathfrak{sp}_4$ was defined. To a $\mathfrak{sp}_4$-tableau $D$
there corresponds a $\mathfrak{gl}_3$-tableau $D^*$ from which the zero in the upper row is removed.
Analogously to the case $\mathfrak{o}_5$ one can prove
\begin{thm}
The Wigner coefficient for $\mathfrak{sp}_4$ and $\mathfrak{gl}_3$ are equal
\begin{equation}
<\bar{D} \begin{pmatrix} j \\ [10]_{\mathfrak{sp}_4} \\ -2
\end{pmatrix} D>=<\bar{D^*} \begin{pmatrix} j \\ [100]_{\mathfrak{gl}_3} \\ -2
\end{pmatrix} D^*>=\begin{vmatrix} j: 3 \\ 2\end{vmatrix}^{[m]_2,[m']_{2}},
\end{equation}
where $[m]_2=[m_{-2,2},m_{-1,2},0]$ и $[m']_2=[m'_{-2,2},m'_{-1,2}]$.
\end{thm}
\begin{cor}
\begin{equation}
< \widetilde{\bar{D}} \mid F_{-1,-2} \mid \widetilde{D}>=\begin{vmatrix} j: 3 \\ 2\end{vmatrix}^{[m]_2,[m']_{2}},
\end{equation}
where $[m]_2=[m_{-2,2},m_{-1,2},0]$ и $[m']_2=[m'_{-2,2},m'_{-1,2}]$.
\end{cor}
\subsubsection{The matrix element $< \widetilde{\bar{D}} \mid F_{-1,-2} \mid \widetilde{D}>$. The case $\mathfrak{o}_4$}
In this case the matrix element equals to a $\mathfrak{o}_4$-Wigner coefficient.
\begin{equation}
< \widetilde{\bar{D}} \mid F_{-1,-2} \mid \widetilde{D}>=<\bar{D} \begin{pmatrix} j \\ [10]_{\mathfrak{o}_4} \\ -2
\end{pmatrix} D>.
\end{equation}
Let us calculate directly the $\mathfrak{o}_4$-Wigner coefficient on the right.
The index $j$ can take values $-2$ and $-1$. If $j=-2$ then
\begin{equation}
\bar{m}_{-2}=m_{-2}+1,\,\,\,\bar{m}_{-1}=m_{-1},\\
\end{equation}
and if $j=-1$ then
\begin{equation}
\bar{m}_{-2}=m_{-2},\,\,\,\bar{m}_{-1}=m_{-1}+1.\\
\end{equation}
For $r_{-2}=m_{-2}-m_{-1}$ and $r_{-1}=m_{-2}+m_{-1}$ (these are
highest weights for two $\mathfrak{sl}_2$ copies), one has in the
case $j=-2$
\begin{equation}
\bar{r}_{-2}=r_{-2}+1,\,\,\,\bar{r}_{-1}=r_{-1}+1,\\
\end{equation}
and in the case $j=-1$
\begin{equation}
\bar{r}_{-2}=r_{-2}-1,\,\,\,\bar{r}_{-1}=r_{-1}-1.\\
\end{equation}
The weights $p$ and $q$ for two copies of $\mathfrak{sl}_2$ are expressed through the elements of a $\mathfrak{o}_4$-tableau using the Proposition \ref{pq}.
Thus one gets the Theorem
\begin{thm}
Put $[m^1]_2=[\frac{m_{-2,2}-m_{-1,2}}{2},0]$, $[m^2]_2=[\frac{m_{-2,2}+m_{-1,2}}{2},0]$.
For $j=-2$ one has
\begin{equation}
<\bar{D} \begin{pmatrix} j \\ [10]_{\mathfrak{o}_4} \\ -2
\end{pmatrix} D>=\begin{vmatrix} \frac{1}{2}: 2 \\ 1\end{vmatrix}^{[m^1]_2, p}\begin{vmatrix} \frac{1}{2}: 2 \\ 1 \end{vmatrix}^{[m^2]_2, q}.
\end{equation}
For $j=-1$ one has
\begin{equation}
<\bar{D} \begin{pmatrix} j \\ [10]_{\mathfrak{o}_4} \\ -2
\end{pmatrix} D>=\begin{vmatrix} \frac{1}{2}: 2 \\ 1 \end{vmatrix}^{[m^1]_2, p}\begin{vmatrix} -\frac{1}{2}: 2 \\ 1\end{vmatrix}^{[m^2]_2, q}.
\end{equation}
\end{thm}
\subsection{Reduced Wigner coefficients}
For the Wigner coefficients \begin{align} \label{vigui}
<(\bar{m})
\begin{pmatrix} j \\ [1\dot{0}]_{n-1} \\ i
\end{pmatrix} (m)>
\end{align}
the following formula takes place
\begin{align} \label{viguii} <(\bar{m})
\begin{pmatrix} j_1 \\ [1\dot{0}]_{n-1} \\ i
\end{pmatrix} (m)>=\begin{pmatrix} j_{-i-1} \\ [1\dot{0}]_{n-i} \\ i \end{pmatrix}\prod_{l=1}^{-i-2}\begin{vmatrix} j_l \\ [1\dot{0}]_{n-l} \\ j_{l+1}
\end{vmatrix}
\end{align}
To calculate all Wigner coefficients we must obtain a formula for the reduced Wigner coefficients $\begin{vmatrix} j_l \\ [1\dot{0}]_{n-l} \\ j_{l+1}
\end{vmatrix} $.
Let s obtain the formula for the reduced Wigner coefficients using the previous calculations. The following theorem takes place.
\begin{thm}
\label{t4}
\begin{align}
\begin{vmatrix}j_1 \\ [1\dot{0}]_{n-1} \\ j_2 \end{vmatrix}=\begin{pmatrix}j_1 \\ [1\dot{0}]_{n-1} \\ -2 \end{pmatrix} <(\bar{m'})_{red} \bar{D}' \mid F_{-2-3} \mid
(m')_{red}D'>_{red},
\end{align}
where $'$ means that we take only the part of the tableaux that correspond to $g_{n-1}$.
\end{thm}
\proof
To prove the theorem let us calculate the matrix element $<(\bar{m})
\mid F_{-1-3} \mid (m)>$, $i<-2$ in two ways.
Firstly apply to the matrix element the Wigner-Eckart theorem and decompose the Wigner coeficient into the product of a reduced Wigner coefficient and a Wigner coefficient.
One has
\begin{align}
<(\bar{m}) \mid F_{-1-3} \mid (m)>=<(\bar{m})_{red}\bar{D} \mid F_{-1-3} \mid (m)_{red}D>_{red}
\begin{vmatrix}j_1 \\ [1\dot{0}]_{n-1} \\ j_2 \end{vmatrix}
\begin{pmatrix}j_2 \\ [1\dot{0}]_{n-2} \\ -3 \end{pmatrix}.
\end{align}
Secondly using the commutation relation
$F_{-1,-3}=[F_{-1,-2},F_{-2,-3}]$ we obtain
\begin{align}\begin{split}&
<(m') \mid F_{-1,-3} \mid (m)>=<(\bar{m})_{red} \bar{D}\mid F_{-1-2} \mid
(m)_{red}D>_{red}
\begin{pmatrix}j_1 \\ [1\dot{0}]_{n-1} \\ -2 \end{pmatrix} \cdot \\& \cdot<(\bar{m'})_{red}\bar{D}' \mid F_{-2-3} \mid
(m')_{red}D'>_{red}
\begin{pmatrix}j_2 \\ [1\dot{0}]_{n-2} \\ -3
\end{pmatrix},\end{split}
\end{align}
where $'$ means that we take only the part of the tableaux that correspond to $g_{n-1}$.
Compare two expressions, one obtains
\begin{align}
\begin{vmatrix}j_1 \\ [1\dot{0}]_{n-1} \\ j_2 \end{vmatrix}=\begin{pmatrix}j_1 \\ [1\dot{0}]_{n-1} \\ -2 \end{pmatrix} <(\bar{m'})_{red} \bar{D}' \mid F_{-2-3} \mid
(m')_{red}D'>_{red}
\end{align}
The theorem is proved
\endproof
The expressions in Theorem \ref{t4} are obtained in previous Sections.
\subsection{Matrix elements}
Using the previous results let us write the formulas for the action of generators of $g_n$ in the base that we have constructed. It is enough to give a formula for the action of $F_{-1,-2}$.
The following theorem takes place.
\begin{thm}
\label{matgn}
\begin{align}
\begin{split} &
<(\bar{m})| F_{-1,-2} |(m)>=<(\bar{m})_{red} \bar{D} \mid F_{-1,-2} \mid
(m)_{red}D>_{red}<(\bar{m}) \begin{pmatrix} j \\ [1\dot{0}]_{n-1} \\ -2
\end{pmatrix} (m)>,
\end{split}
\end{align}
where the expression for the factors are given in theorems \ref{t2gn} and
\ref{t3gn}.
\end{thm}
\proof The theorem is proved by application of the Wigner-Eckart theorem
\endproof
\section{Appendix}
\label{appendi}
\subsection{Lie algebras.}
\label{alglie}
\subsubsection{The symplectic algebra $\mathfrak{sp}_{2n}$}
Take the space $\mathbb{C}^{2n}$. Choose a base and let us index its elements by numbers
$-n,...,-1,1,...,n$. Fix a skew-symmetric form $$\omega=\sum_{i=1}^n x_i\wedge x_{-i}.$$
The group $Sp_{2n}$ consists of isomorphisms of $\mathbb{C}^N$, that preserve this skew-symmetric form. It's Lie algebra is denoted as
$\mathfrak{sp}_{2n}$.
Define the generators of the Lie algebra
$$F_{ij}=E_{ij}-sign (i) sign (j) E_{-j-i}.$$
The only relations between them are
$$F_{ij}=-sign(i)sign(j)F_{-j-i}.$$
The generators $F_{ij} $,
in the case $i>j$,
correspond to negative roots. The generators $F_{ij} $,
in the case $i<j$,
correspond to positive roots. In the case $i=j$ the generator belongs to the Cartan subalgebra.
\subsubsection{The orthogonal algebra $\mathfrak{o}_{N}$}
Take the space $\mathbb{C}^N$. Let $n$ be such that $N=2n$ in the case of even $N$, and $N=2n+1$ in the case of odd $N.$
Choose a base in $\mathbb{C}^N$ and index its elements by $-n,...,-1,1,...,n$ in the case of even $N$ and by numbers
$-n,...,-1,0,1,...,n$ in the case of odd $N$.
In the case of even $N$ take a quadratic form $$x_{-n}x_{n}+...+x_{-1}x_1,$$ в
and in the case of odd $N$ take a quadratic form $$x_{-n}x_n+...+x_{-1}x_1+x_0^2.$$
The group $O_N$ consists of isomorphisms of $\mathbb{C}^N$,
that preserve this quadratic form. It's Lie algebra is denoted as
$\mathfrak{o}_{N}$.
Introduce generators
$$F_{ij}=E_{ij}- E_{-j-i}.$$
The only relations between them are
$$F_{ij}=-F_{-j-i}.$$
The generators $F_{ij} $,
in the case $i>j$,
correspond to negative roots. The generators $F_{ij} $,
in the case $i<j$,
correspond to positive roots. In the case $i=j$ the generator belongs to the Cartan subalgebra.
\subsection{Tensor operators and Wigner coefficients }
In this section the Wigner coefficients are defined, the solution of the multiplicity problem is given, the reduced Wigner coefficients are introduced. We follow the analogous discussion for the case of the algebra
$\mathfrak{gl}_n$ in \cite{1968}.
The Wigner coefficients are closely related with irreducible tensor operators.
\begin{defn}
An irreducible tensor operator of type $[M]_n$,where $[M]_n$ is a dominant $g_n$-weight, is an indexed by vectors $(M)\in
V^{[M]_n}$the set of linear mappings
$$f_{(M)}: V^{[\bar{m}]_{n}} \rightarrow V^{[m]_{n}},$$
which has the following property. For $g\in g_n$ one has
$$[g,f_{(M)}]=f_{g(M)}.$$
\end{defn}
For given $[\bar{m}]_n$, $[m]_n$ and $[M]_n$ the tensor operator $ V^{[\bar{m}]_{n}}
\rightarrow V^{[m]_{n}} $ of type $[M]_n$ is not unique. If it is not unique one says that for this tensor operator the multiplicity problem takes place.
By Wigner-Eckart theorem the matrix elements of a tensor operator decompose into a product of a factor that depents only on highest weights
весов $[m]_{n},[\bar{m}]_{n},[M]_{n}$ (this factor is called the reduced matrix element) and a Wigner coefficient that defines an interwinnig operator $\Phi:
V^{[\bar{m}]_{n}}\rightarrow V^{[M]_{n}}\otimes V^{[m]_{n}}$.
\subsection{ The solution of the multiplicity problem}
As it is know the interwinnig operator $\Phi: V^{[\bar{m}]_{n}}
\rightarrow V^{[M]_{n}}\otimes V^{[m]_{n}}$
is in general not unique. On the other language this means that one irreducible representation $V^{[\bar{m}]_{n}} $ can occur in splitting of tensor product not once. But the parametrization of all interwinnig operators is well-known.
All such operators are indexed by tableaux $(\Gamma)\in V^{[M]_{n}}$,
such that $$[\bar{m}]_n=\cal Delta (\Gamma)+[m]_n,$$ where $\cal Delta (\Gamma)$ is the weight of $v$. This vector $\Phi$ is defined by the operator as follows
$$\Phi((m)_{max})=\cal Delta (\Gamma)\otimes (m)_{max}+l.o.t.,$$ where $l.o.t.$ (lower order terms)denotes a sum of tensor products of weight vectors where the second vector has a weight lower than $[m]_n$.
For the group $U(n)$ this was proved by Biedenharn and Baird in \cite{19633}.
The corresponding Wigner coefficient is denoted as
\begin{equation}\label{viggen}
<\begin{pmatrix} [\bar{m}]_n \\ (m')_{n-1} \end{pmatrix}
\begin{pmatrix} (\Gamma)_{n-1} \\ [ M]_n\\ (M)_{n-1}\end{pmatrix}
\begin{pmatrix}[m]_n\\ (m)_{n-1}\end{pmatrix}>
\end{equation}
This coefficient can be nonzero only if the following equality holds $[\bar{m}]_n=\cal Delta(\Gamma)+[m]_n$.
\subsection{Reduced Wigner coefficients}
Take a Wigner coefficient\eqref{viggen} for the algebra $g_n$.
It also defines a tensor operator for the algebra $g_{n-1}$.
Decompose it into a sum of irreducible tensor operators and apply the Wigner-Eckart theorem. One gets
\begin{align*}
&<\begin{pmatrix} [\bar{m}]_n \\ [\bar{m}']{n} \\ [\bar{m}]_{n}\\
(\bar{m})_{n-2}
\end{pmatrix} \begin{pmatrix} ( \Gamma )_{n-1}\\ [ M]_n\\ (M)_{n-1}
\end{pmatrix} \begin{pmatrix}[m]_n\\ [m']_{n} \\ [m]_n \\ (m)_{n-1}
\end{pmatrix}>=\sum_{ (\gamma)_{n-2}} <\begin{pmatrix} [\bar{m}]_n \\ [\bar{m}']_{n} \\ [\bar{m}]_{n}\\ (\bar{m})_{n-2} \end{pmatrix} \begin{vmatrix} (\Gamma)_{n-1} \\ [ M]_n\\ (\gamma)_{n-1}
\end{vmatrix} \begin{pmatrix}[m]_n\\ [m']_{n} \\ [m]_{n-1} \\ (m)_{n-2}
\end{pmatrix}> \\&
<\begin{pmatrix} [\bar{m}]_{n-1} \\ (\bar{m})_{n-2} \end{pmatrix} \begin{pmatrix} (\gamma)_{n-2} \\ [ M]_{n-1}\\ (M)_{n-2}
\end{pmatrix} \begin{pmatrix}[ [m]_{n-1} \\ (m)_{n-1}
\end{pmatrix}>
\end{align*}
In this formula the first factor on the right is a notation for the reduced matrix element of an irreducible $g_{n-1}$-tensor operator.
This factor is called the reduced Wigner coefficient.
The tableau $(\gamma)_{n-1}$ is obtained by adding to the tableau $(\gamma)_{n-2}$ the rows $[M']_{n}$ and $[M]_{n-1}$.
Note that the reduced matrix element does not depend on the rows $[\bar{m}]_{n-2}$, $[m]_{n-2}$ and below.
Thus the reduced Wigner coefficient can be denoted as
\begin{align}\label{redviggen}
<\begin{pmatrix} [\bar{m}]_n \\ [\bar{m}']{n} \\
[\bar{m}]_{n-1}
\end{pmatrix} \begin{vmatrix} (\Gamma)_{n-1} \\ [ M]_n\\ (\gamma)_{n-1}
\end{vmatrix} \begin{pmatrix}[m]_n\\ [m']_{n} \\ [m]_{n-1}
\end{pmatrix}>
\end{align}
This coefficient is nonzero only if the following holds: $[\bar{m}]_n=\cal Delta(\Gamma)+[m]_n$ и
$[\bar{m}]_{n-1}=\cal Delta((\gamma)_{n-1})+[m]_n$.
\subsection{ Fundamental operators}
In the present paper only the Wigner coefficients are considered for which $[M]_n=[1,0,...,0]=[1 \dot{0}]_{n}$, that is when the tensor factor
$V^{[M]_n}$ is a standard representation. Such Wigner coefficients are called fundamental.
Note that weight vectors $(m)$ of the standard representations are completely defined by their weights $\cal Delta(m)=[0,...,\pm 1,...,0]$, where $\pm 1$ occurs at the place $i$. If $i=0$ then only $1$ is allowed. Thus the Wigner coefficient is of type can be denoted as
\begin{equation}
\begin{pmatrix} i \\ [1 \dot{0}]_{n}\\j \end{pmatrix}
\end{equation}
\subsection{The proof of Propositions \ref{stroka05} and \ref{weightofo5}} \label{sootv}
The Proposition \ref{stroka05} is the following statement.
\begin{prop*}
The number $m_{-2,1}$ in the tableau \eqref{gco5} is the
$\mathfrak{o}_3$-weight of the corresponding vector.
\end{prop*}
\proof Take a polynomial
\begin{equation} \label{monomo5}
\sum_{k,l,r,s}с_{ k,l,r,s } z_{-2,-1}^{k}z_{-2,1}^{l}z_{-1,1}^r z_{-1,0}^s.\end{equation}
Suggest it defines a $\mathfrak{o}_{3}$-highest vector in a
$\mathfrak{o}_{5}$-representaion with the highest weight $[m_{-2},m_{-1}]$, then to each it's monomial there corresponds a function
\begin{align}\begin{split}
\label{al2}
\delta_{-2}^{m_{-2,2}-k-l}\delta_{-1}^{m_{-1,2}+k-2r-s-l}.\end{split}\end{align}
Then the $\mathfrak{o}_{3}$-weight of the corresponding vector equals
$m_{-2}-k-l.$
Also to \eqref{monomo5} there corresponds a polynomial
\begin{equation} \label{monomgl3}
\sum_{k,l,r,s}с_{ k,l,r,s }z_{-2,-1}^{k}z_{-2,1}^{l}z_{-1,1}^r
z_{-1,1}^{[\frac{s}{2}]}.\end{equation}
that defines a $\mathfrak{gl}_{1}$-highest vector in s
$\mathfrak{gl}_{3}$-representaion with the highest weight
$[m_{-2,2},m_{-1,2},0]$, to each it's monomial there corresponds a function
\begin{align}\begin{split}\label{al11} \delta_{-2}^{m_{-2,2}-k-l}\delta_{-1}^{m_{-1,2}+k-r-s}\delta_{1}^{r+l}.\end{split}\end{align}
The weight of the vector that corresponds to this polynomial equals
$m_{-2,2}-k-l.$
Thus the weight of the $\mathfrak{o}_{3}$-highest vector that corresponds to
\eqref{monomo5} equals the weight of the
$\mathfrak{gl}_{1}$-highest vector that corresponds to \eqref{monomgl3}. But the weight of the last vector equals $m_{-2,1}$. The Proposition is proved.
\endproof
Let us prove the Proposition \ref{weightofo5}.
\begin{prop*}
\label{weighto5}
$\cal Delta(m)_{-1}=-2\sum_i m'_{i,2}+\sum_i
m_{i,2}+m_{-2,1}-\sigma_{-2}$
\end{prop*}
\proof
Take a polynomial \eqref{monomo5} that corresponds to a
Gelfand-Tsetlin tableaux. Consider the expression \eqref{al2}. One
obtains
\begin{equation}\label{vesochik}
\cal Delta(m)_{-1}=m_{-1,2}+k-2r-s-l.
\end{equation}
Consider the polynomial the corresponding \eqref{monomgl3} that defines a
$\mathfrak{gl}_1$-highest vector in a
$\mathfrak{gl}_{3}$-representation, its components with indices
$-1$ and $1$ are equal to $m_{-1,2}+k-r-s$ and $r+l$.
Their difference equals $$m_{-1,2}+k-2[\frac{s}{2}]-2-l.$$ This equals
\eqref{vesochik}, if $s$ is even and differs by one from
\eqref{vesochik}, if $s$ is odd. Thus the expression
\begin{equation}
m_{-1,2}+k-2[\frac{s}{2}]-2-l-\sigma_{-1}
\end{equation}
equals \eqref{vesochik}.
Thus $\cal Delta(m)_{-1}$ is a difference of $\mathfrak{gl}_3$-weights with numbers $-1$ and $1$ minus $\sigma_{-1}$.
Note that in the case $\mathfrak{gl}_{3}$ the component of the weight with the index $-1$ equals $\sum_{i} m'_{i,2}-\sum_i m_{i,2}$, and the component of the weight with the index $1$ equals
$m_{-2,1}-\sum_{i} m'_{i,2}$.
\endproof
\subsection{The proof of Proposition \ref{strokan}}
\label{sootvn}
Let us prove the following statement
\begin{prop*}
\begin{align*}
\cal Delta(m)_{-k+n-1}=-2\sum_{i}m'_{i,k}+\sum_{i}m_{i,k}+\sum_{i}m_{i,k-1} \text { in the cases } \mathfrak{sp}_{2n},\mathfrak{o}_{2n},\\
\cal Delta(m)_{-k+n-1}=-2\sum_{i}m'_{i,k}+\sum_{i}m_{i,k}+\sum_{i}m_{i,k-1}-\sigma_{-k}\text
{ in the cases } \mathfrak{o}_{2n+1}
\end{align*}
\end{prop*}
\proof
It is enough to prove the formula for $\cal Delta(m)_{-1}$. One can suggest that the tableau $(\Gamma)$ is maximal with respect to $g_{n-1}$. To this tableau there corresponds a polynomial $f$, the procedure of it's construction is described in Section \ref{indn}.
The polynomial $f$ is of type $$f=cf_0,$$ where $c\in \mathfrak{k}$ (see Definition of the field $\mathfrak{k}$
in Section \ref{indn}), and $f_0\in\mathbb{C}(z_{-3,-1},z_{-3,1},...,z_{-n,-1},z_{-n,1})$.
The weight of the vector corresponding to $f$ is calculated as follows (see \cite{zh}).
To each variable $z_{ij}$
(also to variables from $\mathfrak{k}$) the correspond the multiplicator $\delta_i^{-1}\delta_j$. We suggest that $\delta_0=1$, $\delta_{1}=\delta_{-1}^{-1}$. The multiplicators are multiplied onto the function
$\delta_{-n}^{m_{-n}}...\delta_{-1}^{m_{-1}}$, corresponding to the highest weight. The degree of $\delta_{-1}$ is the weight $\cal Delta(m)_{-1}$.
From the structure of $f$ one sees that the change of transformation of the weight $[m_{-n,n},...,m_{-1,n}]$ under the action of $f$ equals to the sum of transformations under the action of $c$ and $f_0$. The transformation under the action of $c$ was investigated when we considered the case $g_2$, it equals
\begin{align*}
&-2\sum_{i=-2}^{T}m'_{i,k}+\sum_{i=-2}^{-1}m_{i,n}+m_{-2,n-1} \text { where }T=-1\text{ or }-2 \text{ in the case } \mathfrak{sp}_{2n},\mathfrak{o}_{2n},\\&
-2\sum_{i=-2}^{-1}m'_{i,k}+\sum_{i=-2}^{-1}m_{i,n}+m_{-2,n-1}-\sigma_{-n}\text { in the case } \mathfrak{o}_{2n+1}
\end{align*}
Since the transformation of the weight under th action of $f_0$ is the same for all $g_n$ and $\mathfrak{gl}_{n+1}$ then using result for $\mathfrak{gl}_{n+1}$ we obtain that under the action of $f_0$ to $\cal Delta(\Gamma)_{-1}$ the following value is added
\begin{align*}
-2\sum_{i=-n}^{-3}m'_{i,k}+\sum_{i=-n}^{-3}m_{i}+\sum_{i=-n}^{-3}m_{-2,n-1}
\end{align*}
Adding the transformations of the weight corresponding to $c$ and $f_0$ we prove the Proposition.
\endproof
\section{The proof of Proposition \ref{pp}}
\label{redel}
\begin{prop*}
The following equality takes place
\begin{align}
<(\bar{m})_{max} \mid F_{\pm 1,-2} \mid (m)_{max}
>=<(\bar{m})_{red} \mid F_{\pm 1,-2} \mid (m)_{red}>_{red}
\end{align}
\end{prop*}
The equality is proved using the Wigner-Eckart theorem. One has
\begin{align}
\begin{split}&
<(\bar{m})_{max} \mid F_{-1,-2} \mid
(m)_{red}>_{max}=<(\bar{m})_{red}\mid F_{-1,-2} \mid
(m)_{red}>\cdot\\&\cdot<\begin{pmatrix}[\bar{m}]_{n-1}\\max
\end{pmatrix}\mid \begin{pmatrix}j \\ [1\dot{0}]_{n-1}\\-2\end{pmatrix} \mid \begin{pmatrix}(m)_{red}\\max
\end{pmatrix}>,
\end{split}
\end{align}
where the Wigner coefficient equals to $1$ if there exists $j$,
such that $[\bar{m}]_{n-1}=[m]_{n-1}+[0,...,1_{\text{ at the place
}j},...,0]$, and equals zero otherwise. Thus one has
\begin{align} <(\bar{m})_{red} \mid F_{-1,-2} \mid
(m)_{red}>_{red}=<(\bar{m})_{max}\mid F_{-1,-2} \mid (m)_{max}>.
\end{align}
\endproof
\section{The proof of Proposition \ref{predl}}
\label{docpredl}
Let us be given a diagram $(m)$ that define a vector in s $g_{n-1}$-representation
with the highest weight $[m_{-n,n-1},...,m_{-2,n-1}]$.
A polynomial on the group $Z_{n-1}$ corresponds to this vector. Consider it as a polynomial on a bigger group $Z_n$.
\begin{prop*}
$g_{n}$-representation with the highest weight $[m_{-n,n-1},...,m_{-2,n-1},0]$. The corresponding tableau is of
type $\begin{pmatrix}max\\ m\end{pmatrix}$, this is a $g_{n}$-tableau that is obtained form
$(m)$ by adding two maximum row.
\end{prop*}
\proof One has $$(m)=\zeta_1...\zeta_t(max),$$ where $\zeta_i\in
Z^{-}_{n-1}$ and $(max)$ is the highest vector.
Let use the realization on the group $Z$. The highest vector is the function that equals identically to one.
One has $(m)=(
T_{\zeta_1}...T_{\zeta_t}1)(z)=\alpha_{n-1}(\widetilde{z\zeta_1...\zeta_t})$.
Also one has
\begin{align}
\begin{pmatrix} max\\ m \end{pmatrix}=\zeta_1...\zeta_t(max),\end{align} where $\zeta_i\in Z^{-}_{n-1}\subset Z^{-}_n$.
In the space of functions on $Z_n$, one has
\begin{align}
\begin{pmatrix} max\\ m \end{pmatrix}=(T_{\zeta_1}...T_{\zeta_t} 1)(z)=\alpha_{n}(\widetilde{z\zeta_1...\zeta_t} )
\end{align}
If $\alpha_n$ corresponds to the highest weight
$[m_{-n,n-1},m_{-n+1,n-1},...,m_{-2,n-1},0]$, then
$\alpha_{n-1}(\widetilde{z\zeta_1...\zeta_t})=\alpha_{n}(\widetilde{z\zeta_1...\zeta_t}
)$.
Thus the polynomials corresponding to $(m)$ and $\begin{pmatrix} max\\ m
\end{pmatrix}$ coincide.
\endproof
\end{document} | math | 99,142 |
\begin{document}
\title{Total connected domination game}
\author{Csilla Bujt\'as$^{a}$\thanks{Email: \texttt{[email protected]}}
\and Michael A. Henning $^{b}$\thanks{Email: \texttt{[email protected]}}
\and Vesna Ir\v si\v c$^{a,c}$\thanks{Email: \texttt{[email protected]}}
\and Sandi Klav\v zar $^{a,c,d}$\thanks{Email: \texttt{[email protected]}}
}
\date{\today}
\maketitle
\begin{center}
$^a$ Faculty of Mathematics and Physics, University of Ljubljana, Slovenia\\
$^b$ Department of Mathematics and Applied Mathematics, University of Johannesburg, Auckland Park, 2006 South Africa\\
$^c$ Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia\\
$^d$ Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia\\
\end{center}
\begin{abstract}
The (total) connected domination game on a graph $G$ is played by two players, Dominator and Staller, according to the standard (total) domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of $G$. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the (total) connected game domination number ($\gamma_{\rm tcg}(G)$) $\gamma_{\rm cg}(G)$ of $G$. We show that $\gamma_{\rm tcg}(G)\in \{\gamma_{\rm cg}(G), \gamma_{\rm cg}(G) + 1, \gamma_{\rm cg}(G) + 2\}$, and consequently define $G$ as Class~$i$ if $\gamma_{\rm tcg}(G) = \gamma_{\rm cg} + i$ for $i \in \{0,1,2\}$. A large family of Class $0$ graphs is constructed which contains all connected Cartesian product graphs and connected direct product graphs with minumum degree at least $2$. We show that no tree is Class~$2$ and characterize Class~$1$ trees. We provide an infinite family of Class~$2$ bipartite graphs.
\end{abstract}
{\small \textbf{Keywords:} Connected domination game; Total connected domination game; Graph product; Tree} \\
\indent {\small \textbf{AMS Subj.\ Class.\ (2010):} 05C57, 05C69, 91A43}
\section{Introduction}
\label{sec:intro}
The domination game introduced in 2010 in~\cite{bresar-2010}, and the total domination game put forward in 2015 in~\cite{he-kl-ra-2015}, were studied in depth by now, the respective lists of papers~\cite{bujtas-2015, dorbec-2015, james-2019, kinnersley-2013, klavzar-2019, xu-2018} and~\cite{bresar-2017, bujtas-2018, dorbec-2016, henning-2017a, henning-2016, irsic-2019} form just a selection of these studies. The two games are in some respects similar, for instance, they both admit the so-called Continuation Principle, but also significantly different in other respects, say in the conjectured upper bounds on the corresponding invariants in terms of the order of a graph~\cite{henning-2017a, kinnersley-2013}.
Recently, in 2019, the connected domination game was introduced by Borowiecki, Fiedorowicz, and Sidorowicz~\cite{borowiecki-2019} and further investigated in~\cite{bujtas-2019, irsic-2019+}. Although the connected domination game mostly follows the definition of the standard domination game, the new game is significantly different. For instance, on the class of trees, the connected game domination number can be obtained in linear time, while the complexity of determining the game domination number is open and we suspect that it is NP-hard. In other cases, finding optimal stategies appears difficult in both games, but the connected one still being more accessible, as for instance on the games played on Cartesian product graphs, see~\cite{borowiecki-2019, bujtas-2019}.
In this paper we introduce the total version of the connected domination game following the above mentioned pattern of the total domination game~\cite{he-kl-ra-2015} versus the domination game~\cite{bresar-2010}. We proceed as follows. In Section~\ref{S:notation} we give additional definitions needed, in Section~\ref{sec:connected-games} the connected and total connected domination games are described and discussed. In Section~\ref{sec:relation} we prove that $\gamma_{\rm tcg}(G)\in \{\gamma_{\rm cg}(G), \gamma_{\rm cg}(G) + 1, \gamma_{\rm cg}(G) + 2\}$ and consequently partition the class of all graphs into Classes 0, 1, and 2, depending on which of the three possibilities holds. Then, in Section~\ref{sec:classes}, we determine different families of graphs belonging to respective classes. In the concluding section several open problems are listed.
\subsection{Notation}
\label{S:notation}
For graph theory notation and terminology, we generally follow~\cite{HeYe-book-2013}. Specifically, let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and of order $n(G) = |V(G)|$ and size $m(G) = |E(G)|$. We denote the degree of a vertex $v$ in $G$ by $d_G(v)$. The minimum and maximum degrees among the vertices of $G$ are denoted by $\delta(G)$ and $\Delta(G)$, respectively. Two vertices are \emph{neighbors} if they are adjacent. The \emph{open neighborhood} of a vertex $v$ in a graph $G$, denoted $N_G(v)$, is the set of neighbors of $v$ in $G$, while the \emph{closed neighborhood} of $v$ is the set $N_G[v] = N_G(v) \cup \{v\}$. The \emph{open neighborhood} of a set of vertices $S \subseteq V(G)$ is $N_G(S) = \bigcup_{v \in S} N_G(v)$, and the \emph{closed neighborhood} of $S$ is $N_G[S] = \bigcup_{v \in S} N_G[v]$. If the graph $G$ is clear from context, we may omit the subscript in the above definitions.
The \emph{Cartesian product} $G\,\square\, H$ of graphs $G$ and $H$ has the vertex set $V(G)\times V(H)$, vertices $(g,h)$ and $(g',h')$ being adjacent if either $gg'\in E(G)$ and $h=h'$, or $g=g'$ and $hh'\in E(H)$. The \emph{direct product} $G\times H$ has the same vertex set $V(G)\times V(H)$, vertices $(g,h)$ and $(g',h')$ being adjacent if $gg'\in E(G)$ and $hh'\in E(H)$~\cite{hik-2011}.
Let $G$ be a graph with $V(G) = \{v_1,\ldots, v_n\}$ and let ${\cal H} = \{H_1,\ldots, H_n\}$ be a family of pairwise vertex disjoint graphs. Then the {\em generalized corona} $G\odot {\cal H}$ is the graph obtained from the disjoint union of $G, H_1, \ldots, H_n$, by joining, for each $i\in [n]$, every vertex of $H_i$ with the vertex $v_i$ of $G$, cf.~\cite{dettlaff-2016}. If all the graphs $H_i$ are isomorphic to a given graph $H$, then we may write $G\odot H$ instead of $G\odot {\cal H}$. In particular, $G\odot K_1$ is the {\em corona} over $G$.
A set $S \subseteq V(G)$ is a \emph{dominating set} of the graph $G$ if $N[S] = V(G)$. The minimum cardinality of a dominating set is the \emph{domination number} $\gamma(G)$ of $G$. A \emph{connected dominating set} is a dominating set $S$ with the additional property that the subgraph $G[S]$ induced by $S$ is connected. The minimum cardinality of a connected dominating set is the \emph{connected domination number} $\gamma_{\rm c}(G)$ of $G$. Note that $\gamma_{\rm c}$ is defined only for connected graphs.
A \emph{total dominating set} of a graph $G$ is a set $S$ of vertices of $G$ such that every vertex has a neighbor in $S$. The \emph{total domination number} of an isolate-free graph $G$, denoted by $\gamma_t(G)$, is the minimum cardinality of a total dominating set in $G$. A vertex \emph{totally dominates} a vertex if they are neighbors, that is, the vertices totally dominated by a vertex $v$ are the vertices that belong to the open neighborhood $N(v)$ of $v$. We note that if $S$ is a total dominating set of $G$, then every vertex in $G$ is totally dominated by at least one vertex in~$S$.
Finally, for an integer $k \ge 1$ we denote $[k] = \{1,\ldots,k\}$.
\subsection{Connected domination games}
\label{sec:connected-games}
In this section we recall the connected domination game and the connected domination game with Chooser, and introduce the total connected domination game.
The {\em connected domination game} on a (connected) graph $G$ is played by Dominator and Staller. They play in turns, at each move selecting a single vertex of $G$ such that it dominates at least one vertex that is not yet dominated with the previously played vertices and such that at each stage of the game the selected vertices induce a connected subgraph of $G$. If Dominator has the first move, then we speak of a {\em D-game}, otherwise they play an {\em S-game}. When the game is finished, that is, when there is no legal move available, the players have determined a connected dominating set $D$ of $G$. The goal of Dominator is to finish with $|D|$ as small as possible, the goal of Staller is just the opposite. If both players play optimally, then $|D|$ is unique. In the D-game it is called the {\em connected game domination number} $\gamma_{\rm cg}(G)$ of $G$, while when the S-game is played, the corresponding invariant is denoted by $\gamma_{\rm cg}'(G)$. The connected domination game is thus defined as the standard domination game~\cite{bresar-2010} with the additional requirement that the players maintain connectedness of the subgraph induced by the selected vertices at all times. To shorten the presentation we will abbreviate the term ``connected domination game" to \emph{c}-\emph{game}.
The \emph{connected domination game with Chooser}~\cite{borowiecki-2019} has similar rules as the normal game, except that there is another player, Chooser, who can make zero, one, or more moves after any move of Dominator or Staller. The conditions for his move to be legal are the same as for Dominator and Staller. Chooser has no specific goal, he can help Dominator or Staller or none. We recall the Chooser Lemma from~\cite{borowiecki-2019} to be used later on.
\begin{lemma}[Chooser Lemma]
\label{lema:chooser}
Consider the connected domination game with Chooser on a graph $G$. Suppose that in the game Chooser picks $k$ vertices, and that both Dominator and Staller play optimally. Then at the end of the game the number of played vertices is at most $\gamma_{\rm cg}(G) + k$ and at least $\gamma_{\rm cg}(G) - k$.
\end{lemma}
We now introduce the total connected domination game. First, we recall that the \emph{total domination game} is defined analogously as the domination game, except that whenever a player selects a new vertex in the course of the game, the selected vertex must totally dominate at least one vertex that was not totally dominated by vertices previously selected by the players~\cite{he-kl-ra-2015}. The \emph{game total domination number} and the \emph{Staller-start game total domination number} are denoted by $\gamma_{\rm tg}(G)$ and $\gamma_{\rm tg}s(G)$, respectively. The \emph{total connected domination game} is just as the total domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of $G$. The \emph{game total connected domination number} and the \emph{Staller-start game total connected domination number} are denoted by $\gamma_{\rm tcg}(G)$ and $\gamma_{\rm tcg}'(G)$, respectively. For simplicity, we abbreviate the term ``total connected domination game" to \emph{tc}-\emph{game}.
The Chooser Lemma holds also for the total connected domination game. Its proof proceeds along the same lines as the proof of the Chooser Lemma in~\cite{borowiecki-2019}, hence we do not repeat it here.
If $\gamma_{\rm c}(G) = 1$, then $G$ contains a universal vertex and hence $\gamma_{\rm tcg}(G) = 2$. For all the other cases we have the following bounds that can be proved along the same lines as~\cite[Theorem 1]{borowiecki-2019}, see also~\cite[Theorem 2.1]{bujtas-2019} for more detailed arguments.
\begin{proposition}
\label{prop:bounds-tcc}
If $\gamma_{\rm c}(G)\ge 2$, then $\gamma_{\rm c}(G) \le \gamma_{\rm tcg}(G) \le 2\gamma_{\rm c}(G) - 1$.
\end{proposition}
No matter which game is played, we adopt the convention that the consecutive moves of Dominator are denoted by $d_1, d_2, \ldots$, and the consecutive moves of Staller by $s_1, s_2, \ldots$.
\section{Relating $\gamma_{\rm tcg}(G)$ to $\gamma_{\rm cg}(G)$}
\label{sec:relation}
The main result of this section reads as follows.
\begin{theorem}
\label{thm:only-3-classes}
If $G$ is a connected graph, then $\gamma_{\rm cg}(G) \le \gamma_{\rm tcg}(G) \le \gamma_{\rm cg}(G) + 2$.
\end{theorem}
\noindent{\bf Proof.\ }
To prove the lower bound, let a c-game be played on a graph $G$, call it the {\em R-game} (where R stands for ``real"). In parallel, Dominator imagines that also a tc-game is played on $G$, call it the {\em I-game} (where I stands for ``imagined"). In the R-game Staller plays optimally (and Dominator maybe not), while in the I-game Dominator plays optimally (and Staller maybe not). At the beginning, Dominator selects an optimal first move in the I-game and copies this move to the real game. (The first move of Dominator in the R-game might not be optimal.) After Staller replies with her optimal move in the R-game, Dominator copies her move into the I-game. Note that this move is legal in the I-game. Afterwards Dominator replies optimally in the I-game, and copies his move into the R-game. The two games proceed along the same lines until the games are finished. By the strategy of Dominator, the sequences of moves played in the R-game and in the I-game are the same. Consequently, the number of moves, say $s$, is the same in both games. Since Staller played optimally in the R-game (which is a c-game) we have $\gamma_{\rm cg}(G) \le s$, and since Dominator was playing optimally in the I-game (which is a tc-game) we have $\gamma_{\rm tcg}(G) \ge s$. Hence, $\gamma_{\rm cg}(G) \le s \le \gamma_{\rm tcg}(G)$, proving the left inequality.
In order to prove the right inequality, let again a c-game, called R-game, be played on a graph $G$. Now Staller imagines that a parallel tc-game, called I-game, is played on $G$. In this set-up, in the R-game Dominator plays optimally (and Staller maybe not), while in the I-game Staller plays optimally (and Dominator maybe not). The R-game starts with an optimal move $d_1$ of Dominator, and Staller copies this move into the I-game. Then Staller optimally replies in the I-game with the move $s_1$. We now distinguish two cases.
\emph{Case 1: $s_1$ is a legal move in the R-game.} In this case Staller copies it into the R-game and the game continues along the same lines. As above we infer that the sequences of moves played in the R-game and in the I-game are the same, let $s$ be the number of them. Since Dominator played optimally in the R-game we have $\gamma_{\rm cg}(G) \ge s$, and since Staller was playing optimally in the I-game we have $\gamma_{\rm tcg}(G) \le s$. So, $\gamma_{\rm tcg}(G) \le s \le \gamma_{\rm cg}(G)$.
\emph{Case 2: $s_1$ is not a legal move in the R-game.} This situation has happened because $N_G[s_1] \subseteq N_G[d_1]$. In the R-game now, after the move $d_1$, Chooser plays an arbitrary legal move $x$. Staller then imagines in the I-game that $x$ is the second move of Dominator. Note that $x$ is a legal move (of Dominator) in the I-game because in the R-game Chooser dominated at least one vertex not adjacent to $d_1$. Till this moment the sequence of moves played in the R-game is $d_1$, $x$, and the sequence of moves played in the I-game is $d_1$, $s_1$, $x$. In this way the set of vertices dominated in both games is the same. Moreover, in both games it is Staller's turn. Afterwards, both games continue by copying each Staller's move from the I-game to the R-game and by copying each Dominator's move from the R-game to the I-game. Hence in the rest of the games the sequence of moves is the same in both of them. Setting $s$ to be the number of moves played at the end of the R-game, the number of moves played in the I-game is therefore $s+1$. Since the R-game is a c-game in which Dominator played optimally and Chooser played one move, the Chooser Lemma implies that $\gamma_{\rm cg}(G) \ge s - 1$. On the other hand, since Staller played optimally in the I-game, $\gamma_{\rm tcg}(G) \le s + 1$. Therefore, $\gamma_{\rm tcg}(G) \le s + 1 \le \gamma_{\rm cg}(G) + 2$.~
$\square$
In view of Theorem~\ref{thm:only-3-classes}, we say that a graph $G$ is
\begin{itemize}
\item {\em Class~$0$}, if $\gamma_{\rm tcg}(G) = \gamma_{\rm cg}(G)$,
\item {\em Class~$1$}, if $\gamma_{\rm tcg}(G) = \gamma_{\rm cg}(G) + 1$, and
\item {\em Class~$2$}, if $\gamma_{\rm tcg}(G) = \gamma_{\rm cg}(G) + 2$.
\end{itemize}
\section{Families of Class $0$, $1$, and $2$ graphs}
\label{sec:classes}
To describe a large family of Class $0$ graphs, we say that the neighborhoods of a graph $G$ are {\em non-inclusive} if for every pair $u$ and $v$ of distinct vertices of $G$ we have $N[u]\not\subseteq N[v]$. Note that the latter condition is trivially fulfilled if $uv\notin E(G)$, hence an equivalent way to say that the neighborhoods of $G$ are non-inclusive is that for every edge $uv\in E(G)$ we have $N(u) \setminus \{v\} \not\subseteq N(v)$.
\begin{proposition}
\label{prp:not-included-neighborhood}
If the neighborhoods of a connected graph $G$ are non-inclusive, then $G$ is Class~$0$.
\end{proposition}
\noindent{\bf Proof.\ }
This result can be deduced from the proof of Theorem~\ref{thm:only-3-classes} as follows. When proving that $\gamma_{\rm tcg}(G)\le \gamma_{\rm cg}(G) + 2$, the condition that the neighborhoods of $G$ are non-inclusive implies that the move $s_1$ of Staller in the I-game is a legal move in the R-game, hence only Case 1 applies. Therefore, $\gamma_{\rm tcg}(G) \le \gamma_{\rm cg}(G)$ holds and because $\gamma_{\rm cg}(G)\le \gamma_{\rm tcg}(G)$ by the first part of the proof of Theorem~\ref{thm:only-3-classes}, we conclude that $\gamma_{\rm tcg}(G) = \gamma_{\rm cg}(G)$.
$\square$
\vskip -0.25 cm
\begin{figure}
\caption{Two graphs, $F_8$ and $D_{15}
\label{fig:class-0}
\end{figure}
The two graphs in Fig.~\ref{fig:class-0} show that the converse of Proposition~\ref{prp:not-included-neighborhood} is not true. $F_8$ is a cubic graph that contains twin vertices, that is, vertices with the same closed neighborhoods. However, as Dominator may choose an optimal start-vertex from the $6$-cycle that does not have a twin in $F_8$, we have $\gamma_{\rm cg}(F_8)= \gamma_{\rm tcg}(F_8)=4$. Therefore, $F_8$ is Class~$0$. More generally, let $F_{4k}$ be the analogous construction on $4k$ vertices. That is, $F_{4k}$ is obtained from a cycle $C_{3k}$ by adding a twin vertex to every third vertex of the cycle. It is clear that $F_{4k}$ is Class~$0$ for each integer $k \ge 2$.
The other graph $D_{15}$ illustrated in Fig.~\ref{fig:class-0} is also Class~$0$ and its neighborhoods are not non-inclusive since $x$ and $y$ are twins. For every $v \in V(D_{15})$ define the invariants $c(v)$ and $t(v)$ as the number of played vertices in a c-game and tc-game, respectively, where Dominator starts the game by playing $d_1=v$ and, after this move the players follow optimal strategies. Then, $\gamma_{\rm cg}(D_{15})= \min_{v\in V(D_{15})} c(v)$ and $\gamma_{\rm tcg}(D_{15})= \min_{v\in V(D_{15})} t(v)$. It can be checked that $\gamma_{\rm cg}(D_{15}) = \gamma_{\rm tcg}(D_{15}) = 9$, hence $D_{15}$ is Class~$0$. Clearly, if $v \in V(D_{15}) \setminus \{x,y\}$, then $c(v)=t(v)$. The interesting fact here is that it can be checked that $c(x)=t(x)= 10$ and $c(y)=t(y)=10$ also holds, that is, Staller cannot gain any advantage from the fact that she can play a twin in her first move.
By the structure of the Cartesian and the direct product of graphs, Proposition~\ref{prp:not-included-neighborhood} yields the following two consequences. With respect to the second one we recall that the direct product $G\times H$ is connected if and only if both $G$ and $H$ are connected and at least one of them contains an odd cycle~\cite{hik-2011}. Recall that a non-trivial graph has at least two vertices.
\begin{corollary}
\label{cor:Cartesian-direct}
If $G$ and $H$ are non-trivial graphs, then the following holds.
\begin{enumerate}
\item If both $G$ and $H$ are connected, then $G\,\square\, H$ is Class~$0$.
\item If $\delta(G)\ge 2$ and $G\times H$ is connected, then $G\times H$ is Class~$0$.
\end{enumerate}
\end{corollary}
\noindent{\bf Proof.\ }
(a) Let $(g,h)(g',h)\in E(G\,\square\, H)$. Since $H$ is non-trivial, there exists an edge $hh'\in E(H)$. Then $(g,h')\in N_{G\,\square\, H}((g,h))\setminus N_{G\,\square\, H}((g',h))$. A parallel conclusion can be obtained for an edge $(g,h)(g,h')\in E(G\,\square\, H)$. Hence the neighborhoods of $G\,\square\, H$ are non-inclusive and Proposition~\ref{prp:not-included-neighborhood} applies.
(b) Consider an edge $(g,h)(g',h')\in E(G\times H)$. Since $\delta(G) \ge 2$, there exists an edge $gg''\in E(G)$, where $g''\ne g'$. Now $(g'',h')\in N_{G\times H}((g,h))\setminus N_{G\times H}((g',h'))$. Hence Proposition~\ref{prp:not-included-neighborhood} applies again.
$\square$
There exist also direct product graphs that are Class~$0$ but not covered by Corollary~\ref{cor:Cartesian-direct}(b). Let $H$ be the graph obtained from $K_{1,3}$ by adding an arbitrary edge to it ($H$ is known as the paw graph). Then, $\gamma_{\rm cg}(H \times K_2) = \gamma_{\rm tcg}(H \times K_2) = 5$ and, of course, $\delta(H) = \delta(K_2) = 1$. Moreover, $C_{2k+1}\times K_2 = C_{4k+2}$ is Class~0 for every $k\ge 1$. But not all direct products are Class~0. For instance, it can be easily checked that
$$\gamma_{\rm cg}((C_{2k+1}\odot K_1) \times K_2) = 4k+2$$ and
$$\gamma_{\rm tcg}((C_{2k+1}\odot K_1) \times K_2) = 4k+3\,.$$
We next show that generalized coronas over connected graphs are Class~$1$.
\begin{proposition}
\label{prp:generalized-coronas}
If $G$ is a connected graph and ${\cal H} = \{H_1,\ldots, H_{n(G)}\}$, then $G\odot {\cal H}$ is Class~$1$.
\end{proposition}
\noindent{\bf Proof.\ }
Observe first that $\gamma_{\rm c}(G\odot H) = n(G)$. Therefore, $\gamma_{\rm tcg}(G \odot H) \ge \gamma_{\rm cg}(G \odot H) \ge n(G)$. If in the c-game Dominator selects as his first move a vertex of $G$, then Staller must reply with an adjacent vertex of $G$. Proceeding in this way Dominator can ensure that exactly all the vertices of $G$ will be played by the end of the game, hence $\gamma_{\rm cg}(G\odot {\cal H}) \le n(G)$ and so $\gamma_{\rm cg}(G\odot {\cal H}) = n(G)$. In the tc-game, Dominator's optimal strategy is again to play a vertex $v_i$ of $G$. If Staller replies with a vertex of $G$, then, as above, only the vertices of $G$ will be played. Hence an optimal reply of Staller is to play a vertex from $H_i$. After that Dominator can ensure that only the remaining vertices of $G$ will be played, so that $\gamma_{\rm cg}(G\odot {\cal H}) = n(G) + 1$. We conclude that $G\odot {\cal H}$ is Class~$1$.
$\square$
\begin{corollary}
\label{cor:trees}
A tree $T$ with $n(T)\ge 3$ is Class~$1$ if and only if $T = T'\odot {\cal H}$, where $T'$ is a tree and ${\cal H}$ is a family of edge-less graphs. Otherwise, $T$ is Class~$0$.
\end{corollary}
\noindent{\bf Proof.\ }
As observed in~\cite{borowiecki-2019}, $\gamma_{\rm cg}(T) = n(T) - \ell(T)$, where $\ell(T)$ is the number of leaves of $T$. If $T$ contains a vertex $x$ of degree at least $2$ with no leaf attached to it, then Dominator plays $x$ first and in this way guarantees that exactly the non-leaves will be played. Hence, $T$ is Class~$0$ in this case. Otherwise every non-leaf has at least one leaf attached. In this case, $T$ can be represented as $T'\odot {\cal H}$ for some tree $T'$ and a family ${\cal H}$ of edge-less graphs. By Proposition~\ref{prp:generalized-coronas}, $T$ is Class~$1$ in this case.
$\square$
Let $G_r$, $r\ge 3$, be a class of graphs whose formal definition should be clear from Fig.~\ref{fig:cacti-class-2}.
\begin{figure}
\caption{The graph $G_r$}
\label{fig:cacti-class-2}
\end{figure}
\begin{proposition}
\label{prp:cacti-class-2}
If $r\ge 3$, then $G_r$ is Class~$2$.
\end{proposition}
\noindent{\bf Proof.\ }
Let $r\ge 3$ and let the vertices of $G_r$ be labeled as shown in Fig.~\ref{fig:cacti-class-2}. Let $X= \{x_1,x_2, \ldots, x_r\}$ and let $Y = \{y_2,y_3, \ldots, y_r\}$. Every connected dominating set of $G_r$ contains the set $X \cup Y$. On the other hand, $X \cup Y$ is itself a connected dominating set of $G_r$. Consequently, $\gamma_{\rm c}(G_r) = |X \cup Y| = 2r-1$ (and the set $X \cup Y$ is the unique minimum connected dominating set of $G_r$). Hence, $\gamma_{\rm cg}(G_r) \ge \gamma_{\rm c}(G_r) = 2r-1$.
We first consider the c-game played on $G_r$. Set $d_1 = x_2$. Then, $s_1\in \{y_2, y_3\}$. The strategy of Dominator in the rest of the game is the following. Suppose by induction that $y_i$, $2\le i\le r$, is the currently last vertex played by Staller. Then Dominator plays the vertex from $\{x_{i-1}, x_{i}\}$ that has not yet been played. Maintaining this strategy, Dominator ensures that Staller is forced always to play a vertex from $Y$. Moreover, by the end of the game, Dominator will play all vertices from $X$, so that $\gamma_{\rm cg}(G_r) \le 2r-1$ and hence $\gamma_{\rm cg}(G_r) = 2r-1$.
We consider next the tc-game on $G_r$, and show that $\gamma_{\rm tcg}(G) \ge 2r + 1 = \gamma_{\rm cg}(G) + 2$, implying that $G_r$ is Class~$2$. As observed earlier, every connected dominating set of $G_r$ contains the set $X \cup Y$. Hence, it suffices for us to show that Staller has a strategy that forces at least two vertices of $G$ to be played that do not belong to the set $X \cup Y$. In this case, we say that Staller achieves her goal. Let $V_i = \{w_i,x_i,y_i,y_{i+1},z_i\}$ where $i \in [r]$.
Suppose that $d_1 \in V_1$. If $d_1 = y_1$, then Staller plays $s_1 = z_1$, while if $d_1 = z_1$, then Staller plays $s_1 = y_1$. In both cases, two moves not in $X \cup Y$ are played, and Staller achieves her goal. If $d_1 = w_1$, then $s_1 = x_1$, and either $d_2 = y_1$ or $d_2 = y_2$. If $d_2 = y_1$, then Staller immediately achieves her goal, while if $d_2 = y_2$, then Staller plays $s_2 = z_2$ to achieve her goal. If $d_1 = x_1$, then Staller plays $s_1 = w_1$, and either $d_2 = y_1$ or $d_2 = y_2$, and as in the previous case Staller can achieve her goal. Hence, Staller achieves her goal that two vertices are played from outside the set $X \cup Y$ in all cases except possibly in the case when $d_1 = y_2$. Suppose, therefore, that $d_1 = y_2$. In this case, Staller plays $s_1 = z_2$. If $d_2 = z_1$, then Staller immediately achieves her goal. If $d_2 \in \{x_2,y_3\}$, then Staller plays $s_2 = z_1$, and achieves her goal. If $d_2 = x_1$, then Staller plays $s_2 = x_2$, which forces $d_3 = y_3$, and enables Staller to play $s_3 = z_3$, and again she achieves her goal. Hence, if $d_1 \in V_1$, then Staller achieved her goal. We may therefore assume that $d_1 \notin V_1$ and, by symmetry, that $d_1 \notin V_r$. Hence, $d_1 \in V_i$ for some $i \in [r-1] \setminus \{1\}$.
Suppose that $d_1 = w_i$, which forces $s_1 = x_i$, and either $d_2 = y_i$ or $d_2 = y_{i+1}$. If $d_2 = y_i$, then Staller plays $s_2 = y_{i+1}$, while if $d_2 = y_{i+1}$, then Staller plays $s_2 = y_{i}$. Thus, $d_3 \in \{x_{i-1},z_{i-1},x_{i+1},z_{i+1}\}$. If $d_3 \in \{x_{i-1},z_{i-1}\}$, then Staller plays $s_3 = z_{i+1}$, while if $d_3 \in \{x_{i+1},z_{i+1}\}$, then Staller plays $s_3 = z_{i-1}$. In both cases, after her third move Staller already achieve her goal that two vertices are played from outside the set $X \cup Y$.
Suppose that $d_1 = x_i$. In this case, Staller plays $s_1 = w_i$, which forces either $d_2 = y_i$ or $d_2 = y_{i+1}$. Proceeding exactly as in the previous case, if $d_2 = y_i$, then Staller plays $s_2 = y_{i+1}$, while if $d_2 = y_{i+1}$, then Staller plays $s_2 = y_{i}$, thereby achieving her goal as before.
Suppose that $d_1 = y_i$. In this case, Staller plays $s_1 = z_i$. We note that $d_2 \in \{x_{i-1},z_{i-1},x_{i},y_{i+1}\}$. If $d_2 = z_{i-1}$, then already two vertices are played from outside the set $X \cup Y$, and Staller immediately achieves her goal. If $d_2 \in \{x_{i},y_{i+1}\}$, then Staller plays $s_2 = z_{i-1}$ and achieves her goal. Hence, we may assume that $d_2 = x_{i-1}$. In this case, Staller plays $s_2 = x_i$, forcing either $d_3 = y_{i-1}$ or $d_3 = y_{i+1}$. If $d_3 = y_{i-1}$ and $i = 2$, then Staller immediately achieves her goal. If $d_3 = y_{i-1}$ and $i \ge 3$, then Staller plays $s_3 = z_{i-2}$, while if $d_3 = y_{i+1}$, then Staller plays $s_3 = z_{i+1}$, and in both cases she achieves her goal. Analogously, if $d_1 = y_{i+1}$, then Staller achieves her goal.
Suppose finally that $d_1 = z_i$. In this case, Staller plays $s_1 = y_i$. Thus, $d_2 \in \{x_{i-1},z_{i-1},x_{i},y_{i+1}\}$. If $d_2 = z_{i-1}$, then Staller immediately achieves her goal. If $d_2 \in \{x_{i},y_{i+1}\}$, then Staller plays $s_2 = z_{i-1}$ and achieves her goal. Hence, we may assume that $d_2 = x_{i-1}$. In this case, Staller plays $s_2 = x_i$, forcing either $d_3 = y_{i-1}$ or $d_3 = y_{i+1}$. Staller now proceeds therefore exactly as in the previous case to achieve her goal.
$\square$
\section{Concluding remarks}
\label{sec:problems}
In Proposition~\ref{prop:bounds-tcc} the bounds are sharp. The lower bound is attained by trees which are Class~$0$, while the upper bound is attained by classes of Cartesian products $X$ from~\cite{borowiecki-2019, bujtas-2019} for which $\gamma_{\rm cg}(X) = 2\gamma_{\rm c}(X) - 1$. Hence the upper bound in Proposition~\ref{prop:bounds-tcc} is attained because Cartesian products are Class~$0$ graphs.
\begin{problem}
Determine whether in Proposition~\ref{prop:bounds-tcc} all possible values of $\gamma_{\rm tcg}(G)$ are realizable.
\end{problem}
With respect to Corollary~\ref{cor:Cartesian-direct}(b) we pose:
\begin{problem}
Classify direct product graphs into Classes $0$, $1$, and $2$.
\end{problem}
Analyzing the graphs $G_n$ from~\cite{irsic-2019+} (see also Fig.~1 there), it can be demonstrated that for every natural number $k$ there exist graphs $G$ such that $\gamma_{\rm tcg}'(G) - \gamma_{\rm tcg}(G) \ge k$. In this paper we do not however further investigate the S-game, hence the following task remains to be done.
\begin{problem}
Consider the total connected domination S-game. In particular, we suspect that $\gamma_{\rm tcg}'(G) = \gamma_{\rm cg}'(G)$ holds whenever $G$ is not a complete graph.
\end{problem}
We conclude with the following problem that also seems to be interesting.
\begin{problem}
Classify cactus graphs into Classes $0$, $1$, and $2$.
\end{problem}
\end{document} | math | 30,376 |
\begin{document}
\section*{General response}
We thank the reviewers for their insightful comments. After reading all the review we have made the following major changes to make the paper easier to understand: 1) we have rephrased the summary of FOLD-R algorithm with brief comments. 2) to better explain how we use the prefix sum during heuristic calculation, we give the comparison assumption examples and an extra brief explanation for the parameters of the information gain function. 3) we have replaced the ``Titanic Survival'' dataset with the ``Adult Census Income'' dataset (more complicated) to better illustrate the explainability of the FOLD-R++ algorithm. We also give the output of the RIPPER algorithm to show the different rule format of the two algorithms. The github location of the source code is also included.
\section*{Response to Reviewer 1}
We rephrase the summary of Algorithm 1 with some brief comments for better readability. The ADD\_BEST\_LITERAL function calculates information gain for all the possible literals and adds the one with the highest score to the current clause. In the FOLD algorithm, with encoding data preparation, this function only need to deal with categorical values. In FOLD-R algorithm, the function additionally calculates the heuristic for all the possible numerical splits for numerical features. And, this is the FOLD-R for numeric extension. The other functions of FOLD-R are inherited from FOLD.
\section*{Response to Reviewer 2}
The mentioned grammatical mistakes have been fixed. We have rephrased the summary of the FOLD-R algorithm with brief comment for better readability.
The output of the RIPPER algorithm is shown in the updated version, more repeated literals appear in the formulas. Exception learning procedure only focus on smaller hypothesis space other than the whole hypothesis space, the literals that locates the hypothesis space implied by default is therefore unnecessary. For the ``Adult Census Income'' example in the paper, the RIPPER algorithm generates 53 rules with 235 literals while the FOLD-R++ algorithm generates 13 rules with 36 literals by explicitly learning exceptions.
To better explain how the prefix sum technique is used in heuristic calculation, news examples (Table 1) are given for comparison assumption in FOLD-R++ and how the information gain function is called.
The average execution time of FOLD-R for the ``Adult Census Income'' and ``Credit Card Approval'' is estimated with polynomial regression on several rounds of experiments. The number of examples used for the estimation experiments are 100, 200, 300, 500, 800, 1300, 2100, 3400, 5500. The minimum number of the estimation is reported on the Table 4 for each of them (``Adult Census Income'' and ``Credit Card Approval'' dataset).
\section*{Response to Reviewer 3}
Among the three major differences we mentioned in the paper, the first two significantly improves the performance:
1) The introduction of prefix sum for heuristic calculation improve the algorithm in time complexity level. The FOLD-R++ would have taken days to finish the training on the Adult Census Income dataset without the prefix sum technique, in contrast, it takes around 10 second to finish the training with this improvement!!
2) The introduction of exception ratio improves the algorithm on higher level. The FOLD and FOLD-R would start to learn exceptions when it failed to find a literal to avoid falsely covering the negative examples. However, avoiding false covering negative examples by adding literals to the default part would reduce the number of positive examples the rule can imply. Explicitly activating the exception learning procedure with a proportional threshold could increase the number of positive examples a rule can cover while reducing the total number of rules generated. As a result, the interpretability is increased due to fewer rules and literals being generated. For the Adult Census Income dataset, without the hyper-parameter exception ratio, the FOLD-R++ (set ratio as 0) would take around 30 minutes to finish the training and generate more than 500 rules.
3) The FOLD and FOLD-R disabled the negated literals in the default theories to make the generated rules look more elegant (only exceptions are negated literals). However, a negated literal sometimes is the optimal literal (with the most useful information gain). FOLD-R++ enables the negated literals in the default part of the generated rules.
We cannot make sure that FOLD-R++ generates optimal literal combination, because it is a greedy algorithm. Instead of choosing sub-optimal literal in literal selection iteration, finding the optimal literal should be an improvement.
The ratio set as 0.5 means the exception examples a rule can cover should be less than half of the examples covered by default. We introduce the hyper-parameter $ratio$ as a proportional value in order to make the algorithm scalable. In experiment, we intend to show the experiment with only simple setting. This parameter can impact on the result on accuracy and number of rules generated, we can try multiple values for training in practice.
\end{document} | math | 5,186 |
\begin{document}
\newcommand{\mathbb{Z}}{\mathbb{Z}}
\newcommand{\mB}[1]{\mathbf{#1}}
\newcommand{\boldsymbol{\wr}}{\boldsymbol{\wr}}
\newcommand{\textsc{SgpDec}}{\textsc{SgpDec}}
\newcommand{\textsc{Gap}}{\textsc{Gap}}
\newcommand{\textsc{Viz}}{\textsc{Viz}}
\newcommand{\textsc{GraphViz}}{\textsc{GraphViz}}
\pgfdeclarelayer{background layer}
\pgfsetlayers{background layer,main}
\newcommand{\left(}{\left(}
\newcommand{\right)}{\right)}
\title[SgpDec]{SgpDec: Cascade (De)Compositions of Finite Transformation Semigroups and Permutation Groups}
\author{Attila Egri-Nagy$^{1,3}$ \and James D. Mitchell$^{2}$ \and Chrystopher L. Nehaniv$^{3}$}
\address{
Centre for Research in Mathematics\\
School of Computing, Engineering and Mathematics\\
University of Western Sydney, Australia\\
\and
School of Mathematics and Statistics\\
University of St Andrews, United Kingdom\\
\and
Centre for Computer Science \& Informatics Research\\
University of Hertfordshire, United Kingdom}
\email{[email protected],[email protected],[email protected]}
\maketitle
\begin{abstract}
We describe how the \textsc{SgpDec}~computer algebra package can be used for composing and decomposing permutation groups and transformation semigroups hierarchically by directly constructing substructures of wreath products, the so called cascade products.
\keywords{transformation semigroup, permutation group, wreath product, Krohn-Rhodes Theory}
\end{abstract}
\section{Introduction}
Wreath products are widely used theoretical constructions in group and semigroup theory whenever one needs to build a composite structure with hierarchical relations between the building blocks.
However, from a computational and engineering perspective they are less useful since wreath products are subject to combinatorial explosions and we are often interested only in substructures of them.
Cascade products precisely build these substructures by defining the hierarchical connections explicitly.
As input, given a group or a semigroup with unknown internal structure, the goal of cascade decomposition algorithms is to come up with a list of simpler building blocks and put them together in a cascade product, which realizes in some sense the original group or semigroup.
Roughly speaking, for permutation groups, cascade product decompositions can be interpreted as putting the inner workings of the Schreier-Sims algorithm (generalized to any subgroup chain) into an external product form, therefore one can build cascade products isomorphic to the group being decomposed.
For semigroups, Krohn-Rhodes decompositions~\cite{primedecomp65} can be computationally represented by cascade products of transformation semigroups.
In this paper we describe how the \textsc{Gap}~\cite{GAP4} package \textsc{SgpDec}~\cite{sgpdec} implements cascade products and decomposition algorithms and we also give a few simple example computations.
This description of the package only focuses on the core functionality of the package.
A \emph{transformation} is a function $f:X\rightarrow X$ from a set to itself,
and a \emph{transformation semigroup} $(X,S)$ of degree $n$ is a collection $S$ of transformations of $X$ closed under function composition, $|X|=n$. In case $S$ is a group of permutations of $X$, we call $(X,S)$ a \emph{permutation group}.
Using automata theory terminology sometimes we call $X$ the \emph{state set}, often represented as a set of integers $\mB{n}=\{0,\ldots,n-1\}$.
We write $x^s$ to denote the new state resulting from applying a transformation $s\in S$ to a state $x\in X$.
\section{Cascade Product by a Motivating Example}
\begin{figure}
\caption{Action in a cascade product of components $[(X_1,S_1)$, $(X_2,S_2)$, $(X_3,S_3)]$. The current state $(x_1,x_2,x_3)$ (top) is transformed to the new state $\left( x_1^{ s_1}
\label{fig:cascaction}
\end{figure}
To motivate the definition of the cascade product, we consider how the mod-4 counter, the cyclic permutation group $(\mB{4},\mathbb{Z}_4)$, can be constructed from two mod-2 counters.
The direct product $\mathbb{Z}_2\times \mathbb{Z}_2$ contains no element of order 4.
Since Aut$(\mathbb{Z}_2)$ is trivial there is only one semidirect product of $\mathbb{Z}_2$ and $\mathbb{Z}_2$, which equals their direct product. Their wreath product, $\mathbb{Z}_2\wr \mathbb{Z}_2 \cong D_4$,
the dihedral group of the square can be used to emulate a mod-4 counter, since $\mathbb{Z}_4\hookrightarrow D_4$.
But this construction is not efficient, beyond the required rotations the dihedral group has the flip-symmetry as well, doubling the size of the group.
However, we would like to have a product construction that is isomorphic to $(\mB{4},\mathbb{Z}_4)$.
This motivates the definition of \emph{cascade products}: efficient constructions of substructures of wreath products, induced by explicit dependency functions~\cite{cascprod}.
Essentially, cascade products are transformation semigroups glued together by functions in a hierarchical tree.
More precisely, let $\big((X_{1},S_{1}),\ldots,(X_{n},S_{n})\big)$ be a fixed list of transformation semigroups, and dependency functions of the form
\[
d_i: X_1\times\ldots\times X_{i-1}\rightarrow S_i,\quad\text{for } i\in \{1,\ldots,n\}.
\]
A \emph{transformation cascade} is then defined to be an $n$-tuple of dependency functions $(d_1,\ldots,d_n)$, where $d_i$ is a dependency function of level $i$. If no confusion arises, on the top level we can simply write $d_1\in S_1$ instead of $d_1(\varnothing)\in S_1$.
The cascade action is defined coordinatewise by $x_i^{d_i(x_1,\ldots,x_{i-1})}$, applying the results of the evaluated dependency functions (see Fig.\ \ref{fig:cascaction}), so that the cascade product can be regarded as a special transformation representation on the set $X_1\times\ldots\times X_{n}$.
The hierarchical structure allows us to conveniently distribute computation among the components $(X_i, S_i)$, and perform abstractions and approximations of the system modelled as a cascade product.
Then if $W$ is a set of transformation cascades $(X_1,S_1)\wr_W\cdots \wr_W (X_n,S_n)$ denotes the transformation semigroup $(X_1 \times \cdots X_n, \langle W \rangle)$, where $\langle W \rangle$ is the semigroup of transformation cascades generated by $W$.
We can construct $(\mB{4},\mathbb{Z}_4)$ exactly by using two copies of $(\mB{2},\mathbb{Z}_2)$. The generator set contains only one permutation cascade $W=\{(+1,c)\}$, where $+1$ is the generator of $\mathbb{Z}_2$ and $c$ is a dependency function mapping $\mB{2}$ to $\mathbb{Z}_2$ with $c(0) =1_{\mathbb{Z}_2}$, and $c(1)=+1$.
The first dependency is a constant (increment modulo 2) while the second dependency implements the carry.
Therefore, with fewer dependencies than required by the wreath product, the mod-4 counter can be realized by an isomorphic cascade product: $(\mB{2},\mathbb{Z}_2)\wr_W(\mB{2},\mathbb{Z}_2)\cong(\mB{4},\mathbb{Z}_4)$, see Fig.\ \ref{fig:mod4counter}.
\begin{figure}
\caption{Two mod-2 counters cascaded together to build a mod-4 counter.}
\label{fig:mod4counter}
\end{figure}
An immediate consequence of the generality of the cascade product is that
several well-known constructions are special cases of the cascade product, and
as such they are easy to implement.
Direct products consist of all $d=(d_1,\ldots, d_n)$ with each $d_i$ constant. Wreath products consist of all possible dependency functions.
Direct, cascade, and wreath products constructions for transformation semigroups are now available in \textsc{SgpDec}, and iterated wreath products for permutation groups also became a bit more convenient to define.
\section{Functionality}
There are two different basic ways of using the \textsc{SgpDec}~package. Depending on whether the starting point is a complex structure or a set of (simple) building blocks, we can do \emph{decomposition} or \emph{composition}.
\subsection{Composition and Construction}
\label{sect:composition}
The questions we aim to answer by constructing cascade products can be of the following types.
\begin{enumerate}
\item What is the (semi)group generated by a given set of transformation cascades?
\item What can be built from a given set of (simple) components?
\end{enumerate}
The usual scenario is that for a list of components we give a set of cascades as a generating set.
For instance, the quaternion group $Q=\langle i , j \rangle$ is not a semidirect product, but it embeds into the full cascade product $(\mB{2},\mathbb{Z}_2)\boldsymbol{\wr}(\mB{2},\mathbb{Z}_2)\boldsymbol{\wr}(\mB{2},\mathbb{Z}_2)$, a group with 128 elements. Therefore, it can be built from copies of $\mathbb{Z}_2$.
The dependency functions can only have two values, thus to define cascade permutations it is enough to give only those arguments that give $+1$ (the generator of $\mathbb{Z}_2$). A cascade permutation realizing $i$ is defined by the dependency functions $(d_1,d_2,d_3)$ where
$d_2(0)=d_2(1)=d_3(0,0)=d_3(1,1)=+1$
and all other arguments map to the identity.
Similarly, a cascade realizing $j$ is defined by $(d'_1, d'_2,d'_3)$ where $d'_1(\varnothing)=d'_3(0,0)=d'_3(0,1)=+1$, (see Fig.\ \ref{fig:cascquaternion}, note that the state values are shifted by 1).
One can check that these two order 4 elements generate the 8-element quaternion group Q. Therefore by $W=\{(d_1,d_2,d_3),(d'_1,d'_2,d'_3)\}$ we have
$$(Q,Q)\cong (\mB{2},\mathbb{Z}_2)\wr_W(\mB{2},\mathbb{Z}_2)\wr_W(\mB{2},\mathbb{Z}_2).$$
\begin{verbbox}[\small]
gap> Z2:=CyclicGroup(IsPermGroup,2);
Group([ (1,2) ])
gap> d:=Cascade([Z2,Z2,Z2],[[[1],(1,2)],[[2],(1,2)],
[[1,1],(1,2)],[[2,2],(1,2)]]);
<perm cascade with 3 levels with (2, 2, 2) pts, 4 dependencies>
gap> dprime:=Cascade([Z2,Z2,Z2],[[[],(1,2)],[[1,1],(1,2)],[[1,2],(1,2)]]);
<perm cascade with 3 levels with (2, 2, 2) pts, 3 dependencies>
gap> StructureDescription(Group([d,dprime]));
"Q8"
\end{verbbox}
\theverbbox
\begin{figure}
\caption{Generators of a cascade representation of the quaternion group in a tree form. The edge labels are states, while the nodes contain the action. Empty node corresponds to the identity. The gray part of the tree is fixed.}
\label{fig:cascquaternion}
\end{figure}
\subsection{Decomposition}
\begin{enumerate}
\item What are the basic building blocks of a given (semi)group?
\item How can we represent it as a cascade product?
\end{enumerate}
A typical scenario is that for a given composite semigroup or group we choose a decomposition algorithm which returns a cascade product.
\subsubsection{Frobenius-Lagrange Decomposition.}
In the case of groups the decomposition uses the idea behind induction in representation theory (see e.g.\ \cite{AlperinBell}), so it traces back to Frobenius. Indeed, a special case of them comprises the well-known Krasner-Kaloujnine embeddings \cite{KrasnerKaloujnine}. All we need here is just standard group theory, namely the action on cosets, hence the name \emph{Frobenius-Lagrange Decomposition}.
How would someone come up with the generators cascades of the quaternion group in Section \ref{sect:composition}?
The easiest solution is to use this group decomposition.
\noindent\begin{verbbox}[\small]
gap> Q := QuaternionGroup(IsPermGroup,8);
Group([ (1,5,3,7)(2,8,4,6), (1,2,3,4)(5,6,7,8) ])
gap> CQ := FLCascadeGroup(Q);
<cascade group with 2 generators, 3 levels with (2, 2, 2) pts>
\end{verbbox}
\theverbbox
The actual implementation takes a subgroup chain as input (chief series by default) and form the components by examining the coset space actions derived from the chain.
Therefore, the decomposition method can be considered as a generalized Schreier-Sims algorithm~\cite{CGTHandbook}.
Coordinatewise calculation in a cascade product can also be thought of as a sequence of refining approximate solutions.
For instance, each completed step of an algorithm for solving the Rubik's Cube corresponds to calculating the desired value at a hierarchical level of some cascade product representation and it gives a configuration `closer' to the solved state.
\subsubsection{Holonomy Decomposition.}
\begin{figure}
\caption{Tiling picture -- the internal details of the holonomy decomposition of the transformation semigroup $T$ generated by $t_1$ and $t_2$. The numbers on the left denote the hierarchical levels (level 4 consists of singleton sets and it is needed by the holonomy algorithm but not a component of the cascade product). Outer boxes contain subsets that are mutually reachable from each other under the semigroup action. The arrows indicate how a subset is `tiled' by its subsets, the arrow labels contain words (sequences of generators) that take a subset to one of its tiles. Dotted arrow means the tile is not an image. Roughly, the holonomy algorithm finds the components by checking the action of the semigroup on a set of tiles.}
\label{fig:tiling}
\end{figure}
For transformation semigroups the holonomy method \cite{zeiger67a,zeiger68,ginzburg_book68,eilenberg,holcombe_textbook,KRTforCategories,automatanetworks2005} is used.
The holonomy decomposition works by a close analysis of how the semigroup acts on those subsets of the state set which are images of the state set.
As a small example let's define $T$ as the transformation semigroup generated by $t_1=\left(\begin{smallmatrix}1&2&3&4\\3& 2& 4&4\end{smallmatrix}\right)$ and $t_2=\left(\begin{smallmatrix}1&2&3&4\\3&3& 1& 3\end{smallmatrix}\right)$.
Calculating its holonomy decomposition and displaying some information can be done by the following commands:
\noindent\begin{verbbox}[\small]
gap> T:=Semigroup([Transformation([3,2,4,4]),Transformation([3,3,1,3])]);
<transformation semigroup on 4 pts with 2 generators>
gap> HT := HolonomyCascadeSemigroup(T);
<cascade semigroup with 2 generators, 3 levels with (2, 2, 4) pts>
gap> DisplayHolonomyComponents(SkeletonOf(HT));
1: 2
2: 2
3: (2,C2) 2
\end{verbbox}
\theverbbox
\noindent The displayed information tells us that this 13-element semigroup can be realized as the cascade product of four copies of the transformation monoid of constant maps of two points and one instance of $\mathbb{Z}_2$.
The components are put together in a 3-level cascade product.
Holonomy decompositions are useful whenever a finite state-transition model of some process needs to be analyzed (e.g.\ \cite{Dini2013}).
\subsection{Visualization}
\textsc{SgpDec}~uses \textsc{GraphViz}, a widely used graph drawing package \cite{Ellson03graphviz}, for visualisation purposes.
The underlying idea is that a function generates source code in the \texttt{dot} language for the given mathematical object.
Then the actual figure can be generated separately to be included in papers, or using the \textsc{Viz}~package \cite{viz} immediately displayed on screen from the \textsc{Gap}~command line.
Figure \ref{fig:cascquaternion} and \ref{fig:tiling} were both auto-generated using \textsc{GraphViz}.
\section*{Acknowledgment}
The work reported in this article was funded in part by the EU project BIOMICS, contract number CNECT-318202. This support is gratefully acknowledged.
\end{document} | math | 15,132 |
\begin{document}
\title{A remark on subbundles of symplectic and orthogonal vector bundles over curves}
\abstract{We review the notions of symplectic and orthogonal vector bundles over curves, and the connection between principal parts and extensions of vector bundles. We give a criterion for a certain extension of rank $2n$ to be symplectic or orthogonal. We then describe almost all of its rank $n$ vector subbundles using graphs of sheaf homomorphisms, and give criteria for the isotropy of these subbundles.}
\section{Introduction}
Throughout this paper, $X$ is a complex projective smooth curve of genus $g$, with structure sheaf $\strs$ and function field $K(X)$.
\subsection{Symplectic and orthogonal vector bundles}
\textbf{Definition:} Let $K$ be a field and $M$ and $P$ vector spaces of dimensions $m$ and $1$ over $K$ respectively. Let $\alpha \colon M \to \Homom(M,P)$ be a $K$--linear map. Then the \textsl{transpose} of $\alpha$ is the unique linear map $^{t}\alpha \colon M \to \Homom(M,P)$ satisfying
\[ \left( \alpha (m_{1}) \right) (m_{2}) = \left( {^{t}\alpha} (m_{2}) \right) (m_{1}) \]
for all $m_{1}, m_{2} \in M$. If $A$ is the matrix of $\alpha$ with respect to some choice of bases, the matrix of $^{t}\alpha$ is just $^{t}A$. The following proposition shows that this also makes sense for vector bundles.
\begin{prop}
Let $E \to X$ be a vector bundle, $L \to X$ a line bundle and $\alpha \colon E \to \Homom(E,L)$ a vector bundle map over an open set $U \subseteq X$. Then the transpose of $\alpha$ is a well--defined map $E \to \Homom(E,L)$.
\label{gensym}
\end{prop}
\textbf{Proof}\\
Suppose the bundles $E$ and $L$ have transition functions $\{ e_{i,j} \}$ and $\{ l_{i,j} \}$ respectively relative to an open cover $\{ U_{i}: i \in J \}$ of $X$. Then $\alpha$ is given by a cochain $\{ \alpha_{i} \}$ of $n \times n$ matrices which satisfy
\begin{equation} \alpha_{i} e_{i,j} = \left( ^{t}e_{i,j}^{-1} l_{i,j} \right) \alpha_{j} \label{homcond} \end{equation}
over $U \cap U_{i} \cap U_{j}$. If $^{t}\alpha$ exists then it should be given by the cocycle $\{ {^{t}\alpha_i} \}$, by the discussion before this proposition. Taking the transpose of (\ref{homcond}), we have
\[ {^{t}e_{i,j}} \left( ^{t}\alpha_{i} \right) = {^{t}\alpha_j} l_{i,j} e_{i,j}^{-1}, \]
equivalently
\[ {^{t}\alpha_i} e_{i,j} = \left( ^{t}e_{i,j}^{-1} l_{i,j} \right) {^{t}\alpha_j} \]
since the $l_{i,j}$ commute with the other transition functions. This shows that $\{ {^{t}\alpha_i} \}$ does indeed define a map $E \to \Homom(E,L)$ over $U$. \ifhmode\unskip\nobreak\fi\quad\ensuremath\square \\
\\
By this proposition, it makes sense to speak of symmetric and antisymmetric homomorphisms $E \to \Homom(E,L)$. We denote these by $\Sym(E, \Homom(E,L))$ and $\bigwedge(E, \Homom(E,L))$ respectively.\\
\\
Similar statements hold for maps $\Homom(E,L) \to E$.\\
\\
\textbf{Definition:} A vector bundle $W \to X$ is \textsl{symplectic} (resp., \textsl{orthogonal}) if there exists a bilinear nondegenerate antisymmetric (resp., symmetric) form $\theta$ on $W \times W$ with values in a line bundle $L$.\\
\\
Two immediate consequences of the nondegeneracy of $\theta$ are:
\begin{itemize}
\item There is an antisymmetric or symmetric isomorphism
\[ W \xrightarrow{\sim} \Homom(W, L) \]
given by $w \mapsto \theta(w, \cdot)$. In particular, $(\det W)^{2} = L^{\rank W}$.
\item $W$ is symplectic only if it has even rank, since skew--symmetric matrices have even rank.
\end{itemize}
We shall henceforth restrict ourselves to the case where $\rank W = 2n$, even if $W$ is orthogonal.
\par
A subbundle of $W$ is \textsl{isotropic} if $\theta$ restricts to zero on it. For any subbundle $E \subseteq W$, we have the short exact vector bundle sequence
\[ 0 \to E^{\perp} \to W \to \Homom(E, L) \to 0 \label{orthcomp} \]
where the surjection is the map $w \mapsto \theta( w , \cdot )|_E$ and
\[ E^{\perp} = \left\{ w \in W : \theta( w , E ) = 0 \right\} \]
is the \textsl{orthogonal complement} of $E$ with respect to $\theta$. Clearly $E$ is isotropic if and only if $E \subseteq E^{\perp}$; this shows that the rank of an isotropic subbundle is at most $n = \frac{1}{2} \rank W$. An isotropic subbundle of rank $n$ is called a \textsl{Lagrangian} subbundle.
\subsection{Principal parts and extensions}
For vector bundles $E$ and $F$ over $X$, it is well known that an extension
\begin{equation} 0 \to E \to W \to F \to 0 \label{FextE} \end{equation}
is determined up to isomorphism of extensions by its cohomology class $\delta(W) \in H^{1}(X, \Homom(F,E))$. This class can be realised explicitly as follows. There exist trivialisations of $W$ over some open cover $\{ U_{i} \}$ of $X$ whose transition function over the intersection $U_{i} \cap U_j$ is of the form
\[ \begin{pmatrix} e_{i,j} & \delta_{i,j} \\ 0 & f_{i,j} \end{pmatrix} \]
where $e_{i,j}$ and $f_{i,j}$ are transition functions for $E$ and $F$ respectively. Then $\delta(W)$ is the class determined by the $\Homom(F,E)$--valued $1$--cocycle $\{ \delta_{i,j} \}$.
\par
We give another description of $\delta(W)$ following Kempf \cite{Kem1983}. (Here the results are given for extensions of invertible sheaves, but the arguments are readily adapted to the case of arbitrary rank.) Firstly, we fix some notation. We denote the sheaf of regular sections of a vector bundle $W$, $E$, $L$ etc.\ by the corresponding script letter ${\cal W}$, ${\cal E}$, ${\cal L}$ etc. A vector bundle $E \to X$ gives rise to an exact sequence of $\strs$--modules
\[ 0 \to {\cal E} \to \sRat(E) \to \sPrin(E) \to 0 \]
where $\sRat(E)$ is the sheaf of rational sections of $E$ and $\sPrin(E)$ the sheaf of principal parts with values in $E$. We denote their groups of global sections by $\Rat(E)$ and $\Prin(E)$ respectively. The sheaves $\sRat(E)$ and $\sPrin(E)$ are flasque, so we have the cohomology sequence
\begin{equation} 0 \to H^{0}(X, E) \to \Rat(E) \to \Prin(E) \to H^{1}(X, E) \to 0. \label{cohomseq} \end{equation}
We denote $\overline{s}$ the principal part of $s \in \Rat(E)$, and we write $[p]$ for the class in $H^{1}(X, E)$ of $p \in \Prin(E)$.
\par
Now given an extension of the form (\ref{FextE}), it can be shown (see for example \cite{Hit2005}, Lemma 3.1) that there exists a unique principal part $p \in \Prin(\Homom(F,E))$ such that the sheaf of sections ${\cal W}$ is of the form
\[ {\cal W}_{p} := \left\{ (e,f) \in \sRat(E) \oplus \sRat(F) : f \hbox{ is regular and } \overline{e} = p(f) \right\}. \]
Following Kempf \cite{Kem1983}, Chap.\ 6, one proves as for the case where ${\cal E}$ and ${\cal F}$ are invertible that two principal parts define isomorphic extensions if and only if they differ by the principal part $\overline{\alpha}$ of some $\alpha \in \Rat(\Homom(F,E))$, and that the cohomology class $\delta(W)$ is just $[p]$.\\
\par
In the following sections, we give a criterion for a certain extension of vector bundles to be symplectic or orthogonal. We then generalise a result from Mukai \cite{Muk2001} to describe almost all rank $n$ subbundles of such an extension, and give criteria for the isotropy of these subbundles. We conclude by sketching how these results are applicable to the study of moduli spaces of symplectic or orthogonal vector bundles over curves.\\
\par
\textbf{Acknowledgements:} I thank my doctoral supervisors C.\ Pauly, J.\ Bolton, W.\ Klingenberg and W.\ Oxbury for their time and ideas. I am grateful to S.\ Ramanan for showing me Criterion \ref{fundam}, which is central to this work. Thanks to M.\ Reid for his continuing support, mathematical and practical. I acknowledge gratefully the financial support and hospitality of the University of Durham and l'Universit\'e de Nice et Sophia--Antipolis.
\section{Symplectic and orthogonal extensions}
Let $W \to X$ be a symplectic or orthogonal vector bundle of rank $2n$ and let $E \subset W$ be a Lagrangian subbundle. Then $W$ is an extension
\[ 0 \to E \to W \to \Homom(E, L) \to 0. \tag{$\delta(W)$} \]
Conversely, it is natural to ask for which extension classes $\delta(W)$ this sequence is induced by a bilinear antisymmetric or symmetric form. We have
\begin{crit} An extension $0 \to E \to W \to \Homom(E, L) \to 0$ has a symplectic (resp., orthogonal) structure with respect to which $E$ is isotropic if and only if $W$ is isomorphic as a vector bundle to an extension whose cohomology class belongs to $H^{1}(X, \Sym(\Homom(E,L),E))$ (resp., $H^{1}(X, \bigwedge(\Homom(E,L), E))$). \label{fundam}
\end{crit}
\textbf{Proof}\\
We prove the criterion for the symplectic case; the orthogonal case is practically identical.
\par
$\Leftarrow$: Firstly, suppose $\delta(W)$ is actually symmetric. By the discussion in $\S$ 1.2, there exists $p \in \Prin(\Homom(\Homom(E, L), E))$ such that the sheaf ${\cal W}$ is equal to
\begin{equation} \left\{ (f,\phi) \in \sRat(E) \oplus \sRat(\Homom(E,L)) : \phi \hbox{ is regular and } \overline{f} = p(\phi) \right\}. \label{sectW} \end{equation}
To say that $\delta(W) = [p]$ is symmetric is to say that
\[ ^{t}p - p = \overline{\alpha} \]
for some $\alpha \in \Rat( \Homom( \Homom(E,L), E))$. Clearly $\overline{\alpha}$ is antisymmetric; replacing $\alpha$ by $\frac{\alpha - {^{t}\alpha}}{2}$ if necessary, we can assume that $\alpha$ itself is antisymmetric.
\par
Now we define a $\sRat(L)$--valued bilinear antisymmetric form
\[ \underline{\theta} \colon \left( \sRat(E) \oplus \sRat(\Homom(E,L)) \right)^{\times 2} \to \sRat(L) \]
by setting
\[ \underline{\theta} \left( (e_{1},\phi_{1}) , (e_{2},\phi_{2}) \right) = \phi_{1}(e_{2}) - \phi_{2}(e_{1}) - \phi_{2} \left( \alpha (\phi_{1}) \right). \]
By the description in (\ref{sectW}), for any $(e_{1},\phi_{1})$, $(e_{2},\phi_{2}) \in {\cal W}_p$, the principal part
\begin{align*} \overline{\underline{\theta} \left( (e_{1},\phi_{1}) , (e_{2},\phi_{2}) \right) } &= \phi_{1}(p(\phi_{2})) - \phi_{2}(p(\phi_{1})) - \phi_{2} \left( \overline{\alpha} ( \phi_{1}) \right) \\
&= \phi_{2}((^{t}p - p - \overline{\alpha})(\phi_{1}))\\
&= 0
\end{align*}
since $^{t}p - p = \overline{\alpha}$. Hence $\underline{\theta}$ is regular on ${\cal W} \times {\cal W}$. It is clearly $\strs$--bilinear and nondegenerate, and ${\cal E}$ is isotropic. Thus $\underline{\theta}$ induces a global regular symplectic form $\theta \colon W \times W \to L$ with the required properties.
\par
For the general case, we note that this form pulls back to give the required symplectic structure to any vector bundle isomorphic to $W$, which need not be isomorphic as an extension.\\
\par
$\Rightarrow$: Choose transition functions $\{ e_{i,j} \}$ and $\{ l_{i,j} \}$ for $E$ and $L$ respectively over an open cover $\{ U_{i}: i \in J \}$ of $X$. Then the transition functions of $\Homom(E, L)$ are $\{ {^{t}e_{i,j}^{-1}}l_{i,j} \}$ and there exist trivialisations for $W$ over $\{ U_{i} \}$ whose transition functions are of the form
\[ \{ w_{i,j} \} = \left\{ \begin{pmatrix} e_{i,j} & \delta_{i,j} \\ 0 & {^{t}e_{i,j}^{-1}} l_{i,j} \end{pmatrix} \right\}; \]
the cohomology class of the cocycle $\{\delta_{i,j}\}$ is $\delta(W)$.
\par
The symplectic form is given with respect to $\{ U_{i} \}$ by a cochain $\{\Theta_{i}\}$ of antisymmetric matrices which satisfy
\begin{equation} (l_{i,j}^{-1}){^{t}w_{i,j}}\Theta_{i}w_{i,j} = \Theta_{j} \label{coch} \end{equation}
on the intersection $U_{i} \cap U_{j}$ for all $i, j \in J$, since the symplectic form defines a homomorphism $W \to \Homom(W, L)$. We can write
\[ \Theta_{i} = \begin{pmatrix} A_i & B_i \\ -{^{t}B_i} & C_i \end{pmatrix} \]
where $\{A_{i}\}$, $\{B_{i} \}$ and $\{C_i\}$ are $M_{n,n}(\C)$--valued cochains and all the $A_i$ and $C_i$ are antisymmetric. Firstly, we see that every $A_i \equiv 0$ because $E \subset W$ is isotropic.
\par
Expanding condition (\ref{coch}), we see that
\[ B_{i} \left( ^{t}e_{i,j}^{-1} \right) = \left( ^{t}e_{i,j}^{-1} \right) B_{j}, \]
so $\{ B_{i} \}$ defines an endomorphism of $E^{*}$ and also of $\Homom(E, L)$. Since all the $A_{i}$ are zero but the form is nondegenerate, this must be an automorphism. Also by (\ref{coch}), we have
\begin{equation} {^{t}\delta_{i,j}} B_{i} \left( ^{t}e_{i,j}^{-1} \right) - e_{i,j}^{-1} \left(^{t}B_{i} \right) \delta_{i,j} = C_{j} - e_{i,j}^{-1}C_{i} \left( ^{t}e_{i,j}^{-1} l_{i,j} \right), \label{transcomp} \end{equation}
whence we see that the difference between the cocycle $\{ {^{t}B_i} \delta_{i,j} \}$ and its transpose is cohomologically trivial. Hence the cohomology class defined by $\{ {^{t}B_i}\delta_{i,j} \}$ belongs to $H^{1}(X, \Sym(\Homom(E,L),E))$. This belongs to the same orbit as $\delta(W)$ under the action of $\Autom E$ on the extension space $H^{1} \left( X, \Homom(\Homom(E,L),E) \right)$, so defines an extension isomorphic to $W$ as a vector bundle. \ifhmode\unskip\nobreak\fi\quad\ensuremath\square\\
\\
\textbf{Caution:} Strictly, in order to obtain cocycles which map between the correct spaces in (\ref{transcomp}), we should replace ${^{t}\delta_{i,j}} B_{i} (^{t}e_{i,j}^{-1})$ with
\[ l_{i,j}^{-1}(^{t}\delta_{i,j}) B_{i} (^{t}e_{i,j}^{-1} l_{i,j}), \]
but this does not change the value. In any case, by Prop.\ \ref{gensym} the transpose of $\{ e_{i,j}^{-1} (^{t}B_{i}) \delta_{i,j} \}$ is indeed defined by the transposed cocycle $\{ {^{t}\delta_{i,j}} B_{i} (^{t}e_{i,j}^{-1}) \}$.\\
\\
\textbf{Remark:} If $E$ is simple (for example, if $E$ is stable) then the cocycle $\{ \delta_{i,j} \}$ will itself be symmetric because $\{ B_{i} \}$ defines a nonzero homothety.
\section{Vector subbundles and graphs}
Firstly, we recall some linear algebra. Let $K$ be a field and suppose $M$, $N$ and $P$ are vector spaces over $K$ of dimensions $m$, $n$ and $1$ respectively. If $N = \Homom(M,P)$ and
\[ \alpha \colon \Homom(M,P) \to M \]
is an antisymmetric map then then we can define a $P$--valued bilinear nondegenerate antisymmetric form
\[ \theta \colon \left( M \oplus \Homom(M, P) \right)^{\times 2} \to P \]
by
\[ \theta \left( (m_{1},\psi_{1} ),(m_{2},\psi_{2} ) \right) = \psi_{1}(m_{2}) - \psi_{2}(m_{1}) - \psi_{2} \left( \alpha ( \psi_{1} ) \right) \]
The following is a slight generalisation of Mukai \cite{Muk2001}, Example 1.5.
\begin{lemma} \label{ano}
\begin{enumerate}
\renewcommand{$\mathrm{(\roman{enumi})}$}{$\mathrm{(\roman{enumi})}$}
\item There is a bijection between $\Homom_{K}( N , M)$ and the set of $n$--dimensional $K$--vector subspaces of $M \oplus N$ intersecting $M$ in zero, given by associating to a map $\beta$ its graph $\Gamma_{\beta}$.
\item The kernel of $\beta$ is canonically isomorphic to $\Gamma_{\beta} \cap \left( \{ 0 \} \oplus N \right)$.
\item If $N = \Homom(M,P)$ then $\Gamma_{\beta}$ is isotropic with respect to $\theta$ if and only ${^{t}\beta} - \beta = \alpha$.
\end{enumerate}
\end{lemma}
\textbf{Proof}\\
This is straightforward to check. \ifhmode\unskip\nobreak\fi\quad\ensuremath\square \\
\\
Now let $E$ and $F$ be vector bundles of rank $m$ and $n$ respectively over $X$. Consider an extension $0 \to E \to W \to F \to 0$ with sheaf of sections ${\cal W}_p$ and class $\delta(W) = [p] \in H^{1}(X, \Homom(F,E))$. We want to study vector subbundles $G \subset W$ of rank $n$ whose projection to $F$ is generically surjective.\\
\\
\textbf{Notation:} Let $\Lambda$ be a vector space over $K(X)$. By analogy with Hartshorne \cite{Har1977}, p.\ 69, we write $\underline{\Lambda}$ for the constant sheaf on $X$ associated to $\Lambda$.\\
\\
Now $\Rat(E)$ and $\Rat(F)$ are vector spaces of dimensions $m$ and $n$ respectively over $K(X)$, the field of rational functions on $X$. The following theorem, globalising Lemma \ref{ano}, is a generalisation of Mukai \cite{Muk2001}, Example 1.7, to the case where $W$ may be a nontrivial extension.
\begin{thm}
\begin{enumerate}
\renewcommand{$\mathrm{(\roman{enumi})}$}{$\mathrm{(\roman{enumi})}$}
\item There is a bijection
\[ \Homom_{K(X)}(\Rat(F),\Rat(E)) \leftrightarrow \left\{ \begin{array}{c} \hbox{rank $n$ vector subbundles} \\
G \subset W \hbox{ with } G|_{x} \cap E|_{x} = 0 \\
\hbox{ for generic } x \in X \end{array} \right\} \]
given by $\beta \leftrightarrow \underline{\Gamma_{\beta}} \cap {\cal W}_{p} =: {\cal G}_{\beta}$. Moreover, $\cal{G}_{\beta}$ is isomorphic to the kernel of the map
\[ \left( p - \overline{\beta} \right) \colon {\cal F} \to \sPrin(E). \]
\item Let $\widetilde{\beta}$ denote the restriction of $\beta$ to $\Ker ( p - \overline{\beta} ) \subseteq {\cal F}$. Then $\Ker ( \widetilde{\beta} ) \cong ( \underline{\Gamma_{\beta}} \cap {\cal W}_{p} ) \cap (\{ 0 \} \oplus {\cal F})$ (although we will not use this result).
\item Suppose $F = \Homom(E,L)$ and ${^{t}p} - p = \overline{\alpha}$ for some
\[ \alpha \in \Rat \left( \bigwedge \left( \Homom(E,L), E \right) \right), \]
so $W$ carries the symplectic form $\theta$ defined in the proof of Criterion \ref{fundam}. Then $G_{\beta}$ is isotropic with respect to $\theta$ if and only if ${^{t}\beta} - \beta = \alpha$. \label{vbgraphs}
\end{enumerate}
\end{thm}
\textbf{Proof}\\
(i) Let $G \subset W$ be a vector subbundle of rank $n$ intersecting $E$ in zero except at a finite number of points. Then $\Rat(G)$ is a $K(X)$--vector subspace of $\Rat(W) = \Rat(E) \oplus \Rat(F)$ of dimension $m$ which intersects $\Rat(E)$ in zero. By Lemma \ref{ano} and the remarks just before this theorem, $\Rat(G)$ is the graph $\Gamma_{\beta}$ of some uniquely determined
\[ \beta \in \Homom_{K(X)} \left( \Rat(F), \Rat(E) \right).\]
Furthermore, ${\cal G} = \underline{\Gamma_{\beta}} \cap {\cal W}_p$ since a regular section of $G$ is the same thing as a rational section of $G$ which is a regular section of $W$.
\par
Conversely, we claim that the the association
\[ \beta \mapsto \underline{\Gamma_{\beta}} \cap {\cal W}_{p} =: {\cal G}_{\beta} \]
defines a subsheaf of ${\cal W}_p$ which in fact corresponds to a vector subbundle $G_{\beta} \subset W$ with the required properties. By the definitions of $\Gamma_{\beta}$ and ${\cal W}_p$, we have
\[ {\cal G}_{\beta} = \left\{ ( \beta(f) , f ) \in \sRat(E) \oplus {\cal F} : p(f) = \overline{\beta(f)} \right\}. \]
This is clearly isomorphic to the kernel of the map
\[ (p - \overline{\beta}) \colon {\cal F} \to \sPrin(E) \]
via the projection of ${\cal G}_{\beta}$ onto its image in ${\cal F}$. (The inverse map is $f \mapsto (\beta(f), f)$.) But since any principal part is supported at a finite number of points, $({\cal G}_{\beta})_{x} \cong {\cal F}_x$ for all but finitely many $x \in X$. Hence ${\cal G}_{\beta}$ has rank $n$ and projects surjectively to ${\cal F}$ at all but a finite number of points of $X$.
\par
We now check that the inclusion ${\cal G}_{\beta} \hookrightarrow {\cal W}_p$ actually corresponds to a vector bundle injection $G_{\beta} \hookrightarrow W$. We have a short exact sequence of $\strs$--modules
\[ 0 \to {\cal G}_{\beta} \to {\cal W}_{p} \to {\cal Q} \to 0 \]
where ${\cal Q}$ is coherent. Let ${\cal G}^{\prime}$ denote the inverse image in ${\cal W}_p$ of the torsion subsheaf of ${\cal Q}$. Clearly ${\cal G}^{\prime}$ contains ${\cal G}_{\beta}$. Now ${\cal G}^{\prime}$ corresponds to an injection of vector bundles $G^{\prime} \hookrightarrow W$ by Atiyah \cite{Ati1957}, Prop.\ 1, since by construction ${\cal W}_{p}/{\cal G}^{\prime}$ is locally free. But in fact $\Rat(G^{\prime}) = \Gamma_{\beta}$; this is because ${\cal G}^{\prime}$ is contained in ${\cal G}_{\beta}(D)$ for some divisor $D$ on $X$, so they have the same sheaf of rational sections. Hence ${\cal G}^{\prime} \subseteq \Gamma_{\beta} \cap {\cal W}_{p} = {\cal G}_{\beta}$, so ${\cal G}_{\beta} = {\cal G}^{\prime}$ corresponds to a vector subbundle $G_{\beta} \subset W$.
\par
It is not hard to check that these constructions are mutually inverse, so we have a bijection.\\
\\
(ii) Suppose $g \in \Ker (p - \overline{\beta} )$. Then $\widetilde{\beta}(g) = 0$ if and only if
\[ (\beta(g), g) \in (\underline{\Gamma_{\beta}} \cap {\cal W}_{p}) \cap ( \{ 0 \} \oplus {\cal F}). \]
\\
(iii) The symplectic form on $W$ is induced by the restriction of the form defined earlier
\[ \underline{\theta} \colon \sRat \left( E \oplus \Homom(E, L) \right)^{\times 2} \to \sRat(L) \]
to ${\cal W}_{p} \times {\cal W}_p$, so the criterion for isotropy follows from part (iii) of Lemma \ref{ano}. \ifhmode\unskip\nobreak\fi\quad\ensuremath\square \\
\\
We make an observation:
\begin{lemma}
The $K(X)$--linear map $\beta$ is everywhere regular on ${\cal G}_{\beta}$. \label{reg}
\end{lemma}
\textbf{Proof}\\
If the supports of $p$ and $\overline{\beta}$ are disjoint, then this is clear. Suppose the supports coincide at a point $x \in X$. Then the maps
\[ p_{x}, \overline{\beta}_x \hbox{ and } (p - \overline{\beta})_{x} \in \Homom_{\strs_{,x}}(({\cal G}_{\beta})_{x},\sPrin(E_{\beta})_{x}) \]
are given locally by matrices of rational functions on a neighbourhood of $x$. Since $X$ is of dimension $1$, we can assume that the numerators and denominators of each of these functions are relatively prime, and then in fact the denominators determine the maps.
\par
The key point is that by the identity
\[ \frac{a}{f} - \frac{b}{h} = \frac{ah-bf}{fh}, \]
the denominators of the entries of the matrix $(p - \overline{\beta})_x$ are at worst the products of the corresponding entries of $p_x$ and $\overline{\beta}_x$. Since ${\cal G}_{\beta} \cong \Ker(p - \overline{\beta})$, the value $(p - \overline{\beta})_{x}(g)$ is regular for any $g \in ({\cal G}_{\beta})_x$. But for regular functions $a$, $f$ and $h$, if $\frac{a}{fh}$ is regular then so is $\frac{a}{h}$. Hence $\beta$ itself is regular on $({\cal G}_{\beta})_x$. \ifhmode\unskip\nobreak\fi\quad\ensuremath\square\\
\\
Now we examine a useful special case.
\begin{cor} Suppose that $h^{0}(X, \Homom(F,E)) = 0$. Then principal parts defining the cohomology class $\delta(W) = [p]$ are in bijection with elementary transformations of $F$ lifting to rank $n$ subbundles of $W$ via $q \leftrightarrow \Ker \left( q \colon {\cal F} \to \sPrin(E) \right)$. \label{bijection} \end{cor}
\textbf{Proof}\\
If $\delta(W)$ is defined by $q \in \Prin ( \Homom(F,E) )$ then by (\ref{cohomseq}) we have $q = p - \overline{\beta}$ for some global rational section $\beta$ of $\Homom(F,E)$, which is uniquely determined by hypothesis. Then $\Ker ( q ) \subseteq {\cal F}$ lifts to the rank $n$ subsheaf $\Gamma_{\beta} \cap {\cal W}_p$ of ${\cal W}_p$ by the map $f \mapsto (\beta(f), f)$. By the proof of Thm.\ 4 (i) this corresponds to a vector subbundle.
\par
Conversely, suppose $G \subset W$ is a rank $n$ vector subbundle lifting from $F$. By the proof of Thm.\ 4 (i), the sheaf of sections of $G$ is isomorphic to $\Ker ( p - \overline{\beta} )$ for some $\beta \in \Rat(\Homom(F,E))$. But by (\ref{cohomseq}) we have $[ p - \overline{\beta} ] = [p] = \delta(W)$.
\par
It is easy to see that these constructions are mutually inverse. \ifhmode\unskip\nobreak\fi\quad\ensuremath\square
\subsection*{Another criterion for isotropy}
We give a refinement of Theorem 4 (iii) for this case. Let $E$ and $L$ be vector bundles of ranks $n$ and $1$ respectively over $X$ such that $\Homom( \Homom(E, L) , E)$ has no global sections. Let
\[ 0 \to E \to W \to \Homom(E,L) \to 0 \]
be a symplectic extension with sheaf of sections ${\cal W}_p$ where as before ${^{t}p} - p = \overline{\alpha}$. Let $G \subset W$ be a subbundle of rank $n$ which intersects $E$ generically in rank $0$; by Cor.\ \ref{bijection}, the sheaf ${\cal G} = \Ker(q)$ for some $q \in \Prin(\Homom(\Homom(E,L),E))$ such that $\delta(W) = [q]$.
\begin{crit} The subbundle $G \subset W$ is isotropic if and only if $q$ is a symmetric principal part (note that this is a stronger condition than that the cohomology class defined by $q$ be symmetric). \label{isotcrit}
\end{crit}
\textbf{Proof}\\
By Cor.\ \ref{bijection} (i) we have ${\cal G} = \underline{\Gamma_{\beta}} \cap {\cal W}_p$ for some
\[ \beta \in \Rat(\Homom( \Homom(E,L),E)) \]
which, by hypothesis, is determined by the condition $q = p - \overline{\beta}$.
\par
We claim that ${^{t}\beta} - \beta = \alpha$ if and only if $\overline{{^{t}\beta} - \beta} = \overline{\alpha}$. One direction is clear. Conversely, suppose $\overline{{^{t}\beta} - \beta} = \overline{\alpha}$. By (\ref{cohomseq}), the difference ${^{t}\beta} - \beta - \alpha$ is a global regular section of $\Homom( \Homom (E, L) , E)$. But this is zero by hypothesis, so ${^{t}\beta} - \beta = \alpha$.
\par
Thus by Theorem 4 (iii), the subbundle $G$ is isotropic if and only if $\overline{{^{t}\beta} - \beta} = \overline{\alpha}$. Now ${^{t}p} - p = \overline{\alpha}$, so
\[ {^{t}q} - q = {^{t}(p - \overline{\beta})} - (p - \overline{\beta}) = \overline{\alpha} - \left( \overline{^{t}\beta} - \overline{\beta} \right). \]
Hence $G$ is isotropic if and only if $q$ is a symmetric principal part. \ifhmode\unskip\nobreak\fi\quad\ensuremath\square
\section*{The orthogonal case}
There are obvious analogues to these results for the orthogonal case which are proven identically. In Lemma \ref{ano} (iii), we consider a symmetric map $\alpha \colon \Homom(E,L) \to E$ and work with the standard bilinear nondegenerate \textsl{symmetric} form $\theta$ given by
\[ \theta \left( (v_{1},\phi_{1}),(v_{2},\phi_{2}) \right) = \phi_{1}(v_{2}) + \phi_{2}(v_{1}) - \phi_{2} \left( \alpha ( \phi_{1} ) \right) \]
and require ${^{t}\beta} + \beta = \alpha$. In Theorem 4 (iii), we consider an orthogonal extension ${\cal W}_p$ where ${^{t}p} + p = \overline{\alpha}$ and the condition of the criterion is that ${^{t}\beta} + \beta = \alpha$. With this setup, we get an orthogonal version of Criterion \ref{isotcrit}: the sheaf $G$ is isotropic if and only if $q$ is an antisymmetric principal part.
\section{Applications}
\subsection{Bundles of rank 2}
As an example, we show how these results apply to the well--known case where $n=1$.
\par
Firstly, let $W \to X$ be of rank $2$ and trivial determinant. Then any line subbundle $E \subset W$ gives a short exact sequence
\[ 0 \to E \to W \to E^{*} \to 0 \]
and Criterion \ref{fundam} gives another proof of the well--known fact that every such $W$ is symplectic, since in this case $\Sym^{2}E = \Homom(E^{*},E)$. Furthermore, Theorem 4 (iii) then gives another proof that every line subbundle of $W$ is isotropic.
\par
On the other hand, suppose $W$ is an orthogonal vector bundle of rank $2$ and let $E \subset W$ be an isotropic line subbundle. Then $W \cong E \oplus E^{-1}L$ by Criterion \ref{fundam} since $\bigwedge^{2}E = 0$.
\subsection{Covers of moduli spaces}
Let $\Mxn$ denote the moduli space of semistable principal $\Spn$--bundles over $X$. This is an irreducible projective variety of dimension $n(2n+1)(g-1)$; see Ramanathan \cite{Rthn1996a}, \cite{Rthn1996b} for results on moduli of principal $G$--bundles over curves. $\Mxn$ is naturally a moduli space for semistable vector bundles of rank $2n$ which carry an $O_X$--valued symplectic form (see \cite{Hit2005} for details). Following Narasimhan and Ramanan \cite{NR1984}, one might try to construct a generically finite cover of $\Mxn$ by the classifying map from the union of the extension spaces
\[ \mathbb{P} H^{1}(X, \Sym^{2}E) \]
as $E$ varies over some collection of rank $n$ vector bundles over $X$. By Criterion \ref{fundam}, determining the fibre over a bundle $W$ in the image of such a classifying map involves asking for certain Lagrangian subbundles of $W$. If $n \geq 2$ then not every rank $n$ subbundle need be Lagrangian, and Criterion \ref{isotcrit} may be of use in distinguishing those that are.
\par
In a forthcoming paper (and see \cite{Hit2005}, Chaps.\ 4 \& 5), we shall use this technique to construct a cover of $\Mxt$ when $X$ has genus $2$.\\
\\
\textbf{Remark:} The hypothesis $h^{0}(X,\Homom(F, E)) = 0$ of Cor.\ \ref{bijection} is a natural one if we want to build stable bundles. Firstly, if $\degree E \geq \degree F$ then no extension
\[ 0 \to E \to W \to F \to 0 \]
can be stable. If the converse is true and furthermore $E$ and $F$ are themselves stable then indeed $h^{0}(X,\Homom(F, E)) = 0$.
Laboratoire J.--A.\ Dieudonn\'e,\\
Universit\'e de Nice et Sophia--Antipolis,\\
Parc Valrose,\\
06108 Nice CEDEX 02,\\
France.\\
E--mail: \texttt{[email protected]}
\end{document} | math | 28,243 |
\begin{equation}gin{document}
\preprint{APS/123-QED}
\begin{equation}gin{CJK*}{GB}{gbsn}
\title{Partial-Norm of Entanglement: Entanglement Monotones That are not Monogamous \\}
\author{Yu Guo}
\email{[email protected]}
\affiliation{Institute of Quantum Information Science, Shanxi Datong University, Datong, Shanxi 037009, China}
\begin{equation}gin{abstract}
Quantum entanglement is known to be monogamous, i.e., it obeys
strong constraints on how the entanglement can be distributed among multipartite systems. Almost all the entanglement monotones so far are shown to be monogamous. We explore here a family of entanglement monotones with the reduced functions are concave but not strictly concave and show that they are not monogamous. They are defined by four kinds of the ``partial-norm'' of the reduced state, which we call them \textit{partial-norm of entanglement}, minimal partial-norm of entanglement, reinforced minimal partial-norm of entanglement, and \textit{partial negativity}, respectively.
This indicates that, the previous axiomatic definition of the entanglement monotone
needs supplemental agreement that the reduced function should be strictly concave since such a strict concavity can make sure that the corresponding convex-roof extended entanglement monotone is monogamous.
Here, the reduced function of an entanglement monotone refers to the corresponding function on the reduced state for the measure on bipartite pure states.
\end{abstract}
\pacs{03.67.Mn, 03.65.Db, 03.65.Ud.}
\maketitle
\end{CJK*}
Entanglement, as a quintessential manifestation
of quantum mechanics~\cite{Nielsen,Schrodinger1935,Schrodinger1936},
has shown to be a crucial resource in various quantum information processing tasks~\cite{Nielsen,Bennett1992communication,Bennett1993teleporting,Bennett1996prl,Zhang2006experimental}.
The most striking property of entanglement is its distributability,
that is, the impossibility of
sharing entanglement unconditionally across many subsystems
of a composite quantum system~\cite{Osborne,Terhal2004}.
Understanding how entanglement can be quantified and distributed over many parties reveals fundamental
insights into the nature of quantum correlations~\cite{Horodecki2009} and has profound
applications in both quantum communication~~\cite{Bennett2014,Toner,Seevinck} and other area of physics~\cite{streltsov2012are,Augusiak2014pra,Ma2011,Brandao2013,Garcia,Bennett2014,Lloyd}.
Particularly, monogamy law of quantum correlation is the predominant feature
that guarantees the quantum key distribution secure~\cite{Terhal2004,Pawlowski}.
Quantitatively, the monogamy of entanglement is described by an inequality,
involving a bipartite entanglement monotone. The term
``monotone'' refers to the fact that a proper measure of
entanglement cannot increase on average under local operations and classical communication (LOCC)~\cite{Vidal2000,Vedral1998pra,Plenio2005prl}.
Recall that the traditional monogamy relation of entanglement
measure $E$ is quantitatively displayed as an inequality of the following form:
\begin{equation}\langlebel{basic}
E\left(A|BC\right)\geq E\left(AB\right)+E \left(AC\right),
\end{equation}
where the vertical bar indicates the bipartite split
across which the (bipartite) entanglement is measured.
However, Eq.~\eqref{basic} is not valid for many entanglement measures but $E^\alpha$
satisfies the relation for some $\alpha>0$~\cite{Coffman,Osborne,Bai}.
Intense research has been undertaken in this
direction. It has been proved that the squashed entanglement and the one-way distillable entanglement are monogamous~\cite{Koashi}, and almost all the bipartite entanglement measures so far
are monogamous for the multiqubit system or monogamous on pure states~\cite{Coffman,Osborne,streltsov2012are,
Bai,Luo2016pra,Dhar,Hehuan}.
However, for the higher dimensional system, it is difficult to check the monogamy of entanglement measure
according to Eq.~\eqref{basic} in general. Consequently, the definition of the monogamy is then improved as~\cite{GG}:
a measure of entanglement $E$ is monogamous if for any $\rho^{ABC}\in\mathcal{S}^{ABC}$
that satisfies the \textit{disentangling condition}, i.e.,
\begin{equation}\langlebel{cond}
E\left( A|BC\right) =E\left( AB\right),
\end{equation}
we have that $E(AC)=0$, where $\mathcal{S}^X$ denotes the set of all density matrices acting the state space $\mathcal{H}^X$. It is closely related to the traditional one: a continuous measure $E$ is monogamous according to this definition if and only if
there exists $0<\alpha<\infty$ such that
$E^\alpha\left( A|BC\right) \geq E^\alpha\left( AB\right) +E^\alpha\left( AC\right)$,
for all $\rho^{ABC}\in\mathcal{S}^{ABC}$
with fixed $\dim\mathcal{H}^{ABC}=d<\infty$.
Such a definition simplifies the justification of the monogamy of entanglement measure greatly~\cite{GG,GG2019}.
Recall that, a function $E: \mathcal{S}^{AB}\to\mbb{R}_{+}$ is called a
measure of entanglement if {(1)} $E(\sigma^{AB})=0$ for any
separable density matrix $\sigma^{AB}\in\mathcal{S}^{AB}$, and {(2)}
$E$ behaves monotonically under LOCC.
Moreover, convex measures of entanglement that do not increase \emph{on average}
under LOCC are called entanglement monotones~\cite{Vidal2000}.
Let $E$ be a measure of entanglement on bipartite states.
We define
$E_F\left(\rho^{AB}\right)\equiv\min\sum_{j=1}^{n}p_jE\left(|\psi_j\ranglengle\langlengle\psi_j|^{AB}\right)$,
where the minimum is taken over all pure state decompositions of $\rho^{AB}=\sum_{j=1}^{n}p_j|\psi_j\ranglengle\langlengle\psi_j|^{AB}$.
That is, $E_F$ is the convex roof extension of $E$.
Vidal~\cite[Theorem 2]{Vidal2000} showed that for any entanglement measure $E$, $E_F$ above is an entanglement monotone
if
\begin{equation}\langlebel{h}
h\left( \rho^A\right) = E\left( |\psi\ranglengle\langlengle\psi|^{AB}\right)
\end{equation}
is concave, i.e.
$h[\langlembda\rho_1+(1-\langlembda)\rho_2]\geq\langlembda h(\rho_1)+(1-\langlembda)h(\rho_2)$
for any states $\rho_1$, $\rho_2$, and any
$0\leq\langlembda\leq1$.
Hereafter, we call $h$ the \textit{reduced function} of $E$ and $\mathcal{H}^A$ the \textit{reduced subsystem} for convenience.
In Ref.~\cite{GG}, according to definition~\eqref{cond}, we showed that $E_F$ is monogamous whenever $E_F$ is defined via Eq.~\eqref{h} with $h$ is strictly concave additionally. Except for the R\'enyi $\alpha$-entropy of entanglement with $\alpha>1$, all other measures of entanglement, that were studied intensively in literature, correspond on pure bipartite state to strict concave functions of the reduced density matrix. Theses include the original entanglement of formation~\cite{Bennett1996pra}, tangle~\cite{Rungta2003pra}, concurrence\cite{Rungta,Rungta2003pra}, $G$-concurrence~\cite{Gour2005pra}, Tsallis entropy of entanglement~\cite{Kim2010pra}, and the entanglement measures induced by the fidelity distances~\cite{Guo2020qip}.
Nevertheless,
we are not sure yet whether the entanglement monotone is monogamous if the reduced function
is concave but not strictly concave.
The purpose of this Letter is to address such a issue.
We explore the entanglement monotone suggested in Ref.~\cite{Vidal1999pral}, from which we also obtain another two
entanglement monotones. We also investigate the partial negativity which is defined as the norm of the negative part of the state after partial transposition. The reduced functions of theses quantities are not strictly concave, and they are not equivalent to each other. We then show that they are not monogamous. This is the first time to prove that there exist entanglement monotones that are not monogamous in the light of the disentangling condition.
Our results establish a more closer relation between the monogamy of an entanglement monotone and the strict concavity of the reduced function and suggest that we should require the strict concavity of the reduced function for any ``fine'' entanglement monotone.
Let $|\psi\rangle=\sum_{j=1}^r\langlembda_j|e_j\rangle^A|e_j\rangle^B$
be the Schmidt decomposition of $|\psi\rangle\in\mathcal{H}^{AB}$, where $\langlembda_1\geq\langlembda_2\geq\cdots\geq\langlembda_r$, and $r$ is the Schmidt rank of $|\psi\rangle$.
In 1999, Vidal proposed an entanglement monotone in Ref.~\cite{Vidal1999pral}, i.e.,
\begin{equation}gin{eqnarray}
E_k\left( |\psi\rangle\right) =\sum\limits_{i=k}^r\langlembda_i^2,\quad k\geq2.
\end{eqnarray}
In particular,
\begin{equation}gin{eqnarray}
E_2\left( |\psi\rangle\right) =\sum\limits_{i=2}^r\langlembda_i^2=1-\langlembda_1^2=1-\|\rho^A\|,
\end{eqnarray}
where $\rho^A={\rm Tr}_B|\psi\rangle\langle\psi|$, $\|\cdot\|$ is the operator norm, i.e.,
$\|X\|=\sup_{|\psi\rangle}\|A|\psi\rangle\|$.
Hereafter, we call $E_2$ the \textit{partial-norm of entanglement} in the sense that $1-\|\rho^A\|$ counts for only a portion of the norm $\|\rho^A\|$ for the qubit case.
Obviously, $E_2\geq0$ for any $|\psi\rangle\in\mathcal{H}^{AB}$ and
$E_2(|\psi\rangle)=0$ if and only if $|\psi\rangle$ is separable.
For mixed state, $E_2(\rho)$ is defined by the convex-roof extension.
Generally,
$\|A+B\|=\|A\|+\|B\|$ but $A\neq\alpha B$ for hermitian operators $A$ and $B$,
so this reduced function $h(\rho)=1-\|\rho\|$ is not strictly concave.
We next illustrate with counter-examples that $E_2$ is not monogamous.
\begin{equation}gin{theorem}
$E_2$ is not monogamous.
\end{theorem}
Let
\begin{equation}gin{eqnarray}x
|\psi_0\rangle^{AB}&=&a_0|0\rangle^A|0\rangle^B+a_1|1\rangle^A|1\rangle^B+a_2|2\rangle^A|2\rangle^B,\\
|\psi_1\rangle^{AB}&=&a'_0|0\rangle^A|3\rangle^B+a'_1|1\rangle^A|2\rangle^B+a'_2|2\rangle^A|1\rangle^B
\end{eqnarray}x
with $a_0^2={a'}_0^2\geq\frac12$, $a_1'a_2\neq a_1a_2'$, $\sum_i a_i^2=\sum_i {a'}_i^2=1$, $a_0>a_1\geq a_2$, $a'_0>a'_1\geq a'_2$,
and
\begin{equation}gin{eqnarray}\langlebel{eg3}
|\Phi\rangle=\frac{1}{\sqrt{2}}\left(|\psi_0\rangle^{AB}|0\rangle^C+|\psi_1\rangle^{AB}|1\rangle^C\right).
\end{eqnarray}
After tracing over subsystems we are left with
\begin{equation}gin{eqnarray}x
\rho^{AB}&=&\frac12\left(|\psi_0\rangle\langle\psi_0|^{AB}+|\psi_1\rangle\langle\psi_1|^{AB}\right),\\
\rho^{AC}&=&\frac12\left[\left( a_0^2|0\rangle\langle0|^A+a_1^2|1\rangle\langle1|^A+a_2^2|2\rangle\langle2|^A\right)\otimes|0\rangle\langle0|^C\right. \\
&&+\left( {a'}_0^2|0\rangle\langle0|^A+{a'}_1^2|1\rangle\langle1|^A+{a'}_2^2|2\rangle\langle2|^A\right)\otimes|1\rangle\langle1|^C \\
&&+ \left( a_1a'_2|1\rangle\langle2|^A+a_2a'_1|2\rangle\langle1|^A\right)\otimes|0\rangle\langle1|^C\\
&&+ \left. \left( a_1a'_2|2\rangle\langle1|^A+a_2a'_1|1\rangle\langle2|^A\right)\otimes|1\rangle\langle0|^C
\right],\\
\rho_0^A&=&a_0^2|0\rangle\langle0|^A+a_1^2|1\rangle\langle1|^A+a_2^2|2\rangle\langle2|^A,\\
\rho_1^A&=&a_0^2|0\rangle\langle0|^A+{a'}_1^2|1\rangle\langle1|^A+{a'}_2^2|2\rangle\langle2|^A
\end{eqnarray}x
and
\begin{equation}gin{eqnarray}x
\rho^A&=&a_0^2|0\rangle\langle0|^A+\frac12({a}_1^2+{a'}_1^2)|1\rangle\langle1|^A\\
&&\quad\quad+\frac12({a}_2^2+{a'}_2^2)|2\rangle\langle2|^A,
\end{eqnarray}x
where $\rho_{0,1}^A={\rm Tr}_b|\psi_{0,1}\rangle\langle\psi_{0,1}|^{AB}$.
From here it follows that $E_2(|\Phi\rangle^{A|BC})=1-a_0^2$.
We next show that $E_2(\rho^{AB})=E_2(|\Phi\rangle^{A|BC})$ but $E_2(\rho^{AC})>0$, namely,
$E_2$ is not monogamous.
For any pure state ensemble of $\rho^{AB}=\sum_ip_i|\phi_i\rangle\langle\phi_i|^{AB}$,
we have
\begin{equation}gin{eqnarray}x
p_i|\phi_i\rangle^{AB}=\frac{1}{\sqrt{2}}\left( u_{i0}|\psi_0\rangle^{AB}+u_{i1}|\psi_1\rangle^{AB}\right)
\end{eqnarray}x
for any $i$, where $|u_{i0}|^2+|u_{i1}|^2\leq 1$, which yields
the largest eigenvalue of $\sigma_i^A={\rm Tr}_B|\phi_i\rangle\langle\phi|^{AB}$
is always $a_0$. Thus
\begin{equation}gin{eqnarray}x
E_2(\rho^{AB})=E_2(|\Phi\rangle^{A|BC})=1-a_0^2
\end{eqnarray}x as desired.
On the other hand,
we let $|x\rangle^{AC}=a_1|1\rangle^A|0\rangle^C+a_2'|2\rangle^A|1\rangle^C$
and $|y\rangle^{AC}=a_2|2\rangle^A|0\rangle^C+a_1'|1\rangle^A|1\rangle^C$,
then
\begin{equation}gin{eqnarray}x
\rho^{AC}&=&a_0^2|0\rangle\langle0|^A\otimes(|0\rangle\langle0|^C+|1\rangle\langle1|^C)\\
&&+\frac12|x\rangle\langle x|^{AC}+\frac12|y\rangle\langle y|^{AC}.
\end{eqnarray}x
It is easy to see that $\rho_{AC}^{T_A}$ is not positive whenever $a_1'a_2\neq a_1a_2'$,
and thus $E_2(\rho^{AC})>0$, here $T_X$
denotes the partial transpose transformation with respect to
the subsystem $X$.
If the reduced subsystem is two-dimensional, we consider the three-qubit case with no loss of generality.
Any pure state $|\psi\ranglengle$ in $\mathbb{C}^{2}\otimesimes \mathbb{C}^{2}\otimesimes \mathbb{C}^{2} $ can be expressed as~\cite{Acin2000prl}
\begin{equation}gin{eqnarray}x
|\psi\ranglengle^{ABC} &=&\langlembda_{0}|000\ranglengle+\langlembda_{1}e^{{\rm i}\varphi}|100\ranglengle+\langlembda_{2}|101\ranglengle\\
&&+\langlembda_{3}| 110\ranglengle+\langlembda_{4}|111\ranglengle
\end{eqnarray}x
up to local unitary transformation,
where $\langlembda_{i}\geq0$, $0 \leq \varphi \leq \pi$, $\sum_{i}{\langlembda_{i}}^{2}=1$.
The reduced states
$\rho^{AB}=p|x_1\rangle\langle x_1|+(1-p)|x_2\rangle\langle x_2|$ with
$\sqrt{p}|x_1\rangle=\langlembda_2|10\rangle+\langlembda_4|11\rangle$ and $\sqrt{(1-p)}|x_2\rangle=\langlembda_0|00\rangle+\langlembda_1e^{{\rm i}\varphi}|10\rangle+\langlembda_3|11\rangle$,
and
\begin{equation}gin{eqnarray}x
\rho^{A}=
\left(
\begin{equation}gin{array}{cc}
\langlembda_{0}^2 & \langlembda_0\langlembda_1e^{-{\rm i}\varphi}\\
\langlembda_0\langlembda_1e^{{\rm i}\varphi}& \langlembda_1^2+\langlembda_2^2+\langlembda_3^2+\langlembda_4^2
\end{array}
\right).
\end{eqnarray}x
It is straightforward that
(1) $|\psi\rangle$ is genuinely entangled if and only if
$\langlembda_0> 0$, $\langlembda_2^2+\langlembda_4^2>0$ and $\langlembda_3^2+\langlembda_4^2>0$,
(2) $\rho^{AB}$ is separable iff $\langlembda_3=0$,
and (3) $\rho^{AC}$ is separable iff $\langlembda_2=0$.
If $E_2(|\psi\rangle^{A|BC})=E_2(\rho^{AB})$, then
\[E_2(\rho^{AB})=\sum_kp_kE_2(|\phi_k\rangle)\] for any
$\rho^{AB}=\sum_kp_k|\phi_k\rangle\langle\phi_k|$ according to Corollary 5 in Ref.~\cite{GG}.
This leads to
the minimal eigenvalue of
\begin{equation}gin{eqnarray}x
(1-p){\rm Tr}_B|x_2\rangle\langle x_2|=\left(
\begin{equation}gin{array}{cc}
\langlembda_{0}^2 & \langlembda_0\langlembda_1e^{-{\rm i}\varphi}\\
\langlembda_0\langlembda_1e^{{\rm i}\varphi}& \langlembda_1^2+\langlembda_3^2
\end{array}
\right)
\end{eqnarray}x
coincides with that of $\rho^A$, which yields either
$\langlembda_2=\langlembda_4=0$, or $\langlembda_1=0$ and $\langlembda_0\leq \langlembda_3$. That is, $\rho^{AC}$ could be entangled. Therefore
$E_2$ is still not monogamous whenever the reduced subsystem is two dimensional.
Let $\langlembda_{\min}$ be the minimal positive Schmidt number of $|\psi\rangle$.
We define
\begin{equation}gin{eqnarray}
E_{\min}(|\psi\rangle)=\begin{equation}gin{cases}~\langlembda_{\min}^2,& \langlembda_{\min}<1,\\
~0, &\langlembda_{\min}=1
\end{cases}
\end{eqnarray}
for pure state and then define by means of the convex-roof extension for mixed state.
Denoting by
\begin{equation}gin{eqnarray}
\|\rho\|_{\min}=\begin{equation}gin{cases}~\langlembda_{\min}^2,& \langlembda_{\min}<1,\\
~0, &\langlembda_{\min}=1.
\end{cases}
\end{eqnarray}
it turns out that
\begin{equation}gin{eqnarray}x
E_{\min}(|\psi\rangle)=h(\rho^A)=\|\rho^A\|_{\min}.
\end{eqnarray}x
We call $E_{\min}$ the \textit{minimal partial-norm of entanglement}, which reflects as the minimal case of the partial-norm.
It is clear that $E_{\min}(\rho)=0$ iff $\rho$ is separable. Let $\delta(\rho)=(\delta_1, \delta_2, \dots, \delta_d)$ for any state $\rho\in\mathcal{S}$ with $\dim\mathcal{H}=d$, where $\delta_i$s are the eigenvalues,
$\delta_1\geq\delta_2\geq\dots\geq\delta_d$.
The concavity of $h$ is clear since
\begin{equation}gin{eqnarray}x
\delta[t\rho+(1-t)\sigma]\prec t\delta(\rho)+(1-t)\delta(\sigma),
\end{eqnarray}x
which implies
\begin{equation}gin{eqnarray}x
\|t\rho+(1-t)\sigma\|_{\min}\geq t\|\rho\|_{\min}+(1-t)\|\sigma\|_{\min},
\end{eqnarray}x
where ``$\prec$'' is the majorization relation between probability distributions.
\begin{equation}gin{theorem}
$E_{\min}$ is an entanglement monotone.
\end{theorem}
\begin{equation}gin{proof}
We only need to prove $E_{\min}$ is an entanglement monotone on pure states.
In fact, if $E_{\min}$ is an entanglement monotone on pure states,
then, for any given mixed state $\rho\in\mathcal{S}^{AB}$ with $E_{\min}(\rho)=\sum_ip_iE_{\min}(|\psi_i\rangle)$,
we have
\begin{equation}gin{eqnarray}x
E_{\min}(\rho)&=&\sum_ip_iE_{\min}(|\psi_i\rangle)\\
&\geq&\sum_ip_i\left(\sum_kq_kE_{\min}(\sigma_{i,k}) \right)\\
&=&\sum_kq_k\left(\sum_ip_iE_{\min}(\sigma_{i,k}) \right)\\
&\geq&\sum_kq_kE_{\min}(\sigma_{k}),
\end{eqnarray}x
where $\sigma_{k}=\sum_ip_i\sigma_{i,k}$, $\sigma_{i,k}$ is the $k$-th output under some stochastic LOCC $\{\Phi_k\}$.
According to the scenario in Ref.~\cite{Vidal02},
we only need to consider a family $\{\Phi_k\}$ consisting of completely positive linear maps
such that $\Phi_k(\rho)=p_k{\sigma_k}$,
where $\Phi_k(X)=(I^A\otimesimes M_k)X(I^A\otimesimes M_k^\dag)$ transforms pure states to some scalar multiple of pure states,
$\sum_kM_k^\dag M_k= I^B$. For any $|\psi\rangle^{AB}\in\mathcal{H}^{AB}$
with Schmidt decomposition $|\psi\rangle=\sum_{j=1}^r\langlembda_j|e_j\rangle^A|e_j\rangle^B$,
\begin{equation}gin{eqnarray}x
\Phi_k\left( |\psi\rangle\langle\psi|\right) &=&\sum_{ij}\langlembda_i\langlembda_j|e_i\rangle\langle e_j|^A\otimes M_k|e_i\rangle\langle e_j|^BM_k^\dag\\
&=&p_k\sigma_k,
\end{eqnarray}x
this reveals
\begin{equation}gin{eqnarray}x
&&\sum_k\Phi_k(|\psi\rangle\langle\psi|)\\
&=&\sum_{ij}\langlembda_i\langlembda_j|e_i\rangle\langle e_j|^A\otimes ( \sum_kM_k|e_i\rangle\langle e_j|^BM_k^\dag)\\
&=&\sum_kp_k\sigma_k.
\end{eqnarray}x
By $\sum_kp_k\sigma_k^A=\rho^A={\rm Tr}_B|\psi\rangle\langle\psi|$ we have
\begin{equation}gin{eqnarray}x
\|\rho^A\|_{\min}\geq \sum_kp_k\|\sigma_k^A\|_{\min}.
\end{eqnarray}x
That is, $E_{\min}(|\psi\rangle)\geq\sum_kp_kE_{\min}(\sigma_k)$,
which completes the proof.
\end{proof}
The argument above yields that, for any non-negative function defined as
$E(|\psi\rangle)=h(\rho^A)$, $E_F$ is an entanglement monotone if and only if $h$ is concave.
Moreover, we can conclude the following.
\begin{equation}gin{pro}{\langlebel{pro1}}
Let $E$ be an entanglement measure with the reduced function defined as Eq.~\eqref{h}.
If $E$ is an entanglement monotone, then the reduced function is concave and $E_F$ is an entanglement monotone.
\end{pro}
\begin{equation}gin{proof}
Let $\rho$ and $\sigma$ be any given two states in $\mathcal{S}^A$, $0\leq t\leq1$.
Taking $|\psi\rangle^{AB}$ and $|\phi\rangle^{AB}$ in $\mathcal{H}^{AB}$ such that
$\rho={\rm Tr}_B|\psi\rangle\langle\psi|^{AB}$ and $\sigma={\rm Tr}_B|\phi\rangle\langle\phi|^{AB}$,
we let
\begin{equation}gin{eqnarray}x
|\Psi\rangle^{ABC}=\sqrt{t}|\psi\rangle^{AB}|0\rangle^{C}+\sqrt{1-t}|\phi\rangle^{AB}|1\rangle^{C}
\end{eqnarray}x
be a pure state in $\mathcal{H}^{ABC}$.
Consider a LOCC $\{I^A\otimes I^B\otimes |0\rangle\langle0|^C, I^A\otimes I^B\otimes |1\rangle\langle1|^C\}$ acting on $|\Psi\rangle^{ABC}$, we obtain
the output
\begin{equation}gin{eqnarray}x
\left\lbrace t|\psi\rangle\langle\psi|^{AB}\otimes|0\rangle\langle0|^C, (1-t)|\phi\rangle\langle\phi|^{AB}\otimes|1\rangle\langle1|^C\right\rbrace,
\end{eqnarray}x
where $I^{A,B}$ is the identity operator acting on $\mathcal{H}^{A,B}$.
This leads to
\begin{equation}gin{eqnarray}x
E(|\Psi\rangle^{A|BC})\geq tE(|\psi\rangle^{AB}|0\rangle^C)+(1-t)E(|\phi\rangle^{AB}|1\rangle^C)
\end{eqnarray}x
since $E$ is an entanglement monotone, which is equivalent to
\begin{equation}gin{eqnarray}x
h(t\rho+(1-t)\sigma)\geq th(\rho)+(1-t)h(\sigma),
\end{eqnarray}x
that is, $h$ is concave. This completes the proof.
\end{proof}
By Proposition~\ref{pro1}, the convex-roof extension of the negativity $N$~\cite{Lee} is an entanglement monotone since $N$ is an entanglement monotone, where
$N$ is defined as~\cite{Vidal02}
$N(\rho)=\sum_i\mu_i$ with $\mu_i$s are the eigenvalues of the negative part of $\rho^{T_A}$. Note here that, in Ref.~\cite{Lee}, there is a gap in the proof of the concavity of the reduced function: the second inequality of the last part in page 2 is wrong since
${|\phi_k\ranglengle}$ is not necessarily a basis (i.e., it is just an orthogonal set but not complete) in general.
By now, except for the convex-roof extended negativity $N_F$~\cite{Lee}, all the reduced functions of convex-roof extended entanglement monotones are shown to be strictly concave. We show that the reduced function of $N_F$, denoted by $h_N$ is strictly concave as well.
We assume to obtain a contradiction that $h_N$ is not strictly concave.
Then there exists $\rho^A=p\rho_1^A+(1-p)\rho_2^A\in\mathcal{S}^A$ with ${\rm spec}(\rho_1^A)\neq{\rm spec}(\rho_2^A)$, but
$h_N(\rho^A)=ph_N(\rho_1^A)+(1-p)h_N(\rho_2^A)$, here ${\rm spec}(X)$ denotes the spectrum of $X$.
Let
\begin{equation}gin{eqnarray}x
\rho^{AB}=p|\psi_1\rangle\langle\psi_1|^{AB}+(1-p)|\psi_2\rangle\langle\psi_2|^{AB}
\end{eqnarray}x
with $|\psi_i\rangle^{AB}=\sum_j\langlembda_{ij}|e_{ij}\rangle^A|e_{ij}\rangle^B$ is the Schmidt decomposition of $|\psi_i\rangle^{AB}$, $i=1$, 2,
where ${\rm Tr}_B|\psi_i\rangle\langle\psi_i|^{AB}=\rho_i^A$, and
\begin{equation}gin{eqnarray}x
\langle e_{ij}|e_{kl}\rangle^{B}=\delta_{ik}\delta_{jl}.
\end{eqnarray}x
\if false We take
\begin{equation}gin{eqnarray}x
|\varPsi\rangle^{ABC}=\sqrt{p}|\psi_1\rangle^{AB}|0\rangle^C+\sqrt{1-p}|\psi_2\rangle^{AB}|1\rangle^C,
\end{eqnarray}x
then for any ensemble of $\rho^{AB}=\sum_kq_k|\phi_k\rangle\langle\phi_k|^{AB}$,
\begin{equation}gin{eqnarray}x
&&\sum_kq_kN(|\phi_k\rangle^{AB})
=\sum_kq_kh_N(\rho_k^A)\\
&\geq&\sum_k\left[ p|u_{k1}|^2h_N(\rho_1^A)+(1-p)|u_{k2}|^2h_N(\rho_2^A)\right] \\
&=&ph_N(\rho_1^A)+(1-p)h_N(\rho_2^A),
\end{eqnarray}x
where
\begin{equation}gin{eqnarray}x
\sqrt{q_k}|\phi_k\rangle^{AB}=u_{k1}\sqrt{p}|\psi_1\rangle^{AB}+u_{k2}\sqrt{1-p}|\psi_2\rangle^{AB}.
\end{eqnarray}x
\fi
It turns out that
$N(\rho^{AB})=N(|\varPsi\rangle^{A|BC})$.
But $|\varPsi\rangle^{ABC}$ does not admit the form $|\psi\rangle^{AB_1}|\psi\rangle^{B_2C}$ up to some local unitary operation,
where $B_1B_2$ means $\mathcal{H}^B$ has a subspace isomorphic to $\mathcal{H}^{B_1}\otimesimes\mathcal{H}^{B_2}$ and up to local unitary on system $B_1B_2$, which contradicts with Theorem 3 in~\cite{Hehuan}.
Thus $h_N$ is strictly concave.
That is, all the reduced functions of the monogamous entanglement monotones so far are strictly concave.
We now go back to discuss the monogamy of $E_{\min}$. Clearly, if the reduced system is two-dimensional,
then $E_{\min}=E_2$, which is not monogamous.
For higher dimensional case,
we consider a pure state as in Eq.~\eqref{eg3} just by replacing $a_0^2={a'}^2_0\geq\frac12$, $a_0>a_1\geq a_2$, $a'_0>a'_1\geq a'_2$,
with $a_0=a'_0$, $a_1\geq a_2>a_0$, $a'_1\geq a'_2>a'_0$,
from which one can conclude that $E_{\min}$ is not monogamous.
However, $E_{\min}$ does not achieve the maximal value for the maximally entangled state.
For making up the disadvantages, we can define
\begin{equation}gin{eqnarray}
E'_{\min}(|\psi\rangle)=\begin{equation}gin{cases}~\langlembda_{\min}^2S_r(|\psi\rangle),& \langlembda_{\min}<1,\\
~0, &\langlembda_{\min}=1,
\end{cases}
\end{eqnarray}
for pure state and then define by means of the convex-roof extension for mixed state,
where $S_r(|\psi\rangle)$ denotes the Schmidt rank of $|\psi\rangle$. We call it the \textit{reinforced minimal partial-norm of entanglement}.
$E'_{\min}$ is equal to $2E_{\min}$
for any $2\otimes n$ state.
In such a case, $E'_{\min}$ reaches the maximal quantity for the maximally entangled state but not only for these states.
In addition, it is easy to follow that $E'_{\min}$ is also an entanglement monotone and is not monogamous.
Let $|\psi\rangle$, $|\phi\rangle$, $|\varphi\rangle$, $|\xi\rangle$, and $|\zeta\rangle$ be
pure states with the reduced states, respectively, are
$ \mathrm{diag}(2/3, 1/6, 1/6)$, $ \mathrm{diag}(1/3, 1/3, 1/3)$,
$ \mathrm{diag}(3/5, 2/5, 0)$, $ \mathrm{diag}(2/5, 2/5, 15)$,
and $ \mathrm{diag}(4/5, 1/5, 0)$.
Then we arrive at
\begin{equation}gin{eqnarray}x
E_2(|\varphi\rangle)<E_2(|\phi\rangle)
~~\mbox{but} ~~
E_{\min}(|\phi\rangle)<E_{\min}(|\varphi\rangle),
\end{eqnarray}x
\begin{equation}gin{eqnarray}x
E_2(|\varphi\rangle)<E_2(|\xi\rangle)
~~\mbox{but} ~~ E'_{\min}(|\xi\rangle)<E'_{\min}(|\varphi\rangle),
\end{eqnarray}x
\begin{equation}gin{eqnarray}x E_{\min}(|\psi\rangle)<E_{\min}(|\zeta\rangle)
~~\mbox{but} ~~ E'_{\min}(|\zeta\rangle)<E'_{\min}(|\psi\rangle).
\end{eqnarray}x
That is, these three measures are not equivalent to each other.
The maximal value of $E_2$ is $(d-1)/d$.
We thus, in order to get a normalized measure, replace $E_2$ by $dE_2/(d-1)$. Hereafter the notation $E_2$ refers to the normalized one.
For the $2\otimes n$ system, $E_2$ coincides with $E'_{\min}$ but not for $m\otimes n$ system with $2<m\leq n$.
For any pure state $|\psi\rangle$ with Schmidt numbers $p$ and $1-p$ in $2\otimes n$ system, $p\leq1/2$, it is immediate that
\begin{equation}gin{eqnarray}x
E_2(|\psi\rangle)=2E_{\min}(|\psi\rangle)=E'_{\min}(|\psi\rangle)=2p,
\end{eqnarray}x and
\begin{equation}gin{eqnarray}x
\tau(|\psi\rangle)=2p(1-p).
\end{eqnarray}x Here, $\tau$ is the tangle, which is defined as the square of concurrence, i.e.,
$\tau(|\psi\rangle)=2(1-{\rm Tr}\rho_A^2)$, $\rho_A={\rm Tr}_B|\psi\rangle\langle\psi|$.
That is $E_2\geq\tau$ and both of them are monotonically increasing with $0\leq p\leq1/2$.
We now compute these three entanglement monotones for the qutrit-qutrit pure state and then compare them with tangle.
It can be easily calculated since they are homogeneous. We consider $|\psi\rangle\in\mathbb{C}^3\otimes\mathbb{C}^3$ with Schmidt numbers $(\sqrt{2/3-t}, \sqrt{1/3}, \sqrt{t})$, $0\leq t\leq1/3$,
and $|\phi\rangle\in\mathbb{C}^3\otimes\mathbb{C}^3$ with Schmidt numbers $(\sqrt{p}, \sqrt{q}, \sqrt{1-p-q})$ for illustration purposes, $p\geq q$.
The behaviours of theses quantities for these two states are depicted in Fig.~\ref{fig1} and Fig.~\ref{fig2}, respectively. In the case of $t=0$, $E'_{\min}(|\psi\rangle)=2/3=2E_{\min}(|\psi\rangle)$ and $\tau(|\psi\rangle)=8/9$.
For the case of $p+q=1$, $E_2(|\psi\rangle)=3q/2<E'_{\min}(|\psi\rangle)=2q$. That is, $E_{\min}$ and $E'_{\min}$ are not continuous and are not equivalent.
\begin{equation}gin{figure}
\hspace{-3mm}\includegraphics[width=80mm]{fig1.eps}
\caption{\langlebel{fig1}(color online). Comparing $E_2$, $E'_{\min}$ with the tangle $\tau$ for $|\psi\rangle$ with Schmidt numbers $(\sqrt{2/3-t}, \sqrt{1/3}, \sqrt{t})$, $0\leq t\leq1/3$.}
\end{figure}
The upper bounds of these quantities can be easily derived.
Let $\rho$ be a state in $\mathcal{S}^{AB}$, and $E_2(\rho)=\sum_ip_iE_2(|\psi_i\rangle)$,
then
\begin{equation}gin{eqnarray}x &&E_2(\rho)=\sum_ip_iE_2(|\psi_i\rangle)\\
&=&\frac{d}{d-1}\sum_ip_i\left(1- \|\rho_i^A\|\right) \\
&=&\frac{d}{d-1}\left[ 1-\left(\sum_ip_i\|\rho_i^A\|\right)\right] \\
&\leq& \frac{d}{d-1}\left( 1-\|\sum_ip_i\rho_i^A\|\right) \\
&=&\frac{d}{d-1}\left( 1-\|\rho^A\|\right).
\end{eqnarray}x
That is
\begin{equation}gin{eqnarray}
E_2(\rho)\leq\frac{d}{d-1}\min\{1-\|\rho^A\|, 1-\|\rho^B\|\}.
\end{eqnarray}
Analogously,
\begin{equation}gin{eqnarray}
E_{\min}(\rho)\leq \min\{\|\rho^A\|_{\min}, \|\rho^B\|_{\min}\}
\end{eqnarray}
and
\begin{equation}gin{eqnarray}
E'_{\min}(\rho)\leq \min\{r_A\|\rho^A\|_{\min}, r_B\|\rho^B\|_{\min}\},
\end{eqnarray}
where $r_{A,B}$ is the rank of $\rho^{A,B}$.
\begin{equation}gin{figure}
\hspace{-2mm}\includegraphics[width=85mm]{fig2.eps}
\caption{\langlebel{fig2}(color online). Comparing $E_2$, $E'_{\min}$ with the tangle $\tau$ for $|\phi\rangle$ with Schmidt numbers $(\sqrt{p}, \sqrt{q}, \sqrt{1-p-q})$, $p+q<1$. }
\end{figure}
When $k\geq3$, $E_k$ is not a faithful entanglement monotone, and it is not monogamous either.
Another entanglement measure that lack of investigating the monogamy is the Schmidt number, which is regarded as a universal entanglement measure~\cite{Sperling2011}, defined by~\cite{Terhal1999}
\begin{equation}gin{eqnarray}
S_r(\rho)=\min_{p_i,|\psi_i\rangle}\max_{|\psi_i\rangle}S_r(|\psi_i\rangle),
\end{eqnarray}
where the minimum is taken over all decomposition $\rho=\sum_ip_i|\psi_i\rangle\langle\psi_i|$.
It is also not monogamous since both the Schmidt number of $|W\rangle=\frac{1}{\sqrt{3}}(|100\rangle+|010\rangle+|001\rangle)$ and that of its two reduced states are 2.
In addition, let $\rho^{T_A}$ be the partial transpose of $\rho$, one may consider the partial-norm of the negative part of $\rho^{T_A}$, $N\rho^-$. For example,
we take
\begin{equation}gin{eqnarray}
\hat{N}(\rho)=\|N\rho^-\|.
\end{eqnarray}
We call it \textit{partial negativity} hereafter.
Take $\rho=|\psi\rangle\langle\psi|$ with
$|\psi\rangle=\sum_j\langlembda_j|e_j\rangle^A|e_j\rangle^B$ as the
Schmidt decomposition of $|\psi\rangle$.
Then
$\hat{N}(|\psi\rangle)=\langlembda_1\langlembda_2$,
and the corresponding reduced function is
\begin{equation}gin{eqnarray}
\hat{h}(\rho^A)=\sqrt{\delta_1\delta_2},
\end{eqnarray}
where $\delta_1=\langlembda_1^2$, $\delta_2=\langlembda_2^2$.
$\hat{N}$ can still be regarded as a kind of partial norm as $\sqrt{\delta_1\delta_2}\leq\delta_1=\|\rho^A\|$, in other words, $\hat{N}$ is also a kind of partial norm of entanglement.
By definition, $|\psi\rangle^{ab}$ is separable if and only if it is separable, and
\begin{equation}gin{eqnarray}x
0<\hat{N}(\rho)\leq N(\rho)
\end{eqnarray}x
for any non-positive partial transpose state $\rho$.
A simple comparison between $\hat{N}$ and $E_2$, $E_{\min}$, $E'_{\min}$ are given in Fig.~\ref{fig1} and Fig.~\ref{fig2}, which indicate that they are not equivalent to each other.
For the two-qubit case, $2\hat{N}_F$ coincides with the $G$-concurrence~\cite{Gour2005pra}.
We conjecture that $\hat{h}$ is concave~\cite{Suppl}.
$\hat{h}$ is strictly concave on $\mathcal{S}(\mathcal{H})$ with $\dim\mathcal{H}=2$ since it reduced to an elementary symmetric function~\cite[p.~116]{Marshall}, but it is not true for the higher dimensional case.
In order to see this, we take
\begin{equation}gin{eqnarray}x
\rho=\left( \begin{equation}gin{array}{ccc}
1/3&0&0\\
0&1/3&0\\
0&0&1/3
\end{array}\right),\quad
\sigma=\left( \begin{equation}gin{array}{ccc}
1/2&0&0\\
0&1/2&0\\
0&0&0
\end{array}\right),
\end{eqnarray}x
which yields $\hat{h}(\frac12\rho+\frac12\sigma)=\frac12\hat{h}(\rho)+\frac12\hat{h}(\sigma)$.
We now assume that $\hat{N}$ is an entanglement monotone, then we can conclude the following.
\begin{equation}gin{theorem}
$\hat{N}$ and $\hat{N}_F$ are not monogamous whenever the reduced subsystem has dimension greater than 2.
\end{theorem}
We show this statement by a counter-example. Let
\begin{equation}gin{eqnarray}\langlebel{eg2}
|\varOmega\rangle^{ABC}&=&\langlembda_0|0\rangle^A|00\rangle^{BC}+\langlembda_1|1\rangle^A|10\rangle^{BC}\nonumber\\
&&+\langlembda_2|2\rangle^A|11\rangle^{BC},~~~\langlembda_0\geq \langlembda_1\geq \langlembda_2>0,\quad
\end{eqnarray}
it turns out that
\begin{equation}gin{eqnarray}x
\hat{N}(|\varOmega\rangle^{A|BC})=\hat{N}(\rho^{AB})=\langlembda_0\langlembda_1
\end{eqnarray}x
but
\begin{equation}gin{eqnarray}x
\hat{N}(\rho^{BC})=\langlembda_1\langlembda_2>0.
\end{eqnarray}x
That is, $\hat{N}$ is not monogamous.
Moreover, from this example, we can also get $\hat{N}_F$ is not monogamous either in light of $\hat{N}\leq\hat{N}_F$.
Analogous to that of the logarithmic negativity, we define the \textit{logarithmic partial negativity} by
\begin{equation}gin{eqnarray}
\hat{N}_l(\rho)=\log_2[\hat{N}(\rho)+1].
\end{eqnarray}
It is straightforward that $\hat{N}_l$ is not convex.
For any LOCC acting on $\rho^{ab}$ that leaves the output states $\{p_i\sigma_i\}$,
we have
\begin{equation}gin{eqnarray}x
\sum_ip_i\hat{N}_l(\sigma_i)&=&\sum_ip_i\log_2x_i\leq\log_2\sum_ip_ix_i\\
&\leq&\log_2[\hat{N}(\rho)+1]=\hat{N}_l(\rho)
\end{eqnarray}x
since $\log_2$ is concave and $\hat{N}$ is non-increasing on average under LOCC by assumption,
where $x_i=\hat{N}(\sigma_i)+1$.
Therefore it is also an entanglement monotone and is not monogamous (hereafter, we still call it an entanglement monotone even though it is not convex as in Ref.~\cite{Plenio2005prl}).
In sum, for the sake of distinguishing these entanglement monotones so far in the sense of monogamy law, we suggest the term \textit{informationally complete entanglement monotone}, which means that its reduced function is related to all its eigenvalues. For example, the entanglement of formation is informationally complete since the von Neumann entropy is defined on all of the eigenvalues which include all the information of the entanglement, but $E_2$, $E_{\min}$, $E'_{\min}$, $\hat{N}$, $\hat{N}_F$, and $\hat{N}_l$ are not the case except for the two-dimensional case since they just capture ``partial information'' of the entanglement.
The worst one is the Schmidt number, which reflects the least information of the entanglement, and of course is not informationally complete.
Our discussion supports that,
for an entanglement monotone $E_F$ with reduced function $h$,
$E_F$ is monogamous if and only if it is informationally complete, and in turn,
iff $h$ is strictly concave (the ``if'' part is proved~\cite{GG2019}).
So the axiomatic definition of an entanglement monotone should be improved as follows.
Let $E$ be a nonnegative function on $\mathcal{S}^{AB}$ with $E(|\psi\rangle)=h(\rho^A)$ for pure state. We call $E$ a \textit{strict entanglement monotone} if {(i)} $E(\sigma^{AB})=0$ for any
separable density matrix $\sigma^{AB}\in\mathcal{S}^{AB}$, {(ii)}
$E$ behaves monotonically decreasing under LOCC on average, and {(iii)} the reduced function $h$ is strictly concave.
We use henceforth the term strict entanglement monotone to distinguish it from the previous entanglement monotone.
With such a spirit, except for $E_2$, $E_{\rm min}$, $E'_{\rm min}$, $\hat{N}$, $\hat{N}_F$, $\hat{N}_l$ and the Schmidt number, all the previous entanglement monotones that are shown to be monogamous or monogamous on pure states are strict entanglement monotones, these include the original entanglement of formation, negativity, the squashed entanglement~\cite{Christandl2004jmp}, the convex-roof extension of negativity, tangle, concurrence, the relative entropy of entanglement~\cite{Vedral1998pra}, $G$-concurrence, the Tsallis entropy of entanglement, the conditional entanglement of mutual information~\cite{Yang2008prl}, and the entanglement measures induced by the fidelity distances, etc.
However, it still remains unknown that whether or not the non convex-roof extended strict entanglement monotones in literature are monogamous in addition to the squashed entanglement. We conjecture that all the informationally complete entanglement monotones are monogamous.
As a by-product, we can obtain new coherence measures from the reduced function $h$ of $E_2$, $E_{\rm min}$ and $E'_{\rm min}$, respectively. Let
\begin{equation}gin{eqnarray}
C_h(|\psi\rangle)=h(x_0, x_1, \dots, x_{d-1})
\end{eqnarray}
for pure state $|\psi\rangle=\sum_ix_i|i\rangle$ under the reference basis $\{|i\rangle\}_{i=0}^{d-1}$,
and by the convex-roof extension for mixed state, i.e.,
\begin{equation}gin{eqnarray}x
C_h(\rho)=\min_{p_j,|\psi_j\rangle}\sum_jp_jC_h(|\psi_j\rangle),
\end{eqnarray}x
where the minimum is taken over all decomposition $\rho=\sum_jp_j|\psi_j\rangle\langle\psi_j|$.
It turns out that (i) $h(1, 0, \cdots, 0)=0$, (ii) $h(\pi(x_0, x_1, \dots, x_{d-1}))=h(x_0, x_1, \dots, x_{d-1})$
for any permutation $\pi$ and any $(x_0, x_1, \dots, x_{d-1})$, and (iii) $h$ is concave.
This reveals that $C_h$ is a well-defined coherence measure according to Theorem 1 in Ref.~\cite{Du2015qic}.
Also notice here that, the associated function $h$ of all the previous coherence measures defined by means of the convex-roof extension are strictly concave, which are different from $C_h$.
\begin{equation}gin{acknowledgements}
This work is supported by the National Natural Science Foundation of
China under Grant Nos.~11971277 and 12001158, the Fund Program for the Scientific Activities of Selected
Returned Overseas Professionals in Shanxi Province under Grant No.~20220031, and the Scientific Innovation Foundation of the Higher
Education Institutions of Shanxi Province under Grant No.~2019KJ034.
\end{acknowledgements}
\begin{equation}gin{thebibliography} {99}
\bibitem{Nielsen} M.~A.~Nielsen, I.~L.~Chuang, \textit{Quantum Computatation and
Quantum Information}, (Cambridge University Press, Cambridge, 2000).
\bibitem{Schrodinger1935} E. Schr\"{o}dinger, Discussion of Probability Relations between Separated Systems,
Proc. Cambridge Philos. Soc. \textbf{31}, 555 (1935).
\bibitem{Schrodinger1936} E. Schr\"{o}dinger,
Probability relations between separated systems,
Proc. Cambridge Philos. Soc. \textbf{32}, 446
(1936).
\bibitem{Bennett1993teleporting} C. H. Bennett, G. Brassard, C. Cr\'{e}peau, R. Jozsa,
A. Peres, and W. K. Wootters,
Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,
Phys. Rev. Lett. \textbf{70}, 1895 (1993).
\bibitem{Zhang2006experimental} Q. Zhang, A. Goebel, C. Wagenknecht, Y.-A. Chen,
B. Zhao, T. Yang, A. Mair, J. Schmiedmayer, and J.-W. Pan,
Experimental quantum teleportation of a two-qubit composite system,
Nat. Phys. \textbf{2}, 678 (2006).
\bibitem{Bennett1992communication} C. H. Bennett and S. J. Wiesner,
Communication via one-and two-particle operators on Einstein-Podolsky-Rosen states,
Phys. Rev. Lett. \textbf{69}, 2881 (1992).
\bibitem{Bennett1996prl} C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters,
Purification of noisy entanglement and faithful teleportation via noisy channels,
Phys. Rev. Lett. \textbf{76}, 722 (1996).
\bibitem{Terhal2004} B. Terhal,
Is entanglement monogamous?
IBM J. Res. Dev. \textbf{48}, 71 (2004).
\bibitem{Osborne} T. J. Osborne and F. Verstraete,
General monogamy inequality for bipartite qubit entanglement,
Phys. Rev. Lett. \textbf{96}, 220503 (2006).
\bibitem{Horodecki2009} R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki,
Quantum entanglement,
Rev. Mod. Phys. \textbf{81}, 865 (2009).
\bibitem{Bennett2014} C. H. Bennett,
in \textit{Proceedings of the FQXi 4th International Conference, Vieques Island, Puerto Rico, 2014},
http://fqxi.org/conference/talks/2014.
\bibitem{Toner} B. Toner,
Monogamy of non-local quantum correlations,
Proc. R. Soc. A \textbf{465}, 59 (2009).
\bibitem{Seevinck} M. P. Seevinck,
Monogamy of correlations versus monogamy of entanglement,
Quantum Inf. Process. \textbf{9}, 273 (2010).
\bibitem{streltsov2012are} A. Streltsov, G. Adesso, M. Piani, D. Bru\ss,
Are general quantum correlations monogamous?
Phys. Rev. Lett. \textbf{109}, 050503 (2012).
\bibitem{Augusiak2014pra} R. Augusiak, M. Demianowicz, M. Paw\l{}owski, J. Tura, and A. Ac\'{\i}n,
Elemental and tight monogamy relations in nonsignaling theories,
Phys. Rev. A \textbf{90}, 052323 (2014).
\bibitem{Ma2011} X.-s. Ma, B. Dakic, W. Naylor, A. Zeilinger, and P.Walther,
Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems,
Nat. Phys. \textbf{7}, 399 (2011).
\bibitem{Brandao2013} F. G. S. L. Brandao and A.W. Harrow, in \textit{Proceedings
of the 45th Annual ACM Symposium on Theory of Computing,
2013}, http://dl.acm.org/citation.cfm?doid=2488608.2488718.
\bibitem{Garcia} A. Garc\'{\i}a-S\'{a}ez and J. I. Latorre,
Renormalization group contraction of tensor networks in three dimensions,
Phys. Rev. B \textbf{87}, 085130 (2013).
\bibitem{Lloyd} S. Lloyd and J. Preskill,
Unitarity of black hole evaporation in final-state projection models,
J. High Energy Phys. \textbf{08}, 126 (2014).
\bibitem{Pawlowski} M. Paw\l owski,
Security proof for cryptographic protocols based only on the monogamy of Bell's inequality violations,
Phys. Rev. A \textbf{82}, 032313 (2010).
\bibitem{Vidal2000} G. Vidal,
Entanglement monotone,
J. Mod. Opt. \textbf{47}, 355 (2000).
\bibitem{Plenio2005prl} M. B. Plenio,
Logarithmic negativity: a full entanglement monotone that is not convex,
Phys. Rev. Lett. \textbf{95}, 090503 (2005).
\bibitem{Vedral1998pra} V. Vedral and M. B. Plenio,
Entanglement measures and purification procedures,
Phys. Rev. A \textbf{57}, 1619 (1998).
\bibitem{Coffman} V.~Coffman, J.~Kundu, and W.~K.~Wootters,
Distributed entanglement,
Phys. Rev. A \textbf{61}, 052306~(2000).
\bibitem{Bai} Y.-K. Bai, Y.-F. Xu, and Z. D. Wang,
General monogamy relation for the entanglement of formation in multiqubit systems,
Phys. Rev. Lett. \textbf{113}, 100503 (2014).
\bibitem{Koashi} M. Koashi and A. Winter,
Monogamy of quantum entanglement and other correlations,
Phys. Rev. A \textbf{69}, 022309 (2004).
\bibitem{Luo2016pra} Y. Luo, T. Tian, L.-H. Shao, and Y. Li,
General monogamy of Tsallis $q$-entropy entanglement in multiqubit systems,
Phys. Rev. A \textbf{93}, 062340 (2016).
\bibitem{Dhar} H. S. Dhar, A. K. Pal, D. Rakshit, A. S. De, and U Sen,
Monogamy of quantum correlations-a review,
In Lectures on General Quantum Correlations and their Applications, pp. 23-64. Springer, Cham, 2017.
\bibitem{Hehuan} H. He and G. Vidal,
Disentangling theorem and monogamy for entanglement negativity,
Phys. Rev. A \textbf{91}, 012339 (2015).
\bibitem{GG} G. Gour and Y. Guo, Monogamy of entanglement without inequalities,
Quantum \textbf{2}, 81 (2018).
\bibitem{GG2019} Y. Guo and G. Gour, Monogamy of the entanglement of formation,
Phys. Rev. A \textbf{99}, 042305 (2019).
\bibitem{Bennett1996pra} C.~H. Bennett, D.~P.~DiVincenzo, J.~A.~Smolin, and W.~K.~Wootters,
Mixed-state entanglement and quantum error correction,
Phys. Rev. A \textbf{54}, 3824~(1996).
\bibitem{Rungta2003pra} P. Rungta and C. M. Caves,
Concurrence-based entanglement measures for isotropic states,
Phys. Rev. A \textbf{67}, 012307 (2003).
\bibitem{Rungta} P. Rungta, V. Bu\v{z}ek, C. M. Caves, M. Hillery, G. J. Milburn,
Universal state inversion and concurrence in arbitrary dimensions,
Phys. Rev. A \textbf{64}, 042315 (2001).
\bibitem{Gour2005pra} G. Gour. Family of concurrence monotones
and its applications. Phys. Rev. A \textbf{71}, 012318 (2005).
\bibitem{Kim2010pra} J. S. Kim,
Tsallis entropy and entanglement constraints in multiqubit systems,
Phys. Rev. A \textbf{81}, 062328 (2010).
\bibitem{Guo2020qip} Y. Guo, L. Zhang, and H. Yuan,
Entanglement measures induced by fidelity-based distances,
Quant. Inf. Process. \textbf{19}, 1-17 (2020).
\bibitem{Vidal1999pral} G. Vidal,
Entanglement of pure states for a single copy,
Phys. Rev. Lett. \textbf{83}, 1046 (1999).
\bibitem{Acin2000prl} A. Ac\'{\i}n \textit{et al}.,
Generalized Schmidt decomposition and classification of three-quantum-bit states,
Phys. Rev. Lett. \textbf{85}, 1560 (2000).
\bibitem{Vidal02} G.~Vidal and R.~F.~Werner,
Computable measure of entanglement,
Phys. Rev. A \textbf{65}, 032314~(2002).
\bibitem{Lee} S.~Lee, D.~P.~Chi, S.~D.~Oh, and J.~Kim,
Convex-roof extended negativity as an entanglement measure for bipartite quantum systems,
Phys. Rev. A \textbf{68}, 062304~(2003).
\bibitem{Sperling2011} J. Sperling and W. Vogel,
The Schmidt number as a universal entanglement measure,
Phys. Scr. \textbf{83}, 045002 (2011).
\bibitem{Terhal1999} B. M. Terhal and P. Horodecki,
Schmidt number for density matrices,
Phys. Rev. A \textbf{61},
040301(R) (1999).
\bibitem{Suppl} We checked by many examples that it is true. Especially, if $[\rho, \sigma]=0$, $\hat{h}(\rho/2+\sigma/2)\geq\hat{\rho}/2+\hat{h}(\sigma)/2$ is always true.
\bibitem{Marshall} A. W. Marshall, I. Olkin, and B. C. Arnold,
Inequalities: theory of majorization and its applications,
New York: Academic press, 2011.
\bibitem{Christandl2004jmp} M. Christandl and A. Winter,
``Squashed entanglement'': an additive entanglement measure,
J. Math. Phys. (N.Y.) \textbf{45}, 829 (2004).
\bibitem{Yang2008prl} D. Yang, M. Horodecki, and Z. D. Wang,
An additive and operational entanglement measure: conditional entanglement of mutual information,
Phys. Rev. Lett. \textbf{101}, 140501 (2008).
\bibitem{Du2015qic} S. Du, Z. Bai, and X. Qi,
Coherence measures and optimal conversion for coherent states,
Quant. Inf. \& Comput. \textbf{15} (15-16), 1307-1316 (2015).
\end{thebibliography}
\end{document} | math | 44,626 |
\begin{equation}gin{document}
\title{ Noise robustness of the nonlocality of entangled quantum states }
\author{Mafalda L. Almeida$^{1}$, Stefano Pironio$^{1}$, Jonathan Barrett$^2$, G\'eza T\'oth $^{1,3}$, and Antonio Ac\'\i n$^{1,4}$}
\affiliation{ $^1$ICFO-Institut de Ciencies Fotoniques, E-08860
Castelldefels, Barcelona, Spain\\
$^2$Perimeter Institute for Theoretical Physics, 31 Caroline Street N, Waterloo, Ontario, Canada N2L 2Y5\\
$^3$Research Institute for Solid State Physics and Optics,
P.O. Box 49, H-1525 Budapest, Hungary\\
$^4$ICREA-Instituci\'o Catalana de Recerca i Estudis Avan\c{c}ats, Lluis Companys 23, 08010 Barcelona, Spain
}
\date{\today}
\begin{equation}gin{abstract}
We study the nonlocal properties of states resulting from the
mixture of an arbitrary entangled state $\rho$ of two
$d$-dimensional systems and completely depolarized noise, with
respective weights $p$ and $1-p$. We first construct a local model
for the case in which $\rho$ is maximally entangled and $p$ at or
below a certain bound. We then extend the model to arbitrary $\rho$.
Our results provide bounds on the resistance to noise of the
nonlocal correlations of entangled states. For projective
measurements, the critical value of the noise parameter $p$ for
which the state becomes local is at least asymptotically $\log(d)$
larger than the critical value for separability.
\end{abstract}
\pacs{03.65.Ud, 03.65.-w, 03.67.-a}
\maketitle
In 1964, Bell showed that some entangled states are nonlocal, in the
sense that measurements on them yield outcome correlations that
cannot be reproduced by a locally causal model \cite{Bell}. This
nonlocal character of entangled states may be demonstrated through
the violation of Bell inequalities. All pure entangled states
violate such an inequality, hence are nonlocal \cite{Gisin}. For
noisy states, the picture is much subtler. Werner constructed in
1989 a family of bipartite mixed states, which, while being
entangled, return outcome correlations under projective measurements
that can be described by a local model \cite{Werner}. This result
has been extended to general measurements \cite{Barrett} and more
parties \cite{TA}. Thus, while entanglement is necessary for a state
to be nonlocal, in the case of mixed states it is not sufficient.
Beyond these exploratory results, little is known about the relation
between noise, entanglement, and quantum nonlocality. Understanding
this relation, apart from its fundamental interest, is important
from the perspective of Quantum Information Science. In this
context, entanglement is commonly viewed as a useful resource for
various information-processing tasks. Not all entangled states,
however, are useful for every task: for example, quantum computation
with slightly entangled states can be efficiently simulated on a
classical computer \cite{comp}, and bound entangled states are
useless for teleportation \cite{HHH}. For certain tasks, such as
quantum communication complexity problems \cite{commcompl}, or
device-independent quantum key distribution \cite{AGM}, entangled
states are useful only to the extent that they exhibit nonlocal
correlations. Indeed, in these scenarios two (or more) distant
observers, Alice and Bob, directly exploit the correlations
\begin{equation}gin{equation}\label{qcorr}
P_{MN}(a,b)=\mbox{tr}(\rho_{AB}\,M_a\otimes N_b)\,,
\end{equation}
obtained by performing measurements $M$ and $N$ on a distributed
entangled state $\rho_{AB}$ (in the above formula, $M_a$ and $N_b$
are the positive operators associated with the measurement
outcomes $a$ and $b$). If the entangled state $\rho_{AB}$ can be
simulated by a local model, these correlations can be written as
\begin{equation}gin{equation}\label{loccorr}
P_{MN}(a,b)=\int\!\! \mu(d\lambda)\, P_M(a|\lambda)P_N(b|\lambda)\, ,
\end{equation}
where $\lambda$ denotes a shared classical variable distributed
with probability measure $\mu$, and $P_M(a|\lambda)$ and
$P_N(b|\lambda)$ are the local response functions of Alice and
Bob. For all practical purposes then, the entangled state
$\rho_{AB}$ can be replaced by classical correlations, and so does
not provide any improvement over what is achievable using
classical resources \cite{Lluis}.
In this work, we estimate the resistance to noise of the nonlocal
correlations of bipartite entangled states in $\mathbb{C}^d\otimes
\mathbb{C}^d$, where $d$ is the local Hilbert space dimension of each
subspace. To do this, we analyze the nonlocal properties of states
resulting from the mixture of an arbitrary state $\rho$ with
completely depolarized noise,
\begin{equation}gin{equation}\label{noisy_states}
\rho(p)=p\,\rho+(1-p)\frac{\openone}{d^2}\, .
\end{equation}
Our goal is to find the minimal amount of noise that destroys the
nonlocal correlations of any state $\rho$, i.e., the maximal value
$p_L$ such that $\rho(p)$ is local for any $\rho$ when $p\leq p_L$.
Clearly, for sufficiently small values of $p\leq p_S$, the state
$\rho(p)$ becomes separable for any $\rho$ \cite{sep, GB}, thus
local. We give here lower bounds on $p_L$ that are more constraining
than the one obtained from the separability condition. If we
restrict Alice and Bob to perform projective measurements only, the
bound that we obtain for the locality limit is asymptotically
$\log(d)$ larger than the separability limit.
A key step in the proof of our results is the construction of a
local model for states of the form (\ref{noisy_states}) when
$\rho=\ket{\phi_d}\bra{\phi_d}$ is maximally entangled, i.e.
$\ket{\phi_d}=1/\sqrt d\sum_{i=1}^d \ket{ii}$. Thus we also provide
a lower bound on $p_L^{\phi}$, defined as the maximal value of $p$
such that
\begin{equation}gin{equation}
p\,\ket{\phi_d}\bra{\phi_d}+(1-p)\frac{\openone}{d^2}
\end{equation}
is local. This last result implies in particular the existence of
entangled states whose nonlocal correlations are more robust than
those of maximally entangled ones.
The results presented here concern mostly the simpler but
physically relevant case in which Alice and Bob are restricted to
projective measurements. Extensions to completely general
measurements are discussed at the end of the paper. Our results
also provide bounds for the notion of state steerability
introduced in \cite{WJD}.
As mentioned, we start by analyzing the case in which the state $\rho$ in
(\ref{noisy_states}) is maximally entangled.
Such states are
called isotropic states and are the unique ones invariant under
$U\otimes U^*$ transformations for all unitary operators $U$ on
$\mathbb{C}^d$ \cite{HH}. If Alice and Bob each make on these states a
projective measurement, specified by a set of $d$ orthogonal
projectors $Q=\{Q_a\}$ for Alice and $R=\{R_b\}$ for Bob, with
$a,b=1,\ldots,d$, the resulting joint outcome probabilities are
given by
\begin{equation}gin{equation}
\label{quantum prob proj} \frac{p}{d}\mbox{tr}\left(Q_a^T
R_b\right)+\frac{1-p}{d^2}\, .
\end{equation}
Our first aim is to construct a local model for isotropic states,
that is, to write the quantum probabilities \eqref{quantum prob
proj} in the form (\ref{loccorr}) for some value of the noise
parameter $p$.
Our construction is inspired by the model given in Ref.
\cite{Werner} for Werner states, which are $U\otimes U$ invariant,
and which we adapt to the $U\otimes U^*$ symmetry of isotropic
states. The local classical variables $\lambda$ in our model are
taken to be complex $d$-dimensional vectors which we can thus
formally identify with $d$-dimensional quantum states
$\ket{\lambda}$. The probability measure $\mu$ is
the unique measure invariant under all unitary transformations $U$
on $\mathbb{C}^d$. In analogy with the quantum formalism, Alice's
response function is defined as
\begin{equation}gin{equation}
\label{aliceresp} P_Q(a|\lambda)=\bra{\lambda}Q_a^T\ket{\lambda}\,
.
\end{equation}
Bob's response function is suggested by the perfect correlations of maximally entangled states and taken to be
\begin{equation}gin{equation}
\label{bobresp} P_R(b|\lambda)=\begin{equation}gin{cases}
1& \text{if } \bra{\lambda}R_b\ket{\lambda}=\max_{i}{\bra{\lambda}R_i\ket{\lambda}}\\
0& \text{otherwise}\,.
\end{cases}
\end{equation}
It satisfies \begin{equation}\label{Ubob} P_{U^\dagger R
U}(b|\lambda)=P_R(b|U\lambda)\,. \end{equation} To obtain the joint
probabilities predicted by this model, and to compare them with
\eqref{quantum prob proj}, it is necessary to compute the integral
\eqref{loccorr} for our specific choice of measure $\mu$ and
response functions. Following Werner (see \cite{Werner} for
details), one can show that the $U$-invariance of $\mu$, the form
\eqref{aliceresp} of Alice's response function, and the relation
\eqref{Ubob} satisfied by Bob's response function, imply
\begin{equation}gin{equation}\label{integral LHV}
\int\!\!\mu(d\lambda)\,P_Q(a|\lambda)P_R(b|\lambda)=\mbox{tr}\left(Q_a^T\hat
B(b,R)\right)\,,
\end{equation}
where $\hat B(b,R)$ is a positive operator depending on Bob's
response function. One can further show, exploiting the fact that
the relation \eqref{integral LHV} holds for all one-dimensional
projectors $Q_a$ \cite{Werner}, that $\hat B(b,R)=(p^\phi/d)\,
R_b+(1-p^\phi)/d^2\, \openone$, for some $p^\phi\in \mathbb{R}$, and
thus that
\begin{equation}gin{equation}\label{integral LHV 2}
\int\!\!\mu(d\lambda)P_Q(a|\lambda)P_R(b|\lambda)\!=\!\frac{p^\phi}{d}\mbox{tr}\left(Q_a^T
R_b \right)+\frac{1-p^\phi}{d^2}\,.
\end{equation}
These correlations are thus already of the prescribed form
(\ref{quantum prob proj}). To determine the value of $p^\phi$ for
which \eqref{integral LHV 2} holds, it is sufficient to compute
the integral \eqref{integral LHV} in the simplest case where
$Q_a^T=R_b$, which gives
\begin{equation}gin{equation}\label{pcr}
p^\phi=\frac{1}{d-1}\left(-1+d^2\int\!\!
\mu(d\lambda)\,\bra{\lambda}R_b
\ket{\lambda}P_R(b|\lambda)\right)\, .
\end{equation}
It now remains to evaluate this integral for the specific choice
(\ref{bobresp}) for $P_R(b|\lambda)$. After patient algebra, one
obtains
\begin{equation}gin{equation}\label{boundmax}
p^\phi=\frac{1}{d-1}\left(-1+\sum_{k=1}^d\frac{1}{k}\right)
\,\xrightarrow[\mathrm{large}\; d]{}\,\frac{\log(d)}{d}\,.
\end{equation}
For $d=2$, $p^\phi=1/2$ is equal to the critical value for
two-dimensional Werner states, as expected since Werner and
isotropic states are equivalent up to local unitary
transformations when $d=2$. In the limit of large $d$, $p^\phi$ is
asymptotically $\log(d)$ larger than the critical probability
$p^\phi_S=1/(d+1)$ for the separability of isotropic states
\cite{HH}.
Our next goal is to generalize the local model for isotropic states
to mixed states of the form
\begin{equation}gin{equation} \label{pure+noise}
\rho=p\,\proj{\psi}+(1-p)\frac{\openone}{d^2}\, ,
\end{equation}
where $\ket{\psi}$ is an arbitrary pure state in
$\mathbb{C}^d\otimes\mathbb{C}^d$. This automatically also implies a model
for the general states (\ref{noisy_states}), since any mixed state
$\rho$ is a convex combination of pure states. To do this, we
incorporate Nielsen's protocol \cite{nielsen} for the conversion
of bipartite pure states by local operations and classical
communication (LOCC) into our model. Recall that a maximally
entangled state $\ket{\phi_d}$ can be transformed by LOCC in a
deterministic way into an arbitrary state $\ket{\psi}$ by a single
measurement on Alice's particle followed by a unitary operation on
Bob's side, depending on Alice's measurement outcome. Indeed,
consider an arbitrary pure entangled state written in its Schmidt
form $\ket{\psi}=\sum_{j=0}^{d-1}\nu_j\ket{jj}$, and denote by
$D_\nu$ the $d\times d$ diagonal matrix with entries
$(D_\nu)_{jj}=\nu_j$. Taking the $d$ cyclic permutations
$\Pi_i=\sum_{j=0}^{d-1} \ket{j}\bra{j+i \,(\mathrm{mod}\, d)}$,
where $i=0,\ldots,d-1$, it is possible to write
\begin{equation}gin{equation}
\label{nielsenrel} \ket{\psi}=\sqrt {d}(A_i\otimes \Pi_i)
\ket{\phi_d}\quad
\text{for all }
i=0,\ldots,d-1\, ,
\end{equation}
with $A_i=D_\nu\Pi_i$. The operators $W_i=A_i^\dagger A_i$ define a
measurement, since they are positive and sum to the identity,
$\sum_i W_i=\openone$. In order to convert $\ket{\phi_d}$ into
$\ket{\psi}$, Alice first carries out this measurement, obtaining
the outcome $i$ with probability $\bra{\phi_d}W_i\ket{\phi_d}=1/d$.
She then communicates her result to Bob who applies the
corresponding unitary operation $\Pi_i$, the resulting normalized
state being $\ket{\psi}$, as implied by (\ref{nielsenrel}).
The quantum-like properties of our local model, i.e., the fact
that the hidden variable $\ket{\lambda}$ can be thought of as a
quantum state and the quantum form of the response function
(\ref{aliceresp}), allow us to adapt Nielsen's construction to it.
The idea is that at the source, before sending the classical
instructions $\ket{\lambda}$ to each party, a measurement defined
by the operators $A^*_i$ is simulated on $\ket{\lambda}$, giving
outcome $i$ with probability $q_i(\lambda)=\bra{\lambda}A_i^T
A_i^*\ket{\lambda}$. The classical description of the normalized
hidden states $\ket{\lambda^A_i}=A^*_i\ket{\lambda}/\sqrt{q_i}$
and $\ket{\lambda^B_i}=\Pi_i\ket{\lambda}$ are then sent,
respectively, to Alice and Bob, who use them in the response
functions \eqref{aliceresp} and \eqref{bobresp} instead of
$\ket{\lambda}$. The joint probabilities $P_{QR}(a,b)$ predicted
by the model for measurements $Q$ and $R$ are thus given by
\begin{equation}gin{multline}\label{nielsenmodel}
\int\!\! \mu(d\lambda)\, \sum_{i=0}^{d-1} q_i(\lambda) P_Q(a|\lambda^A_i)P_R(b|\lambda^B_i) \\
=\sum_{i=0}^{d-1} \int\!\! \mu(d\lambda)\, \bra{\lambda}A_i^T
Q_a^T A_i^*\ket{\lambda} P_{\Pi^{\dagger}_i R_b \Pi_i}(b|\lambda)
\end{multline}
where we used property \eqref{Ubob}. Replacing the integral in the last expression by the right-hand side of \eqref{integral LHV 2}, we obtain
\begin{equation}gin{equation}
\sum_{i=0}^{d-1}\left(\frac{p^\phi}{d}\mbox{tr}(A_i^T Q_a^TA_i^*\Pi_i^\dagger R_b \Pi_i)+
\frac{1-p^\phi}{d^2}\mbox{tr}(A_i^T Q_a^TA_i^*)\right)\, .
\end{equation}
Using Eqs. (\ref{nielsenrel}) and \eqref{quantum prob proj}, and
the fact that $\sum A_iA_i^\dagger=d\sigma$, where
$\sigma=\mbox{tr}_B\proj{\psi}$, one can check that these probabilities
are equal to the quantum probabilities
$\mbox{tr}\left(\tilde\rho\,Q_a\otimes R_b \right)$ for the state
\begin{equation}gin{equation}\label{noisy_psi}
\tilde\rho=p^\phi\proj{\psi}+(1-p^\phi)\sigma\otimes\frac{\openone}{d}\, .
\end{equation}
Not surprisingly, the measurement at the source modifies the local noise of Alice, which is no longer completely depolarized, and
introduce some bias depending on $\ket{\psi}$.
This result can already be interpreted as a measure of the
robustness of the nonlocal correlations of an arbitrary entangled
state $\ket{\psi}$. By mixing a state-dependent local noise, with
mixing probability $1-p^\phi$, it is always possible to wash out
the nonlocal correlations of the state $\ket{\psi}$.
In order to extend this result to the case of completely
depolarized noise, one can add some extra local noise to Alice
such that the resulting state has the form \eqref{pure+noise},
with the penalty that $p< p^\phi$. Writing the reduced density
matrix $\sigma$ in its diagonal form $\sigma=\sum_j \mu_j^2
\proj{j}$, and defining $\sigma_k=\sum_j \mu_{j+k
\,(\mathrm{mod}\,d)}^2 \proj{j}$, it is clear that the state
\begin{equation}gin{equation}
\label{mixtonoise}
q\tilde\rho+\frac{1-q}{d-1}\sum_{k=1}^{d-1}\sigma_k\otimes\frac{\openone}{d}
\end{equation}
has the form \eqref{pure+noise} for $q(1-p_d)=(1-q)/(d-1)$, in which case the
probability $p$ is given by
\begin{equation}gin{equation}\label{asympt pr}
p^\rho=\frac{p^\phi}{(1-p^\phi)(d-1)+1}\,\xrightarrow[\mathrm{large}\;
d]{}\,\frac{\log(d)}{d^2}\,.
\end{equation}
The state (\ref{mixtonoise}) is clearly local, since it is a
convex combination of local states. We have thus shown that the
noisy states (\ref{noisy_states}) have a local model for
projective measurements whenever $p\leq p^\rho$. The
probabilities $p^\phi$ and $p^\rho$ represent the main results of
this work and provide lower bounds on $p_L^{\phi}$ and $p_L$.
Several implications of our findings are discussed in what
follows.
First of all, one may ask about the tightness of our bound.
Actually, our model is based on Werner's construction, and this
model is known not to be tight in the case $d=2$ \cite{AGT}. Even
if it is not tight, it would be interesting to understand whether
the model predicts the right asymptotic dependence with the
Hilbert-space dimension $d$. An upper bound on $p_L$ follows from
the results of \cite{ADGL}, where it was shown that a state of the
form $\varrho_2=p \ket{\phi_2}\bra{\phi_2}+(1-p)\openone/d^2$, where
$\ket{\phi_2}=1/\sqrt{2}(\ket{00}+\ket{11})$ is a projector onto a
two-qubit maximally entangled state, violates the
Clauser-Horne-Shimony-Holt inequality \cite{CHSH} whenever
$p>p^{\varrho_{2}}$, where
\begin{equation}gin{equation}\label{upprob}
p^{\varrho_2}=\frac{4(d-1)}{(\sqrt 2-1)d^2+4d-4}\,\xrightarrow[\mathrm{large}\; d]{}\,\frac{4}{(\sqrt 2-1)d}\,,
\end{equation}
which tends to zero when $d\to\infty$. This result together with
our previous model thus imply that $p^\rho\leq p_L\leq
p^{\varrho_2}$.
\begin{equation}gin{table*}[t]
\caption{Asymptotic bounds on the critical noise threshold for separability ($p_S$) and locality ($p_L$) for maximally entangled states ($\ket{\phi_d}$) and arbitrary states ($\rho$). For maximally entangled states, $p_S^{\phi}$ is given in \cite{HH}; the lower bounds for $p_L^\phi$ follow from eqs.~\eqref{boundmax} and \eqref{probpovm}; and the upper bounds from \cite{CGLMP}, where $K$ is Catalan's constant. For arbitrary states, bounds for $p_S$ were derived in \cite{GB}; the lower bounds for $p_L$ are obtained from those for the maximally entangled states using eq.~\eqref{asympt pr}; the upper-bounds are those of \cite{ADGL}.
}
\begin{equation}gin{ruledtabular}
\begin{equation}gin{tabular}{c|lll}
state & separability & locality (projective meas.) & locality (general meas.)\\
\hline$\ket{\phi_d}$ & $p^\phi_S=\frac{1}{d+1}$ & $\Theta\left(\frac{\log d}{d}\right)\leq p^\phi_L\leq\frac{\pi^2}{16\,K}\simeq 0.67$ & $\Theta\left(\frac{3}{e\,d}\right)\leq p^\phi_L\ \leq\frac{\pi^2}{16\,K}\simeq 0.67$\\
arbitrary $\rho$ & $\frac{1}{d^2-1}\leq p_S \leq \frac{2}{d^2+2}$ & $\Theta\left(\frac{\log d}{d^2}\right)\leq p_L\leq \Theta\left(\frac{4}{(\sqrt{2}-1)d}\right)$&$\Theta(\frac{3}{e\,d^2})\leq p_L\leq\Theta\left(\frac{4}{(\sqrt{2}-1)d}\right)$
\end{tabular}
\end{ruledtabular}
\end{table*}
Our results, when combined with (\ref{upprob}), also provide a
strict proof of the fact that the nonlocal correlations of
maximally entangled states, under projective measurements, are not
the most robust ones. Indeed, we have a local model for isotropic
states whenever $p\leq p^\phi$, while there exist quantum states
of the form (\ref{noisy_states}) violating a Bell inequality when
$p>p^{\varrho_2}$. For sufficiently large dimension,
$p^{\varrho_2}<p^\phi$, so we have a Bell inequality violation in
a range of $p$ for which we have shown the existence of a local
model for isotropic states.
It is also interesting to compare the bounds derived here for
nonlocality with those known for entanglement. To our knowledge, the
best upper and lower bound on the critical probability $p_S$ such
that the states (\ref{noisy_states}) are guaranteed to be separable were obtained in
Ref. \cite{GB}:
\begin{equation}gin{equation}\label{sep_bounds}
\frac{1}{d^2-1}\leq p_S\leq \frac{2}{d^2+2}\, .
\end{equation}
Interestingly, the upper bound is obtained, as above, for the case
in which the state $\rho$ in (\ref{noisy_states}) is equal to a
projector onto $\ket{\phi_2}$. Comparing with Eq. (\ref{asympt
pr}), we see that the critical noise probability for nonlocality
under projective measurements is, at least, asymptotically
$\log(d)$ larger than the one for separability, as it is for
isotropic states.
Finally, let us briefly mention how the above results can be
extended to the case of general measurements. The idea is, as
above, to start by constructing a model for isotropic states,
adapting the one for Werner states of Ref. \cite{Barrett}. As
noted in \cite{Barrett}, it is sufficient to simulate measurements
$M$ and $N$ defined by operators $M_a=c_a Q_a$ and $N_b=c_b R_b$
proportional to one-dimensional projectors $Q_a$ and $R_b$ to be
able to simulate any measurement by Alice and Bob. In our
corresponding model, the hidden states are again vectors
$\ket{\lambda}$ in $\mathbb{C}^{d}$ chosen with the Haar measure
$\mu$. Alice's response function is basically the same as before,
\begin{equation}gin{equation}\label{alicepovm}
P_M(a|\lambda)=\bra{\lambda}M_a^T\ket{\lambda}\,,
\end{equation}
while Bob's is, taking inspiration from \cite{Barrett}, chosen as
\begin{equation}gin{multline}\label{bobpovm}
P_N(b|\lambda) = \bra{\lambda}N_b\ket{\lambda}
\Theta\!\left(\bra{\lambda}R_b\ket{\lambda}-\frac{1}{d}\right)\\
+ \frac{c_b}{d}\left[1-\sum_k \bra{\lambda}N_k\ket{\lambda}\Theta\!\left(\bra{\lambda}R_k\ket{\lambda}-
\frac{1}{d}\right)\right]\,,
\end{multline}
where $\Theta$ is the Heaviside step function. Evaluation of the
integral \eqref{loccorr} with the definitions \eqref{alicepovm}
and \eqref{bobpovm} can be done along the same steps as in
\cite{Barrett} and yields the joint measurement outcome
probabilities for an isotropic state with the critical value
\begin{equation}gin{equation}\label{probpovm}
\tilde p^\phi=\frac{(3d-1)(d-1)^{d-1}}{(d+1)d^d}\,\xrightarrow[\text{large }d]{}\,\frac{3}{e}\frac{1}{d}\, .
\end{equation}
Since this model has the same quantum-like properties as the one
for projective measurements, cf.~definition~(\ref{alicepovm}), it
can also be extended to arbitrary noisy
states~(\ref{noisy_states}) using Nielsen's protocol. The
corresponding critical probability is given by (\ref{pcr}) with
$p^\phi$ replaced by the above value of $\tilde p^\phi$.
In conclusion, we have obtained bounds on the robustness of the
nonlocal correlations of arbitrary entangled states. Our results
are summarized in Table I. In the particular but interesting case
where the state is maximally entangled, we derived better bounds
by exploiting the symmetry of isotropic states \cite{WJD}. Apart
from their fundamental significance, our results are interesting
from the point of view of the characterization of quantum
information resources: if the noise affecting a state is larger
than our bounds, its outcome correlations for local measurements
can be reproduced by classical means alone.
\textit{Note added.} While completing this work, we learned that
our local model for isotropic states was independently derived in
\cite{WJD} in the context of state steerability. We note that all
our models imply the non-steerability of the corresponding quantum
states because Alice's response function is always quantum (see
\cite{WJD} for details).
\textit{Acknowledgements.} We acknowledge financial support from the
EU Qubit Applications Project (QAP) Contract number 015848, the
Spanish projects FIS2004-05639-C02-02, Consolider QOIT, the Spanish
MEC for ``Ramon y Cajal" and ``Juan de la Cierva" grants, the
Generalitat de Catalunya, the Funda\c{c}\~{a}o para a Ci\^{e}ncia e
a Tecnologia (Portugal) through the grant SFRH/BD/21915/2005, the
National Research Fund of Hungary OTKA under Contract No. T049234,
and the Hungarian Academy of Sciences (Bolyai Programme). Research
at Perimeter Institute for Theoretical Physics is supported in part
by the Government of Canada through NSERC and by the Province of
Ontario through MRI.
\begin{equation}gin{thebibliography}{99}
\bibitem{Bell}
J. S. Bell, Physics {\bf 1}, 195 (1964).
\bibitem{Gisin}
N. Gisin, Phys. Lett. A {\bf 154}, 201 (1991).
\bibitem{Werner}
R. F. Werner, Phys. Rev. A {\bf 40}, 4277 (1989).
\bibitem{Barrett}
J. Barrett, Phys. Rev. A {\bf 65}, 042302 (2002).
\bibitem{TA}
G. T\'oth and A. Ac\'\i n, Phys. Rev. A {\bf 74}, 030306(R) (2006).
\bibitem{comp}
R. Jozsa and N. Linden, quant-ph/0201143; G. Vidal, Phys. Rev.
Lett. {\bf 91}, 147902 (2003).
\bibitem{HHH}
M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev. A {\bf
60}, 1888 (1999).
\bibitem{commcompl}
See for instance G. Brassard, quant-ph/0101005; C.~Brukner, M.
Zukowski, and A. Zeilinger, Phys. Rev. Lett. {\bf 89}, 197901
(2002).
\bibitem{AGM}
A. Ac\'\i n {\sl et al.} quant-ph/0702152; A. Ac\'\i n, N. Gisin,
and L. Masanes, Phys. Rev. Lett. {\bf 97}, 120405 (2006); J.
Barrett, L. Hardy, and A. Kent, {\sl ibid} \textbf{95}, 010503
(2005).
\bibitem{Lluis}
This does not mean that the state can be replaced by shared
randomness in any scenario, see Ll. Masanes, Phys. Rev. Lett. {\bf
96}, 150501 (2006).
\bibitem{sep}
K. Zyczkowski, P. Horodecki, A. Sanpera and M. Lewenstein, Phys.
Rev. A {\bf 58}, 883 (1998); G. Vidal and R. Tarrach, Phys. Rev.
A {\bf 59}, 141 (1999); S. L. Braunstein {\sl et al.}, Phys. Rev.
Lett. {\bf 83}, 1054 (1999).
\bibitem{GB}
L. Gurvits and H. Barnum, Phys. Rev. A {\bf 66}, 062311 (2002).
\bibitem{WJD}
H. M. Wiseman, S. J. Jones and A. C. Doherty, Phys. Rev. Lett.
{\bf 98}, 140402 (2007).
\bibitem{HH}
M. Horodecki and P. Horodecki, Phys. Rev. A {\bf 59}, 4206 (1999).
\bibitem{nielsen}
M. A. Nielsen, Phys. Rev. Lett. {\bf 83}, 436 (1999).
\bibitem{AGT}
A. Ac\'\i n, N. Gisin and B. Toner, Phys. Rev. A \textbf{73},
062105 (2006).
\bibitem{ADGL}
A. Ac\'\i n, T. Durt, N. Gisin and J. I. Latorre, Phys. Rev. A
\textbf{65}, 052325 (2002).
\bibitem{CHSH}
J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Phys. Rev.
Lett. {\bf 23}, 880 (1969).
\bibitem{CGLMP}
D. Collins, N. Gisin, N. Linden, S. Massar and S. Popescu, Phys.
Rev. Lett. \textbf{88}, 040404 (2002).
\end{thebibliography}
\end{document} | math | 25,874 |
\begin{document}
\begin{center} {\bf \Large Complexes of Nonseparating Curves
and}\\
{\bf \Large Mapping Class Groups}
\large {Elmas Irmak} \\
\end{center}
\begin{abstract}Let $R$ be a compact, connected, orientable surface of genus $g$,
$Mod_R^*$ be the extended mapping class group of $R$,
$\mathcal{C}(R)$ be the complex of curves on $R$, and
$\mathcal{N}(R)$ be the complex of nonseparating curves on $R$. We
prove that if $g \geq 2$ and $R$ has at most $g-1$ boundary
components, then a simplicial map $\lambda: \mathcal{N}(R)
\rightarrow \mathcal{N}(R)$ is superinjective if and only if it is
induced by a homeomorphism of $R$. We prove that if $g \geq 2$ and
$R$ is not a closed surface of genus two then
$Aut(\mathcal{N}(R))= Mod_R^*$, and if $R$ is a closed surface of
genus two then $Aut(\mathcal{N}(R))= Mod_R ^* /\mathcal{C}(Mod_R
^*)$. We also prove that if $g=2$ and $R$ has at most one boundary
component, then a simplicial map $\lambda: \mathcal{C}(R)
\rightarrow \mathcal{C}(R)$ is superinjective if and only if it is
induced by a homeomorphism of $R$. As a corollary we prove some
new results about injective homomorphisms from finite index
subgroups to $Mod_R^*$. The last two results complete the author's
previous results to connected orientable surfaces of genus at
least two.
\end{abstract}
{\it MSC}: 57M99, 20F38.
{\it Keywords}: Mapping class groups; Surfaces; Complex of curves.
\section{Introduction}
Let $R$ be a compact, connected, orientable surface of genus $g$
with $p$ boundary components. The extended mapping class group,
$Mod_R^*$, of $R$ is the group of isotopy classes of all
(including orientation reversing) homeomorphisms of $R$. Let
$\mathcal{A}$ denote the set of isotopy classes of nontrivial
simple closed curves on $R$. The \textit{complex of curves},
$\mathcal{C}(R)$, on $R$ is an abstract simplicial complex,
introduced by Harvey \cite{Har}, with vertex set $\mathcal{A}$
such that a set of $n$ vertices $\{{ \alpha_{1}}, {\alpha_{2}},
..., {\alpha_{n}}\}$ forms an $n-1$ simplex if and only if
${\alpha_{1}}, {\alpha_{2}},..., {\alpha_{n}}$ have pairwise
disjoint representatives. Let $\mathcal{B}$ denote the set of
isotopy classes of nonseparating simple closed curves on $R$. The
\textit{complex of nonseparating curves}, $\mathcal{N}(R)$, is the
subcomplex of $\mathcal{C}(R)$ with the vertex set $\mathcal{B}$
such that a set of $n$ vertices $\{{ \beta_{1}}, {\beta_{2}}, ...,
{\beta_{n}}\}$ forms an $n-1$ simplex if and only if ${\beta_{1}},
{\beta_{2}},..., {\beta_{n}}$ have pairwise disjoint
representatives. The main results of the paper are the following:
\begin{theorem} \label{theorem1} Suppose that $g \geq 2$
and $R$ has at most $g-1$ boundary components. Then a simplicial
map $\lambda : \mathcal{N}(R) \rightarrow \mathcal{N}(R)$ is
superinjective if and only if $\lambda$ is induced by a
homeomorphism of $R$.
\end{theorem}
\begin{theorem} \label{theorem2} Suppose that $g \geq 2$. If $R$ is not a closed
surface of genus two, then $Aut(\mathcal{N}(R))= Mod_R^*$. If $R$
is a closed surface of genus 2, then $Aut(\mathcal{N}(R))= Mod_R
^* /\mathcal{C}(Mod_R ^*)$.
\end{theorem}
\noindent \rule{1.5in}{0.3pt}
\noindent \setminusall{Supported by a Rackham Faculty Fellowship, The
Rackham Graduate School, University of Michigan.}
\begin{theorem} \label{theorem3} Suppose $g = 2$ and $p \leq 1$.
A simplicial map $\lambda : \mathcal{C}(R) \rightarrow
\mathcal{C}(R)$ is superinjective if and only if $\lambda$ is
induced by a homeomorphism of $R$.
\end{theorem}
\begin{theorem}
\label{theorem4} Let $K$ be a finite index subgroup of $Mod_R^*$
and $f$ be an injective homomorphism $f:K \rightarrow Mod_R^*$. If
$g = 2$ and $p=1$ then $f$ has the form $k \rightarrow hkh^{-1}$
for some $h \in Mod_R^*$ and $f$ has a unique extension to an
automorphism of $Mod_R^*$. If $R$ is a closed surface of genus 2,
then $f$ has the form $k \rightarrow hkh^{-1} i^{m(k)}$ for some
$h \in Mod_R^*$ where $m$ is a homomorphism $K \rightarrow
\mathbb{Z}_2$ and $i$ is the hyperelliptic involution on $R$.
\end{theorem}
In section 2, we give some properties of the superinjective
simplicial maps of $\mathcal{N}(R)$. In section 3, we give some
properties of the superinjective simplicial maps of
$\mathcal{C}(R)$ and we prove Theorem \ref{theorem3} and Theorem
\ref{theorem4}. In section 4, we prove that a superinjective
simplicial map $\lambda : \mathcal{N}(R) \rightarrow
\mathcal{N}(R)$ extends to a superinjective simplicial map on
$\mathcal{C}(R)$, and we prove Theorem \ref{theorem1} and Theorem
\ref{theorem2}.\\
Theorem \ref{theorem3} and Theorem \ref{theorem4} complete the
author's previous results given in \cite{Ir1} and \cite{Ir2} to
connected orientable surfaces of genus at least two. These
theorems were motivated by the work of Ivanov \cite{Iv1}, and
Ivanov and McCarthy \cite{IMc}. For automorphism groups of some
complexes related to complex of nonseparating curves
$\mathcal{N}(R)$, see \cite{Luo}, \cite{Sc}.
\section{Properties of Superinjective Simplicial Maps of $\mathcal{N}(R)$}
\noindent A \textit{circle} on $R$ is a properly embedded image of
an embedding $S^{1} \rightarrow R$. A circle on $R$ is said to be
\textit{nontrivial} (or \textit{essential}) if it doesn't bound a
disk and it is not homotopic to a boundary component of $R$. Let
$C$ be a collection of pairwise disjoint circles on $R$. The
surface obtained from $R$ by cutting along $C$ is denoted by
$R_C$. A nontrivial circle $a$ on $R$ is called
\textit{nonseparating} if the surface $R_{a}$ is connected. Let
$\alpha$ and $\beta$ be two vertices in $\mathcal{N}(R)$. The
\textit{geometric intersection number} $i(\alpha, \beta)$ is
defined to be the minimum number of points of $a \cap b$ where $a
\in \alpha$ and $b \in \beta$. A simplicial map $\lambda :
\mathcal{N}(R) \rightarrow \mathcal{N}(R)$ is called {\bf
superinjective} if the following condition holds: if $\alpha,
\beta$ are two vertices in $\mathcal{N}(R)$ such that
$i(\alpha,\beta) \neq 0$, then $i(\lambda(\alpha),\lambda(\beta))
\neq 0$.
\begin{lemma}
\label{injective} Suppose $g \geq 2$ and $p \geq 0$. A
superinjective simplicial map $\lambda : \mathcal{N}(R)
\rightarrow \mathcal{N}(R)$ is injective.
\end{lemma}
\begin{proof} Let $\alpha$ and $\beta$ be two distinct vertices
in $\mathcal{N}(R)$. If $i(\alpha, \beta) \neq 0$, then
$i(\lambda(\alpha), \lambda(\beta)) \neq 0$, since $\lambda$
preserves nondisjointness. So, $\lambda(\alpha) \neq
\lambda(\beta)$. If $i(\alpha, \beta) = 0$, then, since $g \geq 2$
and $p \geq 0$, we can choose a vertex $\gamma$ of
$\mathcal{N}(R)$ such that $i(\gamma, \alpha)= 0$ and $i(\gamma,
\beta) \neq 0$. Then $i(\lambda(\gamma), \lambda(\alpha)) = 0$ and
$i(\lambda(\gamma), \lambda(\beta)) \neq 0$. So, $\lambda(\alpha)
\neq \lambda(\beta)$. Hence $\lambda$ is
injective. \end{proof}\\
Let $P$ be a set of pairwise disjoint circles on $R$. $P$ is
called a {\it pair of pants decomposition} of $R$, if $R_P$ is a
disjoint union of genus zero surfaces with three boundary
components, pairs of pants. A pair of pants of a pants
decomposition is the image of one of these connected components
under the quotient map $q:R_P \rightarrow R$. The image of the
boundary of this component is called the \textit{boundary of the
pair of pants}. A pair of pants is called \textit{embedded} if the
restriction of $q$ to the corresponding component of $R_P$ is an
embedding. An ordered set $(a_1, ..., a_{3g-3+p})$ is called an
{\it ordered pair of pants decomposition} of $R$ if $\{a_1, ...,
a_{3g-3+p}\}$ is a pair of pants decomposition of $R$.
\begin{lemma}
\label{imageofpantsdecomp} Suppose $g \geq 2$ and $p \geq 0$. Let
$\lambda : \mathcal{N}(R) \rightarrow \mathcal{N}(R)$ be a
superinjective simplicial map. Let $P$ be a pair of pants
decomposition consisting of nonseparating circles on $R$. Then
$\lambda$ maps the set of isotopy classes of elements of $P$ to
the set of isotopy classes of elements of a pair of pants
decomposition $P'$ of $R$.
\end{lemma}
\begin{proof} The set of isotopy classes of elements of $P$ forms
a top dimensional simplex, $\bigtriangleup$, in $\mathcal{N}(R)$.
Since $\lambda$ is injective, it maps $\bigtriangleup$ to a top
dimensional simplex $\bigtriangleup'$ in $\mathcal{N}(R)$. A set
of pairwise disjoint representatives of the vertices of
$\bigtriangleup'$ is a pair of pants decomposition $P'$ of
$R$.\end{proof}\\
Let $P$ be a pair of pants decomposition of $R$. Let $a$ and $b$
be two distinct elements in $P$. Then $a$ is called {\it adjacent}
to $b$ w.r.t. $P$ iff there exists a pair of pants in
$P$ which has $a$ and $b$ on its boundary.\\
\noindent {\bf Remark}: Let $P$ be a pair of pants decomposition
of $R$. Let $[P]$ be the set of isotopy classes of elements of
$P$. Let $\alpha, \beta \in [P]$. We say that $\alpha$ is adjacent
to $\beta$ w.r.t. $[P]$ if the representatives of $\alpha$ and
$\beta$ in $P$ are adjacent w.r.t. $P$. By Lemma
\ref{imageofpantsdecomp}, $\lambda$ gives a correspondence on the
isotopy classes of elements of pair of pants decompositions
consisting of nonseparating circles on $R$. $\lambda([P])$ is the
set of isotopy classes of elements of a pair of pants
decomposition which corresponds to $P$, under this correspondence.
\begin{lemma}
\label{adjacent} Suppose $g \geq 2$ and $p \geq 0$. Let $\lambda :
\mathcal{N}(R) \rightarrow \mathcal{N}(R)$ be a superinjective
simplicial map. Let $P$ be a pair of pants decomposition
consisting of nonseparating circles on $R$. Then $\lambda$
preserves the adjacency relation for two circles in $P$, i.e. if
$a, b \in P$, $a$ is adjacent to $b$ w.r.t. $P$ and $[a]=\alpha,
[b]=\beta$, then $\lambda(\alpha)$ is adjacent to $\lambda(\beta)$
w.r.t. $\lambda([P])$.\end{lemma}
\begin{proof} Let $P$ be a pair of pants decomposition consisting of
nonseparating circles on $R$. If $g=2$ and $p \leq 1$, then every
element in $\lambda([P])$ is adjacent to any other element in
$\lambda([P])$, so the lemma is clear. For the other cases; Let
$a, b$ be two adjacent circles in $P$ and $[a]=\alpha$,
$[b]=\beta$. By Lemma \ref{imageofpantsdecomp}, we can choose a
pair of pants decomposition, $P'$, such that $\lambda([P])= [P']$.
Let $P_o$ be a pair of pants of $P$, having $a$ and $b$ on its
boundary. $P_o$ is an embedded pair of pants. There are two
possible cases for $P_o$, depending on whether $a$ and $b$ are the
boundary components of another pair of pants or not. For each of
these cases, we show how to choose a circle $c$ which essentially
intersects $a$ and $b$ and does not intersect any other
circle in $P$ in Figure 1.\\
\begin{figure}
\caption{Two possible cases for $P_o$}
\label{picture1}
\end{figure}
Let $\gamma= [c]$. Assume that $\lambda(\alpha)$ and
$\lambda(\beta)$ do not have adjacent representatives. Since
$i(\gamma, \alpha)\neq 0$ and $i(\gamma, \beta) \neq 0$, we have
$i(\lambda (\gamma), \lambda(\alpha)) \neq 0$ and
$i(\lambda(\gamma), \lambda(\beta)) \neq 0$ by superinjectivity.
Since $i(\gamma, [e]) = 0$ for all $e$ in $P \setminus \{a,b\}$,
we have $i(\lambda(\gamma), \lambda([e]))=0$ for all $e$ in $P
\setminus \{a,b\}$. But this is not possible because
$\lambda(\gamma)$ has to intersect geometrically essentially with
some isotopy class other than $\lambda(\alpha)$ and
$\lambda(\beta)$ in the image pair of pants decomposition to be
able to make essential intersections with $\lambda(\alpha)$ and
$\lambda(\beta)$. This is a contradiction to the assumption that
$\lambda(\alpha)$ and $\lambda(\beta)$ do not have adjacent
representatives.\end{proof}\\
Let $P$ be a pair of pants decomposition of $R$. A curve $x \in P$
is called a {\it 4-curve} in $P$ if there exist four distinct
circles in $P$, which are adjacent to $x$ w.r.t. $P$. Note that
in a pants decomposition every curve is adjacent to at most 4
curves.
\begin{lemma}
\label{embedded} Suppose $g \geq 2$ and $p \geq 0$. Let $\lambda :
\mathcal{N}(R) \rightarrow \mathcal{N}(R)$ be a superinjective
simplicial map and $\alpha, \beta, \gamma$ be distinct vertices in
$\mathcal{N}(R)$ having pairwise disjoint representatives which
bound a pair of pants in $R$. Then $\lambda(\alpha),
\lambda(\beta), \lambda(\gamma)$ are distinct vertices in
$\mathcal{N}(R)$ having pairwise disjoint representatives which
bound a pair of pants in $R$.\end{lemma}
\begin{proof} Let $a, b, c$ be pairwise disjoint representatives of
$\alpha, \beta, \gamma$ respectively. If $R$ is a closed surface
of genus two then the statement is obvious. Consider the case when
$g = 2, p = 1$. We complete $\{a, b, c\}$ to a pair of pants
decomposition $P = \{a, b, c, a_1\}$ consisting of nonseparating
circles as shown in Figure 2 (i). Let $P'$ be a pair of pants
decomposition of $R$ such that $\lambda([P]) = [P']$. Let $a', b',
c', a_1'$ be the representatives of $\lambda([a]), \lambda([b]),
\lambda([c]), \lambda([a_1])$ in $P'$ respectively. Since $a$ is
adjacent to $b$ w.r.t. $P$, $a'$ is adjacent to $b'$ w.r.t. $P'$.
Then there is a pair of pants $Q_1$ in $P'$ having $a'$ and $b'$
on its boundary. Let $x$ be the other boundary component of $Q_1$.
If $x = c'$ then we are done. Assume that $x \neq c'$. Then $x$
can't be $a'$, since otherwise $b'$ would be a separating circle
which is a contradiction. Similarly, $x$ can't
be $b'$. Then $x$ is either $a_1'$ or it is the boundary component of $R$.\\
Assume $x$ is a boundary component of $R$. Then, since $a$ is
adjacent to $c, a_1$ w.r.t. $P$, $a'$ is adjacent to $c', a_1'$
w.r.t. $P'$ and so, there is a pair of pants $Q_2$ in $P'$ having
$a'$ and $c', a_1'$ on its boundary. Similarly, since $b$ is
adjacent to $c, a_1$ w.r.t. $P$, $b'$ is adjacent to $c', a_1'$
w.r.t. $P'$ and so, there is a pair of pants $Q_3$ in $P'$ having
$b'$ and $c', a_1'$ on its boundary. Then it is clear that $R =
Q_1 \cup Q_2 \cup Q_3$. Now we consider the circles $a_2, a_3$
which are as shown in Figure 2 (i). Since $a_2$ intersects $b$
essentially and $a_2$ is disjoint from $a$, we can choose a
representative $a_2'$ of $\lambda([a_2])$ such that there exists
an essential arc $w$ of $a_2'$ in $Q_1$ which starts and ends on
$b'$ and which does not intersect $a' \cup \partial R$. Since
$a_3$ is disjoint from $b$ and $a_2$, there exists a
representative $a_3'$ of $\lambda([a_3])$ such that $a_3'$ is
disjoint from $b' \cup w$. But then $a_3'$ could be isotoped so
that it is disjoint from $a'$, since $a'$ is a boundary component
of a regular neighborhood of $b' \cup w$ in $Q_1$. This is a
contradiction since $i([a], [a_3]) \neq 0$ and so $i(\lambda([a]),
\lambda([a_3])) \neq 0$. So, $x$ can not be
a boundary component of $R$.\\
Assume that $x = a_1'$. Then, since $a$ is adjacent to $c$ w.r.t.
$P$, $a'$ is adjacent to $c'$ w.r.t. $P'$ and so, there is a pair
of pants $Q_2$ in $P'$ having $a'$ and $c'$ on its boundary. Let
$y$ be the other boundary component of $Q_2$. If $y = c'$, then
$a'$ would be a separating circle, which is a contradiction. If
$y$ is the boundary component of $R$, then since $c$ is adjacent
to $b, a_1$ w.r.t. $P$, $c'$ is adjacent to $b', a_1'$ in $P'$ and
so, there is a pair of pants $Q_3$ in $P'$ having $b', c', a_1'$
on its boundary. Then it is clear that $R = Q_1 \cup Q_2 \cup
Q_3$. Now we consider the circles $a_2, a_4$ which are as shown in
Figure 2 (i). Since $a_4$ intersects $a$ essentially and $a_4$ is
disjoint from $c$, we can choose a representative $a_4'$ of
$\lambda([a_4])$ such that there exists an essential arc $w$ of
$a_4'$ in $Q_2$ which starts and ends on $a'$ and which does not
intersect $c' \cup \partial R$. Now, we consider $a_2$; $a_2$ is
disjoint from $a$ and $a_4$. Then there exists a representative
$a_2'$ of $\lambda([a_2])$ such that $a_2'$ is disjoint from $a'
\cup w$. But then $a_2'$ could be isotoped so that it is disjoint
from $c'$, since $c'$ is a boundary component of a regular
neighborhood of $a' \cup w$ in $Q_2$. This is a contradiction
since $i([c], [a_2]) \neq 0$ and so $i(\lambda([c]),
\lambda([a_2])) \neq 0$. So, $y$ can not be the boundary component
of $R$. If $y = a_1'$, then there exists a pair of pants $Q_3$
having $b', c'$ and the boundary component of $R$ as its boundary
components. Then we have $R = Q_1 \cup Q_2 \cup Q_3$. Now, we
consider the circles $a_3, a_4$. Since $a_3$ intersects $c$
essentially and $a_3$ is disjoint from $b$, we can choose a
representative $a_3'$ of $\lambda([a_3])$ such that there exists
an essential arc $w$ of $a_3'$ in $Q_3$ which starts and ends on
$c'$ and which does not intersect $b' \cup \partial R$. Now, we
consider $a_4$; $a_4$ is disjoint from $c$ and $a_3$. Then there
exists a representative $a_4'$ of $\lambda([a_4])$ such that
$a_4'$ is disjoint from $c' \cup w$. But then $a_4'$ could be
isotoped so that it is disjoint from $b'$, since $b'$ is a
boundary component of a regular neighborhood of $c' \cup w$ in
$Q_3$. This is contradiction since $i([a_4], [b]) \neq 0$ and so
$i(\lambda([a_4]), \lambda([b])) \neq 0$. Hence, we conclude that
$y = b'$. Then $a', b', c'$ bound a pair of pants
which completes the proof for $g = 2, p = 1$.\\
For the remaining cases we take a subsurface of genus two with two
boundary components, $N$, of $R$, containing nonisotopic circles
$a, b, c, a_1, a_2$ as shown in Figure 2 (ii). Then we complete
$\{a, b, c, a_1, a_2\}$ to a pair of pants decomposition $P$
consisting of nonseparating circles on $R$ in any way we want. Let
$P'$ be a pair of pants decomposition of $R$ such that
$\lambda([P]) = [P']$. Let $a', b', c', a_1', a_2'$ be the
representatives of $\lambda([a]), \lambda([b]), \lambda([c]),
\lambda([a_1]), \lambda([a_2])$ in $P'$ respectively. Since $b$ is
adjacent to $a, c, a_1, a_2$ w.r.t. $P$, $b'$ is adjacent to $a',
c', a_1', a_2'$ w.r.t. $P'$. Then there are two pairs of pants
$Q_1, Q_2$ having $b'$ as boundary components and $Q_1 \cup Q_2$
has $a', c', a_1', a_2'$ on its boundary. W.L.O.G. assume that
$Q_1$ has $a', b'$ on its boundary. Let $x$ be the other boundary
component of $Q_1$. If $x = c'$ then we are done. Assume that $x
\neq c'$. Then $x$ is either $a_1'$ or $a_2'$. Assume that
$x=a_1'$. Since $a_3$ intersects $b$ essentially and $a_3$ is
disjoint from $a \cup a_1$, we can choose a representative $a_3'$
of $\lambda([a_3])$ such that there exists an essential arc $w$ of
$a_3'$ in $Q_1$ which starts and ends on $b'$ and which does not
intersect $a' \cup a_1'$. Now, we consider $a_4$; $a_4$ is
disjoint from $b$ and $a_3$. Then there exists a representative
$a_4'$ of $\lambda([a_4])$ such that $a_4'$ is disjoint from $b'
\cup a_3'$. But then $a_4'$ could be isotoped so that it is
disjoint from $a'$, since $a'$ is a boundary component of a
regular neighborhood of $b' \cup w$ in $Q_1$. This is a
contradiction since $i([a_4], [a]) \neq 0$ and so
$i(\lambda([a_4]), \lambda([a])) \neq 0$. So, $x \neq a_1'$. Now,
assume that $x= a_2'$. We consider the curves $a_5, a_6$ on $N$ as
shown in Figure 2 (iii). Since $a_5$ intersects $c$ essentially
and $a_5$ is disjoint from $a_1, b$, we can choose a
representative $a_5'$ of $\lambda([a_5])$ such that there exists
an essential arc $w$ of $a_5'$ in $Q_2$ which starts and ends on
$c'$ and which does not intersect $a_1' \cup b'$. The circle $a_6$
is disjoint from $c$ and $a_5$. Then there exists a representative
$a_6'$ of $\lambda([a_6])$ such that $a_6'$ is disjoint from $c'
\cup a_5'$. But then $a_6'$ could be isotoped so that it is
disjoint from $a_1'$, since $a_1'$ is a boundary component of a
regular neighborhood of $c' \cup w$ in $Q_2$. This is a
contradiction since $i([a_6], [a_1]) \neq 0$ and so
$i(\lambda([a_6]), \lambda([a_1])) \neq 0$. This completes the
proof of the lemma.\end{proof}\\
Let $\alpha$, $\beta$ be two distinct vertices in
$\mathcal{N}(R)$. We call $(\alpha, \beta)$ to be a
\textit{peripheral pair} in $\mathcal{N}(R)$ if they have disjoint
representatives $x, y$ respectively such that $x, y$ and a
boundary component of $R$ bound a pair of pants in $R$.
\begin{lemma}
\label{peripheral} Suppose $g \geq 2$ and $p \geq 1$. Let $\lambda
: \mathcal{N}(R) \rightarrow \mathcal{N}(R)$ be a superinjective
simplicial map and $(\alpha, \beta)$ be a peripheral pair in
$\mathcal{N}(R)$. Then $(\lambda(\alpha), \lambda(\beta))$ is a
peripheral pair in $\mathcal{N}(R)$.
\end{lemma}
\begin{proof} Let $x, y$ be disjoint representatives of $\alpha, \beta$
respectively such that $x, y$ and a boundary component of $R$
bound a pair of pants in $R$. If $g = 2, p=1$ then we complete $x,
y$ to a pair of pants decomposition $P$ consisting of
nonseparating circles $a, b$ on $R$. Then there are two distinct
pair pants in $P$, one of them has $a, b, x$ on its boundary and
the other has $a, b, y$ on its boundary. Let $P'$ be a pair of
pants decomposition of $R$ such that $\lambda([P]) = [P']$. Let
$x', y', a', b'$ be the representatives of $\lambda([x]),
\lambda([y]), \lambda([a]), \lambda([b]))$ in $P'$ respectively.
By the previous lemma, there exist two pairs of pants $Q_1$, $Q_2$
in $P'$ such that $Q_1$ has $a', b', x'$ on its boundary and $Q_2$
has $a', b', y'$ on its boundary. Then it is clear that $x', y'$
and the boundary component of $R$ bound a pair of pants, which
proves the lemma for this case. If $g \geq 3, p = 1$ then the
proof is similar.\\
If $g = 2, p=2$ then it is easy to see that we can complete $x, y$
to a pair of pants decomposition $P$ consisting of nonseparating
circles $a, b, z$ on $R$ such that $([y], [z])$ is a peripheral
pair, $a, b, x$ bound a pair of pants in $P$ and also $a, b, z$
bound a pair of pants in $P$. Let $P'$ be a pair of pants
decomposition of $R$ such that $\lambda([P]) = [P']$. Let $x', y',
z', a', b'$ be the representatives of $\lambda([x]), \lambda([y]),
\lambda([z]), \lambda([a]), \lambda([b]))$ in $P'$ respectively.
By the previous lemma, there exist two pairs of pants $Q_1$, $Q_2$
in $P'$ such that $Q_1$ has $a', b', x'$ on its boundary and $Q_2$
has $a', b', z'$ on its boundary. Then, since $x$ is adjacent to
$y$ w.r.t. $P$, $x'$ is adjacent to $y'$ w.r.t. $P'$. So, there
exists a pair of pants $Q_3$ in $P'$ having $x', y'$ on its
boundary. Let $w$ be the third boundary component of $Q_3$. It is
easy to see that $Q_1 \cup Q_2 \cup Q_3$ is a genus one surface
with three boundary components, $w, y', z'$. If $w = z'$ then $y'$
would be a separating curve which is a contradiction. If $w = y'$
then $z'$ would be a separating curve which is a contradiction.
Then it is clear that $w$ has to be a boundary component of $R$,
which proves the lemma for this case.\\
Assume that $g = 2, p \geq 3$. We choose distinct essential
circles $z, t, w$ as shown in Figure 3 (i). Then we complete $x,
y, z, t, w$ to a pair of pants decomposition $P$ consisting of
nonseparating circles in any way we like. Let $P'$ be a pair of
pants decomposition of $R$ such that $\lambda([P]) = [P']$. Let
$x', y', z', t', w'$ be the representatives of $\lambda([x]),
\lambda([y]), \lambda([z]), \lambda([t]), \lambda([w])$ in $P'$
respectively. Since $z$ is a 4-curve in $P$, $z'$ is a 4-curve in
$P'$. Since $x, z, w$ are the boundary components of a pair of
pants in $P$, by the previous lemma, $x', z', w'$ are the boundary
components of a pair of pants, $Q_1$, in $P'$. Similarly, since
$z, y, t$ are the boundary components of a pair of pants in $P$,
by the previous lemma, $z', y', t'$ are the boundary components of
a pair of pants, $Q_2$, in $P'$. Since $x$ is adjacent to $y$ in
$P$, $x'$ is adjacent to $y'$ in $P'$. Then there exists a pair of
pants $Q_3$ such that $Q_3$ has $x', y'$ on its boundary. Let $r$
be the boundary component of $Q_3$ different from $x'$ and $y'$.
Suppose that $r$ is not a boundary component of $R$. Then $r$ is
an essential circle. Then $Q_1 \cup Q_2 \cup Q_3$ is a genus one
subsurface with three boundary components $r, w', t'$ which are
nonseparating circles in $R$. Each two of $x', w', t'$ can be
connected by an arc in the complement of $Q_1 \cup Q_2 \cup Q_3$
in $R$. But this would be possible only if genus of $R$ is at
least 3. Since we assumed that $g=2$, we get a contradiction. So,
$r$ has to be a boundary component of $R$, i.e. $(\lambda(\alpha),
\lambda(\beta))$ is a peripheral pair in $\mathcal{N}(R)$.\\
Now, assume that $g=3, p \geq 2$. We choose distinct essential
circles $a_1, ..., a_6$ as shown in Figure 3 (ii). Then we
complete $x, y, a_1, ..., a_6$ to a pair of pants decomposition
$P$ consisting of nonseparating circles in any way we like. Let
$P'$ be a pair of pants decomposition of $R$ such that
$\lambda([P]) = [P']$. Let $x', y', a_1', ..., a_6'$ be the
representatives of $\lambda([x]), \lambda([y]), \lambda([a_1]),
..., \lambda([a_6])$ in $P'$ respectively. Since $a_1$ is a
4-curve in $P$, $a_1'$ is a 4-curve in $P'$. Since $x, a_1, a_3$
are the boundary components of a pair of pants in $P$, by the
previous lemma, $x', a_1', a_3'$ are the boundary components of a
pair of pants, $Q_1$, in $P'$. Similarly, since $y, a_1, a_2$ are
the boundary components of a pair of pants in $P$, by the previous
lemma, $y', a_1', a_2'$ are the boundary components of a pair of
pants, $Q_2$, in $P'$. By using similar arguments we can see that
$a_4'$ is a 4-curve, $a_3', a_4', a_6'$ are the boundary
components of a pair of pants, $Q_3$, in $P'$ and $a_2', a_4',
a_5'$ are the boundary components of a pair of pants, $Q_4$ in
$P'$. Since $x$ is adjacent to $y$ in $P$, $x'$ is adjacent to
$y'$ in $P'$. Then there exists a pair of pants $Q_5$ such that
$Q_5$ has $x', y'$ on its boundary. Let $r$ be the boundary
component of $Q_5$ different from $x'$ and $y'$. Suppose that $r$
is not a boundary component of $R$. Then $r$ is an essential
circle. Then $Q_1 \cup Q_2 \cup Q_3 \cup Q_4 \cup Q_5$ is a genus
two subsurface with three boundary components $r, a_5', a_6'$
which are nonseparating circles in $R$. Each two of $r, a_5',
a_6'$ can be connected by an arc in the complement of $Q_1 \cup
Q_2 \cup Q_3 \cup Q_4 \cup Q_5$ in $R$. But this would be possible
only if genus of $R$ is at least 4. Since we assumed that $g=3$,
we get a contradiction. So, $r$ has to be a boundary component of
$R$, i.e. $(\lambda(\alpha), \lambda(\beta))$ is a peripheral pair
in $\mathcal{N}(R)$. The proof of the case when $g \geq 4, p \geq
2$ is similar.\end{proof}
\begin{lemma}
\label{top} Suppose $g \geq 2$ and $p \geq 0$. Let $\lambda :
\mathcal{N}(R) \rightarrow \mathcal{N}(R)$ be a superinjective
simplicial map. Then $\lambda$ preserves topological equivalence
of ordered pairs of pants decompositions consisting of
nonseparating circles on $R$, (i.e. for a given ordered pair of
pants decomposition $P=(c_1, c_2, ..., c_{3g-3+p})$ of $R$ where
$[c_i] \in \mathcal{N}(R)$, and a corresponding ordered pair of
pants decomposition $P'=(c_1', c_2', ...,$ $ c_{3g-3+p}')$ of $R$,
where $[c_i']= \lambda([c_i])$ $\forall i= 1, 2, ..., 3g-3+p$,
there exists a homeomorphism $H: R \rightarrow R$ such that
$H(c_i)=c_i'$ $\forall i= 1, 2, ..., 3g-3+p$).\end{lemma}
\begin{proof} Let $P = (c_1, c_2, ..., c_{3g-3+p})$ be an ordered pair of pants
decomposition consisting of nonseparating circles on $R$. Let
$c_i' \in \lambda([c_i])$ such that the elements of $\{c_1', c_2',
..., c_{3g-3+p}'\}$ are pairwise disjoint. Then $P'=(c_1', c_2',
..., c_{3g-3+p}')$ is an ordered pair of pants decomposition of
$R$. Let $(B_1, B_2, ..., B_{m})$ be an ordered set of all the
pairs of pants in $P$. By Lemma \ref{embedded} and Lemma
\ref{peripheral}, there is a corresponding, ``image'', ordered
collection of pairs of pants $(B_1', B_2',..., B_{m}')$. A pair of
pants having three essential curves on its boundary corresponds to
a pair of pants having three essential curves on its boundary. A
pair of pants having an inessential boundary component corresponds
to a pair of pants having an inessential boundary component. Then
by the classification of surfaces, there exists an orientation
preserving homeomorphism $h_i : B_i \rightarrow B_i'$, for all $i
=1,..., m$. We can compose each $h_i$ with an orientation
preserving homeomorphism $r_i$ which switches the boundary
components, if necessary, to get $h_i'= r_i \circ h_i$ to agree
with the correspondence given by $\lambda$ on the boundary
components, (i.e. for each essential boundary component $a$ of
$B_i$ for $i=1,...,m$, $\lambda([q(a)])= [q'(h_i'(a))]$ where $q:
R_P \rightarrow R$ and $q': R_{P'} \rightarrow R$ are the natural
quotient maps). Then for two pairs of pants with a common boundary
component, we can glue the homeomorphisms by isotoping the
homeomorphism of the one pair of pants so that it agrees with the
homeomorphism of the other pair of pants on the common boundary
component. By adjusting these homeomorphisms on the boundary
components and gluing them we get a homeomorphism $h : R
\rightarrow R$ such that $h(c_i)=c_i'$ for all $i=1,2, ...,
3g-3+p$.\end{proof}
\begin{lemma} \label{Irmaklemma} Suppose $g \geq 2$ and $p \geq 0$.
Let $\alpha_{1}$ and $\alpha_{2}$ be two vertices in
$\mathcal{N}(R)$. Then $i( \alpha_{1}, \alpha_{2})=1$ if and only
if there exist isotopy classes $\alpha_{3}, \alpha_{4},
\alpha_{5}, \alpha_{6}, \alpha_{7}$ in $\mathcal{N}(R)$ such that
\indent (i) $i(\alpha_{i}, \alpha_{j})=0$ if and only if the
$i^{th}$ and $j^{th}$ circles on Figure 4 are disjoint.
\indent (ii) $\alpha_{1}, \alpha_{3}, \alpha_{5}, \alpha_{6}$ have
pairwise disjoint representatives $a_1, a_3, a_5, a_6$
respectively such that $a_5 \cup a_6$ divides $R$ into two pieces,
one of these is a torus with two holes, $T$, containing some
representatives of the isotopy classes $\alpha_1, \alpha_2$ and
$a_1, a_3, a_5$ bound a pair of pants in $T$ and $a_1, a_3, a_6$
bound a pair of pants in $T$.\end{lemma}
\begin{figure}
\caption{Circles
intersecting once}
\end{figure}
\begin{proof}Let $i(\alpha_{1}, \alpha_{2})=1$. Let $a_1, a_2$ be
representatives of $\alpha_{1}, \alpha_{2}$ respectively such that
$a_1$ intersects $a_2$ transversely once. Let $N$ be a regular
neighborhood of $a_1 \cup a_2$. Then it is easy to see that $N$ is
a genus one surface with one boundary component and since $R$ is a
surface of genus at least 2, there are simple closed curves as
shown in Figure 4 such that their isotopy classes $\alpha_{3},
\alpha_{4}, \alpha_{5}, \alpha_{6}, \alpha_{7}$ satisfy the
properties (i) and (ii). Suppose that there exist isotopy classes
$\alpha_{1}, \alpha_{2}, ..., \alpha_{7}$ in $\mathcal{N}(R)$
satisfying (i) and (ii). Then we have $a_1, a_3, a_5, a_6$ as
pairwise disjoint representatives of $\alpha_{1}, \alpha_{3},
\alpha_{5}, \alpha_{6}$ respectively such that $a_5 \cup a_6$
divides $R$ into two pieces, one of these is a torus with two
holes, $T$, containing some representatives of the isotopy classes
$\alpha_1, \alpha_2$ and $a_1, a_3, a_5$ bound a pair of pants $P$
in $T$, and $a_1, a_3, a_6$ bound a pair of pants $Q$ in $T$. Let
$a_2, a_4, a_7$ be representatives of $\alpha_{2}, \alpha_{4},
\alpha_{7}$ such that all the curves $a_i$, $i=1,...,7$ have
minimal intersection with each other. Then we have $a_4 \cap a_1 =
\emptyset, a_2 \cap a_6 = \emptyset, a_7 \cap a_3 = \emptyset$.
Since $i(\alpha_{4}, \alpha_{1}) = 0$, $i(\alpha_{4}, \alpha_{3})
\neq 0$ and $i(\alpha_{4}, \alpha_{6}) \neq 0$, $a_4 \cap Q$ has
an arc which connects $a_3$ to $a_6$. Since $i(\alpha_{7},
\alpha_{3}) = 0$, $i(\alpha_{7}, \alpha_{1}) \neq 0$ and
$i(\alpha_{7}, \alpha_{6}) \neq 0$, $a_7 \cap Q$ has an arc which
connects $a_1$ to $a_6$. Similarly, since $i(\alpha_{2},
\alpha_{6}) = 0$, $i(\alpha_{2}, \alpha_{1}) \neq 0$ and
$i(\alpha_{2}, \alpha_{3}) \neq 0$, $a_2 \cap Q$ has an arc which
connects $a_1$ to $a_3$. Then, since $a_2, a_4, a_7$ are pairwise
disjoint, we can see that all the arcs of $a_4 \cap Q$ connect
$a_3$ to $a_6$, all the arcs of $a_7 \cap Q$ connect $a_1$ to
$a_6$ and all the arcs of $a_2 \cap Q$ connect $a_1$ to $a_3$. By
using similar arguments, we can see that all the arcs of $a_4 \cap
P$ connect $a_3$ to $a_5$, all the arcs of $a_7 \cap P$ connect
$a_1$ to $a_5$, and all the arcs of $a_2 \cap P$ connect $a_1$ to
$a_3$. Then by looking at the gluing between the arcs in $a_2 \cap
Q$ and the arcs in $a_2 \cap P$ to form $a_2$, we see that $a_2
\cap Q$ has one arc and $a_2 \cap P$ has one arc. Hence,
$i(\alpha_{1}, \alpha_{2}) = 1$.\end{proof}\\
A characterization of geometric intersection one property in
$\mathcal{C}(R)$ was given by Ivanov, in Lemma 1 in \cite{Iv1}.
\begin{lemma} \label{intone} Suppose $g \geq 2$ and $p \geq
0$. Let $\lambda : \mathcal{N}(R) \rightarrow \mathcal{N}(R)$ be a
superinjective simplicial map. Let $\alpha$, $\beta$ be two
vertices of $\mathcal{N}(R)$. If $i(\alpha, \beta)=1$, then
$i(\lambda(\alpha), \lambda(\beta))=1$.\end{lemma}
\begin{proof} Let $\alpha$, $\beta$ be two vertices of $\mathcal{N}(R)$
such that $i(\alpha, \beta)=1$. Then by Lemma \ref{Irmaklemma},
there exist isotopy classes $\alpha_{3}, \alpha_{4}, \alpha_{5},
\alpha_{6}, \alpha_{7}$ in $\mathcal{N}(R)$ such that
$i(\alpha_{i}, \alpha_{j})=0$ if and only if $i^{th}, j^{th}$
circles on Figure 4 are disjoint and $\alpha_{1}, \alpha_{3},
\alpha_{5}, \alpha_{6}$ have pairwise disjoint representatives
$a_1, a_3, a_5, a_6$ respectively such that $a_5 \cup a_6$ divides
$R$ into two pieces, one of these is a torus with two holes, $T$,
containing some representatives of the isotopy classes $\alpha_1,
\alpha_2$ and $a_1, a_3, a_5$ bound a pair of pants in $T$ and
$a_1, a_3, a_6$ bound a pair of pants in $T$. Then, since
$\lambda$ is superinjective, $i(\lambda(\alpha_{i}),
\lambda(\alpha_{j}))=0$ if and only if $i^{th}, j^{th}$ circles on
Figure 4 are disjoint, and by using Lemma \ref{top} and the
properties that $\lambda$ preserves disjointness and
nondisjointness, we can see that there are pairwise disjoint
representatives $a_1', a_3', a_5', a_6'$ of $\lambda(\alpha_{1}),
\lambda(\alpha_{3}), \lambda(\alpha_{5}), \lambda(\alpha_{6})$
respectively, such that $a_5' \cup a_6'$ divides $R$ into two
pieces, one of these is a torus with two holes, $T$, containing
some representatives of the isotopy classes $\lambda(\alpha_1),
\lambda(\alpha_2)$ and $a_1', a_3', a_5'$ bound a pair of pants in
$T$ and $a_1', a_3', a_6'$ bound a pair of pants in $T$. Then by
Lemma \ref{Irmaklemma}, $i(\lambda(\alpha),
\lambda(\beta))=1$.\end{proof}
\section{Superinjective Simplicial Maps of $\mathcal{C}(R)$ and
Injective Homomorphisms of Finite Index Subgroups of $Mod_R^*$}
If $g = 2$, $p \geq 2$ or $g \geq 3$, $p \geq 0$, by the results
given in \cite{Ir1} and \cite{Ir2}, we have the following: A
simplicial map $\lambda : \mathcal{C}(R) \rightarrow
\mathcal{C}(R)$ is superinjective if and only if $\lambda$ is
induced by a homeomorphism of $R$. If $K$ is a finite index
subgroup of $Mod_R^*$ and $f:K \rightarrow Mod_R^*$ is an
injective homomorphism, then $f$ is induced by a homeomorphism of
$R$. In this section we prove similar results, Theorem
\ref{theorem3} and Theorem \ref{theorem4}, when $g = 2, p \leq 1$.\\
\noindent If $g=2$ and $p \leq 1$, and $\lambda : \mathcal{C}(R)
\rightarrow \mathcal{C}(R)$ is a superinjective simplicial map,
then $\lambda$ is an injective simplicial map which maps pair of
pants decompositions of $R$ to pair of decompositions of $R$ and
it preserves adjacency relation. The proofs of these are similar
to the proofs given in Lemma 1.1-1.3. Now, we prove the following
lemma:
\begin{lemma}
\label{embedded2} Suppose $g=2$ and $p \leq 1$. Let $\lambda :
\mathcal{C}(R) \rightarrow \mathcal{C}(R)$ be a superinjective
simplicial map and $\alpha, \beta, \gamma$ be distinct vertices in
$\mathcal{C}(R)$ having pairwise disjoint representatives which
bound a pair of pants in $R$. Then $\lambda(\alpha),
\lambda(\beta), \lambda(\gamma)$ are distinct vertices in
$\mathcal{C}(R)$ having pairwise disjoint representatives which
bound a pair of pants in $R$.\end{lemma}
\begin{proof} Let $a, b, c$ be pairwise disjoint representatives of
$\alpha, \beta, \gamma$ respectively. If $R$ is a closed surface
of genus two then $a, b, c$ is a pair of pants decomposition on
$R$ and since $a, b, c$ bound a pair of pants, they have to be all
nonseparating circles in this case. Let $P'$ be a pair of pants
decomposition of $R$ such that $\lambda([P]) = [P']$. Let $a', b',
c'$ be the representatives of $\lambda([a]), \lambda([b]),
\lambda([c])$ in $P'$ respectively. Since $a$ is adjacent to $b$
and $c$ w.r.t. $P$, $a'$ is adjacent to $b'$ and $c'$ w.r.t. $P'$.
If one of $a', b'$ or $c'$ was a separating curve on $R$ then the
other two wouldn't be adjacent to each other w.r.t. $P'$, which
would give a contradiction. So, each of $a', b', c'$ is a
nonseparating curve. Then clearly they bound a pair of pants on
$R$. Now assume that $g = 2, p = 1$. There are two cases: either
each of $a, b, c$ is a nonseparating circle or exactly one
of $a, b, c$ is a separating circle.\\
Case i: Assume that each of $a, b, c$ is a nonseparating circle
and complete $\{a, b, c\}$ to a pair of pants decomposition $P =
\{a, b, c, a_1\}$ consisting of nonseparating circles as shown in
Figure 5 (i). Let $P'$ be a pair of pants decomposition of $R$
such that $\lambda([P]) = [P']$. Let $a', b', c', a_1'$ be the
representatives of $\lambda([a]), \lambda([b]), \lambda([c]),
\lambda([a_1])$ in $P'$ respectively. Notice that any two curves
in $P$ are adjacent w.r.t. $P$. Since adjacency is preserved, any
two curves in $P'$ must be adjacent w.r.t. $P'$. If one of $a',
b', c', a_1'$ was a separating curve then there would be two
circles in $P'$ which are not adjacent. So, we conclude that all
of $a', b', c', a_1'$ are nonseparating. Then we consider the
curve configuration as shown in Figure 5 (i) and the proof of the
lemma in this case follows as in the proof of Lemma \ref{embedded}
for $g=2, p=1$ case.\\
Case ii: Assume that exactly one of $a, b, c$ is a separating
circle. W.L.O.G. assume that $c$ is a separating circle. Then $c$
separates $R$ into two subsurfaces $R_1, R_2$ as shown in Figure
5 (ii). Let $a_1, a_2, a_3$ be as shown in the figure. Then $a, b,
c, a_1$ is a pants decomposition $P$ on $R$. Let $P'$ be a pair of
pants decomposition of $R$ such that $\lambda([P]) = [P']$. Let
$a', b', c', a_1'$ be the representatives of $\lambda([a]),
\lambda([b]), \lambda([c]), \lambda([a_1])$ in $P'$ respectively.
Every nonseparating circle $x$ on $R$ could be put inside of a
pair of pants decomposition consisting of nonseparating circles,
and using the method given in case (i), we could see that
$\lambda([x])$ has a nonseparating representative. So, since $a,
b, a_1$ are nonseparating, $a', b', a_1'$ are nonseparating. Since
$a$ is adjacent to $c$ w.r.t. $P$, $a'$ is adjacent to $c'$ w.r.t.
$P'$. Then there is a pair of pants $Q_1$ having $a'$ and $c'$ on
its boundary. Let $x$ be the other boundary component of $Q_1$.
Since $a'$ is not a separating circle, $x$ can't be $c'$. If $x$
is the boundary component of $R$, then since $c'$ is adjacent to
$a_1'$ and $b'$, there exists a pair of pants $Q_2$ in $P'$ having
$c'$, $b'$ and $a_1'$ on its boundary. Then, since $a_2$
intersects $a_1$ essentially and $a_2$ is disjoint from $b$ and
$c$, we can choose a representative $a_2'$ of $\lambda([a_2])$
such that there exists an essential arc $w$ of $a_2'$ in $Q_2$
which starts and ends on $a_1'$ and which does not intersect $b'
\cup c'$. Now, we consider $a_3$ which is disjoint from $a_1$ and
$a_2$. Then there exists a representative $a_3'$ of
$\lambda([a_3])$ such that $a_3'$ is disjoint from $a_1' \cup w$.
But then $a_3'$ could be isotoped so that it is disjoint from
$b'$, since $b'$ is a boundary component of a regular neighborhood
of $a_1' \cup w$ in $Q_2$. This is a contradiction since $i([b],
[a_3]) \neq 0$ and so $i(\lambda([b]), \lambda([a_3])) \neq 0$.
So, $x$ can not be the boundary component of $R$. Then there are
three possibilities: $x$ is one of $a', b', a_1'$. If $x = a'$
then $a'$ wouldn't be adjacent to $b'$ w.r.t. $P'$ which is a
contradiction. Assume $x = a_1'$. Then, since $a_2$ intersects
$a_1$ essentially and $a_2$ is disjoint from $a$ and $c$, we can
choose a representative $a_2'$ of $\lambda([a_2])$ such that there
exists an essential arc $w$ of $a_2'$ in $Q_1$ which starts and
ends on $a_1'$ and which does not intersect $a' \cup c'$. Now, we
consider $a_3$ which is disjoint from $a_1$ and $a_2$. Then there
exists a representative $a_3'$ of $\lambda([a_3])$ such that
$a_3'$ is disjoint from $a_1' \cup w$. But then $a_3'$ could be
isotoped so that it is disjoint from $a'$, since $a'$ is a
boundary component of a regular neighborhood of $a_1' \cup w$ in
$Q_1$. This is a contradiction since $i([a ], [a_3]) \neq 0$ and
so $i(\lambda([a]), \lambda([a_3])) \neq 0$. So, $x \neq a_1'$.
Then $x= b'$, and $a', b', c'$ bound a pair of pants in $P'$.\end{proof}\\
Let $\alpha$, $\beta$ be two distinct vertices in
$\mathcal{C}(R)$. We call $(\alpha, \beta)$ to be a
\textit{peripheral pair} in $\mathcal{C}(R)$ if they have disjoint
representatives $x, y$ respectively such that $x, y$ and a
boundary component of $R$ bound a pair of pants in $R$.
\begin{lemma}
\label{peripheral2} Suppose $g = 2$ and $p=1$. Let $\lambda :
\mathcal{C}(R) \rightarrow \mathcal{C}(R)$ be a superinjective
simplicial map and $(\alpha, \beta)$ be a peripheral pair in
$\mathcal{C}(R)$. Then $(\lambda(\alpha), \lambda(\beta))$ is a
peripheral pair in $\mathcal{C}(R)$.
\end{lemma}
\begin{proof} Let $x, y$ be disjoint representatives of $\alpha, \beta$
respectively such that $x, y$ and a boundary component of $R$
bound a pair of pants in $R$. There are two cases to consider:
Case i: $x$ and $y$ are both nonseparating. Case ii: $x$ and $y$
are both separating. The proof in the first case is similar to the
proof given in Lemma \ref{peripheral}. For the second case, we
complete $x, y$ to a pair of pants decomposition $Q$ consisting of
nonseparating circles $a, b$ on $R$ such that $a$ is in the torus
with one hole which comes with the separation by $x$. Then we will
replace $y$ with a nonseparating curve $w$ such that $a, x, b, w$
is a pants decomposition $P$ on $R$, $x, w, b$ bound a pair of
pants and $w, b$ and the boundary component of $R$ bound a pair of
pants on $R$. Let $P'$ be a pair of pants decomposition of $R$
such that $\lambda([P]) = [P']$. Let $x', w', a', b'$ be the
representatives of $\lambda([x]), \lambda([w]), \lambda([a]),
\lambda([b]))$ in $P'$ respectively. Then by using case i and
Lemma \ref{embedded2}, we see that $x'$ is a separating circle of
genus one. Similarly, $y'$ is a separating circle of genus one on
$R$. Then it is easy to see that $x', y'$ and the boundary
component of $R$ bound a pair of pants.\end{proof}
\begin{lemma}
\label{top2} Suppose $g = 2$ and $p \leq 1$. Let $\lambda :
\mathcal{C}(R) \rightarrow \mathcal{C}(R)$ be a superinjective
simplicial map. Then $\lambda$ preserves topological equivalence
of ordered pairs of pants decompositions on $R$, (i.e. for a given
ordered pair of pants decomposition $P=(c_1, c_2, ..., c_{3+p})$
of $R$ where $[c_i] \in \mathcal{C}(R)$, and a corresponding
ordered pair of pants decomposition $P'=(c_1', c_2', ...,$ $
c_{3+p}')$ of $R$, where $[c_i']= \lambda([c_i])$ $\forall i= 1,
2, ..., 3 + p$, there exists a homeomorphism $H: R \rightarrow R$
such that $H(c_i)=c_i'$ $\forall i= 1, 2, ..., 3+p$).\end{lemma}
\begin{proof} First we will consider the case when $g = 2$ and $p=0$.
Let $P$ be a pair of pants decomposition of $R$ and $A$ be a
nonembedded pair of pants in $P$. The boundary of $A$ consists of
the circles $a, c$ where $c$ is a 1-separating circle on $R$ and
$a$ is a nonseparating circle on $R$. Let $b, a_1, a_2$ be as
shown in Figure 5 (iii). Then $P= \{a, b, c\}$ is a pants
decomposition on $R$. Let $P'$ be a pair of pants decomposition of
$R$ such that $\lambda([P]) = [P']$. Let $a', b', c'$ be the
representatives of $\lambda([a]), \lambda([b]), \lambda([c])$ in
$P'$ respectively. Every nonseparating circle $x$ on $R$ could be
put inside of a pair of pants decomposition consisting of
nonseparating circles, and using the method given in case (i) in
Lemma \ref{embedded2}, we could see that $\lambda([x])$ has a
nonseparating representative. So, since $a, b$ are nonseparating,
$a', b'$ are nonseparating. Assume that $c'$ is also
nonseparating. Then there exist two pairs of pants $Q_1, Q_2$ in
$P'$ such that both of them have $a', b', c'$ on their boundary.
Then, since $a_1$ intersects $a$ essentially and $a_1$ is disjoint
from $b$ and $c$, we can choose a representative $a_1'$ of
$\lambda([a_1])$ such that there exists an essential arc $w$ of
$a_1'$ in $Q_1$ which starts and ends on $a'$ and which does not
intersect $b' \cup c'$. Now, we consider $a_2$ which is disjoint
from $a$ and $a_1$. There exists a representative $a_2'$ of
$\lambda([a_2])$ such that $a_2'$ is disjoint from $a_1' \cup a'$.
But then $a_2'$ could be isotoped so that it is disjoint from
$b'$, since $b'$ is a boundary component of a regular neighborhood
of $a' \cup w$ in $Q_1$. This is a contradiction since $i([b],
[a_2]) \neq 0$ and so $i(\lambda([b]), \lambda([a_2])) \neq 0$.
So, $c'$ has to be separating. Then clearly $a'$ and $c'$ are the
boundary components of a nonembedded pair of pants. Hence,
nonembedded pair of pants in $P$ corresponds to a nonembedded pair
of pants in $P'$. When $g = 2$ and $p=1$, this result follows from
Lemma \ref{embedded2} and Lemma \ref{peripheral2} considering the
circles in Figure 5 (ii).\\
Suppose $g = 2$ and $p \leq 1$. Let $P = (c_1, c_2, ..., c_{3+p})$
be an ordered pair of pants decomposition on $R$. Let $c_i' \in
\lambda([c_i])$ such that the elements of $\{c_1', c_2', ...,
c_{3+p}'\}$ are pairwise disjoint. Then $P'=(c_1', c_2', ...,
c_{3+p}')$ is an ordered pair of pants decomposition of $R$. Let
$(B_1, B_2, ..., B_{m})$ be an ordered set of all the pairs of
pants in $P$. By Lemma \ref{embedded2}, Lemma \ref{peripheral2}
and the arguments given above, there is a corresponding,
``image'', ordered collection of pairs of pants $(B_1', B_2',...,
B_{m}')$. Nonembedded pairs of pants correspond to nonembedded
pairs of pants, embedded pairs of pants correspond to embedded
pairs of pants, and a pair of pants having an inessential boundary
component corresponds to a pair of pants having an inessential
boundary component. Then the proof follows as in Lemma \ref{top}.
\end{proof}
\begin{lemma} \label{intone2} Suppose $g = 2$ and $p \leq 1$. Let $\lambda :
\mathcal{C}(R) \rightarrow \mathcal{C}(R)$ be a superinjective
simplicial map. Let $\alpha$, $\beta$ be two vertices of
$\mathcal{C}(R)$. If $i(\alpha, \beta)=1$, then
$i(\lambda(\alpha), \lambda(\beta))=1$.\end{lemma}
\begin{proof} The proof follows as in the proof of Lemma \ref{intone},
using Lemma \ref{top2}.\end{proof}
\begin{lemma}
\label{horver2} Suppose $g = 2$ and $p \leq 1$. Let $\lambda :
\mathcal{C}(R) \rightarrow \mathcal{C}(R)$ be a superinjective
simplicial map. Let $\alpha$ and $\beta$ be two vertices in
$\mathcal{C}(R)$ which have representatives with geometric
intersection 2 and algebraic intersection 0 on $R$. Then
$\lambda(\alpha)$ and $\lambda(\beta)$ have representatives with
geometric intersection 2 and algebraic intersection 0 on $R$.
\end{lemma}
\begin{proof} Assume that $g=2$ and $p=0$. Let $h, v$ be representatives
of $\alpha, \beta$ with geometric intersection 2 and algebraic
intersection 0 on $R$. Let $N$ be a regular neighborhood of $h
\cup v$ in $R$. Then $N$ is a sphere with four boundary
components. Let $c, x, y, z$ be boundary components of $N$ such
that there exists a homeomorphism $\varphi : (N, c, x, y, z, h,
v)$ $\rightarrow (N_o, c_o, x_o, y_o, z_o, h_o, v_o)$ where $N_o$
is a standard sphere with four holes having $c_o, x_o, y_o, z_o$
on its boundary and $h_o, v_o$ (horizontal, vertical) are two
circles as indicated in Figure 6 (i). Since $h$ and $v$ have
geometric intersection 2 and algebraic intersection 0 on $R$, none
of $c, x, y, z$ bound a disk on $R$. There are two cases to
consider: either exactly one of $h$ or $v$ is separating or both
$h$ and $v$ are nonseparating. Case i: W.L.O.G. Assume that $v$ is
separating and $h$ is nonseparating. Then $x$ is isotopic to $z$
and $y$ is isotopic to $c$ and $c, x, h, v$ are as shown in Figure
6 (ii). Let $a_1, a_2$ be as shown in the figure. Let $c', h', x'$
be pairwise disjoint representatives of $\lambda([c]),
\lambda([h]), \lambda([x])$ respectively. Then $\{c', h', x'\}$ is
a pair of pants decomposition on $R$. Since nondisjointness and
intersection one is preserved by $\lambda$, there are disjoint
representatives $a_1', a_2'$ of $\lambda([a_1]), \lambda([a_2])$
respectively such that $a_1'$ intersects each of $c'$ and $h'$
exactly once and $a_1'$ is disjoint from $x'$, and similarly
$a_2'$ intersects each of $x'$ and $h'$ exactly once and $a_2'$ is
disjoint from $c'$. Then, since $v$ is disjoint from $c \cup a_1
\cup a_2 \cup x$, we can choose a representative $v'$ of
$\lambda([v])$ such that it lives inside of the cylinder that we
get after cutting $R$ along $c' \cup a_1' \cup a_2' \cup x'$. Then
it is easy to see that $h'$ and $v'$ intersects geometrically
twice and algebraically zero times.\\
Case ii: Assume that both of $h, v$ are nonseparating. Then $c$
and $z$ are the boundary components of an annulus, $A_1$, in $R
\setminus N$ and x and y are the boundary components of an
annulus, $A_2$, in $R \setminus N$. Let $i$ and $j$ be as shown in
Figure 7 (i). We connect the ends points of $i$ on $N$ with an arc
in $A_1$ to get a circle $w$ such that $w$ intersects each of $c$
and $z$ at only one point. Similarly, we connect the ends points
of $j$ on $N$ with an arc in $A_2$ to get a circle $k$ such that
$k$ intersects each of $x$ and $y$ at only one point. Notice that
$w$ and $k$ intersect at only one point. Let $c', x', h', y', z'$
be pairwise disjoint representatives of the image curves. Let $N'$
be sphere with four holes bounded by $c', x', y', z'$. By Lemma
\ref{top2}, $h'$ gives a pants decompositions on $N'$. Now we
choose minimally intersecting representatives $w', k'$ of
$\lambda([w]), \lambda([k])$ respectively such that each of $w'$
and $k'$ intersects $c', x', y', z'$ minimally. Then, since the
intersection one is preserved, there are arcs $i'$, $j'$ of $w'$,
$k'$ in $N'$ respectively such that $i'$ intersects $j'$ once in
$N'$, $i'$ connects $c'$ to $z'$ and $j'$ connects $x'$ to $y'$.
Since $v$ does not intersect any of $x, c, y, z$, and $v$
intersects $w$ and $k$ exactly once, there is a representative
$v'$ of $\lambda([v])$ in $N'$ such that $v'$ intersects each of
$i'$ and $j'$ exactly once. Notice that $h'$ also intersects each
of $i'$ and $j'$ exactly once. When we cut $N'$ along the arcs of
$i'$ and $j'$ we get a disk. The boundary of the disk either is as
shown in Figure 7 (ii) or it has the similar form where only $x'$
and $y'$ are switched. We will consider the first case (the
arguments follow in the second case similarly). If $v'$ makes its
intersection with $i'$ and $j'$ at the intersection point of $i'$
and $j'$ with each other, then $v'$ has to be one of the arcs
shown in Figure 7 (ii), and by looking at the intersection of $v'$
with the other curves, it is easy to see that $v'$ intersects $h'$
geometrically twice and algebraically 0 times on $R$. Suppose that
$v'$ intersects $i'$ and $j'$ at different points. Then there are
two arcs of $v'$ connecting $i'$ to $j'$ in the disk. Then, since
$v'$ does not intersect any of $x', c', y', z'$ we see that $v'$
has to be one of the curves as shown in Figure 7 (iii), (iv).
Since $v'$ is not isotopic to $h'$, $v'$ has to be the curve shown
in Figure 7 (iii) and hence $v'$ intersects $h'$ geometrically
twice and algebraically 0 times on $R$. If $g = 2$ and $p=1$ the
proof is similar (for Case (i) we use Figure 6
(iii)).\end{proof}\\
The proof of Theorem \ref{theorem3} follows from Lemma \ref{top2},
Lemma \ref{intone2}, Lemma \ref{horver2}, and the techniques given
for the proof of Theorem 1.1 in \cite{Ir1}, \cite{Ir2}.\\
A mapping class $g \in Mod_R^*$ is called \textit{pseudo-Anosov}
if $\mathcal{A}$ is nonempty and if $g ^n (\alpha) \neq \alpha$,
for all $\alpha$ in $\mathcal{A}$ and any $n \neq 0$. $g$ is
called \textit{reducible} if there is a nonempty subset $
\mathcal{B} \subseteq \mathcal{A}$ such that a set of disjoint
representatives can be chosen for $\mathcal{B}$ and
$g(\mathcal{B}) = \mathcal{B}$. In this case, $ \mathcal{B}$ is
called a \textit{reduction system} for $g$. Each element of
$\mathcal{B}$ is called a \textit{reduction class} for $g$. A
reduction class, $\alpha$, for $g$, is called an \textit{essential
reduction class} for $g$, if for each $\beta \in \mathcal{A}$ such
that $i(\alpha, \beta) \neq 0$ and for each integer $m \neq 0$,
$g^m (\beta) \neq \beta$. The set, $\mathcal{B}_g$, of all
essential reduction classes for $g$ is called the
\textit{canonical reduction system} for $g$.\\
Let $\Gamma'=ker(\varphi)$ where $\varphi: Mod_R^* \rightarrow
Aut(H_1(R, \mathbb{Z}_3))$ is the homomorphism defined by the
action of homeomorphisms on the homology. The proofs of the Lemma
\ref{rank=1} and Lemma \ref{reducible} follow by the techniques
given in \cite{Ir1}. Note that we need to use that the maximal
rank of an abelian subgroup of $Mod_R^*$ is $3g-3+p$, \cite{BLM}.
\begin{lemma}
\label{rank=1} Suppose $g=2$ and $p \leq 1$. Let $K$ be a finite
index subgroup of $Mod_R^*$ and $f:K \rightarrow Mod_R^*$ be an
injective homomorphism. Let $\alpha \in \mathcal{A}$. Then there
exists $N \in \mathbb{Z^*}$ such that $rank$ $C(C_{\Gamma'}
(f(t_{\alpha} ^{N})) ) = 1$.\end{lemma}
\begin{lemma}
\label{reducible} Suppose $g=2$ and $p \leq 1$. Let K be a finite
index subgroup of $Mod_R^*$. Let $f:K \rightarrow Mod_R^*$ be an
injective homomorphism. Then there exists $N \in \mathbb{Z^*}$
such that $f(t_{\alpha} ^ N)$ is a reducible element of infinite
order for all $\alpha \in \mathcal{A}$.
\end{lemma}
In the proof of Lemma \ref{reducible}, we use that the centralizer
of a p-Anosov element in the extended mapping class group is a
virtually infinite cyclic group, \cite{Mc2}.
\begin{lemma}
\label{correspondence} Suppose $g=2$ and $p \leq 1$. Let $K$ be a
finite index subgroup of $Mod_R^*$ and $f:K \rightarrow Mod_R^*$
be an injective homomorphism. Then $\forall \alpha \in
\mathcal{A}$, $f( t_\alpha ^N)= t_{\beta(\alpha)}^M$ for some $M,
N \in \mathbb{Z^*}$, $\beta(\alpha) \in \mathcal{A}$.
\end{lemma}
\begin{proof} Let $\Gamma= f^{-1}(\Gamma') \cap \Gamma'$. Since
$\Gamma$ is a finite index subgroup we can choose $N \in Z^*$ such
that $t_\alpha^N \in \Gamma$ for all $\alpha$ in $\mathcal{A}$. By
Lemma \ref{reducible} $f(t_{\alpha} ^ N)$ is a reducible element
of infinite order in $Mod_R^*$. Let $C$ be a realization of the
canonical reduction system of $f(t_{\alpha}^N)$. Let $c$ be the
number of components of $C$ and $r$ be the number of p-Anosov
components of $f(t_{\alpha} ^N)$. Since $t_{\alpha} ^ N \in
\Gamma, f(t_{\alpha} ^ N) \in \Gamma'$. By Theorem 5.9 \cite{IMc},
$C(C_{\Gamma'} (f(t_{\alpha} ^ N )))$ is a free abelian group of
rank $c+r$. By Lemma \ref{rank=1} $c+r=1$. Then either $c=1$,
$r=0$ or $c=0$, $r=1$. Since there is at least one curve in the
canonical reduction system we have $c=1$, $r=0$. Hence, since
$f(t_{\alpha} ^ N) \in \Gamma'$, $f(t_{\alpha} ^{N}) = t_{\beta
({\alpha})}^{M}$ for some $M \in \mathbb{Z^*}$, $\beta(\alpha) \in
\mathcal{A}$, \cite{BLM}, \cite{IMc}.\end{proof}\\
\noindent {\bf Remark:} Suppose that $f(t_{\alpha} ^{M}) = t_\beta
^P$ for some $\beta \in \mathcal{A}$ and $M, P \in \mathbb{Z^*}$
and $f(t_{\alpha} ^{N}) = t_\gamma ^Q$ for some $\gamma \in
\mathcal{A}$ and $N, Q \in \mathbb{Z^*}$. Since $f(t_{\alpha} ^{M
\cdot N}) = f(t_{\alpha} ^{N \cdot M})$, $t_\beta ^{PN} = t_\gamma
^{QM}$, $P, Q, M, N \in \mathbb{Z^*}$. Then $\beta = \gamma$.
Therefore, by Lemma \ref{correspondence}, $f$ gives a
correspondence between isotopy classes of circles and $f$ induces
a map, $f_*: \mathcal{A} \rightarrow \mathcal{A}$,
where $f_*(\alpha) = \beta(\alpha)$.\\
In the following lemma we use a well known fact that $f t_\alpha
f^{-1}=t_{f(\alpha)} ^{\epsilon(f)}$ for all $\alpha$ in
$\mathcal{A}$, $f \in Mod_R^*$, where $\epsilon(f) = 1$ if $f$ has
an orientation preserving representative and $\epsilon(f) = -1$ if
$f$ has an orientation reversing representative.
\begin{lemma}
\label{identity} Let $K$ be a finite index subgroup of $Mod_R^*$.
Let $f:K \rightarrow Mod_R^*$ be an injective homomorphism. Assume
that there exists $N \in \mathbb{Z}^*$ such that $\forall \alpha
\in$ $\mathcal{A}$, $\exists Q \in \mathbb{Z}^*$ such that
$f(t_{\alpha} ^N) = t_{\alpha}^Q$. If $g=2$ and $p=1$ then $f$ is
the identity on $K$. If $g=2$ and $p=0$ then $f(k) = ki^{m(k)}$
where $i$ is the hyperelliptic involution on $R$ and $m(k) \in
\{0,1\}$.\end{lemma}
\begin{proof}
We use Ivanov's trick to see that $f(kt_{\alpha} ^ N k^{-1})=$
$f(t_{k(\alpha)} ^{\epsilon{(k)} \cdot N }) = t_{k(\alpha)} ^{Q
\cdot \epsilon{(k)}}$ and $f(kt_{\alpha} ^ N k^{-1}) = f(k)
f(t_{\alpha} ^N) f(k)^{-1}=$ $f(k) t_{\alpha} ^Q f(k)^{-1} =
t_{f(k)(\alpha)} ^{\epsilon(f(k))\cdot Q}$ $\forall \alpha \in
\mathcal{A}$, $\forall k \in K$. Then we have $t_{k(\alpha)} ^{Q
\cdot \epsilon{(k)}} = t_{f(k)(\alpha)} ^{\epsilon(f(k)) \cdot Q}$
$\forall \alpha \in \mathcal{A}$, $\forall k \in K$. Hence,
$k(\alpha) = f(k)(\alpha)$ $\forall \alpha \in \mathcal{A}$,
$\forall k \in K$. Then $k^{-1}f(k)(\alpha) = \alpha$ $\forall
\alpha \in \mathcal{A}$, $\forall k \in K$. Therefore,
$k^{-1}f(k)$ commutes with $t_{\alpha}$ $\forall \alpha \in
\mathcal{A}$, $\forall k \in K$. Then, since $Mod_R$ is generated
by Dehn twists when $g=2, p \leq 1$, $k^{-1}f(k) \in C(Mod_R)$
$\forall k \in K$. If $g=2$ and $p=1$, then $C(Mod_R)$ is trivial
by 5.3 in \cite{IMc}. So, $k = f(k)$ $\forall k \in K$. Hence,
$f=id_K$. If $g=2$ and $p=0$, then $C(Mod_R) = \{id_R, i\} =
\mathbb{Z}_2$, where $i$ is the hyperelliptic involution on $R$.
Then for each $k \in K$ either $k^{-1}f(k)=id_R$ or $k^{-1}f(k) =
i$. So, $f(k) = ki^{m(k)}$ where $m(k) \in \{0,1\}$.\end{proof}
\begin{coroll}
\label{id} Suppose $g=2$ and $p=1$. Let $h: Mod_R^* \rightarrow
Mod_R^*$ be an isomorphism and $f : Mod_R^* \rightarrow Mod_R^*$
be an injective homomorphism. Assume that there exists $N \in
\mathbb{Z}^*$ such that $\forall \alpha \in$ $\mathcal{A}$,
$\exists Q \in \mathbb{Z}^*$ such that $h(t_{\alpha} ^N) =
f(t_{\alpha}^Q)$. Then $h=f$.
\end{coroll}
\begin{proof} Apply Lemma \ref{identity} to $h^{-1} f$ with $K = Mod_R^*$.
Since for all $\alpha$ in $\mathcal{A}$, $h^{-1} f(t_{\alpha} ^N)
= t_{\alpha} ^{Q}$, we have $h^{-1} f = id_K$. Hence, $h =
f$.\end{proof}\\
By the remark after Lemma \ref{correspondence}, we have that $f: K
\rightarrow Mod_R^*$ induces a map $f_*: \mathcal{A} \rightarrow
\mathcal{A}$, where $K$ is a finite index subgroup of $Mod_R^*$.
In the following lemma we prove that $f_*$ is a superinjective
simplicial map on $\mathcal{C}(R)$.
\begin{lemma}
\label{intersection0} Suppose $g=2$ and $p \leq 1$. Let $f:K
\rightarrow Mod_R^*$ be an injection. Let $\alpha$, $\beta \in
\mathcal{A}$. Then $i(\alpha,\beta)=0 \Leftrightarrow
i(f_{*}(\alpha), f_{*}(\beta))=0$.
\end{lemma}
\begin{proof} There exists $N \in \mathbb {Z^*}$ such that
$t_{\alpha} ^N \in K$ and $t_{\beta} ^N \in K$. Then we have the
following: $i(\alpha, \beta)=0$ $\Leftrightarrow$ $t_{\alpha} ^N
t_\beta ^N = t_{\beta} ^N t_\alpha ^N$ $\Leftrightarrow$
$f(t_{\alpha} ^N) f(t_{\beta} ^ N) = f(t_{\beta} ^ N) f(t_{\alpha}
^ N)$ (since $f$ is injective on K) $\Leftrightarrow$
$t_{f_*(\alpha)} ^P t_{f_*(\beta)}^Q = t_{f_*(\beta)}^Q
t_{f_*(\alpha)} ^P$ where $P = M(\alpha, N), Q = M(\beta, N) \in
\mathbb{Z}^*$ $ \Leftrightarrow i(f_{*}(\alpha),
f_{*}(\beta))=0$.\end{proof}\\
Now, we prove the second main theorem of the section.
\begin{theorem}
\label{main4} Let $K$ be a finite index subgroup of $Mod_R^*$ and
$f$ be an injective homomorphism $f:K \rightarrow Mod_R^*$. If $g
= 2$ and $p=1$ then $f$ has the form $k \rightarrow hkh^{-1}$ for
some $h \in Mod_R^*$ and $f$ has a unique extension to an
automorphism of $Mod_R^*$. If $R$ is a closed surface of genus 2,
then $f$ has the form $k \rightarrow hkh^{-1} i^{m(k)}$ for some
$h \in Mod_R^*$ where $m$ is a homomorphism $K \rightarrow
\mathbb{Z}_2$ and $i$ is the hyperelliptic involution on $R$.
\end{theorem}
\begin{proof} If $g=2$ and $p \leq 1$, by Lemma \ref{intersection0} $f_*$
is a superinjective simplicial map on $\mathcal{C}(R)$. Then by
Theorem \ref{theorem3} $f_*$ is induced by a homeomorphism $h:R
\rightarrow R$, i.e. $f_*(\alpha) = h_\#(\alpha)$ for all $\alpha$
in $\mathcal{A}$, where $h_\#=[h]$. Let $\chi ^ {h\#}: Mod_R^*
\rightarrow Mod_R^*$ be the isomorphism defined by the rule $\chi
^ {h_\#}(k) = h_\#k{h_\#}^{-1}$ for all $k$ in $Mod_R^*$. Then for
all $\alpha$ in $\mathcal{A}$, we have the following:
$\chi ^{h_\# ^{-1}} \circ f ({t_ \alpha} ^N) = \chi ^{h_\# ^{-1}}
(t_{f_*(\alpha)}^ M) = \chi ^{h_\# ^{-1}} (t_{h_\#(\alpha)} ^M) =
h_\#^{-1} t_{h_\#(\alpha)} ^M h_\# = t_ {h_\#^{-1} (h_\#(\alpha))}
^{M \cdot \epsilon{(h_\#^{-1})}} = t_\alpha ^{M \cdot
\epsilon{(h_\#^{-1})}}$.\\
If $g=2$ and $p=1$, then since $\chi ^{h_\#^{-1}} \circ f$ is
injective, $\chi ^{h_\#^{-1}} \circ f = id_K$ by Lemma
\ref{identity}. So, $\chi ^h_\# |_K = f$. Hence, $f$ is the
restriction of an isomorphism which is conjugation by $h_\#$,
(i.e. $f$ is induced by $h$). Suppose that there exists an
automorphism $\tau : Mod_R^* \rightarrow Mod_R^*$ such that $\tau
|_{K}=f$. Let $N \in Z^*$ such that $ t_\alpha ^N \in K$ for all
$\alpha$ in $\mathcal{A}$. Since $\chi ^h_\# |_K = f = \tau |_K$
and $t_\alpha ^N \in K$, $\tau(t_\alpha ^N) = \chi ^h_\#(t_\alpha
^N)$ for all $\alpha$ in $\mathcal{A}$. Then by Corollary
\ref{id}, $\tau = \chi ^ {h_\#}$. Hence, the extension of $f$ is unique.\\
If $g=2$ and $p=0$, then since $\chi ^{h_\#^{-1}} \circ f$ is
injective, $\chi ^{h_\#^{-1}} \circ f (k) = ki^{m(k)}$ where $m(k)
\in \{0,1\}$ by Lemma \ref{identity}. So, $f$ has the form $k
\rightarrow h_\# k h_\# ^{-1} i^{m(k)}$. Since $\chi ^{h_\# ^{-1}}
\circ f (k_1 k_2) = k_1 k_2 i^{m(k_1 k_2)}$ and $\chi ^{h_\#
^{-1}} \circ f (k_1) \chi ^{h^{-1}_\# } \circ f ( k_2)$ $ = k_1
i^{m(k_1)} k_2 i^{m(k_2)} = k_1 k_2 i^{m(k_1)} i^{m(k_2)}$, we
have that $m(k_1 k_2) = m(k_1) + m(k_2)$ for all $k_1, k_2 \in K$.
So, $m: K \rightarrow \mathbb{Z}_2$ is a homomorphism.\end{proof}\\
\noindent {\bf Remark:} Note that $k \rightarrow hkh^{-1}
i^{m(k)}$ defines a homomorphism from $K \rightarrow Mod_R^*$ for
every $h \in Mod_R^*$ and for every homomorphism $m : K
\rightarrow \mathbb{Z}_2$. It is easy to see that $k \rightarrow
hkh^{-1} i^{m(k)}$ is injective if and only if either $i \notin K$
or $i \in Ker(m)$. Inner automorphisms of $K$ act on the set of
injective homomorphisms from $K \rightarrow Mod_R^*$. By using
Theorem \ref{main4} we can see that the orbit space $InjHom(K,
Mod_R^*)/Inn(K)$ of this action is finite. Then we have the
following corollary.
\begin{coroll} Suppose that $g=2$ and $p \leq 1$. Let $K$ be a finite
index subgroup of $Mod_R^*$. Then $Out(K)$ is finite.
\end{coroll}
In the other cases, when $R$ has genus at least two and $K$ is a
finite index subgroup, we have that $Out(K)$ is finite as a
corollary to the main results in \cite{Ir1}, \cite{Ir2}. See
\cite{Mc1} for an explicit description of automorphisms of
$Mod_R^*$ for a closed surface of genus two.
\section{Extending Superinjective Simplicial Maps of $\mathcal{N}(R)$ to
Superinjective Simplicial Maps of $\mathcal{C}(R)$}
In this section we prove Theorem \ref{theorem1} and Theorem
\ref{theorem2}.
\begin{lemma} \label{extension} Suppose that $g \geq 2$ and $p \leq 1$.
Let $\lambda : \mathcal{N}(R) \rightarrow
\mathcal{N}(R)$ be a superinjective simplicial map. Then $\lambda$
extends to a superinjective simplicial map $\lambda_* :
\mathcal{C}(R) \rightarrow \mathcal{C}(R)$.\end{lemma}
\begin{proof} If $x$ is a nonseparating simple closed curve, we define
$\lambda_*([x])=\lambda([x])$. Let $c$ be a separating simple
closed curve on $R$.\\
Case 1: Assume that $R$ is closed. Since $g \geq 2$, $c$ separates
$R$ into two subsurfaces $R_1, R_2$, and both of $R_1, R_2$ have
genus at least one. We take a chain on $R_1$, $\{\alpha_1, ...,
\alpha_m\}$ with $i( \alpha_{i}, \alpha_{i+1})=1$, $i( \alpha_{i},
\alpha_{j})=0$ for $|i-j| > 1$, $[a_i] = \alpha_i \in
\mathcal{N}(R)$, as shown in Figure 8, i (for $g=4$ case when
$R_1$ has genus 3), such that $R_1 \cup \{c\}$ is a regular
neighborhood of $a_1 \cup ... \cup a_n$. Since $\lambda$ preserves
disjointness, nondisjointness and intersection one property, we
can see that the chain $\{\alpha_1, ..., \alpha_n \}$ is mapped by
$\lambda$ into a similar chain, $\{\lambda(\alpha_1), ...,
\lambda(\alpha_m) \}$ with $i( \lambda(\alpha_{i}),
\lambda(\alpha_{i+1}))=1$, $i(\lambda(\alpha_{i}),
\lambda(\alpha_{j}))=0$ for $|i-j| > 1$. Let $a_i' \in
\lambda(\alpha_i)$ such that any two elements in $\{a_1', ...,
a_m'\}$ have minimal intersection with each other. Let $M$ be a
regular neighborhood of $a_1' \cup ... \cup a_n'$. Then it is easy
to see that $M$ is homeomorphic to $R_1 \cup c$. Let $a'$ be the
boundary of $M$. Suppose that we have another chain on $R_1$,
$\{\beta_1, ..., \beta_m \}$ with $i( \beta_{i}, \beta_{i+1})=1$,
$i( \beta_{i}, \beta_{j})=0$ for $|i-j| > 1$, $[b_i] = \beta_i \in
\mathcal{N}(R)$ such that $R_1 \cup \{c\}$ is a regular
neighborhood of $b_1 \cup ... \cup b_m$. Again we see that the
chain $\{\beta_1, ..., \beta_m \}$ is mapped by $\lambda$ into a
similar chain, $\{\lambda(\beta_1), ..., \lambda(\beta_m) \}$ with
$i( \lambda(\beta_{i}), \lambda(\beta_{i+1}))=1$,
$i(\lambda(\beta_{i}), \lambda(\beta_{j}))=0$ for $|i-j| > 1$. Let
$b_i' \in \lambda(\beta_i)$ such that any two elements in $\{b_1',
..., b_m'\}$ have minimal intersection with each other. Let $T$ be
a regular neighborhood of $b_1' \cup ... \cup b_n'$. Then $T$ is
homeomorphic to $R_1 \cup c$. Let $b'$ be the boundary of $T$.\\
Claim: $[a'] = [b']$.\\
Proof: We take a similar chain on $R_2$, $\{\gamma_1, ...,
\gamma_n \}$ with $i(\gamma_{i}, \gamma_{i+1})=1$, $i( \gamma_{i},
\gamma_{j})=0$ for $|i-j| > 1$, $[c_i] = \gamma_i \in
\mathcal{N}(R)$, such that $R_2 \cup \{c\}$ is a regular
neighborhood of $c_1 \cup ... \cup c_n$. This chain is mapped into
a similar chain, $\{\lambda(\gamma_1), ..., \lambda(\gamma_n) \}$.
Let $c_i' \in \lambda(\gamma_i)$ such that any two elements in
$\{a_1', ..., a_n', c_1', ..., c_m'\}$ and $\{b_1', ..., b_n',
c_1', ..., c_m'\}$ have minimal intersection with each other. Then
$a_i'$ is disjoint from $c_j'$ for any $i, j$, and $b_i'$ is
disjoint from $c_j'$ for any $i, j$. Then we can choose a regular
neighborhood $N$ of $c_1' \cup ... \cup c_n'$ in $R$ such that $N$
is disjoint from $M \cup T$. Let $c'$ be the boundary component of
$N$. Then it is easy to see that $N$ is homeomorphic to $R_2 \cup
c$ and the boundary components of $M$ and $N$ are isotopic in $R$.
Similarly, the boundary components of $T$ and $N$ are isotopic in
$R$. Hence, $[a'] = [c'] = [b']$. We define $\lambda_*([c]) = [a']$.\\
Claim: $\lambda_*: \mathcal{C}(R) \rightarrow \mathcal{C}(R)$ is
a simplicial map.\\
Proof: Let $\alpha, \beta$ be two vertices in $\mathcal{C(R)}$
such that $i(\alpha, \beta)=0$. Let $x$ and $y$ be disjoint
representatives of $\alpha$ and $\beta$ respectively. If $x, y$
are two nonseparating simple closed curves then we have
$i(\lambda_*(\alpha), \lambda_*(\beta)) = i(\lambda(\alpha),
\lambda(\beta))=0$. If $x$ is a nonseparating simple closed curve
and $y$ is a separating simple closed curve then $x$ lives in a
subsurface $R_1$ which comes from separation by $y$. Then we could
choose a chain as described above which contains $x$, and then see
that by the construction of the image of $[y]$, we get that
$i(\lambda_*([x]), \lambda_*([y]))=0$. If both $x$ and $y$ are
separating, then it is easy to see that $R \setminus ((R \setminus
{x}) \cap (R \setminus {y}))$ has two connected components $T_1,
T_2$ such that $T_1$ is disjoint from $T_2$ and $y$ is an
essential boundary component of $T_1$ and $x$ is an essential
boundary component of $T_2$. Then we see that the chains which
come from disjoint subsurfaces $T_1$ and $T_2$ which will be used
to define the images of $[x]$ and $[y]$ are disjoint. Since,
$\lambda$ preserves disjointness we see that the ``image" chains
will be disjoint, and hence, $i(\lambda_*([x]),
\lambda_*([y]))=0$.\\
\begin{figure}
\caption{Chains}
\end{figure}
Claim: $\lambda_*: \mathcal{C}(R) \rightarrow \mathcal{C}(R)$ is
a superinjective simplicial map.\\
Proof: Let $\alpha, \beta$ be two vertices in $\mathcal{C(R)}$
such that $i(\alpha, \beta) \neq 0$. Let $x$ and $y$ be
representatives of $\alpha$ and $\beta$ respectively.\\
(i) Assume that $x$ and $y$ are two nonseparating simple closed
curves. Then we have $i(\lambda_*(\alpha), \lambda_*(\beta)) =
i(\lambda(\alpha), \lambda(\beta)) \neq 0$ (since $\lambda$ is
superinjective).\\
(ii) Assume that $y$ is a nonseparating simple closed curve and
$x$ is a separating simple closed curve, and $x$ separates $R$
into two subsurfaces $R_1, R_2$. W.L.O.G. assume that $y$ is in
$R_1$. Let $\{\alpha_1, ..., \alpha_m\}$ be a chain on $R_1$ with
$i( \alpha_{i}, \alpha_{i+1})=1$, $i( \alpha_{i}, \alpha_{j})=0$
for $|i-j| > 1$, $[a_i] = \alpha_i \in \mathcal{N}(R)$ such that
$R_1 \cup \{x\}$ is a regular neighborhood of $a_1 \cup ... \cup
a_m$. Then, since $i(\alpha, \beta) \neq 0$, $i(\beta, \alpha_i)
\neq 0$ for some $i$. Then $i(\lambda_*(\beta),
\lambda_*(\alpha_i)) = i(\lambda(\beta), \lambda(\alpha_i)) \neq
0$ (since $\lambda$ is superinjective). Then it is easy to see
that $i(\lambda_*(\alpha),
\lambda_*(\beta)) \neq 0$.\\
(iii) Assume that both $x$ and $y$ are separating and $x$
separates $R$ into two subsurfaces $R_1, R_2$. Let $\{\alpha_1,
..., \alpha_m\}$ be a chain on $R_1$ with $i( \alpha_{i},
\alpha_{i+1})=1$, $i( \alpha_{i}, \alpha_{j})=0$ for $|i-j| > 1$,
$[a_i] = \alpha_i \in \mathcal{N}(R)$ such that $R_1 \cup \{x\}$
is a regular neighborhood of $a_1 \cup ... \cup a_m$. Then, since
$i(\alpha, \beta) \neq 0$, $i(\beta, \alpha_i) \neq 0$ for some
$i$. Then $i(\lambda_*(\beta), \lambda_*(\alpha_i)) \neq 0$ by
(ii). Then it is easy to see that $i(\lambda_*(\alpha),
\lambda_*(\beta)) \neq 0$. Hence, we have a superinjective
simplicial extension $\lambda_*: \mathcal{C}(R) \rightarrow
\mathcal{C}(R)$ of $\lambda: \mathcal{N}(R) \rightarrow
\mathcal{N}(R)$.\\
Case 2: Assume that $R$ has one boundary component. Then $c$
separates $R$ into two subsurfaces $R_1, R_2$. W.L.O.G. assume
that $R_1$ is a genus $k$ subsurface having $c$ as its boundary.
We consider chains on $R_1$ as in the first case, and chains on
$R_2$ such that regular neighborhoods of the curves coming from
the chains have $c$ as their essential boundary component and the
boundary of $R$ as their inessential boundary component (see
Figure 7, ii). Then it is easy to see that the proof of the lemma
is similar to case 1.\end{proof}\\
\noindent {\bf Remark:} If $g \geq 2$ and $p \geq 2$ and $C$ is
the set of separating circles on $R$ which separate $R$ into two
pieces such that each piece has genus at least one, then by using
chains on these two pieces and following the techniques in the
previous lemma we can extend $\lambda$ to $\lambda_*$ on $C$ and
get a superinjective extension.\\
Let $M$ be a sphere with $k$ holes and $k \geq 5$. A circle $a$ on
$M$ is called an {\it n-circle} if $a$ bounds a disk with $n$
holes on $M$ where $n \geq 2$. A {\it pentagon} in
$\mathcal{C}(M)$ is an ordered 5-tuple $(\alpha_1, \alpha_2,
\alpha_3, \alpha_4, \alpha_5)$, defined up to cyclic permutations,
of vertices of $\mathcal{C}(M)$ such that $i(\alpha_j,
\alpha_{j+1}) = 0$ for $j=1,2,...,5$ and $i(\alpha_j, \alpha_k)
\neq 0$ otherwise, where $\alpha_6 = \alpha_1$. A vertex in
$\mathcal{C}(M)$ is called an {\it n-vertex} if it has a
representative which is an n-circle on $M$.\\
Let $x, y$ be disjoint simple closed curves on $R$ such that
$([x], [y])$ is a peripheral pair, i.e. $x, y$ and a boundary
component of $R$ bound a pair of pants on $R$. $x \cup y$ separate
$R$ into two subsurfaces. Let $R_{x, y}$ be the positive genus
subsurface of $R$ which comes from this separation. We can
identify $\mathcal{N}(R_{x, y})$ with a subcomplex, $L_{x,y}$ of
$\mathcal{N}(R)$. By $\lambda_{x, y}$ we will denote the
restriction of $\lambda$ on $\mathcal{N}(R_{x, y})$. If $k: R
\rightarrow R$ is a homeomorphism then we will use $k_\#$ for the
map induced by $k$ on $\mathcal{N}(R)$ (i.e. $k_\#: \mathcal{N}(R)
\rightarrow \mathcal{N}(R)$ where $k_\# =[k]$).\\
An arc $i$ on $R$ is called \textit{properly embedded} if
$\partial i \subseteq \partial R$ and $i$ is transversal to
$\partial R$. $i$ is called \textit{nontrivial} (or
\textit{essential}) if $i$ cannot be deformed into $\partial R$ in
such a way that the endpoints of $i$ stay in $\partial R$ during
the deformation. If $a$ and $b$ are two disjoint arcs connecting a
boundary component of $R$ to itself, $a$ and $b$ are called {\it
linked} if their end points alternate on the boundary component.
Otherwise, they are called {\it unlinked}. The \textit{complex of
arcs}, $\mathcal{B}(R)$, on $R$ is an abstract simplicial complex.
Its vertices are the isotopy classes of nontrivial properly
embedded arcs $i$ in $R$. A set of vertices forms a simplex if
these vertices can be represented by pairwise disjoint arcs. Let
$i$ be an essential properly embedded arc on $R$. Let $A$ be a
boundary component of $R$ which has one end point of $i$ and $B$
be the boundary component of $R$ which has the other end point of
$i$. Let $N$ be a regular neighborhood of $i \cup A \cup B$ in
$R$. By Euler characteristic arguments, $N$ is a pair of pants.
The boundary components of $N$ are called {\it encoding circles of
$i$ on $R$}. An essential properly embedded arc $i$ on $R$ is
called {\it type 1} if it joins one boundary component
$\partial_k$ of $R$ to itself. $i$ is called {\it nonseparating}
if its complement in $R$ is connected.\\
The mapping class group, $Mod_R$, of $R$ is the group of isotopy
classes of orientation preserving homeomorphisms of $R$. The pure
mapping class group, $PMod_R$, is the subgroup of $Mod_R$
consisting of isotopy classes of homeomorphisms which preserve
each boundary component of $R$.
\begin{lemma}
\label{imp} Suppose $g \geq 3$ and $p=2$. Let $\lambda:
\mathcal{N}(R) \rightarrow \mathcal{N}(R)$ be a superinjective
simplicial map. Assume that for any peripheral pair $([x], [y])$
on $R$ with $x, y$ disjoint, $\lambda_{x, y}$ agrees with a map,
$(g_{x, y})_\# : \mathcal{N}(R_{x, y}) \rightarrow
\mathcal{N}(R_{x', y'})$, which is induced by a homeomorphism
$g_{x, y} : R_{x,y} \rightarrow R_{x', y'}$ where $x', y'$ are
disjoint, $\lambda([x]) = [x']$, $\lambda([y]) = [y']$, $g_{x,
y}(x)= x'$, $g_{x, y}(y)= y'$. Then $\lambda$ agrees with a map
$h_\#: \mathcal{N}(R) \rightarrow \mathcal{N}(R)$ which is induced
by a homeomorphism $h: R \rightarrow R$.\end{lemma}
\begin{figure}
\caption{(i) A dual curve,
(ii) Pants decomposition}
\end{figure}
\begin{proof} Let $x, y$ be disjoint simple closed curves such
that $([x], [y])$ is a peripheral pair, and let $(g_{x, y})_\# :
\mathcal{N}(R_{x, y}) \rightarrow \mathcal{N}(R_{x', y'})$ be a
simplicial map which is induced by a homeomorphism $g_{x, y} :
R_{x,y} \rightarrow R_{x', y'}$ where $x', y'$ are disjoint,
$\lambda([x]) = [x']$, $\lambda([y]) = [y']$, $g_{x, y}(x)= x'$,
$g_{x, y}(y)= y'$ such that $\lambda_{x, y}$ agrees with $(g_{x,
y})_\# $ on $\mathcal{N}(R_{x, y})$. Let $g$ be a homeomorphism of
$R$ which cuts to a homeomorphism $R_{x,y} \rightarrow R_{x', y'}$
which is isotopic to $g_{x, y}$. Then each homeomorphism of $R$
which cuts to a homeomorphism $R_{x, y} \rightarrow R_{x', y'}$
which is isotopic to $g_{x, y}$, is isotopic to an element in the
set $\{gt_x^mt_y^n, m, n \in \mathbb{Z}\}$. It is easy to see that
$\lambda_{x, y}$ agrees with the restriction of $(gt_x^mt_y^n)_\#$
on $\mathcal{N}(R_{x, y})$ for all $m, n \in \mathbb{Z}$. Let $w$
be a simple closed curve which is dual to both of $x$ and $y$ (see
Figure 9 (i)). Since $\lambda$ preserves geometric intersection
one property by Lemma \ref{intone}, $\lambda([w])$ has a
representative which is dual to both of $x'$ and $y'$. Let $P$ be
a regular neighborhood of $x \cup y \cup w$. $P$ is a genus one
surface with two boundary components. Let $t$ be the essential
boundary component of $P$. Let $Q$ be the genus one subsurface
with two boundary components of $R$ which has $g(t)$ as its
boundary. Then by using the properties of $\lambda$ it is easy to
see that $\lambda([w])$ has a representative which lies in the
interior of $Q$, and dual to both of $x'$ and $y'$ and there
exists $m_o, n_o \in \mathbb{Z}$ such that $gt_x^{m_o} t_y^{n_o}$
agrees with $\lambda$ on $[w]$. Let $D_{x,y}$ be the set of
isotopy classes of simple
closed curves which are dual to each of $x$ and $y$ on $R$.\\
Claim 1: $(gt_x^{m_o} t_y^{n_o})_\#$ agrees with $\lambda$ on
$\{[x]\} \cup \{[y]\} \cup L_{x,y} \cup D_{x,y}$.\\
Proof: It is clear that $(gt_x^{m_o} t_y^{n_o})_\# ([x])= \lambda
([x]) = [x']$ and $(gt_x^{m_o} t_y^{n_o})_\# ([y])= \lambda ([y])
= [y']$. Since $\lambda_{x, y}$ agrees with the restriction of
$(gt_x^{m_o} t_y^{n_o})_\#$ on $\mathcal{N}(R_{x, y})$,
$(gt_x^{m_o} t_y^{n_o})_\#$ agrees with $\lambda$ on $L_{x, y}$.
We have seen that $(gt_x^{m_o} t_y^{n_o})_\#$ agrees with
$\lambda$ on $[w]$. Let $w_1$ be a simple closed curve which is
disjoint from $w$ and dual to both of $x$ and $y$. As we described
before, there exists $\tilde{m}, \tilde{n} \in \mathbb{Z}$ such
that $\lambda$ agrees with $(gt_x^{\tilde{m}} t_y^{\tilde{n}})_\#$
on $[w_1]$. Since $w$ and $w_1$ are disjoint nonseparating curves,
$i(\lambda([w]), \lambda([w_1])) = 0$. Then by using the
properties that $\lambda$ preserves disjointness and
nondisjointness, we can see that $m = \tilde{m}$ and $n =
\tilde{n}$. This shows that $(gt_x^{m_o} t_y^{n_o})_\#$ also
agrees with $\lambda$ on $[w_1]$. Given any simple closed curve
$v$ which is dual to both of $x$ and $y$, we can find a sequence
of dual curves to both of $x$ and $y$, connecting $w$ to $v$, such
that each consecutive pair is disjoint, i.e. the isotopy classes
of these curves define a path between $w$ and $v$ in
$\mathcal{N}(R)$. Then using the argument given above and the
sequence, we conclude that $(gt_x^{m_o} t_y^{n_o})_\#$ agrees with
$\lambda$ on $D_{x, y}$. Hence, $(gt_x^{m_o} t_y^{n_o})_\#$ agrees
with $\lambda$ on $\{[x]\} \cup \{[y]\} \cup L_{x, y} \cup D_{x,
y}$. This proves claim 1. Let $h_{x,y} = gt_x^{m_o} t_y^{n_o}$. We
have that $(h_{x,y})_\#$ agrees with $\lambda$ on
$\{[x]\} \cup \{[y]\} \cup L_{x, y} \cup D_{x, y}$.\\
We complete $x, y$ to a pair of pants decomposition $P$ consisting
of nonseparating circles as shown in Figure 9 (ii) for $g=3, p=2$,
(similar configurations can be chosen for $g \geq 3$).
Let $t, z, w, c, k, r$ be as shown in the figure.\\
\begin{figure}
\caption{Arcs and their encoding circles}
\end{figure}
Claim 2: $(h_{x,y})_\#$ agrees with $\lambda$ on $c$.\\
Proof: Let $t', z', x', y', k', r'$ be pairwise disjoint
representatives of $\lambda([t]), \lambda([z]), \lambda([x]),$
$\lambda([y]), \lambda([k]), \lambda([r])$ respectively and let
$w'$ be a representative of $\lambda([w])$ which has minimal
intersection with each of $t', z', x', y', k', r'$. Since $k, r,
z, y, \partial_1$ bound a sphere with five holes, $T$, on $R$,
containing $x$ and $c$, then $k', r', z', y'$ and a boundary
component of $R$ bound a sphere with five holes, $T'$, on $R$ by
Lemma \ref{top}. Since $x$ and $c$ have geometric intersection 2
and algebraic intersection 0 in $T$, it is easy to see that there
exist vertices $\gamma_1, \gamma_2, \gamma_3$ of $\mathcal{C}(T)$
such that $(\gamma_1, \gamma_2, [x], \gamma_3, [c])$ is a pentagon
in $\mathcal{C}(T)$, $\gamma_1$ and $\gamma_3$ are 2-vertices,
$\gamma_2$ is a 3-vertex, and $\{[x], \gamma_3\}$, $\{[x],
\gamma_2\}$, $\{[c], \gamma_3\}$ and $\{\gamma_1, \gamma_2 \}$ are
codimension-zero simplices of $\mathcal{C}(T)$, and each of
$\gamma_i$ has a representative which is nonseparating on $R$.
Since $\lambda$ is superinjective, we can see that
$(\lambda(\gamma_1), \lambda(\gamma_2)$, $\lambda([x]),
\lambda(\gamma_3)$, $\lambda([c]))$ is a pentagon in
$\mathcal{C}(T')$. By Lemma \ref{top}, $\lambda(\gamma_1)$ and
$\lambda(\gamma_3)$ are 2-vertices, and $\lambda(\gamma_2)$ is a
3-vertex in $\mathcal{C}(T')$. Since $\lambda$ is an injective
simplicial map $\{\lambda([x]), \lambda(\gamma_3)\}$,
$\{\lambda([x]), \lambda(\gamma_2)\}$, $\{\lambda([c])$,
$\lambda(\gamma_3)\}$ and $\{\lambda(\gamma_1),
\lambda(\gamma_2)\}$ are codimension-zero simplices of
$\mathcal{C}(T')$. It is easy to see that $\lambda([c])$ has a
representative which is disjoint from $t', z', y'$. Then
$\{\lambda([x]), \lambda([c])\}$ have representatives with
geometric intersection 2 and algebraic intersection 0 in the
sphere with four holes bounded by $t', z', y'$ and the boundary
component of $R$, \cite{K}. We also have that $(h_{x,y})_\# ([c])$
has a representative $c''$ which is disjoint from $t', z', y', w'$
and which intersects $x'$ geometrically twice and algebraically 0
times. Since $(h_{x,y})_\#$ agrees with $\lambda$ on $\{[x]\} \cup
\{[y]\} \cup L_{x, y} \cup D_{x, y}$ it is easy to see that
$[c']=[c'']$, i.e.
$(h_{x,y})_\#$ agrees with $\lambda$ on $c$.\\
Claim 3: $(h_{x,y})_\#$ agrees with $\lambda$ on the class of
every nonseparating circle on $R$.\\
Proof: Let $z$ be a nonseparating simple closed curve on $R$. Let
$t$ be another simple closed curve disjoint from $z$ such that $z,
t$ and $\partial_1$ bound a pair of pants in $R$. Let $i$ and $j$
be nonseparating type 1 arcs connecting $\partial_1$ to itself
such that $i$ has $x, y$ as its encoding circles and $j$ has $z,
t$ as its encoding circles. W.L.O.G. we can assume that $i$ and
$j$ have minimal intersection. By Lemma 3.8 in \cite{Ir2}, there
is a sequence $i = r_0 \rightarrow r_1 \rightarrow ... \rightarrow
r_{n+1}=j$ of essential properly embedded nonseparating type 1
arcs joining $\partial_1$ to itself so that each consecutive pair
is disjoint, i.e. the isotopy classes of these arcs define a path
in $\mathcal{B}(R)$ between $i$ and $j$. Let $x_i, y_i$ be the
encoding circles for $r_i$ for $i=1,..., n$. For the pair of arcs
$i$ and $r_1$, we will consider the following cases:\\
Case i: Assume that $i$ and $r_1$ are linked, i.e their end points
alternate on the boundary component $\partial_1$. Then a regular
neighborhood of $i \cup r_1 \cup \partial_1$ is a genus one
surface with two boundary components $N$, and the arcs $i, r_1$
and their encoding circles $x, y, x_1, y_1$ on $N$ are as shown in
Figure 10 (i). In this case, we complete $\{x, y, x_1, y_1\}$ to a
curve configuration $G$ consisting of nonseparating circles which
is shown in Figure 11 (i), for $g=3, p=3$, see \cite{IMc}, such
that the isotopy classes of Dehn twists about the elements of this
set generate $PMod_R$ and all the curves in this set are (i)
either disjoint from $x, y$ or simultaneous dual to $x, y$, and
(ii) either disjoint from $x_1, y_1$ or simultaneous dual to $x_1,
y_1$. Then, since all the curves in $G$ are either disjoint from
$x, y$ or simultaneous dual to $x, y$, by claim 1 we have that
$(h_{x,y})_\# ([x]) = \lambda ([x])$ for every $x \in G$.
Similarly, since all the curves in $G$ are either disjoint from
$x_1, y_1$ or simultaneous dual to $x_1, y_1$, by claim 1 we have
$(h_{x_1,y_1})_\# ([x]) = \lambda ([x])$ for every $x \in G$.
Hence $(h_{x,y})_\# ([x])= \lambda ([x]) = (h_{x_1,y_1})_\# ([x])$
for every $x \in G$. Then $(h_{x,y}^{-1} h_{x_1,y_1})_\# \in
C_{Mod_R}(PMod_R)$. By Theorem 5.3 in \cite{IMc},
$C_{Mod_R}(PMod_R) = \{1\}$. Hence $(h_{x,y})_\# = (h_{x_1,y_1})_\#$.\\
\begin{figure}
\caption{Encoding circles in a curve
configuration}
\end{figure}
Case ii: Assume that $i$ and $r_1$ are unlinked, i.e their end
points don't alternate on the boundary component $\partial_1$.
Then a regular neighborhood of $i \cup r_1 \cup \partial_1$ is a
sphere with four boundary components $S_4 ^2$, and the arcs $i,
r_1$ and their encoding circles $x, y, x_1, y_1$ on $S_4 ^2$ are
as shown in Figure 10 (ii). Let $w$ be the boundary component of
$S_4 ^2$ which is different from $x_1, y, \partial_1$. If $w$ is a
nonseparating curve, then we complete $\{x, y, x_1, y_1\}$ to a
curve configuration $G$ consisting of nonseparating circles as
shown in Figure 11 (ii) such that the isotopy classes of Dehn
twists about the elements of $G$ generate $PMod_R$. By claim 1 and
claim 2, $(h_{x,y})_\# ([x]) = \lambda ([x])$ for every $x \in G$
and $(h_{x_1,y_1})_\# ([x]) = \lambda ([x])$ for every $x \in G$.
Then $(h_{x,y}^{-1} h_{x_1,y_1})_\# ([x]) = [x]$ for every $x \in
G$. Then $(h_{x,y}^{-1} h_{x_1,y_1})_\# \in C_{Mod_R}(PMod_R)$. By
Theorem 5.3 in \cite{IMc}, $C_{Mod_R}(PMod_R) = \{1\}$. Hence
$(h_{x,y})_\#= (h_{x_1,y_1})_\#$.\\
Suppose that $w$ is a separating curve. By the remark after Lemma
\ref{extension}, we can extend $\lambda$ to a superinjective map
$\lambda_*$ on a subcomplex of $\mathcal{C}(R)$ containing
separating circles on $R$ which separate $R$ into two pieces such
that each piece has genus at least one. Notice that $w$ is such a
circle. For the rest of the proof, we will use $\lambda$ for this
extension. We will do the case when $x_1$ and $y$ are not
isotopic. The other case can be done similarly. Let $M$ be the
subsurface which has $w$ on its boundary and which does not
contain $N$. Let $T$ be the closure of $R \setminus \{M \cup N\}$.
The circles $x_1$ and $y$ are boundary components of $T$. Since
$w$ is an essential separating circle and $p=2$, $M$ has genus at
least one. In Figure 12, we show $M, N, T$ for a special case. By
claim 1, $(h_{x,y})_\# ([x]) = \lambda ([x])$ for every $x \in
\mathcal{N}(T)$. Similarly, $(h_{x_1,y_1})_\# ([x]) = \lambda
([x])$ for every $x \in \mathcal{N}(T)$. Then $(h_{x,y}^{-1}
h_{x_1,y_1})_\# ([x]) = [x]$ for every $x \in \mathcal{N}(T)$.
Then the restriction of $(h_{x,y}^{-1} h_{x_1,y_1})_\#$ on
$\mathcal{N}(T)$ is in $C(PMod_T)$. By Theorem 5.3 in \cite{IMc},
$C(PMod_T) = \{1\}$. Hence $(h_{x,y})_\# = (h_{x_1,y_1})_\#$ on
$\mathcal{N}(T)$. It is easy to see that $(h_{x,y})_\# =
(h_{x_1,y_1})_\#$ on the set $\{[x_1], [w], [y]\}$. Following the
proof of claim 2 (considering that we have the extended
superinjective simplicial map on ``good" separating circles), we
see that $(h_{x,y})_\# = (h_{x_1,y_1})_\#$ on $\{[x], [y_1]\}$.
Then, since Dehn twists about $x$ and $y_1$ generate $PMod_N$, the
restriction of $(h_{x,y}^{-1} h_{x_1,y_1})_\#$ on $\mathcal{C}(N)$
is in $C(PMod_N)$. By Theorem 5.3 in \cite{IMc}, $C(PMod_N) =
\{1\}$. Hence $(h_{x,y})_\# = (h_{x_1,y_1})_\#$ on
$\mathcal{C}(N)$. By claim 1, $(h_{x,y}^{-1} h_{x_1,y_1})_\# ([x])
= [x]$ for every $x \in \mathcal{N}(M)$. Then the restriction of
$(h_{x,y}^{-1} h_{x_1,y_1})_\#$ on $\mathcal{N}(M)$ is in
$C(PMod_M)$. By considering the action on oriented circles and
using Theorem 5.3 in \cite{IMc}, we see that $(h_{x,y})_\# =
(h_{x_1,y_1})_\#$ on $\mathcal{N}(M)$. So we have $(h_{x,y})_\# =
(h_{x_1,y_1})_\#$ on $\mathcal{N}(M) \cup \mathcal{C}(N) \cup
\mathcal{N}(T)$. In Figure 12 we see the curve $c$ which is dual
to each of $x, y, x_1, y_1$. By claim 1, $(h_{x,y})_\# ([c]) =
\lambda ([c]) = (h_{x_1,y_1})_\# ([c])$. Then, since
$(h_{x,y})_\#$ and $(h_{x_1,y_1})_\#$ agree on $[c]$, we have
$(h_{x,y})_\# = (h_{x_1,y_1})_\#$ on $\mathcal{N}(R)$. Then, since
$g \geq 3$, $(h_{x,y})_\# = (h_{x_1,y_1})_\#$. If $w$ is a
boundary component of $R$, then showing that $(h_{x,y})_\# =
(h_{x_1,y_1})_\#$
is similar.\\
\begin{figure}
\caption{Curves
intersecting twice}
\end{figure}
In both cases we have seen that $(h_{x,y})_\# = (h_{x_1,y_1})_\#$.
By using our sequence, with an inductive argument we have $(h_{x,
y})_\# = (h_{z, t})_\#$, and $(h_{x, y})_\# = (h_{z, t})_\# =
\lambda$ on $\{[x]\} \cup \{[y]\} \cup \{[z]\} \cup \{[t]\} \cup
L_{x, y} \cup D_{x, y} \cup L_{z, t} \cup D_{z, t}$. In particular
we see that $(h_{x, y})_\#$ agrees with $\lambda$ on any
nonseparating curve $z$, hence $(h_{x, y})_\#$ agrees with
$\lambda$ on $\mathcal{N}(R)$. This proves the lemma.\end{proof}
\begin{theorem} \label{last} Suppose that the genus of $R$ is
at least two and $R$ has at most $g-1$ boundary components. A
simplicial map $\lambda : \mathcal{N}(R) \rightarrow
\mathcal{N}(R)$ is superinjective if and only if $\lambda$ is
induced by a homeomorphism of $R$.
\end{theorem}
\begin{proof} If $\lambda$ is induced by a homeomorphism of $R$,
then it preserves disjointness and nondisjointness and hence it is
superinjective. Assume that $\lambda$ is superinjective. In the
cases when $R$ is a closed surface or when $R$ has exactly one
boundary component, by Lemma \ref{extension} $\lambda$ extends to
a superinjective simplicial map $\lambda_*$ on $\mathcal{C}(R)$.
Then $\lambda_*$ is induced by a homeomorphism $h:R \rightarrow
R$, i.e. $\lambda_*(\alpha) = h_\#(\alpha)$ for each vertex
$\alpha$ in $\mathcal{C}(R)$, by the main results in \cite{Ir1}
and \cite{Ir2} and by Theorem \ref{theorem3}. Hence $\lambda$ is
induced by the
homeomorphism $h$.\\
Assume that $g \geq 3$ and $p=2$. Let $x$ and $y$ be disjoint
nonseparating circles such that $x, y$ and a boundary component,
$\partial_1$, of $R$ bound a pair of pants on $R$. Let $x', y'$ be
disjoint representatives of $\lambda([x]), \lambda([y])$
respectively. By using Lemma \ref{peripheral} and knowing that
$\lambda$ preserves disjointness and nondisjointness, it is easy
to see that $\lambda$ maps $\mathcal{N}(R_{x \cup y})$ to
$\mathcal{N}(R_{x' \cup y'})$. Since every essential separating
curve on $R_{x \cup y}$ separates $R$ into two subsurfaces each of
which has genus at least one, using chains on these subsurfaces we
could extend $\lambda$ to a superinjective simplicial map
$\lambda_{x, y} : \mathcal{C}(R_{x \cup y}) \rightarrow
\mathcal{C}(R_{x' \cup y'})$ as in case 1 in Lemma
\ref{extension}. Then by the main results in \cite{Ir2}, there
exists a homeomorphism $h : R_{x \cup y} \rightarrow R_{x' \cup
y'}$ such that $h(x) = x'$, $h(y) = y'$ and $\lambda_{x, y}$ is
induced by $h$. Then the proof of the theorem follows from Lemma
\ref{imp}.\\
\begin{figure}
\caption{Cutting $R$ along
peripheral pairs}
\end{figure}
Now assume that $g \geq 4$ and $3 \leq p \leq g-1$. We will give
the proof when $p = g-1$. The proof of the remaining cases is
similar. Let $\{a_1, ..., a_{2(p-1)}\}$ be a set of pairwise
disjoint nonseparating circles such that $(a_{2i+1}, a_{2i+2})$ is
a peripheral pair as shown in Figure 13 for $i=0, ..., p-2$. Let
$R_{a_1 \cup a_2 ... \cup a_{2(p-1)}}$ be the genus two surface
with $2p-1$ boundary components which comes from the separation by
${a_1 \cup a_2 ... \cup a_{2(p-1)}}$. Exactly one of the boundary
components of $R_{a_1 \cup a_2 ... \cup a_{2(p-1)}}$ is a boundary
component of $R$. We identify $\mathcal{N}(R_{a_1 \cup a_2 ...
\cup a_{2(p-1)}})$ with a subcomplex $L_{a_1 \cup a_2 ... \cup
a_{2(p-1)}}$ of $\mathcal{N}(R)$. Let $\lambda_{a_1 \cup a_2 ...
\cup a_{2(p-1)}}$ denote the restriction of $\lambda$ on
$\mathcal{N}({R_{a_1 \cup a_2 ... \cup a_{2(p-1)}}})$. Let $a_1',
... , a_{2(p-1)}'$ be pairwise disjoint representatives of
$\lambda([a_1]),... ,$ $ \lambda([a_{2(p-1)}])$ respectively. By
using Lemma \ref{peripheral} and the properties of $\lambda$, it
is easy to see that $\lambda$ maps $\mathcal{N}({R_{a_1 \cup a_2
... \cup a_{2(p-1)}}})$ to $\mathcal{N}({R_{a_1' \cup a_2' ...
\cup a'_{2(p-1)}}})$. Since every essential separating curve on
$R_{a_1 \cup a_2 ... \cup a_{2(p-1)}}$ separates $R$ into two
subsurfaces each of which has genus at least one, using chains on
these subsurfaces we extend $\lambda$ to a superinjective
simplicial map $\lambda_{a_1 \cup a_2 ... \cup a_{2(p-1)}} :
\mathcal{C}(R_{a_1 \cup a_2 ... \cup a_{2(p-1)}}) \rightarrow
\mathcal{C}(R_{a_1' \cup a_2' ... \cup a'_{2(p-1)}})$ as in Lemma
\ref{extension}. Then by the main results in \cite{Ir2}, there
exists a homeomorphism $h : R_{a_1 \cup a_2 ... \cup a_{2(p-1)}}
\rightarrow R_{a_1' \cup a_2' ... \cup a'_{2(p-1)}}$ such that
$h(a_i) = a_i'$ and $\lambda_{a_1 \cup a_2 ... \cup a_{2(p-1)}}$
is induced by $h$. If we replace $(a_{2p-3}, a_{2p-2})$ with
another peripheral pair $(b_{2p-3}, b_{2p-2})$ where each of
$b_{2p-3}$ and $b_{2p-2}$ is disjoint from ${a_1 \cup a_2 ... \cup
a_{2p-4}}$, then by similar techniques we get a homeomorphism $t :
R_{a_1 \cup a_2 ... \cup a_{2p-4} \cup b_{2p-3} \cup b_{2p-2}}
\rightarrow R_{a_1' \cup a_2' ... \cup a'_{2p-4} \cup b'_{2p-3}
\cup b'_{2p-2}}$ such that $t(a_i) = a_i'$, $t(b_j) = b_j'$, where
$b'_{2p-3}, b'_{2p-2}$ are pairwise disjoint representatives of
$\lambda([b_{2p-3}]), \lambda([b_{2p-2}])$ respectively, and the
map $\lambda_{a_1 \cup a_2 ... \cup a_{2p-4} \cup b_{2p-3} \cup
b_{2p-2}}$ is induced by $t$. Then by following the techniques in
the proof of Lemma \ref{imp}, we see that there exists a
homeomorphism $r : R_{a_1 \cup a_2 ... \cup a_{2p-4}} \rightarrow
R_{a_1' \cup a_2' ... \cup a'_{2p-4}}$ such that $r(a_i) = a_i'$
and $\lambda_{a_1 \cup a_2 ... \cup a_{2p-4}}$ is induced by $r$.
Then by an inductive argument, there exists a homeomorphism $q : R
\rightarrow R$ such that $\lambda$ is induced by $q$.\end{proof}\\
Now we consider the graph $\mathcal{G}(R)$ defined by Schaller.
The vertex set of $\mathcal{G}(R)$ is the set of isotopy classes
of nontrivial nonseparating simple closed curves on $R$. Two
vertices are connected by an edge if and only if their geometric
intersection number is one. In the proof of the following theorem,
we use some of Schaller's results given in \cite{Sc} that if $g
\geq 2$ and $R$ is not a closed surface of genus two, then
$Aut(\mathcal{G}(R))= Mod_R^*$.
\begin{theorem} \label{lastlast} Suppose that $R$ has genus at least two.
If $R$ is not a closed surface of genus two, then
$Aut(\mathcal{N}(R))= Mod_R^*$. If $R$ is a closed surface of
genus two, then $Aut(\mathcal{N}(R))= Mod_R ^* /\mathcal{C}(Mod_R
^*)$.\end{theorem}
\begin{proof} Assume that $R$ is not a closed surface of genus
two. If $[f] \in Mod_R^*$, then $[f]$ induces an automorphism of
$\mathcal{N}(R)$. If $[f]$ fixes the isotopy class of every
nontrivial nonseparating simple closed curve on $R$, then $f$ is
orientation preserving and it can be shown that it is isotopic to
$id_R$. An automorphism $\lambda$ of $\mathcal{N}(R)$ is a
superinjective simplicial map of $\mathcal{N}(R)$. By Lemma
\ref{intone}, it preserves geometric intersection one property,
and hence induces an automorphism of $\mathcal{G}(R)$. An
automorphism of $\mathcal{G}(R)$ is induced by a homeomorphism of
$R$, \cite{Sc} (Note that if $R$ has at most $g-1$ boundary
components, then Theorem \ref{last} also implies that $\lambda$ is
induced by a homeomorphism of $R$). Then it is easy to see that we
get $Aut(\mathcal{N}(R))= Mod_R^*$. Assume that $R$ is a closed
surface of genus two. If $[f] \in Mod_R^*$, then $[f]$ induces an
automorphism of $\mathcal{N}(R)$. If $[f]$ fixes the isotopy class
of every nonseparating simple closed curve on $R$ then $f$ is
either isotopic to $id_R$ or the hyperelliptic involution. An
automorphism of $\mathcal{N}(R)$ is a superinjective simplicial
map of $\mathcal{N}(R)$ and by Theorem \ref{last} it is induced by
a homeomorphism of $R$. Then we have $Aut(\mathcal{N}(R))= Mod_R^*
/\mathcal{C}(Mod_R ^*)$.\end{proof}\\
Note that by the main results of this paper and the results in
\cite{Ir1}, \cite{Ir2}, \cite{Iv1}, \cite{Sc}, we have that if $R$
has genus at least two, then $Aut(\mathcal{N}(R))=
Aut(\mathcal{C}(R)) =
Aut(\mathcal{G}(R))$.\\
{\bf Acknowledgments}\\
We thank John D. McCarthy for his suggestions and many discussions
about this work. We also thank Peter Scott for his comments.
\noindent University of Michigan, Department of Mathematics, Ann
Arbor, MI 48109, USA;
\noindent [email protected]\\
\end{document} | math | 96,340 |
\begin{document}
\title{A weak convergence theorem for mean nonexpansive mappings}
\begin{abstract}
In this paper, we prove first that the iterates of a mean nonexpansive map defined on a weakly compact, convex set converge weakly to a fixed point in the presence of Opial's property and asymptotic regularity at a point. Next, we prove the analogous result for closed, convex (not necessarily bounded) subsets of uniformly convex Opial spaces. These results generalize the classical theorems for nonexpansive maps of Browder and Petryshyn in Hilbert space and Opial in reflexive spaces satisfying Opial's condition.
\varepsilonnd{abstract}
\section{Introduction}
Let $(X,\norm{\cdot})$ be a Banach space. Given $C \subseteq X$, we say a function $T: C \to X$ is \textit{nonexpansive} if
\[
\norm{Tx-Ty}\leq \norm{x-y}
\]
for all $x,y \in C$. It is a well-known application of Banach's contraction mapping theorem that every nonexpansive map $T : C\to C$ has an \textit{approximate fixed point sequence}; that is, a sequence $(u_n)_n$ in $C$ for which $\norm{Tu_n - u_n} \to_n 0$. Also, we say $T:C \to C$ is \textit{asymptotically regular at $x$} if
\[
\lim_{n\to\infty} \norm{T^n x - T^{n+1}x} = 0.
\]
If $T$ is asymptotically regular at every $x \in C$, we simply say $T$ is asymptotically regular. Note that asymptotic regularity at $x$ implies that $(T^n x)_n$ is an approximate fixed point sequence for $T$. Denote the set of all fixed points of $T$ as $F(T)$. That is, $F(T) := \{x \in C : Tx=x\}$.
In 1966, Browder and Petryshyn \cite[Theorem 4]{browderpetryshyn66} proved the following theorem for asymptotically regular nonexpansive mappings on Hilbert space.
\begin{theorem}[Browder and Petryshyn]
Suppose $H$ is a Hilbert space and $T: H \to H$ is nonexpansive and asymptotically regular with $F(T)=\{x_0\}$. Then $(T^nx)_n$ converges weakly to $x_0$.
\varepsilonnd{theorem}
In 1967, Opial \cite{opial67} extended this theorem to spaces satisfying Opial's property. Recall that $C \subseteq X$ has the \textit{Opial property} if, whenever $(u_n)_n$ is a sequence in $C$ converging weakly to some $u \in X$, we have
\[
\liminf_n \norm{u_n - u} < \liminf_n \norm{u_n - v}
\]
for any $v \neq u$. All Hilbert spaces have the Opial property, as do $\varepsilonll^p$ spaces for all $p \in (1, \infty)$. $L^p$ fails to have the Opial property for all $p\neq 2$, however.
We will further extend Opial's result to the class of mean nonexpansive maps, first defined by Goebel and Jap\'on Pineda in 2007 \cite{gjp07}. We say $T : C \to C$ is \textit{mean nonexpansive} (or $\alpha$-nonexpansive) if, for some multi-index $\alpha=(\alpha_1,\ldots,\alpha_n)$ with $\alpha_1, \alpha_n >0$, $\alpha_j \geq 0$ for all $j$, and $\alpha_1+\cdots +\alpha_n = 1$, we have
\[
\sum_{j=1}^n \alpha_j \norm{T^jx - T^jy} \leq \norm{x-y}
\]
for all $x, y \in C$. Goebel and Jap\'on Pineda further suggested the notion of \textit{$(\alpha,p)$}-nonexpansiveness, wherein $T$ would satisfy
\[
\sum_{j=1}^n \alpha_j \norm{T^jx - T^jy}^p \leq \norm{x-y}^p
\]
for some $p \in [1,\infty)$. It is easy to check that any $(\alpha,p)$-nonexpansive map is mean nonexpansive (i.e. $(\alpha,1)$-nonexpansive), but the converse does not necessarily hold \cite{piasecki13}.
To prove our theorem, we will need one further notion. We will use ``$\rightharpoonup$'' to denote weak convergence and ``$\to$'' to denote strong convergence. We say $T: C \to X$ is \textit{demiclosed at $y$} if, whenever $x_n \rightharpoonup x$ in $C$ and $Tx_n \to y$, it follows that $Tx=y$. The present author recently proved \cite[Theorem 4.2]{gallagher16} that if $C \subseteq X$ is closed and convex and has the Opial property, then any mean nonexpansive map $T: C\to C$ is demiclosed at 0. We will use this demiclosedness principle to extend the theorems of Browder and Petryshyn and Opial stated above. First, we will present the results for the simple case of multi-indices of length 2 before proving the full theorem for multi-indices of arbitrary length.
\section{Results for $\alpha=(\alpha_1,\alpha_2)$}
Let us state the main theorem of this section. The proofs of the following theorem and lemmas can be found in the next section.
\begin{theorem}\label{theorem}
Suppose $(X, \norm{\cdot})$ is a Banach space and $C \subseteq X$ is weakly compact, convex, and has the Opial property. Suppose further that $T : C \to C$ is $(\alpha_1,\alpha_2)$-nonexpansive and asymptotically regular at some point $x \in C$. Then $(T^n x)_n$ converges weakly to a fixed point of $T$.
\varepsilonnd{theorem}
To ensure that this theorem is a genuine extension of the classical theorems for nonexpansive maps, we present an example of a $((\frac{1}{2},\frac{1}{2}), 2)$-nonexpansive (hence mean nonexpansive) map defined on $(\varepsilonll^2,\norm{\cdot}_2)$ for which none of its iterates are nonexpansive. The map below is based on an example given by Goebel and Sims \cite{goebelsims10} and can also be found in \cite{gallagher16}; moreover, it is asymptotically regular.
\begin{example}
Let $(\varepsilonll^2,\norm{\cdot}_2)$ be the Hilbert space of square-summable sequences endowed with its usual norm. Let $\tau : [-1,1]\to[-1,1]$ be given by
\[
\tau(t) :=
\begin{cases}
\sqrt{2} \, t + (\sqrt{2}-1) & -1 \leq t \leq -\frac{\sqrt{2}-1}{\sqrt{2}}\\
0 & -\frac{1+\sqrt{2}}{\sqrt{2}} \leq t \leq \frac{1+\sqrt{2}}{\sqrt{2}}\\
\sqrt{2} \, t - (\sqrt{2}-1) & \frac{\sqrt{2}-1}{\sqrt{2}} \leq t \leq 1
\varepsilonnd{cases}
\]
and note the following facts about $\tau$:
\begin{enumerate}
\item $\tau$ is Lipschitz with $k(\tau) = \sqrt{2}$,
\item $|\tau(t)| \leq |t|$ for all $t \in [-1,1]$, and
\varepsilonnd{enumerate}
Let $B_{\varepsilonll^2}$ denote the closed unit ball of $(\varepsilonll^2,\norm{\cdot}_2)$ and for any $x \in \varepsilonll^2$, define $T$ by
\[
T(x_1,x_2,\ldots) := \left(\tau(x_2),\sqrt{\frac{2}{3}}\,x_3, x_4, x_5, \ldots\right)
\]
and
\[
T^2 (x_1,x_2,\ldots) = \left( \tau\left( \sqrt{\frac{2}{3}} \, x_3 \right), \sqrt{\frac{2}{3}} \, x_4, x_5, \ldots \right)
\]
Observe that $|\tau(t)|\leq|t|$ implies that $T(B_{\varepsilonll^2}) \subseteq B_{\varepsilonll^2}$, and $k(T)=\sqrt{2}>1$ and $k(T^j)= \frac{2}{\sqrt{3}}>1$ for all $j\geq 2$. Now, for any $x, y \in B_{\varepsilonll^2}$ we find
\begin{align*}
\frac{1}{2} \norm{Tx-Ty}_2^2 &+ \frac{1}{2}\norm{T^2x-T^2y}_2^2\\
&= \frac{1}{2}\left( |\tau(x_2)-\tau(y_2)|^2 + \frac{2}{3} |x_3 - y_3|^2 + \sum_{j=4}^\infty |x_j - y_j|^2 \right)\\
&+ \frac{1}{2}\left( \Bigg|\tau\left( \sqrt{\frac{2}{3}} \, x_3\right)-\tau\left(\sqrt{\frac{2}{3}} \, y_3\right)\Bigg|^2 + \frac{2}{3} |x_4 - y_4|^2 + \sum_{j=5}^\infty |x_j - y_j|^2 \right)\\
&\leq \frac{1}{2} \left( 2 |x_2 - y_2|^2 + \frac{4}{3}|x_3 - y_3|^2 + \frac{5}{3}|x_4-y_4|^2 + 2 \sum_{j=5}^\infty |x_j-y_j|^2\right)\\
&\leq \norm{x-y}_2^2
\varepsilonnd{align*}
Hence, $T: B_{\varepsilonll^2} \to B_{\varepsilonll^2}$ is a $((\frac{1}{2},\frac{1}{2}), 2)$-nonexpansive map for which each iterate $T^j$ is not nonexpansive.
\varepsilonnd{example}
Before proving the theorem, let's state some preliminary definitions and results. For any $x \in C$, let
\[
\omega_w (x) := \{ y \in C : y \mbox{ is a weak subsequential limit of } (T^nx)_n \}
\]
and note that if $C$ is weakly compact, then $\omega_w (x) \neq \varepsilonmptyset$. Further note that if $I-T$ is demiclosed at $0$ and asymptotically regular at $x$, then $\varepsilonmptyset \neq \omega_w (x) \subseteq F(T)$. We have the following lemma.
\begin{lemma}\label{existence}
Suppose $C$ is weakly compact and convex with the Opial property, and suppose that $T : C \to C$ is $(\alpha_1,\alpha_2)$-nonexpansive and asymptotically regular at some $x \in C$. Then for all $y \in \omega_w (x)$, $\lim_n \norm{T^nx - y}$ exists.
\varepsilonnd{lemma}
Our theorem will be proved if we can show that $\omega_w (x)$ is a singleton. This follows from the fact that $C$ is Opial and the knowledge that $(\norm{T^nx - y})_n$ converges for all $y \in \omega_w (x)$, as summarized in the following lemma.\\
\begin{lemma}\label{uniqueness}
If $C \subseteq X$ is Opial, $T : C\to C$ is a function, and for some $x \in C$, $\lim_n \norm{T^nx - y}$ exists for all $y \in \omega_w (x)$, then $\omega_w (x)$ is empty or consists of a single point.
\varepsilonnd{lemma}
\section{Proofs}
\begin{proof}[Proof of Lemma \ref{existence}]
$C$ closed and convex with the Opial property implies that $I-T$ is demiclosed at 0. That is, whenever $(z_n)_n$ is a sequence in $C$ converging weakly to some $z$ (which is necessarily in $C$ since closed and convex implies weakly closed) for which $\norm{(I-T)z_n} \to_n 0$, it follows that $(I-T)z = 0$.
By the asymptotic regularity of $T$ at $x$, we have that $(T^nx)_n$ is an approximate fixed point sequence for $T$.
Since $y \in \omega_w (x)$ and $I-T$ is demiclosed at $0$, we have that $y$ is a fixed point of $T$ and we see that
\begin{align*}
\alpha_1 \norm{Tx - y} + \alpha_2 \norm{T^{2}x -y } &= \alpha_1 \norm{Tx - Ty} + \alpha_2 \norm{T^{2}x -T^{2}y }\\
&\leq \norm{x - y}
\varepsilonnd{align*}
Hence, at least one of $\norm{Tx-y}$ or $\norm{T^2x-y}$ must be less than or equal to $\norm{x-y}$. Let $k_1 \in \{1,2\}$ be such that $\norm{T^{k_1}x-y} \leq \norm{x-y}$.
Next, we know that
\begin{align*}
\alpha_1 \norm{T^{k_1+1}x - y} + \alpha_2 \norm{T^{k_1+2}x -y } &= \alpha_1 \norm{T^{k_1+1}x - T^{k_1+1}y} + \alpha_2 \norm{T^{k_1+2}x -T^{k_1+2}y }\\
&\leq \norm{T^{k_1}x - T^{k_1} y}\\
&= \norm{T^{k_1}x - y}
\varepsilonnd{align*}
and so one of $\norm{T^{k_1+1}x - y}$ or $\norm{T^{k_1+2}x - y}$ must be less than or equal to $\norm{T^{k_1}x - y}$. As above, let $k_2 \in \{k_1+1, k_1+2\}$ be such that $\norm{T^{k_2}x-y} \leq \norm{T^{k_1}x-y}$.
Inductively, build a sequence $(k_n)_n$ which satisfies
\begin{enumerate}
\item $k_n + 1 \leq k_{n+1} \leq k_n + 2$, and
\item $\norm{T^{k_{n+1}}x-y} \leq \norm{T^{k_n}x-y}$
\varepsilonnd{enumerate}
for all $n \in \mathbb{N}$. Now $(\norm{T^{k_n}x-y})_n$ is a non-increasing sequence in $\mathbb R^+$, and is thus convergent to some $q\in \mathbb R^+$.
Consider the set $M := \mathbb{N} \setminus \{k_n : n \in \mathbb{N} \}$. We have two cases. First, if $M$ is a finite set, then the claim is proved. Second, if $M$ is infinite, write $M = \{ m_n : n \in \mathbb{N} \}$, where $(m_n)_n$ is strictly increasing. Note that, by property (1) of the sequence $(k_n)_n$ above, we must have that for all $n \in \mathbb{N}$, there exists a $j_n \in \mathbb{N}$ for which
\[
m_n = k_{j_n}+1
\]
Also, $(j_n)_n$ is strictly increasing. Asymptotic regularity of $T$ at $x$ and the fact that $\lim_n \norm{T^{k_n}x-y} = q$ gives us that for any $\varepsilon >0$, there is $n$ large enough such that
\begin{enumerate}
\item $\norm{T^{m_n}x - T^{m_n-1}x} < \varepsilon/2$, and
\item $\big|\, \norm{T^{k_{j_n}}x - y} - q \,\big| < \varepsilon/2$.
\varepsilonnd{enumerate}
Thus,
\begin{align*}
\norm{T^{m_n}x - y} - q &\leq \norm{T^{m_n}x - T^{m_n-1}x} + \norm{T^{m_n-1}x-y}-q\\
&= \norm{T^{m_n}x - T^{m_n-1}x} + \norm{T^{k_{j_n}}x - y} - q\\
&< \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon
\varepsilonnd{align*}
Entirely similarly, we have that
\begin{align*}
\norm{T^{m_n}x - y} - q &\geq -\norm{T^{m_n}x - T^{m_n-1}x} + \norm{T^{m_n-1}x-y}-q\\
&= -\norm{T^{m_n}x - T^{m_n-1}x} + \norm{T^{k_{j_n}}x - y} - q\\
&> -\frac{\varepsilon}{2} - \frac{\varepsilon}{2} = -\varepsilon
\varepsilonnd{align*}
Hence, $\big| \, \norm{T^{m_n}x -y } - q \, \big| < \varepsilon$ for $n$ large enough. Since $\{m_n : n \in \mathbb{N} \} \cup \{k_n : n \in \mathbb{N}\} = \mathbb{N}$, we have finally that $\lim_n \norm{T^nx - y}$ exists for any $y \in \omega_w (x)$.
\varepsilonnd{proof}
\begin{remark}\label{remark}
The above argument presented in the proof above actually works for any $y \in F(T)$, but in particular for $y \in \omega_w (x)$. This will be of use to us in Theorem \ref{not bounded}.
\varepsilonnd{remark}
\begin{proof}[Proof of Lemma \ref{uniqueness}]
Suppose for a contradiction that $z$ and $y$ are distinct elements of $\omega_w (x)$. Then there exist $(n_k)_k$ and $(m_k)_k$ for which $T^{n_k}x \rightharpoonup_k z$ and $T^{m_k}x \rightharpoonup_k y$. Thus, using the fact that $C$ is Opial, we have
\begin{align*}
\lim_n \norm{T^nx - y} &= \lim_n \norm{T^{m_k}x-y}\\
&< \lim_n \norm{T^{m_k}x - z}\\
&= \lim_n \norm{T^n x - z}\\
&= \lim_n \norm{T^{n_k}x-z}\\
&< \lim_n \norm{T^{n_k}x-y}\\
&= \lim_n \norm{T^n x - y},
\varepsilonnd{align*}
which is a contradiction. Thus, $\omega_w (x)$ is a singleton.
\varepsilonnd{proof}
\begin{proof}[Proof of Theorem \ref{theorem}]
As stated above, let $\omega_w (x) := \{y \in C : y \mbox{ is a weak subsequential limit of } (T^nx)_n \}$ and note that $\omega_w(x) \neq \varepsilonmptyset$ since $C$ is weakly compact, as well as that the demiclosedness of $I-T$ at $0$ gives us that $\omega_w (x) \subseteq F(T)$. By Lemma \ref{uniqueness}, we know that $\omega_w (x)$ consists of a single point, say $y$. Thus, $T^nx \rightharpoonup_n y$, and the theorem is proved.
\varepsilonnd{proof}
\section{Results for arbitrary $\alpha$}
We have the corresponding theorem for $\alpha$ of arbitrary length.
\begin{theorem}
If $C \subseteq X$ is weakly compact, convex, and has the Opial property, $T: C \to C$ is $\alpha$-nonexpansive and asymptotically regular at some point $x \in C$, then $T^nx$ converges weakly to a fixed point of $T$.
\varepsilonnd{theorem}
The theorem will follow immediately from the analogous lemma concerning convergence of the sequence $(\norm{T^nx-y})_n$ for any $y \in \omega_w (x)$.
\begin{lemma}
Suppose $C$ is weakly compact and convex with the Opial property, and suppose that $T : C \to C$ is $\alpha$-nonexpansive and asymptotically regular at some $x \in C$. Then for all $y \in \omega_w (x)$, $\lim_n \norm{T^nx - y}$ exists.
\varepsilonnd{lemma}
\begin{proof}[Proof of the Lemma]
Let $\alpha = (\alpha_1,\ldots, \alpha_{n_0})$. In the same way as above, we build a sequence $(k_n)_n$ for which
\begin{enumerate}
\item $k_n + 1 \leq k_{n+1} \leq k_n + n_0$, and
\item $\norm{T^{k_{n+1}}x-y} \leq \norm{T^{k_n}x - y}$
\varepsilonnd{enumerate}
Again, as above, let $M = \mathbb{N} \setminus \{ k_n : n \in \mathbb{N} \}$. If $M$ is finite, we are done. If $M$ is infinite, then write the elements of $M$ as $(m_n)_n$, strictly increasing. Note that for all $n \in \mathbb{N}$, there exist $j_n \in \mathbb{N}$ and $i_n \in \{1, \ldots, n_0-1\}$ for which
\[
m_n = k_{j_n} + i_n
\]
Also, $(j_n)_n$ is strictly increasing. Now, for any $\varepsilon>0$, we can find $n$ large enough so that
\[
\norm{T^{m_n - j + 1} x - T^{m_n - j } x} < \frac{\varepsilon}{n_0} \quad \mbox{for all } j = 1, \ldots, n_0-1, \mbox{ and}
\]
\[
\Big| \norm{T^{k_{j_n}}x - y} - q \, \Big|< \frac{\varepsilon}{n_0}, \quad \mbox{where } q = \lim_{n\to\infty} \norm{T^{k_n}x-y}
\]
Thus, for $n$ large, we have
\begin{align*}
\norm{T^{m_n}x - y} - q &\leq \norm{T^{m_n}x - T^{m_n - 1} x} + \cdots + \norm{T^{m_n - i_n + 1}x - T^{m_n - i_n} x} + \norm{T^{m_n - i_n} x - y} - q\\
&= \norm{T^{m_n}x - T^{m_n - 1} x} + \cdots + \norm{T^{m_n - i_n + 1}x - T^{m_n - i_n} x} + \norm{T^{k_{j_n}} x - y} - q\\
&< \underbrace{\frac{\varepsilon}{n_0} + \cdots + \frac{\varepsilon}{n_0}}_{i_n \mbox{ times}} + \frac{\varepsilon}{n_0}\\
&\leq (n_0-1) \frac{\varepsilon}{n_0} + \frac{\varepsilon}{n_0} = \varepsilon
\varepsilonnd{align*}
A similar argument proves that $\big| \norm{T^{m_n}x-y} - q \, \big| < \varepsilon$ for $n$ large, and the lemma is proved.
\varepsilonnd{proof}
\section{Losing boundedness of $C$}
Similar arguments show that, under appropriate circumstances, the assumption of boundedness of $C$ may be dropped. Before we state the theorem, we need the notion of a duality mapping, a lemma due to Opial \cite[Lemma 3]{opial67}, and a theorem of Garc\'ia and Piasecki \cite[Theorem 4.2]{garciapiasecki12}.
\begin{definition}
A mapping $J: X \to X^*$ is called a \textit{duality mapping} of $X$ into $X^*$ with gauge function $\mu$ (that is, $\mu: [0,\infty) \to [0,\infty)$ is strictly increasing, continuous, and $\mu(0)=0$) if, for every $x \in X$, $(Jx)(x) = \norm{Jx}\norm{x} = \mu(\norm{x})\norm{x}$.
\varepsilonnd{definition}
Recall also that a Banach space $(X, \norm{\cdot})$ is called \textit{uniformly convex} if for every $\varepsilon \in (0,2]$, there exists a $\delta \in (0,1)$ such that
\[
\begin{cases}
\norm{u},\norm{v} \leq 1,&\\
\norm{u-v} \geq \varepsilon
\varepsilonnd{cases}
\quad \implies \quad \frac{1}{2} \norm{u +v} \leq 1-\delta.
\]
It is easy to see that this is equivalent to a sequential notion of uniform convexity. That is, $X$ is uniformly convex if and only if for every $R>0$ and for any sequences $(u_n)_n$ and $(v_n)_n$ in $X$,
\[
\begin{cases}
\norm{u_n},\norm{v_n} \leq R \mbox{ for all } $n$, \mbox{ and }&\\
\frac{1}{2}\norm{u_n+v_n} \to R
\varepsilonnd{cases}
\quad \implies \quad \norm{u_n - v_n} \to 0.
\]
Now we have a lemma of Opial describing those uniformly convex spaces which have Opial's property.
\begin{lemma}[Opial]
If $(X, \norm{\cdot})$ is uniformly convex and has a weakly continuous duality mapping, then $(X,\norm{\cdot})$ is Opial.
\varepsilonnd{lemma}
Finally, we state a theorem of Garc\'ia and Piasecki regarding the structure of the set of fixed points for any mean nonexpansive mapping defined in a strictly convex space.
\begin{theorem}[Garc\'ia and Piasecki]
Suppose $C \subseteq X$ is closed and convex and $(X,\norm{\cdot})$ is strictly convex. Then for any mean nonexpansive mapping $T : C\to C$, $F(T)$ is closed and convex.
\varepsilonnd{theorem}
We use the tools above to prove a theorem:
\begin{theorem}\label{not bounded}
Suppose $(X,\norm{\cdot})$ is uniformly convex with a weakly sequentially continuous duality map and $C \subseteq X$ is closed and convex. Assume further that $T : C\to C$ is $\alpha$-nonexpansive, $F(T) \neq \varepsilonmptyset$, and $T$ is asymptotically regular at some $x \in C$. Then $(T^n x)_n$ converges weakly to some $y_0 \in F(T)$.
\varepsilonnd{theorem}
The proof follows largely from the work done above and the original proof for nonexpansive mappings due to Opial \cite[Theorem 1]{opial67}, and we present it here for completeness.
\begin{proof}
By Opial's lemma, $X$ is uniformly convex with a weakly continuous duality map implies that $X$ is Opial. Thus, for every $y \in F(T)$, by the proof of Theorem \ref{existence} and Remark \ref{remark}, we know that $\lim_n \norm{T^nx - y}$ exists. In particular, this implies that $\{T^nx : n \in \mathbb{N} \}$ is bounded. Let $\varphi: F(T) \to [0,\infty)$ be given by $\varphi(y) := \lim_n \norm{T^n x - y}$. For any $r \in [0,\infty)$, consider the set
\begin{align*}
F_r :&= \{ y \in F(T) : \varphi(y) \leq r\}\\
&= \varphi^{-1} [0, r].
\varepsilonnd{align*}
We summarize the relevant facts about $F_r$.
\begin{claim}\label{claim}
The sets $F_r$ satisfy the following four properties:
\begin{enumerate}
\item $F_r$ is nonempty for $r$ sufficiently large,
\item $F_r$ is closed, bounded, and convex for all $r\geq 0$,
\item there is a minimal $r_0$ for which $F_{r_0}$ is nonempty, and
\item $F_{r_0}$ is a singleton.
\varepsilonnd{enumerate}
\varepsilonnd{claim}
\begin{proof}[Proof of Claim \ref{claim}]
(1) and (2) are easy to verify.
(3) follows from the fact that each $F_r$ is weakly compact (since $X$ is reflexive) and $\{F_r : r \geq 0\}$ forms a nested family. Thus, if each $F_r \neq \varepsilonmptyset$ for $r > t$ for some $t \geq 0$, it follows that
\[
F_t = \bigcap_{r>t} F_r \neq \varepsilonmptyset.
\]
(4) follows from uniform convexity. Suppose $u, v \in F_{r_0}$ with $u \neq v$, and let $z := \frac{1}{2}(u+v)$. Note that $z \in F_{r_0}$ since $F_{r_0}$ is convex. Because $r_0$ is minimal for which $F_{r_0} \neq \varepsilonmptyset$, it follows that $\varphi(u)=r_0=\varphi(v)$. We want to show that $\varphi(z) < r_0$. Suppose for a contradiction that $\varphi(z) = r_0$. Then
\[
\lim_n \frac{1}{2} \norm{(T^nx - u) + (T^n x - v)} = \lim_n \norm{T^nx - z} = r_0
\]
and uniform convexity implies that
\[
\lim_n \norm{(T^nx - u) - (T^n x - v)} = \norm{u - v} = 0,
\]
but $\norm{u-v}>0$. This tells us that $\varphi(z) < r_0$, which contradicts the minimality of $r_0$. Hence, $F_{r_0}$ must be a singleton. This completes the proof of the claim.
\varepsilonnd{proof}
Let $F_{r_0} = \{y_0\}$. We aim to show that $T^nx \rightharpoonup y_0$. For a contradiction, suppose this is not the case. Since $\{ T^n x : n \in \mathbb{N} \}$ is bounded and $X$ is reflexive, there is some subsequence $(T^{n_k}x)_k$ converging weakly to some $y \neq y_0$. By asymptotic regularity of $T$ and demiclosedness of $I-T$ at $0$, we know that $\norm{(I-T)T^{n_k}x} \to 0$ yields $Ty=y$. That is, $y \in F(T)$. Thus,
\begin{align*}
r_0 = \varphi(y_0) &= \lim_n \norm{T^nx - y_0}\\
&= \lim_k \norm{T^{n_k}x - y_0}\\
&> \lim_k \norm{T^{n_k}x - y}\\
&= \lim_n \norm{T^nx - y} = \varphi(y),
\varepsilonnd{align*}
which contradicts the minimality of $r_0$. Finally, we have that $T^nx \rightharpoonup y_0$, and the proof is complete.
\varepsilonnd{proof}
\begin{remark}
We note here, just as Opial did, that the same result will hold in any reflexive Opial space where $F(T)$ is convex and $F_{r_0}$ is a singleton. For example, to guarantee that $F(T)$ is convex for a mean nonexpansive map, we need only assume strict convexity of $X$ as opposed to uniform convexity.
\varepsilonnd{remark}
\begin{thebibliography}{20}
\bibitem{browderpetryshyn66}
F.E. Browder, W.V. Petryshyn,
``The solution by iteration of nonlinear functional equations in Banach spaces,''
Bull. Amer. Math. Soc., \textbf{72} (1966), pp. 571-575.
\bibitem{gallagher16}
T.M. Gallagher,
``The demiclosedness principle for mean nonexpansive mappings,''
Journal of Math. Analysis \& Appl., \textbf{439} (2016), pp. 832-842.
\bibitem{garciapiasecki12}
V.P. Garc\'ia and \L. Piasecki,
``On mean nonexpansive mappings and the Lifshitz constant,''
Journal of Math. Analysis \& Appl., \textbf{396} (2012), pp. 448-454.
\bibitem{gjp07}
K. Goebel and M. Jap\'on Pineda,
``A new type of nonexpansiveness,''
Proc. of the 8th International Conference on Fixed Point Theory and Appl.,
Chiang Mai, 2007
\bibitem{goebelsims10}
K. Goebel and B. Sims,
\textit{Mean Lipschitzian Mappings},
Contemp. Math.
\textbf{513}, 157-167 (2010)
\bibitem{opial67}
Z.O. Opial,
``Weak convergence of the sequence of successive approximations for nonexpansive mappings,''
Bull. AMS, \textbf{73} (1967), pp. 591-597
\bibitem{piasecki13}
{\L}. Piasecki,
\textit{Classification of Lipschitz Mappings},
CRC Press,
2013
\varepsilonnd{thebibliography}
\varepsilonnd{document} | math | 22,818 |
\begin{document}
\title[The Spectrum of Eventually Positive Operators]{Towards a Perron--Frobenius theory for Eventually Positive Operators}
\author{Jochen Gl\"uck}
\email{[email protected]}
\address{Jochen Gl\"uck, Institute of Applied Analysis, Ulm University, 89069 Ulm, Germany}
\keywords{Eventual positivity; asymptotic positivity; Perron--Frobenius theory; Kre\u{\i}n--Rutman theorem; spectral radius; positive eigenvectors; peripheral spectrum}
\subjclass[2010]{47B65; 47A10}
\date{\today}
\begin{abstract}
This article is a contribution to the spectral theory of so-called eventually positive operators, i.e.\ operators $T$ which may not be positive but whose powers $T^n$ become positive for large enough $n$. While the spectral theory of such operators is well understood in finite dimensions, the infinite dimensional case has received much less attention in the literature.
We show that several sensible notions of ``eventual positivity'' can be defined in the infinite dimensional setting, and in contrast to the finite dimensional case those notions do not in general coincide. We then prove a variety of typical Perron--Frobenius type results: we show that the spectral radius of an eventually positive operator is contained in the spectrum; we give sufficient conditions for the spectral radius to be an eigenvalue admitting a positive eigenvector; and we show that the peripheral spectrum of an eventually positive operator is a cyclic set under quite general assumptions. All our results are formulated for operators on Banach lattices, and many of them do not impose any compactness assumptions on the operator.
\end{abstract}
\maketitle
\section{Introduction}
The classical theorems of Perron and Frobenius about the spectrum of positive matrices, which were published in \cite{Perron1907, Perron1907a} and \cite{Frobenius1908, Frobenius1909, Frobenius1912}, have had a profound impact on mathematical analysis for more than a century now. Given the intrinsic elegance of these theorems as well as their numerous applications (for an overview of several applications, see for instance the survey article \cite{MacCluer2000}) it is no surprise that many attempts were made to generalise those theorems in various respects. Two of those generalisations serve as a motivation for the present paper:
First, it is possible to prove Perron--Frobenius like theorems on infinite dimensional spaces; here, one replaces positive matrices with positive operators defined on a Banach space that is endowed with some kind of ordering. After pioneering work of Kre{\u\i}n and Rutman on rather general ordered Banach spaces \cite{Krein1948} it was noticed later on that positive operators on \emph{Banach lattices} have a particularly rich spectral theory. For an overview of this theory we refer for example to \cite[Section~V.4 and~V.5]{Schaefer1974}, \cite[Chapter~4]{Meyer-Nieberg1991} and \cite{Grobler1995}. Second, one can prove that Perron--Frobenius theorems hold in fact for a much wider class of matrices than only for positive ones. In particular, it is possible to show Perron--Frobenius type results for matrices whose powers become positive for all sufficiently large exponents. Those matrices are usually called \emph{eventually positive}; for the last two decades a very extensive study of them was performed by many researchers (see below for references).
It might come as surprise that until very recently little effort was made to combine those two approaches, i.e.\ to consider eventually positive operators in infinite dimensions. A step into this direction was made in two recent papers by Daners, Kennedy and the author \cite{Daners2016, Daners2016a} where eventually positive $C_0$-semigroups on infinite dimensional Banach lattices were considered. Spectral theory and Perron--Frobenius type results played an essential role there. Yet, such results were proved under rather strong a priori assumptions on the spectrum in order to obtain \emph{characterisations} of eventual positivity which are well-suited for applications to partial differential equations. In the present paper we go into a different direction: we perform an analysis of the spectral properties of eventually positive operators under very general assumptions. In particular, we do not require much a priori information about the spectrum nor do we need the operators (or their powers) to be compact.
\subsection*{Contributions of this article} We define several notions of eventual positivity for an operator and we demonstrate by a couple of examples that those notions are not equivalent in infinite dimensions (Section~\ref{section:basic-notions-1-eventual-positivity}). We also define a related but more general notion which we call \emph{asymptotic positivity} in Section~\ref{section:basic-notions-2-asymptotic-positivity}. Again, we give several versions of this notion and show that they are not equivalent in general. The rest of the article is devoted to a thorough spectral analysis of such operators. Motivated by the classical theorems of Perron and Frobenius we discuss three different themes: (i) the question whether the spectral radius of such an operator is contained in the spectrum, (ii) sufficient conditions for the existence of a positive eigenvector for the spectral radius and (iii) symmetry properties of the so-called \emph{peripheral spectrum}, i.e.\ the set of all spectral values of maximal modulus.
Question~(i) is treated in Sections~\ref{section:the-spectral-radius-1} and~\ref{section:the-spectral-radius-2}. Among others we prove the following result (see Corollary~\ref{cor:weakly-ev-pos-implies-spec-in-spec}):
\begin{theorem_no_number}
Let $T$ be a continuous linear operator on a complex Banach lattice $E$. Suppose that, for all $0 \le x \in E$ and all $0 \le x' \in E'$, there exists an $n_0 \in \mathbb{N}$ such that $\langle x', T^n x \rangle \ge 0$ for all $n \ge n_0$. Then the spectral radius of $T$ is contained in the spectrum of $T$.
\end{theorem_no_number}
Sufficient conditions for the spectral radius to be an eigenvalue and to admit a positive eigenvector are given in Section~\ref{section:positive-eigenvectors}. Finally, we deal with symmetry properties of the peripheral (point) spectrum in Sections~\ref{section:the-peripheral-spectrum} and~\ref{section:the-periphera-point-spectrum}; more precisely, we give sufficient conditions for this set to be \emph{cyclic}. Recall that a subset $S$ of the complex numbers is called cyclic if $re^{i\theta} \in S$ (where $r \in [0,\infty)$ and $\theta \in \mathbb{R}$) implies that $re^{in\theta} \in S$ for all integers $n \in \mathbb{Z}$. One of our results now reads as follows:
\begin{theorem_no_number}
Let $T$ be a continuous linear operator on a complex Banach lattice, with spectral radius $\spr(T) > 0$. If $T/\spr(T)$ is power bounded and if $T^n \ge 0$ for all sufficiently large $n$, then the peripheral spectrum of $T$ is cyclic.
\end{theorem_no_number}
This result is a simple special case of Theorem~\ref{thm:unif-asympt-pos-adjoint-with-bdd-error-cyclcic-per-spec} below.
\subsection*{On the history of eventual positivity} The literature on eventually positive matrices is vast and we cannot give a complete list of references here. Yet, we want to briefly mention some contributions to the theory. Matrices which have at least \emph{one} positive power were already considered in \cite{Brauer1961}, and \cite[pp 48--54]{Seneta1981} deals with matrices for which some polynomial is positive. Another early paper where eventual positivity occurred is \cite{Friedland1978} where the phenomenon was considered in the context of inverse spectral problems; see also \cite{Zaslavsky1999} for a further paper related to inverse spectral problems. Eventually positive matrices were employed in the study of Perron--Frobenius type properties for matrices with some negative entries in numerous papers such as \cite{Tarazaga2001, Johnson2004, Noutsos2006, Elhashash2008, Elhashash2009}. We refer to \cite{Hogben2009} for an algorithm to determine whether a given matrix is eventually positive and to \cite{CarnochanNaqvi2002, CarnochanNaqvi2004, Hogben2015, Saha2015} for further structure results. Moreover, we also refer to the extensive literature on the relation of eventual positivity and sign patterns of a matrix, see for example \cite{Berman2009, Ellison2009, Catral2012}. We point out once again that these references are far from being complete; the reader can find more information in these articles and in the references therein.
Instead of eventually positive matrices and operators it is also worthwhile studying eventually positive $C_0$-semigroups. In the finite dimensional case such semigroups are, for instance, considered in \cite{Noutsos2008, Olesky2009, Erickson2015} and in \cite[Theorem~2.9]{Ellison2009}. Application to control theory can be found in \cite{Altafini2015, Altafini2016, Sootla2016a}. On infinite dimensional spaces eventually positive $C_0$-semigroups first occurred in the analysis a certain concrete differential operators, namely the bi-Laplace operator \cite{Ferrero2008, Gazzola2008} and the Dirichlet-to-Neumann operator \cite{Daners2014}. The development of a general theory of eventually positive $C_0$-semigroups was recently initiated in \cite{Daners2016, Daners2016a}. It is also worthwhile mentioning the related theory of eventually monotone dynamical systems, see \cite{Sootla2015, Sootla2016}.
\subsection*{Preliminaries} We denote the set of all \emph{strictly} positive integers by $\mathbb{N} := \{1,2,...\}$ and we set $\mathbb{N}_0 := \mathbb{N} \cup \{0\}$. By $\mathbb{T} := \{\lambda \in \mathbb{C}: \; |\lambda| = 1\}$ we denote the complex unit circle. We recall again that a set $S \subseteq \mathbb{C}$ is called \emph{cyclic} if $re^{i\theta} \in S$ ($r \in [0,\infty)$, $\theta \in \mathbb{R}$) implies that $re^{in\theta} \in S$ for all $n \in \mathbb{Z}$. If $S$ is a subset of a given vector space, then we denote by $\linSpan S$ the \emph{linear span} (or \emph{linear hull}) of $S$. If $S$ is a subset of a metric space $(M,d)$ and if $x \in M$, then we denote by $\dist(x,S) := \inf\{d(x,y): \; y \in S\}$ the \emph{distance} of $x$ to $S$. If $I \subseteq \mathbb{R}$ is an interval, then we call a function $\varphi: I \to \mathbb{R}$ \emph{increasing} if $\varphi(s) \le \varphi(t)$ for all $s,t \in I$ fulfilling $s \le t$. For every compact Hausdorff space we denote the space of all real- (or complex-) valued continuous functions on $K$ by $\mathcal{C}(K;\mathbb{R})$ (respectively, by $\mathcal{C}(K;\mathbb{C})$) and we endow this space with the usual supremum norm.
Let $E$ be a real or complex Banach space. We denote the space of all bounded linear operators on $E$ by $\mathcal{L}(E)$ and the \emph{identity operator} on $E$ by $\id_E$. By $\mathcal{K}(E)$ we denote the space of all compact linear operators on $E$. The \emph{dual space} of $E$ is denoted by $E'$. For $x \in E$ and $x' \in E'$ we use the common notation $\langle x', x\rangle := x'(x)$. The \emph{adjoint} of an operator $T \in \mathcal{L}(E)$ is denoted by $T' \in \mathcal{L}(E')$. An operator $T \in \mathcal{L}(E)$ is called \emph{power bounded} if $\sup_{n \in \mathbb{N}_0} \|T^n\| < \infty$.
Let $E$ be a complex Banach space and let $T \in \mathcal{L}(E)$. The \emph{spectrum}, the \emph{point spectrum} and the \emph{approximate point spectrum} of $T$ are denoted by $\sigma(T)$, $\sigma_{\operatorname{pnt}}(T)$ and $\sigma_{\operatorname{appr}}(T)$, respectively. The number $\spr(T) := \sup \{|\lambda|: \; \lambda \in \sigma(T)\}$ denotes the \emph{spectral radius} of $T$ and the two sets
\begin{align*}
\sigma_{\operatorname{per}}(T) & := \{\lambda \in \sigma(T): \; |\lambda| = \spr(T)\} \\
\text{and} \qquad \sigma_{{\operatorname{per}},{\operatorname{pnt}}}(T) & := \{\lambda \in \sigma_{\operatorname{pnt}}(T): \; |\lambda| = \spr(T)\}
\end{align*}
are called the \emph{peripheral spectrum} and the \emph{peripheral point spectrum} of $T$. The number $\spr_{\operatorname{ess}}(T) := \sup\{|\lambda|: \; \lambda - T \text{ is not Fredholm}\}$ is called the \emph{essential spectral radius} of $T$; it coincides with the spectral radius of the congruence class of $T$ in the Calkin algebra $\mathcal{L}(E)/\mathcal{K}(E)$ (this follow from Atkinson's theorem see e.g.\ \cite[Theorem~3.3.2]{Arveson2002}). The \emph{resolvent set} of $T$ is denoted by $\rho(T) := \mathbb{C} \setminus \sigma(T)$ and for every $\lambda \in \rho(T)$ the operator $\mathcal{R}(\lambda,T) := (\lambda - T)^{-1}$ is called the \emph{resolvent} of $T$ at $\lambda$.
The reader is assumed to be familiar with the theory of real and complex Banach lattices; standard references for this theory are for instance \cite{Schaefer1974} and \cite{Meyer-Nieberg1991}. Let $E$ be a complex Banach lattice. We denote by $E_\mathbb{R}$ the underlying real Banach lattice, and by $E_+ := (E_\mathbb{R})_+$ the positive cone. We can decompose every element $x \in E$ as $x = \re x + i\im x$, where $\re x$ and $\im x$ are uniquely determined elements of $E_\mathbb{R}$, called the \emph{real part} and the \emph{imaginary part} of $x$. We call a vector $x \in E$ \emph{positive} if $x \in E_+$ and we denote this by $x \ge 0$; moreover, we write $x > 0$ if $x \ge 0$, but $x \not= 0$. More generally, we write $g \ge f$ (respectively, $g > f$) for two elements $f,g \in E$ if $f,g \in E_\mathbb{R}$ and if $g-f \ge 0$ (respectively, if $g \ge f$ but $g \not= f$). We call an operator $T \in \mathcal{L}(E)$ \emph{positive} if $TE_+ \subseteq E_+$ and we denote this by $T \ge 0$. This terminology differs from what is usually used in the theory of (eventually) positive matrices, but it is very common in the theory of Banach lattices. A linear operator $T \in \mathcal{L}(E)$ is called \emph{real} if $TE_\mathbb{R} \subseteq E_\mathbb{R}$. Note that every positive operator is real.
Let $E$ be a real or complex Banach lattice and let $u \in E_+$. The \emph{principal ideal} generated by $u$ is defined to be the set
\begin{align*}
E_u := \{x \in E: \; \exists c \ge 0 \; |x| \le cu\}
\end{align*}
and for every $x \in E_u$ the \emph{gauge} norm of $x$ with respect to $u$ is defined as
\begin{align*}
\|x\|_u := \inf \{c \ge 0: \; |x| \le cu\}.
\end{align*}
The gauge norm is indeed a norm on the vector space $E_u$. The space $E_u$ is a (real or complex) Banach lattice with respect to the gauge norm $\|\cdot\|_u$ and with respect to the order inherited from $E$ (or $E_\mathbb{R}$, respectively); in fact, $(E_u,\|\cdot\|_u)$ is even an AM-space with unit $u$. These results follow from the corollary to Proposition~II.7.2 in \cite{Schaefer1974}. Kakutani's representation theorem for AM-spaces thus asserts that there exists a compact Hausdorff space $K$ and an isometric Banach lattice isomorphism from $E_u$ to $\mathcal{C}(K;\mathbb{R})$ (respectively, to $\mathcal{C}(K;\mathbb{C})$) which maps $u$ to the constant function with value $1$ (which we denote by $\mathbbm{1}$). For the case of real scalars this theorem can, for instance, be found in \cite[Theorem~II.7.4]{Schaefer1974} or in \cite[Theorem~2.1.3]{Meyer-Nieberg1991}; the case of complex scalars is a simple consequence of the real case.
A particular role in our paper is played by the distance of a vector to the positive cone. Let $E$ be a complex Banach lattice. For every $f \in E$ we denote by $\operatorname{d}_+(f) := \dist(f,E_+) = \inf\{\|f-g\|: \; g \in E_+\}$ the distance of $f$ to the positive cone $E_+$. This notation applies in particular to the complex Banach lattice $\mathbb{C}$ where $\mathbb{C}_+ = \mathbb{R}_+ = [0,\infty)$. The function $E \to \mathbb{R}$, $f \mapsto \operatorname{d}_+(f)$ has a few simple but important properties which we are going to use tacitly throughout: for every $f \in E$ we have $\operatorname{d}_+(f) = 0$ if and only if $f \ge 0$; the function $\operatorname{d}_+(\mathord{\,\cdot\,})$ is continuous (even Lipschitz continuous with Lipschitz constant $1$); we have $\operatorname{d}_+(\alpha f) = \alpha \operatorname{d}_+(f)$ for every $\alpha \in [0,\infty)$ and every $f \in E$; we have $\operatorname{d}_+(f+g) \le \operatorname{d}_+(f) + \operatorname{d}_+(g)$ for all $f,g \in E$; and finally, we have $\operatorname{d}_+(f) \le \|f\|$ for all $f \in E$.
\section{Basic Notions I: Eventual Positivity} \label{section:basic-notions-1-eventual-positivity}
In this section we make precise what me mean by an eventually positive operator. Since we are mainly interested in spectral theory in this paper, we formulate the definition on \emph{complex} Banach lattices.
\begin{definition} \label{def:evtl-pos}
Let $E$ be a complex Banach lattice. An operator $T \in \mathcal{L}(E)$ is called
\begin{itemize}
\item[(a)] \emph{uniformly eventually positive} if there exists an $n_0 \in \mathbb{N}$ such that $T^n \ge 0$ for all $n \ge n_0$.
\item[(b)] \emph{individually eventually positive} if for each $x \in E_+$ there exists an $n_0 \in \mathbb{N}$ such that $T^nx \ge 0$ for all $n \ge n_0$.
\item[(c)] \emph{weakly eventually positive} if for each $x \in E_+$ and each $x' \in E'_+$ there exists an $n_0 \in \mathbb{N}_0$ such that $\langle x', T^nx \rangle \ge 0$ for all $n \ge n_0$.
\end{itemize}
\end{definition}
The definition of uniform and individual eventual positivity was motivated by the papers \cite{Daners2016a} and \cite{Daners2016} where the same terminology was introduced for $C_0$-semigroups; the notion of \emph{weak} eventual positivity seems to be new.
Obviously, every uniformly eventually positive operator is also individually eventually positive, and every individually eventually positive operator is weakly eventually positive. Moreover, since every finite dimensional complex Banach lattice is isomorphic to $\mathbb{C}^d$ (this follows e.g.\ from \cite[Corollary~1 to Theorem~II.3.9]{Schaefer1974}), it is clear that the three notions are in fact equivalent in case that $\dim E < \infty$. Encouraged by this observation one might be tempted to suspect that a similar assertion holds for compact operator, or at least for operators of finite rank, on infinite dimensional Banach lattices. Yet, it turns out that this is not the case as the following two examples show.
\begin{examples} \label{ex:ev-pos-not-equivalent}
(a) Let $E = \mathcal{C}([-1,1];\mathbb{C})$ denote the space of all continuous, complex-valued functions on $[-1,1]$, endowed with the supremum norm $\|\cdot\|_\infty$. There exists an operator $T \in \mathcal{L}(E)$ which has two-dimensional range and which is individually but not uniformly eventually positive.
(b) Let $p \in [1,\infty)$ and set $E := L^p((-1,1);\mathbb{C})$. There exists an operator $T \in \mathcal{L}(E)$ which has two-dimensional range and which is weakly but not individually eventually positive.
\end{examples}
\begin{proof}
We use the following notation: Whenever $E$ is a Banach space, $f \in E$ and $\varphi \in E'$, then we define $f \otimes \varphi \in \mathcal{L}(E)$ by $(f \otimes \varphi)g = \langle \varphi,g\rangle f$ for all $g \in E$.
(a) Let $E = \mathcal{C}([-1,1];\mathbb{C})$. Define $f_1,f_2 \in E$ be $f_1(x) = 1$ and $f_2(x) = x$ for all $x \in [-1,1]$ and define $\varphi_1,\varphi_2 \in E'$ by $\langle \varphi_1, g\rangle = \frac{1}{2} \int_{-1}^1 g(x) \, dx$ and $\langle \varphi_2, g \rangle = \frac{1}{4}[g(1) - g(-1)]$ for all $g \in E$. Note that we have
\begin{align}
\label{eq:dualities-for-counterexample}
\begin{aligned}
\langle \varphi_1, f_1 \rangle = 1, & \qquad \langle \varphi_1, f_2 \rangle = 0,\\
\langle \varphi_2, f_1 \rangle = 0, & \qquad \langle \varphi_2, f_2 \rangle = \frac{1}{2}.
\end{aligned}
\end{align}
We define the operator $T \in \mathcal{L}(E)$ by $T = f_1 \otimes \varphi_1 + f_2 \otimes \varphi_2$. Obviously the range $TE$ of $T$ has dimension $2$.
We have $T^n = f_1 \otimes \varphi_1 + \frac{1}{2^{n-1}} f_2 \otimes \varphi_2$ for all $n \in \mathbb{N}$. Let us show that $T$ is individually eventually positive: for every $0 < g \in E$ we have
\begin{align*}
T^ng = \langle \varphi_1,g \rangle f_1 + \frac{1}{2^{n-1}} \langle \varphi_2, g \rangle \to \langle \varphi_1, g \rangle f_1
\end{align*}
as $n \to \infty$. Since $\langle \varphi_1, g \rangle > 0$, since $f_1$ is the constant function with value $1$ and since $T^ng$ is real-valued for each $n$, it follows that $T^ng \ge 0$ for all sufficiently large $n$; hence, $T$ is individually eventually positive.
Now we show that $T$ is not uniformly eventually positive. For every $\varepsilon > 0$ we choose a function $0 \le g_\varepsilon \in E$ which fulfils $\int_{-1}^1 g_\varepsilon(x) \, dx \le \varepsilon$, $g_\varepsilon(-1) = 1$ and $g_\varepsilon(1) = 0$. Then we obtain for every $n \in \mathbb{N}$ that
\begin{align*}
T^ng_\varepsilon = \frac{1}{2} \int_{-1}^1 g_\varepsilon(x) \, dx f_1 - \frac{1}{2^{n-1}} \frac{1}{4} f_2 \le \frac{1}{2} \varepsilon f_1 - \frac{1}{2^{n+1}} f_2
\end{align*}
and thus,
\begin{align*}
(T^ng_\varepsilon)(-1) \le \frac{1}{2} \varepsilon - \frac{1}{2^{n+1}}.
\end{align*}
For every $n \in \mathbb{N}$ we can therefore find an $\varepsilon > 0$ and a function $g_\varepsilon \ge 0$ such that $(T^ng_\varepsilon)(-1) < 0$. This even proves that $T^n$ is not a positive operator for any $n \in \mathbb{N}$. In particular, $T$ is not uniformly eventually positive.
(b) Now, let $p \in [1,\infty)$ and $E = L^p((-1,1);\mathbb{C})$. Let $f_1,f_2 \in E$ be given by $f_1(x) = 1$ and $f_2(x) = \operatorname{sgn}x \, |x|^{-\frac{1}{2p}}$ for all $x \in (-1,1)$; note that $f_2$ is indeed contained in $L^p$ due to the choice of the exponent $-\frac{1}{2p}$. Define $\varphi_1,\varphi_2 \in E'$ by $\langle \varphi_1, g \rangle = \frac{1}{2} \int_{(-1,1)} g(x) \, dx$ and $\langle \varphi_2, g\rangle = c\int_{(-1,1)} \operatorname{sgn}x \, g(x) \, dx$ for all $g \in E$, where the constant $c > 0$ is chosen such that $\langle \varphi_2, f_2 \rangle = \frac{1}{2}$. Note that $f_1,f_2$ and $\varphi_1, \varphi_2$ fulfil the equations~\eqref{eq:dualities-for-counterexample}.
As above we define $T := f_1 \otimes \varphi_1 + f_2 \otimes \varphi_2$ and we thus obtain
\begin{align*}
T^n = f_1 \otimes \varphi_1 + \frac{1}{2^{n-1}} f_2 \otimes \varphi_2
\end{align*}
for all $n \in \mathbb{N}$. Let us show that $T$ is weakly eventually positive. For every $0 < g \in E$ and and every $0 < \psi \in E'$ we obtain
\begin{align*}
\langle \psi, T^n g\rangle = \langle \varphi_1, g\rangle \langle \psi, f_1\rangle + \frac{1}{2^{n-1}} \langle \varphi_2, g\rangle \langle \psi, f_2 \rangle \to \langle \varphi_1, g \rangle \langle \psi, f_1 \rangle > 0
\end{align*}
as $n \to \infty$. Hence, $\langle \psi, T^n g \rangle$ is positive for all sufficiently large $n$.
Now we show that $T$ is not individually eventually positive. To this end, choose a function $0 < g \in E$ such that $\langle \varphi_2, g \rangle > 0$. We obtain for all $n \in \mathbb{N}$ that
\begin{align*}
T^ng = \langle \varphi_1, g\rangle f_1 + \frac{1}{2^{n-1}} \langle \varphi_2, g\rangle f_2.
\end{align*}
The function $f_1$ is bounded, but the function $f_2$ fulfils $\lim_{x \uparrow 0}f_2(x) = -\infty$. This proves that $T^ng \not \ge 0$ for any $n \in \mathbb{N}$. In particular, $T$ is not individually eventually positive.
\end{proof}
We note in passing that one can also construct examples of $C_0$-semigroups which are, say, individually but not uniformly eventually positive (see \cite[Examples~5.7 and~5.8]{Daners2016}); those examples are, however, a bit more involved, in particular if one wants to ensure certain compactness properties.
\section{Basic Notions II: Asymptotic Positivity} \label{section:basic-notions-2-asymptotic-positivity}
The second notion we deal with in this paper is \emph{asymptotic positivity}. For certain $C_0$-semigroups two versions of this notion were introduced in \cite{Daners2016a}. Here we define three versions of the notion for (powers of) single operators. Recall that for every element $f$ of a complex Banach lattice $E$ the symbol $\operatorname{d}_+(f) := \dist(f,E_+)$ denotes the distance of $f$ to the positive cone.
\begin{definition} \label{def:asymp-pos}
Let $T$ be a bounded linear operator on a complex Banach lattice $E$. Let $r(T) > 0$ and define $S := T/\spr(T)$. We call $T$
\begin{itemize}
\item[(a)] \emph{uniformly asymptotically positive} $\sup_{x \in E_+, \; \|x\| \le 1} \operatorname{d}_+(S^n x) \to 0$ as $n \to \infty$.
\item[(b)] \emph{individually asymptotically positive} if $\operatorname{d}_+(S^nx) \to 0$ as $n \to \infty$ for each $x \in E_+$.
\item[(c)] \emph{weakly asymptotically positive} if $\operatorname{d}_+(\langle x', S^nx \rangle) \to 0$ as $n \to \infty$ for each $x \in E_+$ and each $x' \in E'_+$.
\end{itemize}
\end{definition}
\begin{remarks}
(a) The above definition is rather bold in the following sense. Consider an operator $T$ on a complex Banach lattice $E$ with $r(T) = 1$ and assume that $T$ is uniformly asymptotically positive. In case that $T$ is power bounded uniform asymptotic positivity means that the fraction
\begin{align}
\label{eq:fraction-for-uniform-asympt-pos}
\frac{\sup_{x \in E_+, \; \|x\| = 1} \operatorname{d}_+(T^n x)}{\|T^n\|}
\end{align}
converges to $0$ as $n \to \infty$. If, however, $T$ is not power-bounded, then the condition that $T$ be uniformly asymptotically positive is \emph{stronger} than the condition that~\eqref{eq:fraction-for-uniform-asympt-pos} converge to $0$.
Hence, one could also suggest to call $T$ asymptotically positive if only the fraction~\eqref{eq:fraction-for-uniform-asympt-pos} converges to $0$, and it does not seem to be clear whether one should prefer this definition or the definition that we gave above (and that we use throughout the paper). This is the reason why asymptotic positivity for $C_0$-semigroups was only defined under additional boundedness assumptions in \cite{Daners2016a} (compare also \cite[Problem~(c) in Section~10]{Daners2016a}).
(b) We did not assume the operator $T$ in Definition~\ref{def:asymp-pos} to be real. A simple example for a non-real but uniformly asymptotically positive operator $T$ on the complex Banach lattice $\mathbb{C}^2$ is given by
\begin{align*}
T =
\begin{pmatrix}
1 & 0 \\
0 & \frac{i}{2}
\end{pmatrix}.
\end{align*}
None of our subsequent spectral results for asymptotically positive operators requires the operator to be real. Hence, (the finite dimensional versions of) these results contribute to the Perron--Frobenius theory of matrices with some complex entries, a topic which has already been studied by several authors and from various perspectives (see for instance \cite{Rump2003, Noutsos2012, Tudisco2015}).
\end{remarks}
The following proposition might help to get a better understanding of the distance to the positive cone which plays an important role in Definition~\ref{def:asymp-pos}.
\begin{proposition} \label{prop:formula-for-distance-to-positive-cone}
Let $E$ be a complex Banach lattice and let $x \in E$. Then we have $\operatorname{d}_+(x) = \|x - (\re x)^+\| = \|-(\re x)^- + i \im x\|$.
\end{proposition}
\begin{proof}
Let $x \in E$. The second equality in the assertion is obvious, so let us prove the first equality. We define $\hat x := x - (\re x)^+$. Of course we have $\operatorname{d}_+(x) \le \|\hat x\|$.
In order to prove the converse inequality it suffices to show that $|\hat x| \le |x - y|$ for all $y \in E_+$. This inequality is easy to check in case that $E = \mathbb{C}$ and hence it is also true if $E$ is the space $\mathcal{C}(K;\mathbb{C})$ of continuous complex-valued functions on a compact Hausdorff space $K$. Now, let $E$ be arbitrary and let $y \in E_+$. We set $u := |x| + y \in E_+$. The principal ideal $E_u$ contains both $x$ and $y$ and, when endowed with the gauge norm $\|\cdot\|_u$, it is a complex Banach lattice which is isometrically Banach lattice isomorphic to a $\mathcal{C}(K;\mathbb{C})$-space. Therefore, the inequality $|\hat x| \le |x-y|$ is true in the complex Banach lattice $(E_u,\|\cdot\|_u)$. Since $E_u$ is an ideal in $E$, the complex modulus in $E_u$ coincides with the complex modulus in $E$ for every element of $E_u$. Thus we have $|\hat x| \le |x-y|$ in $E$, as claimed.
\end{proof}
Let us briefly comment on the relation between the three notions in Definition~\ref{def:asymp-pos}:
\begin{proposition} \label{prop:asymp-pos-relation-between-different-versions}
Let $E$ be a complex Banach lattice and let $T \in \mathcal{L}(E)$ be an operator with $\spr(T)$.
\begin{itemize}
\item[(a)] If $T$ is uniformly asymptotically positive, then $T$ is also individually asymptotically positive.
\item[(b)] If $T$ is individually asymptotically positive, then $T$ is also weakly asymptotically positive.
\end{itemize}
\end{proposition}
\begin{proof}
The implication in~(a) is obvious and the implication in~(b) follows from the estimate
\begin{align*}
\operatorname{d}_+(\langle x', y\rangle) \le \inf_{x \in E_+} |\langle x', y \rangle - \langle x',x \rangle| \le \|x'\| \operatorname{d}_+(y),
\end{align*}
which is true for all vectors $y \in E$ and for all functionals $0 \le x' \in E'$.
\end{proof}
As in Section~\ref{section:basic-notions-1-eventual-positivity}, the converse implications are not in general true. We demonstrate this by two simple examples:
\begin{examples} \label{ex:asymp-pos-not-equivalent}
Let $p \in [1,\infty)$ and let $E := \ell^p(\mathbb{N};\mathbb{C})$.
(a) There exists an operator $T \in \mathcal{L}(E)$ with spectral radius $\spr(T) = 1$ which has the following properties: $T$ is not uniformly asymptotically positive, but $T^n$ converges strongly to $0$ as $n \to \infty$ and thus $T$ is individually asymptotically positive (though for trivial reasons).
(b) There exists an operator $T \in \mathcal{L}(E)$ with spectral radius $\spr(T) = 1$ which has the following properties: $T$ is not individually asymptotically positive, but $T^n$ converges weakly to $0$ as $n \to \infty$ and thus $T$ is weakly asymptotically positive (though for trivial reasons).
\end{examples}
\begin{proof}
(a) If we define $T$ to be the multiplication operator with symbol $(-1+\frac{1}{n})_{n \in \mathbb{N}}$, then $T$ clearly has all the claimed properties.
(b) Let $T$ be $-1$ times the right shift operator on $E$; then $T$ has all the properties that we claimed.
\end{proof}
On the other hand weak, individual, and even uniform eventual positivity coincide under sufficiently strong compactness assumptions as we prove in the following proposition.
\begin{proposition} \label{prop:regularity-implies-better-asymp-pos}
Let $E$ be a complex Banach lattice and let $T \in \mathcal{L}(E)$ be an operator with $\spr(T) > 0$ which is weakly asymptotically positive. Denote by $\mathcal{S} := \{(T/\spr(T))^n: \; n \in \mathbb{N}_0\}$ the semigroup in $\mathcal{L}(E)$ generated by the rescaled operator $T/\spr(T)$.
\begin{itemize}
\item[(a)] If $\mathcal{S}$ is relatively compact in $\mathcal{L}(E)$ with respect to the strong operator topology, then $T$ is individually asymptotically positive.
\item[(b)] If $\mathcal{S}$ is relatively compact in $\mathcal{L}(E)$ with respect to the operator norm topology, then $T$ is uniformly asymptotically positive.
\end{itemize}
\end{proposition}
In the situation of the above proposition, define $S := T/\spr(T)$. If the set $\mathcal{S}$ is relatively compact with respect to the strong operator topology, then $S$ is usually said to have \emph{relatively compact orbits} or to be \emph{almost periodic} in the literature. It is well-known that $\mathcal{S} = \{S^n: \; n \in \mathbb{N}_0\}$ is relatively compact in $\mathcal{L}(E)$ with respect to the strong operator topology if and only if, for every $f \in E$, the set $\{S^nf: \; n \in \mathbb{N}_0\}$ is relatively compact in $E$ with respect to the norm topology; see e.g.~\cite[Corollary~A.5]{Engel2000}.
The condition in assertion~(b) that $\mathcal{S}$ be relatively compact with respect to the operator norm topology is for instance fulfilled if $S$ is power-bounded and some power of $T$ is compact.
\begin{proof}[Proof of Proposition~\ref{prop:regularity-implies-better-asymp-pos}]
We may assume throughout the proof that $\spr(T) = 1$.
(a) Let $f \in E_+$. We have to show that $\operatorname{d}_+(T^nf)$ converges to $0$ as $n \to \infty$, and to this end it suffices to prove that every subsequence $(\operatorname{d}_+(T^{n_k}f))_{k \in \mathbb{N}_0}$ of $(\operatorname{d}_+(T^nf))_{n \in \mathbb{N}_0}$ has itself a subsequence which converges to $0$. Since the set $\{T^{n_k}f: \; k \in \mathbb{N}_0\}$ is relatively compact in $E$, the sequence $(T^{n_k}f)_{k \in \mathbb{N}_0}$ has a subsequence $(T^{n_{k_j}})_{j \in \mathbb{N}_0}$ which converges to a vector $g \in E$. Since $T$ is weakly asymptotically positive, one readily obtains that $g \in E_+$. Hence,
\begin{align*}
\operatorname{d}_+(T^{n_{k_j}}f) \le \|T^{n_{k_j}} - g\| \to 0,
\end{align*}
which proves that claim.
(b) Let $\mathcal{L}(E)_+$ denote the set of all positive operators in $\mathcal{L}(E)$. If the assumption of~(b) is fulfilled, then we can show by the same arguments as in~(a) that $\dist(T^n,\mathcal{L}(E)_+) \to 0$ as $n \to \infty$. Now, let $\varepsilon > 0$ and choose $n_0 \ge n$ such that $\dist(T^n,\mathcal{L}(E)_+) < \varepsilon$ for each $n \ge n_0$. Then, for every $n \ge n_0$, we can find an operator $R_n \ge 0$ such that $\|T^n-R_n\| < \varepsilon$. In particular we obtain for every $f \in E_+$ with $\|f\| \le 1$ and every $n \ge n_0$ that
\begin{align*}
\operatorname{d}_+(T^nf) \le \|T^nf - R_nf\| + \operatorname{d}_+(R_nf) \le \|T^n-R_n\| < \varepsilon.
\end{align*}
Thus,
\begin{align*}
\sup_{f \in E_+, \; \|f\| \le 1} \operatorname{d}_+(T^nf) \le \varepsilon
\end{align*}
for all $n \ge n_0$, which proves that $T$ is indeed uniformly asymptotically positive.
\end{proof}
\section{The Spectral Radius I} \label{section:the-spectral-radius-1}
Let $E$ be a complex Banach lattice and let $T \in \mathcal{L}(E)$. If $T$ is positive, then it is well-known that $\spr(T) \in \sigma(T)$, see e.g.\ \cite[Proposition~V.4.1]{Schaefer1974}. In this section we prove that the same is still true for uniformly asymptotically positive operators. Individually asymptotically positive operator, for which the situation is more subtle, and individually eventually positive operators are treated in the next section. Our main result in this section is as follows:
\begin{theorem} \label{thm:spec-rad-in-spec-for-unif-asymp-pos-op}
Let $E$ be a complex Banach lattice and let $T \in \mathcal{L}(E)$ an operator with $\spr(T) > 0$. If $T$ is uniformly asymptotically positive, then $r(T) \in \sigma(T)$.
\end{theorem}
The proof of the above theorem is most easily understood if we recall how the proof for positive operators works. So let $T$ be a positive operator on a complex Banach lattice $E$. A simple application of the Neumann series representation of the resolvent yields the estimate $|\mathcal{R}(\lambda,A)f| \le \mathcal{R}(|\lambda|,A)|f|$ for all $f \in E$ and all $\lambda \in \mathbb{C}$ with $|\lambda| > \spr(T)$. From this resolvent estimate one can easily deduce that $\spr(T) \in \sigma(T)$.
For the proof of Theorem~\ref{thm:spec-rad-in-spec-for-unif-asymp-pos-op} we use a similar approach. Let us begin by showing a version of the estimate $|\mathcal{R}(\lambda,A)f| \le \mathcal{R}(|\lambda|,A)|f|$; since the positivity is only asymptotic now, a certain error term occurs (compare also \cite[Lemma~7.4]{Daners2016}):
\begin{lemma} \label{lem:resolvent-estimate-for-unif-asymp-pos-op}
Let $E$ be a complex Banach lattice and let $T \in \mathcal{L}(E)$ be an operator with $\spr(T) = 1$ which is uniformly asymptotically positive. Then there is a function $\omega: (1,\infty)\times E_+ \to E_+$ with the following properties:
\begin{itemize}
\item[(a)] For each $x \in E_+$ and each $\lambda \in \mathbb{C}$ satisfying $|\lambda| > 1$ we have
\begin{align*}
|\mathcal{R}(\lambda,T)x| \le \re\big(\mathcal{R}(|\lambda|,T)x\big) + \omega(|\lambda|,x) \text{.}
\end{align*}
\item[(b)] We have $\sup_{x \in E_+,\; \|x\| \le 1} (r - 1)\|\omega(r,x)\| \to 0$ as $r \downarrow 1$.
\end{itemize}
\end{lemma}
\begin{proof}
For each $r > 1$ and each $x \in E_+$ we define
\begin{align*}
\omega(r,x) = \sum_{n=0}^\infty \frac{1}{r^{n+1}} \big( |T^nx| - \re (T^n x) \big).
\end{align*}
Let us show that this function fulfils the assertions~(a) and~(b).
(a) For every $x \in E_+$ and every $\lambda \in \mathbb{C}$, $|\lambda| > 1$, we have
\begin{align*}
\omega(|\lambda|,x) & = \sum_{n=0}^\infty |\frac{1}{\lambda^{n+1}} T^n x| - \sum_{n=0}^\infty \frac{1}{|\lambda|^{n+1}} \re(T^n x) \ge \\
& \ge |\sum_{n=0}^\infty \frac{1}{\lambda^{n+1}} T^n x| - \re \sum_{n=0}^\infty \frac{1}{|\lambda|^{n+1}} T^n x.
\end{align*}
The first summand in the latter term equals $|\mathcal{R}(\lambda,T)x|$ and the second summand equals $\re \big(\mathcal{R}(|\lambda|,T)x\big)$, so we obtain (a).
(b) Define $\delta_n := \sup_{x \in E_+, \; \|x\| \le 1} \operatorname{d}_+(T^n x)$ for each $n \in \mathbb{N}_0$. Since $T$ is uniformly asymptotically positive, we have $\delta_n \to 0$ as $n \to \infty$. As explained in the subsequent Remark~\ref{rem:estimate-for-real-part-and-distance-to-positive-cone} we have $\| |y| - \re y \| \le 2 \operatorname{d}_+(y)$ for each $y \in E$. Using this, we obtain that
\begin{align*}
\sup_{x \in E_+, \; \|x\| \le 1} \|\omega(r,x)\| \le \sup_{x \in E_+, \; \|x\| = 1} 2 \sum_{n=0}^\infty \frac{1}{r^{n+1}} \dist(T^nx,E_+) \le 2 \sum_{n=0}^\infty \frac{1}{r^{n+1}} \delta_n \text{,}
\end{align*}
for every $r > 1$.
Using that $\sum_{n=0}^\infty \frac{r-1}{r^{n+1}} = 1$ for all $r > 1$ and that $\delta_n \to 0$ as $n \to \infty$, we can see that $\sum_{n=0}^\infty \frac{r-1}{r^{n+1}}\delta_n \to 0$ as $r \downarrow 1$. Hence, we obtain~(b).
\end{proof}
In the proof of Lemma~\ref{lem:resolvent-estimate-for-unif-asymp-pos-op} we made use of the following observation.
\begin{remark} \label{rem:estimate-for-real-part-and-distance-to-positive-cone}
Let $E$ be a complex Banach lattice. Then we have $\| |x| - \re x \| \le 2 \operatorname{d}_+(x)$ for every $x \in E$.
\end{remark}
\begin{proof}
According to Proposition~\ref{prop:formula-for-distance-to-positive-cone} we have have $\|x - (\re x)^+\| = \operatorname{d}_+(x)$, so it suffices to show that $0 \le |x| - \re x \le 2|x - (\re x)^+|$.
The first equality is obvious since we have $\re x \le |\re x| \le |x|$. In order to prove the second inequality $|x| - \re x \le 2|x - (\re x)^+|$ one argues as in the proof of Proposition~\ref{prop:formula-for-distance-to-positive-cone}: first one checks by a brief computation that the inequality holds if $E = \mathbb{C}$ and hence it also holds if $E$ is the space of continuous complex-valued functions on any compact Hausdorff space. This implies that the inequality is true in the principal ideal $E_{|x|}$ and hence in $E$.
\end{proof}
The last ingredient that we need for the proof of Theorem~\ref{thm:spec-rad-in-spec-for-unif-asymp-pos-op} is the following simple observation about the norm of operators on a complex Banach lattice.
\begin{remark} \label{rem:norm-estimate-on-positive-cone}
Let $E$ be a complex Banach lattice and let $T \in \mathcal{L}(E)$. Then there exists a vector $x \in E_+$ of norm $\|x\| \le 1$ such that $\|Tx\| \ge \frac{1}{8}\|T\|$.
\end{remark}
\begin{proof}
The assertion is obvious of $T = 0$, so assume that $\|T\| > 0$. Then we can find a vector $z \in E$ of norm $\|z\| \le 1$ such that $\frac{1}{2}\|T\| \le \|Tz\|$ and hence,
\begin{align*}
\frac{1}{2}\|Tz\| \le \|T(\re z)^+\| + \|T(\re z)^-\| + \|T(\im z)^+\| + \|T(\im z)^-\|.
\end{align*}
Thus, at least one the latter four summands is $\ge \frac{1}{8}\|Tz\|$.
\end{proof}
The estimate in the above remark is of course not optimal, but it suffices for our purposes. Using the resolvent estimate from Lemma~\ref{lem:resolvent-estimate-for-unif-asymp-pos-op} we can now prove Theorem~\ref{thm:spec-rad-in-spec-for-unif-asymp-pos-op}:
\begin{proof}[Proof of Theorem~\ref{thm:spec-rad-in-spec-for-unif-asymp-pos-op}]
We may assume that $\spr(T) = 1$. Let $\lambda \in \mathbb{C}$ be a spectral value of $T$ of modulus $1$ and choose a sequence $(r_n) \subseteq (1,\infty)$ which converges to $1$. According to Remark~\ref{rem:norm-estimate-on-positive-cone} we can find a sequence of vectors $(x_n) \subseteq E_+$, each of them of norm $\le 1$, such that
\begin{align*}
\|\mathcal{R}(r_n\lambda,T)x_n\| \ge \frac{1}{8} \|\mathcal{R}(r_n\lambda,T)\| \ge \frac{1}{8} \frac{1}{\dist(r_n\lambda,\sigma(T))} = \frac{1}{8(r_n-1)}.
\end{align*}
Let $\omega: (1,\infty) \times E_+ \to E_+$ be as in Lemma~\ref{lem:resolvent-estimate-for-unif-asymp-pos-op}. For each index $n$ we define $\delta_n := \sup_{x \in E_+, \, \|x\| \le 1} \|\omega(r_n,x)\|$. Then
\begin{align*}
(r_n-1) & \|\mathcal{R}(r_n,T)\| \ge (r_n-1)\|\re (\mathcal{R}(r_n,T)x_n)\| \\
& \ge (r_n-1)\|\mathcal{R}(r_n\lambda,T)x_n\| - (r_n-1)\delta_n \ge \frac{1}{8} - (r_n-1)\delta_n \to \frac{1}{8}
\end{align*}
as $n \to \infty$, so $\|\mathcal{R}(r_n,T)\| \to \infty$. Since $(r_n)$ converges to $1$ we conclude that $1 \in \sigma(T)$, as claimed.
\end{proof}
\section{The Spectral Radius II} \label{section:the-spectral-radius-2}
While the spectral radius of a uniformly asymptotically positive operator is always contained in its spectrum according to Theorem~\ref{thm:spec-rad-in-spec-for-unif-asymp-pos-op}, the situation is more subtle for individually and weakly asymptotically positive operators. We first demonstrate by a simple example what is \emph{not} true:
\begin{example}
Let $p \in [1,\infty)$ and let $E := \ell^p(\mathbb{N};\mathbb{C})$. There exists an operator $T \in \mathcal{L}(E)$ with spectral radius $\spr(T) = 1$ which has the following properties: the powers $T^n$ converges strongly to $0$ as $n \to \infty$, so $T$ is individually asymptotically positive; yet, the spectral radius $\spr(T) = 1$ is not contained in the spectrum $\sigma(T)$.
\end{example}
\begin{proof}
Let $T$ be the multiplication operator with symbol $(-1+\frac{1}{n})_{n \in \mathbb{N}}$ (which we have already considered in Example~\ref{ex:asymp-pos-not-equivalent}(a)). Then $T$ fulfils all the properties we claimed.
\end{proof}
The above example raises the question whether at least for weakly \emph{eventually} positive operators the spectral radius is contained in the spectrum. This is indeed the case, and it follows from a more general result: we will see in Theorems~\ref{thm:countable-summability-condition-implies-spec-in-spec} and~\ref{thm:weakly-asymp-pos-with-rate-implies-spec-rad-in-spec} below that the spectral radius of a weakly asymptotically positive operator is automatically contained in the spectrum provided that the sequences $\big(\operatorname{d}_+(\langle x', T^nx \rangle)\big)_{n \in \mathbb{N}_0}$ (for $x \in E_+$, $x' \in E'_+$) do not only converge to $0$, but satisfy a certain decay rate.
Such a result might not come as a complete surprise and it is motivated by the following observation: let $T$ be a continuous linear operator on a, say complex, Banach space $E$. If, for all $x \in E$ and all $x' \in E'$, the sequence $(\langle x', T^nx\rangle)_{n \in \mathbb{N}_0}$ converges to $0$ \emph{with a certain rate}, then the powers $T^n$ actually converge to $0$ with respect to the operator norm; results of this type can for instance be found in \cite{Weiss1989}, \cite{Neerven1995} and \cite{Glueck2015}. Here, we consider an operator $T$ on a complex Banach lattice $E$ such that for all $x,x'\ge 0$ the sequence $\big((\operatorname{d}_+(\langle x',T^nx\rangle)\big)_{n \in \mathbb{N}_0}$ has a certain decay rate. We do not know whether this condition already implies that $T$ is uniformly asymptotically positive, but we are going to show that the condition implies $\spr(T) \in \sigma(T)$. Our main results in this section are the following theorem and its corollaries:
\begin{theorem} \label{thm:countable-summability-condition-implies-spec-in-spec}
Let $E$ be a complex Banach lattice, let $T \in \mathcal{L}(E)$ with $r(T) > 0$ and define $S := T/\spr(T)$. Let $\Phi$ be an at most countable set of increasing functions $\varphi: [0,\infty) \to [0,\infty)$ which fulfil $\varphi(t) > 0$ for all $t > 0$. Suppose that for every $x \in E_+$ with $\|x\| \le 1$ and for every $x \in E'_+$ with $\|x'\| \le 1$ there exists a function $\varphi \in \Phi$ such that
\begin{align*}
\sum_{n=0}^\infty \varphi\big(\operatorname{d}_+(\langle x', S^n x \rangle)\big) < \infty \text{.}
\end{align*}
Then $\spr(T) \in \sigma(T)$.
\end{theorem}
It follows readily from Theorem~\ref{thm:countable-summability-condition-implies-spec-in-spec} that we have $\spr(T) \in \sigma(T)$ in case that the distance of $\langle x', S^n x \rangle$ to the positive real numbers is summable. Let us formulate this as an extra corollary:
\begin{corollary} \label{cor:negative-parts-summable-implies-spec-in-spec}
Let $E$ be a complex Banach lattice, let $T \in \mathcal{L}(E)$ with $r(T) > 0$ and define $S := T/\spr(T)$. Suppose that
\begin{align*}
\sum_{n=0}^\infty \operatorname{d}_+(\langle x', S^n x \rangle) < \infty
\end{align*}
for all $x \in E_+$, $x' \in E'_+$. Then $\spr(T) \in \sigma(T)$.
\end{corollary}
From Theorem~\ref{thm:countable-summability-condition-implies-spec-in-spec} (or from Corollary~\ref{cor:negative-parts-summable-implies-spec-in-spec}) it follows, in particular, that the spectral radius of a weakly eventually positive operators is contained in the spectrum. We state this in an extra corollary, too:
\begin{corollary} \label{cor:weakly-ev-pos-implies-spec-in-spec}
Let $E \not= \{0\}$ be a complex Banach lattice and let $T \in \mathcal{L}(E)$ be weakly eventually positive. Then $\spr(T) \in \sigma(T)$.
\end{corollary}
Finally, we formulate another consequence of Theorem~\ref{thm:countable-summability-condition-implies-spec-in-spec} which is more general then Corollary~\ref{cor:negative-parts-summable-implies-spec-in-spec}:
\begin{corollary} \label{cor:negative-parts-in-l_p-implies-spec-in-spec}
Let $E$ be a complex Banach lattice, let $T \in \mathcal{L}(E)$ with $r(T) > 0$ and define $S := T/\spr(T)$. Suppose that
\begin{align*}
\big(\operatorname{d}_+(\langle x', S^n x \rangle)\big)_{n \in \mathbb{N}_0} \in \bigcup_{1 \le p < \infty} \ell^p(\mathbb{N}_0; \mathbb{R})
\end{align*}
for all $x \in E_+$, $x' \in E'_+$. Then $\spr(T) \in \sigma(T)$.
\end{corollary}
For the proof of Theorem~\ref{thm:countable-summability-condition-implies-spec-in-spec} we employ techniques from \cite{Glueck2015}. We first need to introduce a bit of terminology. By $c_0$ we denote the space of all complex-valued sequences which are indexed over $\mathbb{N}_0$ and which converge to $0$. We endow this space with the supremum norm which renders it a complex Banach lattice. In particular, for every $u \in (c_0)_+$, the principal ideal $(c_0)_u$ is defined as explained in the preliminaries.
For a sequence $0 \le x = (x_n)_{n \in \mathbb{N}_0} \in c_0$ we denote by $x^* := (x_n^*)_{n \in \mathbb{N}_0} \in c_0$ the \emph{decreasing rearrangement} of $x$, i.e.~the sequence consisting of the same entries as $x$ (including multiplicities) which have been rearranged in decreasing order. The following definition is taken from \cite[Definition~3.1]{Glueck2015}.
\begin{definition}
Let $F \subset (c_0)_+$ and let $a = (a_n)_{n \in \mathbb{N}_0}$ be a sequence of complex numbers. We say that $F$ \emph{governs} the sequence $a$ if $a \in c_0$ and if there exists an element $f \in F$ such that the decreasing rearrangement $|a|^*$ of $|a|$ is contained in the principal ideal $(c_0)_f$.
\end{definition}
A corollary of the following quite general result will be the key to give the proof of Theorem~\ref{thm:countable-summability-condition-implies-spec-in-spec} at the end of the section.
\begin{theorem} \label{thm:weakly-asymp-pos-with-rate-implies-spec-rad-in-spec}
Let $E$ be a complex Banach lattice, let $T \in \mathcal{L}(E)$ with $\spr(T) > 0$ and define $S := T/\spr(T)$. Let $f \in (c_0)_+$ and suppose that $\{f\}$ governs the sequence
\begin{align*}
\big(\operatorname{d}_+(\langle x', S^n x \rangle)\big)_{n \in \mathbb{N}_0}
\end{align*}
for each $x \in E_+$ and each $x \in E'_+$. Then $r(T) \in \sigma(T)$.
\end{theorem}
We can prove Theorem~\ref{thm:weakly-asymp-pos-with-rate-implies-spec-rad-in-spec} by a method which was also employed in the proof of \cite[Theorem~3.2]{Glueck2015}:
\begin{proof}[Proof of Theorem~\ref{thm:weakly-asymp-pos-with-rate-implies-spec-rad-in-spec}]
We may assume that $\spr(T) = 1$ and that $f \not= 0$ (if $f = 0$ then we can replace $f$ with an arbitrary function from $(c_0)_+ \setminus \{0\}$). Choose $\mu \in \sigma(T)$ with $|\mu| = 1$.
For every $r > 1$ we define $\alpha(r) := \sum_{n=0}^\infty \frac{f_n}{r^{n+1}} > 0$. Using the Neumann series representation of the resolvent we obtain for all $x \in E_+$, $x' \in E'_+$ and $r \in (1,\infty)$ that
\begin{align*}
|\langle x', \mathcal{R}(r \mu,T)x \rangle| - \re \langle x', & \mathcal{R}(r,T) x \rangle \le \sum_{n=0}^\infty \frac{|\langle x', T^nx \rangle| - \re \langle x', T^nx \rangle}{r^{n+1}} \\
& \le 2 \sum_{n=0}^\infty \frac{\operatorname{d}_+(\langle x', T^nx \rangle)}{r^{n+1}} \le 2 \sum_{n=0}^\infty \frac{\operatorname{d}_+(\langle x', T^nx \rangle)^*}{r^{n+1}},
\end{align*}
where we used Remark~\ref{rem:estimate-for-real-part-and-distance-to-positive-cone} to obtain the inequality between the first and the second line and where an infinite series version of the rearrangement inequality (see e.g.\ \cite[Lemma~3.3]{Glueck2015}) yields the inequality in the second line.
By assumption there exists a number $c \ge 0$ (which might depend on $x$ and $x'$) such that $\operatorname{d}_+(\langle x', T^nx \rangle)^* \le cf_n$ for each $n \in \mathbb{N}_0$, so we conclude that
\begin{align}
\label{eq:resolvent-estimate-ind-asymp-pos}
\begin{aligned}
\re \langle x', \mathcal{R}(r,T)x \rangle & \ge |\langle x', \mathcal{R}(r\mu,T)x \rangle| - 2c \, \sum_{n=0}^\infty \frac{f_n}{r^{n+1}} \\
& = |\langle x', \mathcal{R}(r\mu,T)x \rangle| - 2c \, \alpha(r)
\end{aligned}
\end{align}
for all $r > 1$.
One can easily check that $(r-1)\alpha(r) \to 0$ for $r \downarrow 1$ since $f \in c_0$. Noting that $\|R(r\mu,T)\| \ge \frac{1}{\operatorname{dist}(r\mu, \sigma(T))} = \frac{1}{r-1}$ we thus conclude that $\lim_{r \downarrow 1} \frac{\|R(r\mu,T)\|}{\alpha(r)} = \infty$. Due to the uniform boundedness principle we can therefore find vectors $x \in E_+$ and $x' \in E'_+$ and a sequence of real numbers $r_k \downarrow 1$ such that
\begin{align*}
\lim_{k \to \infty} \frac{|\langle x', R(r_k\mu,T)x \rangle|}{\alpha(r_k)} = \infty \text{.}
\end{align*}
Using~\eqref{eq:resolvent-estimate-ind-asymp-pos} and the fact that $\liminf_{r \downarrow} \alpha(r) > 0$ we thus obtain the estimate
\begin{align*}
\re \langle x', R(r_k,T) x \rangle \ge \alpha(r_k) \Big( \frac{|\langle x', R(r_k\mu,T)x \rangle|}{\alpha(r_k)} - 2c \Big) \overset{k \to \infty}{\to} \infty \text{.}
\end{align*}
Hence, $\lim_{k \to \infty} \|R(r_k,T)\| = \infty$ which proves that $1 \in \sigma(T)$.
\end{proof}
In the following corollary we show that the conclusion of Theorem~\ref{thm:weakly-asymp-pos-with-rate-implies-spec-rad-in-spec} remains true if one allows $f$ to vary within a countable set. The proof is virtually the same as the proof of \cite[Proposition~3.4 and~Corollary~3.5]{Glueck2015}; for the convenience of the reader we include the entire argument here.
\begin{corollary} \label{cor:weakly-asymp-pos-with-countably-varying-rate-implies-spec-rad-in-spec}
Let $E$ be a complex Banach lattice, let $T \in \mathcal{L}(E)$ with $\spr(T) > 0$ and define $S := T/\spr(T)$. Let $F \subseteq (c_0)_+$ be an at most countable set and suppose that $F$ governs the sequence
\begin{align*}
\big(\operatorname{d}_+(\langle x', S^n x \rangle)\big)_{n \in \mathbb{N}_0}
\end{align*}
for each $x \in E_+$ and each $x \in E'_+$. Then $r(T) \in \sigma(T)$.
\end{corollary}
\begin{proof}
It follows from the assumptions that $F \not= \emptyset$; we may assume that $F$ does not contain $0$. Let $(f_n)_{n \in \mathbb{N}}$ be an enumeration of the elements of $F$ (where some of the vectors in $F$ may occur several times in case that $F$ is finite). We define
\begin{align*}
f := \sum_{n=1}^\infty \frac{f_n}{2^n\|f_n\|} \in c_0.
\end{align*}
Then $f$ dominates a multiple of every element of $F$ and hence, $\{f\}$ governs the sequence
\begin{align*}
\big(\operatorname{d}_+(\langle x', S^n x \rangle)\big)_{n \in \mathbb{N}_0}
\end{align*}
for all $x \in E$ and all $x' \in E'$. According to Theorem~\ref{thm:weakly-asymp-pos-with-rate-implies-spec-rad-in-spec} this implies that $\spr(T) \in \sigma(T)$.
\end{proof}
We close this section by demonstrating why Theorem~\ref{thm:countable-summability-condition-implies-spec-in-spec} is a consequence of Corollary~\ref{cor:weakly-asymp-pos-with-countably-varying-rate-implies-spec-rad-in-spec}:
\begin{proof}[Proof of Theorem~\ref{thm:countable-summability-condition-implies-spec-in-spec}]
Let $A$ be a set of real-valued sequences $a = (a_n)_{n \in \mathbb{N}_0} \subseteq [0,\infty)$ and let $\varphi \in \Phi$. It was proved in \cite[Lemma~4.1]{Glueck2015} that if $\sum_{n=0} \varphi(a_n) < \infty$ for each $a \in A$, then there exists a sequence $0 \le f \in c_0$ such that $\{f\}$ governs each element of $A$.
For every $\varphi \in \Phi$ we now define $D_\varphi$ to be the set of all pairs $(x,x') \in E \times E'$ which fulfil $\|x\| \le 1$ and $\|x'\| \le 1$ and for which $\sum_{n=0}^\infty \varphi\big(\operatorname{d}_+(\langle x', S^n x \rangle)\big) < \infty$. As noted at the beginning of the proof we can find, for each $\varphi \in \Phi$, a sequence $0 \le f_\varphi \in c_0$ such that $\{f_\varphi\}$ governs $\big(\operatorname{d}_+(\langle x', S^n x \rangle)\big)_{n \in \mathbb{N}_0}$ for all $(x',x) \in D_\varphi$. By assumption, we have
\begin{align*}
\bigcup_{\varphi \in \Phi} D_\varphi = \{(x,x') \in E' \times E: \; \|x\| \le 1 \text{ and } \|x'\| \le 1 \},
\end{align*}
so $F := \{f_\varphi: \; \varphi \in \Phi\}$ governs $\big(\operatorname{d}_+(\langle x', S^n x \rangle)\big)_{n \in \mathbb{N}_0}$ for all $x \in E$ and $x' \in E'$ of norm $\le 1$. By a simple scaling argument it follows that $F$ governs the sequence $\big(\operatorname{d}_+(\langle x', S^n x \rangle)\big)_{n \in \mathbb{N}_0}$ for actually all $x \in E$ and $x' \in E'$. Since $F$ is at most countable, it follows from Corollary~\ref{cor:weakly-asymp-pos-with-countably-varying-rate-implies-spec-rad-in-spec} that $\spr(T) \in \sigma(T)$.
\end{proof}
\section{Positive Eigenvectors} \label{section:positive-eigenvectors}
In this section we give sufficient conditions for the spectral radius of an operator $T$ to be an eigenvalue of $T$ which admits a positive eigenvector. Even for a positive operator the spectral radius need, in general, not be an eigenvalue at all (the multiplication operator with symbol $(1-\frac{1}{n})$ on $\ell^p(\mathbb{N};\mathbb{C})$ is a counterexample). A very common condition to ensure that a spectral value $\lambda$ of an arbitrary operator $T$ is an eigenvalue is to assume that it is a pole of the resolvent $\mathcal{R}(\mathord{\,\cdot\,},T)$. This is also a common assumption in Perron--Frobenius theory. Under this assumption we obtain the following Kre\u{\i}n--Rutman type result.
\begin{theorem} \label{thm:positive-eigenvectors}
Let $E$ be a complex Banach lattice and let $T \in \mathcal{L}(E)$ with $\spr(T) > 0$. Assume that $T$ is weakly asymptotically positive and that $\spr(T)$ is a spectral value of $T$ and a pole of the resolvent $\mathcal{R}(\mathord{\,\cdot\,},T)$.
Then $\spr(T)$ is an eigenvalue of $T$ and of the adjoint $T'$ and each of the eigenspaces $\ker(\spr(T) - T)$ and $\ker(\spr(T) - T')$ contains a non-zero positive vector.
\end{theorem}
Recall that sufficient conditions for $\spr(T)$ to be a spectral value of $T$ are given in Sections~\ref{section:the-spectral-radius-1} and~\ref{section:the-spectral-radius-2}. Suppose that $0 < \spr(T) \in \sigma(T)$. The assumption that $\spr(T)$ be a pole of $\mathcal{R}(\mathord{\,\cdot\,},T)$ is for example fulfilled if the essential spectral radius $\spr_{\operatorname{ess}}(T)$ is strictly smaller than $\spr(T)$ (see e.g.\ \cite[formula~(1.16) on p.\,249]{Engel2000}). It is also fulfilled if there exists an open neighbourhood $U$ of $\sigma(T)$ and an analytic function $f: U \to \mathbb{C}$ such that $f(T)$ is compact and such that $f(\spr(T)) \not= 0$ \cite[Theorem~5.8-F]{Taylor1958}. In particular, $\spr(T)$ is a pole of $\mathcal{R}(\mathord{\,\cdot\,},A)$ if $T$ or some power of $T$ is compact. Combining these observations with our results from Sections~\ref{section:the-spectral-radius-1} and~\ref{section:the-spectral-radius-2} we obtain, for instance, the following corollaries:
\begin{corollary}
Let $E$ be a complex Banach lattice and let $T \in \mathcal{L}(E)$ be a weakly eventually positive operator with $\spr(T) > 0$. Assume that some power of $T$ is compact or, more generally, that there exists an open neighbourhood $U$ of $\sigma(T)$ and an analytic function $f: U \to \mathbb{C}$ with $f(\spr(T)) \not= 0$ for which $f(T)$ is compact.
Then $\spr(T)$ is an eigenvalue of $T$ and $T'$ and each of the eigenspaces $\ker(\spr(T) - T)$ and $\ker(\spr(T) - T')$ contains a non-zero positive vector.
\end{corollary}
\begin{proof}
It follows from Corollary~\ref{cor:weakly-ev-pos-implies-spec-in-spec} that $\spr(T)$ is a spectral value of $T$. Moreover, as recalled above, $\spr(T)$ is a pole of the resolvent $\mathcal{R}(\mathord{\,\cdot\,},T)$. Hence, the assertion follows from Theorem~\ref{thm:positive-eigenvectors}.
\end{proof}
\begin{corollary}
Let $E$ be a complex Banach lattice and let $T \in \mathcal{L}(E)$ be an operator which fulfils $0 \le \spr_{\operatorname{ess}}(T) < \spr(T)$ and which is weakly asymptotically positive. Suppose that $T/\spr(T)$ is power bounded.
Then $T$ is uniformly asymptotically positive; moreover, $\spr(T)$ is an eigenvalue of $T$ and $T'$ and each of the eigenspaces $\ker(\spr(T) - T)$ and $\ker(\spr(T) - T')$ contains a non-zero positive vector.
\end{corollary}
\begin{proof}
Since $T/\spr(T)$ is power bounded and since $\spr_{\operatorname{ess}}(T) < \spr(T)$, it is easy to see that the set $\{(T/\spr(T))^n: \; n \in \mathbb{N}_0\}$ is relatively compact in $\mathcal{L}(E)$ with respect to the operator norm topology. Hence, it follows from Proposition~\ref{prop:regularity-implies-better-asymp-pos}(b) that $T$ is uniformly asymptotically positive.
Thus, the spectral radius $\spr(T)$ is contained in $\sigma(T)$ according to Theorem~\ref{thm:spec-rad-in-spec-for-unif-asymp-pos-op}. Since $\spr_{\operatorname{ess}}(T) < \spr(T)$ we know that $\spr(T)$ is a pole of the resolvent $\mathcal{R}(\mathord{\,\cdot\,},T)$, so the assertion follows from Theorem~\ref{thm:positive-eigenvectors}.
\end{proof}
The proof of Theorem~\ref{thm:positive-eigenvectors} uses some well-known properties of the Laurent series expansion of the resolvent and is quite elementary:
\begin{proof}[Proof of Theorem~\ref{thm:positive-eigenvectors}]
We may assume that $\spr(T) = 1$. Let us begin with a preliminary observation. For all $x \in E_+$, all $x' \in E'_+$ and all $r > 1$ we have
\begin{align*}
\operatorname{d}_+(\langle x', \mathcal{R}(r,T)x \rangle) \le \sum_{n=0}^\infty \frac{\operatorname{d}_+(\langle x', T^nx \rangle)}{r^{n+1}}
\end{align*}
according to the Neumann series representation of the resolvent. Using that we have $\operatorname{d}_+(\langle x', T^nx \rangle) \to 0$ as $n \to \infty$ and that $(r-1)\sum_{n=0}^\infty \frac{1}{r^{n+1}} = 1$ for all $r > 1$, we thus obtain
\begin{align}
\label{eq:resolvent-weakly-asymptotically-positive}
(r-1)\operatorname{d}_+(\langle x', \mathcal{R}(r,T)x \rangle) \to 0 \qquad \text{as} \qquad r \downarrow 1.
\end{align}
Now, let $m \in \mathbb{N}$ denote the order of $1$ as a pole of the resolvent $\mathcal{R}(\mathord{\,\cdot\,},T)$ and let
\begin{align*}
\mathcal{R}(\lambda,T) = \sum_{n=-m}^\infty Q_n(\lambda - 1)^n
\end{align*}
be the Laurent series expansion of the resolvent about $1$ (where $Q_n \in \mathcal{L}(E)$ for all $n \in \{-m,-m+1,...\}$). Note that the operator $Q_{-m}$ is non-zero and that its range $Q_{-m}E$ is contained in $\ker(1 - T)$ \cite[Theorem~2 in Section~VIII.8]{Yosida1995}. In particular, $1$ is an eigenvalue of $T$.
Moreover, the operator $Q_{-m}$ is positive: indeed, $(r-1)^m \mathcal{R}(r,T)$ converges to $Q_{-m}$ with respect to the operator norm as $r \downarrow 1$, so it follows from~\eqref{eq:resolvent-weakly-asymptotically-positive} that $\operatorname{d}_+(\langle x', Q_{-m}x \rangle) = 0$ for all $x \in E_+$ and all $x' \in E'_+$. Since $Q_{-m}$ is positive and non-zero and since $E = E_+ - E_+$, we can find a vector $x \in E_+$ such that $Q_{-m}x > 0$. Thus, $Q_{-m}x$ is a positive eigenvector of $T$ for the eigenvalue $1$.
Let us now consider the adjoint operator $T'$. It has the same spectrum as $T$ and we have $\mathcal{R}(\lambda,T') = \mathcal{R}(\lambda,T)'$ for all $\lambda \in \sigma(T') = \sigma(T)$. The Laurent series representation of $\mathcal{R}(\mathord{\,\cdot\,},T')$ about $1$ is thus given by
\begin{align*}
\mathcal{R}(\lambda,T') = \sum_{n=-m}^\infty Q_n' (\lambda - 1)^n.
\end{align*}
Since $Q_{-m}' \not= 0$, it follows that $1$ is also a pole of order $m$ of $\mathcal{R}(\mathord{\,\cdot\,},T')$. As above we have $Q_{-m}'E' \subseteq \ker(1 - T')$. Since $Q_{-m}$ is positive, so is $Q_{-m}'$ and hence, there exists a vector $x' \in E'_+$ such that $Q_{-m}' x' > 0$. Thus, $Q_{-m}'x'$ is a positive eigenvector of $T$ for the eigenvalue $1$.
\end{proof}
\section{The Peripheral Spectrum} \label{section:the-peripheral-spectrum}
In this section we turn to the peripheral spectrum $\sigma_{\operatorname{per}}(T)$ of an asymptotically positive operator $T$. We note once again that $\sigma_{\operatorname{per}}(T)$ is defined to be the set of all spectral values of $T$ with maximal modulus.
Recall that an operator $T$ on a complex Banach space $E$ is called \emph{Abel bounded} if $\sup_{\lambda > \spr(T)} (\lambda - \spr(T))\|\mathcal{R}(\lambda,T)\| < \infty$. An easy application of the Neumann series representation of the resolvent shows that an operator $T$ with non-zero spectral radius is automatically Abel bounded in case that $T/\spr(T)$ is power bounded. The converse implication is, however, not true: for instance, an operator is automatically Abel bounded if $\spr(T) \not\in \sigma(T)$; moreover, there exist even positive operators (for which we always have $\spr(T) \in \sigma(T)$) which are Abel bounded, but for which the rescaled operator $T/\spr(T)$ is not power bounded; see \cite[Section~2]{Derriennic1973} for a counterexample.
A deep result in Perron--Frobenius theory asserts that the peripheral spectrum of a positive, Abel-bounded operator $T$ on a complex Banach lattice is automatically cyclic, i.e.\ we have $\spr(T)e^{in\theta} \in \sigma(T)$ for all integers $n \in \mathbb{Z}$ whenever $\spr(T)e^{i\theta} \in \sigma(T)$ ($\theta \in \mathbb{R}$). This was proved independently by Krieger \cite[Folgerung 2.2.1(b)]{Krieger1969} and Lotz \cite[Theorem~4.7]{Lotz1968} in the late 1960s. It is worthwhile pointing out that the question whether the peripheral spectrum of \emph{every} positive operator on a complex Banach lattice is cyclic is an open problem until today; we refer to \cite{Glueck2016} and \cite{GlueckGR} for a detailed discussion of this topic and for some recent partial results.
Here we show that the peripheral spectrum of a uniformly \emph{asymptotically} positive operator $T$ is cyclic in case that $T/\spr(T)$ is power bounded:
\begin{theorem} \label{thm:unif-asympt-pos-adjoint-with-bdd-error-cyclcic-per-spec}
Let $E$ be a complex Banach lattice and let $T \in \mathcal{L}(E)$ with $\spr(T) > 0$. If $T/\spr(T)$ is power-bounded and $T$ is uniformly asymptotically positive, then $\sigma_{\operatorname{per}}(T)$ is cyclic.
\end{theorem}
We have not been able yet to prove or disprove the same assertion for operators which are merely Abel bounded. Similarly to Krieger and Lotz we employ some kind of lifting technique to transform the peripheral spectrum of an operator into point spectrum (more precisely, we use ultra powers of Banach lattices). The rest of our proof is, however, quite different from the arguments used by Krieger and Lotz.
Let us give a very brief reminder of ultra powers of Banach lattices. Let $E$ be a complex Banach lattice and let $\mathcal{U}$ be a free ultra filter on $\mathbb{N}$. By $\ell^\infty(\mathbb{N};E)$ we denote the space of all $E$-valued norm bounded sequences, endowed with the supremum norm; note that $\ell^\infty(\mathbb{N};E)$ is itself a complex Banach lattice. By $c_{0,\mathcal{U}}(\mathbb{N};E)$ we denote the closed ideal in $\ell^\infty(\mathbb{N};E)$ of all sequences which converge to $0$ along $\mathcal{U}$. The quotient space
\begin{align*}
E^\mathcal{U} := \ell^\infty(\mathbb{N};E) / c_{0,\mathcal{U}}(\mathbb{N};E)
\end{align*}
is called an \emph{ultra power} of $E$; it is itself a complex Banach lattice. For every sequence $x = (x_n) \in \ell^\infty(\mathbb{N};E)$ we denote by $x^\mathcal{U} := (x_n)^\mathcal{U}$ the equivalence class of $x$ in $E^\mathcal{U}$; it is not difficult to see that the norm of $x^\mathcal{U}$ in $E^\mathcal{U}$ is given by $\|x^\mathcal{U}\| = \lim_\mathcal{U} \|x_n\|$. Moreover, we have
\begin{align}
\label{eq:distance-to-positive-cone-in-ultra-power}
\operatorname{d}_+(x^\mathcal{U}) = \lim_\mathcal{U} \operatorname{d}_+(x_n)
\end{align}
for all $x^\mathcal{U} = (x_n)^\mathcal{U} \in E^\mathcal{U}$; this follows from Proposition~\ref{prop:formula-for-distance-to-positive-cone}. For every $x \in E$ we denote by $x^\mathcal{U}$ the equivalence class of the constant sequence $(x,x,...)$ in $E^\mathcal{U}$. Note that the mapping $E \to E^\mathcal{U}$, $x \mapsto x^\mathcal{U}$ is an isometric lattice homomorphism.
Now, let $T \in \mathcal{L}(E)$. Then we define $T^\mathcal{U} \in \mathcal{L}(E^\mathcal{U})$ to be the operator given by $T^\mathcal{U} x^\mathcal{U} = (Tx_n)^\mathcal{U}$ for all $x^\mathcal{U} \in E^\mathcal{U}$. The mapping $\mathcal{L}(E) \to \mathcal{L}(E^\mathcal{U})$, $T \mapsto T^\mathcal{U}$ is an isometric Banach lattice homomorphism, and $T^\mathcal{U}$ is positive if and only if $T$ is positive. Moreover, we have $\sigma(T) = \sigma(T^\mathcal{U})$ and $\sigma_{\operatorname{pnt}}(T^\mathcal{U}) = \sigma_{\operatorname{appr}}(T^\mathcal{U}) = \sigma_{\operatorname{appr}}(T)$; in particular, the peripheral spectrum of $T^\mathcal{U}$ consists of eigenvalues of $T^\mathcal{U}$ and coincides with the peripheral spectrum of $T$. For more details we refer to \cite[Section~V.1]{Schaefer1974}, \cite[pp.\,251--253]{Meyer-Nieberg1991} and to the survey article \cite{Heinrich1980}.
In order to give the proof of Theorem~\ref{thm:unif-asympt-pos-adjoint-with-bdd-error-cyclcic-per-spec} we need one further ingredient, namely the next proposition. Let $E$ be a complex Banach lattice, which is by definition the complexification of a real Banach lattice $E_\mathbb{R}$, and let $F \subseteq E$ be a closed vector subspace. We call $F$ a \emph{lattice subspace} of $E$ and the \emph{real part} $F_\mathbb{R} := F \cap E_\mathbb{R}$ of $F$ fulfils $F_\mathbb{R} + iF_\mathbb{R} = E_\mathbb{R}$ and if $F_\mathbb{R}$ is a vector lattice with respect to the order induced by $E_\mathbb{R}$. We also recall that the dual space $E'$ of a complex Banach lattice $E$ is itself a complex Banach lattice; more precisely, if $E$ is a complexification of a real Banach lattice $E_\mathbb{R}$, then $E'$ is a complexification of $E_\mathbb{R}'$ \cite[Corollary~3 to Theorem~IV.1.8]{Schaefer1974}.
\begin{proposition} \label{prop:fixed-space-of-dual-operator}
Let $E$ be a complex Banach lattice and let $T \in \mathcal{L}(E)$ be positive and power bounded. Then the fixed space $F := \ker(1-T')$ of the adjoint operator $T'$ is a lattice subspace of $E'$; moreover, there exists a norm on $F$ which is equivalent to the norm induced by $E'$ and which renders $F$ a complex Banach lattice.
\end{proposition}
\begin{proof}
By definition, $E$ is the complexification of a real Banach lattice $E_\mathbb{R}$; the space $E'$ is the complexification of the dual Banach lattice $E_\mathbb{R}'$. We define $F_\mathbb{R} := F \cap E_\mathbb{R}'$; since $T'$ maps $E_\mathbb{R}'$ to $E_\mathbb{R}'$ we clearly have $F_\mathbb{R} + iF_\mathbb{R} = F$. In order to prove that $F$ is a lattice subspace of $E'$ we thus have to show that $F_\mathbb{R}$ is a vector lattice with respect to the order induced by $E_\mathbb{R}'$. To this end, it suffices to proves that, for every $f \in F_\mathbb{R}'$, there exists a supremum of $f$ and $-f$ in $F_\mathbb{R}$.
So let $f \in F_\mathbb{R}$. We have $T'f = f$ and thus $T'|f| \ge |f|$. Iterating this inequality we obtain that the sequence $((T')^n|f|)_{n \in \mathbb{N}_0}$ is increasing. Note that the sequence is also norm bounded since we assumed $T$ to be power bounded. Hence, $((T')^n|f|)_{n \in \mathbb{N}_0}$ converges to a vector $0 \le g \in E_\mathbb{R}'$ with respect to the weak${}^*$-topology. Since $T'$ is continuous with respect to this topology, $g$ is a fixed point of $T'$ and thus contained in $F_\mathbb{R}$. Since $g \ge |f|$, the vector $g$ is clearly an upper bound of $f$ and $-f$ in $F_\mathbb{R}$. Assume now, on the other hand, that $h \in F_\mathbb{R}$ is another upper bound of $f$ and $-f$ in $F_\mathbb{R}$. Then we have $|f| \le h$ and hence $(T')^n|f| \le (T')^n h = h$ for all $n \in \mathbb{N}_0$. This proves that $g \le h$, so $g$ is indeed the supremum of $f$ and $-f$ in $F_\mathbb{R}$. We have thus proved that $F$ is indeed a lattice subspace of $E'$.
Finally, we denote the modulus of any $f \in F_\mathbb{R}$ in the vector lattice $F_\mathbb{R}$ by $|f|_F$ and we define $\|f\|_F := \| |f|_F\|$ for every $f \in F_\mathbb{R}$. We clearly have $\|f\| \le \|f\|_F$ for all $f \in F_\mathbb{R}$ and from the construction of $|f|_F$ in the above part of the proof we obtain that $\|f\|_F \le \sup_{n \in \mathbb{N}_0} \|T^n\| \|f\|$. Hence, the norm $\|\cdot\|_F$ on $F_\mathbb{R}$ is equivalent to the norm $\|\cdot\|$ induced by $E'$. It is now straightforward to check that $(F_\mathbb{R}, \|\cdot\|_F)$ is a (real) Banach lattice, and from this it readily follows that $F = F_\mathbb{R} + iF_\mathbb{R}$ is a complex Banach lattice with respect to a norm equivalent to the norm induced by $E'$.
\end{proof}
Arguments as used in the above proof are quite common in Perron--Frobenius theory, compare for instance \cite[the proof of Corollary~C-III.4.3(a)]{Arendt1986}. We also refer to \cite[Theorem~2.1 and Corollary~2.2]{Glueck2016} for related results.
The following proof of Theorem~\ref{thm:unif-asympt-pos-adjoint-with-bdd-error-cyclcic-per-spec} exhibits some similarities to the proof of \cite[Theorem~3.2]{Glueck2016}; the technical details are, however, quite different.
\begin{proof}[Proof of Theorem~\ref{thm:unif-asympt-pos-adjoint-with-bdd-error-cyclcic-per-spec}]
We may assume that $\spr(T) = 1$. Let $\lambda \in \sigma_{\operatorname{per}}(T)$, i.e.\ let $\lambda \in \sigma(T)$ and $|\lambda| = 1$. We have to prove that $\lambda^m \in \sigma(T)$ for all $m \in \mathbb{Z}$. Replacing $E$ with an ultra power and $T$ with its lifting to this ultra power we may assume that $\lambda$ is an eigenvalue of $T$ with an eigenvector $z \in E$ (note that the uniform asymptotic positivity of $T$ is conserved if we lift $T$ to an ultra power of $E$; this follows from formula~\eqref{eq:distance-to-positive-cone-in-ultra-power}).
Let us now employ a second ultra power argument. Choose a free ultra filter $\mathcal{U}$ on $\mathbb{N}$ and a sequence of integers $1 \le k_n \to \infty$ such that $\lambda^{k_n} \to 1$ (such a sequence clearly exists). We define two operators $R$ and $S$ on the ultra power $E^\mathcal{U}$ which are given by $Rx^\mathcal{U} = (T^{k_n-1}x_n)^\mathcal{U}$ and $Sx^\mathcal{U} = (T^{k_n}x_n)^{\mathcal{U}}$ for all $x^\mathcal{U} \in E^\mathcal{U}$. Using that $T$ is power bounded it is easy to see that $R$ and $S$ are well-defined. Moreover, the operators $T^\mathcal{U}$, $R$ and $S$ commute, they are power bounded and we have $R T^\mathcal{U} = T^\mathcal{U} R = S$. Since $T$ is uniformly asymptotically positive, it follows from formula~\eqref{eq:distance-to-positive-cone-in-ultra-power} that $R$, $S$ and $S T^\mathcal{U} = T^\mathcal{U} S$ are positive operators on the complex Banach lattice $E^\mathcal{U}$. Note that the vector $z^\mathcal{U}$ is contained in the fixed space $\ker(1-S)$ of $S$ since $\lambda^{k_n} \to 1$.
We now take bi-adjoints; to keep the notation simple, let us define $\hat E := (E^\mathcal{U})''$, $\hat T := (T^\mathcal{U})''$, $\hat R := R''$ and $\hat S := S''$. Then the operators $\hat T$, $\hat R$ and $\hat S$ commute, they are power bounded and we have $\hat R \hat T = \hat T \hat R = \hat S$; moreover, $\hat R$, $\hat S$ and $\hat S \hat T = \hat T \hat S$ are positive. Proposition~\ref{prop:fixed-space-of-dual-operator} shows that the fixed space $F := \ker(1-\hat S)$ of $\hat S$ is a lattice subspace of $\hat E$ and a complex Banach lattice with respect to an equivalent norm. Since $\hat T$ and $\hat R$ commute with $\hat S$, they leave $F$ invariant, and their restrictions to $F$ fulfil $\hat R|_F \hat T|_F = \hat T|_F \hat R_F = \hat S|_F = \id_F$; hence, $\hat T|_F$ and $\hat R|_F$ are inverse to each other. Since $\hat R$ is positive, so is $\hat R|_F$, and since $\hat T \hat S$ is positive, so is $\hat T|_F$. This proves that $\hat T_F$ is actually a Banach lattice isomorphism on the complex Banach lattice $F$.
We consider $E^\mathcal{U}$ as a subspace of $\hat E$ by means of evaluation. Since the eigenvector $z^\mathcal{U}$ of $T^\mathcal{U}$ for the eigenvalue $\lambda$ is contained in $\ker(1-S)$, it is also contained in $F = \ker(1-\hat S)$; so $\lambda$ is an eigenvalue of $\hat T|_F$. Since the point spectrum of every lattice homomorphism on a complex Banach lattice is cyclic \cite[Corollary~2 to Proposition~V.4.2]{Schaefer1974}, it follows that $\lambda^m$ is an eigenvalue of $\hat T|_F$, and thus of $\hat T$, for every $m \in \mathbb{Z}$. Therefore, $\lambda^m$ is a spectral value of $T^\mathcal{U}$, and hence of $T$, for every $m \in \mathbb{Z}$.
\end{proof}
We close this section with a few comments on the assumptions of Theorem~\ref{thm:unif-asympt-pos-adjoint-with-bdd-error-cyclcic-per-spec}:
\begin{remarks}
(a) The assertion of Theorem~\ref{thm:unif-asympt-pos-adjoint-with-bdd-error-cyclcic-per-spec} does not in general remain true if $T$ is only assumed to be individually asymptotically positive instead of uniformly asymptotically positive. A counterexample is again provided by the multiplication operator on $\ell^p(\mathbb{N};\mathbb{C})$ ($1 \le p < \infty$) with symbol $(-1+\frac{1}{n})_{n \in \mathbb{N}}$.
(b) It does not seem to be clear whether the peripheral spectrum of an \emph{individually eventually} positive operator $T$ is cyclic (in case that $T/\spr(T)$ is power bounded). One might conjecture that every individually eventually positive operator $T$ is uniformly asymptotically positive (at least if $T/\spr(T)$ is power bounded), in which case the answer to this question would be positive due to Theorem~\ref{thm:unif-asympt-pos-adjoint-with-bdd-error-cyclcic-per-spec}; yet, it does not seem to be clear either whether such a conjecture is justified (compare the discussion before Theorem~\ref{thm:countable-summability-condition-implies-spec-in-spec}).
\end{remarks}
\section{The Peripheral Point Spectrum} \label{section:the-periphera-point-spectrum}
In this final section we consider the peripheral \emph{point} spectrum rather than the peripheral spectrum. Recall that the peripheral point spectrum $\sigma_{{\operatorname{per}},{\operatorname{pnt}}}(T)$ of an operator $T$ is defined to be the set of all eigenvalues of $T$ with modulus $\spr(T)$. We point out that the peripheral point spectrum can be empty, in general.
While the peripheral spectrum of a positive operator $T$ is always cyclic in case that $T/\spr(T)$ is power bounded, this is not in general true for the peripheral point spectrum; see for instance \cite[Sections~5 and~6]{Glueck2016} where several counterexamples are discussed. On the other hand, the same reference contains many sufficient conditions which ensure that the peripheral point spectrum of a positive operator is indeed cyclic. Here, we adapt one of these conditions (namely \cite[Theorem~5.5]{Glueck2016}) to the case of weakly asymptotically positive operators.
Recall that an operator $S$ on a Banach space $E$ is said to have \emph{relatively weakly compact orbits} if the set $\{S^n: \; n \in \mathbb{N}_0\}$ is relatively compact in $\mathcal{L}(E)$ with respect to the weak operator topology. This is equivalent to the set $\{S^nx: \; n \in \mathbb{N}_0\}$ being relatively compact in $E$ with respect to the weak topology for each $x \in E$ (see e.g.\ \cite[Corollary~A.5]{Engel2000}). Note that every power bounded operator on a reflexive Banach space automatically has relatively weakly compact orbits.
\begin{theorem} \label{thm:criterion-for-cyclic-per-point-spec}
Let $E$ be a complex Banach lattice and let $T \in \mathcal{L}(E)$ with $\spr(T) > 0$. Suppose that the powers of $T/\spr(T)$ are relatively weakly compact and that $T$ is weakly asymptotically positive. Then we have
\begin{align*}
\dim \ker (\spr(T) e^{i\theta} - T) \le \dim \ker (\spr(T) e^{in \theta} - T)
\end{align*}
for all $n \in \mathbb{Z}$ and all $\theta \in \mathbb{R}$. In particular, the peripheral point spectrum of $T$ is cyclic.
\end{theorem}
In the above theorem we understand the dimension of a vector space to be either an integer or $\infty$, i.e.\ we do not distinguish between different infinite cardinalities. The proof of the above theorem relies on the so-called \emph{Jacobs--de Leeuw--Glicksberg decomposition} of an operator (see for instance \cite[Section~2.4]{Krengel1985}, \cite[Section~V.2]{Engel2000} or \cite[Section~16.3]{Eisner2015} for details about this construction) and is very similar to the proof of \cite[Theorem~5.5]{Glueck2016}. The only difference is that we now use a bit more information about the Jacobs--de Leeuw--Glicksberg decomposition to compensate for the fact that the operator might no longer be positive, but only weakly asymptotically positive.
\begin{proof}[Proof of Theorem~\ref{thm:criterion-for-cyclic-per-point-spec}]
We may assume that $\spr(T) = 1$. Let $\mathcal{S}$ denote the closure of the set $\{T^n: \; n \in \mathbb{N}_0\}$ in $\mathcal{L}(E)$ with respect to the weak operator topology. Then $\mathcal{S}$ is a compact commutative semi-topological semigroup with respect to operator multiplication and with respect to the weak operator topology. Hence, the so-called \emph{Sushkevich kernel}
\begin{align*}
\mathcal{K} := \bigcap_{S \in \mathcal{S}} S \mathcal{S}
\end{align*}
is an ideal in the semigroup $\mathcal{S}$ and in fact it is even a group. The neutral element $P$ of the group $\mathcal{K}$ is a projection on $E$; since $P$ commutes with every operator $S \in \mathcal{S}$ it reduces each such $S$. The restriction of $T$ to the range $PE$ of $P$ is an invertible operator in $\mathcal{L}(PE)$ and its inverse $(T|_{PE})^{-1}$ is given by $R|_{PE}$ for some operator $R \in \mathcal{S}$. Moreover, the range of $P$ is the closed linear span of all eigenvectors of $T$ belonging to eigenvalues of modulus $1$; in particular, we have $\ker(e^{i\theta}- T) = \ker(e^{i\theta} - T|_{PE})$ for all $\theta \in \mathbb{R}$. All these results can, for instance, be found in \cite[Section~2.4]{Krengel1985}.
Now we prove that each $K \in \mathcal{K}$ is a positive operator on $E$. To this end, we first note that we have $S\mathcal{S} = \overline{S \{T^n: \; n \in \mathbb{N}_0\} }^{\operatorname{w}}$ for each $S \in \mathcal{S}$ (where $\overline{\mathcal{A}}^{\operatorname{w}}$ denotes the closure of any subset $\mathcal{A} \subseteq \mathcal{L}(E)$ with respect to the weak operator topology). Indeed, the inclusion ``$\subseteq$'' is obvious and the converse inclusion ``$\supseteq$'' follows from the weak compactness of $\mathcal{S}$. Hence, we obtain
\begin{align}
\label{eq:sushkevich-kernel-asymptotic-behaviour}
\mathcal{K} = \bigcap_{S \in \mathcal{S}} \overline{S \{T^n: \; n \in \mathbb{N}_0\} }^{\operatorname{w}} \subseteq \bigcap_{m \in \mathbb{N}_0} \overline{ \{T^{m+n}: \; n \in \mathbb{N}_0\} }^{\operatorname{w}}.
\end{align}
Let $K \in \mathcal{K}$, let $x \in E_+$, $x' \in E'_+$ and let $\varepsilon > 0$. Since $T$ is weakly asymptotically positive, there exists an $m \in \mathbb{N}_0$ such that $\operatorname{d}_+(\langle x', T^{m+n}x \rangle) < \varepsilon$ for all $n \in \mathbb{N}_0$. Moreover, according to~\eqref{eq:sushkevich-kernel-asymptotic-behaviour}, we can find an integer $n \in \mathbb{N}_0$ such that
\begin{align*}
|\langle x', T^{m+n}x\rangle - \langle x', Kx\rangle| < \varepsilon.
\end{align*}
Hence, $\operatorname{d}_+(\langle x', Kx\rangle) < 2\varepsilon$. Since $\varepsilon > 0$ was arbitrary, it follows that $K$ is indeed positive.
This implies in particular that the projection $P$ is positive, so its range is a lattice subspace of $E$ and a complex Banach lattice with respect to some equivalent norm (this is a simple consequence of the same result for positive projections on \emph{real} Banach lattices which can for instance be found in \cite[Proposition~III.11.5]{Schaefer1974}). We have $T|_{PE} = (TP)|_{PE}$; since the operator $TP$ is contained in $\mathcal{K}$, it is positive and hence, so is $T|_{PE}$. Finally, recall that the inverse $(T|_{PE})^{-1}$ is given by $R|_{PE}$ for some $R \in \mathcal{K}$, so it is also positive. This proves that the restricted operator $T|_{PE}$ is a lattice isomorphism on the complex Banach lattice $PE$. We now conclude for all $n \in \mathbb{Z}$ and all $\theta \in \mathbb{R}$ that
\begin{align*}
\dim \ker (e^{i\theta} - T) & = \dim \ker (e^{i\theta} - T|_{PE}) \\
& \le \dim \ker(e^{in\theta} - T|_{PE}) = \dim \ker(e^{in\theta} - T);
\end{align*}
the dimension estimate between the first and the second line is true for every lattice homomorphism as proved in \cite[Proposition~3.1(b)]{Glueck2016}.
\end{proof}
\end{document} | math | 83,275 |
\begin{document}
{\cal S}igmaetlength{\abovedisplayskip}{{\cal S}igmatdskip}
{\cal S}igmaetlength{\belowdisplayskip}{{\cal S}igmatdskip}
\widehat def\title#1{\widehat def\thetitle{#1}}
\widehat def\authors#1{\widehat def\theauthors{#1}}
\widehat def\author#1{\widehat def\theauthors{#1}}
\widehat def\address#1{\widehat def\theaddress{#1}}
\widehat def{\cal S}igmaecondaddress#1{\widehat def\thesecondaddress{#1}}
\widehat def\email#1{\widehat def\theemail{#1}}
\widehat def\url#1{\widehat def\theurl{#1}}
\ |\ ong\widehat def\abstract#1\endabstract{\ |\ ong\widehat def\theabstract{#1}}
\widehat def\primaryclass#1{\widehat def\theprimaryclass{#1}}
\widehat def{\cal S}igmaecondaryclass#1{\widehat def\thesecondaryclass{#1}}
\widehat def\keywords#1{\widehat def\thekeywords{#1}}
\widehat def\ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax}
\ |\ ong\widehat def\maketitlepage{
\vglue 0.2truein
{\parskip=0pt\ |\ eftskip 0pt plus 1fil\widehat def\ \ {\rm and}\ \ {\par{\cal S}igmamallskip}{\Large
\bf\thetitle}\par
}
\vglue 0.15truein
{\parskip=0pt\ |\ eftskip 0pt plus 1fil\widehat def\ \ {\rm and}\ \ {\par}{{\cal S}igmac\theauthors}
\par
}
\vglue 0.1truein
{\parskip=0pt{\cal S}igmamall
{\ |\ eftskip 0pt plus 1fil\widehat def\ \ {\rm and}\ \ {\par}{{\cal S}igmal\theaddress}\par}
\ifundefined{thesecondaddress}\else{\mathrm cl}{and}
{\ |\ eftskip 0pt plus 1fil\widehat def\ \ {\rm and}\ \ {\par}{{\cal S}igmal\thesecondaddress}\par}\fi
\ifundefined{theemail}\else
\vglue 5pt \widehat def\ \ {\rm and}\ \ {\ \ {\rm and}\ \ }
{\mathrm cl}{Email:\ \ \tt\theemail}\fi
\ifundefined{theurl}\else
\vglue 5pt \widehat def\ \ {\rm and}\ \ {\ \ {\rm and}\ \ }
{\mathrm cl}{URL:\ \ \tt\theurl}\fi\par}
\vglue 7pt
{\bf Abstract}
\vglue 5pt
\theabstract
\vglue 7pt
{\bf AMS Classification numbers}\quad Primary:\quad \theprimaryclass
Secondary:\quad \thesecondaryclass
\vglue 5pt
{\bf Keywords:}\quad \thekeywords
}
\title{Miller Spaces and Spherical Resolvability of Finite Complexes}
\authors{Jeffrey Strom}
\address{Dartmouth College, Hanover, NH 03755\ \ {\rm and}\ \
{\tt{[email protected]}} \ \ {\rm and}\ \
{\tt{www.math.dartmouth.edu/\~{}strom/}} }
\abstract
We show that if $K$ is a nilpotent finite complex,
then $\Omega K$ can be built from spheres using fibrations and
homotopy (inverse) limits. This is applied to show
that if ${\mathrm map}_*(X,S^n)$ is weakly contractible
for all $n$, then ${\mathrm map}_*(X,K)$ is weakly contractible for
any nilpotent finite complex $K$.
\endabstract
\primaryclass{55Q05}
{\cal S}igmaecondaryclass{55P50}
\keywords{Miller Spaces, Spherically Resolvable, Resolving Class,
Homotopy Limit, Cone Length, Closed Class}
\maketitlepage
{\cal S}igmaection*{Discussion of Results}
A {\bf Miller space} is a CW complex $X$ with the property that
the space of pointed maps from $X$ to $K$
is weakly contractible for every nilpotent
finite complex $K$, written ${\mathrm map}_*(X,K){\cal S}igmaim *$.
They are named for Haynes Miller, who
proved in \cite{Miller} that the spaces
$B\kern -.25em \ZZ\kern -.25em /p$ are
all Miller spaces; in fact, he proved that
${\mathrm map}_*(B\kern -.19em \ZZ\kern -.25em /p, K)$
is weakly contractible for
every finite dimensional CW complex $K$.
In the stable category, one can define a {\it Miller
spectrum} by requiring that the mapping spectrum
$F(X,K)$ is contractible for every finite spectum $K$.
Since cofibrations and fibrations are
the same in the stable category, a finite spectrum $K$ with $m$
cells is the fiber in a fibration
$K {\mathrm map}rt{} L {\mathrm map}rt{} S^n$ in which $L$ has only $m-1$ cells;
in the terminology of \cite{Cohen,Levi}, this means that
$K$ is {\it spherically resolvable with weight $m$}.
An easy induction shows that $X$ is a Miller spectrum if
and only if $F(X,S^n){\cal S}igmaimeq *$ for every $n$.
Our goal is to prove the following unstable analog of this
observation: if ${\mathrm map}_*(X,S^n){\cal S}igmaim *$, for all $n$,
then $X$ is a Miller space. The proof of the stable version is not
available to us because cofibrations are not fibrations,
unstably. To prove our result, it is necessary
to determine the extent to which a finite complex can
be constructed from spheres in a more general way, i.e.,
by arbitrary homotopy (inverse) limits
\cite{Bousfield-Kan} and extensions by fibrations.
To be more precise, we require some new
terminology. We call a nonempty class ${\cal R}$
of spaces a \term{resolving class} if it is closed under
weak equivalences and pointed homotopy (inverse) limits
(all spaces, maps and homotopy limits will be pointed). It is a
\term{strong resolving class} if it is further closed
under extensions by fibrations, i.e., if whenever
$F{\mathrm map}rt{} E{\mathrm map}rt{}B$ is a fibration with $F,B \in{\cal R}$,
then $E\in{\cal R}$.
Resolving classes are dual to closed classes as defined in
\cite{Chacholski} and \cite[p.\thinspace 45]{EDF}.
Notice that every resolving class ${\cal R}$ contains the one-point
space $*$ (cf. \cite[p.\thinspace 47]{EDF}).
From this, it follows that if $F{\mathrm map}rt{}E{\mathrm map}rt{}B$
is a fibration with $E,B\in{\cal R}$, then $F\in{\cal R}$. Similarly,
if $A_\alpha\in{\cal R}$ for each $\alpha$ then the
{\it categorical product} $\Pi_\alpha A_\alpha\in{\cal R}$ also.
The {\it weak product} $\widetilde \Pi_\alpha A_\alpha$
is the homotopy colimit of the finite subproducts; if
for each $i$ only
finitely many of the groups $\pi_i(A_\alpha)$ are nonzero,
then the weak product has the same weak homotopy type as
the categorical product.
Let ${\cal S}$ be the smallest resolving class that contains $S^n$ for
each $n$, and let $\overline {\cal S}$ be the smallest strong resolving
class that contains $S^n$ for each $n$. We say that a space
$K$ is \term{spherically resolvable} if $\Omega^k K\in \overline {\cal S}$
for some $k$. This concept is related to, but not the same as,
the notion of spherical resolvability described in \cite{Cohen,Levi}.
\rk{Examples}
\begin{enumerate}
\item[(a)] If $f:A{\mathrm map}rt{}B$ is any map then the class of all $f$-local
spaces is a resolving class \cite[p.\thinspace 5]{EDF}. This includes,
for example, the class of all spaces with $\pi_i(X) = 0$ for $i> n$,
or all $h_*$-local spaces, where $h_*$ is a homology theory.
\item[(b)] If $P$ is a set of primes, then the class of all
$P$-local spaces is a strong resolving class.
\item[(c)] If $f: W{\mathrm map}rt{}*$, then the class of all
$f$-local spaces is a strong resolving class \cite[p.\thinspace 5]{EDF}.
This includes, for example, the class $\{ K^+ \}$,
where $K^+$ denotes the Quillen plus construction on $K$
\cite[p.\thinspace 27]{EDF}.
\item[(d)] More generally, if $F$ is a covariant functor that
commutes with homotopy limits (and hence with fibrations) and ${\cal R}$ is a (strong)
resolving class, then the class
$
\{ K \, |\, F(K) \in {\cal R} \}
$
is also a (strong) resolving class. This applies, for example
to the functor $F(K) = {\mathrm map}_*(X,K)$.
\item[(e)] The class $\{ K \, | \, K{\cal S}igmaim * \}$ is a strong resolving class.
\end{enumerate}
Our proofs will proceed by induction on a certain kind of cone length
\cite{A-S-S}.
Let ${\cal F}$ denote the collection of all finite type wedges of
spheres. The {\bf ${\cal F}$-cone length} ${\mathrm cl}_{\cal F}(K)$ of a space $K$ is
the least integer $n$ for which there are cofibrations
$S_i{\mathrm map}rt{} K_i {\mathrm map}rt{} K_{i+1}$,
$0\ |\ eq i < n$, with $K_0{\cal S}igmaimeq *$, $K_n{\cal S}igmaimeq K$
and each $S_i\in {\cal F}$. If no
such $n$ exists, then ${\mathrm cl}_{\cal F}(K) =\infty$.
Clearly every finite complex $K$ has ${\mathrm cl}_{\cal F}(K) < \infty$.
We denote by ${\cal S}igma{\cal F}{\cal S}igmaseq {\cal F}$ the subcollection of all simply-connected
finite type wedges of spheres.
Finally, let ${\cal S}^\vee$ be the smallest strong resolving class
that contains ${\cal S}igma{\cal F}$.
With these preliminaries in place, we can state our main
result.
\begin{thrm}\ |\ abel{thrm:sres}
If $K$ is a nilpotent space with ${\mathrm cl}_{\cal F}(K) =n <\infty $, then
\begin{enumerate}
\item[{\rm (a)}] $ K \in {\cal S}^\vee $,
\item[{\rm (b)}] $\Omega K\in \overline {\cal S} $, and
\item[{\rm (c)}] $\Omega^n K \in {\cal S} $.
\end{enumerate}
In particular, {\rm (b)} implies that
every nilpotent finite complex $K$ is spherically
resolvable in our sense.
\end{thrm}
Our application to Miller spaces follows from the following
more general consequence of Theorem \ref{thrm:sres}.
\begin{thrm}\ |\ abel{thrm:main}
Let ${\cal R}$ be a strong resolving class and let $F$ be a functor
that commutes with homotopy limits.
\begin{enumerate}
\item[{\rm (a)}] Assume that $F(S^n)\in {\cal R}$ for each $n$.
Then $F(\Omega K)\in R$ for each nilpotent space $K$ with ${\mathrm cl}_{\cal F}(K) < \infty$.
\item[{\rm (b)}] Assume that $F(S)\in {\cal R}$ for each $S\in{\cal S}igma{\cal F}$.
Then $F(K) \in {\cal R}$ for each nilpotent space $K$ with ${\mathrm cl}_{\cal F}(K) < \infty$.
\end{enumerate}
\end{thrm}
To apply part (b), we require the following result of Dwyer \cite{Dwyer}.
\begin{prop}\ |\ abel{prop:SinF}
Let $F$ be a functor that commutes with homotopy limits,
let $W$ be a space
and let ${\cal R} = \{ K\, |\, {\mathrm map}_*(W,F(K)) {\cal S}igmaim *\}$.
If $S^n\in{\cal R}$ for each $n$, then ${\cal S}igma{\cal F}{\cal S}igmaseq {\cal R}$.
\end{prop}
Together, Theorem \ref{thrm:main}(b) and Proposition \ref{prop:SinF}
immediately imply the desired statement about Miller spaces.
\begin{cor}\ |\ abel{cor:appl} If ${\mathrm map}_*(X,S^n){\cal S}igmaim *$ for all $n$,
then ${\mathrm map}_*( X,K){\cal S}igmaim *$ for every nilpotent space
$K$ with ${\mathrm cl}_{\cal F}(K) < \infty$. In other words, $X$ is a Miller space.
\end{cor}
Corollary \ref{cor:appl} is by no means the only corollary of
interest. Other consequences are easily obtained by applying
Theorem \ref{thrm:main} to various strong resolving classes. For example,
if ${\mathrm map}_*(X,S^n)$ is $P$-local for all $n$, then
${\mathrm map}_*({\cal S}igma X,K)$ is $P$-local for every nilpotent space
$K$ with ${\mathrm cl}_{\cal F}(K) < \infty$.
If $X$ is simply-connected then Corollary \ref{cor:appl} can be strengthened
somewhat. If $L$ is a space with a nilpotent covering space $K$
having ${\mathrm cl}_{\cal F}(K)<\infty$, then it is easy to see that ${\mathrm map}_*(X,L){\cal S}igmaim *$.
We end by making the surprising observation that a
(non-nilpotent, of course) finite complex can be
a Miller space!
\noindent{\bf Example}\ \
Let $A$ be a connected $2$-dimensional acyclic finite complex.
(The classifying space of the Higman group \cite{Hig} is such a space \cite{D-V};
so is the space obtained by removing a point from a homology $3$-sphere).
Since $\pi_1(A)$ is equal to its commutator subgroup, there are no
nontrivial homomorphisms from $\pi_1(A)$ to any nilpotent group.
It follows that if $f:A{\mathrm map}rt{} K$ with $K$ a nilpotent finite complex,
then $\pi_1(f) =0$ and so $f$ factors through $q:A{\mathrm map}rt{}A/A_1{\cal S}igmaimeq \bigvee S^2$.
Since $[A,S^2] \colon\thinspaceng H^2(A) =0$, we conclude $f{\cal S}igmaimeq *$.
Thus $A$ is a Miller space.
This example shows that the nilpotency hypothesis on the targets
in Corollary \ref{cor:appl} cannot be entirely removed.
It remains possible, however, that if $X$ is simply-connected
and ${\mathrm map}_*(X,S^n){\cal S}igmaim *$ for each $n$, then ${\mathrm map}_*(X,K){\cal S}igmaim *$
for {\it every} finite complex $K$, or even for every finite-dimensional
complex.
{\it Acknowledgements}\ \
I would like to thank Robert Bruner and Charles McGibbon for
suggesting that I think about Miller spaces. This work
owes much to McGibbon in particular -- Corollary \ref{cor:appl}
was conjectured in joint work with him.
Thanks to Bill Dwyer for directing me to the result
of \cite{Hopkins}, which is the key to
Proposition \ref{prop:desuspend}, and for the statement
and proof of Proposition \ref{prop:SinF}; thanks are also due to Daniel
Tanr\'e for bringing Proposition \ref{prop:hofiber}
to my attention.
{\cal S}igmaection{Proof of Theorem \ref{thrm:sres}}
We begin with two supporting results.
\begin{prop}\ |\ abel{prop:desuspend}
Let $K$ be a connected nilpotent space, let ${\cal R}$
be a resolving class and let $F$ be a functor that
commutes with homotopy (inverse) limits.
If $F(\bigvee_{i=1}^m {\cal S}igma K)\in{\cal R}$
for each $m$, then $F(K)\in {\cal R}$.
\end{prop}
\begin{prf}
This follows from a result of Hopkins \cite[p.\thinspace 222]{Hopkins},
which says that $K$ is homotopy equivalent to
the homotopy (inverse) limit of a tower
$$
A_0 {\mathrm map}lft{} A_1 {\mathrm map}lft{} \cdots {\mathrm map}lft{} A_n {\mathrm map}lft{} A_{n+1}{\mathrm map}lft{} \cdots
$$
of spaces, each of which is a homotopy (inverse) limit
of a diagram of spaces of the form $\bigvee_{i=1}^m {\cal S}igma K$.
\end{prf}
\begin{prop}\ |\ abel{prop:hofiber}
Let $A{\mathrm map}rt{}B{\mathrm map}rt{}C$ be a cofibration,
and let $F$ be the homotopy fiber of $B{\mathrm map}rt{}C$.
Then
$$
{\cal S}igma F {\cal S}igmaimeq {\cal S}igma A \vee ({\cal S}igma A {\cal S}igmamsh \Omega C).
$$
\end{prop}
\begin{prf}
Convert the maps $A{\mathrm map}rt{*}C$, $B{\mathrm map}rt{}C$ and
$C{\mathrm map}rt{=}C$ to fibrations. The total spaces and fibers
form the commutative diagram
$$
\xymatrix{
A\times \Omega C \ar[rd]\ar[rr]\ar[dd] && F \ar'[d][dd]\ar[rd]\ \ {\rm and}\ \
& \Omega C \ar'[d][dd]\ar[rr] && {*}\ar[dd]\ \ {\rm and}\ \
A \ar[rr]\ar[rd] && B \ar[rd]\ \ {\rm and}\ \
& {*} \ar[rr] && C, \ \ {\rm and}\ \
\ \ {\rm and}\ \ }
$$
\vskip -.5in
\noindent
in which the bottom square is a homotopy pushout.
A result of V. Puppe \cite{Puppe} shows that the top
square is also a homotopy pushout.
Hence, the cofiber ${\cal S}igma F$ of the map
$F{\mathrm map}rt{} *$ has the same homotopy type as
the cofiber of $A\times \Omega C{\mathrm map}rt{}\Omega C$,
namely ${\cal S}igma A \vee ({\cal S}igma A {\cal S}igmamsh \Omega C)$, as can be seen from
the diagram
$$
\xymatrix{
A\times \Omega C \ar[d]\ar[r]\ar@{}[rd]|{{\mathrm pushout}}
& \Omega C \ar[r]\ar[d] & {\cal S}igma F\ar@{=}[d] \ \ {\rm and}\ \
A \ar[r]^(.4){*} & A * \Omega C\ar[r] & {\cal S}igma A \vee ({\cal S}igma A {\cal S}igmamsh \Omega C) . }
$$
\end{prf}
\noindent{\bf Proof of Theorem \ref{thrm:sres}}\ \
Notice that the assumption on $X$ implies that $X$
is connected; we may therefore assume that $K$ is also
connected.
We prove assertion (a) by induction on
${\mathrm cl}_{\cal F}(K)$. If ${\mathrm cl}_{\cal F}(K)=1$, then $K$ is weakly equivalent
to a connected finite type wedge of spheres.
Therefore each $\bigvee_{i=1}^m {\cal S}igma K\in{\cal S}igma{\cal F}$
and Proposition \ref{prop:desuspend} proves the assertion
in the initial case.
Now assume that the result is known for all nilpotent
spaces with ${\cal F}$-cone length less than $n$, and that $K$ is nilpotent
with ${\mathrm cl}_{\cal F}(K) = n$. Two applications of Proposition \ref{prop:desuspend}
reveal that it is enough to show $\bigvee_{i=1}^m {\cal S}igma^2 K \in{\cal S}^\vee$
for each $m$.
Write $V= \bigvee_{i=1}^m {\cal S}igma^2 K$.
Notice that ${\mathrm cl}_{\cal F}(\bigvee_{i=1}^m K) \ |\ eq {\mathrm cl}_{\cal F}(K)$,
and the double suspension of an ${\cal F}$-cone decomposition
of $\bigvee_{i=1}^m K$ is an ${\cal F}$-cone decomposition of $V$.
Thus we may assume that $V$ has an ${\cal F}$-cone decomposition
$S_i{\mathrm map}rt{} V_i{\mathrm map}rt{} V_{i+1}$, $0\ |\ eq i < n$
with $S_i, V_i\in {\cal S}igma{\cal F}$ for each $i$.
Therefore, we have a cofibration
$L {\mathrm map}rt{} V {\mathrm map}rt{} W$ with $L$ simply-connected,
${\mathrm cl}_{\cal F}(L) < n$ and $W\in{\cal S}igma{\cal F}$.
Let $F$ denote the homotopy fiber of $V{\mathrm map}rt{}W$,
so
$$
F {\mathrm map}rt{}V{\mathrm map}rt{}W
$$
is a fibration.
Since $W \in {\cal S}^\vee$, it suffices to show that $F\in {\cal S}^\vee$.
Now we use Proposition \ref{prop:hofiber} to determine
the homotopy type of ${\cal S}igma F$:
$$
{\cal S}igma F {\cal S}igmaimeq {\cal S}igma L \vee ( L {\cal S}igmamsh {\cal S}igma\Omega W) {\cal S}igmaimeq L{\cal S}igmamsh
\biggl(\bigvee_{\alpha} S^{n_\alpha}\biggr)
$$
which is a finite type wedge of suspensions of $L$. If we smash
an ${\cal F}$-cone length decomposition of $L$ with the space
$\bigvee_{\alpha} S^{n_\alpha}$ we obtain an ${\cal F}$-cone length
decomposition for ${\cal S}igma F$ -- in other words, ${\mathrm cl}_{\cal F}({\cal S}igma F) < n$
and, more importantly, ${\mathrm cl}_{\cal F}(\bigvee_{i=1}^l {\cal S}igma F) < n$ for each $l$.
By the inductive hypothesis, $\bigvee_{i=1}^l {\cal S}igma F \in {\cal S}^\vee$
for each $l$. Since $L, V$
and $W$ are each simply-connected, so is $F$, and Proposition
\ref{prop:desuspend} implies that $F\in {\cal S}^\vee$, as desired.
To prove (b), observe that the collection ${\cal M}$ of
all $K$ with $\Omega K\in\overline {\cal S}$ is a resolving class
that contains ${\cal S}igma F$ by the Hilton-Milnor theorem \cite{G}.
Hence ${\cal M}$ contains all nilpotent spaces $K$ with ${\mathrm cl}_{\cal F}(K)<\infty$
by part (a).
The proof of (c) is similar to the proof of (a).
The initial case of the induction is a special case of (b).
To prove the inductive step, we write
$V= \bigvee_{i=1}^m {\cal S}igma^2 K$ and show that $V\in {\cal S}$. As before, we
consider the cofiber sequence $L{\mathrm map}rt{} V{\mathrm map}rt{}W$ with $W\in{\cal S}igma{\cal F}$ and the
corresponding fibration $F{\mathrm map}rt{}V{\mathrm map}rt{}W$. This gives us
a fibration
$$
\Omega^n V {\mathrm map}rt{} \Omega^n W {\mathrm map}rt{} \Omega^{n-1} F
$$
with $\Omega^n W\in{\cal S}$. It now suffices to prove that $\Omega^{n-1} F\in{\cal S}$,
which follows by induction using Proposition \ref{prop:desuspend}.
$\Box$\par
{\cal S}igmaection{Proof of Theorem \ref{thrm:main} and Proposition \ref{prop:SinF}}
\noindent{\bf Proof of Theorem \ref{thrm:main}}\ \ {\rm and}\ \
Let ${\cal M}$ be the class of all spaces $K$ such that
$F(K) \in {\cal R}$; we have already seen
that ${\cal M}$ is a strong resolving class. For part (a),
$S^n\in {\cal M}$ for each $n$ by assumption, so $\overline {\cal S} {\cal S}igmaseq {\cal M}$.
By Theorem \ref{thrm:sres}(b), ${\cal M}$ contains $\Omega K$
for every nilpotent space $K$ with ${\mathrm cl}_{\cal F}(K) < \infty$.
In part (b), we find that ${\cal S}^\vee{\cal S}igmaseq {\cal M}$, and so
${\cal M}$ contains every nilpotent space $K$ with ${\mathrm cl}_{\cal F}(K) < \infty$.
$\Box$\par
\noindent{\bf Proof of Proposition \ref{prop:SinF}}\ \
Define a relation $<$ on ${\cal S}igma{\cal F}$ as follows: $S<T$ if either
(1) the connectivity of $S$ is greater than the connectivity of $T$,
or
(2) the connectivity of $S$ equals the connectivity of $T$ (say
both are $(n-1)$-connected) and the rank of $\pi_n S$ is less than
the rank of $\pi_n T$.
The key to this proof is the following claim.
{\cal S}igmamallskip
\noindent {{\cal S}igmac Claim}\ \ Suppose that $S\in{\cal F}$ is
$(n-1)$-connected and has $\pi_n S\neq 0$. Then there is a map $f:S{\mathrm map}rt{} S^n$
such that the homotopy fibre $T$ of $f$ belongs to ${\cal F}$ and $T<S$.
\noindent {{\cal S}igmac Proof of Claim}\ \
Write $S{\cal S}igmaim S'\vee S^n$, and let $f:S{\mathrm map}rt{} S^n$ be
the map which collapses $S'$. By \cite{G}, the homotopy fibre of $f$ is
$$
(S'\times\Omega S^n)/(*\times\Omega S^n){\cal S}igmaim S'{\cal S}igmamsh(\Omega S^n)_+ {\cal S}igmaim
\bigvee_{m=0}^\infty {\cal S}igma^{(n-1)m} S' \in{\cal S}igma{\cal F},
$$
using the James splitting of ${\cal S}igma\Omega S^n$.
{\cal S}igmamallskip
Now let $S=S_0 \in {\cal S}igma{\cal F}$. Define $S_{n+1}$ as the
fiber of a map $f:S_n{\mathrm map}rt{}S^{k(n)}$ as in the claim. The result
is a tower of spaces
$$
S_0{\mathrm map}lft{} S_1{\mathrm map}lft{} \cdots{\mathrm map}lft{} S_n{\mathrm map}lft{} S_{n+1}{\mathrm map}lft{}\cdots
$$
with $S_n\in{\cal F}$ and $S_{n+1} < S_n$ for each $n$.
Since spaces in this tower become arbitrarily highly connected
as $n$ increases, ${\mathrm holim}_n\, S_n {\cal S}igmaim *$.
The fibrations $S_{n}{\mathrm map}rt{}S_{n+1}{\mathrm map}rt{} S^{k(n)}$ give rise
to fibrations
$$
{\mathrm map}_* (W, F(S_{n+1})){\mathrm map}rt{}
{\mathrm map}_*(W, F(S_n)) {\mathrm map}rt{}
\overbrace{{\mathrm map}_*(W, F(S^{k(n)}))}^{*}.
$$
It follows by induction that each map $S_n{\mathrm map}rt{} S$ induces a weak equivalence
${\mathrm map}_*(W,F(S_n)){\cal S}igmaim {\mathrm map}_*(W,F(S))$. Finally, we compute
\[
\begin{array}{rcl}
{\mathrm map}_*(W,F(S))
&{\cal S}igmaim&{\mathrm holim}_n\,{\mathrm map}_*(W,F(S))
\ \ {\rm and}\ \
&{\cal S}igmaim&{\mathrm holim}_n\,{\mathrm map}_*(W,F(S_n ))
\ \ {\rm and}\ \
&{\cal S}igmaim& {\mathrm map}_*(W,{\mathrm holim}_n\, F(S_n ))
\ \ {\rm and}\ \
&{\cal S}igmaim&{\mathrm map}_*(W,*)
\ \ {\rm and}\ \
&{\cal S}igmaim& {*}\ \ {\rm and}\ \
\end{array}
\]
$\Box$\par
\end{document} | math | 21,321 |
\begin{document}
\begin{abstract}
We investigate systems of degenerate parabolic equations idealizing reactive solute transport in porous media. Taking advantage of the inherent structure of the system that allows to deduce a scalar Generalized Porous Medium Equation for the sum of the solute concentrations, we show existence of a unique weak solution to the coupled system and derive regularity estimates. We also prove that the system supports solutions propagating with finite speed thus giving rise to free boundaries and interaction of compactly supported initial concentrations of different species.
\mathbb R^{nN}thbb Nd{abstract}
\mathbb R^{nN}ketitle
\section{Introduction}
The transport of pollutants in subsurface environments is a complex process modeled by advection-diffusion-reaction equations that describe the evolution of contaminant concentrations in porous medium through advection, dispersion, diffusion and adsorption. More than often the adsorption, accumulation of a pollutant on the solid matrix at the fluid-solid interface, is in fact the main mechanism responsible for the contaminant transport in soil.
In this work, we address the case of multicomponent contaminant transport by considering a competitive adsorption process of Freundlich type between different species $z_1,\ldots z_N$. For expediency, we write the Freundlich multicomponent equilibrium isotherm as
$$
\mathbb R^{nN}thbf{b}_a(\mathbb R^{nN}thbf{z})= |\mathbb R^{nN}thbf{z}|_1^{p-1}\mathbb R^{nN}thbf{z} \in \mathbb R^{nN}thbb{R}^N\, , \quad p\in(0,1)\, , \quad \mathbb R^{nN}thbf{z}=(z_1,\ldots,z_N)\, ,
$$
and consider the model problem
\begin{equation}
\partial_t\mathbb R^{nN}thbf{b}(\mathbb R^{nN}thbf{z})=\Delta \mathbb R^{nN}thbf{z}+\mathbb R^{nN}thbf{f} ,
\label{eq:model}
\mathbb R^{nN}thbb Nd{equation}
where $|\mathbb R^{nN}thbf{z}|_1=\sum\limits_{i=1}^N |z_i|$ is the usual $l^1$ norm in $\mathbb R^{nN}thbb{R}^N$, $\mathbb R^{nN}thbf{b}(\mathbb R^{nN}thbf{z})=\phi\mathbb R^{nN}thbf{z}+(1-\phi)\mathbb R^{nN}thbf{b}_a(\mathbb R^{nN}thbf{z})$, and $\phi \in [0,1)$ is the medium constant porosity.
For a more thorough discussion on the physical background and derivation of this model, see Section~\ref{sec:phys} below.
While keeping in mind that the $\mathbb R^{nN}thbf{b}$-term in problem \eqref{eq:model} arises from the multicomponent adsorption with the Freundlich isotherm, we will allow for more general nonlinearities and denote hereafter the Freundlich nonlinearity as $\mathbb R^{nN}thbf{b}_f(\mathbb R^{nN}thbf{z})=(\phi+(1-\phi)|\mathbb R^{nN}thbf{z}|_1^{p-1})\mathbb R^{nN}thbf{z}$.
\subsection{General isotherms} \label{sec:ass}
Let us assume that
$$
\mathbb R^{nN}thbf{b}(\mathbb R^{nN}thbf{z})= B(|\mathbb R^{nN}thbf{z}|_1)\, \mathbb R^{nN}thbf{z},
$$
where $B:\mathbb R^{nN}thbb{R}^+\to\mathbb R^{nN}thbb{R}^+$ is such that
$$
\beta(r):=B(|r|)r\in \mathbb R^{nN}thcal{C}(\mathbb R^{nN}thbb{R})\cap\mathbb R^{nN}thcal{C}^{1}(\mathbb R^{nN}thbb{R}\setminus\{0\})
$$
and
\begin{equation}
\beta(0)=0,\quad\beta (\pm\infty)=\pm\infty,\quad\beta'(r)>0\text{ for }r\neq 0.
\label{eq:monotonicity_b}
\tag{$H_1$}
\mathbb R^{nN}thbb Nd{equation}
Note that this nonlinearity includes the Freundlich one $\mathbb R^{nN}thbf{b}_f(\mathbb R^{nN}thbf{z})=(\phi+(1-\phi)|\mathbb R^{nN}thbf{z}|_1^{p-1})\mathbb R^{nN}thbf{z}$, allows for blow up $D_{\mathbb R^{nN}thbf z}\mathbb R^{nN}thbf{b}(0)\sim\infty$, and that by definition $|\mathbb R^{nN}thbf{b}(\mathbb R^{nN}thbf{z})|_1=\beta(|\mathbb R^{nN}thbf{z}|_1)$. Most importantly, this type of nonlinearity possesses a structure which will allow us to derive a particular scalar equation from the system. Since $\beta$ is monotone increasing, the inverse
$$
\Phi=\beta^{-1}
$$
is well-defined and continuous with $\Phi(0)=0$. We further assume that $\Phi\in\mathbb R^{nN}thcal{C}^1(\mathbb R^{nN}thbb{R})$ satisfies the structural conditions
\begin{equation}
s\in \mathbb R^{nN}thbb{R}:\qquad 1 \leq \frac{s\Phi'(s)}{\Phi(s)} \leq \frac{1}{a}
\label{eq:structural_diffusion_hyp}
\tag{$H_2$}
\mathbb R^{nN}thbb Nd{equation}
and
\begin{equation}
s\in \mathbb R^{nN}thbb{R}:\qquad \frac{s\Phi''(s)}{\Phi'(s)} \geq - \frac{1}{a}
\label{eq:convexity_hyp}
\tag{$H_3$}
\mathbb R^{nN}thbb Nd{equation}
for some structural constant $a\in (0,1)$. In the scalar case $\partial_t b(z)=\Delta z+(\ldots)$ the change of variables $u=b(z)$ produces the well-known Generalized Porous Media Equation (GPME) $\partial_tu=\Delta \Phi(u)+(\ldots)$ for which \eqref{eq:structural_diffusion_hyp} is a standard assumption, see~\cite{DK07,Va07} and the references therein. It is well-known that the scalar GPME is degenerate at $u=0$ if $\Phi'(0)=0$ and strictly parabolic if $\Phi'(0)>0$, which can be seen from the divergence form $\Delta\Phi(u)=\textnormal{dist}ve(\Phi'(u)\nabla u)$. Note in particular that the structural lower bound in \eqref{eq:structural_diffusion_hyp} includes both the degenerate \emph{slow diffusion} $\Phi'(0)=0$ and the nondegenerate case $\Phi'(0)>0$, but the assumption $\Phi\in\mathbb R^{nN}thcal{C}^1(\mathbb R^{nN}thbb{R})$ excludes \emph{fast diffusion} $\Phi'(0)=+\infty$. We will shortly perform a similar change of variables $\mathbb R^{nN}thbf{u}=\mathbb R^{nN}thbf{b}(\mathbb R^{nN}thbf{z})$ for the multicomponent problem in order to move the nonlinearity from the time derivative to the spatial ones.
We shall deal with the degenerate and the nondegenerate diffusions simultaneously in a unified framework, except in Section \ref{section:FB} where we discuss the existence of free boundaries and consequently restrict ourselves to \emph{slow diffusions} $\Phi'(0)=0$. Note also that in view of \eqref{eq:structural_diffusion_hyp} the function $\Phi(s)/s$ is continuous monotone and nondecreasing in $\mathbb R^{nN}thbb{R}$ with at most algebraic growth
\begin{equation}
0<s_1<s_2:\qquad \frac{\Phi(s_2)/s_2}{\Phi(s_1)/s_1}\leq \left(\frac{s_2}{s_1}\right)^{\frac{1}{a}-1}\,,
\label{eq:algebraic_growth_Phi(s)/s}
\mathbb R^{nN}thbb Nd{equation}
all properties that will be crucial in the subsequent analysis just as in the standard theory for GPME. The structural assumptions \eqref{eq:monotonicity_b}-\eqref{eq:structural_diffusion_hyp}-\eqref{eq:convexity_hyp} are easily verified for the physical Freundlich isotherm when $p\in (0,1)$. With these structural assumptions the blowup $D_{\mathbb R^{nN}thbf z}\mathbb R^{nN}thbf{b}(0)\sim\infty$ corresponds now to slow diffusion $\Phi'(0)=0$, but linear diffusion $\Phi(s)=s$ is also allowed. In fact, the Freundlich isotherm $\mathbb R^{nN}thbf{b}_f(\mathbb R^{nN}thbf{z})=(\phi +(1-\phi)|\mathbb R^{nN}thbf{z}|_1^{p-1})\mathbb R^{nN}thbf{z}$ behaves like $\phi\mathbb R^{nN}thbf{z}$ for large $|\mathbb R^{nN}thbf{z}|_1$, hence $\beta(r)\sim \phi r$ and $\phi(s)=\beta^{-1}(s)\sim s/\phi$ for large $r,s$. In other words, the system $\partial_t\mathbb R^{nN}thbf{b}_f(\mathbb R^{nN}thbf{z})=\Delta \mathbb R^{nN}thbf{z}+\mathbb R^{nN}thbf{f}$ behaves as $N$ uncoupled linear heat equations $\phi \partial_t \mathbb R^{nN}thbf{z} \approx \Delta \mathbb R^{nN}thbf{z}+\mathbb R^{nN}thbf{f}$ for large $|\mathbb R^{nN}thbf{z}|_1$. From the physical point of view, roughly speaking, this means that for very large concentrations the porous rock matrix saturates and
the adsorption phenomena become negligible compared to inertial effects.
The monotonicity assumption \eqref{eq:monotonicity_b} allows us to invert
$$
\mathbb R^{nN}thbf{b}(\mathbb R^{nN}thbf{z})=\mathbb R^{nN}thbf{u}\,\Leftrightarrow \, \mathbb R^{nN}thbf{z}=\frac{\Phi(|\mathbb R^{nN}thbf{u}|_1)}{|\mathbb R^{nN}thbf{u}|_1}\mathbb R^{nN}thbf{u},
$$
so that problem (\ref{eq:model}) can be recast as a degenerate parabolic system of (generalized) porous medium type
\begin{equation}
\partial_t\mathbb R^{nN}thbf{u} = \Delta \, \left( \frac{\Phi(|\mathbb R^{nN}thbf{u}|_1)}{|\mathbb R^{nN}thbf{u}|_1}\mathbb R^{nN}thbf{u} \right)+\mathbb R^{nN}thbf{f}.
\label{eq:model_PME}
\mathbb R^{nN}thbb Nd{equation}
We shall refer to $\mathbb R^{nN}thbf{u}$ as density whereas we shall speak of concentration when dealing with the original $\mathbb R^{nN}thbf{z}$ variable.\\
\subsection{Systems of decoupled Cauchy-Dirichlet problems}
Let $\Omega\subset \mathbb R^{nN}thbb{R}^n$ be a bounded open set with smooth boundary $\partial\Omega$ and define $Q_T=\Omega\times(0,T)$ and $\Sigma_T=\partial\Omega\times(0,T)$ for fixed $T>0$. For given boundary data $\mathbb R^{nN}thbf{z}^D(x,t)=(z^D_1,\ldots,z^{D}_N)(x,t)$, initial condition $\mathbb R^{nN}thbf{z}^0(x)=(z^0_1,\ldots,z^0_N)(x)$, and the resultant of forcing terms $\mathbb R^{nN}thbf{f}(x,t)=(f_1,\ldots,f_N)(x,t)$, we consider the following two equivalent formulations, the first written for the original concentrations $\mathbb R^{nN}thbf{z}=(z_1,\ldots,z_N)$
\begin{equation}
\left\{
\begin{array}{ll}
\partial_t \mathbb R^{nN}thbf{b}(\mathbb R^{nN}thbf{z})=\Delta \mathbb R^{nN}thbf{z}+\mathbb R^{nN}thbf{f} & \mbox{in } \, Q_T\\
\mathbb R^{nN}thbf{z}(x,0) = \mathbb R^{nN}thbf{z}^0 (x) & \mbox{in }\Omega \\
\mathbb R^{nN}thbf{z} = \mathbb R^{nN}thbf{z}^D & \mbox{in }\Sigma_T
\mathbb R^{nN}thbb Nd{array}
\right.
\label{eq:PB_isotherm_z}
\mathbb R^{nN}thbb Nd{equation}
and the second one for the densities $\mathbb R^{nN}thbf{u}=(u_1,\ldots,u_N)$
\begin{equation}
\left\{
\begin{array}{ll}
\partial_t \mathbb R^{nN}thbf{u}=\Delta\left(\frac{\Phi(|\mathbb R^{nN}thbf{u}|_1)}{|\mathbb R^{nN}thbf{u}|_1}\mathbb R^{nN}thbf{u}\right)+\mathbb R^{nN}thbf{f} & \mbox{in } \, Q_T\\
\mathbb R^{nN}thbf{u}(x,0) = \mathbb R^{nN}thbf{u}^0 (x) & \mbox{in }\Omega \\
\frac{\Phi(|\mathbb R^{nN}thbf{u}|_1)}{|\mathbb R^{nN}thbf{u}|_1}\mathbb R^{nN}thbf{u} = \mathbb R^{nN}thbf{z}^D & \mbox{in }\Sigma_T
\mathbb R^{nN}thbb Nd{array}
\right.
\label{eq:PB_u}
\mathbb R^{nN}thbb Nd{equation}
where $\mathbb R^{nN}thbf{b}(\mathbb R^{nN}thbf{z})=\mathbb R^{nN}thbf{u}\Leftrightarrow \mathbb R^{nN}thbf{z}=\frac{\Phi(|\mathbb R^{nN}thbf{u}_1|)}{|\mathbb R^{nN}thbf{u}|_1}\mathbb R^{nN}thbf{u}$. As usual for the scalar GPME, the boundary conditions in the density formulation \eqref{eq:PB_u} are enforced in terms of the physical concentration $\frac{\Phi(|\mathbb R^{nN}thbf{u}|_1)}{|\mathbb R^{nN}thbf{u}|_1}\mathbb R^{nN}thbf{u}=\mathbb R^{nN}thbf{b}^{-1}(\mathbb R^{nN}thbf{u})=\mathbb R^{nN}thbf{z}^D$ rather than $\mathbb R^{nN}thbf{u}=\mathbb R^{nN}thbf{u}^D=\mathbb R^{nN}thbf{b}(\mathbb R^{nN}thbf{z}^D)$. It is easy to see formally that non-negative data $f_i,u^0_i,z^D_i\geq 0$ should lead to non-negative solutions $u_i\geq 0\iff z_i\geq 0$ and, therefore, we shall only deal with such non-negative data and solutions. This is of course consistent with the fact that $z_i$ represent physical concentrations and should stay non-negative when time evolves. Summing the equations in \eqref{eq:PB_u} we recognize that $w=|\mathbb R^{nN}thbf{u}|_1=u_1+\ldots+u_N$ is a non-negative solution to
\begin{equation}
w=|\mathbb R^{nN}thbf{u}|_1:\qquad
\left\{
\begin{array}{ll}
\partial_t w=\Delta(\Phi(w))+F & \mbox{in } \, Q_T,\\
w(x,0) = w^0(x) & \mbox{in }\Omega, \\
\Phi(w)= g^D & \mbox{in }\Sigma_T\,
\mathbb R^{nN}thbb Nd{array}
\right.
\label{eq:PB_w}
\tag{GPME}
\mathbb R^{nN}thbb Nd{equation}
with $F=f_1+\ldots + f_N\geq 0$, $g^D=|\mathbb R^{nN}thbf{z}^D|_1\geq 0$, and $w^0=|\mathbb R^{nN}thbf{u}^0|_1\geq 0$. Note that the boundary data is written for $\Phi(w)$ rather than for $w$ as is common for the scalar GPME.\\
The initial condition and inhomogeneity should satisfy
\begin{equation}
\forall i=1\ldots N:\ \ 0 \leq u^0_i\leq M \text{ and }0\leq f_i\leq M \quad \text{a.e. }(x,t)\in Q_T
\label{hyp:initial+forcing}
\mathbb R^{nN}thbb Nd{equation}
for some finite $M>0$. The boundary data will always assumed to be non-negative and bounded as well, but we shall sometimes assume the following. If $\gamma:\Omega\to \partial\Omega$ is the usual trace operator then there exists $\mathbb R^{nN}thbf{Z}^D(x,t)$ such that
\begin{equation}
\label{hyp:boundary_data}
\mathbb R^{nN}thbf{z}^D=\gamma(\mathbb R^{nN}thbf{Z}^D):\ \ 0\leq Z_i^D\in L^{\infty}(Q_T)\cap L^{2}(0,T;H^1(\Omega))\text{ and }\partial_t Z_i^D\in L^{\infty}(Q_T)
\mathbb R^{nN}thbb Nd{equation}
(we shall indistinctly write $\mathbb R^{nN}thbf{z}^D$ or $\mathbb R^{nN}thbf{Z}^D$ both for the trace boundary values or their extension to $\Omega$).
\subsection{Main results}
Let us now first introduce our main theorem, which addresses existence, uniqueness, and regularity.
\begin{theorem}
Assume that \eqref{eq:structural_diffusion_hyp} holds. For any data $0\leq u_i^0\leq M$, $0\leq z^D_i\leq M$, and $0\leq f_i\leq M$ there exists a unique non-negative bounded very weak solution $\mathbb R^{nN}thbf{u}$ to \eqref{eq:PB_u}. Moreover, $w=|\mathbb R^{nN}thbf{u}|_1$ is the unique non-negative bounded very weak solution to \eqref{eq:PB_w} and there exist positive constants $\alpha=\alpha(a,n)\in (0,1)$ and $C=C(a,T,n,N,M)$ such that
\begin{equation}
\|\mathbb R^{nN}thbf{u}\|_{C^{\alpha,\alpha/2}(Q')}\leq C(1+1/d'+1/\sqrt{\tau})
\label{eq:uniform_Holder}
\mathbb R^{nN}thbb Nd{equation}
holds in all parabolic subdomains $Q'=\Omega'\times(\tau,T)$ with $0<\tau<T$, $\Omega'\subset\subset \Omega$, and $d'=\textnormal{dist}(\overline{\Omega'},\partial\Omega)$.
Assume in addition that \eqref{eq:convexity_hyp} holds and that the data satisfy \eqref{hyp:initial+forcing}-\eqref{hyp:boundary_data}. Then $w$ is a global weak energy solution to \eqref{eq:Cauchy_PB_w} and $\mathbb R^{nN}thbf{u}$ is a local weak energy solution to \eqref{eq:Cauchy_PB_u} in the sense that
\begin{equation}
\|\nabla(\varrho u_i)\|_{L^2(Q'_T)} \leq C(1+1/d')\qquad \forall\, i=1\ldots N\, ,
\label{eq:energy_very_weak_ui}
\mathbb R^{nN}thbb Nd{equation}
where $\varrho=\frac{\Phi(w)}{w}=\frac{\Phi(|\mathbb R^{nN}thbf{u}|_1)}{|\mathbb R^{nN}thbf{u}|_1}$, holds in any $Q'_T=\Omega'\times(0,T)$ with some constant $C=C(a,T,n,N,M)>0$.
\label{theo:exist_sols_Dirichlet_Cauchy_PB}
\mathbb R^{nN}thbb Nd{theorem}
\noindent
The proof of the theorem can be found from Section~\ref{section:weak_sols}, where one also find definitions of very weak and energy solutions. It is also worth stressing that if the initial and boundary data are compatible, then the local regularity \eqref{eq:uniform_Holder} can be improved to global regularity up to the bottom and lateral boundaries, see Proposition~\ref{prop:boundary_regularity} later on.
Let us immediately comment the content of the theorem. First of all, in the theory of (possibly degenerate) \emph{scalar} diffusion equations such as \eqref{eq:PB_w} the so-called \emph{pressure} variable $p=p(w)=\Phi'(w)$ plays an important role, as can be seen from the divergence form $\Delta\Phi(w)=\textnormal{dist}ve(\Phi'(w)\nabla w)$. In view of \eqref{eq:PB_u} another pressure variable of interest is clearly
\begin{equation}
\varrho=\varrho(w)=\frac{\Phi(w)}{w},\qquad w=|\mathbb R^{nN}thbf{u}|_1.
\label{eq:def_pressure_rho}
\mathbb R^{nN}thbb Nd{equation}
Note that our structural assumption \eqref{eq:structural_diffusion_hyp} bounds the ratio $p/\varrho$ away from zero and from above, roughly meaning that the degeneracy of $p$ in \eqref{eq:PB_w} should be comparable to the degeneracy of $\varrho$ in \eqref{eq:PB_u}. The natural energy space for \eqref{eq:PB_w} is $\nabla \Phi(w)=p\nabla w=\nabla(\varrho w)\in L^2(Q_T)$, whereas that for \eqref{eq:PB_u} is rather here $\nabla (\varrho u_i) \in L^2(Q_T)$. Our structural assumption \eqref{eq:convexity_hyp} will later allow us to derive such an estimate for each $u_i$ from that estimate obtained for $w$. In other words, bounds for the scalar quantity $w$ suffice to control the system in terms of energy considerations. This idea of controlling the vector-valued $\mathbb R^{nN}thbf{u}$ by means of the scalar $w$ will be the cornerstone of our analysis, and will also appear in the study of the H\"older regularity and of the free-boundaries.
Second, concerning the existence, in their celebrated work \cite{AL}, Alt and Luckhaus studied systems of elliptic-parabolic PDEs which included the $\mathbb R^{nN}thbf{b}$-term as in \eqref{eq:model} and also nonlinear $p$-Laplacian type diffusion, Stefan problems, and reaction terms $\mathbb R^{nN}thbf{f}=\mathbb R^{nN}thbf{f}(x,t,\mathbb R^{nN}thbf{z})$. Their analysis requires however a particular monotone structure which restricts the $\mathbb R^{nN}thbf{b}$-term to be of the form $\mathbb R^{nN}thbf{b}=D_{\mathbb R^{nN}thbf{z}}\varphi$ for some convex potential $\varphi$ satisfying certain structural assumptions. In our case, the dependence of the Freundlich nonlinearity $\mathbb R^{nN}thbf{b}_f$ on $\mathbb R^{nN}thbf{z}$ through the $l^1(\mathbb R^{nN}thbb{R}^N)$-norm precludes any such monotonicity and, therefore, the results of \cite{AL} seem to be of no use here. Though system \eqref{eq:Cauchy_PB_u} is formally parabolic for non-negative solutions, it is readily checked that the ellipticity fails for signed solutions due to the dependence on the $l^1(\mathbb R^{nN}thbb{R}^N)$ norm.
A direct approach by Galerkin approximation as in \cite{AL} produces here approximative solutions whose componentwise sign cannot be controlled uniformly. Since ellipticity fails for signed solutions the sequence of projected solutions does not enjoy enough compactness, hence the method from \cite{AL} cannot be adapted. In order to tackle this issue we use instead the specific structure of the system allowing to control each component $u_i$ in terms of the scalar quantity $w=|\mathbb R^{nN}thbf{u}|_1$.
The method of proof for Theorem~\ref{theo:exist_sols_Dirichlet_Cauchy_PB} is classical for scalar problems, but requires technical work for system \eqref{eq:Cauchy_PB_u}: we first establish existence of positive classical solutions $\mathbb R^{nN}thbf{u}^k$ for approximated positive data $u^{0.k},\mathbb R^{nN}thbf{z}^{D,k},\mathbb R^{nN}thbf{f}^k$ and derive some a priori regularity and energy estimates. Taking $k\to\infty$ finally gives the desired solution $\mathbb R^{nN}thbf{u}=\lim \mathbb R^{nN}thbf{u}^k$, which inherits regularity and energy estimates from the previous ones.\\
\subsection{Physical background} \label{sec:phys}
Based on a continuum approach at a macroscopic level by homogenization, the mass conservation written for the concentration of one contaminant component $z=z(t,x)$ can be written as, cf. \cite{BV87},
\begin{equation}
\phi\frac{\partial z}{\partial t} +\rho (1-\phi)\,\frac{\partial b_a}{\partial t} + \phi\, \nabla\cdot (z\mathbb R^{nN}thbf{V} -\, \mathbb R^{nN}thbf{D} \nabla z ) = f\, .
\label{eq:real_model}
\mathbb R^{nN}thbb Nd{equation}
Here, we have made the assumptions of saturated flow and constant porosity $\phi\in(0,1)$,
denoted the advective water flux by $z\mathbb R^{nN}thbf{V}$ ($\mathbb R^{nN}thbf{V}$ is the Darcy velocity), the bulk density of the solid matrix by $\rho>0$, the hydrodynamic dispersion matrix describing both the molecular diffusion and the mechanical dispersion by $\mathbb R^{nN}thbf{D}$, and modeled the source or sink terms by $f$. Moreover, $b_a=b_a(z)$ describes the concentration of contaminant adsorbed on the solid matrix through a reactive adsorption process that can be assumed to be either fast (equilibrium) or slow (non-equilibrium).
An adsorption isotherm $b_a(z)$ relates the concentration of the adsorbed component to its concentration in the fluid phase at constant temperature. One of the most commonly used nonlinear equilibrium isotherms for a single species is the Freundlich isotherm expressed as, cf. \cite{BV87,WMK91}
$$
b_a(z)=K\, z^{p}\, \quad K>0\, , \quad p\in(0,1)\, , \quad z\geq 0\, .
$$
The Freundlich exponent $p\in(0,1)$ makes equation \eqref{eq:real_model} singular at $z=0$ because, at least formally, $\partial_t b_a(z)=b_a^\prime(z)\partial_t z$ and $b_a^\prime(0)=\infty$. The equation may thus exhibit finite speed of propagation of compactly supported initial solutions giving rise to free boundaries that separate the region where the solute concentration vanishes from that with positive concentration. This is in marked contrast with the behavior of solutions when the Freundlich exponent $p$ equals or exceeds one since the equation becomes nonsingular for $p\geq 1$ and the information propagates with infinite speed, as usual for uniformly parabolic equations.
Equation (\ref{eq:real_model}), complemented with suitable initial and boundary conditions, and all its variants arising from different equilibrium and non-equilibrium, linear or non-linear, isotherms, have attracted considerable attention over the last 20 years, both from an analysis and numerical simulation point of view, see, \emph{e.g.}, \cite{vDK91,DvDW94, DvDG96,BK97a,BK97b,KO2000,ADCC01}. It is, however, the above equilibrium Freundlich isotherm which makes the problem most challenging due to the degeneracy and similarity to the porous medium equation.
A competitive adsorption process between different species $z_1,\ldots z_N$, can be modeled by a multicomponent isotherm of Freundlich type, cf. \cite{SRS81,GF93, WKLL02}, and \cite{SMM06} for a review of competitive equilibrium adsorption modeling. As an idealization of the physical model proposed, e.g., in \cite{SRS81}, the multicomponent Freundlich equilibrium isotherm can be expressed as
$$
\mathbb R^{nN}thbf{b}_a(\mathbb R^{nN}thbf{z})= |\mathbb R^{nN}thbf{z}|_1^{p-1}\mathbb R^{nN}thbf{z} \in \mathbb R^{nN}thbb{R}^N\, , \quad p\in(0,1)\, , \quad \mathbb R^{nN}thbf{z}=(z_1,\ldots,z_N)\, ,
$$
where $|\mathbb R^{nN}thbf{z}|_1=\sum\limits_{i=1}^N |z_i|$.
Scaling the density, neglecting the hydrodynamical effects ($\mathbb R^{nN}thbf{V}=0)$, assuming that $\mathbb R^{nN}thbf{D}=\operatorname{Id}$ and writing $\mathbb R^{nN}thbf{b}(\mathbb R^{nN}thbf{z})=\phi\mathbb R^{nN}thbf{z}+(1-\phi)\mathbb R^{nN}thbf{b}_a(\mathbb R^{nN}thbf{z})$, we obtain from (the multicomponent version of) \eqref{eq:real_model} the model problem~\eqref{eq:model}, i.e.,
$$
\partial_t\mathbb R^{nN}thbf{b}(\mathbb R^{nN}thbf{z})=\Delta \mathbb R^{nN}thbf{z}+\mathbb R^{nN}thbf{f} \, .
$$
\subsection{The content}
The paper is organized as follows. In Section \ref{section:smooth_sols} we consider smooth positive data, construct corresponding smooth positive solutions to \eqref{eq:PB_u}, and establish a priori energy as well as H\"older estimates. The H\"older estimates are based on the celebrated method of intrinsic scaling \cite{DBb}, a standard technique at least for scalar problems. In Section~\ref{section:weak_sols} we consider more general data, introduce different notions of weak solutions, and prove Theorem~\ref{theo:exist_sols_Dirichlet_Cauchy_PB}. Approximating the data suitably we show existence of a unique weak solution to problem \eqref{eq:PB_u} which inherits H\"older regularity and energy estimates from the smooth positive solutions constructed in Section~\ref{section:smooth_sols}. Finally in Section~\ref{section:FB} we impose the degeneracy condition $\Phi'(0)=0$ and consider the problem in the whole space without the forcing term and with compactly supported initial data. We show that the
corresponding Cauchy problem is well posed and admits free boundary solutions and, moreover, investigate the finite speed of
propagation of the free boundaries and the evolution and interaction of distinct compactly supported initial concentrations.
\section{smooth positive solutions and a priori estimates}
\label{section:smooth_sols}
We will assume throughout this section that the data is smooth and (componentwise) positive. Solutions of \eqref{eq:PB_u} and \eqref{eq:PB_w} corresponding to such data are shown to be classical and positive and satisfy certain a priori energy and locally uniform H\"older estimates.
\begin{prop}[Existence of positive classical solutions]
Assume that $\mathbb R^{nN}thbf{z}^D$ and $\mathbb R^{nN}thbf{u}^0$ are smooth and positive and $\mathbb R^{nN}thbf{f}$ is smooth and non-negative. Moreover, let $F:=|\mathbb R^{nN}thbf{f}|_1$ and assume that
\begin{align*}
0 < m = \min \left\{ \essinf \limits_{\overline{Q_T}}|\mathbb R^{nN}thbf{u}^0|_1, \quad \essinf \limits_{\overline{\Sigma_T}}|\mathbb R^{nN}thbf{z}^D|_1\right\},\\
0 < M = \mathbb R^{nN}x \left\{ \esssup \limits_{\overline{Q_T}}|\mathbb R^{nN}thbf{u}^0|_1, \quad \esssup \limits_{\overline{\Sigma_T}}|\mathbb R^{nN}thbf{z}^D|_1, \quad \esssup \limits_{\overline{Q_T}}F\right\}.
\mathbb R^{nN}thbb Nd{align*}
Then there exists a classical solution $\mathbb R^{nN}thbf{u}\in \mathbb R^{nN}thcal{C}^{2,1}(\overline{\Omega}\times(0,T))\cap\mathbb R^{nN}thcal{C}^{2,1}(\Omega\times[0,T])\cap \mathbb R^{nN}thcal{C}^{\infty}(Q_T)$ to \eqref{eq:PB_u} with $u_i>0$ on $\overline{Q_T}$. Moreover, defining $w=|\mathbb R^{nN}thbf{u}|_1=u_1+\ldots+u_N$, $w \in \mathbb R^{nN}thcal{C}^{2,1}(\overline{Q_T})\cap \mathbb R^{nN}thcal{C}^{\infty}(Q_T)$ is a classical solution to \eqref{eq:PB_w} and
$$
0 < m \leq w(x,t) \leq M (1+T)\quad \text{in }\overline{Q_T}.
$$
\label{prop:exists_classical_solutions}
\mathbb R^{nN}thbb Nd{prop}
\begin{remark} We do not impose any compatibility conditions on the initial and boundary data at $\partial\Omega\times\{t=0\}$. Although this limits the boundary regularity it has no importance in the sequel. Note also that we could prove uniqueness of positive classical solutions at this stage. However since we will later establish a stronger uniqueness result (within the class of non-negative very weak solutions) we postpone the uniqueness issue until then.
\mathbb R^{nN}thbb Nd{remark}
\begin{proof}
We will exploit the diagonal structure of the system by first showing existence of a classical solution $w$ to \eqref{eq:PB_w}, then reconstructing $\mathbb R^{nN}thbf{u}$ by solving $N$ independent linear parabolic equations for the $u_i$, and finally checking that $w=|\mathbb R^{nN}thbf{u}|_1$ as desired.
\par
For smooth positive data let $w^0:=|\mathbb R^{nN}thbf{u}^0|_1>0$ in $\overline{\Omega}$ and $g^D:=|\mathbb R^{nN}thbf{z}^D|_1>0$ in $\overline{\Sigma_T}$. Write $\Delta\Phi(w)=\textnormal{dist}ve (\Phi'(w)\nabla w)$ and observe that hypothesis \eqref{eq:structural_diffusion_hyp} implies that $\Phi'(w)>0$ is bounded away from zero and from above as long as $0<m \leq w\leq C$ so that equation \eqref{eq:PB_w} is uniformly parabolic for such values of $w$. Therefore, after approximating $\Phi$ by a globally Lipschitz function $\Phi_{\varepsilon}$ such that $\Phi_{\varepsilon}(0)=0$, $\Phi(s)=\Phi_{\varepsilon}(s)$ for $|s|\in(\varepsilon,1/\varepsilon)$ and $\Phi_{\varepsilon}'>0$, well known results for quasilinear parabolic equations (cf. \cite{LSU}) guarantee the existence of a positive classical solution $w_{\varepsilon}(x,t)$ to the $\varepsilon$-problem. A standard comparison principle with $0\leq F\leq M $ and $0<m \leq w^0,g^D\leq M $ shows moreover that
\[
0<m \leq w_{\varepsilon}\leq \mathbb R^{nN}x\left\{\|w^0\|_{L^{\infty}(\Omega)},\|g^D\|_{L^{\infty}(\Sigma_T)}\right\}+T\|F\|_{L^{\infty}(Q^T)}\leq M (1+T)
\]
as in our statement. In particular, for $\varepsilon>0$ small
enough there holds $\Phi_{\varepsilon}(w_{\varepsilon})=\Phi(w_{\varepsilon})$ so that $w_{\varepsilon}$
is in fact a classical solution to the original problem. The argument is standard for scalar equations and we refer, e.g., to \cite{Va07,DK07} for more details.
\par
Once there exists a smooth positive solution $w$ to \eqref{eq:PB_w}, the pressure $\varrho = \frac{\Phi(w)}{w}$ becomes smooth in the interior, belongs to $\mathbb R^{nN}thcal{C}^{2,1}(\overline{\Omega}\times(0,T))\cap\mathbb R^{nN}thcal{C}^{2,1}(\Omega\times[0,T])$, and the bounds
\begin{equation}
0<C_1\leq \varrho \leq C_2\quad \mbox{in }\overline{Q^T}
\label{eq:rho_ui}
\mathbb R^{nN}thbb Nd{equation}
hold for some $C_1,C_2>0$ depending on $a,m ,M ,T$ only. Then standard results \cite{LSU} on \emph{linear} parabolic equations allow us to solve
\begin{equation}
\left\{
\begin{array}{ll}
\partial_t u_i = \Delta(\varrho u_i)+f_i = \textnormal{dist}ve(\varrho \nabla u_i )+ \textnormal{dist}ve(u_i \nabla \varrho)+f_i \qquad & \text{in }Q_T,\\
\varrho u_i = z^D_i & \text{in }\Sigma_T,\\
u_i(x,0) =u^0_i(x) & \text{in }\Omega\, ,
\mathbb R^{nN}thbb Nd{array}
\right.
\label{eq:rho_bnd}
\mathbb R^{nN}thbb Nd{equation}
for fixed $i=1,\ldots N$ and show that $u_i\in \mathbb R^{nN}thcal{C}^{2,1}(\overline{\Omega}\times(0,T))\cap\mathbb R^{nN}thcal{C}^{2,1}(\Omega\times[0,T])\cap \mathbb R^{nN}thcal{C}^{\infty}(Q_T)$ (up to the corners if the data are compatible). Indeed, from \eqref{eq:rho_bnd} it follows that the equation is uniformly parabolic and the boundary condition reads simply as $u_i=\frac{z^D_i}{\varrho}$ on $\Sigma_T$. The assumptions on the data and the strong maximum principle ensure moreover that $u_i>0$ in $\overline{Q_T}$.
\par
Let now $\tilde{w}=|\mathbb R^{nN}thbf{u}|_1$ and observe that $\partial_tw=\Delta\Phi(w)+F=\Delta(\varrho w)+F$. Because $u_i>0$ we can write $\tilde{w}=u_1+\ldots +u_N$. Summing \eqref{eq:rho_ui} over $i=1\ldots N$, we obtain $\partial_t \tilde{w}=\Delta(\varrho \tilde{w})+F$. In other words, $\tilde{w}$ is a positive classical solution to the same equation as $w$ with the same initial and boundary data. By standard uniqueness argument for smooth positive solutions we conclude that $w=\tilde{w}$. In particular $\varrho = \frac{\Phi(w)}{w} = \frac{\Phi(\tilde{w})}{\tilde{w}}=\frac{\Phi(|\mathbb R^{nN}thbf{u}|_1)}{|\mathbb R^{nN}thbf{u}|_1}$ in \eqref{eq:rho_ui} and the proof is complete.
\mathbb R^{nN}thbb Nd{proof}
\begin{prop}[A priori energy estimates]
Assume that hypotheses \eqref{eq:monotonicity_b}, \eqref{eq:structural_diffusion_hyp} and \eqref{eq:convexity_hyp} hold, let $\mathbb R^{nN}thbf{u}\in \mathbb R^{nN}thcal{C}^{2,1}(\overline{Q_T})$ be a classical positive solution corresponding to smooth positive data and assume that
$$
\|\mathbb R^{nN}thbf{u}^0\|_{L^{\infty}(\Omega)} + \|\mathbb R^{nN}thbf{z}^D\|_{L^{\infty}(0,T;H^1(\Omega))}+\|\partial_t \mathbb R^{nN}thbf{z}^D\|_{L^{\infty}(Q_T)}+\|\mathbb R^{nN}thbf{f}\|_{L^{\infty}(Q_T)}\leq M,
$$
for some $M>0$. Then we have the bounds
\begin{equation}
\| \nabla (\varrho w) \|_{L^2(Q_T)}\leq C,
\label{eq:energy_w}
\mathbb R^{nN}thbb Nd{equation}
where $w=|\mathbb R^{nN}thbf{u}|_1$ and $\varrho=\frac{\Phi(w)}{w}$, and
\begin{equation}
\qquad \|\nabla (\varrho u_i)\|_{L^2(Q'_T)}\leq C(1+1/d'), \qquad \forall\,i=1\ldots N \, .
\label{eq:energy_rho_ui}
\mathbb R^{nN}thbb Nd{equation}
\label{prop:energy_estimate}
Here, the constant $C>0$ depends on $a,T,n,N,M$ only, and $Q^\prime_T=\Omega^\prime\times (0,T)$ with
$\Omega'\subset\subset\Omega$ and $d'=\operatorname{dist}(\overline{\Omega'},\partial\Omega)>0$.
\mathbb R^{nN}thbb Nd{prop}
\begin{remark}
We were not able to establish energy estimates for $\nabla(\varrho u_i)$ up to the boundary as the dependence on $1/d^\prime$ in \eqref{eq:energy_rho_ui} shows. This will not be an issue later on since
in Proposition~\ref{prop:uniqueness_weak_sols} we shall prove uniqueness within the class of very weak solutions and estimate \eqref{eq:energy_rho_ui} as well as
assumption \eqref{eq:convexity_hyp},
which is only used in proving \eqref{eq:energy_rho_ui}, can be dispensed with while considering very weak solutions. On the other hand, estimate \eqref{eq:energy_rho_ui} is sufficient for our purposes in Section~\ref{section:FB} where the problem is considered in $\mathbb R^{nN}thbb{R}^n$ with compactly supported initial data.
\label{rmk:cst=1/d'}
Observe also that the validity of estimate \eqref{eq:energy_rho_ui} up to the boundary would directly yield \eqref{eq:energy_w} since $w=|\mathbb R^{nN}thbf{u}|_1=\sum_i u_i$ for non-negative solutions.
\mathbb R^{nN}thbb Nd{remark}
\begin{proof}
We will first establish \eqref{eq:energy_w} for the scalar variable $w$ and then show how the particular structure of system \eqref{eq:PB_u} allows us to derive \eqref{eq:energy_rho_ui} for each component $u_i$. We shall denote by $C$ any positive constant depending, as in the statement, only on $a,T,n,N,M$ whereas the primed constants $C'$ are also allowed to depend on $d'=\text{dist}(\overline{\Omega'},\partial\Omega)$. \\
\noindent
{\bf Step 1.} The assumptions on $\mathbb R^{nN}thbf{u}^0,\mathbb R^{nN}thbf{z}^D,\mathbb R^{nN}thbf{f}$ translate into similar properties for the data $w^0=|\mathbb R^{nN}thbf{u}^0|_1,g^D=|\mathbb R^{nN}thbf{z}^D|_1,F=|\mathbb R^{nN}thbf{f}|_1$ so by the comparison principle for solutions to \eqref{eq:PB_w} we have
$$
0\leq u_i\leq |\mathbb R^{nN}thbf{u}|_1=w\leq C(M,N,T).
$$
Recalling that $\varrho w=\Phi(w)$, inequality \eqref{eq:energy_w} is nothing but the usual (global) energy estimate for the GPME leading to the usual concept of weak \emph{energy} solutions. For smooth positive solutions, bound \eqref{eq:energy_w} is easily derived for $\|\nabla \Phi(w)\|_{L^2(Q_T)}=\|\nabla (\varrho w)\|_{L^2(Q_T)}$ by taking $\varphi=(\Phi(w)-g^D)\in L^2(0,T;H^1_0(\Omega))$ as a test function in \eqref{eq:PB_w}, cf. \cite{DK07,Va07} for further details.\\
\noindent
{\bf Step 2.} Since $0\leq w\leq C$, the structural assumptions imply that
$$
0\leq \varrho =\frac{\Phi(w)}{w}\leq \frac{1}{a}\Phi'(w)\leq C(a,M,T).
$$
The $L^{\infty}(Q_T)$-norm of any term involving $u_i,w,\varrho$ can thus be bounded by a constant $C=C(a,T,n,N,M)>0$ only. Now fix $i\in\{1,,\ldots,N\}$ and choose a cutoff function $\chi=\chi(x)\in \mathbb R^{nN}thcal{C}^{\infty}_c(\Omega)$ such that $0\leq \chi\leq 1$ in $\Omega$, $\chi\equiv 1$ in $\Omega'$ and $|\nabla \chi|\leq 2/d'$ where $\Omega'\subset\subset\Omega$ and $d'=\text{dist}(\overline{\Omega'},\partial\Omega)$. Multiplying the $i$-th equation in \eqref{eq:PB_u} by a test function $\varphi=\chi^2\varrho u_i$, integrating over $Q_T$ and by parts in the Laplacian term, we obtain
\begin{align*}
\int\limits_{Q_T}\chi^2|\nabla (\varrho u_i)|^2\,\mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t & = - 2\int\limits_{Q_T}\varrho u_i \chi \nabla \chi \cdot \nabla (\varrho u_i) \,\mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t \\ & + \int\limits_{Q_T}\chi^2 \varrho u_i f_i \,\mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t
- \int\limits_{Q_T}\chi^2\varrho u_i\partial_t u_i \,\mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t.
\mathbb R^{nN}thbb Nd{align*}
Integrating the last term by parts in $t$ and using $0\leq u_i,\varrho\leq C$ to bound the limit terms at $t=0,T$, gives
\begin{align*}
\int\limits_{Q_T}\chi^2 & |\nabla (\varrho u_i)|^2 \,\mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t \\
& \leq 2 \|\varrho u_i\nabla \chi\|_{L^2(Q_T)} \|\chi \nabla(\varrho u_i)\|_{L^2(Q_T)} + C +\Big(C+\frac{1}{2}\int\limits_{Q_T} \chi^2 u_i^2 \partial_t \varrho\,\mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t\Big)\\
& \leq \frac12 \|\chi \nabla(\varrho u_i)\|_{L^2(Q_T)}^2 + 8 \|\varrho u_i\nabla \chi\|_{L^2(Q_T)}^2+ C + \frac{1}{2}\int\limits_{Q_T} \chi^2 u_i^2 \partial_t \varrho\,\mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t \, ,
\mathbb R^{nN}thbb Nd{align*}
where we have also taken into account that $0\leq \chi^2 \varrho u_if_i\leq C$ and used Young's inequality.
Estimating $\|\varrho u_i\nabla \chi\|_{L^2(Q_T)}^2\leq C/(d')^2\leq C'$ then yields the bound
\begin{equation}
\|\chi \nabla (\varrho u_i)\|^2_{L^2(Q_T)}\leq C' + \underbrace{ \int\limits_{Q_T} \chi^2 u_i^2 \partial_t \varrho\,\mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t}_{:=A}.
\label{eq:estimate_nabla(ro u_i)}
\mathbb R^{nN}thbb Nd{equation}
We exploit now the structure of the system to control $A$. Indeed, since $\varrho =\frac{\Phi(w)}{w}$ one easily computes for smooth positive solutions
$$
\partial_t\varrho = \frac{d}{dw}\left(\frac{\Phi(w)}{w}\right)\partial_t w=\frac{w\Phi'(w)-\Phi(w)}{w^2}\big(\Delta (\varrho w)+F\big).
$$
Thus integrating by parts gives
\begin{align*}
A & =\int\limits_{Q_T}\chi ^2 (\varrho u_i)^2\frac{1}{(\varrho w)^2}\big(w\Phi'(w)-\Phi(w)\big)\, \big(\Delta(\varrho w)+F\big) \,\mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t\\
& =-\int\limits_{Q_T}\nabla(\chi ^2) (\varrho u_i)^2 \frac{1}{(\varrho w)^2} \big(w\Phi'(w)-\Phi(w)\big) \cdot \nabla(\varrho w)\,\mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t\\
& \phantom{=} -\int\limits_{Q_T}\chi ^2 \nabla(\varrho u_i)^2 \frac{1}{(\varrho w)^2} \big(w\Phi'(w)-\Phi(w)\big) \cdot \nabla(\varrho w)\,\mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t\\
& \phantom{=} -\int\limits_{Q_T}\chi ^2 (\varrho u_i)^2 \nabla\left(\frac{1}{(\varrho w)^2}\right) \big(w\Phi'(w)-\Phi(w)\big) \cdot \nabla(\varrho w)\,\mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t\\
& \phantom{=} -\int\limits_{Q_T}\chi ^2 (\varrho u_i)^2 \frac{1}{(\varrho w)^2} \nabla\big(w\Phi'(w)-\Phi(w)\big) \cdot \nabla(\varrho w) \,\mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t \\
& \phantom{=}+\int\limits_{Q_T}\chi ^2 (\varrho u_i)^2\frac{1}{(\varrho w)^2}\big(w\Phi'(w)-\Phi(w)\big)F\,\mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t\\
& =A_1 + A_2 + A_3 + A_4+B.
\mathbb R^{nN}thbb Nd{align*}
Observing that $0\leq \varrho u_i\leq \varrho w =\Phi(w)$ and that the hypothesis
\eqref{eq:structural_diffusion_hyp} implies that $ 0\leq w\Phi'(w)-\Phi(w)\leq C(a)\Phi(w)$,
we can control the first term as
\begin{align*}
A_1 & =-2\int\limits_{Q_T}\chi\nabla\chi\,\cdot (\varrho u_i)^2 \frac{1}{(\varrho w)^2} \big(w\Phi'(w)-\Phi(w)\big) \nabla(\varrho w)\,\mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t\\
& \leq C \|\nabla \chi\|_{L^2(Q_T)} \|\nabla (\varrho w)\|_{L^2(Q_T)}\leq C'.
\mathbb R^{nN}thbb Nd{align*}
\item
The second term is bounded similarly as follows
\begin{align*}
A_2 & = -\, 2\int\limits_{Q_T}\chi^2 (\varrho u_i)\nabla(\varrho u_i)\,\cdot \frac{1}{(\varrho w)^2} \big(w\Phi'(w)-\Phi(w)\big) \nabla(\varrho w)\,\mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t\\
& \leq 2\, \|\chi\nabla (\varrho u_i)\|_{L^2(Q_T)} \, \left\|\chi\frac{\varrho u_i}{\varrho w}\frac{w\Phi'(w)-\Phi(w)}{\varrho w}\nabla (\varrho w)\right\|_{L^2(Q_T)}\\
& \leq C\, \|\chi\nabla (\varrho u_i)\|_{L^2(Q_T)}\, \|\nabla (\varrho w)\|_{L^2(Q_T)}\\
& \leq \frac{1}{2}\, \|\chi\nabla (\varrho u_i)\|_{L^2(Q_T)}^2 + C \|\nabla (\varrho w)\|_{L^2(Q_T)}^2\\
&\leq \frac{1}{2}\, \|\chi\nabla (\varrho u_i)\|_{L^2(Q_T)}^2+C \, ,
\mathbb R^{nN}thbb Nd{align*}
where we have also used the Young's inequality (the first term on the right-hand side will be reabsorbed into \eqref{eq:estimate_nabla(ro u_i)}).
The third quantity is controlled as
\begin{align*}
A_3 & = 2\, \int\limits_{Q_T}\chi^2 (\varrho u_i)^2\frac{\nabla (\varrho w)}{(\varrho w)^3} \, \cdot \big(w\Phi'(w)-\Phi(w)g\big) \nabla(\varrho w) \,\mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t\\
& = 2\int\limits_{Q_T}\chi^2\frac{(\varrho u_i)^2}{(\varrho w)^2} \frac{w\Phi'(w)-\Phi(w)}{\Phi(w)}|\nabla (\varrho w)|^2 \mathbb R^{nN}thrm{d}x \, \mathbb R^{nN}thrm{d}t \leq C \|\nabla (\varrho w)\|_{L^2(Q_T)}^2\leq C.
\mathbb R^{nN}thbb Nd{align*}
In the fourth term we write $\nabla(\varrho w)=\nabla \Phi(w)=\Phi'(w)\nabla w$ and use \eqref{eq:convexity_hyp} to get
\begin{align*}
A_4 & = -\int\limits_{Q_T} \chi^2 (\varrho u_i)^2 \frac{1}{(\varrho w)^2} \nabla \big(w\Phi'(w)-\Phi(w)\big)\,\cdot \nabla(\varrho w) \,\mathbb R^{nN}thrm{d}x\, \mathbb R^{nN}thrm{d}t\\
& = -\int\limits_{Q_T} \chi^2 (\varrho u_i)^2 \frac{1}{(\varrho w)^2} w\Phi''(w)\nabla w\,\cdot \nabla(\varrho w) \,\mathbb R^{nN}thrm{d}x\, \mathbb R^{nN}thrm{d}t\\
& = -\int\limits_{Q_T} \chi^2 (\varrho u_i)^2 \frac{1}{(\varrho w)^2} \frac{w\Phi''(w)}{\Phi'(w)}|\nabla(\varrho w)|^2 \,\mathbb R^{nN}thrm{d}x\, \mathbb R^{nN}thrm{d}t\\
& \leq \frac{1}{a}\|\nabla(\varrho w)\|_{L^2(Q_T)}\leq C.
\mathbb R^{nN}thbb Nd{align*}
\begin{remark}
Note that an upper bound for $A_4$ is obtained here by using the lower bound \eqref{eq:convexity_hyp} for $\frac{s\Phi''(s)}{\Phi'(s)}$. If in particular $\Phi''(s)\geq 0$ for all $s\geq 0$, which is the typical case for the PME nonlinearity $\Phi(s)=|s|^{m-1}s$ in the range $m\geq 1$, then $A_4\leq 0$. This convexity condition is also valid for the Freundlich isotherm since $\beta_f(r)=\phi r+(1-\phi)r^p$ is concave and thus $\Phi_f=\beta_f^{-1}$ is convex.
\mathbb R^{nN}thbb Nd{remark}
For the last term we obtain
$$
B =\int\limits_{Q_T}\chi ^2 (\varrho u_i)^2\frac{1}{(\varrho w)^2}\big(w\Phi'(w)-\Phi(w)\big)F \,\mathbb R^{nN}thrm{d}x\, \mathbb R^{nN}thrm{d}t
\leq
C(a)\int\limits_{Q_T}\Phi(w) F \,\mathbb R^{nN}thrm{d}x\, \mathbb R^{nN}thrm{d}t
\leq C.
$$
Plugging the above estimates back into \eqref{eq:estimate_nabla(ro u_i)} finally yields
$$
\|\nabla (\varrho u_i)\|^2_{L^2(Q'_T)}\leq \|\chi \nabla (\varrho u_i)\|^2_{L^2(Q_T)}\leq C'.
$$
Keeping track of the dependence of the estimates on $d'=\textnormal{dist}\left(\overline{\Omega'},\partial\Omega\right)$ and optimizing all inequalities, one easily sees that $C'=C(1+1/d')$ with $C=C(a,T,n,N,M)$ only and the proof is complete.
\mathbb R^{nN}thbb Nd{proof}
We will next address the regularity issue.
\begin{prop}
Let $\mathbb R^{nN}thbf{u}$ and $M$ be as in Proposition~\ref{prop:exists_classical_solutions}. There exist $\alpha=\alpha(a,n)\in (0,1)$ and $C=C(a,T,n,N,M)>0$ such that the estimate
$$
\|\mathbb R^{nN}thbf{u}\|_{C^{\alpha,\alpha/2}(Q')}\leq C(1+1/d'+1/\sqrt{\tau}),
$$
holds for any parabolic subdomain $Q'=\Omega'\times(\tau,T)$, where $0<\tau<T$, $\Omega'\subset\subset\Omega$, and $d'=\textnormal{dist}(\overline{\Omega'},\partial\Omega)$.
\label{prop:C_alpha_estimate}
\mathbb R^{nN}thbb Nd{prop}
\begin{remark}
We would like to stress that our proof handles both the nondegenerate $\Phi'(0)>0$ and degenerate $\Phi'(0)=0$ cases in a unified framework.
\mathbb R^{nN}thbb Nd{remark}
\begin{proof}
The proof goes in several steps and is based on the intrinsic scaling method, cf. \cite{DBb}. As usual, we can assume after translation that the intrinsic cylinders are centered at the origin $(x_0,t_0)=(0,0)$. We write $w=|\mathbb R^{nN}thbf{u}|_1$ and recall that $w$ is positive and bounded with the $L^{\infty}$ bounds depending on $M,T$ only. We begin by considering the scalar problem.
\\
\noindent {\bf Step 1: Alternatives.} Suppose that $\sup_{Q_r ^\mu} w \leq \mu$ in an intrinsic cylinder
\[
Q_r ^\mu := B_r \times (-\tau_r ^\mu ,0)\,, \qquad \tau_r ^\mu := \frac{\mu}{\Phi(\mu)} r ^2\,,
\]
and define
\[
\widetilde Q_r ^\mu := B_{3 r /4} \times (- \frac34 \tau_r ^\mu , -\frac12 \tau_r ^\mu) \subset Q_r^\mu.
\]
Now consider the following two alternatives
\begin{equation} \label{eq:nondeg}
\left| \widetilde Q_{r }^\mu \cap \{w \leq \mu/2\} \right| \leq \delta | \widetilde Q_{r }^\mu| \, ,
\mathbb R^{nN}thbb Nd{equation}
\begin{equation} \label{eq:deg}
\left| \widetilde Q_{r }^\mu \cap \{w \leq \mu/2\} \right| > \delta | \widetilde Q_{r }^\mu| \, ,
\mathbb R^{nN}thbb Nd{equation}
where $\delta \in (0,1)$ is a small parameter to be fixed shortly.
The first one is the nondegenerate alternative and the second is the degenerate alternative. We will analyze them separately.
\\
\noindent {\bf Step 2: Nondegenerate alternative 1.} Set
\[
r _j = \left(\frac1{4} + \frac1{4^{j+1}} \right) r \,, \qquad k_j := \left(\frac14 + \frac1{4^{j+1}} \right) \mu \,, \qquad w_j := (k_j-w)_+\,.
\]
Set also
\[
\widetilde Q^j := B_{r /4 + r _j} \times \left(- \frac12 \tau_r ^\mu - \tau_r ^\mu \left(\frac{r _j}{r }\right)^2 ,-\frac12 \tau_r ^\mu\right)\,,
\]
and let $\phi_j$ be a cut-off function such that $\phi_j$ is smooth, $0\leq \phi_j \leq 1$, $\phi_j$ vanishes on the parabolic boundary $\partial_p \widetilde Q^j$ (bottom and lateral), is one on $\widetilde Q^{j+1}$ and $|\nabla \phi_j|^2 + (\partial_t \phi_j^2)_+ \leq r ^{-2} 16^{j+1}$.
Since $ w$ solves the equation $\partial_t w - \Delta \Phi(w) = F$ and $F\geq 0$, it is easy to check that $w_j$ is a weak subsolution to the equation $\partial_t w_j - \textnormal{div} (\Phi'(w) \nabla w_j) = 0$, and testing the latter with $w_j \phi_j^2$ leads to the Caccioppoli inequality
\begin{eqnarray}
\notag &&
\sup_{-\tau_r ^\mu<t<0} \mean{B_r } \frac{w_j^2 \phi_j^2}{\tau_r ^\mu} \, dx + \mean{Q_r ^\mu} \Phi'(w) |\nabla(w_j \phi_j)|^2\, dx \, dt
\\ \notag && \qquad \qquad \leq c \mean{Q_r ^\mu} \left[ \Phi'(w) w_j^2 |\nabla \phi_j|^2 + w_j^2 (\partial_t \phi_j^2)_+ \right] \, dx \, dt \,.
\mathbb R^{nN}thbb Nd{eqnarray}
Setting
\[
\bar w_j :=
\begin{cases}
k_j - k_{j+1} & \mbox{if }w \leq k_{j+1} \\
w_j & \mbox{if } w > k_{j+1}\, \\
\mathbb R^{nN}thbb Nd{cases}
\]
and recalling from \eqref{eq:structural_diffusion_hyp} that $\Phi(s)/s$ is monotone non-decreasing with at most algebraic growth, we see that
\[
\frac{r ^2}{\tau_r ^\mu} |\nabla \bar w_j |^2 = \frac{\Phi(\mu)}{\mu} |\nabla \bar w_j |^2 \leq c(a) \frac{\Phi(k_{j+1})}{k_{j+1}} |\nabla \bar w_j |^2 \leq c(a) \Phi'(w) |\nabla w_j|^2
\]
since in the support of $\nabla \bar w_j$ we have $w \geq k_{j+1} \geq \mu/4$. Similarly,
\[
\Phi'(w) w_j^2 |\nabla \phi_j|^2 \leq c(a) 16^j \frac{1}{r ^2} \frac{\Phi(\mu)}{\mu} w_j^2 \leq \frac{c(a) 16^j}{\tau_r ^\mu} \mu^2 \chi_{\{w<k_j\}}\,.
\]
Collecting estimates we arrive at
\[
\sup_{-\tau_r ^\mu<t<0} \mean{B_r } \bar w_j^2 \phi_j^2 \, dx + r ^2 \mean{Q_r ^\mu} |\nabla (\bar w_j \phi_j) |^2 \, dx \, dt \leq c 16^j \mu^2\left( \frac{|\widetilde Q^j \cap \{w<k_j\}|}{|\widetilde Q^j| } \right)
\]
Next, the parabolic Sobolev embedding (see~\cite[Proposition 3.1, p.7]{DBb}) gives us
\begin{eqnarray}
\notag &&
\mean{Q_r ^\mu} (\bar w_j \phi_j)^{2(1+2/n)} \, dx \, dt
\\ \notag && \qquad \leq c(n) \left( \sup_{-\tau_r ^\mu<t<0} \mean{B_r } \bar w_j^2 \phi_j^2 \, dx + r ^2 \mean{Q_r ^\mu} |\nabla (\bar w_j \phi_j) |^2 \, dx \, dt \right)^{1+2/n}\,.
\mathbb R^{nN}thbb Nd{eqnarray}
Since
\[
(\bar w_j \phi_j)^{2(1+2/n)}
\geq 4^{-6j} \mu^{2(1+2/n)} \chi_{\widetilde Q^{j+1} \cap \{w<k_{j+1}\}}\,,
\]
we get
\[
E_{j+1} \leq \bar c(a,n) 4^{8j} E_{j}^{1+2/n}
\qquad \mbox{with} \quad
E_j := \frac{|\widetilde Q^j \cap \{w<k_j\}|}{|\widetilde Q^j| } \,.
\]
A standard iteration lemma on fast geometric convergence of series (\cite[p.12]{DBb})) shows that if
$E_0 \leq \bar c^{-n/2} 4^{-2 n^2}$ then $E_j$ tends to zero as $j \to \infty$. Indeed, choosing $\delta := \bar c^{-n/2} 4^{-2 n^2} $, it follows from~\eqref{eq:nondeg} that $w \geq \mu/4$ in $B_{r /2} \times (-9 \tau_r ^\mu/16,- \tau_r ^\mu/2)$.
\\
\noindent {\bf Step 3: Nondegenerate alternative 2.}
We test $\partial_tw =\textnormal{dist}ve(\Phi'(w)\nabla w)+F$ with $(1/w -4/\mu)_+ \xi^2$, where $\xi \in C_0^\infty(B_{r /2})$, $0 \leq \xi \leq 1$, $\xi \equiv 1$ in $ B_{r /4}$ and $|\nablaÊ\xi| \leq 8/r $. Note that the chosen test function vanishes on $B_{r /2} \times \{- \tau_r ^\mu/2\}$ by Step 2. Taking advantage of $-F\left(\frac{1}{w}-\frac{4}{\mu}\right)_+\xi^2\leq 0$, straightforward manipulations then lead to
\begin{eqnarray}
\notag &&
\sup_{-\tau_r ^\mu/2<t<0} \mean{B_{r /2}} \log \left( \frac{\mu/4}{w} \right)_+ \xi^2 \, dx
\\
\notag && \qquad + \frac{\tau_r ^\mu}{4} \mean{B_{r /2} \times (- \tau_r ^\mu/2,0)} \Phi'(w) \frac{|\nabla (w-\mu/4)_+|^2}{w^2 } \xi^2 \, dx \, dt
\\
\notag && \qquad \qquad \leq \tau_r ^\mu \mean{B_{r /2} \times (- \tau_r ^\mu/2,0)} \Phi'(w) |\nabla \xi|^2 \, dx \, dt + \mean{B_{r /2}} \xi^2 \, dx
\\
\notag && \qquad \qquad \leq \tau_r ^\mu \mean{B_{r /2} \times (- \tau_r ^\mu/2,0)} \frac{1}{a}\frac{\Phi(w)}{w} |\nabla \xi|^2 \, dx \, dt + \mean{B_{r /2}} \xi^2 \, dx
\\
\notag && \qquad \qquad \leq \tau_r ^\mu \frac{1}{a} \frac{\Phi(\mu)}{\mu} \mean{B_{r /2} \times (- \tau_r ^\mu/2,0)} |\nabla \xi|^2 \, dx \, dt + \mean{B_{r /2}} \xi^2 \, dx\, \leq \frac{65}{a},
\mathbb R^{nN}thbb Nd{eqnarray}
where we used successively \eqref{eq:structural_diffusion_hyp}, the monotonicity of $\Phi(s)/s$ with $w\leq \mu$, the definition $\tau_r ^\mu \frac{\Phi(\mu)}{\mu}=r^2$, and the cutoff function properties $|\nabla\xi|\leq 8/r$, $\xi^2\leq 1$. As a consequence, we readily obtain
\[
\sup_{-\tau_r ^\mu/2<t<0} | B_{r /4} \cap \{ w(\cdot,t) < 4^{-1-m}\mu\} | \leq \frac1m \frac{2^n 65}{a \log 4} | B_{r /4} |\,,
\]
and in particular
\[
\frac{|B_{r /4} \times (- \tau_r ^\mu/2,0) \cap \{w< 4^{-1-m}\mu\}|}{ |B_{r /4} \times (- \tau_r ^\mu/2,0)| } \leq \frac1m \frac{2^n 65}{a \log 4} \, ,
\]
for any $m \in \mathbb R^{nN}thbb N$.
Next, redefine $k_j := 4^{-m-1} (2^{-1} + 2^{-1-j}) \mu$ and $r _j = (2^{-3}+2^{-3-j})r $, and set
\[
\hat Q^j := \hat B^j \times (- \tau_r ^\mu/2,0)\,, \qquad \hat B^j := B_{r _j}(0)\,.
\]
Choose $\xi_j \in C_0^\infty(\hat B^j)$ in such a way that $0\leq \xi_j \leq 1$, $\xi_j = 1$ in $\hat B^{j+1} $ and $|\nabla \xi_j| \leq 2^{4+j}/r $.
The Caccioppoli estimate then takes the form
\[
\sup_{-\tau_r ^\mu/2<t<0} \mean{\hat B^j} \frac{w_j^2 \xi_j^2}{\tau_r ^\mu} \, dx + \mean{\hat Q^j } \Phi'(w) |\nabla(w_j \xi_j)|^2\, dx \, dt
\leq c \mean{\hat Q^j } \Phi'(w) w_j^2 |\nabla \xi_j|^2\, dx \, dt \,,
\]
because $\xi_j$ is independent of time and the newly defined $w_j$ vanishes on the initial boundary of $\hat Q^j$ by Step 2.
Since $s \mathbb R^{nN}psto \Phi(s)/s$ is a nondecreasing function and $1/\tau_r ^\mu = \Phi(\mu)/\mu$, it follows that
\[
\frac{\Phi(k_j)}{k_j} \sup_{-\tau_r ^\mu/2<t<0} \mean{\hat B^j} w_j^2 \xi_j^2 \, dx + \mean{\hat Q^j } \Phi'(w) |\nabla(w_j \xi_j)|^2\, dx \, dt \leq \hat c(a) \frac{4^j}{r ^2} \Phi(k_j) k_j \hat E_j\,,
\]
where this time $\hat E_j := | \hat Q^j \cap \{w_j>0\}|/ |\hat Q^j |$. Analogously to Step 2, we then arrive at $\hat E_{j+1} \leq c(n,a) 4^{8j} E_j^{1+2/n}$, and by choosing $m \equiv m(n,a)$
large enough, i.e.
\[
\hat E_0 \leq \frac1m \frac{2^n 65}{a \log 4} \leq \hat c^{-n/2} 4^{-2n^2}\,,
\]
we conclude that $w\geq 4^{-m-2} \mu$ in $Q_{r /8}^\mu$.
\\
\noindent {\bf Step 4: Degenerate alternative.}
Let us then analyze the occurrence of~\eqref{eq:deg}. For this, set $v = \mu/2 - (w-\mu/2)_+ + \|F\|_{L^{\infty}(Q^{\mu}_r )}(t+\tau^{\mu}_r )$, which is a nonnegative weak supersolution to
$\partial_t v - [\Phi(\mu)/\mu] \textnormal{div} (b(x,t) \nabla v) \geq 0 $ in $Q_r ^\mu$ with $b(x,t) := \mu \Phi'(w(x,t)) / \Phi(\mu)$. By definition of $v$ and $\mu=\sup w$ we have in the support of $\nabla v$
\[
\nabla v(x,t) \neq 0 \quad \mathbb R^{nN}thbb{R}ightarrow \quad \mu/2 \leq w\leq \mu \quad \mathbb R^{nN}thbb{R}ightarrow \quad c(a)^{-1} \leq b(x,t) \leq c(a)\,.
\]
Redefining $b$ to be one on $\{\nabla v(x,t) = 0\}$ and scaling $b$ and $v$ as $\bar b(x,t) = b(r x, \tau_r ^\mu t)$ and $\bar v(x,t) = v(r x, \tau_r ^\mu t)$, we see that $\bar v$ is a weak supersolution to the equation $\partial \bar v - \textnormal{div}( \bar b \nabla \bar v) = 0$ in $B_1 \times (-1,0)$ with measurable coefficient $\bar b$ bounded uniformly from below and from above by positive constants depending only on $a$. By the weak Harnack principle we then obtain
\[
\mean{B_{3/4} \times (-3/4,-1/2)} \bar v \, dx \leq c(a) \inf_{B_{1/2} \times (-1/4,0)} \bar v\,.
\]
Scaling back to $v$ and recalling its definition in terms of $w$, we finally get by \eqref{eq:deg}
$$
\sup\limits_{Q^\mu_{r /2}} \, w\leq \mu\left(1-\frac{\delta}{2c}\right) + \tau^{\mu}_r \|F\|_{L^{\infty}(Q^{\mu}_r )}\left(1-\frac{1}{8c}\right).
$$
\noindent {\bf Step 5: Conclusion from alternatives.}
Taking
\[\sigma =\sigma(a,n):= \min\left\{4^{-m-2},\frac{\delta}{4c}, \frac{1 - 16^{-a}}{2} \right\}\,\in(0,1),
\]
we deduce from the previous alternatives that
\[
\mbox{either} \qquad \inf_{Q_{r/8}^{\mu}} w \geq \sigma \mu \qquad \mbox{or} \qquad \sup_{Q_{r/2}^{\mu}} w \leq (1-2\sigma) \mu +
\tau^{\mu}_r |F|_{L^{\infty}(Q^{\mu}_r)}
\]
holds provided that $\sup_{Q_{r}^{\mu}} w \leq \mu$. Choose now any $R>0$ such that $B_R\times(-R^2,0)\subset Q_T$ (after translation), let $R_j = 8^{-j} R$,
\[
\mu_0 := 1 + \Phi^{-1} \left( \sigma^{-1} R^2 \|F\|_{L^{\infty}(B_R \times (-R^2,0 )} \right) + \sup_{B_R \times (-R^2,0 )}w\,,
\]
and then inductively
\[
\mu_{j+1} := (1-2\sigma) \mu_{j} + \frac{\mu_{j}}{\Phi(\mu_{j})} R_j^2 \|F\|_{L^{\infty}(B_R \times (-R^2,0 )}
\]
for $j \geq 0$. Clearly $\mu_j \geq (1-2 \sigma)^j \mu_0$. Using the algebraic growth \eqref{eq:algebraic_growth_Phi(s)/s} leads to
\begin{equation}
\begin{array}{l}
\textnormal{dist}splaystyle \frac{R_j^2 \|F\|_{L^{\infty}(B_R \times (-R^2,0 )} }{\Phi(\mu_{j})} = \frac{\Phi(\mu_0)}{\Phi(\mu_{j})} R_j^2 \frac{\|F\|_{L^{\infty}(B_R \times (-R^2,0 )}}{\Phi(\mu_0)}
\\ \\ \textnormal{dist}splaystyle \qquad \qquad \leq \left( \frac{(1-2\sigma)^{-1/a}}{65} \right)^j \frac{R^2 \|F\|_{L^{\infty}(B_R \times (-R^2,0 )}}{\Phi(\mu_0)} \leq \sigma,
\mathbb R^{nN}thbb Nd{array}
\label{eq:sigma_bnd}
\mathbb R^{nN}thbb Nd{equation}
where the last inequality follows from the definition of $\sigma$ and from the bound
\[
\mu_0\geq \Phi^{-1} \left( \sigma^{-1} R^2 \|F\|_{L^{\infty}}\right) \quad \Longrightarrow \quad \Phi(\mu_0)\geq \sigma^{-1} R^2 \|F\|_{L^{\infty}}\, .
\]
Similarly, we get
\[
\frac{\mu_{j+1}}{\Phi(\mu_{j+1})} \left(\frac{R_j}{8}\right)^2 = \frac1{16} \frac{\mu_{j+1}}{\mu_{j}} \frac{\Phi(\mu_{j})}{\Phi(\mu_{j+1})} \frac{\mu_{j}}{\Phi(\mu_{j})} \left(\frac{R_j}{2}\right)^2 \leq \frac{\mu_{j}}{\Phi(\mu_{j})} \left(\frac{R_j}{2}\right)^2\,.
\]
Therefore, we conclude that
\[
(1-2\sigma) \mu_j \leq \mu_{j+1} \leq (1-\sigma) \mu_j \qquad \mbox{and} \qquad Q_{R_{j+1}}^{\mu_{j+1}} \subset Q_{R_{j}/2}^{\mu_{j}} \,,
\]
and alternatives reduce to
\[
\mbox{either} \qquad \inf_{Q_{R_{j+1}}^{\mu_j}} w \geq \sigma \mu_j \qquad \mbox{or} \qquad \sup_{Q_{R_{j+1}}^{\mu_{j+1}}} w \leq \mu_{j+1}\,,
\]
provided that $\sup_{Q_{R_{j}}^{\mu_{j}}} w \leq \mu_j$. Observe that $\sup_{Q_{R_{0}}^{\mu_{0}}} w \leq \mu_0$ by the definition of~$\mu_0$. Since we are considering positive solutions $w>0$ and $\mu_j \to 0$ as $j\to \infty$ the degenerate alternative can clearly occur at most a finite number of times, thus
\[
\sigma \mu_J \leq \inf_{Q_{R_J/8}^{\mu_J }} w \leq \sup_{Q_{R_J/8}^{\mu_J }} w \leq \mu_J\,\quad \mbox{for some finite } J.
\]
Since $\sigma=\sigma(a,n)$ only, the algebraic growth \eqref{eq:algebraic_growth_Phi(s)/s} then readily implies that
\[
\frac1{c(a,n)} \frac{\Phi(\mu_J)}{\mu_J} \leq \frac{\Phi(w)}{w} \leq c(a,n) \frac{\Phi(\mu_J)}{\mu_J} \qquad \mbox{in} \quad Q_{R_J/8}^{\mu_J }\,.
\]
Scaling as in step 4 and writing $\partial_t u_i=\Delta\varrho u_i+f_i=\textnormal{dist}ve(\varrho\nabla u_i)+(\ldots)$, where $\varrho=\Phi(w)/w$, we see that each scaled component $\overline{u}_i$ solves in $B_{1/8}\times(-1,0)$ a uniformly parabolic linear equation $\partial_t \overline{u}_i=\textnormal{dist}ve (\overline{\varrho}\nabla \overline{u}_i)+(\ldots)$ in divergence form with a measurable coefficient $\overline{\varrho}$ satisfying $c(a,n)\leq \overline{\varrho}\leq c(a,n)^{-1}$. In particular, $\overline{u}_i$ are H\"older continuous and satisfy a DeGiorgi-Nash-Moser oscillation estimate which, scaling back to $u_i$, takes the explicit form
\[
\osc_{Q_{\theta R_J}^{\mu_J }} u_i \leq c \theta^\beta \mu_J + c \frac{\mu_J}{\Phi(\mu_J)} (\theta R_J)^2 \|f_i\|_{L^{\infty}\left(B_R \times (-R^2,0 )\right)} \, ,
\]
for some $\beta=\beta(a,n)$, $c=c(a,n)$ only and for all $\theta \in (0,1)$. Observing that $0\leq f_i\leq |\mathbb R^{nN}thbf{f}|_1=F$ and recalling estimate \eqref{eq:sigma_bnd} which holds for all $j$ and with $\sigma=\sigma(a,n)$, we obtain
\[
\forall \,\theta\in(0,1): \qquad \osc_{Q_{\theta R_J}^{\mu_J }} u_i \leq c\, ( \theta^\beta + \theta^2) \mu_J.
\]
Setting
\[\alpha =\alpha(a,n):= \min\{\beta,- \log(1-\sigma)/\log 8,2\}\] and increasing the constant $c$ by a factor depending only on $n,a$, we conclude that
\[
\osc_{B_r \times (-r^2,0)} u_i \leq c \left( \frac{r}{R} \right)^\alpha \left( \frac{\Phi(\mu_0)}{\mu_0} \right)^{\alpha/2} \mu_0
\]
for all $r \in (0,R]$. This yields the desired interior H\"older continuity estimate after standard manipulations.
\mathbb R^{nN}thbb Nd{proof}
Assuming regularity and compatibility from the data, the solution can be shown to be H\"older continuous up to the boundary.
\begin{prop}
Let $\mathbb R^{nN}thbf{u}$, $M$ be as in Proposition~\ref{prop:exists_classical_solutions} and $\alpha=\alpha(a,n)$ as in Proposition~\ref{prop:C_alpha_estimate}. Assume further that the initial and boundary data are compatible and $\beta$-H\"older continuous with some $\beta\in(0,1)$, i.e. there is $\mathbb R^{nN}thbf{U}\in\mathbb R^{nN}thcal{C}^{\beta,\beta/2}(\overline{Q^T})$ such that $\mathbb R^{nN}thbf{u}^0=\mathbb R^{nN}thbf{U}(\,.\,,0)$ and $\mathbb R^{nN}thbf{z}^D=\frac{\Phi(|\mathbb R^{nN}thbf{U}|_1)}{|\mathbb R^{nN}thbf{U}|_1}\mathbb R^{nN}thbf{U}$ in $\Sigma_T$. Then $\mathbb R^{nN}thbf{u}\in\mathbb R^{nN}thcal{C}^{\gamma,\gamma/2}(\overline{Q^T})$, with $\gamma=\min(\alpha,\beta)$. Moreover, the H\"older norm of $\mathbb R^{nN}thbf{u}$ depends only on $a,n,N,M,T$ and on the $\beta$-H\"older norm of the data.
\label{prop:boundary_regularity}
\mathbb R^{nN}thbb Nd{prop}
\begin{proof}
Our assumptions on the data turn into similar compatibility and regularity conditions for the scalar problem \eqref{eq:PB_w}. A straightforward modification of our interior argument (see, e.g., \cite{DBb,Va07}) shows that $w$ is $\mathbb R^{nN}thcal{C}^{\gamma,\gamma/2}$ up to the boundary, which in turn yields the same regularity for $u_i$ through the linear parabolic equation $\partial_t u_i=\Delta(\varrho u_i)+f_i=\textnormal{dist}ve(\varrho \nabla u_i)+\textnormal{dist}ve(u_i\nabla\varrho)+f_i$.
\mathbb R^{nN}thbb Nd{proof}
\section{Weak solutions}
\label{section:weak_sols}
Let us first introduce different notions of solutions.
\begin{definition}[weak solutions] \
\begin{enumerate}
\item[(i)]
A non-negative function $ w \in L^{\infty}(Q_T)$ is called bounded very weak solution of \eqref{eq:PB_w} if the equality
\begin{equation}
\qquad \int\limits_{Q_T} \{w\partial_t \varphi + \Phi(w) \Delta \varphi + F \varphi \} \,\mathbb R^{nN}thrm{d}x\, \mathbb R^{nN}thrm{d}t
= - \int\limits_{\Omega} w^0(x) \varphi (x,0) \,\mathbb R^{nN}thrm{d}x + \int\limits_{\Sigma_T} g^D\frac{\partial \varphi }{\partial \nu} \,\mathbb R^{nN}thrm{d}x\, \mathbb R^{nN}thrm{d}t
\label{eq:weak_form_w}
\mathbb R^{nN}thbb Nd{equation}
holds for all $\varphi \in\mathbb R^{nN}thcal{C}^{2,1}(\overline{Q_T})$ vanishing on $\Sigma_T$ and in $\Omega\times\{t=T\}$.
\label{def:very_weak_solution_w}
\item[(ii)]
A non-negative function $w \in L^{\infty}(Q_T)$ is called a bounded weak energy solution of \eqref{eq:PB_w} if $\Phi(w)\in L^2(0,T;H^1(\Omega))$, the trace $\gamma(\Phi(w))=g^D$ in $L^{2}(0,T;H^{1/2}(\partial\Omega))$ and the equality
\begin{equation}
\qquad \int\limits_{Q_T} \{w\partial_t \varphi - \nabla \Phi(w) \cdot \nabla \varphi + F \varphi \}\,\mathbb R^{nN}thrm{d}x\mathbb R^{nN}thrm{d}t
= - \int\limits_{\Omega} w^0(x) \varphi (x,0) \,\mathbb R^{nN}thrm{d}x
\label{eq:energy_weak_form_w}
\mathbb R^{nN}thbb Nd{equation}
holds for all $\varphi \in\mathbb R^{nN}thcal{C}^{2,1}(\overline{Q_T})$ vanishing on $\Sigma_T$ and in $\Omega\times\{t=T\}$.
\label{def:weak_energy_solution_w}
\item[(iii)]
A function $\mathbb R^{nN}thbf{u}=(u_1,\ldots,u_N)\in L^{\infty}(Q_T)$ is called a (non-negative) bounded very weak solution of \eqref{eq:PB_u} if $u_i\geq 0$ a.e. in $Q_T$ and the equality
\begin{equation}
\int\limits_{Q_T} \{ u_i \partial_t \varphi + \varrho u_i \Delta \varphi + f_i \varphi \}\,\mathbb R^{nN}thrm{d}x\mathbb R^{nN}thrm{d}t\,
+ \int\limits_{\Omega} u_i^0(x) \varphi (x,0)\,\mathbb R^{nN}thrm{d}x = \int\limits_{\Sigma_T} z^D_i\frac{\partial \varphi }{\partial \nu}\,\mathbb R^{nN}thrm{d}x\mathbb R^{nN}thrm{d}t,
\label{eq:weak_form_u}
\mathbb R^{nN}thbb Nd{equation}
where $\varrho =\frac{\Phi(|\mathbb R^{nN}thbf{u}|_1)}{|\mathbb R^{nN}thbf{u}|_1}$, holds for any $ i=1,\ldots,N$ and for all $\varphi \in\mathbb R^{nN}thcal{C}^{2,1}(\overline{Q_T})$ vanishing on $\Sigma_T$ and in $\Omega\times\{t=T\}$.
\label{def:very_weak_solution_u}
\item[(iv)]
A function $\mathbb R^{nN}thbf{u} \in L^{\infty}(Q_T)$ is called a (non-negative) bounded weak energy solution of \eqref{eq:PB_u} if $u_i\geq 0$ a.e. in $Q_T$, $(\varrho u_i)\in L^2(0,T;H^1(\Omega))$, the trace $\gamma(\varrho u_i)=z^D_i$ in $L^{2}(0,T;H^{1/2}(\partial\Omega))$, and the equality
\begin{equation}
\int\limits_{Q_T} \{w\partial_t \varphi - \nabla (\varrho u_i) \cdot \nabla \varphi + f_i \varphi \}\,\mathbb R^{nN}thrm{d}x\mathbb R^{nN}thrm{d}t
= - \int\limits_{\Omega} z_i^0(x) \varphi (x,0)\,\mathbb R^{nN}thrm{d}x,
\label{eq:energy_weak_form_u}
\mathbb R^{nN}thbb Nd{equation}
where $\varrho=\frac{\Phi\left(|\mathbb R^{nN}thbf{u}|_1\right)}{|\mathbb R^{nN}thbf{u}|_1}$, holds for any $ i=1,\ldots,N$ and for all $\varphi \in\mathbb R^{nN}thcal{C}^{2,1}(\overline{Q_T})$ vanishing on $\Sigma_T$ and in $\Omega\times\{t=T\}$.
\label{def:weak_energy_solution_u}
\mathbb R^{nN}thbb Nd{enumerate}
\label{def:weak_solutions}
\mathbb R^{nN}thbb Nd{definition}
The notion of very weak solutions preserves the diagonal structure of the system as shown in the following lemma.
\begin{lemma}
If $\mathbb R^{nN}thbf{u}$ is a non-negative bounded very weak (resp. energy) solution of system \eqref{eq:PB_u} in the sense of Definition~\ref{def:weak_solutions} then $w=|\mathbb R^{nN}thbf{u}|_1$ is a non-negative bounded very weak (resp. energy) solution to problem \eqref{eq:PB_w} in the sense of Definition~\ref{def:weak_solutions}.
\label{lem:u_solution=>w_solution}
\mathbb R^{nN}thbb Nd{lemma}
\begin{proof}
Sum equalities \eqref{eq:weak_form_u} over $i$ from $1$ to $N$ and observe that by definition $\sum_i \varrho u_i=\varrho \sum_i u_i=\varrho |\mathbb R^{nN}thbf{u}|_1=\Phi(w)$, $\sum_i f_i=F$, $\sum_i u^0_i=|\mathbb R^{nN}thbf{u}^0|_1=w^0$, and $\sum_i z^D_i=|\mathbb R^{nN}thbf{z}^D|_1=g^D$.
\mathbb R^{nN}thbb Nd{proof}
We will now address uniqueness. Note that the following proposition guarantees uniqueness also within the class of energy solutions since weak energy solutions are in particular very weak solutions.
\begin{prop}[Uniqueness]
Given the non-negative and bounded data $\mathbb R^{nN}thbf{f},\mathbb R^{nN}thbf{u}^0,\mathbb R^{nN}thbf{z}^D$ there exists at most one non-negative bounded very weak solution to problem~\eqref{eq:PB_u} in the sense of Definition \ref{def:weak_solutions}.
\label{prop:uniqueness_weak_sols}
\mathbb R^{nN}thbb Nd{prop}
\begin{proof}
Let $\mathbb R^{nN}thbf{u}^1$ and $\mathbb R^{nN}thbf{u}^2$ be two solutions to problem~\eqref{eq:PB_u}, corresponding to the same initial and boundary data. It follows from the previous lemma that $w^1=|\mathbb R^{nN}thbf{u}^1|_1$ and $w^2=|\mathbb R^{nN}thbf{u}^2|_1$ are both bounded very weak solutions to the same Cauchy-Dirichlet problem \eqref{eq:PB_w}. A standard comparison result for such solutions \cite[Theorem 6.5]{Va07} provides uniqueness within this class. Thus $w^1=w^2=w$ and, in particular, the pressures coincide, $\varrho ^1=\varrho ^2=\varrho =\frac{\Phi(w)}{w}$.
Next, we use a duality proof, as in proving the comparison results for GMPE, to show that $\mathbb R^{nN}thbf{u}^1=\mathbb R^{nN}thbf{u}^2$. In fact, the situation here is simpler because we already know that $\varrho ^1=\varrho ^2$. For the sake of completeness, we nonetheless give the details.
\par
Fixing any $i\in 1\ldots N$, denoting $\tilde{u}=u^1_i-u^2_i$, and subtracting the weak formulation \eqref{eq:weak_form_u} satisfied by $u^2$ from that satisfied by $u^1$, we see that
\begin{equation}
\int\limits_{Q_T} \{\tilde{u} \partial_t \varphi + \varrho \tilde{u} \Delta \varphi\}\,\,\mathbb R^{nN}thrm{d}x\mathbb R^{nN}thrm{d}t =0
\label{eq:weak_u1-u2}
\mathbb R^{nN}thbb Nd{equation}
for all $\varphi \in \mathbb R^{nN}thcal{C}^{2,1}(\overline{Q_T})$ vanishing on $\Sigma_T\cup\{t=T\}$. Fix some arbitrary $\theta\in \mathbb R^{nN}thcal{C}^{\infty}_c(Q_T)$, choose $\varepsilon>0$, and let $\varrho _{\varepsilon}=\mathbb R^{nN}x\{\varrho , \varepsilon\}$. Since $\mathbb R^{nN}thbf{u}^1$ and $\mathbb R^{nN}thbf{u}^2$ are bounded so is $\varrho =\varrho ^1=\varrho ^2$, and we can construct a smooth approximation $\{\varrho _{\varepsilon,k}\}_{k\in \mathbb R^{nN}thbb{N}}$ to $\varrho _{\varepsilon}$ such that $\varepsilon\leq \varrho _{\varepsilon,k}\leq C$. For fixed $\varepsilon,k$ we can then solve the approximate dual backward equation
\begin{equation}
\left\{
\begin{array}{ll}
\partial_t \varphi +\varrho _{\varepsilon,k} \Delta \varphi = \theta \qquad & \text{in }Q_T\\
\varphi =0 & \text{in }\Sigma_T\\
\varphi (.,T) =0 & \text{in }\Omega.
\mathbb R^{nN}thbb Nd{array}
\right.
\label{eq:dual}
\mathbb R^{nN}thbb Nd{equation}
for a unique $\varphi =\varphi_{\varepsilon,k}\in \mathbb R^{nN}thcal{C}^{2,1}(\overline{Q_T})\cap \mathbb R^{nN}thcal{C^{\infty}}(Q_T)$. Since $\varphi $ vanishes by construction on $\Sigma_T$ and in $\Omega\times\{t=T\}$ it is admissible as a test function in \eqref{eq:weak_u1-u2}. This gives
\begin{align*}
\left|\int_{Q_T}\tilde{u}\theta \,\mathbb R^{nN}thrm{d}x\mathbb R^{nN}thrm{d}t \right| &= \left|\int_{Q_T} \tilde{u} (\varrho -\varrho _{\varepsilon,k}) \Delta\varphi \,\mathbb R^{nN}thrm{d}x\mathbb R^{nN}thrm{d}t\right|\\
&\leq \left(\, \int \limits_{Q_T} \tilde{u}^2\frac{|\varrho - \varrho _{\varepsilon,k}|^2}{\varrho _{\varepsilon,k}} \,\mathbb R^{nN}thrm{d}x\mathbb R^{nN}thrm{d}t \right)^{1/2}
\left(\, \int\limits_{Q_T}\varrho _{\varepsilon,k} |\Delta \varphi |^2 \,\mathbb R^{nN}thrm{d}x\mathbb R^{nN}thrm{d}t \right)^{1/2}\\
& \leq \frac{C}{\varepsilon^{1/2}}\|\varrho -\varrho _{\varepsilon,k}\|_{L^2(Q_T)} \left(\, \int\limits_{Q_T}\varrho _{\varepsilon,k} |\Delta \varphi |^2 \,\mathbb R^{nN}thrm{d}x\mathbb R^{nN}thrm{d}t \right)^{1/2},
\mathbb R^{nN}thbb Nd{align*}
because $\tilde{u}\in L^{\infty}$ and $\varrho _{\varepsilon,k}\geq \varepsilon$. Since $\varphi $ is smooth, a straightforward computation shows that (cf. \cite[Theorem 6.5]{Va07})
$$
\left(\int\limits_{Q_T}\varrho _{\varepsilon,k} |\Delta \varphi |^2\,\mathbb R^{nN}thrm{d}x\mathbb R^{nN}thrm{d}t\right)^{1/2}\leq C \|\nabla \theta\|_{L^2(Q_T)}
$$
for some $C>0$ independent of $\varepsilon,k,\theta$. For fixed $\varepsilon>0$ we can then choose $k$ large enough such that $|\varrho _{\varepsilon}-\varrho _{\varepsilon,k}|_{L^2(Q_T)}\leq \varepsilon$. By definition of the cutoff function $\varrho _{\varepsilon}$, we have $0\leq \varrho_{\varepsilon}-\varrho\leq\varepsilon$. Hence $|\varrho -\varrho _{\varepsilon}|_{L^2(Q_T)}\leq \varepsilon|Q_T|^{1/2}$ so that $|\varrho -\varrho _{\varepsilon,k}|_{L^2(Q_T)}\leq |\varrho -\varrho _{\varepsilon}|_{L^2(Q_T)}+|\varrho _{\varepsilon}-\varrho _{\varepsilon,k}|_{L^2(Q_T)}\leq C\varepsilon$, and we obtain
$$
\left|\int_{Q_T}\tilde{u}\theta\,\mathbb R^{nN}thrm{d}x\mathbb R^{nN}thrm{d}t\right|\leq C\varepsilon^{1/2}\|\nabla \theta\|_{L^2(Q_T)}.
$$
Because $\theta\in \mathbb R^{nN}thcal{C}^{\infty}_c(Q_T)$ was arbitrary and $\varepsilon$ was independent of $\theta$ we conclude letting $\varepsilon\to 0$ that $\tilde{u}=u^1_i-u^2_i=0$ a.e. in $ Q_T$ and the proof is complete.
\mathbb R^{nN}thbb Nd{proof}
\begin{remark}
The above uniqueness proof does not really require $L^{\infty}(Q_T)$ bounds but merely that $\mathbb R^{nN}thbf{u},\varrho \mathbb R^{nN}thbf{u}\in L^{2}_{loc}(Q_T)$. In fact, scalar parabolic equations such as \eqref{eq:PB_w} benefit usually from smoothing properties that should allow one to extend the theory to $L^{1}$ data. Due to the coupled vectorial nature of the problem and the lack of space we shall not pursue this direction here.
\mathbb R^{nN}thbb Nd{remark}
Theorem~\ref{theo:exist_sols_Dirichlet_Cauchy_PB} allows for the initial data $\mathbb R^{nN}thbf{u}^0$ to vanish identically in some ball $B_r(x_0)\subset \Omega$. As we will see in Section~\ref{section:FB}, this leads to free boundaries in the degenerate case $\Phi'(0)=0$. It will also become clear in the proof that the structural condition \eqref{eq:convexity_hyp} needs to be enforced only to get the estimate \eqref{eq:energy_very_weak_ui}. In fact, this energy estimate plays no role whatsoever in the analysis so one may actually dispense with it and limit oneself to very weak solutions of \eqref{eq:PB_u}.
\begin{proof}[Proof of Theorem~\ref{theo:exist_sols_Dirichlet_Cauchy_PB}]
Uniqueness follows from Proposition~\ref{prop:uniqueness_weak_sols}. The existence argument is based on a ``lifting'' technique, classical for scalar GPME and working here thanks to the diagonal structure of the system.
\par
We first lift and approximate the bounded non-negative data $\mathbb R^{nN}thbf{u}^0,\mathbb R^{nN}thbf{f},\mathbb R^{nN}thbf{z}^D$ component-wise by smooth functions $u_i^{0,k},f_i^k$ and $z_i^{D,k}$ such that $\frac{1}{k}\leq u_i^{0,k},f_i^k,z_i^{D,k}\leq C+\frac{1}{k}$
for some constant $C>0$ depending only on the data, and
$$
\|\mathbb R^{nN}thbf{u}^{0,k}-\mathbb R^{nN}thbf{u}^0\|_{L^{1}(\Omega)} + \|\mathbb R^{nN}thbf{f}^k-\mathbb R^{nN}thbf{f}\|_{L^1(Q_T)} + \|\mathbb R^{nN}thbf{z}^{D,k}-\mathbb R^{nN}thbf{z}^{D}\|_{L^1(\Sigma_T)} \to 0
$$
as $k\to\infty$. By Proposition~\ref{prop:exists_classical_solutions}, given the smooth data $\mathbb R^{nN}thbf{u}^{0,k}, \mathbb R^{nN}thbf{f}^k, \mathbb R^{nN}thbf{z}^{D,k}$ there exists a positive classical solution $\mathbb R^{nN}thbf{u}^k$ to \eqref{eq:PB_u} which is bounded in $Q_T$ uniformly in $k$. By virtue of Proposition~\ref{prop:C_alpha_estimate} $\{\mathbb R^{nN}thbf{u}^k\}_{k}$ is also bounded in $\mathbb R^{nN}thcal{C}^{\alpha,\alpha/2}(Q')$ for any subdomain $Q'$ and for some $\alpha=\alpha(a,n)\in(0,1)$. By diagonal extraction we may then assume that $\mathbb R^{nN}thbf{u}^k\to \mathbb R^{nN}thbf{u}$ in $\mathbb R^{nN}thcal{C}_{\operatorname{loc}}(Q_T)$ with the limit function $\mathbb R^{nN}thbf{u}$ satisfying the local $\mathbb R^{nN}thcal{C}^{\alpha,\alpha/2}$-estimate \eqref{eq:uniform_Holder}. In particular $\mathbb R^{nN}thbf{u}^k(x,t)\to \mathbb R^{nN}thbf{u}(x,t)\geq 0$ pointwise in $Q_T$. From the continuity of $\Phi$ with $\lim\limits_{s\to 0}\frac{\Phi(s)}{s}=\Phi'(0)$ it then follows that $\varrho^k=\frac{\Phi(|\mathbb R^{nN}thbf{u}^k|_1)}{|\mathbb R^{nN}thbf{u}^k|_1)}\to \frac{\Phi(|\mathbb R^{nN}thbf{u}|_1)}{|\mathbb R^{nN}thbf{u}|_1)}=\varrho$ a.e. in $Q_T$.
Since $\mathbb R^{nN}thbf{u}^k,\varrho^k$ are bounded uniformly in $L^{\infty}(Q_T)$
we conclude by dominated convergence that $u^k_i\to u_i$ and $\varrho^k u^k_i\to \varrho u_i$ in $L^p(Q_T)$ for all $p\in [1,\infty)$. Given that $\mathbb R^{nN}thbf{u}^k$ is a smooth positive solution and that for all $i=1,\ldots, N$ it holds
$$
\int\limits_{Q_T}\{ u^k_i \partial_t \varphi + \varrho^k u^k_i \Delta \varphi + f^k_i \varphi \}\,\mathbb R^{nN}thrm{d}x\mathbb R^{nN}thrm{d}t
+ \int\limits_{\Omega} u_i^{0,k}(x) \varphi (x,0)\mathbb R^{nN}thrm{d}x = \int\limits_{\Sigma_T} z^{D,k}_i\frac{\partial \varphi }{\partial \nu}\mathbb R^{nN}thrm{d}x\mathbb R^{nN}thrm{d}t.
$$
The previous strong $L^p(Q_T)$ convergence and convergence of the data allow one to send $k\to\infty$ to obtain \eqref{eq:weak_form_u}. Similarly by Lemma~\ref{lem:u_solution=>w_solution} we see that $w=\lim w^k=\lim |\mathbb R^{nN}thbf{u}^k|_1=|\mathbb R^{nN}thbf{u}|_1$ is a very weak solution to \eqref{eq:PB_w}.
Regarding the energy estimates, if the data satisfy \eqref{hyp:boundary_data} and \eqref{hyp:initial+forcing}, then they can be approximated as before by smooth positive data satisfying in addition
\begin{eqnarray} \label{eq:}
\notag
&& \|\mathbb R^{nN}thbf{u}^{0,k}\|_{L^{\infty}(Q_T)} + \|\mathbb R^{nN}thbf{f}^k\|_{L^{\infty}(Q_T)}
\\ \notag
&& \qquad + \|\mathbb R^{nN}thbf{z}^{D,k}\|_{L^{\infty}(Q_T)} + \|\mathbb R^{nN}thbf{z}^{D,k}\|_{L^2(0,T;H^1(\Omega))} + \|\partial_t \mathbb R^{nN}thbf{z}^{D,k}\|_{L^{\infty}(Q_T)}\leq C.
\mathbb R^{nN}thbb Nd{eqnarray}
By Proposition~\ref{prop:energy_estimate} we get
$$
\|\nabla (\varrho^k w^k)\|_{L^2(Q_T)}\leq C
$$
and
$$
\|\nabla (\varrho^k u_i^k)\|_{L^2(Q'_T)}\leq C(1+1/d') \qquad \forall i=1\ldots N
$$
uniformly in $k$ for some $C=C(a,n,N,T,M)$ only. Since $\varrho^k u_i^k,\varrho^k w^k\to \varrho u_i,\varrho w$ we conclude that $\nabla(\varrho u_i),\nabla(\varrho w)$ satisfy the same $L^2$ bounds and the proof is complete.
\mathbb R^{nN}thbb Nd{proof}
\section{Free boundaries}
\label{section:FB}
In this section we set $Q=\mathbb R^{nN}thbb{R}^n\times (0,\infty)$, $Q^{\tau,T}=\mathbb R^{nN}thbb{R}^n\times(\tau,T)$, $Q^T=\mathbb R^{nN}thbb{R}^n\times(0,T)$, and consider the Cauchy Problem
\begin{equation}
\left\{
\begin{array}{ll}
\partial_t \mathbb R^{nN}thbf{u}=\Delta\left(\frac{\Phi(|\mathbb R^{nN}thbf{u}|_1)}{|\mathbb R^{nN}thbf{u}|_1}\mathbb R^{nN}thbf{u}\right) & \mbox{in } \, \mathbb R^{nN}thbb{R}^n\times(0,\infty)\\
\mathbb R^{nN}thbf{u}(x,0) = \mathbb R^{nN}thbf{u}^0 (x) & \mbox{in }\mathbb R^{nN}thbb{R}^n
\mathbb R^{nN}thbb Nd{array}
\right.
\label{eq:Cauchy_PB_u}
\mathbb R^{nN}thbb Nd{equation}
with a non-negative, bounded and compactly supported initial data $\mathbb R^{nN}thbf{u}^0 $. If the modulus of ellipticity $\varrho=\frac{\Phi(|\mathbb R^{nN}thbf{u}|_1)}{|\mathbb R^{nN}thbf{u}|_1}$ in \eqref{eq:Cauchy_PB_u} vanishes when $|\mathbb R^{nN}thbf{u}|_1=0$ (the degenerate case), compactly supported solutions should evolve from compactly supported initial data. By analogy with scalar equations, the free boundary $\Gamma(t):=\partial \operatorname{supp}\mathbb R^{nN}thbf{u}(\,.\,,t)$ should then propagate with finite speed in the sense that, at any $x_0\notin \operatorname{supp}\,\mathbb R^{nN}thbf{u}(\,.\,,t_0)$ we should have $x_0\notin \operatorname{supp}\,\mathbb R^{nN}thbf{u}(\,.\,,t_0+h)$ for small enough $h>0$. Although this behaviour is well understood for scalar equations, the coupled nature of system \eqref{eq:Cauchy_PB_u} prevents us from just recalling known results. Instead, we will again resort to our central idea, based on the particular structure of the system, that
controlling $w=|\mathbb R^{nN}thbf{u}|_1$ controls each individual species $u_i$. Indeed, $0\leq u_i\leq |\mathbb R^{nN}thbf{u}|_1=w$, thus $\operatorname{supp}\,u_i\subset \operatorname{supp}\,w$ and $\mathbb R^{nN}thbf{u}$ will propagate with finite speed as long as $w$ does. This will in turn be ensured by looking at the scalar problem
\begin{equation}
\left\{
\begin{array}{ll}
\partial_t w=\Delta\Phi(w) & \mbox{in } \, \mathbb R^{nN}thbb{R}^n\times(0,\infty)\\
w(x,0) = w^0 (x) & \mbox{in }\mathbb R^{nN}thbb{R}^n
\mathbb R^{nN}thbb Nd{array}
\right.
\label{eq:Cauchy_PB_w}
\mathbb R^{nN}thbb Nd{equation}
with a non-negative, bounded and compactly supported initial data $w^0=|\mathbb R^{nN}thbf{u}^0|_1$.
Given that assumption \eqref{eq:structural_diffusion_hyp} does not rule out nondegenerate diffusion (we may have $\Phi'(0)>0$) for which the finite speed of propagation obviously fails, we need to impose an extra degeneracy condition. For the scalar Cauchy problem \eqref{eq:Cauchy_PB_w} this is normally done through replacing the structural condition \eqref{eq:structural_diffusion_hyp} by the \emph{slow diffusion} hypothesis
\begin{equation}
s>0:\qquad 1+a \leq \frac{s\Phi'(s)}{\Phi(s)} \leq \frac{1}{a}.
\label{eq:slow_diffusion_Sa}
\tag{$S_a$}
\mathbb R^{nN}thbb Nd{equation}
One readily sees that condition \eqref{eq:slow_diffusion_Sa} implies $\Phi'(0)=0$ and the algebraic behaviour
$ 0 \leq \Phi(1)s^{\frac{1}{a}} \leq \Phi(s) \leq \Phi(1)s^{1+a}$ for small $s$, which usually provides information on the speed of propagation in terms of $a$. However, since our analysis should include the Freundlich isotherm $\mathbb R^{nN}thbf{b}_f(\mathbb R^{nN}thbf{z})=(\phi+(1-\phi)|\mathbb R^{nN}thbf{z}|_1^{p-1})\mathbb R^{nN}thbf{z}$ for which the corresponding $\Phi_f(s)$ behaves linearly at infinity, we cannot assume \eqref{eq:slow_diffusion_Sa} globally in $s>0$ as the lower bound $1<cst\leq \frac{s\Phi'(s)}{\Phi(s)}$ is not admissible for large $s$. In fact, given that the degeneracy is essentially a local feature at the level sets $\{s\approx 0\}$ we could require condition \eqref{eq:slow_diffusion_Sa} to hold only for $0<s\leq s_0$ which is certainly true for our Freundlich isotherm with $1+a=1/p>1$. However, we prefer to avoid this technical path and, instead, impose the less restrictive degeneracy condition
\begin{equation}
\forall s>0:\qquad \int_0^s\Phi(s')ds'\geq c\, \Phi(s)^{\frac{m+1}{m}} \qquad\mbox{for some }m>1.
\tag{$S_m$}
\label{eq:slow_diffusion_Phi_m}
\mathbb R^{nN}thbb Nd{equation}
This implies that $\Phi'(0)=0$ and is valid for the pure PME nonlinearity $\Phi(s)=s^m$ with $c=m+1$ if $m>1$.
For technical reasons it will be convenient to reformulate
\eqref{eq:slow_diffusion_Phi_m} in terms of the original concentration $\mathbb R^{nN}thbf{z}=\mathbb R^{nN}thbf{b}^{-1}(\mathbb R^{nN}thbf{u})$. It is easy to see that the change of variables $r=\Phi(s)=\beta^{-1}(s)$ turns \eqref{eq:slow_diffusion_Phi_m} into the equivalent condition
\begin{equation}
\forall r>0:\qquad f(r):=r\beta(r)-\int_0^r\beta(r')dr'\geq cr^{\frac{m+1}{m}}\qquad\mbox{for some }m>1,
\tag{$S'_m$}
\label{eq:slow_diffusion_beta_m}
\mathbb R^{nN}thbb Nd{equation}
from which it follows that $\beta'(0)=\infty$, as expected since $\beta=\Phi^{-1}$. An explicit computation shows that \eqref{eq:slow_diffusion_beta_m} holds true globally in $r>0$ for the Freundlich isotherm $\beta_f(r)=\phi r+(1-\phi)r^p$ with $c=(1-\phi)^{-1/p}$ and $m=\frac{1}{p}>1$, showing that \eqref{eq:slow_diffusion_Phi_m}, equivalently \eqref{eq:slow_diffusion_beta_m}, is indeed weaker than \eqref{eq:slow_diffusion_Sa}.
\par
In the case of pure PME nonlinearity $\Phi(s)=s^m$ the Cauchy problem \eqref{eq:Cauchy_PB_w} has been widely studied and the qualitative and quantitative theory of free boundaries is now well understood, see e.g. \cite{CVW87,CW90} and references therein. Partial results \cite{dPV91,DR03} also hold for general nonlinearities $\Phi(s)$ but to the best of our knowledge one always assumes the degeneracy condition in the form \eqref{eq:slow_diffusion_Sa}, which fails for the Freundlich isotherm.
We will start our analysis with a standard statement and prove it assuming only the weaker condition \eqref{eq:slow_diffusion_Phi_m}.
\begin{prop}
Assume that conditions \eqref{eq:monotonicity_b}-\eqref{eq:structural_diffusion_hyp} and \eqref{eq:slow_diffusion_Phi_m} hold for some $a\in (0,1)$ and $m>1$, and let the initial datum $0\leq w^0(x)\leq M$ be compactly supported in $ B_{R_0}$ for some $R_0>0$. Then the Cauchy problem \eqref{eq:Cauchy_PB_w} admits a unique weak energy solution $0\leq w(x,t)\leq M$ and $\|\nabla \Phi(w)\|_{L^2(Q)}\leq C(a,R_0,M,n)$. Moreover, $w(\,.\,,t)$ is compactly supported for all $t>0$, the free boundary $\Gamma(t)=\partial \operatorname{supp}w(\,.\,,t)$ propagates with finite speed, and
\begin{equation}
\forall t\geq 0:\qquad \operatorname{supp}\, w(\,.\,,t)\subseteq B_{R(t)}\quad \mbox{with }R(t):=R_0+C_1t^\lambda \, ,
\label{eq:estimate_FB_propagation}
\mathbb R^{nN}thbb Nd{equation}
with some constants $C_1(m,a,M,R_0)>0$ and $\lambda=\lambda(m,n)>0$.
\label{prop:FB_Cauchy_PB_w_Rn}
\mathbb R^{nN}thbb Nd{prop}
\begin{proof}
Existence and uniqueness are proven in \cite{Va07}. Since $0\leq w^0\leq M$, the comparison principle gives $0\leq w\leq M$ in $Q$. The $L^2(Q)$-bound for $\nabla\Phi(u)$ easily follows from letting $t\to\infty$ in the classical energy identity
\begin{equation}
\int_{B_R}\Psi(w(t,x))\,\mathbb R^{nN}thrm{d}x+\int\limits_0^t\int\limits_{B_R}|\nabla \Phi(w(x,\tau))|^2\,\mathbb R^{nN}thrm{d}x\,\mathbb R^{nN}thrm{d}\tau=\int_{B_R}\Psi(w^0(x))\,\mathbb R^{nN}thrm{d}x\,
\label{eq:energy_identity_w}
\mathbb R^{nN}thbb Nd{equation}
where $\Psi(s):=\int_0^s\Phi(s')\,\mathbb R^{nN}thrm{d}s'$ (see \cite{Va07} for details). Indeed with our assumptions $w^0$ is bounded and compactly supported hence $\|\Psi(w^0)\|_{L^1(\mathbb R^{nN}thbb{R}^n)}\leq \Psi(M)\operatorname{meas}(B_{R_0})= C(a,M,R_0,n)$.\par
As for the speed of propagation of the support, we go back to the original concentration formulation and recall from \cite{DV85} two results based on the energy methods introduced in \cite{A81}. To verify that the assumptions in \cite{DV85} are satisfied,
we set $z:= \Phi(w)=\beta^{-1}(w)$ and observe that $\partial_t\beta(z)=\Delta z$ in $Q$. Making the change of variables $r=\Phi(s)$ in \eqref{eq:slow_diffusion_beta_m} shows that $f(z(x,t))=\Psi(w(x,t))$, and the energy identity \eqref{eq:energy_identity_w} readily gives
$$
E(z;t):=\sup\limits_{0\leq \tau\leq t} \int_{\mathbb R^{nN}thbb{R}^n}f(z(x,\tau))\,\mathbb R^{nN}thrm{d}x + \int\limits_0^t\int\limits_{\mathbb R^{nN}thbb{R}^n}|\nabla z(x,\tau)|^2\,\mathbb R^{nN}thrm{d}x\,\mathbb R^{nN}thrm{d}\tau \leq C(a,M,R_0,n).
$$
Defining $j(r):=\int_0^r\beta(r')\,\mathbb R^{nN}thrm{d}r'$ and making the change $r=\Phi(s)$, we obtain for all $0\leq w_1,w_2\leq M$ the bound
\begin{align*}
\left|j(\Phi(w_1))-j(\Phi(w_2))\right| =\left|\int_{\Phi(w_2)}^{\Phi(w_1)} \beta(r)\,\mathbb R^{nN}thrm{d}r\right| &=\left|\int_{w_1}^{w_2}s\Phi'(s)\,\mathbb R^{nN}thrm{d}s\right|\\
& \leq \left|\int_{w_1}^{w_2}\frac{1}{a}\Phi(s)\,\mathbb R^{nN}thrm{d}s\right| \leq \frac{\Phi(M)}{a}|w_1-w_2|.
\mathbb R^{nN}thbb Nd{align*}
Since $0\leq w(x,t)\leq M$ and $z=\Phi(w)\leq \Phi(M)\leq C(a,M)$ we have in particular
$$
\forall t_1,t_2\in [0,T]:\qquad \|j(z(t_1))-j(z(t_2))\|_{L^1(\mathbb R^{nN}thbb{R}^n)}
\leq C\|w(t_1)-w(t_2)\|_{L^1(\mathbb R^{nN}thbb{R}^n)}.
$$
Given our assumptions on $w^0$, we can infer from the classical theory \cite{Va07} for the scalar Cauchy problem \eqref{eq:Cauchy_PB_w} that $w\in \mathbb R^{nN}thcal{C}([0,T];L^1(\mathbb R^{nN}thbb{R}^n))$. This in turn yields $j(z)\in \mathbb R^{nN}thcal{C}(0,T;L^1(\mathbb R^{nN}thbb{R}^n))$.\par
The energy estimate $E(z;t)\leq C$ and the continuity of $j(z)$ allow us to apply \cite[Corollary 3.1]{DV85} and we conclude that $z(\,.\,,t)$ is compactly supported, satisfies \eqref{eq:estimate_FB_propagation}. Similarly, applying \cite[Theorem 3.1]{DV85} shows that the free boundary $\Gamma(t)=\partial \operatorname{supp}w(\,.\,,t)$ propagates with finite speed and the proof is complete.
\mathbb R^{nN}thbb Nd{proof}
We can now establish the corresponding result on the multicomponent Cauchy problem \eqref{eq:Cauchy_PB_u}.
\begin{theorem}[Free Boundary solutions]
Let conditions \eqref{eq:monotonicity_b}-\eqref{eq:structural_diffusion_hyp}-\eqref{eq:convexity_hyp} and \eqref{eq:slow_diffusion_beta_m} hold for some $a\in(0,1)$ and $m>1$. Assume that $\mathbb R^{nN}thbf{u}^0\in L^\infty(\mathbb R^{nN}thbb{R}^n)$ is componentwise non-negative with $w^0=|\mathbb R^{nN}thbf{u}^0|_1\leq M$, and such that $\operatorname{supp}w^0\subseteq B_{R_0}$ for some $R_0>0$. Then there exists a unique non-negative very weak solution $\mathbb R^{nN}thbf{u}\in L^{\infty}(Q)$ to \eqref{eq:Cauchy_PB_u}. Moreover,
\begin{enumerate}
\item[(i)]
$w=|\mathbb R^{nN}thbf{u}|_1$ is the unique weak energy solution to \eqref{eq:Cauchy_PB_w}, $0\leq w\leq M$, and
$$
\|\nabla\Phi(w)\|_{L^2(Q)}\leq C(a,M,R_0)
$$
\item[(ii)]
$\mathbb R^{nN}thbf{u}$ is a local energy solution to \eqref{eq:Cauchy_PB_u} in the sense that for all $T>0$ we have
$$
\|\nabla(\varrho u_i)\|_{L^2(Q^T)}\leq C(a,M,R_0,T)\qquad \forall i=1,\ldots N \, ,
$$
where $\varrho=\frac{\Phi(w)}{w}$.
\item[(iii)]
$\operatorname{supp}\,w(\,.\,,t)$ propagates with finite speed, and
$$
\forall\, t\geq 0,\,i=1\ldots N: \qquad \operatorname{supp}u_i(\,.\,,t)\subseteq \operatorname{supp} w(\,.\,,t)\subseteq B_{R(t)}
$$
with $R(t)=R_0 + C_1t^\lambda$ for some $C_1(m,a,M,R_0)>0$ and $\lambda=\lambda(m,n)>0$ only.
\item[(iv)]
There is $\alpha=\alpha(a,n)\in(0,1)$ such that $\mathbb R^{nN}thbf{u}$ is $(\alpha,\alpha/2)$-H\"older continuous in any strip $Q^{\tau,T}=\mathbb R^{nN}thbb{R}^n\times(\tau,T)$, $0<\tau<T$, and
$$
\|\mathbb R^{nN}thbf{u}\|_{C^{\alpha,\alpha/2}(Q^{\tau,T})}\leq C(1+1/\sqrt{\tau}).
$$
for some $C(a,T,n,N,M)>0$ only.
\item[(v)]
If $\Phi$ is smooth in $\mathbb R^{nN}thbb{R}^+$ then $u_i$ is smooth in $\{u_i>0\}\cap\{t>0\}$.
\mathbb R^{nN}thbb Nd{enumerate}
\label{theo:free_boundaries}
\mathbb R^{nN}thbb Nd{theorem}
\begin{remark} As in Theorem~\ref{theo:exist_sols_Dirichlet_Cauchy_PB}, the structural condition \eqref{eq:convexity_hyp} is only needed to get (ii) and can be relaxed by restricting \eqref{eq:Cauchy_PB_u} to very weak solutions instead of energy solutions. Moreover, as in Proposition~\ref{prop:boundary_regularity}, the H\"older regularity estimate (iv) can be extended up to $t=0^+$ if we assume further $\mathbb R^{nN}thcal{C}^{\beta}(\mathbb R^{nN}thbb{R}^n)$ regularity from $\mathbb R^{nN}thbf{u}^0$. Note in particular that in (iii) we only claim that $w$ has finite speed of propagation but not that the individual species propagate with finite speed. In fact the support of each species should in general be discontinuous in time, see the discussion at the end of this section.
\mathbb R^{nN}thbb Nd{remark}
\begin{proof}
Arguing exactly as in the proof of Proposition~\ref{prop:uniqueness_weak_sols} but choosing now test function $ \theta\in \mathbb R^{nN}thcal{C}^{\infty}_c(\mathbb R^{nN}thbb{R}^n\times(0,\infty))$ it is easy to show uniqueness within the class of very weak solutions. It is therefore enough to prove existence in finite time intervals $[0,T]$ for any fixed $T>0$.
Given that the free boundary should a priori propagate with finite speed, solutions to the Cauchy problem in the whole space should agree with solutions of the Cauchy-Dirichlet problem in $B_R\times(0, T)$ with zero boundary conditions, as long as $R>0$ is large enough so that the free boundary stays at a positive distance from $\partial B_R$ for all $t\leq T$. We should therefore be able to construct solutions to the Cauchy problem in $\mathbb R^{nN}thbb{R}^n\times(0, T)$ by considering auxiliary Dirichlet problems in large balls and using Theorem~\ref{theo:exist_sols_Dirichlet_Cauchy_PB}.\\
From Proposition~\ref{prop:FB_Cauchy_PB_w_Rn} we can define the unique solution $\overline{w}$ of the Cauchy problem \eqref{eq:Cauchy_PB_w} in $\mathbb R^{nN}thbb{R}^n\times(0,\infty)$ with initial data $w^0:=|\mathbb R^{nN}thbf{u}^0|_1$. For any fixed $T>0$ choose $R>0$ large enough so that $R(T)\leq R/2$ and
$$
\forall t\leq T:\qquad \operatorname{supp}\,\overline{w}(\,.\,,t)\subseteq B_{R(T)}\subseteq B_{R/2},
$$
where $R(T)=R_0+C_1T^\lambda$ and the constants $C_1$ and $\lambda$ are as in Proposition~\ref{prop:FB_Cauchy_PB_w_Rn}. Next, let $\mathbb R^{nN}thbf{u}=\mathbb R^{nN}thbf{u}_R$ be the unique solution to problem \eqref{eq:PB_u} in $B_R\times(0,T)$ corresponding to the initial data $\mathbb R^{nN}thbf{u}^0$ and zero boundary values on $\partial B_R$ and given by Theorem~\ref{theo:exist_sols_Dirichlet_Cauchy_PB}. It also follows from Theorem~\ref{theo:exist_sols_Dirichlet_Cauchy_PB} that $w:=|\mathbb R^{nN}thbf{u}|_1$ is a weak solution to the corresponding Cauchy-Dirichlet problem in $B_R\times(0,T)$ with zero boundary values. Given the definition of $R$ we thus see that $\overline{w}$ remains at a distance $R/2$ away from $\partial B_R$. It is easy to see that the restriction $\overline{w}_{|B_R\times(0,T)}$ is also a weak solution to the same Cauchy-Dirichlet problem as $w$. By standard uniqueness theorem for weak solutions of \eqref{eq:PB_w} we conclude that $w=\overline{w}$ in $B_R\times(0,T)$. In particular
$$
\forall\, t\leq T,\,i=1\ldots N: \qquad \operatorname{supp}\,u_i(\,.\,,t)\subseteq\operatorname{supp}\,w(\,.\,,t) = \operatorname{supp}\overline{w}(\,.\,,t)\subseteq B_{R(t)}
$$
where $R(t)=R_0+C_1t^\lambda$ and the distance between the support of $\mathbb R^{nN}thbf{u}$ and $\partial B_R$ is at least $R/2>0$ for all $t\leq T$. Extending $\mathbb R^{nN}thbf{u}$ and $w$ outside $B_{R_0}$ by zero for all $t\in(0,T)$ it is then a simple exercise to verify that these extensions satisfy the weak formulations of \eqref{eq:Cauchy_PB_u} and \eqref{eq:Cauchy_PB_w} in the whole space, whence existence of free-boundary solutions in $\mathbb R^{nN}thbb{R}^n\times(0,T)$ for arbitrary $T>0$. The energy estimate (i) and propagation properties (iii) immediately follow from the definition of $\overline{w}$ and Proposition~\ref{prop:FB_Cauchy_PB_w_Rn}.\\
For fixed $T>0$ take now $R>0$ large enough so that $\mathbb R^{nN}thbf{u}$ stays supported in $B_R$ for all $t\leq T$. Viewing $\mathbb R^{nN}thbf{u}$ as the unique solution to the Cauchy-Dirichlet problem in $B_{R+1}$ and taking $\Omega=B_{R+1}$, $\Omega'=B_R$ in Theorem~\ref{theo:exist_sols_Dirichlet_Cauchy_PB} we have $d'=1$ in \eqref{eq:energy_very_weak_ui}, thus
$$
\|\nabla(\varrho u_i)\|_{L^2(\mathbb R^{nN}thbb{R}^n\times(0,T))}=\|\nabla(\varrho u_i)\|_{L^2(B_R\times(0,T))}\leq C(a,n,N,M,T)
$$
as claimed in (ii).
Assertion (iv) is proven similarly by considering the Cauchy-Dirichlet problem in $B_{R+1}$, choosing $\Omega^\prime=B_{R}\subset B_{R+1}=\Omega$ in Theorem~\ref{theo:exist_sols_Dirichlet_Cauchy_PB} and taking $d'=1$ in estimate \eqref{eq:uniform_Holder}.
To prove (v) we use a local bootstrap argument. If $w>0$ in some $B_r(x_0)\times(t_0-\tau,t_0+\tau)$, $t_0>0$, then in particular the pressure $p=\Phi'(w)>0$ there. Since $w$ is H\"older continuous and $\Phi$ is smooth also $p$ is H\"older continuous. Moreover, $w$ solves a uniformly parabolic equation in divergence form: $\partial_t w=\Delta\Phi(w)=\textnormal{dist}ve(p\nabla w)$. Hence $w\in C^{1+\beta}$ for some $\beta$.
By bootstrapping we immediately see that $w$ is locally smooth. Consider now any species $u_i$ and observe that if $u_i>0$ in $B_r(x_0)\times(t_0-\tau,t_0+\tau)$ then $w=|\mathbb R^{nN}thbf{u}|_1\geq u_i>0$ and also $\varrho=\Phi(w)/w>0$. Since $u_i$ solves $\partial_t u_i=\Delta(\varrho u_i)=\textnormal{dist}ve(\varrho\nabla u_i)+\textnormal{dist}ve(u_i\nabla\varrho)$ with now smooth coefficients we conclude that $u_i$ is smooth and the proof is complete.
\mathbb R^{nN}thbb Nd{proof}
Although degenerate, problem \eqref{eq:Cauchy_PB_w} is nonetheless diffusive in nature and we expect that the information cannot propagate backwards as confirmed by the following result.
\begin{prop}[Persistence property]
Under the hypotheses of Theorem~\ref{theo:free_boundaries}, let us further assume that for any $M >0$ there exists $\overline{a}=\overline{a}(M )>0$ such that
$$
\forall s\in(0,M ]:\qquad 1+\overline{a}\leq \frac{s\Phi'(s)}{\Phi(s)}.
$$
Then the support $\Omega(t):=\{x:\,w(x,t)>0\}$ of $w=|\mathbb R^{nN}thbf{u}|_1$ is non-contracting in time.
\mathbb R^{nN}thbb Nd{prop}
\begin{remark}
This (weaker) degeneracy condition is easily checked for the Freundlich isotherm. In fact, $s\Phi'(s)/\Phi(s)=\beta(r)/(r\beta'(r))$ is strictly greater than one for $\beta_f(r)=\phi r+(1-\phi)r^p$ in any finite interval (but not in the limit $s\to\infty$ because $\Phi_f(s)$ becomes linear).
\mathbb R^{nN}thbb Nd{remark}
\begin{proof}
Dahlberg and Kenig \cite{DK86} proved that nonnegative solutions to \eqref{eq:Cauchy_PB_w} satisfy certain Harnack inequality provided the strong condition \eqref{eq:slow_diffusion_Sa} holds. In this case positivity of $w$ at $(x_0,t_0)$ implies positivity at $(x_0,t)$ for all later $t\geq t_0$ and the support is non-contracting. Unfortunately, as already discussed, \eqref{eq:slow_diffusion_Sa} does not hold globally in $s>0$.\\
In order to tackle this technical detail we recall from Theorem~\ref{theo:free_boundaries} that $0\leq w\leq M$, with $M=\|w^0\|_{L^{\infty}(\mathbb R^{nN}thbb{R}^n)}$.
Now the assumption on $\Phi(s)$ allows one to construct a $\mathbb R^{nN}thcal{C}^1$-function $\overline{\Phi}$ which satisfies \eqref{eq:slow_diffusion_Sa} globally in $s>0$ for some $a=a(M)\in(0,1)$ and such that $\overline{\Phi}(s)=\Phi(s)$ for all $s\in [0,M]$. By construction $w$ is a solution to $\partial_t w=\Delta \overline{\Phi}(w)$, and the assertion follows from \cite{DK86} (for a precise statement see e.g. \cite[Corollary 1.5]{dPV91}).
\mathbb R^{nN}thbb Nd{proof}
We end this section with a ``divide and rule'' result.
\begin{prop}
Assume that \eqref{eq:monotonicity_b}, \eqref{eq:structural_diffusion_hyp}, \eqref{eq:convexity_hyp} and \eqref{eq:slow_diffusion_beta_m} hold true for some $a\in(0,1)$ and $m>1$. Let $k\in \{1,\ldots,N\}$, $\hat{\mathbb R^{nN}thbf{u}}^0=(u_1^0,\ldots,u_k^0)\in L^{\infty}(\mathbb R^{nN}thbb{R}^n;\mathbb R^{nN}thbb{R}^k)$ and $\check{\mathbb R^{nN}thbf{u}}^0=(u_{k+1}^0,\ldots,u_N^0)\in L^{\infty}(\mathbb R^{nN}thbb{R}^n;\mathbb R^{nN}thbb{R}^{N-k})$ be non-negative and compactly supported, with $d=\textnormal{dist}st(\operatorname{supp}(\hat{\mathbb R^{nN}thbf{u}}^0),\operatorname{supp}(\check{\mathbb R^{nN}thbf{u}}^0))>0$. Set $\mathbb R^{nN}thbf{u}^0=(\hat{\mathbb R^{nN}thbf{u}}^0,\check{\mathbb R^{nN}thbf{u}}^0)\in L^{\infty}(\mathbb R^{nN}thbb{R}^n;\mathbb R^{nN}thbb{R}^{k+(N-k)})$. Moreover, let $\hat{\mathbb R^{nN}thbf{u}}(x,t)$ be the unique solution of the $k$-dimensional Cauchy problem \eqref{eq:Cauchy_PB_u} with the initial data $\hat{\mathbb R^{nN}thbf{u}}^0$, $\check{\mathbb R^{nN}thbf{u}}(x,t)$ the unique solution of the $(N-k)$-dimensional Cauchy problem \eqref{eq:Cauchy_PB_u} with the initial data $\check{\mathbb R^{nN}thbf{u}}^0$ and assume finally that $\mathbb R^{nN}thbf{u}(x,t)$ is the unique solution of the $N$-dimensional Cauchy
problem \eqref{eq:Cauchy_PB_u} with the
initial data $\mathbb R^{nN}thbf{u}^0=(\hat{\mathbb R^{nN}thbf{u}}^0,\check{\mathbb R^{nN}thbf{u}}^0)$. Then there exists $T>0$ such that $\mathbb R^{nN}thbf{u}\equiv (\hat{\mathbb R^{nN}thbf{u}},\check{\mathbb R^{nN}thbf{u}})$ in $Q^T=\mathbb R^{nN}thbb{R}^n\times(0,T)$. More precisely, if $\hat{w}$ and $\check{w}$ are the two solutions of the scalar Cauchy problem \eqref{eq:Cauchy_PB_w} with the respective initial data $\hat{w}^0=|\hat{\mathbb R^{nN}thbf{u}}^0|_{l^1(\mathbb R^{nN}thbb{R}^k)}$ and $\check{w}^0=|\check{\mathbb R^{nN}thbf{u}}^0|_{l^1(\mathbb R^{nN}thbb{R}^{N-k})}$, then
$$
T=\inf\{t\geq 0:\ \operatorname{supp} \hat{w}(\,.\,,t)\cap \operatorname{supp} \check{w}(\,.\,,t) \neq \emptyset\}\in(0,\infty].
$$
\label{prop:divide_rule}
\mathbb R^{nN}thbb Nd{prop}
\begin{remark}
As is clear from the above definition, $T$ is the first time when the supports of $\hat{w},\check{w}$ touch. Our statement can be reformulated simply as: if the initial data can be separated into two distinct patches of $k$ and $N-k$ species then it suffices to solve two independent lower dimensional systems of order $k$ and $N-k$ as long as their respective supports do not touch. Note that we do not claim anything for what happens after $t=T$ because when the supports touch the two patches start interacting and the situation becomes more involved.
\mathbb R^{nN}thbb Nd{remark}
\begin{proof}
From Theorem~\ref{theo:free_boundaries} it is clear that the supports of $\hat{\mathbb R^{nN}thbf{u}}(\,.\,,t),\check{\mathbb R^{nN}thbf{u}}(\,.\,,t)$ propagate with finite speed, which implies that $T>0$. Letting now $\tilde{\mathbb R^{nN}thbf{u}}=(\hat{\mathbb R^{nN}thbf{u}},\check{\mathbb R^{nN}thbf{u}})\in L^{\infty}(\mathbb R^{nN}thbb{R}^n\times(0,T);\mathbb R^{nN}thbb{R}^{k+(N-k)})$, we observe that by definition of $T$
$$
(x,t)\in \operatorname{supp}(\hat{\mathbb R^{nN}thbf{u}})\cap\{t\leq T\} \quad \mathbb R^{nN}thbb{R}ightarrow \quad
\left\{
\begin{array}{ll}
\tilde{\mathbb R^{nN}thbf{u}}=(\hat{\mathbb R^{nN}thbf{u}},0)\\
|\tilde{\mathbb R^{nN}thbf{u}}|_{l^1(\mathbb R^{nN}thbb{R}^N)}=|(\hat{\mathbb R^{nN}thbf{u}},0)|_{l^1(\mathbb R^{nN}thbb{R}^{k+(N-k)})}=\left|\hat{\mathbb R^{nN}thbf{u}}\right|_{l^1(\mathbb R^{nN}thbb{R}^{k})}
\mathbb R^{nN}thbb Nd{array}
\right.
$$
and
$$
(x,t)\in \operatorname{supp}(\check{\mathbb R^{nN}thbf{u}})\cap\{t\leq T\} \quad \mathbb R^{nN}thbb{R}ightarrow \quad
\left\{
\begin{array}{ll}
\tilde{\mathbb R^{nN}thbf{u}}=(0,\check{\mathbb R^{nN}thbf{u}})\\
|\tilde{\mathbb R^{nN}thbf{u}}|_{l^1(\mathbb R^{nN}thbb{R}^N)}=|(0,\check{\mathbb R^{nN}thbf{u}})|_{l^1(\mathbb R^{nN}thbb{R}^{k+(N-k)})}=\left|\check{\mathbb R^{nN}thbf{u}}\right|_{l^1(\mathbb R^{nN}thbb{R}^{N-k})}
\mathbb R^{nN}thbb Nd{array}
\right. .
$$
It is then easy to check that $\tilde{\mathbb R^{nN}thbf{u}}=(\hat{\mathbb R^{nN}thbf{u}},\check{\mathbb R^{nN}thbf{u}})$ solves the global $N$-dimensional system in $Q^T$ (but a priori \emph{not} for later times). From uniqueness in Theorem~\ref{theo:exist_sols_Dirichlet_Cauchy_PB} (restricted to finite time intervals) we conclude that $\tilde{\mathbb R^{nN}thbf{u}}=\mathbb R^{nN}thbf{u}$ in $Q^T$.
\mathbb R^{nN}thbb Nd{proof}
One can refine Proposition~\ref{prop:divide_rule} by considering an arbitrary number $j$, say, of initial patches as
follows: 1) as long as the supports do not intersect, solve $j$ independent systems, 2) when two or more patches touch, glue them together into a single higher-dimensional patch and resume with $j'<j$ patches, 3) keep iterating as long as there is more than one patch left.
This ``divide and rule'' behaviour should lead to infinite speed of propagation (discontinuity in time) of the supports of each individual species, even though the support of $w=|\mathbb R^{nN}thbf{u}|_1$ does propagate with finite speed as stated in Theorem~\ref{theo:free_boundaries}. Consider for example the case of only two species $\mathbb R^{nN}thbf{u}=(u_1,u_2)$ with the initial (compact) supports at a positive distance from each other. We know from Proposition~\ref{prop:divide_rule} that we only have to solve two scalar problems as long as the supports do not intersect. Assume that this happens at $t=T$ and that at this particular moment the two supports look like two tangent balls (thus only one species is present in each ball). If at least one of the balls were expanding at time $t=T^-$, then for $t=T^+$ the balls should intersect and the support of $w$ would thus look like an $8$-shaped domain with a thin but non-empty interior neck connecting
the balls. Moreover, the diffusion coefficient
$\varrho =\frac{\Phi(w)}{w}$ becomes positive in the entire $8$-shaped domain, in particular in the neck. Since $\partial_t u_i=\textnormal{dist}ve(\varrho \nabla u_i)+(\ldots)$ with the support of $u_i$ in either of the two balls at time $t=T$, the diffusion occurring in the neck will ensure infinite speed of propagation between the two balls. Thus the support of $u_i$ should jump from only one ball at $t=T$ to the whole $8$-shaped domain at $t=T^+$. This instantaneous invasion phenomenon is beyond the scope of this paper and will be investigated elsewhere.
\begin{thebibliography}{99}
\bibitem{ADCC01} \name[Aizinger, V.] \& \name[Dawson, C.] \& \name[Cockburn, C.] \& \name[Castillo, P.] : The local discontinuous Galerkin method for contaminant transport,
\emph{Adv. Water Res.} 24 (2001), 73-87.
\bibitem {AL} \name[Alt, H.W.] \& \name[Luckhaus, S.]: Quasilinear elliptic-parabolic differential equations. {\em Math. Z.} 183 (1983), 311-341.
\bibitem {A81} \name[Antontsev, S.N. ]: On the localization of solutions of nonlinear degenerate elliptic and parabolic equations. {\em Soviet Math. Dokl. } 24 (1981), 420-424.
\bibitem{BK97a} \name[Barrett, J.W.]\ et \name[Knabner, P.]: Finite element approximation of transport of reactive solutes in porous media. Part 1. Error estimates for equilibrium adsorption processes, \emph{SIAM J. Numer. Anal.} 34 (1997), 201-227.
\bibitem{BK97b} \name[Barrett, J.W.] \& \name[Knabner, P.]: Finite element approximation of transport of reactive solutes in porous media. Part 2. Error estimates for equilibrium adsorption processes, \emph{SIAM J. Numer. Anal.} 34 (1997), 455-479.
\bibitem{BV87}
\name[Bear, J.] \& \name[Verruijt, A.]: \emph{Modeling Groundwater Flow and Pollution}, D. Reidel, Dordrecht, 1987.
\bibitem {DK86} \name[Dahlberg, B.E.J. ] \& \name[Kenig, K.]: Nonnegative solutions of generalized porous medium equations. {\em Rev. Mat. Iber.} 2 (1986), 267-305.
\bibitem {DK07} \name[Daskalopoulos, P.] \& \name[Kenig, C.E.]: \emph{Degenerate Diffusions: Initial Value Problems and Local Regularity Theory}, European Mathematical Society, Z\"urich, 2007.
\bibitem {DR03} \name[Daskalopoulos, P. ] \& \name[Rhee, E.]: Free-boundary regularity for generalized porous media equations. {\em Comm. Pure Appl. Anal. } 2 (2003), 481-494.
\bibitem{DvDG96}
\name[Dawson, C.] \& \name[van Duijn, C.J.] \& \name[Grundy, R.E.]: Large-time asymptotics in
contaminant transport in porous media, \emph{SIAM J. Appl. Math.}
56 (1996), 965-993.
\bibitem{DvDW94}
\name[Dawson, C.] \& \name[van Duijn, C.J.] \& \name[Wheeler, M.F.]: Characteristic-Galerkin
methods for contaminant transport with non-equilibrium adsorption kinetics, \emph{SIAM J. Numer. Anal.} 31 (1994), 982-999.
\bibitem {dPV91} \name[De Pablo, P. ] \& \name[V\'azquez, J.L.]: Regularity of solutions and interfaces of a generalized porous medium equation in $\mathbb R^{nN}thbb{R}^N$. {\em Ann. Mat. Pura Appl. } 158 (1991), 51-74.
\bibitem {DV85} \name[D{\'i}az, J.I. ] \& \name[V{\'e}ron, L.]: Local vanishing properties of solutions of elliptic and parabolic quasilinear equations. {\em Trans. Amer. Math. Soc.} 290 (1985), 787-814.
\bibitem {D} \name[DiBenedetto, E.]: On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, {\em Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV)} 13 (1986), 487-535.
\bibitem {DBb} \name[DiBenedetto, E.]: {\em Degenerate Parabolic Equations}, Springer-Verlag, New
York, 1993.
\bibitem {DBF} \name[DiBenedetto, E.] \& \name[Friedman, A.]: H\"older estimates for nonlinear degenerate parabolic systems, {\em J.~reine~ang.~Math. (Crelles J.)} 357 (1985), 1-22.
\bibitem {DGV1} \name[DiBenedetto, E.] \& \name[Gianazza, U.] \& \name[Vespri, V.]: Alternative forms of the Harnack inequality for non-negative solutions to certain degenerate and singular parabolic equations, {\em Rendiconti Lincei - Matematica e Applicazioni} 20 (2009), 369-377.
\bibitem {DGV2} \name[DiBenedetto, E.] \& \name[Gianazza, U.] \& \name[Vespri, V.]: Harnack Estimates for Quasi-Linear Degenerate Parabolic Differential Equation, {\em Acta Math.} 200 (2008), 181--209.
\bibitem {CVW87} \name[Caffarelli, L.A. ] \& \name[V{\'a}zquez, J.L.] \& \name[Wolanski, N.]: Lipschitz continuity of solutions and interfaces of the N-dimensional porous medium equation. {\em Indiana Univ. Math. J.} 36 (1987), 376-401.
\bibitem {CW90} \name[Caffarelli, L.A. ] \& \name[Wolanski, N.]: $\mathbb R^{nN}thcal{C}^{1,\alpha}$ regularity of the free boundary for the n-dimensional porous media equation. {\em Comm. Pure Appl. Math.} 43 (1990), 885-902.
\bibitem {G} \name[Giusti, E.]: {\em Direct Methods in the Calculus of Variations}, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
\bibitem{GF93} \name[Gutierrez, M.] \& \name[Fuentes, H.R.]: Modeling adsorption in multicomponent
systems using a Freundlich-type isotherm, \emph{J. Contam. Hydrol.} 14 (1993),
247-260.
\bibitem {H} \name[Hungerb\"uhler, N.]: Quasi-linear parabolic systems in divergence form with weak monotonicity, {\em Duke Math. J.} 107(3) (2001), 497--520.
\bibitem{KO2000}
\name[Knabner, P.] \& \name[Otto, F.]: Solute transport in porous media with equilibrium and nonequilibrium multiple-site adsorption: uniqueness of weak solutions, \emph{Nonlinear Anal.} 42 (2000), 381-403.
\bibitem {LSU} \name[Ladyzhenskaia, O.A.] \& \name[Solonnikov, V.A.] \& \name[Ural'ceva, N.N.]: \emph{Linear and Quasi-linear Equations of Parabolic Type}, American Mathematical Society, 1988.
\bibitem {S83} \name[Sacks, P.E.]: Continuity of solutions of a singular parabolic equation. {\em Nonlinear Anal.} 7 (1983), 387--409.
\bibitem{SRS81}
\name[Sheindorf, C.] \& \name[Rebhum, M.] \& \name[Sheintuch, M.]: A
Freundlich type multicomponent isotherm, \emph{J. Colloid.
Interface Sci.} 79 (1981), 136-142.
\bibitem{SMM06} \name[Srivastava, V.C. ] \& \name[Mall, I.D.] \& \name[Mishra, I.M.]: Equilibrium modelling of single and binary adsorption of cadmium and nickel onto bagasse fly ash, \emph{Chem. Engng J.} 46 (2006), 79-91.
\bibitem{vDK91}
\name[van Duijn, C.J.] \& \name[Knabner, P.]: Solute transport in porous media with equilibrium and non-equilibrium multiple-site adsorption: Travelling waves, \emph{ J. reine angew. Math.} 415 (1991), 1-49.
\bibitem {Va07} \name[V\'azquez, J.L.]: \emph{The Porous Medium Equation: Mathematical Theory}, Oxford University Press, New York, 2007.
\bibitem{WMK91}
\name[Weber, J.W.J.] \& \name[McGinley, P.M.] \& \name[Katz, L.E.]: Sorption phenomena in subsurface systems: concepts, models and effects on contaminant fate and transport, \emph{Water Res.}, 25 (1991), 499-528.
\bibitem{WKLL02} \name[Wu, C.-H.] \& \name[Kuo, C.-Y.] \& \name[Lin, C.-F.] \& \name[ Lo, S.-L.]: Modeling competitive adsorption of molybdate, sulfate, selenate, and selenite using a Freundlich-type
multi-component isotherm, \emph{Chemosphere} 47 (2002), 283-292.
\mathbb R^{nN}thbb Nd{thebibliography}
\mathbb R^{nN}thbb Nd{document} | math | 105,441 |
\begin{document}
\title[Compactness issues and bubbling phenomena]
{Compactness issues and bubbling phenomena for the prescribed Gaussian curvature equation on the torus}
\author{Luca Galimberti}
\address[Luca Galimberti]{Departement Mathematik\\ETH-Z\"urich\\mathbbm CH-8092 Z\"urich}
\mathrm email{[email protected]}
\thanks{Supported by SNF grant 200021\_140467 / 1. }
\,\mathrm date{\today}
\begin{abstract}
In the spirit of the previous paper \cite{Borer-Galimberti-Struwe}, where we dealt with the case of a
closed Riemann surface $(M,g_0)$ of genus greater than one, here we study the behaviour of the conformal metrics
$g_\lambda$ of prescribed Gauss curvature $K_{g_\lambda} = f_0 + \lambda$ on the torus, when the parameter $\lambda$ tends to one of the boundary points of the interval of existence of $g_\lambda$, and we characterize their ``bubbling behavior'' as in \cite{Borer-Galimberti-Struwe}.
\mathrm end{abstract}
\maketitle
\section{Introduction}
Consider a closed, connected Riemann surface $M$, whose Euler characteristic $\chi(M)$ is zero,
endowed with a smooth background metric $g_0$. In view of the uniformization theorem, it is possible to
assume that the Gauss curvature $K_{g_0}$ of $g_0$ vanishes identically.
The prescribed Gauss curvature equation, which links the curvature of $g_0$ to the curvature $K_g$ of a conformal metric $g=e^{2u}g_0$, then reads as
\[
K_g = -e^{-2u} \Delta_{g_0}u
\]
Moreover, for convenience, we normalize the volume of $(M,g_0)$ to unity.
Consider a smooth non-constant function $f_0:M\to \mathbbm R$ with $\max_{p\mathrm in M}f_0(p)=0$, all of whose maximum points are non-degenerate, and define for $\lambda \mathrm in \mathbbm R$
\[
f_\lambda := f_0+\lambda
\]
A natural question is to understand for which values $\lambda$ the function $f_\lambda$ is the Gauss curvature of a metric conformal to $g_0$. That is equivalent to ask for which values of $\lambda$, the equation
\begin{equation}\label{eqn: the Gauss curvature}
-\Delta_{g_0}u = f_\lambda e^{2u}
\mathrm end{equation}
admits a solution. The paper \cite{Kazdan-Warner74} completely answers the question, giving
necessary and sufficient conditions for solving the equation above. More precisely, equation (\ref{eqn: the Gauss curvature}) has a solution if and only if
\[
\mathrm int_M f_\lambda d\mu_{g_0} =\mathrm int_M f_0 d\mu_{g_0} + \lambda< 0
\]
and $f_\lambda$ is sign changing. (Recall that the volume is equal to one.)
Thus, by taking account of the assumptions made on $f_0$, we find
that equation (\ref{eqn: the Gauss curvature}) is solvable if and only if
\[
0= -\max_M f_0 < \lambda < -\overline{f_0},
\]
where $\overline{f_0} := \mathrm int_M f_0 d\mu_{g_0}$. Set
\[
\Lambda :=(0,-\overline{f_0}), \;\;\; -\overline{f_0}:= \lambda_{max} .
\]
Our goal in this paper is to study the behaviour of the set of solutions of (\ref{eqn: the Gauss curvature}) when $\lambda$ approches either 0 or $\lambda_{max}$, a problem left open in \cite{Kazdan-Warner74} and which
we solve by means of a blow-up analysis in the spirit of \cite{Borer-Galimberti-Struwe}. Our main results are the following:
\begin{theorem}\label{thm: first main result}
Let $f_0\leq 0$ be a smooth, non-constant function, all of whose maximum points $p_0$
are non-degenerate with $f_0(p_0)=0$, and for $\lambda\mathrm in\mathbbm R$ let $f_{\lambda}=f_0+\lambda$.
Then there exists a sequence $\lambda_n\,\mathrm downarrow 0$, a sequence
$u_n$ of solutions of the equation
\[
-\Delta_{g_0}u_n = f_{\lambda_n} e^{2u_n}
\]
and there exists $I\mathrm in\mathbbm N$ such that, for suitable $p^{(i)}_n\to p^{(i)}_{\mathrm infty}\mathrm in M$ with $f_0(p^{(i)}_{\mathrm infty})=0$, $1\le i \le I$, we obtain $u_n(p_n^{(i)})\to +\mathrm infty$ and one of the following:
\textbf{i)} $u_n\to -\mathrm infty$ locally uniformly on compact domains of
$M_{\mathrm infty}:=M\setminus\{p^{(i)}_{\mathrm infty}; \; 1\leq i \leq I \}$
\textbf{ii)} For suitable $r_n^{(i)}\,\mathrm downarrow 0$, the following holds:
a) We have smooth convergence $u_n\to u_{\mathrm infty}$ locally on
$M_{\mathrm infty}$ and $u_{\mathrm infty}$ induces a complete metric
$g_{\mathrm infty}=e^{2u_{\mathrm infty}}g_0$ on $M_{\mathrm infty}$ of finite total curvature $K_{g_{\mathrm infty}}=f_0$.
b) For each $1\le i \le I$, either 1) there holds
$r_n^{(i)}/\sqrt{\lambda_n}\to 0$ and in local
conformal coordinates around $p^{(i)}_n$ we have
\[w_n(x):=u_n(r^{(i)}_nx)-u_n(0)+ \log 2\to w_{\mathrm infty}(x)
=\log\big(\frac2{1+|x|^2}\big)\]
smoothly locally in $\mathbbm R^2$, where $w_{\mathrm infty}$ induces a spherical metric
$g_{\mathrm infty}=e^{2w_{\mathrm infty}}g_{\mathbbm R^2}$ of curvature $K_{g_\mathrm infty}=1$ on $\mathbbm R^2$,
or 2) we have $r_n^{(i)}=\sqrt{\lambda_n}$,
and in local conformal coordinates around $p^{(i)}_{\mathrm infty}$ with a constant $c^{(i)}_{\mathrm infty}$
there holds
\[w_n(x)=u_n(r^{(i)}_nx)+\log(\lambda_n)+c^{(i)}_{\mathrm infty}\to w_{\mathrm infty}(x)\]
smoothly locally in $\mathbbm R^2$, where the metric
$g_{\mathrm infty}=e^{2w_{\mathrm infty}}g_{\mathbbm R^2}$ on $\mathbbm R^2$ has finite volume and finite total curvature
with $K_{g_\mathrm infty}(x)=1+(Ax,x)$, where $A=\frac12Hess_{f_0}(p^{(i)}_{\mathrm infty})$.
\mathrm end{theorem}
Moreover, we have
\begin{theorem}\label{thm: second main result}
Let $f_0\leq 0$ be a smooth, non-constant function. For $\lambda\mathrm in\mathbbm R$ set
\begin{equation}\label{eqn: l'insieme C lambda}
\mathcal{C_\lambda} := \left\{ u\mathrm in H^1(M;g_0) : \mathrm int_M u d\mu_{g_0} = 0
= \mathrm int_M f_\lambda e^{2u} d\mu_{g_0} \right\}.
\mathrm end{equation}
Then for any arbitrary sequence $(\lambda_n)_n \subset \Lambda$ such that
$\lambda_n\uparrow \lambda_{max}$ for $n\to +\mathrm infty$, we have that:
i)there exists a sequence of minimizers $w_n\mathrm in \mathcal{C}_{\lambda_n}$ of the Dirichlet energy such that:
\[
w_n \to 0 \;\; \mbox{in} \;\; C^{2,\alpha}(M)
\]
for any $\alpha\mathrm in \left[0,1 \right)$.
ii)there exists a sequence of solutions $u_n$ to equation
\[
-\Delta_{g_0}u_n = f_{\lambda_n} e^{2u_n}
\]
such that $u_n\to -\mathrm infty$ uniformly on the whole $M$.
\mathrm end{theorem}
\begin{observation}
We remark that in Theorem \ref{thm: second main result} no assumptions have been made on the nature of the points of maximum of the function $f_0$.
\mathrm end{observation}
\begin{observation}
In contrast to \cite{Borer-Galimberti-Struwe}, in the present paper the monotonicity of the energy of the solutions $u_\lambda$ as a function of $\lambda$ is not obvious. The proof of this fact is perhaps the main new technical achievement in the present work.
\mathrm end{observation}
\section*{Acknowledgments}
I would like to thank Michael Struwe for the guidance through the project which led to this paper.
\section{Some notation and preliminary results}\label{sec:Some notation and preliminary results}
In the following section we will recall some well-known results about the existence of solutions to equation
(\ref{eqn: the Gauss curvature}) and introduce some notation and concepts used through the rest of the paper.
For further details we refer to \cite{Kazdan-Warner74}.
For $\lambda \mathrm in \mathbbm R$ consider the set $\mathcal{C_\lambda}$ defined by (\ref{eqn: l'insieme C lambda}).
Note that for $\lambda\mathrm in (0, -\min_M f_0)$ the function $f_\lambda$ is sign changing and hence
$\mathcal{C_\lambda} \neq \mathrm emptyset$. On the other hand, $\mathcal{C_\lambda} = \mathrm emptyset$ for $\lambda \leq 0$ or $\lambda \geq -\min_M f_0$.
The constraints defining $\mathcal{C_\lambda}$ are natural; the first allows to apply the direct methods, the second one is motivated by the Gauss-Bonnet Theorem.
\begin{lemma}\label{lemma: C lambda e' una varieta'}
For $\lambda\mathrm in (0, -\min_M f_0)$ the set $\mathcal{C_\lambda}$ is a $C^{\mathrm infty}$-Banach manifold.
\mathrm end{lemma}
\begin{proof}
Define $G^\lambda : H^1(M;g_0) \to \mathbbm R^2$ by letting
\[
G^\lambda(u) :=\left( \mathrm int_M u d\mu_{g_0} , \mathrm int_M f_\lambda e^{2u} d\mu_{g_0} \right).
\]
Then $G^\lambda$ is smooth and its first derivative is
\[
DG^\lambda (u)\left[v\right] = \left( \mathrm int_M v d\mu_{g_0} , 2\mathrm int_M f_\lambda v e^{2u} d\mu_{g_0} \right).
\]
Notice that $(G^\lambda)^{-1}(0)= \mathcal{C_\lambda}$. Pick $u\mathrm in \mathcal{C_\lambda}$.
If we compute $DG^\lambda (u)\left[v\right]$ with $v\mathrm equiv 1$ and then with $v=f_\lambda$, we get
two vectors of $\mathbbm R^2$ which are linearly independent; therefore $DG^\lambda(u)$ is surjective.
Since we are in the Hilbert space $H^1(M;g_0)$, we have that it is splitted by the kernel of $DG^\lambda(u)$. It follows that $G^\lambda$ is a submersion at $u\mathrm in\mathcal{C_\lambda}$ and then that $\mathcal{C_\lambda}$ is a
smooth manifold. (For further details we refer to \cite{Zeidler}). The lemma is proved.
\mathrm end{proof}
In order to find solutions to equation (\ref{eqn: the Gauss curvature}) for $\lambda\mathrm in\Lambda$, we minimize the Dirichlet energy $E$
\[
H^1(M;g_0) \ni u \stackrel{E}{\longmapsto } \mathrm int_M |\nabla u|_{g_0}^2 d\mu_{g_0}
\]
in $\mathcal{C_\lambda}$. The energy $E$ is coercive on
$\mathcal{C_\lambda}$ in view of Poincare's inequality and sequentially weakly lower semicontinuous.
Furthermore, $\mathcal{C_\lambda}$ is weakly sequentially closed as can easily be shown by means of Moser-Trudinger's inequality.
Hence the direct method of the calculus of variation applies and for each $\lambda\mathrm in\Lambda$ there
exists a minimizer $w_\lambda\mathrm in \mathcal{C_\lambda}$.
But in the course of the proof of Lemma \ref{lemma: C lambda e' una varieta'} we have seen that $G^\lambda$
is a submersion at any point of $\mathcal{C_\lambda}$: therefore we can apply the Lagrange multipliers rule
and obtain
\begin{equation}\label{eqn: moltiplicatori di Lagrange}
2\mathrm int_M (\nabla w_\lambda,\nabla v)_{g_0} d\mu_{g_0} = \sigma \mathrm int_M v d\mu_{g_0}
+ 2\mu \mathrm int_M f_\lambda v e^{2w_\lambda} d\mu_{g_0}
\mathrm end{equation}
for every $v\mathrm inH^1(M;g_0)$ with suitable $\sigma,\mu\mathrm in\mathbbm R$. Choosing $v\mathrm equiv 1$, we obtain
\[
0 = \sigma \mathrm int_M d\mu_{g_0}
+ 2\mu \mathrm int_M f_\lambda e^{2w_\lambda} d\mu_{g_0} ;
\]
hence $\sigma=0$, because $w_\lambda\mathrm in\mathcal{C_\lambda}$. Notice that, by regularity arguments
(see \cite{Kazdan-Warner74} for the details), $w_\lambda\mathrm in C^{\mathrm infty}(M)$ and hence $v\mathrm equiv e^{-2w_\lambda}\mathrm inH^1(M;g_0)$. For this choice of testing function (\ref{eqn: moltiplicatori di Lagrange}) gives
\[
0 \geq -2 \mathrm int_M |\nabla w_\lambda|_{g_0}^2 e^{-2w_\lambda} d\mu_{g_0} = \mu \mathrm int_M f_\lambda d\mu_{g_0}.
\]
If
\[
\mathrm int_M |\nabla w_\lambda|_{g_0}^2 e^{-2w_\lambda} d\mu_{g_0} =0,
\]
we get $w_\lambda \mathrm equiv constant$, which is a contradiction, since in $\mathcal{C_\lambda}$ there are no
constant functions for $\lambda\mathrm in\Lambda$. Therefore, since $\mathrm int_M f_\lambda d\mu_{g_0}<0$ for
$\lambda\mathrm in\Lambda$, we obtain
\[
\mu=\mu(\lambda) = -2 \frac{\mathrm int_M |\nabla w_\lambda|_{g_0}^2 e^{-2w_\lambda} d\mu_{g_0}}
{\mathrm int_M f_\lambda d\mu_{g_0}}> 0.
\]
As a consequence,
\begin{equation}\label{eqn: definizione della soluzione u lambda}
u_\lambda := w_\lambda + 1/2 \log \mu(\lambda)
\mathrm end{equation}
classically solves (\ref{eqn: the Gauss curvature}).
For the continuation of our analysis and for technical reasons which will become evident later, it is
convenient to introduce for $\lambda\mathrm in \mathbbm R$ the set
\[
\mathcal{E_\lambda} := \left\{ u\mathrm in H^1(M;g_0) : 0 = \mathrm int_M f_\lambda e^{2u} d\mu_{g_0} \right\},
\]
defined by a single constraint only.
As above, it can be seen that $\mathcal{E_\lambda}\neq \mathrm emptyset$ if and only if $\lambda\mathrm in(0,-\min_M f_0)$ and
that it is a $C^{\mathrm infty}$-Banach manifold.
A priori it is not clear if we may expect that the Dirichlet energy $E$ attains a mimimun in $\mathcal{E_\lambda}$; however an elementary argument shows that for $\lambda\mathrm in (0,\lambda_{max})$ we have
\[
E(u_\lambda) = \min_{v\mathrm in \mathcal{E_\lambda}} E(v) = \min_{v\mathrm in \mathcal{C_\lambda}} E(v)
\]
where $u_\lambda$ is defined by (\ref{eqn: definizione della soluzione u lambda}). Indeed, for any
$v\mathrm in \mathcal{E_\lambda}$, we have $v-\overline{v}\mathrm in \mathcal{C_\lambda}$ and $E(v)=E(v-\overline{v})$,
where $\overline{v}:= \mathrm int_M v d\mu_{g_0}$.
Notice finally that for $\lambda= \lambda_{max}$, $u\mathrm equiv constant$ belongs to $\mathcal{E_\lambda}$
and it mimimizes the energy (which is zero). Furthermore, for
$\lambda\mathrm in (\lambda_{max}, -\min_M f_0) $, it is always true that the energy $E$, even though it does not admit a mimimum, is non negative. That suggests to define the following function:
\begin{equation}\label{eqn: definizione di beta lambda}
\beta_\lambda := \left\{
\begin{array}{ll}
\mathrm int_M |\nabla u_\lambda|_{g_0}^2 d\mu_{g_0}=E(u_\lambda) & \mbox{if } \lambda \mathrm in (0, \lambda_{max}) \\
0 & \mbox{if } \lambda\mathrm in \left[\lambda_{max}, -\min_M f_0 \right) .
\mathrm end{array}
\right.
\mathrm end{equation}
In the next sections, we study the properties of $\beta_\lambda$ and use this
information to prove, respectively, Theorem \ref{thm: first main result}
and Theorem \ref{thm: second main result}.
\section{Proof of Theorem \ref{thm: first main result}}
In this section we will analyse the behaviour of the set of solutions to equation
(\ref{eqn: the Gauss curvature}) when the parameter $\lambda$ approaches zero and we will prove Theorem
\ref{thm: first main result}.
The first result is quite elementary but it shows that in an arbitrary neighborhood of zero the function
$\beta_\lambda$ can achieve arbitrarily large values. More precisely, we can state:
\begin{lemma}\label{lemma: the energy blows up}
$\limsup_{\lambda \,\mathrm downarrow 0,\lambda\mathrm in\Lambda} \beta_\lambda= +\mathrm infty$.
\mathrm end{lemma}
\begin{proof}
Assume by contradiction that there exists $\,\mathrm delta\mathrm in\Lambda$ such that
$\sup_{\lambda\mathrm in(0,\,\mathrm delta)} \beta_\lambda < +\mathrm infty$. Choose a sequence $(\lambda_n)_n \subset (0,\,\mathrm delta)$
which converges to zero as $n\to + \mathrm infty$. Thus we have $\mathrm int_M |\nabla w_{\lambda_n}|_{g_0}^2 d\mu_{g_0}
< +\mathrm infty$ uniformly in $n$, where $w_{\lambda_n}\mathrm in\mathcal{C}_{\lambda_{n}}$ is a minimizer of the energy
$E$. Therefore, since the average of $w_{\lambda_{n}}$ is zero, we have, up to subsequences, that
$w_{\lambda_{n}}\rightharpoonup w_0$ weakly in $H^1(M;g_0)$ and $e^{2w_{\lambda_{n}}}\rightarrow e^{2w_0}$
strongly in $L^1$. Thus
\[
0= \mathrm int_M f_{\lambda_{n}} e^{2w_{\lambda_{n}}} d\mu_{g_0} \rightarrow
\mathrm int_M f_0 e^{2w_0} d\mu_{g_0}
\]
and $w_0\mathrm in\mathcal{E}_0=\mathrm emptyset$. The contradiction proves the Lemma.
\mathrm end{proof}
In the following, we are going to construct a suitable comparison function belonging
to the manifold $\mathcal{E_\lambda}$, which will give a control on the rate of blow-up of the mimimum of the
energy. This is the content of the next proposition, but before we need:
\begin{lemma}\label{lemma: determinazione della costante L}
There exists $L>0$ such that for any $\lambda< -\min_M f_0$ and for any $p\mathrm in M$ point of maximum of $f_0$
we have
\begin{enumerate}
\mathrm item $\frac{\sqrt{\lambda}}{L} < 1$
\mathrm item $f_0(x) > -\frac{\lambda}{2}$ on $B_{\frac{\sqrt{\lambda}}{L}}(0) \subset \mathbbm R^2$,
\mathrm end{enumerate}
where $x$ are suitable local conformal coordinates around $p\simeq 0$.
\mathrm end{lemma}
\begin{proof}
Fix a point of maximum $p_i$ of $f_0$. Then, by choosing local conformal coordinates $x$ around $p_i\simeq 0$, we
have
\[
f_0(x) = \frac{1}{2} D^2f_0(0)[x,x] + O(|x|^3) \;\; \mbox{in}\;\; B_1(0)\subset\mathbbm R^2
\]
From the beginning we may assume that $\frac{1}{2} D^2f_0(0)[x,x]\geq -c_1 |x|^2$, where $c_1>0$.
Then, for $x\mathrm in B_1(0)$, we have
\[
f_0(x)\geq -c_1|x|^2 - c_2|x|^3\geq -c(|x|^2 + |x|^3),
\]
with $c_2>0$ and $c:=\max(c_1,c_2)>0$.
Pick $\lambda>0$ and $L_i>0$ to be determined later, such that $\sqrt{\lambda}/L_i<1$, namely $\lambda<L_i^2$.
Then, on the ball $B_{\frac{\sqrt{\lambda}}{L_i}}(0)$ we get
\[
f_0(x) > -c\left( \frac{\lambda}{L_i^2} + \frac{\lambda^{3/2}}{L_i^3}\right) \geq -\frac{\lambda}{2}
\]
where the last inequality holds if we choose $L_i^2 \geq 4c$. Choose $L_i>>0$ so that
$-\min_M f_0 < L_i^2$. Taking $L:= \max_{p_i}L_i$, we obtain the desired result.
\mathrm end{proof}
\begin{proposition}\label{prop: construction of the comparison function}
For any $0<\sigma\leq 1$ there exists $\lambda_\sigma <1$, $\lambda_\sigma\mathrm in\Lambda$, such that for any
$0<\lambda\leq\lambda_\sigma$ there holds:
\begin{equation}\label{eqn: stima del blow-up dell'energia}
\beta_\lambda \leq 2\pi M_0 \left(\sigma +2 \right)^2 \log(1/\lambda)
\mathrm end{equation}
where $M_0$ is a constant which depends only on $(M,g_0)$ and the function $f_0$.
\mathrm end{proposition}
\begin{proof}
Choose $p_0\mathrm in M$ such that $f_0(p_0)=0$ and choose conformal coordinates $x$ as in the previous Lemma so
that
\[
f_0(x) + \lambda \geq \frac{\lambda}{2}, \;\;\; x\mathrm in B_{\frac{\sqrt{\lambda}}{L}}(0)
\]
for any $\lambda < -\min_M f_0$. Locally we can write $g_0= e^{2v_0}g_{\mathbbm R^2}$ where
$v_0\mathrm in C^{\mathrm infty}(\overline{B_1(0)})$ and $v_0(0)=0$. Fix $\lambda\mathrm in\Lambda$ with $\lambda<1$.
Define the function $\varphi(\lambda):M\to \mathbbm R $ as
\begin{equation}\label{eqn: definizione della funzione comparison}
\varphi(\lambda)(x)=
\left\{
\begin{array}{rl}
\log\left(\frac{\sqrt{\lambda}}{L|x|} \right), &
\;\; \frac{\lambda^{3/2}}{L} \leq |x| \leq \frac{\sqrt{\lambda}}{L} \\
\log\left(\frac{1}{\lambda} \right), & \;\; |x| \leq \frac{\lambda^{3/2}}{L} \\
0, & \;\; \frac{\sqrt{\lambda}}{L} \leq |x| \leq 1
\mathrm end{array}
\right.
\mathrm end{equation}
extended to zero on the rest of $M$. We have $\varphi(\lambda)\mathrm inH^1(M;g_0) $ and $f_\lambda$ is positive on the support of $\varphi(\lambda)$.
Consider the continuous function $z:\mathbbm R\to\mathbbm R$ defined by
$z(\alpha)= \mathrm int_M f_\lambda e^{2\alpha\varphi(\lambda)} d\mu_{g_0}$; then $z(0)< 0$ and
$\lim_{\alpha\to +\mathrm infty}z(\alpha) = +\mathrm infty$; thus there exists
$\alpha=\alpha(\lambda)\mathrm in(0,+\mathrm infty)$ where
\[
0 = z(\alpha) =\mathrm int_M f_\lambda e^{2\alpha\varphi(\lambda)} d\mu_{g_0},
\]
that is, $\alpha \varphi(\lambda)\mathrm in \mathcal{E_\lambda} $.
We can give a more precise estimate of $\alpha$, as follows. Recall that $\vol(M;g_0)=1$, therefore
\begin{eqnarray}
0 = \mathrm int_M f_\lambda e^{2\alpha\varphi(\lambda)} d\mu_{g_0}
& \geq & \lambda/2\mathrm int_{B_{\frac{\sqrt{\lambda}}{L}}(0)} e^{2\alpha\varphi(\lambda) } e^{2v_0}dx
-||f_0||_{\mathrm infty} \nonumber \\
& > & \lambda/2\mathrm int_{B_{\frac{\lambda^{3/2}}{L}}(0)} e^{2\alpha\log(1/\lambda) } e^{2v_0}dx
-||f_0||_{\mathrm infty} . \nonumber
\mathrm end{eqnarray}
Let $m_0:= \min_{B_1(0)}e^{2v_0}$ and $M_0:= \max_{B_1(0)}e^{2v_0}$. We obtain:
\[
\frac{m_0\pi}{2} \frac{\lambda^{4-2\alpha}}{L^2} \leq ||f_0||_{\mathrm infty}
\]
or equivalently
\[
0< \alpha \leq \frac{\log\left( \frac{2L^2||f_0||_{\mathrm infty}}{m_0\pi}\right)}{2\log(1/\lambda)} + 2.
\]
Given $0<\sigma\leq 1$, there exists $\lambda_\sigma <1$, $\lambda_\sigma\mathrm in\Lambda$, such that for any
$0<\lambda\leq\lambda_\sigma$ we have
$\frac{\log\left( \frac{2L^2||f_0||_{\mathrm infty}}{m_0\pi}\right)}{2\log(1/\lambda)} <\sigma$. Hence
\[
\alpha^2 \leq \left(\sigma +2 \right)^2.
\]
Next we have:
\[
\mathrm int_M |\nabla \varphi (\lambda)|_{g_0}^2 d\mu_{g_0} =
\mathrm int_{B_{\frac{\sqrt{\lambda}}{L}}(0)\setminus B_{\frac{\lambda^{3/2}}{L}}(0)} |x|^{-2} e^{2v_0} dx;
\]
hence
\begin{equation}\label{eqn:estimates for the Dirichlet energy of the comparison function}
m_0 2\pi \log(1/\lambda) \leq \mathrm int_M |\nabla \varphi (\lambda)|_{g_0}^2 d\mu_{g_0}
\leq M_0 2\pi \log(1/\lambda).
\mathrm end{equation}
We conclude
\[
\beta_\lambda \leq \alpha^2 \mathrm int_M |\nabla \varphi (\lambda)|_{g_0}^2 d\mu_{g_0} \leq
2\pi M_0 \left(\sigma +2 \right)^2 \log(1/\lambda),
\]
which proves the Proposition.
\mathrm end{proof}
From Proposition \ref{prop: construction of the comparison function}, by means of elliptic estimates we obtain uniform $L^{\mathrm infty}$ -bounds for the set of solutions of (\ref{eqn: the Gauss curvature}), away from the boundary of $\Lambda$. More precisely:
\begin{proposition}\label{prop: stime L infinito uniformi}
Fix $0<\sigma\leq 1$ and let $\lambda_\sigma$ be as in Proposition (\ref{prop: construction of the comparison function}). Then for any $\lambda^{\ast}\mathrm in(0,\lambda_\sigma)$ we have
\begin{equation}\label{eqn: stime L infinito uniformi}
\sup_{\lambda^{\ast}\leq\lambda\leq\lambda_\sigma} ||u_\lambda||_{\mathrm infty} < +\mathrm infty .
\mathrm end{equation}
\mathrm end{proposition}
\begin{observation}
Obviously, the estimate above can be improved by replacing the $L^{\mathrm infty}$ norm with "higher" norms (use
a bootstrap argument), but in the rest of the paper the estimate above will turn out to be sufficient for
all our purposes.
\mathrm end{observation}
\begin{proof}[Proof of Proposition \ref{prop: stime L infinito uniformi}]
Because of the Sobolev embedding, it is enough to prove that
\[
\sup_{\lambda^{\ast}\leq\lambda\leq\lambda_\sigma} ||u_\lambda||_{H^2} < +\mathrm infty
\]
For $\lambda\mathrm in \left[\lambda^{\ast}, \lambda_\sigma\right]$ consider the minimizer
$w_\lambda\mathrm in\mathcal{C_\lambda}$, which solves the equation
\[
-\Delta_{g_0}w_\lambda = \mu(\lambda)f_\lambda e^{2w_\lambda}
\]
where $\mu(\lambda)\mathrm in(0,\mathrm infty)$ is a Lagrange multiplier.
From (\ref{eqn: stima del blow-up dell'energia}), we have
\[
\sup_{{\lambda^{\ast}\leq\lambda\leq\lambda_\sigma}} \beta_\lambda
\leq C \log(1/\lambda^{\ast}) .
\]
Hence, using Poincar\'e's inequality, we obtain $||w_\lambda||_{H^1(M;g_0)} < C$ uniformly in $\lambda$.
By the Moser-Trudinger's inequality, for every $p\geq 1$ then there holds:
\[
\sup_{{\lambda^{\ast}\leq\lambda\leq\lambda_\sigma}} \mathrm int_M e^{pw_\lambda} d\mu_{g_0} < C(p)< \mathrm infty.
\]
Our claim thus follows once we can give a lower and an upper bound for $\mu(\lambda)$. Inserting
$v=f_\lambda$ in (\ref{eqn: moltiplicatori di Lagrange}), we obtain
\begin{equation}\label{eqn:integraz. per parti con test f lambda}
\mathrm int_M (\nabla w_\lambda,\nabla f_\lambda)_{g_0} d\mu_{g_0} = \mu(\lambda) \mathrm int_M (f_\lambda)^2 e^{2w_\lambda} d\mu_{g_0} .
\mathrm end{equation}
Since $\lambda_\sigma < -\overline{f_0}$, we have $0<c \leq \left( \mathrm int_M f_\lambda d\mu_{g_0} \right)^2$ uniformly in $\lambda\mathrm in \left[\lambda^{\ast}, \lambda_\sigma\right]$. Thus, by H\"older
\[
c< \mathrm int_M (f_\lambda)^2 e^{2w_\lambda} d\mu_{g_0} \mathrm int_M e^{-2w_\lambda} d\mu_{g_0} .
\]
Applying Moser-Trudinger's inequality, we get
\[
c< C\mathrm int_M (f_\lambda)^2 e^{2w_\lambda} d\mu_{g_0} \mathrm exp\left(\frac{1}{4\pi}\mathrm int_M |\nabla w_\lambda|_{g_0}^2 d\mu_{g_0}\right) .
\]
Thus, we see that $\mathrm int_M (f_\lambda)^2 e^{2w_\lambda} d\mu_{g_0} $ for
$\lambda\mathrm in\left[\lambda^{\ast}, \lambda_\sigma\right]$ is uniformly bounded away from zero and, from
(\ref{eqn:integraz. per parti con test f lambda}), we obtain
\[
\mu(\lambda) \leq C(||w_\lambda||_{H^1(M;g_0)}) < C < \mathrm infty
\]
uniformly in $\lambda$.
To see that $\mu(\lambda)$ is also away from zero, we argue by contradiction.
Assume that $\mathrm inf_{{\lambda^{\ast}\leq\lambda\leq\lambda_\sigma}} \mu (\lambda ) = 0$. Take a sequence $\lambda_n\mathrm in \left[\lambda^{\ast},\lambda_\sigma \right] $ such that:
\[
\mu(\lambda_n ) \to 0
\]
and $\lambda_n \to \lambda\mathrm in \left[\lambda^{\ast},\lambda_\sigma \right]$. From the estimates above,
we can assume, up to subsequences, that $w_{\lambda_n}\rightharpoonup w$ weakly in $H^1(M;g_0)$ and
$e^{2w_{\lambda_n}}\rightarrow e^{2w}$ strongly in $L^1$ as $n\to \mathrm infty$. Recall that we have
\[
\mathrm int_M (\nabla w_{\lambda_n},\nabla v)_{g_0} d\mu_{g_0} =
\mu(\lambda_n) \mathrm int_M f_{\lambda_n} v e^{2w_{\lambda_n}} d\mu_{g_0}
\]
for any $v\mathrm inH^1(M;g_0)$. Passing to the limit $n\to\mathrm infty$ in this equation, we obtain
\[
\mathrm int_M (\nabla w,\nabla v)_{g_0} d\mu_{g_0} = 0
\]
for each $v\mathrm inH^1(M;g_0)$, that is $w$ is harmonic. But then $w\mathrm equiv 0$ which is clearly impossible. Therefore, we have shown that for $\lambda\mathrm in \left[\lambda^{\ast},\lambda_\sigma \right]$, $\mu(\lambda)$
is uniformly away from 0 and infinity.
In conclusion, we get a uniform bound in $\lambda$ for
\[
||\Delta_{g_0}w_{\lambda}||_{L^2} = ||\mu(\lambda)f_\lambda e^{2w_\lambda}||_{L^2}.
\]
Hence, by $L^p$-elliptic estimates (see for instance \cite{Kazdan-Warner74}, p. 24), we have
\[
||w_\lambda||_{H^2} \leq C \left\{ ||w_\lambda||_{H^1} + ||\Delta_{g_0}w_{\lambda}||_{L^2}\right\} < C,
\]
uniformly for $\lambda\mathrm in \left[\lambda^{\ast},\lambda_\sigma \right]$. Recalling equation
(\ref{eqn: definizione della soluzione u lambda}) and the bounds on $\mu(\lambda)$, the bound
(\ref{eqn: stime L infinito uniformi}) follows.
\mathrm end{proof}
\begin{remark}\label{rmk: i massimi delle soluzioni tendono ad esplodere}
The Proposition above is false when $\lambda$ approaches zero. Indeed, an estimate like
$\sup_{0<\lambda\leq\,\mathrm delta} \max_M u_{\lambda} < \mathrm infty$ for some $\,\mathrm delta$ would lead,
in view of Schauder's estimates, to a uniform $C^{2,\alpha}$ bound for $u_\lambda$, which clearly contradicts Lemma \ref{lemma: the energy blows up}.
\mathrm end{remark}
In the following we show that the function $\beta_\lambda$ is monotone decreasing in a suitable right
neighborhood of zero, which is crucial for our argument. As a consequence, $\beta_\lambda$ will be differentiable almost everywhere.
\begin{proposition}\label{prop: teorema strumentale per il teorema di monotonicita'}
There exists $\lambda_0 \leq \min\left\{1/2, -\overline{f_0}/2 \right\}$ such that for any $\lambda^{\ast}\mathrm in (0,\lambda_0)$ there exists $\mathrm ell(\lambda^\ast)\mathrm in (\lambda^{\ast}, -\min_M f_0)$ such that for any
$\lambda\mathrm in (\lambda^\ast, \mathrm ell(\lambda^\ast)) $ we have
\[
\beta_\lambda < \beta_{\lambda^{\ast}}
\]
Furthermore, choosing $\lambda\mathrm in(0,\lambda_0), \lambda > \lambda^{\ast} $,
and defining $\mathrm ell(\lambda)$ as above, we have
$\mathrm ell(\lambda) - \lambda\geq \tau=\tau(\lambda^{\ast})>0 $ where $\tau$ is a constant not depending on $\lambda$.
\mathrm end{proposition}
\begin{corollary}\label{cor: monotonicita' di beta lambda}
There exists $\lambda_0 \leq \min\left\{1/2, -\overline{f_0}/2 \right\}$ such that $\beta_\lambda$ is strictly monotone decreasing on the interval $(0,\lambda_0)$.
\mathrm end{corollary}
In order to prepare for the proof of Proposition \ref{prop: teorema strumentale per il teorema di monotonicita'}, define the map $I:H^1(M;g_0) \to \mathbbm R$ by letting
\begin{equation}\label{eqn: definizione della mappa I}
I(u) := -\frac{\mathrm int_M f_0 e^{2u} d\mu_{g_0}}{\mathrm int_M e^{2u} d\mu_{g_0}}.
\mathrm end{equation}
Note that for any $u\mathrm inH^1(M;g_0)$ there holds
\begin{equation}\label{eqn:proprieta' cardine della mappa I}
u\mathrm in\mathcal{E}_{I(u)} .
\mathrm end{equation}
Moreover, we have $I(u)\mathrm in(0,-min_M f_0) $ and $I$ is smooth with first derivative given by the following expression:
\begin{equation}\label{eqn: differenziale di I}
DI(u)[v] = -2 \,\frac{\mathrm int_M f_{I(u)} v\,e^{2u} d\mu_{g_0}}{\mathrm int_M e^{2u} d\mu_{g_0}}\;\;\;\; u,v\mathrm inH^1(M;g_0) .
\mathrm end{equation}
Fix $0 < \lambda_0 \leq \min\left\{1/2, -\overline{f_0}/2 \right\}$ and for $\lambda^{\ast}\mathrm in(0,\lambda_0)$
let
\begin{equation}\label{eqn: prima funzione ausiliaria}
A(\lambda^{\ast}) := \sup_{\lambda^{\ast}\leq\lambda <\lambda_0} \sup_M e^{2u_\lambda}
\mathrm end{equation}
and
\begin{equation}\label{eqn: seconda funzione ausiliaria}
a(\lambda^{\ast}) := \mathrm inf_{\lambda^{\ast}\leq\lambda <\lambda_0} \mathrm inf_M e^{2u_\lambda}.
\mathrm end{equation}
Observe that in view of Proposition \ref{prop: stime L infinito uniformi}, the above functions are
well defined if $\lambda_0$ is taken small enough, and that $0< a(\lambda^\ast)\leq A(\lambda^{\ast})< +\mathrm infty$.
Finally, $A$ is monotone decreasing and $a$ is monotone increasing in $\lambda^{\ast}$.
We are ready to prove our Proposition.
\subsection{Proof of Proposition \ref{prop: teorema strumentale per il teorema di monotonicita'}}
\begin{proof}
Fix for convenience $\sigma=1$ and let $\lambda_\sigma$ as given in Proposition \ref{prop: construction of the comparison function}. Consider $\min\left\{\lambda_0, \lambda_\sigma \right\}$, which with a little
abuse of notation we will still call $\lambda_0$.
We consider $\lambda^{\ast}\mathrm in(0,\lambda_0)$ and
$\beta_{\lambda^{\ast}}= \mathrm int_M |\nabla u^{\ast}|_{g_0}^2 d\mu_{g_0} $, where we have used the abbreviation
$u^{\ast}\mathrm equiv u_{\lambda^{\ast}}$. We also set $\varphi^{\ast}\mathrm equiv\varphi(\lambda^{\ast})$, where
$\varphi(\lambda^{\ast})$ is the comparison function defined by the equation
(\ref{eqn: definizione della funzione comparison}). (We recall that $\lambda^{\ast} <\lambda_0\leq 1/2<1$, therefore
$\varphi^{\ast}$ is well defined.) Thus, we have inequality
(\ref{eqn:estimates for the Dirichlet energy of the comparison function}) and
\begin{eqnarray*}
\mathrm int_M (\nabla u^{\ast},\nabla \varphi^{\ast})_{g_0} d\mu_{g_0} & = &
\mathrm int_M f_{\lambda^{\ast}} \varphi^{\ast}e^{2u^{\ast}} d\mu_{g_0} \\
& > & \frac{\lambda^{\ast}}{2} \log(1/\lambda^{\ast}) \mathrm int_{B_{\frac{(\lambda^\ast)^{3/2}}{L}}(0)} e^{2u^{\ast} }e^{2v_0}dx ,\nonumber
\mathrm end{eqnarray*}
since $f_{\lambda^{\ast}} \varphi^{\ast} \geq 0$ and since $f_{\lambda^{\ast}} \geq \lambda^{\ast}/2$ in the ball $B_{\frac{\sqrt{\lambda^{\ast}}}{L}}(0)$.
Observing that
\begin{equation}\label{eqn: stima dell'integrale di lambda star alla 3/2}
\mathrm int_{B_{\frac{(\lambda^\ast)^{3/2}}{L}}(0)}e^{2u^{\ast} }e^{2v_0} dx
\geq \frac{m_0\pi}{L^2} (\lambda^{\ast})^3 a(\lambda^{\ast})
\mathrm end{equation}
where $a(\lambda^\ast)$ is defined by (\ref{eqn: seconda funzione ausiliaria}) and $m_0=\min_{B_1(0)}e^{2v_0}$ as above, we obtain
\begin{eqnarray}\label{eqn: l'integrale che coinvolge phi star e' positivo}
\mathrm int_M (\nabla u^{\ast},\nabla \varphi^{\ast})_{g_0} d\mu_{g_0} & = &
\mathrm int_M f_{\lambda^{\ast}} \varphi^{\ast}e^{2u^{\ast}} d\mu_{g_0} \\
& > & \frac{m_0\pi}{2L^2}\log(1/\lambda^{\ast})(\lambda^{\ast})^4 a(\lambda^\ast) \; >0 . \nonumber
\mathrm end{eqnarray}
Moreover, using equations (\ref{eqn: stima del blow-up dell'energia}) and
(\ref{eqn:estimates for the Dirichlet energy of the comparison function}), from H\"older's inequality we deduce
\[
\mathrm int_M (\nabla u^{\ast},\nabla \varphi^{\ast})_{g_0} d\mu_{g_0}\leq 6\pi M_0 \log(1/\lambda^{\ast}).
\]
Hence, defining
\begin{equation}\label{eqn: definzione di epsilon star}
\varepsilon^{\ast} := 2 \frac{\mathrm int_M (\nabla u^{\ast},\nabla \varphi^{\ast})_{g_0} d\mu_{g_0}}
{\mathrm int_M |\nabla \varphi^{\ast}|_{g_0}^2 d\mu_{g_0}}
\mathrm end{equation}
and using inequality (\ref{eqn: l'integrale che coinvolge phi star e' positivo}) and once more
(\ref{eqn:estimates for the Dirichlet energy of the comparison function}), we eventually get
\begin{equation}\label{eqn: bounds for epsilon star}
\frac{m_0}{2 M_0 L^2}(\lambda^{\ast})^4 a(\lambda^{\ast}) < \varepsilon^{\ast} < \frac{6M_0}{m_0} .
\mathrm end{equation}
In particular, $\varepsilon^{\ast}$ is positive. (Recall that $M_0:= \max_{B_1(0)}e^{2v_0}$).
For $\varepsilon\mathrm in [-\varepsilon^{\ast}, \varepsilon^{\ast} ]$ consider the function
$u^{\ast}-\varepsilon \varphi^{\ast}\mathrm inH^1(M;g_0)$. Recall that by (\ref{eqn:proprieta' cardine della mappa I}),
we trivially have
\[
u^{\ast}-\varepsilon \varphi^{\ast}\mathrm in\mathcal{E}_{I(u^{\ast}-\varepsilon \varphi^{\ast})} .
\]
\begin{lemma}\label{lemma: I risultato di monotonicita' di beta lambda}
For $\varepsilon\mathrm in (0, \varepsilon^{\ast} )$ we have
\begin{equation}\label{eqn: prima stima di monotonicita' sui beta}
\beta_{I(u^{\ast}-\varepsilon \varphi^{\ast})} < \beta_{\lambda^{\ast}} .
\mathrm end{equation}
\mathrm end{lemma}
\begin{proof}
By expanding the Dirichlet energy, for $\varepsilon\mathrm in (0, \varepsilon^{\ast} )$ we obtain
\begin{eqnarray*}
\beta_{I(u^{\ast}-\varepsilon \varphi^{\ast})} \leq E(u^{\ast}-\varepsilon \varphi^{\ast}) & = &
E(u^{\ast}) -2\varepsilon \mathrm int_M (\nabla u^{\ast},\nabla \varphi^{\ast})_{g_0} d\mu_{g_0}
+\varepsilon^2 \mathrm int_M |\nabla \varphi^{\ast}|_{g_0}^2 d\mu_{g_0} \\
& = &
E(u^{\ast}) -\varepsilon(\varepsilon^{\ast} - \varepsilon) \mathrm int_M |\nabla \varphi^{\ast}|_{g_0}^2 d\mu_{g_0} \\
& < & E(u^{\ast}) = \beta_{\lambda^{\ast}},
\mathrm end{eqnarray*}
as claimed.
\mathrm end{proof}
The next step is to understand whether the value $I(u^{\ast}-\varepsilon \varphi^{\ast})$ is greater or
smaller than $\lambda^{\ast}=I(u^{\ast})$. In order to do that, we introduce the function
$h:[-\varepsilon^{\ast}, \varepsilon^{\ast} ]\to (0,-min_M f_0)$
given by:
\begin{equation}\label{eqn: funzione ausiliaria h}
h(\varepsilon ) := I(u^{\ast}-\varepsilon \varphi^{\ast}) .
\mathrm end{equation}
By definition of $I$, we have $h\mathrm in C^1(\left[-\varepsilon^{\ast},\varepsilon^{\ast} \right])$; moreover,
there holds:
\begin{lemma}\label{lemma: proprieta' di h}
We have that
\[
h'>0 \;\; \mbox{on} \;\; [0,\varepsilon^{\ast}] .
\]
As a consequence, $h$ is smoothly invertible on $[0, \varepsilon^{\ast}]$.
\mathrm end{lemma}
Postponing the proof of the lemma, we continue with the proof of Proposition
\ref{prop: teorema strumentale per il teorema di monotonicita'}.
In view of Lemma \ref{lemma: proprieta' di h}, we have $h(\varepsilon^{\ast}) > \lambda^{\ast}$.
Furthermore, for any $\lambda\mathrm in(\lambda^{\ast}, h(\varepsilon^{\ast} ))$ there exists a unique
$\varepsilon \mathrm in (0, \varepsilon^{\ast} )$ such that
$h(\varepsilon )= I(u^{\ast}-\varepsilon \varphi^{\ast})=\lambda$. From Lemma
\ref{lemma: I risultato di monotonicita' di beta lambda}, then we get
$\beta_\lambda < \beta_{\lambda^{\ast}}$.
Therefore, setting $\mathrm ell(\lambda^\ast) := h(\varepsilon^{\ast})$, we obtain the first part of Proposition
\ref{prop: teorema strumentale per il teorema di monotonicita'}.
It remains to show the estimate on the length of this interval $(\lambda^{\ast}, \mathrm ell(\lambda^\ast))$
and the relations between it and $(\lambda,\mathrm ell(\lambda))$, for $\lambda>\lambda^{\ast}$.
This will be done in Lemma \ref{lemma: lunghezza dell'intervallo U lambda star e varie}.
\begin{proof}[Proof of Lemma \ref{lemma: proprieta' di h}]
Recall that $h(0)=\lambda^{\ast}$. Compute the first derivative of $h$, using
(\ref{eqn: differenziale di I}):
\[
h'(\varepsilon) = DI(u^{\ast}-\varepsilon \varphi^{\ast})[- \varphi^{\ast}]=
2\,\frac{\mathrm int_M f_{I(u^{\ast}-\varepsilon \varphi^{\ast} )}
\varphi^{\ast} \,e^{2u^{\ast}-2\varepsilon \varphi^{\ast}} d\mu_{g_0}}
{\mathrm int_M e^{2u^{\ast}-2\varepsilon \varphi^{\ast}} d\mu_{g_0}} .
\]
Thus
\[
h'(0) = 2\,\frac{\mathrm int_M f_{\lambda^{\ast}}
\varphi^{\ast} \,e^{2u^{\ast}} d\mu_{g_0}}
{\mathrm int_M e^{2u^{\ast}} d\mu_{g_0}} >0
\]
in view of (\ref{eqn: l'integrale che coinvolge phi star e' positivo}).
By continuity of $h'$, there exists $\varepsilon\mathrm in (0,\varepsilon^{\ast} ]$ such that
$h'>0$ on $[0,\varepsilon )$ and such that $\varepsilon$ is maximal with this property. We claim that
$\varepsilon=\varepsilon^{\ast}$. Suppose by contradiction that $\varepsilon<\varepsilon^{\ast}$. Note
that $h(\varepsilon ) = I(u^{\ast}-\varepsilon \varphi^{\ast})>\lambda^{\ast}=h(0)$, since $h'>0$ on
$[0,\varepsilon )$. Moreover,
\[
\begin{split}
\mathrm int_M f_{h(\varepsilon )}\,
\varphi^{\ast} \,e^{2u^{\ast}-2\varepsilon \varphi^{\ast}} d\mu_{g_0} & =
\mathrm int_{B_{\frac{\sqrt{\lambda^\ast}}{L}}(0) } f_{h(\varepsilon)}\,
\varphi^{\ast} \,e^{2u^{\ast}-2\varepsilon \varphi^{\ast}} e^{2v_0}dx\\
& \geq \frac{h(\varepsilon)}{2} \,
\mathrm int_{B_{\frac{\sqrt{\lambda^\ast}}{L}}(0)}
\varphi^{\ast} \,e^{(2u^{\ast}-2\varepsilon \varphi^{\ast}) }
e^{2v_0}dx
\mathrm end{split}
\]
where in the last inequality we used the fact that
\[
f_{h(\varepsilon )}\geq \frac{h(\varepsilon)}{2} \;\;
\mbox{on} \;\; B_{\frac{\sqrt{h(\varepsilon)}}{L}}(0) \supset B_{\frac{\sqrt{\lambda^{\ast}}}{L}}(0)
\]
(recall Lemma \ref{lemma: determinazione della costante L}). Therefore, we obtain
\[
\begin{split}
\mathrm int_M f_{h(\varepsilon )}\,
\varphi^{\ast} \,e^{2u^{\ast}-2\varepsilon \varphi^{\ast}} d\mu_{g_0} & \geq
\frac{h(\varepsilon )}{2} \log(1/\lambda^{\ast}) \,
\mathrm int_{B_{\frac{(\lambda^\ast)^{3/2}}{L}}(0)} e^{(2u^{\ast}-2\varepsilon \varphi^{\ast}) }
e^{2v_0}dx \\
& >
\frac{\lambda^{\ast}}{2} \log(1/\lambda^{\ast}) \,
\mathrm int_{B_{\frac{(\lambda^\ast)^{3/2}}{L}}(0)} e^{(2u^{\ast}-2\varepsilon^{\ast} \varphi^{\ast}) }
e^{2v_0}dx \\
& =
\frac{(\lambda^{\ast})^{1+2\mathrm epsilon^{\ast} }}{2} \log(1/\lambda^{\ast}) \,
\mathrm int_{B_{\frac{(\lambda^\ast)^{3/2}}{L}}(0)} e^{2u^{\ast} }
e^{2v_0}dx \\
& \geq
\frac{m_0\pi}{2L^2} \log(1/\lambda^{\ast}) (\lambda^\ast)^{4 +2\varepsilon^\ast} a(\lambda^\ast) \;>0
\mathrm end{split}
\]
where in the last line we used (\ref{eqn: stima dell'integrale di lambda star alla 3/2}).
Thus, we have, since $\varepsilon>0$ and $\varphi^{\ast}\geq 0 $,
\begin{eqnarray}\label{eqn: prima stima della derivata di h}
h'(\varepsilon ) & > & \frac{m_0\pi}{L^2} \log(1/\lambda^{\ast})(\lambda^{\ast})^{4+2\mathrm epsilon^{\ast} }
\frac{a(\lambda^\ast)}{\mathrm int_M e^{2u^{\ast}-2\varepsilon \varphi^{\ast}} d\mu_{g_0}} \nonumber \\
& > &
\frac{m_0\pi}{L^2}\log(1/\lambda^{\ast}) (\lambda^{\ast})^{4+2\mathrm epsilon^{\ast} }
\frac{a(\lambda^\ast)}{\mathrm int_M e^{2u^{\ast}} d\mu_{g_0}} > 0,
\mathrm end{eqnarray}
contradicting the maximality of $\varepsilon$.
Furthermore, reasoning as we have just done, we see that the bound (\ref{eqn: prima stima della derivata di h})
holds uniformly on $(0,\varepsilon^{\ast})$. We deduce $h'(\varepsilon^{\ast}) >0$ and the Lemma is proved.
\mathrm end{proof}
\begin{lemma}\label{lemma: lunghezza dell'intervallo U lambda star e varie}
Let $\lambda_0$ be defined as in the proof of Proposition
\ref{prop: teorema strumentale per il teorema di monotonicita'}.
Fix $0<\lambda^\ast < \lambda < \lambda_0$ and consider $\mathrm ell(\lambda)$ given by the first part of Proposition \ref{prop: teorema strumentale per il teorema di monotonicita'}. Then
$$
\mathrm ell(\lambda) - \lambda \geq \tau = \tau(\lambda^\ast) >0
$$
where $\tau$ is a constant not depending on $\lambda$.
\mathrm end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma: lunghezza dell'intervallo U lambda star e varie}]
Let's begin with estimating $\mathrm ell(\lambda^\ast) - \lambda^{\ast}$. We restart from
(\ref{eqn: prima stima della derivata di h}), which holds for $\varepsilon\mathrm in(0,\varepsilon^{\ast})$.
By equation (\ref{eqn: bounds for epsilon star}) and by the fact that $\lambda^{\ast} < 1$, we get
$(\lambda^{\ast})^{4+2\mathrm epsilon^{\ast} } > (\lambda^{\ast})^{4+ \frac{12M_0}{m_0} }$ and
$\log(1/\lambda^{\ast})> \log(1/\lambda_0) $. Recalling the definition of the auxiliary function
$A$ (equation (\ref{eqn: prima funzione ausiliaria})), we can bound
\[
A(\lambda^{\ast}) \geq \mathrm int_M e^{2u^{\ast}} d\mu_{g_0}
\]
and obtain
\[
h'(\varepsilon) >\frac{ m_0\pi}{L^2} \log(1/\lambda_0) (\lambda^{\ast})^{4+ \frac{12M_0}{m_0}}
\frac{a(\lambda^{\ast})}{A(\lambda^{\ast})} .
\]
Recalling once more (\ref{eqn: bounds for epsilon star}), with the constant
$k_0:=\frac{m_0^2\pi}{2M_0L^4}\log(1/\lambda_0)>0 $ we may finally estimate
\begin{eqnarray*}
\mathrm ell(\lambda^\ast) -\lambda^{\ast} = h(\varepsilon^{\ast})-\lambda^{\ast} & = &
\mathrm int_0^{\varepsilon^{\ast}} h'(\varepsilon)d\varepsilon \\
&> & \varepsilon^\ast \frac{ m_0\pi}{L^2} \log(1/\lambda_0) (\lambda^{\ast})^{4+ \frac{12M_0}{m_0}}
\frac{a(\lambda^{\ast})}{A(\lambda^{\ast})} \\
&>&
k_0 (\lambda^{\ast})^{8+ \frac{12M_0}{m_0}}
\frac{(a(\lambda^{\ast}))^2}{A(\lambda^{\ast})} ,
\mathrm end{eqnarray*}
and the function $(\lambda^{\ast})^{8+ \frac{12M_0}{m_0}}\frac{(a(\lambda^{\ast}))^2}{A(\lambda^{\ast})}$
is not decreasing in $\lambda^{\ast}$.
Hence, taking $\lambda\mathrm in (\lambda^{\ast}, \lambda_0)$, we deduce
\[
\mathrm ell(\lambda) - \lambda
> k_0 (\lambda)^{8+ \frac{12M_0}{m_0}}\frac{(a(\lambda))^2}{A(\lambda)}
\geq k_0 (\lambda^{\ast})^{8+ \frac{12M_0}{m_0}}\frac{(a(\lambda^{\ast}))^2}{A(\lambda^{\ast})}
:= \tau>0 .
\]
The Lemma is proved.
\mathrm end{proof}
This concludes the proof of Proposition \ref{prop: teorema strumentale per il teorema di monotonicita'}.
\mathrm end{proof}
\subsection{A bound for the total curvature}\mbox{}
With the help of Corollary \ref{cor: monotonicita' di beta lambda},
Proposition \ref{prop: construction of the comparison function} and following \cite{Struwe93},
it is now quite straightforward to show the following estimate for the derivative of $\beta_\lambda$:
\begin{lemma}\label{lemma: stima per la derivata di beta lambda}
There exists a sequence $(\lambda_n)_n\subset (0,\lambda_0)$ of points of differentiability for $\beta_\lambda$, such
that $\lambda_n \,\mathrm downarrow 0$ as $n\rightarrow \mathrm infty$ and
\[
|\beta'_{\lambda_n} | \leq C_0/\lambda_n
\]
where $C_0$ is a positive constant.
\mathrm end{lemma}
\begin{proof}
By Proposition \ref{prop: construction of the comparison function}, we have
$\beta_\lambda \leq C \log(1/\lambda)$, for any $\lambda<\lambda_0$. Set $C_0:=C+1$ and assume that
exists $\tilde{\lambda}< \lambda_0$ such that for any $\lambda<\tilde{\lambda}$, $\lambda$ point
of differentiability of $\beta_\lambda$, there holds:
\[
|\beta'_{\lambda}| > C_0/\lambda \, .
\]
Then we obtain, by Lebesgue's Theorem, that
\[
\beta_\lambda - \beta_{\tilde{\lambda}} \geq \mathrm int^{\tilde{\lambda}}_\lambda |\beta'_{s}|ds
\]
and hence
\[
C\log(1/\lambda)\geq \beta_\lambda > \beta_{\tilde{\lambda}} + C_0 \log(\tilde{\lambda}/\lambda) .
\]
Thus, we get
\[
\beta_{\tilde{\lambda}} + C_0 \log(\tilde{\lambda}) -\log(\lambda) \leq 0 ,
\]
which, for $\lambda$ small enough, is clearly impossible. The Lemma is proved.
\mathrm end{proof}
We can now prove the analogue of equation (5.1) in \cite{Borer-Galimberti-Struwe}:
\begin{proposition}\label{prop: bound for lambda x volume}
Let $(\lambda_n)_n$ be a sequence like the one given by Lemma \ref{lemma: stima per la derivata di beta lambda}.
and set $u_n:= u_{\lambda_n}$. Then
\begin{equation}\label{eqn: bound for lambda x volume}
\limsup_n \left( \lambda_n \mathrm int_M e^{2u_n} d\mu_{g_0} \right) < \mathrm infty
\mathrm end{equation}
\mathrm end{proposition}
\begin{proof}
Fix $n\mathrm in\mathbbm N$ and set for convenience $\lambda^{\ast}:=\lambda_n
\mathrm in(0,\lambda_0)$ and $u^{\ast}:=u_n$.
Consider the function $h$ defined by equation (\ref{eqn: funzione ausiliaria h}), where
$\varepsilon^{\ast}$ and $\varphi^{\ast}$ are defined as in the proof of Proposition
\ref{prop: teorema strumentale per il teorema di monotonicita'}.
For $\lambda_k \,\mathrm downarrow \lambda^{\ast}$, $\lambda_k<h(\varepsilon^{\ast})$, set
$\varepsilon_k := h^{-1}(\lambda_k)$. By Lemma \ref{lemma: proprieta' di h},
$\varepsilon_k \to 0$ as $k\to \mathrm infty$. Finally, by Lemma \ref{lemma: stima per la derivata di beta lambda},
we may assume that for all $k$
\[
-\frac{\beta_{\lambda^{\ast}}-\beta_{\lambda_k} }{\lambda^{\ast}-\lambda_k} \leq 2C_0/\lambda^{\ast} :=
C/\lambda^{\ast}
\]
where $C_0$ is the constant of Lemma \ref{lemma: stima per la derivata di beta lambda}.
Observe that $\beta_{\lambda^{\ast}}-\beta_{\lambda_k} \geq
E(u^{\ast})- E(u^{\ast}-\varepsilon_k\varphi^{\ast})$, since
$u^{\ast}-\varepsilon_k\varphi^{\ast}\mathrm in\mathcal{E}_{I(u^{\ast}-\varepsilon_k\varphi^{\ast})}=
\mathcal{E}_{\lambda_k}$. Now:
\[
E(u^{\ast})- E(u^{\ast}-\varepsilon_k\varphi^{\ast})=
\mathrm int_M \left(-\varepsilon_k^2 |\nabla \varphi^{\ast}|_{g_0}^2 + 2\varepsilon_k(\nabla u^{\ast},\nabla
\varphi^{\ast} )_{g_0}\right) d\mu_{g_0} .
\]
Hence,
\[
\frac{C}{\lambda^{\ast}} \geq \frac{1}{\lambda_k -\lambda^{\ast}}
\mathrm int_M \left(-\varepsilon_k^2 |\nabla \varphi^{\ast}|_{g_0}^2 + 2\varepsilon_k(\nabla u^{\ast},\nabla
\varphi^{\ast} )_{g_0} \right)d\mu_{g_0} .
\]
Recalling that $\varepsilon_k = h^{-1}(\lambda_k)$, $h^{-1}(\lambda^{\ast})=0$ and using
(\ref{eqn:estimates for the Dirichlet energy of the comparison function}), we have
\begin{eqnarray*}
\frac{1}{\lambda_k -\lambda^{\ast}}\mathrm int_M \varepsilon_k^2 |\nabla \varphi^{\ast}|_{g_0}^2
&\leq & 2\pi M_0\log(1/\lambda^{\ast}) \frac{h^{-1}(\lambda_k)- h^{-1}(\lambda^{\ast})}{\lambda_k
-\lambda^{\ast}} \varepsilon_k \\
& \to & 0
\mathrm end{eqnarray*}
as $k\rightarrow \mathrm infty$, since $h^{-1}$ is differentiable at $\lambda^{\ast}$ and $\varepsilon_k$ goes to
zero. Therefore, we may write, with an error term $o(1)$ as $k\rightarrow \mathrm infty$, that
\[
\frac{C}{\lambda^{\ast}} \geq
\frac{2\varepsilon_k}{\lambda_k -\lambda^{\ast}} \mathrm int_M (\nabla u^{\ast},\nabla
\varphi^{\ast} )_{g_0} d\mu_{g_0} + o(1) .
\]
Thus, when $k\to\mathrm infty$, we obtain
\begin{eqnarray*}
\frac{C}{\lambda^{\ast}} &\geq &
2(h^{-1})'(\lambda^{\ast}) \mathrm int_M (\nabla u^{\ast},\nabla
\varphi^{\ast} )_{g_0} d\mu_{g_0}\\
&=& \frac{2}{h'(0)} \mathrm int_M f_{\lambda^{\ast}}
\varphi^{\ast} \,e^{2u^{\ast}} d\mu_{g_0} \\
&=& \mathrm int_M e^{2u^\ast} d\mu_0
\mathrm end{eqnarray*}
where in the last line we have used the explicit expression of $h'(0)$. Going back to the original
notation, we have for any $n\mathrm in\mathbbm N$
\[
\mathrm int_M e^{2u_n} d\mu_{g_0} \leq C/\lambda_n ,
\]
which is nothing but equation (\ref{eqn: bound for lambda x volume}). The Proposition is proved.
\mathrm end{proof}
As a consequence of Proposition \ref{prop: bound for lambda x volume} and the Gauss-Bonnet identity
$0= \mathrm int_M f_{\lambda_n} e^{2u_n} d\mu_{g_0}$, we deduce the uniform bound
\[
\sup_{n\mathrm in\mathbbm N} \mathrm int_M (|f_0|+\lambda_n)e^{2u_n} d\mu_{g_0} < \mathrm infty
\]
for the total curvature of $g_n=e^{2u_n}g_0$.
\subsection{Blow-up analysis}\mbox{}
In this subsection we complete the Proof of Theorem \ref{thm: first main result}. For the rest of this part,
let $(\lambda_n)_n$ be a sequence like the one given by Lemma \ref{lemma: stima per la derivata di beta lambda} and set $u_n:=u_{\lambda_n}$. We follow closely Section 5 of \cite{Borer-Galimberti-Struwe}.
As shown by Ding-Liu \cite{Ding-Liu95}, we obtain for any open domain
$\,\mathcal Omega\subset\subset M^-:=\left\{p\mathrm in M: f_0(p)< 0 \right\}$,
$\mathrm int_\,\mathcal Omega (|\nabla u_n^+|^2_{g_0} + |u_n^+|^2) d\mu_{g_0} \leq C(\,\mathcal Omega)$, where $t^+ =\max
\left\{t,0 \right\}$, $t\mathrm in\mathbbm R$,
and hence, as proved in \cite{Borer-Galimberti-Struwe}, that
\begin{equation}\label{eqn:local pointwise upper bound for the solutions}
u_n \leq C'(\,\mathcal Omega) .
\mathrm end{equation}
Thus, if a sequence $(u_n)_n$ blows up near a point $p_0\mathrm in M$ in the sense that for every
$r>0$ there holds $\sup_{B_r(p_0)} u_n\to +\mathrm infty$ (and we know that it is always the case
in view of Remark \ref{rmk: i massimi delle soluzioni tendono ad esplodere}),
necessarily $f_0(p_0)=0$. Moreover, there exists a sequence of points $p_n\to p_0$ such that
for some $r>0$, $u_n(p_n)=\sup_{B_r(p_0)}u_n$.
Let $p_0$ be such a blow-up point for a sequence of solutions $u_n$. We introduce local isothermal coordinates $x$ on $B_r(p_0)$ around $p_0=0$. We can write $g_0 = e^{2v_0}g_{\mathbbm R^2}$ for some smooth function $v_0$. Setting
$v_n:= u_n + v_0$, we get
\[
-\Delta v_n = (f_0(x) + \lambda_n)e^{2v_n} \;\;\; \mbox{on} \;\;\; B_R(0)
\]
for some $R>0$ and there is a sequence $x_n\to 0$ so that
\[
v_n(x_n)=\sup_{|x|\leq R}v_n(x)\to +\mathrm infty
\]
as $n\to +\mathrm infty$. Moreover, $\Delta v_n(x_n)\leq 0$ and thus $f_0(x_n)+\lambda_n\geq 0$, which leads to
\[
|x_n|^2 \leq C \lambda_n
\]
for some constant $C>0$.
We observe that in the present case we do not have available a uniform global lower bound for the
sequence of solutions $u_n$ (and hence for $v_n$) of the kind present in \cite{Borer-Galimberti-Struwe}.
But we can still show that the analogue of Lemma 5.2 \cite{Borer-Galimberti-Struwe} holds
true. Indeed, a careful inspection shows that a uniform lower bound is not needed in the proof of
Lemma 5.2 \cite{Borer-Galimberti-Struwe}.
\begin{lemma}
For every $r>0$, that holds
\[
\limsup_n \mathrm int_{B_r(0)} (f_0+\lambda_n)^+ e^{2v_n} dx \geq 2\pi \, .
\]
\mathrm end{lemma}
In order to prove Theorem \ref{thm: first main result}, as regards part ii),
we would like to imitate the proof of Theorem 1.4 \cite{Borer-Galimberti-Struwe}. To do that and to show the
convergence results therein, the last ingredient we need is at least a local lower bound for our sequence
of solutions $u_n$.
The next Lemma shows that either the sequence degenerates or that we have a local lower bound.
After this Lemma, we will obtain part i) of Theorem \ref{thm: first main result}. To prove part ii), it will
be sufficient to repeat the same reasoning as after Lemma 5.2. in \cite{Borer-Galimberti-Struwe}.
\begin{lemma}\label{lemma: local lower bound for the sequence of solutions or convergence to minus inf}
Let $(\lambda_n)_n$ and $(u_n)_n$ be defined as above and set
\[
M_\mathrm infty:= M \setminus \left\{p_{\mathrm infty}^{(1)}, \cdots ,p_{\mathrm infty}^{(I)} \right\} .
\]
where $p_{\mathrm infty}^{(1)}, \cdots ,p_{\mathrm infty}^{(I)}$ are blow-up points. Then, up to subsequences, either\\
i) $u_n \to -\mathrm infty$ locally uniformly on compact domains of $M_\mathrm infty$, or\\
ii) for any compact domain $\,\mathcal Omega\subset\subset M_\mathrm infty$, there exists a constant $C=C(\,\mathcal Omega)\mathrm in\mathbbm R$ such that
\[
u_n\big |_{\,\mathcal Omega} > C(\,\mathcal Omega)
\]
uniformly in $n$.
\mathrm end{lemma}
\begin{proof}
We fix two open domains $\,\mathcal Omega\subset\subset\tilde{\,\mathcal Omega}\subset\subset M_\mathrm infty$. From
(\ref{eqn:local pointwise upper bound for the solutions}),
for any $n$ we get that $u_n\big |_{\tilde{\,\mathcal Omega}} \leq C(\tilde{\,\mathcal Omega})$.
We pick an arbitrary point $p\mathrm in \overline{\,\mathcal Omega}$ and $r_p>0$ so that
$B_{r_p}(p)\subset\tilde{\,\mathcal Omega}$. If needed, we choose a smaller radius and we consider a conformal chart
$\Psi: B_{r_p}(p) \to B_1(0)\subset\mathbbm R^2 $ with coordinates $x$ so that locally we have
$g_0= e^{2v_0}g_{\mathbbm R^2}$ with $v_0\mathrm in C^{\mathrm infty}(\overline{B_1(0)})$. Setting
$v_n := u_n + v_0$, we obtain
\[
-\Delta v_n = (f_0(x)+\lambda_n)e^{2v_n} \;\;\; \mbox{on} \;\;\; B_1(0) .
\]
Split $v_n= v_n^{(0)}+v_n^{(1)}$, where $v_n^{(1)}\mathrm in H^1_0(B_1(0))$ solves the boundary value problem
\[
\left\{
\begin{array}{ll}
-\Delta v_n^{(1)} = (f_0(x)+\lambda_n) e^{2v_n} & \mbox{in} \;\; B_1(0), \\
v_n^{(1)} = 0 & \mbox{on} \;\; \partial B_1(0).
\mathrm end{array}
\right.
\]
and $v_n^{(0)}$ is harmonic. Hence it follows, uniformly in $n$,
\[
||\Delta v_n^{(1)}||_{L^{p}(B_1(0))}\leq||\Delta v_n^{(1)}||_{L^{\mathrm infty}(B_1(0))} \leq C
\]
for any $p\geq 1$. Fixing $p>1$, from elliptic regularity theory we obtain that
$(v_n^{(1)})_n$ is bounded in $W^{2,p}(B_1(0))\hookrightarrow C^0({\overline{B_1(0)}})$. From the local upper bound on $\tilde{\,\mathcal Omega}$ for the sequence $(u_n)_n$ (and hence for $(v_n)_n$), we infer that for
any $x\mathrm in \overline{B_1(0)}$,
\[
v_n^{(0)}(x) \leq ||v_n^{(1)}||_{L^{\mathrm infty}(B_1(0))} + C(\tilde{\,\mathcal Omega}) \leq C
\]
uniformly in $n$. Therefore, Harnack's inequality implies that
\[
\sup_{B_{1/2}(0)} v_n^{(0)} \leq C_1\mathrm inf_{B_{1/2}(0)} v_n^{(0)} + C_2
\]
for suitable constants $C_1>0$ and $C_2\mathrm in\mathbbm R$ depending on $B_{1/2}(0)$ but not on $n$.
We see that we have two mutually disjoint cases (up to subsequences):
\begin{enumerate}
\mathrm item $\mathrm inf_{B_{1/2}(0)} v_n^{(0)}\to -\mathrm infty$, as $n\to +\mathrm infty$
\mathrm item $\mathrm inf_{B_{1/2}(0)} v_n^{(0)} \geq -C$, uniformly in $n$.
\mathrm end{enumerate}
In the first case, it follows, recalling that $(v_n^{(1)})_n$ is bounded in $L^{\mathrm infty}(B_1(0))$, that
\[
v_n \to -\mathrm infty
\]
uniformly in $\overline{B_{1/2}(0)}$.
In the second case, we deduce $C < v_n\big |_{\overline{B_{1/2}(0)}}$ uniformly in $n$.
Since $\overline{\,\mathcal Omega} $ is connected, we conclude that either on $\overline{\,\mathcal Omega}$ the sequence of solutions $u_n$ goes uniformly to $-\mathrm infty$ or that there exists $C=C(\,\mathcal Omega)$ such that
$u_n\big |_{\,\mathcal Omega} > C$ for any $n$. The Lemma is proved.
\mathrm end{proof}
\section{Proof of Theorem \ref{thm: second main result}}
In this section, we will analyze the asymptotic behaviour of the set of solutions to the
prescribed Gaussian curvature equation, when the parameter $\lambda\uparrow -\overline{f_0}=\lambda_{max}$.
The main content of this section is the proof of Theorem \ref{thm: second main result}.
\begin{proposition}\label{prop: the energy vanishes}
Let $\beta_\lambda$ be defined by equation (\ref{eqn: definizione di beta lambda}). Then
$\beta_\lambda \to 0$ as $\lambda \uparrow \lambda_{max}$.
\mathrm end{proposition}
In preparation for the proof of the Proposition, consider the Hilbert space $H^1(M;g_0) \times \mathbbm R$ endowed with the natural scalar product and consider the set
\begin{equation} \label{eqn: the hyperset C}
\mathcal{C} := \left\{(u,\lambda) \mathrm in H^1(M;g_0) \times \mathbbm R : \mathrm int_M u \,d\mu_{g_0} = 0 = \mathrm int_M f_\lambda e ^{2u}\,
d\mu_{g_0}\right\} .
\mathrm end{equation}
We claim that $\mathcal{C}$ is a $C^{\mathrm infty}$-Banach manifold. Indeed, we define
$G:H^1(M;g_0) \times \mathbbm R \rightarrow \mathbbm R^2$ as:
\[
G(u,\lambda) := \left( \mathrm int_M u \,d\mu_{g_0} \, ;\mathrm int_M f_\lambda e ^{2u}\,d\mu_{g_0} \right) .
\]
Then
\[
G^{-1}((0,0)) = \mathcal{C}
\]
and $G\mathrm in C^{\mathrm infty}$ with first Frechet derivative
\[
DG(u,\lambda)\left[v ,t\right] =
\left( \mathrm int_M v \,d\mu_{g_0} ; 2\mathrm int_M f_{\lambda} v e ^{2u}\,d\mu_{g_0} + t \mathrm int_M e^{2u} \, d\mu_{g_0} \right)
\]
for any $(v,t)\mathrm inH^1(M;g_0)\times\mathbbm R$.
For any $(u, \lambda) \mathrm in \mathcal{C}$, letting $DG(u,\lambda)$ act on $(1,0)$ and (0,1), we obtain
respectively the vectors $(1,0)$ and $ \left(0 , \mathrm int_M e^{2u}\, d\mu_{g_0} \right)$, which are clearly
a basis for $\mathbbm R^2$. Moreover, the kernel of $DG(u,\lambda)$ splits $H^1(M;g_0)\times\mathbbm R$. Thus, $\mathcal{C}$ is a smooth manifold of codimension equal to 2.
Define
\[
\mathcal{\tilde C}_\lambda := \mathcal{C} \cap \left\{(w,\mu)\mathrm in H^1(M;g_0) \times \mathbbm R : \mu=\lambda \right\}
\]
that is, the slice of $\mathcal{C}$ determined by the hyperplane in
$H^1(M;g_0) \times \mathbbm R$ of equation $\mu=\lambda$. We observe that this set is not empty for $\lambda\mathrm in(0,-\min_M f_0)$.
\begin{lemma}\label{lemma: riscrittura della slice}
There exist a function $s: \mathcal{C}_{\lambda_{max}} \to \mathbbm R$ and a map
$\Theta : \mathcal{C}_{\lambda_{max}}\times (0,-\min_M f_0) \to H^1(M;g_0)$ such that for any
$(u,\lambda)\mathrm in \mathcal{C}_{\lambda_{max}}\times (0,-\min_M f_0)$ we have
\[
u + s(u)(\lambda -\lambda_{max})(f_0 - \overline{f_0}) + \Theta(u, \lambda) \mathrm in \mathcal{C}_{\lambda}
\]
and with the property that for any fixed $u\mathrm in \mathcal{C}_{\lambda_{max}}$
\[
|| \Theta(u,\lambda)||_{H^1(M;g_0)} = o(\lambda - \lambda_{max})
\]
as $\lambda\to\lambda_{max}$.
\mathrm end{lemma}
\begin{proof}
We take $u\mathrm in\mathcal{C}_{\lambda_{max}}$, $\lambda \mathrm in (0, -\min_M f_0)$ and consider the vector $\left(s(f_0-\overline{f_0}),1\right)\mathrm inH^1(M;g_0) \times \mathbbm R $ where $s\mathrm in\mathbbm R$. We want to find a suitable $s=s(u)$
such that the vector $\left(s(f_0-\overline{f_0}),1\right)$ belongs to the tangent space
$T_{(u,\lambda_{max})} \mathcal{C}$.
That amounts to impose
\[
DG(u,\lambda_{max})\left[s(f_0-\overline{f_0}),1\right] = (0,0)
\]
that is,
\[
\left(
\begin{array}{c}
s \mathrm int_M (f_0-\overline{f_0}) \, d\mu_{g_0} \\
2s \mathrm int_M (f_0-\overline{f_0})^2 e^{2u} \, d\mu_{g_0} + \mathrm int_M e^{2u}\, d\mu_{g_0}
\mathrm end{array}
\right) =
\left(
\begin{array}{c}
0 \\
0
\mathrm end{array}
\right)
\]
Since $ \mathrm int_M (f_0-\overline{f_0}) \, d\mu_{g_0} =0$, we get from the second equation that
\[
s(u)= - \frac{\mathrm int_M e^{2u}\, d\mu_{g_0}}{2\mathrm int_M (f_0-\overline{f_0})^2 e^{2u} \, d\mu_{g_0}} <0 .
\]
In view of the differentiable structure of $\mathcal{C}$, there exists
$\Theta : \mathcal{C}_{\lambda_{max}}\times (0,-\min_M f_0) \to H^1(M;g_0)$ such that
\[
\left( u + s(u)(\lambda -\lambda_{max})(f_0 - \overline{f_0}) + \Theta(u, \lambda) ; \lambda \right)
\mathrm in \mathcal{\tilde C}_\lambda
\]
and $|| \Theta(u,\lambda)||_{H^1(M;g_0)} = o(\lambda - \lambda_{max})$ as $\lambda\to\lambda_{max}$.
The result follows.
\mathrm end{proof}
\begin{proof}[Proof of Proposition \ref{prop: the energy vanishes}]
We choose $u\mathrm equiv 0\mathrm in \mathcal{C}_{\lambda_{max}}$, $\lambda \mathrm in (0, -\min_M f_0)$ and compute $s$ and
$\Theta $ accordingly. Thus,
$v_\lambda := s(0)(\lambda -\lambda_{max})(f_0 - \overline{f_0}) + \Theta(0, \lambda)
\mathrm in \mathcal{C}_\lambda$; we evaluate its $H^1(M;g_0)$ norm
\begin{equation*}
\begin{split}
||s(0)(\lambda -\lambda_{max})(f_0 - \overline{f_0}) + \Theta(0, \lambda)||_{H^1(M;g_0)} \leq \\ \leq
|s(0)|\,|\lambda -\lambda_{max}|\,\,||f_0 - \overline{f_0} ||_{H^1(M;g_0)} + o(\lambda - \lambda_{max})
\mathrm end{split}
\mathrm end{equation*}
and see that it goes to zero as $\lambda \to \lambda_{max}$.
Since for $\lambda<\lambda_{max}$ we have by definition $\beta_\lambda \leq E(v_\lambda)$, it follows $\beta_\lambda \to 0$ as $\lambda \uparrow \lambda_{max}$.
\mathrm end{proof}
\begin{proof}[Proof of Theorem \ref{thm: second main result} (completed)]
Let $w_\lambda\mathrm in \mathcal{C}_\lambda$ be a minimizer
for $\lambda\mathrm in\Lambda$, as the one given in Section \ref{sec:Some notation and preliminary results}:
then, since $\overline{w}_\lambda=0$ and $||\nabla w_\lambda||_{L^2(M)}^2=\beta_\lambda
\to 0$ when $\lambda \uparrow \lambda_{max}$, it follows by Poincar\'{e} -Wirtinger's inequality that
$w_\lambda \to 0$ in $H^1(M;g_0)$.
Applying Moser-Trudinger's inequality, we also have
$e^{2w_\lambda}\to 1$ in $L^p(M)$ for any $p\mathrm in \left[1,\mathrm infty \right)$. Therefore, by H\"older's inequality, we obtain that for any $v\mathrm inH^1(M;g_0)$
\[
\mathrm int_M f_\lambda v e^{2w_\lambda} d\mu_{g_0} \to \mathrm int_M (f_0-\overline{f_0}) v d\mu_{g_0}
\]
when $\lambda\uparrow \lambda_{max}$. We recall that, for any $\lambda\mathrm in\Lambda$, $w_\lambda$ solves
\[
\mathrm int_M (\nabla w_\lambda,\nabla v)_{g_0} d\mu_{g_0} =
\mu(\lambda) \mathrm int_M f_\lambda v e^{2w_\lambda} d\mu_{g_0}, \;\;\; v\mathrm inH^1(M;g_0)
\]
where $\mu(\lambda)>0$ is a Lagrange multiplier. Choosing $v=f_0-\overline{f_0}$, we obtain for
$\lambda \uparrow \lambda_{max}$
\[
0 = \lim_{\lambda \uparrow \lambda_{max}} \mu(\lambda) \;\mathrm int_M (f_0-\overline{f_0})^2 d\mu_{g_0}
\]
and therefore $\lim_{\lambda \uparrow \lambda_{max}} \mu(\lambda) =0 $.
Thus, using $L^p$-estimates, we obtain
\[
||w_\lambda||_{H^2(M)} \leq c\left(||\Delta w_\lambda ||_{L^2(M)} + ||w_\lambda||_{H^1(M)} \right) .
\]
Since
\[
||\mu(\lambda)f_\lambda e^{2w_\lambda} ||_{L^2(M)} \leq
\mu(\lambda) ||f_\lambda||_{\mathrm infty} \left[\mathrm int_M e^{4w_\lambda}d\mu_{g_0} \right]^{1/2}
\]
and $e^{4w_\lambda}\to 1$ in $L^1$ as $\lambda \uparrow \lambda_{max}$, it follows that
$||\Delta w_\lambda ||_{L^2(M)}\to 0$ and hence $w_\lambda$ converges to zero in $H^2(M,g_0)$.
By Sobolev's embedding results, we also have for any $\alpha\mathrm in \left[0,1 \right)$
\[
w_\lambda \to 0 \;\;\mbox{in} \;\; C^{0,\alpha}(M)
\]
when $\lambda \uparrow \lambda_{max}$.
Thus, using the bootstrap method and Schauder's estimates, we obtain $C^{2,\alpha}$ convergence as well.
Finally, we obtain that
\[
u_\lambda := w_\lambda + 1/2 \log \mu(\lambda),
\]
solution to equation (\ref{eqn: the Gauss curvature}), goes uniformly to $-\mathrm infty$ on $M$ when
$\lambda \uparrow \lambda_{max}$ and therefore it can can not admit any convergent subsequence.
This concludes the proof of Theorem \ref{thm: second main result}.
\mathrm end{proof}
\begin{remark}
Because of the conformal invariance of the Dirichlet energy and from convergence
$||\nabla u_\lambda||_{L^2(M)}^2\to 0$ as $\lambda \uparrow -\overline{f_0}=\lambda_{max}$, it follows that no
``fine structure'' can appear in the ``limit'' geometry of the surfaces $\left(M,e^{2u_\lambda}g_0\right)$,
independently of how we blow up the scale.
\mathrm end{remark}
\begin{thebibliography}{1}
\bibitem{Aubin91} Aubin, Thierry: {\mathrm it Some Nonlinear Problems in Riemannian Geometry}.
Springer-Verlag Berlin Heidelberg (1998)
\bibitem{Aubin94} Aubin, Thierry: {\mathrm it Sur le probl\`eme de la courbure scalaire prescrite}.
(French. English, French summary) [On the problem of prescribed scalar curvature]
Bull. Sci. Math. 118 (1994), no. 5, 465-474.
\bibitem{Berger71} Berger, Melvyn S.:
{\mathrm it Riemannian structures of prescribed Gaussian curvature for compact $2$-manifolds}.
J. Differential Geometry 5 (1971), 325-332.
\bibitem{Bismuth2000} Bismuth, Sophie: {\mathrm it Prescribed scalar curvature on a
$C^\mathrm infty$ compact Riemannian manifold of dimension two}.
Bull. Sci. Math. 124 (2000), no. 3, 239-248.
\bibitem{Borer-Galimberti-Struwe} Borer, Franziska; Galimberti, Luca; Struwe, Michael:
{\mathrm it ``Large'' conformal metrics of prescribed Gauss curvature on surfaces of higher genus}.
Comm. Math. Helv. (to appear)
\bibitem{Brezis-Merle91} Brezis, Haim; Merle, Frank: {\mathrm it Uniform estimates and blow-up
behavior for solutions of $-\Delta u=V(x)e^u$ in two dimensions}.
Comm. Partial Differential Equations 16 (1991), no. 8-9, 1223-1253.
\bibitem{Chang2004} Chang, Sun-Yung Alice:
{\mathrm it Non-linear elliptic equations in conformal geometry}.
Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Z\"urich, 2004.
\bibitem{Chang-Liu92} Chang, Kung Ching; Liu, Jia Quan:
{\mathrm it A Morse-theoretic approach to the prescribing Gaussian curvature problem.}
Variational methods in nonlinear analysis (Erice, 1992), 55-62, Gordon and Breach, Basel, 1995.
\bibitem{Chen-Li91} Chen, Wen Xiong; Li, Congming:
{\mathrm it Classification of solutions of some nonlinear elliptic equations}.
Duke Math. J. 63 (1991), 615-622.
\bibitem{Cheng-Lin97} Cheng, Kuo-Shung; Lin, Chang-Shou:
{\mathrm it On the asymptotic behavior of solutions of the conformal Gaussian curvature equations in
$\mathbbm R^2$}. Math. Ann. 308 (1997), 119-139.
\bibitem{Ding-Liu95} Ding, Wei Yue; Liu, Jiaquan:
{\mathrm it A note on the prescribing Gaussian curvature on surfaces},
Trans. Amer. Math. Soc. 347 (1995), 1059-1066.
\bibitem{Huber57} Huber, Alfred:
{\mathrm it On subharmonic functions and differential geometry in the large}.
Comment. Math. Helv. 32 (1957), 13-72.
\bibitem{Kazdan-Warner74} Kazdan, Jerry L.; Warner, F. W.: {\mathrm it
Curvature functions for compact $2$-manifolds}.
Ann. of Math. (2) 99 (1974), 14-47.
\bibitem{Kazdan-Warner75} Kazdan, Jerry L.; Warner, F. W.: {\mathrm it
Scalar curvature and conformal deformation of Riemannian structure}.
J. Differential Geometry 10 (1975), 113-134.
\bibitem{Li-Shafrir94} Li, Yan Yan; Shafrir, Itai: {\mathrm it
Blow-up analysis for solutions of $-\Delta u=V(x)e^u$ in dimension two}.
Indiana Univ. Math. J. 43 (1994), no. 4, 1255-1270.
\bibitem{Martinazzi2009} Martinazzi, Luca: {\mathrm it Concentration-compactness phenomena
in the higher order Liouville's equation}. J. Funct. Anal. 256 (2009), no. 11, 3743-3771.
\bibitem{Moser73} Moser, J.: {\mathrm it
On a nonlinear problem in differential geometry}.
Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 273-280.
Academic Press, New York, 1973.
\bibitem{Struwe88a} Struwe, Michael:
{\mathrm it Critical points of embeddings of $H^{1,n}_0$ into Orlicz spaces}.
Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire 5 (1988), no. 5, 425-464.
\bibitem{Struwe88b}Struwe, Michael:
{\mathrm it The existence of surfaces of constant mean curvature with free boundaries}.
Acta Math. 160 (1988), no. 1-2, 19-64.
\bibitem{Struwe93} Struwe, Michael:
{\mathrm it Une estimation asymptotique pour le mod\`ele de Ginzburg-Landau}.
[An asymptotic estimate for the Ginzburg-Landau model]
C. R. Acad. Sci. Paris Sr. I Math. 317 (1993), no. 7, 677-680.
\bibitem{Struwe-Tarantello} Struwe, Michael; Tarantello, Gabriella:
{\mathrm it On multivortex solutions in Chern-Simons gauge theory}.
Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), no. 1, 109-121.
\bibitem{Zeidler} Zeidler, Eberhard: {\mathrm it Nonlinear Functional Analysis and its Applications III}.
Springer-Verlag New york (1985)
\mathrm end{thebibliography}
\mathrm end{document} | math | 66,059 |
\begin{document}
\title{Semilinear automorphisms of reductive algebraic groups}
\author[1]{Thierry Stulemeijer\thanks{Postdoctoral fellow at Justus-Liebig Universität Giessen}}
\affil[1]{\small{Justus-Liebig Universität Giessen, 35392 Giessen, Germany}}
\date{March 01, 2019}
\maketitle
\begin{abstract}
Let $ G $ be a connected reductive algebraic group over a field $ k $. We study the group of semilinear automorphisms $ \Aut (G\to \Spec k) $ consisting of algebraic automorphisms of $ G $ over automorphisms of $ k $. We focus on the exact sequence $ 1\to \Aut G\to \Aut (G\to \Spec k)\to \Aut_{G}(k)\to 1 $. When $G$ is quasi-split, we show that $\Aut_{G}(k)$ is isomorphic to $\Aut_{\mathcal{R}(G)}(k)$, where $\mathcal{R}(G)$ denotes the scheme of based root datum of $G$. Furthermore, the exact sequence $ 1\to \Aut G\to \Aut (G\to \Spec k)\to \Aut_{G}(k)\to 1 $ splits if and only if the exact sequence $ 1\to \Aut \mathcal{R}(G) \to \Aut (\mathcal{R}(G) \to \Spec k)\to \Aut_{\mathcal{R}(G)}(k)\to 1 $ splits. As a corollary, we get many examples of algebraic groups $ G $ over $ k $ whose group of abstract automorphisms does not decompose as the semidirect product of $ \Aut G $ with $ \Aut_G(k) $. We also study the same questions for inner forms of $ \SL_n $ over a local field.
\end{abstract}
\tableofcontents
\section{Introduction}\label{Sec:introchapsemilinear}
The so-called abstract automorphisms of (the rational points of) a reductive algebraic group have been studied extensively in the literature. In 1955, J.\ Dieudonn\'e wrote a comprehensive treatise (see \cite{Di71} for the third edition) covering the case of classical groups, and even going beyond the world of algebraic groups (since he also considers classical groups over division algebras that are infinite dimensional over their center). Many results in this area have been subsumed in the famous article \cite{BT73} by A.\ Borel and J.\ Tits. To wit, here is one of their results:
\begin{theorem*}[{\cite{BT73}*{excerpt of Corollaire~8.13}}]\label{Thm:abstract homo.}
Let $ k $ be an infinite field, and let $ G $ be an absolutely simple algebraic group over $ k $, which is assumed to be isotropic and adjoint. Furthermore, if $ k $ is of characteristic $ 2 $ or $ 3 $, assume that $ k $ is not perfect. Let $ \alpha $ be an automorphism of $ G(k) $, considered as an abstract group. Then there exists a unique automorphism $ \varphi \colon k\to k $ and a unique semilinear automorphism $ f_{\varphi} \colon G\to G $ over $ \varphi $ such that for $ g\in G(k) = \Hom_{\text{k-schemes}}(\Spec k,G) $, we have $ \alpha (g) = f_{\varphi}\circ g\circ (\Spec \varphi)^{-1} $.
\end{theorem*}
By a \textbf{semilinear automorphism} $ f_{\varphi} $ of a $ k $-group scheme $ G $ over an automorphism $ \varphi \colon k\to k $, we mean that we have the following commutative diagram in the category of group schemes
\begin{center}
\begin{tikzpicture}[->]
\node (1) [anchor=east] {$ G $};
\node (2) [right=1.5cm of 1] {$ G $};
\node (3) [below=0.5cm of 1] {$ \Spec k $};
\node (4) [below=0.5cm of 2] {$ \Spec k $};
\path [every node/.style={font=\sffamily\small}]
(1) edge node [above] {$ f_{\varphi} $} (2)
edge node {} (3)
(2) edge node {} (4)
(3) edge node [above] {\footnotesize{$ \Spec \varphi $}} (4);
\end{tikzpicture}
\end{center}
where the vertical arrows are the structural morphisms of the $ k $-scheme $ G $. Note that if $G$ is realised as a matrix group, and if $g=(g_{ij})\in G(k)$, then $ f_{\varphi}\circ g\circ (\Spec \varphi)^{-1} $ is given by the matrix whose $ij$-th coefficient is $\varphi^{-1}((f_{\varphi})_{ij}(g))$, so that our convention differs from \cite{BT73} (see Remark~\ref{Rem:difference between B-T and me} for a discussion of this difference).
Another way to phrase the Borel--Tits theorem is to say that the group $ \Aut_{\text{abstract}}(G(k)) $ of abstract automorphisms of $ G(k) $ fits in the exact sequence $ 1\to \Aut G\to \Aut_{\text{abstract}}(G(k))\to \Aut (k) $. Letting $ \Aut_G(k) $ denote the image of $ \Aut_{\text{abstract}}(G(k)) $ in $ \Aut(k) $, it is natural to wonder what $ \Aut_G(k) $ is, and whether the group of abstract automorphisms of $ G(k) $ splits as the semi-direct product $ (\Aut G)\rtimes \Aut_G(k) $. While this semi-direct product decomposition does hold when G is $ k $-split, it turns out that it may fail in general, even if G is quasi-split. One of the aims of the present paper is to address this issue.
To illustrate some of our results in the most concrete way, here is a corollary of our results (we refer the reader to Corollary~\ref{Cor:very explicit non-splitting} for a proof of this statement and to Remark~\ref{Rem:an explicit description of quasi-split PGU} for an explicit realisation of the algebraic group appearing in this corollary) :
\begin{corollary*}[of Theorem~\ref{Thm: MainThm2}]
Let $ G $ be the quasi-split, absolutely simple, adjoint algebraic group of type $ ^2A_{n-1} $ over a field $ k $ with corresponding quadratic separable extension $ l $. Let $ k_0 $ be the field of rational numbers $ ( $respectively the field of $ p $-adic numbers for some prime $ p) $ and assume that $ k $ and $ l $ are possibly infinite $ ( $respectively finite$ ) $ Galois extensions of $ k_0 $. Then $ \Aut_G(k) = \Gal (k/k_0) $ and the short exact sequence $ 1\to \Aut G\to \Aut_{\text{abstract}}(G(k))\to \Aut_G(k)\to 1 $ splits if and only if the short exact sequence of abstract groups $ 1\to \Gal (l/k)\to \Gal (l/k_0)\to \Gal (k/k_0)\to 1 $ splits.
\end{corollary*}
We now recast the problem in a more useful way for us. Let $ \Aut (G\to \Spec k) $ denotes the group of semilinear automorphisms of $ G $. We can then rephrase the Borel--Tits theorem as saying that under the assumptions of the theorem, the natural homomorphism $ \Aut (G\to \Spec k)\to \Aut_{\text{abstract}} (G(k)) $ is an isomorphism. The rest of the paper is concerned with the study of $ \Aut (G\to \Spec k) $.
Given a $ k $-group scheme $ G $ (in this paper the letter $ k $ is exclusively used to designate a field), we have a homomorphism $ \Aut (G\to \Spec k)\to \Aut (k)\colon f_{\varphi}\mapsto \varphi^{-1} $. Let $ \Aut_{G}(k) $ denotes the image of this homomorphism, and let $ \Aut G $ denotes the group of $ k $-algebraic automorphisms of $ G $, or in other words the kernel of $ \Aut (G\to \Spec k)\to \Aut (k) $. In summary, we defined the exact sequence $ 1\to \Aut G\to \Aut (G\to \Spec k)\to \Aut_G (k)\to 1 $.
It seems that not much is known about the subgroup $\Aut_G(k)\leq \Aut (k)$ in general (in \cite{Di71}*{p. 18, last paragraph}, J.\ Dieudonn\'e makes a related comment, though not exactly about $ \Aut_G(k) $). Actually, for $G = \textbf{SL}_n(D)$ with $D$ a non-commutative finite dimensional central division algebra, $\Aut_G(k)$ is isomorphic to the group of outer automorphisms of $D$ (or to a degree $ 2 $ extension of this group). But computing the latter is probably hard (for example, in \cite{Ha05} T.\ Hanke computes explicitly an outer automorphism of a division algebra of degree $3$ over a number field, and then uses it to construct a non-crossed product division algebra over $\mathbf{Q}(\!(t)\!)$).
Nonetheless, in some special cases, $\Aut_G(k)$ is easily understood. For example, when $G$ is a split connected reductive group, then $\Aut_G(k) = \Aut (k)$ (this follows directly from the fact that split connected reductive groups are defined over the prime field of $k$). The next easiest case should be to compute $ \Aut_G(k) $ for quasi-split groups, and this is indeed feasible.
\begin{theorem}\label{Thm: MainThm1}
Let $G$ be a connected reductive $k$-group and let $\mathcal{R}(G)$ be its $k$-scheme of based root datum. There exists a homomorphism $\Aut (G\to \Spec k)\to \Aut (\mathcal{R}(G)\to \Spec k) $ preserving the underlying automorphism of $k$ and whose kernel is $ (\Ad G)(k) $. Hence $ \Aut_G(k)\leq \Aut_{\mathcal{R}(G)}(k) $. If $G$ is quasi-split, the corresponding exact sequence $1\to (\Ad G)(k)\to \Aut (G\to \Spec k)\to \Aut (\mathcal{R}(G)\to \Spec k)$ splits. Hence if $ G $ is quasi-split $\Aut_G(k) = \Aut_{\mathcal{R}(G)}(k)$.
\end{theorem}
The $k$-scheme of based root datum $\mathcal{R}(G)$ appearing in this result is an object (see Definition~\ref{Def:scheme of based root datum}) which can be described as a variation on the scheme of Dynkin diagram as defined in \cite{Gil-Pol11}*{Exposé 24, Section 3}. In the end, the construction of the $k$-scheme of based root datum is a restatement of \cite{Gil-Pol11}*{Exposé 24, Théorème 3.11}. Let us directly give a brief description of what $\mathcal{R}(G)$ is in concrete terms: for $G_0$ a split connected reductive $k$-group, $\mathcal{R}(G_0)$ is defined to be the constant object on $\Spec k$ associated to the based root datum $R = (M,M^{\ast},R,R^{\ast},\Delta )$ of $G_0$. Now given a $k_s/k$-form $G$ of $G_0$, one can choose a cocycle $c\colon \Gal (k_s/k)\to \Aut((G_0)_{k_s})$ defining $G$ and compose it with the projection $\Aut((G_0)_{k_s})\to \Aut R$ to obtain a twist of $\mathcal{R}(G_0)$ which we define to be the $k$-scheme of based root datum of $G$. In light of this description, it is clear that $\mathcal{R}(G) \cong \mathcal{R}(G')$ (in the category of based root datum over $k$) if and only if $G$ and $G'$ are inner forms of each other.
The proof of Theorem~\ref{Thm: MainThm1} consists mainly of a straightforward adaptation of \cite{Gil-Pol11}*{Exposé 24, Théorème 3.11} to the semilinear situation. One crucial step is to show that taking the scheme of based root datum of $ G $ commutes with base change (see Lemma~\ref{Lem:base change commute with taking root datum}). We chose to prove this in concrete terms, by using cocycle computations adapted to the semilinear situation. Those cocycle computations also lead to the following Galois cohomological formulation.
\begin{theorem}\label{Thm: MainThm3}
Let $ G $ be a connected reductive $k$-group and let $ \mathcal{R}(G) $ be its $ k $-scheme of based root datum. We have two exact sequences
\begin{align*}
1\to (\Ad G)&(k)\to \Aut (G\to \Spec k)\to \Aut (\mathcal{R}(G)\to \Spec k)\to H^1(k,(\Ad G)(k_s))\\
&1\to \Aut G\to \Aut (G\to \Spec k)\to \Aut (k)\to H^1(k,\Aut G_{k_s}).
\end{align*}
\end{theorem}
We refer the reader to Section~\ref{Sec:Semilinear Galois cohomology} for the definition of the coboundary maps involved in those exact sequences. After proving Theorem~\ref{Thm: MainThm3}, we illustrate how it can be used by computing $ \Aut_G(k) $ when $ G\cong \textbf{SL}_1(D) $ and $ D $ is a division algebra of degree $ 3 $ over $ k $. In doing so, we recover some of the results in \cite{Ha07} without needing to introduce bycyclic algebras (see Lemma~\ref{Lem:Hanke result part 1}, Lemma~\ref{Lem:Hanke result part 2} and Remark~\ref{Rem:applicability of these computations}).
Once we have some control over $\Aut_G(k)$, it is natural to wonder whether the exact sequence $ 1\to \Aut G \to \Aut (G\to \nolinebreak \Spec k)\to \Aut_G (k)\to 1 $ splits. Again, for $G$ a split connected reductive algebraic group, this exact sequence does split (a statement already made in \cite{Tits74}*{Corollary~5.10}) because such a group is defined over its prime field. But somewhat surprisingly, this is not any more the case for a general quasi-split group.
\begin{theorem}[The bowtie theorem]\label{Thm: MainThm2}
Let $ G $ be a connected reductive $k$-group which is quasi-split, and let $\mathcal{R}(G)$ be its $k$-scheme of based root datum. Then the short exact sequence $$ 1\to \Aut G \to \Aut (G\to \Spec k)\to \Aut_G (k)\to 1 $$ splits if and only if the short exact sequence $$ 1\to \Aut \mathcal{R}(G)\to \Aut (\mathcal{R}(G)\to \Spec k)\to \Aut_{\mathcal{R}(G)} (k)\to 1 $$ splits.
\end{theorem}
As it turns out, the bowtie theorem (whose name is due to the diagram appearing in its proof) is a direct corollary of Theorem~\ref{Thm: MainThm1} (see the end of Section~\ref{Sec:RD's scheme} for the proof).
When $G$ is absolutely simple, we can identify the short exact sequence $ 1\to \Aut \mathcal{R}(G)\to \Aut (\mathcal{R}(G)\to \Spec k)\to \Aut_{\mathcal{R}(G)} (k)\to 1 $ as a sequence involving various automorphism groups of fields naturally associated with $ G $ (see Proposition~\ref{Prop:epplicit SES for Dyn}). Using this description, we can then give many explicit examples of absolutely simple, quasi-split algebraic $ k $-groups $G$ for which $ \Aut (G\to \Spec k) $ is not a split extension of $ \Aut_G(k) $. The corollary given at the beginning of this introduction illustrates this in the most concrete way.
In the last section of the paper, we also explore the same questions when $ G $ is an inner form of $ \SL_n $ over a local field $ K $ (see the beginning of Section~\ref{Sec:SL_n(D)} for a precise definition of what we mean by a local field). The first step is to get some control over $ \Aut_G(K) $. Actually, a direct corollary of the results of T.\ Hanke in \cite{Ha07} is that in this case, $ \Aut_G(K) = \Aut (K) $.
\begin{theorem}[corollary of \cite{Ha07}]\label{Thm:Main Thm 2.1}
Let $ K $ be a local field, let $ D $ be a central division algebra over $K$ of degree $ d $, and consider the algebraic group $ G = \textbf{SL}_n(D) $. Then $ \Aut_{G}(K) = \Aut (K) $.
\end{theorem}
In \cite{Ha07}, T.\ Hanke does not mention local fields, but he gives an algorithm over an arbitrary field to compute outer automorphisms of cyclic division algebras, and this implies the result for local fields. In fact, we only need to use the simplest version of his algorithm and for the ease of the reader we give a self contained proof of Theorem~\ref{Thm:Main Thm 2.1} in Corollary~\ref{Cor:etxending auto to D}. Also note that in characteristic $0$, this result has probably been known for a long time since it is a direct consequence of \cite{EML48}*{Corollary~7.3}.
Finally, we obtain an explicit characterisation for the splitting of the exact sequence $ 1\to \Aut G\to \Aut (G\to \Spec K)\to \Aut _G(K)\to 1 $ for $ G $ an inner form of $ \SL_n $ over a local field.
\begin{theorem}\label{Thm:Main Thm 2.2}
Let $ K $ be a local field, let $ D $ be a central division algebra over $K$ of degree $ d $, and consider the algebraic $K$-group $ G = \textbf{SL}_n(D) $. The short exact sequence $ 1\to \Aut G\to \Aut (G\to \Spec K)\to \Aut _G(K)\to 1 $ splits if and only if $ \gcd (nd,[K:K']) $ divides $ n $ for all subfields $ K'\leq K $ such that $ K/K' $ is finite Galois.
\end{theorem}
As we prove in Proposition~\ref{Prop:existence of Galois subfield of some degree}, for $ K = \mathbf{F}_{p^i}(\!(T)\!) $, the condition ``$ \gcd (nd,[K:K']) $ divides $ n $ for all subfields $ K'\leq K $ such that $ K/K' $ is finite Galois'' is equivalent to requiring that $ \gcd (d,p) = 1 $ and that $ \gcd (nd,i(p^i-1)) $ divides $ n $, so that this criterion is very explicit in characteristic $p$. On the other hand, in characteristic $0$ note that $\mathbf{Q}_p$ is rigid (see Definition~\ref{Def:strongly rigid field} and Lemma~\ref{Lem:Qp is rigid}), so that the condition is a finite one. See also Remark~\ref{Rem:Vivid example of splitting for divison algebras} for a vivid illustration of Theorem~\ref{Thm:Main Thm 2.2} in characteristic $ 0 $.
The necessity of the condition is proved in Corollary~\ref{Cor:non-splitting condition for SLn(D)}. The hard part of Theorem~\ref{Thm:Main Thm 2.2} is to show that our explicit criterion is sufficient. Whilst in characteristic $0$ this follows from Galois descent (see Theorem~\ref{Thm:splitting in char. 0 for SL_n(D)}), no descent technique can be used in characteristic $p>0$ since $\textbf{SL}_n(D)$ is not defined over the fixed field of $\Aut (K)$ (which is just $\mathbf{F}_p$) when $D$ is non commutative. Hence we have to work by hand and give the splitting explicitly. In order to achieve this, for $ K = \mathbf{F}_{p^i}(\!(T)\!) $, we decompose $ \Aut (K) $ as $ (J(K)\rtimes \mathbf{F}_{p^i}^{\times})\rtimes \Gal (K/\mathbf{F}_{p}(\!(T)\!)) $ (see Definition~\ref{Def:J(K) and C(K)} for the definition of $ J(K) $). Fortunately, it is easy enough to find an explicit section of $ \Aut (G\to \Spec K)\to \Aut _G(K) $ for the $ J(K) $ components, and the theory of Galois descent predicts when a section to $ \Aut (G\to \Spec K)\to \Aut _G(K) $ exists for the component $ \mathbf{F}_{p^i}^{\times} \rtimes \Gal (K/\mathbf{F}_{p}(\!(T)\!)) $. It thus suffices to compute explicitly those sections predicted by Galois descent, and to check that the formulas we found on each component can be glued together.
\section{Semilinear automorphisms and Galois descent}\label{Sec:Semiauto and Galois descent}
For the rest of the paper, the letter $ k $ stands for an arbitrary field. By a $ k $-group scheme, we mean an affine group scheme of finite type over $ k $. A smooth $ k $-group scheme is called an algebraic group. Given an object $ X $ in a category, we write $ \Aut X $ for the automorphisms of $ X $ in that category. Also given a $ k $-scheme $ X $, we denote by \underline{$ \Aut $} $ X $ its $ k $-group functor of automorphisms (i.e.\ for any $ k $-algebra $ R $, (\underline{$ \Aut $} $ X)(R) = \Aut X_R $). With these conventions, for $ G $ a $k$-group scheme, $ \Aut G $ is the automorphism functor of $ G $ evaluated at $ k $, i.e.\ $ \Aut G = $ (\underline{$ \Aut $} $G)(k) $.
Let $ G $ be a $ k $-group scheme. We gave in the introduction the definition of a semilinear automorphism of $ G $. The vocabulary ``semilinear automorphism'' is already used in the literature (see for example \cite{FSS98}*{Section~1.2}). It has the same meaning than our usage, except that in those references, the underlying automorphisms of the base field are assumed to fix a subfield $ k_0 $ such that $ k/k_0 $ is Galois. We do not make this assumption, and for example in Section~\ref{Sec:SL_n(D)}, we consider the case of arbitrary automorphisms of $ k = \mathbf{F}_p(\!(T)\!) $, which is a more general situation.
In the literature, the notation $ \text{SAut}(G_{k_s}) $ is used for the group of semilinear automorphisms (see for examples \cite{BKLR12}*{Section~3.2} and also the references therein). We prefer to use the notation $ \Aut (G\to \Spec k) $ so that the ground field explicitly appears in the notation.
\begin{remark}
It is tempting to define a ``semilinear automorphism sheaf of $ k $-algebras'' such that $ \Aut (G\to \Spec k) $ would be its $k$-rational points. Unfortunately, this is not possible, because we do not know how to extend functorially automorphisms of $ k $ to automorphisms of an arbitrary $ k $-algebra $ R $.
\end{remark}
Let us continue by recalling some standard vocabulary.
\begin{definition}\label{Def:Notation Aut(l/k) and base change}
Let $ \varphi \colon k\to l $ be a field homomorphism (if $ l $ is a field containing $ k $, we take $ \varphi $ to be the identity), let $ G, G' $ be $ k $-group schemes and let $ H $ be an $l$-group scheme.
\begin{enumerate}
\item The group of automorphisms of $ l $ whose restriction to $ \varphi (k) $ is trivial is denoted $ \Aut (l/k) $.
\item We set $ \varphi^{\ast} = \Spec \varphi $. We denote the base change of $ G $ along $ \Spec l\xrightarrow{\varphi^{\ast}} \Spec k $ either by $ G_{l} $ or by $ ^{\varphi}G $. If $ G_{l} $ is isomorphic to $ H $ (as an $l$-group scheme), we say that $ G $ is an $ l/k$-\textbf{form of $ H $} (or just a form of $H$ if the field extension is understood from the context). If there exists an $ l/k $-form of $ H $, we say that $ H $ is \textbf{defined over $ k $}.
\item For $ f\colon G\to G' $ a homomorphism of $ k $-group schemes, we denote by $ ^{\varphi}f\colon\,^{\varphi}G\to\,^{\varphi}G' $ the base change of $ f $ along $ \varphi^{\ast} $.
\end{enumerate}
\end{definition}
\begin{remark}\label{Rem:difference between B-T and me}
Having set up our notations, let us elucidate the difference between our conventions and the conventions in \cite{BT73}. Given a $ k $-group scheme $ G $, a $ k' $-group scheme $ G' $ and given an abstract homomorphism $ \alpha \colon G(k)\to G'(k') $, A.\ Borel and J.\ Tits aim to obtain a field homomorphism $ \varphi \colon k\to k' $ and an isogeny $ \beta \colon \,^{\varphi}G\to G' $ such that for $ g\in G(k) = \Hom_{\text{k-schemes}}(\Spec k,G) $, $ \alpha (g) = \beta \circ \,^{\varphi}g $. The following commutative diagram summarises the situation:
\begin{center}
\begin{tikzcd}[row sep=2.5em, column sep=3em]
G' \arrow{dr} &^{\varphi}G \arrow[l,swap,"\beta"]\arrow{r}\arrow[d,shift right=1.5ex] & G \arrow[d,shift right=1.5ex]\\
& \Spec k' \arrow[r,"\Spec \varphi "] \arrow[u,swap ,"^{\varphi}g"] & \Spec k \arrow[u,swap ,"g"]
\end{tikzcd}
\end{center}
On the other hand, the present paper focuses entirely on the group of semilinear automorphisms of $ G $. To keep the Borel--Tits convention, one should define this group as $ \lbrace \Isom_{k\text{-grp schemes}}(~^{\varphi}G, G)~\vert~ \linebreak \varphi \in \Aut (k)\rbrace $. We prefer to use the more natural definition that a semilinear automorphism over $ \varphi \in \Aut (k) $ is a commutative diagram of the following kind (note that either one of the red arrows determines the other):
\[
\begin{tikzcd}[row sep=2.5em]
G \arrow[drr,red,"f_{\varphi}"] \arrow[dr,red] \arrow[ddr] & & \\
&^{\varphi}G \arrow[r] \arrow[d] & G \arrow[d]\\
&\Spec k \arrow[r,"\Spec \varphi"] & \Spec k
\end{tikzcd}
\]
where $ f_{\varphi} $ and $ \Spec \varphi $ are both automorphisms of group schemes (but they are not automorphisms of $ k $-group schemes when $ \varphi $ is not the identity). In this setting, there are two ways (admittedly not as natural as in the Borel--Tits setting) to obtain an abstract automorphism of $ G(k) $. Either we define this abstract automorphism proceeding ``from right to left'', in which case we would obtain the map $ G(k)\to G(k)\colon g\mapsto f_{\varphi}^{-1}\circ g\circ \Spec \varphi $. Or we proceed ``from left to right'', in which case we obtain the map $ G(k)\to G(k)\colon g\mapsto f_{\varphi}\circ g\circ (\Spec \varphi)^{-1} $. We chose the latter option.
\end{remark}
The following elementary observation plays a fundamental role in this work.
\begin{lemma}\label{Lem:field of definition are in AutG(k)}
Let $ k\leq l $ be a field extension of $ k $, let $ G $ be an $ l $-group scheme and assume that $ G $ is defined over $k$. Then there exists a homomorphism $ \Aut (l/k)\to \Aut (G \to \Spec l) $ whose composition with $ \Aut (G\to \Spec l)\to \Aut_G(l) $ is the identity on $ \Aut (l/k) $. In particular, $ \Aut_G(l) $ contains $ \Aut (l/k) $.
\end{lemma}
\begin{proof}
Let $ H $ be an $ l/k $-form of $ G $. For $ \varphi \in \Aut (l/k) $, we define $$ f_{\varphi}=\Id_H \times \varphi^*\colon H\times_{\Spec k} \Spec l\to H\times_{\Spec k} \Spec l. $$ The map $ \Aut (l/k)\to \Aut (G\to \Spec l)\colon \varphi \mapsto f_{\varphi^{-1}} $ is a homomorphism. Furthermore, its composition with $ \Aut (G\to \Spec l)\to \Aut_G(l) $ is the identity on $ \Aut (l/k) $, as wanted.
\end{proof}
In fact, if the field extension $ l/k $ appearing in Lemma~\ref{Lem:field of definition are in AutG(k)} is finite Galois, then we have a converse to Lemma~\ref{Lem:field of definition are in AutG(k)} by the theory of Galois descent.
\begin{theorem}[Galois descent]\label{Thm:Galois descent}
Let $ k\leq l $ be a field extension of $ k $ such that $ l/k $ is a finite Galois extension and let $ G $ be an $ l $-group scheme. If there exists a homomorphism $ \Gal (l/k)\to \Aut (G\to \Spec l) $ whose composition with $ \Aut (G\to \nolinebreak \Spec l)\to \Aut_G(l) $ is the identity on $ \Gal (l/k) $, then $ G $ is defined over $ k $.
\end{theorem}
\begin{proof}
This is a classical result from descent theory, see \cite{Poo17}*{Section~4.4}. Note that giving such a homomorphism is the same as giving a descent datum on $ G $ by \cite{Poo17}*{Proposition~4.4.2}, so that the result holds by \cite{Poo17}*{Corollary~4.4.6}.
\end{proof}
\begin{remark}
One could also treat the case of infinite Galois extensions by adding a continuity assumption as in \cite{FSS98}*{Remark~1.15}, but we do not need it in our work. See also \cite{Poo17}*{Remark~4.4.8} for how to deal with infinite Galois extensions.
\end{remark}
In view of the strong link between Galois descent and semilinear automorphisms, it seems natural that there should be a cocycle interpretation of semilinear automorphisms. We now take some time to set up this formalism in detail.
\begin{definition}\label{Def:gammatilde and Idgamma}
\begin{enumerate}
\item Let $ k\leq l $ be a field extension of $ k $, let $ G_0 $ be an $l$-group scheme and let $G$ be an $l/k$-form of $G_0$. Choose an isomorphism $ G_0\cong G_l $, or in other words choose an exact diagram
\begin{center}
\begin{tikzpicture}[->]
\node (1) [anchor=east] {$ G_0 $};
\node (2) [right=1.5cm of 1] {$ G $};
\node (3) [below=0.5cm of 1] {$ \Spec l $};
\node (4) [below=0.52cm of 2] {$ \Spec k $};
\path [every node/.style={font=\sffamily\small}]
(1) edge node [above] {$ \pi_1 $} (2)
edge node [left] {$ t $} (3)
(2) edge node [right] {$ s $} (4)
(3) edge node [above] {\footnotesize{$ \pi_0 $}} (4);
\end{tikzpicture}
\end{center}
For any $\gamma \in \Aut (l/k)$, by the definition of base change there exists a unique isomorphism of $G_0$ above $\gamma $ such that the following diagram commutes:
\[
\begin{tikzcd}[row sep=2.5em]
&&
G_0 \arrow[dd,swap,"t" near start] \arrow[dr,"\pi_1"] \\
& G_0 \arrow[rr,crossing over,swap,"\pi_1" near start] \arrow[ur,crossing over,"\tilde{\gamma}_{G}"] &&
G \arrow[dd,"s"] \\
&& \Spec l \arrow[dr,"\pi_0"] \\
& \Spec l \arrow[rr,"\pi_0"] \arrow[ur,"\gamma^*"] \arrow[uu,<-,crossing over,"t"]&& \Spec k
\end{tikzcd}
\]
We denote this isomorphism $\tilde{\gamma}_{G}$.
\item For $G_0$ a split connected reductive $l$-group we assume that an isomorphism with $ H_l $ has been chosen, where $ H $ is a split algebraic group over the prime field of $ l $. Now in this special situation, for $\gamma \in \Aut (l)$, instead of
$\tilde{\gamma}_{H}$
we use the more suggestive notation $\Id_{\gamma}$.
\end{enumerate}
\end{definition}
\begin{remark}
\begin{enumerate}[(i)]
\item Note that when $l/k$ is a finite Galois extension, the collection $\lbrace $
$\tilde{\gamma}_{G}$
$\rbrace_{\gamma \in \Gal(l/k)}$ is nothing but a descent datum on $G_0$ (as defined in \cite{Poo17}*{Proposition~4.4.2~(i)}) which descends to $G$.
\item Note that for $G_0$ a split connected reductive $ l $-group and $\gamma \in \Aut (l)$, if we choose a realisation of $G_0$ as a matrix group such that the realisation is defined over the prime field of $l$, then for $g=(g_{ij})\in G_0(l)$ and $\gamma \in \Aut (l)$, $ \Id_{\gamma} \circ \, g \circ (\gamma^*)^{-1} \in G_0(l) $ is given by the matrix whose $ij$-th coefficient is $\gamma^{-1}(g_{ij})$. This explains why we prefer to use the notation $\Id_{\gamma}$ in this situation.
\end{enumerate}
\end{remark}
We now study the behaviour of $\tilde{\gamma}_{G}$ under base change.
\begin{lemma}\label{Lem:choosing a good base change}
Let $k \leq l$ be a field extension of $ k $, let $ G_0 $ be an $ l $-group scheme and let $G$ be a $l/k$-form of $G_0$. Fix an isomorphism $ G_l \cong G_0 $, or in other words fix an exact diagram
\begin{center}
\begin{tikzpicture}[->]
\node (1) [anchor=east] {$ G_0 $};
\node (2) [right=1.5cm of 1] {$ G $};
\node (3) [below=0.5cm of 1] {$ \Spec l $};
\node (4) [below=0.52cm of 2] {$ \Spec k $};
\path [every node/.style={font=\sffamily\small}]
(1) edge node [above] {$ \pi_1 $} (2)
edge node [left] {$ t $} (3)
(2) edge node [right] {$ s $} (4)
(3) edge node [above] {\footnotesize{$ \pi_0 $}} (4);
\end{tikzpicture}
\end{center}
Let $\alpha \in \Aut (k)$ and let $\beta \in \Aut (l)$ be such that $\beta |_{k} = \alpha $. Further assume that $G_0$ is split connected reductive. Then there exists a unique map $\pi_{\beta}\colon G_0\to G $ such that the following diagram commutes
\[
\begin{tikzcd}[row sep=2.5em]
G_0 \arrow[rr,"\Id_{\beta}"] \arrow[dr,swap,"\pi_{\beta}"] \arrow[dd,swap,"t"] &&
G_0 \arrow[dd,swap,"t" near start] \arrow[dr,"\pi_1"] \\
& G \arrow[rr,crossing over,"\Id_{G}" near start] &&
G \arrow[dd,"s"] \\
\Spec l \arrow[rr,"\beta^*" near end] \arrow[dr,swap,"\pi_0"] && \Spec l \arrow[dr,swap,"\pi_0"] \\
& \Spec k \arrow[rr,"\alpha^*"] \arrow[uu,<-,crossing over,"(\alpha^*)^{-1}s" near end]&& \Spec k
\end{tikzcd}
\]
Furthermore, all squares appearing in this diagram are exact.
\end{lemma}
\begin{proof}
The existence and uniqueness of $ \pi_{\beta} $ follows from the fact that the front square of the diagram is a base change. The fact that all squares are exact is a straightforward verification, using the fact that $ \alpha^* $ and $ \beta^* $ are isomorphisms.
\end{proof}
\begin{lemma}\label{Lem:the descent datum of a base change}
Keep the notations of Lemma~\ref{Lem:choosing a good base change}, so that in particular we chose an isomorphism $G_0\cong(\,^{\alpha }G)_l$ via the exact diagram
\begin{center}
\begin{tikzpicture}[->]
\node (1) [anchor=east] {$ G_0 $};
\node (2) [right=1.5cm of 1] {$ ^{\alpha}G=G $};
\node (3) [below=0.5cm of 1] {$ \Spec l $};
\node (4) [below=0.515cm of 2] {$ \Spec k $};
\path [every node/.style={font=\sffamily\small}]
(1) edge node [above] {$ \pi_{\beta} $} (2)
edge node [left] {$ t $} (3)
(2) edge node [right] {$ (\alpha^{\ast})^{-1} s $} (4)
(3) edge node [above] {\footnotesize{$ \pi_0 $}} (4);
\end{tikzpicture}
\end{center}
With these identifications of base change, for all $\gamma \in \Aut (l/k)$ we have $ \tilde{\gamma}_{\,^{\alpha}G} = \Id_{\beta}^{-1}(\widetilde{\beta^{-1}\gamma \beta})_{G}\Id_{\beta} $.
\end{lemma}
\begin{proof}
The proof follows from the commutativity of the following diagram
\[
\begin{tikzcd}[row sep=2.5em]
& G_0 \arrow[rrr,swap,"\Id_{\beta}"] \arrow[dr,swap,"\pi_{\beta}"] \arrow[ddd,swap,"t" near start] &&&
G_0 \arrow[ddd,"t" near start] \arrow[dl,"\pi_1"] & \\
&& G \arrow[r,crossing over,"\Id_{G}"] \arrow[ddd,swap,"(\alpha^*)^{-1}s"] &
G \arrow[ddd,"s"] & & \\
G_0 \arrow[rrrrr,crossing over, "\Id_{\beta}"] \arrow[uur,"\tilde{\gamma}_{\alpha_{G}}"] \arrow[urr,crossing over,"\pi_{\beta}" near end] \arrow[ddd,swap,"t"] & & & & & G_0 \arrow[uul,swap,"(\widetilde{\beta^{-1}\gamma \beta})_{G}"] \arrow[ull,crossing over,swap,"\pi_{1}" near end] \arrow[ddd,"t"] \\
& \Spec l \arrow[rrr,swap,"\beta^*"] \arrow[dr,swap,"\pi_0"] & & & \Spec l \arrow[dl,"\pi_0"] & \\
& & \Spec k \arrow[r,"\alpha^*"] & \Spec k & & \\
\Spec l \arrow[uur,swap,"\gamma^*"] \arrow[rrrrr,"\beta^*"] \arrow[urr,swap,"\pi_0" near end] & & & & & \Spec l \arrow[uul,"(\beta^{-1}\gamma\beta)^*"] \arrow[ull,"\pi_0" near end]
\end{tikzcd}
\]
Indeed, $ \tilde{\gamma}_{\alpha_G} $ is defined to be the unique map such that the left hand side of the diagram commutes. But the front side and the back side of the diagram commutes by the definition of $ \pi_{\beta} $ (see Lemma~\ref{Lem:choosing a good base change}), whilst the right hand side of the diagram commutes by definition of $ (\widetilde{\beta^{-1}\gamma \beta})_{G} $. Also note that $ (\beta^{-1}\gamma\beta)^* = \beta^*\gamma^*(\beta^{-1})^* $, so that the bottom square of the diagram commutes as well. A diagram chase then shows that $ \Id_{\beta}^{-1}(\widetilde{\beta^{-1}\gamma \beta})_{G}\Id_{\beta} $ satisfies the property uniquely defining $ \tilde{\gamma}_{\alpha_G} $, as was to be shown.
\end{proof}
We can now state a clean descent formula for semilinear automorphisms in terms of cocycles. In this formula, we use the fact that for $G_0$ a split connected reductive $l$-group, $\Aut (G_0\to \Spec l)\cong \Aut G_0\rtimes \Aut (l)$, where the splitting of the exact sequence $ 1\to \Aut G_0\to \Aut (G_0\to \Spec l)\to \Aut (l) $ is realised by $\gamma \mapsto \Id_{\gamma}^{-1}$ (see Definition~\ref{Def:gammatilde and Idgamma} for the notation $\Id_{\gamma}$). This thus defines a (left) action of $\Aut (l)$ on $\Aut G_0$ that we denote $^{\gamma}f$ (for $\gamma \in \Aut (l)$ and $f\in \Aut G_0$). Explicitly, we have $^{\gamma}f = \Id_{\gamma}^{-1}f\Id_{\gamma} $.
\begin{lemma}\label{Lem:Condition for a semilinear auto. to descend}
Let $G_0$ be a split connected reductive $ l $-group, let $ k $ be a subfield of $ l $ such that $ l/k $ is a $ ( $possibly infinite$ ) $ Galois extension, and let $ G $ be a $ l/k $-form of $ G_0 $. Let $\beta \in \Aut (l) $ be such that $\beta(k)=k $, and let $\alpha = \beta_{\vert_{k}}\in \Aut (k) $. Fix an isomorphism $ G_l\cong G_0 $ and let $ c\colon \Gal (l/k)\to \Aut G_0 $ be the corresponding cocycle. Finally, let $b\in \Aut G_0 $. Then $b\Id_{\beta}\in \Aut (G_0\to \Spec l)$ descends to a semilinear automorphism of $ G $ over $\alpha$ if and only if $c_{\beta^{-1}\gamma \beta}\,^{\beta^{-1}\gamma \beta}b\,^{\beta^{-1}}c_{\gamma}^{-1} = b $ for all $\gamma \in \Gal (l/k)$.
\end{lemma}
\begin{proof}
Recall that a morphism of $G_0$ over $\beta$ is equivalent to an $ l $-morphism from $G_0$ to $^{\beta}G_0$. In this correspondence, $b\Id_{\beta}$ corresponds to $^{\beta}b\in \Aut G_0$, as can be seen directly from the diagram
\[
\begin{tikzcd}[row sep=2.5em]
G_0 \arrow[dr,"b\Id_{\beta}"] \arrow[d,swap," \Id_{\beta}^{-1}b\Id_{\beta} = \,^{\beta}b"] & \\
^{\beta}G_0 = G_0 \arrow[r,swap,"\Id_{\beta}"] \arrow[d,swap,"t"] & G_0 \arrow[d,"t"]\\
\Spec k \arrow[r,"\beta^*"] & \Spec k
\end{tikzcd}
\] Now by the general theory for morphisms between schemes with a descent datum, the $ l $-morphism $^{\beta}b\colon G_0\to \,^{\beta}G_0$ descends to a $ k $-morphism $ G\to \,^{\alpha}G $ if and only if $(\tilde{\gamma}_{^{\alpha}G})^{-1}(\,^{\beta}b)\tilde{\gamma}_{G} = \,^{\beta}b $ for all $\gamma \in \Gal (l/k) $ (technically speaking, this is only true for finite Galois extensions, but our schemes are of finite type. Hence $ ^{\beta}b $ is defined over a finite Galois extension of $ k $ and we are just trying to descend from there). Using Lemma~\ref{Lem:the descent datum of a base change}, we get that $^{\beta}b$ descends if and only $\Id_{\beta}^{-1}(\widetilde{\beta^{-1}\gamma^{-1} \beta})_{G} \Id_{\beta}(\,^{\beta}b)\tilde{\gamma}_{G} = \,^{\beta}b$. Finally, to transfer this to a cocycle condition, recall that in the correspondence between descent datum and cocycle, we have (in our notations) $\tilde{\gamma}_{G} = c_{\gamma^{-1}}\Id_{\gamma}$ (this of course relies on the fact that we used the same isomorphism $G_0\cong G_l$ to define $c_{\gamma}$ and $\tilde{\gamma}$). Furthermore, by definition $c\colon \Gal(l/k)\to \Aut G_0 $ is a cocycle for the Galois action introduced before the statement of the theorem, i.e.\ for $\gamma , \delta \in \Gal(l/k)$ we have $c_{\gamma \delta} = c_{\gamma }\,^{\gamma}c_{\delta} = c_{\gamma}\Id_{\gamma}^{-1}c_{\delta}\Id_{\gamma}$. Hence, the conclusion of the theorem readily follows.
\end{proof}
\begin{remark}
In our conventions, if $ \gamma \in \Gal(l/k) $ appears in exponent, then it acts on the element appearing below it on the right. So if one wishes to put more parenthesis in the formula $ c_{\beta^{-1}\gamma \beta}\,^{\beta^{-1}\gamma \beta}b\,^{\beta^{-1}}c_{\gamma}^{-1} $, the unique way to do so respecting this convention is by writing $ c_{\beta^{-1}\gamma \beta}(\,^{\beta^{-1}\gamma \beta}b)(\,^{\beta^{-1}}c_{\gamma}^{-1}) $. Note also that $ \beta \in \Aut (k) $ acts by group automorphisms on $ \Aut G $, so that $ ^{\beta^{-1}}(c_{\gamma}^{-1}) = (\,^{\beta^{-1}}c_{\gamma})^{-1} $, i.e.\ there is no need for any parenthesis to distinguish the two.
\end{remark}
\begin{remark}
If $\beta $ is the identity, our formula specialises to the usual condition for $b$ to descend to a $ k $-automorphism of $ G $, namely $c_{\gamma}\,^{\gamma}bc_{\gamma}^{-1}=b $ for all $\gamma \in \Gal(l/k)$. Also note that for $\gamma \in \Gal(l/k)$, the automorphism $\tilde{\gamma}_{G} = c_{\gamma^{-1}} \Id_{\gamma} \in \Aut (G_0\to \Spec l)$ satisfies the condition to descend (of course, it descends to the trivial automorphism of $ G $).
\end{remark}
\section{Schemes of based root datum}\label{Sec:RD's scheme}
In \cite{Gil-Pol11}*{Expos\'e~$ 24 $, section~$ 3 $}, the authors define what they call a ``Dynkin's scheme'' of a reductive group $ G $. The strategy is to first define this Dynkin's scheme for split reductive groups, and then to use descent. The Dynkin's scheme is well suited to describe quasi-split semisimple groups that are adjoint or simply connected. Since there is not much more work to define a scheme of based root datum and since this allows us to treat the more general case of quasi-split reductive groups, we decided it was worth doing it.
In order to define a scheme of based root datum, we need the notion of a $ \Z $-module scheme and of perfect duality between two $ \Z $-module schemes. Recall that throughout the paper, the letter $ k $ stands for a field.
\begin{definition}
Let $ R $ be a $ k $-algebra and let $ M $ be a $ R $-scheme. $ M $ is called a $ \Z $-\textbf{module }$R$-\textbf{scheme} if $ M $ is a (non necessarily affine) commutative $ R $-group scheme.
\end{definition}
Recall that given any set $ E $ and a $ k $-algebra $ R $, we can consider the \textbf{constant object on} $E$ which is defined to be the $ R $-scheme $ E_R = \coprod \limits_{e\in E}\Spec R $. This defines a fully faithfull functor from the category of Sets to the category of $ R $-schemes, called the \textbf{constant object functor}. The constant object functor commutes with forming finite products (see \cite{Gil-Pol11a}*{Exposé~1, Section~1.8}). Hence given a $ \Z $-module $ M $, the constant scheme $ M_R $ acquires the structure of a $ \Z $-module $ R $-scheme. We can now define the notion of perfect pairing for $ \Z $-module $ k $-schemes.
\begin{definition}
Let $M,M'$ be two $\Z$-module $k$-schemes.
\begin{enumerate}
\item The \textbf{dual of} $M$, denoted $M^t$, is defined to be the functor from the category of $k$-algebras to the category of sets sending a $ k $-algebra $R$ to $\Hom_{R}(M_R,\Z_R)$.
\item We say that $M$ and $M'$ are in \textbf{duality} if there exists $f\in \Hom_k (M\times M'\to \Z_k)$. We say that the duality $f$ is \textbf{perfect} if for all $ k $-algebra $R$, the map $M'(R)\to M^t(R)\colon m\mapsto (f(.\,,m)\colon M_R\to \Z_R) $ is an isomorphism (here, for any $R$-algebra $R'$ and $n\in M_R(R')$, $f(.\,,m)(n) = f(n,m_{R'})$, with $m_{R'}$ being the image of $m$ under $M'(R)\to M'(R')$).
\end{enumerate}
\end{definition}
\begin{remark}
As usual in this situation, one should restrict the categories under considerations to avoid set theoretic problems. One way to do so is by using universe.
\end{remark}
Note that $M^t$ is a commutative group functor, and hence $M^t$ is a $\Z$-module $ k $-scheme if $M^t$ is representable. Also note that given $f\in \Hom_k(M,M')$ we can define $f^t\in \Hom_{k\text{-functors}}({M'}^t,M^t)$ by mimicking the definition for $\Z$-modules. Namely for all $ k $-algebras $ R $, $ f^t(R)\colon {M'}^t(R)\to M^t(R)\colon \alpha \mapsto \alpha \circ f_R $, where $ f_R\colon M_R\to M'_R $ denotes the base change of $ f $ to $ R $. We call $ f^t $ the \textbf{dual of} $f$.
By a \textbf{reduced based root datum} $R$, we mean a reduced root datum $ (M,M^*,\Phi ,\Phi^*)$ (as defined in \cite{Gil-Pol11}*{Exposé~21, Définition~1.1.1 and Definition~2.1.3}) together with a choice of simple roots $\Delta \subset \Phi $. We can finally give the definition of a $k$-scheme of root datum.
\begin{definition}\label{Def:scheme of based root datum}
\begin{enumerate}
\item A $k$-\textbf{scheme of based root datum} is a $5$-tuple $\mathcal{R} = (\mathcal{M},\mathcal{M}^*, \Psi ,\Psi^*, \Gamma )$ where:
\begin{enumerate}
\item $\mathcal{M}$ and $ \mathcal{M^*} $ are $ \Z $-module $ k $-schemes in perfect duality.
\item $ \Psi ,\Psi^* $ and $ \Gamma $ are finite $k$-schemes, and there are closed immersions $\Gamma \hookrightarrow \Psi \hookrightarrow \mathcal{M}$ and $ \Psi^*\hookrightarrow \mathcal{M}^* $.
\item There is an isomorphism of $ k $-schemes $ \Psi \cong \Psi^* $.
\item There exists a reduced based root datum $R = (M,M^*,\Phi ,\Phi^*, \Delta)$ and a finite Galois extension $l/k$ together with an isomorphism of $\Z$-module $l$-schemes $f\colon M_l\to \mathcal{M}_l$ such that $f$ induces an isomorphism of $ l $-schemes $\Phi_l\cong \Psi_l $, $f^t$ induces an isomorphism of $ l $-schemes $ {\Psi }_l^* \cong \Phi_l^*$, $f(\Delta_l) = \Gamma_l $ and the composition $ \Phi_l\cong \Psi_l\cong \Psi^*_l\cong \Phi_l^* $ induces the bijection $ \Phi \to \Phi^*\colon \alpha \mapsto \alpha^* $ given in the definition of $ R $. In this case, we say that $ \mathcal{R} $ is \textbf{of type} $ R $.
\end{enumerate}
\item Let $R = (M,M^*,\Phi ,\Phi^*, \Delta)$ be a reduced based root datum. Using the constant object functor, the $5$-tuple $R_k = (M_k,M^*_k,\Phi_k ,\Phi^*_k, \Delta_k)$ has a natural structure of a $ k $-scheme of based root datum. We call it the \textbf{split} $k$\textbf{-scheme of based root datum of type} $R$. A \textbf{split} $k$-\textbf{scheme of based root datum} is a split $k$-scheme of based root datum of type $R$ for some reduced based root datum $R$.
\item Given $\mathcal{R} = (\mathcal{M},\mathcal{M}^*, \Psi ,\Psi^*, \Gamma )$ and $\mathcal{R}' = (\mathcal{M}',{\mathcal{M}'}^*, \Psi ',{\Psi '}^*, \Gamma ')$ two $k$-schemes of based root datum, a $ k $-morphism $\mathcal{R}\to \mathcal{R}'$ is a $ k $-homomorphism $ f\colon \mathcal{M}\to \mathcal{M}' $ such that $f$ induces two $ k $-isomorphisms $\Psi\cong \Psi ' $ and $ \Gamma \cong \Gamma ' $, and such that $f^t$ induces a $ k $-isomorphism $ {\Psi '}^* \cong \Psi^*$.
\end{enumerate}
\end{definition}
\begin{remark}
Let us stress that with this definition, if a scheme of based root datum is of type $ R $, then $ R $ is a \textit{reduced} based root datum. It would be safer (but more tedious) to call these objects ``$ k $-schemes of reduced based root datum''.
\end{remark}
\begin{remark}\label{Rem:Root datum and Galois descent}
Let $ k_s $ be a separable closure of $ k $, let $R_k = (M_k,M^*_k,\Phi_k ,\Phi^*_k, \Delta_k)$ be a split $k$-scheme of based root datum of type $R$ and let $E = \Aut R$ (see \cite{Gil-Pol11}*{Exposé~21, Définition~6.1.1} for the definition of the morphisms in the category of based root datum). Then $ \Aut R_{k_s} = E$, and the action of $\Gal (k_s/k)$ on $E$ is trivial. Hence elements of $H^1(k,E)$ are continuous homomorphisms $\Gal(k_s/k)\to E$ up to conjugation. Also recall that $H^1(k,E)$ classifies $k$-schemes of based root datum of type $R$. Indeed, Galois descent for $M_k$ and $M_k^*$ is effective because they can be covered by quasi-affine open that are stable under $ \Gal(k_s/k) $ (hence effectivenes is ensured by \cite{Poo17}*{Theorem~4.3.5}), and the other structures (the duality $ M_k\times M^*_k\to \Z_k $, the closed immersions $ \Delta_k\hookrightarrow \Phi_k\hookrightarrow M_k $, $ \Phi^*_k\hookrightarrow M^*_k $ and the $ k $-isomorphism $ \Phi_k\cong \Phi_k^* $) will descend as well.
\end{remark}
\begin{remark}
If $R = (M,M^*,\Phi ,\Phi^*, \Delta)$ is a based root datum such that $\Phi$ is empty, then $\Aut R \cong GL_n(\Z )$ where $n$ is the rank of $M$. Hence in this case, $k$-schemes of based root datum of type $R$ classify $k$-tori of rank $n$.
\end{remark}
Note that given $ \alpha \in \Aut (k) $ and a $ k $-scheme of based root datum $ \mathcal{R} $, we have an obvious notion of base change of $ \mathcal{R} $ along $ \alpha $, and we denote this base change by $ ^{\alpha}\mathcal{R} $.
\begin{definition}\label{Def:semi-auto of root datum}
Let $ \mathcal{R} $ be a $k$-scheme of based root datum, and let $ \alpha \in \Aut (k) $. A \textbf{semilinear automorphism of} $ \mathcal{R} $\textbf{ over} $ \alpha $ is an isomorphism of $k$-schemes of based root datum $ f_{\alpha}\colon \mathcal{R}\to \,^{\alpha}\mathcal{R} $. Given a semilinear automorphism $ f_{\alpha} $ (respectively $f'_{\beta}$) over $ \alpha $ (respectively $ \beta $) in $ \Aut (k) $, their composition is $ (\, ^{\alpha}f'_{\beta})f_{\alpha} $, which is a semilinear automorphism over $ \alpha \beta $. We denote the group of semilinear automorphisms of $ \mathcal{R} $ by $ \Aut (\mathcal{R}\to \Spec k) $.
\end{definition}
As in the case of $ k $-group schemes, for $ \mathcal{R} $ a $ k $-scheme of based root datum, we have a homomorphism $ \Aut (\mathcal{R} \to \Spec k)\to \Aut (k)\colon f_{\alpha}\mapsto \alpha ^{-1} $. We let $ \Aut_{\mathcal{R}}(k) $ be the image of this homomorphism. Furthermore denoting the $ k $-automorphisms of the based root datum $ \mathcal{R} $ by $ \Aut \mathcal{R} $ (or also (\underline{$ \Aut $} $ \mathcal{R})(k) $, following the conventions discussed at the start of Section~\ref{Sec:Semiauto and Galois descent}), we get a short exact sequence $ 1\to \Aut \mathcal{R}\to \Aut (\mathcal{R} \to \Spec k)\to \Aut_{\mathcal{R}}(k)\to 1 $.
We now discuss how to associate functorially a $ k $-scheme of based root datum to a connected reductive $ k $-group. One possible approach would be to take an inductive limit of based root datum in the split case, and then descend this canonical object to any form. This would lead to the same construction as the one we now explain.
Actually, it suffices to incorporate our definition of the $k$-scheme of based root datum in \cite{Gil-Pol11}*{Exposé~24, Théorème~3.11}, by replacing principal Galois cover of group $E = \Aut R$ with the objects over $k$ that they classify (i.e.\ $k$-schemes of based root datum). As a corollary, we will get the definition of the $ k $-scheme of based root datum of a connected reductive $ k $-group. First, we recall the definition of the group scheme of exterior isomorphisms.
\begin{definition}[\cite{Gil-Pol11}*{Exposé~24, Corollaire~1.10}]
Let $G,G'$ be two connected reductive $ k $-group of type $R$, for some reduced based root datum $R$. Then $ \Ad G $ acts freely (on the right) on the $k$-group functor $ \underline{\Isom}_{k\text{-gr.}}(G,G') $. We define the $k$-\textbf{group functor of exterior isomorphisms between} $G$ \textbf{and }$G'$ to be the quotient sheaf $ \underline{\Extisom} (G,G') = \underline{\Isom}_{k\text{-gr.}}(G,G')/\Ad G $.
\end{definition}
\begin{remark}\label{Rem:computing Extisom}
Actually, \cite{Gil-Pol11}*{Exposé~24, Corollaire~1.10} asserts that this quotient is representable. Since it will be useful later, here is an explicit description of $ \underline{\Extisom} (G,G') $ using cocycles: let $k_s$ be a separable closure of $k$, let $ E = \Aut R $, let $ G_0 $ be the (split) connected reductive $ k_s $-group of type $ R $ and identify $ \underline{\Aut}~G_0 $ with $ \Ad G_0\rtimes E_{k_s} $ (after a choice of pinning for $ G_0 $, see \cite{Gil-Pol11}*{Exposé~24, Théorème~1.3}). Fix an isomorphism $G_0\cong G_{k_s} $ (respectively $ G_0\cong G'_{k_s}$), denote by $c$ (respectively $ c' $) the corresponding cocycle $ \Gal (k_s/k)\to \Aut G_0 $ and let $ \tilde{c} $ (respectively $ \tilde{c}' $) be the composition of this cocycle with $ \Aut G_0 \to E $. Further assume that the pinning of $ G_0 $ is defined over $ k $, so that the Galois action on $ \Aut R_{k_s} = E $ is trivial. Then $ \underline{\Extisom} (G,G') $ is a $ k_s/k $-form of $ E_{k_s} $ given by the Galois action $ \gamma .f = \tilde{c}'_{\gamma}f\tilde{c}_{\gamma}^{-1} $ (for all $ f\in E_{k_s} $ and for all $ \gamma \in \Gal (k_s/k) $). This follows directly from the Galois condition for an automorphism of $ G_0 $ to descend to an isomorphism $ G\to G' $, together with the fact that we are moding out by adjoint automorphisms.
\end{remark}
Let us also recall the notion of a quasi-pinning.
\begin{definition}\label{Def:quasi-split groups}
Let $ G $ be a connected reductive $ k $-group. If it exists, a \textbf{quasi-pinning} of $ G $ is:
\begin{enumerate}
\item A choice of a Borel subgroup $ B $ containing a maximal torus $ T $ of $ G $. Once this is chosen, let $ k_s $ be a separable closure of $ k $, let $ \Delta $ be the fundamental roots of $ G_{k_s} $ corresponding to the pair $ (T_{k_s},B_{k_s}) $ and for $ \alpha \in \Delta $, let $ \mathfrak{g}_{\alpha} $ be the corresponding one dimensional subspace of $ \Lie (G_{k_s}) $.
\item A choice of a nontrivial element $ X_{\alpha}\in \mathfrak{g}_{\alpha} $ for all $ \alpha \in \Delta $ such that for all $ \gamma \in \Gal(k_s/k) $, $ \gamma .X_{\alpha} = X_{\gamma (\alpha)} $.
\end{enumerate}
If $ G $ has a quasi-pinning, we say that $ G $ is \textbf{quasi-split}.
\end{definition}
The more classical definition for a connected reductive $ k $-group to be quasi-split is that it possesses a Borel subgroup. It is well-known that this definition agrees with Definition~\ref{Def:quasi-split groups} (and the equivalence is proved in a more general setting in \cite{Gil-Pol11}*{Exposé~24, Proposition~3.9.1}). For the convenience of the reader, let us reprove this fact.
\begin{lemma}
Let $ G $ be a connected reductive $ k $-group. If $ G $ has a borel subgroup, then $ G $ has a quasi-pinning.
\end{lemma}
\begin{proof}
Let $ B $ be a borel subgroup of $ G $. Then it contains a maximal torus $ T $ of $ G $ (see for example \cite{Gil-Pol11}*{Exposé~22, Corollaire~5.9.7}). Let $ k_s, \Delta, \alpha \in \Delta $ and $ \mathfrak{g}_{\alpha} $ be as in Definition~\ref{Def:quasi-split groups} for the pair ($ T,B $). Let $ H\leq \Gal (k_s/k) $ be the stabiliser of $ \alpha $, and let $ k_{\alpha} $ be the subfield of $ k_s $ fixed by $ H $. Then there exists an $ H $-equivariant isomorphism $ k_{\alpha}\otimes_{k_{\alpha}}k_s\cong \mathfrak{g}_{\alpha} $ (this holds because all $ k_s/k_{\alpha} $-forms of the vector space $ k_s $ are equivalent by Hilbert's 90). Set $ X_\alpha = 1\in k_{\alpha}\subset \mathfrak{g}_{\alpha} $. Now, for $ \beta $ in the $ \Gal (k_s/k) $-orbit of $ \alpha $, we set $ X_{\beta} = \gamma .X_{\alpha} $ where $ \gamma \in \Gal (k_s/k) $ is any element such that $ \gamma (\alpha) = \beta $. The point is that $ X_{\beta} $ does not depend on a choice of $ \gamma \in \Gal(k_s/k) $ such that $ \gamma (\alpha )=\beta $ because $ H $ acts trivially on $ X_{\alpha} $. Doing so for each orbit of $ \Gal (k_s/k) $ on $ \Delta $ concludes the proof.
\end{proof}
\begin{theorem}[\cite{Gil-Pol11}*{Exposé~24, Théorème~3.11}]\label{Thm:Quasi-split groups and BRD}
Let $ R = (M,M^*,\Phi ,\Phi^* ,\Delta) $ be a reduced based root datum. Consider the following categories:
\begin{enumerate}
\item The category $ \underline{\BRD} $ of $k$-schemes of based root datum of type $ R $. The morphisms are the isomorphisms of $ k $-schemes of based root datum.
\item The category $ \underline{\RedExt} $ of connected reductive $k$-groups of type $R$. The morphisms between $G$ and $G'$ are elements of the group $\underline{\Extisom} (G,G')(k)$.
\item The category $ \underline{\QsPin} $ of connected reductive quasi-split $k$-groups of type $R$, together with a choice of quasi-pinning. The morphisms are the isomorphisms preserving the quasi-pinning.
\end{enumerate}
These three categories are equivalent. More precisely, we have a diagram of functors between categories
\[
\begin{tikzcd}[row sep=2.5em]
\underline{\BRD} \arrow[rr,"\text{qspin}"] &&
\underline{\QsPin} \arrow[dl,"\iota"] \\
& \underline{\RedExt} \arrow[ul,"\text{brd}"] &
\end{tikzcd}
\]
such that the composition of those three functors $ ( $starting with anyone of them$ ) $ is naturally isomorphic to the identity.
\end{theorem}
\begin{proof}
We follow the proof given in \cite{Gil-Pol11}*{Exposé~24, Section~3.11}, with the advantage that we can work with the more concrete Galois descent, and that the notion of pinning is simpler over fields.
Set $ E = \Aut R $. Let $ k_s $ be a separable closure of $ k $ and let $ G_0 $ be the (split) connected reductive $ k_s $-group of type $ R $. For the proof, we choose a pinning for $G_0$ which is defined over $ k $, i.e. we choose a pinning for the split connected reductive $ k $-group of type $ R $ and we base change it to a pinning of $ G_0 $. In particular we choose a torus $ T_0 $ contained in a Borel subgroup $ B_0 $ (both defined over $ k $), and we get an identification $ \underline{\Aut}~G_0 \cong \Ad G_0\rtimes E_{k_s} $ (where the $ \Gal (k_s/k) $-action on $ E_{k_s} $ is trivial), and in particular an embedding $E= E_{k_s}(k_s)\to (\underline{\Aut}~G_0)(k_s)=\Aut G_0 $.
\begin{enumerate}
\item The functor $\iota$. On objects, $\iota (G) $ is the natural inclusion whilst for $f\in \Mor_{\QsPin}(G,G')$, $\iota (f)$ is the projection of $f$ in $ \underline{\Extisom} (G,G')(k) = (\underline{\Isom}_{k\text{-gr.}}(G,G')/\Ad G)(k) $.
\item The functor qspin. Let $\mathcal{R}$ be a $k$-scheme of based root datum of type $R$. Choose an isomorphism $R_l\cong \mathcal{R}_l$ for some finite Galois extension $ l/k $ and let $c\colon \Gal (k_s/k)\to E $ be the corresponding cocycle. The quasi-split group qspin($ \mathcal{R} $) is the $ k_s/k $-form of $G_0$ defined by the cocycle $ c\colon \Gal (k_s/k)\to E \to \Aut G_0 $. Note that this cocycle preserves $T_0$ and $B_0$, so that qspin($ \mathcal{R} $) is indeed quasi-split. We choose for quasi-pinning on qspin($ \mathcal{R} $) the pair $ (T_0,B_0) $ descended to $ k $, and for $ \alpha \in \Delta $, we choose the element $ X_{\alpha}\in \Lie (G_0) $ to be the same as the one appearing in the pinning of $ G_0 $. Since the pinning of $ G_0 $ is defined over $ k $ by assumption, this indeed constitutes a quasi-pinning of qspin($ \mathcal{R} $). Finally, for a morphism $ f\in \Mor_{\BRD}(\mathcal{R},\mathcal{R}') $, qspin($ f $) is defined to be the descent of $f_{k_s}\in \Mor (\mathcal{R}_{k_s},\mathcal{R}'_{k_s})\cong E\leq \Aut G_0$ to an isomorphism qspin($ \mathcal{R} $)$ \to $qspin($ \mathcal{R}' $).
\item The functor brd. For $G$ a connected reductive group of type $R$, choose an isomorphism $ G_0\cong G_{k_s} $ and let $c\colon \Gal (k_s/k)\to \Aut G_0 $ be the corresponding cocycle. Consider $\tilde{c}$, the composition of $c$ with the projection $ \Aut G_0\to E $. Now brd($G$) is defined to be the $k_s/k$-form of the split $k_s$-scheme of root datum $R_{k_s}$ obtained by Galois descend using the cocycle $ \tilde{c} $. Whilst for a morphism $ f\in \Mor_{\RedExt}(G,G') $, $ f_{k_s}\in \underline{\Extisom}(G_{k_s},G_{k_s}')(k_s)\cong \Aut R_{k_s} $, and brd($ f $) is defined to be the descent of $ f_{k_s} $ to an isomorphism brd($ G $)$ \to $brd($ G' $).
\end{enumerate}
We now check that the composition of those three functors (starting with anyone of them) is naturally isomorphic to the identity.
\begin{enumerate}
\item brd$\, \circ \, i \, \circ \, $qspin $ \cong \Id_{\BRD} $. Let $c'\colon \Gal (k_s/k)\to E $ be the cocycle arising from a choice of $ R_l\cong \mathcal{R}_l $. By definition of qspin, a choice of isomorphism $ G_0\cong \text{qspin}(\mathcal{R})_{k_s} $ gives rise to a cocycle $c$ which is cohomologous to $c'$ (as cocycles with values in $ \Aut G_0 $), hence the $\tilde{c}$ appearing in the definition of brd is cohomologous to $ c' $ as well (as cocycles with values in $ E $).
\item qspin $ \circ $ brd$\, \circ \, i \cong \Id_{\QsPin} $. We need to check that given a quasi-split group $ G $ together with a choice of isomorphism $ G_0\cong G_{k_s} $ and corresponding cocycle $ c\colon \Gal(k_s/k)\to \Aut G_0 $, then $ c $ is cohomologous to $ c $ composed with $ \Aut G_0\to E\to \Aut G_0 $. The quasi-pinning on $ G $ gives a pinning of $ G_{k_s} $, which is sent by $ G_0\cong G_{k_s} $ to a pinning of $ G_0 $. Up to conjugation by $g\in G_0(k_s)$, which has the effect of replacing $c$ by a cohomologous cocycle, we can assume that this pinning of $ G_0 $ is the one we chose from the outset. Because the pinning of $ G_0 $ is defined over $ k $, it is invariant under the action of $ \Gal (k_s/k) $. Hence, the cocycle $ c $ has values in $ E $, as wanted.
\item $ i\, \circ $ qspin $ \circ $ brd $ \cong \Id_{\RedExt} $. Let $ G $ be a connected reductive group of type $ R $. We want to check that $ \underline{\Extisom} (G,G')(k)\neq \emptyset $, where $ G' = (i\, \circ $ qspin $ \circ $ brd)$ (G) $. Let $ G_0\cong G_{k_s} $ be the chosen isomorphism to define brd$(G)$, with corresponding cocycle $ c $, and let $ \tilde{c} $ be the projection of $ c $ under $ \Aut G_0\to E $. By definition, a cocycle defining $ G' $ is cohomologous to $ \tilde{c} $, so we can assume that $ G' $ is defined by $ \tilde{c} $. Now, by Remark~\ref{Rem:computing Extisom}, the identity on $ G_0 $ descends to an element of $ \underline{\Extisom} (G,G')(k) $, concluding the proof. \qedhere
\end{enumerate}
\end{proof}
In view of Theorem~\ref{Thm:Quasi-split groups and BRD}, one can attach in a functorial way a $k$-scheme of based root datum to any connected reductive $k$-group.
\begin{definition}
Let $ G $ be a connected reductive $k$-group. The $k$-\textbf{scheme of based root datum associated to} $ G $ is brd$ (G) $, where brd is the functor appearing in Theorem~\ref{Thm:Quasi-split groups and BRD}. We denote it $ \mathcal{R}(G) $.
\end{definition}
The crucial input is that taking the scheme of based root datum commutes with base change.
\begin{lemma}\label{Lem:base change commute with taking root datum}
Let $ G $ be a connected reductive $k$-group and let $ \alpha $ be an automorphism of $ k $. Then $ \mathcal{R}(\,^{\alpha}G) \cong \,^{\alpha}\mathcal{R}(G) $, naturally in $ G $.
\end{lemma}
\begin{proof}
Let $R$ be the type of $ G $, let $ k_s $ be a separable closure of $ k $, and let $ G_0 $ be the (split) connected $ k_s $-group of type $ R $. Let $ \beta $ be an extension of $ \alpha $ to $ k_s $, choose an isomorphism $ G_0\cong G_{k_s} $, and let $ G_0\cong \,(^{\alpha}G)_{k_s} $ be the corresponding isomorphism defined in Lemma~\ref{Lem:choosing a good base change}. Now by Lemma~\ref{Lem:the descent datum of a base change}, if $ c $ denotes the cocycle defining $ G $, then the corresponding cocycle $ ^{\alpha}c $ defining $ ^{\alpha}G $ is given by $ (\, ^{\alpha}c)_{\gamma} = \Id_{\beta }^{-1}c_{\beta^{-1}\gamma \beta}\Id_{\beta} $, for all $ \gamma \in \Gal (k_s/k) $. Finally, we choose a pinning of $G_0$ defined over the prime field of $k$ (so that we can identify $\Aut G_0\cong (\Ad G_0)(k_s)\rtimes \Aut R $), and we let $ \tilde{c} $ (respectively $^{\alpha }\tilde{c}$) be the projection of $ c $ (respectively $ ^{\alpha}c $) under $ \Aut G_0\to \Aut R $. Note that since the Galois action on $ \Aut R $ is trivial, $ (^{\alpha}\tilde{c})_{\gamma} $ is just the projection of $ c_{\beta^{-1}\gamma \beta} $ onto $ \Aut R $.
On the other side, let $ R_{k_s} $ be the split $ k $-scheme of based root datum of type $ R $. The (choice of) cocycle defining $ \mathcal{R}(G) $ is $ \tilde{c}\colon \Gal(k_s/k)\to \Aut R $. Now exactly the same computation as for algebraic groups (i.e.\ repeating Lemma~\ref{Lem:choosing a good base change} and Lemma~\ref{Lem:the descent datum of a base change} in the category of schemes of based root datum) shows that a cocycle defining $ ^{\alpha}\mathcal{R}(G) $ is given by $\gamma \mapsto \Id_{\beta }^{-1}\tilde{c}_{\beta^{-1}\gamma \beta}\Id_{\beta} = \tilde{c}_{\beta^{-1}\gamma \beta} $. But this a also the chosen cocycle defining $ \mathcal{R}(^{\alpha}G) $, as was to be shown. The naturality in $ G $ of this isomorphism is straightforward.
\end{proof}
\begin{remark}
Of course, for this whole section, we did not need the fact that the base scheme is the spectrum of a field, and for example, Lemma~\ref{Lem:base change commute with taking root datum} should be true over any base scheme, and under any base change. The advantage of working over a field is that the notion of pinning is simpler, and that Galois descent is more concrete than fppf descent.
\end{remark}
The proof of Theorem~\ref{Thm: MainThm1} now follows easily from Theorem~\ref{Thm:Quasi-split groups and BRD} and Lemma~\ref{Lem:base change commute with taking root datum}.
\begin{proof}[Proof of Theorem~\ref{Thm: MainThm1}]
Recall that to give an automorphism $ f_{\alpha} $ of $ G $ over $\alpha \in \Aut (k)$ is equivalent to give an isomorphism of $ k $-group schemes $ f\colon G\to \,^{\alpha}G $. Hence, projecting $ f $ to an element $ \bar{f}\in \underline{\Extisom}(G,\,^{\alpha}G)(k) $ and using the functor brd defined in Theorem~\ref{Thm:Quasi-split groups and BRD}, we get an isomorphism brd$ (\bar{f})\colon \mathcal{R}(G)\to \mathcal{R}(\,^{\alpha}G)\cong \,^{\alpha}\mathcal{R}(G) $ (where we used Lemma~\ref{Lem:base change commute with taking root datum} for the last isomorphism). Now since brd is a functor, and because the isomorphism $ \mathcal{R}(\,^{\alpha}G)\cong \,^{\alpha}\mathcal{R}(G) $ is natural in $ G $, the map $ f_{\alpha}\mapsto $brd$ (\bar{f}) $ is a group homomorphism which is natural in $ G $. Furthermore, the underlying automorphism of the field is preserved by this homomorphism. To conclude the first part of the proof, note that $ f_{\alpha} $ is in the kernel of this homomorphism if and only if $ \alpha $ is trivial and $ \bar{f} $ is trivial in $ \underline{\Extisom} (G, G)(k) $, which is to say that $ f\in (\Ad G)(k) $.
For the last assertion, assume that $ G $ is quasi-split and choose a quasi-pinning of it. Define the subgroup $ H=\lbrace f_{\alpha}\in \Aut (G\to \Spec k)~\vert~ f_{\alpha } \text{ preserves the quasi-pinning of } G\rbrace $. Seeing $ f_{\alpha} $ as an isomorphism $ f $ from $ G $ to $ ^{\alpha}G $, the condition for $ f $ to belong to $ H $ is that it preserves the quasi-pinnings (where $ ^{\alpha}G $ is endowed with the quasi-pinning on $G$ based changed to $ ^{\alpha}G $). Now the fact that $ H $ maps isomorphically onto $ \Aut(\mathcal{R}(G)\to \Spec k) $ under $ \Aut (G\to \Spec k)\to \Aut (\mathcal{R}(G)\to \Spec k) $ is a direct consequence of Lemma~\ref{Lem:base change commute with taking root datum} and of the equivalence of categories $ \underline{\BRD} $ and $ \underline{\QsPin} $ in Theorem~\ref{Thm:Quasi-split groups and BRD}.
\end{proof}
\begin{remark}
For $ G $ a connected reductive $ k $-group which is not quasi-split, the decomposition $ \Aut G\cong (\Ad G)(k)\rtimes \Out G $ as a semidirect product is usually destroyed. Similarly, one should not expect to obtain a semidirect decomposition of $ \Aut (G \to \Spec k) $ for a general connected reductive $ k $-group. Investigating a possible semidirect decomposition of the group of semilinear automorphisms of simple algebraic groups is an entirely different matter when $ G $ is not quasi-split, as is illustrated by our treatment of the $ \SL_n(D) $ case in Section~\ref{Sec:SL_n(D)}.
\end{remark}
As a corollary of Theorem~\ref{Thm: MainThm1}, we obtain a proof of Theorem~\ref{Thm: MainThm2}.
\begin{proof}[Proof of Theorem~\ref{Thm: MainThm2}]
By Theorem~\ref{Thm: MainThm1}, $ \Aut_{\mathcal{R}(G)} (k) = \Aut_G(k) $. We thus obtain the following commutative diagram:
\begin{center}
\begin{tikzcd}[row sep=2.5em, column sep=3em] 1 \arrow{dr} &1 \arrow[d] & &1&1\\ & (\Ad G)(k) \arrow{dd} \arrow{dr} & & \Aut_{\mathcal{R}(G)} (k) \arrow{u} \arrow{ur} & \\ & & \Aut (\! G \to \Spec k\! ) \arrow[dr,swap, "\pi"'] \arrow[ur,"p_1"] & & \\ & \Aut G \arrow[d] \arrow{ur} & & \Aut (\mathcal{R}(G)\to \Spec k) \arrow[uu,"p_2"]\arrow[ul, shift left=1.5ex,swap, "\iota"'] \arrow{dr} & \\ 1 \arrow{ur} & Out G \arrow[d] & & \Aut \mathcal{R}(G) \arrow{u} & 1 \\ &1 & &1 \arrow{u} &
\end{tikzcd}
\end{center}
where all diagonal lines and vertical lines are exact. Here, $ \pi $ denotes the homomorphism provided by Theorem~\ref{Thm: MainThm1}, and $ \iota $ is a section of $ \pi $ (which exists, again by Theorem~\ref{Thm: MainThm1}). Note that in particular, $ \iota $ preserves the underlying field automorphism, i.e.\ $ p_1\circ \iota = p_2 $.
We thus conclude that the short exact sequence $ 1\to \Aut G\to \Aut (G \to \nolinebreak \Spec k)\to \Aut_{G}(k)\to 1 $ splits if and only if the short exact sequence involving $ k $-schemes of based root datum $ 1\to \Aut \mathcal{R}(G) \to \Aut (\mathcal{R}(G) \to \Spec k)\to \Aut_{\mathcal{R}(G)}(k)\to 1 $ does, as was to be shown.
\end{proof}
\section{Semilinear automorphisms and Galois cohomology}\label{Sec:Semilinear Galois cohomology}
We have just proved that for any connected reductive algebraic $ k $-group $ G $, we have a natural exact sequence $ 1\to (\Ad G)(k)\to \Aut (G\to \Spec k)\to \Aut (\mathcal{R}(G)\to \Spec k) $. It would be nice to be able to express the failure of surjectivity on the right using Galois cohomology. We explain in this section how to do so.
In this section, $ k_s $ denotes a separable closure of $ k $ with Galois group $ \Gamma = \Gal (k_s/k) $, $ R $ is a reduced based root datum, $ G_0 $ is a (split) connected reductive $ k_s $-group of type $ R $ with a choice of pinning defined over the base field of $ k $, and $ R_{k_s} $ is the split $ k_s $-scheme of based root datum of type $ R $. We furthermore set $ E = \Aut R $ and we let $ E_{k_s} $ be the corresponding constant object over $ k_s $. Also, we again use the convention that $ G_0 $ comes together with a preferred split form of it over the prime field of $ k $. In particular, we get a decomposition $ \Aut (G_0\to \Spec k_s)\cong \Aut G_0\rtimes \Aut (k_s) $, where the splitting $ \Aut (k_s)\to \Aut (G_0\to \Spec k_s) $ is given by $ \beta^{-1} \mapsto \Id_{\beta} $ (see Definition~\ref{Def:gammatilde and Idgamma}).
\begin{definition}\label{Def:semilinear auto. preserving k}
\begin{enumerate}
\item Given a field extension $l\geq k$, we set $ \Aut (l\geq k) = \lbrace \alpha \in \Aut (l)~\vert~\alpha (k) = k\rbrace $
\item We set $ \Aut (G_0\to \Spec k_s\geq k) = $ $$ \lbrace f_{\alpha}\in \Aut (G_0\to \Spec k_s)~\vert~\text{the underlying } \alpha \in \Aut (k_s) \text{ belongs to } \Aut (k_s\geq k)\rbrace. $$
\item We denote an element of $ \Aut (G_0\to \Spec k_s\geq k)\cong \Aut G_0\rtimes \Aut (k_s\geq k) $ by $ b\Id_{\beta} $ (where $ b \in \Aut G_0 $ and $ \beta \in \Aut (k_s\geq k) $).
\end{enumerate}
\end{definition}
\begin{remark}
In Definition~\ref{Def:semilinear auto. preserving k}, $ \alpha $ is required to globally preserves $ k $, but its restriction to $ k $ can be non-trivial. Also, we will use the fact that $ \Aut (l/k) $ (see Definition~\ref{Def:Notation Aut(l/k) and base change}) is a normal subgroup of $ \Aut (l\geq k) $.
\end{remark}
\begin{definition}\label{Def:semilinear Galois action}
Let $ G $ be a connected reductive $ k $-group of type $ R $, choose an isomorphism $ G_0\cong G_{k_s} $ and let $ c\colon \Gamma \to \Aut G_0 $ be the corresponding Galois cocycle. We define the \textbf{semilinear Galois action corresponding to }$ c $ (we also say corresponding to $ G_0\cong G_{k_s} $) on $ \Aut (G_0\to \Spec k_s\geq k) \cong \Aut G_0\rtimes \Aut (k_s) $ as follows:
\begin{equation*}
\text{ For all } b\in \Aut G_0,~\beta \in \Aut (k_s\geq k) \text{ and } \gamma \in \Gamma,~~~\gamma .(b\Id_{\beta}) = c_{\beta^{-1}\gamma \beta}\,^{\beta^{-1}\gamma \beta}b\,^{\beta^{-1}}c_{\gamma}^{-1}\Id _{\beta}.
\end{equation*}
\end{definition}
\begin{remark}
In view of Lemma~\ref{Lem:Condition for a semilinear auto. to descend}, an element of $ \Aut (G_0\to \Spec k_s\geq k) $ descends to an element of $ \Aut (G\to \Spec k) $ if and only if it is Galois invariant. This is the origin of Definition~\ref{Def:semilinear Galois action}.
\end{remark}
It is important to notice that in general, the $ \Gamma $-action on $ \Aut (G_0\to \Spec k_s\geq k) $ does not preserve the group structure. Let us prove some elementary properties of this action.
\begin{lemma}\label{Lem:computing with the semilinear action}
Keep the notations of Definition~\ref{Def:semilinear Galois action}. Let $ \gamma ,\gamma_1,\gamma_2 \in \Gamma $ and let $ b\Id_{\beta}$, $b_1\Id_{\beta_1}$, $ b_2\Id_{\beta_2} \in \Aut (G_0\to \Spec k_s\geq k)\cong \Aut G_0\rtimes \Aut (k_s\geq k) $.
\begin{enumerate}
\item $\gamma_1\gamma_2 . (b\Id_{\beta}) = \gamma_1.(\gamma_2.(b\Id_{\beta})) $
\item $ \gamma .(b_1\Id_{\beta_1} b_2\Id_{\beta_2}) = \Big( \beta_2^{-1}\gamma \beta_2.(b_1\Id_{\beta_1})\Big) \Big( \gamma.(b_2\Id_{\beta_2})\Big) $
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item $\begin{aligned}[t]
\gamma_1\gamma_2 . (b\Id_{\beta})
&= c_{\beta^{-1}\gamma_1\gamma_2 \beta}\,^{\beta^{-1}\gamma_1\gamma_2 \beta}b\,^{\beta^{-1}}c_{\gamma_1\gamma_2}^{-1}\Id _{\beta}\\
&= c_{\beta^{-1}\gamma_1\beta \beta^{-1} \gamma_2 \beta}\,^{\beta^{-1}\gamma_1\gamma_2 \beta}b\,^{\beta^{-1}}(c_{\gamma_1}\,^{\gamma_1}c_{\gamma_2})^{-1}\Id _{\beta}\\
&= c_{\beta^{-1}\gamma_1\beta}\,^{\beta^{-1} \gamma_1 \beta}(c_{\beta^{-1} \gamma_2 \beta}\,^{\beta^{-1} \gamma_2 \beta}b\,^{\beta^{-1}}c_{\gamma_2}^{-1})\,^{\beta^{-1}}c_{\gamma_1}^{-1}\Id _{\beta}\\
&= \gamma_1.(c_{\beta^{-1} \gamma_2 \beta}\,^{\beta^{-1} \gamma_2 \beta}b\,^{\beta^{-1}}c_{\gamma_2}^{-1}\Id_{\beta})\\
&= \gamma_1.(\gamma_2.(b\Id_{\beta}))
\end{aligned}$
\item $\begin{aligned}[t]
&\gamma .(b_1\Id_{\beta_1} b_2\Id_{\beta_2}) = \gamma .(b_1 \,^{\beta_1^{-1}}b_2 \Id_{\beta_2\beta_1})\\
&= c_{(\beta_2\beta_1)^{-1}\gamma \beta_2\beta_1}\,^{(\beta_2\beta_1)^{-1}\gamma \beta_2\beta_1}(b_1 \,^{\beta_1^{-1}}b_2)\,^{(\beta_2\beta_1)^{-1}}c_{\gamma}^{-1}\Id _{\beta_2\beta_1}\\
&= c_{\beta_1^{-1}\beta_2^{-1}\gamma \beta_2\beta_1}\,^{\beta_1^{-1}\beta_2^{-1}\gamma \beta_2\beta_1}b_1\,^{\beta_1^{-1}}c_{\beta_2^{-1}\gamma \beta_2}^{-1}\,^{\beta_1^{-1}}c_{\beta_2^{-1}\gamma \beta_2} \,^{(\beta_2\beta_1)^{-1}\gamma \beta_2}b_2\,^{(\beta_2\beta_1)^{-1}}c_{\gamma}^{-1}\Id _{\beta_2 \beta_1}\\
&=\Big( \beta_2^{-1}\gamma \beta_2 . (b_1\Id_{\beta_1})\Big) \Id_{\beta_1}^{-1}\,^{\beta_1^{-1}}c_{\beta_2^{-1}\gamma \beta_2} \,^{\beta_1^{-1}\beta_2^{-1}\gamma \beta_2}b_2\,^{\beta_1^{-1}\beta_2^{-1}}c_{\gamma}^{-1}\Id _{\beta_1}\Id_{\beta_2}\\
&= \Big( \beta_2^{-1}\gamma \beta_2.(b_1\Id_{\beta_1})\Big) \Big( \gamma.(b_2\Id_{\beta_2})\Big)
\end{aligned}$
\end{enumerate}
\end{proof}
\end{lemma}
\begin{lemma}
Keep the notations of Definition~\ref{Def:semilinear Galois action}. The set of elements in $ \Aut (G_0\to \Spec k_s\geq k) $ that are fixed by the $ \Gamma $ action is a subgroup.
\end{lemma}
\begin{proof}
Let $b_1\Id_{\beta_1}$, $ b_2\Id_{\beta_2} \in \Aut (G_0\to \Spec k_s\geq k)\cong \Aut G_0\rtimes \Aut (k_s\geq k) $ be elements that are fixed by the $ \Gamma $ action. For $ \gamma \in \Gamma $, we have $ \gamma . (b_1\Id_{\beta_1} b_2\Id_{\beta_2}) = \Big( \beta_2^{-1}\gamma \beta_2.(b_1\Id_{\beta_1})\Big) \Big( \gamma.(b_2\Id_{\beta_2})\Big) $ by Lemma~\ref{Lem:computing with the semilinear action}. Hence $ b_1\Id_{\beta_1} b_2\Id_{\beta_2} $ is $ \Gamma $ invariant as well.
Similarly, if $ b\Id_{\beta}$ is $\Gamma $ invariant, for all $ \gamma \in \Gamma $ we have $ \Id_{G_0} = \gamma . \Id_{G_0} = \gamma .((b\Id_{\beta})^{-1}b\Id_{\beta}) = \Big( \beta^{-1}\gamma \beta .((b\Id_{\beta})^{-1})\Big) \Big( \gamma.(b\Id_{\beta})\Big) $. Hence, since $ b\Id_{\beta} $ is $ \Gamma $ invariant, we get $ \beta^{-1}\gamma \beta .((b\Id_{\beta})^{-1}) = (b\Id_{\beta})^{-1} $ for all $ \gamma \in \Gamma $, and hence $ (b\Id_{\beta})^{-1} $ is $ \Gamma $ invariant as well.
\end{proof}
\begin{definition}
In the notations of Definition~\ref{Def:semilinear Galois action}, the subgroup of elements of $ \Aut (G_0\to \Spec k_s\geq k) $ that are fixed by $ \Gamma $ is denoted $ \Aut (G_0\to \Spec k_s\geq k)^{\Gamma} $.
\end{definition}
We now aim to state that the group $ \Aut (G\to \Spec k) $ is the group $ \Aut (G_0\to \Spec k_s\geq k)^{\Gamma} $ modulo the Galois group. So we need to embed the Galois group as a normal subgroup of $ \Aut (G_0\to \Spec k_s\geq k)^{\Gamma} $.
\begin{definition}\label{Def:embedding Gamma in semilinear automorphisms}
Consider the homomorphism $ \Gamma \to \Aut (G_0\to \Spec k_s\geq k)^{\Gamma}\colon \gamma \mapsto c_{\gamma}\Id_{\gamma}^{-1} $. We denote the image of $ \Gamma $ by $ \tilde{\Gamma} $.
\end{definition}
\begin{remark}
Note that for $ \gamma \in \Gamma $, $ c_{\gamma}\Id_{\gamma}^{-1} $ is an invariant element of $ \Aut (G_0\to \Spec k_s\geq k) $. Indeed, for $ \delta \in \Gamma $ we have $ \delta .(c_{\gamma}\Id_{\gamma}^{-1}) = c_{\gamma \delta \gamma^{-1}}\,^{\gamma \delta \gamma^{-1}}c_{\gamma}\,^{\gamma}c_{\delta}^{-1}\Id_{\gamma}^{-1} = c_{\gamma \delta }\,^{\gamma \delta}c_{\delta^{-1}}\Id_{\gamma}^{-1} = c_{\gamma}\Id_{\gamma}^{-1} $.
\end{remark}
\begin{remark}
If we denote $ \tilde{\gamma} = c_{\gamma}\Id_{\gamma}^{-1} $, it is unfortunate that $ \tilde{\gamma} $ is the inverse of the element $ \tilde{\gamma}_G = c_{\gamma^{-1}}\Id_{\gamma} $ appearing in Definition~\ref{Def:gammatilde and Idgamma}. In the language of descent datum, $ \gamma \mapsto \tilde{\gamma}_G $ is traditionally required to be an anti-homomorphism, whereas it felt more natural to use a homomorphism in Definition~\ref{Def:embedding Gamma in semilinear automorphisms}, so we indulge in this inconsistency.
\end{remark}
\begin{lemma}\label{Lem:Identification of Aut(G Spec k)}
Keeping the notations of Definition~\ref{Def:semilinear Galois action}, $ \Aut (G\to \Spec k) $ is naturally isomorphic to $ \Aut (G_0\to \Spec k_s\geq k)^{\Gamma}/\tilde{\Gamma} $.
\end{lemma}
\begin{proof}
We have a homomorphism $ \Aut (G_0\to \Spec k_s\geq k)^{\Gamma}\to \Aut (G\to \Spec k) $, which maps an invariant element of $ \Aut (G_0\to \Spec k_s\geq k) $ to its descent in $ \Aut (G\to \Spec k) $ (see Lemma~\ref{Lem:Condition for a semilinear auto. to descend}). Note that $ \tilde{\Gamma} $ is in the kernel of this map (since $ c_{\gamma^{-1}}\Id_{\gamma} = \tilde{\gamma}_{G} $ arises as a choice of base change for the identity). Now if $ b\Id_\beta $ is in the kernel of this homomorphism, then $ \beta $ acts trivially on $ k $, i.e.\ $ \beta = \gamma $ for some $ \gamma \in \Gamma $, and $ b\Id_{\beta}c_{\gamma}\Id_{\gamma}^{-1}\in \Aut G_0 $ descends to the identity in $ \Aut G $. But this holds if and only if $ b\Id_{\beta}c_{\gamma}\Id_{\gamma}^{-1} $ is already the identity on $ G_0 $. Since $ b\Id_{\beta}c_{\gamma}\Id_{\gamma}^{-1} = b\,^{\gamma^{-1}}c_{\gamma} = bc_{\gamma^{-1}}^{-1} $, we conclude that $ b\Id_{\beta} $ descends to the identity if and only if it is equal to $ c_{\gamma^{-1}}\Id_{\gamma} $ for some $ \gamma \in \Gamma $, i.e.\ if and only if it belongs to $ \tilde{\Gamma} $.
It remains to check that the homomorphism $ \Aut (G_0\to \Spec k_s\geq k)^{\Gamma}\to \Aut (G\to \Spec k) $ is surjective as well. But this follows from the fact that any automorphism of $ k $ can be extended to an automorphism of $ k_s $.
\end{proof}
Note that there was nothing special about the category of algebraic $k$-groups, and we could as well repeat this construction for other algebraic categories over $ k $ for which descent is effective. In particular, we can repeat everything we did so far for $ k $-schemes of based root datum. Also recall that in Theorem~\ref{Thm:Quasi-split groups and BRD}, still keeping the notations of Definition~\ref{Def:semilinear Galois action}, the cocycle defining $ \mathcal{R}(G) $ is obtained from $ c $ by projecting via $ \Aut G_0\to E $. Recalling that the Galois action on the split $ k_s $-scheme of based root datum is trivial, this gives the following result.
\begin{lemma}\label{Lem:semilinear Galois cohomology for root datum}
Keep the notations of Definition~\ref{Def:semilinear Galois action}. Let $ \tilde{c} $ be the projection of $ c $ under $ \Aut G_0\to E $. Define a semilinear Galois action on $ \Aut (R_{k_s}\to \Spec k_{s}\geq k)\cong E\times \Aut (k_s\geq k) $ as follows:
\begin{equation*}
\text{ For all } b\in E,~\beta \in \Aut (k_s\geq k) \text{ and } \gamma \in \Gamma,~~~\gamma .(b
\Id_{\beta}) = \tilde{c}_{\beta^{-1}\gamma \beta}b\tilde{c}_{\gamma}^{-1}\Id _{\beta}.
\end{equation*}
Define $ \tilde{\Gamma}\leq \Aut (R_{k_s}\to \Spec k_{s}\geq k)^{\Gamma} $ to be the image of the
homomorphism $ \Gamma \to \Aut (R_{k_s}\to \Spec k_{s}) \colon \gamma \mapsto \tilde{c}
_{\gamma}\Id_{\gamma}^{-1} $. Then $ \Aut (\mathcal{R}(G)\to \Spec k) $ is naturally isomorphic to
$ \Aut (R_{k_s}\to \Spec k_{s}\geq k)^{\Gamma}/\tilde{\Gamma} $. Furthermore, the homomorphism
$ \Aut (G_0\to \Spec k_s)\to \Aut (R_{k_s}\to \Spec k_s) $ induces a homomorphism $ \Aut
(G_0\to \Spec k_s\geq k)^{\Gamma}/\tilde{\Gamma}\to \Aut (R_{k_s}\to \Spec k_{s}\geq
k)^{\Gamma}/\tilde{\Gamma} $.
\end{lemma}
We can now formulate the failure of surjectivity of the map $ \Aut (G\to \Spec k)\to \Aut_G (k) $ and of the map $ \Aut (G\to \Spec k)\to \Aut (\mathcal{R}(G)\to \Spec k ) $ using a variant of Galois cohomology. We first give an approximation of this.
\begin{proposition}\label{Prop:Galois cohomology for semilinear auto}
Keep the notations of Lemma~\ref{Lem:semilinear Galois cohomology for root datum}. Endow $ \Aut G_0 $ and $ (\Ad G_0) (k_s) $ with the Galois action given by restricting the semilinear Galois action on $ \Aut (G_0\to \Spec k_s\geq k) $ $ ( $this is just the Galois action corresponding to the form $ G_0\cong G_{k_s}) $.
\begin{enumerate}
\item There exists a coboundary map $ \Aut (k_{s}\geq k)\xrightarrow{\partial } H^1(\Gamma , \Aut G_0) $ such that the sequence
\begin{equation*}
1\to (\Aut G_0)^{\Gamma}\to \Aut (G_0\to \Spec k_s\geq k)^{\Gamma}\to \Aut (k_{s}\geq k)\xrightarrow{\partial } H^1(\Gamma , \Aut G_0)
\end{equation*}
is exact.
\item There exists a coboundary map $ \Aut (R_{k_s}\to \Spec k_{s}\geq k)^{\Gamma}\xrightarrow{\partial } H^1(\Gamma , (\Ad G_0) (k_s)) $ such that the sequence
\begin{align*}
1\to (\Ad G_0)(k_s)^{\Gamma}\to \Aut (G_0\to &\Spec k_s\geq k)^{\Gamma}\to \\
&\Aut (R_{k_s}\to \Spec k_{s}\geq k)^{\Gamma}\xrightarrow{\partial } H^1(\Gamma , (\Ad G_0) (k_s))
\end{align*}
is exact.
\end{enumerate}
\end{proposition}
\begin{proof}
It is important not to confuse the two Galois actions we are considering on $ \Aut G_0 $. One arises from the fact that we chose a form of $ G_0 $ over the prime field of $ k $, and the other is the Galois action arising from $ G_0\cong G_{k_s} $. For $ \gamma \in \Gamma $ and $ b\in \Aut G_0 $, recall that the former is denoted $ ^{\gamma}b = \Id_{\gamma}^{-1}b\Id_{\gamma} $ whilst the latter is denoted $ \gamma .b = c_{\gamma}\,^{\gamma}bc_{\gamma}^{-1} $.
We have a similar remark for $ E = \Aut R_{k_s} $: for $ \gamma \in \Gamma $ and $ b\in E $, we denote $ ^{\gamma}b = \Id_{\gamma}^{-1}b\Id_{\gamma} = b $ (the latter equality holds because any automorphism of $ R_{k_s} $ is defined over $ k $) and $ \gamma .b = \tilde{c}_{\gamma}\,^{\gamma}b\tilde{c}_{\gamma}^{-1} = \tilde{c}_{\gamma}b\tilde{c}_{\gamma}^{-1} $.
\begin{enumerate}
\item Let $ \beta^{-1} \in \Aut (k_s\geq k) $. As usual in this situation, we set $ \partial (\beta^{-1} )\colon \Gamma \to \Aut G_0\colon \gamma \mapsto \partial (\beta^{-1} )_{\gamma} $ where $ \partial (\beta^{-1} )_{\gamma} $ is defined by the equality (in $\Aut (G_0\to \Spec k_s\geq k)$) $ \gamma .\Id_{\beta} = \Id_{\beta }\partial (\beta^{-1} )_{\gamma} $. In other words, $ \partial (\beta^{-1} )_{\gamma} = \Id_{\beta }^{-1}\gamma .\Id_{\beta} $. Hence $ \partial (\beta^{-1} )_{\gamma} $ clearly belongs to $ \Aut G_0 $. The fact that it is a cocycle follows directly from Lemma~\ref{Lem:computing with the semilinear action}. Indeed, $ \partial (\beta^{-1} )_{\gamma \gamma '} = \Id_{\beta }^{-1}\gamma \gamma ' .\Id_{\beta} = \Id_{\beta }^{-1}\gamma.(\Id_{\beta}\Id_{\beta}^{-1} \gamma ' .\Id_{\beta}) = \Id_{\beta }^{-1}\gamma.(\Id_{\beta})\gamma .(\Id_{\beta}^{-1} \gamma ' .\Id_{\beta}) = \partial (\beta^{-1} )_{\gamma} \gamma .\partial (\beta^{-1} )_{\gamma '} $.
The sequence $ 1\to \Aut G_0\to \Aut (G_0\to \Spec k_s\geq k)\to \Aut (k_{s}\geq k) $ is a $ \Gamma $-equivariant exact sequence (where we endow $ \Aut (k_{s}\geq k) $ with the trivial $ \Gamma $ action). Hence, taking $ \Gamma $-invariant elements, it remains exact. So we just need to check exactness at $ \Aut (k_{s}\geq k) $.
Let $ \beta^{-1} \in \Aut (k_s\geq k) $. Then $ \partial (\beta^{-1} ) $ is trivial in $ H^1(\Gamma , \Aut G_0) $ if and only if there exists $ b\in \Aut G_0 $ such that for all $ \gamma \in \Gamma $, $ b^{-1}\Id_{\beta }^{-1}(\gamma .\Id_{\beta})(\gamma .b) = 1 $. By Lemma~\ref{Lem:computing with the semilinear action}, $ b^{-1}\Id_{\beta }^{-1}(\gamma .\Id_{\beta})(\gamma .b) = (\Id_{\beta }b)^{-1}\gamma .(\Id_{\beta }b) $, and hence $ \partial (\beta^{-1} ) $ is trivial if and only if there exists $b\in \Aut G_0 $ such that $ \Id_{\beta}b $ is a $ \Gamma $-invariant element of $ \Aut (G_0\to \Spec k_s\geq k) $, which proves exactness at $ \Aut (k_s\geq k) $.
\item Let $ b\Id_{\beta} \in \Aut (R_{k_s}\to \Spec k_{s}\geq k)^{\Gamma}\cong (E\times \Aut (k_s\geq k))^{\Gamma} $. As usual in this situation, we set $ \partial (b\Id_{\beta} )\colon \Gamma \to \Aut G_0\colon \gamma \mapsto \partial (b\Id_{\beta} )_{\gamma} $ where $ \partial (b\Id_{\beta} )_{\gamma} $ is defined by the equality (in $\Aut (G_0\to \Spec k_s\geq k)$) $ \gamma .(b\Id_{\beta}) = b\Id_{\beta }\partial (b\Id_{\beta} )_{\gamma} $. In other words, $ \partial (b\Id_{\beta} )_{\gamma} = (b\Id_{\beta })^{-1}\gamma .(b\Id_{\beta}) $. Let us check that $ \partial (b\Id_{\beta} )_{\gamma} $ belongs to $ (\Ad G_0) (k_s) $. For $ \gamma \in \Gamma $, define $ c'_{\gamma} = c_{\gamma}\tilde{c}_{\gamma}^{-1} $. Now $ \partial (b\Id_{\beta} )_{\gamma} = (b\Id_{\beta })^{-1}c_{\beta^{-1}\gamma \beta}\,^{\beta^{-1}\gamma \beta}b\,^{\beta^{-1}}c_{\gamma}^{-1}\Id_{\beta} =$ $(b\Id_{\beta })^{-1}c'_{\beta^{-1}\gamma \beta}b\Id_{\beta}{c'}_{\gamma}^{-1} $ because $ b\Id_{\beta}\in \Aut (R_{k_s}\to \Spec k_{s}\geq k)^{\Gamma} $, hence we conclude that $ \partial (b\Id_{\beta} )_{\gamma} \in (\Ad G_0) (k_s) $ because $ c'_{\gamma }\in (\Ad G_0)(k_s) $ and $ (\Ad G_0)(k_s) $ is a normal subgroup of $ \Aut (G_0\to \Spec k_s\geq k) $.
The sequence $ 1\to (\Ad G_0)(k_s)\to \Aut (G_0\to \Spec k_s\geq k)\to \Aut (R_{k_s}\to \Spec k_{s}\geq k) $ is a $ \Gamma $-equivariant exact sequence. Hence, taking $ \Gamma $-invariant elements, it remains exact. So we just need to check exactness at $\Aut (R_{k_s}\to \Spec k_{s}\geq k)^{\Gamma} $.
Let $ b\id_{\beta} \in \Aut (R_{k_s}\to \Spec k_{s}\geq k)^{\Gamma}\cong (E\times \Aut (k_s\geq k))^{\Gamma} $. Then $ \partial ( b\id_{\beta}) $ is trivial in $ H^1(\Gamma , (\Ad G_0)(k_s)) $ if and only if there exists $ g\in (\Ad G_0)(k_s) $ such that for all $ \gamma \in \Gamma $, $ g^{-1}(b\id_{\beta})^{-1}\gamma .(b\id_{\beta})\gamma .g = 1 $. But $ g^{-1}(b\id_{\beta})^{-1}\gamma .(b\id_{\beta})\gamma .g = (b\Id_{\beta }g)^{-1}\gamma .(b\Id_{\beta }g) $ by Lemma~\ref{Lem:computing with the semilinear action}, and hence $ \partial (b\id_{\beta}) $ is trivial if and only if there exists $g\in (\Ad G_0)(k_s) $ such that $ b\Id_{\beta }g $ is a $ \Gamma $-invariant element of $ \Aut (G_0\to \Spec k_s\geq k) $, which proves exactness at $ \Aut (R_{k_s}\to \Spec k_{s}\geq k)^{\Gamma} $. \qedhere
\end{enumerate}
\end{proof}
In order to prove Theorem~\ref{Thm: MainThm3}, it suffices now to observe that in Proposition~\ref{Prop:Galois cohomology for semilinear auto}, the image of $ \tilde{\Gamma} $ under the coboundary operator is trivial.
\begin{lemma}\label{Lem:triviality of Gamma under cocoundary}
Keep the notation of Proposition~\ref{Prop:Galois cohomology for semilinear auto}.
\begin{enumerate}
\item The image of $ \Gamma \unlhd \Aut(k_s\geq k) $ under $ \Aut(k_s\geq k)\xrightarrow{\partial} H^1(\Gamma ,\Aut G_0) $ is trivial.
\item The image of $ \tilde{\Gamma}\unlhd \Aut (R_{k_s}\to \Spec k_s\geq k)^{\Gamma} $ under the coboundary map $ \Aut (R_{k_s}\to \Spec k_s\geq k)^{\Gamma}\xrightarrow{\partial} H^1(\Gamma ,(\Ad G_0)(k_s)) $ is trivial.
\end{enumerate}
\end{lemma}
\begin{proof}
The proof is just a straightforward computation, using directly the definition of the semilinear $ \Gamma $-action.
\begin{enumerate}
\item Let $ \gamma \in \Gamma $. We want to check that the cocycle $ \Gamma \to \Aut G_0\colon \sigma \mapsto \Id_{\gamma}^{-1}\sigma .\Id_{\gamma} $ is trivial. By definition, $ \Id_{\gamma}^{-1}\sigma .\Id_{\gamma} = \Id_{\gamma}^{-1} c_{\gamma^{-1}\sigma\gamma}\,^{\gamma^{-1}}c_{\sigma}^{-1}\Id_{\gamma} = \,^{\gamma}c_{\gamma^{-1}\sigma\gamma}c_{\sigma}^{-1} = \,^{\gamma}c_{\gamma^{-1}}c_{\sigma}\,^{\sigma}c_{\gamma}c_{\sigma}^{-1} = c_{\gamma}^{-1}\sigma.c_{\gamma} $. Since $ c_{\gamma} $ belongs to $ \Aut G_0 $, this indeed shows that $ \partial (\gamma^{-1}) $ is cohomologous to the trivial cocycle.
\item Let $ \tilde{c}_{\gamma}\Id_{\gamma}^{-1} \in \tilde{\Gamma} \unlhd \Aut (R_{k_s}\to \Spec k_{s}\geq k)^{\Gamma}\cong (E\times \Aut (k_{s}\geq k))^{\Gamma} $. By definition, $ \partial (\tilde{c}_{\gamma}\Id_{\gamma}^{-1})_{\sigma } = (\tilde{c}_{\gamma}\Id_{\gamma}^{-1})^{-1}\sigma .(\tilde{c}_{\gamma}\Id_{\gamma}^{-1}) $. Plugging the definition of the semilinear action, we get
\begin{align*}
(\tilde{c}_{\gamma}\Id_{\gamma}^{-1})^{-1}\sigma .(\tilde{c}_{\gamma}\Id_{\gamma}^{-1}) &= \Id_{\gamma}\tilde{c}_{\gamma}^{-1}c_{\gamma \sigma\gamma^{-1}}\,^{\gamma \sigma\gamma^{-1}}\tilde{c}_{\gamma}\,^{\gamma}c_{\sigma}^{-1}\Id_{\gamma}^{-1}\\
& = \,^{\gamma^{-1}}(\tilde{c}_{\gamma}^{-1}c_{\gamma \sigma\gamma^{-1}})\,^{\sigma\gamma^{-1}}\tilde{c}_{\gamma}c_{\sigma}^{-1}\\
& = \,^{\gamma^{-1}}(\tilde{c}_{\gamma}^{-1}c_{\gamma})c_{\sigma}\,^{\sigma}c_{\gamma^{-1}}\,^{\sigma\gamma^{-1}}\tilde{c}_{\gamma}c_{\sigma}^{-1}\\
& = \,^{\gamma^{-1}}\tilde{c}_{\gamma}^{-1}c_{\gamma^{-1}}^{-1}c_{\sigma}\,^{\sigma}(c_{\gamma^{-1}}\,^{\gamma^{-1}}\tilde{c}_{\gamma})c_{\sigma}^{-1}\\
& = (c_{\gamma^{-1}}\,^{\gamma^{-1}}\tilde{c}_{\gamma})^{-1} \sigma .(c_{\gamma^{-1}}\,^{\gamma^{-1}}\tilde{c}_{\gamma})\\
& = (c_{\gamma^{-1}} \tilde{c}_{\gamma})^{-1} \sigma .(c_{\gamma^{-1}} \tilde{c}_{\gamma})
\end{align*}
where the last equality holds because the Galois action on $ E = \Aut R_{k_s} $ is trivial. Since $ c_{\gamma^{-1}} \tilde{c}_{\gamma} $ belongs to $ (\Ad G_0)(k_s) $, this indeed shows that $ \partial (\tilde{c}_{\gamma}\Id_{\gamma}^{-1}) $ is cohomologous to the trivial cocycle. \qedhere
\end{enumerate}
\end{proof}
\begin{corollary}
Keep the notations of Proposition~\ref{Prop:Galois cohomology for semilinear auto}.
\begin{enumerate}
\item The exact sequence
\begin{equation*}
1\to (\Aut G_0)^{\Gamma}\to \Aut (G_0\to \Spec k_s\geq k)^{\Gamma}\to \Aut (k_{s}\geq k)\xrightarrow{\partial } H^1(\Gamma , \Aut G_0)
\end{equation*}
induces an exact sequence
\begin{equation*}
1\to (\Aut G_0)^{\Gamma}\to \Aut (G_0\to \Spec k_s\geq k)^{\Gamma}/\tilde{\Gamma} \to \Aut (k_{s}\geq k)/\Gamma \xrightarrow{\partial } H^1(\Gamma , \Aut G_0)
\end{equation*}
\item The exact sequence
\begin{align*}
1\to (\Ad G_0)(k_s)^{\Gamma}\to \Aut (G_0\to &\Spec k_s\geq k)^{\Gamma}\to \\
&\Aut (R_{k_s}\to \Spec k_{s}\geq k)^{\Gamma}\xrightarrow{\partial } H^1(\Gamma , (\Ad G_0) (k_s))
\end{align*}
induces an exact sequence
\begin{align*}
1\to (\Ad G_0)(k_s)^{\Gamma}\to \Aut (G_0\to &\Spec k_s\geq k)^{\Gamma}/\tilde{\Gamma}\to \\ &\Aut (R_{k_s}\to \Spec k_{s}\geq k)^{\Gamma}/\tilde{\Gamma}\xrightarrow{\partial } H^1(\Gamma , (\Ad G_0) (k_s))
\end{align*}
\end{enumerate}
\end{corollary}
\begin{proof}
Note that in both cases, moding out by $ \tilde{\Gamma} $ does not modify exactness on the left. Hence the results follows directly from Lemma~\ref{Lem:triviality of Gamma under cocoundary}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{Thm: MainThm3}]
Note that by Lemma~\ref{Lem:Identification of Aut(G Spec k)}, $ \Aut (G_0\to \Spec k_s\geq k)^{\Gamma}/\tilde{\Gamma} $ is naturally isomorphic to $ \Aut (G\to \Spec k) $. Similarly, by Lemma~\ref{Lem:semilinear Galois cohomology for root datum}, $ \Aut (R_{k_s}\to \Spec k_{s}\geq k)^{\Gamma}/\tilde{\Gamma}\cong \Aut (\mathcal{R}(G)\to \Spec k) $. Also note that the restriction of the semilinear $ \Gamma $-action on $ \Aut (G_0\to \Spec k_s\geq k)^{\Gamma}/\tilde{\Gamma} $ to $ \Aut G_0 $ (respectively $ (\Ad G_0)(k_s) $) is the natural Galois action on $ \Aut G_{k_s} $ (respectively $ (\Ad G)(k_s) $). In particular, the $ \Gamma $ invariant elements are the elements of $ \Aut G $ (respectively $ (\Ad G)(k) $). Finally, noting that $ \Aut (k) \cong \Aut (k_s\geq k)/\Gamma $ and that all those identifications are natural enough, we get the result.
\end{proof}
We now describe how the coboundary map of the exact sequence $ 1\to \Aut G\to \Aut (G\to \Spec k)\to \Aut (k)\to H^1(k,\Aut G_{k_s}) $ can be used to compute $ \Aut_G(k) $. To illustrate this, we set the following notations for the rest of the section:
\begin{definition}\label{Def:setting notation for explicit computation with division algebras}
\begin{enumerate}
\item $ D $ denotes a central division algebra of degree $ 3 $ over $ k $ (hence by a theorem of Wedderburn, $ D $ is cyclic). We fix a maximal Galois subfield $ l $ of $ D $ so that $ \Gal (l/k) $ is cyclic of order $ 3 $ (which exists because $ D $ is cyclic). We choose a generator of $ \Gal (l/k) $ that we denote $ \gamma $. Choosing an element $ u\in D $ normalising $ l $ and such that its action by conjugation on $ l $ generates $ \Gal (l/k) $, we set $ a = u^3\in k $. We set $ G := \textbf{SL}_1(D) $ to be the corresponding algebraic $ k $-group.
\item Set $ G_0 := \SL_3 $ (that we consider over $ k_s $, as in the beginning of this section). Recall that $ \Ad \SL_3 = \PGL_3 $. We denote elements of $ \PGL_3(k_s) $ as $ \begin{bsmallmatrix}
g_{11} & g_{12} & g_{13} \\
g_{21} & g_{22} & g_{23} \\
g_{31} & g_{32} & g_{33}
\end{bsmallmatrix} $, which is to be read as ``the equivalence class corresponding to the matrix $ \begin{psmallmatrix}
g_{11} & g_{12} & g_{13} \\
g_{21} & g_{22} & g_{23} \\
g_{31} & g_{32} & g_{33}
\end{psmallmatrix}\in GL_3(k_s) $''.
\item We choose the usual pinning of $ \SL_3 $ where the pair $ (T,B) $ consists of diagonal matrices and of upper triangular matrices, and where we choose some generators of the corresponding ``basic root groups''. Let $ R $ be the corresponding based root datum. Note that $ \Aut R $ is of order $ 2 $, and that if our choice of generators for the ``basic root groups'' is sensible enough, the splitting of $ \Aut \SL_3\to \Aut R $ is given by the automorphism $ \SL_3\to \SL_3\colon g\mapsto \,^{\text{at}}g^{-1} $, where $ ^{\text{at}}g $ denotes the anti-transposed of $ g $, i.e.\ ``the transposed of $ g $ along the anti-diagonal''. More formally, for $ i,j\in \lbrace 1, 2, 3\rbrace $, $ (^{\text{at}}g)_{ij} = g_{4-j;4-i} $. Note that taking anti-transpose commutes with taking inverse, so that there is no ambiguity in the notation $ ^{\text{at}}g^{-1} $.
\item Consider the homomorphism $ f\colon \Gal (l/k)\to PGL_3(l)\colon \gamma \mapsto \begin{bsmallmatrix}
0 & 0 & a \\
1 & 0 & 0 \\
0 & 1 & 0
\end{bsmallmatrix} $ (where $ a\in k $ and $ \gamma \in \Gal (l/k) $ have been defined in the first item of these definitions). We choose the cocycle $ c\colon \Gamma \to \Aut G_0 $ defining $ G = \textbf{SL}_1(D) $ over $ k $ to be the composition $ \Gamma \to \Gal (l/k)\xrightarrow{f}\PGL_3(l)\to \Aut G_0 $.
\end{enumerate}
\end{definition}
Having set those notations, we are ready to start computing. The following lemmas are two special cases of \cite{Ha07} that we recover without using any theory of division algebras.
\begin{lemma}\label{Lem:Hanke result part 1}
Keep the notations of Definition~\ref{Def:setting notation for explicit computation with division algebras} and let $ \beta \in \Aut (k_s\geq k) $ be such that $ \beta (l) = l $. Let $ \alpha $ be the restriction of $ \beta $ to $ k $. Then $ \alpha \in \Aut_G(k) $ if and only if there exists $ \lambda \in l^* $ such that either $ \frac{\alpha(a)}{a} = \,^{\gamma^2}\lambda \,^{\gamma}\lambda \lambda $ or $ \alpha (a)a = \,^{\gamma^2}\lambda \,^{\gamma}\lambda \lambda $.
\end{lemma}
\begin{proof}
Note that $ \alpha \in \Aut_G(k) $ if and only if $ \alpha^{-1}\in \Aut_G(k) $. Using the coboundary map of Proposition~\ref{Prop:Galois cohomology for semilinear auto}, for all $ \sigma \in \Gamma $ we have $ \partial (\beta^{-1})_{\sigma} = \Id_{\beta}^{-1}\sigma .\Id_{\beta} = \Id_{\beta}^{-1} c_{\beta^{-1}\sigma \beta}\,^{\beta^{-1}}c_{\sigma}^{-1} \Id_{\beta} = \,^{\beta}c_{\beta^{-1}\sigma \beta}c_{\sigma}^{-1} $. Hence by Proposition~\ref{Prop:Galois cohomology for semilinear auto}, $ \alpha \in \Aut_G(k) $ if and only if the cocycle $ \sigma \mapsto \,^{\beta}c_{\beta^{-1}\sigma \beta}c_{\sigma}^{-1} $ is trivial in $ H^1(k,\Aut SL_3) $, i.e.\ if and only if there exists $ g \in \PGL_3(k_s) $ and $ e\in \Aut R\leq \Aut \SL_3 $ such that $ (ge)^{-1}\,^{\beta}c_{\beta^{-1}\sigma \beta}c_{\sigma}^{-1} \sigma .(ge) = 1 $. But since $ \gamma $ generates $ \Gal (l/k) $ and since $ c_{\sigma}=1 $ for all $ \sigma $ acting trivially on $ l $, this is equivalent to the existence of $ g \in \PGL_3(l) $ and $ e\in \Aut R\leq \Aut \SL_3 $ such that $ (ge)^{-1}\,^{\beta}c_{\beta^{-1}\gamma \beta}c_{\gamma}^{-1} \gamma .(ge) = 1 $. Recalling that $ \gamma .(ge) = c_{\gamma }\,^{\gamma}(ge)c_{\gamma }^{-1} = c_{\gamma }\,^{\gamma}g\,ec_{\gamma }^{-1} $, this is equivalent to $ \,^{\beta}c_{\beta^{-1}\gamma \beta}\,^{\gamma}g\,ec_{\gamma }^{-1}e^{-1} = g $. Also note that $ ec_{\gamma }^{-1}e^{-1} = \begin{cases}
c_{\gamma} \text{ if } e \text{ is non-trivial}\\
c_{\gamma }^{-1} \text{ otherwise}\end{cases} $.
First assume that $ \beta^{-1}\gamma \beta = \gamma $. Then $ \partial (\beta^{-1}) $ is trivial if and only if there exists $ g\in \PGL_3(l) $ such that either $ \,^{\beta}c_{\gamma}\,^{\gamma}gc_{\gamma}^{-1} = g $ or $ \,^{\beta}c_{\gamma}\,^{\gamma}gc_{\gamma} = g $. Letting $ g = \begin{bsmallmatrix}
g_{11} & g_{12} & g_{13} \\
g_{21} & g_{22} & g_{23} \\
g_{31} & g_{32} & g_{33}
\end{bsmallmatrix} \in \PGL_3(l) $, we have $ \,^{\beta}c_{\gamma}\,^{\gamma}gc_{\gamma}^{-1} = \begin{bsmallmatrix}
\,^{\gamma}g_{33}\frac{\beta (a)}{a} & \,^{\gamma}g_{31}\beta (a) & \,^{\gamma}g_{32}\beta (a) \\
\,^{\gamma}g_{13}a^{-1} & \,^{\gamma}g_{11} & \,^{\gamma}g_{12} \\
\,^{\gamma}g_{23}a^{-1} & \,^{\gamma}g_{21} & \,^{\gamma}g_{22}
\end{bsmallmatrix} $. Since $ g $ is invertible, one of $ g_{13} $, $ g_{23} $ and $ g_{33} $ is non-zero. Let us for example assume that $ g_{33}\neq 0 $. Now $ \,^{\beta}c_{\gamma}\,^{\gamma}gc_{\gamma}^{-1} = g $ if and only if there exists $ \lambda \in l^* $ such that $ \,^{\gamma}g_{33}\frac{\beta (a)}{a} = g_{11}\lambda $, $ \,^{\gamma}g_{11} = g_{22}\lambda $ and $ \,^{\gamma}g_{22} = g_{33}\lambda $. Hence $ \dfrac{^{\gamma}(\,^{\gamma}g_{33}\frac{\beta (a)}{a})}{^{\gamma}\lambda} = g_{22}\lambda $, and furthermore $ \dfrac{^{\gamma^2}(\,^{\gamma}g_{33}\frac{\beta (a)}{a})}{^{\gamma^2}\lambda\,^{\gamma}\lambda} = g_{33}\lambda $. Since the computation is similar if $ g_{13}\neq 0 $ or $ g_{23}\neq 0 $ instead, we conclude that if such a $ g $ exists, then there exists $ \lambda \in l $ such that $ \frac{\beta (a)}{a} = \,^{\gamma^2}\lambda\,^{\gamma}\lambda \lambda $ (because $ \gamma $ acts trivially on $ a\in k $ and $ \gamma^3 $ acts trivially on $ g_{33}\in l $). Conversely, if there exists $ \lambda \in l^* $ such that $ \frac{\beta (a)}{a} = \,^{\gamma^2}\lambda\,^{\gamma}\lambda \lambda $, then $ g = \begin{bsmallmatrix}
\,^{\gamma^2}\lambda \,^{\gamma}\lambda &0 & 0 \\
0 & \,^{\gamma^2}\lambda & 0 \\
0 &0 & 1
\end{bsmallmatrix} $ is such that $ \,^{\beta}c_{\gamma}\,^{\gamma}gc_{\gamma}^{-1} = g $. A similar computation shows that there exists $ g\in \PGL_3(l) $ such that $ \,^{\beta}c_{\gamma}\,^{\gamma}gc_{\gamma} = g $ if and only if there exists $ \lambda \in l^* $ such that $ \beta (a)a = \,^{\gamma^2}\lambda\,^{\gamma}\lambda \lambda $.
Now assuming that $ \beta^{-1}\gamma \beta = \gamma^{-1} $, $ \partial (\beta^{-1}) $ is trivial if and only if there exists $ g\in \PGL_3(l) $ such that either $ \,^{\beta}c_{\gamma^{-1}}\,^{\gamma}gc_{\gamma}^{-1} = g $ or $ \,^{\beta}c_{\gamma^{-1}}\,^{\gamma}gc_{\gamma} = g $. Or equivalently if and only if there exists $ g\in \PGL_3(l) $ such that either $ \,^{\beta}c_{\gamma}\,^{\gamma^{-1}}gc_{\gamma} = g $ or $ \,^{\beta}c_{\gamma}\,^{\gamma^{-1}}gc_{\gamma}^{-1} = g $. Hence we get the same setting as for the case $ \beta^{-1}\gamma \beta = \gamma $ up to replacing $ ^{\gamma}g_{ij} $ with $ \,^{\gamma^{-1}}g_{ij} $, which leaves the conclusion unchanged.
\end{proof}
\begin{lemma}\label{Lem:Hanke result part 2}
Keep the notations of Definition~\ref{Def:setting notation for explicit computation with division algebras} and let $ \beta \in \Aut (k_s\geq k) $ be such that $ \beta (l) = l' \neq l $. Let $ \delta = \beta \gamma \beta^{-1} $ $ ( $a generator of $ \Gal (l'/k))$, let $ K $ be the compositum $ ll'\le k_s $ and let $ \Gal (K/k) = \Gal (l/k)\times \Gal (l'/k) $ be the corresponding decomposition of $ \Gal (K/k) $. Let $ \alpha $ be the restriction of $ \beta $ to $ k $. Then $ \alpha \in \Aut_G(k) $ if and only if there exist $ \lambda, \mu \in K^* $ such that
\begin{enumerate}
\item either $ \,^{\gamma^2}\lambda\,^{\gamma}\lambda \lambda = \frac{1}{a} $, $ \,^{\delta^2}\mu\,^{\delta}\mu \mu = \beta (a) $ and $ \frac{^{\delta}\lambda}{\lambda} = \frac{^{\gamma}\mu}{\mu} $,
\item or $ \,^{\gamma^2}\lambda\,^{\gamma}\lambda \lambda = a $, $ \,^{\delta^2}\mu\,^{\delta}\mu \mu = \beta (a) $ and $ \frac{^{\delta}\lambda}{\lambda} = \frac{^{\gamma}\mu}{\mu} $.
\end{enumerate}
\end{lemma}
\begin{proof}
Arguing as in the beginning of the proof of Lemma~\ref{Lem:Hanke result part 1}, $ \alpha \in \Aut_G(k) $ if and only if there exists $ g \in \PGL_3(K) $ and $ e\in \Aut R\leq \Aut \SL_3 $ such that
\begin{equation}\label{Eqn 1}
^{\beta}c_{\beta^{-1}\sigma \beta} \,^{\sigma}g ec_{\sigma}^{-1}e^{-1} = g \text{ for all } \sigma \in \Gal (K/k)
\end{equation}
We first do the case $ e=1 $. Taking $ \sigma = \gamma $ in Equation~\ref{Eqn 1}, we get $ ^{\gamma}gc_{\gamma}^{-1} = g $. On the other hand, taking $ \sigma = \delta $ in Equation~\ref{Eqn 1}, we get $ ^{\beta}c_{\gamma}\,^{\delta}g = g $. Hence Equation~\ref{Eqn 1} implies that
\begin{equation*}
\begin{bsmallmatrix}
\frac{^{\gamma}g_{13}}{a} & ^{\gamma}g_{11} & ^{\gamma}g_{12} \\
\frac{^{\gamma}g_{23}}{a} & ^{\gamma}g_{21} & ^{\gamma}g_{22} \\
\frac{^{\gamma}g_{33}}{a} & ^{\gamma}g_{31} & ^{\gamma}g_{32}
\end{bsmallmatrix} = \begin{bsmallmatrix}
g_{11} & g_{12} & g_{13} \\
g_{21} & g_{22} & g_{23} \\
g_{31} & g_{32} & g_{33}
\end{bsmallmatrix} = \begin{bsmallmatrix}
^{\delta}g_{31}\beta (a) & ^{\delta}g_{32}\beta (a) & ^{\delta}g_{33}\beta (a) \\
^{\delta}g_{11} & ^{\delta}g_{12} & ^{\delta}g_{13} \\
^{\delta}g_{21}& ^{\delta}g_{22} & ^{\delta}g_{23}
\end{bsmallmatrix}
\end{equation*}
To fix ideas, assume $ g_{11}\neq 0 $ (the computation is similar if $ g_{21}\neq 0 $ or $ g_{31}\neq 0 $ instead). So we can further assume that $ g_{11} = 1 $. Hence there exists $ \lambda , \mu \in K^* $ such that
\begin{center}
\begin{tikzpicture}[->]
\node (1) at (-0.3,0) {$ \lambda^{-1}\begin{pmatrix}
\frac{^{\gamma}(^{\gamma}\lambda^{-1}\lambda^{-1})}{a} & 1 & ^{\gamma}\lambda^{-1} \\
\frac{^{\gamma}g_{23}}{a} & ^{\gamma}g_{21} & ^{\gamma}g_{22} \\
\frac{^{\gamma}g_{33}}{a} & ^{\gamma}g_{31} & ^{\gamma}g_{32}
\end{pmatrix} $};
\node (2) at (7.5,0) {$ \mu^{-1}\begin{pmatrix}
^{\delta}(^{\delta}\mu^{-1}\mu^{-1})\beta (a) & ^{\delta}g_{32}\beta (a) & ^{\delta}g_{33}\beta (a) \\
1 & ^{\delta}g_{12} & ^{\delta}g_{13} \\
^{\delta}\mu^{-1}& ^{\delta}g_{22} & ^{\delta}g_{23}
\end{pmatrix} $} ;
\node (3) at (3.5,-2) {$ \begin{pmatrix}
1 & \lambda^{-1} & ^{\gamma}\lambda^{-1}\lambda^{-1} \\
\mu^{-1} & g_{22} & g_{23} \\
^{\delta}\mu^{-1}\mu^{-1} & g_{32} & g_{33}
\end{pmatrix} $};
\node (4) at (1.7,-1.1) [rotate = -45] {$ = $};
\node (5) at (5.3,-1.1) [rotate = 45] {$ = $};
\end{tikzpicture}
\end{center}
By looking at the $ 11 $ coefficient, this already implies that $ ^{\gamma^2}\lambda\,^{\gamma}\lambda \lambda = a^{-1} $ and $ ^{\delta^2}\mu\,^{\delta}\mu \mu = \beta (a) $. Finally, by looking at the central coefficient, we see that $ \lambda^{-1}\,^{\gamma}\mu^{-1} = \mu^{-1}\,^{\delta}\lambda^{-1} $. Conversely, if there exists $ \lambda ,\mu \in K^* $ such that $ \,^{\gamma^2}\lambda\,^{\gamma}\lambda \lambda = \frac{1}{a} $, $ \,^{\delta^2}\mu\,^{\delta}\mu \mu = \beta (a) $ and $ \frac{^{\delta}\lambda}{\lambda} = \frac{^{\gamma}\mu}{\mu} $, one can check that $ g = \begin{bmatrix}
1 & \lambda^{-1} & ^{\gamma}\lambda^{-1}\lambda^{-1} \\
\mu^{-1} & \lambda^{-1}\,^{\gamma}\mu^{-1} & \lambda^{-1}\,^{\gamma}(\lambda^{-1}\,^{\gamma}\mu^{-1}) \\
^{\delta}\mu^{-1}\mu^{-1} & \lambda^{-1}\,^{\gamma}(^{\delta}\mu^{-1}\mu^{-1}) & \lambda^{-1}\,^{\gamma} (\lambda^{-1}\,^{\gamma}(^{\delta}\mu^{-1}\mu^{-1}))
\end{bmatrix} $ satisfies Equation~\ref{Eqn 1}.
In the case where $ e $ is the non-trivial automorphism of $ R $, one uses the fact $ ec_{\gamma}e^{-1} = c_{\gamma}^{-1} $ and then imitates the above computation to get the second condition on $ \lambda ,\mu $ (to carry out this computation most easily, assume that $ g_{11} = 1 $ and for the last condition, consider the $ 23 $ coefficient).
\end{proof}
\begin{remark}\label{Rem:applicability of these computations}
The kind of computations we perform in Lemma~\ref{Lem:Hanke result part 1} can easily be adapted for any cyclic division algebra, and the computations in Lemma~\ref{Lem:Hanke result part 2} can easily be adapted to any cyclic division algebras of prime degrees. When the degree is not prime, $ l \cap \beta (l) $ might be a non-trivial extension of $ k $ (for $ l $ a maximal cyclic subfield of $ D $), and we did not try to overcome this complication using our methods. Note that in \cite{Ha07}*{Section~3}, T.\ Hanke deals with this extra difficulty very efficiently.
\end{remark}
\begin{remark}
In light of the results in \cite{Ha07}, we proved that $ \alpha \in \Aut_G(k) $ if and only if $ ^{\alpha}D\cong D $ or $ ^{\alpha}D\cong D^{\text{opp}} $ (as $ k $-algebras). This is consistent with the fact that $ ^{\alpha }\textbf{SL}_1(D) \cong \textbf{SL}_1(\,^{\alpha}D) $, and that for two finite dimensional central division algebras $ D_1,D_2 $ over $ k $, $ \textbf{SL}_1(D_1)\cong \textbf{SL}_1(D_2) $ (as algebraic $ k $-groups) if and only if $ D_1\cong D_2 $ or $ D_1\cong D_2^{\text{opp}} $ (as $ k $-algebras).
\end{remark}
\section{Semilinear automorphisms of based root datum}\label{Sec:computing the group of semilinear automorphism of based root datum}
We aim to give an explicit description of the short exact sequence $ 1\to \Aut \mathcal{R}(G) \to \Aut (\mathcal{R}(G) \to \Spec k)\to \Aut_{\mathcal{R}(G)}(k)\to 1 $. We base this computation on Lemma~\ref{Lem:semilinear Galois cohomology for root datum}.
Recall that for $ R_{k_s} $ a split $ k $-scheme of based root datum of type $ R $, the $ \Gal (k_s/k) $-action on $ \Aut R_{k_s} $ is trivial, so that $ H^1(k_s/k, \Aut R_{k_s}) $ is isomorphic to the set of continuous homomorphisms $ \Hom (\Gal (k_s/k),\Aut R) $ up to conjugation.
\begin{definition}
Let $ \mathcal{R} $ be a $ k $-scheme of based root datum of type $ R $.
\begin{enumerate}
\item Fix an isomorphism $ \mathcal{R}_{k_s}\cong R_{k_s} $ and let $ \tilde{c}\colon \Gal (k_s/k)\to \Aut R $ be the corresponding cocycle. Let $ N\unlhd \Gal (k_s/k) $ be the kernel of the homomorphism $ \tilde{c} $, and let $ l $ be the Galois extension of $ k $ fixed by $ N $. We call $ l $ \textbf{the classifying field of} $ \mathcal{R} $. Once a separable closure of $ k $ has been fixed, the classifying field of $ \mathcal{R} $ is uniquely determined by $ \mathcal{R} $.
\item We say that $ \mathcal{R} $ (or $ R $) is \textbf{semisimple} (respectively \textbf{simply connected}, respectively \textbf{adjoint}, respectively \textbf{simple}) if the split connected reductive group of type $ R $ is semisimple (respectively simply connected, respectively adjoint, respectively simple).
\end{enumerate}
\end{definition}
\begin{remark}
We use the following terminology: a connected reductive $ k $-group is simple if it is non-abelian and has no non-trivial connected closed normal subgroup (some author prefer to call such groups quasi-simple).
\end{remark}
\begin{lemma}\label{Lem:classification of Dynkin diagrams}
Let $R $ be a simple reduced based root datum and let $ k_s $ be a separable closure of $ k $. The map which associates to a $ k_s/k $-form of $ R_{k_s} $ its classifying field is a bijection between $ k $-schemes of based root datum of type $ R $ (up to $ k $-isomorphism) and subfields $ l\leq k_s $ such that $ l $ is Galois over $ k $ and $ \Gal (l/k) $ is isomorphic to a subgroup of $ \Aut R $.
\end{lemma}
\begin{proof}
Let $ D $ be the Dynkin diagram associated to $ R $. Since $ R $ is semisimple and reduced, $ \Aut R\leq \Aut D $. Furthermore, since $ R $ is simple $ \Aut D $ is either trivial, $ \Z/2\Z $ or $ S_3 $. Hence, if two subgroups of $ \Aut R $ are isomorphic, they are actually conjugate. The result follows from the fact that the Galois action on $ \Aut R $ is trivial, and hence $ H^1(k_s/k,\Aut R_{k_s}) $ is isomorphic to the set of continuous homomorphisms $ \Hom (\Gal (k_s/k),\Aut R) $ modulo conjugation.
\end{proof}
\begin{remark}
In the notations of the proof of Lemma~\ref{Lem:classification of Dynkin diagrams}, one might wonder when the inclusion $ \Aut R\leq \Aut D $ is an equality. This is always the case, except possibly when $ R $ is not simply connected or adjoint and is of type $ D_{2n} $. See \cite{Con14}*{Proposition~1.5.1} for a precise statement.
\end{remark}
\begin{definition}
Let $ \mathcal{R} $ be a simple $ k $-scheme of based root datum, and let $ k_s $ be a separable closure of $ k $. We define the \textbf{Tits index of} $ \mathcal{R} $ to be $ ^{g}X_{n,l} $ where
\begin{enumerate}
\item $ l\leq k_s $ is the classifying field of $ \mathcal{R} $ (hence $ l $ is a finite Galois extension of $ k $).
\item $ X_n $ is the label of the Dynkin diagram associated to $ R $.
\item $ g $ is the order of the Galois group $ \Gal (l/k) $.
\end{enumerate}
\end{definition}
\begin{lemma}\label{Lem:descriptionofabstractautomoprhismsofDyn}
Let $ \mathcal{R} $ be a simple $ k $-scheme of based root datum of type $ R $ with index $ ^gX_{n,l} $.
\begin{enumerate}
\item $ g \in \lbrace 1,2,3,6\rbrace $.
\item If $ g=1 $, $ \Aut (\mathcal{R} \to \Spec k)\cong \Aut R\times \Aut (k) $, and this isomorphism restricts to $ \Aut \mathcal{R} \cong \Aut R $.
\item If $g=2$ or $g=3$, $ \Aut (\mathcal{R} \to \Spec k)\cong \Aut (l\geq k) $, and this isomorphism restricts to $ \Aut \mathcal{R}\cong \Gal (l/k) $.
\item If $g=6$, $ \Aut (\mathcal{R} \to \Spec k)\cong \Aut (l_3\geq k) $, where $ l_3 $ is any non-normal cubic subextension of $ l/k $. Furthermore, $ \Aut \mathcal{R} $ is trivial.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item Let $ D $ be the Dynkin diagram associated to $ R $. Since $ R $ is semisimple and reduced, $ \Aut R\leq \Aut D $. Furthermore, since $ R $ is simple $ \Aut D $ is either trivial, $ \Z/2\Z $ or $ S_3 $. It follows that $ g\in \lbrace 1,2,3,6\rbrace $.
\item The case $ g=1 $ means that $ \mathcal{R} $ is a split $ k $-scheme of based root datum. Hence, $ \Aut \mathcal{R}\cong \Aut R $ (because the functor of constant objects is fully faithfull). Furthermore the short exact sequence $ 1\to \Aut \mathcal{R}\to \Aut (\mathcal{R} \to \Spec k)\to \Aut (k)\to 1 $ splits. Also note that $ \Aut (k) $ acts trivially on $ \Aut R $, so that the result follows.
\item Recall that by Lemma~\ref{Lem:semilinear Galois cohomology for root datum}, $ \Aut (\mathcal{R} \to \Spec k)\cong (\Aut R\times \Aut (k_s\geq k))^{\Gamma}/\tilde{\Gamma} $, where $ \Aut R\times \Aut (k_s\geq k) $ is endowed with a semilinear $ \Gamma $-action arising from a choice of cocycle $ \tilde{c}\colon \Gamma \to \Aut R $ defining $ \mathcal{R} $. For $ \beta \in \Aut (l\geq k) $, let $ \tilde{\beta} $ denote an extension of $ \beta $ to an element of $ \Aut (k_s\geq k) $. Let also $ s\in \Aut R $ be an element of order $ 2 $ (note that the only case where the existence of such an element is not clear is in type $ ^3D_4 $, in which case it follows from Lemma~\ref{Lem:elements of order 2 in Aut R}). We define a map
\begin{align*}
\Phi \colon \Aut (l\geq k)&\to (\Aut R \times \Aut (k_s\geq k))^{\Gamma}/\tilde{\Gamma}\\
\beta &\mapsto \begin{cases} \Id_{\tilde{\beta}}^{-1} \text{ if } \beta\gamma \beta^{-1} = \gamma \text{ for all } \gamma \in \Gal (l/k).\\
s\Id_{\tilde{\beta}}^{-1} \text{ if } \beta \gamma \beta^{-1} \neq \gamma \text{ for } \gamma \text{ a generator of } \Gal (l/k).
\end{cases}
\end{align*}
We first check that $ \Phi $ is well-defined. If $ \tilde{\beta} $ and $ \tilde{\beta}' $ are two extensions of $ \beta $, we have to check that $ \Id_{ \tilde{\beta}'}\Id_{\tilde{\beta}}^{-1} $ belongs to $ \tilde{\Gamma} $. But $ (\tilde{\beta}')^{-1}\tilde{\beta} $ acts trivially on $ l $, hence the image of $ (\tilde{\beta}')^{-1}\tilde{\beta} $ under $ \Gamma \to (\Aut R \times \Aut (k_s\geq k))^{\Gamma}\colon \delta \mapsto \tilde{c}_{\delta}\Id_{\delta}^{-1} $ is indeed equal to $ \Id_{(\tilde{\beta}')^{-1}\tilde{\beta}}^{-1} = \Id_{ \tilde{\beta}'}\Id_{\tilde{\beta}}^{-1} $.
We now check that the image of $ \Phi $ is $ \Gamma $-invariant. Let $ \delta \mapsto \bar{\delta} $ denotes the projection $ \Aut (k_s/k)\to \Aut (l/k) $. When $ \beta \gamma \beta^{-1} = \gamma $ for all $ \gamma \in \Gal (l/k) $, $ \delta .\Id_{\tilde{\beta}}^{-1} = \tilde{c}_{\tilde{\beta} \delta \tilde{\beta}^{-1}}\tilde{c}_{\delta}^{-1}\Id_{\tilde{\beta}}^{-1} = \tilde{c}_{\beta \bar{\delta} \beta^{-1}}\tilde{c}_{\bar{\delta}}^{-1}\Id_{\tilde{\beta}}^{-1} = \tilde{c}_{\bar{\delta}}\tilde{c}_{\bar{\delta}}^{-1}\Id_{\tilde{\beta}}^{-1} = \Id_{\tilde{\beta}}^{-1}$ for all $ \delta \in \Gal (k_s/k) $. On the other hand, when $ \beta \gamma \beta^{-1} \neq \gamma $ for $ \gamma $ a generator of $ \Gal (l/k) $, we have $$ \delta .(s\Id_{\tilde{\beta}}^{-1}) = \tilde{c}_{\tilde{\beta} \delta \tilde{\beta}^{-1}}s\tilde{c}_{\delta}^{-1}\Id_{\tilde{\beta}}^{-1} = \tilde{c}_{\beta \bar{\delta} \beta^{-1}}s\tilde{c}_{\bar{\delta}}^{-1}\Id_{\tilde{\beta}}^{-1} = s\tilde{c}_{\bar{\delta}}\tilde{c}_{\bar{\delta}}^{-1}\Id_{\tilde{\beta}}^{-1} = s\Id_{\tilde{\beta}}^{-1}$$ for all $ \delta \in \Gal (k_s/k) $.
It is readily checked that $ \Phi $ is a homomorphism, so it remains to check that $ \Phi $ is bijective. If $ \Phi (\beta) $ is trivial, then $ \Id_{\tilde{\beta}}^{-1} $ or $ s \Id_{\tilde{\beta}}^{-1} $ belongs to $ \tilde{\Gamma} $, i.e.\ there exists $ \delta \in \Gamma $ such that either $ \Id_{\tilde{\beta}}^{-1} $ or $ s \Id_{\tilde{\beta}}^{-1}$ is equal to $ \tilde{c}_{\delta}\Id_{\delta}^{-1} $. This implies that $ \tilde{c}_{\delta} $ is trivial, so that $ \delta $ acts trivially on $ l $, and hence $ \beta $ is trivial.
Finally, we check that $ \Phi $ is surjective. Let $ b\Id_{\tilde{\beta}} \in (\Aut R \times \Aut (k_s\geq k))^{\Gamma} $. We claim that $ \tilde{\beta} $ preserves $ l $. Indeed, for all $ \delta \in \Gamma $, $ \delta . (b\Id_{\tilde{\beta}}) = \tilde{c}_{\tilde{\beta}^{-1}\delta \tilde{\beta}}b\tilde{c}_{\delta}^{-1}\Id_{\tilde{\beta}} $. Hence $ b\Id_{\tilde{\beta}} $ is $ \Gamma $-invariant if and only if $ b\tilde{c}_{\delta} = \tilde{c}_{\tilde{\beta}^{-1} \delta \tilde{\beta}}b $ for all $ \delta \in \Gamma $. But if $ \tilde{\beta} $ does not preserve $ l $, there exists $ \delta \in \Gamma $ such that $ \tilde{c}_{\delta} = 1 \neq \tilde{c}_{\tilde{\beta}^{-1} \delta \tilde{\beta}} $, a contradiction. Hence the claim is proved. To conclude, note that if $ b\Id_{\tilde{\beta}} $ is $ \Gamma $-invariant, $ \tilde{\beta} $ preserves $ l $ and hence up to an element in the image of $ \Phi $, we can assume that $ \tilde{\beta} $ acts trivially on $ l $. Hence, since $ b\Id_{\tilde{\beta}} $ is $ \Gamma $-invariant, either $ b $ is trivial, or $ b $ commutes with the image of $ \tilde{c} $, and hence belongs to the image of $ \tilde{c} $. In the first case, $ b\Id_{\tilde{\beta}} = \Id_{\tilde{\beta}} $ is in $ \tilde{\Gamma} $. In the second case, $ b\Id_{\tilde{\beta}} = \tilde{c}_{\gamma}\Id_{\tilde{\gamma}}^{-1}\Id_{\tilde{\gamma}}\Id_{\tilde{\beta}} $ for some $ \gamma \in \Gal (l/k) $. But this is in the image of $\Phi$ because $ \tilde{c}_{\tilde{\gamma}}\Id_{\tilde{\gamma}}^{-1} $ and $ \Id_{\tilde{\beta}} $ are in $ \tilde{\Gamma} $, whilst $ \Id_{\tilde{\gamma}} = \Phi (\gamma^{-1}) $.
For the last statement, note that under the isomorphism $ \Aut (l\geq k)\cong \Aut (\mathcal{R}\to \Spec k) $, the algebraic automorphisms are the one acting trivially on $ k $, i.e.\ we have $ \Aut \mathcal{R}\cong \Aut (l/k) $.
\item We begin by proving the following claim.
\begin{claim}\label{Claim:1}
Any automorphism $ \beta \in \Aut (l_3\geq k) $ has a unique extension $ \beta_0 \in \Aut (l\geq k) $ such that for all $ \gamma \in \Gal (l/k) $, $ \beta_0^{-1}\gamma \beta_0 = \gamma $.
\end{claim}
\begin{claimproof}
Let $ \beta \in \Aut (l_3\geq k) $ and let $ \tilde{\beta}\in \Aut (k_s/k) $ be an extension to $ k_s $. Since $ \beta $ preserves $ l_3 $ and since $ l $ is the normal closure of $ l_3 $, $ \tilde{\beta} $ preserves $ l $. Let $ l_3' $ and $ l_3'' $ be the two other degree $ 3 $ extension of $ k $ contained in $ l $. Either $ \tilde{\beta}(l_3') = l_3' $, or $ \tilde{\beta}(l_3') = l_3'' $. In the latter case, replace $ \tilde{\beta} $ by $ \tilde{\beta}\gamma_0 $, where $ \gamma_0 \in \Gal(l/k) $ acts trivially on $ l_3 $ and exchanges $ l_3' $ and $ l_3'' $. Hence we can assume that $ \tilde{\beta} $ preserves $ l_3 $, $ l_3' $ and $ l_3'' $. But now the restriction of $ \tilde{\beta} $ to $ l $ has the desired property. For uniqueness, note that if $ \beta_0' $ is another such extension, then $ \beta_0'\beta_0^{-1} $ is an element of $ \Gal (l/k) $ preserving $ l_3 $, $ l_3' $ and $ l_3'' $, hence $ \beta_0'\beta_0^{-1} =1 $.
\end{claimproof}
For $ \beta \in \Aut (l_3\geq k) $, we denote by $ \beta_0 $ the unique extension of $ \beta $ to an element of $ \Aut (l\geq k) $ provided by Claim~\ref{Claim:1}, and by $ \tilde{\beta_0} $ an extension of $ \beta_0 $ to $ \Aut (k_s/k) $. Now the proof follows the same line as the previous proof of the previous item, and we discuss it more briefly. We define a map
\begin{align*}
\Phi \colon \Aut (l_3\geq k)&\to (\Aut R \times \Aut (k_s\geq k))^{\Gamma}/\tilde{\Gamma}\\
\beta &\mapsto \Id_{\tilde{\beta}_0}^{-1}
\end{align*}
The proof that $ \Phi (\beta) $ does not depend on a lift of $ \beta_0 $ and that $ \Id_{\tilde{\beta}_0}^{-1} $ is $ \Gamma $-invariant follows the same line as in the previous item. Furthermore, $ \Phi $ is clearly a homomorphism.
Assume now that $ \Phi (\beta) $ is trivial. Hence there exists $ \delta \in \Gamma $ such that $ \Id_{\tilde{\beta}_0}^{-1} = \tilde{c}_{\delta}\Id_{\delta}^{-1} $. Hence $ \tilde{c}_{\delta} $ is trivial, which implies that $ \delta $ acts trivially on $ l $, so that $ \beta $ was trivial. Hence $ \Phi $ is injective. Let us now prove surjectivity. Let $ b\Id_{\tilde{\beta}} \in (\Aut R \times \Aut (k_s\geq k))^{\Gamma} $. Since $ \tilde{c}\colon \Gal (k_s/k)\to \Aut R $ is surjective and since we are working modulo $ \tilde{\Gamma} $, we can assume that $ b=1 $. We claim that $ \tilde{\beta} $ preserves $ l $ and that $ \tilde{\beta}^{-1}\gamma \tilde{\beta} = \gamma $ for all $ \gamma \in \Aut (l/k) $. Indeed, for all $ \delta \in \Gamma $, $ \delta . \Id_{\tilde{\beta}} = \tilde{c}_{\tilde{\beta}^{-1}\delta \tilde{\beta}}\tilde{c}_{\delta}^{-1}\Id_{\tilde{\beta}} $. Hence $ \Id_{\tilde{\beta}} $ is $ \Gamma $-invariant if and only if $ \tilde{c}_{\delta} = \tilde{c}_{\tilde{\beta}^{-1} \delta \tilde{\beta}} $ for all $ \delta \in \Gamma $. But if $ \tilde{\beta} $ does not preserve $ l $, there exists $ \delta \in \Gamma $ such that $ \tilde{c}_{\delta} = 1 \neq \tilde{c}_{\tilde{\beta}^{-1} \delta \tilde{\beta}} $, a contradiction. The fact that $ \tilde{\beta}^{-1}\gamma \tilde{\beta} = \gamma $ for all $ \gamma \in \Aut (l/k) $ also follows directly, and the claim is proved.
To conclude, note that the claim implies that $ \tilde{\beta} $ preserves $ l_3 $, and hence up to an element in the image of $ \Phi $, we can assume that $ \tilde{\beta} $ acts trivially on $ l $, so that $ \Id_{\tilde{\beta}} $ is trivial modulo $ \tilde{\Gamma} $, as wanted. \qedhere
\end{enumerate}
\end{proof}
In the proof of Lemma~\ref{Lem:descriptionofabstractautomoprhismsofDyn}, we needed the following lemma.
\begin{lemma}\label{Lem:elements of order 2 in Aut R}
Let $ \mathcal{R} $ be a simple based root datum of type $ R $ with Tits index $ ^3D_{4,l} $. Then $ \mathcal{R} $ is simply connected or adjoint, and hence $ \Aut R $ contains an element of order $ 2 $.
\end{lemma}
\begin{proof}
If $ R $ is neither simply connected nor adjoint, the corresponding split connected reductive group is the split $ \SO_8 $ (there are actually three proper subgroups in the center of the split $ \Spin_8 $, but the corresponding intermediate quotients are all isomorphic). But the split $ \SO_8 $ does not have an outer automorphism of order $ 3 $, contradicting the fact that the Tits index of $ \mathcal{R} $ is $ ^3D_{4,l} $. The last part of the lemma follows from the fact that if $ R $ is simply connected or adjoint, $ \Aut R = \Aut D_4 $ (see \cite{Con14}*{Proposition~1.5.1}) and the fact that $ \Aut D_4 = S_3 $.
\end{proof}
\begin{corollary}\label{Cor:field auto preserving k}
Let $ \mathcal{R} $ be a simple $ k $-scheme of based root datum with classifying field $ l $. If $ \Aut (l/k)\ncong S_3 $, then $ \Aut_{\mathcal{R}}(k) \cong \lbrace \alpha \in \Aut (k)~\vert~ \text{there exists } \tilde{\alpha }\in\Aut (l) \text{ extending } \alpha \rbrace $. While if $ \Aut (l/k)\cong S_3 $, then $ \Aut_{\mathcal{R}}(k) \cong \lbrace \alpha \in \Aut (k)~\vert~ \text{there exists } \tilde{\alpha }\in\Aut (l_3) \text{ extending } \alpha \rbrace $, where $ l_3 $ is a chosen non-normal cubic subextension of $ l/k $.
\end{corollary}
\begin{proof}
This follows from the surjectivity of $ \Aut (\mathcal{R} \to \Spec k)\to \Aut_{\mathcal{R}}(k) $ and from the description of $ \Aut (\mathcal{R} \to \Spec k) $ contained in Lemma~\ref{Lem:descriptionofabstractautomoprhismsofDyn}.
\end{proof}
In view of Corollary~\ref{Cor:field auto preserving k}, it is useful to introduce the following notation.
\begin{definition}
Let $ l\geq k $ be a field extension of $ k $. We denote by $ \Aut_l (k) $ the group of automorphisms of $ k $ which extend to an automorphism of $ l $, i.e.\ $ \Aut_l (k) = \lbrace \alpha \in \Aut (k)~\vert~ \text{there exists } \tilde{\alpha}\in \Aut (l) \text{ extending } \alpha \rbrace $.
\end{definition}
Using the identifications we made in Lemma~\ref{Lem:descriptionofabstractautomoprhismsofDyn} and Corollary~\ref{Cor:field auto preserving k}, we can rewrite in a very explicit form the short exact sequence $ 1\to \Aut \mathcal{R}(G) \to \Aut (\mathcal{R}(G) \to \Spec k)\to \Aut_{\mathcal{R}(G)}(k)\to 1 $.
\begin{proposition}\label{Prop:epplicit SES for Dyn}
Let $ \mathcal{R} $ be a simple $ k $-scheme of based root datum of type $ R $ with Tits index $ ^gX_{n,l} $.
\begin{enumerate}
\item If $ g=1 $, the short exact sequence $ 1\to \Aut \mathcal{R} \to \Aut (\mathcal{R} \to \Spec k)\to \Aut_{\mathcal{R}}(k)\to 1 $ is isomorphic to the short exact sequence $ 1\to \Aut R\to \Aut R\times \Aut (k)\to \Aut (k)\to 1 $. In particular, it always splits.
\item If $ g=2 $ or $ g=3 $, the short exact sequence $ 1\to \Aut \mathcal{R} \to \Aut (\mathcal{R} \to \Spec k)\to \Aut_{\mathcal{R}}(k)\to 1 $ is isomorphic to $ 1\to \Gal (l/k)\to \Aut (l\geq k)\to \Aut_l (k)\to 1 $.
\item If $ g = 6 $, let $ l_3 $ be a $ ( $non normal$ ) $ cubic subextension of $ l/k $. The short exact sequence $ 1\to \Aut \mathcal{R} \to \Aut (\mathcal{R} \to \Spec k)\to \Aut_{\mathcal{R}}(k)\to 1 $ is isomorphic to $ 1\to 1\to \Aut (l_3\geq k)\to \Aut_{l_3} (k)\to 1 $. In particular, it always splits.
\end{enumerate}
\end{proposition}
\begin{proof}
This is a direct consequence of Lemma~\ref{Lem:descriptionofabstractautomoprhismsofDyn} and Corollary~\ref{Cor:field auto preserving k}. Note that in each case, the map $ \Aut (l\geq k)\to \Aut_l (k) $ is given by restriction to $ k $. Also note that when $ g=6 $, and since $ l_3 $ is a non normal cubic extension of $ k $, the group $ \Aut (l_3/k) $ is trivial, and $ \Aut (l_3\geq k)\cong \Aut_{l_3}(k) $.
\end{proof}
We end this discussion with examples where the short exact sequence $ 1\to \Aut \mathcal{R} \to \Aut (\mathcal{R} \to \Spec k)\to \Aut_{\mathcal{R}}(k)\to 1 $ does not split.
\begin{definition}\label{Def:strongly rigid field}
The field $ k $ is called \textbf{rigid} if for any finite Galois extension $ k' $ of $ k $ such that $ k' $ is not algebraically closed, every automorphism of $ k' $ fixes $ k $ pointwise.
\end{definition}
\begin{definition}
A \textbf{prime field} is either the field of rational numbers of a finite field of order $ p $ for some prime $ p $.
\end{definition}
Examples of rigid fields include prime fields and $ \mathbf{Q}_p $ (the field of $ p $-adic numbers) for any prime $ p $. Let us give a reference for this latter assertion.
\begin{lemma}\label{Lem:Qp is rigid}
Let $ p $ be a prime number and let $ \mathbf{Q}_p $ be the field of $ p $-adic numbers. Let $ \mathbf{Q}_p\leq k' $ be a finite Galois extension. Then every automorphism of $ k' $ fixes pointwise $ \mathbf{Q}_p $.
\end{lemma}
\begin{proof}
The field $ k' $ is complete and non algebraically closed. Hence by \cite{Sch33}, all complete norms on $ k' $ are equivalent. Hence an automorphism of $ k' $ has to preserve the norm, which is to say that it has to be continuous. But since any automorphism acts trivially on $ \mathbf{Q} $, by continuity it also has to act trivially on $ \mathbf{Q}_p $.
\end{proof}
\begin{corollary}\label{Cor:very explicit non-splitting}
Assume that $ k $ is a finite $ ( $respectively possibly infinite$ ) $ Galois extension of a rigid $ ( $respectively prime$ ) $ field $ k_0 $. Let $ G $ be a connected reductive $ k $-group which is quasi-split and absolutely simple. Assume that $ \mathcal{R}(G) $ has Tits index $ ^{g}X_{n,l} $, with $ g = 2 $ or $ g = 3 $. Further assume that $ l $ is a Galois extension of $ k_0 $. Then $ \Aut_G(k) = \Aut (k) $ and the short exact sequence $ 1\to \Aut G\to \Aut (G \to \Spec k)\to \Aut_G(k)\to 1 $ splits if and only if $ 1\to \Gal (l/k)\to \Gal (l/k_0)\to \Gal (k/k_0)\to 1 $ splits.
\end{corollary}
\begin{proof}
In view of Theorem~\ref{Thm: MainThm2} and Proposition~\ref{Prop:epplicit SES for Dyn}, the short exact sequence $ 1\! \to \! \Aut G\to \Aut (G \to \Spec k)\to \Aut_G(k)\to 1 $ splits if and only if the short exact sequence $ 1\to \Gal (l/k)\to \Aut (l\geq k)\to \Aut_l (k)\to 1 $ splits. Since $ k_0 $ is rigid (or even prime if $ k $ is an infinite Galois extension) and $ k $ is a normal extension, $ \Aut (l\geq k) = \Gal (l/k_0) $. Furthermore, $ \Aut (k) = \Gal (k/k_0) $, and since $ l/k_0 $ is Galois, every element of $ \Gal (k/k_0) $ extends to $ \Gal (l/k_0) $. Hence $ \Aut_l (k) = \Gal (k/k_0) $, as wanted.
\end{proof}
\begin{remark}\label{Rem:an explicit description of quasi-split PGU}
Corollary~\ref{Cor:very explicit non-splitting} directly implies the corollary stated at the beginning of the
introduction of this paper. Indeed, $ \mathbf{Q} $ is a prime field and $ \mathbf{Q}_p $ is rigid by Lemma~\ref{Lem:Qp is rigid}. Furthermore, $ \Aut_{\text{abstract}}(G(k)) = \Aut (G\to \Spec k) $ by the Borel--Tits theorem that we stated at the very beginning of the introduction. For the ease of non-expert readers, let us also give an explicit realisation of the quasi-split, absolutely simple, adjoint algebraic $ k $-group of type $ ^2A_{n-1} $ with corresponding quadratic separable extension $ l $: denote the Galois conjugation on $ l $ by $ x\mapsto \bar{x} $, and for $ g\in \PGL_n(l) $, set $ (^{\text{at}}\bar{g})_{ij} = \bar{g}_{n+1-j;n+1-i} $ (i.e.\ the anti-transposed conjugated matrix). We define $ \PGU_n(k) = \lbrace g\in \PGL_n(l)~\vert~^{\text{at}}\bar{g}g = 1\rbrace $. This is easily interpreted as the $ k $-rational points of an algebraic $ k $-group, and one readily sees that this algebraic $ k $-group is the quasi-split, absolutely simple, adjoint algebraic $ k $-group of type $ ^2A_{n-1} $ with corresponding quadratic separable extension $ l $ (because the corresponding cocycle is $ g\mapsto \,^{\text{at}}g^{-1} $, which is an outer automorphism of $ \PGL_n $ preserving its Borel subgroup consisting of upper triangular matrices).
\end{remark}
\section{The \texorpdfstring{$\textbf{SL}_n(D)$}{SLn(D)} case over a local field}\label{Sec:SL_n(D)}
\subsection[Outer automorphisms of CSA over local fields]{Outer automorphisms of finite dimensional central simple algebras over local fields}
We now explore the same question for algebraic groups of the form $ \SL_n(D) $. First, we need to be a bit more precise and make a distinction between the algebraic $ k $-group and its group of $ k $-rational points.
\begin{definition}\label{Def:algebraic SL_n(D)}
Let $ A $ be a finite dimensional central simple $ k $-algebra. Following the notation of \cite{KMRT98}, we denote the corresponding algebraic $ k $-group of ``reduced norm $ 1 $ elements'' by $ \textbf{SL}_1(A) $. The $ k $-rational points of $ \textbf{SL}_1(A) $ are the elements of $ A $ of reduced norm $ 1 $, and we denote this group by $ \SL_1(A) $. When $ A = M_n(A') $ for some finite dimensional central simple $ k $-algebra $ A' $, we also denote $ \textbf{SL}_1(A) $ (respectively $ \SL_1(A) $) by $ \textbf{SL}_n(A') $ (respectively $ \SL_n(A') $).
\end{definition}
\begin{remark}
Note that for $ A $ a finite dimensional central simple $ k $-algebra and $ \alpha \in \Aut (k) $, $ ^{\alpha}\textbf{SL}_n(A) $ is naturally isomorphic (as an algebraic $ k $-group) to $\textbf{SL}_n(\,^{\alpha}A) $. Hence by (a slightly enhanced version of) \cite{KMRT98}*{Remark~26.11}, $ \alpha \in \Aut_{\textbf{SL}_n(A)}(k) $ if and only if $ A\cong \,^{\alpha}A $ or $ A^{opp}\cong \,^{\alpha}A $ (as $ k $-algebras).
\end{remark}
We will restrict ourselves to working over a local field. For us, a \textbf{local field} is a non-archimedean non-discrete topological field which is locally compact (or equivalently, a field isomorphic to $ \mathbf{F}_{p^n}(\!(T)\!) $ or a finite extension of $ \mathbf{Q}_p $ for some prime number $ p $). For the rest of the paper, the letter $ K $ exclusively stands for a local field. Let us begin by recalling the classification of central simple algebras over local fields.
\begin{definition}\label{Def:the cyclic algebra A_(l/k, sigma , a)}
Let $ k $ be a field and let $ l/k $ be a finite cyclic extension of degree $d$. Let $ \sigma \in \Gal (l/k) $ be a generator of the cyclic group $\Gal (l/k)$, let $a\in k$ and let $ u $ be an abstract symbol. The cyclic algebra $ A(l/k,\sigma ,a, u) $ is defined as follows: as a $ k $-vector space, $ A(l/k,\sigma ,a, u)\cong \bigoplus \limits_{i=0}^{d-1}u^il $, and the multiplication is defined by using the relations $ u^{d} = a $ and $ u^{-1}xu=\sigma (x) $ for all $ x\in l $. We also denote it $ A(l/k,\sigma ,a) $.
\end{definition}
We recall that the algebra $ A(l/k,\sigma ,a, u) $ of Definition~\ref{Def:the cyclic algebra A_(l/k, sigma , a)} is always central simple over $k$, and that it is isomorphic to the $ k $-algebra $ M_n(k) $ if and only if $a$ is the norm of an element in $l$.
\begin{definition}\label{Def:the CSA A_(d,r)}
Let $ K $ be a local field and let $ d,r \in \mathbf{N} $ with $ d\geq 1 $. Let $ K_d $ be the unramified extension of $ K $ of degree $ d $, let $ \sigma \in \Gal (K_d/K) $ be the Frobenius automorphism (i.e.\ the automorphism inducing the Frobenius automorphism on $ \Gal (\overline{K_d}/\overline{K}) $), and let $ \pi $ be a uniformiser of $K$. We define $ A(d,r) $ to be the cyclic algebra $ A(K_d/K,\sigma ,\pi^{r}) $.
\end{definition}
Note that up to isomorphism, $ A(d,r) $ does not depend on the choice of $ \pi $. In fact, given two uniformisers $ \pi $ and $ \tilde{\pi} $, an explicit isomorphism $ (K_d/K,\sigma ,\pi^{r})\cong (K_d/K,\sigma ,\tilde{\pi}^{r}) $ having the same form as the one appearing in Lemma~\ref{Lem:automorphisms of A(d,r)} can be given.
\begin{lemma}\label{Lem:automorphisms of A(d,r)}
Let $ K $ be a local field. Let $ A = A(d,r) $ and $ K_d, \sigma , \pi $ be as in Definition~\ref{Def:the CSA A_(d,r)}. Let $ \alpha $ be an automorphism of $ K_d $ such that $ \alpha (K) = K $, and assume that there exists an element $ x $ in $ K_d $ such that $ N_{K_d/K}(x) = \frac{\alpha (\pi^{r})}{\pi^{r}} $. Then the map $ \phi (\alpha ,x) \colon A\to A\colon \sum \limits_{i=0}^{d-1}u^ia_i\mapsto \sum \limits_{i=0}^{d-1}(ux)^i\alpha (a_i) $ is a ring automorphism of $ A $.
\end{lemma}
\begin{proof}
We view $ A $ as a quotient of the twisted polynomial ring $ K_d[u; \sigma ] $ (see \cite{Jac96}*{Section~1.1} for the definition of a twisted polynomial ring) modulo the relation $ u^d = \pi^{r} $. Given an automorphism $ \alpha $ in $ \Aut (K_d) $, we can define a map $ f_{\alpha }\colon K_d[u;\sigma ]\to K_d[u;\sigma ]\colon \begin{cases}
u\mapsto ux\\
a\mapsto \alpha (a) \text{ for all } a \in K_d
\end{cases} $. By \cite{Jac96}*{Proposition~4.6.20}, $ f_{\alpha} $ is a ring automorphism as soon as $ \alpha \sigma = \sigma \alpha $. Recall that by assumption, $ \alpha (K) = K $. Hence $ \sigma^{-1}\alpha\sigma\alpha^{-1} $ belongs to $ \Gal (K_d/K) $, and its induced automorphism on the residue field $ \overline{K_d} $ is a commutator in $ \Aut (\overline{K_d}) $, thus trivial (note that since every automorphism of a local field is continuous, it always induces an automorphism of the residue field). We conclude that $ \sigma^{-1}\alpha\sigma\alpha^{-1} $ itself was trivial by \cite{S79}*{Chapter III, \S 5, Theorem 3}. Hence, $ f_{\alpha} $ is indeed a ring automorphism.
Furthermore, if it passes to the quotient, $ f_{\alpha} $ induces the automorphism $ \phi (\alpha ,x) $. Hence it suffices to check that $ f_{\alpha} $ preserves the relation. But we have $f_{\alpha}(u^d-\pi^{r}) = (ux)^d-\alpha (\pi^r) = u^dN_{K_d/K}(x)-\alpha (\pi^r) = (u^d-\pi^r)\frac{\alpha (\pi^r)}{\pi^r} $, as wanted.
\end{proof}
For $ \alpha $ an automorphism of a (non-necessarily commutative) ring $ R $, we denote by $ \tilde{\alpha} $ the corresponding automorphism of $ M_n(R) $ (the algebra of $ n\times n $ matrices with coefficient in $ R $) obtained by applying $ \alpha $ coefficient by coefficient. Also, for $ A $ a finite dimensional central simple algebra over a field $ k $, we denote by $ \Nrd \colon A\to k $ its reduced norm.
\begin{lemma}\label{Lem:Automatic preservation of the reducednorm}
Let $k$ be a field and let $A$ be a central simple $k$-algebra. For every ring automorphism $\alpha$ of $A$ and $x\in A$,
\[
\Nrd(\alpha(x))=\alpha(\Nrd(x)).
\]
\end{lemma}
\begin{proof}
Let $k_s$ be a separable closure of $k$. Every automorphism $\alpha$
of $A$ preserves the center $k$; the restriction $\alpha\rvert_k$
extends to an automorphism $\beta$ of $k_s$, and we may consider the
tensor product
\[
\alpha\otimes\beta\colon A\otimes_kk_s\to A\otimes_kk_s.
\]
Since $k_s$ splits $A$, we may also consider an isomorphism of
$k_s$-algebras $f\colon A\otimes_kk_s\to M_d(k_s)$. The ring
automorphism $f\circ(\alpha\otimes\beta)\circ f^{-1}$ of $M_d(k_s)$
restricts to $\beta$ on the center $k_s$, hence
$f\circ(\alpha\otimes\beta)\circ f^{-1}\circ \tilde\beta^{-1}$ is the
identity on $k_s$. Since every $k_s$-automorphism of $M_d(k_s)$ is
inner, we may find $g\in \GL_d(k_s)$ such that
\[
f\circ(\alpha\otimes\beta)\circ f^{-1}\circ\tilde\beta^{-1} = \intaut (g).
\]
The following diagram then commutes:
\[
\begin{tikzcd}
A\otimes_kk_s \ar[r, "f"] \ar[d,swap, "\alpha\otimes\beta"] & M_d(k_s)
\ar[r,"\det"] \ar[d,swap, "\intaut (g)\circ\tilde\beta"]
&
k_s^\times\ar[d,"\beta"]
\\
A\otimes_kk_s \ar[r,swap, "f"]&M_d(k_s)\ar[r,swap, "\det"] & k_s^\times
\end{tikzcd}
\]
Since $\Nrd=\det\circ f$, the lemma follows.
\end{proof}
We set some notations that we use for the rest of the paper.
\begin{definition}\label{Def:notation for automorphism of SLn(D)}
Let $ K $ be a local field. Let $ A(d,r) $ and $ K_d, \sigma , \pi $ be as in Definition~\ref{Def:the CSA A_(d,r)}. Let $ \alpha $ be an automorphism of $ K_d $ such that $ \alpha (K) = K $, and assume that there exists an element $ x $ in $ K_d $ such that $ N_{K_d/K}(x) = \frac{\alpha (\pi^{r})}{\pi^{r}} $. The map $ \tilde{\phi}(\alpha ,x)\colon M_n(A)\to M_n(A) $ corresponding to the automorphism $ \phi (\alpha ,x)\colon A\to A $ from Lemma~\ref{Lem:automorphisms of A(d,r)} preserves elements of reduced norm $ 1 $ by Lemma~\ref{Lem:Automatic preservation of the reducednorm}. We again denote its restriction to $ \SL_n(A) $ by $ \tilde{\phi}(\alpha ,x) $.
\end{definition}
\begin{remark}\label{Rem:abstract automorphisms are algebraic}
In the notations of Definition~\ref{Def:notation for automorphism of SLn(D)}, $ \tilde{\phi}(\alpha ,x) $ is an isomorphism of $ k $-algebras $ M_n(A)\cong \,^{\alpha^{-1}}M_n(A) $. Hence, by \cite{KMRT98}*{Theorem~26.9}, $ \tilde{\phi}(\alpha ,x) $ corresponds to a unique $ k $-isomorphism of algebraic groups $\textbf{SL}_n(A)\cong \,^{\alpha^{-1}}\textbf{SL}_n(A) $. More concretely, this can also be seen by using a representation of $ A $ in $ M_{d^2}(K) $ (where $ d $ is the degree of $ A $).
\end{remark}
The following observation explains in part why the local field case is so much simpler than say the global field case (see also the end of Remark~\ref{Rem:litterature on outer automorphism of D}).
\begin{lemma}\label{Lem:extending to unramfiied extensions}
Let $ K $ be a local field and let $ K_d $ be a finite dimensional unramified extension of $ K $. Any automorphism of $ K $ extends to an automorphism of $ K_d $.
\end{lemma}
\begin{proof}
Let $ \alpha \in \Aut (K) $. There exists an extension $ \beta $ of $ \alpha $ to the separable closure of $ K $. Note that if $ K_d \cong K[X]/(f) $, then $ \beta (K_d)\cong K[X]/(\,^{\alpha}f) $, where $ ^{\alpha}f $ is the polynomial obtained from $ f $ by applying $ \alpha $ to its coefficients. But $ \alpha $ is continuous, and an extension is unramified if and only if it is isomorphic to $ K[X]/(g) $ for some polynomial $ g $ whose coefficients are all of valuation $ 0 $. Hence by uniqueness of unramified extensions of a given degree, $ \beta $ preserves $ K_d $.
\end{proof}
\begin{corollary}\label{Cor:etxending auto to D}
Let $ A $ be a finite dimensional central simple algebra over a local field $ K $. Every automorphism of $ K $ extends to an automorphism of $ A $. Hence, $ \Aut_{\textbf{SL}_n(A)}(K)= \Aut (K) $.
\end{corollary}
\begin{proof}
By Theorem~\ref{Thm:classificatio nof CSA over local fields}, the central simple algebra $ A $ is an algebra of the form $ A(d,r) $, i.e.\ a cyclic algebra of the form $ (K_d/K, \sigma ,\pi^{r}) $ with $ K_d, \sigma , \pi $ as in Definition~\ref{Def:the CSA A_(d,r)}.
Let $ \alpha \in \Aut (K) $. By Lemma~\ref{Lem:extending to unramfiied extensions}, there exists $ \beta \in \Aut (K_d) $ extending $ \alpha $. Also, by \cite{S79}*{Chapter V,\S 2, Corollary}, $ N_{K_d/K} $ is surjective on $ \mathcal{O}_{K}^{\times} $. Furthermore, any automorphism of a local field preserves the valuation. Hence there exists $ x\in K_d $ such that $ N_{K_d/K}(x) = \frac{\alpha (\pi^{r})}{\pi^{r}} $. Then the automorphism $ \phi (\beta ,x) $ defined in Lemma~\ref{Lem:automorphisms of A(d,r)} is an extension of $ \alpha $ to $ A $. Finally, $ \tilde{\phi}(\beta ,x) $ from Definition~\ref{Def:notation for automorphism of SLn(D)} is defined over $ \alpha^{-1} $, so that the last claim follows from Remark~\ref{Rem:abstract automorphisms are algebraic}.
\end{proof}
\begin{remark}\label{Rem:litterature on outer automorphism of D}
If $ \alpha \in \Aut (K) $ is of finite order, the result in Corollary~\ref{Cor:etxending auto to D} asserting that $ \alpha $ extends to an automorphism of $ A $ is an old result. Indeed, using Lemma~\ref{Lem:Base change of CSA}, it is a direct consequence of \cite{EML48}*{Corollary~7.3} (see also \cite{Han07}*{Theorem~5.6}) and the fact that $ A = M_n(D) $ for some division algebra $ D $. This already settles the question in characteristic $ 0 $. In positive characteristic, Lemma~\ref{Cor:etxending auto to D} can be seen as a direct corollary of the results in \cite{Ha07}. Note that the fact that any extension of $ \alpha \in \Aut (K) $ to the separable closure of $ K $ preserves $ K_d $ simplifies matters (compare with Lemma~\ref{Lem:Hanke result part 2} when the extension $ \beta $ does not preserve the chosen maximal subfield $ l $).
\end{remark}
\subsection[Condition for the exact sequence not to split]{Sufficient condition for the exact sequence not to split}
We turn to the splitting question for the exact sequence $ 1\to \Aut G\to \Aut (G\to \Spec k)\to \Aut_G(k)\to 1 $, still assuming $ G = \textbf{SL}_n(A) $ over a local field. Let us introduce another notation for a subgroup of the group of semilinear automorphisms, which allow us to introduce a ``ground field''.
\begin{definition}
Let $ G $ be a $ k $-group scheme. Let $ k' $ be a subfield of $ k $. We denote by $ \Aut (G\to \Spec k/k') $ the subgroup of $ \Aut (G\to \Spec k) $ consisting of semilinear automorphisms over an automorphism $ \alpha $ belonging to $ \Aut (k/k') $. Furthermore, we denote by $ \Aut_G(k/k') $ the image of $ \Aut (G\to \Spec k/k') $ under the map $ \Aut (G\to \Spec k)\to \Aut_G(k) $.
\end{definition}
\begin{theorem}\label{Thm:local non-splitting for SLn(D)}
Let $ D $ be a central division algebra of degree $ d $ over a local field $ K $ and let $ G = \textbf{SL}_n(D) $. Let $ K' $ be a subfield of $ K $ such that $ K/K' $ is a finite Galois extension. Then the short exact sequence $ 1\to \Aut G\to \Aut (G\to \Spec K/K')\to \Aut_G(K/K')\to 1 $ splits if and only if $ \gcd (nd,[K:\nolinebreak K']) $ divides $ n $.
\end{theorem}
\begin{proof}
By Corollary~\ref{Cor:etxending auto to D}, $ Aut_G(K)=Aut(K) $. Hence, since $ \Gal (K/K') $ is contained in $ Aut_G(K) $, the short exact sequence splits if and only if $ G $ is defined over $ K' $ (see Theorem~\ref{Thm:Galois descent}). Let $ H $ be this hypothetical form of $ G $ over $ K' $.
The case $ d=1 $ being obviously true, let us assume that $ d\geq 2 $. Now, by the classification of simple groups over local fields (see \cite{Tits77}*{Section~4.2 and 4.3}), the Tits index of $ H $ is of the form $ ^{1}A^{(d')} $ or $ ^{2}A^{(1)} $, since these are the only groups of type $ A $ over local fields. Note that a distinguished orbit has to remain distinguished after scalar extension, because a non-trivial root remains non-trivial after scalar extension. Hence $ H $ cannot be of type $ ^{2}A^{(1)} $, because groups of type $ ^{2}A^{(1)} $ have extremal roots that are distinguished, whereas $ G $ has undistinguished extremal roots when $ d\geq 2 $. But the only groups of type $ ^{1}A^{(d')} $ are groups of the form $ \textbf{SL}_{n'}(D') $ where $ n'\geq 1 $ and $ D' $ is a division algebra over $ K' $. So we conclude that $ H $ is of this form.
We use the notation $ \inv $ for the map classifying division algebras over local fields (see Theorem~\ref{Thm:classificatio nof CSA over local fields} for a precise definition of $ \inv $). Let $ d' $ be the degree of $ D' $ over $ K' $, and let $ r' $ be such that $ [\frac{r'}{d'}] = \inv ([D']) $ in $ \mathbf{Q}/\mathbf{Z} $. Also, let $ a = \gcd (d',[K:K']) $. The base change of $ \textbf{SL}_{n'}(D') $ from $ K' $ to $ K $ is the algebraic group $ \textbf{SL}_{an'}(A(\frac{d'}{a},\frac{[K:K']}{a}r')) $ by Proposition~\ref{Prop:base change of SL_n(A)}. Since $ H $ is isomorphic to $ G $ over $ K $, $ an' = n $ and $ ad = d' $. Hence, $ a = \gcd (ad,[K:K']) $, which implies that $ \gcd (adn',[K:K']) $ divides $ an' $. Now, the equation $ an' = n $ already proves that if $ H $ exists, then $ \gcd (nd,[K:K']) $ divides $ n $.
Conversely, let $ a = \gcd (nd,[K:K']) $, and assume that $ a $ divides $ n $. We then set $ n' = \frac{n}{a} $, $ d' = ad $ and $ r' $ such that $ \frac{[K:K']}{a}r' - r \in d\mathbf{Z} $ (such an $ r' $ exists because $ \frac{[K:K']}{a} $ is prime to $ d $). With those parameters, the algebraic group $ \textbf{SL}_{n'}(A(d',r')) $ is a form of $ G $ over $ K' $, as wanted.
\end{proof}
\begin{remark}
The condition that $ \gcd (nd,[K:\nolinebreak K']) $ divides $ n $ is equivalent to require that for all primes $ p $ dividing $ d $, the $ p $-adic valuation of $ [K:K'] $ is less than or equal to the $ p $-adic valuation of $ n $.
\end{remark}
\begin{corollary}\label{Cor:non-splitting condition for SLn(D)}
Let $ D $ be a central division algebra of degree $ d $ over a local field $ K $ and let $ G = \textbf{SL}_n(D) $. The short exact sequence $ 1\to \Aut G\to \Aut (G\to \Spec K)\to \Aut_G(K)\to 1 $ does not split if there exists a subfield $ K'\leq K $ such that $ K/K' $ is finite Galois and $ \gcd (nd,[K:K']) $ does not divide $ n $.
\end{corollary}
\begin{proof}
$ 1\to \Aut G\to \Aut (G\to \Spec K/K')\to \Aut_G(K/K')\to 1 $ does not split by Theorem~\ref{Thm:local non-splitting for SLn(D)}, hence neither does $ 1\to \Aut G\to \Aut (G\to \Spec K)\to \Aut_G(K)\to 1 $.
\end{proof}
\subsection{Sufficient condition for the exact sequence to split}
In characteristic $ 0 $, it is actually straightforward to prove the converse of Corollary~\ref{Cor:non-splitting condition for SLn(D)}.
\begin{theorem}\label{Thm:splitting in char. 0 for SL_n(D)}
Let $ D $ be a central division algebra of degree $ d $ over a local field $ K $ of characteristic $ 0 $ and let $ G = \textbf{SL}_n(D) $. The short exact sequence $ 1\to \Aut G\to \Aut (G\to \Spec K)\to \Aut_G(K)\to 1 $ does not split only if there exists a subfield $ K'\leq K $ such that $ K/K' $ is finite Galois and $ \gcd (nd,[K:K']) $ does not divide $ n $.
\end{theorem}
\begin{proof}
By Corollary~\ref{Cor:etxending auto to D}, $ Aut_G(K)=Aut(K) $. Since $ K $ is of characteristic $ 0 $, it is a finite extension of $ \mathbf{Q}_p $ for some prime $ p $. But every automorphism of $ K $ acts trivially on $ \mathbf{Q}_p $ by Lemma~\ref{Lem:Qp is rigid}. Hence, by Galois theory, $ \Aut (K) $ is a finite group. Furthermore, letting $ K^{\Aut (K)} $ be the subfield of $ K $ fixed by $ \Aut (K) $, the extension $ K/K^{\Aut (K)} $ is Galois with Galois group $ \Aut (K) $.
Let $ a = \gcd (nd,[K:K^{\Aut (K)}]) $. Assuming that there does not exist a subfield $ K'\leq K $ such that $ K/K' $ is finite Galois and such that $ \gcd (nd,[K:K']) $ does not divide $ n $, we have in particular that $ a $ divides $ n $. Also, let $ r\in \mathbf{N} $ be such that $ [\frac{r}{d}] = \inv ([D]) $. Since $ \frac{[K:K^{\Aut (K)}]}{a} $ is prime to $ d $, there exists $ r'\in \mathbf{N} $ such that $ \frac{[K:K^{\Aut (K)}]}{a}r' - r \in d\mathbf{Z} $. Hence, by Proposition~\ref{Prop:base change of SL_n(A)}, the algebraic group $ \SL_{\frac{n}{a}}(A(ad,r')) $ is a form of $ G $ over $ K^{\Aut (K)} $, because $ \gcd (ad,[K:K^{\Aut (K)}]) = a $. But in view of Lemma~\ref{Lem:field of definition are in AutG(k)}, this implies that the homomorphism $ \Aut (G\to \Spec K)\to \Aut (K) = \Gal (K/K^{\Aut( K)}) $ has a section, as wanted.
\end{proof}
\begin{remark}\label{Rem:Vivid example of splitting for divison algebras}
Putting Corollary~\ref{Cor:non-splitting condition for SLn(D)} and Theorem~\ref{Thm:splitting in char. 0 for SL_n(D)} together already proves Theorem~\ref{Thm:Main Thm 2.2} in characteristic $ 0 $. In particular, the sequence always splits for $ K = \mathbf{Q}_p $ (this actually directly follows from the rigidity of $ \mathbf{Q}_p $, which was used in the proof of Theorem~\ref{Thm:splitting in char. 0 for SL_n(D)}). For a more interesting example, if $ K $ is a Galois extension of $ \mathbf{Q}_p $ of degree $ p^i $ for some prime $ p $ and some $ i\in \N $, then Theorem~\ref{Thm:Main Thm 2.2} asserts that the following are equivalent:
\begin{enumerate}
\item The sequence $ 1\to \Aut \textbf{SL}_n(D)\to \Aut (\textbf{SL}_n(D)\to \Spec K)\to \Aut_{\textbf{SL}_n(D)}(K)\to 1 $ splits.
\item If $ n $ is not divisible by $ p^i $, the degree of $ D $ is not divisible by $ p $.
\end{enumerate}
\end{remark}
We now aim to prove an analogue of Theorem~\ref{Thm:splitting in char. 0 for SL_n(D)} but in positive characteristic. When $ K $ is of positive characteristic, the fixed field $ K^{\Aut (K)} $ is finite and $ K/K^{\Aut( K)} $ is not Galois. Thus we cannot use the same method than in characteristic $ 0 $.
Instead, the strategy goes as follows: we decompose $ \Aut( K) $ in various pieces, we give a section of $ \Aut (\textbf{SL}_n(D)\to \Spec K)\to \Aut (K) $ separately for each pieces and then we check that everything can be glued. Let us begin by decomposing $ \Aut (K) $.
\begin{lemma}
Let $ K = \mathbf{F}_{p^i}(\!(T)\!) $. Since $ \mathbf{F}_{p^i} $ is the algebraic closure in $ K $ of the prime field of $ K $, $ \mathbf{F}_{p^i} $ is preserved by any automorphism of $ K $. Let $ N(K) = \lbrace \alpha \in \Aut (K)~\vert~\alpha $ acts trivially on $ \mathbf{F}_{p^i} \rbrace $. We have $ \Aut (K) \cong N(K)\rtimes \Gal (K/\mathbf{F}_{p}(\!(T)\!)) $.
\end{lemma}
\begin{proof}
We want to show that the short exact sequence $ 1\to N(K)\to \Aut (K)\xrightarrow{f} \Gal (\mathbf{F}_{p^i}/\mathbf{F}_p)\to 1 $ splits. But by \cite{S79}*{Chapter III, \S 5, Theorem 3}, $ f $ maps $ \Gal (K/\mathbf{F}_{p}(\!(T)\!)) $ isomorphically onto $ \Gal (\mathbf{F}_{p^i}/\mathbf{F}_p) $, hence the result.
\end{proof}
We furthermore decompose the group $ N(K) $. Since automorphisms of $ K $ are continuous, an element $ \alpha $ of $ N(K) $ is therefore determined by its action on $ T $, and we have $ \alpha (T) = \sum \limits_{j=1}^{\infty}a_jT^j $, where $ a_1 \in \mathbf{F}_{p^i}^{\times} $ and $ a_j \in \mathbf{F}_{p^i} $ for all $ j\geq 2 $.
\begin{definition}\label{Def:J(K) and C(K)}
Let $ J(K) = \lbrace \alpha \in N(K)~\vert~\alpha (T) = T + \sum \limits_{j=2}^{\infty}a_jT^j,~a_j \in \mathbf{F}_{p^i}\rbrace $ and let $ C_{p^i-1} = \lbrace \alpha \in N(K)~\vert~\alpha (T) = aT,~a \in \mathbf{F}_{p^i}^{\times}\rbrace $. With those notations, the group $ N(K) $ is isomorphic to $ J(K)\rtimes \mathbf{F}_{p^i}^{\times} $. For $ x\in \mathbf{F}_{p^i}^{\times} $, we denote by $ \ev (xT) $ the corresponding element of $ \Aut (K) $.
\end{definition}
In summary, we have decomposed $ \Aut (K) $ as the group $ (J(K)\rtimes \mathbf{F}_{p^i}^{\times})\rtimes \Gal (K/\mathbf{F}_{p}(\!(T)\!)) $. We go on by giving a section to $ \Aut (\textbf{SL}_n(D)\to \Spec K)\to \Aut (K) $ for each component of $ \Aut (K) $, one at a time. In doing so, we will at the same time take care that the given section glues well with the other sections (though each are studied separately). Hence, a given formula for a section on one component of $ \Aut (K) $ will at times be slightly more complicated than a formula one would naturally consider if one was not aiming for a global section. In each case, we write a remark to explain how the given formula could be simplified if not aiming for a global splitting.
We need to set a few notations.
\begin{definition}\label{Def:notation for splitting explicitly}
\begin{enumerate}
\item Let $ \mathbf{F}_p^{\Alg} $ be an algebraic closure of $ \mathbf{F}_p $. We denote by $ F $ the Frobenius automorphism of $ \mathbf{F}_p^{\Alg}(\!(T)\!) $ (i.e.\ the automorphism of $ \mathbf{F}_p^{\Alg}(\!(T)\!) $ fixing $ \mathbf{F}_p(\!(T)\!) $ and inducing the Frobenius automorphism on $ \mathbf{F}_p^{\Alg} $). For any finite extension extension $ L $ of $ \mathbf{F}_p $, we also denote by $ F $ the restriction of $ F $ to $ L $ and to $ L (\!(T)\!) $.
\item We fix $ p $ a prime number, $i,n,d,r\in \mathbf{N}_{>0} $ such that $ \gcd (d,r) = 1 $ and two symbols $u,T $. We set $ K = \mathbf{F}_{p^i}(\!(T)\!) $, $ E = \mathbf{F}_{p^{id}}(\!(T)\!) $ and $ D = (E/K,F^i ,T^r, u) $ (a cyclic division algebra of degree $ d $ over $ K $ with symbol $ u $ as in Definition~\ref{Def:the CSA A_(d,r)}). Furthermore, we let $ G $ be the algebraic $ K $-group $ \textbf{SL}_n(D) $.
\item For $ \alpha \in N(K) $ we define its extension $ \alpha_{E} $ to $ \Aut (E) $ as follows: $ \alpha_{E} $ acts trivially on the residue field, while $ \alpha_E(T) = \alpha (T) $. We thus get an injective homomorphism $ N(K)\to N(E)\colon \alpha \mapsto \alpha_{E} $. Abusing notations, we again denote $ \alpha_{E} $ by $ \alpha $.
\item \label{Def:decomposing C(pi-1)} We fix $ a, b\in \mathbf{N} $ such that $ ab = p^i-1 $, $ \gcd (d^{p^i-1},p^i-1) = \gcd (d^{b},b) = b $ and $ \gcd (d,a) = 1 $.
\item \label{Def:decomposing C(i)} We fix $ a', b'\in \mathbf{N} $ such that $ a'b' = i $, $ \gcd (d^{i},i) = \gcd (d^{i},b') = b' $ and $ \gcd (d,a') = 1 $.
\item We choose a generator $ \zeta $ of the multiplicative group $ \mathbf{F}_{p^{i}}^{\times} $.
\item For $ g\in \PGL_n(D) $, we denote by $ \intaut (g) $ the automorphism by conjugation of $ g $ on $ G $, i.e.\ $ \intaut (g)\colon G\to G\colon h\mapsto ghg^{-1} $.
\end{enumerate}
\end{definition}
\begin{remark}\label{Rem:on the notation for explicit splitting}
The natural numbers $ a,b,a',b' $ are uniquely determined by their definition. Note that we have in particular $ \gcd(a,b) = 1 = \gcd (a', b') $. Also note that by Definition~\ref{Def:the CSA A_(d,r)}, $ u^{-1}xu = F^i(x) $ for all $ x\in E $.
\end{remark}
We will further make use of the following notation: if $ l,m\in \mathbf{N} $ and $ A_1,\dots , A_l $ are $ m\times m $ matrices, we denote $ \diag (A_1,\dots , A_l) $ the corresponding block diagonal $ lm\times lm $ matrix. Furthermore, the $ m\times m $ identity matrix is denoted $ \Id_m $. We will denote the cyclic group of order $ m $ by $ C_m $.
\begin{proposition}\label{Prop:constructing fJK}
Keep the notations of Definition~\ref{Def:notation for splitting explicitly}. Assume that $ \gcd (p,d) = 1 $ and that $ bb' $ divides $ n $. For $ \alpha \in J(K) $, there exists a unique $ x_{\alpha} \in 1+T\mathbf{F}_{p^i}[\![T]\!] $ such that $ x_{\alpha}^{db'} = \frac{\alpha (T^{r})}{T^{r}} $. Let $ M = \diag (\Id_b, x_{\alpha}\Id_b, x_{\alpha}^2\Id_b,\dots ,x_{\alpha}^{b'-1}\Id_b) $ $ ( $so that $ M $ is a $ bb'\times bb' $ matrix which is block diagonal$ ) $ and let $ X_{\alpha} = \diag (M,\dots ,M) $ where we have $ \frac{n}{bb'} $ terms $ ( $so that $ X_{\alpha} $ is a $ n\times n $ matrix which is block diagonal with coefficients in $ 1+T\mathbf{F}_{p^i}[\![T]\!]) $. Recalling the notation introduced in Remark~\ref{Rem:abstract automorphisms are algebraic}, the map
\begin{align*}
f_{J(K)}\colon J(K)&\to \Aut (G\to \Spec K)\\
\alpha &\mapsto \intaut (X_{\alpha})\tilde{\phi}(\alpha ,x_{\alpha}^{b'})
\end{align*}
is a homomorphism whose composition with the map $ \Aut (\! G\to \Spec K\! )\! \to \Aut_G(K) $ is the identity on $ J(K) $.
\end{proposition}
\begin{proof}
Note that $ \gcd(p,d) = 1 $ and $ \gcd (d^{b'},b') = b' $ implies $ \gcd (p,b') = 1 $, so that $ \gcd (p,db') =1 $. Hence, for $ \alpha \in J(K) $ the existence and uniqueness of $ x_{\alpha} $ in $ 1+T\mathbf{F}_{p^i}[\![T]\!] $ such that $ x_{\alpha}^{db'} = \frac{\alpha (T^{r})}{T^{r}} $ follows directly from Hensel's lemma. We claim that for $ \alpha ,\beta \in J(K) $, $ x_{\beta \circ \alpha} = x_{\beta}.\beta (x_{\alpha}) $. By uniqueness, this equation holds if and only if $ \frac{(\beta \circ \alpha )(T^{r})}{T^{r}}=[x_{\beta}.\beta (x_{\alpha})]^{db'} $. But the right hand side is equal to $ \frac{\beta (T^{r})}{T^r}.\beta (\frac{\alpha (T^{r})}{T^r}) $, which is indeed equal to $ \frac{(\beta \circ \alpha )(T^{r})}{T^{r}} $. Checking that $ f_{J(K)} $ is a homomorphism is now straightforward:
\begin{align*}
\intaut (X_{\beta})\tilde{\phi}(\beta ,x_{\beta}^{b'}) \circ \intaut (X_{\alpha})\tilde{\phi}(\alpha ,x_{\alpha}^{b'})&=\intaut (X_{\beta}\beta (X_{\alpha}))\tilde{\phi}(\beta \circ \alpha , x_{\beta}^{b'}.\beta (x_{\alpha}^{b'}))\\
&=\intaut (X_{\beta \circ \alpha})\tilde{\phi}(\beta \circ \alpha , x_{\beta \circ \alpha}^{b'}). \qedhere
\end{align*}
\end{proof}
Note that if we were not aiming to define a global section of $ \Aut (G\to \Spec K)\to \Aut_G(K) $, we could just as well get rid of the factor $ \intaut(X_{\alpha}) $ and hence we would not need the assumption that $ bb' $ divides $ n $. In light of this, the next proposition really is a converse to Proposition~\ref{Prop:constructing fJK}.
\begin{proposition}
Keep the notations of Definition~\ref{Def:notation for splitting explicitly}. If $ \gcd (p,d) \neq 1 $, there does not exist a homomorphism $ J(K)\to \Aut (G\to \Spec K) $ whose composition with $ \Aut (G\to \Spec K)\to \Aut_G(K) $ is the identity on $ J(K) $.
\end{proposition}
\begin{proof}
By Theorem~\ref{Thm:local non-splitting for SLn(D)}, it suffices to prove that there exists $ K'\leq K $ such that $ K/K' $ is finite Galois with $ \Gal (K/K')\leq J(K) $ and such that $ \gcd (nd,[K:K']) $ does not divide $ n $. Let $ H $ be a group of order $ p^n $. By \cite{C97}*{Theorem 3}, there exists an injective homomorphism $ H\hookrightarrow J(\mathbf{F}_{p}(\!(T)\!)) $. Also note that $ J(\mathbf{F}_{p}(\!(T)\!)) $ can be seen as a subgroup of $ J(K) $ in a natural way, so that $ J(K) $ has a subgroup of order $ p^n $, that we again denote by $ H $. Now, let $ K' = K^H = \lbrace x\in K~\vert~\alpha (x) = x \text { for all } \alpha \in H\rbrace $. Hence, $ K/K' $ is a Galois extension with $ \Gal (K/K') = H\leq J(K) $ and $ \gcd (nd,[K:K']) =\gcd (nd,p^n) $ does not divide $ n $ because $ \gcd (p,d) \neq 1 $, as wanted.
\end{proof}
We now construct a section of $ \Aut (G\to \Spec K)\to \Aut (K) $ for $ \mathbf{F}_{p^i}^{\times} $. In fact, using the same line of argument as for Theorem~\ref{Thm:local non-splitting for SLn(D)}, we know that a section for $ \mathbf{F}_{p^i}^{\times} $ exists if and only if $ \gcd (nd,p^{i}-1) $ divides $ n $ (where $ d $ and $ n $ appear in the form of $ G = \SL_n(D) $, $ d $ denoting as usual the degree of $ D $). But we need to have an explicit formula, since we want to ensure that it glues well with the map $ f_{J(K)} $ constructed in Proposition~\ref{Prop:constructing fJK}. We found those explicit formulas by working out by hand some low degree examples for which we could follow explicitly what the theory was predicting, and then by generalising our findings to any degree. Again, we give a section which is going to be slightly more complicated than necessary, because we aim to define a global section in the end.
We first need a $ (db') $-th root of $ \zeta^{br} $.
\begin{lemma}\label{Lem:db' root of zeta}
Keep the notations of Definition~\ref{Def:notation for splitting explicitly} and let $ C_a $ be the group generated by $ \zeta^b $ in $ \mathbf{F}_{p^i}^{\times} $. There exists a unique $ z\in C_a $ such that $ z^{db'} = \zeta^{br} $.
\end{lemma}
\begin{proof}
Note that $ \gcd(d,a) = 1 $ and $ \gcd (d^{b'},b') = b' $ implies $ \gcd (b',a) = 1 $, so that $ \gcd (db',a) =1 $. Hence the result follows from the fact that $ db' $ is invertible in the cyclic group of order $ a $.
\end{proof}
\begin{proposition}\label{Prop:constructing fCl(d)}
Keep the notations of Definition~\ref{Def:notation for splitting explicitly} and of Lemma~\ref{Lem:db' root of zeta}, so that $ z^{db'} = \zeta^{br} $. Assume that $ bb' $ divides $ n $. Let $ M = \diag (\Id_b, z\Id_b, z^2\Id_b,\dots , z^{b'-1}\Id_b) $ $ ( $so that $ M $ is a $ bb'\times bb' $ matrix which is block diagonal$ ) $ and let $ Z = \diag (M,\dots ,M) $ where we have $ \frac{n}{bb'} $ terms $ ( $so that $ Z $ is a $ n\times n $ matrix which is block diagonal with coefficients in $ \mathbf{F}_{p^i} )$. Recalling the notation introduced in Remark~\ref{Rem:abstract automorphisms are algebraic}, the map
\begin{align*}
f_{C_a}\colon C_a&\to \Aut (G\to \Spec K)\\
\ev (\zeta^{bj}T) &\mapsto \intaut (Z^j)\tilde{\phi}(\ev (\zeta^{bj}T) ,z^{b'j})
\end{align*}
is a homomorphism whose composition with the map $ \Aut (\! G\to \Spec K\! )\! \to \Aut_G(K) $ is the identity on $ C_a $.
\end{proposition}
\begin{proof}
Note that $ \frac{\ev(\zeta^{bj}T)(T^{r})}{T^{r}}=\zeta^{brj}=(z^{b'j})^d=N_{E/K}(z^{b'j}) $, so that we can indeed use Definition~\ref{Def:notation for automorphism of SLn(D)}. With these definitions, for all $ j,j' \in \mathbf{N} $, we have
\begin{align*}
\intaut (Z^j)\tilde{\phi}(\ev (\zeta^{bj}T) ,z^{b'j}) \circ \intaut (Z^{j'})\tilde{\phi}(\ev (\zeta^{bj'}T) ,z^{b'j'})&=\intaut (Z^jZ^{j'})\tilde{\phi}(\ev (\zeta^{bj}T) \circ \ev (\zeta^{bj'}T) , z^{b'j}.z^{b'j'})\\
&=\intaut (Z^{j+j'})\tilde{\phi}(\ev (\zeta^{b(j+j')}T) ,z^{b'(j+j')}).
\end{align*}
Hence the fact that $ f_{C_a} $ is well-defined follows from $ z^a = 1 $ (which holds because $ z^a $ is the unique $ (db') $-th root in $ C_a $ of $ \zeta^{abr} = 1 $).
\end{proof}
\begin{remark}
In the proof of Proposition~\ref{Prop:constructing fCl(d)}, we needed to show that $ f_{C_a}(\ev (\zeta^{b}T))^a $ is a trivial element of $ \Aut (G\to \Spec K) $. We proved it by showing that this algebraic automorphism of $ \textbf{SL}_n(D) $ induces a trivial automorphism of $ SL_n(D) $, hence is itself trivial by the density of rational points for $ G $. This will be used repeatedly to show that $ K $-automorphisms of $ G $ are trivial. In using that argument, it is also important to notice that a semilinear automorphism of the form $ \intaut (g)\tilde{\phi}(\alpha ,x) $ is algebraic if and only if $ \alpha $ acts trivially on $ K $.
\end{remark}
\begin{remark}
Note that in Proposition~\ref{Prop:constructing fCl(d)}, the factor $ \intaut (Z^j) $ is unnecessary if one is just interested in a section defined on $ C_a $ alone. Hence, a section of $ \Aut (G\to \Spec K)\to \Aut_G(K) $ only defined on $ C_a $ always exists (i.e.\ one does not need to assume that $ bb' $ divides $ n $).
\end{remark}
\begin{proposition}\label{Prop:constructing fCk(d)}
Keep the notations of Definition~\ref{Def:notation for splitting explicitly} and assume that $ b $ divides $ n $. Let $ C_b $ be the group generated by $ \zeta^a $ in $ \mathbf{F}_{p^i}^{\times} $. There exists an element $ y\in \mathbf{F}_{p^{idb}} $ such that $ \frac{F^{id}(y)}{y} = \zeta^{ar} $. Choosing a $ \mathbf{F}_{p^{id}} $-basis of $ \mathbf{F}_{p^{idb}} $, we obtain an embedding $ \varphi \colon \mathbf{F}_{p^{idb}}\to M_b(\mathbf{F}_{p^{id}}) $. Let $ g = \varphi (y^{-1}) $ and let $ Y = \diag (g,\dots ,g) $ where we have $ \frac{n}{b} $ terms $ ( $so that $ Y $ is a $ n\times n $ matrix which is block diagonal with coefficients in $ \mathbf{F}_{p^{id}} )$. Recalling the notation introduced in Remark~\ref{Rem:abstract automorphisms are algebraic}, the map
\begin{align*}
f_{C_b}\colon C_b&\to \Aut (G\to \Spec K)\\
\ev (\zeta^{aj}T) &\mapsto \intaut (Y^j)\tilde{\phi}(\ev (\zeta^{aj}T) ,(\frac{F^i(y)}{y})^j)
\end{align*}
is a homomorphism whose composition with the map $ \Aut (\! G\to \Spec K\! )\! \to \Aut_G(K) $ is the identity on $ C_b $.
\end{proposition}
\begin{proof}
For the existence of $ y\in \mathbf{F}_{p^{idb}} $ such that $ \frac{F^{id}(y)}{y} = \zeta^{ar} $, note that $ N_{\mathbf{F}_{p^{idb}}/\mathbf{F}_{p^{id}}}(\zeta^{ar}) = \zeta^{abr} = 1 $. Also note that the extension $ \mathbf{F}_{p^{idb}}/\mathbf{F}_{p^{id}} $ is Galois cyclic, and that $ F^{id} $ generates its Galois group. Hence, by Hilbert's Theorem~90, there indeed exists $ y\in \mathbf{F}_{p^{idb}} $ such that $ \frac{F^{id}(y)}{y} = \zeta^{ar} $. For the rest of the proof, we choose such an $ y $.
From $ \frac{F^{id}(y)}{y} = \zeta^{ar} $, it readily follows that $ F^i(y)y^{-1} $ and $ y^b $ belong to $ \mathbf{F}_{p^{id}} $, since they are both invariant under $ F^{id} $. Note that $ \frac{\ev(\zeta^{aj}T)(T^{r})}{T^{r}}=\zeta^{ajr}=(\frac{F^{id}(y)}{y})^j=N_{E/K}((\frac{F^i(y)}{y})^j) $, so that we can indeed use Definition~\ref{Def:notation for automorphism of SLn(D)}.
It remains to check that $ f_{C_b} $ is well-defined and is a homomorphism. Note that for all $ j,j' \in \mathbf{N} $, we have
\begin{align*}
&\intaut (Y^j)\tilde{\phi}(\ev (\zeta^{aj}T) ,(\frac{F^i(y)}{y})^j) \circ \intaut (Y^{j'})\tilde{\phi}(\ev (\zeta^{aj'}T) ,(\frac{F^i(y)}{y})^{j'})\\
&=\intaut (Y^jY^{j'})\tilde{\phi}(\ev (\zeta^{aj}T) \circ \ev (\zeta^{aj'}T) ,(\frac{F^i(y)}{y})^{j+j'})\\
&=\intaut (Y^{j+j'})\tilde{\phi}(\ev (\zeta^{a(j+j')}T) ,(\frac{F^i(y)}{y})^{j+j'}).
\end{align*}
Hence, it suffices to check that $ \intaut (Y^b)\tilde{\phi}(\ev (\zeta^{ab}T) ,(F^i(y)y^{-1})^b) $ is the identity on $ \SL_n(D) $. But since $ y^b\in \mathbf{F}_{p^{id}} $, $ g^b $ and $ Y^b $ are diagonal and by definition of $ Y $, $ y^{-b}Y^{-b} = 1 $. Furthermore, recall that $ u^{-1}Y^bu = F^i(Y^b) $ (see Remark~\ref{Rem:on the notation for explicit splitting}). Hence $ \intaut (Y^b)\tilde{\phi}(1,(F^i(y)y^{-1})^b) (u.\Id_n) = Y^b (u(F^i(y)y^{-1})^b \Id_n) Y^{-b} = u F^i(Y^b)F^i(y^b)y^{-b}Y^{-b} = u.\Id_n $. Since $ \intaut (Y^b)\tilde{\phi}(1,(F^i(y)y^{-1})^b) $ also acts trivially on $ \SL_n(E)\leq \SL_n(D) $, this concludes the proof.
\end{proof}
\begin{remark}
When trying to find a section of $ \Aut (G\to \Spec K)\to \Aut_G(K) $ only defined on $ C_b $, this is the only formula we could come up with. Otherwise stated, the complicatedness of the formula defining $ f_{C_b} $ does not come from the need to adjust it to other partial sections of $ \Aut (G\to \Spec K)\to \Aut_G(K) $.
\end{remark}
Though not needed, we check that the automorphism $ \intaut (Y^j)\tilde{\phi}(\ev (\zeta^{aj}T) ,(F^i(y)y^{-1})^j) $ appearing in Proposition~\ref{Prop:constructing fCk(d)} does not depend on the choice of $ y $.
\begin{lemma}\label{Lem:the splitting on Cb does not depend on the choice of y}
Keep the notations of Proposition~\ref{Prop:constructing fCk(d)}. Let $ y'\in \mathbf{F}_{p^{idb}} $ such that $ \frac{F^{id}(y')}{y'} = \zeta^{ar} $. Let $ g' = \varphi ({y'}^{-1}) $ and let $ Y' = \diag (g',\dots ,g') $ where we have $ \frac{n}{b} $ terms. Then $ f_{C_b}(\ev (\zeta^{aj}T)) = \intaut ({Y'}^j)\tilde{\phi}(\ev (\zeta^{aj}T) ,(\frac{F^i(y')}{y'})^j) $.
\end{lemma}
\begin{proof}
Those two elements of $ \Aut (G\to \Spec K) $ differ by $ \intaut (Y^j{Y'}^{-j})\tilde{\phi}(1,(\frac{F^i(y)y'}{F^i(y')y'})^j) $. Let $ x = y{y'}^{-1} $. Note that $ x $ belongs to $ \mathbf{F}_{p^{id}} $ because $ x $ is invariant under $ F^{id} $, and hence $ Y^j{Y'}^{-j} = \diag (g{g'}^{-1}, \dots ,g{g'}^{-1})^j $ is actually the diagonal matrix $ x^j.\Id_n $.
Hence $ \intaut (Y^j{Y'}^{-j})\tilde{\phi}(1,(\frac{F^i(y)y'}{F^i(y')y'})^j) = \intaut (x^j.\Id_n)\tilde{\phi}(1,(\frac{F^i(x)}{x})^j) $. But this automorphism is trivial, since $ \intaut (x^j.\Id_n)\tilde{\phi}(1,(\frac{F^i(x)}{x})^j) (u.\Id_n) = x^j.u(\frac{F^i(x)}{x})^j.x^{-j}\Id_n = u.\Id_n $.
\end{proof}
As before, we can also prove a converse to Proposition~\ref{Prop:constructing fCk(d)}.
\begin{proposition}
Keep the notations of Proposition~\ref{Prop:constructing fCk(d)}. If $ b $ does not divide $ n $, there does not exist a homomorphism $ C_b\to \Aut (G\to \Spec K) $ whose composition with $ \Aut (G\to \Spec K)\to \Aut_G(K) $ is the identity on $ C_b $.
\end{proposition}
\begin{proof}
By Theorem~\ref{Thm:local non-splitting for SLn(D)}, it suffices to prove that there exists $ K'\leq K $ such that $ K/K' $ is finite Galois with $ \Gal (K/K')\leq C_b $ and such that $ \gcd (nd,[K:K']) $ does not divide $ n $. Recall (see Definition~\ref{Def:notation for splitting explicitly}) that $ a $ is prime to $ b $ with $ ab = p^i-1 $ so that $ \zeta^a $ is a $ b $-th primitive root of unity of $ K $. Hence, $ K' = \mathbf{F}_{p^i}(\!(T^b)\!) $ is such that $ K/K' $ is Galois of degree $ b $, and $ \Gal (K/K') $ is generated by the automorphism of $ K $ sending $ T $ to $ \zeta^a T $, so that $ \Gal (K/K') = C_b $. Finally, $ \gcd (nd,[K:K']) = \gcd (nd,b) $ does not divide $ n $ because by definition $ b = \gcd (d^b,b) $.
\end{proof}
Finally, we construct a section to $ \Aut (G\to \Spec K)\to \Aut (K) $ for $ \Gal (\mathbf{F}_{p^i}(\!(T)\!)/\mathbf{F}_{p}(\!(T)\!)) $.
\begin{proposition}\label{Prop:constructing fCa'}
Keep the notations of Definition~\ref{Def:notation for splitting explicitly}. Let $ C_{a'} $ be the group generated by $ F^{b'} $ in $ \Gal (K/\mathbf{F}_{p}(\!(T)\!)) $. Let $ c\in \mathbf{N} $ be such that $ ca'+1\in d\mathbf{Z} $ $ ( $which exists because $ \gcd (a',d) = 1) $. Recalling the notation introduced in Remark~\ref{Rem:abstract automorphisms are algebraic}, the map
\begin{align*}
f_{C_{a'}}\colon C_{a'}&\to \Aut (G\to \Spec K)\\
F^{b'j} &\mapsto \tilde{\phi}(F^{j(ci+b')},1)
\end{align*}
is a homomorphism. Furthermore, its composition with the map $ \Aut (\! G\! \to \Spec K)\to \Aut_G(K) $ is the identity on $ C_{a'} $.
\end{proposition}
\begin{proof}
Since $ F^i $ acts trivially on $ K $, $ F^{j(ci+b')} $ is indeed an extension of $ F^{jb'} $ (seen as restricted to $ K $) to $ E $. Furthermore, $ F^{a'(ci+b')} $ acts trivially on $ E = \mathbf{F}_{p^{id}}(\!(T)\!) $, because $ a'(ci+b') = i(ca'+1)\in id\Z $ by definition of $ c $. Hence, $ f_{C_{a'}} $ is indeed a well-defined homomorphism.
\end{proof}
We need one more bit of notation before defining the last portion of the section $ \Aut (G\to \Spec K)\to \Aut_G(K) $ on $ \Gal (K/\mathbf{F}_p(\!(T)\!)) $.
\begin{definition}\label{Def:the matrix w}
With the notations of Definition~\ref{Def:notation for splitting explicitly}, let $ 0_b $ be the zero $ b\times b $ matrix, and let $ w $ be the following $ bb'\times bb' $ matrix:
$$ w = \begin{pmatrix}
0_b & 0_b & 0_b &\dots & 0_b & u\Id_b\\
\Id_b & 0_b & 0_b & \dots & 0_b & 0_b \\
0_b & \Id_b & 0_b &\dots & 0_b & 0_b \\
0_b & 0_b &\Id_b&\dots & 0_b & 0_b \\
\vdots &\vdots & \vdots & \ddots & \vdots &\vdots \\
0_b & 0_b & 0_b &\dots &\Id_b& 0_b \\
\end{pmatrix} $$
\end{definition}
\begin{proposition}\label{Prop:constructing fCb'}
Keep the notations of Definition~\ref{Def:notation for splitting explicitly}. Assume that $ bb' $ divides $ n $. Let $ C_{b'} $ be the group generated by $ F^{a'} $ in $ \Gal (K/\mathbf{F}_{p}(\!(T)\!)) $. With the notations of Definition~\ref{Def:the matrix w}, let $ W = \diag (w,\dots ,w) $ where we have $ \frac{n}{bb'} $ terms $ ( $so that $ W $ is a $ n\times n $ matrix which is block diagonal with coefficients in the set $ \lbrace 0,1,u\rbrace )$. Recalling the notation introduced in Remark~\ref{Rem:abstract automorphisms are algebraic}, the map
\begin{align*}
f_{C_{b'}}\colon C_{b'}&\to \Aut (G\to \Spec K)\\
F^{a'j} &\mapsto \intaut (W^j)\tilde{\phi}(F^{a'j},1)
\end{align*}
is a homomorphism. Furthermore, its composition with the map $ \Aut (\! G\! \to \Spec K)\to \Aut_G(K) $ is the identity on $ C_{b'} $.
\end{proposition}
\begin{proof}
The only assertion that requires a justification is that the map is well-defined, i.e.\ we have to check that $ (\intaut (W)\tilde{\phi}(F^{a'},1))^{b'} $ is the identity on $ \SL_n(D) $. By definition, $ W^{b'} $ is the scalar matrix $ u.\Id_n $. Hence $ (\intaut (W)\tilde{\phi}(F^{a'},1))^{b'} = \intaut(W^{b'})\tilde{\phi}(F^{a'b'},1) = \intaut(u.\Id_n)\tilde{\phi}(F^{i},1) $, which indeed acts as the identity on $ \SL_n(D) $ because for all $ x\in E $, $ uxu^{-1} = F^{-i}(x) $ (see Remark~\ref{Rem:on the notation for explicit splitting}).
\end{proof}
\begin{remark}
Note that if one is just interested in a section defined on $ C_{b'} $ alone, one can take $ b=1 $ in Proposition~\ref{Prop:constructing fCb'}. Hence, a section of $ \Aut (G\to \Spec K)\to \Aut_G(K) $ only defined on $ C_{b'} $ exists if and only if $ b' $ divides $ n $ (i.e.\ the stronger assumption that $ bb' $ divides $ n $ is there to ensure that we can glue $ f_{C_{b'}} $ with $ f_{C_b} $).
\end{remark}
As before, we have a converse to Proposition~\ref{Prop:constructing fCb'}.
\begin{proposition}
Keep the notations of Proposition~\ref{Prop:constructing fCb'}. If $ b' $ does not divide $ n $, there does not exist a homomorphism $ C_{b'}\to \Aut (G\to \Spec K) $ whose composition with $ \Aut (G\to \Spec K)\to \Aut_G(K) $ is the identity on $ C_{b'} $.
\end{proposition}
\begin{proof}
By Theorem~\ref{Thm:local non-splitting for SLn(D)}, it suffices to prove that there exists $ K'\leq K $ such that $ K/K' $ is finite Galois, $ \Gal (K/K')\leq C_{b'} $ and $ \gcd (nd,[K:K']) $ does not divide $ n $. But $ K' = \mathbf{F}_{p^{a'}}(\!(T)\!) $ is such a subfield.
\end{proof}
We can finally glue all the previous constructions to obtain a global splitting of the initial short exact sequence.
\begin{theorem}\label{Thm:splitting for SLn(D) in char. p}
Keep the notations of Definition~\ref{Def:notation for splitting explicitly}. Assume that for all subfields $ K'\leq K $ such that $ K/K' $ is finite Galois, $ \gcd (nd,[K:K']) $ divides $ n $. Then the short exact sequence $ 1\to \Aut G\to \Aut (G\to \Spec K)\to \Aut_G(K)\to 1 $ splits.
\end{theorem}
\begin{proof}
In view of Proposition~\ref{Prop:existence of Galois subfield of some degree}, the hypotheses imply that $ \gcd(d,p)=1 $ and $ \gcd (nd,i(p^i-1)) $ divides $ n $. Hence $ bb' $ divides $ n $ and we can apply Propositions~\ref{Prop:constructing fJK},~\ref{Prop:constructing fCl(d)},~\ref{Prop:constructing fCk(d)},~\ref{Prop:constructing fCa'} and~\ref{Prop:constructing fCb'}. For the rest of the proof, we strictly adhere to the notations that are introduced in the statements of those propositions.
Recall that $ \Aut_G(K) = \Aut (K) $ (Corollary~\ref{Cor:etxending auto to D}). Also recall that we decomposed $ \Aut (K) $ as $ (J(K)\rtimes (C_a\times C_b))\rtimes (C_{a'}\times C_{b'}) $, where
\begin{enumerate}[(i)]
\item $ C_a\leq \mathbf{F}_{p^i}^{\times} $ is generated by $ \zeta^b $.
\item $ C_b\leq \mathbf{F}_{p^i}^{\times} $ is generated by $ \zeta^a $.
\item $ C_{a'}\leq Gal (K/\mathbf{F}_p(\!(T)\!)) $ is generated by $ F^{b'} $ restricted to $ K $.
\item $ C_{b'}\leq Gal (K/\mathbf{F}_p(\!(T)\!)) $ is generated by $ F^{a'} $ restricted to $ K $.
\end{enumerate}
We define a map
\begin{align*}
f\colon &(J(K)\rtimes (C_a\times C_b))\rtimes (C_{a'}\times C_{b'})\to \Aut (G\to \Spec K)\\
&(g_1 ,g_2,g_3,g_4,g_5)\mapsto f_{J(K)}(g_1)f_{C_a}(g_2)f_{C_b}(g_3)f_{C_{a'}}(g_4)f_{C_{b'}}(g_5)
\end{align*}
We claim that $ f $ is a homomorphism. To prove this claim, it suffices to compute various commutators in $ \Aut (G\to \Spec K) $. To carry the computation, we pick $ j,j' \in \mathbf{N} $.
\begin{enumerate}
\item The images of $ f_{C_a} $ and $ f_{C_b} $ commute. Indeed, $ \intaut (Z^j)\tilde{\phi}(\ev (\zeta^{bj}T) ,z^{b'j})$ readily commutes with $ \intaut (Y^{j'})\tilde{\phi}(\ev (\zeta^{aj'}T) ,(\frac{F^i(y)}{y})^{j'}) $ (note that $ Y $ and $ Z $ are both $ b\times b $ block diagonal matrices, and that the blocks defining $ Z $ are scalars).
\item Let $ \alpha \in J(K) $. We compute $ f_{C_a}(\ev (\zeta^{bj}T))f_{J(K)}(\alpha)f_{C_a}(\ev (\zeta^{-bj}T)) $:
\begin{align*}
&\intaut (Z^j)\tilde{\phi}(\ev (\zeta^{bj}T),z^{b'j})\circ \intaut (X_{\alpha})\tilde{\phi}(\alpha ,x_{\alpha}^{b'})\circ \intaut (Z^{-j})\tilde{\phi}(\ev (\zeta^{-bj}T),z^{-b'j}) \\
&= \intaut (Z^j \ev (\zeta^{bj}T)(X_{\alpha}) \alpha (Z^{-j})) \tilde{\phi}(\ev (\zeta^{bj}T)\circ \alpha \circ \ev (\zeta^{-bj}T) ,z^{b'j}\ev (\zeta^{bj}T)(x_{\alpha}^{b'}) \alpha (z^{-b'j})) \\
&= \intaut (\ev (\zeta^{bj}T)(X_{\alpha})) \tilde{\phi}(\ev (\zeta^{bj}T)\circ \alpha \circ \ev (\zeta^{-bj}T) ,\ev (\zeta^{bj}T)(x_{\alpha})^{b'})
\end{align*}
where the last equality follows from the fact that $ Z $ and $ X_{\alpha} $ commutes because they are block diagonal matrices, together with the equality $ \alpha (Z^{-j}) = Z^{-j} $ which holds because $ Z $ has coefficients in $ \mathbf{F}_{p^i} $.
But $ \ev (\zeta^{bj}T) (x_{\alpha}) = x_{\ev (\zeta^{bj}T)\circ \alpha \circ \ev (\zeta^{-bj}T)} $, because $ \ev (\zeta^{bj}T) (x_{\alpha}) $ belongs to $ 1+T\mathbf{F}_{p^i}[\![T]\!] $, and
\begin{align*}
\ev (\zeta^{bj}T)(x_{\alpha})^{b'd} &= \ev (\zeta^{bj}T)(\frac{\alpha (T^r)}{T^r}) \\
&= \zeta^{-bj}.\dfrac{\ev (\zeta^{bj}T)\alpha (T^r)}{T^r} \\
&= \dfrac{(\ev (\zeta^{bj}T)\circ \alpha \circ \ev (\zeta^{-bj}T)) (T^r)}{T^r}
\end{align*}
Hence $ f_{C_a}(\ev (\zeta^{bj}T))f_{J(K)}(\alpha)f_{C_a}(\ev (\zeta^{-bj}T)) = f_{J(K)}(\ev (\zeta^{bj}T)\circ \alpha \circ \ev (\zeta^{-bj}T)) $.
\item The equality $ f_{C_b}(\ev (\zeta^{aj}T))f_{J(K)}(\alpha)f_{C_b}(\ev (\zeta^{-aj}T)) =f_{J(K)}(\ev (\zeta^{aj}T)\circ \alpha \circ \ev (\zeta^{-aj}T)) $ is proved by doing a similar computation than in the previous item.
\item The images of $ f_{C_{a'}} $ and $ f_{C_{b'}} $ commute. Indeed, $ \tilde{\phi}(F^{j(ci+b')},1) $ readily commutes with $ \intaut (W^{j'}) \tilde{\phi}(F^{a'j'},1) $ (recall that $ W $ has coefficients in $ \lbrace 0,1,u\rbrace $).
\item We check that $ f_{C_{a'}}(F^{b'j})f_{C_b}(\ev (\zeta^{aj'}T))f_{C_{a'}}(F^{-b'j}) = f_{C_b}(F^{b'j} \circ \ev (\zeta^{aj'}T)\circ F^{-b'j}) $. We have
\begin{align*}
&\tilde{\phi}(F^{j(ci+b')},1)\circ \intaut (Y^{j'})\tilde{\phi}(\ev (\zeta^{aj'}T) ,(\frac{F^i(y)}{y})^{j'})\circ \tilde{\phi}(F^{-j(ci+b')},1)\\
&= \intaut (F^{j(ci+b')}(Y^{j'}))\tilde{\phi}(F^{j(ci+b')}\circ \ev (\zeta^{aj'}T)\circ F^{-j(ci+b')},F^{j(ci+b')}((\frac{F^i(y)}{y})^{j'}))
\end{align*}
Noting that $ F^{j(ci+b')}\circ \ev (\zeta^{aj'}T)\circ F^{-j(ci+b')} = \ev (F^{j(ci+b')}(\zeta^{aj'})T) $, the desired equality follows from the fact that the Frobenius automorphism on $ \mathbf{F}_{p^{id}} $ is just elevating to the power $ p $.
\item One readily check that $ f_{C_{a'}}(F^{b'j})f_{C_a}(\ev (\zeta^{bj'}T))f_{C_{a'}}(F^{-b'j}) = f_{C_a}(F^{b'j} \circ \ev (\zeta^{bj'}T)\circ F^{-b'j}) $.
\item We have $ f_{C_{a'}}(F^{b'j})f_{J(K)}(\alpha)f_{C_{a'}}(F^{-b'j}) = f_{J(K)}(F^{b'j} \circ \alpha\circ F^{-b'j}) $. Indeed, $ F^{j(ci+b')}(x_{\alpha}) $ belongs to $ 1+T\mathbf{F}_{p^i}[\![T]\!] $, and $ F^{j(ci+b')}(x_{\alpha})^{b'd} \! =\! F^{j(ci+b')}(\frac{\alpha (T^r)}{T^r}) = \dfrac{F^{j(ci+b')}\alpha F^{-j(ci+b')} (T^r)}{T^r} $ because $ F $ acts trivially on $ T $. Hence
\begin{align*}
&\tilde{\phi}(F^{j(ci+b')},1)\intaut (X_{\alpha})\tilde{\phi}(\alpha ,x_{\alpha}^{b'})\tilde{\phi}(F^{-j(ci+b')},1) \\
&=\intaut (F^{j(ci+b')}(X_{\alpha})) \tilde{\phi}(F^{j(ci+b')}\alpha F^{-j(ci+b')} ,F^{j(ci+b')}(x_{\alpha}^{b'}))\\
&= \intaut (X_{F^{j(ci+b')}\alpha F^{-j(ci+b')}})\tilde{\phi}(F^{j(ci+b')}\alpha F^{-j(ci+b')} ,x^{b'}_{F^{j(ci+b')}\alpha F^{-j(ci+b')}})
\end{align*}
as wanted.
\item We check that $ f_{C_{b'}}(F^{a'j})f_{C_b}(\ev (\zeta^{aj'}T))f_{C_{b'}}(F^{-a'j}) = f_{C_b}(F^{a'j} \circ \ev (\zeta^{aj'}T)\circ F^{-a'j}) $. It is obviously enough to check this when $ j=j'=1 $. Then
\begin{align*}
&\intaut (W)\tilde{\phi}(F^{a'},1)\circ \intaut (Y)\tilde{\phi}(\ev (\zeta^{a}T) ,\frac{F^i(y)}{y})\circ \intaut (W^{-1})\tilde{\phi}(F^{-a'},1) = \\
&\intaut (WF^{a'}(Y.\tilde{\phi}(\ev (\zeta^{a}T) ,\frac{F^i(y)}{y})(W^{-1})))\tilde{\phi}(F^{a'}\circ \ev (\zeta^{a}T)\circ F^{-a'},F^{a'}(\frac{F^i(y)}{y}))
\end{align*}
Again, since $ F^{a'}\circ \ev (\zeta^{a}T)\circ F^{-a'} = \ev (F^{a'}(\zeta^{a})T) $, it suffices to show that the matrix appearing in the argument of $ \intaut () $ is equal to $ F^{a'}(Y) $. Since $ W $ is made up of $ bb'\times bb' $ block matrices, whilst $ Y $ is made up of $ b\times b $ block matrices, we can work with one block at a time. Otherwise stated, we may assume that $ bb' = n $. Now $ \tilde{\phi}(\ev (\zeta^{a}T) ,\frac{F^i(y)}{y})(W^{-1}) $ is the matrix
\begin{equation*}
\begin{pmatrix}
0_b & \Id_b & 0_b & 0_b & \dots & 0_b \\
0_b & 0_b & \Id_b & 0_b &\dots & 0_b \\
0_b & 0_b & 0_b &\Id_b&\dots & 0_b \\
\vdots & \vdots &\vdots & \vdots & \ddots &\vdots \\
0_b & 0_b & 0_b & 0_b &\dots &\Id_b \\
(u\frac{F^i(y)}{y})^{-1}.\Id_b & 0_b & 0_b & 0_b &\dots & 0_b
\end{pmatrix}
\end{equation*} so that the matrix appearing in the argument of $ \intaut () $ is the $ n\times n $ block diagonal matrix $ \diag (uF^{a'}(g(u\frac{F^i(y)}{y})^{-1}),F^{a'}(g),\dots ,F^{a'}(g)) $. But $ uF^{a'}(g(u\frac{F^i(y)}{y})^{-1}) = F^{a'}(ug\frac{y}{F^i(y)}u^{-1}) $. Recalling the embedding $ \varphi \colon \mathbf{F}_{p^{idb}}\to M_b(\mathbf{F}_{p^{id}}) $ of Proposition~\ref{Prop:constructing fCk(d)}, we have
\begin{align*}
ug\frac{y}{F^i(y)}u^{-1} &= u\varphi(y^{-1}) \varphi (y) F^i(\varphi (y)^{-1})u^{-1}\\
&= F^{-i}(F^i(g)) = g
\end{align*}
We conclude that the argument of $ \intaut () $ is indeed $ \diag (F^{a'}(g),F^{a'}(g),\dots ,F^{a'}(g)) = F^{a'}(Y) $, as wanted.
\item Let us check that $ f_{C_{b'}}(F^{a'j})f_{C_a}(\ev (\zeta^{bj'}T))f_{C_{b'}}(F^{-a'j}) = f_{C_a}(F^{a'j} \circ \ev (\zeta^{bj'}T)\circ F^{-a'j}) $. It is obviously enough to check this when $ j=j'=1 $. Then
\begin{align*}
&\intaut (W)\tilde{\phi}(F^{a'},1)\intaut (Z)\circ \tilde{\phi}(\ev (\zeta^{b}T) ,z^{b'})\intaut (W^{-1})\tilde{\phi}(F^{-a'},1) = \\
&\intaut (WF^{a'}(Z.\tilde{\phi}(\ev (\zeta^{b}T) ,z^{b'})(W^{-1})))\tilde{\phi}(F^{a'}\circ \ev (\zeta^{b}T)\circ F^{-a'},F^{a'}(z^{b'}))
\end{align*}
Again, since $ F^{a'}\circ \ev (\zeta^{b}T)\circ F^{-a'} = \ev (F^{a'}(\zeta^{b})T) $, it suffices to show that the matrix appearing in the argument of $ \intaut () $ is equal to $ F^{a'}(Z) $. As in the previous item, we can assume that $ bb'=n $, and doing the same kind of computation as in the previous item, we find that the matrix in the argument of $ \intaut () $ is $ F^{a'}(\diag (uz^{-1}u^{-1}\Id_b,\Id_b,z\Id_b,\dots ,z^{b'-2}\Id_b)) $. Since $ z\in \mathbf{F}_{p^{id}} $, $ uz^{-1}u^{-1} = z^{-1} $, and we conclude that multiplication by the scalar matrix $ z\Id_n $ (which belongs to the center of $ \GL_n(D) $ because $ z\in \mathbf{F}_{p^{i}} $), the argument appearing in $ \intaut () $ is equal to $ F^{a'}(Z) $, as wanted.
\item Checking that $ f_{C_{b'}}(F^{a'j})f_{J(K)}(\alpha)f_{C_{b'}}(F^{-a'j}) = f_{J(K)}(F^{a'j} \circ \alpha\circ F^{-a'j}) $ is a similar computation than in the previous item.
\end{enumerate}
We conclude that $ f $ is indeed a homomorphism. The fact that $ f $ is a splitting of the short exact sequence in the statement of the proposition follows from the fact that the restriction of $ f $ to each component is locally a section of $ \Aut (G\to \Spec K)\to \Aut_G(K) $.
\end{proof}
The first step in the proof of Theorem~\ref{Thm:splitting for SLn(D) in char. p} is to translate the existence of Galois subfields of some degree into some divisibility relations between $ p,d,i $ and $ p^i-1 $. We now prove the ad hoc proposition. We warn the reader that the notations of Definition~\ref{Def:notation for splitting explicitly} (and of the subsequent propositions) are not in use any more.
\begin{proposition}\label{Prop:existence of Galois subfield of some degree}
Let $ K = \mathbf{F}_{p^i}(\!(T)\!) $, let $ q $ be a prime number and let $ a\in \N $. There exists a subfield $ K' $ such that $ K/K' $ is finite Galois and $ q^a $ divides $[K:K'] $ if and only if $ q=p $ or $ q^a $ divides $ i(p^i-1) $.
\end{proposition}
\begin{proof}
First assume that such a $ K' $ exists. Since $ K/K' $ is Galois and $ q^a$ divides $[K:K'] $, there exists $ \tilde{K} $ such that $ K/\tilde{K} $ is Galois and $ [K:\tilde{K}] = q^a $. Up to replacing $ K' $ by $ \tilde{K} $, we can thus assume that $ [K:K'] = q^a $. Let also $ K'_{ur} $ be the maximal unramified extension of $ K' $ inside $ K $.
Note that $ K' $ and $ K'_{ur} $ are local fields, so that in particular $ K'\cong \mathbf{F}_{p^j}(\!(T)\!) $ and $ K'_{ur}\cong \mathbf{F}_{p^i}(\!(T)\!) $. Since $ [K'_{ur}:K'] $ divides $ q^a $, there exists $ a_1 $ such that $ q^{a_1} = \frac{i}{j} $. Letting $ a_2 = a-a_1 $, we have that $ K/K'_{ur} $ is a totally ramified extension of degree $ q^{a_2} $.
If $ p=q $, the proposition is proved, hence there just remains to investigate the case $ p\neq q $. In this case, $ K $ is a tamely totally ramified extension of $ K'_{ur}$. Thus, $ K $ is isomorphic to $ K'_{ur}[X]/(X^{q^{a_2}}-\pi) $ for some uniformiser $ \pi\in \mathbf{F}_{p^i}(\!(T)\!) $. But $ K $ is a Galois extension, and hence this implies that $ \mathbf{F}_{p^i}(\!(T)\!) $ has a primitive $ q^{a_2} $-th root of unity, so that $ q^{a_2} $ divides $ p^i-1 $, as wanted.
To prove the converse, we use a classical fact from local class field theory: there exists an extension $ K_{\pi} $ of $ K $ which is Galois and totally ramified, and such that $ \Gal (K_{\pi}/K)$ is isomorphic to the group of invertible elements $ \mathbf{F}_{p^i}[\![T]\!]^{\times} $ of $ \mathbf{F}_{p^i}[\![T]\!] $ (see for example \cite{Iwa86}*{Section~5.3}). Note that the degree of $ \mathbf{F}_{p^i}^{\times}+T^{a+1}\mathbf{F}_{p^i}[\![T]\!] $ in $ \mathbf{F}_{p^i}[\![T]\!]^{\times} $ is equal to $ p^a $. Let $ L_1 $ be the Galois extension of $ K $ corresponding to $ \mathbf{F}_{p^i}^{\times}+T^{a+1}\mathbf{F}_{p^i}[\![T]\!] $. Let also $ L_2 $ be the splitting field of $ X^{p^i-1}-T $ over $ \mathbf{F}_{p}(\!(T)\!) $. For $ j=1$ or $ 2 $, $ L_j $ is totally ramified of finite degree over $ K $, so that there exists an isomorphism $ \phi_j \colon K\to L_j $. Hence $ K_1 = \phi_1^{-1}(K) $ (respectively $ K_2 = \phi_2^{-1}(\mathbf{F}_{p}(\!(T)\!)) $) is such that $ K/K_1 $ (respectively $ K/K_2 $) is Galois, and $ [K:K_1] = p^a $ (respectively $ [K:K_2] = i(p^i-1) $), which concludes the proof.
\end{proof}
\appendix
\section{Base change of the algebraic group \texorpdfstring{$\textbf{SL}_n(D)$}{SLn(D)}}
We begin by recalling some classical facts about finite dimensional central simple algebras over local fields.
\begin{theorem}\label{Thm:classificatio nof CSA over local fields}
Let $ K $ be a local field. Every central simple algebra over $ K $ is isomorphic to an algebra of the form $ A(d,r) $ as in Definition~\ref{Def:the CSA A_(d,r)}. Furthermore, the map $ \inv \colon Br(K)\to \mathbf{Q}/\mathbf{Z}\colon [A(d,r)]\mapsto [\frac{r}{d}] $ is an isomorphism of groups.
\end{theorem}
\begin{proof}
See for example \cite{Mor97}*{Theorem~8} for the first assertion, while the second is precisely the content of \cite{Pie82}*{Chapter~17, \S 10, Theorem}.
\end{proof}
\begin{corollary}\label{Cor:CSA over local fields and Wedderburn}
Let $ K $ be a local field and let $ d,r \in \mathbf{N} $ with $ d\geq 1 $. Let $ a = \gcd (d,r) $. Then $ A(d,r) $ is a division algebra if and only if $ a=1 $, and $ A(d,r)\cong M_a(A(\frac{d}{a},\frac{r}{a})) $.
\end{corollary}
\begin{proof}
The central simple algebra $ A(d,r) $ is a division algebra if and only if all central simple algebras over $ K $ in the same Brauer class have a higher degree. In view of Theorem~\ref{Thm:classificatio nof CSA over local fields}, it readily implies that $ A(d,r) $ is a division algebra if and only if $ a=1 $. Furthermore, by Wedderburn's theorem, $ A(d,r) $ is isomorphic to $ M_n(D) $ for some division algebra $ D $ and some $ 1\leq n \in \mathbf{N} $, and by definition of the Brauer group, $ [D] = [A(d,r)] $. Hence, using the first part of the Theorem, $ D\cong A(\frac{d}{a},\frac{r}{a}) $. Now, comparing degrees readily imply that $ n=a $, and the result is proved.
\end{proof}
We now study the base change of the algebraic group $ SL_n(A) $.
\begin{lemma}\label{Lem:base change of SL_1}
Let $ A $ be a central simple algebra over a field $ k $, and let $ \textbf{SL}_1(A) $ be the corresponding algebraic $ k $-group (see Definition~\ref{Def:algebraic SL_n(D)}). For $ k' $ a field extension of $ k $, $ \textbf{SL}_1(A)_{k'} = \textbf{SL}_1(A\otimes_k k') $.
\end{lemma}
\begin{proof}
Let $ \overline{k'} $ be an algebraic closure of $ k' $. Since $ \overline{k'} $ splits $ A $, the reduced norm is the map $ f\colon A\to A\otimes_k \overline{k'}\cong M_n(\overline{k'})\xrightarrow[]{\det} \overline{k'} $. Let $ \varphi $ denotes the isomorphism $ A\otimes_k \overline{k'}\cong M_n(\overline{k'}) $. If we take a $ k $-basis of $ A $ to get coordinates on $ A\otimes_k \overline{k'} $, the map $ \det \circ \varphi $ is actually a polynomial map on $ A\otimes_k \overline{k'} $ with coefficients in $ k $, by \cite{Bourb73}*{Chapitre~VIII, \S 12, Proposition~11}.
Hence, $ f_{\overline{k'}} = \det \circ \varphi $. This implies that $ f_{k'}\colon A\otimes_k k' \to k' $ is just the composition $ A\otimes_k k' \to A\otimes_k \overline{k'}\cong M_n(\overline{k'})\xrightarrow[]{\det} \overline{k'} $, i.e.\ $ f_{k'} $ is the reduced norm map of the algebra $ A\otimes_k k' $, as wanted.
\end{proof}
Before giving the formula for the base change of $ \SL_n(A) $, we recall the effect of extending scalars for central simple algebras over local fields.
\begin{lemma}\label{Lem:Base change of CSA}
Let $ K $ be a local field and let $ A(d,r) $ be the central simple algebra over $ K $ defined in Definition~\ref{Def:the CSA A_(d,r)}. Let $ L $ be a finite extension of $ K $. Then $ A(d,r)\otimes_K L\cong A(d,r[L:K]) $.
\end{lemma}
\begin{proof}
By Wedderburn's theorem, a central simple algebra over a field is uniquely determined by its degree and its Brauer class. By \cite{Pie82}*{Chapter~17, Section~17.10, Proposition}, we have $ \inv([A(d,r)\otimes_K L]) = [L:K].\inv([A(d,r)] $. Hence $ A(d,r[L:K]) $ and $ A(d,r)\otimes_K L $ are in the same Brauer class. Since they have the same degree as well, this concludes the proof.
\end{proof}
\begin{proposition}\label{Prop:base change of SL_n(A)}
Let $ A(d',r') $ be a division algebra over a local field $ K' $ as in Definition~\ref{Def:the CSA A_(d,r)}. Let $ K/K' $ be a finite field extension and let $ a = \gcd (d',[K,K']) $. Then the base change of $ \textbf{SL}_{n'}(A(d',r'))$ to $ K $ is isomorphic to $ \textbf{SL}_{an'}(A(\frac{d'}{a},\frac{[K:K']}{a}r')) $.
\end{proposition}
\begin{proof}
The base change of $ \textbf{SL}_{n'}(A(d',r')) = \textbf{SL}_{1}(M_{n'}(A(d',r'))) $ to $ K $ is isomorphic to the algebraic group $ \textbf{SL}_{1}(M_{n'}(A(d',r'))\otimes_{K'} K) \cong \textbf{SL}_{n'}(A(d',r')\otimes_{K'} K) $ by Lemma~\ref{Lem:base change of SL_1}. But by Corollary~\ref{Cor:CSA over local fields and Wedderburn} and Lemma~\ref{Lem:Base change of CSA}, $ A(d',r')\otimes_{K'} K\cong M_a(A(\frac{d'}{a},\frac{[K:K']}{a}r')) $. To conclude, note that for any central simple algebra $ A $, $ \textbf{SL}_{n'}(M_a(A))\cong \textbf{SL}_{an'}(A) $.
\end{proof}
\end{document} | math | 187,100 |
\begin{document}
\begin{abstract}
An extremal curve germ is the analytic germ of a threefold with terminal singularities
along a reduced complete curve admitting a contraction
whose fibers have dimension at most one.
The aim of the present paper
is to review the results concerning those contractions whose central fiber is irreducible
and contains only one non-Gorenstein point.
\operatorname{e}nd{abstract}
{\operatorname{m}}aketitle
\tableofcontents
\section{Introduction}
One of the most important problems in the three-dimensional birational
geometry is to describe explicitly all the steps of the Minimal Model Program (MMP).
These steps consist of certain maps, called divisorial, flipping, and fiber-type contractions (Mori contractions).
The structure of these maps is still unknown in its complete generality, though
much progress has been made in this direction.
We refer to \cite{CKM} for an introduction to the subject.
The aim of the present paper
is to review the results concerning those contractions whose fibers have dimensioв специальном случаеn
at most one. The project was started in the initial paper \cite{Mori:flip} where
the minimal model problem was solved in the three-dimensional case.
To study Mori contractions in this situation one needs to work in the analytic
category and analytic counterparts of the corresponding notions are needed.
The central objects of this paper are so-called \operatorname{e}mph{extremal curve germs}.
An extremal curve germ is the analytic germ of a threefold with terminal singularities
along a reduced complete curve admitting a contraction
whose fibers have dimension at most one.
The present paper is a survey of known results on the classification of objects of this type.
Basically we concentrate on the case of irreducible central fiber with only
one non-Gorenstein point. In this case the results are complete, however
they are scattered in the literature. This is the main reason to write
this survey.
The classification of extremal curve germs is done in terms of a general element
$H$ of the linear system $|{{\operatorname{m}}athscr{O}}_X|$ of
trivial Cartier divisors containing $C$. In many cases this element $H$ is a
normal surface
and then the threefold can be viewed as a one-parameter deformation of $H$.
A birational extremal curve germ $f: (X, C)\to (Z,o)$ is said to be
\operatorname{e}mph{semistable}, if for a general member $D\in |-K_Z|$, the germ
$D_Z:=\operatorname{Spec}_{Z}f_*{{\operatorname{m}}athscr{O}}_D$ is Du Val of type~\type{A} \cite{KM92}. The
semistable case is subdivided into two cases \typec{k1A} and \typec{k2A} according to the
number of non-Gorenstein points of $X$ on $C$. Other cases are called
\operatorname{e}mph{exceptional}. It turns out that treating semistable and exceptional germs
uses different approaches. For example, in the exceptional flipping case,
\cite{KM92} provides relatively simple computations of flipped variety. For
semistable germs these computations become more explicit; in the \typec{k2A}
case, from the general member $H\in |{{\operatorname{m}}athscr{O}}_X|$ one can decide whether $(X,C)$ is
flipping or divisorial \cite[Corollary 4.1]{Mori:ss} and furthermore describe
the flipped variety \cite[Theorem 4.7]{Mori:ss} and $Z$ \cite[Theorem
4.5]{Mori:ss}, respectively; the \typec{k1A} case is similarly treated by
\cite{HTU} under additional assumption ``$b_2(X_s)=1$'' (see \ref{HTU}). According to
local classification (see Propositions \xref{prop:local-primitive} and
\xref{prop-imp-types}), a semistable extremal curve germ of type \typec{k1A} can be
of type \typec{IA^\vee} or \typec{IA}. They are treated in Sect.
\xref{sect:index2}, \xref{sect:imprimitive}, and \xref{sect:IA}.
Here are summary of some of the results.
\begin{theorem}
Let $f:(X,C)\to (Z,o)$ be a flipping extremal curve germ
with irreducible central fiber $C$
and let $H_Z\in |{{\operatorname{m}}athscr{O}}_Z|$ be a general hyperplane section containing $C$.
Then $H_Z$ and $f^{-1}(H_Z)$ are normal and have rational singularities. The
singularity $(H_Z,o)$
is log terminal except for the cases described in \xref{cD/3:flip:.3.2)}
and \xref{IIA:flip:iP=2}. Moreover, $(H_Z,o)$ is a cyclic quotient singularity
if and only if $(X,C)$ is semistable.
\operatorname{e}nd{theorem}
The case where $(X,C)$ is semistable follows from Lemma \xref{lemma:lc}.
If $(X,C)$ has only one non-Gorenstein point (i.e. of type \typec{k1A}), we can use
explicit classification \xref{index2flipping}, \xref{imprimitiveIA},
\xref{IC-main}, \xref{theorem-main-birational}, \xref{cD/3:thm)}, \xref{IIA:thm}.
For the remaining \typec{kAD} case we refer to \cite[\S 9]{KM92}.
\begin{theorem}
Let $f:(X,C)\to (Z,o)$ be a divisorial extremal curve germ
with irreducible central fiber $C$
and let $H_Z\in |{{\operatorname{m}}athscr{O}}_Z|$ be a general hyperplane section containing $o$.
Then $(H_Z,o)$ is either a Du Val point,
a rational log canonical point of type \type{\tilde D} \textup(in the case \xref{imprimitiveII}\textup),
or a cyclic quotient singularity of class \type{T}.
Moreover, the last two possibilities occurs only if $(X,C)$ has a locally imprimitive point
or $(X,C)$ is semistable and has two non-Gorenstein points whose indices are not coprime.
\operatorname{e}nd{theorem}
Moreover, by Theorem \xref{thm:div:Q-Cartier} the singularity
$(Z,o)$ is terminal. If $(X,C)$ has no covers \'etale in codimension one, then
$(Z,o)$ is of index one and $H_Z$ is Du Val. In the semistable case
the assertion as above follows from Lemma \xref{lemma:lc}.
It remains to consider locally imprimitive cases \typec{IA^\vee} and \typec{II^\vee}
(see \xref{imprimitiveII} and \xref{imprimitiveIA}). Note that for a divisorial curve germ
the surface $H:=f^{-1}(H_Z)$ can be non-normal (see e.g. Example~\ref{ex:IIA-n-normal}).
For the ${\operatorname{m}}athbb{Q}$-conic bundle $f:(X,C)\to (Z,o)$
we can show that the base is Du Val of type \type{A}
(Corollary \xref{base}).
The proof uses the existence of a Du Val member $D\in |-K_X|$, see
\cite{MP:cb1}, \cite{MP:cb3}, and Theorem \xref{thm:ge}.
The paper was written during the second author's stay at RIMS, Kyoto University.
The author is very grateful to the institute for
the invitation, hospitality and good working environment.
\section{Preliminaries}
\subsection{Threefold terminal singularities}
\label{3terminal}
Recall that a three-dimensional terminal singularity of index $m$ is a
quotient of an isolated hypersurface singularity by a cyclic group ${\operatorname{m}}umu_m$ of
order $m$.
More precisely, let $(X, P)$ be an analytic germ of a three-dimensional
terminal singularity of index $m$. Then there exists a terminal singularity
$(X^\sharp, P^\sharp)$ of index $1$ and a cyclic ${\operatorname{m}}umu_m$-cover
\begin{equation*}
(X^\sharp, P^\sharp) \longrightarrow (X, P)
\operatorname{e}nd{equation*}
which is \'etale outside $P$ \cite{Reid:Pagoda}.
Moreover, the singularity $(X^\sharp, P^\sharp)$ can be embedded to $({\operatorname{m}}athbb{C}^4,
0)$
so that its general hyperplane section is a surface Du Val singularity
(thus $(X^\sharp, P^\sharp)$ is so-called \type{cDV} singularity).
A detailed classification of all possibilities for equations of
$X^\sharp\subset {\operatorname{m}}athbb{C}^4$ and the action of ${\operatorname{m}}umu_m$ was obtained in
\cite{Mori:sing} (see also \cite{Reid:YPG}, \cite{KSh88}).
Assume that $m > 1$. Then the ${\operatorname{m}}umu_m$-action on
$(X^\sharp,P^\sharp)$ will be analyzed. We
fix a character $\chi$ generating $\operatorname{Hom}({\operatorname{m}}umu_m, {\operatorname{m}}athbb{C}^*) ={\operatorname{m}}athbb{Z}/m{\operatorname{m}}athbb{Z}$.
For a ${\operatorname{m}}umu_m$-semi-invariant $z$, we
write
\[
\operatorname{wt}(z)\operatorname{e}quiv a {\operatorname{m}}od m
\]
if $g(z) = \chi(g)^a z$ for all $g\in
{\operatorname{m}}umu_m$.
\begin{stheorem}[\cite{Mori:sing}]
\label{clasifiction-terminal}
In the above notation the singularity $(X^\sharp, P^\sharp)$ is
${\operatorname{m}}umu_m$-isomorphic to a hypersurface ${{p}_{\operatorname{a}}}hi = 0$ in $({\operatorname{m}}athbb{C}^4_{x_1,\dots,x_4},
0)$ such that
for some $a, b\in {\operatorname{m}}athbb{Z}$ prime to $m$ one of the following holds:
\begin{enumerate}
\item \label{classification-singularities-m}
$\operatorname{wt}(x, {{p}_{\operatorname{a}}}hi) \operatorname{e}quiv (a, b, - a, 0, 0) {\operatorname{m}}od m$;
\item \label{classification-singularities-cAx/4}
$m = 4$, and $\operatorname{wt}(x,{{p}_{\operatorname{a}}}hi) \operatorname{e}quiv (a, b, - a, 2, 2) {\operatorname{m}}od m$.
\operatorname{e}nd{enumerate}
In the case \xref{classification-singularities-cAx/4}
we say that $(X,P)$ is a point of type \type{cAx/4}.
\operatorname{e}nd{stheorem}
Thus the locus $\Upsilon \subset{\operatorname{m}}athbb{C}^4$ of the points at which ${\operatorname{m}}umu_m$-action is not free
is a coordinate axis which is not contained in $X^\sharp$.
The following number
\begin{equation*}
\operatorname{aw}(X,P):={\operatorname{m}}ult_0({{p}_{\operatorname{a}}}hi|_{\Upsilon})
\operatorname{e}nd{equation*}
is well defined
and called the \operatorname{e}mph{axial multiplicity} of $(X,P)$.
\subsection{}
Recall that a \operatorname{e}mph{contraction} is a proper surjective morphism $f:X\to Z$ of
normal varieties such that $f_*{{\operatorname{m}}athscr{O}}_X={{\operatorname{m}}athscr{O}}_Z$.
\begin{sdefinition}
Let $(X,C)$ be the analytic germ
of a threefold with terminal singularities along a reduced complete curve. We
say that $(X,C)$ is an \operatorname{e}mph{extremal curve germ} if there is a contraction
\[
f: (X,C)\to (Z,o)
\]
such that $C=f^{-1}(o)_{\operatorname{red}}$ and $-K_X$ is $f$-ample.
Furthermore, $f$ is called \operatorname{e}mph{flipping} if its exceptional locus coincides
with $C$ and \operatorname{e}mph{divisorial} if its exceptional locus is two-dimensional. If
$f$ is not birational, then $Z$ is a surface and $(X,C)$ is said to be a
\operatorname{e}mph{${\operatorname{m}}athbb{Q}$-conic bundle germ} \cite{MP:cb1}.
\operatorname{e}nd{sdefinition}
In general, we do not assume that $X$ is ${\operatorname{m}}athbb{Q}$-factorial.
This is because the ${\operatorname{m}}athbb{Q}$-factoriality is not a local condition
in the analytic category (see \cite[\S 1]{Kaw:Crep}).
For future references we need the following easy example.
\begin{example}
\label{ex-toric}
Consider the following action of ${\operatorname{m}}umu_m$ on ${\operatorname{m}}athbb{P}^1_x\times
{\operatorname{m}}athbb{C}^2_{u,v}$:
\begin{equation*}
(x;u,v) \longmapsto(\varepsilon^a x; \varepsilon u,
\varepsilon^{-1} v),
\operatorname{e}nd{equation*}
where $\varepsilon$ is a primitive $m$-th root of unity and ${\operatorname{g}}cd
(m,a)=1$. Let $X:={\operatorname{m}}athbb{P}^1\times{\operatorname{m}}athbb{C}^2/{\operatorname{m}}umu_m$,
$Z:={\operatorname{m}}athbb{C}^2/{\operatorname{m}}umu_m$ and let $f: X\to Z$ be the natural
projection. Since ${\operatorname{m}}umu_m$ acts freely in codimension one, $-K_X$ is
$f$-ample. The images of two fixed points on ${\operatorname{m}}athbb{P}^1\times {\operatorname{m}}athbb{C}^2$ are terminal cyclic
quotient singularities of types $\frac1m({{p}_{\operatorname{a}}}m a,1,-1)$ on $X$. Hence, $f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle.
A ${\operatorname{m}}athbb{Q}$-conic bundle germ
biholomorphic to $f$ as above is called
\operatorname{e}mph{toroidal}.
\operatorname{e}nd{example}
The following key fact is an immediate consequence of the Kawamata-Viehweg
vanishing theorem.
\begin{theorem}
\label{th-vanish}
Let $f: (X,C)\to (Z,o)$ be an extremal curve germ. Then
$R^if_*{{\operatorname{m}}athscr{O}}_X=0$ for $i>0$.
\operatorname{e}nd{theorem}
\begin{scorollary}[cf. {\cite[Remark 1.2.1, Cor. 1.3]{Mori:flip}}]
\label{cor-C-pa=0}
\begin{enumerate}
\item\label{cor-C-pa=0a}
If ${{\operatorname{m}}athcal{J}}JJ$ is an ideal such that ${\operatorname{Supp}}({{\operatorname{m}}athscr{O}}_X/{{\operatorname{m}}athcal{J}}JJ)\subset C$, then
$H^1({{\operatorname{m}}athscr{O}}_X/{{\operatorname{m}}athcal{J}}JJ)=0$.
\item\label{cor-C-pa=0b}
${{p}_{\operatorname{a}}}(C)=0$ and $C$ is a union of smooth rational curves.
\item\label{cor-C-pa=0c}
$\operatorname{Pic} X\simeq H^2(C,{\operatorname{m}}athbb{Z})\simeq {\operatorname{m}}athbb{Z}^\uprho$, where $\uprho$ is the number
of irreducible components of $C$.
\operatorname{e}nd{enumerate}
\operatorname{e}nd{scorollary}
\begin{sremark}
\label{rem-prel-extr-nbd}
If $C$ is reducible, then for every proper curve
$C'\subsetneq C$, the germ $(X,C')$ is also an extremal curve germ.
\operatorname{e}nd{sremark}
\begin{lemma}\label{lemma:base}
Let $f: (X,C)\to (Z,o)$ be an extremal curve germ.
\begin{enumerate}
\item \label{lemma:base-1}
If $f$ is birational, then on $Z$ there exists an effective ${\operatorname{m}}athbb{Q}$-divisor
$B$ such that the pair $(Z,B)$ has only canonical singularity at $o$.
If moreover $f$ is flipping, then the singularity of $(Z,B)$ at $o$ is terminal.
\item \label{lemma:base-2}
If $f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle, then $Z$ has a log terminal singularity at $o$.
\operatorname{e}nd{enumerate}
\operatorname{e}nd{lemma}
\begin{proof}
Take $n{\operatorname{g}}g 0$ so that the divisor $nK_X$ is Cartier and the linear
system $|-nK_X|$ is base point free. Let $H\in |-nK_X|$ be a general member.
Then $H$ is a smooth surface meeting the components of $C$ transversally.
For \xref{lemma:base-1}, put $D:=\frac 1n H$ and $B:=f_*D$.
Then the singularities of the pair $(X,D)$ are terminal.
Since $f$ is crepant with respect to $K_X+D$ and does not
contract components of $D$, we see that
the singularities of $(Z,B)$ are canonical \cite[Lemma 3.38]{KM:book}.
To show \xref{lemma:base-2} we note that
the restriction $f_H: H\to Z$ is a finite morphism.
Thus $(Z, o)$ is a log terminal singularity \cite[Prop.~5.20]{KM:book}.
\operatorname{e}nd{proof}
Note however that in \xref{lemma:base-1} we do not assert that the
point $(Z,o)$ is ${\operatorname{m}}athbb{Q}$-Gorenstein, even in the divisorial case,
see Theorem~\xref{thm:div:Q-Cartier}.
The result of \xref{lemma:base-2} is significantly improved in \xref{corollary:cyclic}
and \cite[1.2.7]{MP:cb1}.
\subsection{General member of $|{{\operatorname{m}}athscr{O}}_X|$}
\label{H}
Let $f: (X,C)\to (Z,o)$ be an extremal curve germ. If $f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle, then we assume that $(Z,o)$ is smooth. We denote by $|{{\operatorname{m}}athscr{O}}_Z|$ the linear system of Cartier divisors (hyperplane sections) passing through $o$ and $|{{\operatorname{m}}athscr{O}}_X|:=f^*|{{\operatorname{m}}athscr{O}}_Z|$. Let $H$ be a general member of $|{{\operatorname{m}}athscr{O}}_X|$ and let $H_Z=f(H)$. Let $H^{{\operatorname{m}}athrm{n}}\to H$ be the normalization (we put $H^{{\operatorname{m}}athrm{n}}= H$ if $H$ is normal). By \cite[5.25]{KM:book} and Lemma \xref{lemma:base} both $H_Z$ and $H$ are Cohen-Macaulay. Hence by Bertini's theorem $H_Z$ is normal. Then the composition map $H^{{\operatorname{m}}athrm{n}}\to H_Z$ has connected fibers. Moreover, it is a rational curve fibration if $\dim Z=2$; a birational contraction to a point $(H_Z, o)$ if $f$ is birational. Thus in the ${\operatorname{m}}athbb{Q}$-conic bundle case $H^{{\operatorname{m}}athrm{n}}$ has only rational singularities. The same is true in the birational case if the singularity $(H_Z,o)$ is rational.
\subsection{Notation on dual graphs}
Let $S$ be a normal surface and let $C\subset S$ be a curve. Suppose that on the minimal resolution of $S$ the exceptional divisors and the proper transform of $C$ form a normal crossing divisor, say $R$. We use the usual notation of dual graphs ${\operatorname{D}}elta (S,C)$ of $R$: each $\diamond$ corresponds to an irreducible component of $C$ and each $\circ$ corresponds to an exceptional divisor, and we may use $\bullet$ instead of $\diamond$ if we want to emphasize that it is a complete $(-1)$-curve. A number attached to a vertex denotes the minus self-intersection number. For short, we may omit $2$ if the self-intersection equals $-2$.
\begin{sproposition}[{\cite{Cutkosky-1988}}]
\label{prop:Gor}
\begin{enumerate}
\item \label{prop:Gor-cb}
Let $f: X\to Z$ be a ${\operatorname{m}}athbb{Q}$-conic bundle. If $X$ is Gorenstein
\textup(and terminal\textup), then $Z$ is smooth and there is a
vector bundle ${\operatorname{m}}athscr{E}$ of rank $3$ on $Z$ and an embedding
$X\hookrightarrow {\operatorname{m}}athbb{P}({\operatorname{m}}athscr{E})$ such that every scheme fiber $X_z$,
$z\in Z$ is a conic in ${\operatorname{m}}athbb{P}({\operatorname{m}}athscr{E})_z$.
\item \textup(see also \cite[4.7.2]{KM92}\textup)
\label{prop:Gor-bir}
Let $f: (X,C)\to (Z,o)$ be a birational curve germ such that $X$ is Gorenstein.
Then $f$ is divisorial, $(Z,o)$ is smooth, $C$ is irreducible, and
$f$ is the blowup of a curve $B\subset Z$
having only planar singularities. Moreover, $X$ has exactly one singular point
which is of type~\type{cA} and for a general member $H\in |{{\operatorname{m}}athscr{O}}_X|$
the graph ${\operatorname{D}}elta(H,C)$ has the form
\begin{equation*}
\bullet\text{---}\underbrace{\circ\text{---}\cdots\text{---}\circ}_m
\operatorname{e}nd{equation*}
\operatorname{e}nd{enumerate}
\operatorname{e}nd{sproposition}
The following fact is a particular case of \cite[Theorem~4.9]{KM92}.
\begin{theorem}
Let $f:(X,C)\to (Z,o)$ be a divisorial extremal curve germ.
Let $E$ be its exceptional locus \textup(with reduced structure\textup) and let $B:=f(E)_{\operatorname{red}}$.
Assume that $K_Z$ is ${\operatorname{m}}athbb{Q}$-Cartier \textup(this automatically
holds if $C$ is irreducible, see \xref{thm:div:Q-Cartier}\textup).
Then the following holds.
\begin{enumerate}
\item
The set $E$ of $f$ is purely two-dimensional and
is a ${\operatorname{m}}athbb{Q}$-Cartier divisor, and the singularity $(Z,o)$ is terminal.
\item
The variety $X$ is the symbolic blowup of $B$, that is,
\[
X=\operatorname{Proj}_Z \bigoplus_{m=0}^{\infty} {{\operatorname{m}}athcal{I}}_B^{(m)},
\]
where ${{\operatorname{m}}athcal{I}}_B$ is the ideal sheaf of $B$,
and ${{\operatorname{m}}athcal{I}}_B^{(m)}$ denotes its symbolic power. In particular, $X$ is uniquely determined by
$B\subset Z$.
\operatorname{e}nd{enumerate}
\operatorname{e}nd{theorem}
It is possible to study divisorial curve germs algebraically,
by scrupulous analysis of the curve $B$ and its embedding $B\subset X$
(see \cite{Tzi:03}, \cite{Tzi:05D}, \cite{Kaw:div},
\cite{Tzi:10}, \cite[\S~6.1]{Prokhorov-Reid}, \cite{Ducat:16}). This method
is completely different from our approach.
\section{Basic techniques}
\subsection{}
Let ${{\operatorname{m}}athcal{I}}_C\subset {{\operatorname{m}}athscr{O}}_X$ be the
ideal sheaf of $C$ and let ${{\operatorname{m}}athcal{I}}_C^{(n)}$ be its symbolic $n$th power, that is, the saturation of ${{\operatorname{m}}athcal{I}}_C^n$ in ${{\operatorname{m}}athscr{O}}_X$. Put
\begin{equation*}
\operatorname{gr}_C^n{{\operatorname{m}}athscr{O}}:={{\operatorname{m}}athcal{I}}_C^{(n)}/{{\operatorname{m}}athcal{I}}_C^{(n+1)}.
\operatorname{e}nd{equation*}
Further, let $F^n\upomega_X$ be
the saturation of ${{\operatorname{m}}athcal{I}}_C^n\upomega_X$ in $\upomega_X$ and let
\begin{equation*}
\operatorname{gr}_C^n\upomega:=F^n\upomega_X/F^{n+1}\upomega_X.
\operatorname{e}nd{equation*}
Let $m$ be the index
of $K_X$. We have natural homomorphisms
\begin{equation*}
\begin{array}{lllll}
\alpha_1 &:& \bigwedge^2 \operatorname{gr}_C^1{{\operatorname{m}}athscr{O}} &\longrightarrow&
\HHom_{{{\operatorname{m}}athscr{O}}_C}(\Omega_C^1,\operatorname{gr}_C^0\upomega),
\\[7pt]
\beta_0&:& (\operatorname{gr}_C^0\upomega)^{\otimes m} &\longrightarrow&
(\upomega_X^{\otimes m})^{\vee\vee}\otimes {{\operatorname{m}}athscr{O}}_C.
\operatorname{e}nd{array}
\operatorname{e}nd{equation*}
Denote
\begin{equation}
\label{equation:iP-wP}
i_P(1):=\operatorname{len}_P\operatorname{Coker} (\alpha_1),{\operatorname{m}}athbin{\sim_{\scriptscriptstyle{\QQ}}}uad w_P(0):=\operatorname{len}_P\operatorname{Coker} (\beta_0)/m.
\operatorname{e}nd{equation}
To study extremal germs more carefully,
Mori \cite{Mori:flip} introduced also series of local invariants $i_P(n)$,
$w_P(n)$, $w^*_P(n)$ similar to $i_P(1)$ and $w_P(0)$.
We do not define them here.
Assume that $C\simeq {\operatorname{m}}athbb{P}^1$. Then we have by \cite[2.3.1]{Mori:flip}
\begin{eqnarray}
\label{eq-grw-w}
&&
-\deg \operatorname{gr}_C^0\upomega=-K_X\cdot C+\sum_P w_P(0),
\\
\label{eq-grO-iP1}
&&
2+\deg \operatorname{gr}_C^0\upomega-\deg \operatorname{gr}_C^1{{\operatorname{m}}athscr{O}}=\sum_P i_P(1).
\operatorname{e}nd{eqnarray}
Since $\operatorname{rk} \operatorname{gr}_C^1{{\operatorname{m}}athscr{O}}=2$, taking \xref{th-vanish} into account we obtain
\begin{equation}
\label{eq-grO-iP1-1}
\deg \operatorname{gr}_C^1{{\operatorname{m}}athscr{O}}{\operatorname{g}}e -2,
\operatorname{e}nd{equation}
\begin{equation}
\label{eq-grO-iP1-2}
4{\operatorname{g}}e -\deg \operatorname{gr}_C^0\upomega+\sum_P i_P(1)= -K_X\cdot C+\sum_P
w_P(0)+\sum_P i_P(1).
\operatorname{e}nd{equation}
\begin{sremark}
\label{remark:grw}
In the case where $f$ is birational, by the Grauert-Riemenshneider
vanishing, one has $\operatorname{gr}_C^0\upomega={{\operatorname{m}}athscr{O}}_{C}(-1)$ (see
\cite[2.3]{Mori:flip}). This is no longer true for ${\operatorname{m}}athbb{Q}$-conic
bundles: in the toroidal example \xref{ex-toric} easy computations
show
$\deg \operatorname{gr}_C^0\upomega= -2$ (see \operatorname{e}qref {eq-grw-w}). Similarly, in the case
\xref{item-main-th-impr-barm=1} we also have $\deg \operatorname{gr}_C^0\upomega=
-2$.
We will show below that these two examples are the only
exceptions (see Corollaries \xref{cor-prop-grw=2-2-points-prim} and
\xref{corollary:gr-w}).
\operatorname{e}nd{sremark}
\subsection{}
Let $(X,P)$ be a germ of threefold terminal singularity.
Throughout this paper $(X^\sharp, P^\sharp)\to (X,P)$
denotes the index-one cover. For any object $V$ on $X$
we denote by $V^\sharp$ the pull-back of $V$ on $X^\sharp$.
\begin{lemma}[{\cite[2.16]{Mori:flip}}]
\label{equation-iP}
In the above notation, assume that $C^\sharp$ is smooth. Denote
\begin{equation*}
\operatorname{e}ll(P):=\operatorname{len}_P {{\operatorname{m}}athcal{I}}_C^{\sharp (2)}/{{\operatorname{m}}athcal{I}}_C^{\sharp 2},
\operatorname{e}nd{equation*}
where ${{\operatorname{m}}athcal{I}}_C^\sharp$ is the ideal of $C^\sharp$ in $X^\sharp$.
Then
\begin{equation}\label{equation-iP-lP}
i_P(1)=
\begin{cases}
\operatorname{e}ll(P)&\text{if $m=1$},
\\
\lfloor(\operatorname{e}ll(P)+6)/4\rfloor&\text{if $(X,P)$ is of type~\type{cAx/4}},
\\
\lfloor\operatorname{e}ll(P)/m\rfloor+1&\text{if $(X,P)$ is not as above}.
\operatorname{e}nd{cases}
\operatorname{e}nd{equation}
\operatorname{e}nd{lemma}
\begin{lemma}[{\cite[2.10, 2.15]{Mori:flip}}]
\label{lemma:iP-wP}
If $(X,P)$ is singular, then $i_P(1){\operatorname{g}}e 1$.
If $(X,P)$ is not Gorenstein, then $w_P(0)>0$.
\operatorname{e}nd{lemma}
Then from \operatorname{e}qref{eq-grO-iP1-2} we obtain
\begin{scorollary}
An extremal curve germ $(X,C\simeq{\operatorname{m}}athbb{P}^1)$ has at most three
singular points.
\operatorname{e}nd{scorollary}
\subsection{}
Let $(X,C)$ be an extremal curve germ.
By Lemma \xref{cor-C-pa=0}\xref{cor-C-pa=0a} we have
$H^1 (\operatorname{gr}^1_ C{{\operatorname{m}}athscr{O}}) = 0$.
From the
standard exact sequence
\begin{equation*}
0\xrightarrow{\hspace*{20pt}} {{\operatorname{m}}athcal{I}}_C^{(n+1)} \xrightarrow{\hspace*{20pt}} {{\operatorname{m}}athcal{I}}_C^{(n)} \xrightarrow{\hspace*{20pt}} \operatorname{gr}_C^n{{\operatorname{m}}athscr{O}}\xrightarrow{\hspace*{20pt}} 0.
\operatorname{e}nd{equation*}
we obtain the following easy but useful fact.
\begin{slemma}\label{lemma-grC}
The following assertions hold.
\begin{enumerate}
\item \label{lemma-grC-1}
If $H^1\bigl(\operatorname{gr}_C^n{{\operatorname{m}}athscr{O}}\bigr)=0$ and the
map $H^0\bigl({{\operatorname{m}}athcal{I}}_C^{(n)}\bigr)\to H^0\bigl(\operatorname{gr}_C^n{{\operatorname{m}}athscr{O}}\bigr)$ is surjective, then
$H^1\bigl({{\operatorname{m}}athcal{I}}_C^{(n+1)}\bigr)\simeq H^1\bigl({{\operatorname{m}}athcal{I}}_C^{(n)}\bigr)$. In particular, $H^1(I)=0$ from the case $n=0$.
\item \label{lemma-grC-2}
If for all $i<n$ one has $H^1(\operatorname{gr}_C^i{{\operatorname{m}}athscr{O}})=0$ and the
map $H^0({{\operatorname{m}}athcal{I}}_C^{(i)})\to H^0(\operatorname{gr}_C^i{{\operatorname{m}}athscr{O}})$ is surjective, then
$H^1({{\operatorname{m}}athcal{I}}_C^{(n)})\simeq H^1(\operatorname{gr}_C^n{{\operatorname{m}}athscr{O}})=0$.
\item \label{lemma-grC-3}
If
$H^0(\operatorname{gr}_C^1{{\operatorname{m}}athscr{O}})=0$, then
$H^1({{\operatorname{m}}athcal{I}}_C^{(2)})= H^1(\operatorname{gr}_C^2{{\operatorname{m}}athscr{O}})=0$.
\operatorname{e}nd{enumerate}
In particular, if a general member $H\in |{{\operatorname{m}}athscr{O}}_X|$ is normal, then
$H^0(\operatorname{gr}_C^1{{\operatorname{m}}athscr{O}}){\operatorname{m}}athrm{n}eq0$.
\operatorname{e}nd{slemma}
Note however that this is necessary but not sufficient condition
for normality of $H$ \cite{MP:IA}.
\subsection{Sheaves $\operatorname{gr}_C^n\upomega$}
\begin{slemma}
\label{lemma-omega-main}
Let $f: (X,C) \to (Z,o)$ be an extremal curve germ.
\begin{enumerate}
\item \textup(\cite[1.2]{Mori:flip}\textup)\label{lemma-omega-main-1}
If $f$ is birational, then $R^if_*\upomega_X=0$ for $i>0$.
\item \textup(\cite[Lemma 4.1]{MP:cb1}\textup)\label{lemma-omega-main-2}
If $f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle
and $Z$ is smooth, then there is a canonical
isomorphism
$R^1f_*\upomega_X\simeq \upomega_Z$.
\operatorname{e}nd{enumerate}
\operatorname{e}nd{slemma}
\begin{proof}
\xref{lemma-omega-main-1} follows from the Grauert-Riemenshneider vanishing.
Let us prove \xref{lemma-omega-main-2}.
Let $g: W\to X$ be a resolution. By {\cite[Prop.
7.6]{Kollar-1986-I}} we have $R^1(f \comp g)_* \upomega_W=\upomega_Z$.
Since $X$ has only terminal singularities, $g_*\upomega_W=\upomega_X$
and by the Grauert-Riemenshneider vanishing, $R^ig_* \upomega_W=0$ for
$i>0$. Then the Leray spectral sequence gives us $R^1f_*
\upomega_X=R^1(f \comp g)_* \upomega_W= \upomega_Z$.
\operatorname{e}nd{proof}
We also have the following useful fact
\begin{corollary}
\label{cor-gr-omega-=0}
Let $f: (X,C\simeq {\operatorname{m}}athbb{P}^1) \to (Z,o)$ be an extremal curve germ.
\begin{enumerate}
\item \label{cor-gr-omega-=0a}
If $f$ is birational, then $\deg \operatorname{gr}_C^0\upomega=-1$.
\item\label{cor-gr-omega-=0b}
Assume that $f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle with smooth base. If
$\deg \operatorname{gr}_C^0\upomega{\operatorname{m}}athrm{n}eq-1$, then $f^{-1}(o)=C$ \textup(as a scheme\textup).
\operatorname{e}nd{enumerate}
\operatorname{e}nd{corollary}
\begin{proof}[Sketch of the proof]
For \xref{cor-gr-omega-=0a} we note that by \xref{lemma-omega-main}\xref{lemma-omega-main-1}
for an arbitrary ideal ${{\operatorname{m}}athcal{J}}JJ$ such that ${\operatorname{Supp}} ({{\operatorname{m}}athscr{O}}_X/{{\operatorname{m}}athcal{J}}JJ)\subset C$
we have $H^1 (\upomega_X/{{\operatorname{m}}athcal{J}}JJ\upomega_X)) = 0$. Hence,
$H^1(\operatorname{gr}_C^0\upomega)=0$ in this case. On the other hand, $\deg \operatorname{gr}_C^0\upomega<0$ by
\operatorname{e}qref{eq-grw-w}.
For \xref{cor-gr-omega-=0b} we
apply \cite[Theorem~4.4]{MP:cb1} with $J={{\operatorname{m}}athcal{I}}_C$.
\operatorname{e}nd{proof}
\begin{slemma}[{\cite[Cor.~1.15]{Mori:flip}}, {\cite[Prop. 4.2]{Kollar-1999-R}}, {\cite[Lemma 4.4.2]{MP:cb1}}]
\label{lemma-int-non-Gor}
Let $f: (X,C) \to (Z,o)$ be an extremal curve germ.
Suppose that $C$ is reducible and let $P$ be a singular
point of $C$.
If $X$ is Gorenstein at $P$, then
$f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle and $C$ has two components meeting at $P$.
If moreover $(Z,o)$ is smooth, then $X$ is Gorenstein
\textup(see \xref{prop:Gor}\xref{prop:Gor-cb}\textup).
\operatorname{e}nd{slemma}
We will show below in \ref{cor:int-non-Gor} that in the above assumptions $(Z,o)$ is smooth
automatically.
\begin{proof}
By Corollary \ref{cor-C-pa=0}
there are at least two components, say $C_1,\, C_2\subset C$ passing through $P$.
Replacing $(X,C)$ with $(X,C_1\cup C_2)$ we may assume that $C=C_1\cup C_2$
(see Remark \xref{rem-prel-extr-nbd}). Since the point $P\in X$ is Gorenstein,
the sheaf $\operatorname{gr}_C^0\upomega=\upomega_X\otimes
{{\operatorname{m}}athscr{O}}_C$ is invertible at $P$. Consider the injection
\[
\varphi:\operatorname{gr}_C^0\upomega\hookrightarrow \operatorname{gr}_{C_{1}}^0\upomega \oplus
\operatorname{gr}_{C_{2}}^0\upomega.
\]
Recall that $(X,C_i)$ is a (birational) extremal curve
germ
by Remark \xref{rem-prel-extr-nbd}. Then by \xref{cor-gr-omega-=0}\xref{cor-gr-omega-=0a}
we have $\operatorname{gr}_{C_{i}}^0\upomega={{\operatorname{m}}athscr{O}}_{C_i}(-1)$, so
$H^0(\operatorname{Coker} (\varphi))= H^1(\operatorname{gr}_C^0\upomega)$. On the other hand,
$\operatorname{Coker} (\varphi)$ is a sheaf of finite length supported at $P$. Since
$\operatorname{gr}_C^0\upomega$ is invertible, $\operatorname{Coker} (\varphi)$ is non-trivial. So,
$H^1(\operatorname{gr}_C^0\upomega){\operatorname{m}}athrm{n}eq 0$ and by Corollary \xref{cor-gr-omega-=0}
the contraction $f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle. Moreover, if the base $(Z,o)$ is smooth,
then again by Corollary \xref{cor-gr-omega-=0} we have
$C=f^{-1}(o)$ (scheme-theoretically). Hence $P$ is the only
singular point of $X$ and
are done.
\operatorname{e}nd{proof}
\section{Topological observations}
Let $\operatorname{Cl}sc(X)$ be the subgroup of the divisor class group $\operatorname{Cl}(X)$
consisting of Weil divisor classes which are ${\operatorname{m}}athbb{Q}$-Cartier.
We will use the following easy consequence of the classification of terminal singularities
without additional reference.
\begin{proposition}[{\cite[Lemma~5.1]{Kaw:Crep}}]
\label{Clsc}
Let $(X,P)$ be an \textup(analytic\textup) germ of three-dimensional terminal singularity of index $m$.
Then
\begin{equation}\label{eq:Clsc}
\operatorname{Cl}sc(X,P)\simeq \uppi_1(X\setminus \{P\})\simeq {\operatorname{m}}athbb{Z}/m{\operatorname{m}}athbb{Z}.
\operatorname{e}nd{equation}
\operatorname{e}nd{proposition}
\begin{definition}[{\cite[(0.4.16), (1.7)]{Mori:flip}}]
\label{splitting}
Let $(X,P)$ be a terminal three-dimensional singularity of index $m$
and let $C\subset X$ be a smooth curve passing through $P$. We say
that $C$ is (locally) \operatorname{e}mph{primitive} at $P$ if the natural map
\begin{equation*}
\varrho : {\operatorname{m}}athbb{Z}\simeq \uppi_1(C\setminus \{P\})\longrightarrow
\uppi_1(X\setminus \{P\})\simeq {\operatorname{m}}athbb{Z}/m{\operatorname{m}}athbb{Z}
\operatorname{e}nd{equation*}
is surjective and \operatorname{e}mph{imprimitive}
at $P$
otherwise. The order $s$ of
$\operatorname{Coker} (\varrho)$ is called the \operatorname{e}mph{splitting degree} and the
number $\bar m=m/s$ is called the \operatorname{e}mph{subindex} of $P\in C$.
\operatorname{e}nd{definition}
It is easy to see that the splitting degree coincides
with the number of
irreducible components of the preimage $C^\sharp$ of $C$ under the
index-one cover $X^\sharp\to X$ near $P$. If $P$ is primitive, we
put $s=1$ and $\bar m=m$.
\subsection{}
In the above notation it is easy to show that for any Weil divisor class $\xi\in \operatorname{Cl}sc(X,P)$
there exists an effective Weil ${\operatorname{m}}athbb{Q}$-Cartier divisor $D$ whose class in
$\operatorname{Cl}sc(X,P)$
equals $\xi$ and such that $D\cap C=\{P\}$. Then one can define the intersection number $\xi\cdot C:= (D\cdot C)_P{\operatorname{m}}od {\operatorname{m}}athbb{Z}$.
Hence there exists a well-defined homomorphism
\begin{equation*}
{\operatorname{m}}athrm{cl}: \operatorname{Cl}sc(X,P) \longrightarrow \textstyle{\frac 1m} {\operatorname{m}}athbb{Z}/{\operatorname{m}}athbb{Z}\subset {\operatorname{m}}athbb{Q}/{\operatorname{m}}athbb{Z},{\operatorname{m}}athbin{\sim_{\scriptscriptstyle{\QQ}}}uad \xi
\longmapsto \xi\cdot C.
\operatorname{e}nd{equation*}
The curve $C$ is locally primitive at $P$ if and only if the map ${\operatorname{m}}athrm{cl}$ is an
isomorphism.
In general, the splitting degree equals the order of
the kernel of ${\operatorname{m}}athrm{cl}$ \cite[1.7]{Mori:flip}.
\subsection{}
Let $(X,C)$ be an extremal curve germ with irreducible central fiber $C$.
Let $P_1,\dots, P_n$ be all the non-Gorenstein points of $X$
and let $m_1,\dots, m_n$ be their indices.
We have the following exact sequence
\begin{equation}
\label{exact-Clcs}
\vcenter{
\xymatrix@R=-4pt{
0\ar[r] & \operatorname{Pic}(X)\ar[r] & \operatorname{Cl}sc (X)\ar[r]& \oplus \operatorname{Cl}sc(X,P_i)\ar[r] & 0
\\
& \rotatebox{90}{$\simeq$}& &\rotatebox{90}{$\simeq$}&
\\
& {\operatorname{m}}athbb{Z}^{\uprho(X)}&& \oplus {\operatorname{m}}athbb{Z}/m_i{\operatorname{m}}athbb{Z}
}}
\operatorname{e}nd{equation}
\begin{scorollary}\label{corollary:cover}
In the above notation assume that $C$ is irreducible.
Let $D_i$, $n=1,\dots,n$ be an effective Weil ${\operatorname{m}}athbb{Q}$-Cartier divisor whose class generates $\operatorname{Cl}sc(X, P_i)$ and
let $H$ be an effective Cartier divisor such that $H\cdot C=1$. Then the following holds.
\begin{enumerate}
\item \label{corollary:cover0}
The group $\operatorname{Cl}sc(X)$ is generated by the classes of $H$, $D_1,\dots, D_n$.
\item \label{corollary:cover-i}
If the point $P_i$ is imprimitive of splitting degree $s_i$ and subindex $\bar m_i$, then
the class of $H-\bar m_i D_i$ is an $s_i$-torsion element in $\operatorname{Cl}sc(X)$.
\item \label{corollary:cover-p}
If $(X,C)$ is locally primitive at distinct points
$P_i, P_j\in C$ and ${\operatorname{g}}cd(m_i,m_j)=d{\operatorname{m}}athrm{n}eq 1$, then
the class of $\frac {m_i}d D_i-\frac {m_j}d D_j$ is a $d$-torsion element in $\operatorname{Cl}sc(X)$.
\operatorname{e}nd{enumerate}
\operatorname{e}nd{scorollary}
\begin{construction}
\label{base-change}
Let $f: (X,C)\to (Z,o)$ be an extremal curve germ
and let $\theta :(X^{\flat},C^{\flat})\to (X,C)$ be a finite cover which is \'etale in codimension one.
Clearly, $\theta$ must be \'etale over the Gorenstein locus of $X$.
The Stein factorization gives us the following diagram.
\begin{equation}
\label{eq:base-change}
\vcenter{
\xymatrix{
(X^{\flat},C^{\flat})\ar[r]^{\theta}\ar[d]^{f^{\flat}}& (X,C)\ar[d]^f
\\
(Z^{\flat},o^{\flat})\ar[r]^{}&(Z,o)
}}
\operatorname{e}nd{equation}
where $(Z^{\flat},o^{\flat})\to (Z,o)$ is a finite cover
which is \'etale over $Z\setminus \{o\}$.
We have $K_{X^{\flat}}=\theta^*K_X$ and
singularities of $X^{\flat}$ are terminal. In particular, $(X^{\flat},C^{\flat})$ is an extremal curve germ.
Note that in our situation
$X^{\flat}$ is the normalization of $X\times_Z Z^{\flat}$ and
$C^{\flat}:=f^{\flat -1}(C)_{\operatorname{red}}$.
Conversely, if $f: (X,C)\to (Z,o)$ is an extremal curve germ
and $(Z^{\flat},o^{\flat})\to (Z,o)$ is a finite cover
which is \'etale over $Z\setminus \{o\}$. Then the base change produces
the diagram \operatorname{e}qref{eq:base-change}, where $X^{\flat}$ is the normalization of $X\times_Z Z^{\flat}$,
$(X^{\flat},C^{\flat})$ is an extremal curve germ, and $\theta$ is \'etale in codimension one.
\operatorname{e}nd{construction}
\begin{definition}\label{torsion-free-cover}
Let $(X,C)$ be an extremal curve germ. By the above construction \xref{base-change}
the torsion part $\operatorname{Cl}(X)_{{\operatorname{m}}athrm{tors}}\subset \operatorname{Cl}(X)$ defines an abelian Galois cover
\begin{equation}
\label{eq:torsion-free-cover}
\tau :(X',C')\longrightarrow (X,C)
\operatorname{e}nd{equation}
which is \'etale over the Gorenstein locus of $X$. We call this map
the \operatorname{e}mph{torsion free cover} of $(X,C)$
and the degree of this cover we call the \operatorname{e}mph{topological index} of $(X,C)$.
Similar to \operatorname{e}qref{eq:base-change} we have the diagram
\begin{equation}
\label{eq:base-cover}
\vcenter{
\xymatrix{
(X',C')\ar[r]^{\tau}\ar[d]^{f'}& (X,C)\ar[d]^f
\\
(Z',o')\ar[r]^{}&(Z,o)
}}
\operatorname{e}nd{equation}
Hence $(X',C')$ is also an extremal curve germ.
Clearly, $\operatorname{Cl}(X')$ is torsion free.
\operatorname{e}nd{definition}
\begin{slemma}\label{lemma:cyclic}
Let $(X,C)$ be an extremal curve germ and let $\theta:(X^\flat,C^\flat)\to (X,C)$
be a finite cover which is \'etale in codimension two. Then $\theta$ is a cyclic cover.
\operatorname{e}nd{slemma}
\begin{proof}
We may assume that the cover $\theta$ is Galois with group $G$
and it is sufficient to show that the group $G$ is cyclic.
By taking composition with the torsion free cover, we may assume also that
$\operatorname{Cl}sc(X^\flat)$ is torsion free.
By the construction $G$ effectively acts on $C^\flat=\cup C_i^\flat$ (because $X$ has only isolated singularities).
Since $C^\flat$ is a tree of smooth
rational curves, it is easy to prove by induction
on the number of components of $C^\flat$ that $G$ has either an
invariant component $C_i^\flat\subset C^\flat$ or a fixed point $P^\flat\in \operatorname{Sing}(C^\flat)$.
In the latter case, let $P=\theta(P^\flat)$. There is a surjection $\uppi_1(X\setminus
\{P\})\twoheadrightarrow G$. Since $\uppi_1(U\setminus
\{P\})$ is cyclic (see \xref{Clsc}), we are done.
In the former case, let $C_i:=\theta (C^\flat_i)$. By Remark \ref{rem-prel-extr-nbd}
we may replace $(X^\flat, C^\flat)$ with $(X^\flat, C^\flat_i)$
and $(X, C)$ with $(X, C_i)$. Thus
$C=C_i$, $C^\flat=C^\flat_i$, and $C^\flat/G=C\simeq {\operatorname{m}}athbb{P}^1$. Assume that the group $G$ is not cyclic.
Then there is no fixed points on $C^\flat$.
If $X^\flat$ has a point of index $m>1$, then its orbit contains
at least two points of the same index. By \operatorname{e}qref{exact-Clcs}
the torsion part of the group $\operatorname{Cl}sc(X^\flat)$ is non-trivial. This contradicts the assumption above.
Thus $X^\flat$ is Gorenstein.
Let $P_1,\dots, P_n\in C$ be all branch points of $C^\flat\to C$
and let $m_1,\dots, m_n$ be their ramification indices.
By the Hurwitz formula we can write
\[
\frac 1{|G|} \bigl (2 g(C^\flat_i)-2\bigr)= 2g(C_i)-2 +\sum_{i=1}^n \left (1-\frac 1{ m_i} \right)
\]
Hence, $\sum 1/m_i >n- 2$.
Since the group $G$ is not cyclic, we have $n>2$.
The index of the point $P_i\in X$ is equal to $m_i$.
By \operatorname{e}qref{equation:iP-wP} and Lemma \ref{lemma:iP-wP} we
have $w_{P_i}(0){\operatorname{g}}e 1/m_i$ and $i_{P_i}(1){\operatorname{g}}e 1$.
Therefore, $\sum w_{P_i}(0) >1$ and $\deg \operatorname{gr}_C^0\upomega=-1$ by \operatorname{e}qref{eq-grO-iP1-2}.
Then we get a contradiction by \operatorname{e}qref{eq-grw-w}.
\operatorname{e}nd{proof}
\begin{scorollary}
\label{prop-cyclic-quo}
Let $(X,C)$ be an extremal curve germ.
Then the torsion part $\operatorname{Cl}(X)_{{\operatorname{m}}athrm{tors}}\subset \operatorname{Cl}(X)$ is a cyclic group.
Hence the torsion free cover \operatorname{e}qref{eq:torsion-free-cover}
is cyclic.
Moreover,
$X',C')$ has no finite cover which is \'etale in codimension one.
\operatorname{e}nd{scorollary}
\begin{scorollary}[{\cite[Lemma 1.10]{P97}}]
\label{corollary:cyclic}
If $f: (X,C)\to (Z,o)$ is a ${\operatorname{m}}athbb{Q}$-conic bundle germ, then $(Z,o)$ is a
cyclic quotient singularity.
\operatorname{e}nd{scorollary}
\begin{proof}
Follows from Lemma~\xref{lemma:cyclic} and \xref{base-change}.
\operatorname{e}nd{proof}
\subsection{}
From now on we assume that $f: (X,C)\to (Z,o)$ is an extremal curve germ with $C\simeq{\operatorname{m}}athbb{P}^1$.
Assume that the torsion part $\operatorname{Cl}(X)_{{\operatorname{m}}athrm{tors}}={\operatorname{m}}athbb{Z}/d{\operatorname{m}}athbb{Z}$ is non-trivial
and consider the torsion free cover \operatorname{e}qref{eq:torsion-free-cover}.
Thus $(X,C)=(X', C')/G$
and $(Z,o)=(Z',o')/G$, where $G={\operatorname{m}}umu_d$ acts on $Z'\setminus \{o'\}$
and $X'\setminus \tau^{-1}\left(\operatorname{Sing}(X)\right)$ freely.
We distinguish two cases (cf.~\cite[(1.12)]{Mori:flip}):
\begin{scase}\label{case-prim-1}
{\bf Case: $C'$ is irreducible.}
Then $G={\operatorname{m}}umu_d$ has exactly two fixed points $P_1'$ and $P_2'$ on $C'\simeq {\operatorname{m}}athbb{P}^1$.
They give us two points $P_i:=\tau(P_i')$ on $C$ whose indices are divisible by $d$.
The germ $(X,C)$ is locally primitive along $C$.
\operatorname{e}nd{scase}
\begin{scase}\label{top:imprim}
{\bf Case: $C'=\cup_{i=1}^s C_i'$, where $s>1$ and $C'_i\simeq {\operatorname{m}}athbb{P}^1$.}
In this case, $G$ acts on $\{C'_1,\dots, C_s'\}$ transitively.
Since ${{p}_{\operatorname{a}}}(C')=0$, each
component $C_i'$ meets the closure of $C'\setminus C_i'$ at one point.
Therefore, in this case,
all the irreducible components
$C_i'$ pass through one point $P'$ and do not meet each other
elsewhere. In this case $(X,C)$ is imprimitive at $\tau(P')$ of splitting degree $s$
and has no other locally imprimitive points.
\operatorname{e}nd{scase}
It is worthwhile to mention in the case \xref{top:imprim} that
$\tau(P')$ is the only non-Gorenstein point of $X$
and $d=s$ (see \cite[Th.~6.7, 9.4]{Mori:flip} and \cite[\S~7]{MP:cb1}).
\begin{scorollary}[{\cite[(1.10)]{Mori:flip}}]
Let $(X,C\simeq {\operatorname{m}}athbb{P}^1)$ be an extremal curve germ.
Let $P_1,\dots, P_n$ be all the non-Gorenstein points of $X$.
The following
are equivalent:
\begin{enumerate}
\item
$D\cdot C= 1/m_1\cdots m_n$ for some $D\in\operatorname{Cl}sc(X)$,
\item
$\operatorname{Cl}sc(X)\simeq {\operatorname{m}}athbb{Z}$,
\item
$\operatorname{Cl}sc(X)$ is torsion-free,
\item
$(X, C)$ is locally primitive and ${\operatorname{g}}cd(m_i,m_j)=1$, $i{\operatorname{m}}athrm{n}eq j$.
\operatorname{e}nd{enumerate}
\operatorname{e}nd{scorollary}
\begin{proof}
Follows from Lemma \xref{lemma:cyclic} and \operatorname{e}qref{exact-Clcs}.
\operatorname{e}nd{proof}
\begin{scorollary}[cf. {\cite[Lemma~2.8]{MP:cb1}}]
\label{lemma:KC}
Let $(X,C\simeq {\operatorname{m}}athbb{P}^1)$ be an extremal curve germ. Let
$d$ be the topological index of $(X,C)$ and let $m_1,\dots,m_r$ be
indices of all the non-Gorenstein points.
Assume that $(X,C)$ is either divisorial or a ${\operatorname{m}}athbb{Q}$-conic bundle which is not
toroidal \xref{ex-toric}. Then
\begin{equation}
\label{eq:KC}
-K_X\cdot C=d/m_1\cdots m_r.
\operatorname{e}nd{equation}
\operatorname{e}nd{scorollary}
\begin{proof}
It follows from \operatorname{e}qref{exact-Clcs} that for the ample generator $D$ of
the group $\operatorname{Cl}sc(X)/{\operatorname{e}quiv}$
one has $D\cdot C=d/m_1\cdots m_r$. Write $-K_X\operatorname{e}quiv a D$ for some $a\in {\operatorname{m}}athbb{Z}$.
Intersecting $D$ and $K_X$ with a general one-dimensional fiber $L$,
we obtain $-K_X\cdot L=D\cdot L$ and $a=1$.
\operatorname{e}nd{proof}
Now we can strengthen the assertion of Lemma \xref{lemma-int-non-Gor}.
\begin{scorollary}
\label{cor:int-non-Gor}
Let $f: (X,C) \to (Z,o)$ be an extremal curve germ.
Suppose that $C$ is reducible and let $P$ be a singular
point of $C$.
If $X$ is Gorenstein at $P$, then $(Z,o)$ is smooth and $f$ is a Gorenstein conic bundle.
\operatorname{e}nd{scorollary}
\begin{proof}
By Lemma \xref{lemma-int-non-Gor} \ $f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle and
$(Z,o)$ is singular. Recall (see Lemma \ref{lemma:base}) that $(Z,o)$ is a quotient singularity.
Thus there is a finite Galois \'etale over $Z\setminus \{o\}$ cover $(Z^{\flat},o^{\flat})\to(Z,o)$
where $(Z^{\flat},o^{\flat})$ is smooth. Then we can consider the base change (see
\operatorname{e}qref{eq:base-change}). Thus $X=X^{\flat}/G$, where $G$ is a finite group acting on $X^{\flat}$
freely outside finite number of points.
Since $X$ is Gorenstein at
$P$, so is $X^{\flat}$ at all the points $P_i^{\flat}\in \theta^{-1}(P)$. Moreover, $\theta$
is \'etale over $P$ by \operatorname{e}qref{eq:Clsc}. Hence, the central curve $C^{\flat}$ is singular at
$P_i^{\flat}$. By Lemma \ref{lemma-int-non-Gor} the variety $X^{\flat}$ is Gorenstein and by Corollary
\xref{prop:Gor} the contraction $f^{\flat}: X^{\flat}\to Z^{\flat}$ is a standard Gorenstein conic
bundle. In particular, $C^{\flat}$ is a plane conic.
Thus $C^{\flat}$ has two components meeting at one point $\theta^{-1}(P)$
which must be fixed by $G$. Again by \operatorname{e}qref{eq:Clsc}
the group $G$ is trivial, a contradiction.
\operatorname{e}nd{proof}
\begin{scorollary}
\label{zam-imp-Gor-fac}
Let $f: (X,C\simeq {\operatorname{m}}athbb{P}^1) \to (Z,o)$ be an extremal curve germ.
Assume that $(X,C)$ is locally imprimitive at $P$.
If the subindex of $P$ equals $1$, then $f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle and
in the diagram \operatorname{e}qref{eq:base-cover} the contraction $f'$ is a Gorenstein conic bundle.
\operatorname{e}nd{scorollary}
${\operatorname{m}}athbb{Q}$-conic bundles which are quotients of Gorenstein conic bundles by a finite group were
described in \cite[\S~2]{P97}. It turns out that such a ${\operatorname{m}}athbb{Q}$-conic bundle
is locally imprimitive if and only if it is of type~\xref{item-main-th-impr-barm=1}.
\begin{scorollary}[cf. {\cite[Prop. 1.14]{Mori:flip}}]
\label{cor-prop-grw=2-2-points-prim}
Let $f: (X,C)\to (Z,o)$ be an extremal curve germ.
Assume that $C$ is irreducible. If $\operatorname{gr}_C^0\upomega {\operatorname{m}}athrm{n}ot\simeq
{{\operatorname{m}}athscr{O}}_C(-1)$, then $f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle and
in notation of \operatorname{e}qref{eq:base-cover} we
have $f'^{-1}(o')=C'$. If furthermore $(X,C)$ is locally primitive,
then it is toroidal \textup(see \xref{ex-toric}\textup).
\operatorname{e}nd{scorollary}
\begin{proof}
By Remark \ref{remark:grw} the contraction $f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle.
Apply the construction \operatorname{e}qref{eq:base-cover}.
Since
\[
H^1(\operatorname{gr}_{C}^0\upomega)=H^1(\operatorname{gr}_{C'}^0\upomega)^{{\operatorname{m}}umu_d},
\]
we have $H^1(\operatorname{gr}_{C'}^0\upomega){\operatorname{m}}athrm{n}eq 0$.
By Corollary~\xref{cor-gr-omega-=0} $C'=f'^{-1}(o')$. If $f$ is locally
primitive, $C'$ is irreducible (see \xref{case-prim-1}). So $C'\simeq
{\operatorname{m}}athbb{P}^1$ and $X'$ is smooth. Up to analytic isomorphism we may assume
that there exists a ${\operatorname{m}}umu_d$-equivariant
decomposition $X'\simeq Z'\times {\operatorname{m}}athbb{P}^1$. So, $f$ is toroidal \xref{ex-toric},
\cite[\S~2]{P97}.
\operatorname{e}nd{proof}
\begin{proposition}[{\cite[Lemma 9.2.3]{MP:cb1}}, {\cite[0.4.13.3]{Mori:flip}}]
\label{lem-prop-3points}
An extremal curve germ $(X,C\simeq{\operatorname{m}}athbb{P}^1)$ has at most two
non-Gorenstein points.
\operatorname{e}nd{proposition}
\begin{proof}
Assume that $P_1$, $P_2$, $P_3\in X$ are singular points of indices $m_1$,
$m_2$, $m_3 >1$.
If $(X,C)$ is locally imprimitive at some point, then
the torsion free cover $\tau: (X',C')\to(X,C)$ has the form \ref{top:imprim},
i.e. $C'$ is a union of $s$ components $C_1',\dots, C_s'$ passing through one point, say $P'$,
and $\tau$ is \'etale over $X'\setminus \{P'\}$. By Corollary \xref{zam-imp-Gor-fac}
the point $P'\in X'$ is not Gorenstein.
Thus for any component $C_i'$ the germ $(X,C_i')$ has at least three non-Gorenstein
points.
Replacing $(X,C)$ with $(X',C_i')$
we may assume that $(X,C)$ is
locally primitive, i.e. the maps $\uppi_1(C\setminus \{P_i\})\to \uppi_1(U_i\setminus \{P_i\})$
are surjective, where $U_i\subset X$ is a small neighborhood of $P_i$.
Then using Van Kampen's theorem and \operatorname{e}qref{eq:Clsc} it is easy to compute
the fundamental group of $X\setminus\{P_1,\, P_2,\, P_3\}$:
\begin{equation*}
\uppi_1(X\setminus\{P_1,\, P_2,\, P_3\})= \langle
\upsigma_1,\upsigma_2,\upsigma_3\rangle/ \{\upsigma_1^{m_1}=\upsigma_2^{m_2}=
\upsigma_3^{m_3}=\upsigma_1\upsigma_2\upsigma_3=1\}.
\operatorname{e}nd{equation*}
This group has a finite quotient group $G$ in which the images of
$\upsigma_1$, $\upsigma_2$, $\upsigma_3$ are exactly of order $m_1$, $m_2$ and
$m_3$, respectively (see, e.g., \cite{Feuer-1971}). The above quotient
defines a finite Galois cover $\tau: (X',C')\to (X,C)$ with non-abelian Galois group $G$.
This contradicts Lemma~\xref{lemma:cyclic}.
\operatorname{e}nd{proof}
\section{Local description}
\subsection{Notation}\label{notation:terminal}
Let $(X,P)$ be a threefold terminal singularity of index $m$ and
let $C \subset (X, P)$ be a smooth
curve such that $P$ has subindex $\bar m$ and splitting degree $s$
(see \xref{splitting}).
We use the notation of \xref{3terminal}. Put $C^\sharp:={{p}_{\operatorname{a}}}i^{-1}(C)$. Then $C^\sharp$ has $s$
irreducible components.
Let $(C^\dag, P^\dag)$ be the normalization
of an irreducible component $C^\sharp(i)\subset C^\sharp$, $1\le i\le s$
and let $\tau: (X',C')\to (X,C)$ be the torsion free cover (see \operatorname{e}qref{eq:base-cover}).
Then ${\operatorname{m}}umu_m$ naturally acts on $(X^\sharp, P^\sharp)$ and $(C^\sharp,
P^\sharp)$, and
so does ${\operatorname{m}}umu_{\bar m}$ on $(C',P')$. Let
\begin{equation*}
\operatorname{e}ta: {{\operatorname{m}}athscr{O}}_{X^\sharp, P^\sharp} \longrightarrow {{\operatorname{m}}athscr{O}}_{X^\dag, P^\dag}
\operatorname{e}nd{equation*}
be the natural map. Since $(X, P)$ and $(C, P)$ are normal, one has
\begin{equation*}
{{\operatorname{m}}athscr{O}}_{X,P}= \left({{\operatorname{m}}athscr{O}}_{X^\sharp, P^\sharp}\right)^{{\operatorname{m}}umu_m}\quad\text{and}
\quad
{{\operatorname{m}}athscr{O}}_{C,P}= \left({{\operatorname{m}}athscr{O}}_{C^\dag, P^\dag}\right)^{{\operatorname{m}}umu_{\bar m}}.
\operatorname{e}nd{equation*}
Since ${\operatorname{m}}umu_m$ acts freely on $X^\sharp\setminus \{P^\sharp\}$, so it does
on $C^\sharp\setminus \{P^\sharp\}$ and hence ${\operatorname{m}}umu_{\bar m}$ on
$C^\dag\setminus \{P^\dag\}$.
Hence ${{\operatorname{m}}athscr{O}}_{C^\dag, P^\dag}$ has a uniformizing parameter, say $t$,
such that $t$ is a ${\operatorname{m}}umu_{\bar m}$-semi-invariant. Let $\chi$ be a generator
of
$\operatorname{Hom}({\operatorname{m}}umu_m, {\operatorname{m}}athbb{C}^*) = {\operatorname{m}}athbb{Z}/m{\operatorname{m}}athbb{Z}$ whose restriction $\bar \chi$ to
${\operatorname{m}}umu_{\bar m}$ is the character associated to $t$. Then
${{\operatorname{m}}athscr{O}}_{C,P}={\operatorname{m}}athbb{C}\{t\}^{{\operatorname{m}}umu_{\bar m}}$.
For a semi-invariant $z{\operatorname{m}}athrm{n}eq 0$, let $C^\sharp\text{-}\operatorname{wt}(z)$(or simply $\operatorname{wt}(z)$
if there is no confusion)
be $n\in {\operatorname{m}}athbb{Z}/m{\operatorname{m}}athbb{Z}$ such that $n\chi$ is the
character associated to $z$.
For a ${\operatorname{m}}umu_{m}$-semi-invariant $z\in {{\operatorname{m}}athscr{O}}_{X^\sharp,P^\sharp}$, let
\begin{equation*}
C^\sharp\text{-}\operatorname{ord}(z):= \sup\left\{n\in {\operatorname{m}}athbb{Z}_{{\operatorname{g}}e 0} {\operatorname{m}}id \operatorname{e}ta(z)\in
t^n{\operatorname{m}}athbb{C}\{t\}\right\}.
\operatorname{e}nd{equation*}
We also write $\operatorname{ord}(z)$, if it does not
cause confusion. Let
\begin{equation*}
\operatorname{ow}(z):=(\operatorname{ord}(z), \operatorname{wt}(z)).
\operatorname{e}nd{equation*}
We define semigroups
\begin{eqnarray*}
\operatorname{ord}(C^\sharp) &:= &\left\{\operatorname{ord}(z) {\operatorname{m}}id z\in {{\operatorname{m}}athscr{O}}_{C^\sharp, P^\sharp},\ z{\operatorname{m}}athrm{n}eq
0\right\}\subset {\operatorname{m}}athbb{Z}_{>0},
\\
\operatorname{ow}(C^\sharp) &:= &\left\{\left(\operatorname{ord}(z), \operatorname{wt}(z)\right) {\operatorname{m}}id z\in
{{\operatorname{m}}athscr{O}}_{C^\sharp, P^\sharp},\ z{\operatorname{m}}athrm{n}eq 0\right\}
\subset {\operatorname{m}}athbb{Z}_{{\operatorname{g}}e 0}\times {\operatorname{m}}athbb{Z}/m{\operatorname{m}}athbb{Z}.
\operatorname{e}nd{eqnarray*}
One can show that in some coordinates $C^\sharp$ can be given
by a monomial parametrization (see \cite[Lemma~2.7]{Mori:flip} for the precise
statement).
\subsection{Notation}
\label{not-nazalo-loc}
Let $f: (X,C\simeq{\operatorname{m}}athbb{P}^1)\to (Z,o)$ be an
extremal curve germ and let $P\in C$ be a point of index $m{\operatorname{g}}e 1$.
Let $s$ and $\bar
m$ be the splitting degree and subindex, respectively.
Consider the index-one ${\operatorname{m}}umu_m$-cover ${{p}_{\operatorname{a}}}i:
(X^\sharp,P^\sharp)\to (X,P)$ and let $C^\sharp:={{p}_{\operatorname{a}}}i^{-1}(C)$. Take
normalized $\operatorname{e}ll$-coordinates $(x_1,\dots,x_4)$ and let
${{p}_{\operatorname{a}}}hi$ be an $\operatorname{e}ll$-equation of $X\supset C{\operatorname{m}}athrm{n}i P$ (see
\cite[2.6]{Mori:flip}). Put $a_i:=\operatorname{ord}(x_i)$.
Note that $a_i<\infty$ and $\operatorname{wt}(x_i)\operatorname{e}quiv a_i{\operatorname{m}}od \bar m$.
The following is the key fact in the local classification of
possible singularities of extremal curve germs.
\begin{lemma}[{\cite[3.8, 4.2]{Mori:flip}}, {\cite[\S~5]{MP:cb1}}]
\label{lemma:planar}
In the above notation, assume that $P$ is not of type~\typec{IE^\vee} below.
Then $\operatorname{ow}(C^\sharp)$ is generated by $\operatorname{ow}(x_1)$ and $\operatorname{ow}(x_2)$.
In particular, $C^\sharp$ is a planar curve.
\operatorname{e}nd{lemma}
This lemma allows to obtain a local classification of
possible singularities. We reproduce this classification below.
We start with the primitive case.
\begin{proposition}[{\cite[Prop. 4.2]{Mori:flip}}, {\cite[Prop. 5.2.1]{MP:cb1}}]
\label{prop:local-primitive}
Let $f: (X,C\simeq{\operatorname{m}}athbb{P}^1)\to (Z,o)$ be an
extremal curve germ and let $P\in C$ be a primitive point of index $m{\operatorname{g}}e 1$.
Then modulo permutations of $x_i$'s, the
semigroup $\operatorname{ord}(C^\sharp)$ is generated by $a_1$ and $a_2$. Moreover,
exactly one of the following holds:
\begin{enumerate}
\item [\typec{IA}]
$a_1+a_3\operatorname{e}quiv 0{\operatorname{m}}od m$, $a_4=m$, $m\in {\operatorname{m}}athbb{Z}_{>0} a_1+{\operatorname{m}}athbb{Z}_{>0} a_2$,
where we may still permute $x_1$ and $x_3$ if $a_2=1$,
\item [\typec{IB}]
$a_1+a_3\operatorname{e}quiv 0{\operatorname{m}}od m$,\ $a_2=m$,\ $a_1{\operatorname{g}}e 2$,
\item [\typec{IC}]
$a_1+a_2=a_3=m$,\ $a_4{\operatorname{m}}athrm{n}ot\operatorname{e}quiv a_1,\, a_2 {\operatorname{m}}od m$,\ $2\le a_1<a_2$, $m{\operatorname{g}}e 5$,
\item [\typec{IIA}]
$m=4$, $P$ is of type~\type{cAx/4}, and $\operatorname{ord}(x)=(1,1,3,2)$,
\item [\typec{IIB}]
$m=4$, $P$ is of type~\type{cAx/4}, and $\operatorname{ord}(x)=(3,2,5,5)$,
\item [\typec{III}]
$m=1$, $X=X^\sharp$, $C=C^\sharp$, and $P\in X$ is a \type{cDV} point.
\operatorname{e}nd{enumerate}
\operatorname{e}nd{proposition}
Now consider the locally imprimitive case.
\begin{proposition}[{\cite[Prop. 4.2]{Mori:flip}}, {\cite[Prop. 5.3.1]{MP:cb1}}]
\label{prop-imp-types}
Let $f: (X,C\simeq{\operatorname{m}}athbb{P}^1)\to (Z,o)$ be an
extremal curve germ and let $P\in C$ be an imprimitive point of index $m$.
Modulo permutations of
$x_i$'s and changes of $\operatorname{e}ll$-characters, the semigroup $\operatorname{ow}(C^\sharp)$ is generated by
$\operatorname{ow}(x_1)$ and $\operatorname{ow}(x_2)$ except for the
case \typec{IE^{\vee}} below. Moreover, exactly one of the following
holds:
\begin{enumerate}
\item[\typec{IA^{\vee}}]
$\bar m>1$, $\operatorname{wt}(x_1)+\operatorname{wt}(x_3)\operatorname{e}quiv 0{\operatorname{m}}od m$, $\operatorname{ow} x_4=(\bar m,0)$,
$\operatorname{ow}(C^\sharp)$ is generated by $\operatorname{ow}(x_1)$ and $\operatorname{ow}(x_2)$, and
$w_P(0){\operatorname{g}}e 1/2$.
\item[\typec{IC^{\vee}}]
$s=2$, $\bar m$ is an even integer ${\operatorname{g}}e 4$, and
\begin{equation*}
\begin{array}{cccccc}
&x_1&x_2&x_3&x_4&
\\
\operatorname{wt}&1&-1&0&\bar m+1&{\operatorname{m}}od m
\\
\operatorname{ord}&1&\bar m-1&\bar m&\bar m+1&
\operatorname{e}nd{array}
\operatorname{e}nd{equation*}
\item[\typec{II^{\vee}}]
$\bar m=s=2$, $P$ is of type~\type{cAx/4}, and
\begin{equation*}
\begin{array}{cccccc}
&x_1&x_2&x_3&x_4&
\\
\operatorname{wt}&1&3&3&2&{\operatorname{m}}od 4
\\
\operatorname{ord}&1&1&1&2&
\operatorname{e}nd{array}
\operatorname{e}nd{equation*}
\item[\typec{ID^{\vee}}]
$\bar m=1$, $s=2$, $P$ is of type~\type{cA/2} or \type{cAx/2}, and
\begin{equation*}
\begin{array}{cccccc}
&x_1&x_2&x_3&x_4&
\\
\operatorname{wt}&1&1&1&0&{\operatorname{m}}od 2
\\
\operatorname{ord}&1&1&1&1&
\operatorname{e}nd{array}
\operatorname{e}nd{equation*}
\item[\typec{IE^{\vee}}]
$\bar m=2$, $s=4$, $P$ is of type~\type{cA/8}, and
\begin{equation*}
\begin{array}{cccccc}
&x_1&x_2&x_3&x_4&
\\
\operatorname{wt}&5&1&3&0&{\operatorname{m}}od 8
\\
\operatorname{ord}&1&1&1&2&
\operatorname{e}nd{array}
\operatorname{e}nd{equation*}
\operatorname{e}nd{enumerate}
Moreover, cases \typec{ID^{\vee}} and \typec{IE^{\vee}} occurs if
and only if $f$ is
a ${\operatorname{m}}athbb{Q}$-conic bundle and $C'=f'^{-1}(o')$. In these cases, $P$ is the only
non-Gorenstein point.
\operatorname{e}nd{proposition}
Proofs are based on very careful local computations. We do not present them here. See
Example~\xref{ex:ICdual}
for sample of computations.
\begin{scorollary}
\label{corollary:gr-w}
Let $(X,C\simeq{\operatorname{m}}athbb{P}^1)$ be an extremal curve germ. Assume that $\operatorname{gr}_C^0\upomega{\operatorname{m}}athrm{n}ot\simeq {{\operatorname{m}}athscr{O}}(-1)$.
Then $(X,C)$ is a ${\operatorname{m}}athbb{Q}$-conic bundle germ which is either toroidal or the only
non-Gorenstein point of $(X,C)$ is of type~\typec{ID^\vee}.
\operatorname{e}nd{scorollary}
\begin{proof}
By Corollary \xref{cor-prop-grw=2-2-points-prim}\
$(X,C)$ is ${\operatorname{m}}athbb{Q}$-conic bundle and $C'=f'^{-1}(o')$.
Assume that it is not a toroidal.
Then again by \xref{cor-prop-grw=2-2-points-prim} it is locally imprimitive
and by the proposition above $(X,C)$ has a unique non-Gorenstein point, say $P$,
which is of type~\typec{ID^\vee} or \typec{IE^\vee}. In the case \typec{IE^\vee}
we have $-K_X\cdot C=1/2$ (see \xref{lemma:KC}). Easy computations show that $w_P(0)=1/2$
and so $\deg \operatorname{gr}_C^0\upomega=-1$ by \operatorname{e}qref{eq-grw-w}.
\operatorname{e}nd{proof}
\subsection{}
By Lemma \xref{lemma:planar} there exist monomials
$\lambda_3$ and $\lambda_4$ in $x_1$, $x_2$ such that
$x_3= \lambda_3(x_1, x_2)$ and $x_4= \lambda_4(x_1, x_2)$ on $C^\sharp$.
Then
\begin{equation*}
x_1^{sa_2}-x_2^{sa_1},{\operatorname{m}}athbin{\sim_{\scriptscriptstyle{\QQ}}}uad x_3-\lambda_3,{\operatorname{m}}athbin{\sim_{\scriptscriptstyle{\QQ}}}uad x_4-\lambda_4
\operatorname{e}nd{equation*}
generate the defining ideal $I^\sharp$ of $C^\sharp\subset {\operatorname{m}}athbb{C}^4$.
Then the equation of $X^\sharp$ can be written as follows
\begin{equation*}
{{p}_{\operatorname{a}}}hi= (x_1^{sa_2}-x_2^{sa_1}){{p}_{\operatorname{a}}}hi_2+(x_3-\lambda_3){{p}_{\operatorname{a}}}hi_3+(x_4-\lambda_4){{p}_{\operatorname{a}}}hi_4
\operatorname{e}nd{equation*}
for some semi-invariant ${{p}_{\operatorname{a}}}hi_i\in {\operatorname{m}}athbb{C}\{x_1,\dots,x_4\}$ with suitable weights.
\begin{lemma}
\label{lemma:local-eq}
Under the notation of \xref{prop:local-primitive} and \xref{prop-imp-types}, one has
\begin{enumerate}
\item \label{lemma:local-eq1}
if $P$ is of type~\typec{IC} or \typec{IC^\vee}, then
$(X^\sharp, P^\sharp)$ is smooth and $I^\sharp = (x_1^{sa_2}-x_2^{sa_1},\, x_4-\lambda_4,{{p}_{\operatorname{a}}}hi)$;
\item\label{lemma:local-eq2}
if $P$ is of type~\typec{IIB} or \typec{II^\vee}, then
$I^\sharp = (x_3-\lambda_3, x_4-\lambda_4, {{p}_{\operatorname{a}}}hi)$.
\operatorname{e}nd{enumerate}
\operatorname{e}nd{lemma}
\begin{proof}
Let us consider for example the case \typec{IC}. Then $\lambda_3$ must be $x_1x_2$,
and so
\begin{equation*}
{{p}_{\operatorname{a}}}hi= (x_1^{sa_2}-x_2^{sa_1}){{p}_{\operatorname{a}}}hi_2+(x_3-x_1x_2){{p}_{\operatorname{a}}}hi_3+(x_4-\lambda_4){{p}_{\operatorname{a}}}hi_4.
\operatorname{e}nd{equation*}
Since $P$ is
of type~\typec{IC}, one sees that $m = a_1 + a_2 > 4$, ${{p}_{\operatorname{a}}}hi_2\in (x)$, and that
${{p}_{\operatorname{a}}}hi_4,\, \lambda_4\in (x)^2$ because
$\operatorname{wt}(x_4){\operatorname{m}}athrm{n}ot\operatorname{e}quiv 0, {{p}_{\operatorname{a}}}m \operatorname{wt}(x_1), {{p}_{\operatorname{a}}}m \operatorname{wt}(x_2) {\operatorname{m}}od m$.
Since $m{\operatorname{g}}e 5$, by the classification of terminal singularities
either $x_1 x_2$ or $x_3$ must appear in
the power series expansion. Since $a_1,\, a_2 {\operatorname{g}}e 2$, this is only possible if ${{p}_{\operatorname{a}}}hi_3$
is a unit.
\operatorname{e}nd{proof}
\begin{example}
\label{ex:ICdual}
According to Lemma \xref{lemma:local-eq}\xref{lemma:local-eq1}
a point $P\in (X,C)$ of type \typec{IC^\vee} can be written as follows:
\begin{equation*}
(X,C,P)=\bigl({\operatorname{m}}athbb{C}^3_{x_1,x_2,x_4}, \{x_4=x_2^2-x_1^{2\bar{m}-2}=0\}, 0\bigr)/{\operatorname{m}}umu_{2\bar m}(1,-1,\bar{m}+1)
\operatorname{e}nd{equation*}
We have $C\simeq \{x_4=x_2-x_1^{\bar{m}-1}=0\}/{\operatorname{m}}umu_{\bar{m}}$ and a local uniformizing
parameter on $C$ is $x_1^{\bar{m}}$. Hence, ${{\operatorname{m}}athscr{O}}_{C,P}={\operatorname{m}}athbb{C}\{x_1^{\bar{m}}\}$.
Furthermore,
\begin{eqnarray*}
{{\operatorname{m}}athscr{O}}_C(mK_X)&=&{{\operatorname{m}}athscr{O}}_C(d x_1\wedge d x_2\wedge d x_4)^m,
\\
\operatorname{gr}_C^0\upomega &=& {{\operatorname{m}}athscr{O}}_C(x_1^{\bar{m}-1}d x_1\wedge d x_2\wedge d x_4),
\\
\operatorname{gr}_C^1{{\operatorname{m}}athscr{O}}&=&{{\operatorname{m}}athscr{O}}_C(x_1^{\bar{m}-1}x_4)\oplus{{\operatorname{m}}athscr{O}}_C(x_1^2(x_2^2-x_1^{2\bar{m}-2})),
\\
w_P(0)&=&(\bar{m}-1)/\bar{m},\quad i_p(1)=2.
\operatorname{e}nd{eqnarray*}
\operatorname{e}nd{example}
\section{Deformations}
In this section we discuss deformations of extremal curve germs.
It is known that a small deformation of a terminal singularity is again terminal
(see e.g. \cite[Theorem~9.1.14]{Ishii:book}). Moreover, any three-dimensional
terminal singularity admits a ${\operatorname{m}}athbb{Q}$-smoothing, i.e. a deformation to
a collection of cyclic quotient singularities \cite[6.4A]{Reid:YPG}.
To study extremal curve germs it is very convenient to deform
an original germ to more general one. For example sometimes
this procedure increases the number of singular points and
it can be used to derive a contradiction.
\begin{definition}
Let $X$ be a threefold
with at worst terminal singularities.
We say $X$ is \operatorname{e}mph{ordinary} at $P$ (or $P$ is an ordinary point) if $(X, P)$
is either a cyclic quotient singularity or an ordinary double point.
\operatorname{e}nd{definition}
First, we note that for extremal curve germs deformations are unobstructed:
\begin{proposition}[{\cite[1b.8.2]{Mori:flip}}, {\cite[11.4.2]{KM92}},
{\cite[6.1]{MP:cb1}}]
\label{prop-cor-def-f}
Let $f : (X, C) \to (Z,o)$ be an extremal curve germ
and let $P\in C$. Then
every deformation of germs $(X,P)\supset (C,P)$ can be extended to a
deformation of $(X, C)$ so that the deformation is trivial outside
some small neighborhood of $P$.
\operatorname{e}nd{proposition}
\begin{proof}
Let $P_i\in X$ be singular points. Consider the natural morphism
\begin{equation*}
\Psi: \operatorname{Def} (X) \longrightarrow {{p}_{\operatorname{a}}}rod\operatorname{Def} (X,P_i).
\operatorname{e}nd{equation*}
It is sufficient to show that $\Psi$ is smooth (in particular,
surjective). The obstruction to globalizing a deformation in
${{p}_{\operatorname{a}}}rod\operatorname{Def} (X,P_i)$ lies in $R^2 f_*T_X$. Since $f$ has only
one-dimensional fibers, $R^2 f_*T_X=0$.
Alternate, more explicit proof can be found in {\cite[1b.8.2]{Mori:flip}}.
\operatorname{e}nd{proof}
The following theorem was proved in {\cite[Th. 3.2]{MP:IA}} for $f$ divisorial;
in {\cite[(11.4)]{KM92}} for $f$ flipping; or
in {\cite[(6.2)]{MP:cb1}} for $f$ a ${\operatorname{m}}athbb{Q}$-conic bundle.
\begin{theorem}[{\cite[(6.2)]{MP:cb1}}]
\label{theorem-main-def}
Let $f: (X,C) \to (Z,o)$ be a divisorial
\textup{(}resp. flipping, ${\operatorname{m}}athbb{Q}$-conic bundle\textup{)} curve germ,
where $C$ is not necessarily irreducible.
Let $ {{{p}_{\operatorname{a}}}i}: {\operatorname{m}}athfrak{X} \to ({\operatorname{m}}athbb{C}_\lambda^1,0)$ be a flat deformation of
$X= {\operatorname{m}}athfrak{X}_0:= {{{p}_{\operatorname{a}}}i}^{-1}(0)$
over a germ $({\operatorname{m}}athbb{C}_\lambda^1,0)$ with a flat closed subspace
${\operatorname{m}}athfrak{C}\subset{\operatorname{m}}athfrak{X}$ such that $C= {\operatorname{m}}athfrak{C}_0$. Then there exists a
flat deformation ${\operatorname{m}}athfrak{Z} \to ({\operatorname{m}}athbb{C}_\lambda^1,0)$ and
a proper ${\operatorname{m}}athbb{C}_\lambda^1$-morphism ${\operatorname{m}}athfrak{f}: {\operatorname{m}}athfrak{X} \to {\operatorname{m}}athfrak{Z}$
such that $f= {\operatorname{m}}athfrak{f}_0$ and
\[
{\operatorname{m}}athfrak{f}_\lambda: ({\operatorname{m}}athfrak{X}_\lambda,
{\operatorname{m}}athfrak{f}_\lambda^{-1}(o_\lambda)_{\operatorname{red}})
\to ({\operatorname{m}}athfrak{Z}_\lambda,o_\lambda)
\]
is a divisorial
\textup{(}resp. flipping, ${\operatorname{m}}athbb{Q}$-conic bundle\textup{)} extremal curve germ
for every small $\lambda$, where
$o_\lambda:={\operatorname{m}}athfrak{f}_\lambda({\operatorname{m}}athfrak{C}_\lambda)$.
\operatorname{e}nd{theorem}
Note however that the deformations do not preserve
irreducibility of the central fiber: one can easily construct
an example of
an extremal curve germ $(X,C\simeq {\operatorname{m}}athbb{P}^1)$ whose deformation
$({\operatorname{m}}athfrak{X}_\lambda, {\operatorname{m}}athfrak{f}_\lambda^{-1}(o_\lambda)_{\operatorname{red}})$
has reducible central fiber.
In practice, we often pick up a suitable irreducible component
of ${\operatorname{m}}athfrak{f}_\lambda^{-1}(o_\lambda)_{\operatorname{red}}$ and obtain
an extremal curve germ whose central fiber is irreducible (see Remark~\xref{rem-prel-extr-nbd}).
\begin{scase}
Let $f: (X,C) \to (Z,o)$ be an extremal curve germ with a singular point $P\in
C$ of of index
$m$, and let $P_1,\dots, P_r$ be all the other singular points of $X$ on $C$.
Let $(X^\sharp, P^\sharp)\to (X,P)$ be the index one cover
and let $(X^\sharp, P^\sharp)\subset ({\operatorname{m}}athbb{C}^4_{x_1,\dots,x_4},0)$ be an equivariant embedding as in \xref{clasifiction-terminal}.
Let ${{p}_{\operatorname{a}}}hi=0$ be an equation of $X^\sharp$.
We will choose semi-invariant ${{p}_{\operatorname{a}}}si\in {\operatorname{m}}athbb{C}\{x_1,\dots,x_4\}$
with $\operatorname{wt}({{p}_{\operatorname{a}}}si) \operatorname{e}quiv \operatorname{wt}({{p}_{\operatorname{a}}}hi) {\operatorname{m}}od m$
such that
\begin{equation*}
X_{\lambda,\operatorname{e}psilon}:=\{(x_1,\dots,x_4){\operatorname{m}}id {{p}_{\operatorname{a}}}hi+\lambda{{p}_{\operatorname{a}}}si=0,\ |x_i|<\operatorname{e}psilon\}/{\operatorname{m}}umu_m
\subset {\operatorname{m}}athbb{C}^4/{\operatorname{m}}umu_m
\operatorname{e}nd{equation*}
has only terminal singularities
for $|\lambda|\ll \operatorname{e}psilon\ll 1$.
\operatorname{e}nd{scase}
\begin{sproposition}[{\cite[4.7]{Mori:flip}}]
\label{prop:localDeformations}
For suitable choice of ${{p}_{\operatorname{a}}}si$, each nearby extremal
curve germ $X_\lambda^o\supset C_\lambda\simeq {\operatorname{m}}athbb{P}^1$ contains $P,P_1, \dots
P_r$
so that $(X_{\lambda}^o, P_i) \supset (C_{\lambda}, P_i)$ is naturally
isomorphic to $(X, P_i) \supset(C, P_i)$ for all $i$ and
$X_{\lambda,\operatorname{e}psilon} \supset C_{\lambda,\operatorname{e}psilon}$
contains all
the singularities ($\in C_\lambda$) of $X_\lambda\supset C_\lambda$ other than
$P_1, \dots P_r$. All the singularities of $X_{\lambda,\operatorname{e}psilon} \supset C_{\lambda,\operatorname{e}psilon}$
are ordinary.
If $P$ is a primitive \textup(resp. an imprimitive\textup) point, then
$X_{\lambda,\operatorname{e}psilon} \supset C_{\lambda,\operatorname{e}psilon}$ is locally primitive
\textup(resp. $P$ is an imprimitive point of $X_{\lambda,\operatorname{e}psilon} \supset
C_{\lambda,\operatorname{e}psilon}$
with the same subindex and splitting degree as $X \supset C {\operatorname{m}}athrm{n}i P$\textup).
Depending on the type
of $X \supset C {\operatorname{m}}athrm{n}i P$, one has
{{p}_{\operatorname{a}}}ar{\operatorname{m}}edskip{\operatorname{m}}athrm{n}oindent
{\rm
\begin{tabularx}{1\textwidth}{l|l|l|l|X}
{\operatorname{m}}ultirow{2}{*}{type}&{\operatorname{m}}ulticolumn{4}{c}{$X_{\lambda,\operatorname{e}psilon} \supset
C_{\lambda,\operatorname{e}psilon}$}
\\\cline{2-5}
& & type & index & $w_{P}(0)$, $w_{P_i}(0)$ on $X_\lambda$
\\\hline
\typec{IA}&$P$ & \typec{IA} & $m$ & the same as for $X$
\\
\typec{IA^\vee}& $P$ & \typec{IA^\vee} & $m$ & the same as for $X$
\\
\typec{IIA} &
$P$ & \typec{IA} & $m$ &the same as for $X$
\\
\typec{II^\vee} & $P$ &\typec{IA^\vee}& $m$ &the same as for $X$
\\
\typec{IB} &
$a_1$ points & \typec{IA}& $m$ & the same as for $X$
\\
\typec{IIB}&$P$ and $Q$ & \typec{IA}&$4$ and $2$&
\\
\typec{III}&$i_P(1)$ points& \typec{III} & $1$ &$0$
\operatorname{e}nd{tabularx}}
{{p}_{\operatorname{a}}}ar{\operatorname{m}}edskip{\operatorname{m}}athrm{n}oindent
In the case \typec{III}, one can also make $X_{\lambda,\operatorname{e}psilon}$ smooth by
choosing
some other suitable ${{p}_{\operatorname{a}}}si$.
\operatorname{e}nd{sproposition}
\begin{scorollary}
Arbitrary extremal curve germ $(X,C)$ can be deformed to an extremal curve germ
$(X^{o},C^{o})$ with only ordinary points.
\operatorname{e}nd{scorollary}
\begin{scorollary}\label{corollary:flipGorenstein}
A flipping extremal curve germ $(X,C)$ has at least one non-Gorenstein point.
\operatorname{e}nd{scorollary}
\begin{proof}
Assume that $(X,C)$ has only type~\typec{III}
singular points. Applying smoothings
as in \xref{prop:localDeformations} repeatedly at type~\typec{III} points,
one obtains a flipping extremal curve germ $(X^{o},C^{o})$ such that
$X^{o}$ is smooth.
Thus by \operatorname{e}qref{eq-grO-iP1} and \xref{cor-prop-grw=2-2-points-prim}
for the normal bundle of $C^{o}$ one has
\begin{equation*}
\deg {\operatorname{m}}athbb{N}N_{C^{o}/X^{o}} = -\deg\operatorname{gr}^1_{C^{o}} {{\operatorname{m}}athscr{O}} = -1.
\operatorname{e}nd{equation*}
Hence the space of deformation of $C^{o}$ in $X^{o}$ has dimension ${\operatorname{g}}e 1$. This means
that $C^{o}$ moves inside $X^{o}$.
This contradicts our assumption that $(X,C)$ is flipping.
\operatorname{e}nd{proof}
If $f: X\to Z$ is a $K$-negative extremal divisorial contraction from a
variety $X$ with terminal ${\operatorname{m}}athbb{Q}$-factorial singularities, then
the target variety $Z$ is also terminal.
This is no longer true for divisorial extremal curve germs.
The problem is that the exceptional locus of $f$ is not necessarily
a divisor in this case (because the ${\operatorname{m}}athbb{Q}$-factoriality is not assumed).
Nevertheless we have the following.
\begin{theorem}[{\cite[Th.~3.1]{MP:IA}}]
\label{thm:div:Q-Cartier}
Let $f:(X,C)\to (Z,o)$ be a three-dimensional divisorial
extremal curve germ,
where $C$ is not necessarily irreducible,
and let $E$ be its exceptional locus.
Then the divisorial part of $E$ is a ${\operatorname{m}}athbb{Q}$-Cartier divisor.
If furthermore $C$ is irreducible, then
$E$ is ${\operatorname{m}}athbb{Q}$-Cartier and
$(Z,o)$ is a terminal singularity.
\operatorname{e}nd{theorem}
\begin{corollary}
\label{theorem-main-Q-Cartier-i}
Let $f:(X,C\simeq {\operatorname{m}}athbb{P}^1)\to (Z,o)$ be a three-dimensional birational
extremal curve germ.
Then $f$ is divisorial if and only if
$(Z,o)$ is a terminal singularity.
\operatorname{e}nd{corollary}
The proof uses deformation techniques.
\begin{scorollary}[{\cite[Th. 6.3]{Mori:flip}}, {\cite[Prop.~8.3]{MP:cb1}}]
\label{corollary:IB}
An extremal curve germ $(X,C)$ cannot have a point of type~\typec{IB}.
\operatorname{e}nd{scorollary}
\begin{proof}
Assume that $(X,C)$ has a type~\typec{IB} point $P$.
We may apply deformations to $(X,C)$ and obtain
an extremal curve germ $(X^{o},C^{o}\simeq {\operatorname{m}}athbb{P}^1)$ with only ordinary singular points which has at
least two points $P^{o}$ and $Q^{o}$ of type~\typec{IA} with the same index $m$ ($> 1$).
By Proposition \xref{lem-prop-3points} \ $P^{o}$ and $Q^{o}$ are the only
non-Gorenstein points of $X^{o}$, and $P$ is the only
non-Gorenstein point of $X$. Thus by \operatorname{e}qref{exact-Clcs} we have $\operatorname{Cl}sc(X^o)\simeq {\operatorname{m}}athbb{Z}\oplus {\operatorname{m}}athbb{Z}/m{\operatorname{m}}athbb{Z}$
and so there exists
an \'etale outside $\{P^{o},\, Q^{o}\}$ cyclic cover $(X',C')\to (X^{o},C^{o})$ of degree $m$
such that $(X',C')$ is again an extremal curve germ with $C'\simeq {\operatorname{m}}athbb{P}^1$.
By Corollary \xref{corollary:flipGorenstein} the germ $(X',C')$ and $(X^{o},C^{o})$ cannot be flipping.
Assume that $(X^{o},C^{o})$ is divisorial and let $f^o: (X^{o},C^{o}) \to (Z^{o}, o^{o})$
be the corresponding contraction. By Theorem \xref{thm:div:Q-Cartier}
the point $(Z^{o}, o^{o})$ is terminal and the construction
\operatorname{e}qref{base-change} shows that $(Z^{o}, o^{o})$ is of index $m$.
According to \cite{Kawamata:discr} the exists an exceptional divisor, say $E$,
with center $o^{o}$ whose discrepancy equals $a(E,Z^{o})=1/m$. On the other hand,
since $E$ is not $f^o$-exceptional and the contraction $f^o$ is $K$-negative, we have
$a(E,Z^{o})>a(E,Z^{o})=1/m$, a contradiction.
Finally, assume that $(X^{o},C^{o})$ is a ${\operatorname{m}}athbb{Q}$-conic bundle. Since
$(X,C)$ has exactly one non-Gorenstein point which is locally primitive, the base
$(S,o)$ is smooth. Since $(X^{o},C^{o})$ has two points of the same index $>1$,
the base $(Z^{o}, o^{o})$ is singular. By \xref{theorem-main-def} there exists a
deformation family whose general fiber is $(Z^{o}, o^{o})$ and the special fiber is $(S,o)$
(in this case a general fiber ${\operatorname{m}}athfrak{f}_\lambda^{-1}(o_\lambda)_{\operatorname{red}}$
must be irreducible).
This is impossible.
\operatorname{e}nd{proof}
\subsection{}
\label{def:existence}
Deformation arguments are also used to show the existence of extremal curve germs.
Suppose we are given a normal surface germ $(H, C)$ along a curve $C\simeq {\operatorname{m}}athbb{P}^1$
and a contraction $f_H: H\to H_Z$ such that $C$ is a fiber.
Let $P_1, \dots, P_r\in H$ be singular points. Assume also that near each point $P_i$
there exists a small one-parameter deformation $H_t^{i}$
of $H \cap U_{P_i}$, where $U_{P_i}$ is a neighborhood of $P_i$, such that
the total space $V^{i} = \cup H_t^{i}$ has terminal singularity at $P_i$.
Further, by the arguments similar to that in Proposition \xref{prop-cor-def-f} we see that the
natural morphism $\operatorname{Def} H \to{{p}_{\operatorname{a}}}rod \operatorname{Def}(H, P_i)$ is
smooth.
Hence there exists a global one-parameter deformation $H_t$ of $H$
which induces a local deformation of $H_t^{i}$ near each $P_i$.
Then we construct a threefold $X$ as a total one-parameter deformation
space $X = \cup H_t$.
This shows the existence of $X\supset C$ with $H\in |{{\operatorname{m}}athscr{O}}_X |$ and such that
$P_i\in C\cap U_{P_i}\subset U_{P_i}$ has the desired structure.
Note however
that $H$ may not be \operatorname{e}mph{general} in $|{{\operatorname{m}}athscr{O}}_X |$.)
The contraction $f : X \to Z$ exists by arguments similar to
\cite[11.4.1]{KM92} and Theorem~\xref{theorem-main-def}.
The contraction is birational (resp. ${\operatorname{m}}athbb{Q}$-conic bundle)
if $H_Z$ is a surface (resp. a curve).
\section{General member of $|-K_X|$}
\label{sect:elephant}
\subsection{}
Let $X$ be a threefold having terminal singularities only and let $D$
be an effective integral ${\operatorname{m}}athbb{Q}$-Cartier divisor on $X$. Then $D$ is
Cohen-Macaulay \cite[Cor. 5.25]{KM:book}. Therefore, we have
\begin{itemize}
\item
if $D$ has only isolated
singularities, then $D$ is normal;
\item
if $D\sim -K_X$, then $D$ is Gorenstein.
\operatorname{e}nd{itemize}
In certain situations we can say more:
\begin{theorem}[{\cite[(6.4B)]{Reid:YPG}}]
\label{thm:elephant}
Let $(X, P)$ be a three-dimensional terminal singularity. Then a general
member of $|- K_X|$ has at most a Du Val singularity at $P$.
\operatorname{e}nd{theorem}
\begin{scase}\label{local:elephant}
Depending on the types of terminal singularities, a general member
$D\in |-K_X|$ and its preimage $D^\sharp$ under the index-one cover are described
below (see {\cite[(6.4B)]{Reid:YPG}}).
{{p}_{\operatorname{a}}}ar{\operatorname{m}}edskip{\operatorname{m}}athrm{n}oindent
\begin{center}
\begin{tabularx}{1\textwidth}{c|c|c|c|c}
name & equation of $D^\sharp$&${\operatorname{m}}umu_m$-action &cover $D^\sharp\to D$& $\operatorname{aw}(X,P)$
\\\hline
\type{cA/m} & $xy+z^{k}$ &$(1,-1,0)$&\type{A_{k-1}}$\overset{m:1}\longrightarrow$ \type{A_{km-1}}&$k$
\\
\type{cAx/4} & $x^2+y^2+z^{2k-1}$ &$(1,3,2)$&\type{A_{2k-2}}$\overset{4:1}\longrightarrow$ \type{D_{2k+1}} &$2k-1$
\\
\type{cD/3} & $x^2+y^3+z^3$ &$(0,1,2)$&\type{D_4}$\overset{3:1}\longrightarrow$ \type{E_6} &$2$
\\
\type{cAx/2} &$x^2+y^2+z^{2k}$&$(0,1,1)$ & \type{A_{2k-1}}$\overset{2:1}\longrightarrow$ \type{D_{k+2}} &$2$
\\
\type{cD/2} & $x^2+y^2z+z^k$ &$(1,1,0)$ &\type{D_{k+1}}$\overset{2:1}\longrightarrow$ \type{D_{2k}} &$k$
\\
\type{cE/2} & $x^2+y^3+z^4$&$(1,0,1)$ &\type{E_6}$\overset{2:1}\longrightarrow$ \type{E_7} &$3$
\operatorname{e}nd{tabularx}
\operatorname{e}nd{center}
\operatorname{e}nd{scase}
M. Reid conjectured that an analog of \xref{thm:elephant} holds for any $K$-negative contraction of terminal threefolds (general elephant). The conjecture is very important in birational geometry.
The following theorem shows that this conjecture is true for extremal curve germs. Different parts of this theorem were proved in \cite{Mori:flip}, \cite{KM92} \cite{MP:cb1}, \cite{MP:cb3}.
\begin{theorem}
\label{thm:ge}
Let $(X,C\simeq{\operatorname{m}}athbb{P}^1)$ be an extremal curve germ. Then a general member of the linear system $|-K_X|$
is normal and has only Du Val singularities.
\operatorname{e}nd{theorem}
Note that by the inversion of adjunction \cite[\S~3]{Sh:flips}, \cite[Ch.~17]{Utah} the Du Val property of a general member $D\in |-K_X|$
is equivalent to the plt property of the pair $(X,D)$.
All the possibilities for general members of $|-K_X|$ have been classified,
see \cite{KM92} and \cite{MP:cb3}.
Below we reproduce this classification in the case where $(X,C\simeq{\operatorname{m}}athbb{P}^1)$
has only one non-Gorenstein point.
\begin{theorem}
\label{thm:ge1}
Let $f:(X,C\simeq{\operatorname{m}}athbb{P}^1)\to (Z,o)$ be an extremal curve germ.
Assume that $(X,C)$
has only one non-Gorenstein point $P$.
Let $D\in |-K_{X}|$ be a general member.
If $f$ is birational, we let $D_Z:= f(D)$ which is a general member of $|-K_Z|$.
If $f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle, we let $D_{Z} :=\operatorname{Spec}_Z f_*{{\operatorname{m}}athscr{O}}_{D}$.
Then $D$ and $D_Z$ have only
Du Val singularities and the morphism $f_D: D\to D_Z$ is birational and crepant.
Moreover, only one of the following possibilities holds.
\begin{emptytheorem}[{\cite[(2.2.1), (2.2.1${}'$)]{KM92}, \cite[(1.2.3)--(1.2.6)]{MP:cb1}}]
\label{ge:simple}
We have $D\cap C=\{P\}$ and $f_D: D\to D_Z$ is an isomorphism.
In this case,
$D$ induces a general member of $|-K_{(X,P)}|$, and
${\operatorname{D}}elta(D)$ is described by \xref{local:elephant}.
\operatorname{e}nd{emptytheorem}
\begin{emptytheorem}[{\cite[(2.2.2)]{KM92}, {\cite[1.3.1]{MP:cb3}}}]
\label{ge:IC}
$P\in (X,C)$ is of type \typec{IC},
\begin{equation*}
{\operatorname{D}}elta (D,C):{\operatorname{m}}athbin{\sim_{\scriptscriptstyle{\QQ}}}uad
\vcenter{
\xymatrix@R=-1pt{
&\circ \ar@{-}[d]
\\
{\underbrace{\circ -\cdots - \circ}_{m-3}} \ar@{-}[r]&\circ \ar@{-}[r]&\diamond,
}}
\operatorname{e}nd{equation*}
where $m$, the index of $(X,P)$, is odd and $m{\operatorname{g}}e 5$.
\operatorname{e}nd{emptytheorem}
\begin{emptytheorem}[{\cite[(2.2.2${}'$)]{KM92}, {\cite[1.3.2]{MP:cb3}}}]
\label{ge:IIB}
$P\in (X,C)$ is of type \typec{IIB},
\begin{equation*}
{\operatorname{D}}elta (D,C):{\operatorname{m}}athbin{\sim_{\scriptscriptstyle{\QQ}}}uad
\vcenter{
\xymatrix@R=8pt{
&&\circ \ar@{-}[d]
\\
\circ\ar@{-}[r]&\circ \ar@{-}[r]&\circ \ar@{-}[r]& \circ \ar@{-}[r]&\diamond.
}}
\operatorname{e}nd{equation*}
\operatorname{e}nd{emptytheorem}
\operatorname{e}nd{theorem}
In some cases of \xref{ge:simple} there are additional restrictions on the general member $D\in |-K_X|$.
For example, in the case where $f$ is birational and $(X,P)$ is of type \type{cAx/2},
the general $D\in |-K_X|$ is of type \type{D_4} \cite[4.8.5.7]{KM92}.
A lot of restrictions are imposed on imprimitive ${\operatorname{m}}athbb{Q}$-conic bundles
(see \xref{th:imprimitive}).
\subsection{}
Let us outline the main ideas of the proof in the case where $(X,C)$ has
only one non-Gorenstein point. Thus, let $(X,C)$ be an extremal curve germ
with a unique non-Gorenstein point $P$.
\begin{slemma}[see {\cite[Theorem~7.3]{Mori:flip}},
{\cite[\S~7, 8.6.1]{MP:cb1}}]
\label{lemma-ge}
In the notation of \xref{thm:ge1}
and with the symbols in
Propositions \xref{prop:local-primitive} for primitive points and
\xref{prop-imp-types} for imprimitive points,
we have.
\begin{enumerate}
\item
\label{lemma-ge1}
If $P\in (X,C)$ is of type \typec{IA}, \typec{IIA}, \typec{IA^\vee}, \typec{IIA^\vee},
\typec{ID^\vee}, or \typec{IE^\vee}, then for a general member $D\in |-K_X|$
we have $D\cap C=\{P\}$.
\item
\label{lemma-ge2}
If $P\in (X,C)$ is of type \typec{IC} or \typec{IIB}, then for a general member $S\in |-2K_X|$
we have $S\cap C=\{P\}$. Moreover, the pair $(X,\frac12 S)$ is klt.
\operatorname{e}nd{enumerate}
\operatorname{e}nd{slemma}
\begin{proof}[Sketch of the proof]
Consider the case where $P$ is of type \typec{IA}.
Take ${{p}_{\operatorname{a}}}si:=x_2+{{p}_{\operatorname{a}}}si_\bullet$,
where ${{p}_{\operatorname{a}}}si_\bullet\in {\operatorname{m}}athbb{C}\{x_1,\dots,x_4\}$ is a sufficiently general semi-invariant
with $\operatorname{wt}({{p}_{\operatorname{a}}}si_\bullet)\operatorname{e}quiv \operatorname{wt}(x_2)$ and let $D:=\{{{p}_{\operatorname{a}}}si=0\}/{\operatorname{m}}umu_m$.
Then $D\cap C=\{P\}$ and $\operatorname{wt}({{p}_{\operatorname{a}}}si)\operatorname{e}quiv \operatorname{wt}(\varOmega)$, where
$\varOmega={\operatorname{Res}}({{p}_{\operatorname{a}}}hi^{-1}d x_1\wedge\dots \wedge d x_4)$.
Therefore, ${{p}_{\operatorname{a}}}si\varOmega^{-1} \in {{\operatorname{m}}athscr{O}}_X(-K_X)$ in a neighborhood of $P$.
Since $P$ is the only non-Gorenstein point,
this implies that $K_X+D$ is
Cartier globally by \operatorname{e}qref{exact-Clcs}.
On the other hand, $D\cdot C=\frac 1m \operatorname{ord}{{{p}_{\operatorname{a}}}si}=\frac {a_2}m <1$
and $-K_X\cdot C<1$ (see \operatorname{e}qref{eq-grO-iP1} and \xref{corollary:gr-w}). Therefore, $K_X+D\operatorname{e}quiv 0$.
Then Corollary \xref{cor-C-pa=0}\xref{cor-C-pa=0c} implies $K_X+D\sim 0$.
Other cases are treated similarly.
\operatorname{e}nd{proof}
Thus in the cases of \xref{lemma-ge}\xref{lemma-ge1} we are done.
Cases \typec{IC} and \typec{IIB} are much more delicate.
Rough idea of proof in these cases is to use surjectivity
of the restriction map
\begin{equation*}
H^0(X,{{\operatorname{m}}athscr{O}}_X(-K_X)) \longrightarrow H^0(S,{{\operatorname{m}}athscr{O}}_S(-K_X))
\operatorname{e}nd{equation*}
and extend a ``good'' member of $|-K_X|\bigr|_S$ to $X$.
\subsection{}
Kawamata \cite{Kaw:Crep} had shown that
Theorem \xref{thm:ge} in the flipping case is a sufficient condition for the existence of flips.
Indeed,
applying a Bertini type arguments (see \cite[Corollary~2.33]{KM:book})
one can show that, for a general member $S\in |-2K_X|$, the pair $(X,\frac 12 S)$ is klt.
Consider the double cover $(X^\flat,C^\flat)\to (X,C)$ branched over $S$.
Then $X^\flat$ has only canonical singularities (see \cite[5.20]{KM:book}, \cite[7.2]{Mori:flip})
and admits a flopping contraction of $C^\flat$.
Then the existence of flip for $(X,C)$ follows from the existence of flop for
$(X^\flat,C^\flat)$.
\subsection{}
As a corollary of Theorem \xref{thm:ge} we have the following fact which was conjectured by
V. Iskovskikh \cite{I96}.
\begin{scorollary}[{\cite{P97}}, {\cite{MP:cb1}}]
\label{base}
Let $f:(X,C)\to (Z,o)$ be a ${\operatorname{m}}athbb{Q}$-conic bundle germ.
Then $(Z,o)$ is either smooth or a Du Val singularity of type \type{A}.
\operatorname{e}nd{scorollary}
The corollary has important applications in birational geometry of conic bundles
(see \cite{I96}, \cite{P17}).
\section{Index two germs}
\label{sect:index2}
In this section we discuss extremal curve germs
having index two points only. The methods are different from those used in other
sections. Throughout this section we do not assume that the central
curve of an extremal curve germ is irreducible.
\begin{proposition}[{\cite[4.6]{KM92}}]
\label{index:2}
Let $(X,C)$ be an extremal curve germ of index two.
If $(X,C)$ is a ${\operatorname{m}}athbb{Q}$-conic bundle germ, then we assume that
the base surface is smooth. Then we have the following.
\begin{enumerate}
\item \label{index:2a}
If $P$ is a point of index two, then $P$ is the only non-Gorenstein point, all the components of $C$ pass through $P$ and they do not
meet each other elsewhere.
\item \label{index:2b}
Each germ $(X,C_i)$ is of type \typec{IA} at $P$.
\item \label{index:2c}
A general member $F\in |-K_X|$ satisfies $F\cap
C=\{P\}$ and has only Du Val singularity at $P$.
\operatorname{e}nd{enumerate}
\operatorname{e}nd{proposition}
\begin{proof}
By Lemma \xref{lemma-int-non-Gor} it is sufficient to show that every irreducible component of $C$ has at most one non-Gorenstein point. Assume the converse: a component $C_i\subset C$ contains two points $P$ and $Q$ of index two. If $\operatorname{gr}_{C_i}^0{\operatorname{m}}athrm{n}ot \simeq {{\operatorname{m}}athscr{O}}_C(-1)$, then $(X,C_i)$ is a ${\operatorname{m}}athbb{Q}$-conic bundle germ by Corollary~\xref{cor-gr-omega-=0}\xref{cor-gr-omega-=0a}. In this case, $C=C_i$ and $(X,C_i)$ is primitive (because the base is smooth). This contradicts Corollary~\xref{cor-prop-grw=2-2-points-prim}. Thus $\operatorname{gr}_{C_i}^0\simeq {{\operatorname{m}}athscr{O}}_C(-1)$. Since the numbers $K_X\cdot C_i$, $w_P(0)$, and $w_Q(0)$ are strictly positive and contained in $\frac12 {\operatorname{m}}athbb{Z}$, we get a contradiction by \operatorname{e}qref{eq-grw-w}.
\xref{index:2b} follows from
\xref{prop:local-primitive} (the case \typec{IB} is excluded
by Corollary~\xref{corollary:IB}).
\xref{index:2c} is proved as in Sect. \xref{sect:elephant}.
\operatorname{e}nd{proof}
First, we consider the birational case following \cite[\S 4]{KM92}.
\begin{theorem}[{\cite[4.7]{KM92}}]
\label{thm:index2}
Let $(X,C)$ be a birational extremal curve germ of index two. Let $P\in X$ be a non-Gorenstein point. Then a general member $H\in |{{\operatorname{m}}athscr{O}}_X|$ is normal and has only rational singularities. The following are the only possibilities for the dual graph ${\operatorname{D}}elta(H,C)$, where $\overset{3}\circ \text{---}\underbrace{\circ\text{---}\cdots\text{---}\circ}_{n-2}\text{---}\overset{3}\circ$ should be replaced with $\overset{4}\circ$ if $n=1$.
The following is the only flipping case.
\begin{emptytheorem}
\label{index2flipping}
Then $C\simeq{\operatorname{m}}athbb{P}^1$, the singularity $(X,P)$ is of type \type{cA/2}, $(H_Z,o)$ is of type $\frac1{2n+1}(1, 2n-1)$,
and
\begin{equation*}
{\operatorname{D}}elta(H,C):{\operatorname{m}}athbin{\sim_{\scriptscriptstyle{\QQ}}}uad
\bullet\text{---}\overset{3}\circ \text{---}\underbrace{\circ\text{---}\cdots\text{---}\circ}_{n-2} \text{---}\overset{3}\circ
\operatorname{e}nd{equation*}
\operatorname{e}nd{emptytheorem}
In the remaining cases $(X,C)$ is divisorial.
Then $(Z,o)$ is a \type{cDV} point and $(H_Z,o)$ is a Du Val singularity.
If we say that $(H_Z,o)$ is of type \type{A_0}, this means that it is smooth.
\begin{longtable}{l|p{43pt}|l|c}
No. & $(X,P)$ & $(H_Z,o)$ &{\operatorname{m}}ulticolumn{1}{c}{${\operatorname{D}}elta(H,C)$}
\\\hline
\operatorname{e}ndfirsthead
No. & $(X,P)$ & $(H_Z,o)$ &{\operatorname{m}}ulticolumn{1}{c}{${\operatorname{D}}elta(H,C)$}
\\{{p}_{\operatorname{a}}}agebreak\hline{{p}_{\operatorname{a}}}agebreak
\operatorname{e}ndhead
{\operatorname{m}}ulticolumn{4}{c}{$C$ has one component}
\\\hline
{\operatorname{m}}athrm{n}om
\label{KM:4.7.3.1.1} & \type{cA/2} & \type{A_1}
&$
\circ\text{---}\bullet\text{---}\overset{3}\circ \text{---}\underbrace{\circ\text{---}\cdots\text{---}\circ}_{n-2}\text{---}\overset{3}\circ
$
\\
{\operatorname{m}}athrm{n}om
\label{KM:4.7.3.1.2}& \type{cA/2}& \type{A_0}
&$
\circ\text{---}\circ\text{---}\bullet\text{---}\overset{4}\circ
$
\\ {\operatorname{m}}athrm{n}om
\label{KM:4.7.3.1.3}& \type{cA/2} & \type{A_2}
&$
\vcenter{
\xymatrix@R=3pt@C=16pt{
\overset{3}\circ\ar@{-}[r]&\circ\ar@{-}[r]\ar@{-}[d]&\overset{3}\circ
\\
&\bullet&
}}
$
\\
{\operatorname{m}}athrm{n}om
\label{KM:4.7.3.1.4} & \type{cA/2}& \type{A_0}
&$
\vcenter{
\xymatrix@R=3pt@C=16pt{
\overset{3}\circ\ar@{-}[r]&\circ\ar@{-}[r]\ar@{-}[d]&\circ\ar@{-}[r]&\overset{3}\circ
\\
&\bullet&
}}
$
\\\hline
{\operatorname{m}}ulticolumn{4}{c}{$C$ has two components}
\\ \hline
{\operatorname{m}}athrm{n}om \label{KM:4.7.3.2.1}& \type{cA/2}& \type{A_m}
&$
\bullet\text{---}\overset{3}\circ \text{---}\underbrace{\circ\text{---}\cdots\text{---}\circ}_{n-2}\text{---}\overset{3}\circ\text{---}\bullet
$
\\ {\operatorname{m}}athrm{n}om
\label{KM:4.7.3.2.2}& \type{cA/2}& \type{A_0}
&$
\circ\text{---}\bullet\text{---}\overset{3}\circ \text{---}\underbrace{\circ\text{---}\cdots\text{---}\circ}_{n-2}\text{---}\overset{3}\circ\text{---}\bullet
$
\\ {\operatorname{m}}athrm{n}om
\label{KM:4.7.3.2.3}& \type{cA/2}& \type{A_1}
&$
\vcenter{
\xymatrix
@R=-7pt@C=13pt
{
\bullet\ar@{-}[dr]
\\
&\overset{3}\circ\ar@{-}[r] &
{\underbrace{\circ\text{---}\cdots\text{---}\circ}_{n-2}} &\overset{3}\circ\ar@{-}[l]
\\
\bullet\ar@{-}[ur]
}}
$
\\ {\operatorname{m}}athrm{n}om
\label{KM:4.7.3.2.4}& \type{cA/2}& \type{A_0}
&$
\vcenter{
\xymatrix@R=3pt@C=16pt{
\overset{3}\circ\ar@{-}[r]&\circ\ar@{-}[r]\ar@{-}[d]&\overset{3}\circ\ar@{-}[r]&\bullet
\\
&\bullet&
}}
$
\\\hline
{\operatorname{m}}ulticolumn{4}{c}{$C$ has three components}
\\ \hline
{\operatorname{m}}athrm{n}om \label{KM:4.7.3.3.1}& \type{cA/2}& \type{A_0}
&$
\vcenter{
\xymatrix
@R=-7pt@C=13pt
{
\bullet\ar@{-}[dr]
\\
&\overset{3}\circ\ar@{-}[r] &{\underbrace{\circ\text{---}\cdots\text{---}\circ}_{n-2}}
&\overset{3}\circ\ar@{-}[l]&\bullet\ar@{-}[l]
\\
\bullet\ar@{-}[ur]
}}
$
\\\hline
{\operatorname{m}}ulticolumn{4}{c}{$C$ has one component}
\\\hline
{\operatorname{m}}athrm{n}om \label{KM:4.7.4} & \type{cAx/2}, \type{cD/2} or \type{cE/2}
& \type{D_4}
&$
\vcenter{
\xymatrix@R=10pt@C=16pt{
\circ\ar@{-}[r]&\overset{3}\circ\ar@{-}[r]&\circ\ar@{-}[r]&\bullet
\\
\circ\ar@{-}[ru]&&\circ\ar@{-}[lu]
}}
$
\\ {\operatorname{m}}athrm{n}om
\label{KM:4.7.5} & \type{cD/2} or \type{cE/2}& \type{D_{n+4}}
&$
\vcenter{
\xymatrix
@R=-7pt@C=13pt
{
\circ\ar@{-}[dr]&&&&\circ\ar@{-}[dl]
\\
&\circ\ar@{-}[r]
&{\underbrace{\circ\text{---}\cdots\text{---}\circ}_{n-1}}
&\overset 3 \circ\ar@{-}[l]
\\
\circ\ar@{-}[ur]&&&&\circ\ar@{-}[ul]&\bullet\ar@{-}[l]
}}
$
\\
&{\operatorname{m}}ulticolumn{3}{c}{where $n{\operatorname{g}}e 1$ and $n=1$ if $(X,P)$ is of type \type{cE/2}}
\\ {\operatorname{m}}athrm{n}om
\label{KM:4.7.6} & \type{cE/2}& \type{E_6}
&$
\vcenter{
\xymatrix@R=6pt@C=16pt{
\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ\ar@{-}[r]\ar@{-}[d]&\circ\ar@{-}[r]&\circ
\\
\bullet\ar@{-}[r]&\circ\ar@{-}[r]&\underset{3}\circ&
}}
$
\operatorname{e}nd{longtable}
\operatorname{e}nd{theorem}
Note that the singularities of $H$ are log terminal
in all the cases except for \xref{KM:4.7.4}, \xref{KM:4.7.5}, \xref{KM:4.7.6}.
In the cases \xref{KM:4.7.4} and \xref{KM:4.7.5} the singularities of $H$ are log canonical.
\begin{stheorem}[{\cite[4.2]{KM92}}]
\label{flip-index=2}
In the notation of Theorem \xref{thm:index2} assume that $f$ is flipping
\textup(see \xref{index2flipping}\textup)
and let $(X,C)\dashrightarrow (X^+,C^+)$ be the corresponding flip.
Then the following hold.
\begin{enumerate}
\item
\label{KM(4.2.1)} In appropriate
coordinates the point $(X{\operatorname{m}}athrm{n}i P)$
is given by
\begin{equation*}
\left\{x_1x_2 +p(x_3^2, x_4) = 0\right\}/{\operatorname{m}}umu_2(1, 1,1, 0)
\operatorname{e}nd{equation*}
and $C$ is the $x_1$-axis.
\item
\label{KM(4.2.2)}
$X^+$ has at most one singular point which is isolated \type{cDV} with equation
$x_1x_2 +p(x_3, x_4) = 0$
and $C^+$ is the $x_1$-axis.
\item
\label{KM(4.2.3)}
$(Z,o)$ is a rational triple point given by the $2\times 2$-minors
of the matrix
\begin{equation*}
\begin{pmatrix}
z_1& z_2& z_3
\\
z_2& z_5 &p(z_1,z_4)
\operatorname{e}nd{pmatrix}
\operatorname{e}nd{equation*}
\operatorname{e}nd{enumerate}
\operatorname{e}nd{stheorem}
The proofs use the following standard construction.
\begin{sconstruction}
\label{index2:constr1}
Let $C_i\subset C$ be the
irreducible
components of $C$. Since $X$ has only points of index one and two,
$m_i = -2K_X\cdot C_i$ is a positive integer. Let $E_i\subset X$ be the union of $m_i$ disjoint
discs transversal
to $C_i$ and let $E = \sum E_i$. Then $E \in | - 2K_X|$. Hence we can
take the corresponding
double cover $X'\to X$ branched over $E$. Here $X'$ has only
index one terminal singularities. Let $E' \subset X'$ be the preimage of $E$. The
natural map $E'\to E$ is an isomorphism.
The Stein factorization induces the following diagram
\begin{equation*}
\xymatrix{
E\subset X\supset C\ar[d]^{f}&E'\subset X'\supset C'\ar[l]\ar[d]^{f'}
\\
D\subset Z{\operatorname{m}}athrm{n}i o& D'\subset Z'{\operatorname{m}}athrm{n}i o'\ar[l]
}
\operatorname{e}nd{equation*}
where $D:= f(E)$ and $D':= f(E')$. Here $Z'\to Z$ is a double
cover branched over $D$.
By construction,
$f'$ is crepant with respect to $K_{X'}$ and the fibers of $f'$ have dimension $\le 1$.
Therefore, $Z'$ has
\type{cDV} points only (if $f$ is divisorial, then $Z'$ has
a double curve).
\operatorname{e}nd{sconstruction}
\begin{proof}[Sketch of the proof of \xref{thm:index2} and \xref{flip-index=2}.]
The above construction defines a ${\operatorname{m}}umu_2$-action on $X'/Z'$ and the
quotient is $X/Z$. The fixed point set of the action on $Z'$ is precisely
$D'$. Since $Z'{\operatorname{m}}athrm{n}i o'$ is a \type{cDV} point, it is a hypersurface in ${\operatorname{m}}athbb{C}^4$, thus it
can be written down explicitly. This will enable us to get equations for $X$ and
$Z$. We have an ${\operatorname{m}}umu_2$-equivariant embedding $(Z',o')\subset ({\operatorname{m}}athbb{C}^4_{y_1,\dots, y_4},0)$,
and we may assume that the coordinates are eigenvectors and $y_1,\dots,y_j$
are those of weights $1$. Thus $D'=\{y_1 =\cdots=y_j = 0\}\cap Z'$. Hence $j = 1$ or $2$.
\begin{sclaim}
\begin{enumerate}
\item
If $D'$ is Cartier, then $f$ is divisorial and $D$ is singular
along $f(E)$, where $E$ is the $f$-exceptional divisor.
\item
If $D'$ is not Cartier, then $f$ is flipping, $D$ is smooth
and $C$ is irreducible.
\operatorname{e}nd{enumerate}
\operatorname{e}nd{sclaim}
\begin{proof}
If $j = 1$ then $D'$ is Cartier. In this case, $f$ must be divisorial. Indeed,
otherwise since $f'$ is an isomorphism
outside the origin and $E'$ is $f'$-ample, $D'$ cannot be Cartier.
Hence $f$ contracts an exceptional divisor $E \subset X$. Then for a general fiber $l$ of
$E$ we have $K_X\cdot l=-1$. Hence $E \cdot F = 2$. Therefore, $D$ has a double
curve along the image of $E$ and is smooth elsewhere. If $E$ is chosen
generically, then $D$ has an ordinary double curve along the image of $E$.
Assume that $j = 2$. Then $\{y_1 = y_2 = 0\}$ must be contained in $Z'$.
Furthermore, $D'$ is irreducible and this implies
that $C$ is irreducible.
Then $E \to D$ is an isomorphism outside the origin, in
fact, it turns out to be an isomorphism. In particular, $D$ is smooth.
This implies that $f$ is flipping.
\operatorname{e}nd{proof}
First consider the flipping case.
Since $\{y_1 =y_2 = 0\} \subset Z'$, the equation of $Z'$ can be written in the form
$y_1{{p}_{\operatorname{a}}}hi_1 + y_2{{p}_{\operatorname{a}}}hi_2 = 0$. If $\operatorname{wt}({{p}_{\operatorname{a}}}hi_1) = \operatorname{wt}({{p}_{\operatorname{a}}}hi_2) = 1$,
then $y_1{{p}_{\operatorname{a}}}hi_1 + y_2{{p}_{\operatorname{a}}}hi_2 \in (y_1,y_2)^2$, which
implies that $Z'$ is singular along $\{y_1 =y_2 = 0\}$. This is impossible. Thus
$\operatorname{wt}({{p}_{\operatorname{a}}}hi_1) = \operatorname{wt}({{p}_{\operatorname{a}}}hi_2) = 1$. Since $Z'$ is a double point, either ${{p}_{\operatorname{a}}}hi_1$
or ${{p}_{\operatorname{a}}}hi_2$ must contain a
linear term. Assume that ${{p}_{\operatorname{a}}}hi_1$ contains $y_j$. By wt reasons $j = 3$ or $4$.
Now we can rewrite the equation in the following form
\begin{equation*}
y_1y_3+y_2p(y_2^2,y_4)=0.
\operatorname{e}nd{equation*}
With this explicit equation we can easily compute everything. The variety $X$ is obtained by blowing up $\{y_2 = y_3 = 0\}$ and taking quotient by the group action. This gives us one singular point with the required equation. The flipped variety $X^+$ is obtained by blowing up $\{y_1 = y_2 = 0\}$ and taking quotient by the group action. To get equations for $(Z,o)$, we note that the invariants of the ${\operatorname{m}}umu_2$-action on ${\operatorname{m}}athbb{C}\{y_1,\dots, y_4\}$ are
\begin{equation*}
z_1=y_2^2,\quad
z_2=y_1y_2,\quad
z_3=y_3,\quad
z_4=y_4,\quad
z_5=y_1^2.
\operatorname{e}nd{equation*}
We get exactly the equations given by the minors of the matrix in the assertion of the theorem.
A hyperplane section given by $z_4 =cz_1$.
Now consider the divisorial case.
Then $D' \subset Z'$ is Cartier and
the ${\operatorname{m}}umu_2$-action is given by $\operatorname{wt}(y)= (0,0,0,1)$.
Let $D'$ be given by $y_4 = {{p}_{\operatorname{a}}}si(y_1,y_2,y_3) = 0$.
Thus we can
write the equation of $Z'$ in the form
\begin{equation*}
y_4^2{{p}_{\operatorname{a}}}hi(y_1,\dots,y_4)+{{p}_{\operatorname{a}}}si(y_1,y_2,y_3)=0.
\operatorname{e}nd{equation*}
Since $f'$ is crepant, $Z'$ cannot be smooth, in particular, ${\operatorname{m}}ult_0({{p}_{\operatorname{a}}}si) {\operatorname{g}}e 2$.
The equation of $Z$ is now given by
\begin{equation}
\label{index2equationZ}
t{{p}_{\operatorname{a}}}hi(y_1,y_2,y_3,t)+{{p}_{\operatorname{a}}}si(y_1,y_2,y_3)=0,\quad (t=y_4^2).
\operatorname{e}nd{equation}
In particular, this shows that $(Z,o)$ is an (isolated) \type{cDV} point
(cf. \xref{theorem-main-Q-Cartier-i}).
Now the proof proceeds by a careful analysis of the equations.
See \cite[\S~4]{KM92} for details.
\operatorname{e}nd{proof}
\subsection{}
Now we consider ${\operatorname{m}}athbb{Q}$-conic bundles.
The case of singular base surface is easy:
\begin{sproposition}[{\cite[\S~3]{P97}}, {\cite{MP:cb2}}]
A ${\operatorname{m}}athbb{Q}$-conic bundle of index two over a
singular base is either of type \xref{item-main-th-impr-barm=1} or toroidal \xref{ex-toric}.
\operatorname{e}nd{sproposition}
Index two ${\operatorname{m}}athbb{Q}$-conic bundles over a smooth base were classified in \cite[\S~3]{P97}
and \cite{MP:cb1}. Similar to birational case these are quotients of some elliptic fibrations by an
involution.
On the other hand, one can note that there exists an embedding to a
relative weighted projective space:
\begin{theorem}
\label{th-index=2}
Let $f: (X,C)\to (Z,o)$ be a ${\operatorname{m}}athbb{Q}$-conic bundle germ of index
two. Assume that $(Z,o)$ is smooth. Fix an isomorphism $(Z,o)\simeq
({\operatorname{m}}athbb{C}^2,0)$. Then there is an embedding
\begin{equation}
\label{eq-diag-last-2}
\vcenter{
\xymatrix{X \ar@{^{(}->}[r] \ar[rd]_{f}& {\operatorname{m}}athbb{P}(1,1,1,2)\times {\operatorname{m}}athbb{C}^2
\ar[d]^{p}
\\
&{\operatorname{m}}athbb{C}^2}}
\operatorname{e}nd{equation}
such that $X$ is given by two equations
\begin{equation}
\label{eq-eq-index2}
\begin{array}{l}
q_1(y_1,y_2,y_3)-{{p}_{\operatorname{a}}}si_1(y_1,\dots,y_4;u,v)=0,
\\
q_2(y_1,y_2,y_3)-{{p}_{\operatorname{a}}}si_2(y_1,\dots,y_4;u,v)=0,
\operatorname{e}nd{array}
\operatorname{e}nd{equation}
where ${{p}_{\operatorname{a}}}si_i$ and $q_i$ are weighted quadratic in $y_1,\dots,y_4$
with respect to $\operatorname{wt}(y_1,\dots,y_4)=(1,1,1,2)$ and
${{p}_{\operatorname{a}}}si_i(y_1,\dots,y_4;0,0)=0$. The only non-Gorenstein point of $X$
is $(0,0,0,1; 0,0)$. Up to projective transformations, the
following are the possibilities for $q_1$ and $q_2$:
{{p}_{\operatorname{a}}}ar{\operatorname{m}}edskip{\operatorname{m}}athrm{n}oindent
\begin{tabularx}{1\textwidth}{l|X|X|X}
\hline
{\rm no.}& $q_1$ & $q_2$ & $f^{-1}(o)$
\\[5pt]\hline
{\operatorname{m}}athrm{n}om\label{cla-index-2-4} &
$y_1^2-y_2^2$ & $y_1y_2-y_3^2$ &
$C_1+C_2+C_3+C_4$
\\
{\operatorname{m}}athrm{n}om\label{cla-index-2-22}&
$y_1y_2$ & $(y_1 +y_2)y_3$ & $2C_1+C_2+C_3$
\\
{\operatorname{m}}athrm{n}om\label{cla-index-2-1-3} &
$y_1y_2 - y_3^2$ & $y_1y_3$ & $3C_1+C_2$
\\
{\operatorname{m}}athrm{n}om\label{cla-index-2-2-2} &
$y_1^2-y_2^2$ & $y_3^2$ & $2C_1+2C_2$
\\
{\operatorname{m}}athrm{n}om\label{cla-index-4a} &
$y_1y_2 - y_3^2$ & $y_1^2$ & $4C_1$
\\
{\operatorname{m}}athrm{n}om\label{cla-index-4b} &
$y_1^2$ & $y_2^2$ & $4C_1$
\operatorname{e}nd{tabularx}
{{p}_{\operatorname{a}}}ar{\operatorname{m}}edskip{\operatorname{m}}athrm{n}oindent
Conversely, if $X\subset{\operatorname{m}}athbb{P}(1,1,1,2)\times {\operatorname{m}}athbb{C}^2$ is given by
equations of the form \operatorname{e}qref{eq-eq-index2} and singularities of $X$
are terminal, then the projection $f: (X,f^{-1}(0)_{\operatorname{red}}) \to
({\operatorname{m}}athbb{C}^2,0)$ is a ${\operatorname{m}}athbb{Q}$-conic bundle of index two.
\operatorname{e}nd{theorem}
\begin{sremark}
A general member $H\in |{{\operatorname{m}}athscr{O}}_X|$ is normal in the case \xref{cla-index-4a}
and non-normal in the case \xref{cla-index-4b}.
\operatorname{e}nd{sremark}
\begin{proof}[Sketch of the proof]
First we prove the last statement. By our assumption $X$ has only terminal
singularities. Then $X$ does not contain the surface
$\{y_1=y_2=y_3=0\} = \operatorname{Sing}({\operatorname{m}}athbb{P}\times {\operatorname{m}}athbb{C}^2)$ (otherwise both ${{p}_{\operatorname{a}}}si_1$
and ${{p}_{\operatorname{a}}}si_2$ do not depend on $y_4$). By the adjunction formula,
$K_X=-L|_X$, where $L$ is a Weil divisor on ${\operatorname{m}}athbb{P}\times {\operatorname{m}}athbb{C}^2$ such
that the restriction $L|_{{\operatorname{m}}athbb{P}}$ is ${{\operatorname{m}}athscr{O}}_{{\operatorname{m}}athbb{P}}(1)$. Therefore, $X\to
{\operatorname{m}}athbb{C}^2$ is a ${\operatorname{m}}athbb{Q}$-conic bundle. It is easy to see that the only
non-Gorenstein point of $X$ is $(0,0,0,1;0,0)$ and it is of index
two.
Now let $f: (X,C)\to (Z,o)\simeq ({\operatorname{m}}athbb{C}^2,0)$ be a ${\operatorname{m}}athbb{Q}$-conic
bundle germ of index two. Let $P\in X$ be a point of index two.
Let ${{p}_{\operatorname{a}}}i: (X^\sharp,P^\sharp)\to (X,P)$ be the
index-one cover.
We need the following lemma.
\begin{slemma}[{\cite[12.1.9]{MP:cb1}}]
\label{lemma-12-fiber-new}
Let $F^\sharp={{p}_{\operatorname{a}}}i^{-1} (F)_{\operatorname{red}}$
be the pull-back of $F$.
Let $\Gamma:=f^{-1}(o)$ be the scheme fiber and let
$\Gamma^\sharp={{p}_{\operatorname{a}}}i^{-1} (\Gamma)$. Then we have
\begin{equation*}
{{\operatorname{m}}athscr{O}}_{F^\sharp \cap \Gamma^\sharp} \simeq {\operatorname{m}}athbb{C}[x,y]/(xy,\, x^2+y^2).
\operatorname{e}nd{equation*}
Furthermore, the ${\operatorname{m}}umu_2$-action is given by $\operatorname{wt}(x,y) \operatorname{e}quiv (1,1) {\operatorname{m}}od 2$.
\operatorname{e}nd{slemma}
Using this lemma one can apply
arguments of \cite[pp. 631--633]{Mori:ci}
to get the desired embedding $X\subset {\operatorname{m}}athbb{P}(1,1,1,2)\times Z$
considering the graded anti-canonical ${{\operatorname{m}}athscr{O}}_Z$-algebra
\begin{equation*}
{\operatorname{m}}athbb{R}R:=\bigoplus_{i{\operatorname{g}}e 0}{\operatorname{m}}athbb{R}R_i,\quad{\operatorname{m}}box{where}
\quad {\operatorname{m}}athbb{R}R_i:= H^0({{\operatorname{m}}athscr{O}}_X(-iK_X)).
\operatorname{e}nd{equation*}
We sketch the main idea.
Let $w$ be a local generator of ${{\operatorname{m}}athscr{O}}_{X^\sharp} (-K_{X})$ at
$P^\sharp$, let $u$, $v$
be coordinates on $Z={\operatorname{m}}athbb{C}^2$,
and let $z=0$ be the local equation of $F^\sharp$ in
$(X^\sharp, P^\sharp)$.
Using the vanishing of $H^1({{\operatorname{m}}athscr{O}}_X(-K_X))$ for $i>0$
and the exact sequence
\begin{equation*}
0 \to {{\operatorname{m}}athscr{O}}_X(-(i-1)K_X) \to {{\operatorname{m}}athscr{O}}_X(-iK_X)\to
{{\operatorname{m}}athscr{O}}_F(-iK_X)\to 0
\operatorname{e}nd{equation*}
one can see
\begin{equation*}
{\operatorname{m}}athbb{R}R_i/(zw){\operatorname{m}}athbb{R}R_{i-1}\simeq H^0({{\operatorname{m}}athscr{O}}_F(-iK_X)), \quad i>0.
\operatorname{e}nd{equation*}
Therefore,
\begin{equation*}
{\operatorname{m}}athbb{R}R_i/(zw){\operatorname{m}}athbb{R}R_{i-1}+(u,\, v){\operatorname{m}}athbb{R}R_i
=\bigl({{\operatorname{m}}athscr{O}}_{F^\sharp\cap \Gamma^\sharp}(-iK_X)\bigr)^{{\operatorname{m}}umu_2}.
\operatorname{e}nd{equation*}
By Lemma \xref{lemma-12-fiber-new} we have an embedding
\begin{equation*}
{\operatorname{m}}athbb{R}R/(zw,\, u,\, v){\operatorname{m}}athbb{R}R
\hookrightarrow
\left({\operatorname{m}}athbb{C}[x,\, y,\, w]/(xy,\, x^2+y^2)\right)^{{\operatorname{m}}umu_2}.\operatorname{e}nd{equation*}
Using ${\operatorname{m}}athbb{R}R_0/(u,v){\operatorname{m}}athbb{R}R_0={\operatorname{m}}athbb{C}$, one can easily see that
\begin{equation*}
{\operatorname{m}}athbb{R}R/(zw,\, u,\, v){\operatorname{m}}athbb{R}R=
{\operatorname{m}}athbb{C}[y_1,y_2,y_4]/(y_1y_2,\, y_1^2+y_2^2),
\operatorname{e}nd{equation*}
where $y_1=xw$, $y_2=yw$, $y_4=w^2$.
Put $y_3:=zw$. Then similar to \cite[pp. 631--633]{Mori:ci}
we obtain
\begin{equation*}
{\operatorname{m}}athbb{R}R \simeq
{{\operatorname{m}}athscr{O}}_Z[y_1,y_2,y_3,y_4]/{{\operatorname{m}}athcal{I}}II,
\operatorname{e}nd{equation*}
where ${{\operatorname{m}}athcal{I}}II$ is generated by the following regular
sequence
\begin{equation*}
\begin{array}{ll}
y_1y_2+y_3\operatorname{e}ll_1(y_1,\dots,y_3)&+{{p}_{\operatorname{a}}}si_1(y_1,\dots,y_4;u,v),
\\[5pt]
y_1^2+y_2^2+y_3\operatorname{e}ll_2(y_1,\dots,y_3)&+{{p}_{\operatorname{a}}}si_2(y_1,\dots,y_4;u,v)
\operatorname{e}nd{array}
\operatorname{e}nd{equation*}
with ${{p}_{\operatorname{a}}}si_i(y_1,\dots,y_4;0,0)=0$.
\operatorname{e}nd{proof}
Note also that the construction \xref{index2:constr1}
in the ${\operatorname{m}}athbb{Q}$-conic bundle case produces an elliptic fibration.
It can be used for classification (see \cite[\S~3]{P97}).
\begin{sexample}
Let $X\subset{\operatorname{m}}athbb{P}(1,1,1,2)\times{\operatorname{m}}athbb{C}^2_{u,v}$
is given by the equations
\begin{eqnarray*}
y_1y_2&=&(au+bu^2+cuv)y_4,
\\
(y_1+y_2+y_3)y_3&=&vy_4,
\operatorname{e}nd{eqnarray*}
where $a, b, c \in {\operatorname{m}}athbb{C}$ are constants.
It is easy to check that the projection $X\to {\operatorname{m}}athbb{C}^2$ is a ${\operatorname{m}}athbb{Q}$-conic bundle
as in \xref{cla-index-2-4}. The only singular point
is of type \type{cA/2}.
If $a{\operatorname{m}}athrm{n}e 0$, then this point is a cyclic quotient of type
$\frac{1}{2}(1,1,1)$.
\operatorname{e}nd{sexample}
\begin{sexample}
Let $X\subset{\operatorname{m}}athbb{P}(1,1,1,2)\times{\operatorname{m}}athbb{C}^2_{u,v}$
is given by the equations
\begin{eqnarray*}
y_1^2&=&uy_3^2+vy_4
\\
y_2^2&=&uy_4+vy_3^2
\operatorname{e}nd{eqnarray*}
Then the projection $X\to {\operatorname{m}}athbb{C}^2$ is a ${\operatorname{m}}athbb{Q}$-conic bundle of type \xref{cla-index-4b}
containing
one singular point of type $\frac{1}{2}(1,1,1)$ and two ordinary double
points.
\operatorname{e}nd{sexample}
More examples are given in \cite[\S~7 and Remark~6.7.1]{MP:IA}, and \cite[\S~3]{P97}.
It can be shown \cite[\S 7]{MP:IA} that every type of terminal index two singularity
can occur on some index two ${\operatorname{m}}athbb{Q}$-conic bundle
as in~\xref{cla-index-4a} or \xref{cla-index-4b}.
\section{Locally imprimitive germs}
\label{sect:imprimitive}
In this section we collect the results concerning extremal curve germs
with a locally imprimitive point.
Note that in this case the imprimitive point is unique
and the splitting cover is locally primitive along
arbitrary irreducible component of the central curve (see
Corollary \ref{prop-cyclic-quo}). Moreover, one can show that the imprimitive point
is the only
non-Gorenstein point, see
\cite[Th.~6.7, 9.4]{Mori:flip} and \cite[\S~7]{MP:cb1}.
The following theorem summarizes the results contained in \cite{Mori:flip},
{\cite{KM92}}, {\cite{MP:cb1}}, {\cite{MP:IA}}.
\begin{theorem}
\label{th:imprimitive}
Let $f: (X, C\simeq {\operatorname{m}}athbb{P}^1)\to (Z,o)$ be an extremal curve germ such that
$(X,C)$ is locally imprimitive.
Let $P\in X$ be the imprimitive
point and let $m$, $s$ and $\bar m$ be its index, splitting degree
and subindex, respectively. In this case, $P$ is the only
non-Gorenstein point and $X$ has at most one type~\typec{III}
point. Then one of the following holds.
\begin{emptytheorem}[{\cite[1.2.3]{MP:cb1}}]
\label{item-main-th-impr-barm=2-s=4}
$f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle, $(X,C)$ is of type~\typec{IE^\vee} at $P$, $(Z,o)$
is Du Val of type~\type{A_3}, $X$ has a cyclic quotient singularity $P$
of type $\frac18(5,1,3)$ and has no other singular points.
Furthermore, $(X,C)$ is the quotient of the index-two ${\operatorname{m}}athbb{Q}$-conic
bundle germ given by the following two equations in
${\operatorname{m}}athbb{P}(1,1,1,2)_{y_1,\dots,y_4}\times {\operatorname{m}}athbb{C}^2_{u,v}$
\begin{equation*}
\label{eq-imp-exc-8-eq-a}
\begin{array}{lll}
y_1^2-y_2^2&=&u {{p}_{\operatorname{a}}}si_1(y_1,\dots,y_4;u,v)+v{{p}_{\operatorname{a}}}si_2(y_1,\dots,y_4;u,v),
\\[5pt]
y_1y_2-y_3^2&=&u {{p}_{\operatorname{a}}}si_3(y_1,\dots,y_4;u,v)+v{{p}_{\operatorname{a}}}si_4(y_1,\dots,y_4;u,v)
\operatorname{e}nd{array}
\operatorname{e}nd{equation*}
by ${\operatorname{m}}umu_{4}$-action:
\begin{equation*}
y_1{\operatorname{m}}apsto -\operatorname{i} y_1,\quad y_2{\operatorname{m}}apsto \operatorname{i} y_2,\quad y_3{\operatorname{m}}apsto - y_3,\quad
y_4{\operatorname{m}}apsto
\operatorname{i} y_4,\quad u{\operatorname{m}}apsto \operatorname{i} u,\quad v{\operatorname{m}}apsto -\operatorname{i} v
\operatorname{e}nd{equation*}
\textup(as an example one can take ${{p}_{\operatorname{a}}}si_1={{p}_{\operatorname{a}}}si_4=y_4$, ${{p}_{\operatorname{a}}}si_2={{p}_{\operatorname{a}}}si_3=0$\textup).
\operatorname{e}nd{emptytheorem}
\begin{emptytheorem}[{\cite[1.2.4]{MP:cb1}}]
\label{item-main-th-impr-barm=1}
$f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle, $(X,C)$ is of type~\typec{ID^\vee} at $P$, $(Z,o)$
is Du Val of type~\type{A_1}, $(X,C)$ is a quotient of a Gorenstein conic
bundle given by the following equation in ${\operatorname{m}}athbb{P}^2_{y_1,y_2,y_3}\times
{\operatorname{m}}athbb{C}^2_{u,v}$
\begin{equation*}
y_1^2+y_2^2+{{p}_{\operatorname{a}}}si(u,v)y_3^2=0, {\operatorname{m}}athbin{\sim_{\scriptscriptstyle{\QQ}}}uad {{p}_{\operatorname{a}}}si(u,v)\in{\operatorname{m}}athbb{C}\{u^2,\, v^2,\, uv\}
\operatorname{e}nd{equation*}
by ${\operatorname{m}}umu_{2}$-action:
\begin{equation*}
u{\operatorname{m}}apsto -u,\quad v{\operatorname{m}}apsto -v,\quad y_1{\operatorname{m}}apsto -y_1,\quad y_2{\operatorname{m}}apsto y_2,\quad
y_3{\operatorname{m}}apsto y_3.
\operatorname{e}nd{equation*}
Here ${{p}_{\operatorname{a}}}si(u,v)$ has no multiple factors. In this case, $(X,P)$ is
the only singular point and it is of type~\type{cA/2} or \type{cAx/2}.
\operatorname{e}nd{emptytheorem}
\begin{emptytheorem}[{\cite[1.2.5]{MP:cb1}}]
\label{item-main-th-impr-barm=2-s=2-cycl}
$f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle, $(X,C)$ is of type~\typec{IA^\vee} at $P$ with $\bar
m=2$, $s=2$, $(Z,o)$
is Du Val of type~\type{A_1}, $(X,P)$ is a cyclic quotient singularity
of type $\frac{1}{4}(1,1,3)$,
and $(X,C)$ is the quotient of the index-two ${\operatorname{m}}athbb{Q}$-conic bundle germ
given by the following two equations in
${\operatorname{m}}athbb{P}(1,1,1,2)_{y_1,\dots,y_4}\times {\operatorname{m}}athbb{C}^2_{u,v}$
\begin{equation*}
\begin{array}{lll}
y_1^2-y_2^2&=&u {{p}_{\operatorname{a}}}si_1(y_1,\dots,y_4;u,v)+v{{p}_{\operatorname{a}}}si_2(y_1,\dots,y_4;u,v),
\\[5pt]
y_3^2&=&u {{p}_{\operatorname{a}}}si_3(y_1,\dots,y_4;u,v)+v{{p}_{\operatorname{a}}}si_4(y_1,\dots,y_4;u,v)
\operatorname{e}nd{array}
\operatorname{e}nd{equation*}
by ${\operatorname{m}}umu_{2}$-action:
\begin{equation*}
y_1{\operatorname{m}}apsto y_1,\quad y_2{\operatorname{m}}apsto - y_2,\quad y_3{\operatorname{m}}apsto y_3,\quad y_4{\operatorname{m}}apsto -
y_4,\quad u{\operatorname{m}}apsto - u,\quad v{\operatorname{m}}apsto - v.
\operatorname{e}nd{equation*}
As an example one can take ${{p}_{\operatorname{a}}}si_1={{p}_{\operatorname{a}}}si_4=y_4$, ${{p}_{\operatorname{a}}}si_2=0$, ${{p}_{\operatorname{a}}}si_3=uy_2^2+\lambda y_1y_2$,
where $\lambda$ is a constant.
If $\lambda{\operatorname{m}}athrm{n}eq 0$, then $P$ is the only singular point. If
$\lambda=0$, then $X$
has also a type~\typec{III} point.
\operatorname{e}nd{emptytheorem}
\begin{emptytheorem}[{\cite[1.2.6]{MP:cb1}}]
\label{item-main-th-impr-barm=2-s=2-cAx/4}
$f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle, $(X,C)$ is of type~\typec{II^\vee} at $P$, $(Z,o)$
is Du Val of type~\type{A_1},
and $(X,C)$ is the quotient of the same form as in
\xref{item-main-th-impr-barm=2-s=2-cycl}.
As an example one can take ${{p}_{\operatorname{a}}}si_1=u^2y_4$, ${{p}_{\operatorname{a}}}si_2={{p}_{\operatorname{a}}}si_4=y_4$,
${{p}_{\operatorname{a}}}si_3=uy_2^2+\lambda y_1y_2$,
where $\lambda$ is a constant.
\operatorname{e}nd{emptytheorem}
\begin{emptytheorem}[{\cite[Theorem~4.11.2]{KM92}}]
\label{imprimitiveII}
$f$ is divisorial, $(X,C)$ is of type~\typec{II^\vee} at $P$, a general member
$H\in |{{\operatorname{m}}athscr{O}}_X|$ is normal.
The graph ${\operatorname{D}}elta(H,C)$ is of the form
\begin{equation*}
\xymatrix@R=3pt@C=17pt{
\circ\ar@{-}[r]&\circ\ar@{-}[r]&\cdots\ar@{-}[r]&\overset{4}\circ\ar@{-}[rr]
&&\cdots
\ar@{-}[r]&\circ\ar@{-}[r]&\circ
\\
&\circ\ar@{-}[u]&&\circ\ar@{-}[u]\ar@{-}[r]&\circ\ar@{-}[r]&\bullet&\circ\ar@{-}
[u]
}
\operatorname{e}nd{equation*}
In this case $(X,C)$ is a quotient of an index two divisorial curve germ $(\bar X,\bar C)$
by ${\operatorname{m}}umu_2$ that acts freely outside $P$ and switches two components of $\bar C$.
The point $(Z,o)$ is terminal of index two given by
\begin{equation*}
\{t{{p}_{\operatorname{a}}}hi(y_1,t)+y_3^2- y_2^2=0\}/{\operatorname{m}}umu_2(1,1,0,1)
\operatorname{e}nd{equation*}
\textup(cf. \operatorname{e}qref{index2equationZ}\textup) where the image of the exceptional divisor
is the curve $\{y_2=y_3=t=0\}/{\operatorname{m}}umu_2$.
\operatorname{e}nd{emptytheorem}
\begin{emptytheorem}[{\cite[Theorem~1.9]{MP:IA}}]
\label{imprimitiveIA}
$f$ is birational, a general member
$H\in |{{\operatorname{m}}athscr{O}}_X|$ is normal and has only log terminal singularities of
class~\type{T} \textup(see \xref{typeT} below\textup).
The graph ${\operatorname{D}}elta(H,C)$ is of the form
\begin{equation}
\label{imprimitiveIA-graph}
\vcenter{
\xymatrix@R=5pt{\overset{c_1}\circ\ar@{-}[r]&\overset{c_2}\circ\ar@{-}[r]
&\cdots\ar@{}[r]&\cdots\ar@{-}[r]&
\overset{c_r}\circ\ar@{-}[r]&\cdots\ar@{-}[r]&
\overset{c_n}\circ
\\
&\circ\ar@{-}[r]&\circ\ar@{-}[r]&\cdots\ar@{-}[r]&\bullet\ar@{-}[u]
}}
\operatorname{e}nd{equation}
Here $r{\operatorname{m}}athrm{n}eq 1,\, n$ and the chain $[c_1,\dots,c_n]$ corresponds to the non-Du
Val singularity
$(H,P)$ of class~\type{T}. The chain of $(-2)$-vertices in the bottom line
corresponds to a Du Val
point $(H,Q)$. It is possible that this chain is empty \textup(i.e., $(H,Q)$ is
smooth\textup).
The germ $(X,C)$ is of type~\typec{IA^\vee} and $C^\sharp$ explained in
\xref{splitting-degree} is reducible.
The contraction $f$ is divisorial or flipping according as $(H'_{Z'},o')$
explained in \xref{divisorial-or-flipping} is Gorenstein or not.
\operatorname{e}nd{emptytheorem}
\operatorname{e}nd{theorem}
\begin{scase}
\label{typeT}
Recall that a surface log terminal singularity $(H,P)$ is called a singularity of class \type{T}, if it
admits a one-parameter smoothing $\{H_t\}$, $H_0=H$ whose total space $X=\cup H_t$ is ${\operatorname{m}}athbb{Q}$-Gorenstein
\cite{LW86}, \cite{KSh88}.
By the inversion of adjunction
this total space must be terminal.
Any singularity of class \type{T} is either Du Val or a cyclic quotient
\begin{equation*}
\frac{1}{m^2d}(1,\, mdt-1),{\operatorname{m}}athbin{\sim_{\scriptscriptstyle{\QQ}}}uad {\operatorname{g}}cd(m,t)=1.
\operatorname{e}nd{equation*}
There is an explicit characterization of such singularities in terms of minimal resolutions,
see \cite[\S~3]{KSh88} for details.
\operatorname{e}nd{scase}
\subsection{}
The rough idea of the proof of Theorem \xref{th:imprimitive} is to apply the construction \operatorname{e}qref{eq:base-cover}.
Then $(X,C)$ can be viewed as a quotient of an extremal curve germ $(X',C')$
with reducible central fiber by ${\operatorname{m}}umu_s$.
In the case \typec{ID^\vee} we have $\bar m=1$. Hence, $(X',C')$ is a Gorenstein conic bundle
germ \xref{prop:Gor}\xref{prop:Gor-cb}. Then it is easy to write
down the action explicitly \cite[\S~2]{P97}. Similarly, in the cases \typec{IE^\vee}
and \typec{II^\vee} we have $\bar m=2$. Then $(X',C')$ is an extremal
curve germ of index two and we can apply the results of Sect. \xref{sect:index2}.
The case \typec{IC^\vee} does not occur \cite[Th.~6.1(i)]{Mori:flip}, \cite[7.3]{MP:cb1}.
\subsection{}
Consider the case \typec{IA^\vee}.
We need
the following helpful observation which allows to study a general hyperplane section
$H\in |{{\operatorname{m}}athscr{O}}_X|$. It will also be used below in the case \typec{IA}.
\begin{slemma}\label{lemma:lc}
Let $(Z,o)$ be a normal threefold singularity and let
$D_Z\in |-K_X|$ be a general member. Assume that $(D_Z,o)$
is a Du Val singularity of type \type{A}. Then for
a general hyperplane section $H_Z$, the pair $(X, H_Z+D_Z)$
is lc. In particular, $(H_Z, o)$ is a cyclic quotient singularity.
\operatorname{e}nd{slemma}
\begin{proof}
Clearly,
$H_Z\cap D_Z$ is general hyperplane section of $(D_Z,o)$ and so
$H_Z\cap D_Z=\Gamma_1+\Gamma_2$ for some irreducible curves $\Gamma_i$
such that the pair $(D_Z, \Gamma_1+\Gamma_2)$ is lc.
By the inversion of adjunction so is the pair $(Z,D_Z+H_Z)$
\cite[\S 3]{Sh:flips},
\cite{Kawakita2007}.
Hence $(H_Z,\Gamma_1+\Gamma_2)$ is lc and
$(H_Z,o)$ is a cyclic quotient singularity
(see, e.g., \cite[Ch. 3]{Utah}).
\operatorname{e}nd{proof}
\begin{sproposition}
\label{prop:lc}
Let $f: (X,C)\to (Z,o)$ be an extremal curve germ
\textup($C$ is not necessarily irreducible\textup).
Let $D\in |{-}K_X|$ and $H\in |{{\operatorname{m}}athscr{O}}_X|$ be general members.
Let $\Lambda$ be the non-normal locus of $H$ and let ${\operatorname{m}}athrm{n}u: H^{\operatorname{m}}athrm{n}\to H$ be the normalization
\textup(if $H$ is normal we put $\Lambda=\operatorname{e}mptyset$ and
${\operatorname{m}}athrm{n}u=\operatorname{id}$\textup).
Assume that $D\cap C$ is a point $P$ such that
$(D,P)$ is a Du Val singularity of type~\type{A}.
Then the pairs $(X, \, D+H)$ and $(H^{\operatorname{m}}athrm{n}, \, {\operatorname{m}}athrm{n}u^{-1}(D)+{\operatorname{m}}athrm{n}u^{-1}(\Lambda))$ are log canonical.
In particular, $H$ has only normal crossings in codimension one.
If $f$ is birational, then the pair $(Z,D_Z+H_Z)$ is also log canonical, where $D_Z=f(D)\in |{-}K_Z|$ and $H_Z:=f(H)\in |{{\operatorname{m}}athscr{O}}_Z|$.
In this case, $(H_Z,o)$ is a cyclic quotient singularity.
\operatorname{e}nd{sproposition}
\begin{proof}
First we consider the case where $f$ is birational.
Then $(D_Z,o)\simeq (D,P)$ is a Du Val singularity of type~\type{A}.
By Lemma \xref{lemma:lc} the pair $(X, H_Z+D_Z)$
is lc.
Take $H:=f^*H_Z$. Then $K_X+D+H=f^*(K_Z+D_Z+H_Z)$, i.e., the contraction
$f$ is $K_X+D+H$-crepant. Hence the pair $(X,D+H)$ is lc
and so is the pair $(H^{\operatorname{m}}athrm{n}, \, {\operatorname{m}}athrm{n}u^{-1}(D)+{\operatorname{m}}athrm{n}u^{-1}(\Lambda))$ again
by the inversion of adjunction.
Now consider the case where $Z$ is a surface.
First we claim that $(X, \, D+H)$ is lc near $D$.
Consider the restriction
$\varphi=f_D: (D,P) \to (Z,o)$.
Let $\Xi\subset Z\simeq {\operatorname{m}}athbb{C}^2$ be the branch divisor of $\varphi$.
By the Hurwitz formula we can write
$K_{D}=\varphi^*\bigl(K_Z+\frac12 \Xi\bigr)$.
Hence,
\begin{equation*}
K_{D}+H|_D=\varphi^*{\operatorname{m}}athbf{B}igl(K_Z+\frac12 \Xi+H_Z{\operatorname{m}}athbf{B}igr).
\operatorname{e}nd{equation*}
Using this and the inversion of adjunction we get the following equivalences:
$(X, \, D+H)$ is lc near $D$ $\Longleftrightarrow$
$(D, H|_D=\varphi^*H_Z)$ is lc $\Longleftrightarrow$
$(Z={\operatorname{m}}athbb{C}^2, \frac 12 \Xi+H_Z)$ is lc.
Thus it is sufficient to show that $(Z, \frac 12 \Xi+H_Z)$ is lc.
Let $\xi(u,v)=0$ be the equation of $\Xi\subset {\operatorname{m}}athbb{C}^2$.
Then $(D,P)$ is given by the equation
$w^2=\xi(u,v)$ in ${\operatorname{m}}athbb{C}^3_{u,v,w}$.
By the classification of Du Val singularities we can choose
coordinates $u$, $v$ so that
$\xi=u^2+v^{n+1}$.
Take $H_Z:=\{v-u=0\}$.
Then $\operatorname{ord}_0 \xi(u,v)|_{H_Z}=2$.
By the inversion of adjunction the pair $(Z,H_Z+\frac12 \Xi)$ is lc.
Thus we have shown that $(X,D+H)$ is lc near $D$.
Assume that $(X,D+H)$ is not lc at some point $Q\in C$.
By the above, $Q{\operatorname{m}}athrm{n}otin D$.
Note that $H$ is smooth outside $C$ by Bertini's theorem.
If $H$ is normal, then we have an immediate contradiction by
a connectedness result \cite[Th.~6.9]{Sh:flips}
applied to $(H,D|_H)$.
If $H$ is not normal, we can apply the same result on the normalization.
\operatorname{e}nd{proof}
\begin{scase}
\label{pf:IAdual:normality}
We claim that $H$ is normal.
Assume the converse, i.e. $H$ is singular along $C$.
The lemma above implies that in our situation $C$ is the minimal
log canonical center of $(X,H)$ \cite{Kaw97}.
Now let $\tau: (X',C')\to (X,C)$ be the torsion free cover \xref{torsion-free-cover}
and let $H':=\tau^*H$. Then the pair $(X',C')$ is log canonical
and $C'$ is its minimal log canonical center \cite[20.4]{Utah}.
Since the minimal log canonical center is normal \cite{Kaw97},
we conclude that $C'$ is irreducible. This contradicts imprimitivity of $(X,C)$ at $P$.
\operatorname{e}nd{scase}
\begin{scase}
\label{pf:IAdual:normal}
Thus $H$ is normal and then $P$ is the only log canonical center of the pair
$(X,H+D)$. This implies that the pair $(X,H)$ is plt.
Since $H$ is a Cartier divisor, the singularities of $H$ are of class \type{T}
\textup(see \xref{typeT}\textup).
This gives very strong restriction to the dual graph of the minimal resolution.
If $(X,C)$ is a ${\operatorname{m}}athbb{Q}$-conic bundle germ, then
using completely combinatorial techniques one can show that for
${\operatorname{D}}elta(H,C)$ there is only one possibility (cf. \cite{P04:s}):
\begin{equation*}
\xymatrix{\overset4\circ\ar@{-}[r]&\bullet\ar@{-}[r]&\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ}
\operatorname{e}nd{equation*}
But in this case the pair $(H,C)$ is plt, hence so is $(H',C')$,
whence $C'$ cannot split and this case does not occur.
In the birational case we obtain \operatorname{e}qref{imprimitiveIA-graph}.
Since $(H,C)$ cannot be plt as above, we have $r{\operatorname{m}}athrm{n}eq 1,\, n$.
This concludes the explanation of the proof of Theorem \xref{th:imprimitive}.
\operatorname{e}nd{scase}
\subsection{}\label{splitting-degree}
To decide whether an extremal curve germ $(X,C)$ is locally imprimitive at $P$,
one needs to compute the inverse image $C^\sharp$ of $C$ in the index-one cover
$(X^\sharp,P^\sharp)$ which can be computed within $H^\sharp$ the pull back of
$H$ as in \xref{ex:IAdual}, once the diagram like \operatorname{e}qref{imprimitiveIA-graph} is
exhibited. Indeed, $P$ is imprimitive if and only if the splitting degree $s>1$, which is
equal to the number of irreducible components of $C^\sharp$, see \xref{splitting}.
\subsection{}
\label{divisorial-or-flipping}
To distinguish divisorial and flipping contractions in the case \xref{imprimitiveIA}
one can use the following arguments.
Let $f':(X',C')\to (Z',o')$ be the torsion free cover \operatorname{e}qref{eq:base-cover}.
By \xref{theorem-main-Q-Cartier-i}
the germ $(X,C\simeq {\operatorname{m}}athbb{P}^1)$ is divisorial if and only if the point $(Z,o)$ is terminal
and if and only if the point $(Z',o')$ is terminal of index one (i.e. isolated \type{cDV}).
Note that in our case $(H_Z,o)$ is a cyclic quotient singularity
and so is its pull-back $(H_Z',o')$.
Hence the divisoriality of $(X,C)$ is equivalent to that $(H_Z',o')$ is
Gorenstein, that is, Du Val singularity in our case.
Once $(H,C\simeq {\operatorname{m}}athbb{P}^1)$ is given, one can find its splitting cover $(H',C')$
and so the surface germ $(H_Z',o')$ can be computed.
\begin{example}
\label{ex:IAdual}
Consider the quotient surface singularity
\begin{equation*}
(H,P)=({\operatorname{m}}athbb{C}^2_{u,v},0)/{\operatorname{m}}umu_{m^2}(1,m-1),{\operatorname{m}}athbin{\sim_{\scriptscriptstyle{\QQ}}}uad m{\operatorname{g}}e 3.
\operatorname{e}nd{equation*}
It is of class \type{T} and for its index-one cover we have
\begin{equation*}
(H^\sharp,P^\sharp)=({\operatorname{m}}athbb{C}^2,0)/{\operatorname{m}}umu_{m}(1,m-1).
\operatorname{e}nd{equation*}
Hence it is Du Val of type \type{A_{m-1}}. Consider the ${\operatorname{m}}umu_{m}$-equivariant curve
\begin{equation*}
C^\sharp=\{u^{m-2}-v^{m+2}=0\}/{\operatorname{m}}umu_{m}\subset H^\sharp
\operatorname{e}nd{equation*}
and $C=C^\sharp/{\operatorname{m}}umu_{m}$.
Then $C^\sharp$ is irreducible (resp. has two irreducible components)
if $m$ is odd (resp. $m$ is even)
and it is easy to see that $C$ is smooth.
Now consider the weighted
$\frac1{m^2}(1, m-1)$-blowup of $(H,P)$.
In the chart $v{\operatorname{m}}athrm{n}eq 0$ the origin is a Du Val point ${\operatorname{m}}athbb{C}^2/{\operatorname{m}}umu_{m-1}(-1,m^2)$
of type \type{A_{m-2}}, the exceptional divisor $\Lambda$ is $v'=0$, and the proper transform $\hat C$ of $C$
is given by $v'=u'^{m-2}$. Hence, on the minimal resolution of the
\type{A_{m-2}}-point, both $\Lambda$ and $\hat C$ meet the same end of the chain.
Therefore, the dual graph ${\operatorname{D}}elta(H,C)$ is of the form
\begin{equation*}
\xymatrix@R=0pt{
\overset{m+2}\circ\ar@{-}[r]&\circ\ar@{-}[r]&{\overbrace{\circ\text{---}\cdots\text{---}\circ}^{m-3}}
\\
&\bullet\ar@{-}[u]
}
\operatorname{e}nd{equation*}
Now suppose that $C$ is a compact curve, $C\simeq {\operatorname{m}}athbb{P}^1$ and consider a surface germ $(H,C)$
whose minimal resolution has the above form.
It is easy to see that $K_H\cdot C=-2/m$ and $C$ can be contracted to a cyclic quotient singularity $(H_Z,o)$
of type $\frac14(1,1)$.
There is a Gorenstein threefold germ $X^\sharp$ with ${\operatorname{m}}umu_{m}$-action containing $H^\sharp$
as a ${\operatorname{m}}umu_{m}$-stable hypersurface.
According to \xref{def:existence} (and \cite[\S~3]{KSh88}) the germ $(H,C)$ has a smoothing in a
${\operatorname{m}}athbb{Q}$-Gorenstein family. Thus there exists a ${\operatorname{m}}athbb{Q}$-Gorenstein threefold $X$ containing $H$
as a Cartier divisor.
By the inversion of adjunction (see \cite[\S~3]{Sh:flips}, \cite[Ch.~17]{Utah})
$X$ has only terminal singularities. By arguments similar to \xref{theorem-main-def}
we see that there exists a birational contraction $f:X\to Z$ extending $H\to H_Z$.
We note that $C^\sharp$ can be identified with the pull back of $C$ by the splitting cover of $(X,C)$ at $P$.
Now we distinguish two cases according to the parity of $m$.
a) $m$ is even. Then $(X,C)$ is imprimitive of splitting degree $2$ at $P$.
Since $(H_Z,o)$ is a type~\type{T} singularity of index $2$,
its pull-back $(H_Z',o')$ in the torsion free (degree $2$) cover \operatorname{e}qref{eq:base-cover}
is Du Val and so $(Z',o')$ is a \type{cDV} point.
Hence, both contractions $f'$ and $f$ are divisorial.
b) $m$ is odd. Then $(X,C)$ is primitive
and the contraction is flipping by \operatorname{e}qref{eq:KC}.
Note that in this case the singularity $(Z,o)$ is not
${\operatorname{m}}athbb{Q}$-Gorenstein.
On the other hand, since $(H_Z,o)$ is a singularity of class \type{T}, it has a ${\operatorname{m}}athbb{Q}$-Gorenstein
smoothing. This smoothing belongs to a component of the versal deformation space
which is different from that corresponding to $(Z,o)$ \cite[3.9]{KSh88}.
\operatorname{e}nd{example}
\section{Cases \typec{IC} and \typec{IIB}}
In this section we consider curve germs of types \typec{IC} and \typec{IIB}.
\subsection{Case \typec{IIB}}
\label{IIB-local-description}
Let $(X,P)$ be the germ of a three-dimensional terminal singularity and let $C\subset (X,P)$ be
a smooth curve. Recall that the triple $(X,C,P)$ is said to be of type~\typec{IIB} if
$(X,P)$ is a terminal singularity of type~\type{cAx/4} and
there are analytic isomorphisms
\begin{eqnarray*}
(X,P) &\simeq& \{y_1^2-y_2^3+\alpha =0\}/{\operatorname{m}}umu_4\subset {\operatorname{m}}athbb{C}^4_{y_1,\dots,y_4} /{\operatorname{m}}umu_4(3,2,1,1),
\\
C&=& \{y_1^2-y_2^3=y_3=y_4=0\}/{\operatorname{m}}umu_4,
\operatorname{e}nd{eqnarray*}
where $\alpha=\alpha(y_1,\dots,y_4)\in (y_3,\, y_4)$
is a semi-invariant with $\operatorname{wt}(\alpha)\operatorname{e}quiv 2{\operatorname{m}}od 4$ and the quadratic part
$\alpha_2$ of $\alpha(0,0,y_3,y_4)$ is not zero
(see \cite[A.3]{Mori:flip}).
We say that $(X,P)$ is a \operatorname{e}mph{simple} (resp. \operatorname{e}mph{double}) \type{cAx/4}-point if
$\operatorname{rk} \alpha_2=2$
(resp. $\operatorname{rk} \alpha_2=1$).
\begin{theorem}[{\cite{MP:ICIIB}}]
\label{main-IIB}
Let $f:(X,C\simeq{\operatorname{m}}athbb{P}^1)\to (Z,o)$ be an extremal curve germ.
Suppose that $X$ contains a point $P$ of type~\typec{IIB}.
Then $(X,C)$ is not flipping \cite[Th.~4.5]{KM92} and $P\in X$ is the unique singular point of $X$ on $C$.
Furthermore, a general member $H\in |{{\operatorname{m}}athscr{O}}_X|$ is normal, smooth outside $P$,
and has only rational singularities. The following are the only possibilities
for the dual graphs of $(H,C)$ and $H_Z:=f(H)$:
{\rm
\begin{longtable}{c|c|p{0.25\textheight}|c|c}
{\rm No.} & \type{cAx/4}-point & {\operatorname{m}}ulticolumn{1}{c|}{${\operatorname{D}}elta(H,C)$} &${\operatorname{D}}elta(H_Z,o)$
\\\hline
\operatorname{e}ndhead
{\operatorname{m}}athrm{n}om
\label{IIB:thm-A2case-simple}
&simple&
$
\xymatrix@R=0pt@C=12pt{
\overset{3}\circ\ar@{-}[r]&\overset{4}\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ
\\
&\underset{3}\circ\ar@{-}[u]&\circ\ar@{-}[r]\ar@{-}[u]&\bullet
}
$&\type{A_2}& \type{d}
\\\hline
{\operatorname{m}}athrm{n}om
\label{IIB:thm-smooth-case-simple}
&simple&
$
\xymatrix@R=0pt@C=11pt{
\overset{3}\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ\ar@{-}[r]
&\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ
\\
&&\underset3\circ\ar@{-}[r]&\underset4\circ\ar@{-}[u]&&\bullet \ar@{-}[u]
}
$&\type{A_0}& \type{d}
\\\hline
{\operatorname{m}}athrm{n}om
\label{IIB:thm-D4case-double}
&double&
$
\xymatrix@R=3pt@C=12pt{
\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ\ar@{-}[r]&\overset4\circ\ar@{-}[r]&\overset3\circ\ar@{-}[r]&\circ
\\
&\bullet\ar@{-}[r]&\circ\ar@{-}[u]&&\circ\ar@{-}[u]&\circ\ar@{-}[ul]
}
$& \type{D_4}&\type{d}
\\\hline
{\operatorname{m}}athrm{n}om
\label{IIB-theorem-conic-bundle-case-double}
&double&
{\operatorname{m}}ulticolumn{2}{l|}{$
\xymatrix@R=0pt@C=8pt{
\circ\ar@{-}[r]&\overset{3}\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ\ar@{-}[r]&\bullet
\\
&\circ\ar@{-}[u]&&&\underset{4}\circ\ar@{-}[u]
}$}
& \type{c}
\operatorname{e}nd{longtable}}
The last column indicates if the germ is divisorial \typec{d} or ${\operatorname{m}}athbb{Q}$-conic bundle \typec{c};
and the column ${\operatorname{D}}elta(H_Z,o)$ is not used in the latter case \typec{c}.
\operatorname{e}nd{theorem}
An example of divisorial contraction of type \xref{IIB:thm-A2case-simple}
is given in \cite[4.12]{KM92}.
The case \xref{IIB:thm-smooth-case-simple} was studied also by T.~Ducat
\cite[Thm. 4.1(2b)]{Ducat:16} in terms of symbolic blowups of smooth
threefolds.
\begin{proof}[Sketch of the proof]
In our case a general member $D\in |-K_X|$ contains $C$, has only Du Val singularities,
and the graph ${\operatorname{D}}elta(D,C)$ has the form \xref{ge:IIB}.
Under the identifications
of \xref{IIB-local-description},
a general member $D\in |-K_X|$ near $P$ is given by $\lambda y_3+{\operatorname{m}}u y_4=0$
for some $\lambda,\, {\operatorname{m}}u \in {{\operatorname{m}}athscr{O}}_X$
such that $\lambda(0)$, ${\operatorname{m}}u(0)$ are general in ${\operatorname{m}}athbb{C}^*$
\cite[2.11]{KM92}, \cite[\S 4]{MP:cb3}.
Let $\Gamma:=H\cap D$.
By \cite[Th.~4.5]{KM92} the contraction $f$ is not flipping.
If $f$ is divisorial, we put $D_Z:=f(D)$ and $\Gamma_Z:=f(\Gamma)$.
Then $D_Z\in |-K_Z|$, $H_Z$ is a general hyperplane section of
$(Z,o)$, and $\Gamma_Z$ is a general hyperplane section of $D_Z$.
If $f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle, we put $D_Z:=\operatorname{Spec}_{Z}f_*{{\operatorname{m}}athscr{O}}_D$
(the Stein factorization) and let $\Gamma_Z\subset D_Z$ be the image of $\Gamma$.
In both cases $D_Z$ is a Du Val singularity of type \type{E_6} by \xref{ge:IIB}.
We claim that $\Gamma=C+\Gamma_1$ \textup(as a scheme\textup), where $\Gamma_1$ is a reduced irreducible curve,
and $H$ is normal, smooth outside $P$, and has only rational singularities.
Consider two cases:
\begin{scase}\label{case-IIB-H-divisorial}
{\bf Case: $f$ is divisorial.}
Since the point $(Z,o)$ is terminal of index $1$, the germ $(H_Z,o)$ is a Du Val singularity.
Since $\Gamma_Z$ is a general hyperplane section of $D_Z$
we see that the graph ${\operatorname{D}}elta(D,\Gamma)$ has the form
\begin{equation}
\label{equation-IIB-graph-E6}
\vcenter{
\xymatrix@R=0pt{
&&\overset{2}\circ\ar@{-}[d]&\vartriangle\ar@{-}[l]
\\
\underset{1}\circ\ar@{-}[r]&\underset{2}\circ\ar@{-}[r]&\underset{3}\circ\ar@{-}[r]&\underset{2}\circ\ar@{-}[r]&\underset{1}{\diamond}
}}
\operatorname{e}nd{equation}
where $\vartriangle$ corresponds to the proper transform of $\Gamma_Z$
and numbers attached to vertices are coefficients of corresponding exceptional curves in the pull-back of $\Gamma_Z$.
By Bertini's theorem $H$ is smooth outside $C$.
Since the coefficient of $C$ equals $1$, $D\cap H=C+\Gamma_1$ (as a scheme), so $H$ is smooth
outside $P$. In particular, $H$ is normal.
Since $f_H: H\to H_Z$ is a birational contraction and $(H_Z,o)$
is a Du Val singularity, the singularities of $H$ are rational.
\operatorname{e}nd{scase}
\begin{scase}\label{case-IIB-H-conic-bundle}
{\bf Case: $f$ is a ${\operatorname{m}}athbb{Q}$-conic bundle.}
We may assume that, in a suitable coordinate system,
the germ $(D_Z, o_Z)$ is given by
$x^2+y^3+z^4=0$ and the double cover $(D_Z, o_Z) \longrightarrow (Z,o)$
is just the projection to the $(y,z)$-plane.
Then $\Gamma_Z$ is given by $z=0$.
As in the case above we see that the graph ${\operatorname{D}}elta(D,\Gamma)$ has the form
\operatorname{e}qref{equation-IIB-graph-E6}. Therefore, $H$ is smooth outside $P$.
The restriction $f_H: H\to H_Z$ is a rational curve fibration.
Hence $H$ has only rational singularities. This proves our claim.
\operatorname{e}nd{scase}
\begin{scase}
Further, $\operatorname{gr}_C^1{{\operatorname{m}}athscr{O}}\simeq {{\operatorname{m}}athscr{O}}_{{\operatorname{m}}athbb{P}^1}(d_1)\oplus {{\operatorname{m}}athscr{O}}_{{\operatorname{m}}athbb{P}^1}(d_2)$ for some
$d_1{\operatorname{g}}e d_2$. Since $H^1(\operatorname{gr}_C^1{{\operatorname{m}}athscr{O}})=0$ by Corollary \ref{cor-C-pa=0}\ref{cor-C-pa=0a},
we have $d_2{\operatorname{g}}e -1$.
Since $H$ is normal, we have $d_1{\operatorname{g}}e 0$ (see Lemma
\ref{lemma-grC}).
On the other hand,
$\deg \operatorname{gr}_C^1{{\operatorname{m}}athscr{O}}=1-i_P(1)$ by \operatorname{e}qref{eq-grO-iP1}.
One can compute $i_P(1)=2$ from \cite[(2.12)]{Mori:flip} for $P$
of type (IIB) described in \ref{IIB-local-description}.
Therefore,
\[
\operatorname{gr}_C^1{{\operatorname{m}}athscr{O}}\simeq {{\operatorname{m}}athscr{O}}_{{\operatorname{m}}athbb{P}^1}\oplus {{\operatorname{m}}athscr{O}}_{{\operatorname{m}}athbb{P}^1}(-1)
\]
and ${{\operatorname{m}}athscr{O}}_C(-H)={{\operatorname{m}}athscr{O}}\subset \operatorname{gr}_C^1{{\operatorname{m}}athscr{O}}$, i.e. the local equation of $H$
must be a generator of ${{\operatorname{m}}athscr{O}}\subset \operatorname{gr}_C^1{{\operatorname{m}}athscr{O}}$.
In the notation of \xref {IIB-local-description} the surface
$H\subset X$ is locally near $P$ given by
the equation $y_3 v_3+ y_4 v_4=0$, where $v_3,\, v_4\in {{\operatorname{m}}athscr{O}}_{P^\sharp, X^\sharp}$
are semi-invariants with $\operatorname{wt}(v_i)\operatorname{e}quiv 3$
and at least one of $v_3$ or $v_4$ contains a linear term in $y_1$.
Therefore,
the surface germ $(H,P)$ can be given in ${\operatorname{m}}athbb{C}^4/{\operatorname{m}}umu_4(3,2,1,1)$ by
two equations:
\begin{eqnarray*}
\label{equation-IIB-H}
y_1^2-y_2^3+\operatorname{e}ta(y_3,y_4)+ {{p}_{\operatorname{a}}}hi(y_1,y_2,y_3,y_4)&=&0,
\\
y_1l(y_3,y_4)+y_2q(y_3,y_4)+\xi(y_3,y_4)+{{p}_{\operatorname{a}}}si(y_1,y_2,y_3,y_4) &=&0,
\operatorname{e}nd{eqnarray*}
where $\operatorname{e}ta$, $l$, $q$ and $\xi$ are homogeneous polynomials
of degree $2$, $1$, $2$ and $4$, respectively, $\operatorname{e}ta{\operatorname{m}}athrm{n}eq 0$, $l{\operatorname{m}}athrm{n}eq 0$,
${{p}_{\operatorname{a}}}hi, \, {{p}_{\operatorname{a}}}si \in (y_3,\, y_4)$, $\sigma{\operatorname{m}}box{-}\operatorname{ord} {{p}_{\operatorname{a}}}hi{\operatorname{g}}e 3/2$, $\sigma{\operatorname{m}}box{-}\operatorname{ord} {{p}_{\operatorname{a}}}si{\operatorname{g}}e 2$.
Moreover, $\operatorname{rk} \operatorname{e}ta=2$
(resp. $\operatorname{rk} \operatorname{e}ta=1$) if $(X,P)$ is a simple (resp. double) \type{cAx/4}-point.
Then considering the weighted $\frac14(3,2,1,1)$-blowup and using
rationality of $(H,P)$, as well as, generality of $H$ in $|{{\operatorname{m}}athscr{O}}_X|$
one can obtain the possibilities in Theorem \xref{main-IIB}. See \cite[\S~3]{MP:ICIIB}
for details.\qedhere
\operatorname{e}nd{scase}
\operatorname{e}nd{proof}
\subsection{Case \typec{IC}}
Let $(X,C\simeq {\operatorname{m}}athbb{P}^1)$ be an extremal curve germ. Assume that $(X,C)$ has
a type~\typec{IC} point $P$ of index $m$. Then $P$ is the only singular point of $X$,
$m$ is odd ${\operatorname{g}}e 5$,
and $w_p (0) = (m - 1)/m$.
Moreover, $i_P(1) = a_1=2$ and
\begin{equation*}
(X,C,P)\simeq \bigl({\operatorname{m}}athbb{C}_{y_1,y_2,y_4}^3,\{y_1^{m-2}-y_2^2=y_4=0\}, 0\bigr)/{\operatorname{m}}umu_m(2, m - 2, 1).
\operatorname{e}nd{equation*}
(see Lemma \xref{lemma:local-eq}\xref{lemma:local-eq1} and \cite[5.5, 6.5, A.3]{Mori:flip}).
Below is a complete classification of extremal curve germs of type \typec{IC}:
\begin{theorem}[{\cite[\S 8]{KM92}}, {\cite{MP:ICIIB}}]
\label{IC-main}
Let $f:(X,C\simeq {\operatorname{m}}athbb{P}^1)\to (Z,o)$ be an extremal curve germ of type~\typec{IC}.
Let $P\in X$ be \textup(a unique\textup) singular point
and let $m$ be its index.
Then a general member $H\in |{{\operatorname{m}}athscr{O}}_X|$ is normal, smooth outside $P$,
has only rational singularities.
Moreover, $(X,C)$ is not divisorial and
we have one of the following:
\begin{emptytheorem}
$(X,C)$ is flipping and the following are the only possibilities
for the dual graphs of $(H,C)$ and $H_Z=f(H)$\textup:
\begin{equation*}
\xymatrix@R=-2pt@C=10pt{
&\bullet\ar@{-}[d]&\overset{(m+3)/2}\circ\ar@{-}[d]&
&\circ\ar@{-}[d]&
&&&\circ\ar@{-}[d]&
\\
\circ\ar@{-}[r]&
\circ\ar@{-}[r]&
\underset{3}\circ\ar@{-}[r]&
{\operatorname{m}}box{$\underset{(m-7)/2}{\underbrace{\circ\text{---} \cdots\text{---}\circ}}$}\ar@{-}[r]
&
\underset{3}\circ\ar@{-}[r]&
\circ
&&
\underset{4}\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ
}
\operatorname{e}nd{equation*}
\begin{equation*}
\xymatrix@R=1pt@C=10pt{
&&\circ\ar@{-}[d]&\overset{(m+3)/2}\circ\ar@{-}[l]&
&\circ\ar@{-}[d]&
&&\quad&\circ\ar@{-}[d]&
\\
\bullet\ar@{-}[r]&\circ\ar@{-}[r]&
\circ\ar@{-}[r]&
\underset{3}\circ\ar@{-}[r]&
{\operatorname{m}}box{$\underset{(m-7)/2}{\underbrace{\circ\text{---} \cdots\text{---}\circ}}$}\ar@{-}[r]
&
\underset{3}\circ\ar@{-}[r]&
\circ
&&
\underset{3}\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ
}
\operatorname{e}nd{equation*}
where
{\operatorname{m}}box{$\underset{3}\circ\text{---}\underset{(m-7)/2}{\underbrace{\circ\text{---} \cdots\text{---}\circ}}\text{---}\underset{3}\circ$}
must be replaced with $\underset {4}\circ$ in the case $m=5$.
\operatorname{e}nd{emptytheorem}
\begin{emptytheorem}
\label{IC-(8.3.2)}
$(X,C)$ is a ${\operatorname{m}}athbb{Q}$-conic bundle, $m=5$,
and ${\operatorname{D}}elta(H,C)$ has the form\textup:
\begin{equation*}
\xymatrix@R=-2pt@C=14pt{
&&&\circ\ar@{-}[d]&&\circ\ar@{-}[r]&\overset3\circ
\\
\bullet\ar@{-}[r]&\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ\ar@{-}[r]&
\underset3\circ\ar@{-}[rd]\ar@{-}[ru]
\\
&&&&&\underset3\circ\ar@{-}[r]&\circ
}
\operatorname{e}nd{equation*}
\operatorname{e}nd{emptytheorem}
\operatorname{e}nd{theorem}
In the flipping case, the general member $H^+\in |{{\operatorname{m}}athscr{O}}_{X^+}|$
of the flipped variety is also computed. Here $X^+$ is either
of index two or Gorenstein \cite[A.3]{KM92}.
\section{Case \typec{IA}}
\label{sect:IA}
\subsection{}
\label{IA:setup}
An extremal curve germ $(X, C\simeq {\operatorname{m}}athbb{P}^1)$ is said to be of type~\typec{IA} if it
contains exactly one non-Gorenstein point $P$ which is of type~\typec{IA}.
For readers' convenience, we note the following characterization
for an extremal curve germ $(X, C\simeq {\operatorname{m}}athbb{P}^1)$
to be of type~\typec{IA} with $D\in |-K_X |$ a general member (see \xref{prop:local-primitive}
and \xref{thm:ge1}):
\begin{quote}
$(X, C)$ is of type~\typec{IA} if and only if (i) $P$ is locally primitive, (ii)
$D \cap C$ is a single point, and (iii) $(X,P)$ is not of
type~\type{cAx/4}.
\operatorname{e}nd{quote}
\begin{scase}\label{IA:setup:c}
From now on we assume that the germ $(X, C\simeq {\operatorname{m}}athbb{P}^1)$
satisfies the assumptions of \xref{IA:setup}.
The following are the only possibilities for the singularity
$(X,P)$:
\begin{enumerate}
\item
$(X,P)$ is of type \type{cA/m}, in this case $(X,C)$ is said to be of type \typec{k1A}
according to \cite{KM92};
\item
$(X,P)$ is of type \type{cD/3};
\item
$(X,P)$ is of type \type{cAx/2}, \type{cD/2} or \type{cE/2}.
\operatorname{e}nd{enumerate}
Thus in our case of type \typec{IA}, $(X,C)$ is semistable if and only if it is of type \typec{k1A}.
Extremal curve germs of index two are classified in
Sect.~\xref{sect:index2}. Thus we discuss here cases \typec{k1A} and \type{cD/3}.
We start with ${\operatorname{m}}athbb{Q}$-conic bundles:
\operatorname{e}nd{scase}
\begin{theorem} [{\cite[1.6]{MP:IA}}]
\label{thm:IA:cb}
Let $(X,C\simeq{\operatorname{m}}athbb{P}^1)$ be a ${\operatorname{m}}athbb{Q}$-conic bundle germ of index $m>2$ and
of type~\typec{IA}.
Let $P\in X$ be the non-Gorenstein point.
Then $(X,P)$ is a point of type~\type{cA/m}
and a general member $H\in |{{\operatorname{m}}athscr{O}}_X|$ is not normal.
Furthermore, the dual graph of $(H^{{\operatorname{m}}athrm{n}},C^{{\operatorname{m}}athrm{n}})$, the normalization $H^{{\operatorname{m}}athrm{n}}$
and the inverse image $C^{{\operatorname{m}}athrm{n}}$ of $C$, has the form:
\begin{equation}
\label{IA-n-normal-graph:cb}
\underbrace{\overset{a_r}\circ\text{---}\cdots\text{---}\overset{a_1}\circ}_{\varDelta_1}
\text{---}\bullet\text{---}
\underbrace{\overset{b_1}\circ\text{---}\cdots\text{---}\overset{b_s}\circ}_{\varDelta_2}
\operatorname{e}nd{equation}
{{p}_{\operatorname{a}}}ar{\operatorname{m}}athrm{n}oindent
\textup(in particular, $C^{{\operatorname{m}}athrm{n}}$ is irreducible\textup).
Here
the chain $\varDelta_1$ \textup(resp., $\varDelta_2$\textup)
corresponds to the singularity of type $\frac1m(1,a)$
\textup(resp., $\frac1m(1,-a)$\textup)
for some integer $a$ \ \textup($1\le a<m$\textup) relatively prime to $m$.
The germ $(H,C)$ is analytically isomorphic to the germ
along the line $y=z=0$
of the hypersurface given by the following weighted polynomial of degree $2m$
in variables $x$, $y$, $z$, $u$:
\begin{equation*}
{{p}_{\operatorname{a}}}hi:= x^{2m-2a}y^2+x^{2a}z^2+yzu
\operatorname{e}nd{equation*}
in ${\operatorname{m}}athbb{P}(1,a,m-a,m)$.
Furthermore $(X,C)$ is given as an analytic germ of a subvariety of
${\operatorname{m}}athbb{P}(1,a,m-a,m) \times {\operatorname{m}}athbb{C}_t$ along $C \times 0$ given by
\begin{equation*}
{{p}_{\operatorname{a}}}hi+ \alpha_1x^{2m-a}y+\alpha_2x^{m-a}uy+\alpha_3x^{2m}+\alpha_4x^{m}u+\alpha_5u^{2}=0
\operatorname{e}nd{equation*}
for some $\alpha_1,\ldots \alpha_5\in t{{\operatorname{m}}athscr{O}}_{0,{{\operatorname{m}}athbb{C}_t}}$ and there is a ${\operatorname{m}}athbb{Q}$-conic
bundle structure $X \to {\operatorname{m}}athbb{C}^2$ through which the second projection $X \to {\operatorname{m}}athbb{C}_t$ factors.
\operatorname{e}nd{theorem}
\begin{theorem}[{\cite[1.9]{MP:IA}}, see also \cite{Tzi:05}]
\label{theorem-main-birational}
Let $(X,C)$ be a birational extremal curve germ of type~\typec{k1A}.
Let $P\in X$ be the point of index $m{\operatorname{g}}e 2$.
\begin{emptytheorem}
\label{thm:IA-normal}
If a general element $H$ is normal, then the graph ${\operatorname{D}}elta(H,C)$ has
the same form as in \operatorname{e}qref{imprimitiveIA-graph}, however the cases $r=1$ and $r=n$ are not excluded.
\operatorname{e}nd{emptytheorem}
\begin{emptytheorem}
\label{thm:IA-not-normal}
If every member of $|{{\operatorname{m}}athscr{O}}_X|$ is non-normal, then the dual graph of the normalization $(H^{{\operatorname{m}}athrm{n}},C^{{\operatorname{m}}athrm{n}})$
is of the form
\begin{equation}
\label{IA-n-normal-graph}
\underbrace{\overset{a_r}\circ\text{---}\cdots\text{---}\overset{a_1}\circ}_{\varDelta_1}
\text{---}\bullet\text{---}
\underbrace{\overset{c_1}\circ\text{---}\cdots\text{---}\overset{c_l}\circ}_{\varDelta_3}
\text{---}\diamond\text{---}
\underbrace{\overset{b_1}\circ\text{---}\cdots\text{---}\overset{b_s}\circ}_{\varDelta_2}
\operatorname{e}nd{equation}
\textup(in particular, $C^{{\operatorname{m}}athrm{n}}$ is reducible\textup).
The chain $\varDelta_1$ \textup(resp., $\varDelta_2$\textup)
corresponds to the singularity of type $\frac1m(1,a)$
\textup(resp., $\frac1m(1,-a)$\textup) for some $a$ with ${\operatorname{g}}cd(m,a)=1$
and the chain $\varDelta_3$
corresponds to the point $(H^{{\operatorname{m}}athrm{n}},Q^{{\operatorname{m}}athrm{n}})$, where $Q^{{\operatorname{m}}athrm{n}}=C_1^{{\operatorname{m}}athrm{n}}\cap C_2^{{\operatorname{m}}athrm{n}}$.
Moreover,
\begin{equation*}
\sum (c_i-2)\le 2 \quad\text{and}\quad \widetilde C_1^2 +\widetilde C_2^2 +5-\sum(c_i-2){\operatorname{g}}e 0,
\operatorname{e}nd{equation*}
where $\widetilde C=\widetilde C_1 +\widetilde C_2$ is the proper transform of $C$ on the minimal resolution
$\widetilde H$.
Both components of $\widetilde C$ are contracted on the minimal model
of $\widetilde H$.
In this case,
\begin{equation*}
(X,C,P)\simeq \bigl(\{\alpha(x_1,\dots,x_4)=0\}, \text{$x_1$-axis}, 0\bigr)/{\operatorname{m}}umu_m(1,a,-a,0),
\operatorname{e}nd{equation*}
where ${\operatorname{g}}cd(m,a)=1$ and $\alpha=0$ is the equation of
a terminal \type{cA/m}-point in ${\operatorname{m}}athbb{C}^4/{\operatorname{m}}umu_m$.
\textup(In particular, $(X,C)$ is of type~\typec{IA}\textup).
\operatorname{e}nd{emptytheorem}
Conversely, for any germ $(H,C\simeq {\operatorname{m}}athbb{P}^1)$ of the form
\xref{thm:IA-normal} or \xref{thm:IA-not-normal}
admitting
a birational contraction $(H,C)\to (H_Z,o)$ there exists
a threefold birational contraction
$f: (X,C)\to (Z,o)$ as in \xref{IA:setup} of type~\typec{IA}
such that $H\in |{{\operatorname{m}}athscr{O}}_X|$.
\operatorname{e}nd{theorem}
\begin{scase}
To study a general member $H\in |{{\operatorname{m}}athscr{O}}_X|$
we can use Lemma \xref{prop:lc}.
However we cannot assert as in \xref{pf:IAdual:normality}
that $H$ is normal. In fact, arguments similar to
\xref{pf:IAdual:normal} show that the case of normal $H$ does not occur
if $(X,C)$ is a ${\operatorname{m}}athbb{Q}$-conic bundle.
\operatorname{e}nd{scase}
\subsection{}
Let us outline the proofs of Theorems \ref{thm:IA:cb} and \ref{theorem-main-birational}.
The case where $H$ is normal is teated in the same way as \xref{imprimitiveIA} (see \xref{pf:IAdual:normal})
and $X$ can be recovered as a one-parameter deformation space by \xref{def:existence}.
Examples are given in \xref{ex:IA-n} below.
Suppose that $H$ is not normal.
Let ${\operatorname{m}}athrm{n}u: H^{\operatorname{m}}athrm{n}\to H$ be the normalization and let $C^{\operatorname{m}}athrm{n}\subset H^{\operatorname{m}}athrm{n}$ be the inverse image of $C$.
By the inversion of adjunction the pair $(H,C)$ is slc, the pair $(H^{\operatorname{m}}athrm{n},C^{\operatorname{m}}athrm{n})$ is lc,
and the point $P\in (H,C)$ is slt \cite[16.9]{Utah}.
In particular, $H$ is a generically normal crossing divisor.
At certain (finite number) of dissident points $H$ may have singularities worse than
just normal crossing points.
\begin{scase}\label{IA:H:points}
Since $H$ has ${\operatorname{m}}athbb{Q}$-Gorenstein smoothing,
by \cite[Theorem 4.24, 5.2]{KSh88} the only possibilities are:
\begin{itemize}
\item
Pinch point: $\{x^2-y^2z = 0\} \subset {\operatorname{m}}athbb{C}^3$.
\item
Degenerate cusp of embedding dimension at most $4$, where a degenerate cusp
is a non-normal Gorenstein singularity having a semi-resolution whose
exceptional divisor is a cycle of smooth rational curves or a rational
nodal curve (see \cite{SB:degen}).
\item
Slt singularity of the form
\begin{equation*}
\{xy = 0\}/{\operatorname{m}}umu_m(a,-a, 1),\quad {\operatorname{g}}cd(a, n) = 1\}
\operatorname{e}nd{equation*}
(this point corresponds to $P\in H$).
\operatorname{e}nd{itemize}
\operatorname{e}nd{scase}
\begin{scase}
The restriction ${\operatorname{m}}athrm{n}u_C: C^{\operatorname{m}}athrm{n}\to C$ of the normalization to the
inverse image of $C$ is a double cover.
We distinguish two possibilities:
\begin{enumerate}
\item
\label{case:n-normal:irre}
$C^{\operatorname{m}}athrm{n}$ is smooth irreducible and ${\operatorname{m}}athrm{n}u_C$ is branched
at two points,
\item
\label{case:n-normal:red}
$C^{\operatorname{m}}athrm{n}$ has two irreducible components meeting at one point and
the restriction of ${\operatorname{m}}athrm{n}u_C$ to each of them is an isomorphism.
\operatorname{e}nd{enumerate}
A detailed analysis (see \cite{MP:IA} and also \cite{Tzi:05}) shows that
\xref{case:n-normal:irre} leads to the ${\operatorname{m}}athbb{Q}$-conic bundle case \operatorname{e}qref{IA-n-normal-graph:cb}
while \xref{case:n-normal:red} leads to the birational case \operatorname{e}qref{IA-n-normal-graph}.
In both cases the subgraphs $\varDelta_1$ and $\varDelta_2$ correspond to points $P^n_1,\, P^{\operatorname{m}}athrm{n}_2\in H^{\operatorname{m}}athrm{n}$
lying over $P\in H$.
\qed
\operatorname{e}nd{scase}
\begin{scase}
To recover $X$ as a one-parameter deformation space
we also can apply arguments as in \xref{def:existence}.
However, in the case of non-normal surface $H$, it needs some restriction to singularities and
additional technical tools \cite{Tziolas2009}.
Fortunately, the results of \cite{Tziolas2009} are applicable if $H$ has singularities described above.
Moreover, the miniversal deformation family of $(H,C)$ in the ${\operatorname{m}}athbb{Q}$-conic bundle case
is computed explicitly \cite[6.8.3]{MP:IA}. \qed
\operatorname{e}nd{scase}
\begin{scase}
\label{IA:check-divisoriality}
To check divisoriality one can use the criterion
\xref{theorem-main-Q-Cartier-i}. Indeed, if $f$ is divisorial, then
$(Z,o)$ is a terminal point and its index equals $1$ because
$(X,C)$ is primitive (see \xref{base-change}). Therefore,
its general hyperplane section $(H_Z,o)$ must be a Du Val singularity.
If on the contrary $f$ is flipping, then $(Z,o)$ is not ${\operatorname{m}}athbb{Q}$-Gorenstein and
$(H_Z,o)$ cannot be Du Val. Given a graph ${\operatorname{D}}elta(H, C)$ of type \operatorname{e}qref{imprimitiveIA-graph}
one can easily draw the graph ${\operatorname{D}}elta(H_Z)$ contracting
black vertices successfully. Thus the Du Val condition
of $(H_Z,o)$ can be checked in purely combinatorial terms.
\operatorname{e}nd{scase}
\begin{sremark}
Assume that in the assumptions of \xref{thm:IA-normal} and \operatorname{e}qref{imprimitiveIA-graph}
we have $r=1$ or $r=n$.
Then the graph ${\operatorname{D}}elta(H,C)$ is a chain. In this case there exists
an element $D\in |-K_X|$ \operatorname{e}mph{containing} $C$ and having Du Val singularities only.
This is a particular case of the situation considered in \cite{Mori:ss}
where a powerful algorithm to construct $(X,C)$ was obtained.
\operatorname{e}nd{sremark}
\begin{scase}
\label{HTU}
One special case of Theorem \xref{theorem-main-birational} was studied in details in \cite{HTU}.
There the authors assumed that the nearby fiber $H_t$ of the one-parameter deformation
$\cup H_t=X$ has $b_2(H_t)=1$. This strong assumption is equivalent to that
$H$ is normal and has so-called \operatorname{e}mph{Wahl singularity} at $P$: $(H,P)\simeq {\operatorname{m}}athbb{C}^2/{\operatorname{m}}umu_{m^2}(1, ma-1)$.
Under this assumption, it is shown that birational germs of this type
belong to the same deformation family as those of \typec{k2A} studied in \cite{Mori:ss},
constructed the universal family, and the algorithm \cite{Mori:ss} of computing
flips was extended.
\operatorname{e}nd{scase}
\begin{sexamples}
\label{ex:IA-n}
Consider several examples of extremal germs of type \xref{thm:IA-normal}:
\begin{enumerate}
\item
The index two germs \xref{index2flipping}-\xref{KM:4.7.3.1.4}
are of type \typec{IA}. By using arguments of \xref{IA:check-divisoriality}
one can conclude that the germ as in \xref{index2flipping} is flipping and those in
\xref{KM:4.7.3.1.1}-\xref{KM:4.7.3.1.4} are divisorial.
\item
Let ${\operatorname{D}}elta(H,C)$ be of the form
\begin{equation*}
\xymatrix{
\bullet\ar@{-}[r]&\overset{c_1}\circ\ar@{-}[r]&\cdots\ar@{-}[r]&\overset{c_n}\circ
}
\operatorname{e}nd{equation*}
where the white vertices form a dual graph of a non-Du Val \type{T}-singularity \textup(see \xref{typeT} \textup).
It is easy to see that $C$ can be contracted to
a cyclic quotient non-Du Val point.
Therefore, the one-parameter deformation
produces a flipping contraction. Since $(H,C)$ is plt, the contraction is primitive.
\item
Let ${\operatorname{D}}elta(H,C)$ be of the form
\begin{equation*}
\xymatrix@R=1pt{
\overset{3}\circ\ar@{-}[r]&\overset{5}\circ\ar@{-}[r]&\circ
\\
&\bullet\ar@{-}[u]\ar@{-}[r]&\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ
}
\operatorname{e}nd{equation*}
This is an example of a divisorial contraction to a smooth point.
\item
A series of examples were given in \xref{ex:IAdual} b).
\operatorname{e}nd{enumerate}
\operatorname{e}nd{sexamples}
For completeness, we provide an example of birational curve germ
with non-normal $H$.
\begin{sexample}[{\cite[Ex. 2]{Tzi:05}}, {\cite[Ex. 6.10.3]{MP:IA}}]
Consider a surface $\tilde H$ containing a configuration with the following graph
\[
\xymatrix{
\overset{}\circ\ar@{-}[r]&
\overset{4}{\circ} \ar@{-}[r]&
\underset{C_1} {\overset{4}\diamond}\ar@{-}[r]&
\underset{C_2}\bullet\ar@{-}[r]&
\overset{}\circ\ar@{-}[r]&
\overset{}\circ\ar@{-}[r]&
\overset{3}\circ
}
\]
Contracting all curves except those marked by $C_1$ and $C_2$, we obtain a
normal surface $H^n$ having two cyclic quotient singularities
$P_1$ and $P_2$ of types $\frac17(1,2)$ and $\frac 17(1,-2)$.
Identifying the curves $C_1$ and $C_2$ we obtain a non-normal surface
$H$ so that the map ${\operatorname{m}}athrm{n}u: H^n\to H$ is the normalization.
The dissident singularities of $H$ are a degenerate cusp of multiplicity $2$ and embedding dimension $3$
at ${\operatorname{m}}athrm{n}u(C_1\cap C_2)$, and one point of type
$\{xy = 0\}/{\operatorname{m}}umu_7(2,-2, 1)$. The results of \cite{Tziolas2009} are applicable here
and so there exists a one-parameter smoothing $X\supset H\supset C$
which is a divisorial curve germ and $H$ is general in $|{{\operatorname{m}}athscr{O}}_X|$,
see \cite[Prop. 6.3 and Th. 6.10]{MP:IA}.
\operatorname{e}nd{sexample}
\subsection{Points of type~\type{cD/3}.}
Let $(X,C,P)$ be a triple of type~\typec{IA}, where $(X,P)$ is
a singularity of type~\type{cD/3} \cite{Mori:sing}, \cite{Reid:YPG}.
These triples are described as follows
(see \cite[6.5]{KM92}).
Put $\sigma:=(1,1,2,3)$.
Up to coordinate
change the point $(X,C,P)$ is given in ${\operatorname{m}}athbb{C}^4_{y_1,\dots,y_4}$ as follows
\begin{equation}
\label{eq:cD/3}
\begin{array}{l}
(X,C,P)=\bigl(\{\alpha=0\}, \ \{\text{$y_1$-axis}\},\ 0\bigr)/{\operatorname{m}}umu_3(1,1,2,0),
\\[1pt]
\alpha=y_4^2+y_3^3+\delta_3(y_1,y_2)
+(\text{terms of degree ${\operatorname{g}}e 4$}),
\operatorname{e}nd{array}
\operatorname{e}nd{equation}
where
$\delta_3{\operatorname{m}}athrm{n}eq 0$ is homogeneous of degree $3$ and
$\alpha$ is invariant. Moreover,
\begin{equation*}
\alpha\operatorname{e}quiv y_1^\operatorname{e}ll y_i {\operatorname{m}}od (y_2, y_3,y_4)^2,
\operatorname{e}nd{equation*}
where
$\operatorname{e}ll=\operatorname{e}ll(P)$ and $i =2$ (resp. $3$, $4$) if $\operatorname{e}ll\operatorname{e}quiv 2$ (resp. $1$, $0$) ${\operatorname{m}}od 3$
\cite[(2.16)]{Mori:flip}.
If $\delta_3(y_1,y_2)$ is square free (resp. has a double factor, is a cube of a linear form),
then $(X,P)$ is said to be a \operatorname{e}mph{simple} (resp. \operatorname{e}mph{double}, \operatorname{e}mph{triple}) \type{cD/3} point.
Extremal curve germs containing a terminal
singular point of type~\type{cD/3}
are described by the following theorem.
\begin{stheorem}[{\cite[Th.~6.2-6.3]{KM92}}, {\cite[Th.~4.5, 4.8]{MP:IA}}]
\label{cD/3:thm)}
Let {\operatorname{m}}box{$f:(X,C\simeq {\operatorname{m}}athbb{P}^1)\to (Z,o)$} be an extremal curve germ
having a point $P$ of type~\type{cD/3}.
Then $f$ is a birational contraction, not a ${\operatorname{m}}athbb{Q}$-conic bundle.
General members $H\in |{{\operatorname{m}}athscr{O}}_X|$ and $H_Z=f(H)\in |{{\operatorname{m}}athscr{O}}_Z|$ are normal and
have only rational singularities.
We have the following possibilities for graphs ${\operatorname{D}}elta(H,C)$ and ${\operatorname{D}}elta(H_Z,o)$
and local invariants.
{{p}_{\operatorname{a}}}ar{\operatorname{m}}edskip{\operatorname{m}}athrm{n}oindent
{\rm
\begin{longtable}{c|c|c|p{0.2\textheight}|l|c}
{\rm No.} & $\operatorname{e}ll(P)$ & $i_P(1)$ & ${\operatorname{D}}elta(H,C)$ &${\operatorname{D}}elta(H_Z,o)$
\\\hline
\operatorname{e}ndhead
{\operatorname{m}}ulticolumn{5}{c}{Cases of simple \type{cD/3} point $P$}
\\\hline
{\operatorname{m}}athrm{n}om
\label{cD/3:flip:.3.1)}
& $2$& $1$ &
$\xymatrix@R=0pt@C=15pt{
\bullet\ar@{-}[r]&\overset3\circ\ar@{-}[r]&\circ\ar@{-}[r]&\overset3\circ
\\
&&\underset3\circ\ar@{-}[u]
}$
&
$\xymatrix@R=0pt@C=10pt{
&\overset3\circ\ar@{-}[d]
\\
\circ\ar@{-}[r]&\circ\ar@{-}[r]&\underset3\circ
}$ & \type{f}
\\\hline
{\operatorname{m}}athrm{n}om
\label{cD/3:thm:A2}
& $2$ & $1$ &
$
\xymatrix@R=3pt@C=15pt{
\bullet\ar@{-}[r]&\overset{3}\circ\ar@{-}[r]&\circ\ar@{-}[r]&\overset{3}\circ
\\
\circ\ar@{-}[u]&&\underset{3}\circ\ar@{-}[u]&
}
$
&\type{A_2} & \type{d}
\\\hline
{\operatorname{m}}ulticolumn{5}{c}{Cases of double \type{cD/3} point $P$}
\\\hline
{\operatorname{m}}athrm{n}om
\label{cD/3:flip:.3.2)}
& $2$ & $1$ &
$\xymatrix@R=7pt@C=15pt{
\bullet\ar@{-}[d]&&\circ\ar@{-}[d]
\\
\underset3\circ\ar@{-}[r]&\circ\ar@{-}[r]&\underset3\circ\ar@{-}[r]&\circ
\\
&&\circ\ar@{-}[u]
}$
&
$\xymatrix@R=7pt@C=10pt{
\circ\ar@{-}[d]&\circ\ar@{-}[d]
\\
\circ\ar@{-}[r]&\underset3\circ\ar@{-}[r]&\circ
\\
&\circ\ar@{-}[u]
}$ & \type{f}
\\\hline
{\operatorname{m}}athrm{n}om
\label{cD/3:thm:D4}
& $2$ & $1$ &
$
\xymatrix@R=9pt@C=15pt{
\bullet\ar@{-}[d]&\circ\ar@{-}[l]&\circ\ar@{-}[d]&
\\
\underset{3}\circ
\ar@{-}[r]&\circ\ar@{-}[r]&\underset{3}\circ\ar@{-}[r]&\circ
\\
&&\circ\ar@{-}[u]&
}
$
& \type{D_4} & \type{d}
\\\hline
{\operatorname{m}}athrm{n}om
\label{cD/3:flip:iP=34)}
& $3$, $4$ & $2$ &
$
\xymatrix@R=9pt@C=15pt{
&\bullet\ar@{-}[d]&\circ\ar@{-}[d]&\overset3\circ
\\
&\circ\ar@{-}[r]&\underset3\circ\ar@{-}[r]&\circ\ar@{-}[u]
\\
&&\circ\ar@{-}[u]
}
$
&
$
\xymatrix@R=7pt@C=15pt{
\circ\ar@{-}[d]
\\
\circ\ar@{-}[r]&\circ\ar@{-}[r]&\underset3\circ
\\
\circ\ar@{-}[u]
}
$ & \type{f}
\\\hline
{\operatorname{m}}ulticolumn{5}{c}{Case of triple \type{cD/3} point $P$}
\\\hline
{\operatorname{m}}athrm{n}om
\label{cD/3:thm:E6}
&$3$, $4$& $2$ &
$
\xymatrix@R=9pt@C=15pt{
\bullet\ar@{-}[d]&&\circ\ar@{-}[d]\ar@{-}[r]&\circ
\\
\circ\ar@{-}[r]&\circ\ar@{-}[r]&\overset{3}\circ\ar@{-}[r]&\circ
\\
&&\circ\ar@{-}[u]&\circ\ar@{-}[l]
}
$& \type{E_6} & \type{d}
\operatorname{e}nd{longtable}}
In the cases
\xref{cD/3:flip:.3.1)}, \xref{cD/3:flip:.3.2)}, \xref{cD/3:flip:iP=34)}, and \xref{cD/3:thm:E6}
the variety $X$ is smooth outside $P$ and in the cases \xref{cD/3:thm:A2} and \xref{cD/3:thm:D4}\ $X$
may has at most one type \typec{III} point.
The last column indicates if the germ is flipping \typec{f} or divisorial \typec{d}.
\operatorname{e}nd{stheorem}
Note that \cite[\S~6]{KM92} and \cite[\S~4]{MP:IA} provide much more information about
these contractions:
infinitesimal structure, criterion for an arbitrary germ to be of the corresponding type,
and computations of flipped varieties \cite[A.1]{KM92}. Flipping contractions
can be constructed explicitly by patching certain open subsets:
\begin{sexample}[{\cite[6.11]{KM92}}]
\label{ex:cD/3:flip}
Let $V\supset C$ be a germ of a smooth threefold along $C\simeq {\operatorname{m}}athbb{P}^1$
such that ${\operatorname{m}}athbb{N}N_{C/V}\simeq {{\operatorname{m}}athscr{O}}_C\oplus{{\operatorname{m}}athscr{O}}_C$.
Pick a point $P\in C$ and let $(v_1, v_2, v_3)$ be coordinates at
$(V, P)$ such that $(C, P) = \{\text{$v_1$-axis}\}$. Let $(X,C, P)$ be a \type{cD/3} point as in
\operatorname{e}qref{eq:cD/3} with $\operatorname{e}ll=2$.
For suitable $\varepsilon_1$ and $\varepsilon_2$ such that $0<\varepsilon_1<\varepsilon_1\ll 1$,
$(y_1^3, y_4, y_1y_3)$ form
coordinates for $U = (X, P)\cap \{\varepsilon_1 < | y_1^3| < \varepsilon_2\}$ by the implicit function theorem.
Thus
$v_1 = y_1^3$, $v_2 = y_4$, and $v_3 =y_1y_3$
patch $(X, P)$ and $V\setminus (V, P) \cap \{|v_1| < \varepsilon_1\}$ along $U$.
By \cite[6.2.4]{KM92} the germ $(X,C)$ is a flipping
curve germ of type \type{cD/3} as in \xref{cD/3:flip:.3.1)}
or \xref{cD/3:flip:.3.2)} (depending on the choice of $\delta_3$ in \operatorname{e}qref{eq:cD/3}).
\operatorname{e}nd{sexample}
More examples of flipping contractions
are given in \cite[6.17 and 6.21]{KM92}.
To show that all the possibilities in Theorem~\xref{cD/3:thm)} occur one can also use
the deformation arguments \xref{def:existence}:
\begin{sexample}
\label{ex:cD/3:E6}
Consider the surface contraction $f_H: H\to H_Z$ with dual graph
\xref{cD/3:thm:E6} and consider the following triple of germs:
\begin{equation*}
(X, H, P) = \bigl(\{y_2^3+y_3^3+y_3y_1^4+y_4^2 \}, \{y_4=y_1y_3\}, 0\bigr) /{\operatorname{m}}umu_3(1,1,2,0),
\operatorname{e}nd{equation*}
where $H$ is cut out by $y_4=y_1y_3$.
Here $(X,P)$ is a triple \type{cD/3}-singularity (see
\xref{eq:cD/3}).
By \cite[4.12]{MP:IA} the dual graph of the
minimal resolution of $(H,P)$ is the same as that in \xref{cD/3:thm:E6}.
By \xref{def:existence} one obtains a
birational contraction $f: X\to Z$ extending $f_H: H\to H_Z$,
which is as in \xref{cD/3:thm:E6}. Examples similar to \xref{cD/3:thm:A2} and
\xref{cD/3:thm:D4} are given in \cite[4.14]{MP:IA}.
\operatorname{e}nd{sexample}
Divisorial contractions of type \xref{cD/3:thm:E6} were studied also in
\cite[5.1(2)]{Tzi:10} by a different method.
\section{Case \typec{IIA}}
\begin{setup}
\label{Set-up}
Let $(X,C)$ be an extremal
curve germ and let {\operatorname{m}}box{$f: (X, C)\to (Z,o)$} be the corresponding
contraction.
Assume that $(X,C)$ has a point $P$ of type~\typec{IIA}.
Then by \cite[6.7, 9.4]{Mori:flip} and
\cite[8.6, 9.1, 10.7]{MP:cb1} $P$ is the only non-Gorenstein point of $X$
and $(X,C)$ has at most one Gorenstein singular point $R$ \cite[6.2]{Mori:flip},
\cite[9.3]{MP:cb1}. Since $P\in (X,C)$ is locally primitive,
the topological index of $(X,C)$ equals $1$.
Hence the base $(Z,o)$ is smooth
in the ${\operatorname{m}}athbb{Q}$-conic bundle case,
and
is a \type{cDV} point (or smooth) in the divisorial case
(cf. \xref{thm:div:Q-Cartier}).
\operatorname{e}nd{setup}
\subsection{}
\label{(7.5)}
According to \cite[A.3]{Mori:flip} we can express the \typec{IIA} point as
\begin{equation}
\label{equation-XC}
\begin{array}{rcl}
(X, P)&=&
\{\alpha=0\}/{\operatorname{m}}umu_4(1, 1, 3, 2)\subset{\operatorname{m}}athbb{C}^4_{y_1,\dots, y_4}/{\operatorname{m}}umu_4(1, 1, 3, 2),
\\[1pt]
C&=&\{y_1\text{-axis}\}/{\operatorname{m}}umu_4,
\operatorname{e}nd{array}
\operatorname{e}nd{equation}
where $\alpha=\alpha(y_1,\dots, y_4)$ is a semi-invariant such that
\begin{equation}
\label{equation-alpha}
\operatorname{wt}\alpha\operatorname{e}quiv 2{\operatorname{m}}od 4,{\operatorname{m}}athbin{\sim_{\scriptscriptstyle{\QQ}}}uad \alpha\operatorname{e}quiv y_1^{\operatorname{e}ll(P)}y_j{\operatorname{m}}od (y_2, y_3, y_4)^2,
\operatorname{e}nd{equation}
where $j= 2$ (resp. $3$, $4$)\ if $\operatorname{e}ll(P)\operatorname{e}quiv 1$ (resp. $3$, $0$) ${\operatorname{m}}od 4$
(see \operatorname{e}qref{equation-iP-lP}) and $({{\operatorname{m}}athcal{I}}_C^\sharp)^{(2)}=(y_j)+({{\operatorname{m}}athcal{I}}_C^\sharp)^{2}$.
Moreover, $y_2^2,\, y_3^2\in \alpha$ (because $(X,P)$ is a terminal point of type
\type{cAx/4}).
Note that $\operatorname{e}ll(P){\operatorname{m}}athrm{n}ot\operatorname{e}quiv 2{\operatorname{m}}od 4$ because
of the lack of a variable with $\operatorname{wt}\operatorname{e}quiv 0{\operatorname{m}}od 4$.
\begin{theorem}[{\cite[7.2-7.4]{KM92}}, {\cite{MP:IIA-1}}, {\cite{MP:IIA-2}}]
\label{IIA:thm}
Let $f:(X,C\simeq {\operatorname{m}}athbb{P}^1)\to (Z,o)$ be an extremal curve germ
having a point $P$ of type~\typec{IIA}.
We have the following possibilities for graphs ${\operatorname{D}}elta(H,C)$ and ${\operatorname{D}}elta(H_Z,o)$
and local invariants.
{{p}_{\operatorname{a}}}ar{\operatorname{m}}edskip{\operatorname{m}}athrm{n}oindent
{\rm
\begin{longtable}{c|c|c|p{0.4\textwidth}|l|c}
{\rm No.} & $i_P(1)$ & $\operatorname{e}ll(P)$ & {\operatorname{m}}ulticolumn{1}{c}{${\operatorname{D}}elta(H,C)$} &{\operatorname{m}}ulticolumn{1}{|c|}{${\operatorname{D}}elta(H_Z,o)$} &
\\\hline\operatorname{e}ndhead
{\operatorname{m}}ulticolumn{6}{c}{\rm Cases: $H$ is normal}
\\\hline
{\operatorname{m}}athrm{n}om
\label{IIA:flip:iP=1a}
&$1$& $1$ &
$\xymatrix@R=0pt@C=13pt{
&&\overset{4}\circ\ar@{-}[d]
\\
\bullet\ar@{-}[r]&\underset{4}\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ
}$
&
$\xymatrix@R=0pt@C=13pt{
&\overset{4}\circ\ar@{-}[d]
\\
\underset{3}\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ
}$
&
\type{f}
\\\hline
{\operatorname{m}}athrm{n}om
\label{IIA:flip:iP=1b}
&$1$& $1$ &
$\xymatrix@R=0pt@C=13pt{
\circ\ar@{-}[d] &&\overset{4}\circ\ar@{-}[d]
\\
\bullet\ar@{-}[r]&\underset{4}\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ
}$
&
$\xymatrix@R=0pt@C=13pt{
&\overset{4}\circ\ar@{-}[d]
\\
\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ
}$
&
\type{f}
\\\hline
{\operatorname{m}}athrm{n}om
\label{IIA:flip:iP=2}
&$2$& $3$, $4$ &
$\xymatrix@R=10pt@C=13pt{
\circ\ar@{-}[r]&\circ\ar@{-}[rd]&\circ\ar@{-}[d]
\\
&&\overset{4}{\circ}\ar@{-}[r]&\circ
\\
\bullet\ar@{-}[r]&\circ\ar@{-}[ru]&\circ\ar@{-}[u]
}$
&
$\xymatrix@C=13pt@R=10pt{
\circ\ar@{-}[d] &\circ\ar@{-}[d]
\\
\circ\ar@{-}[r]&\overset3\circ\ar@{-}[r]&\circ
\\
&\circ\ar@{-}[u]
}$
&
\type{f}
\\\hline
{\operatorname{m}}athrm{n}om\label{IIA:normalA1:a}
&$1$& $1$ &
$
\xymatrix@R=3pt@C=10pt{
\circ\ar@{-}[r]\ar@{-}[d]&\circ&\circ\ar@{-}[d]
\\
\bullet\ar@{-}[r]&\underset4\circ\ar@{-}[r]
&\circ\ar@{-}[r]&\underset4\circ
}
$&\type{A_1}
&
\type{d}
\\\hline
{\operatorname{m}}athrm{n}om\label{IIA:normalA1:b}
&$1$& $1$ &
$
\xymatrix@R=3pt@C=10pt{
\circ\ar@{-}[d]&&\overset3\circ\ar@{-}[d]&\overset4\circ\ar@{-}[d]
\\
\bullet\ar@{-}[r]&\underset3\circ\ar@{-}[r]
&\circ\ar@{-}[r]&\circ
}
$& \type{A_1} &\type{d}
\\\hline
{\operatorname{m}}athrm{n}om\label{IIA:normalD5}
&$2$ & $3$, $5$ &
$
\xymatrix@R=7pt@C=10pt{
\bullet\ar@{-}[d]&\circ\ar@{-}[d]&\circ\ar@{-}[d]&\circ\ar@{-}@/^1pt/[dl]
\\
\circ\ar@{-}[r]&\circ\ar@{-}[r]
&\underset{4}\circ\ar@{-}[r]&\circ\ar@{-}[r] &\circ
}
$&\type{D_{5}}&\type{d}
\\\hline
{\operatorname{m}}athrm{n}om\label{IIA:normal:cb}
& $2$ & $4$, $5$&
{\operatorname{m}}ulticolumn{1}{l|}{
$
\xymatrix@R=3pt@C=13pt{
\bullet\ar@{-}[r]&\circ\ar@{-}[d]&&\circ\ar@{-}[d]&\circ\ar@{-}[d]
\\
\circ\ar@{-}[r]&\circ\ar@{-}[r]
&\underset3\circ\ar@{-}[r]&\circ\ar@{-}[r]&\underset3\circ\ar@{-}[r]&\circ
}
$}&&\type{c}
\\\hline
{\operatorname{m}}ulticolumn{5}{c}{Cases: $H$ is not normal}
\\\hline
{\operatorname{m}}athrm{n}om\label{IIA-n-normal-div}
\footnote{This case was erroneously
omitted in \cite[Th. 3.6 and Cor. 3.8]{Tzi:05D}.}
&& &
$
\xymatrix@R=7pt@C=13pt{
&\circ\ar@{-}[d]
\\
\underset {} \bullet \ar@{-}[r]
&\underset 3\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ
\\
&\circ\ar@{-}[u]
}
$&\type{D_{5}}&\type{d}
\\\hline
{\operatorname{m}}athrm{n}om\label{IIA:n-normal-cb}
&& &
{\operatorname{m}}ulticolumn{1}{l|}{$
\xymatrix@R=7pt@C=11pt{
&\circ\ar@{-}[r]&\overset {3}\circ\ar@{-}[d]\ar@{-}[r]&\circ
\\
\bullet \ar@{-}[r] &\underset {}\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ
}
$}&&\type{c}
\operatorname{e}nd{longtable}}
The variety $X$ can have \textup(at most one\textup) extra
type \typec{III} singular point in all cases except for
\xref{IIA:flip:iP=1a},
\xref{IIA:flip:iP=2},
\xref{IIA:normalD5}, and
\xref{IIA:normal:cb} where the singular point is unique.
\operatorname{e}nd{theorem}
Examples of flipping contractions can be constructed similar to~\xref{ex:cD/3:flip}.
\begin{example}[{\cite[7.6.4]{KM92}}]
\label{ex:IIA:flip}
Let $V\supset C$ be a germ of a smooth threefold along $C\simeq {\operatorname{m}}athbb{P}^1$
such that ${\operatorname{m}}athbb{N}N_{C/V}\simeq {{\operatorname{m}}athscr{O}}_C\oplus{{\operatorname{m}}athscr{O}}_C$.
Pick a point $P\in C$ and let $(v_1, v_2, v_3)$ be coordinates at
$(V, P)$ such that $(C, P) = \{\text{$v_1$-axis}\}$.
Let $(X,C, P)$ be a \typec{IIA}-point as in
\operatorname{e}qref{equation-XC}-\operatorname{e}qref{equation-alpha} with $\alpha\operatorname{e}quiv y_1y_2 {\operatorname{m}}od (y_2, y_3, y_4)^2$.
For suitable $\varepsilon_1$ and $\varepsilon_2$ such that $0<\varepsilon_1<\varepsilon_1\ll 1$,
$(y_1^4, y_1^2y_4, y_1y_3)$ form
coordinates for $U = (X, P)\cap \{\varepsilon_1 < | y_1^4| < \varepsilon_2\}$ by the implicit function theorem.
Thus
$v_1 = y_1^4$, $v_2 =y_1^2 y_4$, and $v_3 =y_1y_3$
patch $(X, P)$ and $V\setminus (V, P) \cap \{|v_1| < \varepsilon_1\}$ along $U$.
By \cite[7.2.4]{KM92} the germ $(X,C)$ is a flipping
curve germ of type \typec{IIA} as in \xref{IIA:flip:iP=1a}.
See \cite[7.9.4, 7.12.5]{KM92} for more examples of flipping contractions.
\operatorname{e}nd{example}
The existence in the above theorem in the case where $H$ is normal can be established
by using arguments of \xref{def:existence}.
In the case \xref{IIA:normalD5} we have also explicit example:
\begin{example}[{\cite[6.6]{MP:IIA-1}}]
\label{ex:IIA-normal}
Let $Z \subset{\operatorname{m}}athbb{C}^5_{z_1,\ldots,z_5}$ be defined by two equations:
\begin{eqnarray*}
0&=& z_2^2+z_3+z_4z_5^k+z_1^3,{\operatorname{m}}athbin{\sim_{\scriptscriptstyle{\QQ}}}uad k{\operatorname{g}}e 1,\\
0&=& z_1^2z_2^2 + z_4^2-z_3z_5+z_1^3z_2+cz_1^2z_4.
\operatorname{e}nd{eqnarray*}
By eliminating $z_3$ using the first equation,
one sees easily that $(Z,0)$ is a
threefold singularity of type~\type{cD_{5}}.
Let $B \subset Z$ be the $z_5$-axis, and
let $f: X \to Z$ be the weighted blowup of $B$ with
weight $(1,1,4,2,0)$.
By an easy computation one sees that
$C:=f^{-1}(0)_{\operatorname{red}} \simeq {{\operatorname{m}}athbb{P}}^1$
and $X$ is covered by two charts: $z_1$-chart
and $z_3$-chart. The origin
of the $z_3$-chart
is a type~\typec{IIA} point $P$ with $\operatorname{e}ll(P)=3$:
\begin{equation*}
\{y_1^3y_3+y_2^2+y_3^2+y_4(y_1^2y_2^2+y_4^2+y_1^3y_2+c y_1^2 y_4)^k=0\}/{\operatorname{m}}umu_{4}(1,1,3,2),
\operatorname{e}nd{equation*}
where $(C,P)$ is the $y_1$-axis.
Moreover, $X$ is smooth outside $P$.
Thus $X\to Z$ is a divisorial contraction of type~\xref{IIA:normalD5}.
See also {\cite[8.3.3]{MP:IIA-1}} for an example with $\operatorname{e}ll(P)=5$.
\operatorname{e}nd{example}
The case \xref{IIA:normalD5} was also studied by N.~Tziolas \cite[Th. 3.6]{Tzi:05D}.
The existence of \xref{IIA-n-normal-div} can be shown similar to Example~\xref{ex:IIA-normal}:
\begin{example}[{\cite[3.6]{MP:IIA-2}}]
\label{ex:IIA-n-normal}
Let $Z \subset {{\operatorname{m}}athbb{C}}^5_{z_1,\ldots,z_5}$ be
defined by
\begin{eqnarray*}
0&=& z_2^2+z_3+z_4z_5^k-z_1^3,{\operatorname{m}}athbin{\sim_{\scriptscriptstyle{\QQ}}}uad k{\operatorname{g}}e 1,\\
0&=& z_1^2z_2^2+z_4^2-z_3z_5.
\operatorname{e}nd{eqnarray*}
Then $(Z,0)$ is a
threefold singularity of type~\type{cD_{5}}.
Let $B \subset Z$ be the $z_5$-axis and
let $f: X \to Z$ be the weighted $(1,1,4,2,0)$-blowup.
The origin
of the $z_3$-chart
is a type~\typec{IIA} point $P$ with $\operatorname{e}ll(P)=3$:
\begin{equation*}
\{-y_1^3y_3+y_2^2+y_3^2+y_4(y_1^2y_2^2+y_4^2)^k=0\}/{\operatorname{m}}umu_{4}(1,1,3,2),
\operatorname{e}nd{equation*}
where $(C,P)$ is the $y_1$-axis.
In the $z_1$-chart we have a type~\typec{III} point.
See also \cite[3.7]{MP:IIA-2} for an example of
a divisorial germ as in \xref{IIA-n-normal-div} whose singular locus
consists of a single \typec{IIA} point $P$ with $\operatorname{e}ll(P)=7$.
\operatorname{e}nd{example}
\begin{example}[{\cite[4.8]{MP:IIA-2}}]
\label{example-conic-bundle-lP=4+III}
Let $X$ be the the hypersurface of weighted degree $10$ in
the weighted projective space ${\operatorname{m}}athbb{P}(1,1,3,2,4)_{x_1,\dots, x_4, w}$
given by the equation
\begin{equation*}w{{p}_{\operatorname{a}}}hi_6 -x_1^6{{p}_{\operatorname{a}}}hi_4=0,\quad \text{where}\quad
\begin{array}{lll}
{{p}_{\operatorname{a}}}hi_6&:=&x_1^4x_4+x_3^2+x_2^2w+\delta x_4^3,
\\
{{p}_{\operatorname{a}}}hi_4&:=&x_4^2+{\operatorname{m}}athrm{n}u x_2x_3+\operatorname{e}ta x_1^2x_4+{\operatorname{m}}u x_1^3x_2
\operatorname{e}nd{array}
\operatorname{e}nd{equation*}
(for simplicity we assume that the coefficients $\delta$, ${\operatorname{m}}athrm{n}u$, $\operatorname{e}ta$ are
general).
Regard $X$ as a small analytic neighborhood of $C$.
In the affine chart $U_w:=\{w{\operatorname{m}}athrm{n}eq 0\}\simeq {\operatorname{m}}athbb{C}^4/{\operatorname{m}}umu_{4}(1,1,3,2)$
the variety $X$ is given by
\begin{equation*}
{{p}_{\operatorname{a}}}hi_6(y_1,y_2,y_3,y_4, 1) - y_1^6{{p}_{\operatorname{a}}}hi_4(y_1,y_2,y_3,y_4, 1)=0
\operatorname{e}nd{equation*}
and $C$ is the $y_1$-axis.
Clearly, it has the form \operatorname{e}qref{equation-alpha}.
So, the origin $P\in (X,C)$ is a type~\typec{IIA} point with $\operatorname{e}ll(P)=4$.
In the affine chart $U_1:=\{x_1{\operatorname{m}}athrm{n}eq 0\}\simeq {\operatorname{m}}athbb{C}^4$
the variety $X$ is defined by
\begin{equation*}
w{{p}_{\operatorname{a}}}hi_6(1,z_2,z_3,z_4, w) - {{p}_{\operatorname{a}}}hi_4(1,z_2,z_3,z_4, w)=0.
\operatorname{e}nd{equation*}
If ${\operatorname{m}}u{\operatorname{m}}athrm{n}eq 0$, then $X$ is smooth outside $P$, i.e. $(X,C)$ is as
in the case \xref{IIA:normal:cb}. If ${\operatorname{m}}u=0$, then
$(X,C)$ has a type~\typec{III} point at $(0,0,0,\operatorname{e}ta)$.
We claim that $(X,C)$ admits a structure of a ${\operatorname{m}}athbb{Q}$-conic bundle germ
as in \xref{IIA:n-normal-cb} (resp. \xref{IIA:normal:cb}) if
${\operatorname{m}}u=0$ (resp. ${\operatorname{m}}u{\operatorname{m}}athrm{n}eq 0$).
\operatorname{e}nd{example}
\begin{proof}
Consider the surface $H=\{{{p}_{\operatorname{a}}}hi_6={{p}_{\operatorname{a}}}hi_4=0\}\subset X$.
Let ${{p}_{\operatorname{a}}}si: H^{{\operatorname{m}}athrm{n}}\to H$ be the normalization (we put $H^{{\operatorname{m}}athrm{n}}=H$ if $H$ is normal)
and let $C^{{\operatorname{m}}athrm{n}}:={{p}_{\operatorname{a}}}si^{-1}(C)$.
One can explicitly check that $H$ is normal and smooth outside $P$ if ${\operatorname{m}}u {\operatorname{m}}athrm{n}eq 0$
and $H$ is singular along $C$, the curve
$C^{{\operatorname{m}}athrm{n}}$ is irreducible and rational, and ${{p}_{\operatorname{a}}}si_C:= C^{{\operatorname{m}}athrm{n}}\to C$ is a double cover
if ${\operatorname{m}}u =0$.
Moreover, the singularities of $H^{{\operatorname{m}}athrm{n}}$ are rational.
Note that $H$ is a fiber of the fibration ${{p}_{\operatorname{a}}}i: X\to D$ over a small disk around the origin
given by the rational function $ {{p}_{\operatorname{a}}}hi_4/w ={{p}_{\operatorname{a}}}hi_6/x_1^6$
which is regular in a neighborhood of $C$.
Analyzing the minimal resolution
one can show that
there exists a rational curve fibration $f_H: H\to B$, where $B\subset {\operatorname{m}}athbb{C}$
is a small disk around the origin, such that $C=f_H^{-1}(0)_{\operatorname{red}}$.
Now the existence of a contraction is a consequence of the following.
\operatorname{e}nd{proof}
\begin{sclaim}
\label{IIA:claim:exist}
\begin{enumerate}
\item \label{IIA:claim:exist1}
$H^1(\hat X,{{\operatorname{m}}athscr{O}}_{\hat X})=0$,
where $\hat X$ denotes the completion of $X$ along $C$.
\item \label{IIA:claim:exist2}
The contraction {\operatorname{m}}box{$f_H: H\to B$} extends to a contraction {\operatorname{m}}box{$\hat f: \hat X\to \hat Z$}.
\item \label{IIA:claim:exist3}
There exists a contraction {\operatorname{m}}box{$f:X\to Z$} that approximates {\operatorname{m}}box{$\hat f: \hat X\to \hat Z$}.
\operatorname{e}nd{enumerate}
\operatorname{e}nd{sclaim}
\begin{proof}
For \xref{IIA:claim:exist1} we refer to \cite[4.8.4]{MP:IIA-2}.
\xref{IIA:claim:exist2}
Since $H^1({{\operatorname{m}}athscr{O}}_{\hat X})=0$, from the exact sequence
\begin{equation*}
0 \xrightarrow{\hspace*{20pt}} {{\operatorname{m}}athscr{O}}_X \xrightarrow{\hspace*{20pt}} {{\operatorname{m}}athscr{O}}_X (H) \xrightarrow{\hspace*{20pt}} {{\operatorname{m}}athscr{O}}_H (H)\xrightarrow{\hspace*{20pt}} 0
\operatorname{e}nd{equation*}
we see that the map $H^0({{\operatorname{m}}athscr{O}}_{\hat X} (\hat H))\to H^0({{\operatorname{m}}athscr{O}}_{\hat H} (\hat H))$
is surjective.
Hence there exists a divisor $\hat H_1\in |{{\operatorname{m}}athscr{O}}_{\hat X}|$ such that $\hat H_1|_{\hat H}=\hat {\operatorname{m}}athbb{C}C$. Then the divisors
$\hat H$ and $\hat H_1$ define a contraction $\hat f: \hat X\to \hat Z$.
\xref{IIA:claim:exist3}
Let $F$ be the scheme fiber of $f_H: H\to B$ over the origin.
The above arguments shows that the deformations of $F$ are unobstructed.
Therefore the corresponding component of the Douady space is smooth and two-dimensional.
This allow us to produce a contraction on $X$.
\operatorname{e}nd{proof}
\begin{example}[{\cite[4.9]{MP:IIA-2}}]
\label{example-conic-bundle-lP=8}
Similarly to Example \xref{example-conic-bundle-lP=4+III}, let
$X\subset {\operatorname{m}}athbb{P}(1,1,3,2,4)$ be a small analytic neighborhood of $C=
\{\text{$(x_1,w)$-line}\}$ given by the equation
$x_1^6{{p}_{\operatorname{a}}}hi_4-w {{p}_{\operatorname{a}}}hi_6 =0$, where
\begin{eqnarray*}
{{p}_{\operatorname{a}}}hi_6&:=&x_3^2+x_2^2w+\delta x_4^3+cx_1^2x_4^2,
\\
{{p}_{\operatorname{a}}}hi_4&:=&x_4^2+{\operatorname{m}}athrm{n}u x_2x_3+\operatorname{e}ta x_1^2x_4.
\operatorname{e}nd{eqnarray*}
It is easy to check that $P:=(0:0:0:0:1)$ is the only
singular point of $X$ on $C$ and it is a type~\typec{IIA} point with $\operatorname{e}ll(P)=8$.
The rational function ${{p}_{\operatorname{a}}}hi_4/w={{p}_{\operatorname{a}}}hi_6/x_1^6$ near $C$ defines a fibration
whose central fiber $H$ is given by ${{p}_{\operatorname{a}}}hi_4={{p}_{\operatorname{a}}}hi_6 =0$ such that ${\operatorname{D}}elta(H,C)$ is
of type \xref{IIA:n-normal-cb}.
The existence of a contraction
$f: X\to Z$ can be shown similar to Claim \xref{IIA:claim:exist}.
\operatorname{e}nd{example}
\begin{example}[{\cite[4.9.1]{MP:IIA-2}}]
\label{example-conic-bundle-normal-H}
In a similar way we can construct an example of a ${\operatorname{m}}athbb{Q}$-conic bundle with $\operatorname{e}ll(P)=5$
and normal $H$ as in \xref{IIA:normal:cb}.
Consider $X\subset {\operatorname{m}}athbb{P}(1,1,3,2,4)$ given by $w{{p}_{\operatorname{a}}}hi_6-x_1^6{{p}_{\operatorname{a}}}hi_4=0$, where
\begin{eqnarray*}
{{p}_{\operatorname{a}}}hi_6&:=&x_1^5 x_2+x_2^2w+x_3^2+\delta x_4^3+cx_1 ^2 x_4^2
\operatorname{e}nd{eqnarray*}
and ${{p}_{\operatorname{a}}}hi_4$ is as in \xref{example-conic-bundle-lP=4+III}.
In the affine chart $U_w\simeq {\operatorname{m}}athbb{C}^4/{\operatorname{m}}umu_{4}(1,1,3,2)$
the origin $P\in (X,C)$ is a type~\typec{IIA} point with $\operatorname{e}ll(P)=5$.
It is easy to see that $X$ is smooth outside $P$.
The rational function ${{p}_{\operatorname{a}}}hi_4/w={{p}_{\operatorname{a}}}hi_6/x_1^6$ defines
a fibration on $X$ near $C$ with central fiber $H=\{{{p}_{\operatorname{a}}}hi_4={{p}_{\operatorname{a}}}hi_6=0\}$.
\operatorname{e}nd{example}
\appendix
\renewcommand{\Alph{section}.\arabic{subsection}.\arabic{equation}}{{\operatorname{A}}lph{section}.\arabic{subsection}.\arabic{equation}}
\renewcommand{\Alph{section}.\arabic{subsection}}{{\operatorname{A}}lph{section}.\arabic{subsection}}
\section{A remark on divisorial contractions}
\begin{proposition}\label{prop:mult}
Let $(X,C\simeq {\operatorname{m}}athbb{P}^1)$ be a divisorial curve germ
with one non-Gorenstein point which is not of type~\type{cA/m} with $m>2$
and let
$f : (X, C) \to (Z,o)$ be corresponding contraction.
Let $E\subset X$ be the exceptional divisor and let $B:=f(E)$ be the
blowup curve. Then the multiplicity ${\operatorname{m}}ult_{o}(B)$ is given by the following table.
{\rm
\begin{center}
\begin{tabularx}{1\textwidth}{lXcll}
$(X,C)$ & ${\operatorname{D}}elta(H,C)$ &{\operatorname{m}}box{${\operatorname{m}}ult_{o}(B)$}&$H_Z$&$D_Z$
\\
\hline
\\[-9pt]
\typec{IIA}&\xref{IIA:normalA1:a}, \xref{IIA:normalA1:b} & 3
&\type{A_1}&\type{D_{2n+1}}
\\
\typec{IIA}&\xref{IIA:normalD5}, \xref{IIA-n-normal-div} &1
&\type{D_5}&\type{D_{2n+1}}
\\
\type{(IIB)}& \xref{IIB:thm-A2case-simple}, \xref{IIB:thm-D4case-double} & 2
&\type{A_2}, \type{D_4}&\type{E_6}
\\
\type{(IIB)}& \xref{IIB:thm-smooth-case-simple} & 5
&\type{A_0}&\type{E_6}
\\
\type{cD/3}& \xref{cD/3:thm:A2}, \xref{cD/3:thm:D4} & 2
&\type{A_2}, \type{D_4}&\type{E_6}
\\
\type{cD/3}& \xref{cD/3:thm:E6} & 1
&\type{E_6}&\type{E_6}
\\
\type{cA/2}&\xref{KM:4.7.3.1.1}& $n$&\type{A_1}&\type{A}
\\
\type{cA/2}&\xref{KM:4.7.3.1.2}& 3&\type{A_0}&\type{A}
\\
\type{cA/2}& \xref{KM:4.7.3.1.3} & 1&\type{A_2}&\type{A}
\\
\type{cA/2}& \xref{KM:4.7.3.1.4}& 4&\type{A_0}&\type{A}
\\
\type{cAx/2}, \type{cD/2}&\xref{KM:4.7.4}-\xref{KM:4.7.5} &1&\type{D}&\type{D_{}}
\\
\type{cE/2}& \xref{KM:4.7.4}-\xref{KM:4.7.6} &1&\type{D}, \type{E_6}&\type{E_7}
\operatorname{e}nd{tabularx}
\operatorname{e}nd{center}}
{\operatorname{m}}athrm{n}oindent
where $H_Z$ is a general hyperplane section of $(Z,o)$
and $D_Z$ is a general hyperplane section of $(Z,o)$ passing through $B$.
In the \type{cA/2}-case the meaning of $n$ is the same as in \xref{KM:4.7.3.1.1}.
\operatorname{e}nd{proposition}
The cases with ${\operatorname{m}}ult_{o}(B)=1$, i.e. those with smooth $B$, were studied
in details by N.~Tziolas
\cite{Tzi:03}, \cite{Tzi:05}, \cite{Tzi:05D}, \cite{Tzi:10}.
\begin{proof}
Recall that $Z$ is ${\operatorname{m}}athbb{Q}$-Gorenstein and $E$ is ${\operatorname{m}}athbb{Q}$-Cartier divisor
(Theorem~\xref{thm:div:Q-Cartier}).
By classification in all our cases $Z$ is in fact Gorenstein
(that is, $H_Z$ has at worst Du Val singularity).
Hence, $E\in |K_X|$.
Let $H:=f^*(H_Z)$. Let $D\in |-K_X|$ be a general member.
We have
$-K_X\cdot C=1/m$,
where
$m$ is the index of the non-Gorenstein point (see \xref{lemma:KC}).
For simplicity assume that $H$ is normal.
The case \xref{IIA-n-normal-div} can be treated in a similar way.
\begin{lemma}[{\cite[Lemma~5.1]{Tzi:05}}]
If in the above notation $H$ is normal, then
\begin{equation}
{\operatorname{m}}ult_o(B)=-\frac{(K_C\cdot C)^2}{(C^2)_H}=-\frac{1}{m^2(C^2)_H}.
\operatorname{e}nd{equation}
\operatorname{e}nd{lemma}
Now let ${{p}_{\operatorname{a}}}si: \hat H\to H$ be the minimal resolution.
Write ${{p}_{\operatorname{a}}}si^* C= \hat C+ \Theta$,
where ${\operatorname{Supp}}(\Theta)\subset \operatorname{Exc}({{p}_{\operatorname{a}}}si)$ and $\Theta=\sum \theta_i\Theta_i$.
Since $\hat C^2=-1$, we have
\begin{equation}
\label{equation-proposition-C2}
C^2= -1+\hat C\cdot \Theta=-1+\sum{\operatorname{m}}athrm{n}olimits' \theta_i,
\operatorname{e}nd{equation}
where $\sum'$ runs through all the components $\Theta_i$ meeting $\hat C$.
The coefficients $\theta_i$ are computed from the standard system
of linear equations:
\begin{equation*}
0=-\Theta_j\cdot{{p}_{\operatorname{a}}}si^* C=\Theta_j\cdot \hat C+\sum_i \theta_i\Theta_j\cdot \Theta_i.
\operatorname{e}nd{equation*}
Now ${\operatorname{m}}ult_{o}(B)$ can be computed by using \operatorname{e}qref{equation-proposition-C2}.
Consider for example the cases
\xref{IIA:normalA1:a}, \xref{IIA:normalA1:b} and \xref{IIA:normalD5} of Theorem \xref{IIA:thm}
(other cases are similar).
In the graphs below
we attach the coefficients $\theta_i$ of $\Theta={{p}_{\operatorname{a}}}si^*C-\hat C$
to the corresponding vertices and indicate the value of $C^2$. This immediately
gives us the values of ${\operatorname{m}}ult_o(B)$ as desired.\qedhere
\begin{align*}\text{\xref{IIA:normalA1:a}}&
\xymatrix@R=0pt@C=17pt{
&&&& &\overset{2/16}\circ\ar@{-}[d]
\\
&\underset{1/3}\circ\ar@{-}[r]&\underset{2/3}\circ\ar@{-}[r]&\underset{1}\bullet\ar@{-}[r]&
\underset{5/16}\circ\ar@{-}[r]
&\underset{4/16}\circ\ar@{-}[r]&\underset{1/16}\circ&
}
&
\scriptstyle C^2=-1/48
\\
\text{{\xref{IIA:normalA1:b}}}&
\xymatrix@R=0pt@C=17pt{
&&&& &\overset{7/48}{\circ}\ar@{-}[d]
\\
&&\underset{1/2}\circ\ar@{-}[r]&\underset{1}\bullet\ar@{-}[r]&\underset{23/48}\circ\ar@{-}[r]
&\underset{7/16}\circ\ar@{-}[r]&\underset{1/4}\circ\ar@{-}[r]&\underset{1/16}\circ
}
&
\scriptstyle C^2=-1/48
\\
\text{\xref{IIA:normalD5}}&
\vcenter{
\xymatrix@R=0pt@C=17pt{
&&&\overset{7/16}\circ\ar@{-}[d]&\overset{3/16}\circ\ar@{-}[d]&
\overset{3/16}\circ\ar@{-}@/^7pt/[dl]
\\
&\underset 1 \bullet\ar@{-}[r]&\underset{15/16}\circ\ar@{-}[r]&\underset{7/8}\circ\ar@{-}[r]
&\underset{3/8}\circ\ar@{-}[r]&\underset{1/4}\circ\ar@{-}[r]&\underset{1/8}\circ
}}
&\scriptstyle C^2=-1/16
\operatorname{e}nd{align*}
\operatorname{e}nd{proof}
\def$'${$'$}
\begin{thebibliography}{CKM88}
\bibitem[CKM88]{CKM}
Herbert Clemens, J{\'a}nos Koll{\'a}r, and Shigefumi Mori.
{\operatorname{m}}athrm{n}ewblock {Higher-dimensional complex geometry}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Ast{\'e}risque}, (166):144 pp. (1989), 1988.
\bibitem[Cut88]{Cutkosky-1988}
Steven Cutkosky.
{\operatorname{m}}athrm{n}ewblock {Elementary contractions of {G}orenstein threefolds}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Math. Ann.}, 280(3):521--525, 1988.
\bibitem[Duc16]{Ducat:16}
Tom Ducat.
{\operatorname{m}}athrm{n}ewblock {Divisorial extractions from singular curves in smooth 3-folds}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Internat. J. Math.}, 27(1):1650005, 23, 2016.
\bibitem[Feu71]{Feuer-1971}
R.~D. Feuer.
{\operatorname{m}}athrm{n}ewblock {Torsion-free subgroups of triangle groups}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Proc. Amer. Math. Soc.}, 30:235--240, 1971.
\bibitem[HTU17]{HTU}
Paul Hacking, Jenia Tevelev, and Giancarlo Urz{\'u}a.
{\operatorname{m}}athrm{n}ewblock {Flipping surfaces}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m J. Algebraic Geom.}, 26(2):279--345, 2017.
\bibitem[Ish14]{Ishii:book}
Shihoko Ishii.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m {Introduction to singularities}}.
{\operatorname{m}}athrm{n}ewblock Springer, Tokyo, 2014.
\bibitem[Isk96]{I96}
V.~A. Iskovskikh.
{\operatorname{m}}athrm{n}ewblock {On a rationality criterion for conic bundles}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Sbornik: Math.}, 187(7):1021--1038, 1996.
\bibitem[Kaw88]{Kaw:Crep}
Yujiro Kawamata.
{\operatorname{m}}athrm{n}ewblock {Crepant blowing-up of {$3$}-dimensional canonical singularities and
its application to degenerations of surfaces}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Ann. of Math. (2)}, 127(1):93--163, 1988.
\bibitem[Kaw92]{Kawamata:discr}
Yujiro Kawamata.
{\operatorname{m}}athrm{n}ewblock {The minimal discrepancy coefficients of terminal singularities in
dimension three. {A}ppendix to {V}.{V}. {S}hokurov's paper {"}3-fold log
flips{"}}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Russ. Acad. Sci., Izv., Math.}, 40(1):95--202, 1992.
\bibitem[Kaw96]{Kaw:div}
Yujiro Kawamata.
{\operatorname{m}}athrm{n}ewblock {Divisorial contractions to {$3$}-dimensional terminal quotient
singularities}.
{\operatorname{m}}athrm{n}ewblock In {\operatorname{e}m {Higher-dimensional complex varieties (Trento, 1994)}}, pages
241--246. de Gruyter, Berlin, 1996.
\bibitem[Kaw97]{Kaw97}
Yujiro Kawamata.
{\operatorname{m}}athrm{n}ewblock {On {F}ujita's freeness conjecture for {$3$}-folds and {$4$}-folds}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Math. Ann.}, 308(3):491--505, 1997.
\bibitem[Kaw07]{Kawakita2007}
Masayuki Kawakita.
{\operatorname{m}}athrm{n}ewblock {Inversion of adjunction on log canonicity}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Invent. Math.}, 167:129--133, 2007.
\bibitem[KM92]{KM92}
J{\'a}nos Koll{\'a}r and Shigefumi Mori.
{\operatorname{m}}athrm{n}ewblock {Classification of three-dimensional flips}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m J. Amer. Math. Soc.}, 5(3):533--703, 1992.
\bibitem[KM98]{KM:book}
J{\'a}nos Koll{\'a}r and Shigefumi Mori.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m {Birational geometry of algebraic varieties}}, volume 134 of
{\operatorname{e}m {Cambridge Tracts in Mathematics}}.
{\operatorname{m}}athrm{n}ewblock Cambridge University Press, Cambridge, 1998.
{\operatorname{m}}athrm{n}ewblock With the collaboration of C. H. Clemens and A. Corti, Translated from
the 1998 Japanese original.
\bibitem[Kol86]{Kollar-1986-I}
J{\'a}nos Koll{\'a}r.
{\operatorname{m}}athrm{n}ewblock {Higher direct images of dualizing sheaves. {I}}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Ann. of Math. (2)}, 123(1):11--42, 1986.
\bibitem[Kol92]{Utah}
J{\'a}nos Koll{\'a}r, editor.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m {Flips and abundance for algebraic threefolds}}.
{\operatorname{m}}athrm{n}ewblock Soci{\'e}t{\'e} Math{\'e}matique de France, Paris, 1992.
{\operatorname{m}}athrm{n}ewblock Papers from the Second Summer Seminar on Algebraic Geometry held at
the University of Utah, Salt Lake City, Utah, August 1991, Ast{\'e}risque No.
211 (1992).
\bibitem[Kol99]{Kollar-1999-R}
J{\'a}nos Koll{\'a}r.
{\operatorname{m}}athrm{n}ewblock {Real algebraic threefolds. {III}. {C}onic bundles}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m J. Math. Sci. (New York)}, 94(1):996--1020, 1999.
{\operatorname{m}}athrm{n}ewblock Algebraic geometry, 9.
\bibitem[KSB88]{KSh88}
J.~Koll{\'a}r and N.~I. Shepherd-Barron.
{\operatorname{m}}athrm{n}ewblock Threefolds and deformations of surface singularities.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Invent. Math.}, 91(2):299--338, 1988.
\bibitem[LW86]{LW86}
Eduard Looijenga and Jonathan Wahl.
{\operatorname{m}}athrm{n}ewblock {Quadratic functions and smoothing surface singularities}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Topology}, 25(3):261--291, 1986.
\bibitem[Mor75]{Mori:ci}
Shigefumi Mori.
{\operatorname{m}}athrm{n}ewblock {On a generalization of complete intersections}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m J. Math. Kyoto Univ.}, 15(3):619--646, 1975.
\bibitem[Mor85]{Mori:sing}
Shigefumi Mori.
{\operatorname{m}}athrm{n}ewblock {On {$3$}-dimensional terminal singularities}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Nagoya Math. J.}, 98:43--66, 1985.
\bibitem[Mor88]{Mori:flip}
Shigefumi Mori.
{\operatorname{m}}athrm{n}ewblock {Flip theorem and the existence of minimal models for {$3$}-folds}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m J. Amer. Math. Soc.}, 1(1):117--253, 1988.
\bibitem[Mor02]{Mori:ss}
Shigefumi Mori.
{\operatorname{m}}athrm{n}ewblock {On semistable extremal neighborhoods}.
{\operatorname{m}}athrm{n}ewblock In {\operatorname{e}m {Higher dimensional birational geometry (Kyoto, 1997)}},
volume~35 of {\operatorname{e}m {Adv. Stud. Pure Math.}}, pages 157--184. Math. Soc. Japan,
Tokyo, 2002.
\bibitem[MP08a]{MP:cb1}
Shigefumi Mori and Yuri Prokhorov.
{\operatorname{m}}athrm{n}ewblock {On {${\operatorname{m}}athbf Q$}-conic bundles}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Publ. Res. Inst. Math. Sci.}, 44(2):315--369, 2008.
\bibitem[MP08b]{MP:cb2}
Shigefumi Mori and Yuri Prokhorov.
{\operatorname{m}}athrm{n}ewblock {On {${\operatorname{m}}athbf Q$}-conic bundles. {II}}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Publ. Res. Inst. Math. Sci.}, 44(3):955--971, 2008.
\bibitem[MP09]{MP:cb3}
S.~Mori and Yu. Prokhorov.
{\operatorname{m}}athrm{n}ewblock {On {${\operatorname{m}}athbf Q$}-conic bundles, {III}}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Publ. Res. Inst. Math. Sci.}, 45(3):787--810, 2009.
\bibitem[MP11]{MP:IA}
S.~Mori and Yu. Prokhorov.
{\operatorname{m}}athrm{n}ewblock {Threefold extremal contractions of type {IA}}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Kyoto J. Math.}, 51(2):393--438, 2011.
\bibitem[MP14]{MP:ICIIB}
S.~Mori and Yu. Prokhorov.
{\operatorname{m}}athrm{n}ewblock {Threefold extremal contractions of types {(IC)} and {(IIB)}}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Proc. Edinburgh Math. Soc.}, 57(1):231--252, 2014.
\bibitem[MP16]{MP:IIA-1}
S.~Mori and Yu. Prokhorov.
{\operatorname{m}}athrm{n}ewblock {Threefold extremal contractions of type {(IIA)}, {I}}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Izv. Math.}, 80(5):884--909, 2016.
\bibitem[MP17]{MP:IIA-2}
Shigefumi Mori and Yuri Prokhorov.
{\operatorname{m}}athrm{n}ewblock {Threefold extremal contractions of type {(IIA)}, {II}}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m ArXiv e-print}, 1707.05047, 2017.
{\operatorname{m}}athrm{n}ewblock to appear in Festschrift in honour of Nigel Hitchin, (2018), Oxford
University Press.
\bibitem[PR16]{Prokhorov-Reid}
Yuri Prokhorov and Miles Reid.
{\operatorname{m}}athrm{n}ewblock {On {Q}-{F}ano threefolds of {F}ano index 2}.
{\operatorname{m}}athrm{n}ewblock In {\operatorname{e}m {Minimal Models and Extremal Rays (Kyoto 2011)}}, volume~70
of {\operatorname{e}m {Adv. Stud. Pure Math.}}, pages 397--420. Mathematical Society of
Japan, Kinokuniya, Tokyo, 2016.
\bibitem[Pro97]{P97}
Yu. Prokhorov.
{\operatorname{m}}athrm{n}ewblock {On the existence of complements of the canonical divisor for {M}ori
conic bundles}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Russian Acad. Sci. Sb. Math.}, 188(11):1665--1685, 1997.
\bibitem[Pro04]{P04:s}
Yu. Prokhorov.
{\operatorname{m}}athrm{n}ewblock {On semistable {M}ori contractions}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Izv. Math.}, 68(2):147--158, 2004.
\bibitem[Pro18]{P17}
Yuri Prokhorov.
{\operatorname{m}}athrm{n}ewblock The rationality problem for conic bundles.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Russian Math. Surveys}, 73(3), 2018.
\bibitem[Rei83]{Reid:Pagoda}
Miles Reid.
{\operatorname{m}}athrm{n}ewblock {Minimal models of canonical {$3$}-folds}.
{\operatorname{m}}athrm{n}ewblock In {\operatorname{e}m {Algebraic varieties and analytic varieties (Tokyo, 1981)}},
volume~1 of {\operatorname{e}m {Adv. Stud. Pure Math.}}, pages 131--180. North-Holland,
Amsterdam, 1983.
\bibitem[Rei87]{Reid:YPG}
Miles Reid.
{\operatorname{m}}athrm{n}ewblock Young person's guide to canonical singularities.
{\operatorname{m}}athrm{n}ewblock In {\operatorname{e}m {Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine,
1985)}}, volume~46 of {\operatorname{e}m {Proc. Sympos. Pure Math.}}, pages 345--414. Amer.
Math. Soc., Providence, RI, 1987.
\bibitem[SB83]{SB:degen}
N.~I. Shepherd-Barron.
{\operatorname{m}}athrm{n}ewblock {Degenerations with numerically effective canonical divisor}.
{\operatorname{m}}athrm{n}ewblock In {\operatorname{e}m {The birational geometry of degenerations ({C}ambridge,
{M}ass., 1981)}}, volume~29 of {\operatorname{e}m {Progr. Math.}}, pages 33--84.
Birkh{\"a}user Boston, Boston, MA, 1983.
\bibitem[Sho93]{Sh:flips}
V.~V. Shokurov.
{\operatorname{m}}athrm{n}ewblock {3-fold log flips}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Russ. Acad. Sci., Izv., Math.}, 40(1):95--202, 1993.
\bibitem[Tzi03]{Tzi:03}
Nikolaos Tziolas.
{\operatorname{m}}athrm{n}ewblock {Terminal 3-fold divisorial contractions of a surface to a curve. I.}
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Compos. Math.}, 139(3):239--261, 2003.
\bibitem[Tzi05a]{Tzi:05D}
Nikolaos Tziolas.
{\operatorname{m}}athrm{n}ewblock {Families of {$D$}-minimal models and applications to 3-fold
divisorial contractions}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Proc. Lond. Math. Soc., III. Ser.}, 90(2):345--370, 2005.
\bibitem[Tzi05b]{Tzi:05}
Nikolaos Tziolas.
{\operatorname{m}}athrm{n}ewblock {Three dimensional divisorial extremal neighborhoods}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Math. Ann.}, 333(2):315--354, 2005.
\bibitem[Tzi09]{Tziolas2009}
Nikolaos Tziolas.
{\operatorname{m}}athrm{n}ewblock {Q-{G}orenstein deformations of nonnormal surfaces}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Amer. J. Math.}, 131(1):171--193, 2009.
\bibitem[Tzi10]{Tzi:10}
N.~Tziolas.
{\operatorname{m}}athrm{n}ewblock {Three-fold divisorial extremal neighborhoods over {$cE_7$} and
{$cE_6$} compound {D}u{V}al singularities}.
{\operatorname{m}}athrm{n}ewblock {\operatorname{e}m Internat. J. Math.}, 21(1):1--23, 2010.
\operatorname{e}nd{thebibliography}
\operatorname{e}nd{document} | math | 199,577 |
\begin{document}
\title{My Publication Title --- Single Author}
\begin{abstract}
Multimodal traffic flow can reflect the health of the transportation system, and its prediction is crucial to urban traffic management. Recent works overemphasize spatio-temporal correlations of traffic flow, ignoring the physical concepts that lead to the generation of observations and their causal relationship. Spatio-temporal correlations are considered unstable under the influence of different conditions, and spurious correlations may exist in observations. In this paper, we analyze the physical concepts affecting the generation of multimode traffic flow from the perspective of the observation generation principle and propose a Causal Conditional Hidden Markov Model (CCHMM) to predict multimodal traffic flow. In the latent variables inference stage, a posterior network disentangles the causal representations of the concepts of interest from conditional information and observations, and a causal propagation module mines their causal relationship. In the data generation stage, a prior network samples the causal latent variables from the prior distribution and feeds them into the generator to generate multimodal traffic flow. We use a mutually supervised training method for the prior and posterior to enhance the identifiability of the model. Experiments on real-world datasets show that CCHMM can effectively disentangle causal representations of concepts of interest and identify causality, and accurately predict multimodal traffic flow.
\end{abstract}
\section{Introduction}
\label{Introduction}
Urban transportation systems are generally multimodal in nature, consisting of several interconnected subsystems representing different modes of transportation, such as bike, taxi, bus and car. They aim to meet diverse travel demands and provide residents with a variety of travel options\cite{liang2021joint}. Multimodal traffic flow can reflect the health of the transportation system. Urban traffic managers can formulate corresponding management strategies according to the traffic flow in different environments to improve the smoothness of urban operation. Therefore, multimodal traffic flow prediction is a key part of urban traffic management, providing important data support for traffic guidance \cite{liang2021joint}.
\begin{figure}
\caption{Multimodal traffic flow in different regions.}
\label{fig:1}
\end{figure}
Most methods only predict a certain traffic flow (e.g., taxi demand or speed)\cite{bai2020adaptive, li2021dynamic, guo2021hierarchical, wu2020connecting, ye2021coupled, han2021dynamic}. They are only partial observations of the traffic system and cannot truly reflect the real situation in real-world scenarios. In contrast, the existing multimodal traffic prediction methods often take different traffic flows as the channel expansion of input data\cite{IOT2021,partc2021,zhou2021modeling, liang2021fine}, or integrate the feature representation of different flows in the model\cite{ye2019co, deng2021pulse}. They implicitly extract the so-called spatio-temporal correlations while lacking the description of causality. However, more input information cannot improve the prediction ability of the model. Instead, it will introduce a large number of confounding factors and extract spurious correlations in observations\cite{comprehensive2021towards, liu2021learning, deng2021comprehensive}, resulting in the decline of model performance.
Nowadays, traffic flow prediction method overemphasizes spatio-temporal correlations of traffic flow\cite{liu2021community, bai2020adaptive, li2021dynamic, ye2021coupled, han2021dynamic}, ignoring the physical concepts that lead to the generation of observations and the causal relationship between these concepts. Spatio-temporal correlations are considered unstable under the influence of different conditions, and spurious correlations may exist in observations. Causality is necessary when we delve into the generation principle of observation. For example, researchers\cite{ye2019co, deng2021pulse} believe that there is a certain correlation between taxi and bike flow, and that they can boost each other in terms of multi-task learning. As shown in Fig. \ref{fig:1}(b), the flow of taxis and bikes seems to be correlated under normal conditions. Since people's demand for arriving or leaving a region is consistent during rush hours, the tends are similar. However, when it rains (marked in red), the demand for bikes decreases due to weather changes, but the demand for taxis increases, with diametrically opposite trends during the same period. This indicates a spurious correlation between taxis and bike flow under the influence of weather. Second, we believe that the regional attribute has a strong causal relationship with people's travel demand. As shown in Fig. \ref{fig:1}(b) and (c), the area has a strong regional attraction under the influence of the hospital attribute, leading to a large demand for people, so it has obvious rush hours in morning and noon. In addition, this area is chronically congested due to high demand(marked in green). Beijing Financial Street is the main working area with a large number of enterprises, so it has obvious morning and evening rush hours (marked in green). We provide more examples of regional POI elements affecting travel demand in appendix. Finally, the demand for taxis may have an impact on the traffic speed. As shown in Fig.\ref{fig:1}(c). The taxi demand can be inferred from flow. The larger the taxi flow, the more vehicles on the road, and the slower traffic speed (marked in blue). By contrast, High bus demand does not mean a large number of buses on the road, so there is little causal relationship between bus demand and speed.
According to the above analysis, the essential factor affecting multimodal traffic observations is the causal relationship with physical concepts, and excessive attention to the correlation will lead to unstable prediction results. we rethink the generation process of multimodal traffic flow, and explicitly separate the core physical concepts affecting the observation generation into three groups: 1) The attraction factor of the region to people in different time periods. 2) The demand factor (including bikes, taxis and buses) of people choosing different transportation modes under different conditions, and 3) The speed factor affected by the number of vehicles on the road. Our primary task is to disentangle the causal representation of these concepts from conditional information and observations, and further explore their causal relationship.
In this paper, we regard the spatio-temporal multimodal traffic sequence generation process as a Conditional Markov Process, and propose a Causal Conditional Hidden Markov Model (CCHMM). We disentangle the underlying explanatory factors by means of Variational inference, and establish the causal relationship between latent variables by using the Structural Causal Model (SCM)\cite{pearl2009causality, scholkopf2022causality}. Compared with the existing work, instead of building a complex adjacency graph between regions to extract the spatio-temporal correlations in the observation data, we model multimodal traffic flow prediction from a causal perspective. The theoretical innovation in the field of traffic forecasting is as follows: Based on the idea of causality, we model the operation process of multimodal traffic systems from the perspective of the observation generation principle, while the existing methods do not focus on causality in the observation data. We propose a causal graph (shown in Fig. \ref{fig:2}) to describe the operation of multimodal traffic systems, on which we define a joint distribution (shown in Eq. \ref{equ:1}) that describes the principle of observation data generation. Specifically, first, the posterior network infers the disentangled representation of concepts of interest from conditional information and observation data and learns the variational posterior distribution. Then, the prior network models the natural physical laws that existed in the system from the conditional information and learns the prior distribution of the concepts of interest. Third, the causal propagation module mines the causal effects and transforms the exogenous variables inferred from the prior and posterior networks into causal endogenous variables. Finally, The causal endogenous variables are fed into the generator to generate multimodal traffic flow and regarded as the prediction results. The main contributions of this work are as follows:
\begin{itemize}
\item We analyze the core physical concepts that affect the multimodal traffic flow generation process, disentangle the causal representations of concepts of interest, and further explore their causal relationship.
\item We reform the previous prediction methods and innovatively propose a Causal Conditional Hidden Markov Model (CCHMM) to predict multimodal traffic flow from the perspective of observation generation principle.
\item We propose a mutually supervised training method for the prior and posterior to capture physical rules of concepts and enhance the causal identifiability of the model.
\item Extensive experiments on real-world datasets show that CCHMM comprehensively outperforms state-of-the-art methods for multimodal traffic flow prediction.
\end{itemize}
\section{Related works}
\textbf{Multimodal traffic flow prediction.} Giving the increasing availability of diverse data sources, most recent studies has focused on the multimodal fusion in traffic flow prediction. Researchers construct models based on multi-task learning framework to forecast traffic flow and speed simultaneously\cite{IOT2021,partc2021}. Ye \cite{ye2019co} et al. decompose spatial traffic flow with a convolutional autoencoder and implement heterogeneous LSTM for predicting traffic flow of three traffic modes simultaneously. Deng \cite{deng2021pulse} et al. learn multi-view representations for single-modal traffic flow and introduce a cross-view self-attention mechanism to capture the co-evolution correlation between different traffic modes. Most of these works implemented Multilayer Perception (MLP) for encoding conditional information(e.g. weather and POI)\, utilized CNN\cite{liang2021fine, cao2021bert} or Graph Convolutional networks (GCN)\cite{graphwavenet2019, han2021dynamic} for capturing spatial features and used RNN for temporal features\cite{ye2021coupled, li2021dynamic,bai2020adaptive}. Finally, the fused features are fed into downstream prediction network. However, these models do not distinguish the features related to different tasks, which make models learn spurious correlations during the training process. The spurious correlations make models difficult to generalize beyond their training distribution.
\textbf{Causal disentangled representation learning.} In representation learning, the observation $x$ is generated by a two-step generative process. First, the latent variable $z$ is sampled from a prior distribution $p(z)$, and then the observation $x$ is sampled from the conditional distribution $p(x|z)$\cite{locatello2019challenging}. Disentangled representation learning aims to learn separable latent variables $z=\left\{z_1,z_2,\dots,z_n\right\}$. Most existing methods rely on the independency assumption of latent variables which is potentially unrealistic\cite{khemakhem2020variational}. In fact, there is generally a complex causal relationship between latent variables\cite{yang2021causalvae}. To address this issue, recent works are proposed to combine SCM \cite{pearl2009causality, scholkopf2022causality} with deep learning models.
CasualVAE \cite{yang2021causalvae} proposes a model with causal layer to transform exogenous factors into causal endogenous ones that correspond to causally related concepts in data. Shen \cite{shen2020disentangled}et al. use a SCM as the prior for bidirectional generative model which can generate data from any desired interventional distributions of the latent factors. Different from above works, our model focused on causal disentangled representation learning on spatial-temporal series. Li \cite{li2021causal} et al. propose a time series disease forecasting method based on HMM. Although this method can disentangle the latent variables that are related to disease, while ignores the casual relationship among factors. In our model, we construct a comprehensive temporal causal graph for conditional information, latent variables and observation data. To the best of our knowledge, our work is the first one that successfully applies the structural causal model to traffic prediction problems.
\section{Methodology}
\label{Methodology}
\subsection{Problem Definition}
\label{Problem Definition}
\begin{figure}
\caption{The causal graph of multimodal transportation systems}
\label{fig:2}
\end{figure}
We define the generation process of multimodal traffic flow as a Conditional Markov Process, illustrated as a Directed Acyclic Graph(DAG), as shown in Fig \ref{fig:2}. For the latent variable inference stage at time step $t$, the conditional information $\mathbf{C}_t$ composed of $POI$, time position $TP_t$ and weather $WX_t$ reflects the current system external status. The conditional information is combined with causal endogenous latent variables $\mathbf{z}_{t-1}$ from the previous time step $t-1$ to extract independent exogenous variables $ \boldsymbol{ \epsilon}_{t}=\left[ \boldsymbol{\epsilon}_{t}^{p o i}, \boldsymbol{\epsilon}_{t}^{{bike }}, \boldsymbol{\epsilon}_{t}^{{taxi }}, \boldsymbol{\epsilon}_{t}^{b u s}, \boldsymbol{\epsilon}_{t}^{{v }}\right] $, which is determined by the system external status and are not affected by observations. Then, the Structural Causal Model (SCM) $\mathbf{z}_i \leftarrow f(pa(\mathbf{z}_i), \boldsymbol{\epsilon}_i) $\cite{scholkopf2022causality} assigns the generative mechanism of each latent endogenous variable, where $pa(\mathbf{z}_i)$ denotes the set of parent nodes of $\mathbf{z}_i$. It transforms the independent exogenous variables to causal endogenous variables $\mathbf{z}_{t}=\left[ \mathbf{z}_{t}^{p o i}, \mathbf{z}_{t}^{{bike }}, \mathbf{z}_{t}^{{taxi }}, \mathbf{z}_{t}^{{bus }}, \mathbf{z}_{t}^{ {v }}\right]$.
The causal endogenous latent variables $\mathbf{z}_{t}$ are regarded as an approximate representation of a series of concepts of interest, where the elements represent the regional attraction factor, bike demand factor, taxi demand factor, bus demand factor and speed factor at time $t$, respectively. Since these latent variables evolve as intrinsic drivers for the progression of multimodal traffic observations, the prior distribution of the latent variables has Markov property and is defined as $p( \boldsymbol{ \epsilon}_t, \mathbf{z}_t|\mathbf{z}_{t-1}, \mathbf{C}_t)=p( \boldsymbol{ \epsilon}_t|\mathbf{z}_{t-1}, \mathbf{C}_t) * p(\mathbf{z}_t| \boldsymbol{ \epsilon}_t)$.
For the data generation stage at time step $t$, the exogenous latent variables $\boldsymbol{ \epsilon}_t$ are sampled from the prior distribution $p( \boldsymbol{ \epsilon}_t|\mathbf{z}_{t-1}, \mathbf{C}_t)$, The causal endogenous latent variables $\mathbf{z}_{t}$ are generated using a SCM. Finally, the observations $\mathbf{x}_{t}=\left[\mathbf{x}_{t}^{b i k e}, \mathbf{x}_{t}^{t a x i}, \mathbf{x}_{t}^{b u s}, \mathbf{x}_{t}^{{v }}\right]$ are generated from the conditional distribution $p(\mathbf{x}_t|\mathbf{z}_t)$.
\subsection{A Probabilistic Generative Model for CCHMM}
\label{A Probabilistic Generative Model for CCHMM}
We give the joint distribution definition of the probabilistic generative model of CCHMM and factorize it according to the DAG (Fig. \ref{fig:2}) and Causal Markov Condition\cite{pearl2009causality}:
\begin{equation}\label{equ:1}
\begin{aligned}
& p_{\theta}\left(\mathbf{x}_{t<T}, \boldsymbol{ \epsilon}_{t<T}, \mathbf{z}_{t<T} \mid \mathbf{C}_{t<T}\right) \\
&=\prod_{t=1}^{T-1} p_{\theta}\left( \boldsymbol{ \epsilon}_{t}, \mathbf{z}_{t} \mid \mathbf{z}_{t-1}, \mathbf{C}_{t}\right) * p_{\theta}\left(\mathbf{x}_{t} \mid \mathbf{z}_{t}\right)
\end{aligned}
\end{equation}
The first term is the prior model, which can be further factored into the generative mechanism of exogenous and endogenous variables based on the causal relationship:
\begin{equation}\label{equ:2}
\begin{aligned}
p_{\theta}\left( \boldsymbol{ \epsilon}_{t}, \mathbf{z}_{t} \mid \mathbf{z}_{t-1}, \mathbf{C}_{t}\right)=p_{\theta}\left( \boldsymbol{ \epsilon}_{t} \mid \mathbf{z}_{t-1}, \mathbf{C}_{t}\right)* p_{\theta}\left(\mathbf{z}_{t} \mid \boldsymbol{ \epsilon}_{t}\right)
\end{aligned}
\end{equation}
The second item is the generative model, which can be further factored into generative models for each modality depending on endogenous variables corresponding to concepts of interest:
\begin{equation}\label{equ:3}
\begin{aligned}
p_{\theta}\left(\mathbf{x}_{t} \mid \mathbf{z}_{t}\right)
&=p_{\theta}\left(\mathbf{x}_{t}^{b i k e} \mid \mathbf{z}_{t}^{b i k e}\right) * p_{\theta}\left(\mathbf{x}_{t}^{t a x i} \mid \mathbf{z}_{t}^{t a x i}\right) \\
& * p_{\theta}\left(\mathbf{x}_{t}^{b u s} \mid \mathbf{z}_{t}^{b u s}\right) * p_{\theta}\left(\mathbf{x}_{t}^{v} \mid \mathbf{z}_{t}^{v}\right)
\end{aligned}
\end{equation}
We apply variational Bayes to learn a tractable distribution $q_{\phi}$ to approximate the true posterior $p_{\theta}$, defined as follows:
\begin{equation}\label{equ:4}
\begin{aligned}
q_{\phi}\left( \boldsymbol{ \epsilon}_{t<T}, \mathbf{z}_{t<T} \mid \mathbf{x}_{t<T}, \mathbf{C}_{t<T}\right) &=\prod_{t=1}^{T-1} q_{\phi}\left( \boldsymbol{ \epsilon}_{t} \mid \mathbf{z}_{t-1}, \mathbf{x}_{t}, \mathbf{C}_{t}\right) \\
&* q_{\phi}\left(\mathbf{z}_{t} \mid \boldsymbol{ \epsilon}_{t}\right)
\end{aligned}
\end{equation}
\subsection{Causal Conditional Hidden Markov Model}
\label{Causal Conditional Hidden Markov Model}
\begin{figure*}
\caption{The architecture of CCHMM.}
\label{fig:3}
\end{figure*}
To model Causal Conditional Hidden Markov Model based on the above probabilistic generative model, as shown in Fig. \ref{fig:3}, our main tasks are as follows: (1) In the latent variable inference stage, a deep neural network is used to fit the prior distributions $p_{\theta}\left( \boldsymbol{ \epsilon}_{t}, \mathbf{z}_{t} \mid \mathbf{z}_{t-1}, \mathbf{C}_{t}\right)$ and posterior distributions $q_{\phi}\left( \boldsymbol{ \epsilon}_{t}, \mathbf{z}_{t} \mid \mathbf{z}_{t-1}, \mathbf{x}_{t}, \mathbf{C}_{t}\right)$ of latent variables to disentangle the causal representations of concepts affecting the generation of multimodal traffic observations. (2) A causal propagation module is proposed to mine the causal relationship between endogenous latent variables through a trainable causal graph, and propagate the causal effect according to the causal order. (3) In the observation data generation stage, the generator is established to approximate the conditional generation distribution $p_\theta(\mathbf{x}_t|\mathbf{z}_t)$. We utilize learnable variational distributions to approximate the true data distribution, with the aim of disentangling causal representations of physical concepts using variational inference. Compared to traditional VAE, we explicitly endow latent variables with real semantic information (i.e., causal representations of physical concepts).
\subsection{Posterior Network}
\label{Posterior Network}
We use conditional information and observations to build a PosteriorNet, whose purpose is to approximate the true posterior distribution of latent variables by learning a variational posterior distribution $q_{\phi}\left( \boldsymbol{ \epsilon}_{t}, \mathbf{z}_{t} \mid \mathbf{z}_{t-1}, \mathbf{x}_{t}, \mathbf{C}_{t}\right)$ using neural networks. As shown in the yellow part of Fig. \ref{fig:3}, it consists of Graph Gated Recurrent Unit (GraphGRU) and Causal Propagation Module.
\paragraph{GraphGRU}
The progression of multimodal traffic flow has Markov property, and the evolution of latent variables is the intrinsic drivie for the spatio-temporal dependencies of multimodal traffic observations. Therefore, we use the GraphGRU to model the evolution process of system status, capturing the spatio-temporal dependencies into the exogenous latent variables. We build parameter-independent GraphGRU to learn mode-specific patterns for each traffic modes, defined as follows:
\begin{equation}\label{equ:5}
\begin{aligned}
\mathbf{s}_{t}^{po,i} &= \operatorname{FC}(\mathbf{C}_t||\mathbf{x}_t^i) \\
{\mathbf{r}_{t}^{po,i}} & = \sigma ({\mathbf{W}_{r}^i}{{\star }_{G}}({\mathbf{s}_{t}^{po,i}}||{\mathbf{z}_{t-1}^{po,i}})+{\mathbf{b}_{r}^i}) \\
{{\mathbf{u}}_{t}^{po,i}} & = \sigma ({\mathbf{W }_{u}^i}{{\star }_{G}}({\mathbf{s}_{t}^{po,i}}||{\mathbf{z}_{t-1}^{po,i}})+{\mathbf{b}_{u}^i}) \\
{\mathbf{\tilde{h}}_{t}^{po,i}} & = \operatorname{tanh} ({\mathbf{W }_{h}^i}{{\star }_{G}}({\mathbf{s}_{t}^{po,i}}||(\mathbf{r}_{t}^{po,i}\odot {\mathbf{z}_{t-1}^{po,i}}))+{\mathbf{b}_{h}^i}) \\
{\boldsymbol{ \epsilon}_{t}^{po,i}} & = {{\mathbf{u}}_{t}^{po,i}}\odot {\mathbf{z}_{t-1}^{po,i}}+(1-{\mathbf{u}}_{t}^{po,i})\odot {\mathbf{\tilde{h}}_{t}^{po,i}}
\end{aligned}
\end{equation}
where $i \in \left \{poi, bike, taxi, bus, v \right \}$ denotes physical concept of interest, $||$ denotes concatenate operation, $\sigma$ denotes sigmoid funtion, $\mathbf{C}_t \in \mathbb{R}^{N\times c_c} $ is conditional information, $\mathbf{x}_t^i \in \mathbb{R}^{N\times c_i}$ is the observation of the $i$-th mode, $c_i$ is the number of traffic flow channels of the $i$-th mode, ${\mathbf{z}_{t-1}^{po,i}} \in \mathbb{R}^{N\times d}$ is the posterior endogenous latent variable at $t-1$, $\boldsymbol{ \epsilon}_{t}^{po,i} \in \mathbb{R}^{N\times d}$ is posterior exogenous latent variable at $t$, and $\mathbf{W}, \mathbf{b}$ are the parameters of graph convolution. The graph convolution defined by $\mathbf{W}\star_{G}(\mathbf{X}) + \mathbf{b} = (\mathbf{I}+\mathbf{D}^{-1/2}\mathbf{G}\mathbf{D}^{-1/2})\mathbf{X}\mathbf{W}+\mathbf{b} $, where $\mathbf{G} \in \mathbb{R}^{N \times N}$ is the distance adjacency matrix of regions, ${{\mathbf{D}}_{ii}}=\sum\nolimits_{j}{{{\mathbf{G}}_{ij}}}$ and $N$ is the number of regions. Then, we calculate the mean and log-variance of $ \boldsymbol{ \epsilon}_{t}^{po}$ by using separate fully connected layers for each traffic modes to obtain the posterior distribution of exogenous latent variables $q_{\phi}\left(\boldsymbol{ \epsilon}_{t} \mid \mathbf{z}_{t-1}, \mathbf{x}_{t}, \mathbf{C}_{t}\right)$.
\paragraph{Causal Propagation Module}
Concepts that affect the generation of observations are naturally causally related. Therefore, endogenous latent variables, as semantic representations of concepts, also have causal relationships. We propose a causal propagation module to transform independent exogenous variables into causal endogenous variables and leverage a learnable causal graph to mine their causal relationships.
The linear SCM is defined as $\mathbf{z}={{\mathbf{A}}^{T}}\mathbf{z}+\boldsymbol{\epsilon }={{(\mathbf{I-}{{\mathbf{A}}^{T}})}^{-1}}\boldsymbol{\epsilon }$. We add parameter-independent nonlinear transformations for each traffic modes to improve the representation ability. In this paper, the causal propagation module is defined as:
\begin{equation}\label{equ:6}
\begin{aligned}
\tilde{\mathbf{A}} &=\operatorname{ReLU}\left(\tanh \left(\alpha \mathbf{W}_{\mathbf{A}}\right)\right) \in \mathbb{R}^{5 \times 5} \\
\mathbf{h}_{t}^{p o} &=\left(\mathbf{I}-\tilde{\mathbf{A}}^{T}\right)^{-1} \boldsymbol{\epsilon}_{t}^{p o} \in \mathbb{R}^{N \times 5 \times d} \\
\mathbf{z}_{t}^{p o, i} &=f_{i}\left(\left[\mathbf{h}_{t}^{p o}\right]_{\left[:,i,:\right]}\right)=\mathrm{FC}_{i}\left(\tanh \left(\mathrm{FC}_{i}\left(\left[\mathbf{h}_{t}^{p o}\right]_{\left[:,i,:\right]}\right)\right)\right)
\end{aligned}
\end{equation}
where $ \mathbf{W}_{\mathbf{A}} \in \mathbb{R}^{5 \times 5} $ denotes a learnable parameter, $\alpha$ is a hyper-parameter for controlling the saturation rate of the activation function. ReLU regularizes the parameter matrix to ensure sparsity and non-negativity. $\tilde{\mathbf{A}}$ is the causal graph of endogenous latent variables, where $\tilde{\mathbf{A}}_{ij}$ represents the causal effect of the parent variable $\mathbf{z}_i$ on the child variable $\mathbf{z}_j$. Therefore, when the graph nodes are sorted in topological order, the matrix $\tilde{\mathbf{A}}$ is strictly upper triangular. Then, we calculate the mean and log-variance by using separate fully connected layers for each traffic modes.
\subsection{Prior Network}
\label{Prior Network}
Previous unsupervised disentangled representation learning based on VAE regularizes the posterior of the latent variables with a standard Multivariate Gaussian prior, which greatly limits the expression ability of the model. Unsupervised disentangled representation learning can not guarantee the model identifiability due to the lack of inductive bias \cite{locatello2019challenging}. To improve the identifiability of the model, we build a PriorNet based on conditional information, which aims to model the physical rules of the concepts of interest that naturally exist in the system, and use a learnable prior distribution to approximate this rules. As shown in the pink part of Fig. \ref{fig:3}, the PirorNet is similar in structure to the PosteriorNet, which is composed of GraphGRU and causal propagation module.
\paragraph{GraphGRU}
The PriorNet only inputs the conditional information of the current system, calculates prior exogenous latent variables $ \boldsymbol{ \epsilon}_{t}^{pr}$ according to Eq.\ref{equ:5}, and then obtains the prior distribution of exogenous latent variables $p_{\theta}\left(\boldsymbol{ \epsilon}_{t} \mid \mathbf{z}_{t-1}, \mathbf{C}_{t}\right)$ by calculating the mean and log-variance.
\paragraph{Causal Propagation Module}
The PriorNet and the PosteriorNet share a causal propagation module. We argue that causality is a stable natural phenomenon that does not change with time or space, thus globally sharing a causal graph and nonlinear transformation. We calculate the prior endogenous latent variables $\mathbf{z}_{t}^{p r}$ according to Eq.\ref{equ:6}, and then obtain the prior distribution of the endogenous latent variables $p_{\theta}\left(\mathbf{z}_{t} \mid \boldsymbol{\epsilon}_t\right)$ by calculating the mean and log-variance.
\subsection{Generator}
\label{Generator}
We build the generator using two fully connected layers to parameterize the conditional distribution of the generative models $p_{\theta}\left(\mathbf{x}_{t} \mid \mathbf{z}_{t}\right)$ defined in Eq.\ref{equ:3}. As shown in Figure 3, a generator is globally shared. The generator is shared globally as shown in Fig. \ref{fig:3}. The results of generative models have different meanings depending on the type of $\mathbf{z}$.
\paragraph{Reconstruction}
As shown by the yellow arrow in Fig. \ref{fig:3}, the PosteriorNet takes the current observations as part of the input. So when generating data using the posterior endogenous latent variables $\mathbf{z}_{t}^{p o}$, the output is the reconstruction result, represented as $\mathbf{\hat{x}}_{t}^{i,rec}=\operatorname{Generator}_i (\mathbf{z}_{t}^{po,i})$, $i \in \left \{bike, taxi, bus, v \right \}$.
\paragraph{Prediction}
The PriorNet only utilizes the current conditional information to fit the prior distribution and does not involve current observations. Therefore, when generating data using the prior endogenous latent variables $\mathbf{z}_{t}^{p r}$, the output is the prediction result. Based on the Markov property of sequence generation, we leverage a simple attention mechanism to weight the current prior endogenous latent variables with the previous posterior, which can further improve the effect of prediction. The attention mechanism is defined as:
\begin{equation}\label{equ:7}
\begin{aligned}
\mathbf{\tilde{z}}_{t}^{pr}=\operatorname{softmax}\left( \mathbf{z}_{t-1}^{po}{{\mathbf{W}}_{att}}{{\left( \mathbf{z}_{t}^{pr} \right)}^{T}} \right)\mathbf{z}_{t}^{pr} \in \mathbb{R}^{N \times 5 \times d}
\end{aligned}
\end{equation}
where $\mathbf{W}_{att} \in \mathbb{R}^{ d \times d}$ is the learnable parameter. Then $\mathbf{\tilde{z}}_{t}^{pr}$ is fed into the generator to obtain the prediction result, represented as $\mathbf{\hat{x}}_{t}^{i,pred}=\operatorname{Generator}_i (\mathbf{\tilde z}_{t}^{pr,i})$, $i \in \left \{bike, taxi, bus, v \right \}$.
\subsection{Learning Strategy}
\label{Learning Strategy}
We propose a mutually supervised training method for the PriorNet and PosteriorNet, which benefits the model to approximate the physical rules of concepts of interest, while helping the to identifiably disentangle causal representations. Based on variational inference, we use a neural network to learn a tractable distribution $q_{\phi}$ to approximate the true posterior distribution $p_{\theta}$. Given a dataset $\mathcal{D}$, the Evidence Lower Bound (ELBO) of CCHMM is as follows:
\begin{small}
\begin{equation}\label{equ:8}
\begin{aligned}
\mathcal{L}_{ELBO}&={{\mathbb{E}}_\mathcal{D}}\left[ {{\mathbb{E}}_{{{q}_{\phi }}}}\left[ \log \left( \frac{{{p}_{\theta }}({{\mathbf{x}}_{t<T}},{{\boldsymbol{\epsilon }}_{t<T}},{{\mathbf{z}}_{t<T}}|\mathbf{C}_{t<T})}{{{q}_{\phi }}({{\boldsymbol{\epsilon }}_{t<T}},{{\mathbf{z}}_{t<T}}|{{\mathbf{x}}_{t<T}},\mathbf{C}_{t<T})} \right) \right] \right] \\
&= {{\mathbb{E}}_\mathcal{D}}\left[ \sum\limits_{t=1}^{T-1}\mathcal{L}_t^{q_\phi, p_\theta } \right]\\
\mathcal{L}_t^{q_\phi, p_\theta }&= {{{\mathbb{E}}_{{{q}_{\phi }}\left( {{\boldsymbol{\epsilon }}_{t}},{{\mathbf{z}}_{t}}|{{\mathbf{z}}_{t-1}},{{\mathbf{x}}_{t}},{{\mathbf{C}}_{t}} \right)}}}\left[ \log \left( {{p}_{\theta }}({{\mathbf{x}}_{t}}|{{\mathbf{z}}_{t}}) \right) \right]\\
&-{{D}_{KL}}\left[ {{q}_{\phi }}\left( {{\boldsymbol{\epsilon }}_{t}},{{\mathbf{z}}_{t}}|{{\mathbf{z}}_{t-1}},{{\mathbf{x}}_{t}},{{\mathbf{C}}_{t}} \right)||{{p}_{\theta }}\left( {{\boldsymbol{\epsilon }}_{t}},{{\mathbf{z}}_{t}}|{{\mathbf{z}}_{t-1}},\mathbf{C}_{t} \right) \right]
\end{aligned}
\end{equation}
\end{small}
We rewrite Eq. \ref{equ:6} as ${{\mathbf{z}}_{t}}={{\varphi }_{\mathbf{w}}}({{\mathbf{\epsilon }}_{t}})$, where $\mathbf{w}$ is the parameter of the causal propagation module and ${\varphi }_{\mathbf{w}}$ is invertible. Therefore, we reformulate the of the prior and posterior distributions with the Dirac delta function $\delta (\cdot )$, represented as follows:
\begin{small}
\begin{equation}\label{equ:9}
\begin{aligned}
{{q}_{\phi }}\left( {\boldsymbol{\epsilon}_{t}},{{\mathbf{z}}_{t}}|{{\mathbf{z}}_{t-1}},{{\mathbf{x}}_{t}},{{\mathbf{C}}_{t}} \right)&={{q}_{\phi }}\left( {\boldsymbol{\epsilon}_{t}}|{{\mathbf{z}}_{t-1,}}{{\mathbf{x}}_{t}},{{\mathbf{C}}_{t}} \right)\delta \left( {{\mathbf{z}}_{t}}={{\varphi }_{\mathbf{w}}}({\boldsymbol{\epsilon}_{t}}) \right)\\
&={{q}_{\phi }}\left( {{\mathbf{z}}_{t}}|{{\mathbf{z}}_{t-1,}}{{\mathbf{x}}_{t}},{{\mathbf{C}}_{t}} \right)\delta \left( {\boldsymbol{\epsilon}_{t}}=\varphi_{\mathbf{w}}^{\text{-}1}({{\mathbf{z}}_{t}}) \right) \\
{{p}_{\theta }}\left( {\boldsymbol{\epsilon}_{t}},{{\mathbf{z}}_{t}}|{{\mathbf{z}}_{t-1}},{{\mathbf{C}}_{t}} \right)&={{p}_{\theta }}\left( {\boldsymbol{\epsilon}_{t}}|{{\mathbf{z}}_{t-1}},{{\mathbf{C}}_{t}} \right)\delta \left( {{\mathbf{z}}_{t}}={{\varphi }_{\mathbf{w}}}({\boldsymbol{\epsilon}_{t}}) \right)\\
&={{p}_{\theta }}\left( {{\mathbf{z}}_{t}}|{{\mathbf{z}}_{t-1}},{{\mathbf{C}}_{t}} \right)\delta \left( {\boldsymbol{\epsilon}_{t}}=\varphi _{\mathbf{w}}^{\text{-}1}({{\mathbf{z}}_{t}}) \right)
\end{aligned}
\end{equation}
\end{small}
We substitute the prior and the posterior distributions in Eq. \ref{equ:9} and reformulate $\mathcal{L}_t^{q_\phi, p_\theta }$ as:
\begin{equation}\label{equ:10}
\begin{aligned}
\mathcal{L}_t^{q_\phi, p_\theta }= &{{{\mathbb{E}}_{{{q}_{\phi }}\left( {{\mathbf{z}}_{t}}|{{\mathbf{z}}_{t-1}},{{\mathbf{x}}_{t}},{{\mathbf{C}}_{t}} \right)}}}\left[ \log \left( {{p}_{\theta }}({{\mathbf{x}}_{t}}|{{\mathbf{z}}_{t}}) \right) \right]\\
&-{{D}_{KL}}\left[ {{q}_{\phi }}\left( {{\boldsymbol{\epsilon}}_{t}}|{{\mathbf{z}}_{t-1}},{{\mathbf{x}}_{t}},{{\mathbf{C}}_{t}} \right)||{{p}_{\theta }}\left( {{\boldsymbol{\epsilon}}_{t}}|{{\mathbf{z}}_{t-1}},\mathbf{C}_{t} \right) \right] \\
&-{{D}_{KL}}\left[ {{q}_{\phi }}\left( {{\mathbf{z}}_{t}}|{{\mathbf{z}}_{t-1}},{{\mathbf{x}}_{t}},{{\mathbf{C}}_{t}} \right)||{{p}_{\theta }}\left( {{\mathbf{z}}_{t}}|{{\mathbf{z}}_{t-1}},\mathbf{C}_{t} \right) \right]
\end{aligned}
\end{equation}
where, the first term is the reconstruction loss, and the last two terms are the KL divergence of the exogenous and endogenous latent variables, respectively.
Since the causal graph has the property of being acyclic, it is necessary to increase the acyclic constraint \cite{yu2019dag} of $\tilde{\mathbf{A}}$, expressed as $h(\tilde{\mathbf{A}}) = \operatorname{tr}\left[(I+\tilde{\mathbf{A}} \circ \tilde{\mathbf{A}})^{n}\right]-n$. In addition, we use L2-norm as the predicted loss, defined as $\mathcal{L}^{{pred }}=\mathbb{E}_{\mathcal{D}}\left[\sum_{t=1}^{T-1}\left\|\hat{\mathbf{x}}_{t}^{pred }-\mathbf{x}_{t}\right\|_{2}^{2}\right]$. In summary, the total loss function of CCHMM is defined as follows:
\begin{equation}\label{equ:11}
\begin{aligned}
\mathcal{L}=-\mathcal{L}_{ELBO}+\mathcal{L}_{pred}+\lambda h(\tilde{\mathbf{A}})
\end{aligned}
\end{equation}
where $\lambda$ is hyper-parameter for controlling the loss balance.
\section{Experiments}
\label{Experiments}
We evaluate the performance of our model on real world traffic datasets and compare with some recent compelling methods for traffic flow prediction\footnote{\url{https://github.com/EternityZY/CCHMM}}. Further, a comprehensive ablation study shows the effectiveness of each component of our model.
\begin{table*}
\caption{Performance comparison with other models.}
\label{tab3}
\resizebox{\hsize}{!}{
\begin{tabular}{c|ccc|ccc|ccc|ccc}
\hline
\multirow{2}{*}{Models} & \multicolumn{3}{c|}{Bike} & \multicolumn{3}{c|}{Taxi} & \multicolumn{3}{c|}{Bus} & \multicolumn{3}{c}{Speed} \\ \cline{2-13}
& MAE & RMSE & MAPE & MAE & RMSE & MAPE & MAE & RMSE & MAPE & MAE & RMSE & MAPE \\ \hline
HMM & 6.3239 & 13.1028 & 24.8122\% & 5.0146 & 8.6146 & 26.8858\% & 6.7976 & 13.0332 & 21.2048\% & 1.3766 & 2.2260 & 4.3131\% \\
HGCN & 5.6612 & 10.5526 & 23.2818\% & 4.8627 & 8.5967 & 25.4933\% & 7.5434 & 14.7726 & 22.4939\% & 1.9353 & 2.8477 & 5.9296\% \\
CCRNN & 5.3530 & 11.34111 & 21.1122\% & 4.7581 & 8.7107 & 24.7395\% & 6.6719 & 13.4522 & 20.2636\% & 1.5560 & 2.5605 & 4.8916\% \\
DMSTGCN & 5.2675 & 9.9759 & 21.5044\% & 4.5879 & 7.9499 & 24.2042\% & 6.3610 & 12.1108 & 19.9572\% & 1.4072 & 2.2511 & 4.3595\% \\
AGCRN & 5.0185 & 9.3577 & 20.3816\% & 4.5611 & 7.8992 & 23.9883\% & 6.5580 & 12.5084 & 19.9864\% & 1.3678 & 2.1587 & 4.2762\% \\
DGCRN & 4.9378 & 9.1436 & 20.3287\% & 4.5360 & 7.8984 & 23.9846\% & 6.4283 & 12.2228 & 19.6494\% & 1.4154 & 2.2878 & 4.4090\% \\
\textbf{CCHMM(our)} & \textbf{4.6418} & \textbf{8.5213} & \textbf{19.4286\%} & \textbf{4.4150} & \textbf{7.6262} & \textbf{23.5661\%} & \textbf{6.2450} & \textbf{11.8570} & \textbf{19.2033\%} & \textbf{1.2433} & \textbf{1.9943} & \textbf{3.8588\%} \\ \hline
\end{tabular}}
\end{table*}
\subsection{Dataset}
\label{Dataset}
\textbf{XC-Trans}:The XC-Trans dataset contains order records of three traffic modes(bike, bus and taxi) from June 1st 2021 to December 31th 2021 in Xicheng District, Beijing. The researched region is split into 175 non-overlapping subregions. We statistics the inflow and outflow for each traffic modes in all of the subregions.
\textbf{XC-Speed}:The XC-speed dataset contains speed records of main roads from June 1st 2021 to December 31th 2021 in Xicheng District, Beijing. We use the average speed of road segments within each region to represent the regional speed in every 30 minutes.
Besides, corresponding meteorological information, time position and POI data are collected as conditional information. We split this dataset with a 30-minute interval to obtain 11753 samples. we use three-hour historical data to predict the next 30-minute data. 60\% of the data is used for training, 20\% is used for validating and the rest is used for testing.
\subsection{Experimental settings}
We compare our framework with the following methods.
1)\textbf{HMM}\cite{li2021causal}: It uses multimodal information to achieve robust prediction of irreversible disease at an early stage.
2) \textbf{AGCRN}\cite{bai2020adaptive}: It employs an adaptive graph and integrates GRU with graph convolutions.
3) \textbf{CCRNN}\cite{ye2021coupled}: It employs coupled layer-wise graph convolution layer to capture the multi-level spatial dependence and temporal dynamics simultaneously.
4) \textbf{DGCRN}\cite{li2021dynamic}: It generates a dynamic graph by combining the predefined adjacency matrix and input features.
5) \textbf{HGCN}\cite{guo2021hierarchical}: It constructs road graph and region graph from micro and macro view respectively.
6) \textbf{DMSTGCN}\cite{han2021dynamic}: It designs an adaptive graph construction method to learn the time-specific spatial dependencies of road segments.
\subsection{Overall Comparison}
\label{Overall Comparison}
We evaluate the performance of methods with Mean Absolute Error (MAE), Root Mean Square Error (RMSE) and Mean Absolute Percentage Error(MAPE). Table \ref{tab3} presents the overall prediction performances which are the averaged results over three independent experiments. There is no method compatible with all traffic modes except us.
The baseline models focus on adaptively or dynamically generating graph structures, while our model pay more attention to modeling the causality between latent semantic variables in traffic system. Due to the lack of modeling spatial dependency and causality, the HMM model shows the worst performance. The model based on dynamic graph(e.g. DGCRN) perform better than models based on adaptive graph(e.g. AGCRN). Besides, it can be observed that our model outperforms baseline models consistently and overwhelmingly. Especially in speed prediction, our CCHMM brings about 10\% improvements to the best results in all metrics due to the causality of speed factor is more clear
\subsection{Ablation Study}\label{Ablation Study}
\begin{table*}[]
\caption{Results of ablation study.}
\label{tab4}
\resizebox{\hsize}{!}{
\begin{tabular}{ccccc|ccc|ccc|ccc}
\hline
\multirow{2}{*}{Category} & \multirow{2}{*}{Models} & \multicolumn{3}{c|}{Bike} & \multicolumn{3}{c|}{Taxi} & \multicolumn{3}{c|}{Bus} & \multicolumn{3}{c}{Speed} \\ \cline{3-14}
& & MAE & RMSE & MAPE & MAE & RMSE & MAPE & MAE & RMSE & MAPE & MAE & RMSE & MAPE \\ \hline
\multirow{4}{*}{PriorNet} & w/o GRU & 10.4177 & 22.7940 & 39.8982\% & 9.4037 & 17.6233 & 47.0230\% & 10.7402 & 21.5010 & 28.9726\% & 1.9951 & 3.0643 & 6.4062\% \\
& w/o GCN & 6.3110 & 12.6767 & 24.8612\% & 5.2715 & 9.3825 & 27.1115\% & 6.7583 & 12.9373 & 20.9419\% & 1.4340 & 2.3319 & 4.5195\% \\
& w/o Cond & 5.7693 & 11.1578 & 22.9648\% & 5.5484 & 9.9345 & 28.4668\% & 7.1146 & 13.8885 & 21.5514\% & 1.4798 & 2.3978 & 4.6165\% \\
& w/o Prior & \multicolumn{1}{l}{5.4904} & \multicolumn{1}{l}{10.6453} & \multicolumn{1}{l|}{21.8347\%} & 5.0789 & 9.0566 & 26.1980\% & 6.8861 & 13.2697 & 21.1046\% & 1.4735 & 2.4378 & 4.5712\% \\ \hline
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Causal Propagation \\ Module\end{tabular}} & Entangle & 5.5317 & 10.4375 & 22.5408\% & 5.1505 & 9.0548 & 26.9062\% & 6.9659 & 13.1916 & 21.7755\% & 1.5514 & 2.5234 & 4.8609\% \\
& w/o SCM & 5.1907 & 9.5034 & 21.6663\% & 4.9019 & 8.4785 & 26.3348\% & 6.6320 & 12.6021 & 20.4787\% & 1.4398 & 2.3503 & 4.4778\% \\
& w/o non-linear & 4.9508 & 8.7033 & 20.9008\% & 4.5376 & 7.7674 & 24.2565\% & 6.5281 & 12.4300 & 20.1166\% & 1.3443 & 2.1769 & 4.1931\% \\ \hline
Our & CCHMM & \textbf{4.6418} & \textbf{8.5213} & \textbf{19.4286\%} & \textbf{4.4150} & \textbf{7.6262} & \textbf{23.5661\%} & \textbf{6.2450} & \textbf{11.8570} & \textbf{19.2033\%} & \textbf{1.2433} & \textbf{1.9943} & \textbf{3.8588\%} \\ \hline
\end{tabular}}
\end{table*}
To evaluate the effectiveness of key components, we conduct comprehensive ablation experiments. For PriorNet, we design four variants: 1)w/o GRU: This variant replaces GraphGRU with GCN. The prior of latent variables is only generated from conditional information, which means discarding long-term temporal dependencies. 2)w/o GCN: This variant removes GCN in GraphGRU, which means discarding spatial dependencies. 3) w/o Cond: This variant removes conditional information. Note that we consider $\epsilon$ as exogenous variables which are relevant to conditional information. Removing conditional information is equivalent to removing PriorNet and generating latent variables directly from observation data in PostierNet. 4) w/o Prior: This variant removes the PriorNet but retains conditional information. Different from variant 3, the latent variables are generated from both conditional information and observation with SCM.
For the causal propagation module, we design three variants: 5) Entangle: There is only one latent variable in this variant. 6) w/o SCM: This variant removes SCM, which means the latent variables are directly generated from conditional information and observation. 7) w/o non-linear: This variant replaces the non-linear transformation with linear transformation in SCM.
Note that except for variant 3 and variant 4 that use additional FC layers for prediction, other networks use generator to obtain prediction results.
\begin{figure}
\caption{Comparison of Reconstruction performance of w/o Prior and CCHMM.}
\label{fig9}
\end{figure}
The performance of ablation experiments is shown in Table \ref{tab4}. We can find that variant 1 and 2 perform worst of all due to the lack of spatial and temporal dependencies. The performance of variant 3 shows the necessity of conditional information. In fact, the exogenous variables only affect the system, but are not constrained by the system. It means that we can only determine them by conditional information. Eventually, the model without conditional information degenerates into an ordinary sequence disentangled representation learning model. In variant 4, we drop the PriorNet. The role of the PriorNet is to obtain the stable rules of physical concepts, while the PosteriorNet is designed for obtaining disentangled representations from observation data and conditional information. Posterior collapse may occur in the absence of prior supervision, resulting in failure to obtain a stable and effective causal representation. An evidence is shown in Fig. \ref{fig9}, the reconstruction loss of variant 4 is generally lower than our CCHMM, which means that the model prefers to learn a representation for reconstruction rather than disentangling.
For causal propagation module, the model with disentangle latent variables perform better than the entangle one, which means that VAE-based structures decouple the latent variables to some extent. Since there is no restriction of causal structure, it suffers from spurious correlation. The most obvious consequence is that the speed prediction performance is reduced by 15\%. Besides, the performance of variant 7 linear model is insufficient to express causal relationships in complex scenarios.
\begin{figure}
\caption{Visual comparisons of learned causal graphs}
\label{fig10}
\end{figure}
In addition, for each model with causal propagation module, we initialize the causal graph as an upper triangular matrix subject to standard normal distribution. As shown in Fig. \ref{fig10}, it can be observed that the variant 1 failed to learn a stable causal relationship. The model without GCN and the one without non-linear transformation learnt a causal graph similar to our CCHMM. Particularly, the model without PriorNet learnt an causal graph with large diagonal elements. It means that the model failed to learn representations of physical concepts that conform to causality.
\section{Conclusion}
\label{Conclusion}
In this paper, we analyze the core physical concepts affecting the generation of multimodal traffic flow and disentangle the concepts of interest into three groups: regional attraction factor, the transportation demand factor and traffic speed factor. We infer causal representations of these concepts from conditional information and observations of the current system based on variational inference and structural causal model, and mine their causal relationships by using learnable causal graphs. For the data generation stage, we feed the prior causal representation into the generator to generate predictions. Extensive experiments show that all metrics of CCHMM are optimal, which reveal that it is crucial to introduce causal theory into spatio-temporal sequence analysis. In future work we further explore causal discovery and refine causal relationships in multimodal traffic systems.
\end{document} | math | 48,680 |
\begin{document}
\title{Geometric properties of \reliability polynomials}
\begin{abstract}
Geometric modeling of multivariate reliability polynomials is
based on algebraic hypersurfaces, constant level sets, rulings etc.
The solved basic problems are: (i) find the reliability polynomial using the Maple and Matlab software environment;
(ii) find restrictions of reliability polynomial via equi-reliable components;
(iii) how should the reliability components linearly depend on time, so that the reliability
of the system be linear in time?
The main goal of the paper is to find geometric
methods for analysing the reliability of electric systems used
inside aircrafts.
\end{abstract}
{\bf Keywords}: reliability polynomial; ruled hypersurfaces; electric systems inside aircrafts.
{\bf 2010 Mathematics Subject Clasification}: 60K10, 62N05, 90B25.
\section{Introduction}
During the decades following the war, many research laboratories
and universities developed and initiated programs to study life
testing and reliability problems \cite{Fw, AB}.
Numerous such research topics focus on the study of different
types of reliability systems \cite{HZ, HZ1, HZ2}, like serial, parallel,
serial-parallel, parallel-serial, and complex,
which have considerable impact on different life fields \cite{Fw}-\cite{RS}.
Since there exist different important available systems, the
researchers attempted to find more than one method to solve these complex systems,
and determine the optimal ones \cite{Fw, HZ1, IY}.\par
In the present work, we change the classical view, by trying to
get information from the differential geometry naturally related to the stochastic models.
Of particular interest is the study of reliability hypersurface
and establishing the number of straight lines situated on this set. For further ideas, see \cite{CU}.
\section{Some definitions and basic terminology}
We shall present first the concepts in network topology and
in graph theory which are needed
to calculate the network reliability \cite{Fw}-\cite{SA}.\par
\begin{defn}
A graph $G = (V,E)$,
where $V$ is the set of vertices (or nodes) and $E$ the set of
edges (or arcs), is called a network.\par
\end{defn}
\noindent
{\bf The network model}.
We describe our system as a directed network consisting of nodes
and arcs, as illustrated in Fig. 1.
One node is considered as the \emph{source} (node $A$ in the
figure), and a second node is considered as a \emph{sink} (node $D$).
Each component of the network is identified as an arc passing from
one node to another.
The arcs are numbered for identification. A {\em failure of a
component} is equivalent to an arc being removed or
cut out from the network.
The system is {\em successful} if there exists a valid path from
the source to the sink. The system is said to be {\em failed} if no such path exists. The
{\em reliability of the system} is the probability that there
exist one or more successful paths from the source to the sink \cite{Lj, ML}.
\centerline{\includegraphics[scale=.34]{1}}
\centerline{{\sc Fig. 1}. A bridge network. }
\noindent
\begin{defn}
A set of components is called a {cut}
if, when all the components in this set fail, the
system will fail, even if all other components are successful. \par
\end{defn}
\begin{defn}
A cut, such that any removal of
one component from it causes the resulting
set do not be a cut, is called a {minimal cut}.
\end{defn}
The set of all components is a cut.
In the network a minimal cut breaks all
simple paths from the source to the sink. In Fig. 1, we observe
that the minimal cuts are: $\{1, 2\}$, $\{1, 5\}$, $\{2, 3, 4\}$,
and $\{4, 5\}$.
\section{Complex reliability systems (network model)}
We introduce a graphical network model in which it is possible to
determine whether a system is working correctly by determining
whether a successful path exists in the system.
The system fails when no such path exists.\par
The system in Fig. 2 cannot be split into a group of series
and parallel systems.
\centerline{\includegraphics[scale=.44]{2}}
\centerline{{\sc Fig. 2}. A complex system (network model). }
This is primarily due to the fact that the components $A$ and $D$
each allow two paths emerging from them, whereas $B$ has only one;
$S_1$, $S_2$, $S_3$, $S_4$ and $S_5$ are called subsystems
or arcs.
\subsection{Minimal cut method}
There exist several methods for obtaining the reliability
of a complex system, as, for example, {\em minimal cut method}.
The minimal cut method is proper for systems which are connected in the form of a bridge.
When we apply this method to the system in Fig. 2, we should pursue the
following steps:
\begin{enumerate}[a)]
\item we enumerate all the minimal cut-sets in the system;
\item the failure of all components in a minimal
cut-set causes system failure;
\item this implies \emph{parallel} connections among these
components;
\item each minimal cut set determines the system failure;
\item this implies \emph{series} connections among the minimal cut
sets;
\item we draw an equivalent system and use the
\emph{parallel/seies method}
to compute the system reliability.
\end{enumerate}
\begin{thm}If $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ are arcs (paths) in a bridge system (Fig. 2),
then the reliability $R_{Mc}(t)$ of all system is
$$R_{Mc}(t)=R_1(t)R_4(t) + R_2(t)R_5(t) + R_2(t)R_3(t)R_4(t) \eqno(3.1)$$
$$
- R_1(t)R_2(t)R_3(t)R_4(t) - R_1(t)R_2(t)R_4(t)R_5(t)
$$
$$
- R_2(t)R_3(t)R_4(t)R_5(t) + R_1(t)R_2(t)R_3(t)R_4(t)R_5(t).
$$
\end{thm}
\begin{proof} By using minimal cut method, we have
$$
\hbox{Minimal cut-set} = \{ (S_{1}, S_{2}), (S_{4}, S_{5}),(S_{2}, S_{4}), (S_{1}, S_{3}, S_{5}) \},
$$
and then Fig. 2 can be replaced by Fig. 3, which will represent the reliability of a
parallel-series system ~\cite{HZ}, as follows:
\centerline{\includegraphics[scale=.44]{3}}
\centerline{{\sc Fig. 3}. A parallel-series system. }
We shall assume that $R_{i}(t)$ represents the reliability of the
$S_{i}$-th component (probability that the component $S_i$ to be functional on whole interval $[0,t]$)
in a cut set $M_{Cj}$, $j\in\{1, 2, 3, 4\}$. Therefore, there exist four possibilities
of cut sets, and their representation shows as a \emph{parallel-series} system, as
shown in Fig. 3, and any failure which occurs in a cut set that will cause the system failure.
A symbolic expression for reliability of such a complex system is evaluated by applying Boolean Function (BF) Technique.
The probability that each cut set $M_{Cj}$ fails is
$$
M_{C1}(t)=1- [(1-R_{1}(t))(1-R_{2}(t))]
$$
$$
M_{C2}(t) = 1- [(1-R_{4}(t))(1-R_{5}(t))]
$$
$$
M_{C3}(t) = 1- [(1-R_{2}(t))(1-R_{4}(t))]
$$
$$
M_{C4}(t)= 1- [(1-R_{1}(t))(1-R_{3}(t))(1-R_{5}(t))].
$$
The cut sets are incompatible. That is why, the reliability of
the system is
$$
R_{Mc}(t)=M_{C1}(t)M_{C2}(t)M_{C3}(t)M_{C4}(t),
$$
where the computations have probabilistic-boolean sense, i.e.,
$R_i^2(t)$ is formally replaced by $R_i(t)$. We find the expression (3.1) which is the pullback
of the reliability polynomial (see also \cite{HZ,JC} and the Section 4).
\end{proof}
\section{Reliability-Geometry transfer}
The basic ingredient is a vector of probabilities $$(R_1(t),R_2(t),R_3(t),R_4(t),R_5(t))$$
and associated relation with $R_{Mc}(t)$. From the pullback we go to the polynomial equation and conversely.
Also, via a geometric interpretation, we find new properties of any reliability polynomial.
Here, geometric modeling means: (i) to change probability functions into variables
whose values are in the interval $[0,1]$, and then to variables in the interval $(-\infty,\infty)$, (ii) to analyse
and identify a body of techniques that can model certain classes
of piecewise parametric surfaces, subject to particular conditions of shape and smoothness, and (iii) to come back
into the context of probability variables, reinterpreting the geometric results.
These stochastic results are read within the (uni-, bi-,..., and six-dimensional) unit cube $[0,1], [0,1]^2$, ..., $[0,1]^6$.
\subsection{Multivariate reliability polynomial}
\begin{defn}
The multivariate polynomial
$$
R_{Mc}=R_1R_4 + R_2R_5 + R_2R_3R_4 - R_1R_2R_3R_4 \eqno(3.2)
$$
$$ - R_1R_2R_4R_5 - R_2R_3R_4R_5 + R_1R_2R_3R_4R_5
$$
is called {\it reliability polynomial}.
\end{defn}
The critical points of the polynomial ($3.2$) determine a {\it
variety} described by the system
$$\frac{\partial R}{\partial R_1}=0,\frac{\partial R}{\partial R_2}=0,
\frac{\partial R}{\partial R_3}=0,\frac{\partial R}{\partial
R_4}=0,\frac{\partial R}{\partial R_5}=0.$$
To solve this system, we can use Maple or Matlab procedures
$$\begin{array}{lr}
\hbox{solve}(\{
&x_4 - x_2x_3x_4 - x_2x_4x_5 + x_2x_3x_4x_5 = 0,
\\
&x_5 + x_3x_4 - x_1x_3x_4 - x_1x_4x_5 - x_3x_4x_5 + x_1x_3x_4x_5 = 0,
\\
&x_2x_4 - x_1x_2x_4 - x_2x_4x_5 + x_1x_2x_4x_5 = 0,
\\
&x_1 + x_2x_3 - x_1x_2x_3 - x_1x_2x_5 - x_2x_3x_5 + x_1x_2x_3x_5 = 0,
\\
&x2 - x_1x_2x_4 - x_2x_3x_4 + x_1x_2x_3x_4=0
\\
&\},\;\;[x_1, x_2, x_3, x_4, x_5])\end{array}$$
which lead to, e.g., the obvious solutions $\{(0,0,x_3,0,0)\;|\;x_3\in \mathbb{R}\}$,
which proves the nontrivial compatibility of the system.
\begin{thm} All critical points of reliability polynomial
are saddle points. \end{thm}
\begin{proof} We compute the second order differential, which turns
out to be non-definite.
It follows that the extrema points of
interest are only on the boundary of a compact set, as for example
$[0,1]^6$.
Consequently the significant optimization problems involving the
previous polynomial are of the type $\min\max$, $\max\min$ or
optimizations with constraints. In these cases we can find
solutions in the $6$-dimensional interval $[0,1]^6$.
\end{proof}
\noindent
An example for locating such solutions, using Maple, is described below:
$$\begin{array}{l}>\hbox{with(Optimization)};
\\
>\hbox{Minimize}(x1*x4+x2*x5+x2*x3*x4-x1*x2*x3*x4
\\
\qquad\qquad\qquad-x1*x2*x4*x5-x2*x3*x4*x5+x1*x2*x3*x4*x5,
\\
\qquad\qquad\qquad {0. <= x1-2*x2},\hbox{assume = nonnegative)};
\end{array}$$
$$\begin{array}{l}
\begin{array}{l}[0., [x1 = 0.684438040345821397e-1, x2 = 0., x3 = 1., x4 = 0.,\\
\qquad\qquad x5 = .913400576368876061]];\end{array}
\\
\begin{array}{l}[ 0.0,[x1= 0.0684438040345821397, x2= 0.0, x3= 1.0, x4= 0.0,\\
\qquad\qquad x5= 0.913400576368876061]\end{array}\end{array}$$
$$\begin{array}{l}> \hbox{Maximize}(x1*x4+x2*x5+x2*x3*x4-x1*x2*x3*x4\\
\qquad\qquad-x1*x2*x4*x5 -x2*x3*x4*x5+x1*x2*x3*x4*x5,\\
\qquad\;\;\;\qquad x1 = 0 .. 1, x2 = 0 .. 1, x3 = 0 .. 1, x4 = 0 .. 1, x5 = 0 .. 1,\hbox{location})\end{array}$$
\subsection{Restrictions of reliability polynomial via \\equi-reliable components}
\begin{thm} There are $1, 15, 50, 60, 120$ diagonal
polynomials induced by 3.2, corresponding to $1, ... , 5$ variables.
\end{thm}
\begin{proof} These restrictions are counted as:
$1$ variable: $1$ polynomial.
$2$ variables: if one variable is $x$, and the other four are $y$,
we have five polynomials; if two variables are $x$ and the other
three are $y$, we have $C_5^2=10$ polynomials.
$3$ variables: if one variable is $x$, another is $y$ and the
other three are $z$, the we have $20$ polynomials; if one variable
is $x$, other two are $y$, and other two $z$, then we have $30$
polynomials.
$4$ variables: if one variable is $x$, another is $y$, another is
$z$ and the other two are $w$, then we have $60$ polynomials.
$5$ variables: the number of polynomials is permutations of $5$,
i.e. $120$.
For example, if we take $R_{1}=R_{2}= ...= R_{5}= x$, with
independent identical units, we get one diagonal (univariate) polynomial
\begin{equation}
y=2x^2+ x^3- 3x^4+x^5.
\end{equation}
Let $a$ be a parameter. For $a= \min y$, we have a double
positive root; for $a \in (\min y, 0)$, we have two positive
roots; for $a=0$, we have three roots, zero and two positive
roots; for $a \in (0, \max y)$, we have three positive roots; for
$a=\max y$, we have three roots, $y_{\max}$ as double and one
positive root; for $a$ greater than $\max y$, only one positive
root. \end{proof}
\begin{pro} The graph of the restriction of the polynomial
$$y=2x^2+ x^3- 3x^4+x^5$$
to $[0,1]$ looks like "standard logistic sigmoid function graph" and particularly
like "{stress-strain curve for low-carbon steel" }.
\end{pro}
In the case of two variables, the following particular cases appear: \par
\noindent
(i) the substitutions $R_1=x, R_2=R_3=R_4=R_5=y$ produce the diagonal polynomial (there are $5$ such polynomials):
$$P(x,y)=xy - 2xy^3 + xy^4 + y^2 + y^3 - y^4;$$
(ii) the substitutions $R_1=R_2=x, R_3=R_4=R_5=y$ produce the diagonal polynomial (there are $C_5^2=10$
such polynomials)
$$Q(x,y)= x^2y^3 - 2x^2y^2 - xy^3 + xy^2 + 2xy.$$
\subsection{Straight lines contained in the reliability hypersurface}
The graph of the reliability polynomial is a hypersurface in $R^6$
of Cartesian explicit equation 3.2, called {\it reliability
hypersurface}. Our aim is to solve the following problem:
{\bf Problem.} How should the components $R_i$ linearly depend on time,
so that the reliability of the system be linear in time?
Geometrically, this means to find all the straight lines
which are contained in the reliability hypersurface.
\begin{thm}
The family of straight lines in the reliability
hypersurface depends on at least five and at most six parameters.
\end{thm}
\begin{proof} Let us find the number of essential parameters such
that the family of straight lines
$$R_1=a_1 t+b_1, R_2=a_2 t+b_2, R_3=a_3 t+b_3,$$
$$ R_4=a_4 t+b_4, R_5=a_5 t+b_5, R(S)= a_6t+b_6,$$
with $a_1^2+...+a_6^2>0$, are included in reliability hypersurface. Here,
$t$ is a parameter on the line. All reasonings remain similar if we
replace $a_it+b_i$ by $a_ie^{-b_i t},\, a_i, \,b_i >0$.
First, we compute the following products:
\begin{multline*}\begin{split}
R_1R_4=&a_1a_4 t^2 + (a_1b_4+b_1a_4)t + b_1b_4,\\
R_2R_5=&a_2a_5 t^2 + (a_2b_5+b_2a_5)t + b_2b_5,\\
R_2R_3R_4=&(a_2a_3a_4)t^3 + (a_2a_3b_4 + a_2a_4b_3 + a_3a_4b_2)t^2\\&+ (a_2b_3b_4 + a_3b_2b_4
+ a_4b_2b_3)t + b_2b_3b_4,\\
R_1R_2R_3R_4 =
& (a_1a_2a_3a_4)t^4 + (a_1a_2a_3b_4 + a_1a_2a_4b_3 +a_1a_3a_4b_2\\& + a_2a_3a_4b_1)t^3
+ (a_1a_2b_3b_4 + a_1a_3b_2b_4+ a_1a_4b_2b_3\\
&+ a_2a_3b_1b_4 + a_2a_4b_1b_3 + a_3a_4b_1b_2)t^2\\
&+ (a_1b_2b_3b_4 + a_2b_1b_3b_4 + a_3b_1b_2b_4 + a_4b_1b_2b_3)t\\&+b_1b_2b_3b_4,
\end{split}\end{multline*}
\begin{multline*}\begin{split}
R_1R_2R_4R_5=
&(a_1a_2a_4a_5)t^4 + (a_1a_2a_4b_5 + a_1a_2a_5b_4 +a_1a_4a_5b_2\\& + a_2a_4a_5b_1)t^3
+ (a_1a_2b_4b_5 + a_1a_4b_2b_5+ a_1a_5b_2b_4\\&+ a_2a_4b_1b_5 + a_2a_5b_1b_4+ a_4a_5b_1b_2)t^2+ (a_1b_2b_4b_5 \\&
+ a_2b_1b_4b_5 + a_4b_1b_2b_5 + a_5b_1b_2b_4)t+ b_1b_2b_4b_5,
\end{split}\end{multline*}
\begin{multline*}\begin{split}
R_2R_3R_4R_5 =& (a_2a_3a_4a_5)t^4 + (a_2a_3a_4b_5 + a_2a_3a_5b_4 \\
&+a_2a_4a_5b_3 + a_3a_4a_5b_2)t^3\\
& + (a_2a_3b_4b_5 + a_2a_4b_3b_5+ a_2a_5b_3b_4 \\
&+ a_3a_4b_2b_5 + a_3a_5b_2b_4+ a_4a_5b_2b_3)t^2\\
& + (a_2b_3b_4b_5 + a_3b_2b_4b_5 + a_4b_2b_3b_5 \\&+ a_5b_2b_3b_4)t+ b_2b_3b_4b_5,\\
R_1R_2R_3R_4R_5=&(a_1a_2a_3a_4a_5)t^5 + (a_1a_2a_3a_4b_5 +a_1a_2a_3a_5b_4 + a_1a_2a_4a_5b_3\\
& + a_1a_3a_4a_5b_2 +a_2a_3a_4a_5b_1)t^4 + (a_1a_2a_3b_4b_5 + a_1a_2a_4b_3b_5\\
& +a_1a_2a_5b_3b_4 + a_1a_3a_4b_2b_5 + a_1a_3a_5b_2b_4 +a_1a_4a_5b_2b_3\\
& + a_2a_3a_4b_1b_5 + a_2a_3a_5b_1b_4 +a_2a_4a_5b_1b_3 + a_3a_4a_5b_1b_2)t^3\\
& + (a_1a_2b_3b_4b_5 +a_1a_3b_2b_4b_5 + a_1a_4b_2b_3b_5 + a_1a_5b_2b_3b_4\\
& +a_2a_3b_1b_4b_5 + a_2a_4b_1b_3b_5 + a_2a_5b_1b_3b_4 +a_3a_4b_1b_2b_5\\
& + a_3a_5b_1b_2b_4 + a_4a_5b_1b_2b_3)t^2 +(a_1b_2b_3b_4b_5 + a_2b_1b_3b_4b_5\\
& + a_3b_1b_2b_4b_5 +a_4b_1b_2b_3b_5 + a_5b_1b_2b_3b_4)t + b_1b_2b_3b_4b_5.
\end{split}\end{multline*}
By replacement, ordering by powers of $t$ and identifying,
we obtain a system whose solutions describe the number of straight lines situated on the reliability hypersurface.
We write the system ordering by the coefficients of the powers of degree from zero to five, relative to $t$:
\begin{multline*}\begin{split}
b_6=&b_1b_4 + b_2b_5 + b_2b_3b_4 - b_1b_2b_3b_4 - b_1b_2b_4b_5 -b_2b_3b_4b_5 + b_1b_2b_3b_4b_5, \\
a_6=&a_1b_4 + a_4b_1 + a_2b_5 +a_5b_2 + a_2b_3b_4 + a_3b_2b_4 + a_4b_2b_3 - a_1b_2b_3b_4\\
& -a_2b_1b_3b_4 - a_3b_1b_2b_4 - a_4b_1b_2b_3 - a_1b_2b_4b_5 -a_2b_1b_4b_5 - a_4b_1b_2b_5\\
& - a_5b_1b_2b_4 - a_2b_3b_4b_5 -a_3b_2b_4b_5 - a_4b_2b_3b_5 - a_5b_2b_3b_4 + a_1b_2b_3b_4b_5\\
& +a_2b_1b_3b_4b_5 + a_3b_1b_2b_4b_5 + a_4b_1b_2b_3b_5 +a_5b_1b_2b_3b_4;\\
0=&a_1a_4 + a_2a_5 + a_2a_3b_4 + a_2a_4b_3 + a_3a_4b_2 -a_1a_2b_3b_4 - a_1a_3b_2b_4\\
&- a_1a_4b_2b_3 - a_2a_3b_1b_4 -a_2a_4b_1b_3 - a_3a_4b_1b_2 \\
&- a_1a_2b_4b_5 - a_1a_4b_2b_5 -a_1a_5b_2b_4\\
& - a_2a_4b_1b_5 - a_2a_5b_1b_4 - a_4a_5b_1b_2\\
& -a_2a_3b_4b_5 - a_2a_4b_3b_5 - a_2a_5b_3b_4
\end{split}\end{multline*}
\begin{multline*}\begin{split}
& - a_3a_4b_2b_5 -a_3a_5b_2b_4 - a_4a_5b_2b_3 \\
&+ a_1a_2b_3b_4b_5 + a_1a_3b_2b_4b_5 +a_1a_4b_2b_3b_5\\
& + a_1a_5b_2b_3b_4 + a_2a_3b_1b_4b_5 +a_2a_4b_1b_3b_5 \\
&+ a_2a_5b_1b_3b_4 + a_3a_4b_1b_2b_5\\
& +a_3a_5b_1b_2b_4 + a_4a_5b_1b_2b_3;\\
0=&a_2a_3a_4 - a_1a_2a_3b_4 - a_1a_2a_4b_3 - a_1a_3a_4b_2 \\
&-a_2a_3a_4b_1 - a_1a_2a_4b_5 - a_1a_2a_5b_4\\
& - a_1a_4a_5b_2 -a_2a_4a_5b_1 - a_2a_3a_4b_5 - a_2a_3a_5b_4\\
& - a_2a_4a_5b_3 -a_3a_4a_5b_2\\
& + a_1a_2a_3b_4b_5 + a_1a_2a_4b_3b_5 +a_1a_2a_5b_3b_4 \\
&+ a_1a_3a_4b_2b_5 + a_1a_3a_5b_2b_4\\
& +a_1a_4a_5b_2b_3 + a_2a_3a_4b_1b_5 + a_2a_3a_5b_1b_4 \\
&+a_2a_4a_5b_1b_3 + a_3a_4a_5b_1b_2;\\
0=&a_1a_2a_3a_4b_5 - a_1a_2a_4a_5 - a_2a_3a_4a_5 - a_1a_2a_3a_4\\
& + a_1a_2a_3a_5b_4 + a_1a_2a_4a_5b_3\\
& + a_1a_3a_4a_5b_2 + a_2a_3a_4a_5b_1,\\
0=&a_{1}a_{2}a_{3}a_{4}a_{5}.\end{split}\end{multline*}
Starting from the last equation, at least one of the numbers $a_i,
i=1,...,5$ must be zero (number of cases:
$C_5^1+C_5^2+C_5^3+C_5^4=30$). So the straight-lines are parallel
to some hyperplane of coordinates. The first equation shows that
at $t=0$, the point $(b_1,...,b_6)$ is on the reliability
hypersurface. This remark requires the following procedure: we
choose arbitrarily $b_1,...,b_5$, and compute $b_6$. We replace the
values $b_1,...,b_5$ in the remaining equations. If the new system,
in unknown $(a_1,...,a_6)$, has a solution with
at least non-zero component, then there exists one straight line
passing through the point $(b_1,...,b_6)$ and lying on the
reliability hypersurface.
Explicitly, after solving the algebraic system, we have the following cases:\par
\noindent
{\small
\textbf{Case 1} ($a_1=0$):
\begin{multline*}\begin{split}
b_6 =&b_1b_4 + b_2b_5 + b_2b_3b_4 - b_1b_2b_3b_4 - b_1b_2b_4b_5\\
& -b_2b_3b_4b_5 + b_1b_2b_3b_4b_5,\\
a_6=& a_4b_1 + a_2b_5 + a_5b_2 + a_2b_3b_4 + a_3b_2b_4 + a_4b_2b_3\\
&- a_2b_1b_3b_4 - a_3b_1b_2b_4 - a_4b_1b_2b_3- a_2b_1b_4b_5 \\
&-a_4b_1b_2b_5 - a_5b_1b_2b_4 - a_2b_3b_4b_5 - a_3b_2b_4b_5\\
& -a_4b_2b_3b_5 - a_5b_2b_3b_4 + a_2b_1b_3b_4b_5\\
& + a_3b_1b_2b_4b_5 +a_4b_1b_2b_3b_5+ a_5b_1b_2b_3b_4,
\end{split}\end{multline*}
\begin{multline*}\begin{split}
0 =& a_2a_5 + a_2a_3b_4 + a_2a_4b_3 + a_3a_4b_2\\
& - a_2a_3b_1b_4 -a_2a_4b_1b_3 - a_3a_4b_1b_2\\
& - a_2a_4b_1b_5 - a_2a_5b_1b_4 -a_4a_5b_1b_2\\
& - a_2a_3b_4b_5 - a_2a_4b_3b_5 - a_2a_5b_3b_4\\
& -a_3a_4b_2b_5 - a_3a_5b_2b_4 - a_4a_5b_2b_3 \\
&+ a_2a_3b_1b_4b_5 +a_2a_4b_1b_3b_5 + a_2a_5b_1b_3b_4\\
& + a_3a_4b_1b_2b_5 +a_3a_5b_1b_2b_4 + a_4a_5b_1b_2b_3,\\
0 =&-a_2a_3a_4 - a_2a_3a_4b_1 - a_2a_4a_5b_1 \\
&- a_2a_3a_4b_5 -a_2a_3a_5b_4 - a_2a_4a_5b_3 - a_3a_4a_5b_2\\
& + a_2a_3a_4b_1b_5 +a_2a_3a_5b_1b_4 + a_2a_4a_5b_1b_3 + a_3a_4a_5b_1b_2,\\
0=&a_2a_3a_4a_5(b_1- 1).
\end{split}\end{multline*}
i) ($a_1=0$ and $b_1=1$):
\begin{multline*}\begin{split}
b_6 =&b_4 + b_2b_5 - b_2b_4b_5,\\
a_6=& a_4 + a_2b_5 + a_5b_2 - a_2b_4b_5 - a_4b_2b_5 - a_5b_2b_4,\\
0 =& a_2a_5 - a_2a_4b_5 - a_2a_5b_4 - a_4a_5b_2,\\
0 =&a_2a_4a_5.\end{split}\end{multline*}
In this case, for an arbitrary point $(b_1=1,b_2,b_3,b_4,b_5,b_6)$,
the solution $(a_1=0,a_2,a_3$, $a_4,a_5,a_6)$ depends on six parameters (a family of straight lines).
All the foregoing straight lines are in the plane $R_1=1$.
In this case the reliability hypersurface is a fiber bundle (ruled hypersurface).
ii) ($a_1=a_2=0$):
\begin{multline*}\begin{split}
b_6 =&b_1b_4 + b_2b_5 + b_2b_3b_4 - b_1b_2b_3b_4 - b_1b_2b_4b_5\\
& -b_2b_3b_4b_5 + b_1b_2b_3b_4b_5,\\
a_6=& a_4b_1 + a_5b_2 + a_3b_2b_4 + a_4b_2b_3 - a_3b_1b_2b_4\\
& -a_4b_1b_2b_3 - a_4b_1b_2b_5- a_5b_1b_2b_4 \\
& - a_3b_2b_4b_5 -a_4b_2b_3b_5 - a_5b_2b_3b_4 \\
&+ a_3b_1b_2b_4b_5 + a_4b_1b_2b_3b_5+ a_5b_1b_2b_3b_4,\\
0 =& a_3a_4b_2 - a_3a_4b_1b_2 - a_4a_5b_1b_2 \\
&- a_3a_4b_2b_5 -a_3a_5b_2b_4 - a_4a_5b_2b_3 \\
&+ a_3a_4b_1b_2b_5 + a_3a_5b_1b_2b_4+ a_4a_5b_1b_2b_3,\\
0 =&a_3a_4a_5b_2(b_1- 1).
\end{split}\end{multline*}
iii) ($a_1=a_3=0$):
\begin{multline*}\begin{split}
b_6 =&b_1b_4 + b_2b_5 + b_2b_3b_4 - b_1b_2b_3b_4 \\
&- b_1b_2b_4b_5 -b_2b_3b_4b_5 + b_1b_2b_3b_4b_5,\\
a_6=& a_4b_1 + a_2b_5 + a_5b_2 + a_2b_3b_4 + a_4b_2b_3\\
& -a_2b_1b_3b_4 - a_4b_1b_2b_3\\
\end{split}\end{multline*}
\begin{multline*}\begin{split}
& - a_2b_1b_4b_5 - a_4b_1b_2b_5 -a_5b_1b_2b_4 \\
&- a_2b_3b_4b_5 - a_4b_2b_3b_5 - a_5b_2b_3b_4\\
& +a_2b_1b_3b_4b_5 + a_4b_1b_2b_3b_5 + a_5b_1b_2b_3b_4,\\
0 =& a_2a_5 + a_2a_4b_3 - a_2a_4b_1b_3 - a_2a_4b_1b_5\\
& -a_2a_5b_1b_4 - a_4a_5b_1b_2 - a_2a_4b_3b_5\\
& - a_2a_5b_3b_4 -a_4a_5b_2b_3 + a_2a_4b_1b_3b_5\\
& + a_2a_5b_1b_3b_4 + a_4a_5b_1b_2b_3,\\
0 =&a_2a_4a_5(b_1b_3 - b_3 - b_1).
\end{split}\end{multline*}
iv) ($a_1a_4=0$):
\begin{multline*}\begin{split}
b_6 =&b_1b_4 + b_2b_5 + b_2b_3b_4 - b_1b_2b_3b_4\\
& - b_1b_2b_4b_5 -b_2b_3b_4b_5 + b_1b_2b_3b_4b_5,\\
a_6=& a_2b_5 + a_5b_2 + a_2b_3b_4 + a_3b_2b_4 - a_2b_1b_3b_4\\
& -a_3b_1b_2b_4 - a_2b_1b_4b_5- a_5b_1b_2b_4\\
& - a_2b_3b_4b_5 -a_3b_2b_4b_5 - a_5b_2b_3b_4 + a_2b_1b_3b_4b_5\\
& + a_3b_1b_2b_4b_5+ a_5b_1b_2b_3b_4,\\
0 =& a_2a_5 + a_2a_3b_4 - a_2a_3b_1b_4 - a_2a_5b_1b_4\\
& -a_2a_3b_4b_5 - a_2a_5b_3b_4- a_3a_5b_2b_4\\
& + a_2a_3b_1b_4b_5 +a_2a_5b_1b_3b_4 + a_3a_5b_1b_2b_4,\\
0=&a_2a_3a_5b_4(b_1-1).
\end{split}\end{multline*}
v) ($a_1a_5=0$):
\begin{multline*}\begin{split}
b_6 =&b_1b_4 + b_2b_5 + b_2b_3b_4 - b_1b_2b_3b_4\\
& - b_1b_2b_4b_5 -b_2b_3b_4b_5 + b_1b_2b_3b_4b_5,\\
a_6=& a_4b_1 + a_2b_5 + a_2b_3b_4 + a_3b_2b_4 \\
&+ a_4b_2b_3 -a_2b_1b_3b_4 - a_3b_1b_2b_4\\
& - a_4b_1b_2b_3 - a_2b_1b_4b_5 -a_4b_1b_2b_5 \\
&- a_2b_3b_4b_5 - a_3b_2b_4b_5 - a_4b_2b_3b_5\\
& +a_2b_1b_3b_4b_5 + a_3b_1b_2b_4b_5 + a_4b_1b_2b_3b_5,\\
0 =& a_2a_3b_4 + a_2a_4b_3 + a_3a_4b_2 - a_2a_3b_1b_4 \\
&-a_2a_4b_1b_3 - a_3a_4b_1b_2 - a_2a_4b_1b_5\\
& - a_2a_3b_4b_5 -a_2a_4b_3b_5 - a_3a_4b_2b_5 \\
&+ a_2a_3b_1b_4b_5 + a_2a_4b_1b_3b_5 + a_3a_4b_1b_2b_5,\\
0 =&a_2a_3a_4(1-b_1-b_5+b_1b_5).
\end{split}\end{multline*}
In case v), for an arbitrary point $(b_1,b_2,b_3,b_4,b_5,b_6)$,
the solution $(a_1=0,a_2,a_3$, $a_4,a_5=0,a_6)$ depends on six parameters (a family of straight lines).
In this case the reliability hypersurface is a fiber bundle (ruled hypersurface).
The situations ii)-iv) are similar.
\textbf{Case 2.} ($a_1=0, a_2=0$):
\begin{multline*}\begin{split}
b_6 =&b_1b_4 + b_2b_5 + b_2b_3b_4 - b_1b_2b_3b_4 \\
&- b_1b_2b_4b_5 -b_2b_3b_4b_5 + b_1b_2b_3b_4b_5,\\
a_6=& a_4b_1 + a_5b_2 + a_3b_2b_4 + a_4b_2b_3 \\
&- a_3b_1b_2b_4 -a_4b_1b_2b_3 - a_4b_1b_2b_5- a_5b_1b_2b_4\\
& - a_3b_2b_4b_5 -a_4b_2b_3b_5 - a_5b_2b_3b_4 \\
&+ a_3b_1b_2b_4b_5 + a_4b_1b_2b_3b_5+ a_5b_1b_2b_3b_4,\\
0 =& a_3a_4b_2 - a_3a_4b_1b_2 - a_4a_5b_1b_2 \\
&- a_3a_4b_2b_5 -a_3a_5b_2b_4 - a_4a_5b_2b_3\\
& + a_3a_4b_1b_2b_5 + a_3a_5b_1b_2b_4+ a_4a_5b_1b_2b_3,\\
0 =&a_3a_4a_5b_2(b_1- 1).
\end{split}\end{multline*}
i) ($a_1=0,a_2=0$ and $b_1=1$):
\begin{multline*}\begin{split}
b_6 =&b_4 + b_2b_5 - b_2b_4b_5,\\
a_6=& a_4 + a_5b_2 - a_4b_2b_5 - a_5b_2b_4,\\
0 =& a_2a_5 - a_2a_4b_5 - a_2a_5b_4 - a_4a_5b_2,\\
0 =&a_4a_5b_2 .
\end{split}\end{multline*}
ii) ($a_1=0,a_2=0, a_3=0$):
\begin{multline*}\begin{split}
b_6 =&b_1b_4 + b_2b_5 + b_2b_3b_4 - b_1b_2b_3b_4 \\
&- b_1b_2b_4b_5 -b_2b_3b_4b_5 + b_1b_2b_3b_4b_5,\\
a_6=& a_4 + a_5b_2 - a_4b_2b_5 - a_5b_2b_4,\\
0 =& a_4b_1 + a_5b_2 + a_4b_2b_3 - a_4b_1b_2b_3 \\
&- a_4b_1b_2b_5 -a_5b_1b_2b_4 - a_4b_2b_3b_5- a_5b_2b_3b_4\\
& + a_4b_1b_2b_3b_5 +a_5b_1b_2b_3b_4\\
0 =&a_4a_5(b_1b_2b_3 - b_2b_3 - b_1b_2).
\end{split}\end{multline*}
iii) ($a_1=0,a_2=0, a_4=0$):
\begin{multline*}\begin{split}
b_6 =&b_1b_4 + b_2b_5 + b_2b_3b_4 - b_1b_2b_3b_4\\
& - b_1b_2b_4b_5 -b_2b_3b_4b_5 + b_1b_2b_3b_4b_5,\\
a_6=& a_5b_2 + a_3b_2b_4 - a_3b_1b_2b_4 \\
&- a_5b_1b_2b_4 -a_3b_2b_4b_5 - a_5b_2b_3b_4\\
& + a_3b_1b_2b_4b_5 + a_5b_1b_2b_3b_4,\\
0 =&a_3a_5b_2b_4(b_1-1).
\end{split}\end{multline*}
\textbf{Case 3.} ($a_1=0, a_2=0, a_3=0$):
\begin{multline*}\begin{split}
b_6 =&b_1b_4 + b_2b_5 + b_2b_3b_4 \\
&- b_1b_2b_3b_4 - b_1b_2b_4b_5 - b_2b_3b_4b_5 + b_1b_2b_3b_4b_5,\\
a_6=& a_4b_1 + a_5b_2 + a_4b_2b_3 - a_4b_1b_2b_3 - a_4b_1b_2b_5 -
a_5b_1b_2b_4 - a_4b_2b_3b_5\\& - a_5b_2b_3b_4 + a_4b_1b_2b_3b_5 +
a_5b_1b_2b_3b_4,\\
0 =&a_4a_5(b_1b_2b_3-b_2b_3-b_1b_2).
\end{split}\end{multline*}
i) ($a_1=0,a_2=0,a_3=0, a_4=0$):
\begin{multline*}\begin{split}
b_6 =&b_1b_4 + b_2b_5 + b_2b_3b_4 \\
&- b_1b_2b_3b_4 - b_1b_2b_4b_5 - b_2b_3b_4b_5 + b_1b_2b_3b_4b_5,\\
a_6=&a_5b_2(1-b_1b_4-b_3b_4+b_1b_3b_4).
\end{split}\end{multline*}
ii) ($a_1=0,a_2=0,a_3=0, a_5=0$):
\begin{multline*}\begin{split}
b_6 =&b_1b_4 + b_2b_5 + b_2b_3b_4 - b_1b_2b_3b_4 - b_1b_2b_4b_5 -
b_2b_3b_4b_5 + b_1b_2b_3b_4b_5,\\
a_6=& a_4b_1 + a_4b_2b_3 - a_4b_1b_2b_3 - a_4b_1b_2b_5 -
a_4b_2b_3b_5 + a_4b_1b_2b_3b_5.
\end{split}\end{multline*}
\textbf{Case 4.} ($a_1=0, a_2=0, a_3=0, a_4=0$):
\begin{multline*}\begin{split}
b_6 =&b_1b_4 + b_2b_5 + b_2b_3b_4 - b_1b_2b_3b_4 - b_1b_2b_4b_5 -
b_2b_3b_4b_5 + b_1b_2b_3b_4b_5,\\
a_6=& a_5b_2(1 - b_1b_4 - b_3b_4 + b_1b_3b_4).
\end{split}\end{multline*}
The rest of cases are similar.
}
For each case, using the Jacobian matrix and its rank, we count the number of essential parameters.
\end{proof}
\subsection{Returning to the probability framework}
In order to return to the probability ansatz, we must assume that the
coefficients $a_i, b_i,i=1,...,6$ satisfy the conditions imposed by the
assumption that each function $a_i t + b_i$, $i=1,...,6,$ is a probability,
i.e., $0\leq a_i t + b_i \leq 1, i=1,...,6$. If $a_k=0$, then $0\leq b_k\leq 1$.
We further assume that all $a_i$ are different from zero.
(i) If $a_i>0$, then we find the intervals\ $I_i:\,\,-\frac{b_i}{a_i}\leq t \leq \frac{1-b_i}{a_i}, i=1,...,6$.
(ii) If there exists $a_k<0$, then a non-void interval is $I_k:\,\, \frac{1-b_k}{a_k}\leq t\leq -\frac{b_k}{a_k}$.
Suppose we have a non-void intersection $I=\cap I_i$.
Consequently, the significant parts from probabilistic point of view are
segments of straight lines included in the interval $[0,1]^6$.
\begin{thm}
Let us consider the vector of probabilities
$$(R_1(t),R_2(t),R_3(t),R_4(t),R_5(t)).$$
The most plausible situation is that
which imposes a maximum number of parameters in the family of
straight lines on the reliability hypersurface. \end{thm}
\begin{proof} In this case we have maximum degrees of freedom
(number of parameters). \end{proof}
\begin{rem} We can reiterate the process, by replacing this time the affine
framework with an exponential or a logarithmic one.
\end{rem}
\subsection{Equi-reliable hypersurfaces}
We further consider in $\mathbb{R}^5$ the constant level algebraic hypersurfaces
of the reliability polynomial (the {\it equi-reliable hypersurfaces}):
\begin{equation}\label{1c}\begin{split}c=&R_1R_4 + R_2R_5 + R_2R_3R_4- R_1R_2R_3R_4\\
& - R_1R_2R_4R_5 -R_2R_3R_4R_5 + R_1R_2R_3R_4R_5.\end{split}\end{equation}
{\bf Open problem.} How many straight lines are included in each equi-reliable hypersurface?
As an example, the constant level zero hypersurface contains the linear varieties
$OR_3R_4R_5: R_1=0,R_2=0$; $OR_1R_3R_5: R_2=0,R_4=0$; $OR_2R_3: R_1=0, R_4=0,R_5=0$.
Indeed, we have
$$R_{Mc}=R_1(R_4 - R_2R_3R_4 - R_2R_4R_5+R_2R_3R_4R_5)+ R_2(R_5 + R_3R_4- R_3R_4R_5).$$
{\bf Acknowledgements.} Partially supported by University Politehnica of Bucharest and by Academy of Romanian Scientists.
The first author would like to acknowledge the financial support of the Ph.D. studies
by the Iraqi Ministry of Higher Education and Scientific Research, and to thank to
Professor Fouad A. Majeed from University of Babylon for the helpful discussions.
PhD student Zahir Abdul Haddi Hassan\\
Department of Mathematics and Informatics,\\
Faculty of Applied Sciences,\\
University {\sc Politehnica} of Bucharest, \\Splaiul Independentei
313, RO-060042, Bucharest, Romania; \\mathbb{D}epartment of Mathematics, College for Pure Sciences,\\
University of {\sc Babylon}, Babylon, Iraq. \\E-mail: {\tt
zaher\[email protected]}, {\tt [email protected]}\\
Prof. Emeritus Dr. Constantin {\sc Udri\c ste}\\
Department of Mathematics and Informatics, \\Faculty of Applied
Sciences, \\University {\sc Politehnica} of Bucharest, \\Splaiul
Independentei 313, RO-060042, Bucharest, Romania.\\
E-mail: {\tt [email protected]}\\
Prof. Dr. Vladimir {\sc Balan}\\
Department of Mathematics and Informatics, \\Faculty of Applied
Sciences, \\University {\sc Politehnica} of Bucharest, \\Splaiul
Independentei 313, RO-060042, Bucharest, Romania.\\
E-mail: {\tt [email protected]}
\end{document} | math | 30,602 |
\begin{document}
\title{Repeat-Until-Success quantum computing using stationary and
flying qubits}
\author{Yuan Liang Lim}
\affiliation{Blackett Laboratory, Imperial College London, Prince
Consort Road, London SW7 2BZ, United Kingdom}
\author{Sean D. Barrett}
\affiliation{Hewlett-Packard Laboratories, Filton Road, Stoke Gifford,
Bristol BS34 8QZ, United Kingdom}
\author{Almut Beige}
\affiliation{Blackett Laboratory, Imperial College London, Prince
Consort Road, London SW7 2BZ, United Kingdom}
\author{Pieter Kok}
\affiliation{Hewlett-Packard Laboratories, Filton Road, Stoke Gifford,
Bristol BS34 8QZ, United Kingdom}
\author{Leong Chuan Kwek}
\affiliation{National Institute of Education, Nanyang Technological
University, Singapore 63 9798, Singapore}
\affiliation{Department of Physics, National University of Singapore,
Singapore 11 7542, Singapore}
\date{\today}
\begin{abstract}
We introduce an architecture for robust and scalable quantum
computation using both stationary qubits (e.g.~single photon sources
made out of trapped atoms, molecules, ions, quantum dots, or defect
centers in solids) and flying qubits (e.g.~photons). Our scheme
solves some of the most pressing problems in existing non-hybrid
proposals, which include the difficulty of scaling conventional
stationary qubit approaches, and the lack of practical means for
storing single photons in linear optics setups. We combine elements
of two previous proposals for distributed quantum computing, namely
the efficient photon-loss tolerant build up of cluster states by
Barrett and Kok [Phys. Rev. A {\bf 71}, 060310 (2005)] with the
idea of Repeat-Until-Success (RUS) quantum computing by Lim {\em et
al.} [Phys. Rev. Lett. {\bf 95}, 030505 (2005)]. This idea can be
used to perform eventually deterministic two-qubit logic gates on
spatially separated stationary qubits via photon pair
measurements. Under non-ideal conditions, where photon loss is a
possibility, the resulting gates can still be used to build graph
states for one-way quantum computing. In this paper, we describe the
RUS method, present possible experimental realizations, and analyse
the generation of graph states.
\end{abstract}
\pacs{03.67.Lx, 42.50.Dv}
\maketitle
\section{Introduction}
\noindent
Quantum computing offers a way to realize certain algorithms
exponentially more efficiently than with the best known classical
solutions \cite{shor,deutsch}. A substantial effort has therefore been
made to develop the corresponding quantum
technologies. Proof-of-principle experiments demonstrating the
feasibility of quantum computing have already been performed: Using
nuclear magnetic resonance techniques, Vandersypen {\em et al.}
\cite{chuang} realized a simple instance of Shor's algorithm by
factoring $15 = 3 \times 5$. A two qubit gate has been implemented in
a color center in diamond, utilizing the electron spin state of the
nitrogen-vacancy defect center together with a nearby nuclear spin as
qubits \cite{jelezko2004b}. Groups in Innsbruck and Boulder
implemented a universal two-qubit gate in an ion trap
\cite{blatt,wineland}, and the three-qubit teleportation protocol
\cite{blatt2,wineland2}. Adding more qubits to this ``proto quantum
computer'' will increase the density of the motional states used for
the two-qubit interaction. Consequently, it will become even harder to
implement clean two-qubit gates. Scaling ion trap quantum computers
much further therefore seems to require some form of distributed
quantum information processing, possibly involving ion transport
\cite{kielpinski}.
The schemes mentioned above are based on manipulating {\em stationary}
qubits such as atoms, molecules, or trapped ions. An alternative route
to finding a feasible and scalable technology for building quantum
computers is based on {\em flying} qubits, such as photons. The main
advantage of photons is their extremely long coherence time: In vacuum
and in simple dielectric media, photons do not interact with their
environment, and hence do not lose their quantum information. This is
why photons are usually the qubits of choice for quantum communication
\cite{bennett,ekert}. However, at the same time this lack of
interaction means that it is very hard to create two-photon entangling
gates. It therefore came as a surprise that the bosonic symmetry
requirement of the electromagnetic field, together with photon
counting and proper single-photon sources, is sufficient for
implementing scalable quantum computing \cite{KLM}. The overhead cost
for Linear Optical Quantum Computing (LOQC) has subsequently been
brought down significantly. In particular, the one-way or
cluster-state model for quantum computing \cite{raussendorf} has
allowed for drastic improvements in the scalability
\cite{yoran03,nielsen04,browne04}. Recently, a four-qubit cluster
state was realized experimentally by Walther {\em et al}.\
\cite{walther}. The main drawbacks of LOQC are the difficulties of
maintaining interferometric stability, the lack of practical `on
demand' single-photon sources and the lack of quantum memories for
photonic qubits \cite{gingrich03}.
\begin{figure}
\caption{Experimental realization of a universal two-qubit gate for
the considered network of single photon sources (stationary
qubits). This requires the generation of a photon
within each of the sources involved. The two photons then pass within
their coherence time through a linear optics network, which performs
a certain photon pair measurement.}
\end{figure}
In this paper, we consider the practical advantage of combining
stationary and flying qubits for the realisation of scalable quantum
computing. The stationary qubits (single photon sources) are arranged
in a network of nodes with each node processing and storing a small
number of qubits. To achieve scalability, the concept of distributed
quantum computing was introduced and it was proposed that distant
qubits communicate with each other through the means of flying qubits
(i.e.~photons) \cite{Grover,Cirac}. Initial schemes for the
implementation of this idea relied on entangled ancillas as a resource
\cite{Grover,Cirac,Jens,duan04,taylor04}. Others required that the
photon from one source is fed into another source
\cite{cirac97,Enk,Molmer,meier04,Guo,Zhou} or a photon-mediated
interaction between two fiber-coupled distant cavities needed to be
established \cite{Bose}. More hybrid approaches to quantum computing
can be found in Refs.~\cite{franson,bill}.
Other authors developed schemes for the probabilistic generation of
highly entangled states between distant single photon sources
\cite{cabrillo,bose99,zoller,last,browne,Simon,lim,Zou,chim}. In these
schemes, one generates a photon in each of the sources and then
performs an entangling photon measurement. By virtue of entanglement
swapping, this results in entangled stationary qubits. It has been
shown that similar ideas can also result in the implementation of
probabilistic remote two-qubit gates \cite{grangier}. At this point,
it was believed that scalable quantum computing with distant photon
sources requires additional resources such as local entangling gates
\cite{duan04} or entangled ancillas in order to become deterministic.
The concrete setup that we consider in this paper has recently been
introduced by Lim {\em et al.} \cite{moonlight} and allows for the
more efficient implementation of universal two-qubit gates than
previous proposals. The presented scheme
consists of a network of single stationary qubits (like trapped atoms,
molecules, ions, quantum dots and nitrogen vacancy color centers)
inside optical cavities, which act as a source for the generation of
single photons on demand. Read-out measurements and single qubit
rotations can be performed on the stationary qubits using laser pulses
and standard quantum optics techniques as employed in the recent ion
trap experiments in Innsbruck and Boulder \cite{blatt,wineland}.
The main building block for the realization of a two-qubit gate, which
qualifies the setup for universal quantum computing, is shown in
Figure \ref{moon}. It requires the simultaneous generation of a photon
in each source involved in the operation. Afterwards the photons
should pass through a linear optics setup, where a pair measurement is
performed in the output ports. This photon pair measurement results
either in the completion of the gate or indicates the presence of the
original qubits. In the later event, the gate should be repeated. The
qubits are never lost in the computation and the presented scheme has
therefore been called {\em Repeat-Until-Success} quantum
computing \cite{moonlight}.
Under realistic conditions, i.e.~in the presence of finite
detector efficiencies and finite success rates for the
generation of a single photon on demand, the setup in
Figure 1 can still be used for the implementation of
probabilistic gates with a very high fidelity. As shown
recently by Barrett and Kok \cite{Sean}, it is possible to use
probabilistic gates to efficiently generate graph states
for one-way quantum computing \cite{raussendorf}. Both
schemes, \cite{moonlight} and \cite{Sean} overcome the
limitations to scalable quantum computing faced before when
using the same resources. In Ref.~\cite{moonlight} this is
achieved with an eventually deterministic gate.
Ref.~\cite{Sean} introduced a so-called double-heralding
scheme, in which the entangling photo-detection stage was
employed twice to eliminate unwanted separable
contributions to the density matrix.
In this paper, we combine the ideas presented in our previous work
\cite{moonlight,Sean}. In this way, we obtain a truly scalable design
for quantum computing, i.e.~even in the presence of imperfect
components, with several key advantages:
\begin{enumerate}
\item Since our system uses {\em no direct qubit-qubit interactions},
the qubits can be well isolated. Not only does this allow us to
address the individual qubits easily, it also means that there are
fewer decoherence channels and hence fewer errors in the computation.
\item We achieve {\em robustness to photon loss}. In the
presence of photon loss, the two qubit gates become
non-deterministic. However, the gate failures are
heralded, and so the gates can still be used to build
high-fidelity entangled states, albeit in a
non-deterministic manner. Photon loss thus increases the
overall overhead cost associated with the scheme, but does
not directly reduce the fidelity of the computation. When
realistic photo-detectors and optical elements are used,
photon loss is inevitable and this built-in robustness is
essential.
\item Our scheme largely relies only on {\em components that have
been demonstrated in experiments} like atom-photon entanglement
\cite{Monroe,weber}. Apart from linear optics, we require only
relatively good sources for the generation of single photons on
demand \cite{Kuhn2,Yamamoto,Mckeever,Lange04,Kuzmich}, preferrably
at a high rate \cite{grangierscience}, and relatively efficient but
not necessarily number resolving photon detectors
\cite{Rosenberg}. Combining these in a working
quantum computer will be challenging, but the basic physics has
been shown to be correct.
\item The photon pair measurement is {\em interferometrically
stable}. Since each generated photon contributes equally to the
detection of a photon in the linear optics setup, fluctuations in the
length between the photon source and the detectors can at most result
in an overall phase factor with no physical consequences. This
constitutes a significant advantage compared to previous schemes based on
one-photon measurements (the only interferometrically stable schemes
are \cite{last,Simon,lim,moonlight}), since the
photons do not need to arrive
simultaneously in the detectors as long as they overlap within their
coherence time in the setup.
\item The basic ideas presented in this paper are {\em
implementation independent} and the stationary qubits can be realised
in a variety of ways. Any system with the right energy-level
structure and able to produce encoded flying qubits may be used.
\item Our scheme is inherently {\em distributed}. Hence, it can be
used in applications which integrate both quantum
computation and quantum communication. We show that entanglement
can be generated directly between any two
stationary qubits in the physical quantum computer. This
significantly reduces the computational cost compared to
architectures involving only nearest-neighbor interactions between
the qubits \cite{gottesman2000}.
\end{enumerate}
This paper is organized as follows. In the next Section, we give an
overview on the basic principles of measurement-based quantum
computing, since the described hybrid approach to quantum computing
constitutes a novel implementation of these ideas. Section III details
the general principle of a remote two-qubit gate implementation. In
Section \ref{rea}, we discuss possible gate implementations with
polarization and time-bin encoded photons. In Section \ref{cluster},
we describe how to overcome imperfections of inefficient photon
generation and detection with the help of
pre-fabricated graph states. Finally, we state our conclusions in
Section \ref{conc}.
\section{Measurement-based quantum computing} \langlebel{sectionii}
\noindent
One condition for the successful implementation of a measurement-based
quantum gate is that the measurement outcome is {\em mutually
unbiased} \cite{Wootters} with respect to the computational
basis. In this way, an observer does not learn anything about the
state of the qubits and the information might remain stored inside the
computer. To avoid the destruction of qubits, it is not allowed to
measure on the qubits directly. Measurements should only be performed
on ancillas, which have interacted and therefore share entanglement
with the qubits. These ancillas can be of the same physical
realisation as the computational qubits
\cite{nine,fifteen,raussendorf,leung} but they might also be realised
differently. If the stationary qubits are atoms, the ancilla can be
the quantised field mode inside an optical cavity \cite{beige}, a
common vibrational mode \cite{beige2} or newly generated photons, as
in the setup considered here. Vice versa, it has been found
advantageous to use collective atomic states as ancillas for photonic
qubits \cite{franson,bill}.
Let us now briefly describe the principles of measurement-based
quantum computing in a more formal way. Using the terms `qubits' and
`ancillas' provides a convenient picture, which is especially suited
for the description of hybrid approaches, where the qubits may remain
encoded in the same physical qubits instead of being assigned
dynamically as the computation proceeds. As in Ref.~\cite{kok}, we
consider two systems, $s$ and $a$, that are initially in the state
($c_n \in \mathbb{C}$)
\begin{equation}
|\Psi\ranglengle_s |A_0\ranglengle_a \equiv \sum_n c_n |\psi_n\ranglengle_s \otimes
|A_0\ranglengle_a \, .
\end{equation}
After some interaction, the joint system evolves into
\begin{equation}
|\Psi\ranglengle_s |A_0\ranglengle_a \rightarrow \sum_n c_n |\psi_n\ranglengle_s
\otimes |A_n\ranglengle_a \equiv |\Phi\ranglengle \, ,
\end{equation}
where the $|A_n\ranglengle_a$ are the eigenstates of an observable
$\mathsf{A}$. We can then measure $\mathsf{A}$, which
will reveal the state of the system $s$. This can be interpreted as a
quantum non-demolition measurement of $s$ but this is not what we are
interested in here.
In this article we will instead consider measurements of an
observable $\mathsf{B}$, as shown in Figure \ref{minkslevel}, that is
complementary to $\mathsf{A}$. In other words, the eigenvectors of
$\mathsf{A}$ and $\mathsf{B}$ form a so-called mutually unbiased basis
of the Hilbert space of system $a$. A specific outcome labelled $k$ of
such a measurement corresponds to the application of the projection
operator $\hat{B}_k$ (associated with the $k^{\mathrm{th}}$
eigenvector of $\mathsf{B}$), and the state of system $s$ is then
given by
\begin{equation}
|\Upsilon_k\ranglengle_s = \frac{ \text{Tr}_a \left[ \langlengle\Phi| \mbox{\small 1} \!\! \mbox{1}
\otimes \hat{B}_k |\Phi\ranglengle \right]}{\text{Tr}_{sa} \left[
\langlengle\Phi| \mbox{\small 1} \!\! \mbox{1} \otimes \hat{B}_k |\Phi\ranglengle \right]} \, .
\end{equation}
This can be generalized to situations where $\hat{B}_k$ is a
multi-rank projector or a Positive Operator Valued Measure (POVM). The
conditions for the evolution $|\psi_n\ranglengle_s \to |\Upsilon_k
\ranglengle_s$ to be a unitary transformation on system $s$ are presented
in Lapaire {\em et al.} \cite{kok}. If $s$ describes the qubits and
$a$ the ancilla, they guarantee, as mentioned above, that the
detection of $\hat{B}_k$ does not reveal any information about the
qubits.
Especially, in the setup considered in this paper the system $s$
consists of a set of $N$ stationary qubits occupying a Hilbert space
of size $2^N$, and system $a$ consists of $N$ flying quantum systems
occupying a Hilbert space of dimension $d \geq 2^N$. A measurement of
the observable $\mathsf{B}$ on the flying qubits will result in a
multi-qubit (entangling) operation on the stationary qubits. We are
interested in the case where the projector $\hat{B}_k$ induces a {\em
unitary} transformation on the stationary qubits,
\begin{equation}
|\Upsilon_k\ranglengle_s = U_k |\Psi\ranglengle_s \, ,
\end{equation}
which means that $\hat{B}_k$ is a proper projector. There are two
interesting cases to consider:
\begin{enumerate}
\item The set $\{\hat{B}_k\}_a$ corresponds to a basis of states
that induces a complete set $\{ U_k\}_a$ of entangling quantum
gates. As a result, finding any measurement outcome $k$ will induce
a unitary entangling gate operation on the stationary qubits.
\item The set $\{\hat{B}_k\}_a$ corresponds to a basis that can be
divided into two sets of states: Some of the projectors will induce
a unitary entangling gate $U_k$ on the stationary qubits, while the
remaining projectors induce a transformation that is locally
equivalent to the identity map $\mbox{\small 1} \!\! \mbox{1}$.
\end{enumerate}
\begin{figure}
\caption{Measurement-based quantum computing. The input state
$|\Psi\ranglengle_s$ and the auxiliary state $|A_0\ranglengle_a$ are transformed
in an $N$-port that induces a unitary transformation $U_{sf}
\end{figure}
The second setup is interesting for the following reason: Suppose that
system $s$ consists of $N$ non-interacting (e.g., well-separated)
stationary qubits with long decoherence times. If this system can
generate flying qubits in the manner described above, we can perform a
measurement of the observable $\mathsf{B}$ and entangle the
non-interacting stationary qubits. When not all measurement outcomes
produce an entangling gate on the stationary qubits (some yield
instead the identity operator), then the unitary gate is applied only
part of the time. However, due to the fact that a gate failure corresponds to
an identity operation (or something locally equivalent), we can again
prepare the flying qubits in the state $|A_0\ranglengle_a$. This allows us
to repeat the protocol
until the entangling gate is applied successfully, which is why we
called this idea Repeat-Until-Success quantum computing \cite{moonlight}.
\section{Remote two-qubit phase gates}
\noindent
One of the requirements for universal quantum computing is the ability
to perform an entangling two-qubit gate operation, like a controlled
phase gate \cite{divincenzo}. In this Section we describe the general
concept for the implementation of this gate between two distant single
photon sources. Note that our method of distributed quantum computing
allows to realize entangling operations, since the measurement on a
photon pair can imprint a phase on the state of its sources although
it cannot change the distribution of their populations. The first step
for the implementation of a two-qubit gate is the generation of a
photon within each respective source, which encodes the information of
the stationary qubit.
\subsection{Encoding}
\noindent
Let us denote the states of the photon sources, which encode the
logical qubits $|0 \ranglengle_{\rm L}$ and $|1 \ranglengle_{\rm L}$ as $|0
\ranglengle$ and $|1 \ranglengle$, respectively. An arbitrary pure state of
two stationary qubits can then be written as
\begin{equation} \langlebel{original}
\ket{\psi_{\rm in}}=\alpha \, \ket{00} + \beta\, \ket{01} + \gamma \,
\ket{10} + \delta \, \ket{11} \, ,
\end{equation}
where $\alpha$, $\beta$, $\gamma$ and $\delta$ are
complex coefficients with $|\alpha|^2 + |\beta|^2 + |\gamma|^2 +
|\delta|^2=1$. Suppose that a photon is generated in each of the two
sources, whose state (i.e.~its polarization or generation
time) depends on the state of the respective source. In
the following we assume that source 1 prepared in $|i \ranglengle$ leads
to the creation of one photon in state $|{\sf x}_i \ranglengle$, while
source 2 prepared in $|i \ranglengle$ leads to the creation of one photon
in state $|{\sf y}_i \ranglengle$,
\begin{equation} \langlebel{enc}
\ket{i}_1 \rightarrow \ket{i ,{\sf x}_i}_1 \, , ~~ \ket{i}_2
\rightarrow \ket{i ,{\sf y}_i}_2 \, .
\end{equation}
The simultaneous creation of a photon in both sources then transfers
the initial state (\ref{original}) into
\begin{eqnarray} \langlebel{theencoding}
\ket{\psi_{\rm enc}} &=& \alpha \, \ket{00 ,{\sf x}_0{\sf y}_0} + \beta
\, \ket{01 ,{\sf x}_0{\sf y}_1} + \gamma \, \ket{10 ,{\sf x}_1{\sf
y}_0} \nonumber \\ && +\delta \, \ket{11 ,{\sf x}_1{\sf y}_1} \, .
\end{eqnarray}
Note that the generation of photons whose state depends on the states
of the stationary qubits is a highly non-linear process. The
preparation of the generally entangled state (\ref{theencoding}) is
indeed the key step which allows the completion of an
eventually deterministic two-qubit gate with otherwise nothing else
than linear optics and photon pair measurements. The way the encoding step
(\ref{enc}) can be realized experimentally is discussed in Section
\ref{rea}. In this section we focus on the general ideas underlying
Repeat-Until-Success quantum computing.
\subsection{Mutually Unbiased Basis}
\noindent
Once the photons have been created, an entangling phase gate can be
implemented by performing an absorbing measurement on the photon
pair. Thereby, it is important to choose the photon measurement such
that none of the possible outcomes reveals any information about the
coefficients $\alpha$, $\beta$, $\gamma$ and $\delta$, as mentioned in
Section II. This can be achieved with a
photon pair measurements in a basis mutually unbiased \cite{Wootters} with respect to
the computational basis given by the states $\{\ket{{\sf x}_0{\sf
y}_0}, \, \ket{{\sf x}_0{\sf y}_1}, \, \ket{{\sf x}_1{\sf y}_0}, \,
\ket{{\sf x}_1{\sf y}_1} \}$. More concretely, all possible outcomes
of the photon measurement should be of the form
\begin{equation} \langlebel{unbiased}
\ket{\Phi} = {\textstyle {1 \over 2}} \big[ \ket{{\sf x}_0{\sf y}_0}
+ {\rm e}^{{\rm i} \varphi_1} \, \ket{{\sf x}_0{\sf y}_1} + {\rm
e}^{{\rm i} \varphi_2} \, \ket{{\sf x}_1{\sf y}_0} + {\rm e}^{{\rm
i} \varphi_3} \, \ket{{\sf x}_1{\sf y}_1} \big] \, .
\end{equation}
As we see below, a complete set of basis states of this form can be
found. Any bias in the
amplitudes would yield information about $\alpha$, $\beta$, $\gamma$,
and $\delta$, and would therefore not induce a unitary gate on the
stationary qubits. Detecting the state (\ref{unbiased}) and absorbing
the two photons in the process transfers the encoded state
(\ref{theencoding}) into
\begin{eqnarray} \langlebel{out}
\ket{\psi_{\rm out}} &=& \alpha \, \ket{00} + {\rm e}^{-{\rm i}
\varphi_1} \, \beta \, \ket{01} + {\rm e}^{-{\rm i} \varphi_2} \,
\gamma \, \ket{10} \nonumber \\ && + {\rm e}^{-{\rm i} \varphi_3} \,
\delta \, \ket{11} \, .
\end{eqnarray}
Note that the output state (\ref{out}) differs from the initial state
(\ref{original}) by a two-qubit phase gate.
Let us now consider the angle
\begin{equation}
\varphi_3 = \varphi_1 + \varphi_2 \, .
\end{equation}
In this case, the state $\ket{\Phi}$ is a product state and the output
(\ref{out}) differs from the initial state (\ref{original}) only by
local operations. However, if
\begin{equation} \langlebel{phasecondition}
\varphi_3 = \varphi_1 + \varphi_2 + \pi \, ,
\end{equation}
the state (\ref{unbiased}) becomes a maximally entangled state, as it
becomes obvious when writing $\ket{\Phi}$ as
\begin{equation}
|\Phi \ranglengle = {\textstyle {1 \over 2}} \big[ \ket{{\sf x}_0} (
\ket{{\sf y}_0} + {\rm e}^{{\rm i} \varphi_1} \, \ket{{\sf y}_1} )+
{\rm e}^{{\rm i} \varphi_2} \, \ket{{\sf x}_1} (\ket{{\sf y}_0}
- {\rm e}^{{\rm i} \varphi_1} \, \ket{{\sf y}_1}) \big] \, .
\end{equation}
The detection of a photon pair in this maximally entangled state
results in the completion of a phase gate with maximum entangling
power on the stationary qubit. Vice versa, maximum entanglement of the
state (\ref{unbiased}) also automatically implies
Eq.~(\ref{phasecondition}) as one can show by calculating the
entanglement of formation of the state (\ref{unbiased}).
\subsection{A deterministic gate}
\noindent
In the following, we denote the states of the measurement basis,
i.e. the mutually unbiased basis, by $\{ \ket{\Phi_i} \}$. In order to
find a complete Bell basis with all states of form (\ref{unbiased}),
we define
\begin{eqnarray} \langlebel{base}
\ket{\Phi_1} &\equiv & {\textstyle {1 \over \sqrt{2}}} \big[ \ket{{\sf
a}_1 {\sf b}_1}+\ket{{\sf a}_2 {\sf b}_2} \big] \, , \nonumber
\\ \ket{\Phi_2} &\equiv & {\textstyle {1 \over \sqrt{2}}} \big[
\ket{{\sf a}_1 {\sf b}_1}-\ket{{\sf a}_2 {\sf b}_2} \big] \, ,
\nonumber \\
\ket{\Phi_3} &\equiv & {\textstyle {1 \over
\sqrt{2}}} \big[ \ket{{\sf a}_1 {\sf b}_2}+\ket{{\sf a}_2 {\sf
b}_1} \big] \, , \nonumber \\ \ket{\Phi_4} &\equiv & {\textstyle
{1 \over \sqrt{2}}} \big[ \ket{{\sf a}_1 {\sf b}_2}-\ket{{\sf
a}_2 {\sf b}_1} \big] \, ,
\end{eqnarray}
where the states $|{\sf a}_i \ranglengle$ describe photon 1 and the $|{\sf
b}_i \ranglengle$ describe photon 2. Assuming orthogonality,
i.e.~$\langlengle {\sf a}_1 | {\sf a}_2 \ranglengle =0$ and $\langlengle {\sf
b}_1 | {\sf b}_2 \ranglengle = 0$, one can write the photon states on
the right hand side of Eq.~(\ref{base}) without loss of generality as
\begin{eqnarray} \langlebel{ab}
\ket{{\sf a}_1} &=& c_1 \, \ket{{\sf x}_0} + {\rm e}^{{\rm i}
\vartheta_1} s_1 \, \ket{{\sf x}_1} \, , \nonumber \\ \ket{{\sf
a}_2} &=& {\rm e}^{-{\rm i} \xi_1}( {\rm e}^{-{\rm i} \vartheta_1}
s_1\, \ket{{\sf x}_0} - c_1 \, \ket{{\sf x}_1}) \, ,\nonumber \\
\ket{{\sf b}_1} &=& c_2 \, \ket{{\sf y}_0} + {\rm e}^{{\rm i}
\vartheta_2} s_2 \,\ket{{\sf y}_1} \, , \nonumber \\ \ket{{\sf b}_2}
&=& {\rm e}^{-{\rm i} \xi_2}({\rm e}^{-{\rm i} \vartheta_2} s_2 \,
\ket{{\sf y}_0} - c_2 \, \ket{{\sf y}_1})
\end{eqnarray}
with
\begin{eqnarray}
s_i \equiv \sin{\theta_i} \, , ~~ c_i \equiv \cos{\theta_i} \, .
\end{eqnarray}
Inserting this into Eq.~(\ref{base}), we find
\begin{widetext}
\begin{eqnarray} \langlebel{base2}
\ket{\Phi_1} &=& {\textstyle {1 \over \sqrt{2}}} \big[ \big
(c_1c_2+{\rm e}^{-{\rm i}(\vartheta_1+\vartheta_2)} {\rm e}^{-{\rm
i}(\xi_1+\xi_2)} s_1s_2 \big) \ket{{\sf x}_0{\sf y}_0} + \big( {\rm
e}^{{\rm i}\vartheta_2}c_1s_2 - {\rm e}^{-{\rm i}\vartheta_1} {\rm
e}^{-{\rm i}(\xi_1+\xi_2)} s_1c_2 \big) \ket{{\sf x}_0{\sf y}_1}
\nonumber \\ &&+ \big( {\rm e}^{{\rm i}\vartheta_1}s_1c_2-{\rm
e}^{-{\rm i}\vartheta_2} {\rm e}^{-{\rm i}(\xi_1+\xi_2)} c_1s_2
\big) \ket{{\sf x}_1{\sf y}_0}+ \big( {\rm e}^{{\rm
i}(\vartheta_1+\vartheta_2)}s_1s_2+ {\rm e}^{-{\rm i}(\xi_1+\xi_2)}
c_1c_2 \big) \ket{{\sf x}_1{\sf y}_1} \big], \nonumber \\
\ket{\Phi_2} &=& {\textstyle {1 \over \sqrt{2}}} \big[
\big(c_1c_2-{\rm e}^{-{\rm i}(\vartheta_1+\vartheta_2)} {\rm
e}^{-{\rm i}(\xi_1+\xi_2)} s_1s_2 \big)\ket{{\sf x}_0{\sf y}_0} +
\big({\rm e}^{{\rm i}\vartheta_2}c_1s_2+{\rm e}^{-{\rm
i}\vartheta_1} {\rm e}^{-{\rm i}(\xi_1+\xi_2)} s_1c_2 \big)
\ket{{\sf x}_0{\sf y}_1} \nonumber \\ && + \big( {\rm e}^{{\rm
i}\vartheta_1}s_1c_2+{\rm e}^{-{\rm i}\vartheta_2} {\rm e}^{-{\rm
i}(\xi_1+\xi_2)} c_1s_2 \big) \ket{{\sf x}_1{\sf y}_0}+ \big( {\rm
e}^{{\rm i}(\vartheta_1+\vartheta_2)}s_1s_2-{\rm e}^{-{\rm
i}(\xi_1+\xi_2)} c_1c_2 \big) \ket{{\sf x}_1{\sf y}_1} \big],
\nonumber \\ \ket{\Phi_3} &=&{\textstyle {1 \over \sqrt{2}}}\big[
\big({\rm e}^{-{\rm i}\vartheta_2} {\rm e}^{-{\rm
i}\xi_2}c_1s_2+{\rm e}^{-{\rm i}\vartheta_1} {\rm e}^{-{\rm i}\xi_1}
s_1c_2 \big)\ket{{\sf x}_0{\sf y}_0}- \big({\rm e}^{-{\rm i}\xi_2}
c_1c_2-{\rm e}^{-{\rm i}(\vartheta_1-\vartheta_2)} {\rm e}^{-{\rm
i}\xi_1}s_1s_2 \big) \ket{{\sf x}_0{\sf y}_1} \nonumber \\ &&+
\big({\rm e}^{{\rm i}(\vartheta_1-\vartheta_2)} {\rm e}^{-{\rm
i}\xi_2} s_1s_2- {\rm e}^{-{\rm i}\xi_1} c_1c_2 \big) \ket{{\sf
x}_1{\sf y}_0} - \big({\rm e}^{{\rm i}\vartheta_1} {\rm e}^{-{\rm
i}\xi_2} s_1c_2+{\rm e}^{{\rm i}\vartheta_2} {\rm e}^{-{\rm
i}\xi_1}c_1s_2 \big) \ket{{\sf x}_1{\sf y}_1} \big] , \nonumber \\
\ket{\Phi_4} &=&{\textstyle {1 \over \sqrt{2}}}\big[ \big({\rm
e}^{-{\rm i}\vartheta_2} {\rm e}^{-{\rm i}\xi_2} c_1s_2-{\rm
e}^{-{\rm i}\vartheta_1} {\rm e}^{-{\rm i}\xi_1}s_1c_2 \big)
\ket{{\sf x}_0{\sf y}_0} - \big({\rm e}^{-{\rm i}\xi_2} c_1c_2+{\rm
e}^{-{\rm i}(\vartheta_1-\vartheta_2)} {\rm e}^{-{\rm i}\xi_1}
s_1s_2 \big) \ket{{\sf x}_0{\sf y}_1} \nonumber \\ && + \big({\rm
e}^{{\rm i}(\vartheta_1-\vartheta_2)} {\rm e}^{-{\rm i}\xi_2}
s_1s_2+ {\rm e}^{-{\rm i}\xi_1} c_1c_2 \big) \ket{{\sf x}_1{\sf
y}_0}- \big({\rm e}^{{\rm i}\vartheta_1} {\rm e}^{-{\rm i}\xi_2}
s_1c_2-{\rm e}^{{\rm i}\vartheta_2} {\rm e}^{-{\rm i}\xi_1} c_1s_2
\big) \ket{{\sf x}_1{\sf y}_1} \big] . \nonumber \\
\end{eqnarray}
\end{widetext}
These states are of the form (\ref{unbiased}), if the amplitudes are
all of the same size, which yields the conditions
\begin{eqnarray} \langlebel{constraint1}
&& \hspace*{-1.6cm} \big| c_1c_2\pm{\rm e}^{-{\rm
i}(\vartheta_1+\vartheta_2+\xi_1+\xi_2)}s_1s_2 \big| \nonumber
\\ &=& \big|c_1s_2\pm{\rm e}^{-{\rm
i}(\vartheta_1+\vartheta_2+\xi_1+\xi_2)}s_1c_2 \big| =
\textstyle{1 \over \sqrt{2}} \, ,
\end{eqnarray}
and
\begin{eqnarray} \langlebel{constraint2}
&& \hspace*{-1.6cm} \big|c_1s_2\pm{\rm e}^{-{\rm i}(\vartheta_1 -
\vartheta_2+\xi_1-\xi_2)}s_1c_2 \big| \nonumber \\ &=&
\big|c_1c_2\pm{\rm e}^{-{\rm
i}(\vartheta_1-\vartheta_2+\xi_1-\xi_2)}s_1s_2| = \textstyle{1
\over \sqrt{2}} \, .
\end{eqnarray}
The only solution of the constraints (\ref{constraint1}) and
(\ref{constraint2}) is
\begin{equation} \langlebel{con}
\cos(2\theta_1)\cos(2\theta_2)=\cos(\vartheta_1\pm\vartheta_2+\xi_1
\pm \xi_2)=0
\end{equation}
provided that neither
$\cos(2\theta_1)$ nor $\cos(2\theta_2)$ equals 1. In the special case,
where either $\cos(2\theta_1)=1$ or $\cos(2\theta_2)=1$, condition
(\ref{con}) simplifies to $\cos(2\theta_1)\cos(2\theta_2)=0$ with no
restrictions in the angles $\vartheta_1$, $\vartheta_2$, $\xi_1$ and
$\xi_2$.
One particular way to fulfill the restrictions (\ref{con}) is to set
\begin{equation} \langlebel{angels}
\xi_2=-{\textstyle {1 \over 2}} \pi \, ,
~\xi_1=\vartheta_1=\vartheta_2=0 ~~ {\rm and} ~~\theta_1 = \theta_2 =
{\textstyle {1 \over 4}} \pi \, ,
\end{equation}
which corresponds to the choice (c.f.~\cite{moonlight})
\begin{eqnarray} \langlebel{angelencoding}
\ket{{\sf a}_{1}} &=& {\textstyle {1 \over \sqrt{2}}} ( \ket{{\sf
x}_0} + \, \ket{{\sf x}_1}) \, , \nonumber \\ \ket{{\sf a}_{2}}
&=& {\textstyle {1 \over \sqrt{2}}} ( \ket{{\sf x}_0} - \,
\ket{{\sf x}_1}) \, , \nonumber \\ \ket{{\sf b}_{1}} &=&
{\textstyle {1 \over \sqrt{2}}} ( \ket{{\sf y}_0} + \ket{{\sf
y}_1}) \, , \nonumber \\ \ket{{\sf b}_{2}} &=& {\textstyle {{\rm
i} \over \sqrt{2}}} ( \ket{{\sf y}_0} - \ket{{\sf y}_1})
\end{eqnarray}
and yields
\begin{eqnarray}
\ket{\Phi_1} &= & {\textstyle {1 \over 2}}{\rm e}^{{\rm i}\pi/4} \big[
\ket{\sf x_0y_0}-{\rm i}\ket{\sf x_0y_1}-{\rm i}\ket{\sf
x_1y_0}+\ket{\sf x_1y_1} \big] \, , \nonumber \\ \ket{\Phi_2} &= &
{\textstyle {1 \over 2}}{\rm e}^{-{\rm i}\pi/4} \big[ \ket{\sf
x_0y_0}+{\rm i}\ket{\sf x_0y_1}+{\rm i}\ket{\sf x_1y_0}+\ket{\sf
x_1y_1} \big] \, , \nonumber \\ \ket{\Phi_3} &= & {\textstyle {1
\over 2}}{\rm e}^{{\rm i}\pi/4} \big[ \ket{\sf x_0y_0}-{\rm
i}\ket{\sf x_0y_1}+{\rm i}\ket{\sf x_1y_0}-\ket{\sf x_1y_1} \big] \,
, \nonumber \\ \ket{\Phi_4} &= & -{\textstyle {1 \over 2}}{\rm
e}^{-{\rm i}\pi/4} \big[ \ket{\sf x_0y_0}+{\rm i}\ket{\sf
x_0y_1}-{\rm i}\ket{\sf x_1y_0}-\ket{\sf x_1y_1} \big] \,
. \nonumber \\
\end{eqnarray}
To find out which gate operation the detection of the corresponding
maximally entangled states (\ref{base}) combined with a subsequent
absorption of the photon pair results into, we write the input
state (\ref{theencoding}) as
\begin{equation} \langlebel{bla}
\ket{\psi_{\rm enc}} = {\textstyle {1 \over 2}} \sum_i^4
\ket{\psi_i} \otimes \ket{\Phi_i}
\end{equation}
and determine the states $\ket{\psi_i}$ of the
stationary qubits. Using the notation
\begin{eqnarray} \langlebel{CZ}
U_{\rm CZ} &\equiv& \ket{00}\bra{00} + \ket{01}\bra{01} +
\ket{10}\bra{10} - \ket{11}\bra{11} ~~~
\end{eqnarray}
for the controlled two-qubit phase gate (the CZ gate) and the notation
\begin{equation}
Z_i(\phi) \equiv \ket{0}_{\rm ii}\bra{0}+{\rm e}^{-{\rm
i}\phi}\ket{1}_{\rm ii}\bra{1}
\end{equation}
for the local controlled-Z gate on photon source $i$ \cite{endnote},
we find
\begin{eqnarray}
\ket{\psi_1} &=& \exp \big({- \textstyle {1 \over 4}} {\rm i} \pi
\big) \, Z_2 \big(- {\textstyle {1 \over 2}} \pi \big) \, Z_1 \big(-
{\textstyle {1 \over 2}} \pi \big) \, U_{\rm CZ} \, \ket{\psi_{\rm
in}} \, , \nonumber \\ \ket{\psi_2} &=& \exp \big({\textstyle {1
\over 4}} {\rm i} \pi \big) \, Z_2 \big({\textstyle {1 \over 2}} \pi
\big) \,Z_1 \big( {\textstyle {1 \over 2}} \pi \big) \, U_{\rm CZ} \,
\ket{\psi_{\rm in}} \, , \nonumber \\ \ket{\psi_3}&=& \exp \big({-
\textstyle {1 \over 4}} {\rm i} \pi \big) \, Z_2 \big(-{\textstyle {1
\over 2}} \pi \big) \, Z_1 \big({\textstyle {1 \over 2}} \pi \big)
\, U_{\rm CZ} \, \ket{\psi_{\rm in}} \, , \nonumber \\
\ket{\psi_4} &=& -\exp \big({\textstyle {1 \over 4}} {\rm i} \pi
\big) \, Z_2 \big({\textstyle {1 \over 2}} \pi \big) \, Z_1
\big(-{\textstyle {1 \over 2}} \pi \big) \, U_{\rm CZ} \,
\ket{\psi_{\rm in}} \, . \nonumber \\
\end{eqnarray}
From this we see that one can indeed obtain the CZ gate operation
(\ref{CZ}) up to local unitary operations upon the detection of any of
the four Bell states $\ket{\Phi_{i}}$, as it has been pointed out
already by Protsenko {\em et al.} \cite{grangier}.
\subsection{Repeat-Until-Success quantum computing} \langlebel{insurance}
\noindent
When implementing distributed quantum computing with photons as
flying qubits, the problem arises that it is impossible to perform a
complete deterministic Bell measurement on the photons using only
linear optics elements. As it has been shown
\cite{Lutkenhaus}, in the best case, one can distinguish two of the
four Bell states. Since the construction of efficient non-linear optical
elements remains experimentally challenging, the above described phase
gate could therefore be operated at most with success rate ${1 \over 2}$.
What must be done to solve this problem is to choose the photon pair measurement
basis $\{ |\Phi_i \ranglengle \}$ such that two of the basis states are
maximally entangled while the other two basis states are product
states. Most importantly, all basis states must be mutually unbiased
with respect to the computational basis and information will not be
destroyed at any stage of the computation. In the following we
choose $\ket{\Phi_3}$ and $\ket{\Phi_4}$ as in Eq.~(\ref{base}) and
$\ket{\Phi_1}$ and $\ket{\Phi_2}$ as product states such that
\begin{eqnarray}\langlebel{twoproduct}
\ket{\Phi_1} &=& \ket{{\sf a_1b_1}} \, , ~~ \ket{\Phi_2}=\ket{{\sf
a_2b_2}} \, , \nonumber \\
\ket{\Phi_3} &\equiv & {\textstyle {1 \over
\sqrt{2}}} \big[ \ket{{\sf a}_1 {\sf b}_2}+\ket{{\sf a}_2 {\sf
b}_1} \big] \, , \nonumber \\ \ket{\Phi_4} &\equiv & {\textstyle
{1 \over \sqrt{2}}} \big[ \ket{{\sf a}_1 {\sf b}_2}-\ket{{\sf
a}_2 {\sf b}_1} \big] \, .
\end{eqnarray}
The aim of this is (see Section \ref{sectionii}) that in the event of
the ``failure'' of the gate implementation (i.e.~in case of the detection of
$\ket{\Phi_1}$ or $\ket{\Phi_2}$) the system remains, up to a local
phase gate, in the original qubit state. This means that the initial
state (\ref{original}) can be restored and the described protocol can
be repeated, thereby eventually resulting in the performance of the
universal controlled phase gate (\ref{CZ}). The probability for the
realization of the gate operation within one step equals ${1 \over 2}$ and
the final completion of a quantum phase gate therefore requires on
average {\em two} repetitions of the above described photon pair
generation and detection process.
Let us now determine the conditions under which the states $\{ |\Phi_i
\ranglengle \}$ are of the form (\ref{unbiased}). Proceeding as above,
we find that the angles $\vartheta_i$, $\xi_i$ and $\theta_i$ in
Eq.~(\ref{ab}) should fulfill, for example, Eq.~(\ref{angels}). In
analogy to Eqs.~(\ref{constraint1}) and (\ref{constraint2}), we find
that $\ket{\Phi_1}$ and $\ket{\Phi_2}$ are mutually unbiased, if
\begin{eqnarray}
\big|c_1c_2 \big| = \big| c_1s_2 \big| = \big|s_1c_2 \big| =
\big|s_1s_2 \big| = \textstyle{1 \over 2} \, ,
\end{eqnarray}
which also holds for the parameter choice in Eq.~(\ref{angels}).
Using Eq.~(\ref{angelencoding}), one can
easily verify that with the above choice the basis (\ref{twoproduct})
becomes
\begin{eqnarray} \langlebel{xxx}
\ket{\Phi_1}&=& {\textstyle {1 \over 2}} \big[ \ket{\sf
x_0y_0}+\ket{\sf x_0y_1}+\ket{\sf x_1y_0}+\ket{\sf x_1y_1} \big]
\, , \nonumber \\ \ket{\Phi_2}&=& {\textstyle {{\rm i} \over 2}}
\big[ \ket{\sf x_0y_0}-\ket{\sf x_0y_1}-\ket{\sf x_1y_0}+\ket{\sf
x_1y_1} \big] \, , \nonumber \\
\ket{\Phi_3} &= & {\textstyle {1
\over 2}}{\rm e}^{{\rm i}\pi/4} \big[ \ket{\sf x_0y_0}-{\rm
i}\ket{\sf x_0y_1}+{\rm i}\ket{\sf x_1y_0}-\ket{\sf x_1y_1} \big] \,
, \nonumber \\ \ket{\Phi_4} &= & -{\textstyle {1 \over 2}}{\rm
e}^{-{\rm i}\pi/4} \big[ \ket{\sf x_0y_0}+{\rm i}\ket{\sf
x_0y_1}-{\rm i}\ket{\sf x_1y_0}-\ket{\sf x_1y_1} \big] \, . \nonumber \\
\end{eqnarray}
Choosing the states $|{\sf a}_i \ranglengle$ and $|{\sf
b}_i \ranglengle$ as in Eq.~(\ref{angelencoding}) allows to implement the
gate operation (\ref{CZ}) eventually deterministically.
Finally, we determine the gate operations corresponding to the
detection of a certain measurement outcome $|\Phi_i \ranglengle$. To do
this, we decompose the input state (\ref{theencoding}) again into a state
of the form (\ref{bla}). Proceeding as in the previous subsection, we
find
\begin{eqnarray} \langlebel{PossibleOutcomesOfCZ}
\ket{\psi_1} &=& \ket{\psi_{\rm in}} \, , \nonumber \\ \ket{\psi_2}
&=& -{\rm i} \, Z_2 \big( \pi \big) \, Z_2 \big( \pi \big) \,
\ket{\psi_{\rm in}} \, , \nonumber \\ \ket{\psi_3}&=& \exp \big({-
\textstyle {1 \over 4}} {\rm i} \pi \big) \, Z_2 \big(-{\textstyle {1
\over 2}} \pi \big) \, Z_1 \big({\textstyle {1 \over 2}} \pi \big)
\, U_{\rm CZ} \, \ket{\psi_{\rm in}} \, , \nonumber \\
\ket{\psi_4} &=& -\exp \big({\textstyle {1 \over 4}} {\rm i} \pi
\big) \, Z_2 \big({\textstyle {1 \over 2}} \pi \big) \, Z_1
\big(-{\textstyle {1 \over 2}} \pi \big) \, U_{\rm CZ} \,
\ket{\psi_{\rm in}} \, . \nonumber \\
\end{eqnarray}
Again one obtains the CZ gate operation
(\ref{CZ}) up to local unitary operations upon the detection of either
$\ket{\Phi_{3}}$ or $\ket{\Phi_{4}}$. In the event of the detection of the
product states $\ket{\Phi_1}$ or $\ket{\Phi_2}$, the initial state can
be restored with the help of one-qubit phase gates, which then allows us
to repeat the operation until success.
It should be emphasized that there are other possible encodings that
yield a universal two-qubit phase gate upon the detection of a
Bell-state, but where the original state is destroyed upon the
detection of a product state (see e.g.~\cite{Zou}). This happens when
the product states are not mutually unbiased and their detection
erases the qubit states in the respective photon sources. To achieve the effect of
an {\em insurance} against failure, the encoding (\ref{enc}) should be
chosen as described in this Section.
\section{Possible experimental realizations} \langlebel{rea}
\noindent
Possible experimental realizations of the above described eventually
deterministic quantum phase gate consist of two basic steps. Firstly,
the information of the stationary qubits involved in the operation has
to be redundantly encoded in the states of two newly generated ancilla
photons. Afterwards, a measurement is performed on the photon pair
resulting with probability ${1 \over2}$ in the desired gate
operation. Depending on the type of the photon source, one can choose
different types of encoding. There are also different possibilities
how to perform the photon pair measurement. Examples are given below.
\subsection{Redundant encoding}
\noindent
In order to obtain robust qubits, the states $|0 \ranglengle$ and $|1
\ranglengle$ should be two different longliving ground states of the
single photon source. Each photon source carries one qubit. Depending
on its level structure (see Figure \ref{photongun}), it might be
advantageous to realise the encoding step (\ref{theencoding}) either
by generating photons with different polarisations (polarisation
encoding) or photons that agree in all degrees of freedom apart from
their creation time (time bin encoding). Note that different encodings
can easily be transformed into each other using linear optics elements
like a polarising beam splitters and delaying photons in time.
\begin{figure}
\caption{Schematic view of a single photon (a) polarization encoder,
Ê(b) time-bin encoder and level configuration of the sourceÊ
Êcontaining the qubit.}
\end{figure}
{\em Polarization encoding.} Suppose, the photon source contains an
atomic double $\Lambda$ level configuration as shown in Figure
\ref{photongun}(a) (see also Ref.~\cite{Gheri}). A single photon can
then be created by simultaneously applying a laser pulse with
increasing Rabi frequency to the $0$-$e_0$ transition and the
$1$-$e_1$ transition of the atomic system. Thereby, the atom goes to
the ground state $|v_0 \ranglengle$ and $|v_1 \ranglengle$, respectively,
depending on whether its initial state equalled $|0 \ranglengle$ or $|1
\ranglengle$ due to the coupling of the $e_0$-$v_0$ transition and the
$e_1$-$v_1$ transition to the cavity mode. It has been shown in the
past that this technique \cite{Law} is very well suited to place
exactly one excitation into the field of an optical resonator, from
where it can leak out \cite{Kuhn2}.
If the two transitions, $e_0$-$v_0$ and $e_1$-$v_1$, couple to the two
different polarisation modes ${\sf h}$ and ${\sf v}$, in the cavity
field, the photon generation results effectively, for example, in the
operation
\begin{equation}
\ket{0}_i \rightarrow \ket{0 ,{\sf h}}_i \, , ~~ \ket{1}_i \rightarrow
\ket{1 ,{\sf v}}_i
\end{equation}
once atom $i$ has been repumped into its initial state $|0 \ranglengle_i$
and $|1 \ranglengle_i$, respectively. Finally we remark that the encoding
does not affect the coefficients $\alpha$, $\beta$, $\gamma$ and
$\delta$ of the initial state (\ref{original}). As long as no
measurement is performed on the system, all coherences are preserved.
{\em Time-bin encoding.}
Alternatively, if the photon sources possess a level structure like
the one shown in Figure \ref{photongun}(b), one can redundantly encode
the information contained in the qubits into time bin encoded photons,
\begin{equation} \langlebel{higgs}
\ket{0}_i \rightarrow \ket{0 ,{\sf E}}_i \, , ~~ \ket{1}_i \rightarrow
\ket{1 ,{\sf L}}_i \, .
\end{equation}
This encoding is simpler and may therefore find realizations not only in
atoms but also in quantum dots and nitrogen vacancy color centers.
In Eq.~(\ref{higgs}), $|{\sf E} \ranglengle$ and $|{\sf L} \ranglengle$ denote
a single photon generated at an {\em early} and a {\em later} time,
respectively. The above operation can be achieved by first coupling a
laser field with increasing Rabi frequency to the $1-e_1$ transition,
while the cavity mode couples to the $e_1-v_1$ transition. Once the
excitation has been placed into the cavity mode and leaked out through
the outcoupling mirror, the atom can be repumped into $|0
\ranglengle$. Afterwards, one should swap the states $\ket{0}$ and
$\ket{1}$ and repeat the process. This results in the generation of a
late photon, if the system was initially prepared in $|1 \ranglengle$. To
complete the encoding, the states $\ket{0}$ and $\ket{1}$ have to be
swapped again.
\subsection{Photon pair measurement}\langlebel{rea2}
\noindent
We now give two examples how to perform a photon pair measurement of
the mutually unbiased basis (\ref{xxx}). The first method is suitable
for polarization encoded photons, the second one for dual-rail
encoded photons. If the qubits have initially been time bin-encoded,
their encoding should be transformed first using standard linear
optics techniques.
{\em Polarization encoding.}
It is well known that sending two
polarization encoded photons through the different input ports of a
50:50 beam splitter together with polarization sensitive measurements
in the $\ket{\sf h}/\ket{\sf v}$-basis in the output ports would
result in a measurement of the states ${\textstyle {1 \over {\sqrt
2}}}(\ket{\sf hv}\pm\ket{\sf vh})$, $\ket{\sf hh}$ and $\ket{\sf vv
}$. To measure the states (\ref{twoproduct}), we therefore propose to
proceed as shown in Figure \ref{moon3}(a) \cite{moonlight} and to
perform the mapping
\begin{eqnarray}
U_1 &=& |{\sf h} \ranglengle \langlengle {\sf a_1}| + |{\sf v} \ranglengle \langlengle
{\sf a_2}| \, , \nonumber \\
U_2 &=& |{\sf h} \ranglengle \langlengle {\sf b_1}| + |{\sf v} \ranglengle \langlengle
{\sf b_2}|
\end{eqnarray}
on the photon coming from source $i$. Using Eq.~(\ref{angelencoding}),
we see that this corresponds to the single qubit rotations
\begin{eqnarray}
U_1 &=& {\textstyle {1 \over \sqrt{2}}} \, \big[ \, |{\sf h} \ranglengle
\big( \langlengle {\sf h}| + \langlengle {\sf v}| \big) + |{\sf v} \ranglengle
\big( \langlengle {\sf h}| - \langlengle {\sf v}| \big) \, \big] \, ,
\nonumber \\ U_2 &=& {\textstyle {1 \over \sqrt{2}}} \, \big[ \,
|{\sf h} \ranglengle \big( \langlengle {\sf h}| + \langlengle {\sf v}| \big) -
{\rm i} \, |{\sf v} \ranglengle \big( \langlengle {\sf h}| - \langlengle {\sf
v}| \big) \, \big] \, .
\end{eqnarray}
After leaving the beam splitter, the photons should be detected in the
$\ket{\sf h}/\ket{\sf v}$-basis. A detection of two ${\sf h}$ and two ${\sf
v}$ polarized photons indicates a measurement of $|\Phi_1 \ranglengle$
and $|\Phi_2 \ranglengle$, respectively. Finding two photons of different
polarization in the same or in different detectors corresponds to a
detection of $|\Phi_3 \ranglengle$ or $|\Phi_4 \ranglengle$.
{\em Dual-rail encoding.}
Alternatively, one can redirect the generated photons to the different
input ports of a $4 \times 4$ Bell multiport beam splitter as shown in
Figure \ref{moon3}(b). If $a_n^\dagger$ and $b_n^\dagger$ denotes the
creation operator for a photon in input and output port $n$,
respectively, the effect of the multiport can be summarized as
\cite{multi}
\begin{equation} \langlebel{scatter}
a_n^\dagger \to \sum_m U_{mn} b_m^\dagger
\end{equation}
with
\begin{equation}
U_{mn} = {\textstyle {1 \over 2}} \, {\rm exp} \big( {\rm i} \pi
(n-1)(m-1) \big) \, .
\end{equation}
A Bell multiport redirects each incoming photon with equal probability
to any of the possible output ports, thereby erasing
the which-way information of the incoming photons. One way to measure
in the mutually unbiased basis (\ref{xxx}) is to direct the $\ket{\sf
x_0}$ photon from source 1 to input port 1, the $\ket{\sf x_1}$ photon
from source 1 to input port 3 and to direct the $\ket{\sf y_0}$ photon
and the $\ket{\sf y_1}$ photon from source 2 to input port 2 and 4,
respectively. If $\ket{\rm vac}$ denotes the state with no photons in
the setup, this results in the conversion
\begin{eqnarray}
&& |{\sf x_0y_0} \ranglengle \to a_1^\dagger a_2^\dagger \, \ket{{\rm vac}} \, , ~~
|{\sf x_0y_1} \ranglengle \to a_1^\dagger a_4^\dagger \, \ket{{\rm vac}}
\, , \nonumber \\
&& |{\sf x_1y_0} \ranglengle \to a_2^\dagger a_3^\dagger \, \ket{{\rm vac}} \, , ~~
|{\sf x_1y_1} \ranglengle \to a_3^\dagger a_4^\dagger \, \ket{{\rm vac}} \, .
\end{eqnarray}
This conversion should be realized such that the photons enter
the multiport at the same time. Using Eq.~(\ref{scatter}) one can show
that the network transfers the basis states (\ref{xxx}) according to
\begin{eqnarray} \langlebel{multiportresult}
|\Phi_1 \ranglengle & \to & {\textstyle {1 \over 2}} \, \big(
b_1^{\dagger \, 2} - b_3^{\dagger \, 2} \big) \, \ket{\rm vac} \, ,
\nonumber \\ |\Phi_2 \ranglengle & \to & - {\textstyle {1 \over 2}} \,
\big( b_2^{\dagger \, 2} - b_4^{\dagger \, 2} \big) \, \ket{\rm vac}
\, , \nonumber \\ |\Phi_3 \ranglengle & \to & {\textstyle {1 \over {\sqrt
2}}} \, \big( b_1^\dagger b_4^\dagger - b_2^\dagger
b_3^\dagger \big) \, \ket{\rm vac} \, , \nonumber \\ |\Phi_4
\ranglengle & \to & - {\textstyle {1 \over {\sqrt 2}}} \, \big(
b_1^\dagger b_2^\dagger - b_3^\dagger b_4^\dagger \big) \, \ket{\rm
vac} \, .
\end{eqnarray}
Finally, detectors measure the presence of photons in each of the
possible output ports. The detection of two photons in the same output
port, namely in 1 or 3 and in 2 or 4, corresponds to a measurement of
the state $|\Phi_1 \ranglengle$ and $|\Phi_2 \ranglengle$, respectively. The
detection of a photon in ports 1 and 4 or in 2 and 3 indicates a
measurement of the state $\ket{\Phi_3}$, while a photon in the ports 1
and 2 or in 3 and 4 indicates the state $\ket{\Phi_4}$.
\begin{figure}
\caption{Linear optics networks for the realization of a measurement
of the basis states (\ref{base}
\end{figure}
Any unknown fixed (or slowly varying with respect to the coherence length of
the photon pulse) phase factor introduced along the photon paths contributes
at most to a global phase factor to the input state (\ref{theencoding}), which
is also a feature of the schemes outlined in
Refs.~\cite{last,Simon,lim,moonlight}. The implementation of
Repeat-Until-Success quantum computing therefore does not require
interferometric stability. It requires only overlapping of the photons within
their coherence length within the linear optics setup.
\section{Scalable quantum computation in the presence of
inefficient photon generation and detection}\langlebel{cluster}
\noindent
In this section, we discuss the possibility of implementing scalable
quantum computation using the Repeat-Until-Success quantum gate
described in the previous sections. The implementation of this gate
requires the generation of single photons on demand and linear optical
elements together with absorbing quantum measurements. In the limit
of perfect photon emission, collection, and detection efficiency,
two-qubit CZ gates can be performed deterministically, as described
above. In real systems however, photon emission, collection, and
detection is not perfect \cite{kok01}. In existing experiments, all
of these processes have significant inefficiencies, which means that
there is a finite probability that two photons will not be observed in
the photon measurement. The failure to observe two photons in an
attempted CZ operation means that the static qubits are left in an
unknown state, which constitutes a correlated two qubit error. If such
losses are sufficiently small (e.g. less than $\sim 10^{-2}$), the
resulting gate failures can be dealt with using existing fault
tolerance techniques \cite{steane2003,knill2005}. Recently, much
higher fault tolerance levels of up to 50\% were found in linear
optical quantum computing \cite{varnava,gilchrist}.
More concretely, the highest reported photon detection efficiency for
single photon detection with photon number resolution is about $88\%$
\cite{Takeuchi99,Rosenberg}. A recent experiment by McKeever {\em
et al.} \cite{Mckeever} involving an atom-cavity system for
the generation of single photons yields a photon generation efficiency of
nearly 70\%, limited only by passive cavity loss. The lifetime
of the atom in the cavity was $0.14\,$s, allowing for as many as $1.4 \cdot 10^4$ photon
generation events. Moreover, Legero {\em et al}.~\cite{Legero} demonstrated perfect time-resolved interference with two photons of different frequencies. Time-resolved detection acts as a temporal
filter to erase the which-way information which is important to any scheme
involving photon interference. This suggests that strictly
identical single photon sources are not required for attaining high fidelities
in the state preparation. The cost of this high fidelity is a lower
probability of success.
Fortunately, scalable quantum computing is possible, even in the presence of
large errors, as long as no errors imply a very high fidelity and the occurrence of an error is \emph{heralded}:
if fewer than two photons are detected, we know that the attempted CZ operation has failed. Only when the detectors have a substantial amount of dark
counts, we cannot rely on this error detection mechanism. However,
comercially available silicon avalanche photodetectors are available with a
detection efficiency of 65\%, and a dark count rate of $\Gamma_{dc} \le
25\,$s$^{-1}$ \cite{PED}. Assuming, a photon regeneration rate of
$105\,$s$^{-1}$ gives a clock time of $10^{-5}$ s. The total dark count
probability is then $p_{dc} \approx 10^{-4}$ per clock cycle, which is small
enough to be dealt with using existing error correction techniques.
Moreover, if one could experiment with detectors like the one reported by
Rosenberg {\em et al}.~\cite{Rosenberg}, dark count rate effects would be
negligible.
In the case of
an error, the state of the static qubits can be determined by
subsequently performing measurements on the sources, which allows the
sources to be re-prepared in a known state. In earlier work, we have
shown that scalable quantum computation can be performed in the
presence of significant heralded error rates, by first using a
non-deterministic entangling operation to create \emph{cluster states}
of many qubits \cite{Sean}, and subsequently implementing scalable
quantum computation via the `one-way quantum computer'
\cite{raussendorf}. Given cluster states of many qubits, the one-way
quantum computer can be implemented by single qubit measurements
alone. This technique permits fully scalable quantum computation,
albeit with a fixed overhead per two qubit gate in the algorithm,
which we calculate below. We briefly review how one way quantum
computing can proceed within our scheme, and then provide an estimate
of the overhead costs involved.
\subsection{One way quantum computation}
\noindent
One way quantum computation \cite{raussendorf} proceeds by first
creating a graph state of many qubits, and subsequently performing
single qubit measurements on the graph state \cite{rbb,hein,weinstein}. Graph
states may be represented as a graph comprising set of qubit `nodes'
connected by `edges' which may be understood as `bonds' between the
qubits. The quantum state corresponding to such a graph may be
defined (and also implemented) by the following procedure: (i) prepare
each qubit in the state $|+\ranglengle = (|0\ranglengle +
|1\ranglengle)/\sqrt{2}$, and then (ii) for each bond in the corresponding
graph, apply a deterministic CZ operation (see Eq. \ref{CZ}) between
the relevant qubits. In this work, we will restrict our attention to
the rectangular lattice graph states of the form shown in Figure
\ref{fig:cluster1} (hereafter referred to simply as \emph{cluster
states}), which are sufficient for simulating arbitrary logic
networks, and hence universal quantum computations
\cite{nielsen04}. It is worth noting however that straightforward
generalizations of the procedure described below allow us to scalably
generate \emph{arbitrary} graph states. This may be useful in that it
might result in reduced costs for implementing certain
algorithms.
\begin{figure}
\caption{A rectangular lattice cluster state. Each circle
represents a physical qubit, and each line represents a `bond'
between qubits. These states are sufficient for simulating
arbitrary logic networks \cite{nielsen04}
\end{figure}
In these clusters, each horizontal row of physical qubits
represents a single logical qubit in the logic network being
simulated. Two qubit operations are implemented by the vertical
bonds acting between rows. We also permit bonds between
non-adjacent rows, which permits highly non-local two qubit gates
to be implemented. Note also that the location of the qubits
within the cluster is notional, and need not correspond to the
physical location of the static qubit (the mapping between the
notional qubit positions within the cluster, and the actual
physical location of the qubits can be stored in a classical
computer). After making the state, quantum computation proceeds by
performing a sequence of single qubit measurements on the static
qubits, with each measurement performed in a particular basis so
as to implement a given sequence of one- and two-qubit gates
\cite{raussendorf,nielsen04}. At each time step, a whole column of
physical qubits in the cluster is measured. The measurements are
performed in order, starting with the column at left side of the
cluster, and proceeding rightwards across the cluster. In general,
the basis of the measurements made at a given time step will
depend on the outcomes of earlier measurements. Once a physical
qubit has been measured, that qubit is disentangled from the
cluster state and so may be re-initialized in a particular state
and subsequently used later in the computation.
We assume that single-shot single qubit measurements and single
qubit unitary operations on the static qubits can be implemented
using standard techniques. Implementing one-way quantum
computation in our scheme therefore reduces to the problem of
scalably generating cluster states using the heralded,
non-deterministic CZ operation. We outline the general procedure
here, and give a more detailed description in the subsequent
section.
In our scheme, cluster states can be generated by attempting to
bond qubits using the non-deterministic CZ operation. This
operation has three possible outcomes: `success', `insurance', or
`failure'. In the case of observing two photons, one of the gates
of Eq.(\ref{PossibleOutcomesOfCZ}) is implemented, and subsequent
application of appropriate single qubit unitaries implements
either the CZ operation (denoting a `success'), or the identity
operation (denoting `insurance'). In the case of insurance, the CZ
operation can simply be reattempted. Observing fewer than two
photons denotes a failure. In this case, the static qubits are
left in an unknown state. However, this damage can be repaired as
follows. Firstly, each of the two qubits involved in the failed
gate can be measured in the computational basis to determine the
nature of the error. If either qubit was already part of a cluster
state, the bonds to its neighbors within the cluster are also
destroyed. However, the remainder of the cluster state can be
recovered by applying appropriate single qubit unitary operations
to these neighboring qubits, conditional on the outcome of the
measurement on the qubit involved in the failed CZ gate.
Therefore, the cluster state can grow, shrink, or remain the same
size, depending whether the CZ operation was successful, failed,
or failed with insurance. The key to scalably generating cluster
states is to attempt CZ operations between qubits in a sequence
order such that the cluster state grows on average. We give such a
sequence in Sec. \ref{Overhead costs}.
\begin{figure}
\caption{Dynamically growing clusters during a computation. The
cluster contains three regions: the active region (M), at the
left of the cluster, in which the logic gate networks are being
simulated via single qubit measurements; the buffer region; and
the connection region, where new cluster fragments are added to
the right edge of the main cluster. The connecting cluster chains
have a buffer length $L_a$ to accommodate the probabilistic
entangling operation.}
\end{figure}
We conclude this section by noting that it is not necessary
to build the whole cluster required for simulating a
particular algorithm before commencing the single qubit
measurement part of the computation. It is possible to
build a partial cluster, and then to simultaneously perform
single qubit measurements on one part of the cluster, while
adding new qubits to another region in the cluster. In this
approach to one-way quantum computing, one can think of the
cluster as being split into three regions, as shown in Figure
\ref{fig:cluster2}. The \emph{active region}, to the left
of the cluster, contains the part of the cluster where the
logic gate networks are being simulated via single qubit
measurements. At the right of the cluster, the
\emph{connection region} comprises of several horizontal
dangling linear chains which extend from the right edge of
the main cluster, each corresponding to a logical qubit. In
this region, nondeterministic CZ operations are applied in
order to add further cluster sections to the main section.
These additional sections are manufactured separately, as
described in Sec. \ref{Overhead costs}. Between the active
region and the connection region, the \emph{buffer region}
comprises a quiescent region which suffices to protect the
active region in the event of a long sequence of failed CZ
operations; this would lead to the right edge of the
cluster running back into the active region, damaging the
logical computation. The depth of the buffer region should
be chosen such that the probability of erasing a logical
qubit is sufficiently small that it can be handled with
existing fault tolerance techniques
\cite{steane2003,knill2005}.
There are several advantages to this approach. Firstly,
fewer physical qubits are needed, because qubits that have
already been measured at the left edge of the cluster can
be recycled and added to the right hand side of the
cluster. Secondly, preparing the whole cluster initially
means that some of the qubits will spend a lot of time in
an `idle' state before they are involved in the
computation; any errors accumulated in these idle qubits
due to decoherence will degrade the fidelity of the
computation \cite{raussendorf}. This is crucial if fault
tolerant quantum computation is to be implemented within
the cluster model, as such schemes require a source of
fresh ancilla qubits throughout the algorithm. Thirdly, the
overhead costs for this approach can be reduced, because it
is not necessary to prepare the \emph{whole} cluster with a
total success probability close to one; the probability for
erasing a given logical qubit need only be made smaller
than the error threshold required for fault tolerance.
\subsection{Overhead costs} \langlebel{Overhead costs}
\noindent
A number of authors have considered efficient cluster state generation
using non-deterministic, but heralded, Entangling Operations (EOs)
\cite{yoran03,nielsen04,browne04,Benjamin,Sean,DuanRaussendorf,benjamin04,chen}.
References \cite{yoran03,nielsen04,browne04} calculated explicit costs
for making cluster states of optical qubits in the ideal case
(i.e. neglecting photon loss). Subsequently, Barrett and Kok
\cite{Sean} showed that, in the case of hybrid matter-optical systems
(such as those considered in this work), arbitrarily small EO success
probabilities could be tolerated. They provided a `divide and conquer'
algorithm for building linear clusters, which has moderate costs even
for small success probability. An efficient algorithm for building two
dimensional clusters, capable of simulating arbitrary logic networks
was also given in \cite{Sean}. More recently, in
Ref. \cite{DuanRaussendorf}, a similar algorithm for building linear
clusters was proposed, which made more use of recycling, and hence has
a lower overhead cost. Ref. \cite{DuanRaussendorf} also gives an
alternative algorithm for making 2-dimensional clusters, and
explicitly calculates the associated overhead costs. In
Ref. \cite{benjamin04}, some elegant cost reducing improvements to the
scheme proposed in Ref. \cite{Sean} were suggested, utilizing the
redundantly encoded qubits inherent in the original scheme.
In this work, we will combine elements of the approaches taken in
Refs. \cite{browne04,Sean,DuanRaussendorf} to provide a simple upper
bound for the scaling costs for building cluster states using our
scheme. This estimate is based on an explicit procedure, and we do not
claim that it is optimal; an improved algorithm may yield
substantially reduced costs. Nevertheless, the procedure given here
allows a straightforward calculation of the overhead costs. Despite
its apparent similarity to Refs.~\cite{Sean,DuanRaussendorf}, there is
a crucial difference: in the scheme under consideration in this paper,
there is the possibility of obtaining the `insurance' outcome. In
general, this leads to a reduction in costs relative to schemes in
which there is no insurance outcome.
In the presence of imperfect photon emission, detection, and
collection, the performance of the CZ operation can be
characterized by three probabilities:
\begin{itemize}
\item the probability of successfully implementing the CZ
operation on the input qubits (up to local operations),
$p_s$;
\item the probability of obtaining the `insurance' outcome
in which known local operations are applied to the qubits,
$p_i$;
\item the probability of failure due to failing to emit, collect,
or detect one or more photons during the remote gate operation,
$p_f$.
\end{itemize}
These probabilities are determined by the physics of the sources
and detectors.
Calculating the total cost of growing cluster states can be
simplified by noting that, in the case of obtaining the
`insurance' outcome, after applying the necessary single qubit
corrections, one simply attempts the gate operation again. This
process is repeated until a definite outcome (success or failure)
is obtained. Thus, we can define \emph{total} success and failure
probabilities, $P_s$ and $P_f$, of the corresponding definite
outcomes after an (arbitrarily long) sequence of `insurance'
outcomes. These probabilities are given by $P_s =
\sum_{j=0}^{\infty} p_i^j p_s = p_s/(1-p_i)$ and $P_f =
\sum_{j=0}^{\infty} p_i^j p_f = p_f/(1-p_i)$. The average number
of attempted CZ operations required before we obtain a definite
outcome is $N_{\mathrm{av}} = 1/(1-p_i)$.
The overhead cost for making cluster
states is then found using similar calculations to those presented
in Refs. \cite{Sean,DuanRaussendorf}. We first calculate the cost
(i.e. the number of attempted CZ operations per qubit in the final
cluster) of generating linear clusters. If a CZ gate is repeatedly
applied between the end qubits of two linear chains, each of
length $L_k$, either the gate is (ultimately) successful, in which
case the total length of the new cluster is $2 L_k$, or the gate
(ultimately) fails, in which case, the length of the original
clusters shrinks by one qubit each. Repeatedly applying this
procedure until a successful outcome is obtained (or until both
original clusters are destroyed) \cite{DuanRaussendorf} gives the
expected length $L_{k+1} = \sum_{i=0}^{L_k} 2 (L_k - i) P_s P_f^i
\approx 2 L_k - 2p_f/p_s$. Denoting the average number of attempts
to create a chain of length $L_k$ by $N_k$, we also have $N_{k+1}
= 2 N_k + 1/p_s$. Solving these recursion relations gives a total
cost
\begin{equation}
N(L) = \frac{\left(N_0+ \frac{1}{p_s}\right) \left(L -
\frac{2p_f}{p_s}\right) }{ \left(L_0 - \frac{2p_f}{p_s} \right)} -
\frac{1}{p_s}, \langlebel{TotalCost}
\end{equation}
where $N_0$ denotes the cost of growing a short cluster of length
$L_0$. Note that for the average cluster length to grow on each
round of the protocol, we require $L_1 > L_0$, which implies that
the length of the short chains should satisfy $L_0 > 2p_f/p_s$.
\begin{figure}
\caption{Creating vertical bonds. a) We start out with two
sufficiently long cluster chains, and we wish to create a vertical
bond between the two qubits on the left. b) We apply a $\sigma_x$
measurement to the two adjacent qubits and a Hadamard operation
$H$ on the next. c) This will result in a
redundant encoding of the qubits we wish to bond together. d)
Applying the entangling operation to the dangling ``cherries''
will create the vertical bond. Note that we also removed the
qubits in the vertical bond by applying another $\sigma_x$
measurement, resulting in another redundant encoding. If this
procedure fails, we are left with a shorter chain, and we can try
to create a vertical bond again.}
\end{figure}
Chains of fixed length $L_0$ can be grown independently using the
probabilistic CZ operation, by joining sub-chains together.
Growing these short chains adds a constant overhead cost to the
cluster generation process. We use a `divide and conquer' approach
to making these short chains \cite{Sean,DuanRaussendorf}, in
which, on each round of the protocol, we attempt to join equal
length pairs of linear clusters using the probabilistic CZ
operation. If we obtain the `insurance' outcome on any such
attempt, we try the operation again, whereas if we fail, we assume
(for ease of calculation) that the short chains are discarded. On
the $k$th round of this protocol, the length of the chains is $l_k
= 2^k$, and the number of attempted CZ operations is given by the
recursion relation $n_k = 2 n_{k-1}/P_s + N_{\mathrm{av}}/P_s$.
Solving these relations gives
\begin{equation}
N_0(L_0) = N_{\mathrm{av}}\sum_{i=1}^{\log_2 L_0}
\frac{2^{i-1}}{P_s^i} \,. \langlebel{OffLineCost}
\end{equation}
Combining Eq. (\ref{OffLineCost}) and Eq.
(\ref{TotalCost}), one can calculate the total cost of
growing linear clusters for given values of $p_f$, $p_s$
and $p_i$. For instance, taking $p_f = 0.6$, $p_i = p_s =
0.2$, we require $L_0 > 6$. Taking $L_0 = 2^3 = 8$, the
total cost for making a linear cluster of length $L$ is
found to be $N(L) = 185 L - 1115$ attempted CZ operations.
A moderate increase in success probability can dramatically
decrease the cost: taking $p_f = 0.4$, $p_i = p_s = 0.3$,
we require $L_0 > 2.67$, and taking $L_0=2^2=4$, we find
the total cost to be $N(L) = 16\frac{2}{3} L -
47\frac{7}{9}$. Note that the negative constant term in
these expressions is an artefact of joining small numbers
of chains together to make an isolated chain of length $L$.
This is an `edge effect' which should be neglected when
considering the asymptotic cost of making long chains.
\begin{table}[t]
\caption{The average number of entangling operations per vertical
bond, given by $N_{\rm bond} = 2 N(M) + (1-p_i)/p_s$. Here $p_s$,
$p_f$ and $p_i$ are the success, failure and insurance probabilities,
respectively. $L_0=2^n$ is the length of the chain that is needed to
obey the growth requirement, and $N_0$ is the number of EOs needed to
achieve this length. $M$ is the average cluster chain consumed by the
forging of a vertical bond, and $N(M)$ is the number of EOs needed to
achieve this length.}
\langlebel{tab:cost}
\begin{ruledtabular}
\begin{tabular}{lllcrlcr}
$p_s$ & $p_f$ & $p_i$ & $L_0$ & $N_0\phantom{\frac{1}{2}}$ & $M$ &
$N(M)$ & $N_{\rm bond}$ \\
\hline
0.2 & 0.6 & 0.2 & $2^3=8$ & 365$\phantom{\frac{1}{2}}$ & 9 & $185
M$ & 3334$\phantom{\frac{1}{2}}$ \\
0.3 & 0.4 & 0.3 & $2^2=4$ & 18$\frac{8}{9}$ & 5$\frac{2}{3}$ &
$16\frac{2}{3} M$ & 191$\frac{2}{9}$ \\
0.4 & 0.2 & 0.4 & $2^1=2$ & 2$\frac{1}{2}$ & 3 & $5 M$ &
32$\frac{1}{2}$ \\
0.5 & 0.5 & 0 & $2^2=4$ & 10$\phantom{\frac{1}{2}}$ & 5 & $6
M$ & 62$\phantom{\frac{1}{2}}$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
Linear clusters are not sufficient for simulating arbitrary
logic networks \cite{Nielsen2005}, and therefore it is
necessary to generate more general graph states. A variety
of techniques for making such states using probabilistic
entangling operations have been proposed, which include
linking linear clusters using independently prepared `I'
shaped clusters \cite{Sean}, using micro-clusters
\cite{nielsen04}, using redundantly encoded qubits
\cite{browne04}, or by making use of `+' shaped clusters
\cite{DuanRaussendorf}. Here, we propose a relatively
efficient method for creating vertical bonds between linear
cluster chains.
We employ a technique based on that introduced by Browne
and Rudolph \cite{browne04}, which involves four steps as
shown in Figure~\ref{fig:cluster3}.
\begin{enumerate}
\item[a)] First, we assume that we have sufficiently long linear
cluster chains. These can be produced efficiently in the manner
outlined above. In order to establish the amount of resources
needed to create a vertical bond, we will count the number of
qubits that are utilized on average in this process, as well as the
average number of entangling operations.
\item[b)] Secondly, we identify the two qubits that we wish to
entangle with a vertical bond (in Figure~\ref{fig:cluster3} the two
left-most qubits). The qubits directly on the right of
these qubits are then measured in the $\sigma_x$ basis. A Hadamard
operation on the third qubit in each chain returns the overal state to a
graph state.
\item[c)] This will result dangling bonds, or {\em cherries}
\cite{tom} hanging from the two qubits that are to be
connected. This is a form of redundant encoding, and it allows us
to apply the entangling operation to the two cherries. In case of a
failure, the entangling operation will {\em not} break the linear
cluster chains. It will destroy only the cherries and as a result
both chains are shortened by two qubits. Steps (b) and
(c) can then be repeated.
\item[d)] When the entangling operation succeeds, we have forged a
vertical bond between the two qubits chosen in step a). The
vertical link is itself a chain of two qubits. These are typically
not wanted, so we can remove one of them with a $\sigma_x$
measurement, creating another cherry in the other qubit in the
chain. This redundancy can be pruned, but may also be
useful for creating additional bonds, or may even be
useful for error correction.
\end{enumerate}
We will now estimate the cost of this procedure. Since two qubits are
burnt in each step, and we need to repeat the process $P_s^{-1}$
times, the average length of each chain that is consumed in the
bonding process is
\begin{equation}
M = 2 P_s^{-1} + 1 = \frac{2 (1-p_i)}{p_s} + 1\; ,
\end{equation}
where the extra $+1$ counts the qubits that will establish the
vertical link. The number of entangling operations needed to make a
vertical bond is then
\begin{equation}
N_{\rm bond} = 2 N(M) + P_s^{-1} = 2 N(M) + \frac{(1-p_i)}{p_s}\; ,
\end{equation}
where the extra $P_s^{-1}$ takes into account the number of entangling
operations that are needed to link the cherries together into a
vertical bond. In Table~\ref{tab:cost}, we calculated the
number of entangling operations that are needed to forge a vertical
bond, given several specific values for the success, failure, and
insurance probabilities.
\section{Conclusions } \langlebel{conc}
\noindent
We analysed a hybrid architecture for quantum computing using
stationary and flying qubits, which is based on our earlier work \cite{moonlight,Sean}, in detail.
It was shown that this new approach solves some of the most pressing
problems that arise in non-hybrid architectures. Our system is
scalable, even with non-ideal components, and more importantly, it
uses no direct qubit-qubit interactions. This means that the qubits
will be subject to less decoherence and fewer control errors. When realistic
photo-detectors are used, photon loss will affect only the efficiency
of the scheme. Furthermore, our system relies on components that have
been demonstrated in experiment, and is largely implementation
independent. Despite the no-go
theorem for optical Bell-state measurements, it is in principle
possible to implement a deterministic gate between distant
qubits.
However, when losses are taken into account, the gate becomes
necessarily probabilistic. In order to achieve robustness against
general decoherence and to guarantee high fidelities, we showed how to construct cluster- or
graph-states using the two-qubit gate. Our entangling operation, which produces the bonds in the
graph states, is not limited to physically adjacent matter qubits. As
a consequence, no extensive swapping operations need to be taken into
account in the production of nontrivial graph states. This
architecture for quantum computation is inherently distributed, and
hence can be used for integrated quantum computation and communication
purposes. \\[0.5cm]
{\em Acknowledgment.} YLL, AB and LCK thank H.J. Briegel,
A. Browaeys, D.E. Browne, T. Durt, A.K. Ekert, P. Grangier, M. Jones, and
P.L. Knight for stimulating discussions. PK and SDB thank S. Benjamin,
J. Eisert, B. Lovett, M.A. Nielsen, T. Stace for fruitful discussions
and particularly D. Browne for giving them a deeper insight into
cluster state generation. YLL acknowledges funding from the DSO
National Laboratories in Singapore and AB acknowledges support from
the Royal Society and the GCHQ. This work was supported in part by the
European Union (RAMBOQ, QGATES) and the UK Engineering and Physical
Sciences Research Council (QIP IRC).
\end{document} | math | 78,621 |
\begin{document}
\title{Comment on (t, n) Threshold d-level Quantum Secret Sharing}
\author{Shih-Hung Kao and Tzonelih Hwang\thanks{Corresponding author\protect \\
[email protected]\protect \\
Department of Computer Science and Information Engineering, National
Cheng Kung University, No. 1, University Rd., Tainan City, 701, Taiwan,
R.O.C.}}
\maketitle
\begin{abstract}
This comment points out a problem in Song et al.'s (t, n) threshold
quantum secret sharing {[}Scientific Reports, Vol. 7, No. 1 (2017),
pp. 6366{]}, indicating that the agent is unable to obtain the expected
information.
\end{abstract}
\paragraph{$ $}
In 2017, Song et al. {[}1{]} proposed an interesting way to perform
the summation of Lagrange interpolation formula in Shamir's secret
sharing (SS) by using quantum mechanics. Based on this method, the
agents in the QSS can obtain the boss's secret without announcing
any information. However, this study points out a calculation problem,
which indicates that the agent is unable to obtain the boss's secret
information without publishing any information.
\paragraph{Song et al.'s Secret Reconstruction Protocol.}
In Song et al.'s QSS, the boss uses Shamir's SS to generate the polynomial
$f\left(x\right)=a_{0}+a_{1}x+a_{2}x^{2}+...+a_{t-1}x^{t-1}$ and
$n$ distinct points of $\left(x_{i},\text{ }f\left(x_{i}\right)\right)$,
where $a_{0}$, $a_{t-1}\in Z_{d}$ and $i=1\sim n$. When $t$ Bobs
(the agents) want to recover the boss's secret, $a_{0}$, they will
execute the secret reconstruction protocol to compute the Lagrange
interpolation formula. Without loss of generality, assume that Bob$_{2}$
\textasciitilde{} Bob$_{t}$ help Bob$_{1}$ to obtain $a_{0}$. The
secret reconstruction protocol is as follows:
\begin{description}
\item [{Step1}] Bob$_{1}$ generates the d-level quantum state $\left|\varphi\right\rangle =\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}\left|kk...k\right\rangle _{12...t}$,
where the subscript 1 (2, 3, .., t) denotes the first (second, third,
..., t-th) qubit, respectively. Bob$_{1}$ sends the r-th qubit to
Bob$_{r}$ via the authenticated quantum channel, where $r=2\sim t$.
\item [{Step2}] Every participant computes $s_{r}=f\left(x_{r}\right)\prod_{1\leq j\leq t,j\neq r}\frac{x_{j}}{x_{j}-x_{r}}\text{ }\text{mod}\text{ }d$
according to Lagrange interpolation formula, where $r=1\sim t$. Then,
Bob$_{r}$ performs $U_{0,s_{r}}$ on their r-th qubit, where $r=1\sim t$
and $U_{0,s_{r}}$ is defined as:
\begin{equation}
U_{0,s_{r}}=\sum_{k=0}^{d-1}\omega^{s_{r}\cdot k}\left|k\right\rangle _{rr}\left\langle k\right|,
\end{equation}
where $\omega=e^{2\pi i/d}$. After these operations, the entire quantum
system will become:
\begin{equation}
\begin{array}{lll}
\left|\varphi^{\prime}\right\rangle & = & \frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}\omega^{s_{1}\cdot k}\left|k\right\rangle _{1}\omega^{s_{2}\cdot k}\left|k\right\rangle _{2}...\omega^{s_{t}\cdot k}\left|k\right\rangle _{t}\\
& = & \frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}\omega^{\left(\sum_{r=1}^{t}s_{r}\right)\cdot k}\left|k\right\rangle _{1}\left|k\right\rangle _{2}...\left|k\right\rangle _{t}
\end{array}
\end{equation}
\item [{Step3}] Without receiving any information from the other agents,
Bob$_{1}$ can perform inverse quantum Fourier transform ($QFT^{-1}$)
on his first qubit and measures it with the basis $\left\{ \left|0\right\rangle ,\left|1\right\rangle ,...,\left|d-1\right\rangle \right\} $.
The measurement result, $a_{0}^{\prime}$, should be equal to the
boss's secret information, $a_{0}$.
\end{description}
\paragraph{The Problem.}
In Step 3, Song et al. claimed that when Bob$_{1}$ performs $QFT^{-1}$
on the first qubit, the quantum system will be
\begin{equation}
\begin{array}{lll}
QFT^{-1}\left(\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}\omega^{\left(\sum_{r=1}^{t}s_{r}\right)\cdot k}\left|k\right\rangle _{1}\right) & =\frac{1}{\sqrt{d}} & \sum_{k=0}^{d-1}QFT^{-1}\left(\omega^{\left(\sum_{r=1}^{t}s_{r}\right)\cdot k}\left|k\right\rangle _{1}\right)\\
& = & \left|\sum_{r=1}^{t}s_{r}\text{ }\text{mod}\text{ }d\right\rangle _{1}\\
& = & \left|a_{0}\text{ }\text{mod}\text{ }d\right\rangle _{1}.
\end{array}
\end{equation}
However, Bob$_{1}$'s qubit is actually entangled with the other participants'
qubits and the derivation should include the entire quantum system,
which should be written as follows:
\begin{equation}
\begin{array}{ll}
QFT^{-1}\otimes I\otimes...\otimes I\left(\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}\omega^{\left(\sum_{r=1}^{t}s_{r}\right)\cdot k}\left|k\right\rangle _{1}\left|k\right\rangle _{2}...\left|k\right\rangle _{t}\right) & =\\
\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}QFT^{-1}\left(\omega^{\left(\sum_{r=1}^{t}s_{r}\right)\cdot k}\left|k\right\rangle _{1}\right)\left|k\right\rangle _{2}...\left|k\right\rangle _{t}
\end{array}
\end{equation}
It can be seen that $QFT^{-1}\left(\omega^{\left(\sum_{r=1}^{t}s_{r}\right)\cdot k}\left|k\right\rangle _{1}\right)$
cannot be summed up together because $\left|k\right\rangle _{2}...\left|k\right\rangle _{t}$
are not equal to one another, where $k=0\sim d-1$. Hence, when Bob$_{1}$
measures his qubit, he cannot obtain the boss's secret information
$a_{0}$. In other words, Bob$_{1}$ cannot recover the boss's secret
without receiving any information from the other agents.
Let us take an example to explain the problem. Let $d=4$, $t=3$,
and $a_{0}=3$. After all Bobs' encoding, the quantum state is as
follows:
\begin{equation}
\begin{array}{lll}
\left|\varphi^{\prime}\right\rangle & = & \frac{1}{2}\left(\omega^{3\cdot0}\left|0\right\rangle _{1}\left|0\right\rangle _{2}\left|0\right\rangle _{3}+\omega^{3\cdot1}\left|1\right\rangle _{1}\left|1\right\rangle _{2}\left|1\right\rangle _{3}+\right.\\
& & \left.\omega^{3\cdot2}\left|2\right\rangle _{1}\left|2\right\rangle _{2}\left|2\right\rangle _{3}+\omega^{3\cdot3}\left|3\right\rangle _{1}\left|3\right\rangle _{2}\left|3\right\rangle _{3}\right)
\end{array}
\end{equation}
When Bob$_{1}$ performs the $QFT^{-1}$ on the first qubit, the system
becomes:
\begin{equation}
\begin{array}{lll}
QFT^{-1}\otimes I\otimes I\left|\varphi^{\prime}\right\rangle & = & \frac{1}{2}\left(QFT^{-1}\left(\omega^{3\cdot0}\left|0\right\rangle _{1}\right)\left|0\right\rangle _{2}\left|0\right\rangle _{3}+\right.\\
& & \left.QFT^{-1}\left(\omega^{3\cdot1}\left|1\right\rangle _{1}\right)\left|1\right\rangle _{2}\left|1\right\rangle _{3}+\right.\\
& & \left.QFT^{-1}\left(\omega^{3\cdot2}\left|2\right\rangle _{1}\right)\left|2\right\rangle _{2}\left|2\right\rangle _{3}+\right.\\
& & \left.QFT^{-1}\left(\omega^{3\cdot3}\left|3\right\rangle _{1}\right)\left|3\right\rangle _{2}\left|3\right\rangle _{3}\right)
\end{array}
\end{equation}
To expand Eq. (6), the $QFT^{-1}\left(\omega^{3\cdot0}\left|0\right\rangle _{1}\right)$
, $QFT^{-1}\left(\omega^{3\cdot1}\left|1\right\rangle _{1}\right)$,
$QFT^{-1}\left(\omega^{3\cdot2}\left|2\right\rangle _{1}\right)$,
and $QFT^{-1}\left(\omega^{3\cdot3}\left|3\right\rangle _{1}\right)$
in this equation can be written as follows, where $\omega^{x}=e^{2\pi ix/4}$.
\begin{equation}
\begin{array}{l}
QFT^{-1}\left(\omega^{3\cdot0}\left|0\right\rangle \right)=\frac{1}{2}\left(\left|0\right\rangle +\left|1\right\rangle +\left|2\right\rangle +\left|3\right\rangle \right)\\
QFT^{-1}\left(\omega^{3\cdot1}\left|1\right\rangle \right)=\frac{1}{2}\left(-i\left|0\right\rangle -\left|1\right\rangle +i\left|2\right\rangle +\left|3\right\rangle \right)\\
QFT^{-1}\left(\omega^{3\cdot2}\left|2\right\rangle \right)=\frac{1}{2}\left(-\left|0\right\rangle +\left|1\right\rangle -\left|2\right\rangle +\left|3\right\rangle \right)\\
QFT^{-1}\left(\omega^{3\cdot3}\left|3\right\rangle \right)=\frac{1}{2}\left(i\left|0\right\rangle -\left|1\right\rangle -i\left|2\right\rangle +\left|3\right\rangle \right)
\end{array}
\end{equation}
According to Eq. (7), Eq. (6) becomes:
\begin{equation}
QFT^{-1}\otimes I\otimes I\left|\varphi^{\prime}\right\rangle =\frac{1}{4}\left(\begin{array}{l}
\left(\left|0\right\rangle +\left|1\right\rangle +\left|2\right\rangle +\left|3\right\rangle \right)_{1}\left|00\right\rangle _{23}+\\
\left(-i\left|0\right\rangle -\left|1\right\rangle +i\left|2\right\rangle +\left|3\right\rangle \right)_{1}\left|11\right\rangle _{23}+\\
\left(-\left|0\right\rangle +\left|1\right\rangle -\left|2\right\rangle +\left|3\right\rangle \right)_{1}\left|22\right\rangle _{23}+\\
\left(i\left|0\right\rangle -\left|1\right\rangle -i\left|2\right\rangle +\left|3\right\rangle \right)_{1}\left|33\right\rangle _{23}
\end{array}\right)
\end{equation}
If the superposition states of the first qubit could be summed up
together, then the state of the first qubit would be $\left|3\right\rangle $,
which is the information that Bob$_{1}$ wants to obtain. However,
because the superposition states of the second and the third qubits
are different from the other superposition states (that is, $\left|00\right\rangle _{23}$,
$\left|11\right\rangle _{23}$, $\left|22\right\rangle _{23}$, and
$\left|33\right\rangle _{23}$), the superposition states of the first
qubit cannot be summed up and hence it is not equal to $\left|3\right\rangle $.
\section*{Acknowledgment}
This research is supported by the Ministry of Science and Technology,
Taiwan, R.O.C., under the Contract No. MOST 105-2221-E-006 -162 -MY2.
\section*{References}
{[}1{]} X.-L. Song, Y.-B. Liu, H.-Y. Deng, and Y.-G. Xiao, \textquotedblleft (t,
n) threshold d-level quantum secret sharing,\textquotedblright{} Scientific
Reports, vol. 7, no. 1, p. 6366, 2017.
\end{document} | math | 9,534 |
\begin{document}
\begin{abstract}
Let $(U_n)_{n\in \mathbb{N}}$ be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants $B$ and $N_0$ such that for any $b,c\in \mathbb{Z}$ with $b> B$ the equation $U_n - b^m = c$ has at most two distinct solutions $(n,m)\in \mathbb{N}^2$ with $n\geq N_0$ and $m\geq 1$. Moreover, we apply our result to the special case of Tribonacci numbers given by $T_1= T_2=1$, $T_3=2$ and $T_{n}=T_{n-1}+T_{n-2}+T_{n-3}$ for $n\geq 4$. By means of the LLL-algorithm and continued fraction reduction we are able to prove $N_0=1.1\cdot 10^{37}$ and $B=e^{438}$. The corresponding reduction algorithm is implemented in Sage.
\end{abstract}
\maketitle
\section{Introduction}
In the last couple of years investigating Pillai-type problems with linear recurrence sequences has been very popular (see Table \ref{table:refs}).
\begin{table}[h]
\caption{Overview of results on $U_n-V_m=c$}\label{table:refs}
\begin{tabular}{p{0.3\textwidth} p{0.3\textwidth} p{0.3\textwidth}}
\hline
$U_n$ & $V_m$ & authors \\ \hline
Fibonacci numbers & powers of 2 & Ddamulira, Luca, Rakotomalala \cite{DdamuliraLucaRakotomalala2017} \\
Fibonacci numbers & Tribonacci numbers & Chim, Pink, Ziegler \cite{ChimPinkZiegler2017} \\
Tribonacci numbers & powers of 2 & Bravo, Luca, Yaz\'{a}n \cite{BravoLucaYazan2017} \\
$k$-Fibonacci number & powers of 2 & Ddamulira, G\'{o}mez, Luca \cite{DdamuliraGomezCarlosLuca2018} \\
Pell numbers & powers of 2 & Hernane, Luca, Rihane, Togb\'{e} \cite{HernaneLucaRihaneTogbe2018} \\
Tribonacci numbers & powers of 3 & Ddamulira \cite{Ddamulira2019Tribos} \\
Fibonacci numbers & Pell numbers & Hern\'{a}ndez, Luca, Rivera \cite{HernandezLucaRivera2019} \\
Padovan numbers & powers of 2 & Lomel\'{\i}, Hern\'{a}ndez \cite{LomeliHernandez2019} \\
Padovan numbers & powers of 3 & Ddamulira \cite{Ddamulira2019} \\
Padovan numbers & Tribonacci numbers & Lomel\'{\i}, Hern\'{a}ndez, Luca \cite{LomeliHernandezLuca2019Indian} \\
Fibonacci numbers & Padovan numbers & Lomel\'{\i}, Hern\'{a}ndez, Luca \cite{LomeliHernandezLuca2019} \\
Fibonacci numbers & powers of 3 & Ddamulira \cite{Ddamulira2020} \\
$k$-Fibonacci numbers & powers of 3 & Ddamulira, Luca \cite{DdamuliraLuca2020} \\
$X$-coordinates of Pell equations & powers of 2 & Erazo, G\'{o}mez, Luca \cite{ErazoGomezLuca2021} \\
$k$-Fibonacci numbers & Pell numbers & Bravo, D\'{\i}az, G\'{o}mez \cite{BravoDiazGomez2021} \\ \hline
\end{tabular}
\end{table}
This trend was started in 2017 by Ddamulira, Luca, Rakotomalala \cite{DdamuliraLucaRakotomalala2017}, who proved that the only integers $c$ having at least two representations of the form $F_n - 2^m$ are $c\in \{0,1,-1,-3,5,-11, -30,85\}$ (here $F_n$ is the $n$-th Fibonacci number).
This problem was inspired by a result due to S. S. Pillai. In 1936 Pillai \cite{Pillai1936, Pillai1937} proved that if $a$ and $b$ are coprime integers, then there exists a constant $c_0(a,b)$ depending on $a$ and $b$ such that for any $c>c_0(a,b)$ the equation
\begin{equation}\label{eq:Pillai}
a^n-b^m=c
\end{equation}
has at most one solution $(n,m)\in \mathbb{Z}_{>0}^2$.
A natural generalisation of this problem is to replace $a^n$ and $b^m$ by other linear recurrence sequences. This is what the authors in \cite{DdamuliraLucaRakotomalala2017} did, and also what all the other authors in Table~\ref{table:refs} have done. All these results use lower bounds for linear forms in logarithms and reduction methods. Moreover, there exists a general result:
Chim, Pink and Ziegler \cite{ChimPinkZiegler2018} proved that for two fixed linear recurrence sequences $(U_n)_{n\in \mathbb{N}}$, $(V_n)_{n\in \mathbb{N}}$ (with some restrictions) the equation
\begin{equation*}
U_n - V_m = c
\end{equation*}
has at most one solution $(n,m)\in \mathbb{Z}_{>0}^2$ for all $c\in \mathbb{Z}$,
except if $c$ is in a finite and effectively computable set $\mathcal{C} \subset \mathbb{Z}$ that depends on $(U_n)_{n\in \mathbb{N}}$ and $(V_n)_{n\in \mathbb{N}}$.
In this paper, we would like to generalize that result by ``unfixing'' one of the linear recurrence sequences.
In the classical setting, it is possible to ``unfix'' $a$ and $b$ completely: Bennett \cite{Bennett2001} proved that for any integers $a,b\geq 2$ and $c\geq 1$ Equation~\eqref{eq:Pillai} has at most two solutions $(n,m)\in \mathbb{Z}_{>0}^2$.
Moreover, he conjectured that in fact the equation has at most one solution $(n,m)$ for all but 11 specific exceptional triples $(a,b,c)$.
Of course, we cannot simply say that $(U_n)_{n\in \mathbb{N}}$ and $(V_n)_{n\in \mathbb{N}}$ should be completely arbitrary. However, there already exist results where the linear recurrence sequence $(U_n)_{n\in \mathbb{N}}$ is not entirely fixed: In Table~\ref{table:refs} there are some results involving $k$-Fibonacci numbers \cite{DdamuliraLuca2020, BravoDiazGomez2021}, where $k$ is variable.
Now what we will do is fix $(U_n)_{n\in \mathbb{N}}$ and let $V_m=b^m$ with variable $b$. Our main result will be that for fixed $(U_n)_{n\in \mathbb{N}}$ (with some restrictions) the equation
\[
U_n - b^m = c
\]
has at most two distinct solutions $(n,m) \in \mathbb{Z}_{>0}^2$ for any $(b,c)\in \mathbb{Z}^2$ with only finitely many exceptions $b \in \mathcal{B}$, where $\mathcal{B}$ is an effectively computable set. Allowing two solutions (instead of one solution) is the price we have to pay for letting $b$ vary. The second solution is needed for technical reasons, but we believe that the result might also be true if we only allow at most one solution.
Finally, note that our method does not enable us to solve the problem for a specific sequence $(U_n)_{n\in \mathbb{N}}$ completely. We will show how far we can get by computing the effective bounds for the Tribonacci numbers and reducing the bounds as far as possible.
Let us outline the rest of this paper. The next section contains some notations and our results: Theorem~\ref{thm:mainthm} is the main theorem, Theorem~\ref{thm:Tribos} shows what happens if we apply our methods to the Tribonacci numbers. Moreover, we make several remarks on the assumptions in Theorem~\ref{thm:mainthm} and pose some open problems regarding Theorem~\ref{thm:Tribos}. Section~\ref{sec:diophApprox} is a collection of rather well known results from Diophantine approximation.
Section~\ref{sec:proofMainThm} is devoted to the proof of Theorem~\ref{thm:mainthm} and Section~\ref{sec:Tribos} is devoted to the proof of Theorem~\ref{thm:Tribos}.
Beforehand, in Section~\ref{sec:overviewProof}, we give an overview of the two proofs. In particular, we point out the parallels and differences between the two proofs.
\section{Notation and results}
A linear recurrence sequence $ (U_n)_{n \in \mathbb{N}} $ is given by finitely many initial values together with a recursive formula of the shape
\begin{equation*}
U_{n+\ell} = w_{\ell-1} U_{n+\ell-1} + \cdots + w_0 U_n.
\end{equation*}
We say that such a recurrence sequence is defined over the integers if the coefficients $ w_0, \ldots, w_{\ell-1} $ as well as the initial values are all integers.
In this situation all elements of the sequence are integers.
It is well known that any such linear recurrence sequence can be written in its Binet representation
\begin{equation*}
U_n = a_1(n) \alpha_1^n + \cdots + a_k(n) \alpha_k^n,
\end{equation*}
where the characteristic roots $ \alpha_1, \ldots, \alpha_k $ are algebraic integers and the coefficients $ a_1(n), \ldots, a_k(n) $ are polynomials in $ n $ with coefficients in $ \mathbb{Q}(\alpha_1, \ldots, \alpha_k) $.
The recurrence sequence is called simple if $ a_1(n), \ldots, a_k(n) $ are all constant, i.e.\ independent of $ n $.
Moreover, $ \alpha_1 $ is called the dominant root if $ \abs{\alpha_1} > \abs{\alpha_i} $ for all $ i = 2, \ldots, k $.
Our result is now the following theorem:
\begin{mythm}
\label{thm:mainthm}
Let $ (U_n)_{n \in \mathbb{N}} $ be a simple linear recurrence sequence defined over the integers with Binet representation
\begin{equation*}
U_n = a \alpha^n + a_2 \alpha_2^n + \cdots + a_k \alpha_k^n
\end{equation*}
and irrational dominant root $ \alpha > 1 $.
Assume further that $ a > 0 $, that $ a $ and $ \alpha $ are multiplicatively independent, and that the equation
\begin{equation}
\label{eq:techcond}
\alpha^z - 1 = a^x \alpha^y
\end{equation}
has no solutions with $ z \in \mathbb{N} $, $ x,y \in \mathbb{Q} $ and $ -1 < x < 0 $.
Then there exist effectively computable constants $ B \geq 2 $ and $ N_0 \geq 2 $ such that the equation
\begin{equation}
\label{eq:centraleq}
U_n - b^m = c
\end{equation}
has for any integer $ b > B $ and any $ c \in \mathbb{Z} $ at most two distinct solutions $ (n,m) \in \mathbb{N}^2 $ with $ n \geq N_0 $ and $ m \geq 1 $.
\end{mythm}
Let us give some remarks regarding the technical condition involving Equation~\eqref{eq:techcond} in the above theorem:
\begin{myrem}
The technical condition containing Equation \eqref{eq:techcond}
can be effectively checked for any given recurrence sequence $ (U_n)_{n \in \mathbb{N}} $:
First note that by construction $ \alpha $ is an algebraic integer.
Moreover, note that the ideals $ (\alpha) $ and $ (\alpha^z-1) $ with $ z \in \mathbb{N} $ have no common prime ideals in their factorisations.
Let $ \mathfrak{P}_1, \ldots, \mathfrak{P}_n $ be the prime ideals that appear in the prime ideal factorisation of $ (a) $.
If $ \mathfrak{P}_i $ is not a prime factor of $ (\alpha) $, then let $ k_i $ be the order of $ \alpha $ modulo $ \mathfrak{P}_i $, i.e.\ $ k_i $ is minimal such that $ \mathfrak{P}_i $ is a prime factor of $ (\alpha^{k_i}-1) $.
Note that if $ \mathfrak{P}_i $ lies above $ (p_i) $ and $ f_i $ is the inertia degree, then $ k_i \mid p_i^{f_i}-1 $, so the $k_i$ are bounded.
Thus we can compute the maximum of all these orders $ k_0 := \max k_i $.
By Schinzel's theorem on primitive divisors \cite{Schinzel1974}, there exists an effectively computable number $ n_0 $ such that $ \alpha^z - 1 $ has a primitive divisor for any $ z > n_0 $.
This means that for $ z > \max \set{k_0, n_0} $ the ideal $ (\alpha^z-1) $ has a primitive divisor which is not a divisor of $ (a) $.
Since $ (\alpha) $ and $ (\alpha^z-1) $ have no common divisors, it is impossible that $ \alpha^z - 1 = a^x \alpha^y $ for $ z > \max \set{k_0,n_0} $.
For each $ z = 1, \ldots, \max \set{k_0,n_0} $ one can check whether $ \alpha^z - 1 = a^x \alpha^y $ has a solution with $ x,y \in \mathbb{Q} $ and $ -1 < x < 0 $ by looking at the primes of $ \alpha^z-1 $, $ a $ and $ \alpha $.
\end{myrem}
\begin{myrem}
\label{rem:easierCond}
Let $ \mathfrak{P}_1, \ldots, \mathfrak{P}_n $ be all prime ideals that appear in the prime ideal factorisations of $ (a) $ and $ (\alpha) $.
Then we can write
\begin{align*}
(a) &= \mathfrak{P}_1^{a_1} \cdots \mathfrak{P}_n^{a_n}, \\
(\alpha) &= \mathfrak{P}_1^{b_1} \cdots \mathfrak{P}_n^{b_n},
\end{align*}
where the $ a_i $ and $ b_i $ are integers.
The following two conditions are relatively easy to check and each of them implies the technical condition containing Equation \eqref{eq:techcond}:
\begin{enumerate}[I)]
\item
\label{it:condDet}
There are $ i,j \in \set{1,\ldots,n} $ such that
\begin{equation*}
\det
\begin{pmatrix}
a_i & b_i \\
a_j & b_j
\end{pmatrix}
= \pm 1.
\end{equation*}
\item
\label{it:condUnit}
$ \alpha $ is a unit and there is an index $ i \in \set{1,\ldots,n} $ with $ a_i = \pm 1 $.
\end{enumerate}
\end{myrem}
\begin{proof}
If \eqref{eq:techcond} is satisfied, then the factorisation of $ (\alpha^z-1) $ contains also only the prime ideals $ \mathfrak{P}_1, \ldots, \mathfrak{P}_n $ and we can write
\begin{equation*}
(\alpha^z-1) = \mathfrak{P}_1^{z_1} \cdots \mathfrak{P}_n^{z_n}
= (\mathfrak{P}_1^{a_1} \cdots \mathfrak{P}_n^{a_n})^x (\mathfrak{P}_1^{b_1} \cdots \mathfrak{P}_n^{b_n})^y,
\end{equation*}
which implies
\begin{equation*}
z_i = a_i x + b_i y
\end{equation*}
for $ i = 1, \ldots, n $.
Note that all the $ z_i, a_i, b_i $ are integers.
Therefore if \ref{it:condDet}) is satisfied, then it follows that $ x $ and $ y $ are integers as well and in particular we do not have $ -1 < x < 0 $.
If \ref{it:condUnit}) is satisfied, then $ b_1 = \cdots = b_n = 0 $, so $ a_i = \pm 1 $ implies that $ x $ is an integer and again we do not have $ -1 < x < 0 $.
\end{proof}
\begin{myrem}
Theorem \ref{thm:mainthm} can be applied to the Fibonacci numbers.
Here we have $ a = \frac{1}{\sqrt{5}} $ and $ \alpha = \frac{1+\sqrt{5}}{2} $, i.e.\ $ \alpha $ is a unit and $ (a) = (\sqrt{5})^{-1} $. Thus condition \ref{it:condUnit}) in Remark \ref{rem:easierCond} is satisfied.
\end{myrem}
\begin{myrem}
Theorem \ref{thm:mainthm} can be applied to the linear recurrence sequence given by $ U_0 = 0 $, $ U_1 = 1 $ and $ U_{n+2} = U_{n+1} + 3 U_n $ for $ n \geq 0 $.
Here we have $ a = \frac{1}{\sqrt{13}} $ and $ \alpha = \frac{1+\sqrt{13}}{2} $, i.e.\ $ (\alpha) = \left( \frac{1+\sqrt{13}}{2} \right)^1 $ is prime (it lies over $ p=3 $) and $ (a) = (\sqrt{13})^{-1} $. Thus condition \ref{it:condDet}) in Remark \ref{rem:easierCond} is satisfied.
\end{myrem}
\begin{myrem}
If we weaken Theorem \ref{thm:mainthm} in the sense that we prove the existence of at most three solutions, then an inspection of the proof shows that the technical condition containing Equation \eqref{eq:techcond} is not needed any more.
Furthermore, it is not clear, whether all assumptions in Theorem \ref{thm:mainthm} are really necessary for the statement to be true or only required for our proof to work.
\end{myrem}
As a special case of Theorem \ref{thm:mainthm} we get the following result for the Tribonacci numbers, where the technical condition is checked directly in the proof (Section \ref{sec:Tribos}). Note that the case of Fibonacci numbers has recently been considered by Batte et al.~\cite{BatteEtAl2022}.
\begin{mythm}\label{thm:Tribos}
Let $(T_n)_{n\in \mathbb{N}}$ be the Tribonacci sequence given by $T_1=1, T_2=1, T_3=2$ and $T_{n}=T_{n-1}+T_{n-2}+T_{n-3}$ for $n\geq 4$.
If for some integers $b\geq 2$ and $c$ the equation $T_n-b^m=c$ has at least three solutions in positive integers $n,m$ given by $(n_1,m_1),(n_2,m_2),(n_3,m_3)$ with $n_1>n_2>n_3\geq 2$, then
\[
\log b \leq 438
\quad \text{and} \quad
150<n_1\leq 1.1\cdot 10^{37}.
\]
\end{mythm}
\begin{myrem}
The assumption $n_1>n_2>n_3\geq 2$ in the above theorem is natural because of $T_1=T_2$.
\end{myrem}
In view of this result, the following question remains:
\begin{myproblem}\label{problem:Tribos-complete}
Do there exist any pairs $(b,c)$ such that the equation $T_n-b^m=c$ has at least three solutions?
\end{myproblem}
Moreover, in the proof of Theorem \ref{thm:Tribos} (Section \ref{sec:Tribos}, \hyperlink{step:smallSols}{``small solutions''}) we will search for $b$'s and $c$'s such that $T_n-b^m=c$ has at least two small solutions. We will only find two solutions for
\begin{equation}
\label{eq:bc}
\begin{split}
(b,c) \in
\{&(2, -8), (2, -3), (2, -1), (2, 0), (2, 5), (3, -2), (3, 4), (5, -121), \\
&(5, -1), (5, 19), (7, -5), (17, -15), (54, 220), (641, -137)\}.
\end{split}
\end{equation}
Thus the following question remains:
\begin{myproblem}
Except for the pairs from \eqref{eq:bc}, do there exist any further $(b,c)$ such that $T_n-b^m=c$ has at least two solutions?
\end{myproblem}
\begin{myrem}
In the proof of Theorem \ref{thm:Tribos} (Section \ref{sec:Tribos}, \hyperlink{step:smallSols}{``small solutions''}) we search for all $2\leq n_2 < n_1 \leq 150$ such that the difference of the corresponding Tribonacci numbers can be written in the form $T_{n_1}-T_{n_2}=b^{m_1}-b^{m_2}$ with $b\geq 2$ and $m_1>m_2\geq 1$.
The bound 150 is chosen because the computations only take a few minutes and the proof of the upper bound in Theorem \ref{thm:Tribos} is easier if we assume $n_1>150$.
In fact, the authors also ran the computations further and checked if there are $2\leq n_2 < n_1 \leq 350$ such that $T_{n_1}-T_{n_2}=b^{m_1}-b^{m_2}$. These computations took about a week on a usual computer using 4 cores. No further solutions than those in \eqref{eq:bc} were found. At this point, the computations start taking pretty long because the factorisation of huge $T_{n_1}-T_{n_2}$ is expensive.
\end{myrem}
\section{Results from Diophantine Approximation}\label{sec:diophApprox}
In this section we state all results from Diophantine approximation, that will be used in the proofs below. In particular, we will use lower bounds for linear forms in logarithms, i.e.\ Baker-type bounds for expressions of the form $|\Lambda|=|b_1 \log \eta_1 + \dots + b_t \log \eta_t|$. These linear forms will be coming from expressions of the form $|\eta_1^{b_1}\cdots \eta_t^{b_t}-1|$ and we will switch between these expressions via the following lemma.
\begin{mylemma}
\label{lemma:linsmall}
Let $ \lambda $ be a real number with $ \abs{\lambda} \leq 1 $.
Then we have the inequality
\begin{equation*}
\frac{1}{4} \abs{\lambda} \leq \abs{e^{\lambda} - 1} \leq 2 \abs{\lambda}.
\end{equation*}
\end{mylemma}
\begin{proof}
The proof for these bounds is implied by a straight-forward calculation.
For the upper bound we have
\begin{align*}
\abs{e^{\lambda} - 1} &= \abs{\sum_{t=1}^{\infty} \frac{\lambda^t}{t!}}
\leq \sum_{t=1}^{\infty} \frac{\abs{\lambda}^t}{t!}
= \abs{\lambda} \cdot \sum_{t=0}^{\infty} \frac{\abs{\lambda}^t}{(t+1)!} \\
&\leq \abs{\lambda} \cdot \sum_{t=0}^{\infty} \frac{1}{(t+1)!}
= \abs{\lambda} \cdot (e-1) \leq 2 \abs{\lambda}
\end{align*}
and for the lower bound we have
\begin{align*}
\abs{e^{\lambda} - 1} &= \abs{\sum_{t=1}^{\infty} \frac{\lambda^t}{t!}}
\geq \abs{\lambda} - \abs{\sum_{t=2}^{\infty} \frac{\lambda^t}{t!}}
\geq \abs{\lambda} - \sum_{t=2}^{\infty} \frac{\abs{\lambda}^t}{t!} \\
&= \abs{\lambda} - \abs{\lambda} \cdot \sum_{t=1}^{\infty} \frac{\abs{\lambda}^t}{(t+1)!}
\geq \abs{\lambda} - \abs{\lambda} \cdot \sum_{t=1}^{\infty} \frac{1}{(t+1)!} \\
&= \abs{\lambda} \cdot (3-e) \geq \frac{1}{4} \abs{\lambda}.
\end{align*}
\end{proof}
Also, we will use the following simple fact.
\begin{mylemma}
\label{lemma:linbig}
Let $ \lambda $ be a real number with $ \abs{\lambda} > 1 $.
Then we have the inequality
\begin{equation*}
\abs{e^{\lambda} - 1} \geq \frac{3}{5}.
\end{equation*}
\end{mylemma}
\begin{proof}
Since $ \abs{\lambda} > 1 $ we either have $ e-1 \leq e^{\lambda} - 1 $ or $ e^{\lambda} - 1 \leq \frac{1}{e} - 1 $.
Thus we get
\begin{equation*}
\abs{e^{\lambda} - 1} \geq \min \setb{e-1, 1-\frac{1}{e}} \geq \frac{3}{5}.
\end{equation*}
\end{proof}
Before we state some lower bounds for linear forms in logarithms, let us recall the definition of the logarithmic height.
Let $ \eta $ be an algebraic number of degree $ d $ over the rationals, with minimal polynomial
\begin{equation*}
a_0 (x - \eta_1) \cdots (x - \eta_d) \in \mathbb{Z}[x].
\end{equation*}
Then the absolute logarithmic height of $ \eta $ is given by
\begin{equation*}
h(\eta) := \frac{1}{d} \left( \log a_0 + \sum_{i=1}^{d} \log \max \setb{1, \abs{\eta_i}} \right).
\end{equation*}
This height function satisfies some basic properties, which are all well-known (see e.g.\ \cite{Zannier2009} for a reference). Namely for all $ \eta, \gamma \in \overline{\mathbb{Q}} $ and all $ z \in \mathbb{Z} $ we have:
\begin{align*}
h(\eta + \gamma) &\leq h(\eta) + h(\gamma) + \log 2, \\
h(\eta \gamma) &\leq h(\eta) + h(\gamma), \\
h(\eta^z) &= \abs{z} h(\eta), \\
h(z) &= \log |z| \qquad \text{for } z\neq 0.
\end{align*}
The following lower bound for linear forms in logarithms is well-known and follows from Matveev's bound \cite{Matveev2000}.
\begin{myprop}[Matveev]\label{prop:Matveev}
Let $\eta_1, \ldots, \eta_t$ be positive real algebraic numbers in a number field $K$ of degree $D$, let $b_1, \ldots b_t$ be rational integers and assume that
\[
\Lambda := b_1 \log \eta_1 + \dots + b_t \log \eta_t \neq 0.
\]
Then
\[
\log |\Lambda|
\geq -1.4 \cdot 30^{t+3} \cdot t^{4.5} \cdot D^2 (1 + \log D) (1 + \log B) A_1 \cdots A_t,
\]
where
\begin{align*}
B &\geq \max \setb{\abs{b_1}, \ldots, \abs{b_t}},\\
A_i &\geq \max \setb{D h(\eta_i), \abs{\log \eta_i}, 0.16}
\quad \text{for all }i = 1,\ldots,t.
\end{align*}
Moreover, the assumption $\Lambda\neq 0$ is equivalent to $e^\Lambda-1 \neq 0$ and the following bound holds as well:
\begin{align*}
\log |e^\Lambda-1|
&= \log |\eta_1^{b_1}\cdots \eta_t^{b_t} -1 |\\
&\geq -1.4 \cdot 30^{t+3} \cdot t^{4.5} \cdot D^2 (1 + \log D) (1 + \log B) A_1 \cdots A_t.
\end{align*}
\end{myprop}
\begin{proof}
The bound for $|\Lambda|$ follows immediately from \cite[Corollary 2.3]{Matveev2000}. In fact, we have that
\begin{equation}\label{eq:Matveev-proof}
\log |\Lambda|
\geq - \frac{e}{2} \cdot 30^{t+3} \cdot t^{4.5} \cdot D^2 (1 + \log D) (1 + \log B) A_1 \cdots A_t,
\end{equation}
and we can estimate $e/2$ by $1.4$.
The bound for $|e^\Lambda-1|$ now follows with Lemma~\ref{lemma:linsmall} and Lemma \ref{lemma:linbig}: If $|\Lambda|>1$, then $|e^\Lambda-1|\geq 3/5$, so $\log |e^\Lambda-1|\geq -0.52$ and the bound is trivially fulfilled. If $|\Lambda|\leq 1$, then we have $|e^\Lambda-1|\geq \frac{1}{4} |\Lambda|$ and we obtain
\begin{align*}
\log |e^\Lambda-1|
\geq \log |\Lambda| - \log 4.
\end{align*}
Here the bound for $|e^\Lambda-1|$ follows from \eqref{eq:Matveev-proof} since we can omit the $\log 4$ when estimating $e/2$ by $1.4$.
\end{proof}
\begin{myrem}
The lower bound for linear forms in logarithms due to Baker and Wüstholz \cite{bawu93} played a significant role in the development of linear forms in logarithms.
The final structure of the lower bound for linear forms in logarithms without an explicit determination of the constant involved has been established by Wüstholz \cite{wu88}, and the precise determination of that constant is the central aspect of \cite{bawu93}.
However, slightly sharper bounds are obtained by using Matveev's result \cite{Matveev2000} instead.
Let us note that using the highly technical result due to Mignotte \cite{Mignotte:kit} for linear forms in three logarithms would yield even smaller bounds, but to avoid technical difficulties we refrain from applying this result.
\end{myrem}
For linear forms in only two logarithms, there exist results with much smaller constants. In Section \ref{sec:Tribos} we will use the following bound, which follows immediately from \cite{Laurent2008}.
\begin{myprop}[Laurent]\label{prop:Laurent}
Let $\eta_1, \eta_2 > 0$ be two real multiplicatively independent algebraic numbers, let $b_1, b_2 \in \mathbb{Z}$ be nonzero integers and let
\[
\Lambda := b_1 \log \eta_1 + b_2 \log \eta_2.
\]
Then
\[
\log |\Lambda|
\geq - 20.3 D^2 (\max( \log b' + 0.38, 18/D, 1))^2 \log A_1 \log A_2,
\]
where
\begin{align*}
D&=[\mathbb{Q}(\eta_1,\eta_2):\mathbb{Q}],\\
b'&=\frac{|b_1|}{\log A_2} + \frac{|b_2|}{\log A_1},\\
\log A_i &\geq \max ( D h(\eta_i),\abs{\log \eta_i},1)
\qquad \text{for }i=1,2.
\end{align*}
\end{myprop}
\begin{proof}
The bound follows immediately from the bound \cite[Corollary 2]{Laurent2008} with $m=18$ (we have only hidden two of the $D$'s inside $\log A_1$ and $\log A_2$).
As for the assumptions, note that Laurent additionally supposes that $b_1$ is positive and $b_2$ is negative, and that $\log \eta_1$ and $\log \eta_2$ are positive. However, changing the signs of both $b$'s does not change $|\Lambda|$ and changing the sign of only one $b$ makes $\abs{\Lambda}$ larger. Thus it is enough to assume that $b_1,b_2$ are nonzero.
If $\log \eta_i$ is negative, one can simply swap $\eta_i$ for $\eta_i^{-1}$ and $b_i$ for $-b_i$ without changing the value of $\Lambda$ and still apply Laurent's result. Thus we can also allow the $\log \eta$'s to be negative. Moreover, we do not need to explicitly exclude $\log \eta_i=0$ as in that case we would have $\eta_i=1$, so the $\eta$'s would be multiplicatively dependent.
\end{proof}
Finally, at the end of Section \ref{sec:Tribos} we will apply the LLL-algorithm to reduce some of the obtained bounds.
The next lemma is an immediate variation of \cite[Lemma~VI.1]{Smart1998}.
\begin{mylemma}[LLL reduction]\label{lem:LLL}
Let $\gamma_1,\gamma_2,\gamma_3$ be positive real numbers and $x_1,x_2,x_3$ integers, and let
\[
0 \neq |\Lambda|
:= |x_1 \log \gamma_1 + x_2 \log \gamma_2 + x_3 \log \gamma_3|.
\]
Assume that the $x_i$ are bounded in absolute values by some constant $M$ and choose a constant $C>M^3$.
Consider the matrix
\[
A= \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
[C \log \gamma_1] & [C \log \gamma_2] &[C \log \gamma_3]
\end{pmatrix},
\]
where $[x]$ denotes the nearest integer to $x$. The columns of $A$ form a basis of a lattice. Let $B$ be the matrix that corresponds to the LLL-reduced basis and let $B^*$ be the matrix corresponding to the Gram-Schmidt basis constructed from $B$. Let $c$ be the Euclidean norm of the smallest column vector of $B^*$ and set $S:=2M^2$ and $T:=(1+3M)/2$.
If $c^2 > T^2 + S$, then
\[
|\Lambda|
> \frac{1}{C}\left( \sqrt{c^2-S} - T \right).
\]
\end{mylemma}
\section{Overview of the proofs}\label{sec:overviewProof}
The main idea in the proofs of Theorem~\ref{thm:mainthm} and Theorem~\ref{thm:Tribos} is to consider equations of the form $U_{n_1}-b^{m_1}=U_{n_2}-b^{m_2}$ and note that they are very roughly of the form $a \alpha^{n_1} - b^{m_1}= a \alpha^{n_2} - b^{m_2}$. Then one can do the usual tricks for solving Pillai-type equations via lower bounds for linear forms in logarithms: shifting expressions, estimating and applying e.g.\ Matveev's lower bound. This needs to be done in several steps. The main problem in our situation is that $b$ is not fixed, which is why we need to eliminate it at some point and why we assume the existence of three solutions.
We split up the proof of Theorem~\ref{thm:mainthm} into several lemmas and the proof of Theorem~\ref{thm:Tribos} into several steps.
Table \ref{table:overview} shows a simplified overview of the main lemmas/steps (the cases which allow us to skip some steps are left out in the table). In each lemma/step a new bound is obtained. Note that the symbol $\ll$ stands for ``$\leq$ up to some effectively computable constant''. Lemmas and steps which use the same linear form in logarithms are displayed side by side. Steps marked with a *star do not involve linear forms in logarithms but simply combine previous bounds.
\begin{table}[h]
\caption{Overview of the proofs}\label{table:overview}
\begin{tabular}{l|l}
\hline
\textbf{Proof of Theorem \ref{thm:mainthm}} & \textbf{Proof of Theorem \ref{thm:Tribos}} \\
(general result) & (Tribonacci numbers) \\
\hline
\textbf{Lemma \ref{lemma:step1}:} & \textbf{Step \ref{step:Step1}:}
\\
$\min \setb{n_1-n_2, (m_1-m_2) \log b} \ll \log n_1 \log b$ & $n_1-n_2 \ll \log n_1 \log b$
\\
\textbf{Lemma \ref{lemma:step2}:} &
\\
$\max \setb{n_1-n_2, (m_1-m_2) \log b} \ll (\log n_1 \log b)^2$ & \\
\textbf{Lemma \ref{lemma:step3}:} & \textbf{Step \ref{step:Step2}:} \\
$n_1 \ll (\log n_1 \log b)^3$ & $n_1 \ll (\log n_1 \log b)^2$ \\ \cline{1-1}
\textbf{Lemma \ref{lemma:step5}:} & \textbf{Step \ref{step:Step3}:} \\
$n_2-n_3 \ll \log n_1$ & $n_2-n_3 \ll (\log n_1)^2$
\\
\textbf{Lemma \ref{lemma:step6}:} & \textbf{Step \ref{step:Step4}:} \\
$\log b \ll (\log n_1)^2$ & $n_1-n_2 \ll (\log n_1)^2$ \\
& \textbf{*Step \ref{step:Step5}:} \\
& $\log b \ll (\log n_1)^2$ \\
\textbf{*End:} & \textbf{*Step \ref{step:Step6}:} \\
$n_1 \ll (\log n_1)^9$ & $n_1 \ll (\log n_1)^8$ \\
& \textbf{Reduction Steps} \\ \hline
\end{tabular}
\end{table}
In the proof of Theorem~\ref{thm:mainthm} in the first three lemmas we only assume the existence of two solutions. We do that because we want to show how far we can get with our method if we aim at proving a stronger result. Below the horizontal line in the middle of Table~\ref{table:overview} we assume the existence of three solutions.
In the proof of Theorem~\ref{thm:Tribos} we assume the existence of three solutions from the beginning. This saves us one step and thus helps to keep the constants and exponents smaller (see Step~\ref{step:Step2} in the table). The attentive reader will notice that the bound in Step~\ref{step:Step3} has a larger exponent than the bound in Lemma ~\ref{lemma:step5}. This is because the corresponding linear form has only two logarithms, so in the Tribonacci setting we apply Laurent's lower bound instead of Matveev's, trading the exponent for a much better constant.
In the end, in both proofs we obtain an absolute upper bound for $n_1$ from an inequality of the form $n_1 \ll (\log n_1)^k$. Since the constant is very large, the bound for $n_1$ is also very large. In the Tribonacci setting we try to reduce it. Unfortunately, since $b$ is not fixed and the bound for $b$ is very large, we cannot use the linear forms which contain $b$ for the reduction. This is why we are not able to solve Problem~\ref{problem:Tribos-complete}. However, we can still use the linear forms in logarithms which do not contain $b$ and reduce the initial bound for $n_1$ significantly.
\section{Proof of Theorem \ref{thm:mainthm}}\label{sec:proofMainThm}
Since the proof is a little bit lengthy, we will split it into several lemmas.
All assumptions listed in Theorem \ref{thm:mainthm} are general assumptions in this section.
During the whole section we will denote by $ C_i $ effectively computable positive constants which are independent of $ n,m,b,c $.
We may assume that $ b \geq 2 $ without loss of generality.
Let us now fix a Galois automorphism $ \sigma $
on the splitting field $\mathbb{Q}(\alpha, \alpha_2 , \ldots , \alpha_k)$
with the property $ \abs{\sigma(\alpha)} < \alpha $.
This is always possible because $ \alpha $ is irrational and the dominant root of $ U_n $.
The effectively computable constant $ N_0 $ will only depend on the characteristic roots and coefficients of the recurrence $ U_n $ as well as on the chosen automorphism $ \sigma $.
Although it possibly will be updated at some points in the proof, we will always denote it by the same label $ N_0 $.
As a first step we will choose $ N_0 $ large enough such that $ U_n $ is positive and strictly increasing for all $ n \geq N_0 $.
This is possible since $ U_n $ has a dominant root $ \alpha > 1 $ with positive coefficient.
In addition we can assume that $ \abs{\alpha_2} \geq \abs{\alpha_i} $ for $ i = 2,\ldots,k $ without loss of generality.
Assuming that Equation \eqref{eq:centraleq} has three distinct solutions $ (n_1,m_1) $, $ (n_2,m_2) $ and $ (n_3,m_3) $, we can write
\begin{equation*}
U_{n_1} - b^{m_1} = U_{n_2} - b^{m_2} = U_{n_3} - b^{m_3}
\end{equation*}
with $ n_1 > n_2 > n_3 \geq N_0 $ and $ m_1 > m_2 > m_3 \geq 1 $.
When working only with two of those solutions we often may write
\begin{equation}
\label{eq:twosols}
U_{n_1} - U_{n_2} = b^{m_1} - b^{m_2}.
\end{equation}
Let us fix
\begin{equation*}
\varepsilon := \frac{1}{2} \cdot \frac{\alpha-1}{\alpha+1} > 0.
\end{equation*}
Recalling that $ \alpha $ is the dominant root of $ U_n $, we get
\begin{equation*}
\abs{\frac{U_n}{a \alpha^n} - 1} \leq \varepsilon
\end{equation*}
for $ n \geq N_0 $ (where we might have updated $N_0$). Thus for $ n \geq N_0 $ we have
\begin{equation*}
\abs{U_n - a \alpha^n} \leq \varepsilon a \alpha^n,
\end{equation*}
which implies the bounds
\begin{equation}
\label{eq:unrange}
C_1 a \alpha^n \leq U_n \leq C_2 a \alpha^n
\end{equation}
for $ C_1 = 1 - \varepsilon $ and $ C_2 = 1 + \varepsilon $.
The choice of $ \varepsilon $ guarantees that $ C_3 := C_1 \alpha - C_2 > 0 $.
Therefore by using Equation \eqref{eq:twosols} and Inequality \eqref{eq:unrange} we get
\begin{align*}
b^{m_1} &\geq b^{m_1} - b^{m_2} = U_{n_1} - U_{n_2} \geq C_1 a \alpha^{n_1} - C_2 a \alpha^{n_2} \\
&= C_1 \alpha a \alpha^{n_1-1} - C_2 a \alpha^{n_2} \geq C_3 a \alpha^{n_1-1} = C_4 \alpha^{n_1}
\end{align*}
as well as
\begin{equation*}
C_2 a \alpha^{n_1} \geq U_{n_1} \geq U_{n_1} - U_{n_2} = b^{m_1} - b^{m_2} \geq \left( 1 - \frac{1}{b} \right) b^{m_1} \geq \frac{1}{2} b^{m_1}.
\end{equation*}
These two inequality chains yield
\begin{equation}
\label{eq:relleadterm}
C_4 \alpha^{n_1} \leq b^{m_1} \leq C_5 \alpha^{n_1}.
\end{equation}
Note that these bounds are valid for any solution of \eqref{eq:centraleq} provided that there is a further smaller solution.
Applying the logarithm to Equation \eqref{eq:relleadterm} gives us
\begin{equation*}
m_1 \leq \frac{\log C_5}{\log b} + n_1 \cdot \frac{\log \alpha}{\log b} < n_1
\end{equation*}
if $ b $ is large enough.
Thus for our purpose we may assume that $ n_1 > m_1 $.
The next big intermediate result is to prove that, if there exist at least two distinct solutions $ (n_1,m_1) $ and $ (n_2,m_2) $ to Equation \eqref{eq:centraleq}, then for the larger one $ (n_1,m_1) $ the bound
\begin{equation}
\label{eq:logbound}
n_1 \leq C_6 (\log n_1 \log b)^3
\end{equation}
holds.
This will be done by the following three lemmas.
\begin{mylemma}
\label{lemma:step1}
Assume that Equation \eqref{eq:centraleq} has at least two distinct solutions $ (n_1,m_1) $ and $ (n_2,m_2) $ with $ n_1 > n_2 $ as considered in Theorem \ref{thm:mainthm}.
Then we have
\begin{equation*}
\min \setb{n_1-n_2, (m_1-m_2) \log b} \leq C_7 \log n_1 \log b.
\end{equation*}
\end{mylemma}
\begin{proof}
Inserting the Binet representation of the linear recurrence sequence into Equation \eqref{eq:twosols} and regrouping terms yields
\begin{align*}
\abs{a \alpha^{n_1} - b^{m_1}} &= \abs{a \alpha^{n_2} + \sum_{l=2}^{k} a_l \alpha_l^{n_2} - \sum_{l=2}^{k} a_l \alpha_l^{n_1} - b^{m_2}} \\
&\leq a \alpha^{n_2} + C_8 \abs{\alpha_2}^{n_2} + C_9 \abs{\alpha_2}^{n_1} + b^{m_2},
\end{align*}
and, dividing both sides by $ b^{m_1} $, we get
\begin{align}
\abs{\frac{a \alpha^{n_1}}{b^{m_1}} - 1} &\leq \frac{1}{b^{m_1}} \cdot \left( a \alpha^{n_2} + C_8 \abs{\alpha_2}^{n_2} + C_9 \abs{\alpha_2}^{n_1} + b^{m_2} \right) \nonumber \\
&= \frac{a \alpha^{n_2}}{b^{m_1}} + C_8 \frac{\abs{\alpha_2}^{n_2}}{b^{m_1}} + C_9 \frac{\abs{\alpha_2}^{n_1}}{b^{m_1}} + b^{m_2 - m_1} \nonumber \\
\label{eq:chain_step1}
&\leq \frac{a \alpha^{n_2}}{C_4 \alpha^{n_1}} + C_8 \frac{\abs{\alpha_2}^{n_2}}{C_4 \alpha^{n_1}} + C_9 \frac{\abs{\alpha_2}^{n_1}}{C_4 \alpha^{n_1}} + b^{m_2 - m_1} \\
&\leq \frac{a}{C_4} \alpha^{n_2-n_1} + \frac{C_8}{C_4} \alpha^{n_2-n_1} + \frac{C_9}{C_4} \abs{\alpha_2}^{n_1-n_2} \alpha^{n_2-n_1} + b^{m_2 - m_1} \nonumber \\
&\leq C_{10} \max \setb{\alpha^{n_2-n_1}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_2-n_1}, b^{m_2 - m_1}},\nonumber
\end{align}
where in the third line we have used Inequality \eqref{eq:relleadterm}.
Note that we can either have $ \abs{\alpha_2} \leq 1 $ or $ \abs{\alpha_2} > 1 $ and that the bound may be simplified if we work with a concrete recurrence sequence where the size of $ \alpha_2 $ is known.
We aim for applying Proposition~\ref{prop:Matveev} to get also a lower bound.
Therefore let us set $ t=3 $, $ K = \mathbb{Q}(a,\alpha) $, $ D = [K:\mathbb{Q}] $ as well as
\begin{align*}
\eta_1 &= a, \quad &
\eta_2 &= \alpha, \quad &
\eta_3 &= b,\\
b_1 &= 1, \quad &
b_2 &= n_1, \quad &
b_3 &= -m_1.
\end{align*}
Further we can put
\begin{align*}
A_1 &= \max \setb{Dh(a), \abs{\log a}, 0.16}, \\
A_2 &= \max \setb{Dh(\alpha), \log \alpha, 0.16}, \\
A_3 &= D \log b, \\
B &= n_1.
\end{align*}
Finally, we have to show that the expression
\begin{equation*}
\Lambda := a \alpha^{n_1} b^{-m_1} - 1
\end{equation*}
is nonzero.
Assuming the contrary, we would have $ a \alpha^{n_1} = b^{m_1} $
and by applying $ \sigma $ we would get the equality $ a \alpha^{n_1} = \sigma(a) \sigma(\alpha)^{n_1} $.
Taking absolute values this implies
\begin{equation*}
\frac{\abs{\sigma(a)}}{a} = \left( \frac{\alpha}{\abs{\sigma(\alpha)}} \right)^{n_1},
\end{equation*}
which is a contradiction for $ n_1 \geq N_0 $ (where, as always, we may have updated $N_0$).
Hence we have $ \Lambda \neq 0 $.
Now Proposition~\ref{prop:Matveev} states that
\begin{equation*}
\log \abs{\Lambda} \geq -C_{11} (1 + \log n_1) \log b \geq -C_{12} \log n_1 \log b.
\end{equation*}
Comparing this lower bound with the upper bound coming from the first paragraph of this proof gives us
\begin{equation*}
-C_{12} \log n_1 \log b \leq \log C_{10} + \log \max \setb{\alpha^{n_2-n_1}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_2-n_1}, b^{m_2 - m_1}}
\end{equation*}
which immediately implies
\begin{equation*}
\min \setb{n_1-n_2, (m_1-m_2) \log b} \leq C_7 \log n_1 \log b.
\end{equation*}
This proves the lemma.
\end{proof}
\begin{mylemma}
\label{lemma:step2}
Assume that Equation \eqref{eq:centraleq} has at least two distinct solutions $ (n_1,m_1) $ and $ (n_2,m_2) $ with $ n_1 > n_2 $ as considered in Theorem \ref{thm:mainthm}.
Then we have
\begin{equation*}
\max \setb{n_1-n_2, (m_1-m_2) \log b} \leq C_{13} (\log n_1 \log b)^2
\end{equation*}
or
\begin{equation*}
n_1 \leq C_{14} (\log n_1 \log b)^2.
\end{equation*}
\end{mylemma}
\begin{proof}
We distinguish between two cases according to the statement of Lemma \ref{lemma:step1}.
Let us first assume that
\begin{equation*}
\min \setb{n_1-n_2, (m_1-m_2) \log b} = n_1-n_2.
\end{equation*}
Then we have
\begin{equation}
\label{eq:s2case1}
n_1-n_2 \leq C_7 \log n_1 \log b.
\end{equation}
Inserting the Binet representation of the linear recurrence sequence into Equation~\eqref{eq:twosols} and regrouping terms yields
\begin{align*}
\abs{a \alpha^{n_2} (\alpha^{n_1-n_2} - 1) - b^{m_1}} &= \abs{\sum_{l=2}^{k} a_l \alpha_l^{n_2} - \sum_{l=2}^{k} a_l \alpha_l^{n_1} - b^{m_2}} \\
&\leq C_8 \abs{\alpha_2}^{n_2} + C_9 \abs{\alpha_2}^{n_1} + b^{m_2}.
\end{align*}
Now we divide this inequality by $ b^{m_1} $.
If $ \abs{\alpha_2} \leq 1 $ we get
\begin{equation*}
\abs{\frac{a \alpha^{n_2} (\alpha^{n_1-n_2} - 1)}{b^{m_1}} - 1} \leq \frac{C_8 + C_9 + b^{m_2}}{b^{m_1}} \leq C_{15} b^{m_2-m_1},
\end{equation*}
and if $ \abs{\alpha_2} > 1 $ we get
\begin{align*}
\abs{\frac{a \alpha^{n_2} (\alpha^{n_1-n_2} - 1)}{b^{m_1}} - 1} &\leq \frac{C_{16} \abs{\alpha_2}^{n_1} + b^{m_2}}{b^{m_1}} \leq \frac{C_{16} \abs{\alpha_2}^{n_1}}{C_4 \alpha^{n_1}} + b^{m_2-m_1} \\
&\leq C_{17} \max \setb{\left( \frac{\abs{\alpha_2}}{\alpha} \right)^{n_1}, b^{m_2-m_1}},
\end{align*}
where for the second inequality we have used \eqref{eq:relleadterm}.
So in both subcases we have
\begin{equation}
\label{eq:s2c1upper}
\abs{\frac{a \alpha^{n_2} (\alpha^{n_1-n_2} - 1)}{b^{m_1}} - 1} \leq C_{18} \max \setb{\left( \frac{\abs{\alpha_2}}{\alpha} \right)^{n_1}, b^{m_2-m_1}}.
\end{equation}
We aim for applying Proposition~\ref{prop:Matveev} to get a lower bound.
Therefore let us set $ t=3 $, $ K = \mathbb{Q}(a,\alpha) $, $ D = [K:\mathbb{Q}] $ as well as
\begin{align*}
\eta_1 &= a(\alpha^{n_1-n_2} - 1), \quad &
\eta_2 &= \alpha, \quad &
\eta_3 &= b, \\
b_1 &= 1, \quad &
b_2 &= n_2, \quad &
b_3 &=-m_1.
\end{align*}
Using Inequality \eqref{eq:s2case1} gives us
\begin{align*}
h(\eta_1) &\leq h(a) + h(\alpha^{n_1-n_2} - 1) \leq h(a) + h(\alpha^{n_1-n_2}) + h(1) + \log 2 \\
&\leq C_{19} + (n_1-n_2) h(\alpha) \leq C_{20} \log n_1 \log b
\end{align*}
and an analogous bound for $ \log \eta_1 $.
Thus we can put
\begin{align*}
A_1 &= C_{21} \log n_1 \log b, \\
A_2 &= \max \setb{Dh(\alpha), \log \alpha, 0.16}, \\
A_3 &= D \log b, \\
B &= n_1.
\end{align*}
Finally, we have to show that the expression
\begin{equation*}
\Lambda := a(\alpha^{n_1-n_2} - 1) \alpha^{n_2} b^{-m_1} - 1
\end{equation*}
is nonzero.
Assuming the contrary, we would have $ a (\alpha^{n_1-n_2} - 1) \alpha^{n_2} = b^{m_1} $ and by applying $ \sigma $ the equality $ a (\alpha^{n_1-n_2} - 1) \alpha^{n_2} = \sigma(a) (\sigma(\alpha)^{n_1-n_2} - 1) \sigma(\alpha)^{n_2} $.
Taking absolute values this implies
\begin{equation*}
\abs{\frac{\sigma(a) (\sigma(\alpha)^{n_1-n_2} - 1)}{a (\alpha^{n_1-n_2} - 1)}} = \left( \frac{\alpha}{\abs{\sigma(\alpha)}} \right)^{n_2},
\end{equation*}
which is a contradiction for $ n_2 \geq N_0 $ since the left hand side is bounded above by a constant.
Hence we have $ \Lambda \neq 0 $.
Now Proposition~\ref{prop:Matveev} states that
\begin{equation*}
\log \abs{\Lambda} \geq -C_{22} (1 + \log n_1) \log n_1 \log b \log b \geq -C_{23} (\log n_1 \log b)^2.
\end{equation*}
Comparing this lower bound with the upper bound \eqref{eq:s2c1upper} gives us
\begin{equation*}
-C_{23} (\log n_1 \log b)^2 \leq \log C_{18} + \log \max \setb{\left( \frac{\abs{\alpha_2}}{\alpha} \right)^{n_1}, b^{m_2-m_1}}
\end{equation*}
which immediately implies
\begin{equation*}
\min \setb{n_1, (m_1-m_2) \log b} \leq C_{24} (\log n_1 \log b)^2.
\end{equation*}
Then, depending on which of the two expressions is the minimum, we either have
\begin{equation*}
n_1 \leq C_{14} (\log n_1 \log b)^2
\end{equation*}
or
\begin{equation*}
\max \setb{n_1-n_2, (m_1-m_2) \log b} = (m_1-m_2) \log b \leq C_{13} (\log n_1 \log b)^2.
\end{equation*}
This concludes the first case.
In the second case we assume that
\begin{equation*}
\min \setb{n_1-n_2, (m_1-m_2) \log b} = (m_1-m_2) \log b.
\end{equation*}
Thus we have
\begin{equation}
\label{eq:s2case2}
(m_1-m_2) \log b \leq C_7 \log n_1 \log b.
\end{equation}
Inserting the Binet representation of the linear recurrence sequence into Equation~\eqref{eq:twosols} and regrouping terms yields
\begin{align*}
\abs{a \alpha^{n_1} - b^{m_2} (b^{m_1-m_2} - 1)} &= \abs{a \alpha^{n_2} + \sum_{l=2}^{k} a_l \alpha_l^{n_2} - \sum_{l=2}^{k} a_l \alpha_l^{n_1}} \\
&\leq a \alpha^{n_2} + C_8 \abs{\alpha_2}^{n_2} + C_9 \abs{\alpha_2}^{n_1} \\
&\leq C_{25} \alpha^{n_2} + C_9 \abs{\alpha_2}^{n_1},
\end{align*}
and, dividing both sides by $ b^{m_1} - b^{m_2} $, we get
\begin{align}
\abs{\frac{a \alpha^{n_1}}{b^{m_2} (b^{m_1-m_2} - 1)} - 1} &\leq \frac{C_{25} \alpha^{n_2} + C_9 \abs{\alpha_2}^{n_1}}{b^{m_1} - b^{m_2}} \nonumber \\
&\leq \frac{C_{25} \alpha^{n_2} + C_9 \abs{\alpha_2}^{n_1}}{\frac{1}{2} b^{m_1}} \nonumber \\
&\leq \frac{C_{25} \alpha^{n_2} + C_9 \abs{\alpha_2}^{n_1}}{\frac{1}{2} C_4 \alpha^{n_1}} \nonumber \\
&\leq C_{26} \alpha^{n_2-n_1} + C_{27} \abs{\alpha_2}^{n_1-n_2} \alpha^{n_2-n_1} \nonumber \\
\label{eq:s2c2upper}
&\leq C_{28} \max \setb{\alpha^{n_2-n_1}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_2-n_1}},
\end{align}
where we have used the inequalities $ b^{m_1} - b^{m_2} \geq \frac{1}{2} b^{m_1} $ and \eqref{eq:relleadterm}.
We aim for applying Proposition~\ref{prop:Matveev} to get a lower bound.
Therefore let us set $ t=3 $, $ K = \mathbb{Q}(a,\alpha) $, $ D = [K:\mathbb{Q}] $ as well as
\begin{align*}
\eta_1 &= (b^{m_1-m_2} - 1)/a, \quad &
\eta_2 &= \alpha, \quad &
\eta_3 &= b, \\
b_1 &= -1, \quad &
b_2 &= n_1, \quad &
b_3 &= -m_2.
\end{align*}
Using Inequality \eqref{eq:s2case2} as well as $ h(b) = \log b $ gives us
\begin{align*}
h(\eta_1) &\leq h(a) + h(b^{m_1-m_2} - 1) \leq h(a) + h(b^{m_1-m_2}) + h(1) + \log 2 \\
&\leq C_{29} + (m_1-m_2) h(b) \leq C_{30} \log n_1 \log b
\end{align*}
and an analogous bound for $ \log \eta_1 $.
Thus we can put
\begin{align*}
A_1 &= C_{31} \log n_1 \log b, \\
A_2 &= \max \setb{Dh(\alpha), \log \alpha, 0.16}, \\
A_3 &= D \log b, \\
B &= n_1.
\end{align*}
Finally, we have to show that the expression
\begin{equation*}
\Lambda := \alpha^{n_1} b^{-m_2} ((b^{m_1-m_2} - 1)/a)^{-1} - 1
\end{equation*}
is nonzero.
Assuming the contrary, we would have $ a \alpha^{n_1} = b^{m_1} - b^{m_2} $ and by applying $ \sigma $ we would get the equality $ a \alpha^{n_1} = \sigma(a) \sigma(\alpha)^{n_1} $.
Taking absolute values this implies
\begin{equation*}
\frac{\abs{\sigma(a)}}{a} = \left( \frac{\alpha}{\abs{\sigma(\alpha)}} \right)^{n_1},
\end{equation*}
which is a contradiction for $ n_1 \geq N_0 $.
Hence we have $ \Lambda \neq 0 $.
Now Proposition~\ref{prop:Matveev} states that
\begin{equation*}
\log \abs{\Lambda} \geq -C_{32} (1 + \log n_1) \log n_1 \log b \log b \geq -C_{33} (\log n_1 \log b)^2.
\end{equation*}
Comparing this lower bound with the upper bound \eqref{eq:s2c2upper} gives us
\begin{equation*}
-C_{33} (\log n_1 \log b)^2 \leq \log C_{28} + \log \max \setb{\alpha^{n_2-n_1}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_2-n_1}}
\end{equation*}
which immediately implies
\begin{equation*}
\max \setb{n_1-n_2, (m_1-m_2) \log b} = n_1-n_2 \leq C_{13} (\log n_1 \log b)^2.
\end{equation*}
This concludes the proof of the lemma.
\end{proof}
\begin{mylemma}
\label{lemma:step3}
Assume that Equation \eqref{eq:centraleq} has at least two distinct solutions $ (n_1,m_1) $ and $ (n_2,m_2) $ with $ n_1 > n_2 $ as considered in Theorem \ref{thm:mainthm}.
Then we have
\begin{equation*}
n_1 \leq C_6 (\log n_1 \log b)^3.
\end{equation*}
\end{mylemma}
\begin{proof}
By Lemma \ref{lemma:step2} we can assume that
\begin{equation}
\label{eq:s2bound}
\max \setb{n_1-n_2, (m_1-m_2) \log b} \leq C_{13} (\log n_1 \log b)^2
\end{equation}
since the other case is trivial.
Inserting the Binet representation of the linear recurrence sequence into Equation \eqref{eq:twosols} and regrouping terms yields
\begin{align*}
\abs{a \alpha^{n_2} (\alpha^{n_1-n_2} - 1) - b^{m_2} (b^{m_1-m_2} - 1)} &= \abs{\sum_{l=2}^{k} a_l \alpha_l^{n_2} - \sum_{l=2}^{k} a_l \alpha_l^{n_1}} \\
&\leq C_8 \abs{\alpha_2}^{n_2} + C_9 \abs{\alpha_2}^{n_1}.
\end{align*}
Dividing both sides by $ b^{m_1} - b^{m_2} $ then gives us
\begin{align*}
\abs{\frac{a \alpha^{n_2} (\alpha^{n_1-n_2} - 1)}{b^{m_2} (b^{m_1-m_2} - 1)} - 1} &\leq \frac{C_8 \abs{\alpha_2}^{n_2} + C_9 \abs{\alpha_2}^{n_1}}{b^{m_1} - b^{m_2}} \\
&\leq \frac{C_8 \abs{\alpha_2}^{n_2} + C_9 \abs{\alpha_2}^{n_1}}{\frac{1}{2} b^{m_1}} \\
&\leq \frac{C_8 \abs{\alpha_2}^{n_2} + C_9 \abs{\alpha_2}^{n_1}}{\frac{1}{2} C_4 \alpha^{n_1}} \\
&\leq C_{34} \max \setb{\alpha^{-n_1}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{-n_1}},
\end{align*}
where we have used the inequalities $ b^{m_1} - b^{m_2} \geq \frac{1}{2} b^{m_1} $ and \eqref{eq:relleadterm}.
Note that the maximum occurs in view of the distinction between $ \abs{\alpha_2} \leq 1 $ and $ \abs{\alpha_2} > 1 $.
We aim for applying Proposition~\ref{prop:Matveev} to get a lower bound.
Therefore let us set $ t=3 $, $ K = \mathbb{Q}(a,\alpha) $, $ D = [K:\mathbb{Q}] $ as well as
\begin{align*}
\eta_1 &= \frac{a(\alpha^{n_1-n_2} - 1)}{b^{m_1-m_2} - 1}, \quad &
\eta_2 &= \alpha, \quad &
\eta_3 &= b,\\
b_1 &= 1, \quad &
b_2 &= n_2, \quad &
b_3 &= -m_2.
\end{align*}
Using the bound \eqref{eq:s2bound} as well as $ h(b) = \log b $ gives us
\begin{align*}
h(\eta_1) &\leq h(a) + h(\alpha^{n_1-n_2} - 1) + h(b^{m_1-m_2} - 1) \\
&\leq h(a) + h(\alpha^{n_1-n_2}) + h(1) + \log 2 + h(b^{m_1-m_2}) + h(1) + \log 2 \\
&\leq C_{35} + (n_1-n_2) h(\alpha) + (m_1-m_2) h(b) \leq C_{36} (\log n_1 \log b)^2
\end{align*}
and an analogous bound for $ \abs{\log \eta_1} $.
Thus we can put
\begin{align*}
A_1 &= C_{37} (\log n_1 \log b)^2, \\
A_2 &= \max \setb{Dh(\alpha), \log \alpha, 0.16}, \\
A_3 &= D \log b, \\
B &= n_1.
\end{align*}
Finally, we have to show that the expression
\begin{equation*}
\Lambda := \alpha^{n_2} b^{-m_2} a(\alpha^{n_1-n_2} - 1)/(b^{m_1-m_2} - 1) - 1
\end{equation*}
is nonzero.
Assuming the contrary, we would have $ a (\alpha^{n_1-n_2} - 1) \alpha^{n_2} = b^{m_1} - b^{m_2} $ and by applying $ \sigma $ we get the equality $ a (\alpha^{n_1-n_2} - 1) \alpha^{n_2} = \sigma(a) (\sigma(\alpha)^{n_1-n_2} - 1) \sigma(\alpha)^{n_2} $.
Taking absolute values this implies
\begin{equation*}
\abs{\frac{\sigma(a) (\sigma(\alpha)^{n_1-n_2} - 1)}{a (\alpha^{n_1-n_2} - 1)}} = \left( \frac{\alpha}{\abs{\sigma(\alpha)}} \right)^{n_2},
\end{equation*}
which is a contradiction for $ n_2 \geq N_0 $ since the left hand side is bounded above by a constant.
Hence we have $ \Lambda \neq 0 $.
Now Proposition~\ref{prop:Matveev} states that
\begin{equation*}
\log \abs{\Lambda} \geq -C_{38} (1 + \log n_1) (\log n_1 \log b)^2 \log b \geq -C_{39} (\log n_1 \log b)^3.
\end{equation*}
Comparing this lower bound with the upper bound coming from the first paragraph of this proof gives us
\begin{equation*}
-C_{39} (\log n_1 \log b)^3 \leq \log C_{34} + \log \max \setb{\alpha^{-n_1}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{-n_1}}
\end{equation*}
which immediately implies
\begin{equation*}
n_1 \leq C_6 (\log n_1 \log b)^3.
\end{equation*}
Thus the lemma is proven.
\end{proof}
We have now reached our first milestone, the bound \eqref{eq:logbound} is proven.
The second milestone is to prove that if there are at least three distinct solutions $ (n_1,m_1) $, $ (n_2,m_2) $ and $ (n_3,m_3) $, then for the largest one we have
\begin{equation}
\label{eq:bbound}
\log b \leq C_{40} (\log n_1)^2.
\end{equation}
Again we will split this part into some lemmas.
\begin{mylemma}
\label{lemma:step4}
Assume that Equation \eqref{eq:centraleq} has at least two distinct solutions $ (n_1,m_1) $ and $ (n_2,m_2) $ with $ n_1 > n_2 $ as considered in Theorem \ref{thm:mainthm}.
Then we have
\begin{equation*}
n_1 \log \alpha - C_{41} \leq m_1 \log b \leq n_1 \log \alpha + C_{42}.
\end{equation*}
\end{mylemma}
\begin{proof}
This follows immediately by applying the logarithm to Inequality \eqref{eq:relleadterm} from above.
\end{proof}
\begin{mylemma}
\label{lemma:step5}
Assume that Equation \eqref{eq:centraleq} has at least three distinct solutions $ (n_1,m_1) $, $ (n_2,m_2) $ and $ (n_3,m_3) $ with $ n_1 > n_2 > n_3 $ as considered in Theorem \ref{thm:mainthm}.
Then at least one of the following inequalities holds:
\begin{enumerate}[(i)]
\item $ \log b \leq C_{43} \log n_1 $,
\item $ n_2-n_3 \leq C_{44} \log n_1 $.
\end{enumerate}
\end{mylemma}
\begin{proof}
Let us recall Inequality \eqref{eq:chain_step1} from the proof of Lemma \ref{lemma:step1}, where we obtained
\begin{equation*}
\abs{\frac{a \alpha^{n_1}}{b^{m_1}} - 1} \leq C_{10} \max \setb{\alpha^{n_2-n_1}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_2-n_1}, b^{m_2 - m_1}}.
\end{equation*}
Now we define the linear form
\begin{equation*}
\Lambda_{12} := n_1 \log \alpha - m_1 \log b + \log a = \log \frac{a \alpha^{n_1}}{b^{m_1}}
\end{equation*}
and distinguish between two cases.
Let us first assume that $ \abs{\Lambda_{12}} > 1 $.
Then by Lemma \ref{lemma:linbig} we have
\begin{equation*}
\frac{3}{5} \leq \abs{e^{\Lambda_{12}} - 1} \leq C_{10} \max \setb{\alpha^{n_2-n_1}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_2-n_1}, b^{m_2 - m_1}}.
\end{equation*}
If the maximum is either $ \alpha^{n_2-n_1} $ or $ \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_2-n_1} $, this implies an upper bound $ n_1-n_2 \leq C_{45} $.
We will come back to this later.
If the maximum is $ b^{m_2 - m_1} $, then the inequality implies
\begin{equation*}
\log b \leq (m_1-m_2) \log b \leq C_{46}
\end{equation*}
and we are done.
Therefore we will now assume that $ \abs{\Lambda_{12}} \leq 1 $.
Since we are now working with three solutions, we analogously get the upper bound
\begin{equation*}
\abs{\frac{a \alpha^{n_2}}{b^{m_2}} - 1} \leq C_{10} \max \setb{\alpha^{n_3-n_2}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_3-n_2}, b^{m_3 - m_2}}
\end{equation*}
and define the linear form
\begin{equation*}
\Lambda_{23} := n_2 \log \alpha - m_2 \log b + \log a = \log \frac{a \alpha^{n_2}}{b^{m_2}}.
\end{equation*}
Once again we distinguish between two cases and assume first that $ \abs{\Lambda_{23}} > 1 $.
Then by Lemma \ref{lemma:linbig} we have
\begin{equation*}
\frac{3}{5} \leq \abs{e^{\Lambda_{23}} - 1} \leq C_{10} \max \setb{\alpha^{n_3-n_2}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_3-n_2}, b^{m_3 - m_2}}.
\end{equation*}
If the maximum is either $ \alpha^{n_3-n_2} $ or $ \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_3-n_2} $, this implies an upper bound $ n_2-n_3 \leq C_{45} $.
If the maximum is $ b^{m_3 - m_2} $, then the inequality implies
\begin{equation*}
\log b \leq (m_3-m_2) \log b \leq C_{46}.
\end{equation*}
In both situations we are done.
Therefore we will now assume that $ \abs{\Lambda_{23}} \leq 1 $.
As we have $ \abs{\Lambda_{12}} \leq 1 $ as well as $ \abs{\Lambda_{23}} \leq 1 $, we can apply Lemma \ref{lemma:linsmall} to both linear forms, which yields
\begin{align*}
\abs{\Lambda_{12}} &\leq 4 \abs{e^{\Lambda_{12}} - 1} \leq C_{47} \max \setb{\alpha^{n_2-n_1}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_2-n_1}, b^{m_2 - m_1}}, \\
\abs{\Lambda_{23}} &\leq 4 \abs{e^{\Lambda_{23}} - 1} \leq C_{47} \max \setb{\alpha^{n_3-n_2}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_3-n_2}, b^{m_3 - m_2}}.
\end{align*}
As the next step we define a further linear form by
\begin{equation*}
\Lambda := m_2 \Lambda_{12} - m_1 \Lambda_{23} = (m_2 n_1 - m_1 n_2) \log \alpha + (m_2 - m_1) \log a.
\end{equation*}
Using the upper bounds for $ \abs{\Lambda_{12}} $ and $ \abs{\Lambda_{23}} $ from above we get
\begin{align}
\abs{\Lambda} &\leq m_2 \abs{\Lambda_{12}} + m_1 \abs{\Lambda_{23}} \nonumber \\
\label{eq:s5bound}
&\leq C_{48} n_1 \max \left( \alpha^{n_2-n_1}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_2-n_1}, b^{m_2 - m_1}, \right. \\
&\hspace{4cm} \left. \alpha^{n_3-n_2}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_3-n_2}, b^{m_3 - m_2} \right). \nonumber
\end{align}
We aim for applying Proposition \ref{prop:Matveev} to get a lower bound.
Therefore let us set $ t=2 $, $ K = \mathbb{Q}(a,\alpha) $, $ D = [K:\mathbb{Q}] $ as well as
\begin{align*}
\eta_1 &= \alpha, \quad &
\eta_2 &= a,\\
b_1 &= m_2 n_1 - m_1 n_2, \quad &
b_2 &= m_2 - m_1.
\end{align*}
Further we can put
\begin{align*}
A_1 &= \max \setb{Dh(\alpha), \log \alpha, 0.16}, \\
A_2 &= \max \setb{Dh(a), \abs{\log a}, 0.16}, \\
B &= n_1^2.
\end{align*}
Finally, we have to show that $ \Lambda $ is nonzero.
But this follows immediately from $ m_1 \neq m_2 $ and the assumption that $ \alpha $ and $ a $ are multiplicatively independent.
Now Proposition \ref{prop:Matveev} states that
\begin{equation*}
\log \abs{\Lambda} \geq -C_{49} (1 + \log (n_1^2)) \geq -C_{50} \log (n_1^2) \geq -C_{51} \log n_1.
\end{equation*}
Comparing this lower bound with the upper bound \eqref{eq:s5bound} gives us
\begin{align*}
-C_{51} \log n_1 &\leq \log C_{48} + \log n_1 + \log \max \left( \alpha^{n_2-n_1}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_2-n_1}, b^{m_2 - m_1}, \right. \\
&\hspace{5cm} \left. \alpha^{n_3-n_2}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_3-n_2}, b^{m_3 - m_2} \right)
\end{align*}
which immediately implies
\begin{equation}
\label{eq:s5minbound}
\min \setb{n_1-n_2, n_2-n_3, (m_1-m_2) \log b, (m_2-m_3) \log b} \leq C_{52} \log n_1.
\end{equation}
We have now to handle three cases.
If the minimum in Inequality \eqref{eq:s5minbound} is either $ (m_1-m_2) \log b $ or $ (m_2-m_3) \log b $, then we get
\begin{equation*}
\log b \leq (m_i-m_{i+1}) \log b \leq C_{52} \log n_1
\end{equation*}
for an $ i \in \set{1,2} $ and are done.
If the minimum in Inequality \eqref{eq:s5minbound} is $ n_2-n_3 $, we have case (ii) of the lemma.
So it remains to consider the case when the minimum in Inequality \eqref{eq:s5minbound} is $ n_1-n_2 $.
This can be handled together with the still open case $ n_1-n_2 \leq C_{45} $ from above.
Hence let us assume that $ n_1-n_2 \leq C_{53} \log n_1 $.
By Lemma \ref{lemma:step4} we get
\begin{align*}
C_{53} \log n_1 &\geq n_1-n_2 \\
&\geq \frac{1}{\log \alpha} (m_1 \log b - C_{42}) - \frac{1}{\log \alpha} (m_2 \log b + C_{41}) \\
&= \frac{1}{\log \alpha} \left( (m_1-m_2) \log b - C_{41} - C_{42} \right)
\end{align*}
and thus
\begin{equation*}
\log b \leq (m_1-m_2) \log b \leq C_{54} \log n_1
\end{equation*}
which concludes the proof of the lemma.
\end{proof}
\begin{mylemma}
\label{lemma:step6}
Assume that Equation \eqref{eq:centraleq} has at least three distinct solutions $ (n_1,m_1) $, $ (n_2,m_2) $ and $ (n_3,m_3) $ with $ n_1 > n_2 > n_3 $ as considered in Theorem \ref{thm:mainthm}.
Then we have
\begin{equation*}
\log b \leq C_{40} (\log n_1)^2.
\end{equation*}
\end{mylemma}
\begin{proof}
From Lemma \ref{lemma:step5} we get that either (i) or (ii) holds.
Since in the case (i) there is nothing to do, we may assume that we are in the case (ii) and therefore have the bound
\begin{equation}
\label{eq:s6bound}
n_2-n_3 \leq C_{44} \log n_1.
\end{equation}
Recall the linear form
\begin{equation*}
\Lambda_{12} = n_1 \log \alpha - m_1 \log b + \log a
\end{equation*}
from the proof of Lemma \ref{lemma:step5}. Note that it is enough to consider the situation $ \abs{\Lambda_{12}} \leq 1 $, which by Lemma \ref{lemma:linsmall} implied
\begin{equation*}
\abs{\Lambda_{12}} \leq 4 \abs{e^{\Lambda_{12}} - 1} \leq C_{47} \max \setb{\alpha^{n_2-n_1}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_2-n_1}, b^{m_2 - m_1}},
\end{equation*}
since the situation $ \abs{\Lambda_{12}} > 1 $ led to case (i).
Taking a look at the proof of Lemma \ref{lemma:step2} and noting that we are working with three solutions, we recall from \eqref{eq:s2c1upper} the upper bound
\begin{equation*}
\abs{\frac{a \alpha^{n_3} (\alpha^{n_2-n_3} - 1)}{b^{m_2}} - 1} \leq C_{18} \max \setb{\left( \frac{\abs{\alpha_2}}{\alpha} \right)^{n_2}, b^{m_3-m_2}}
\end{equation*}
and define the linear form
\begin{equation*}
\widetilde{\Lambda_{23}} := n_3 \log \alpha - m_2 \log b + \log (a(\alpha^{n_2-n_3} - 1)).
\end{equation*}
Again we distinguish between two cases and assume first that $ \abs{\widetilde{\Lambda_{23}}} > 1 $.
Then by Lemma \ref{lemma:linbig} we have
\begin{equation*}
\frac{3}{5} \leq \abs{e^{\widetilde{\Lambda_{23}}} - 1} \leq C_{18} \max \setb{\left( \frac{\abs{\alpha_2}}{\alpha} \right)^{n_2}, b^{m_3-m_2}}.
\end{equation*}
If the maximum is $ \left( \frac{\abs{\alpha_2}}{\alpha} \right)^{n_2} $, this implies an upper bound $ n_2 \leq C_{55} $.
We will come back to this later.
If the maximum is $ b^{m_3 - m_2} $, then the inequality implies
\begin{equation*}
\log b \leq (m_2-m_3) \log b \leq C_{56}
\end{equation*}
and we are done.
Therefore we will now assume that $ \abs{\widetilde{\Lambda_{23}}} \leq 1 $.
In this situation we can apply Lemma \ref{lemma:linsmall} which yields
\begin{equation*}
\abs{\widetilde{\Lambda_{23}}} \leq 4 \abs{e^{\widetilde{\Lambda_{23}}} - 1} \leq C_{57} \max \setb{\left( \frac{\abs{\alpha_2}}{\alpha} \right)^{n_2}, b^{m_3-m_2}}.
\end{equation*}
In the next step we define a further linear form by
\begin{align*}
\Lambda :=\ &m_2 \Lambda_{12} - m_1 \widetilde{\Lambda_{23}} \\
=\ &(m_2 n_1 - m_1 n_3) \log \alpha + (m_2 - m_1) \log a - m_1 \log (\alpha^{n_2-n_3}-1).
\end{align*}
Using the upper bounds for $ \abs{\Lambda_{12}} $ and $ \abs{\widetilde{\Lambda_{23}}} $ from above we get
\begin{align}
\abs{\Lambda} &\leq m_2 \abs{\Lambda_{12}} + m_1 \abs{\widetilde{\Lambda_{23}}} \nonumber \\
\label{eq:s6linbound}
&\leq C_{58} n_1 \max \setb{\alpha^{n_2-n_1}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_2-n_1}, b^{m_2 - m_1}, \left( \frac{\abs{\alpha_2}}{\alpha} \right)^{n_2}, b^{m_3-m_2}}.
\end{align}
We aim for applying Proposition \ref{prop:Matveev} to get a lower bound.
Therefore let us set $ t=3 $, $ K = \mathbb{Q}(a,\alpha) $, $ D = [K:\mathbb{Q}] $ as well as
\begin{align*}
\eta_1 &= \alpha, \quad &
\eta_2 &= a, \quad &
\eta_3 &= \alpha^{n_2-n_3}-1,\\
b_1 &= m_2 n_1 - m_1 n_3, \quad &
b_2 &= m_2 - m_1, \quad &
b_3 &= -m_1.
\end{align*}
Using the bound \eqref{eq:s6bound} gives us
\begin{align*}
h(\eta_3) &\leq h(\alpha^{n_2-n_3}) + h(1) + \log 2 = (n_2-n_3) h(\alpha) + h(1) + \log 2 \\
&\leq C_{59} (n_2-n_3) \leq C_{60} \log n_1
\end{align*}
and an analogous bound for $ \abs{\log \eta_3} $.
Thus we can put
\begin{align*}
A_1 &= \max \setb{Dh(\alpha), \log \alpha, 0.16}, \\
A_2 &= \max \setb{Dh(a), \abs{\log a}, 0.16}, \\
A_3 &= C_{61} \log n_1, \\
B &= n_1^2.
\end{align*}
Finally, we have to show that $ \Lambda $ is nonzero.
Assume the contrary.
Since $ m_1 \neq m_2 $, this means that $ \alpha $, $ a $ and $ \alpha^{n_2-n_3}-1 $ are multiplicatively dependent.
Thus there exist rational numbers $ x,y \in \mathbb{Q} $ such that
\begin{equation*}
\alpha^{n_2-n_3} - 1 = a^x \alpha^y
\end{equation*}
because $ \alpha $ and $ a $ are multiplicatively independent by assumption.
Then the linear form $ \Lambda $ becomes
\begin{equation*}
\Lambda = (m_2 n_1 - m_1 n_3 - m_1 y) \log \alpha + (m_2 - m_1 - m_1 x) \log a.
\end{equation*}
Again using that $ \alpha $ and $ a $ are multiplicatively independent by assumption, $ \Lambda = 0 $ in particular implies
\begin{equation*}
m_2 - m_1 - m_1 x = 0
\end{equation*}
which yields
\begin{equation*}
m_2 = m_1 (1+x)
\end{equation*}
and hence $ -1 < x < 0 $ since $ m_1 > m_2 \geq 1 $.
But this was excluded in the theorem.
Therefore we have $ \Lambda \neq 0 $.
Now Proposition \ref{prop:Matveev} states that
\begin{equation*}
\log \abs{\Lambda} \geq -C_{62} (1 + \log (n_1^2)) \log n_1 \geq -C_{63} \log (n_1^2) \log n_1 \geq -C_{64} (\log n_1)^2.
\end{equation*}
Comparing this lower bound with the upper bound \eqref{eq:s6linbound} gives us
\begin{align*}
-C_{64} (\log n_1)^2 &\leq \log C_{58} + \log n_1 \\
&\hspace{0.5cm}+ \log \max \setb{\alpha^{n_2-n_1}, \left( \frac{\alpha}{\abs{\alpha_2}} \right)^{n_2-n_1}, b^{m_2 - m_1}, \left( \frac{\abs{\alpha_2}}{\alpha} \right)^{n_2}, b^{m_3-m_2}}
\end{align*}
which immediately implies
\begin{equation}
\label{eq:s6minbound}
\min \setb{n_1-n_2, n_2, (m_1-m_2) \log b, (m_2-m_3) \log b} \leq C_{65} (\log n_1)^2.
\end{equation}
We have now to handle three cases.
If the minimum in Inequality \eqref{eq:s6minbound} is either $ (m_1-m_2) \log b $ or $ (m_2-m_3) \log b $, then we get
\begin{equation*}
\log b \leq (m_i-m_{i+1}) \log b \leq C_{65} (\log n_1)^2
\end{equation*}
for an $ i \in \set{1,2} $ and are done.
If the minimum in Inequality \eqref{eq:s6minbound} is $ n_1-n_2 $, then we have
\begin{equation*}
n_1-n_2 \leq C_{65} (\log n_1)^2
\end{equation*}
which yields, by using Lemma \ref{lemma:step4}, analogously to the end of the proof of Lemma \ref{lemma:step5} the bound
\begin{equation*}
\log b \leq (m_1-m_2) \log b \leq C_{66} (\log n_1)^2
\end{equation*}
and we are done as well.
So it remains to consider the case when the minimum in Inequality \eqref{eq:s6minbound} is $ n_2 $.
This can be handled together with the still open case $ n_2 \leq C_{55} $ from above.
Hence let us assume that $ n_2 \leq C_{67} (\log n_1)^2 $.
By Lemma \ref{lemma:step4} we get
\begin{equation*}
C_{67} (\log n_1)^2 \geq n_2 \geq \frac{1}{\log \alpha} (m_2 \log b - C_{42}) \geq \frac{1}{\log \alpha} (\log b - C_{42})
\end{equation*}
and thus
\begin{equation*}
\log b \leq C_{68} (\log n_1)^2
\end{equation*}
which concludes the proof of the lemma.
\end{proof}
We have now reached our second milestone, the bound \eqref{eq:bbound} is proven.
So if there are at least three distinct solutions $ (n_1,m_1) $, $ (n_2,m_2) $ and $ (n_3,m_3) $ with $ n_1 > n_2 > n_3 $ as considered in Theorem \ref{thm:mainthm}, then we have the two bounds \eqref{eq:logbound} and \eqref{eq:bbound}.
Inserting \eqref{eq:bbound} into \eqref{eq:logbound} gives
\begin{equation*}
n_1 \leq C_{69} (\log n_1)^9
\end{equation*}
and therefore the absolute bound
\begin{equation*}
n_1 \leq C_{70}.
\end{equation*}
Now we insert this into Inequality \eqref{eq:bbound} and get
\begin{equation*}
b \leq C_{71}.
\end{equation*}
This proves Theorem \ref{thm:mainthm}.
\section{Proof of Theorem \ref{thm:Tribos}}\label{sec:Tribos}\label{sec:proofTribos}
Recall that the Tribonacci sequence is given by $T_1=1, T_2=1, T_3=2$ and $T_{n}=T_{n-1}+T_{n-2}+T_{n-3}$ for $n\geq 4$.
Then one can compute the roots $\alpha,\beta,\gamma$ of $f(X)=X^3-X^2-X-1$ and the coefficients $a,b,c$ such that
\[
T_n= a\alpha^n + b\beta^n + c\gamma^n.
\]
Let $\sigma$ be the Galois automorphism on the splitting field $K$ of $f$ that maps $\alpha \mapsto \beta$. It turns out that
\[
a=\frac{1}{-\alpha^2 + 4\alpha -1}, \quad
b=\sigma(a),\quad
c=\sigma^2(a).
\]
We fix $\alpha,\beta, \gamma$ such that
\begin{align*}
\alpha &\approx 1.839, \quad &
\beta &\approx -0.42 + 0.61 i, \quad &
\gamma &\approx -0.42 - 0.61 i,\\
a &\approx 0.336, \quad &
b &\approx -0.17 - 0.20 i, \quad &
c &\approx -0.17 + 0.20i.
\end{align*}
Note that
\[
|\beta|=|\gamma|\leq 0.74
\quad \text{and} \quad
|b|=|c|\leq 0.26,
\]
so we can estimate
\begin{equation}\label{eq:trib:Tn}
T_n= a \alpha^n + L(0.52 \cdot 0.74^n).
\end{equation}
Here the $L$-notation means the following: For functions $f(n)$, $g(n)$ with $g(n)>0 $ for $n\geq 1$ we write
\[
f(n) = L(g(n))
\quad \text{if} \quad
|f(n)|\leq g(n).
\]
We check that all assumptions in Theorem \ref{thm:mainthm} are fulfilled: First, $\alpha$ is indeed an irrational dominant root larger than 1 and $a>0$.
Second, we check hat $a$ and $\alpha$ are multiplicatively independent.
Assume that they are not, then there exist nonzero integers $x,y$ such that $a^x \alpha^y=1$. Since $|a|<1$ and $|\alpha|>1$, the integers $x$ and $y$ must have the same sign. However, if we apply $\sigma$, we get $b^x\beta^y=1$, but since $|b|<1$ and $|\beta|<1$ this is impossible if $x$ and $y$ have the same sign. Therefore, $a$ and $\alpha$ are multiplicatively independent. Finally, we check that there are no unwanted solutions to \eqref{eq:techcond}. Assume that
\[
\alpha^z - 1 = a^x \alpha^y
\]
with $ z \in \mathbb{N} $, $ x,y \in \mathbb{Q} $ and $ -1 < x < 0 $.
Then we have for the norms
\[
N_{K/\mathbb{Q}}(\alpha^z-1)
= N_{K/\mathbb{Q}}(a^x \alpha^y)
= N_{K/\mathbb{Q}}(a)^x N_{K/\mathbb{Q}}(\alpha)^y
= (1/44)^x1^y
=44^{-x}.
\]
Now since $\alpha$ and therefore $\alpha^z -1$ are algebraic integers, $N_{K/\mathbb{Q}}(\alpha^z-1)=44^{-x}$ has to be an integer. But this is impossible for a rational $x$ between $-1$ and $0$.
Assume that we have three solutions $T_{n_1}-b^{m_1}=T_{n_2}-b^{m_2}=T_{n_3}-b^{m_3}=c$ with $n_1>n_2>n_3\geq 2$.
We do some preliminary estimations.
From \eqref{eq:trib:Tn} we have
\begin{align}\label{eq:trib:eqationL}
a \alpha^{n_1} - b^{m_1} - (a \alpha^{n_2} - b^{m_2})\nonumber
&= L(0.52 \cdot 0.74^{n_1}) + L(0.52 \cdot 0.74^{n_2})\\
&= L(0.91 \cdot 0.74^{n_2})
\end{align}
and analogously
\begin{equation}\label{eq:trib:equationL2}
a \alpha^{n_2} - b^{m_2} - (a \alpha^{n_3} - b^{m_3})
= L(0.91 \cdot 0.74^{n_3}).
\end{equation}
Moreover, we have
\begin{multline*}
b^{m_1} \nonumber
\geq b^{m_1} - b^{m_2}
= T_{n_1} - T_{n_2}
= a \alpha^{n_1} - a \alpha^{n_2} + L(0.91 \cdot 0.74^{n_2}) \\
\geq a(1-\alpha^{-1})\alpha^{n_1} - 0.91 \cdot 0.74^{n_2}
\geq 0.15 \alpha^{n_1} - 0.68.
\end{multline*}
Now note that on the one hand for $n_1 \geq 6$ we have $0.68/\alpha^{n_1} \leq 0.02$ and on the other hand $b^{m_1}\geq 2^2=4\geq 0.13 \alpha^{n_1}$ is trivially fulfilled for $n_1\leq 5$. Thus we have in any case
\begin{equation}\label{eq:trib:balpha}
b^{m_1}\geq 0.13 \alpha^{n_1}
\quad
\text{and analogously}
\quad
b^{m_2}\geq 0.13 \alpha^{n_2}.
\end{equation}
Estimating from the other side, we have
\begin{align*}
0.5 \alpha^{n_1}
&\geq a \alpha^{n_1} + L(0.52\cdot 0.74^{n_1})
= T_{n_1}
\geq T_{n_1}-T_{n_2}\\
&= b^{m_1}-b^{m_2}
\geq 0.5 b^{m_1}
\end{align*}
and the same argument works for the second and third solution. So we have
\begin{equation}\label{eq:trib:alphab}
\alpha^{n_1} \geq b^{m_1}
\quad \text{and} \quad
\alpha^{n_2} \geq b^{m_2}.
\end{equation}
In particular, \eqref{eq:trib:balpha} and \eqref{eq:trib:alphab} imply the bounds
\begin{equation}\label{eq:trib:nlogalphamlogb}
\begin{split}
&m_1 \log b
\leq n_1 \log \alpha
\leq m_1 \log b + 2.1,\\
&m_2 \log b
\leq n_2 \log \alpha
\leq m_2 \log b + 2.1
\end{split}
\end{equation}
and in particular
\[
n_1 \geq m_1.
\]
\step{step:smallSols}{Small solutions:} First, we check that there are no solutions with $n_1\leq 150$.
To that end we simply search for all differences of two Tribonacci numbers that can be written in the form $T_{n_1}-T_{n_2} = b^{m_1}-b^{m_2}$ with $b\geq 2$ and $m_1>m_2\geq 1$.
With the help of Sage \cite{sagemath} this is not difficult (see \nameref{sec:appendix} for the code):
For each pair $2\leq n_2 < n_1 \leq 150$, we compute the prime factorisation of $T_{n_1}-T_{n_2}=p_1^{k_1}\cdots p_l^{k_l}$.
Now we need to check if there exist $b\geq 2$ and $x,y\geq 1$
such that $p_1^{k_1}\cdots p_l^{k_l}=b^x(b^y-1)$.
Since $\gcd(b^x,b^y-1)=1$, we can simply try
$b=p_{i_1}^{k_{i_1}/d}\cdots p_{i_t}^{k_{i_t}/d}$ for each subset $\{p_{i_1},\ldots p_{i_t}\}\subset\{p_1,\ldots,p_l\}$ and $d=\gcd(k_{i_1},\ldots,k_{i_t})$, and check if $(T_{n_1}-T_{n_2})/b^d$ can be written in the form $(b^y -1)$.
It turns out that this only happens on 14 occasions:
\begin{align*}
T_{ 4 } - T_{ 3 } = 2 ^{ 1 } ( 2 ^{ 1 }-1), \quad c&= 0 = T_{ 4 } - 2 ^{ 2 } = T_{ 3 } - 2 ^{ 1 }; \\
T_{ 5 } - T_{ 2 } = 2 ^{ 1 } ( 2 ^{ 2 }-1), \quad c&= -1 = T_{ 5 } - 2 ^{ 3 } = T_{ 2 } - 2 ^{ 1 }; \\
T_{ 5 } - T_{ 2 } = 3 ^{ 1 } ( 3 ^{ 1 }-1), \quad c&= -2 = T_{ 5 } - 3 ^{ 2 } = T_{ 2 } - 3 ^{ 1 }; \\
T_{ 6 } - T_{ 2 } = 2 ^{ 2 } ( 2 ^{ 2 }-1), \quad c&= -3 = T_{ 6 } - 2 ^{ 4 } = T_{ 2 } - 2 ^{ 2 }; \\
T_{ 6 } - T_{ 5 } = 2 ^{ 1 } ( 2 ^{ 2 }-1), \quad c&= 5 = T_{ 6 } - 2 ^{ 3 } = T_{ 5 } - 2 ^{ 1 }; \\
T_{ 6 } - T_{ 5 } = 3 ^{ 1 } ( 3 ^{ 1 }-1), \quad c&= 4 = T_{ 6 } - 3 ^{ 2 } = T_{ 5 } - 3 ^{ 1 }; \\
T_{ 7 } - T_{ 4 } = 5 ^{ 1 } ( 5 ^{ 1 }-1), \quad c&= -1 = T_{ 7 } - 5 ^{ 2 } = T_{ 4 } - 5 ^{ 1 }; \\
T_{ 8 } - T_{ 3 } = 7 ^{ 1 } ( 7 ^{ 1 }-1), \quad c&= -5 = T_{ 8 } - 7 ^{ 2 } = T_{ 3 } - 7 ^{ 1 }; \\
T_{ 8 } - T_{ 7 } = 5 ^{ 1 } ( 5 ^{ 1 }-1), \quad c&= 19 = T_{ 8 } - 5 ^{ 2 } = T_{ 7 } - 5 ^{ 1 }; \\
T_{ 11 } - T_{ 3 } = 17 ^{ 1 } ( 17 ^{ 1 }-1), \quad c&= -15 = T_{ 11 } - 17 ^{ 2 } = T_{ 3 } - 17 ^{ 1 }; \\
T_{ 12 } - T_{ 4 } = 5 ^{ 3 } ( 5 ^{ 1 }-1), \quad c&= -121 = T_{ 12 } - 5 ^{ 4 } = T_{ 4 } - 5 ^{ 3 }; \\
T_{ 12 } - T_{ 7 } = 2 ^{ 5 } ( 2 ^{ 4 }-1), \quad c&= -8 = T_{ 12 } - 2 ^{ 9 } = T_{ 7 } - 2 ^{ 5 }; \\
T_{ 15 } - T_{ 11 } = 54 ^{ 1 } ( 54 ^{ 1 }-1), \quad c&= 220 = T_{ 15 } - 54 ^{ 2 } = T_{ 11 } - 54 ^{ 1 }; \\
T_{ 23 } - T_{ 12 } = 641 ^{ 1 } ( 641 ^{ 1 }-1), \quad c&= -137 = T_{ 23 } - 641 ^{ 2 } = T_{ 12 } - 641 ^{ 1 }.
\end{align*}
The only $c$ that appears twice is $c=-1$, but on the two occasions the $b$'s are distinct ($2$ and $5$). So there is no $c$ with three solutions with $n_1\leq 150$.
The computations only took a couple of minutes on a usual laptop.
Let us from now on assume that
\[
n_1>150.
\]
Finally, a little remark on notation: The constants in this section will be numbered starting with $C_{100}$ to set them apart from the previous constants.
\begin{mystep}\label{step:Step1}
From \eqref{eq:trib:eqationL} we obtain
\[
|a \alpha^{n_1} - b^{m_1}|
= |a \alpha^{n_2} - b^{m_2} + L(0.91 \cdot 0.74^{n_2})|
\leq \max( a \alpha^{n_2}, b^{m_2} )
\leq \alpha^{n_2},
\]
where for the first inequality we used the fact that $a\alpha^{n_2}$ and $b^{m_2}$ are both positive and larger than the $L$-terms and for the second estimation we used $a<1$ and \eqref{eq:trib:alphab}.
Dividing by $b^{m_1} \geq 0.13 \alpha^{n_1}$ (see \eqref{eq:trib:balpha}) we obtain
\begin{equation}\label{eq:trib:Lambda1}
|\Lambda_1|
:= \left| \frac{a\alpha^{n_1}}{b^{m_1}}-1 \right|
\leq \frac{\alpha^{n_2}}{0.13 \alpha^{n_1}}
\leq 7.7 \alpha^{-(n_1-n_2)}.
\end{equation}
We check that $\Lambda_1 \neq 0$: If $\Lambda_1=0$, then $a \alpha^{n_1} =b^{m_1} \in \mathbb{Z}$, so an application of the Galois automorphism $\sigma$ does not change $a \alpha^{n_1}$, i.e.\ we get $a \alpha^{n_1} = \sigma (a \alpha^{n_1}) =b \beta^{n_1}$.
Taking absolute values and estimating we get $0.33 \cdot 1.83^{n_1} \leq |a \alpha^{n_1}| = |b \beta^{n_1}|\leq 0.26 \cdot 0.74^{n_1}$,
which is impossible.
Now we apply Proposition~\ref{prop:Matveev} with
$t=3$, $K=\mathbb{Q}(a,\alpha)=\mathbb{Q}(\alpha)$, $D=3=[K:\mathbb{Q}]$ and
\begin{align*}
\eta_1 &= \alpha, \quad &
\eta_2 &= b, \quad &
\eta_3 &=a, \\
b_1 &= n_1, \quad &
b_2 &= -m_1, \quad &
b_3 &= 1.
\end{align*}
We put
\begin{align*}
A_1&=0.7\geq \log \alpha = \max(D h(\alpha),\log \alpha,0.16),\\
A_2&=3 \log b= \max(D h(b),\abs{\log b},0.16),\\
A_3&= 3.8 \geq \log 44 =3 h(a)= \max(D h(a),\abs{\log a},0.16),\\
B&=n_1.
\end{align*}
Thus we get that
\[
\log|\Lambda_1|
\geq - C_{100} \cdot \log n_1 \cdot \log b,
\]
where
\[
C_{100} = 2.6 \cdot 10^{13}
> 1.4 \cdot 30^6 \cdot 3^{4.5} \cdot 3^2 (1+ \log 3) \cdot 1.2 \cdot 0.7 \cdot 3 \cdot 3.8.
\]
Note that the factor $1.2$ comes from the estimate $1+\log B = 1+\log n_1 \leq 1.2 \log n_1$, since $n_1>150$.
Together with \eqref{eq:trib:Lambda1} this implies
\[
-C_{100}\cdot \log n_1 \cdot \log b
\leq \log 7.7 - (n_1-n_2)\log \alpha,
\]
which yields
\begin{equation}\label{eq:trib:boundn1-n2}
(n_1-n_2)\log \alpha
\leq C_{100}\cdot \log n_1 \cdot \log b.
\end{equation}
Here we omitted the term $\log 7.7$. We can do this because we estimated quite generously when computing the constant $C_{100}=2.6\cdot 10^{13}$.
\end{mystep}
\begin{mystep}{}\label{step:Step2}
From \eqref{eq:trib:eqationL} we obtain
\[
\left| (a \alpha^{n_1} - a \alpha^{n_2}) - (b^{m_1} - b^{m_2}) \right|
= L(0.91 \cdot 0.74^{n_2})
\leq 0.5.
\]
Dividing by $b^{m_1} - b^{m_2} \geq 0.5 b^{m_1}$ we obtain
\begin{equation}\label{eq:trib:Lambda2}
|\Lambda_2|
:= \left| \frac{a \alpha^{n_2}(\alpha^{n_1-n_2}-1)}{b^{m_2} (b^{m_1-m_2}-1)}-1\right|
\leq b^{-m_1}.
\end{equation}
We check that $\Lambda_2 \neq 0$: If $\Lambda_{2}=0$, then $a (\alpha^{n_1}-\alpha^{n_2}) \in \mathbb{Z}$, so we must have $a (\alpha^{n_1}-\alpha^{n_2})= \sigma (a (\alpha^{n_1}-\alpha^{n_2})) = b(\beta^{n_1}-\beta^{n_2})$. Taking absolute values and estimating we get
\begin{align*}
0.15 \cdot 1.83^{n_1}
&\leq 0.33 \alpha^{n_1} (1-\alpha^{-1})
\leq \left| a (\alpha^{n_1}-\alpha^{n_2}) \right|
= \left| b (\beta^{n_1}-\beta^{n_2}) \right|\\
&\leq 0.26 (|\beta|^{n_1} + |\beta|^{n_2})
\leq 0.26 (0.74^{n_1} + 0.74^{n_2}),
\end{align*}
which is impossible for $n_1>150$.
Now we apply Proposition~\ref{prop:Matveev} just like in Step \ref{step:Step1}, except that now $b_1=n_2$, $b_2=-m_2$ and $\eta_3=\frac{a(\alpha^{n_1-n_2}-1)}{b^{m_1-m_2}-1}$. In order to find an $A_3$ we estimate the height of $\eta_3$:
\begin{align*}
h\left( \frac{a(\alpha^{n_1-n_2}-1)}{b^{m_1-m_2}-1} \right)
&\leq h(a) + h(\alpha^{n_1-n_2}-1) + h(b^{m_1-m_2}-1)\\
&\leq \frac{1}{3} \log 44 + (n_1-n_2)h(\alpha) + \log 2 + \log (b^{m_1-m_2}-1)\\
&\leq 1.96 + (n_1-n_2) \frac{\log \alpha}{3} + (m_1-m_2)\log b\\
&\leq 1.96 + (n_1-n_2) \frac{\log \alpha}{3} + (n_1-n_2)\log \alpha + 2.1\\
&\leq 4.9 (n_1-n_2),
\end{align*}
where we used \eqref{eq:trib:nlogalphamlogb} to estimate $(m_1-m_2)\log b \leq (n_1-n_2)\log \alpha + 2.1$.
Thus we can set $A_3= 14.7 (n_1-n_2) \geq \max(D h(\eta_3), \abs{\log \eta_3}, 0.16)$ and we obtain analogously to the application of Proposition~\ref{prop:Matveev} in Step~\ref{step:Step1}
\[
\log|\Lambda_2|
\geq - C_{101} \cdot \log n_1 \cdot \log b \cdot (n_1-n_2),
\]
where
\[
C_{101} = 1.1 \cdot 10^{14}
> 1.4 \cdot 30^6 \cdot 3^{4.5} \cdot 3^2 (1+ \log 3) \cdot 1.2 \cdot 0.7 \cdot 3 \cdot 14.7.
\]
Together with \eqref{eq:trib:Lambda2} this implies
\[
m_1 \log b
\leq C_{101} \cdot \log n_1 \cdot \log b \cdot (n_1-n_2).
\]
Using \eqref{eq:trib:nlogalphamlogb} and \eqref{eq:trib:boundn1-n2} from Step~\ref{step:Step1}, we obtain
\begin{multline*}
n_1 \log \alpha
\leq m_1 \log b + 2.1
\leq C_{101} \cdot \log n_1 \cdot \log b \cdot (n_1-n_2)\\
\leq C_{101} \cdot \log n_1 \cdot \log b \cdot (\log \alpha)^{-1} \cdot C_{100}\cdot \log n_1 \cdot \log b,
\end{multline*}
where we omitted the constant $ 2.1 $ because $C_{101}$ was estimated roughly.
Thus, we end up with
\begin{equation}\label{eq:trib:boundStep2}
n_1 \leq C_{102} \cdot (\log n_1 \cdot \log b)^2,
\end{equation}
where
\[
C_{102} = 7.8 \cdot 10^{27} > C_{100}\cdot C_{101} \cdot (\log \alpha)^{-2
}.
\]
\end{mystep}
\begin{mystep}\label{step:Step3}
Recall the bound \eqref{eq:trib:Lambda1}:
\[
|\Lambda_1|
=\left| \frac{a\alpha^{n_1}}{b^{m_1}}-1 \right|
\leq 7.7 \alpha^{-(n_1-n_2)}.
\]
Assume for a moment that $|\Lambda_1|\geq 0.5$. Then $n_1-n_2 \leq 4$, we get that $\log b \leq m_1 \log b - m_2 \log b \leq (n_1-n_2)\log \alpha + 2.1 \leq 4.6$ and we can immediately skip to Step~\ref{step:Step6}.
Therefore, we may assume that $|\Lambda_1|< 0.5$. Since $\abs{\log x} \leq 2 |x-1|$ for $|x-1|<0.5$, we obtain
\[
|\Lambda_1'|
:= \abs{\log a + n_1 \log \alpha - m_1 \log b}
\leq 15.4 \alpha^{-(n_1-n_2)}.
\]
Now we consider the second and the third solution and obtain from \eqref{eq:trib:equationL2}
\[
|a \alpha^{n_2}-b^{m_2}|
=|a\alpha^{n_3}-b^{m_3}+L(0.91\cdot 0.74^{n_3})|
\leq \max(a \alpha^{n_3}, b^{m_3}).
\]
Dividing by $b^{m_2} \geq 0.13 \alpha^{n_2}$ (by \eqref{eq:trib:balpha})
we obtain
\[
|\Lambda_{12}|
:=\left| \frac{a\alpha^{n_2}}{b^{m_2}}-1 \right|
\leq \max\left( \frac{a \alpha^{n_3}}{0.13 \alpha^{n_2}}, \frac{b^{m_3}}{b^{m_2}}\right)
\leq \max(2.6 \alpha^{-(n_2-n_3)}, b^{-(m_2-m_3)}).
\]
Assume for a moment that $|\Lambda_{12}|\geq 0.5$. Then we either get $n_2-n_3\leq 2$ or $b=2$. In the first case we can immediately skip to the next step. In the second case we can immediately skip to Step~\ref{step:Step6}. Thus we may assume $|\Lambda_{12}|<0.5$ and we get that
\[
|\Lambda_{12}'|
:= \abs{\log a + n_2 \log \alpha - m_2 \log b}
\leq \max(5.2 \alpha^{-(n_2-n_3)}, 2b^{-(m_2-m_3)}).
\]
Thus for the linear form $\Lambda_3':= m_2 \Lambda_1' - m_1 \Lambda_{12}'$ we get the upper bound
\begin{align}\label{eq:trib:Lambda3}
|\Lambda_{3}'|
&=|(n_1m_2 - n_2m_1)\log \alpha + (m_2-m_1)\log a| \nonumber\\
&\leq m_2 \cdot 15.4 \alpha^{-(n_1-n_2)} +
m_1 \cdot \max(5.2 \alpha^{-(n_2-n_3)}, 2b^{-(m_2-m_3)}) \nonumber\\
&\leq 20.6 n_1 \max( \alpha^{-(n_1-n_2)}, \alpha^{-(n_2-n_3)},b^{-(m_2-m_3)} ).
\end{align}
Now we have a linear form in only two logarithms, so we can use Laurent's bound instead of Matveev's.
We set
\begin{align*}
\eta_1 &= \alpha, \quad b_1 = n_1m_2 - n_2 m_1,\\
\eta_2 &= a, \quad b_2 = m_2-m_1,\\
D &= 3, \\
\log A_1 &= 1 = \max(3h(\alpha),\log \alpha,1),\\
\log A_2 &= 3.8 \geq \log 44 = 3h(a) = \max(Dh(a),\abs{\log a}, 1),\\
b' &= \frac{|n_1 m_2 - n_2 m_1|}{3.8} + \frac{|m_2-m_1|}{1}
\leq {n_1^2}/{3.7}.
\end{align*}
We estimate the factor
\[
\max( \log b' + 0.38, 18/D, 1)
\leq \max (2 \log n_1 - \log 3.7 + 0.38, 6, 1)
\leq 2 \log n_1.
\]
In order to apply Proposition \ref{prop:Laurent}, we need to check if $a$ and $\alpha$ are multiplicatively independent, and if $b_1$ and $b_2$ are nonzero.
We have already checked at the beginning of this section that $a$ and $\alpha$ are multiplicatively independent and we are assuming that $n_1>n_2$, which implies $m_1>m_2$, so $b_2=m_2-m_1$ is nonzero. Assume for a moment that $b_1=0$. Then we have
\begin{align*}
1
<\abs{\log a}
\leq |b_2 \log a|
=|\Lambda_3'|,
\end{align*}
so
\[
\log |\Lambda_3'|
>0,
\]
which is much better than what we will obtain from the application of Proposition~\ref{prop:Laurent}. So let us now apply Proposition \ref{prop:Laurent}. We obtain
\[
\log |\Lambda_{3}'|
\geq -C_{103} (\log n_1)^2,
\]
where
\[
C_{103}
= 2778
> 20.3 \cdot 3^2 \cdot 2^2 \cdot 1 \cdot 3.8.
\]
Together with \eqref{eq:trib:Lambda3} this implies
\begin{multline*}
- 2778 \cdot (\log n_1)^2 \\
\leq \log 20.6 + \log n_1 - \min((n_1-n_2)\log \alpha, (n_2-n_3)\log \alpha, (m_2-m_3)\log b)
\end{multline*}
and we get
\[
\min((n_1-n_2)\log \alpha, (n_2-n_3)\log \alpha, (m_2-m_3)\log b)
\leq 2780 \cdot (\log n_1)^2.
\]
If the minimum is realised by $(n_1-n_2)\log \alpha$, then we can immediately skip to Step~\ref{step:Step5}. If it is realised by $(n_2-n_3)\log \alpha$, then we have
\begin{equation}\label{eq:trib:boundn2-n3}
n_2-n_3 \leq 4563 \cdot (\log n_1)^2
\end{equation}
and go to the next step. If it is realised by $(m_2-m_3)\log b$, we get that $\log b \leq (m_2-m_3)\log b \leq 2780 \cdot (\log n_1)^2$ and we can skip to Step~\ref{step:Step6}.
\end{mystep}
\begin{mystep}\label{step:Step4}
First, recall from the previous step the bound
\[
|\Lambda_1'|
= \abs{\log a + n_1 \log \alpha - m_1 \log b}
\leq 15.4 \alpha^{-(n_1-n_2)}.
\]
Second, we generate a new linear form in logarithms by starting from \eqref{eq:trib:equationL2}:
\[
|a \alpha^{n_3} (\alpha^{n_2-n_3} - 1) - b^{m_2}|
= |b^{m_3} + L(0.91\cdot 0.74^{n_3})|
\leq b^{m_3} + 0.7.
\]
Dividing by $b^{m_2}$ we obtain
\[
|\Lambda_{41}|
:=\abs{\frac{a \alpha^{n_3} (\alpha^{n_2-n_3} - 1)}{b^{m_2}} - 1}
\leq b^{-(m_2-m_3)} + 0.7 b^{-m_2}
\leq 1.7 b^{-(m_2-m_3)}.
\]
If $|\Lambda_{41}|\geq 0.5$, then we immediately get $b\leq 3$ and we can skip to Step~\ref{step:Step6}. Let us assume $|\Lambda_{41}|< 0.5$. Then we obtain
\[
|\Lambda_{41}'|
:= \abs{\log a + n_3 \log \alpha + \log (\alpha^{n_2-n_3}-1) - m_2 \log b}
\leq 3.4 b^{-(m_2-m_3)}.
\]
Hence we have for the linear form $\Lambda_4':= m_2 \Lambda_1' -m_1 \Lambda_{41}'$ the upper bound
\begin{align}\label{eq:trib:Lambda4}
|\Lambda_4'|
&= |(m_2-m_1)\log a + (m_2n_1 - m_1n_3) \log \alpha - m_1 \log(\alpha^{n_2-n_3}-1)| \nonumber\\
&\leq m_2 \cdot 15.4 \alpha^{-(n_1-n_2)} + m_1 \cdot 3.4 b^{-(m_2-m_3)} \nonumber\\
&\leq 18.8 \cdot n_1 \max( \alpha^{-(n_1-n_2)}, b^{-(m_2-m_3)}).
\end{align}
Now $\Lambda_4'\neq 0$ because the technical condition involving Equation \eqref{eq:techcond} is fulfilled, and we can apply Proposition \ref{prop:Matveev}. We set $t=3$, $K=\mathbb{Q}(a,\alpha)=\mathbb{Q}(\alpha)$, $D=3=[K:\mathbb{Q}]$, as well as
\begin{align*}
\eta_1&=a, \quad &
\eta_2&=\alpha, \quad &
\eta_3&=\alpha^{n_2-n_3}-1,\\
b_1 &=m_2-m_1, \quad &
b_2 &=m_2n_1 - m_1n_3, \quad &
b_3 &= - m_1.
\end{align*}
Further, we can put
\begin{align*}
A_1&=3.8 \geq \log 44= \max(Dh(a),\abs{\log a}, 0.16),\\
A_2&=0.7 \geq \log \alpha =\max(Dh(\alpha),\log \alpha, 0.16),\\
A_3&= 2.7 (n_2-n_3)
\geq 3 ( (n_2-n_3)h(\alpha)+ \log 2)
\geq \max(Dh(\eta_3),\abs{\log\eta_3}, 0.16),\\
B&= n_1^2.
\end{align*}
Thus we get the lower bound
\[
\log |\Lambda_4'|
\geq - C_{104} (n_2-n_3) \log n_1,
\]
where
\[
C_{104}
= 4.3 \cdot 10^{13}
> 1.4 \cdot 30^6 \cdot 3^{4.5} \cdot 3^2 \cdot (1+ \log 3) \cdot 2.2 \cdot 3.8 \cdot 0.7 \cdot 2.7.
\]
Note that we used the estimate $1+ \log B = 1+ \log (n_1^2) \leq 2.2 \log n_1$ for $n_1 > 150$.
Together with \eqref{eq:trib:Lambda4} this implies
\[
- C_{104} (n_2-n_3) \log n_1
\leq \log 18.8 + \log n_1 - \min ((n_1-n_2) \log \alpha, (m_2-m_3)\log b),
\]
from which we get, omitting small terms because $C_{104}$ was roughly estimated,
\[
\min ((n_1-n_2) \log \alpha, (m_2-m_3)\log b)
\leq C_{104} (n_2-n_3) \log n_1.
\]
If the minimum is realised by $(n_1-n_2) \log \alpha$, then using \eqref{eq:trib:boundn2-n3} from the previous step we obtain
\begin{align}\label{eq:trib:boundStep5}
n_1-n_2
&\leq (\log \alpha)^{-1} \cdot C_{104} (n_2-n_3) \log n_1 \nonumber\\
& \leq (\log \alpha)^{-1} \cdot C_{104} \cdot \log n_1 \cdot 4563 \cdot (\log n_1)^2 \nonumber\\
& \leq C_{105} (\log n_1)^3,
\end{align}
with
\[
C_{105}
= 3.3\cdot 10^{17}
> (\log \alpha)^{-1} \cdot C_{104} \cdot 4563.
\]
If the minimum is realised by $(m_2-m_3)\log b$, then in a similar way we obtain
\begin{align*}
\log b
\leq (m_2-m_3)\log b
\leq C_{104} \cdot \log n_1 \cdot 4563 \cdot (\log n_1)^2
\leq 2 \cdot 10^{17} \cdot (\log n_1)^3
\end{align*}
and we can skip to Step \ref{step:Step6}.
\end{mystep}
\begin{mystep}\label{step:Step5}
We now use \eqref{eq:trib:nlogalphamlogb} and \eqref{eq:trib:boundStep5} to obtain a bound for $\log b$:
\begin{multline*}
\log b
\leq (m_1-m_2)\log b
= m_1 \log b- m_2 \log b\\
\leq n_1 \log \alpha - n_2 \log \alpha + 2.1
= (n_1-n_2)\log \alpha + 2.1
\leq C_{105} (\log n_1)^3 \log \alpha,
\end{multline*}
where we omitted the constant $ 2.1 $ because $C_{105}$ came from a rough estimation.
Thus we have
\begin{equation}\label{eq:trib:boundStep6}
\log b
\leq C_{106} (\log n_1)^3,
\end{equation}
with
\[
C_{106}
= 2.1\cdot 10^{17}
> C_{105} \log \alpha.
\]
\end{mystep}
\begin{mystep}\label{step:Step6}
Finally, we combine \eqref{eq:trib:boundStep2} from Step \ref{step:Step2} with \eqref{eq:trib:boundStep6}:
\begin{align*}
n_1 \leq C_{102}\cdot (\log n_1 \cdot \log b)^2
&\leq C_{102}\cdot \left(\log n_1 \cdot C_{106} (\log n_1)^3\right)^2\\
&\leq C_{107} \cdot (\log n_1)^8,
\end{align*}
with
\[
C_{107}
= 3.5 \cdot 10^{62}
> C_{102} \cdot C_{106}^2 = 7.8 \cdot 10^{27} \cdot \left(2.1 \cdot 10^{17}\right)^2.
\]
Solving the inequality $n_1 \leq 3.5 \cdot 10^{62} (\log n_1)^8$ numerically yields
\[
n_1 \leq 5 \cdot 10^{80}.
\]
\end{mystep}
We have finally found an explicit upper bound for $n_1$. Next we want to reduce this bound. Since the bound for $b$ is extremely large, we cannot use the linear forms from Steps \ref{step:Step1} and \ref{step:Step2} for the reduction process. Instead, we will do four Reduction Steps A--D corresponding to the Steps 3--6 and reduce the bounds as far as we can.
\begin{redstep}[Step \ref{step:Step3}]\label{step:RedstepA}
Recall from \eqref{eq:trib:Lambda3} that
\begin{multline*}
|\Lambda_3'|
=|(n_1m_2 - n_2m_1)\log \alpha - (m_1-m_2)\log a|\\
\leq 20.6 n_1 \max( \alpha^{-(n_1-n_2)}, \alpha^{-(n_2-n_3)},b^{-(m_2-m_3)} ).
\end{multline*}
Note that $m_1-m_2 < m_1 \leq n_1 \leq 5 \cdot 10^{80}$.
We compute the continued fraction expansion of $\log a/ \log \alpha$ and find the first convergent $p/q$ such that $q\geq 5 \cdot 10^{80}$.
Then by the best approximation property of continued fractions it turns out that
\[
4.4 \cdot 10^{-82}
\leq |p \log \alpha - q \log a|
\leq |\Lambda_3'|.
\]
This implies
\[
\max( \alpha^{-(n_1-n_2)}, \alpha^{-(n_2-n_3)},b^{-(m_2-m_3)} )
\geq 4.4 \cdot 10^{-82} \cdot \frac{1}{20.6 n_1}
\geq 4.2 \cdot 10^{-164}.
\]
\textit{Case 1:} $\max( \alpha^{-(n_1-n_2)}, \alpha^{-(n_2-n_3)},b^{-(m_2-m_3)} )=\alpha^{-(n_1-n_2)}$. Then we get that
\[
n_1 - n_2
\leq - \log (4.2 \cdot 10^{-164}) /\log \alpha
< 618
\]
and we can skip to Reduction Step \ref{step:RedstepC}.
\textit{Case 2:} $\max( \alpha^{-(n_1-n_2)}, \alpha^{-(n_2-n_3)},b^{-(m_2-m_3)} )=\alpha^{-(n_2-n_3)}$. Then we get
\[
n_2 - n_3
< 618
\]
and go to the next step.
\textit{Case 3:} $\max( \alpha^{-(n_1-n_2)}, \alpha^{-(n_2-n_3)},b^{-(m_2-m_3)} )=b^{-(m_2-m_3)}$. Then we get
\[
\log b
\leq (m_2-m_3)\log b
\leq -\log(4.2 \cdot 10^{-164})
\leq 377
\]
and we can skip to Reduction Step \ref{step:RedstepD}.
\end{redstep}
\begin{redstep}[Step \ref{step:Step4}]\label{step:RedstepB}
Recall from \eqref{eq:trib:Lambda4} that
\begin{multline*}
|\Lambda_4'|
= |(m_2-m_1)\log a + (m_2n_1 - m_1n_3) \log \alpha - m_1 \log(\alpha^{n_2-n_3}-1)|\\
\leq 18.8 n_1 \max( \alpha^{-(n_1-n_2)}, b^{-(m_2-m_3)})
\end{multline*}
and all coefficients are bounded by $n_1^2\leq (5 \cdot 10^{80})^2\leq 2.5\cdot 10^{161}=:M$.
Now for each $n_2-n_3\in \{1,2,\ldots, 617\}$ we apply the LLL-algorithm to find an absolute lower bound for $|\Lambda_4'|$ as described in Lemma \ref{lem:LLL}.
To obtain the matrices $B$ and $B^*$ we use the matrix attributes \verb|LLL()| and \verb|gram_schmidt()| in Sage \cite{sagemath}.
In each case, we try $C\approx M^3$ and if the algorithm fails (i.e.\ $c^2\leq T^2+S$), we increase $C$ by a factor of 10.
Indeed, in each of the cases the LLL reduction works after at most three tries and we get a lower bound for $|\Lambda_4'|$ in that case.
As an overall lower bound we obtain
\[
3.7 \cdot 10^{-326}
\leq |\Lambda_4'|.
\]
This implies
\[
\max( \alpha^{-(n_1-n_2)}, b^{-(m_2-m_3)})
\geq 3.7 \cdot 10^{-326} \cdot \frac{1}{18.8 n_1}
\geq 3.9 \cdot 10^{-408}.
\]
\textit{Case 1:} $\max(\alpha^{-(n_1-n_2)}, b^{-(m_2-m_3)}) = \alpha^{n_1-n_2}$: Then
\[
n_1-n_2
\leq -\log (3.9 \cdot 10^{-408})/\log \alpha
< 1540
\]
and we go to the next step.
\textit{Case 2:} $\max(\alpha^{-(n_1-n_2)}, b^{-(m_2-m_3)}) = b^{-(m_2-m_3)}$: Then
\[
\log b
\leq (m_2-m_3)\log b
\leq - \log (3.9 \cdot 10^{-408})
\leq 939
\]
and we can skip to Reduction Step \ref{step:RedstepD}.
\end{redstep}
\begin{redstep}[Step \ref{step:Step5}]\label{step:RedstepC}
Now we can compute a small bound for $\log b$ like in Step \ref{step:Step5}:
\[
\log b
\leq (n_1-n_2)\log \alpha + 2.1
\leq 1539 \cdot \log \alpha + 2.1
\leq 940.
\]
\end{redstep}
\begin{redstep}[Step \ref{step:Step6}]\label{step:RedstepD}
From \eqref{eq:trib:boundStep2} we get
\[
n_1
\leq 7.8 \cdot 10^{27} (\log b)^2 (\log n_1)^2
\leq 7.8 \cdot 10^{27} \cdot 940^2 (\log n_1)^2
\leq 6.9 \cdot 10^{33} \cdot (\log n_1)^2.
\]
Solving this inequality numerically, we obtain
\[
n_1 \leq 5.3\cdot 10^{37}.
\]
\end{redstep}
\step{step:repeating}{Repeating the reduction steps:}
With this smaller bound for $n_1$ we can now repeat the Reduction Steps A--D (see Table \ref{table:redSteps}). The Sage code is included in the \nameref{sec:appendix}.
\begin{table}[h]
\caption{Repeating the reduction steps}\label{table:redSteps}
\begin{tabular}{rcccc}
\hline
\multicolumn{1}{l}{} & \textbf{1\ts{st} round} & \textbf{2\ts{nd} round} & \textbf{3\ts{rd} round} & \textbf{4\ts{th} round} \\ \hline
$n_1\leq \ldots$ & $5 \cdot 10^{80}$ & $5.3\cdot 10^{37}$ & $1.2\cdot 10^{37}$ & $1.1\cdot 10^{37}$ \\ \hline
\multicolumn{1}{l}{\textbf{Step A}} & & & & \\
$n_1-n_2\leq \ldots$ & 617 & 292 & 288 & 288 \\
$n_2-n_3\leq \ldots$ & 617 & 292 & 288 & 288 \\
$\log b \leq \ldots$ & 377 & 179 & 176 & 176 \\ \hline
\multicolumn{1}{l}{\textbf{Step B}} & & & & \\
$n_1-n_2\leq \ldots$ & 1539 & 729 & 719 & 715 \\
$\log b \leq \ldots$ & 939 & 445 & 439 & 437 \\ \hline
\multicolumn{1}{l}{\textbf{Step C}} & & & & \\
$\log b \leq \ldots$ & 940 & 447 & 441 & 438 \\ \hline
\multicolumn{1}{l}{\textbf{Step D}} & & & & \\
$n_1\leq \ldots$ & $5.3\cdot 10^{37}$ & $1.2\cdot 10^{37}$ & $1.1\cdot 10^{37}$ & $1.1\cdot 10^{37}$ \\ \hline
\end{tabular}
\end{table}
After that, we are not able to reduce the bound significantly any further.
From the bounds in the table one can see that we have proven the bounds from Theorem~\ref{thm:Tribos}, i.e. overall we have proven that under the assumptions of Theorem~\ref{thm:Tribos} we have
\[
\log b \leq 438
\quad \text{and} \quad
150<n_1\leq 1.1\cdot 10^{37}.
\]
\section*{Appendix}\label{sec:appendix}
Below we have enclosed the Sage code that was used to determine the \hyperlink{step:smallSols}{small solutions} to $T_{n_1}-T_{n_2}=b^{m_1}-b^{m_2}$ in Section \ref{sec:Tribos}, as well as the code that was used to determine the bounds for the reduction rounds in Table \ref{table:redSteps}.
\begin{lstlisting}[language=Python]
# small solutions
# start computing Tribonacci numbers T_n1
t1_veryold = 0; t1_old = 0; t1 = 1 # starting values
n1 = 1
while n1 < 150:
# compute next Tribonacci number T_n1
temp = t1 + t1_old + t1_veryold
t1_veryold = t1_old
t1_old = t1
t1 = temp
n1 = n1 + 1
# start computing Tribonacci numbers T_n2 < T_n1
t2_veryold = 0; t2_old = 0; t2 = 1 # starting values
n2 = 1
while t2 < t1_old: # because we increase at the beginning
# compute next Tribonacci number T_n2
temp = t2 + t2_old + t2_veryold
t2_veryold = t2_old
t2_old = t2
t2 = temp
n2 = n2 + 1
# check representation
diff = t1 - t2
primefactorisation = list(factor(diff))
factors_for_b = list(Combinations(primefactorisation))
del factors_for_b[0] # exclude b = 1
for factors in factors_for_b:
x = gcd([k for [p,k] in factors])
b = prod(p^(k/x) for [p,k] in factors)
y = round(log(1 + diff/(b^x))/log(b))
if b^x * (b^y - 1) == diff:
m2 = x
m1 = x + y
c = t2 - b^x
print("T_{",n1,"} - T_{",n2,"}
=",b,"^{",m2,"} (",b,"^{",y,"}-1), \\quad
c&=", c, "= T_{",n1,"} -",b,"^{",m1,"}
= T_{",n2,"} -",b,"^{",m2,"}; \\\\")
\end{lstlisting}
\begin{lstlisting}[language=Python]
# reduction steps
alpha = n(solve(x^3 - x^2 - x - 1 == 0, x,
solution_dict=True)[2][x], digits=2000)
a = 1/(-alpha^2 + 4*alpha - 1)
#################################################################
print("Step A:")
bound_n1 = 5.3*10^37 # replace for each round
c = continued_fraction(log(a)/log(alpha))
i = 1
while c.denominator(i) < bound_n1:
i = i + 1
p = c.numerator(i)
q = c.denominator(i)
lowerbound_linform = abs(p*log(alpha) - q*log(a))
lowerbound = lowerbound_linform/(20.6*bound_n1)
n2n3max = floor(-log(lowerbound)/log(alpha))
print("bound n_1 - n_2 and n_2 - n_3:", n2n3max)
print("bound log(b):", -log(lowerbound))
#################################################################
print("Step B:")
M = bound_n1^2
C0 = 10^int(3*log(M)/log(10)) # approx. M^3 but with full precision
lowerBound = 1
for n2n3 in range(1, n2n3max+1):
# loop in order to find a C that works
C = C0
Cmax = C0*10^30
done = False
while not done:
C = C*10
A = Matrix([[1, 0, round(C*log(a))],
[0, 1, round(C*log(alpha))],
[0, 0, round(C*log(alpha^n2n3 - 1))]])
B = A.LLL()
Bstar, mu = B.gram_schmidt()
c = min([norm(N(b)) for b in Bstar])
S = 2*M^2
T = (1 + 3*M)/2
if c^2 > T^2 + S:
lowerBound = min(lowerBound, 1/C * (sqrt(c^2 - S) - T))
done = True
elif C == Cmax:
print('did not work for n2n3 =', n2n3)
break
lowerBound2 = lowerBound/(18.8*bound_n1)
n1n2max = floor(-log(lowerBound2)/log(alpha))
print("bound n_1 - n_2:", n1n2max)
print("bound log b:", -log(lowerBound2))
#################################################################
print("Step C:")
logbmax = n1n2max*log(alpha) + 2.1
print("bound log b:", logbmax.n())
#################################################################
print("Step D:")
logbmax = max(logbmax, -log(lowerBound2))
newbound_n1 = find_root(x - 7.8*10^27*logbmax^2 * log(x)^2,
7.8*10^27*logbmax^2, (7.8*10^27*logbmax^2)^2)
print("new bound for n_1:", newbound_n1)
\end{lstlisting}
\end{document} | math | 98,882 |
\begin{document}
\maketitle
\centerline{\scshape Bogdan-Vasile Matioc}
{\footnotesize
\centerline{University of Vienna}
\centerline{Nordbergstra\ss e 15}
\centerline{1090, Vienna, Austria}
}
\begin{abstract}
In this paper we present a characterization of the symmetric rotational periodic gravity water
waves of finite depth and without stagnation points in terms of the underlying flow.
Namely, we show that such a wave is symmetric and has a single crest and trough per period if and only if
there exists a vertical line within the fluid domain such that all
the fluid particles located on that line minimize there simultaneously their distance to the fluid bed
as they move about.
Our analysis uses the moving plane method, sharp elliptic maximum principles, and the principle of analytic continuation.
\end{abstract}
\section{Introduction}\label{S:1}
When the wind blows over a still water surface it generates waves which in time evolve into nice regular wave trains, that is, they
become two-dimensional periodic waves that are symmetric about the crest and trough lines and propagate at constant speed and
without change of shape.
The study of the a priori symmetry properties of gravity water waves goes back to Garabedian \cite{PG65}
who used a variational approach to show that Stokes waves for which each streamline has a unique maximum and minimum per period
-- located beneath the crest and trough, respectively -- are symmetric.
The proof of Garabedian was simplified later on by Toland \cite{TO00}.
Assuming merely that the free surface of the irrotational flow has a single crest per period Okamoto and Sh{\=o}ji \cite{OS01} have established
the symmetry of the wave, improving thus upon the previous symmetry results.
The approach presented in \cite{OS01} is based on maximum principles and the moving planes method, cf. \cite{BN88, JS71}, which have been applied also with
success when studying the symmetry properties of solitary waves with and without vorticity, cf. \cite{CS88, H08, MM12}.
Concerning rotational flows, the characterization of the symmetric waves found in \cite{OS01} was extended
by Constantin and Escher to finite depth \cite{CE1} and deep water \cite{CE2} flows
without stagnation points and possessing a continuously differentiable vorticity function, under some restrictions on the
derivative of the vorticity function.
For finite depth waves without stagnation points, the latter restrictions were eliminate in \cite{CEW07}, the authors taking advantage in their
analysis of the formulation of the problem in terms of the height function.
Sufficient conditions which ensure the symmetry of stratified water waves can be found in \cite{Wa09}.
Concerning flows with stagnation points, it is shown in the constant vorticity case \cite{T12} that small-amplitude gravity waves with a single crest per period are
symmetric.
Still in the context of rotational water waves of finite depth without stagnation points and with a continuous vorticity function Hur \cite{H07} has shown that if:
$(i)$ the wave possesses a unique global minimum per period; $(ii)$ all the streamlines attain beneath the global trough their global minimum;
and $(iii)$ the wave profile is monotone near the global trough; then the wave is symmetric and has a single crest and trough per period.
The later result was improved in \cite{MM13} where the condition $(i)$ was dropped and just one sided monotonicity was required instead of $(iii)$.
In this paper we show that in fact both conditions $(i)$ and $(iii)$ of \cite{H07} can be omitted, and we are left with a characterization
of the symmetry of the free surface wave in terms of the underlying flow.
This is intriguing because the wave pattern observed at the water surface and the underlying flow are two different aspects of the water
motion which are coupled in an extremely intricate way.
Let us emphasize that the only known explicit solution of the water wave problem is due to Gerstner \cite{Ger} (see also \cite{C4, H4}):
it is a symmetric deep water wave with only one crest per period and with all the particles located beneath a trough
attaining there their maximal distance to a fixed reference plane above the fluid about.
This suggests that our result could be extended to deep-water waves.
The characterization that we found is no longer valid if we allow for stagnation points: there are symmetric waves which present a Kelvin cat's eye pattern and for which
there is no vertical line within the fluid domain with the property that all
the fluid particles located on that line minimize there their distance to the fluid bed, cf. \cite{CV11, W09}.
The outline of the paper is as follows: in Section \ref{S:2} we present the physical setting that we consider and three equivalent mathematical formulations.
At the end of the section we present our main result Theorem \ref{T:MT}, which is proven in Section \ref{S:3}.
\section{The main result}\label{S:2}
We consider a Cartesian coordinate system with the $x-$axis being the direction of wave propagation, the $y$-axis pointing vertically upwards, and
we assume that the water
flow is independent of the $z$-coordinate.
From a reference frame which moves in the same direction with the wave and with the wave speed $c$, the flow beneath the steady wave is described by
the steady Euler equations
\begin{subequations}\label{eq:P}
\begin{equation}\label{eq:Euler}
\left\{
\begin{array}{rllll}
({u}-c) { u}_x+{ v}{ u}_y&=&-{ P}_x,\\
({ u}-c) { v}_x+{ v}{ v}_y&=&-{ P}_y-g,\\
{ u}_x+{v}_y&=&0
\end{array}
\right.\qquad \text{in $\Omega_\eta.$}
\end{equation}
We have taken the fluid to be inviscid and its density equal to 1.
Moreover, the free surface of the wave is assumed to be the graph $y=\eta(x) $ and that the flat fluid bed is located at $y=-d$, meaning that
the two-dimensional fluid domain $\Omega_\eta $ is
\[
\Omega_\eta:=\{(x,y)\,:\,\text{$ x\in{\mathbb R} $,\, $-d<y<\eta(x)$}\}.
\]
Furthermore, the positive constant $d$ is the average mean depth of the flow, property which implies that $\eta$ has integral mean equal to zero.
In order to complete the mathematical system describing the motion of gravity water waves, we impose the standard boundary conditions
\begin{equation}\label{eq:BC}
\left\{
\begin{array}{rllll}
P&=&{P}_0&\text{on $ y=\eta(x)$},\\
v&=&( u-c) \eta'&\text{on $ y=\eta(x)$},\\
v&=&0 &\text{on $ y=-d$},
\end{array}
\right.
\end{equation}
cf. \cite{ConBook, John, Kins}.
Hereby, $ P_0$ denotes the constant atmospheric pressure.
We shall restrict our considerations to flows for which the horizontal
velocity of each water particle is less than
the wave speed
\begin{equation}\label{SC}
u<c\qquad\text{in $\overline \Omega_\eta$.}
\end{equation}
This excludes the presence of stagnation points and enables us to use equivalent formulations of the hydrodynamical problem and recent regularity properties of such flows
\cite{CE3, EM13}.
Finally, the vorticity of the two-dimensional flow described by the system \eqref{eq:P} is identified with the scalar function
\begin{equation*}
\omega:= { u}_y-{ v}_x\qquad\text{in $\overline\Omega_\eta$.}
\end{equation*}
\end{subequations}
The solutions considered in this paper correspond to periodic water waves--meaning that $\eta, u,v,P$ are periodic in $x$ and have
the same period--and possess the following
regularity
\begin{equation}\label{Reg}
\eta\in C^2_{\rm per}({\mathbb R}), \qquad u,v,P\in C^1_{\rm per}(\overline\Omega_\eta).
\end{equation}
The conservation of mass equation allows us to introduce the stream function $\partialsi\in C^2_{\rm per}(\overline\Omega_\eta)$ via the relations
$\nabla \partialsi=(-v,u-c)$ in $\overline\Omega_\eta$ and $\partialsi=0$ on $y=\eta(x)$.
Because of \eqref{SC}, one can show, cf. e.g. \cite{CS2}, that the dependence of the vorticity on its variables takes the form
$\omega(x,y)=\gamma(-\partialsi(x,y))$ for all $(x,y)\in\overline\Omega_\eta.$
Hereby, $\gamma\in C([p_0,0])$ is the vorticity function and $p_0<0$ is the relative mass flux.
With these observations, the formulation \eqref{eq:P} can be recast as the following problem
\begin{subequations}\label{eq:SP}
\begin{equation}\label{eq:psi}
\left\{
\begin{array}{rllll}
\Delta \partialsi&=&\gamma(-\partialsi)&\text{in}&\Omega_\eta,\\
\displaystyle|\nabla\partialsi|^2+2g(y+d)&=&Q&\text{on} &y=\eta(x),\\
\partialsi&=&0&\text{on}&y=\eta(x),\\
\partialsi&=&-p_0&\text{on} &y=-d,
\end{array}
\right.
\end{equation}
and
\begin{equation}\label{SC1}
\partialsi_y<0\qquad\text{in $\overline \Omega_\eta$.}
\end{equation}
\end{subequations}
The real constant $Q $ is related to the total energy of the fluid.
The equivalence of the velocity formulation \eqref{eq:P} and of the stream function formulation \eqref{eq:SP} in the setting of classical solutions
has been established in \cite{ConBook}.
The formulation \eqref{eq:SP} is useful because it reduces the number of unknowns, but it also helps us to identify some properties of the underlying flow.
Particularly, because the time is eliminated from the problem the particle paths in the moving frame, that is the solutions of the ODE system
\[
\left\{
\begin{array}{lll}
x'&=u(x,y)-c,\\
y'&=v(x,y),
\end{array}
\right.
\]
are also the streamlines of the steady flow and they coincide with the level curves of $\partialsi$.
Moreover, in view of \eqref{SC1}, they can be parametrized as graphs of periodic $C^2$- functions and they foliate the entire fluid domain.
We emphasize that due to \eqref{SC} the semi-hodograph transformation
$\Phi:\overline\Omega_\eta\to\overline\Omega$ given by
\begin{equation}\label{semH}
\Phi(x,y):=(q,p)(x,y):=(x,-\partialsi(x,y))\qquad \text{for $(x,y)\in\overline\Omega_\eta$},
\end{equation}
where $\Omega:={\mathbb R}\times(p_0,0),$ is a $C^2$-diffeomorphism.
Introducing the height function $h\in C^2_{\rm per}(\overline \Omega)$ via the relation
\begin{equation}\label{hodo}
h(q,p):=y+d \qquad\text{for $(q,p)\in\overline\Omega$},
\end{equation}
we obtain a second equivalent formulation of \eqref{eq:P}, cf. \cite{ConBook},
\begin{subequations}\label{eq:HP}
\begin{equation}\label{PB}
\left\{
\begin{array}{rllll}
(1+h_q^2)h_{pp}-2h_ph_qh_{pq}+h_p^2h_{qq}-\gamma h_p^3&=&0&\text{in $\Omega$},\\
\displaystyle 1+h_q^2+(2gh-Q)h_p^2&=&0&\text{on $p=0$},\\
h&=&0&\text{on $ p=p_0,$}
\end{array}
\right.
\end{equation}
the condition \eqref{SC} being re-expressed as
\begin{equation}\label{PBC}
\min_{\overline \Omega}h_p>0.
\end{equation}
\end{subequations}
The latter formulation enables us to compare different solutions of \eqref{eq:P} and it also helps us to parametrize the
streamlines of the flow, as any streamline is the graph of one of the mappings $h(\cdot,p)-d\in C^2_{\rm per}({\mathbb R}),$ $p\in[p_0,0].$
Particularly, the flat bed corresponds to the choice $p=p_0$ and the free surface of the wave to $p=0.$
The main result of this paper is the following theorem.
\begin{thm} \label{T:MT}
Consider a periodic gravity water wave solution of \eqref{eq:P} satisfying \eqref{Reg} and having a non-flat free surface.
Moreover, assume that the vorticity function is Lipschitz continuous in $[p_1,0] $ for some $p_1\in(p_0,0). $
Then, the wave profile is symmetric and has only one crest and trough per period if and only if there exists a vertical line within the fluid domain such that all
the fluid particles located on that line minimize there their distance to the fluid bed.
\end{thm}
\begin{rem}
The Theorem \ref{T:MT} concerns waves whose profiles are symmetric with respect to the crest and to the trough lines, the wave profile being strictly monotone between consecutive crests and troughs.
In fact, the symmetry of the wave profile ensures that $u$ is symmetric and $v$ is anti-symmetric with respect to the crest and to the trough lines.
These claims can be easily deduced from properties derived in the proof of Lemma \ref{L:1}.
\end{rem}
\begin{rem}
The Lipschitz continuity of the vorticity function on $[p_1,0],$ for some $p_1\in(p_0,0) $ which can be chosen as close to $0$ as we want,
is needed only once in the proof of Lemma \ref{L:2} (the case $(D)$), and it seems to us that
it cannot be omitted.
\end{rem}
\section{Proof of the main result}\label{S:3}
Our approach uses the moving plane method: we reflect the wave surface with respect to vertical lines close to line on which the fluid particles attain their minimal distance to the bed and move the line of reflection
to the right until an extremal position is reached.
Then, we show that the limiting line is the unique crest line and the wave is symmetric with respect to it.
Besides using recent regularity results for the streamlines and the profile of gravity waves, cf. \cite{EM13}, in our analysis we employ
sharp maximum principles for elliptic partial differential equations.
For completeness, we state the following result.
\begin{lemma}[Serrin's corner point lemma]\label{L:S}
Let
\[R=\{(q,p)\in{\mathbb R}^2\,:\, a<q<b,\, p_0<p<f(q)\}
\] where $a<b$, $f \in C^2([a,b], (p_0,\infty))$ and $f'(a)=0$ [resp. $f'(b)=0$].
Let further $H\in C^2(\overline R)$ satisfy $\mathcal{L}H\geq 0$ in $R$ for a uniformly elliptic operator
$\mathcal{L}=a_{ij}\partial_{ij}+b_i\partial_i$ with continuous coefficients in $\overline R.$
Additionally, assume that there exists a positive constant $K$ such that
\begin{equation}\label{SL}
|a_{12}(q,p)|\leq K (q-a) \qquad\text{[resp. $|a_{12}(q,p)|\leq K (b-q)$]}
\end{equation}
for all $(q,p)\in R$.
If the corner point $P=(a,f(a))$ [resp. $P=(b,f(b))$] satisfies $H(P)=0$ and if $H<0$ in $R$, then either
\[
\frac{\partial w}{\partial s}<0\qquad\text{or}\qquad \frac{\partial^2 w}{\partial s^2}<0 \qquad\text{at $P$,}
\]
where $s\in R^2$ is any direction at $P$ that enters $R$ non-tangentially.
\end{lemma}
\begin{proof} See \cite[Lemma 2]{JS71}.
\end{proof}
In order to prove the main result, let us first observe that the assumption
that all the fluid particles located on a vertical line minimize there their distance to the fluid bed is equivalent
to saying that all the mappings $h(\cdot,p), p\in[p_0,0],$ attain their global minimum on the same vertical line.
Whence, Theorem \ref{T:MT} is an improvement upon \cite[Theorem 3.1]{MM13} where an additional monotonicity assumption on the wave profile close to the
global trough was made.
\begin{lemma}[The necessity] \label{L:1} Assume that $(\eta,u,v,P)$ is a solution of \eqref{eq:P} satisfying \eqref{Reg}.
If $\eta$ is symmetric and has only one crest and trough per period, then all the particles located on the trough line minimize there their distance to the fluid bed.
\end{lemma}
\begin{proof} Let $T>0$ be the minimal period of the wave.
Without restricting the generality, we may assume that $x=0 $ is the trough line, so that the crest line is the vertical line $x=T/2.$
Because the particle paths coincide with the streamlines, it suffices to show that the map $h(\cdot,p)$ is symmetric with respect to $q=T/2$ and strictly increasing on $[0,T/2]$ for each $p\in[p_0,0].$
The assumption $\eta=\eta(T-\cdot)$ implies that $h(\cdot,0)=h(T-\cdot,0).$
Taking into account that $\widetilde h:=h(T-\cdot,\cdot)$ is also a solution of \eqref{eq:HP},
we find that $H=\widetilde h-h\in C^2_{\rm per}(\overline\Omega)$ satisfies $H=0$ on $\partial\Omega$ and solves the elliptic equation $\mathcal{L}H=0$ in $\overline\Omega$, whereby
\begin{equation}\label{EO}
\mathcal{L}H:=(1+\widetilde h_q^2)H_{pp}-2\widetilde h_p\widetilde h_qH_{pq}+\widetilde h_p^2H_{qq}+a_1H_q+a_2H_p
\end{equation}
and
\begin{equation}\label{coeff}
\begin{aligned}
a_1&:=(h_q+\widetilde h_q) h_{pp}-2h_p h_{pq},\\
a_2&:=(h_p+\widetilde h_p) h_{qq}-2\widetilde h_q h_{pq}-\gamma(h_p^2+h_p\widetilde h_p+\widetilde h^2_p).
\end{aligned}
\end{equation}
The weak elliptic maximum principle ensures that $h =h(T-\cdot,\cdot) $ in $\overline\Omega,$ so that all the streamlines are symmetric with respect to the crest line $x=T/2.$
Consequently, $h_q(0,p)=h_q(T/2,p)=0$ for every $p\in[p_0,0].$
Altogether, we find that $h_q\geq0$ on $\partial R,$ whereby $R:=(0,T/2)\times (p_0,0).$
Our final goal is to use elliptic maximum principles and to show that $h_q>0$ in $R$.
Therefore, we infer from \cite[Proposition 2.1]{EM13} (see also \cite{CS1}), that the distributional derivatives
$\partial_q^m h\in C^1_{\rm per}(\overline\Omega)$ for all integers $m\geq1$, with $w:=h_q$ being the weak solution of the elliptic equation
\begin{equation}\label{EL}
\left(\frac{1}{h_p}\partial_q w\right)_q-\left(\frac{h_q}{h_p^2}\partial_p w\right)_q-\left(\frac{h_q}{h_p^2}\partial_q w\right)_p+
\left(\frac{1+h^2_q}{h_p^3}\partial_p w\right)_p=0\qquad \text{in $\Omega$.}
\end{equation}
But then, it is not difficult to see these properties together with $h\in C^2_{\rm per}(\overline\Omega)$ yield additional regularity $h_q\in C^2_{\rm per}(\overline\Omega),$
meaning that $h_q$ is a classical solution of \eqref{EL}.
The weak and strong elliptic maximum principles, cf. e.g. \cite{GT01}, ensure then that the derivative $h_q$ is strictly positive in $R$,
which is the desired claim.
\end{proof}
We are left to prove the sufficiency claim.
\begin{lemma}[The sufficiency]\label{L:2}
Consider a periodic gravity water wave solution $(\eta,u,v,P)$ of \eqref{eq:P} satisfying \eqref{Reg} and having a non-flat surface.
Moreover, assume that the vorticity function is Lipschitz continuous in $[p_1,0] $, whereby $p_1\in(p_0,0). $
If there exists a vertical line within the fluid domain such that all
the fluid particles located on that line minimize there their distance to the fluid bed, then the wave
has a single crest and trough per period and is symmetric with respect to the crest line.
\end{lemma}
\begin{proof} Without loss of generality, we may assume that $q=0$ is the point where all the mappings $h(\cdot,p), p\in[p_0,0],$ attain their global minimum.
Because $h\in C ^2_{\rm per}(\overline\Omega)$ is a classical solution of \eqref{eq:HP}, it can be also interpreted as a weak solution of this problem,
that is $h$ satisfies the first equation of \eqref{PB} in the following weak sense
\begin{equation}\label{PB1}
\int_\Omega\frac{h_q}{h_p}\varphi_q-\left(\Gamma+\frac{1+h_q^2}{2h_p^2}\right)\varphi_p\, d(q,p)=0\qquad\text{for all $\varphi\in C^1_0(\Omega)$},
\end{equation}
cf. \cite{CS1, EM13}.
Hereby, $\Gamma\in C^1([p_0,0])$ is the anti-derivative of $\gamma$ satisfying $\Gamma(0)=0$, and $ C^1_0(\Omega)$ is
the space containing continuously differentiable functions with compact support in $\Omega.$
Whence, we find all the assumptions of \cite[Theorem 3.1]{EM13} satisfied, fact which ensures us that the functions $h(\cdot,p)$ are real-analytic for all $p\in[p_0,0].$
Since $\eta=h(\cdot,0)-d,$ the principle of analytic continuation implies that there must exist an $\varepsilon>0$ with $\eta'>0$ on $(0,\varepsilon).$
Therefore, for each $\lambda\in(0,\varepsilon/2],$ the reflection $\Omega_\lambda^r$ of
\[\Omega_\lambda:=\{(x,y)\,:\, \text{$0<x<\lambda$,\, $-d<y<\eta(x)$}\}\]
with respect to $x=\lambda$ is a subset of $\Omega_\eta.$
Let $\Lambda:=\max\{\lambda>0\,:\, \Omega_\lambda ^r\subset \Omega_\eta\}.$
Then, one of the following two cases occurs: $(1)$ $\Lambda=T/2;$ $(2)$ the reflection of free wave surface with respect to $x=\Lambda$ intersects the wave surface tangentially
(say at $(x_0,\eta(x_0))$ with $x_0\in[\Lambda, 2\Lambda]$).
We define now $H:\overline R\to{\mathbb R}$ by setting $H(q,p)=h(2\Lambda-q,p)-h(q,p)$ for all $(q,p)$ in the rectangle $R:=(\Lambda, 2\Lambda)\times (p_0,0),$ and prove
in a first step that $H\equiv 0$.
Indeed, our assumption ensures that $H\leq 0$ on $q=2\Lambda.$
Moreover, with our choice of $\Lambda$ we also have that $H\leq 0 $ on $p=0.$
The last equation of \eqref{PB} and the definition of $H$ combine now to give $H\leq0$ on $\partial R.$
Moreover, as $h(2\Lambda-\cdot,\cdot)$ is also a solution of \eqref{eq:HP}, we find similarly as in the proof of Lemma \ref{L:1}
that $\mathcal{L}H=0$ in $\overline R$, whereby $\mathcal{L}$ is the uniformly elliptic operator \eqref{EO}, with the modification that the
coefficients of $\mathcal{L}$ are as in \eqref{EO} and \eqref{coeff}, but with $\widetilde h:=h(2\Lambda-\cdot,\cdot).$
Let as assume by contradiction that $H\not\equiv0.$
There are several cases to be considered separately.
{$(A)$} \, Assume first that the case $(1)$ occurs.
Letting $P:=(2\Lambda,0)$, the weak and strong maximum principles yield that $H(P)=0=\max_R H$ and $H<H(P)$ in $R$.
Since $\widetilde h_q(2\Lambda,p)=-h_q(0,p)=0$ for any $p\in[p_0,0],$ there exists a positive constant $K$ with
\begin{align*}
| (\widetilde h_q\widetilde h_p)(q,p) |\leq K|\widetilde h_q(q,p)- \widetilde h_q (2\Lambda,p)|\leq K(2\Lambda-q)
\end{align*}
for all $(q,p)\in R$.
We are thus in the position of applying Lemma \ref{L:S} at the corner $P$.
On the other hand, since $\Lambda=T/2,$ the periodicity of $H$ implies that $H_q=H_p=H_{qq}=H_{pp}=0$ at $P$.
Moreover, since $h_q(0,p)=0$ for all $p\in[p_0,0],$ we additionally have that $h_{pq}(P)=0,$ and therefore $H_{pq}(P)=-2h_{pq}(P)=0.$
This is in contradiction with Lemma \ref{L:S}, meaning that $H\equiv 0$ in this case.
Let us assume in the following that $\Lambda<T/2$ and that only $(2)$ occurs.
This means that at the point $(x_0,\eta(x_0))$ where the wave surface intersects its reflection across $x=\Lambda$ we have
\begin{equation}\label{refl}
\begin{aligned}
&h(x_0,0)=h(2\Lambda-x_0,0),\\
&h_q(x_0,0)=-h_q(2\Lambda-x_0,0), \\
&h_p(x_0,0)=h_p(2\Lambda-x_0,0),
\end{aligned}
\end{equation}
the last relation following from the previous two and the second equation of \eqref{PB}.
We distinguish the following sub-cases.
{$(B)$} \, Assume now that we are in the case $(2)$ and that $x_0=2\Lambda$.
Again, we choose $P:=(2\Lambda,0).$
Then, since $H\not\equiv 0,$ we obtain that $H(P)=0 $ and $H<H(P)$ in $R$.
In order to use Serrin's corner point lemma, the same arguments as in the previous case show
\begin{align*}
| \widetilde h_q\widetilde h_p (q,p)|\leq K(2\Lambda-q)
\end{align*}
for all $(q,p)\in R$, with $K$ being a positive constant.
Clearly, $H_q(P)=H_p(P)=0,$ so that Serrin's lemma ensures us that
\[
\frac{\partial^2 H}{\partial s^2}(P)<0
\]
for all directions $s$ at $P$ that enter $R$ non-tangentially.
Therefore, for the choice $s:=( -\widetilde h_p(P),-1),$ it must hold
\begin{equation}\label{PME}
\frac{\partial^2 H}{\partial s^2}(P)=H_{pp}+2\widetilde h_p H_{pq}+ \widetilde h_p^2 H_{qq}<0 \qquad\text{at $P$}.
\end{equation}
On the other hand, if we differentiate the second relation of \eqref{PB} with respect to $q$, we find that
\begin{equation}\label{REL}
h_qh_{qq}+(2gh-Q)h_ph_{pq}+ gh_qh_p^2=0\qquad\text{on $p=0$},
\end{equation}
and therefore $h_{pq}(P)=h_{pq}(0,0)=0,$ as additionally to \eqref{refl} we know also that $h_{q}(P)=0$ in this case.
Hence, we get $H_{pq}(P)=-h_{pq}(0,0)-h_{pq}(P)=0$.
Recalling that $\mathcal{L}H=0$ in $\overline R,$ we have
\begin{equation*}
H_{pp}(P)+ \widetilde h_p^2(P) H_{qq}(P)=0,
\end{equation*}
and together with $H_{pq}(P)=0$ we obtained a contradiction to \eqref{PME}.
Therefore, $H\equiv 0$ also in this case.
{$(C)$} \, Let us now consider the case $(2)$ when $x_0\in(\Lambda, 2\Lambda)$.
We define $P:=(x_0,0)$ and notice that $H\not\equiv 0$ implies that $H(P)=0>H$ in $R.$
Using Hopf's lemma, it follows that $H_p(P)>0.$
However, a direct computation shows
\[
H_p(P)=h_p(2\Lambda-x_0,0)-h_p(x_0,0)=0,
\]
cf. \eqref{refl}, which is a contradiction.
Hence, we conclude that $H\equiv 0$ in $R$.
{$(D)$} \, We are finally left with the case $(2)$ when $x_0=\Lambda$.
In this situation we choose $P:=(\Lambda,0)$ and note that the weak and strong maximum principles and $H\not\equiv 0$ yield $H(P)=0>H$ in $R$, cf. \eqref{refl}.
Therefore, $h(2\Lambda-q,p)<h(q,p)$ for all $(q,p)\in R.$
We cannot apply Serrin's corner point lemma at this stage because the relation \eqref{SL} is not satisfied by the coefficient $a_{12}$ of $\mathcal{L}.$
Instead, we define $\Psi:\overline{ \Omega_\Lambda ^r}\to {\mathbb R}$ by setting
\[
\Psi(x,y)=\partialsi(2\Lambda-x,y)-\partialsi(x,y)\qquad\text{for $(x,y)\in\overline{ \Omega_\Lambda ^r}$}.
\]
Setting $\widetilde P:= (\Lambda, \eta(\Lambda)),$ it is obvious that $\Psi(\widetilde P)=0.$
We now claim that $\Psi<0$ in $\Omega_\Lambda ^r.$
Indeed, pick $(x,y)\in \Omega_\Lambda ^r$ and let $p_1, p_2\in(p_0,0)$ satisfy $p_1:=-\partialsi(2\Lambda-x,y)$ and $p_2:=-\partialsi(x,y),$ or equivalently
$y=h (2\Lambda-x,p_1)-d=h(x,p_2)-d,$ cf. \eqref{semH} and \eqref{hodo}.
But, since $(x,p_1)\in R,$ $h(x,p_1)>h(2\Lambda-x,p_1)=h(x,p_2)$, and in view of \eqref{PBC} we find that $p_1>p_2.$
Consequently, $\Psi(x,y)=p_2-p_1<0,$ which is exactly what we claimed.
We use now the Lipschitz continuity of $\gamma$ on $[p_1,0]$, with $p_1\in(p_0,0)$, to write
\begin{align*}
\Delta\Psi(x,y)&=\Delta \partialsi(2\Lambda-x,y)-\Delta \partialsi(x,y)=\gamma(-\partialsi(2\Lambda-x,y))-\gamma(-\partialsi(x,y))\\
&=c(x,y)\Psi(x,y)
\end{align*}
for all $(x,y)$ in $\overline{{\mathbb B}\cap\Omega_\Lambda ^r},$ whereby ${\mathbb B}$ is
a small ball centered at the crest $(\Lambda,\eta(\Lambda)).$
Hereby, $c:\overlineerline{{\mathbb B}\cap\Omega_\Lambda ^r}\to{\mathbb R}$ is the function
\[
c(x,y)=
\left\{
\begin{array}{llll}
\displaystyle\frac{\gamma(-\partialsi(2\Lambda-x,y))-\gamma(-\partialsi(x,y))}{\partialsi(2\Lambda-x,y)-\partialsi(x,y)}, & \text{if $\partialsi(x,y)\neq \partialsi(2\Lambda-x,y))$},\\
0, & \text{if $\partialsi(x,y)=\partialsi(2\Lambda-x,y))$,}
\end{array}
\right.
\]
and it is bounded if ${\mathbb B}$ is chosen sufficiently small.
Finally, we introduce the function $G: \overlineerline{{\mathbb B}\cap\Omega_\Lambda ^r}\to{\mathbb R}$ by $G(x,y):=e^{\beta x}\Psi(x,y) $, where $\beta$ is a positive constant.
We notice that $G(\widetilde P)=0> G$ in ${\mathbb B}\cap\Omega_\Lambda ^r$, and
\begin{equation}\label{SF}
\Delta G-2\beta G_x=(c-\beta^2) G\geq0\qquad\text{in $\overline{{\mathbb B}\cap\Omega_\Lambda ^r}$,}
\end{equation}
if we choose $\beta^2\geq\sup_{{\mathbb B}\cap\Omega_\Lambda ^r} c.$
Because of the special form of the elliptic operator in \eqref{SF}, we are in the position of applying Lemma \ref{L:S} and
deduce that at least one of the first or second order partial derivatives of $G$ does not vanish at $\widetilde P.$
To finish, let us observe from the definition of $\Psi$ that $\Psi=\Psi_y=\Psi_{xx}=\Psi_{yy}=0$ at $\widetilde P.$
Moreover, as $\partialsi(x,\eta(x))=0$ for all $x\in{\mathbb R},$ we also have that $\partialsi_x(\widetilde P)=0,$ and therewith $\Psi_x(\widetilde P)=0.$
Additionally, differentiating the second equation of \eqref{eq:psi} yields
\[
\partialsi_x\partialsi_{xx}+\eta'\partialsi_x\partialsi_{xy}+\partialsi_y\partialsi_{xy}+\eta'\partialsi_y\partialsi_{yy}+g\eta'=0\qquad \text{at $\widetilde P,$}
\]
and, since $\partialsi_y<0,$ we find that $\Psi_{xy}(\widetilde P)=-2\partialsi_{xy}(\widetilde P)=0.$
These relations imply that all first and second order partial derivatives of $G$ vanish at $\widetilde P,$ which is a contradiction.
Consequently, $H\equiv0$ in $R.$
We infer now from the analysis in $(A)$, $(B)$, $(C)$, and $(D)$ that $H\equiv 0$ in $R$, that is $h(2\Lambda-q,p)=h(q,p)$ for all $(q,p)\in R,$
with $(\Lambda,\eta(\Lambda))$ being the first local maximum of the wave surface.
We note that the wave profile being real-analytic is also strictly increasing on $[0,\Lambda].$
Let us show that $(\Lambda,\eta(\Lambda))$ this is the only maximum of $\eta$ within a minimal period.
Indeed, let $\widehat h:\overline\Omega\to{\mathbb R}$ be the $2\Lambda-$periodic extension of the restriction $h\big|{[0,2\Lambda]\times [p_0,0]}.$
By assumption, $h_q(0,p)=0$ for all $p\in[p_0,0]$, so that we also have $h_{qp}(0,p)=0$ for each $p\in[p_0,0].$
These relations together with $h(2\Lambda-q,p)=h(q,p)$ for all $(q,p)\in R $
ensure that $\widehat h\in C^2_{\rm per}(\overline\Omega)$ is also a solution of \eqref{eq:HP}.
But, the same argument as in the first part of the proof shows that $\widehat h(\cdot,p)$ is a real-analytic map for all $p\in[p_0,0].$
Particularly, since $\widehat h=h$ on $[0,2\Lambda]\times [p_0,0],$ the principle of analytic continuation yields that the minimal period of $\eta=h(\cdot,0)-d$ is $2\Lambda$.
Moreover, $\eta$ has a unique maximum (and minimum) per period and is symmetric with respect to the crest (and trough) line.
This completes the proof.
\end{proof}
\section*{Acknowledgments} I am grateful for the suggestions and the comments made by the anonymous referees.
\end{document} | math | 28,827 |
\begin{document}
\title{\LARGE \bf Supplementary Material to: Passive Controller Realization of a Biquadratic Impedance with Double Poles and Zeros
as a Seven-Element Series-Parallel Network for Effective Mechanical Control}
\thispagestyle{empty}
\begin{abstract}
This report presents some supplementary material to the paper entitled ``Passive controller realization of a biquadratic impedance with double poles and zeros
as a seven-element Series-parallel network for effective mechanical control'' \cite{WCLC18}.
\noindent{\em Keywords:} Passive mechanical control, passive network synthesis, inerters, biquadratic impedances, series-parallel networks.
\end{abstract}
\IEEEpeerreviewmaketitle
\section{Introduction}
This report presents some supplementary material to the paper entitled ``Passive controller realization of a biquadratic impedance with double poles and zeros as a seven-element Series-parallel network for effective mechanical control'' \cite{WCLC18}, which are omitted from the paper for brevity.
\section{Preliminaries of Passive Network Synthesis}
A two-terminal electrical network is defined to be \textit{passive} if $\int_{-\infty}^{T} i(t)v(t) dt \geq 0$ for all $T$ and for all admissible current $i(t)$ and voltage $v(t)$ \cite{AV73}.
A real-rational function $F(s)$ is \textit{positive real} if $F(s)$ is analytic and $\mathfrak{R}(F(s)) \geq 0$ for any $\mathfrak{R}(s) > 0$ \cite{Gui57}. An \textit{impedance} (resp. \textit{admittance}) is defined to be $Z(s) = V(s)/I(s)$ (resp. $Y(s) = I(s)/V(s)$), where $V(s)$ and $I(s)$ are voltages and currents of the port of a two-terminal network, where the network is said to \textit{realize} (or is a realization of) its impedance (resp. admittance).
A two-terminal electrical network is passive if and only if its impedance (resp. admittance) is positive real,
and any positive-real impedance (resp. admittance) is realizable as a two-terminal passive $RLC$ network \cite{Gui57}.
Any positive-real impedance (resp. admittance) is realizable as a two-terminal passive $RLC$ network.
A \textit{reactive element} is an inductor or capacitor, and a resistor is also called as a \textit{resistive element}.
\section{Definitions of the network duality and the frequency inverse}
Any two-terminal passive $RLC$ network $N$ can be regarded as a \textit{one-terminal-pair labeled graph} $\mathcal{N}$ with two distinguished \textit{terminal vertices} (see \cite[pg.~14]{SR61}), in which the labels designate passive circuit elements regardless of element values, namely resistors, capacitors, and inductors, which are labeled as $R_i$, $C_i$, and $L_i$, respectively.
Two natural maps acting on the labeled graph are defined as follows:
\begin{enumerate}
\item $\text{GDu} :=$ Graph duality, which takes the one-terminal-pair graph into its dual (see \cite[Definition~3-12]{SR61}) while preserving the labeling.
\item $\text{Inv} :=$ Inversion, which preserves the graph but interchanges the reactive elements, that is, capacitors to inductors and inductors to capacitors with their labels $C_i$ to $L_i$ and $L_i$ to $C_i$.
\end{enumerate}
Consequently, one defines
\begin{equation*}
\text{Dual} := \text{network duality of one-terminal-pair labeled graph} := \text{GDu} \circ \text{Inv} = \text{Inv} \circ \text{GDu}.
\end{equation*}
Consider a network $N$ whose one-terminal-pair labeled graph is $\mathcal{N}$.
Denote $\text{Inv}(N)$ as the network whose one-terminal-pair labeled graph is $\text{Inv}(\mathcal{N})$, resistors are of the same values as those of $N$, and inductors (resp. capacitors) are replaced by capacitors (resp. inductors) with reciprocal values, which is called the \textit{frequency inverse network} of $N$.
Denote $\text{GDu}(N)$ as the network whose one-terminal-pair labeled graph is $\text{GDu}(\mathcal{N})$ and elements are of the reciprocal values to those of $N$, which is called the \textit{frequency inverse dual network} of $N$.
Denote $\text{Dual}(N)$ as the network whose one-terminal-pair labeled graph is $\text{Dual}(\mathcal{N})$, resistors are of reciprocal values to those of $N$, and inductors (resp. capacitors) are replaced by capacitors (resp. inductors) with same values, which is called the \textit{dual network} of $N$.
It can be proved that $Z(s)$ (resp. $Y(s)$) is realizable as the impedance (resp. admittance)
of a network $N$ whose one-terminal-pair labeled graph is $\mathcal{N}$,
if and only if $Z(s^{-1})$ (resp. $Y(s^{-1})$) is realizable as the impedance (resp. admittance) of $\text{Inv}(N)$ whose one-terminal-pair labeled graph is $\text{Inv}(\mathcal{N})$, if and only if
$Z(s^{-1})$ (resp. $Y(s^{-1})$) is realizable as the admittance (resp. impedance) of $\text{GDu}(N)$ whose one-terminal-pair labeled graph is $\text{GDu}(\mathcal{N})$, and if and only if
it is realizable as
the admittance (resp. impedance) of $\text{Dual}(N)$ whose one-terminal-pair labeled graph is $\text{Dual}(\mathcal{N})$.
Therefore, if a necessary and sufficient condition is derived for
$H(s) =
\sum_{k=0}^m a_k s^k/\sum_{k=0}^m b_k s^k$ to be realizable as the impedance (resp. admittance) of a two-terminal network whose one-terminal-pair labeled graph is $\mathcal{N}$, then the corresponding condition for $\text{Inv}(\mathcal{N})$ can be obtained from that for $\mathcal{N}$ through conversion $a_k \leftrightarrow a_{m-k}$ and $b_k \leftrightarrow b_{m-k}$ for $k = 0,1,...,\lfloor m/2 \rfloor$ (the \textit{principle of frequency inversion}).
The corresponding condition for $\text{GDu}(\mathcal{N})$ can be obtained from that for $\mathcal{N}$ through conversion $a_k \leftrightarrow b_{m-k}$ for $k = 0,1,...,m$ (the \textit{principle of frequency-inverse duality}).
Furthermore, the corresponding condition for $\text{Dual}(\mathcal{N})$ can be obtained from that for $\mathcal{N}$ through conversion $a_k \leftrightarrow b_k$ for $k = 0, 1,..., m$ (the \textit{principle of duality}).
Specifically, based on the principle of frequency inversion,
a necessary and sufficient condition for $Z(s)$ in the form of (1) with $k$, $z$, $p$ $> 0$ to be realizable as the impedance of a two-terminal network whose one-terminal-pair labeled graph is $\text{Inv}(\mathcal{N})$ can be obtained from that for $\mathcal{N}$ through $z \leftrightarrow z^{-1}$ and $p \leftrightarrow p^{-1}$.
Based on the principle of frequency-inversion duality, a necessary and sufficient condition for $Z(s)$ in the form of (1) with $k$, $z$, $p$ $> 0$ to be realizable as the impedance of a two-terminal network whose one-terminal-pair labeled graph is $\text{GDu}(\mathcal{N})$ can be obtained from that for $\mathcal{N}$ through $p \leftrightarrow z^{-1}$ and $z \leftrightarrow p^{-1}$.
Based on the principle of duality, a necessary and sufficient condition for $Z(s)$ in the form of (1) with $k$, $z$, $p$ $> 0$ to be realizable as the impedance of a two-terminal network whose one-terminal-pair labeled graph is $\text{Dual}(\mathcal{N})$ can be obtained from that for $\mathcal{N}$ through $p \leftrightarrow z$.
\section{Realizability as Series-Parallel Networks with No More Than Five Elements}
\begin{theorem} \label{theorem: fewer than four elements}
{A biquadratic impedance $Z(s)$ in the form of (1), that is,
\begin{equation*}
Z(s) = k \frac{(s + z)^2}{(s + p)^2},
\end{equation*}
where $k$, $z$, $p$ $> 0$ and $p \neq z$, cannot be realized with fewer than four elements.}
\end{theorem}
\begin{proof}
Let $A = kx$, $B = 2kzx$, $C = k z^2x$, $D = x$, $E = 2px$, and $F = p^2x$ for $x > 0$. This theorem can be proved from the realizability conditions of a general biquadratic impedance in the form of (2), that is,
\begin{equation} \label{eq: general biquadratic impedances}
Z(s) = \frac{A s^2 + B s + C}{D s^2 + E s + F},
\end{equation}
where $A$, $B$, $C$, $D$, $E$, $F$ $> 0$, with at most three elements in \cite{WCH14}.
\end{proof}
\begin{theorem} \label{theorem: four elements}
{A biquadratic impedance $Z(s)$ in the form of (1), where $k$, $z$, $p$ $> 0$ and $p \neq z$, is realizable as a four-element series-parallel network if and only if $p = z/3$ or $p = 3z$.}
\end{theorem}
\begin{proof}
Let $A = kx$, $B = 2kzx$, $C = k z^2x$, $D = x$, $E = 2px$, and $F = p^2x$ for $x > 0$. This condition can be derived from the realizability conditions of a general biquadratic impedance in the form of (2) where $A$, $B$, $C$, $D$, $E$, $F$ $> 0$ as a four-element network in \cite{WCH14}, where it is obvious that any four-element network must be series-parallel.
\end{proof}
\begin{theorem} \label{theorem: five elements}
{A biquadratic impedance $Z(s)$ in the form of (1), where $k$, $z$, $p$ $> 0$ and $p \neq z$, is realizable as a five-element series-parallel network if and only if $p/z \in (1/3, 3)$, $p = (2 + \sqrt{2})z$, or $p = z/(2 + \sqrt{2})$.}
\end{theorem}
\begin{proof}
Let $A = kx$, $B = 2kzx$, $C = k z^2x$, $D = x$, $E = 2px$, and $F = p^2x$ for $x > 0$. This condition can be derived from the realizability conditions of a general biquadratic impedance in the form of (2) where $A$, $B$, $C$, $D$, $E$, $F$ $> 0$ as a five-element series-parallel network in \cite{JS11}.
\end{proof}
\section{Some Basic Lemmas}
\begin{lemma}
\label{lemma: biquadratic impedances Z2 three elements}
{Consider a biquadratic impedance $F(s)$ in the form of
\begin{equation} \label{eq: biquadratic impedances Z2}
F(s) = \frac{\alpha s^2 + \beta s + \gamma}{(s + p)^2},
\end{equation}
where $\alpha$, $\beta$, $\gamma$ $\geq 0$, and $p > 0$.
Then, $F(s)$ is realizable as a three-element series-parallel network if and only if at least one of the following conditions holds: 1. $\alpha \gamma = 0$; 2. $\beta = 0$ and $\alpha p^2 - \gamma = 0$; 3. $\gamma = 0$ and $\alpha p - 2 \beta = 0$; 4. $\alpha = 0$ and $2 \beta p - \gamma = 0$; 5. $\alpha p^2 - \beta p + \gamma = 0$.
}
\end{lemma}
\begin{proof}
Since it is obvious that $F(s)$ in the form of \eqref{eq: biquadratic impedances Z2} is not a \textit{reactance function} \cite[Definition~3.1]{Bah84}, there is at least one resistor, which means that there are at most two reactive elements.
When the number of reactive elements is at most one, the \textit{degree} \cite[Section~3.6]{AV73} of $F(s)$ cannot exceed one by \cite[Theorem~4.4.3]{AV73}.
Therefore, there must exist at least one common factor between $(\alpha s^2 + \beta s + \gamma)$ and $(s + p)^2$, which holds if and only if
Condition~5 is satisfied.
When the number of reactive elements is two, there is one resistor. Based on the method of enumeration, one can obtain
Conditions~1--4.
\end{proof}
\begin{lemma}
\label{lemma: biquadratic impedances Z2 Four elements}
{Consider a biquadratic impedance $F(s)$ in the form of
\eqref{eq: biquadratic impedances Z2},
where $\alpha$, $\beta$, $\gamma$ $\geq 0$, $p > 0$, assuming that the condition of Lemma~\ref{lemma: biquadratic impedances Z2 three elements} does not hold. Then, $F(s)$ is realizable as a four-element series-parallel network if and only if at least one of the six conditions holds:
1. $\alpha = 0$ and $\gamma < 2 \beta p$; 2. $\gamma = 0$ and $\alpha p < 2 \beta$; 3. $\alpha$, $\beta$, $\gamma$ $> 0$ and $\alpha p^2 - \gamma = 0$; 4. $\alpha$, $\beta$, $\gamma$ $> 0$, $\alpha p^2 < \gamma$, and
$3 \alpha p^2 + \gamma - 2\beta p = 0$ or $\beta^2 p^2 + \gamma^2 - \alpha \gamma p^2 - 2 \beta \gamma p = 0$ holds; 5. $\alpha$, $\beta$, $\gamma$ $> 0$, $\alpha p^2 > \gamma$, and $\alpha p^2 + 3 \gamma - 2\beta p = 0$
or $\alpha^2 p^2 + \beta^2 - 2 \alpha \beta p - \alpha \gamma = 0$ holds; 6. $\alpha$, $\beta$, $\gamma$ $> 0$ and $\alpha^2 p^4 - 2\alpha \beta p^3 + 6 \alpha \gamma p^2 - 2 \beta \gamma p + \gamma^2 = 0$.
Moreover, if one of the above six conditions holds, then $F(s)$ is realizable as a two-reactive four-element series-parallel network.
}
\end{lemma}
\begin{proof}
Let $A = \alpha x$, $B = \beta x$, $C = \gamma x$, $D = x$, $E = 2px$, and $F = p^2x$ for any $x > 0$.
A necessary and sufficient condition for $Z(s)$ in the form of (2) with $A$, $B$, $C$, $D$, $E$, $F$ $\geq 0$ to be positive real is $(\sqrt{AF} - \sqrt{CD})^2 \leq BE$ \cite{CS09(2),Fos62}.
Since any positive-real biquadratic impedance with zero coeffients is realizable as a two-reactive four-element series-parallel network \cite[Lemma~8]{JS11}, one obtains Conditions~1 and 2 together with Lemma~\ref{lemma: biquadratic impedances Z2 three elements}.
Now, it remains to considering the case of $\alpha$, $\beta$, $\gamma$ $> 0$. By \cite[Theorem~5]{WCH14}, one obtains Conditions~3--6.
By the covering configurations in \cite[Figs.~4--6]{WCH14}, the number of reactive elements for realizations is two.
\end{proof}
\begin{lemma}
\label{lemma: biquadratic impedances Z2 Five elements Two reactive}
{Consider a biquadratic impedance $F(s)$ in the form of
\eqref{eq: biquadratic impedances Z2},
where $\alpha$, $\beta$, $\gamma$, $p > 0$, and neither the condition of Lemmas~\ref{lemma: biquadratic impedances Z2 three elements} nor the condition of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements} holds. Then, $F(s)$ is realizable as a two-reactive five-element series-parallel network if and only if at least one of the following conditions holds: 1. $\alpha p^2 > \gamma$ and $\alpha p^2 + 3\gamma - 2\beta p < 0$; 2. $\alpha p^2 > \gamma$ and $\alpha^2 p^2 + \beta^2 - 2\alpha \beta p - \alpha \gamma < 0$; 3. $\alpha p^2 < \gamma$ and $3 \alpha p^2 + \gamma - 2 \beta p < 0$; 4. $\alpha p^2 < \gamma$ and $\beta^2 p^2 + \gamma^2 - \alpha \gamma p^2 - 2 \beta \gamma p < 0$.
}
\end{lemma}
\begin{proof}
A necessary and sufficient condition for $Z(s)$ in the form of (2) where $A$, $B$, $C$, $D$, $E$, $F$ $> 0$ to be realizable as a two-reactive five-element series-parallel network is presented in \cite[Theorem~1]{JS11}.
Letting $A = \alpha x$, $B = \beta x$, $C = \gamma x$, $D = x$, $E = 2px$, and $F = p^2x$ for any $x > 0$, this lemma can be proved together with the assumption that neither the condition of Lemma~\ref{lemma: biquadratic impedances Z2 three elements} nor the condition of Lemma~ does not hold.
\end{proof}
\section{Supplementary Lemmas of Three-Reactive Seven-Element Series-Parallel Realizations for the Proof of Lemma~2}
\begin{lemma}
\label{lemma: Three-Reactive-Structure}
{Consider a biquadratic impedance $Z(s)$ in the form of \eqref{eq: general biquadratic impedances}
with $A$, $B$, $C$, $D$, $E$, $F$ $> 0$ that cannot be realized as a series-parallel network containing fewer than seven elements. If $Z(s)$ is realizable as a three-reactive seven-element series-parallel network as shown in Fig.~2(a),
where $N_1$ is a three-element series-parallel network
and $N_2$ is a four-element series-parallel network, then $Z(s)$ is also realizable as shown in Fig.~\ref{fig: Equivalent structure}, where $N_b$ is a two-reactive five-element series-parallel network. }
\end{lemma}
\begin{figure}
\caption{A class of three-reactive seven-element series-parallel networks, where $N_b$ is a two-reactive five-element series-parallel network (Fig.~7).}
\label{fig: Equivalent structure a}
\label{fig: Equivalent structure b}
\label{fig: Equivalent structure}
\end{figure}
\begin{proof}
By \cite[Lemma~2]{WCH14}, $Z(s)$ cannot be realized as the series connection of two networks, one of which only contains reactive elements.
Therefore, $N_1$ must contain one or two reactive elements.
If $N_1$ contains two reactive elements, then $N_2$ contains one reactive element. Therefore, the \textit{degree} \cite[Section~3.6]{AV73} of $Z_2(s)$ cannot exceed one by \cite[Theorem~4.4.3]{AV73}, where $Z_2(s)$ denotes the impedance of $N_2$.
By the discussion in \cite{WCH14}, $Z_2(s)$ is realizable as a one-reactive three-element series-parallel network, which implies that $Z(s)$ is realizable with a series-parallel network containing fewer than seven elements. This contradicts the assumption.
If $N_1$ contains one reactive element, then the \textit{degree} \cite[Section~3.6]{AV73} of $Z_1(s)$ cannot exceed one by \cite[Theorem~4.4.3]{AV73}, where $Z_1(s)$ denotes the impedance of $N_1$. By the discussion in \cite{WCH14}, $Z_1(s)$ is realizable as a one-reactive three-element series-parallel network, which is equivalent to one of configurations in Fig.~\ref{fig: configurations Lemma1}. Regarding the series connection of $R_2$ and $N_2$ as $N_b$, this lemma is proved.
\end{proof}
\begin{figure}
\caption{Configurations for $N_1$ mentioned in the proof of
Lemma~\ref{lemma: Three-Reactive-Structure}
\label{fig: configurations Lemma1 a}
\label{fig: configurations Lemma1 b}
\label{fig: configurations Lemma1}
\end{figure}
\section{Complete Proof of Lemma~2}
Assume that $Z(s)$ is realizable as in Fig.~\ref{fig: Equivalent structure}(a), where $N_b$ is a two-reactive five-element series-parallel network.
Let $Z(s) = Z_a(s) + Z_b(s)$, where $Z_a(s)$ is the impedance of the parallel connection of $R_1$ and $C_1$ and $Z_b(s)$ is the impedance of $N_b$. It is clear that $Z_a(s)$ can be written in the form of
\begin{equation*}
Z_a(s) = \frac{m}{s+p},
\end{equation*}
where $m > 0$.
Since the condition of Theorem~1 does not hold, $N_b$ cannot be equivalent to a network containing fewer than five elements. Therefore, $Z_b(s)$ can be expressed in the form of
\begin{equation*}
Z_b(s) = \frac{\alpha s^2 + \beta s + \gamma}{(s+p)^2},
\end{equation*}
where $\alpha$, $\beta$, $\gamma$ $> 0$, and the condition of
Lemma~\ref{lemma: biquadratic impedances Z2 Five elements Two reactive}
holds. Since
\begin{equation*}
Z_a(s) + Z_b(s) = \frac{\alpha s^2 + (\beta + m) s + (\gamma + mp)}{(s+p)^2},
\end{equation*}
it follows that
$\alpha s^2 + (\beta + m) s + (\gamma + mp) = k(s+z)^2$.
Therefore,
\begin{equation} \label{eq: N1 two N2 five 01}
\beta = 2 \alpha z - m, ~~~~~ \gamma = \alpha z^2 - mp.
\end{equation}
Since $\beta$, $\gamma$ $> 0$, it follows from \eqref{eq: N1 two N2 five 01} that
\begin{equation} \label{eq: N1 two N2 five 02}
\alpha > \max\left\{ \frac{m}{2z}, \frac{mp}{z^2} \right\}.
\end{equation}
If $Z_b(s)$ satisfies Condition~1 of Lemma~\ref{lemma: biquadratic impedances Z2 Five elements Two reactive}, then $\alpha p^2 > \gamma$ and $\alpha p^2 + 3\gamma - 2\beta p < 0$. Together with \eqref{eq: N1 two N2 five 01}, one obtains
\begin{align}
(p-z)(p+z) \alpha + mp &> 0, \label{eq: N1 two N2 five 03} \\
(p-z)(p-3z) \alpha - mp &< 0. \label{eq: N1 two N2 five 04}
\end{align}
If $z < p \leq 3z$, then it is obvious that the condition of Theorem~1 holds.
If $p > 3z$ or $p < z$, then it follows from \eqref{eq: N1 two N2 five 04} that $\alpha < mp/((p-z)(p-3z))$.
Together with \eqref{eq: N1 two N2 five 02}, one obtains
$mp/z^2 < mp/((p-z)(p-3z))$,
which implies $(2 - \sqrt{2})z < p < (2 + \sqrt{2})z$. Therefore, the condition of Theorem~1 holds.
If $Z_b(s)$ satisfies Condition~2 of Lemma~\ref{lemma: biquadratic impedances Z2 Five elements Two reactive}, then $\alpha p^2 > \gamma$ and $\alpha^2 p^2 + \beta^2 - 2\alpha \beta p - \alpha \gamma < 0$. Together with \eqref{eq: N1 two N2 five 01}, one obtains \eqref{eq: N1 two N2 five 03} and
\begin{equation} \label{eq: N1 two N2 five 05}
(p-z)(p-3z)\alpha^2 + m(3p-4z)\alpha + m^2 < 0.
\end{equation}
It follows from \eqref{eq: N1 two N2 five 05} that $p < 3z$. If $z < p < 3z$, then it is obvious that the condition of Theorem~1 holds. If $p < z$, then
\eqref{eq: N1 two N2 five 03} implies that
\begin{equation} \label{eq: N1 two N2 five 06}
\alpha < -\frac{mp}{(p-z)(p+z)}.
\end{equation}
From \eqref{eq: N1 two N2 five 02} and \eqref{eq: N1 two N2 five 06}, it follows that $m/(2z) < -mp/((p-z)(p+z))$,
which implies $p > z/(1+\sqrt{2})$. Therefore, the condition of Theorem~1 holds.
If $Z_b(s)$ satisfies Condition~3 of Lemma~\ref{lemma: biquadratic impedances Z2 Five elements Two reactive}, then $\alpha p^2 < \gamma$ and $3 \alpha p^2 + \gamma - 2 \beta p < 0$.
Together with \eqref{eq: N1 two N2 five 01}, one obtains
\begin{align}
(p-z)(p+z)\alpha + mp &< 0, \label{eq: N1 two N2 five 07} \\
(3p-z)(p-z)\alpha + mp &< 0. \label{eq: N1 two N2 five 08}
\end{align}
It follows from \eqref{eq: N1 two N2 five 08} that
$z/3 < p < z$. Therefore, the condition of Theorem~1 holds.
If $Z_b(s)$ satisfies Condition~4 of Lemma~\ref{lemma: biquadratic impedances Z2 Five elements Two reactive}, then $\alpha p^2 < \gamma$ and $\beta^2 p^2 + \gamma^2 - \alpha \gamma p^2 - 2 \beta \gamma p < 0$. Together with \eqref{eq: N1 two N2 five 01}, one obtains \eqref{eq: N1 two N2 five 07} and
\begin{equation} \label{eq: N1 two N2 five 09}
z^2 (3p - z)(p - z)\alpha^2 + p^3 m \alpha < 0.
\end{equation}
It follows from \eqref{eq: N1 two N2 five 09} that $z/3 < p < z$. Therefore, the condition of Theorem~1 holds.
This means that $Z(s)$ cannot be realized as in Fig.~\ref{fig: Equivalent structure}(a), where $N_b$ is a two-reactive five-element series-parallel network.
It is clear that any network in Fig.~\ref{fig: Equivalent structure}(b)
can be a frequency inverse network of another one in
Fig.~\ref{fig: Equivalent structure}(a). Therefore, by the principle of frequency inverse (Section~III), $Z(s)$ cannot be realized as in
Fig.~\ref{fig: Equivalent structure}(b), where $N_b$ is a two-reactive five-element series-parallel network.
By Lemma~\ref{lemma: biquadratic impedances Z2 Five elements Two reactive}, $Z(s) \in \mathcal{Z}_{p2,z2}$ cannot be realized as a three-reactive seven-element series-parallel network as shown in Fig.~2(a), where $N_1$
is a three-element series-parallel network and $N_2$ is a four-element series-parallel network. Since any network in Fig.~2(b), where $N_1$
is a three-element series-parallel network and $N_2$ is a four-element series-parallel network, can be a dual network of the case of Fig.~2(a), by the principle of duality (Section~III), this lemma can be proved.
\section{Supplementary Lemmas of Four-Reactive Seven-Element Series-Parallel Realizations for the Proof of Lemma~3}
\begin{lemma}
\label{lemma: Possible configurations of N1}
{Consider the four-reactive seven-element series-parallel network in Fig.~2, realizing a biquadratic impedance $Z(s)$ in the form of \eqref{eq: general biquadratic impedances} with $A$, $B$, $C$, $D$, $E$, $F$ $> 0$, where $N_1$ is a two-reactive three-element series-parallel network and $N_2$ is a two-reactive four-element series-parallel network.
If $Z(s)$ cannot be realized as a series-parallel network containing fewer than seven elements, then $N_1$ must be
one of the configurations in Fig.~\ref{fig: Two-reactive configurations N1}.}
\end{lemma}
\begin{figure}
\caption{Two-reactive three-element configurations for the $N_1$ mentioned in Lemma~\ref{lemma: Possible configurations of N1}
\label{fig: Two-reactive configurations N1 a}
\label{fig: Two-reactive configurations N1 b}
\label{fig: Two-reactive configurations N1 c}
\label{fig: Two-reactive configurations N1 d}
\label{fig: Two-reactive configurations N1}
\end{figure}
\begin{proof}
Let $\mathcal{C}(a,a')$ denote the \textit{cut-set} \cite[pg.~28]{SR61} separating a one-terminal-pair labeled graph of a network into two connected subgraphs containing two terminal vertices $a$ and $a'$, respectively.
By \cite[Lemma~1]{WCH14}, for any realization of $Z(s)$ there is no cut-set $\mathcal{C}(a,a')$ corresponding to only one kind of reactive elements, where $a$ and $a'$ denote two terminals.
Since $N_1$ contains three elements, all the possible \textit{network graphs} \cite{CWSL13}
of $N_1$ are listed as in Fig.~\ref{fig: Three-element graphs}. By the method of enumeration, this lemma can be proved.
\end{proof}
\begin{figure}
\caption{Possible network graphs for three-element networks.}
\label{fig: Three-element graphs a}
\label{fig: Three-element graphs b}
\label{fig: Three-element graphs c}
\label{fig: Three-element graphs d}
\label{fig: Three-element graphs}
\end{figure}
\begin{lemma} \label{lemma: 01 N1}
{A biquadratic impedance $Z(s) \in \mathcal{Z}_{p2,z2}$ cannot be realized as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(a)
and $N_2$ is a two-reactive four-element series-parallel network.}
\end{lemma}
\begin{proof}
By calculation, the impedance of the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(a) is obtained as
\begin{equation*}
Z_1(s) = \frac{R_1L_1 s}{R_1L_1C_1 s^2 + L_1 s + R_1}.
\end{equation*}
Since by assumption the condition of Theorem~1 does not hold, $N_2$ cannot be equivalent to
a series-parallel network containing fewer than four elements. If
$Z(s)$ is realizable as in
Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(a) and $N_2$ is a two-reactive four-element series-parallel network, then
the impedance of $N_1$ is in the form of
\begin{equation*}
Z_1(s) = \frac{m s}{(s + p)^2},
\end{equation*}
where $m > 0$, and the impedance of $N_2$ is in the form of
\begin{equation} \label{eq: 01 N1 00 Z2}
Z_2(s) = \frac{\alpha s^2 + \beta s + \gamma}{(s + p)^2},
\end{equation}
where $\alpha$, $\beta$, $\gamma$ $> 0$, and moreover the condition of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements} holds.
Since
$\alpha s^2 + (\beta + m) s + \gamma = k(s + z)^2$,
it follows that
\begin{align}
\beta + m &= 2 z \alpha, \label{eq: 01 N1 01} \\
\gamma &= z^2 \alpha. \label{eq: 01 N1 02}
\end{align}
Since $\alpha$, $\gamma$ $> 0$, $Z_2(s)$
satisfies neither Condition~1 nor Condition~2 of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements}.
If $Z_2(s)$ satisfies Condition~3 of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements}, then $\alpha p^2 - \gamma = 0$. Together with \eqref{eq: 01 N1 02}, it follows that $p = z$, which contradicts the assumption.
If $Z_2(s)$ satisfies Condition~4 of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements}, then $\alpha p^2 < \gamma$ and
either
$3 \alpha p^2 + \gamma - 2\beta p = 0$ or $\beta^2 p^2 + \gamma^2 - \alpha \gamma p^2 - 2 \beta \gamma p = 0$ holds. For the case of $3 \alpha p^2 + \gamma - 2\beta p = 0$,
together with \eqref{eq: 01 N1 01} and \eqref{eq: 01 N1 02}, one obtains
$\alpha = - 2mp/((3p-z)(p-z))$, $\beta = -m(3p^2 + z^2)/((3p-z)(p-z))$, and $\gamma = -2mz^2p/((3p-z)(p-z))$,
which implies $z/3 < p < z$ by $\alpha$, $\beta$, $\gamma$ $> 0$.
Thus, the condition of Theorem~1 holds.
For the case of $\beta^2 p^2 + \gamma^2 - \alpha \gamma p^2 - 2 \beta \gamma p = 0$, together with \eqref{eq: 01 N1 01} and \eqref{eq: 01 N1 02}, one obtains
\begin{equation} \label{eq: 01 N1 03}
\alpha = \frac{mp}{z(3p-z)}, ~~~ \beta = -\frac{m(p-z)}{3p-z}, ~~~
\gamma = \frac{mzp}{3p - z},
\end{equation}
or
\begin{equation} \label{eq: 01 N1 04}
\alpha = \frac{mp}{z(p - z)}, ~~~ \beta = \frac{m(p+z)}{p-z}, ~~~
\gamma = \frac{mzp}{p-z}.
\end{equation}
Because
$\alpha$, $\beta$, $\gamma$ $> 0$, it follows from \eqref{eq: 01 N1 03} that $z/3 < p < z$, which satisfies the condition of Theorem~1.
Substituting \eqref{eq: 01 N1 04} into $\alpha p^2 < \gamma$ yields
$(p+z)pm/z < 0$, which is impossible.
If $Z_2(s)$ satisfies Condition~5 of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements}, then $\alpha p^2 > \gamma$ and either $\alpha p^2 + 3 \gamma - 2\beta p = 0$
or $\alpha^2 p^2 + \beta^2 - 2 \alpha \beta p - \alpha \gamma = 0$ holds. For the case of $\alpha p^2 + 3 \gamma - 2\beta p = 0$, together with \eqref{eq: 01 N1 01} and \eqref{eq: 01 N1 02}, one
obtains
$\alpha = -2mp/((p-z)(p-3z))$, $\beta = -m(p^2 + 3z^2)/((p-z)(p-3z))$, and $\gamma = -2mz^2p/((p-z)(p-3z))$,
which implies $z < p < 3z$ because $\alpha$, $\beta$, $\gamma$ $> 0$. Thus, the condition of Theorem~1 holds.
For the case of
$\alpha^2 p^2 + \beta^2 - 2 \alpha \beta p - \alpha \gamma = 0$,
together with \eqref{eq: 01 N1 01} and \eqref{eq: 01 N1 02}, one
obtains
\begin{equation} \label{eq: 01 N1 05}
\alpha = -\frac{m}{p-3z}, ~~~ \beta = -\frac{m(p-z)}{p-3z}, ~~~
\gamma = - \frac{mz^2}{p - 3z},
\end{equation}
or
\begin{equation} \label{eq: 01 N1 06}
\alpha = - \frac{m}{p - z}, ~~~ \beta = - \frac{m(p + z)}{p - z}, ~~~ \gamma = - \frac{mz^2}{p - z}.
\end{equation}
It follows from \eqref{eq: 01 N1 05} that $z < p < 3z$
by
$\alpha$, $\beta$, $\gamma$ $> 0$. Thus, the condition of Theorem~1 holds. Substituting \eqref{eq: 01 N1 06} into $\alpha p^2 > \gamma$ yields $-m(p+z) > 0$, which is impossible.
If $Z_2(s)$ satisfies Condition~6 of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements}, then $\alpha^2 p^4 + 6 \alpha \gamma p^2 + \gamma^2 - 2\alpha \beta p^3 - 2 \beta \gamma p = 0$.
Together with \eqref{eq: 01 N1 01} and \eqref{eq: 01 N1 02}, one
obtains $\alpha = 0$, $\beta = -m$, and $\gamma = 0$ or $\alpha = - 2mp(z^2 + p^2)/(p-z)^4$, $\beta = - m(p^4 + 6z^2p^2 + z^4)/(p-z)^4$, and $\gamma = - 2mz^2p(z^2+p^2)/(p-z)^4$,
which contradicts the assumption that $\alpha$, $\beta$, $\gamma$ $> 0$.
As a conclusion, this lemma is proved.
\end{proof}
\begin{lemma} \label{lemma: 02 N1}
{If a biquadratic impedance $Z(s) \in \mathcal{Z}_{p2,z2}$ is realizable as in Fig.~2(a), where $N_1$ is one of the configurations in Figs.~\ref{fig: Two-reactive configurations N1}(b), \ref{fig: Two-reactive configurations N1}(c), and \ref{fig: Two-reactive configurations N1}(d),
and $N_2$ is a two-reactive four-element series-parallel network, then the condition of Lemma~1 holds.}
\end{lemma}
\begin{proof}
First, the case where $N_1$ is a configuration in Fig.~\ref{fig: Two-reactive configurations N1}(b) and $N_2$ is a two-reactive four-element series-parallel network will be discussed.
It is calculated that the impedance of the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(b) is in the form of
\begin{equation} \label{eq: Z1 b configuration}
Z_1(s) = \frac{R_1L_1C_1 s^2 + R_1}{L_1C_1 s^2 + R_1C_1 s + 1}.
\end{equation}
Since it is assumed that the condition of Theorem~1 does not hold, $N_2$ cannot be equivalent to
a series-parallel network containing fewer than four elements. If $Z(s)$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(b)
and $N_2$ is a two-reactive four-element series-parallel network,
then the impedance of $N_1$ is in the form of
\begin{equation*}
Z_1(s) = \frac{m (s^2 + p^2)}{(s + p)^2},
\end{equation*}
where $m > 0$, and the impedance of $N_2$ is in the form of \eqref{eq: 01 N1 00 Z2}, where $\beta > 0$, $\alpha$, $\gamma$ $\geq 0$, and the condition of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements} holds.
Since
$(m + \alpha) s^2 + \beta s + (mp^2 + \gamma) = k (s + z)^2$,
it follows that
\begin{align}
\beta &= 2z(m + \alpha), \label{eq: 02 N1 01} \\
mp^2 + \gamma &= z^2(m + \alpha). \label{eq: 02 N1 02}
\end{align}
If $Z_2(s)$ satisfies Condition~1 of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements}, then $\beta$, $\gamma$ $> 0$, $\alpha = 0$, and $\gamma < 2 \beta p$. Together with \eqref{eq: 02 N1 01} and \eqref{eq: 02 N1 02}, one obtains
\begin{equation} \label{eq: 02 N1 03}
\beta = 2mz, ~~~ \gamma = -m(p-z)(p+z),
\end{equation}
which implies $p < z$ by $\beta$, $\gamma$ $> 0$.
Substituting \eqref{eq: 02 N1 03} into $\gamma < 2 \beta p$ yields
$m(p^2 + 4zp - z^2) > 0$.
This implies $z/(2 + \sqrt{5}) < p < z$, which satisfies the condition of Lemma~1.
If $Z_2(s)$ satisfies Condition~2 of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements}, then $\alpha$, $\beta$ $> 0$, $\gamma = 0$, and $\alpha p < 2 \beta$. Together with \eqref{eq: 02 N1 01} and \eqref{eq: 02 N1 02}, one obtains
\begin{equation} \label{eq: 02 N1 04}
\alpha = \frac{m(p-z)(p+z)}{z^2}, ~~~ \beta = \frac{2mp^2}{z},
\end{equation}
which implies $p > z$ by $\alpha$, $\beta$ $> 0$.
Substituting \eqref{eq: 02 N1 04} into $\alpha p < 2 \beta$ yields
$mp(p^2 - 4zp - z^2)/z^2 < 0$.
This further implies $z < p < (2 + \sqrt{5})z$. Therefore, the condition of Lemma~1 holds.
If $Z_2(s)$ satisfies Condition~3 of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements}, then $\alpha$, $\beta$, $\gamma$ $> 0$ and $\alpha p^2 - \gamma = 0$. Together with \eqref{eq: 02 N1 01} and \eqref{eq: 02 N1 02}, one obtains $\alpha = -m$, $\beta = 0$, and $\gamma = -mp^2$, which contradicts the assumption.
If $Z_2(s)$ satisfies Condition~4 of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements}, then $\alpha$, $\beta$, $\gamma$ $> 0$, $\alpha p^2 < \gamma$, and
either $3 \alpha p^2 + \gamma - 2\beta p = 0$ or $\beta^2 p^2 + \gamma^2 - \alpha \gamma p^2 - 2 \beta \gamma p = 0$. For the case of $3 \alpha p^2 + \gamma - 2\beta p = 0$, together with \eqref{eq: 02 N1 01} and \eqref{eq: 02 N1 02}, one obtains
\begin{equation} \label{eq: 02 N1 05}
\alpha = \frac{m(p^2 + 4zp - z^2)}{(3p-z)(p-z)}, ~~~
\beta = \frac{8mzp^2}{(3p-z)(p-z)}, ~~~
\gamma = - \frac{mp^2(3p^2 - 4zp - 3z^2)}{(3p-z)(p-z)}.
\end{equation}
Substituting \eqref{eq: 02 N1 05} into $\alpha p^2 < \gamma$ yields
$4mp^2(p+z)/(3p-z) < 0$,
which implies $z/(2 + \sqrt{5}) < p < z/3$ together with \eqref{eq: 02 N1 05} by $\alpha$, $\beta$, $\gamma$ $> 0$. Thus, the condition of Lemma~1 holds.
For the case of $\beta^2 p^2 + \gamma^2 - \alpha \gamma p^2 - 2 \beta \gamma p = 0$, together with \eqref{eq: 02 N1 01} and \eqref{eq: 02 N1 02}, one obtains
\begin{equation} \label{eq: 02 N1 06}
\alpha = -m, ~~~ \beta = 0, ~~~ \gamma = -mp^2,
\end{equation}
or
\begin{equation} \label{eq: 02 N1 07}
\alpha = - \frac{m(p^2 + 2zp - z^2)^2}{z^2(3p-z)(p-z)}, ~~~
\beta = - \frac{2mp^2(p^2 + 4zp - z^2)}{z(3p-z)(p-z)}, ~~~
\gamma = - \frac{4mp^4}{(3p-z)(p-z)}.
\end{equation}
By $\alpha$, $\beta$, $\gamma$ $> 0$, \eqref{eq: 02 N1 06} is impossible. It is implied from \eqref{eq: 02 N1 07} that
$z/3 < p < z$, which satisfies the condition of Lemma~1.
If $Z_2(s)$ satisfies Condition~5 of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements}, then $\alpha$, $\beta$, $\gamma$ $> 0$, $\alpha p^2 > \gamma$, and either $\alpha p^2 + 3 \gamma - 2\beta p = 0$ or
$\alpha^2 p^2 + \beta^2 - 2 \alpha \beta p - \alpha \gamma = 0$. For the case of $\alpha p^2 + 3 \gamma - 2\beta p = 0$,
together with \eqref{eq: 02 N1 01} and \eqref{eq: 02 N1 02}, one obtains
\begin{equation} \label{eq: 02 N1 08}
\alpha = \frac{m(3p^2 + 4zp - 3z^2)}{(p-z)(p-3z)}, ~~~
\beta = \frac{8mzp^2}{(p-z)(p-3z)}, ~~~
\gamma = -\frac{mp^2(p^2 - 4zp - z^2)}{(p-z)(p-3z)}.
\end{equation}
Substituting \eqref{eq: 02 N1 08} into $\alpha p^2 > \gamma$ yields
$4mp^2(p+z)/(p-3z) > 0$,
which implies $3z < p < (2 + \sqrt{5})z$ together with \eqref{eq: 02 N1 08} by $\alpha$, $\beta$, $\gamma$ $> 0$.
For the case of $\alpha^2 p^2 + \beta^2 - 2 \alpha \beta p - \alpha \gamma = 0$, together with \eqref{eq: 02 N1 01} and \eqref{eq: 02 N1 02}, one obtains
\begin{equation} \label{eq: 02 N1 09}
\alpha = -m, ~~~ \beta = 0, ~~~ \gamma = - m p^2,
\end{equation}
or
\begin{equation} \label{eq: 02 N1 10}
\alpha = - \frac{4mz^2}{(p-z)(p-3z)}, ~~~
\beta = \frac{2mz(p^2 - 4zp - z^2)}{(p-z)(p-3z)}, ~~~
\gamma = - \frac{m(p^2 - 2zp - z^2)^2}{(p-z)(p-3z)}.
\end{equation}
It is obvious that
\eqref{eq: 02 N1 09} contradicts the assumption that $\alpha$, $\beta$, $\gamma$ $> 0$.
Moreover, it is implied from \eqref{eq: 02 N1 10} that
$z < p < 3z$, which satisfies the condition of Theorem~1.
If $Z_2(s)$ satisfies Condition~6 of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements}, then $\alpha$, $\beta$, $\gamma$ $> 0$ and $\alpha^2 p^4 + 6 \alpha \gamma p^2 + \gamma^2 - 2\alpha \beta p^3 - 2 \beta \gamma p = 0$. Together with \eqref{eq: 02 N1 01} and \eqref{eq: 02 N1 02}, one obtains
\begin{equation} \label{eq: 02 N1 11}
\begin{split}
\alpha &= \frac{m(3p^4 - 2 z^2p^2 + 4z^3p - z^4 + 2p^2\sqrt{2p^4 + 2 z^4})}{(p-z)^4}, \\
\beta &= \frac{4mzp^2(2p^2 - 2zp + 2z^2 +\sqrt{2p^4 + 2z^4})}{(p-z)^4}, \\
\gamma &= \frac{mp^2(-p^4 + 4zp^3 - 2z^2p^2 + 3z^4 + 2z^2\sqrt{2p^4 + 2z^4})}{(p-z)^4},
\end{split}
\end{equation}
or
\begin{equation} \label{eq: 02 N1 12}
\begin{split}
\alpha &= \frac{m(3p^4 - 2 z^2p^2 + 4z^3p - z^4 - 2p^2\sqrt{2p^4 + 2 z^4})}{(p-z)^4}, \\
\beta &= \frac{4mzp^2(2p^2 - 2zp + 2z^2 -\sqrt{2p^4 + 2z^4})}{(p-z)^4}, \\
\gamma &= \frac{mp^2(-p^4 + 4zp^3 - 2z^2p^2 + 3z^4 - 2z^2\sqrt{2p^4 + 2z^4})}{(p-z)^4}.
\end{split}
\end{equation}
Consider the solutions in \eqref{eq: 02 N1 11}.
Assume that $p \geq (2 + \sqrt{5})z$. Then, $\gamma < 0$ since
$-p^4 + 4zp^3 - 2z^2p^2 + 3z^4 < 0$ and
$(2z^2\sqrt{2p^4 + 2z^4})^2 - (-p^4 + 4zp^3 - 2z^2p^2 + 3z^4)^2 = -(p+z)(p^2 - 4zp - z^2)(p-z)^5 \leq 0$. This contradicts the assumption. Assume that $p \leq z/(2 + \sqrt{5})$. Then, $\alpha < 0$ since $3p^4 - 2 z^2p^2 + 4z^3p - z^4 < 0$ and $(2p^2\sqrt{2p^4 + 2z^4})^2 - (3p^4 - 2z^2p^2 + 4z^3p - z^4)^2 = -(p+z)(p^2 + 4zp - z^2)(p-z)^5 \leq 0$. This also contradicts the assumption.
Consider the solutions in \eqref{eq: 02 N1 12}. Assume that $p \geq (2 + \sqrt{5})z$. Then, $\gamma < 0$ because of $-p^4 + 4zp^3 - 2z^2p^2 + 3z^4 < 0$. This contradicts the assumption. Assume that $p \leq z/(2 + \sqrt{5})$. Then, $\alpha < 0$ because of $3p^4 - 2 z^2p^2 + 4z^3p - z^4 < 0$. This also contradicts the assumption.
Therefore, it can be proved that if
a biquadratic impedance $Z(s) \in \mathcal{Z}_{p2,z2}$ is realizable as in Fig.~2(a),
where $N_1$ is one of the configurations in Fig.~\ref{fig: Two-reactive configurations N1}(b)
and $N_2$ is a two-reactive four-element series-parallel network, then the condition of Lemma~1 holds.
Then, it turns to the case where $N_1$ is a configurations in Figs.~\ref{fig: Two-reactive configurations N1}(c) and $N_2$ is a two-reactive four-element series-parallel network.
It is calculated that the impedance of the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(c) is in the form of
\begin{equation} \label{eq: Z1 c configuration}
Z_1(s) = \frac{s(R_1L_1C_1 s + L_1)}{L_1C_1 s^2 + R_1C_1 s + 1}.
\end{equation}
Since it is assumed that the condition of Theorem~1 does not hold, $N_2$ cannot be equivalent to
a series-parallel network containing fewer than four elements.
If $Z(s)$ is realizable as in Fig.~2(a),
where $N_1$ is the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(c)
and $N_2$ is a two-reactive four-element series-parallel network, then
the impedance of $N_1$ is in the form of
\begin{equation*}
Z_1(s) = \frac{ms(s + p/2)}{(s + p)^2},
\end{equation*}
where $m > 0$, and the impedance of $N_2$ is in the form of
\eqref{eq: 01 N1 00 Z2},
where $\beta$, $\gamma$ $> 0$, $\alpha$ $\geq 0$, and the condition of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements} holds. Since
$(m + \alpha) s^2 + (mp/2 + \beta) s + \gamma
= k(s + z)^2$,
it follows that
\begin{align}
\frac{mp}{2} + \beta &= 2(m+\alpha)z, \label{eq: 03 N1 01} \\
\gamma &= (m+\alpha)z^2. \label{eq: 03 N1 02}
\end{align}
If $Z_2(s)$ satisfies Condition~1 of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements}, then
$\beta$, $\gamma$ $> 0$, $\alpha = 0$, and $\gamma < 2 \beta p$. Together with \eqref{eq: 03 N1 01} and \eqref{eq: 03 N1 02}, one obtains
\begin{equation} \label{eq: 03 N1 03}
\beta = - \frac{1}{2}m(p-4z), ~~~ \gamma = mz^2,
\end{equation}
which implies $p < 4z$ by $\beta > 0$.
Substituting \eqref{eq: 03 N1 03} into $\gamma < 2 \beta p$ yields
$m(p^2 - 4zp + z^2) < 0$,
which further implies $z/(2 + \sqrt{3}) < p < (2 + \sqrt{3})z$ together with \eqref{eq: 03 N1 03}. Thus, the condition of Lemma~1 holds.
Since $\gamma > 0$, $Z_2(s)$ cannot satisfy Condition~2 of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements}.
If $Z_2(s)$ satisfies Condition~3 of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements}, then $\alpha$, $\beta$, $\gamma$ $> 0$ and $\alpha p^2 - \gamma = 0$. Together with \eqref{eq: 03 N1 01} and \eqref{eq: 03 N1 02}, one obtains
\begin{equation*}
\alpha = \frac{mz^2}{(p-z)(p+z)}, ~~~
\beta = -\frac{mp(p^2 - 4zp - z^2)}{2(p-z)(p+z)}, ~~~
\gamma = \frac{mz^2p^2}{(p-z)(p+z)},
\end{equation*}
which implies $z < p < (2 + \sqrt{5})z$ by $\alpha$, $\beta$, $\gamma$ $> 0$. Thus, the condition of Lemma~1 holds.
If $Z_2(s)$ satisfies Condition~4 of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements}, then $\alpha$, $\beta$, $\gamma$ $> 0$, $\alpha p^2 < \gamma$, and
either
$3 \alpha p^2 + \gamma - 2\beta p = 0$ or $\beta^2 p^2 + \gamma^2 - \alpha \gamma p^2 - 2 \beta \gamma p = 0$.
For the case of $3 \alpha p^2 + \gamma - 2\beta p = 0$, together with \eqref{eq: 03 N1 01} and \eqref{eq: 03 N1 02}, one obtains
\begin{equation*}
\alpha = - \frac{m(p^2 - 4zp + z^2)}{(3p - z)(p - z)}, ~~~
\beta = - \frac{mp(3p^2 - 12zp + z^2)}{2(3p-z)(p-z)}, ~~~
\gamma = \frac{2mz^2p^2}{(3p-z)(p-z)},
\end{equation*}
which implies $z/(2 + \sqrt{3}) < p < z/3$ or $z < p < (2 + \sqrt{3})z$ by $\alpha$, $\beta$, $\gamma$ $> 0$. Thus, the condition of Lemma~1 holds.
For the case of $\beta^2 p^2 + \gamma^2 - \alpha \gamma p^2 - 2 \beta \gamma p = 0$, together with \eqref{eq: 03 N1 01} and \eqref{eq: 03 N1 02}, one obtains
\begin{equation} \label{eq: 03 N1 07}
\begin{split}
\alpha &= \frac{m(2p^3 - 8zp^2 + 8z^2p - 2z^3 + p^2\sqrt{(p-z)(p-3z)})}{2z(3p-z)(p-z)}, \\
\beta &= \frac{mp(p^2 - z^2 + 2p\sqrt{(p-z)(p-3z)})}{2(3p-z)(p-z)}, \\
\gamma &= \frac{mzp^2(2p-2z+\sqrt{(p-z)(p-3z)})}{2(3p-z)(p-z)},
\end{split}
\end{equation}
or
\begin{equation} \label{eq: 03 N1 08}
\begin{split}
\alpha &= \frac{m(2p^3 - 8zp^2 + 8z^2p - 2z^3 - p^2\sqrt{(p-z)(p-3z)} )}{2z(3p-z)(p-z)}, \\
\beta &= \frac{mp(p^2 - z^2 - 2p\sqrt{(p-z)(p-3z)})}{2(3p-z)(p-z)}, \\
\gamma &= \frac{2p - 2z - \sqrt{(p-z)(p-3z)}}{2(3p-z)(p-z)},
\end{split}
\end{equation}
where $p < z$ or $p > 3z$ must hold to guarantee the existence of the solutions.
Consider the solutions in \eqref{eq: 03 N1 07}. Substituting \eqref{eq: 03 N1 07} into $\alpha p^2 < \gamma$ yields
\begin{equation} \label{eq: 03 N1 09}
\frac{mp^2(2p(p-3z) + (p+z)\sqrt{(p-z)(p-3z)})}{2(3p-z)z} < 0.
\end{equation}
It is further implied from \eqref{eq: 03 N1 09} that $p < z$. Moreover, since $((p+z)\sqrt{(p-z)(p-3z)})^2 - (2p^2 - 6zp)^2 = -(3p-z)(p-3z)(p^2 - 4zp - z^2)$, it is implied that $z/3 < p < z$, which satisfies the condition of Theorem~1.
Consider the solutions in \eqref{eq: 03 N1 08}. The assumption that $\beta$, $\gamma$ $> 0$ implies $z/3 < p < z$ or $p > 3z$. Assume that $p \geq (2 + \sqrt{5})z$. Then, $\beta \leq 0$ since $p^2 - z^2 > 0$ and $(p^2 - z^2)^2 - (2p\sqrt{(p-z)(p-3z)})^2 =
-(p-z)(3p-z)(p^2 - 4zp - z^2) \leq 0$.
This contradicts the assumption.
If $Z_2(s)$ satisfies Condition~5 of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements}, then
$\alpha$, $\beta$, $\gamma$ $> 0$, $\alpha p^2 > \gamma$, and either $\alpha p^2 + 3 \gamma - 2\beta p = 0$
or $\alpha^2 p^2 + \beta^2 - 2 \alpha \beta p - \alpha \gamma = 0$.
For the case of $\alpha p^2 + 3 \gamma - 2\beta p = 0$, together with \eqref{eq: 03 N1 01} and \eqref{eq: 03 N1 02}, one obtains
\begin{equation} \label{eq: 03 N1 12}
\begin{split}
\alpha &= \frac{m(-p^2 + 6zp - 7z^2 + \sqrt{-z^2(p^2 - 4zp - z^2)})}{2(p-z)(p-3z)}, \\
\beta &= \frac{m(-p^3 + 6zp^2 - 7z^2p - 2z^3 + 2z\sqrt{-z^2(p^2 - 4zp - z^2)})}{2(p-z)(p-3z)}, \\
\gamma &= \frac{mz^2(p^2 - 2zp - z^2 + \sqrt{-z^2(p^2 - 4zp - z^2)})}{2(p-z)(p-3z)},
\end{split}
\end{equation}
or
\begin{equation} \label{eq: 03 N1 13}
\begin{split}
\alpha &= \frac{m(-p^2 + 6zp - 7z^2 - \sqrt{-z^2(p^2 - 4zp - z^2)})}{2(p-z)(p-3z)}, \\
\beta &= \frac{m(-p^3 + 6zp^2 - 7z^2p - 2z^3 - 2z\sqrt{-z^2(p^2 - 4zp - z^2)})}{2(p-z)(p-3z)}, \\
\gamma &= \frac{mz^2(p^2 - 2zp - z^2 - \sqrt{-z^2(p^2 - 4zp - z^2)})}{2(p-z)(p-3z)},
\end{split}
\end{equation}
where $p \leq (2 + \sqrt{5})z$ must hold to guarantee the existence of the solutions. Assume that $p = (2 + \sqrt{5})z$. It is implied from \eqref{eq: 03 N1 12} and \eqref{eq: 03 N1 13} that $\beta = 0$, which contradicts the assumption.
Assume that $p \leq z/(2 + \sqrt{5})$. For \eqref{eq: 03 N1 12}, it is derived that $\alpha < 0$ since $(-z^2(p^2 - 4zp - z^2))^2 - (-p^2 + 6zp - 7z^2)^2 = -(p-z)(p-3z)(p-4z)^2 < 0$. This contradicts the assumption. For \eqref{eq: 03 N1 13}, it is derived that $\alpha$, $\beta$, $\gamma$ $< 0$ since $-p^2 + 6zp - 7z^2 < 0$, $-p^3 + 6zp^2 - 7z^2p - 2z^3 < 0$, and $p^2 - 2zp - z^2 < 0$.
This also contradicts the assumption.
If $Z_2(s)$ satisfies Condition~6 of Lemma~\ref{lemma: biquadratic impedances Z2 Four elements}, then $\alpha$, $\beta$, $\gamma$ $> 0$ and $\alpha^2 p^4 + 6 \alpha \gamma p^2 + \gamma^2 - 2\alpha \beta p^3 - 2 \beta \gamma p = 0$.
Together with \eqref{eq: 03 N1 01} and \eqref{eq: 03 N1 02}, one obtains
\begin{equation} \label{eq: 03 N1 14}
\alpha = -m, ~~~ \beta = -\frac{mp}{2}, ~~~ \gamma = 0,
\end{equation}
or
\begin{equation} \label{eq: 03 N1 15}
\begin{split}
\alpha &= - \frac{mz^2(p^2 - 4zp + z^2)}{(p-z)^4}, \\
\beta &= - \frac{mp(p^4 - 8zp^3 + 22z^2p^2 - 24z^3p + z^4)}{2(p-z)^4}, \\
\gamma &= \frac{mz^2p^2(p^2 - 4zp + 5z^2)}{(p-z)^4}.
\end{split}
\end{equation}
It is obvious that \eqref{eq: 03 N1 14} is impossible. For \eqref{eq: 03 N1 15}, one implies $z/(2 + \sqrt{3}) < p < (2 + \sqrt{3})z$ by $\alpha$, $\beta$, $\gamma$ $> 0$. Thus, the condition of Lemma~1 holds.
Therefore, it can be proved that if
a biquadratic impedance $Z(s) \in \mathcal{Z}_{p2,z2}$ is realizable as in Fig.~2(a),
where $N_1$ is one of the configurations in Fig.~\ref{fig: Two-reactive configurations N1}(c)
and $N_2$ is a two-reactive four-element series-parallel network, then the condition of Lemma~1 holds.
It is clear that any network in Fig.~2(a), where $N_1$ is a configuration in Fig.~\ref{fig: Two-reactive configurations N1}(d) and $N_2$ is a two-reactive four-element series-parallel network can be a frequency inverse network of the case where $N_1$ is a configuration in Fig.~\ref{fig: Two-reactive configurations N1}(c). By the principle of frequency inverse,
if
a biquadratic impedance $Z(s) \in \mathcal{Z}_{p2,z2}$ is realizable as in Fig.~2(a),
where $N_1$ is one of the configurations in Fig.~\ref{fig: Two-reactive configurations N1}(d)
and $N_2$ is a two-reactive four-element series-parallel network, then the condition of Lemma~1 holds.
\end{proof}
\begin{lemma}
\label{lemma: Possible configurations of N1 one-reactive three-element and N2 three-reactive four-element}
{Consider the four-reactive seven-element series-parallel network in Fig.~2(a), realizing a biquadratic impedance $Z(s)$ in the form of \eqref{eq: general biquadratic impedances} with $A$, $B$, $C$, $D$, $E$, $F$ $> 0$, where $N_1$ is a one-reactive three-element series-parallel network and $N_2$ is a three-reactive four-element series-parallel network. If $Z(s)$ cannot be realized as a series-parallel network containing fewer than seven elements, then $N_1$ will be equivalent to one of the configurations in Fig.~\ref{fig: One-reactive Three-element N1}, and $N_2$ will be equivalent to one of the configurations in Fig.~\ref{fig: Three-reactive Four-element N2}.}
\end{lemma}
\begin{proof}
For any realization of $Z(s)$,
there is no cut-set $\mathcal{C}(a,a')$ corresponding to one kind of reactive elements, where $a$ and $a'$ denote two terminal vertices,
by \cite[Lemma~1]{WCH14}.
The possible network graphs for subnetworks $N_1$ and $N_2$ are listed as in Figs.~\ref{fig: Three-element graphs} and \ref{fig: Four-element graphs}, respectively. Based on the method of enumeration and the equivalence in \cite[Lemma~11]{JS11}, $N_1$ can be equivalent to one of configurations in Fig.~\ref{fig: One-reactive Three-element N1}, and $N_2$ can be equivalent to one of configurations in Fig.~\ref{fig: Three-reactive Four-element N2}.
\end{proof}
\begin{figure}
\caption{One-reactive three-element series-parallel configurations for the $N_1$ mentioned in Lemma~\ref{lemma: Possible configurations of N1 one-reactive three-element and N2 three-reactive four-element}
\label{fig: One-reactive Three-element N1 a}
\label{fig: One-reactive Three-element N1 b}
\label{fig: One-reactive Three-element N1}
\end{figure}
\begin{figure}
\caption{Three-reactive four-element series-parallel configurations for the $N_2$ mentioned in Lemma~\ref{lemma: Possible configurations of N1 one-reactive three-element and N2 three-reactive four-element}
\label{fig: Three-reactive Four-element N2 a}
\label{fig: Three-reactive Four-element N2 b}
\label{fig: Three-reactive Four-element N2 c}
\label{fig: Three-reactive Four-element N2 d}
\label{fig: Three-reactive Four-element N2 e}
\label{fig: Three-reactive Four-element N2 f}
\label{fig: Three-reactive Four-element N2 g}
\label{fig: Three-reactive Four-element N2 h}
\label{fig: Three-reactive Four-element N2}
\end{figure}
\begin{figure}
\caption{Possible network graphs for four-element networks.}
\label{fig: Four-element graphs a}
\label{fig: Four-element graphs b}
\label{fig: Four-element graphs c}
\label{fig: Four-element graphs d}
\label{fig: Four-element graphs e}
\label{fig: Four-element graphs f}
\label{fig: Four-element graphs g}
\label{fig: Four-element graphs h}
\label{fig: Four-element graphs i}
\label{fig: Four-element graphs j}
\label{fig: Four-element graphs}
\end{figure}
\begin{lemma} \label{lemma: four-reactive seven-element configurations not realizable}
A biquadratic impedance $Z(s) \in \mathcal{Z}_{p2,z2}$ cannot be realized as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: One-reactive Three-element N1}(a),
and $N_2$ is one of the configurations in Figs.~\ref{fig: Three-reactive Four-element N2}(a) and \ref{fig: Three-reactive Four-element N2}(c)--\ref{fig: Three-reactive Four-element N2}(e).
\end{lemma}
\begin{proof}
By calculation, the impedance of the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(a) is obtained as
\begin{equation} \label{eq: 01 Noa configurations not realizable}
Z_2(s) = \frac{R_{21}L_{21}C_{22} s^2 + R_{21}}{R_{21}L_{21}C_{21}C_{22} s^3 + L_{21}C_{22} s^2 + R_{21}(C_{21} + C_{22}) s + 1}.
\end{equation}
If $Z(s)$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: One-reactive Three-element N1}(a) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(a), then the impedance of $N_1$ is in the form of
\begin{equation} \label{eq: Z1 one-reactive three-element N1}
Z_1(s) = \frac{ms + q}{s + p_1},
\end{equation}
where $m$, $q$, $p_1$ $> 0$ and
\begin{equation} \label{eq: Realizability of N1 one-reactive three-element}
q - mp_1 > 0
\end{equation}
holds, and moreover the impedance of $N_2$ is in the form of
\begin{equation} \label{eq: 02 Noa configurations not realizable}
Z_2(s) = \frac{\alpha s^2 + \gamma}{(s+p_1)(s+p)^2},
\end{equation}
where $\alpha$, $\gamma$ $> 0$. Then, it follows from \eqref{eq: 01 Noa configurations not realizable} and \eqref{eq: 02 Noa configurations not realizable} that
\begin{align}
\frac{1}{C_{21}} &= \alpha,
\label{eq: 03 Noa configurations not realizable} \\
\frac{1}{L_{21}C_{21}C_{22}} &= \gamma,
\label{eq: 04 Noa configurations not realizable} \\
\frac{1}{R_{21} C_{21}} &= 2p + p_1,
\label{eq: 05 Noa configurations not realizable} \\
\frac{C_{21} + C_{22}}{L_{21} C_{21} C_{22}} &= p(p+2p_1),
\label{eq: 06 Noa configurations not realizable} \\
\frac{1}{R_{21} L_{21} C_{21} C_{22}} &= p_1p^2.
\label{eq: 07 Noa configurations not realizable}
\end{align}
From \eqref{eq: 03 Noa configurations not realizable}, one obtains
\begin{equation} \label{eq: C21 Noa configurations not realizable}
C_{21} = \frac{1}{\alpha}.
\end{equation}
It follows from \eqref{eq: 05 Noa configurations not realizable} and \eqref{eq: C21 Noa configurations not realizable} that
\begin{equation} \label{eq: R21 Noa configurations not realizable}
R_{21} = \frac{\alpha}{2p+p_1}.
\end{equation}
By \eqref{eq: 04 Noa configurations not realizable}, \eqref{eq: 06 Noa configurations not realizable}, and \eqref{eq: C21 Noa configurations not realizable}, one obtains
\begin{equation} \label{eq: C22 Noa configurations not realizable}
C_{22} = \frac{\alpha p^2 + 2\alpha p_1 p - \gamma}{\alpha \gamma}.
\end{equation}
By \eqref{eq: 04 Noa configurations not realizable}, \eqref{eq: C21 Noa configurations not realizable}, and \eqref{eq: C22 Noa configurations not realizable}, one obtains
\begin{equation} \label{eq: L21 Noa configurations not realizable}
L_{21} = \frac{\alpha^2}{\alpha p^2 + 2 \alpha p_1 p - \gamma}.
\end{equation}
Substituting \eqref{eq: C21 Noa configurations not realizable}--\eqref{eq: L21 Noa configurations not realizable} into \eqref{eq: 07 Noa configurations not realizable} gives
\begin{equation} \label{eq: Condition01 Noa configurations not realizable}
\alpha p_1 p^2 - 2\gamma p - \gamma p_1 = 0.
\end{equation}
The assumption that $C_{22} > 0$ and $L_{21} > 0$ gives
\begin{equation} \label{eq: Condition02 Noa configurations not realizable}
\alpha p^2 + 2\alpha p_1 p - \gamma > 0.
\end{equation}
Based on \eqref{eq: Z1 one-reactive three-element N1} and \eqref{eq: 02 Noa configurations not realizable}, calculation yields
\begin{equation} \label{eq: Z1 + Z2 Noa configurations not realizable}
\begin{split}
&Z(s) = Z_1(s) + Z_2(s) = \\
&\frac{ms^3 + (2mp + \alpha + q)s^2 + p(mp + 2q)s + (qp^2 + \gamma)}{(s+p_1)(s+p)^2}.
\end{split}
\end{equation}
Comparing (1) with \eqref{eq: Z1 + Z2 Noa configurations not realizable}, one obtains
\begin{align}
m &= k, \label{eq: 101 Noa configurations not realizable} \\
2mp + \alpha + q &= k(p_1 + 2z), \label{eq: 102 Noa configurations not realizable} \\
p(mp + 2q) &= kz(z+2p_1), \label{eq: 103 Noa configurations not realizable} \\
qp^2 + \gamma &= kp_1z^2. \label{eq: 104 Noa configurations not realizable}
\end{align}
Then, \eqref{eq: 101 Noa configurations not realizable} and \eqref{eq: 103 Noa configurations not realizable} together yield
\begin{equation} \label{eq: q Noa configurations not realizable}
q = \frac{-k(p^2 - 2p_1z - z^2)}{2p}.
\end{equation}
By \eqref{eq: 101 Noa configurations not realizable}, \eqref{eq: 102 Noa configurations not realizable}, and \eqref{eq: q Noa configurations not realizable}, one obtains
\begin{equation} \label{eq: alpha Noa configurations not realizable}
\alpha = \frac{-k(p-z)(3p - z - 2p_1)}{2p}.
\end{equation}
By \eqref{eq: 101 Noa configurations not realizable} and \eqref{eq: 104 Noa configurations not realizable}, one obtains
\begin{equation} \label{eq: gamma Noa configurations not realizable}
\gamma = \frac{1}{2}k(p-z)(p^2 + zp - 2zp_1).
\end{equation}
Together with \eqref{eq: 101 Noa configurations not realizable} and \eqref{eq: q Noa configurations not realizable}--\eqref{eq: gamma Noa configurations not realizable}, condition~\eqref{eq: Realizability of N1 one-reactive three-element} is equivalent to
$p < z$, and \eqref{eq: Condition01 Noa configurations not realizable} and \eqref{eq: Condition02 Noa configurations not realizable} are equivalent to
\begin{align}
(p+z)p_1^2 - 2p(p-z)p_1 - p^2(p+z) &= 0,
\label{eq: Equivalent Condition01 Noa configurations not realizable} \\
p^2 + p_1p - (p_1^2 + zp_1) &> 0,
\label{eq: Equivalent Condition02 Noa configurations not realizable}
\end{align}
respectively.
Then, it follows from \eqref{eq: Equivalent Condition01 Noa configurations not realizable} and \eqref{eq: Equivalent Condition02 Noa configurations not realizable} that $-p_1(p-z)^2/(p+z) > 0$, which is impossible.
Therefore, $Z(s) \in \mathcal{Z}_{p2,z2}$ cannot be realized as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: One-reactive Three-element N1}(a),
and $N_2$ is the configurations in Fig.~\ref{fig: Three-reactive Four-element N2}(a).
It is calculated that the impedance of the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(c) is
\begin{equation} \label{eq: 01 Noc configurations not realizable}
Z_2(s) = \frac{R_{21} L_{21} C_{22} s^2 + L_{21} s}{R_{21} L_{21} C_{21} C_{22} s^3 + L_{21} (C_{21} + C_{22}) s^2 + R_{21} C_{22} s + 1}.
\end{equation}
If $Z(s)$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: One-reactive Three-element N1}(a)
and $N_2$ is the configuration in
Fig.~\ref{fig: Three-reactive Four-element N2}(c), then the impedance of $N_1$ is in the form of \eqref{eq: Z1 one-reactive three-element N1}, where $m$, $q$, $p_1$ $> 0$ and \eqref{eq: Realizability of N1 one-reactive three-element} holds, and the impedance of $N_2$ is in the form of
\begin{equation} \label{eq: 02 Noc configurations not realizable}
Z_2(s) = \frac{\alpha s^2 + \beta s}{(s+p_1)(s+p)^2},
\end{equation}
where $\alpha$, $\beta$ $> 0$. Then, it follows from \eqref{eq: 01 Noc configurations not realizable} and \eqref{eq: 02 Noc configurations not realizable} that
\begin{align}
\frac{1}{C_{21}} &= \alpha,
\label{eq: 03 Noc configurations not realizable} \\
\frac{1}{R_{21} C_{21} C_{22}} &= \beta,
\label{eq: 04 Noc configurations not realizable} \\
\frac{C_{21} + C_{22}}{R_{21} C_{21} C_{22}} &= 2p + p_1,
\label{eq: 05 Noc configurations not realizable} \\
\frac{1}{L_{21} C_{21}} &= p(p+2p_1),
\label{eq: 06 Noc configurations not realizable} \\
\frac{1}{R_{21} L_{21} C_{21} C_{22}} &= p_1p^2.
\label{eq: 07 Noc configurations not realizable}
\end{align}
It follows from \eqref{eq: 03 Noc configurations not realizable} that
\begin{equation} \label{eq: C21 Noc configurations not realizable}
C_{21} = \frac{1}{\alpha}.
\end{equation}
By \eqref{eq: 06 Noc configurations not realizable} and \eqref{eq: C21 Noc configurations not realizable}, one obtains
\begin{equation} \label{eq: L21 Noc configurations not realizable}
L_{21} = \frac{\alpha}{p(p+2p_1)}.
\end{equation}
By \eqref{eq: 04 Noc configurations not realizable}, \eqref{eq: 05 Noc configurations not realizable}, and \eqref{eq: C21 Noc configurations not realizable}, one obtains
\begin{equation} \label{eq: C22 Noc configurations not realizable}
C_{22} = \frac{2\alpha p + \alpha p_1 - \beta}{\alpha \beta}.
\end{equation}
Then, \eqref{eq: 04 Noc configurations not realizable}, \eqref{eq: C21 Noc configurations not realizable}, and \eqref{eq: C22 Noc configurations not realizable} yield
\begin{equation} \label{eq: R21 Noc configurations not realizable}
R_{21} = \frac{\alpha^2}{2\alpha p + \alpha p_1 - \beta}.
\end{equation}
Then, it follows from \eqref{eq: 07 Noc configurations not realizable}--\eqref{eq: R21 Noc configurations not realizable} that
\begin{equation} \label{eq: Condition01 Noc configurations not realizable}
(\beta - \alpha p_1)p + 2\beta p_1 = 0.
\end{equation}
The assumption that
$R_{21} > 0$ and $C_{22} > 0$ gives
\begin{equation} \label{eq: Condition02 Noc configurations not realizable}
2\alpha p + \alpha p_1 - \beta > 0.
\end{equation}
By \eqref{eq: Z1 one-reactive three-element N1} and \eqref{eq: 02 Noc configurations not realizable}, $Z(s)$ is calculated as
\begin{equation} \label{eq: Z1 + Z2 Noc configurations not realizable}
Z(s) = Z_1(s) + Z_2(s) =
\frac{ms^3 + (2mp + \alpha + q)s^2 + (mp^2 + 2qp + \beta)s + qp^2}{(s+p_1)(s+p)^2}.
\end{equation}
Comparing (1) with \eqref{eq: Z1 + Z2 Noc configurations not realizable}, one obtains
\begin{align}
m &= k, \label{eq: 101 Noc configurations not realizable} \\
2mp + \alpha + q &= k(p_1 + 2z), \label{eq: 102 Noc configurations not realizable} \\
mp^2 + 2qp + \beta &= kz(z+2p_1), \label{eq: 103 Noc configurations not realizable} \\
qp^2 &= kp_1z^2. \label{eq: 104 Noc configurations not realizable}
\end{align}
Then, \eqref{eq: 104 Noc configurations not realizable} yields
\begin{equation} \label{eq: q Noc configurations not realizable}
q = \frac{kp_1z^2}{p^2}.
\end{equation}
By \eqref{eq: 101 Noc configurations not realizable}, \eqref{eq: 102 Noc configurations not realizable}, and \eqref{eq: q Noc configurations not realizable}, one obtains
\begin{equation} \label{eq: alpha Noc configurations not realizable}
\alpha = -\frac{k(p-z)(2p^2-p_1(p+z))}{p^2}.
\end{equation}
It follows from \eqref{eq: 101 Noc configurations not realizable}, \eqref{eq: 103 Noc configurations not realizable}, and \eqref{eq: q Noc configurations not realizable} that
\begin{equation} \label{eq: beta Noc configurations not realizable}
\beta = - \frac{k(p-z)(p^2 + z(p-2p_1))}{p}.
\end{equation}
Together with \eqref{eq: 101 Noc configurations not realizable} and \eqref{eq: q Noc configurations not realizable}--\eqref{eq: beta Noc configurations not realizable}, condition~\eqref{eq: Realizability of N1 one-reactive three-element} is equivalent to $p < z$, and \eqref{eq: Condition01 Noc configurations not realizable} and $\alpha > 0$ are equivalent to
\begin{equation} \label{eq: Equivalent Condition01 Noc configurations not realizable}
(p - 3z)p_1^2 + p^2(p+z) = 0
\end{equation}
and
\begin{equation} \label{eq: alpha positive Noc configurations not realizable}
p_1 < \frac{2p^2}{p+z},
\end{equation}
respectively. Combining \eqref{eq: Equivalent Condition01 Noc configurations not realizable} and \eqref{eq: alpha positive Noc configurations not realizable}, one obtains $p^2(5p+z)(p-z)^2/(p+z)^2 < 0$, which is impossible.
Therefore, $Z(s) \in \mathcal{Z}_{p2,z2}$ cannot be realized as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: One-reactive Three-element N1}(a),
and $N_2$ is the configurations in Fig.~\ref{fig: Three-reactive Four-element N2}(c).
It is calculated that the impedance of the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(d) is
\begin{equation} \label{eq: 01 Nod configurations not realizable}
Z_2(s) = \frac{L_{21} L_{22} s^2 + R_{21} L_{21} s}{L_{21} L_{22} C_{21} s^3 + R_{21} L_{21} C_{21} s^2 + (L_{21} + L_{22}) s + R_{21}}.
\end{equation}
If $Z(s)$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: One-reactive Three-element N1}(a) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(d), then the impedance of $N_1$ is in the form of \eqref{eq: Z1 one-reactive three-element N1}, where $m$, $q$, $p_1$ $> 0$ and \eqref{eq: Realizability of N1 one-reactive three-element} holds, and the impedance of $N_2$ is in the form of
\eqref{eq: 02 Noc configurations not realizable} where $\alpha$, $\beta$ $> 0$. Then, it follows from \eqref{eq: 02 Noc configurations not realizable} and \eqref{eq: 01 Nod configurations not realizable} that
\begin{align}
\frac{1}{C_{21}} &= \alpha, \label{eq: 03 Nod configurations not realizable} \\
\frac{R_{21}}{L_{22} C_{21}} &= \beta, \label{eq: 04 Nod configurations not realizable} \\
\frac{R_{21}}{L_{22}} &= 2p + p_1, \label{eq: 05 Nod configurations not realizable} \\
\frac{L_{21}+L_{22}}{L_{21} L_{22} C_{21}} &= p(p+2p_1), \label{eq: 06 Nod configurations not realizable} \\
\frac{R_{21}}{L_{21} L_{22} C_{21}} &= p_1p^2. \label{eq: 07 Nod configurations not realizable}
\end{align}
It follows from \eqref{eq: 03 Nod configurations not realizable} that
\begin{equation} \label{eq: C21 Nod configurations not realizable}
C_{21} = \frac{1}{\alpha}.
\end{equation}
By \eqref{eq: 04 Nod configurations not realizable}, \eqref{eq: 05 Nod configurations not realizable}, and \eqref{eq: C21 Nod configurations not realizable}, one obtains
\begin{equation} \label{eq: Condition01 Nod configurations not realizable}
2 \alpha p + \alpha p_1 - \beta = 0.
\end{equation}
Then, \eqref{eq: 05 Nod configurations not realizable}, \eqref{eq: 07 Nod configurations not realizable}, and \eqref{eq: C21 Nod configurations not realizable} yield
\begin{equation} \label{eq: L21 Nod configurations not realizable}
L_{21} = \frac{\alpha (2p + p_1)}{p_1 p^2}.
\end{equation}
It follows from \eqref{eq: 06 Nod configurations not realizable}, \eqref{eq: C21 Nod configurations not realizable}, and \eqref{eq: L21 Nod configurations not realizable} that
\begin{equation} \label{eq: L22 Nod configurations not realizable}
L_{22} = \frac{\alpha (2p + p_1)}{2p(p+p_1)^2}.
\end{equation}
By \eqref{eq: 05 Nod configurations not realizable} and \eqref{eq: L22 Nod configurations not realizable}, one obtains
\begin{equation}
R_{21} = \frac{\alpha (2p + p_1)^2}{2p(p+p_1)^2}.
\end{equation}
By \eqref{eq: Z1 one-reactive three-element N1} and \eqref{eq: 02 Noc configurations not realizable}, $Z(s)$ is calculated as
\eqref{eq: Z1 + Z2 Noc configurations not realizable}. Then, one obtains \eqref{eq: 101 Noc configurations not realizable}--\eqref{eq: beta Noc configurations not realizable}. Furthermore, it follows from conditions~\eqref{eq: Realizability of N1 one-reactive three-element} and \eqref{eq: Condition01 Nod configurations not realizable} that $z/3 < p < z$, which satisfies the condition of Theorem~1.
Therefore, $Z(s) \in \mathcal{Z}_{p2,z2}$ cannot be realized as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: One-reactive Three-element N1}(a),
and $N_2$ is the configurations in Fig.~\ref{fig: Three-reactive Four-element N2}(d).
It is calculated that the impedance of the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(e) is
\begin{equation} \label{eq: 01 Noe configurations not realizable}
Z_2(s) = \frac{R_{21} L_{21} C_{22} s^2 + L_{21} s + R_{21}}{R_{21} L_{21} C_{21} C_{22} s^3 + L_{21} (C_{21} + C_{22}) s^2 + R_{21} C_{21} s + 1}.
\end{equation}
If $Z(s)$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: One-reactive Three-element N1}(a) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(e), then the impedance of $N_1$ is in the form of \eqref{eq: Z1 one-reactive three-element N1}, where $m$, $q$, $p_1$ $> 0$ and \eqref{eq: Realizability of N1 one-reactive three-element} holds, and the impedance of $N_2$ is in the form of
\begin{equation} \label{eq: 02 Noe configurations not realizable}
Z_2(s) = \frac{\alpha s^2 + \beta s + \gamma}{(s+p_1)(s+p)^2},
\end{equation}
where $\alpha$, $\beta$, $\gamma$ $> 0$. Then, it follows from \eqref{eq: 01 Noe configurations not realizable} and \eqref{eq: 02 Noe configurations not realizable} that
\begin{align}
\frac{1}{C_{21}} &= \alpha, \label{eq: 03 Noe configurations not realizable} \\
\frac{1}{R_{21} C_{21} C_{22}} &= \beta, \label{eq: 04 Noe configurations not realizable} \\
\frac{1}{L_{21} C_{21} C_{22}} &= \gamma, \label{eq: 05 Noe configurations not realizable} \\
\frac{C_{21} + C_{22}}{R_{21}C_{21}C_{22}} &= 2p + p_1, \label{eq: 06 Noe configurations not realizable} \\
\frac{1}{L_{21} C_{22}} &= p(p+2p_1), \label{eq: 07 Noe configurations not realizable} \\
\frac{1}{R_{21}L_{21}C_{21}C_{22}} &= p_1p^2. \label{eq: 08 Noe configurations not realizable}
\end{align}
It follows from \eqref{eq: 03 Noe configurations not realizable} that
\begin{equation} \label{eq: C21 Noe configurations not realizable}
C_{21} = \frac{1}{\alpha}.
\end{equation}
By \eqref{eq: 04 Noe configurations not realizable}, \eqref{eq: 06 Noe configurations not realizable}, and \eqref{eq: C21 Noe configurations not realizable}, one obtains
\begin{equation} \label{eq: C22 Noe configurations not realizable}
C_{22} = \frac{2\alpha p + \alpha p_1 - \beta}{\alpha \beta}.
\end{equation}
By \eqref{eq: 04 Noe configurations not realizable}, \eqref{eq: C21 Noe configurations not realizable}, and \eqref{eq: C22 Noe configurations not realizable}, one obtains
\begin{equation} \label{eq: R21 Noe configurations not realizable}
R_{21} = \frac{\alpha^2}{2\alpha p + \alpha p_1 - \beta}.
\end{equation}
Then, \eqref{eq: 05 Noe configurations not realizable}, \eqref{eq: C21 Noe configurations not realizable}, and \eqref{eq: C22 Noe configurations not realizable} yield
\begin{equation} \label{eq: L21 Noe configurations not realizable}
L_{21} = \frac{\alpha^2 \beta}{\gamma (2\alpha p + \alpha p_1 - \beta)}.
\end{equation}
It follows from \eqref{eq: 07 Noe configurations not realizable}, \eqref{eq: C22 Noe configurations not realizable}, and \eqref{eq: L21 Noe configurations not realizable} that
\begin{equation} \label{eq: Condition01 Noe configurations not realizable}
\alpha p^2 + 2\alpha p_1 p - \gamma = 0.
\end{equation}
Moreover, substituting \eqref{eq: C21 Noe configurations not realizable}--\eqref{eq: L21 Noe configurations not realizable} into \eqref{eq: 08 Noe configurations not realizable} yields
\begin{equation} \label{eq: Condition02 Noe configurations not realizable}
\alpha^2 p_1 p^2 - 2\alpha \gamma p - \gamma (\alpha p_1 - \beta) = 0.
\end{equation}
The assumption that $R_{21} > 0$, $L_{21} > 0$, and $C_{22} > 0$ gives
\begin{equation} \label{eq: Condition03 Noe configurations not realizable}
2\alpha p + \alpha p_1 - \beta > 0.
\end{equation}
By \textcolor[rgb]{0.98,0.00,0.00}{(28)} and \eqref{eq: 02 Noe configurations not realizable}, $Z(s)$ is calculated as
\begin{equation} \label{eq: Z1 + Z2 Noe configurations not realizable}
Z(s) = Z_1(s) + Z_2(s) =
\frac{ms^3 + (2mp+\alpha + q)s^2 + (mp^2 + 2qp + \beta)s + (qp^2 + \gamma)}{(s+p_1)(s+p)^2}.
\end{equation}
Comparing (1) with \eqref{eq: Z1 + Z2 Noe configurations not realizable}, one obtains
\begin{align}
m &= k, \label{eq: 101 Noe configurations not realizable} \\
2mp + \alpha + q &= k(p_1 + 2z), \label{eq: 102 Noe configurations not realizable} \\
mp^2 + 2q p + \beta &= kz(z+2p_1), \label{eq: 103 Noe configurations not realizable} \\
q p^2 + \gamma &= kp_1z^2. \label{eq: 104 Noe configurations not realizable}
\end{align}
Then, \eqref{eq: 101 Noe configurations not realizable} and \eqref{eq: 102 Noe configurations not realizable} yield
\begin{equation} \label{eq: alpha + q Noe configurations not realizable}
\alpha + q = k(p_1 + 2z - 2p).
\end{equation}
By \eqref{eq: Condition01 Noe configurations not realizable} and \eqref{eq: 104 Noe configurations not realizable}, one obtains
\begin{equation} \label{eq: alpha + q 02 Noe configurations not realizable}
(p^2 + 2p_1p)\alpha + p^2 q = kp_1z^2.
\end{equation}
By \eqref{eq: alpha + q Noe configurations not realizable} and \eqref{eq: alpha + q 02 Noe configurations not realizable}, one obtains
\begin{equation} \label{eq: alpha Noe configurations not realizable}
\alpha = \frac{k(p-z)(2p^2-p_1p-p_1z)}{2p_1p}
\end{equation}
and
\begin{equation} \label{eq: q Noe configurations not realizable}
q = -\frac{k(2p^3+(3p_1-2z)p^2-(2p_1^2+4p_1z)p+p_1z^2)}{2p_1p}.
\end{equation}
Then, \eqref{eq: 101 Noe configurations not realizable}, \eqref{eq: 103 Noe configurations not realizable}, and \eqref{eq: q Noe configurations not realizable} yield
\begin{equation} \label{eq: beta Noe configurations not realizable}
\beta = \frac{2k(p-z)(p^2+p_1p-p_1z-p_1^2)}{p_1}.
\end{equation}
It follows from \eqref{eq: 104 Noe configurations not realizable} and \eqref{eq: q Noe configurations not realizable} that
\begin{equation} \label{eq: gamma Noe configurations not realizable}
\gamma = \frac{k(p+2p_1)(p-z)(2p^2-p_1p-p_1z)}{2p_1}.
\end{equation}
Based on \eqref{eq: 101 Noe configurations not realizable} and \eqref{eq: q Noe configurations not realizable}, one implies that
\eqref{eq: 01 N1 00 Z2} is equivalent to
\begin{equation} \label{eq: equivalent Condition01 Noe configurations not realizable}
(p-z)(2p^2 + 3p_1p - p_1z) < 0,
\end{equation}
which implies $p < z$. Since the condition of Theorem~1 does not hold, one only needs to consider to the case of $p \leq z/(2 + \sqrt{5})$. Furthermore, \eqref{eq: Condition03 Noe configurations not realizable} is equivalent to
\begin{equation} \label{eq: equivalent Condition02 Noe configurations not realizable}
(p-z)(4p^2 - (3p_1 + 2z)p + p_1z) < 0,
\end{equation}
by \eqref{eq: alpha Noe configurations not realizable} and \eqref{eq: beta Noe configurations not realizable}.
Combining \eqref{eq: equivalent Condition01 Noe configurations not realizable} and \eqref{eq: equivalent Condition02 Noe configurations not realizable}, one obtains
\begin{equation*}
\frac{2zp-4p^2}{z-3p} < p_1 < \frac{2p^2}{z-3p},
\end{equation*}
which implies $p > z/3$. Therefore, the condition of Theorem~1 holds.
Therefore, $Z(s) \in \mathcal{Z}_{p2,z2}$ cannot be realized as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: One-reactive Three-element N1}(a),
and $N_2$ is the configurations in Fig.~\ref{fig: Three-reactive Four-element N2}(e).
\end{proof}
\begin{lemma} \label{lemma: four-reactive seven-element configurations lemma 1}
If a biquadratic impedance $Z(s) \in \mathcal{Z}_{p2,z2}$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: One-reactive Three-element N1}(a)
and $N_2$ is one of the configurations in Figs.~\ref{fig: Three-reactive Four-element N2}(b) and \ref{fig: Three-reactive Four-element N2}(f), then the condition of Lemma~1 holds.
\end{lemma}
\begin{proof}
It is calculated that the impedance of the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(b) is
\begin{equation} \label{eq: 01 Nob configurations not realizable}
Z_2(s) = \frac{R_{21} L_{21} L_{22} C_{21} s^3 + R_{21} L_{21} s}{L_{21} L_{22} C_{21} s^3 + R_{21} C_{21} (L_{21} + L_{22}) s^2 + L_{21} s + R_{21}}.
\end{equation}
If $Z(s)$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: One-reactive Three-element N1}(a) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(b), then the impedance of $N_1$ is in the form of \eqref{eq: Z1 one-reactive three-element N1}, where $m$, $q$, $p_1$ $> 0$ and \eqref{eq: Realizability of N1 one-reactive three-element}
holds, and the impedance of $N_2$ is in the form of
\begin{equation} \label{eq: 02 Nob configurations not realizable}
Z_2(s) = \frac{\alpha s^3 + \gamma s}{(s+p_1)(s+p)^2},
\end{equation}
where $\alpha$, $\gamma$ $> 0$. Then, it follows from \eqref{eq: 01 Nob configurations not realizable} and \eqref{eq: 02 Nob configurations not realizable} that
\begin{align}
R_{21} &= \alpha, \label{eq: 03 Nob configurations not realizable} \\
\frac{R_{21}}{L_{22} C_{21}} &= \gamma, \label{eq: 04 Nob configurations not realizable} \\
\frac{R_{21} (L_{21} + L_{22})}{L_{21} L_{22}} &= 2p + p_1, \label{eq: 05 Nob configurations not realizable} \\
\frac{1}{L_{22} C_{21}} &= p(p+2p_1), \label{eq: 06 Nob configurations not realizable} \\
\frac{R_{21}}{L_{21} L_{22} C_{21}} &= p_1p^2. \label{eq: 07 Nob configurations not realizable}
\end{align}
By \eqref{eq: 03 Nob configurations not realizable}, \eqref{eq: 06 Nob configurations not realizable}, and \eqref{eq: 07 Nob configurations not realizable}, one obtains
\begin{equation} \label{eq: L21 Nob configurations not realizable}
L_{21} = \frac{\alpha (p+2p_1)}{p_1p}.
\end{equation}
Then, \eqref{eq: 03 Nob configurations not realizable}, \eqref{eq: 05 Nob configurations not realizable}, and \eqref{eq: L21 Nob configurations not realizable} yield
\begin{equation} \label{eq: L22 Nob configurations not realizable}
L_{22} = \frac{\alpha (p + 2p_1)}{2(p+p_1)^2}.
\end{equation}
By \eqref{eq: 06 Nob configurations not realizable} and \eqref{eq: L22 Nob configurations not realizable}, one obtains
\begin{equation} \label{eq: C21 Nob configurations not realizable}
C_{21} = \frac{2(p+p_1)^2}{\alpha p (p + 2p_1)^2}.
\end{equation}
Then, \eqref{eq: 03 Nob configurations not realizable}, \eqref{eq: 04 Nob configurations not realizable}, \eqref{eq: L22 Nob configurations not realizable}, and \eqref{eq: C21 Nob configurations not realizable} yield
\begin{equation} \label{eq: Condition01 Nob configurations not realizable}
\alpha p^2 + 2\alpha p_1 p - \gamma = 0.
\end{equation}
By \eqref{eq: Z1 one-reactive three-element N1} and \eqref{eq: 02 Nob configurations not realizable}, $Z(s)$ is calculated as
\begin{equation} \label{eq: Z1 + Z2 Nob configurations not realizable}
Z(s) = Z_1(s) + Z_2(s) = \frac{(m+\alpha) s^3 + (2mp + q) s^2 + (mp^2 + 2qp + \gamma) s + q p^2 }{(s+p_1)(s+p)^2}.
\end{equation}
Comparing (1) with \eqref{eq: Z1 + Z2 Nob configurations not realizable}, one obtains \begin{align}
m + \alpha &= k, \label{eq: 101 Nob configurations not realizable} \\
2mp + q &= k(p_1 + 2z), \label{eq: 102 Nob configurations not realizable} \\
mp^2 + 2qp + \gamma &= kz(z+2p_1), \label{eq: 103 Nob configurations not realizable} \\
qp^2 &= kp_1z^2. \label{eq: 104 Nob configurations not realizable}
\end{align}
By \eqref{eq: 104 Nob configurations not realizable}, one obtains
\begin{equation} \label{eq: q Nob configurations not realizable}
q = \frac{kp_1z^2}{p^2}.
\end{equation}
Then, \eqref{eq: 102 Nob configurations not realizable} and \eqref{eq: q Nob configurations not realizable} yield
\begin{equation} \label{eq: m Nob configurations not realizable}
m = \frac{k(p_1p^2 + 2zp^2 - p_1z^2)}{2p^3}.
\end{equation}
By \eqref{eq: 101 Nob configurations not realizable} and \eqref{eq: m Nob configurations not realizable}, one obtains
\begin{equation} \label{eq: alpha Nob configurations not realizable}
\alpha = \frac{k(p-z)(2p^2 - p_1 p - p_1 z)}{2p^3}.
\end{equation}
It follows from \eqref{eq: 103 Nob configurations not realizable}, \eqref{eq: q Nob configurations not realizable}, and \eqref{eq: m Nob configurations not realizable} that
\begin{equation} \label{eq: gamma Nob configurations not realizable}
\gamma = \frac{-k(p-z)(p_1p + 2zp - 3zp_1)}{2p}.
\end{equation}
Together with \eqref{eq: q Nob configurations not realizable}--\eqref{eq: gamma Nob configurations not realizable}, condition~\eqref{eq: Realizability of N1 one-reactive three-element}
is equivalent to
$p < z$, and \eqref{eq: Condition01 Nob configurations not realizable} and
$m > 0$ are equivalent to
\begin{equation} \label{eq: Equivalent Condition01 Nob configurations not realizable}
(p+z)p_1^2 - 2p(p-z)p_1 - p^2(p+z) = 0
\end{equation}
and
\begin{equation} \label{eq: m positive Nob configurations not realizable}
p_1 < \frac{2zp^2}{z^2 - p^2},
\end{equation}
respectively.
Combining \eqref{eq: Equivalent Condition01 Nob configurations not realizable} and \eqref{eq: m positive Nob configurations not realizable}, one obtains
$z/(2 + \sqrt{3}) < p < z$, which satisfies the condition of Lemma~1.
Therefore, if a biquadratic impedance $Z(s) \in \mathcal{Z}_{p2,z2}$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: One-reactive Three-element N1}(a)
and $N_2$ is one of the configurations in Figs.~\ref{fig: Three-reactive Four-element N2}(b), then the condition of Lemma~1 holds.
It is calculated that the impedance of the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(f) is
\begin{equation} \label{eq: 01 Nof configurations not realizable}
Z_2(s) = \frac{R_{21} L_{21} C_{22} s^2 + L_{21} s + R_{21}}{R_{21} L_{21} C_{21} C_{22} s^3 + L_{21} C_{21} s^2 + R_{21} (C_{21} + C_{22}) s + 1}.
\end{equation}
If $Z(s)$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: One-reactive Three-element N1}(a) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(f), then the impedance of $N_1$ is in the form of
\eqref{eq: Z1 one-reactive three-element N1}, where $m$, $q$, $p_1$ $> 0$ and \eqref{eq: Realizability of N1 one-reactive three-element} holds, and the impedance of $N_2$ is in the form of
\eqref{eq: 02 Noe configurations not realizable},
where $\alpha$, $\beta$, $\gamma$ $> 0$. Then, it follows from \eqref{eq: 02 Noe configurations not realizable} and \eqref{eq: 01 Nof configurations not realizable} that
\begin{align}
\frac{1}{C_{21}} &= \alpha, \label{eq: 03 Nof configurations not realizable} \\
\frac{1}{R_{21} C_{21} C_{22}} &= \beta, \label{eq: 04 Nof configurations not realizable} \\
\frac{1}{L_{21} C_{21} C_{22}} &= \gamma, \label{eq: 05 Nof configurations not realizable} \\
\frac{1}{R_{21} C_{22}} &= 2p + p_1, \label{eq: 06 Nof configurations not realizable} \\
\frac{C_{21} + C_{22}}{L_{21} C_{21} C_{22}} &= p(p+2p_1), \label{eq: 07 Nof configurations not realizable} \\
\frac{1}{R_{21} L_{21} C_{21} C_{22}} &= p_1p^2. \label{eq: 08 Nof configurations not realizable}
\end{align}
It follows from \eqref{eq: 03 Nof configurations not realizable} that
\begin{equation} \label{eq: C21 Nof configurations not realizable}
C_{21} = \frac{1}{\alpha}.
\end{equation}
Then, \eqref{eq: 05 Nof configurations not realizable}, \eqref{eq: 07 Nof configurations not realizable}, and \eqref{eq: C21 Nof configurations not realizable} yield
\begin{equation} \label{eq: C22 Nof configurations not realizable}
C_{22} = \frac{\alpha p^2 + 2\alpha p_1 p - \gamma}{\alpha \gamma}.
\end{equation}
By \eqref{eq: 06 Nof configurations not realizable} and \eqref{eq: C22 Nof configurations not realizable}, one obtains
\begin{equation} \label{eq: R21 Nof configurations not realizable}
R_{21} = \frac{\alpha \gamma}{(\alpha p^2 + 2\alpha p_1 p - \gamma)(2 p + p_1)}.
\end{equation}
Then, it follows from \eqref{eq: 05 Nof configurations not realizable}, \eqref{eq: C21 Nof configurations not realizable}, and \eqref{eq: C22 Nof configurations not realizable} that
\begin{equation} \label{eq: L21 Nof configurations not realizable}
L_{21} = \frac{\alpha^2}{\alpha p^2 + 2\alpha p_1 p - \gamma}.
\end{equation}
By \eqref{eq: 04 Nof configurations not realizable} and \eqref{eq: C21 Nof configurations not realizable}--\eqref{eq: R21 Nof configurations not realizable}, one obtains
\begin{equation} \label{eq: Condition01 Nof configurations not realizable}
2 \alpha p - \beta + \alpha p_1 = 0.
\end{equation}
The assumption that $R_{21} > 0$, $L_{21} > 0$, and $C_{22} > 0$ gives
\begin{equation} \label{eq: Condition02 Nof configurations not realizable}
\alpha p^2 + 2\alpha p_1 p - \gamma > 0.
\end{equation}
Then, \eqref{eq: 08 Nof configurations not realizable}--\eqref{eq: L21 Nof configurations not realizable} yield
\begin{equation} \label{eq: Condition03 Nof configurations not realizable}
2\alpha p p_1^2 + (4\alpha p^2 - \gamma) p_1 + 2p(\alpha p^2 - \gamma) = 0.
\end{equation}
By (28) and \eqref{eq: 02 Noe configurations not realizable}, $Z(s)$ is calculated as
\begin{equation} \label{eq: Z1 + Z2 Nof configurations not realizable}
Z(s) = Z_1(s) + Z_2(s) =
\frac{ms^3 + (2mp + \alpha + q)s^2 + (mp^2 + 2qp + \beta)s + (qp^2 + \gamma)}{(s+p_1)(s+p)^2}.
\end{equation}
Comparing (1) with \eqref{eq: Z1 + Z2 Nof configurations not realizable}, one obtains
\begin{align}
m &= k, \label{eq: 101 Nof configurations not realizable} \\
2mp + \alpha + q &= k(p_1 + 2z), \label{eq: 102 Nof configurations not realizable} \\
mp^2 + 2q p + \beta &= kz(z+2p_1), \label{eq: 103 Nof configurations not realizable} \\
q p^2 + \gamma &= kp_1z^2. \label{eq: 104 Nof configurations not realizable}
\end{align}
Then, it follows from \eqref{eq: Condition01 Nof configurations not realizable} and \eqref{eq: 101 Nof configurations not realizable}--\eqref{eq: 103 Nof configurations not realizable} that
\begin{align}
\alpha &= \frac{k(p-z)(3p-z-2p_1)}{p_1},
\label{eq: alpha Nof configurations not realizable} \\
q &= -\frac{k(3p^2 - 4zp - p_1^2 + z^2)}{p_1}.
\label{eq: q Nof configurations not realizable}
\end{align}
By \eqref{eq: 104 Nof configurations not realizable} and \eqref{eq: q Nof configurations not realizable}, one obtains
\begin{equation} \label{eq: gamma Nof configurations not realizable}
\gamma = \frac{k(p-z)((3p-z)p^2 - (p+z)p_1^2)}{p_1}.
\end{equation}
Then, \eqref{eq: Condition01 Nof configurations not realizable} and \eqref{eq: alpha Nof configurations not realizable} yield
\begin{equation} \label{eq: beta Nof configurations not realizable}
\beta = \frac{k(p-z)(3p-z-2p_1)(2p+p_1)}{p_1}.
\end{equation}
Together with \eqref{eq: 101 Nof configurations not realizable} and \eqref{eq: q Nof configurations not realizable}, condition~\eqref{eq: Realizability of N1 one-reactive three-element} is equivalent to $z/3 < p < z$, which satisfies the condition of Lemma~1.
Therefore, if a biquadratic impedance $Z(s) \in \mathcal{Z}_{p2,z2}$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: One-reactive Three-element N1}(a)
and $N_2$ is one of the configurations in Figs.~\ref{fig: Three-reactive Four-element N2}(f), then the condition of Lemma~1 holds.
\end{proof}
\begin{lemma} \label{lemma: configuration Nog realizable}
A biquadratic impedance $Z(s) \in \mathcal{Z}_{p2,z2}$ can be realized as the configuration in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: One-reactive Three-element N1}(a)
and $N_2$ is the configurations in Figs.~\ref{fig: Three-reactive Four-element N2}(g) (that is, the configuration in Fig.~3(a)), if and only if
\begin{align}
(p-z)(p-3z) &> 0,
\label{lemma: condition01 configuration Nog realizable} \\
p^4 - 6zp^3 + 6z^2p^2 - 14z^3p + 5z^4 &< 0.
\label{lemma: condition configuration Nog realizable}
\end{align}
\end{lemma}
\begin{proof}
The realizability condition of a general biquadratic impedance $Z(s)$ in the form of \eqref{eq: general biquadratic impedances} as a configuration that is equivalent to Fig.~3(a) is available in \cite[Table~I]{JZ14}. Letting $A = kx$, $B = 2kzx$, $C = k z^2x$, $D = x$, $E = 2px$, and $F = p^2x$ for $x > 0$, the realizability condition for such a specific biquadratic impedance can be derived.
The element values can be derived as $R_1 = q/p_1$, $R_2 = mq/(q-mp_1)$, $C_1 = (q - mp_1)/q^2$, $R_{21} = \alpha$, $L_{21} = \alpha/(2p + p_1)$, $L_{22} = \alpha \beta/\gamma$, and $C_{21} = 1/\beta$, where $\alpha = k(p-z)(2p+p_1)(p^2+zp-2zp_1)/(2p^4)$, $\beta = (2k(p-z)(-zp_1^2+p(p-z)p_1+zp^2))/p^3$, $\gamma = (kp_1(p-z)(p^2+zp-2zp_1))/(2p^2)$, and $p_1$ is a positive root of $(3z-p)p_1^2 - 2p(p-z)p_1 + p^2(p-3z) = 0$.
\end{proof}
\begin{lemma} \label{lemma: configuration Noh realizable}
If a biquadratic impedance $Z(s) \in \mathcal{Z}_{p2,z2}$ can be realized as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: One-reactive Three-element N1}(a) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(h), then the condition of Lemma~\ref{lemma: configuration Nog realizable} holds.
\end{lemma}
\begin{proof}
By calculation,
the impedance of the configuration in
Fig.~\ref{fig: Three-reactive Four-element N2}(h) is obtained as
\begin{equation} \label{eq: 01 Noh configurations not realizable}
Z_2(s) = \frac{s(R_{21}L_{21}L_{22}C_{21} s^2 + L_{21} L_{22} s + R_{21} L_{21})}{L_{21} L_{22} C_{21} s^3 + R_{21} (L_{21} + L_{22})C_{21} s^2 + L_{22} s + R_{21}}.
\end{equation}
If $Z(s)$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: One-reactive Three-element N1}(a) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(h), then the impedance of $N_1$ is in the form of \eqref{eq: Z1 one-reactive three-element N1}, where $m$, $q$, $p_1$ $> 0$ and \eqref{eq: Realizability of N1 one-reactive three-element} holds, and moreover the impedance of $N_2$ is in the form of
\begin{equation} \label{eq: 02 Noh configurations not realizable}
Z_2(s) = \frac{s(\alpha s^2 + \beta s + \gamma)}{(s+p_1)(s+p)^2},
\end{equation}
where $\alpha$, $\beta$, $\gamma$ $> 0$. Consequently, it follows from \eqref{eq: 01 Noh configurations not realizable} and
\eqref{eq: 02 Noh configurations not realizable} that
\begin{align}
R_{21} &= \alpha, \label{eq: 03 Noh configurations not realizable} \\
\frac{1}{C_{21}} &= \beta, \label{eq: 04 Noh configurations not realizable} \\
\frac{R_{21}}{L_{22} C_{21}} &= \gamma, \label{eq: 05 Noh configurations not realizable} \\
\frac{R_{21} (L_{21} + L_{22})}{L_{21} L_{22}} &= 2p + p_1, \label{eq: 06 Noh configurations not realizable} \\
\frac{1}{L_{21} C_{21}} &= p(p+2p_1), \label{eq: 07 Noh configurations not realizable} \\
\frac{R_{21}}{L_{21} L_{22} C_{21}} &= p_1p^2. \label{eq: 08 Noh configurations not realizable}
\end{align}
Thus, \eqref{eq: 04 Noh configurations not realizable} yields
\begin{equation} \label{eq: C21 Noh configurations not realizable}
C_{21} = \frac{1}{\beta}.
\end{equation}
By \eqref{eq: 05 Noh configurations not realizable} and \eqref{eq: 08 Noh configurations not realizable}, one obtains
\begin{equation} \label{eq: L21 Noh configurations not realizable}
L_{21} = \frac{\gamma}{p_1p^2}.
\end{equation}
It follows from \eqref{eq: 03 Noh configurations not realizable}, \eqref{eq: 05 Noh configurations not realizable}, and \eqref{eq: C21 Noh configurations not realizable} that
\begin{equation} \label{eq: L22 Noh configurations not realizable}
L_{22} = \frac{\alpha \beta}{\gamma}.
\end{equation}
By \eqref{eq: 03 Noh configurations not realizable}, \eqref{eq: 06 Noh configurations not realizable}, \eqref{eq: L21 Noh configurations not realizable}, and \eqref{eq: L22 Noh configurations not realizable}, one obtains
\begin{equation} \label{eq: Condition01 Noh configurations not realizable}
(\alpha p^2 - \gamma)\beta p_1 + \gamma(\gamma - 2\beta p) = 0.
\end{equation}
It follows from \eqref{eq: 07 Noh configurations not realizable}, \eqref{eq: C21 Noh configurations not realizable}, and \eqref{eq: L21 Noh configurations not realizable} that
\begin{equation} \label{eq: Condition02 Noh configurations not realizable}
(\beta p - 2\gamma)p_1 - \gamma p = 0.
\end{equation}
Based on \eqref{eq: Z1 one-reactive three-element N1} and \eqref{eq: 02 Noh configurations not realizable}, calculation yields
\begin{equation} \label{eq: Z1 + Z2 Noh configurations not realizable}
\begin{split}
&Z(s) = Z_1(s) + Z_2(s) = \\
&\frac{(m+\alpha)s^3 + (2mp+\beta+q)s^2 + (mp^2 + 2qp + \gamma)s + qp^2}{s^3 + (2p+p_1)s^2 + p(p+2p_1)s + p_1p^2}.
\end{split}
\end{equation}
Comparing (1) with \eqref{eq: Z1 + Z2 Noh configurations not realizable}, one obtains
\begin{align}
m + \alpha &= k, \label{eq: 101 Noh configurations not realizable} \\
2mp + \beta + q &= k(p_1 + 2z), \label{eq: 102 Noh configurations not realizable} \\
mp^2 + 2qp + \gamma &= kz(z+2p_1), \label{eq: 103 Noh configurations not realizable} \\
q p^2 &= kp_1z^2. \label{eq: 104 Noh configurations not realizable}
\end{align}
It follows from \eqref{eq: 104 Noh configurations not realizable} that \begin{equation} \label{eq: q Noh configurations not realizable}
q = \frac{kz^2p_1}{p^2}.
\end{equation}
Thus, \eqref{eq: Condition02 Noh configurations not realizable}, \eqref{eq: 102 Noh configurations not realizable}, \eqref{eq: 103 Noh configurations not realizable}, and \eqref{eq: q Noh configurations not realizable} together yield
\begin{equation} \label{eq: beta Noh configurations not realizable}
\beta = \frac{k(p-z)(p+2p_1)((p-3z)p_1+2zp)}{p^3},
\end{equation}
and
\begin{equation} \label{eq: m Noh configurations not realizable}
m = \frac{-k((p-z)(p-3z)p_1^2-z^2p^2)}{p^4}.
\end{equation}
By \eqref{eq: 101 Noh configurations not realizable} and
\eqref{eq: m Noh configurations not realizable}, one obtains
\begin{equation} \label{eq: alpha Noh configurations not realizable}
\alpha = \frac{k(p-z)((p-3z)p_1^2+p^2(p+z))}{p^4}.
\end{equation}
It follows from \eqref{eq: Condition02 Noh configurations not realizable} and \eqref{eq: beta Noh configurations not realizable} that
\begin{equation} \label{eq: gamma Noh configurations not realizable}
\gamma = \frac{kp_1(p-z)((p-3z)p_1+2zp)}{p^2}.
\end{equation}
Together with \eqref{eq: q Noh configurations not realizable} and \eqref{eq: m Noh configurations not realizable},
condition~\eqref{eq: Realizability of N1 one-reactive three-element} is equivalent to \eqref{lemma: condition01 configuration Nog realizable}. Together with \eqref{eq: beta Noh configurations not realizable}, \eqref{eq: alpha Noh configurations not realizable}, and \eqref{eq: gamma Noh configurations not realizable},
condition~\eqref{eq: Condition01 Noh configurations not realizable} is equivalent to
\begin{equation} \label{eq: equivalent Condition01 Noh configurations not realizable}
(5z-3p)p_1^2 + (p-3z)p^2 = 0.
\end{equation}
From \eqref{eq: m Noh configurations not realizable}, it follows that $m > 0$ is equivalent to $p_1^2 < z^2p^2/((p-z)(p-3z))$, which, in turn, is equivalent to $p^2 - 5zp + 2z^2 < 0$, together with \eqref{lemma: condition01 configuration Nog realizable} and \eqref{eq: equivalent Condition01 Noh configurations not realizable}. Therefore, the condition of Lemma~\ref{lemma: configuration Nog realizable} must hold.
\end{proof}
\section{Supplementary Lemmas of Five-Reactive Seven-Element Series-Parallel Realizations for the Proof of Lemma~4}
\begin{lemma}
\label{lemma: Possible configurations of N1 and N2 two-reactive three-element and N2 three-reactive four-element}
{Consider the five-reactive seven-element series-parallel network in Fig.~2(a), realizing a biquadratic impedance $Z(s)$ in the form of \eqref{eq: general biquadratic impedances} with $A$, $B$, $C$, $D$, $E$, $F$ $> 0$, where $N_1$ is a three-element series-parallel network and $N_2$ is a four-element series-parallel network. If $Z(s)$ cannot be realized as a series-parallel network containing fewer than seven elements, then $N_1$ is one of the configurations in Figs.~\ref{fig: Two-reactive configurations N1}(b)--\ref{fig: Two-reactive configurations N1}(d) and $N_2$ will be equivalent to one of the configurations in Fig.~\ref{fig: Three-reactive Four-element N2}.}
\end{lemma}
\begin{proof}
By \cite[Lemma~2]{WCH14}, $Z(s)$ cannot be realized as the series connection of two networks, one of which contains only reactive elements.
Therefore, $N_1$ can only contain two reactive elements.
For any realization of $Z(s)$,
there is no cut-set $\mathcal{C}(a,a')$ corresponding to one kind of reactive elements, where $a$ and $a'$ denote two terminals,
by \cite[Lemma~1]{WCH14}.
The possible network graphs for subnetworks $N_1$ and $N_2$ are listed in Figs.~\ref{fig: Three-element graphs} and \ref{fig: Four-element graphs}, respectively. Based on the method of enumeration and the equivalence in \cite[Lemma~11]{JS11}, $N_1$ is one of the configurations in Figs.~\ref{fig: Two-reactive configurations N1}(b)--\ref{fig: Two-reactive configurations N1}(d) and $N_2$ can be equivalent to one of the configurations in Fig.~\ref{fig: Three-reactive Four-element N2}.
\end{proof}
\begin{lemma} \label{lemma: configurations not realizable}
A biquadratic impedance $Z(s) \in \mathcal{Z}_{p2,z2}$
not satisfying the condition of Lemma~3
cannot be realized as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(b) and $N_2$ is one of the configurations in Figs.~\ref{fig: Three-reactive Four-element N2}(a)--\ref{fig: Three-reactive Four-element N2}(d), \ref{fig: Three-reactive Four-element N2}(f), and \ref{fig: Three-reactive Four-element N2}(h).
\end{lemma}
\begin{proof}
It has been shown that the impedance of the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(b) is in the form of
\begin{equation} \label{eq: Z1 b configuration}
Z_1(s) = \frac{R_1L_1C_1 s^2 + R_1}{L_1C_1 s^2 + R_1C_1 s + 1},
\end{equation}
and the impedance of the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(a) is in the form of \eqref{eq: 01 Noa configurations not realizable}. Since it is assumed that the condition of Lemma~3
does not hold, $Z(s) \in \mathcal{Z}_{p2,z2}$ cannot be realized with fewer than five reactive elements.
If $Z(s)$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(b) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(a), then the impedance of $N_1$ is of degree two and is in the form of
\begin{equation} \label{eq: Z1 No2 (s+p1)(s+p)}
Z_1(s) = \frac{m(s^2 + pp_1)}{(s+p_1)(s+p)},
\end{equation}
where $m$, $p_1$, $p$ $> 0$, and the impedance of $N_2$ is of degree three and is in the form of
\eqref{eq: 02 Noa configurations not realizable}, where $\alpha$, $\gamma$ $> 0$, and \eqref{eq: Condition01 Noa configurations not realizable} and \eqref{eq: Condition02 Noa configurations not realizable} hold. Furthermore,
\begin{equation} \label{eq: Z1 + Z2 No2 and Noa configurations}
\begin{split}
&Z(s) = Z_1(s) + Z_2(s) =\\
& \frac{m s^3 + (m p + \alpha)s^2 + m p_1 p s + (mp_1p^2 + \gamma)}{(s+p_1)(s+p)^2}.
\end{split}
\end{equation}
Comparing (1) with \eqref{eq: Z1 + Z2 No2 and Noa configurations}, one obtains
\begin{align}
m &= k, \label{eq: 101 Z1 + Z2 No2 and Noa configurations} \\
mp + \alpha &= k(p_1 + 2z), \label{eq: 102 Z1 + Z2 No2 and Noa configurations} \\
mp_1p &= kz(2p_1+z), \label{eq: 103 Z1 + Z2 No2 and Noa configurations} \\
mp_1p^2 + \gamma &= kz^2p_1. \label{eq: 104 Z1 + Z2 No2 and Noa configurations}
\end{align}
By \eqref{eq: 101 Z1 + Z2 No2 and Noa configurations}--\eqref{eq: 103 Z1 + Z2 No2 and Noa configurations}, one obtains
\begin{equation} \label{eq: p_1 Z1 + Z2 No2 and Noa configurations}
p_1 = \frac{z^2}{p-2z}.
\end{equation}
By \eqref{eq: 101 Z1 + Z2 No2 and Noa configurations}, \eqref{eq: 104 Z1 + Z2 No2 and Noa configurations}, and \eqref{eq: p_1 Z1 + Z2 No2 and Noa configurations}, one obtains
\begin{equation} \label{eq: gamma Z1 + Z2 No2 and Noa configurations}
\gamma = \frac{-kz^2(p-z)(p+z)}{p-2z}.
\end{equation}
It follows from \eqref{eq: p_1 Z1 + Z2 No2 and Noa configurations} and $p_1 > 0$ that
$p > 2z$, which further implies that $\gamma < 0$ by \eqref{eq: gamma Z1 + Z2 No2 and Noa configurations}. This is impossible.
Therefore, $Z(s) \in \mathcal{Z}_{p2,z2}$ cannot be realized as in Fig.~2(a), where $N_1$ is the configuration in
Fig.~\ref{fig: Two-reactive configurations N1}(b) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(a).
It is clear that any network in Fig.~2(a), where $N_1$ is the configuration in
Fig.~\ref{fig: Two-reactive configurations N1}(b) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(b), can be a frequency inverse dual network of another one in Fig.~2(a), where $N_1$ is the configuration in
Fig.~\ref{fig: Two-reactive configurations N1}(b) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(a). Based on the principle of frequency inverse, $Z(s) \in \mathcal{Z}_{p2,z2}$ cannot be realized as such a network.
It has been shown that the impedance of the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(b) is in the form of \eqref{eq: Z1 b configuration}, and the impedance of the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(c) is in the form of $Z(s) = (R_{21}L_{21}C_{22}s^2 + L_{21}s)/(R_{21}L_{21}C_{21}C_{22}s^3 + L_{21}(C_{21}+C_{22})s^2 + R_{21}C_{22}s + 1)$.
If $Z(s)$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(b) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(c), then the impedance of $N_1$ is of degree two and is in the form of
\eqref{eq: Z1 No2 (s+p1)(s+p)},
where $m$, $p_1$, $p$ $> 0$, and the impedance of $N_2$ is of degree three and is in the form of
$Z(s) = (\alpha s^2 + \beta s)/((s+p_1)(s+p)^2)$, where $\alpha$, $\beta$ $> 0$. Furthermore,
\begin{equation} \label{eq: Z1 + Z2 No2 and Noc configurations}
\begin{split}
&Z(s)
= Z_1(s) + Z_2(s)= \\
& \frac{ms^3 + (mp+\alpha)s^2 + (mp_1p+\beta)s + mp_1p^2}{(s+p_1)(s+p)^2}.
\end{split}
\end{equation}
Comparing (1) with \eqref{eq: Z1 + Z2 No2 and Noc configurations}, one obtains $m = k$ and $mp_1p^2 = kz^2p_1$, which further implies $p=z$. This contradicts the assumption.
Therefore, $Z(s) \in \mathcal{Z}_{p2,z2}$ cannot be realized as in Fig.~2(a), where $N_1$ is the configuration in
Fig.~\ref{fig: Two-reactive configurations N1}(b) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(c).
It is clear that any network in Fig.~2(a), where $N_1$ is the configuration in
Fig.~\ref{fig: Two-reactive configurations N1}(b) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(d), can be a frequency inverse dual network of another one in Fig.~2(a), where $N_1$ is the configuration in
Fig.~\ref{fig: Two-reactive configurations N1}(b) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(c). Based on the principle of frequency inverse, $Z(s) \in \mathcal{Z}_{p2,z2}$ cannot be realized as such a network.
It has been shown that the impedance of the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(b) is in the form of
\eqref{eq: Z1 b configuration}, and the impedance of the configuration in Fig.~8(f) is in the form of
\eqref{eq: 01 Nof configurations not realizable}.
If $Z(s)$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(b) and $N_2$ is the configuration in Fig.~8(f), then the impedance of $N_1$ is of degree two and is in the form of
\eqref{eq: Z1 No2 (s+p1)(s+p)},
where $m$, $p_1$, $p$ $> 0$ and the impedance of $N_2$ is degree three and is in the form of \eqref{eq: 02 Noe configurations not realizable}, where $\alpha$, $\beta$, $\gamma$ $> 0$ and \eqref{eq: Condition01 Nof configurations not realizable}--\eqref{eq: Condition03 Nof configurations not realizable} hold.
Furthermore, one obtains
\eqref{eq: Z1 + Z2 No2 and Noe configurations}--\eqref{eq: gamma No2 and Noe configurations}.
Substituting \eqref{eq: 101 No2 and Noe configurations} and \eqref{eq: alpha No2 and Noe configurations}--\eqref{eq: gamma No2 and Noe configurations} into \eqref{eq: Condition01 Nof configurations not realizable}--\eqref{eq: Condition03 Nof configurations not realizable} yields
\begin{align}
p_1^2 + 2pp_1 - (2p^2-4zp+z^2) = 0,
\label{eq: equivalent Condition01 Nof configurations not realizable} \\
2pp_1^2 + z(4p-z)p_1 - p^2(p-2z) > 0,
\label{eq: equivalent Condition02 Nof configurations not realizable} \\
2pp_1^3 + (3p^2+4zp-z^2)p_1^2 + 2zp(4p-z)p_1 - 2p^3(p-2z) = 0,
\label{eq: equivalent Condition03 Nof configurations not realizable}
\end{align}
respectively. By \eqref{eq: equivalent Condition01 Nof configurations not realizable}, \eqref{eq: equivalent Condition03 Nof configurations not realizable} can be further equivalent to
\begin{equation} \label{eq: equivalent02 Condition03 Nof configurations not realizable}
(p^2 - 4zp + z^2)p_1^2 - 4p^3p_1 + 2p^3(p-2z) = 0.
\end{equation}
It is calculated that the resultant of \eqref{eq: equivalent Condition01 Nof configurations not realizable} and \eqref{eq: equivalent02 Condition03 Nof configurations not realizable} in $p_1$ is $8p^8 + 48zp^7 - 312z^2p^6 + 624z^3p^5 - 617z^4p^4 + 336z^5p^3 - 102z^6p^2 + 16z^7p - z^8$.
Since the condition of Lemma~3 does not hold, there exists at least one common root between \eqref{eq: equivalent Condition01 Nof configurations not realizable} and \eqref{eq: equivalent02 Condition03 Nof configurations not realizable} in $p_1$ if and only if
\begin{equation} \label{eq: Condition five-reactive-element configurations realizable 02}
\begin{split}
8p^8 + 48zp^7 &- 312z^2p^6 + 624z^3p^5 - 617z^4p^4 \\
&+ 336z^5p^3- 102z^6p^2 + 16z^7p - z^8 = 0
\end{split}
\end{equation}
holds with $p < z/(2 + \sqrt{5})$, which implies that the condition of Lemma~3 holds.
One further implies $p_1 > 0$.
By \eqref{eq: equivalent Condition01 Nof configurations not realizable}, it is implied that \eqref{eq: equivalent Condition02 Nof configurations not realizable} is equivalent to
\begin{equation} \label{eq: equivalent02 Condition02 Nof configurations not realizable}
p_1 < \frac{p(3p^2 - 6zp + 2z^2)}{(2p-z)^2}.
\end{equation}
By \eqref{eq: equivalent02 Condition03 Nof configurations not realizable}, \eqref{eq: equivalent02 Condition02 Nof configurations not realizable} is further equivalent to $(p-z)(7p^5 + 63zp^4 - 174z^2p^3 + 134z^3p^2 - 40z^4p + 4z^5) < 0$, which is verified to be satisfied.
It is clear that any network in Fig.~2(a), where $N_1$ is the configuration in
Fig.~\ref{fig: Two-reactive configurations N1}(b) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(h), can be a frequency inverse dual network of another one in Fig.~2(a), where $N_1$ is the configuration in
Fig.~\ref{fig: Two-reactive configurations N1}(b) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(f). Based on the principle of frequency inverse, $Z(s) \in \mathcal{Z}_{p2,z2}$ cannot be realized as such a network either.
\end{proof}
\begin{lemma} \label{lemma: five-reactive-element configurations realizable 01}
A biquadratic impedance $Z(s) \in \mathcal{Z}_{p2,z2}$ not satisfying the condition of Lemma~3 is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(b) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(e) (that is, the configuration in Fig.~4(a) whose one-terminal-pair labeled graph is $N_{4a}$), if and only if
\begin{equation} \label{eq: Condition five-reactive-element configurations realizable 01}
16 p^4 - 40 z p^3 + 31 z^2 p^2 - 10 z^3 p + z^4 = 0
\end{equation}
and $p < z/(2 + \sqrt{5})$
(it can be verified that there is only one distinct root of the equation $16 \eta^4 - 40 \eta^3 + 31 \eta^2 - 10 \eta + 1 = 0$ for $\eta \in (0, 1/(2 + \sqrt{5}))$).
\end{lemma}
\begin{proof}
\textit{Necessity.}
It has been shown that the impedance of the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(b)
is in the form of
\eqref{eq: Z1 b configuration}, and the impedance of the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(e) is in the form of
\eqref{eq: 01 Noe configurations not realizable}.
Since it is assumed that the condition of Lemma~3 does not hold, $Z(s) \in \mathcal{Z}_{p2,z2}$ cannot be realized with fewer than five reactive elements.
If $Z(s)$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(b) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(e), then the impedance of $N_1$ is of degree two and is in the form of
\eqref{eq: Z1 No2 (s+p1)(s+p)},
where $m$, $p_1$, $p$ $> 0$, and the impedance of $N_2$ is of degree three and is in the form of \eqref{eq: 02 Noe configurations not realizable}, where $\alpha$, $\beta$ $> 0$ and \eqref{eq: Condition01 Noe configurations not realizable}--\eqref{eq: Condition03 Noe configurations not realizable} hold.
Furthermore, one obtains
\begin{equation} \label{eq: Z1 + Z2 No2 and Noe configurations}
Z(s) = Z_1(s) + Z_2(s) =
\frac{ms^3 + (mp+\alpha)s^2 + (mp_1p+\beta)s + (mp_1p^2 + \gamma)}{(s+p_1)(s+p)^2}.
\end{equation}
Comparing (1) with \eqref{eq: Z1 + Z2 No2 and Noe configurations}, one obtains
\begin{align}
m &= k,
\label{eq: 101 No2 and Noe configurations} \\
mp + \alpha &= k(p_1 + 2z),
\label{eq: 102 No2 and Noe configurations} \\
mp_1p + \beta &= kz(2p_1+z),
\label{eq: 103 No2 and Noe configurations} \\
mp_1p^2 + \gamma &= kz^2p_1.
\label{eq: 104 No2 and Noe configurations}
\end{align}
Then, \eqref{eq: 101 No2 and Noe configurations} and \eqref{eq: 102 No2 and Noe configurations} yield
\begin{equation} \label{eq: alpha No2 and Noe configurations}
\alpha = -k(p-p_1-2z).
\end{equation}
By \eqref{eq: 101 No2 and Noe configurations} and \eqref{eq: 103 No2 and Noe configurations}, one obtains
\begin{equation} \label{eq: beta No2 and Noe configurations}
\beta = -k(p_1p - 2zp_1 - z^2).
\end{equation}
It follows from \eqref{eq: 101 No2 and Noe configurations} and \eqref{eq: 104 No2 and Noe configurations} that
\begin{equation} \label{eq: gamma No2 and Noe configurations}
\gamma = -kp_1(p-z)(p+z).
\end{equation}
Substituting \eqref{eq: 101 No2 and Noe configurations} and
\eqref{eq: alpha No2 and Noe configurations}--\eqref{eq: gamma No2 and Noe configurations} into \eqref{eq: Condition01 Noe configurations not realizable}--\eqref{eq: Condition03 Noe configurations not realizable} yields
\begin{equation} \label{eq: equivalent Condition01 No2 and Noe configurations not realizable}
2pp_1^2 + z(4p-z)p_1 - p^2(p-2z) = 0,
\end{equation}
\begin{equation} \label{eq: equivalent Condition02 No2 and Noe configurations not realizable}
\begin{split}
(2p^2 - z^2)p_1^2 &+ 2pz(2p-z)p_1 \\
&- (p^4 -5z^2p^2 + 4z^3p - z^4) = 0,
\end{split}
\end{equation}
\begin{equation} \label{eq: equivalent Condition03 No2 and Noe configurations not realizable}
p_1^2 + 2pp_1 - (2p^2 - 4zp + z^2) > 0,
\end{equation}
respectively. It is calculated that the resultant of \eqref{eq: equivalent Condition01 No2 and Noe configurations not realizable} and \eqref{eq: equivalent Condition02 No2 and Noe configurations not realizable} in $p_1$ is $z^2(p+z)(p-z)^3(16p^4 - 40zp^3 + 31z^2p^2 - 10z^3p + z^4)$.
Since the condition of Lemma~3 does not hold, there exists at least one common root between \eqref{eq: equivalent Condition01 No2 and Noe configurations not realizable} and \eqref{eq: equivalent Condition02 No2 and Noe configurations not realizable} in $p_1$ if and only if (6) holds with $p < z/(2 + \sqrt{5})$. One implies $p_1 > 0$.
By \eqref{eq: equivalent Condition01 No2 and Noe configurations not realizable}, it is implied that \eqref{eq: equivalent Condition03 No2 and Noe configurations not realizable} is equivalent to
\begin{equation}
\label{eq: equivalent02 Condition03 No2 and Noe configurations not realizable}
p_1 > \frac{p(3p^2 - 6zp + 2z^2)}{(2p - z)^2}.
\end{equation}
Based on \eqref{eq: equivalent Condition02 No2 and Noe configurations not realizable}, one implies that \eqref{eq: equivalent02 Condition03 No2 and Noe configurations not realizable} is equivalent to $(p^2+2zp-z^2)(p-z)^3(2p^3+10zp^2-7z^2p+z^3)>0$, which can be verified to be satisfied.
\textit{Sufficiency.}
Based on the discussion in the necessity part, there exists $p_1 > 0$ such that \eqref{eq: equivalent Condition01 No2 and Noe configurations not realizable}--\eqref{eq: equivalent Condition03 No2 and Noe configurations not realizable} hold.
Let $m$, $\alpha$, $\beta$, and $\gamma$ satisfy \eqref{eq: 101 No2 and Noe configurations} and \eqref{eq: alpha No2 and Noe configurations}--\eqref{eq: gamma No2 and Noe configurations}, which obviously implies that $\alpha$, $\beta$, $\gamma$, $m$ $> 0$.
Therefore, \eqref{eq: Condition01 Noe configurations not realizable}--\eqref{eq: Condition03 Noe configurations not realizable}
and \eqref{eq: 102 No2 and Noe configurations}--\eqref{eq: 104 No2 and Noe configurations} hold. Therefore, $Z(s)$ can be written in the form of \eqref{eq: Z1 + Z2 No2 and Noe configurations}. Decompose
$Z(s)$ as $Z(s) = Z_1(s) + Z_2(s)$, where $Z_1(s)$ is in the form of \eqref{eq: Z1 No2 (s+p1)(s+p)} where $m$, $p_1$, $p$ $> 0$ and $Z_2(s)$ is in the form of \eqref{eq: 02 Noe configurations not realizable} where $\alpha$, $\beta$ $> 0$.
By letting $R_1 = m$, $L_1 = m/(p+p_1)$, and $C_1 = (p+p_1)/(mpp_1)$,
$Z_1(s)$ is realizable as in Fig.~\ref{fig: Two-reactive configurations N1}(b).
Let $C_{21}$, $C_{22}$, $R_{21}$, and $L_{21}$ satisfy \eqref{eq: C21 Noe configurations not realizable}--\eqref{eq: L21 Noe configurations not realizable}. Since \eqref{eq: Condition03 Noe configurations not realizable} holds, $C_{21}$, $C_{22}$, $R_{21}$, $L_{21}$ $> 0$. Based on the discussion in the proof of Lemma~11, it follows that \eqref{eq: 03 Noe configurations not realizable}--\eqref{eq: 08 Noe configurations not realizable} hold because \eqref{eq: Condition01 Noe configurations not realizable} and \eqref{eq: Condition02 Noe configurations not realizable} are satisfied.
Therefore, $Z_2(s)$ can be realized as in Fig.~\ref{fig: Three-reactive Four-element N2}(e).
The sufficiency part is proved.
\end{proof}
\begin{lemma}
\label{lemma: five-reactive-element configurations realizable 03}
A biquadratic impedance $Z(s) \in \mathcal{Z}_{p2,z2}$ not satisfying the condition of Lemma~3 cannot be realized
as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(c) and $N_2$ is one of the configurations in Fig.~\ref{fig: Three-reactive Four-element N2}(a) and \ref{fig: Three-reactive Four-element N2}(f).
\end{lemma}
\begin{proof}
It has been shown that the impedance of the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(c) is in the form of $Z_1(s) = s(R_1L_1C_1 s + L_1)/(L_1C_1 s^2 + R_1C_1 s + 1)$, and the impedance of the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(a) is in the form of
\eqref{eq: 01 Noa configurations not realizable}.
Since it is assumed that the condition of Lemma~3
does not hold, $Z(s) \in \mathcal{Z}_{p2,z2}$ cannot be realized with fewer than five reactive elements.
If $Z(s)$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(c) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(a), then the impedance of $N_1$ is in the form of
\begin{equation} \label{eq: Z1 No3 (s+p1)(s+p)}
Z_1(s) = \frac{ms(s + q)}{(s+p_1)(s+p)},
\end{equation}
where $m$, $p_1$, $p$ $> 0$ and $q = p_1p/(p_1 + p)$, and the impedance of $N_2$ is in the form of \eqref{eq: 02 Noa configurations not realizable}, where $\alpha$, $\gamma$ $> 0$, and moreover \eqref{eq: Condition01 Noa configurations not realizable} and \eqref{eq: Condition02 Noa configurations not realizable} hold. Furthermore,
\begin{equation} \label{eq: Z1 + Z2 No3 and Noa configurations}
\begin{split}
&Z(s) = Z_1(s) + Z_2(s) = \\
& \frac{ms^3 + \frac{mp^2 + (2mp_1+\alpha)p + \alpha p_1}{p_1 + p}s^2 + \frac{mp_1p^2}{p_1+p}s + \gamma}{(s+p_1)(s+p)^2}.
\end{split}
\end{equation}
Comparing (1) with \eqref{eq: Z1 + Z2 No3 and Noa configurations}, one obtains \begin{align}
m &= k,
\label{eq: 101 No3 and Noa configurations} \\
\frac{mp^2 + (2mp_1+\alpha)p + \alpha p_1}{p + p_1} &= k(p_1 + 2z),
\label{eq: 102 No3 and Noa configurations} \\
\frac{mp_1p^2}{p + p_1} &= kz(2p_1+z),
\label{eq: 103 No3 and Noa configurations} \\
\gamma &= kp_1z^2.
\label{eq: 104 No3 and Noa configurations}
\end{align}
Then, it follows from \eqref{eq: 101 No3 and Noa configurations} and \eqref{eq: 102 No3 and Noa configurations} that
\begin{equation} \label{eq: alpha No3 and Noa configurations}
\alpha = \frac{k(p_1^2 - (p-2z)p_1 - p(p-2z))}{p+p_1}.
\end{equation}
Substituting \eqref{eq: 101 No3 and Noa configurations}, \eqref{eq: 104 No3 and Noa configurations}, and \eqref{eq: alpha No3 and Noa configurations} into \eqref{eq: Condition01 Noa configurations not realizable}, \eqref{eq: Condition02 Noa configurations not realizable}, and \eqref{eq: 103 No3 and Noa configurations} yields
\begin{equation} \label{eq: equivalent Condition01 No3 and Noa configurations not realizable}
\begin{split}
2pp_1^3 &- (p^2 - 4zp + z^2)p_1^2 \\
&- p(3p^2 - 6zp + z^2)p_1 - p^3(p-2z) > 0,
\end{split}
\end{equation}
\begin{equation} \label{eq: equivalent Condition02 No3 and Noa configurations not realizable}
\begin{split}
(p-z)(p+z)p_1^2 - p(p^2 &- 2zp + 3z^2)p_1 \\
&- p^2(p^2 - 2zp + 2z^2) = 0,
\end{split}
\end{equation}
\begin{equation} \label{eq: equivalent Condition03 No3 and Noa configurations not realizable}
\begin{split}
2zp_1^2 - (p^2 - 2zp - z^2)p_1 + z^2p = 0,
\end{split}
\end{equation}
respectively. By \eqref{eq: alpha No3 and Noa configurations}, the assumption of $\alpha > 0$ implies
\begin{equation} \label{eq: equivalent Condition04 No3 and Noa configurations not realizable}
p_1^2 - (p-2z)p_1 - p(p-2z) > 0.
\end{equation}
By calculation,
the resultant of \eqref{eq: equivalent Condition02 No3 and Noa configurations not realizable} and \eqref{eq: equivalent Condition03 No3 and Noa configurations not realizable} in $p_1$ is
$-p^4(p^6 - 8zp^5 + 20z^2p^4 - 28z^3p^3 + 21z^4p^2 - 12z^5p + 2z^6)$.
Since the condition of
Lemma~3 does not hold, there exists at least one common root in $p_1$ between \eqref{eq: equivalent Condition02 No3 and Noa configurations not realizable} and \eqref{eq: equivalent Condition03 No3 and Noa configurations not realizable} if and only if
\begin{equation}
\label{eq: Condition five-reactive-element configurations realizable 03}
p^6 - 8zp^5 + 20z^2p^4 -28z^3p^3 + 21z^4p^2 -12z^5p + 2z^6 = 0
\end{equation}
holds with $p > (2 + \sqrt{5})z$, which implies that the condition of Lemma~3 holds.
This further implies that $p_1 > 0$.
From \eqref{eq: equivalent Condition03 No3 and Noa configurations not realizable}, it follows that \eqref{eq: equivalent Condition01 No3 and Noa configurations not realizable} is equivalent to
\begin{equation}
\label{eq: equivalent02 Condition01 No3 and Noa configurations not realizable}
p_1 > \frac{zp(2p-3z)}{(p-z)(p-3z)}.
\end{equation}
By \eqref{eq: equivalent Condition02 No3 and Noa configurations not realizable}, one has that \eqref{eq: equivalent02 Condition01 No3 and Noa configurations not realizable} is further equivalent to
$p^{10} - 12zp^9 + 54z^2p^8 - 114z^3p^7 + 100z^4p^6 + 40z^5p^5 - 164z^6p^4 + 142z^7p^3 - 65z^8p^2 + 16z^9p - 2z^{10} > 0$, which can indeed be verified to be true.
It has been shown that the impedance of the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(c) is in the form of
\eqref{eq: Z1 c configuration}, and the impedance of the configuration in Fig.~8(f) is in the form of
\eqref{eq: 01 Nof configurations not realizable}.
If $Z(s)$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(c) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(f), then the impedance of $N_1$ is in the form of \eqref{eq: Z1 No3 (s+p1)(s+p)}, where $m$, $p_1$, $p$ $> 0$ and $q = p_1p/(p_1 + p)$, and the impedance of $N_2$ is in the form of \eqref{eq: 02 Noe configurations not realizable}, where $\alpha$, $\beta$, $\gamma$ $> 0$ and \eqref{eq: Condition01 Nof configurations not realizable}--\eqref{eq: Condition03 Nof configurations not realizable} hold. Furthermore, one obtains \eqref{eq: Z1 + Z2 No3 and Noe configurations}--\eqref{eq: 104 No3 and Noe configurations}. It follows from \eqref{eq: Condition03 Nof configurations not realizable} and \eqref{eq: 104 No3 and Noe configurations} that
\begin{equation} \label{eq: alpha No3 and Nof configurations}
\alpha = \frac{kp_1z^2(p_1+2p)}{2p(p+p_1)^2}.
\end{equation}
Then, \eqref{eq: Condition01 Nof configurations not realizable} and \eqref{eq: alpha No3 and Nof configurations} yield
\begin{equation} \label{eq: beta No3 and Nof configurations}
\beta = \frac{kp_1z^2(p_1 + 2p)^2}{2p(p+p_1)^2}.
\end{equation}
Substituting \eqref{eq: 101 No3 and Noe configurations}, \eqref{eq: 104 No3 and Noe configurations}, \eqref{eq: alpha No3 and Noe configurations}, and \eqref{eq: beta No3 and Nof configurations} into \eqref{eq: Condition02 Nof configurations not realizable}, \eqref{eq: 102 No3 and Noe configurations}, and \eqref{eq: 103 No3 and Noe configurations} gives
\begin{align}
\frac{kp_1^2z^2p}{2(p+p_1)^2} &> 0,
\label{eq: equivalent Condition01 No3 and Nof configurations not realizable} \\
2pp_1^3 + z(4p - z)p_1^2 - 2p(2p^2 - 4zp + z^2)p_1 - 2p^3(p-2z) &= 0,
\label{eq: equivalent Condition02 No3 and Nof configurations not realizable} \\
z(4p-z)p_1^3 - 2p(p^2 - 4zp + z^2)p_1^2 - 2p^3(p-2z)p_1 + 2z^2p^3 &= 0,
\label{eq: equivalent Condition03 No3 and Nof configurations not realizable}
\end{align}
respectively. It is obvious that \eqref{eq: equivalent Condition01 No3 and Nof configurations not realizable} holds.
It is calculated that the resultant of \eqref{eq: equivalent Condition02 No3 and Nof configurations not realizable} and \eqref{eq: equivalent Condition03 No3 and Nof configurations not realizable} in
$p_1$ is $-4p^6z^4(2p^4 - 12zp^3 + 18z^2p^2 - 8z^3p + z^4)^2$.
Since the condition of
Lemma~3 does not hold,
it is implied that there exists at least one common root between
\eqref{eq: equivalent Condition02 No3 and Nof configurations not realizable} and \eqref{eq: equivalent Condition03 No3 and Nof configurations not realizable} if and only if
\begin{equation} \label{eq: Condition five-reactive-element configurations realizable 05}
2p^4 - 12zp^3 + 18z^2p^2 - 8z^3p + z^4 = 0
\end{equation}
holds with $p < z/(2 + \sqrt{5})$, which implies that the condition of Lemma~3 holds.
This further implies that $p_1 > 0$.
\end{proof}
\begin{lemma}
\label{lemma: five-reactive-element configurations realizable 04}
A biquadratic impedance $Z(s) \in \mathcal{Z}_{p2,z2}$ not satisfying the condition of Lemma~3 is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(c) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(e) (that is, the configuration in Fig.~5(a) whose one-terminal-pair labeled graph is $N_{5a}$), if and only if
\begin{equation} \label{eq: Condition five-reactive-element configurations realizable 04}
\begin{split}
p^{10} &- 16zp^9 + 118z^2p^8 - 476z^3p^7
\\
&+ 1066z^4p^6 - 1372z^5p^5
+ 1064z^6p^4 \\
&- 524z^7p^3+ 161z^8p^2 - 28z^9p + 2z^{10} = 0
\end{split}
\end{equation}
and $p < z/(2 + \sqrt{5})$ (it can be verified that the equation $\eta^{10} - 16 \eta^9 + 118 \eta^8 - 476 \eta^7
+ 1066 \eta^6 - 1372 \eta^5 + 1064 \eta^4 - 524 \eta^3 + 161 \eta^2 - 28 \eta + 2 = 0$ has only one distinct root for $\eta \in (0, 1/(2 + \sqrt{5}))$).
\end{lemma}
\begin{proof}
\textit{Necessity.}
It has been shown that the impedance of the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(c) is in the form of
\eqref{eq: Z1 c configuration}, and the impedance of the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(e) is in the form of
\eqref{eq: 01 Noe configurations not realizable}.
Since it is assumed that the condition of Lemma~3
does not hold, $Z(s) \in \mathcal{Z}_{p2,z2}$ cannot be realized with fewer than five reactive elements.
If $Z(s)$ is realizable as in Fig.~2(a), where $N_1$ is the configuration in Fig.~\ref{fig: Two-reactive configurations N1}(c) and $N_2$ is the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(e), then the impedance of $N_1$ is in the form of \eqref{eq: Z1 No3 (s+p1)(s+p)},
where $m$, $p_1$, $p$ $> 0$ and $q = p_1p/(p_1 + p)$, and the impedance of $N_2$ is in the form of \eqref{eq: 02 Noe configurations not realizable}, where $\alpha$, $\beta$, $\gamma$ $> 0$ and
\eqref{eq: Condition01 Noe configurations not realizable}--\eqref{eq: Condition03 Noe configurations not realizable} hold.
Furthermore, one obtains
\begin{equation} \label{eq: Z1 + Z2 No3 and Noe configurations}
Z(s) = Z_1(s) + Z_2(s) = \frac{ms^3 + \frac{mp^2 + (2mp_1+\alpha)p + \alpha p_1}{p_1 + p}s^2 + \frac{mp_1p^2 + \beta p + \beta p_1}{p_1+p}s + \gamma}{(s+p_1)(s+p)^2}.
\end{equation}
Combining (1) with \eqref{eq: Z1 + Z2 No3 and Noe configurations}, one obtains
\begin{align}
m &= k,
\label{eq: 101 No3 and Noe configurations} \\
\frac{mp^2 + (2mp_1+\alpha)p + \alpha p_1}{p + p_1} &= k(p_1 + 2z),
\label{eq: 102 No3 and Noe configurations} \\
\frac{mp_1p^2 + \beta p + \beta p_1}{p + p_1} &= kz(2p_1+z),
\label{eq: 103 No3 and Noe configurations} \\
\gamma &= kp_1z^2.
\label{eq: 104 No3 and Noe configurations}
\end{align}
It follows from \eqref{eq: Condition01 Noe configurations not realizable} and \eqref{eq: 104 No3 and Noe configurations} that
\begin{equation} \label{eq: alpha No3 and Noe configurations}
\alpha = \frac{kp_1z^2}{p(p+2p_1)}.
\end{equation}
By \eqref{eq: Condition02 Noe configurations not realizable}, \eqref{eq: 104 No3 and Noe configurations}, and \eqref{eq: alpha No3 and Noe configurations}, one obtains
\begin{equation} \label{eq: beta No3 and Noe configurations}
\beta = \frac{2kp_1z^2(p+p_1)^2}{(p+2p_1)^2p}.
\end{equation}
Substituting \eqref{eq: 101 No3 and Noe configurations} and \eqref{eq: 104 No3 and Noe configurations}--\eqref{eq: beta No3 and Noe configurations} into \eqref{eq: Condition03 Noe configurations not realizable}, \eqref{eq: 102 No3 and Noe configurations}, and \eqref{eq: 103 No3 and Noe configurations} gives
\begin{align}
\frac{kp_1^2z^2}{(p+2p_1)^2} &> 0,
\label{eq: equivalent Condition01 No3 and Noe configurations not realizable} \\
2pp_1^3 - (p^2-4zp+z^2)p_1^2 - p(3p^2-6zp+z^2)p_1 - p^3(p-2z) &= 0, \label{eq: equivalent Condition02 No3 and Noe configurations not realizable}
\end{align}
and
\begin{equation} \label{eq: equivalent Condition03 No3 and Noe configurations not realizable}
\begin{split}
2z(4p-z)p_1^4 -2p(2p^2-8zp+z^2)p_1^3 &- 2p^2(2p^2-5zp-z^2)p_1^2 \\
&- p^3(p+z)(p-3z)p_1 + z^2p^4 = 0,
\end{split}
\end{equation}
respectively. It is obvious that \eqref{eq: equivalent Condition01 No3 and Noe configurations not realizable} holds. It is calculated that the resultant of \eqref{eq: equivalent Condition02 No3 and Noe configurations not realizable} and \eqref{eq: equivalent Condition03 No3 and Noe configurations not realizable} in $p_1$ is
$-4z^3p^{10}(4p-z)(p^{10} - 16zp^9 + 118z^2p^8 - 476z^3p^7 + 1066z^4p^6 - 1372z^5p^5 + 1064z^6p^4 - 524z^7p^3 + 161z^8p^2 - 28z^9p + 2z^{10})$. Since the condition of
Lemma~3 does not hold, it is implied that there exists at least one common root between \eqref{eq: equivalent Condition02 No3 and Noe configurations not realizable} and \eqref{eq: equivalent Condition03 No3 and Noe configurations not realizable} if and only if \eqref{eq: Condition five-reactive-element configurations realizable 04} holds with $p < z/(2 + \sqrt{5})$.
This further implies that $p_1 > 0$.
\textit{Sufficiency.}
Based on the discussion in the necessity part, there exists $p_1 > 0$ such that \eqref{eq: equivalent Condition01 No3 and Noe configurations not realizable}--\eqref{eq: equivalent Condition03 No3 and Noe configurations not realizable} hold. Let $m$, $\gamma$, $\alpha$, and $\beta$ satisfy \eqref{eq: 101 No3 and Noe configurations} and \eqref{eq: 104 No3 and Noe configurations}--\eqref{eq: beta No3 and Noe configurations}, which implies $\alpha$, $\beta$, $\gamma$, $m$ $> 0$. Therefore, \eqref{eq: Condition01 Noe configurations not realizable}--\eqref{eq: Condition03 Noe configurations not realizable},
\eqref{eq: 102 No3 and Noe configurations}, and \eqref{eq: 103 No3 and Noe configurations} hold. Therefore,
$Z(s)$ can be written in the form of \eqref{eq: Z1 + Z2 No3 and Noe configurations}. Decompose $Z(s)$ as $Z(s) = Z_1(s) + Z_2(s)$, where $Z_1(s)$ is in the form of \eqref{eq: Z1 No3 (s+p1)(s+p)} with $m$, $p_1$, $p$ $> 0$ and $q = p_1p/(p_1 + p)$, and $Z_2(s)$ is in the form of \eqref{eq: 02 Noe configurations not realizable} with $\alpha$, $\beta$, $\gamma$ $> 0$.
By letting $R_1 = m$, $C_1 = 1/(mq)$, and $L_1 = m/(p_1 + p)$, $Z_1(s)$ is realizable as in Fig.~\ref{fig: Two-reactive configurations N1}(c).
Let $C_{21}$, $C_{22}$, $R_{21}$, and $L_{21}$ satisfy \eqref{eq: C21 Noe configurations not realizable}--\eqref{eq: L21 Noe configurations not realizable}. Since \eqref{eq: Condition03 Noe configurations not realizable} hold, $C_{21}$, $C_{22}$, $R_{21}$, $L_{21}$ $> 0$. Based on the discussion in the proof of Lemma~11, \eqref{eq: 03 Noe configurations not realizable}--\eqref{eq: 08 Noe configurations not realizable} hold. Therefore, $Z_2(s)$ can be realized as the configuration in Fig.~\ref{fig: Three-reactive Four-element N2}(e). The sufficiency part is proved.
\end{proof}
\ifCLASSOPTIONcaptionsoff
\fi
\end{document} | math | 126,671 |
\begin{document}
\title{Quantum correlations across two octaves from combined up and down conversion}
\author{Jingyan Li$^{1,2}$ and M.~K. Olsen$^{2,3}$}
\affiliation{\\
$^{1}$School of Electronic and Electrical Engineering, Wuhan Textile
University, Wuhan 430079, China\\
}
\affiliation{$^{2}$Quantum Science Otago and Dodd-Walls Centre for Photonic and Quantum
Technologies, Department of Physics, University of Otago, Dunedin, New
Zealand\\
}
\affiliation{$^{3}$School of Mathematics and Physics, University of Queensland, Brisbane,
Queensland 4072, Australia\\
}
\date{\today}
\begin{abstract}
We propose and analyse a cascaded optical parametric system which involves three interacting modes across two octaves of frequency difference. Our system, combining degenerate optical parametric oscillation (OPO) with second harmonic generation (SHG), promises to be a useful source of squeezed and entangled light at three differing frequencies. We show how changes in damping rates and the ratio of the two concurrent nonlinearities affect the quantum correlations in the output fields. We analyse the threshold behaviour, showing how the normal OPO threshold is changed by the addition of the SHG interactions. We also find that the inclusion of the OPO interaction removes the self-pulsing behaviour found in normal SHG. Finally, we show how the Einstein-Podolsky-Rosen correlations can be controlled by the injection of a coherent seed field at the lower frequency.
\mbox{e}nd{abstract}
\maketitle
\section{Introduction}
\label{sec:intro}
The theory of the interaction of light fields at one frequency with
nonlinear materials to produce fields at different frequencies goes back at
least to Armstrong {\mbox{e}m et al. } and their seminal work which included downconversion and second and third harmonic generation~\cite{Armstrong}. Since the publication of that work, the optical parametric oscillator (OPO) in both its degenerate and nondegenerate forms~\cite{Giordmaine,Nassau,Yariv} has become a standard workhorse for quantum optics and quantum information, especially with respect to the Einstein Podolsky Rosen paradox~\cite{RMPMargaret}. The related process of intracavity second harmonic generation (SHG) has also long been known to produce quantum states of the optical field~\cite{SHGPereira}.
In the degenerate OPO, any entanglement will necessarily be across one octave, with the same being true of
SHG~\cite{sumdiff,PingKoy}. In this work we combine these two processes in either a cascaded or concurrent manner, to produce entangled beams and states exhibiting EPR steering across two octaves of frequency difference. Such a difference in frequencies has previously been predicted for a system which cascades two SHG processes to produce entangled outputs at three different frequencies, with both bipartite~\cite{4HG} and tripartite correlations~\cite{4HGtri}. The three level system we analyse here differs essentially only in the choice of cavity field which is externally pumped. In these previous two octave systems, this was the field at the lowest frequency. In this work it is the field at the intermediate frequency which is pumped. Just as with the normal OPO and SHG processes, this small change leads to markedly different behaviours.
The system we analyse has the potential to provide enhanced flexibility for quantum interfaces between light and atomic ensembles, quantum state engineering, multiplexing in quantum communications~\cite{multiplex}, the entanglement of atomic ensembles, and quantum teleportation~\cite{Hammerer}. The availability of entanglement and EPR-steering over such a large frequency range will bring further flexibility to the linking of quantum processes at different wavelengths, for example the telecommunications frequencies and atomic systems used in quantum information processing, particularly with regard to quantum memory~\cite{Julsgaard}.
In this article we first provide the Hamiltonian, then develop the equations of motion in the positive-P representation~\cite{P+}. These equations are then solved numerically to find the time evolution of the intracavity fields. We check the full quantum numerical results against those found analytically for the classical steady states, finding that these agree in most parameter regimes. One regime where they do not agree is that in which the classical solutions exhibit self pulsing behaviour. In other regimes we use the steady state solutions for a linearised fluctuation analysis. This allows us to find the oscillation threshold, which is changed from that in the standard OPO. Using the standard input-output relations~\cite{mjc}, we are able to calculate the expressions for squeezing and both bipartite and tripartite EPR steering and inseparability in the output modes. In cases where the output expressions are rather simple, we give these analytically. In other cases the results are produced graphically. We look at the effects of changing the ratio of the two nonlinearities and the cavity damping rates. Finally we examine the effects of an injected signal at the lowest frequency. The range of interesting quantum states found suggests that this system shows promise for emerging quantum technological applications.
\section{Hamiltonian and equations of motion}
\label{sec:Ham}
The system we investigate here uses two $\chi^{(2)}$ nonlinear interactions within the same pumped optical cavity which is resonant for all three frequencies of interest. These could be either two crystals or one customised dielectric~\cite{Zhu} which converts the input field via both up and down conversion. The three interacting electromagnetic fields are the central externally pumped field at frequency $\omega_{2}$, and two others at $\omega_{1}$ and $\omega_{3}$. The field at $\omega_{2}$ interacts via a nonlinearity represented by $\kappa_{1}$ to produce a downconverted field at $\omega_{1}$, where $\omega_{2}=2\omega_{1}$. It also interacts via the nonlinearity represented by $\kappa_{2}$ to produce an upconverted field at $\omega_{3} (=2\omega_{2})$. This field is therefore the fourth harmonic of $\omega_{1}$, with the interacting fields spanning two octaves of frequency difference.
The low frequency field at $\omega _{1}$, is
represented by the bosonic operator $\hat{a}_{1}$. The second harmonic, at $\omega _{2}=2\omega
_{1}$, which will be externally pumped, is represented by $\hat{a}_{2}$, and
the fourth harmonic, at $\omega _{3}=4\omega _{1}$, is represented by $\hat{a}_{3}$. The unitary interaction Hamiltonian in a rotating frame is then
written as
\begin{equation}
\mathcal{H}_{int}=\frac{i\hbar }{2}\left[ \kappa _{1}(\hat{a}_{1}^{2}\hat{a}
_{2}^{\dag }-\hat{a}_{1}^{\dag \,2}\hat{a}_{2})+\kappa _{2}(\hat{a}_{2}^{2}
\hat{a}_{3}^{\dag }-\hat{a}_{2}^{\dag \,2}\hat{a}_{3})\right] .
\label{eq:UHam}
\mbox{e}nd{equation}
Since we are analysing the intracavity configuration, we also have the pumping Hamiltonian,
\begin{equation}
\mathcal{H}_{pump}=i\hbar \left( \mbox{e}psilon _{2}\hat{a}_{2}^{\dag }-\mbox{e}psilon
_{2}^{\ast }\hat{a}_{2}\right) , \label{eq:Hpump}
\mbox{e}nd{equation}
where $\mbox{e}psilon _{2}$ represents an external pumping field which is usually
taken as coherent, although this is not necessary~\cite{Liz}. The damping of
the cavity into a zero temperature Markovian reservoir is described by the Lindblad
superoperator
\begin{equation}
\mathcal{L}\rho =\sum_{i=1}^{3}\gamma _{i}\left( 2\hat{a}_{i}\rho \hat{a}
_{i}^{\dag }-\hat{a}_{i}^{\dag }\hat{a}_{i}\rho -\rho \hat{a}_{i}^{\dag }
\hat{a}_{i}\right) ,
\label{eq:Lindblad}
\mbox{e}nd{equation}
where $\rho $ is the system density matrix and $\gamma _{i}$ is the cavity
loss rate at $\omega _{i}$. In this work we will treat all three optical
fields as being at resonance with the optical cavity. While including detuning is possible, this makes analytical results very difficult to obtain, so we will stick to the simplest case here. In general, any detuning acts to degrade the correlations used to measure squeezing and entanglement in a $\chi^{(2)}$ system~\cite{Granja}.
In order to analyse this system, we will use the well known and exact quantum phase space method, the positive-P representation~\cite{P+}, which allows us to readily calculate any time-normally-ordered operator moments.
Following the usual procedures~\cite{DFW}, we derive equations of motion
in the positive-P representation~\cite{P+},
\begin{eqnarray}
\frac{d\alpha _{1}}{dt} &=&-\gamma _{1}\alpha _{1}+\kappa _{1}\alpha
_{1}^{+}\alpha _{2}+\sqrt{\kappa _{1}\alpha _{2}}\,\mbox{e}ta _{1}, \notag \\
\frac{d\alpha _{1}^{+}}{dt} &=&-\gamma _{1}^{+}\alpha _{1}^{+}+\kappa
_{1}\alpha _{1}\alpha _{2}^{+}+\sqrt{\kappa _{1}\alpha _{2}^{+}}\,\mbox{e}ta _{2},
\notag \\
\frac{d\alpha _{2}}{dt} &=&\mbox{e}psilon _{2}-\gamma _{2}\alpha _{2}+\kappa
_{2}\alpha _{2}^{+}\alpha _{3}-\frac{\kappa _{1}}{2}\alpha _{1}^{2}+\sqrt{
\kappa _{2}\alpha _{3}}\,\mbox{e}ta _{3}, \notag \\
\frac{d\alpha _{2}^{+}}{dt} &=&\mbox{e}psilon _{2}^{\ast }-\gamma _{2}\alpha
_{2}^{+}+\kappa _{2}\alpha _{2}\alpha _{3}^{+}-\frac{\kappa _{1}}{2}\alpha
_{1}^{+\,2}+\sqrt{\kappa _{2}\alpha _{3}^{+}}\,\mbox{e}ta _{4}, \notag \\
\frac{d\alpha _{3}}{dt} &=&-\gamma _{3}\alpha _{3}-\frac{\kappa _{2}}{2}
\alpha _{2}^{2}, \notag \\
\frac{d\alpha _{3}^{+}}{dt} &=&-\gamma _{3}\alpha _{3}^{+}-\frac{\kappa _{2}}{2}\alpha _{2}^{+\,2}.
\label{eq:Pplus}
\mbox{e}nd{eqnarray}
It should be noted that these have the same form in either It\^{o} or
Stratonovich calculus \cite{SMCrispin}. In the above, the complex variable
pairs $(\alpha _{i},\alpha _{j}^{+})$ correspond to the operator pairs $(\hat{a}_{i},\hat{a}_{j}^{\dag })$ in the sense that stochastic averages of
products converge to normally-ordered operator expectation values, e.g. $\overline{\alpha _{i}^{+\,m}\alpha _{j}^{n}}\rightarrow \langle \hat{a}_{i}^{\dag \,m}\hat{a}_{j}^{n}\rangle $. The $\mbox{e}ta _{j}$ are Gaussian noise
terms with the properties $\overline{\mbox{e}ta _{i}}=0$ and $\overline{\mbox{e}ta
_{j}(t)\mbox{e}ta _{k}(t^{Phys. Rev.\ ime })}=\delta _{jk}\delta (t-t^{Phys. Rev.\ ime })$. Although there can be divergence problems with the positive-P representation, it is known to be accurate where it converges, which is the case with all results presented here.
\section{Steady-state and threshold properties}
\label{sec:cavidade}
In order to obtain analytical steady-state results for the intracavity intensities and amplitudes, we solve the semi-classical equivalents of Eq.~\ref{eq:Pplus}, simply obtained by removing the noise terms. The results thus obtained can be checked against stochastic integration of the full equations. This procedure also allows us to calculate the threshold pumping value at which the downconversion process begins to produce non-zero amplitudes in the low frequency mode. This threshold behaviour is well known from the theory of the optical parametric oscillator (OPO)~\cite{DMW,Arabe}. A stability analysis of the system allows the threshold pumping amplitude to be calculated as
\begin{equation}
\mbox{e}psilon _{2}^{c}=\frac{\gamma _{1}\gamma _{2}}{\kappa _{1}}+\frac{\gamma
_{1}^{3}\kappa _{2}^{2}}{2\gamma _{3}\kappa _{1}^{3}}.
\label{eq:critpump}
\mbox{e}nd{equation}
We immediately see that this is higher than the threshold for isolated downconversion, where the threshold is
$\gamma _{1}\gamma _{2}/\kappa _{1}$. The increased pump power is required because the upconversion process to produce the mode at $\omega_{3}$ also depletes the pump in our system.
The steady state amplitudes for the three modes can be found in the two different cases:\\
(i) below threshold $\mbox{e}psilon _{2}<\mbox{e}psilon _{2}^{c}$,
\begin{eqnarray}
\alpha _{1}^{ss} &=&0, \nonumber \\
\alpha _{2}^{ss} &=&\frac{\xi }{3\kappa _{2}^{2}}-\frac{2\gamma _{2}\gamma
_{3}}{\xi }, \nonumber \\
\alpha _{3}^{ss} &=&-\frac{\kappa _{2}\left( \alpha _{2}^{ss}\right) ^{2}}{2\gamma _{3}},
\label{eq:belowcrit}
\mbox{e}nd{eqnarray}
where
\begin{equation}
\xi =\left( 27\mbox{e}psilon _{2}\gamma _{3}\kappa _{2}^{4}+3\sqrt{3}\sqrt{8\gamma _{2}^{3}\gamma _{3}^{3}\kappa _{2}^{6}+27\mbox{e}psilon _{2}^{2}\gamma
_{3}^{2}\kappa _{2}^{8}}\right) ^{1/3},
\label{eq:xi}
\mbox{e}nd{equation}
and \\
(ii) above threshold $\mbox{e}psilon _{2}>\mbox{e}psilon _{2}^{c}$,
\begin{eqnarray}
\alpha _{1}^{ss} &=&\pm \frac{2}{\kappa _{1}}\left( \mbox{e}psilon
_{2}-\mbox{e}psilon _{2}^{c}\right) , \nonumber \\
\alpha _{2}^{ss} &=&\frac{\gamma _{1}}{\kappa _{1}}, \nonumber \\
\alpha _{3}^{ss} &=& -\frac{\gamma _{1}^{2}\kappa _{2}}{2\kappa _{1}^{2}\gamma
_{3}}.
\label{eq:abovecrit}
\mbox{e}nd{eqnarray}
As with the standard OPO, the system exhibits similar behaviour to a second-order phase transition at
$\mbox{e}psilon _{2}=\mbox{e}psilon _{2}^{c}$. When the pumping is above threshold, the
below-threshold solution for the fundamental frequency field $\alpha _{1}^{ss}=0$
becomes unstable and the system moves onto a new stable
branch witht two solutions of the the fundamental field
having equal amplitude and opposite phase.
The steady amplitudes of the central frequency $\omega _{2}$ and higher frequency $\omega _{3}$
modes have opposite phases whether the system is running below or above threshold. What is noticeable is that the steady state solutions above threshold for $\alpha_{2}$ and $\alpha_{3}$ have no dependence on the pump power. Once the cavity is being pumped above the oscillation threshold, these two fields do not change with changes in the pumping.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.85\columnwidth]{F1.eps}
\mbox{e}nd{center}
\caption{(colour online) The intracavity intensities calculated via $4\times 10^{5}$ trajectories of the positive-P equations are shown as the solid lines. The dashed lines are the analytical steady-state expressions. The parameters used are $\gamma_{j}=1$, $\kappa_{1}=\kappa_{2}=10^{-2}$, and $\mbox{e}psilon=1.5\mbox{e}psilon_{c}$. Averaging errors are smaller than the plotted linewidths. All quantities plotted in this and subsequent graphics are dimensionless.}
\label{fig:intensities}
\mbox{e}nd{figure}
The time development of the intensities above threshold is shown in Fig.~\ref{fig:intensities} in the fully
quantum picture with the positive-P equations integrated over $4\times 10^{5}$ stochastic trajectories.
With $\mbox{e}psilon _{2}=1.5\mbox{e}psilon _{2}^{c}$ we see that the analytical steady-state values, plotted as dashed lines, are in good agreement with the quantum solutions.
It is also well known that in normal second harmonic generation (SHG) there is a pumping threshold above which the output intensities exhibit a periodic pulsing behaviour~\cite{pulse,Bache}. In the present case the classical behaviour of the system is similar and a hard mode transition can be found above which self-pulsing occurs. However, this does not survive the full quantum treatment, with the oscillations disappearing completely. A less pronounced damping of self-pulsing oscillations has recently been found in a full quantum treatment of other cascaded systems~\cite{3HG,4HG} and shows the dangers of relying on classical analyses of quantum optical systems. The canonical method to calculate self pulsing in SHG is to numerically integrate the classical equations with a small complex seed in one or both the modes. Without this seed, the self-pulsing is not found, although it appears with integration of the positive-P equations without needing any seed at all. For our system, small complex seeds in the initial condition of the classical simulations gives self-pulsing, as shown in Fig.~\ref{fig:selfpulse}. On the other hand, the quantum solution diverges from this at short times, to enter a steady state with a much lower average value. The reason for this is that the classical solutions stay on the unstable branch of the solutions for $\alpha_{1}$, remaining at zero. The classical solution is unphysical. In the quantum case, spontaneous downconversion early in the evolution leads to stimulated downconversion and the steady state remains on the stable branch. A small injected signal $\mbox{e}psilon_{1}$ in the classical integration will also push the solutions onto the stable branch, and in this case self-pulsing is found neither classically nor quantum mechanically.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.85\columnwidth]{selfpulse.eps}
\mbox{e}nd{center}
\caption{(colour online) The classical and quantum solutions for $N_{2}$, with the same parameters as Fig.~\ref{fig:intensities} except for $\mbox{e}psilon_{2}=5\mbox{e}psilon_{2}^{c}$. Both integrations have a small complex seed in the initial conditions, with $\alpha_{1}(0)=0$, $\alpha_{2}(0)=1+2i$ and $\alpha_{3}(0)=1-2i$.}
\label{fig:selfpulse}
\mbox{e}nd{figure}
\section{Ornstein-Uhlenbeck analysis and fluctuation spectra}
\label{sec:OU}
When nonlinear optical media are held inside a pumped optical cavity, the
accessible observables are usually the output spectral correlations, which
are accessible using homodyne measurement techniques~\cite{mjc}. These are
readily calculated in the steady state by treating the system as an
Ornstein-Uhlenbeck process~\cite{SMCrispin}. In order to do this, we begin
by expanding the positive-P variables into their steady-state expectation
values plus delta-correlated Gaussian fluctuation terms, e.g.
\begin{equation}
\alpha_{ss} \rightarrow \langle\hat{a}\rangle_{ss}+\delta\alpha.
\label{eq:fluctuate}
\mbox{e}nd{equation}
Given that we can calculate the $\langle\hat{a}\rangle_{ss}$, we may now
write the equations of motion for the fluctuation terms. The resulting
equations are written for the vector of fluctuation terms as
\begin{equation}
d\delta \vec{\alpha}=-A\delta \vec{\alpha}dt+Bd\vec{W},
\label{eq:OEeqn}
\mbox{e}nd{equation}
where $A$ is the drift matrix containing the steady-state solution, $B$ is
found from the factorisation of the drift matrix of the original
Fokker-Planck equation, $D=BB^{T}$, with the steady-state values substituted
in, and $d\vec{W}$ is a vector of Wiener increments. As long as the matrix $A
$ has no eigenvalues with negative real parts, this method may be used to
calculate the intracavity spectra via
\begin{equation}
S(\omega) = (A+i\omega)^{-1}D(A^{\mbox{\small{T}}}-i\omega)^{-1},
\label{eq:Sout}
\mbox{e}nd{equation}
from which the output spectra are calculated using the standard input-output
relations~\cite{mjc}.
In this case, $A$ is found as
\begin{equation}
A =
\begin{bmatrix}
\gamma_{1} & -\kappa_{1}\alpha_{2} & -\kappa_{1}\alpha_{1}^{\ast} & 0 & 0 & 0
\\
-\kappa_{1}\alpha_{2}^{\ast} & \gamma_{1} & 0 & -\kappa_{1}\alpha_{1} & 0 & 0
\\
\kappa_{1}\alpha_{1} & 0 & \gamma_{2} & -\kappa_{2}\alpha_{3} &
-\kappa_{2}\alpha_{2}^{\ast} & 0 \\
0 & \kappa_{1}\alpha_{1}^{\ast} & -\kappa_{2}\alpha_{3}^{\ast} & \gamma_{2}
& 0 & -\kappa_{2}\alpha_{2} \\
0 & 0 & \kappa_{2}\alpha_{2} & 0 & \gamma_{3} & 0 \\
0 & 0 & 0 & \kappa_{2}\alpha_{2}^{\ast} & 0 & \gamma_{3}
\mbox{e}nd{bmatrix}
\label{eq:Amat}
\mbox{e}nd{equation}
and $D$ is a $6\times 6$ matrix with
$\left[\kappa_{1}\alpha_{2},\kappa_{1}\alpha_{2}^{\ast},\kappa_{2}\alpha_{3},\kappa_{2}\alpha_{3}^{\ast},0,0\right]$
on the diagonal. In the above, the $\alpha_{j}$ should be read as their
steady-state mean values, so that $\alpha _{j}^{\ast }=\overline{\alpha _{j}^{+}}$, for example.
These are now complex numbers that are the
averages of the positive-P stochastic variables. Because we have parametrised
our system using $\gamma_{1}=1$, the frequency $\omega$ is in units of $\gamma_{1}$.
$S(\omega)$ is now in terms of quadratic products of the fluctuation operators such as
$\delta\alpha_{i}\delta\alpha_{j}$ and $\delta\alpha_{i}^{\ast}\delta\alpha_{j}^{\ast}$.
Since quadrature properties are what is measured by homodyne detection, we define the amplitude and phase quadrature operators as
\begin{equation}
\begin{array}[b]{l}
\hat{X}_{j}=\hat{a}_{j}+\hat{a}_{j}^{\dag }, \\
\hat{Y}_{j}=-i\left( \hat{a}_{j}-\hat{a}_{j}^{\dag }\right) .
\mbox{e}nd{array}
\mbox{e}nd{equation}
We note here that other definitions are sometimes used in the literature and that this changes the numerical value of the Heisenberg uncertainty principle. Our choice gives $V(\hat{X}_{j})V(\hat{Y}_{j}\geq 1$ and means that squeezing in a particular quadrature exists whenever its variance is found to be less than 1.
To express the fluctuation expressions in terms of the canonical quadratures, we calculate
\begin{equation}
S^{q}\left( \omega \right) =QSQ^{T},
\label{eq:quadtransform}
\mbox{e}nd{equation}
where $Q$ is the block diagonal $6\times 6$ matrix constructed from
\begin{equation}
q=\left[
\begin{array}{cc}
1 & 1 \\
-i & i
\mbox{e}nd{array}
\right].
\mbox{e}nd{equation}
$S^{q}\left( \omega \right)$ gives us the products from which we construct the
output variances and covariances for modes $i$ and $j$ as,
\begin{eqnarray}
V\left( \hat{X}_{i},\hat{X}_{j}\right) &=& \delta _{ij}+\sqrt{\gamma _{i}\gamma
_{j}}\left( S_{2i-1,2j-1}^{q}+S_{2j-1,2i-1}^{q}\right), \nonumber \\
V\left( \hat{Y}_{i},\hat{Y}_{j}\right) &=& \delta _{ij}+\sqrt{\gamma _{i}\gamma
_{j}}\left( S_{2i,2j}^{q}+S_{2j,2i}^{q}\right),
\label{eq:quadvars}
\mbox{e}nd{eqnarray}
in which the variances and covariances are defined as
$V\left( \hat{X}_{i}\right) =\left\langle \hat{X}_{i}^{2}\right\rangle -\left\langle
\hat{X}_{i}\right\rangle^{2}$ and $V\left( \hat{X}_{i},\hat{X}_{j}\right) =\left\langle \hat{X}_{i}\hat{X}_{j}\right\rangle -\left\langle \hat{X}_{i}\right\rangle \left\langle \hat{X}
_{j}\right\rangle.$
\section{Steady state bipartite correlations}
\label{sec:sscorrelations}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.85\columnwidth]{squeeze.eps}
\mbox{e}nd{center}
\caption{(colour online) Quadrature variances for the three squeezed quadratures below threshold, with
$\kappa _{1}=\kappa _{2}=0.01$, $\gamma _{1}=\gamma _{2}=\gamma _{3}=1$, and
$\mbox{e}psilon _{2}=0.9\mbox{e}psilon _{2}^{c}$. The frequency axis is in units of the linewidth of the
fundamental, $\gamma _{1}$.}
\label{fig:squeeze}
\mbox{e}nd{figure}
The squeezing in the amplitude and phase
quadrature for the three different modes can be calculated analytically following
from Eq.~\ref{eq:belowcrit} and Eq.~\ref{eq:quadtransform}.
Since the fundamental mode has a mean amplitude of zero
below threshold, this simplifies the drift matrix and we can
derive the below threshold output squeezing spectra as
\begin{eqnarray}
S_{1\pm }\left( \omega \right) &=&1\pm \frac{4\gamma _{1}\kappa
_{1}\alpha _{2}}{\omega ^{2}+\left( \gamma _{1}\mp \kappa _{1}\alpha
_{2}\right) ^{2}}, \nonumber \\
S_{2\pm }\left( \omega \right) &=&1\pm 4\gamma _{2}\kappa _{2}\alpha
_{3}\left( \omega ^{2}+\gamma _{3}^{2}\right) \mbox{e}ta _{\pm }\left( \omega
\right), \nonumber \\
S_{3\pm }\left( \omega \right) &=&1\pm 4\gamma _{3}\alpha
_{2}^{2}\alpha _{3}\kappa _{2}^{3}\mbox{e}ta _{\pm }\left( \omega \right),
\label{eq:quadspekbelow}
\mbox{e}nd{eqnarray}
where
\begin{equation}
\mbox{e}ta _{\pm }\left( \omega \right) =\frac{1}{\omega^{2}\left( \gamma
_{2}+\gamma _{3}\mp \kappa _{2}\alpha _{3}\right)^{2}+\left( -\omega
^{2}+\kappa _{2}^{2}\alpha _{2}^{2}+\gamma _{2}\gamma _{3}\mp \gamma
_{3}\kappa _{2}\alpha _{3}\right)^{2}},
\label{eq:eta}
\mbox{e}nd{equation}
and
$S_{j+}\left(\omega \right) =S\left(X_{j}\right) ,S_{j-}\left(\omega
\right)=S\left(Y_{j}\right).$ The spectral variances of the squeezed quadratures are shown in Fig.~\ref{fig:squeeze}, for $\mbox{e}psilon_{2}=0.9\mbox{e}psilon_{2}^{c}$. We note here that all spectra shown are symmetric about zero frequency. What we notice is that the quadratures which exhibit squeezing are those we expect from parametric downconversion, with $\hat{Y}_{1}$ being squeezed, and from second harmonic generation, with both $\hat{X}_{2}$ and $\hat{X}_{3}$ being squeezed.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.85\columnwidth]{squeezeabove.eps}
\mbox{e}nd{center}
\caption{(colour online) Quadrature variances for the three squeezed quadratures above threshold, with
$\kappa _{1}=\kappa _{2}=0.01$, $\gamma _{1}=\gamma _{2}=\gamma _{3}=1$, and
$\mbox{e}psilon _{2}=1.5\mbox{e}psilon _{2}^{c}$. The dotted line at one is a guide to the eye and the frequency axis is in units of the linewidth of the
fundamental, $\gamma _{1}$.}
\label{fig:squeezeabove}
\mbox{e}nd{figure}
Above threshold the analytical expressions for the output squeezing are quite lengthy, mainly due to that fact that the low frequency mode now has a non-zero solution. We will not give these here, but will illustrate the results in Fig.~\ref{fig:squeezeabove}, for $\mbox{e}psilon_{2}=1.5\mbox{e}psilon_{2}^{c}$. We see that the same quadratures are squeezed as below threshold, but that the degree of squeezing has been reduced.
The next question we raise is whether any of the possible bipartitions will exhibit the Einstein-Podolsky-Rosen paradox~\cite{EPR}, now commonly known as EPR steering~\cite{Erwin,Jonesteer}. In the continuous variable case, this is usually measured using the Reid inequalities for the inferred variances~\cite{EPRMDR,ZYOu}. This is written for the output spectral variances as
\begin{equation}
EPR_{ij}(\omega) = S^{inf}(\hat{X}_{i})S^{inf}(\hat{Y}_{i})\geq 1,
\label{eq:eprMDR}
\mbox{e}nd{equation}
where
\begin{eqnarray}
S_{inf}(\hat{X}_{i}) &=& S(\hat{X}_{i})-\frac{[S(\hat{X}_{i},\hat{X}_{j})]^{2}}{S(\hat{X}_{j})}, \nonumber \\
S_{inf}(\hat{Y}_{i}) &=& S(\hat{Y}_{i})-\frac{[S(\hat{Y}_{i},\hat{Y}_{j})]^{2}}{S(\hat{Y}_{j})}.
\label{eq:EPRdef}
\mbox{e}nd{eqnarray}
In the language of EPR-steering, $EPR_{ij}<1$ shows that mode $i$ can be steered by measurements of mode $j$. In some cases asymmetric steering is possible, where $EPR_{ij}<1$ while $EPR_{ji}>1$. The question as to whether this was possible was first raised by Wiseman {\mbox{e}m et al. }s~\cite{Wiseman}, and answered in the affirmative for Gaussian measurements by Olsen and Bradley~\cite{SFG}, Midgley {\mbox{e}m et al. }s~\cite{sapatona}, and H\"andchen {\mbox{e}m et al. }s~\cite{Handchen}. It has since been shown that asymmetric steering is generally possible~\cite{Bowles}, without any restriction on measurements.
Because EPR steerable states are a strict subset of the entangled states,
both symmetric and asymmetric steering demonstrate that the two modes concerned
are fully bipartite entangled. We will therefore use the Reid inequalities to demonstrate both EPR steering and bipartite entanglement.
We obtain the below threshold covariances between each pair of modes as
\begin{eqnarray}
S(\hat{X}_{1},\hat{X}_{2}) &=& S(\hat{X}_{1},\hat{X}_{3})=0, \nonumber \\
S(\hat{Y}_{1},\hat{Y}_{2}) &=& S(\hat{Y}_{1},\hat{Y}_{3})=0, \nonumber \\
S(\hat{X}_{2},\hat{X}_{3}) &=& -4\alpha _{2}\alpha _{3}\gamma _{3}\sqrt{\gamma _{2}\gamma _{3}}\kappa _{2}^{2}\mbox{e}ta _{+}\left( \omega \right) ,
\nonumber \\
S(\hat{Y}_{2},\hat{Y}_{3}) &=& 4\alpha _{2}\alpha _{3}\gamma _{3}\sqrt{\gamma
_{2}\gamma _{3}}\kappa _{2}^{2}\mbox{e}ta _{-}\left( \omega \right).
\label{eq:covariances}
\mbox{e}nd{eqnarray}
Since the covariances between modes 1 and 2 and modes 1 and 3 are zero, we
can easily find four of the possible EPR correlations as
\begin{eqnarray}
EPR_{12} &=& EPR_{13} = S_{1+}\left( \omega \right)S_{1-}\left( \omega \right), \nonumber \\
EPR_{21} &=& S_{2+}\left( \omega \right) S_{2-}\left( \omega \right), \nonumber \\
EPR_{31} &=& S_{3+}\left( \omega \right) S_{3-}\left( \omega \right).
\label{eq:EPRanalytic}
\mbox{e}nd{eqnarray}
It is obvious that none of these bipartitions can exhibit EPR steering below threshold, due to to Heisenberg Uncertainty Principal. An interesting result is that, although $EPR_{21}$ and $EPR_{31}$ are products of variances for different modes, they have equal values, with neither falling below one. This is not the case above threshold, where these two are no longer equal.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.85\columnwidth]{EPR23below.eps}
\mbox{e}nd{center}
\caption{(colour online) $EPR_{23}$ and $EPR_{32}$ for
$\kappa _{1}=\kappa _{2}=0.01$, $\gamma _{1}=\gamma _{2}=\gamma _{3}=1$, and
$\mbox{e}psilon _{2}=0.9\mbox{e}psilon _{2}^{c}$. The dotted line at one is a guide to the eye.}
\label{fig:EPR23below}
\mbox{e}nd{figure}
The case for modes 2 and 3, however, is different. A complicated analytical expression tells us that
$EPR_{32}=EPR_{23}$, so that any EPR steering here is completely symmetric. The result for the same parameters as in Fig.~\ref{fig:squeeze} is shown in Fig.~\ref{fig:EPR23below}. We see that the Reid inequalities are violated over a range near zero frequency, meaning that modes 2 and 3 are genuinely bipartite entangled.
Above threshold, the analytical expressions for all bipartitions become extremely complicated, and are best represented graphically.
We will begin with $\kappa_{1}=\kappa_{2}$ and all cavity loss rates being equal, showing the effects of varying these later in the article.
We find that modes 1 and 2 exhibit symmetric EPR steering over a broad range, while 1 and 3 exhibit completely asymmetric EPR steering over a narrower range of frequencies. The two higher frequency modes, which exhibit EPR steering below threshold, lose this property completely as the solution for $\alpha_{1}$ moves onto the stable branch where it has non-zero amplitude. In terms of entanglement and EPR steering properties, the system changes completely at threshold.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.85\columnwidth]{EPRabove.eps}
\mbox{e}nd{center}
\caption{(colour online) The EPR correlations which violate the inequality above threshold, for
$\kappa _{1}=\kappa _{2}=0.01$, $\gamma _{1}=\gamma _{2}=\gamma _{3}=1$, and
$\mbox{e}psilon _{2}=1.5\mbox{e}psilon _{2}^{c}$. The dotted line at one is a guide to the eye.}
\label{fig:EPRabove}
\mbox{e}nd{figure}
We find that the symmetry or asymmetry of the EPR steering between the output modes above threshold can be
simply controlled by the ratio of loss rates and the ratio of nonlinearities.
Firstly, in Fig.~\ref{fig:EPRsmallg2}, we show the results of a loss rate for the middle frequency which is one tenth of that for the other two, i.e. $\gamma_{2}=0.1\gamma_{1}=0.1\gamma_{3}$. Whereas modes 1 and 2 exhibited symmetric steering for equal loss rates, their steering is now asymmetric. The opposite has happened with modes 1 and 3, with their steering now being symmetric.
The symmetry properties of the EPR steering can be controlled by adjusting the cavity loss rates, as was also found with intracavity second harmonic generation~\cite{SHGEPR}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.85\columnwidth]{EPRsmallg2.eps}
\mbox{e}nd{center}
\caption{(colour online) The EPR correlations which violate the inequality above threshold, for
$\kappa _{1}=\kappa _{2}=0.01$, $\gamma _{1}=\gamma _{3} = 1 = 10\gamma _{2}$, and
$\mbox{e}psilon _{2}=1.5\mbox{e}psilon _{2}^{c}$. The dotted line at one is a guide to the eye.}
\label{fig:EPRsmallg2}
\mbox{e}nd{figure}
Changing the ratio $\kappa_{1}/\kappa_{2}$ also has an effect on the EPR steering properties above threshold. We can see in Fig.~\ref{fig:EPRkappa} that this can result in asymmetric steering in the bipartition of modes 1 and 2, with this swapping over at a certain frequency. Below $\omega \approx 2.1\gamma_{1}$, mode 1 can steer mode 2, while above this frequency there is a small violation of the inequality by $EPR_{12}$. The pairing of 1 and 3 exhibits both symmetric and asymmetric EPR steering as the measurement frequency changes.
We did not find any any steering involving the pair of fields at $\omega_{2}$ and $\omega_{3}$,
for the whole parameter range investigated with this ratio of the nonlinearities.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.85\columnwidth]{EPRkappa.eps}
\mbox{e}nd{center}
\caption{(colour online) The EPR correlations which violate the inequality above threshold, for
$\gamma _{j} = 1 \forall j$,
$\mbox{e}psilon _{2}=1.5\mbox{e}psilon _{2}^{c}$ and $\kappa_{2}=1.5\kappa_{1}$, with $\kappa_{1}=0.01$. The dotted line at one is a guide to the eye.}
\label{fig:EPRkappa}
\mbox{e}nd{figure}
\section{Tripartite correlations}
\label{sec:tri}
There are several methods of detecting tripartite inseparability and entanglement, with one common technique being based on inequalities developed by van Loock and Furusawa (vLF)~\cite{vLF}. These have proven useful for other cascaded
systems~\cite{AxMuzzJPB,AxMuzz}. The spectral inequalities we will use here are the set
\begin{equation}
S_{ijk} = S(\hat{X}_{i}-\frac{\hat{X}_{j}+\hat{X}_{k}}{\sqrt{2}})+S(\hat{Y}_{i}+\frac{\hat{Y}_{j}+\hat{Y}_{k}}{\sqrt{2}}) \geq 4,
\label{eq:VLFijk}
\mbox{e}nd{equation}
the violation of any one of which is sufficient to prove bipartite inseparability.
Following the work of Teh and Reid~\cite{Teh&Reid}, any one of these less than $2$ demonstrates genuine tripartite entanglement, while one of them less than $1$ demonstrates genuine tripartite EPR steering. We did not find a violation of these inequalities below threshold. Above threshold we found that some, but not all, of the set of inequalities are violated for particular parameter regimes, as shown in Fig.~\ref{fig:tripart}, where we have divided the values of $S_{312}$ by four so as to be directly comparable with the tripartite EPR steering inequality to be described below. This value of $S_{312}$ demonstrates tripartite inseparability for the system.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.85\columnwidth]{tripart.eps}
\mbox{e}nd{center}
\caption{(colour online) The spectral tripartite correlations which violate the inequalities above threshold, for the parameters
$\gamma _{j} = 1 \forall j$,
$\mbox{e}psilon _{2}=1.5\mbox{e}psilon _{2}^{c}$ and $\kappa_{2}=\kappa_{1}=0.01$. The dotted line at one is a guide to the eye.}
\label{fig:tripart}
\mbox{e}nd{figure}
With our three mode system, investigating tripartite EPR-steering is also of interest. It has been shown by Wang {\mbox{e}m et al. }s~\cite{Wang} that, in a multipartite system,
the steering of a given quantum mode is allowed when not less than half of the total number
of modes take part in the steering group. In a tripartite system, this means that measurements on
two of the modes are needed to steer the third.
In order to quantify this, we will use the correlation
functions developed by Olsen, Bradley, and Reid~\cite{OBR}.
With spectral tripartite inferred variances defined as
\begin{eqnarray}
S_{inf }^{(t)}\left( \hat{X}_{i}\right) &=& S\left( \hat{X}_{i}\right) -\frac{\left [ S\left( \hat{X}_{i},\hat{X}_{j}\pm \hat{X}_{k}\right)
\right]^{2} }{S\left( \hat{X}_{j}\pm \hat{X}_{k}\right) }, \nonumber \\
S_{inf }^{(t)}\left( \hat{Y}_{i}\right) &=& S\left( \hat{Y}_{i}\right) -\frac{\left [ S\left( \hat{Y}_{i},\hat{Y}_{j}\pm \hat{Y}_{k}\right)
\right ]^{2} }{S\left( \hat{Y}_{j}\pm \hat{Y}_{k}\right) },
\label{eq:OBRinf}
\mbox{e}nd{eqnarray}
we define
\begin{equation}
OBR_{ijk}=S_{inf }^{(t)}\left( \hat{X}_{i}\right) S_{inf }^{(t)}\left(
\hat{Y}_{i}\right),
\label{eq:OBRproduct}
\mbox{e}nd{equation}
so that a value of less than one means that there is an inferred violation of the Heisenberg uncertainty principal and mode $i$ can be steered by the combined forces of modes $j$ and $k$.
According to the work of He and Reid~\cite{HeReid}, genuine tripartite steering is demonstrated whenever
\begin{equation}
OBR_{ijk}+OBR_{jki}+OBR_{kij} < 1.
\label{eq:genuinetristeer}
\mbox{e}nd{equation}
We did not find genuine tripartite steering for this system. As shown in Fig.~\ref{fig:tripart}, we found that modes 1 and 2 could combine for some parameters to steer mode 3. We investigated a wide parameter regime numerically, but did not find any for which more than one of the modes could be steered by the remaining pair simultaneously.
\section{An injected signal at the lower frequency}
\label{sec:inject}
It is also possible to pump one of the cavity modes other than that at $\omega_{2}$. The process of optical parametric downconversion with an injected signal has been experimentally and theoretically studied in some depth~\cite{Bjorkholm,Haub,Hovde,Plusquellic}, with the injected signal often used for frequency stabilisation. An injected signal has also been shown to have a strong effect on any quantum correlations~\cite{kaled}, both changing the quadratures where squeezing is found and allowing for control of the asymmetry of EPR steering~\cite{signal}. For these reasons, we will examine here the effects of injecting a coherent signal at $\omega_{1}$. Theoretically, this involves another term in the pumping Hamiltonian, so that
\begin{equation}
{\cal H}_{pump}^{(s)} = {\cal H}_{pump}+i\hbar \left( \mbox{e}psilon _{1}\hat{a}_{1}^{\dag }-\mbox{e}psilon
_{1}^{\ast }\hat{a}_{1}\right),
\label{eq:Hpumpsignal}
\mbox{e}nd{equation}
where ${\cal H}_{pump}^{(s)}$ is the pumping Hamiltonian with injected signal. This change means that the equations of motion for $\alpha_{1}$ and $\alpha_{1}^{+}$ will have $\mbox{e}psilon_{1}$ and $\mbox{e}psilon_{1}^{\ast}$ added to them.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.85\columnwidth]{finject.eps}
\mbox{e}nd{center}
\caption{(colour online) The minima of the spectral bipartite EPR steering correlations with injected signal which violate the inequality, for parameters
$\gamma _{j} = 1 \forall j$,
$\mbox{e}psilon _{2}=0.9\mbox{e}psilon _{2}^{c}$, and $\kappa_{2}=\kappa_{1}=0.01$. The dotted line at one is a guide to the eye.}
\label{fig:finject}
\mbox{e}nd{figure}
The immediate effect of an injected signal is to change the threshold properties of the system, with the low frequency mode developing a steady-state non-zero amplitude for all finite values of $\mbox{e}psilon_{1}$. There is now no critical pump value for $\mbox{e}psilon_{2}$, with the solutions remaining on the stable branch for all pumping values. The injected signal has an even more dramatic effect on the EPR steering properties of the system. As seen above in Fig.~\ref{fig:EPR23below} the only two modes exhibiting EPR steering below threshold without injected signal were modes 2 and 3. With injected signal, the EPR steering of this bipartition soon vanishes as the signal is increased, which can be seen on the left hand side of Fig.~\ref{fig:finject}, which shows the $EPR_{ij}$ results for steerable bipartitions as the amplitude of the injected signal is increased. The quantities plotted are the minimum values of the Reid EPR correlations across all frequencies, ($0\leq \omega \leq 6$ numerically), so that a value of one means that the values near the carrier frequency can actually be larger than one. The addition of even a small injected signal (by comparison with $\mbox{e}psilon_{2}$) has a dramatic effect on the $(1,2)$ and $(1,3)$ bipartitions, These become highly steerable for small injection and then less so as $\mbox{e}psilon_{1}$ is increased. While $(1,2)$ exhibits symmetric steering, $(1,3)$ is totally asymmetric for these parameters, with $EPR_{31}\leq 1 \leq EPR_{13}$ across the whole range shown. The steerability of $(2,3)$ disappears on the same sort of scale of injection with which the others increase.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.85\columnwidth]{EPRinject.eps}
\mbox{e}nd{center}
\caption{(colour online) The spectral bipartite EPR steering correlations with injected signal, for parameters
$\gamma _{j} = 1 \forall j$,
$\mbox{e}psilon _{2}=0.9\mbox{e}psilon _{2}^{c}$, $\mbox{e}psilon_{1}=0.1\mbox{e}psilon_{2}$, and $\kappa_{2}=\kappa_{1}=0.01$. The dotted line at one is a guide to the eye.}
\label{fig:EPRinject}
\mbox{e}nd{figure}
The spectral values of the Reid EPR correlations for the bipartitions which exhibit steering for similar parameters as in
Fig.~\ref{fig:finject}, but at a fixed $\mbox{e}psilon_{1}=0.1\mbox{e}psilon_{2}$, are shown in Fig.~\ref{fig:EPRinject}. The asymmetry of the EPR steering demonstrated by $EPR_{31}$ and $EPR_{13}$ is clearly shown. Nevertheless, this result shows that modes 1 and 3 are entangled across two octaves of frequency difference, and that this system is therefore a potentially important resource for any quantum processes linking resources over a large bandwidth. The injection of the coherent signal allows for a simple means of control over the entanglement properties of the system.
\section{Conclusion}
In conclusion, the proposed system is a good candidate for novel quantum technologies which need squeezed and entangled optical states spanning a wide range of frequencies. With a single cavity input field it produces three output fields which are quadrature squeezed and different pairs of modes which are EPR steerable, with selection of the desired pairs being possible either by increasing the pump power or by injected signal. The quantum correlations of interest change depending on whether the system is being operated above or below the oscillation threshold, with good EPR steering being available in both regimes. The tripartite entanglement inequalities are only violated above threshold, where the lowest frequency mode develops a non-zero mean amplitude. An injected signal at the lowest frequency removes the threshold altogether and can provide either symmetric EPR steering across one octave or asymmetric EPR steering across two octaves. The flexibility and easy controllability of this system make it an attractive candidate for experimental investigation and future technological use.
\acknowledgments
J.Y. Li was supported by the National Natural Science Foundation of China
(Grant No. 11704287).
\begin{thebibliography}{99}
\bibitem{Armstrong} {J.A. Armstrong, N. Bloembergen, J. Ducuing, and P.S.
Pershan, Phys. Rev.\ \textbf{127}, 1918 (1962).}
\bibitem{Giordmaine}{J.A. Giordmaine and R.C. Miller, Phys. Rev.\ l {\bf 14}, 973 (1965).}
\bibitem{Nassau}{K. Nassau and H.J. Levinstein, Appl. Phys. Lett. {\bf 7}, 69 (1965).}
\bibitem{Yariv}{A. Yariv and W.H. Louisell, IEEE J. Quantum Elect. QE{\bf 2}, 418 (1966).}
\bibitem{RMPMargaret}{M.D. Reid, P.D. Drummond, W.P. Bowen, E.G. Cavalcanti, P.K. Lam, H.A. Bachor, U.L. Andersen, and G. Leuchs, \rmp {\bf 81}, 1727 (2009).}
\bibitem{SHGPereira}{S.F. Pereira, Min Xiao, H.J. Kimble, and J.L. Hall, Phys. Rev.\ a {\bf 38}, 4931 (1988).}
\bibitem{sumdiff}{M.K. Olsen and R.J. Horowicz, \oc {\bf 168}, 135 (1999).}
\bibitem{PingKoy}{N.B. Grosse, S. Assad, M. Mehmet, R. Schnabel, T. Symul, and P.K. Lam, Phys. Rev.\ l {\bf 100}, 243601 (2008).}
\bibitem{4HG}{M.K. Olsen, arXiv:1707.02537.}
\bibitem{4HGtri}{M.K. Olsen, \oc {\mbox{e}m in press}, https://doi.org/10.1016/j.optcom.2017.09.090}
\bibitem{multiplex} {C. Baune, J. Gniesmer, S. Kocsis, C.E. Vollmer, P. Zell,
J. Fiurasek, and R. Schnabel, Phys. Rev.\ a {\bf 93}, 010302 (2016).}
\bibitem{Hammerer}{K. Hammerer, A.S. S\o rensen, and E.S. Polzik, \rmp {\bf 82}, 1041 (2010).}
\bibitem{Julsgaard}{B. Julsgaard, J. Sherson, I. Cirac, L. Fiur\'a\u{s}ek, and E.S. Polzik, Nature {\bf 432}, 482 (2004).}
\bibitem{P+} {P.D. Drummond and C.W. Gardiner, J. Phys. A\ {\bf 13}, 2353
(1980).}
\bibitem{mjc} {C.W. Gardiner and M.J. Collett, Phys. Rev.\ a {\bf 31}, 3761 (1985).}
\bibitem{Zhu}{S. Zhu, Y. Zhu, and N. Ming, Science {\bf 278}, 843 (1997).}
\bibitem{Liz} {E. Marcellina, J .F.Corney, and M. K. Olsen \oc {\bf 309}, 9
(2013).}
\bibitem{Granja}{M.K. Olsen, S.C.G. Granja, and R.J. Horowicz, \oc {\bf 165}, 293 (1999).}
\bibitem{DFW} {D.F. Walls and G.J. Milburn, \mbox{e}mph{Quantum Optics}
(Springer, Berlin, 1995).}
\bibitem{SMCrispin} {C.W. Gardiner, \mbox{e}mph{Handbook of Stochastic Methods\/},
(Springer, Berlin, 1985).}
\bibitem{DMW}{P.D. Drummond, K.J. McNeil, and D.F. Walls, Optica Acta {\bf 28}, 211 (1981).}
\bibitem{Arabe}{S. Chaturvedi, K. Dechoum, and P.D. Drummond, Phys. Rev.\ a {\bf 65}, 033805 (2002).}
\bibitem{pulse}{K.J. McNeil, P.D. Drummond, and D.F. Walls, \oc {\bf 27}, 292 (1978).}
\bibitem{Bache}{M. Bache, P. Lodahl, A.V. Mamaev, M. Marcus, and M. Saffman, Phys. Rev.\ a {\bf 65}, 033811 (2002).}
\bibitem{3HG}{M.K. Olsen, arXiv:1706.05174.}
\bibitem{EPR} {A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev.\ {\bf 47}, 777 (1937).}
\bibitem{Erwin} {E. Schr\"odinger, Proc. Cam. Philos. Soc. {\bf 31}, 555
(1935).}
\bibitem{Jonesteer} {S.J. Jones, H.M. Wiseman, and A.C. Doherty, Phys. Rev.\ a {\bf
76}, 052116 (2007).}
\bibitem{EPRMDR} {M.D. Reid, Phys. Rev.\ a {\bf 40}, 913 (1989).}
\bibitem{ZYOu}{Z.Y. Ou, S.F. Pereira, H.J. Kimble, and K.C. Peng, Phys. Rev.\ l {\bf 68}, 3663 (1992).}
\bibitem{Wiseman}{H.M. Wiseman, S.J. Jones, and A.C. Doherty, Phys. Rev.\ l {\bf 98}, 140402 (2007).}
\bibitem{SFG}{M.K. Olsen and A.S. Bradley, Phys. Rev.\ a {\bf 77}, 023813 (2008).}
\bibitem{sapatona}{S.L.W Midgley, A.J. Ferris, and M.K. Olsen, Phys. Rev.\ a {\bf 81}, 022101 (2010).}
\bibitem{Handchen}{V. H\"andchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R.F. Werner and R. Schnabel, Nat. Photon. {\bf 6}, 596 (2012).}
\bibitem{Bowles}{J. Bowles, T. V\'ertesi, M.T. Quintino, and N. Brunner, Phys. Rev.\ l {\bf 112}, 200402 (2014).}
\bibitem{SHGEPR}{M.K. Olsen, Phys. Rev.\ a {\bf 88}, 051802 (2013).}
\bibitem{vLF}{P. van Loock and A. Furusawa, Phys. Rev.\ a {\bf 67}, (2003) 052315.}
\bibitem{AxMuzzJPB}{M.K. Olsen and A.S. Bradley, J. Phys. B\ {\bf 39}, 127 (2005).}
\bibitem{AxMuzz}{M.K. Olsen and A.S. Bradley, Phys. Rev.\ a {\bf 74} 063809 (2006).}
\bibitem{Teh&Reid}{R.Y. Teh and M.D. Reid, Phys. Rev.\ a {\bf 90}, 062337 (2014).}
\bibitem{Wang} {M. Wang, Y. Xiang, Q. He, Q. Gong, Phys. Rev.\ a {\bf 91}, 012112 (2015).}
\bibitem{OBR} {M.K. Olsen, A.S. Bradley, M.D. Reid, J. Phys. B\ {\bf 39}, 2515 (2006).}
\bibitem{HeReid}{Q.Y. He and M.D. Reid, Phys. Rev.\ l {\bf 111}, 250403 (2013).}
\bibitem{Bjorkholm}{J.E. Bjorkholm and H.G. Danielmeyer,
Appl. Phys. Lett. {\bf 15}, 171 (1969).}
\bibitem{Haub}{J.G. Haub, M.J. Johnson, B.J. Orr, and R. Wallenstein, Appl. Phys. Lett. {\bf 58}, 1718 (1991).}
\bibitem{Hovde}{ D.C. Hovde, J.H. Timmermans, G. Scoles, and K.K.
Lehmann, \oc {\bf 86}, 294 (1991).}
\bibitem{Plusquellic}{D.F. Plusquellic, O. Votava, and D.J. Nesbitt, Appl. Optics {\bf 35}, 1464 (1996). }
\bibitem{kaled}{M.K. Olsen, K. Dechoum, and L.I. Plimak, \oc {\bf 223}, 123 (2003).}
\bibitem{signal} {M.K. Olsen, Phys. Rev.\ l {\bf 119}, 160501 (2017).}
\mbox{e}nd{thebibliography}
\mbox{e}nd{document} | math | 46,656 |
\begin{document}
\title{Finding minimum Tucker submatrices}
\author{J\'an Ma\v nuch \inst{1,2} \and Arash Rafiey\inst{2}}
\institute{Department of Computer Science, UBC, Vancouver, BC, Canada
\and Department of Mathematics, Simon Fraser University,
Burnaby, BC, Canada\\
\email{[email protected],[email protected]}
}
\maketitle
\begin{abstract}
A binary matrix has the Consecutive Ones Property (C1P) if its columns can be ordered in such a way that all 1s on each row are consecutive.
These matrices are used for DNA physical mapping and ancestral genome reconstruction in computational biology on the
other hand they represents a class of convex bipartite graphs and are of interest of algorithm graph theory researchers.
Tucker gave a forbidden submartices characterization of matrices that have C1P property in 1972. Booth and Lucker (1976) gave a
first linear time recognition algorithm for matrices with C1P property and then in 2002, Habib, et al. gave a simpler linear time recognition algorithm.
There has been substantial amount of works on efficiently finding minimum size forbidden submatrix.
Our algorithm is at least $n$ times faster than the existing algorithm where $n$ is the number of columns of the input matrix.
\end{abstract}
\section{Introduction and Preliminaries}
\label{sec:preliminaries}
A binary matrix has the Consecutive Ones Property (C1P) if its columns
can be ordered in such a way that all ones in each row are
consecutive. Deciding if a matrix has the C1P can be done in
linear-time and
space~\cite{BoothLueker1976,habib-lex,hsu-simple,mcconnell-certifying,meidanis-on}.
The problem of deciding if a matrix has the C1P has been considered in
genomic, for problems such as physical
mapping~\cite{alizadeh-physical,lu-test} or ancestral
genome reconstruction~\cite{adam-modelfree,chauve-methodological,ma-reconstructing}.
Let $M$ be a $m\times n$ binary matrix. Let $\mathbf{R}=
\{r_{i}:\ i = 1,\dots,m\}$ be the set of its rows and $\mathbf{C}= \{c_{j}:\
j = 1,\dots,n\}$ the set of its columns. Its \emph{corresponding
bipartite graph} $G(M) = (V_{M},E_{M})$ is defined as follows:
$V_{M} = \R\cup \C$, and two vertices $r_{i}\in R$ and $c_{j}\in \C$
are connected by an edge if and only if $M[i,j] = 1$. We will refer to
the partition $\R$ and $\C$ of $G(M)$ as black and white vertices,
respectively. The set of neighbors of a vertex $x$ will be denoted by
$N(x)$. The $i$-the neighborhood of $x$, denoted by $N_{i}(x)$, is the
set of vertices distance $i$ from $x$. All these sets, for a fixed
$x$, can be computed in time $O(e)$ using the bread-first search
algorithm. A subgraph of $G(M)$ induces by vertices
$x_{1},\dots,x_{k}$ will be denoted by $G(M)[x_{1},\dots,x_{k}]$. A
set of edges of bipartite graph is called \emph{induced matching} if
the set of endpoints of these edges induces this matching in the
graph. For example, two edges $\{u,v\}$ and $\{u',v'\}$, where $u,u'$
are in the same partition form an induced matching if $\{u,v'\}$ and
$\{u',v\}$ are not edges of the graph.
An \emph{asteroidal triple} is an independent set of three vertices such that
each pair is connected by a path that avoids the neighborhood of the
third vertex. A \emph{white asteroidal triple} is an asteroidal triple
on white (column) vertices.
The following result of Tucker links the C1P of matrices to asteroidal
triples of their bipartite graphs.
\begin{theorem}[\cite{Tucker1972}]
A binary matrix has the C1P if and only if its corresponding
bipartite graph does not contain any white asteroidal
triples.
\end{theorem}
\begin{theorem}[\cite{Tucker1972}]
A binary matrix has the C1P
if and only
if its corresponding bipartite graph does not contain any of the
forbidden subgraphs in $T =
\{G_{\mathrm{I}_{k}},G_{\mathrm{II}_{k}},G_{\mathrm{III}_{k}}:\ k\ge
1\} \cup \{G_{\mathrm{IV}},G_{\mathrm{V}}\}$, depicted in
Figure~\ref{fig:forbidden-subgraphs}. We will refer to these
subgraphs as the type I, II, III, IV and V, respectively.
\end{theorem}
\begin{figure}
\caption{The set of Tucker's forbidden subgraphs.}
\label{fig:forbidden-subgraphs}
\end{figure}
The author in \cite{DBLP:conf/wg/LindzeyM13} developed an algorithm for finding one of the obstructions
in linear time. However, their algorithm does not guarantee the minimum size obstruction.
The characterization can be used to determine whether a given binary
matrix has the C1P in time $O(\Delta mn^{2} + n^{3})$, where $\Delta $
is the maximum number of ones per row, i.e., the maximum degree of
black vertices in $G(M)$, as explained by the following result in
\cite{DomGuoNiedermeier2010}.
\begin{lemma}[\cite{DomGuoNiedermeier2010}]
\label{l:asteroidal}
A white asteroidal triple $u,v,w$ with the smallest sum of the three
paths (avoiding the third neighborhood) can be computed in time
$O(\Delta mn^{2} + n^{3})$.
\end{lemma}
For practical purposes, there is a much faster algorithm that uses
PQ-trees for determining whether a binary matrix has the C1P,
cf. \cite{BoothLueker1976}. Tucker's interest was in finding the
smallest submatrix of a non-C1P binary matrix which makes this matrix
non-C1P. He further refined his asteroidal triple characterization
using a set of \emph{forbidden submatrices}. We will state this
results in terms of \emph{forbidden subgraphs}.
We will consider two problems: (1) detected a smallest forbidden
subgraph of each type (Section~\ref{sec:detect-small-forb-each}), and
(2) detecting a smallest forbidden subgraph of any type
(Section~\ref{sec:detect-small-forb-all}).
We use the followings to improve the complexity :
\begin{itemize}
\item In our computation we use degree of each vertex instead the
maximum degree $\Delta$.
\item
We compute some of the necessary sets in advance.
\item
In our analysis we use the minimum obstruction assumption and
explore the connection of vertices around a minimum obstruction with
it.
\end{itemize}
\setlength{\tabcolsep}{5pt}
\begin{table}
\centering
\begin{tabular}{c | l l l}
Subgraph type & \multicolumn{3}{c}{Time complexity} \\
& Previous result & Our result (Exact) & Our result\\
\hline
I & $O(\Delta^{4}m^{3})$ \cite{BlinRizziVialette2012} &
$O(\Delta e^{2}) = O(\Delta^{3}m^{2})$ &
$O(n^{2}e)$ \cite{DomGuoNiedermeier2010}\\
II & $O(\Delta^{4}m^{3}) = O(ne^{3})$ \cite{BlinRizziVialette2012} &
--- &
$O(n^{2}e)$ \cite{DomGuoNiedermeier2010} \\
III & $O(\Delta^{2}m^{2}n^{2})$ \cite{BlinRizziVialette2012} &
$O(e^{3}) = O (\Delta^{3}m^{3})$ &
$O(ne^{2})$ \\
IV & $O(\Delta^{3}m^{2}n^{3})$ \cite{DomGuoNiedermeier2010} &
$O(m^{3}e) = O(\Delta m^{4})$ &
$O(n^{3}e)$ \\
V & $O(\Delta^{4}m^{2}n )$ \cite{DomGuoNiedermeier2010} &
$O(m^{3}e) = O(\Delta m^{4})$ &
$O(n^{3}e)$ \\[3mm]
Any & $O(\Delta^{3}m^{2}(\Delta m + n^{3}))$ &
\multicolumn{2}{l}{
$O(ne(n^{2} + e)) = O(\Delta mn(\Delta m + n^{2}))$}
\end{tabular}
\caption{Comparison of our results with the previous results.}
\label{tab:comparison}
\end{table}
Note that without loss of generality we can assume that $M$ does not
contain any all-zero columns or rows, as such columns does not affect whether
the matrix has the C1P or the forbidden submatrices of $M$. It follows
that $\Delta m\ge n$. We will use this assumption throughout this
paper. Also note that the number of edges in $G(M)$ is the same as the
number of ones in $M$, which we denote as $e$. Note that $e = O(\Delta
m)$ and that $e\ge m,n$ (since we assume that there are no all-zero
columns or rows in $M$).
We will use the following auxiliary lemma.
\begin{lemma}\label{l:induced-matching-2}
Given a bipartite graph $G$ with $e$ edges and partitions of size
$m$ and $n$, picking an induced matching of size two of $G$ or
determining that no such induced matching exists can be done in time
$O(e + m + n)$.
\end{lemma}
\begin{proof}
Let $U$ be the partition of size $n$. Order vertices of $U$ by their
degrees:
$\deg (u_{1})\le \deg (u_{2})\le\dots \le \deg (u_{n})$. For every
$i = 1,\dots,n - 1$, check if $N(u_{i})\setminus N(u_{i + 1})$ is
non-empty. If for some $i$, $N(u_{i})\setminus N(u_{i + 1})\ne
\emptyset $, then also $N(u_{i + 1})\setminus N(u_{i})\ne \emptyset$.
In this case, we can pick any $a\in N(u_{i})\setminus N(u_{i + 1})$
and any $b\in N(u_{i + 1})\setminus N(u_{i})$, and return
$\{u_{i},a\}$ and $\{u_{i + 1},b\}$, as it forms an induced
matching of $G$.
Now, assume that for every $i$, $N(u_{i})\setminus N(u_{i + 1}) =
\emptyset $, i.e., $N(u_{i})\subseteq N(u_{i + 1})$. We will show
that there is no induced matching of $G$ of size two. Assume for
contradiction that $\{u_{i},a\}$ and $\{u_{j},b\}$, where $i < j$,
is such an induced matching. We have $N(u_{i})\subseteq N(u_{i +
1})\subseteq\dots \subseteq N(u_{j})$, i.e., $a\in N(u_{j})$, a
contradiction. Hence, in this case we can report that there is no
such matching.
Vertices of $U$ can be sorted by their degrees in time $O(n + m)$
using a count sort. For each $i$, checking if $N(u_{i})\setminus N(u_{i +
1})$ is non-empty can be done in time $O(\deg (u_{1}))$, hence,
the total time spent on checking is $O(\sum_{i = 1}^{n - 1} \deg
(u_{i})) = O(e)$.
\end{proof}
\section{Detection of smallest forbidden subgraphs for each type}
\label{sec:detect-small-forb-each}
We will present four algorithms which find a smallest subgraph of type
I, III, IV and V, respectively, each improving the complexity of the
best known such algorithm, cf.~\cite{BlinRizziVialette2012}. For type
II, we refer reader to the $O(ne^{3})$ algorithm\footnote{The authors of
\cite{BlinRizziVialette2012} showed that the complexity of their
algorithm is $O(\Delta^{4}m^{3})$, however, it is easy to check that
their algorithm works in time $O(ne^{3})$.}
in~\cite{BlinRizziVialette2012}.
\subsection{Type I}
\label{sec:type-i}
Algorithm~\ref{alg:type1} finds a smallest forbidden subgraph of type I
in time $O(\Delta e^{2})$.
\begin{algorithm}
\caption{Find a smallest $G_{\mathrm{I}_{k}}$ subgraph.}
\label{alg:type1}
\scriptsize
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input {$G(M)$}
\Output {A smallest subgraph $G_{\mathrm{I}_{k}}$ of $G(M)$}
\BlankLine
\For{$w\in \R$}{
\For{$x,y\in N(w)$}{
construct the subgraph $G_{w,x,y}$ of $G(M)$ induced by
vertices $(\R\setminus (N(x)\cap N(y)))\cup (\C\setminus
N(w))\cup \{x,y\}$\;
find a shortest path between $x$ and $y$ in $G_{w,x,y}$\;
\If {the length of the path is smaller than any observed so
far}{
remember $w$ and the vertices of the path\;
}
}
}
\Return {subgraph of $G(M)$ induced by the remembered set of
vertices (if any)}
\end{algorithm}
\emph{Correctness of Algorithm~\ref{alg:type1}.} We are looking for
induced cycles of length 6 or more. For each black vertex $w$ and its
two neighbors $x,y$, we find a shortest induced cycle of length at
least 6. Such cycle cannot contain any vertex incident with $w$ other
than $x$ and $y$, and any vertex incident with both $x$ and $y$ other
than $w$. Hence, a shortest such cycle $c$ can be obtained from the a
shortest $x - y$ path $p$ in $G_{w,x,y}$ by adding two edges $\{x,w\}$
and $\{y,w\}$. This cycle cannot be of length $4$, otherwise $p$ would
contain a vertex in $N(x)\cap N(y)$. It remains to show that $c$ is
induced. Assume that there is a chord $\{u,v\}$ in $c$. Since $p$ does
not contain $N(w)\setminus \{x,y\}$, $u,v\ne w$. Hence, we could use
the chord as a shortcut to find a shorter cycle containing edges
$\{x,w\}$ and $\{y,w\}$, and hence, a shorter path between $x$ and $y$
in $G_{w,x,y}$, a contradiction.
\emph{Complexity of Algorithm~\ref{alg:type1}.} We will show that the
complexity of Algorithm~\ref{alg:type1} is $O(\Delta e^{2}) =
O(\Delta^{3}m^{2})$. The first loop executes $m$ times and the second
$\deg (w)^{2}$ times. Hence, the body of the second loop executes
$\sum_{w\in \mathbf{R}} \deg (w)^{2} = O(\Delta e)$ times. Constructing graph
$G_{w,x,y}$ takes time $O(e)$ and finding a shortest path in
$G_{w,x,y}$ can be done in time $O(e)$ using the Breadth-first search
algorithm.
\subsection{Type III}
\label{sec:type-iii}
Algorithm~\ref{alg:type3a} finds a smallest forbidden subgraph of type
II in time $O(e^{3})$.
\begin{algorithm}
\caption{Find a smallest $G_{\mathrm{III}_{k}}$ subgraph.}
\label{alg:type3a}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\scriptsize
\Input {$G(M)$}
\Output {A smallest subgraph $G_{\mathrm{III}_{k}}$ of $G(M)$}
\BlankLine
\For{$\{x,w\}\in E_{M}$, where $x\in \C$ and $w\in \R$}{
\For{$\{y,a\}$, where $y\in \C\setminus N(w)$ and $a\in \mathbf{R}\setminus N(x)$}{
construct the subgraph $G_{x,w,y,a}$ of $G(M)$ induced by vertices
$(\R\setminus N(x)\setminus N(y))\cup \{a\} \cup (N(w)\setminus
\{x\})\cup (\mathbf{C}\setminus N(a))$\;
find a shortest path between $a$ and set $\C\setminus
N(w)\setminus N(a)$ in $G_{x,w,y,a}$\;\label{3a-path}
\If{if the path exists and is shorter than any observed
so far}{
remember $w,x,y$ and the path\;
}
}
}
\Return {subgraph of $G(M)$ induced by the remembered set of
vertices (if any)}
\end{algorithm}
\emph{Correctness of Algorithm~\ref{alg:type3a}.} Let us first verify
that the vertices of a shortest path found in line~\ref{3a-path} and
$w,x,y$ induce a subgraph of type III. Obviously, $x$ is connected
only to $w$, $w$ is not connected to $y$ and the last vertex $z$ of
the path. On the other hand, $w$ must be connected to all other white
vertices on the path, since any such white vertex that is not in
$N(w)$ is in $\mathbf{C}\setminus N(a)$ and hence, also $\mathbf{C}\setminus
N(w)\setminus N(a)$, i.e., we would have a shorter path ending at this
vertex. Since the path is a shortest path, all black vertices on the
path are connected only to its predecessor and successor on the
path. In addition $a$ is connected to $y$ and no other black vertex on
the path is connected to $y$ since $G_{x,w,y,a}$ does not contain any
other neighbors of $y$. It follows that the vertices $w,x,y$ and the
vertices of a shortest path induce a subgraph of type III.
Second, consider a smallest subgraph of type III in $G(M)$. We will
show it is considered by the algorithm. Assume the algorithm is in the
cycle, where it picked edges $\{x,w\}$ and $\{y,a\}$ of this
subgraph. Then the rest of the vertices must lie in $G_{x,w,y,a}$: the
remaining black vertices are not connected to $x$ and $y$ and the
remaining white vertices are either in $N(w)\setminus \{x\}$ and $z$
is $\mathbf{C}\setminus N(a)$. These vertices together with $a$ must form a
shortest path from $a$ to $\mathbf{C}\setminus N(w)\setminus N(a)$ in
$G_{x,w,y,a}$, hence, Algorithm~\ref{alg:type3a} finds this subgraph
or a subgraph with the same number of vertices.
\emph{Complexity of Algorithm~\ref{alg:type3a}.} We will show that the
complexity of Algorithm~\ref{alg:type3a} is $O(e^{3}) =
O(\Delta^{3}m^{3})$. The first loop executes $e$ times. The second
loop executes $O(e)$ times. Constructing graph $G_{x,w,y,a}$ takes
time $O(e)$. Finding a shortest path in $G_{x}$ can be done in time
O(e) using a breadth-first search algorithm.
\subsection{Type IV}
\label{sec:type-iv}
Algorithm~\ref{alg:type4} determines if $G(M)$ contains a forbidden
subgraph of type IV in time $O(m^{3}e)$.
\begin{algorithm}
\caption{Find a $G_{\mathrm{IV}}$ subgraph.}
\label{alg:type4}
\scriptsize
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input {$G(M)$}
\Output {A subgraph $G_{\mathrm{IV}}$ of $G(M)$}
\BlankLine
\For{distinct $a,b,c,d\in R$}{
find $UX = N(a)\setminus (N(b)\cup N(c))$\;
find $VY = N(b)\setminus (N(a)\cup N(c))$\;
find $WZ = N(c)\setminus (N(a)\cup N(b))$\;
find $U = UX\cap N(d)$ and $X = UX\setminus N(d)$\;
find $V = VY\cap N(d)$ and $Y = VY\setminus N(d)$\;
find $W = WZ\cap N(d)$ and $Z = WZ\setminus N(d)$\;
\If{each of the sets $X,Y,Z,U,V,W$ is non-empty}{
pick any $x\in X,y\in Y,z\in Z,u\in U,v\in V,w\in W$\;
\Return{$G(M)[a,b,c,d,x,y,z,u,v,w]$}
}
}
\Return {not found}
\end{algorithm}
\emph{Correctness of Algorithm~\ref{alg:type4}.} It is easy to see
that once $a,b,c,d$ are picked, each of $x,y,z,u,v,w$ has to belong to
computed set $X,Y,Z,U,V,W$, respectively, and that once they are
picked from those sets, the returned vertices induce
$G_{\mathrm{IV}}$.
\emph{Complexity of Algorithm~\ref{alg:type4}.} We will show that the
complexity of Algorithm~\ref{alg:type4} is $O(m^{3}e) = O(\Delta
m^{4})$. The time complexity of the steps inside the loop depends on
degrees of nodes $a,b,c,d$, i.e., it is $O(\deg (a) + \deg (b) + \deg (c)
+ \deg (d))$. Hence, the overall complexity is $\sum_{a,b,c,d\in R}
O(\deg (a) + \deg (b)\deg (c) + \deg (d)) = 4\sum_{a,b,c,d\in R}
O(\deg (d)) = 4\sum_{a,b,c\in R} O(e) = m^{3}e$.
\subsection{Type V}
\label{sec:type-v}
Algorithm~\ref{alg:type5} determines if $G(M)$ contains a forbidden
subgraph of type V in time $O(m^{3}e)$.
\begin{algorithm}
\caption{Find a $G_{\mathrm{V}}$ subgraph.}
\label{alg:type5}
\scriptsize
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input {$G(M)$}
\Output {A subgraph $G_{\mathrm{V}}$ of $G(M)$}
\BlankLine
\For{distinct $a,b,c,d\in \R$}{
find $UY = N(b)\cap N(d)\setminus N(c)$\;
find $VZ = N(b)\cap N(c)\setminus N(d)$\;
find $U = UY\cap N(a)$ and $Y = UY\setminus N(a)$\;
find $V = VZ\cap N(a)$ and $Z = VZ\setminus N(a)$\;
find $X = N(a)\setminus (N(b)\cup N(c)\cup N(d))$\;
\If{each of the sets $X,Y,Z,U,V$ is non-empty}{
pick any $x\in X,y\in Y,z\in Z,u\in U,v\in V$\;
\Return{$G(M)[a,b,c,d,x,y,z,u,v]$}
}
}
\Return {not found}
\end{algorithm}
\emph{Correctness of Algorithm~\ref{alg:type5}.} It is easy to see
that once $a,b,c,d$ are picked, each of $x,y,z,u,v$ has to belong to
computed set $X,Y,Z,U,V$, respectively, and that once they are picked
from those sets, the returned vertices induce $G_{\mathrm{V}}$.
\emph{Complexity of Algorithm~\ref{alg:type5}.} The complexity of
Algorithm~\ref{alg:type5} is $O(m^{3}e) = O(\Delta m^{4})$. This
follows by the same argument as for Algorithm~\ref{alg:type5}.
\section{Detection of a smallest forbidden subgraph}
\label{sec:detect-small-forb-all}
Overall, we will use Dom et al. (\cite{DomGuoNiedermeier2010})
approach to find the smallest forbidden subgraph in $G(M)$. We will
first find a shortest-paths (the sum of the lengths of the three paths)
white asteroidal triple $A$ in time $O(n^{2}e) = O(\Delta mn^{2})$
using the algorithm in \cite{DomGuoNiedermeier2010}.
A shortest-paths white asteroidal triple $A$
must be in $T$, but does not need to be a smallest forbidden
subgraph. Let $\ell $ be the sum of the lengths of the three paths of
$A$. If $A$ is of
\begin{itemize}
\item type I or II, then it contains $\ell $ vertices;
\item type III, it contains $\ell - 5$ vertices;
\item type IV, it contains $10 = \ell - 8$ vertices;
\item type V, it contains $9 = \ell - 1$ vertices.
\end{itemize}
It follows that if one of the smallest forbidden subgraphs is of type
I or II, then each shortest-paths asteroidal triple is of type I or II and
is a smallest forbidden subgraph. For the remaining cases, we need
to determine the smallest forbidden subgraphs of type III, IV and
V. However, we only need to find a smallest subgraph of type X if it is a
smallest forbidden subgraph. Hence, for types IV and V, if we find
during the search that there is a smaller forbidden subgraph of some other
type, we can stop searching for this type. For type III, since it has
a variable size, we cannot stop searching, however, we can abandon the
branch which would yield a larger or even the same size subgraph of
type III than we have observed. We will use this in what follows to
obtain faster algorithms for types III, IV and V than the ones
presented in the previous section.
\subsection{Type III}
\label{sec:type-iii-all}
Algorithm~\ref{alg:type3-all} guarantees to find a smallest subgraph
of type III \textbf{if} it is smaller than other types of forbidden
subgraphs in time $O(ne^{2})$. If there is a smaller subgraph of type
I or there is a smaller of same size subgraph of type V in $G(M)$, it
either reports that or it could report a subgraph of type III which is
not the smallest. It will first determine whether
$G_{\mathrm{III}_{1}}$ is a subgraph of $G(M)$. If not it continues to
the second phase, where it assumes that the smallest subgraph of type
III (if it exists) has at least 9 vertices.
\begin{algorithm}[H]
\caption{Find a smallest $G_{\mathrm{III}_{k}}$ subgraph if it is
smaller than other types of subgraphs.}
\label{alg:type3-all}
\scriptsize
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input {$G(M)$}
\Output {A smallest subgraph $G_{\mathrm{III}_{k}}$ of $G(M)$ or
report there is a subgraph of other type (I or V) of equal
or smaller size}
\BlankLine
\For{$w\in \mathbf{R}$}{
\For{$x,u\in N(w)$}{\label{3-all-ph1-b}
construct the subgraph $G_{x,w,u}$ of $G(M)$ induced by vertices
$N(u)\setminus N(x)\cup \mathbf{C}\setminus N(w)$\;
find induced matching of size two using
Lemma~\ref{l:induced-matching-2}\;
\If{induced matching exists}{
\Return subgraph of $G(M)$ induced by $x,w,u$ and the induced
matching ($G_{\mathrm{III}_{1}}$)
}\label{3-all-ph1-e}
}
}
\tcc{We can now assume that there is no $G_{\mathrm{III}_{1}}$ in
$G(M)$}
set $i_{min} = \infty $\;
\For{$\{x,w\}\in E_{M}$, where $x\in \C$ and $w\in \R$}{
find $D = N_{2}(w)\setminus N(x)$ and $Y = N(D)\setminus N(w)$\;
\For{$y\in$ Y}{\label{3-all-ph2-y-b}
construct the subgraph $G_{x,w,y}$ of $G(M)$ induced by vertices
$N(w)\setminus \{x\}\cup \{y\} \cup D$\;
find $D_{i} = N_{i}(y)$ in $G_{x,w,y}$, for $i\ge 1$\;
find $Y' = \{y'\in Y:\ D_{1}\setminus N(y')\ne \emptyset \} $
and $D' = D\cap N(Y')$\;
find smallest odd $i\ge 3$ such that $D_{i}\cap D'\ne \emptyset $
(if possible)\;
\If{found}{
pick any $d_{i}\in D_{i}\cap D'$, any $y'\in Y'\cap N(d)$ \;
find a path $P$ from $d_{i}$ to some $d_{1}\in D_{1}$ in
$G_{w,w,y}$ of length $i - 1$\;
\If{$\{y',d_{1}\} \notin E(M)$ and $i < i_{min}$}{
set $i_{min}$ to $i$\;
remember $x,w,y,y'$ and vertices of $P$\;
}
}
}\label{3-all-ph2-y-e}
}
\eIf{$i_{min} = \infty $}{
\Return subgraph of type III not found or there is a subgraph of
type I or V of the size at most the size of the smallest type III subgraph
}{
\Return {subgraph of $G(M)$ induced by remembered set of vertices}
}
\end{algorithm}
\emph{Correctness of Algorithm~\ref{alg:type3-all}.} It is easy to
check that the first phase of the algorithm finds
$G_{\mathrm{III}_{1}}$ subgraph if it exists in $G(M)$. Assume that
$G_{\mathrm{III}_{1}}$ is not an induced subgraph of $G(M)$., i.e.,
that a smallest subgraph of type III (if it exists) has at least 9
vertices. The algorithm continues to the second phase.
First, assume that $i$ is not found, i.e., for all odd $i\ge 3$,
$D_{i}\cap D' = \emptyset $. This implies that any path starting at
$y$ in $G_{x,w,y}$ cannot be extended with a white vertex $y'$ that is
not adjacent to $w$ and not adjacent to the second vertex $d_{1}\in
D_{1}$ of this path. Hence, the algorithm correctly continues with
examining another selection of vertices $x,w,y$. Assume that $i$ was
found. Now, assume that $G(M)$ does not contain edge
$\{y',d_{1}\}$. Let us verify that vertices $x,w,y,y'$ and the
vertices of $P$ induce $G_{\mathrm{III}_{(i - 1)/2}}$. It is clear
that $x$ is connected only to $w$ and $w$ only to white vertices on
$P$ except the first vertex $y$. By the construction, each vertex on
$P$ can be adjacent only to its predecessor or successor on $P$. Since
$i$ is the smallest odd integer larger than two such that $D_{i}\cap
D'\ne \emptyset $, $y'$ is not adjacent to any black vertex on the
path other than the last one. Hence, the vertices induce a subgraph of
type III. Finally, assume that $\{y',d_{1}\}\in E(M)$. If $i\ge 5$,
then vertices of $P$ without $y$ and $y'$ induce a cycle of length $i
+ 1$, i.e., a subgraph $G_{\mathrm{I}_{(i - 3)/2}}$, which is smaller
than a subgraph of type III we could get for this selection of $x,w,y$
(by choosing a different $d_{i}$, $y'$ or path $P$, or searching for
another odd $i$ such that $D_{i}\cap D'\ne \emptyset $). If $i = 3$,
consider $d_{1}'\in D_{1}$ that is not adjacent to $y'$ and let $P =
y,d_{1},u,d_{3}$. If $d_{1}'$ is adjacent to $u$, vertices
$x,w,u,d_{1}',y,d_{3},y'$ induce $G_{\mathrm{III}_{1}}$, a
contradiction. Hence, assume $\{d_{1}',u\} \notin E(M)$. Since
$d_{1}'\in D\subseteq N_{2}(w)$, there exists $u'\in N(w)$ adjacent to
$d_{1}'$. If $\{d_{1},u'\} \in E(M)$, then vertices
$x,w,u,u',d_{1},d_{1}',d_{3},y,y'$ induce $G_{\mathrm{V}}$. Otherwise,
vertices $w,u,d_{1},y,d_{1}',u'$ induce a cycle of length 6. In any
case, there exists a subgraph of other type of size equal or smaller
than it would be possible to find for this choice of $x,w,y$, hence,
the algorithm correctly moves to the next choice.
\emph{Complexity of Algorithm~\ref{alg:type3-all}.} We will show that
the complexity of Algorithm~\ref{alg:type3-all} is $O(ne^{2}) =
O(\Delta^{2}m^{2})$. The body of the loop in
lines~\ref{3-all-ph1-b}--\ref{3-all-ph1-e} will execute $O(\Delta e)$
times and each step of the body take $O(e)$ time. Hence, the
complexity of the first phase is $O(\Delta e^{2}) = O(ne^{2})$.
The main loop of the second phase will execute $O(e)$
times. Determining $D$ and $Y$ takes time $O(e)$. The nested loop in
lines~\ref{3-all-ph2-y-b}--\ref{3-all-ph2-y-e} will execute $O(n)$
times. Each step of the body of this loop will take time
$O(e)$. Hence, the complexity of the second phase is $O(ne^{2})$.
\subsection{Type IV}
\label{sec:type-iv-all}
Algorithm~\ref{alg:type4a-all} finds the subgraph $G_{\mathrm{IV}}$ in
time $O(n^{3}e)$, if it exists and if it is a smallest forbidden
subgraph. If there is a smaller forbidden subgraph of type I or III,
it might find an instance of $G_{\mathrm{IV}}$ or it might report
that there is a smaller forbidden subgraph instead.
\begin{algorithm}
\caption{Find a $G_{\mathrm{IV}}$ subgraph or report that there is a
smaller subgraph of type I or III.}
\label{alg:type4a-all}
\scriptsize
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input {$G(M)$}
\Output {A subgraph $G_{\mathrm{IV}}$ of $G(M)$ or report that
$G_{\mathrm{IV}}$ is not a smallest subgraph}
\BlankLine
\For{distinct $x,y,z\in \C$}{
find $A = N(x)\setminus (N(y)\cup N(z))$\;
find $B = N(y)\setminus (N(x)\cup N(z))$\;
find $C = N(z)\setminus (N(x)\cup N(y))$\;
find $D = \C\setminus (N(x)\cup N(y)\cup N(z))$\;
find $U = N(A)\setminus \{x,y,z\}$\;
find $V = N(B)\setminus \{x,y,z\}$\;
find $W = N(C)\setminus \{x,y,z\}$\;
\If{all sets $A,B,C,D,U,V,W$ are non-empty}{
\For{$d\in D$}{
\If{there exists distinct $u\in U\cap N(d)$, $v\in V\cap
N(d)$ and $w\in W\cap N(d)$}{
find $a\in A\cap N(u)$, $b\in B\cap N(v)$ and $c\in C\cap
N(w)$\;
\eIf{none of the edges
$\{a,v\},\{a,w\},\{b,u\},\{b,w\},\{c,u\},\{c,v\}$
exists}{
\Return {$G(M)[x,y,z,u,v,w,a,b,c,d] = G_{\mathrm{IV}}$}
}{
\Return {there is a smaller subgraph of type I or III}
}
}
}
}
}
\Return {not found}
\end{algorithm}
\emph{Correctness of Algorithm~\ref{alg:type4a-all}.}
Correctness of the algorithm follows by the following lemma.
\begin{lemma}
\label{l:cross-edges}
Consider a subgraph $G'$ of $G(M)$ induced by vertices
$x,y,z,u,v,w,a,b,c,d$ that contains edges
\begin{equation*}
\{x,a\} ,\{y,b\} ,\{z,c\} ,\{a,u\} ,\{b,v\} ,\{c,w\} ,\{u,d\} ,\{v,d\} ,\{w,d\} \,,
\end{equation*}
and does not contain edges
\begin{equation*}
\{x,d\} ,\{y,d\} ,\{z,d\} \,.
\end{equation*}
Then either $G'$ is an instance of $G_{\mathrm{IV}}$ or $G'$ contains
either $G_{\mathrm{I}_{1}}$, $G_{\mathrm{III}_{1}}$ or
$G_{\mathrm{III}_{2}}$ as an induced subgraph.
\end{lemma}
\begin{proof}
We will use the following two partial maps: $R(x) = a$, $R(y) = b$,
$R(z) = c$, $R(a) = u$, $R(b) = v$ and $R(c) = w$, and $L = R^{-1}$.
If none of the edges in
$E' = \{\{a,v\},\{a,w\},\{b,u\},\{b,w\},\{c,u\},\{c,v\}\}$ is present, then
$G'$ is isomorphic to $G_{\mathrm{IV}}$.
If exactly one edge $e$ in $E'$ is present, we have an induced
subgraph $G_{\mathrm{III}_{1}}$ centered at the vertex $r = e\cap
\{u,v,w\}$. In particular, vertices $d,r,L(r),L(L(r)),\ell ,L(\ell
),z$, where $\ell = e \cap \{a,b,c\}$ and $z\in \{u,v,w\} \setminus
\{r,R(\ell )\}$, induce $G_{\mathrm{III}_{1}}$.
We can assume that there are at least two edges in $E'$ present. We
will distinguish two cases. Either (i) there exists two edges $e$ and
$e'$ in $E'$ present such that $e\cap e'\ne \emptyset $, or (ii) for
each pair of such edges $e\cap e' = \emptyset $.
First, consider case (i) and let $e,e'$ be such that $e\cap e'\ne \emptyset
$. Depending on whether the intersection lies in $\{a,b,c\}$ or
$\{u,v,w\}$, we have two cases:
\begin{enumerate}
\item $e\cap e'\in \{a,b,c\}$ (``edges joing on the left''), then
vertices $V(G')\setminus \{e\cap e'\}$ induce
$G_{\mathrm{III}_{2}}$;
\item $e\cap e'\in \{u,v,w\}$ (``edges joing on the right''), then
vertices $x,y,z,a,b,c,e\cap e'$ induce $G_{\mathrm{III}_{1}}$.
\end{enumerate}
Now, consider case (ii). Note the number of edges in $E'$ present is
at most three. We will consider two cases depending on the number of
such edges:
\begin{enumerate}
\item $|E'\cap E(G')| = 2$: Without loss of generality we can assume
that $e\cap \{a,b,c\} = L(e'\cap \{u,v,w\})$ for $e,e'\in E'$
present in $G'$. Then the same collection of vertices as in the
case of one edge $e$ induces $G_{\mathrm{III}_{1}}$, since one end
of $e'$ lies outside of this collection.
\item $|E'\cap E(G')| = 3$: Then the vertices $a,b,c,u,v,w$ induce
$C_{6}$, i.e., $G_{\mathrm{I}_{1}}$.
\end{enumerate}
\end{proof}
\emph{Complexity of Algorithm~\ref{alg:type4a-all}.}
We will show that the complexity of Algorithm~\ref{alg:type4a-all} is
$O(n^{3}e) = O(\Delta mn^{3})$. The first loop executes
$O(n^{3})$ times, determining $A,B,C,D$ takes time $O(m)$, determining
sets $U,V,W$ time $O(e)$. The loop for $d\in D$ is executed $O(m)$
times and each execution takes time $O(deg(d))$, i.e., the total time
spent in this loop is $\sum_{d\in D} O(\deg (d)) = O(e)$.
\subsection{Type V}
\label{sec:type-v-all}
Algorithm~\ref{alg:type5a-all} find the subgraph $G_{\mathrm{V}}$ in
time $O(n^{3}e)$, if it exists and if it is a smallest forbidden
subgraph. If there is a smaller forbidden subgraph of type I or
III, it might find an instance of $G_{\mathrm{V}}$ or it might
report that there is a smaller forbidden subgraph instead.
\begin{algorithm}
\caption{Find a $G_{\mathrm{V}}$ subgraph or report that there is a
smaller subgraph of type I or III.}
\label{alg:type5a-all}
\scriptsize
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input {$G(M)$}
\Output {A subgraph $G_{\mathrm{V}}$ of $G(M)$ or report that
$G_{\mathrm{V}}$ is not a smallest subgraph}
\BlankLine
\For{distinct $x,y,z\in \C$}{
find $A = N(x)\setminus (N(y)\cup N(z))$\;
find $B = N(y)\setminus (N(x)\cup N(z))$\;
find $C = N(z)\setminus (N(x)\cup N(y))$\;
find $D = (N(y)\cap N(z))\setminus N(x)$\;
pick any $u\in N(A)\cap N(B)\cap N(D)$ if possible\;
pick any $v\in N(A)\cap N(C)\cap N(D)$ if possible\;
\If{$u$ and $v$ has been picked}{
\If{$u\in N(C)$ or $v\in N(B)$}{
\Return there is a smaller subgraph of type III ($G_{\mathrm{III}_{1}}$)\label{5a-all-III}
}
find $A' = A\cap N(u)\cap N(v)$ and $D' = D\cap N(u)\cap N(v)$\;
\If{$A' = \emptyset $ or $D' = \emptyset $}{
\Return there is a smaller subgraph of type I
($G_{\mathrm{I}_{1}}$ or $G_{\mathrm{I}_{2}}$)\label{5a-all-I}
}
pick any $a\in A'$, $b\in B\cap N(u)$, $c\in C\cap N(v)$ and
$d\in D'$\;
\Return $G(M)[x,y,z,u,v,a,b,c,d]$\label{5a-all-V}
}
}
\Return {not found}
\end{algorithm}
\emph{Correctness of Algorithm~\ref{alg:type5a-all}.}
The algorithm is able
to reduce time complexity by avoiding trying all possible choices for
$u,v$ and $a,b,c,d$, but rather picking one choice (if
possible), and then either finding $G_{\mathrm{V}}$ or a smaller
forbidden subgraph. Let us verify that decisions algorithm makes are
correct:
\begin{itemize}
\item
First, assume that the algorithm stops in
line~\ref{5a-all-III}. Then there exists $w\in N(A)\cap N(B)\cap
N(C)\cap N(D)$ (either $u$ or $v$). Then there exists $a\in A\cap
N(w)$, $b\in B\cap N(w)$ and $c\in C\cap N(w)$. Vertices
$x,y,z,a,b,c,w$ induce $G_{\mathrm{III}_{1}}$.
\item Assume that the algorithm stops in line~\ref{5a-all-I}. If $A' =
\emptyset $ and $D' = \emptyset $, there exists $a\in A\cap N(u)$,
$a'\in A\cap N(v)$, $d\in D\cap N(u)$ and $d'\in D\cap N(v)$. Note
that $a\ne a'$, $d\ne d'$, $a,d\notin N(v)$ and $a',d'\not in$
N(u). It is easy to check that vertices $x,a,u,d,y,d',v,a'$ induce
$C_{8}$. Similarly, if either $A' = \emptyset $ or $D' = \emptyset
$, we can find vertices that induce $C_{6}$.
\item
Finally, it is easy to check that if the algorithm outputs an
induced subgraph in line~\ref{5a-all-V}, it is $G_{\mathrm{V}}$.
\end{itemize}
On the other hand, if $G_{\mathrm{V}}$ is a smallest forbidden
subgraph of $G(M)$, then the algorithm cannot finish in
lines~\ref{5a-all-III} and~\ref{5a-all-I}, and
hence, it will eventually output $G_{\mathrm{IV}}$ in
line~\ref{5a-all-V}.
\emph{Complexity of Algorithm~\ref{alg:type5a-all}.} We will show
that the complexity of Algorithm~\ref{alg:type5a-all} is $O(n^{3}e) =
O(\Delta mn^{3})$. The first loop executes $O(n^{3})$ times,
determining $A,B,C,D$ takes time $O(m)$, picking $u,v$ time
$O(e)$, picking $a,b,c,d$ time $O(m)$. Hence, the total time
used by the algorithm is $O(n^{3}(O(m) + O(e))) = O(n^{3}e)$.
\subsection{Main algorithm}
\label{sec:main-algorithm}
Algorithm~\ref{alg:Tucker} finds a smallest forbidden subgraph
using the three algorithms described above.
\begin{algorithm}
\caption{Find a smallest forbidden Tucker subgraph.}
\label{alg:Tucker}
\scriptsize
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input {$G(M)$}
\Output {A smallest forbidden subgraph of $G(M)$}
\BlankLine
find a smallest white asteroidal triple $A$ using
Lemma~\ref{l:asteroidal}\;
let $\ell $ be the sum of the lengths of three paths of $A$\;
find a smallest subgraph of types III, IV and V (using the
procedures described above)\;
let $s_{\mathrm{III}},s_{\mathrm{IV}},s_{\mathrm{V}}$ be
the sizes of these subgraphs (or $\infty $ if not found), respectively\;
\eIf {$\ell = \min\{\ell ,s_{\mathrm{III}},s_{\mathrm{IV}},s_{\mathrm{V}}\}$}
{\Return {$A$}}
{let $s_{X} =
\min\{\ell ,s_{\mathrm{III}},s_{\mathrm{IV}},s_{\mathrm{V}}\}$\;
\Return {the smallest subgraph of type $X$}}
\end{algorithm}
To verify the correctness of Algorithm~\ref{alg:Tucker}, first
consider that one of the smallest forbidden subgraphs of $G(M)$ is of
type I or II. By the above argument, asteroidal triple $A$ is of type
I or II with size $\ell $, and since it is a smallest forbidden
subgraph, we have $\ell =
\min\{\ell,s_{\mathrm{III}},s_{\mathrm{IV}},s_{\mathrm{V}}\}$. Hence,
the algorithm correctly outputs one of the smallest forbidden
subgraphs. Second, assume that all smallest forbidden subgraphs of
$G(M)$ are of type III, IV and V. Let $s =
\min\{s_{\mathrm{III}},s_{\mathrm{IV}},s_{\mathrm{V}}\}$. If $A$ is of
type I or II, then the size of $A$ is $\ell $, and hence, $\ell > s$
and $s_{X} =
\min\{\ell,s_{\mathrm{III}},s_{\mathrm{IV}},s_{\mathrm{V}}\}$. If $A$
is of type III, IV or V, then $\ell \ge s + 1$, and hence again $s_{X}
= \min\{\ell,s_{\mathrm{III}},s_{\mathrm{IV}},s_{\mathrm{V}}\}$. It
follows that Algorithm~\ref{alg:Tucker} correctly outputs one of the
smallest forbidden subgraphs.
It follows from Algorithm~\ref{alg:Tucker} that we do not need a
special detection algorithms for type I and II forbidden
subgraphs. However, in some applications, there might be a need to
determine a smallest forbidden subgraph of each type. Therefore, we
present such algorithms for these two types of forbidden subgraphs as
well.
\end{document} | math | 37,126 |
\begin{document}
\title{Demonstration of a universal one-way quantum quadratic phase gate}
\date{\today}
\author{Yoshichika Miwa}
\affiliation{\affA}
\author{Jun-ichi Yoshikawa}
\affiliation{\affA}
\author{Peter van Loock}
\affiliation{\affB}
\author{Akira Furusawa}
\affiliation{\affA}
\begin{abstract}
We demonstrate a quadratic phase gate
for one-way quantum computation in the continuous-variable regime.
This canonical gate, together with phase-space displacements
and Fourier rotations, completes the set of universal gates
for realizing any single-mode Gaussian transformation such as
arbitrary squeezing. As opposed to previous implementations of
measurement-based squeezers, the current gate is fully controlled
by the local oscillator phase of the homodyne detector.
Verifying this controllability, we give an experimental
demonstration of the principles of one-way quantum computation
over continuous variables. Moreover, we can observe sub-shot-noise
quadrature variances in the output states, confirming that
nonclassical states are created through cluster computation.
\end{abstract}
\pacs{03.67.Lx, 42.50.Dv, 42.50.Ex}
\maketitle
Measurement-based one-way quantum computation~\cite{Raussendorf01.prl},
using an offline prepared, multi-party entangled cluster state,
is a conceptually interesting alternative to the
standard unitary circuit model of quantum computation~\cite{Nielsen2000}.
In the cluster-model, universality is achieved through
different choices of measurement bases, while the cluster state
remains fixed. Unitary gates are effectively applied at each measurement
step, corresponding to elementary teleportations~\cite{Zhou00.pra,Nielsen2006}
for propagating and manipulating a quantum state through the cluster.
The cluster model also turned out to provide new,
potentially more efficient approaches to the experimental
realization of quantum logical gates, especially in the
quantum optical setting~\cite{Nielsen2004,Browne2005}.
A translation of the circuit model for quantum computation
over continuous variables (CV)~\cite{lloyd,SamPvLRMP} to universal cluster
computation with CV was given in ~Ref.~\cite{clusterMenicucci}. The canonical,
universal gate set for CV is $\{\hat U_3(\lambda),C\}$, where
$C=\{\hat Z(s),\hat U_2(\kappa),\hat F,C_Z\}$ with the momentum shift operator
$\hat Z(s) = \exp(2 i s \hat x)$, the phase gates
$\hat U_l(\kappa_l) = \exp(i\kappa_l \hat x^l)$, the Fourier transform operator
$\hat F$, and the controlled-$Z$ gate
$C_Z = \exp(2i \hat x\otimes\hat x)$~\cite{Bartlett}.
Through concatenation, the full set enables one to simulate any
Hamiltonian in terms of arbitrary polynomials of the position $\hat x$ and
the momentum $\hat p$ to any precision~\cite{lloyd}.
The same set without the cubic gate $\hat U_3$, i.e., the set $C$,
is still universal for realizing any quadratic Hamiltonian,
that is, the whole group of Gaussian unitary transformations, the analogue
to the Clifford group for discrete variables (DV).
In the case of DV, for example, single-qubit Clifford transformations
are fully covered by the Hadamard gate $\hat H$ and the
`$\pi/4$'-phase gate $\hat U_{\pi/4}$ acting upon the qubit Pauli operators as
$\hatd U_{\pi/4}Z\hat U_{\pi/4}= Z$ and
$\hatd U_{\pi/4}X\hat U_{\pi/4}=-i X Z=-Y$; full universality
for single-qubit transformations would then require, in addition, the
well-known `$\pi/8$'-phase gate~\cite{Nielsen2000}, the analogue to the
cubic phase gate $\hat U_3$ for CV. Focussing on CV, the quadratic gate
from the universal set $C$ for all Gaussian transformations, maps the
Weyl-Heisenberg displacement operators $\hat Z(s)$ and
$\hat X(s)=\exp(-2 i s \hat p)$ into
\begin{align}
&\hatd U_2(\kappa) \hat Z(s) \hat U_2(\kappa) = \hat Z(s) \nonumber\\
&\hatd U_2(\kappa) \hat X(s) \hat U_2(\kappa) = e^{-i \kappa s^2} \hat X(s) \hat Z(-\kappa s),
\end{align}
in analogy to the qubit `$\pi/4$'-phase gate $\hat U_{\pi/4}$.
The effect of the phase gate may be more conveniently expressed
in terms of the generators of the Weyl-Heisenberg group,
$\hatd U_2(\kappa) \hat x \hat U_2(\kappa) = \hat x$,
$\hatd U_2(\kappa) \hat p \hat U_2(\kappa) = \hat p + \kappa \hat x$.
In quantum optics, it is well-know that there is an exact,
finite decomposition of any quadratic unitary into single-mode squeezers
and beam splitters~\cite{Braunstein05.pra,Reck}.
In this quantum optical language, the quadratic phase gate
$\hat U_2$, together with the Fourier transform $\hat F$,
provides single-mode squeezing, and the two-mode gate
$C_Z$ involves beam splitting modulo single-mode squeezing.
In the cluster-based one-way model,
the quadratic gate can be fully controlled through the local oscillator
phase of the homodyne detector~\cite{clusterMenicucci}. Here,
we experimentally demonstrate this controllability,
with a fixed, offline two-mode cluster state.
We show that a large set of squeezing transformations can be achieved
by means of this one-way phase gate; sequential application of the gate
would lead to universal single-mode Gaussian transformations
(where changes of the 1st moments in phase space require, in addition,
$p$-displacements $\hat Z(s)$, trivially realizable through
a cluster state for CV~\cite{clusterMenicucci}).
The output states
of our elementary cluster computations exhibit sub-shot-noise
quadrature variance; thus, nonclassical states are created
deterministically through cluster computation with the
degree of nonclassicality fully
controlled by the measurement apparatus.
Therefore, our demonstration differs from previous implementations of
universal offline squeezing~\cite{Filip05.pra,Yoshikawa07.pra},
in which different squeezing transformations require
different beam splitter transformations to achieve universality.
The elementary teleportation step for the
case of CV~\cite{clusterMenicucci} is described as follows.
First, in the ideal scheme (Fig.~1(a)),
an arbitrary input state is coupled
to a single-mode, infinitely squeezed state (a position eigenstate $\ket{x = 0}$),
$\hat{U}_{\mathrm{QND}} \ket{\psi}_\mathrm{in}\ket{x=0}_{\mathrm{A}}$. This results in
$e^{-2i\hat{x}_\mathrm{in}\hat{p}_\mathrm{A}} \int dx\, \psi (x)
\ket{x}_\mathrm{in} \int dp\,\ket{p}_{\mathrm{A}}/\sqrt{\pi}
= \int dx \psi(x)\ket{x}_{\mathrm{in}}\ket{x}_{\mathrm{A}}$,
where the subscripts `$\mathrm{in}$' and `$\mathrm{A}$' denote the input and ancilla modes, respectively.
Up to local Fourier rotations, the resulting state corresponds to a perfect two-mode
cluster state, already carrying the quantum information to be processed through
the cluster (i.e., the quantum state $\ket{\psi}_\mathrm{in}$).
Next, we measure the observable $\hatd{U}(\hat{x})\hat{p}\hat{U}(\hat{x})$ of mode 1, where
$\hat{U}(\hat{x})\equiv \exp[i f(\hat x)]$ is diagonal in the position basis and
$\hat{p}$ is the conjugate momentum to $\hat{x}$ ($[\hat{x},\hat{p}]=i/2$).
The quantum state after the measurement with outcome $p_0$ is
$\sqrt{\pi}\,{_\mathrm{in}\!\bradmket{p_0}{\hat{U}(\hat{x}_\mathrm{in} ) \int \psi(x)}{x}_\mathrm{in}}\ket{x}_{\mathrm{A}} dx
= \sqrt{\pi} \int {_\mathrm{in}\!\bracket{p_0}{x}_\mathrm{in}} U(x) \psi (x) \ket{x}_{\mathrm{A}} dx
= \hat{Z}(-p_0)\hat{U}(\hat{x}_\mathrm{A} )\ket{\psi}_\mathrm{A}$.
After correcting the displacement $\hat{Z}(-p_0)$,
we obtain the desired state $\hat{U}(\hat{x})\ket{\psi}$ in the ancilla mode.
Through this scheme, in principle, we can apply an arbitrary unitary
operator $\hat{U}(\hat{x})$ to $\ket{\psi}_\mathrm{in}$;
for nonlinear gates such as the cubic gate $\hat U_3$, however, this would require
measuring a nonlinear observable. Here, we consider detection of the whole range of rotated
quadratures (all linear combinations of $\hat x$ and $\hat p$), effectively applying the
quadratic phase gate $\hat U_2(\kappa)=\exp\left(i\kappa\hat{x}^2\right)$
to $\ket{\psi}_\mathrm{in}$, up to a phase-space displacement
depending on the measurement result $p_0$.
In our optical realization, $\hat{x}$ and $\hat{p}$ are quadrature operators,
for the mode operator $\hat{a}=\hat{x}+i\hat{p}$. The quadratic gate $\hat U_2(\kappa)$ corresponds to a sequence of rotation,
squeezing, and rotation~\cite{Braunstein05.pra}, with $\hat{x}_\mathrm{out} = \hat{x}_\mathrm{in}$ and
$\hat{p}_\mathrm{out} = \hat{p}_\mathrm{in} +\kappa \hat{x}_\mathrm{in}$.
Thus, the required measurement corresponds to measuring~\cite{clusterMenicucci}
$\hat{p} + \kappa \hat{x}=
\sqrt{1+\tan^2 \theta}\left(\hat{p}\cos \theta + \hat{x}\sin \theta \right)$ with
$\kappa = \tan\theta$.
Using homodyne detection and setting the phase of the local oscillator (LO) to $\theta$, we can measure $(\hat{p}\cos\theta + \hat{x}\sin\theta)$.
Appropriate electric amplification of the homodyne results with gain $(1+\tan^2 \theta)^{1/2}$ leads to the desired measurement of $\hatd{U}\hat{p}\hat{U}$.
We show this for several values of $\kappa$: $0$, $\pm 1.0$, $\pm 1.5$, $\pm 2.0$, with coherent-state inputs.
The corresponding LO phases are $0^\circ$ , $\pm 45^\circ$, $\pm 56.3^\circ$, and $\pm 63.4^\circ $, respectively.
\begin{figure}
\caption{Schematic of a one-way quantum gate and our experimental setup.
OPO: optical parametric oscillator, LO: optical local oscillator, and EOM: electro-optic modulator.}
\label{fig:whole}
\end{figure}
In our optical demonstration, we use three squeezed-vacuum ancillae.
One ancilla is coupled to the input via a QND gate (denoted by subscript $\mathrm{A}$). The QND gate itself requires two additional squeezed vacuum states (denoted by subscripts $\mathrm{B}$, $\mathrm{C}$). For the QND gate, we employ the scheme of Refs.~\cite{Filip05.pra,Yoshikawa08.prl}.
The full input-output relations of the scheme including finite-squeezing resources are
\begin{align}
\hat{x}_\mathrm{out} &= \hat{x}_\mathrm{in} +\hat{x}_\mathrm{A}^{(0)}e^{-r_\mathrm{A}} -\frac{\sqrt{5}-1}{2\sqrt[4]{5}}\hat{x}_\mathrm{B}^{(0)}e^{-r_\mathrm{B}}, \notag \\
\hat{p}_\mathrm{out} &= \hat{p}_\mathrm{in} +\kappa \hat{x}_{\mathrm{in}} + \frac{1}{\sqrt[4]{5}}\kappa \hat{x}_\mathrm{B}^{(0)}e^{-r_\mathrm{B}} + \frac{\sqrt{5}+1}{2\sqrt[4]{5}}\hat{p}_\mathrm{C}^{(0)}e^{-r_\mathrm{C}}.
\label{eq:input-output with excess noise}
\end{align}
Even with the excess noise from the finite squeezing of the ancillae,
we are able to observe sub-shot-noise quadrature squeezing for sufficiently large $\kappa$.
In the remainder of the letter, we shall describe the experimental details and present the results of the experiment.
{\it Experimental setup.}---
A schematic of the experimental setup is illustrated in Fig.~1(b).
The original source of light is a continuous wave (CW) Ti:sapphire laser, whose output is 860~nm in wavelength and 1.5~W in power.
Quantum states at the 1.34 MHz sideband are used in our demonstration.
The experimental setup consists of the following parts: preparation of the input and ancilla states, the QND coupling gate, measurement, feedforward, and, finally, the verification measurement.
The input state, a coherent state at the 1.34~MHz sideband, is generated by modulating a weak laser beam of about 10~$\mu$W using electro-optic modulators (EOMs).
We prepare three types of coherent states $\ket{\alpha}$:
$\alpha = x_\mathrm{in}$, $\alpha = i p_\mathrm{in}$, and $\alpha = 0$, corresponding to phase modulation, amplitude modulation, and no modulation of the laser beam, respectively.
In order to prepare the ancilla states, there are three sub-threshold optical parametric oscillators (OPOs), each generating a single-mode squeezed state, whose squeezing level is $-4.3$dB, $-4.9$dB, and $-5.2$dB.
An OPO is a bow-tie shaped cavity of 500~mm in length, containing a PPKTP crystal~\cite{Suzuki06.apl}.
The second harmonic (430~nm in wavelength) of Ti:sapphire output is divided into three beams in order to pump the OPOs.
The QND gate basically consists of a Mach-Zehnder interferometer with a single-mode squeezing gate in each arm~\cite{Yoshikawa08.prl}. Each single-mode squeezing gate contains a squeezed vacuum ancilla, homodyne detection, and feedforward~\cite{Filip05.pra, Yoshikawa07.pra}.
The variable beam splitters in the QND gate are composed of two polarizing beam splitters and a half-wave plate.
We can eliminate the QND gate and just measure the input states by setting the transmittances of the variable beam splitters to unity.
At each beam splitter, we lock the relative phase of the two input beams by means of active feedback to a piezoelectric transducer.
For this purpose, two modulation sidebands of 154~kHz and 107~kHz are used as phase references.
For the homodyne detection, the LO phase is adjusted in accordance to the desired $\kappa$ value; the feedforward
displacement is carried out with the right gain depending on $\kappa$.
To verify the output state, we employ another homodyne detection.
As is well known from optical homodyne tomography, we can reconstruct the quantum state from the marginal distributions for various phases~\cite{Lvovsky09.rmp}.
We slowly scan through the LO phase and perform a series of homodyne measurements.
The 1.34 MHz component of the homodyne signal is extracted by means of lock-in detection: it is mixed with a reference signal and then sent through a 30~kHz low pass filter.
Finally, it is analog-to-digital converted where the sampling rate is 300,000 samples per second.
The powers of the LOs are about 3~mW.
The detector's quantum efficiencies are greater than 99\%, the interference visibilities to the LOs are on average 98\%, and the dark noise of each homodyne detector is about 17~dB below the optical shot noise level produced by the LO.
Propagation losses of our whole setup are about 7\%.
\begin{figure}
\caption{Input and output states with several $\kappa$. Left figures show raw data of marginal distributions and right ones show the Wigner functions, reconstructed via maximum-likelihood method~\cite{Lvovsky04}
\label{fig:results}
\end{figure}
{\it Experimental results.}---
As mentioned earlier, we carry out the experiment with three types of input coherent states $\ket{\alpha}$:
$\alpha = x_\mathrm{in}$ ($x_\mathrm{in} = 1.4$), $\alpha = i p_\mathrm{in}$ ($p_\mathrm{in} = 1.3$), and $\alpha = 0$.
For each input state, we demonstrate the gate for seven different $\kappa$ values: $0$, $\pm 1.0$, $\pm 1.5$, and $\pm 2.0$.
Fig.~2 shows the raw data of marginal distributions and the Wigner functions reconstructed via maximum-likelihood method~\cite{Lvovsky04}.
We show the results for the input state with the amplitude in $x$ as an example.
Each scan contains about 80,000 data points which are uniformly distributed in phase from 0 to $2\pi$, and every 20 points are plotted in the figure (about 4,000 data points).
For $\kappa = 0$ (Fig.~2(b)), the input state is regenerated at the output except for some excess noise.
For nonzero $\kappa$ (Fig.~2(c, d)), we can see that the distribution of the $p$ variable is shifted proportional to $x$, with a proportionality factor $\kappa$. As a result, the output states are squeezed and rotated.
\begin{figure}
\caption{(color). Input coherent state (black circle) and output states for several $\kappa$.
We assume Gaussian distributions, and show averaged amplitudes, variances.
(a, c, d):~Experimental results for three types of input coherent state $\ket{\alpha}
\label{fig:fit2G}
\end{figure}
In Fig.~3, the elliptic output Wigner functions for $\kappa = 0, \pm 1.0, \pm 2.0$ are shown,
where the position, size, and shape of each ellipse correspond to the averaged amplitudes and variances.
Fig.~3(a, b) are for the case of $\alpha = x_\mathrm{in}$: (a) experimental results and (b) theoretical, ideal operation.
They agree well in positions and inclinations of ellipses, although the ellipses in Fig.~3(a) are thermalized because of the
finite squeezing of the ancilla states.
We estimate the experimentally obtained $\kappa$ via $\kappa _{\mathrm{act}} = \avg{\hat{p}_\mathrm{out}}/\avg{\hat{x}_\mathrm{in}}$, and the values obtained are $\kappa _{\mathrm{act}} = 0.00$, $0.95$, $-1.04$, $1.94$, and $-2.02$ for theoretical values $\kappa _{\mathrm{th}}= 0$, $\pm 1.0$, and $\pm 2.0$, respectively.
The differences in inclinations between experimental and ideal Wigner functions are less than $3^\circ$.
The experimental results for the other input states are shown in Fig.~3(c, d).
The change of the amplitude in the input states only affects the positions of the ellipses; the shapes and inclinations of the ellipses remain the same.
We can see in Fig~3(d) that the input amplitude in the $p$ quadrature ($p_\mathrm{in}$) is simply reproduced at the output and is otherwise not affected for any $\kappa$.
All of these results are in good agreement with the theoretical input-output relations.
In Fig.~4(a), the fidelities of the experimental output states compared to the ideal pure output states (i.e. without excess noise) are plotted. The fidelity quantifies the overlap between two quantum states, and it can be calculated as $_\mathrm{in}\bra{\psi}\hatd{U}\hat{\rho}_\mathrm{out}\hat{U}\ket{\psi}_\mathrm{in}$.
In the case of infinitely squeezed ancillae, unit fidelity is achieved.
In the experiment, excess noises due to finitely squeezed ancillae lead to non-unit fidelities.
Without quantum resources (i.e., using vacuum states for ancillary inputs), the experimental fidelity is $0.62 \pm 0.01$ for $\kappa = 0$, which agrees with the theoretical result $0.63$ derived from Eq.~\eqref{eq:input-output with excess noise}.
With squeezed-vacuum ancillae, the experimental fidelity is $0.81 \pm 0.01$ for $\kappa = 0$, which is much better than the case without nonclassical resources.
For nonzero $\kappa$, the fidelities decrease as $|\kappa |$ increases, because the squeezing of the ideal output state grows compared to that
used in the ancillary states.
Experimental results are very close to the theoretical curves which are calculated from the experimentally obtained squeezing levels of the ancillae.
In Fig.~4(b), the quadrature squeezing of our setup is plotted. Note that the squeezed quadratures are fragile and easily degraded by excess noise.
In the case of infinitely squeezed ancillae, squeezing is obtained for any nonzero $\kappa$; for $\kappa = 0$, on the other hand, the variance of the input coherent state is preserved.
With finitely squeezed ancillae, the excess noises are added to the variances of the ideal outputs.
Without nonclassical resources, squeezing below the SNL is, of course, not obtained for any $\kappa$.
In the case of a squeezing level of the ancillae below -2.9 dB relative to the SNL, the output state is squeezed for sufficiently large $|\kappa |$.
We can observe a noise suppression below the SNL by $0.3 \pm 0.1$~dB for $\kappa = \pm 1.0$, $0.8 \pm 0.1$~dB for $\kappa = \pm 1.5$, and $1.0 \pm 0.1$~dB for $\kappa = \pm 2.0$.
\begin{figure}
\caption{Fidelities of output states and variances of squeezed quadrature.
(i):~experimental results with squeezed ancillae and their theoretical curves derived from Eq.~\eqref{eq:input-output with excess noise}
\label{fig:sq_fi}
\end{figure}
In conclusion, we have experimentally demonstrated the canonical quadratic phase gate for CV
in a small cluster computation. The gate is fully controlled by the local oscillator phase
of the homodyne detector. We demonstrated controllability for a set of coherent input states and
we observed sub-shot-noise quadrature variances in the output states, verifying that our measurement-based
gate creates nonclassical states. Concatenating this scheme would enable one to realize any
single-mode Gaussian transformation, efficiently applicable to arbitrary input states including
non-Gaussian states.
This work was partly supported by SCF, GIA, G-COE, and PFN commissioned by the MEXT of Japan, the Research Foundation for Opt-Science and Technology, and SCOPE program of the MIC of Japan.
P.~v.~L. acknowledges support from the
Emmy Noether programme of the DFG in Germany.
\end{document} | math | 19,988 |
\begin{document}
\title{\bf The asymptotic behavior of globally smooth solutions of bipolar non-isentropic compressible
Euler-Maxwell system for plasma }
\author{ Shu Wang$^{1}$, Yuehong Feng$^{1}$ and Xin Li$^2$ }
\date{}
\maketitle \markboth{S. Wang, Y. H. Feng and X. Li} { Bipolar
non-isentropic compressible
Euler-Maxwell system }
\begin{center}
{\small $^1$College of Applied Sciences, Beijing University of Technology, Beijing 100124, China \\[2mm]
$^2$Department of Mathematics and computer Science, Xinyang
Vocational and Technical College, Xinyang, 464000, China \\[2mm]
Email~:[email protected],\hspace{1mm}[email protected],\hspace{1mm}[email protected]}
\end{center}
\begin{center}
\begin{minipage}{14cm}
{\bf Abstract.} {\small The bipolar non-isentropic compressible Euler-Maxwell system is
investigated in $R^3$ in the present paper, and the $L^q$ time decay
rate for the global smooth solution is established. It is shown that
the total densities, total temperatures and magnetic field of two
carriers converge to the equilibrium states at the same rate
$(1+t)^{-\frac{3}{2}+\frac{3}{2q}}$ in $L^q$ norm. But, both the
difference of densities and the difference of temperatures of two
carriers decay at the rate $(1+t)^{-2-\frac{1}{q}}$, and the
velocity and electric field decay at the rate
$(1+t)^{-\frac{3}{2}+\frac{1}{2q}}$. This phenomenon on the charge
transport shows the essential difference between the non-isentropic
unipolar Euler-Maxwell and the bipolar isentropic Euler-Maxwell
system. }
\end{minipage}
\end{center}
\noindent {\bf Keywords:} Bipolar non-isentropic Euler-Maxwell
equations, Plasma, Globally smooth solution, Asymptotic behavior
\noindent {\bf AMS Subject Classification (2000)~:} 35A01, 35L45,
35L60, 35Q35
\section{Introduction and main results}
The Euler-Maxwell system is used to model and simulate the transport
of charged particles in
plasma\cite{Chen84,Dink05,Jero03,Jero05,YW11}. Usually, it takes the
form of compressible non-isentropic Euler equations forced by the
electromagnetic field, which is governed by the self-consistent
Maxwell equation. In present paper, we consider the Cauchy problem
for the bipolar non-isentropic Euler-Maxwell system
\begin{equation}
\label{0}
\left\{\begin{aligned}
&\partialrtial_t {n_e} + \nabla\cdot(n_e u_e )=0, \\
&\partialrtial_t (n_e u_e) +\nabla\cdot (n_e u_e\otimes u_e) +\nabla p_e
= -n_e(E+ u_e \times B)-n_e u_e
, \\
&\partialrtial_t(n_e \mathcal {E}_e)+\nabla\cdot (n_e u_e\mathcal
{E}_e+u_e p_e)=-n_e u_e E-n_e |u_e|^2-n_e(\theta_e-1),
\\
&\partialrtial_t {n_i} + \nabla\cdot(n_i u_i )=0, \\
&\partialrtial_t (n_i u_i) +\nabla\cdot (n_i u_i\otimes u_i) +\nabla p_i
= n_i(E+ u_i \times B)-n_i u_i
, \\
&\partialrtial_t(n_i \mathcal {E}_i)+\nabla\cdot (n_i u_i\mathcal
{E}_i+u_i p_i)=n_i u_i E-n_i |u_i|^2-n_i(\theta_i-1),
\\
& \partialrtial_t E-\nabla\times B= n_e u_e-n_i u_i ,
\\
& \partialrtial_t B+\nabla\times E=0,\\
& \nabla\cdot E=n_i-n_e ,\quad \nabla\cdot B=0,\quad
(t,x)\in(0,\infty)\times\mathbb{R}^3,
\end{aligned} \right.
\end{equation}
where the unknowns are the density $n_\mu>0$, the velocity
$u_\mu=(u_\mu^1,u_\mu^2,u_\mu^3)$, the absolute temperature
$\theta_\mu>0$, the total energy $\mathcal
{E}_\mu=\displaystyle\frac{1}{2}|u_\mu|^2+C_\nu\theta_\mu$, the
pressure function $p_\mu=R_\mu n_\mu\theta_\mu$ for $\mu=e,i$, the
electronic field $E$ and magnetic field $B$. Furthermore, the
constants $C_\nu>0$, $R_\nu>0$ are the heat capacity at constant
volume and the coefficient of heat conductivity respectively.
Throughout this paper, we set $C_\nu=R_\nu=1$ without loss of
generality. Then, the system \eqref{0} is equivalent to
\begin{equation}
\label{1.1}
\left\{\begin{aligned}
&\partialrtial_t {n_e} + \nabla\cdot(n_e u_e )=0, \\
&\partialrtial_t u_e +(u_e\cdot\nabla) u_e +\frac{\theta_e}{n_e}\nabla
n_e+\nabla\theta_e= - (E+ u_e \times B)- u_e
, \\
&\partialrtial_t {\theta_e} + \nabla\cdot(\theta_e u_e )+(\theta_e-1)=0,
\\
&\partialrtial_t {n_i} + \nabla\cdot(n_i u_i )=0, \\
&\partialrtial_t u_i +(u_i\cdot\nabla) u_i +\frac{\theta_i}{n_i}\nabla
n_i+\nabla\theta_i= (E+ u_i \times B)- u_i
, \\
&\partialrtial_t {\theta_i} + \nabla\cdot(\theta_i u_i )+(\theta_i-1)=0,
\\
& \partialrtial_t E-\nabla\times B= n_e u_e-n_i u_i ,
\\
& \partialrtial_t B+\nabla\times E=0,\\
& \nabla\cdot E=n_i-n_e ,\quad \nabla\cdot B=0,\quad
(t,x)\in(0,\infty)\times\mathbb{R}^3.
\end{aligned} \right.
\end{equation}
Initial data is given as
\begin{equation}
\label{1.2}
(n_\mu, u_\mu, \theta_\mu, E,B)|_{t=0} = (n_{\mu0}, u_{\mu0},
\theta_{\mu0}, E_0,B_0),\quad x\in
\mathbb{R}^3,
\end{equation}
with the compatible condition
\begin{equation}
\label{1.3}
\nabla\cdot E_0 = n_{i0} - n_{e0}, \quad \nabla\cdot B_0 = 0, \quad x \in \;
\mathbb{R}^3.
\end{equation}
The Euler-Maxwell system (\ref{1.1}) is a symmetrizable hyperbolic
system for $n_\mu,\theta_\mu >0$. Then the Cauchy problem
(\ref{1.1})-(\ref{1.2}) has a local smooth solution when the initial
data are smooth. In a simplified one dimensional isentropic
Euler-Maxwell system, the global existence of entropy solutions has
been given in \cite{{CJW00}} by the compensated compactness method.
For the three dimensional isentropic Euler-Maxwell system, the
existence of global smooth solutions with small amplitude to the
Cauchy problem in the whole space and to the periodic problem in the
torus is established by Peng et al in \cite{PWG11} and Ueda et al in
\cite{UKW10} respectively, and the decay rate of the smooth solution
when t goes to infinity is obtained by Duan in \cite{Duan11} and
Ueda et al in \cite{UK11}. For asymptotic limits with small
parameters, see \cite{PW08a,PW08b} and references therein. For the
three dimensional bipolar isentropic Euler-Maxwell system, the
global existence and the asymptotic behavior of the smooth solution
is also obtained by Duan et al in \cite{Duan11b}. Recently, Yang et
al in \cite{YW11} consider the diffusive relaxation limit of the
three dimensional unipolar non-isentropic Euler-Maxwell system, and
Wang et al asymptotics and global existence in \cite{FWK11}.
However, there is no analysis on the asymptotics and global
existence for the bipolar non-isentropic Euler-Maxwell system in
three space dimensions yet. Therefore, the goal of the present paper
is to establish the global existence of smooth solutions around a
equilibrium solution of system (\ref{1.1}) and the decay rate of the
smooth solution as $t\rightarrow\infty$.
The main result of this paper can be stated as follows.
\begin{theorem}\label{thm1.1}
Assume \eqref{1.3} hold. If $\left\|
{[n_{\mu0}-1,u_{\mu0},\theta_{\mu0}-1,E_0,B_0] } \right\|_s\leq
\delta_0$ for $s\geq 4$. Then, there is a unique global solution
$[n_\mu(t,x)$, $u_\mu (t,x),$ $\theta_\mu(t,x), E(t,x), B(t,x)]$ to
the initial value problem \eqref{1.1}- \eqref{1.2} which satisfies
$$[n_\mu -1,u_\mu,\theta_\mu-1, E, B]
\in C^1\big([0,T);H^{s-1}(\mathbb{R}^3)\big) \cap
C\big([0,T);H^s(\mathbb{R}^3)\big)$$ and
$$
\mathop {\sup }\limits_{t \geqslant 0}\left\|
{[n_\mu(t)-1,u_\mu(t),\theta_\mu(t)-1,E(t),B(t)] } \right\|_s\leq
C_0\left\| {[n_{\mu0}-1,u_{\mu0},\theta_{\mu0}-1,E_0,B_0] }
\right\|_s,
$$
where $\delta_0, C_0 >0$ are constants independent of time.
Moreover, if $\left\| {[n_{\mu0}-1,u_{\mu0},\theta_{\mu0}-1,E_0,B_0]
} \right\|_{L^1 \cap H^{13} } \leq \delta_1$,
then the solution
$[n_\mu (t,x)$, $u_\mu (t,x)$,
$\theta_\mu(t,x)$, $E(t,x)$, $B(t,x)]$ satisfies
\begin{equation}\label{1.4}
\left\|\left[ {n_e(t)-n_i(t), \theta_e(t)-\theta_i(t) }\right]
\right\|_{L^q}\leq C_1 (1+t)^{-2-\frac{1}{q}},
\end{equation}
\begin{equation}\label{1.5}
\left\|\left[ {n_e(t)+n_i(t)-2, \theta_e(t)+\theta_i(t)-2 }\right]
\right\|_{L^q}\leq C_1 (1+t)^{-\frac{3}{2}+\frac{3}{2q}},
\end{equation}
\begin{equation}\label{1.6}
\left\| {u_e(t)\pm u_i(t), E(t) } \right\|_{L^q}\leq C_1
(1+t)^{-\frac{3}{2}+\frac{1}{2q}},
\end{equation}
\begin{equation}\label{1.7}
\left\|
{B}(t) \right\|_{L^q}\leq C_1 (1+t)^{-\frac{3}{2}+\frac{3}{2q}},
\end{equation} for any $t\geq 0$ and $2\leq q \leq\infty$.
Where, constants $\delta_1, C_1 >0$ are also independent of time.
\end{theorem}
\begin{remark}It should be emphasized that both the velocity and temperature relaxation term of the
bipolar non-isentropic Euler-Maxwell system \eqref{1.1} plays a key
role in the proof of Theorem \ref{thm1.1}.
\end{remark}
\noindent {\bf Notations.}
In this paper, $f \sim
g$ means $\gamma a\leq b\leq \frac{1}{\gamma}$ for a constant
$0<\gamma<1$. $H^s$ denotes the standard Sobolev space
$W^{s,2}({{\mathbb R}^3})$. We use $\dot{H}^s$ to denote the
corresponding $s$-order homogeneous Sobolev space. Set $L^2=H^0$. The
norm of $H^s$ is denoted by $\left\| {\cdot} \right\|_s$ with
$\left\| {\cdot} \right\|=\left\| {\cdot} \right\|_0$, and
$\langle\cdot,\cdot\rangle$ denotes the inner product over
$L^2({\mathbb R}^3)$. For the multi-index $\alpha
=(\alpha_1,\alpha_2,\alpha_3)$, we denote $\partialrtial^\alpha$
$=\partialrtial_{x_1}^{\alpha_1}
\partialrtial_{x_2}^{\alpha_2}\partialrtial_{x_3}^{\alpha_3}$ $=\partialrtial_{ 1}^{\alpha_1}
\partialrtial_{ 2}^{\alpha_2}\partialrtial_{ 3}^{\alpha_3}$ and
$|\alpha|=\alpha_1+\alpha_2+\alpha_3$. For an integrable function
$f:{\mathbb R}^3\rightarrow{\mathbb R}$, its Fourier transform is
defined by
$$ \hat{f}(k)=\int_{{\mathbb R}^3}e^{-ix\cdot k}f(x)dx,\ \ x\cdot k:=
\sum\limits_{j=1}^{3} {x_j k_j},\ \ k\in {\mathbb R}^3,
$$
where $i=\sqrt{-1}\in \mathbb C$ is the imaginary unit.
The rest of the paper is arranged as follows. In Section 2, the
transformation of the initial value problem and the proof of the
global existence and uniqueness of solutions are presented. In
Section 3, we study the linearized homogeneous equations to get the
$L^p-L^q$ decay property and the explicit representation of
solutions. In the last Section 4, we investigate the decay rates of
solutions to the transformed nonlinear equations and complete the
proof of Theorem \ref{thm1.1}.
\section{Global solutions for equations \eqref{1.1}}
\subsection{Preliminary}
Suppose $[n_\mu (t,x), u_\mu (t,x),\theta_\mu (t,x),E(t,x),B(t,x)]$
be a smooth solution of the initial value problem for the bipolar
non-isentropic Euler-Maxwell equations \eqref{1.1} with initial data
\eqref{1.2} which satisfies \eqref{1.3}. Set
\begin{equation}\label{2.1}
n_\mu (t,x)=1+\rho_\mu (t,x), \theta_\mu (t,x)=1+\mathbb{T}heta_\mu (t,x).
\end{equation}
Thus, we can rewrite the system \eqref{1.1}-\eqref{1.3} as
\begin{equation}
\label{2.2}
\left\{\begin{aligned}
&\partialrtial_t {\rho_e} + \nabla\cdot((1+\rho_e) u_e )=0, \\
&\partialrtial_t u_e +(u_e\cdot\nabla) u_e
+\frac{1+\mathbb{T}heta_e}{1+\rho_e}\nabla \rho_e +\nabla\mathbb{T}heta_e= - (E+ u_e
\times B)- u_e
, \\
&\partialrtial_t {\mathbb{T}heta_e} + \nabla\cdot((1+\mathbb{T}heta_e) u_e )+\mathbb{T}heta_e=0,
\\
&\partialrtial_t {\rho_i} + \nabla\cdot((1+\rho_i) u_i )=0, \\
&\partialrtial_t u_i +(u_i\cdot\nabla) u_i
+\frac{1+\mathbb{T}heta_i}{1+\rho_i}\nabla \rho_i+\nabla\mathbb{T}heta_i= (E+ u_i
\times B)- u_i
, \\
&\partialrtial_t {\mathbb{T}heta_i} + \nabla\cdot((1+\mathbb{T}heta_i) u_i )+\mathbb{T}heta_i=0,
\\
& \partialrtial_t E-\nabla\times B -u_e+ u_i= \rho_e u_e-\rho_i u_i ,
\\
& \partialrtial_t B+\nabla\times E=0,\\
& \nabla\cdot E=\rho_i-\rho_e ,\quad \nabla\cdot B=0,\quad
(t,x)\in(0,\infty)\times\mathbb{R}^3,
\end{aligned} \right.
\end{equation}
with initial data
\begin{equation}\label{2.3}
U|_{t=0}=U_0:=[\rho_{\mu 0},u_{\mu 0},\mathbb{T}heta_{ \mu0},E_0,B_0],\ x
\in{\mathbb R}^3,
\end{equation}
which satisfies the compatible condition
\begin{equation}\label{2.4}
\nabla\cdot E_0 = \rho_{i0}-\rho_{e0}, \quad \nabla\cdot B_0 = 0, \quad x \in \;
\mathbb{R}^3.
\end{equation}
Here, $\rho_{\mu 0}=n_{\mu 0}-1.$
In the following, we usually assume $s\geq4$. Moreover, for
$U=[\rho_{\mu }$, $u_{\mu }$, $\mathbb{T}heta_{\mu }$, $E$, $B]$, we use
$\mathcal {E}_s(U(t))$, $\mathcal {E}_s^h(U(t))$, $\mathcal
{D}_s(U(t))$ and $\mathcal {D}_s^h(U(t))$ to define the energy
functional, the high-order energy functional, the dissipation rate
and the high-order dissipation rate as
\begin{equation}\label{2.5}
\mathcal {E}_s(U(t))\sim \left\| {[\rho_\mu,u_\mu,\mathbb{T}heta_\mu,E,B]}
\right\|_s^2,
\end{equation}
\begin{equation}\label{2.6}
\mathcal {E}_s^h(U(t))\sim \left\|
{\nabla[\rho_\mu,u_\mu,\mathbb{T}heta_\mu,E,B]} \right\|_{s-1}^2,
\end{equation}
\begin{equation}\label{2.7}
\begin{split}
\mathcal {D}_s(U(t))\sim & \left\| {\nabla[\rho_e,\rho_i]}
\right\|_{s-1}^2 +\left\| {[u_e,u_i,\mathbb{T}heta_e,\mathbb{T}heta_i]}
\right\|_{s}^2\\
&+\left\| {E} \right\|_{s-1}^2+\left\| {\nabla B}
\right\|_{s-2}^2+\left\| {\rho_e-\rho_i} \right\|^2
\end{split}
\end{equation}
and
\begin{equation}\label{2.8}
\begin{split}
\mathcal {D}^h_s(U(t))\sim & \left\| {\nabla^2[\rho_e,\rho_i]}
\right\|_{s-2}^2 +\left\| {\nabla[u_e,u_i,\mathbb{T}heta_e,\mathbb{T}heta_i]}
\right\|_{s-1}^2\\
&+\left\| {\nabla E} \right\|_{s-2}^2+\left\| {\nabla^2 B}
\right\|_{s-3}^2+\left\| {\nabla [\rho_e-\rho_i]} \right\|^2,
\end{split}\end{equation}
respectively. Now, concerning the transformed initial value problem
\eqref{2.2}-\eqref{2.3}, we have the global existence result as
follows.
\begin{prop}\label{prop2.1}
Assume that $U_0=[\rho_{\mu0},u_{\mu0},\mathbb{T}heta_{\mu0},E_0,B_0]$
satisfies the compatible condition \eqref{2.4}. If \ $ \mathcal
{E}_s(U_0) $ is small enough, then, for any $t \geq0$, the initial
value problem \eqref{2.2}-\eqref{2.3} has a unique global nonzero
solution $U =[\rho_\mu,u_\mu,\mathbb{T}heta_\mu,E,B]$ which satisfies
\begin{equation}\label{2.9}
U\in C^1\big([0,T);H^{s-1}(\mathbb{R}^3)\big) \cap
C\big([0,T);H^s(\mathbb{R}^3)\big),
\end{equation}
and
\begin{equation}\label{2.10}
\mathcal {E}_s(U(t))+\lambda\int_0^t\mathcal {D}_s(U(s))ds\leq
\mathcal {E}_s(U_0).
\end{equation}
\end{prop}
Obviously, from the Proposition \ref{prop2.1}, it is straightforward
to get the existence result of Theorem \ref{thm1.1}. Furthermore,
solutions of Proposition \ref{prop2.1} really decay under some extra
conditions on $U_0=[\rho_{\mu0},u_{\mu0},\mathbb{T}heta_{\mu0},E_0,B_0]$.
For this purpose, we define $\omega_s(U_0)$ as
\begin{equation}\label{2.11}
\omega_s(U_0)=\left\|{U_0}\right\|_s+\left\|{[\rho_{\mu0}, u_{\mu0},
\mathbb{T}heta_{\mu0}, E_0,B_0]}\right\|_{L^1}
\end{equation}
for $s\geq4.$ Then, we obtain the following decay results.
\begin{prop}\label{prop2.2}
Assume that $U_0=[\rho_{\mu0}$, $u_{\mu0}$, $\mathbb{T}heta_{\mu0}$, $E_0$,
$B_0]$ satisfies \eqref{2.4}. If $ \omega_{s+2}(U_0)$ is
sufficiently small, then system \eqref{2.2}-\eqref{2.4} has a
solution $U =[\rho_\mu$, $u_\mu$, $\mathbb{T}heta_\mu$, $E$, $B ]$
satisfying
\begin{equation}\label{2.12}
\left\|{U(t)}\right\|_s\leq C \omega_{s+2}(U_0)(1+t)^{-\frac{3}{4}}
\end{equation}
for any $t\geq0$. Moreover, if $\omega_{s+6}(U_0)$ is sufficiently
small, then, for any $t \geq0$, the solution also satisfies
\begin{equation}\label{2.13}
\left\|{\nabla U(t)}\right\|_{s-1}\leq C
\omega_{s+6}(U_0)(1+t)^{-\frac{5}{4}}.
\end{equation}
\end{prop}
Thus, one can obtain the decay rates \eqref{1.4}-\eqref{1.7}
through the method of bootstrap and the Proposition stated above.
\subsection{Weighted energy estimates.} In this subsection, we shall give the
proof of Proposition \ref{prop2.1} for the global existence and
uniqueness of solutions to the initial value problem
\eqref{2.2}-\eqref{2.3}. Since hyperbolic equations \eqref{2.2} is
quasi-linear symmetrizable, thus one has the local existence of
smooth solutions to \eqref{2.2} as follows.
\begin{lemma}
\label{L2.1}
(Local existence of smooth solutions, see \cite{Ka75,Ma84})
Let $s > \frac{5}{2}$ and $(\rho_{\mu 0}$, $u_{\mu 0}$, $\mathbb{T}heta_{\mu
0}$, $E_0$, $B_0)$ $\in H^s(\mathbb{R}^3)$. Then there exist $T> 0$ and a unique smooth solution
$(n_\mu$, $u_\mu$, $\theta_\mu$, $E$, $B)$ to the Cauchy problem
(\ref{1.1})-(\ref{1.2}) satisfying $(\rho_{\mu}$, $u_{\mu}$, $\mathbb{T}heta_{\mu}$, $E$, $B)\in
C^1\big([0,T);H^{s-1}(\mathbb{R}^3)\big) \cap
C\big([0,T);H^s(\mathbb{R}^3)\big)$.
\end{lemma}
Then, with the help of the continuity argument, the global existence
of solutions satisfying \eqref{2.9} and \eqref{2.10} follows by
combing Lemma \ref{L2.1} and a priori estimate as follows.
\begin{theorem}\label{thm2.1}
Assume that
$U$ $=[\rho_\mu$, $u_\mu$, $\mathbb{T}heta_\mu$, $E$, $B ]$ $\in
C^1\big([0,T);H^{s-1}(\mathbb{R}^3)\big) \cap
C\big([0,T);H^s(\mathbb{R}^3)\big)$ is smooth for $T>0$ with
\begin{equation}\label{2.14}
\mathop {\sup }\limits_{0\leq t\leq T
}\left\|{U(t)}\right\|_s\leq\delta
\end{equation}
for $\delta\leq \delta_0$ with $\delta_0$ sufficiently small and
suppose $U$ to be the solution of the equations \eqref{2.2} for
$t\in(0,T)$. Then, for a constant $0<\gamma<1$ and any $0\leq t\leq
T$, it holds that
\begin{equation}\label{2.15}
\frac{d}{dt}\mathcal {E}_s(U(t))+\gamma \mathcal {D}_s(U(t))\leq
C[\mathcal {E}_s(U(t))^\frac{1}{2}+\mathcal {E}_s(U(t))]\mathcal
{D}_s(U(t)).
\end{equation}
\end{theorem}
\noindent \emph{Proof.} We will use five steps to finish the proof
as follows. In step 1, we establish the estimate of Euler part and
Maxwell part of the system \eqref{2.2} by using weighted energy
estimate method. In the following steps $2-4$, we utilize the
skew-symmetric structure of the system \eqref{2.2} to get the
dissipative estimates for $\rho_\mu$, $E$ and $B$. \\
\emph{Step 1.}
It holds that
\begin{equation}\label{2.16}
\frac{d} {{dt}}\left\| U \right\|_s^2 + \left\| {\left[
{{u_e},{u_i},{\mathbb{T}heta _e},{\mathbb{T}heta _i}} \right]} \right\|_s^2
\leqslant C{\left\| U \right\|_s}\left( {\left\| {\left[
{{u_e},{u_i},{\mathbb{T}heta _e},{\mathbb{T}heta _i}} \right]} \right\|_s^2 +
\left\| {\nabla \left[ {{\rho _e},{\rho _i}} \right]} \right\|_{s -
1}^2} \right).
\end{equation}
In fact, from the first six equations of \eqref{2.2}, weighted
energy estimate on $\partialrtial^\alpha \rho_\mu$, $\partialrtial^\alpha
u_\mu$ and $\partialrtial^\alpha \mathbb{T}heta_\mu$ with $|\alpha|\leq s$ imply
\begin{equation}\label{2.17}
\begin{split} \frac{1} {2}&\frac{d} {{dt}} \sum\limits_{\mu = e,i} {\left(
{\left\langle {\frac{{1 + {\mathbb{T}heta _\mu }}} {{1 + {\rho _\mu
}}},{{\left| {{\partialrtial ^\alpha }{\rho _\mu }} \right|}^2}}
\right\rangle + \left\langle {1 + {\rho _\mu },{{\left| {{\partialrtial
^\alpha }{u_\mu }} \right|}^2}} \right\rangle + \left\langle
{\frac{{1 + {\rho _\mu }}}
{{1 + {\mathbb{T}heta _\mu }}},{{\left| {{\partialrtial ^\alpha }{\mathbb{T}heta _\mu }} \right|}^2}} \right\rangle } \right)} \\
& + \sum\limits_{\mu = e,i} {\left( {\left\langle {1 + {\rho _\mu },{{\left| {{\partialrtial ^\alpha }{u_\mu }}
\right|}^2}} \right\rangle + \left\langle {\frac{{1 + {\rho _\mu }}}
{{1 + {\mathbb{T}heta _\mu }}},{{\left| {{\partialrtial ^\alpha }{\mathbb{T}heta _\mu }} \right|}^2}} \right\rangle } \right)}
+ \left\langle {\left( {1 + {\rho _e}} \right){\partialrtial ^\alpha }E,{\partialrtial ^\alpha }{u_e}} \right\rangle \\
& - \left\langle {\left( {1 + {\rho _i}} \right){\partialrtial ^\alpha
}E,{\partialrtial ^\alpha }{u_i}} \right\rangle
= - \sum\limits_{\beta < \alpha } {C_\beta ^\alpha {I_{\alpha ,\beta }}(t) + {I_1}(t).}
\end{split}\end{equation}
Where, ${I_{\alpha ,\beta }}(t)={I^e_{\alpha ,\beta
}}(t)+{I^i_{\alpha ,\beta }}(t)$, $I_1(t)=I_1^e(t)+I_1^i(t)$ with
\begin{equation}\notag\begin{split}
I_{\alpha ,\beta }^e(t) =& \left\langle {\frac{{1 + {\mathbb{T}heta _e}}}
{{1 + {\rho _e}}}{\partialrtial ^{\alpha - \beta }}{\rho _e}\nabla
{\partialrtial ^\beta }{u_e},{\partialrtial ^\alpha }{\rho _e}} \right\rangle
+ \left\langle {\frac{{1 + {\mathbb{T}heta _e}}} {{1 + {\rho _e}}}{\partialrtial
^{\alpha - \beta }}{u_e}\nabla {\partialrtial ^\beta }
{\rho _e},{\partialrtial ^\alpha }{\rho _e}} \right\rangle \\
& + \left\langle {\frac{{1 + {\rho _e}}} {{1 + {\mathbb{T}heta
_e}}}{\partialrtial ^{\alpha - \beta }}{u_e}\nabla {\partialrtial ^\beta
}{\mathbb{T}heta _e},{\partialrtial ^\alpha }{\mathbb{T}heta _e}} \right\rangle +
\left\langle {\frac{{1 + {\rho _e}}} {{1 + {\mathbb{T}heta _e}}}{\partialrtial
^{\alpha - \beta }}{\mathbb{T}heta _e}\nabla {\partialrtial ^\beta }{u_e},
{\partialrtial ^\alpha }{\mathbb{T}heta _e}} \right\rangle \\
& + \left\langle {\left( {1 + {\rho _e}} \right){\partialrtial ^{\alpha - \beta }}{u_e}\nabla
{\partialrtial ^\beta }{u_e},{\partialrtial ^\alpha }{u_e}} \right\rangle + \left\langle
{\left( {1 + {\rho _e}} \right){\partialrtial ^{\alpha - \beta }}\left( {\frac{{1 + {\mathbb{T}heta _e}}}
{{1 + {\rho _e}}}} \right)\nabla {\partialrtial ^\beta }{\rho _e},{\partialrtial ^\alpha }{u_e}} \right\rangle \\
& + \left\langle {\left( {1 + {\rho _e}} \right){\partialrtial ^{\alpha - \beta }}{u_e} \times
{\partialrtial ^\beta }B,{\partialrtial ^\alpha }{u_e}} \right\rangle,
\end{split}\end{equation}
\begin{equation}\notag\begin{split}
I_{\alpha ,\beta }^i(t) =& \left\langle {\frac{{1 + {\mathbb{T}heta _i}}}
{{1 + {\rho _i}}}{\partialrtial ^{\alpha - \beta }}{\rho _i}\nabla
{\partialrtial ^\beta }{u_i},{\partialrtial ^\alpha }{\rho _i}} \right\rangle
+ \left\langle {\frac{{1 + {\mathbb{T}heta _i}}} {{1 + {\rho _i}}}{\partialrtial
^{\alpha - \beta }}{u_i}\nabla {\partialrtial ^\beta }
{\rho _i},{\partialrtial ^\alpha }{\rho _i}} \right\rangle \\
& + \left\langle {\frac{{1 + {\rho _i}}} {{1 + {\mathbb{T}heta
_i}}}{\partialrtial ^{\alpha - \beta }}{u_i}\nabla {\partialrtial ^\beta
}{\mathbb{T}heta _i},{\partialrtial ^\alpha }{\mathbb{T}heta _i}} \right\rangle +
\left\langle {\frac{{1 + {\rho _i}}} {{1 + {\mathbb{T}heta _i}}}{\partialrtial
^{\alpha - \beta }}{\mathbb{T}heta _i}\nabla {\partialrtial ^\beta }{u_i},
{\partialrtial ^\alpha }{\mathbb{T}heta _i}} \right\rangle \\
& + \left\langle {\left( {1 + {\rho _i}} \right){\partialrtial ^{\alpha - \beta }}{u_i}\nabla
{\partialrtial ^\beta }{u_i},{\partialrtial ^\alpha }{u_i}} \right\rangle + \left\langle
{\left( {1 + {\rho _i}} \right){\partialrtial ^{\alpha - \beta }}\left( {\frac{{1 + {\mathbb{T}heta _i}}}
{{1 + {\rho _i}}}} \right)\nabla {\partialrtial ^\beta }{\rho _i},{\partialrtial ^\alpha }{u_i}} \right\rangle \\
& - \left\langle {\left( {1 + {\rho _i}} \right){\partialrtial ^{\alpha - \beta }}{u_i} \times
{\partialrtial ^\beta }B,{\partialrtial ^\alpha }{u_i}} \right\rangle,
\end{split}\end{equation}
and
\begin{equation}\notag\begin{split}
I_1^e(t) = &~~\frac{1}
{2}\left\langle {{\partialrtial _t}\left( {\frac{{1 + {\mathbb{T}heta _e}}}
{{1 + {\rho _e}}}} \right),{{\left| {{\partialrtial ^\alpha }{\rho _e}} \right|}^2}}
\right\rangle + \left\langle {\nabla {\mathbb{T}heta _e}{\partialrtial ^\alpha }{u_e},{\partialrtial ^\alpha }{\rho _e}}
\right\rangle + \left\langle {\nabla {\rho _e}{\partialrtial ^\alpha }{u_e},{\partialrtial ^\alpha }{\mathbb{T}heta _e}}
\right\rangle \\
& + \frac{1}
{2}\left\langle {\nabla \cdot \left( {\frac{{1 + {\mathbb{T}heta _e}}} {{1
+ {\rho _e}}}{u_e}} \right),{{\left| {{\partialrtial ^\alpha }{\rho _e}}
\right|}^2}} \right\rangle + \frac{1} {2}\left\langle {{\partialrtial
_t}\left( {\frac{{1 + {\rho _e}}}
{{1 + {\mathbb{T}heta _e}}}} \right),{{\left| {{\partialrtial ^\alpha }{\mathbb{T}heta _e}} \right|}^2}} \right\rangle \\
& + \frac{1}
{2}\left\langle {\nabla \cdot \left( {\frac{{1 + {\rho _e}}}
{{1 + {\mathbb{T}heta _e}}}{u_e}} \right),{{\left| {{\partialrtial ^\alpha }{\mathbb{T}heta _e}} \right|}^2}} \right\rangle
- \left\langle {\left( {1 + {\rho _e}} \right){u_e} \times {\partialrtial ^\alpha }B,{\partialrtial ^\alpha }{u_e}}
\right\rangle ,
\end{split}\end{equation}
\begin{equation}\notag\begin{split}
I_1^i(t) = &~~\frac{1}
{2}\left\langle {{\partialrtial _t}\left( {\frac{{1 + {\mathbb{T}heta _i}}} {{1 +
{\rho _i}}}} \right),{{\left| {{\partialrtial ^\alpha }{\rho _i}}
\right|}^2}}
\right\rangle + \left\langle {\nabla {\mathbb{T}heta _i}{\partialrtial ^\alpha }{u_i},{\partialrtial ^\alpha }{\rho _i}}
\right\rangle + \left\langle {\nabla {\rho _i}{\partialrtial ^\alpha }{u_i},{\partialrtial ^\alpha }{\mathbb{T}heta _i}}
\right\rangle \\
& + \frac{1}
{2}\left\langle {\nabla \cdot \left( {\frac{{1 + {\mathbb{T}heta _i}}} {{1
+ {\rho _i}}}{u_i}} \right),{{\left| {{\partialrtial ^\alpha }{\rho _i}}
\right|}^2}} \right\rangle + \frac{1} {2}\left\langle {{\partialrtial
_t}\left( {\frac{{1 + {\rho _i}}}
{{1 + {\mathbb{T}heta _i}}}} \right),{{\left| {{\partialrtial ^\alpha }{\mathbb{T}heta _i}} \right|}^2}} \right\rangle \\
& + \frac{1}
{2}\left\langle {\nabla \cdot \left( {\frac{{1 + {\rho _i}}} {{1 +
{\mathbb{T}heta _i}}}{u_i}} \right),{{\left| {{\partialrtial ^\alpha }{\mathbb{T}heta
_i}} \right|}^2}} \right\rangle
+\left\langle {\left( {1 + {\rho _i}} \right){u_i} \times {\partialrtial ^\alpha }B,{\partialrtial ^\alpha }{u_i}}
\right\rangle ,
\end{split}\end{equation}
where we have used integration by parts. When $|\alpha|=0$, one has
\begin{equation}\notag\begin{split}
{I_1}(t) =& I_1^e(t) + I_1^i(t) \\
= &\sum\limits_{\mu = e,i} {\left( {\frac{1}
{2}\left\langle {{\partialrtial _{{\mathbb{T}heta _\mu }}}\left( {\frac{{1 +
{\mathbb{T}heta _\mu }}} {{1 + {\rho _\mu }}}} \right){\partialrtial _t}{\mathbb{T}heta
_\mu } + {\partialrtial _{{\rho _\mu }}}\left( {\frac{{1 + {\mathbb{T}heta _\mu
}}} {{1 + {\rho _\mu }}}} \right){\partialrtial _t}{\rho _\mu }, {{\left|
{{\rho _\mu }} \right|}^2}} \right\rangle
+ \left\langle {\nabla {\mathbb{T}heta _\mu }{u_\mu },{\rho _\mu }} \right\rangle } \right.} \\
& + \frac{1}
{2}\left\langle {{\partialrtial _{{\mathbb{T}heta _\mu }}}\left( {\frac{{1 +
{\mathbb{T}heta _\mu }}} {{1 + {\rho _\mu }}}{u_\mu }} \right)\nabla {\mathbb{T}heta
_\mu } + {\partialrtial _{{u_\mu }}}\left( {\frac{{1 + {\mathbb{T}heta _\mu }}}
{{1 + {\rho _\mu }}}{u_\mu }} \right)\nabla \cdot {u_\mu } +
{\partialrtial _{{\rho _\mu }}}\left( {\frac{{1 + {\mathbb{T}heta _\mu }}}
{{1 + {\rho _\mu }}}{u_\mu }} \right)\nabla {\rho _\mu },{{\left| {{\rho _\mu }} \right|}^2}} \right\rangle \\
& + \frac{1}
{2}\left\langle {{\partialrtial _{{\mathbb{T}heta _\mu }}}\left( {\frac{{1 +
{\rho _\mu }}} {{1 + {\mathbb{T}heta _\mu }}}} \right){\partialrtial _t}{\mathbb{T}heta
_\mu } + {\partialrtial _{{\rho _\mu }}}\left( {\frac{{1 + {\rho _\mu }}}
{{1 + {\mathbb{T}heta _\mu }}}} \right){\partialrtial _t}{\rho _\mu },{{\left| {{\mathbb{T}heta _\mu }} \right|}^2}} \right\rangle
+ \left\langle {\nabla {\rho _\mu }{ }{u_\mu },{ }{\mathbb{T}heta _\mu }} \right\rangle \\
& \left. { + \frac{1}
{2}\left\langle {{\partialrtial _{{\mathbb{T}heta _\mu }}}\left( {\frac{{1 +
{\rho _\mu }}} {{1 + {\mathbb{T}heta _\mu }}}{u_\mu }} \right)\nabla {\mathbb{T}heta
_\mu } + {\partialrtial _{{u_\mu }}}\left( {\frac{{1 + {\rho _\mu }}} {{1
+ {\mathbb{T}heta _\mu }}}{u_\mu }} \right)\nabla \cdot {u_\mu } +
{\partialrtial _{{\rho _\mu }}}\left( {\frac{{1 + {\rho _\mu }}}
{{1 + {\mathbb{T}heta _\mu }}}{u_\mu }} \right)\nabla {\rho _\mu },{{\left| {{\mathbb{T}heta _\mu }}
\right|}^2}} \right\rangle } \right) \\
&- \left\langle {\left( {1 + {\rho _e}} \right){u_e} \times B,{u_e}} \right\rangle
+\left\langle {\left( {1 + {\rho _i}} \right){u_i} \times B,{u_i}}
\right\rangle
\end{split}\end{equation}
\begin{equation}\notag\begin{split}
=& \sum\limits_{\mu = e,i} {\left( { - \frac{1}
{2}\left\langle {{\partialrtial _{{\mathbb{T}heta _\mu }}}\left( {\frac{{1 +
{\mathbb{T}heta _\mu }}} {{1 + {\rho _\mu }}}} \right)\nabla \cdot \left(
{{u_\mu }\left( {1 + {\mathbb{T}heta _\mu }} \right)} \right) + {\partialrtial
_{{\rho _\mu }}}\left( {\frac{{1 + {\mathbb{T}heta _\mu }}} {{1 + {\rho _\mu
}}}} \right)\nabla \cdot \left( {{u_\mu }\left( {1 + {\rho _\mu }}
\right)} \right),
{{\left| {{\rho _\mu }} \right|}^2}} \right\rangle } \right.} \\
& + \frac{1}
{2}\left\langle {{\partialrtial _{{\mathbb{T}heta _\mu }}}\left( {\frac{{1 +
{\mathbb{T}heta _\mu }}} {{1 + {\rho _\mu }}}{u_\mu }} \right)\nabla {\mathbb{T}heta
_\mu } + {\partialrtial _{{u_\mu }}}\left( {\frac{{1 + {\mathbb{T}heta _\mu }}}
{{1 + {\rho _\mu }}}{u_\mu }} \right)\nabla \cdot {u_\mu } +
{\partialrtial _{{\rho _\mu }}}\left( {\frac{{1 + {\mathbb{T}heta _\mu }}}
{{1 + {\rho _\mu }}}{u_\mu }} \right)\nabla {\rho _\mu },{{\left| {{\rho _\mu }}
\right|}^2}} \right\rangle \\
& - \frac{1}
{2}\left\langle {{\partialrtial _{{\mathbb{T}heta _\mu }}}\left( {\frac{{1 +
{\rho _\mu }}} {{1 + {\mathbb{T}heta _\mu }}}} \right)\nabla \cdot \left(
{{u_\mu }\left( {1 + {\mathbb{T}heta _\mu }} \right)} \right) + {\partialrtial
_{{\rho _\mu }}}\left( {\frac{{1 + {\rho _\mu }}}
{{1 + {\mathbb{T}heta _\mu }}}} \right)\nabla \cdot \left( {{u_\mu }\left( {1 + {\rho _\mu }}
\right)} \right),{{\left| {{\mathbb{T}heta _\mu }} \right|}^2}} \right\rangle \\
& + \frac{1}
{2}\left\langle {{\partialrtial _{{\mathbb{T}heta _\mu }}}\left( {\frac{{1 +
{\rho _\mu }}} {{1 + {\mathbb{T}heta _\mu }}}{u_\mu }} \right)\nabla {\mathbb{T}heta
_\mu } + {\partialrtial _{{u_\mu }}}\left( {\frac{{1 + {\rho _\mu }}} {{1
+ {\mathbb{T}heta _\mu }}}{u_\mu }} \right)\nabla \cdot {u_\mu } +
{\partialrtial _{{\rho _\mu }}}\left( {\frac{{1 + {\rho _\mu }}}
{{1 + {\mathbb{T}heta _\mu }}}{u_\mu }} \right)\nabla {\rho _\mu },{{\left| {{\mathbb{T}heta _\mu }} \right|}^2}} \right\rangle \\
& \left. { + \left\langle {\nabla {\rho _\mu }{ }{u_\mu },{ }{\mathbb{T}heta _\mu }}
\right\rangle + \left\langle {\nabla {\mathbb{T}heta _\mu }{u_\mu },{\rho _\mu }} \right\rangle } \right)
- \left\langle {\left( {1 + {\rho _e}} \right){u_e} \times B,{u_e}} \right\rangle
+ \left\langle {\left( {1 + {\rho _i}} \right){u_i} \times B,{u_i}} \right\rangle
\end{split}\end{equation}
\begin{equation}\notag\begin{split}
\leqslant & C\left\| {{\rho _\mu }} \right\|{\left\| {{\rho _\mu }} \right\|_{{L^\infty }}}\left
\{ {{{\left\| {{\partialrtial _{{\mathbb{T}heta _\mu }}}\left( {\frac{{1 + {\mathbb{T}heta _\mu }}}
{{1 + {\rho _\mu }}}} \right)} \right\|}_{{L^\infty }}}\left( {{{\left\| {1 + {\mathbb{T}heta _\mu }}
\right\|}_{{L^\infty }}}\left\| {\nabla \cdot {u_\mu }} \right\| + {{\left\| {\nabla {\mathbb{T}heta _\mu }}
\right\|}_{{L^\infty }}}\left\| {{u_\mu }} \right\|} \right)} \right. \\
& + {\left\| {{\partialrtial _{{\rho _\mu }}}\left( {\frac{{1 + {\mathbb{T}heta _\mu }}}
{{1 + {\rho _\mu }}}} \right)} \right\|_{{L^\infty }}}\left(
{{{\left\| {1 + {\rho _\mu }} \right\|}_{{L^\infty }}}\left\|
{\nabla \cdot {u_\mu }} \right\| + {{\left\| {\nabla {\rho _\mu }}
\right\|}_{{L^\infty }}}\left\| {{u_\mu }} \right\|} \right) +
{\left\| {{\partialrtial _{{\mathbb{T}heta _\mu }}}\left( {\frac{{1 + {\mathbb{T}heta
_\mu }}}
{{1 + {\rho _\mu }}}{u_\mu }} \right)} \right\|_{{L^\infty }}} \\
& \left. {\left\| {\nabla {\mathbb{T}heta _\mu }} \right\| + {{\left\| {{\partialrtial _{{u_\mu }}}\left(
{\frac{{1 + {\mathbb{T}heta _\mu }}}
{{1 + {\rho _\mu }}}{u_\mu }} \right)} \right\|}_{{L^\infty
}}}\left\| {\nabla \cdot {u_\mu }} \right\| + {{\left\| {{\partialrtial
_{{\rho _\mu }}}\left( {\frac{{1 + {\mathbb{T}heta _\mu }}}
{{1 + {\rho _\mu }}}{u_\mu }} \right)} \right\|}_{{L^\infty }}}\left\| {\nabla {\rho _\mu }} \right\|} \right\} \\
& + C\left\| {{\mathbb{T}heta _\mu }} \right\|{\left\| {{\mathbb{T}heta _\mu }} \right\|_{{L^\infty }}}\left\{ {{{\left\|
{{\partialrtial _{{\mathbb{T}heta _\mu }}}\left( {\frac{{1 + {\rho _\mu }}}
{{1 + {\mathbb{T}heta _\mu }}}} \right)} \right\|}_{{L^\infty }}}\left( {{{\left\| {1 + {\mathbb{T}heta _\mu }}
\right\|}_{{L^\infty }}}\left\| {\nabla \cdot {u_\mu }} \right\| + {{\left\| {\nabla {\mathbb{T}heta _\mu }}
\right\|}_{{L^\infty }}}\left\| {{u_\mu }} \right\|} \right)} \right. \\
& + {\left\| {{\partialrtial _{{\rho _\mu }}}\left( {\frac{{1 + {\rho _\mu }}}
{{1 + {\mathbb{T}heta _\mu }}}} \right)} \right\|_{{L^\infty }}}\left(
{{{\left\| {1 + {\rho _\mu }} \right\|}_{{L^\infty }}}\left\|
{\nabla \cdot {u_\mu }} \right\| + {{\left\| {\nabla {\rho _\mu }}
\right\|}_{{L^\infty }}}\left\| {{u_\mu }} \right\|} \right) +
{\left\| {{\partialrtial _{{\mathbb{T}heta _\mu }}}\left( {\frac{{1 + {\rho _\mu
}}}
{{1 + {\mathbb{T}heta _\mu }}}{u_\mu }} \right)} \right\|_{{L^\infty }}} \\
& \left. { \left\| {\nabla {\mathbb{T}heta _\mu }} \right\| + {{\left\| {{\partialrtial _{{u_\mu }}}\left(
{\frac{{1 + {\rho _\mu }}}
{{1 + {\mathbb{T}heta _\mu }}}{u_\mu }} \right)} \right\|}_{{L^\infty
}}}\left\| {\nabla \cdot {u_\mu }} \right\| + {{\left\| {{\partialrtial
_{{\rho _\mu }}}\left( {\frac{{1 + {\rho _\mu }}}
{{1 + {\mathbb{T}heta _\mu }}}{u_\mu }} \right)} \right\|}_{{L^\infty }}}\left\|
{\nabla {\rho _\mu }} \right\|} \right\} \\
& + C\left\| {\nabla {\rho _\mu }} \right\|\left\| {{u_\mu }} \right\|{\left\|
{{\mathbb{T}heta _\mu }} \right\|_{{L^\infty }}} + C\left\| {\nabla {\mathbb{T}heta _\mu }}
\right\|\left\| {{u_\mu }} \right\|{\left\| {{\rho _\mu }} \right\|_{{L^\infty }}}
+ C{\left\| {1 + {\rho _\mu }} \right\|_{{L^\infty }}}\left\| {{u_\mu }} \right\|\left\| B \right\|{
\left\| {{u_\mu }} \right\|_{{L^\infty }}}
\end{split}\end{equation}
\begin{equation}\notag\begin{split}
\leqslant & C\left( {\left\| {\nabla {u_\mu }} \right\| + \left\| {{u_\mu }} \right\| +
\left\| {\nabla {\mathbb{T}heta _\mu }} \right\| + \left\| {\nabla {\rho _\mu }} \right\|} \right)\left(
{\left\| {{\rho _\mu }} \right\|{{\left\| {\nabla {\rho _\mu }} \right\|}_1} + \left\| {{\mathbb{T}heta _\mu }}
\right\|{{\left\| {\nabla {\mathbb{T}heta _\mu }} \right\|}_1}} \right) \\
& + C\left\| {\nabla {\rho _\mu }} \right\|\left\| {{u_\mu }} \right\|\left\| {\nabla {\mathbb{T}heta _\mu }} \right\|
+ \left\| {\nabla {\mathbb{T}heta _\mu }} \right\|\left\| {{u_\mu }} \right\|{\left\| {\nabla {\rho _\mu }} \right\|_1}
+ C\left\| {{u_\mu }} \right\|\left\| B \right\|{\left\| {\nabla {u_\mu }} \right\|_1} \\
\leqslant & C\left\| {\left[ {{\rho _\mu },{u_\mu },{\mathbb{T}heta _\mu },B} \right]} \right\|\left( {\left\|
{\nabla {\rho _\mu }} \right\|_1^2 + \left\| {{u_\mu }} \right\|_2^2 + \left\| {\nabla {\mathbb{T}heta _\mu }}
\right\|_1^2} \right),
\end{split}\end{equation}
which will further be bounded by the right hand side term of
\eqref{2.16}, and where we have used \eqref{2.14}. When
$|\alpha|\geq 1$, similarly as before, one has
\[{I_{\alpha ,\beta }}(t) + {I_1}(t) \leqslant C{\left\| {\left[ {{\rho _\mu },{u_\mu },{\mathbb{T}heta _\mu },B}
\right]} \right\|_N}\left( {\left\| {\nabla {\rho _\mu }} \right\|_{N - 1}^2 + \left\|
{\left[ {{u_\mu },{\mathbb{T}heta _\mu }} \right]} \right\|_N^2} \right),\]
which will also be bounded by the right hand side term of
\eqref{2.16}.
Besides, for$|\alpha|\leq s$, standard energy estimates on
$\partialrtial^\alpha E$ and $\partialrtial^\alpha B$ from \eqref{2.2} yield
\begin{equation}\label{2.18}
\begin{split}
&\frac{1}
{2}\frac{d}
{{dt}}\left( {{{\left\| {{\partialrtial ^\alpha }E} \right\|}^2} + {{\left\| {{\partialrtial ^\alpha }B}
\right\|}^2}} \right) - \left\langle {\left( {1 + {\rho _e}} \right){\partialrtial ^\alpha }{u_e} -
\left( {1 + {\rho _i}} \right){\partialrtial ^\alpha }{u_i},{\partialrtial ^\alpha }E} \right\rangle \\
& = \left\langle {{\partialrtial ^{\alpha - \beta }}{\rho _e}{\partialrtial ^\alpha }{u_e} - {\partialrtial ^{\alpha
- \beta }}{\rho _i}{\partialrtial ^\alpha }{u_i},{\partialrtial ^\alpha }E} \right\rangle \\
& \leqslant
C{\left\| E \right\|_s}\left( {\left\| {{u_\mu }} \right\|_s^2 + \left\| {\nabla {\rho _\mu }}
\right\|_{s - 1}^2} \right),
\end{split}\end{equation}
which will be bounded by the right hand side term of \eqref{2.16}.
Then, with the help of \eqref{2.14}, the summation \eqref{2.17} and
\eqref{2.18} over $|\alpha|\leq s$, one has \eqref{2.16}.
\noindent \emph{Step 2.} It holds that
\begin{equation}\label{2.19}
\begin{split}
\frac{d}
{{dt}} & \sum\limits_{\left| \alpha \right| \leqslant s - 1}
{\sum\limits_{\mu = e,i} {\left\langle {{\partialrtial ^\alpha }{u_\mu
},\nabla {\partialrtial ^\alpha }{\rho _\mu }} \right\rangle } }
+ \gamma \left( {\left\| {\nabla \left[ {{\rho _e},{\rho _i}} \right]} \right\|_{s- 1}^2
+ {{\left\| {{\rho _e} - {\rho _i}} \right\|}^2}} \right) \\
& \leqslant C {\left( {\left\| {{u_\mu }} \right\|_s^2
+ \left\| {\left[ {{\rho _\mu },{u_\mu },{\mathbb{T}heta _\mu },B} \right]} \right\|_s^2\left( {\left\|
{\nabla {\rho _\mu }} \right\|_{s - 1}^2 + \left\| {\left[ {{u_\mu },{\mathbb{T}heta _\mu }} \right]} \right\|_s^2}
\right)} \right)} .
\end{split}\end{equation}
In fact, we can rewrite the equations \eqref{2.2} as
\begin{equation}
\label{2.20}
\left\{\begin{aligned}
&\partialrtial_t {\rho_e} +\nabla\cdot u_e =g_{1e}, \\
&\partialrtial_t u_e +\nabla\rho_e+\nabla\mathbb{T}heta_e+ u_e+E=g_{2e}, \\
&\partialrtial_t {\mathbb{T}heta_e} +\nabla\cdot u_e+ \mathbb{T}heta_e=g_{3e},
\\
&\partialrtial_t {\rho_i} +\nabla\cdot u_i =g_{1i}, \\
&\partialrtial_t u_i +\nabla\rho_i+\nabla\mathbb{T}heta_i+u_i-E=g_{2i}
,\\
&\partialrtial_t {\mathbb{T}heta_i} +\nabla\cdot u_i+ \mathbb{T}heta_i=g_{3i},\\
& \partialrtial_t E-\nabla\times B -u_e+ u_i= g_{4e}-g_{4i} ,
\\
& \partialrtial_t B+\nabla\times E=0,\\
& \nabla\cdot E=\rho_i-\rho_e ,\quad \nabla\cdot B=0,\quad
(t,x)\in(0,\infty)\times\mathbb{R}^3,
\end{aligned} \right.
\end{equation}
where
\begin{equation}
\label{2.21}
\left\{\begin{aligned}
&g_{1e}=-\rho_e \nabla\cdot u_e -u_e \nabla\rho_e, \\
&g_{2e}= -(u_e\cdot\nabla) u_e
-(\frac{\mathbb{T}heta_e+1}{1+\rho_e}-1)\nabla \rho_e- u_e \times B
, \\
&g_{3e}=-\mathbb{T}heta_e\nabla\cdot u_e-u_e\nabla\mathbb{T}heta_e,
\\
&g_{4e}=\rho_e
u_e, \\
&g_{1i}=-\rho_i \nabla\cdot u_i -u_i \nabla\rho_i, \\
&g_{2i}= -(u_i\cdot\nabla) u_i
-(\frac{\mathbb{T}heta_i+1}{1+\rho_i}-1)\nabla \rho_i+ u_i \times B
,\\
&g_{3i}=-\mathbb{T}heta_i\nabla\cdot u_i-u_i\nabla\mathbb{T}heta_i,\\
&g_{4i}= \rho_i u_i.
\end{aligned} \right.
\end{equation}
Let $\left| \alpha \right| \leqslant s - 1$. Utilizing
$\partialrtial^\alpha$ to the second equation of \eqref{2.20},
multiplying it by $\nabla\partialrtial^\alpha\rho_e$, integrating over
$\mathbb{R}^3$ and using the last equation in \eqref{2.2}, replacing
$\partialrtial_t {\rho_e}$ from the first equation of \eqref{2.20}
implies
\[\begin{gathered}
\frac{d}
{{dt}}\left\langle {{\partialrtial ^\alpha }{u_e},\nabla {\partialrtial ^\alpha }{\rho _e}} \right\rangle
+ {\left\| {\nabla {\partialrtial ^\alpha }{\rho _e}} \right\|^2} + {\left\| {{\partialrtial ^\alpha }{\rho _e}}
\right\|^2} - \left\langle {{\partialrtial ^\alpha }{\rho _i},{\partialrtial ^\alpha }{\rho _e}} \right\rangle
+ \left\langle {\nabla {\partialrtial ^\alpha }{\mathbb{T}heta _e},\nabla {\partialrtial ^\alpha }{\rho _e}} \right\rangle
\\
= {\left\| {{\partialrtial ^\alpha }\nabla \cdot {u_e}} \right\|^2} + \left\langle {{\partialrtial ^\alpha }
\nabla {\rho _e},{\partialrtial ^\alpha }{g_{2e}}} \right\rangle - \left\langle {{\partialrtial ^\alpha }{u_e},
\nabla {\partialrtial ^\alpha }{\rho _e}} \right\rangle - \left\langle {{\partialrtial ^\alpha }\nabla \cdot {u_e},
{\partialrtial ^\alpha }{g_{1e}}} \right\rangle.
\\
\end{gathered} \]
Similarly as before, from the fourth and fifth equations of
\eqref{2.20}, we have
\[\begin{gathered}
\frac{d}
{{dt}}\left\langle {{\partialrtial ^\alpha }{u_i},\nabla {\partialrtial ^\alpha }{\rho _i}} \right\rangle
+ {\left\| {\nabla {\partialrtial ^\alpha }{\rho _i}} \right\|^2} + {\left\| {{\partialrtial ^\alpha }{\rho _i}}
\right\|^2} - \left\langle {{\partialrtial ^\alpha }{\rho _i},{\partialrtial ^\alpha }{\rho _e}} \right\rangle
+ \left\langle {\nabla {\partialrtial ^\alpha }{\mathbb{T}heta _i},\nabla {\partialrtial ^\alpha }{\rho _i}} \right\rangle
\\
= {\left\| {{\partialrtial ^\alpha }\nabla \cdot {u_i}} \right\|^2} + \left\langle {{\partialrtial ^\alpha }\nabla
{\rho _i},{\partialrtial ^\alpha }{g_{2i}}} \right\rangle - \left\langle {{\partialrtial ^\alpha }{u_i},\nabla
{\partialrtial ^\alpha }{\rho _i}} \right\rangle - \left\langle {{\partialrtial ^\alpha }\nabla \cdot {u_i},
{\partialrtial ^\alpha }{g_{1i}}} \right\rangle.
\\
\end{gathered} \]
Furthermore, the summation of the two equations above gives
\begin{equation}\notag
\begin{split}
\frac{d}
{{dt}}&\left( {\left\langle {{\partialrtial ^\alpha }{u_e}, \nabla
{\partialrtial ^\alpha }{\rho _e}} \right\rangle + \left\langle
{{\partialrtial ^\alpha }{u_i},\nabla {\partialrtial ^\alpha }{\rho _i}}
\right\rangle } \right) + {\left\| {\nabla {\partialrtial ^\alpha }{\rho _e}} \right\|^2}
+ {\left\| {\nabla {\partialrtial ^\alpha }{\rho _i}} \right\|^2} + {\left\| {{\partialrtial ^\alpha }
\left( {{\rho _e} - {\rho _i}} \right)} \right\|^2} \\
&= {\left\| {{\partialrtial ^\alpha }\nabla \cdot {u_e}} \right\|^2} + {\left\| {{\partialrtial ^\alpha }
\nabla \cdot {u_i}} \right\|^2} - \left\langle {\nabla {\partialrtial ^\alpha }{\mathbb{T}heta _i},
\nabla {\partialrtial ^\alpha }{\rho _i}} \right\rangle - \left\langle {\nabla {\partialrtial ^\alpha }
{\mathbb{T}heta _e},\nabla {\partialrtial ^\alpha }{\rho _e}} \right\rangle \\
&\quad + \left\langle {{\partialrtial ^\alpha }
\nabla {\rho _e},{\partialrtial ^\alpha }{g_{2e}}} \right\rangle - \left\langle {{\partialrtial ^\alpha }{u_e},
\nabla {\partialrtial ^\alpha }{\rho _e}} \right\rangle
- \left\langle {{\partialrtial ^\alpha }\nabla \cdot {u_e},{\partialrtial ^\alpha }{g_{1e}}}
\right\rangle\\
&\quad+ \left\langle {{\partialrtial ^\alpha }\nabla {\rho _i},{\partialrtial ^\alpha }{g_{2i}}} \right\rangle
- \left\langle {{\partialrtial ^\alpha }{u_i},\nabla {\partialrtial ^\alpha }{\rho _i}} \right\rangle
- \left\langle {{\partialrtial ^\alpha }\nabla \cdot {u_i},{\partialrtial ^\alpha }{g_{1i}}}
\right\rangle.
\end{split}\end{equation}
Therefore, after using Cauchy-Schwarz inequality, one has
\begin{equation}\label{2.22}
\begin{split}
\frac{d}
{{dt}}&\left( {\left\langle {{\partialrtial ^\alpha }{u_e},\nabla
{\partialrtial ^\alpha } {\rho _e}} \right\rangle + \left\langle
{{\partialrtial ^\alpha }{u_i},\nabla {\partialrtial ^\alpha } {\rho _i}}
\right\rangle } \right) + \lambda \left( {{{\left\| {\nabla
{\partialrtial ^\alpha }{\rho _e}} \right\|}^2} + {{\left\| {\nabla
{\partialrtial ^\alpha }{\rho _i}} \right\|}^2} + {{\left\|
{{\partialrtial ^\alpha }\left( {{\rho _e} - {\rho _i}} \right)} \right\|}^2}} \right) \\
&\leqslant C {\left( {{{\left\| {{\partialrtial ^\alpha }\nabla
\cdot {u_\mu }} \right\|}^2} + {{\left\| {{\partialrtial ^\alpha }{u_\mu }} \right\|}^2}
+ {{\left\| {{\partialrtial ^\alpha }\nabla {\mathbb{T}heta _\mu }} \right\|}^2} + {{\left\|
{{\partialrtial ^\alpha }{g_{1\mu }}} \right\|}^2} + {{\left\| {{\partialrtial ^\alpha }{g_{2\mu }}}
\right\|}^2}} \right)}.
\end{split}\end{equation}
From the definition of $g_{j\mu}$, $(j=1,2)$, one can check that
\begin{equation}\notag
{\left\| {{\partialrtial ^\alpha }{g_{1\mu }}} \right\|^2} + {\left\|
{{\partialrtial ^\alpha }{g_{2\mu }}} \right\|^2} \leqslant C\left\|
{\left[ {{\rho _\mu },{u_\mu },{\mathbb{T}heta _\mu },B} \right]}
\right\|_s^2\left( {\left\| {\nabla {\rho _\mu }} \right\|_{s - 1}^2
+ \left\| {{u_\mu }} \right\|_s^2 + \left\| {{\mathbb{T}heta _\mu }}
\right\|_s^2} \right),
\end{equation}
Putting this into \eqref{2.22}, then, \eqref{2.19} follows by taking
summation over $\left| \alpha \right| \leqslant s - 1$.
\noindent \emph{Step 3.} It holds that
\begin{equation}\label{2.23}
\begin{split}
\frac{d}
{{dt}}\sum\limits_{\left| \alpha \right| \leqslant s - 1}
{\left\langle {{\partialrtial ^\alpha }\left( {{u_e}
- {u_i}} \right),{\partialrtial ^\alpha }E} \right\rangle } + \gamma \left\| E \right\|_{s - 1}^2
\leqslant& C \left\| {\left[ {{u_\mu },{\mathbb{T}heta _\mu }} \right]}
\right\|_s^2 + C \|\nabla \rho_\mu \|_{s-1}^2+ C \left\| {{u_\mu }}
\right\|_s \\
\cdot{{\left\| {\nabla B} \right\|}_{s - 2}}
&+ C \left\| U \right\|_s^2\left( {\left\|
{\nabla {\rho _\mu }} \right\|_{s - 1}^2 + \left\| {\left[ {{u_\mu },{\mathbb{T}heta _\mu }} \right]} \right\|_s^2}
\right) .
\end{split}
\end{equation}
In fact, for $|\alpha|\leq s-1$, from the second and fifth equation
of \eqref{2.20}, one has
\begin{equation}\label{2.24}
\begin{split}{\partialrtial _t}\left( {{u_e} - {u_i}} \right) + \nabla \left( {{\rho
_e} - {\rho _i}} \right) + \nabla \left( {{\mathbb{T}heta _e} - {\mathbb{T}heta _i}}
\right) + 2E = {g_{2e}} - {g_{2i}} - \left( {{u_e} - {u_i}} \right).
\end{split}
\end{equation}
Utilizing $\partialrtial^\alpha$ to \eqref{2.24}, multiplying it by
$\partialrtial^\alpha E$, integrating over $\mathbb{R}^3$ and replacing
$\partialrtial_t E$ from the seventh equation of \eqref{2.2} implies
\begin{equation}\notag
\begin{split}
\frac{d}
{{dt}}&\left\langle {{\partialrtial ^\alpha }\left( {{u_e} - {u_i}}
\right),{\partialrtial ^\alpha }E} \right\rangle + {\left\| {{\partialrtial
^\alpha }\left( {{\rho _e} - {\rho _i}} \right)} \right\|^2} +
2{\left\| {{\partialrtial ^\alpha }E}
\right\|^2} \\
&= - \left\langle {{\partialrtial ^\alpha }\left( {{\mathbb{T}heta _e} - {\mathbb{T}heta _i}} \right),{\partialrtial ^\alpha }\left( {{\rho _e}
- {\rho _i}} \right)} \right\rangle + \left\langle {{\partialrtial ^\alpha }\left( {{u_e} - {u_i}} \right),
{\partialrtial ^\alpha }E} \right\rangle + \left\langle {{\partialrtial ^\alpha }\left( {{u_e} - {u_i}} \right),
\nabla \times {\partialrtial ^\alpha }B} \right\rangle \\
&\quad+ {\left\| {{\partialrtial ^\alpha }\left( {{u_e} - {u_i}} \right)} \right\|^2} + \left\langle {{\partialrtial ^\alpha }
\left( {{u_e} - {u_i}} \right),{\partialrtial ^\alpha }\left( {{\rho _e}{u_e} - {\rho _i}{u_i}} \right)} \right\rangle
+ \left\langle {{\partialrtial ^\alpha }\left( {{g_{2e}} - {g_{2i}}} \right),{\partialrtial ^\alpha }E}
\right\rangle,
\end{split}
\end{equation}
Therefore, after using Cauchy-Schwarz inequality, one has
\begin{equation}\notag
\begin{split}
\frac{d}
{{dt}} & \left\langle {{\partialrtial ^\alpha }\left( {{u_e} - {u_i}}
\right),{\partialrtial ^\alpha }E}
\right\rangle + \gamma {\left\| {{\partialrtial ^\alpha }E} \right\|^2} \\
\leq & C\left( {{{\left\| {{\partialrtial ^\alpha } {{u_\mu} } } \right\|}^2}
+ {{\left\| {{\partialrtial ^\alpha }{{\mathbb{T}heta _\mu} }} \right\|}^2}}+
{{\left\| {{\partialrtial ^\alpha }{{\nabla\rho _\mu} }} \right\|}^2}
\right) + C{\left\| {\left[ {{u_e},{u_i}} \right]} \right\|_s}{\left\| {\nabla B} \right\|_{s - 2}} \\
& + C\left\| {\left[ {{\rho _\mu },{u_\mu },{\mathbb{T}heta _\mu },B} \right]} \right\|_s^2\left( {\left\|
{\nabla {\rho _\mu }} \right\|_{s - 1}^2 + \left\| {\left[ {{u_\mu },{\mathbb{T}heta _\mu }} \right]}
\right\|_s^2} \right).
\end{split}
\end{equation}
Thus, with help of the summation of the previous estimate over
$|\alpha|\leq s-1$, one can obtain \eqref{2.23}.
\noindent \emph{Step 4.} It holds that
\begin{equation}\label{2.25}
\begin{split}
\frac{d} {{dt}}\sum\limits_{\left| \alpha \right| \leqslant s - 2}
{\left\langle {{\partialrtial ^\alpha }E, - \nabla \times {\partialrtial
^\alpha }B} \right\rangle + \gamma \left\| {\nabla B} \right\|_{s -
2}^2}
\leqslant C {(\left\| {\left[
{{u_\mu },E} \right]} \right\|_{s - 1}^2 + } \left\| {\nabla {\rho
_\mu }} \right\|_{s - 1}^2\left\| {{u_\mu }} \right\|_s^2 ).
\end{split}\end{equation}
In fact, for $|\alpha|\leq s - 2$, applying $\partialrtial^\alpha$ to the
seventh equation of \eqref{2.2}, multiplying it by
$-\partialrtial^\alpha\nabla\times B $, integrating over $\mathbb{R}^3$
and then utilizing the eighth equation of \eqref{2.2} gives
\begin{equation}\notag
\begin{split}
\frac{d}
{{dt}}&\sum\limits_{\left| \alpha \right| \leqslant s - 2}
{\left\langle {{\partialrtial ^\alpha }E, - \nabla
\times {\partialrtial ^\alpha }B} \right\rangle + {{\left\| {\nabla \times {\partialrtial ^\alpha }B} \right\|}^2}} \\
&= {\left\| {\nabla \times {\partialrtial ^\alpha }E} \right\|^2} - \left\langle {{\partialrtial ^\alpha }\left( {{u_e}
- {u_i}} \right),\nabla \times {\partialrtial ^\alpha }B} \right\rangle + \left\langle {{\partialrtial ^\alpha }
\left( {{\rho _e}{u_e} - {\rho _i}{u_i}} \right), - \nabla \times {\partialrtial ^\alpha }B} \right\rangle
\end{split}
\end{equation}
Furthermore, with the help of Cauchy-Schwarz inequality and the
summation over $|\alpha|\leq s - 2$, we yield \eqref{2.25}. Where
we have used
$$\left\|{\partialrtial^\alpha\partialrtial_i B}\right\|= \left\|{\partialrtial_i\triangle^{-1}\nabla\times(\nabla\times
\partialrtial^\alpha B)}\right\|
\leq C\left\|{\nabla\times\partialrtial^\alpha B}\right\|$$ for $1\leq
i\leq 3$, due to $\nabla\cdot B=0$ and the fact that
$\partialrtial_i\triangle^{-1}\nabla$ is bounded from $L^p$ to $L^p$ with
$1<p<\infty$, see \cite{Stein}.\\
\emph{Step 5.} Now, based on the four previous steps, we will search
\eqref{2.15}. We define the energy functional as
\begin{equation}\notag
\begin{split}
\mathcal{E}_s(U(t))=& \left\| U \right\|_s^2 +\mathcal {K}_1 \sum\limits_{\left| \alpha \right| \leqslant s - 1}
{\sum\limits_{\mu = e,i}
{\left\langle {{\partialrtial ^\alpha }{u_\mu },\nabla {\partialrtial ^\alpha }{\rho _\mu }} \right\rangle } } \\
&
+\mathcal {K}_2 \sum\limits_{\left| \alpha \right| \leqslant s - 1}
{\left\langle {{\partialrtial ^\alpha }\left( {{u_e}
- {u_i}} \right),{\partialrtial ^\alpha }E} \right\rangle }
+\mathcal {K}_3 \sum\limits_{\left| \alpha \right| \leqslant s - 2}
{\left\langle {{\partialrtial ^\alpha }E, - \nabla \times {\partialrtial ^\alpha }B} \right\rangle } ,
\end{split}
\end{equation}
for constants $0<\mathcal {K}_3\ll\mathcal {K}_2\ll\mathcal {K}_1\ll
1$ to be chosen later. Notice that as soon as $0<\mathcal {K}_j
\ll1$ $_{(1\leq j \leq 3)}$ is sufficiently small, then $\mathcal
{E}_s(U(t))\sim ||U||^2_s$ holds true. Furthermore, the summation of
\eqref{2.16}, \eqref{2.19}$\times\mathcal {K}_1$,
\eqref{2.23}$\times\mathcal {K}_2$ and \eqref{2.25}$\times\mathcal
{K}_3$ implies that there is $0<\gamma<1$ such that
\begin{equation}\notag
\begin{split}
\frac{d}{dt}&\mathcal
{E}_s(U(t))+\|[u_e,u_i,\mathbb{T}heta_e,\mathbb{T}heta_i]\|_s^2 + \gamma\mathcal
{K}_1 (\|\nabla[\rho_e,\rho_i]\|_{s-1}^2+\|\rho_e-\rho_i\|^2)\\
&\quad+ \gamma\mathcal {K}_2\|E\|_{s-1}^2+ \gamma\mathcal {K}_3
\left\| {\nabla B}
\right\|_{s - 2}^2 \\
&\leq
C[\mathcal {E}_s(U(t))^\frac{1}{2}+\mathcal {E}_s(U(t))]\mathcal
{D}_s(U(t))+C \mathcal {K}_1 \|u_\mu\|_s^2+C\mathcal
{K}_2\left(\|[u_\mu,\mathbb{T}heta_\mu]\|_s^2+\|\nabla\rho_\mu\|_{s-1}
^2\right)\\
&\quad +C
\mathcal {K}_2 \|u_\mu\|_s\|\nabla B\|_{s-2} +C \mathcal {K}_3 \left\| {[u_\mu,E]} \right\|_{s - 1}^2\\
&\leq
C[\mathcal {E}_s(U(t))^\frac{1}{2}+\mathcal {E}_s(U(t))]\mathcal
{D}_s(U(t))+C \mathcal {K}_1 \|u_\mu\|_s^2+C\mathcal
{K}_2\left(\|[u_\mu,\mathbb{T}heta_\mu]\|_s^2+\|\nabla\rho_\mu\|_{s-1}
^2\right)\\
&\quad + \frac{1}{2}C\left(\mathcal {K}_2^\frac{1}{2}\left\|{u_\mu
}\right\|_s^2+\mathcal {K}_2^\frac{3}{2}\left\|{\nabla
B}\right\|_{s-2}^2\right) +C \mathcal {K}_3 \left\| {[u_\mu,E]}
\right\|_{s - 1}^2.
\end{split}
\end{equation}
By letting
$0<\mathcal {K}_3\ll\mathcal {K}_2\ll\mathcal {K}_1\ll 1$ be
sufficiently small with $\mathcal {K}_2^{\frac{3}{2}}\ll\mathcal
{K}_3$, we obtain \eqref{2.15}. Now, we complete the proof of the
Theorem \ref{thm2.1}.
$\Box$
\section{Linearized homogeneous equations}
In this section, for searching the time-decay property of solutions
to the nonlinear equations \eqref{2.2} in the last section, we have
to consider the decay properties of the linearized equations
\eqref{2.20}. Let us introduce the transformation
\begin{equation}\label{3.1}
\rho_1=\frac{\rho_e - \rho_i}{2},~u_1=\frac{u_e -
u_i}{2},~\mathbb{T}heta_1=\frac{\mathbb{T}heta_e - \mathbb{T}heta_i}{2}.
\end{equation}
Then, from system \eqref{2.2}, $U_1=[\rho_1,~u_1,~\mathbb{T}heta_1,~E,~B]$
satisfies
\begin{equation}\label{3.2}
\left\{\begin{split}
&{\partialrtial _t}{\rho _1} + \nabla \cdot {u_1} = \frac{1}
{2}\left( {{g_{1e}} - {g_{1i}}} \right), \\
& {\partialrtial _t}{u_1} + \nabla {\rho _1} + \nabla {\mathbb{T}heta _1} + E + {u_1} = \frac{1}
{2}\left( {{g_{2e}} - {g_{2i}}} \right), \\
&{\partialrtial _t}{\mathbb{T}heta _1} + \nabla \cdot {u_1} + {\mathbb{T}heta _1} = \frac{1}
{2}\left( {{g_{3e}} - {g_{3i}}} \right), \\
& {\partialrtial _t}E - \nabla \times B - 2{u_1} = {g_{4e}} - {g_{4i}}, \\
& {\partialrtial _t}B + \nabla \times E = 0, \\
& \frac{1}
{2}\nabla \cdot E = - {\rho _1},\nabla \cdot B = 0,\quad
(t,x)\in(0,\infty)\times\mathbb{R}^3,
\end{split}\right. \end{equation}
with initial value ${U_1}{|_{t = 0}} = {U_{1,0}}: = \left[ {{\rho
_{1,0}},{u_{1,0}},{\mathbb{T}heta _{1,0}},{E_0},{B_0}} \right],x \in
{\mathbb{R}^3}$ which satisfies the compatibility conditions
$\displaystyle\frac{1} {2}\nabla \cdot E_0 = - {\rho
_{1,0}},\nabla \cdot B_0 = 0.$ Where, $\left[ {{\rho
_{1,0}},{u_{1,0}},{\mathbb{T}heta _{1,0}}} \right]$ is given from $\left[
{{\rho _{\mu0}},{u_{\mu0}},{\mathbb{T}heta _{\mu0}}} \right]$ from the
transformation \eqref{3.1}.
Moreover, we introduce another transformation
\begin{equation}\label{3.3}
\rho_2=\frac{\rho_e + \rho_i}{2},~u_2=\frac{u_e +
u_i}{2},~\mathbb{T}heta_2=\frac{\mathbb{T}heta_e + \mathbb{T}heta_i}{2}.
\end{equation}
Then $U_2=[\rho_2,~u_2,~\mathbb{T}heta_2]$ satisfies
\begin{equation}\label{3.4}
\left\{\begin{split}
& {\partialrtial _t}{\rho _2} + \nabla \cdot {u_2} = \frac{1}
{2}\left( {{g_{1e}} + {g_{1i}}} \right), \\
& {\partialrtial _t}{u_2} + \nabla {\rho _2} + \nabla {\mathbb{T}heta _2} + {u_2} = \frac{1}
{2}\left( {{g_{2e}} + {g_{2i}}} \right),\\
& {\partialrtial _t}{\mathbb{T}heta _2} + \nabla \cdot {u_2} + {\mathbb{T}heta _2} =
\frac{1} {2}\left( {{g_{3e}} + {g_{3i}}} \right),\quad
(t,x)\in(0,\infty)\times\mathbb{R}^3,
\end{split}\right. \end{equation}
with initial value ${U_2}{|_{t = 0}} = {U_{2,0}}: = \left[ {{\rho
_{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}} \right],x \in {\mathbb{R}^3}$,
where $\left[ {{\rho _{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}} \right]$ is
from the transformation \eqref{3.3}.
Therefore, one can define the solution
$U_1=[\rho_1,~u_1,~\mathbb{T}heta_1,~E,~B]$ and
$U_2=[\rho_2,~u_2,~\mathbb{T}heta_2]$, respectively, as follows
\begin{equation}\label{3.5}
{U_1}(t) = {e^{tL_1}}{U_{1,0}} + \frac{1}
{2}\int_0^t {{e^{\left( {t - y} \right)L_1}}\left[ {{g_{1e}} -
{g_{1i}},{g_{2e}} - {g_{2i}},{g_{3e}}
- {g_{3i}},2\left( {{g_{4e}} - {g_{4i}}} \right)} \right](y)} dy,
\end{equation}
and
\begin{equation}\label{3.6}
{U_2}(t) = {e^{tL_2}}{U_{2,0}} + \frac{1}
{2}\int_0^t {{e^{\left( {t - y} \right)L_2}}\left[ {{g_{1e}} +
{g_{1i}},{g_{2e}} + {g_{2i}},{g_{3e}}
+ {g_{3i}}} \right](y)} dy,
\end{equation}
where $e^{tL_1}U_{1,0}$ and $e^{tL_2}U_{2,0}$, respectively, denote
the solution of the following hohomogeneous initial value problems
\eqref{3.7}-\eqref{3.8} and \eqref{3.10}-\eqref{3.11}, which will be
given as follows:
The linearized homogeneous equations corresponding to \eqref{3.2} is
\begin{equation}\label{3.7}
\left\{\begin{split}
&{\partialrtial _t}{\rho _1} + \nabla \cdot {u_1} = 0, \\
& {\partialrtial _t}{u_1} + \nabla {\rho _1} + \nabla {\mathbb{T}heta _1} + E + {u_1} = 0, \\
&{\partialrtial _t}{\mathbb{T}heta _1} + \nabla \cdot {u_1} + {\mathbb{T}heta _1} = 0, \\
& {\partialrtial _t}E - \nabla \times B - 2{u_1} = 0, \\
& {\partialrtial _t}B + \nabla \times E = 0, \\
& \frac{1}
{2}\nabla \cdot E = - {\rho _1},\nabla \cdot B = 0,\quad
(t,x)\in(0,\infty)\times\mathbb{R}^3,
\end{split}\right. \end{equation}
with initial value
\begin{equation}\label{3.8}
{U_1}{|_{t = 0}} = {U_{1,0}}: = \left[ {{\rho
_{1,0}},{u_{1,0}},{\mathbb{T}heta _{1,0}},{E_0},{B_0}} \right],x \in
{\mathbb{R}^3}\end{equation}
which satisfies the compatible
conditions
\begin{equation}\label{3.9}
\displaystyle\frac{1} {2}\nabla \cdot E_0 = - {\rho _{1,0}},\nabla
\cdot B_0 = 0.
\end{equation}
And the linearized homogeneous equations corresponding to
\eqref{3.7} is
\begin{equation}\label{3.10}
\left\{\begin{split}
& {\partialrtial _t}{\rho _2} + \nabla \cdot {u_2} = 0, \\
& {\partialrtial _t}{u_2} + \nabla {\rho _2} + \nabla {\mathbb{T}heta _2} + {u_2} = 0,\\
& {\partialrtial _t}{\mathbb{T}heta _2} + \nabla \cdot {u_2} + {\mathbb{T}heta _2}
=0,\quad (t,x)\in(0,\infty)\times\mathbb{R}^3,
\end{split}\right. \end{equation}
with initial value
\begin{equation}\label{3.11}
{U_2}{|_{t = 0}} = {U_{2,0}}: = \left[ {{\rho
_{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}} \right],x \in
{\mathbb{R}^3}.\end{equation}
Here $\left[ {{\rho _{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}} \right]$ is
from the transform \eqref{3.6}.
In the sequel, we usually denote $U_1$ $=[\rho_1$, $u_1$,
$\mathbb{T}heta_1$, $E$, $B]$ as the solution of the linearized homogeneous
equations \eqref{3.7}, and $U_2$ $=[\rho_2$, $u_2$, $\mathbb{T}heta_2]$ as
the one of \eqref{3.10}.
Firstly, for the linearized homogeneous system
\eqref{3.7}-\eqref{3.8}, similarly as \cite{FWK11}, we obtain the
$L^p-L^q$ decay property as follows
\begin{prop}\label{prop3.1}
Assume $U_1(t)=e^{tL_1}U_{1,0}$ is the solution to the initial value
problem \eqref{3.7}-\eqref{3.8} with ${U_{1,0}}$ $= [ {\rho
_{1,0}}$, ${u_{1,0}}$, ${\mathbb{T}heta _{1,0}}$, ${E_0}$, ${B_0} ]$ which
satisfies \eqref{3.8}. Then, for any $t\geq0$, $U_1$ $=[\rho_1$,
$u_1$, $\mathbb{T}heta_1$, $E$, $B]$ satisfies
\begin{equation}\label{3.12}
\left\{
\begin{split}
&\left\| [{\rho_{1}(t),~ \mathbb{T}heta_{1} \left( t \right)}] \right\| \leqslant C{e^{ - \frac{t}
{2}}}\left\| {\left[ {{\rho _{1,0}},{u_{1,0}},{\mathbb{T}heta _{1,0}}} \right]} \right\|, \\
& \left\| {u_1\left( t \right)} \right\| \leqslant C{e^{ - \frac{t}
{2}}}\left\| {\left[ {{\rho _{1,0}},{\mathbb{T}heta _{1,0}}} \right]}
\right\| + C{\left( {1 + t} \right)^{ - \frac{5}
{4}}}{\left\| {\left[ {{u_{1,0}},{E_0},{B_0}} \right]} \right\|_{{L^1} \cap {{\dot H}^2}}}, \\
& \left\| {E\left( t \right)} \right\| \leqslant C{\left( {1 + t} \right)^{ - \frac{5}
{4}}}{\left\| {\left[ {{u_{1,0}},{\mathbb{T}heta_{1,0}},{E_0},{B_0}} \right]} \right\|_{{L^1} \cap {{\dot H}^3}}}, \\
& \left\| {B\left( t \right)} \right\| \leqslant C{\left( {1 + t} \right)^{ - \frac{3}
{4}}}{\left\| {\left[ {{u_{1,0}},{E_0},{B_0}} \right]}
\right\|_{{L^1} \cap {{\dot H}^2}}},
\end{split}\right.
\end{equation}
\begin{equation}\label{3.13}
\left\{\begin{split}& {\left\| [{\rho_{1}(t),~\mathbb{T}heta_{1} \left( t
\right)}] \right\|_{{L^\infty }}} \leqslant C{e^{ - \frac{t}
{2}}}{\left\| {\left[ {{\rho _{1,0}},{u_{1,0}},{\mathbb{T}heta _{1,0}}}
\right]} \right\|_{{L^2} \cap
{{\dot H}^2}}},\\
& {\left\| {u_1 \left( t \right)}
\right\|_{{L^\infty }}} \leqslant C{e^{ - \frac{t} {2}}}{\left\|
{\left[ {{\rho _{1,0}},{\mathbb{T}heta _{1,0}}} \right]} \right\|_{{L^1}
\cap {{\dot H}^2}}} + C{\left( {1 + t} \right)^{ - 2}}{\left\|
{\left[ {{u_{1,0}},{E_0},{B_0}} \right]} \right\|_{{L^1}
\cap {{\dot H}^5}}}, \\
& {\left\| {E\left( t \right)} \right\|_{{L^\infty }}} \leqslant
C{\left( {1 + t} \right)^{ - 2}}
{\left\| {\left[ {{u_{1,0}},{\mathbb{T}heta_{1,0}},{E_0},{B_0}} \right]} \right\|_{{L^1} \cap {{\dot H}^6}}}, \\
& {\left\| {B\left( t \right)} \right\|_{{L^\infty }}} \leqslant
C{\left( {1 + t} \right)^{ - \frac{3} {2}}}{\left\| {\left[
{{u_{1,0}},{E_0},{B_0}} \right]} \right\|_{{L^1} \cap {{\dot
H}^5}}},
\end{split}\right.
\end{equation}
and
\begin{equation}\label{3.14}
\left\{\begin{split}
& \left\| {\nabla B\left( t \right)} \right\| \leqslant C{\left( {1 + t} \right)^{ - \frac{5}
{4}}}{\left\| {\left[ {{u_{1,0}},{E_0},{B_0}} \right]} \right\|_{{L^1} \cap {{\dot H}^4}}}, \\
& \left\| {{\nabla ^s}\left[ {E\left( t \right),B\left( t \right)} \right]} \right\| \leqslant
C{\left( {1 + t} \right)^{ - \frac{5}
{4}}}{\left\| {\left[ {{u_{1,0}},\mathbb{T}heta_{1,0},{E_0},{B_0}} \right]}
\right\|_{{L^2} \cap {{\dot H}^{s + 3}}}}.
\end{split}\right.
\end{equation}
\end{prop}
\subsection{Explicit solutions of \eqref{3.10}-\eqref{3.11}}
Firstly, let us search the explicit Fourier transform solution
$U_2=[\rho_2,~u_2,~\mathbb{T}heta_2]$ of the initial value problem
\eqref{3.10}-\eqref{3.11}.
From the three equations of \eqref{3.10}, one has
\begin{equation}\label{3.15}
{\partialrtial _{ttt}}{\rho _2} + 2{\partialrtial _{tt}}{\rho _2} - 2\Delta {\partialrtial _t}{\rho _2}
+ {\partialrtial _t}{\rho _2} - \Delta {\rho _2} = 0,\end{equation}
with initial value
\begin{equation}\label{3.16}
\left\{
\begin{split}
& {\rho _2}\left| {_{t = 0}} \right. = \rho _{2,0}, \\
&{\partialrtial _t}{\rho _2}\left| {_{t = 0}} \right. = - \nabla \cdot {u_{2,0}}, \\
&{\partialrtial _{tt}}{\rho _2}\left| {_{t = 0}} \right. = \Delta {\rho _{2,0}} + \nabla \cdot {u_{2,0}} + \Delta {\mathbb{T}heta _{2,0}}.
\end{split} \right.\end{equation}
After taking the Fourier transform on \eqref{3.15} and \eqref{3.16},
it follows that
\begin{equation}\label{3.17}
{\partialrtial _{ttt}}{\hat{\rho} _2} + 2{\partialrtial _{tt}}{\hat{\rho} _2}
+ (1+2|k|^2){\partialrtial _t}{\hat{\rho} _2} +|k|^2 {\hat{\rho} _2} = 0,\end{equation}
with initial value
\begin{equation}\label{3.18}
\left\{
\begin{split}
& {\hat{\rho} _2}\left| {_{t = 0}} \right. = \hat{\rho} _{2,0}, \\
&{\partialrtial _t}{\hat{\rho} _2}\left| {_{t = 0}} \right. = - i|k|\tilde{k} \cdot {\hat{u}_{2,0}}, \\
&{\partialrtial _{tt}}{\hat{\rho} _2}\left| {_{t = 0}} \right. = -|k|^2 {\hat{\rho} _{2,0}} + i|k|\tilde{k} \cdot {\hat{u}_{2,0}}
-|k|^2{\hat{\mathbb{T}heta} _{2,0}},
\end{split} \right.\end{equation}
in this paper, we set $\tilde{k}=\frac{k}{|k|}.$ The characteristic
equation of \eqref{3.17} is
$$ F(\mathcal {X}):=\mathcal {X}^3+2\mathcal {X}^2+\left(1+2|k|^2\right)\mathcal {X}+|k|^2=0.$$
For the roots of the previous characteristic equation and their
properties, we obtain
\begin{lemma}\label{L3.1}
Assume $|k|\neq0.$ Then, $F(\mathcal {X})=0,$ $\mathcal
{X}\in\mathbb{C}$ has a real root
$\sigma=\sigma(|k|)\in(-\frac{1}{2},0)$ and two conjugate complex
roots $\mathcal {X}_\pm=\beta\pm i\omega$ with
$\beta=\beta(|k|)\in(-1,-\frac{3}{4})$ and
$\omega=\omega(|k|)\in(0,+\infty)$ which satisfy the following
properties:
\begin{equation}\label{3.19}
\beta=-1-\frac{\sigma}{2},~
\omega=\frac{1}{2}\sqrt{3\sigma^2+4\sigma+8|k|^2}.
\end{equation}
$\sigma,\beta,\omega$ are smooth in $|k|>0$, and $\sigma(|k|)$ is
strictly decreasing over $|k|>0$, with
$$ \lim_{|k|\longrightarrow0}\sigma(|k|)=0,~\lim_{|k|\longrightarrow\infty}\sigma(|k|)=-\frac{1}{2}.$$
Furthermore, the asymptotic behavior as follows hold true:
$$\sigma(|k|)=-O(1)|k|^2,~\beta(|k|)=-1+O(1)|k|^2,~\omega(|k|)=O(1)|k| $$
whenever $|k|\leq1$ is sufficiently small, and
$$\sigma(|k|)=-\frac{1}{2}+O(1)|k|^{-2},~\beta(|k|)=-\frac{3}{4}-O(1)|k|^{-2},~\omega(|k|)=O(1)|k|$$
whenever $|k|\geq1$ is sufficiently large. Here and in the sequel
$O(1)$ means strictly positive constant.
\end{lemma}
\noindent \emph{Proof.} Assume $|k|\neq0.$ First of all, we look for
the possibly existing real root for $F(\mathcal {X})=0$ over
$\mathcal {X}\in \mathbb{R}$. Since
$$F'(\mathcal {X})=3\mathcal {X}^2+4\mathcal {X}+1+2|k|^{2}>0,$$
and
$F(-\frac{1}{2})=-\frac{1}{8}<0,~F(0)=|k|^{2}>0,$
then equation $F(\mathcal {X})=0$ really has one and only one real
root defined as $\sigma=\sigma(|k|)$ which satisfies $-\frac{1}{2}<\sigma<0.$
After taking derivative of $F(\sigma(|k|))=0$ in $|k|$, one has
$$\sigma'(|k|)=\frac{-|k|\left( 2+4\sigma\right)}{3\sigma^2+4\sigma+1+2|k|^{2}}<0,$$
so that $\sigma(\cdot)$ is strictly decreasing in $|k|>0.$ Since
$F(\sigma)=0$ can be re-written as
$$\sigma\left[ \frac{\sigma(\sigma+2)}{1+2|k|^{2}}+1\right]=-\frac{|k|^{2}}{1+2|k|^{2}},$$
then $\sigma$ has limits $0$ and $-\frac{1}{2}$ as $|k|\rightarrow
0$ and $|k|\rightarrow \infty$, respectively.
$F(\sigma(|k|))=0$ is also equivalent with
$$\sigma+\frac{1}{2}=\frac{\frac{1}{2}(\sigma+1)^2}{(\sigma+1)^2+2|k|^{2}}$$
Therefore, it follows that $\sigma(|k|)=-O(1)|k|^2$ whenever $|k|<1$
is small enough and $\sigma(|k|)=-\frac{1}{2}+O(1)|k|^{-2} $
whenever $|k|\geq 1$ is large enough. Next, let us search roots of
$F(\mathcal {X})=0$ on $\mathcal {X}\in \mathbb{C}$. Since
$F(\sigma)=0$ with $\sigma\in \mathbb{R}$, $F(\mathcal {X})=0$ can
be split up into
$$ F(\mathcal {X})=(\mathcal {X}-\sigma)\left[\left(\mathcal {X}+1+\frac{\sigma}{2} \right)^2+\frac{3}{4}\sigma^2
+\sigma+2|k|^2 \right]=0.$$ Therefore, there are two conjugate
complex roots $\mathcal {X}_\pm=\beta\pm i\omega$ which satisfy
$$ \left(\mathcal {X}+1+\frac{\sigma}{2} \right)^2+\frac{3}{4}\sigma^2+\sigma+2|k|^2=0.$$
After solving the above equation, one can get that
$\beta=\beta(|k|),~\omega=\omega(|k|)$ take the form of
\eqref{3.19}. From the asymptotic behavior of $\sigma(|k|)$ at
$|k|=0$ and $\infty$, one can directly acquire that of
$\beta(|k|),~\omega(|k|)$. Now, we complete the proof of Lemma
\ref{L3.1}.
$\Box$
Based on Lemma \ref{L3.1}, one can define the solution of
\eqref{3.17} as
\begin{equation}\label{3.20}
\hat{\rho}_2 (t,k)=c_1(k)e^{\sigma t}+e^{\beta t}\left(
c_2(k)\cos\omega t+c_3(k)\sin\omega t \right),
\end{equation}
where $c_i(k),~1\leq i \leq 3,$ is to be ascertained by \eqref{3.18}
later.
In fact,
\eqref{3.18} implies
\begin{equation}\label{3.21}
\left[ {\begin{array}{*{20}{c}}
{\hat \rho_2 {|_{t = 0}}} \\
{{\partialrtial _t}\hat \rho_2 {|_{t = 0}}} \\
{{\partialrtial _{tt}}\hat \rho_2 {|_{t = 0}}} \\
\end{array} } \right] = A\left[ {\begin{array}{*{20}{c}}
{{c_1}} \\
{{c_2}} \\
{{c_3}} \\
\end{array} } \right],\quad A = \left[ {\begin{array}{*{20}{c}}
1 & 1 & 0 \\
\sigma & \beta & \omega \\
{{\sigma ^2}} & {{\beta ^2} - {\omega ^2}} & {2\beta \omega } \\
\end{array} } \right].\end{equation}
It is directly to check that
$$\det A = \omega \left[ {{\omega ^2} + {{\left( {\sigma - \beta } \right)}^2}} \right]
= \omega \left( {3{\sigma ^2} + 4\sigma + 1 + 2{{\left|
k \right|}^2}} \right) > 0
$$
and
\[{A^{ - 1}} = \frac{1}
{{\det A}}\left[ {\begin{array}{*{20}{c}}
{\left( {{\beta ^2} + {\omega ^2}} \right)\omega } & { - 2\beta \omega } & \omega \\
{\sigma \left( {\sigma - 2\beta } \right)\omega } & {2\beta \omega } & { - \omega } \\
{\sigma \left( {{\beta ^2} - {\omega ^2} - \sigma \beta } \right)} & {{\omega ^2} + {\sigma ^2}
- {\beta ^2}} & {\beta - \sigma } \\
\end{array} } \right].\]
Notice that \eqref{3.21} together with \eqref{3.18} gives
\begin{equation}\notag
\begin{split}
[{c_1},~ & {c_2},~ {c_3}]^T =\frac{1} {{3{\sigma ^2} +
4\sigma + 1 + 2 {{\left| k \right|}^2}}}\\
&\left[ {\begin{array}{*{20}{c}}
{{\beta ^2} + {\omega ^2} - { {{\left| k \right|}^2}} } & { i\left| k \right|\left( {2\beta + 1} \right)} & { - {{\left| k \right|}^2}} \\
{{\sigma ^2} - 2\sigma \beta + { {{\left| k \right|}^2}} } & {-i\left| k \right|\left( {2\beta + 1} \right)} & {{{\left| k \right|}^2}} \\
{\frac{{\sigma \left( {{\beta ^2} - {\omega ^2} - \sigma \beta } \right) - \left( {\beta - \sigma } \right) {{{\left| k \right|}^2}} }}
{\omega }} & {\frac{{i\left| k \right|}} {\omega }\left( {{\beta ^2}
- {\sigma ^2} - {\omega ^2} + \beta - \sigma } \right)} &
{\frac{{\sigma - \beta }}
{\omega }{{\left| k \right|}^2}} \\
\end{array} } \right]\left[ {\begin{array}{*{20}{c}}
{{{\hat \rho }_{2,0}}} \\
{\tilde k \cdot {{\hat u}_{2,0}}} \\
{{{\hat \mathbb{T}heta }_{2,0}}} \\
\end{array} } \right].
\end{split}\end{equation}
Here, we utilize $[\cdot]^T$ to denote the transpose of any vector.
Substituting the form of $\beta$ and $\omega$, and making further
simplifications, we obtain
\begin{equation}\label{3.22}
\begin{split}
[&{c_1},~ {c_2},~ {c_3}]^T =\frac{1} {{3{\sigma ^2} +
4\sigma + 1 + 2 {{\left| k \right|}^2}}}\\
&\left[ {\begin{array}{*{20}{c}}
{{{\left( {\sigma + 1} \right)}^2} + {{\left| k \right|}^2}} & { - i\left| k \right|\left( {\sigma + 1} \right)} & { - {{\left| k \right|}^2}} \\
{2( \sigma + 1) + {{\left| k \right|}^2}} & {i\left| k \right|\left( {\sigma + 1} \right)} & {{{\left| k \right|}^2}} \\
{\frac{{{\sigma ( \sigma + 1)} + (1- \frac{1}
{2}\sigma) {{\left| k \right|}^2}}} {\omega }} & {\frac{{i\left| k
\right|}} {\omega }\left( {\frac{3} {2}{\sigma ^2} + \frac{3}
{2}\sigma + 2{{\left| k \right|}^2}} \right)} &
{\frac{{1 + \frac{3} {2}\sigma }}
{\omega }{{\left| k \right|}^2}} \\
\end{array} } \right]\left[ {\begin{array}{*{20}{c}}
{{{\hat \rho }_{2,0}}} \\
{\tilde k \cdot {{\hat u}_{2,0}}} \\
{{{\hat \mathbb{T}heta }_{2,0}}} \\
\end{array} } \right].
\end{split}\end{equation}
Similarly, from the three equations of \eqref{3.10}, one has
\begin{equation}\label{3.23}
\partialrtial_{ttt}\hat{\mathbb{T}heta}_2
+2\partialrtial_{tt}\hat{\mathbb{T}heta}_2+\left(1+2|k|^2\right)\partialrtial_{t}\hat{\mathbb{T}heta}_2
+|k|^2\hat{\mathbb{T}heta}_2=0,
\end{equation}
with initial value
\begin{equation}\label{3.24}
\left\{\begin{split}
&\hat{\mathbb{T}heta}_2|_{t=0}=\hat{\mathbb{T}heta}_{2,0},\\
&\partialrtial_{t}\hat{\mathbb{T}heta}_2|_{t=0}=-i|k|\tilde{k}\cdot \hat{u}_{2,0}-\hat{\mathbb{T}heta}_{2,0},\\
&\partialrtial_{tt}\hat{\mathbb{T}heta}_2|_{t=0}=-|k|^2\hat{\rho}_{2,0}+2i|k|\tilde{k}\cdot
\hat{u}_{2,0}+\left(1-|k|^2\right)\hat{\mathbb{T}heta}_{2,0}.
\end{split}
\right.\end{equation} Based on Lemma \ref{L3.1}, one can also set
the solution of \eqref{3.23} as
\begin{equation}\label{3.25}
\hat{\mathbb{T}heta}_2(t,k)=c_4(k)e^{\sigma t}+e^{\beta t}\left(
c_5(k)\cos\omega t+c_6(k)\sin\omega t \right),
\end{equation}
where $c_i(k),~4\leq i \leq 6,$ is to be ascertained by \eqref{3.24}
later. In fact, after tenuous computation, \eqref{3.24} implies
\begin{equation}\label{3.26}
\begin{split}
[&{c_4},~ {c_5},~ {c_6}]^T =\frac{1} {{3{\sigma ^2} +
4\sigma + 1 + 2 {{\left| k \right|}^2}}} \\
& \left[ {\begin{array}{*{20}{c}}
{- {{\left| k \right|}^2}} & {-i\left| k
\right|\left( {1 + \sigma } \right)} & {(1 + \sigma) \sigma+{{\left| k \right|}^2}} \\
{{{\left| k \right|}^2}} & { i\left| k
\right|\left( {1 + \sigma } \right)} & {(1 + 2\sigma)(1 + \sigma) +{{\left| k \right|}^2} } \\
{\frac{\frac{3}{2}\sigma +1 } {\omega }|k|^2} &
{\frac{-i|k|} {\omega }(\frac{3}{2}\sigma(\sigma+2)+1+2|k|^2)} & {-\frac{|k|^2+\frac{1}{2}\sigma(|k|^2+1+\sigma) }
{\omega }} \\
\end{array} } \right]\left[ {\begin{array}{*{20}{c}}
{{{\hat \rho }_{2,0}}} \\
{\tilde k \cdot {{\hat u}_{2,0}}} \\
{{{\hat \mathbb{T}heta }_{2,0}}} \\
\end{array} } \right].
\end{split}\end{equation}
Similarly, again from the three equations of \eqref{3.10}, we also
have
\begin{equation}\label{3.27}
{\partialrtial _{ttt}} ({\tilde{k}\cdot \hat{u}_2}) + 2{\partialrtial _{tt}}(\tilde{k}\cdot \hat{u}_2)+
(1+2|k|^2)\partialrtial_t ({\tilde{k}\cdot \hat{u}_2})+|k|^2({\tilde{k}\cdot \hat{u}_2})= 0,
\end{equation}
with initial value
\begin{equation}\label{3.28}
\left\{\begin{split}
&{\tilde{k}\cdot \hat{u}_2}|_{t=0}={\tilde{k}\cdot \hat{u}}_{2,0},\\
&\partialrtial_{t}{(\tilde{k}\cdot \hat{u}_2)}|_{t=0}=-i|k|{\hat
\rho _{2,0}} - \tilde k \cdot {\hat u_{2,0}} - i\left| k \right|{\hat \mathbb{T}heta
_{2,0}},
\\
&\partialrtial_{tt}{(\tilde{k}\cdot \hat{u}_2)}|_{t=0}=i|k|{\hat \rho _{2,0}} +(1-2{\left| k \right|^2})\tilde k \cdot {\hat u_{2,0}} + 2i\left|
k \right|{\hat \mathbb{T}heta _{2,0}}.
\end{split}
\right.\end{equation}
From Lemma \ref{L3.1}, one can also check that the
solution of \eqref{3.27} has the form
\begin{equation}\label{3.29}
\tilde{k}\cdot \hat{u}_2(t,k)=c_7(k)e^{\sigma t}+e^{\beta t}\left(
c_8(k)\cos\omega t+c_9(k)\sin\omega t \right),
\end{equation}
with
\begin{equation}\label{3.30}
\begin{split} [&{c_7},~ {c_8},~ {c_9}]^T =\frac{1}
{{3{\sigma ^2} +
4\sigma + 1 + 2 {{\left| k \right|}^2}}} \\
& \left[ {\begin{array}{*{20}{c}}
{-i\left| k
\right|\left( {1 + \sigma } \right)} & {\sigma\left( {1 + \sigma } \right) } & {-i\left| k
\right| \sigma } \\
{i\left| k
\right|\left( {1 + \sigma } \right)} & { \left( {1 + \sigma } \right)\left( {1 + 2\sigma } \right)+2|k|^2} & {i\left| k
\right| \sigma } \\
{\frac{-i|k|} {\omega }(\frac{3}{2}\sigma(\sigma+1)-2|k|^2)} &
{ \frac{-\sigma \left( {1 + \sigma } -2|k|^2 \right) }{2\omega} } & {\frac{i|k|} {\omega }(-\frac{3}{2}\sigma (\sigma+2)+2|k|^2-1) }
\\
\end{array} } \right]\left[ {\begin{array}{*{20}{c}}
{{{\hat \rho }_{2,0}}} \\
{\tilde k \cdot {{\hat u}_{2,0}}} \\
{{{\hat \mathbb{T}heta }_{2,0}}} \\
\end{array} } \right].
\end{split}\end{equation}
Furthermore, after taking the curl for the second equation of
\eqref{3.10} and making the Fourier transform in $x$, we have
\begin{equation}\label{3.31}
\partialrtial_t \left(\tilde{k}\times (\tilde{k}\times \hat{u}_2)\right)
+ \tilde{k}\times (\tilde{k}\times \hat{u}_2)=0,
\end{equation}
with initial value
\begin{equation}\label{3.32}
\tilde{k}\times (\tilde{k}\times \hat{u}_2)\left|_{t=0}\right.=
\tilde{k}\times (\tilde{k}\times \hat{u}_{2,0}).
\end{equation}
After solving \eqref{3.31}-\eqref{3.32}, we have
\begin{equation}\label{3.33}
\tilde{k}\times (\tilde{k}\times \hat{u}_2)
=e^{-t}\left(\tilde{k}\times (\tilde{k}\times \hat{u}_{2,0})\right).
\end{equation}
Now, we can obtain the explicit Fourier transform solution $U_2=[\rho_2,~u_2,~\mathbb{T}heta_2]$ as follows
from the above computations.
\begin{theorem}\label{thm3.1}
Assume $U_2=[\rho_2,~u_2,~\mathbb{T}heta_2]$ be the solution of the initial
value problem \eqref{3.10}-\eqref{3.11} on the linearized
homogeneous equations. For $(t,k)\in(0,\infty)\times\mathbb{R}^3$
with $|k|\neq0$, we obtain
\begin{equation}\label{3.34}
\left[ {\begin{array}{*{20}c}
{\hat{\rho }_2 (t,k)} \\
{\hat{u}_2 (t,k)} \\
{\hat{\mathbb{T}heta}_2 (t,k)} \\
\end{array}} \right] = \left[ {\begin{array}{*{20}c}
{\hat{\rho}_2 (t,k)} \\
{\hat{u}_{ 2||} (t,k)} \\
{\hat{\mathbb{T}heta}_2 (t,k)} \\
\end{array}} \right] + \left[ {\begin{array}{*{20}c}
0 \\
{\hat{u}_{ 2 \bot } (t,k)} \\
0\\
\end{array}} \right].
\end{equation}
Here $\hat{u}_{ 2||},\hat{u}_{ 2\bot }$ are defined by
$$\hat{u}_{2
||}=\tilde{k}\tilde{k}\cdot\hat{u}_2,\ ~ \hat{u}_{ 2\bot
}=-\tilde{k}\times(\tilde{k}\times\hat{u}_2)=(I_3-\tilde{k}\otimes\tilde{k})
\hat{u}_2.$$
Then, there exit matrices $G^I_{5\times5}(t,k)$ and
$G^{II}_{3\times3}(t,k)$ such that
\begin{equation}\label{3.35}
\left[ {\begin{array}{*{20}c}
{\hat{\rho_2} (t,k)} \\
{\hat{u}_{2||} (t,k)} \\
{\hat{\mathbb{T}heta}_2 (t,k)} \\
\end{array}} \right] = G_{5 \times 5}^I (t,k)\left[ {\begin{array}{*{20}c}
{\hat{\rho} _{2,0} (k)} \\
{\hat{u}_{2||,0} (k)} \\
{\hat{\mathbb{T}heta} _{2,0} (k)} \\
\end{array}} \right]
\end{equation}
and
\begin{equation}\label{3.36}
{\begin{array}{*{20}c}
{\hat{u}_{ 2\bot } (t,k)} \\
\end{array}} = G_{3 \times 3}^{II} (t,k) {\begin{array}{*{20}c}
{\hat{u}_{ 2\bot,0 } (k)} \\
\end{array}} ,
\end{equation}
where $ G_{5 \times 5}^I$ is explicitly ascertained by
representations \eqref{3.20}, \eqref{3.29}, \eqref{3.25} for
$\hat{\rho}_2(t,k)$, $\hat{u}_{2||}(t,k)$, $\hat{\mathbb{T}heta}_2(t,k)$
with $c_i(k)$, $(1\leq i\leq 9)$ are defined as \eqref{3.22},
\eqref{3.30}, \eqref{3.26} in terms of $\hat{\rho}_{2,0}(k)$,
$\hat{u}_{2||,0}(k)$, $\hat{\mathbb{T}heta}_{2,0}(k)$; and $G_{3 \times
3}^{II} $ is chosen by the representations \eqref{3.33} for
$\hat{u}_{ 2\bot } (t,k)$ in terms of $\hat{u}_{ 2\bot,0 } (k)$.
\end{theorem}
\subsection{$L^p-L^q$ decay
property.} In this subsection, we use Theorem \ref{thm3.1} to obtain
$L^p-L^q$ decay property for every component of the solution $U_2$
$=[\rho_2$, $u_2$, $\mathbb{T}heta_2]$. For this aim, we first search the
rigorous time-frequency estimates on $\hat{U}_2$ $=[\hat{\rho}_2$,
$\hat{u}_2$, $\hat{\mathbb{T}heta}_2]$ as follws
\begin{lemma}\label{L3.2}
Assume $U_2=[\rho_2,~u_2,~\mathbb{T}heta_2]$ be the solution to the initial
value problem \eqref{3.10}-\eqref{3.11} on the linearized
homogeneous equations. Then, there are constants $\gamma>0,C>0$ such
that for all $(t,k)\in(0,\infty)\times\mathbb{R}^3$,
\begin{equation}\label{3.37}
\begin{split}
\left|\hat{\rho}_2 (t,k) \right| \leqslant
C\left| {\left[ {{{\hat \rho}_{2,0}}(t,k),{{\hat u}_{2,0}}(t,k),{{\hat \mathbb{T}heta}_{2,0}}(t,k)} \right]} \right| \cdot
\left\{ {\begin{array}{*{20}{c}}
{{e^{ - \gamma t}} + {e^{ - \gamma {{\left| k \right|}^2}t}}}
& {if~\left| k \right| \leqslant 1,} \\
{{e^{ - \gamma t}} +
{e^{\frac{{ - \gamma }}
{{{{\left| k \right|}^2}}}t}}} & {if~\left| k \right| > 1,} \\
\end{array} } \right.
\end{split}
\end{equation}
\begin{equation}\label{3.38}
\begin{split}
\left| {\hat u_2(t,k)} \right| \leqslant
C\left| {\left[ {{{\hat \rho}_{2,0}}(k),{{\hat u}_{2,0}}(k),{{\hat \mathbb{T}heta}_{2,0}}(k)} \right]} \right|
\cdot
\left\{ {\begin{array}{*{20}{c}}
{{e^{ - \gamma t}} + \left| k \right|{e^{ - \gamma {{\left| k \right|}^2}t}}}
& {if~\left| k \right| \leqslant 1,} \\
{{|k|^{-1}e^{ - \gamma t}} +
{e^{\frac{{ - \gamma }}
{{{{\left| k \right|}^2}}}t}}} & {if~\left| k \right| > 1,} \\
\end{array} } \right.
\end{split}
\end{equation}
and
\begin{equation}\label{3.39}
\begin{split}
\left|\hat{\mathbb{T}heta}_2 (t,k) \right| \leqslant
C\left| {\left[ {{{\hat \rho}_{2,0}}(t,k),{{\hat u}_{2,0}}(t,k),{{\hat \mathbb{T}heta}_{2,0}}(t,k)} \right]} \right| \cdot
\left\{ {\begin{array}{*{20}{c}}
{{e^{ - \gamma t}} + {e^{ - \gamma {{\left| k \right|}^2}t}}}
& {if~\left| k \right| \leqslant 1,} \\
{{e^{ - \gamma t}} +
{e^{\frac{{ - \gamma }}
{{{{\left| k \right|}^2}}}t}}} & {if~\left| k \right| > 1,} \\
\end{array} } \right.
\end{split}
\end{equation}
\end{lemma}
\noindent \emph{Proof.}~ Firstly, let us look for the upper bound
of $\hat{\rho}_2$ defined as \eqref{3.37}.
In fact, from Lemma \ref{L3.1}, it is directly to check
\eqref{3.22} to get
\[\left[ {\begin{array}{*{20}{c}}
{{c_1}} \\
{{c_2}} \\
{{c_3}} \\
\end{array} } \right] = \left[ {\begin{array}{*{20}{c}}
{O(1){}} & { - O(1){{\left| k \right|}}i} & { - O(1){{\left| k \right|}^2}} \\
{O(1)} & {O(1){{\left| k \right|}}i} & {O(1){{\left| k \right|}^2}} \\
{O(1)\left| k \right|} & { - O(1)\left| k \right|^2i} & { O(1){{\left| k \right|}}} \\
\end{array} } \right]\left[ {\begin{array}{*{20}{c}}
{{{\hat \rho }_{2,0}}} \\
{\tilde k \cdot {{\hat u}_{2,0}}} \\
{{{\hat \mathbb{T}heta }_{2,0}}} \\
\end{array} } \right]\]
as $|k|\rightarrow0$, and
\[\left[ {\begin{array}{*{20}{c}}
{{c_1}} \\
{{c_2}} \\
{{c_3}} \\
\end{array} } \right] = \left[ {\begin{array}{*{20}{c}}
{O(1)} & { - O(1){{\left| k \right|}^{ - 1}}i} & { - O(1)} \\
{O(1)} & {O(1){{\left| k \right|}^{ - 1}}i} & {O(1)} \\
{O(1){{\left| k \right|}^{ - 1}}} & { - O(1)i} & {O(1){{\left| k \right|}^{ - 1}}} \\
\end{array} } \right]\left[ {\begin{array}{*{20}{c}}
{{{\hat \rho }_{2,0}}} \\
{\tilde k \cdot {{\hat u}_{2,0}}} \\
{{{\hat \mathbb{T}heta }_{2,0}}} \\
\end{array} } \right]\]
as $|k|\rightarrow\infty$.
Therefore, after putting the previous computations into
\eqref{3.20}, it holds that
\[\begin{split}
\hat \rho_2 (t,k) =& \left( {O(1){{}}{{\hat \rho }_{2,0}} - O(1)
{{\left| k \right|}}i\tilde k \cdot {{\hat u}_{2,0}} - O(1){{\left| k \right|}^2}
{{\hat \mathbb{T}heta }_{2,0}}} \right){e^{\sigma t}} \\
&+ \left( {O(1){{\hat \rho }_{2,0}} + O(1){{\left| k \right|}}i\tilde k \cdot {{\hat u}_{2,0}}
+ O(1){{\left| k \right|}^2}{{\hat \mathbb{T}heta }_{2,0}}} \right){e^{\beta t}}\cos \omega t \\
&+ \left( {O(1)|k|{{\hat \rho }_{2,0}} - O(1)\left| k \right|^2i\tilde k \cdot {{\hat u}_{2,0}} + O(1)
{{\left| k \right|}}{{\hat \mathbb{T}heta }_{2,0}}} \right){e^{\beta t}}\sin \omega t, \\
\end{split} \]
as $|k|\rightarrow0$, and
\[\begin{split}
\hat \rho_2 (t,k) =& \left( {O(1){{\hat \rho }_{2,0}} - O(1){{\left| k \right|}^{ - 1}}i\tilde k \cdot
{{\hat u}_{2,0}} - O(1){{\hat \mathbb{T}heta }_{2,0}}} \right){e^{\sigma t}} \\
&+ \left( {O(1){{\hat \rho }_{2,0}} + O(1){{\left| k \right|}^{ - 1}}i\tilde k \cdot {{\hat u}_{2,0}}
+ O(1){{\hat \mathbb{T}heta }_{2,0}}} \right){e^{\beta t}}\cos \omega t \\
&+ \left( {O(1){{\left| k \right|}^{ - 1}}{{\hat \rho }_{2,0}} - O(1)i\tilde k \cdot {{\hat u}_{2,0}}
+ O(1){{\left| k \right|}^{ - 1}}{{\hat \mathbb{T}heta }_{2,0}}} \right){e^{\beta t}}\sin \omega t, \\
\end{split} \]
as $|k|\rightarrow\infty$.
Based on Lemma \ref{L3.1}, we find that there is $\gamma>0$ such
that
\begin{equation}\notag\left\{\begin{split}
& {\sigma (k) \leqslant - \gamma {{\left| k \right|}^2},}\quad {\beta (k) = - 1 - \frac{\sigma }
{2} \leqslant - \gamma }\quad {{over}\left| k \right| \leqslant 1,} \\
&{\sigma (k) \leqslant - \gamma ,} \quad {\beta (k) = - 1 - \frac{\sigma }
{2} \leqslant - \frac{\gamma } {{{{\left| k \right|}^2}}}}
\quad{{over}\left| k \right| \geqslant 1.}
\end{split} \right.\end{equation}
Thus, one can obtain, for $|k|\leq 1$
\[\left| {{{\hat \rho }_2}\left( {t,k} \right)} \right| \leqslant
C\left( {{e^{ - \gamma t}} + {e^{ - \gamma {{\left| k
\right|}^2}t}}} \right)\left| {\left[ {{{\hat \rho }_{2,0}},\tilde k
\cdot {{\hat u}_{2,0}},{{\hat \mathbb{T}heta }_{2,0}}} \right]} \right|,\]
and for $|k|\geq 1$
\[\left| {{{\hat \rho }_2}\left( {t,k} \right)} \right| \leqslant
C\left( {{e^{ - \gamma t}} + {e^{ - \frac{\gamma}{{\left| k
\right|}^2} {}t}}} \right)\left| {\left[ {{{\hat \rho
}_{2,0}},\tilde k \cdot {{\hat u}_{2,0}},{{\hat \mathbb{T}heta }_{2,0}}}
\right]} \right|.\]
Furthermore, one has
\begin{equation}\notag
\left| {{{\hat \rho }_2}\left( {t,k} \right)} \right| \leqslant C
\left| {\left[ {{{\hat \rho }_{2,0}}, {{\hat u}_{2,0}},{{\hat
\mathbb{T}heta }_{2,0}}} \right]} \right|\cdot \left\{\begin{split}\left(
{{e^{ - \gamma t}} + {e^{ - \gamma {{\left| k
\right|}^2}t}}} \right) & \quad {if}\quad |k|\leq 1, \\
\left( {{e^{ - \gamma t}} + {e^{ - \frac{\gamma}{{\left| k
\right|}^2} {}t}}} \right) & \quad {if}\quad |k|\geq 1.
\end{split} \right.\end{equation}
Similarly, we obtain \eqref{3.38} and \eqref{3.39}. Now, we complete
the proof of Lemma \ref{L3.2}.
$\Box$
From Lemma \ref{L3.2}, it is straightforward to acquire the decay
property for every component of the solution $U_2$ $=[\rho_2$,
$u_2$, $\mathbb{T}heta_2]$. So that we omitted the details of proof for
briefness. See for instance \cite{FWK11}.
\begin{theorem}\label{thm3.2}
Assume $1\leq p,r\leq 2\leq q\leq\infty,l\geq0$ and an integer
$m\geq0$. Suppose
$U_2(t)$ $=e^{tL_2}U_{2,0}$ be the solution of the initial value
problem \eqref{3.10}-\eqref{3.11}. Then, for any $t\geq0$, $U_2$
$=[\rho_2$, $u_2$, $\mathbb{T}heta_2]$ satisfies decay property as follows
\begin{equation}\label{3.40}
\begin{split}
{\left\| {{\nabla ^m}{\rho _2}\left( t \right)} \right\|_{L^q}} \leqslant & C{\left( {1 + t} \right)^{ - \frac{3}
{2}\left( {\frac{1} {p} - \frac{1} {q}} \right) - \frac{m}
{2}}}{\left\| {\left[ {{\rho _{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}} \right]} \right\|_{{L^p}}} \\
&+ C{\left( {1 + t} \right)^{ - \frac{l}
{2}}}{\left\| {{\nabla ^{m + {{\left[ {l + 3\left( {\frac{1} {r} -
\frac{1} {q}} \right)} \right]}_ + }}}\left[ {{\rho
_{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}} \right]} \right\|_{{L^r}}},
\end{split}\end{equation}
\begin{equation}\label{3.41}
\begin{split}
{\left\| {{\nabla ^m}u_2\left( t \right)} \right\|_{{L^q}}} \leqslant& C{\left( {1 + t} \right)^{ - \frac{3}
{2}\left( {\frac{1} {p} - \frac{1} {q}} \right) - \frac{{m + 1}}
{2}}}{\left\| {\left[ {{\rho _{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}} \right]} \right\|_{{L^p}}} \\
& + C{\left( {1 + t} \right)^{ - \frac{{l }}
{2}}}{\left\| {{\nabla ^{m + {{\left[ {l + 3\left( {\frac{1} {r} -
\frac{1}
{q}} \right)} \right]}_ + }}}\left[ {\rho _{2,0},{u_{2,0}},{\mathbb{T}heta _{2,0}}} \right]} \right\|_{{L^r}}}, \\
\end{split}
\end{equation}
and
\begin{equation}\label{3.42}
\begin{split}
{\left\| {{\nabla ^m}{\mathbb{T}heta _2}\left( t \right)} \right\|_{L^q}} \leqslant & C{\left( {1 + t} \right)^{ - \frac{3}
{2}\left( {\frac{1} {p} - \frac{1} {q}} \right) - \frac{m}
{2}}}{\left\| {\left[ {{\rho _{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}} \right]} \right\|_{{L^p}}} \\
&+ C{\left( {1 + t} \right)^{ - \frac{l}
{2}}}{\left\| {{\nabla ^{m + {{\left[ {l + 3\left( {\frac{1} {r} -
\frac{1} {q}} \right)} \right]}_ + }}}\left[ {{\rho
_{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}} \right]} \right\|_{{L^r}}},
\end{split}
\end{equation}
where
\begin{equation}\notag
{[l + 3(\frac{1}{r} - \frac{1}{q})]_ + } = \left\{
{\begin{array}{*{20}{c}}
l & { \mbox{if} ~~ r = q = 2\ \mbox{ and }{\rm{ }}l{\mbox{ is an integer,}}} \\
{{{[l + 3(\frac{1}{r} - \frac{1}{q})]}_ - } + 1} & \mbox{otherwise,} \\
\end{array}} \right.
\end{equation}
where, we use $[\cdot]_-$ to denote the integer part of the
argument.
\end{theorem}
From Theorem \ref{thm3.2}, let us list some particular cases as
follows for later use.
\begin{coro}\label{Corollary3.1}
Let $U_2(t)=e^{tL_2}U_{2,0}$ be the solution of the initial value
problem \eqref{3.10}-\eqref{3.11}. Then, for any $t\geq0$, $U_2$
$=[\rho_2$, $u_2$, $\mathbb{T}heta_2]$ satisfies
\begin{equation}\label{3.43}
\left\{
\begin{split}
&\left\| {{\rho _2}\left( t \right)} \right\| \leqslant C{\left( {1 + t} \right)^{ - \frac{3}
{4}}}{\left\| {\left[ {{\rho _{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}}
\right]} \right\|_{{L^1} \cap {{\dot H}^2}}}, \\
& \left\| {{u _2}\left( t \right)} \right\| \leqslant C{\left( {1 + t} \right)^{ - \frac{5}
{4}}}{\left\| {\left[ {{\rho _{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}}
\right]} \right\|_{{L^1} \cap {{\dot H}^3}}}, \\
& \left\| {{\mathbb{T}heta _2}\left( t \right)} \right\| \leqslant C{\left( {1 + t} \right)^{ - \frac{3}
{4}}}{\left\| {\left[ {{\rho _{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}}
\right]} \right\|_{{L^1} \cap {{\dot H}^2}}},
\end{split}\right.
\end{equation}
\begin{equation}\label{3.44}
\left\{\begin{split}& \left\| {\nabla {\rho _2}\left( t \right)}
\right\| \leqslant C{\left( {1 + t} \right)^{ - \frac{5}
{4}}}{\left\| {\left[ {{\rho _{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}}
\right]} \right\|_{{L^1} \cap {{\dot H}^4}}},
\\
& \left\| {\nabla {u _2}\left( t \right)}
\right\| \leqslant C{\left( {1 + t} \right)^{ - \frac{7}
{4}}}{\left\| {\left[ {{\rho _{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}}
\right]} \right\|_{{L^1} \cap {{\dot H}^5}}}, \\
& \left\| {\nabla {\mathbb{T}heta _2}\left( t \right)} \right\| \leqslant
C{\left( {1 + t} \right)^{ - \frac{5} {4}}}{\left\| {\left[ {{\rho
_{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}} \right]} \right\|_{{L^1} \cap
{{\dot H}^4}}}
\end{split}\right.
\end{equation}
and
\begin{equation}\label{3.45}
\left\{
\begin{split}
&\left\| {{\rho _2}\left( t \right)} \right\|_{L^\infty} \leqslant C{\left( {1 + t} \right)^{ - \frac{3}
{2}}}{\left\| {\left[ {{\rho _{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}}
\right]} \right\|_{{L^1} \cap {{\dot H}^5}}}, \\
& \left\| {{u _2}\left( t \right)} \right\|_{L^\infty} \leqslant C{\left( {1 + t} \right)^{ - 2}}
{\left\| {\left[ {{\rho _{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}}
\right]} \right\|_{{L^1} \cap {{\dot H}^6}}}, \\
& \left\| {{\mathbb{T}heta _2}\left( t \right)} \right\|_{L^\infty} \leqslant C{\left( {1 + t} \right)^{ - \frac{3}
{2}}}{\left\| {\left[ {{\rho _{2,0}},{u_{2,0}},{\mathbb{T}heta _{2,0}}}
\right]} \right\|_{{L^1} \cap {{\dot H}^5}}}.
\end{split}\right.
\end{equation}
\end{coro}
\section{ Decay rates for system \eqref{2.2}}
\subsection{Decay rates for the energy functional.} In
this subsection, we will prove the decay rate \eqref{2.12} in
Proposition \ref{prop2.2} for the energy
$\left\|{U(t)}\right\|^2_s$. We begin with the Lemma as follows
which can be seen directly from the proof of Theorem \ref{thm2.1}.
\begin{lemma}\label{L4.1}
Assume that $U=[\rho_\mu,~u_\mu,~\mathbb{T}heta_\mu,~E,~B]$ is the solution of the
initial value
problem \eqref{2.2}-\eqref{2.3} with $U_0$ $=[\rho_{\mu0}$, $u_{\mu0}$, $\mathbb{T}heta_{\mu0}$, $E_0,~B_0]$
which satisfies \eqref{2.4}. If $\mathcal {E}_s(U_0)$
is small enough, then, for any $t \geq 0$
\begin{equation}\label{4.1}
\frac{d}{dt}\mathcal {E}_s(U(t))+\lambda \mathcal {D}_s(U(t))\leq 0.
\end{equation}
\end{lemma}
From Lemma \ref{L4.1}, we can check that
\begin{equation}\notag
\begin{split}
(1+t)^l\mathcal {E}_s(U(t))+& \gamma\int_0^t(1+y)^l\mathcal
{D}_s(U(y))dy\\
&\leq \mathcal {E}_{s}(U_0)+l\int_0^t(1+y)^{l-1}\mathcal
{E}_s(U(y)) dy\\
&\leq \mathcal {E}_{s}(U_0)+C l\int_0^t(1+y)^{l-1}\left(\left\|{B(y)
}\right\|^2+\left\|{(\rho_e+\rho_i)(y) }\right\|^2+\mathcal
{D}_{s+1}(U(y)) \right)dy,
\end{split}
\end{equation}
where we have used $\mathcal {E}_s(U(t))\leq \left\|{B(t)
}\right\|^2+\left\|{(\rho_e+\rho_i)(t) }\right\|^2+\mathcal
{D}_{s+1}(U(t))$. Using \eqref{4.1} again, we have
$$\mathcal {E}_{s+2}(U(t))+\gamma\int_0^t \mathcal {D}_{s+2}(U(y))dy\leq \mathcal {E}_{s+2}(U_0)$$
and
\begin{equation}\notag
\begin{split}
(1+t)^{l-1}\mathcal {E}_{s+1}(U(t))+&
\gamma\int_0^t(1+y)^{l-1}\mathcal
{D}_{s+1}(U(y))dy\\
\leq \mathcal
{E}_{s+1}(U_0)&+C(l-1)\int_0^t(1+y)^{l-2}\left(\left\|{B(y)
}\right\|^2+\left\|{( \rho_e+\rho_i ) (y) }\right\|^2+\mathcal
{D}_{s+2}(U(y)) \right)dy.
\end{split}
\end{equation}
Then, by iterating the previous estimates, we obtain
\begin{equation}\label{4.2}
\begin{split}
(1+t)^l\mathcal {E}_s(U(t))+& \gamma\int_0^t(1+y)^l\mathcal
{D}_s(U(y))dy\\
&\leq C\mathcal
{E}_{s+2}(U_0)+C\int_0^t(1+y)^{l-1}\left(\left\|{B(y)
}\right\|^2+\left\|{( \rho_e+\rho_i ) (y) }\right\|^2\right)dy
\end{split}
\end{equation}
for $1<l<2.$
Now, let us establish the estimate on the integral term on the right
hand side of \eqref{4.2}. Applying the estimate on $B$ in
\eqref{3.12} and the estimate on $\rho_2$ in \eqref{3.43} to
\eqref{3.5} and \eqref{3.6}, respectively,
we have \begin{equation}\label{4.3}
\begin{split}\left\| {B\left( t \right)}
\right\| \leqslant & C {\left( {1 + t} \right)^{ - \frac{3}
{4}}}{\left\| {\left[ {{u_{1,0}},{E_0},{B_0}} \right]}
\right\|_{{L^1}
\cap {{\dot H}^2}}}\\
& + C\int_0^t {{{\left( {1 + t - y} \right)}^{ - \frac{3}
{4}}}{{\left\| {\left[
{{g_{2e}}(y)-{g_{2i}}(y)},~{{g_{4e}}(y)-{g_{4i}}(y)} \right]}
\right\|}_{{L^1} \cap {{\dot H}^2}}}} dy, \end{split}\end{equation}
\begin{equation}\label{4.4}
\begin{split}
\left\| {\left( {{\rho _e} + {\rho _i}} \right)\left( t \right)}
\right\| \leqslant & C\left\| {{\rho _2}\left( t \right)} \right\| \leqslant C{\left( {1 + t} \right)^{ - \frac{3}
{4}}}{\left\| {\left[ {{\rho _{\mu 0}},{u_{\mu 0}},{\mathbb{T}heta _{\mu 0}}} \right]} \right\|_{{L^1} \cap {{\dot H}^2}}} \\
&\quad + C\int_0^t {{{\left( {1 + t - y} \right)}^{ - \frac{3}
{4}}}{{\left\| {\left[ {{g_{1e}} + {g_{1i}},{g_{2e}} +
{g_{2i}},{g_{3e}} + {g_{3i}}} \right](y)} \right\|}_{{L^1}
\cap {{\dot H}^2}}}dy.}
\end{split}\end{equation}
It is directly to check that for any $0\leq y \leq t$,
\[{\left\| {\left[
{{g_{2e}}(y)-{g_{2i}}(y)},~{{g_{4e}}(y)-{g_{4i}}(y)} \right]}
\right\|_{{L^1} \cap {{\dot H}^2}}}
\leqslant C{\mathcal {E}_s}\left( {U(y)} \right) \leqslant
C{\left( {1 + y} \right)^{ - \frac{3}
{2}}}{\mathcal {E} _{s,\infty }}\left( {U(t)} \right),\]
\[{\left\| {\left[ {{g_{1e}} + {g_{1i}},{g_{2e}} + {g_{2i}},{g_{3e}} + {g_{3i}}} \right](y)}
\right\|_{{L^1} \cap {{\dot H}^2}}}
\leqslant C{\mathcal {E}_s}\left( {U(y)} \right) \leqslant
C{\left( {1 + y} \right)^{ - \frac{3}
{2}}}{\mathcal {E} _{s,\infty }}\left( {U(t)} \right),\]
where ${\mathcal {E} _{s,\infty }}\left(
{U(t)} \right): = \mathop {\sup }\limits_{0 \leqslant y \leqslant t}
{\left( {1 + y} \right)^{\frac{3} {2}}}{\mathcal {E}_s}\left( {U(y)}
\right).$ Plugging the two previous inequalities into \eqref{4.3}
and \eqref{4.4} respectively implies
\begin{equation}\label{4.5}
\left\| {B\left( t \right)} \right\| \leqslant C {\left( {1 + t}
\right)^{ - \frac{3} {4}}}\left( {{{\left\| {\left[
{{u_{\mu0}},{E_0},{B_0}} \right]} \right\|}_{{L^1} \cap {{\dot
H}^2}}} + {\mathcal {E} _{s,\infty }}\left( {U(t)} \right) }
\right)\end{equation}
and
\begin{equation}\label{4.6}
\left\| {\left( {{\rho _e} + {\rho _i}} \right)\left( t \right)}
\right\| \leqslant C {\left( {1 + t}
\right)^{ - \frac{3} {4}}}\left( {{{\left\| {\left[
{{\rho_{\mu0}},{u_{\mu0}},{\mathbb{T}heta_{\mu0}}} \right]} \right\|}_{{L^1}
\cap {{\dot H}^2}}} + {\mathcal {E} _{s,\infty }}\left( {U(t)}
\right) } \right).\end{equation}
Next, we search the uniform bound of
${\mathcal {E} _{s,\infty }}\left( {U(t)} \right)$ which will imply
the decay rates of ${\mathcal {E} _{s }}\left( {U(t)} \right)$ or
equivalent to $\|U(t)\|_s^2$. In fact, after choosing
$l=\frac{3}{2}+\varepsilon$ in \eqref{4.2} with $\varepsilon>0$
sufficiently small and utilizing \eqref{4.5} and \eqref{4.6}, one
obtain
\begin{equation}\notag\begin{split}
{\left( {1 + t} \right)^{\frac{3}
{2} + \varepsilon }}{\mathcal {E} _s} & \left( {U(t)} \right) +
\gamma {\int_0^t {\left( {1 + y} \right)} ^{\frac{3}
{2} + \varepsilon }}{\mathcal {D}_s}\left( {U(y)} \right)dy \\
& \leqslant C{\mathcal {E} _{s + 2}}\left( {{U_0}} \right) +
C{\left( {1 + t} \right)^\varepsilon }\left( {\left\| {\left[
{{\rho_{\mu0}},{u_{\mu0}},{\mathbb{T}heta_{\mu0}},{E_0},{B_0}} \right]}
\right\|_{_{{L^1} \cap {{\dot H}^2}}}^2 + {{\left[ {{\mathcal {E}
_{s,\infty }}\left( {U(t)} \right)} \right]}^2}} \right),
\end{split} \end{equation}
which implies
$${\left(
{1 + t} \right)^{\frac{3} {2}}}{\mathcal {E}_s}\left( {U(t)} \right)
\leqslant C\left( {{\mathcal {E}_{s + 2}}\left( {{U_0}} \right) +
\left\| {\left[
{{\rho_{\mu0}},{u_{\mu0}},{\mathbb{T}heta_{\mu0}},{E_0},{B_0}} \right]}
\right\|_{{L^1}}^2 + {{\left[ {{\mathcal {E} _{s,\infty }}\left(
{U(t)} \right)} \right]}^2}} \right),
$$
and therefore,
$$
{\mathcal {E} _{s,\infty }}\left( {U(t)} \right) \leqslant C\left(
{{\omega _{s + 2}}
{{\left( {{U_0}} \right)}^2} + {{\left[ {{\mathcal {E}_{s,\infty }}\left( {U(t)} \right)}
\right]}^2}} \right), $$
since $\omega _{s + 2}\left( {{U_0}} \right)>0 $ is small enough, it
holds that ${\mathcal {E} _{s,\infty }}\left( {U(t)} \right)
\leqslant C{\omega _{s + 2}}{\left( {{U_0}} \right)^2} $ for any
$t\geq 0,$ which implies ${\left\| {U(t)} \right\|_s} \leqslant
C{\mathcal {E}_s}{\left( {U(t)} \right)^{\frac{1} {2}}} \leqslant
C{\omega _{s + 2}}\left( {{U_0}} \right){\left( {1 + t} \right)^{ -
\frac{3} {4}}}$, that is \eqref{2.12}.
\subsection{Decay rate for high-order energy
functional.} In this subsection, we will look for the decay estimate
of the high-order energy $\|\nabla U(t)\|^2_{s-1}$, that is
\eqref{2.13} of Proposition \ref{prop2.2}. We begin with the
following Lemma.
\begin{lemma}\label{L4.2}
Assume $U=[\rho_\mu,~u_\mu,~\mathbb{T}heta_\mu,~E,~B]$ is the solution of
the initial value problem \eqref{2.2}-\eqref{2.3} with $U_0$
$=[\rho_{\mu0}$, $u_{\mu0}$, $\mathbb{T}heta_{\mu0}$, $E_0$, $B_0]$ which
satisfies \eqref{2.4} in the sense of Proposition \ref{prop2.1}. If
$\mathcal {E}_s(U_0)$ is small enough, then, there exist the
high-order energy function $\mathcal {E}_s^h(\cdot) $ and the
high-order dissipative rate $\mathcal {D}_s^h(\cdot) $ such that
\begin{equation}\label{4.7}
\frac{d}{dt}\mathcal {E}_s^h(U(t))+\gamma \mathcal {D}_s^h(U(t))\leq
0,
\end{equation}
holds for any $t\geq 0.$
\end{lemma}
\noindent \emph{Proof.} The proof is very similar to the proof of
Theorem \ref{thm2.1}. In fact, by letting $|\alpha|\geq 1$, then
corresponding to \eqref{2.16}, \eqref{2.19}, \eqref{2.23} and
\eqref{2.25}, it can also be checked that
\begin{equation}\notag
\frac{d} {{dt}}\left\|\nabla U \right\|_{s-1}^2 + \left\|
{\nabla\left[ {{u_e},{u_i},{\mathbb{T}heta _e},{\mathbb{T}heta _i}} \right]}
\right\|_{s-1}^2 \leqslant C{\left\| U \right\|_s} { \left\| {\nabla
\left[ {{\rho _e},{\rho _i},{u_e},{u_i},{\mathbb{T}heta _e},{\mathbb{T}heta _i}}
\right]} \right\|_{s - 1}^2},
\end{equation}
\begin{equation}\notag
\begin{split}
\frac{d}
{{dt}} \sum\limits_{1 \leq\left| \alpha \right| \leqslant s - 1}
{\sum\limits_{\mu = e,i} {\left\langle {{\partialrtial ^\alpha }{u_\mu
},\nabla {\partialrtial ^\alpha }{\rho _\mu }} \right\rangle } }
&+ \gamma \left( {\left\| {\nabla^2 \left[ {{\rho _e},{\rho _i}} \right]} \right\|_{s - 2}^2
+ {{\left\| \nabla[{{\rho _e} - {\rho _i}}] \right\|}^2}} \right) \\
& \leqslant C {\left( {\left\| {\nabla{u_\mu }} \right\|_{s-1}^2
+ \left\| {U} \right\|_s^2 {\left\|
{\nabla [{\rho _\mu },{u_\mu },{\mathbb{T}heta _\mu }]} \right\|_{s - 1}^2 }
} \right)},
\end{split}\end{equation}
\begin{equation}\notag
\begin{split}
\frac{d}
{{dt}}&\sum\limits_{1\leq\left| \alpha \right| \leqslant s - 1}
{\left\langle {{\partialrtial ^\alpha }\left( {{u_e}
- {u_i}} \right),{\partialrtial ^\alpha }E} \right\rangle } + \gamma \left\| \nabla E \right\|_{s - 2}^2 \\
& \leqslant C \left( {\left\| {\nabla\left[ {{u_\mu },{\mathbb{T}heta _\mu }} \right]} \right\|_{s-1}^2
+ \left\|\nabla^2 \rho_\mu \right\|_{s-2}^2
+ \left\| {\nabla{u_\mu }}
\right\|_{s-1}{{\left\| {\nabla^2 B} \right\|}_{s - 3}}} + \left\| U \right\|_s^2 {\left\|
\nabla [{\rho _\mu },{u_\mu },{\mathbb{T}heta _\mu }] \right\|_{s - 1}^2 }
\right),
\end{split}
\end{equation}
and
\begin{equation}\notag
\begin{split}
\frac{d} {{dt}}\sum\limits_{1\leq\left| \alpha \right| \leqslant s
- 2} & { \left\langle {{\partialrtial ^\alpha }E, - \nabla \times
{\partialrtial ^\alpha }B} \right\rangle + \gamma \left\| {\nabla^2 B}
\right\|_{s - 3}^2}\\
& \leqslant C {(\left\| { {\nabla E} } \right\|_{s - 2}^2 +
\left\| {\nabla {{u_\mu }} } \right\|_{s - 1}^2 } +\left\| {\nabla
[{\rho _\mu },{u _\mu }]} \right\|_{s- 1}^2\left\| {{U }}
\right\|_s^2 ).
\end{split}\end{equation}
Now, similarly done as that in
\emph{Step 5} of Theorem \ref{thm2.1}. Let us define the high-order
energy functional as
\begin{equation}\label{4.8}
\begin{split}
\mathcal{E}_s(U(t))=& \left\|\nabla U \right\|_{s-1}^2 +\mathcal {K}_1 \sum\limits_{1\leq \left| \alpha \right|
\leqslant s - 1}
{\sum\limits_{\mu = e,i}
{\left\langle {{\partialrtial ^\alpha }{u_\mu },\nabla {\partialrtial ^\alpha }{\rho _\mu }} \right\rangle } } \\
&
+\mathcal {K}_2 \sum\limits_{1\leq \left| \alpha \right| \leqslant
s - 1} {\left\langle {{\partialrtial ^\alpha }\left( {{u_e}
- {u_i}} \right),{\partialrtial ^\alpha }E} \right\rangle }
+\mathcal {K}_3 \sum\limits_{1\leq \left| \alpha \right| \leqslant
s - 2}
{\left\langle {{\partialrtial ^\alpha }E, - \nabla \times {\partialrtial ^\alpha }B} \right\rangle } ,
\end{split}
\end{equation}
Similarly, one can take $0<\mathcal {K}_3\ll\mathcal
{K}_2\ll\mathcal {K}_1\ll 1$ be sufficiently small with $\mathcal
{K}_2^{\frac{3}{2}}\ll\mathcal {K}_3$, such that
$\mathcal{E}_s^h(U(t))\sim \|\nabla U(t)\|_{s-1}^2 $, that is
$\mathcal{E}_s^h(\cdot)$ is really a high-order energy functional
which satisfies \eqref{2.6}, and moreover, the sum of the four
previously estimates with coefficients corresponding to \eqref{4.8}
gives \eqref{4.7}. Now, we complete the proof of Lemma \ref{L4.2}.
$\Box$
Based on Lemma \ref{L4.2}, one can check that
\begin{equation}\notag
\frac{d}{dt}\mathcal {E}_s^h(U(t))+\gamma \mathcal {E}_s^h(U(t))\leq
C\left(\|\nabla
B\|^2+\|\nabla^s[E,B]\|^2+\|\nabla(\rho_e+\rho_i)\|^2\right),
\end{equation}
which implies
\begin{equation}\label{4.9}
\begin{split}
\mathcal {E}_s^h(U(t))\leq & e^{-\gamma t}\mathcal {E}_s^h(U_0)
\\&
+ C\int_0^t{e^{-\gamma (t-y)}\left(\|\nabla B(y)\|^2+\|\nabla^s
[E,~B](y) \|^2 + \|\nabla (\rho_e+\rho_i)(y) \|^2 \right)}dy.
\end{split}
\end{equation}
Now, let us estimate the time integral term on the right hand side of the
previous inequality. Noting that the equations of $E$ and $B$ in
bipolar non-isentropic Euler-Maxwell system are the same as that in
bipolar isentropic Euler-Maxwell system, similarly as that in
\cite{Duan11b}, we obtain
\begin{lemma}\label{L4.3}
Assume $U=[\rho_\mu,~u_\mu,~\mathbb{T}heta_\mu,~E,~B]$ is the solution of
the initial value problem \eqref{2.2}-\eqref{2.3} with $U_0$
$=[\rho_{\mu0}$, $u_{\mu0}$, $\mathbb{T}heta_{\mu0}$, $E_0$, $B_0]$ which
satisfies \eqref{2.4} in the sense of Proposition \ref{prop2.1}. If
$\omega_{s+6}(U_0)$ is small enough, then, for any $t\geq 0$
\begin{equation}\label{4.10}
{\left\| {\nabla B(t)} \right\|^2} + {\left\| {{\nabla ^s}\left[
{E(t),B(t)} \right]} \right\|^2}+\|\nabla (\rho_e+\rho_i)(t) \|^2
\leqslant C{\omega _{s + 6}}{({U_0})^2}{(1 + t)^{ - \frac{5} {2}}}.
\end{equation}
\end{lemma}
\noindent \emph{Proof.} Utilize the estimate \eqref{3.14} to
\eqref{3.5} of the solution $U_1(t)$ so that
\begin{equation}\notag
\begin{split}
\left\| {\nabla B\left( t \right)} \right\| & \leqslant C{\left( {1 + t} \right)^{ - \frac{5}
{4}}}{\left\| {\left[ {{u_{\mu0}},{E_0},{B_0}} \right]}
\right\|_{{L^1} \cap {{\dot H}^4}}}\\
&\quad + C\int_0^t{\left( {1 +t- y} \right)^{ - \frac{5}
{4}}}{\left\| {\left[ {{g_{2e}(y)-g_{2i}(y)},{g_{4e}(y)-g_{4i}(y)}} \right]} \right\|_{{L^1} \cap {{\dot H}^4}}}dy \\
& \leqslant C{\left( {1 + t} \right)^{ - \frac{5}
{4}}}{\left\| {\left[ {{u_{\mu0}},{E_0},{B_0}} \right]}
\right\|_{{L^1} \cap {{\dot H}^4}}}+ C\int_0^t{\left( {1 +t- y}
\right)^{ - \frac{5}
{4}}}{\left\| {U(y)} \right\|^2_{\max\{5,s\} }}dy\\
& \leqslant C{\left( {1 + t} \right)^{ - \frac{5}
{4}}}{\left\| {\left[ {{u_{\mu0}},{E_0},{B_0}} \right]}
\right\|_{{L^1} \cap {{\dot H}^4}}}+ C\int_0^t{\left( {1 +t- y}
\right)^{ - \frac{5} {4}}}\omega_{s+6}(U_0)^2 \left( {1 + y}
\right)^{ - \frac{3}
{2}} dy\\
& \leqslant C\omega_{s+6}(U_0) \left( {1 + t} \right)^{ - \frac{5}
{4}}
\end{split}
\end{equation}
and
\begin{equation}\notag
\begin{split}
& \left\| {{\nabla ^s}\left[ {E\left( t \right),B\left( t \right)} \right]}
\right\|\\
& \leqslant
C{\left( {1 + t} \right)^{ - \frac{5}
{4}}}{\left\| {\left[ {{u_{\mu0}},\mathbb{T}heta_{\mu0},{E_0},{B_0}}
\right]} \right\|_{{L^2} \cap {{\dot H}^{s + 3}}}}\\&
\quad+C\int_0^t{\left( {1 + t-y} \right)^{ - \frac{5} {4}}}{\left\|
{\left[
{{g_{2e}(y)-g_{2i}(y)},g_{3e}(y)-g_{3i}(y),{g_{4e}(y)-g_{4i}(y)}}
\right]} \right\|_{{L^2} \cap {{\dot
H}^{s + 3}}}}dy\\
& \leqslant
C{\left( {1 + t} \right)^{ - \frac{5}
{4}}}{\left\| {\left[ {{u_{\mu0}},\mathbb{T}heta_{\mu0},{E_0},{B_0}}
\right]} \right\|_{{L^2} \cap {{\dot H}^{s + 3}}}}+C\int_0^t{\left(
{1 + t-y} \right)^{ - \frac{5} {4}}}{\left\| { {U(y)} } \right\|^2_{
{{s +
4}}}}dy\\
& \leqslant
C{\left( {1 + t} \right)^{ - \frac{5}
{4}}}{\left\| {\left[ {{u_{\mu0}},\mathbb{T}heta_{\mu0},{E_0},{B_0}}
\right]} \right\|_{{L^2} \cap {{\dot H}^{s + 3}}}}+C\int_0^t{\left(
{1 + t-y} \right)^{ - \frac{5} {4}}}\omega_{s+6}(U_0)^2 \left( {1
+ y} \right)^{ - \frac{3}
{2}} dy\\
& \leqslant C\omega_{s+6}(U_0) \left( {1 + t} \right)^{ - \frac{5}
{4}}.
\end{split}
\end{equation}
Similarly, utilizing the estimate on $\rho_2$ in
\eqref{3.44} to \eqref{3.6} of the solution $U_2(t)$, we obtain
\begin{equation}\notag
\begin{split}
& \left\| {{\nabla }\left( {\rho_e+\rho_i} \right)\left( t \right)}
\right\|\\
& \leqslant
C{\left( {1 + t} \right)^{ - \frac{5}
{4}}}{\left\| {\left[ {\rho_{\mu0},{u_{\mu0}},\mathbb{T}heta_{\mu0}}
\right]} \right\|_{{L^1} \cap {{\dot H}^{4}}}}\\&
\quad+C\int_0^t{\left( {1 + t-y} \right)^{ - \frac{5} {4}}}{\left\|
{\left[
{g_{1e}(y)+g_{1i}(y)},{{g_{2e}(y)+g_{2i}(y)},g_{3e}(y)+g_{3i}(y)}
\right]} \right\|_{{L^1} \cap {{\dot
H}^{4}}}}dy\\
& \leqslant
C{\left( {1 + t} \right)^{ - \frac{5}
{4}}}{\left\| {\left[ {\rho_{\mu0},{u_{\mu0}},\mathbb{T}heta_{\mu0}}
\right]} \right\|_{{L^1} \cap {{\dot H}^{4}}}}+C\int_0^t{\left(
{1 + t-y} \right)^{ - \frac{5} {4}}}{\left\| { {U(y)} } \right\|^2_{\max\{5,s\} }}dy\\
& \leqslant
C{\left( {1 + t} \right)^{ - \frac{5}
{4}}}{\left\| {\left[ {\rho_{\mu0},{u_{\mu0}},\mathbb{T}heta_{\mu0}}
\right]} \right\|_{{L^2} \cap {{\dot H}^{s + 3}}}}+C\int_0^t{\left(
{1 + t-y} \right)^{ - \frac{5} {4}}}\omega_{s+6}(U_0)^2 \left( {1
+ y} \right)^{ - \frac{3}
{2}} dy\\
& \leqslant C\omega_{s+6}(U_0) \left( {1 + t} \right)^{ - \frac{5}
{4}}.
\end{split}
\end{equation}
Where we have used \eqref{2.12} and the smallness of
$\omega_{s+6}(U_0) $. Now, we complete the proof of Lemma
\ref{L4.3}.
$\Box$\\
Then, after plugging \eqref{4.10} into \eqref{4.9},
we have
\begin{equation}\notag
\mathcal {E}_s^h(U(t))\leq e^{-\gamma t}\mathcal {E}_s^h(U_0)+ C
\omega_{s+6}(U_0)^2(1+t)^{-\frac{5}{2}}.
\end{equation}
Since $\mathcal {E}^h_s(U(t))\sim \|\nabla U(t) \|^2_{s-1}$ holds
true for any $t\geq 0$, \eqref{2.13} follows. Now, we finish the
proof of Proposition \ref{2.2}.
\subsection{Decay rate in $L^q$ .} In this subsection, we are to look for the
decay rates of solutions $U$ $=[\rho_\mu$, $u_\mu$, $\mathbb{T}heta_\mu$,
$E,~B]$ in $L^q$ $_{(2\leq q\leq +\infty)}$ of the initial value
problem \eqref{2.2}-\eqref{2.3} by proving the second part of
Theorem \ref{thm1.1}. Throughout this subsection, we usually suppose
that $\omega_{13}(U_0)$ is small enough. Firstly, for $s\geq 4$,
Proposition \ref{prop2.2} shows that if $\omega_{s+2}(U_0)$ is small
enough,
\begin{equation}\label{4.11}
\|U(t)\|_s\leq C \omega_{s+2}(U_0)(1+t)^{-\frac{3}{4}},
\end{equation}
and if $\omega_{s+6}(U_0)$ is small enough,
\begin{equation}\label{4.12}
\|\nabla U(t)\|_{s-1}\leq C \omega_{s+6}(U_0)(1+t)^{-\frac{5}{4}}.
\end{equation}
Now, let us establish the estimates on $B$, $[u_e-u_i,~E]$,
$u_e+u_i$, $[\rho_e-\rho_i,~\mathbb{T}heta_e-\mathbb{T}heta_i]$ and
$[\rho_e+\rho_i,~\mathbb{T}heta_e+\mathbb{T}heta_i]$
as follows.\\
\noindent \emph{Estimate on $\|B\|_{L^q} $.} For $L^2$ rate, it is
directly from \eqref{4.11} to get
$$\|B(t)\|\leq C \omega_{6} (U_0)(1+t)^{-\frac{3}{4}}.$$
For $L^\infty$ rate, by applying $L^\infty$ estimate on $B$
of \eqref{3.13} to \eqref{3.5}, we obtain
\begin{equation}\notag
\begin{split}
{\left\| {B(t)} \right\|_{{L^\infty }}} \leqslant & C{(1 + t)^{ -
\frac{3} {2}}}{\left\| {[{u_{\mu0}},{E_0},{B_0}]} \right\|_{{L^1}
\cap {{\dot H}^5}}} \\
& + C\int_0^t {{{(1 + t - y)}^{ - \frac{3} {2}}}} {\left\|
{[{g_{2e}}-{g_{2i}},{g_{4e}}-{g_{4i}}](y)} \right\|_{{L^1} \cap
{{\dot H}^5}}}dy.
\end{split}
\end{equation}
Because of \eqref{4.11},
\begin{equation}\notag
\begin{split}
{\left\| {[{g_{2e}}-{g_{2i}},{g_{4e}}-{g_{4i}}](t)} \right\|_{{L^1}
\cap {{\dot H}^5}}} \leqslant C\left\|
{U(t)} \right\|_6^2 \leqslant C{\omega _8}{({U_0})^2}{(1 + t)^{ - \frac{3}
{2}}},\end{split}
\end{equation}
we have
$${\left\| {B(t)} \right\|_{{L^\infty }}} \leqslant C\omega_8 ({U_0}) {(1 + t)^{ -
\frac{3}
{2}}}.$$ Therefore, by $L^2-L^\infty$ interpolation
\begin{equation}\label{4.13}
\|B(t)\|_{L^q}\leq C \omega_8 ({U_0}) {(1 + t)^{ - \frac{3}
{2}+\frac{3}{2q}}},
\end{equation}
for $2\leq q \leq \infty.$\\
\noindent\emph{Estimate on $\|[u_e-u_i,E]\|_{L^q}$.} For $L^2$ rate,
applying the $L^2$ estimate on $u_e-u_i$ and $E$ in \eqref{3.12} to
\eqref{3.5}, one has
\begin{equation}\notag
\begin{split}
\left\| {(u_e-u_i)\left( t \right)} \right\| \leqslant & C{\left( {1 + t} \right)^{ - \frac{5}
{4}}}\left( {\left\| {\left[ {{\rho _{\mu0}},{\mathbb{T}heta _{\mu0}}}
\right]} \right\| + {{\left\|
{\left[ {{u_{\mu0}},{E_0},{B_0}} \right]} \right\|}_{{L^1} \cap {{\dot H}^2}}}} \right) \\
& + C\int_0^t {{\left( {1 + t - y} \right)}^{ - \frac{5}
{4}}} \left\| {\left[ {{g_{1e}-g_{1i}},{g_{3e}-g_{3i}}} \right](y)}
\right\|dy
\\ & + C\int_0^t {{\left( {1 + t - y} \right)}^{ - \frac{5}
{4}}} {{\left\| {\left[ {{g_{2e}-g_{2i}},{g_{4e}-g_{4i}}}
\right](y)} \right\|}_{{L^1} \cap {{\dot H}^2}}}
dy
\end{split}
\end{equation}
and
\begin{equation}\notag
\begin{split}
\left\| {E\left( t \right)} \right\| \leqslant & C{\left( {1 + t}
\right)^{ - \frac{5} {4}}}{\left\| {\left[
{{u_{\mu0}},\mathbb{T}heta_{\mu0},{E_0},{B_0}} \right]} \right\|_{{L^1} \cap
{{\dot
H}^3}}}\\
& + C\int_0^t {{{\left( {1 + t - y} \right)}^{ - \frac{5}
{4}}}{{\left\| {\left[
{{g_{2e}-g_{2i}},{g_{3e}-g_{3i}},{g_{4e}-g_{4i}}} \right](y)}
\right\|}_{{L^1} \cap {{\dot H}^3}}}dy.}
\end{split}
\end{equation}
Since by \eqref{4.11},
\begin{equation}\notag
\begin{split}
\left\| {\left[ {{g_{1e}-g_{1i}}, {g_{3e}-g_{3i}}} \right](t)}
\right\| + {\left\| {\left[
{{g_{2e}-g_{2i}},{g_{3e}-g_{3i}},{g_{4e}-g_{4i}}}
\right](t)} \right\|_{{L^1} \cap {{\dot H}^3}}}\\
\leqslant C\left\| {U(t)} \right\|_4^2 \leqslant C{\omega
_6}{({U_0})^2}{(1 + t)^{ - \frac{3} {2}}},\end{split}
\end{equation}
which implies that
\begin{equation}\label{4.14}
\left\| [u_e-u_i,~E]( t )\right\|\leqslant C{\omega _6}({U_0}){\left( {1 + t} \right)^{ -
\frac{5} {4}}}.
\end{equation}
For $L^\infty$ rate, utilize the $L^\infty$ estimates on $u_e-u_i$ and $E$ in \eqref{3.13} to \eqref{3.5},
we have
\begin{equation}\notag
\begin{split}
{\left\| {(u_e-u_i)\left( t \right)} \right\|_{{L^\infty }}} \leqslant& C{\left( {1 + t} \right)^{ - 2}}
\left( {{{\left\| {\left[ {{\rho _{\mu0}},{\mathbb{T}heta _{\mu0}}} \right]} \right\|}_{{L^1} \cap {{\dot H}^2}}}
+ {{\left\| {\left[ {{u_{\mu0}},{E_0},{B_0}} \right]} \right\|}_{{L^1} \cap {{\dot H}^5}}}} \right) \\
& + C\int_0^t {{\left( {1 + t - y} \right)}^{ -2}} \left\| {\left[ {{g_{1e}-g_{1i}},{g_{3e}-g_{3i}}}
\right](y)} \right\|_{{L^1} \cap {{\dot H}^2}}dy
\\ & + C\int_0^t {{\left( {1 + t - y} \right)}^{ - 2}} {{\left\| {\left[ {{g_{2e}-g_{2i}},{g_{4e}-g_{4i}}}
\right](y)} \right\|}_{{L^1} \cap {{\dot H}^5}}}
dy
\end{split} \end{equation}
and
\begin{equation}\notag
\begin{split}{\left\| {E\left( t \right)} \right\|_{{L^\infty }}}
\leqslant & C{\left( {1 + t} \right)^{ - 2}}{\left\| {\left[
{{u_{\mu0}},{\mathbb{T}heta_{\mu0}},{E_0},{B_0}} \right]} \right\|_{{L^1}
\cap {{\dot
H}^6}}}\\
& + C\int_0^t {{{\left( {1 + t - y} \right)}^{ - 2}}{{\left\|
{\left[ {{g_{2e}-g_{2i}},{g_{3e}-g_{3i}},{g_{4e}-g_{4i}}}
\right](y)} \right\|}_{{L^1} \cap {{\dot H}^6}}}dy}.
\end{split} \end{equation}
Since
\begin{equation}\notag
\begin{split} \|[ {g_{1e}-g_{1i}},{g_{2e}-g_{2i}},&{g_{3e}-g_{3i}},{g_{4e}-g_{4i}}
](t)\|_{L^1} \\& \leq C \|U(t)\|(\|(u_e-u_i)(t)\|+\|\ U(t)\|+\|\nabla U(t)\|)\\
&\leq \omega_{10}(U_0)^2(1+t)^{-\frac{3}{2}},
\end{split} \end{equation}
and
\begin{equation}\notag
\begin{split}
\|[ {g_{1e}-g_{1i}},{g_{2e}-g_{2i}},&{g_{3e}-g_{3i}},{g_{4e}-g_{4i}}
](t)\|_{{{\dot H}^5} \cap {{\dot H}^6}}
\leqslant C\left\| {\nabla U(t)} \right\|_6^2 \leqslant {\omega _{13}}{({U_0})^2}{(1 + t)^{ - \frac{5}
{2}}},
\end{split} \end{equation}
then, one has
$$\|[u_e(t)-u_i(t),~E(t)]\|_{L^\infty}\leq C \omega_{13}(U_0)^2(1+t)^{-\frac{3}{2}}.$$
Therefore, by $L^2-L^\infty$ interpolation
\begin{equation}\label{4.15}
\|[u_e(t)-u_i(t),~E(t)]\|_{L^q}\leq C \omega_{13} ({U_0}) {(1 + t)^{
- \frac{3}{2}+\frac{1}{2q}}},
\end{equation}
for $2\leq q \leq \infty.$
\noindent\emph{Estimate on $\|u_e+u_i\|_{L^q}$.} For $L^2$ rate,
utilizing the $L^2$ estimates on $u_e+u_i$ in \eqref{3.43} to
\eqref{3.6}, we have
\begin{equation}\notag
\begin{split}
\left\| {(u_e+u_i)\left( t \right)} \right\| \leqslant & C{\left( {1 + t} \right)^{ - \frac{5}
{4}}} {\left\| {\left[ {{\rho _{\mu0}},{u _{\mu0}},{\mathbb{T}heta
_{\mu0}}} \right]} \right\|_{{L^1} \cap {{\dot H}^3}} } \\
& + C\int_0^t {{\left( {1 + t - y} \right)}^{ - \frac{5}
{4}}} \left\| {\left[
{{g_{1e}+g_{1i}},g_{2e}+g_{2i},{g_{3e}+g_{3i}}} \right](y)}
\right\|_{{L^1} \cap {{\dot H}^3}}dy.
\end{split}
\end{equation}
Since by \eqref{4.11},
\begin{equation}\notag
\begin{split} \left\| {\left[
{{g_{1e}+g_{1i}},g_{2e}+g_{2i},{g_{3e}+g_{3i}}} \right](t)}
\right\|_{{L^1} \cap {{\dot H}^3}}
\leq C \|U(t)\|^2_4 \leq \omega_{6}(U_0)^2(1+t)^{-\frac{3}{2}},
\end{split} \end{equation}
it follows that
\begin{equation}\notag
\left\| (u_e+u_i)( t )\right\|\leqslant C{\omega _6}({U_0}){\left( {1 + t} \right)^{ -
\frac{5} {4}}}.
\end{equation}
For $L^\infty$ rate, utiliz the $L^\infty$ estimates on $u_e+u_i$ in
\eqref{3.45} to \eqref{3.6}, we have
\begin{equation}\notag
\begin{split}
{\left\| {(u_e+u_i)\left( t \right)} \right\|_{{L^\infty }}} \leqslant& C{\left( {1 + t} \right)^{ - 2}}
{\left\| {\left[ {{\rho _{\mu0}},{u _{\mu0}},{\mathbb{T}heta
_{\mu0}}} \right]} \right\|_{{L^1} \cap {{\dot H}^6}} } \\
& + C\int_0^t {{\left( {1 + t - y} \right)}^{ -2}} \left\| {\left[
{{g_{1e}+g_{1i}},g_{2e}+g_{2i},{g_{3e}+g_{3i}}} \right](y)}
\right\|_{{L^1} \cap {{\dot H}^6}}dy
\end{split} \end{equation}
Since by \eqref{4.11},
\begin{equation}\notag
\begin{split} \|[ {g_{1e}+g_{1i}},{g_{2e}+g_{2i}},&{g_{3e}+g_{3i}}
](t)\|_{{L^1} \cap {{\dot H}^6}} \leq C \|U(t)\|^2_7\leq
\omega_{9}(U_0)^2(1+t)^{-\frac{3}{2}},
\end{split} \end{equation}
it follows that
$$\|u_e(t)+u_i(t)\|_{L^\infty}\leq C \omega_{9}(U_0) (1+t)^{-\frac{3}{2}}.$$
Therefore, by $L^2-L^\infty$ interpolation
\begin{equation}\label{4.16}
\|u_e(t)+u_i(t)\|_{L^q}\leq C \omega_{9} ({U_0}) {(1 + t)^{ -
\frac{3}{2}+\frac{1}{2q}}},
\end{equation}
for $2\leq q \leq \infty.$
Then from \eqref{4.15} and \eqref{4.16} we have
\begin{equation}\label{4.17}
\|u_\mu (t)\|_{L^q}\leq C \omega_{13} ({U_0}) {(1 + t)^{ -
\frac{3}{2}+\frac{1}{2q}}},
\end{equation}
for $2\leq q \leq \infty.$
\noindent\emph{Estimate on
$\|[\rho_e-\rho_i,\mathbb{T}heta_e-\mathbb{T}heta_i]\|_{L^q}$ and
$\|[\rho_e+\rho_i,\mathbb{T}heta_e+\mathbb{T}heta_i]\|_{L^q}$.} For $L^2$ rate,
utilizing the $L^2$ estimates on $\rho_e-\rho_i$ and
$\mathbb{T}heta_e-\mathbb{T}heta_i$ in \eqref{3.12} to \eqref{3.5}, we have
\begin{equation}\label{4.18}
\begin{split}
& \left\| {\left[ {\rho_e-\rho_i}, {\mathbb{T}heta_e-\mathbb{T}heta_i}\right]\left( t \right)}
\right\| \\
&\quad \leqslant
C{e^{ - \frac{t}
{2}}}{\left\| {\left[ {\rho_{\mu0},{u_{\mu0}},\mathbb{T}heta_{\mu0}}
\right]} \right\|} +C\int_0^t{e^{ - \frac{{t - y}} {2}}}{\left\|
{\left[ {g_{1e} -g_{1i} },{{g_{2e} -g_{2i} },g_{3e} -g_{3i}}
\right](y)} \right\|}dy.
\end{split}
\end{equation}
Because of
\begin{equation}\notag
\begin{split}
& \| {\left[ {g_{1e} -g_{1i} },{{g_{2e} -g_{2i} },g_{3e} -g_{3i}}
\right](t)} \|\\
& \leqslant C\left(\left\| {\nabla U(t)}
\right\|_1^2 + \left\| {(u_e+u_i)(t)} \right\|
{\left\| { {B(t)} } \right\|_{{L^\infty }}}\right)
\leqslant C{\omega _{10}}{({U_0})^2}{\left( {1 + t} \right)^{ - \frac{5}
{2}}},
\end{split}
\end{equation}
where \eqref{4.12}, \eqref{4.13} and \eqref{4.16} were used. Then
\eqref{4.18} yields the decay estimate
\begin{equation}\label{4.19}
\begin{split}\left\| {\left[ {\rho_e-\rho_i}, {\mathbb{T}heta_e-\mathbb{T}heta_i}\right]\left( t \right)}
\right\|\leqslant C
{\omega _{10}}{({U_0})}{\left( {1 + t} \right)^{ - \frac{5}
{2}}}.\end{split}
\end{equation}
Similarly for $\|[\rho_e-\rho_i,\mathbb{T}heta_e-\mathbb{T}heta_i]\|$, by utilizing
the ${L^2}$ estimate on $[\rho_e+\rho_i,\mathbb{T}heta_e+\mathbb{T}heta_i]$ in
\eqref{3.43} to \eqref{3.6}, we obtain the decay estimate
\begin{equation}\label{4.20}
\begin{split}\left\| {\left[ {\rho_e+\rho_i}, {\mathbb{T}heta_e+\mathbb{T}heta_i}\right]\left( t \right)}
\right\|\leqslant C
{\omega _{6}}{({U_0})}{\left( {1 + t} \right)^{ - \frac{3}
{4}}}.\end{split}
\end{equation}
Combining \eqref{4.19} and \eqref{4.20}, we obtain
\begin{equation}\label{4.21}
\begin{split}\left\| {\left[ {\rho_\mu}, {\mathbb{T}heta_\mu}\right]\left( t \right)}
\right\|\leqslant C
{\omega _{10}}{({U_0})}{\left( {1 + t} \right)^{ - \frac{3}
{4}}}.\end{split}
\end{equation}
For $L^{\infty}$
rate, by utilizing the $L^{\infty}$ estimate on
$[\rho_e-\rho_i,\mathbb{T}heta_e-\mathbb{T}heta_i]$ in \eqref{3.13} to \eqref{3.5},
we have the decay estimate
\begin{equation}\label{4.22}
\begin{split}
\left\| {\left[ {\rho_e-\rho_i}, {\mathbb{T}heta_e-\mathbb{T}heta_i}\right]\left( t \right)}
\right\|_{L^\infty} \leqslant &
C{e^{ - \frac{t}
{2}}}{\left\| {\left[ {\rho_{\mu0},{u_{\mu0}},\mathbb{T}heta_{\mu0}}
\right]} \right\|_{L^2\cap\dot{H}^2}} \\
& +C\int_0^t{e^{ - \frac{{t -
y}} {2}}}{\left\| {\left[ {g_{1e} -g_{1i} },{{g_{2e} -g_{2i}
},g_{3e} -g_{3i}} \right](y)} \right\|_{L^2\cap\dot{H}^2}}dy.
\end{split}
\end{equation}
Notice that one can check
\begin{equation}\label{4.23}
\begin{split}
&{{\left\| {\left[ {g_{1e} -g_{1i} },{{g_{2e} -g_{2i}
},g_{3e} -g_{3i}} \right](t)} \right\|}_{{L^2}\cap {{\dot H}^2}}}\\
&\quad\quad \leqslant C{\left\|
{\nabla U(t)} \right\|_4}\left({\left\| {\left[ {\rho_\mu (t),\mathbb{T}heta_\mu (t)} \right]} \right\|}+
\left\| {{u_\mu (t)} }
\right\|+
{{{\left\| {\left[ {u_\mu (t),B(t)} \right]} \right\|}_{{L^\infty }}} } \right) \\
&\quad\quad \leq C{\omega _{13}}{({U_0})^2}{(1 +
t)^{ - 2}},\end{split}
\end{equation}
where we have used \eqref{4.12}, \eqref{4.13}, \eqref{4.17} and
\eqref{4.21}. Which implies from \eqref{4.22} that
\[\left\| {\left[ {\rho_e-\rho_i}, {\mathbb{T}heta_e-\mathbb{T}heta_i}\right]\left( t \right)}
\right\|_{L^\infty} \leqslant C{\omega _{13}}{({U_0})}(1 +
t)^{ - 2}.\]
Therefore, by $L^2-L^\infty$ interpolation
\begin{equation}\label{4.24}
\|\left[ {\rho_e-\rho_i}, {\mathbb{T}heta_e-\mathbb{T}heta_i}\right]\|_{L^q}\leq C
\omega_{13} ({U_0}) {(1 + t)^{ -2- \frac{1}{q}}},
\end{equation}
for $2\leq q \leq \infty.$
For $\|\left[ {\rho_e+\rho_i},
{\mathbb{T}heta_e+\mathbb{T}heta_i}\right]\|_{L^\infty}$, by utilizing the
$L^\infty$ estimate on $ \left[ {\rho_e+\rho_i},
{\mathbb{T}heta_e+\mathbb{T}heta_i}\right] $ in \eqref{3.45} to \eqref{3.6}, we have
the decay estimate
\begin{equation}\label{4.25}
\begin{split}\left\| {\left[ {\rho_e+\rho_i}, {\mathbb{T}heta_e+\mathbb{T}heta_i}\right]\left( t \right)}
\right\|_{L^\infty} \leqslant C
{\omega _{8}}{({U_0})}{\left( {1 + t} \right)^{ - \frac{3}
{2}}}.\end{split}
\end{equation}
Then from \eqref{4.20} and \eqref{4.25} we have
\begin{equation}\label{4.26}
\begin{split}\left\| {\left[ {\rho_e+\rho_i}, {\mathbb{T}heta_e+\mathbb{T}heta_i}\right]\left( t \right)}
\right\|_{L^q} \leqslant C
{\omega _{8}}{({U_0})}{\left( {1 + t} \right)^{ - \frac{3}
{2}+\frac{3}{2q}}}.\end{split}
\end{equation}
Thus, \eqref{4.24}, \eqref{4.26}, \eqref{4.15}-\eqref{4.16} and
\eqref{4.13} give \eqref{1.4}, \eqref{1.5}, \eqref{1.6} and \eqref{1.7}, respectively.
Now, we complete the proof of Theorem \ref{thm1.1}.
$\Box$\\
\noindent {\bf Acknowledgments}
This work is supported by the NSFC (Grant no. 11071009), BSF (Grant
no. 1082001), the fund of Beijing education committee of China, and the Foundation
Project of Doctor Graduate Student Innovation of Beijing University
of Technology of China.
\end{document} | math | 118,882 |
\begin{equation}gin{document}
\begin{equation}gin{center} {\bf {{\mathcal L}arge Standard dilations
of $q$-commuting tuples}}
{\epsilon}nd{center}
\begin{equation}gin{center}
{\sc Santanu Dey }
{\bf October 24, 2003}
{\epsilon}nd{center}
\begin{equation}gin{center}
{\underline {\bf Abstract}}
{\epsilon}nd{center}
Here we study dilations of $q$-commuting tuples. In [BBD] the
authors gave the correspondence between the two standard dilations
of commuting tuples and here these results have been extended to
$q$-commuting tuples. We are able to do this when $q$-coefficients
`$q_{ij}$' are of modulus one. We introduce `maximal $q$-commuting
subspace ' of a $n$-tuple of operators and `standard q-commuting
dilation'. Our main result is that the maximal $q$-commuting
subspace of the standard noncommuting dilation of $q$-commuting
tuple is the `standard q-commuting dilation'. We also introduce
$q$-commuting Fock space as the maximal $q$-commuting subspace of
full Fock space and give a formula for projection operator onto
this space. This formula for projection helps us in working with
the completely positive maps arising in our study. The first
version of the Main Theorem (Theorem 19) of the paper for normal
tuples using some tricky norm estimates and then use it to prove
the general version of this theorem.
----------------------------------------------------------------------
\noindent {\sc Key words}: Dilation, $q$-Commuting Tuples,
Complete Positivity
\noindent {\sc Mathematics Subject Classification}: 47A45, 47A20
\setcounter{equation}{0}
\begin{equation}gin{section}{Introduction}
A generalization of contraction operator in multivariate operator
theory is a contractive $n$-tuple which is defined as follows:
\begin{equation}gin{Definition}
{{\epsilon}m A $n$-tuple $\underline{T}=(T_1,\ldots ,T_n)$ of bounded
operators on a Hilbert space ${\mathcal H}$ such that $T_1T_1^*+
\cdots +T_nT_n^*\leq I$ is a {{\epsilon}m contractive $n$-tuple,\/} or a {{\epsilon}m row
contraction\/}.}
{\epsilon}nd{Definition}
Along the lines of [BBD], we will study the dilation of a class of
operator tuples defined as follows:
\begin{equation}gin{Definition}
{{\epsilon}m A $n$-tuple ${\underline{T}} =(T_1,\ldots, T_n)$ is said to be {{\epsilon}m $q$-commuting \/}if $T_j T_i =
q_{ij} T_i T_j$ for all $1 \leq i < j \leq n $, where $q_{ij}$ are
complex numbers.}
{\epsilon}nd{Definition}
Such operator tuples appear often in Quantum Theory ([C] [M]
[Pr]). Here we introduce `maximal $q$-commuting piece' and using
this and a particular representation of permutation group we give
a definition for $q$-commuting Fock space when $q$-coefficients
`$q_{ij}$' are of modulus one. We have this condition for
$q$-coefficient for almost all the results here. This q-commuting
Fock space is different from the twisted Fock space of M.
Bo\.zejko and R. Speicher ([BS1]) or that of P. E. T. Jorgensen
([JSW]). In section 2 we give formula for the projection of full
Fock space onto this space. We obtain a special tuple of
$q$-commuting operators and show that it is unitarily equivalent
to the tuple of shift operators of [BB]. We are able to show that
the range of the operator $A$ defined in equation (2.4) gives an
isometry onto the $q$-commuting Fock Space tensored with a Hilbert
space when ${\underline{T}}$ is a pure contractive tuple (this operator were
used by Popescu and Arveson in [Po3], [Po4], [Ar2] and for
$q$-commuting case by Bhat and Bhattacharyya in [BB]). Using this
we are able to give a condition equivalent to the assertion of the
Main Theorem to hold for $q$-commuting purely contractive tuple.
In section 3 the proof of the particular case of Theorem 19 where
${\underline{T}}$ is also $q$-spherical unitary (introduced in section 3) is
more difficult than the version for commuting tuple and we had to
carefully choose the terms and proceed in a way that `$q_{ij}$' of
the $q$-commuting tuples get absorbed or cancel out when we
simplify the terms. Also unlike [BBD] we had to use an inequality
related to completely positive map before getting the result
through norm estimates. We are not able to generalize section 4 of
[BBD]. In the last section here we calculate the distribution of
$S_i+S_i^*$ with respect to the vacuum expectation and study some
properties of the related operator spaces.
For operator tuples $(T_1, \ldots, T_n)$, we need to consider the
products of the form $T_{\alpha _1}T_{\alpha _2}\cdots$ $
T_{\alpha _m}$, where each $\alpha _k\in \{ 1, 2, \ldots , n\}$.
We would have the following a notation for such products. Let
${\mathcal L}ambda $ denote the set $\{ 1, 2, \ldots , n\}$ and ${\mathcal L}ambda ^m$
denote the $m$-fold cartesian product of ${\mathcal L}ambda $ for $m\geq 1.$
Given $\alpha =(\alpha _1, \ldots , \alpha _m)$ in ${\mathcal L}ambda ^m$,
${\underline{T}} ^{\alpha }$ will mean the operator $T_{\alpha _1}T_{\alpha
_2}\cdots T_{\alpha _m}$. Let ${\overline{t}}ilde{{\mathcal L}ambda}$ denote
$\cup_{n=0}^{\infty} {\mathcal L}ambda^n$, where ${\mathcal L}ambda ^0$ is just the
set $\{ 0\}$ by convention and by ${\underline{T}} ^0$ we would mean the
identity operator of the Hilbert space where $T_i$'s are acting.
Let ${\mathcal {S}}_m$ denote the group of permutation on $m$ symbols
$\{1,2,\cdots, m\}$. For a $q$-commuting tuple
$\underline{T}=(T_1,\ldots ,T_n),$ consider the product
$T_{x_1}T_{x_2}...T_{x_m}$ where $1 \leq x_i \leq n. $ If we
replace a consecutive pair say $T_{x_i}T_{x_{i+1}}$ of operators
in the above product by $q_{x_{i+1}x_i}T_{x_{i+1}}T_{x_i}$ and do
finite number of such operations with different choices of
consecutive pairs of these operators appearing in the subsequent
product of operators after each such operation, we will get a
permutation $\sigma \in {\mathcal {S}}_m$ such that the final product of
operators can be written as
$kT_{x_{\sigma^{-1}(1)}}T_{x_{\sigma^{-1}(2)}}
...T_{x_{\sigma^{-1}(m)}}$ for some $k \in \C,$ i. \@ e.,
$T_{x_1}T_{x_2}...T_{x_m}=
kT_{x_{\sigma^{-1}(1)}}T_{x_{\sigma^{-1}(2)}}
...T_{x_{\sigma^{-1}(m)}}.$ For defining $q$-commuting tuple in
definition 2 we needed the known fact that this $k$ depends only
on $\sigma$ and $x_i,$ and not on the different choice of above
operations that give rise to the same final product of operators
$T_{x_{\sigma^{-1}(1)}}T_{x_{\sigma^{-1}(2)}}
...T_{x_{\sigma^{-1}(m)}}$. It also follows from the Proposition 5
in section 2.
\begin{equation}gin{Definition}
{{\epsilon}m Let ${\mathcal H}, {\mathcal L}$ be two Hilbert spaces such
that ${\mathcal H}$ be a closed subspace of ${\mathcal L}$ and let
${\underline{T}}, {\underline{R}}$ are $n$-tuples of bounded operators on
${\mathcal H}$, ${\mathcal L}$ respectively. Then ${\underline{R}} $ is called
a {{\epsilon}m dilation\/} of ${\underline{T}} $ if
$$R_i^*u=T_i^*u $$
for all $u\in {\mathcal H}, 1\leq i\leq n.$ In such a case ${\underline{T}}$
is called a {{\epsilon}m piece\/} of ${\underline{R}} .$ If ${\underline{T}}$ is a
$q$-commuting tuple ( i.\@ e., $T_jT_i=q_{ij} T_iT_j,$ for all
$i,j$), then it is called a $q$-commuting piece of ${\underline{R}} .$ A
dilation ${\underline{R}} $ of ${\underline{T}}$ is said to be a {{\epsilon}m minimal dilation\/ }
if ${\overline {\mbox span}}\{ {\underline{R}} ^{\alpha }h: \alpha \in {{\overline{t}}ilde
{\mathcal L}ambda}, h\in {\mathcal {H}} \}={\mathcal {L}} .$ And if ${\underline{R}}$ is a tuple of
$n$ isometries and is a minimal dilation of ${\underline{T}}$, then it is
called the {{\epsilon}m minimal isometric dilation \/} or the {{\epsilon}m
standard noncommuting dilation \/} of ${\underline{T}}$.}
{\epsilon}nd{Definition}
A presentation of the standard noncommuting dilation taken from [Po1]
has been used here to proof the main Theorem.
All Hilbert spaces that we consider will be complex and separable.
For a subspace ${\mathcal {H}}$ of a Hilbert space, $P_{{\mathcal {H}}}$ will denote the
orthogonal projection onto ${\mathcal {H}}$. Standard noncommuting dilation
of $n$-tuple of bounded operators,
is unique upto unitary equivalence (refer [Po1-4]). Extensive
study of standard noncommuting dilation was carried out by Popescu.
He generalized many one variable results to multivariable case. It is easy to
see that if ${\underline{R}}$ is a dilation of ${\underline{T}}$ then
\begin{equation}gin{equation}
{\underline{T}} ^{\alpha }({\underline{T}} ^{\begin{equation}ta })^*= P_{{\mathcal H}}{\underline{R}} ^{\alpha
}({\underline{R}} ^{\begin{equation}ta })^*|_{{\mathcal H}},
{\epsilon}nd{equation}
and for any polynomials $p, q$ in $n$-noncommuting variables
$$p({\underline{T}})(q({\underline{T}}))^*=P_{{\mathcal H}}p({\underline{R}} )(q({\underline{R}}))^*|_{{\mathcal
H}}.$$ For a $n$-tuple ${\underline{R}} $ of bounded operators on a Hilbert
space ${\mathcal {M}}$, consider
$${\mathcal C}^q ({\underline{R}} )=\{ {\mathcal {N}} : R_i ~~\mbox {leaves ${\mathcal {N}}$ invariant },
~~R_i^*R_j^*h={\overline{q}}_{ij}R_j^*R_i^*h, \forall h\in {\mathcal {N}}, \forall i,j\}.$$
It is a complete lattice, in the sense that
arbitrary intersections and span closures of arbitrary unions of
such spaces are again in this collection. So it has a
maximal element and we denote it by ${\mathcal {M}} ^q({\underline{R}})$ (or by ${\mathcal {M}}^q $
when the tuple under consideration is clear).
\begin{equation}gin{Definition}
{{\epsilon}m Let ${\underline{R}} $ be a $n$-tuple of operators on a Hilbert space
${\mathcal {M}}$. The $q$-commuting piece ${\underline{R}} ^q=(R_1^q, \ldots , R_n^q)$
obtained by compressing ${\underline{R}}$ to the maximal element ${\mathcal {M}} ^q({\underline{R}})$
of ${\mathcal C}^q ({\underline{R}})$ is called the {{\epsilon}m maximal $q$-commuting
piece\/ } of ${\underline{R}}$. The maximal $q$-commuting piece is said to be
{{\epsilon}m trivial \/ } if ${\mathcal {M}} ^q({\underline{R}})$ is the zero space.}
{\epsilon}nd{Definition}
For any Hilbert space ${\mathcal {K}}$, we have the full Fock space over
${\mathcal {K}}$ denoted by $\Gamma ({\mathcal {K}})$ as,
$$\Gamma ({\mathcal {K}})=\mathbb{C}\oplus {\mathcal {K}} \oplus {\mathcal {K}}^{\otimes ^2}\oplus \cdots
\oplus {\mathcal {K}} ^{\otimes ^m}\oplus \cdots ,
$$
We denote the vacuum vector $1\oplus 0\oplus \cdots $ by
$\omega$. For fixed
$n\geq 2,$ let ${\mathbb{C}}^n$ be the $n$-dimensional complex
Euclidian space with usual inner product and
$\Gamma({\mathbb{C}}^n)$ be the full Fock space over
${\mathbb{C}}^n$. Let $\{e_1, \ldots , e_n\}$ be the standard
orthonormal basis of ${\mathbb{C}}^n$. For $\alpha \in {{\overline{t}}ilde {\mathcal L}ambda}$, $e^{\alpha }$ will
denote the vector $e_{\alpha _1}\otimes e_{\alpha _2}\otimes
\cdots \otimes e_{\alpha _m}$ in the full Fock space $\Gamma
(\mathbb{C}^n)$ and $e^0$ will denote the vacuum vector $\omega $. Then the (left) creation
operators $V_i$ on $\Gamma({\mathbb{C}^n})$ are defined by
\[ V_i x = e_i \otimes x \] where $1 \leq i \leq n$ and $x
\in \Gamma({\mathbb{C}}^n)$ ( here $e_i\otimes \omega$ is
interpreted as $e_i$). It is obvious that the tuple
$\underline{V}= (V_1, \ldots , V_n)$ consists of isometries with
orthogonal ranges and $\sum V_iV_i^*=I-I_0$, where $I_0$ is the
projection on to the vacuum space. Let us define {{\epsilon}m
$q$-commuting Fock space} as the subspace $(\Gamma (\mathbb
{C}^n))^q ({\underline{V}})$ and let it be denoted by $\Gamma_q (\mathbb
{C}^n)$.
Let ${\underline{S}} =(S_1,
\ldots , S_n)$ be the tuple of operators on $\Gamma
_q(\mathbb{C}^n)$ where $S_i$ is the compression of $V_i$ to
$\Gamma _q(\mathbb{C}^n)$:
$$S_i=P_{\Gamma _q(\mathbb{C}^n)}V_i|_{\Gamma _q(\mathbb{C}^n)}.$$
Clearly each $V_i^*$ leaves $\Gamma _q(\mathbb{C}^n)$ invariant.
Then it is easy to see that ${\underline{S}}$ satisfies $\sum
S_iS_i^*=I^q-I^q_0$ (where $I^q, I^q_0$ are identity, projection
onto vacuum space respectively in $\Gamma _q(\mathbb{C}^n)$). So ${\underline{V}}$ and
${\underline{S}}$ are contractive tuples, $S_jS_i=q_{ij}S_iS_j$ for all $1\leq
i, j\leq n,$ and $S_i^*x=V_i^*x$, for $x\in \Gamma
_q(\mathbb{C}^n)$.
The following result gives a description for maximal $q$-commuting
piece.
\begin{equation}gin{Proposition}
Let ${\underline{R}}=(R_1,...,R_n)$ be a $n$-tuple of bounded operators on a
Hilbert space ${\mathcal M}$, ${\mathcal {K}} _{ij}= \overline{span}\{ {\underline{R}}
^{\alpha }(q_{ij} R_iR_j-R_jR_i)h: h\in {\mathcal {M}} , \alpha \in {\overline{t}}ilde
{{\mathcal L}ambda }\}$ for all $1\leq i, j\leq n$, and ${\mathcal {K}}=\overline{span}
\{\cup_{i, j =1}^n {\mathcal {K}} _{ij} \}$. Then ${\mathcal {M}} ^q({\underline{R}}) = {\mathcal {K}} ^{\perp
}$ and ${\mathcal {M}} ^q({\underline{R}})=\{ h\in {\mathcal {M}}: ({\overline{q}}_{ij}
R_j^*R_i^*-R_i^*R_j^*)({\underline{R}} ^\alpha )^*h=0, \forall 1\leq i, j\leq
n, \alpha\in {{\overline{t}}ilde {\mathcal L}ambda}\}.$
{\epsilon}nd{Proposition}
The above Proposition can be easily proved using arguement similar
to the proof of Proposition 4 of [BBD].
\begin{equation}gin{Corollary}
Suppose ${\underline{R}} $, ${\underline{T}} $ are $n$-tuples of operators on two Hilbert
spaces ${\mathcal {L}} , {\mathcal {M}}$. Then the maximal $q$-commuting piece of
$(R_1\oplus T_1, \ldots , R_n\oplus T_n)$ acting on ${\mathcal {L}}\oplus
{\mathcal {M}}$ is $(R_1^q \oplus T_1^q, \ldots , R_n^q\oplus T_n^q)$ acting
on ${\mathcal {L}} ^q\oplus {\mathcal {M}} ^q$ and the maximal $q$-commuting piece of
$(R_1\otimes I, \ldots , R_n\otimes I)$ acting on ${\mathcal {L}} \otimes
{\mathcal {M}}$ is $(R_1^q\otimes I, \ldots , R_n^q\otimes I)$ acting on ${\mathcal {L}}
^q\otimes {\mathcal {M}}.$
{\epsilon}nd{Corollary}
\noindent {\sc Proof:} Clear from Proposition 6.
${\rm I\kern-.25em B}ox$
\begin{equation}gin{Proposition}
Let ${\underline{T}}, {\underline{R}}$ are $n$-tuples of bounded operators on ${\mathcal
H}$, ${\mathcal L}$, with ${\mathcal {H}} \subseteq {\mathcal {L}}$, such that ${\underline{R}}$ is
a dilation of ${\underline{T}}$. Then ${\mathcal {H}} ^q({\underline{T}})={\mathcal {L}} ^q({\underline{R}})\bibitemgcap {\mathcal {H}}$ and
${\underline{R}} ^q$ is a dilation of ${\underline{T}} ^q$.
{\epsilon}nd{Proposition}
\noindent {\sc Proof:} This can be using arguements similar to
proof of Proposition 7 of [BBD].
${\rm I\kern-.25em B}ox$
{\epsilon}nd{section}
\begin{equation}gin{section}{A $q$-Commuting Fock Space}
\setcounter{equation}{0}
For a $q$-commuting $n$-tuple ${\underline{T}}$ on a finite dimensional Hilbert space ${\mathcal {H}}$
say of dimension $m,$ because of the relation
$$\mbox{Spectrum} (T_iT_j)\cup \{0\}=\mbox{Spectrum} (T_jT_i)
\cup \{0\}=\mbox{Spectrum} (q_{ij}T_iT_j)\cup \{0\},$$
we get $q_{ij}$ is either $0$ or $m^{\mbox{th}}$-root of unity.
Here after whenever we deal with $q$-commuting tuples we would
have another condition on the tuples that $|q_{ij}|=1$ for $1\leq
i, j \leq n.$ However Proposition 5, Proposition 6 and Corollary 7 does not
need this assumption. Let $\underline{T}=(T_1,\ldots ,T_n)$ be a $q$-commuting tuple
and consider the product $T_{x_1}T_{x_2}...T_{x_m}$ where $1 \leq
x_i \leq n. $ Let $\sigma \in {\mathcal {S}}_m.$ As transpositions of the type
$(k,k+1), 1 \leq k \leq m-1$ generates ${\mathcal {S}}_m,$ Let $\sigma^{-1}$ be $ {\overline{t}}au_1 \ldots
{\overline{t}}au_s$ where for each $1 \leq i \leq s$ there exist
$k_i$ such
that $1 \leq k_i \leq m-1$ and ${\overline{t}}au_i$ is a transposition of the
form $(k_i, k_i+1).$ Let ${\overline{t}}ilde{\sigma}_i = {\overline{t}}au_{i+1} {\overline{t}}au_{i}
\ldots {\overline{t}}au_{s}$ for $1 \leq i \leq s-1$ and ${\overline{t}}ilde{\sigma}_s$ be
the identity permutation. Let us define $y_i=x_{{\overline{t}}ilde{\sigma}_{i}(k_{i})}$
and $z_i=x_{{\overline{t}}ilde{\sigma}_{i}(k_{i}+1)}$. If we substitute
$T_{y_s}T_{z_s}$ by
$q_{z_sy_s}T_{z_s}T_{y_s}$ corresponding to
${\overline{t}}au_s,$ substitute
$T_{y_{s-1}}T_{z_{s-1}}$ by $q_{z_{s-1}y_{s-1}} T_{z_{s-1}}T_{y_{s-1}}$
corresponding to ${\overline{t}}au_{s-1},$
and so on till we substitute the corresponding
term for ${\overline{t}}au_1,$ we would get
$q^\sigma_1(x) \ldots
q^\sigma_s(x)T_{x_{\sigma^{-1}(1)}}
T_{x_{\sigma^{-1}(2)}} \cdots T_{x_{\sigma^{-1}(m)}}$ where
$q^{\sigma}_i(x) =
q_{z_iy_i}.$ That is
$T_{x_1}T_{x_2} \cdots T_{x_m}=q^\sigma_1(x) \ldots
q^\sigma_s(x)T_{x_{\sigma^{-1}(1)}}
T_{x_{\sigma^{-1}(2)}} \cdots T_{x_{\sigma^{-1}(m)}}.$ Let
$q^{\sigma}(x) = q^\sigma_1(x) \ldots q^\sigma_s(x)$
where $q^{\sigma}_i(x) = q_{z_iy_i}$.
\begin{equation}gin{Proposition}
Let $\underline{T}=(T_1,\ldots ,T_n)$ be a $q$-commuting tuple and
consider the product $T_{x_1}T_{x_2}...T_{x_m}$ where $1 \leq x_i
\leq n. $ Suppose $\sigma \in {\mathcal {S}}_m$ and $ q^\sigma (x)$ be as
defined above. Then
$$q^{\sigma}(x)=\prod q_{x_{\sigma^{-1}(k)}x_{\sigma^{-1}(i)}},$$
where product is over $\{(i,k): 1\leq i < k \leq m, \sigma^{-1}(i)
> \sigma^{-1}(k) \}.$ Moreover $q^{\sigma}(x)$ does not depend on
the choice of $\sigma.$
{\epsilon}nd{Proposition}
\noindent {\sc Proof:} We have
$$q^{\sigma}(x) = q^\sigma_1(x) \ldots q^\sigma_s(x)$$
where $q^{\sigma}_i(x) = q_{z_iy_i}$.
For a pair $i,k$ such that $1\leq i < k \leq
m$ let $k'=\sigma^{-1}(k)$ and $i'=\sigma^{-1}(i).$ Let $\sigma=
{\overline{t}}au_1 \cdots, {\overline{t}}au_s$ and ${\overline{t}}ilde{\sigma}_i$ be as defined above.
If $i'
> k'$ then there are odd number of transpositions ${\overline{t}}au_r$ for $1
\leq r \leq m$ such that they interchange the positions of $i'$
and $k'$ in the image of ${\overline{t}}ilde{\sigma}_r$ when we consider the
composition ${\overline{t}}au_r {\overline{t}}ilde{\sigma}_r.$ And for $1\leq i < k \leq
m$ if $i' < k'$ then there are even number of transpositions
${\overline{t}}au_r$ for $1 \leq r \leq m$ such that they interchange the
positions of $i'$ and $k'$ in the image of ${\overline{t}}ilde{\sigma}_r$ when
we consider the composition ${\overline{t}}au_r{\overline{t}}ilde{\sigma}_r.$ For the
first transposition in ${\overline{t}}au_r$ that interchanges $i'$ and $k',$
the corresponding factor in $q^\sigma(x)$ say $q^\sigma_r(x)$ is
$q_{x_{k'}x_{i'}},$ for the second transposition that interchanges
$i'$ and $k',$
the corresponding factor is
$q_{x_{i'}x_{k'}},$ for the third transposition that interchanges $i'$ and $k',$
the corresponding factor is
$q_{x_{k'}x_{i'}},$ and so on. But
$(q_{x_{i'}x_{k'}})^{-1}=q_{x_{k'}x_{i'}}$ and so
$$q^\sigma(x)= \prod q_{x_{\sigma^{-1}(i)}x_{\sigma^{-1}(k)}},$$ where
product is over $\{(i,k): 1\leq i < k \leq m, \sigma^{-1}(i) >
\sigma^{-1}(k) \}.$
${\rm I\kern-.25em B}ox$
Following similar arguements it is easy to see
that if there exist $\sigma \in {\mathcal {S}}_m$ such that $(x_1,\cdots,x_n)
=(x_{\sigma^{-1}(1)},\cdots,x_{\sigma^{-1}(n)}),$ then $q^\sigma (x)=1.$
Let
$U^{m,q}_{\sigma}$ be defined on $(\mathbb {C}^n)^{\otimes ^m}$ by
\begin{equation}gin{equation}
U^{m,q}_{\sigma}( e_{x_1} \otimes \ldots \otimes e_{x_m} ) =
q^{\sigma}(x)e_{x_{\sigma^{-1}(1)}} \otimes \ldots \otimes
e_{x_{\sigma^{-1}(m)}}
{\epsilon}nd{equation}
on the standard basis vectors and extended
linearly on $(\mathbb {C}^n)^{\otimes ^m}$. As $|q_{ij}|=1$ for $1
\leq i,j \leq n,~ $ $U^m_{\sigma}$ is unitary and $U^m_{\sigma}$
extends uniquely to a unitary operator on $(\mathbb
{C}^n)^{\otimes ^m}$.
Let $$(\mathbb{C}^n)^{{\bigcirc \!\!\!\!q} ^m} = \{u \in (\mathbb
{C}^n)^{\otimes ^m}: U^{m,q}_{\sigma} u = u ~ \forall \sigma \in
{\mathcal {S}}_m \}$$ and $(\mathbb {C}^n)^{{\bigcirc \!\!\!\!q} ^0} = \mathbb{C}$
\begin{equation}gin{Lemma}
The map defined from ${\mathcal {S}} _m$ to $B(({\mathbb C}^n)^{\otimes^m})$
defined by $\sigma {\overline{t}}o U^{m,q}_{\sigma}$ for all $\sigma \in {\mathcal {S}}_m
$ is a representation.
{\epsilon}nd{Lemma}
\noindent {\sc Proof:} Let $\otimes_{i=1}^m
e_{x_i}, \otimes_{i=1}^m e_{y_i} \in (\mathbb {C}^n)^{\otimes
^m}, 1 \leq x_i, y_i \leq n.$ Suppose there exist $\sigma \in {\mathcal {S}}_m$ such that
$\otimes_{i=1}^m e_{y_i} = \otimes_{i=1}^m
e_{x_{\sigma^{-1}(i)}}.$ Then
$\langle U^{m,q}_{\sigma}(\otimes_{i=1}^m e_{x_i}), \otimes_{i=1}^m e_{y_i}
{\rightarrow}ngle = q^\sigma(x)$ and $\langle \otimes_{i=1}^m e_{x_i},
U^{m,q}_{\sigma^{-1}}(\otimes_{i=1}^m
e_{y_i}){\rightarrow}ngle=\overline{q^{({\sigma}^{-1})}(y)}.$ Also
$$q^{(\sigma^{-1})}(y)=\prod q_{y_{\sigma(k)}y_{\sigma(i)}}
= \prod q_{x_{k}x_{i}} $$ where the products are over $\{(i,k):
1\leq i < k \leq m, \sigma(i) > \sigma(k) \}.$ If we substitute
$k=\sigma^{-1}(i')$ and $i=\sigma^{-1}(k')$ in the last term we
get
$$q^{(\sigma^{-1})}(y)= \prod q_{x_{\sigma^{-1}(i')}x_{\sigma^{-1}(k')}}=(\prod
q_{x_{\sigma^{-1}(k')}x_{\sigma^{-1}(i')}})^{-1}=(q^\sigma (x))^{-1}$$
where the products are
over $\{(i',k'): 1\leq i'< k' \leq m, \sigma^{-1}(i')>
\sigma^{-1}(k') \}.$ So
$$q^\sigma(x)=(q^{({\sigma}^{-1})}(y))^{-1}=\overline{q^{({\sigma}^{-1})}(y)}.$$
The last equality holds as $|q_{ij}|=1.$ This implies $\langle
U^{m,q}_{\sigma}(\otimes_{i=1}^m e_{x_i}), \otimes_{i=1}^m e_{y_i}
{\rightarrow}ngle = \langle \otimes_{i=1}^m e_{x_i},
U^{m,q}_{\sigma^{-1}}(\otimes_{i=1}^m e_{y_i}){\rightarrow}ngle.$ If there
does not exist any $\sigma \in {\mathcal {S}}_m$ such that $\otimes_{i=1}^m
e_{y_i} = \otimes_{i=1}^m e_{x_{\sigma^{-1}(i)}}$ then
$$\langle U^{m,q}_{\sigma'}(\otimes_{i=1}^m e_{x_i}), \otimes_{i=1}^m
e_{y_i} {\rightarrow}ngle = 0 = \langle \otimes_{i=1}^m e_{x_i},
U^{m,q}_{(\sigma')^{-1}}(\otimes_{i=1}^m e_{y_i}){\rightarrow}ngle$$ for all
$\sigma' \in {\mathcal {S}}_m.$ So
$(U^{m,q}_{\sigma})^*=U^{m,q}_{\sigma^{-1}}$ for $\sigma \in
{\mathcal {S}}_m,$ when acting on the basis elements of the
$(\mathbb{C}^n)^{\otimes^m},$ and hence is true for all elements
$(\mathbb{C}^n)^{\otimes^m}.$
Next let $\sigma \in {\mathcal {S}}_m$ be equal to $\sigma_1 \sigma_2$ for
some $\sigma_1, \sigma_2 \in {\mathcal {S}}_m.$ We would show that
$U^{m,q}_{\sigma}=U^{m,q}_{\sigma_1}U^m_{\sigma_2}.$ Let $e_x=
e_{x_1} \otimes \ldots \otimes e_{x_m}$ where $x_j \in
\{1,...,n\}$ for $1 \leq j \leq m.$ Let $\sigma_1^{-1} = {\overline{t}}au_1
\ldots {\overline{t}}au_r$ and $\sigma_2^{-1} = {\overline{t}}au_{r+1} \ldots {\overline{t}}au_s$
where for each $1 \leq i \leq s$, there exist $k_i$ such that $1
\leq k_i \leq m-1$ and ${\overline{t}}au_i$ is a transposition of the form
$(k_i, k_i+1).$
$$ U^{m,q}_{\sigma_1}U^{m,q}_{\sigma_2}( e_{x_1} \otimes \ldots \otimes e_{x_m} )
=U^{m,q}_{\sigma_1} (q^{\sigma_2}(x)e_{x_{\sigma_2^{-1}(1)}}
\otimes \ldots \otimes e_{x_{\sigma_2^{-1}(m)}}) = q^{\sigma_1}(z)
q^{\sigma_2}(x)e_{x_{\sigma_2^{-1}\sigma_1^{-1}(1)}} \otimes
\ldots \otimes e_{x_{\sigma_2^{-1}\sigma_1^{-1}(m)}}$$ where
$e_z=e_{z_1} \otimes \ldots \otimes e_{z_m},$ i.\@e,
$z_i=x_{\sigma_2^{-1}(i)}.$ But as $\sigma={\overline{t}}au_1 \ldots {\overline{t}}au_r
{\overline{t}}au_{r+1} \ldots {\overline{t}}au_s$ it is easy to see that
$q^{\sigma}(x)=q^{\sigma_1}(z)q^{\sigma_2}(x)$ using the
definition of $q^{\sigma}(x).$ So we get
$$ U^{m,q}_{\sigma_1}U^{m,q}_{\sigma_2}( e_{x_1} \otimes \ldots \otimes e_{x_m} )
= q^{\sigma}(x) e_{x_{\sigma^{-1}(1)}} \otimes \ldots \otimes
e_{x_{\sigma^{-1}(m)}} = U^{m,q}_{\sigma}( e_{x_1} \otimes \ldots
\otimes e_{x_m} ). $$ And hence $U^{m,q}_{\sigma_1
\sigma_2}=U^{m,q}_{\sigma_1} U^{m.q}_{\sigma_2}.$
${\rm I\kern-.25em B}ox$
Now if we use $Y^{m,q}_{\sigma}$ also to denote a operator in
$\Gamma ({\mathbb C}^n)$ which acts as $U^{m,q}_{\sigma}$ on
$(\mathbb {C}^n)^{\otimes ^m}$ and $I$ on the orthogonal, we get a
representation of $S_m$ on $B(\Gamma(\mathbb{C}^n)).$ In the next
Lemma and Proposition we derive a formula for the projection
operator onto the $q$-commuting Fock space.
\begin{equation}gin{Lemma}
Let $P_m$ be a operator on $(\mathbb {C}^n)^{\otimes ^m}$
defined by
\begin{equation}gin{equation}
P_m = \frac{1}{m !} \sum_{\sigma \in {\mathcal {S}}_m} U^{m,q}_{\sigma}.
{\epsilon}nd{equation}
Then $P_m $ is a projection of $(\mathbb
{C}^n)^{\otimes ^m}$ onto $(\mathbb {C}^n)^{{\bigcirc \!\!\!\!q} ^m}$.
{\epsilon}nd{Lemma}
\noindent {\sc Proof:} First we see that
$$ P_m^* = \frac{1}{m !} \sum_{\sigma \in {\mathcal {S}}_m}
(U^{m,q}_{\sigma})^* = \frac{1}{m !} \sum_{\sigma \in {\mathcal {S}}_m}
U^{m,q}_{\sigma^{-1}} = P_m,$$
Consider a permutation $\sigma' \in {\mathcal {S}}_m.$
$$ P_m U^{m,q}_{\sigma'} = \frac{1}{m !} \sum_{\sigma \in {\mathcal {S}}_m}
U^{m,q}_{\sigma\sigma'}= \frac{1}{m !} \sum_{\sigma \in {\mathcal {S}}_m}
U^{m,q}_{\sigma}= P_m.$$ Similarly $ U^{m,q}_{\sigma'} P_m = P_m.$
So $P^2_m=P_m$ and hence $P_m$ is a projection.
${\rm I\kern-.25em B}ox$
\begin{equation}gin{Proposition}
$\oplus_{m=0}^{\infty} (\mathbb {C}^n) ^{{\bigcirc \!\!\!\!q} ^m} = \Gamma_q
(\mathbb {C}^n)$
{\epsilon}nd{Proposition}
\noindent {\sc Proof:} Let $Q=\oplus_{m=0}^{\infty} P_{m}$ be the
projection of $\Gamma (\mathbb {C}^n)$ onto $\oplus_{m=0}^{\infty}
(\mathbb {C}^n) ^{{\bigcirc \!\!\!\!q} ^m}$ where $P_m$ is a defined in Lemma
9. For transposition $(1,2),$ let us define $U^q_{(1,2)}$ as
$\oplus_{m=o}^{\infty} U^{m,q}_{(1,2)}$ where $U^{0,q}_{(1,2)}=I$
and $U^{1'q}_{(1,2)}=I.$ Let $\otimes_{i=1}^k e_{x_i} \in
(\mathbb {C}^n)^{\otimes ^k}, 1 \leq x_i \leq n.$ Then
$$U^q_{(1,2)}
V_jV_i (\otimes_{i=1}^k e_{x_i})= U^q_{(1,2)} \{e_j \otimes e_i \otimes
(\otimes_{i=1}^k e_{x_i})\} =q_{ij} e_i \otimes e_j \otimes
(\otimes_{i=1}^k e_{x_i})=q_{ij} V_iV_j (\otimes_{i=1}^k e_{x_i}).$$
Next we would show that
$\oplus_{m=0}^{\infty} (\mathbb {C}^n) ^{{\bigcirc \!\!\!\!q} ^m}$ is
left invariant by $V_i^*.$ Let $\otimes_{j=1}^m
e_{x_j} \in (\mathbb {C}^n)^{\otimes ^m},1 \leq x_j \leq n.$
Then $V_i^*\{P_m (\otimes_{j=1}^m e_{x_j})\}$
is zero if none of $x_j$ is equal to $i.$ Otherwise $V_i^* \{ P_m ( \otimes_{j=1}^m e_{x_j})\}$
is some non-zero element belonging to $\oplus_{m=0}^{\infty} (\mathbb {C}^n)
^{{\bigcirc \!\!\!\!q} ^{(m-1)}}$ because of the following.
Let $x_j=i$ iff $j \in \{i_1,...,i_p\},$ and let ${\mathcal {A}}_k$ be the set of all
$\sigma \in S_m$ such that $\sigma^{-1}$ sends $1$ to $i_k, 1 \leq k \leq p,$ then each
element of ${\mathcal {A}}_k$ is a composition ${\overline{t}}au \sigma'$ where ${\overline{t}}au$ is the transposition $(1,i_k)$
and a permutation $\sigma'$ for which $({\sigma'})^{-1}$ keeps $1$ fixed and permutes
rest of the $m-1$ symbols. As $V_i$ are isometries with orthogonal ranges,
\begin{equation}gin{eqnarray*}
V_i^*\{P_m (\otimes_{j=1}^m e_{x_j})\} &=&V_i^*\{\frac{1}{m !}
\sum_{\sigma \in {\mathcal {S}}_m} U^{m,q}_{\sigma}(\otimes_{j=1}^m
e_{x_j})\}
=\frac{1}{m !} \sum_{k=1}^p V_i^*(\sum_{\sigma \in {\mathcal {A}}_{i_k}} U^{m,q}_{\sigma}e_{x_j})\\
&=&\frac{1}{m !} \sum_{k=1}^p V_i^*\{\sum_{{\overline{t}}au \sigma' \in {\mathcal {A}}_{i_k}}
U^{m,q}_{{\overline{t}}au}U^{m,q}_{\sigma'} (\otimes_{j=1}^m e_{x_j})\} \\
&=& \sum_{k=1}^p a_k (x) P_{m-1} (\otimes_{j=1}^m
e_{x_1} \otimes \cdots \otimes \hat{e}_{x_{i_k}} \otimes \cdots \otimes e_{x_{m}})
{\epsilon}nd{eqnarray*}
where $a_k(x)$ are constants and $\hat{e}_{x_{p}}$ denotes the term
$e_{x_1} \otimes \cdots \otimes e_{x{p-1}} \otimes e_{x_{p+1}} \otimes \cdots \otimes e_{x_{m}}.$ This shows that
$\oplus_{m=0}^{\infty} (\mathbb {C}^n) ^{{\bigcirc \!\!\!\!q} ^m}$ is
left invariant by $V_i^*.$
Using these and the results of Lemma 9 we have the following.
Taking $R_i=QV_iQ$ for $\alpha \in {\mathcal L}ambda^m$ we get
\begin{equation}gin{eqnarray*}
R_i R_j R^\alpha \omega = QV_iV_j V^\alpha \omega = QU^{m+2,q}_{(1,2)}
q_{ji}V_jV_i V^\alpha \omega = q_{ji} QV_jV_i V^\alpha \omega
= q_{ji} (R_j)(R_i) R^\alpha \omega.
{\epsilon}nd{eqnarray*}
So $( QV_1Q, \ldots, QV_n Q )$ is a $q$-commuting piece of ${\underline{V}}$.
To show maximality we make use of Proposition 6. Suppose $x \in
\Gamma ({\mathbb{C}}^n)$ and $\langle x, {\underline{V}}
^\alpha(q_{ij}V_iV_j-V_jV_i)y{\rightarrow}ngle =0$ for all $\alpha \in
{\overline{t}}ilde{{\mathcal L}ambda }, 1\leq i,j\leq n$ and $y\in \Gamma
({\mathbb{C}}^n)$. We wish to show that $x\in \Gamma
_q({\mathbb{C}}^n)$. Suppose $x_m$ is the $m$-particle component
of $x$, i.\@e., $x=\oplus _{m\geq 0}x_m$ with $x_m \in
(\mathbb{C}^n)^{{\otimes}^m}$ for $m\geq 0$. For $m\geq 2$ and any
permutation $\sigma $ of $\{1,2, \ldots ,m\}$ we need to show that
the unitary $U^{m,q}_{\sigma }:(\mathbb{C}^n)^{{\otimes}^m} {\overline{t}}o
(\mathbb{C}^n)^{{\otimes}^m}$, defined by equation (2.1) leaves
$x_m$ fixed. Since ${\mathcal {S}}_m$ is generated by the set of
transpositions $\{(1,2), \ldots , (m-1, m)\}$ it is enough to
verify $U^{m,q}_{\sigma }(x_m)=x_m$ for permutations $\sigma $ of
the form $(i, i+1)$. So fix $m$ and $i$ with $m\geq 2$ and $1\leq
i\leq (m-1).$ We have
\begin{equation}gin{equation}
\langle \oplus _px_p, {\underline{V}} ^\alpha(q_{kl}V_kV_l-V_lV_k){\underline{V}}^\begin{equation}ta
\omega {\rightarrow}ngle =0,
{\epsilon}nd{equation}
for every $\begin{equation}ta \in {\overline{t}}ilde{{\mathcal L}ambda}, 1\leq k,l\leq n.$ This implies that
$$\langle x_m, e^\alpha \otimes (q_{kl}e_k\otimes e_l- e_l\otimes e_k)
\otimes e^\begin{equation}ta {\rightarrow}ngle =0$$
for any $\alpha \in {\mathcal L}ambda^{i-1}, \begin{equation}ta \in {\mathcal L}ambda^{m-i-1}.$ So if
$$x_m=\sum a(s,t,\alpha,\begin{equation}ta )e^\alpha
\otimes e_s \otimes e_t \otimes e^\begin{equation}ta$$ where the sum is over
$\alpha \in {\mathcal L}ambda^{i-1}, \begin{equation}ta \in {\mathcal L}ambda^{m-i-1}$ and $1\leq
s,t\leq n,$ and $a(s,t,\alpha,\begin{equation}ta )$ are constants,then for
fixed $\alpha$ and $\begin{equation}ta$ it follows from equation (2.3) that
${\overline{q}}_{kl}a(k,l,\alpha,\begin{equation}ta )=a(l,k,\alpha,\begin{equation}ta )$ or
$q_{lk}a(k,l,\alpha,\begin{equation}ta )=a(l,k,\alpha,\begin{equation}ta ).$ Therefore
\begin{equation}gin{eqnarray*}
U^{m,q}_\sigma ( a(k,l,\alpha,\begin{equation}ta )e^\alpha \otimes e_k \otimes
e_l \otimes e^\begin{equation}ta +a(l,k,\alpha,\begin{equation}ta )e^\alpha
\otimes e_l \otimes e_k \otimes e^\begin{equation}ta ) \\
= q_{lk}a(k,l,\alpha,\begin{equation}ta )e^\alpha \otimes e_l \otimes e_k \otimes e^\begin{equation}ta
+q_{kl}a(l,k,\alpha,\begin{equation}ta )e^\alpha \otimes e_k \otimes e_l \otimes e^\begin{equation}ta\\
= a(l,k,\alpha,\begin{equation}ta )e^\alpha
\otimes e_l \otimes e_k \otimes e^\begin{equation}ta+ a(k,l,\alpha,\begin{equation}ta )e^\alpha \otimes e_k \otimes e_l \otimes e^\begin{equation}ta
{\epsilon}nd{eqnarray*}
This clearly implies $U{m,q}_{\sigma }(x_m)=x_m$, for $\sigma =(i,
i+1)$.
${\rm I\kern-.25em B}ox$
Let ${\mathcal {P}}$ be the vector space of all polynomials in $q$-commuting
variables $z_1, \ldots, z_n$ that is $z_jz_i=q_{ij}z_iz_j.$ Any
multi-index ${\underline{k}}$ is a ordered $n$-tuple of non-negative integers
$(k_1, \ldots,k_n)$. We shall write $k_1+ \ldots + k_n$ as
$|{\underline{k}}|$. The special multi-index which has $0$ in all positions
except the $i^{th}$ one, where it has $1$, is denoted by ${\underline{e}}_i$.
For any non-zero multi-index ${\underline{k}}$ the monomial $z_1^{k_1} \ldots
z_n^{k_n}$ will be denoted by ${\underline{z}}^{{\underline{k}}}$ and for the multi-index
${\underline{k}}= (0, \ldots, 0),$ let ${\underline{z}}^{{\underline{k}}}$ be the complex number $1$.
Let us have the following inner product with it. Declare
${\underline{z}}^{{\underline{k}}}$ and ${\underline{z}}^{{\underline{l}}}$ orthogonal if ${\underline{k}}$ is not the same as
${\underline{l}}$ as ordered multi-indices. Let $$\| {\underline{z}}^{{\underline{k}}} \|^2 =
\frac{k_1 ! \cdots k_n ! }{|k|!}.$$ Note that the following
inner-product is also refered in [BB] in Definition (1.1) in
general case. Now define ${\mathcal {H}}'$ to be the closure of ${\mathcal {P}}$ with
respect to this inner product. Define a tuple ${\underline{S}}'= (S_1',
\ldots, S_n')$ where each $S_i'$ is defined for $f \in {\mathcal {P}}$ by
$$ S_i' f(z_1, \ldots, z_n) = z_i f(z_1, \ldots, z_n)$$
and $S_i$ is linearly extended to ${\mathcal {H}}'$. In the case of our standard
$q$-commuting $n$-tuple ${\underline{S}}$ of operators on $\Gamma_q (\mathbb
{C}^n)$, when ${\underline{k}}=(k_1, \ldots,k_n)$ let ${\underline{S}}^{{\underline{k}}}= S_1^{k_1}
\ldots S_n^{k_n}$ and when ${\underline{k}}= (0, \ldots, 0)$ let ${\underline{S}}^{{\underline{k}}} = 1.$
Using (2.2) and the fact that $V_i$'s are isometries
with orthogonal ranges for ${\underline{k}}=(k_1, \ldots,k_n),|{\underline{k}}|=m$ we get
$$\|{\underline{S}}^{{\underline{k}}}\omega\|= \langle P_m {\underline{V}}^{{\underline{k}}} \omega, {\underline{V}}^{{\underline{k}}} \omega {\rightarrow}ngle =
\langle \frac{1}{|{\underline{k}}| !} \sum_{\sigma \in {\mathcal {S}}_m} U^{m,q}_{\sigma}
{\underline{V}}^{{\underline{k}}} \omega, {\underline{V}}^{{\underline{k}}} \omega {\rightarrow}ngle = \frac{k_1 ! \cdots
k_n !}{|{\underline{k}}|!}.$$ If we denote ${\underline{V}}^{{\underline{k}}} \omega$ by $e_{x_1}
\otimes \cdots \otimes e_{x_m}, 1 \leq x_i \leq n,$ then to get
the last term of the above equation we used the fact that there
are $k_1 ! \cdots k_n !$ permutations $\sigma \in {\mathcal {S}}_m$ such that
$e_{x_1} \otimes \cdots \otimes e_{x_m} = e_{x_{\sigma^{-1}(1)}}
\otimes \cdots \otimes e_{x_{\sigma^{-1}(m)}}$ .
Next we show that the above tuples ${\underline{S}}'$
and ${\underline{S}}$ are unitarily equivalent.
\begin{equation}gin{Proposition} Let ${\underline{S}}'= (S_1', \ldots, S_n')$
be the operator tuples on ${\mathcal {H}}'$ as introduced above and let ${\underline{S}}=
(S_1, \ldots, S_n)$ be the standard $q$-commuting tuple of
operators on $\Gamma_q (\mathbb {C}^n)$. Then there exist unitary
$U: {\mathcal {H}}' {\overline{t}}o {\mathcal {H}}$ such that $US_i'= S_i U$ for $1 \leq i \leq n$.
{\epsilon}nd{Proposition}
\noindent {\sc Proof :} Define $U: {{\mathcal {P}}} {\overline{t}}o \Gamma_q (\mathbb
{C}^n)$ as
$$ U ( \sum_{|{\underline{k}}| \leq s } b_{{\underline{k}}} {\underline{z}}^{{\underline{k}}}) = \sum_{ |{\underline{k}}| \leq
s} b_{{\underline{k}}} {\underline{S}}^{{\underline{k}}}\omega $$
where $b_{{\underline{k}}} {\underline{z}}^{{\underline{k}}} \in {\mathcal {P}},$ $b_{{\underline{k}}}$ are constants.
As $\|{\underline{z}}^{{\underline{k}}} \|=\| {\underline{S}}^{{\underline{k}}}\omega \|$
we have $$\|\sum_{|{\underline{k}}| \leq s } b_{{\underline{k}}} {\underline{z}}^{{\underline{k}}} \|^2= \sum_{|{\underline{k}}| \leq s }
|b_{{\underline{k}}}|^2 \|{\underline{z}}^{{\underline{k}}} \|^2 = \sum_{|{\underline{k}}| \leq s } |b_{{\underline{k}}}|^2 \| {\underline{S}}^{{\underline{k}}}\omega \| ^2
= \|\sum_{|{\underline{k}}| \leq s } b_{{\underline{k}}} {\underline{S}}^{{\underline{k}}}\omega \|^2.$$
So we can extend it
linearly to ${\mathcal H}'$ and $U$ is a unitary.
\begin{equation}gin{eqnarray*}
U S_i'(\sum_{|{\underline{k}}| \leq s } b_{{\underline{k}}} {\underline{z}}^{{\underline{k}}})
&=& U (z_i\sum_{|{\underline{k}}| \leq s } b_{{\underline{k}}} {\underline{z}}^{{\underline{k}}})= q_{1i}^{k_1} \cdots q_{i-1i}^{k_{i-1}}
U(\sum_{|{\underline{k}}| \leq s } b_{{\underline{k}}} {\underline{z}}^{{\underline{k}} + {\underline{e}}_i})\\
&=& q_{1i}^{k_1} \cdots q_{i-1i}^{k_{i-1}} \sum_{|{\underline{k}}| \leq s } b_{{\underline{k}}} {\underline{S}}^{{\underline{k}} + {\underline{e}}_i}
\omega = S_i (\sum_{|{\underline{k}}| \leq s } b_{{\underline{k}}} {\underline{S}}^{{\underline{k}}} \omega )\\
&=& S_i U (\sum_{|{\underline{k}}| \leq s } b_{{\underline{k}}} {\underline{z}}^{{\underline{k}}}),
{\epsilon}nd{eqnarray*}
i.\@ e., $U S_i'= S_i U$ for $1 \leq i \leq n$.
${\rm I\kern-.25em B}ox$
For any complex number $z$, the $z$-commutator of two operators
$A, B$ is defined as:
$$[A, B]_z = AB - zBA.$$ The following Lemma holds for ${\underline{S}}$
as ${\underline{S}}'$ and ${\underline{S}}$ are unitarily equivalent and
the same properties have been proved for ${\underline{S}}'$ in [BB].
\begin{equation}gin{Lemma}
\begin{equation}gin{enumerate}
\item Each monomial ${\underline{S}} ^{{\underline{k}}} \omega$ is an eigenvector for $\sum
S_i^*S_i - I$, so that it is a diagonal operator on the standard
basis. In fact,
$$ \sum_{i=1}^n S_i^* S_i ({\underline{S}}^{\underline{k}} \omega) =
\left(\sum_{i=1}^n \frac{\|{\underline{S}}^{{\underline{k}} + {\underline{e}}_i}\omega \|^2}
{\|{\underline{S}}^{\underline{k}} \omega \|^2} \right) {\underline{S}}^{\underline{k}} \omega .$$ Also $\sum
S_i^*S_i - I$ is compact.
\item The commutator $[S_i^* , S_i]$ is as follows:
$$[S_i^* , S_i] {\underline{S}}^{\underline{k}} \omega = \left( \frac{\|{\underline{S}}^{{\underline{k}} + {\underline{e}}_i}\omega\|^2
}{\|{\underline{S}}^{\underline{k}} \omega \|^2} - \frac{\|{\underline{S}}^{\underline{k}} \omega \|^2}
{\|{\underline{S}}^{{\underline{k}} - {\underline{e}}_i} \omega \|^2} \right) {\underline{S}}^{\underline{k}} \omega, \mbox{
when } k_i \neq 0.$$
If $k_i = 0$, then $[S_i^* , S_i] {\underline{S}}^{\underline{k}} \omega =
S_i^*S_i {\underline{S}}^{\underline{k}} \omega= \frac{\|{\underline{S}}^{{\underline{k}} + {\underline{e}}_i} \omega
\|^2}{\|{\underline{S}}^{\underline{k}} \omega \|^2} {\underline{S}}^{\underline{k}} \omega.$
\item $[S_i^*, S_j]_{q_{ij}}$ is compact for all
$1\leq i, j \leq n$.
{\epsilon}nd{enumerate}
{\epsilon}nd{Lemma}
The map $U^{m,q}:{\mathcal {S}}_m {\overline{t}}o \Gamma(\mathbb {C}^n)$ given by
$$ U^{m,q}(\sigma) = U^{m,q}_{\sigma}$$
gives the representation of ${\mathcal {S}}_m$ on $\Gamma(\mathbb{C}^n).$ It
is easy to see that for all $q=(q_{ij})_{n {\overline{t}}imes n}, |q_{ij}|=
1,$ the representations are isomorphic or similar by checking the
characters of the representaions. They have same characters. But
for the representations of permuation groups it follows that they
are unitarily equivalent representations. So there exist unitary
$W^q : \Gamma (\mathbb {C}^n) {\overline{t}}o \Gamma (\mathbb {C}^n)$ such
that
\begin{equation}gin{equation}
W^q P_{\Gamma_S (\mathbb{C}^n)} =P_{\Gamma_q (\mathbb{C}^n)}W^q.
{\epsilon}nd{equation}
This $W^q$ is not unique as for $k \in \C$ such that $|k|=1,$ the
operator $k W^q$ is also a unitary which satisfy equation (2.4).
We will give one such $W^q$ explicitely.
For $m \in {\rm I\kern-.23em N}, y_i \in {\mathcal L}ambda$ define $W^{q,m}$ over
$(\C^n)^{\otimes ^m}$ as
$$W^{q,m} ( e_{y_1}\otimes
\ldots \otimes e_{y_m})
= q^{\sigma^{-1}}(x) e_{y_1}\otimes \ldots
\otimes e_{y_m}.$$ where $x=(x_1,\cdots,x_m)$ is the tuple got by
rearranging $(y_1,\cdots,y_m)$ in nondecreasing order and
$\sigma\in {\mathcal {S}}_m$ such that $y_i=x_{\sigma(i)}.$ From Proposition
8 its clear that $q^{\sigma^{-1}}(x)$ does not depend upon the
choice of $\sigma.$ And
\begin{equation}gin{eqnarray*}
W^{m,q} P_{\Gamma _S(\mathbb{C}^n)} (e_{y_1}\otimes \ldots \otimes
e_{y_m}) &=& W^{m,q}(\frac{1}{m!} \sum_{{\overline{t}}au \in {\mathcal {S}}_m }
e_{y_{{\overline{t}}au^{-1} (1)}}\otimes \ldots
\otimes e_{y_{{\overline{t}}au^{-1}(m)}})\\
&= & \frac{1}{m!} \sum_{{\overline{t}}au \in {\mathcal {S}}_m } q^{({\overline{t}}au^{-1}
\sigma)^{-1}}(x) e_{y_{{\overline{t}}au^{-1} (1)}}
\otimes \ldots \otimes e_{y_{{\overline{t}}au^{-1} (m)}}\\
&=& \frac{1}{m!} \sum_{{\overline{t}}au \in {\mathcal {S}}_m } q^{\sigma^{-1} {\overline{t}}au}(x)
e_{y_{{\overline{t}}au^{-1}(1)}}\otimes
\ldots \otimes e_{y_{{\overline{t}}au^{-1} (m)}} \\
&= & \frac{1}{m!} \sum_{{\overline{t}}au \in {\mathcal {S}}_m }
q^{{\overline{t}}au}(x_{\sigma(1)},\cdots,x_{\sigma(m)})q^{ \sigma^{-1}}(x)
e_{y_{{\overline{t}}au^{-1}(1)}}\otimes
\ldots \otimes e_{y_{{\overline{t}}au^{-1} (m)}}\\
&= & P_{\Gamma _q(\mathbb {C}^n)} q^{\sigma^{-1}}(x)
e_{y_1}\otimes \ldots \otimes
e_{y_m}\\
& = & P_{\Gamma _q(\mathbb {C}^n)}W^{m,q} (e_{y_1}\otimes \ldots
\otimes e_{y_m}).
{\epsilon}nd{eqnarray*}
So, $W^{m,q} P_{\Gamma _S(\mathbb {C}^n)}=P_{\Gamma _q(\mathbb
{C}^n)}W^{m,q}$ and $W^q= \oplus^\infty_{m=0} W^{m,q}$ gives the
required unitary which satisfy equation (2.4)(here $W^{0,q}=I$).
Also note that for $\Gamma_q(\C^n)$ and $\Gamma_{q'}(\C^n)$ we
have unitary $W^{q'}(W^{q})^*$ such that
$$W^{q'}(W^{q})^* P_{\Gamma_q(\mathbb {C}^n)}=
P_{\Gamma_{q'}(\mathbb {C}^n)} W^{q'}(W^{q})^*$$
{\epsilon}nd{section}
\begin{equation}gin{section}{Dilation of $q$-Commuting Tuples and the Main Theorem}
\setcounter{equation}{0}
\begin{equation}gin{Definition}
{{\epsilon}m Let ${\underline{T}} =(T_1, \ldots , T_n)$ be a contractive tuple on a
Hilbert space ${\mathcal {H}} .$ The operator ${\rm I\kern-.25em D}elta _{{\underline{T}}}=
[I-(T_1T_1^*+\cdots +T_nT_n^*)]^{\frac{1}{2}}$ is called the {{\epsilon}m
defect operator \/} of ${\underline{T}}$ and the subspace $\overline {{\rm I\kern-.25em D}elta
_{\underline{T}}({\mathcal {H}})}$ is called the {{\epsilon}m defect space\/} of ${\underline{T}} .$ The
tuple ${\underline{T}} $ is said to be {{\epsilon}m pure \/ } if $\sum _{\alpha \in
{\mathcal L}ambda ^m}{\underline{T}}^{\alpha }({\underline{T}}^{\alpha})^*$ converges to zero in
strong operator topology as $m$ tends to infinity. }
{\epsilon}nd{Definition}
When $\sum T_iT_i^*=I$, we have $\sum _{\alpha \in {\mathcal L}ambda
^m}{\underline{T}}^{\alpha }({\underline{T}}^{\alpha})^* =I$ for all $m$ and hence ${\underline{T}}$
is not pure. Let ${\underline{T}} $ be a pure contractive tuple on ${\mathcal {H}} .$
Take ${\overline{t}}ilde {{\mathcal {H}}}=\Gamma (\mathbb{C}^n)\otimes \overline{{\rm I\kern-.25em D}elta
_{{\underline{T}}}({\mathcal {H}})},$ and define an operator $A:{\mathcal {H}} {\overline{t}}o {\overline{t}}ilde {{\mathcal {H}}}$ by
\begin{equation}gin{equation}
Ah= \sum _{\alpha }e^{\alpha }\otimes {\rm I\kern-.25em D}elta _{{\underline{T}}}({\underline{T}} ^{\alpha
})^*h,
{\epsilon}nd{equation}
where the sum is taken over all $\alpha \in {\overline{t}}ilde {{\mathcal L}ambda}$
(this operator was used by Popescu and Arveson in [Po3], [Po4],
[Ar2] and for $q$-commuting case by Bhat and Bhattacharyya in
[BB]). $A$ is an isometry and we have ${\underline{T}} ^\alpha =A^*({\underline{V}}
^\alpha \otimes I)A$ for all $\alpha \in {{\overline{t}}ilde {\mathcal L}ambda}$ (see
[Po4]). Also the tuple ${{\overline{t}}ilde {\underline{V}}}= (V_1\otimes I, \ldots ,
V_n\otimes I)$ of operators on ${{\overline{t}}ilde {\mathcal {H}} }$ is a realization of
the minimal noncommuting dilation of ${\underline{T}}$.
Let $C^*({\underline{V}})$, and $C^*({\underline{S}})$ be unital $C^*$-algebras generated
by tuples ${\underline{V}}$ and ${\underline{S}}$ (defined in the Introduction) on Fock spaces $\Gamma
({\mathbb{C}}^n)$ and $ \Gamma _q({\mathbb{C}}^n)$ respectively.
For any $\alpha , \begin{equation}ta \in
{{\overline{t}}ilde {\mathcal L}ambda }$, ${\underline{V}} ^{\alpha }(I-\sum V_iV_i^*) ({\underline{V}} ^{\begin{equation}ta
})^*$ is the rank one operator $x\mapsto \langle e^{\begin{equation}ta },
x{\rightarrow}ngle e^\alpha ,$ formed by basis vectors $e^{\alpha },
e^{\begin{equation}ta }$ and so $C^*({\underline{V}})$ contains all compact operators. Similarly
we see that $C^*({\underline{S}})$ also contains all compact
operators of $\Gamma _q({\mathbb{C}}^n)$. As $V_i^*V_j=\delta
_{ij}I,$ it is easy to see that $C^*({\underline{V}})= ~~\overline {\mbox
{span}}~~\{{\underline{V}}^\alpha ({\underline{V}}^\begin{equation}ta)^*: \alpha , \begin{equation}ta \in {{\overline{t}}ilde
{\mathcal L}ambda}\}.$ As $q_{ij}$-commutators $[S_i^*, S_j]_{q_{ij}}$ are
compact for all $i,j$, we can also get
$C^*({\underline{S}})=~~\overline {\mbox {span}}~~\{{\underline{S}}^\alpha ({\underline{S}}^\begin{equation}ta)^*:
\alpha , \begin{equation}ta \in {{\overline{t}}ilde {\mathcal L}ambda}\}.$
Consider a contractive tuple ${\underline{T}} $ on a Hilbert space ${\mathcal {H}}$.
For $0< r <1 $ the tuple $r{\underline{T}}= (rT_1, \ldots , rT_n)$ is clearly
a pure contraction. So by equation (2.4) we have an isometry $A_r: {\mathcal {H}} {\overline{t}}o
\Gamma ({\mathbb{C}}^n)\otimes \overline {{\rm I\kern-.25em D}elta _r({\mathcal {H}})}$ defined
by
$$A_rh = \sum _{\alpha }e^{\alpha }\otimes {\rm I\kern-.25em D}elta _r((r{\underline{T}}) ^\alpha )^*h, ~~h\in {\mathcal {H}} ,$$
where ${\rm I\kern-.25em D}elta _r=(I-r^2\sum T_iT_i^*)^{\frac{1}{2}}.$ So for every
$0<r<1$ we have a completely positive map $\psi _r: C^*({\underline{V}}){\overline{t}}o
{\mathcal {B}}({\mathcal {H}})$ defined by
$\psi _r(X)=A_r^*(X\otimes I)A_r, ~~X\in C^*({\underline{V}}).$
By taking limit as $r$ increases to 1 (See [Po1-4] for details),
we get a unital completely positive map $\psi $ from $C^*({\underline{V}})$
to ${\mathcal {B}}({\mathcal {H}})$ (Popescu's Poisson transform) satisfying
$$\psi ({\underline{V}}^\alpha ({\underline{V}}^\begin{equation}ta)^*)= {\underline{T}}^\alpha ({\underline{T}}^\begin{equation}ta )^* ~~\mbox{for} ~ \alpha , \begin{equation}ta \in
{{\overline{t}}ilde {\mathcal L}ambda}.$$ As $C^*({\underline{V}})= ~~\overline {\mbox
{span}}~~\{{\underline{V}}^\alpha ({\underline{V}}^\begin{equation}ta)^*: \alpha , \begin{equation}ta \in {{\overline{t}}ilde
{\mathcal L}ambda}\},$ $\psi $ is the unique such completely positive map.
Let the minimal Stinespring dilation of $\psi $ be unital
$*$-homomorphism $\pi : C^*({\underline{V}}){\overline{t}}o {\mathcal {B}}({{\overline{t}}ilde {\mathcal {H}} })$ where
${{\overline{t}}ilde {\mathcal {H}}}$ is a a Hilbert space containing ${\mathcal {H}} $, and
$$\psi (X)=P_{{\mathcal {H}}}\pi(X)|_{{\mathcal {H}}} ~~\forall X\in C^*({\underline{V}}),$$
and $\overline {\mbox {span}}~\{ \pi(X)h: X\in C^*({\underline{V}}), h\in
{\mathcal {H}}\}= {{\overline{t}}ilde {\mathcal {H}}}.$ Let ${{\overline{t}}ilde {\underline{V}}}=({{\overline{t}}ilde V_1}, \ldots ,
{{\overline{t}}ilde V_n})$ where ${{\overline{t}}ilde V_i}= \pi(V_i)$ and so ${{\overline{t}}ilde {\underline{V}}}$
is the unique standard noncommuting dilation of ${\underline{T}}$ and clearly
${\overline{t}}ilde {(V_i)}^*$ leaves ${\mathcal {H}} $ invariant. If ${\underline{T}}$ is $q$-commuting,
by considering $C^*({\underline{S}})$ instead of
$C^*({\underline{V}})$, and restricting $A_r$ in the range to $\Gamma
_q({\mathbb{C}}^n)$, and taking limits as $r$ increases to 1 as before we
would get the unique unital completely positive map $\phi :
C^*({\underline{S}}){\overline{t}}o {\mathcal {B}} ({\mathcal {H}}),$ (also see [BB]) satisfying
\begin{equation}gin{equation}
\phi ({\underline{S}}^\alpha ({\underline{S}}^\begin{equation}ta)^*)= {\underline{T}}^\alpha ({\underline{T}}^\begin{equation}ta )^* ~~~~~
\alpha , \begin{equation}ta \in {{\overline{t}}ilde {\mathcal L}ambda}.
{\epsilon}nd{equation}
\begin{equation}gin{Definition}
{{\epsilon}m Let ${\underline{T}}$ be a $q$-commuting tuple. Then we have a unique
unital completely positive map $\phi : C^*({\underline{S}}){\overline{t}}o {\mathcal {B}} ({\mathcal {H}})$
satisfying equation (3.2). Consider the minimal Stinespring
dilation of $\phi .$ Here we have a Hilbert space ${\mathcal {H}} _1 $
containing ${\mathcal {H}}$ and a unital $*$-homomorphism $\pi _1 :
C^*({\underline{S}}){\overline{t}}o {\mathcal {B}}({\mathcal {H}} _1),$ such that
$$ \phi (X)=P_{{\mathcal {H}}}\pi _1(X)|_{{\mathcal {H}}} ~~~~\forall X\in C^*({\underline{S}}),$$
and $\overline {\mbox {span}}~\{ \pi _1(X)h: X\in C^*({\underline{S}}), h\in
{\mathcal {H}} \}= { {\mathcal {H}} _1}.$ Let ${\overline{t}}ilde{S_i}=\pi_1(S_i)$ and
${\overline{t}}ilde{{\underline{S}}}=({\overline{t}}ilde{S_1},\ldots,{\overline{t}}ilde{S_n}).$ Then ${\overline{t}}ilde
{{\underline{S}}}$ is called the {{\epsilon}m standard $q$-commuting dilation \/} of
${\underline{T}}. $}
{\epsilon}nd{Definition}
Standard $q$-commuting dilation is also unique up to
unitary equivalence as minimal Stinespring dilation is unique up to unitary
equivalence.
\begin{equation}gin{Lemma} Suppose ${\underline{T}}=(T_1, \cdots , T_n)$ is a $q$-commuting tuple on a
Hilbert Space ${\mathcal {H}}$ and let $A$ be the operator introduced in
Equation (3.1). Then there exist a Hilbert space $K$ such that
$(S_1\otimes I_{{\mathcal {K}}}, \ldots , S_n\otimes I_{{\mathcal {K}}})$ is a dilation
of ${\underline{T}}$ and dim $({\mathcal {K}})=$ rank $~({\rm I\kern-.25em D}elta _{{\underline{T}}}).$
{\epsilon}nd{Lemma}
\noindent {\sc Proof:} $A(h)=\sum_\alpha e^\alpha \otimes {\rm I\kern-.25em D}elta_{\underline{T}} ({\underline{T}}^\alpha)^*h $ for $h \in {\mathcal {H}}$
where the sum is over $\alpha \in {\overline{t}}ilde{{\mathcal L}ambda}.$
For a given ${\underline{k}}=(k_1,\cdots,k_n)$ such that $|{\underline{k}}|=m$ let us denote
$e_1^{k_1} \otimes \cdots \otimes e_n^{k_n}$ by $e_{x_1}\otimes \cdots \otimes
e_{x_m}, 1 \leq x_m \leq n$ in the following calculation.
\begin{equation}gin{eqnarray*}
A(h)&=& \sum_{m=0}^\infty \sum_{\sigma \in {\mathcal {S}}_m}
e_{x_{\sigma^{-1}(1)}} \cdots e_{x_{\sigma^{-1}(m)}} \otimes
{\rm I\kern-.25em D}elta_{\underline{T}} ( T_{x_{\sigma^{-1}(1)}} \cdots T_{x_{\sigma^{-1}(m)}})^*h\\
&=& \sum_{m=0}^\infty \sum_{\sigma \in {\mathcal {S}}_m}
e_{x_{\sigma^{-1}(1)}} \cdots e_{x_{\sigma^{-1}(m)}} \otimes
{\rm I\kern-.25em D}elta_{\underline{T}} \overline{(q^\sigma (x))^{-1}}( T_{x_1} \cdots T_{x_m})^*h \\
&=& \sum_{m=0}^\infty \sum_{\sigma \in {\mathcal {S}}_m} q^\sigma (x)
e_{x_{\sigma^{-1}(1)}} \cdots e_{x_{\sigma^{-1}(m)}} \otimes
{\rm I\kern-.25em D}elta_{\underline{T}} ( T_{x_1} \cdots T_{x_m})^*h \\
&=& \sum_{m=0}^\infty (m!) P_m e_{x_1} \cdots e_{x_m} \otimes
{\rm I\kern-.25em D}elta_{\underline{T}} ( T_{x_1} \cdots T_{x_m})^*h
{\epsilon}nd{eqnarray*}
So the range of $A$ is contained in ${{\overline{t}}ilde
{\mathcal {H}}}_q = \Gamma _q({\mathbb{C}}^n)\otimes \overline {{\rm I\kern-.25em D}elta
_{{\underline{T}}}({\mathcal {H}})}$. In other words now ${\mathcal {H}} $ can be considered as a
subspace of ${{\overline{t}}ilde {\mathcal {H}}}_q$. Moreover, ${\overline{t}}ilde
{{\underline{S}}}=(S_1\otimes I, \ldots , S_n\otimes I)$, as a tuple of
operators in ${{\overline{t}}ilde {\mathcal {H}}}_q$ is the standard $q$-commuting
dilation of $(T_1, \ldots T_n).$ More abstractly we can get a
Hilbert space ${\mathcal {K}}$ such that ${\mathcal {H}}$ can be isometrically embedded
in $\Gamma _q(\mathbb{C}^n)\otimes {\mathcal {K}}$ and $(S_1\otimes I_{{\mathcal {K}}},
\ldots , S_n\otimes I_{{\mathcal {K}}})$ is a dilation of ${\underline{T}}$ and
$\overline {\mbox {span}}\{({\underline{S}} ^\alpha\otimes I_{{\mathcal {K}}})h: h\in
{\mathcal {H}}, \alpha \in {\overline{t}}ilde {{\mathcal L}ambda }\}=\Gamma _q(\mathbb{C}^n)\otimes
{\mathcal {K}}$. There is a unique such dilation and up to unitary equivalence
and dim $({\mathcal {K}})=$ rank $~({\rm I\kern-.25em D}elta _{{\underline{T}}}).$
${\rm I\kern-.25em B}ox$
\begin{equation}gin{Theorem}
Let ${\underline{T}} $ be a pure contractive tuple on a Hilbert space ${\mathcal {H}}$.
\begin{equation}gin{enumerate}
\item Then the maximal $q$-commuting piece ${{\overline{t}}ilde {\underline{V}}}^q$ of the
standard noncommuting dilation ${{\overline{t}}ilde {\underline{V}}}$ of ${\underline{T}}$ is a
realization of the
standard $q$-commuting dilation of ${\underline{T}} ^q$ if and only
if $\overline {{\rm I\kern-.25em D}elta _{\underline{T}}({\mathcal {H}})}=\overline {{\rm I\kern-.25em D}elta _{\underline{T}}({\mathcal {H}}
^q({\underline{T}}))}.$ And if $\overline {{\rm I\kern-.25em D}elta _{\underline{T}}({\mathcal {H}})}=\overline {{\rm I\kern-.25em D}elta _{\underline{T}}({\mathcal {H}}
^q({\underline{T}}))}$ then rank $({\rm I\kern-.25em D}elta _{{\underline{T}}})=$ rank $({\rm I\kern-.25em D}elta
_{{\underline{T}}^q})=$ rank $({\rm I\kern-.25em D}elta _{{{\overline{t}}ilde {\underline{V}}}})=$ rank $({\rm I\kern-.25em D}elta
_{{{\overline{t}}ilde {\underline{V}}}^q}).$
\item Let the standard noncommuting dilation of ${\underline{T}} $ be ${{\overline{t}}ilde {\underline{V}}}$.
If rank ${\rm I\kern-.25em D}elta _{{\underline{T}}}$
and rank ${\rm I\kern-.25em D}elta _{{\underline{T}} ^q}$ are finite and equal then ${{\overline{t}}ilde {\underline{V}}}^q$
is a realization of the standard $q$-commuting dilation of ${\underline{T}} ^q$.
{\epsilon}nd{enumerate}
{\epsilon}nd{Theorem}
\noindent {\sc Proof:}
The proof is similar to the proofs of that of Theorem 10 and Remark 11
of [BBD].
${\rm I\kern-.25em B}ox$
If the ranks of both ${\rm I\kern-.25em D}elta _{{\underline{T}}}$
and ${\rm I\kern-.25em D}elta _{{\underline{T}} ^q}$ are infinite then we can not ensure that
$\overline {{\rm I\kern-.25em D}elta _{\underline{T}}({\mathcal {H}})}=\overline {{\rm I\kern-.25em D}elta _{\underline{T}}({\mathcal {H}}
^q({\underline{T}}))}$ and hence can not ensure the converse of the last Theorem,
as seen by the following example.
For any $n \geq 2$ consider the Hilbert space ${\mathcal {H}} _0=\Gamma_q
(\mathbb {C}^n) \otimes {\mathcal {M}}$ where ${\mathcal {M}}$ is of infinite dimension
and let ${\underline{R}}=(S_1 \otimes I,\cdots, S_n \otimes I)$ be a $q$-commuting
pure contractive $n$-tuple. Infact one can take any ${\underline{R}}$ to be any $q$-commuting
pure $n$-tuple on some Hilbert space ${\mathcal {H}}_0$ with
$\overline {{\rm I\kern-.25em D}elta _{\underline{R}}({\mathcal {H}} _0)}$ of infinite dimensional. Suppose
$P_k=(p^k_{ij})_{n {\overline{t}}imes n}$ for $1 \leq k \leq n$ are $n {\overline{t}}imes n$ matrices
with complex entries such that
$$p^k_{ij}= \left \{
\begin{equation}gin{array}{ccc}
t_k & \mbox {~if~} i=k,j=k+1 \\
0 & \mbox{~otherwise~}
{\epsilon}nd{array} \right . \mbox{~for~} 1 \leq k < n \mbox{~and~}
p^n_{ij}= \left\{ \begin{equation}gin{array}{ccc}
t_n & \mbox {~if~}i=n,j=1\\
0 & \mbox{~otherwise~}
{\epsilon}nd{array} \right .$$
where $t_k$'s are complex numbers satisfying $0< |t_k| <1.$
Let ${\mathcal {H}} ={\mathcal {H}} _0\oplus \mathbb{C}^n$. Take ${\underline{T}}=(T_1, \cdots, T_n)$ where $T_k$
for $1 \leq k \leq n$ be operators on ${\mathcal {H}} $ defined by
$$T_k=\left[\begin{equation}gin{array}{ccc}
R_k & \\
& P_k
{\epsilon}nd{array}\right] \mbox{~for~} 1\leq k \leq n.$$
So ${\underline{T}}$
is a pure contractive tuple, the maximal $q$-commuting piece of
${\underline{T}}$ is ${\underline{R}}$ and ${\mathcal {H}} ^q({\underline{T}})={\mathcal {H}} _0$ (by Corollary 7). Here
rank$~({\rm I\kern-.25em D}elta _{{\underline{T}}^q})=$ rank$~({\rm I\kern-.25em D}elta _{{\underline{T}}})=\infty$ but
$\overline {{\rm I\kern-.25em D}elta _{\underline{T}}({\mathcal {H}})}=\overline {{\rm I\kern-.25em D}elta _{\underline{R}}({\mathcal {H}}
_0)}\oplus \mathbb{C}^n.$ But the converse of Theorem 18 holds
when rank of ${\rm I\kern-.25em D}elta_{{\underline{T}}}$ is finite.
Consider the case when ${\underline{T}} $ is a
$q$-commuting tuple on Hilbert space ${\mathcal {H}} $ satisfying $\sum
T_iT_i^* =I$.
As $C^*({\underline{S}})$ contains the ideal of all
compact operators by standard $C^*$-algebra theory we have a
direct sum decomposition of $\pi _1$ as follows. Take
${\mathcal H}_1 = {\mathcal H}_{1C} \oplus {\mathcal H}_{1N}$
where ${\mathcal H}_{1C} = \overline{\mbox{span}}\{\pi_1(X)h :
h\in {\mathcal H}, X \in C^*(\underline{S})$ and $X$ is
compact$\}$ and ${\mathcal H}_{1N}$ is the orthogonal complement of it.
Clearly ${\mathcal H}_{1C}$ is a reducing subspace for $\pi _1$.
Therefore $\pi _1=\pi _{1C}\oplus \pi _{1N}$ where $\pi_{1C}(X) =
P_{{\mathcal H}_{1C}} \pi _1(X) P_{{\mathcal H}_{1C}}$,
$\pi_{1N}(X) = P_{{\mathcal H}_{1N}} \pi _1(X) P_{{\mathcal
H}_{1N}}$. Also $\pi _{1C}(X)$ is just the identity
representation with some multiplicity. Infact ${\mathcal {H}} _{1C}$
can be written as ${\mathcal {H}} _{1C}= \Gamma _q({\mathbb{C}}^n)\otimes
\overline {{\rm I\kern-.25em D}elta _{{\underline{T}}}({\mathcal {H}})}$ (see Theorem 4.5 of [BB]) and $\pi
_{1N}(X)=0$ for compact $X$. But ${\rm I\kern-.25em D}elta _{{\underline{T}}}({\mathcal {H}})=0$ and commutators
$[S_i^*,S_i]$ are compact. So if we take $W_i=\pi_{1N}(S_i)$,
$\underline {W}=(W_1, \ldots , W_n)$ is a tuple of normal operators.
It follows that the standard $q$-commuting dilation of ${\underline{T}} $ is a tuple of normal operators.
\begin{equation}gin{Definition}
{{\epsilon}m A $q$-commuting $n$-tuple ${\underline{T}} =(T_1, \ldots , T_n) $ of operators on a
Hilbert space ${\mathcal {H}} $ is called a $q$-{spherical unitary \/} if
each $T_i$ is normal and $T_1T_1^*+\cdots
+T_nT_n^*=I.$}
{\epsilon}nd{Definition}
If ${\mathcal {H}} $ is a finite dimensional Hilbert space and ${\underline{T}} $ is a
$q$-commuting tuple on ${\mathcal {H}} $ satisfying $\sum T_iT_i^* =I$, then
${\underline{T}} $ a spherical unitary because each $T_i$ would be subnormal
and all finite dimensional subnormal operators are normal (see [Ha]).
\begin{equation}gin{Theorem} (Main Theorem)
Let ${\underline{T}}$ is a $q$-commuting contractive tuple on a Hilbert
space ${\mathcal {H}} .$ Then the maximal $q$-commuting piece of the standard
noncommuting dilation of ${\underline{T}} $ is a realization of the standard
$q$-commuting dilation of ${\underline{T}}$.
{\epsilon}nd{Theorem}
\noindent {\sc Proof of the theorem 19:} Let ${\overline{t}}ilde{{\underline{S}}}$ denote
the standard $q$-commuting dilation of ${\underline{T}} $ on a Hilbert space
${\mathcal {H}} _1$ and we follow the notations as in section 2. As ${\underline{S}} $
is also a contractive tuple, we have a unique unital completely
positive map ${\epsilon}nd{Theorem}a : C^*({\underline{V}}){\overline{t}}o C^*({\underline{S}})$, satisfying
$${\epsilon}nd{Theorem}a ({\underline{V}}^\alpha ({\underline{V}}^\begin{equation}ta)^*)= {\underline{S}}^\alpha ({\underline{S}}^\begin{equation}ta )^* ~~~~ \alpha , \begin{equation}ta \in
{{\overline{t}}ilde {\mathcal L}ambda}.$$ It is easy to see that $\psi = \phi \circ {\epsilon}nd{Theorem}a
$. Let unital $*$-homomorphism $\pi _2 : C^*({\underline{V}}){\overline{t}}o {\mathcal {B}}({\mathcal {H}} _2)$
for some Hilbert space ${\mathcal {H}} _2 $ containing ${\mathcal {H}} _1,$ be the
minimal Stinespring dilation of the map $\pi _1\circ {\epsilon}nd{Theorem}a :
C^*({\underline{V}}){\overline{t}}o {\mathcal {B}}({\mathcal {H}} _1)$ such that $\pi _1\circ {\epsilon}nd{Theorem}a (X)=P_{{\mathcal {H}}
_1 }\pi _2(X)|_{{\mathcal {H}} _1}, ~~ ~~\forall X\in C^*({\underline{V}}),$ and
$\overline {\mbox {span}}~\{ \pi _2(X)h: X\in C^*({\underline{V}}), h\in {\mathcal {H}}
_1\}= {{\mathcal {H}} _2}.$ We get the following commuting diagram.
\hskip1.5in \begin{equation}gin{picture}(200,115)
\put(0,20){$C^*({\underline{V}})$}
\put(40,21){$\longrightarrow$}
\put(70,20){$C^*({\underline{S}})$}
\put(110,21){$\longrightarrow$}
\put(140,20){$\mathcal{B}({\mathcal {H}})$}
\put(140,60){$\mathcal{B}({\mathcal {H}}_1)$}
\put(140,100){$\mathcal{B} ({\mathcal {H}}_2)$}
\put(40,30){\vector(4,3){80}}
\put(110,30){\vector(4,3){20}}
\put(150,40){$\downarrow $}
\put(150,80){$\downarrow $}
\put(45,10){${\epsilon}nd{Theorem}a$}
\put(115,10){$\phi$}
\put(110,40){$\pi_1$}
\put(70,70){$\pi_2$}
{\epsilon}nd{picture}
\noindent where all the down arrows are compression maps,
horizontal arrows are unital completely positive maps and diagonal
arrows are unital $*$-homomorphisms. Let ${\hat {\underline{V}}}=({\hat V_1},
\ldots , {\hat V_n})$ where $ {\hat V_i}=\pi _2(V_i).$ We would
show that ${\hat {\underline{V}}}$ is the standard noncommuting dilation of
${\underline{T}}.$ We have this result if we can show that $\pi _2$ is a
minimal dilation of $\psi=\phi \circ{\epsilon}nd{Theorem}a $ as minimal
Stinespring dilation is unique up to unitary equivalence. For this
first we show that ${{\overline{t}}ilde {{\underline{S}} }} =(\pi _1(S_1), \ldots , \pi
_1(S_n))$ is the maximal $q$-commuting piece of $\hat {{\underline{V}}}$.
First we consider a
particular case when ${\underline{T}} $ is a $q$-spherical unitary on a Hilbert space ${\mathcal {H}} .$
In this case we would show that standard commuting dilation and
the maximal $q$-commuting piece of the standard noncommuting
dilation of ${\underline{T}} $ is itself.
We have $\phi({\underline{S}}^\alpha (I-\sum S_iS_i^*)({\underline{S}} ^\begin{equation}ta )^*) ={\underline{T}} ^\alpha
(I-\sum T_iT_i^*)({\underline{T}} ^\begin{equation}ta)^*=0$ for any $\alpha , \begin{equation}ta \in
{{\overline{t}}ilde {\mathcal L}ambda}.$ This forces that $\phi (X)=0$ for any compact
operator $X$ in $C^*({\underline{S}}).$ Now as the $q_{ij}$-commutators
$[S_i^*, S_j]_{q_{ij}}$ are all compact we see that $\phi $ is a
unital $*$-homomorphism. So the minimal Stinespring dilation of
$\phi $ is itself and standard commuting dilation of ${\underline{T}} $ is itself.
Next we would show that
the maximal $q$-commuting piece of the standard noncommuting
dilation of ${\underline{T}} $ is itself. The presentation of the standard
noncommuting dilation which we would use is taken from [Po1]. The dilation space
${\overline{t}}ilde {{\mathcal {H}}} $ can be decomposed as $ {\overline{t}}ilde {{\mathcal {H}} } ={\mathcal H} \oplus
(\Gamma({\mathbb{C}}^n)\otimes\mathcal{D})$ where $\mathcal{D}$ is
the closure of the range of operator
\[ D:\underbrace{{\mathcal H}\oplus \cdots \oplus {\mathcal H}}_{n~
copies} \rightarrow \underbrace{{\mathcal H} \oplus \cdots \oplus
{\mathcal H}}_{n ~copies} \] and $D$ is the positive square root
of
\[ D^2=[\delta_{ij}I-T_i^*T_j]_{n {\overline{t}}imes n}. \]
For convenience, at some places we would identify
$\underbrace{{\mathcal H}\oplus \cdots \oplus {\mathcal
H}}_{n~copies}$ with ${\mathbb{C}}^n\otimes {\mathcal H}$ so that
$(h_1, \ldots ,h_n)= \sum_{i=1}^n e_i \otimes h_i.$
Then
\begin{equation}gin{equation}
D(h_1, \ldots ,h_n)= D(\sum_{i=1}^n e_i \otimes h_i)=\sum_{i=1}^n
e_i\otimes (h_i - \sum _{j=1}^nT^*_iT_j h_j)
{\epsilon}nd{equation}
and the standard noncommuting dilation ${\overline{t}}ilde{V_i}$
\begin{equation}gin{equation}
{\overline{t}}ilde{V_i}(h\oplus \sum_{\alpha \in
{{\overline{t}}ilde{{\mathcal L}ambda}}}e^{\alpha}\otimes d_{\alpha})= T_ih\oplus
D(e_i\otimes h )\oplus e_i\otimes(\sum_{\alpha \in
{{\overline{t}}ilde{{\mathcal L}ambda}}} e^{\alpha}\otimes d_{\alpha})
{\epsilon}nd{equation}
for $h \in {\mathcal {H}}$,
$d_{\alpha} \in
\mathcal{D}$ for $\alpha \in {\overline{t}}ilde{{\mathcal L}ambda}$, and $1 \leq i \leq
n$ ($\mathbb{C}^n\omega \otimes {\mathcal {D}}$ has been identified with
${\mathcal {D}}$).
We have $$T_iT_i^*=
T_i^*T_i \mbox {~and~} T_jT_i= q_{ij}T_iT_j \forall 1 \leq i, j
\leq n .$$ Also by Fuglede-Putnam Theorem ([Ha] [Pu])
$$T_j^*T_i={\overline{q}}_{ij} T_iT_j^* = q_{ji} T_i T_j^* \mbox {~and~}
T_j^*T_i^*= q_{ij}T_i^*T_j^* \forall 1 \leq i, j \leq n .$$
As $\sum T_iT_i^*=I$, by direct computation
$D^2$ is seen to be a projection. So, $D=D^2$. Note that $q_{ij}{\overline{q}}_{ij} = 1,$
i.\@ e., ${\overline{q}}_{ij} = q_{ji}.$ Then
we get
\begin{equation}gin{eqnarray}
D(h_1, \ldots ,h_n)
&= & \sum_{i,j=1}^n e_i \otimes T_j (T_j^*h_i - {\overline{q}}_{ji}T_i^*h_j)
= \sum_{i,j=1}^n e_i \otimes T_j (h_{ij})
{\epsilon}nd{eqnarray}
where $h_{ij} = T_j^*h_i - {\overline{q}}_{ji} T_i^*h_j = T_j^*h_i - q_{ij}
T_i^*h_j$ for $1 \leq i, j \leq n$. Note that $h_{ii}=0$ and
$h_{ji}=-{\overline{q}}_{ij}h_{ij}.$
As clearly ${\mathcal {H}} \subseteq {\overline{t}}ilde{{\mathcal {H}}}^q({\underline{V}}),$ lets begin with
$y\in {\mathcal {H}} ^{\perp}\bibitemgcap {\overline{t}}ilde {{\mathcal {H}} }^q({\overline{t}}ilde {{\underline{V}}}). $ We wish
to show that $y=0$. Decompose $y$ as $y=0\oplus \sum _{\alpha \in {\overline{t}}ilde {{\mathcal L}ambda }}
e^{\alpha }\otimes y_{\alpha },$ with $y_{\alpha }\in {\mathcal {D}} .$ We assume $y\neq 0$
and arrive at a contradiction. If for some $\alpha $, $y_{\alpha }\neq 0$, then
$\langle \omega \otimes y_{\alpha }, ({\overline{t}}ilde {{\underline{V}}}^{\alpha })^*y{\rightarrow}ngle
=\langle e^{\alpha }\otimes y_{\alpha }, y{\rightarrow}ngle =
\langle y_{\alpha }, y_{\alpha }{\rightarrow}ngle \neq 0.$
Since $({\overline{t}}ilde {{\underline{V}}}^{\alpha })^*y\in {\overline{t}}ilde {{\mathcal {H}} }^q({\overline{t}}ilde {{\underline{V}}}),$
we can assume $\|y_0\|=1$ without loss of generality.
Taking ${\overline{t}}ilde {y}_m =\sum _{\alpha \in {\mathcal L}ambda ^m}e^{\alpha
}\otimes y_{\alpha}$, we get $y=0\oplus \oplus _{m\geq 0}{\overline{t}}ilde
{y}_m.$ $D$ being a projection its range is closed and as $y_0 \in
{\mathcal D}$, there exist some $(h_1, \ldots ,h_n)$ such that
$y_0=D(h_1, \ldots ,h_n)$. Let ${\overline{t}}ilde{x_0}={\overline{t}}ilde{y_0}=y_0$,
${\overline{t}}ilde{x_1}= \sum_{i,j=1}^n e_i \otimes D(e_j\otimes h_{ij}),$
and for $m\geq 1$,
$$ {\overline{t}}ilde{x}_m = \sum_{i_1, \ldots ,i_{m-1},i,j=1}^n e_{i_1}
\otimes \cdots \otimes e_{i_{m-1}} \otimes e_i \otimes D(e_j
\otimes (\prod_{1 \leq r < s \leq
m-1}q_{i_ri_s})(\prod_{k=1}^{m-1}q_{i_ki}q_{i_kj}) T_{i_1}^*
\ldots T_{i_{m-1}}^* h_{ij}).$$
So ${\overline{t}}ilde{x}_m \in ({\mathbb
C}^n)^{\otimes m} \otimes {\mathcal D}$ for all $m \in
{\mathbb{N}}$. As ${\underline{T}}$ is $q$-commutating $n$-tuple and $D$ is a
projection, we have
\begin{equation}gin{eqnarray*}
\sum_{1\leq i < j \leq n} (q_{ij}{\overline{t}}ilde{V}_i {\overline{t}}ilde{V}_j -
{\overline{t}}ilde{V}_j {\overline{t}}ilde{V}_i)q_{ji}h_{ij}
&=&\sum _{1\leq i<j\leq n}(q_{ij}T_iT_j-T_jT_i)q_{ji}h_{ij}\\
& &+\sum_{1\leq i < j \leq n}D(e_i\otimes T_j h_{ij} -
q_{ji}e_j\otimes T_i h_{ij})\\
& &
+\sum_{1\leq i < j \leq n}(e_i\otimes D(e_j\otimes h_{ij})
-q_{ji}e_j\otimes D(e_i \otimes h_{ij}))\\
& =& 0 + D(\sum _{i,j = 1}^n e_i\otimes T_jh_{ij}) + \sum_{ i,j=1}^n e_i\otimes D(e_j\otimes h_{ij})\\
& =& D^2(h_1, \ldots ,h_n) + \sum_{ i,j=1}^n e_i\otimes
D(e_j\otimes h_{ij})\\
&=& {\overline{t}}ilde{x}_0 +{\overline{t}}ilde{x}_1.
{\epsilon}nd{eqnarray*}
So by Proposition 6, $\langle y, {\overline{t}}ilde{x}_0 +{\overline{t}}ilde{x}_1 {\rightarrow}ngle
=0$ . Next let $m \geq 2.$
\begin{equation}gin{eqnarray*}
& & \sum_{i_1, \ldots ,i_{m-1}=1}^n {\overline{t}}ilde{V}_{i_1} \ldots
{\overline{t}}ilde{V}_{i_{m-1}}\{\sum_{i,j=1}^n (q_{ij}
{\overline{t}}ilde{V}_i{\overline{t}}ilde{V}_j-{\overline{t}}ilde{V}_j{\overline{t}}ilde{V}_i)(\prod_{1 \leq r < s
\leq m-1}q_{i_ri_s})(\prod_{k=1}^{m-2}q_{i_kj})(T_i^* T_{i_1}^*
\ldots T_{i_{m-2}}^* h_{i_{m-1}j})\}\\
& =& \sum_{i_1, \ldots ,i_{m-1}=1}^n e_{i_1} \otimes \ldots
\otimes e_{i_{m-1}}\otimes [\sum_{i,j=1}^n D((\prod_{1 \leq r < s
\leq m-1} q_{i_ri_s})(\prod_{k=1}^{m-2}q_{i_kj})(q_{ij} e_i\otimes
T_jT_i^*T_{i_1}^*\ldots T_{i_{m-2}}^*h_{i_{m-1}j}\\
& & - e_j\otimes T_iT_i^*T_{i_1}^*\ldots
T_{i_{m-2}}^*h_{i_{m-1}j})) + \sum_{i,j=1}^n(\prod_{1 \leq r < s
\leq m-1}\ q_{i_ri_s})(\prod_{k=1}^{m-2}q_{i_kj}) \{q_{ij} e_i
\otimes D(e_j \otimes \\
& & T_i^*T_{i_1} ^* \ldots T_{i_{m-2}}^*h_{i_{m-1}j}) - e_j\otimes
D(e_i \otimes T_i^*T_{i_1}^*\ldots
T_{i_{m-2}}^*h_{i_{m-1}j})\}]\\
&= &- \sum_{i_1,\ldots
,i_{m-1}=1}^n e_{i_1} \otimes \cdots \otimes e_{i_{m-1}} \otimes
\{(\sum_{j=1}^n (\prod_{1 \leq r < s \leq m-1}
q_{i_ri_s})(\prod_{k=1} ^{m-2}q_{i_kj}) D(e_j \otimes
T_{i_1}^* \ldots T_{i_{m-2}}^* h_{i_{m-1}j})\}\\
& & + \sum_{i_1,\ldots ,i_{m-1}=1}^n e_{i_1} \otimes \ldots
\otimes e_{i_{m-1}} \otimes \{\sum_{i,j=1}^n e_i \otimes D(e_j
\otimes q_{ij} (\prod_{1 \leq r
< s \leq m-1}q_{i_ri_s})(\prod_{k=1}^{m-2} q_{i_kj})\\
& & (T_i^*T_{i_1}^* \ldots T_{i_{m-2}}^* h_{i_{m-1}j}))
-\sum_{i,j=1}^n e_i \otimes D(e_j\otimes (\prod_{1 \leq r < s \leq
m-1}q_{i_ri_s})(\prod_{k=1}^{m-2} q_{i_ki})
(T_j^*T_{i_1}^* \ldots T_{i_{m-2}}^* h_{i_{m-1}i}))\}\\
& & \mbox{(in the term above, $i$ and $j$ have been interchanged
in the last summation)}\\
&= &- \sum_{i_1,\ldots ,i_{m-1}=1}^n
e_{i_1} \otimes \cdots \otimes e_{i_{m-1}} \otimes \{\sum_{j=1}^n
(\prod_{1 \leq r < s \leq m-2} q_{i_ri_s})(\prod_{k=1}
^{m-2}q_{i_ki}q_{i_kj}) D(e_j \otimes
T_{i_1}^* \ldots T_{i_{m-2}}^* h_{ij})\}\\
& & + \sum_{i_1,\ldots ,i_{m-1}=1}^n e_{i_1} \otimes \ldots
\otimes e_{i_{m-1}}\otimes \{\sum_{i,j=1}^n e_i \otimes D(e_j\\
& & \otimes (\prod_{1 \leq r < s \leq m-1} q_{i_ri_s})
q_{ij}(\prod_{k=1}^{m-2} q_{i_kj}) (T_i^*T_{i_1}^* \ldots
T_{i_{m-2}}^*T_j^* h_{i_{m-1}}
-q_{i_{m-1}j}T_i^*T_{i_1}^* \ldots T_{i_{m-2}}^*T_{i_{m-1}}^* h_j )\\
& & -(\prod_{1 \leq r < s \leq m-2}q_{i_ri_s})(\prod_{k=1}^{m-2}
q_{i_ki}) (T_j^*T_{i_1}^* \ldots T_{i_{m-2}}^* T_i^*h_{i_{m-1}}-
q_{i_{m-1}i}T_j^*T_{i_1}^* \ldots T_{i_{m-2}}^* T_{i_{m-1}}^*h_i ))\}\\
& & \mbox{(in the term above, index $i_{m-1}$ has been replaced
by $i$ in the first summation)}\\
& =& - \sum_{i_1,\ldots ,i_{m-2},i,j=1}^n e_{i_1} \otimes \cdots
\otimes e_{i_{m-2}} \otimes e_i \otimes (\prod_{1 \leq r < s \leq
m-2} q_{i_ri_s})(\prod_{k=1}^{m-2}q_{i_ki}q_{i_kj})
D(e_j \otimes T_{i_1}^* \ldots T_{i_{m-2}}^* h_{ij})\\
& & + \sum_{i_1,\ldots ,i_{m-1},i,j=1}^n e_{i_1} \otimes \cdots
\otimes e_{i_{m-1}} \otimes e_i \otimes (\prod_{1 \leq r < s \leq
m-1}q_{i_ri_s})(\prod_{k=1}^{m-1}q_{i_ki}q_{i_kj})D(e_j\otimes
T_{i_1}^* \ldots T_{i_{m-1}}^* h_{ij})\\
& = & -{\overline{t}}ilde{x}_{m-1} +
{\overline{t}}ilde{x}_m.
{\epsilon}nd{eqnarray*}
Hence by proposition 6, $\langle y, {\overline{t}}ilde{x}_{m-1} - {\overline{t}}ilde{x}_m{\rightarrow}ngle =0.$
Further we compute $\|{\overline{t}}ilde{x}_{m}\| $ for all $m \in
{\mathbb{N}}$.
\begin{equation}gin{eqnarray*}
\|{\overline{t}}ilde{x}_{m}\|^2 & =& \langle \sum_{i_1,\ldots
,i_{m-1},i,j=1}^n e_{i_1} \otimes \cdots \otimes e_{i_{m-1}}
\otimes e_i \otimes D(e_j\otimes (\prod_{1 \leq r < s \leq
m-1}q_{i_ri_s})(\prod_{k=1}^{m-1}q_{i_ki}q_{i_kj})T_{i_1}^* \ldots
T_{i_{m-1}}^* h_{ij}),\\
& & \sum_{i_1',\ldots ,i_{m-1}',i',j'=1}^n e_{i_1'} \otimes \cdots
\otimes e_{i_{m-1}'} \otimes e_{i'} \otimes D(e_{j'}\otimes
(\prod_{1 \leq r' < s' \leq
m-1}q_{i_{r'}i_{s'}})\\
& & (\prod_{k'=1}^{m-1}q_{i'_{k'}i'}
q_{i'_{k'}j'})T_{i_1'}^*
\ldots T_{i_{m-1}'}^* h_{i'j'}){\rightarrow}ngle\\
&= & \sum_{i_1,\ldots ,i_{m-1},i=1}^n \langle \sum_{j=1}^n
D(e_j\otimes (\prod_{1 \leq r < s \leq m-1}q_{i_ri_s}
)(\prod_{k=1}^{m-1}q_{i_ki}q_{i_kj})T_{i_1}^* \ldots
T_{i_{m-1}}^* h_{ij}),\\
& & \sum_{j'=1}^n D(e_{j'}\otimes (\prod_{1 \leq r' < s' \leq
m-1}q_{i_{r'}i_{s'}})(\prod_{k'=1}^{m-1}
q_{i_{k'}i}q_{i_{k'}j'})T_{i_1}^* \ldots T_{i_{m-1}}^*
h_{ij'}){\rightarrow}ngle\\ &= & \sum_{i_1,\ldots ,i_{m-1},i=1}^n \langle
D(\sum_{j=1}^n e_j\otimes (\prod_{1 \leq r < s \leq m-1}q_{i_ri_s}
)(\prod_{k=1}^{m-1}q_{i_ki}q_{i_kj})T_{i_1}^* \ldots T_{i_{m-1}}^*
h_{ij}),\\
& & \sum_{j'=1}^n e_{j'}\otimes (\prod_{1 \leq r' < s'\leq
m-1}q_{i_{r'}i_{s'}}
)(\prod_{k'=1}^{m-1}q_{i_{k'}i}q_{i_{k'}j'})T_{i_1}^*
\ldots T_{i_{m-1}}^* h_{ij'}{\rightarrow}ngle\\
& =&
\sum_{i_1,..,i_{m-1},i=1}^n \langle ( \prod_{1 \leq r < s \leq
m-1}q_{i_ri_s} )(\prod_{k=1}^{m-1}q_{i_ki}q_{i_kj})\{
\sum_{j,l=1}^n e_j\otimes T_l (T_l^*T_{i_1}^* \ldots T_{i_{m-1}}^*
h_{ij}- \\
& & q_{jl}T_j^* T_{i_1}^*\ldots T_{i_{m-1}}^* h_{il})\},
\sum_{j'=1}^n (\prod_{1 \leq l' < s'\leq m-1}q_{i_{l'}i_{s'}}
)(\prod_{k'=1}^{m-1}q_{i_{k'}i}q_{i_{k'}j'}) e_{j'}\otimes
T_{i_1}^* \ldots T_{i_{m-1}}^* h_{ij'}{\rightarrow}ngle\\
& =& \sum_{i_1,...,i_{m-1},i,j=1}^n \langle ( \prod_{1 \leq r < s
\leq m-1}q_{i_ri_s}
)(\prod_{k=1}^{m-1}q_{i_ki}q_{i_kj})\sum_{l=1}^n T_l
(T_l^*T_{i_1}^* \ldots T_{i_{m-1}}^* h_{ij}- q_{jl}T_j^*
T_{i_1}^*\ldots T_{i_{m-1}}^* h_{il}), \\
& & (\prod_{1 \leq r' < s'\leq m-1}q_{i_{r'}i_{s'}}
)(\prod_{k'=1}^{m-1}q_{i_{k'}i}q_{i_{k'}j})
T_{i_1}^* \ldots T_{i_{m-1}}^* h_{ij}{\rightarrow}ngle\\
& =& \sum_{i,j=1}^n \langle h_{ij},h_{ij}{\rightarrow}ngle -
\sum_{i_1,...,i_{m-1},i,j,l=1}^n \langle T_{i_{m-1}} \ldots
T_{i_1} T_j^* T_l T_{i_1}^* \ldots T_{i_{m-1}}^*
h_{il}),h_{ij}{\rightarrow}ngle
{\epsilon}nd{eqnarray*}
Let ${\overline{t}}au : {\mathcal {B}}({\mathcal {H}}) {\overline{t}}o {\mathcal {B}}({\mathcal {H}})$ be defined by ${\overline{t}}au (X) =
\sum_{i=1}^n T_i X T_i^*$ for all $X \in {\mathcal {B}}({\mathcal {H}}),$ and let
${\overline{t}}ilde{{\overline{t}}au}^m : M_n ({\mathcal {B}}({\mathcal {H}})) {\overline{t}}o M_n ({\mathcal {B}}({\mathcal {H}}))$ be defined by
${\overline{t}}ilde{{\overline{t}}au}^m(X)= ({\overline{t}}au^m(X_{ij}))_{n {\overline{t}}imes n}$ for all $X =
(X_{ij})_{n {\overline{t}}imes n} \in M_n ({\mathcal {B}}({\mathcal {H}})).$ As ${\overline{t}}au$ is a
completely positive map, ${\overline{t}}ilde{{\overline{t}}au}^m$ is also a completely
positive map. \\
So we have ${\overline{t}}ilde{{\overline{t}}au}^m(D) \leq I$ and
\begin{equation}gin{eqnarray*}
\|{\overline{t}}ilde{x}_m \|^2 &=& \sum_{r=1}^n \langle {\overline{t}}ilde{{\overline{t}}au}^m(D)
(h_{r1}\ldots h_{rn}) , (h_{r1}\ldots
h_{rn}){\rightarrow}ngle\\
& \leq & \sum_{r=1}^n \langle (h_{r1}\ldots h_{rn}) , (h_{r1}\ldots
h_{rn}){\rightarrow}ngle\\
& = & \sum_{r,i}^n \langle h_{ri} , h_{ri}{\rightarrow}ngle
=\sum_{i,r=1}^n \langle T_i^* h_r - {\overline{q}}_{ir} T_r^* h_i,
T_i^* h_r - {\overline{q}}_{ir} T_r^* h_i {\rightarrow}ngle\\
& =&\sum_{i,r=1}^n \{ \langle T_i^*T_ih_r - T_r^*T_ih_i,
h_r{\rightarrow}ngle - \langle T_i^*T_rh_r- T_r^*T_rh_i ,
h_i{\rightarrow}ngle \}\\
& = &
\sum_{r=1}^n \langle h_r -\sum_{i=1}^n T_r^*T_ih_i,
h_r{\rightarrow}ngle - \sum_{i=1}^n \langle \sum_{r=1}^n T_i^*T_rh_r-h_i ,
h_i{\rightarrow}ngle \\
& =& 2 \sum_{r=1}^n \langle h_r -\sum_{i=1}^n T_r^*T_ih_i,
h_r{\rightarrow}ngle
= 2 \langle D(h_1,\ldots ,h_n),(h_1,\ldots ,h_n) {\rightarrow}ngle
= 2 \| {\overline{t}}ilde{x}_0 \| ^2 =2.
{\epsilon}nd{eqnarray*}
As $\langle y, {\overline{t}}ilde{x}_0 +{\overline{t}}ilde{x}_1{\rightarrow}ngle =0$ and $\langle y,
{\overline{t}}ilde{x}_{m-1} -{\overline{t}}ilde{x}_m {\rightarrow}ngle =0 $ for $m + 1 \in
{\mathbb{N}}$,
we get $\langle y, {\overline{t}}ilde{x}_0 +{\overline{t}}ilde{x}_{m} {\rightarrow}ngle =0$
for $m \in {\mathbb{N}}$. So $1=\langle {\overline{t}}ilde {y}_0,
{\overline{t}}ilde {y_0}{\rightarrow}ngle = \langle {\overline{t}}ilde{y}_0,{\overline{t}}ilde{x}_0 {\rightarrow}ngle = -
\langle {\overline{t}}ilde{y}_{m}, {\overline{t}}ilde{x}_{m} {\rightarrow}ngle$. By Cauchy-Schwarz
inequality, $1 \leq \| {\overline{t}}ilde{y}_m \| \| {\overline{t}}ilde{x}_m \|$ , which implies
$\frac{1}{\sqrt{2}} \leq \| {\overline{t}}ilde{y}_m \|$ for $m \in
{\mathbb{N}}$. This is a contradiction as $y=0\oplus \oplus
_{m\geq 0}{\overline{t}}ilde {y}_m$ is in the Hilbert space ${\overline{t}}ilde {{\mathcal {H}} }$.
This proves the particular case.
Using arguements similar to that of Theorem 13 of [BBD], the proof
of the general case (that is when $T_i$ is not necessarily normal)
and the proof of ``${\overline{t}}ilde{{\underline{V}}}$ is the standard noncommuting
dilation of ${\underline{T}}$", both follows .
${\rm I\kern-.25em B}ox$
{\epsilon}nd{section}
\begin{equation}gin{section}{Distribution of $S_i + S_i^*$ and Related Operator Spaces}
\setcounter{equation}{0}
Let ${\mathcal {R}}$ be the von Neumann algebra generated by $G_i=S_i +
S_i^*$ for all $1 \leq i \leq n.$ We are interested in
calculating the moments of $S_i + S_i^*$ with respect to the
vaccum state and inferring about the distribution. The vacuum
expectation is given by ${\epsilon}psilon (T)=\langle \omega,T \omega
{\rightarrow}ngle$ where $T \in {\mathcal {R}}.$ So,
$${\epsilon}psilon ((S_i + S_i^*)^n)= \langle \omega, (S_i + S_i^*
)^n \omega {\rightarrow}ngle = \left \{ \begin{equation}gin{array}{ccc}
0 & \mbox {~if n is odd} \\
C_n = \frac{1}{n+1} (^n_{\frac{n}{2}}) & \mbox{~otherwise}
{\epsilon}nd{array}\right .$$
where $C_n $ the catalan number (refer [Com]). This shows that
$S_i + S_i^*$ has semicircular distribution. Further this vaccum
expectation is not tracial on ${\mathcal {R}}$ for $n \geq 2$ as
\begin{equation}gin{eqnarray*}
{\epsilon}psilon(G_2G_2G_1G_1)&=&\langle \omega, (S_2 + S_2^* )(S_2 +
S_2^*)(S_1 + S_1^* )(S_1 + S_1^*) \omega {\rightarrow}ngle\\
&=&\langle \omega, S_2^* S_2^*S_1S_1 + S_2^* S_2S_1^*S_1\omega {\rightarrow}ngle = 1 \\
{\epsilon}psilon(G_2G_1G_1G_2) &=&\langle \omega, (S_2 +
S_2^*)(S_1 + S_1^* )(S_1 + S_1^*)(S_2 + S_2^* ) \omega {\rightarrow}ngle\\
&=&\langle \omega, S_2^* S_1^*S_1S_2 + S_2^* S_1S_1^*S_2\omega
{\rightarrow}ngle = \frac{1}{2}
{\epsilon}nd{eqnarray*}
We would now investigate using arguements of theory of operator
spaces introduced by Effros and Ruan [ER]. Here we follow the
ideas of [BS2] and [HP]. Operator spaces which are Hilbert spaces
are called Hilbertian operator spaces. For some Hilbert space
${\overline{t}}ilde{{\mathcal {H}}}$ and $a_i \in B({\overline{t}}ilde{{\mathcal {H}}}), 1\leq i \leq n $ define
$$ \|(a_1,\cdots,a_n)\|_{max} = \mbox{max} (\| \sum^n_{i=1} a_ia^*_i\|^{\frac{1}{2}},
\| \sum^n_{i=1} a^*_ia_i\|^{\frac{1}{2}}).$$ Let us denote the
operator space
$$\left \{ \left( \begin{equation}gin{array}{cccc} r_1& 0&\cdots&0\\
. &&& .\\
. &&& .\\
. &&& .\\
r_n & 0 & \cdots& 0
{\epsilon}nd{array}\right) \oplus
\left (\begin{equation}gin{array}{ccc} r_1& \cdots&r_n\\
0 && 0\\
. && .\\
. && .\\
. && .\\
0 & \cdots& 0 {\epsilon}nd{array}\right) | r_1,\cdots, r_n \in \C \right\}
\subset M_n \oplus M_n$$ by $E_n.$ Let $\{e_{ij}:1 \leq i,j \leq n
\}$ denote the standard basis of $M_n$ and $\delta_i=e_{i1}\oplus
e_{1i}.$ Then one has
$$ \|\sum^n_{i=1}a_i\otimes \delta_i\|_{B({\overline{t}}ilde{{\mathcal {H}}})\otimes M_n}=\|(a_1,\cdots,a_n)\|_{max}.$$
\begin{equation}gin{Theorem}
The operator space generated by $G_i, ~~1 \leq i \leq n$ is
completely isomorphic to $E_n.$
{\epsilon}nd{Theorem}
\noindent{\sc Proof:} Its enough to show that for $a_i \in
B({\overline{t}}ilde{{\mathcal {H}}}), 1\leq i \leq n$ we have
$$ \|(a_1,\cdots,a_n)\|_{max}\leq \| \sum^n_{i=1}
a_i \otimes G_i\|_{{\overline{t}}ilde{{\mathcal {H}}}\otimes \Gamma_q(\mathbb{C}^n)} \leq
2 \|(a_1,\cdots,a_n)\|_{max}$$
\begin{equation}gin{eqnarray*}
\| \sum^n_{i=1} a_i \otimes S^*_i\|_{{\overline{t}}ilde{{\mathcal {H}}}\otimes
\Gamma_q(\mathbb{C}^n)}&=&\| \sum^n_{i=1} (a_i \otimes 1)(1
\otimes
S^*_i)\|_{{\overline{t}}ilde{{\mathcal {H}}}\otimes \Gamma_q(\mathbb{C}^n)}\\
& \leq & \| \sum^n_{i=1} a_i a^*_i\|^{\frac{1}{2}}_{{\overline{t}}ilde{{\mathcal {H}}}}
\|\sum^n_{i=1} S_i S^*_i\|^{\frac{1}{2}}_{\Gamma_q(\mathbb{C}^n)}
\leq \| \sum^n_{i=1} a_i
a^*_i\|^{\frac{1}{2}}_{\Gamma_q(\mathbb{C}^n)}
{\epsilon}nd{eqnarray*}
Similarly
\begin{equation}gin{eqnarray*}
\|\sum^n_{i=1} a_i \otimes S_i\|_{{\overline{t}}ilde{{\mathcal {H}}}\otimes
\Gamma_q(\mathbb{C}^n)}&=&\| \sum^n_{i=1} (1 \otimes
S_i)(a_i \otimes 1)\|_{{\overline{t}}ilde{{\mathcal {H}}}\otimes \Gamma_q(\mathbb{C}^n)}\\
&\leq & \| \sum^n_{i=1} a^*_i
a_i\|^{\frac{1}{2}}_{\Gamma_q(\mathbb{C}^n)}
{\epsilon}nd{eqnarray*}
So
$$ \| \sum^n_{i=1}a_i \otimes G_i\|_{{\overline{t}}ilde{{\mathcal {H}}}\otimes
\Gamma_q(\mathbb{C}^n)} \leq 2 \|(a_1,\cdots,a_n)\|_{max}$$ Let
${\mathcal {S}}$ denote the set of all states on $B({\overline{t}}ilde{{\mathcal {H}}}).$ Now using
the fact that ${\epsilon}psilon(G_iG_j)=\langle \omega, S^*_i S_j\omega
{\rightarrow}ngle= \delta_{ij} $ we get
\begin{equation}gin{eqnarray*}
\| \sum^n_{i=1}a_i \otimes G_i\|^2_{{\overline{t}}ilde{{\mathcal {H}}}\otimes
\Gamma_q(\mathbb{C}^n)} & \geq & \underbrace{\mbox{sup}}_{{\overline{t}}au \in
{\mathcal {S}}} ({\overline{t}}au \otimes {\epsilon}psilon)[(\sum^n_{i=1}a_i \otimes
G_i)^*\sum^n_{j=1}a_j \otimes G_j]\\
& = & \underbrace{\mbox{sup}}_{{\overline{t}}au \in {\mathcal {S}}} {\overline{t}}au (\sum^n_{i=1}
a^*_ia_i) = \| \sum^n_{i=1} a^*_ia_i\|
{\epsilon}nd{eqnarray*}
Using similar arguements
$$\| \sum^n_{i=1}a_i \otimes
G_i\|^2_{{\overline{t}}ilde{{\mathcal {H}}}\otimes \Gamma_q(\mathbb{C}^n)} \geq \|
\sum^n_{i=1} a_ia^*_i\|.$$
${\rm I\kern-.25em B}ox$
\noindent{\bf Acknowledgements:} The author is
supported by a research fellowship from the Indian Statistical
Institute. The author is thankful to B. V. Rajarama Bhat and
Tirthankar Bhattacharyya for many helpful discussions.
{\epsilon}nd{section}
\begin{equation}gin{center}
{\bf References}
{\epsilon}nd{center}
\begin{equation}gin {itemize}
\item [{[A]}] J. Agler: The Arveson extension theorem and
coanalytic models, Integral Equations and Operator Theory,
{\bf 5} (1982), 608-631, {\bf MR 84g:47011}.
\item [{[AP1]}] Arias, A.; Popescu, G.: Noncommutative interpolation and
Poisson transforms, Israel J. Math., {\bf 115}(2000), 205-234, {\bf
MR 2001i:47021}.
\item [{[Ar1]}] Arveson, W. B.: {{\epsilon}m An Invitation to $C^*$-algebras,\/ }
Graduate Texts in Mathematics, No. 39, Springer-Verlag, New York-Heidelberg
(1976). {\bf MR 58$\#$23621}.
\item [{[Ar2]}] Arveson, W. B.: Subalgebras of $C^*$-algebras III, Multivariable
operator theory, Acta Math., {\bf 181}(1998), no. 2, 159-228. {\bf MR 2000e:47013}.
\item [{[At1]}] Athavale, A. : On the intertwining of joint isometries,
J. Operator Theory, {\bf 23}(1990), 339-350. {\bf MR 91i:47029}.
\item [{[At2]}] Athavale, A.: Model theory on the unit ball in $\mathbb{C}^m$,
J. Operator Theory, {\bf 27}(1992), 347-358. {\bf MR 94i:47011}
\item [{[BB]}] Bhat, B. V. Rajarama; Bhattacharyya, T. : A model theory
for $q$-commuting contractive tuples, J. Operator Theory, {\bf 47}
(2002), 97-116.
\item [ { [BBD]}] Bhat, B. V. Rajarama; Bhattacharyya, T.; Dey,
Santanu : Standard noncommuting and commuting dilations of
commuting tuples, to appear in Trans. Amer. Math. Soc.
\item [ {[B]}] Bhattacharyya, T. : Dilation of contractive tuples: a survey; to appear in
{{\epsilon}m Survey of Analysis and Operator Theory }, Proceedings of CMA, Volume 40.
\item [{[BS1]}] Bo\.zejko, M., Speicher, R.: An example of a
generalized Brownian motion, Commun. Math. Phys. {\bf 137} (1991),
no. 3, 519-531 {\bf MR 92m:46096}.
\item [{[BS2]}] Bo\.zejko, M., Speicher, R.: Completely positive maps on Coxeter groups,
deformed commutation relations, and operator spaces , Math. Ann. {\bf 300} (1994),
no. 1, 97-120 {\bf MR 95g:46105}.
\item [{[Bu]}] Bunce, J. W.: Models for $n$-tuples of noncommuting
operators, J. Funct. Anal., {\bf 57}(1984), 21-30. {\bf MR
85k:47019}.
\item [{[Con]}] Connes, A.:{{\epsilon}m Noncommutative Geometry,}
Academic Press, (1994) {\bf MR 95j:46063}.
\item [{[Com]}] Comtet, L.:{{\epsilon}m Advanced combinatorics,} D. Reidel Publishing Co., (1974)
{\bf MR 57 \#124}.
\item [{[Cu]}] Cuntz, J. : Simple $C^*$-algebras generated by isometries,
Commun. Math. Phys., {\bf 57} (1977),173-185. {\bf MR 57$\# $7189}.
\item [{[Da]}] Davis, C.: Some dilation and representation theorems.
Proceedings of the Second International Symposium in West Africa on
Functional Analysis and its Applications (Kumasi, 1979), 159-182.
{\bf MR 84e:47012}.
\item [{[DKS]}] Davidson K. R.; Kribs, D. W.; Shpigel, M.E.:
Isometric dilations of non-commuting finite rank $n$-tuples,
Canad. J. Math., {\bf 53} (2001) 506-545. {\bf MR 2002f:47010}.
\item [{[ER]}] Effros, E. G.; Ruan, Z. J.:
{{\epsilon}m Operator spaces,\/} London Mathematical Society Monographs.
New Series, {\bf 23},(2000) {\bf MR 2002a:46082}.
\item [{[Fr]}] Frazho, A. E.: Models for noncommuting operators, J.
Funct. Anal., {\bf 48} (1982), 1-11. {\bf MR 84h:47010}.
\item [{[Ha]}] Halmos, P. R.: {{\epsilon}m A Hilbert Space Problem Book,\/} Second Edition,
Graduate Texts in Mathematics {\bf 19}, (Springer-Verlag, New
York-Berlin 1982) {\bf MR 84e:47001}.
\item [{[HP]}] Haagerup, U.; Pisier, G.: Bounded linear operators between
$C *$-algebras, Duke Math. J., {\bf 71} (1993), no. 3, 889--925.
{\bf 94k:46112}.
\item [{[JSW]}] Jorgensen, P. E. T.; Schmitt, L. M.; Werner, R.
F.: $q$-canonical commutation relations and stability of the Cuntz
algebra, Pacific J. Math. {\bf 165} (1994), no. 1, 131-151. {\bf
MR 95g:46116}.
\item [{[M]}] Majid, S.: {{\epsilon}m Foundations of Quantum Group Theory,\/} Cambridge
University Press, (1995) {\bf MR 97g:17016 }.
\item [{[Pa]}] Parrott, S.: Unitary dilations for commuting contractions,
Pacific J. Math. {\bf 34} (1970), 481-490. {\bf MR 42$\#$3607}.
\item [{[Po1]}] Popescu, G.: Isometric dilations for infinite
sequences of noncommuting operators, Trans. Amer. Math. Soc., {\bf
316}(1989), 523-536. {\bf MR 90c:47006}.
\item [{[Po2]}]
Popescu, G.: Characteristic functions for infinite
sequences of noncommuting operators, J. Operator Theory, {\bf 22} (1989),
51-71. {\bf MR 91m:47012}.
\item [{[Po3]}] Popescu, G.: Poisson transforms on some $C\sp *$-algebras
generated by isometries. J. Funct. Anal. 161 (1999), 27-61. {\bf MR 2000m:46117}.
\item [{[Po4]}] Popescu, G.: Curvature invariant for Hilbert modules
over free semigroup algebras, Advances in Mathematics,
{\bf 158} (2001), 264-309. {\bf MR 2002b:46097}.
\item [{[Pr]}] Prugovecki, E.: {{\epsilon}m Quantum Mechanics in Hilbert Space, \/}
Academic Press, Second Edition(1981) {\bf MR 84k:81005 }.
\item [{[Pu]}] Putnam, C. R.: {{\epsilon}m Commutation properties of Hilbert space
Operators and Related Topics,\/} Springer-Verlag (1967). {\bf MR 36$\#$707}.
{\epsilon}nd{itemize}
\noindent {\sc Santanu Dey} \\ Indian Statistical Institute,
R. V. College Post, Bangalore 560059, India.\\
e-mail: {\sl [email protected] }\\
{\epsilon}nd{document} | math | 84,923 |
\betagin{document}
\title[Effective {\L}ojasiewicz gradient inequality]{Effective {\L}ojasiewicz gradient inequality\\ and finite determinacy of non-isolated Nash function singularities}
\subjclass[2000]{14R99, 11E25, 14P05, 32S70.}
\keywords{Semialgebraic function, Nash function, {\L}ojasiewicz gradient inequality, {\L}ojasiewicz exponent.}
\author[B. Osi\'nska-Ulrych]{Beata Osi\'nska-Ulrych}
\address{Beata Osi\'nska-Ulrych, Faculty of Mathematics and Computer Science, University of \L \'od\'z,
S. Banacha 22, 90-238 \L \'od\'z, Poland}
\email{[email protected]}
\author[G. Skalski]{Grzegorz Skalski}
\address{Grzegorz Skalski, Faculty of Mathematics and Computer Science, University of \L \'od\'z,
S. Banacha 22, 90-238 \L \'od\'z, Poland}
\email{[email protected]}
\author[S. Spodzieja]{Stanis{\l}aw Spodzieja}
\address{Stanis{\l}aw Spodzieja, Faculty of Mathematics and Computer Science, University of \L \'od\'z,
S. Banacha 22, 90-238 \L \'od\'z, Poland}
\email{[email protected]}
\date{\today}
\betagin{abstract}
Let $X\subset \mathbf{\mathbb{R}}^n$ be a compact semialgebraic set and let $f:X\to \mathbf{\mathbb{R}}$ be a nonzero Nash function. We give a Solern\'o and D'Acunto-Kurdyka type estimation of the exponent $\varrho\in[0,1)$ in the
{\L}ojasiewicz gradient inequality $|\gammarad f(x)|\gammae C|f(x)|^\varrho$ for $x\in X$, $|f(x)|<\varepsilon$ for some constants $C,\varepsilon>0$, in terms of the degree of a polynomial $P$ such that $P(x,f(x))=0$, $x\in X$. As a corollary we obtain an estimation of the degree of sufficiency of non-isolated Nash functions singularities.
\end{abstract}
\maketitle
\section{Introduction}
{\L}ojasiewicz inequalities are important tools in various branches of mathematics: differential equations, singularity theory and optimization (for more detailed references, see for example \cite{KMP}, \cite{KS1}, \cite{KS2}, \cite{LejeuneTeissier} and
\cite{RS3}).
Quantitative aspects, like estimates (or exact computation), of these exponents are subject of intensive study in real and complex algebraic geometry (see for instance \cite{KS1}, \cite{KS2}, \cite{KSS} and \cite{RS2}).
Our main goal is to give, in terms of the {\L}ojasiewicz inequality, an effective sufficient condition for Nash function germs of non-isolated singularity at zero to be isotopical (Theorem \ref{mainsuffjetLojineq}). The main tool in the proof is an effective estimation of the exponent in the {\L}ojasiewicz gradient inequality (Theorems \ref{maintwr} and \ref{maintwrIII}).
Determinacy of jets of functions with isolated singularity at zero was investigated by many authors, including N. H.~Kuiper \cite{Kui}, T. C.~Kuo \cite{Kuo}, J.~Bochnak and S. \L ojasiewicz \cite{BL} for real functions and S. H.~Chang and Y. C.~Lu \cite{CL}, B.~Teissier \cite{T} and J. Bochnak and W.~Kucharz \cite{BK} for complex functions. Similar investigations were also carried out for functions in a neighbourhood of infinity by P. Cassou-Nogu\`es and H. H. Vui \cite{CH} (see also \cite{RS4}, \cite{Sk}). The case of real jets with non-isolated singularities was studied among others by V. Grandjean \cite{G} and
X. Xu \cite{Xu}, and for complex functions by D.~Siersma \cite{Si2} and R. Pellikaan \cite{P}. In the case of nondegenerate analytic functions $f$, $g$, a condition for topological triviality of deformations $f+tg$, $t\in[0,1]$ in terms of Newton polyhedra was obtained by J. Damon and T. Gaffney \cite{DG}, and for blow analytic triviality by T. Fukui and E.~~Yoshinaga \cite{FY}. Some algebraic conditions for finite determinacy of a smooth function jet were obtained by L. Kushner \cite{Kne}.
\subsection{{\L}ojasiewicz gradient inequality}
Let $U\subset \mathbf{\mathbb{R}}^n$ be an open set and let $a\in U$.
Let $f,F: U \to \mathbf{\mathbb{R}}$ be continuous semialgebraic functions such that $a\in F^{-1}(0) \subset f^{-1}(0)\subset U$.
Then
the following \emph{{\L}ojasiewicz inequality} holds:
\betagin{equation}\lambdabel{Lojineq01}
|F(x)|\gammae C|f(x)|^\eta \; \hbox{ in a neighbourhood of $a\in\mathbf{\mathbb{R}}^n$ for some constant $C>0$.}
\end{equation}
The lower bound of the exponents $\eta$ in \eqref{Lojineq01} is called the \emph{{\L}ojasiewicz exponent} of the pair $(F,f)$ \emph{at} $a$ and is denoted by ${\mathcal{L}}_a(F,f)$. It is known that ${\mathcal{L}}_a(F,f)$ is a rational number (see \cite{BR}) and the
inequality \eqref{Lojineq01} holds actually with $\eta= {\mathcal{L}}_a(F,f)$ on some neighbourhood of the point $a$ for some positive constant $C$
(see for instance \cite{Sp1}).
An asymptotic estimate for ${\mathcal{L}}_a(F,f)$ was obtained by Solern\'o \cite{Solerno}:
\betagin{equation}\lambdabel{eqsolerno}\tag{S}
{\mathcal{L}}_a(F,f) \le D^{M^{c\ell}},
\end{equation}
where $D$ is a bound for the degrees of the polynomials involved in a description of $F$, $f$ and $U$; $M$ is the number of variables
in these formulas; $\ell$ is the maximum number of alternating blocs of quantifiers in these formulas; and $c$ is an unspecified universal constant.
In this paper, we consider the case when $F$ is equal to the gradient $\nabla f:=\left(\frac{\partial f}{\partial x_1},\ldots,\frac{\partial f}{\partial x_n}\right):U\to\mathbf{\mathbb{R}}^n $ of a Nash function $f$ in $x=(x_1,\ldots,x_n)$. Recall that semialgebraic and analytic functions are called \emph{Nash functions}.
Our main goal is to obtain an effective estimate for the exponent $\varrho\in[0,1)$ in the following \emph{{\L}ojasiewicz gradient inequality} (see \cite{Lo1} or \cite{Lo2}, cf. \cite{T}):
\betagin{equation}\lambdabel{Lojineq1}\tag{{\rm \L}}
|\nabla f(x)|\gammae C|f(x)|^\varrho \hbox{ in a neighbourhood of $a\in\mathbf{\mathbb{R}}^n$ for some constant $C>0$}
\end{equation}
for an arbitrary Nash function $f:U\to \mathbf{\mathbb{R}}$, where $f(a)=0$, in terms of the degree of a polynomial $P\in\mathbf{\mathbb{R}}[x,y]$ describing the graph of $f$.
We denote by $|\nabla f(x)|$ the Euclidean norm of $\nabla f(x)$, i.e. $|\nabla f(x)|^2=\left(\frac{\partial f}{\partial x_1}(x)\right)^2+\cdots+\left(\frac{\partial f}{\partial x_n}(x)\right)^2$.
The smallest exponent $\varrho$ in \eqref{Lojineq1}, denoted by $\varrho_a(f)$, is called the \emph{{\L}ojasiewicz exponent in the gradient inequality} at $a$. It is known that \eqref{Lojineq1} holds with $\varrho=\varrho_a(f)$.
In the case of a polynomial function $f:\mathbf{\mathbb{R}}^n\to\mathbf{\mathbb{R}}$ of degree $d>0$ such that $0$ is an isolated point of $f^{-1}(0)$, J.~Gwo\'zdziewicz \cite{Gw} (cf. \cite{K1}) proved that
\betagin{equation}\lambdabel{Gw2}\tag{G2}
\varrho_0(f)\le 1-\frac{1}{(d-1)^n+1},
\end{equation}
and in the general case of an arbitrary polynomial $f$,
D. D'Acunto and K.~Kurdyka \cite{DK} (cf. \cite{DK1}, \cite{Gabrielov} and \cite{TienSon}) showed that
\betagin{equation}\lambdabel{DKineq}\tag{DK}
\varrho_0(f)\le 1-\frac{1}{d(3d-3)^{n-1}},\quad\hbox{provided}\quad d\gammae 2.
\end{equation}
If $f$ is a rational function of the form $f={p}\slash{q}$,
where $p,\,q\in \mathbf{\mathbb{R}}[x]$, $p(0)=0$ and $q(0)\ne 0$, then $\varrho_0(f)=\varrho_0(p)$, so \eqref{Gw2} and \eqref{DKineq} hold with $d=\deg p$.
The aim of this paper is to show generalizations of the above estimates
for Nash functions
(see Theorems \ref{maintwr} and \ref{maintwrIII} in Section \ref{Lojgradineqsect}). More precisely, let $U\subset \mathbf{\mathbb{R}}^n$ be a neighbourhood of $a\in\mathbf{\mathbb{R}}^n$ and let $f:U\to \mathbf{\mathbb{R}}$ be a nonzero Nash function. We give a Solern\'o and D'Acunto-Kurdyka type estimation of the exponent $\varrho\in[0,1)$ in the
{\L}ojasiewicz gradient inequality \eqref{Lojineq1} in terms of the degree $d$ of a nonzero polynomial $P$ such that $P(x,f(x))=0$, $x\in U$. Namely, in Theorem \ref{maintwrIII} we obtain
$$
\varrho_a(f)\leq 1-\frac{1}{2(2d-1)^{3n+1}}.
$$
If additionally $n\gammae 2$ and $\frac{\partial P}{\partial y}(x,f(x))\ne 0$ for $x\in U$, then in Theorem \ref{maintwr} we obtain
$$
\varrho_a(f)\leq 1-\frac{1}{d(3d-2)^{n}+1}, \quad\hbox{provided}\quad d\gammae 2.
$$
The above estimates are comparable with the Solern\'o estimate \eqref{eqsolerno}, but our estimates are explicit.
As a corollary, we obtain the following inequality (see Corollary \ref{wn012}):
\betagin{equation}\lambdabel{eq052}
|\nabla f(x)|\gammae C \operatorname{dist} (x,f^{-1}(0))^{2(2d-1)^{3n+1}-1}\quad \hbox{in a neighbourhood of $a$}.
\end{equation}
If additionally $n\gammae 2$ and $\frac{\partial P}{\partial y}(x,f(x))\ne 0$ for $x\in U$, then
\betagin{equation}\lambdabel{eq051}
|\nabla f(x)|\gammae C \operatorname{dist} (x,f^{-1}(0))^{d(3d-2)^{n}}\quad \hbox{in a neighbourhood of $a$}.
\end{equation}
The inequalities \eqref{eq052}, \eqref{eq051} are essential points in the effective estimate of the degree of sufficiency of non-isolated Nash function singularities given in the next section. The proof of these inequalities is based on Theorem \ref{maintwrIII} and estimates of the length of trajectories of the vector field $\nabla f$ in $U\setminus f^{-1}(0)$ (see Theorem \ref{lengthtrajectory}).
\subsection{Sufficiency of non-isolated Nash function singularities}
Let ${\mathscr{C}}^k_a(n)$ denote the set of ${\mathscr{C}}^k$ real functions defined in neighbourhoods of $a\in \mathbf{\mathbb{R}}^n$.
By a $k$-\emph{jet} at $a\in \mathbf{\mathbb{R}}^n$ in the class ${\mathscr{C}}^\ell$ we mean a family of functions $w\subset {\mathscr{C}}^\ell_a(n)$, called \emph{${\mathscr{C}}^\ell$-realizations} of this jet, possessing the same Taylor polynomial of degree $k$ at $a$. We also say that $f$ \emph{determines} a $k$-jet at $a$ in ${\mathscr{C}}^\ell$ if $f$ is a ${\mathscr{C}}^\ell$-realization of this jet.
For a function $f\in {\mathscr{C}}^k_a(n)$, we denote by $j^kf(a)$ the $k$-jet at $a$ (in ${\mathscr{C}}^k$) determined by $f$.
Let $Z\subset \mathbf{\mathbb{R}}^n$ be a set such that $0\in Z$ and let $k\in\mathbf{\mathbb{Z}}$, $k>0$. By a $k$-$Z$-\emph{jet in the class ${\mathscr{C}}^k$},
or briefly a $k$-$Z$-\emph{jet}, we mean an equivalence class $w\subset {\mathscr{C}}^k_0(n)$ of the following equivalence relation: $f\sim g$ iff for some neighbourhood $U\subset \mathbf{\mathbb{R}}^n$ of the origin, $j^kf(a)=j^kg(a)$ for $a\in Z\cap U$ (cf. \cite{MigusRS}, \cite{Xu}). The functions $f\in w$ are called \emph{${\mathscr{C}}^k$-$Z$-realizations} of the jet $w$ and we write $w=j^{k}_{Z}f$. The set of all jets $j^{k}_{Z}f$ is denoted by $J^{k}_Z(n)$.
The $k$-$Z$-jet $w\in J^{k}_Z(n)$ is said to be ${\mathscr{C}}^r$-$Z$-\emph{sufficient} (resp. $Z$-$v$-\emph{sufficient}) in the class ${\mathscr{C}}^k$ if for every of its ${\mathscr{C}}^k$-$Z$-realizations $f$ and $g$ there exist sufficiently small neighbourhoods $U_1,\,U_2\subset\mathbf{\mathbb{R}}^n$ of $0$, and a ${\mathscr{C}}^r$ diffeomorphism $\varphi:U_1\to U_2$, such that $f\circ \varphi=g$ in $U_1$ (resp. there exists a homeomorphism $\varphi: [f^{-1}(0)\cup Z]\cap U_1 \to[g^{-1}(0)\cup Z]\cap U_2$ with $\varphi(0)=0$ and $\varphi(Z\cap U_1)= Z\cap U_2$).
The classical and significant result on sufficiency of jets is the following:
\betagin{twr}[Kuiper, Kuo, Bochnak-{\L}ojasiewicz]\lambdabel{KKBL}
Let $w$ be a $k$-jet at $0\in \mathbf{\mathbb{R}}^n$ and let $f$ be its ${\mathscr{C}}^k$-realization. If $f(0)=0$ then the following conditions are equivalent:
\betagin{itemize}
\item[(a)] $w$ is ${\mathscr{C}}^0$-sufficient in ${\mathscr{C}}^k$,
\item[(b)] $w$ is $v$-sufficient in ${\mathscr{C}}^k$,
\item[(c)] $|\nabla f(x)|\gammae C|x|^{k-1}$ in a neighbourhood of the origin for some $C>0$.
\end{itemize}
\end{twr}
The implication (c)$\Rightarrow$(a) was proved by N. H.~Kuiper \cite{Kui} and T. C.~Kuo \cite{Kuo}, (b)$\Rightarrow$(c) by J.~Bochnak and S. \L ojasiewicz \cite{BL}, and (a)$\Rightarrow$(b) is obvious (cf. \cite{OSS}).
Let us recall the notions of isotopy and topological triviality.
Let $\Omega\subset\mathbf{\mathbb{R}}^n$ be a neighbourhood of $0\in\mathbf{\mathbb{R}}^n$ and let $Z\subset \mathbf{\mathbb{R}}^n$ with $0\in Z$.
A continuous mapping $H\colon\Omega\times [0,1]\to\mathbf{\mathbb{R}}^n$ is called an \emph{isotopy near $Z$ at zero} if:
\noindent{\rm (a)} $H_0(x)=x$ for $x\in \Omega$ and $H_t(x)=x$ for $t\in[0,1]$ and $x\in \Omega\cap Z$,\\
{\rm (b)} for any $t$ the mapping $H_t:\Omega\to \mathbf{\mathbb{R}}^n$ is a homeomorphism onto $H_t(\Omega)$,\\
where $H_t(x)=H(x,t)$ for $x\in\Omega$, $t\in [0,1]$.
Functions $f:\Omega_1\to\mathbf{\mathbb{R}} $, $g:\Omega_2\to \mathbf{\mathbb{R}}$, where $\Omega_1,\Omega_2\subset\mathbf{\mathbb{R}}^n$ are neighbourhoods of $0\in\mathbf{\mathbb{R}}^n$, are
called \emph{isotopical near $Z$ at zero} if there exists an isotopy near $Z$ at zero,
$H:\Omega\times [0,1]\to\mathbf{\mathbb{R}}^n$, with $\Omega\subset\Omega_1\cap \Omega_2$, such that
$f(H_1(x))=g(x)$, $x\in\Omega$.
A deformation $f+tg$ is called \emph{topologically trivial near $Z$} along $[0,1]$ if there exists an isotopy near $Z$ at zero, $H:\Omega \times [0,1]\to \mathbf{\mathbb{R}}^n$, with $\Omega \subset \Omega_1\cap\Omega_2$, such that $f(H(t,x))+tg(H(t,x))$ does not depend on $t$.
Theorem \ref{KKBL} concerns the case of an isolated singularity of $f$ at $0$, i.e. $0$ is an isolated zero of $\nabla f$. In the case of a non-isolated singularity of $f$ at $0$, from \cite[Theorems 1.3 and 1.4]{MigusRS} (cf. \cite{Xu}) we have the following criterion
for sufficiency of jets.
\betagin{twr}\lambdabel{XuXu}
Let $f\in {\mathscr{C}}_0^k(n)$ be a ${\mathscr{C}}^k$-$Z$-realization of a $k$-$Z$-jet $w\in J^{k}_Z(n)$, where $k>1$ and $Z=f^{-1}(0)$, $0\in Z$, and suppose $(\nabla f)^{-1}(0)\subset Z$. Then the following conditions are equivalent:
\noindent{\rm (a)} The $k$-$Z$-jet $w$ is ${\mathscr{C}}^0$-$Z$-sufficient in ${\mathscr{C}}^k$.
\noindent{\rm (b)} For any ${\mathscr{C}}^k$-$Z$-realizations $f_1,\,f_2$ of $w$, the deformation $f_1+t(f_2-f_1)$, $t\in\mathbf{\mathbb{R}}$, is topologically trivial along $[0,1]$.
\noindent{\rm (c)} Any two ${\mathscr{C}}^k$-$Z$-realizations
of $w$ are isotopical at zero.
\noindent{\rm(d)} The $k$-$Z$-jet $w$ is $Z$-$v$-sufficient in ${\mathscr{C}}^k$.
\noindent{\rm(e)} There exists a positive constant $C$ such that
\betagin{equation*}\lambdabel{KKcondition}
|\nabla f(x)|\gammae C\operatorname{dist}(x,Z)^{k-1}\quad \hbox{in a neighbourhood of the origin}.
\end{equation*}
\end{twr}
Let $f:U\to \mathbf{\mathbb{R}}$ be a Nash function, where $U\subset \mathbf{\mathbb{R}}^n$ is a neighbourhood of the origin, let $Z=f^{-1}(0)$, and suppose $0\in Z$.
The main result of this paper is the following corollary from Theorem \ref{XuXu} and inequality \eqref{eq052}.
\betagin{twr}\lambdabel{mainsuffjetLojineq}
Let $k=2(2d-1)^{3n+1}$, where $d=\deg_0 f$, and let $w\in J^{k}_Z(n)$ be the $k$-$Z$-jet for which $f$ is a ${\mathscr{C}}^k$-$Z$-realization. Then:
\noindent{\rm(a)} The $k$-$Z$-jet $w$ is ${\mathscr{C}}^0$-$Z$-sufficient in ${\mathscr{C}}^k$.
\noindent{\rm(b)} For any ${\mathscr{C}}^k$-$Z$-realizations $f_1,\,f_2$ of $w$, the deformation $f_1+t(f_2-f_1)$, $t\in\mathbf{\mathbb{R}}$, is topologically trivial along $[0,1]$.
\noindent{\rm(c)} Any two ${\mathscr{C}}^k$-$Z$-realizations
of $w$ are isotopical at zero.
\noindent{\rm(d)} The $k$-$Z$-jet $w$ is $Z$-$v$-sufficient in ${\mathscr{C}}^k$.
\end{twr}
Under additional assumption on $f$, from Theorem \ref{XuXu} and inequality \eqref{eq051}, we obtain
\betagin{twr}\lambdabel{mainsuffjetLojineqII}
Assume that there exists a nonzero polynomial $P\in\mathbf{\mathbb{R}}[x,y]$ such that $P(x,f(x))=0$ and $\frac{\partial P}{\partial y}(x,f(x))\ne 0$ for $x\in U$. Then the assertion of Theorem \ref{mainsuffjetLojineq} holds with $k=d(3d-2)^{n}+1$, where $d=\deg P$.
\end{twr}
\betagin{remark}\lambdabel{rem1}
If $f$ is a polynomial of degree $d>1$ or a rational function $f={p}\slash{q}$, where $p(0)=0$, $q(0)\ne 0$ and $d=\deg p$, then from Theorem \ref{XuXu} and by \eqref{DKineq}, the assertion of Theorem \ref{mainsuffjetLojineq} holds with $k=d(3d-3)^{n-1}$. If additionally the origin is an isolated zero of $f$, then by \eqref{Gw2} the assertion of Theorem \ref{mainsuffjetLojineq} holds with $k=(d-1)^n+1$.
\end{remark}
\section{{\L}ojasiewicz gradient inequality}\lambdabel{Lojgradineqsect}
Let $f:U \to \mathbf{\mathbb{R}}$, where $U\subset \mathbf{\mathbb{R}}^n$ is a connected neighbourhood of $a\in\mathbf{\mathbb{R}}^n$, be a Nash function.
Let $P\in\mathbf{\mathbb{R}}[x,y]$ be the unique irreducible real polynomial such that
\betagin{equation}\lambdabel{eqP}
P(x,f(x))=0\quad \hbox{for }x\in U,
\end{equation}
and let
$$
d=\deg P.
$$
We will call this number $d$ the \emph{degree of the Nash function} $f$ \emph{at} $a$ and denote it by $\deg_a f$. Obviously $d=\deg_a f > 0$ is uniquely determined. For $d=1$, the function $f$ is linear and \eqref{Lojineq1} holds with $\varrho=0$, so we will assume that $d>1$. We will also assume that $\nabla f(a)=0$, because in the opposite case \eqref{Lojineq1} holds with $\varrho=0$.
Put
$$
\mathcal{R}(n,d)=\max\{2d(2d-1),
d(3d-2)^n\}+1.
$$
The main result of this section is the following theorem.
\betagin{twr}\lambdabel{maintwr}
Let $f: U\to \mathbf{\mathbb{R}}$ be a nonzero Nash function such that $f(a)=0$ and $\nabla f(a)=0$. Assume that for the unique polynomial $P$ satisfying \eqref{eqP} we have
\betagin{equation}\lambdabel{geberalassumption}
\frac{\partial P}{\partial y}(x,f(x))\ne 0\quad\hbox{for }x\in U.
\end{equation}
Then $\varrho_a(f)\le 1-\frac{1}{\mathcal{R}(n,d)}$. Moreover, for $\varrho=1-\frac{1}{\mathcal{R}(n,d)}$ and some constants $C,\varepsilon>0$,
\betagin{equation}\lambdabel{Lojineqmain}
|\nabla f(x)|\gammae C|f(x)|^\varrho\quad\hbox{for}\quad |x-a|<\varepsilon,\quad |f(x)|<\varepsilon.
\end{equation}
\end{twr}
Without the assumption \eqref{geberalassumption}, we have a somewhat weaker estimation of the exponent $\varrho_a(f)$ than in Theorem \ref{maintwr}. Namely, let
$$
\mathcal{S}(n,d)=
2(2d-1)^{3n+1}.
$$
\betagin{twr}\lambdabel{maintwrIII}
Let $f: U\to \mathbf{\mathbb{R}}$ be a nonzero Nash function such that $f(a)=0$ and $\nabla f(a)=0$ and let $P$ be the unique polynomial satisfying \eqref{eqP}.
Then $\varrho_a(f)\le 1-\frac{1}{\mathcal{S}(n,d)}$. Moreover, \eqref{Lojineqmain} holds actually
with $\varrho=1-\frac{1}{\mathcal{S}(n,d)}$.
\end{twr}
Theorems \ref{maintwr} and \ref{maintwrIII} are generalizations for Nash functions of the above mentioned results by J. Gwo\'zdziewicz and D.~D'Acunto and K.~Kurdyka in the polynomial function case.
They are also comparable with Solern\'o's estimate \eqref{eqsolerno}, but our estimates are explicit. In the case of Nash functions with isolated singularity at zero, a similar result was obtained in \cite{KOSS}.
We give the proofs of Theorems \ref{maintwr} and \ref{maintwrIII} in Section \ref{roz2}.
\section{{\L}ojasiewicz inequality}
Let $X\subset \mathbf{\mathbb{R}}^n$ be a compact semialgebraic set and let $f:X\to\mathbf{\mathbb{R}}$ be a Nash function. Then $f$ is defined in a neighbourhood of $X$. So, there exists a compact semialgebraic set $Y\subset \mathbf{\mathbb{R}}^n$ such that $X\subset \operatorname{Int} Y$ and $f$ is defined on $Y$.
The \emph{degree} of $f$ is defined to be $\sup\{\deg_a f:a\in X\}$ and is denoted by $\deg_X f$. In fact, $\deg_X f=\max\{\deg_a f:a\in X\}$. Moreover, one can assume that $Y$ was chosen in such a manner that $\deg _X f=\deg_Y f$.
Let $\operatorname{dist} (x,V)$ denote the distance of a point $x\in\mathbf{\mathbb{R}}^n$ to a set $V\subset \mathbf{\mathbb{R}}^n$ in the Euclidean norm (with $\operatorname{dist} (x,V)=1$ if $V=\emptyset$).
\subsection{Global gradient £ojasiewicz inequality}
Theorems \ref{maintwr} and \ref{maintwrIII} have a local character. From these theorems we obtain a \emph{global {\L}ojasiewicz gradient inequality}.
\betagin{cor}\lambdabel{corLojgradglobal}
Let $d=\deg_X f$. If $(\nabla f)^{-1}(0)\subset f^{-1}(0)$ then for some positive constant $C$,
\betagin{equation}\lambdabel{eqLojgradglobal}
|\nabla f(x)|\gammae C|f(x)|^\varrho\quad\hbox{for}\quad x\in X
\end{equation}
with $\varrho=1-\frac{1}{\mathcal{S}(n,d)}$. If additionally there exists a polynomial $P\in\mathbf{\mathbb{R}}[x,y]$ such that $P(x,f(x))=0$ and $\frac{\partial P}{\partial y}(x,f(x))\ne 0$ for $x\in X$ and $d_1=\deg P$, then \eqref{eqLojgradglobal} holds with $\varrho=1-\frac{1}{\mathcal{R}(n,d_1)}$.
\end{cor}
Denote by $\varrho_X(f)$ the smallest exponent $\varrho$ for which \eqref{eqLojgradglobal} holds. We call it the \emph{{\L}ojasiewicz exponent in the gradient inequality} on $X$. It is known that the inequality \eqref{eqLojgradglobal} holds with $\varrho=\varrho_X(f)$. So, from Corollary \ref{corLojgradglobal} we obtain
\betagin{cor}\lambdabel{corestLojexp}
$\varrho_X(f)\leq 1-\frac{1}{S(n,d)}$.
\end{cor}
\subsection{Length of trajectory}\lambdabel{lenghtsect}
Let $f:X\to\mathbf{\mathbb{R}}$ be
a nonzero Nash function such that $(\nabla f)^{-1}(0)\subset f^{-1}(0)$, let $\varrho\in(0,1)$ and $C>0$ be such that the global inequality \eqref{eqLojgradglobal} in Corollary \ref{corLojgradglobal} holds in $X$,
and let $V=f^{-1}(0)$. Then $\nabla f(x)\ne 0$ for $x\in X\setminus V$.
Let $\varphi(t)=|t|^{1-\varrho}$ for $t\in \mathbf{\mathbb{R}}$. By the same argument as in the proof of \cite[Proposition 1]{KS1} we obtain (cf. \cite{KMP})
\betagin{prop}[Kurdyka-{\L}ojasiewicz inequality]\lambdabel{KurdykaLojasiewicz}
Under the above notations,
\betagin{equation*}
|\nabla (\varphi\circ f)(x)|\gammae (1-\varrho)C \quad\hbox{for $x\in X\setminus V$}.
\end{equation*}
\end{prop}
We will also assume that $\overline{\operatorname{Int} X\setminus V}=X$. Let
$$
U_{X,f}=\left\{x\in \operatorname{Int} X:
\frac{1}{C(1-\varrho)}|f(x)|^{1-\varrho}
<\operatorname{dist} (x,\mathbf{\mathbb{R}}^n\setminus X) \right\}.
$$
Then $U_{X,f}\subset X$ is a neighbourhood of $(\operatorname{Int} X)\cap V$.
Take a global trajectory $\gammaamma:[0,s)\to U_{X,f}\setminus V$ of the vector field
\betagin{equation*}\lambdabel{eq3H}
H(x)=-\,{\rm sign}\, f(x)\frac{\nabla f(x)}{|\nabla f(x)|} \quad\hbox{for $x\in U_{X,f}\setminus V$}.
\end{equation*}
Then the function $f\circ \gammaamma$ is monotonic,
so the limit $\lim_{t\to s}f\circ \gammaamma(t)$ exists.
Let $\operatorname{length} \gammaamma$ denote the length of $\gammaamma$. Since $|\gammaamma'(t)|=1$, we have $\operatorname{length} \gammaamma=s$.
The following generalization of \cite[Theorem 1]{KS1} has a similar proof.
\betagin{twr}\lambdabel{lengthtrajectory} The limit $\lim_{t\to s}\gammaamma(t)$ exists and belongs to $V$. Moreover,
\betagin{equation*}
\operatorname{dist} (\gammaamma(0),V)\le\operatorname{length} \gammaamma\le \frac{1}{(1-\varrho)C}|f(\gammaamma(0))|^{1-\varrho}.
\end{equation*}
\end{twr}
\betagin{proof}
Let $s_1\in [0,s)$ and $\gammaamma_{s_1}=\gammaamma|_{[0,s_1]}$. Then $\operatorname{length} \gammaamma_{s_1}=s_1$.
Since $\frac{\nabla f}{|\nabla f|}\,{\rm sign}\, f(x)=\frac{\nabla (\varphi \circ f)}{|\nabla(\varphi \circ f)|}$ for $x\in U\setminus V$,
it follows that
\betagin{equation*}
(\varphi\circ f\circ \gammaamma)'=\lambdangle \nabla (\varphi \circ f)\circ \gammaamma,\gammaamma '\rangle =-|\nabla (\varphi \circ f)\circ \gammaamma|,
\end{equation*}
where $\lambdangle \cdot,\cdot\rangle$ denotes the standard scalar product in
$\mathbf{\mathbb{R}}^n$, and Proposition \ref{KurdykaLojasiewicz} gives
\betagin{equation*}
\betagin{split}
\varphi(f(\gammaamma(0)))-\varphi(f(\gammaamma(s_1)))&=-s_1(\varphi\circ f\circ \gammaamma)'(t)=s_1 |\nabla (\varphi\circ f)\circ \gammaamma(t)|\\
&\gammae
(1-\varrho)C\operatorname{length} \gammaamma_{s_1}
\end{split}
\end{equation*}
for some $t\in [0,s_1]$. Then, letting $s_1\to s$, from the definition of $\varphi$ we have
$$
\operatorname{length} \gammaamma \le \frac{1}{(1-\varrho)C}(|f(\gammaamma(0))|^{1-\varrho}-\alpha)\le \frac{1}{(1-\varrho)C}|f(\gammaamma(0))|^{1-\varrho},
$$
where $\alpha=\lim_{s_1\to s}|f(\gammaamma(s_1))|^{1-\varrho}\gammae 0$.
Since $\gammaamma(0)\in U_{X,f}$, we see that $\operatorname{length}\gammaamma< \operatorname{dist}(\gammaamma(0),\mathbf{\mathbb{R}}^n\setminus X)$, so the limit $\lim_{t\to s}\gammaamma(t)$ certainly exists and belongs to $U_{X,f}$. Consequently, $\lim_{t\to s}\gammaamma(t)\in V$ and $\operatorname{length} \gammaamma\gammae \operatorname{dist} (\gammaamma(0),V)$. This gives the assertion.
\end{proof}
From Theorem \ref{lengthtrajectory}
we have
\betagin{cor}\lambdabel{wn01} Under the assumptions and notations of Theorem \ref{lengthtrajectory},
\betagin{equation*}\lambdabel{eq-15}
|f(x)|\gammae \left(C(1-\varrho)\right)^{1\slash(1-\varrho)} \operatorname{dist} (x,V)^{1\slash(1-\varrho)}, \quad x\in U_{X,f},
\end{equation*}
and
\betagin{equation*}\lambdabel{eq05}
|\nabla f(x)|\gammae \left(C(1-\varrho)\right)^{\varrho\slash(1-\varrho)} \operatorname{dist} (x,V)^{\varrho\slash(1-\varrho)},\quad x\in U_{X,f}.
\end{equation*}
\end{cor}
Similarly to \cite{KS1}, we obtain a version of the above corollary in the complex case with the same formulation.
From Corollaries \ref{corLojgradglobal}, \ref{wn01} and Theorem \ref{maintwrIII}, we immediately obtain
\betagin{cor}\lambdabel{wn012} Let $d=\deg_X f$. Then there exists a positive constant $C$ such that
\betagin{equation*}\lambdabel{eq-15}
|f(x)|\gammae C \operatorname{dist} (x,V)^{2(2d-1)^{3n+1}}, \quad x\in X,
\end{equation*}
and
\betagin{equation*}\lambdabel{eq05}
|\nabla f(x)|\gammae C \operatorname{dist} (x,V)^{2(2d-1)^{3n+1}-1},\quad x\in X.
\end{equation*}
If additionally $n\gammae 2$ and there exists a polynomial $P\in\mathbf{\mathbb{R}}[x,y]$ such that $P(x,f(x))=0$ and $\frac{\partial P}{\partial y}(x,f(x))\ne 0$ for $x\in X$, and $d=\deg P$, then
\betagin{equation*}\lambdabel{eq-15}
|f(x)|\gammae C \operatorname{dist} (x,V)^{d(3d-2)^{n}+1}, \quad x\in X,
\end{equation*}
and
\betagin{equation*}\lambdabel{eq05}
|\nabla f(x)|\gammae C \operatorname{dist} (x,V)^{d(3d-2)^{n}},\quad x\in X.
\end{equation*}
\end{cor}
\betagin{proof}
Take a compact semialgebraic set $Y\subset \mathbf{\mathbb{R}}^n$ such that $X\subset \operatorname{Int} Y$ and $Y\subset \{x\in\mathbf{\mathbb{R}}^n:\operatorname{dist}(x,X)<\varepsilon\}$. If $\varepsilon$ is sufficiently small, then we can consider the function $f$ on $Y$. Then we may assume that $\deg _Y f=\deg_X f$ and $(\nabla f)^{-1}(0)\subset f^{-1}(0)$ after extending $f$
onto~$Y$. So, the assertions of Theorem \ref{lengthtrajectory} and Corollary \ref{wn01} hold with $\varrho =1-\frac{1}{S(n,d)}$ on the set $U_{Y,f}$. Hence the assertions hold for $x\in X\cap U_{Y,f}$. By the definition of $U_{Y,f}$, we see that $X\setminus U_{Y,f}$ is a compact set and $\min \{|x-y|:x\in V,\;y\in X\setminus U_{Y,f}\}>0$. So, diminishing $C$ if necessary, we obtain the first part of the assertion. The second part is proved analogously.
\end{proof}
\subsection{{\L}ojasiewicz exponent}
Corollary \ref{wn01} implies the known fact that the exponents $\alpha>0$ in the inequality
\betagin{equation}\lambdabel{eq-15}
|f(x)|\gammae C\operatorname{dist} (x,V)^\alpha,\quad x\in X,
\end{equation}
for some positive constant $C$, are bounded below. The inequality \eqref{eq-15} is called the \emph{{\L}ojasiewicz inequality for $f$ on} $X$ and the lower bound of the exponents $\alpha>0$ is the \emph{{\L}ojasiewicz exponent} of $f$ on $X$, denoted by $\wll_X(f)$. It is known that \eqref{eq-15} holds with $\alpha=\wll_X(f)$ and some positive constant $C$.
From Theorem \ref{lengthtrajectory} we obtain
\betagin{cor}\lambdabel{corrhoLojexp}
$\wll_X(f)\leq \frac{1}{1-\varrho_X(f)}$.
\end{cor}
Corollary \ref{wn01} implies
\betagin{cor}\lambdabel{corLojexp} If $d=\deg_X f$, then
$\wll_X(f)\leq 2(2d-1)^{3n+1}$.
\end{cor}
For $n\gammae 4$ the above estimate is sharper than the one given in \cite{KSS} for continuous semialgebraic functions: $\wll_X(f)\leq d(6d-3)^{n+r-1}$, where $r\leq\frac{n(n+1)}{2}$ is the degree of complexity of $f$, equal to the number of inequalities necessary to define the graph of $f$, and $d$ is the maximal degree of polynomials describing the graph of~$f$. Consequently, this gives the estimate
$
\wll_X(f)\leq d(6d-3)^{n+n(n+1)/2-1}
$
in terms of the degree only. So, the estimate in Corollary \ref{corLojexp} is more exact than the one above for $n\gammae 4$.
\section{Total degree of algebraic sets}
Let $\mathbf{\mathbb{C}}[x]$ denote the ring of complex polynomials in $x=(x_1,\ldots,x_n)$.
Let $f=(f_1,\ldots,f_r):\mathbf{\mathbb{C}}^n\to\mathbf{\mathbb{C}}^r$ be a polynomial mapping with $\deg f_i >0$ for $i=1,\ldots,r$. Let $V=f^{-1}(0)\subset \mathbf{\mathbb{C}}^n$.
The \emph{total degree} of $V$ is the number
$$
\delta (V)=\deg V_1+\cdots+\deg V_s,
$$
where $V=V_1\cup\cdots\cup V_s$ is the decomposition into irreducible components (see \cite{Lo3}).
We have the following useful fact (see \cite{Lo3}).
\betagin{fact}\lambdabel{deltainters}
If $V,W\subset \mathbf{\mathbb{C}}^n$ are algebraic sets, then
$$
\delta(V\cap W)\le \delta(V)\delta(W).
$$
\end{fact}
From Fact \ref{deltainters} and the definition of total degree of algebraic sets we have the following two facts (cf. \cite{Lo3}).
\betagin{fact}\lambdabel{Fact1}$\delta(V)\le \deg f_1\cdots \deg f_r$.
In particular, for any irreducible component $V_j$ of $V$ we have
$$
\deg V_j\le \deg f_1\cdots \deg f_r.
$$
\end{fact}
\betagin{fact}\lambdabel{Fact2} Let $L:\mathbf{\mathbb{C}}^n\to\mathbf{\mathbb{C}}^k$ be a linear mapping. Then
$$
\delta(\overline{L(V)})\le \delta(V).
$$
\end{fact}
We will need the following lemma (see \cite[Lemma 3.20]{KOSS}).
\betagin{lem}\lambdabel{Fact3} Let $V_j$ be an irreducible component of the set $V$, and suppose $\dim V_j\gammae 1$. Then for a generic linear mapping $L=(L_1,\ldots,L_{n-1}):\mathbf{\mathbb{C}}^r\to\mathbf{\mathbb{C}}^{n-1}$ the set $V_j$ is an irreducible component of the set of common zeros of the system of equations
$$
L_i\circ f=0,\quad i=1,\ldots, n-1.
$$
In particular,
$$
\deg V_j\le \deg (L_1\circ f)\cdots\deg (L_{n-1}\circ f).
$$
Moreover, we can take $L_1(y_1,\ldots,y_r)=y_1$.
\end{lem}
\section{Proofs of Theorems \ref{maintwr} and \ref{maintwrIII}}\lambdabel{roz2}
The idea of the proofs is similar to that in \cite[proof of Theorem 1.2]{KOSS}.
Without loss of generality, we may assume that $a=0$.
Let $f:U \to \mathbf{\mathbb{R}}$ be a nonzero Nash function defined in an open neighbourhood $U\subset \mathbf{\mathbb{R}}^n$ of the origin such that $f(0)=0$ and $\nabla f(0)=0$. Let $P\in\mathbf{\mathbb{R}}[x,y]$ be the unique irreducible polynomial satisfying \eqref{eqP} and let $d=\deg P$.
Since the set of critical values of a differentiable semialgebraic function is finite, we have
\betagin{fact}\lambdabel{Fact4} There exists $\varepsilon>0$ such that $f$ has no critical values in the interval $(-\varepsilon,\varepsilon)$ except $0$.
\end{fact}
Let $\varepsilon>0$ be as in Fact \ref{Fact4}.
Take $r>0$. Denote by $\Omega$ the closed ball
$$
\Omega:=\{x\in\mathbf{\mathbb{R}}^n:|x|\le r\}
$$
and by $\partial \Omega$ the sphere $\{x\in\mathbf{\mathbb{R}}^n:|x|=r\}$.
Suppose that
$\Omega\subset U$.
Define a semialgebraic set $\Gamma\subset \Omega$ by
$$
\Gamma:=\{x\in \Omega:\forall_{\taueta\in \Omega}\;f(x)=f(\taueta)\;\Rightarrow\;|\nabla f(x)|\le|\nabla f(\taueta)|\}.
$$
Then by the definition of $\Gamma$ we have
\betagin{fact}\lambdabel{Factproof1}
Let $\varrho\in\mathbf{\mathbb{R}}$ and let $C>0$. If $|\nabla f(x)|\gammae C|f(x)|^\varrho$ for $x\in \Gamma$ such that $|f(x)|<\varepsilon$, then
$|\nabla f(x)|\gammae C|f(x)|^\varrho$ for $x\in\Omega$, $|f(x)|<\varepsilon$.
\end{fact}
Let $\varrho_0=\varrho_0(f)$. Then, decreasing $r$ if necessary, we can assume that
\betagin{equation}\lambdabel{Lojgradinequality}
|\nabla f(x)|\gammae C|f(x)|^{\varrho_0} \quad \hbox{for $x\in\Omega$ and some constant $C>0$}.
\end{equation}
Let us fix such an $r$.
Consider the case $n=1$. Denote by $\operatorname{ord}_0 f$ the order of $f$ at zero.
Then $f$ has an isolated zero and singularity at zero, $\operatorname{ord}_0 f>0$ and the inequality \eqref{Lojineqmain} holds with
\betagin{equation}\lambdabel{caen1}
\varrho_0(f)=\frac{\operatorname{ord}_0 f-1}{\operatorname{ord}_0 f}=1-\frac{1}{\operatorname{ord}_0 f}.
\end{equation}
Let the polynomial $P$ be of the form $
P(x_1,y)=p_0(x_1)y^d+p_1(x_1)y^{d-1}+\cdots +p_d(x_1)$,
where $p_0,\ldots,p_d\in\mathbf{\mathbb{R}}[x_1]$. As $P$ is irreducible, $p_d\ne 0$ and $\operatorname{ord}_0 p_d\le d$. Since
$$
-p_d(x_1)=f(x_1)(p_0(x_1)(f(x_1))^{d-1}+p_1(x_1)(f(x_1))^{d-2}+\cdots +p_{d-1}(x_1)),
$$
we have $\operatorname{ord}_0 f\le \operatorname{ord}_0 p_d\le d$. Together with \eqref{caen1} this gives \eqref{Lojineqmain} with $\varrho_0(f)=1-\frac{1}{d}$ and the assertions of Theorems \ref{maintwr} and \ref{maintwrIII} in the case $n=1$.
In the remainder of this article we will assume that $n>1$.
By \eqref{Lojgradinequality} and the Curve Selection Lemma, there exists an analytic curve $\varphi:[0,1)\to \Omega$ for which $f(\varphi(0))=0$, $f(\varphi(\xi))\ne 0$ for $\xi\in(0,1)$ and for some constant $C_{1}>0$,
\betagin{equation}\lambdabel{Lojgradinequality2}
C|f(\varphi(\xi))|^{\varrho_0}\le |\nabla f(\varphi(\xi))|\le C_{1}|f(\varphi(\xi))|^{\varrho_0},\quad \xi\in [0,1)
\end{equation}
(cf. \cite{Sp1}).
By Fact \ref{Factproof1} we may assume that $\varphi ([0,1))\subset \Gamma$. Then we have two cases:
I. $\varphi\big(( 0,1)\big)\subset\operatorname{Int}\Omega$,
II. $\varphi\big([ 0,1)\big)\subset\partial\Omega$.
We will use the Lagrange multipliers theorem to describe the relation between the values $y=f(x)$ and $u=|\nabla f(x)|^2$ for $x\in\Gamma$, so we put
$$
\Gamma_I =\{x\in\Omega:\exists_{\lambda\in\mathbf{\mathbb{R}}}\, \nabla|\nabla f(x)|^2-\lambda\nabla f(x)=0\},
$$
$$
{\Gamma}_{II} =\{x\in\partial \Omega:|f(x)|<\varepsilon\;\lambdand\;\exists_{\lambda_1,\lambda_2\in\mathbf{\mathbb{R}}}\, \nabla|\nabla f(x)|^2-\lambda_1\nabla f(x)-2\lambda_2x=0\}.
$$
To fulfill the assumptions of the Lagrange theorem we will need
\betagin{lem}\lambdabel{ggg}
There exists $\varepsilon>0$ such that for every $x\in \partial\Omega$ and every $y\in\mathbf{\mathbb{R}}$ such that $0<|y|<\varepsilon$ and $y=f(x)$, the vectors $
\nabla \big(|x|^2-r^2\big)$ and $\nabla f(x)$ (that is, $2x$ and $\nabla f(x)$) are linearly independent.
\end{lem}
\betagin{proof}
If $f|_{\partial \Omega}$ is a constant function then the assertion is obvious. Assume that $f$ is not constant on $\partial \Omega$. Then, by Fact \ref{Fact4}, there exists $\varepsilon>0$ such that $\nabla f(x)\ne 0$ for $x\in\partial \Omega$, $0<|f(x)|<\varepsilon$.
Suppose to the contrary that for any $\varepsilon>0$ there exist $x\in\partial\Omega$ and $y_{\varepsilon}\in\mathbf{\mathbb{R}}$ with $0<|y_{\varepsilon}|<\varepsilon$ such that $y_{\varepsilon}=f(x)$ and
$\nabla f(x)=\xi\cdot 2x$ for some $\xi\in\mathbf{\mathbb{R}}\setminus\{0\}$.
Then by the Curve Selection Lemma there exist analytic curves $\gammaamma:[ 0,1)\to \partial\Omega$ with $\gammaamma((0,1))\subset\Omega\setminus f^{-1}(0)$ and $f(\gammaamma(0))=0$, and $\alpha:[ 0,1)\to\mathbf{\mathbb{R}} $,
such that for $t\in (0,1)$,
\betagin{equation*}\lambdabel{gx}
\nabla f\big(\gammaamma(t)\big)=\alpha(t)\cdot 2\gammaamma(t).
\end{equation*}
Then
$$
(f\circ \gammaamma)'(t)=\lambdangle \nabla f(\gammaamma(t)),\gammaamma'(t)\rangle=\alpha(t)\lambdangle \gammaamma(t),\gammaamma'(t)\rangle=0,
$$
and consequently $f\circ \gammaamma$ is a constant function equal to $0$. This contradicts the choice of $\gammaamma$ and ends the proof.
\end{proof}
By the Lagrange multipliers theorem, Fact \ref{Fact4} and Lemma \ref{ggg} we obtain
\betagin{fact}\lambdabel{fact1IandIIl}
Let $\varepsilon>0$ fulfill Fact \ref{Fact4} and Lemma \ref{ggg}. Take a point $x_0\in \Omega$ such that $0<|f(x_0)|<\varepsilon$.
{\rm (a)} If $x_0\in\Gamma\cap\operatorname{Int} \Omega$ then $x_0$ is a
lower critical point of the function $\Omega\ni x\mapsto |\nabla f(x)|^2\in\mathbf{\mathbb{R}}$ on the set $f^{-1}(f(x_0))\cap \Omega$. In particular, $\Gamma\cap \operatorname{Int} \Omega\subset \Gamma_I$.
{\rm (b)} If $n\gammae 3$, $x_0\in\Gamma\cap \partial \Omega$ then $x_0$ is a lower critical point of the function $\partial \Omega\ni x\mapsto |\nabla f(x)|^2\in\mathbf{\mathbb{R}}$ on the set $f^{-1}(f(x_0))\cap \partial \Omega$. In particular, $\Gamma \cap \partial \Omega\subset \Gamma_{II}$.
\end{fact}
Let $\mathbf{\mathbb{M}}=\mathbf{\mathbb{C}}^n\times\mathbf{\mathbb{C}}\times\mathbf{\mathbb{C}}\times\mathbf{\mathbb{C}}^n\times\mathbf{\mathbb{C}}^n$, and let $\mathbf{\mathbb{X}}\subset \mathbf{\mathbb{M}}$ be the Zariski closure of the set
$$
\{(x,f(x),|\nabla f(x)|^2,\nabla f(x),\nabla |\nabla f(x)|^2)\in \mathbf{\mathbb{M}}:x\in\Omega\}.
$$
We will determine polynomials describing a certain algebraic set $\mathbf{\mathbb{Y}}\subset \mathbf{\mathbb{M}}$ containing $\mathbf{\mathbb{X}}$ as an irreducible component. Let $G\in \mathbf{\mathbb{C}}[x,y,u]$, where $u$ is a variable, be the polynomial defined by
\betagin{equation}\lambdabel{eqdefG}
G(x,y,u)=\sum_{i=1}^n\left(\frac{\partial P}{\partial x_i}(x,y)\right)^2-\left(\frac{\partial P}{\partial y}(x,y)\right)^2 \cdot u.
\end{equation}
It is easy to observe that
$G(x,f(x),|\nabla f(x)|^2)=0$ for $x\in \Omega$. In particular, the polynomial $G$ vanishes on $\mathbf{\mathbb{X}}$.
Take systems of variables $t=(t_1,\ldots,t_n)$, $z=(z_1,\ldots,z_n)$, and let
$G_1,G_{2,i},G_{3,i} \in\mathbf{\mathbb{C}}[x,y,u,t,z]$ be defined by
\betagin{align}
G_1(x,y,u)&=u-t_1^2-\cdots- t_n^2, \,,\nonumber&\\%\numberthis\lambdabel{eqdefG5}&\\
G_{2,i}(x,y,t)&=\frac{\partial P}{\partial x_i}(x,y)+\frac{\partial P}{\partial y}(x,y)t_i \,,& 1\le i\le n,\nonumber\\%\lambdabel{eqG2i}\\
G_{3,i}(x,y,u,t,z)&=\frac{\partial G}{\partial x_i}\left(x,y,u\right)+\frac{\partial G}{\partial y}\left(x,y,u\right)t_i \nonumber
&\\
&\mathbf{\mathbb{Q}}uad\mathbf{\mathbb{Q}}uad\mathbf{\mathbb{Q}}uad-\left(\frac{\partial P}{\partial y}\left(x,y\right)\right)^2\cdot z_i\, ,& 1\le i\le n.\nonumber
\end{align}
Let $\mathbf{\mathbb{Y}}\subset \mathbf{\mathbb{M}}$ be the closure of the
constructible set
\betagin{multline*}
\mathbf{\mathbb{Y}}^0=\{w=(x,y,u,t,z)\in \mathbf{\mathbb{M}}: P(x,y)=0,\;\frac{\partial P}{\partial y}(x,y)\ne 0,\; G_1(x,y,u)=0,\\
G_{2,i}(x,y,t)=0,\; G_{3,i}(w)=0,\;1\le i\le n\}.
\end{multline*}
Obviously $\mathbf{\mathbb{X}}\subset \mathbf{\mathbb{Y}}$, and locally $\mathbf{\mathbb{Y}}^0$ is the graph of a complex Nash mapping (i.e., a holomorphic mapping with semialgebraic graph). Moreover, we have
\betagin{lem}\lambdabel{factY0smooth}
The set $\mathbf{\mathbb{X}}$ is an irreducible component of $\,\mathbf{\mathbb{Y}}$. Moreover, $\mathbf{\mathbb{Y}}^0$ is a Zariski open and dense subset of $\mathbf{\mathbb{Y}}$, and any point $w=(x_0,y_0,u_0,t_0,z_0)\in\mathbf{\mathbb{Y}}^0$ has a neighbourhood $B\subset \mathbf{\mathbb{M}}$ such that $\mathbf{\mathbb{Y}}\cap B=\mathbf{\mathbb{Y}}^0\cap B$ and
\betagin{equation*}\lambdabel{eqformY0graph}
\mathbf{\mathbb{Y}}^0\cap B=\left\{w=\left(x,g(x),h(x),\nabla g(x),\nabla h(x)\right)\in\mathbf{\mathbb{M}}:x\in \Delta\right\}
\end{equation*}
for some holomorphic function $g:\Delta\to \mathbf{\mathbb{C}}$, where $\Delta\subset \mathbf{\mathbb{C}}^n$ is a neighbourhood of $x_0$, and $h(x)=\left(\frac{\partial g}{\partial x_1}(x)\right)^2+\cdots+\left(\frac{\partial g}{\partial x_n}(x)\right)^2$.
\end{lem}
\betagin{proof}
Since $P$ is an irreducible polynomial, $\frac{\partial P}{\partial y}$ does not vanish on $\mathbf{\mathbb{X}}$. So, by the Implicit Function Theorem,
$
\{w=(x,y,u,t,z)\in \mathbf{\mathbb{X}}:\frac{\partial P}{\partial y}(x,y)\ne0\}
$
is an open and dense subset of $\mathbf{\mathbb{X}}$, and moreover it is a smooth and connected submanifold of~$\mathbf{\mathbb{Y}}^0$. Consequently, $\mathbf{\mathbb{X}}$ is an irreducible component of $\mathbf{\mathbb{Y}}$. The ``moreover'' part of the assertion follows immediately from the Implicit Function Theorem.
\end{proof}
Define $G_0,G_{4,i,j},G_{4,i,j,k}\in\mathbf{\mathbb{C}}[x,y,u,t,z]$ by
\betagin{align}
G_0(x)&=x_1^2+\cdots+x_n^2-r^2,\nonumber\\
G_{4,i,j}(t,z)&=\det\left[\betagin{matrix}
t_i&z_i\\
t_j&z_j \end{matrix}\right]
\,,& 1\le i<j\le n,\nonumber\\%\numberthis\lambdabel{eqdefG4ij}\\%\nonumber \\
G_{4,i,j,k}(x,t,z)&=\det\left[\betagin{matrix}
t_i&z_i&x_i\\
t_j&z_j&x_j\\
t_k&z_k&x_k
\end{matrix}\right]\,,& 1\le i<j<k\le n,\nonumber
\end{align}
where the polynomials $G_{4,i,j,k}$ are defined if $n\gammaeq 3$. Put
\betagin{align}
\mathbf{\mathbb{X}}_I&=\{w=(x,y,u,t,z)\in \mathbf{\mathbb{X}}:G_{4,i,j}(t,z)= 0,\;1\le i<j\le n\},&\nonumber\\
\mathbf{\mathbb{X}}_{II}&=\{w=(x,y,u,t,z)\in \mathbf{\mathbb{X}}:G_0(x)=0,\;G_{4,i,j,k}(x,t,z)=0,\; 1\le i<j<k\le n\},&\nonumber\\
\mathbf{\mathbb{L}}_I&=\{(w,\lambda)=(x,y,u,t,z,\lambda)\in \mathbf{\mathbb{X}}\times \mathbf{\mathbb{C}}: z= \lambda t \},&\nonumber\\
\mathbf{\mathbb{L}}_{II}&=\{(w,\lambda_1,\lambda_2)=(x,y,u,t,z,\lambda_1,\lambda_2)\in \mathbf{\mathbb{X}}\times \mathbf{\mathbb{C}}\times \mathbf{\mathbb{C}}: G_0(x)=0,\;z=\lambda_1 t+\lambda_2 x\},&\nonumber\\
\mathbf{\mathbb{Y}}_I&=\{w=(x,y,u,t,z)\in \mathbf{\mathbb{Y}}:G_{4,i,j}(t,z)= 0,\;1\le i<j\le n\},&\nonumber\\
\mathbf{\mathbb{Y}}_{II}&=\{w=(x,y,u,t,z)\in \mathbf{\mathbb{Y}}:G_0(x)=0,\,G_{4,i,j,k}(x,t,z)=0,\; 1\le i<j<k\le n\},&\nonumber\\
{\mathbf{\mathcal{Z}}}_{I}&=\{w=(x,y,u,t,z)\in \mathbf{\mathbb{X}}:x\in \Gamma_I\},&\nonumber\\
{\mathbf{\mathcal{Z}}}_{II}&=\{w=(x,y,u,t,z)\in \mathbf{\mathbb{X}}:x\in\Gamma_{II}\},\nonumber&\\
\mathbf{\mathbb{F}}F&=\{w=(x,y,u,t,z)\in \mathbf{\mathbb{X}}:x\in \varphi((0,1))\},&\nonumber
\end{align}
where the sets $\mathbf{\mathbb{X}}_{II}$, $\mathbf{\mathbb{L}}_{II}$ and $\mathbf{\mathbb{Y}}_{II}$ are defined for $n\gammaeq 3$.
Obviously $\mathbf{\mathbb{X}}_I\subset \mathbf{\mathbb{Y}}_I$ and $\mathbf{\mathbb{X}}_{II}\subset\mathbf{\mathbb{Y}}_{II}$. Moreover, any irreducible component of $\mathbf{\mathbb{X}}_I$ is an irreducible component of $\mathbf{\mathbb{Y}}_I$. The same holds for $\mathbf{\mathbb{X}}_{II}$ and $\mathbf{\mathbb{Y}}_{II}$. Additionally, by the Lagrange multipliers theorem and Facts \ref{Fact4}, \ref{fact1IandIIl} we immediately obtain
\betagin{fact}\lambdabel{factZaropensubs}
{\rm (a)} Let
$$
A_I=
\left\{w\in \mathbf{\mathbb{X}}:\exists_{\lambda\in\mathbf{\mathbb{C}}}\;(w,\lambda)\in \mathbf{\mathbb{L}}_I\right\}.
$$
If $\varphi((0,1))\subset \operatorname{Int} \Omega$ then $\mathbf{\mathbb{F}}F\subset {\mathbf{\mathcal{Z}}}_I\subset A_I
\subset \mathbf{\mathbb{X}}_I\subset \mathbf{\mathbb{Y}}_I$ and there exists an irreducible component $\mathbf{\mathbb{X}}_{I,*}$ of $\overline{A_I}$
which contains $\mathbf{\mathbb{F}}F$ and is an irreducible component of $\mathbf{\mathbb{X}}_I$.
{\rm (b)} Let
$$
A_{II}=
\left\{w\in \mathbf{\mathbb{X}}:\exists_{\lambda_1,\lambda_2\in\mathbf{\mathbb{C}}}\;(w,\lambda_1,\lambda_2)\in \mathbf{\mathbb{L}}_{II}\right\}.
$$
If $\varphi((0,1))\subset \partial \Omega$ then $\mathbf{\mathbb{F}}F\subset {\mathbf{\mathcal{Z}}}_{II}\subset A_{II}
\subset \mathbf{\mathbb{X}}_{II}\subset \mathbf{\mathbb{Y}}_{II}$ and there exists an irreducible component $\mathbf{\mathbb{X}}_{II,*}$ of $\overline{A_{II}}$
which contains $\mathbf{\mathbb{F}}F$ and is an irreducible component of $\mathbf{\mathbb{X}}_{II}$.
\end{fact}
\betagin{proof}
From
Fact \ref{fact1IandIIl}(a) we have $\mathbf{\mathbb{F}}F\subset \{(x,y,u,t,z)\in \mathbf{\mathbb{X}}:x\in\Gamma_I\}\subset A_I$.
Since all the polynomials $G_{4,i,j}$ vanish on $\mathbf{\mathbb{X}}_I$, the vectors $t,\,z$ are linearly dependent provided $(x,y,u,t,z)\in\mathbf{\mathbb{X}}_I$ for some $x,y,u$. So $\mathbf{\mathbb{X}}_I=\mathcal{X}_I\cup A_I$,
where
$$
\mathcal{X}_I=\{w=(x,y,u,t,z)\in \mathbf{\mathbb{X}}_I: t=0\}.
$$
Obviously, the set $\mathcal{X}_I$ is contained in the hyperplane $H$ defined by $t=0$, and by Fact \ref{Fact4} we have $\mathbf{\mathbb{F}}F\setminus H\ne \emptyset$, so $\overline{A_I}$
has an irreducible component containing $\mathbf{\mathbb{F}}F$ which is an irreducible component of $\mathbf{\mathbb{X}}_I$. This gives assertion (a).
Analogously, from Fact \ref{fact1IandIIl}(b) we obtain $\mathbf{\mathbb{F}}F\subset A_{II}$.
Moreover, the vectors $x,t,z$ are linearly dependent provided $(x,y,u,t,z)\in \mathbf{\mathbb{X}}_{II}$ for some $y,u$, so $\mathbf{\mathbb{X}}_{II}=\mathcal{X}_{II}\cup A_{II}$,
where
$$
\mathcal{X}_{II}=\{w=(x,y,u,t,z)\in \mathbf{\mathbb{X}}_I: G_0(x)=0,\; G_{4,i,j}(x,t)=0,\;1\le i<j\le n\}.
$$
Obviously, $\mathcal{X}_{II}$ is contained in the set $W$ defined by $G_{4,i,j}(x,t)=0$, $1\le i<j\le n$. By Lemma \ref{ggg} we have $\mathbf{\mathbb{F}}F\setminus W\ne \emptyset$, so as above, the set $A_{II}$
has an irreducible component satisfying (b).
\end{proof}
From Fact \ref{factZaropensubs} and Lemmas \ref{Fact3} and \ref{factY0smooth} and the definition of $\mathbf{\mathbb{Y}}$ we have
\betagin{fact}\lambdabel{factdegrees}
$\delta(\mathbf{\mathbb{X}}_{I,*})\le \delta(\mathbf{\mathbb{Y}}_I)\le 2(2d-1)^{3n+1}$ and
$\delta(\mathbf{\mathbb{X}}_{II,*})\le \delta(\mathbf{\mathbb{Y}}_{II})\le 2(2d-1)^{3n+1}$.
\end{fact}
The proofs of Theorems \ref{maintwr} and \ref{maintwrIII} consist in showing that the projections of the sets $\mathbf{\mathbb{X}}_{I,*}$ and $\mathbf{\mathbb{X}}_{II,*}$ onto the space of $(y,u)\in \mathbf{\mathbb{C}}^2$ are proper algebraic subsets of $\mathbf{\mathbb{C}}^2$, since we have
\betagin{lem}\lambdabel{lemfinishing}
If $Q\in\mathbf{\mathbb{C}}[y,u]$ is a nonzero polynomial of degree $D$ such that
$$
Q(f(\varphi(t)),|\nabla f(\varphi(t))|^2)=0\quad\hbox{for }t\in[0,1),
$$
where $\varphi$ is the curve fulfilling \eqref{Lojgradinequality2}, then
{\rm (a)} $\varrho_0(f)\leq 1-\frac{1}{D}$ if $D$ is even,
{\rm (b)} $\varrho_0(f)\leq 1-\frac{1}{D+1}$ if $D$ is odd.
\end{lem}
\betagin{proof}
Let $\operatorname{ord}_0 (f\circ \varphi)=M$ and $\operatorname{ord}_0 |\nabla f\circ\varphi|^2=K$. Then $M,K>0$ and
\betagin{equation*}\lambdabel{eqQfnabla1}
\operatorname{ord}_0 (f\circ \varphi)^K=\operatorname{ord}_0|\nabla f\circ \varphi|^{2M},
\end{equation*}
i.e., $|f\circ \varphi|^{\frac{K}{2M}}\sim |\nabla f\circ\varphi|$ near zero\footnote{That is, there are $C_1,C_2>0$ such that $C_1|f\circ \varphi|^{\frac{K}{2M}}\le |\nabla f\circ\varphi|\le C_2|f\circ \varphi|^{\frac{K}{2M}}$ near zero.}, so by \eqref{Lojgradinequality2} we have
\betagin{equation}\lambdabel{eqro0}
\varrho_0(f)=\frac{K}{2M}.
\end{equation}
Then, by definitions of $M$ and $K$
there exists a pair of different monomials $\alpha u^N y^S$ and $\betata u^{N_1}y^{S_1}$ of the polynomial $Q$ such that
$$
N+S\le D \quad\hbox{and}\quad N_1+S_1\le D,
$$
and
$$
NK+SM=N_1K+S_1 M.
$$
Hence $N-N_1\ne 0$, $S_1-S\ne 0$, and
$$
\frac{K}{2M}=\frac{S_1-S}{2(N-N_1)}.
$$
Since $M>0$, we have $\operatorname{ord}_0\nabla f\circ \varphi\le M-1$, and so $K\le 2M-2$, and $\frac{K}{2M}<1$.
On the other hand, $|S_1-S|, |N-N_1|\in \{1,\ldots, D\}$, so by \eqref{eqro0}, $\varrho_0(f)$ is estimated from above by the maximal possible rational number less than $1$ with numerator from the set $ \{1,\ldots, D\}$ and denominator from $ \{2,4,\ldots, 2D\}$. Consequently, we obtain the assertion.
\end{proof}
\subsection{Proof of Theorem \ref{maintwr} in case I when $\varphi((0,1))\subset \operatorname{Int} \Omega$}
By the assumption \eqref{geberalassumption},
in the definition of $\mathbf{\mathbb{Y}}$ one can take the polynomials
\betagin{multline}\lambdabel{eqformG3i}
K_{3,i}(x,y,u,z)
=\frac{\partial G}{\partial x_i}\left(x,y,u\right)\frac{\partial P}{\partial y}(x,y)-\frac{\partial G}{\partial y}\left(x,y,u\right) \frac{\partial P}{\partial x_i}(x,y) \\
\mathbf{\mathbb{Q}}uad-\left(\frac{\partial P}{\partial y}\left(x,y\right)\right)^3\cdot z_i
\end{multline}
instead of $G_{3,i}$, $1\le i\le n$; also
in the definitions of $\mathbf{\mathbb{X}}_I$ and $\mathbf{\mathbb{Y}}_I$ one can take
\betagin{equation*}\lambdabel{eqnewG4ij}
K_{4,i,j}(x,y,u)=\frac{\partial P}{\partial x_i}(x,y)\frac{\partial G}{\partial x_j}(x,y,u)\\-\frac{\partial P}{\partial x_j}(x,y)\frac{\partial G}{\partial x_i}(x,y,u)
\end{equation*}
instead of $G_{4,i,j}$, $1\le i<j\le n$.
From the above and Fact \ref{factZaropensubs}
we obtain the following fact.
\betagin{fact}\lambdabel{factKonv} For $x\in \Gamma_I$ and $v=(x,y,u)=(x,f(x),|\nabla f(x)|^2)$ we have
\betagin{align}\lambdabel{eqK1}
P(v)&=0,&\\
\lambdabel{eqK2}
G(v)&=0,&\\
K_{4,i,j}(v)&=0,&\quad 1\le i<j\le n.\lambdabel{eqK4ij}
\end{align}
\end{fact}
Let $\mathbf{Y}_{I,0}\subset \bf{M}$, where ${\bf M}= \mathbf{\mathbb{C}}^n\times\mathbf{\mathbb{C}}\times\mathbf{\mathbb{C}}$, be an algebraic set defined by
the system of equations \eqref{eqK1}--\eqref{eqK4ij}, and let
\betagin{align}\nonumber
\mathbf{\mathbb{Y}}^0_{I}&=\left\{(x,y,u,t,z)\in \mathbf{\mathbb{Y}}_{I}:\frac{\partial P}{\partial y}(x,y)\ne 0\right\},\\
\nonumber
\mathbf{Y}^0_{I}&=\left\{(x,y,u)\in \mathbf{Y}_{I,0}:\frac{\partial P}{\partial y}(x,y)\ne 0\right\},\\
\nonumber
\mathbf{Y}_I&=\overline{\mathbf{Y}_{I}^0}.
\end{align}
We have the following fact (cf. \cite[Fact 2.11]{KOSS}).
\betagin{fact}\lambdabel{factWprojV}
The mapping
\betagin{equation*}\lambdabel{eqWprojV1}
\mathbf{\mathbb{Y}}_I^0 \ni (x,y,u,t,z)\mapsto (x,y,u)\in\mathbf{Y}^0_{I}
\end{equation*}
is a bijection.
\end{fact}
\betagin{proof}
Taking any $(x,y,u,t,z)\in \mathbf{\mathbb{Y}}^0_{I}$ (respectively $(x,y,u)\in\mathbf{Y}^0_{I}$), by the Implicit Function Theorem there are a neighbourhood $\Delta\subset \mathbf{\mathbb{C}}^n$ of $x$, a holomorphic function $g:\Delta\to\mathbf{\mathbb{C}}$
and neighbourhoods $U_1\subset \mathbf{\mathbb{C}}\times\mathbf{\mathbb{C}}\times\mathbf{\mathbb{C}}^n\times\mathbf{\mathbb{C}}^n$ and $U_2\subset \mathbf{\mathbb{C}}\times\mathbf{\mathbb{C}}$ of $(y,u,t,z)$ and $(y,u)$ respectively such that
\betagin{align}
\nonumber
\mathbf{\mathbb{Y}}_I^0\cap(\Delta\times U_1)&=\{(\taueta,g(\taueta),h(\taueta),\nabla g(\taueta),\nabla h(\taueta))\in\mathbf{\mathbb{M}}:\taueta\in\Delta\cap V\},\\
\nonumber
\mathbf{Y}_I^0\cap(\Delta\times U_2)&=\{(\taueta,g(\taueta),h(\taueta))\in {\bf M}:
\taueta\in\Delta\cap V\},
\end{align}
where $h(\taueta)=\left(\frac{\partial g}{\partial x_1}(\taueta)\right)^2+\cdots+\left(\frac{\partial g}{\partial x_n}(\taueta)\right)^2$, and
$$
V=\{\taueta\in\Delta:K_{4,i,j}(\taueta,g(\taueta),h(\taueta))=0,\;1\le i<j\le n\}.
$$
In particular, $g(x)=y$, $u=h(x)$,
$t=\nabla g(x)$ and $z=\nabla h(x)$. Thus, we obtain the assertion.
\end{proof}
Let $\mathbf{L}_I\subset {\bf M}\times \mathbf{\mathbb{C}}$
be the Zariski closure of the set
\betagin{equation*}
\mathbf{L}_{I,0}=\{(x,y,u,\lambda)\in \Omega\times \mathbf{\mathbb{R}}\times\mathbf{\mathbb{R}}\times \mathbf{\mathbb{R}}:y=f(x),\; u=|\nabla f(x)|^2,\\%\exists_{\lambda\in\mathbf{\mathbb{R}}}\,
\nabla|\nabla f(x)|^2=\lambda \nabla f(x)\}.
\end{equation*}
From Fact \ref{factZaropensubs}(a) we obtain
\betagin{fact}\lambdabel{factEILagrmult}
There exists an irreducible component $\mathbf{L}_{I,*}$ of $\mathbf{L}_I$ which contains a Zariski open and dense subset $\mathcal{U}$
such that for any $(x,y,u,\lambda)\in \mathcal{U}$ there exist $t,z\in\mathbf{\mathbb{C}}^n$ such that $(x,y,u,t,z)\in \mathbf{\mathbb{X}}_{I,*}$ and in particular $z=\lambda t$.
\end{fact}
\betagin{proof}
The set $\mathbf{L}_I$ is the projection of the union of some irreducible components of $\mathbf{\mathbb{L}}_I$ onto $(x,y,u,\lambda)\in{\bf M}\times\mathbf{\mathbb{C}}$.
So by Fact \ref{factZaropensubs}(a) we obtain the assertion.
\end{proof}
Let
$$
\pi:{\bf M}\times \mathbf{\mathbb{C}}
\ni (x,y,u,\lambda)\mapsto (x,y,u)\in {\bf M},
$$
let $\mathbf{L}_{I,*}$ be an irreducible component of $\mathbf{L}_I$ as in Fact \ref{factEILagrmult} and let
$$
\mathbf{L}E_I:=\overline{\pi(\mathbf{L}_{I,*})}.
$$
\betagin{lem}\lambdabel{factcalV1proj}
The set $\mathbf{L}E_I$ is an irreducible component of the algebraic set $\mathbf{Y}_{I}$. Moreover, $\mathbf{L}E_I$ contains a Zariski open and dense subset $\mathcal{U}_I$ such that $\mathcal{U}_I\subset\mathbf{Y}_{I}^0\cap \pi(\mathbf{L}_{I,*})$, and any point $(x_0,y_0,u_0)\in\mathcal{U}_I$ has a neighbourhood $B\subset {\bf M}$
such that $\mathbf{Y}_I\cap B=\mathcal{U}_I\cap B$ and
\betagin{equation}\lambdabel{eqformVgraph}
\mathcal{U}_I\cap B=\left\{\left(x,g(x),\left(\frac{\partial g}{\partial x_1}(x)\right)^2+\cdots+\left(\frac{\partial g}{\partial x_n}(x)\right)^2\right):x\in \Delta\cap V\right\}
\end{equation}
for some analytic set $V\subset \Delta$ with $x_0\in V$ and a holomorphic function $g:\Delta\to \mathbf{\mathbb{C}}$, where $\Delta\subset \mathbf{\mathbb{C}}^n$ is a neighbourhood of $x_0$.
\end{lem}
\betagin{proof}
By Facts \ref{factZaropensubs}, \ref{factWprojV} and \ref{factEILagrmult} we have $\pi(\mathbf{L}_{I,0})\subset \mathbf{Y}_{I}$, so $\mathbf{L}E_I\subset \mathbf{Y}_{I}$
and $\mathbf{L}E_I$ is an algebraic subset of $\mathbf{Y}_{I}$. Since any irreducible component of $\mathbf{\mathbb{X}}_I$ is an irreducible component of $\mathbf{\mathbb{Y}}_I$, the same holds for $\pi(\mathbf{L}_I)$ and $\mathbf{Y}_I$, because these sets are projections onto the space ${\bf M}$
of some collections of irreducible components of $\mathbf{\mathbb{X}}_I$ and $\mathbf{\mathbb{Y}}_I$, respectively. In particular, this holds for $\mathbf{L}E_I$ and $\mathbf{Y}_I$. This gives the first part of the assertion. We prove the ``moreover'' part analogously to Fact \ref{factWprojV}.
\end{proof}
Let
$$
\pi_y:\mathbf{L}E_I\ni v=(x,y,u)\mapsto y\in\mathbf{\mathbb{C}},
$$
$$
\pi_u:\mathbf{L}E_I\ni v= (x,y,u)\mapsto u\in \mathbf{\mathbb{C}}.
$$
We have the following lemma (cf. \cite[Lemmas 2.12, 2.14]{KOSS}).
\betagin{lem}\lambdabel{lemfinitenesscriticalvalues}
For generic $y_0\in \mathbf{\mathbb{C}}$, i.e., for any $y_0\in\mathbf{\mathbb{C}}$ off a finite set, the function $\pi_u$
is constant on each connected component of $(\pi_y)^{-1}(y_0)$.
\end{lem}
\betagin{proof}
If $\dim \mathbf{L}E_I=0$ or $\dim (\pi_y)^{-1}(y)\leq 0$ for generic $y\in\mathbf{\mathbb{C}}$, then the assertion holds. Assume that $\dim \mathbf{L}E_I>0$ and $\dim(\pi_y)^{-1}(y)>0$ for generic $y\in\mathbf{\mathbb{C}}$. Then by Lemma \ref{factcalV1proj}, and under the notations of this lemma, we have
$\overline{\pi_y(\mathcal{U}_I)}=\overline{\pi_y(\mathbf{L}E_I)}=\mathbf{\mathbb{C}}$ and $(\pi_y)^{-1}(y)\cap \mathcal{U}_I\ne \emptyset$ for generic $y\in \mathbf{\mathbb{C}}$.
Take any $y_0\in \mathbf{\mathbb{C}}$ such that $(\pi_y)^{-1}(y_0)\cap \mathcal{U}_I\ne \emptyset$. Take any $x_0\in\mathbf{\mathbb{C}}^n$ and $u_0\in\mathbf{\mathbb{C}}$ such that $(x_0,y_0,u_0)\in \mathcal{U}_I$. By Lemma \ref{factcalV1proj} there exist a neighbourhood $B\subset {\bf M}$
of $(x_0,y_0,u_0)$ and a holomorphic function $g:\Delta\to\mathbf{\mathbb{C}}$, where $\Delta\subset \mathbf{\mathbb{C}}^n$ is a neighbourhood of $x_0$, such that \eqref{eqformVgraph} holds for some analytic set $V\subset \Delta$.
Take any smooth curve $\gammaamma:[0,1]\to \Delta\cap V$ such that $g(\gammaamma(t))=y_0$ for $t\in [0,1]$. Let $h(x)=\left(\frac{\partial g}{\partial x_1}(x)\right)^2+\cdots+\left(\frac{\partial g}{\partial x_n}(x)\right)^2 $ for $x\in \Delta$ and take a function $u:[0,1]\to\mathbf{\mathbb{C}}$ defined by
$$
u(t)=h\circ \gammaamma(t).
$$
Observe that the function $u$ is constant. Indeed, by definition of $\mathcal{U}_I$ we see that for any $x\in \Delta\cap V$ there exists $\lambda_x\in\mathbf{\mathbb{C}}$ such that
$$
\nabla h(x)=\lambda_x \nabla g(x).
$$
So,
\betagin{equation*}\lambdabel{equprime}
u'(t)=\lambda_{\gammaamma(t)}\lambdangle \nabla g(\gammaamma(t)),\overline{\gammaamma'(t)}\rangle\quad\hbox{for }t\in[0,1],
\end{equation*}
where $\lambdangle \cdot,\cdot\rangle$ denotes the standard scalar product in $\mathbf{\mathbb{C}}^n$.
Since $g(\gammaamma(t))=y_0$ for $t\in[0,1]$, we have $\lambdangle \nabla g(\gammaamma(t)),\overline{\gammaamma'(t)}\rangle=0$, and consequently $u'(t)=0$ for $t\in[0,1]$ and $u$ is constant. Summing up, the function $\pi_u$ is constant on each connected component of $(\pi_y)^{-1}(y_0)\cap \mathcal{U}_I$.
Since $\mathcal{U}_I$ is a Zariski open and dense subset of $\mathbf{L}E_I$, any irreducible component of $\mathbf{L}E_I\setminus \mathcal{U}_I$ has dimension smaller than the dimension of
$\mathbf{L}E_I$, and for generic $y\in\mathbf{\mathbb{C}}$ any irreducible component $A$ of the fibre $\pi_y^{-1}(y)$ has a dense subset of the form $A\cap \mathcal{U}_I$ (see \cite[Chapter 3]{Mumford}). Then by the above we obtain the assertion.
\end{proof}
Since $\Gamma$ is an infinite set, it follows that $\dim\mathbf{L}_{I,0}\gammae 1$, so by Fact \ref{factWprojV}, $\dim {\mathbf{L}_I}\gammae 1$, and since $d=\deg P\gammae 2$, Lemma \ref{Fact3} and the definition of $\mathbf{Y}_I$ yield
$\delta (\mathbf{L}E_I)\le d(3d-2)^n$, where $\delta(\mathbf{L}E_I)$ is the total degree of $\mathbf{L}E_I$.
So, from Lemma \ref{lemfinitenesscriticalvalues}, the closure of the projection of $\mathbf{L}E_I$, $W=\overline{\{(y,u)\in\mathbf{\mathbb{C}}^2:\exists_{x\in\mathbf{\mathbb{C}}^n}\;(x,y,u)\in \mathbf{L}E_{I}\}}$, is a proper algebraic subset of $\mathbf{\mathbb{C}}^2$ and by Fact \ref{Fact2}, $\delta(W)\leq \delta(\mathbf{L}E_I)$. Then
there exists a nonzero polynomial $Q\in\mathbf{\mathbb{C}}[y,u]$ such that
\betagin{equation*}\lambdabel{eqestdegQ}
\deg Q\le d(3d-2)^n\leq \mathcal{R}(n,d)-1
\end{equation*}
and $Q(y,u)=0$ for $(x,y,u)\in \mathbf{L}E_I$. In particular, $Q(f(\varphi(t)),|\nabla f(\varphi(t))|^2)=0$ for $t\in [0,1)$.
Since $D=d(3d-2)^n$ may be odd, by Lemma \ref{lemfinishing}(b) we obtain the assertion of Theorem \ref{maintwr} in case I.
\subsection{Proof of Theorem \ref{maintwr} in case II when $\varphi\big([ 0,1)\big)\subset\partial\Omega$}
For any $x\in\partial \Omega\setminus f^{-1}(0)$ sufficiently close to $f^{-1}(0)$ the tangent spaces to $\partial \Omega$ and $f^{-1}(f(x))$ are transversal, as shown in Lemma \ref{ggg}.
We will
prove Theorem \ref{maintwr} in two dimensions and in the multidimensional case separately.
\noindent{\it Proof of Theorem \ref{maintwr} in case II for $n=2$.} Take a polynomial $G\in \mathbf{\mathbb{C}}[x,y,u]$, where $x=(x_1,x_2)$ and $y$, $u$ are single variables, defined by \eqref{eqdefG}, i.e., $
G(x,y,u)=\sum_{i=1}^2\left(\frac{\partial P}{\partial x_i}(x,y)\right)^2-\left(\frac{\partial P}{\partial y}(x,y)\right)^2 \cdot u$.
Let
\betagin{align}\nonumber
\mathbf{Y}_{II,0}&=\{(x,y,u)\in \mathbf{\mathbb{C}}^2\times \mathbf{\mathbb{C}}\times \mathbf{\mathbb{C}}:P(x,y)=0,\;G_0(x)=0,\;G(x,y,u)=0\},\\
\nonumber
\mathbf{Y}_{II}^0&=\left\{(x,y,u)\in \mathbf{Y}_{II,0}:\frac{\partial P}{\partial y}(x,y)\ne 0\right\},\\
\nonumber
\mathbf{Y}_{II}&=\overline{\mathbf{Y}^0_{II}}.
\end{align}
Then for any $x\in \Gamma\cap\partial \Omega$ we have $(x,f(x),|\nabla f(x)|^2)\in \mathbf{Y}_{II}$. Consequently,
$$
(\varphi(t),f(\varphi(t)),|\nabla f(\varphi(t))|^2)\in \mathbf{Y}_{II}\quad\hbox{for }t\in [0,1).
$$
In particular, $\dim \mathbf{Y}_{II}\gammae 1$ and by Fact \ref{Fact1} we have $\delta (\mathbf{Y}_{II})\le 2d(2d-1)$.
Since $P$ is an irreducible polynomial of positive degree with respect to $y$, for any $y\in\mathbf{\mathbb{C}}\setminus\{0\}$ sufficiently close to $0$ the set $\{x\in\mathbf{\mathbb{C}}^2:P(x,y)=0,\,G_0(x)=0\}$ is finite, so the set
$
\{(x,u)\in \mathbf{\mathbb{C}}^2\times \mathbf{\mathbb{C}}:(x,y,u)\in \mathbf{Y}_{II}\}
$
is also finite. Then the projection
$$
W=\{(y,u)\in\mathbf{\mathbb{C}}^2:\exists_{x\in\mathbf{\mathbb{C}}^2}(x,y,u)\in \mathbf{Y}_{II}\}
$$
is contained in a proper algebraic subset of $\mathbf{\mathbb{C}}^2$. By Fact \ref{Fact2},
$$
\delta(\overline{W})\le 2d(2d-1)\le \mathcal{R}(n,d).
$$
Then there exists a nonzero polynomial $Q\in\mathbf{\mathbb{C}}[y,u]$ of degree $\deg Q\le \delta(\overline{W})\le \mathcal{R}(n,d)$ which vanishes on $W$. Since $2d(2d-1)$ is even, by Lemma \ref{lemfinishing}(a) we obtain the assertion of Theorem \ref{maintwr} in case II for $n=2$.
$\square$
Let us consider the case $n\gammae 3$.
Let $\varepsilon>0$ be as in Lemma \ref{ggg}.
By the assumption \eqref{geberalassumption},
in the definition of the set $\mathbf{\mathbb{Y}}$ one can take the polynomials $K_{3,i}$ of the form \eqref{eqformG3i} instead of $G_{3,i}$; also, in the definitions of $\mathbf{\mathbb{X}}_{II}$ and $\mathbf{\mathbb{Y}}_{II}$,
one can take the polynomials
\betagin{equation*}
K_{4,i,j,k}(x,y,u)=\left|\betagin{matrix}
\frac{\partial P}{\partial x_i}(x,y)&\frac{\partial G}{\partial x_i}\left(x,y,u\right)& x_i\\
\frac{\partial P}{\partial x_j}(x,y)&\frac{\partial G}{\partial x_j}\left(x,y,u\right)& x_j\\
\frac{\partial P}{\partial x_k}(x,y)&\frac{\partial G}{\partial x_k}\left(x,y,u\right) &x_k
\end{matrix}\right|,
\end{equation*}
instead of $G_{4,i,j,k}$ for $1\le i<j<k\le n $, where $G$ is defined in \eqref{eqdefG}.
Then
\betagin{align}
\mathbf{\mathbb{X}}_{II}&=\{w=(x,y,u,t,z)\in \mathbf{\mathbb{X}}:G_0(x)=0,\;K_{4,i,j,k}(x,y,u)=0,\; 1\le i<j<k\le n\},\nonumber\\
\mathbf{\mathbb{Y}}_{II}&=\{w=(x,y,u,t,z)\in \mathbf{\mathbb{Y}}:G_0(x)=0,\,K_{4,i,j,k}(x,y,u)=0,\; 1\le i<j<k\le n\}.\nonumber
\end{align}
Let $\mathbf{Y}_{II,0}\subset {\bf M}$, where ${\bf M}=\mathbf{\mathbb{C}}^n\times\mathbf{\mathbb{C}}\times\mathbf{\mathbb{C}}$,
be the algebraic set defined by
\betagin{multline*}
\mathbf{Y}_{II,0}=\{(x,y,u)\in {\bf M} :
P(x,y)=0,\;G_0(x)=0,\;G(x,y,u)=0,\\
K_{4,i,j,k}(x,y,u)=0,\;1\le i<j<k\le n \}
\end{multline*}
and let
\betagin{align}
\nonumber
\mathbf{\mathbb{Y}}^0_{II}&=\left\{(x,y,u,t,z)\in \mathbf{\mathbb{Y}}_{II}:\frac{\partial P}{\partial y}(x,y)\ne 0\right\},\\
\nonumber
\mathbf{Y}^0_{II}&=\left\{(x,y,u)\in \mathbb{V}_{II,0}:\frac{\partial P}{\partial y}(x,y)\ne 0\right\},\\
\nonumber
\mathbf{Y}_{II}&=\overline{\mathbf{Y}_{II}^0}.
\end{align}
By an analogous argument to the proof of Fact \ref{factWprojV} we obtain
\betagin{fact}\lambdabel{factWprojVII}
The mapping
\betagin{equation*}\lambdabel{eqWprojV1}
\mathbf{\mathbb{Y}}_{II}^0 \ni (x,y,u,t,z)\mapsto (x,y,u)\in\mathbf{Y}^0_{II}
\end{equation*}
is a bijection.
\end{fact}
Let $\mathbf{L}_{II}\subset {\bf M}\times\mathbf{\mathbb{C}}^2$
be the Zariski closure of the set
\betagin{multline*}
\mathbf{L}_{II,0}=\{(x,y,u,(\lambda_1,\lambda_2))\in \partial \Omega\times \mathbf{\mathbb{R}}\times\mathbf{\mathbb{R}}\times \mathbf{\mathbb{R}}^2:y=f(x),\; u=|\nabla f(x)|^2,\\%\exists_{\lambda\in\mathbf{\mathbb{R}}}\,
\nabla|\nabla f(x)|^2=\lambda_1 \nabla f(x)+\lambda_2 x\}.
\end{multline*}
By a similar argument to the proof of Fact \ref{factEILagrmult}, from Fact \ref{factZaropensubs}(b) we obtain
\betagin{fact}\lambdabel{factEILagrmultII}
There exists an irreducible component $\mathbf{L}_{II,*}$ of $\mathbf{L}_{II}$ which contains a Zariski open, dense subset $\mathcal{U}$
such that for any $(x,y,u,\lambda_1,\lambda_2)\in \mathcal{U}$ there exist $t,z\in\mathbf{\mathbb{C}}^n$ such that $(x,y,u,t,z)\in \mathbf{\mathbb{X}}_{II,*}$ and in particular $z=\lambda_1 t+\lambda_2 x$.
\end{fact}
Let
$$
\pi':{\bf M}
\times \mathbf{\mathbb{C}}^2\ni (x,y,u,(\lambda_1,\lambda_2))\mapsto (x,y,u)\in {\bf M},
$$
and let
$$
\mathbf{L}E_{II}=\overline{\pi'(\mathbf{L}_{II,*})}.
$$
By an analogous argument to the proof of Lemma \ref{factcalV1proj} we obtain
\betagin{lem}\lambdabel{factcalV1projII}
The set $\mathbf{L}E_{II}$
is an
irreducible component of the algebraic set $\mathbf{Y}_{II}$. Moreover, $\mathbf{L}E_{II}$
contains a Zariski open and dense subset $\mathcal{U}_{II}$
such that $\mathcal{U}_{II}\subset\mathbf{Y}^0_{II}\cap \pi'(\mathbf{L}_{II,*})$ and any point $(x_0,y_0,u_0)\in\mathcal{U}_{II}$ has a neighbourhood $B\subset {\bf M}$
such that $\mathbf{Y}_{II}\cap B=\mathcal{U}_{II}\cap B$ and
\betagin{equation}\lambdabel{eqformVgraphII}
\mathcal{U}_{II}\cap B=\left\{\left(x,g(x),\left(\frac{\partial g}{\partial x_1}(x)\right)^2+\cdots+\left(\frac{\partial g}{\partial x_n}(x)\right)^2\right):x\in \Delta\cap V\right\}
\end{equation}
for some analytic set $V\subset \Delta$, where $x_0\in V$ and $G_0$ vanishes on $V$, and a holomorphic function $g:\Delta\to \mathbf{\mathbb{C}}$, where $\Delta\subset \mathbf{\mathbb{C}}^n$ is a neighbourhood of $x_0$.
\end{lem}
Let
$$
\pi_y:\mathbf{L}E_{II}\ni v=(x,y,u)\mapsto y\in\mathbf{\mathbb{C}},
\quad\
\pi_u:\mathbf{L}E_{II}\ni v= (x,y,u)\mapsto u\in \mathbf{\mathbb{C}}.
$$
We have the following lemma (cf. Lemma \ref{lemfinitenesscriticalvalues} and \cite[Lemmas 2.12, 2.14]{KOSS}).
\betagin{lem}\lambdabel{lemfinitenesscriticalvaluesII}
For generic $y_0\in \mathbf{\mathbb{C}}$ the function $\pi_u$
is constant on each connected component of $(\pi_y)^{-1}(y_0)$.
\end{lem}
\betagin{proof} As in the proof of Lemma \ref{lemfinitenesscriticalvalues}, we may assume that $\dim \mathbf{L}E_{II}>0$ and $\dim\,(\pi_y)^{-1}(y)>0$ for generic $y\in\mathbf{\mathbb{C}}$. Then by Lemma \ref{factcalV1projII}, and under the notations of that lemma,
$\overline{\pi_y(\mathcal{U}_{II})}=\overline{\pi_y(\mathbf{L}E_{II})}=\mathbf{\mathbb{C}}$ and $(\pi_y)^{-1}(y)\cap \mathcal{U}_{II}\ne \emptyset$ for generic $y\in \mathbf{\mathbb{C}}$.
Take any $y_0\in \mathbf{\mathbb{C}}$ such that $(\pi_y)^{-1}(y_0)\cap \mathcal{U}_{II}\ne \emptyset$. Take any $x_0\in\mathbf{\mathbb{C}}^n$ and $u_0\in\mathbf{\mathbb{C}}$ such that $(x_0,y_0,u_0)\in \mathcal{U}_{II}$. By Lemma \ref{factcalV1projII} there exist a neighbourhood $B\subset \mathbf{\mathbb{C}}^n\times\mathbf{\mathbb{C}}\times\mathbf{\mathbb{C}}$ of $(x_0,y_0,u_0)$ and a holomorphic function $g:\Delta\to\mathbf{\mathbb{C}}$, where $\Delta\subset \mathbf{\mathbb{C}}^n$ is a neighbourhood of $x_0$, such that \eqref{eqformVgraphII} holds for some analytic set $V\subset \Delta$ such that $G_0$ vanishes on $V$.
Take a smooth curve $\gammaamma=(\gammaamma_1,\ldots,\gammaamma_n):[0,1]\to \Delta\cap V$ such that $g(\gammaamma(t))=y_0$. Then
\betagin{equation}\lambdabel{eqgammagammaprime}
G_0(\gammaamma(t))=0 \quad\hbox{for }t\in [0,1] .
\end{equation}
Let $h(x)=\left(\frac{\partial g}{\partial x_1}(x)\right)^2+\cdots+\left(\frac{\partial g}{\partial x_n}(x)\right)^2$, $x\in \Delta$. Take a function $u:[0,1]\to\mathbf{\mathbb{C}}$ defined by
$$
u(t)=h\circ \gammaamma(t),\quad t\in [0,1].
$$
Observe that the function $u$ is constant. Indeed, by definition of $\mathcal{U}_{II}$, for any $x\in \Delta\cap V$ there exist $\lambda_{1,x},\lambda_{2,x}\in\mathbf{\mathbb{C}}$ such that
$$
\nabla h(x)=\lambda_{1,x} \nabla g(x)+\lambda_{2,x}x.
$$
So
\betagin{equation*}\lambdabel{equprime}
u'(t)=\lambda_{1,\gammaamma(t)}\lambdangle \nabla g(\gammaamma(t)),\overline{\gammaamma'(t)}\rangle+\lambda_{2,\gammaamma(t)}\lambdangle \gammaamma(t),\overline{\gammaamma'(t)}\rangle \quad\hbox{for }t\in[0,1].
\end{equation*}
Since $g(\gammaamma(t))=y_0$, we have $ \lambdangle \nabla g(\gammaamma(t)),\overline{\gammaamma'(t)}\rangle=0$ for $t\in[0,1]$. Moreover, by \eqref{eqgammagammaprime} we have $\lambdangle \gammaamma(t),\overline{\gammaamma'(t)}\rangle=0$ for $t\in[0,1]$.
Consequently, $u'(t)=0$ for $t\in[0,1]$ and $u$ is constant. Summing up, the function $\pi_u$ is constant on each connected component of $(\pi_y)^{-1}(y_0)\cap \mathcal{U}_{II}$. Since $\mathcal{U}_{II}$ is a dense subset of $\mathbf{L}E_{II}$, we obtain the assertion.
\end{proof}
Since $\Gamma$ is an infinite set, we have $\dim\mathbf{L}_{II,0}\gammae 1$, so by Fact \ref{factWprojVII}, $\dim {\mathbf{L}_{II}}\gammae 1$, and since $d=\deg P\gammae 2$, Lemma \ref{Fact3} and the definition of $\mathbf{Y}_{II}$ yield $\delta (\mathbf{L}E_{II})\le d(3d-2)^n$. So, from Lemma \ref{lemfinitenesscriticalvaluesII}, the closure of the projection of $\mathbf{L}E_{II}$,
$$W=\overline{\{(y,u)\in\mathbf{\mathbb{C}}^2:\exists_{x\in\mathbf{\mathbb{C}}^n}\;(x,y,u)\in \mathbf{L}E_{II}\}},$$
is a proper algebraic subset of $\mathbf{\mathbb{C}}^2$ and $\delta({W})\leq \delta(\mathbf{L}E_{II})$. Then there exists a nonzero polynomial $Q\in\mathbf{\mathbb{C}}[y,u]$ such that
$\deg Q\le 2(3d-2)^n \leq\mathcal{R}(n,d)-1$
and $Q(y,u)=0$ for $(x,y,u)\in \mathbf{L}E_{II}$. Since $D=2(3d-2)^n$ is an even number, by Lemma \ref{lemfinishing}(a) we obtain the assertion of Theorem \ref{maintwr} in case II.
\subsection{Proof of Theorem \ref{maintwrIII}}\lambdabel{roz2III}
Analogously to the proof of Lemma \ref{lemfinitenesscriticalvalues},
we prove that the set
$$
W=\overline{\{(y,u)\in\mathbf{\mathbb{C}}^2:\exists_{x\in\mathbf{\mathbb{C}}^n}\;\exists_{t\in\mathbf{\mathbb{C}}^n}\;\exists_{z\in\mathbf{\mathbb{C}}^n}\;(x,y,u,t,z)\in \mathbf{\mathbb{Y}}_{I}\}}
$$
is a proper algebraic subset of $\mathbf{\mathbb{C}}^2$. Moreover, by Fact \ref{factdegrees}
we have $\delta(W)\le \delta(\mathbf{\mathbb{Y}}_{I})\le 2d(2d-1)$ if $n=1$ and $\delta(W)\le \delta(\mathbf{\mathbb{Y}}_I)\le 2(2d-1)^{3n+1}$ for $n\gammae 2$. Then by Lemma \ref{lemfinishing}(a) we obtain the assertion of Theorem \ref{maintwrIII} in case I. An analogous argument gives the assertion in case II.
\betagin{thebibliography}{99}
\bibitem{BK} J. Bochnak, W. Kucharz,
\emph{Sur les germes d'applications diff\'erentiables \`a singularit\'es isol\'ees}.
{Trans. Amer. Math. Soc.} 252 (1979), 115--131.
\bibitem{BL} J. Bochnak, S. {\L}ojasiewicz,
\emph{A converse of the
Kuiper-Kuo theorem}. In: Proc. of Liverpool Singularities-Symposium I
(1969\slash70), Lecture Notes in Math., vol. 192, Springer, Berlin, 1971, 254--261.
\bibitem{BR} J. Bochnak, J. J. Risler, \emph{Sur les exposants de {\L}ojasiewicz}, Comment. Math. Helv. 50 (1975), 493--507.
\bibitem{CH} P. Cassou-Nogu\`es, H. H. Vui, \emph{Th\'eor\`eme de Kuiper-Kuo-Bochnak-Lojasiewicz \`a l'infini}. {Ann. Fac. Sci. Toulouse Math}. (6) 5 (1996), 387--406.
\bibitem{CL}
S. S. Chang, Y. C. Lu, \emph{On $C^{0}$-sufficiency of complex jets}.
{Canad. J. Math}. 25 (1973), 874--880.
\bibitem{DK} D. D'Acunto, K. Kurdyka, \emph{Explicit bounds for the {\L}ojasiewicz exponent in the gradient inequality for polynomials}. Ann. Polon. Math. 87 (2005), 51--61.
\bibitem{DK1} D. D'Acunto, K. Kurdyka, \emph{Bounds for gradient trajectories and geodesic diameter of real algebraic sets}. Bull. London Math. Soc. 38 (2006), 951--965.
\bibitem{DG} J. Damon, T. Gaffney, \emph{Topological triviality of deformations of functions and Newton filtrations}.
Invent. Math. 72 (1983), 335--358.
\bibitem{FY} T. Fukui, E. Yoshinaga, \emph{The modified analytic trivialization of family of real analytic functions}.
Invent. Math. 82 (1985), 467--477.
\bibitem{Gabrielov} A. Gabrielov, \emph{Multiplicities of Pfaffian intersection, and the {\L}ojasiewicz inequality}. Selecta Math. (N.S.) 1 (1995), 113--127.
\bibitem{G} V. Grandjean, \emph{Finite determinacy relative to closed and finitely generated ideals}. {Manuscripta Math}. 103 (2000), 313--328.
\bibitem{Gw} J. Gwo\'zdziewicz, \emph{The {\L}ojasiewicz exponent of an analytic function at an isolated zero}. Comment. Math. Helv. 74 (1999), 364--375.
\bibitem{K1} J. Koll\'ar, \emph{An effective {\L}ojasiewicz inequality for real polynomials}. Period. Math. Hungar. 38 (1999), 213--221.
\bibitem{Kui}N. H. Kuiper, \emph{$C^1$-equivalence of functions near
isolated critical points}. In: Proc. Sympos. in Infinite Dimensional
Topology (Baton Rouge, 1967), Ann. of Math. Stud. 69,
Princeton Univ. Press, Princeton, NJ, 1972, 199--218.
\bibitem{Kuo}T. C. Kuo, \emph{On $C^0$-sufficiency of jets of potential
functions}. Topology 8 (1969), 167--171.
\bibitem{KMP} K. Kurdyka, T. Mostowski, A. Parusi\'nski, \emph{Proof of the gradient conjecture of R. Thom}.
Ann. of Math. (2) 152 (2000), 763--792.
\bibitem{KOSS}
K. Kurdyka, B. Osi\'nska-Ulrych, G. Skalski, S. Spodzieja, \emph{Effective {\L}ojasiewicz gradient inequality for generic Nash functions with isolated singularity}. Bull. Soc. Sci. Lett. {\L}\'od\'z S\'er. Rech. D\'eform. 66 (2016),47--64.
\bibitem{KS1} K. Kurdyka, S. Spodzieja, \emph{Separation of real algebraic sets and the {\L}ojasiewicz exponent}. Proc. Amer. Math. Soc. 142 (2014), 3089--3102.
\bibitem{KS2} K. Kurdyka, S. Spodzieja, \emph{Convexifying positive polynomials and sums of squares approximation}. SIAM J. Optim. 25 (2015), 2512–-2536.
\bibitem{KSS} K. Kurdyka, S. Spodzieja, A. Szlachci\'nska, \emph{Metric properties of semialgebraic sets}. Discrete Comput. Geom. 55 (2016), 786--800.
\bibitem{Kne} L. Kushner, \emph{Finite determination on algebraic sets}.
Trans. Amer. Math. Soc. 331 (1992), 553--561.
\bibitem{LejeuneTeissier}
M. Lejeune-Jalabert, B. Teissier, \emph{Cl\^{o}ture int\'egrale des id\'eaux et \'equisingularit\'e}. Centre de Math\'ematiques Ecole Polytechnique Palaiseau, 1974.
\bibitem{Lo1} S. {\L}ojasiewicz, \emph{Ensembles semi-analytiques}. Preprint IHES, 1965.
\bibitem{Lo2} S. {\L}ojasiewicz, \emph{Sur les trajectoires du gradient d'une fonction analytique}. In: Geometry Seminars, 1982-1983, Univ. Stud. Bologna, Bologna, 1984, 115--117.
\bibitem{Lo3} S. {\L}ojasiewicz, \emph{Introduction to complex analytic geometry}. Birkh\"auser Verlag, Basel, 1991.
\bibitem{Merle}
M. Merle, \emph{Invariants polaires des courbes planes}. Invent. Math. 41 (1977), 103--111.
\bibitem{MigusRS}
P. Migus, T. Rodak, S. Spodzieja, \emph{Finite determinacy of non-isolated singularities}. Ann. Polon. Math. 117 (2016), 197--206.
\bibitem{Mumford}
D. Mumford, \emph{Algebraic Geometry I. Complex Projective Varieties}. Springer, Berlin, 1976.
\bibitem{OSS} B. Osi\'nska-Ulrych, G. Skalski, S. Spodzieja, \emph{On $C^0$-sufficiency of jets}. Analytic and algebraic geometry, {Faculty of Mathematics and Computer Science, University of {\L}\'od\'z, {\L}\'od\'z}, 2013, 95--113.
\bibitem{P} R. Pellikaan, \emph{Finite determinacy of functions with non-isolated singularities}. {Proc. London Math. Soc}. (3) 57 (1988), 357--382.
\bibitem{TienSon} T. S. Pham, \emph{An explicit bound for the {\L}ojasiewicz exponent of real polynomials}. Kodai Math. J. 35 (2012), 311--319.
\bibitem{Ra} P. J. Rabier, \emph{Ehresmann fibrations and Palais-Smale conditions for morphisms of Finsler manifolds}. Ann. of Math. (2) 146 (1997), 647--691.
\bibitem{RS2}
T. Rodak, S. Spodzieja, \emph{Effective formulas for the local {\L}ojasiewicz exponent}. Math. Z. 268 (2011), 37--44.
\bibitem{RS3}
T. Rodak, S. Spodzieja, \emph{{\L}ojasiewicz exponent near the fibre of a mapping}. Proc. Amer. Math. Soc. 139 (2011), 1201--1213.
\bibitem{RS4} T. Rodak, S. Spodzieja, \emph{Equivalence of mappings at infinity}. Bull. Sci. Math. 136 (2012), 679-–686.
\bibitem{Si2} D. Siersma, \emph{Singularities with critical locus a 1-dimensional complete intersection and transversal type $A_1$}. {Topology Appl}. 27 (1987), 51–73.
\bibitem{Sk} G. Skalski, \emph{On analytic equivalence of functions at infinity}. {Bull. Sci. Math}. 135 (2011), 517--530.
\bibitem{Solerno}
P. Solern\'o, \emph{Effective {\L}ojasiewicz inequalities in semialgebraic geometry}. Appl. Algebra Engrg. Comm. Comput. 2 (1991), 2-–14.
\bibitem{Sp1} S. Spodzieja, \emph{The {\L}ojasiewicz exponent of subanalytic sets}. Ann. Polon. Math. 87 (2005), 247--263.
\bibitem{T} B. Teissier, \emph{Vari\'et\'es polaires. I. Invariants polaires des singularit\'es d'hypersurfaces}. Invent. Math. 40 (1977), 267--292.
\bibitem{Xu} X. Xu, \emph{$C^0$-sufficiency, Kuiper-Kuo and Thom conditions for non-isolated singularity}. {Acta Math. Sin. (Engl. Ser.)} 23 (2007), 1251–1256.
\end{thebibliography}
\end{document}
\pagebreak
\betagin{center}Efektywna nier\'owno\'s\'c {\L}ojasiewicza z gradientem\\ dla generycznej funkcji Nasha z izolowan\k{a} osobliwo\'sci\k{a}
\end{center}
\betagin{abstract} Niech $\Omega$ bêdzie kul¹ domkniêt¹ w $\mathbf{\mathbb{R}}^n$ o œrodku w pocz¹tku uk³adu wspó³rzêdnych i niech $f:\Omega \to \mathbb{R}$ bêdzie funkcj¹ Nasha. Wówczas istnieje nierozk³adalny wielomian $P\in \mathbb{R}[x,y]$ zmiennych $x=(x_1,\ldots,x_n)$ i $y$ taki, ¿e $P(x,f(x))=0$. Podajemy oszacowanie typu D'Acunto-Kurdyki wyk³adnika $\varrho\in[0,1)$ w nierównoœci £ojasiewicza z gradientem: $|\gammarad f(x)|\gammae C|f(x)|^\varrho$ dla $x\in \Omega$, $|f(x)|<\varepsilon$, gdzie $C,\varepsilon>0$ s¹ pewnymi sta³ymi, w terminach stopnia $P$, przy za³o¿eniu, ¿e $f(0)=0$, $f$ ma osobliwoœæ izolowan¹ w zerze i $\nabla P (x,f(x))\ne 0$ dla $x\in\Omega$.
\end{abstract}
\noindent{\bf S³owa kluczowe:} {Funkcja semialgebraiczna, Funkcja Nasha, Nierównoœæ {\L}ojasiewicza z gradientem, Wyk³adnik {\L}ojasiewicza.}
\end{document}
\title[Efektywna nier\'owno\'s\'c {\L}ojasiewicza z gradientem]{Efektywna nier\'owno\'s\'c {\L}ojasiewicza z gradientem\\ dla generycznej funkcji Nasha z izolowan\k{a} osobliwo\'scio\k{a}}
\subjclass[2000]{14R99, 11E25, 14P05, 32S70.}
\keywords{Funkcja semialgebraiczna, Funkcja Nasha,Nierównoœæ {\L}ojasiewicza z gradientem, Wyk³adnik {\L}ojasiewicza.}
\betagin{abstract} Niech $\Omega$ bêdzie kul¹ domkniêt¹ w $\mathbf{\mathbb{R}}^n$ o œrodku w pocz¹tku uk³adu wspó³rzêdnych i niech $f:\Omega \to \mathbb{R}$ bêdzie funkcj¹ Nasha. Wówczas istnieje nierozk³adalny wielomian $P\in \mathbb{R}[x,y]$ zmiennych $x=(x_1,\ldots,x_n)$ i $y$ taki, ¿e $P(x,f(x))=0$. Podajemy oszacowanie w rodzaju D'Acunta-Kurdyki wyk³adnika $\varrho\in[0,1)$ w nierównoœci £ojasiewicza z gradientem: $|\gammarad f(x)|\gammae C|f(x)|^\varrho$ dla $x\in \Omega$, $|f(x)|<\varepsilon$, gdzie $C,\varepsilon>0$ s¹ pewnymi sta³ymi, w terminach stopnia $P$, przy za³o¿eniu, ¿e $f(0)=0$, $f$ ma osobliwoœæ izolowan¹ w zerze i $\nabla P (x,f(x))\ne 0$ dla $x\in\Omega$.
\end{abstract}
{\small
\addcontentsline{toc}{section}{References}
\betagin{thebibliography}{CK}
\bibitem[1]{A}
E. Artin, \emph{\it \"{U}ber die Zerlegung definiter Funktionen in Quadrate}, Abh. Math. Sem. Univ. Hamburg 5, (1927), 100--115.
\bibitem[2]{BCR} J. Bochnak, M. Coste, M.-F. Roy, {\it Real Algebraic Geometry}, Springer-Verlag, Berlin, 1998.
\bibitem[3]{Brownawell}
W. D. Brownawell, \emph{\it Bounds for the degrees in the Nullstellensatz}, Ann. of Math. (2) 126 (1987), no. 3, 577--591.
\bibitem[4]{BSS} S. Burgdorf, C. Scheiderer, M. Schweighofer, \emph{Pure states, nonnegative polynomials and sums of squares}. Comment. Math. Helv. 87 (2012), no. 1, 113�140.
\bibitem{CH} P. Cassou-Nogues, H. H. Vui, \emph{Theoreme de
Kuiper-Kuo-Bochnak-Lojasiewicz a l'infini}.,
{Ann. Fac. Sci. Toulouse Math}. (6) 5, no. 3, 387--406, (1996)
\bibitem[5]{C}
E. Cygan, \emph{\it A note on separation of algebraic sets and the \L ojasiewicz exponent for polynomial mappings}, Bull. Sci. Math. 129 (2005), no. 2, 139--147.
\bibitem[6]{Dubois} D. W. Dubois, \emph{Anullstellensatz for ordered fields.}
Ark.Mat. 8 1969 111�114 (1969).
\bibitem[7]{ZJ2}
Z. Jelonek, \emph{\it On the \L ojasiewicz exponent}, Hokkaido Math. J. 35 (2006), no. 2, 471--485.
\bibitem[8]{JKS}
S. Ji, J. Koll�r, B. Shiffman, \emph{\it A global \L ojasiewicz inequality for algebraic varieties}, Trans. Amer. Math. Soc. 329 (1992), no. 2, 813--818.
\bibitem[9]{JK}
J. Koll\'ar, \emph{\it Sharp effective Nullstellensatz}, J. Amer. Math. Soc. 1 (1988), no. 4, 963--975.
\bibitem[10]{JK2}
J. Koll�r, \emph{\it Effective Nullstellensatz for arbitrary ideals}, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 3, 313--337.
\bibitem{11]{KOSS}
K. Kurdyka, B. Osiñska-Ulrych, G. Skalski, S. Spodzieja, \emph{Effective £ojasiewicz gradient inequality for generic Nash functions with isolated singularity}. Bull. Soc. Sci. Lett. £ódŸ Sér. Rech. Déform. 66 (2016), no. 3, 47–64.
\bibitem[11]{KS}
K. Kurdyka, S. Spodzieja, \emph{\it Separation of real algebraic sets and �ojasiewicz exponent}, Faculty of Math., Univ. of ��d�, preprint (http://www.math.uni.lodz.pl/preprints,all.html) (2011).
\bibitem[12]{Krivine} J.-L. Krivine, \emph{Anneaux pr�ordonn�s}. J. Analyse Math. 12 1964 307�326.
\bibitem[13]{L}
S. �ojasiewicz, \emph{\it Introduction to complex analytic geometry}, Birkh\"auser Verlag, Basel Boston Berlin, 1991.
\bibitem[14]{Marshall}
M. Marshall, \emph{Positive polynomials and sums of squares}. Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008.
\bibitem[15]{Motzkin} T. S. Motzkin, \emph{The arithmetic-geometric inequality}, In: Inequalities (Ed. O. Shisha) Academic Press (1967), 205-224
\bibitem[16]{O}
B. Osi\'nska, \emph{\it Extensions of regular mappings and the �ojasiewicz exponent at infinity}, Bull. Sci. Math. 135, (2011), 215-229.
\bibitem[17]{OSS}
B. Osi\'nska, G. Skalski, S. Spodzieja, \emph{Extensions of regular mappings and the {\L}ojasiewicz exponent at infinity}, Faculty of Math., Univ. of ��d�, preprint (http://www.math.uni.lodz.pl/preprints,all) (2012).
\bibitem[18]{PD}
A. Prestel, Ch. N. Delzell, \emph{Positive polynomials. From Hilbert's 17th problem to real algebra}. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2001.
\bibitem[19]{Risler} J. J. Risler, \emph{Une caract\'erisation des id\'eaux des vari\'et\'es alg\'ebriques r\'eelles}. C.R. Acad. Sci. Paris 27/, 1 171- 1173(1970).
\bibitem[20]{RS2}
T. Rodak, S. Spodzieja, \emph{ \it {\L}ojasiewicz exponent near the fibre of a mapping}, Proc. Amer. Math. Soc. 139 (2011), 1201-1213.
\bibitem[21]{SaSc} Y. Savchuk, K. Schm�dgen, \emph{Positivstellens�tze for algebras of matrices}. Linear Algebra Appl. 436 (2012), no. 3, 758�788.
\bibitem[22]{Sch1}
C. Scheiderer, \emph{Sums of squares on real algebraic surfaces}. Manuscripta Math. 119 (2006), no. 4, 395�410.
\bibitem[23]{Sch}
C. Scheiderer, \emph{\it Positivity and sums of squares: a guide to recent results}, Emerging applications of algebraic geometry, 271�324, IMA Vol. Math. Appl., 149, Springer, New York, 2009.
\bibitem[24]{Schmudgen}
K. Schm\"udgen, \emph{The $K$-moment problem for compact semi-algebraic sets}. Math. Ann. 289 (1991), no. 2, 203�206.
\bibitem[25]{Schweigh} M. Schweighofer, \emph{Global optimization of polynomials using gradient tentacles and sums of squares}. SIAM J. Optim. 17 (2006), no. 3, 920�942
\bibitem[26]{Spodzieja1}
S. Spodzieja, \emph{\it The \L ojasiewicz exponent at infinity for overdetermined polynomial mappings}, Ann. Polon. Math. 78 (2002), no. 1, 1--10.
\bibitem[27]{SS}
S. Spodzieja, A. Szlachci�ska \emph{{\L}ojasiewicz exponent of overdetermined mappings}, Faculty of Math., Univ. of ��d�, preprint (http://www.math.uni.lodz.pl/preprints,all) (2012).
\bibitem[28]{Netzer1} T. Netzer, \emph{An elementary proof of Schm�dgen's theorem on the moment problem of closed semi-algebraic sets}. Proc. Amer. Math. Soc. 136 (2008), no. 2, 529�537
\end{thebibliography}
}
\end{document}
\noindent Krzysztof Kurdyka \newline
Laboratoire de Mathematiques (LAMA) Universi\'e de Savoie, UMR-5127 de CNRS\\%\linebreak
73-376 Le Bourget-du-Lac cedex FRANCE\newline
\indent E-mail: [email protected]
\noindent
Beata Osi�ska-Ulrych\\
Faculty of Mathematics and Computer Science, University of ��d�\\
S. Banacha 22, 90-238 ��d�, Poland\\
\indent E-mail: [email protected]
\noindent
Grzegorz Skalski\\
Faculty of Mathematics and Computer Science,
University of ��d�\\
S. Banacha 22,
90-238 ��d�, Poland\\
\indent E-mail: [email protected]
\noindent
Stanis�aw Spodzieja\\
Faculty of Mathematics and Computer Science,
University of ��d�\\
S. Banacha 22,
90-238 ��d�, Poland\\
\indent E-mail: [email protected]
\end{document} | math | 84,313 |
\begin{document}
\twocolumn[
\icmltitle{SPDY: Accurate Pruning with Speedup Guarantees}
\begin{icmlauthorlist}
\icmlauthor{Elias Frantar}{ist}
\icmlauthor{Dan Alistarh}{ist,neuralmagic}
\end{icmlauthorlist}
\icmlaffiliation{ist}{IST Austria}
\icmlaffiliation{neuralmagic}{Neural Magic}
\icmlcorrespondingauthor{Elias Frantar}{[email protected]}
\icmlcorrespondingauthor{Dan Alistarh}{[email protected]}
\icmlkeywords{Machine Learning, ICML}
\vskip 0.2in
]
\printAffiliationsAndNotice{}
\begin{abstract}
The recent focus on the efficiency of deep neural networks (DNNs) has led to significant work on model compression approaches, of which \emph{weight pruning} is one of the most popular.
At the same time, there is rapidly-growing computational support for efficiently executing the unstructured-sparse models obtained via pruning.
Yet, most existing pruning methods minimize just the number of remaining weights, i.e. the size of the model, rather than optimizing for inference time.
We address this gap by introducing SPDY, a new compression method which automatically determines layer-wise sparsity targets achieving a desired inference speedup on a given system, while minimizing accuracy loss. SPDY is the composition of two new techniques.
The first is an efficient and general dynamic programming algorithm for solving constrained layer-wise compression problems, given a set of layer-wise error scores.
The second technique is a local search procedure for automatically determining such scores in an accurate and robust manner.
Experiments across popular vision and language models show that SPDY guarantees speedups while recovering higher accuracy relative to existing strategies, both for one-shot and gradual pruning scenarios, and is compatible with most existing pruning approaches.
We also extend our approach to the recently-proposed task of pruning with very little data, where we achieve the best known accuracy recovery when pruning to the GPU-supported 2:4 sparsity pattern.
\end{abstract}
\section{Introduction}
Increasing the efficiency of deep neural networks (DNNs) has the potential to not just reduce the cost of compute- and energy-hungry models, but also to make them more readily available and privacy-conscious, by allowing model execution on end-devices.
There are many approaches for compressing DNNs for increased efficiency~\cite{hoefler2021sparsity, gholami2021survey}. The one we focus on in this paper is \textit{sparsity}, i.e. setting to zero a large fraction of the values in the big parameter matrices of a neural network. Sparsification of neural networks has a long history~\cite{lecun1990optimal}, and current pruning techniques can shrink models by more than an order of magnitude, while largely preserving accuracy~\cite{hoefler2021sparsity}.
Pruning methods are usually categorized by the granularity at which they are applied: \emph{Structured pruning} aims to drop groups of related weights, e.g. filters in a convolutional layer, which directly leads to improved inference speeds, but is usually limited in the compression amount before significant accuracy loss occurs.
\emph{Unstructured pruning} methods have traditionally been focused on reducing model size, as they drop individual weights in often random patterns, which is harder to translate into faster execution. More recent \emph{semi-structured} methods~\cite{NVIDIASparse, zhou2021learning, lagunas21block} trade off additional structure in the zeroed weights, e.g. small rectangular blocks, with higher accuracy loss relative to unstructured pruning.
On the runtime side, increasingly advanced algorithms have been introduced to provide computational speedup also for \emph{unstructured} sparse models, whether executed on CPUs \cite{pmlr-v119-kurtz20a, deepsparse, elsen2020fast}, GPUs \cite{sgk_sc2020}, or specialized hardware~\cite{han2015learning, dave2021hardware}.
Currently, unstructured pruning provides some of the best compression-to-accuracy trade-offs among existing approaches~\cite{hoefler2021sparsity}, and efficient sparse inference on CPUs, which is our main focus, is particularly interesting in terms of accessibility and cost. Further, several commodity CPUs, e.g. current AMD models, do not support efficient quantized arithmetic, making sparsity their primary means of accelerating inference.
One key practical issue is that state-of-the-art \emph{unstructured} pruning methods, e.g.~\cite{evci2020rigging, singh2020woodfisher, schwarz2021powerpropagation, peste2021ac}, do not directly take the behavior of acceleration methods into account, while existing speedup-aware \emph{structured} pruning methods are not straightforward to adapt to the unstructured case. This means that sparsifying a model to reach a certain speed, rather than a certain size, with minimal accuracy loss, is still an extremely laborious process.
\textbf{Contribution.} We provide a general solution to this problem, called \emph{learned efficient Sparsity Profiles via DYnamic programming search (SPDY)}. SPDY automatically determines how much to prune each individual network layer to achieve a desired speedup on a given inference engine and hardware combination, while minimizing accuracy loss.
The underlying optimization problem solved by SPDY is very general, as it also occurs in the context of structured pruning~\cite{he2018amc}, non-uniform layer quantization \cite{hubara2021accurate, yao2021hawq}, low-rank decomposition \cite{liebenwein2021compressing}, or even gradient compression \cite{markov2021project}.
While we focus on unstructured pruning here, as it is an under-explored area, our approach should extend to these other settings as well.
First, unstructured pruning with a speedup target can be viewed as constrained optimization problem in terms of layer-wise execution timings, and so-called layer-wise ``error scores,'' and we propose an essentially exact dynamic-programming solver for this problem. This algorithm is extremely efficient: it has \emph{linear complexity} in the number of layers times the number of possible sparsity choices per layer, and can be easily scaled to very large models.
Second, we address how to reliably determine layer-wise error scores for unstructured sparsity. We first observe that known metrics, e.g. (normalized) weight magnitudes, do not correlate consistently with superior accuracy of the resulting unstructured sparse models.
We then introduce a new approach which \emph{learns} the layer-wise error-scores automatically, based on the network's global pruning behavior on a small set of input data. This relaxes the strictly layer-wise problem and thus makes it possible to account for global cross-layer effects, while still utilizing the advantages of the original formulation.
Specifically, SPDY determines good layer-wise error scores via local search,
which assesses the quality of profiles determined through our DP algorithm by how well a ``reconstructed'' version of the sparse model behaves on calibration data. For sparse model reconstruction, we leverage a new variant of the AdaPrune one-shot pruning technique~\cite{hubara2021accelerated}.
\begin{figure}
\caption{Speedup and relative performance measure drop trade-off after gradual pruning for SPDY and baselines on various models.}
\label{fig:rel-drops}
\end{figure}
We apply SPDY to determine
``optimal'' layer-wise compression levels for a sparsity-aware CPU inference engine~\cite{deepsparse}, both for \emph{one-shot} and \emph{gradual} pruning.
We first show that SPDY sparsity profiles can be used to compress a larger model, e.g. ResNet101, to match the inference speed of a smaller model, e.g. ResNet50, \emph{in a single compression step, without finetuning}, and still maintain an accuracy advantage.
We then show that the layer-wise sparsity targets found by SPDY result in models with significantly better accuracy-vs-speed trade-offs than existing baselines, also when applying state-of-the-art \emph{gradual pruning} methods, as shown in Figure \ref{fig:rel-drops}. Further, SPDY can be used in conjunction with quantization-aware training.
As a second application, we consider GPU acceleration, where we propose an enhancement of the AdaPrune method~\cite{hubara2021accelerated}, which we call \emph{global AdaPrune (gAP)}. We apply gAP to generate models with the GPU-supported 2:4 sparsity pattern in the ``post-training pruning'' setting of~\cite{hubara2021accelerated}, where only a small amount of data is available for re-calibrating the model.
Our extension significantly outperforms the original AdaPrune: the gAP models with 2:4 sparsity have higher accuracy than AdaPrune models imposing the much less stringent 4:8 sparsity pattern.
In sum, our work introduces two new techniques for accuracy-aware acceleration of deep learning models, applicable to both CPU and GPU-based inference, which can produce state-of-the-art results in both settings.
While we focused on unstructured and semi-structured pruning in our experiments, our approaches should be directly extensible to other settings.
We provide efficient implementations of our methods at \url{https://github.com/IST-DASLab/spdy}.
\section{Related Work}
Existing pruning techniques range from simple approaches like gradual magnitude pruning \cite{hagiwara1994,zhu2017prune}, which periodically drops the fraction of the weights with lowest magnitude, followed by model finetuning, to dynamic techniques like Soft Threshold Reparametrization \cite{kusupati2020soft}, Movement Pruning \cite{2020-sanh}, or Rigging the Lottery~\cite{evci2020rigging}, which adapt mask selection during training itself. Surprisingly, properly-tuned gradual magnitude pruning is often competitive with more complex methods~\cite{singh2020woodfisher, evci2020rigging, frantar2021m, peste2021ac}. None of the above pruning techniques specifically optimize for fast inference.
However, most can be easily modified to work with layer-wise sparsity targets, and are thus complementary to the methods we introduce; we illustrate this in our experiments by employing a range of different pruning techniques for accuracy recovery.
While structured pruning is primarily concerned with inference speed, many works focus instead on pruning FLOPs. Yet,~\citet{liu2021group} found that this is sometimes less correlated with real speedups than just reducing model size. This highlights the need to operate directly on \emph{timing data}. Related structured pruning works are Constraint-Aware Importance Estimation (CAIE) \cite{wu2020constraint} and Knapsack Pruning (KP) \cite{aflalo2020knapsack}. The former greedily ranks filters by considering their contributions towards multiple resource constraints, while the latter directly solves for the filters to select under a single constraint, a 0/1 knapsack problem, via dynamic programming. The optimal solution to this knapsack problem can be approximated with a greedy approach similar to single-constraint CAIE or \cite{yang2020automatic}. However, in our constrained optimization problem (see Section \ref{sec:optimization-problem}) the choices, i.e. the sparsity levels, are more than binary per ``item''/layer, and in practice layer speedups under sparsity are also non-linear, which makes an accurate approximation of the optimal solution more difficult. Furthermore, we show that the weight magnitudes and individual loss impact metrics, used by CAIE and KP, are not reliable for fine-grained unstructured pruning under constraints. Instead, our approach ``learns'' the error metrics automatically, allowing it to also inject global information into the layer-wise problem, which we find to be crucial for reliably finding good higher speedup solutions.
The idea of learning the layer sensitivities is related to Automated Model Compression (AMC) \cite{he2018amc} which uses one-shot pruning performance as a proxy to train a reinforcement learning agent that predicts layer-wise (structured) sparsity targets. The idea of performing layer reconstruction via linear regression, introduced by \cite{he2017channel} and further explored by~\cite{evci2018mean}, is a precursor to the AdaPrune method, upon which our accuracy evaluation is based.
However, we directly solve for the speedup constraint with an efficient algorithm, whereas AMC enforces the constraint only implicitly by restricting the action space of the agent.
Yet, this and similar reinforcement learning approaches~\cite{ashok2018n2n} suffer from high tuning complexity, making them difficult to apply to new models, and have not found widespread adoption. In contrast, we demonstrate how SPDY, with a fixed set of hyper-parameters, works reliably across tasks and models.
Another related research direction is \emph{speedup-aware model quantization}, in the form of recent approaches like AdaQuant \cite{hubara2021accurate}, HAWQv3 \cite{yao2021hawq} and BRECQ \cite{li2021brecq}. These approaches solve constrained optimization problems to identify optimal per-layer bit-widths. A key simplifying feature is that these problems only have a very small number of possible choices per layer, as there are few default precision levels. Thus, the resulting problem can be solved fast without custom algorithms. This is not the case for unstructured pruning, where there are significantly more pruning choices. In addition, methods for accurate ``post training quantization'', i.e. with little available training data, like AdaQuant and AdaRound \cite{nagel2020up}, inspired the pruning equivalent AdaPrune \cite{hubara2021accelerated}. These techniques all perform a layer-wise optimization of the compressed weights to produce outputs as close as possible to the original model.
We use AdaPrune as a basis for our one-shot pruning approach, but also extend it significantly.
In general, from the perspective of model compression under a target constraint, SPDY can be seen as a fusion of global search based approaches like AMC and layer-wise constraint-solver methods like AdaQuant; combining the respective advantages of both schemes.
\section{Methods}
\subsection{Pruning for Speed: The Abstract Problem}
\label{sec:optimization-problem}
There are two core problems when trying to search for a fast \textit{unstructured} sparse model: (a) each individual weight (of which there are typically many millions) can either be dropped or kept, resulting in an enormous search space, and (b) the change in execution time for deleting a single weight is practically too small to be measured. The former challenge makes directly-extending filter-level optimization approaches \cite{aflalo2020knapsack} for structured pruning difficult, while the latter requires ranking-based approaches \cite{wu2020constraint} to rely on usually-inaccurate FLOP proxies~\cite{liu2021group} instead of actual timings.
However, both of these problems can be circumvented by leveraging the fact that the unstructured sparsity masks produced by established pruning methods have close to random structure \cite{sgk_sc2020}. Thus, unstructured acceleration techniques cannot rely on specific patterns in the layer sparsity, and have similar performance for masks of the same layer with the same sparsity level. Thus, we can reduce the overall problem of ``pruning for speed'' to identifying target sparsity values $s_\ell$ for each layer $1 \leq \ell \leq L$. Further, we can accurately estimate the runtime of such a \textit{sparsity profile} by simply imposing the corresponding sparsities with random masks, without actually running any pruning algorithms. Yet, it is critical to explicitly time different layers and sparsity levels, since acceleration rates due to unstructured sparsity are typically \emph{non-linear}: at low sparsity, sparse execution may be slower than dense execution, while speedup curves tend to flatten at high sparsities.
We wish to solve the problem of finding the ``best'' sparsity profile, in the sense of yielding the smallest possible model accuracy drop, while matching a threshold execution time $T$. To make this problem tractable, we require additional approximations. First, we assume that the overall execution time is given by the sum of the individual layer runtimes $t_{\ell}^s$ for the chosen sparsities. This is not always exact, since inference runtimes may perform layer fusion, but it is generally a good estimate, as shown in e.g.~\cite{cai2018proxylessnas}. Next, we assume that pruning a layer $\ell$ to sparsity $s$ ultimately incurs some model error $e^s_{\ell}$, and that those errors are \emph{additive}. This is a strong assumption, but one which is common in literature \cite{yao2021hawq, hubara2021accurate, aflalo2020knapsack}. Finally, we assume that the set of sparsity choices $S$ is discrete, which is reasonable in practice, as very small sparsity increments usually only lead to negligible performance gains. Under these assumptions, we can state the following constrained optimization problem:
\begin{align}
\text{min}_{s_1, \dots, s_L \in S} \, \sum_{\ell = 1}^L e^{s_\ell}_{\ell} \quad \text{s.t.} \quad \sum_{\ell = 1}^L t_{\ell}^{s_{\ell}} \leq T.
\label{eq:opt-problem}
\end{align}
This problem can be formulated as an integer linear program (ILP) and solved with specialized software~\cite{yao2021hawq, hubara2021accurate}. However, since each additional option per layer adds $L$ variables to the ILP, whose solving time usually increases exponentially in the number of variables, running an off-the-shelf solver would be prohibitively slow, for any model of interest, as a relatively fine-grained $S$ is needed. We note that the \textit{execution time target} $T$ corresponding to a \textit{speedup target} $X$ can be calculated as $T_\text{dense} / X - T_\text{base}$ where $T_\text{dense}$ and $T_\text{base}$ are the original dense runtime and the total runtime of operations that are unaffected by pruning, respectively.
\subsection{Efficiently Solving the Optimization Problem}
\label{sec:dp-algorithm}
In its most general form, the constrained optimization problem stated in Equation (\ref{eq:opt-problem}) is NP-hard and thus (most probably) requires exponential time to solve. However, if time is integer-valued, then, as we will show, the problem is actually solvable efficiently in time $O(|S| \cdot LT)$ with $O(LT)$ memory. In our use-case, we can discretize time into $B$ buckets of width $T / B$, which means that $T = B$. Since we will be interested in large enough discretization $B$, e.g. $B = 10^4$, the inherent randomness in the individual timings will usually exceed the error incurred due to discretization, thus making this extra approximation negligible.
The key to efficiently solving the discrete version of problem~(\ref{eq:opt-problem}) is the following observation: the lowest possible error achievable in the first $\ell$ layers while taking exactly time $t$ to execute and choosing sparsity $s$ at layer $\ell$, denoted by $E_{\ell}^t(s)$, is the error caused by sparsity $s$, i.e. $e^s_{\ell}$, plus the minimum error achievable with all $\ell - 1$ previous layers while taking time exactly $t$ minus the time the choice of sparsity $s$ takes at layer $\ell$. Then, the dependence of $E_\ell^t(s)$ on $s$ can be eliminated by simply taking the minimum, leading to the following recursion:
\begin{align}
E_\ell^t &= \text{min}_{s \in S} \, E_{\ell - 1}^{t - t^s_\ell} + e_\ell^s \\
E_1^t &= \text{min}_{s \in S'} \, e^s_1 \,\, \text{if} \,\, S' = \{s \, | \, t^s_1 = t \} \neq \emptyset \,\, \text{else} \,\, \infty.
\end{align}
Using dynamic programming (DP), i.e. by caching intermediate values, the final solution $\text{min}_{t \leq T} \, E^t_L$ can be computed efficiently by taking the minimum over $|S|$ options at each of the $LT$ values $E^t_\ell$. This leads to the linear memory and compute costs claimed previously. Appendix \ref{app:dp-implementation} shows a complete bottom-up DP implementation including the book-keeping to reconstruct the optimal sparsity profile.
In practice, this approach is highly-efficient. For example, when $T = 10^4$, $|S| = 42$ and $L = 52$ (a ResNet50 configuration) the DP algorithm finds the optimal solution in less than 100 milliseconds on a CPU and scales linearly for each parameter individually. Finally, we emphasize that this method could be used to solve any kind of layer-wise optimization problem written as (\ref{eq:opt-problem}). This includes, for instance, non-uniform layer quantization \cite{hubara2021accurate, yao2021hawq}, where the choices are quantization levels, low-rank decomposition \cite{liebenwein2021compressing}, where choices are simply ranks, or gradient compression \cite{markov2021project}, where the choices are the gradient bit-width.
The challenge for adapting the procedure to a new setting is to find a robust and accurate layer scoring metric.
Next, we derive such a metric for unstructured pruning.
\subsection{Learning the Error Metric}
\label{sec:error-metric-learning}
For the optimal solution of optimization problem (\ref{eq:opt-problem}) to be useful, we need meaningful error metric values $e_\ell^s$. However, especially in the context of unstructured pruning, it is quite challenging to define a general such metric, which works well across different DNNs. This is illustrated by Table~\ref{tab:metrics-problem} comparing the one-shot accuracy of profiles generated via the DP algorithm in combination with common metrics such as squared weights \cite{yao2021hawq}, squared weights normalized \cite{liu2021group} or the layer-wise loss change \cite{hubara2021accurate}. More details and additional experiments can be found in Appendix \ref{app:ablation}. While designing a robust and precise pruning error metric from first principles is an interesting direction for future work, the alternative we propose is to circumvent this issue by ``learning'' an appropriate metric automatically, which we find to consistently outperform manual options, as shown in Table \ref{tab:metrics-problem}. This also has the additional advantage that global information, e.g. if two consecutive layers should not both be heavily pruned, is integrated as well. We emphasize that this approach is only enabled by the high efficiency of our DP algorithm, since in order to reliably learn a metric, we will have to solve the constrained optimization problem a large number of times.
\begin{table}[h!]
\centering
\scalebox{.75}{
\begin{tabular}{|l|c|c|c|c|c|}
\toprule
\multirow{2}{*}{Method} & \multicolumn{2}{c|}{ResNet34} & \multicolumn{2}{c|}{BERT} \\
& $2.50\times$ & $3.50\times$ & $2.50\times$ & $3.50\times$ \\
\midrule
Uniform & 64.65 & 42.38 & 57.42 & 09.00 \\
DP + squared & 54.18 & 31.59 & 71.20 & 07.88 \\
DP + squared norm. & 56.32 & 09.15 & 58.77 & 06.17 \\
DP + layer-wise loss & 65.92 & 46.11 & 30.59 & 06.13 \\
\midrule
\textbf{SPDY} & \textbf{68.23} & \textbf{51.68} & \textbf{74.59} & \textbf{18.83} \\
\bottomrule
\end{tabular}
}
\caption{One-shot accuracy comparison of SPDY search with DP using several common error metrics.}
\label{tab:metrics-problem}
\end{table}
Generally, the difficulty of pruning a layer depends on how sparse a layer already is and how much of the remaining weights are to be pruned: pruning 10\% of the remaining weights will be easier than pruning 20\%, and pruning a layer that is 50\% sparse will be a lot easier than if the layer is 90\% sparse. This suggests that we should view the error in log-sparsity space, i.e. in steps of pruning $\delta$ percent of the remaining weights and, further, that the error still increases considerably faster than linearly in this parametrization. For this reason, we suggest the simple quadratic error model for each layer shown in (\ref{eq:error-approx}) where the scalar coefficient $c_\ell$ controls how quickly the error increases, i.e. it represents the sensitivity of the corresponding layer, and $i \in \{0, \dots, |S| - 1\}$. We choose a quadratic approximation because this is the ``simplest'' function with the intuitive properties discussed above. In addition, the sensitivity coefficients can be interpreted as the curvature of the errors.
\begin{equation}
\label{eq:error-approx}
e_\ell^s = c_\ell \cdot \Big(\frac{i}{|S| - 1}\Big)^2, \quad s = 1 - (1 - \delta)^i.
\end{equation}
Given layer-wise timings $t^s_\ell$ for all sparsity levels, and using the error definition (\ref{eq:error-approx}), the DP algorithm will produce a valid profile with the desired speedup for any sensitivity coefficient vector $\mathbf{c} = (c_1, \dots, c_L) \in [0, 1]^L$. This means that we have reduced our original \emph{constrained} problem to the \emph{unconstrained} optimization problem of finding a $\mathbf{c}$ for which the DP algorithm will return a good sparsity profile with the target speedup.
We now need an efficient procedure to determine how ``good'' a given profile is, which we discuss in Section \ref{sec:assessing-profile-quality}.
Assuming such a procedure is available, we can then optimize $\textbf{c}$ with some heuristic optimization method, e.g. local search. It may seem that this reformulation has not lead to significant progress: the goal remains to find one choice per layer (now a sensitivity rather than a sparsity) which ultimately yields a good profile. However, we have actually eliminated the speedup constraint, which is automatically taken care of by the embedded DP algorithm. Further, various useful priors are directly enforced in a principled way, e.g. layers with poor acceleration are only pruned strongly for low sensitivity values, and high sparsities are chosen only if they are required to reach the target speedup. As shown by comparing with a direct search (without reparametrization) and with a genetic programming method \cite{guo2020single} (Figure~\ref{fig:search-comparison-new}), the SPDY reparametrization leads to better solutions with fewer model evaluations and reduced variance. Appendix \ref{app:search} provides more details and additional analysis.
\begin{figure}
\caption{Comparison of SPDY with a direct search and genetic programming.}
\label{fig:search-comparison-new}
\end{figure}
For the heuristic coefficient optimization, we have found the following \emph{randomized neighborhood-shrinking local search} to work well, and therefore use it in all our experiments.
\begin{enumerate}
\item Sample 100 random candidates for $\mathbf{c}$ and choose the one with the best resulting profile as initial $\mathbf{c}^*$.
\item Copy $\mathbf{c}^*$ while uniformly resampling $k = \lceil 0.1 \cdot L \rceil$ random elements and replace the original $\mathbf{c}^*$ with the copy if the resulting profile is better. Stop the process if there was no improvement in 100 trials.
\item Decrement $k$ by 1 and repeat step 2. If $k = 0$, then return the current $\mathbf{c}^*$ as the final solution.
\end{enumerate}
\subsection{Quickly Assessing the Quality of a Sparsity Profile}
\label{sec:assessing-profile-quality}
We are now left with the problem of efficiently evaluating the quality of a given sparsity profile, that is, predicting the accuracy of the resulting sparse model after fine-tuning.
Intuitively, the main issue here is that, while applying a single pruning step, e.g. removing a fraction of weights by magnitude, may be very fast, pruned model accuracy collapses dramatically even at moderate sparsities, if pruning is applied in one-shot, and may not correlate well with model accuracy after fine-tuning.
Recently, approaches such as second-order pruning~\cite{singh2020woodfisher,frantar2021m}, as well as AdaPrune (AP) \cite{hubara2021accelerated}, have dramatically improved one-shot pruning performance, mostly by ``adapting'' the remaining unpruned weights to reduce the loss increase due to weights being removed at a step.
We will leverage this idea to address our estimation problem, in particular via the AdaPrune approach.
Specifically, assuming that the layer $\ell$ output function is $f_\ell(X, W)$, where $X$ are the sample inputs of a small calibration data set and $W$ are weight values, AP optimizes the sparse weights remaining after pruning $W^s$ to best reconstruct the ``functionality'' of the original dense weights $W$, independently for each layer, by minimizing the squared error between the dense and the sparse layer output:
\begin{equation}
\label{eq:adaprune}
\text{argmin}_{W^s} \, ||f_\ell(X, W) - f_\ell(X, W^s)||^2_2, \textnormal{ for layer $\ell$.}
\end{equation}
We leverage AdaPrune to evaluate the quality of a profile by building a \emph{layer reconstruction database}. This database stores for each layer $\ell$ and each sparsity $s$ the ``reconstructon'' of the remaining weights after sparsity $s$ has been imposed via AdaPrune. Then, we can check the quality of an arbitrary \emph{non-uniform} sparsity profile by performing two steps.
First, we query the database for the corresponding reconstructed weights of each layer, each at its target sparsity.
Second, we ``stitch together'' the resulting model from the reconstructed weights, and evaluate it on a given small validation set. In practice, we use the same data for validation as for the AP; similar to~\cite{hubara2021accelerated}, we do not observe any overfitting.
The loss of the model on this calibration data is a proxy for the ``quality'' of the sparsity profile.
Our results show that this ``reconstruction database'' approach provides significant improvements when estimating the quality of a given profile relative to previous approaches, such as measuring the loss directly after one-shot magnitude pruning~\cite{he2018amc}.
Yet, we found that we can still enhance its accuracy (see Appendix \ref{fig:oneshot-comparison} for experiments), especially when targeting high sparsity values, by performing this estimation \emph{iteratively}.
Specifically, we start from the observation that solving the AP optimization problem in~(\ref{eq:adaprune}) can significantly alter the magnitudes of the unpruned weights. Practically, the weights chosen to be pruned if we applied a single high-sparsity step, e.g. 0\% to 60\%, could be quite different from those that would be selected if we pruned in several smaller increments, of e.g. 6 steps of 10\%, after each of which we update the remaining weights.
Following this intuition, our method performs iterative pruning via the AdaPrune approach, where the target sparsity is reached in several smaller pruning steps with AP optimization in between.
For the reconstruction database generation, this just means that we bootstrap the pruning process of the next sparsity level with the result of the previous one, which incurs no extra cost. Finally, we illustrate the key role of the reconstruction database for SPDY in Appendix \ref{app:ablation}.
\subsection{The SPDY Method: Overview}
\label{sec:spdy-overview}
We now summarize the full SPDY method which is the combination of all the techniques discussed so far. A visual summary is given by Figure \ref{fig:SPDY-summary} and corresponding pseudo code can be found in Appendix \ref{app:pseudocode}. Our system takes as input a target execution time $T$, which is easily calculated from a target speedup $X$ (see Section \ref{sec:optimization-problem}), and a set of possible sparsity choices $S$. We then generate timing data for each layer and each sparsity choice. Additionally, we precompute a reconstruction database for fast and accurate one-shot pruning (see Section \ref{sec:assessing-profile-quality}). The output sparsity profile is then determined by a cyclic search procedure which finds sensitivity coefficients that impose a layer-wise error metric (see Section \ref{sec:error-metric-learning}). Together with the previously computed timing data, these error values are passed to a dynamic programming solver (see Section \ref{sec:dp-algorithm}) which determines a sparsity profile with target execution time $T$ and minimal total error. The actual quality of this profile is then determined by stitching together layers from the reconstruction database (see Section \ref{sec:assessing-profile-quality}) and computing the loss of the composite model on a small calibration set. The search procedure (see Section \ref{sec:error-metric-learning}) attempts to minimize this loss by adapting the layer-wise sensitivity coefficients and ultimately outputs the sparsity profile determined by the DP algorithm when applied to the best found sensitivities.
\begin{figure*}
\caption{A visual overview of the full SPDY method.}
\label{fig:SPDY-summary}
\end{figure*}
\subsection{Post Training Pruning via global AdaPrune (gAP)}
An appealing but challenging setup is \textit{post training quantization}~\cite{nagel2020up, hubara2021accurate}, where acceleration should be done with a small amount of calibration data, without any finetuning.
The \emph{pruning} version of this problem is an interesting target for SPDY, since architectures such as AMD CPUs do not have quantization support, but have good speedups even at moderate sparsities.
\citet{hubara2021accelerated} showed the first \textit{post training pruning} results with acceptable accuracy drops when applied to the 4:8 semi-structured sparsity pattern using their AdaPrune (AP) approach. However, for higher sparsities, the accuracy drop of AP becomes too large to be directly useful. To address this, we introduce an extension of AP we call \textit{global AdaPrune}, or gAP for short, which boosts AP accuracy and thereby extends the practicality of post training pruning. We note that this technique is orthogonal to the SPDY method introduced in the previous sections, but can be very useful as an additional step for post-training applications.
The main motivation behind gAP is that standard AP optimizes each layer independently, thus not taking compounding errors into account, and also not considering that a slightly larger error on one layer might allow significantly reducing the error on another one. Thus, we complement AP with an optimization process over the small calibration set which (globally) optimizes the full model, in order to minimize the sum of relative layer-wise errors. We normalize the error of each layer by the squared magnitude of the output of the original dense model. This is important to ensure that all layers influence the objective equally. Specifically, let $f_\ell(X_\ell, W_\ell)$ be the output of layer $\ell$ in the original dense model and let $f_\ell(X_\ell^s, W_\ell^s)$ be the output of layer $\ell$ in the sparse model. Then, the gAP loss is written as (\ref{eq:gap-loss}) and can be minimized by gradient-based optimization:
\begin{equation}
\label{eq:gap-loss}
\mathcal{L}_{\text{gAP}}(W^s) = \sum_{\ell = 1}^L \frac{||f_\ell(X_\ell, W_\ell) - f_\ell(X^s_\ell, W^s_\ell)||^2_2}{||f_\ell(X_\ell, W_\ell)||^2_2}.
\end{equation}
\section{Experiments}
\label{sec:experiments}
\paragraph{Setup.} We now describe our experimental setup. In all our experiments, we use the same set of sparsity targets for each layer $S = \{0\} \cup \{ 1 - (1 - 0.4) \cdot \delta^i \, | \, i = 0, \dots, 40\}$ with $\delta = ((1 - 0.99) / (1 - 0.4))^{1 / 40}$. That is, we either set a layer to dense, or to one of 41 sparsity levels, each of which prunes an additional $\approx 10\%$ of the remaining weights.
The $0.4$ sparsity lower bound is the minimum sparsity at which the inference runtime provides some acceleration over dense execution, while $0.99$ is an upper limit to prevent model breakdown.
For time discretization, we always use $B = 10^4$ buckets as individual units of time.
For experiments on ImageNet~\cite{deng2009imagenet} we follow ~\cite{hubara2021accelerated}, by defining the calibration set for AP, gAP and the profile search to contain exactly one randomly-selected training image per class. For other tasks, we select 1000 training samples at random for the calibration set. The reconstruction database generation performs 10 epochs of optimization over this calibration set, using Adam \cite{kingma2014adam} with batchsize 32 and learning rate $10^{-3}$ per sparsity level while gAP runs for 100 epochs with learning rate $10^{-5}$ and frozen batch norms. The profile search follows Section \ref{sec:error-metric-learning}. With these settings, applying SPDY takes, for instance, 16min (13 database + 3 search) for ResNet18 or 51min (29 + 23) for ResNet50. This is executed on a single
NVIDIA 3090 GPU, and can be significantly optimized.
We measure speedups and execute inference on the publicly-available DeepSparse v0.9.1 CPU inference engine \cite{deepsparse, pmlr-v119-kurtz20a}, which is competitive when executing dense models with the standard ONNX and OpenVINO runtimes, but can additionally leverage unstructured sparsity for speedup.
DeepSparse is currently the most mature CPU engine with sparsity support; yet, we emphasize that alternatives exist~\cite{elsen2020fast, sgk_sc2020}, and that our techniques are independent of the runtime.
We compare against two baseline profiles: the \emph{uniform} profile of minimal sparsity which matches the target execution time, and the profile determined by pruning using \emph{global magnitude pruning (GMP)} until the target speedup is reached.
Both are direct speedup-aware adaptations of widely-used pruning strategies, which result in reasonable profiles without manual tuning \cite{he2019filter, liu2021group}.
Additionally, following other works \cite{elsen2020fast, jayakumar2021top}, we always keep the first and the last layer dense, since those typically have disproportionately large effects on the model accuracy while taking only a very small fraction of the total computation time. (The only exception is YOLO, where the 3 output layers are compute-heavy, so we only skip the input.) We emphasize that this helps the baseline profiles (uniform and GMP) significantly. (For instance, \citet{peste2021ac} report 73.14\% accuracy for a 95\% sparse ResNet50 versus 74.16\% for one with first and last layer skipped.) This choice has little effect on SPDY since our method generally detects the compute/sensitivity imbalance automatically, and skips those layers. We also experimented with a GMP version that normalizes magnitude scores by the corresponding FLOPs. However, this did not provide improvements without model-specific manual tweaking of layer-wise normalizers, e.g. on ResNet50 it would prune all the early layers to 99\% sparsity. To obtain a consistent AP reconstruction database and timing data, we round our baselines to the fixed sparsities in $S$.
There are some unstructured pruning methods which relate their results to speed/efficiency: STR \cite{kusupati2020soft}, WoodFisher FLOPs \cite{singh2020woodfisher} and AMC \cite{he2018amc}. The comparison in Figure \ref{fig:other-comp} shows that they all perform similar or worse than our simpler uniform and GMP baselines. Additionally, some of these methods do not provide mechanisms to control the target speedup. Thus, we focus on comparisons with uniform and GMP profiles.
Layer-wise timings for the AMD system are collected on an Amazon AWS c5a.8xlarge machine with 16 cores, while for Intel CPUs we use a c5.9xlarge server with 18 cores.
The listed speedups are for batchsize 64, except for BERT \cite{devlin2018bert}, which uses batchsize 16.
The speedups given in the tables below are calculated in terms of the layer-wise timing data, which usually slightly underestimates the real speedups when running the engine with all extra optimizations turned on. This allows comparing different profile generation methods in isolation of highly engine-specific timing inaccuracies and will also make it easy for future work to directly compare with our results using the published timing data. We provide real speedup information for our most interesting profiles in Appendix \ref{app:real-timings}, which demonstrates that the layer-wise approximation is indeed quite accurate in most cases.
\begin{figure}
\caption{Comparison with results of speedup-related unstructured pruning methods on ResNet50 and MobileNetV1 in terms of relative accuracy drop.}
\label{fig:other-comp}
\end{figure}
\textbf{Post Training Pruning.}
We begin in the post-training pruning setting, and consider the torchvision \cite{marcel2010torchvision} implementation of ResNets~\cite{he2016deep} running on an AMD CPU. These models are popular, and they can be well-accelerated by DeepSparse.
We report performance directly after stitching the model from the AP database (labelled AP) as well as after a subsequent run of global AP (labelled +gAP). We consider the use-case of pruning each network to approximately the same inference speed as the next smallest variant available in torchvision (e.g. ResNet101 to ResNet50, see Table \ref{tab:oneshot-next}).
\begin{figure*}\label{tab:oneshot-next}
\label{tab:rn-nm}
\end{figure*}
Table \ref{tab:oneshot-next} clearly shows that ResNets post training pruned with SPDY and global AdaPrune always exceed the accuracy of the smaller dense variant with approximately the same speed. In several cases, the difference is significant, e.g. $\approx 1.7\%$ Top-1 for RN34 or $\approx 1.5\%$ for RN18. Further, the SPDY profile almost always performs best. The only exception is the very large ResNet-152 with a low speedup target, where GMP performs $\approx 0.1\%$ better. Meanwhile, the SPDY profile outperforms the baselines by $\approx 1\%$ on RN50 and even by $\approx 2.5\%$ on RN50 with doubled width (RN50w2). We get similar results when pruning all models to the same $1.5\times$ speedup (see Appendix \ref{app:additional-experiments}) but with smaller relative accuracy drops. Note that uniform pruning outperforms GMP at smaller models and higher speedups, while the situation is reversed for larger models and lower speedups.
\textbf{Compounding Pruning and Quantization.}
Quantization is an alternative compression technique, which is popular with both GPUs and end devices~\cite{gholami2021survey}, as well as high-end Intel CPUs. The DeepSparse runtime
supports both techniques with speedup in complementary fashion. To leverage this, we set a pruning speedup target of 2x for ResNet50, and then run SPDY vs. uniform pruning in a post-training setting, followed by Quantization Aware Training (QAT) \cite{nagel2021white} for 6 epochs to quantize weights and recover additional accuracy.
Quantization provides an additional $2.6\times$ speedup, so the end model is $5.2\times$ faster than the dense full-precision version, on an AWS c5.12xlarge instance. SPDY's accuracy improvement after pruning is clear: we have 73.33\% vs. 71.94\% top-1 accuracy, for the semi-structured pruning pattern with blocks of 4 consecutive zeros required by DeepSparse. This directly leads to improved 75.00\% vs. 74.33\% accuracy after the quantization-aware finetuning stage, in favor of SPDY.
\textbf{Post Training N:M Sparsity.}
To gain additional insight into the accuracy difference between our global layer reconstruction approach relative to existing methods, we investigate the performance of global AdaPrune in the context of post training pruning models to full N:M sparsity, a type of semi-structured sparsity that can be well accelerated on the newer NVIDIA GPUs. \citet{hubara2021accelerated} were the first to demonstrate that achieving N:M sparsity in the post training setting with acceptable accuracy loss is possible using AdaPrune and batchnorm tuning (A + B). Unfortunately, their results are somewhat theoretical, as they only targeted 4:8 sparsity, which to our knowledge is not yet supported by hardware. We now show that by applying global AP (after standard AP), we can exceed their 4:8 accuracy results with the significantly more challenging 2:4 sparsity pattern that is already well supported by current NVIDIA GPUs. A comparison for the ResNet model family can be found in Table \ref{tab:rn-nm}. We note that further improvements might be possible through optimized channel permutations \cite{pool2021channel}. Additionally, in Appendix \ref{app:additional-experiments}, we present results for post training 2:4 pruning YOLOv5 models and BERT.
\textbf{Gradual Pruning Results.}
\label{sec:gradual-pruning}
We now present our main results, generating speedup profiles and then performing \emph{gradual pruning} using state-of-the-art pruning methods under their default parameters, with these layer-wise sparsity targets. We consider the standard ResNet50 model \cite{he2016deep}, as well as the hard-to-compress MobileNetV1 architecture~\cite{howard2017mobilenets, kusupati2020soft}, the Small and the Medium versions of the popular YOLOv5\footnote{We evaluate using the author's validation script with default parameters and report the pycocotools [email protected].} \cite{yolov5} object detector, and the widely used BERT-base for language modelling \cite{devlin2018bert} on the SQuAD dataset \cite{rajpurkar2016squad}. ResNet50 is gradual pruned with AC/DC \cite{peste2021ac} for 100 epochs, MobileNetV1 with M-FAC \cite{frantar2021m} for 100 epochs and YOLOv5 and BERT with gradual magnitude pruning (using the setup of \cite{kurtic2022optimal} but performing a pruning step every $0.01$ epochs) for 240 and 30 epochs, respectively. The results are summarized in Table \ref{tab:gradual-results}.
\begin{table}[h]
\begin{minipage}[c]{\linewidth}
\centering
\scalebox{.75}{
\begin{tabular}{|l|c|c|c|c|c|c|}
\toprule
Model & Dense & Speed. & CPU & \textbf{SPDY} & Uni. & GMP \\
\midrule
ResNet50 & 76.13 & $2.00\times$ & AMD & \textbf{76.39} & 76.01 & 75.85 \\
ResNet50 & 76.13 & $2.50\times$ & AMD & \textbf{75.56} & 75.12 & 74.76 \\
ResNet50 & 76.13 & $3.00\times$ & AMD & \textbf{74.75} & 74.02 & 73.44 \\
ResNet50 & 76.13 & $3.50\times$ & AMD & \textbf{73.06} & 71.62 & 70.22 \\
\midrule
MobileNetV1 & 71.95 & $1.50\times$ & Intel & \textbf{71.38} & 61.33 & 70.63 \\
\midrule
YOLOv5s & 56.40 & $1.50\times$ & Intel & \textbf{55.90} & 54.70 & 55.00 \\
YOLOv5s & 56.40 & $1.75\times$ & Intel & \textbf{53.10} & 50.90 & 47.20 \\
YOLOv5m & 64.20 & $1.75\times$ & Intel & \textbf{62.50} & 61.70 & 61.50 \\
YOLOv5m & 64.20 & $2.00\times$ & Intel & \textbf{60.70} & 58.30 & 57.20 \\
\midrule
BERT SQuAD & 88.54 & $3.00\times$ & Intel & \textbf{88.53} & 88.22 & 87.98 \\
BERT SQuAD & 88.54 & $3.50\times$ & Intel & \textbf{87.56} & 87.23 & 87.22 \\
BERT SQuAD & 88.54 & $4.00\times$ & Intel & \textbf{86.44} & 85.63 & 85.13 \\
BERT SQuAD$^*$ & 88.54 & $4.00\times$ & Intel & \textbf{87.14} & 86.37 & 86.39 \\
\bottomrule
\end{tabular}
}
\captionof{table}{Comparing accuracy metrics for sparsity profiles after gradual pruning models with respective state-of-the-art methods.}
\label{tab:gradual-results}
\end{minipage}
\end{table}
The SPDY profiles appear to yield more accurate final weights across all models and speedups, often with a large gap to the next best method. At higher speedups, the advantage of SPDY seems to increase further, with $1.4$ points better accuracy for ResNet50 and even $> 2$ points better [email protected] for YOLO models. In the case of YOLO, it should also be mentioned that the profile generation optimizes not the relatively complex training loss but simply the squared error of the final output to the one of the original dense model. This suggests that our method can be applied successfully even if the calibration set is \emph{unlabelled}.
In some experiments, the improvement via SPDY is relatively small, e.g. for the lower speedup BERT runs we get only $\approx 0.3$ higher F1. However, we note that a) the performance improvement of SPDY is larger than the difference between uniform and GMP in those cases and b) this occurs after extensive finetuning, which usually narrows accuracy gaps.
The fact that these differences are always consistent can be seen as an indication in favor of our method.
Finally, in these experiments all profiles were generated initially, based on the dense model. Potentially, results could be improved further by rerunning SPDY from a model that has already been pruned to an intermediate sparsity.
We test this by generating new $4\times$ speedup profiles based on the gradually-pruned $3\times$ BERT models which are also the starting points for subsequent gradual pruning (indicated by ``BERT SQuAD$^*$'' in the table). Although the absolute gap seems to stay the same, all models are noticeably more accurate, so the relative gap in terms of F1-drop is bigger, suggesting that iterating SPDY can bring additional gains.
We also highlight the ResNet50 results, achieved with AC/DC, a method that trains and prunes the model \textit{from scratch}. This implies that the sparsity profiles found by SPDY are not just fitted to a particular set of dense weights but also possess some architecture-specific generalization properties. In that context, the results of our experiments reinforce the idea that better one-shot performance is a good indicator of performance after finetuning~\cite{he2018amc}. They seem to suggest that good sparsity profiles can ``transfer'' also when running gradual pruning, which may be loosely connected to ``lottery ticket'' approaches~\cite{frankle2018lottery}. We discuss some of the profiles produced by SPDY in more detail in Appendix \ref{app:profiles}. In general, SPDY profiles achieve the same speedup with a slightly lower overall sparsity than uniform and GMP, due to prioritizing execution time (see also Appendix \ref{app:sparsity}), which likely contributes to the higher accuracies after extensive finetuning. However, our approach can be adapted to any additive constraint, such as parameters, FLOPs, or energy consumption.
\section{Conclusion \& Future Work}
We introduced SPDY, a method for automatically determining sparsity profiles optimized for a particular acceleration setup. The resulting profiles consistently perform better than ones determined by conventional methods, without any model-specific parameter tuning, both in one-shot and gradual-pruning scenarios. An extension of our layer reconstruction method can also provide improvements for post-training 2:4 pruning via global AdaPrune.
Our ideas can be extended to other settings: (1) a combination of a dynamic program constraint solver and automated learning of sensitivity scores could also be applied to determine compression profiles for low-rank approximation; (2) it could be interesting to apply our techniques to neural architecture search super-networks as a replacement for our one-shot pruning reconstruction database. In sum, our techniques provide notable improvements relative to current methods, and have significant potential for follow-up work.
\begin{thebibliography}{55}
\providecommand{\natexlab}[1]{#1}
\providecommand{\url}[1]{\texttt{#1}}
\expandafter\ifx\csname urlstyle\endcsname\relax
\providecommand{\doi}[1]{doi: #1}\else
\providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi
\bibitem[Aflalo et~al.(2020)Aflalo, Noy, Lin, Friedman, and
Zelnik]{aflalo2020knapsack}
Aflalo, Y., Noy, A., Lin, M., Friedman, I., and Zelnik, L.
\newblock Knapsack pruning with inner distillation.
\newblock \emph{arXiv preprint arXiv:2002.08258}, 2020.
\bibitem[Ashok et~al.(2018)Ashok, Rhinehart, Beainy, and Kitani]{ashok2018n2n}
Ashok, A., Rhinehart, N., Beainy, F., and Kitani, K.~M.
\newblock {N2N} learning: Network to network compression via policy gradient
reinforcement learning.
\newblock In \emph{International Conference on Learning Representations
(ICLR)}, 2018.
\bibitem[Cai et~al.(2018)Cai, Zhu, and Han]{cai2018proxylessnas}
Cai, H., Zhu, L., and Han, S.
\newblock {ProxylessNAS}: Direct neural architecture search on target task and
hardware.
\newblock In \emph{International Conference on Learning Representations
(ICLR)}, 2018.
\bibitem[Cai et~al.(2019)Cai, Gan, Wang, Zhang, and Han]{cai2019once}
Cai, H., Gan, C., Wang, T., Zhang, Z., and Han, S.
\newblock {Once-for-All}: Train one network and specialize it for efficient
deployment.
\newblock In \emph{International Conference on Learning Representations
(ICLR)}, 2019.
\bibitem[Dave et~al.(2021)Dave, Baghdadi, Nowatzki, Avancha, Shrivastava, and
Li]{dave2021hardware}
Dave, S., Baghdadi, R., Nowatzki, T., Avancha, S., Shrivastava, A., and Li, B.
\newblock Hardware acceleration of sparse and irregular tensor computations of
{ML} models: A survey and insights.
\newblock \emph{Proceedings of the IEEE}, 109\penalty0 (10):\penalty0
1706--1752, 2021.
\bibitem[Deng et~al.(2009)Deng, Dong, Socher, Li, Li, and
Fei-Fei]{deng2009imagenet}
Deng, J., Dong, W., Socher, R., Li, L.-J., Li, K., and Fei-Fei, L.
\newblock {ImageNet}: A large-scale hierarchical image database.
\newblock In \emph{Conference on Computer Vision and Pattern Recognition
(CVPR)}, 2009.
\bibitem[Devlin et~al.(2019)Devlin, Chang, Lee, and Toutanova]{devlin2018bert}
Devlin, J., Chang, M.-W., Lee, K., and Toutanova, K.
\newblock {BERT}: Pre-training of deep bidirectional transformers for language
understanding.
\newblock In \emph{North American Chapter of the Association for Computational
Linguistics (NAACL)}, 2019.
\bibitem[Elsen et~al.(2020)Elsen, Dukhan, Gale, and Simonyan]{elsen2020fast}
Elsen, E., Dukhan, M., Gale, T., and Simonyan, K.
\newblock Fast sparse convnets.
\newblock In \emph{Conference on Computer Vision and Pattern Recognition
(CVPR)}, 2020.
\bibitem[Evci et~al.(2018)Evci, Le~Roux, Castro, and Bottou]{evci2018mean}
Evci, U., Le~Roux, N., Castro, P., and Bottou, L.
\newblock Mean replacement pruning.
\newblock 2018.
\bibitem[Evci et~al.(2020)Evci, Gale, Menick, Castro, and
Elsen]{evci2020rigging}
Evci, U., Gale, T., Menick, J., Castro, P.~S., and Elsen, E.
\newblock Rigging the lottery: Making all tickets winners.
\newblock In \emph{International Conference on Machine Learning (ICML)}. PMLR,
2020.
\bibitem[Frankle \& Carbin(2019)Frankle and Carbin]{frankle2018lottery}
Frankle, J. and Carbin, M.
\newblock The lottery ticket hypothesis: Finding sparse, trainable neural
networks.
\newblock In \emph{International Conference on Learning Representations
(ICLR)}, 2019.
\bibitem[Frantar et~al.(2021)Frantar, Kurtic, and Alistarh]{frantar2021m}
Frantar, E., Kurtic, E., and Alistarh, D.
\newblock {M-FAC}: Efficient matrix-free approximations of second-order
information.
\newblock In \emph{Conference on Neural Information Processing Systems
(NeurIPS)}, 2021.
\bibitem[Gale et~al.(2020)Gale, Zaharia, Young, and Elsen]{sgk_sc2020}
Gale, T., Zaharia, M., Young, C., and Elsen, E.
\newblock Sparse {GPU} kernels for deep learning.
\newblock In \emph{International Conference for High Performance Computing,
Networking, Storage and Analysis (SC)}, 2020.
\bibitem[Gholami et~al.(2021)Gholami, Kim, Dong, Yao, Mahoney, and
Keutzer]{gholami2021survey}
Gholami, A., Kim, S., Dong, Z., Yao, Z., Mahoney, M.~W., and Keutzer, K.
\newblock A survey of quantization methods for efficient neural network
inference.
\newblock \emph{arXiv preprint arXiv:2103.13630}, 2021.
\bibitem[Guo et~al.(2020)Guo, Zhang, Mu, Heng, Liu, Wei, and
Sun]{guo2020single}
Guo, Z., Zhang, X., Mu, H., Heng, W., Liu, Z., Wei, Y., and Sun, J.
\newblock Single path one-shot neural architecture search with uniform
sampling.
\newblock In \emph{European Conference on Computer Vision (ECCV)}, 2020.
\bibitem[Hagiwara(1994)]{hagiwara1994}
Hagiwara, M.
\newblock A simple and effective method for removal of hidden units and
weights.
\newblock \emph{Neurocomputing}, 6\penalty0 (2):\penalty0 207 -- 218, 1994.
\newblock ISSN 0925-2312.
\newblock Backpropagation, Part IV.
\bibitem[Han et~al.(2015)Han, Pool, Tran, and Dally]{han2015learning}
Han, S., Pool, J., Tran, J., and Dally, W.~J.
\newblock Learning both weights and connections for efficient neural networks.
\newblock In \emph{Conference on Neural Information Processing Systems
(NeurIPS)}, 2015.
\bibitem[He et~al.(2016)He, Zhang, Ren, and Sun]{he2016deep}
He, K., Zhang, X., Ren, S., and Sun, J.
\newblock Deep residual learning for image recognition.
\newblock In \emph{Conference on Computer Vision and Pattern Recognition
(CVPR)}, 2016.
\bibitem[He et~al.(2017)He, Zhang, and Sun]{he2017channel}
He, Y., Zhang, X., and Sun, J.
\newblock Channel pruning for accelerating very deep neural networks.
\newblock In \emph{International Conference on Computer Vision (ICCV)}, 2017.
\bibitem[He et~al.(2018)He, Lin, Liu, Wang, Li, and Han]{he2018amc}
He, Y., Lin, J., Liu, Z., Wang, H., Li, L.-J., and Han, S.
\newblock {AMC}: {AutoML} for model compression and acceleration on mobile
devices.
\newblock In \emph{European Conference on Computer Vision (ECCV)}, 2018.
\bibitem[He et~al.(2019)He, Liu, Wang, Hu, and Yang]{he2019filter}
He, Y., Liu, P., Wang, Z., Hu, Z., and Yang, Y.
\newblock Filter pruning via geometric median for deep convolutional neural
networks acceleration.
\newblock In \emph{Conference on Computer Vision and Pattern Recognition
(CVPR)}, 2019.
\bibitem[Hoefler et~al.(2021)Hoefler, Alistarh, Ben-Nun, Dryden, and
Peste]{hoefler2021sparsity}
Hoefler, T., Alistarh, D., Ben-Nun, T., Dryden, N., and Peste, A.
\newblock Sparsity in deep learning: Pruning and growth for efficient inference
and training in neural networks.
\newblock \emph{arXiv preprint arXiv:2102.00554}, 2021.
\bibitem[Howard et~al.(2017)Howard, Zhu, Chen, Kalenichenko, Wang, Weyand,
Andreetto, and Adam]{howard2017mobilenets}
Howard, A.~G., Zhu, M., Chen, B., Kalenichenko, D., Wang, W., Weyand, T.,
Andreetto, M., and Adam, H.
\newblock Mobilenets: Efficient convolutional neural networks for mobile vision
applications.
\newblock \emph{arXiv preprint arXiv:1704.04861}, 2017.
\bibitem[Hubara et~al.(2021{\natexlab{a}})Hubara, Chmiel, Island, Banner, Naor,
and Soudry]{hubara2021accelerated}
Hubara, I., Chmiel, B., Island, M., Banner, R., Naor, S., and Soudry, D.
\newblock Accelerated sparse neural training: A provable and efficient method
to find {N:M} transposable masks.
\newblock In \emph{Conference on Neural Information Processing Systems
(NeurIPS)}, 2021{\natexlab{a}}.
\bibitem[Hubara et~al.(2021{\natexlab{b}})Hubara, Nahshan, Hanani, Banner, and
Soudry]{hubara2021accurate}
Hubara, I., Nahshan, Y., Hanani, Y., Banner, R., and Soudry, D.
\newblock Accurate post training quantization with small calibration sets.
\newblock In \emph{International Conference on Machine Learning (ICML)}. PMLR,
2021{\natexlab{b}}.
\bibitem[Jayakumar et~al.(2021)Jayakumar, Pascanu, Rae, Osindero, and
Elsen]{jayakumar2021top}
Jayakumar, S.~M., Pascanu, R., Rae, J.~W., Osindero, S., and Elsen, E.
\newblock {Top-KAST}: {Top-K} always sparse training.
\newblock \emph{arXiv preprint arXiv:2106.03517}, 2021.
\bibitem[Jocher(2022.)]{yolov5}
Jocher, G.
\newblock {YOLOv5}, 2022.
\newblock URL \url{https://github.com/ultralytics/yolov5}.
\bibitem[Kingma \& Ba(2015)Kingma and Ba]{kingma2014adam}
Kingma, D.~P. and Ba, J.
\newblock Adam: A method for stochastic optimization.
\newblock In \emph{International Conference on Learning Representations
(ICLR)}, 2015.
\bibitem[Kurtic et~al.(2022)Kurtic, Campos, Nguyen, Frantar, Kurtz, Fineran,
Goin, and Alistarh]{kurtic2022optimal}
Kurtic, E., Campos, D., Nguyen, T., Frantar, E., Kurtz, M., Fineran, B., Goin,
M., and Alistarh, D.
\newblock The {Optimal BERT Surgeon}: Scalable and accurate second-order
pruning for large language models.
\newblock \emph{arXiv preprint arXiv:2203.07259}, 2022.
\bibitem[Kurtz et~al.(2020)Kurtz, Kopinsky, Gelashvili, Matveev, Carr, Goin,
Leiserson, Moore, Nell, Shavit, and Alistarh]{pmlr-v119-kurtz20a}
Kurtz, M., Kopinsky, J., Gelashvili, R., Matveev, A., Carr, J., Goin, M.,
Leiserson, W., Moore, S., Nell, B., Shavit, N., and Alistarh, D.
\newblock Inducing and exploiting activation sparsity for fast inference on
deep neural networks.
\newblock In \emph{International Conference on Machine Learning (ICML)}, 2020.
\bibitem[Kusupati et~al.(2020)Kusupati, Ramanujan, Somani, Wortsman, Jain,
Kakade, and Farhadi]{kusupati2020soft}
Kusupati, A., Ramanujan, V., Somani, R., Wortsman, M., Jain, P., Kakade, S.,
and Farhadi, A.
\newblock Soft threshold weight reparameterization for learnable sparsity.
\newblock In \emph{International Conference on Machine Learning (ICML)}, 2020.
\bibitem[Lagunas et~al.(2021)Lagunas, Charlaix, Sanh, and Rush]{lagunas21block}
Lagunas, F., Charlaix, E., Sanh, V., and Rush, A.
\newblock Block pruning for faster transformers.
\newblock In \emph{Conference on Empirical Methods in Natural Language
Processing (EMNLP)}, 2021.
\bibitem[LeCun et~al.(1990)LeCun, Denker, and Solla]{lecun1990optimal}
LeCun, Y., Denker, J.~S., and Solla, S.~A.
\newblock Optimal brain damage.
\newblock In \emph{Conference on Neural Information Processing Systems
(NeurIPS)}, pp.\ 598--605, 1990.
\bibitem[Li et~al.(2016)Li, Kadav, Durdanovic, Samet, and Graf]{li2016pruning}
Li, H., Kadav, A., Durdanovic, I., Samet, H., and Graf, H.~P.
\newblock Pruning filters for efficient convnets.
\newblock \emph{arXiv preprint arXiv:1608.08710}, 2016.
\bibitem[Li et~al.(2021)Li, Gong, Tan, Yang, Hu, Zhang, Yu, Wang, and
Gu]{li2021brecq}
Li, Y., Gong, R., Tan, X., Yang, Y., Hu, P., Zhang, Q., Yu, F., Wang, W., and
Gu, S.
\newblock {BRECQ}: Pushing the limit of post-training quantization by block
reconstruction.
\newblock In \emph{International Conference on Learning Representations
(ICLR)}, 2021.
\bibitem[Liebenwein et~al.(2021)Liebenwein, Maalouf, Feldman, and
Rus]{liebenwein2021compressing}
Liebenwein, L., Maalouf, A., Feldman, D., and Rus, D.
\newblock Compressing neural networks: Towards determining the optimal
layer-wise decomposition.
\newblock \emph{Conference on Neural Information Processing Systems (NeurIPS)},
2021.
\bibitem[Liu et~al.(2021)Liu, Zhang, Kuang, Zhou, Xue, Wang, Chen, Yang, Liao,
and Zhang]{liu2021group}
Liu, L., Zhang, S., Kuang, Z., Zhou, A., Xue, J.-H., Wang, X., Chen, Y., Yang,
W., Liao, Q., and Zhang, W.
\newblock Group fisher pruning for practical network compression.
\newblock In \emph{International Conference on Machine Learning (ICML)}, 2021.
\bibitem[Marcel \& Rodriguez(2010)Marcel and Rodriguez]{marcel2010torchvision}
Marcel, S. and Rodriguez, Y.
\newblock Torchvision the machine-vision package of torch.
\newblock In \emph{ACM International Conference on Multimedia}, 2010.
\bibitem[Markov et~al.(2021)Markov, Ramezani, and Alistarh]{markov2021project}
Markov, I., Ramezani, H., and Alistarh, D.
\newblock Project {CGX}: Scalable deep learning on commodity {GPUs}.
\newblock \emph{arXiv preprint arXiv:2111.08617}, 2021.
\bibitem[Mishra et~al.(2021)Mishra, Latorre, Pool, Stosic, Stosic, Venkatesh,
Yu, and Micikevicius]{NVIDIASparse}
Mishra, A., Latorre, J.~A., Pool, J., Stosic, D., Stosic, D., Venkatesh, G.,
Yu, C., and Micikevicius, P.
\newblock Accelerating sparse deep neural networks.
\newblock \emph{arXiv preprint arXiv:2104.08378}, 2021.
\bibitem[Nagel et~al.(2020)Nagel, Amjad, Van~Baalen, Louizos, and
Blankevoort]{nagel2020up}
Nagel, M., Amjad, R.~A., Van~Baalen, M., Louizos, C., and Blankevoort, T.
\newblock Up or down? adaptive rounding for post-training quantization.
\newblock In \emph{International Conference on Machine Learning (ICML)}, 2020.
\bibitem[Nagel et~al.(2021)Nagel, Fournarakis, Amjad, Bondarenko, van Baalen,
and Blankevoort]{nagel2021white}
Nagel, M., Fournarakis, M., Amjad, R.~A., Bondarenko, Y., van Baalen, M., and
Blankevoort, T.
\newblock A white paper on neural network quantization.
\newblock \emph{arXiv preprint arXiv:2106.08295}, 2021.
\bibitem[NeuralMagic(2021)]{deepsparse}
NeuralMagic.
\newblock {DeepSparse}, 2021.
\newblock URL \url{https://github.com/neuralmagic/deepsparse}.
\bibitem[Peste et~al.(2021)Peste, Iofinova, Vladu, and Alistarh]{peste2021ac}
Peste, A., Iofinova, E., Vladu, A., and Alistarh, D.
\newblock Ac/dc: Alternating compressed/decompressed training of deep neural
networks.
\newblock In \emph{Conference on Neural Information Processing Systems
(NeurIPS)}, 2021.
\bibitem[Pool \& Yu(2021)Pool and Yu]{pool2021channel}
Pool, J. and Yu, C.
\newblock Channel permutations for {N:M} sparsity.
\newblock In \emph{Conference on Neural Information Processing Systems
(NeurIPS)}, 2021.
\bibitem[Rajpurkar et~al.(2016)Rajpurkar, Zhang, Lopyrev, and
Liang]{rajpurkar2016squad}
Rajpurkar, P., Zhang, J., Lopyrev, K., and Liang, P.
\newblock {SQuAD}: 100,000+ questions for machine comprehension of text.
\newblock In \emph{Conference on Empirical Methods in Natural Language
Processing (EMNLP)}, 2016.
\bibitem[Sanh et~al.(2020)Sanh, Wolf, and Rush]{2020-sanh}
Sanh, V., Wolf, T., and Rush, A.~M.
\newblock Movement pruning: Adaptive sparsity by fine-tuning.
\newblock \emph{arXiv preprint arXiv:2005.07683}, 2020.
\bibitem[Schwarz et~al.(2021)Schwarz, Jayakumar, Pascanu, Latham, and
Teh]{schwarz2021powerpropagation}
Schwarz, J., Jayakumar, S., Pascanu, R., Latham, P., and Teh, Y.
\newblock Powerpropagation: A sparsity inducing weight reparameterisation.
\newblock In \emph{Conference on Neural Information Processing Systems
(NeurIPS)}, 2021.
\bibitem[Singh \& Alistarh(2020)Singh and Alistarh]{singh2020woodfisher}
Singh, S.~P. and Alistarh, D.
\newblock Woodfisher: Efficient second-order approximation for neural network
compression.
\newblock In \emph{Conference on Neural Information Processing Systems
(NeurIPS)}, 2020.
\bibitem[Wu et~al.(2020)Wu, Liu, Chen, and Chien]{wu2020constraint}
Wu, Y.-C., Liu, C.-T., Chen, B.-Y., and Chien, S.-Y.
\newblock Constraint-aware importance estimation for global filter pruning
under multiple resource constraints.
\newblock In \emph{Conference on Computer Vision and Pattern Recognition (CVPR)
Workshops}, 2020.
\bibitem[Yang et~al.(2020)Yang, Gui, Zhu, and Liu]{yang2020automatic}
Yang, H., Gui, S., Zhu, Y., and Liu, J.
\newblock Automatic neural network compression by sparsity-quantization joint
learning: A constrained optimization-based approach.
\newblock In \emph{Conference on Computer Vision and Pattern Recognition
(CVPR)}, 2020.
\bibitem[Yang et~al.(2021)Yang, Liao, and Sze]{yang2021netadaptv2}
Yang, T.-J., Liao, Y.-L., and Sze, V.
\newblock {NetAdaptV2}: Efficient neural architecture search with fast
super-network training and architecture optimization.
\newblock In \emph{Conference on Computer Vision and Pattern Recognition
(CVPR)}, 2021.
\bibitem[Yao et~al.(2021)Yao, Dong, Zheng, Gholami, Yu, Tan, Wang, Huang, Wang,
Mahoney, et~al.]{yao2021hawq}
Yao, Z., Dong, Z., Zheng, Z., Gholami, A., Yu, J., Tan, E., Wang, L., Huang,
Q., Wang, Y., Mahoney, M., et~al.
\newblock {HAWQ-v3}: Dyadic neural network quantization.
\newblock In \emph{International Conference on Machine Learning (ICML)}, 2021.
\bibitem[Zhou et~al.(2021)Zhou, Ma, Zhu, Liu, Zhang, Yuan, Sun, and
Li]{zhou2021learning}
Zhou, A., Ma, Y., Zhu, J., Liu, J., Zhang, Z., Yuan, K., Sun, W., and Li, H.
\newblock Learning {N:M} fine-grained structured sparse neural networks from
scratch.
\newblock In \emph{International Conference on Learning Representations
(ICLR)}, 2021.
\bibitem[Zhu \& Gupta(2017)Zhu and Gupta]{zhu2017prune}
Zhu, M. and Gupta, S.
\newblock To prune, or not to prune: exploring the efficacy of pruning for
model compression.
\newblock \emph{arXiv preprint arXiv:1710.01878}, 2017.
\end{thebibliography}
\section{DP Algorithm Implementation}
\label{app:dp-implementation}
\begin{algorithm}[h!]
\caption{We efficiently compute the optimal layer-wise sparsity profile with execution time at most $T$ given $S$, $e^s_\ell$, $t^s_\ell$ and assuming that time is discretized, using bottom-up dynamic programming.}
\label{alg:dp}
\begin{algorithmic}
\STATE $\mathbf{D} \gets L \times (T + 1)$ matrix filled with $\infty$
\STATE $\mathbf{P} \gets L \times (T + 1)$ matrix
\FOR {$s \in S$}
\IF {$e^s_1 < \mathbf{D}[1,t^s_1]$}
\STATE $\mathbf{D}[1,t^s_1] \gets e^s_1$; \, $\mathbf{P}[1,t^s_1] \gets s$
\ENDIF
\ENDFOR
\FOR {$\ell = 2, \dots, L$}
\FOR {$s \in S$}
\FOR {$t = t^s_\ell + 1, \dots, T$}
\IF {$e^s_\ell + \mathbf{D}[\ell - 1, t - t^s_\ell] < \mathbf{D}[\ell, t]$}
\STATE $\mathbf{D}[\ell, t] \gets e^s_\ell + \mathbf{D}[\ell - 1, t - t^s_\ell]; \, \mathbf{P}[\ell, t] \gets s$
\ENDIF
\ENDFOR
\ENDFOR
\ENDFOR
\STATE {$t \gets \text{argmin}_t \, \mathbf{D}[L, t]$} // return $\mathbf{D}[L, t]$ as optimal error
\FOR {$\ell = L, \dots, 1$}
\STATE {$s \gets \mathbf{P}[\ell, t]$ // return $s$ as optimal sparsity for layer $\ell$}
\STATE {$t \gets t - t^s_\ell$}
\ENDFOR
\end{algorithmic}
\end{algorithm}
We note that, in practice, the innermost loop over $t$ can easily be vectorized (with corresponding acceleration through specialized CPU / GPU code) via shifting of $\mathbf{D}[\ell - 1, :]$ by $t^s_\ell$, thereby making the overall algorithm highly efficient even for very fine discretization.
\section{SPDY Pseudocode}
\label{app:pseudocode}
This section provides additional details about the overall SPDY framework described in Section \ref{sec:spdy-overview} in form of pseudo code. Specifically, Algorithm \ref{alg:collect-timings} and Algorithm \ref{alg:create-db} illustrate the initial timing collection and iterative AdaPrune reconstruction database generation. Meanwhile, Algorithm \ref{alg:spdy-search} presents the SPDY search for the optimal coefficients $\mathbf{c^*}$.
\begin{algorithm}[h!]
\caption{Collect layer-wise timings $t^s_\ell$.}
\label{alg:collect-timings}
\begin{algorithmic}
\FOR {$\ell = 1, \dots, L$}
\FOR {$s \in S$}
\STATE $W_\ell^s \gets$ randomly prune $W_\ell$ to sparsity $s$
\STATE $t_\ell^s \gets$ collect timing for layer $\ell$ using weights $W_\ell^s$
\ENDFOR
\ENDFOR
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[h!]
\caption{Generate reconstruction database entries $W^s_\ell$.}
\label{alg:create-db}
\begin{algorithmic}
\STATE $\mathbf{s} \gets$ sorted vector of sparsities in $S$
\FOR {$\ell = 1, \dots, L$}
\STATE $W^{s_0}_\ell \gets$ dense weights $W_L$
\FOR {$i = 1, \dots |S|$}
\STATE $W_\ell^{s_i} \gets$ AdaPrune for target $s_i$ with weights $W_\ell^{s_{i - 1}}$
\ENDFOR
\ENDFOR
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[h!]
\caption{SPDY search for optimal sensitivity values $\mathbf{c^*}$. We use $k = 100$ and $\delta = 0.1$ in our experiments.}
\label{alg:spdy-search}
\begin{algorithmic}
\STATE \textbf{function} $eval(\mathbf{c})$
\STATE \quad $e^s_\ell \gets$ compute by formula (\ref{eq:error-approx}) using $\mathbf{c}$ for all $\ell$
\STATE \quad $s_\ell \gets$ run DP algorithm with $e^s_\ell$ for all $\ell$
\STATE \quad $M \gets $ stitch model for $s_\ell$ from database
\STATE \quad Return calibration loss of $M$.
\STATE $\mathbf{c^*} \gets$ sample uniform vector in $[0, 1]^L$
\FOR {$k$ times}
\STATE $\mathbf{c} \gets$ sample uniform vector in $[0, 1]^L$
\IF {$eval(\mathbf{c}) < eval(\mathbf{c^*})$}
\STATE $\mathbf{c^*} \gets \mathbf{c}$
\ENDIF
\ENDFOR
\FOR {$d = \lceil \delta \cdot L \rceil, \dots, 1$}
\FOR {$k$ times}
\STATE $\mathbf{c} \gets \mathbf{c^*}$
\STATE Randomly resample $d$ items of $\mathbf{c}$ in $[0, 1]$
\IF {$eval(\mathbf{c}) < eval(\mathbf{c^*})$}
\STATE $\mathbf{c^*} \gets \mathbf{c}$
\ENDIF
\ENDFOR
\ENDFOR
\end{algorithmic}
\end{algorithm}
\section{One-Shot Comparison}
\label{app:one-shot-comparison}
Figure \ref{fig:oneshot-comparison} demonstrates how iterative AdaPrune significantly improves over standard AdaPrune and GMP.
\begin{figure}
\caption{Comparison of magnitude pruning, AdaPrune and iterative AdaPrune in terms of one-shot performance.}
\label{fig:oneshot-comparison}
\end{figure}
\begin{table*}[h]
\begin{minipage}[c]{\linewidth}
\centering
\scalebox{.75}{
\begin{tabular}{|l|c|c|c|c|c|c|c|}
\toprule
& \multicolumn{2}{c|}{ResNet34} & \multicolumn{2}{c|}{ResNet50} & \multicolumn{2}{c|}{BERT SQuaD} \\
& $2.00\times$ & $3.00\times$ & $2.00\times$ & $3.00\times$ & $2.50\times$ & $3.50\times$ \\
\midrule
DP + squared weights & 54.18 & 31.59 & 47.44 & 03.82 & 71.20 & 07.88 \\
DP + squared weights normalized & 56.32 & 09.15 & 35.42 & 01.20 & 58.77 & 06.17 \\
DP + custom norm based & 66.64 & 49.17 & 66.82 & 11.11 & 72.34 & 13.64 \\
DP + layer-wise loss / no reconstruction & 54.51 & 10.32 & 66.49 & 08.85 & 53.98 & 06.85\\
DP + layer-wise loss / iterative AP & 65.92 & 46.11 & 69.26 & 20.22 & 30.59 & 06.13 \\
SPDY / no reconstruction & 53.38 & 22.18 & 64.59 & 00.67 & 62.41 & 07.58 \\
\textbf{SPDY} & \textbf{68.23} & \textbf{51.68} & \textbf{71.17} & \textbf{26.81} & \textbf{74.59} & \textbf{18.83} \\
\midrule
Uniform & 64.65 & 42.38 & 64.65 & 05.84 & 57.42 & 09.00 \\
Global magnitude & 39.41 & 22.80 & 62.89 & 01.76 & 58.06 & 06.85 \\
\bottomrule
\end{tabular}
}
\captionof{table}{A comparison of one-shot accuracies for profiles generated by DP in combination with various metrics, SPDY and uniform / global magnitude pruning.}
\label{tab:metrics-ablation}
\end{minipage}
\end{table*}
\section{Ablation Studies}
\label{app:ablation}
In Section \ref{sec:error-metric-learning}, we briefly discussed the difficulty of designing a reliable error metric for the use in conjunction with our DP algorithm. We now present a more detailed investigation of this topic including additional experimental results. Concretely, we consider the following 5 error metrics:
\begin{itemize}
\item \textbf{Squared Weights} --- Perhaps the most obvious error score candidate is a natural extension of the popular magnitude pruning criterion \cite{zhu2017prune}, the sum of the squared magnitudes of all pruned weights. It can also be interpreted as the squared norm of the weight perturbation, which has been used in the context of quantization \cite{yao2021hawq}.
\item \textbf{Squared Weights Normalized} --- A potential problem of the previous metric is that smaller layers, which are often quite sensitive, might be pruned too strongly as their sums are inherently smaller than those of bigger layers. A simple way to address this is to normalize each sum by the number of elements in the corresponding layer. Such normalization is for example used for structured pruning in \cite{liu2021group}.
\item \textbf{Custom Norm Based} --- Another possible shortcoming of both metrics mentioned so far is that the sums for high sparsity choices generally differ only in a small number elements and can thus be quite similar. This means the fact that pruning typically becomes a lot more difficult at high sparsities may not be very well reflected in those error metrics. Hence, we developed a custom norm based metric to address this. Let $\text{max}_{w \not \in W^s} \, |w|$ be the magnitude of the largest dropped weight when pruning a layer to sparsity $s$, then the error is defined as:
\begin{equation*}
e^s = \frac{\text{max}_{w \not \in W^s} \, |w|}{1 - s}.
\end{equation*}
The numerator ensures that the influence of the weight magnitudes does not decrease for high sparsities and the division by the remaining density that the error always increases strongly as a layer gets very sparse.
\item \textbf{Layer-wise Loss No Reconstruction} --- One more obvious option for a layer-wise error is simply the change in loss on a small calibration set that is the result of pruning exactly this layer to a certain sparsity. This metric measures the loss after standard magnitude pruning, without any layer-wise reconstruction, similar to \cite{li2016pruning}.
\item \textbf{Layer-wise Loss Iterative AP} --- Finally, we combine the layer-wise loss with the techniques we introduce in Section \ref{sec:assessing-profile-quality}, i.e. using iterative AdaPrune to reconstruct the pruned layer before measuring the loss. This is similar in spirit to the AdaQuant loss used as an error metric for quantization \cite{hubara2021accurate}.
\end{itemize}
We now generate profiles using DP in combination with each of the 5 described metrics, for 3 different models at 2 speedups each, a lower one where AP pruning still gives reasonable accuracies and a higher one which could be used as the starting point for extended gradual pruning as in Section \ref{sec:gradual-pruning}. We consider two ResNets to study the consistency of metrics for models of the same family and a very different BERT model. For computational reasons, we consider the accuracy after one-shot pruning by stitching together layers from the reconstruction database described in Section \ref{sec:assessing-profile-quality}, which \cite{he2018amc} as well as our main experimental results show to be decent proxies for post finetuning / gradual pruning accuracy. We compare with SPDY as well as the standard uniform and global magnitude strategies (see Section \ref{sec:experiments}), input and output layers are always skipped. All results are summarized in Table \ref{tab:metrics-ablation}.
First, we can see that the squared weights based metrics fail to beat the best non-DP profile in all but one case. The same is true for the layer-wise loss without any reconstruction. Clearly, those metrics are unsuitable for generating effective sparsity profiles. Meanwhile, the layer-wise loss with iterative AP reconstruction seems to perform quite well relative to the non-DP results on the ResNet models but performs poorly on BERT where the losses are apparently so misleading that it is beaten even by the no-reconstruction version. A closer look reveals that the layer-wise losses do not all increase strongly with higher sparsities on this large model, which leads to very aggressive pruning of several layers compounding to large accuracy drops (that are not properly reflected in the single-layer losses). Additionally, it should be noted that the layer-wise loss works better than the custom metric on ResNet50 but worse on ResNet34. This means the best performing metric does not even seem to be consistent within a single model family. In general, the custom metric appears to perform reasonably in all 6 scenarios, outperforming the non-DP profiles, sometimes even by sizable margins. However, there is still a significant gap to the SPDY results in several cases. In fact, we found that the custom metric profiles often only lead to minimal 0.1 -- 0.2 improvements over uniform ones when running extensive state-of-the-art gradual pruning, meaning that the extra improvements of SPDY are crucial to achieve the notable gaps we do in Section \ref{sec:gradual-pruning}. Overall, these experiments demonstrate that manually designing a reliable layer-wise error metric for unstructured pruning is indeed a quite challenging problem, as the automatically learned error metric of SPDY consistently beats hand-crafted metrics of varying complexity.
Finally, we also briefly study the importance of the enhanced one-shot pruning performance through our reconstruction database. For that purpose, we run SPDY with one-shot magnitude pruning without any reconstruction to find profiles (the reported accuracy of course still stitches the resulting profile from the reconstruction database for comparability). As Table \ref{tab:metrics-ablation} shows, this dramatically worsens results, especially at high sparsities where simple one-shot pruning typically completely crashes the model, making it essentially impossible for SPDY to find useful sensitivity coefficients.
In summary, the experiments covered in this section strongly suggest that both the search procedure as well as the reconstruction database are indeed key components of our system and are not easily replaced by simpler means.
\section{Search Procedure Discussion}
\label{app:search}
In the main text, we briefly mentioned how the integration of the DP algorithm makes the search significantly easier. We now provide additional discussion and more details on our experimental setup.
While our search only ever considers solutions that lie directly on the constraint boundary, other approaches \cite{cai2019once, yang2021netadaptv2, li2021brecq} typically have to evaluate many candidates for which the constraint is either violated or not tight (in which case the solution cannot be optimal) in order to make progress. Additionally, the DP solving always ensures proper utilization of the layer-wise speedups. For illustration, a layer with poor acceleration will not be pruned much for most sensitivity values. Similarly, due to the quadratic nature of our errors, most layers will only be pruned to very high sparsities, where speedup curves typically flatten, if this is really necessary to reach the overall speedup target. As a consequence, non-key layers will often have large ranges of acceptable sensitivity scores that all lead to a very similar sparsity choice. This makes finding a good profile in terms of $\mathbf{c}$ easier than by direct optimization in the huge space of all $|S|^L$ possible layer-wise sparsity configurations.\footnote{For example, the architecture search space in \cite{cai2019once} is $\approx 10^{19}$ whereas ours is $\approx 10^{80}$ for the ResNet50 model.}
We now demonstrate these claims empirically by comparing the average over 5 runs of our reparametrized unconstrained search using the DP algorithm, and a direct (constrained) search in the original sparsity space. For comparability, we use exactly the same local search method detailed in Section \ref{sec:error-metric-learning} for the direct search, just with resampling (uniformly in $S$) changes to the current solution until one is found that satisfies the speedup constraint. Additionally, we also test a genetic algorithm, implented exactly as described in \cite{guo2020single} but using $2.5 \times$ more iterations for a fair comparison. Figure \ref{fig:search-comparison} shows the evolution of the objective value relative to the dominating runtime factor, the number of stitched model evaluations (calibration loss computations), for a $2.5\times$ ResNet50 profile.
\begin{figure}
\caption{Comparison of our reparametrized unconstrained search and a constrained direct search.}
\label{fig:search-comparison}
\end{figure}
\begin{figure*}
\caption{Accuracy after one-shot pruning vs. speedup for all profile generation methods on ResNet50. For improved visibility, as the y-axes are very different, the plot is split in two showing $1\times$ to $2\times$ speedup on the left and $2\times$ to $3\times$ on the right.}
\label{fig:rn50-speedup-oneshot}
\end{figure*}
The plot clearly shows how the reparametrized search finds profiles with lower calibration loss, does so with much fewer evaluations ($\approx 10\times$ here) and exhibits significantly reduced variance in solution quality over multiple runs. Overall, this much increased efficiency suggests that the reparametrized DP search could probably also be used in conjunction with better but more expensive evaluation strategies. Interestingly, the genetic algorithm seems to perform noticeably worse than the simple local search. We suspect this is due to fact that in our huge search space, getting stuck in local minima seems to be less of a problem than actually reaching such a minimum quickly. Thus, focusing on mutation evaluations of just the current best solution leads to better results faster than distributing them across several candidates, as is done by genetic programming.
\section{Additional Experiments}
\label{app:additional-experiments}
To investigate the one-shot performance of the different profile generation methods more closely, we prune RN50 to various speedup targets in steps of $0.25$ and plot the results in Figure \ref{fig:rn50-speedup-oneshot}.
First, this plot shows that the graphs of uniform and GMP cross somewhere between $1.5\times$ and $1.75\times$ speedup. We can also observe how the relative order of the accuracies directly after AP stitching is preserved even through big accuracy increases by running global AP. Finally, we can clearly see how the gap between SPDY (after AP but also after gAP) and other methods rapidly grows as the speedup target is increased, up to over $10\%$ accuracy at $3\times$ speedup. Overall, it can be said that SPDY appears to work well in the challenging one-shot setting.
\subsection{Global AdaPrune 2:4}
Additionally, we now present results for post training 2:4 pruning various YOLOv5 models and BERT finetuned on SQuAD in Table \ref{tab:ptq-yolo}. Although the mAP drops for YOLO are quite a bit bigger than the accuracy drops for ResNet, the results are still several points above the next smaller and roughly $2\times$ faster model (a similar acceleration is promised by 2:4 sparsity) and could thus be relevant in practice.
\begin{table}[h]
\begin{minipage}[c]{\linewidth}
\centering
\scalebox{.75}{
\begin{tabular}{|l|c|c|c|}
\toprule
\multirow{2}{*}{Model} & \multirow{2}{*}{Dense} & \multicolumn{2}{c|}{2:4} \\
& & AP & \textbf{+gAP} \\
\midrule
YOLOv5n & 46.20 & 13.40 & \textbf{37.10} \\
YOLOv5s & 56.40 & 40.70 & \textbf{51.60} \\
YOLOv5m & 64.20 & 54.00 & \textbf{61.20} \\
YOLOv5l & 67.40 & 59.80 & \textbf{65.40} \\
\midrule
BERT SQuAD & 88.54 & 84.75 & 87.41 \\
\bottomrule
\end{tabular}
}
\captionof{table}{Global AdaPrune performance for 2:4 pruning the YOLOv5 model family as well as BERT-base on SQuAD. All input and output layers are skipped.}
\label{tab:ptq-yolo}
\end{minipage}
\end{table}
\section{Real Timings}
\label{app:real-timings}
\begin{table}[h]
\begin{minipage}[c]{\linewidth}
\centering
\scalebox{.75}{
\begin{tabular}{|l|c|c|c|}
\toprule
Model & Target & CPU & Real \\
\midrule
ResNet50 & $1.00\times$ & AMD & 0.478s -- $1.00\times$ \\
ResNet50 & $2.00\times$ & AMD & 0.234s -- $2.04\times$ \\
ResNet50 & $2.50\times$ & AMD & 0.178s -- $2.69\times$ \\
ResNet50 & $3.00\times$ & AMD & 0.143s -- $3.34\times$ \\
ResNet50 & $3.50\times$ & AMD & 0.121s -- $3.95\times$ \\
\midrule
MobileNetV1 & $1.00\times$ & Intel & 0.045s -- $1.00\times$ \\
MobileNetV1 & $1.50\times$ & Intel & 0.031s -- $1.45\times$ \\
\midrule
YOLOv5s & $1.00\times$ & Intel & 0.641s -- $1.00\times$ \\
YOLOv5s & $1.50\times$ & Intel & 0.449s -- $1.43\times$ \\
YOLOv5s & $1.75\times$ & Intel & 0.380s -- $1.68\times$ \\
YOLOv5m & $1.00\times$ & Intel & 1.459s -- $1.00\times$ \\
YOLOv5m & $1.75\times$ & Intel & 0.848s -- $1.72\times$ \\
YOLOv5m & $2.00\times$ & Intel & 0.725s -- $2.01\times$ \\
\midrule
BERT SQuAD & $1.00\times$ & Intel & 0.969s -- $1.00\times$ \\
BERT SQuAD & $3.00\times$ & Intel & 0.320s -- $3.03\times$ \\
BERT SQuAD & $3.50\times$ & Intel & 0.271s -- $3.58\times$ \\
BERT SQuAD & $4.00\times$ & Intel & 0.223s -- $4.35\times$ \\
\bottomrule
\end{tabular}
}
\captionof{table}{Real timings of the SPDY profiles in Table \ref{tab:gradual-results} using the DeepSparse v0.9.1 engine. All timings are for batchsize 64, except for BERT which uses batchsize 16.}
\label{tab:real-timings}
\end{minipage}
\end{table}
Table \ref{tab:real-timings} shows the real timings of the final pruned models resulting from the SPDY profiles in our gradual pruning experiments in Section \ref{sec:gradual-pruning}. We can see that the speedup predictions of the additive model are in most cases very accurate, especially considering typical performance fluctuations with varying operating temperature and that even different CPUs of the same type can have small performance differences. Only for the higher speedup ResNet50 models and $4.00\times$ BERT, the true speedups are noticeably underestimated. This is most likely due to special engine optimizations (we run with all of them turned on here) that are not active in the layer-wise timing mode. With sufficient knowledge about the engine internals, it would probably be possible to account for such effects in the DP algorithm. However, optimizing our methods for one particular inference engine was not the goal of this work.
\section{Overall Sparsities}
\label{app:sparsity}
As our focus are real model speedups, our results in the main paper are all given with respect to those rather than parameter counts. Nevertheless, for completeness we now provide overall sparsity values (with respect to all pruned layers) corresponding to our main gradual pruning results in Table \ref{tab:gradual-results}.
\begin{table}[h]
\begin{minipage}[c]{\linewidth}
\centering
\scalebox{.75}{
\begin{tabular}{|l|c|c|c|c|c|}
\toprule
Model & Speed. & CPU & SPDY & Uni. & GMP \\
\midrule
ResNet50 & $2.00\times$ & AMD & 70.31 & 80.54 & 85.77 \\
ResNet50 & $2.50\times$ & AMD & 86.15 & 88.33 & 92.30 \\
ResNet50 & $3.00\times$ & AMD & 91.78 & 93.01 & 96.24 \\
ResNet50 & $3.50\times$ & AMD & 95.22 & 96.58 & 98.26 \\
\midrule
MobileNetV1 & $1.50\times$ & Intel & 49.13 & 65.88 & 77.16 \\
\midrule
YOLOv5s & $1.50\times$ & Intel & 59.83 & 70.69 & 70.84 \\
YOLOv5s & $1.75\times$ & Intel & 85.16 & 85.68 & 91.60 \\
YOLOv5m & $1.75\times$ & Intel & 79.81 & 82.42 & 88.20 \\
YOLOv5m & $2.00\times$ & Intel & 90.42 & 91.42 & 94.29 \\
\midrule
BERT SQuAD & $3.00\times$ & Intel & 82.24 & 84.14 & 83.98 \\
BERT SQuAD & $3.50\times$ & Intel & 88.89 & 90.49 & 90.18 \\
BERT SQuAD & $4.00\times$ & Intel & 94.15 & 94.86 & 95.29 \\
\bottomrule
\end{tabular}
}
\captionof{table}{Overall sparsities of profiles corresponding to Table \ref{tab:gradual-results}, calculated with respect to all pruned layers.}
\label{tab:overall-sparsities}
\end{minipage}
\end{table}
As can be seen in Table \ref{tab:overall-sparsities}, SPDY profiles generally achieve the same speedup with a lower overall sparsity. Similarly, the relative difference in the number of remaining weights tends to increase for higher speedups. This is expected as our method directly takes execution information into account and can thus e.g. prioritize pruning layers that provide good speedups while leaving more weights on other (perhaps bigger) layers with worse acceleration behavior. However, we note that our techniques are quite general and can be easily adapted (by replacing the timings in the DP problem with the number of remaining parameters) to optimize for overall sparsity.
\section{Profile Visualizations}
\label{app:profiles}
Figure \ref{fig:profile-visualizations} displays visualizations of some of the SPDY profiles in Section \ref{sec:gradual-pruning}. We will now analyze those briefly.
Starting with ResNet50, we can see how the last convolution in a residual-block is typically pruned less than the others, with this effect being most pronounced in the early layers. Further, we can observe how the first conv is pruned more than the second one early on with the roles seemingly switching in the later layers.
Next, for MobileNetV1, we can see that SPDY keeps all but the very last depth-wise convolution dense since those allow almost no acceleration while at the same being very sensitive. For the standard convolutions, SPDY seems to do the most pruning in the middle layers. YOLOv5s is a quite complex model and features also a correspondingly complex profile. We can see that the first conv of the pathway leading to the eventual output maps ``mconv1'' is typically pruned less than the other layers while the convolution following it ``mconv2'' is typically amongst the most strongly pruned ones. Additionally, convolutions not in a residual block ``conv'' are also pruned quite heavily in most cases. At last, one can notice that the 3 output layers are pruned to relatively high levels, verifying that those should not be skipped like for other models. In the BERT profile, the first projection after attention ``attention.output.dense'' is generally not pruned very strongly and the query and value matrices typically hover around the middle in terms of the assigned sparsities. Meanwhile, the fully-connected layers ``intermediate.dense'' and ``output.dense'' are usually amongst the most strongly pruned parts, which is to be expected as those make up a big portion of the overall run-time. All in all, the profiles found by SPDY exhibit various interesting patterns and carefully balance the layer-wise speed-sensitivity trade-offs in ways that seem very difficult to derive manually.
\begin{figure*}
\caption{Sample visualizations of SPDY profiles.}
\label{fig:profile-visualizations}
\end{figure*}
\end{document} | math | 95,018 |
\begin{document}
\begin{abstract}
In this note, we consider the width of a supercritical random graph according to some commonly studied width measures. We give short, direct proofs of results of Lee, Lee and Oum, and of Perarnau and Serra, on the rank- and tree-width of the random graph $G(n,p)$ when $p= \frac{1+\epsilon}{n}$ for $\epsilon > 0$ constant. Our proofs avoid the use, as a black box, of a result of Benjamini, Kozma and Wormald on the expansion properties of the giant component in this regime, and so as a further benefit we obtain explicit bounds on the dependence of these results on $\epsilon$. Finally, we also consider the width of the random graph in the \emph{weakly supercritical regime}, where $\epsilon = o(1)$ and $\epsilon^3n \to \infty$. In this regime, we determine, up to a constant multiplicative factor, the rank- and tree-width of $G(n,p)$ as a function of $n$ and $\epsilon$.
\end{abstract}
\title{A note on the width of sparse random graphs}
\section{Introduction}
\subsection{Background and motivation}
The \emph{tree-width} of a graph $G$, which we denote by $\textrm{tw}(G)$, is a parameter which broadly measures how similar the global structure of $G$ is to that of a tree, via the existence of a \emph{tree-decomposition}, which displays the structure of $G$ in a tree-like fashion. Tree-decompositions were originally studied by Halin \cite{Halin}, and later independently by Robertson and Seymour as an integral tool in their proof of Wagner's conjecture \cite{RobertsonXX}, and have since then been the subject of much study.
Beyond their usefulness as a tool in structural graph theory, tree-decompositions have also turned out to be important in the field of computational complexity. Many problems which are computationally hard on general graphs can be solved in polynomial time on graphs of bounded tree-width (see Bodlaender \cite{Bodlaender}). As the subject developed, many other notions of `width' for graphs, and other combinatorial structures, have been considered, each with their own applications to structural problems, as well as computational problems. One particular example that we will consider in this note is the \emph{rank-width} of a graph $G$, which we denote by $\textrm{rw}(G)$. Roughly, if low tree-width implies that a graph has a `tree-like' decomposition over small vertex cuts, where small here is measured in terms of their cardinality, low rank-width implies that the graph has a `tree-like' decomposition over \emph{simple} edge cuts, where the simplicity of these edge cuts is not measured in terms of just the number of edges crossing them, but rather in terms of an algebraic measure more akin to the complexity of the edge cut. In particular, both very sparse and very dense edge cuts are simple in this way, and so, unlike the case of tree-width, there can be quite dense graphs with low rank-width. In this way, the graphs of low rank-width are a broader class of `low-complexity' graphs than the graphs of low tree-width, and in particular, Oum \cite{Oum} showed that the rank-width of a graph is always at most one more than the tree-width. We will give precise definitions of the relevant terms later in Section \ref{s:Pre}; for more background on tree-width and other width parameters see the survey of Hlin{\v{e}}n{\`y}, Oum, Seese and Gottlob \cite{Hlineny}.
In this paper, we are interested in the width of \emph{random graphs}. The \emph{binomial random graph model} $G(n,p)$, introduced by Gilbert \cite{Gilbert}, is a random variable distributed on the subgraphs of the complete graph $K_n$ where we retain each edge independently with probability $p$. The tree-width of $G(n,p)$ was first considered by Kloks \cite{Kloks}, who showed that if $p = \frac{d}{n}$ for $d \geq 2.36$, then whp\footnote{With probability tending to one as $n \to \infty$.} the tree-width of $G(n,p)$ is at least $r n$ for some $r= r(d)$. Furthermore, he showed that one can take $r(d) \rightarrow 1$ as $d \rightarrow \infty$, which can be seen to be optimal, as the tree-width of a graph on $n$ vertices cannot be larger than $n$. Later, Gao \cite{Gao} improved the lower bound on $d$ to $2.16$, and asked if this could be further improved to $d \geq 1$.
This is a natural question to ask, due to the \emph{phase transition} which $G(n,p)$ undergoes at the critical point $p=\frac{1}{n}$. More explicitly, work of Erd\H{o}s and R\'{e}nyi \cite{E-R} implies that for $p= \frac{d}{n}$ with $d < 1$, whp $G(n,p)$ only has \emph{small} components, of logarithmic order, each of which is a tree or unicyclic. Since all trees and unicyclic graphs have tree-width at most two, it follows that whp in this regime $G(n,p)$ has tree-width at most two as well. However, when $p= \frac{d}{n}$ with $d > 1$, whp $G(n,p)$ contains a unique \emph{giant} component, whose order is linear in $n$ and which is known to have quite a complex structure; see \cite{BollobasBook,Janson,Frieze} for a more detailed introduction to random graphs.
This question of Gao \cite{Gao} was finally answered by Lee, Lee and Oum \cite{LeeLeeOum}, who instead considered the rank-width of the random graph $G(n,p)$ and gave the following, almost complete, description of how this parameter behaves for different ranges of $p$.
\begin{theorem}[{\cite[Theorem 1.1]{LeeLeeOum}}]\label{t:rankwidthrandom}
For a random graph $G:=G(n,p)$, whp the following statements hold:
\begin{enumerate}[(i)]
\item If $p \in (0,1)$ is constant, then $\textrm{rw}(G) = \lceil \frac{n}{3} \rceil - O(1)$;
\item If $\frac{1}{n} \ll p \leq \frac{1}{2}$, then $\textrm{rw}(G) = \lceil \frac{n}{3} \rceil - o(n)$;
\item\label{i:superrank} If $p = \frac{d}{n}$ and $d> 1$, then $\textrm{rw}(G) \geq r n$ for some $r = r(d)$;
\item If $p=\frac{1}{n}$, then whp $\textrm{rw}(G) = O\left(n^{\frac{2}{3}}\right)$;
\item If $p \leq \frac{d}{n}$ and $d <1$, then $\textrm{rw}(G) \leq 2$.
\end{enumerate}
\end{theorem}
Note that, since the rank-width of an $n$-vertex graph can be shown to be at most $\frac{n}{3}$, the first result is tight up to a lower-order additive term for larger $p$. Furthermore, by the aforementioned result of Oum \cite{Oum} that for any graph $G$
\begin{equation}\label{e:twrw}
\textrm{rw}(G) \leq \textrm{tw}(G) +1,
\end{equation}
Theorem \ref{t:rankwidthrandom} \ref{i:superrank} implies that the tree-width of $G\left(n,\frac{d}{n}\right)$ is linear in $n$ for any $d>1$, giving a positive answer to the question of Gao \cite{Gao}.
Later, Perarnau and Serra \cite{Serra} gave a more direct proof of this, and also considered more carefully the tree-width of the critical random graph.
\begin{theorem}[\cite{Serra}]\label{t:treewidthrandom}
For a random graph $G:=G(n,p)$ whp the following hold:
\begin{enumerate}[(i)]
\item\label{i:supertree} If $p = \frac{d}{n}$ and $d> 1$, then $\textrm{tw}(G) \geq r' n$ for some $r' = r'(d)$;
\item\label{i:critical} If $p = \frac{1}{n}$, then $\textrm{tw}(G) = O(1)$.
\end{enumerate}
\end{theorem}
We note that their proof of Theorem \ref{t:treewidthrandom} \ref{i:critical} in fact holds for any $p$ inside the critical window.
From Theorems \ref{t:rankwidthrandom} and \ref{t:treewidthrandom}, together with \eqref{e:twrw}, it follows that both the rank- and tree-width of $G(n,p)$ are bounded above by a constant when $p$ is below or inside the critical window, i.e., when there exists a constant $\lambda > 0$ such that $p \leq \frac{1+ \lambda n^{-\frac{1}{3}}}{n}$, but that both the rank- and tree-width are linearly large in the supercritical regime. This raises the natural question of whether this transition happens \emph{smoothly}.
Both Theorem \ref{t:rankwidthrandom} and Theorem \ref{t:treewidthrandom} rely on a deep result of Benjamini, Kozma and Womald \cite{Bejamini2014} on the \emph{expansion} properties of the giant component of $G(n,p)$ in the supercritical regime. A graph is an expander if it satisfies a type of discrete isoperimetric inequality. Notions of graph expansion have turned out to be of fundamental importance to various topics in combinatorics and computer science. For a comprehensive introduction to expander graphs, see the survey of Hoory, Linial and Widgerson \cite{HLW06}. In particular, graph expansion has been used as a tool in the study of random structures, see, for example, the survey paper of Krivelevich \cite{K19}.
In \cite[Section 4]{LeeLeeOum} it is explained how the results in \cite{Bejamini2014} imply the following theorem. Given a set of vertices $U \subseteq V(H)$ in a graph $H$, let us write $d(U) = \sum_{v \in U} d(v)$. The \emph{Cheeger constant} of a graph $H$ is \[
\Phi (H) := \min_{S \subset V(H)} \frac{e(S,S^c)}{\min \left\{ d(S), d(S^c) \right\} }
\]
where $S^c = V(H) \setminus S$. A graph $H$ with $\Phi(H) \geq \alpha$ is an \emph{$\alpha$-edge-expander}.
\begin{theorem}[\cite{Bejamini2014}]\label{t:expansion}
Let $d >1$ and let $p=\frac{d}{n}$. Then there exist $\alpha,\delta >0$ such that whp $G(n,p)$ contains a connected subgraph $H$ with $\Phi(H) \geq \alpha$ and $|V(H)| \geq \delta n$.
\end{theorem}
Perarnau and Serra \cite{Serra} note that the proof of Theorem \ref{t:rankwidthrandom} \ref{i:superrank} in \cite{LeeLeeOum} shows that
\begin{equation}\label{e:LLO}
r(d) \geq \frac{\alpha \delta}{M^2}
\end{equation}
where $M^2$ is some implicit constant, which can be shown to grow like $\Omega\left(\log \frac{1}{d-1} \right)$ as $d \rightarrow 1$. On the other hand they note that their own proof of Theorem \ref{t:treewidthrandom} \ref{i:supertree} gives
\begin{equation}\label{e:PS}
r'(d) \geq \frac{(\alpha \delta)^2}{9e^3d^2}.
\end{equation}
However, it is not clear how the constants $\alpha$ and $\delta$ in Theorem \ref{t:expansion} behave as a function of $d$ as $d \rightarrow 1$, so it is difficult to compare these two bounds.
An alternative proof of Theorem \ref{t:expansion} can be derived from the work of Krivelevich\footnote{In fact, Krivelevich shows the existence of a linear sized bounded degree vertex-expander, although in this range of $p$ it is relatively easy to move between the two types of expansion.} \cite{Krivelevich}, a careful reading of which leads to the following bounds for the parameters $\alpha$ and $\delta$:
\[
\alpha = \exp \left( - \Omega\left((d-1)^{-2}\right)\right) \qquad \text{ and } \qquad \delta = \left( \frac{(d-1)^3}{\left(\log \frac{1}{d-1}\right)^2}\right).
\]
However, we should note that there does not seem to be any attempt in \cite{Krivelevich} to optimise the arguments with respect to these parameters, and it seems unlikely that the above bounds are optimal. In particular, known results about the structure of the giant component in this regime \cite{DLP14} suggest that the real growth rates should be closer to $\alpha=\Omega\left(d-1\right)$ and $\delta = \Omega\left((d-1)^2\right)$.
\subsection{Main results}
Using a technique of Luczak and McDiarmid \cite{LuczakMcdiarmid}, we are able to give short direct proofs of these results which avoid using the theorem of Benjamini, Kozma and Womald \cite{Bejamini2014}, and so in particular give an explicit lower bound on the rank- and tree-width of $G(n,p)$ for $p=\frac{d}{n}$ for any $d > 1$. Note, in the following theorems, for ease of presentation, we parameterize our results in terms of $\epsilon = d-1$.
\begin{theorem}\label{t:rankwidthdirect}
Let $\epsilon >0$ be sufficiently small and let $p=\frac{1+\epsilon}{n}$. Then whp
$$\textrm{rw}\left(G\left(n,p\right)\right) = \Omega\left( \frac{\epsilon^3}{\left(\log \frac{1}{\epsilon}\right)^3} \right) n.$$
\end{theorem}
From this it is possible to give a corresponding lower bound on the tree-width of $G(n,p)$, however with a more direct argument we can remove some of the polylogarithmic factors from the result.
\begin{theorem}\label{t:treewidthdirect}
Let $\epsilon>0$ be sufficiently small and let $p=\frac{1+\epsilon}{n}$. Then whp
$$\textrm{tw}\left(G\left(n,p\right)\right) = \Omega\left( \frac{\epsilon^3}{\log \frac{1}{\epsilon}} \right) n.$$
\end{theorem}
Since we are interested in the behaviour of these parameters as functions of $\epsilon$ as $\epsilon \to 0$, we have stated the above results only for the \emph{barely supercritical regime}. However, we note that, since tree-width is an increasing graph parameter, Theorem \ref{t:treewidthdirect} clearly implies Theorem \ref{t:treewidthrandom} \ref{i:supertree}. In the case of Theorem \ref{t:rankwidthdirect} it is not immediate, but it is a simple exercise to adapt the proof to cover the range of $p$ in Theorem \ref{t:rankwidthrandom} \ref{i:superrank}. In fact, it can be seen that both of these proofs are even effective all the way to the critical window, holding in the \emph{weakly supercritical regime} where $\epsilon^3 n \to \infty$.
However, neither Theorem \ref{t:rankwidthdirect} nor \ref{t:treewidthdirect} is optimal in terms of their dependence on $\epsilon$ when $\epsilon=o(1)$. Nevertheless, using a different method, also based on the expansion properties of $G(n,p)$, we can remove the extra polylogarithmic terms in Theorems \ref{t:rankwidthdirect} and \ref{t:treewidthdirect}, and obtain an asymptotically optimal bound on the rank- and tree-width of $G(n,p)$ in this regime.
\begin{theorem}\label{t:weaklysup}
Let $\epsilon=\epsilon(n) >0$ be such that $\epsilon=o(1)$ and $\epsilon^3 n \to \infty$, and let $p = \frac{1+\epsilon}{n}$. Then whp
\[
\textrm{rw}\left(G\left(n,p\right)\right) = \Theta\left(\epsilon^3 n \right) \qquad \text{ and } \qquad \textrm{tw}\left(G\left(n,p\right)\right) = \Theta\left(\epsilon^3 n \right) .\]
\end{theorem}
We note that the upper bounds in Theorem \ref{t:weaklysup} follow quite easily from relatively standard facts about $G(n,p)$. Indeed, it is well-known that in this regime of $p$ whp the largest component $L_1$ of $G(n,p)$ has $\Theta(\epsilon n)$ vertices and $\Theta(\epsilon^3 n)$ \emph{excess} edges, and all other components are trees or unicyclic. In particular, the tree-width of all components but the largest is at most one, and so the tree-width of $G(n,p)$ is at most $\max\{ 1, \textrm{tw}(L_1) \}$. The bound then follows from the fact that tree-width of a graph is at most the number of excess edges.
\subsection{Techniques and outline of the paper}
A well-known result in structural graph theory says that the tree-width of a graph $G$ can be bounded from below by the size of the smallest \emph{balanced separation}, a partition of $V(G)$ into three pieces $(A,S,B)$ with $\min |A|,|B| \geq \frac{|V(G)\setminus S|}{3}$ and $e(A,B)=0$, i.e., there are no edges between $A$ and $B$. The \emph{size} of such a separation is the order $|S|$ of the separator. The idea of the proofs of Kloks \cite{Kloks} and of Gao \cite{Gao} is to bound the tree-width by showing the likely non-existence of small separators using a union bound, which turns out to only be effective when $\epsilon = np-1$ is sufficiently large, due to the large number of possible separators.
A useful trick that can reduce the possible number of separators to consider, following a technique of Luczak and McDiarmid \cite{LuczakMcdiarmid}, is to first find a spanning tree of the graph, and since tree-width is decreasing under taking subgraphs, it suffices to first find a large tree in the random graph. Now, classical results on the phase transition already imply that whp a supercritical random graph contains a tree $T$ on $\Omega(\epsilon n)$ vertices, and with a bit of care it is possible to control the maximum degree of this tree (as a function of $\epsilon$).
Restricting our attention to the vertex set $V(T)$, we can show with a sprinkling argument that it is very unlikely that any fixed partition of $V(T)$ forms a balanced separation. Unfortunately, since each subset $S \subseteq V(T)$ could potentially be the separator in many balanced separations, we cannot naively complete the argument with a union bound.
However, since $T$ spans $V(T)$, any subset $S\subseteq V(T)$ splits $T$ into at most $\Delta(T) |S|$ many components, and hence there are at most $2^{\Delta(T) |S|}$ separations whose separator is $S$. This rather simple observation turns out to be powerful enough to use a union bound to show the likely non-existence of small balanced separators.
We note that, in a similar manner, the existence of such a tree reduces the possible number of small balanced edge cuts of $V(T)$, an idea already used by Spencer and T\'{o}th \cite{SpencerToth} in their work on the crossing number of random graphs. This in turn, in a similar manner as for tree-width, can be used to give a lower bound for the rank-width of $G(n,p)[V(T)]$, allowing us to use this technique to also prove Theorem \ref{t:rankwidthdirect}.
For the proof of Theorem \ref{t:weaklysup}, we follow a similar approach to that of Lee, Lee and Oum \cite{LeeLeeOum}, and of Pernarnau and Serra \cite{Serra}, and consider the expansion properties of the giant component of $G(n,p)$. However, unlike in the supercritical regime, it is not the case that the giant component typically contains a large expanding subgraph in this regime of $p$. Instead, using standard estimates on the degree distribution of the kernel of $G(n,p)$, we show that whp the kernel contains as an \emph{induced topological minor} a large subgraph $H$ which is distributed as a random $3$-regular graph, which is known to whp have good expansion properties. This implies that $H$ does not contain any small balanced edge cuts, which, together with the fact that $\Delta(H) =3$, is sufficient to bound the rank-width of $H$ from below. Finally, since rank-width can be shown to be decreasing under taking induced topological minors, we can conclude that the rank-width of $G(n,p)$ is at least the rank-width of $H$.
The paper is structured as follows. In Section \ref{s:Pre} we give preliminary definitions and results. We give our proofs of Theorems \ref{t:rankwidthdirect} and \ref{t:treewidthdirect} in Section \ref{s:direct} and then prove Theorem \ref{t:weaklysup} in Section \ref{s:weakly}. Some final discussion of open and related problems is given in Section \ref{s:discussion}.
\subsection*{Notation}
Unless otherwise specified, all logarithms will be the natural logarithm. Throughout the paper we will suppress floor and ceiling signs for ease of presentation.
\section{Preliminaries}\label{s:Pre}
A \textit{tree-decomposition} of a graph $G$ is a pair $(T, \mathcal{V})$ where $T$ is a tree and $\mathcal{V} = \{ V_x \subseteq V(G) \colon x \in V(T) \}$ is a family of subsets of $V(G)$,
such that the following conditions hold:
\begin{itemize}
\item For all $v\in V(G)$, the set $\left\{x\in V(T): v\in V_x\right\}$ induces a non-empty subtree of $T$;
\item For all $uv\in E(G) $, there is some bag $V_x$ containing both $u$ and $v$.
\end{itemize}
The \textit{width} of $(T, \mathcal{V})$ is $\max \{ |V_x| - 1 \colon x \in V(T) \}$. The \textit{tree-width} $\text{tw}(G)$ of $G$ is the minimum width over all tree-decompositions of $G$.
We say that a set $S\subseteq V(G)$ is a \emph{$(k,\alpha)$-separator} if $|S| = k$ and every component of $G \setminus S$ has size at most $\alpha|V(G) \setminus S|$. The \emph{$\alpha$-separation number} $\text{sep}_{\alpha}(G)$ of $G$ is the smallest size $k$ of a $(k,\alpha)$-separation in $G$. A classic result of Robertson and Seymour \cite{Robertson} bounds the tree-width of a graph from below by the size of the smallest $\left(k,\frac{1}{2}\right)$-separation, although it is phrased in different terms.
\begin{lemma}[See \cite{Harvey}]\label{l:twsep}
For any graph $G$, $\textrm{tw}(G) \geq \text{sep}_{\frac{1}{2}}(G) -1$.
\end{lemma}
For ease of presentation we will want to work with a slightly different notion of a balanced separation. We call a partition $(A,S,B)$ of $V(G)$ a \emph{ $\left(k,\frac{1}{2}\right)$-balanced partition} if $|S| \leq k$, $e(A,B)=0$, and $\frac{1}{3} |V(G) \setminus S| \leq |A|,|B| \leq \frac{2}{3} |V(G) \setminus S|$.
\begin{lemma}\label{l:balancedsep}
Let $G=(V,E)$ be a graph and let $S$ be a $\left(k,\frac{1}{2}\right)$-separator of $G$. Then there exist $A,B \subseteq V$ (not necessarily unique) such that $(A,S,B)$ is a $\left(k,\frac{1}{2}\right)$-balanced partition.
\end{lemma}
\begin{proof}
Let $G'=G[V \setminus S]$ and let $A_1,A_2, \ldots, A_s$ be the vertex sets of the components of $G'$, in non-increasing order of size. Let $j$ be minimal such that $\left| \bigcup_{i=1}^j A_i \right| \geq \frac{1}{3}|G'|$. Since $S$ is a $\left(k,\frac{1}{2}\right)$-separator, $|A_1| \leq \frac{1}{2} |G'| \leq \frac{2}{3} |G'|$, and if $j \neq 1$ then $|A_j| \leq |A_1| \leq \frac{1}{3} |G'|$ and so $\left| \bigcup_{i=1}^j A_i \right| \leq \frac{2}{3} |G'|$. In either case it follows that $A = \bigcup_{i=1}^j A_i$ and $B = \bigcup_{i=j+1}^s A_i$ is the desired partition.
\end{proof}
Given a graph $G$ and two subsets $V_1,V_2 \subseteq V(G)$, we let $N_{V_1,V_2}$ be the adjacency matrix whose rows are labelled by $V_1$ and whose columns are labelled by $V_2$, so that the entry $\left(N_{V_1,V_2}\right)_{v_1,v_2} = 1$ if and only if $v \in V_1$ and $v_2 \in V_2$ are adjacent, and otherwise it is $0$. The \emph{cutrank} of $V_1$ and $V_2$, which we denote by $\rho_G(V_1,V_2)$, is the rank of $N_{V_1,V_2}$ over $\mathbb{F}_2$, which we write as $\textrm{rank}(N_{V_1,V_2})$. A tree is \emph{subcubic} if every vertex has degree one or three. A \emph{rank-decomposition} of a graph $G$ is a pair $(T,f)$ where $T$ is a subcubic tree and $f$ is a bijection from $V(G)$ to the set of leaves of $T$.
Deleting an edge $e=uv$ of $T$ splits $T$ into two components $C_u$ and $C_v$ containing $u$ and $v$, respectively. If we let $A_{uv} = f^{-1}(C_u)$ and $B_{uv} = f^{-1}(C_v)$, then we can define the \emph{rank-width} of $G$ to be
\[
\textrm{rw}(G) = \min_{(T,f)} \max_{uv \in E(T)} \rho_G(A_{uv},B_{uv}),
\]
where we take the minimum over all rank-decompositions $(T,f)$ of $G$.
We will use the following lemmas of Lee, Lee and Oum \cite{LeeLeeOum}. The first is a relatively standard statement bounding a width parameter from below by the non-existence of a balanced separation of low order.
\begin{lemma}[{\cite[Lemma 2.1]{LeeLeeOum}}]\label{l:smallcut}
Let $G$ be a graph with at least two vertices. If the rank-width of $G$ is at most $k$, then there exists a partition $V(G) = A \cup B$ such that $|A|,|B| \geq \frac{|G|}{3}$ and $\rho_G(V_1,V_2) \leq k$.
\end{lemma}
The second is a useful technical lemma which bounds the rank of a matrix in terms of the size of its support.
\begin{lemma}[{\cite[Lemma 4.3]{LeeLeeOum}}]\label{l:rank}
Let $A$ be a matrix over $\mathbb{F}_2$ with at least $n$ non-zero entries. If each row and column contains at most $M$ non-zero entries then $\textrm{rank}(A) \geq \frac{n}{M^2}$.
\end{lemma}
The \emph{bisection width} $b(G)$ of a graph $G$ is the minimum of $e(A,B)$ over all partitions $(A,B)$ of $V(G)$ such that $\frac{1}{3}|G|\leq |A|, |B|\leq \frac{2}{3}|G|$. We note that, Lemmas \ref{l:smallcut} and \ref{l:rank} allow us to bound the rank-width of a graph in terms of its bisection width and maximum degree.
Also, the following bound relating the rank- and tree-width of a graph will be useful.
\begin{theorem}[{\cite[Theorem 3]{Oum}}]\label{t:rwtw}
For every graph $G$
\[
\textrm{rw}(G) \leq \textrm{tw}(G) +1.
\]
\end{theorem}
If $G$ is a graph and $v \in V(G)$, then the \emph{local complementation of $G$ at $v$}, denoted by $G*v$, is the graph whose vertex set is $V(G)$ and whose edge set is the same as $E(G)$, except adjacency and non-adjacency are reversed in $N(v)$, see Figure \ref{f:complementation}. Two graphs are \emph{locally equivalent} if one can be obtained from the other by a sequence of local complementations. A graph $H$ is a \emph{vertex-minor} of $G$ if $H$ is an induced subgraph of a graph which is locally equivalent to $G$.
\begin{figure}
\caption{The local complementation $G*v$ of a graph $G$ at a vertex $v$.}
\label{f:complementation}
\end{figure}
It is shown in Oum \cite{Oum2005} that two locally equivalent graphs have the same rank-width, and so it is easy to see that
\begin{equation}\label{e:vtxminor}
\text{if $H$ is a vertex-minor of $G$ then $\textrm{rw}(H) \leq \textrm{rw}(G)$,}
\end{equation}
since rank-width is non-increasing when taking induced subgraphs. A simple consequence of this fact and the following lemma is that rank-width is also non-increasing when taking induced topological minors.
\begin{lemma}\label{l:indtopminorvtxminor}
If $H$ is an induced topological minor of $G$ then $H$ is a vertex-minor of $G$.
\end{lemma}
\begin{proof}
It is clearly sufficient to prove the lemma in the case where $G$ is obtained from $H$ by subdividing a single edge $e = uv$ by a new vertex $x$. However, since $N_G(x) = \{u,v\}$ and $uv \not\in E(G)$, it follows that $E(G*x) = E(G) \cup \{uv\}$ and so $H = G*x[V(H)]$. In particular, $H$ is a vertex-minor of $G$.
\end{proof}
We will use the following lemma of Krivelevich \cite{Krivelevich} on high degree vertices in $G(n,p)$.
\begin{lemma}[{\cite[Proposition 2]{Krivelevich}}]\label{l:highdegree}
Let $d \geq 0$, let $p = \frac{d}{n}$ and let $\delta >0$ be sufficiently small. Then whp every set of $\frac{\delta}{\log \frac{1}{\delta}} n$ vertices in $G(n,p)$ touches at most $\delta n$ edges.
\end{lemma}
We will need the following simple bound on the expectation of a truncated binomial distribution.
\begin{lemma}[{\cite[Lemma 2.5]{Erde}}]\label{l:restricted}
Let $X \sim \text{Bin}(n,p)$ be a binomial random variable with $2enp < K$ for some constant $K>0$. If $Y = \min\{ X , K\}$, then
\[
\mathbb{E}(Y) \geq np - K2^{-K}.
\]
\end{lemma}
We will use the following Chernoff-type bound on the tail probabilities of the binomial distribution, see e.g., \cite[Appendix A]{Alon}.
\begin{lemma}\label{l:chernoff}
Let $n \in \mathbb{N}$, let $p \in [0,1]$, and let $X \sim \text{Bin}(n,p)$. Then for every positive $a$ with $a \leq \frac{np}{2}$,
\[
\mathbb{P}\left(\left|X -np \right| > a\right) < 2 \exp\left(-\frac{a^2}{4np} \right).
\]
\end{lemma}
We will also need the following generalised Chernoff-type bound, due to Hoeffding.
\begin{lemma}[\cite{H63}]\label{l:hoeffding}
Let $K>0$ be a constant and let $X_1,\ldots, X_n$ be independent random variables such that $0\leq X_i \leq K$ for each $1\leq i \leq n$. If $X = \sum_{i=1}^n X_i$ and $t\geq 0$ then
\[
\mathbb{P}\left(|X-\mathbb{E}(X)| \geq t\right) \leq 2 \exp\left(-\frac{t^2}{nK^2}\right).
\]
\end{lemma}
\section{General bounds: Proofs of Theorems \ref{t:rankwidthdirect} and \ref{t:treewidthdirect}}\label{s:direct}
We start with the proof of Theorem \ref{t:rankwidthdirect}, which is slightly simpler. We note that the proof essentially has two ingredients: The first is using the technique of Luczak and McDiarmid \cite{LuczakMcdiarmid} to bound from below the bisection width of a linear sized subgraph of $G(n,p)$, which already appears in the paper of Spencer and T\'{o}th \cite{SpencerToth}. The second is to use Lemmas \ref{l:smallcut} and \ref{l:rank} to turn this into a bound on the rank-width of $G(n,p)$. However, in order to do so we first need to delete a small number of edges to reduce the maximum degree. This part is similar to the argument of Lee, Lee and Oum \cite{LeeLeeOum}, who instead use the result of Benjamini, Kozma and Wormald \cite{Bejamini2014} to bound from below the bisection width of an appropriate subgraph.
\begin{proof}[Proof of Theorem \ref{t:rankwidthdirect}]
We argue via a sprinkling argument, generating two random graphs $G(n,p_1)$ and $G(n,p_2)$ independently such that $(1-p_1)(1-p_2) = 1-p$ so that their union $G(n,p_1) \cup G(n,p_2)$ has the same distribution as $G(n,p)$. Explicitly we take $p_1 = \frac{1+\frac{\epsilon}{2}}{n}$ and $p_2 = \frac{p-p_1}{1-p_1} \geq \frac{\epsilon}{2n}$. Let us fix $c = \frac{\alpha}{\log \frac{1}{\epsilon}}$for some sufficiently small $\alpha>0$ that we will choose later.
By standard results, see for example \cite[Lemma 5.4]{Janson}, whp the largest component in $G(n,p_1)$ has size at least $\frac{\epsilon}{2} n$ and so whp $G(n,p_1)$ contains a tree $T$ of size $m = \frac{\epsilon}{2} n$. We claim there are at most $\binom{m}{i}2^i$ many partitions $(A,B)$ of $V(T)$ with exactly $i$ \emph{crossing edges}, i.e., edges in $T$ between $A$ and $B$. Indeed, any partition $(A,B)$ of $V(T)$ induces an orientation of the crossing edges in $T$, by orienting them from the vertex in $A$ to the vertex in $B$, and this orientation determines the partition of $V(T)$. Since each of the $\binom{m-1}{i} \leq \binom{m}{i}$ subsets of $E(T)$ of size $i$ admits at most $2^i$ orientations, the bound follows.
We will also use the following bound on the sum of binomial coefficients, which says for all $m$ and $k \leq \frac{m}{2}$,
\begin{equation}\label{e:entropy}
\sum_{i=1}^k \binom{m}{i} \leq 2^{h\left(\frac{k}{m}\right)m}
\end{equation}
where $h(x) = -x \log_2 x - (1-x) \log_2 (1-x)$ is the binary entropy function. We note that
\begin{equation}\label{e:approx}
h(x) \approx x \log_2 \frac{1}{x}
\end{equation}
for sufficiently small $x$.
It follows that the total number of partitions with at most $c \epsilon^3 n$ crossing edges in $T$ is at most
\begin{align*}
\sum_{i=1}^{c\epsilon^3 n} \binom{m}{i}2^i &\leq 2^{ c\epsilon^3 n}\sum_{i=1}^{c\epsilon^3 n} \binom{m}{i}\\
&\leq 2^{c\epsilon^3 n +h\left(\frac{c\epsilon^3 n}{m}\right) m } \tag{by \eqref{e:entropy}}\\
&\leq 2^{c\epsilon^3 n +h\left(2c\epsilon^2\right) \frac{\epsilon n}{2}} \tag{since $m=\frac{\epsilon n}{2}$}\\
&\leq 2^{c\epsilon^3 n + 3c \epsilon ^3 n \log \left(\frac{1}{\epsilon} \right)} \tag{by \eqref{e:approx}}\\
&\leq 2^{4\alpha \epsilon ^3 n } \tag{since $c = \frac{\alpha}{\log \frac{1}{\epsilon}}$}.
\end{align*}
Now, if we fix a partition $V(T) = V_1 \cup V_2$ with $ |V_1|,|V_2| \geq \frac{m}{3}$, then there are $|V_1|\cdot |V_2| \geq \frac{2m^2}{9} = \frac{\epsilon^2 n^2}{18}$ potential edges between $V_1$ and $V_2$, and we expect at least $\frac{\epsilon^3}{36} n$ of them to appear in $G(n,p_2)$, see Figure \ref{f:crossingedges}. Hence, by Lemma \ref{l:chernoff} the probability that fewer than $c\epsilon^3 n$ of these edges are in $G(n,p_2)$ is at most $ \exp \left( -3 \alpha \epsilon^3n\right)$, for $\alpha$ sufficiently small (independent of $\epsilon$).
\begin{figure}
\caption{Whp for every partition $(V_1,V_2)$ of the vertex set of the tree $T \subseteq G(n,p_1)$, which we draw in red, with few crossing edges in $T$, there are many crossing edges in $G(n,p_2)$, drawn in blue.}
\label{f:crossingedges}
\end{figure}
Hence, by the union bound, the probability that there exists some partition of $V(T)$ with fewer than $c\epsilon^3 n$ crossing edges in $T \cup G(n,p_2) \subseteq G(n,p)$ is at most $\exp \left( - \alpha \epsilon^3n\right) = o(1)$. In particular, it follows that
\begin{equation}\label{e:bisectionwidth}
\text{whp the bisection width of $H = G(n,p)[V(T)]$ satisfies $b(H) \geq c\epsilon^3 n$.}
\end{equation}
We would like to have a bound on the maximum degree of $G(n,p)$ in order to apply Lemma \ref{l:rank}, but the naive bound of $O(\log n)$ will not be sufficient. However, we note that by Lemma \ref{l:highdegree} there are very few edges in $G(n,p)$ incident with vertices of high degree. Indeed, the number of edges incident with the set $X'$ of the $\frac{\delta}{\log \frac{1}{\delta}}n$ vertices of highest degree in $G(n,p)$, where $\delta$ is sufficiently small, is whp at most $\delta n$ by Lemma \ref{l:highdegree}. In particular, at least one vertex in $X$ must have degree at most $2 \log \frac{1}{\delta}$, since otherwise there would be too many edges incident with $X'$. So, taking $\delta = \frac{c \epsilon^3}{2}$, we see that if we let $X \subseteq X'$ be the set of vertices of degree at least
\[
M = 2 \log \frac{2}{c\epsilon^3} \leq 10 \log \frac{1}{\epsilon},
\]
then whp the number of edges incident with $X$ in $G(n,p)$ is at most $\frac{c\epsilon^3}{2} n$.
Then, given any partition $(W_1,W_2)$ of $W = V(H)$ such that $|W_1|,|W_2| \geq \frac{m}{3}$ we have
\[
e(W_1,W_2) \geq b(H) \geq c\epsilon^3 n.
\]
Furthermore, the number of edges incident with $X$ is at most $\frac{c\epsilon^3}{2} n$, and so if we let $W'_i = W_i \setminus X$ for $i=1,2$, then we have $e(W_1',W_2')\geq \frac{c\epsilon^3}{2} n$.
Then, the adjacency matrix $N_{W_1',W_2'}$ between $W_1'$ and $W_2'$ has at least $\frac{c\epsilon^3}{2} n$ non-zero entries and at most $M$ non-zero entries per row, and hence by Lemma \ref{l:rank} it follows that
\[
\rho_H(W_1,W_2) \geq \rho_H(W_1',W_2') \geq \frac{c\epsilon^3}{2M^2} n.
\]
It follows from Lemma \ref{l:smallcut} that $\textrm{rw}(H)\geq \frac{c\epsilon^3}{2M^2} n$ and hence, since $H$ is an induced subgraph of $G(n,p)$, by \eqref{e:vtxminor},
\[
\textrm{rw}(G(n,p))\geq \textrm{rw}(H)\geq \frac{c\epsilon^3}{2M^2} n = \Omega\left(\frac{\epsilon^3}{ \left(\log \frac{1}{\epsilon}\right)^3} n \right).
\]
\end{proof}
In order to argue about the tree-width of $G(n,p)$, it will not be sufficient to just find a large tree in $G(n,p)$ in the first sprinkling step. In order to bound the number of balanced separators efficiently, we will need to control the maximum degree of this tree.
For this reason, we will need the following result, asserting the existence of a large bounded degree tree in a supercritical random graph. The proof is relatively standard, and so we relegate it to Appendix \ref{a:largetree}
\begin{theorem}\label{t:largetree}
Let $\delta >0$ be sufficiently small, let $p = \frac{1 + \delta}{n}$. Then whp there exists a tree $T$ in $G(n,p)$ such that $|T| = \Omega(\delta n)$ and $\Delta(T) = O\left(\log \frac{1}{\delta}\right)$.
\end{theorem}
\begin{proof}[Proof of Theorem \ref{t:treewidthdirect}]
We argue via a sprinkling argument with $p_1 = \frac{1+ \frac{\epsilon}{2}}{n}$ and $p_2 = \frac{p-p_1}{1-p_1} \geq \frac{\epsilon}{2n}$. By Theorem \ref{t:largetree} there exist constants $c_1,c_2 >0$ such that whp there exists a tree $T$ in $G(n,p_1)$ with vertex set $V(T)$ such that $|T| = c_1\epsilon n$ and $\Delta(T) \leq c_2 \log \frac{1}{\epsilon}$. Let us also set $c := \frac{c_3}{\log \frac{1}\epsilon}$, where we will choose $c_3>0$ sufficiently small later
We now sprinkle onto the edges of $V(T)$ with probability $p_2$, and claim that whp there are no $\left( c\epsilon^3 n, \frac{1}{2}\right)$-balanced partitions of $G(n,p)[V(T)]$. Given a set $S$ of $c\epsilon^3 n$ vertices in $T$, there are at most $\Delta(T)c\epsilon^3n$ components of $T \setminus S$ and so at most $2^{ \Delta(T)c\epsilon^3 n}$ partitions $(A,S,B)$ of $V(T)$ giving rise to a separation of $T$ with separator $S$, even without considering which are balanced.
For each $\left(c\epsilon^3 n, \frac{1}{2}\right)$-balanced partition $(A,S,B)$ of $T$, since
\[
\frac{n}{3}\left( c_1 \epsilon- c\epsilon^3 \right) \leq |A|,|B| \leq \frac{2n}{3}\left(c_1 \epsilon - c\epsilon^3 \right) \qquad \text{ and } \qquad |A|+|B| = n(c_1\epsilon - c\epsilon^3),
\]
there are $|A|\cdot |B| \geq\frac{2n^2}{9} \left(c_1 \epsilon - c\epsilon^3\right)^2$ edges between $A$ and $B$, and so the probability that none of these edges are present after sprinkling is at most
\[
(1-p_2)^{\frac{2n^2}{9} \left(c_1 \epsilon - c\epsilon^3\right)^2} \leq \exp \left( -\frac{\epsilon n}{9}\left(c_1 \epsilon - c\epsilon^3\right)^2\right) \leq \exp \left( -\frac{c_1^2 \epsilon^3 n}{36}\right)
\]
since $c\epsilon^3 \leq \frac{c_1 \epsilon}{2}$.
In particular, the expected number of $\left(c\epsilon^3 n, \frac{1}{2}\right)$-balanced partitions of $G(n,p)[V(T)]$ will be at most
\begin{align*}
\binom{n}{c\epsilon^3 n} 2^{ \Delta(T)c\epsilon^3n} \exp \left( -\frac{c_1^2 \epsilon^3 n}{36}\right) &\leq \left( \frac{e}{c\epsilon^3}\right)^{c\epsilon^3n} \exp \left( \Delta(T) c\epsilon^3n - \frac{c_1^2 \epsilon^3 n}{36}\right)\\
&\leq
\exp \left( n\left( c\epsilon^3\left(\log \frac{1}{c\epsilon^3} + 1\right) + c_2 \log \frac{1}{\epsilon} c\epsilon^3 - \frac{c_1^2 \epsilon^3}{36}\right) \right)\\
&= \exp \left( - \Omega\left(\epsilon^3 n\right)\right),
\end{align*}
as long as $c_3$ is sufficiently small.
Since whp there are no $\left(c\epsilon^3n, \frac{1}{2}\right)$-balanced partitions of $G(n,p)[V(T)]$, it follows from Lemma \ref{l:twsep} that whp \[
\textrm{tw}(G(n,p)) \geq \textrm{tw}(G(n,p)[V(T)]) \geq c\epsilon^3n = \Omega\left(\frac{ \epsilon^3 n}{\log \frac{1}{\epsilon}} \right).
\]
\end{proof}
\section{The weakly supercritical regime: Proof of Theorem \ref{t:weaklysup}}\label{s:weakly}
In the weakly subcritical regime, we can in fact show a better bound on the rank- and tree-width of $G(n,p)$ by considering more carefully the expansion properties of the giant component $L_1$. We note however, that a naive analogue of Theorem \ref{t:expansion} will not hold in this regime of $p$, since the giant component is likely too sparse to contain a large expanding \emph{subgraph}. However, it will still whp contain a large expanding substructure whose existence will bound the rank-width of $G(n,p)$ from below. We say that a graph $H$ is an \emph{induced topological minor} of $G$ if there is an induced subgraph $K$ of $G$ which is a subdivision of $H$.
\begin{theorem}\label{t:weakexpander}
Let $\epsilon = \epsilon(n)>0$ be such that $\epsilon^3n \rightarrow \infty$ and $\epsilon = o(1)$, and let $p=\frac{1+\epsilon}{n}$. Then there exist constants $\alpha,\delta >0$ such that whp $G(n,p)$ contains some graph $H$ as an induced topological minor such that
\begin{enumerate}
\item $|V(H)| \geq \delta \epsilon^3 n$;
\item $H$ is a $3$-regular $\alpha$-expander.
\end{enumerate}
\end{theorem}
We note that it is likely that this could also be derived from work of Ding, Kim, Lubetzky and Peres \cite{Ding}, who give a remarkable description of a simple model contiguous to the giant component in the weakly supercritical regime. However, our exposition is relatively straightforward and short, and so we opt for a direct argument.
To prove Theorem \ref{t:weakexpander} we will use some structural properties of the \emph{$2$-core} of $L_1$.
The $2$-core of a graph is the unique maximal subgraph of minimum degree at least two. The \emph{kernel} of a graph is the graph obtained from the $2$-core by deleting all isolated cycles and contracting all \emph{bare paths}, paths in which all internal vertices have degree two, see Figure \ref{f:corekernel}.
\begin{figure}
\caption{A graph $G$, its 2-core $C$, and its kernel $K$.}
\label{f:corekernel}
\end{figure}
It is easy to see that if we condition on the degree sequence of the $2$-core $C$ of $L_1$, then $C$ is uniformly distributed over all simple graphs with this degree sequence. However, the degree distribution of the $2$-core is well understood. The following bounds follow from work of {\L}uczak \cite{Luczakcycle}.
\begin{lemma}\label{l:luczak}
Let $p=\frac{1+\epsilon}{n}$ be such that $\epsilon=o(1)$ and $\epsilon^3 n \to \infty$, and for $i\geq 2$, let $D_i$ be the random variable which counts the number of vertices of degree $i$ in the $2$-core of the largest component $L_1$ of $G(n,p)$. Then whp
\begin{enumerate}[(i)]
\item $D_2= (1+o(1))2\epsilon^2n$;
\item\label{i:degthree} $D_3= (1+o(1))\frac{4}{3}\epsilon^3n$;
\item\label{i:degfour} $\sum_{i\geq 4} i D_i = o(\epsilon^3 n)$;
\item\label{i:lambda} $\lambda = \frac{\sum_i D_i i(i-1)}{2 \sum_i D_i i} = O(1)$.
\end{enumerate}
\end{lemma}
Hence, it follows from Lemma \ref{l:luczak} \ref{i:lambda} and \cite[Theorem 10.3]{Frieze} that any property which holds whp in the \emph{configuration model} for any degree sequence ${\bf d}$ satisfying the conclusions of Lemma \ref{l:luczak}, also holds whp in the $2$-core. For more details on the configuration model, see for example \cite{Frieze}.
Given a degree sequence ${\bf d} \in \mathbb{N}^s$, the configuration model constructs a multigraph $G^*({\bf d})$ as follows: We start with a set of \emph{cells} $\mathcal{W}({\bf d})=\{W_1,\ldots, W_s\}$ where $(|W_1|,|W_2|,\ldots, |W_s|) = {\bf d}$. We call the points in the $W_i$ \emph{half-edges} and we say the \emph{degree} of a cell $W_i$ is $|W_i|$. A \emph{configuration} is a partition $M$ of $W := \bigcup_{i \in [s]} W_i$ into pairs, which we think of as a perfect matching on the set of half-edges. The graph $G^*({\bf d})$ is formed by choosing a configuration $M$ uniformly at random and taking the graph $G(\mathcal{W},M)$ whose vertex set is $[s]$ and the number of edges between $i$ and $j$ is the number of half-edges in $W_i$ which are matched to a half-edge in $W_j$ for each $i \neq j$, and the number of loops at $i$ is half the number of half-edges in $W_i$ which are matched to other half-edges in $W_i$. If ${\bf{d}}=(d,d,\ldots,d) \in \mathbb{N}^m$, then $G^*(\bf{d})$ is a random $d$-regular multigraph on $m$ vertices, which we will denote by $G^*(m,d)$.
The genesis of the following argument already appears in work of {\L}uczak \cite{Luczakcycle}, although he does not make the following claim explicit.
\begin{lemma}\label{l:indtopmin}
Let ${\bf d} \in \mathbb{N}^s$ be a degree sequence such that $\min_i d_i \geq 2$. Then $G^*({\bf d})$ can be coupled with some pair $(m,G^*(m,3))$ such that with probability one
\begin{enumerate}[(i)]
\item $m \geq D_3 - \sum_{i \geq 4} i D_i$;
\item $G^*(m,3)$ is an induced topological minor of $G^*({\bf d})$.
\end{enumerate}
\end{lemma}
\begin{proof}
The idea behind the construction is simple. We expose our multigraph $G^*({\bf d})$ and first delete all the vertices of degree at least four. We then take the $2$-core of this graph, which we can form by recursively deleting leaves, leaving us with an induced subgraph in which all degrees are two or three, which will contain the desired cubic graph as an induced topological minor after contracting all maximal bare paths. It remains to show that the subgraph we obtain in this way is distributed as $G^*(m,3)$ for some appropriate $m$.
In order to show this, we will utilise the \emph{principle of deferred decisions} to partially expose the configuration giving rise to $G^*({\bf d})$, allowing us to assume that the remaining part of the configuration is still uniformly distributed on the unmatched half-edges.
Let $\mathcal{W}({\bf d}) = \{W_i \colon i \in [s]\}$ be the set of cells used to construct $G^*({\bf d})$. Given a partial matching $E$ of the half-edges in $W := \bigcup_{i \in [s]} W_i$ we say that a half-edge $e \in W_i$ is a \emph{leaf with respect to $E$} if it is the only half-edge in $W_i$ which is unmatched in $E$.
We perform the following steps:
\begin{enumerate}[S1)]
\item\label{i:one} We expose the matching edges containing each half-edge which is contained in a cell of degree at least $4$. Let $M_1$ be the set of edges contained in this partial matching.
\item\label{i:two} We initially set $M_2 = M_1$ and then recursively do the following, as long as there is some half-edge $e$ which is a leaf with respect to $M_2$: Let $e$ be a leaf with respect to $M_2$. We expose the matching edge containing $e$ and add this edge to $M_2$.
\end{enumerate}
Let $\mathcal{W}_1 =\{ W_i \setminus \bigcup M_2 \colon i \in [s] \}$ and let $\mathcal{W}_2 = \{W'_1,\ldots, W'_r\}$ be the set of non-empty cells in $\mathcal{W}_1$. We note the following properties of $\mathcal{W}_2$:
\begin{enumerate}[a)]
\item\label{i:degree} Each cell in $\mathcal{W}_2$ has size two or three;
\item\label{i:cubic} The number of cells of degree three in $\mathcal{W}_2$ is at least $D_3 - \sum_{i \geq 4} i D_i$;
\item\label{i:uniform} Conditioned on the outcome $M_2$ of Steps \ref{i:one} and \ref{i:two} the remaining configuration $M' = M \setminus M_2$ is uniformly distributed over all configurations on $\mathcal{W}_2$;
\item\label{i:subgraph} The graph $G(\mathcal{W}_2,M')$ is an induced subgraph of $G(\mathcal{W},M)$.
\end{enumerate}
Indeed, the first is clear by construction. To see that the second holds, we note that, since $M_1$ contains at most $\sum_{i \geq 4} i D_i$ edges and all cells initially have size at least two, the total number of cells of size three which contain a half-edge in $M_1$ plus the total number of leaves with respect to $M_1$ is at most $\sum_{i \geq 4} i D_i$. However in each recursion step in Step \ref{i:two} this property is maintained for $M_2$, and hence the total number of cells of degree three in $\mathcal{W}_2$ is at least $D_3 - \sum_{i \geq 4} i D_i$. The third holds as mentioned by the principle of deferred decisions, and the fourth holds by the construction of the graphs $G(\mathcal{W}_2,M')$ and $G(\mathcal{W},M)$.
\begin{figure}
\caption{The effect of contracting an edge $m$ in a configuration on a set of cells $\mathcal{V}
\label{f:contraction}
\end{figure}
Modelling the contraction of bare paths in the configuration model is a little fiddly. Let $\mathcal{V}$ be an arbitrary collection of cells and let $V_i$ be a cell in $\mathcal{V}$. Let $e$ be an arbitrary half-edge in $V_i$ and let $m=\{e,f\}$ be a matching edge containing $e$, where $f \in V_j$. Let $\mathcal{V}'$ be the set of cells obtained from $\mathcal{V}$ by replacing $V_i$ and $V_j$ with a single cell $V_{ij} = (V_i \cup V_j) \setminus \{e,f\}$, see Figure \ref{f:contraction}. We say $\mathcal{V}'$ is obtained from $\mathcal{V}$ by \emph{contracting} $m$. Then, it is clear that
\begin{align}
&\text{for any configuration $M$ on $\mathcal{V}$ containing $m$, the graph $G(\mathcal{V}', M \setminus \{m\})$ is the minor of $G(\mathcal{V},M)$}\nonumber \\
&\text{obtained by contracting an edge between $i$ and $j$. Furthermore, if the degree of $V_i$ is two,} \nonumber \\
&\text{then $G(\mathcal{V},M)$ is obtained from $G(\mathcal{V}', M \setminus \{m\})$ by subdividing an edge } \footnotemark. \tag{$\dagger$}\label{e:contract}
\end{align}\footnotetext{Or adding an isolated loop, in the case where $m$ is fully contained in $V_i$.}
We then perform the following final step on $\mathcal{W}_2$:
\begin{enumerate}[S3)]
\item\label{i:three} Recursively, as long as there is some cell of size two, let $W$ be a cell of size two and let $e$ be an arbitrary half-edge in $W$. We expose the matching edge $m$ containing $e$ and we contract $m$.
\end{enumerate}
Let the set of contracted edges be $M_3$ and the resulting set of (non-empty) cells be $\mathcal{W}_3$. Since each cell in $\mathcal{W}_2$ has size two or three, each contraction decreases the number of cells of size two by one, but does not change the number of cells of size three. Hence, every cell in $\mathcal{W}_3$ has size three, and $\mathcal{W}_3$ contains at least $D_3 - \sum_{i \geq 4} i D_i$ many cells. Since $\bigcup \mathcal{W}_3 \cup \bigcup M_3 = \bigcup \mathcal{W}_2$, conditioned on the outcome $M_3$ of Step \ref{i:three} the remaining configuration $M'' = M' \setminus M_3$ is uniformly distributed over all configurations on $\mathcal{W}_3$.
Furthermore, by \eqref{e:contract} it follows that $G(\mathcal{W}_2,M')$ is a subdivision of $G(\mathcal{W}_3,M'')$, and hence $G(\mathcal{W}_3,M'')$ is an induced topological minor of $G(\mathcal{W},M)$. However, since $M''$ is uniformly distributed over all configurations on $\mathcal{W}_3$, each cell of $\mathcal{W}_3$ has size three and there are at least $D_3 - \sum_{i \geq 4} i D_i$ many cells in $\mathcal{W}_3$, the theorem follows.
\end{proof}
It is known that $G^*(m,3)$ has good expansion properties. For example, the following theorem is a consequence of work of Bollob\'{a}s \cite{Bollobasisometric}, although its proof is just a, by now relatively standard, first moment calculation.
\begin{theorem}\label{t:regularexpander}
For any $d \geq 3$ there exists an $\alpha:=\alpha(d) >0$ such that whp\footnote{With probability tending to $1$ as $m \to \infty$.} $G^*(m,d)$ is an $\alpha$-expander.
\end{theorem}
We now have the necessary ingredients to complete the proof of Theorem \ref{t:weakexpander}.
\begin{proof}[Proof of Theorem \ref{t:weakexpander}]
Let $C$ be the $2$-core of the largest component of $G(n,p)$, which we note is an induced subgraph of $G(n,p)$. Let us condition on the degree sequence ${\bf d}$ of $C$, which we may assume satisfies the conclusions of Lemma \ref{l:luczak}. In particular, by the comment after Lemma \ref{l:luczak}, any property which holds in $G^*({\bf d})$ whp also holds in $C$ whp. Let us write $D_i$ for the number of entries in ${\bf d}$ which are equal to $i$.
By Lemma \ref{l:indtopmin} there exists some random variable $m$ such that we can couple $G^*(\bf{d})$ with the pair ${(m, G^*(m,3))}$ such that with probability one,
\[
m \geq D_3 - \sum_{i \geq 4}i D_i =\left(\frac{4}{3} +o(1)\right)\epsilon^3 n := \delta \epsilon^3 n
\]
and $G^*(m,3)$ is an induced topological minor of $G^*(d)$. Since, by Theorem \ref{t:regularexpander}, whp $G^*(m,3)$ is an $\alpha$-expander, where $\alpha:=\alpha(3)$ is as in Theorem \ref{t:regularexpander}, it follows that whp $G^*(\bf{d})$ contains an induced topological minor of a $3$-regular graph on at least $\delta\epsilon^3 n$ vertices which is an $\alpha$-expander. It follows that whp the same is true for $C$ and so, since $C$ is an induced subgraph of $G(n,p)$, the conclusion holds.
\end{proof}
Note that our bounds on $\alpha$ and $\delta$ can be made relatively explicit: Lemma \ref{l:luczak} implies that we can take $\delta \approx \frac{4}{3}$, and $\alpha$ can be chosen to be the appropriate $\alpha(3)$ from Theorem \ref{t:regularexpander}, which is shown in \cite{Bollobasisometric} to be at least $\frac{2}{11}$. The current best known bound is $\alpha(3) \geq \frac{1}{4.95}$, which was shown by Kostochka and Melnikov \cite{Kostochka}.
\begin{theorem}\label{t:rw}
Let $\epsilon = \epsilon(n)>0$ be such that $\epsilon^3n \rightarrow \infty$ and $\epsilon = o(1)$, and let $p=\frac{1+\epsilon}{n}$. Then whp \[
\textrm{rw}(G(n,p)) = \Omega(\epsilon^3 n).
\]
\end{theorem}
\begin{proof}
By Theorem \ref{t:weakexpander} there exist constants $\alpha,\delta >0$ such that whp $G(n,p)$ contains an induced topological minor of a graph $H$ with $|V(H)| \geq \delta \epsilon^3 n:=m$ which is a $3$-regular $\alpha$-expander. By Lemma \ref{l:indtopminorvtxminor}, $H$ is a vertex-minor of $G$ and hence $\textrm{rw}(G) \geq \textrm{rw}(H)$.
Let $S$ be an arbitrary subset of $V(H)$ such that $ \frac{m}{3} \leq |S|\leq \frac{m}{2}$. Since $H$ is a 3-regular $\alpha$-expander it follows that
\[
e_H(S,S^c) \geq \alpha d_H(S) \geq \alpha m.
\]
Hence, by Lemma \ref{l:rank}, we can conclude that $\rho_H(S,S^c) \geq \frac{\alpha m }{9}$. Since $S$ was arbitrary, it is easy to conclude from Lemma \ref{l:smallcut} that
\[
\textrm{rw}(G) \geq \textrm{rw}(H) \geq \frac{\alpha m }{9} = \frac{\alpha \delta}{9} \epsilon^3 n.
\]
\end{proof}
\begin{corollary}\label{c:tw}
Let $\epsilon = \epsilon(n)>0$ be such that $\epsilon^3n \rightarrow \infty$ and $\epsilon = o(1)$ and let $p=\frac{1+\epsilon}{n}$. Then whp \[
\textrm{tw}(G(n,p)) = \Omega(\epsilon^3 n).
\]
\end{corollary}
\begin{proof}[Proof of Theorem \ref{t:weaklysup}]
The lower bounds follow immediately from Theorem \ref{t:rw} and Corollary \ref{c:tw}.
For the upper bounds, we note that it is relatively easy to see that the tree-width of a connected graph can be bounded from above by one plus its \emph{excess}, the number of edges minus the number of vertices, see \cite[Proposition 4.3]{Serra}. Indeed, every graph with excess $-1$ is a tree, which has tree-width one, and adding a single edge to a graph can increase the tree-width by at most one.
However, it is known, see for example \cite[Theorem 5.13]{Janson}, that if $\epsilon = \epsilon(n)>0$ is such that $\epsilon^3n \rightarrow \infty$ and $\epsilon = o(1)$, and $p=\frac{1+\epsilon}{n}$, then whp the excess of the largest component $L_1$ of $G(n,p)$ is $\Theta(\epsilon^3 n)$, and every other component has excess at most $0$. In particular, it follows that whp
\begin{align*}
\textrm{tw}(G(n,p)) &= \max\{ \textrm{tw}(K) \colon K \text{ a component of } G(n,p) \}\\
&\leq \max\{ \textrm{excess}(K) \colon K \text{ a component of } G(n,p) \} \\
&\leq \textrm{excess}(L_1) = O(\epsilon^3 n).
\end{align*}
In particular, it follows that whp $\textrm{tw}(G(n,p)) = O\left(\epsilon^3 n\right)$, and so, by Theorem \ref{t:rwtw}, the corresponding upper bound also holds for the rank-width.
\end{proof}
\section{Discussion}\label{s:discussion}
As mentioned in the introduction, it is relatively easy to see that the proof of Theorem \ref{t:rankwidthdirect} also works without changes in the weakly supercritical regime. However, for the proof of Theorem \ref{t:treewidthdirect} we used Theorem \ref{t:largetree}, which was only stated for $\epsilon >0$ constant. Nevertheless, we note that a similar result can be seen to hold in the weakly supercritical regime, although since the bounds in Theorem \ref{t:treewidthdirect} are superseded by the bounds in Theorem \ref{t:weaklysup} in this regime, and, whilst the idea is simple, the technical overhead of implementing it is high, we will just give a short sketch of the idea below. Using such a result it can again be seen that the proof of Theorem \ref{t:treewidthdirect} also works in the weakly supercritical regime.
Indeed, standard results on the distribution of the kernel and $2$-core of the largest component of $G(n,p)$ when $p = \frac{1+\delta}{n}$ with $\delta^3 n \to \infty$ and $\delta=o(1)$ imply that whp there is an almost spanning connected subgraph $\tilde{H}$ of the $2$-core, which has order $\Omega(\delta^2 n)$ and maximum degree three.
Furthermore, if we let $L_1$ be the giant component of $G(n,p)$ and $C$ be the $2$-core of $L_1$, then it follows from work of Ding, Kim, Lubetzky and Peres \cite{Ding} that the graph $R:= L_1 - E(C)$ is contiguous to a graph built by choosing, for each vertex $v \in C$ independently, a tree $T_v$ which is distributed as a Poisson$(1-\delta')$ Galton-Watson tree, where $\delta'$ is the smallest solution to $(1-\delta')e^{-(1-\delta')} = (1+\delta)e^{1+\delta}$, and satisfies $\delta' = 1 - \delta + O(\delta^2)$. Hence, standard bounds on the distribution of Galton-Watson trees imply that with a positive probability the order of $T_v$ is $\Omega\left( \frac{1}{\delta}\right)$, and whp the maximum degree in $T_v$ is $O\left( \frac{1}{\log \frac{1}{\delta}} \right)$, and so whp a positive proportion of the trees $T_v$ with roots in $\tilde{H}$ will satisfy these two properties. Taking the union of these trees together with $\tilde{H}$ itself will lead to the claimed subgraph.
Whilst Theorem \ref{t:weaklysup} is optimal in terms of its dependence on $\epsilon$ and $n$, it seems unlikely that Theorems \ref{t:treewidthdirect} and \ref{t:rankwidthdirect} are. In fact, it seems reasonable to expect that the methods we use to prove Theorem \ref{t:weaklysup} should extend to prove a similar bound in the barely supercritical regime. Indeed, it is likely that the methods of {\L}uczak \cite{Luczak90} can give bounds of a similar nature to Lemma \ref{l:luczak} on the likely degree sequence of the $2$-core of the largest component of $G(n,p)$, in particular that the number of edges incident with vertices of degree four or larger is much smaller than the number of vertices of degree three, which is of order $\Theta(\epsilon^3n)$. It would then follow by the same arguments as in Section \ref{s:weakly} that whp $\textrm{rw}(G(n,p)) = \Omega(\epsilon^3 n)$. However, since the technical details in \cite{Luczak90} are quite involved, we leave this as an open question, and note that perhaps the main interest in Theorems \ref{t:treewidthdirect} and \ref{t:rankwidthdirect} lies in the simplicity and directness of the proofs, rather than their asymptotic dependence on $\epsilon$.
\begin{question}
Let $\epsilon >0$ be a sufficiently small constant and let $p=\frac{1+\epsilon}{n}$. Is there a constant $c>0$ (independent of $n$ and $\epsilon$) such that whp
\[
\textrm{tw}(G(n,p)) \geq c \epsilon^3 n?
\]
\end{question}
We note that, in the barely supercritical regime there is also a natural upper bound on tw$(G(n,p))$ of $O\left(\epsilon^3 n\right)$ given by the likely excess of the giant component, together with the fact that whp all other components are trees or unicyclic.
For large enough $\epsilon$, the fact that rank- and tree-width are vertex-Lipschitz functions imply that these parameters are tightly concentrated about their mean, for example using the Azuma-Hoeffding inequality, see e.g., \cite{Frieze}.
However, for smaller $\epsilon$ is it not clear if these parameters are concentrated in a range of values of size $o\left(\epsilon^3 n\right)$. It would also be interesting to know, if the parameters are tightly concentrated, what the correct leading constant should be.
We note that the upper and lower bounds we get for the rank- and tree-width of $G(n,p)$ in the weakly supercritical regime are not \emph{so} far apart, differing by less than a factor of $100$. Explicitly, using the bounds on $\alpha$ and $\delta$ from Theorem \ref{t:weakexpander}, and the fact that whp the excess of $G(n,p)$ is $(1+o(1)) \frac{2}{3} \epsilon^3 n$ we can deduce that whp
\[
0.02\leq \frac{\textrm{tw}(G)}{\epsilon^3 n} , \frac{\textrm{rw}(G)}{\epsilon^3 n} \leq 1.34.
\]
\begin{question}
Let $\epsilon=\epsilon(n)$ be such that $\epsilon=o(1)$ and $\epsilon^3 n \to \infty$. Is it the case that whp
\[
\textrm{tw}(G(n,p)) = (1+o(1)) c_t \epsilon^3 n \qquad \text{ and }\qquad \textrm{rw}(G(n,p)) = (1+o(1)) c_r \epsilon^3 n
\]
for some constants $c_t, c_r >0$ and can we determine the value of the constants $c_t$ and $c_r$?
\end{question}
Spencer and T\'{o}th \cite{SpencerToth} used the lower bound on the bisection width of a subgraph of $G(n,p)$ from \eqref{e:bisectionwidth} to bound the \emph{crossing number} of $G(n,p)$ in the barely supercritical regime. We say a \emph{drawing} of a graph $G$ is a mapping which assigns to each vertex of $G$ a point in the plane and to each edge a continuous arc, such that no arc passes through a vertex and no three arcs meet at a point. A \emph{crossing point} in a drawing is a point where two arcs meet. Then, the crossing number $\textrm{\footnotesize CR} (G)$ of a graph $G$ is the minimum number of crossing points in a drawing of $G$.
Leighton \cite{L84} used the well-known Lipton-Tarjan planar separator theorem \cite{LT79} to give a connection between the crossing number and the bisection width $b(G)$ of a graph. Explicitly, it was shown by Pach, Shahrokhi and Szegedy \cite[Theorem 2.1]{PSS96} that for every graph $G$
\begin{equation}\label{e:CRB}
b(G) \leq 2 \sqrt{16 \textrm{\footnotesize CR} (G) + \sum_{v\in V(G)} d(v)^2}.
\end{equation}
Using this and \eqref{e:bisectionwidth}, Spencer and T\'{o}th \cite{SpencerToth} showed the following:
\begin{theorem}[\cite{SpencerToth}]
Let $d > 1$ and let $p=\frac{d}{n}$. Then
\[
\mathbb{E}\big(\textrm{\footnotesize CR} (G(n,p))\big) = \Omega\left(n^2\right).
\]
\end{theorem}
We note that, using some of our intermediary results we can extend this into the weakly supercritical regime.
\begin{theorem}
Let $\epsilon:=\epsilon(n) > 0$ be such that $\epsilon^3 n \to \infty$ and $\epsilon = o(1)$, and let $p=\frac{1+\epsilon}{n}$. Then whp
\[
\textrm{\footnotesize CR} (G(n,p)) = \Omega\left(\epsilon^6 n^2\right).
\]
\end{theorem}
\begin{proof}
We first note that the crossing number is an increasing graph property, and furthermore it is clear that if $H$ is a subdivision of $G$, then $\textrm{\footnotesize CR} (H) = \textrm{\footnotesize CR} (G)$.
Hence, by Theorem \ref{t:weakexpander} there exist $\alpha,\delta >0$ such that whp $\textrm{\footnotesize CR} (G(n,p)) \geq \textrm{\footnotesize CR} (H)$ where $H$ is a $3$-regular $\alpha$-expander on $m \geq \delta \epsilon^3 n$ vertices. In particular, it follows from the definition of an $\alpha$-expander that $b(H) \geq \alpha m$.
Therefore, since $H$ is $3$-regular, it follows from \eqref{e:CRB} that
\[
\textrm{\footnotesize CR} (G) \geq \textrm{\footnotesize CR}(H) \geq \frac{ b(H)^2 - 4 \sum_{v\in V(H)} d(v)^2}{16} = \Omega\left(m^2\right) = \Omega\left(\epsilon^6 n^2\right).
\]
\end{proof}
Finally, we note that Pernarnau and Serra \cite{Serra} also bound from below the tree-width of random $d$-regular graphs $G(n,d)$ for sufficiently large $d$.
\begin{theorem}[{\cite[Proposition 4.6]{Serra}}]
There is a constant $d_0$ such that for every $d \geq d_0$ whp the random $d$-regular graph $G(n,d)$ has linear tree-width.
\end{theorem}
Their proof essentially shows that such graphs are whp $\alpha(d)$-expanders for some constant $\alpha(d)$ by using the Cheeger bound, which bounds the expansion of the graph in terms of its second largest eigenvalue. They then show that all $\alpha(d)$-expanders have tree-width linear in their size.
This spectral argument is only effective for sufficiently large $d$, however we note that Bollob\'{a}s \cite{Bollobasisometric} showed (see Theorem \ref{t:regularexpander}), that for each $d \geq 3$ there exists an $\alpha =\alpha(d)$ such that whp $G(n,d)$ is an $\alpha$-expander. The exact same proof as in \cite[Proposition 4.6]{Serra} then shows the following.
\begin{corollary}
For every $d \geq 3$ whp the random $d$-regular graph $G(n,d)$ has linear tree-width.
\end{corollary}
\begin{appendix}
\section{Proof of Theorem \ref{t:largetree}}\label{a:largetree}
Let $G:= G(n,p)$ and let $K:= 4 \log \frac{1}{\delta}$. The basic idea is to analyse a restricted breadth first search process on the graph, where we limit each vertex to have at most $K$ neighbours. In this case, Lemma \ref{l:restricted} will imply that we expect each vertex to have more than one neighbour, and so we should expect this process to grow to a large size. In practice, we have to first build a large tree `by hand', so as to guarantee that the correct growth happens with high probability. This strategy to build a large bounded degree tree already appears in the work of Erde, Kang and Krivelevich \cite{Erde}, where they use similar arguments to build such a tree with large excess in an arbitrary random subgraph.
Throughout the process we will maintain a set of vertices $X$ which will include the vertices of the tree we are building, together with a small set of vertices which we have discarded during an initial step. We will work under the assumption that $|X| \leq \frac{\delta n}{4}$, and later verify that this holds throughout our process. Let us write $U := [n] \setminus X$.
In the initial step, we will build a \emph{partial binary tree} of order $N_1 = 4 \log \log \log n + 1$ via a greedy process. By a partial binary tree we mean a rooted tree, rooted at a vertex $v$ of degree two, in which all other vertices have degree three or one, such that there is some integer $\ell$ such that every leaf is at distance $\ell$ or $\ell-1$ from $v$. Note that, since there are at least $2^{\ell-1}$ leaves in such a tree, and at most $2^{\ell+1}- 1$ vertices, there will be at least $N_2 := \log \log \log n $ leaves in such a partial binary tree of order $N_1$.
To build our partial binary tree of order $N_1$, we will make a series of \emph{attempts}. In the $i$th attempt we start by picking an arbitrary root vertex $v_i \in [n] \setminus X$ of a tree $\hat{T}_i$ and adding it to $X$, and we grow $\hat{T}_i$ by recursively exposing the neighbours in $U$ of some leaf $w$ in $\hat{T}_i$ at a minimal depth. If $w$ has at least two neighbours, we choose two arbitrarily and add them to $\hat{T}_i$ as children of $w$, and to the set $X$. If $w$ has less than two neighbours, the attempt is considered a \emph{failure} and we choose an arbitrary set of size $N_1 - |V(\hat{T}_i)|$ in $U$, add it to $X$ and begin the $(i+1)$th attempt. Otherwise, when $|V(\hat{T}_i)| = N_1$ then we consider the attempt a \emph{success} and begin the $(i+1)$th attempt.
Whenever we expose the neighbours of a vertex $w$ there are $|U| \geq \left(1-\frac{\delta}{4} \right)n$ many possible neighbours and so the probability that we do not find at least two neighbours is at most
\begin{align*}
\mathbb{P}\left( \text{Bin}\left(\left(1-\frac{\delta}{4} \right)n , p \right) < 2 \right) &= (1-p)^{\left(1-\frac{\delta}{4} \right)n} + \left(1-\frac{\delta}{4} \right)np(1-p)^{\left(1-\frac{\delta}{4} \right)n-1}\\
&\leq \left(1 - p + \left(1-\frac{\delta}{4} \right)\left(1+\delta\right)\right)\text{exp}\left(-(1+\delta)\left(1-\frac{\delta}{4} \right) + p\right)\\
&\leq \left(2 + \frac{3 \delta}{4}\right) e^{-1-\frac{\delta}{2}}=: 1-\gamma < 1.
\end{align*}
It follows that the probability that an attempt is successful is at least $\gamma^{N_1}$. In particular, since these probabilities are independent for each attempt, if we make $k = \gamma^{-{N_1}}N_1$ attempts, then whp there will be at least one successful attempt, that is, some $\hat{T}_j$ such that $|V(\hat{T}_j)| = N_1$. Note that, at the end of this initial stage $|X| \leq \gamma^{-{N_1}}N_1^2 = o(n)$.
Let us set $T_0 = \hat{T}_j$ and $S_0$ to be the set of leaves of $T_0$. Note that $|S_0| \geq N_2$. We will build a sequence of trees $T_i$ together with a specified set of leaves $S_i \subseteq V(T_i)$ such that for each $i \geq 0$
\begin{itemize}
\item $T_{i+1} \supseteq T_{i}$;
\item $\Delta(T_i) \leq K$;
\item $|S_{i+1}| \geq \left(1 + \frac{\delta}{2}\right)|S_{i}|$.
\end{itemize}
To do so, let $|S_i| = s_i$ and let us enumerate $S_i = \{ v_1,v_2, \ldots v_{s_i} \}$. We sequentially expose the neighbours of $v_j$ in $U$, the vertices which are neither discarded, nor in the current tree, for each $1 \leq j \leq s_i$. We choose an arbitrary subset $N_j \subseteq N(v_j) \cap U$ of size $\eta_j:= \min \{ |N(v_j) \cap U|, K\}$, add these vertices as children of $v_j$ in the tree $S_{i+1}$, and add them to $X$.
Note that, as long as the tree we are building has not yet grown to size $\frac{\delta n}{8}$, $|X| \leq \frac{\delta n}{4}$ and so the random variables $(\eta_1, \eta_2, \ldots, \eta_{s_i})$ are stochastically dominated by an i.i.d sequence of random variables $(Y_1, Y_2 \ldots, Y_{s_i})$ where each $Y_j \sim Y$ with $Y= \min \left \{ \text{Bin}\left( \left(1-\frac{\delta}{4}\right)n, p\right), K \right\}$.
Then, by Lemma \ref{l:restricted}
\begin{align*}
\mathbb{E}(Y)\geq \left(1-\frac{\delta}{4}\right)np-K2^{-K}=\left(1-\frac{\delta}{4}\right)(1+\delta)-4\delta^4\log \frac{1}{\delta}=1+\frac{3\delta}{4}-\frac{\delta^2}{4}-4\delta^4\log \frac{1}{\delta}\geq 1+\frac{\delta}{2},
\end{align*}
if $\delta$ is sufficiently small. Hence, $\mathbb{E}(|S_{i+1}|) \geq |S_i|\mathbb{E}(Y) \geq \left(1+\frac{\delta}{2}\right)|S_i|$ and so by Lemma \ref{l:hoeffding}
\[
\mathbb{P}\left( |S_{i+1}| \leq \left(1+\frac{\delta}{4}\right)|S_i| \right) \leq 2\exp\left(- \frac{ \delta^2|S_i|}{16K^2} \right).
\]
Therefore, the probability that, whilst $|X| \leq \frac{\delta}{4}n,$ there is some $i$ such that $|S_{i+1}| \leq \left(1+\frac{\delta}{4}\right)|S_i|$ is at most
\[
\sum_{i \colon |T_i| \leq \frac{\delta}{8}n} 2\exp\left(- \frac{ \delta^2|S_i|}{16K^2} \right) \leq \sum_{t=N_2}^{\infty}2\exp\left(- \frac{ \delta^2t}{16K^2} \right) = o(1).
\]
In particular, whp there is some $j$ such that $|T_j| \geq \frac{\delta}{8}$, and this tree satisfies the conclusion of the lemma.
\end{appendix}
\end{document} | math | 67,560 |
\begin{equation}gin{document}
\numberwithin{equation}{section}
\title[Root-counting measures and quadratic differentials]
{Root-counting measures of Jacobi polynomials \\ and topological types and critical
geodesics\\
of related quadratic differentials}
\author[B.~Shapiro]{Boris Shapiro}
\address{Department of Mathematics, Stockholm University, SE-106 91
Stockholm,
Sweden}
\varepsilonmail{[email protected]}
\author[A.~Solynin]{Alexander Solynin}
\address{Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, TX 79409, USA}
\varepsilonmail{[email protected]}
\date{\today}
\keywords{Jacobi polynomials, asymptotic root-counting measure,
quadratic differentials, critical trajectories}
\subjclass[2010]{30C15, 31A35, 34E05}
\begin{equation}gin{abstract}
Two main topics of this paper are asymptotic distributions of zeros of
Jacobi polynomials and topology of critical trajectories of
related quadratic differentials. First, we will discuss recent
developments and some new results concerning the limit of the
root-counting measures of these polynomials. In particular, we
will show that the support of the limit measure sits on the
critical trajectories of a quadratic differential of the form
$Q(z)\,dz^2=\frac{az^2+bz+c}{(z^2-1)^2}\,dz^2$. Then we will give
a complete classification, in terms of complex parameters $a$,
$b$, and $c$, of possible topological types of
critical geodesics for the quadratic differential of this type.
\varepsilonnd{abstract}
\dedicatory {To Mikael Passare, in memoriam}
\maketitle
\section{Introduction: From Jacobi polynomials to quadratic
differentials} \lambdabel{Section-1}
Two main themes of this work are asymptotic behavior of zeros of
certain polynomials and topological properties of related
quadratic differentials. The study of asymptotic root
distributions of hypergeometric, Jacobi, and Laguerre polynomials
with variable real parameters, which grow linearly with degree,
became a rather hot topic in recent publications, which attracted
attention of many authors \cite{CC}, \cite{DMO}, \cite{DrDu},
\cite{DrJo}, \cite{DuGu}, \cite{KuMF}, \cite{MFMGO}, \cite{MMO},
\cite{QW}.
In this paper, we survey some known results in this area and
present some new results keeping focus on Jacobi polynomials.
Recall that the Jacobi polynomial $P_n^{(\alpha,\begin{equation}ta)}(z)$ of degree $n$
with complex parameters $\alpha,\begin{equation}ta$ is defined by
$$P_n^{(\alpha,\begin{equation}ta)}(z)=2^{-n}\sum_{k=0}^n\binom {n+\alpha}{n-k}\binom{n+\begin{equation}ta}{k}(z-1)^k(z+1)^{n-k},$$
where $\binom{\gamma}{k}=\frac{\gamma(\gamma-1)\dots (\gamma-k+1)}{k!}$ with a non-negative integer $k$
and an arbitrary complex number $\gamma$.
Equivalently, $P_n^{(\alpha,\begin{equation}ta)}(z)$ can be defined by the well-known Rodrigues
formula:
$$
P_n^{(\alpha,\begin{equation}ta)}(z)=\frac{1}{2^nn!}(z-1)^{-\alpha}(z+1)^{-\begin{equation}ta}
\left(\frac{d}{dz}\right)^n[(z-1)^{n+\alpha}(z+1)^{n+\begin{equation}ta}].
$$
The following statement, which can be found, for instance, in
\cite[Proposition 2]{MFMGO}, gives an important characterization
of Jacobi polynomials as solutions of second order differential
equation.
\begin{equation}gin{proposition}\lambdabel{Proposition-1}
For arbitrary fixed complex numbers $\alpha$ and $\begin{equation}ta$, the
differential equation
$$
(1-z^2)y^{\prime\prime}+(\begin{equation}ta -\alpha-(\alpha+\begin{equation}ta+2)z)y'+\lambda y=0
$$
with a spectral parameter $\lambda$ has a non-trivial polynomial
solution of degree $n$ if and only if $\lambda=n(n+\alpha+\begin{equation}ta+1)$. This
polynomial solution is unique (up to a constant factor) and
coincides with $P_n^{(\alpha,\begin{equation}ta)}(z)$.
\varepsilonnd{proposition}
Working with root distributions of polynomials, it is convenient
to use root-counting measures and their Cauchy transforms, which
are defined as follows.
\begin{equation}gin{definition}
For a polynomial $p(z)$ of degree $n$ with
(not necessarily distinct) roots $\xi_1,...,\xi_n$, its {\it
root-counting measure} $\mu_p$ is defined as
$$\mu_p=\frac{1}{n}\sum_{i=1}^n\delta_{\xi_i},$$
where
$\delta_\xi$ is the Dirac measure supported at $\xi$.
\varepsilonnd{definition}
\begin{equation}gin{definition}
Given a finite complex-valued Borel measure $\mu$ compactly supported in $\mathbb C,$
its {\it Cauchy transform} $\mathcal C_\mu$ is defined as
\begin{equation}gin{equation} \lambdabel{1.1}
\mathcal C_\mu(z) =\int_{\mathbb C} \frac{d\mu(\xi)}{z-\xi}.
\varepsilonnd{equation}
and its {logarithmic potential} $u_\mu$ is defined as
$$u_\mu(z) =\int_{\mathbb C} \log|{z-\xi}|{d\mu(\xi)}.$$
\varepsilonnd{definition}
We note that the integral in (\ref{1.1}) converges for all $z$,
for which the Newtonian potential $U_{|\mu|}(z)=\int_{\mathbb C}
\frac{d|\mu|(\xi)}{|\xi-z|}$ of $\mu$ is finite, see e.g.
\cite[Ch. 2]{Ga}.
In case when $\mu=\mu_p$ is the root-counting measure of a polynomial $p(z)$,
we will write $\mathcal C_p$ instead of $\mathcal C_{\mu_p}$. It follows from
Definitions~1 and 2 that the Cauchy transform $\mathcal C_p(z)$ of the
root-counting measure of a monic polynomial $p(z)$ of degree $n$
coincides
with the normalized logarithmic derivative of $p(z)$; i.e.,
\begin{equation}gin{equation}\lambdabel{Cauchy}
\mathcal C_p(z)=\frac{p'(z)}{np(z)}=\int_{\mathbb C} \frac{d\mu_p(\xi)}{z-\xi},
\varepsilonnd{equation}
and its logarithmic potential $u_p(z)$ is given by the formula:
\begin{equation}gin{equation}\lambdabel{log}
u_p(z)= \frac{1}{n}\log |p(z)|=\int_{\mathbb C} \log|{z-\xi}|{d\mu_p(\xi)}.\varepsilonnd{equation}
Let $\{p_n(z)\}$ be a sequence of Jacobi polynomials
$p_n(z)=P_n^{(\alpha_n,\begin{equation}ta_n)}(z)$ and let $\{\mu_n\}$ be the
corresponding sequence of their root-counting
measures. The main question we are going to address in this paper is the following:
\begin{equation}gin{problem}
{ Assuming that the sequence $\{\mu_n\}$ weakly converges to a measure $\mu$ compactly supported in
$\mathbb C$, what can be said about properties of the support of the measure $\mu$
and about its Cauchy transform $\mathcal C_\mu$?}
\varepsilonnd{problem}
Regarding the Cauchy transform $\mathcal C_\mu$, our main result in this
direction is the following theorem.
\begin{equation}gin{theorem}\lambdabel{Theorem-1} Suppose that a sequence $\{p_n(z)\}$ of Jacobi polynomials $p_n(z)=P_n^{(\alpha_n,\begin{equation}ta_n)}(z)$ satisfies conditions:
\noindent
{\rm(a)} the limits $A=\lim_{n\to\infty}\frac{\alpha_n}{n}$ and $B=\lim_{n\to\infty}\frac{\begin{equation}ta_n}{n}$
exist, and $1+A+B\neq 0$;
\noindent
{\rm(b)}
the sequence $\{\mu_n\}$ of the root-counting
measures converges weakly to a probability measure
$\mu$, which is compactly supported in $\mathbb C$.
Then the Cauchy transform $\mathcal C_\mu$ of the limit measure $\mu$
satisfies almost everywhere in $\mathbb C$ the quadratic
equation:
\begin{equation}gin{equation}\lambdabel{1.2}
(1-z^2)\mathcal C^2_\mu-((A+B)z+A-B)\mathcal C_\mu+A+B+1=0.
\varepsilonnd{equation}
\varepsilonnd{theorem}
The proof of Theorem~\ref{Theorem-1} given in Section~2 consists
of several steps. Our arguments in Section~2 are similar to the
arguments used in a number of earlier papers on root asymptotics
of orthogonal polynomials.
Equation (\ref{1.2}) of Theorem~1 implies that the support of the
limit measure $\mu$ has a remarkable structure described by
Theorem~2 below. And this is exactly the point where quadratic
differentials, which are the second main theme of this paper,
enter into the play.
\begin{equation}gin{theorem}\lambdabel{Theorem-2}
In notation of Theorem~\ref{Theorem-1}, the support of $\mu$
consists of finitely many
trajectories of the quadratic differential
$$
Q(z)\,dz^2=-\frac{(A+B+2)^2z^2+2(A^2-B^2)z+(A-B)^2-4(A+B+1)} {(z-1)^2(z+1)^2}\,
dz^2
$$
and their end points.
\varepsilonnd{theorem}
Thus, to understand geometrical structure of the support of $\mu$ we have to
study geometry of critical trajectories, or more generally critical geodesics of the quadratic
differential $Q(z)\,dz^2$ of Theorem~\ref{Theorem-1}.
We will consider a slightly more general family of quadratic differentials
$Q(z;a,b,c)\,dz^2$ depending on three complex parameters
$a,b,c\in \mathbb{C}$, $a\not=0$,
where
\begin{equation}gin{equation} \lambdabel{1.4}
Q(z;a,b,c)\,dz^2=\frac{az^2+bz+c}{(z-1)^2(z+1)^2}\,dz^2.
\varepsilonnd{equation}
It is well-known that quadratic differentials appear in many areas
of mathematics and mathematical physics such as moduli spaces of
curves, univalent functions, asymptotic theory of linear ordinary
differential equations, spectral theory of Schr\"odinger
equations, orthogonal polynomials, etc. Postponing necessary
definitions and basic properties of quadratic differentials till
Section~3, we recall here that any meromorphic quadratic
differential $Q(z)\,dz^2$ defines the so-called {\it $Q$-metric}
and therefore it defines \varepsilonmph{$Q$-geodesics} in appropriate
classes of curves.
Motivated by the fact that the family of quadratic differentials (\ref{1.4})
naturally appears in the study of the root
asymptotics for sequences of Jacobi polynomials and
is one of very few examples allowing detailed and explicit investigation in terms of its
coefficients,
we will consider the following two basic questions:
\begin{equation}gin{enumerate}
\item[1)]
How many simple critical $Q$-geodesics may exist for a quadratic
differential $Q(z)\,dz^2$ of the form (\ref{1.4})?
\item[2)]
For given $a,b,c\in\mathbb{\mathbb{C}}$, $a\not=0$, describe
topology of all simple critical $Q$-geodesics.
\varepsilonnd{enumerate}
A complete description of topological structure of trajectories of
quadratic differentials (\ref{1.4}) which, in particular, answers
questions 1) and 2), is given by lengthy Theorem~5 stated in
Section~9.
The rest of the paper consists of two parts and is structured as
follows. The first part, which is the area of expertise of the
first author, includes Sections~\ref{Section-2},~\ref{sq-roots},
and 5.
Section~\ref{Section-2} contains the proof of Theorem~1 and
related results. The material presented in Section~\ref{sq-roots}
is mostly borrowed from a recent paper \cite{BoSh} of the first
author. It contains some general results connecting signed
measures, whose Cauchy transforms satisfy quadratic equations, and
related quadratic differentials in $\mathbb C$. In particular, these
results imply Theorem~2 as a special case. In Section~5,
we formulate a number of general
conjectures about the type of convergence of root-counting
measures of polynomial solutions of a special class of linear
differential equations with polynomial coefficients, which
includes Riemann's differential equation.
Remaining sections constitute the second part, which is the area
of expertise of the second author. In Section~\ref{Section-4}, we
recall basic information about quadratic differentials, their
critical trajectories and geodesics. This information is needed
for presentation of our results in Sections~6--10. In Section~6,
we describe possible domain configurations for the quadratic
differentials (\ref{1.4}). Then, in Section~7, we describe
possible topological types of the structure of critical
trajectories of quadratic differentials of the form (\ref{1.4}).
Finally in Sections~8--10, we identify sets of parameters
corresponding to each topological type. The latter allows us to
answer some related questions.
We note here that our main proofs presented in Sections~6--10 are
geometrical based on general facts of the theory of quadratic
differentials. Thus, our methods can be easily adapted to study
trajectory structure of many quadratic differentials other then
quadratic differential~(\ref{1.4}).
Section~11 is our Figures Zoo, it contains many figures
illustrating our results presented in Sections 6--10.
\noindent {\it Acknowledgements.} The authors want to acknowledge
the hospitality of the Mittag-Leffler Institute in Spring 2011
where this project was initiated. The first author is also
sincerely grateful to R.~B\o gvad, A.~Kuijlaars,
A.~Mart\'inez-Finkelshtein, and A.~Vasiliev for many useful
discussions.
\section{Proof of Theorem~1}
\lambdabel{Section-2}
To settle Theorem~1 we will need several auxiliary
statements. Lemma~\ref{lm:basic} below can be found as Theorem 7.6 of \cite {Ba} and apparently was originally proven by F.~Riesz.
\begin{equation}gin{lemma}\lambdabel{lm:basic} If a sequence $\{\mu_n\}$ of Borel probability measures in $\mathbb C$ weakly converges to a
probability measure $\mu$ with a compact support, then
the sequence $\{\mathcal C_{\mu_n}(z)\}$ of its Cauchy transforms
converges to $\mathcal C_{\mu}(z)$ in $L^1_{loc}$. Moreover there exists a subsequence of $\{\mathcal C_{\mu_n}(z)\}$ which converges to $\mathcal C_{\mu}(z)$ pointwise almost everywhere.
\varepsilonnd{lemma}
The next result is recently obtained by the first author jointly with R.B\o gvad and D.~Khavinsion, see Theorem~1 of \cite{BRSh} and has an independent interest.
\begin{equation}gin{proposition}\lambdabel{lm:bound} Let $\{p_m\}$ be any sequence of polynomials satisfying the following
conditions:
\noindent 1. $n_m:=\deg p_m \to \infty$ as $m \to \infty$,
\noindent 2. almost all roots of all $p_m$ lie in a bounded convex
open $\Omegaega\subset \mathbb C$ when $n\to \infty$. (More exactly, if
$In_m$ denotes the number of roots of $p_m$ counted with
multiplicities which are located in $\Omegaega,$ then $\lim_{m\to
\infty}\frac{In_m}{n_m}=1$), then for any $\varphirepsilons>0$, $$\lim_{m\to
\infty}\frac{In'_m(\varphirepsilons)}{n_m}=1,$$ where $In'_m(\varphirepsilons)$ is the
number of roots of $p_m'$ counted with multiplicities which are
located inside $\Omegaega(\varphirepsilons)$, the latter set being the
$\varphirepsilons$-neighborhood of $\Omegaega$ in $\mathbb C$.
\varepsilonnd{proposition}
The next statement is a strengthening of Lemma~8 of \cite{BR}
based on Proposition~\ref{lm:bound}.
\begin{equation}gin{lemma} \lambdabel{lm:add2} Let $\{p_m\}$ be any sequence of polynomials satisfying the following
conditions:
\noindent 1. $n_m:=\deg p_m \to \infty$ as $m \to \infty$,
\noindent 2. the sequence $\{\mu_m\}$ {\rm(}resp. $\{\mu'_m\}${\rm)} of the
root-counting measures of $\{p_m\}$ {\rm(}resp. $\{p'_m\}${\rm)} weakly
converges to compactly supported measures $\mu$ {\rm(}resp
$\mu'${\rm)}.
Then $u$ and $u'$ satisfy the inequality $u \ge u'$ with
equality on the unbounded component of $\mathbb C \setminus
supp(\mu)$. Here $u$ {\rm(}resp. $u'${\rm)} is the logarithmic potential of
the limiting measure $\mu$ {\rm(}resp. $\mu'${\rm)}.
\varepsilonnd{lemma}
\begin{equation}gin{proof}
Without loss of generality, we can assume that all $p_{m}$ are
monic. Let $K$ be a compact convex set containing almost all the
zeros of the sequences $\{p_{m}\}$ and $\{p_m'\}$, i.e.,
$\lim_{m\to \infty} \frac{In_m(K)}{n_m}=\lim_{m\to \infty}
\frac{In^\prime_m(K)}{n_m}=1$. By \varepsilonqref{log} we have
\begin{equation}gin{equation*}
u(z) = \lim_{m \to \infty}\frac{1}{n_{m}}\log|p_{m}(z)|
\varepsilonnd{equation*}
and
\begin{equation}gin{equation*}
u'(z) = \lim_{m\to \infty}
\frac{1}{n_{m}-1}\log\left|\frac{p_{m}'(z)}{n_{m}}\right| =
\lim_{m\to
\infty} \frac{1}{n_{m}}\log\left|\frac{p_{m}'(z)}{n_{m}}\right|
\varepsilonnd{equation*}
with convergence in $L^1_{loc}$. Hence by \varepsilonqref{Cauchy},
\begin{equation}gin{equation}\lambdabel{eq2.2}
u'(z) - u(z) = \lim_{m \to \infty}
\frac{1}{n_{m}}\log\left|\frac{p_{m}'(z)}{n_{m}p_{m}(z)}\right| =
\lim_{m\to\infty}\frac{1}{n_{m}}\log\left|\int\frac{d\mu_{m}(\zetata)}{z
-\zetata} \right|.
\varepsilonnd{equation}
Now, if $\varphirphii$ is a positive compactly supported test function, then
\begin{equation}gin{equation}\lambdabel{eq2.3}
\begin{equation}gin{split}
\int\varphirphii(z)(u'(z) - u(z))\,dA(z) &=
\lim_{m\to\infty}\frac{1}{n_{m}}\int\varphirphii(z)\log\left|\int
\frac{d\mu_{m}(\zetata)}{z-\zetata}\right|\,dA(z)\\
&\leq \lim_{m\to \infty}\frac{1}{n_{m}}\int\varphirphii(z) \int
\frac{d\mu_{m}(\zetata)}{|z-\zetata|}\,dA(z)\\
&=\lim_{m\to
\infty}\frac{1}{n_{m}}\iint\frac{\varphirphii(z)\,dA(z)}{|z-\zetata|}
\,d\mu_{m}(\zetata)
\varepsilonnd{split}
\varepsilonnd{equation}
where $dA$ denotes Lebesgue measure in the complex plane.
Since $1/|z|$ is locally integrable, the function
$\int\varphirphii(z)|z-\zetata|^{-1}\,dA(z)$ is continuous, and hence
bounded by a constant $M$ for all $z$ in $K$. Since
asymptotically almost all zeros of $\{p_m\}$ belong to $K$, the
last expression in
\varepsilonqref{eq2.3} tends to $0$ when $m\to \infty$.
This proves that $u' \leq u$.
In the complement of $\operatorname{supp} \mu$, $u$ is harmonic and $u'$ is
subharmonic, hence $u' - u$ is a negative subharmonic function.
Moreover, in the complement of $\operatorname{supp} \mu$, $p_{m}'/(n_{m}p_{m})$
converges
to the Cauchy transform $\mathcal C(z)$ of $\mu$ a.e. in $\mathbb C$.
Since $\mathcal C(z)$ is a nonconstant holomorphic function in the
unbounded component of $\mathbb C\smallsetminus \operatorname{supp} \mu$, it follows
from \varepsilonqref{eq2.2} that $u' - u \varepsilonquiv 0$ there.
\varepsilonnd{proof}
Notice that Lemma~\ref{lm:add2} implies the following interesting
fact.
\begin{equation}gin{corollary}\lambdabel{cor:path-conn}
In notation of Lemma~\ref{lm:add2}, if $\operatorname{supp} \mu$ has Lebesque
area 0 and the complement $\mathbb C\smallsetminus \operatorname{supp} \mu$ is
path-connected, then $\mu=\mu'$. In particular, in this case the
whole sequence $\{\mu'_m\}$ weakly converges to $\mu$.
\varepsilonnd{corollary}
In general, however $\mu\neq \mu'$ as shown by a trivial example
of the sequence $\{z^n-1\}_{n=1}^\infty$. Also even if
$\mu=\lim_{m\to \infty}\mu_n$ exists the limit $\lim_{m\to
\infty} \mu'_n$ does not have to exist for the whole sequence. An
example of this kind is the sequence $\{p_n(z)\}$ where
$p_{2l}(z)=z^{2l}-1$ and $p_{2l+1}(z)=z^{2l+1}-z$, $l=1,2,\dots$.
Luckily, the latter phenomenon can never occur for
sequences of Jacobi polynomials, see Proposition~\ref{lm:higher} below.
(Apparently it can not occur for a much more general class of
polynomial sequences introduced in \S~\ref{Riemann}.)
\begin{equation}gin{lemma}\lambdabel{lm:basic2}
If the sequence $\{\mu_n\}$ of the root-counting measures of a
sequence of Jacobi polynomials
$\{p_n(z)\}=\{P_n^{(\alpha_n,\begin{equation}ta_n)}(z)\}$ weakly converges to a
measure $\mu$ compactly supported in $\mathbb C,$ and the sequence $\{\mu'_n\}$ of the root-counting measures of a
sequence
$\{p'_n(z)\}$ weakly converges to a
measure $\mu'$ compactly supported in $\mathbb C,$
then one of the
following alternatives holds:
\noindent {\rm (i)} the sequences
$\left\{\frac{\alpha_n+\begin{equation}ta_n}{n}\right\}$ and
$\left\{\frac{\begin{equation}ta_n-\alpha_n}{n}\right\}$ {\rm (}and, therefore, the
sequences $\left\{\frac{\alpha_n}{n}\right\}$ and
$\left\{\frac{\begin{equation}ta_n}{n}\right\}${\rm )} are bounded;
\noindent {\rm (ii)} the sequence
$\left\{\frac{\alpha_n+\begin{equation}ta_n}{n}\right\}$ is unbounded and the
sequence $\left\{\frac{\begin{equation}ta_n-\alpha_n}{n}\right\}$ is bounded, in
which case $\{\mu_n\}\to \delta_0$ where $\delta_0$ is the unit
point mass at $z=0$ (or, equivalently, $\mathcal C_{\delta_0}(z)=1/z$);
\noindent {\rm (iii)} both sets
$\left\{\frac{\alpha_n+\begin{equation}ta_n}{n}\right\}$ and
$\left\{\frac{\begin{equation}ta_n-\alpha_n}{n}\right\}$ are unbounded, in which
case, there exists at least one $\kappa \in \mathbb C$ and a
subsequence $\{n_m\}$ such that $\lim_{m\to
\infty}\frac{\begin{equation}ta_{n_m}-\alpha_{n_m}}{\alpha_{n_m}+\begin{equation}taa_{n_m}}=\kappa$
and $\{\mu_{n_m}\}\to \delta_\kappa, $ where $\delta_\kappa$ is
the unit point mass at $z=\kappa $ (or, equivalently,
$\mathcal C_{\delta_\kappa}(z)=1/(z-\kappa)$).
\varepsilonnd{lemma}
\begin{equation}gin{proof} Indeed, assume that the alternative (i) does not hold.
Then there is a subsequence $\{n_m\}$ such that at least one of
$\left|\frac{\alpha_{n_m}+\begin{equation}ta_{n_m}}{n_m}\right|, \; \left|\frac{\begin{equation}ta_{n_m}-\alpha_{n_m}}{n_m}\right|$
is unbounded along this subsequence.
By our assumptions $\mu_n\to \mu$ and $\mu_n'\to \mu'$ weakly. Hence, by
Lemma~\ref{lm:basic}, there exists a subsequence of indices along which $\mathcal C_{\mu_n}:=\frac{p_n^\prime}{np_n}$
pointwise converges
to $\mathcal C_\mu$ and $\mathcal C_{\mu'_n}:=\frac{p_n^{\prime\prime}}{(n-1)p'_n}$
pointwise converges
to $\mathcal C_{\mu'}$ a.e. in $\mathbb C$.
Consider the sequence of differential equations satisfied by
$\{p_n\}$ and divided termwise by $n(n-1)p_n$:
\begin{equation}gin{equation}\lambdabel{eq:temp}
\begin{equation}gin{split}
(1-z^2)\frac{p_n^{\prime\prime}}{(n-1)p^\prime_n} \cdot
\frac{p_n^\prime}{np_n}&+ \left(
\frac{(\begin{equation}ta_n-\alpha_n)-(\alpha_n+\begin{equation}ta_n+2)z}{n-1}\right)
\frac{p_n^\prime}{np_n} \\
{ }&+\frac{n+\alpha_n+\begin{equation}ta_n+1}{n-1}=0.
\varepsilonnd{split}
\varepsilonnd{equation}
If for a subsequence of indices, $\left|\frac{\begin{equation}ta_n-\alpha_n}{n}\right|\to \infty$ while
$\left|\frac{\alpha_n+\begin{equation}ta_n}{n}\right|$ stays bounded, then the
Cauchy transform $\mathcal C_\mu$ of the limiting (along this subsequence) measure $\mu$ must
vanish identically in order for \varepsilonqref{eq:temp} to hold in the
limit $n\to \infty$. But $\mathcal CC_\mu\varepsilonquiv 0$ is obviously
impossible.
On the other hand, if for a subsequence of indices,
$\left|\frac{\alpha_n+\begin{equation}ta_n}{n}\right|\to \infty$ while
$\left|\frac{\begin{equation}ta_n-\alpha_n}{n}\right|$ stays bounded, then the
limit of \varepsilonqref{eq:temp} when $n\to \infty$ coincides with
$-z\mathcal C_\mu+1=0 \Leftrightarrow \mathcal C_\mu=\frac{1}{z}$ implying
$\mu=\delta_0$. Thus in Case (ii), the sequence $\{\mu_n\}$
converges to $\delta_0$.
Now assume, that or a subsequence of indices, both $\left|\frac{\alpha_n+\begin{equation}ta_n}{n}\right|$ and
$\left|\frac{\begin{equation}ta_n-\alpha_n}{n}\right|$ tend to $\infty$. Then
dividing \varepsilonqref{eq:temp} by $\frac{\alpha_n+\begin{equation}ta_n}{n}$ and letting
$n\to \infty,$ we conclude that the sequence
$\left\{\frac{\begin{equation}ta_n-\alpha_n}{\alpha_n+\begin{equation}ta_n}\right\}$ must be
bounded. Therefore there exists its subsequence which converges to
some $\kappa\in \mathbb C$. Taking the limit along this subsequence, we
obtain
$$(z-\kappa)\mathcal C_\mu=1.$$
This is true for all $z,$ for which the Cauchy transform
converges, i.e. almost everywhere outside the support of $\mu$.
Using the main results of \cite{BBB, BB} claiming that the
support of $\mu$ consists of piecewise smooth compact curves
and/or isolated points together with the fact that $\mathcal C_\mu$ must
have a discontinuity along every curve in its support, we conclude
that the support of $\mu$ is the point $z=\kappa$. Thus in Case
(iii), the sequence $\{\mu_{n_m}\}$ converges to $\delta_\kappa$.
\varepsilonnd{proof}
The next statement provides more information about Case (i) of
Lemma~\ref{lm:basic2}.
\begin{equation}gin{proposition}\lambdabel{lm:higher} Assume that the sequence $\{\mu_n\}$ of the root-counting measures for a sequence of Jacobi polynomials $\{p_n(z)=P_n^{(\alpha_n,\begin{equation}ta_n)}(z)\}$ weakly converges to a compactly supported measure $\mu$ in $\mathbb C.$ Assume additionally that
$\lim_{n\to\infty}\frac{\alpha_n}{n}=A$ and $\lim_{n\to \infty} \frac {\begin{equation}ta_n}{n}=B$ with $1+A+B\neq 0$. Then, for any positive integer $j,$ the sequence $\{\mu_n^{(j)}\}$ of the root-counting measures for the sequence $\{p^{(j)}_n(z)\}$ of the $j$-th derivatives converges to the same measure $\mu$.
\varepsilonnd{proposition}
\begin{equation}gin{proof} Observe that if an arbitrary polynomial sequence $\{p_m\}$ of
increasing degrees has almost all roots in a convex bounded set
$\Omegaega\subset \mathbb C$, then, by Proposition~\ref{lm:bound}, almost all
roots of $\{p'_m\}$ are in $\Omegaega_\varphirepsilons$, for any $\varphirepsilons>0$.
Therefore, if the sequence $\{\mu_m\}$ of the root-counting
measures of $\{p_m\}$ weakly converges to a compactly supported
measure $\mu,$ then there exists at least one weakly converging
subsequence of $\{\mu'_m\}$. Additionally, by the Gauss-Lucas Theorem,
the support of its limiting measure belongs to the (closure of
the) convex hull of the support of $\mu$. Thus the weak
convergence of $\{\mu_m\}$ implies the existence of a weakly
converging subsequence $\{\mu^\prime_{n_m}\}$.
Proposition~\ref{lm:higher} is obvious in Cases (ii) and (iii) of Lemma~\ref{lm:basic2}. Let us concentrate on the remaning Case (i). Our assumptions imply that along a subsequence of the sequence $\left\{\frac{p'_n}{np_n}\right\}$ of Cauchy transforms
of polynomials $p_n$
converges
pointwise almost
everywhere. We first show that the above sequence
$\left\{\frac{p'_n}{np_n}\right\}$ can not converge to $0$ on a
set of positive measure.
Indeed, the differential equation
satisfied by $p_n$ after its division by $n(n-1)p_n$ is given
by \varepsilonqref{eq:temp}.
Since the
sequences $\left\{\frac{\alpha_n+\begin{equation}ta_n}{n}\right\}$ and
$\left\{\frac{\begin{equation}ta_n-\alpha_n}{n}\right\}$ converge and $1+A+B\neq 0$, equation \varepsilonqref{eq:temp} shows that $\frac{p'_n} {np_n}$
cannot converge to $0$
on a set of positive measure.
Analogously, we see that
$\frac{p^{\prime\prime}_n}{(n-1)p'_n}$ cannot converge to 0 on a
set of positive measure either. Indeed, differentiating \varepsilonqref{eq:temp}, we get that $p_n'$ satisfies the equation
$$(1-z^2)p_n^{\prime\prime\prime}+((\begin{equation}ta_n-\alpha_n)-(\alpha_n+\begin{equation}ta_n+4)z)p_n^{\prime\prime}+(n(n+\alpha_n+\begin{equation}ta_n+1)+(\alpha_n+\begin{equation}ta_n+2))p_n' =0.$$
Using the same analysis as for $p_n$, we can conclude
that the limit $\frac{p^{\prime\prime}_n}{n(n-1)p_n}$ along a subsequence exists
pointwise and is non-vanishing almost everywhere.
Denote the logarithmic potentials of the root-counting measures
associated to $p_n$ and $p'_n$ by $u_n$ and $u'_n$ respectively. Denote
their limits
by $u$ and $u'$ (where $u'$ apriori is a limit only along some subsequence). With a slight abuse of notation, the following holds
$$|u - u'| =\lim_{n\to\infty} |u_n - u'_n | = \lim_{n\to\infty} \frac{1}{n} \log \left | \frac{p^{\prime\prime}_n} {n(n - 1)p_n} \right |= 0$$ due to the above claim about $\frac{p^{\prime\prime}_n}{n(n-1)p_n}$ . But since $u \ge u'$ by Lemma~\ref{lm:add2}, we see that $u = u'$ and, in particular $u'$ exists as a limit over the whole sequence. Hence the asymptotic root-counting measures of $\{p_n\}$ and $\{p'_n\}$ actually coincide. Similar arguments apply to higher derivatives of the sequence $\{p_n\}$.
\varepsilonnd{proof}
\begin{equation}gin{proof}[Proof of Theorem~\ref{Theorem-1}] The polynomial $p_n(z)=P_n^{ (\alpha_n,\begin{equation}ta_n) }(z)$ satisfies the equation~\varepsilonqref{eq:temp}.
By Proposition~\ref{lm:higher} we know that, under the assumptions of Theorem~\ref{Theorem-1}, if $ \left\{\frac {p_n'}{np_n}\right\}$ converges to $\mathcal C_\mu$ a.e. in $\mathbb C,$ then the sequence $ \left\{\frac {p_n^{\prime\prime}}{np_n^\prime}\right\}$ also converges to the same $\mathcal C_\mu$ a.e. in $\mathbb C$. Therefore, the expression $\frac{p_n^{\prime\prime}}{n^2p_n}=\frac{p_n^{\prime\prime}p_n^\prime}{n^2p_np_n^\prime}$ converges to $\mathcal C_\mu^2$ a.e. in $\mathbb C$. Thus $\mathcal C_\mu$ (which is well-defined a.e. in $\mathbb C$) should satisfy the equation
$$(1-z^2)\mathcal C_{\mu}^2-((A+B)z+A-B)\mathcal C_\mu+A+B+1=0,$$
where $A=\lim_{n\to \infty}\frac{\alpha_n}{n}$ and $B=\lim_{n\to \infty}\frac{\begin{equation}ta_n}{n}$.
\varepsilonnd{proof}
\begin{equation}gin{remark} Apparently the condition that the sequences $\left\{\frac{\alpha_n}{n}\right\}$ and
$\left\{\frac{\begin{equation}ta_n}{n}\right\}$ are bounded should be enough for the conclusion of Theorem~\ref{Theorem-1}. (The existence of the limits $\lim \frac{\alpha_n}{n}$ and $\lim \frac{\begin{equation}ta_n}{n}$ should follow automatically with some weak additional restriction.) Indeed, since the sequences $\left\{\frac{\alpha_n}{n}\right\}$ and
$\left\{\frac{\begin{equation}ta_n}{n}\right\}$ are bounded, we can find at
least one subsequence $\{n_m\}$ of indices along which both
sequences of quotients converge. Assume that we have two possible
distinct (pairs of) limits $(A_1,B_1)$ and $(A_2,B_2)$ along
different subsequences. But then the same complex-analytic function $\mathcal C_\mu(z)$ should satisfy a.e. two different algebraic equations of the form \varepsilonqref{1.2} which is impossible at least for generic $(A_1,B_1)$ and $(A_2,B_2)$.
\varepsilonnd{remark}
\section{Preliminaries on quadratic differentials} \lambdabel{Section-4}
\setcounter{equation}{0}
In this section, we recall some definitions and results of the
theory of quadratic differentials on the complex sphere
$\overline{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$. Most of these
results remain true for quadratic differentials defined on any
compact Riemann surface. But for the purposes of this paper, we
will focus on results concerning the domain structure and
properties of geodesics of quadratic differentials defined on
$\overline{\mathbb{C}}$. For more information on quadratic
differentials in general, the interested reader may consult
classical monographs of Jenkins \cite{Je} and Strebel \cite{Str}
and papers \cite{S1} and \cite{S2}.
A quadratic differential on a domain $D\subset
\overline{\mathbb{C}}$ is a differential form $Q(z)\,dz^2$ with
meromorphic $Q(z)$ and with conformal transformation rule
\begin{equation}gin{equation} \lambdabel{4.1}
Q_1(\zetata)\,d\zetata^2=Q(\varphirphi(z))\left(\varphirphi'(z)\right)^2\,dz^2,
\varepsilonnd{equation}
where $\zetata=\varphirphi(z)$ is a conformal map from $D$ onto a domain
$G\subset \overline{\mathbb{C}}$. Then zeros and poles of $Q(z)$
are critical points of $Q(z)\,dz^2$, in particular, zeros and
simple poles are finite critical points of $Q(z)\, dz^2$. Below we
will use the following notations. By $H_p$, $C$, and $H$ we
denote, respectively, the set of all poles, set of all finite
critical points, and set of all infinite critical points of
$Q(z)\,dz^2$. Also, we will use the following notations:
$\mathbb{C}'=\overline{\mathbb{C}}\setminus H$,
$\mathbb{C}''=\overline{\mathbb{C}}\setminus H_p$,
$\mathbb{C}'''=\overline{\mathbb{C}}\setminus (C\cup H)$.
A trajectory (respectively, orthogonal trajectory) of $Q(z)\,dz^2$
is a closed analytic Jordan curve or maximal open analytic arc
$\gamma\subset D$ such
that
$$
Q(z)\,dz^2>0 \quad {\mbox{along $\gamma$}} \quad \quad
({\mbox{respectively}}, Q(z)\,dz^2<0 \quad {\mbox{along
$\gamma$}}).
$$
A trajectory $\gamma$ is called \varepsilonmph{critical} if at least one of
its end points is a finite critical point of $Q(z)\,dz^2$. By a
closed critical trajectory we understand a critical trajectory
together with its end points $z_1$ and $z_2$ (not necessarily
distinct), assuming that these end points exist.
Let $\overline{\Phi}$ denote the closure of the set of points of
all critical trajectories of $Q(z)\,dz^2$. Then, by Jenkins' Basic
Structure Theorem \cite[Theorem~3.5]{Je}, the set
$\overline{\mathbb{C}}\setminus \overline{\Phi}$ consists of a
finite number of \varepsilonmph{circle, ring, strip and end domains}. The
collection of all these domains together with so-called
\varepsilonmph{density domains} constitute the so-called \varepsilonmph{domain
configuration} of $Q(z)\,dz^2$. Here, we give definitions of
circle domains and strip domains only; these two types will appear
in our classification of possible domain configurations in
Section~5. Fig. 1--4 show several domain configurations with
circle and strip domains. For the definitions of other domains, we
refer to \cite[Ch.~3]{Je}.
We recall that a \varepsilonmph{circle domain} of $Q(z)\,dz^2$ is a
simply connected domain $D$ with the following properties:
\begin{equation}gin{enumerate}
\item[1)]
$D$ contains exactly one critical point $z_0$, which is a second
order pole,
\item[2)]
the domain $D\setminus \{z_0\}$ is swept out by trajectories of
$Q(z)\,dz^2$ each of which is a Jordan curve separating $z_0$ from
the boundary $\partial D$,
\item[3)]
$\partial D$ contains at least one finite critical
point.
\varepsilonnd{enumerate}
Similarly, a strip domain of $Q(z)\,dz^2$ is a simply connected
domain $D$ with the following properties:
\begin{equation}gin{enumerate}
\item[1)]
$D$ contains no critical points of $Q(z)\,dz^2$,
\item[2)]
$\partial D$ contains exactly two boundary points $z_1$ and $z_2$
belonging to the set $H$ (these boundary points may be situated at
the same point of $\overline{\mathbb{C}}$),
\item[3)]
the points $z_1$ and $z_2$ divide $\partial D$ into two boundary
arcs each of which contains at least one finite critical point,
\item[4)]
$D$ is swept out by trajectories of $Q(z)\,dz^2$ each of which is
a Jordan arc connecting points $z_1$ and $z_2$.
\varepsilonnd{enumerate}
As we mentioned in the Introduction, every quadratic differential
$Q(z)dz^2$ defines the so-called (singular) $Q$-metric with the
differential element $|Q(z)|^{1/2}\,|dz|$. If $\gamma$ is a
rectifiable arc in $D$ then its $Q$-length is
defined by
$$
|\gamma|_Q=\int_\gamma |Q(z)|^{1/2}\,|dz|.
$$
According to their $Q$-lengthes, trajectories of $Q(z)\,dz^2$ can
be of two types. A trajectory $\gamma$ is called \varepsilonmph{finite} if
its $Q$-length is finite, otherwise $\gamma$ is called
\varepsilonmph{infinite}. In particular, a critical trajectory $\gamma$ is
finite if and only if it has two end points each of which is a
finite critical point.
An important property of quadratic differentials is that
transformation rule (\ref{4.1}) respects trajectories and
orthogonal trajectories and their $Q$-lengthes, as well as it
respects critical points together with their multiplicities and
trajectory structure nearby.
\begin{equation}gin{definition} \lambdabel{Definition-4.1}
A locally rectifiable (in the spherical metric) curve
$\gamma\subset \mathbb{C}'$ is called \varepsilonmph{ a $Q$-geodesic} if
it is locally shortest in the $Q$-metric.
\varepsilonnd{definition}
Next, given a quadratic differential $Q(z)\,dz^2$, we will discuss
geodesics in homotopic classes. For any two points $z_1,z_2\in
\mathbb{C}'$, let $\mathcal{H}^J=\mathcal{H}^J(z_1,z_2)$ denote
the set of all homotopic classes $H$ of Jordan arcs $\gamma\subset
\mathbb{C}'$ joining $z_1$ and $z_2$. Here the letter $J$ stands
for "Jordan". It is well-known that there is a countable number of
such homotopic classes. Thus, we may write
$\mathcal{H}^J=\{H_k^J\}_{k=1}^\infty$.
Every class $H_k^J$ can be extended to a larger class $H_k$ by
adding non-Jordan continuous curves $\gamma$ joining $z_1$ and
$z_2$, each of which is homotopic on $\mathbb{C}'$ to some curve
$\gamma_0\in H_k^J$ in the following sense.
There is a continuous function $\varphirphi(t,\tau)$ from the square
$I^2:=[0,1]\times [0,1]$ to $\mathbb{C}'$ such that
\begin{equation}gin{enumerate}
\item[1)]
$\varphirphi(0,\tau)=z_1$, $\varphirphi(1,\tau)=z_2$ for all $0\le \tau\le
1$, \item[2)]
$\gamma_0=\{z=\varphirphi(t,0):\,0\le t\le 1\}$,
\item[3)]
$\gamma=\gamma_1=\{z=\varphirphi(t,1):\,0\le t\le 1\}$,
\item[4)]
For every fixed $\tau, 0<\tau<1$, the curve
$\gamma_\tau=\{z=\varphirphi(t,\tau):\,0\le t\le 1\}$ is in the class
$H_k^J$.
\varepsilonnd{enumerate}
The following proposition is a special case of a well-known result
about geodesics, see e.g. \cite[Theorem~18.2.1]{Str}.
\begin{equation}gin{proposition} \lambdabel{Proposition-4.1}
For every $k$, there is a unique curve $\gamma'\in H_k$, called
\varepsilonmph{$Q$-geodesic} in $H_k$, such that $|\gamma'|_Q< |\gamma|_Q$
for all
$\gamma\in H_k$, $\gamma\not=\gamma'$. This geodesic is not necessarily a Jordan arc.
\varepsilonnd{proposition}
A $Q$-geodesic from $z_1$ to $z_2$ is called \varepsilonmph{simple} if
$z_1\not=z_2$ and $\gamma$ is a Jordan arc on $\mathbb{C}'''$
joining $z_1$ and $z_2$. A $Q$-geodesic is called \varepsilonmph{critical}
if both its end points belong to the set of finite critical points
of $Q(z)\,dz^2$.
\begin{equation}gin{proposition} \lambdabel{Proposition-4.2}
Let $Q(z)\,dz^2$ be a quadratic differential on
$\overline{\mathbb{C}}$. Then for any two points $z_1,z_2\in
\mathbb{C}'$ and every continuous rectifiable curve $\gamma$ on
$\mathbb{C}'''$ joining the points $z_1$ and $z_2$ there is a
unique shortest curve $\gamma_0$ belonging to the homotopic class
of $\gamma$.
Furthermore, $\gamma_0$ is a
geodesic in this class.
\varepsilonnd{proposition}
\begin{equation}gin{definition} \lambdabel{Definition-4.2}
Let $z_0\in \mathbb{C}'$. A geodesic ray from $z_0$ is a maximal
simple rectifiable arc $\gamma:[0,1) \to \mathbb{C}'''\cup\{z_0\}$
with $\gamma(0)=z_0$ such that for every $t$, $0<t<1$, the arc
$\gamma((0,1))$ is a geodesic from $z_0$ to $z=\gamma(t)$.
\varepsilonnd{definition}
\begin{equation}gin{lemma} \lambdabel{Lemma-4.1}
Let $D$ be a circle domain of $Q(z)\,dz^2$ centered at $z_0$ and
let $\gamma_a:[0,1)\to \mathbb{C}'''\cup\{a\}$ be a geodesic ray
from $a\in
\partial D$ such that $\gamma_a([0,t_0])\subset \overline{D}$ for
some $t_0>0$.
Then either $\gamma_a$ enters into $D$ through the point $a$ and
then approaches to $z_0$ staying in $D$ or $\gamma_a$ is an
arc of some critical trajectory $\gamma\subset \partial D$.
\varepsilonnd{lemma}
\begin{equation}gin{lemma} \lambdabel{Lemma-4.2}
Let $a$ be a second order pole of $Q(z)\,dz^2$ and let $\Gamma$ be
the homotopic class of closed curves on $\mathbb{C}''$ separating
$a$ from $H_p\setminus \{a\}$. Then there is exactly one real
$\theta_0$, $0\le \theta_0<2\pi$, such that the quadratic
differential $e^{i\theta_0}Q(z)\,dz^2$ has a circle domain, say
$D_0$, centered at $a$. Furthermore, the boundary $\partial D_0$
is the only critical $Q$-geodesic (non-Jordan in general) in the
class $\Gamma$.
In particular, $\Gamma$ may contain at most one critical geodesic
loop.
\varepsilonnd{lemma}
We will need some simple mapping properties of the canonical
mapping related to the quadratic differential $Q(z)\,dz^2$, which
is
defined by
$$
F(z)=\int_{z_0} \sqrt{Q(z)}\,dz
$$
with some $z_0\in \overline{\mathbb{C}}$ and some fixed branch of
the radical. A simply connected domain $D$ without critical points
of $Q(z)\,dz^2$ is called a $Q$-rectangle if the boundary of $D$
consists of two arcs of trajectories of $Q(z)\,dz^2$ separated by
two arcs of orthogonal trajectories of this quadratic
differential. As well a canonical mapping $F(z)$ maps any
$Q$-rectangle conformally onto a geometrical rectangle in the
plane with two sides parallel to the horizontal axis.
\section {Cauchy transforms satisfying quadratic equations and quadratic differentials} \lambdabel{sq-roots}
Below we relate the question for which triples of polynomials
$(P,Q,R)$ the equation
\begin{equation}gin{equation}\lambdabel{quadr}
P(z)\mathcal C^2+Q(z)\mathcal C+R(z)=0,
\varepsilonnd{equation}
with $\deg P=n+2,\;\deg Q\le n+1,\;\deg R\le n$
admits a compactly supported signed measure $\mu$ whose Cauchy transform satisfies
\varepsilonqref{quadr} almost everywhere in $\mathbb C$ to a certain problem about rational quadratic differentials. We call such measure $\mu$ a {\it motherbody measure} for \varepsilonqref{quadr}.
For a given quadratic differential $\Psi$ on a compact surface $\mathcal R,$ denote by
$K_\Psi\subset \mathcal R$ the union of all its critical
trajectories and critical points. (In general, $K_\Psi$ can be very
complicated. In particular, it can be dense in some subdomains of
$\mathcal R$.) We denote by $DK_\Psi\subseteq K_\Psi$ (the closure
of) the set of finite critical trajectories of
\varepsilonqref{eq:maindiff}. (One can show that $DK_\Psi$ is an imbedded
(multi)graph in $\mathcal R$. Here by a {\it multigraph} on a
surface we mean a graph with possibly multiple edges and loops.)
Finally, denote by $DK^0_\Psi\subseteq DK_\Psi$ the subgraph of
$DK_\Psi$ consisting of (the closure of) the set of finite
critical trajectories whose both ends are zeros of $\Psi$.
A non-critical trajectory $\gamma_{z_0}(t)$ of a meromorphic
$\Psi$ is called \varepsilonmph{closed} if $\varepsilonxists \ T>0$ such that
$\gamma_{z_0}(t+T)=\gamma_{z_0}(t)$ for all $t\in\mathbb{R}$. The
least such $T$ is called the \varepsilonmph{period} of $\gamma_{z_0}$. A
quadratic differential $\Psi$ on a compact Riemann surface
$\mathcal R$ without boundary is called \varepsilonmph{Strebel} if the set
of its closed trajectories covers $\mathcal R$ up to a set of
Lebesgue measure
zero.
Going back to Cauchy transforms, we formulate the following necessary condition of the existence
of a motherbody measure for \varepsilonqref{quadr}.
\begin{equation}gin{proposition}\lambdabel{pr:stand}
Assume that equation \varepsilonqref{quadr} admits a signed motherbody measure $\mu$. Denote by $D(z)=Q^2(z)-4P(z)R(z)$ the discriminant of equation \varepsilonqref{quadr}.
Then the following two conditions hold:
\noindent {\rm (i)} any connected smooth curve in the support of $\mu$
coincides with a horizontal trajectory of the quadratic
differential
\begin{equation}gin{equation}\lambdabel{eq:maindiff}
{\Bbb T}heta=-\frac{D(z)}{P^2(z)}dz^2=\frac{4P(z)R(z)-Q^2(z)}{P^2(z)}dz^2.
\varepsilonnd{equation}
\noindent
{\rm (ii)} the support of $\mu$ includes all branching points of \varepsilonqref{quadr}.
\varepsilonnd{proposition}
\noindent {\it Remark.} Observe that if $P(z)$ and $Q(z)$ are
coprime, the set of all branching points coincides with the set of
all zeros of $D(z)$. In particular, in this case part (ii) of
Proposition~\ref{pr:stand} implies that the set $DK_{\Bbb T}heta^0$ for
the differential ${\Bbb T}heta$ should contain all zeros of $D(z)$.
\noindent {\it Remark.} Proposition~\ref{pr:stand} applied to
quadratic differential $Q(z)\,dz^2$ of Theorem~\ref{Theorem-1}
implies Theorem~\ref{Theorem-2}.
\begin{equation}gin{proof} The fact that every curve in $\text{supp}(\mu)$ should coincide with some horizontal trajectory of \varepsilonqref{eq:maindiff} is well-known and follows from the Plemelj-Sokhotsky's formula. It is based on the local observation that if a real measure $\mu=\frac{1}{\pi}\frac{\partial \mathcal C}{\partial \bar z}$ is supported
on a smooth curve $\ga$, then the tangent to $\gamma$ at any point $z_0\in \gamma$ should be perpendicular to $\overline {\mathcal C_1(z_0)}- \overline{\mathcal C_2(z_0)}$ where $\mathcal C_1$ and $\mathcal C_2$ are the one-sided limits of $\mathcal C$ when $z\to z_0$, see e.g. \cite{BR}. (Here\; $\bar {}$\; stands for the usual complex conjugation.) Solutions of \varepsilonqref{quadr} are given by $$\mathcal C_{1,2}=\frac{-Q(z)\pm \sqrt{Q^2(z)-4P(z)R(z)}}{2P(z)},$$ their difference being $$\mathcal C_1-\mathcal C_2=\frac{\sqrt{Q^2(z)-4P(z)R(z)}}{P(z)}.$$
Since the tangent line to the support of the real motherbody measure
$\mu$ satisfying \varepsilonqref{quadr} at its arbitrary smooth point
$z_0$, is orthogonal to $\overline{ \mathcal C_1(z_0)}-\overline
{\mathcal C_2(z_0)},$ it is exactly given by the condition
$\frac{4P(z_0)R(z_0)-Q^2(z_0)}{P^2(z_0)}dz^2>0$. The latter
condition defines the horizontal trajectory of ${\Bbb T}heta$ at
$z_0$.
Finally the observation that $\text{supp }\mu$ should contain all branching points of \varepsilonqref{quadr} follows immediately from the fact that $\mathcal C_\mu$ is a well-defined univalued function in $\mathbb C\setminus \text{supp }\mu$.
\varepsilonnd{proof}
In many special cases statements similar to
Proposition~\ref{pr:stand} can be found in the literature, see
e.g. recent \cite {AMFMGT} and references therein.
Proposition~\ref{pr:stand} allows us, under mild nondegeneracy
assumptions, to formulate necessary and sufficient conditions for
the existence of a motherbody measure for \varepsilonqref{quadr} which
however are difficult to verify. Namely, let $\Ga\subset
\mathbb CP^1\times \mathbb CP^1$ with affine coordinates $(\mathcal C,z)$ be the algebraic
curve given by (the projectivization of) equation \varepsilonqref{quadr}.
$\Ga$ has bidegree $(2,n+2)$ and is hyperelliptic. Let
$\pi_z:\Ga\to \mathbb C$ be the projection of $\Ga$ on the $z$-plane
$\mathbb CP^1$ along the $\mathcal C$-coordinate. From \varepsilonqref{quadr} we observe
that $\pi_z$ induces a branched double covering of $\mathbb CP^1$ by
$\Ga$. If $P(z)$ and $Q(z)$ are coprime and if $\deg D(z)=2n+2$,
the set of all branching points of $\pi_z: \Ga\to \mathbb CP^1$
coincides with the set of all zeros of $D(z)$. (If $\deg
D(z)<2n+2,$ then $\infty$ is also a branching pont of $\pi_z$ of
multiplicity $2n+2-\deg D(z)$.) We need the following lemma.
\begin{equation}gin{lemma}\lambdabel{lm:poles} If $P(z)$ and $Q(z)$ are coprime,
then at each pole of \varepsilonqref{quadr} i.e. at each zero of $P(z)$,
only one of two branches of $\Ga$ goes to $\infty$. Additionally the residue
of this branch at this zero equals that of $-\frac {Q(z)}{P(z)}$.
\varepsilonnd{lemma}
\begin{equation}gin{proof} Indeed if $P(z)$ and $Q(z)$ are coprime, then no zero $z_0$ of $P(z)$
can be a branching point of \varepsilonqref{quadr} since $D(z_0)\neq 0$.
Therefore only one of two branches of $\Ga$ goes to $\infty$ at $z_0$.
More exactly, the branch
$\mathcal C_1=\frac{-Q(z)+ \sqrt{Q^2(z)-4P(z)R(z)}}{2P(z)}$ attains a finite value at $z_0$
while the branch $\mathcal C_2=\frac{-Q(z)- \sqrt{Q^2(z)-4P(z)R(z)}}{2P(z)}$ goes to $\infty$
where we use the agreement that $\lim_{z\to z_0} \sqrt{Q^2-4P(z)R(z)}=Q(z_0)$.
Now consider the residue of the branch $\mathcal C_2$ at $z_0$. Since residues depend continuously
on the coefficients $(P(z),Q(z),R(z))$ it suffices to consider only the case when $z_0$
is a simple zero of $P(z)$. Further if $z_0$ is a simple zero of $P(z),$ then
$$Res(\mathcal C_2,z_0)= \frac {-2Q(z_0)} {2P^\prime(z_0)}= Res \left(-\frac{Q(z)}{P(z)}, z_0\right),$$
which completes the proof. \varepsilonnd{proof}
By Proposition~\ref{pr:stand} (besides the obvious condition that \varepsilonqref{quadr} has
a real branch near $\infty$ with the asymptotics $\frac{\alpha}{z}$ for some $\alpha\in \mathbb R$)
the necessary condition for \varepsilonqref{quadr} to admit a motherbody measure is that
the set $DK_{\Bbb T}heta^0$ for the differential \varepsilonqref{eq:maindiff} contains all branching points
of \varepsilonqref{quadr}, i.e. all zeros of $D(z)$. Consider $\Ga_{cut}:=\Ga\setminus \pi_z^{-1}(DK_{\Bbb T}heta^0)$.
Since $DK_{\Bbb T}heta^0$ contains all branching points of $\pi_z$, $\Ga_{cut}$ consists
of some number of open sheets, each projecting diffeomorphically on its image
in $\mathbb CP^1\setminus DK^0_{\Bbb T}heta$. (The number of sheets in $\Ga_{cut}$ equals
to twice the number of connected components in $\mathbb C\setminus DK_{\Bbb T}heta^0$.)
Observe that since we have chosen a real branch of \varepsilonqref{quadr} at infinity
with the asymptotics $\frac{\alpha}{z}$, we have a marked point $p_{br}\in \Ga$ over $\infty$.
If we additionally assume that $\deg D(z)=2n+2,$ then $\infty$ is not a branching point of $\pi_z$
and therefore $p_{br}\in \Ga_{cut}$.
\begin{equation}gin{lemma}\lambdabel{lm:cut} If $\deg D(z)=2n+2$, then any choice of a
spanning (multi)subgraph $G\subset DK_{\Bbb T}heta^0$ with no isolated
vertices induces the unique choice of the section $S_G$ of $\Ga$ over $\mathbb CP^1\setminus G$ which:
\noindent a) contains $p_{br}$; b) is discontinuous at any point
of $G$; c) is projected by $\pi_z$ diffeomorphically onto
$\mathbb CP^1\setminus G$.
\varepsilonnd{lemma}
Here by a spanning subgraph we mean a subgraph containing all the
vertices of the ambient graph. By a section of $\Ga$ over
$\mathbb CP^1\setminus G$ we mean a choice of one of two possible values
of $\Ga$ at each point in $\mathbb CP^1\setminus G$. After these
clarifications the proof is evident.
Observe that the section $S_G$ might attain the value $\infty$ at
some points, i.e. contain some poles of \varepsilonqref{quadr}. Denote the
set of poles of $S_G$ by $Poles_G$. Now we can formulate our
necessary and sufficient conditions.
\begin{equation}gin{theorem}\lambdabel{th:necsuf} Assume that the following conditions are valid:
\noindent {\rm (i)} equation \varepsilonqref{quadr} has a real branch near
$\infty$ with the asymptotic behavior $\frac{\alpha}{z}$ for some
$\alpha\in \mathbb R$;
\noindent {\rm (ii)} $P(z)$ and $Q(z)$ are coprime, and the
discriminant $D(z)=Q^2(z)-4P(z)R(z)$ of equation \varepsilonqref{quadr}
has degree $2n+2$;
\noindent {\rm (iii)} the set $DK_{\Bbb T}heta^0$ for the quadratic
differential ${\Bbb T}heta$ given by \varepsilonqref{eq:maindiff} contains all
zeros of $D(z)$;
\noindent {\rm (iv)} ${\Bbb T}heta$ has no closed horizontal
trajectories.
Then \varepsilonqref{quadr} admits a real motherbody measure if and only if
there exists a spanning (multi)subgraph $G\subseteq DK^0_{\Bbb T}heta$
with no isolated vertices, such that all poles in $Poles_g$ are
simple and all their residues are real, see notation above.
\varepsilonnd{theorem}
\begin{equation}gin{proof} Indeed assume that $\varepsilonqref{quadr}$ satisfying {\rm(ii)}
admits a real motherbody measure $\mu$. Assumption {\rm(i)}
is obviously neccesary for the existence of a real motherbody measure
and the necessity of assumption {\rm(iii)} follows
from Proposition~\ref{pr:stand} if {\rm(ii)} is satisfied.
The support of $\mu$ consists of a finite number of curves
and possibly a finite number of isolated points. Since each curve
in the support of $\mu$ is a trajectory of ${\Bbb T}heta$ and ${\Bbb T}heta$
has no closed trajectories, then the whole support of $\mu$ consists
of finite critical trajectories of ${\Bbb T}heta$ connecting its zeros,
i.e. belongs to $DK_{\Bbb T}heta^0$. Moreover the support of $\mu$
should contain sufficently many finite critical trajectories
of ${\Bbb T}heta$ such that they include all the branching points
of \varepsilonqref{quadr}. By {\rm(ii)} these are exactly all zeros of $D(z)$.
Therefore the union of finite critical trajectories of ${\Bbb T}heta$
belonging to the support of $\mu$ is a spanning (multi)graph of $DK^0_{\Bbb T}heta$
without isolated vertices. The isolated points in the support of $\mu$
are necessarily the poles of \varepsilonqref{quadr}. Observe that the Cauchy transform
of any (complex-valued) measure can only have simple poles (as opposed
to the Cauchy transform of a more general distribution).
Since $\mu$ is real the residue of its Cauchy transform at each pole must be real as well.
Therefore the existence of a real motherbody under the assumptions {\rm (i)}--{\rm(iv)}
implies the existence of a spanning (multi)graph $G$ with the above properties.
The converse is also immediate.
\varepsilonnd{proof}
\noindent {\it Remark.} Observe that if {\rm (i)} is valid, then
assumptions {\rm (ii)} and {\rm (iv)} are generically satisfied.
Notice however that {\rm (iv)} is violated in the special case
when $Q(z)$ is absent. Additionally, if {\rm (iv)} is satisfied,
then the number of possible motherbody measures is finite. On the
other hand, it is the assumption {\rm (iii)} which imposes severe
additional restrictions on admissible triples $(P(z),Q(z),R(z))$.
At the moment the authors have no information about possible
cardinalities of the sets $Poles_G$ introduced above. Thus it is
difficult to estimate the number of conditions required for
\varepsilonqref{quadr} to admit a motherbody measure.
Theorem~\ref{th:necsuf} however leads to the following sufficient condition for the existence of a real motherbody measure for \varepsilonqref{quadr}.
\begin{equation}gin{corollary}\lambdabel{cor:suf} If, additionally to assumptions {\rm (i)}--{\rm (iii)} of Theorem~\ref{th:necsuf}, one assumes that all roots of $P(z)$ are simple and all residues of $\frac{Q(z)}{P(z)}$ are real, then \varepsilonqref{quadr} admits a real motherbody measure.
\varepsilonnd{corollary}
\begin{equation}gin{proof} Indeed if all roots of $P(z)$ are simple and all residues of $\frac{Q(z)}{P(z)}$ are real, then all poles of \varepsilonqref{quadr} are simple with real residues. In this case for any choice of a spanning (multi)subgraph $G$ of $DK_{\Bbb T}heta^0$, there exists a real motherbody measure whose support coincides with $G$ plus possibly some poles of \varepsilonqref{quadr}. Observe that if all roots of $P(z)$ are simple and all residues of $\frac{Q(z)}{P(z)}$ are real one can omit assumption {\rm (iv)}. In case when ${\Bbb T}heta$ has no closed trajectories, then all possible real motherbody measures are in a bijective correspondence with all spanning (multi)subgraphs of $DK_{\Bbb T}heta^0$ without isolated vertices. In the opposite case such measures are in a bijective correspondence with the unions of a spanning (multi)subgraph of $DK_{\Bbb T}heta^0$ and an arbitrary (possibly empty) finite collection of closed trajectories.
\varepsilonnd{proof}
\section{Does weak convergence of Jacobi polynomials imply stronger forms of convergence?}\lambdabel{Riemann}
Observe that, if one considers an arbitrary sequence
$\{s_n(z)\},\;n=0,1,\dots$ of monic univariate polynomials of
increasing degrees, then even if the sequence $\{\theta_n\}$ of
their root-counting measures weakly converges to some limiting
probability measure ${\Bbb T}heta$ with compact support in $\mathbb C$, in
general, it is not true that the roots of $s_n$ stay on some
finite distance from $\operatorname{supp} {\Bbb T}heta$ for all $n$ simultaneously.
Similarly nothing can be said in general about the weak
convergence of the sequence $\{\theta^\prime_n\}$ of the
root-counting measures of $\{s^\prime_n(z)\}$. However we have
already seen that the situation with sequences of Jacobi
polynomials seems to be different, comp.
Proposition~\ref{lm:higher}.
In the present appendix we formulate a general conjecture (and
give some evidence of its validity) about sequences of Jacobi
polynomials as well as sequences of more general polynomial
solutions of a special class of linear differentials equations
which includes Riemann's differential equation.
Consider a linear ordinary differential operator
\begin{equation}gin{equation}\lambdabel{eq:oper}
\dq=\sum_{i=1}^kQ_j(z)\frac{d^{j}}{dz^j}
\varepsilonnd{equation}
with polynomial coefficients. We say that \varepsilonqref{eq:oper} is {\it exactly solvable} if a) $\deg Q_j\le j,$ for all $j=1,\dots, k$; b) there exists at least one value $j_0$ such that $\deg Q_{j_0}(z)=j_0$. We say that an exactly solvable operator \varepsilonqref{eq:oper}
is {\it non-degenerate} if $\deg Q_k=k$.
Observe that any exactly solvable operator $\dq$ has a unique (up
to a constant factor) eigenpolynomial of any sufficiently large
degree, see e.g. \cite {BR}. Fixing an arbitrary monic polynomial
$Q_k(z)$ of degree $k$, consider the family $\mathcal F_{Q_k}$ of
all exactly solvable operators of the form \varepsilonqref{eq:oper} whose
leading term is $Q_k(z)\frac{d^k}{dz^k}$. ($\mathcal F_{Q_k}$ is a
complex affine space of dimension $\binom {k+1}{2}-1$.) Given a
sequence $\{\dqn\}$ of exactly solvable operators from $\mathcal
F_{Q_k}$ of the form
$$\dqn= Q_k(z)\frac{d^k}{dz^k}+\sum_{i=1}^{k-1}Q_{j,n}(z)\frac{d^{j}}{dz^j},$$
we say that this sequence has a {\it moderate growth} if, for
each $j=1,\dots, k-1,$ the sequence of polynomials
$\left\{\frac{Q_{j,n}(z)}{n^{k-j}}\right\}$ has all bounded
coefficients. (Recall that $\forall n$, $\deg Q_{j,n}\le j$.)
\begin{equation}gin{conjecture}\lambdabel{conj:bounded}
For any sequence $\{\dqn\}$ of exactly solvable operators of
moderate growth, the union of all roots of all the
eigenpolynomials of all $\dqn$ is bounded in $\mathbb C$.
\varepsilonnd{conjecture}
Now take a sequence $\{s_n(z)\},\; \deg s_n=n$ of polynomial
eigenfunctions of the sequence of operators $\dqn\in \mathcal
F_{Q_k}$. (Observe that, in general, we have a different
exactly solvable operator for each eigenpolynomial but with the
same leading term.)
\begin{equation}gin{conjecture}\lambdabel{conj:Main} In the above notation, assume that $\{\dqn\}$
is a sequence of exactly solvable operators
of moderate growth and that $\{s_n(z)\}$ is the sequence of their
eigenpolynomials (i.e $s_n(z)$ is the eigenpolynomial of $\dqn$ of
degree $n$) such that:
\noindent
{\rm a)} the limits $\widetilde Q_{j}(z):=\lim_{n\to \infty} \frac{1}{n^{k-j}}Q_{j,n}(z),\;
j=1,\dots, k-1$ exist;
\noindent {\rm b)} the sequence $\{\theta_n\}$ of the
root-counting measures of $\{s_n(z)\}$ weakly converges to a
compactly supported probability measure ${\Bbb T}heta$ in $\mathbb C$,
\noindent then
\noindent {\rm (i)} the Cauchy transform $\mathcal C_{\Bbb T}heta$ of ${\Bbb T}heta$
satisfies a.e. in $\mathbb C$ the algebraic equation
\begin{equation}gin{equation}\lambdabel{genCauchy}
Q_k(z)\left(\frac{\mathcal C_{{\Bbb T}heta}}{\ga}\right)^k+\sum_{j=1}^{k-1}\widetilde
Q_j(z)\left(\frac{\mathcal C_{\Bbb T}heta}{\ga}\right)^j=1,
\varepsilonnd{equation}
where $\ga=\lim_{n\to \infty}\frac{\root k \of {\lambda_n}}{n},$ \quad $\lambda_n$
being the eigenvalue of $s_n(z)$.
\noindent {\rm (ii)} for any positive $\varphirepsilons>0,$ there exist
$n_\varphirepsilons$ such that, for $n\ge n_{\varphirepsilons}$, all roots of all
eigenpolynomials $s_n(z)$ are located within $\varphirepsilons$-neighborhood
of $\operatorname{supp} {\Bbb T}heta$, i.e., the weak convergence of $\theta_n\to
{\Bbb T}heta$ implies a stronger form of this convergence.
\varepsilonnd{conjecture}
Certain cases of Part (i) of the above Conjecture are settled in
\cite {BR} and \cite {BBS} and a version of Part (ii) is discussed
in an unpublished preprint \cite {BoPi}.
Now we present some partial confirmation of the above conjectures.
Consider the family of linear differential operators of second
order depending on parameter $\lambda$ and given by
\begin{equation}gin{equation}\lambdabel{eq:pencil2}
T_\lambda = Q_2(z)\frac{d^2}{dz^2} + (Q_1(z)\lambda + P_1(z))\frac{d}{dz}+
(\lambda^2 + p\lambda + q)Q_0,
\varepsilonnd{equation}
where $Q_2(z)$ is a quadratic polynomial in $z$, $Q_1(z)$ and
$P_1(z)$ are polynomials in $z$ of degree at most $1,$ and $Q_0$
is a non-vanishing constant. (Observe that our use of parameter
$\lambda$ here is the same as of the parameter $\ga$ in the latter
Conjecture.)
Denote $Q_i(z) =\sum_{j=0}^iq_{ji}z^j,\; i = 0,1, 2$ and put
$P_1 = p_{11}z + p_{01}$. The quadratic polynomial
\begin{equation}gin{equation}\lambdabel{eq:charac}
q_{22}+ q_{11}t+q_{00}t^2
\varepsilonnd{equation}
is called the {\it characteristic polynomial} of $T_\lambda$. Here
$q_{22}\neq 0$ and $q_{00}=Q_0\neq 0$.
\begin{equation}gin{definition}
We say that the family $T_\lambda$ has a {\it generic type} if the
roots of \varepsilonqref{eq:charac} have distinct arguments (and in
particular $0$ is not a root of \varepsilonqref{eq:charac} which is
guaranteed by $q_{22}\neq 0$ together with $q_{00}\neq 0$), comp.
\cite{BBS}.
\varepsilonnd{definition}
Below we will denote the roots of characteristic polynomial \varepsilonqref{eq:charac}
by $\alpha_1$ and $\alpha_2$. Thus $T_\lambda$ has a generic type if and only if $\arg \alpha_1\neq \arg \alpha_2$.
\begin{equation}gin{lemma}\lambdabel{gentype}
Equation~\varepsilonqref{eq:charac} has two roots with the same arguments
if and only if $q_{22}q_{00}=\rho q_{11}^2,$ where $0\le \rho\le
\frac{1}{4}$.
\varepsilonnd{lemma}
\begin{equation}gin{proof} Straightforward calculation, see Example 1 of \cite{BBSh1}. \varepsilonnd{proof}
\begin{equation}gin{lemma}\lambdabel{pr:basic} In the above notation, for a family $T_\lambda$ of generic type,
there exists a positive integer $N$ such that, for any integer
$n\ge N,$ there exist two eigenvalues $\lambda_{1,n}$ and $\lambda_{2,n}$
such that the differential equation
\begin{equation}gin{equation}\lambdabel{eq:triv}
T_\lambda(y) =0
\varepsilonnd{equation}
has a polynomial solution of degree $n$. Moreover, $\lim_{n\to
\infty}\frac{\lambda_{i,n}}{n}=\alpha_i$
where $\alpha_1, \alpha_2$ are the roots of the characteristic polynomial of $T_\lambda$.
\varepsilonnd{lemma}
\begin{equation}gin{proof}
Observe that for any $\lambda\in \mathbb C$, the operator $T_\lambda$ acts on
each linear space $Pol_n$ of all polynomials of degree at most
$n$, $n=0,1,2,\dots,$ and its matrix presentation
$(c_{ij})^n_{i,j=0}$ in the standard monomial basis $(1, z,
z^2,..., z^n)$ of $Pol_n$ is an upper-triangular matrix with
diagonal entries $$c_{jj} = j(j-1)q_{22} +jq_{11} +q +(jq_{11}
+p)\lambda+q_{00}\lambda^2.$$ Therefore, for any given non-negative
integer $n$, we have a (unique) polynomial solution of
\varepsilonqref{eq:triv} of degree $n$ if and only if $c_{nn} = 0$ but
$c_{jj}\neq 0$ for $0 \le j < n$. The asymptotic formula for
$\lambda_{i,n}$ follows from the form of the equation $c_{nn} = 0$.
The genericity assumption that the equations
$$n(n-1)q_{22}+nq_{11}+q+(nq_{11}+p)\lambda+q_{00}\lambda^2 =0 $$ and
$$j(j - 1)q_{22} + jq_{11} + q + (jq_{11} + p)\lambda + q_{00}\lambda^2=0$$
should not have a common root, for $0 \le j < n$ and $n$
sufficiently large, is clearly satisfied if we
assume that the characteristic equation does not have two roots with the same argument.
\varepsilonnd{proof}
We can now prove the following stronger result.
\begin{equation}gin{proposition}\lambdabel{pr:local} For a general type family of differential operators
$T_\lambda$ of the form \varepsilonqref{eq:pencil2}, all roots of all polynomial solutions
of $T_\lambda(p) = 0,\; \lambda\in \mathbb C$ are located in some compact set
$K \subset \mathbb C$.
\varepsilonnd{proposition}
\begin{equation}gin{proof} Since $T_\lambda$ is assumed to be of general type, one gets $Q_0\neq 0$.
Therefore, without loss of generality
we can assume that $Q_0 = 1$ in \varepsilonqref{eq:triv}. Let $\{p_n\},
\deg(p_n) = n$ be a sequence of eigenpolynomials for
\varepsilonqref{eq:triv}, and assume that $\lim_{n\to \infty}
\frac{\lambda_n}{n}=\alpha$. (By Lemma~\ref{pr:basic}, $\alpha$ equals
either $\alpha_1$ or $\alpha_2$.) Define $w_n = \frac{p'_n}{\lambda_np_n}$
and notice that $p_n = e^{\lambda_n\int w_n dz}$.
We then have
$$p'_n = \lambda_nw_np_n;\; p^{\prime\prime}_n = (\lambda_n^2w_n^2 + \lambda_nw'_n)p_n.$$
Substituting these expressions in \varepsilonqref{eq:triv}, we obtain:
$$p_n(Q_2(z)(\lambda_n^2w_n^2(z) +\lambda_nw'_n(z)) + \lambda_n^2Q_1(z)w_n(z) + P_1(z)\lambda_nw_n(z) +\lambda^2_n + p\lambda_n + q
= 0.$$ For each fixed $n$, near $z = \infty$ we can conclude
that
$$Q_2(z)(\lambda_n^2w_n^2(z)+\lambda_nw'_n (z))+\lambda_n^2Q_1(z)w_n(z)+P_1(z)\lambda_nw_n(z)+\lambda_n^2+p\lambda_n+q = 0.$$
This relation defines a rational function $w_n$ near infinity. We
will show that the sequence $\{w_n\}$ converges uniformly to an
analytic function $w$ in a sufficiently small disc around
$\infty$. Moreover $w$ does not vanish identically.
Proposition~\ref{pr:local} will immediately follow from this
claim. Introducing $t = \frac{1}{z},$ one obtains
$$\widetilde Q_2\left(\left(\frac{w_n}{t}\right)^2-\frac{1}{\lambda_n}w'_n\right)+
\widetilde Q_1\left(\frac{w_n}{t}\right)
+\frac{1}{\lambda_n}\widetilde P_1\left(\frac{w_n}{t}\right)+ 1
+\frac{ p}{\lambda_n}+\frac{ q}{\lambda^2_n}= 0,$$ where $\widetilde
Q_2(t) := t^2Q_2(1/t),\; \widetilde Q_1(t) := tQ_1(1/t)$ and
$\widetilde P_1(t) := tP_1(1/t)$. Expand $w_n = c_1t + c_2t^2 +
...$ in a power series around $\infty$, i.e. around $t = 0$. (By
a slight abuse of notation, we temporarily disregard the fact
that the coefficients $c_k$ depend on $n$ until we make their
proper estimate.) Set $(w_n/t)^2 = b_0 + b_1t +\dots$. Then $$b_k
= c_1c_{k+1} + c_2c_k +...+ c_kc_2 + c_{k+1}c_1.$$ Finally,
introduce $\varphirepsilons_n = 1/\lambda_n$. Using these notations we obtain the
following system of recurrence relations for the coefficients
$c_k$:
$$q_{22}c^2_1 + (q_{11} - \varphirepsilons_nq_{22} + \varphirepsilons_np_{11})c_1 + 1 + \varphirepsilons_n p + \varphirepsilons^2_n q = 0,$$
$$q_{22}(b_1 - 2\varphirepsilons_nc_2) + q_{12}(b_0 - \varphirepsilons_nc_1) + (q_{11} + \varphirepsilons_np_{11})c_2 +
(q_{01} + \varphirepsilons_np_{01})c_1 = 0,$$
$$q_{22}(b_2-3\varphirepsilons_nc_3)+q_{12}(b_1-2\varphirepsilons_n c_2)+q_{02}(b_0-\varphirepsilons_n c_1)+
(q_{11}+\varphirepsilons_np_{11})c_3+(q_{01}+\varphirepsilons_np_{01})c_2 = 0,$$
and, more generally,
$$q_{22}(b_k - (k + 1)\varphirepsilons_n c_{k+1}) + q_{12}(b_{k-1} - k\varphirepsilons_n c_k) +
q_{02}(b_{k-2} - (k-1)\varphirepsilons_n c_{k-1})+
(q_{11} + \varphirepsilons_np_{11})c_{k+1}$$ $$ + (q_{01} + \varphirepsilons_np_{01})c_k =
0\quad \text{for} \quad k\ge 2.$$
Therefore, for any given $n$, we get $2$ possible values for
$c_1(n)$, which tend to the roots of $q_{22}t^2 + q_{11}t + 1 =
0$ as $n\to \infty$. Notice that $c_1(n) \to \frac{ 1}{\alpha}$ as
$n \to \infty$. Choosing one of two possible values for $c_1,$
we uniquely determine the remaining coefficients (as
rational functions of the previously calculated coefficients). Introducing
$\tilde b_k =b_k - 2c_1c_{k+1},$ we can observe that $\tilde b_k$
is independent of $c_{k+1}$ and we obtain the
following explicit formulas:
$$c_2 = -\frac{ q_{12}(c^2_1 - \varphirepsilons_n c_1) + (q_{01} + \varphirepsilons_np_{01})c_1}
{(2c_1 - 2\varphirepsilons_n)q_{22} + q_{11} + \varphirepsilons_np_{11}},$$
$$c_3 = -
\frac{q_{22}\tilde b_2 + q_{12}(b_1 - 2\varphirepsilons_n c_2) + q_{02}(b_0 -
\varphirepsilons_n c_1) + (q_{01} + \varphirepsilons_np_{01})c_2} {(2c_1 - 3\varphirepsilons_n)q_{22}
+ q_{11} + \varphirepsilons_np_{11}},$$ and more generally,
\[
\begin{equation}gin{split}
c_k =& - \frac{q_{22}\tilde b_{k-1} + q_{12}(b_{k-2} - (k -
1)\varphirepsilons_n c_{k-1})}{(2c_1 - k\varphirepsilons_n)q_{22} + q_{11} + \varphirepsilons_n
p_{11}} \\
&+ \frac{q_{02}(b_{k-2} - (k - 3)\varphirepsilons_n c_{k-3}) + (q_{01} +
\varphirepsilons_n p_{01})c_{k-1}}{(2c_1 - k\varphirepsilons_n)q_{22} + q_{11} +
\varphirepsilons_n p_{11}}.
\varepsilonnd{split}
\]
We will now include the dependence of $c_k$ on $n$ and show that
the coefficients $c_k(n)$ are majorated by the coefficients of a
convergent power series independent of $n$. First we show that
the denominators in these recurrence relations are bounded from
below. Notice that under our assumption, the rational functions
$w_n$ exist and have a power series expansion near $z = \infty$
with coefficients given by the above recurrence relations.
Therefore the denominators in these recurrences do not vanish.
Notice also that $\varphirepsilons_n \sigmameq \frac{c_1(n)}{n}$ asymptotically.
For fixed $k, $ it is therefore clear that the limits
$$\lim_{n\to\infty} (2c_1(n) - k\varphirepsilons_n)q_{22} + q_{11} + \varphirepsilons_np_{11} =
\lim_{n\to \infty} 2c_1(n)q_{22} + q_{11}$$
vanish if and only if the characteristic polynomial
\varepsilonqref{eq:charac} has a double root. We must however find a
uniform bound for $c_k(n)$ valid for all $k$ simultaneously.
Indeed, there might exist a subsequence $I \subset \mathcal NN$ of $k_n$
such that
\begin{equation}gin{equation}\lambdabel{eq:limit}
\lim_{n\in I;n\to \infty}(2c_1(n) - k_n\varphirepsilons_n)q_{22} + q_{11} +
\varphirepsilons_np_{11} = 0.
\varepsilonnd{equation}
(1) But this implies, using the asymptotics of $c_1(n)$ and
$\varphirepsilons_n$, the existence of a real number $r$ such that $\frac{1 -
r}{\alpha} = -\frac{q_{22}}{2q_{11}}$ which is clearly impossible if
the characteristic equation does not have two roots with the same
argument. Thus we have established a positive lower bound for the
absolute value of the denominators in the recurrence relations for
the coefficients $c_k$. The latter circumstance gives us a
possibility of majorizing the coefficients $c_k(n)$ independently
of $k$ and $n$. Namely, if there is a unbounded sequence
$k_n\varphirepsilons_n,$ then we can factor it out from the rational
functions in the recurrence. The existence of the sequence
mentioned above follow from an elementary lemma stated below,
which we leave without a proof. Thus, Proposition~\ref{pr:local}
is now settled.
\varepsilonnd{proof}
\begin{equation}gin{lemma}\lambdabel{lm:add} Consider a recurrence relation $c_{m+1} = P_m(c_1,..., c_m)$ where each
$P_m$ is a polynomial and assume that $d_{m+1} = Q_m(d_1,...,
d_m)$ is a similar recurrence relation whose polynomials have
all positive coefficients. If the polynomials under consideration
satisfy the inequalities
$$|P_m(z_1,..., z_m)| \le Q_m(|z_1|,..., |z_m|),$$
then the power series $\sum c_iz^i$ is dominated by the series
$\sum d_iz^i$ whenever $d_1\ge |c_1|$.
\varepsilonnd{lemma}
\section{Domain configurations of normalized quadratic differentials} \lambdabel{Section-6}
\setcounter{equation}{0}
Let $Q(z;a,b,c)\,dz^2$ be a quadratic differential of the form
(\ref{1.4}). Multiplying $Q(z;a,b,c)\,dz^2$ by a non-zero constant
$A\in \mathbb{C}$, we rescale the corresponding $Q$-metric
$|Q|^{1/2}\,|dz|$ by a positive constant $|A|^{1/2}$. Hence
$A\,Q(z;a,b,c)\,dz^2$ has the same geodesics as the quadratic
differential $Q(z;a,b,c)\,dz^2$ has. Obviously, multiplication
does not affect the homotopic classes.
Thus, while studying geodesics of the quadratic differential $Q(z;a,b,c)\,dz^2$,
we may assume without loss of generality that it
has the form
\begin{equation}gin{equation} \lambdabel{6.1}
Q(z)\,dz^2=-\frac{(z-p_1)(z-p_2)}{(z-1)^2(z+1)^2}\,dz^2.
\varepsilonnd{equation}
In Sections 6--9, we will work with the generic case; i.e we assume that
\begin{equation}gin{equation} \lambdabel{6.1.1}
p_1\not= \pm 1, \quad p_2\not= \pm 1, \quad p_1\not = p_2,
\varepsilonnd{equation}
unless otherwise is mentioned. Some typical configurations in the
limit (or non-generic) cases are shown in Fig.~5a--5g. Expanding
$Q(z)$ into Laurent series at $z=\infty$, we
obtain
\begin{equation}gin{equation} \lambdabel{6.2}
Q(z)=-\frac{1}{z^2}+{\mbox{higher degrees of $z$}} \quad \quad
{\mbox{as $z\to \infty$.}}
\varepsilonnd{equation}
Since the leading coefficient in the series expansion (\ref{6.2})
is real and negative it follows that $Q(z)\,dz^2$ has a circle
domain $D_\infty$ centered at $z=\infty$. The boundary
$L_\infty=\partial D_\infty$ of $D_\infty$ consists of a finite
number of critical trajectories of the quadratic differential
$Q(z)\,dz^2$ and therefore $L_\infty$ contains at least one of the
zeros $p_1$ and $p_2$ of $Q(z)\,dz^2$.
Next, we will discuss possible trajectory structures of
$Q(z)\,dz^2$ on the complement $D_0=\mathbb{C}\setminus
\overline{D}_\infty$. As we have mentioned in Section~3,
according to the Basic Structure Theorem, \cite[Theorem~3.5]{Je},
the domain configuration of a quadratic differential $Q(z)\,dz^2$
on $\overline{\mathbb{C}}$, which will be denoted by
$\mathcal{D}_Q$, may include circle domains, ring domains, strip
domains, end domains, and density domains. For the quadratic
differential (\ref{6.1}), by the Three Pole Theorem \cite[Theorem
3.6]{Je}, there are no density domains in its domain configuration
$\mathcal{D}_Q$. In addition, since $Q(z)\,dz^2$ has only three
poles of order two each, the domain configuration $\mathcal{D}_Q$
does not contain end domains and may contain at most three circle
domains centered at $z=\infty$, $z=-1$, and $z=1$.
We note here that $\mathcal{D}_Q$ may have strip domains (also
called \varepsilonmph{bilaterals}) with vertices at the double poles
$z=-1$ and $z=1$ but $\mathcal{D}_Q$ does not have ring domains.
Indeed, if there were a ring domain $\widehat{D}\subset D_0$ with
boundary components $l_1$ and $l_2$ then, by the Basic Structure
Theorem, each component must contain a zero of $Q(z)\,dz^2$. In
particular, $p_1\not=p_2$ in this case. Suppose that $l_1$
contains a zero $p_1$ and that $p_1\in L_\infty$. Then $L_\infty$
contains a critical trajectory $\gamma'$, which has both its end
points at $p_1$. There is one more critical trajectory $\gamma''$,
which has one of its end points at $p_1$. This trajectory
$\gamma''$ is either lies on the boundary of the circle domain
$D_\infty$ or it lies on the boundary of the ring domain
$\widehat{D}$. Therefore the second end point of $\gamma''$ must
be at a zero of $Q(z)\,dz^2$. Since the only remaining zero is
$p_2$, which lies on the boundary component $l_2$ not intersecting
$l_1$, we obtain a contradiction with our assumption. The latter
shows that $\mathcal{D}_Q$ does not have ring domains.
Next, we will classify topological types of domain configurations according
to the number of circle domains in $\mathcal{D}_Q$. The first
digit in our further classifications stands for the section where
this classification is introduced. The second and further digits
will denote the case under consideration.
\textbf{6.1.} Assume first that $\mathcal{D}_Q$ contains three
circle domains $D_\infty\ni \infty$, $D_{-1}\ni -1$, and $D_1\ni
1$. Then, of course, there are no strip domains in
$\mathcal{D}_Q$. In this case, the domains $D_\infty,D_{-1},D_1$
constitute an extremal configuration of the Jenkins extremal
problem for the weighted sum of reduced moduli with appropriate
choice of positive weights $\alphapha_\infty$, $\alphapha_{-1}$, and
$\alphapha_1$; see, for example, \cite{Str}, \cite{S1}, \cite{S2}.
More precisely, the
problem is to find all possible configurations realizing the following maximum:
\begin{equation}gin{equation} \lambdabel{6.3}
\max \
\left(\alphapha_\infty^2m(B_\infty,\infty)+\alphapha_{-1}^2m(B_{-1},-1)+\alphapha_1^2m(B_1,1)\right)
\varepsilonnd{equation}
over all triples of non-overlapping
simply connected domains $B_\infty\ni \infty$, $B_{-1}\ni -1$, and
$B_1\ni 1$. Here, $m(B,z_0)$ stands for the reduced module of a
simply connected domain $B$ with respect to the point $z_0\in B$;
see \cite[p.24]{Je}.
Since the extremal configuration of problem~(\ref{6.3}) is unique
it follows that the domains $D_\infty$, $D_{-1}$, and $D_1$ are
symmetric with respect to the real axis. In particular, the zeros
$p_1$ and $p_2$ are either both real or they are complex
conjugates of each other. Of course, this symmetry property of
zeros can be derived directly from the fact that the leading
coefficient of the Laurent expansion of $Q(z)$ at each its pole is
negative in the case under consideration. We have three
essentially different possible positions for the zeros:
\begin{equation}gin{enumerate}
\item[\textbf{(a)}]
$-1<p_2<p_1<1$,
\item[\textbf{(b)}]
$1<p_2<p_1$ or $p_1<p_2<-1$,
\item[\textbf{(c)}]
$p_1=\overline{p}_2=p$, where $\Im p>0$.
\varepsilonnd{enumerate}
We note here that in the case when $-1<p_2<1$ and, in addition,
$p_1>1$ or $p_1<-1$ the domain configuration $\mathcal{D}_Q$ must
contain a strip domain.
Case \textbf{(a)}.
The trajectory structure of $Q(z)\,dz^2$ corresponding to this
case is shown in Fig.~1a. There are three critical trajectories:
$\gamma_{-1}$, which is on the boundary of $D_{-1}$ and has both
its end points at $z=p_2$; $\gamma_1$, which is on the boundary of
$D_1$ and has both its end points at $z=p_1$, and $\gamma_0$,
which is the segment $[p_2,p_1]$.
Case \textbf{(b)}. An example of a domain configuration for the
case $1<p_2<p_1$ is shown in Fig.~1b. The boundary of $D_1$
consists of a single critical trajectory $\gamma_1$ having both
end points at $p_2$. The boundary of $D_{-1}$ consists of
critical trajectories $\gamma_\infty$, $\gamma_1$, and $\gamma_0$,
which is the segment $[p_2,p_1]$. In the case $p_1<p_2<-1$, the
domain configuration is similar.
Case \textbf{(c)}.
Since the domain configuration is symmetric, $p_1$ and $p_2$ both
belong to the boundary of $D_\infty$. Furthermore, there are three
critical trajectories: $\gamma_{-1}$, which joins $p_1$
and $p_2$ and intersects the real axis at some point $d_{-1}<-1$, $\gamma_1$, which joins $p_1$
and $p_2$ and intersects the real axis at some point $d_1>1$, and $\gamma^0$, which joins $p_1$
and $p_2$ and intersects the real axis at some point $d_0$,
$-1<d_0<1$. In this case, $\gamma_1\cup \gamma_0\subset \partial D_1$,
$\gamma_{-1}\cup \gamma_0\subset \partial D_{-1}$. An example of
a domain configuration of this type is shown in Fig.~1c.
\textbf{6.2.} Next we consider the case when $\mathcal{D}_Q$ has
exactly two circle domains. Suppose that these domains are
$D_\infty\ni \infty$ and $D_{-1}\ni -1$. In this case it is not
difficult to see that $L_\infty$ contains exactly one zero.
Indeed, if $p_1,p_2\in L_\infty$, then $L_\infty$ must contain one
or two critical trajectories joining $p_1$ and $p_2$. Suppose that
$L_\infty$ contains one such trajectory, call it $\gamma_0$. Since
$p_1,p_2\in L_\infty$ the boundary of $D_\infty$ must contain a
trajectory $\gamma_1$, which has both its end points at $p_1$ and
a trajectory $\gamma_{-1}$, which has both its end points at
$p_2$. Thus, $\gamma_1\cup\{p_1\}$ and $\gamma_{-1}\cup\{p_2\}$
each surrounds a simply connected domain, which must contain a
critical point of $Q(z)\,dz^2$. This implies that $z=-1$ and $z=1$
are centers of circle domains of $Q(z)\,dz^2$, which is the case
considered in part \textbf{ 6.1(a)}.
If $L_\infty$ contains two critical trajectories joining $p_1$ and
$p_2$, then there are critical trajectories $\gamma'$ having one
of its end points at $p_1$ and $\gamma''$ having one of its end
points at $p_2$. If $\gamma'=\gamma''$, then $D_0\setminus
\gamma'$ consists of two simply connected domains, which in this
case must be circle domains of $Q(z)\,dz^2$ as it is shown in
Fig.~1c.
If $\gamma'\not=\gamma''$, then each of these trajectories must
have its second end point at one of the poles $z=-1$ or $z=1$.
Moreover, if $\gamma'$ has an end point at $z=-1$ then $\gamma''$
must have its end point at $z=1$. Thus, there is no second circle
domain of $Q(z)\,dz^2$ in this case. Instead, there is one circle
domain $D_\infty$ and a strip domain, call it $G_2$, as it shown
in Fig. 3a-3e.
Now, let $p_1$ be the only zero of $Q(z)\,dz^2$ lying on $L_\infty$.
Then $L_\infty$ consists of a single critical trajectory of
$Q(z)\,dz^2$, call it $\gamma_\infty$, together with its end
points, each of which is at $p_1$. There is one more critical
trajectory, call it $\gamma_1^+$, that has one of its end points
at $p_1$. Then the second end point of $\gamma_1^+$ is either at
the point $p_2$ or at the second order pole at $z=1$.
If $\gamma_1^+$ terminates at $p_2$, then there is one more
critical trajectory, call it $\gamma_2$, having one of its end
points at $p_2$. Since $D_{-1}$ is a circle domain and $\partial
D_{-1}$ contains at least one zero of $Q(z)\,dz^2$ it follows that
$\gamma_2$ belongs to the boundary of $D_{-1}$. Since $\gamma_2$
lies on the boundary of $D_{-1}$ it have to terminate at a finite
critical point of $Q(z)\,dz^2$ and the only possibility for this
is that $\gamma_2$ terminates at $p_2$. In this case,
$\gamma_\infty$, $\gamma_1^+$, and $\gamma_2$ divide
$\overline{\mathbb{C}}$ into three circle domains, the case which
was already discussed in part \textbf{6.1(b)}.
Suppose that $\gamma_1^+$ joins the points $z=p_1$ and $z=1$. Then
$\mathcal{D}_Q$ contains a strip domain $G_1$. Since $z=1$ is the
only second order pole of $Q(z)\,dz^2$, which has a non-negative
non-zero leading coefficient, the strip domain $G_1$ has both its
vertices at the point $z=1$. Furthermore, one side of $G_1$
consists of two critical trajectories $\gamma_\infty$ and
$\gamma_1^+$. Therefore there is a critical trajectory, call it
$\gamma_1^-$ of $Q(z)\,dz^2$ lying on $\partial G_1$, which joins
$z=1$ and $z=p_2$. Now, the remaining possibility is that the
boundary of $D_{-1}$ consists of a single critical trajectory
$\gamma_{-1}$, which has both its end points at $p_2$. Then $G_1$
is the only strip domain in $\mathcal{D}_Q$ and the second side of
$G_1$ consists of the critical trajectories $\gamma_1^-$ and
$\gamma_{-1}$. Two examples of a domain configuration of this
type, symmetric and non-symmetric, are shown in Fig.~2a and
Fig.~2b.
\textbf{6.3.} Finally, we consider the case when $D_\infty$ is
the only circle domain of $Q(z)\,dz^2$. We consider two
possibilities.
Case \textbf{(a)}. Suppose that both zeros $p_1$ and $p_2$ belong
to the boundary of $D_\infty$. As we have found in part\textbf{
6.2} above, the domain configuration in this case consists of the
circle domain $D_\infty$ and the strip domain $G_2$. The boundary
of $D_\infty$ consists of two critical trajectories
$\gamma_\infty^+$ and $\gamma_\infty^-$ and their end points,
while the boundary of $G_2$ consists of the trajectories
$\gamma_\infty^+$, $\gamma_\infty^-$, $\gamma_1$, and
$\gamma_{-1}$ and their end points, as it is shown in Fig.~3a-3c.
Case \textbf{(b)}. Suppose that the boundary $L_\infty$ of
$D_\infty$ contains only one zero $p_1$. Then there is a critical
trajectory $\gamma_\infty$ having both its end points at $p_1$
such that $L_\infty=\gamma_\infty\cup \{p_1\}$. Since $p_1$ is a
simple zero of $Q(z)\,dz^2$ there is one more critical trajectory
having one of its end points at $p_1$. The second end point of
this trajectory is either at the pole $z=1$, or at the pole
$z=-1$, or at the zero $z=p_2$. Depending on which of these
possibilities is realized, this trajectory will be denoted by
$\gamma_1$, or $\gamma_{-1}$, or $\gamma_0$, respectively. Thus,
we have two essentially different subcases.
Case \textbf{(b1)}. Suppose that there is a critical trajectory
$\gamma_0$ joining the zeros $p_1$ and $p_2$. Then there are two
critical trajectories, call them $\gamma_1$ and $\gamma_{-1}$,
each of which has one of its end point at $p_2$. We note that
$\gamma_1\not=\gamma_{-1}$. Indeed, if $\gamma_1=\gamma_{-1}$,
then the closed curve $\gamma_1\cup \{p_2\}$ must enclose a
bounded circle domain of $Q(z)\,dz^2$, which does not exist.
Furthermore, $\gamma_1$ and $\gamma_{-1}$ both cannot have their
second end points at the same pole at $z=1$ or $z=-1$. If this
occurs then again $\gamma_1$ and $\gamma_{-1}$ will enclose a
simply connected domain having a single pole of order $2$ on its
boundary, which is not possible. The remaining possibility is that
one of these critical trajectories, let assume that $\gamma_1$,
joins the zero $z=p_2$ and the pole at $z=1$ while $\gamma_{-1}$
joins $z=p_2$ and $z=-1$.
In this case the domain configuration $\mathcal{D}_Q$ consists of
the circle domain $D_\infty$ and the strip domain $G_2$; see
Fig.~3d- and Fig.~3e. The boundary of $G_2$ consists of two sides,
call them $l_1$ and $l_2$. The side $l_1$ is the set of boundary
points of $G_2$ traversed by the point $z$ moving along $\gamma_1$
from $z=1$ to $z=p_2$ and then along $\gamma_{-1}$ from the point
$z=p_2$ to $z=-1$. The side $l_2$ is the set of boundary points of
$G_2$ traversed by the point $z$ moving along $\gamma_1$ from
$z=1$ to $z=p_2$, then along $\gamma_0$ from $z=p_2$ to $z=p_1$,
then along $\gamma_\infty$ from $z=p_1$ to the same point $z=p_1$,
then along $\gamma_0$ from $z=p_1$ to $z=p_2$, and finally along
$\gamma_{-1}$ from $z=p_2$ to $z=-1$.
Case \textbf{(b2)}. Suppose that there is a critical trajectory
$\gamma_1$ joining the zero $p_1$ and the pole $z=1$. Then there
is a strip domain, call it $G_1$, which has both its vertices at
the pole $z=1$ and has the critical trajectories $\gamma_1$ and
$\gamma_\infty$ on one of its sides, call it $l_1^1$. More
precisely, the side $l_1^1$ is the set of boundary points of $G_1$
traversed by the point $z$ moving along $\gamma_1$ from $z=1$ to
$z=p_1$, then along $\gamma_\infty$ from $z=p_1$ to the same
point $z=p_1$, and then along $\gamma_1$ from $z=p_1$ to $z=1$.
Let $l_1^2$ denote the second side of $G_1$. Since a side of a
strip domain always has a finite critical point it follows that
$l_1^2$ contains two critical trajectories, call them $\gamma_0^+$
and $\gamma_0^-$, which join the pole $z=1$ with zero $z=p_2$.
There is one critical trajectory of $Q(z)\,dz^2$, call it
$\gamma_{-1}$, which has one of its end points at $z=p_2$. Since
$z=-1$ is a second order pole, which is not the center of a circle
domain, there should be at least one critical trajectory of
$Q(z)\,dz^2$ approaching $z=-1$ at least in one direction. Since
the end points of all critical trajectories, except $\gamma_{-1}$,
are already identified and they are not at $z=-1$, the remaining
possibility is that $\gamma_{-1}$ has its second end point at
$z=-1$. In this case there is one more strip domain, call it
$G_2$, which has vertices at the poles $z=1$ and $z=-1$ and sides
$l_2^1$ and $l_2^2$. Two examples of configurations with one
circle domain and two strip domains, symmetric and non-symmetric,
are shown in Fig.~4a and Fig.~4b. Now we can identify all sides of
$G_1$ and $G_2$. The side $l_1^2$ is the set of boundary points of
$G_1$ traversed by the point $z$ moving along $\gamma_0^+$ from
$z=1$ to $z=p_2$ and then along
$\gamma_0^-$ from $z=p_2$ to $z=1$.
The side $l_2^1$ is the set of boundary points of $G_2$ traversed
by the point $z$ moving along $\gamma_0^+$ from $z=1$ to $z=p_2$
and then along $\gamma_{-1}$ from $z=p_2$ to $z=-1$. Finally, the
side $l_2^2$ is the set of boundary points of $G_2$ traversed by
the point $z$ moving along $\gamma_0^-$ from $z=1$ to $z=p_2$ and
then along $\gamma_{-1}$ from $z=p_2$ to $z=-1$; see Fig.~4a and
Fig.~4b.
Case \textbf{(b3)}. In the case when there is a critical
trajectory joining the zero $p_1$ and the pole $z=-1$, the domain
configuration is similar to one described above, we just have to
switch the roles of the poles at $z=1$ and $z=-1$.
\begin{equation}gin{remark} \lambdabel{Remark-2}
We have described above all possible configurations in the generic
case; i.e. under conditions (\ref{6.1.1}). The remaining special
cases can be obtained from the generic case as limit cases when
$p_2\to -1$, when $p_2\to p_1$; etc. In the case $p_1=p_2$,
possible configurations are shown in Fig.~5a-5c.
In the case when $p_2=-1$, $p_1\not=\pm 1$, possible
configurations are shown in Fig.~5d-5g.
In the case when $p_1=p_2=1$, the limit position of critical
trajectories is just a circle centers at $z=-1$ with radius
$2$configuration and in the case when $p_1=1$, $p_2=-1$ there is
one critical trajectory which is an open interval from $z=-1$ to
$z=1$.
\varepsilonnd{remark}
\section{How parameters determine the type of domain configuration}
\setcounter{equation}{0}
Our goal in this section is to identify the ranges of the
parameters $p_1$ and $p_2$ corresponding to topological types
discussed in Section~6. For a fixed $p_1$ with $\Im p_1\not=0$, we
will define four regions of the parameter $p_2$. These regions and
their boundary arcs will correspond to domain configurations with
specific properties; see Fig.~6.
It will be useful to introduce the following notation. For $a\in
\mathbb{C}$ with $\Im a\not=0$, by $L(a)$ and $H(a)$ we denote,
respectively, an ellipse and hyperbola with foci at $z=1$ and
$z=-1$, which pass through the point $z=a$. If $\Im a\not= 0$,
then the set $\mathbb{C}\setminus (L(a)\cup H(a))$ consists of
four connected components, which will be denoted by $E_1^+(a)$,
$E_1^-(a)$, $E_{-1}^+(a)$, and $E_{-1}^-(a)$. We assume here that
$1\in E_1^+(a)$, $-1\in E_{-1}^+(a)$, $E_1^-(a)\cap
\mathbb{R}_+\not= \varepsilonmptyset$, and $E_{-1}^-(a)\cap
\mathbb{R}_-\not=\varepsilonmptyset$. Furthermore, assuming that $\Im
a\not=\varepsilonmptyset$, we define the following open arcs:
$L^+(a)=(L(a)\cap
\partial E_1^+(a))\setminus\{a,\bar a\}$, $L^-(a)=(L(a)\cap \partial E_{-1}^+(a))
\setminus\{a,\bar a\}$, $H^+(a)=(H(a)\cap
\partial E_1^+(a))\setminus\{a,\bar a\}$, $H^-(a)=(H(a)\cap \partial E_1^-(a))
\setminus\{a,\bar a\}$. Let $l_1(a)$ and $l_{-1}(a)$ be straight
lines passing through the points $1$ and $\bar a$ and $-1$ and
$\bar a$, respectively. Let $l_1^+(a)$ and $l_{-1}^+(a)$ be open
rays issuing from the points $z=1$ and $z=-1$, respectively, which
pass through the point $z=\overline{a}$ and let $l_1^-(a)$ and
$l_{-1}^-(a)$ be their complementary rays. The line $l_1(a)$
divides $\mathbb{C}$ into two half-planes, we call them $P_1$ and
$P_2$ and enumerate such that $P_1\ni 2$. Similarly, the line
$l_{-1}(a)$ divides $\mathbb{C}$ into two half-planes $P_3$ and
$P_4$, where $P_3\ni -2$.
Before we state the main result of this section, we recall the
reader that the local structure of trajectories near a pole $z_0$
is completely determined by the leading coefficient of the Laurent
expansion of $Q(z)$ at $z_0$, see \cite[Ch.~3]{Je}.
In particular, for the quadratic differential $Q(z)\,dz^2$ defined by (\ref{6.1}) we have
\begin{equation}gin{equation} \lambdabel{7.1.1}
Q(z)=-\frac{1}{4}\frac{C_1}{ (z-1)^2}+{\mbox{higher degrees of
$(z- 1)$ \quad as $z\to 1$}}
\varepsilonnd{equation}
and
$$
Q(z)=-\frac{1}{4}\frac{C_{-1}}{ (z+1)^2}+{\mbox{higher degrees of
$(z+ 1)$ \quad as $z\to -1$.}}
$$
Then, assuming that $p_1\not= \pm 1$, $p_2\not= \pm 1$, we find
\begin{equation}gin{equation} \lambdabel{5.1}
C_1=(p_1-1)(p_2-1)\not= 0 \quad {\mbox{and}} \quad C_{-1}=(p_1+1)(p_2+1)\not =0.
\varepsilonnd{equation}
A complete description of sets of pairs $p_1$, $p_2$ with $\Im
p_1>0$ corresponding to all possible types of domain
configurations discussed in Section~6 is given by the following
theorem.
\begin{equation}gin{theorem} \lambdabel{Theorem 5.1}
Let $p_1$ with $\Im p_1>0$ be fixed. Then the following holds.
\textbf{7.A}. The types of domain configurations $\mathcal{D}_Q$
correspond to the following sets of the parameter~$p_2$.
\begin{equation}gin{enumerate}
\item[\textbf{(1)}]
If $p_2=\bar p_1$, then the domain configuration $\mathcal{D}_Q$
is of the type \textbf{6.1(c)}.
\item[\textbf{(2)}]
If $p_2\in l_1^+(p_1)\setminus\{\bar p_1\}$, then $\mathcal{D}_Q$
has the type~\textbf{6.2} with circle domains $D_\infty\ni\infty$
and $D_1\ni 1$. Furthermore, if $p_2\in l_1^+(p_1)\cap
E_1^+(p_1)$, then $p_1\in \partial D_\infty$ and if $p_2\in
l_1^+(p_1)\cap E_{-1}^-(p_1)$, then $p_2\in \partial D_\infty$.
If $p_2\in l_{-1}^+(p_1)\setminus\{\bar p_1\}$, then
$\mathcal{D}_Q$ has the type~\textbf{6.2} with circle domains
$D_\infty\ni\infty$ and $D_{-1}\ni -1$. Furthermore, if $p_2\in
l_{-1}^+(p_1)\cap E_{-1}^+(p_1)$, then $p_1\in \partial D_\infty$
and if $p_2\in l_{-1}^+(p_1)\cap E_{-1}^-(p_1)$, then $p_2\in
\partial D_\infty$.
\item[\textbf{(3a)}]
If $p_2\in L(a)\setminus\{p_1,\bar p_1\}$, then the domain
configuration $\mathcal{D}_Q$ has type \textbf{6.3(a)}.
Furthermore, if $p_2\in L^+(p_1)$, then there is a critical
trajectory having one end point at $p_2$, which in other direction
approaches the pole $z=1$. Similarly, if $p_2\in L^-(p_1)$, then
there is a critical trajectory having one end point at $p_2$,
which in other direction
approaches the pole $z=-1$.
\item[\textbf{(3b1)}]
If $p_2\in H(p_1)\setminus\{p_1,\bar p_1\}$, then $\mathcal{D}_Q$
has type \textbf{6.3(b1)}. Furthermore, if $p_2\in H^+(p_1)$, then
there is a critical trajectory having both end points at $p_1$.
If $p_2\in H^-(p_1)$, then there is a critical trajectory having
both end points at $p_2$.
\item[\textbf{(3b2)}]
In all remaining cases, i.e. if $p_2\not\in L(p_1)\cup H(p_1)\cup
l_1^+(p_1)\cup l_{-1}^+(p_1)\cup\{-1,1\}$, the domain
configuration $\mathcal{D}_Q$
belongs to type \textbf{6.3(b2)}.
Furthermore, if $p_2\in (E_1^+(p_1)\cup E_{-1}^+(p_1))\setminus
(l_1^+(p_1)\cup l_{-1}^+(p_1)\cup\{-1,1\})$, then $p_1\in \partial
D_\infty$ and if $p_2\in (E_1^-(p_1)\cup E_{-1}^-(p_1))\setminus
(l_1^+(p_1)\cup l_{-1}^+(p_1))$, then $p_2\in \partial D_\infty$.
In addition, if $p_2\in E_1^+(p_1)\setminus
(l_1^+(p_1)\cup\{1\})$, then the pole $z=1$ attracts only one
critical trajectory of the quadratic differential (\ref{6.1}),
which has its second end point at $z=p_2$ and if $p_2\in
E_{-1}^-(p_1)\setminus (l_1^+(p_1))$, then the pole $z=1$ attracts
only one critical trajectory of the quadratic differential
(\ref{6.1}), which has its second end point at $z=p_1$. If $p_2\in
E_{-1}^+(p_1)\setminus (l_{-1}^+(p_1)\cup\{-1\})$, then the pole
$z=-1$ attracts only one critical trajectory of the quadratic
differential (\ref{6.1}), which has its second end point at
$z=p_2$ and if $p_2\in E_1^-(p_1)\setminus (l_{-1}^+(p_1))$, then
the pole $z=-1$ attracts only one critical trajectory of the
quadratic differential (\ref{6.1}), which has its second end point
at $z=p_1$.
\varepsilonnd{enumerate}
\textbf{7.B}. The local behavior of the trajectories near the
poles $z=1$ and $z=-1$ is controlled by the position of the zero
$p_2$ with respect to the lines $l_1(p_1)$ and $l_{-1}(p_1)$.
Precisely,
we have the following possibilities.
\begin{equation}gin{enumerate}
\item[\textbf{(1)}]
If $p_2\in l_1^-(p_1)$ or, respectively, $p_2\in l_{-1}^-(p_1)$,
then $Q(z)\,dz^2$ has radial structure of trajectories near the
pole $z=1$ or, respectively, near the pole $z=-1$.
\item[\textbf{(2)}]
If $p_2\in P_1$ or, respectively, $p_2\in P_2$, then the
trajectories of $Q(z)\,dz^2$ approaching the pole $z=1$ spiral
counterclockwise or, respectively, clockwise.
If $p_2\in P_3$ or, respectively, $p_2\in P_4$, then the
trajectories of $Q(z)\,dz^2$ approaching the pole $z=-1$ spiral
counterclockwise or, respectively, clockwise.
\varepsilonnd{enumerate}
\varepsilonnd{theorem}
\noindent
\varepsilonmph{Proof}.
\textbf{7.A(1).} We have shown in Section~6 that a domain
configuration $\mathcal{D}_Q$ of the type~\textbf{6.1(c)} occurs if
and only if $p_2=\bar p_1$. Thus, we have to consider cases \textbf{7.A(2)}
and \textbf{7.A(3)}.
We first prove statements about positions of zeros $p_1$ and
$p_2$ for each of these cases. Then we will turn to statements
about critical trajectories.
\textbf{7.A(2).} A domain configuration $\mathcal{D}_Q$ contains
exactly two circle domains centered at $z=\infty$ and $z=-1$ if
and only if
$C_{-1}>0$ and $C_1$ is not a positive real number. This is equivalent to the following conditions:
\begin{equation}gin{equation} \lambdabel{5.2}
\arg (p_1+1)=-\arg(p_2+1) \quad \mod(2\pi),
\varepsilonnd{equation}
\begin{equation}gin{equation} \lambdabel{5.3}
\arg (p_1-1)\not=-\arg(p_2-1) \quad \mod(2\pi).
\varepsilonnd{equation}
Geometrically, equations (\ref{5.2}) and (\ref{5.3}) mean that the
points $p_1$ and $p_2$ lie on the rays issuing from the pole
$z=-1$, which are symmetric to each other with respect to the real
axis. Furthermore, each ray contains one of these points and
$p_1\not= \bar p_2$.
Assuming (\ref{5.2}), (\ref{5.3}), we claim that $p_1\in
\partial D_\infty$ if and only if $|p_2+1|<|p_1+1|$.
First we prove that the claim is true for all $p_2$ sufficiently
close to $z=-1$ if $p_1$ is fixed. Arguing by contradiction,
suppose that there is a sequence $s_k\to -1$ such that
$\arg(s_k+1)=-\arg(p_1+1)$ and $p_1\in
\partial D_{-1}^k$, $s_k\in \partial D_\infty^k$ for all $k=1,2,\ldots$ Here
$D_{-1}^k\ni -1$ and $D_\infty^k\ni \infty$ denote the
corresponding circle domains of the quadratic
differential
\begin{equation}gin{equation} \lambdabel{5.4}
Q_k(z)\,dz^2=-\frac{(z-p_1)(z-s_k)}{(z-1)^2(z+1)^2}\,dz^2.
\varepsilonnd{equation}
Changing variables in (\ref{5.4}) via $z=(s_k+1)\zetata-1$ and then
dividing the resulting quadratic differential by
$\delta_k=|s_k+1|$, we obtain the following quadratic
differential:
\begin{equation}gin{equation} \lambdabel{5.5}
\widehat{Q}_k(\zetata)\,d\zetata^2=\frac{\zetata-1}{\zetata^2}\frac{|1+p_1|-\delta_k^{-1}(s_k+1)^2\zetata}{(2-(s_k+1)\zetata)^2}
\,d\zetata^2.
\varepsilonnd{equation}
We note that the trajectories of $Q_k(z)\,dz^2$ correspond under
the mapping $z=(s_k+1)\zetata-1$ to the trajectories of the
quadratic differential $\widehat{Q}_k(\zetata)\,d\zetata^2$. Thus,
$\widehat{Q}_k(\zetata)\,d\zetata^2$ has two circle domains
$\widehat{D}_{k,\infty}\ni \infty$ and $\widehat{D}_{k,0}\ni 0$.
The zeros of $\widehat{Q}_k(\zetata)\,d\zetata^2$ are at the points
\begin{equation}gin{equation} \lambdabel{5.6}
\zetata'_k=1\in \partial \widehat{D}_{k,\infty}, \quad
\zetata''_k=\delta_k|1+p_1|(s_k+1)^{-2}\in \partial
\widehat{D}_{k,0}.
\varepsilonnd{equation}
From (\ref{5.5}), we find that
\begin{equation}gin{equation} \lambdabel{5.7}
\widehat{Q}_k(\zetata)\,d\zetata^2\to
\widehat{Q}(\zetata)\,d\zetata^2:=\frac{|1+p_1|}{4}\frac{\zetata-1}{\zetata^2}\,d\zetata^2,
\varepsilonnd{equation}
where convergence is uniform on compact subsets of
$\mathbb{C}\setminus\{0\}$. Since
$$
\widehat{Q}(\zetata)=-(|1+p_1|/4)\zetata^{-2}+\cdots \quad {\mbox{ as
$\zetata\to 0$}}
$$
the quadratic differential $\widehat{Q}(\zetata)\,d\zetata^2$ has a
circle domain $\widehat{D}$ centered at $\zetata=0$. Let
$\widehat{\gamma}$ be a trajectory of
$\widehat{Q}(\zetata)\,d\zetata^2$ lying in $\widehat{D}$ and let
$\hat\gamma_k$ be an arbitrary trajectory of
$\widehat{Q}_k(\zetata)\,d\zetata^2$ lying in the circle domain
$\widehat{D}_{k,0}$. Since
$\hat\gamma_k$ is a $\widehat{Q}_k$-geodesic in its class and by (\ref{5.7}) we have
\begin{equation}gin{equation} \lambdabel{5.8}
|\hat\gamma_k|_{\widehat{Q}_k}\le
|\widehat{\gamma}|_{\widehat{Q}_k}\to
|\widehat{\gamma}|_{\widehat{Q}}=|1+p_1|^{1/2} \quad {\mbox {as
$k\to \infty$.}}
\varepsilonnd{equation}
On the other hand, conditions (\ref{5.6}) imply that for every
$R>1$ there is $k_0$ such that for every $k\ge k_0$ there is an
arc $\tau_k$ joining the circles $\{\zetata:\,|\zetata|=1\}$ and
$\{\zetata:\,|\zetata|=R\}$, which lies on regular trajectory of the
quadratic differential $\widehat{Q}_k(\zetata)\,d\zetata^2$ lying in
the circle domain $\widehat{D}_{k,0}$. Then, using (\ref{5.5}), we
conclude that there is a constant $C>0$ independent on $R$ and
$k$ such that
$$
|\hat\gamma_k|_{\widehat{Q}_k}\ge
|\tau_k|_{\widehat{Q}_k}=\int_{\tau_k}\left|\widehat{Q}_k(\zetata)\right|^{1/2}\,|d\zetata|\ge
C\,\int_1^R \frac{\sqrt{|\zetata|-1}}{|\zetata|}\,d|\zetata|
$$
for all $k\ge k_0$. Since $\int_1^R x^{-1}\sqrt{x-1}\,dx\to
\infty$ as $R\to \infty$, the latter equation contradicts equation
(\ref{5.8}). Thus, we have proved that if $p_1$ is fixed and $p_2$
is sufficiently close to $z=-1$ then $p_1\in \partial D_\infty$
and $p_2\in \partial D_{-1}$.
Now, we fix $p_1$ with $\Im p_1\not =0$ and consider the set $A$
consisting of all points $p'_2$ on the ray $r=\{z: \, \arg(z+1)
=-\arg(p_1+1)\}$ such that $p_1\in
\partial D_\infty(p_1,p_2)$ and $p_2\in \partial
D_{-1}(p_1,p_2)$ for all $p_2\in r$ such that
$|p_2+1|<|p'_2+1|$. Here $D_\infty(p_1,p_2)$ and $D_{-1}(p_1,p_2)$
are corresponding circle domains of the quadratic differential
(\ref{6.1}). Our argument above shows that $A\not= \varepsilonmptyset$. Let
$p_2^m\in r$ be such that
$$
|p_2^m+1|=sup_{p_2\in A} \,|p_2+1|.
$$
Consider the quadratic differential $Q(z;p_1,p_2^m)\,dz^2$ of the
form (\ref{6.1}) with $p_2$ replaced by $p_2^m$. Let
$D_\infty(p_1,p_2^m)\ni \infty$ and $D_{-1}(p_1,p_2^m)\ni -1$ be
the corresponding circle domains of $Q(z;p_1,p_2^m)\,dz^2$. Since
the quadratic differential (\ref{6.1}) depends continuously on the
parameters $p_1$ and $p_2$, it is not difficult to show, using our
definition of $p_2^m$, that both zeros of $Q(z;p_1,p_2^m)\,dz^2$
belong to the boundary of each of the domains $D_{-1}(p_1,p_2^m)$
and $D_\infty(p_1,p_2^m)$. But, as we have shown in part
\textbf{6.2} of Section~6, in this case the domain configuration
of $Q(z;p_1,p_2^m)\,dz^2$ must consist of three circle domains.
Therefore, as we have shown in part~\textbf{6.1} of Section~6, we
must have $p_1^m=\bar p_1$.
Thus, we have shown that $p_2\in \partial D_{-1}$ if $p_1$ and
$p_2$ satisfy (\ref{5.2}) and $|p_2+1|<|p_1+1|$. The M\"{o}bius
map $w=\frac{3-z}{1+z}$ interchanges the poles $z=\infty$ and
$z=-1$ of the quadratic differential (\ref{6.1}) and does not
change the type of its domain configuration. Therefore, our
argument shows also that $p_1\in \partial D_\infty$ if
$|p_2+1|<|p_1+1|$. This complete the proof of our claim that
$p_1\in
\partial D_\infty$ if and only if $|p_2+1|<|p_1+1|$.
Similarly, if $Q(z)\,dz^2$ has exactly two circle domains
$D_\infty\ni \infty$ and $D_1\ni 1$, then $p_2\in \partial D_1$
and $p_1\in \partial D_\infty$ if and only if
$$
\arg(p_1-1)=-\arg(p_2-1) \ \ \mod 2\pi \quad {\mbox{and}} \quad |p_2-1|<|p_1-1|.
$$
\textbf{7.A(3).} In this part, we will discuss cases \textbf{6.3(a)}, \textbf{6.3(b1)}, and \textbf{6.3(b2)} discussed in Section~6.
A domain configuration $\mathcal{D}_Q$ contains
exactly one circle domains centered at $z=\infty$ if
and only if neither $C_1$ or $C_{-1}$ is a positive real number.
As we have found in Section~6, in this case there exist one or
two strip domains $G_1$ and $G_2$ having their vertices at the
poles $z=1$ and $z=-1$. In what follows, we will use the notion
of the \varepsilonmph{normalized height} $h$ of a strip domain $G$, which is
defined as
$$
h=\frac{1}{2\pi} \Im\,\int_\gamma \sqrt{Q(z)}\,dz>0,
$$
where the integral is taken over any rectifiable arc
$\gamma\subset G$ connecting the sides of~$G$.
The sum of \varepsilonmph{normalized heights }in the $Q$-metric of
the strip domains, which have a vertex at the pole $z=1$ or at
the pole $z=-1$ can be found using integration over
circles $\{z:\,|z-1|=r\}$ and $\{z:\,|z+1|=r\}$ of radius $r$, $0<r<1$,
as follows:
\begin{equation}gin{equation} \lambdabel{5.10}
h_+=\frac{1}{2\pi}\Im \int_{|z-1|=r} \sqrt{Q(z)}\,dz=\frac{1}{2}
\Im\sqrt{C_1}=\frac{1}{2} \Im
\sqrt{(p_1-1)(p_2-1)}
\varepsilonnd{equation}
if $z=1$ and
\begin{equation}gin{equation} \lambdabel{5.11}
h_-=\frac{1}{2\pi}\Im \int_{|z+1|=r} \sqrt{Q(z)}\,dz=\frac{1}{2}
\Im\sqrt{C_{-1}}=\frac{1}{2} \Im
\sqrt{(p_1+1)(p_2+1)}
\varepsilonnd{equation}
if $z=-1$. The branches of the radicals in (\ref{5.10}) and
(\ref{5.11}) are chosen such that $h_+\ge 0$, $h_-\ge 0$.
Also, we assume here that if a strip domain has both vertices at
the same pole then its height is counted twice.
Comparing $h_+$ and $h_-$, we find three possibilities:
\begin{equation}gin{itemize}
\item[1)]
If $h_+=h_-$, then the domain configuration $\mathcal{D}_Q$ has
only one strip domain $G_2$. This is the case discussed in parts
\textbf{6.3(a) }and \textbf{6.3(b1)} in Section~6.
\item[2)]
The case $h_+>h_-$ corresponds to the configuration with two strip
domains $G_1$ and $G_2$ discussed in part \textbf{6.3(b2)} in
Section~6. In this case, the normalized heights $h_1$ and $h_2$
of the strip domains $G_1$ and $G_2$ can be calculated as follows:
\begin{equation}gin{equation} \lambdabel{7.9.1}
h_1=\frac{1}{2}\left(h_+-h_-\right), \quad h_2=h_-.
\varepsilonnd{equation}
\item[3)]
The case $h_+<h_-1$ corresponds to the configuration with two
strip
domains mentioned in part \textbf{6.3(b3)} in Section~6.
\varepsilonnd{itemize}
Next, we will identify pairs $p_1$, $p_2$, which correspond to
each of the cases \textbf{6.3(a)}, \textbf{6.3(b1)}, and
\textbf{6.3(b2)}. The domain configuration $\mathcal{D}_Q$ has
exactly one strip domain if and only if $h_+=h_-$. Now,
(\ref{5.10}) and (\ref{5.11}) imply that the latter equation is
equivalent to the
following equation:
$$
\begin{equation}gin{array}{l}
\left(\sqrt{(p_1-1)(p_2-1)}-\sqrt{(\bar p_1-1)(\bar
p_2-1)}\right)^2= \\%
\left(\sqrt{(p_1+1)(p_2+1)}-\sqrt{(\bar
p_1+1)(\bar p_2+1)}\right)^2.
\varepsilonnd{array}
$$
Simplifying this equation, we conclude that $h_+=h_-$ if and
only if $p_1$ and $p_2$ satisfy the following equation:
\begin{equation}gin{equation} \lambdabel{5.12}
p_1+\bar p_1+p_2+\bar p_2+|p_1-1||p_2-1|-|p_1+1||p_2+1|=0
\varepsilonnd{equation}
We claim that for a fixed $p_1$ with $\Im p_1\not=0$, the pair
$p_1$, $p_2$ satisfies equation (\ref{5.12}) if and only if
$p_2\in L(p_1)$ or $p_2\in H(p_1)$. Indeed, $p_2\in L(p_1)$ if and
only if
\begin{equation}gin{equation} \lambdabel{5.13}
|p_1-1|+|p_1+1|=|p_2-1|+|p_2+1|.
\varepsilonnd{equation}
Similarly, $p_2\in H(p_1)$ if and
only if
\begin{equation}gin{equation} \lambdabel{5.14}
|p_1-1|-|p_1+1|=|p_2-1|-|p_2+1|.
\varepsilonnd{equation}
Multiplying equations (\ref{5.13}) and (\ref{5.14}), after
simplification we again obtain equation (\ref{5.12}). Therefore,
$p_2\in L(p_1)$ or $p_2\in H(p_1)$ if and only if the pair $p_1$,
$p_2$ satisfy equation (\ref{5.12}). Thus, $\mathcal{D}_Q$ has
only one strip domain if and only if $p_2\in
L(p_1)\setminus\{p_1,\bar p_1\}$ or $p_2\in
H(p_2)\setminus\{p_1,\bar p_1\}$. This proves the first parts of
statements \textbf{6.3(a)} and \textbf{6.3(b1)}.
Now, we will prove that $p_1\in \partial D_\infty$ for all $p_2\in
E_{-1}^+(p_1)$. First, we claim that $p_1\in
\partial D_\infty$ for all $p_2$ sufficiently close to $-1$.
Arguing by contradiction, suppose that there is a sequence $s_k\to
-1$ such that $s_k\in \partial D_\infty^k$ for all
$k=1,2,\ldots$ Here $D_\infty^k\ni \infty$ denotes the
corresponding circle domain of the quadratic differential
$Q_k(z)\,dz^2$ having the form (\ref{5.4}).
From (\ref{5.4}) we find that
$$
Q_k(z)\,dz^2\to
\widehat{Q}(z)\,dz^2:=-\frac{z-p_1}{(z+1)(z-1)^2}\,dz^2,
$$
where convergence is uniform on compact subsets of
$\mathbb{C}\setminus\{-1,1\}$. Since the residue of
$\widehat{Q}(z)$ at $z=\infty$ equals $1$, the quadratic
differential $\widehat{Q}(z)\,dz^2$ has a circle domain
$\widehat{D}_\infty\ni \infty$ and if $\gamma\subset
\widehat{D}_\infty$ is a closed trajectory of
$\widehat{Q}(z)\,dz^2$, then $|\gamma|_{\widehat{Q}}=2\pi$.
Let us show that the boundary of $\widehat{D}_\infty$ consists of
a single critical trajectory $\widehat{\gamma}_\infty$ of
$\widehat{Q}(z)\,dz^2$, which has both its end points at $z=p_1$.
Indeed, $\partial \widehat{D}_\infty$ consists of a finite number
of critical trajectories of $\widehat{Q}(z)\,dz^2$, which have
their end points at finite critical points. Therefore, if $-1\in
\partial \widehat{D}_\infty$, then $\partial \widehat{D}_\infty$
contains a critical trajectory, call it $\widehat{\gamma}_1$,
which joins $z=-1$ and $z=p_1$. Some notations used in this part
of the proof are shown in Fig.~7a. This figure shows the limit
configuration, which is, in fact, impossible as we explain below.
In this case, $\partial \widehat{D}_\infty$ must contain a second
critical trajectory, call it $\widehat{\gamma}_2$, which has both
its end points at $z=p_1$. This implies that $z=1$ is the only
pole of $\widehat{Q}(z)\,dz^2$ lying in a simply connected domain,
call it $\widehat{D}_1$, which is bounded by critical
trajectories. Hence, $\widehat{D}_1$ must be a circle domain of
$\widehat{Q}(z)\,dz^2$. Furthermore, the domain configuration
$\mathcal{D}_{\widehat{Q}}$ consists of two circle domains
$\widehat{D}_1$, $\widehat{D}_\infty$, which in this case must be
the extremal domains of Jenkins module problem on the following
maximum of the
sum of reduced moduli:
$$
m(B_\infty,\infty)+t^2 m(B_1,1) \quad {\mbox{with some fixed $t>0$,}}
$$
where the maximum is taken over all pairs of simply connected
non-overlapping domains $B_\infty\ni \infty$ and $B_1\ni 1$. It is
well known that such a pair of extremal domains is unique; see for
example, \cite{S1}. Therefore, $\widehat{D}_1$ and
$\widehat{D}_\infty$ must be symmetric with respect to the real
line (as is shown, for instance, in Fig.~5d), which is not the
case since $\widehat{Q}(z)\,dz^2$ has only one zero $p_1$ with
$\Im p_1>0$.
Thus, $\partial \widehat{D}_\infty=\widehat{\gamma}_\infty\cup
\{p_1\}$ and $z=-1$ lies in the domain complementary to the
closure of $\widehat{D}_\infty$. Fig.~7b illustrates notations
used further on in this part of the proof.
Let $\tilde \gamma_{-1}$ denote the $\widehat{Q}$-geodesic in the
class of all curves having their end points at $z=-1$, which
separate the points $z=1$ and $z=p_1$ from $z=\infty$. Since
$-1\not \in \partial \widehat{D}_\infty$ it follows that
\begin{equation}gin{equation} \lambdabel{5.16}
|\tilde
\gamma_{-1}|_{\widehat{Q}}>|\widehat{\gamma}_\infty|_{\widehat{Q}}=2\pi.
\varepsilonnd{equation}
Let $\varphirepsilon>0$ be such that
\begin{equation}gin{equation} \lambdabel{5.17}
\varphirepsilon<\frac{1}{4}\left(|\tilde{\gamma}_{-1}|_{\widehat{Q}}-2\pi\right).
\varepsilonnd{equation}
Let $r>0$ be sufficiently small such that
\begin{equation}gin{equation} \lambdabel{5.18}
|[-1,-1+re^{i\theta}]|_{\widehat{Q}}<\varphirepsilon/8 \quad
{\mbox{for all $0\le \theta<2\pi$.}}
\varepsilonnd{equation}
Now let $\tilde \gamma_r$ be the shortest in the
$\widehat{Q}$-metric among all arcs having their end points on the
circle $C_r(-1)=\{z:\,|z+1|=r\}$ and separating the points $z=1$
and $z=p_1$ from the point $z=\infty$ in the exterior of the
circle $C_r(-1)$. It is not difficult to show that there is at
least one such curve $\tilde\gamma_r$. It follows from
(\ref{5.18}) that
\begin{equation}gin{equation} \lambdabel{5.19}
|\tilde\gamma_r|_{\widehat{Q}}>|\tilde\gamma_{-1}|_{\widehat{Q}}-\varphirepsilon/4.
\varepsilonnd{equation}
Since $s_k\to -1$, $s_k\in \partial D_\infty^k$, and $p_1\not \in
D_\infty^k$, it follows that for every sufficiently large $k$
there is a regular trajectory $\gamma(k)$ of $Q_k(z)\,dz^2$
intersecting the circle $C_r(-1)$ and such that the arc
$\gamma'(k)=\gamma(k)\setminus \{z:\,|z+1|\le r\}$ separates the
points $z=1$ and $z=p_1$ from $z=\infty$ in the exterior of
$C_r(-1)$. Since $|\gamma(k)|_{Q_k}=2\pi$ for all $k$ and since
every quadratic differential $Q_k(z)\,dz^2$ has second order poles
at $z=1$ and $z=\infty$ it follows from (\ref{5.4}) that there is
$r_0>0$ small enough such that $\gamma'(k)$ lies on the compact
set $K_0=\{z:\,|z|\le
1/r_0\}\setminus\left(\{z:\,|z-1|<r_0\}\cup\{z:\,|z+1|<r\}\right)$
for all $k$ sufficiently large. We note also that $Q_k(z)\to
\widehat{Q}(z)$ uniformly on $K_0$. This implies, in particular,
that for all $k$ the Euclidean lengthes of
$\gamma'(k)$ are bounded by the same constant and that
\begin{equation}gin{equation} \lambdabel{5.20}
|\gamma'(k)|_{Q_k}\ge |\gamma'(k)|_{\widehat{Q}}-\varphirepsilon/4
\varepsilonnd{equation}
for all $k$ sufficiently large.
Combining (\ref{5.16})--(\ref{5.20}), we obtain the following
relations:
$$
\begin{equation}gin{array}{rl}
2\pi=&|\gamma(k)|_{Q_k}\ge |\gamma'(k)|_{Q_k}\ge
|\gamma'(k)|_{\widehat{Q}}-\varphirepsilon/4\ge
|\tilde\gamma_r|_{\widehat{Q}}-\varphirepsilon/4 \\
>&|\tilde\gamma_{-1}|_{\widehat{Q}}-\varphirepsilon/2>|\tilde\gamma_{-1}|_{\widehat{Q}}
-\frac{1}{2}\left(|\tilde\gamma_{-1}|_{\widehat{Q}}-2\pi\right) \\
=&
\frac{1}{2}\left(|\tilde\gamma_r|_{\widehat{Q}}+2\pi\right)>2\pi,
\varepsilonnd{array}
$$
which, of course, is absurd. Thus, $p_2\in \partial D_\infty$ for
all $p_2$ sufficiently close to $-1$.
Let $\mathcal Delta\not= \varepsilonmptyset$ be the set of all $p_2\in
E_{-1}^+(p_1)$ such that $p_1\in \partial D_\infty$. To prove
that $\mathcal Delta=E_{-1}^+(p_1)\setminus\{-1\}$, it is sufficient to
show that $\mathcal Delta$ is closed and open in $E_{-1}^+(p_1)$. Arguing
by contradiction, we suppose that there is a sequence of poles
$s_k:=p_2^k\in E_{-1}^+(p_1)$, $k=1,2,\ldots,$ such that $s_k\to
s_0:=p_2^0\in E_{-1}^+(p_1)$ and $p_1\in
\partial D_\infty^k$ for all $k=1,2,\ldots$ but $p_1\not\in \partial D_\infty^0$.
In this part of the proof, the index $k=0,1,2,\ldots$, used in the
notations $D_\infty^k$, $\tilde\gamma_k$, etc., will denote
domains, trajectories, and other objects corresponding to the
quadratic differential $Q_k(z)\,dz^2$ defined by (\ref{5.4}).
Since $\partial D_\infty^0$ contains a critical point and
$p_1\not\in \partial D_\infty^0$, we must have $p_2^0\in
\partial D_\infty^0$.
Fig.~7c illustrates some notations used in this part of the proof.
In this case, the boundary $\partial D_\infty^0$ consists of a
single critical trajectory $\gamma_\infty^0$ and its end points,
each of which is at $z=p_2^0$. In addition, there is a critical
trajectory of infinite $Q^0$-length, called it $\hat\gamma$, which
has one end point at $p_2^0$ and which approaches to the pole
$z=-1$ or the pole $z=1$ in the other direction. Let $P_0$ be a
point on $\hat \gamma$ such that the $Q^0$-length of the arc
$\hat\gamma_0$ of $\hat\gamma$ joining $p_2^0$ and $P_0$ equals
$L$, where $L>0$ is sufficiently large. For $\delta>0$
sufficiently small, let $\gamma_1^\perp$ and $\gamma_2^\perp$
denote disjoint open arcs on the orthogonal trajectory of
$Q^0(z)\,dz^2$ passing through $P_0$ such that each of
$\gamma_1^\perp$ and $\gamma_2^\perp$ has one end point at $P_0$
and each of them has $Q^0$-length equal to $\delta$. If $\delta$
is small enough, then there is an arc of a trajectory of
$Q^0(z)\,dz^2$, call it $\tilde\gamma$, which connects the second
end point of $\gamma_1^\perp$ with the second end point of
$\gamma_2^\perp$. Now, let $D(\delta)$ be the domain, the boundary
of which consists of the arcs $\gamma_\infty^0$, $\hat\gamma_0$,
$\gamma_1^\perp$, $\gamma_2^\perp$, and their end points. In the
terminology explained in Section~3, the domain $D(\delta)$ is a
$Q^0$-rectangle of $Q^o$-height $\delta$.
If
$\delta>0$ is sufficiently small, then $p_1$ belong to the bounded
component of $\mathbb{C}\setminus \overline{D(\delta)}$. Let
$\tilde\gamma_1$ be the arc of a trajectory of $Q^0(z)\,dz^2$,
which divide $D(\delta)$ into two $Q^0$-rectangles, each of which
has the $Q^0$-height equal to $\delta/2$. Since $p_1\in \partial
D_k$ for all $k$ and $p_1$ belongs to the bounded component of
$\mathbb{C}\setminus \overline{D(\delta)}$, it follows that, for
each $k=1,2,\ldots,$ there is a closed trajectory $\hat\gamma_k$
of $Q_k(z)\,dz^2$ lying in $D_\infty^k$, which intersects
$\tilde\gamma_1$ at some point $\tilde z_k\in D(\delta)$.
Since $Q_k(z)\to Q^0(z)$ it follows that, for all sufficiently
large $k$, the trajectory $\hat\gamma_k$ has an arc
$\tilde\gamma_k$ such that $\tilde\gamma_k\subset D(\delta)$ and
$\tilde\gamma_k$ has one end point on each of the arcs
$\gamma_1^\perp$ and $\gamma_2^\perp$.
Now, since $Q_k(z)\to Q^0(z)$ uniformly on $\overline{D(\delta)}$
it follows that
$$
|\hat\gamma_k|_{Q_k}\ge |\tilde\gamma_k|_{Q_k}\to
|\tilde\gamma_1|_{Q^0}=|\gamma_\infty^0|_{Q^0}+2|\hat\gamma_0|_{Q^0}=2\pi+2L,
$$
contradicting to the fact that $|\hat\gamma_k|_{Q_k}=2\pi$. The latter fact
follows from the assumption that $\hat \gamma_k$ is a closed
trajectory of $Q_k(z)\,dz^2$, which lies in a circle domain $D_\infty^k$.
Thus, we have proved that $\mathcal Delta$ is closed in $E_{-1}^+(p_1)$. A
similar argument can be used to show that $\mathcal Delta$ is open in
$E_{-1}^+(p_1)$. The difference is that to construct a domain
$D(\delta)$, we now use an arc $\tilde\gamma_1$ of a critical
trajectory $\hat \gamma_1$, which has one of its end points at the
pole $p_1$ and not at the pole $p_1^0$ as we had in the previous
case.
Therefore, we have proved that if $p_2\in E_{-1}^+(p_1)$, then
$p_1\in
\partial D_\infty$. The same argument can be used to prove that if
$p_2\in E_1^+(p_1)$, then $p_1\in \partial D_\infty$.
Finally, if $p_2\in E_1^-(p_1)$ or $p_2\in E_{-1}^-(p_1)$, then we
can switch the roles of the poles $p_1$ and $p_2$ in our previous
proof and conclude that $p_2\in \partial D_\infty$ in these cases.
This proves the first part of statement \textbf{6.3(b2)}.
Now, possible positions of zeros $p_1$ and $p_2$ on boundaries of
the corresponding circle and strip domains are determined for all
cases. Next, we will discuss limiting behavior of critical
trajectories. We will give a proof for the most general case when
the domain configuration consists of a circle domain $D_\infty$
and strip domains $G_1$ and $G_2$. In all other cases proofs are
similar.
Let $\mathcal Delta$ denote the set of pairs $(p_1,p_2)$, for which the
limiting behavior of critical trajectories is shown in Fig.~4a or
in more general case in Fig.~4b. That is when $\gamma_1$ joins $p_1\in
\partial D_\infty\cap \partial G_1$ and $z=1$, $\gamma_{-1}$ joins
$p_2\in
\partial G_1\cap \partial G_2$ and $z=-1$, and $\gamma_0^+$ and
$\gamma_0^-$ each joins $p_2$ and $z=1$. First, we note that
$\mathcal Delta$ is not empty since $(p_1,p_2)\in \mathcal Delta$ when $p_1>1$
and $-p_1<p_2<-1$. In this case the intervals $(p_2,-1)$ and
$(1,p_1)$ represent critical trajectories $\gamma_1$ and
$\gamma_{-1}$ and critical trajectories $\gamma_0^+$ and
$\gamma_0^-$ connect a zero at $p_2$ with a pole at $z=1$; see
Fig.~4a.
We claim that $\mathcal Delta$ is open. To prove this claim, suppose that
$(p_1^0,p_2^0)\in \mathcal Delta$ and that $(p_1^k,p_2^k)\to
(p_1^0,p_2^0)$ as $k\to \infty$, $k=1,2,\ldots$ Fix $\varphirepsilon>$
small enough and consider the arc
$\gamma_1^0(\varphirepsilon)=\gamma_1^0\setminus
\{z:\,|z-1|<\varphirepsilon\}$ of the critical trajectory
$\gamma_1^0$, which goes from $p_1^0$ to the pole $z=1$. Since
$(p_1^k,p_2^k)\to (p_1^0,p_2^0)$ it follows that for all $k$
sufficiently big there is a critical trajectory $\gamma_1^k$
having one point at $p_1^k$ which has a subarc
$\gamma_1^k(\varphirepsilon)$ which lies in the
$\varphirepsilon/10$-neighborhood of the arc
$\gamma_1^0(\varphirepsilon)$. In particular, eventually,
$\gamma_1^k(\varphirepsilon)$ enters the disk
$\{z:\,|z-1|<\varphirepsilon\}$. Therefore, it follows from the
standard continuity argument and Lemma~4 that $\gamma_1^k$
approaches the pole $z=1$. The same argument works for all other
critical trajectories of the quadratic differential (\ref{6.1})
with $p_1=p_1^k$, $p_2=p_2^k$. Thus, we have proved that $\mathcal Delta$
is open.
Same argument can be applied to show that all other sets of points
$(p_1,p_2)$ responsible for different types of limiting behavior
of critical trajectories mentioned in part \textbf{6.3(b2)} of
Theorem~\ref{Theorem 5.1} are also nonempty and open. The latter
implies that each of these sets must coincide with some connected
component of the set $\mathbb{C}\setminus (L(p_1)\cup H(p_1))$.
This proves the desired statement in the case under consideration.
\textbf{7.B.} The local behavior of trajectories near second order
poles at $z=1$ and $z=-1$ is controlled by Laurent coefficients
$C_1$ and $C_{-1}$, respectively, which are given by formula
(\ref{5.1}). The radial structure near $z=1$ or near $z=-1$ occurs
if and only if $C_1<0$ or $C_{-1}<0$, respectively. The latter
inequalities are equivalent to the following relations:
\begin{equation}gin{equation} \lambdabel{5.23}
\arg(p_1-1)=-\arg(p_2-1)+\pi
\varepsilonnd{equation}
or
\begin{equation}gin{equation} \lambdabel{5.24}
\arg(p_1+1)=-\arg(p_2+1)+\pi.
\varepsilonnd{equation}
Now, statement \textbf{(1)} about radial behavior follows from
(\ref{5.23}) and (\ref{5.24}).
Next, trajectories of $Q(z)\,dz^2$ approaching the pole $z=1$
spiral clockwise if and only if $0<\arg C_1<\pi$. The latter is
equivalent to the inequalities:
$$
-\arg{p_1-1}<\arg(p_2-1)<-\arg(p_1-1)+\pi,
$$
which imply the desired statement for the case when trajectories
of $Q(z)\,dz^2$ approaching $z=1$ spiral clockwise. In the
remaining cases the proof is similar.
The proof of Theorem~\ref{Theorem 5.1} is now complete.
$\Box$
\begin{equation}gin{remark} \lambdabel{Remark-1}
The case when $\Im p_1=0$ but $\Im p_2\not=0$ can be reduced to
the case covered by Theorem~\ref{Theorem 5.1} by changing
numeration of zeros. In the remaining case when $\Im p_1=0$ and
$\Im p_2=0$, the domain configurations are rather simple; they are
symmetric with respect to the real axis as it is shown in
Figures~1a, 1b, 2a, 3a, and some other figures.
\varepsilonnd{remark}
\section{Identifying simple critical geodesics and critical loops} \lambdabel{Section-7}
\setcounter{equation}{0}
Topological information obtained in Section~6 is sufficient to
identify all critical geodesics and all critical geodesic loops of
the quadratic differential (\ref{6.1}) in all cases. In
particular, we can identify all simple geodesics.
Cases \textbf{6.1(a)} and \textbf{6.1(b)}; see Fig.~1a and
Fig.~1b. Let $\gamma$ be a geodesic joining $p_1$ and $p_2$. Since
$D_\infty$, $D_1$, and $D_{-1}$ are simply connected and $p_1\in
\partial D_\infty\cap
\partial D_1$ and $p_2\in \partial D_\infty\cap \partial D_{-1}$
it follows from Lemma~4 that $\gamma$ does not intersect
$D_\infty$, $D_1$, and $D_{-1}$. In this case, $\gamma$ must be
composed of a finite numbers of copies of $\gamma_0$, a finite
number of copies of $\gamma_1$, and a finite number of copies of
$\gamma_{-1}$. Therefore the only simple geodesic joining $p_1$
and $p_2$ in this case is the segment $\gamma_0=[p_2,p_1]$.
In addition, by Lemma~\ref{Lemma-4.2}, $\gamma_1$ is the only
simple non-degenerate geodesic from the point $p_1$ to itself and
$\gamma_{-1}$ is the only short geodesic from $p_2$ to $p_2$.
Case \textbf{6.1(c)}; see Fig.~1c.
As in the previous case, any geodesic $\gamma$ joining $p_1$ and
$p_2$ must be composed of a finite number of copies of $\gamma_0$,
a finite number of copies of $\gamma_1$, and a finite number of
copies of $\gamma_{-1}$. Thus, in this case there exist exactly
three simple geodesics joining $p_1$ and $p_2$, which are
$\gamma_0$, $\gamma_1$, and $\gamma_{-1}$. By
Lemma~\ref{Lemma-4.2}, there are no geodesic loops in this case.
Case \textbf{6.2}; see Fig.~2a, 2b.
Suppose that $\mathcal{D}_Q$ consists of circle domains $D_\infty$
and $D_{-1}$ and
a strip domain $G_1$. Let $\gamma$ be a geodesic joining $p_1$
and $p_2$. If $\gamma$ contains a point $\zetata\in \gamma_{-1}$ or
a point $\zetata\in \gamma_\infty$, then it follows from
Lemma~\ref{Lemma-4.1} that $\gamma_{-1}$ or, respectively, $\gamma_\infty$ is a subarc
of $\gamma$. Thus, $\gamma$ is not simple in these cases.
Suppose now that $\gamma\subset G_1\cup \gamma_1^+\cup
\gamma_1^-$.
Since $G_1$ is a strip domain the function $w=F(z)$ defined by
\begin{equation}gin{equation} \lambdabel{4.1}
F(z)=\frac{1}{2\pi}\,\int_{p_1}^z \sqrt{Q(z)}\,dz,
\varepsilonnd{equation}
with an appropriate choice of the radical, maps $G_1$ conformally and
one-to-one onto the horizontal strip $S_{h_1}$, where $S_h=\{w:\,0<\Im
w<h_1\}$,
in such a way
that the trajectory $\gamma_\infty$ is mapped onto an interval
$(x_1,x_1')\subset \mathbb{R}$ with $x_1=0$ and $x'_1=1$. Here
$h_1$ is the normalized height of the strip domain $G_1$ defined
by (\ref{7.9.1}).
Fig.~8a and Fig.~9a illustrate some notions relevant to Case
\textbf{6.2}.
To simplify notations in our figures, we will use the same
notations for $Q$-geodesics (such as $\gamma_\infty$,
$\gamma_{11}$, $\gamma'_{12}$, etc.) in the $z$-plane and for
their images under the mapping $w=F(z)$ in the $w$-plane.
The indefinite integral $\Phi(z)=\frac{1}{2\pi}\int\sqrt{Q(z)}\,dz$ can be
expressed explicitly in terms of elementary functions as follows:
\begin{equation}gin{equation} \lambdabel{7.9.2}
\begin{equation}gin{array}{ll}
\hspace{-0.2cm}\Phi(z)\hspace{-0.25cm}&=\frac{1}{4\pi i}\left(
\sqrt{(p_1-1)(p_2-1)}\log(z-1)-\sqrt{(p_1+1)(p_2+1)}\log(z+1)
\right. \\
{}&+4\log(\sqrt{z-p_1}+\sqrt{z-p_2})\\
{}&+2\sqrt{(p_1+1)(p_2+1)} \log(\sqrt{(p_1+1)(z-p_2)}-\sqrt{(p_2+1)(z-p_1)}) \\
{}&-\left. 2\sqrt{(p_1-1)(p_2-1)}
\log(\sqrt{(p_1-1)(z-p_2)}-\sqrt{(p_2-1)(z-p_1)})\right).
\varepsilonnd{array}
\varepsilonnd{equation}
Equation (\ref{7.9.2}) can be verified by straightforward
differentiation. Alternatively, it can be verified with
\varepsilonmph{Mathematica} or \varepsilonmph{Maple}. With
(\ref{7.9.2}) at hands, the function $F(z)$ can be written as
\begin{equation}gin{equation} \lambdabel{7.9.3}
F(z)=\Phi(z)-\Phi(p_1),
\varepsilonnd{equation}
where
\begin{equation}gin{equation} \lambdabel{7.9.4}
\Phi(p_1)=\frac{1}{4\pi
i}\left(2+\sqrt{(p_1-1)(p_2-1)}-\sqrt{(p_1+1)(p_2+1)}\right)\log(p_1-p_2).
\varepsilonnd{equation}
Calculating $\Phi(p_2)$,
after some algebra, we find that:
\begin{equation}gin{equation} \lambdabel{7.9.5}
F(p_2)=\frac{1}{2}+\frac{1}{4}\left(\sqrt{(p_1-1)(p_2-1)}-\sqrt{(p_1+1)(p_2+1)}\right).
\varepsilonnd{equation}
Of course, all branches of the radicals and logarithms in
(\ref{7.9.2})--(\ref{7.9.5}) have to be appropriately chosen.
To explain more precisely our choice of branches of multi-valued
functions in (\ref{7.9.2})--(\ref{7.9.5}), we note that the points
$p_1$, $p_2$ and points of the arcs $\gamma_1^+$ and $\gamma_1^-$
each represents two distinct boundary points of $G_1$ and
therefore every such point has two images under the mapping
$F(z)$. These images will be denoted by $x_1(\zetata)$ and
$x'_1(\zetata)$ if $\zetata\in \gamma_1^+\cup\{p_1\}$ and by
$x_2(\zetata)+ih_1$ and $x'_2(\zetata)+ih_1$ if $\zetata\in
\gamma_1^-\cup \{p_2\}$. We assume here that
$x_1(\zetata)<x'_1(\zetata)$ for all $\zetata\in \gamma_1^+\cup \{p_1\}$
and $x_2(\zetata)<x'_2(\zetata)$ for all $\zetata\in \gamma_1^-\cup
\{p_2\}$. In accordance with our notation above, $x_1(p_1)=x_1=0$
and $x'_1(p_1)=x'_1=1$. We also will abbreviate $x_2(p_2)$ and
$x'_2(p_2)$ as $x_2$ and $x'_2$, respectively.
For every $\zetata\in \gamma_1^+$, the segments $[x_1(\zetata),x_1]$
and $[x'_1,x'_1(\zetata)]$ are the images of the same arc on
$\gamma_1^+$. Therefore they have equal lengthes. Similarly, the
segments $[x_2(\zetata)+ih_1,x_2+ih_1]$ and
$[x'_2+ih_1,x'_2(\zetata)+ih_1]$ have equal
lengthes. Thus, for every $\zetata\in \gamma_1^+$ and every $\zetata\in \gamma_1^-$, we have, respectively:
\begin{equation}gin{equation} \lambdabel{4.2}
x_1-x_1(\zetata)=x'_1(\zetata)-x'_1 \quad {\mbox{and}} \quad
x_2-x_2(\zetata)=x'_2(\zetata)-x'_2.
\varepsilonnd{equation}
We know that the preimage under the mapping $F(z)$ of every
straight line segment is a geodesic. This immediately implies that
in the case under consideration there exist four simple critical
geodesics, which are the following preimages:
\begin{equation}gin{equation} \lambdabel{4.3}
\begin{equation}gin{array}{ll} \gamma_{12}=F^{-1}((x_1,x_2+ih_1)),& \quad
\gamma'_{12}=F^{-1}((x_1,x'_2+ih_1)),\\
\gamma_{21}=F^{-1}((x'_1,x_2+ih_1)),& \quad
\gamma'_{21}=F^{-1}((x'_1,x'_2+ih_1)).
\varepsilonnd{array}
\varepsilonnd{equation}
The geodesic loops $\gamma_\infty$ and $\gamma_{-1}$ are the
following preimeges:
\begin{equation}gin{equation} \lambdabel{4.4}
\gamma_\infty=F^{-1}((x_1,x_1)), \quad
\gamma_{-1}=F^{-1}((x_2+ih_1,x'_2+ih_1)).
\varepsilonnd{equation}
We claim that there is no other simple geodesic joining the points
$p_1$ and $p_2$. Fig.~9a illustrates some notation used in the
proof of this claim. Suppose that $\tau$ is a geodesic ray issuing
from $p_1$ into the region $G_1$. Let $\tau_k$, $k=1,\ldots, N$,
be connected components of the intersection $\tau\cap G_1$
enumerated in their natural order on $\tau$. In particular,
$\tau_1$ starts at $p_1$. We may have finite or infinite number of
such components. Thus, $N$ is a finite number or $N=\infty$. Let
$l_k=F(\tau_k)$. Since all $\tau_k$ lie on the same geodesic it
follows that $l_k$ are parallel line intervals in $S$ joining the
real axis and the horizontal line $L_{h_1}$, where $L_h =\{w:\,
\Im w=h\}$. Let $v'_k$ and $v''_k$ be the initial point and
terminal point of $l_k$, respectively. Then $v'_k=e'_k$ and
$v''_k=e''_k+ih_1$ with real $e'_k$ and $e''_k$ if $k$ is odd and
$v'_k=e'_k+ih_1$, $v''_k=e''_k$ with real $e'_k$ and $e''_k$ if
$k$ is even.
The interval $l_1$ may start at $x_1$ or at $x'_1$. To be
definite, suppose that $e'_1=x_1$. For the position of $e''_1$ we
have the
following possibilities:
\begin{equation}gin{enumerate}
\item[(a)]
$e''_1=x_2$ or $e''_1=x'_2$. In this case, $\tau_1=\gamma_{12}$ or
$\tau_1=\gamma'_{12}$. Thus we obtain two out of four
geodesics in (\ref{4.3}).
\item[(b)]
$x_1<e''_1<x'_1$. In this case, $\tau_1$ has its end point on
$\gamma_{-1}$. By Lemma~\ref{Lemma-4.1}, the continuation of
$\tau_1$ as a geodesic will stay in $D_{-1}$ and will approach to
the pole $z=-1$. Thus, $\tau$ is not a geodesic from $p_1$ to
$p_2$ or a geodesic loop from $p_1$ to itself in
this case.
\item[(c)]
$e''_1>x'_2$. Let $d=e''_1-x'_2$. It follows from (\ref{4.2}) that
$e'_2=x_2-d$. Then $e''_2=x_1-d$. In general,
$e'_{2k-1}=x'_1+(k-1)d$, $e''_{2k-1}=x'_2+kd$ for $k=1,2,\ldots$,
and $e'_{2k}=x_2-kd$, $e''_{2k}=x_1-kd$ for $k=1,2,\ldots$. Thus,
$\tau$ cannot terminate at $p_1$ or $p_2$. Instead, $\tau$
approaches to the pole at $z=1$ as a logarithmic spiral.
\item[(d)]
$e''_1<x_2$. Let $d_0=x_2-e''_1$. Then $e'_2=x'_2+d_0$ by
(\ref{4.2}). For the position of $e''_2$ we have three
possibilities.
\begin{equation}gin{itemize}
\item[$(\alphapha$)]
$x_1<e''_2<x'_1$. In this case by Lemma~\ref{Lemma-4.1}, the
continuation of $\tau_2$ as a geodesic ray will stay in $D_\infty$
and will approach to the pole $z=\infty$. Thus, $\tau$ is not a
geodesic from $p_1$ to $p_2$ or a geodesic loop in this
case.
\item[$(\begin{equation}taa$)]
$e''_2=x'_1$. In this case, $\tau$ is a critical geodesic loop
$\gamma_{11}=F^{-1}((x_1,v''_1]\cup [v'_2,x'_1))$ from $p_1$ to
itself. We emphasize here, that since the segments $l_1$ and $l_2$
are parallel a critical geodesic loop from $p_1$ to itself occurs
if and only if
$|\gamma_\infty|_Q=x'_1-x_1>x'_2-x_2=|\gamma_{-1}|_Q$. If $|\gamma_\infty|_Q<|
\gamma_{-1}|_Q$, then there is a critical geodesic loop $\gamma_{22}$ with end points at $p_2$.
\item[$(\gamma$)]
$e''_2>x'_1$. Let $d=e''_2-x'_1$. Then, as in the case c), we
obtain that $e'_{2k+1}=x_1-kd$, $e''_{2k+1}=x_2-d_0-kd$ for
$k=1,2,\ldots$, and $e'_{2k}=x'_2+d_0+kd$, $e''_{2k}=x'_1+kd$ for
$k=1,2,\ldots$. Therefore, $\tau$ does not terminate at $p_1$ or
$p_2$. Instead, $\tau$ approaches to the pole at $z=1$ as a
logarithmic spiral.
\varepsilonnd{itemize}
\varepsilonnd{enumerate}
If $l_1$ has its initial point at $x'_1$, the same argument shows
that there are exactly two geodesics joining $p_1$ and $p_2$,
which are the geodesics $\gamma_{21}$ and $\gamma'_{21}$ defined
by (\ref{4.3}).
Combining our findings for Case \textbf{6.2}, we conclude that in
this case there exist exactly four distinct geodesics joining
$p_1$ and $p_2$, which are given by (\ref{4.3}). The geodesic
loops $\gamma_\infty$ and $\gamma_{-1}$ are given by (\ref{4.4}).
In addition, if $|\gamma_\infty|_Q\not=|\gamma_{-1}|_Q$, then
there is exactly one geodesic loop containing the pole $z=1$ in
its interior domain, which has its end points at a zero of
$Q(z)\,dz^2$. This loop has the pole $z=1$ in its interior domain,
which does not contain other critical points of $Q(z)\,dz^2$, and
has both its end points at $p_1$ or at $p_2$, if
$|\gamma_\infty|_Q>|\gamma_{-1}|_Q$ or
$|\gamma_\infty|_Q<|\gamma_{-1}|_Q$, respectively.
Finally, if $|\gamma_\infty|_Q=|\gamma_{-1}|_Q$, then the
geodesics $\gamma_{12}$ and $\gamma'_{21}$ together with points
$z=p_1$ and $p_2$ form a boundary of a simply connected bounded
domain, which contains the pole $z=1$ and does not contain other
critical points of $Q(z)\,dz^2$. There are no geodesic loops
containing $z=1$ in its interior domain in this case.
The argument based on the construction of parallel segments
divergent to $\infty$, which was used above to prove non-existence
of some geodesics, will be used for the same purpose in several
other cases considered below. Since the detailed construction is
rather lengthy, the detailed exposition will be given for one more
case when we have two strip domains. In other cases, we will just
refer to this argument (which actually is rather standard, see
\cite[Ch. IV]{Str}) and call it the ``proof by
construction of divergent geodesic segments''.
Case \textbf{6.3(a)}; see Fig.~8b. In this case, the domain
configuration $\mathcal{D}_Q$ consists of a circle domain
$D_\infty$ and a strip domain $G_2$ having its vertices at the
poles $z=1$ and $z=-1$. The function $F(z)$ defined by (\ref{4.1})
maps $G_2$ conformally and one-to-one onto the strip $S_{h_1}$
such that the trajectory $\gamma_\infty^+$ is mapped onto the
interval $(x_1,x_2)\subset \mathbb{R}$ with $x_1=0$ and some
$x_2$, $0<x_2<1$. The points $z=p_1$ and $z=p_2$ each has two
images under the mapping $F(z)$. Let $x_1=0$ and $x'_1+ih_1$ with
some real $x'_1$ be the images of $p_1$ and let $x_2$ and
$x'_2+ih_1$ with $x'_2=x'_1+(1-x_2)$ be the images of $p_2$.
Arguing as in Case \textbf{6.2}, one can easily find four distinct
simple geodesics joining the points $p_1$ and $p_2$. These
geodesics are:
$$
\begin{equation}gin{array}{ll}
\gamma_{12}=F^{-1}((x_1,x_2))=\gamma_\infty^+, &\quad
\gamma'_{12}=F^{-1}((x'_1+ih_1,x'_2+ih_1))=\gamma_\infty^-, \\
\gamma_{21}=F^{-1}((x_1,x'_2+ih_1)), &\quad
\gamma'_{21}=F^{-1}((x_2,x'_1+ih_1)).
\varepsilonnd{array}
$$
In addition, there are two critical geodesic loops:
$$
\gamma_{11}=F^{-1}((x_1,x'_1+ih_1)) \quad {\mbox{and}} \quad
\gamma_{22}=F^{-1}((x_2,x'_2+ih_1)).
$$
It follows from Lemma~\ref{Lemma-4.2} that there are no other such
loops.
Using the proof by construction of divergent geodesic segments as
in Case \textbf{6.2}, we can show that there are no other simple
geodesics joining $p_1$ and $p_2$.
Case \textbf{6.3(b1)}; see Fig.~8c. We still have a circle domain
$D_\infty$ and a strip domain $G_2$. In this case, the function
$F(z)$ defined by (\ref{4.1}) as in Case \textbf{6.2} maps $G_2$
conformally and one-to-one onto $S_{h_1}$ such that
$\gamma_\infty$ is mapped onto the interval $(x_1,x'_1)\subset
\mathbb{R}$, where $x_1=0$ and $x'_1=1$. The difference is that
that now the point $p_2$ represents three boundary points of
$G_2$. Two of them belong to the side $l_2$ and the third point
belongs to the side $l_1$. Accordingly, there are three images of
$p_2$ under the mapping $F(z)$, which we will denote by
$x_2+ih_1$, $x'_2$, and $x''_2$. Here $x_2$ may be any real number
while $x'_2$ and $x''_2$ satisfy the following conditions:
$$
x'_2>x'_1, \quad x''_2<x_1, \quad {\mbox{and}} \quad x'_2-x'_1=x_1-x''_2.
$$
In this case, there are three short geodesics, which are the
following
preimages:
$$
\gamma_0=F^{-1}((x''_1,x_1))=F^{-1}((x'_1,x'_2))
$$
and
$$
\gamma_{12}=F^{-1}((x_1,x_2+ih_1)), \quad
\gamma'_{12}=F^{-1}((x'_1,x_2+ih_1)).
$$
In addition, there are three geodesic loops:
$$
\gamma_\infty=F^{-1}((x_1,x'_1)),
\quad \gamma'_{22}=F^{-1}((x_2+ih_1,x'_2)), \quad
\gamma''_{22}=F^{-1}((x_2+ih_1,x''_2)).
$$
Using the proof by construction of divergent segments as above, it
is not difficult to show that there are no other simple geodesics
joining the points $p_1$ and $p_2$.
Case \textbf{6.3(b2)}. This is the most general case with many
subcases illustrated in Fig.~10a-10i. In this case we have a
circle domain $D_\infty$ and two strip domains $G_1$ and $G_2$. We
assume that $\mathcal{D}_Q$ has topological type shown in
Fig.~4b. In other cases the proof follows same lines. The function
$F(z)$ defined by (\ref{4.1}) maps $G_1$ conformally and
one-to-one onto the strip $S_{h_1}$ such that $\gamma_\infty$ is
mapped onto the interval $(x_1,x'_1)\subset \mathbb{R}$, where
$x_1=0$ and $x'_1=1$. The point $p_2$ represents one boundary
point of $G_1$ and two boundary points of $G_2$. Let $x_2+ih_1$ be
the image of $p_2$ considered as a boundary point of $G_1$. Then
the trajectory $\gamma_0^+$ considered as boundary arc of $G_1$ is
mapped onto the ray $r_1=\{w=t+ih_1:\,t<x_2\}$, while the
trajectory $\gamma_0^-$ is mapped onto the ray
$r_2=\{w=t+ih_1:\,t>x_2\}$. The function $F(z)$ can be continued
analytically through the trajectory $\gamma_0^+$. The continued
function (for which we keep our previous notation $F(z)$) maps
$G_2$ conformally and one-to-one onto the strip $S(h_1,h)= \{w:\,
h_1<\Im w<h\}$ with $h=h_1+h_2$, where $h_1$ and $h_2$ are defined
by (\ref{7.9.1}).
Two boundary points of $G_2$ situated at $p_2$ are mapped onto the
points $x_2+ih_1$ and $x'_2+ih$ with some $x'_2\in \mathbb{R}$.
Thus, the domain $\widetilde{D}=G_1\cup G_2\cup \gamma_0^+$ is
mapped by $F(z)$ conformally and one-to-one onto the slit strip
$\widehat{S}(h_1,h)=\{w: \, 0<\Im w <h\}\setminus
\{w=t+ih_1:\,t\ge x_2\}$.
We note that every boundary point $\zetata\in
\gamma_1\cup\gamma_{-1}\cup \gamma_0^-$ under the mapping $F(z)$
has two images $w_1(\zetata)$ and $w_2(\zetata)$, which satisfy the
following conditions similar to conditions (\ref{4.2}):
\begin{equation}gin{equation} \lambdabel{4.8}
x_1-w_1(\zetata)=w_2(\zetata)-x'_1>0 \quad {\mbox{if $\zetata\in
\gamma_1$}},
\varepsilonnd{equation}
\begin{equation}gin{equation} \lambdabel{4.9}
w_1(\zetata)=u_1(\zetata)+ih, \ \ w_2(\zetata)=u_2(\zetata)+ih_1,
\varepsilonnd{equation}
where $x'_2-u_1(\zetata)=u_2(\zetata)-x_2>0$ if $\zetata\in \gamma_0^-$,
and
$$
w_1(\zetata)=u_1(\zetata)+ih, \ \ w_2(\zetata)=u_2(\zetata)+ih_1,
$$
where $u_1(\zetata)-x'_2=u_2(\zetata)-x_2>0$ if $\zetata\in
\gamma_{-1}$.
Consider four straight lines $P_k$, $k=1,2,3,4$, where $P_2$
passes through $x'_1$ and $x_2+ih_1$, $P_3$ passes through $x_1$
and $x_2+ih_1$, $P_1$ passes through $x_1$ and is parallel to
$P_2$, and $P_4$ passes through $x'_1$ and is parallel to $P_3$.
Let $u_k+ih$ denote the point of intersection of $P_k$ and the
horizontal line $L(h)$, where $L(m)$ stands for the line
$\{w:\,\Im w=m\}$. Then the points $u_k+ih$, $k=1,2,3,4$, are
ordered in the positive direction on $L(h)$; see Fig.~10a.
Next, we consider five possible positions for $x'_2$, which
correspond to ``non-degenerate'' cases and four positions
corresponding to ``degenerate'' cases. Fig.~10a--10i illustrate
our constructions of critical geodesics and critical geodesic
loops in all these cases. First, we will work with non-degenerate
cases, which are cases (a), (c), (e), (g), and (i)
and after that we will briefly mention degenerate cases (b), (d), (f), and (h).
\begin{equation}gin{itemize}
\item[(a)]
$x'_2<u_1$. Then the slit strip $S_1$ contains four intervals:
$(x_1,x_2+ih_1)$, $(x'_1,x_2+ih_1)$, $(x_1,x'_2+ih)$, and
$(x'_1,x'_2+ih)$. Therefore the preimages of these intervals under
the mapping
$F(z)$ provide four distinct geodesics joining the points $p_1$ and $p_2$:
\begin{equation}gin{equation} \lambdabel{4.11}
\begin{equation}gin{array}{ll}
\gamma_{12}=F^{-1}((x_1,x_2+ih_1)), &\quad
\gamma'_{12}=F^{-1}((x'_1,x_2+ih_1)), \\
\gamma_{21}=F^{-1}((x_1,x'_2+ih)), &\quad
\gamma'_{21}=F^{-1}((x'_1,x'_2+ih)). \\
\varepsilonnd{array}
\varepsilonnd{equation}
In addition, there are two critical geodesic loops:
\begin{equation}gin{equation} \lambdabel{4.12}
\gamma_\infty=F^{-1}((x_1,x'_1)) \quad {\mbox{and }} \quad
\gamma_{22}=F^{-1}((x_2+ih_1,x'_2+ih)).
\varepsilonnd{equation}
The curve $\gamma_{22}\cup\{p_2\}$ bounds a simply connected
domain, call it $D_{-1}$, which contains the trajectory $\gamma_2$
and the pole $z=-1$.
One more critical geodesic loop can be found as follows. Let $P_5$
be the line through $x'_2+ih$ that is parallel to $P_1$ and let
$u'_5$ be the point of intersection of $P_5$ with the real axis.
It follows from elementary geometry that there exists a point
$u_5$, $u'_5<u_5<x_1$ such that the line segments $[x'_2+ih,u_5]$
and $[u_6, x_2+ih_1]$ with $u_6=x'_1+x_1-u_5$ are parallel to each
other. Therefore, it follows from equation (\ref{4.8}) that the
preimage $\gamma'_{22}=F^{-1}((x'_2+ih,u_5]\cup [u_6,x_2+ih_1))$
is a geodesic loop from $p_2$ to $p_2$ containing the pole $z=1$
in its interior domain.
We claim that there no other simple critical geodesics in this
case. The proof is by the method of construction of divergent
geodesic segments. An example of such construction for the case
under consideration is shown in Fig.~9b.
Suppose that $\tau$ is a geodesic ray issuing from $p_1$ into the
region $\widetilde{G}$. Let $\tau_k$, $k=1,\ldots,N$, where $N$ is
a finite integer or $N=\infty$, be connected component of
$\tau\cap \widetilde{G}$ enumerated in the natural order on
$\tau$. Let $l_k=F(\tau_k)$ and let $e'_k$ and $e''_k$ be the
initial and terminal points of $l_k$, respectively.
The interval $l_1$ may start at $x_1$ or at $x'_1$. To be
definite, assume that $e'_1=x_1$. Then for $e''_1$ we have the
following cases:
\begin{equation}gin{itemize}
\item[$(\alphapha$)]
$e''_1=x'_2-d_1+ih$ with some $d_1>0$,
\item[$(\begin{equation}taa$)]
$e''_1=x'_2+d_1+ih$ with some $d_1>0$,
\item[$(\gamma$)]
$e''_1=x_2+d_1+ih_1$ with some $d_1>0$.
\varepsilonnd{itemize}
We give a proof for the case $\alphapha$). In two other case the
proof is similar. By (\ref{4.9}), $e'_2=x_2+d_1+ih_1$ and
$e''_2>x'_1$. Let $d=e''_2-x'_1$. Continuing, we find the
following expressions for
the end points of the segments $l_k$:
$$
\begin{equation}gin{array}{ll}
e'_{2k-1}=x_1+(k-1)d, &\quad e''_{2k-1}=x'_2+d_1+(k-1)d+ih,\\
e'_{2k}=x_2+d_1+(k-1)d+ih_1, &\quad e''_{2k}=x'_1+kd.
\varepsilonnd{array}
$$
Thus, in this case $\tau$ cannot terminate at $p_2$. Instead, it
approaches to the pole $z=1$ as a logarithmic spiral.
\item[(c)]
$u_1<x'_2<u_2$. In this case we still have geodesics (\ref{4.11})
and loops (\ref{4.12}). The only difference is that we cannot
construct the loop $\gamma'_{22}$ as in part (a). Instead, we can
construct a loop $\gamma'_{11}$ from $p_1$ to $p_1$. Indeed, using
elementary geometry, we easily find that there is a point $u_7+ih$
with $u_7<x'_2$ such that the segments $[x_1,u_7+ih]$ and
$[u_8+ih_1,x'_1]$ with $u_8=x_2+x'_2-u_7$ are parallel. Therefore
using (\ref{4.9}), we conclude that
$\gamma'_{11}=F^{-1}((x_1,u_7+ih]\cup [u_8+ih_1,x_1))$ is a
critical geodesic loop.
\item[(e)]
$u_2<x'_2<u_3$. We still have geodesics $\gamma_{12}$,
$\gamma'_{12}$, and $\gamma_{21}$ given by (\ref{4.11}) and the
loops $\gamma_\infty$, $\gamma_{22}$, and $\gamma'_{11}$ as in the
case c). But the geodesic $\gamma'_{21}$ in (\ref{4.11}) should be
replaced with a geodesic constructed as follows. From elementary
geometry we find that there is $u_9>x_2$ such that the segments
$[x'_1,u_9+ih_1]$ and $[u_{10}+ih,x_2+ih_1]$ with
$u_{10}=x'_2-u_9+x_2$ are parallel. Using (\ref{4.9}), we conclude
that the arc $\gamma'_{21}=F^{-1}((x'_1,u_9+ih_1]\cup
[u_{10}+ih,x_2+ih_1))$ is a geodesic from $p_1$ to $p_2$.
\item[(g)]
$u_3<x'_2<u_4$. The geodesics $\gamma_{12}$, $\gamma'_{12}$, and
$\gamma'_{21}$ and all three critical geodesic loops can be
constructed as in part (e). The geodesic $\gamma_{21}$ in this
case can be constructed as follows. Using elementary geometry one
can find that there is $u_{11}>x_2$ such that the segments
$[x_1,u_{11}+ih_1]$ and $[u_{12}+ih,x_2+ih_1]$ with
$u_{12}=x'_2+x_2-u_{11}$ are parallel. Using (\ref{4.9}) we
conclude that the arc $\gamma_{21}=F^{-1}((x_1,u_{11}+ih_1]\cup
[u_{12}+ih,x_2+ih_1))$ is a geodesic from $p_1$ to $p_2$.
\item[(i)]
$x'_2>u_4$. The geodesics from $p_1$ to $p_2$ can be constructed
as in case (g). Of course, we still have loops (\ref{4.12}). The
third geodesic critical loop can be obtained as follows. For
$u_{13}<x_1=0$, let $l^1$ be the line segment joining the real
axis and the line $L(h)$, which has its initial point at
$z=u_{13}$ and passes through $z=x_2+i$. Let $z=u_{14}+ih$ be the
terminal point of $l^1$ on $L(h)$. We consider only those values
of $u_{13}$, for which $u_{14}<x'_2$. Let $d=x'_2-u_{14}$ and let
$l^2$ be a line segment joining the real axis and $L(h_1)$, which
is parallel to $l^1$ and has its initial point at $u_{15}=x'_1+d$.
Let $z=u_{16}+ih_1$ be the terminal point of $l^2$ on $L(h_1)$. It
follows from elementary geometry that
we can find a unique value of $u_{13}$ such that for this value
$u_{16}-x_2=x'_2-u_{14}$.
It follows from our construction and from the identification
properties (\ref{4.8}) and (\ref{4.9}) that the preimage
$$
\gamma'_{22}=F^{-1}([u_{13},x_2+ih_1)\cup(x_2+ih_1,u_{14}+ih]\cup[u_{15},u_{16}+ih_1])
$$
is a geodesic loop from the point $p_2$ to itself. In addition,
this loop contains the pole $z=1$ in its interior, which does not
contain other critical points.
\varepsilonnd{itemize}
Now we consider four ``degenerate'' cases.
\begin{equation}gin{enumerate}
\item[(b)]
If $x'_2=u_1$, then we still have critical geodesics (\ref{4.11})
and critical geodesic
loops (\ref{4.12}). But there is no critical geodesic loop
separating the pole $z=1$ from other critical points. Instead,
the boundary of a simply connected domain having $z=1$ inside
and bounded by critical geodesics will consist of geodesics
$\gamma'_{12}$ and $\gamma_{22}$.
\item[(d)]
If $x'_2=u_2$, then we have all critical geodesic loops and
geodesics $\gamma_{12}$, $\gamma'_{12}$, and $\gamma_{21}$ as in
the case $u_1<x'_2<u_2$ but instead of geodesic $\gamma'_{21}$
we have a non-simple geodesic, which is the union
$\gamma'_{12}\cup \gamma_{22}$.
\item[(f)]
If $x'_2=u_3$, then we have all critical geodesic loops
and geodesics $\gamma_{12}$, $\gamma'_{12}$, and $\gamma'_{21}$ as
in the case $u_2<x'_2<u_3$ but instead of geodesic $\gamma_{21}$
we have a non-simple geodesic, which is the union
$\gamma_{12}\cup\gamma_{22}$.
\item[(h)]
If $x'_2=u_4$, then we have all geodesics and loops
$\gamma_\infty$, $\gamma_{22}$ constructed as in the case
$u_3<x'_2<u_4$ but instead of the loop $\gamma'_{11}$ we will have
non-simple critical geodesic separating the pole $z=1$ from all
other critical points. This non-simple critical geodesic is the
union $\gamma_{12}\cup \gamma'_{21}$.
\varepsilonnd{enumerate}
Using the proof by construction of divergent geodesic segments one
can show that in all cases considered above there are no any other
critical
geodesics or critical geodesic loops.
Quadratic differentials defined by formula (\ref{6.1}) depend on
four real parameters which are real parts and imaginary parts of
zeroes $p_1$ and $p_2$. As the reader may noticed in the generic
case configurations shown in Figures~10 also depend on four real
parameters which are $x_2$, $x'_2$, $h_1$, and $h$. This is not a
coincidence; in fact, the set of pairs $(p_1,p_2)$ is in a
one-to-one correspondence with the set of these diagrams. To
explain how this one-to-one correspondence works, we will show
three basic steps. To be definite, we assume that the domain
configuration consists of a circle domain $D_\infty$ and strip
domains $G_1$ and $G_2$. Thus, we will consider diagrams shown in
Figures~10.
\begin{equation}gin{enumerate}
\item[$\bullet$]
As we described above, for any given $p_1$ and $p_2$, the function
$F(z)$ defined by (\ref{4.1}) maps $G_1$ and $G_2$ onto horizontal
strips shown in Figures~10. Furthermore, for fixed $p_1$ and
$p_2$, the values of the parameters $x_2$, $x'_2$, $h_1$, and $h$
are uniquely defined via function $F(z)$.
\item[$\bullet$]
To prove that different pairs $(p_1,p_2)$ define different
diagrams, we argue by contradiction. Suppose that mappings
$F_1(z)$ and $F_2(z)$ constructed by formula (\ref{4.1}) for
distinct pairs $(p_1^1,p_2^1)$ and $(p_1^2,p_2^2)$ produce
identical diagrams of the form shown in Figures~10. Then the
composition $\varphirphi=F_1^{-1}\circ F_2$ is well-defined and
defines a one-to-one meromorphic mapping from
$\overline{\mathbb{C}}$ onto itself. Since $\varphirphi(1)=1$,
$\varphirphi(-1)=-1$, and $\varphirphi(\infty)=\infty$ we conclude that
$\varphirphi$ is the identity mapping. Thus, $\varphirphi(z)
\varepsilonquiv z$ and therefore $p_1^1=p_1^2$ and $p_2^1=p_2^2$.
\item[$\bullet$]
Now, we want to show that every diagram of the form shown in
Fig.~10a--10i corresponds via a mapping defined by formula
(\ref{4.1}) to a quadratic differential of the form (\ref{6.1})
with some $p_1$ and $p_2$.
To show this, we will construct a compact Riemann surface
$\mathcal{R}$ using identification of appropriate edges of the
diagram. For more general quadratic differentials, similar
construction was used in \cite{S2}.
To be definite, we will give detailed construction for the diagram
shown in Fig.~10a. In all other cases constructions of an
appropriate Riemann surface follow same lines.
Consider a domain $\Omegaega$ defined by
$$
\begin{equation}gin{array}{ll}
\Omegaega =&\{w:\,x_1<{\Bbb R}e w<x'_1,\, \Im w\le 0\} \cup \\
{}&\{w:\,0<\Im
w<h\} \setminus \{w=t+ih_1:\,t\ge x_2\}.
\varepsilonnd{array}
$$
Thus, $\Omegaega$ is a slit horizontal strip shown in Fig.~10a with a
vertical half strip $\{w:\,x_1<{\Bbb R}e w<x'_1,\, \Im w\le 0\}$
attached to this horizontal strip along the interval $(x_1,x'_1)$;
see Fig.~11. To construct a Riemann surface $\mathcal{R}$
mentioned above, we
identify boundary points of $\Omegaega$ as follows:
\begin{equation}gin{equation} \lambdabel{8.01}
\begin{equation}gin{array}{rll}
iy &\sigmameq 1+iy & {\mbox{for $y\le 0$,}} \\
-x&\sigmameq 1+x & {\mbox{for $x\ge 0$,}} \\
x+x_2+i(h_1-0)&\sigmameq -x+x'_2+ih&
{\mbox{for $x\ge 0$,}} \\
x+x_2+i(h_1+0)&\sigmameq x+x'_2+ih&
{\mbox{for $x\ge 0$.}}
\varepsilonnd{array}
\varepsilonnd{equation}
After identifying points by rules (\ref{8.01}), we obtain a
surface, which is homeomorphic to a complex sphere
$\overline{\mathbb{C}}$ punctured at three points. These punctures
correspond boundary points of $\Omegaega$ situated at $\infty$. One
puncture corresponds to the point of $\partial \Omegaega$, we call it
$b_1$, which is accessible along the path $\{z=\frac{1}{2}+it\}$
as $t\to -\infty$. Second puncture corresponds to a point $b_2$ in
$\partial \Omegaega$, which is accessible along the path
$\{z=t+i\frac{h_1+h}{2}\}$ as $t \to \infty$. The third puncture
corresponds to two boundary points of $\Omegaega$; one of them, we
call it $b_3^1$, is accessible along the path $\{z=t+ih_1\}$ as
$t\to -\infty$ and the other one, we call it $b_3^2$, is
accessible along the path $\{z=t+\frac{h_1}{2}\}$ as $t \to
\infty$. Adding these three punctures, we obtain a compact surface
$\mathcal{R}$ which is homeomorphic to a sphere
$\overline{\mathbb{C}}$.
Next, we introduce a complex structure on $\mathcal{R}$ as
follows. Every point of $\mathcal{R}$ corresponding to a point of
$\Omegaega$ inherits its complex structure from $\Omegaega$ as a subset
of $\mathbb{C}$. A point of $\mathcal{R}$ corresponding to $iy$
inherits its complex structure from two half-disks
$\{z:\,|z-iy|<\varphirepsilon, -\pi/2\le \arg(z-iy)\le \pi/2\}$ and
$\{z:\,|z-(1+iy)|<\varphirepsilon, \pi/2\le \arg(z-iy)\le 3\pi/2\}$.
Similarly, every point of $\mathcal{R}$ corresponding to a finite
boundary point of $\Omegaega$, except those which corresponds to the
points $x_1$, and $x_2+ih_1$, inherits its complex structure from
the corresponding boundary half-disks.
Now we assign complex charts for five remaining special points.
For a point $x_1\sigmameq x'_1$ a complex chart can be assigned as
follows:
\begin{equation}gin{equation} \lambdabel{8.02}
\zetata=\left\{ \begin{equation}gin{array}{ll}
(w-1)^{\frac{2}{3}} & {\mbox{if $|w-1|<\varphirepsilon$, $0\le \arg w
\le \frac{3\pi}{2}$,}} \\
(-w)^{\frac{2}{3}} & {\mbox{if $|w|<\varphirepsilon$,
$-\frac{\pi}{2}\le \arg w \le \pi$,}}
\varepsilonnd{array}
\right.
\varepsilonnd{equation}
where the branches of the radicals are taken such that
$\zetata(w)>0$ when $w$ is real such that $w>1$ or $w<0$.
Similarly, to assign a complex chart to a point $x_2+ih_1\sigmameq
x'_2+i h$, we use the following mapping:
\begin{equation}gin{equation} \lambdabel{8.03}
\zetata=\left\{ \begin{equation}gin{array}{ll}
(w-(x_2+ih_1))^{\frac{2}{3}} & {\mbox{if
$|w-(x_2+ih_1)|<\varphirepsilon$,}} \\
{} &{\ \ \mbox{ $0\le \arg (w-(x_2+ih_1))
\le 2\pi$,}} \\
(w-(x'_2+i h))^{\frac{2}{3}} & {\mbox{if
$|w-(x'_2+ih)|<\varphirepsilon$,}} \\
{} & {\ \ \ \mbox{$\pi\le \arg (w-(x'_2+ih)) \le 2\pi$,}}
\varepsilonnd{array}
\right.
\varepsilonnd{equation}
with appropriate branches of the radicals.
To a point of $\mathcal{R}$ corresponding to an infinite boundary
point
$b_1$, a complex chart can be assigned via the function
\begin{equation}gin{equation} \lambdabel{8.04}
\zetata=\varepsilonxp(-2\pi i w) \quad {\mbox{for $w$ such that $0\le {\Bbb R}e
w\le 1$, $\Im w<0$,}}
\varepsilonnd{equation}
which maps the half-strip $\{w:\,0\le {\Bbb R}e w\le 1,\, \Im w<0\}$
onto the unit disc punctured at $\zetata=0$. This mapping respects
the first identification rule in (\ref{8.01}) and the origin
$\zetata=0$ represents the point $b_1$.
To assign a complex chart to a puncture corresponding to a pair of
boundary points $b_3^1$ and $b_3^2$, we will work with horizontal
half-strips $H_3^1$ and $H_3^2$ defined as follows. The boundary
of $H_3^1$ consists of two horizontal rays $\{w:\,w=t:\,t\ge
u_6\}$ and $\{w=t+ih_1:\,t\ge x_2\}$ and a line segment
$[u_6,x_2+ih_1]$; the boundary of $H_3^2$ consists of two
horizontal rays $\{w:\,w=t:\,t\le u_5\}$ and $\{w=t+ih:\,t\le
x'_2\}$ and a line segment $[u_5,x'_2+ih]$. To construct a
required chart, we rotate the half-strip $H_3^1$ by angle $\pi$
with respect to the point $w=1/2$ and then we glue the result to
the half-strip $H_3^2$ along the interval $(-\infty,u_5)$. As a
result, we obtain a wider half-strip $\widetilde{H}_3$ the
boundary of which consists of horizontal rays
$\{w=t+ih:\,t<x'_2\}$ and $\{w=t-ih_1:\, t<1-x_2\}$ and a line
segment $[1-x_2-ih_1,x'_2+ih]$. After that we map an obtained
wider half-strip $\widetilde{H}_3$ conformally onto the unit disk
in such a way that horizontal rays are mapped onto appropriate
logarithmic spirals. The conformal mapping just described can be
expressed explicitly in the following form:
\begin{equation}gin{equation} \lambdabel{8.05}
\zetata=\left\{ \begin{equation}gin{array}{ll}
\varepsilonxp(2\pi i C_3(1-u_5-w)) & {\mbox{if $w\in H_3^1$,}} \\
\varepsilonxp(2\pi i C_3w) & {\mbox{if $w\in H_3^2$,}}
\varepsilonnd{array}
\right.
\varepsilonnd{equation}
where
$$
C_3=\frac{(x_2+x'_2-1)-i(h+h_1)}{|(x_2+x'_2-1)-i(h+h_1)|^2}.
$$
In a similar way we can assign a complex chart to the puncture
corresponding to the boundary point $b_2$. In this case, we use
the following mapping from the horizontal half-strip $H_2$, the
boundary of which consists of the rays $\{w=t+ih_1:\,t\ge x_2\}$
and $\{w=t+ih:\,t\ge x'_2\}$ and a line segment
$[x_2+ih_1,x'_2+ih]$,
onto the unit disk:
\begin{equation}gin{equation} \lambdabel{8.06}
\zetata=\varepsilonxp(-2\pi i C_2(w-(x_2+ih_1))) \quad {\mbox{for $w\in
H_2$,}}
\varepsilonnd{equation}
where
$$
C_2=\frac{(x'_2-x_2)-i(h-h_1)}{|(x'_2-x_2)-i(h-h_1)|^2}.
$$
\varepsilonnd{enumerate}
Now, our compact surface $\mathcal{R}$ with conformal structure
introduced above is conformally equivalent to the Riemann sphere
$\overline{\mathbb{C}}$. Let $\Phi(w)$ be a conformal mapping
from $\mathcal{R}$ onto $\overline{\mathbb{C}}$ uniquely determined by conditions
$$
\Phi(b_1)=\infty, \quad \Phi(b_2)=1, \quad
\Phi(b_3^1)=\Phi(b_3^2)=-1.
$$
Next, we consider a quadratic differential $\mathcal{Q}(w)\,dw^2$
on $\mathcal{R}$ defined by
\begin{equation}gin{equation} \lambdabel{8.13}
\mathcal{Q}(w)\, dw^2 =1\cdot dw^2
\varepsilonnd{equation}
if $w$ is finite and $w\not= x_1$ and $w\not=x_2+ih_1$. This
quadratic differential can be extended to the points $w=x_1$ and
$w=x_2+ih_1$ as a quadratic differential having simple zeroes at
these points in terms of the local parameters defined by formulas
(\ref{8.02}) and (\ref{8.03}), respectively.
Similarly, using local parameters defined by formulas
(\ref{8.04}), (\ref{8.05}), and (\ref{8.06}), we can extend
quadratic differential (\ref{8.13}) to the points of $\mathcal{R}$
corresponding to the infinite boundary points of $\Omegaega$ situated
at $b_1$ $b_2$, and $b_3^1\sigmameq b_3^2$, respectively.
We note that the horizontal strips $\{w:\,0<\Im w<h_1\}$ and
$\{w:\,h_1<\Im w<h\}$ are strip domains of the quadratic
differential (\ref{8.13}), while the half-strip $\{w:\,0\le {\Bbb R}e
w\le 1,\, \Im w<0\}$, which boundary points are identified by the
first rule in (\ref{8.01}), defines a circle domain of this
quadratic differential.
Now, when the quadratic differential (\ref{8.13}) have been
extended to a quadratic differential defined on the whole Riemann
surface $\mathcal{R}$, we may use conformal mapping $z=\Phi(w)$ to
transplant this quadratic differential to get a quadratic
differential $\widehat{Q}(z)\,dz^2$ defined on
$\overline{\mathbb{C}}$. Since critical points of a quadratic
differential are invariant under conformal mapping, it follows
that $\widehat{Q}(z)\,dz^2$ has second order poles at the points
$z=\infty$, $z=1$ and $z=-1$ and it has simple zeroes at the
images $\Phi(x_1)$ and $\Phi(x_2+ih_1)$ of the points $w=x_1$ and
$w=x_2+ih_1$.
Furthermore, the pole $z=\infty$ belongs to a circle domain of
$\widehat{Q}(z)\,dz^2$ and every trajectory in this circle domain
has length $1$. Using the above information, we conclude that
$\widehat{Q}(z)\,dz^2=\frac{1}{4\pi^2}Q(z)\,dz^2$, where
$Q(z)\,dz^2$ is given by formula (\ref{6.1}) with $p_1=\Phi(x_1)$
and $p_2=\Phi(x_2+ih_1)$.
Combining our observations made in this section, we conclude the
following:
\varepsilonmph{Every quadratic differential of the form (\ref{6.1}) having
two strip domains generates a diagram of the type shown in
Fig.~10a--10i and every diagram of this type corresponds to one
and only one quadratic differential with two strip domains in its
domain configuration of the form (\ref{6.1}).}
\section{How parameters count critical geodesics and critical loops} \lambdabel{Section-9}
\setcounter{equation}{0}
In Section~8, we described $Q$-geodesics corresponding to the
quadratic differential (\ref{6.1}) in terms of Euclidean geodesics
in the $w$-plane. In this section, we explain how this information
can be used to find the number of short geodesics and geodesic
loops for each pair of zeros $p_1$ and $p_2$.
To be definite, we will work with the case \textbf{6.3(b2)} of
Theorem~4 assuming that
\begin{equation}gin{equation} \lambdabel{9.1}
\Im p_1>0,\quad {\mbox{and $p_2\in
E_{-1}^+(p_1)$.}}
\varepsilonnd{equation}
In all other cases, the number of short geodesics and geodesic
loops can be found similarly.
Under conditions (\ref{9.1}), the domain configuration of the
quadratic differential (\ref{6.1}) consists of domains $D_\infty$,
$G_1$, and $G_2$ as it is shown in Fig.~4a and Fig.~4b and
possible configurations of images of $G_1$ and $G_2$ under the
mapping (\ref{4.1}) are shown in Fig.~10a-10i.
Let $\varphirepsilon>0$ be sufficiently small and let
$dz_\varphirepsilon^+$ denote a tangent vector to the trajectory of
the quadratic differential (\ref{6.1}) at $z=1+\varphirepsilon$, which
can be found from the equation $Q(z)\,dz^2>0$. Using (\ref{7.1.1})
and (\ref{5.1}), we find that
\begin{equation}gin{equation} \lambdabel{9.2}
\arg(dz_\varphirepsilon^+)=\frac{\pi}{2}-\frac{1}{2}\arg
C_1+o(1)=\frac{\pi}{2}-\frac{1}{2} \arg((p_1-1)(p_2-1))+o(1),
\varepsilonnd{equation}
where $o(1)\to 0$ as $\varphirepsilon\to 0$. We assume here that
$-\frac{\pi}{2}\le \arg(dz_\varphirepsilon^+)\le \frac{\pi}{2}$.
If $1+\varphirepsilon\in \gamma_1$ then the tangent vector
$dz_\varphirepsilon^+$ corresponds to the direction on $\gamma_1$ from
$z=1$ to $z=p_1$. Let $\alphapha_\varphirepsilon^+=\alphapha^++o(1)$, where
$\alphapha^+$ is a constant such that $0\le \alphapha^+\le \pi$, denote
the angle formed at the point $1+\varphirepsilon \in \gamma_1$ by
$dz_\varphirepsilon ^+$ and the vector $\overrightarrow{v}=-i$, which
is tangent to the circle $\{z:\,|z-1|=\varphirepsilon\}$ at
$z=1+\varphirepsilon$. It follows from (\ref{9.2}) that
\begin{equation}gin{equation} \lambdabel{9.3}
\alphapha^+=\pi-\frac{1}{2}\arg C_1=\pi-\frac{1}{2}\arg((p_1-1)(p_2-1)).
\varepsilonnd{equation}
Similarly, if $dz_\varphirepsilon^-$ denote the tangent vector to the
trajectory of the quadratic differential (\ref{6.1}) at
$z=-1+\varphirepsilon$, then
\begin{equation}gin{equation} \lambdabel{9.4}
\arg(dz_\varphirepsilon^-)=\frac{\pi}{2}-\frac{1}{2}\arg
C_{-1}+o(1)=\frac{\pi}{2}-\frac{1}{2} \arg((p_1+1)(p_2+1))+o(1).
\varepsilonnd{equation}
Suppose that $1+\varphirepsilon\in \gamma_{-1}$ and that
$d_\varphirepsilon^-$ shows direction on $\gamma_{-1}$ from $z=-1$ to
$z=p_2$. As before we can find constant $\alphapha^-$, $0\le \alphapha^-
\le \pi$, such that the angle formed at $z=-1+\varphirepsilon\in
\gamma_{-1}$ by the vectors $dz_\varphirepsilon^+$ and
$\overrightarrow{v}=-i$ is equal to $\alphapha^-+o(1)$, where
$o(1)\to 0$ as $\varphirepsilon\to 0$ and
\begin{equation}gin{equation} \lambdabel{9.5}
\alphapha^-=\pi-\frac{1}{2}\arg C_{-1}=\pi-\frac{1}{2}\arg((p_1+1)(p_2+1)).
\varepsilonnd{equation}
To relate angles $\alphapha^+$ and $\alphapha^-$ to geometric
characteristics of diagrams in Fig.~10a-10i, we recall that
geodesics are conformally invariant and that for small
$\varphirepsilon>0$ a geodesic loop $\gamma_\varphirepsilon^+$ which
passes through the point $z=1+\varphirepsilon$ and surrounds the pole
$z=1$ is an infinitesimal circle. Therefore the angle formed by
the vector $dz_\varphirepsilon^+$ and the tangent vector to
$\gamma_\varphirepsilon^+$ at $z=1+\varphirepsilon$ equals
$\alphapha^++o(1)$.
Similarly, the angle formed by the vector $dz_\varphirepsilon^-$ and
the tangent vector to the corresponding geodesic loop
$\gamma_\varphirepsilon^-\ni -1+\varphirepsilon$ surrounding the pole at
$z=-1$ is equal to $\alphapha^-+o(1)$.
Since geodesics are conformally invariant and since conformal
mappings preserve angles, we conclude that trajectories of the
quadratic differential $\mathcal{Q}(w)\,dw^2$ defined in
Section~8 (see formula (\ref{8.13}) ) form angles of opening
$\alphapha^+$ or $\alphapha^-$ with the images of the corresponding
geodesic loops $\gamma_\varphirepsilon^+$ or $\gamma_\varphirepsilon^-$,
respectively. Since the metric defined by the quadratic
differential (\ref{8.13}) is Euclidean, it follows that the
corresponding images of geodesic loops are line segments joining
pairs of points identified by relations (\ref{8.01}).
Using this observation and identification rule $-x+x'_2+ih\sigmameq
x+x_2+ih_1$, we conclude that the segment $[x_2+ih_1,x'_2+ih]$
forms an angle $\pi-\alphapha^-$ with the positive real axis; i.e.,
\begin{equation}gin{equation} \lambdabel{9.6}
\pi-\alphapha^-=\arg((x'_2-x_2)+i(h-h_1)).
\varepsilonnd{equation}
To find an equation for the angle $\alphapha^+$, we will use the
half-strip $\widetilde{H}_3$ constructed at the end of Section~8,
which is related to a conformal mapping defined by
formula~(\ref{8.05}). In this case, $\pi-\alphapha^+$ is equal to the
angle formed by the segment $[1-x_2-ih_1,x'_2+ih]$ with the
positive real axis; i.e.,
\begin{equation}gin{equation} \lambdabel{9.7}
\pi-\alphapha^+=\arg((x_2+x'_2-1)+i(h+h_1)).
\varepsilonnd{equation}
Equating the right-hand sides of equations (\ref{9.3}) and
(\ref{9.4}) to the right-hand sides of equations (\ref{9.7}) and
(\ref{9.6}), respectively, we obtain two equations, which relate
parameters $x_2$, $x'_2$, $h_1$, and $h$. Combining this with
equations (\ref{5.10})--(\ref{7.9.1}), we obtain the following
system of four equations:
$$
\begin{equation}gin{array}{l}
\arg((x_2+x'_2-1)+i(h+h_1))=\frac{1}{2}\arg ((p_1-1)(p_2-1)) \\
\arg((x'_2-x_2)+i(h-h_1))=\frac{1}{2}\arg ((p_1+1)(p_2+1)) \\
h_1=\frac{1}{4} \Im \left(\sqrt{(p_1-1)(p_2-1)}-\sqrt{(p_1+1)(p_2+1)}\right) \\
h=\frac{1}{4} \Im \left(\sqrt{(p_1-1)(p_2-1)}+\sqrt{(p_1+1)(p_2+1)}\right).
\varepsilonnd{array}
$$
This system of equations can be solved to obtain the following:
\begin{equation}gin{equation} \lambdabel{9.9}
\begin{equation}gin{array}{l}
x_2+ih_1=\frac{1}{2}+\frac{1}{4}
\left(\sqrt{(p_1-1)(p_2-1)}-\sqrt{(p_1+1)(p_2+1)}\right), \\
x'_2+ih=\frac{1}{2}+\frac{1}{4}
\left(\sqrt{(p_1-1)(p_2-1)}+\sqrt{(p_1+1)(p_2+1)}\right).
\varepsilonnd{array}
\varepsilonnd{equation}
Now, when the points $x_2+ih_1$ and $x'_2+ih$ are determined, we
can give explicit conditions on the zeros $p_1$ and $p_2$ which
correspond to all subcases (a)--(i) of the case \textbf{6.3(b2)}
discussed in Section~8.
\begin{equation}gin{theorem} \lambdabel{Theorem-5}
Suppose that zeros $p_1$ and $p_2$ satisfy conditions (\ref{9.1}).
Then the number of short geodesics and geodesic loops and their
topology are determined by the following inequalities, which
corresponds to the subcases (a)--(i) of Case \textbf{6.3(b2)} described in Section~8 and shown in Fig.~10a--10i:
Case {\rm{(a)}} with four short geodesics and three critical geodesic loops occurs if the following conditions are satisfied:
$$
\begin{equation}gin{array}{ll}
0&<\arg(-\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}-\sqrt{(p_1+1)(p_2+1)})
\\
{}&<\arg(\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}+\sqrt{(p_1+1)(p_2+1)})<\pi.
\varepsilonnd{array}
$$
Case {\rm{(b)}} with four short geodesics and two critical geodesic loops occurs if the following conditions are satisfied:
$$
\begin{equation}gin{array}{ll}
0&<\arg(-\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}-\sqrt{(p_1+1)(p_2+1)})
\\
{}&=\arg(\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}+\sqrt{(p_1+1)(p_2+1)})<\pi.
\varepsilonnd{array}
$$
Case {\rm{(c)}} with four short geodesics and three critical geodesic loops occurs if the following conditions are satisfied:
$$
\begin{equation}gin{array}{ll}
0&<\arg(\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}+\sqrt{(p_1+1)(p_2+1)})
\\
{}&<\arg(-\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}-\sqrt{(p_1+1)(p_2+1)})<\pi,
\varepsilonnd{array}
$$
$$
\begin{equation}gin{array}{ll}
0&<\arg(-\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}-\sqrt{(p_1+1)(p_2+1)})
\\
{}&<\arg(-\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}+\sqrt{(p_1+1)(p_2+1)})<\pi.
\varepsilonnd{array}
$$
Case {\rm{(d)}} with three short geodesics and three critical geodesic loops occurs if the following conditions are satisfied:
$$
\begin{equation}gin{array}{ll}
0&<\arg(-\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}-\sqrt{(p_1+1)(p_2+1)})
\\
{}&=\arg(-\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}+\sqrt{(p_1+1)(p_2+1)})<\pi.
\varepsilonnd{array}
$$
Case {\rm{(e)}} with four short geodesics and three critical geodesic loops occurs if the following conditions are satisfied:
$$
\begin{equation}gin{array}{ll}
0&<\arg(-\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}+\sqrt{(p_1+1)(p_2+1)})
\\
{}&<\arg(-\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}-\sqrt{(p_1+1)(p_2+1)})<\pi,
\varepsilonnd{array}
$$
$$
\begin{equation}gin{array}{ll}
0&<\arg(\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}-\sqrt{(p_1+1)(p_2+1)})
\\
{}&<\arg(\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}+\sqrt{(p_1+1)(p_2+1)})<\pi.
\varepsilonnd{array}
$$
Case {\rm{(f)}} with three short geodesics and three critical geodesic loops occurs if the following conditions are satisfied:
$$
\begin{equation}gin{array}{ll}
0&<\arg(\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}-\sqrt{(p_1+1)(p_2+1)})
\\
{}&=\arg(\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}+\sqrt{(p_1+1)(p_2+1)})<\pi.
\varepsilonnd{array}
$$
Case {\rm{(g)}} with four short geodesics and three critical geodesic loops occurs if the following conditions are satisfied:
$$
\begin{equation}gin{array}{ll}
0&<\arg(\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}+\sqrt{(p_1+1)(p_2+1)})
\\
{}&<\arg(\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}-\sqrt{(p_1+1)(p_2+1)})<\pi,
\varepsilonnd{array}
$$
$$
\begin{equation}gin{array}{ll}
0&<\arg(\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}-\sqrt{(p_1+1)(p_2+1)})
\\
{}&<\arg(-\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}+\sqrt{(p_1+1)(p_2+1)})<\pi.
\varepsilonnd{array}
$$
Case {\rm{(h)}} with four short geodesics and two critical geodesic loops occurs if the following conditions are satisfied:
$$
\begin{equation}gin{array}{ll}
0&<\arg(\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}-\sqrt{(p_1+1)(p_2+1)})
\\
{}&=\arg(-\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}+\sqrt{(p_1+1)(p_2+1)})<\pi.
\varepsilonnd{array}
$$
Case {\rm{(i)}} with four short geodesics and three critical geodesic loops occurs if the following conditions are satisfied:
$$
\begin{equation}gin{array}{ll}
0&<\arg(-\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}+\sqrt{(p_1+1)(p_2+1)})
\\
{}&<\arg(\frac{1}{2}+\frac{1}{4}(\sqrt{(p_1-1)(p_2-1)}-\sqrt{(p_1+1)(p_2+1)})<\pi.
\varepsilonnd{array}
$$
\varepsilonnd{theorem}
\section{Some related questions} \lambdabel{Section-9}
\setcounter{equation}{0}
Our results presented in Sections~6-9 provide complete information
concerning critical trajectories and $Q$-geodesic of the quadratic
differential (\ref{6.1}). This allows us to answer many related
questions. As an example, we will discuss three questions
originated in the study of limiting distributions of zeros of
Jacobi polynomials.
Below, we suppose that $p_1,p_2\in \mathbb{C}$ are fixed. Then
we consider the family of quadratic differentials $Q_s(z)\,dz^2$
depending on the real parameter $s$, $0\le s<2\pi$, such
that
\begin{equation}gin{equation} \lambdabel{10.1}
Q_s(z)\,dz^2:=e^{-is}Q(z)\,dz^2=-e^{-is}\frac{(z-p_1)(z-p_2)}{(z-1)^2(z+1)^2}\,dz^2.
\varepsilonnd{equation}
\begin{equation}gin{enumerate}
\item[1)]
For how many values of
$s$, $0\le s<2\pi$, the quadratic differential $Q_s(z)\,dz^2$ has
a trajectory loop with end points at $p_1$ and for how many
values of $s$
$Q_s(z)\,dz^2$ has a trajectory loop with end points at $p_2$?
\item[2)]
For how many values of
$s$, $0\le s<2\pi$, the corresponding quadratic differential
$Q_s(z)\,dz^2$ has a short critical trajectory?
\item[3)]
How we can find the values of
$s$, $0\le s<2\pi$, mentioned in questions stated above?
\varepsilonnd{enumerate}
To answer these questions we need two simple facts:
\begin{equation}gin{enumerate}
\item[(a)]
First, we note that $\gamma$ is a short trajectory loop or,
respectively, a short critical trajectory for the quadratic
differential (\ref{10.1}) with some $s$ if and only if $\gamma$ is
a short geodesic loop or, respectively, a short geodesic joining
points $p_1$ and $p_2$ for the quadratic differential (\ref{6.1}).
Thus, the numbers of values $s$ in question (1) and question (2),
respectively, are bounded by the number of short geodesic loops
and the number of short geodesics, respectively. In the most
general case with one circle domain and two strip domains, these
short geodesic loops and short geodesics were described in
Theorem~\ref{Theorem-5} and their images under the canonical
mapping were shown in Fig.~10a-10i. Of course, one value of $s$
can correspond to more than one short geodesic loop and more than
one short geodesic.
\item[(b)]
To find the values of $s$ in question~3), we use the following
observation. If $l$ is a straight line segment in the image domain
$\Omegaega$ forming an angle $\alphapha$, $0\le \alphapha<\pi$, with the
direction of the positive real axis, then $l$ is an image under
the canonical mapping (\ref{4.1}) of an arc of a trajectory of
the quadratic differential (\ref{10.1})
with
\begin{equation}gin{equation} \lambdabel{10.2}
s=2\alphapha.
\varepsilonnd{equation}
\varepsilonnd{enumerate}
We will use (\ref{10.2}) to find values of $s$ which turn short
geodesic loops and short geodesics into short trajectory loops and
short trajectories, respectively. It is convenient to introduce
notations $\alphapha_\infty$, $\alphapha_{12}$, $\alphapha'_{12}$,
$\alphapha_{22}$, $\alphapha'_{22}$, $\alphapha''_{22}$, and so on, to
denote the angles formed by corresponding geodesics
$\gamma_\infty$, $\gamma_{12}$, $\gamma'_{12}$, $\gamma_{22}$,
$\gamma'_{22}$, $\gamma''_{22}$, and so on (considered in the
$w$-plane) with the positive direction of the real axis.
Furthermore, we will use notations ${\mathcal{A}}(6.1)$,
${\mathcal{A}}(6.1(a))$, ${\mathcal{A}}(6.2)$,
${\mathcal{A}}(6.3(a))$, ${\mathcal{A}}(6.3(b1))$,
${\mathcal{A}}(6.3(b2)(a))$, and so on, to denote the sets of all
angles introduced above in the cases under consideration; i.e. in
the cases $\mathbf{6.1}$, $\mathbf{6.2}$, $\mathbf{6.3(a)}$,
$\mathbf{6.3(b_1)}$, $\mathbf{6.3(b2)}(a)$, and so on.
Now, we are ready to answer questions stated above. We proceed
with two steps. First, we identify the type of domain
configuration $\mathcal{D}_Q$. This will provide us with the first
portion of necessary information. We recall that in general there
are at most three geodesic loops centered at $z=\infty$, $z=1$,
and $z=-1$. Thus, the maximal number of values $s$ in question~1)
is at most three. Then we identify which of the schemes
corresponds to the parameters $p_1$, $p_2$ (in the most general
case these schemes are shown in Fig.~10a-10i). This will provide
us with the remaining portion of necessary information.
$\bullet$ \ Suppose that $\mathcal{D}_Q$ has type~\textbf{6.1}.
Then we already have three circle domains and therefore $s=0$ is
the only value for which $Q_sz)\,dz^2$ may have short trajectory
loops. In case \textbf{6.1(a)}, we have short trajectory loops
centered at $z=1$ and $z=-1$ and no other such loops. In case
\textbf{6.1(b)} with $1<p_2<p_1$ (respectively with $p_1<p_2<-1$),
we have short trajectory loops centered at $z=\infty$ and $z=1$
(respectively, at $z=\infty$ and $z=-1$). In case \textbf{6.1(c)},
there are no short geodesic loops.
As concerns short critical trajectories for domain configuration
of type \textbf{6.1}, again $s=0$ is the only value for which
there are such trajectories. This follows from the fact discussed
in Section~8 that in case~\textbf{6.1} there are no other simple
geodesics joining $p_1$ and $p_2$. In cases \textbf{6.1(a)} and
\textbf{6.1(b)}, there is a single short critical trajectory which
is the interval $\gamma_0=(p_2,p_1)$. In case \textbf{6.1(c)},
there are three short critical trajectories which are arcs
$\gamma_0$, $\gamma_1$, and $\gamma_{-1}$ shown in Fig.~1c.
$\bullet$ \ Next, we consider the case when $\mathcal{D}_Q$ has
type \textbf{6.2}. For $s=0$, we have two short trajectory loops.
As before, we assume that these loops surround points $z=-1$ and
$z=\infty$. In other cases discussion is similar, we just have to
switch roles of the poles of the quadratic differential
(\ref{10.1}).
In this case,
${\mathcal{A}}(6.2)=\{0,\alphapha_{11},\alphapha_{12},\alphapha'_{12},\alphapha_{21},\alphapha'_{21}\}$.
One more value of $s$, for which we may have a short trajectory
loop (centered at $z=1$) may occur for
$s=2\alphapha_{11}=-\arg((1-p_1)(1-p_2))$. If
$|\gamma_\infty|_Q>|\gamma_{-1}|_Q$ then we will have a short
geodesic loop from $p_1$ to $p_1$. This loop corresponds to a
geodesic $\gamma_{11}$ in Fig.~8a. If
$|\gamma_\infty|_Q<|\gamma_{-1}|_Q$, then we will have a similar
short geodesic loop from $p_2$ to $p_2$. In the case
$|\gamma_\infty|_Q=|\gamma_{-1}|_Q$, we have
$\alphapha_{11}=\alphapha_{12}=\alphapha'_{21}$. In this case, we do not
have the third short geodesic loop. Instead, we have two short
critical trajectories joining $p_1$ and $p_2$.
By (\ref{10.2}), the value of $s$, which corresponds to the third
loop (if it exists) is equal to $2\alphapha_{11}$. As concerns
values of $s$ corresponding to short critical trajectories, in
case \textbf{6.2} with $|\gamma_\infty|_Q\not=|\gamma_{-1}|_Q$ we
have four such values. These values are $2\alphapha_{12}$,
$2\alphapha'_{12}$, $2\alphapha_{21}$, and $2\alphapha'_{21}$ (see
Fig.~8a).
If $|\gamma_\infty|_Q=|\gamma_{-1}|_Q$, then there are three
values of $s$, which produce short geodesics from $p_1$ to $p_2$.
Two of these values, $s=2\alphapha'_{12}$ and $s=2\alphapha_{21}$,
generate one short critical trajectory each. The third value
$s=2\alphapha_{12}$ generates two short critical trajectories.
$\bullet$ \ Turning to the most general case \textbf{6.3}, we
will give detailed account for subcases \textbf{6.3(b1)} and
\textbf{6.3(b2)}(i), in all other subcases consideration is
similar.
First, we consider the subcase \textbf{6.3(b1)} when the domain
configuration ${\mathcal{D}}_Q$ consists of one circle domain and
one strip domain; see Fig.~3a--3e. In this case,
${\mathcal{A}}(6.3(b1))=\{0,\alphapha'_{22},\alphapha''_{22},\alphapha_{12},\alphapha'_{12}\}$.
The value $s=0$ generates one short trajectory loop and one short
trajectory. The values $s=2\alphapha'_{22}$ and $s=2\alphapha''_{22}$
generate one short trajectory loop each and the values
$s=2\alphapha_{12}$ and $s=2\alphapha'_{12}$ generate one short
trajectory each.
Let us consider case \textbf{6.3(b2)}(i) shown in Fig.~10i. We
have
${\mathcal{A}}(6.3(b2)(i))=\{0,\alphapha_{22},\alphapha'_{22},\alphapha_{12},\alphapha'_{12},\alphapha_{21},\alphapha'_{21}\}$
where all angles are distinct. The values $s=0$,
$s=2\alphapha_{22}$, and $s=2\alphapha'_{22}$ generate short trajectory
loops $\gamma_{\infty}$, $\gamma_{22}$, and $\gamma''_{22}$,
respectively. Remaining values $s=2\alphapha_{12}$,
$s=2\alphapha'_{12}$, $s=2\alphapha_{21}$, $s=2\alphapha'_{21}$ generate
short trajectories $\gamma_{12}$, $\gamma'_{12}$, $\gamma_{21}$,
and $\gamma'_{21}$, respectively.
Finally, we note that position of points $x_1$, $x'_1$,
$x_2+ih_1$, and $x'_2+ih$ are given explicitly; see formulas
(\ref{9.9}). Using these formulas one can find explicit
expressions for all angles $\alphapha_{12}$, $\alphapha'_{12}$,
$\alphapha_{21}$, $\alphapha'_{21}$, and so on, in all possible cases.
\section{Figures Zoo}
\setcounter{equation}{0}
{\cal F}loatBarrier
This section contains all our figures. For convenience, we divide
the set of all figures in eleven groups.
\textbf{I.} Configurations with three circle domains.
\begin{equation}gin{figure}
$$\includegraphics[scale=.65,angle=0]{Boris-1.png} $$
\caption{1a. Three circle domains. Case \textbf{6.1(a)}.}
\varepsilonnd{figure}
\begin{equation}gin{figure}
$$\includegraphics[scale=.75,angle=0]{Fig-1b.png} $$
\caption{1b. Three circle domains. Case \textbf{6.1(b)}.}
\varepsilonnd{figure}
\begin{equation}gin{figure}[b]
$$\includegraphics[scale=.75,angle=0]{Fig-1c.png} $$
\caption{1c. Three circle domains. Case \textbf{6.1(c)}.}
\varepsilonnd{figure}
{\cal F}loatBarrier
\textbf{II.} Configurations with two circle domains.
\begin{equation}gin{figure}[h]
$$\includegraphics[scale=.7,angle=0]{Fig-2a.png} $$
\caption{2a. Two circle domains. Case \textbf{6.2} with symmetric domains.}
\varepsilonnd{figure}
\begin{equation}gin{figure}[h]
$$\includegraphics[scale=.75,angle=0]{Fig-2b.png} $$
\caption{2b. Two circle domains. Case \textbf{6.2} with non-symmetric domains.}
\varepsilonnd{figure}
{\cal F}loatBarrier
\textbf{III.} Configurations with one circle
domain and one
strip domain.
\begin{equation}gin{figure}
$$\includegraphics[scale=.6,angle=0]{Fig-3a.png} $$
\caption{3a. One circle domain. Case \textbf{6.3(a)} with axial symmetry.}
\varepsilonnd{figure}
\begin{equation}gin{figure}
$$\includegraphics[scale=.7,angle=0]{Fig-3b.png} $$
\caption{3b. One circle domain. Case \textbf{6.3(a)} with central symmetry.}
\varepsilonnd{figure}
\begin{equation}gin{figure}
$$\includegraphics[scale=.7,angle=0]{Fig-3c.png} $$
\caption{3c. One circle domain. Case \textbf{6.3(a)} with non-symmetric domains.}
\varepsilonnd{figure}
\begin{equation}gin{figure}
$$\includegraphics[scale=.65,angle=0]{Fig-3d.png} $$
\caption{3d. One circle domain. Case \textbf{6.3(b1)} with symmetric domains.}
\varepsilonnd{figure}
\begin{equation}gin{figure}
$$\includegraphics[scale=.65,angle=0]{Fig-3e.png} $$
\caption{3e. One circle domain. Case \textbf{6.3(b1)} with non-symmetric domains.}
\varepsilonnd{figure}
\textbf{IV.} Configurations with one circle domain and two
strip domains.
{\cal F}loatBarrier
\begin{equation}gin{figure}
$$\includegraphics[scale=.65,angle=0]{Fig-3f.png} $$
\caption{4a. One circle domain. Case \textbf{6.3(b2)} with symmetric domains.}
\varepsilonnd{figure}
\begin{equation}gin{figure}
$$\includegraphics[scale=.65,angle=0]{Fig-3g-2.png} $$
\caption{4b. One circle domain. Case \textbf{6.3(b2)} with non-symmetric domains.}
\varepsilonnd{figure}
\textbf{V.} Degenerate configurations.
{\cal F}loatBarrier
\begin{equation}gin{figure}
$$\includegraphics[scale=.55,angle=0]{Fig-6b-2.png} $$
\caption{5a. Degenerate case with $-1<p_1=p_2<1$.}
\varepsilonnd{figure}
\begin{equation}gin{figure}
$$\includegraphics[scale=.6,angle=0]{Fig-6c-2.png} $$
\caption{5b. Degenerate case with $p_1=p_2>1$.}
\varepsilonnd{figure}
\begin{equation}gin{figure}
$$\includegraphics[scale=.75,angle=0]{Fig-6a-1.png} $$
\caption{5c. Degenerate case with $p_1=p_2$, $\Im p_1>0$.}
\varepsilonnd{figure}
\begin{equation}gin{figure}
$$\includegraphics[scale=.75,angle=0]{Fig-4a.png} $$
\caption{5d. Degenerate case with $p_2=-1$, $-1<p_1<1$.}
\varepsilonnd{figure}
\begin{equation}gin{figure}
$$\includegraphics[scale=.6,angle=0]{Fig-4e-1.png} $$
\caption{5e. Degenerate case with $p_2=-1$, $p_1<-1$.}
\varepsilonnd{figure}
\begin{equation}gin{figure}[p]
$$\includegraphics[scale=.6,angle=0]{Fig-4f.png} $$
\caption{5f. Degenerate case with $p_2=-1$, $p_1>1$.}
\varepsilonnd{figure}
\begin{equation}gin{figure}[p]
$$\includegraphics[scale=.65,angle=0]{Fig-4c.png} $$
\caption{5g. Degenerate case with $p_2=-1$, $\Im p_1>0$.}
\varepsilonnd{figure}
{\cal F}loatBarrier
\textbf{VI.} Type regions.
\begin{equation}gin{figure}[h]
\centering
\hspace{-5.3cm}
\begin{equation}gin{minipage}{0.6\linewidth}
\begin{equation}gin{tikzpicture}
[inner sep=1mm,
place/.style={circle,draw=blue!50,
fill=blue!20,thick},
scale=2.2]
\draw [black, very thick](0,0) ellipse (4/3 and 0.7453559923);
\node at (1,0) [] {$\bullet$};
\node at (1.03,0) [below] {$1$};
\node at (-1,0) [] {$\bullet$};
\node at (-1.1,0) [below] {$-1$};
\node at (1.187,0.341) [] {$\bullet$};
\node at (1.24,0.341) [right] {$q_1$};
\node at (-1.317,0.132) [] {$\bullet$};
\node at (-1.33,0.05) [left] {$q_2$};
\draw [black, very thick] plot[domain=-1.55:1.55] ({(1/2)*(exp(\x)+exp(-\x))/2},{sqrt(3)/2*(exp(\x)-exp(-\x))/2});
\draw [->,>=stealth,black, very thick]
(1,0) to (1-3*0.3722022787,-3*0.6575636373);
\draw [->,>=stealth,dashed]
(1,0)to (1+2.7*0.3722022787,2.7*0.6575636373);
\draw [->,>=stealth,black, very thick]
(-1,0)to (-1+2.1*1.627797721,-2.1*0.6575636373);
\draw [->,>=stealth,dashed]
(-1,0)to (-1-0.7*1.627797721,0.7*0.6575636373);
\node at (0.6277977213,-0.6575636373) {$\bullet$};
\node at (0.6277977213,-0.6575636373-0.15) [below] {$\overline{p}_1$};
\node at (0.6277977213,0.6575636373) {$\bullet$};
\node at (0.6277977213-0.05,0.6575636373+0.05) [above] {$p_1$};
\node at (0.17,-0.2) [above] {$H^+(p_1)$};
\node at (0.8,1.6) [above] {$H^-(p_1)$};
\node at (0.8,-1.6) [below] {$H^-(p_1)$};
\node at (4/3,0) [right] {$L^+(p_1)$};
\node at (-1,0.6) [above] {$L^-(p_1)$};
\node at (0.81,0.15) [above] {$E_1^+(p_1)$};
\node at (-0.4,0.15) [above] {$E_{-1}^+(p_1)$};
\node at (1.7,-0.25) [below] {$E_1^-(p_1)$};
\node at (-0.3,1) [above] {$E_{-1}^-(p_1)$};
\node at (0.2,-1.4) [left] {$l_1^+(p_1)$};
\node at (1.77,1.4) [left] {$l_1^-(p_1)$};
\node at (1.7,-1.05) [above] {$l_{-1}^+(p_1)$};
\node at (-1.7,0.3) [above] {$l_{-1}^-(p_1)$};
\node at (2.1,1) [] {$P_1$};
\node at (1.1,1) [] {$P_2$};
\node at (-1.6,1.2) [] {$P_4$};
\node at (-1.6,-0.6) [] {$P_3$};
\varepsilonnd{tikzpicture}
\varepsilonnd{minipage}
\caption{6. Type regions.}
\varepsilonnd{figure}
{\cal F}loatBarrier
\textbf{VII.} Figures for the proof of Theorem~4.
\begin{equation}gin{figure}[h]
$$\includegraphics[scale=.23,angle=0]{Fig-6.png} $$
\caption{7a. Proof of Theorem~4: Impossible limit configuration.}
\varepsilonnd{figure}
\begin{equation}gin{figure}
$$\includegraphics[scale=.25,angle=0]{Fig-7.png}$$
\caption{7b. Proof of Theorem~4: Limit configuration.}
\varepsilonnd{figure}
\begin{equation}gin{figure}[b]
$$\includegraphics[scale=.25,angle=0]{Fig-8.png}$$
\caption{7c. Proof of Theorem~4: $Q^0$-rectangle $D(\delta)$ with trajectories.}
\varepsilonnd{figure}
{\cal F}loatBarrier
\textbf{VIII.} Geodesics and loops in simple cases.
\begin{equation}gin{figure*}[h]
\centering
\hskip -4cm
\begin{equation}gin{minipage}{0.60\linewidth}
\begin{equation}gin{tikzpicture}
[inner
sep=1mm,place/.style={circle,draw=blue!50,fill=blue!20,thick}]
\draw (-3,0)--(8,0);
\draw (-3,4)--(8,4);
\draw (4,0)--(5.5,4);
\draw[blue,ultra thick] (0,0)--(2.5,4);
\draw [blue,ultra thick](4,0)--(4.5,4);
\draw [blue,ultra thick] (0,0)--(2/3*4.5-0.1,2/3*4-0.1);
\draw [blue,ultra thick] (2/3*4.5+0.1,2/3*4+0.1)--(4.5,4);
\draw [blue,ultra thick](4,0)--(2.5,4);
\draw [black,ultra thick] (2.5,4)--(4.5,4);
\draw [black,ultra thick] (0,0)--(4,0);
\node at (0,0) [] (name1){\small{$\bullet$}};
\node at (1.5,4) [] (name2) {\small{$\bullet$}};
\node at (0,-0.4) {$x_1$};
\node at (1.35,4.3) {$v'_2$};
\node at (5.65,4.3) {$v_1''$};
\node at (4,0) [] (name3) {\small{$\bullet$}} ;
\node at (4,-0.4) {$x_1'$};
\node at (5.5,4) [] (name4) {\small{$\bullet$}} ;
\node at (2.5,4) [] (name5) {\small{$\bullet$}} ;
\node at (4.5,4) [] (name6) {\small{$\bullet$}} ;
\draw [-] (0,0) to node[auto] {$\gamma_{11}$} (1.5,4);
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\draw [-] (name3) to node[right] {$\gamma_{11}$} (name4);
\draw [-] (name2) to node[below] {$\gamma_{-1}$} (name4);
\draw [-] (2.5,4) to node[above] {$x_2+ih_1$} (name5);
\draw [-] (4.5,4) to node[above] {$x'_2+ih_1$} (name6);
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\node at (1.7,2.15) {$\gamma_{12}$};
\node at (3.95,2.15) {$\gamma'_{21}$};
\node at (1.7,1.1) {$\gamma'_{12}$};
\node at (3.25,1.1) {$\gamma_{21}$};
\varepsilonnd{tikzpicture}
\varepsilonnd{minipage}
\caption{8a. Geodesics and loops. Case \textbf{6.2}.}
\varepsilonnd{figure*}
\begin{equation}gin{figure*}[h]
\centering
\hskip -4cm
\begin{equation}gin{minipage}{0.60\linewidth}
\begin{equation}gin{tikzpicture}
[inner
sep=1mm,place/.style={circle,draw=blue!50,fill=blue!20,thick}]
\draw (-3,0)--(8,0);
\draw (-3,4)--(8,4);
\draw[black,ultra thick] (0,0)--(2.5,4);
\draw [black,ultra thick](4,0)--(4.5,4);
\draw [blue,ultra thick] (0,0)--(2/3*4.5-0.1,2/3*4-0.1);
\draw [blue,ultra thick] (2/3*4.5+0.1,2/3*4+0.1)--(4.5,4);
\draw [blue,ultra thick](4,0)--(2.5,4);
\draw [blue,ultra thick] (2.5,4)--(4.5,4);
\draw [blue,ultra thick] (0,0)--(4,0);
\node at (0,0) [] (name1){\small{$\bullet$}};
\node at (0,-0.4) {$x_1$};
\node at (4,0) [] (name3) {\small{$\bullet$}} ;
\node at (4,-0.4) {$x_2$};
\node at (2.5,4) [] (name5) {\small{$\bullet$}} ;
\node at (4.5,4) [] (name6) {\small{$\bullet$}} ;
\draw [-] (name2) to node[below] {$\gamma_{\infty}^-$} (name4);
\draw [-] (2.5,4) to node[above] {$x'_1+ih_1$} (name5);
\draw [-] (4.5,4) to node[above] {$x'_2+ih_1$} (name6);
\draw [-] (name1) to node[below] {$\gamma_{\infty}^+$} (name3);
\node at (1.7,2.15) {$\gamma_{11}$};
\node at (3.95,2.15) {$\gamma_{22}$};
\node at (1.7,1.1) {$\gamma_{21}$};
\node at (3.25,1.1) {$\gamma'_{21}$};
\varepsilonnd{tikzpicture}
\varepsilonnd{minipage}
\caption{8b. Geodesics and loops. Case \textbf{6.3(a)}.}
\varepsilonnd{figure*}
\begin{equation}gin{figure*}[h]
\centering
\hskip -4cm
\begin{equation}gin{minipage}{0.60\linewidth}
\begin{equation}gin{tikzpicture}
[inner
sep=1mm,place/.style={circle,draw=blue!50,fill=blue!20,thick}]
\draw (-3,0)--(8,0);
\draw (-3,4)--(8,4);
\draw[black,ultra thick] (-2,0)--(2.5,4);
\draw [black,ultra thick](6,0)--(2.5,4);
\draw [blue,ultra thick] (0,0)--(2.5,4);
\draw [blue,ultra thick](4,0)--(2.5,4);
\draw [black,ultra thick] (0,0)--(4,0);
\draw [blue,ultra thick] (-2,0)--(0,0);
\draw [blue,ultra thick] (4,0)--(6,0);
\node at (0,0) [] (name1){\small{$\bullet$}};
\node at (-2,0) {\small{$\bullet$}};
\node at (6,0) {\small{$\bullet$}};
\node at (0,-0.4) {$x_1$};
\node at (-2,-0.1) [below] {$x''_2$};
\node at (6,-0.1) [below] {$x'_2$};
\node at (4,0) [] (name3) {\small{$\bullet$}} ;
\node at (4,-0.4) {$x'_1$};
\node at (2.5,4) [] (name5) {\small{$\bullet$}} ;
\draw [-] (2.5,4) to node[above] {$x_2+ih_1$} (name5);
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\node at (1.5,1.8) {$\gamma_{12}$};
\node at (4.7,2.0) {$\gamma'_{22}$};
\node at (-0.5,2.0) {$\gamma''_{22}$};
\node at (3.0,1.8) {$\gamma'_{12}$};
\node at (-1.0,-0.3) {$\gamma_0$};
\node at (5.0,-0.3) {$\gamma_0$};
\varepsilonnd{tikzpicture}
\varepsilonnd{minipage}
\caption{8c. Geodesics and loops. Case \textbf{6.3(b1)}.}
\varepsilonnd{figure*}
{\cal F}loatBarrier
\textbf{IX.} Divergent segments.
\begin{equation}gin{figure*}[h]
\centering
\begin{equation}gin{tikzpicture}
[inner sep=1mm,
place/.style={circle,draw=blue!50,
fill=blue!20,thick},
scale=0.8]
\draw (-5,0)--(10,0);
\draw (-5,3)--(10,3);
\draw [black,ultra thick] (1.5,3)--(3.5,3);
\draw [black,ultra thick] (0,0)--(4.5,0);
\node at (0,0) [] (name1){\small{$\bullet$}};
\node at (0,-0.4) {$x_1$};
\node at (4.5,0) [] (name3) {\small{$\bullet$}} ;
\node at (4.5,-0.4) {$x_1'$};
\node at (1.5,3) [] (name5) {\small{$\bullet$}} ;
\node at (3.5,3) [] (name6) {\small{$\bullet$}} ;
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\draw [-] (0.5,3) to node[below] {$\gamma_{-1}$} (4.5,3);
\draw [-] (1.5,3) to node[above] {$x_2+ih_1$} (name5);
\draw [-] (3.5,3) to node[above] {$x'_2+ih_1$} (name6);
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\draw [->] (0,0) to node[right] {$l_1$} (-1,3);
\draw [->] (-1,0) to node[right] {$l_3$} (-2,3);
\draw [->] (-2,0) to node[right] {$l_5$} (-3,3);
\draw [->] (4.5,3) to node[left] {$l_2$} (5.5,0);
\draw [->] (5.5,3) to node[left] {$l_4$} (6.5,0);
\draw [->] (6.5,3) to node[left] {$l_6$} (7.5,0);
\node at (-3.5,1.5) [] {$\cdots$};
\node at (7.5,1.5) [] {$\cdots$};
\varepsilonnd{tikzpicture}
\caption{9a. Divergent segments. Case \textbf{6.2}.}
\varepsilonnd{figure*}
\begin{equation}gin{figure}[h]
\centering
\hspace{-4.5cm}
\begin{equation}gin{minipage}{0.60\linewidth}
\begin{equation}gin{tikzpicture}
[inner sep=1mm,
place/.style={circle,draw=blue!50,
fill=blue!20,thick},
scale=0.75]
\draw (-6.5,0)--(10,0);
\draw [red,very thick] (-6.5,4)--(-1,4);
\draw [green,very thick] (-1,4)--(10,4);
\draw [dashed] (-6.5,1.8)--(3.2,1.8);
\draw [red,very thick] (3,1.75)--(10,1.75);
\draw [green,very thick] (3,1.85)--(10,1.85);
\draw [black,ultra thick] (0,0)--(4.5+2.5*1.8/4-1,0);
\node at (0,0) [] (name1){{$\bullet$}};
\node at (0,-0.4) {$x_1$};
\node at (4.5+2.5*1.8/4-1,0) [] (name3) {\small{$\bullet$}} ;
\node at (4.5+2.5*1.8/4-1,-0.4) {$x_1'$};
\node at (3,1.8) [] (name8) {\lambdarge{$\bullet$}} ;
\node at (-1,4) [] {\small{$\bullet$}} ;
\node at (9.2,0) [below] {$\gamma_{1}$} ;
\node at (9.2,4) [below] {$\gamma_{-1}$} ;
\node at (9.2,1.8) [below] {$\gamma_{0}^-$} ;
\node at (9.2,1.83) [above] {$\gamma_{-1}$} ;
\node at (-5.5,1.9) [above] {\tiny{$\bullet$ \,$\bullet$ \,$\bullet$}};
\node at (9,0.8) [] {\tiny{$\bullet$ \,$\bullet$ \,$\bullet$}};
\node at (-6,0) [below] {$\gamma_{1}$} ;
\node at (-6,1.8) [below] {$\gamma_{0}^+$} ;
\node at (-6,4) [below] {$\gamma_{0}^-$} ;
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\node at (-0.4,4.25) {$x'_2+ih$} ;
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\node at (3.2,2.07) {$x_2+ih_1$};
\draw [->] (0,0) to node[right] {$l_1$} (-2.5,4);
\draw [->] (-1,0) to node[right] {$l_3$} (-3.5,4);
\draw [->] (-2,0) to node[right] {$l_5$} (-4.5,4);
\draw [->] (-3,0) to node[right] {$l_7$} (-5.5,4);
\draw [->] (4.5,1.78) to node[left] {$l_2$} (4.5+2.5*1.8/4,0);
\draw [->] (5.5,1.78) to node[left] {$l_4$} (4.5+2.5*1.8/4+1,0);
\draw [->] (6.5,1.78) to node[left] {$l_6$} (4.5+2.5*1.8/4+2,0);
\draw [->] (7.5,1.78) to node[left] {$l_8$} (4.5+2.5*1.8/4+3,0);
\varepsilonnd{tikzpicture}
\varepsilonnd{minipage}
\caption{9b. Divergent segments. Case \textbf{6.3(b2)}.}
\varepsilonnd{figure}
{\cal F}loatBarrier
{\cal F}loatBarrier
\textbf{X.} Geodesics and loops in the most general case.
{\cal F}loatBarrier
\begin{equation}gin{figure}[h]
\centering
\hspace{-4.5cm}
\begin{equation}gin{minipage}{0.60\linewidth}
\begin{equation}gin{tikzpicture}
[inner sep=1mm,
place/.style={circle,draw=blue!50,
fill=blue!20,thick},
scale=0.75]
\draw (-5,0)--(11,0);
\draw [red,very thick] (-5,4)--(-3,4);
\draw [green,very thick] (-3,4)--(11,4);
\draw [dashed] (-5,1.8)--(3.2,1.8);
\draw [red,very thick] (3,1.75)--(11,1.75);
\draw [green,very thick] (3,1.85)--(11,1.85);
\draw[blue,ultra thick] (0,0)--(-3,4);
\draw [blue,ultra thick](4,0)--(3,1.8);
\draw [black,ultra thick] (-3,4)--(3,1.8);
\draw [dashed] (3,1.8)--(12/1.8,4);
\draw [dashed] (4,0)--(4+12/1.8,4);
\draw [dashed] (3,1.8)--(4-4/1.8,4);
\draw [dashed] (0,0)--(-4/1.8,4);
\draw [blue,ultra thick](4,0)--(-3,4);
\draw [blue,ultra thick](2.15,2.15*1.8/3)--(3,1.8);
\draw [blue,ultra thick](0,0)--(1.8,1.8*1.8/3);
\draw [black,ultra thick] (-3,4)--(-0.5,0);
\draw [black,ultra thick] (0,0)--(4,0);
\draw [black,ultra thick] (4.5,0)--(3,1.8);
\node at (0,0) [] (name1){\small{$\bullet$}};
\node at (-3,4) [] (name2) {\small{$\bullet$}};
\node at (0.3,-0.4) {$x_1$};
\node at (4,0) [] (name3) {\small{$\bullet$}} ;
\node at (4,-0.4) {$x_1'$};
\node at (4+12/1.8,4) [] (name4) {\small{$\bullet$}} ;
\node at (12/1.8,4) [] {\small{$\bullet$}} ;
\node at (-3,4) [] (name5) {\small{$\bullet$}} ;
\node at (3,1.8) [] (name8) {\lambdarge{$\bullet$}} ;
\node at (-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (4-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (-0.5,0) [] {\small{$\bullet$}} ;
\node at (4.5,0) [] {\small{$\bullet$}} ;
\node at (3.7+12/1.8,4) [above] {$u_4+ih$} ;
\node at (12/1.8,4) [above] {$u_3+ih$} ;
\node at (4-4/1.8,4) [above] {$u_2+ih$} ;
\node at (8.2,0) [below] {$\gamma_{1}$} ;
\node at (8.2,4) [below] {$\gamma_{-1}$} ;
\node at (8.2,1.8) [below] {$\gamma_{0}^-$} ;
\node at (8.2,1.83) [above] {$\gamma_{-1}$} ;
\node at (-4.2,0) [below] {$\gamma_{1}$} ;
\node at (-4.2,1.8) [below] {$\gamma_{0}^+$} ;
\node at (-4.2,4) [below] {$\gamma_{0}^-$} ;
\node at (-0.5,-0.4) {$u_5$} ;
\node at (4.5,-0.4) {$u_6$} ;
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\draw [-,green,thick] (name2) to node[below] {} (name4);
\node at (-3.7,4.25) {$x'_2+ih$} ;
\node at (-1.7,4.25) {$u_1+ih$} ;
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\node at (4.07,2.07) {$x_2+ih_1$};
\varepsilonnd{tikzpicture}
\varepsilonnd{minipage}
\caption{10a. Critical geodesics and loops. Case
\textbf{6.3(b2)}(a).}
\varepsilonnd{figure}
\begin{equation}gin{figure}[h]
\centering
\hspace{-4.5cm}
\begin{equation}gin{minipage}{0.60\linewidth}
\begin{equation}gin{tikzpicture}
[inner sep=1mm,
place/.style={circle,draw=blue!50,
fill=blue!20,thick},
scale=0.75]
\draw (-5,0)--(11,0);
\draw [red,very thick] (-5,4)--(-4/1.8,4);
\draw [green,very thick] (-4/1.8,4)--(11,4);
\draw [dashed] (-5,1.8)--(3.2,1.8);
\draw [red,very thick] (3,1.75)--(11,1.75);
\draw [green,very thick] (3,1.85)--(11,1.85);
\draw[blue,ultra thick] (0,0)--(-4/1.8,4);
\draw [blue,ultra thick](4,0)--(3,1.8);
\draw [black,ultra thick] (-4/1.8,4)--(3,1.8);
\draw [dashed] (3,1.8)--(12/1.8,4);
\draw [dashed] (4,0)--(4+12/1.8,4);
\draw [dashed] (3,1.8)--(4-4/1.8,4);
\draw [dashed] (0,0)--(-4/1.8,4);
\draw [blue,ultra thick](4,0)--(-4/1.8,4);
\draw [blue,ultra thick](2.15,2.15*1.8/3)--(3,1.8);
\draw [blue,ultra thick](0,0)--(1.8,1.8*1.8/3);
\draw [black,ultra thick] (0,0)--(4,0);
\node at (0,0) [] (name1){\small{$\bullet$}};
\node at (0.3,-0.4) {$x_1$};
\node at (4,0) [] (name3) {\small{$\bullet$}} ;
\node at (4,-0.4) {$x_1'$};
\node at (4+12/1.8,4) [] (name4) {\small{$\bullet$}} ;
\node at (12/1.8,4) [] {\small{$\bullet$}} ;
\node at (3,1.8) [] (name8) {\lambdarge{$\bullet$}} ;
\node at (-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (4-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (3.7+12/1.8,4) [above] {$u_4+ih$} ;
\node at (12/1.8,4) [above] {$u_3+ih$} ;
\node at (4-4/1.8,4) [above] {$u_2+ih$} ;
\node at (8.2,0) [below] {$\gamma_{1}$} ;
\node at (8.2,4) [below] {$\gamma_{-1}$} ;
\node at (8.2,1.8) [below] {$\gamma_{0}^-$} ;
\node at (8.2,1.83) [above] {$\gamma_{-1}$} ;
\node at (-4.2,0) [below] {$\gamma_{1}$} ;
\node at (-4.2,1.8) [below] {$\gamma_{0}^+$} ;
\node at (-4.2,4) [below] {$\gamma_{0}^-$} ;
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\node at (-4/1.8,4.25) {$x'_2+ih$} ;
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\node at (4.07,2.07) {$x_2+ih_1$};
\varepsilonnd{tikzpicture}
\varepsilonnd{minipage}
\caption{10b. Critical geodesics and loops. Case
\textbf{6.3(b2)}(b).}
\varepsilonnd{figure}
\begin{equation}gin{figure}[h]
\centering
\hspace{-4.5cm}
\begin{equation}gin{minipage}{0.60\linewidth}
\begin{equation}gin{tikzpicture}
[inner sep=1mm,
place/.style={circle,draw=blue!50,
fill=blue!20,thick},
scale=0.75]
\draw (-5,0)--(11,0);
\draw [red,very thick] (-5,4)--(-0.5,4);
\draw [green,very thick] (-0.5,4)--(11,4);
\draw [dashed] (-5,1.8)--(3.2,1.8);
\draw [red,very thick] (3,1.75)--(11,1.75);
\draw [green,very thick] (3,1.85)--(11,1.85);
\draw[blue,ultra thick] (0,0)--(-0.5,4);
\draw [blue,ultra thick](4,0)--(3,1.8);
\draw [black,ultra thick] (-0.5,4)--(3,1.8);
\draw [dashed] (3,1.8)--(12/1.8,4);
\draw [dashed] (4,0)--(4+12/1.8,4);
\draw [dashed] (3,1.8)--(4-4/1.8,4);
\draw [dashed] (0,0)--(-4/1.8,4);
\draw [blue,ultra thick](4,0)--(-0.5,4);
\draw [blue,ultra thick](2.55,2.55*1.8/3)--(3,1.8);
\draw [blue,ultra thick](0,0)--(2.18,2.18*1.8/3);
\draw [black,ultra thick] (-1,4)--(0,0);
\draw [black,ultra thick] (0,0)--(4,0);
\draw [black,ultra thick] (4,0)--(3.5,1.8);
\node at (0,0) [] (name1){\lambdarge{$\bullet$}};
\node at (-0.5,4) [] (name2) {\small{$\bullet$}};
\node at (0.3,-0.4) {$x_1$};
\node at (4,0) [] (name3) {\lambdarge{$\bullet$}} ;
\node at (4,-0.4) {$x_1'$};
\node at (4+12/1.8,4) [] (name4) {\small{$\bullet$}} ;
\node at (12/1.8,4) [] {\small{$\bullet$}} ;
\node at (-0.5,4) [] (name5) {\small{$\bullet$}} ;
\node at (3,1.8) [] (name8) {\lambdarge{$\bullet$}} ;
\node at (-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (4-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (-1,4) [] {\small{$\bullet$}} ;
\node at (3.5,1.75) [] {\small{$\bullet$}} ;
\node at (3.7+12/1.8,4) [above] {$u_4+ih$} ;
\node at (12/1.8,4) [above] {$u_3+ih$} ;
\node at (4-4/1.8,4) [above] {$u_2+ih$} ;
\node at (8.2,0) [below] {$\gamma_{1}$} ;
\node at (8.2,4) [below] {$\gamma_{-1}$} ;
\node at (8.2,1.8) [below] {$\gamma_{0}^-$} ;
\node at (8.2,1.83) [above] {$\gamma_{-1}$} ;
\node at (-4.2,0) [below] {$\gamma_{1}$} ;
\node at (-4.2,1.8) [below] {$\gamma_{0}^+$} ;
\node at (-4.2,4) [below] {$\gamma_{0}^-$} ;
\node at (-1.3,4) [above] {$u_7$} ;
\node at (4,1.76) [below] {$u_8$} ;
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\draw [-,green,thick] (name2) to node[below] {} (name4);
\node at (-0.1,4.25) {$x'_2+ih$} ;
\node at (-2.7,4.25) {$u_1+ih$} ;
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\node at (4.11,2.07) {$x_2+ih_1$};
\varepsilonnd{tikzpicture}
\varepsilonnd{minipage}
\caption{10c. Critical geodesics and loops. Case
\textbf{6.3(b2)}(c).}
\varepsilonnd{figure}
\begin{equation}gin{figure}[h]
\centering
\hspace{-4.5cm}
\begin{equation}gin{minipage}{0.60\linewidth}
\begin{equation}gin{tikzpicture}
[inner sep=1mm,
place/.style={circle,draw=blue!50,
fill=blue!20,thick},
scale=0.75]
\draw (-5,0)--(11,0);
\draw [red,very thick] (-5,4)--(4-4/1.8,4);
\draw [green,very thick] (4-4/1.8,4)--(11,4);
\draw [dashed] (-5,1.8)--(3.2,1.8);
\draw [red,very thick] (3,1.75)--(11,1.75);
\draw [green,very thick] (3,1.85)--(11,1.85);
\draw[blue,ultra thick] (0,0)--(4-4/1.8,4);
\draw [blue,ultra thick](4,0)--(3,1.8);
\draw [black,ultra thick] (4-4/1.8,4)--(3,1.8);
\draw [dashed] (3,1.8)--(12/1.8,4);
\draw [dashed] (4,0)--(4+12/1.8,4);
\draw [dashed] (3,1.8)--(4-4/1.8,4);
\draw [dashed] (0,0)--(-4/1.8,4);
\draw [blue,ultra thick](0,0)--(3,1.8);
\draw [black,ultra thick] (0,0)--(4-4/1.8-1.27,4);
\draw [black,ultra thick] (0,0)--(4,0);
\draw [black,ultra thick] (4,0)--(3+1.27,1.8);
\node at (0,0) [] (name1){\lambdarge{$\bullet$}};
\node at (0.3,-0.4) {$x_1$};
\node at (4,0) [] (name3) {\lambdarge{$\bullet$}} ;
\node at (4,-0.4) {$x_1'$};
\node at (4+12/1.8,4) [] (name4) {\small{$\bullet$}} ;
\node at (12/1.8,4) [] {\small{$\bullet$}} ;
\node at (3,1.8) [] (name8) {\lambdarge{$\bullet$}} ;
\node at (-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (4-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (4-4/1.8-1.27,4) [] {\small{$\bullet$}} ;
\node at (3+1.27,1.75) [] {\small{$\bullet$}} ;
\node at (3.7+12/1.8,4) [above] {$u_4+ih$} ;
\node at (12/1.8,4) [above] {$u_3+ih$} ;
\node at (8.2,0) [below] {$\gamma_{1}$} ;
\node at (8.2,4) [below] {$\gamma_{-1}$} ;
\node at (8.2,1.8) [below] {$\gamma_{0}^-$} ;
\node at (8.2,1.83) [above] {$\gamma_{-1}$} ;
\node at (-4.2,0) [below] {$\gamma_{1}$} ;
\node at (-4.2,1.8) [below] {$\gamma_{0}^+$} ;
\node at (-4.2,4) [below] {$\gamma_{0}^-$} ;
\node at (4-4/1.8-1.27,4) [above] {$u_7$} ;
\node at (3+1.55,1.77) [below] {$u_8$} ;
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\node at (4-4/1.8+0.4,4.25) {$x'_2+ih$} ;
\node at (-2.7,4.25) {$u_1+ih$} ;
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\node at (4.5,2.07) {$x_2+ih_1$};
\varepsilonnd{tikzpicture}
\varepsilonnd{minipage}
\caption{10d. Critical geodesics and loops. Case
\textbf{6.3(b2)}(d).}
\varepsilonnd{figure}
\begin{equation}gin{figure}[h]
\centering
\hspace{-4.5cm}
\begin{equation}gin{minipage}{0.60\linewidth}
\begin{equation}gin{tikzpicture}
[inner sep=1mm,
place/.style={circle,draw=blue!50,
fill=blue!20,thick},
scale=0.75]
\draw (-4,0)--(11,0);
\draw [red,very thick] (-4,4)--(3.2,4);
\draw [green,very thick] (3.2,4)--(11,4);
\draw [dashed] (-4,1.8)--(3.2,1.8);
\draw [red,very thick] (3,1.75)--(11,1.75);
\draw [green,very thick] (3,1.85)--(11,1.85);
\draw[blue,ultra thick] (0,0)--(3.2,4);
\draw [blue,ultra thick](4,0)--(3,1.8);
\draw [black,ultra thick] (3.2,4)--(3,1.8);
\draw [dashed] (3,1.8)--(12/1.8,4);
\draw [dashed] (4,0)--(4+12/1.8,4);
\draw [dashed] (3,1.8)--(4-4/1.8,4);
\draw [dashed] (0,0)--(-4/1.8,4);
\draw [blue,ultra thick](4,0)--(3,1.8);
\draw [black,ultra thick](3,1.8)--(3.2,4);
\draw [blue,ultra thick](0,0)--(3,1.8);
\draw [black,ultra thick] (1.5,4)--(0,0);
\draw [black,ultra thick] (0,0)--(4,0);
\draw [black,ultra thick] (4,0)--(4.7,1.8);
\draw [blue,ultra thick] (4,0)--(3.6,1.77);
\draw [blue,ultra thick](2.68,1.8+2.2*0.32/0.4)--(2.6,4);
\draw [blue,ultra thick](3,1.8)--(2.73,1.8+2.2*0.27/0.4);
\node at (0,0) [] {\lambdarge{$\bullet$}};
\node at (2.6,4) [] {\small{$\bullet$}};
\node at (0.3,-0.4) {$x_1$};
\node at (4,0) [] (name3) {\lambdarge{$\bullet$}} ;
\node at (4,-0.4) {$x_1'$};
\node at (4+12/1.8,4) [] (name4) {\small{$\bullet$}} ;
\node at (12/1.8,4) [] {\small{$\bullet$}} ;
\node at (3.2,4) [] (name5) {\small{$\bullet$}} ;
\node at (3,1.8) [] (name8) {\lambdarge{$\bullet$}} ;
\node at (-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (4-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (1.5,4) [] {\small{$\bullet$}} ;
\node at (4.7,1.75) [] {\small{$\bullet$}} ;
\node at (3.6,1.75) [] {\small{$\bullet$}} ;
\node at (3.7+12/1.8,4) [above] {$u_4+ih$} ;
\node at (12/1.8,4) [above] {$u_3+ih$} ;
\node at (4.3-4/1.8-0.2,4) [above] {$u_2+ih$} ;
\node at (8.2,0) [below] {$\gamma_{1}$} ;
\node at (8.2,4) [below] {$\gamma_{-1}$} ;
\node at (8.2,1.8) [below] {$\gamma_{0}^-$} ;
\node at (8.2,1.83) [above] {$\gamma_{-1}$} ;
\node at (-3.2,0) [below] {$\gamma_{1}$} ;
\node at (-3.2,1.8) [below] {$\gamma_{0}^+$} ;
\node at (-3.2,4) [below] {$\gamma_{0}^-$} ;
\node at (2.3,4) [below] {$u_{10}$} ;
\node at (5,1.77) [below] {$u_8$} ;
\node at (1.1,4) [below] {$u_{7}$} ;
\node at (4,1.77) [below] {$u_9$} ;
\node at (4,4.25) {$x'_2+ih$} ;
\node at (-2,4.25) {$u_1+ih$} ;
\node at (4.4,2.07) {$x_2+ih_1$};
\varepsilonnd{tikzpicture}
\varepsilonnd{minipage}
\caption{10e. Critical geodesics and loops. Case
\textbf{6.3(b2)}(e).}
\varepsilonnd{figure}
\begin{equation}gin{figure}[h]
\centering
\hspace{-4.5cm}
\begin{equation}gin{minipage}{0.60\linewidth}
\begin{equation}gin{tikzpicture}
[inner sep=1mm,
place/.style={circle,draw=blue!50,
fill=blue!20,thick},
scale=0.75]
\draw (-4,0)--(11,0);
\draw [red,very thick] (-4,4)--(12/1.8,4);
\draw [green,very thick] (12/1.8,4)--(11,4);
\draw [dashed] (-4,1.8)--(3.2,1.8);
\draw [red,very thick] (3,1.75)--(11,1.75);
\draw [green,very thick] (3,1.85)--(11,1.85);
\draw[blue,ultra thick] (3,1.8)--(12/1.8-2.2,4);
\draw[blue,ultra thick] (4,0)--(3+2.2,1.77);
\draw [blue,ultra thick](4,0)--(3,1.8);
\draw [black,ultra thick] (3,1.8)--(12/1.8,4);
\draw [dashed] (4,0)--(4+12/1.8,4);
\draw [dashed] (3,1.8)--(4-4/1.8,4);
\draw [dashed] (0,0)--(-4/1.8,4);
\draw [blue,ultra thick](0,0)--(3,1.8);
\draw [black,ultra thick] (0,0)--(12/1.8-2.9,4);
\draw [black,ultra thick] (0,0)--(4,0);
\draw [black,ultra thick] (4,0)--(3+2.9,1.77);
\node at (0,0) [] (name1){\lambdarge{$\bullet$}};
\node at (0.3,-0.4) {$x_1$};
\node at (4,0) [] (name3) {\lambdarge{$\bullet$}} ;
\node at (4,-0.4) {$x_1'$};
\node at (4+12/1.8,4) [] (name4) {\small{$\bullet$}} ;
\node at (12/1.8,4) [] {\small{$\bullet$}} ;
\node at (3+2.2,1.77) [] {\small{$\bullet$}} ;
\node at (3,1.8) [] (name8) {\lambdarge{$\bullet$}} ;
\node at (-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (4-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (12/1.8-2.2,4) [] {\small{$\bullet$}} ;
\node at (12/1.8-2.9,4) [] {\small{$\bullet$}} ;
\node at (5.9,1.75) [] {\small{$\bullet$}} ;
\node at (3.7+12/1.8,4) [above] {$u_4+ih$} ;
\node at (4-4/1.8,4) [above] {$u_2+ih$} ;
\node at (8.2,0) [below] {$\gamma_{1}$} ;
\node at (8.2,4) [below] {$\gamma_{-1}$} ;
\node at (8.2,1.8) [below] {$\gamma_{0}^-$} ;
\node at (8.2,1.83) [above] {$\gamma_{-1}$} ;
\node at (-3.2,0) [below] {$\gamma_{1}$} ;
\node at (-3.2,1.8) [below] {$\gamma_{0}^+$} ;
\node at (-3.2,4) [below] {$\gamma_{0}^-$} ;
\node at (12/1.8-2.9,4) [above] {$u_7$} ;
\node at (6.2,1.77) [below] {$u_8$} ;
\node at (3+1.7,1.77) [below] {$u_9$} ;
\node at (12/1.8-2.2,4) [above] {$u_{10}$} ;
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\node at (12/1.8,4.25) {$x'_2+ih$} ;
\node at (-2,4.25) {$u_1+ih$} ;
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\node at (4.55,2.07) {$x_2+ih_1$};
\varepsilonnd{tikzpicture}
\varepsilonnd{minipage}
\caption{10f. Critical geodesics and loops. Case
\textbf{6.3(b2)}(f).}
\varepsilonnd{figure}
{\cal F}loatBarrier
\begin{equation}gin{figure}[h]
\centering
\hspace{-4.5cm}
\begin{equation}gin{minipage}{0.60\linewidth}
\begin{equation}gin{tikzpicture}
[inner sep=1mm,
place/.style={circle,draw=blue!50,
fill=blue!20,thick},
scale=0.75]
\draw (-2.8,0)--(12,0);
\draw [red,very thick] (-2.8,4)--(9.2,4);
\draw [green,very thick] (9.2,4)--(12,4);
\draw [dashed] (-2.8,1.8)--(3.2,1.8);
\draw [red,very thick] (3,1.75)--(12,1.75);
\draw [green,very thick] (3,1.85)--(12,1.85);
\draw[black,ultra thick] (3,1.8)--(9.2,4);
\draw [blue,ultra thick](4,0)--(3,1.8);
\draw [blue,ultra thick] (3,1.8)--(8.2,4);
\draw [dashed] (3,1.8)--(12/1.8,4);
\draw [dashed] (4,0)--(4+12/1.8,4);
\draw [dashed] (3,1.8)--(4-4/1.8,4);
\draw [dashed] (0,0)--(-4/1.8,4);
\draw [blue,ultra thick](4,0)--(3,1.8);
\draw [blue,ultra thick](3,1.8)--(9.2-3.2,4);
\node at (9.2-3.2,4) [] {\small{$\bullet$}} ;
\draw [blue,ultra thick](0,0)--(3,1.8);
\draw [black,ultra thick] (0,0)--(9.2-3.7,4);
\node at (9.2-3.7,4) [] {\small{$\bullet$}} ;
\draw [black,ultra thick] (0,0)--(4,0);
\draw [black,ultra thick] (4,0)--(6.7,1.77);
\node at (6.7,1.77) [] {\small{$\bullet$}} ;
\draw [blue,ultra thick] (4,0)--(6.2,1.77);
\node at (6.2,1.77) [] {\small{$\bullet$}} ;
\draw [blue,ultra thick](3.35,3.35*1.75/4)--(4,1.75);
\draw [blue,ultra thick](0,0)--(3.05,3.05*1.75/4);
\node at (4.1,1.77) [] {\small{$\bullet$}} ;
\node at (0,0) [] {\lambdarge{$\bullet$}};
\node at (0.3,-0.4) {$x_1$};
\node at (4,0) [] (name3) {\lambdarge{$\bullet$}} ;
\node at (4,-0.4) {$x_1'$};
\node at (4+12/1.8,4) [] (name4) {\small{$\bullet$}} ;
\node at (12/1.8,4) [] {\small{$\bullet$}} ;
\node at (3,1.8) [] (name8) {\Large{$\bullet$}} ;
\node at (-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (4-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (9.2,4) [] {\small{$\bullet$}} ;
\node at (8.2,4) [] {\small{$\bullet$}} ;
\node at (4.5+12/1.8+0.3,4) [above] {$u_4+ih$} ;
\node at (4.3-4/1.8,4) [above] {$u_2+ih$} ;
\node at (4.5+12/1.8,0) [below] {$\gamma_{1}$} ;
\node at (4.5+12/1.8,4) [below] {$\gamma_{-1}$} ;
\node at (4.5+12/1.8,1.8) [below] {$\gamma_{0}^-$} ;
\node at (4.5+12/1.8,1.83) [above] {$\gamma_{-1}$} ;
\node at (-1.4,0) [below] {$\gamma_{1}$} ;
\node at (-1.4,1.8) [below] {$\gamma_{0}^+$} ;
\node at (-1.4,4) [below] {$\gamma_{0}^-$} ;
\node at (9.2-3.05,4) [above] {$u_{10}$} ;
\node at (7,1.75) [below] {$u_8$} ;
\node at (9.2-3.9,4) [above] {$u_{7}$} ;
\node at (6.2,1.8) [above] {$u_9$} ;
\node at (7.9,4) [above] {$u_{12}$} ;
\node at (4.3,1.77) [below] {$u_{11}$} ;
\node at (9.4,4.25) {$x'_2+ih$} ;
\node at (-2,4.25) {$u_1+ih$} ;
\node at (1.7,2.1) {$x_2+ih_1$};
\varepsilonnd{tikzpicture}
\varepsilonnd{minipage}
\caption{10g. Critical geodesics and loops. Case
\textbf{6.3(b2)}(g).}
\varepsilonnd{figure}
\begin{equation}gin{figure}
\centering
\hspace{-4.5cm}
\begin{equation}gin{minipage}{0.60\linewidth}
\begin{equation}gin{tikzpicture}
[inner sep=1mm,
place/.style={circle,draw=blue!50,
fill=blue!20,thick},
scale=0.75]
\draw (-2.8,0)--(12,0);
\draw [red,very thick] (-2.8,4)--(4+12/1.8,4);
\draw [green,very thick] (4+12/1.8,4)--(12,4);
\draw [dashed] (-2.8,1.8)--(3.2,1.8);
\draw [red,very thick] (3,1.75)--(12,1.75);
\draw [green,very thick] (3,1.85)--(12,1.85);
\draw [blue,ultra thick](4,0)--(3,1.8);
\draw [black,ultra thick] (3,1.8)--(4+12/1.8,4);
\draw [blue,ultra thick] (3,1.8)--(12/1.8,4);
\draw [blue,ultra thick] (3,1.8)--(2+12/1.8,4);
\node at (2+12/1.8,4) [] {\small{$\bullet$}};
\draw [blue,ultra thick] (3.5,3.5*1.77/5)--(5,1.77);
\draw [blue,ultra thick] (0,0)--(3.15,3.15*1.77/5);
\draw [dashed] (4,0)--(4+12/1.8,4);
\draw [dashed] (3,1.8)--(4-4/1.8,4);
\draw [dashed] (0,0)--(-4/1.8,4);
\draw [blue,ultra thick](0,0)--(3,1.8);
\draw [black,ultra thick] (0,0)--(4,0);
\draw [blue,ultra thick] (4,0)--(7,1.77);
\node at (0,0) [] (name1){\lambdarge{$\bullet$}};
\node at (0.3,-0.4) {$x_1$};
\node at (4,0) [] (name3) {\lambdarge{$\bullet$}} ;
\node at (4,-0.4) {$x_1'$};
\node at (4+12/1.8,4) [] (name4) {\small{$\bullet$}} ;
\node at (12/1.8,4) [] {\small{$\bullet$}} ;
\node at (7,1.77) [] {\small{$\bullet$}} ;
\node at (3,1.8) [] (name8) {\Large{$\bullet$}} ;
\node at (-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (4-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (5,1.75) [] {\small{$\bullet$}} ;
\node at (4-4/1.8,4) [above] {$u_2+ih$} ;
\node at (4.5+12/1.8,0) [below] {$\gamma_{1}$} ;
\node at (4.5+12/1.8,4) [below] {$\gamma_{-1}$} ;
\node at (4.5+12/1.8,1.8) [below] {$\gamma_{0}^-$} ;
\node at (4.5+12/1.8,1.83) [above] {$\gamma_{-1}$} ;
\node at (-1.4,0) [below] {$\gamma_{1}$} ;
\node at (-1.4,1.8) [below] {$\gamma_{0}^+$} ;
\node at (-1.4,4) [below] {$\gamma_{0}^-$} ;
\node at (7.2,1.77) [below] {$u_9$} ;
\node at (3+2.3,1.76) [below] {$u_{11}$} ;
\node at (12/1.8,4) [above] {$u_{10}$} ;
\node at (12/1.8+2,4) [above] {$u_{12}$} ;
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\node at (3.7+12/1.8,4.25) {$x'_2+ih$} ;
\node at (-2,4.25) {$u_1+ih$} ;
\draw [-] (name1) to node[below] {$\gamma_{\infty}$} (name3);
\node at (1.9,2.07) {$x_2+ih_1$};
\varepsilonnd{tikzpicture}
\varepsilonnd{minipage}
\caption{10h. Critical geodesics and loops. Case
\textbf{6.3(b2)}(h).}
\varepsilonnd{figure}
\begin{equation}gin{figure}
\centering
\hspace{-4.5cm}
\begin{equation}gin{minipage}{0.60\linewidth}
\begin{equation}gin{tikzpicture}
[inner sep=1mm,
place/.style={circle,draw=blue!50,
fill=blue!20,thick},
scale=0.75]
\draw (-2.8,0)--(13,0);
\draw [red,very thick] (-2.8,4)--(12,4);
\draw [green,very thick] (12,4)--(13,4);
\draw [dashed] (-2.8,1.8)--(3.2,1.8);
\draw [red,very thick] (3,1.75)--(13,1.75);
\draw [green,very thick] (3,1.85)--(13,1.85);
\draw[black,ultra thick] (3,1.8)--(12,4);
\draw [blue,ultra thick](4,0)--(3,1.8);
\draw [blue,ultra thick] (3,1.8)--(12-4.4,4);
\node at (12-4.4,4) [] {\small{$\bullet$}};
\node at (12-4.1,4) [above] {$u_{10}$};
\draw [black,ultra thick] (3,1.8)--(12-4.9,4);
\node at (12-4.9,4) [] {\small{$\bullet$}};
\node at (12-4.9,4) [above] {$u_{14}$};
\draw [black,ultra thick] (3-4.1*1.8/2.2,0)--(3,1.8);
\node at (3-4.1*1.8/2.2,0) [] {\small{$\bullet$}};
\node at(3-4.1*1.8/2.2-0.2,-0.4) [] {$u_{13}$};
\draw [black,ultra thick] (4-3+4.1*1.8/2.2,0)--(8.1,1.77);
\node at (4-3+4.1*1.8/2.2,0) [] {\small{$\bullet$}};
\node at (4-3+4.1*1.8/2.2+0.3,-0.4) [] {$u_{15}$};
\node at (8.5,1.77) [below] {$u_{16}$};
\node at (8.1,1.77) [] {\small{$\bullet$}};
\draw [dashed] (3,1.8)--(12/1.8,4);
\draw [dashed] (4,0)--(4+12/1.8,4);
\draw [dashed] (3,1.8)--(4-4/1.8,4);
\draw [dashed] (0,0)--(-4/1.8,4);
\draw [dashed] (3,1.8)--(4-4/1.8,4);
\draw [blue,ultra thick](3,1.8)--(12-3.3,4);
\draw [blue,ultra thick](0,0)--(3,1.8);
\draw [black,ultra thick] (0,0)--(4,0);
\node at (7.6,1.77) [] {\small{$\bullet$}} ;
\draw [blue,ultra thick] (4,0)--(3+4.6,1.77);
\node at (8,1.8) [above] {$u_9$};
\node at (6.2,1.77) [] {\small{$\bullet$}} ;
\draw [blue,ultra thick](3.6,3.6*1.76/6.3)--(3+3.3,1.76);
\draw [blue,ultra thick](0,0)--(3.3,3.3*1.76/6.3);
\node at (0,0) [] {\lambdarge{$\bullet$}};
\node at (0.3,-0.4) {$x_1$};
\node at (4,0) [] (name3) {\lambdarge{$\bullet$}} ;
\node at (3.8,-0.4) {$x_1'$};
\node at (4+12/1.8,4) [] (name4) {\small{$\bullet$}} ;
\node at (12/1.8,4) [] {\small{$\bullet$}} ;
\node at (3,1.8) [] (name8) {\Large{$\bullet$}} ;
\node at (-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (4-4/1.8,4) [] {\small{$\bullet$}} ;
\node at (12,4) [] {\small{$\bullet$}} ;
\node at (8.6,4) [] {\small{$\bullet$}} ;
\node at (4.3-4/1.8,4) [above] {$u_2+ih$} ;
\node at (12.5,0) [below] {$\gamma_{1}$} ;
\node at (12.5,4) [below] {$\gamma_{-1}$} ;
\node at (12.5,1.77) [below] {$\gamma_{0}^-$} ;
\node at (12.5,1.83) [above] {$\gamma_{-1}$} ;
\node at (-1.4,0) [below] {$\gamma_{1}$} ;
\node at (-1.4,1.8) [below] {$\gamma_{0}^+$} ;
\node at (-1.4,4) [below] {$\gamma_{0}^-$} ;
\node at (6.2,1.8) [above] {$u_{11}$} ;
\node at (8.9,4) [above] {$u_{12}$} ;
\node at (12,4.25) {$x'_2+ih$} ;
\node at (-2,4.25) {$u_1+ih$} ;
\node at (1.8,2.1) {$x_2+ih_1$};
\varepsilonnd{tikzpicture}
\varepsilonnd{minipage}
\caption{10i. Critical geodesics and loops. Case
\textbf{6.3(b2)}(i).}
\varepsilonnd{figure}
{\cal F}loatBarrier
\textbf{XI.} Identification rules.
{\cal F}loatBarrier
\begin{equation}gin{figure}[h]
\centering
\hspace{-4.5cm}
\begin{equation}gin{minipage}{0.60\linewidth}
\begin{equation}gin{tikzpicture}
[inner sep=1mm,
place/.style={circle,draw=blue!50,
fill=blue!20,thick},
scale=0.75]
\draw [dashed](0,0)--(4,0);
\draw [red,very thick] (-5,4)--(-1,4);
\draw [green,very thick] (-1,4)--(11,4);
\draw [dashed] (-5,1.8)--(3.2,1.8);
\draw [red,very thick] (3,1.75)--(11,1.75);
\draw [green,very thick] (3,1.85)--(11,1.85);
\draw[blue,ultra thick] (0,0)--(0,-3);
\draw [blue,ultra thick](4,0)--(4,-3);
\draw [black,ultra thick] (-5,0)--(0,0);
\draw [black,ultra thick] (4,0)--(11,0);
\node at (0,0) [] (name1){\small{$\bullet$}};
\node at (-1,4) [] (name2) {\small{$\bullet$}};
\node at (0.4,-0.4) {$x_1$};
\node at (4,0) [] (name3) {\small{$\bullet$}} ;
\node at (3.7,-0.4) {$x_1'$};
\node at (3,1.8) [] (name8) {\lambdarge{$\bullet$}} ;
\node at (-2,0) [] {\small{$\bullet$}} ;
\node at (-2,0) [below] {$-x$} ;
\node at (6,0) [] {\small{$\bullet$}} ;
\node at (6,0) [below] {$1+x$} ;
\node at (0,-2) [] {\small{$\bullet$}} ;
\node at (0,-2) [left] {$iy$} ;
\node at (4,-2) [] {\small{$\bullet$}} ;
\node at (4,-2) [right] {$1+iy$} ;
\node at (-4,4) [] {\small{$\bullet$}} ;
\node at (-4,4) [above] {$-x+x'_2+ih$} ;
\node at (2,4) [] {\small{$\bullet$}} ;
\node at (2,4) [above] {$x+x'_2+ih$} ;
\node at (6,1.7) [] {\small{$\bullet$}} ;
\node at (6,1.7) [below] {$x+x_2+h_1$} ;
\node at (6,1.9) [] {\small{$\bullet$}} ;
\node at (8.2,0) [below] {$\gamma_{1}$} ;
\node at (8.2,4) [below] {$\gamma_{-1}$} ;
\node at (8.2,1.8) [below] {$\gamma_{0}^-$} ;
\node at (8.2,1.83) [above] {$\gamma_{-1}$} ;
\node at (-4.2,0) [below] {$\gamma_{1}$} ;
\node at (-4.2,1.8) [below] {$\gamma_{0}^+$} ;
\node at (-4.2,4) [below] {$\gamma_{0}^-$} ;
\draw [-,green,thick] (name2) to node[below] {} (name4);
\node at (-1,4.45) {$x'_2+ih$} ;
\node at (3.5,2.3) {$x_2+ih_1$};
\node at (2,-0.4) {$\gamma_\infty$};
\node at (0,2.3) [above] {{\mbox{\Huge $\Omegaega$}}};
\varepsilonnd{tikzpicture}
\varepsilonnd{minipage}
\caption{11. Domain $\Omegaega$ and identification rules. }
\varepsilonnd{figure}
{\cal F}loatBarrier
\begin{equation}gin{thebibliography}{30}
\bibitem{AMFMGT} M.~J.~Atia, A.~Mart\'inez--Finkelshtein, P.~Mart\'inez--Gonz\'alez, and F.~Thabet, Quadratic differentials and asymptotics of Laguerre polynomials with varying complex coefficients, arXiv: 1311.0372.
\bibitem{AS} M.~Abramowitz, I.~A.~Stegun, eds. (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, ISBN 978-0-486-61272-0.
\bibitem{Ba} R.~G.~Bartle, The elements of integration. John Wiley \& Sons, Inc., New York-London-Sydney 1966 x+129 pp.
\bibitem{Ber} T.~Bergkvist, {\varepsilonm On asymptotics of polynomial eigenfunctions for exactly solvable differential operators}, J. Approx. Theory 149(2), (2007), 151--187.
\bibitem{BR} T.~Bergkvist and H.~Rullg\aa rd, {\varepsilonm On polynomial
eigenfunctions for a class of differential operators}, Math. Res.
Lett., vol.9, (2002), 153--171.
\bibitem{Beu} A.~Beurling, Sur la g\'eom\'etrie m\'etrique des surfaces \`a courbure totale $\le 0$. (French) Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952, (1952). Tome Suppl\'ementaire, 7--11.
\bibitem{BBB} J.-E.~Bj\"ork, J.~Borcea, R.~B\o gvad, {\varepsilonm Subharmonic Configurations and
Algebraic Cauchy Transforms of Probability Measures.} Notions of
Positivity and the Geometry of Polynomials Trends in Mathematics
2011, pp 39--62.
\bibitem{BB} J.~Borcea, R.B\o gvad, {\varepsilonm Piecewise harmonic subharmonic functions and positive Cauchy transforms.} Pacific J. Math., Vol. 240 (2009), No. 2, 231--265.
\bibitem {BBS} J.~ Borcea, R.~B\o gvad and B.~Shapiro, Homogenized spectral pencils for exactly solvable operators: asymptotics of polynomial eigenfunctions, Publ. RIMS, vol 45 (2009) 525--568.
\bibitem{BBSh1} J.~Borcea, R.~B\o gvad, and B.~Shapiro, On rational approximation of algebraic functions, Adv. Math. vol 204, issue 2 (2006) 448--480.
\bibitem{BoPi} J.~Borrego Morell, H.~Pijeira Cabrera, Strong asymptotic behavior of eigenpolynomials for a class of linear differential operators, preprint, Nov. 15, (2012).
\bibitem{BoSh} R.~B\o gvad, B.~Shapiro, On motherbody measures with algebraic Cauchy transform, arXiv:1406.1972, submitted.
\bibitem{BRSh} R.~B\o gvad, D.~Khavinson, B.~Shapiro, On asymptotic Gauss-Lucas theorem, arXiv:1510.02339, submiited.
\bibitem {CC} M.~Charalambides, G.~Csordas, The distribution of zeros of a class of Jacobi polynomials. Proc. Amer. Math. Soc. 138 (2010), no. 12, 4345--4357.
\bibitem{DMO} C.~D\'iaz Mendoza, R. A.~Orive, The Szeg\"o curve and Laguerre polynomials with large negative parameters. J. Math. Anal. Appl. 379 (2011), no. 1, 305--315,
\bibitem {DrDu} K. A.~Driver, P.~Duren, Asymptotic zero distribution of hypergeometric
polynomials, Numerical Algorithms 21 (1999) 147--156.
\bibitem {DrJo} K. A.~Driver, S.~J.~Johnston, Asymptotic zero distribution of a class of hypergeometric polynomials. Quaest. Math. 30 (2007), no. 2, 219--230.
\bibitem{DuGu} P.~Duren, B.~Guillou, Asymptotic properties of zeros of hypergeometric polynomials, JAT 111 (2001) 329--343.
\bibitem{Ga} J.~Garnett, {\varepsilonm Analytic capacity and measure.} Lecture Notes in Mathematics, Vol. 297. Springer-Verlag, Berlin-New York, 1972. iv+138 pp.
\bibitem{GoRa} A.A.~Gonchar, E.A.~Rakhmanov, Equilibrium distributions and the rate of rational approximation of analytic functions, Math.USSR-Sb.62 (2) (1989) 305--348.
\bibitem{Je} J.~A.~Jenkins,
{\varepsilonm Univalent functions and conformal mapping.} Ergebnisse der
Mathematik und ihrer Grenzgebiete. Neue Folge, Heft 18. Reihe:
Moderne Funktionentheorie Springer-Verlag,
Berlin-G\"ottingen-Heidelberg, (1958), vi+169 pp.
\bibitem {KuMF} A. B. J.~Kuijlaars, A.~Mart\'inez-Finkelshtein,
Strong asymptotics for Jacobi polynomials with varying nonstandard
parameters. J. Anal. Math. 94 (2004), 195--234.
\bibitem {KUMFO} A. B. J.~Kuijlaars, A.~Mart\'inez-Finkelshtein, R. A.~Orive, Orthogonality of Jacobi polynomials with general parameters. Electron. Trans. Numer. Anal. 19 (2005), 1--17.
\bibitem {MFMGO} A.~Mart\'inez-Finkelshtein, P.~Mart\'inez-Gonz\'alez and R. A.~Orive, Zeros of Jacobi polynomials with varying non-classical parameters, in Special functions (Hong Kong, 1999), 98--113, World Sci. Publishing, River Edge, NJ, 2000.
\bibitem {MMO} A.~Mart\'inez-Finkelshtein, P.~Mart\'inez-Gonz\'alez and R. A.~Orive, On asymptotic zero distribution of Laguerre and generalized
Bessel polynomials with varying parameters, Journal of
Computational and Applied Mathematics 133 (2001) 477--487.
\bibitem {Po} G.~P\'olya, {\varepsilonm Sur un th\'eoreme de Stieltjes},
C.~R.~Acad.~Sci Paris vol. 155, (1912), 767-769.
\bibitem {QW} W.-Y.~Qiu, R.~Wong, Global asymptotic expansions of the Laguerre polynomials, a Riemann-Hilbert approach. Numer. Algorithms 49 (2008), no. 1--4, 331--372.
\bibitem{Sh} B.~Shapiro, Algebro-geometric aspects of Heine-Stieltjes theory, J. London Math. Soc. 83(1) (2011) 36--56.
\bibitem {ShTaTa} B.~Shapiro, K.~Takemura, M.~Tater, {\varepsilonm On spectral polynomials of the Heun equation. II}, Comm. Math. Phys. 311(2) (2012), 277--300.
\bibitem{S1}
A.~Yu.~Solynin, Modules and extremal metric problems.
\textit{Algebra i Analiz} \textbf{11} (1999), no. 1, 3--86;
English transl. \textit{St. Petersburg Math. J.} \textbf{11}
(2000), 1--65.
\bibitem{S2}
A.~Yu.~Solynin, Quadratic differentials and weighted graphs on
compact surfaces. \varepsilonmph{Analysis and mathematical physics},
473�505, Trends Math., Birkh\"{a}user, Basel, 2009.
\bibitem{Sta} H.~Stahl, Orthogonal polynomials with complex-valued weight function I and II, Constr.Approx.2 (1986) 225--240, 241--251.
\bibitem {Str} K.~Strebel, {\varepsilonm Quadratic differentials.} Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 5. Springer-Verlag, Berlin, (1984), xii+184
pp.
\bibitem{Sz} G.~Szeg\"o, Orthogonal polynomials. Fourth edition. American Mathematical Society, Colloquium Publications, Vol. XXIII. American Mathematical Society, Providence, R.I., 1975. xiii+432 pp.
\varepsilonnd{thebibliography}
\varepsilonnd{document} | math | 226,755 |
\begin{document}
\title{Inviscid incompressible limits under mild stratification: A rigorous derivation of the Euler-Boussinesq system}
\author{Eduard Feireisl \thanks{The research of E.F. leading to these results has received funding from the European Research Council under the European Union's Seventh Framework
Programme (FP7/2007-2013)/ ERC Grant Agreement 320078.} \and Anton\' \i n Novotn\' y \thanks{The work was supported by the MODTERCOM project within the APEX programme of the region Provence-Alpe-C\^ote d'Azur and by RVO: 67985840}}
\maketitle
\centerline{Institute of Mathematics of the Academy of Sciences of
the Czech Republic}
\centerline{\v Zitn\' a 25, 115 67 Praha 1,
Czech Republic}
\centerline{IMATH, Universit\' e du Sud Toulon-Var}
\centerline{BP 20139, 839 57 La Garde, France}
\begin{abstract}
We consider the full Navier-Stokes-Fourier system in the singular regime of small
Mach and large Reynolds and P{\' e}clet numbers, with ill prepared initial data on
an unbounded domain$\Omega \subset R^3$ with a compact boundary. We perform the singular limit in the framework of
weak solutions and identify
the Euler-Boussinesq system as the target problem.
\end{abstract}
\tableofcontents
\section{Introduction}
\label{i}
The present paper is an extension of our previous results concerning the inviscid incompressible limit
of the Navier-Stokes-Fourier system \cite{FeiNov12}. In contrast with \cite{FeiNov12}, where the problem is considered on the whole space $R^3$
without any driving force imposed, we consider a more realistic situation when the fluid is subject to a gravitational force due to the physical objects placed
\emph{outside} the fluid domain. Accordingly, we shall assume that the fluid occupies an unbounded \emph{exterior} domain $\Omega \subset R^3$ with smooth
(compact) boundary. Such a situation is interesting from the point of view of possible applications in various meteorological models as the singular limit in the
low Mach, Froude, and large Reynolds and P\' eclet numbers leads to a target system driven by the buoyancy force proportional to temperature deviations.
In particular, we provide a rigorous justification of the so-called Euler-Boussinesq approximation. Our approach is based on the recently discovered
relative entropy inequality \cite{FeiNov10} and the related concept of \emph{dissipative solution} for the Navier-Stokes-Fourier system.
In comparison with \cite{FeiNov12}, the present problem features some additional mathematical difficulties related to the geometry of the underlying spatial domain
and the presence of a driving force. In particular, we have to handle perturbations of weakly
stratified equilibrium states, whereas those are simply constant in \cite{FeiNov12}.
We consider the motion of a compressible, viscous and heat conducting fluid, with the density $\varrho = \varrho(t,x)$, the velocity $\vc{u} = \vc{u}(t,x)$, and
the absolute temperature
$\vartheta = \vartheta(t,x)$ governed by the scaled \emph{Navier-Stokes-Fourier system}:
\bFormula{i1}
\partial_t \varrho + {\rm div}_x (\varrho \vc{u}) = 0,
\end{equation}
\bFormula{i2}
\partial_t (\varrho \vc{u}) + {\rm div}_x (\varrho \vc{u} \otimes \vc{u}) + \frac{1}{\varepsilon^2} \nabla_x p(\varrho, \vartheta) = \varepsilon^a {\rm div}_x \tn{S} (\vartheta, \nabla_x \vc{u}) +{\frac 1\varepsilon}\varrho \nabla_x F,
\end{equation}
\bFormula{i3}
\partial_t (\varrho s(\varrho , \vartheta)) + {\rm div}_x (\varrho s(\varrho, \vartheta) \vc{u}) + \varepsilon^\beta {\rm div}_x \left( \frac{\vc{q}(\vartheta, \nabla_x \vartheta)}{\vartheta} \right) =
\frac{1}{\vartheta} \left( \varepsilon^{2 + a} \tn{S} (\vartheta, \nabla_x \vc{u}) : \nabla_x \vc{u} - \varepsilon^b \frac{\vc{q}(\vartheta, \nabla_x \vartheta) \cdot
\nabla_x \vartheta }{\vartheta} \right),
\end{equation}
where $p = p(\varrho,\vartheta)$ is the pressure, $s = s(\varrho, \vartheta)$ the specific entropy, the symbol $\tn{S}(\vartheta, \nabla_x \vc{u})$ denotes
the viscous stress satisfying \emph{Newton's law}
\bFormula{i3a}
\tn{S}(\vartheta, \nabla_x \vc{u}) = \mu(\vartheta) \left( \nabla_x \vc{u} + \nabla_x^t \vc{u} - \frac{2}{3} {\rm div}_x \vc{u} \right) + \eta(\vartheta) {\rm div}_x \vc{u} \tn{I},
\end{equation}
and $\vc{q} = \vc{q}(\vartheta, \nabla_x \vartheta)$ is the heat flux determined by \emph{Fourier's law}
\bFormula{i3b}
\vc{q}(\vartheta, \nabla_x \vartheta) = - \kappa (\vartheta) \nabla_x \vartheta,
\end{equation}
where the quantities $\mu$, $\eta$, $\kappa$ are temperature dependent transport coefficients.
The fluid occupies an exterior domain $\Omega \subset R^3$, with impermeable, thermally insulating and frictionless boundary, specifically,
\bFormula{i3c}
\vc{u} \cdot \vc{n} = [\tn{S}(\vartheta, \nabla_x \vc{u}) \cdot \vc{n}]_{\rm tan}|_{\partial \Omega} = 0, \ \nabla_x \vartheta \cdot \vc{n}|_{\partial \Omega} = 0.
\end{equation}
In addition, we consider the far field boundary conditions
\bFormula{i5}
\varrho \to \Ov{\varrho}, \ \vartheta \to \Ov{\vartheta}, \ \vc{u} \to 0 \ \mbox{as} \ |x| \to \infty,
\end{equation}
where $\Ov{\varrho}$, $\Ov{\vartheta}$ are positive constants.
The scaling in (\ref{i1} - \ref{i3}), expressed by means of a single (small) parameter $\varepsilon$, corresponds to:
\noindent
Mach number \dotfill $\varepsilon$,
\noindent
Froude number \dotfill $\varepsilon^{1/2}$,
\noindent
Reynolds number \dotfill $\varepsilon^{-a}$,
\noindent
P\' eclet number \dotfill $\varepsilon^{-b}$.
In accordance with the previous discussion, we consider the driving force induced by a potential
\bFormula{potential1}
F(x) = \int_{R^3} \frac{1}{x - y} m(y) {\rm d}y,\ m \geq 0, \ {\rm supp}[m] \subset R^3 \setminus \Omega,
\end{equation}
meaning the fluid is driven by the gravitational force of objects lying outside the fluid domain.
Finally, the initial data are taken in the form
\bFormula{i4}
\varrho(0, \cdot) = \varrho_{0,\varepsilon} = \Ov{\varrho}_\varepsilon + \varepsilon \varrho^{(1)}_{0,\varepsilon},\
\vartheta(0, \cdot) = \vartheta_{0,\varepsilon} = \Ov{\vartheta} + \varepsilon \vartheta^{(1)}_{0,\varepsilon},\ \vc{u}(0, \cdot) = \vc{u}_{0, \varepsilon},
\end{equation}
where $(\overline\varrho_\varepsilon, \overline\vartheta)$ is the equilibrium solution associated with the far field values of $\Ov{\varrho}$, $\Ov{\vartheta}$, namely
\begin{equation}\label{equilibrium}
\nabla_x p(\overline\varrhoe,\overline\vartheta) = \varepsilon\overline\varrhoe\nabla_x F,\quad\overline\varrhoe\to\overline\varrho\quad\mbox{as $|x|\to \infty$}.
\end{equation}
The limit (target) problem can be formally identified as the incompressible \emph{Euler-Boussinesq system}:
\bFormula{l1}
{\rm div}_x \vc{v} = 0,
\end{equation}
\bFormula{l2}
\partial_t \vc{v} + \vc{v} \cdot \nabla_x \vc{v} + \nabla_x \Pi = - a (\Ov{\varrho}, \Ov{\vartheta} )\theta\nabla_x F,
\end{equation}
\bFormula{l3}
c_p(\Ov{\varrho}, \Ov{\vartheta}) \left( \partial_t \theta + \vc{v} \cdot \nabla_x \theta \right) - \Ov{\vartheta} a(\Ov{\varrho}, \Ov{\vartheta}) \vc{v} \cdot \nabla_x F = 0,
\end{equation}
where we have denoted
\noindent
thermal expansion coefficient \dotfill $a(\Ov{\varrho}, \Ov{\vartheta})$,
\noindent
specific heat at constant pressure \dotfill $c_p(\Ov{\varrho}, \Ov{\vartheta})$,
\noindent
cf. \cite[Chapter 5]{FeNo6} and \cite{FeiNov12}.
Here, the function $\vc{v}$ is the limit velocity, while $\theta$ is associated with the asymptotic temperature
(entropy) deviations
\[
\theta \approx \frac{\vartheta_\varepsilon - \Ov{\vartheta}}{\varepsilon}.
\]
The
exact statement of our results including the initial data for the target system (\ref{l1} - \ref{l3}) will be specified in Theorem \ref{Tm1} below.
We address the problem in the framework of weak solutions for the Navier-Stokes-Fourier system (\ref{i1} - \ref{i3}),
developed in \cite{FeNo6}, and later extended to problems on unbounded domains in \cite{JeJiNo}. The main advantage of this approach is
the convergence towards the target system on any time interval $[0,T]$, on which the Euler-Boussinsesq system (\ref{l1}), (\ref{l2})
possesses a regular solution. We refer to Masmoudi \cite{MAS5} for related results on the compressible barotropic Navier-Stokes system
in the whole space $R^3$, see also the survey \cite{MAS1}. An alternative approach to singular limits, proposed in the seminal paper by
Klainerman and Majda \cite{KM1}, uses the strong solutions for both the primitive and the target system that may exist, however, only
on a possible very short time interval. Using the { same} framework,
Alazard \cite{AL2}, \cite{AL1}, \cite{AL} addresses several
singular limits of the compressible Euler and/or Navier-Stokes-Fourier system, in the absence of external forcing. The present setting, where
the action of the gravitation gives rise to the buoyancy force proportional to $-\theta\nabla_x F$, represents a stronger coupling between the equations,
typical for certain models used in meteorology and physics of the atmosphere, see Klein \cite{KL1}, \cite{Klein}, Zeytounian \cite{ZEY1}.
The necessary preliminary material including various concepts of weak solutions to
the Navier-Stokes-Fourier system is collected in Section \ref{v}. Section \ref{m} contains the main result on the asymptotic limit for $\varepsilon \to 0$, the
proof of which is the main objective of the remaining part for the paper. In Section \ref{b}, the relative entropy inequality is
used to establish the necessary uniform bounds independent of $\varepsilon \to 0$. The problem of propagation and dispersion of the associated
acoustic waves is discussed in Section \ref{r}. The proof of convergence towards the limit system is completed in Section \ref{c}.
\section{Preliminaries, weak solutions to the Navier-Stokes-Fourier system}
\label{v}
Motivated by \cite{FeiNov10}, we introduce the \emph{relative entropy functional}
\bFormula{r0}
\mathcal{E}_\varepsilon \left( \varrho, \vartheta, \vc{u} \Big| r , \Theta, \vc{U} \right) =
\intO{ \left[ \frac{1}{2} \varrho |\vc{u} - \vc{U} |^2 + \frac{1}{\varepsilon^2} \left( H_\Theta (\varrho, \vartheta) -
\frac{\partial H_\Theta (r, \Theta)}{\partial\varrho}(\varrho - r)- H_\Theta (r, \Theta) \right)
\right] },
\end{equation}
where
\bFormula{bfe}
H_\Theta (\varrho, \vartheta) = \varrho \Big( e(\varrho, \vartheta) - \Theta s(\varrho,\vartheta) \Big)
\end{equation}
is the ballistic free energy. We say that a trio of functions $\{ \varrho, \vartheta, \vc{u} \}$ represents a
\emph{dissipative weak solution} of the Navier-Stokes-Fourier system (\ref{i1} - \ref{i5}) in $(0,T) \times \Omega$ if:
\begin{itemize}
\item $\varrho \geq 0$, $\vartheta > 0$ a.a. in $(0,T) \times \Omega$,
\[
( \varrho - \Ov{\varrho}_\varepsilon ) \in L^\infty(0,T; L^2 + L^{5/3}(\Omega)),\ (\vartheta - \Ov{\vartheta}) \in L^\infty(0, T; L^2 + L^4 (\Omega)),
\]
\[
\nabla_x \vartheta , \ \nabla_x \log(\vartheta) \in L^2(0,T; L^2(\Omega;R^3)),
\]
\[
\vc{u} \in L^2(0,T; W^{1,2}(\Omega; R^3)),\ \vc{u} \cdot \vc{n}|_{\partial \Omega} = 0,
\]
where $[\Ov{\varrho}_\varepsilon, \Ov{\vartheta}]$ stands for the equilibrium solution introduced in (\ref{equilibrium});
\item the equation of continuity (\ref{i1}) holds as a family of integral identities
\bFormula{v1}
\int_{\Omega} \Big[ \varrho(\tau, \cdot) \varphi (\tau, \cdot) - \varrho_{0,\varepsilon} \varphi (0, \cdot) \Big] \ {\rm d} {x}
= \int_0^\tau \int_{R^3} \Big( \varrho \partial_t \varphi + \varrho \vc{u} \cdot \nabla_x \varphi \Big) \ {\rm d} {x}dt
\end{equation}
for any $\tau \in [0,T]$ and any test function $\varphi \in C^\infty_c([0,T] \times \Ov{\Omega})$;
\item the momentum equation (\ref{i2}), together with the initial condition (\ref{i4}), are satisfied in the sense of distributions,
\bFormula{v2}
\int_{\Omega} \Big[ \varrho \vc{u} (\tau, \cdot) \cdot \varphi (\tau, \cdot) - \varrho_{0,\varepsilon} \vc{u}_{0,\varepsilon}
\varphi(0, \cdot) \Big] \ {\rm d} {x}
\end{equation}
\[
= \int_0^\tau \int_{\Omega} \Big( \varrho \vc{u} \cdot \partial_t \varphi + \varrho \vc{u} \otimes \vc{u} : \nabla_x \varphi +
\frac{1}{\varepsilon^2} p(\varrho,\vartheta) {\rm div}_x \varphi - \varepsilon^{a} \tn{S}(\vartheta, \nabla_x \vc{u}) : \nabla_x \varphi + \frac{1}{\varepsilon} \nabla_x F \cdot \varphi \Big) \ {\rm d} {x}dt
\]
for any $\tau \in [0,T]$, and any $\varphi \in C^\infty_c([0,T] \times \Ov{\Omega}; R^3)$, $\varphi \cdot \vc{n}|_{\partial \Omega} = 0$;
\item the entropy production equation (\ref{i3}) is relaxed to the entropy inequality
\bFormula{v3}
\int_{\Omega} \Big[ \varrho_{0,\varepsilon} s(\varrho_{0,\varepsilon}, \vartheta_{0, \varepsilon} ) \varphi(0, \cdot) -
\varrho s(\varrho, \vartheta) (\tau, \cdot) \varphi(\tau, \cdot) \Big] \ {\rm d} {x}
\end{equation}
\[
+ \int_0^\tau \int_{\Omega} \frac{1}{\vartheta} \left( \varepsilon^{2 + a} \tn{S}(\vartheta, \nabla_x \vc{u}) : \nabla_x \vc{u} - \varepsilon^b
\frac{\vc{q}(\vartheta, \nabla_x \vartheta) \cdot \nabla_x \vartheta }{\vartheta} \right) \varphi \ {\rm d} {x}dt
\]
\[
\leq - \int_0^\tau \int_{\Omega} \left( \varrho s(\varrho,\vartheta) \partial_t \varphi + \varrho s(\varrho, \vartheta) \vc{u} \cdot \nabla_x \varphi +
\varepsilon^b \frac{ \vc{q}(\vartheta, \nabla_x \vartheta) }{\vartheta} \cdot \nabla_x \varphi \right) \ {\rm d} {x}dt
\]
for a.a. $\tau \in [0,T]$ and any test function $\varphi \in C^\infty_c([0,T] \times \Ov{\Omega})$, $\varphi \geq 0$;
\item the \emph{relative entropy inequality}
\bFormula{r1}
\left[ \mathcal{E}_\varepsilon \left( \varrho, \vartheta, \vc{u} \Big| r , \Theta, \vc{U} \right) \right]_{t = 0}^\tau
+ \int_0^\tau \intO{ \frac{\Theta}{\vartheta} \left( \varepsilon^a \tn{S} (\vartheta, \nabla_x \vc{u}) : \nabla_x \vc{u} -
\varepsilon^{b-2} \frac{\vc{q}(\vartheta, \nabla_x \vartheta) \cdot \nabla_x \vartheta }{\vartheta} \right) } \ {\rm d} t
\end{equation}
\[
\leq \int_0^\tau \intO{ \Big( \varrho \Big( \partial_t \vc{U} +
\vc{u}\cdot \nabla_x \vc{U} \Big) \cdot (\vc{U} - \vc{u}) + \varepsilon^a
\tn{S}(\vartheta, \nabla_x \vc{u}): \nabla_x \vc{U} \Big) } \ {\rm d} t
\]
\[
+\frac
1{\varepsilon^2}\int_0^\tau\intO{\Big[\Big(p(r,\Theta)-p(\varrho,\vartheta)\Big){\rm
div}\vc U +\frac\varrho {r}(\vc U-\vc{u})\cdot\nabla_x
p(r,\Theta)\Big]}{\rm d}t
\]
\[
- \frac{1}{\varepsilon^2} \int_0^\tau \intO{ \left( \varrho \Big( s(\varrho,\vartheta) - s(r, \Theta) \Big) \partial_t \Theta +
\varrho \Big( s(\varrho,\vartheta) - s(r, \Theta) \Big) \vc{u} \cdot \nabla_x \Theta + \varepsilon^b \frac{\vc{q}(\vartheta, \nabla_x \vartheta) }{\vartheta} \cdot
\nabla_x \Theta \right) } \ {\rm d} t
\]
\[
+ \frac{1}{\varepsilon^2} \int_0^\tau \intO{ \frac{r - \varrho}
{r}\Big(
\partial_t p(r, \Theta) + \vc U \cdot \nabla_x p(r,
\Theta)\Big) } \ {\rm d} t { -\frac 1\varepsilon\int_0^\tau\int_{\Omega}\varrhoe\nabla_x F\cdot (\vc U_\varepsilon-\vc{u}_\varepsilon){\rm d}x}.
\]
holds for a.a. $t\in (0,T)$ and for
any trio of continuously differentiable ``test'' functions defined on $[0,T] \times \Ov{\Omega}$,
\[
r > 0 , \ \Theta > 0 , \ r \equiv \Ov{\varrho} , \ \Theta \equiv \Ov{\vartheta} \ \mbox{outside a compact subset of}\ \Ov{\Omega},
\]
\[
\vc{U} \in C([ 0, T]; W^{k,2}(\Omega;R^3)),\ \partial_t \vc{U} \in \ C([0,T]; W^{k-1,2}(\Omega;R^3)),\ k > \frac{5}{2}, \ \vc{U} \cdot \vc{n}|_{\partial
\Omega} = 0.
\]
\end{itemize}
\bRemark{i1}
Note that the above definition of dissipative weak solutions on \emph{unbounded} domains, proposed in \cite{JeJiNo}, is different from
that on bounded domains introduced in \cite{FeiNov10}. In \cite{FeiNov10}, the relative entropy inequality (\ref{r1}) is replaced by the total
energy balance, whereas (\ref{r1}) is automatically satisfied for any weak solution to the Navier-Stokes-Fourier system. The
weak solutions introduced in this paper can be therefore viewed as ``very weak dissipative solutions'' of the primitive system.
\end{Remark}
\subsection{Structural restrictions imposed on constitutive relations}
We study our singular limit problem under certain physically motivated restrictions imposed on constitutive equations. They are basically the same
as required by the existence theory developed in \cite[Chapter 3]{FeNo6}. Although they might be slightly relaxed if only the convergence towards the target
system is studied, we list them in the form presented in \cite[Chapter 3]{FeNo6}, where the interested reader may find more information
concerning the physical background as well as possible generalizations.
The pressure $p=p(\varrho,\vartheta)$ is given by the formula
\bFormula{i10}
p(\varrho, \vartheta) = \vartheta^{5/2} P \left( \frac{\varrho}{\vartheta^{3/2}} \right) + \frac{a}{3} \vartheta^4 , \ a > 0;
\end{equation}
the specific internal energy $e = e(\varrho,\vartheta)$ and the specific entropy $s = s(\varrho, \vartheta)$ read
\bFormula{i11}
e(\varrho, \vartheta) = \frac{3}{2} \vartheta \frac{ \vartheta^{3/2} }{\varrho} P \left( \frac{\varrho}{\vartheta^{3/2}} \right) + {a} \vartheta^4
\end{equation}
\bFormula{i12}
s(\varrho, \vartheta) = S \left( \frac{\varrho}{\vartheta^{3/2}} \right) + \frac{4a}{3} \frac{\vartheta^3}{\varrho},
\end{equation}
where
\bFormula{i13}
P \in C^1[0, \infty) \cap C^3(0,\infty), \ P(0) = 0 ,\ P'(Z) > 0 \ \mbox{for all}\ Z \geq 0,
\end{equation}
\bFormula{i14}
\lim_{Z \to \infty} \frac{P(Z)}{Z^{5/3}} = P_\infty > 0,
\end{equation}
\bFormula{i15}
0 < \frac{ \frac{5}{3} P(Z) - P'(Z) Z }{Z} < c \ \mbox{for all} \ Z > 0,
\end{equation}
and
\bFormula{i16}
S'(Z) = - \frac{3}{2} \frac{ \frac{5}{3} P(Z) - P'(Z) Z }{Z^2} , \ \lim_{Z \to \infty} S(Z) = 0.
\end{equation}
The relation (\ref{i15}) expresses positivity and uniform boundedness of the specific heat at constant volume.
The transport coefficients $\mu$, $\eta$, and $\kappa$ are effective functions of the temperature,
\bFormula{i8}
\mu, \ \eta \in C^1[0, \infty) \ \mbox{are globally Lipschitz in,} \ [0, \infty), \ 0 < \underline{\mu} (1 + \vartheta) \leq \mu(\vartheta),\
\eta(\vartheta) \geq 0,
\ \mbox{for all}\ \vartheta \geq 0,
\end{equation}
\bFormula{i9}
\kappa \in C^1[0, \infty),\ 0 < \underline{\kappa}(1 + \vartheta^3) \leq \kappa (\vartheta) \leq \Ov{\kappa}(1 + \vartheta^3) \ \mbox{for all}\ \vartheta \geq 0.
\end{equation}
\subsection{Target system}
As noted in the introduction, the expected limit is the Euler-Boussinesq system (\ref{l1}- \ref{l3}) endowed with the
initial data
\begin{equation}\label{inittarget}
\theta_0(0,\cdot)=\theta_0,\;\vc{v}(0, \cdot) = \vc{v}_0.
\end{equation}
In agreement with the nowadays standard theory of well-posedness for hyperbolic systems, see e.g. Kato \cite{Kato},
we suppose that the system (\ref{l1}- \ref{l3}), endowed with the initial data
\begin{equation}\label{initdata}
(\theta_0, \vc{v}_0)\in W^{k,2}(\Omega;R^4), \ \|(\theta_0,\vc{v}_0)\|_{W^{k,2}(\Omega;R^4)}\le D, \ {\rm div}_x \vc{v}_0 = 0,\
\vc{v}_0 \cdot \vc{n}|_{\partial \Omega} = 0, \ k > \frac{5}{2},
\end{equation}
possesses a regular solution $(\theta,\vc{v})$,
\bFormula{euler} (\theta,\vc{v}) \in C([0, T_{\rm max}); W^{k,2}(\Omega;R^4)),
\ (\partial_t \vc{v},\,\nabla_x \Pi)\in C([0, T_{\rm max}); W^{k-1,2}(\Omega;R^6)),
\end{equation} defined on a maximal time interval $[0, T_{\rm max})$, $T_{\rm
max} = T_{\rm max}(D)$.
\subsection{Equilibrium state}
We finish this preliminary part by recalling the basic properties of the equilibrium solution $(\Ov{\varrho}_\varepsilon, \Ov{\vartheta})$.
Since the potential $F$ is given by (\ref{potential1}), it is easy to check that
\bFormula{p2} \partial_\varrho H_{\overline\vartheta}(\overline\varrhoe,\overline\vartheta) = \varepsilon F +
\partial_\varrho H_{\overline\vartheta}(\overline\varrho,\overline\vartheta); \end{equation}
whence, under the assumptions (\ref{i10}), (\ref{i13}--\ref{i15}),
\bFormula{p3}
\overline\varrhoe\in C^3(\Omega),\
\left| \frac{\overline{\varrho}_\varepsilon (x) - \overline{\varrho}}{\varepsilon} \right| \leq c F (x) \quad \left| \nabla_x \varrhoe(x) \right| \leq \varepsilon |\nabla_x F(x)| ,\ x \in \Omega.
\end{equation}
The reader can consult \cite{FeiSch} for details.
\section{Main result}
\label{m}
For a vector field $\vc{U} \in L^2(\Omega;R^3)$, we denote by $\vc{H}[\vc{U}]$ the standard \emph{Helmholtz projection}
on the space of solenoidal functions.
We are ready to state the main result of this paper.
\bTheorem{m1}
Let the thermodynamic functions
$p$, $e$, $s$, and the transport coefficients $\mu$, $\eta$, $\kappa$ satisfy the hypotheses (\ref{i10} - \ref{i16}),
(\ref{i8}), (\ref{i9}). Let the potential force $F$ be given by (\ref{potential1}). Let the exponents $a,b$, determining the
Reynold and P\'eclet number scales, satisfy
\bFormula{cof}
b > 0, \ 0 < a < \frac{10}{3}.
\end{equation}
Next, let
the initial data (\ref{i4}) be chosen in such a way that
\bFormula{m1}
\{ \varrho^{(1)}_{0, \varepsilon} \}_{\varepsilon > 0},\ \{ \vartheta^{(1)}_{0,\varepsilon} \}_{\varepsilon > 0} \ \mbox{are bounded in}\
L^2 \cap L^\infty (\Omega), \ \varrho^{(1)}_{0, \varepsilon} \to \varrho^{(1)}_0,\ \vartheta^{(1)}_{0, \varepsilon} \to \vartheta^{(1)}_0
\ \mbox{in}\ L^2(\Omega),
\end{equation}
\bFormula{m2}
\{ \vc{u}_{0,\varepsilon} \}_{\varepsilon > 0} \ \mbox{is bounded in}\ L^2(\Omega;R^3), \ \vc{u}_{0, \varepsilon} \to \vc{u}_0 \ \mbox{in}\ L^2(\Omega;R^3),
\end{equation}
where
\bFormula{HYP}
\varrho^{(1)}_0 , \ \vartheta^{(1)}_0 \in W^{1,2} \cap W^{1,\infty}(\Omega),\ \vc{H}[\vc{u}_0] = \vc{v}_0 \in W^{k,2}(\Omega;R^3) \ \mbox{for a certain}\ k > \frac{5}{2}.
\end{equation}
Suppose that the Euler-Boussinesq system (\ref{l1}--\ref{l3}), endowed with the initial data
\bFormula{m6}
\vc{v}_0 = \vc{H}[ \vc{u}_0 ], \ \theta_0 = \frac{\Ov{\vartheta}}{c_p(\Ov{\varrho}, \Ov{\vartheta})} \left( \frac{\partial s(\Ov{\varrho}, \Ov{\vartheta})}{
\partial \varrho} \varrho^{(1)}_0 + \frac{\partial s(\Ov{\varrho}, \Ov{\vartheta})}{
\partial \vartheta} \vartheta^{(1)}_0 \right) ,
\end{equation}
admits a regular solution $[\vc{v}, \theta]$ in the class (\ref{euler}) defined on a maximal time interval $[0, T_{\rm max})$.
Finally, let $\{ \varrhoe, \vartheta_\varepsilon, \vc{u}_\varepsilon \}$ be a dissipative weak solution of the Navier-Stokes-Fourier system (\ref{i1} - \ref{i5}) in
$(0,T) \times R^3$, $T < T_{\rm max}$.
Then
\bFormula{m3}
{\rm ess} \sup_{t \in (0,T)} \| \varrhoe (t, \cdot) - \Ov{\varrho} \|_{L^{5/3}_{\rm loc}(\Ov{\Omega})} \leq \varepsilon c,
\end{equation}
\bFormula{m4}
\sqrt{ {\varrhoe} } \vc{u}_\varepsilon \to \sqrt{ \Ov{\varrho} } \ \vc{v} \ \mbox{in}\ L^\infty_{\rm loc}((0,T]; L^2_{\rm loc} (\Ov{\Omega};R^3))
\ \mbox{and weakly-(*) in} \ L^\infty(0,T; L^2(\Omega;R^3)),
\end{equation}
and
\bFormula{m5}
\frac{ \vartheta_\varepsilon - \Ov{\vartheta} }{\varepsilon} \to \theta \;
\mbox{in}\ L^\infty_{\rm loc}((0,T]; L^2_{\rm loc} (\Ov{\Omega})),
\ \mbox{and weakly-(*) in}\ L^\infty(0,T; L^2(\Omega)).
\end{equation}
\end{Theorem}
\noindent
\bRemark{m1}
Under the hypotheses (\ref{i10} - \ref{i9}), the existence of
dissipative weak solutions to the Navier-Stokes-Fourier system in $(0,T)
\times \Omega$ was shown in \cite{JeJiNo}.
\end{Remark}
The rest of the paper is devoted to the proof of Theorem \ref{Tm1}.
\section{Uniform bounds}
\label{b}
In this section, we derive uniform bounds on the family of solutions $[\varrhoe, \vc{u}_\varepsilon, \vartheta_\varepsilon]$ \emph{independent} of the scaling parameter $\varepsilon \to 0$.
\subsection{Energy bounds}
Taking $r=\overline\varrhoe$, $\Theta=\overline\vartheta$,
$\vc U=0$ as test functions in the relative entropy inequality (\ref{r1}) we obtain
\bFormula{v4}
\int_{\Omega} \left[ \frac{1}{2} \varrhoe |\vc{u}_\varepsilon|^2 + \frac{1}{\varepsilon^2} \left( H_{\Ov{\vartheta}} (\varrhoe, \vartheta_\varepsilon) -
\frac{\partial H_{\Ov{\vartheta}} (\Ov{\varrho}_\varepsilon, \Ov{\vartheta})}{\partial\varrho}(\varrhoe - \Ov{\varrho}_\varepsilon )- H_{\Ov{\vartheta}} (\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) \right) \right]
{\rm d} x
\end{equation}
\[
+ \Ov{\vartheta} \int_0^\tau \int_{\Omega} \frac{1}{\vartheta_\varepsilon} \left( \varepsilon^{a} \tn{S}(\vartheta_\varepsilon, \nabla_x \vc{u}_\varepsilon) : \nabla_x \vc{u}_\varepsilon - \varepsilon^{b - 2} \frac{\vc{q}(\vartheta_\varepsilon, \nabla_x \vartheta_\varepsilon) \cdot \nabla_x \vartheta_\varepsilon
}{\vartheta} \right) \ {\rm d} {x}dt
\]
\[
\leq \int_{\Omega} \left[ \frac{1}{2} \varrho_{0,\varepsilon} |\vc{u}_{0,\varepsilon}|^2 + \frac{1}{\varepsilon^2} \left( H_{\Ov{\vartheta}} (\varrho_{0,\varepsilon}, \vartheta_{0,\varepsilon}) -
\frac{\partial H_{\Ov{\vartheta}} (\Ov{\varrho}_{\varepsilon}, \Ov{\vartheta})}{\partial\varrho}(\varrho_{0,\varepsilon} - \Ov{\varrho}_\varepsilon )- H_{\Ov{\vartheta}} (\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) \right)\right] \ {\rm d} {x}
\]
for a.a. $\tau \in [0,T]$. Note that such a choice of test functions can be justified by means of a density argument.
Thanks to the hypotheses (\ref{m1}), (\ref{m2}), the integral on the right-hand side of (\ref{v4}) remains bounded uniformly for $\varepsilon \to 0$.
In accordance with the structural properties of the thermodynamic functions imposed through (\ref{i10} - \ref{i16}),
the ballistic free energy enjoys the following properties:
For any compact $K \subset (0, \infty)^2$ and $
(r, \Theta) \in K$,
there exists a strictly positive constant $c(K)$, depending only on $K$ and the structural properties of $P$, such that
\bFormula{b1}
\left( H_\Theta (\varrho, \vartheta) -
\frac{\partial H_\Theta (r, \Theta)}{\partial\varrho}(\varrho - r)- H_\Theta (r, \Theta) \right) \geq c(K) \left( |\varrho - r |^2 + |\vartheta - \Theta |^2 \right) \ \mbox{if}
\ (\varrho, \vartheta) \in K,
\end{equation}
\bFormula{b2}
\left( H_\Theta (\varrho, \vartheta) -
\frac{\partial H_\Theta (r, \Theta)}{\partial\varrho}(\varrho - r)- H_\Theta (r, \Theta) \right) \ge c(K) \Big(1+\varrho^\gamma +\vartheta^4\Big) \ \mbox{if}
\ (\varrho, \vartheta) \in (0,\infty)^2 \setminus K.
\end{equation}
Similarly to \cite[Chapter 4.7]{FeNo6}, we introduce a decomposition of a function $h$:
\[
h = [h]_{\rm ess} + [h]_{\rm res} \ \mbox{for a measurable function} \ h,
\]
where
\[
[h]_{\rm ess} = h \ 1_{ \{ \Ov{\varrho} / 2 < \varrhoe < 2 \Ov{\varrho} ; \ \Ov{\vartheta}/2 < \vartheta_\varepsilon < 2 \Ov{\vartheta} \} },\
[h]_{\rm res} = h - h_{\rm ess}.
\]
Combining (\ref{v4}), (\ref{b1}), (\ref{b2}), (\ref{p3}) with the hypotheses (\ref{i10} - \ref{i9}) we deduce the following estimates:
\bFormula{b3}
{\rm ess} \sup_{t \in (0,T)} \| \sqrt{\varrhoe} \vc{u}_\varepsilon (t, \cdot) \|_{L^2(\Omega;R^3)} \leq c,
\end{equation}
\bFormula{b4}
{\rm ess} \sup_{t \in (0,T)} \left\| \left[ \frac{\varrhoe - \Ov{\varrho}_\varepsilon }{\varepsilon}(t, \cdot) \right]_{\rm ess} \right\|_{L^2(\Omega;R^3)} +
{\rm ess} \sup_{t \in (0,T)} \left\| \left[ \frac{\vartheta_\varepsilon - \Ov{\vartheta} }{\varepsilon} (t, \cdot) \right]_{\rm ess} \right\|_{L^2(\Omega;R^3)} \leq c ,
\end{equation}
\bFormula{b5}
{\rm ess} \sup_{t \in (0,T)} \int_{\Omega} \left( \left[ \varrhoe^{5/3}(t, \cdot) \right]^{5/3}_{\rm res} + \left[ \vartheta_\varepsilon (t, \cdot) \right]^4_{\rm res} + 1_{\rm res}(t, \cdot) \right) \ {\rm d} {x} \leq \varepsilon^2 c,
\end{equation}
and
\bFormula{b6}
\left\| \varepsilon^{a/2} \vc{u}_\varepsilon \right\|_{L^2(0,T; W^{1,2}(\Omega;R^3))} \leq c,
\end{equation}
\bFormula{b7}
\left\| \varepsilon^{(b-2)/2} \left( \vartheta_\varepsilon - \Ov{\vartheta} \right) \right\|_{L^2(0,T; W^{1,2}(\Omega;R^3))}
+ \left\| \varepsilon^{(b-2)/2} \left( \log(\vartheta_\varepsilon) - \log(\Ov{\vartheta}) \right) \right\|_{L^2(0,T; W^{1,2}(\Omega;R^3))} \leq c,
\end{equation}
where the symbol $c$ stands for a generic constant independent of $\varepsilon$. We remark that (\ref{b6}) follows from the generalized Korn's inequality
$
{ (\int_\Omega \varrho_\varepsilon\vc w^2{\rm d} x)^{1/2}+}\| \nabla_x \vc{w} + \nabla_x^t \vc{w} - \frac{2}{3} {\rm div}_x \vc{w} \tn{I} \|_{L^2} \geq c
\| \nabla_x \vc{w} \|_{L^2}$ for $\vc{w} \in W^{1,2}$,
combined with the estimates (\ref{b3}), (\ref{b5}). Similar arguments based on the Sobolev inequality and (\ref{b4}), (\ref{b5})
yield (\ref{b7}).
\subsection{Convergence}
To begin, we denote
\begin{equation}\label{l4}
\alpha = \frac{1}{\Ov{\varrho}} \frac{\partial p (\Ov{\varrho}, \Ov{\vartheta})}{\partial \varrho}, \
\beta = \frac{1}{\Ov{\varrho}} \frac{\partial p (\Ov{\varrho}, \Ov{\vartheta})}{\partial \vartheta},\
\delta = \Ov{\varrho} \frac{\partial s (\Ov{\varrho}, \Ov{\vartheta})}{\partial \vartheta}, \ a(\Ov{\varrho}, \Ov{\vartheta}) = \frac{1}{\Ov{\varrho}} \frac{\beta}{\alpha}.
\end{equation}
It follows from (\ref{b4}--\ref{b5}) and the structural assumptions on the pressure (\ref{i10}), (\ref{i13}--\ref{i15}) that
\bFormula{Z1-}
\Big[\frac{\varrhoe-\overline\varrho_\varepsilon}\varepsilon\Big]_{\rm res} \to 0 \ \mbox{in}\ L^\infty(0,T; L^{5/3}(\Omega)),\
\Big[{ \frac{\vartheta_\varepsilon-\overline\vartheta}\varepsilon}\Big]_{\rm res} \to 0 \ \mbox{in}\ L^\infty(0,T; L^{4}(\Omega)).
\end{equation}
Next, writing
\[
\frac{1}{\varepsilon} \nabla_x p(\varrhoe, \vartheta_\varepsilon) - \varrhoe \nabla_x F = \frac{1}{\varepsilon} \nabla_x p(\varrhoe, \vartheta_\varepsilon) - \Ov{\varrho}_\varepsilon \nabla_x F + \varepsilon \frac{\Ov{\varrho}_\varepsilon - \varrhoe}{\varepsilon} \nabla_x F
= \frac{1}{\varepsilon} \nabla_x \left( p(\varrhoe, \vartheta_\varepsilon) - p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) \right) + \varepsilon \frac{\Ov{\varrho}_\varepsilon - \varrhoe}{\varepsilon} \nabla_x F,
\]
we deduce from the momentum balance (\ref{v2}) that
\begin{equation}\label{Z1}
\alpha\Big[\frac{\varrhoe-\overline\varrho_\varepsilon}\varepsilon\Big]_{\rm ess} +
\beta\Big[\frac{\vartheta_\varepsilon-\overline\vartheta}\varepsilon\Big]_{\rm ess} \to 0 \ \mbox{weakly-(*) in}\ L^\infty (0,T; L^2(\Omega)).
\end{equation}
Finally, we use (\ref{b3}--\ref{b5}) to show that
\bFormula{conv1}
\varrhoe \vc{u}_\varepsilon \to \Ov{ \varrho \vc{u} } \ \mbox{weakly-(*) in}\ L^\infty(0,T; L^2 + L^{5/4}(\Omega;R^3)),
\end{equation}
where, passing to the limit in the continuity equation (\ref{v1}), we may infer that
\bFormula{conv2}
{\rm div}_x (\Ov{ \varrho \vc{u} }) = 0.
\end{equation}
\section{Acoustic and thermal energy transport equations}
\label{r}
Similarly to \cite{FeiNov12}, our aim is to use the relative entropy inequality (\ref{r1}) to deduce the convergence to the target system.
To this end, we take
\[
\varrho = \varrhoe,\ \vartheta = \vartheta_\varepsilon, \ \vc{u} = \vc{u}_\varepsilon
\]
and choose the test functions $\{ r, \Theta, \vc{U} \}$ in the
following way:
\bFormula{Ap}
r = r_\varepsilon = \Ov{\varrho}_\varepsilon + \varepsilon R_\varepsilon ,\
\Theta = \Theta_\varepsilon = \Ov{\vartheta} + \varepsilon T_\varepsilon , \ \vc{U} = \vc{U}_\varepsilon
= \vc{v} + \nabla_x \Phi_\varepsilon; \end{equation}
where $\vc{v}$ is the velocity component of the solution to
the incompressible Euler-Boussinesq system (\ref{l1})-(\ref{l3}), with the
initial condition (\ref{m6}), and the functions $R_\varepsilon$, $T_\varepsilon$, and $\Phi_\varepsilon$
satisfy the \emph{acoustic equation}:
\bFormula{r3} \varepsilon \partial_t
(\alpha R_\varepsilon + \beta T_\varepsilon ) + \omega {\cal D}elta \Phi_\varepsilon = 0, \end{equation}
\bFormula{r4} \varepsilon \partial_t \nabla_x \Phi_\varepsilon + \nabla_x (\alpha R_\varepsilon
+ \beta T_\varepsilon ) = 0, \ \nabla_x \Phi_\varepsilon \cdot \vc{n}|_{\partial \Omega} = 0,
\end{equation}
with the initial values determined by
\bFormula{r4b} R_\varepsilon (0, \cdot) = R_{0},\ T_\varepsilon (0, \cdot) =
T_{0},\ \Phi_{\varepsilon}(0, \cdot) = \Phi_{0}, \end{equation}
and
the constants $\alpha$, $\beta$ defined in (\ref{l4}),
\[
\omega = \Ov{\varrho} \left( \alpha + \frac{\beta^2}{\delta} \right).
\]
The first equation in (\ref{r3}) is nothing other than a linearization of the continuity equation, while the second
equation is a linearization of the momentum equation projected onto the space of gradients.
In order to determine $R_\varepsilon$ and $T_\varepsilon$ in a unique way, we require
$\delta R_\varepsilon-\beta T_\varepsilon$, with $\delta$ defined in (\ref{l4}),
to satisfy the \emph{transport equation}
\bFormula{r4a}
\partial_t (\delta T_\varepsilon - \beta R_\varepsilon) + \vc{U}_\varepsilon \cdot \nabla_x \left( \delta T_\varepsilon - \beta R_\varepsilon - \frac{\beta}{\alpha} F \right) = 0,
\end{equation}
where the initial data are determined by (\ref{r4b}). Equation (\ref{r4a}) is obviously related to the limit equation (\ref{l3}).
Observe that the system of linear equations (\ref{r3}--\ref{r4a}) is well-posed.
\subsection{Initial data}
In view of the future application of the relative entropy inequality (\ref{r1}), the initial data for the test functions must be taken is such a way that
\bFormula{data}
\vc{v}(0, \cdot) = \vc{v}_0 = \vc{H}[\vc{u}_0],\
\Phi_\varepsilon (0, \cdot) = \Phi_{0, \eta}, \ \nabla_x \Phi_{0, \eta} \to \vc{H}^\perp [\vc{u}_0] \ \mbox{in}\ L^2(\Omega;R^3)\ \mbox{as}\ \eta \to 0,
\end{equation}
\bFormula{data1}
R_\varepsilon (0, \cdot) = R_{0, \eta}, \ \| R_{0, \eta} \|_{L^\infty(\Omega)} < c(\eta),\ R_{0, \eta} \to \varrho^{(1)}_0 \ \mbox{in}\ L^2(\Omega)
\ \mbox{as}\ \eta \to 0,
\end{equation}
and
\bFormula{data2}
T_\varepsilon (0, \cdot) = T_{0, \eta}, \ \| T_{0, \eta} \|_{L^\infty(\Omega)} < c(\eta),\ T_{0, \eta} \to \vartheta^{(1)}_0 \ \mbox{in}\ L^2(\Omega)
\ \mbox{as}\ \eta \to 0.
\end{equation}
Note that (\ref{data} - \ref{data2}) imply that
\bFormula{data3}
\mathcal{E}_\varepsilon \left( \varrho_{0,\varepsilon}, \vartheta_{0, \varepsilon}, \vc{u}_{0,\varepsilon} \Big| r_\varepsilon(0,\cdot), \Theta_\varepsilon (0, \cdot), \vc{U}(0, \cdot) \right)
\to \chi(\eta) \ \mbox{as} \ \varepsilon \to 0,
\end{equation}
where
\[
\chi(\eta) \to 0 \ \mbox{as}\ \eta \to 0.
\]
Our next goal is to choose suitable approximations for the initial data.
Following \cite{FeNoSun2}, we consider the Neumann Laplacean ${\cal D}elta_N$,
\[
\mathcal{D}({\cal D}elta_N) = \left\{ v \in L^2(\Omega) \ \Big| \ \nabla_x v \in L^2(\Omega;R^3), \ \intO{ \nabla_x v \cdot \nabla_x \varphi } =
\intO{ g \varphi } \right.
\]
\[
\left.
\mbox{for any}\ \varphi \in C^\infty_c(\Ov{\Omega}) \ \mbox{and a certain}\ g \in L^2(\Omega) \right\},
\]
together with a family of regularizing operators
\bFormula{c1--}
[v]_{\eta} = G_\eta(\sqrt{- {\cal D}elta_N}) [ \psi_{1/ \eta} v ],
\end{equation}
with the cut-off functions
\begin{equation}\label{cutoff}
\psi_\eta(x)=\psi(x/\eta);\; \psi\in C^\infty_c(R),\;0\le\psi\le 1,\; \psi(x)=
\left\{\begin{array}{c}
1\;\mbox{si $|x|\le 1$, }\\
0\;\mbox{si $|x|\ge 2$}
\end{array}\right\},
\end{equation}
\[
G_\eta \in C^\infty_c(R),\
0 \leq G_\eta \leq 1, \ G_\eta (-z) = G_\eta (z),
\]
\[
G_\eta (z) = 1 \ \mbox{for}\ z \in \left( - \frac{1}{\eta}, - \eta \right) \cup \left( {\eta},
\frac{1}{\eta}
\right), \ G_\eta(z) = 0 \ \mbox{for}\ z \in \left(-\infty, - \frac{2}{\eta} \right) \cup
\left(- \frac{\eta}{2}, \frac{\eta}{2} \right) \cup \left( \frac{2}{ \eta}, \infty \right),
\]
where the linear operator $G_\eta(\sqrt{- {\cal D}elta_N})$ is defined by means of the standard spectral theory associated to ${\cal D}elta_N$.
Accordingly, we consider regularized initial data in the form
\bFormula{d1}
R_{0, \eta} = [ \varrho^{(1)}_{0} ]_{\eta} ,\ T_{0, \eta} = [ \vartheta^{(1)}_{0}]_\eta,
\end{equation}
and
\bFormula{d2}
\Phi_{0, \eta} = \Big[ {\cal D}elta^{-1}_N {\rm div}_x [\vc{u}_{0}] \Big]_\eta , \ \mbox{with}\
\nabla_x {\cal D}elta^{-1}_N {\rm div}_x [\vc{u}_{0}] \equiv \vc{H}^\perp [ \vc{u}_{0} ].
\end{equation}
To avoid excessive notation, we omit writing the parameter $\eta$ in the course of the limit passage $\varepsilon \to 0$.
\subsection{Dispersive estimates for the wave equation}
The acoustic equation (\ref{r3} - \ref{r4b}) has been studied in detail in \cite{FeNoSun2}. In particular,
we report the following estimates
(\cite[estimates (6.6), (6.8)]{FeNoSun2}:
\bFormula{a1}
\sup_{t \in [0,T]} \left( \left\| \nabla_x \Phi_{\varepsilon, \eta} \right\|_{W^{k,2} \cap W^{k,\infty}(\Omega:R^3)} +
\left\| (\alpha R_{\varepsilon, \eta} + \beta T_{\varepsilon, \eta}) (t,\cdot) \right\|_{W^{k,2} \cap W^{k,\infty}(\Omega:R^3)} \right)
\end{equation}
\[
\leq c(k, \eta) \left(
\left\| \nabla_x \Phi_{0, \eta } \right\|_{L^2(\Omega;R^3)} + \left\| \alpha R_{0,\eta} + \beta T_{0, \eta} \right\|_{L^2(\Omega)} \right),\
\]
for any $k=0,1,\dots$, $\eta > 0$; and the dispersive estimates
\bFormula{a2}
\int_0^T \left( \left\| \nabla_x \Phi_{\varepsilon, \eta} \right\|_{W^{k,\infty}(\Omega:R^3)} +
\left\| (\alpha R_{\varepsilon, \eta} + \beta T_{\varepsilon, \eta}) (t,\cdot) \right\|_{W^{k,\infty}(\Omega:R^3)} \right) \ {\rm d} t
\end{equation}
\[
\leq \omega(\varepsilon, \eta, k) \left(
\left\| \nabla_x \Phi_{0,\eta} \right\|_{L^2(\Omega;R^3)} + \left\| \alpha R_{0,\eta} + \beta T_{0, \eta} \right\|_{L^2(\Omega)} \right)
\]
where
\[
\omega(\varepsilon, \eta,k) \to 0 \ \mbox{as}\ \varepsilon \to 0 \ \mbox{for any fixed}\ \eta > 0, \ k \geq 0.
\]
The relation (\ref{a2}) represents \emph{dispersive} estimates for the wave equation (\ref{r3}), (\ref{r4}). Note that both (\ref{a1}) and (\ref{a2})
apply to the regularized initial data, meaning for a fixed $\eta > 0$; they in fact blow up when $\eta \to 0$.
Moreover, as shown in \cite[Section 5.3]{EF100},
\bFormula{c1b--}
|x|^s |\partial^k_x [h]_\eta (x) | \leq c(s,k, \eta) \| h \|_{L^2(\Omega)} \ \mbox{for all}\
x \in \Omega, \ s \geq 0, k \geq 0,
\end{equation}
therefore the functions $\Phi_{\varepsilon, \eta}$, $(\alpha R_{\varepsilon, \eta} + \beta T_{\varepsilon, \eta}) $ decay fast for $|x| \to \infty$ as long as $\eta > 0$ is fixed.
\bRemark{r11}
As a matter of fact, the results of \cite{FeNoSun2} are stated for the domain $\Omega$ - a perturbed half-space. However, as pointed out in
\cite{FeNoSun2}, the same holds for a larger class
of domains on which ${\cal D}elta_N$, among which the exterior domains in $R^3$. Alternatively, we may also use the dispersive estimates established
by Isozaki \cite{Isoz}.
\end{Remark}
\subsection{$L^p$ estimates for the transport equation}
For fixed $\eta > 0$, the initial data for the transport equation (\ref{r4a}) enjoy the decay properties (\ref{c1b--}).
Consequently, in view of (\ref{a1}), (\ref{a2}), the solutions of the transport equation (\ref{r4a}) admit the estimates
\bFormula{a3} \sup_{t \in [0,T]} \left\| \delta
T_{\varepsilon, \eta} - \beta R_{\varepsilon, \eta} \right\|_{W^{k,q}(\Omega)} \leq c(\eta, k,F) \left(1 +
\left\| \delta T_{0,\eta} - \beta R_{0, \eta}
\right\|_{L^2(\Omega)} \right),\ k=0,1,\ 1 \leq q \leq \infty, \end{equation}
and the family
\bFormula{a4}
\left\{ \delta
T_{\varepsilon, \eta} - \beta R_{\varepsilon, \eta} \right\}_{\varepsilon > 0} \ \mbox{is precompact in} \ C([0,T]; W^{k,q}(\Omega)), \ k=0,1,\ 1 \leq q \leq \infty.
\end{equation}
Consequently, combining (\ref{a2}), (\ref{a3}), (\ref{a4}) we can let $\varepsilon \to 0$ to obtain
\begin{equation}\label{a5}
T_{\varepsilon, \eta} \to\ T_\eta \;\mbox{ strongly in $L^\infty_{\rm loc}((0,T]; W^{k,p}(\Omega))$, \ $p>2$, and weakly$-(*)$ in $L^\infty(0,T; W^{k,2}(\Omega))$,
\ $k=0,1$,}
\ \mbox{as}\ \varepsilon \to 0,
\end{equation}
\begin{equation}\label{a5aa}
R_{\varepsilon, \eta} \to\ R_\eta \;\mbox{ strongly in $L^\infty_{\rm loc}((0,T]; W^{k,p}(\Omega))$, \ $p>2$, and weakly$-(*)$ in
$L^\infty(0,T; W^{k,2}(\Omega))$,\ $k=0,1$,}
\ \mbox{as}\ \varepsilon \to 0,
\end{equation}
where $T_\eta$ satisfies
\bFormula{a5A}
c_p(\Ov{\varrho}, \Ov{\vartheta}) \left( \partial_t T_\eta + \vc{v} \cdot \nabla_x T_\eta \right) - \Ov{\vartheta} a(\Ov{\varrho}, \Ov{\vartheta}) \vc{v} \cdot \nabla_x F = 0,
\end{equation}
with the initial data
\bFormula{a5B}
T_\eta (0, \cdot) = \frac{\Ov{\vartheta}}{c_p(\Ov{\varrho}, \Ov{\vartheta})} \left( \frac{\partial s(\Ov{\varrho}, \Ov{\vartheta})}{
\partial \varrho} [\varrho^{(1)}_0]_{\eta} + \frac{\partial s(\Ov{\varrho}, \Ov{\vartheta})}{
\partial \vartheta} [\vartheta^{(1)}_0]_\eta \right).
\end{equation}
\section{Convergence}
\label{c}
In this section, we use the test functions
(\ref{Ap}) in the relative entropy inequality (\ref{r1}). Fixing $\eta > 0$ we perform the limit for $\varepsilon \to 0$.
This will be carried over in several steps in the spirit of \cite{FeiNov12}. We omit the subscript $\eta$ whenever no confusion arises.
\subsection{{ Viscous and heat conducting terms}}
We show by direct calculation, splitting the terms in their essential and residual parts
and using assumptions (\ref{i8}--\ref{i9}), uniform bounds (\ref{b5}--\ref{b7}), regularity (\ref{euler}), and estimates (\ref{a1}--\ref{a4})
that the dissipative terms related to the viscosity and to
the heat conductivity on the right-hand side of (\ref{r1}) become
negligible as $\varepsilon \to 0$. More precisely:
\[
\varepsilon^a \tn{S}(\vartheta_\varepsilon, \nabla_x \vc{u}_\varepsilon) : \nabla_x \vc{U}_\varepsilon \to 0 \ \mbox{in} \ L^2((0,T) \times \Omega)+
L^2(0,T; L^{4/3}(\Omega;R^3)) \ \mbox{as}\ \varepsilon
\to 0,
\]
and
\[
\varepsilon^{b - 2} \frac{ \vc{q}(\vartheta_\varepsilon, \nabla_x \vartheta_\varepsilon) \cdot \nabla_x \Theta_\varepsilon }{\vartheta_\varepsilon} \to 0
\ \mbox{in}\ L^2((0,T) \times \Omega) + L^1((0,T) \times \Omega) \ \mbox{as}\ \varepsilon
\to 0.
\]
Consequently, combining the previous observation with (\ref{data3}), we can write the relative entropy inequality (\ref{r1})
as
\bFormula{r9}
\mathcal{E}_\varepsilon \left( \varrhoe, \vartheta_\varepsilon, \vc{u}_\varepsilon \Big| r_\varepsilon , \Theta_\varepsilon , \vc{U}_\varepsilon \right) (\tau)
\end{equation}
\[
\leq \chi (\varepsilon, \eta) + \int_0^\tau \intO{ \varrhoe \Big( \partial_t \vc{U}_\varepsilon + \vc{u}_\varepsilon
\cdot\nabla_x \vc{U}_\varepsilon \Big)\cdot (\vc{U}_\varepsilon - \vc{u}_\varepsilon) } \ {\rm d} t
\]
\[
- \frac{1}{\varepsilon} \int_0^\tau \intO{ \left( \varrhoe \Big( s(\varrhoe,\vartheta_\varepsilon) - s(r_\varepsilon, \Theta_\varepsilon) \Big) \partial_t T_\varepsilon + \varrhoe \Big( s(\varrhoe,\vartheta_\varepsilon) - s(r_\varepsilon, \Theta_\varepsilon) \Big) \vc{u}_\varepsilon \cdot \nabla_x T_\varepsilon \right) } \ {\rm d} t
\]
$$
+\frac
1{\varepsilon^2}\int_0^\tau\intO{\Big[\Big(p(r_\varepsilon,\Theta_\varepsilon)-p(\varrhoe,\vartheta_\varepsilon)\Big){\rm
div}\vc U_\varepsilon +\frac\varrhoe {r_\varepsilon}(\vc U_\varepsilon-\vc{u}_\varepsilon)\cdot\nabla_x
p(r_\varepsilon,\Theta_\varepsilon)\Big]}{\rm d}t
$$
\[
+ \frac{1}{\varepsilon^2} \int_0^\tau \intO{ \frac{r_\varepsilon-\varrhoe}{r_\varepsilon}\Big(
\partial_t p(r_\varepsilon , \Theta_\varepsilon ) +\vc U_\varepsilon \cdot
\nabla_x p(r_\varepsilon , \Theta_\varepsilon ) \Big) } \ {\rm d} t -\frac 1\varepsilon\int_0^\tau\int_{R^3}\varrhoe\nabla_x F\cdot(\vc U_\varepsilon-\vc u_\varepsilon){\rm d}x{\rm d}t,
\]
where $\chi$ { denotes a generic function} satisfying
\begin{equation}\label{chi}
{ \lim_{\eta \to 0} \Big( \lim_{\varepsilon \to 0} \chi(\varepsilon, \eta) \Big) }= 0.
\end{equation}
\subsection{Velocity dependent terms}
Our next goal is to handle the expression
\[
\int_0^\tau \intO{ \Big[ \varrhoe (\vc{U}_\varepsilon - \vc{u}_\varepsilon ) \cdot
\partial_t \vc{U}_\varepsilon + \varrhoe (\vc{U}_\varepsilon - \vc{u}_\varepsilon) \otimes \vc{u}_\varepsilon : \nabla_x \vc{U}_\varepsilon \Big] } \ {\rm d} t =
\]
\[
\int_0^\tau \intO{ \varrhoe (\vc{U}_\varepsilon - \vc{u}_\varepsilon ) \otimes
(\vc{u}_\varepsilon - \vc{U}_\varepsilon ) : \nabla_x \vc{U}_\varepsilon } \ {\rm d} t
\]
\[ + \int_0^\tau \intO{ \varrhoe (\vc{U}_\varepsilon - \vc{u}_\varepsilon) \cdot \Big( \partial_t \vc{v} + \vc{v}
\cdot \nabla_x \vc{v} \Big) } \ {\rm d} t + \int_0^\tau \intO{ \varrhoe (\vc{U}_\varepsilon - \vc{u}_\varepsilon)
\cdot \partial_t \nabla_x \Phi_\varepsilon } \ {\rm d} t
\]
\[
+ \int_0^\tau \intO{ \varrhoe (\vc{U}_\varepsilon - \vc{u}_\varepsilon ) \otimes
\nabla_x \Phi_\varepsilon : \nabla_x \vc{v}} + \int_0^\tau \intO{ \varrhoe (\vc{U}_\varepsilon - \vc{u}_\varepsilon ) \otimes
\vc{v} : \nabla_x^2 \Phi_\varepsilon } \ {\rm d} t
\]
\[
+\frac{1}{2} \int_0^\tau \intO{ \varrhoe (\vc{U}_\varepsilon - \vc{u}_\varepsilon) \cdot \nabla_x |\nabla_x \Phi_\varepsilon |^2 } \ {\rm d} t .
\]
Thanks to (\ref{euler}), (\ref{a1}), (\ref{a2}) and the energy bounds established in (\ref{b3} - \ref{b7}), the first integral on the right hand side
can be dominated by the expression
$$
\chi (\varepsilon, \eta) + c \int_0^\tau{\cal E}\Big(\varrhoe,\vartheta_\varepsilon, \vc{u}_\varepsilon \Big|r_\varepsilon,\Theta_\varepsilon,\vc U_\varepsilon\Big){\rm d}t,
$$
with $c$ independent of $\varepsilon$, $\eta$.
The second term reads
\[
\int_0^\tau \int_{\Omega}{ \varrhoe \vc{u}_\varepsilon \cdot \nabla_x \Pi } \ {\rm d} t
- \int_0^\tau \int_{\Omega}{
\varrhoe (\vc{v} + \nabla_x \Phi_\varepsilon ) \cdot \nabla_x \Pi } \ {\rm d} t
\]
$$
+\frac 1{\overline\varrho} \frac {\beta}{\alpha }\int_0^\tau \int_{\Omega}{ \theta\varrhoe \vc{u}_\varepsilon \cdot \nabla_x F } \ {\rm d} t
- \frac 1{\overline\varrho}\frac {\beta}{\alpha}\int_0^\tau \int_{\Omega}
\theta\varrhoe (\vc{v} + \nabla_x \Phi_\varepsilon ) \cdot \nabla_x F \ {\rm d} t
$$
$$
=\frac 1{\overline\varrho}\frac {\beta}{\alpha}\int_0^\tau \intO{ \theta\varrhoe \vc{u}_\varepsilon \cdot \nabla_x F } \ {\rm d} t
- \frac 1{\overline\varrho}\frac {\beta}{\alpha}\int_0^\tau \int_{\Omega}
\theta\varrhoe \vc{v} \cdot \nabla_x F \ {\rm d} t
+\chi(\varepsilon,\eta)
$$
where we have used the equations (\ref{l1}--\ref{l2}), formulas (\ref{conv1}--\ref{conv2}), the dispersive estimates (\ref{a2}), and relation (\ref{euler}).
Next, using the equation (\ref{r4}), we may write the third integral in the form
\[
- \int_0^\tau \intO{ \varrhoe \vc{u}_\varepsilon \cdot \partial_t \nabla_x \Phi_\varepsilon } \ {\rm d} t
-\int_0^\tau \intO{ \frac{\varrhoe - \Ov{\varrho}}{\varepsilon} \vc{v} \cdot \nabla_x \left( \alpha R_\varepsilon + \beta T_\varepsilon \right) } \ {\rm d} t
\]
\[
-
\int_0^\tau \intO{ \frac{\varrhoe - \Ov{\varrho}}{\varepsilon} \nabla_x \Phi_\varepsilon \cdot \nabla_x (\alpha R_\varepsilon + \beta T_\varepsilon) } \ {\rm d} t +
\frac{1}{2} \int_0^\tau \intO{ \Ov{\varrho} \partial_t | \nabla_x \Phi_\varepsilon |^2 } \ {\rm d} t
\]
\[
=- \int_0^\tau \intO{ \varrhoe \vc{u}_\varepsilon \cdot \partial_t \nabla_x \Phi_\varepsilon } \ {\rm d} t +
\frac{1}{2} \int_0^\tau \intO{ \Ov{\varrho} \partial_t | \nabla_x \Phi_\varepsilon |^2 } \ {\rm d} t +{ \chi(\varepsilon,\eta)}.
\]
where we have used wave equation (\ref{r3}--\ref{r4}), estimates (\ref{b3}--\ref{b5}), (\ref{cutoff}), regularity of $\vc v$ stated (\ref{euler}),
the relation (\ref{p3}),
and dispersive estimates (\ref{a2}).
Finally, in view of the uniform bounds (\ref{euler}), (\ref{b3} - \ref{b5}), and the {dispersive estimates} stated in (\ref{a2}),
the last three integrals tend to zero for $\varepsilon \to 0$, uniformly with respect to $\tau$.
Resuming, we obtain
\[
\int_0^\tau \intO{ \Big[ \varrhoe (\vc{U}_\varepsilon - \vc{u}_\varepsilon ) \cdot
\partial_t \vc{U}_\varepsilon + \varrhoe (\vc{U}_\varepsilon - \vc{u}_\varepsilon) \otimes \vc{u}_\varepsilon : \nabla_x \vc{U}_\varepsilon \Big] } \ {\rm d} t
\]
$$
\le \chi (\varepsilon, \eta) + c\int_0^\tau{\cal E}\Big(\varrhoe,\vartheta_\varepsilon,\vc{u}_\varepsilon\Big|r_\varepsilon,\Theta_\varepsilon,\vc U_\varepsilon\Big){\rm d} t +
\frac 1{\overline\varrho}\frac {\beta}{\alpha}\int_0^\tau \intO{ \theta \varrhoe\vc{u}_\varepsilon \cdot \nabla_x F } \ {\rm d} t
$$
$$
- \int_0^\tau \intO{ \varrhoe \vc{u}_\varepsilon \cdot \partial_t \nabla_x \Phi_\varepsilon } \ {\rm d} t +
\frac{1}{2} \int_0^\tau \intO{ \Ov{\varrho} \partial_t | \nabla_x \Phi_\varepsilon |^2 } \ {\rm d} t
$$
$$
- \frac 1{\overline\varrho}\frac {\beta}{\alpha}\int_0^\tau \int_{\Omega}
\theta\varrhoe \vc{v} \cdot \nabla_x F \ {\rm d} t ;
$$
whence relation (\ref{r9}) becomes
\bFormula{r10}
\mathcal{E}_\varepsilon \left( \varrhoe, \vartheta_\varepsilon, \vc{u}_\varepsilon \Big| r_\varepsilon , \Theta_\varepsilon , \vc{U}_\varepsilon \right) (\tau) \leq \chi(\varepsilon, \eta) +
c \int_0^\tau \mathcal{E}_\varepsilon \left( \varrhoe, \vartheta_\varepsilon, \vc{u}_\varepsilon \Big| r_\varepsilon , \Theta_\varepsilon , \vc{U}_\varepsilon \right) \ {\rm d} t
\end{equation}
\[
+ \left[ \intO{ \Ov{\varrho} \frac{1}{2}|\nabla_x \Phi_\varepsilon |^2 } \right]_{t = 0}^{t = \tau} -
\int_0^\tau \intO{ \varrhoe \vc{u}_\varepsilon \cdot \partial_t \nabla_x \Phi_\varepsilon } \ {\rm d} t
\]
\[
- \frac{1}{\varepsilon} \int_0^\tau \intO{ \left[ \varrhoe \Big( s(\varrhoe,\vartheta_\varepsilon) - s(r_\varepsilon, \Theta_\varepsilon) \Big) \partial_t T_\varepsilon + \varrhoe \Big( s(\varrhoe,\vartheta_\varepsilon) - s(r_\varepsilon, \Theta_\varepsilon) \Big) \vc{u}_\varepsilon \cdot \nabla_x T_\varepsilon \right] } \ {\rm d} t
\]
\[
+ \frac{1}{\varepsilon^2} \int_0^\tau \intO{ \left( ( r_\varepsilon - \varrhoe ) \frac{1}{r_\varepsilon } \partial_t p(r_\varepsilon , \Theta_\varepsilon ) - \frac{\varrhoe}{r_\varepsilon} \vc{u}_\varepsilon \cdot
\nabla_x p(r_\varepsilon , \Theta_\varepsilon ) \right) } \ {\rm d} t - \frac{1}{\varepsilon^2} \int_0^\tau \intO{ \Big( p(\varrhoe, \vartheta_\varepsilon) - p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) \Big)
{\cal D}elta \Phi_\varepsilon } \ {\rm d} t
\]
\[
-\frac 1\varepsilon\int_0^\tau\int_{\Omega}\varrhoe\nabla_x F\cdot(\vc{v} -\vc u_\varepsilon) \ {\rm d} {x}dt
+
\frac 1{\overline\varrho}\frac {\beta}{\alpha}\int_0^\tau \int_{\Omega}{ \theta\varrhoe \vc{u}_\varepsilon \cdot \nabla_x F }{\rm d} x \ {\rm d} t
- \frac 1{\overline\varrho}\frac {\beta}{\alpha}\int_0^\tau \int_{\Omega}
\theta\varrhoe \vc{v} \cdot \nabla_x F \ {\rm d} {x}dt.
$$
In the above, we have used the identity
\[
\intO{ \left[ \Big( p(r_\varepsilon, \Theta_\varepsilon) - p(\varrhoe, \vartheta_\varepsilon) \Big) {\rm div}_x \vc{U}_\varepsilon + \left( 1 - \frac{\varrhoe}{r_\varepsilon} \right) \vc{U}_\varepsilon
\cdot \nabla_x p(r_\varepsilon, \Theta_\varepsilon) + \frac{\varrhoe}{r_\varepsilon} (\vc{U}_\varepsilon - \vc{u}_\varepsilon) \cdot \nabla_x p(r_\varepsilon, \Theta_\varepsilon) \right] }
\]
\[
= - \intO{ p(\varrhoe, \vartheta_\varepsilon) {\cal D}elta \Phi_\varepsilon} - \intO{ \frac{\varrhoe}{r_\varepsilon} \vc{u}_\varepsilon \cdot \nabla_x p(r_\varepsilon, \Theta_\varepsilon) },
\]
together with
\[
-\frac 1\varepsilon\int_0^\tau\int_{\Omega}\varrhoe\nabla_x F\cdot(\vc U_\varepsilon-\vc u_\varepsilon) \ {\rm d} {x}dt =
-\frac 1\varepsilon\int_0^\tau\int_{\Omega}\varrhoe\nabla_x F\cdot(\vc{v}-\vc u_\varepsilon) \ {\rm d} {x}dt - \frac 1\varepsilon\int_0^\tau\int_{\Omega}\varrhoe\nabla_x F\cdot\nabla_x \Phi_\varepsilon \ {\rm d} {x}dt
\]
\[
= { \chi(\varepsilon, \eta)} -\frac 1\varepsilon\int_0^\tau\int_{\Omega}\varrhoe\nabla_x F\cdot(\vc{v}-\vc u_\varepsilon) \ {\rm d} {x}dt + \frac{1}{\varepsilon^2} p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) {\cal D}elta \Phi_\varepsilon
\ {\rm d} {x}dt.
\]
Recall that $\nabla_x \Phi_\varepsilon(t, \cdot)$ decays fast as $|x| \to \infty$ and ${\rm div}_x \vc{v} = 0$, which justifies the by-parts integration.
\subsection{Pressure dependent terms}
We write
\[
\frac{1}{\varepsilon^2} \frac{\varrhoe}{r_\varepsilon} \vc{u}_\varepsilon \cdot \nabla_x p(r_\varepsilon,
\Theta_\varepsilon) = \frac{1}{\varepsilon^2} \frac{\varrhoe}{r_\varepsilon} \vc{u}_\varepsilon \cdot \nabla_x \Big( p(r_\varepsilon, \Theta_\varepsilon) - p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) \Big) +
\frac{1}{\varepsilon^2} \frac{\varrhoe}{r_\varepsilon} \vc{u}_\varepsilon \cdot \nabla_x p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta})
\]
\[
= \frac{1}{\varepsilon^2} \frac{\varrhoe}{r_\varepsilon} \vc{u}_\varepsilon \cdot \nabla_x \left( p(r_\varepsilon, \Theta_\varepsilon) - \frac{\partial p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) }
{\partial \varrho} \varepsilon R_\varepsilon - \frac{\partial p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) }
{\partial \vartheta} \varepsilon T_\varepsilon - p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) \right)
\]
\[
+\frac{1}{\varepsilon} \frac{\varrhoe}{r_\varepsilon} \vc{u}_\varepsilon \cdot \nabla_x \left( \frac{\partial p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) }
{\partial \varrho} R_\varepsilon + \frac{\partial p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) }
{\partial \vartheta} T_\varepsilon \right) + \frac{1}{\varepsilon} \frac{\Ov{\varrho}_\varepsilon} {r_\varepsilon} \varrhoe \vc{u}_\varepsilon \cdot \nabla_x F.
\]
Next, we use the decay properties of the equilibrium density profile $\Ov{\varrho}_\varepsilon$ stated in (\ref{p3}), together with (\ref{a5}), (\ref{a5aa})
to observe that
\[
\frac{1}{\varepsilon^2 r_\varepsilon} \nabla_x
\left( p(r_\varepsilon, \Theta_\varepsilon) - \frac{\partial p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) }
{\partial \varrho} \varepsilon R_\varepsilon - \frac{\partial p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) }
{\partial \vartheta} \varepsilon T_\varepsilon - p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) \right) \to \nabla_x H \ \mbox{in}\ L^p (0,T; (L^2 \cap L^q)(\Omega;R^3)) ,\ p \geq 1,\ q > 2,
\]
where the right-hand side is a gradient of a certain function $H$. Consequently, using (\ref{conv1}), (\ref{conv2}) we may infer that
\[
\int_0^\tau \intO{ \frac{1}{\varepsilon^2} \frac{\varrhoe}{r_\varepsilon} \vc{u}_\varepsilon \cdot \nabla_x \left( p(r_\varepsilon, \Theta_\varepsilon) - \frac{\partial p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) }
{\partial \varrho} \varepsilon R_\varepsilon - \frac{\partial p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) }
{\partial \vartheta} \varepsilon T_\varepsilon - p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) \right) } \ {\rm d} t = { \chi (\varepsilon, \eta)}.
\]
Moreover, by the same token, we obtain
\[
\int_0^\tau \intO{
\frac{1}{\varepsilon} \frac{\varrhoe}{r_\varepsilon} \vc{u}_\varepsilon \cdot \nabla_x \left( \frac{\partial p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) }
{\partial \varrho} R_\varepsilon + \frac{\partial p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) }
{\partial \vartheta} T_\varepsilon \right) } \ {\rm d} t = \eta(\varepsilon, \delta) + \int_0^\tau \intO{
\frac{1}{\varepsilon} {\varrhoe} \vc{u}_\varepsilon \cdot \nabla_x \left( \alpha
R_\varepsilon + \beta
T_\varepsilon \right) } \ {\rm d} t .
\]
Making use of the identity
\[
\int_0^\tau \intO{
\frac{1}{\varepsilon} {\varrhoe} \vc{u}_\varepsilon \cdot \nabla_x \left( \alpha
R_\varepsilon + \beta
T_\varepsilon \right) } \ {\rm d} t = - \int_0^\tau\int_{R^3}\varrhoe \vc{u}_\varepsilon \cdot \partial_t \nabla_x \Phi_\varepsilon{\rm d} x{\rm d} t
\]
we may rewrite (\ref{r10}) in the form
\bFormula{r11-}
\mathcal{E}_\varepsilon \left( \varrhoe, \vartheta_\varepsilon, \vc{u}_\varepsilon \Big| r_\varepsilon , \Theta_\varepsilon , \vc{U}_\varepsilon \right) (\tau) \leq \chi(\varepsilon, \eta) +
c \int_0^\tau \mathcal{E}_\varepsilon \left( \varrhoe, \vartheta_\varepsilon, \vc{u}_\varepsilon \Big| r_\varepsilon , \Theta_\varepsilon , \vc{U}_\varepsilon \right) \ {\rm d} t
+ \left[ \intO{ \Ov{\varrho} \frac{1}{2}|\nabla_x \Phi_\varepsilon |^2 } \right]_{t = 0}^{t = \tau}
\end{equation}
\[
- \frac{1}{\varepsilon} \int_0^\tau \intO{ \left[ \varrhoe \Big( s(\varrhoe,\vartheta_\varepsilon) - s(r_\varepsilon, \Theta_\varepsilon) \Big) \partial_t T_\varepsilon + \varrhoe \Big( s(\varrhoe,\vartheta_\varepsilon) - s(r_\varepsilon, \Theta_\varepsilon) \Big) \vc{u}_\varepsilon \cdot \nabla_x T_\varepsilon \right] } \ {\rm d} t
\]
\[
+ \frac{1}{\varepsilon^2} \int_0^\tau \intO{ ( r_\varepsilon - \varrhoe ) \frac{1}{r_\varepsilon } \partial_t p(r_\varepsilon , \Theta_\varepsilon ) } \ {\rm d} t - \frac{1}{\varepsilon^2} \int_0^\tau \intO{ \Big( p(\varrhoe, \vartheta_\varepsilon) - p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) \Big)
{\cal D}elta \Phi_\varepsilon } \ {\rm d} t
\]
\[
+ \int_0^\tau\int_{\Omega}\frac{R_\varepsilon}{r_\varepsilon} \varrhoe \vc{u}_\varepsilon \cdot \nabla_x F \ {\rm d} {x}dt - \int_0^\tau \intO{ \frac{\varrhoe - \Ov{\varrho}_\varepsilon }{\varepsilon} \vc{v} \cdot \nabla_x F
} \ {\rm d} t
\]
\[
+
\frac 1{\overline\varrho}\frac {\beta}{\alpha}\int_0^\tau \int_{\Omega}{ \theta\varrhoe \vc{u}_\varepsilon \cdot \nabla_x F }{\rm d} x \ {\rm d} t
- \frac 1{\overline\varrho}\frac {\beta}{\alpha}\int_0^\tau \int_{\Omega}
\theta\varrhoe \vc{v} \cdot \nabla_x F \ {\rm d} {x}dt.
\]
Finally, we use the fact that
\bFormula{brum}
\alpha R_{\eta} + \beta T_{\eta} = 0,
\end{equation}
and that $T_\eta$ and $\theta$ satisfy the same equation { (see (\ref{a5A}) and (\ref{l3}))} with the initial data given by
(\ref{a5B}), (\ref{m6}), respectively, to deduce that
\bFormula{r11}
\mathcal{E}_\varepsilon \left( \varrhoe, \vartheta_\varepsilon, \vc{u}_\varepsilon \Big| r_\varepsilon , \Theta_\varepsilon , \vc{U}_\varepsilon \right) (\tau) \leq \chi(\varepsilon, \eta) +
c \int_0^\tau \mathcal{E}_\varepsilon \left( \varrhoe, \vartheta_\varepsilon, \vc{u}_\varepsilon \Big| r_\varepsilon , \Theta_\varepsilon , \vc{U}_\varepsilon \right) \ {\rm d} t
+ \left[ \intO{ \Ov{\varrho} \frac{1}{2}|\nabla_x \Phi_\varepsilon |^2 } \right]_{t = 0}^{t = \tau}
\end{equation}
\[
- \frac{1}{\varepsilon} \int_0^\tau \intO{ \left[ \varrhoe \Big( s(\varrhoe,\vartheta_\varepsilon) - s(r_\varepsilon, \Theta_\varepsilon) \Big) \partial_t T_\varepsilon + \varrhoe \Big( s(\varrhoe,\vartheta_\varepsilon) - s(r_\varepsilon, \Theta_\varepsilon) \Big) \vc{u}_\varepsilon \cdot \nabla_x T_\varepsilon \right] } \ {\rm d} t
\]
\[
+ \frac{1}{\varepsilon^2} \int_0^\tau \intO{ ( r_\varepsilon - \varrhoe ) \frac{1}{r_\varepsilon } \partial_t p(r_\varepsilon , \Theta_\varepsilon ) } \ {\rm d} t - \frac{1}{\varepsilon^2} \int_0^\tau \intO{ \Big( p(\varrhoe, \vartheta_\varepsilon) - p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) \Big)
{\cal D}elta \Phi_\varepsilon } \ {\rm d} t
\]
\[
- \int_0^\tau \intO{ \frac{\varrhoe - \Ov{\varrho}_\varepsilon }{\varepsilon} \vc{v} \cdot \nabla_x F
} \ {\rm d} t
- \frac 1{\overline\varrho}\frac {\beta}{\alpha}\int_0^\tau \int_{\Omega}
\theta\varrhoe \vc{v} \cdot \nabla_x F \ {\rm d} {x}dt.
\]
\subsection{Replacing velocity in the entropy convective term}\label{7.4}
Our intention in this section is to ``replace'' $\vc{u}_\varepsilon$ by $\vc{U}_\varepsilon$ in the remaining (last) convective term in (\ref{r11}). To this end,
we write
\[
\int_0^\tau \intO{ \varrhoe \frac{ s(\varrhoe, \vartheta_\varepsilon) - s(r_\varepsilon, \Theta_\varepsilon ) }{ \varepsilon } \vc{u}_\varepsilon \cdot \nabla_x T_\varepsilon } \ {\rm d} t
\]
\[
=
\int_0^\tau \intO{ \varrhoe \frac{ s(\varrhoe, \vartheta_\varepsilon) - s(r_\varepsilon , \Theta_\varepsilon ) }{ \varepsilon } \vc{U}_\varepsilon \cdot \nabla_x T_\varepsilon } \ {\rm d} t +
\int_0^\tau \intO{ \varrhoe \frac{ s(\varrhoe, \vartheta_\varepsilon) - s(r_\varepsilon, \Theta_\varepsilon ) }{ \varepsilon } (\vc{u}_\varepsilon - \vc{U}_\varepsilon) \cdot \nabla_x T_\varepsilon } \ {\rm d} t ,
\]
where
\[
\left| \int_0^\tau \intO{ \varrhoe \left[ \frac{ s(\varrhoe, \vartheta_\varepsilon) - s(r_\varepsilon , \Theta_\varepsilon ) }{ \varepsilon } \right]_{\rm ess} (\vc{u}_\varepsilon - \vc{U}_\varepsilon) \cdot \nabla_x T_\varepsilon } \ {\rm d} t \right|
\]
\[
\leq A(\eta) \int_0^\tau \intO{ \left( \varrhoe | \vc{u}_\varepsilon - \vc{U}_\varepsilon |^2
+ \left| \left[ \frac{\varrhoe - r_\varepsilon}{\varepsilon} \right]_{\rm ess} \right|^2 + \left| \left[ \frac{\vartheta_\varepsilon - \Theta_\varepsilon}{\varepsilon} \right]_{\rm ess} \right|^2 \right)
}\ {\rm d} t
\]
$$
\le c \int_0^\tau{\cal E}\Big(\varrhoe,\vartheta_\varepsilon,\vc{u}_\varepsilon\Big|r_\varepsilon,\Theta_\varepsilon,\vc U_\varepsilon\Big){\rm d}t
$$
and
\[
+ \int_0^\tau \intO{ \varrhoe \left[ \frac{ s(\varrhoe, \vartheta_\varepsilon) - s(r_\varepsilon , \Theta_\varepsilon ) }{ \varepsilon } \right]_{\rm res} (\vc{u}_\varepsilon - \vc{U}_\varepsilon) \cdot \nabla_x T_\varepsilon } \
{\rm d} t =\chi(\varepsilon ,\eta)\;\mbox{provided $0<a<10/3$.}
\]
When estimating the residual component, we have first deduced from (\ref{i12} - \ref{i16}) the inequality
\begin{equation}\label{*entropy}
\varrho | s(\varrho, \vartheta) | \leq c\left( \vartheta^3 + \varrho |\log(\varrho)| + \varrho [\log(\vartheta)]^+ \right)
\end{equation}
and then employed the estimates (\ref{b5}--\ref{b6}) for $\varrhoe$, $\vartheta_\varepsilon$, together with the estimates (\ref{a2}--\ref{a4}) for $R_\varepsilon$, $T_\varepsilon$,
$\nabla_x\Phi_\varepsilon$, and (\ref{euler}) for $\vc v$.
Consequently, we can can rewrite inequality (\ref{r11}) in the form
\bFormula{r11+}
\mathcal{E}_\varepsilon \left( \varrhoe, \vartheta_\varepsilon, \vc{u}_\varepsilon \Big| r_\varepsilon , \Theta_\varepsilon , \vc{U}_\varepsilon \right) (\tau) \leq \chi(\varepsilon, \eta) +
c \int_0^\tau \mathcal{E}_\varepsilon \left( \varrhoe, \vartheta_\varepsilon, \vc{u}_\varepsilon \Big| r_\varepsilon , \Theta_\varepsilon , \vc{U}_\varepsilon \right) \ {\rm d} t
+ \left[ \intO{ \Ov{\varrho} \frac{1}{2}|\nabla_x \Phi_\varepsilon |^2 } \right]_{t = 0}^{t = \tau}
\end{equation}
\[
- \frac{1}{\varepsilon} \int_0^\tau \intO{ \left[ \varrhoe \Big( s(\varrhoe,\vartheta_\varepsilon) - s(r_\varepsilon, \Theta_\varepsilon) \Big) \partial_t T_\varepsilon + \varrhoe \Big( s(\varrhoe,\vartheta_\varepsilon) - s(r_\varepsilon, \Theta_\varepsilon)
\Big) \vc{U}_\varepsilon \cdot \nabla_x T_\varepsilon \right] } \ {\rm d} t
\]
\[
+ \frac{1}{\varepsilon^2} \int_0^\tau \intO{ ( r_\varepsilon - \varrhoe ) \frac{1}{r_\varepsilon } \partial_t p(r_\varepsilon , \Theta_\varepsilon ) } \ {\rm d} t - \frac{1}{\varepsilon^2} \int_0^\tau \intO{ \Big( p(\varrhoe, \vartheta_\varepsilon) - p(\Ov{\varrho}_\varepsilon, \Ov{\vartheta}) \Big)
{\cal D}elta \Phi_\varepsilon } \ {\rm d} t
\]
\[
- \int_0^\tau \intO{ \frac{\varrhoe - \Ov{\varrho}_\varepsilon }{\varepsilon} \vc{v} \cdot \nabla_x F
} \ {\rm d} t
- \frac 1{\overline\varrho}\frac {\beta}{\alpha}\int_0^\tau \int_{\Omega}
\theta\varrhoe \vc{v} \cdot \nabla_x F \ {\rm d} {x}dt.
\]
\subsection{The entropy and the pressure}
{ \subsubsection{Handling the residual component}}
To begin, we observe
that the residual components of all integrals on the second and third line of inequality (\ref{r11+}) are negligible.
To this end, we first use the estimates (\ref{a2} - \ref{a4}), (\ref{a5}), (\ref{a5aa}), together with the equations (\ref{r3} - \ref{r4a}),
to deduce
\bFormula{pom1}
\sup_{t \in [0,T]} \varepsilon \| \partial_t R_\varepsilon (t, \cdot) \|_{L^\infty(R^3)} ,\
\sup_{t \in [0,T]} \varepsilon \| \partial_t T_\varepsilon (t, \cdot) \|_{L^\infty(R^3)} \leq A(\eta),
\end{equation}
\bFormula{pom2}
\varepsilon \| \partial_t R_\varepsilon (t, \cdot) \|_{L^\infty(R^3)} \to 0 , \ \varepsilon \| \partial_t T_\varepsilon (t, \cdot) \|_{L^\infty(R^3)} \to 0\ \mbox{for any} \ t > 0.
\end{equation}
Now, we employ these relations in combination with the uniform estimates (\ref{b5}); after a long but straightforward calculation,
we finally get the desired result, namely
\begin{equation}\label{*1-}
- \frac{1}{\varepsilon} \int_0^\tau \intO{ \left[ \Big[\varrhoe \Big( s(\varrhoe,\vartheta_\varepsilon)
- s(r_\varepsilon, \Theta_\varepsilon) \Big) \partial_t T_\varepsilon + \varrhoe \Big(
s(\varrhoe,\vartheta_\varepsilon) - s(r_\varepsilon, \Theta_\varepsilon) \Big) \vc U_\varepsilon \cdot \nabla_x
T_\varepsilon\Big]_{\rm res} \right] } \ {\rm d} t
\end{equation}
$$
- \frac{1}{\varepsilon^2} \int_0^\tau \intO{ \Big[\frac{\varrho_\varepsilon - r_\varepsilon}
{r_\varepsilon}\partial_t p(r_\varepsilon , \Theta_\varepsilon ) \Big]_{\rm res} } \ {\rm d} t -\frac 1{\varepsilon^2}
\int_0^\tau
\intO{\Big[\Big(p(\varrhoe,\vartheta_\varepsilon)-p(\overline\varrho_\varepsilon,\overline\vartheta)\Big){\cal D}elta\Phi_\varepsilon\Big]_{\rm res}}\ {\rm
d}t=\chi(\varepsilon,\eta)
$$
\subsubsection{Handling the essential component}
In view of the preceding Section, we have to handle solely the essential part of the integrals at the first and second line of
formula (\ref{r11+}) whose integrands can be, roughly speaking, replaced by their linearization
at $\overline\varrho_\varepsilon$, $\overline\vartheta$. Since we already know that the corresponding residual components are negligible,
we may omit the symbol $[\cdot]_{\rm ess}$ in all integrands.
We check that
\begin{equation}\label{*1}
- \frac{1}{\varepsilon} \int_0^\tau \intO{ \left[ \varrhoe \Big( s(\varrhoe,\vartheta_\varepsilon)
- s(r_\varepsilon, \Theta_\varepsilon) \Big) \partial_t T_\varepsilon + \varrhoe \Big(
s(\varrhoe,\vartheta_\varepsilon) - s(r_\varepsilon, \Theta_\varepsilon) \Big) \vc U_\varepsilon \cdot \nabla_x
T_\varepsilon \right] } \ {\rm d} t
\end{equation}
$$
- \frac{1}{\varepsilon^2} \int_0^\tau \intO{ \frac{\varrho_\varepsilon - r_\varepsilon}
{r_\varepsilon}\partial_t p(r_\varepsilon , \Theta_\varepsilon ) } \ {\rm d} t -\frac 1{\varepsilon^2}
\int_0^\tau
\intO{\Big(p(\varrhoe,\vartheta_\varepsilon)-p(\overline\varrho_\varepsilon,\overline\vartheta)\Big){\cal D}elta\Phi_\varepsilon}\ {\rm
d}t
$$
$$
= -\int_0^\tau\intO{\Big(\delta\frac{\vartheta_\varepsilon-\Theta_\varepsilon}\varepsilon
-\beta\frac{\varrho_\varepsilon-r_\varepsilon}\varepsilon\Big)\Big(\partial_t T_\varepsilon + \vc
U_\varepsilon\cdot\nabla_x T_\varepsilon\Big)}\ {\rm d}t
$$
$$
-\int_0^\tau\intO{\frac{\varrhoe- r_\varepsilon}\varepsilon\partial_t\Big(\alpha
R_\varepsilon+\beta T_\varepsilon\Big)}\ {\rm d}t +\int_0^\tau
\intO{\frac{\delta}{\beta^2+\alpha\delta}\Big(\alpha\frac{\varrhoe
-\overline\varrho_\varepsilon}\varepsilon+\beta\frac{\vartheta_\varepsilon-\overline\vartheta}\varepsilon\Big)
\partial_t\Big(\alpha R_\varepsilon+\beta T_\varepsilon\Big)}\ {\rm d}t
$$
\[
+ \int_0^\tau \intO{ \frac{1}{\varepsilon} \left( \frac{\partial p(\Ov{\varrho}, \Ov{\vartheta}) }{\partial \varrho} -
\frac{\partial p(\Ov{\varrho}_\varepsilon , \Ov{\vartheta}) }{\partial \varrho} \right) \frac{\varrhoe - \Ov{\varrho}_\varepsilon }{\varepsilon} {\cal D}elta \Phi_\varepsilon } \ {\rm d} t
+ \int_0^\tau \intO{ \frac{1}{\varepsilon} \left( \frac{\partial p(\Ov{\varrho}, \Ov{\vartheta}) }{\partial \vartheta} -
\frac{\partial p(\Ov{\varrho}_\varepsilon , \Ov{\vartheta}) }{\partial \vartheta} \right) \frac{\vartheta_\varepsilon - \Ov{\vartheta} }{\varepsilon} {\cal D}elta \Phi_\varepsilon } \ {\rm d} t
+ \chi(\varepsilon, \eta),
\]
where, in accordance with the dispersive estimates (\ref{a1}), (\ref{a2}) and (\ref{p3}),
\[
\int_0^\tau \intO{ \frac{1}{\varepsilon} \left( \frac{\partial p(\Ov{\varrho}, \Ov{\vartheta}) }{\partial \varrho} -
\frac{\partial p(\Ov{\varrho}_\varepsilon , \Ov{\vartheta}) }{\partial \varrho} \right) \frac{\varrhoe - \Ov{\varrho}_\varepsilon }{\varepsilon} {\cal D}elta \Phi_\varepsilon } \ {\rm d} t
+ \int_0^\tau \intO{ \frac{1}{\varepsilon} \left( \frac{\partial p(\Ov{\varrho}, \Ov{\vartheta}) }{\partial \vartheta} -
\frac{\partial p(\Ov{\varrho}_\varepsilon , \Ov{\vartheta}) }{\partial \vartheta} \right) \frac{\vartheta_\varepsilon - \Ov{\vartheta} }{\varepsilon} {\cal D}elta \Phi_\varepsilon } \ {\rm d} t
= \chi(\varepsilon, \eta).
\]
Consequently, we get
\bFormula{1---}
- \frac{1}{\varepsilon} \int_0^\tau \intO{ \left[ \varrhoe \Big( s(\varrhoe,\vartheta_\varepsilon)
- s(r_\varepsilon, \Theta_\varepsilon) \Big) \partial_t T_\varepsilon + \varrhoe \Big(
s(\varrhoe,\vartheta_\varepsilon) - s(r_\varepsilon, \Theta_\varepsilon) \Big) \vc U_\varepsilon \cdot \nabla_x
T_\varepsilon \right] } \ {\rm d} t
\end{equation}
$$
- \frac{1}{\varepsilon^2} \int_0^\tau \intO{ \frac{\varrho_\varepsilon - r_\varepsilon}
{r_\varepsilon}\partial_t p(r_\varepsilon , \Theta_\varepsilon ) } \ {\rm d} t -\frac 1{\varepsilon^2}
\int_0^\tau
\intO{\Big(p(\varrhoe,\vartheta_\varepsilon)-p(\overline\varrho_\varepsilon,\overline\vartheta)\Big){\cal D}elta\Phi_\varepsilon}\ {\rm
d}t
$$
$$
= \int_0^\tau\intO{\Big(\delta T_\varepsilon-\beta R_\varepsilon\Big)\partial_t
T_\varepsilon}{\rm d} t + \int_0^\tau\intO{R_\varepsilon\partial_t\Big(\alpha
R_\varepsilon+\beta T_\varepsilon\Big)}\ {\rm d}t
$$
$$
-\left[ \int_0^\tau\intO{\Big(\delta\frac{\vartheta_\varepsilon-\overline\vartheta}\varepsilon
-\beta\frac{\varrho_\varepsilon-\overline\varrho_\varepsilon}\varepsilon\Big)\partial_t T_\varepsilon}\ {\rm d}t
+
\int_0^\tau\intO{\Big(\frac{\beta^2}{\beta^2+\alpha\delta}\frac{\varrhoe-\overline\varrho_\varepsilon}\varepsilon
-\frac{\beta\delta}{\beta^2+\alpha\delta}\frac{\vartheta_\varepsilon-\overline\vartheta}\varepsilon\Big)
\partial_t\Big(\alpha
R_\varepsilon+\beta T_\varepsilon\Big)}\ {\rm d}t \right]
$$
$$
- \int_0^\tau\intO{\Big(\delta\frac{\vartheta_\varepsilon-\Theta_\varepsilon}\varepsilon
-\beta\frac{\varrho_\varepsilon-r_\varepsilon}\varepsilon\Big)\vc U_\varepsilon\cdot\nabla_x T_\varepsilon} \
{\rm d}t + \chi(\varepsilon, \eta).
$$
In the next steps, we use the identities
\bFormula{ident}
(\beta^2+\alpha\delta) T=\beta(\alpha R+\beta T)+\alpha(\delta
T-\beta R),
\
(\beta^2+\alpha\delta) R=\delta(\alpha R+\beta T)-\beta(\delta
T-\beta R),
\end{equation}
to compute,
\begin{equation}\label{*2}
\int_0^\tau\intO{\Big(\delta T_\varepsilon-\beta R_\varepsilon\Big)\partial_t
T_\varepsilon}\ {\rm d} t + \int_0^\tau\intO{R_\varepsilon\partial_t\Big(\alpha
R_\varepsilon+\beta T_\varepsilon\Big)}\ {\rm d}t
\end{equation}
$$
=
\int_{0}^\tau\int_{\Omega}\Big[\frac\beta{\beta^2+\alpha\delta}\Big(\delta
T_\varepsilon-\beta R_\varepsilon\Big)\partial_t\Big(\alpha R_\varepsilon+\beta T_\varepsilon\Big)
+ \frac\alpha{\beta^2+\alpha\delta}\Big(\delta T_\varepsilon-\beta
R_\varepsilon\Big)\partial_t\Big(\delta T_\varepsilon-\beta R_\varepsilon\Big)
$$
$$
+\frac\delta{\beta^2+\alpha\delta}\Big(\alpha R_\varepsilon+\beta
T_\varepsilon \Big)\partial_t\Big(\alpha R_\varepsilon+\beta T_\varepsilon \Big) -
\frac\beta{\beta^2+\alpha\delta}\Big(\delta T_\varepsilon-\beta
R_\varepsilon\Big)\partial_t\Big(\alpha R_\varepsilon+\beta T_\varepsilon\Big)
\Big]{\rm d}x\ {\rm d}t
$$
$$
=\frac12\frac\delta{\beta^2+\alpha\delta} \left[\int_{\Omega}|\alpha
R_\varepsilon+\beta T_\varepsilon|^2{\rm d} x\right]_0^\tau+
\frac12\frac\alpha{\beta^2+\alpha\delta} \left[\int_{\Omega}|\delta
T_\varepsilon-\beta R_\varepsilon|^2{\rm d} x\right]_0^\tau ,
$$
where we have used (\ref{r3}).
Similarly, we get
\begin{equation}\label{*3}
- \int_0^\tau\intO{\Big(\delta\frac{\vartheta_\varepsilon-\overline\vartheta}\varepsilon
-\beta\frac{\varrho_\varepsilon-\overline\varrho}\varepsilon\Big)\partial_t T_\varepsilon}\ {\rm d}t
-
\int_0^\tau\intO{\Big(\frac{\beta^2}{\beta^2+\alpha\delta}\frac{\varrhoe-\overline\varrho}\varepsilon
-\frac{\beta\delta}{\beta^2+\alpha\delta}\frac{\vartheta_\varepsilon-\overline\vartheta}\varepsilon\Big)
\partial_t\Big(\alpha
R_\varepsilon+\beta T_\varepsilon\Big)}\ {\rm d}t
\end{equation}
$$
=
-\frac{\alpha}{\beta^2+\alpha\delta}\int_0^\tau\intO{\Big(\delta\frac{\vartheta_\varepsilon-\overline\vartheta}\varepsilon-\beta\frac{\varrhoe-\overline\varrho}\varepsilon\Big)
\partial_t\Big(\delta T_\varepsilon-\beta R_\varepsilon\Big)}\ {\rm d} t
$$
Finally, the last line on the right-hand side of (\ref{1---}) reads
\begin{equation}\label{*4}
-\int_0^\tau\intO{\Big(\delta\frac{\vartheta_\varepsilon-\Theta_\varepsilon}\varepsilon
-\beta\frac{\varrho_\varepsilon-r_\varepsilon}\varepsilon\Big)\vc U_\varepsilon\cdot\nabla_x T_\varepsilon}\ {\rm
d}t
\end{equation}
$$=
-\frac\beta{\beta^2+\alpha\delta}\int_0^\tau\intO{\Big(\delta\frac{\vartheta_\varepsilon-\Theta_\varepsilon}\varepsilon
-\beta\frac{\varrho_\varepsilon-r_\varepsilon}\varepsilon\Big)\vc U_\varepsilon\cdot\nabla_x\Big( \alpha
R_\varepsilon +\beta T_\varepsilon\Big)}\ {\rm d}t
$$
$$
-
\frac\alpha{\beta^2+\alpha\delta}\int_0^\tau\intO{\Big(\delta\frac{\vartheta_\varepsilon-\Theta_\varepsilon}\varepsilon
-\beta\frac{\varrho_\varepsilon-r_\varepsilon}\varepsilon\Big)\vc U_\varepsilon\cdot\nabla_x\Big( \delta
T_\varepsilon -\beta R_\varepsilon\Big)}\ {\rm d}t
$$
$$
= -
\frac\alpha{\beta^2+\alpha\delta}\int_0^\tau\intO{\Big(\delta\frac{\vartheta_\varepsilon-\Theta_\varepsilon}\varepsilon
-\beta\frac{\varrho_\varepsilon-r_\varepsilon}\varepsilon\Big)\vc U_\varepsilon\cdot\nabla_x\Big( \delta
T_\varepsilon -\beta R_\varepsilon\Big)}\ {\rm d}t
+\chi(\varepsilon,\eta),
$$
where we have used the dispersive estimates (\ref{a2}).
Summing up the previous integrals and using equation (\ref{r4a}) we may infer that
\begin{equation}\label{r12}
- \frac{1}{\varepsilon} \int_0^\tau \intO{ \left[ \varrhoe \Big( s(\varrhoe,\vartheta_\varepsilon)
- s(r_\varepsilon, \Theta_\varepsilon) \Big) \partial_t T_\varepsilon + \varrhoe \Big(
s(\varrhoe,\vartheta_\varepsilon) - s(r_\varepsilon, \Theta_\varepsilon) \Big) \vc U_\varepsilon \cdot \nabla_x
T_\varepsilon \right] } \ {\rm d} t
\end{equation}
$$
- \frac{1}{\varepsilon^2} \int_0^\tau \intO{ \frac{\varrho_\varepsilon - r_\varepsilon}
{r_\varepsilon}\partial_t p(r_\varepsilon , \Theta_\varepsilon ) } \ {\rm d} t -\frac 1{\varepsilon^2}
\int_0^\tau
\intO{\Big(p(\varrhoe,\vartheta_\varepsilon)-p(\overline\varrho,\overline\vartheta)\Big){\cal D}elta\Phi_\varepsilon}\ {\rm
d}t
$$
$$
= \frac12\frac\delta{\beta^2+\alpha\delta} \left[\intO{|\alpha
R_\varepsilon+\beta T_\varepsilon|^2}\right]_0^\tau+
\frac12\frac\alpha{\beta^2+\alpha\delta} \left[\intO{|\delta
T_\varepsilon-\beta R_\varepsilon|^2}\right]_0^\tau
$$
$$
-\frac\beta{\beta^2+\alpha\delta}\int_0^\tau\intO{\Big(\delta\frac{\vartheta_\varepsilon-\overline\vartheta}\varepsilon
-\beta\frac{\varrho_\varepsilon-\overline\varrho}\varepsilon \Big)\vc v\cdot\nabla_x F_\varepsilon}{\rm d}t +\chi(\varepsilon,\eta)
$$
Finally, we use relation (\ref{Z1}) to obtain that
\[
-\frac\beta{\beta^2+\alpha\delta}\int_0^\tau\intO{\Big(\delta\frac{\vartheta_\varepsilon-\overline\vartheta}\varepsilon
-\beta\frac{\varrho_\varepsilon-\overline\varrho}\varepsilon \Big)\vc v\cdot\nabla_x F_\varepsilon}{\rm d}t = \int_0^\tau \intO{ \frac{\varrhoe - \Ov{\varrho}_\varepsilon}{\varepsilon} \vc{v} \cdot \nabla_x F } \ {\rm d} t +
\chi(\varepsilon, \eta),
\]
while { due to (\ref{r4a}) and (\ref{brum})}
\[
\frac12\frac\alpha{\beta^2+\alpha\delta} \left[\intO{|\delta
T_\varepsilon-\beta R_\varepsilon|^2}\right]_0^\tau = \frac{\beta}{\alpha} \int_0^\tau \intO{ T_\eta \vc{v} \cdot \nabla_x F } \ {\rm d} t + \chi(\varepsilon, \eta).
\]
As $\theta$ and $T_{\eta}$ satisfy the \emph{same} transport equation and the acoustic system (\ref{r3}), (\ref{r4}) conserves the total energy,
we may use the previous estimates to rewrite (\ref{r11+}) in the final form:
\bFormula{rfinal}
\mathcal{E}_\varepsilon \left( \varrhoe, \vartheta_\varepsilon, \vc{u}_\varepsilon \Big| r_\varepsilon , \Theta_\varepsilon , \vc{U}_\varepsilon \right) (\tau) \leq \chi(\varepsilon, \eta) +
c \int_0^\tau \mathcal{E}_\varepsilon \left( \varrhoe, \vartheta_\varepsilon, \vc{u}_\varepsilon \Big| r_\varepsilon , \Theta_\varepsilon , \vc{U}_\varepsilon \right) \ {\rm d} t ,
\end{equation}
which, performing the limit (i) for $\varepsilon \to 0$, and then (ii) $\eta \to 0$, yields the conclusion of Theorem \ref{Tm1}.
\def\cprime{$'$} \def\ocirc#1{\ifmmode\setbox0=\hbox{$#1$}\dimen0=\ht0
\advance\dimen0 by1pt\rlap{\hbox to\wd0{\hss\raise\dimen0
\hbox{\hskip.2em$\scriptscriptstyle\circ$}\hss}}#1\else {\accent"17 #1}\fi}
\end{document} | math | 75,406 |
\begin{document}
\nocite{*}
\title{Sphere theorems for $\RCD$ and stratified spaces}
\begin{abstract}
We prove topological sphere theorems for $\ensuremath{\mathbb R}CD(n-1, n)$ spaces which generalize Colding's results and Petersen's result to the $\ensuremath{\mathbb R}CD$ setting. We also get an improved sphere theorem in the case of Einstein stratified spaces.
\end{abstract}
\ensuremath{\mathbb S}ection*{Introduction}
In \cite{Colding} Colding proved that if a closed $n$-dimensional Riemannian manifold $(M^n, g)$ with $\mathrm{Ric}_{M^n}^g\ge n-1$ satisfies that the radius is close to $\pi$, then $M^n$ is homeomorphic to the standard $n$-dimensional unit sphere $\mathbb{S}^n$, where the \textit{radius} $\mathrm{rad}(X, d)$ of a metric space $(X, d)$ is defined by:
\begin{equation}
\mathrm{rad} (X, d) = \inf_{x \in X} \ensuremath{\mathbb S}up_{y \in X} d(x, y).
\end{equation}
Thanks to Cheeger-Colding's work \cite{CheegerColding1}, it is known that this homeomorphism can be improved to the diffeomorphism.
Our main results generalize the previous to a large class of singular spaces, the so-called \textit{$\ensuremath{\mathbb R}CD$ metric measure spaces}, or \textit{$\ensuremath{\mathbb R}CD$ spaces} for short, whose study is now quickly developping (see for instance \cite{Ambrosio} by Ambrosio for a survey).
Roughly speaking, a metric measure space $(X, d, m)$ is said to be $\ensuremath{\mathbb R}CD(K, N)$ if, in a generalized sense, the Ricci curvature is bounded below by $K$, the dimension is bounded above by $N$ and the space carries some Riemannian structure (we refer to the first section for a precise definition, see Definition \ref{def:rcd}). One of the typical examples can be found in \textit{weighted Riemannian manifolds} $(M^n, d_g, e^{-f}\mu_g)$, where $d_g$ denotes the distance defined by the Riemannian metric $g$, $\mu_g$ is the Riemannian volume measure and $f$ is a smooth function on $M^n$. In fact $(M^n, d_g, e^{-f}\mu_g)$ is a $\ensuremath{\mathbb R}CD(K, N)$ space if and only if $n\le N$ and
$$
\mathrm{Ric}_{M^n}^g +\mathrm{Hess}_f^g-\frac{df \otimes df}{N-n}\ge Kg
$$
hold. As easily noticed from this example, in the $\ensuremath{\mathbb R}CD$ theory, there is a \textit{flexibility} on the choice of reference measures even if the base metric space $(X, d)$ is fixed. In particular, the measure $m$ is not necessarily the Hausdorff measure associated to the distance.
Other examples of $\ensuremath{\mathbb R}CD(K,N)$ spaces are given by compact \emph{stratified spaces} $(X^n, d_g,\mu_g)$ endowed with the distance and measure associated to an iterated edge metric $g$, under the suitable assumptions on $g$. Stratified spaces are singular manifolds with \emph{iterated conical singularities}, isolated or not. When the metric $g$ has Ricci tensor bounded from below on the regular set and angles along the codimension 2 singular set are smaller than $2\pi$, $(X^n,d_g,\mu_g)$ is a $\ensuremath{\mathbb R}CD$ space, as proven in \cite{BKMR} by Bertrand-Ketterer-Richard and the second author. In this work, as a consequence of our main results, we will obtain a sphere theorem for \textit{Einstein} stratified spaces. It is worth pointing out that examples of Einstein stratified spaces occur in various branches of geometry: for instance in mathematical physics, the singular space associated to a static triple \cite{Ambrozio}; in Kähler geometry, Kähler-Einstein manifolds with edge singularities of cone angle smaller than $2\pi$ along a smooth divisor \cite{JMR}.
We are now in a position to state the main result of the paper:
\begin{thmA}[Topological sphere theorem for $\ensuremath{\mathbb R}CD$ spaces, I]\label{mthm}
For all $n \in \mathbb{N}_{\ge 2}$ there exists a positive constant $\ensuremath{\varepsilon}ilon_n>0$ such that if a compact metric space $(X, d)$ satisfies that $\mathrm{rad}(X, d) \ge \pi- \ensuremath{\varepsilon}ilon_n$ and that $(X, d, \mathfrak{m})$ is a $\ensuremath{\mathbb R}CD (n-1, n)$ space for some Borel measure $\mathfrak{m}$ on $X$ with full support, then $X$ is homeomorphic to the $n$-dimensional sphere.
\end{thmA}
This seems the first topological sphere theorem in the $\ensuremath{\mathbb R}CD$ theory. We emphasize again that the theorem states that although there is a flexibility on the choice of $\mathfrak{m}$, the topological structure is uniquely determined. Note that in the previous theorem one cannot replace the radius by the diameter of the space. Indeed, for any $\varepsilon > 0$ Anderson constructed in \cite{Anderson} manifolds of even dimension $n \geq 4$, with Ricci tensor bounded below by $n-1$ and diameter larger than $\pi - \ensuremath{\varepsilon}ilon$, which are \emph{not} homeomorphic to the sphere. Similar examples can be found in \cite{Otsu} by Otsu.
In order to introduce an application, let us recall a result of Petersen \cite{Petersen}; for a closed $n$-dimensional Riemannian manifold $(M^n ,g)$ with $\mathrm{Ric}_{M^n}^g\ge n-1$ the following two conditions are equivalent quantitatively:
\begin{enumerate}
\item The $(n+1)$-th eigenvalue of the Laplacian is close to $n$
\item The radius is close to $\pi$.
\end{enumerate}
In particular if the one of them above holds, then $M^n$ is diffeomorphic to $\mathbb{S}^n$.
Note that even in the $\ensuremath{\mathbb R}CD$-setting, the above equivalence is justified by the spectral convergence result of Gigli-Mondino-Savar\'e \cite{GigliMondinoSavare} and the rigidity results of Ketterer \cite{Ketterer2}. In particular we have the following;
\begin{corA}[Topological sphere theorem for $\ensuremath{\mathbb R}CD$ spaces, II]\label{thm:eigenhomeo}
For all $n \in \mathbb{N}_{\ge 2}$ there exists a positive constant $\ensuremath{\varepsilon}ilon_n>0$ such that if a $\ensuremath{\mathbb R}CD(n-1, n)$ space $(X, d, \mathfrak{m})$ satisfies
\begin{equation}\label{eq:maxeig}
\lambda_{n+1} \le n+\ensuremath{\varepsilon}ilon_n,
\end{equation}
then $X$ is homeomorphic to $\mathbb{S}^n$, where $\lambda_k:=\lambda_k(X, d, \mathfrak{m})$ denotes the $k$-th eigenvalue of the (minus) Laplacian $-\Delta$ on $(X, d, \mathfrak{m})$.
\end{corA}
In the second section, we give a proof for reader's convenience.
In Corollary \ref{thm:eigenhomeo} it is known that if $(X, d, \mathfrak{m})$ is a Riemannian manifold, that is, $(X, d, \mathfrak{m})$ is isometric to $(M^n, d_g, \mu_g)$ for a closed Riemannian manifold $(M^n, g)$, then the assumption (\ref{eq:maxeig}) can be replaced by a weaker one;
\begin{equation}
\lambda_n \le n+\ensuremath{\varepsilon}ilon_n.
\end{equation}
See \cite{Aubry} by Aubry (see also \cite{Bert} by Bertrand and \cite{Honda09} by the first author). However in the $\ensuremath{\mathbb R}CD$ setting we can not get such improvement. In fact, the $n$-dimensional unit hemisphere $\mathbb{S}^n_+$ with the standard Riemannian measure is a $\ensuremath{\mathbb R}CD(n-1, n)$ space with $\lambda_k=n$ for all $1\le k \le n$, but it is not homeomorphic to $\mathbb{S}^n$. Thus Corollary \ref{thm:eigenhomeo} is sharp in this sense.
Let us explain how to prove the main theorem.
For that, we recall the original proof by Colding. First, he proved that if the radius is close to $\pi$, then the volume is almost maximal. Perelman's topological sphere theorem \cite{Perelman} for almost maximal volume then allows Colding to conclude.
We follow a similar argument. However, the almost maximality of the volume does not make sense in the general setting of RCD spaces, because, as we pointed out above, there is flexibility in the choice of measures. In order to overcome this difficulty, the assumption in Theorem \ref{mthm} allows us to get rid of such flexibility of the measure $\mathfrak{m}$, in the following sense: we start by proving
\begin{equation}
\mathfrak{m}=\frac{\mathfrak{m}(X)}{\mathcal{H}^n(X)} \mathcal{H}^n,
\end{equation}
where $\mathcal{H}^n$ is the $n$-dimensional Hausdorff measure. This is justified by using a recent result of the first author \cite{Honda19} which confirms a conjecture by De Philippis-Gigli (see Remark 1.9 in \cite{DePhilippisGigli}) in the compact setting.
Then by combining this with a compactness result for non-collapsed $\ensuremath{\mathbb R}CD$ spaces by De Philippis-Gigli \cite{DePhilippisGigli} and Ketterer's rigidity \cite{Ketterer}, we can show that our situation is reduced to the study of the following measured Gromov-Hausdorff convergent sequence of $\ensuremath{\mathbb R}CD(n-1, n)$ spaces:
$$
(X_i, d_i, \mathcal{H}^n) \ensuremath{\mathbb S}tackrel{mGH}{\to} (\mathbb{S}^n, d_{\mathbb{S}^n}, \mathcal{H}^n).
$$
Then we can follow an argument similar to Colding's proof by using the intrinsic Reifenberg theorem \cite{CheegerColding1} by Cheeger-Colding instead of using Perelman's topological sphere theorem \cite{Perelman}.
One step in the previous proof consists in showing that the almost maximality of the Hausdorff measure implies that the space is homeomorphic to the sphere (see Theorem \ref{prop:toprigidity} in the following). The same assumption in the case of an \emph{Einstein} stratified space actually allows us to get a stronger sphere theorem, independently of the $\ensuremath{\mathbb R}CD$ theory:
\begin{corA}[Sphere theorem for Einstein stratified spaces]
For all $n \in \mathbb{N}_{\ge 2}$ there exists a positive constant $\ensuremath{\varepsilon}ilon_n>0$ such that the following holds. Let $(X, g)$ be compact $n$-dimensional stratified space endowed with an iterated edge metric $g$ such that $\mathrm{Ric}_g \equiv n-1$ on the regular set. Assume that there is no singular stratum of codimension 2. If $\mu_g(X) \ge (1-\ensuremath{\varepsilon}ilon_n)\mathcal{H}^n(\mathbb{S}^n)$, then $(X, d_g)$ is isometric to $(\mathbb{S}^n, d_{\mathbb{S}^n})$.
\end{corA}
The assumption on the codimension 2 stratum cannot be dropped. Indeed, consider a compact Riemannian surface $(X,g)$ with sectional curvature equal to one away from a finite number of isolated conical singularities of angles smaller than $2\pi$. Thanks to \cite{BKMR}, such surface is a $\ensuremath{\mathbb R}CD(1,2)$ space, and therefore the almost maximality of its volume implies that $(X,g)$ is homeomorphic to $\ensuremath{\mathbb S}^2$ and that the angles at the singularities are close to $2\pi$. Nevertheless, this cannot be improved to an isometry. The strategy of our proof actually consists in showing that there cannot be any singularity of codimension strictly greater than 2. Moreover, a local almost maximality for the volume allows one to control the regularity in a neighbourhood of a point (see Corollary 3.5), even if the space is not compact. Both of these proofs do not depend on the space being a $\ensuremath{\mathbb R}CD$ space.
The paper is organized as follows: in the next section we give a quick introduction about $\ensuremath{\mathbb R}CD$ spaces and recall related results we need later.
In Section $2$, after preparing few technical results, we prove the main results. The last section is devoted to showing Corollary C in the case of Einstein stratified spaces, for which we state the basic notions that we need.
After we finalized this article we learned that the paper \cite{KM} by Kapovitch-Mondino contains the same results as in Corollary \ref{thm:eigenhomeo} and Theorem \ref{prop:toprigidity} as an independent work. Their proofs are close to ours, but the scopes of our works differ: while they are mainly concerned with the topology and the boundary of non-collapsed $\ensuremath{\mathbb R}CD$ spaces, our work also intends to discuss sphere theorems in the more specific setting of stratified spaces.
\ensuremath{\mathbb S}mallskip\noindent
\textbf{Acknowledgement.}
The authors would like to thank Takumi Yokota for informing the smooth version of Theorem \ref{thm:strati} and Alexander Lytchak for his comments concerning Alexandrov spaces. The second author would like to thank Erwann Aubry for useful discussions. The first author acknowledges supports of the Grantin-Aid
for Young Scientists (B) 16K17585 and Grant-in-Aid for Scientific Research (B) of 18H01118.
\ensuremath{\mathbb S}ection{Preliminary}
We say that a triple $(X, d, \mathfrak{m})$ is a \textit{metric measure space} if $(X, d)$ is a complete separable metric space and $\mathfrak{m}$ is a Borel measure on $X$ with full support, that is, $\mathfrak{m} (B_r(x))>0$ holds for all $x \in X$ and all $r>0$, where $B_r(x)$ denotes the open ball centered at $x$ of radius $r$.
Throughout this section we fix $K \in \mathbb{R}$, $N \in (1, \infty)$ and $n \in \mathbb{N}_{\ge 2}$.
\ensuremath{\mathbb S}ubsection{General $\ensuremath{\mathbb R}CD$ space}
Let us fix a metric measure space $(X, d, \mathfrak{m})$.
The goal of this section is to give a quick introduction on $\ensuremath{\mathbb R}CD$ spaces with their fundamental properties.
The Cheeger energy
$\mathrm{Ch}=:L^2(X,\mathfrak{m})\to [0,+\infty]$ is a convex and $L^2(X,\mathfrak{m})$-lower semicontinuous functional defined as follows:
\begin{equation}\label{eq:defchp}
\mathrm{Ch}(f):=\inf\left\{\liminf_{n\to\infty}\frac{1}{2}\int_X(\mathrm{Lip} f_n)^2d\mathfrak{m}:\ \text{$f_n\in\mathrm{Lip}_b (X,d)\cap L^2(X, \mathfrak{m})$, $\|f_n-f\|_{L^2}\to 0$}\right\},
\end{equation}
where $\mathrm{Lip} f$ denotes the local Lipschitz constant and $\mathrm{Lip}_b(X,d)$ is the
space of bounded Lipschitz functions.
The Sobolev space $H^{1,2}(X,d,\mathfrak{m})$ then coincides with the domain of the Cheeger energy, that is $\{f\in L^2(X,m):\ \mathrm{Ch}(f)<+\infty\}$. When endowed with the norm
$$
\|f\|_{H^{1,2}}:=\left(\|f\|_{L^2(X,\mathfrak{m})}^2+2\mathrm{Ch}(f)\right)^{1/2}
$$
this space is Banach and
separable Hilbert if $\mathrm{Ch}$ is a quadratic form (see \cite{AmbrosioGigliSavare14}).
According to the terminology introduced in \cite{Gigli1}, we say that a
metric measure space $(X,d,\mathfrak{m})$ is \textit{infinitesimally Hilbertian} if $\mathrm{Ch}$ is a quadratic form.
By looking at minimal relaxed slopes and by a polarization procedure, one can then define a {\it carr\'e du champ}
$$
\Gamma:H^{1,2}(X,d,\mathfrak{m})\times H^{1,2}(X,d,\mathfrak{m})\rightarrow L^1(X,\mathfrak{m})
$$
playing in this abstract theory the role of the scalar product between gradients. In infinitesimally Hilbertian metric measure
spaces, the $\Gamma$ operator
satisfies all natural symmetry, bilinearity, locality and chain rule properties, and provides integral representation to $\mathrm{Ch}$:
$$ 2\mathrm{Ch}(f)=\int_X \Gamma(f,f)\,d\mathfrak{m},
$$
for all $f\in H^{1,2}(X,d,\mathfrak{m})$.
We can now define a densely
defined operator $\Delta:D(\Delta)\to L^2(X,\mathfrak{m})$ whose domain consists of all functions $f\in H^{1,2}(X,d,\mathfrak{m})$
satisfying
$$
\int_X hgd\mathfrak{m}=-\int_X \Gamma(f,h)d\mathfrak{m}\quad\qquad\forall h\in H^{1,2}(X,d,\mathfrak{m})
$$
for some $g\in L^2(X,\mathfrak{m})$. The unique $g$ with this property is then denoted by $\Delta f$ (see \cite{AmbrosioGigliSavare13}).
We are now in a position to introduce the definition of $\ensuremath{\mathbb R}CD$ spaces.
\begin{definition}[$\ensuremath{\mathbb R}CD$ spaces]\label{def:rcd}
For $K \in \mathbb{R}$ and $\hat{N} \in [1, \infty]$, $(X, d, \mathfrak{m})$ is said to be a \textit{$\ensuremath{\mathbb R}CD(K, \hat{N})$ space} if the following are satisfied:
\begin{enumerate}
\item Infinitesimally Hilbertian: it is inifinitesimally Hilbertian;
\item Volume growth: there exist $x \in X$ and $c>1$ such that $\mathfrak{m} (B_r(x)) \le Ce^{Cr^2}$ for all $r>0$;
\item Sobolev-to-Lipschitz property : any
$f\in H^{1,2}(X,d,\mathfrak{m})$ with $\Gamma (f, f) \leq 1$ $\mathfrak{m}$-a.e. in $X$
has a $1$-Lipschitz representative.
\item Bakry-\'Emery inequality : for all $f\in D(\Delta)$ with $\Delta f\in H^{1,2}(X,d,\mathfrak{m})$,
\begin{equation}\label{eq:boch}
\frac{1}{2}\int_X \Gamma (f, f)\Delta \phi d\mathfrak{m}\ge
\int_X\phi\left(\frac{(\Delta f)^2}{\hat N}+ \Gamma (f, \Delta f)+ K\Gamma (f, f)\right)d\mathfrak{m}
\end{equation}
for all $\phi\in D(\Delta) \cap L^{\infty}(X, \mathfrak{m})$ with $\phi\geq 0$ and $\Delta\phi\in L^\infty(X,\mathfrak{m})$.
\end{enumerate}
\end{definition}
It is worth pointing out that a $\ensuremath{\mathbb R}CD(K,N)$ space was originally defined as a metric measure space which is infinitesimally Hilbertian and satisfies the $\ensuremath{\mathbb C}D(K,N)$ condition in the sense of Lott-Sturm-Villani (see
\cite{AmbrosioGigliSavare14, AmbrosioMondinoSavare, ErbarKuwadaSturm, Gigli1, LottVillani, Sturm1, Sturm2}). Such definition has been proven to be equivalent to the formulation given by Definition \ref{def:rcd} (see also \cite{CavMil} for the equivalence between $\ensuremath{\mathbb R}CD$ and $\ensuremath{\mathbb R}CD^*$).
In order to keep our presentation short, we skip the definitions of \textit{pointed measured Gromov-Hausdorff convergence} (pmGH), of \textit{measured Gromov-Hausdorff convergence} (mGH), and of \textit{pointed Gromov-Hausdorff convergence} (pGH). We refer to \cite{CheegerColding1, Fukaya, GigliMondinoSavare, Sturm1, Sturm2} for the precise definitions. Note that the radius is continuous with respect to the Gromov-Hausdorff convergence.
Let us introduce a compactness result for $\ensuremath{\mathbb R}CD$ spaces with respect to the pmGH convergence, which follows from \cite[Cor.3.22, Thm.7.2]{GigliMondinoSavare}.
\begin{theorem}[Compactness of $\ensuremath{\mathbb R}CD$ spaces]\label{thm:comprcd}
Let $(X_i, d_i, \mathfrak{m}_i, x_i) (i=1, 2, \ldots)$ be a sequence of pointed $\ensuremath{\mathbb R}CD(K, N)$ spaces. If there exist $v_i >0(i=1, 2)$ with $v_1\le v_2$ such that $v_1\le \mathfrak{m}_i(B_1(x_i))\le v_2$ holds for all $i$, then there exist a subsequence $(X_{i(j)}, d_{i(j)}, \mathfrak{m}_{i(j)}, x_{i(j)})$ and a pointed $\ensuremath{\mathbb R}CD (K, N)$ space $(X, d, \mathfrak{m}, x)$ such that
$$(X_{i(j)}, d_{i(j)}, \mathfrak{m}_{i(j)}, x_{i(j)}) \ensuremath{\mathbb S}tackrel{pmGH}{\to} (X, d, \mathfrak{m}, x),$$
that is, $(X_{i(j)}, d_{i(j)}, \mathfrak{m}_{i(j)}, x_{i(j)})$ pmGH converge to $(X, d, \mathfrak{m}, x)$.
\end{theorem}
The next definition is a key notion of the paper. Inspired by the stratification of Ricci limit spaces given by Cheeger-Colding theory, one can define a $k$-regular set as the set of points for which the tangent cone, in the sense of pmGH convergence, is the Euclidean space $\ensuremath{\mathbb R}^k$. More precisely we have:
\begin{definition}[Regular set]
Let $(X, d, \mathfrak{m})$ be a $\ensuremath{\mathbb R}CD (K, N)$ space and let $k \in \mathbb{N}$.
Then the \textit{$k$-dimensional regular set $\mathcal{R}_k=\mathcal{R}_k(X)$} is defined by the set of all points $x \in X$ satisfying that
\begin{equation}
(X, r^{-1}d, (\mathfrak{m} (B_r(x)))^{-1}\mathfrak{m}, x) \ensuremath{\mathbb S}tackrel{pmGH}{\to} (\mathbb{R}^k, d_{\mathbb{R}^k}, \omega_k^{-1}\mathcal{L}^k, 0_k) \quad (r \to 0^+).
\end{equation}
\end{definition}
By the Bishop-Gromov inequality (see \cite{LottVillani, Sturm1, Sturm2, Villani}) it is easy to check that $\mathcal{R}_k=\emptyset$ for all $k \in (N, \infty) \cap \mathbb{N}$.
\begin{remark}
It is proved that for $x \in X$ and $k \in \mathbb{N}$, if $(X, r^{-1}d, x) \ensuremath{\mathbb S}tackrel{pGH}{\to} (\mathbb{R}^k, d_{\mathbb{R}^k}, 0_k)$, then $x \in \mathcal{R}_k$. The proof is same as the one of \cite[Prop.1.35]{CheegerColding1}. Thus the $k$-dimensional regular set is a purely metric notion in this sense.
\end{remark}
The following theorem is proved in \cite[Thm.0.1]{BrueSemola} (after \cite[Cor.1.2]{MondinoNaber}). It generalizes a result of \cite[Thm.1.18]{ColdingNaber} to $\ensuremath{\mathbb R}CD$ spaces and allows one to define a unique \emph{essential dimension} for a $\ensuremath{\mathbb R}CD$ space.
\begin{theorem}[Essential dimension]
Let $(X, d, \mathfrak{m})$ be a $\ensuremath{\mathbb R}CD (K, N)$ space. Assume that $X$ is not a single point.
Then there exists a unique $k:=\mathrm{dim}_{d, \mathfrak{m}}(X) \in \mathbb{N} \cap [1, N]$ such that $\mathfrak{m} (X \ensuremath{\mathbb S}etminus \mathcal{R}_k)=0$. We call it the \textit{essential dimension} of $(X, d, \mathfrak{m})$.
\end{theorem}
We end this subsection by introducing a fundamental property on the essential dimension proved in \cite[Thm.1.5]{Kita};
\begin{theorem}[Lower semicontinuity of essential dimensions]\label{thm:lower}
The essential dimension is lower semicontinuous with respect to the pointed measured Gromov-Hausdorff convergence of $\ensuremath{\mathbb R}CD(K, N)$ spaces.
\end{theorem}
\ensuremath{\mathbb S}ubsection{Rigidity for positively Ricci curved $\ensuremath{\mathbb R}CD$ spaces}
It is worth pointing out that in general if a $\ensuremath{\mathbb R}CD(K ,N)$ space $(X, d, \mathfrak{m})$ has a bounded diameter, then $(X, d)$ must be compact and the spectrum of the (minus) Laplacian $-\Delta$ is discrete and unbounded;
\begin{equation}
0=\lambda_0<\lambda_1 \le \lambda_2 \le \cdots \to \infty,
\end{equation}
where $\lambda_i$ denotes the $i$-th eigenvalue counted with multiplicities (see for instance \cite{GigliMondinoSavare} or \cite{AmbrosioHondaTewodrosePortegies}).
Let us introduce some properties of $\ensuremath{\mathbb R}CD(N-1, N)$ spaces with the rigidity results we will use later (\cite[Cor.1.3, Cor.1.6]{Ketterer}, \cite[Thm.1.4]{Ketterer2}).
\begin{theorem}[Rigidity to the sphere]\label{thm:ket}
Let $(X, d, \mathfrak{m})$ be a $\ensuremath{\mathbb R}CD(N-1, N)$ space. Then the following are satisfied:
\begin{enumerate}
\item The diameter $\mathrm{diam}(X, d)$ is at most $\pi$ (in particular $\mathrm{rad}(X, d) \le \pi$) and the first positive eigenvalue $\lambda_1$ is at least $N$.
\item If $\mathrm{rad}(X, d)=\pi$ or $\lambda_{[N]+1}=N$, where $[N]$ is the integer part of $N$, then $N$ must be an integer, $(X, d)$ is isometric to $(\mathbb{S}^N, d_{\mathbb{S}^N})$ and $\mathfrak{m}=a \mathcal{H}^N$ for some $a>0$.
\end{enumerate}
\end{theorem}
The next two corollaries give us reasons for only discussing the case of integer $n$ in Theorem \ref{mthm} and Corollary \ref{thm:eigenhomeo}.
\begin{cor}
For all $N \in [1, \infty) \ensuremath{\mathbb S}etminus \mathbb{N}$, there exists a positive constant $\tau_N>0$ such that any $\ensuremath{\mathbb R}CD(N-1, N)$ space $(X, d, \mathfrak{m})$ satisfies
$\lambda_{[N]+1} \geq N + \tau_N$.
\end{cor}
\begin{proof}
The proof is done by a contradiction. If the statement is not satisfied, then there exists a sequence of $\ensuremath{\mathbb R}CD(N-1, N)$ spaces $(X_i, d_i, \mathfrak{m}_i)$ such that
\begin{equation}\label{eq:gap}
\lim_{i \to \infty}\lambda_{[N]+1}(X_i, d_i, \mathfrak{m}_i)=N.
\end{equation}
By Theorem \ref{thm:comprcd} with no loss of generality we can assume that $(X_i, d_i, \mathfrak{m}_i)$ mGH-converge to a $\ensuremath{\mathbb R}CD(N-1, N)$ space $(X, d, \mathfrak{m})$. Then the spectral convergence result proved in \cite[Thm.7.8]{GigliMondinoSavare} with (\ref{eq:gap}) yields $$\lambda_{[N]+1}(X, d, \mathfrak{m})=\lim_{i \to \infty}\lambda_{[N]+1}(X_i, d_i, \mathfrak{m}_i)=N.$$ Theorem \ref{thm:ket} shows that $N$ must be an integer, which is a contradiction.
\end{proof}
Similarly we have the following.
\begin{cor}
For all $N \in [1, \infty) \ensuremath{\mathbb S}etminus \mathbb{N}$, there exists a positive constant $\delta_N>0$ such that any $\ensuremath{\mathbb R}CD(N-1, N)$ space $(X, d, \mathfrak{m})$ satisfies
$\mathrm{rad}(X, d) \leq \pi - \delta_N$.
\end{cor}
\ensuremath{\mathbb S}ubsection{Non-collapsed $\ensuremath{\mathbb R}CD$ space}
Let us introduce a special class of $\ensuremath{\mathbb R}CD$ spaces introduced in \cite{DePhilippisGigli}, that is non-collapsed and weakly non-collapsed $\ensuremath{\mathbb R}CD$ spaces. The non-collapsing assumption means that the measure $\mathfrak{m}$ is chosen to coincide, or to be absolutely continuous, with respect to the Hausdorff measure. More precisely we have:
\begin{definition}[Non-collapsed $\ensuremath{\mathbb R}CD$ space]
Let $(X, d, \mathfrak{m})$ be a $\ensuremath{\mathbb R}CD(K, N)$ space.
\begin{enumerate}
\item $(X, d, \mathfrak{m})$ is called \textit{non-collapsed} if $\mathfrak{m}=\mathcal{H}^N$.
\item $(X, d, \mathfrak{m})$ is called \textit{weakly non-collapsed} if $\mathfrak{m} \ll \mathcal{H}^N$.
\end{enumerate}
\end{definition}
The following fundamental results for non-collapsed $\ensuremath{\mathbb R}CD(K, N)$ spaces are proved in \cite{DePhilippisGigli};
\begin{theorem}[Fine properties of non-collapsed $\ensuremath{\mathbb R}CD$ spaces]\label{thm:noncol}
Let $(X, d, \mathcal{H}^N)$ be a non-collapsed $\ensuremath{\mathbb R}CD(K, N)$ space. Then the following are satisfied;
\begin{enumerate}
\item $N$ must be an integer;
\item For all $x \in X$ the Bishop inequality holds in the sense of
\begin{equation}\label{eq:bishop0}
\lim_{r \to 0^+}\frac{\mathcal{H}^N(B_r(x))}{\mathrm{Vol}_{K, N}(r)}\le 1,
\end{equation}
where $\mathrm{Vol}_{K, N}(r)$ denotes the volume of a ball of radius $r$ in the $N$-dimensional space form whose Ricci curvature is constant equal to $K$. Moreover the equality in (\ref{eq:bishop0}) holds if and only if $x \in \mathcal{R}_N$.
\end{enumerate}
\end{theorem}
Let us give several remarks on the theorem above. The first property (1) is also true for weakly non-collapsed $\ensuremath{\mathbb R}CD(K, N)$ spaces.
The second property (2) can be regarded as a rigidity result. Moreover it is proven that the \textit{almost} rigidity of (\ref{eq:bishop0}) also holds, and this will play a role in the next section. See \cite[Thm.1.6]{DePhilippisGigli} for the precise statement (see also \cite[Prop.6.6]{AmbrosioHondaTewodrosePortegies}).
The next theorem follows from a combination of \cite{Kita} and \cite{DePhilippisGigli}.
For the reader's convenience we give a proof:
\begin{theorem}\label{thm:topdim}
For $n \in \mathbb{N}_{\ge 2}$, a $\ensuremath{\mathbb R}CD(K, n)$ space $(X, d, \mathfrak{m})$ is weakly non-collapsed if and only if $\mathcal{R}_n \neq \emptyset$.
\end{theorem}
\begin{proof}
We check only the ``if'' part because the ``only if'' part is a direct consequence of \cite[Thm.1.10]{DePhilippisGigli}.
Let $x \in \mathbb{R}_n$. Then since Theorem \ref{thm:lower} yields
$$
\liminf_{r \to 0^+}\mathrm{dim}_{ r^{-1}d, (\mathfrak{m} (B_r(x)))^{-1}\mathfrak{m}}(X) \ge \mathrm{dim}_{d_{\mathbb{R}^n}, \omega_n^{-1}\mathcal{L}^n}(\mathbb{R}^n)=n,
$$
we have $\mathrm{dim}_{d, \mathfrak{m}}(X)=n$ because $\mathrm{dim}_{d, \mathfrak{m}}(X)=\mathrm{dim}_{ r^{-1}d, (\mathfrak{m} (B_r(x)))^{-1}\mathfrak{m}}(X)$ for all $r>0$.
Then \cite[Thm.1.10]{DePhilippisGigli} yields that $(X, d, \mathfrak{m})$ is weakly non-collapsed.
\end{proof}
Let us introduce one of the main results of \cite{Honda19} which confirmed a conjecture raised in \cite{DePhilippisGigli} in the compact case (\cite[Cor.1.4]{Honda19});
\begin{theorem}[``Weakly non-collapsed'' implies ``non-collapsed'']\label{thm:weak}
Let $(X, d, \mathfrak{m})$ be a compact weakly non-collapsed $\ensuremath{\mathbb R}CD(K, n)$ space. Then
$$
\mathfrak{m} =\frac{\mathfrak{m} (X)}{\mathcal{H}^n(X)}\mathcal{H}^n.
$$
\end{theorem}
To conclude this section, we introduce a compactness result for non-collapsed $\ensuremath{\mathbb R}CD$ spaces with respect to pmGH convergence, which is proved in \cite[Thm.1.2, Thm.1.3]{DePhilippisGigli} (Compare with Theorem \ref{thm:comprcd});
\begin{theorem}[Compactness of non-collapsed $\ensuremath{\mathbb R}CD$ spaces]\label{thm:compnon}
Let $(X_i, d_i, \mathcal{H}^n, x_i) (i=1, 2, \ldots)$ be a sequence of pointed non-collapsed $\ensuremath{\mathbb R}CD(K, n)$ spaces with $$\liminf_{i \to \infty}\mathcal{H}^n(B_1(x_i))>0.$$ Then there exist a subsequence $(X_{i(j)}, d_{i(j)}, \mathcal{H}^n, x_{i(j)})$ and a pointed non-collapsed $\ensuremath{\mathbb R}CD (K, n)$ space $(X, d, \mathcal{H}^n, x)$ such that
$$(X_{i(j)}, d_{i(j)}, \mathcal{H}^n, x_{i(j)}) \ensuremath{\mathbb S}tackrel{pmGH}{\to} (X, d, \mathcal{H}^n, x).$$
\end{theorem}
\ensuremath{\mathbb S}ection{Proof of main results}
Throughout the section we fix $n \in \mathbb{N}_{\ge 2}$ and $K \in \mathbb{R}$ too. In the next proposition let us consider Sturm's $\mathbb{D}$-distance between compact $\ensuremath{\mathbb R}CD(K, n)$ spaces, that is, ``$\ensuremath{\varepsilon}ilon$-mGH-close'' means ``$\mathbb{D}<\ensuremath{\varepsilon}ilon$'' below. Note that the convergence with respect to $\mathbb{D}$ is equivalent to the mGH convergence for compact $\ensuremath{\mathbb R}CD(K, N)$ spaces (see \cite{Sturm1, Sturm2}).
\begin{proposition}[Noncollapsed $\ensuremath{\mathbb R}CD$ is an open condition]\label{prop:open}
Let $(X, d_X, \mathfrak{m}_X)$ be a compact weakly non-collapsed $\ensuremath{\mathbb R}CD(K, n)$ space. Then there exists a positive constant $\ensuremath{\varepsilon}ilon_0=\ensuremath{\varepsilon}ilon_0(X, d_X, \mathfrak{m}_X)>0$ such that if a compact $\ensuremath{\mathbb R}CD(K, n)$ space $(Y, d_Y, \mathfrak{m}_Y)$ is $\ensuremath{\varepsilon}ilon_0$-mGH-close to $(X, d_X, \mathfrak{m}_X)$, then
\begin{equation}\label{eq:measrigid}
\mathfrak{m}_Y=\frac{\mathfrak{m}_Y(Y)}{\mathcal{H}^n(Y)}\mathcal{H}^n.
\end{equation}
\end{proposition}
\begin{proof}
Let us prove the proposition by contradiction. If the statement does not hold, then there exists a mGH convergent sequence $(X_i, d_i, \mathfrak{m}_i)$ of $\ensuremath{\mathbb R}CD(K, n)$ spaces to $(X, d_X, \mathfrak{m}_X)$ such that
\begin{equation}\label{eq:contr}
\mathfrak{m}_{X_i}\neq \frac{\mathfrak{m}_i(X_i)}{\mathcal{H}^n(X_i)}\mathcal{H}^n.
\end{equation}
However since Theorem \ref{thm:lower} states
$$
\liminf_{i \to \infty}\mathrm{dim}_{d_i, \mathfrak{m}_i}(X_i) \ge \mathrm{dim}_{d_X, \mathfrak{m}_X}(X)=n,
$$
we see that $\mathrm{dim}_{d_i, \mathfrak{m}_i}(X_i)=n$ for any sufficiently large $i$.
In particular Theorem \ref{thm:topdim} yields that $(X_i, d_i, \mathfrak{m}_i)$ is weakly non-collapsed $\ensuremath{\mathbb R}CD(K, n)$ space for such $i$. Then by applying Theorem \ref{thm:weak} to such $(X_i, d_i, \mathfrak{m}_i)$ we obtain that $\mathfrak{m}_i=\frac{\mathfrak{m}_i(X_i)}{\mathcal{H}^n(X_i)}\mathcal{H}^n$ which contradicts (\ref{eq:contr}).
\end{proof}
Let us remark that the Bishop inequality (\ref{eq:bishop0}) implies
\begin{equation}\label{eq:bishop}
\mathcal{H}^n(X)\le \mathcal{H}^n(\mathbb{S}^n)
\end{equation}
for all non-collapsed $\ensuremath{\mathbb R}CD(n-1, n)$ space $(X, d, \mathcal{H}^n)$. Since the equality easily implies $\mathrm{rad} (X, d)=\pi$, thus by Theorem (\ref{thm:ket}) the equality holds if and only if the space is isometric to the round sphere $\mathbb{S}^n$.
\begin{theorem}[Topological sphere theorem for $\ensuremath{\mathbb R}CD$ spaces, III]\label{prop:toprigidity}
There exists a positive constant $\ensuremath{\varepsilon}ilon_n>0$ such that if a compact non-collapsed $\ensuremath{\mathbb R}CD(n-1, n)$ space $(X, d, \mathcal{H}^n)$ satisfies $\mathcal{H}^n(X) \ge (1- \ensuremath{\varepsilon}ilon_n)\mathcal{H}^n(\mathbb{S}^n)$, then $X$ is homeomorphic to $\mathbb{S}^n$.
\end{theorem}
\begin{remark}
We point out that the previous result is known for Alexandrov spaces, see for example Proposition A.9 in the work of Yamaguchi \cite{Yamaguchi}. It can also be easily proved by contradiction, by combining the rigidity of Bishop-Gromov inequality and the topological stability theorem \cite{Per} for Alexandrov spaces (see \cite{KV07}). Moreover for $n=2$, the work \cite{LS18} showed that a non-collapsed $\ensuremath{\mathbb R}CD(K,2)$ space is an Alexandrov space with curvature at least $K$. As a consequence, our result directly holds in dimension $2$. We give here the proof in full generality.
\end{remark}
\begin{proof}
The proof is done by contradiction. Assume that the theorem does not hold, then there exist sequences $\ensuremath{\varepsilon}ilon_i \rightarrow 0$ and $(X_i, d_i, \mathcal{H}^n)$ of $\ensuremath{\mathbb R}CD(n-1, n)$ spaces such that $\mathcal{H}^n(X_i)\geq (1-\ensuremath{\varepsilon}ilon_i)\mathcal{H}^n(\ensuremath{\mathbb S}^n)$ and $X_i$ is not homeomorphic to $\mathbb{S}^n$. Then we have
\begin{equation}\label{eq:volconv}
\lim_{i \to \infty}\mathcal{H}^n(X_i)=\mathcal{H}^n(\mathbb{S}^n).
\end{equation}
and it is not difficult to check that (\ref{eq:volconv}) implies that $\mathrm{rad}(X_i, d_i) \to \pi$. Applying Theorem \ref{thm:ket} with Theorem \ref{thm:compnon} yields
\begin{equation}
(X_i, d_i, \mathcal{H}^n) \ensuremath{\mathbb S}tackrel{mGH}{\to} (\mathbb{S}^n, d_{\mathbb{S}^n}, \mathcal{H}^n).
\end{equation}
On the other hand the inequality
\begin{equation}\label{eq:max}
\frac{\mathcal{H}^n(X_i)}{\mathcal{H}^n(\mathbb{S}^n)} \ge 1- \ensuremath{\varepsilon}ilon_i,
\end{equation}
together with the Bishop (\ref{eq:bishop0}) and the Bishop-Gromov inequalities implies that for all $x \in X_i$ we have
\begin{equation}\label{eq:maxball}
\frac{\mathcal{H}^n(B_r(x))}{\mathcal{H}^n(B_r(p))}\ge 1- \ensuremath{\varepsilon}ilon_i, \quad \forall r \in (0, \pi],\, \forall p \in \mathbb{S}^n.
\end{equation}
Applying this observation with the almost rigidity on the Bishop inequality \cite[Thm.1.5]{DePhilippisGigli} (see also \cite[Prop.6.6]{AmbrosioHondaTewodrosePortegies}) implies that for all $\ensuremath{\varepsilon} >0$ there exists a positive integer $i_0 \in \ensuremath{\mathbb N}$ such that for all $i \geq i_0$ and $r \in (0,\pi]$
\begin{equation}
d_{GH}(B_{r/2}(x_i),B_{r/2}(0^n))\leq \ensuremath{\varepsilon} r,
\end{equation}
where $d_{GH}$ denotes the Gromov-Hausdorff distance. This means that for all $i \geq i_0$, the metric spaces $\{(X_i, d_i)\}_i$ are uniformly Reifenberg flat. Then applying the intrinsic Reifenberg theorem \cite[Thm.A.1.2 and Thm.A.1.3]{CheegerColding1} yields that $X_i$ is homeomorphic to $\mathbb{S}^n$ for any sufficiently large $i$, which is a contradiction.
\end{proof}
We are in position to prove our main result.
\begin{proof}[Proof of Theorem \ref{mthm}]
The proof is done by a contradiction. Assume that the statement is false: then there exists a sequence $(X_i, d_i, \mathfrak{m}_i)$ of $\ensuremath{\mathbb R}CD(n-1, n)$ spaces such that $\mathrm{rad}(X_i, d_i) \to \pi$, $\mathfrak{m}_i(X_i)=1$ and $X_i$ is not homeomorphic to $\mathbb{S}^n$. By Theorem \ref{thm:ket} with no loss generality we can assume that
$$
(X_i, d_i, \mathfrak{m}_i) \ensuremath{\mathbb S}tackrel{mGH}{\to} (\mathbb{S}^n, d_{\mathbb{S}^n}, \mathfrak{m})
$$
for some Borel probability measure $\mathfrak{m}$ on $\mathbb{S}^n$.
Then Proposition \ref{prop:open} shows that $(X_i, d_i, \mathcal{H}^n) $ is also a $\ensuremath{\mathbb R}CD(n-1, n)$ space for any sufficiently large $i$. Moreover,
$$
(X_i, d_i, \mathcal{H}^n) \ensuremath{\mathbb S}tackrel{mGH}{\to} (\mathbb{S}^n, d_{\mathbb{S}^n}, \mathcal{H}^n).
$$
Therefore for any sufficiently large $i$ we obtain $\mathcal{H}^n(X)\geq (1-\ensuremath{\varepsilon}ilon_n)\mathcal{H}^n(\ensuremath{\mathbb S}^n)$ and Theorem \ref{prop:toprigidity} implies that $X_i$ is homeomorphic to $\mathbb{S}^n$, which is a contradiction.
\end{proof}
We can now prove Corollary \ref{thm:eigenhomeo};
\begin{proof}[Proof of Corollary \ref{thm:eigenhomeo}]
The proof follows the same lines as Theorem \ref{mthm}. Assume that the statement does not hold, then there exists a sequence $(X_i, d_i, \mathfrak{m}_i)$ of $\ensuremath{\mathbb R}CD(n-1, n)$ spaces with $\mathfrak{m}_i(X_i)=1$, such that $X_i$ is not homeomorphic to $\mathbb{S}^n$ and
$$
\lim_{i \to \infty}\lambda_{n+1}(X_i, d_i, \mathfrak{m}_i)= n.
$$
With no loss of generality we can assume that the sequence $(X_i, d_i, \mathfrak{m}_i)$ mGH-converges to a $\ensuremath{\mathbb R}CD(n-1, n)$ space $(X, d, \mathfrak{m})$.
Then the spectral convergence result proved in \cite[Thm.7.8]{GigliMondinoSavare} shows that
$$
\lambda_{n+1}(X, d, \mathfrak{m})=\lim_{i \to \infty}\lambda_{n+1}(X_i, d_i, \mathfrak{m}_i)=n.
$$
Then Theorem \ref{thm:ket} yields that $(X, d)$ is isometric to $(\mathbb{S}^n, d_{\mathbb{S}^n})$. In particular since $\mathrm{rad}(X_i, d_i) \to \mathrm{rad}(\mathbb{S}^n, d_{\mathbb{S}^n})=\pi$, Theorem \ref{mthm} yields that $X_i$ is homeomorphic to $\mathbb{S}^n$ for any sufficiently large $i$, which is a contradiction.
\end{proof}
\begin{remark}
Thanks to the intrinsic Reifenberg theorem \cite[Thm.A.1.2 and Thm.A.1.3]{CheegerColding1}, it is easy to check that all topological sphere theorems stated above can be improved to ``bi-H\"older homeomorphism''. Moreover we can choose \textit{any} $\alpha \in (0, 1)$ as a H\"older exponent. For reader's convenience let us write down the precise statement.
For all $n \in \mathbb{N}$, $r>0$ and $\ensuremath{\varepsilon}ilon >0$, let us denote by $\mathcal{M}(n, r, \ensuremath{\varepsilon}ilon)$ the set of all isometry classes of compact metric spaces $(X, d)$ with $d_{GH}(B_t(x), B_t(0_n))\le \ensuremath{\varepsilon}ilon t$ for all $x \in X$ and all $t\le r$.
Then we have:
\begin{itemize}
\item for all $\alpha \in (0, 1)$ there exist positive constants $\ensuremath{\varepsilon}ilon_0:=\ensuremath{\varepsilon}ilon_0(n ,\alpha, r)>0$ and $\delta_0(n, \alpha, r)>0$ such that if two compact metric space $Z_i \in \mathcal{M}(n, r, \ensuremath{\varepsilon}ilon_0)$ satisfies $d_{GH}(Z_1, Z_2)<\delta_0$, then there exists a homeomorphism $\Phi:Z_1 \to Z_2$ such that $\Phi$ and $\Phi^{-1}$ are $\alpha$-H\"older continuous maps.
\end{itemize}
Although we used the almost maximality of the volume (\ref{eq:max}) in the proof of Theorem \ref{prop:toprigidity} in order to simplify our argument, by an argument similar to the proof of \cite[Thm.5.11]{CheegerColding1} with corresponding almost rigidity results in \cite{DePhilippisGigli}, we see that if a non-collapsed $\ensuremath{\mathbb R}CD(K, n)$ spaces $(X, d_X, \mathcal{H}^n)$ satisfies $X=\mathcal{R}_n$, then for all $\ensuremath{\varepsilon}ilon>0$ there exist positive constants $r>0$ and $\delta>0$ such that if a non-collapsed $\ensuremath{\mathbb R}CD(K, n)$ space $(Y, d_Y, \mathcal{H}^n)$ satisfies $d_{GH}(X, Y)\le \delta$, then $(Y, d_Y) \in \mathcal{M}(n, r, \ensuremath{\varepsilon}ilon)$. See also \cite{KM}.
\end{remark}
\ensuremath{\mathbb S}ection{Improvement to Einstein stratified spaces}
The previous results can be improved to the case of certain (smoothly) stratified spaces. In order to do that, we briefly recall some notions about such spaces, by mostly referring to \cite{MondPhD} and \cite{BKMR} for the precise definitions.
A (compact) stratified space $X$ is a (compact) topological space which admits a decomposition in strata
$$X = \bigsqcup_{j=0}^n \Sigma^{j}(X)$$
such that for each $j=0,\ldots n$, $\Sigma^{j}(X)$ is a smooth manifold of dimension $j$, $\Sigma^n(X)$ is open and dense in $X$ and $\Sigma^{n-1}(X)=\emptyset$. We denote the higher dimension stratum $\Sigma^n(X)$ as $X^{\mbox{\tiny{reg}}}$, the regular set of $X$, and refer to $n$ as the dimension of $X$. We define the singular set of $X$:
$$ X^{\mbox{\tiny{sing}}}= \bigsqcup_{j=0}^{n-2}\Sigma^{j}(X).$$
For $j < (n-1)$, $\Sigma^{j}(X)$ is called the singular stratum of dimension $j$. For each point in $\Sigma^{j}(X)$ there exists a neighbourhood $\mathcal{U}_x$ homeomorphic to the product of an Euclidean ball in $\ensuremath{\mathbb R}^j$ and a truncated cone over a compact stratified space $B_r(0_j)\times C_{[0,r)}(Z_j)$. We refer to $Z_j$ as the \emph{link} of the stratum $\Sigma^j(X)$.
By induction on the dimension, a stratified space $X$ can be endowed with an iterated edge metric $g$, which is a Riemannian metric on $X^{\mbox{\tiny{reg}}}$ with the appropriate asymptotics close to each singular stratum: by denoting $k_j$ an iterated edge metric on the link $Z_j$, there exist positive constants $\Lambda$ and $\gamma$ such that in a neighbourhood $\mathcal{U}_x$ of a point $x \in \Sigma^{j}(X)$ we have
\begin{equation}
\label{eq:ItEdge}
|\varphi_x^*g-(h+dr^2+r^2k_j)| \leq \Lambda r^{\gamma},
\end{equation}
where $h$ is the standard Riemannian metric on $\ensuremath{\mathbb R}^j$ and $\varphi_x$ is the homeomorphism between the product $B_r(0_j)\times C_{[0,r)}(Z_j)$ and the neighbourhood $\mathcal{U}_x$.
In the case of the codimension $2$ stratum, the link is a compact stratified space of dimension $1$, thus a circle. As a consequence, for each point $x \in \Sigma^{n-2}(X)$ there exists $\alpha_x \in (0, +\infty)$ such that the metric $g$ is asymptotic, in the sense of \eqref{eq:ItEdge}, to:
$$ h+ dr^2+\left(\frac{\alpha_x}{2\pi} \right)r^2d\theta^2,$$
on $\ensuremath{\mathbb R}^{n-2}\times C(\ensuremath{\mathbb S}^1)$. We refer to $\alpha_x$ as the angle of $\Sigma^{n-2}(X)$ at $x$.
The iterated edge metric $g$ gives rise to a length structure and a distance $d_g$ on $X$, and to a Riemannian measure $\mu_g$ which in the compact case is Ahlfors-regular and finite. Note that the measure of the singular strata is zero and, as in the case of smooth manifolds, $\mu_g$ coincides with the Hausdorff measure.
Moreover, thanks to the definition of the distance and iterated edge metric, we know that each point $x \in \Sigma^{j}(X)$ admits a unique tangent cone $C(S_x)$ over the tangent sphere at $x$. It is defined by the pGH limit as $\ensuremath{\varepsilon}$ goes to zero:
$$(X, \ensuremath{\varepsilon}^{-1}d_g, x) \ensuremath{\mathbb S}tackrel{pGH}\longrightarrow (C(S_x), d_C, o),$$
where $o$ is the vertex of the cone. The tangent sphere is a compact stratified space of dimension $(n-1)$ given by the $(j-1)$-spherical suspension of the link
$$S_x=\left[0, \frac{\pi}{2} \right] \times \ensuremath{\mathbb S}^{j-1} \times Z_j.$$
Since the convergence is smooth on the regular sets, the tangent sphere is endowed with a double warped product metric:
\begin{equation}\label{eq:double}
h_x= d\psi^2+\ensuremath{\eta_{j}}s^2(\psi) g_{\ensuremath{\mathbb S}^{j-1}}+\ensuremath{\mathbb S}in^2(\psi) k_j.
\end{equation}
Moreover, the smoothness of the convergence and the fact that singular sets have null measures imply that pGH-convergence can be replaced by \textit{pmGH-convergence}.
Note that for $x \in \Sigma^{0}(X)$, the tangent sphere coincides with the link of $\Sigma^{0}(X)$. If $x$ belongs to $\Sigma^{1}(X)$ then $S_x$ is a spherical suspension of the form $[0,\pi] \times Z_1$ endowed with the warped product metric $h_x=d\psi^2+\ensuremath{\mathbb S}in^2(\psi)k_1$.
Also observe that a point belongs to the regular set if and only if its tangent sphere is isometric to the round sphere $\ensuremath{\mathbb S}^{n-1}$.
In view of the following, we need information about how the regularity of $X$ affects the regularity of tangent spheres. This is stated in the following.
\begin{lemma}
\label{lem:Codim2sing}
Let $(X, g)$ be an $n$-dimensional compact stratified space endowed with an iterated edge metric $g$. If $\Sigma^{n-2}(X)=\emptyset$, then for any $x \in X$ the tangent sphere $S_x$ does not carry a singular stratum of codimension $2$.
\end{lemma}
\begin{proof}
Assume $x \in \Sigma^j(X)$ for $j \in \{0,\ldots n-3\}$ and consider $Z_j$ the link of $\Sigma^j(X)$. Denote by $d_j=n-j-1$ its dimension and by $\varphi_x$ the homeomorphism between a neighbourhood of $x$ and the product $B_r(0_j)\times C_{[0,r)}(Z_j)$. We first observe that $Z_j$ does not carry any singular stratum of codimension $2$. Assume by contradiction that there exists $z \in \Sigma^{d_j-2}(Z_j)$: then $z$ has a neighbourhood homeomorphic to $B_s(0_{d_j-2})\times C_{[0,s)}(\ensuremath{\mathbb S}^1)$. Denote by $\bar{p}$ the point of coordinates $(v,s,z)$ in $B_r(0_{j})\times C_{[0,r)}(Z_j)$ and $\bar x =\varphi_x(\bar p) \in X$. Then $\bar x$ has a neighbourhood homeomorphic to $B_{\rho}(0_{n-2})\times C(\ensuremath{\mathbb S}^1)$. As a consequence, $\bar x$ belongs to $\Sigma^{n-2}(X)$, which contradicts the assumption $\Sigma^{n-2}(X)=\emptyset$.
We next show that $Z_j$ not having any singular stratum of codimension $2$, the same holds for the tangent sphere $S_x$. If $j=0$, $S_x$ coincides with $Z_j$ and thus have the same singular set. If $j=1$, $S_x$ is the warped product $([0,\pi] \times Z_1, d\psi^2+\ensuremath{\mathbb S}in^2(\psi)k_1)$. Its singular set $S_x^{\mbox{\tiny{sing}}}$ is composed of the subsets $(0,\pi)\times Z_1^{\mbox{\tiny{sing}}}$ and $\{0,\pi\}\times Z_1$. As for $(0,\pi)\times Z_1^{\mbox{\tiny{sing}}}$, it only generates singularities of the same codimension as the ones of $Z_1^{\mbox{\tiny{sing}}}$, that is at least 3. The singularities at $\{0\} \times Z_1$ and $\{\pi\}\times Z_1$ carry a neighbourhood homeomorphic to $C(Z_1)$ and as a consequence have codimension 0 in $S_x$. Therefore $\Sigma^{n-2}(S_x)=\emptyset$. Similarly, for $j\in \{2, \ldots , n-3\}$, the singular set of $S_x$ is given by :
\begin{itemize}
\item[•] the product $(0, \pi/2)\times Z_j^{\mbox{\tiny{sing}}}$ which has codimension at least 3 because we already know $\Sigma^{d_j-2}(Z_j)=\emptyset$;
\item[•] $\{0\} \times \ensuremath{\mathbb S}^{j-1} \times Z^j$: any point in this product has a neighbourhood homeomorphic to $B(0_j)\times Z_j$, then for the same reason singularities have codimension at least 3;
\item[•] $\{\pi/2\} \times \ensuremath{\mathbb S}^{j-1} \times Z^j$: any point belonging to this set has a neighbourhood homeomorphic to $\ensuremath{\mathbb S}^{j-1}\times C(Z_j)$. This gives singularities of codimension $n-j$ in $S_x$ and since $j \leq n-3$, the codimension is at least 3.
\end{itemize}
We have shown that for any $x \in X$ we have $\Sigma^{n-2}(S_x)=\emptyset$, as we wished. \end{proof}
Thanks to \cite{BKMR} we know the following:
\begin{theorem*}
A compact stratified space $(X, d_g, \mu_g)$ of dimension $n$ endowed with an iterated edge metric $g$ is a $\ensuremath{\mathbb R}CD(K,N)$ space for $K\in \ensuremath{\mathbb R}$, $N \geq 1$, if and only if $n \leq N$, $\mathrm{Ric}_g \geq K$ on $X^{\mbox{\tiny{reg}}}$ and the angles along the singular stratum of codimension $2$ are smaller than or equal to $2\pi$.
\end{theorem*}
Since $\mu_g$ is the Hausdorff measure, a $\ensuremath{\mathbb R}CD(K,n)$ compact stratified space of dimension $n$ is also a \emph{non-collapsed} $\ensuremath{\mathbb R}CD$ space.
An easy consequence of the definition of the iterated edge metric is the following:
\begin{lemma}
\label{lem:RicTg}
Let $(X, d_g, \mu_g)$ be a $\ensuremath{\mathbb R}CD(K,n)$ compact stratified space of dimension $n$ endowed with an iterated edge metric $g$. Then for every $x \in X$, the tangent sphere $(S_x, d_{h_x}, \mu_{h_x})$ at $x$ is a non-collapsed $\ensuremath{\mathbb R}CD(n-2,n-1)$ space. Moreover, if for $K \geq 0$ $||\mathrm{Ric}_g||\leq K$ holds on the regular set, we also have $\mathrm{Ric}_{h_x}=(n-2)$ on $S_x^{\mbox{\tiny{reg}}}$.
\end{lemma}
\begin{proof}
Because of the definition of the tangent cone, we know that if $\mathrm{Ric}_g \geq K$ on $X^{\mbox{\tiny{reg}}}$, then $(C(S_x), ds^2+s^2h_x)$ has non-negative Ricci tensor on its regular part. As a consequence, for each tangent sphere $\mathrm{Ric}_{h_x}\geq (n-2)$ on $S_x^{\mbox{\tiny{reg}}}$ (see Lemma 2.1 in \cite{Mondello}). Analogously, if $||\mathrm{Ric}_g|| \leq K$ on $X^{\mbox{\tiny{reg}}}$ the tangent cone is Ricci flat and $\mathrm{Ric}_{h_x}=(n-2)$ on $S_x^{\mbox{\tiny{reg}}}$.
For every $x$, $S_x$ cannot carry a stratum of codimension $2$ with angles larger than $2\pi$. The argument is the same as in Lemma \ref{lem:Codim2sing}. If for every $x \in \Sigma^{n-2}(X)$ the angle $\alpha_x$ is smaller than or equal to $2\pi$, then the same holds for every link $Z_j$ of dimension $d_j$: the angles along $\Sigma^{d_j-2}(Z_j)$ belongs to $(0,2\pi]$. Now, if $x \in \Sigma^j(X)$, singularities of codimension $2$ of $S_x$ are determined by the singularities of codimension $2$ in $Z_j$: as a consequence, if $p \in \Sigma^{n-3}(S_x)$, then $\alpha_p \in (0, 2\pi]$.
Then for any $x \in X$ the tangent sphere at $x$ satisfies the assumptions of the previous theorem and $(S_x,d_{h_x}, \mu_{h_x})$ is a $\ensuremath{\mathbb R}CD(n-1,n-2)$ space.
\end{proof}
We are now in position to prove the following:
\begin{theorem}[Sphere theorem for Einstein stratified spaces]\label{thm:strati}
For all $n \in \mathbb{N}_{\ge 2}$ there exists a positive constant $\ensuremath{\varepsilon}ilon_n>0$ such that the following holds. Let $(X, g)$ be compact $n$-dimensional stratified space endowed with an iterated edge metric $g$ satisfying $\mathrm{Ric}_g \equiv n-1$ on the regular set. Assume that $\Sigma^{n-2}(X) = \emptyset$. If $\mu_g(X) \ge (1-\ensuremath{\varepsilon}ilon_n)\mathcal{H}^n(\mathbb{S}^n)$, then $(X, d_g)$ is isometric to $(\mathbb{S}^n, d_{\mathbb{S}^n})$.
\end{theorem}
\begin{proof}
Let us first check the theorem in the case of $(X, g)$ not having a singular set, that is $(X, g)$ is a smooth manifold. The proof is done by a contradiction. If the assertion does not hold, then there exists a sequence of $n$-dimensional closed Riemannian manifolds $(M^n_i, g_i)$ such that $\mathrm{Ric}_{M^n_i}^{g_i} \equiv n-1$, that $\mu_{g_i}(M^n_i)\to \mathcal{H}^n(\mathbb{S}^n)$ and that $(M^n_i, g_i)$ is not isometric to $(\mathbb{S}^n, g_{\mathbb{S}^n})$. Then applying the smooth convergence result \cite[Thm.7.3]{CheegerColding1} yields that $(M^n_i, g_i)$ converge smoothly to $(\mathbb{S}^n, g_{\mathbb{S}^n})$. In particular $(M^n_i, g_i)$ is simply connected and has positive curvature operator for suficiently large $i$. Then by a theorem of \cite{Tachibana} $(M^n_i, g_i)$ has constant sectional curvature. Thus $(M^n_i, g_i)$ is isometric to $(\mathbb{S}^n, g_{\mathbb{S}^n})$, which is a contradiction.
For all $n \in \mathbb{N}_{\geq 2}$ take a positive constant $\ensuremath{\varepsilon}ilon_n>0$ such that the smooth version of Theorem \ref{thm:strati} holds.
Next let us fix a compact $n$-dimensional stratified space $(X, g)$ satisfying $\mathrm{Ric}_g \equiv n-1$ on the regular set and $\Sigma^{n-2}(X) = \emptyset$. Our goal is to prove that if $\mu_g (X) \geq (1-\hat{\ensuremath{\varepsilon}ilon}_n)\mathcal{H}^n(\mathbb{S}^n)$, then $(X, d_g)$ is isometric to $(\mathbb{S}^n, d_{\mathbb{S}^n})$, where $\hat{\ensuremath{\varepsilon}ilon}_n:=\min \{\ensuremath{\varepsilon}ilon_{i}, 1 \le i \le n\}$ and $\ensuremath{\varepsilon}ilon_1:=0$.
The proof is done by induction.
For $n=2$, our assumption $\Sigma^{n-2}(X)=\emptyset$ yields that $(X, g)$ has no singular set, thus $(X, g)$ is a smooth Riemannian manifold. By definition of $\ensuremath{\varepsilon}ilon_2$ we get the desired statement.
As for $n \geq 3$, let $x \in X$ and consider the tangent cone $(C(S_x), ds^2+s^2h_x, o)$ at $x$. By an argument similar to (\ref{eq:maxball}), we see that for all points $x \in X$ and $r\in (0,\pi]$ we have $\mu_g(B_r(x))=\mathcal{H}^n(B_r(x))\geq (1-\hat{\ensuremath{\varepsilon}ilon}_n)\omega_n r^n$. By rescaling the metric by a factor $r^{-2}$ and by letting $r$ go to zero, $(X,r^{-2}g, \mathcal{H}^n, x)$ pmGH-converges to the tangent cone $(C(S_x), ds^2+s^2h_x, \mathcal{H}^n, o)$. As a consequence, we have $\mathcal{H}^n(B_1(o)) \geq (1-\hat{\ensuremath{\varepsilon}ilon}_n)\omega_n$. Since $\mu_{h_x}(S_x)=n\mathcal{H}^n(B_1(o))$, we obtain
$$\mu_{h_x}(S_x) \geq (1-\hat{\ensuremath{\varepsilon}ilon}_{n})\mathcal{H}^{n-1}(\ensuremath{\mathbb S}^{n-1})\geq (1-\hat{\ensuremath{\varepsilon}ilon}_{n-1})\mathcal{H}^{n-1}(\ensuremath{\mathbb S}^{n-1}).$$
On the other hand, by Lemma \ref{lem:RicTg}, we also know that $(S_x,h_x)$ is a compact stratified space of dimension $(n-1)$ with $\mathrm{Ric}_{h_x}\equiv (n-2)$ on its regular set. Moreover, since $\Sigma^{n-2}(X)=\emptyset$, Lemma \ref{lem:Codim2sing} ensures that $S_x$ does not have any singular stratum of codimension $2$. Then by the assumption on the induction, $(S_x, d_{h_x})$ is isometric to the round sphere $(\mathbb{S}^{n-1}, d_{\mathbb{S}^{n-1}})$. As a consequence we have proven that for any $x \in X$, $x$ is a regular point. This proves that $X^{\mbox{\tiny{sing}}}=\emptyset$, thus $(X, g)$ is a smooth Riemannian manifold. By definition of $\ensuremath{\varepsilon}ilon_n$, $(X, d_g)$ is isometric to $(\mathbb{S}^n, d_{\mathbb{S}^n})$.
\end{proof}
\begin{remark} In the theorem above the assumption $\Sigma^{n-2}(X)=\emptyset$ is essential. Consider for all $a \in (0,1)$, the $(n-2)$-spherical suspension of $\ensuremath{\mathbb S}^1(a):= (\ensuremath{\mathbb S}^1,a^2d\theta^2)$ defined by:
$$ X =\left[0, \frac{\pi}{2} \right] \times \ensuremath{\mathbb S}^{n-2} \times \ensuremath{\mathbb S}^1(a),$$
endowed with the double warped product metric $g$ as in (\ref{eq:double}). $(X, g)$ is a compact $n$-dimensional stratified space with the iterated edge metric $g$ and $\mathrm{Ric}_g \equiv (n-1)$ on $X^{\mbox{\tiny{reg}}}$ and a non-empty, codimension $2$ singular stratum given by $\{\pi/2\} \times \ensuremath{\mathbb S}^{n-2} \times \ensuremath{\mathbb S}^1(a)$. If the angle $\alpha = 2\pi a$ along $\Sigma^{n-2}(X)$ is close to $2\pi$, then the volume of $X$ is close to the one of $\ensuremath{\mathbb S}^n$, but $(X, g)$ cannot be isometric to $(\ensuremath{\mathbb S}^n, d_{\ensuremath{\mathbb S}^n})$ because of its singular stratum of codimension $2$.
\end{remark}
We now consider stratified spaces that are not necessarily compact. Note that even in this case tangent spheres are compact stratified space defined as above, and Lemmas \ref{lem:Codim2sing} and \ref{lem:RicTg} still hold. In presence of a codimension $2$ stratum and a two-side Ricci bound, the previous theorem allows us to obtain the following:
\begin{cor}\label{cor:reg}
For all $n \in \mathbb{N}_{\ge 2}$ there exists a positive constant $\ensuremath{\varepsilon}ilon_n>0$ such that the following holds. Let $(X, g)$ be an $n$-dimensional stratified space endowed with an iterated edge metric $g$ such that $\mathrm{Ric}_g$ is two-side bounded on the regular set of $X$. If a point $x \in X$ satisfies
$$
\lim_{r \to 0^+}\frac{\mu_g (B_r(x))}{\omega_nr^n}\ge 1-\ensuremath{\varepsilon}ilon_n,
$$
then either $x$ belongs to $X^{\mbox{\tiny{reg}}}$, or $x \in \Sigma^{n-2}(X)$ and $\alpha_x \geq 2\pi(1 -\ensuremath{\varepsilon}ilon_n)$.
\end{cor}
\begin{proof}
Fix $\ensuremath{\varepsilon}ilon_n$ as in Theorem $\ref{thm:strati}$ and assume that
\begin{equation}
\label{eq:limVol}
\lim_{r \to 0^+}\frac{\mu_g (B_r(x))}{\omega_n r^n}\geq 1-\ensuremath{\varepsilon}ilon_n.
\end{equation}
If $x$ belongs to $\Sigma^{n-2}(X)$, thanks to the definition of the tangent sphere and its metric we know that
$$\lim_{r\rightarrow 0^+}\frac{\mu_g(B_r(x))}{\omega_n r^n}=\frac{\alpha_x}{2\pi}.$$
Therefore if (\ref{eq:limVol}) holds, we obtain $\alpha_x \geq 2\pi(1-\ensuremath{\varepsilon}ilon_n)$.
Consider a point $x \in X \ensuremath{\mathbb S}etminus \Sigma^{n-2}(X)$. Observe that the tangent sphere $(S_x,h_x)$ at $x$ is a compact stratified space of dimension $(n-1)$, and with the same argument as in Lemma $\ref{lem:RicTg}$, the metric $h_x$ satisfies $\mathrm{Ric}_{h_x}\equiv (n-2)$. Moreover, since $x$ does not belong to $\Sigma^{n-2}(X)$, Lemma \ref{lem:Codim2sing} can be easily adapted to show that $S_x$ does not carry any singular stratum of codimension $2$. By the same argument as in the previous theorem, letting $r$ go to zero in $(B_r(x),r^{-2}g,x)$ leads to the volume estimate $\mathcal{H}^n(B_1(o))\geq (1- \ensuremath{\varepsilon}ilon_{n})\omega_{n}$ and as above we obtain:
$$\mu_{h_x}(S_x)\geq (1-\ensuremath{\varepsilon}ilon_{n})\mathcal{H}^{n-1}(\ensuremath{\mathbb S}^{n-1}).$$
As a consequence, Theorem \ref{thm:strati} applied to $S_x$ shows that the tangent sphere at $x$ is isometric to $\ensuremath{\mathbb S}^{n-1}$ and $x$ belongs to the regular set.
\end{proof}
\begin{remark}
In Corollary \ref{cor:reg}
the assumption on two side bounds on the Ricci curvature on the regular set is essential because for any $n \in \mathbb{N}_{\ge 3}$, taking $r \in (0, 1)$ which is sufficiently close to $1$, let us consider a compact $n$-dimensional stratified space $X:=[0,\pi] \times \mathbb{S}^{n-1}$ with the warped product metric $g=d\varphi^2+\ensuremath{\mathbb S}in^2(\varphi)r^2g_{\ensuremath{\mathbb S}^{n-1}}$. Then it satisfies
\begin{itemize}
\item for all $x \in X$, $\lim_{t \to 0^+}\frac{\mu_g (B_t(x))}{\omega_nt^n}$ is close to $1$;
\item $\mathrm{Ric}_g \ge n-1$ on the regular set;
\item there is no upper bound on $\mathrm{Ric}_g$.
\end{itemize}
In this case, all singular points of $X$ belongs to $\Sigma^{0}(X)$.
\end{remark}
\end{document} | math | 56,477 |
\begin{document}
\title{On the Equivalence of Youla, System-level and Input-output Parameterizations}
\author{Yang Zheng, Luca Furieri, Antonis Papachristodoulou, Na Li, and Maryam Kamgarpour
\thanks{This work is supported by NSF career 1553407, AFOSR Young Investigator Program, and ONR Young Investigator Program. A. Papachristodoulou is supported by the EPSRC Grant EP/M002454/1.
L. Furieri and M. Kamgarpur are gratefully supported by ERC Starting Grant CONENE. }
\thanks{Y. Zheng and N. Li are with SEAS and CGBC, Harvard
University, Cambridge, MA 02138. (E-mails: [email protected]; [email protected]). }
\thanks{L. Furieri and M. Kamgarpour are with the Automatic Control Laboratory, ETH Zurich, Switzerland. (E-mails: \{furieril, mkamgar\}@control.ee.ethz.ch).}
\thanks{A. Papachristodoulou is with the Department of Engineering Science , University of
Oxford, United Kingdom. (E-mail: [email protected]).
}
}
\maketitle
\begin{abstract}
A convex parameterization of internally stabilizing controllers is fundamental for many controller synthesis procedures. The celebrated Youla parameterization relies on a doubly-coprime factorization of the system, while the recent system-level and input-output characterizations require no doubly-coprime factorization but a set of equality constraints for achievable closed-loop responses. In this paper, we present explicit affine mappings among Youla, system-level and input-output parameterizations. Two direct implications of the affine mappings are 1) any convex problem in Youla, system level, or input-output parameters can be equivalently and convexly formulated in any other one of these frameworks,
including the convex system-level synthesis (SLS); 2) the condition of quadratic invariance (QI) is sufficient and necessary for the classical distributed control problem to admit an equivalent convex reformulation in terms of Youla, system-level, or input-output parameters.
\end{abstract}
\begin{IEEEkeywords}
Stabilizing controller, Youla parameterization, System-level synthesis, Quadratic invariance.
\end{IEEEkeywords}
\section{Introduction}
\label{sec:introduction}
One of the most fundamental problems in control theory is to design a feedback controller that stabilizes a dynamical system. Additionally, one can further design an optimal controller by optimizing a certain performance measure~\cite{zhou1996robust}. It is well-known that the set of stabilizing controllers is in general non-convex, and hence, hard to optimize directly over.
Many optimal controller synthesis procedures first parameterize all stabilizing controllers and the corresponding closed-loop responses in a convex way, and then minimize relevant performance measures over the new parameter(s).
For finite dimensional linear-time-invariant (LTI) systems, the set of LTI stabilizing feedback controllers is fully characterized by the celebrated \emph{Youla parameterization}~\cite{youla1976modern}, where a doubly coprime factorization of the system is used. In~\cite{youla1976modern}, it is shown that the Youla parameterization allows for optimizing the Youla parameter (or system response) directly, instead of the controller itself, leading to a convex problem. Also, customized performance specifications on the closed-loop system can be incorporated with Youla parameterization via convex optimization~\cite{boyd1991linear}. Moreover, the foundational results of robust and optimal control are built on the Youla parameterization~\cite{francis1987course,zhou1996robust}. Note that a doubly-coprime factorization of the system must be computed as a preliminary step in Youla parameterization. Recently, a system-level parameterization~\cite{wang2019system} and an input-output parameterization~\cite{furieri2019input} were introduced to characterize the set of all LTI stabilizing controllers, with no need of computing a doubly-coprime factorization of the system \emph{a priori}. Similar to Youla, the system-level and input-output parameterizations treat certain closed-loop responses as design parameters. The controller synthesis is thus shifted from designing a controller to designing the closed loop responses directly.
This idea of synthesizing closed-loop responses in a convex way was extensively discussed as \emph{closed-loop convexity} in~\cite[Chapter 6]{boyd1991linear}.
The Youla~\cite{youla1976modern}, system-level~\cite{wang2019system}, input-output~\cite{furieri2019input} parameterizations are equivalent since they characterize the same set of stabilizing controllers. However, their explicit relationships have not been fully established before. The main objective of this paper is to reveal an explicit equivalence of Youla, system-level, and input-output parameterizations. In particular, we present explicit \emph{affine mappings} among the Youla parameter, system-level parameters, and input-output parameters. One direct consequence is that any convex problem in terms of Youla, system-level, input-output parameters can be equivalently and convexly formulated into any other one of these three frameworks.
Therefore, the so-called convex system-level synthesis (SLS)~\cite{wang2019system} admits an equivalent convex formulation in terms of Youla or input-output parameters. Another consequence is that if one controller synthesis task does not allow for an equivalent convex reformulation in Youla, a convex reformulation in the system-level or input-output parameterizations is not possible either. Consider the classical distributed controller synthesis task where a subspace constraint is imposed on the controller~\cite{rotkowitz2006characterization}. It has been shown that a notion of quadratic invariance (QI) is sufficient and necessary for the distributed control problem to admit an equivalent convex reformulation in the Youla parameter~\cite{rotkowitz2006characterization,lessard2015convexity}. Accordingly, the QI condition is also sufficient and necessary when using the system-level and input-output parameterizations. For systems with constraints beyond QI, we highlight that a notion of sparsity invariance (SI)~\cite{Furieri2019Sparsity} can be used to derive convex inner-approximations using Youla, system-level, or input-output characterizations.
The rest of this paper is organized as follows. We introduce some preliminaries in Section~\ref{Section:preliminaries}, and review the Youla, system-level, and input-output parameterizations in Section~\ref{Section:paramterization}. Explicit affine relationships and their implication with QI are presented in Section~\ref{Section:equivalence}. We discuss distributed controller synthesis with non-QI constraints in Section~\ref{Section:specialcase}, and conclude the paper in Section~\ref{section:conclusion}.
\noindent\emph{Notation:} We use lower and upper case letters (\emph{e.g.} $x$ and $A$) to denote vectors and matrices, respectively. Lower and upper case boldface letters (\emph{e.g.} $\mathbf{x}$ and $\mathbf{G}$) are used to denote signals and transfer matrices, respectively. For clarity, we consider discrete-time LTI systems only, but unless stated otherwise, all results can be extended to the continuous-time setting. We denote the set of real-rational proper stable transfer matrices as $\mathcal{RH}_{\infty}$. $\mathbf{G} \in \frac{1}{z}\mathcal{RH}_{\infty}$ means $\mathbf{G}$ is stable and strictly proper.
\section{Preliminaries}~\label{Section:preliminaries}
\subsection{System model}
We consider discrete-time LTI systems of the form
\begin{equation} \label{eq:LTI}
\begin{aligned}
{x}[t+1] &= A x[t] + B_1 w[t] + B_2u[t], \\
z[t] &= C_1 x[t] + D_{11}w[t] + D_{12}u[t], \\
y[t] &= C_2x[t] + D_{21}w[t] + D_{22} u[t],
\end{aligned}
\end{equation}
where $x[t],u[t],w[t],y[t],z[t]$ are the state vector, control action, external disturbance, measurement, and regulated output at time $t$, respectively. System~\eqref{eq:LTI} can be written as
$$
\mathbf{P} = \left[ \begin{array}{c|cc} A & B_1 & B_2 \\
\hline
C_1 & D_{11} & D_{12} \\
C_2 & D_{21} & D_{22} \end{array} \right]
= \begin{bmatrix} \mathbf{P}_{11} & \mathbf{P}_{12} \\\mathbf{P}_{21} & \mathbf{P}_{22} \end{bmatrix},
$$
where $\mathbf{P}_{ij} = C_i(zI - A)^{-1}B_j +D_{ij}$. We refer to $\mathbf{P}$ as the open-loop plant model.
Consider a dynamic output feedback controller
$
\mathbf{u} = \mathbf{K}\mathbf{y},
$
where $\mathbf{K}$ has a state space realization
\begin{equation} \label{eq:ControllerLTI}
\begin{aligned}
{\xi}[t+1] &= A_k \xi[t] + B_k y[t], \\
u[t] &= C_k \xi[t] + D_{k}y[t],
\end{aligned}
\end{equation}
with $\xi$ as the internal state of controller $\mathbf{K}$. We have $\mathbf{K} = C_k(zI - A_k)^{-1}B_k + D_k$. Figure~\ref{fig:LTI} shows a schematic diagram of the interconnection of plant $\mathbf{P}$ and controller $\mathbf{K}$. Throughout the paper, we make the following standard assumptions.
\begin{assumption}
Both the plant and controller realizations are stabilizable and detectable, \emph{i.e.}, $(A, B_2)$ and $(A_k, B_k)$ are stabilizable, and $(A,C_2)$ and $(A_k,C_k)$ are detectable.
\end{assumption}
\begin{assumption}
The interconnection in Fig.~\ref{fig:LTI} is well-posed, \emph{i.e.}, $I - D_{22}D_k$ is invertible.
\end{assumption}
\subsection{Stabilization and optimal control}
\begin{definition}
The system in Fig.~\ref{eq:LTI} is \emph{internally stable} if it is well-posed, and the states $(x[t],x_k[t])$ converge to zero as $t\rightarrow \infty$ for all initial states $(x[0],x_k[0])$ when $w[t] = 0, \forall t$.
\end{definition}
We say the controller $\mathbf{K}$ \emph{internally stabilizes} the plant $\mathbf{P}$ if the interconnected system in Fig.~\ref{eq:LTI} is {internally stable}. The set of all stabilizing controllers is defined as
\begin{equation}
\mathcal{C}_{\text{stab}} := \{\mathbf{K} \mid \mathbf{K} \; \text{internally stabilizes} \; \mathbf{P}\}.
\end{equation}
In addition to stability, it is desirable to find a controller $\mathbf{K}$ that minimizes a suitable norm (\emph{e.g.}, $\mathcal{H}_2$ or $\mathcal{H}_{\infty}$) of the closed-loop transfer matrix from $\mathbf{w}$ to $\mathbf{z}$. This amounts to solving the following optimal control formulation~\cite{zhou1996robust}:
\begin{equation} \label{eq:OCP}
\begin{aligned}
\min_{\mathbf{K}} \quad &\|f(\mathbf{P},\mathbf{K})\| \\
\text{subject to} \quad & \mathbf{K} \in \mathcal{C}_{\text{stab}},
\end{aligned}
\end{equation}
where $f(\mathbf{P},\mathbf{K}) = \mathbf{P}_{11} + \mathbf{P}_{12}\mathbf{K}(I - \mathbf{P}_{22}\mathbf{K})^{-1}\mathbf{P}_{21}$. It is known that set $\mathcal{C}_{\text{stab}}$ is non-convex. One can construct explicit examples where $\mathbf{K}_1, \mathbf{K}_2 \in \mathcal{C}_{\text{stab}}$ but $\frac{1}{2}(\mathbf{K}_1+ \mathbf{K}_2) \notin \mathcal{C}_{\text{stab}}$. Also, $f(\mathbf{P},\mathbf{K})$ is in general a non-convex function of $\mathbf{K}$. Therefore, problem~\eqref{eq:OCP} is non-convex in the current form.
\begin{figure}
\caption{Interconnection of the plant $\mathbf{P}
\label{fig:LTI}
\end{figure}
\section{Parameterization of Stabilizing Controllers}\label{Section:paramterization}
To solve the optimal control problem~\eqref{eq:OCP}, one common method is to derive an equivalent convex formulation via a suitable change of variables. A classical technique is the Youla parameterization~\cite{youla1976modern}. Two recent approaches are the so-called system-level parameterization (SLP)~\cite{wang2019system}, and input-output parameterization (IOP)~\cite{furieri2019input}. A common idea among these three approaches is the parameterization of all stabilizing controllers $\mathcal{C}_{\text{stab}}$ using certain closed-loop responses. We review their main results in this section.
\subsection{Youla parameterization}
The classical Youla parameterization is based on a doubly-coprime factorization of the plant $\mathbf{P}_{22}$, defined as follows.
\begin{definition}
A collection of stable transfer matrices, $\mathbf{U}_l, \mathbf{V}_l,\mathbf{N}_l,\mathbf{M}_l,\mathbf{U}_r, \mathbf{V}_r,\mathbf{N}_r,\mathbf{M}_r \in \mathcal{RH_{\infty}}$ is called a doubly-coprime factorization of $\mathbf{P}_{22}$ if
$
\mathbf{P}_{22} = \mathbf{N}_r\mathbf{M}_r^{-1} = \mathbf{M}_l^{-1}\mathbf{N}_l
$
and
$$
\begin{bmatrix} \mathbf{U}_l & -\mathbf{V}_l \\ -\mathbf{N}_l & \mathbf{M}_l\end{bmatrix}
\begin{bmatrix} \mathbf{M}_r & \mathbf{V}_r \\ \mathbf{N}_r & \mathbf{U}_r\end{bmatrix} = I.
$$
\end{definition}
Such doubly-coprime factorizations can always be computed if $\mathbf{P}_{22}$ is stabilizable and detectable~\cite{zhou1996robust}. The authors in~\cite{youla1976modern} established the following equivalence
\begin{equation} \label{eq:youla}
\mathcal{C}_{\text{stab}} = \{\mathbf{K} = (\mathbf{V}_r - \mathbf{M}_r\mathbf{Q})(\mathbf{U}_r - \mathbf{N}_r\mathbf{Q})^{-1} \mid \mathbf{Q} \in \mathcal{RH}_{\infty} \}\footnote{Equivalently, $\mathcal{C}_{\text{stab}}=\{(\mathbf{U}_l-\mathbf{QN}_l)^{-1}(\mathbf{V}_l-\mathbf{QM}_l)|~\mathbf{Q} \in \mathcal{RH}_\infty\}$.},
\end{equation}
where $\mathbf{Q}$ is called the Youla parameter. Using the change of variables
$\mathbf{K} = (\mathbf{V}_r - \mathbf{M}_r\mathbf{Q})(\mathbf{U}_r - \mathbf{N}_r\mathbf{Q})^{-1}$, it is not difficult to derive
$$
f(\mathbf{P},\mathbf{K}) = \mathbf{T}_{11} + \mathbf{T}_{12}\mathbf{Q}\mathbf{T}_{21},
$$
where $\mathbf{T}_{11} = \mathbf{P}_{11} + \mathbf{P}_{12} \mathbf{V}_{r}\mathbf{M}_{l} \mathbf{P}_{21}, \mathbf{T}_{12} = -\mathbf{P}_{12}\mathbf{M}_{r}$, and $\mathbf{T}_{21} = \mathbf{M}_{l}\mathbf{P}_{21}$. Consequently, Problem~\eqref{eq:OCP} can be equivalently reformulated in terms of the Youla parameter as
\begin{equation} \label{eq:OCPYoula}
\begin{aligned}
\min_{\mathbf{Q}} \quad &\|\mathbf{T}_{11} + \mathbf{T}_{12}\mathbf{Q}\mathbf{T}_{21}\| \\
\text{subject to} \quad & \mathbf{Q} \in \mathcal{RH}_{\infty}.
\end{aligned}
\end{equation}
One direct benefit is that~\eqref{eq:OCPYoula} is convex with respect to the Youla parameter $\mathbf{Q}$.
\subsection{System-level parameterization (SLP)}
In~\cite{wang2019system}, the authors proposed a system-level parameterization for $\mathcal{C}_{\text{stab}}$. This approach is based on the closed-loop maps from process and measurement disturbances to state and control action. In particular, assuming a strictly proper plant $\mathbf{P}_{22},$ \emph{i.e.}, $D_{22} = 0$, we use ${\delta_x}[t] = B_1 w[t]$ to denote the disturbance on the state and $\delta_y[t] = D_{21}w[t]$ to denote the disturbance on the measurement. The dynamics of plant~\eqref{eq:LTI} can be written as
\begin{equation*}
\begin{aligned}
{x}[t+1] &= A x[t] + B_2u[t] + \delta_x[t], \\
y[t] &= C_2x[t] + \delta_y[t].
\end{aligned}
\end{equation*}
Then, with a stabilizing controller $\mathbf{u} = \mathbf{K}\mathbf{y}$, the system responses
from perturbations $(\mathbf{\delta_x}, \mathbf{\delta_y})$ to $(\mathbf{x},\mathbf{u})$ are
\begin{equation} \label{eq:LTIsls}
\begin{aligned}
\begin{bmatrix} \mathbf{x} \\\mathbf{u} \end{bmatrix} = \begin{bmatrix} \mathbf{R} & \mathbf{N}\\ \mathbf{M} & \mathbf{L} \end{bmatrix} \begin{bmatrix} \mathbf{\delta_x} \\ \mathbf{\delta_y} \end{bmatrix},
\end{aligned}
\end{equation}
where the system responses $\{\mathbf{R}, \mathbf{M}, \mathbf{N}, \mathbf{L}\}$ are in the following affine subspace~\cite{wang2019system}
\begin{subequations} \label{eq:slp}
\begin{align}
\begin{bmatrix}zI - A & -B_2 \end{bmatrix}\begin{bmatrix} \mathbf{R} & \mathbf{N}\\ \mathbf{M} & \mathbf{L} \end{bmatrix} & = \begin{bmatrix}I & 0\end{bmatrix}, \label{eq:slp_s1}\\
\begin{bmatrix} \mathbf{R} & \mathbf{N}\\ \mathbf{M} & \mathbf{L} \end{bmatrix} \begin{bmatrix}zI - A \\ -C_2 \end{bmatrix} & = \begin{bmatrix}I \\ 0\end{bmatrix}, \label{eq:slp_s2} \\
\mathbf{R}, \mathbf{M}, \mathbf{N} \in \frac{1}{z} \mathcal{RH}_{\infty}, \quad& \mathbf{L} \in \mathcal{RH}_{\infty}. \label{eq:slp_s3}
\end{align}
\end{subequations}
In~\cite{wang2019system}, it is proved that for strictly proper\footnote{The equivalence~\eqref{eq:sls} only holds for strictly proper plants, \emph{i.e.}, $D_{22} = 0$. For a general proper plant $D_{22} \neq 0$, the authors in~\cite{wang2019system} present another controller implementation that internally stabilizes the system; see~\cite[Section III.D]{wang2019system}. Instead, Youla~\eqref{eq:youla} and input-output~\eqref{eq:iop} parameterizations work for both strictly proper and general proper plants. Throughout the paper, we assume a strictly proper plant for the system-level parameterization. } plant $\mathbf{P}_{22}$, the set of all internally stabilizing controllers can be written as
\begin{equation} \label{eq:sls}
\begin{aligned}
\mathcal{C}_{\text{stab}} = \{\mathbf{K} = \mathbf{L} - \mathbf{M}\mathbf{R}^{-1}&\mathbf{N} \mid \mathbf{R}, \, \mathbf{M}, \, \mathbf{N}, \, \mathbf{L} \; \text{are in the } \\
&\text{affine subspace~\eqref{eq:slp_s1}-\eqref{eq:slp_s3}} \}.
\end{aligned}
\end{equation}
Also, the cost function $f(\mathbf{P},\mathbf{K})$ can be expressed in terms of the system responses $\mathbf{R}, \mathbf{M}, \mathbf{N}, \mathbf{L}$. In particular, Problem~\eqref{eq:OCP} can be equivalently reformulated as
\begin{equation} \label{eq:OCPsls}
\begin{aligned}
\min_{\mathbf{R}, \mathbf{M}, \mathbf{N}, \mathbf{L}} \quad &\left\| \begin{bmatrix} C_1 & D_{12} \end{bmatrix}\begin{bmatrix} \mathbf{R} & \mathbf{N}\\ \mathbf{M} & \mathbf{L} \end{bmatrix} \begin{bmatrix} B_1 \\ D_{21} \end{bmatrix} + D_{11} \right\| \\
\text{subject to} \quad & \eqref{eq:slp_s1}-\eqref{eq:slp_s3}.
\end{aligned}
\end{equation}
It is easy to see that~\eqref{eq:OCPsls} is convex in terms of $\mathbf{R}, \mathbf{M}, \mathbf{N}, \mathbf{L}$.
\subsection{Input-output parameterization (IOP)}
\begin{figure}
\caption{Input-output stability.}
\label{fig:IOS}
\end{figure}
Recently, an input-output parameterization for $\mathcal{C}_{\text{stab}}$ was introduced in~\cite{furieri2019input}. As shown in Fig.~\ref{fig:IOS}, the idea is based on a classical internal stability result in terms of the following closed-loop responses
\begin{equation} \label{eq:LTIio}
\begin{aligned}
\begin{bmatrix} \mathbf{y} \\\mathbf{u} \end{bmatrix} = \begin{bmatrix} \mathbf{Y} & \mathbf{W}\\ \mathbf{U} & \mathbf{Z}\end{bmatrix} \begin{bmatrix} \mathbf{\delta_y} \\ \mathbf{\delta_u} \end{bmatrix},
\end{aligned}
\end{equation}
where $\delta_{\mathbf{u}}$ is a disturbance in the input, \emph{i.e.}, $\mathbf{u} = \mathbf{K}y + \delta_{\mathbf{u}}$.
Under Assumption 1, it is known that $\mathbf{K}$ internally stabilizes $\mathbf{P}$ if and only if the four transfer matrices in~\eqref{eq:LTIio} are stable~\cite{francis1987course}. With a stabilizing controller $\mathbf{K}$, the closed-loop responses $\mathbf{Y}, \mathbf{U}, \mathbf{W}, \mathbf{Z}$ are in the following affine subspace~\cite{furieri2019input}
\begin{subequations} \label{Eq:Param}
\begin{align}
&\begin{bmatrix}
I&-\mathbf{P}_{22}
\end{bmatrix}\begin{bmatrix}
\mathbf{Y}&\mathbf{W}\\\mathbf{U}&\mathbf{Z}
\end{bmatrix}=\begin{bmatrix}
I&0
\end{bmatrix}\,, \label{eq:aff1}\\
& \begin{bmatrix}
\mathbf{Y}&\mathbf{W}\\\mathbf{U}&\mathbf{Z}
\end{bmatrix}\begin{bmatrix}
-\mathbf{P}_{22}\\I
\end{bmatrix}=\begin{bmatrix}
0\\I
\end{bmatrix}\label{eq:aff2}\,,\\
&\begin{matrix}
\mathbf{Y}, \mathbf{U}, \mathbf{W}, \mathbf{Z}\in \mathcal{RH}_\infty.
\end{matrix}\label{eq:aff3}
\end{align}
\end{subequations}
It is shown in~\cite{furieri2019input} that the set of all internally stabilizing controllers can be represented as
\begin{equation} \label{eq:iop}
\begin{aligned}
\mathcal{C}_{\text{stab}} = \{\mathbf{K} = \mathbf{U}\mathbf{Y}^{-1} \mid \mathbf{Y}, &\mathbf{U}, \mathbf{W}, \mathbf{Z} \; \text{are in the} \\
&\text{affine subspace~\eqref{eq:aff1}-\eqref{eq:aff3}} \}.
\end{aligned}
\end{equation}
Furthermore, we have $f(\mathbf{P},\mathbf{K}) = \mathbf{P}_{11} + \mathbf{P}_{12}\mathbf{U}\mathbf{P}_{21}$~\cite{furieri2019input}. Accordingly, Problem~\eqref{eq:OCP} can be equivalently reformulated in terms of the system responses as
\begin{equation} \label{eq:OCPiop}
\begin{aligned}
\min_{\mathbf{Y}, \mathbf{U}, \mathbf{W}, \mathbf{Z}} \quad &\left\| \mathbf{P}_{11} + \mathbf{P}_{12}\mathbf{U}\mathbf{P}_{21}\right\| \\
\text{subject to} \quad & ~\eqref{eq:aff1}-\eqref{eq:aff3}.
\end{aligned}
\end{equation}
It is easy to see that~\eqref{eq:OCPiop} is convex in terms of $\mathbf{Y}, \mathbf{U}, \mathbf{W}, \mathbf{Z}$.
\section{Explicit Equivalence of Youla Parameterization, SLP, and IOP}
\label{Section:equivalence}
As discussed in the last section, the set of all stabilizing controllers $\mathcal{C}_{\text{stab}}$ can be parameterized in three different ways, \emph{i.e.},~\eqref{eq:youla},~\eqref{eq:sls}, and~\eqref{eq:iop}, and the optimal control problem~\eqref{eq:OCP} admits three equivalent convex reformulations, \emph{i.e.},~\eqref{eq:OCPYoula},~\eqref{eq:OCPsls}, and~\eqref{eq:OCPiop}. Implicitly,~\eqref{eq:youla},~\eqref{eq:sls}, and~\eqref{eq:iop} are equivalent. However, an explicit relationship between Youla parameterization, SLP, and IOP is not clear from their definitions.
In this section, we present explicit \emph{affine mappings} among Youla parameterization, SLP, and IOP. The consequences are as follows: 1) any convex system-level synthesis (SLS) introduced by~\cite{wang2019system} can be equivalently reformulated into a convex problem in terms of the Youla parameter $\mathbf{Q}$ or the input-output parameters $\mathbf{Y}, \mathbf{U}, \mathbf{W}, \mathbf{Z}$, and vice versa; 2) building on the explicit affine mappings, we show that the notion of quadratic invariance~\cite{rotkowitz2006characterization} allows for equivalent convex reformulations of classical distributed optimal control in either Youla parameterization, SLP, or IOP.
\subsection{Explicit equivalence}
The explicit equivalence between Youla parameterization and IOP is presented in~\cite{furieri2019input}:
\begin{theorem}[\!\cite{furieri2019input}]
\label{th:Youla_eq}
Let $\mathbf{U}_r,\mathbf{V}_r,\mathbf{U}_l,\mathbf{V}_l,\mathbf{M}_r,\mathbf{M}_l,\mathbf{N}_r,\mathbf{N}_l$ be any doubly-coprime factorization of $\mathbf{P}_{22}$. The following statements hold.
\begin{enumerate}
\item For any $\mathbf{Q} \in \mathcal{RH}_\infty$, the following transfer matrices
\begin{subequations} \label{eq:youla_iop}
\begin{align}
&\mathbf{Y}=(\mathbf{U}_r-\mathbf{N}_r\mathbf{Q})\mathbf{M}_l\,, \label{eq:Q_to_X_1}\\
&\mathbf{U}=(\mathbf{V}_r-\mathbf{M}_r\mathbf{Q})\mathbf{M}_l\,,\\
&\mathbf{W}=(\mathbf{U}_r-\mathbf{N}_r\mathbf{Q})\mathbf{N}_l\,,\\
&\mathbf{Z}=I+(\mathbf{V}_r-\mathbf{M}_r\mathbf{Q})\mathbf{N}_l\,,\label{eq:Q_to_X_2}
\end{align}
\end{subequations}
belong to the affine subspace \eqref{eq:aff1}-\eqref{eq:aff3} and are such that $\mathbf{U}\mathbf{Y}^{-1}=(\mathbf{V}_r-\mathbf{M}_r\mathbf{Q})(\mathbf{U}_r-\mathbf{N}_r\mathbf{Q})^{-1}$.
\item For any $(\mathbf{Y},\mathbf{U},\mathbf{W},\mathbf{Z})$ in the affine subspace (\ref{eq:aff1})-(\ref{eq:aff3}), the transfer matrix
\begin{equation}
\label{eq:Youla_with_XYWZ}
\mathbf{Q}=\mathbf{V}_l\mathbf{Y}\mathbf{U}_r-\mathbf{U}_l\mathbf{U}\mathbf{U}_r-\mathbf{V}_l\mathbf{W}\mathbf{V}_r+\mathbf{U}_l\mathbf{Z}\mathbf{V}_r-\mathbf{V}_l\mathbf{U}_r\,,
\end{equation}
is such that $\mathbf{Q} \in \mathcal{RH}_\infty$ and $(\mathbf{V}_r-\mathbf{M}_r\mathbf{Q})(\mathbf{U}_r-\mathbf{N}_r\mathbf{Q})^{-1}=\mathbf{U}\mathbf{Y}^{-1}$.
\end{enumerate}
\end{theorem}
Theorem~\ref{th:Youla_eq} presents explicit {affine mappings} between Youla parameterization and IOP: {any element in the Youla parameterization~\eqref{eq:youla} corresponds to an element in the IOP~\eqref{eq:iop}, and they represent the same controller.} The following result presents explicit {affine mappings} between SLP and IOP.
\begin{theorem}
\label{th:slp_eq}
Consider a strictly proper plant $\mathbf{P}_{22}$, \emph{i.e.}, $D_{22} = 0$. The following statements hold.
\begin{enumerate}
\item For any $\mathbf{R}, \mathbf{M}, \mathbf{N}, \mathbf{L}$ satisfying the affine subspace~\eqref{eq:slp_s1}-\eqref{eq:slp_s3}, the transfer matrices
\begin{subequations} \label{eq:slp-iop}
\begin{align}
\mathbf{Y} &= C_2\mathbf{N} + I, \label{eq:slp-iop1}\\
\mathbf{U} &= \mathbf{L}, \label{eq:slp-iop2}\\
\mathbf{W} &= C_2\mathbf{R}B_2, \label{eq:slp-iop3}\\
\mathbf{Z} &= \mathbf{M}B_2 + I, \label{eq:slp-iop4}
\end{align}
\end{subequations}
belong to the affine subspace~\eqref{eq:aff1}-\eqref{eq:aff3} and are such that
$
\mathbf{L} - \mathbf{M}\mathbf{R}^{-1}\mathbf{N} = \mathbf{U}\mathbf{Y}^{-1}.
$
\item For any $\mathbf{Y}, \mathbf{U}, \mathbf{W}, \mathbf{Z}$ satisfying the affine subspace~\eqref{eq:aff1}-\eqref{eq:aff3}, the transfer matrices
\begin{subequations} \label{eq:iop-sls}
\begin{align}
\mathbf{R} &= (zI - A)^{-1} + (zI - A)^{-1}B_2\mathbf{U}C_2(zI - A)^{-1} \label{eq:iop-sls1} \\
\mathbf{M} & = \mathbf{U}C_2(zI - A)^{-1}, \label{eq:iop-sls2}\\
\mathbf{N} &= (zI - A)^{-1}B_2\mathbf{U}, \label{eq:iop-sls3} \\
\mathbf{L} &= \mathbf{U}, \label{eq:iop-sls4}
\end{align}
\end{subequations}
belong to the affine subspace~\eqref{eq:slp_s1}-\eqref{eq:slp_s3} and are such that
$
\mathbf{U}\mathbf{Y}^{-1} = \mathbf{L} - \mathbf{M}\mathbf{R}^{-1}\mathbf{N}.
$
\end{enumerate}
\end{theorem}
\begin{proof}
\emph{Statement 1}: considering the affine relationships~\eqref{eq:slp_s1}-\eqref{eq:slp_s3} and $\mathbf{P}_{22} = C_2(zI - A)^{-1}B_2$, we have the following algebraic equalities:
$$
\begin{aligned}
\mathbf{Y} - \mathbf{P}_{22}\mathbf{U} &= C_2\mathbf{N} + I - \mathbf{P}_{22}\mathbf{L} \\
&= C_2(\mathbf{N} - (zI - A)^{-1}B_2\mathbf{L}) + I = I, \\
\mathbf{W} - \mathbf{P}_{22}\mathbf{Z} &= C_2\mathbf{R}B_2 - \mathbf{P}_{22}( \mathbf{M}B_2 + I) \\
&= C_2\left(\mathbf{R} - (zI - A)^{-1}B_2\mathbf{M} -(zI - A)^{-1}\right)B_2 \\
&= 0, \\
\mathbf{Y}\mathbf{P}_{22} - \mathbf{W} &= (C_2\mathbf{N} + I)\mathbf{P}_{22} - C_2\mathbf{R}B_2 \\
&= C_2\left( \mathbf{N}C_2(zI - A)^{-1} + (zI - A)^{-1} - \mathbf{R}\right)B_2 \\
&= 0,\\
\end{aligned}
$$
$$
\begin{aligned}
-\mathbf{U}\mathbf{P}_{22} + \mathbf{Z} &= -\mathbf{L}\mathbf{P}_{22} + \mathbf{M}B_2 + I \\
& = (-\mathbf{L}C_2(zI-A)^{-1} + \mathbf{M})B_2 + I \\
&= I.
\end{aligned}
$$
Therefore,~\eqref{eq:aff1} and~\eqref{eq:aff2} are satisfied. Obviously, the transfer matrices in~\eqref{eq:slp-iop} are stable, \emph{i.e.}~\eqref{eq:aff3} is satisfied.
In addition, from~\eqref{eq:slp_s2}, we have $\mathbf{R} = (I + \mathbf{N}C_2)(zI - A)^{-1}$. Then,
$$
\begin{aligned}
\mathbf{L} - \mathbf{M}\mathbf{R}^{-1}\mathbf{N} &= \mathbf{L} - \mathbf{M}(zI - A)(I + \mathbf{N}C_2)^{-1}\mathbf{N} \\
&= \mathbf{L} - \mathbf{L}C_2(I + \mathbf{N}C_2)^{-1}\mathbf{N} \\
&= \mathbf{L} - \mathbf{L}C_2\mathbf{N}(I + C_2\mathbf{N})^{-1} \\
& = \mathbf{L}(I + C_2\mathbf{N})^{-1} \\
& = \mathbf{U}\mathbf{Y}^{-1}.
\end{aligned}
$$
\emph{Statement 2}:
In the Appendix~\ref{Sec:stable}, we verify algebraically that the transfer matrices $\mathbf{R},\mathbf{M},\mathbf{N},\mathbf{L}$ defined in~\eqref{eq:iop-sls} are exactly the closed-loop responses in~\eqref{eq:LTIsls} with controller $\mathbf{K} = \mathbf{U}\mathbf{Y}^{-1}$.
Since $\mathbf{K} = \mathbf{U}\mathbf{Y}^{-1}$ is internally stabilizing $\mathbf{P}_{22}$, the transfer matrices $\mathbf{R},\mathbf{M},\mathbf{N},\mathbf{L}$ defined in~\eqref{eq:iop-sls} naturally satisfy the constraints~\eqref{eq:slp_s1}-\eqref{eq:slp_s3}.
Finally, we can check that the transfer matrices $\mathbf{R},\mathbf{M},\mathbf{N}, \mathbf{L}$ defined in~\eqref{eq:iop-sls} satisfy
$
\mathbf{U}\mathbf{Y}^{-1} = \mathbf{L} - \mathbf{M}\mathbf{R}^{-1}\mathbf{N}.
$
This completes the proof.
\end{proof}
Combining Theorems~\ref{th:Youla_eq} and~\ref{th:slp_eq}, we arrive at the explicit affine mappings between Youla parameterization and SLP, which was not provided in~\cite{wang2019system}.
\begin{theorem}
\label{th:Youla_sls}
Let $\mathbf{U}_r,\mathbf{V}_r,\mathbf{U}_l,\mathbf{V}_l,\mathbf{M}_r,\mathbf{M}_l,\mathbf{N}_r,\mathbf{N}_l$ be any doubly-coprime factorization of the strictly proper system $\mathbf{P}_{22}$. The following statements hold.
\begin{enumerate}
\item For any $\mathbf{Q} \in \mathcal{RH}_\infty$, the following transfer matrices
\begin{subequations} \label{eq:youla_slp}
\begin{align}
\mathbf{R} &= (zI - A)^{-1} + \nonumber\\
(zI - &A)^{-1}B_2(\mathbf{V}_r-\mathbf{M}_r\mathbf{Q})\mathbf{M}_lC_2(zI - A)^{-1}, \label{eq:youla_slp1} \\
\mathbf{M} & = (\mathbf{V}_r-\mathbf{M}_r\mathbf{Q})\mathbf{M}_lC_2(zI - A)^{-1}, \label{eq:youla_slp2}\\
\mathbf{N} &= (zI - A)^{-1}B_2(\mathbf{V}_r-\mathbf{M}_r\mathbf{Q})\mathbf{M}_l, \label{eq:youla_slp3}\\
\mathbf{L} &= (\mathbf{V}_r-\mathbf{M}_r\mathbf{Q})\mathbf{M}_l, \label{eq:youla_slp4}
\end{align}
\end{subequations}
belong to the affine subspace \eqref{eq:slp_s1}-\eqref{eq:slp_s3} and are such that $\mathbf{L} - \mathbf{M}\mathbf{R}^{-1}\mathbf{N}=(\mathbf{V}_r-\mathbf{M}_r\mathbf{Q})(\mathbf{U}_r-\mathbf{N}_r\mathbf{Q})^{-1}$.
\item For any $(\mathbf{R},\mathbf{M},\mathbf{N},\mathbf{L})$ in the affine subspace \eqref{eq:slp_s1}-\eqref{eq:slp_s3}, the transfer matrix
\begin{equation} \label{eq:Youla_with_RMNL}
\begin{aligned}
\mathbf{Q} =&\mathbf{V}_lC_2\mathbf{N}\mathbf{U}_r-\mathbf{U}_l\mathbf{L}\mathbf{U}_r-\mathbf{V}_l C_2\mathbf{R}B_2\mathbf{V}_r \\
& \qquad \qquad \qquad \qquad +\mathbf{U}_l\mathbf{M}B_2\mathbf{V}_r + \mathbf{U}_l\mathbf{V}_r
\end{aligned}
\end{equation}
is such that $\mathbf{Q} \in \mathcal{RH}_\infty$ and $(\mathbf{V}_r-\mathbf{M}_r\mathbf{Q})(\mathbf{U}_r-\mathbf{N}_r\mathbf{Q})^{-1}=\mathbf{L} - \mathbf{M}\mathbf{R}^{-1}\mathbf{N}$.
\end{enumerate}
\end{theorem}
\begin{proof}
Statement 1 directly follows by combining the statement 1 of Theorem~\ref{th:Youla_eq} with the statement 2 of Theorem~\ref{th:slp_eq}.
Combining the statement 2 of Theorem~\ref{th:Youla_eq} with the statement 1 of Theorem~\ref{th:slp_eq} leads to
\begin{equation*}
\begin{aligned}
\mathbf{Q} &=\mathbf{V}_l(C_2\mathbf{N} + I)\mathbf{U}_r-\mathbf{U}_l\mathbf{L}\mathbf{U}_r-\mathbf{V}_l C_2\mathbf{R}B_2\mathbf{V}_r \\
& \qquad +\mathbf{U}_l(\mathbf{M}B_2 + I)\mathbf{V}_r-\mathbf{V}_l\mathbf{U}_r\,, \\
&=\mathbf{V}_lC_2\mathbf{N}\mathbf{U}_r-\mathbf{U}_l\mathbf{L}\mathbf{U}_r-\mathbf{V}_l C_2\mathbf{R}B_2\mathbf{V}_r \\
& \qquad +\mathbf{U}_l\mathbf{M}B_2\mathbf{V}_r + \mathbf{U}_l\mathbf{V}_r.
\end{aligned}
\end{equation*}
This completes the proof.
\end{proof}
An overview of the equivalence of Youla parameterization, SLP, and IOP is shown in Fig.~\ref{Fig:Equivalence}.
\begin{remark}[Closed-loop convexity]
In both SLP and IOP, the parameters $(\mathbf{R},\mathbf{M},\mathbf{N},\mathbf{L})$ and $(\mathbf{Y},\mathbf{U},\mathbf{W},\mathbf{Z})$ have explicit and distinct physical interpretations as corresponding closed-loop transfer matrices. Also, the Youla parameter $\mathbf{Q}$ can be viewed as a closed-loop transfer matrix when the plant $\mathbf{P}_{22}$ is stable (see Remark~\ref{remark:stableQ}). In this sense, Youla parameterization, SLP, and IOP all shift the controller synthesis task from the design of a controller in~\eqref{eq:OCP}, which is non-convex, to the design of closed loop responses, resulting in convex formulations~\eqref{eq:OCPYoula},~\eqref{eq:OCPsls}, and~\eqref{eq:OCPiop}. Note that this idea of closed-loop convexity has been extensively discussed in the book~\cite{boyd1991linear}, and a comprehensive historical note is given in~\cite[Chapter 16.3]{boyd1991linear}.
\end{remark}
\begin{remark}[Numerical computation]
After computing a doubly-coprime factorization of the plant, the Youla parameter $\mathbf{Q}$ is free in $\mathcal{RH}_{\infty}$ for parameterizing $\mathcal{C}_{\emph{\text{stab}}}$, and there are no equality constraints for achievable closed-loop responses. This feature allows to reformulate Problem~\eqref{eq:OCP} as a model matching problem~\eqref{eq:OCPYoula}, which can be reduced to the Nehari problem and then solved via the state-space method in~\cite{francis1987course}. Instead, both SLP and IOP do not require to compute a doubly-coprime factorization, but have explicit affine constraints for achievable closed-loop responses.
Since the decision variables in constraints \eqref{eq:slp_s1}-\eqref{eq:slp_s3} and~\eqref{eq:aff1}-\eqref{eq:aff3} are infinite dimensional, there is no immediately efficient numerical method for solving~\eqref{eq:OCPsls} or~\eqref{eq:OCPiop}. The Ritz approximation~\cite[Chapter 15]{boyd1991linear} is one method for solving infinite dimensional optimization problems. Specifically, for discrete-time systems, the finite impulse response (FIR) approximation is a practical choice~\cite{wang2019system,furieri2019input}.
\end{remark}
\begin{figure}
\caption{Equivalence of Youla paramterization, System-level parameterization (SLP), and Input-output parameterization (IOP).}
\label{Fig:Equivalence}
\end{figure}
\subsection{Convex system-level synthesis}
In~\cite{wang2019system}, the authors introduced a general framework of system-level synthesis (SLS), which defines ``the broadest
known class of constrained optimal control problems that can be solved using convex programming'' (cf.~\cite{anderson2019system}).
Thanks to the full equivalence in Theorems~\ref{th:Youla_eq}-\ref{th:Youla_sls}, we can show that 1) any SLS problem can be equivalently formulated in the Youla or input-output framework, 2) any convex SLS can be addressed by solving a convex problem in terms of Youla parameter $\mathbf{Q}$ or input-output parameters $\mathbf{Y}, \mathbf{U}, \mathbf{W}, \mathbf{Z}$.
Let $g(\cdot)$ be a functional capturing a desired measure of the performance of the plant $\mathbf{P}_{22}$, and let $\mathcal{S}$ be a system-level constraint. The SLS problem in~\cite{wang2019system} is posed as
\begin{equation} \label{eq:slsproblem}
\begin{aligned}
\min_{\mathbf{R}, \mathbf{M}, \mathbf{N}, \mathbf{L}} \quad & g(\mathbf{R}, \mathbf{M}, \mathbf{N}, \mathbf{L}) \\
\text{subject to} \quad & \eqref{eq:slp_s1}-\eqref{eq:slp_s3}, \\
& \begin{bmatrix} \mathbf{R} & \mathbf{N} \\
\mathbf{M} & \mathbf{L} \end{bmatrix} \in \mathcal{S}.
\end{aligned}
\end{equation}
We refer the interested reader to~\cite{wang2019system} for a detailed discussion of SLS. Then, we have the following result.
\begin{theorem} \label{theo:equivalence}
Let $\mathbf{U}_r,\mathbf{V}_r,\mathbf{U}_l,\mathbf{V}_l,\mathbf{M}_r,\mathbf{M}_l,\mathbf{N}_r,\mathbf{N}_l$ be any doubly-coprime factorization of the strictly proper system $\mathbf{P}_{22}$. The following statements hold.
\begin{enumerate}
\item The SLS problem~\eqref{eq:slsproblem} is equivalent to the following problem in Youla parameter $\mathbf{Q}$,
\begin{equation} \label{eq:sls-youla}
\begin{aligned}
\min_{\mathbf{Q}} \quad\; & g_1(\mathbf{Q}) \\
\text{subject to} \quad & \begin{bmatrix} f_1(\mathbf{Q}) & f_3(\mathbf{Q}) \\
f_2(\mathbf{Q}) & f_4(\mathbf{Q}) \end{bmatrix} \in \mathcal{S},
\end{aligned}
\end{equation}
where $f_1(\mathbf{Q}), f_2(\mathbf{Q}), f_3(\mathbf{Q}), f_4(\mathbf{Q})$ are defined by~\eqref{eq:youla_slp1} -\eqref{eq:youla_slp4}, respectively, and $$
g_1(\mathbf{Q}) := g\left(f_1(\mathbf{Q}), f_2(\mathbf{Q}), f_3(\mathbf{Q}), f_4(\mathbf{Q})\right).
$$
\item The SLS problem~\eqref{eq:slsproblem} is equivalent to the following problem in input-output parameters $\mathbf{Y}, \mathbf{U}, \mathbf{W}, \mathbf{Z}$,
\begin{equation} \label{eq:sls-iop}
\begin{aligned}
\min_{\mathbf{Y},\mathbf{U},\mathbf{W},\mathbf{Z}} \quad\; & \hat{g}_1(\mathbf{U}) \\
\text{subject to} \quad & ~\eqref{eq:aff1}-\eqref{eq:aff3}\\
\quad &\begin{bmatrix} \hat{f}_1(\mathbf{U}) & \hat{f}_3(\mathbf{U}) \\
\hat{f}_2(\mathbf{U}) & \hat{f}_4(\mathbf{U}) \end{bmatrix} \in \mathcal{S},
\end{aligned}
\end{equation}
where $\hat{f}_1(\mathbf{U}), \hat{f}_2(\mathbf{U}), \hat{f}_3(\mathbf{U}), \hat{f}_4(\mathbf{U})$ are defined by~\eqref{eq:iop-sls1} -\eqref{eq:iop-sls4}, respectively, and $$
\hat{g}_1(\mathbf{U}) := g\left(\hat{f}_1(\mathbf{U}), \hat{f}_2(\mathbf{U}), \hat{f}_3(\mathbf{U}), \hat{f}_4(\mathbf{U})\right).
$$
\item If the SLS problem~\eqref{eq:slsproblem} is convex, then Problems~\eqref{eq:sls-youla} and~\eqref{eq:sls-iop} are both convex.
\end{enumerate}
\end{theorem}
\begin{proof}
The first two statements directly follow from Theorems~\ref{th:slp_eq} and~\ref{th:Youla_sls}.
The last statement follows from the facts that $f_i(\mathbf{Q})$ and $\hat{f}_i(\mathbf{Q}), i = 1, \ldots, 4$, are all affine. Then, if $\mathcal{S}$ is a convex set and $g(\cdot)$ is a convex functional, the constraint in~\eqref{eq:sls-youla} (resp. ~\eqref{eq:sls-iop}) defines a convex set in $\mathbf{Q}$ (resp. $\mathbf{Y}, \mathbf{U}, \mathbf{W}, \mathbf{Z}$), and $g_1(\cdot)$ (or $\hat{g}_1(\cdot)$) is convex.
\end{proof}
\subsection{Distributed optimal control and quadratic invariance (QI)}
Unlike SLS, which impose constraints on closed-loop responses (see~\eqref{eq:slsproblem}), the classical distributed optimal control problem typically considers a subspace constraint $\mathcal{L}$ on the controller $\mathbf{K}$, which is formulated as~\cite{rotkowitz2006characterization, sabuau2014youla, qi2004structured}
\begin{equation} \label{eq:OCPsparsity}
\begin{aligned}
\min_{\mathbf{K}} \quad &\|f(\mathbf{P},\mathbf{K})\| \\
\text{subject to} \quad & \mathbf{K} \in \mathcal{C}_{\text{stab}} \cap \mathcal{L}.
\end{aligned}
\end{equation}
It is shown in~\cite{rotkowitz2006characterization, sabuau2014youla} that if the subspace constraint $\mathcal{L}$ is \emph{quadratically invariant} (QI) under $\mathbf{P}_{22}$ (\emph{i.e.}, $\mathbf{K}\mathbf{P}_{22}\mathbf{K} \in \mathcal{L}, \forall \mathbf{K} \in \mathcal{L}$), then we have
$$
\begin{aligned}
\mathcal{C}_{\text{stab}} \cap \mathcal{L} = \{\mathbf{K} = &(\mathbf{V}_r - \mathbf{M}_r\mathbf{Q})(\mathbf{U}_r - \mathbf{N}_r\mathbf{Q})^{-1} \mid \\
& (\mathbf{V}_r-\mathbf{M}_r\mathbf{Q})\mathbf{M}_l \in \mathcal{L}, \mathbf{Q} \in \mathcal{RH}_{\infty}\}.
\end{aligned}
$$
Problem~\eqref{eq:OCPsparsity} can thus be equivalently formulated as a convex problem in $\mathbf{Q}$~\cite{rotkowitz2006characterization, sabuau2014youla},
\begin{equation} \label{eq:OCPYoula_sparsity}
\begin{aligned}
\min_{\mathbf{Q}} \quad &\|\mathbf{T}_{11} + \mathbf{T}_{12}\mathbf{Q}\mathbf{T}_{21}\| \\
\text{subject to} \quad & (\mathbf{V}_r-\mathbf{M}_r\mathbf{Q})\mathbf{M}_l \in \mathcal{L}, \\
& \mathbf{Q} \in \mathcal{RH}_{\infty}.
\end{aligned}
\end{equation}
Considering the equivalence shown in Theorems~\ref{th:Youla_eq} and~\ref{th:Youla_sls}, the following corollaries are immediate.
\begin{corollary}[QI with IOP]
\label{prop:iop_structured}
If $\mathcal{L}$ is QI under $\mathbf{P}_{22}$, then
\begin{enumerate}
\item We have $$
\begin{aligned}
\mathcal{C}_{\text{stab}} \cap \mathcal{L} = \{\mathbf{K} = &\mathbf{U}\mathbf{Y}^{-1} \mid \mathbf{Y}, \mathbf{U}, \mathbf{W}, \mathbf{Z} \; \text{are in the} \\
&\quad\text{affine subspace~\eqref{eq:aff1}-\eqref{eq:aff3}}, \mathbf{U} \in \mathcal{L} \}.
\end{aligned}
$$
\item Problem~\eqref{eq:OCPsparsity} can be equivalently formulated as a convex problem
\begin{equation} \label{eq:OCPiop_s}
\begin{aligned}
\min_{\mathbf{Y}, \mathbf{U}, \mathbf{W}, \mathbf{Z}} \quad &\left\| \mathbf{P}_{11} + \mathbf{P}_{12}\mathbf{U}\mathbf{P}_{21}\right\| \\
\text{subject to} \quad &\eqref{eq:aff1}-\eqref{eq:aff3}, \\
& \; \mathbf{U} \in \mathcal{L}.
\end{aligned}
\end{equation}
\end{enumerate}
\end{corollary}
\begin{corollary}[QI with SLA] \label{prop:sls_structured}
If $\mathcal{L}$ is QI under $\mathbf{P}_{22}$, then
\begin{enumerate}
\item We have $$
\begin{aligned}
\mathcal{C}_{\text{stab}} \cap \mathcal{L} = \{\mathbf{K} &= \mathbf{L} - \mathbf{M}\mathbf{R}^{-1}\mathbf{N} \mid \mathbf{R}, \, \mathbf{M}, \, \mathbf{N}, \, \mathbf{L} \; \text{are } \\
&\text{in the affine subspace~\eqref{eq:slp_s1}-\eqref{eq:slp_s3}}, \mathbf{L} \in \mathcal{L} \}.
\end{aligned}
$$
\item Problem~\eqref{eq:OCPsparsity} can be equivalently formulated as a convex problem
\end{enumerate}
\begin{equation} \label{eq:OCPiop_s}
\begin{aligned}
\min_{\mathbf{R}, \mathbf{M}, \mathbf{N}, \mathbf{L}} \quad &\left\| \begin{bmatrix} C_1 & D_{12} \end{bmatrix}\begin{bmatrix} \mathbf{R} & \mathbf{N}\\ \mathbf{M} & \mathbf{L} \end{bmatrix} \begin{bmatrix} B_1 \\ D_{21} \end{bmatrix} + D_{11}\right\| \\
\text{subject to} \quad & \eqref{eq:slp_s1}-\eqref{eq:slp_s3}, \\
& \mathbf{L} \in \mathcal{L}.
\end{aligned}
\end{equation}
\end{corollary}
Corollary~\ref{prop:iop_structured} is the same as Theorem~3 of \cite{furieri2019input} and Corollary~\ref{prop:sls_structured} is consistent with Theorem 3 of~\cite{wang2019system}.
One main insight is that the specialized proofs in \cite{furieri2019input,wang2019system} may be not needed anymore, thanks to the explicit affine mappings between Youla, SLP and IOP.
We also note that the original proof of Theorem 3 in~\cite{wang2019system} is not complete: it relies on that the affine mapping $\mathbf{L} = (\mathbf{V}_r-\mathbf{M}_r\mathbf{Q})\mathbf{M}_l$ is invertible. However, given $\mathbf{L} \in \mathcal{RH}_{\infty}$, it is not immediate to see that $\mathbf{Q} = \mathbf{M}_r^{-1}(\mathbf{V}_r-\mathbf{L}\mathbf{M}_r^{-1})$ is stable. We complete this fact via the construction of $\mathbf{Q}$ in~\eqref{eq:Youla_with_RMNL}.
\begin{remark}
It should be noted that SLS~\eqref{eq:slsproblem} and the classical distributed control problem~\eqref{eq:OCPsparsity} are two distinct formulations: 1) the former imposes constraints on closed-loop responses while the latter imposes a constraint on controller $\mathbf{K}$; 2) feasibility of the former does not imply feasibility of the latter, and vice-versa. Only when the QI property holds, can Problem~\eqref{eq:OCPsparsity} be equivalently reformulated into a convex problem in terms of Youla, system-level, or input-output parameters. Based on the results in~\cite{lessard2015convexity}, QI is necessary for the existence of such equivalent convex reformulation. For systems with QI constraints, SLS~\eqref{eq:slsproblem} can be equivalent to the classical problem~\eqref{eq:OCPsparsity}, as shown in Corollary~\ref{prop:sls_structured}; for the cases beyond QI, they are not directly comparable.
\end{remark}
\section{Distributed optimal control with non-QI constraints}
\label{Section:specialcase}
In this section, we highlight that for systems with non-QI constraints, we may derive convex approximations of~\eqref{eq:OCPsparsity} using Youla, system-level, or input-output parameters. In certain cases, a globally optimal solution can still be obtained. Our approximation procedure is consistent with the idea of sparsity invariance (SI)~\cite{Furieri2019Sparsity}.
In particular, we consider Example 1 in~\cite{wang2019system, anderson2019system}. We first present simplified versions of Youla, system-level, and input-output parameterizations for special cases of state feedback (for completeness, other simplified versions for stable plants are presented in Appendix~\ref{section:special}). Then, we show that Example 1 can be solved exactly using Youla, system-level, or input-output parameters via convex optimization.
\subsection{Simplified parameterizations for state feedback}
In~\cite{wang2019system}, it is shown that for state feedback where $C_2 = I, D_{22} = 0$, the set of internally stabilizing controllers is
\begin{equation} \label{eq:slp-stable-state}
\begin{aligned}
\mathcal{C}_{\text{stab}} = \{\mathbf{K} = \mathbf{M}\mathbf{R}^{-1} \bigm| & \begin{bmatrix} (zI -A) & -B_2 \end{bmatrix} \begin{bmatrix} \mathbf{R} \\ \mathbf{M} \end{bmatrix} = I, \\
&\qquad \qquad \mathbf{M},\mathbf{R} \in \frac{1}{z}\mathcal{RH}_{\infty}
\}.
\end{aligned}
\end{equation}
The proof in~\cite{wang2019system} is directly based on the definition of internal stability. As expected, this special case~\eqref{eq:slp-stable-state} can be reduced from the general case~\eqref{eq:sls} from purely algebraic operations. We provide this alternative proof in Appendix~\ref{section:proofB}.
For IOP and Youla parameterization, simplifications are possible with further assumptions.
\begin{corollary}[Input-output parameterization] \label{coro:iopstate}
Suppose $C_2 = I, D_{22}= 0$ and $B_2$ is invertible. We have
\begin{equation} \label{eq:iop-state}
\begin{aligned}
\mathcal{C}_{\text{stab}} = \bigg\{\mathbf{K} = (\mathbf{Z}-I)&\mathbf{W}^{-1} \bigm|
\begin{bmatrix} I & -\mathbf{P}_{22} \end{bmatrix} \begin{bmatrix} \mathbf{W} \\ \mathbf{Z} \end{bmatrix} = 0 \\ & \mathbf{Z} \in \mathcal{RH}_{\infty}, \mathbf{W} \in \frac{1}{z}\mathcal{RH}_{\infty}\bigg\}.
\end{aligned}
\end{equation}
\end{corollary}
\begin{proof}
We show that any controller in~\eqref{eq:iop-state} is an internally stabilizing controller in~\eqref{eq:iop}. The other direction is similar. Given any $\mathbf{W}, \mathbf{Z}$ satisfying the constraints in~\eqref{eq:iop-state}, we define
$
\mathbf{U} = (\mathbf{Z}-I)B_2^{-1}(zI - A) \in \mathcal{RH}_{\infty},
\mathbf{Y} = \mathbf{W}B_2^{-1}(zI - A) \in \mathcal{RH}_{\infty}.
$
Then, we can easily verify
$$
\begin{aligned}
\mathbf{Z}-\mathbf{U}\mathbf{P}_{22} = I, \mathbf{W}-\mathbf{Y}\mathbf{P}_{22} = 0,
\mathbf{Y}-\mathbf{P}_{22}\mathbf{U} = I.
\end{aligned}
$$
Thus, $\mathbf{Y},\mathbf{U},\mathbf{W},\mathbf{Z} $ above satisfy~\eqref{eq:aff1}-\eqref{eq:aff3}.
We also have
$$
\begin{aligned}
\mathbf{U}\mathbf{Y}^{-1} &= (\mathbf{Z}-I)B_2^{-1}(zI - A)(\mathbf{W}B_2^{-1}(zI - A))^{-1} \\
&= (\mathbf{Z}-I)\mathbf{W}^{-1}.
\end{aligned}
$$
This completes the proof.
\end{proof}
\begin{corollary}[Youla parameterization]
\label{Corollary:YoulaState}
Suppose $C_2 = B_2 = I, D_{22} = 0$. We have\footnote{Note that Corollary~\ref{Corollary:YoulaState} is only valid in discrete-time systems, since the doubly-coprime factorization~\eqref{eq:YoulaCo_state} has no counterpart in continuous time.}
$$
\begin{aligned}
\mathcal{C}_{\text{stab}} = \bigg\{\mathbf{K} = \left(-A - (I- \frac{1}{z}A)\mathbf{Q}\right)&\left(I - \frac{1}{z}\mathbf{Q}\right)^{-1}, \\
&\qquad \mathbf{Q} \in \mathcal{RH}_{\infty} \bigg\}
\end{aligned}
$$
\end{corollary}
\begin{proof}
The proof directly follows by choosing the following doubly-coprime factorization:
\begin{equation} \label{eq:YoulaCo_state}
\begin{aligned}
\mathbf{U}_l &= I, \mathbf{V}_l = -A, \mathbf{N}_l = \frac{1}{z}I, \mathbf{M}_l = I - \frac{1}{z}A,\\
\mathbf{U}_r &= I, \mathbf{V}_r = -A, \mathbf{N}_r = \frac{1}{z}I, \mathbf{M}_r = I - \frac{1}{z}A.\\
\end{aligned}
\end{equation}
\end{proof}
\subsection{Example 1 in~\cite{wang2019system, anderson2019system}}
Consider the following optimal control problem, which is Example 1 in~\cite{wang2019system, anderson2019system},
\begin{equation} \label{eq:Example1}
\begin{aligned}
\min_{\mathbf{K}} \quad &\lim_{T \rightarrow \infty}\frac{1}{T}\sum_{t=0}^T \mathbb{E}||x[t]||_2^2\\
\text{subject to}\quad &~x[t+1]=Ax[t]+u[t]+w[t],\\
\quad &~\mathbf{u}=\mathbf{K}\mathbf{x},
\end{aligned}
\end{equation}
where disturbance $w[t] \stackrel{\text{i.i.d}}{\sim} \mathcal{N}(0,I)$. It can be verified (\emph{e.g.}, via solving the discrete-time algebraic Riccati equation) that the globally optimal solution is the static feedback given by $ \mathbf{K}= -A$.
Assume that $A$ is sparse and let its supports define the adjacency matrix of a graph $\mathcal{G}$. Then, the optimal controller has a particular structure according to $\mathcal{G}$.
Now suppose that we attempt to solve problem~\eqref{eq:Example1} by converting it to
its equivalent $\mathcal{H}_2$ optimal control problem in the form of~\eqref{eq:OCPsparsity}, where the constraint $\mathcal{L}$ corresponds to the sparsity pattern of $A$ (see the Example 1 in~\cite{wang2019system, anderson2019system} for a precise definition). Since~\eqref{eq:OCPsparsity} is not convex in its present form, a certain reformulation is required for numerical computation, \emph{e.g.}, using Youla parameterization, SLP, or IOP.
\begin{proposition}
If the graph $\mathcal{G}$ is strongly connected, then Problem~\eqref{eq:Example1} with a sparsity constraint $\mathbf{K} \in \mathcal{L}$ in the form of~\eqref{eq:OCPsparsity} does not admit any equivalent convex reformulation in Youla, or SLP, or IOP.
\end{proposition}
\begin{proof}
If $\mathcal{G}$ is strongly connected, then the sparsity constraint $\mathcal{L}$ is not QI under $\mathbf{P}_{22} = (zI - A)^{-1}$, since $\mathbf{P}_{22}$ is a dense transfer matrix and it fails to satisfy $\mathbf{K}\mathbf{P}_{22}\mathbf{K} \in \mathcal{L}, \forall \mathbf{K} \in\mathcal{L}$. According to~\cite{lessard2015convexity}, QI is necessary for the existence of an \emph{equivalent} convex reformulation in Youla parameter $\mathbf{Q}$ for~\eqref{eq:OCPsparsity}. The equivalence in Theorem~\ref{theo:equivalence} prevents any \emph{equivalent} convex reformulation via SLP or IOP as well.
\end{proof}
Although there is no equivalent convex reformulation when $\mathcal{G}$ is strongly connected, we could still develop a certain \emph{convex approximation} of~\eqref{eq:OCPsparsity} in Youla parameterization, SLP, or IOP. In the following, we use $\mathcal{I}$ to denote a diagonal structure.
\begin{enumerate}
\item \emph{SLP:}
As suggested by~\cite{wang2019system}, we can add the constraints
$
\mathbf{M} \in \mathcal{L}, \mathbf{R} \in \mathcal{I}
$
to Problem~\eqref{eq:OCPsls}, leading to a convex approximation of~\eqref{eq:OCPsparsity}. It can be checked that $\mathbf{R}= \frac{1}{z}I$ and $\mathbf{M}= -\frac{1}{z}A$ is the optimal solution, recovering the globally optimal controller $\mathbf{K} = \mathbf{M}\mathbf{R}^{-1} = -A$.
\item \emph{Youla parameterization:} We use the simplified Youla parameterization in Corollary~\ref{Corollary:YoulaState}, and add the following constraints
$
-A - (I- \frac{1}{z}A)\textbf{Q} \in \mathcal{L}, I - \frac{1}{z}\textbf{Q} \in \mathcal{I},
$
to Problem~\eqref{eq:OCPYoula}. This leads to a convex program. We can check that the optimal solution is $\textbf{Q} = 0$, leading to
$$
\textbf{K} = \left(-A - (I- \frac{1}{z}A)\textbf{Q}\right)\left(I - \frac{1}{z}\textbf{Q}\right)^{-1} = -A.
$$
\item \emph{IOP:} Since $C_2 = I, B_2 = I$ is invertible, we can use the result in Corollary~\ref{coro:iopstate}. Then, we introduce constraints
$
\mathbf{Z} - I \in \mathcal{L}, \mathbf{W} \in \mathcal{I}
$ to Problem~\eqref{eq:OCPiop}, leading to a convex program. We can check that the solution $\textbf{W} = \frac{1}{z}I$ and $\textbf{Z} = I - \frac{1}{z}A$ is optimal. Then,
$
\textbf{K} = (\textbf{Z}- I)\textbf{W}^{-1} = -A.
$
\end{enumerate}
\begin{remark}[Sparsity invariance and beyond QI]
In the procedures above, we choose separate subspace constraints for the factors of $\mathbf{K}$ in the following form
\begin{equation} \label{eq:SI}
\mathbf{S} \in \mathcal{L}, \mathbf{T} \in \mathcal{I} \quad \Rightarrow \quad \mathbf{K} = \mathbf{S}\mathbf{T}^{-1} \in \mathcal{L},
\end{equation}
where $\mathbf{S}, \mathbf{T}$ denote appropriate transfer matrices in Youla, system-level, and input-output parameterizations. Obviously, this choice leads to a convex inner-approximation of~\eqref{eq:OCPsparsity} since the feasible region of $\mathbf{K}$ is narrowed. For this simple instance, the globally optimal solution is parameterized when using~\eqref{eq:SI}. Thus, the globally optimal controller can be found using Youla, system-level or input-output parameters via convex optimization. However, as observed in~\cite{Furieri2019Sparsity}, this procedure has no guarantee of optimility for general constraints beyond QI using either of Youa parameterization, SLP or IOP.
We note that the property~\eqref{eq:SI} is a special case of sparsity invariance (SI)~\cite{Furieri2019Sparsity}. There may exist other subspace choices for $\mathbf{S}, \mathbf{T}$ satisfying $\mathbf{S}\mathbf{T}^{-1} \in \mathcal{L}$, which still return a structured controller $\mathbf{K} \in \mathcal{L}$. Indeed, the notion of SI goes beyond QI for sparsity constraints, as it includes QI as a special case. We refer the interested reader to~\cite{Furieri2019Sparsity} for details.
\end{remark}
\section{Conclusion} \label{section:conclusion}
In this paper, we have presented an explicit equivalence of Youla, system-level, and input-output parameterizations for the set of internally stabilizing controllers. A doubly-coprime factorization of the system can be considered as a way to eliminate the explicit equality constraints in SLP and IOP. Indeed, both SLP and IOP have four parameters; but due to the equality constraints, SLP and IOP have the same degree of freedom as Youla parameterization.
We remark that the equivalence of Youla, SLP, and IOP does not indicate they offer the same computational features. One parameterization may be better suited for a particular context. For instance, it seems that SLP is more convenient for the case of state feedback, which has found applications in quantifying sample complexity of LQR problems~\cite{dean2017sample}; IOP seems to better suit for the case of output feedback as it exclusively deals with the maps from inputs to outputs without explicitly touching the system state; and Youla parameterization is more convenient when a doubly-coprime factorization is available \emph{a priori}. It is interesting to investigate whether there exist other parameterizations of stabilizing controllers that suit for a particular control application. Finally, we note that Youla, SLP, and IOP naturally suit for parameterizing dynamical controllers, but none of them can parameterize the set of static stabilizing controllers in a convex way. Thus, QI is not relevant for structured static controller synthesis, and this problem deserves further investigations.
\appendix
\subsection{Proof of Statement 2 in Theorem~\ref{th:slp_eq}} \label{Sec:stable}
Given any $\mathbf{Y}, \mathbf{U}, \mathbf{W}, \mathbf{Z}$ satisfying the affine subspace~\eqref{eq:aff1}-\eqref{eq:aff3}, we know that $\mathbf{K} = \mathbf{U}\mathbf{Y}^{-1}$ internally stabilizes the plant $\mathbf{P}_{22}$. In the following, we verify that the transfer matrices $\mathbf{R},\mathbf{M},\mathbf{N},\mathbf{L}$ defined in~\eqref{eq:iop-sls} are exactly the closed-loop responses in~\eqref{eq:LTIsls} with controller $\mathbf{K} = \mathbf{U}\mathbf{Y}^{-1}$.
Recall that $\mathbf{P}_{22}$ is strictly proper, \emph{i.e.}, $\mathbf{P}_{22} = C_2(zI - A)^{-1}B_2$. Then, we can verify the following equation:
$$
\begin{aligned}
\mathbf{R} &= (zI - A)^{-1} + (zI - A)^{-1}B_2\mathbf{U}C_2(zI - A)^{-1} \\
&=\left[(I + B_2\mathbf{U}C_2(zI - A)^{-1})^{-1}(zI - A)\right]^{-1}
\\
&= \left[ zI - A - (I + B_2\mathbf{U}C_2(zI - A)^{-1})^{-1}B_2\mathbf{U}C_2\right]^{-1}\\
&= \left[ zI - A - B_2\mathbf{U}(I + C_2(zI - A)^{-1}B_2\mathbf{U})^{-1}C_2\right]^{-1} \\
&= \left(zI - A - B_2\mathbf{U}\mathbf{Y}^{-1}C_2\right)^{-1}\\
&= (zI - A - B_2\mathbf{K}C_2)^{-1}
\end{aligned}
$$
Also, we can verify
$$
\begin{aligned}
\mathbf{M} & = \mathbf{U}C_2(zI - A)^{-1}=(I - \mathbf{K}\mathbf{P}_{22})^{-1}\mathbf{K}C_2(zI - A)^{-1} \\
&=\mathbf{K}C_2(zI - A)^{-1}(I - B_2\mathbf{K}C_2(zI - A)^{-1})^{-1} \\
&=\mathbf{K}C_2(zI-A - B_2\mathbf{K}C_2)^{-1} \\
&=\mathbf{K}C_2\mathbf{R}
\end{aligned}
$$
Similarly, we have
$ \mathbf{N} = (zI - A)^{-1}B_2\mathbf{U} = \mathbf{R}B_2\mathbf{K}
\nonumber,
\mathbf{L} = \mathbf{U} = \mathbf{K}(I - \mathbf{P}_{22}\mathbf{K})^{-1}. $
Then, the transfer matrices $\mathbf{R},\mathbf{M},\mathbf{N},\mathbf{L}$ are exactly the closed-loop responses in~\eqref{eq:LTIsls} with $\mathbf{K} = \mathbf{U}\mathbf{Y}^{-1}$.
\subsection{Proof of~\eqref{eq:slp-stable-state}} \label{section:proofB}
We show that any controller in~\eqref{eq:slp-stable-state} is an internally stabilizing controller in~\eqref{eq:sls}. The other direction is similar.
Consider any $\mathbf{R}, \mathbf{M}\in \frac{1}{z}\mathcal{RH}_{\infty}$ satisfying
$$ \begin{bmatrix} (zI -A) & -B_2 \end{bmatrix} \begin{bmatrix} \mathbf{R} \\ \mathbf{M} \end{bmatrix} = I.
$$
Upon defining $ \mathbf{L} = \mathbf{M}(zI - A),
\mathbf{N}=\mathbf{R}(zI - A) - I,
$
it is easy to see $ \mathbf{N}, \mathbf{L} \in \mathcal{RH}_{\infty}$. Also, one can straightforwardly verify that $\mathbf{N}, \mathbf{L}$ and $\mathbf{R}, \mathbf{M}$ above satisfy~\eqref{eq:slp_s1}-\eqref{eq:slp_s2} when $C_2 = I$. It is routinely to verify that
$
\textbf{L} - \textbf{M}\textbf{R}^{-1}\textbf{N} = \textbf{M}(sI - A) - \textbf{M}\textbf{R}^{-1}(\textbf{R}(sI - A) - I)
= \textbf{M}\textbf{R}^{-1}.
$
It remains to check that $\mathbf{N}$ defined above is strictly proper. This fact follows from~\eqref{eq:slp_s2} that
$
\mathbf{N} = (zI - A)^{-1}B_2 \mathbf{L},
$
indicating that~\eqref{eq:slp_s3} also hold. Thus, the general parameterization~\eqref{eq:sls} can be reduced to~\eqref{eq:slp-stable-state}.
\subsection{Stable plants} \label{section:special}
When $\mathbf{P}_{22} \in \mathcal{RH}_{\infty}$, we show that Youla, SLP, and IOP can be simplified, and only two paramters are required in SLP/IOP.
\begin{proposition} \label{prop:stable}
If $\mathbf{P}_{22} \in \mathcal{RH}_{\infty}$, we have:
\begin{enumerate}
\item Youla parameterization can be reduced to
\begin{equation} \label{eq:youla-stable}
\mathcal{C}_{\text{stab}} = \{\mathbf{K} = - \mathbf{Q}(I - \mathbf{P}_{22}\mathbf{Q})^{-1} \mid \mathbf{Q} \in \mathcal{RH}_{\infty} \}.
\end{equation}
\item For strictly proper $\mathbf{P}_{22}$, SLP can be reduced to
\begin{equation} \label{eq:slp-stable}
\begin{aligned}
\mathcal{C}_{\text{stab}} = \bigg\{\mathbf{K} = &\mathbf{L}(C_2\mathbf{N} + I)^{-1} \bigm| \mathbf{L} \in \mathcal{RH}_{\infty}, \\
&\begin{bmatrix} (zI -A) & -B_2 \end{bmatrix} \begin{bmatrix} \mathbf{N} \\ \mathbf{L} \end{bmatrix} = 0
\bigg \}.
\end{aligned}
\end{equation}
\item IOP can be reduced to
\begin{equation} \label{eq:iop-stable}
\begin{aligned}
\mathcal{C}_{\text{stab}} = \bigg\{\mathbf{K} = \mathbf{U}\mathbf{Y}^{-1} \bigm| \begin{bmatrix} I & -\mathbf{P}_{22} \end{bmatrix} \begin{bmatrix} \mathbf{Y} \\ \mathbf{U} \end{bmatrix} = I, \\
\mathbf{U} \in \mathcal{RH}_{\infty}
\bigg\}.
\end{aligned}
\end{equation}
\end{enumerate}
\end{proposition}
\emph{Proof:} The proof is directly from the following observations.
\begin{enumerate}
\item If $\mathbf{P}_{22} \in \mathcal{RH}_{\infty}$, a doubly-coprime factorization of $\mathbf{P}_{22}$ can be trivially chosen as $
\mathbf{U}_l = I, \mathbf{V}_l = 0, \mathbf{N}_l = \mathbf{P}_{22}, \mathbf{M}_l = I,
\mathbf{U}_r = I, \mathbf{V}_r = 0, \mathbf{N}_r = \mathbf{P}_{22}, \mathbf{M}_r = I.
$
Then, the parameterization~\eqref{eq:youla} is reduced to~\eqref{eq:youla-stable}.
\item Given $\mathbf{N}, \mathbf{L}$ in~\eqref{eq:slp-stable}, we define
$
\mathbf{R} = (zI -A)^{-1} + \mathbf{N}C_2(zI - A)^{-1},
\mathbf{M} = \mathbf{L}C_2(zI - A)^{-1}.
$
Considering $(zI - A)^{-1} \in \mathcal{RH}_{\infty}$, if $\mathbf{L} \in \mathcal{RH}_{\infty}$, we have $ \mathbf{N}, \mathbf{R}, \mathbf{M} \in \frac{1}{z}\mathcal{RH}_{\infty}$. It can be verified that the $\mathbf{R}, \mathbf{M}, \mathbf{N}, \mathbf{L}$ above satisfies~\eqref{eq:slp_s1}-\eqref{eq:slp_s3} when~\eqref{eq:slp-stable} holds. Also, we have
$$
\begin{aligned}
\mathbf{L} - \mathbf{M}\mathbf{R}^{-1}\mathbf{N}
& = \mathbf{L} -\mathbf{L}C_2(I + \mathbf{N}C_2)^{-1}\mathbf{N} \\
& = \mathbf{L}(C_2\mathbf{N} + I)^{-1}.
\end{aligned}
$$
Thus,~\eqref{eq:slp_s1}-\eqref{eq:slp_s3} can be reduced to~\eqref{eq:slp-stable}.
\item Upon defining
$
\mathbf{Y} = I + \mathbf{P}_{22}\mathbf{U},
\mathbf{Z} = I + \mathbf{U}\mathbf{P}_{22},
\mathbf{W} = \mathbf{P}_{22} \mathbf{Z},
$
we have $\mathbf{Y}, \mathbf{W}, \mathbf{Z} \in \mathcal{RH}_{\infty}$
if $\mathbf{P}_{22}, \mathbf{U} \in \mathcal{RH}_{\infty}$. Also, the $\mathbf{Y}, \mathbf{U}, \mathbf{W}, \mathbf{Z}$ above satisfies~\eqref{eq:aff1}-\eqref{eq:aff2} if~\eqref{eq:iop-stable} holds. Thus,~\eqref{eq:aff1}-\eqref{eq:aff3} can be reduced to~\eqref{eq:iop-stable}.
\end{enumerate}
\begin{remark}\label{remark:stableQ}
The first statement in Proposition~\ref{prop:stable} is a classical result~\cite[Corollary 5.5]{zhou1996robust}. We note that parameterizations~\eqref{eq:slp-stable} and~\eqref{eq:iop-stable} are identical to~\eqref{eq:youla-stable} by noticing that
$
\mathbf{L} = \mathbf{U} = - \mathbf{Q}.
$ They are all reduced to the same form.
\end{remark}
\end{document} | math | 64,157 |
\begin{document}
\title{Computing the Cassels-Tate Pairing for Genus Two Jacobians with Rational Two Torsion Points}
\begin{abstract}
In this paper, we give an explicit formula as well as a practical algorithm for computing the Cassels-Tate pairing on $\text{Sel}^{2}(J) \times \text{Sel}^{2}(J)$ where $J$ is the Jacobian variety of a genus two curve under the assumption that all points in $J[2]$ are $K$-rational. We also give an explicit formula for the Obstruction map $\text{Ob}: H^1(G_K, J[2]) \rightarrow \text{Br}(K)$ under the same assumption. Finally, we include a worked example demonstrating we can indeed improve the rank bound given by a 2-descent via computing the Cassels-Tate pairing. \\
\end{abstract}
\section{Introduction}
For any principally polarized abelian variety $A$ defined over a number field $K$, Cassels and Tate \cite{cassels1} \cite{cassels2} and \cite{tate} constructed a pairing
$$\Sh(A) \times \Sh(A) \rightarrow \mathbb{Q}/\mathbb{Z},$$
that is nondegenerate after quotienting out the maximal divisible subgroup of $\Sh(A)$. This pairing is called the Cassels-Tate pairing and it naturally lifts to a pairing on Selmer groups. One application of this pairing is in improving the bound on the Mordell-Weil rank $r(A)$ obtained by performing a standard descent calculation. More specifically, if $\Sh(A)$ is finite or if all the $n$-torsion points of $A$ are defined over $K$, the kernel of the Cassels-Tate pairing on $\text{Sel}^n(A) \times \text{Sel}^n(A)$ is equal to the image of the natural map $\text{Sel}^{n^2}(A) \rightarrow \text{Sel}^n(A)$ induced from the map $A[n^2] \xrightarrow{n} A[n]$, see \cite[Proposition 1.9.3]{thesis} for details. This shows that carrying out an $n$-descent and computing the Cassels-Tate pairing on $\text{Sel}^n(A) \times \text{Sel}^n(A)$ gives the same rank bound as obtained from $n^2$-descent where $\text{Sel}^{n^2}(A)$ needs to be computed. \\
There have been many results on computing the Cassels-Tate pairing in the case of elliptic curves, such as \cite{cassels98} \cite{steve} \cite{binary quartic} \cite{monique} \cite{3 isogeny} \cite{platonic} \cite{3 selmer}. We are interested in the natural problem of generalizing the different algorithms for computing the Cassels-Tate pairing for elliptic curves to compute the pairing for abelian varieties of higher dimension. \\
In Section 2, we give the preliminary results needed for the later sections, including the homogeneous space definition of the Cassels-Tate pairing. In Section 3, we
state and prove an explicit formula for the pairing $\langle \;, \; \rangle_{CT}$ on $\text{Sel}^2(J) \times \text{Sel}^2(J)$ where $J$ is the Jacobian variety of a genus two curve under the assumption that all points in $J[2]$ are $K$-rational. This formula is analogous to that in the elliptic curve case in \cite{cassels98}. In Section 4, we describe a practical algorithm for computing the pairing $\langle \;, \; \rangle_{CT}$ using the formula in Section 3. In section 5, we also give an explicit formula for the Obstruction map $\text{Ob}: H^1(G_K, J[2]) \rightarrow \text{Br}(K)$ under the assumption that all points in $J[2]$ are defined over $K$ generalizing the result in the elliptic curve case \cite [Proposition 3.4]{oneil}, \cite[Theorem 6]{clark}. Finally, in Section 7, we include a worked example demonstrating that computing the Cassels-Tate pairing can indeed turn a 2-descent to a 4-descent and improve the rank bound given by a 2-descent. The content of this paper is based on Chapter 4 of the thesis of the author \cite{thesis}.\\
\section{Preliminary Results}
\subsection{The set-up}\label{sec:the-set-up}
In this paper, we are working over a number field $K$. For any field $k$, we let $\bar{k}$ denote its algebraic closure and let $\mu_n \subset \bar{k}$ denote the $n^{th}$ roots of unity in $\bar{k}$. We let $G_k$ denote the absolute Galois group $\text{Gal}(\bar{k}/k)$.\\
Let $\mathcal{C}$ be a general \emph{genus two curve} defined over $K$, which is a smooth projective curve. It can be given in the following hyperelliptic form:
$$\mathcal{C}: y^2 = f(x)= f_6x^6 +f_5x^5 +f_4x^4 +f_3x^3 +f_2x^2 +f_1x +f_0,$$
where $f_i \in K$, $f_6 \neq 0$ and the discriminant $\triangle(f) \neq 0$, which implies that $f$ has distinct roots in $\bar{K}$.\\
We let $J$ denote the \emph{Jacobian variety} of $\mathcal{C}$, which is an abelian variety of dimension two defined over $K$ that can be identified with $\text{Pic}^0(\mathcal{C})$. We denote the identity element of $J$ by $\mathcal{O}_J$. Via the natural isomorphism $\text{Pic}^2(\mathcal{C}) \rightarrow \text{Pic}^0(\mathcal{C})$ sending $[P_1+P_2] \mapsto [P_1 + P_2 - \infty^+ - \infty^-]$, a point $P \in J$ can be identified with an unordered pair of points of $\mathcal{C}$, $\{P_1, P_2\}$. This identification is unique unless $P= \mathcal{O}_J$, in which case it can be represented by any pair of points on $\mathcal{C}$ in the form $\{(x, y), (x, -y)\}$ or $\{\infty^+, \infty^-\}$. Suppose the roots of $f$ are denoted by $\omega_1, ..., \omega_6$. Then $J[2] = \{\mathcal{O}_J, \{(\omega_i, 0), (\omega_j, 0)\}\text{ for } i\neq j\}$. Also, for a point $P \in J$, we let $\tau_P:J \rightarrow J$ denote the translation by $P$ on $J$.\\
As described in \cite[Chapter 3, Section 3]{the book}, suppose $\{P_1, P_2\} $and $\{Q_1, Q_2\}$ represent $P, Q \in J[2]$ where $P_1, P_2, Q_1, Q_2$ are Weierstrass points, then $$e_2(P, Q)= (-1)^{|\{P_1, P_2\} \cap \{Q_1, Q_2\} |}.$$\\
\subsection{Theta divisor and Kummer surface}\label{sec:theta-divisor-and-kummer-surface}
The \emph{theta divisor}, denoted by $\Theta$, is defined to be the divisor on $J$ that corresponds to the divisor $\{P\}\times \mathcal{C}+ \mathcal{C} \times \{P\}$ on $\mathcal{C} \times \mathcal{C}$ under the birational morphism $\text{Sym}^2\mathcal{C} \rightarrow J$, for some Weierstrass point $P \in \mathcal{C}$. The Jacobian variety $J$ is principally polarized abelian variety via $\lambda: J \rightarrow J^{\vee}$ sending $P$ to $[\tau_P^*\Theta-\Theta].$\\
The \emph{Kummer surface}, denoted by $\mathcal{K}$, is the quotient of $J$ via the involution $[-1]: P \mapsto -P$. The fixed points under the involution are the 16 points of order 2 on $J$ and these map to the 16 nodal singular points of $\mathcal{K}$ (the $\emph{nodes}$). General theory, as in \cite[Theorem 11.1]{abelian varieties}, \cite[page 150]{theta}, shows that the linear system of $n\Theta$ of $J$ has dimension $n^2$. Moreover, $|2\Theta|$ is base point free and $|4\Theta|$ is very ample.\\
\subsection{Explicit embeddings of $J$ and $\mathcal{K}$}\label{sec:explicit-embeddings-of-j-and-mathcalk}
Denote a generic point on the Jacobian $J$ of $\mathcal{C}$ by $\{(x, y), (u, v)\}$. Then, following \cite[Chapter 3, Section 1]{the book}, the morphism from $J$ to $\mathbb{P}^3$ is given by $$k_1 = 1, k_2 = (x+u), k_3 = xu, k_4 = \beta_0,$$ where
$$\beta_0 = \frac{F_0(x, u)-2yv}{(x-u)^2}$$ with $F_0(x, u)= 2f_0 + f_1(x+u) +2f_2(xu) + f_3(x+u)(xu) + 2f_4(xu)^2 + f_5(x+u)(xu)^2 + 2f_6(xu)^3.$ \\
We denote the above morphism by $J \xrightarrow{|2\Theta|} \mathcal{K} \subset \mathbb{P}^3$ and it maps $\mathcal{O}_J$ to $(0:0:0:1)$. It is known that its image in $\mathbb{P}^3$ is precisely the Kummer surface $\mathcal{K}$ and is given by the vanishing of the quartic $G(k_1, k_2, k_3, k_4)$ with explicit formula given in \cite[Chapter 3, Section 1]{the book}. Therefore, the Kummer surface $\mathcal{K} \subset \mathbb{P}^3_{k_i}$ is defined by $G(k_1, k_2, k_3, k_4) =0.$\\
\begin{remark}\label{rem: translation by 2 torsion is linear on K}
Suppose $P \in J[2]$. We know $\tau_P^*(2\Theta) \sim 2\Theta$ via the polarization. This implies that translation by $P$ on $J$ induces a linear isomorphism on $\mathcal{K} \subset \mathbb{P}^3$.\\
\end{remark}
We now look at the embedding of $J$ in $\mathbb{P}^{15}$ induced by $|4\Theta|$. Let $k_{ij} = k_ik_j$, for $1 \leq i\leq j \leq 4$. Since $\mathcal{K}$ is irreducible and defined by a polynomial of degree 4, $k_{11}, k_{12}, ..., k_{44}$ are 10 linearly independent even elements in $\mathcal{L}(2\Theta^+ +2 \Theta^-)$. The six odd basis elements in $\mathcal{L}(2\Theta^+ + 2\Theta^-)$ are given explicitly in \cite[Section 3]{explicit twist}. A function $g$ on $J$ is \emph{even} when it is invariant under the involution $[-1]: P \mapsto -P$ and is \emph{odd} when $g\circ [-1]=-g$. \\
Unless stated otherwise, we will use the basis $k_{11},k_{12}, ..., k_{44}, b_1, ..., b_6$ for $\mathcal{L}(2\Theta^+ + 2\Theta^-)$, to embed $J$ in $\mathbb{P}^{15}$. The following theorem gives the defining equations of $J$.\\
\begin{theorem}\label{theorem: 72}({\cite[Therorem 1.2]{72 theorem}, \cite[Therorem 1.2]{the gp law paper}}) Let $J$ be the Jacobian variety of the genus two curve $\mathcal{C}$ defined by $y^2=f_6x^6+... +f_1x + f_0$. The $72$ quadratic forms over $\mathbb{Z}[f_0, . . . , f_6]$ given in \cite[Appendix A]{72 theorem} are a set of defining equations for the projective variety given by the embedding of $J$ in $\mathbb{P}^{15}$ induced by the basis of $\mathcal{L}(2\Theta^++2\Theta^-)$ with explicit formulae given in \cite[Definition 1.1]{72 theorem} or \cite[Definition 1.1]{the gp law paper}. The change of basis between this basis of $\mathcal{L}(2\Theta^++2\Theta^-)$ and $k_{11}, k_{12}, ..., k_{44}, b_1, ..., b_6$ is given in \cite[Section 3]{explicit twist}.\\
\end{theorem}
\subsection{Principal homogeneous space and 2-coverings}\label{sec:principal-homogeneous-space-and-2-coverings}
A \emph{principal homogeneous space} or \emph{torsor} for $J$ defined over a field $K$ is a variety $V$ together with a morphism $\mu : J \times V \rightarrow V$, both defined over $K$, that induces a simply transitive action on the $\bar{K}$-points.\\
We say $(V_1, \mu_1)$ and $(V_2, \mu_2)$ are isomorphic over a field extension $K_1$ of $K$ if there is an isomorphism $\phi: V_1 \rightarrow V_2$ defined over $K_1$ that respects the action of $J$.\\
\begin{comment}
In this section, we state some definitions concerning the twists of the Jacobian variety $J$ as well as some useful propositions needed later in the paper. Similar results in the elliptic curves are in \cite{many authors paper 1}. Note, unless stated otherwise, a twist in this thesis means $\bar{K}/K$ twist, that is an isomorphic variety defined over $K$ with the isomorphism defined over $\bar{K}$. \\
A \emph{principal homogeneous space or a torsor} for an abelian variety $A$ defined over a field $K$ is a variety $V$ together with a morphism $\mu : A \times V \rightarrow V$, both defined over $K$, that induces a simply transitive action of $A(\bar{K})$ on $V(\bar{K})$.\\
We say $(V_1, \mu_1)$ and $(V_2, \mu_2)$ are isomorphic over a field extension $K_1$ of $K$ if there is an isomorphism $\phi: V_1 \rightarrow V_2$ defined over $K_1$ such that the following diagram commutes.
\[ \xymatrix{ A\,\,\, \ar@<-1.5em>[d]^{=} \times
\,\,\, V_1 \ar[r]^-{\mu_1} \ar@<1.5em>[d]^{\phi}
& V_1 \ar[d]^{\phi} \\
A \,\,\, \times \,\,\, V_2 \ar[r]^-{\mu_2} & V_2 }
\]
We observe that $(A, +)$ is a trivial principal homogeneous space for $A$ and we have the following lemma.\\
\begin{lemma}\label{lem: trvial torsor}
Let $A$ be an abelian variety defined over $K$. The only isomorphism $\phi: A\rightarrow A$ as the trivial torsor is translation by some $P \in A$ .\\
\end{lemma}
\begin{proof}
Consider the following commutative diagram.
\[ \xymatrix{ A\,\,\, \ar@<-1.5em>[d]^{=} \times
\,\,\, A \ar[r]^-{+} \ar@<1.5em>[d]^{\phi}
& A \ar[d]^{\phi} \\
A \,\,\, \times \,\,\, A \ar[r]^-{+} & A }
\]
We have $Q+\phi(\operatorname{id})=\phi(Q)$, for any $Q \in A$. Here, $\operatorname{id}$ denotes the identity element in $A$ \\
\end{proof}
For simplicity, we sometimes denote $(V, \mu)$ by $V$ and we have the following proposition whose proof we omit.\\
\begin{proposition}\cite[Proposition 4]{sha1}\label{prop: torsors biject H^1}
Let $A$ be an abelian variety defined over $K$. There is a canonical bijection between $H^1(G_K, A)$ and the set of isomorphism classes of principal homogeneous spaces for $A$ over $K$.\\
\end{proposition}
\begin{remark}\label{rem:torsor isomorphism and bijection to H^1}
$\;$\\
\begin{enumerate}[label=(\roman*)]
\item From the proof of the above proposition, we know that for a principal homogeneous space $V$ of $A$ with a morphism $\mu: A \times V \rightarrow V$ and a point $P \in V(\bar{K})$, the map $Q \mapsto \mu(Q, P)$ for all $Q \in A$ gives an isomorphism of principal homogeneous spaces defined over $K(P)$. Denote the inverse of this isomorphism by $\phi: V \rightarrow A$, then we have $\phi \cdot (\phi^{-1})^{\sigma}$ is a translation by a point $P_{\sigma} \in A$ and $(\sigma \mapsto P_{\sigma})$ gives the corresponding element in $H^1(G_K, A)$. Note that different choices of the point $P$ give the same cocycle up to coboundaries.\\
\item
Let $V$ be a principal homogeneous space of $A$ with an isomorphism $\phi: V \rightarrow A$ such that $\phi (\phi^{-1})^{\sigma}$ is translation by $\epsilon_{\sigma} \in J$ and $(\sigma \mapsto \epsilon_{\sigma})$ is a cocycle representing $\epsilon$. Then for $P \in J$, $\tau_P \circ \phi: V \rightarrow A$ is an isomorphism such that it induces the cocycle representation of $\epsilon$ that differs from $(\sigma \mapsto \epsilon_{\sigma})$ by a coboundary of $(\sigma \mapsto \sigma(P)-P)$.\\
\end{enumerate}
\end{remark}
\begin{remark}\label{torsion_group}
With this geometric interpretation, $H^1(G_K, A)$ is called the \emph{Weil-Chatelet group}. By \cite[Proposition 5]{sha1}, we know the Weil-Chatelet group is a torsion group, i.e. every element of it has finite order.\\
\end{remark}
\begin{corollary}\label{cor: PHS trivial iff has a point}
Let $A$ be an abelian variety defined over $K$. $V$ is a trivial principal homogeneous space if and only if $V$ has a $K$-rational point.\\
\end{corollary}
\begin{proof}
If $V$ is a trivial principal homogeneous space, that is, there exists an isomorphism $A \cong V$ over $K$, then the image of the identity element of $A$ in $ V$ is defined over $K$. On the other hand, if there exists $P \in V(K)$, then by Proposition \ref{prop: torsors biject H^1} and Remark \ref{rem:torsor isomorphism and bijection to H^1} (i), we know that the map $Q \mapsto \mu(Q, P)$ for all $ Q \in A$ gives an isomorphism of principal homogeneous spaces $A \cong V$ defined over $K(P)=K$, where $\mu: A \times V \rightarrow V$ is the action associated to $V$.\\
\end{proof}
\end{comment}
A \emph{2-covering} of $J$ is a variety $X$ defined over $K$ together with a morphism $\pi : X \rightarrow J$ defined over $K$, such that there exists an isomorphism $\phi : X \rightarrow J$ defined over $\bar{K}$ with $\pi = [2] \circ \phi$. An isomorphism $(X_1, \pi_1) \rightarrow (X_2, \pi_2)$ between two 2-coverings is an isomorphism $h: X_1 \rightarrow X_2$ defined over $K$ with $\pi_1 = \pi_2 \circ h$. We sometimes denote $(X, \pi)$ by $X$ when the context is clear.\\
\begin{comment}
In the proposition below, we show that an $n$-covering of an abelian variety $A$ is a principal homogeneous space of $A$.\\
\begin{proposition}\label{prop: 2 covering is PHS}
Let $A$ be an abelian variety defined over $K$ and $(X, \pi)$ be an $n$-covering of $A$ for some $n$. $X$ is a principal homogeneous space of $A$ with the simply transitive action $\mu: (P, Q) \mapsto \phi^{-1}(P + \phi(Q))$ for any $P \in A, Q \in X$ and isomorphism $\phi: X \rightarrow A$ such that $[n] \circ \phi=\pi$.\\
\end{proposition}
\begin{proof}
Since $(X, \pi)$ is an $n$-covering of $A$, there exists an isomorphism $\phi: X \rightarrow A$ defined over $\bar{K}$ such that $[n] \circ \phi= \pi$. Since $\phi$ is an isomorphism, we know $\mu: (P, Q) \mapsto \phi^{-1}(P + \phi(Q))$ for any $P \in A, Q \in X$ induces a simply transitive action of $A(\bar{K})$ on $X(\bar{K})$. \\
It suffices to show that $\mu$ is defined over $K$. Since $\pi = [n] \circ \phi$ is defined over $K$, we know $[n]= [n] \circ \phi(\phi^{-1})^{\sigma}$ for all $\sigma \in G_K$. So fix $\sigma \in G_K$, we have a morphism $\alpha_{\sigma}: A \rightarrow A[n]$ that sends $P$ to $\phi(\phi^{-1})^{\sigma}(P)-P$. Since $\alpha_{\sigma}$ is continuous and $A[n]$ is a discrete set, we know $\alpha_{\sigma}$ is locally constant. The connectedness of $A$ implies that $\alpha_{\sigma}$ is indeed a constant morphism. Hence, there exists $P_{\sigma} \in A[n]$ such that $\phi(\phi^{-1})^{\sigma}(P) = P + P_{\sigma}$. Now to show $\mu^{\sigma}(P, Q)= \mu(P, Q)$, it suffices to show $P +\phi(Q)=\phi (\phi^{-1})^{\sigma}(P + \phi^{\sigma}(Q))$. Since the right hand side is equal to $\phi (\phi^{-1})^{\sigma}(P + \phi^{\sigma}\phi^{-1}\phi(Q))$ and $\phi(\phi^{-1})^{\sigma}= \tau_{P_{\sigma}},$ we are done.\\
\end{proof}
\end{comment}
It can be checked that a 2-covering is a principal homogeneous space. The short exact sequence $0 \rightarrow J[2] \rightarrow J \xrightarrow{2} J \rightarrow 0$ induces the connecting map in the long exact sequence
\begin{equation}\label{eqn: connecting map}
\delta: J(K) \rightarrow H^1(G_K,J[2]).
\end{equation}
The following two propositions are proved in \cite{explicit twist}.\\
\begin{proposition} \label{proposition_2_covering} \cite[Lemma 2.14]{explicit twist}
Let $(X, \pi)$ be a $2$-covering of an abelian variety $J$ defined over $K$ and choose an isomorphism $\phi:X \rightarrow J$ such that $\pi = [2] \circ \phi$. Then for each $\sigma \in G_K$, there is a unique point $P \in J[2](\bar{K})$ satisfying $\phi \circ \sigma(\phi^{-1}) = \tau_P$. The map $\sigma \mapsto P$ is a cocycle whose class in $H^1(G_K, J[2])$ does not depend on the choice of $\phi$. This yields a bijection between the set of isomorphism classes of $2$-coverings of $J$ and the set $H^1(G_K,J[2])$.\\
\end{proposition}
\begin{proposition}\label{prop:2 covering has a point}\cite[Proposition 2.15]{explicit twist} Let $X$ be a $2$-covering of $J$ corresponding to the cocycle class $\epsilon \in H^1(G_K, J[2])$. Then $X$ contains a $K$-rational point (equivalently $X$ is a trivial principle homogeneous space) if and only if $\epsilon$ is in the image of the connecting map $\delta$ in \eqref{eqn: connecting map}. \\
\end{proposition}
We also state and prove the following proposition which is useful for the computation of the Cassels-Tate pairing later in Sections \ref{sec:explicit-computation-of-d} and \ref{sec:explicit-computation-of-dp-dq-dr-ds}. A \emph{Brauer-Severi} variety is a variety that is isomorphic to a projective space over $\bar{K}$.\\
\begin{proposition}\label{prop:BS diagram}
Let $(X, \pi)$ be a $2$-covering of J, with $\phi \circ [2]= \pi$. Then the linear system $|\phi^*(2\Theta)|$ determines a map $X \rightarrow S$ defined over $K$, where $S$ is a Brauer-Severi variety. Also, there exists an isomorphism $\psi$ defined over $\bar{K}$ making the following diagram commute:
\begin{equation}\label{diagram: BS def}
\begin{tikzcd}
X \arrow[r, "|\phi^*(2\Theta)|"] \arrow[d, "\phi"]&S \arrow[d, "\psi"]\\
J \arrow[r, "|2\Theta|"]& \mathbb{P}^{3}.
\end{tikzcd}
\end{equation}
In particular, if $(X, \pi)$ corresponds to a Selmer element via the correspondence in Proposition~ \ref{proposition_2_covering}, then the Brauer-Severi variety $S$ is isomorphic to $\mathbb{P}^{3}$. \\
\end{proposition}
\begin{proof}
Since $(X, \pi)$ is a $2$-covering of $J$, by Proposition \ref{proposition_2_covering}, we have that for each $\sigma \in G_K$, $\phi \circ (\phi^{-1})^{\sigma} = \tau_{P}$ for some $P \in J[2]$. The principal polarization gives $\tau_P^*(2\Theta) \sim 2\Theta$ which implies that $\phi^*(2\Theta) \sim (\phi^{\sigma})^*(2\Theta)$, hence the morphism induced by $|\phi^*(2\Theta)|$ is defined over $K$. \\
Now if $(X, \pi)$ corresponds to a Selmer element, then $X$ everywhere locally has a point by Proposition \ref{prop:2 covering has a point}, and hence $S$ everywhere locally has a point. Since the Hasse principle holds for Brauer-Severi varieties by \cite[Corollary 2.6]{bs hasse}, we know that $S$ has a point over $K$ and hence it is isomorphic to $\mathbb{P}^{3}$ by \cite[Therem 5.1.3]{QA}. \\
\end{proof}
We now make some observations and give some notation.\\
\begin{remark}\label{rem: diagram of twited kummer}
Let $\epsilon \in \text{Sel}^2(J)$, and let $(J_{\epsilon}, \pi_{\epsilon})$ denote the 2-covering corresponding to $\epsilon$. There exists an isomorphism $\phi_{\epsilon}$ defined over $\bar{K}$ such that $[2] \circ \phi_{\epsilon}=\pi_{\epsilon}$. Then, by Proposition \ref{prop:BS diagram}, we have the following commutative diagram:
\begin{equation}\label{diagram: BS diagram}
\begin{tikzcd}
J_{\epsilon} \arrow[r, "|\phi_{\epsilon}^*(2\Theta)|"] \arrow[d, "\phi_{\epsilon}"]&\mathcal{K}_{\epsilon} \subset \mathbb{P}^3 \arrow[d, "\psi_{\epsilon}"]\\
J \arrow[r, "|2\Theta|"]& \mathcal{K} \subset \mathbb{P}^3.
\end{tikzcd}
\end{equation}
The image of $J_{\epsilon}$ under the morphism induced by $|\phi_{\epsilon}^*(2\Theta)|$ is a surface, denoted by $\mathcal{K}_{\epsilon}$, which we call the \emph{twisted Kummer surface} corresponding to $\epsilon$. Also $\psi_{\epsilon}$ is a linear isomorphism $\mathbb{P}^3 \rightarrow \mathbb{P}^3$ defined over $\bar{K}$. \\
\end{remark}
\begin{notation}\label{notation: involution}
Suppose $(J_{\epsilon}, \pi_{\epsilon})$ is the 2-covering of $J$ corresponding to $\epsilon \in H^1(G_K, J[2])$. The involution $[-1]: P \mapsto -P$ on $J$ induces an involution $\iota_{\epsilon}$ on $J_{\epsilon}$ such that $\phi_{\epsilon} \circ \iota_{\epsilon}=[-1] \circ \phi_{\epsilon}$, where $[2] \circ \phi_{\epsilon}=\pi_{\epsilon}$. Moreover, the degree 2 morphism $J_{\epsilon} \xrightarrow{|\phi_{\epsilon}^*(2\Theta)|} \mathcal{K}_{\epsilon} \subset \mathbb{P}^3$ in \eqref{diagram: BS diagram} is precisely the quotient by $\iota_{\epsilon}$ and so an alternative definition of $\mathcal{K}_{\epsilon}$ is as the quotient of $J_{\epsilon}$ by $\iota_{\epsilon}$. We call a function $g$ on $J_{\epsilon}$ even if it is invariant under $\iota_{\epsilon}$ and odd if $g \circ \iota_{\epsilon}=-g$.\\
\end{notation}
\subsection{Definition of the Cassels-Tate Pairing}\label{sec:definition-of-the-cassels-tate-pairing}
There are four equivalent definitions of the Cassels-Tate pairing stated and proved in \cite{poonen stoll}. In this paper we will only be using the homogeneous space definition of the Cassels-Tate pairing. Suppose $a , a' \in \Sh(J)$. Via the polarization $\lambda$, we get $a' \mapsto b$ where $b \in \Sh(J^{\vee}).$ Let $X$ be the (locally trivial) principal homogeneous space defined over $K$ representing $a$. Then $\text{Pic}^0(X_{\bar{K}})$ is canonically isomorphic as a $G_K$-module to $\text{Pic}^0(J_{\bar{K}}) = J^{\vee}(\bar{K}).$ Therefore, $b \in \Sh(J^{\vee}) \subset H^1(G_K, J^{\vee})$ represents an element in $H^1(G_K, \text{Pic}^0(X_{\bar{K}}))$. \\
Now consider the exact sequence:
$$0 \rightarrow \bar{K}(X)^*/\bar{K}^* \rightarrow \text{Div}^0(X_{\bar{K}}) \rightarrow \text{Pic}^0(X_{\bar{K}}) \rightarrow 0.$$
We can then map $b$ to an element $b' \in H^2(G_K, \bar{K}(X)^*/\bar{K}^*)$ using the long exact sequence associated to the short exact sequence above. Since $H^3(G_K, \bar{K}^*) = 0$, $b'$ has a lift $f' \in H^2(G_K, \bar{K}(X)^*) $ via the long exact sequence induced by the short exact sequence $0 \rightarrow \bar{K}^* \rightarrow \bar{K}(X)^* \rightarrow \bar{K}(X)^*/\bar{K}^* \rightarrow 0:$
\begin{equation}\label{eqn:def}
H^2(G_K, \bar{K}^*) \rightarrow H^2(G_K, \bar{K}(X)^*)\rightarrow H^2(G_K, \bar{K}(X)^*/\bar{K}^*) \rightarrow H^3(G_K, \bar{K}^*)=0.
\end{equation}
Next we show that $f'_v \in H^2(G_{K_v}, \bar{K_v}(X)^*)$ is the image of an element $c_v \in H^2(G_{K_v}, \bar{K_v}^*).$ This is because $b \in \Sh(J^{\vee})$ is locally trivial which implies its image $b'$ is locally trivial. Then the statement is true by the exactness of local version of sequence \eqref{eqn:def}. \\
We then can define
$$\langle a, b\rangle = \sum_v \text{inv}_v(c_v) \in \mathbb{Q}/\mathbb{Z}.$$
The Cassels-Tate pairing $\Sh(J) \times \Sh(J) \rightarrow \mathbb{Q}/\mathbb{Z}$ is defined by
$$\langle a, a' \rangle_{CT} := \langle a, \lambda(a')\rangle.$$
We sometimes refer to $\text{inv}_v(c_v)$ above as the local Cassels-Tate pairing between $a, a' \in \Sh(J)$ for a place $v$ of $K$. Note that the local Cassels-Tate pairing depends on the choice of $f'\in H^2(G_K, \bar{K}(X)^*)$. We make the following remarks that are useful for the computation for the Cassels-Tate pairing.\\
\begin{remark}\label{rem:CT_Selmer}
$\;$\\
\begin{enumerate}[label=(\roman*)]
\item By \cite{poonen stoll}, we know the homogeneous space definition of the Cassels-Tate pairing is independent of all the choices we make.\\
\item Via the map $\text{Sel}^2(J) \rightarrow \Sh(J)[2]$, the definition of the Cassels-Tate pairing on $\Sh(J)[2] \times \Sh(J)[2]$ naturally lifts to a pairing on $\text{Sel}^2(J) \times \text{Sel}^2(J)$. In fact, from now on, we will only be considering $\langle \epsilon, \operatorname{et}a \rangle_{CT}$ for $\epsilon, \operatorname{et}a \in \text{Sel}^2(J)$. The principal homogeneous space $X$ in the definition is always taken to be the $2$-covering of $J$ corresponding to $\epsilon$. One can compute $c_v$ by evaluating $f_v'$ at a point in $X(K_v)$ provided that one avoids the zeros and poles of $f_v'$. Note that $X(K_v) \neq \emptyset$ by Proposition \ref{prop:2 covering has a point}.\\
\end{enumerate}
\end{remark}
\begin{comment}
By the Mordell-Weil Theorem, the set of $K$-rational points of any abelian variety defined over a number field $K$ is a finitely generated abelian group. This implies that the rank of $J(K)$, denoted by $r$, is finite. Computing the rank is difficult but there are ways to compute upper bounds. The standard method is descent calculation. One application of the Cassels-Tate pairing is that it can potentially improve the rank bound of an abelian variety obtained by a standard descent calculation. \\
Via the Mordell-Weil Theorem and the structure theorem, we deduce $$\left|\frac{J(K)}{2J(K)}\right|=2^r|J(K)[2]|.$$ From the short exact sequence
$$0 \rightarrow J(K)/2(J(K)) \xrightarrow {\delta} \text{Sel}^ 2(J)\rightarrow \Sh(J)[2] \rightarrow 0,$$
we get $|J(K)/2J(K)| \le |\text{Sel}^2(A)|$and hence $$2^{r} \leq \frac{|\text{Sel}^{2}(J)|}{|J(K)[2]|}.$$ \\
\begin{proposition}\label{prop: CT_bound}
Suppose all points in $J[2]$ are defined over the base field $K$. Then carrying out an $2$-descent and computing the Cassels-Tate pairing on $\text{Sel}^2(J) \times \text{Sel}^2(J)$ gives the same rank bound as obtained from a $4$-descent where $\text{Sel}^{4}(J)$ needs to be computed.\\
More explicitly, the kernel of the Cassels-Tate pairing $\langle\;, \;\rangle_{CT}$ on $ \text{Sel}^2(J) \times \text{Sel}^2(J)$ is equal to the image of the natural map $\alpha: \text{Sel}^{4}(J) \rightarrow \text{Sel}^2(J)$ induced from $J[4] \xrightarrow{ \cdot 2} J[2].$\\
\end{proposition}
\begin{proof}
Via a direct check, the following sequence is exact.
$$0 \rightarrow J[2](K) \rightarrow J[4](K) \xrightarrow{\cdot 2} J[2](K) \rightarrow \text{Sel}^2(J) \rightarrow {Sel}^4(J) \rightarrow \text{Sel}^2(J).$$\\
We will show that the kernel of the Cassels-Tate pairing $\langle\;, \;\rangle_{CT}$ on $ \text{Sel}^2(J) \times \text{Sel}^2(J)$ is equal to the image of the natural map $\alpha: \text{Sel}^{4}(J) \rightarrow \text{Sel}^2(J)$ induced from $J[4] \xrightarrow{ \cdot 2} J[2].$ Then, via the above exact sequence, we know that carrying out an $2$-descent and computing the Cassels-Tate pairing on $\text{Sel}^2(J) \times \text{Sel}^2(J)$, together with computing $J[2](K)$ and $J[4](K)$, gives the size of $\text{Sel}^{4}(J)$. This implies that we would get the same rank bound as obtained from a $4$-descent. \\
We note that $\Ima \alpha \subset \ker \langle\; , \;\rangle_{CT}$. Indeed, suppose $a=\alpha(b) \in \text{Sel}^2(J)$ where $b \in \text{Sel}^{4}(J)$. Denote the image of $a$ in $\Sh[2] \subset H^1(G_K, J)$ by $a'$ and the image of $b$ in $\Sh[4] \subset H^1(G_K, J)$ by $b'$. Then we have $2b'=a'$ and $\langle a', c\rangle_{CT} = \langle nb', c\rangle_{CT}= \langle b', nc\rangle_{CT}= \langle b', 0\rangle_{CT} = 0, \text{ for any } c \in \Sh[2].$ \\
Now to prove $\ker \langle \;, \;\rangle_{CT} \subset \Ima \alpha$, we first show that for any $b \in \text{Sel}^2(J)$, there exists $b_1 \in H^1(K, J[4])$ mapping to $b$. Aster that, the rest of the proof is essentially the same as the proof in the elliptic curve case in \cite[Theorem 3]{platonic}.\\
Since all points in $J[2]$ are defined over $K$, we have $H^2(G_K, J[2]) \cong (H^2(G_K, \mu_2))^{4}\cong (\text{Br}(K)[2])^{4} $ and similarly $H^2(G_{K_v}, J[2]) \cong (\text{Br}(K_v)[2])^{4} $. Hence, via the injection of $\text{Br}(K) \rightarrow \bigoplus_v \text{Br}(K_v)$, we have $H^2(G_K, J[2]) \xrightarrow{res} \prod_{v} H^2(G_{K_v}, J[2])$ is injective.\\
Consider the following commutative diagram of short exact sequences.
\[
\begin{tikzcd}
0 \arrow[r]& J[2] \arrow[r] \arrow[d, "="]&J[4] \arrow[r, "2"] \arrow[d, "inc"]&J[2] \arrow[r] \arrow[d, "inc"]& 0\\
0 \arrow[r]& J[2] \arrow[r]&J \arrow[r, "2"]&J \arrow[r]& 0\\
\end{tikzcd}
\]
We then obtain the following commutating diagram of long exact sequences along the rows by taking Galois cohomology.
\[
\begin{tikzcd}
H^1(G_K, J[4]) \arrow[r, "2"] \arrow[d]&H^1(G_K, J[2]) \arrow[r] \arrow[d, "b \mapsto c"]& H^2(G_K, J[2]) \arrow[d, "="]\\
H^1(G_K, J) \arrow[r, "2"] \arrow[d, "res"]&H^1(G_K, J) \arrow[r] \arrow[d, "res"]&H^2(G_K, J[2]) \arrow[d, "inj"]\\
\prod_v H^1(G_{K_v}, J) \arrow[r, "2"] & \prod_v H^1(G_{K_v}, J) \arrow[r] & \prod_v H^2(G_{K_v}, J[2])\\
\end{tikzcd}
\]
Since $b \in \text{Sel}^2(J)$, its image $c \in H^1(G_K, J)$ is locally trivial. Hence, its image is also trivial in $\prod_v H^2(G_{K_v}, J[2])$. Via the injectivity of the map $ H^2(G_K, J[2]) \rightarrow \prod_v H^2(G_{K_v}, J[2]) $, we get that $b \mapsto 0 \in H^2(G_K, J[2]).$ Thus $b$ has a lift $b_1 \in H^1(G_K, J[4])$, as required. \\
\end{proof}
\end{comment}
\subsection{Explicit 2-coverings of $J$}\label{sec:explicit-2-coverings-of-j}
Let $\Omega$ represent the set of 6 roots of $f$, denoted by $\omega_1, ..., \omega_6$. Recall, as in Proposition \ref{proposition_2_covering}, the isomorphism classes of 2-coverings of $J$ are parameterized by $H^1(G_K, J[2])$. For the explicit computation of the Cassels-Tate pairing, we need the following result on the explicit 2-coverings of $J$ corresponding to elements in $\text{Sel}^2(J)$. We note that this theorem in fact works over any field of characteristic different from 2.\\
\begin{theorem}\cite[Proposition 7.2, Theorem 7.4, Appendix B]{explicit twist}\label{theorem: explicit twist of J} Let $J$ be the Jacobian variety of a genus two curve defined by $y^2=f(x)$ where $f$ is a degree 6 polynomial and $\epsilon \in \mathrm{Sel}^2(J)$. Embed $J$ in $\mathbb{P}^{15}$ via the coordinates $k_{11}, k_{12}, ..., k_{44}, b_1, ..., b_6$. There exists $J_{\epsilon} \subset \mathbb{P}^{15}$ defined over $K$ with Galois invariant coordinates $u_0, ..., u_9, v_1, ..., v_6$ and a linear isomorphism $\phi_{\epsilon}: J_{\epsilon} \rightarrow J$ such that $(J_{\epsilon}, [2] \circ \phi_{\epsilon})$ is a $2$-covering of $J$ whose isomorphism class corresponds to the cocycle class $\epsilon$. Moreover, $\phi_{\epsilon}$ can be explicitly represented by the $16\times 16$ matrix $R=
\begin{bmatrix}
R_1&0\\
0&R_2\\
\end{bmatrix}$ for some $10 \times 10$ matrix $R_1$ and some $6 \times 6$ matrix $R_2$.\\
\end{theorem}
\begin{remark}\label{rem: explicit twist of J formula}
The explicit formula for $\phi_{\epsilon}$ is given in the beginning of \cite[Section 7]{explicit twist} and depends only on $\epsilon$ and the underlying genus two curve. Note that the coordinates $u_0, ..., u_9, v_1, ..., v_6$ are derived from another set of coordinates $c_0, ..., c_9, d_1, ..., d_6$ defined in \cite[Definitions 6.9, 6.11]{explicit twist} where $c_0, ..., c_9$ are even and $d_1, ..., d_6$ are odd. This set of coordinates are in general not Galois invariant, however, they are in the case where all points of $J[2]$ are defined over the base field.\\
\end{remark}
\begin{comment}
\begin{remark}\label{rem: explicit twist formula of J}
Moreover, $\phi_{\epsilon}$ can be explicitly represented by the $16\times 16$ matrix $R=
\begin{bmatrix}
R_1&0\\
0&R_2\\
\end{bmatrix}$ for some $10 \times 10$ matrix $R_1$ and some $6 \times 6$ matrix $R_2$ given in the remark below.\\
Since $R$ is block diagonal, $u_0, ..., u_9$ are 10 even elements and $v_1, ..., v_6$ are 6 odd elements on $J_{\epsilon}$. Here, the parity is corresponding to the induced involution $\iota_{\epsilon}$ on $J_{\epsilon}$ as defined in Notation \ref{notation: involution}. We now give the explicit fomulae for $R$ in the above theorem following the proof in \cite{explicit twist}. Suppose $\epsilon \in P^1(G_K, J[2])$ is represented by $(\delta, n)$ for $\delta \in L^*$ with $N(\delta)=n^2$ as in Corollary $\ref{cor: notation of selmer}$. Define $\zeta \in \bar{L}$ such that $\zeta^2=\delta$ and $N(\zeta)=n$. Let $I_1, ..., I_{10}$ be the 10 different subsets of $\Omega$ of size 3. Let $T_1$ be the diagonal matrix whose $r^{th}$ diagonal entry is $\prod_{\omega \in I_r} \zeta(\omega) + \prod_{\omega \in \Omega \setminus I_r} \zeta(\omega)$ for $r=1, ..., 10$. Since $K$ is infinite, we can assume the entries of $T_1$ are nonzero by carefully picking $\zeta, \delta$ and $n$ representing $\epsilon$. Let $T_2$ be the diagonal matrix whose $r^{th}$ entry is $\zeta(\omega_r)$ for $r=1, ..., 6$. Then define $S=\begin{bmatrix}1& 1& ... & 1\\
\omega_1 & \omega_2&...&\omega_6\\
\vdots&\vdots &&\vdots\\
\omega_1^5&\omega_2^5& ...&\omega_6^5\\
\end{bmatrix}$. Let $G$ be the matrix whose $r^{th}$ row is $$\frac{1}{4}\big(\prod_{\omega \in I_r}\prod_{\psi \in \Omega \setminus I_r}(\psi -\omega)^{-1}\big) \cdot (\lambda_{11}(I_r) \;\;\lambda_{12}(I_r) \;\;...\;\; \lambda_{44}(I_r) ),$$
with the explicit formulae of $\lambda_{ij}$ in \cite[Definition 6.11]{explicit twist} which are defined over $L_1$, the splitting field of $f$. Then $R_1=G^{-1}T_1G$ and $R_2=ST_2S^{-1}$. We note the explicit formula for $\phi_{\epsilon}$ is defined over $L_1(\sqrt{a_1}, ..., \sqrt{a_6})$ where $a_i=\delta(\omega_i)$. In fact, the formula for $\phi_{\epsilon}$ given in Theorem \ref{theorem: explicit twist of J} is a conjugation of the original construction given in \cite[Proposition 7.2, Theorem 7.4]{explicit twist} where the coordinates for $J, J_{\epsilon} \subset \mathbb{P}^{15}$ are given in \cite[Definition 6.9, Definition 6.11]{explicit twist} and the matrix representing the twist is $\begin{bmatrix}
T_1&0\\
0&T_2\\
\end{bmatrix}$. Note that this set of coordinates in general are not Galois invariant.\\
\end{remark}
\end{comment}
\section{Formula for the Cassels-Tate Pairing }\label{sec:formula-for-the-cassels-tate-pairing}
From now on, we always assume that the genus two curve $\mathcal{C}$ is defined by $y^2=f(x)$ such that all roots of $f$ are defined over $K$. Note that this implies that all points in $J[2]$ are defined over $K$ which is equivalent to all the Weierstrass points defined over $K$. In this section, under the above assumption, we state and prove an explicit formula for the Cassels-Tate pairing on $\text{Sel}^{2}(J) \times \text{Sel}^2(J)$. \\
Let the genus two curve $\mathcal{C}$ be of the form
$$\mathcal{C}: y^2 = \lambda(x-\omega_1)(x-\omega_2)(x-\omega_3)(x-\omega_4)(x-\omega_5)(x-\omega_6), $$
where $\lambda, \omega_i \in K$ and $\lambda \neq 0$. Its Jacobian variety is denoted by $J$.\\
The 2-torsion subgroup $J[2]$ has basis
\begin{align*}
P =\{(\omega_1, 0), (\omega_2, 0)\}, & \;\;\;Q =\{(\omega_1, 0), (\omega_3, 0)\},\\
R =\{(\omega_4, 0), (\omega_5, 0)\}, & \;\;\;S=\{(\omega_4, 0), (\omega_6, 0)\}.\\
\end{align*}
By the discussion at the end of Section \ref{sec:the-set-up}, the Weil pairing is given relative to this basis by:
\begin{equation}\label{equation: WP matrix}
W=\begin{bmatrix}
1&-1&1&1\\
-1&1&1&1\\
1&1&1&-1\\
1&1&-1&1\\
\end{bmatrix}.
\end{equation}
More explicitly, $W_{ij}$ denotes the Weil pairing between the $i^{th}$ and $j^{th}$ generators. \\
We now show that this choice of basis determines an isomorphism $H^1(G_K, J[2]) \cong (K^*/(K^*)^2)^4$. Consider the map $ J[2] \xrightarrow{w_2} (\mu_2(\bar{K}))^4, $ where $w_2$ denotes taking the Weil pairing with $P, Q, R, S$. Since $P, Q, R, S$ form a basis for $J[2]$ and the Weil pairing is a nondegerate bilinear pairing, we get that $w_2$ is injective. This implies that $w_2$ is an isomorphism as $|J[2]|= |(\mu_2(\bar{K}))^4|=16$. We then get
\begin{equation}\label{eq: isomorphism of H^1(J[2])}
H^1(G_K, J[2]) \xrightarrow{w_{2, *}}H^1(G_K, (\mu_2(\bar{K}))^4) \cong (K^*/(K^*)^2)^4,
\end{equation}
where $w_{2, *}$ is induced by $w_2$ and $\cong$ is the Kummer isomorphism derived from Hilbert's Theorem 90. Since the map \eqref{eq: isomorphism of H^1(J[2])} is an isomorphism, we can represent elements in $H^1(G_K, J[2])$ by elements in $(K^*/(K^*)^2)^4$.\\
\begin{comment}
\begin{remark}\label{rem: places for selmer 2}
Let $S= \{\text{places of bad reduction for } \mathcal{C}\} \cup \{\text{places dividing 2}\} \cup \{\text{infinite places}\}$. Define $K(S, 2)=\{x \in K^*/(K^*)^2: \text{ord}_v(x) \text{ is even for all } v\notin S \}$. We get that $\ker\big((K^*/(K^*)^2) \rightarrow \prod_{v \notin S}K_v^{nr *}/(K_v^{nr *})^2 \big)=K(S, 2)$. Define $$H^1(G_K, J[2]; S) := \ker \left(H^1(G_K, J[2]) \rightarrow \prod_{v \notin S} H^1(G_{K_v^{nr}}, J[2])\right).$$
By \cite[Chapter I, Section 6]{arithmetic duality}, and \cite[Section 3]{selmer}, we have
$\text{Sel}^{2}(J) \subset H^1(G_K, J[2]; S).$ Hence, if we embed $H^1(G_K, J[2]) \xrightarrow{w_{2, *}}(K^*/(K^*)^2)^4$, then the image of any Selmer element is in $K(S, 2)^4$.\\
\end{remark}
\end{comment}
Before stating and proving the formula for the Cassels-Tate pairing, we first state and prove the following lemma. \\
\begin{lemma}\label{lem: existence of rational divisor D_T}
For $\epsilon \in \mathrm{Sel}^2(J)$, let $(J_{\epsilon}, \pi_{\epsilon})$ denote the corresponding $2$-covering of $J$. Hence, there exists an isomorphism $\phi_{\epsilon}: J_{\epsilon} \rightarrow J$ defined over $\bar{K}$ such that $[2] \circ \phi_{\epsilon}= \pi_{\epsilon}$. Suppose $T \in J(K)$ and $T_1 \in J(\bar{K})$ satisfy $2T_1=T$. Then
\begin{enumerate}[label=(\roman*)]
\item There exists a $K$-rational divisor $D_T$ on $J_{\epsilon}$ which represents the divisor class of $\phi_{\epsilon}^*(\tau_{T_1}^*(2\Theta))$.
\item Let $D$ and $D_T$ be $K$-rational divisors on $J_{\epsilon}$ representing the divisor class of $\phi_{\epsilon}^*(2\Theta)$ and $\phi_{\epsilon}^*(\tau_{T_1}^*(2\Theta))$ respectively. Then $D_T-D \sim \phi_{\epsilon}^*(\tau_T^*\Theta-\Theta)$. Suppose $T$ is a two torsion point. Then $2D_T -2D$ is a $K$-rational principal divisor. Hence, there exists a $K$-rational function $f_T$ on $J_{\epsilon}$ such that $\text{div}(f_T)= 2D_T -2D$. \\
\end{enumerate}
\end{lemma}
\begin{proof}
By definition of a 2-covering, $[2] \circ \phi_{\epsilon}= \pi_{\epsilon}$ is a morphism defined over $K$. Also, by Proposition \ref{proposition_2_covering}, $\phi_{\epsilon} \circ (\phi_{\epsilon}^{-1})^{\sigma} = \tau_{\epsilon_{\sigma}}$ for all $\sigma \in G_K$, where $(\sigma \mapsto \epsilon_{\sigma})$ is a cocycle representing $\epsilon$. Since $[2] \circ \tau_{T_1} \circ \phi_{\epsilon}= \tau_T \circ [2]\circ \phi_{\epsilon}=\tau_T \circ \pi_{\epsilon}$ and $\tau_T$ is defined over $K$, $(J_{\epsilon}, \tau_T \circ \pi_{\epsilon})$ is also a 2-covering of $J$. We compute $\tau_{T_1} \circ \phi_{\epsilon} \circ ((\tau_{T_1} \circ \phi_{\epsilon} )^{-1})^{\sigma }= \tau_{T_1} \circ \phi_{\epsilon} \circ (\phi_{\epsilon}^{-1})^{\sigma} \circ \tau_{-\sigma(T_1)}=\tau_{\epsilon_{\sigma}} \circ \tau_{T_1} \circ \tau_{-\sigma(T_1)},$ for all $\sigma \in G_K$. This implies the 2-covering $(J_{\epsilon}, \tau_T \circ \pi_{\epsilon})$ corresponds to the element in $ H^1(G_K, J[2])$ that is represented by the cocycle $(\sigma \mapsto \epsilon_{\sigma} + T_1 -\sigma(T_1))$. Hence, $(J_{\epsilon}, \tau_T \circ \pi_{\epsilon})$ is the 2-covering of $J$ corresponding to $\epsilon+ \delta(T)$, where $\delta$ is the connecting map as in \eqref{eqn: connecting map}. By Proposition \ref{prop:BS diagram}, there exists a commutative diagram:
\[\begin{tikzcd}[column sep=large]
J_{\epsilon} \arrow[r, "|\phi_{\epsilon}^*(\tau_{T_1}^*(2\Theta))|"] \arrow[d, "\tau_{T_1} \circ \phi_{\epsilon}"]&\mathbb{P}^3 \arrow[d, "\psi_{\epsilon}"]\\
J \arrow[r, "|2\Theta|"]& \mathbb{P}^3,
\end{tikzcd}
\]
where the morphism $J_{\epsilon} \xrightarrow{|\phi_{\epsilon}^*(\tau_{T_1}^*(2\Theta))|}\mathbb{P}^3$ is defined over $K$. So the pull back of a hyperplane section via this morphism gives us a rational divisor $D_T$ representing the divisor class of $\phi_{\epsilon}^*(\tau_{T_1}^*(2\Theta))$ as required by (i).\\
Since the polarization $\lambda: J \rightarrow J^{\vee}$ is an isomorphism and $2T_1=T$, we have $\phi_{\epsilon}^*(\lambda(T)) = [\phi_{\epsilon}^*(\tau_T^*\Theta- \Theta)] = [\phi_{\epsilon}^*(\tau_{T_1}^*(2\Theta))] - [\phi_{\epsilon}^*(2\Theta)]=[D_T]-[D]$. The fact that $T$ is a two torsion point implies that $2\phi_{\epsilon}^*(\lambda(P))= 0$. Hence, $2D_T-2D$ is a $K$-rational principal divisor which gives (ii).\\
\end{proof}
The following remark explains how we will use Lemma \ref{lem: existence of rational divisor D_T} in the formula for the Cassels-Tate pairing on $\text{Sel}^2(J) \times \text{Sel}^2(J)$.\\
\begin{remark}\label{rem: DP, DQ, DR, DS}
Applying Lemma \ref{lem: existence of rational divisor D_T}(i) with $T = \mathcal{O}_J, P, Q, R, S \in J[2]$ gives divisors $D=D_{\mathcal{O}_J}$ and $D_P, D_Q, D_R D_S$. Then by Lemma \ref{lem: existence of rational divisor D_T}(ii), there exist $K$-rational functions $f_P, f_Q, f_R, f_S$ on $J_{\epsilon}$ such that $\text{div}(f_T)= 2D_T -2D$ for $T=P, Q, R, S$.\\
\end{remark}
\begin{comment}
\begin{lemma}\label{lem: fpfqfrfs}
Suppose $D$ is a $K$-rational divisor on $J$ representing the divisor class of $\phi$Follow the notation in Remark \ref{rem: DP, DQ, DR, DS}. $2D_P-2D$ is a $K$-rational principal divisor. Hence there exists a $K$-rational function $f_P$ on $J_{\epsilon}$ such that $(f_P)= 2D_P -2D$. We have similar results for $Q, R, S$ and the corresponding $K$-rational functions are denoted by $f_Q, f_R, f_S$.\\
\end{lemma}
\begin{proof}
Note that by definition, $\phi^*(\lambda(P)) = [\phi^*(\tau_P(\Theta)- \Theta)] = [\phi^*(\tau_{P_1}^*(2\Theta))] - [\phi^*(2\Theta)],$ where $P_1\in J(\bar{K})$ such that $2P_1 = P.$
This is because the polarization $\lambda: J \rightarrow J^{\vee}$ is an isomorphism. The fact that $P$ is a two torsion point implies that $2\phi^*(\lambda(P))= 0$. Hence $2D_P-2D$ is $K$-rational and represents the trivial divisor class as required.\\
\end{proof}
\end{comment}
\begin{theorem}\label{thm: 1}
Let $J$ be the Jacobian variety of a genus two curve $\mathcal{C}$ defined over a number field $K$ where all points in $J[2]$ are defined over $K$. For any $\epsilon, \operatorname{et}a \in \mathrm{Sel}^2(J)$, let $(J_{\epsilon}, [2]\circ \phi_{\epsilon})$ be the $2$-covering of $J$ corresponding to $\epsilon$ where $\phi_{\epsilon}: J_{\epsilon} \rightarrow J$ is an isomorphism defined over $\bar{K}$. Fix a choice of basis $P, Q, R, S$ for $J[2]$, with the Weil pairing given by matrix \eqref{equation: WP matrix}. Let $(a, b, c, d)$ denote the image of $\operatorname{et}a$ via $H^1(G_K, J[2]) \cong(K^*/(K^*)^2)^4$, where this is the isomorphism induced by taking the Weil pairing with $P, Q, R, S$. Let $f_P, f_Q, f_R, f_S$ be the $K$-rational functions on $J_{\epsilon}$ defined in Remark \ref{rem: DP, DQ, DR, DS}. Then the Cassels-Tate pairing $\langle\;,\; \rangle_{CT}: \mathrm{Sel}^2(J) \times \mathrm{Sel}^2(J) \rightarrow \{\pm 1\}$ is given by
$$\langle \epsilon, \operatorname{et}a\rangle_{CT} = \prod_{\text{place } v}(f_P(P_v), b)_v(f_Q(P_v), a)_v(f_R(P_v), d)_v(f_S(P_v), c)_v,$$
where $(\;,\;)_v$ denotes the Hilbert symbol for a given place $v$ of $K$ and $P_v$ is an arbitrary choice of a local point on $J_{\epsilon}$ avoiding the zeros and poles of $f_P, f_Q, f_R, f_S$.\\
\end{theorem}
\begin{comment}
\subsection{Proof of the formula}\label{sec:proof-of-the-formula}
In this section, we give a proof for Theorem \ref{thm: 1}. We need to first quote the following lemma which can be proved via explicitly computing the difference of the two cocycles as a coboundary element.\\
\begin{lemma}\cite[Chapter XIV, Section 2, Proposition 5]{local fields}\label{lem: br element}
Let $a, b \in \bar{K}^*$ for some perfect field $K$. The following two cocycles represent the same element in $H^2(G_K, \bar{K}^*)$:\\
\begin{enumerate}[label=(\roman*)]
\item \[ (\sigma, \tau) \mapsto \left\{
\begin{array}{ll}
b & \text{ if } \sigma(\sqrt{a})/\sqrt{a}= \tau(\sqrt{a})/\sqrt{a}= -1, \\
1 & \text{ otherwise}
\end{array} \right.
\]
\item \[ (\sigma, \tau) \mapsto \left\{
\begin{array}{ll}
-1 & \text{ if } \sigma(\sqrt{a})/\sqrt{a}= \tau(\sqrt{b})/\sqrt{b}= -1, \\
1 & \text{ otherwise}
\end{array} \right.
\]
\end{enumerate}
Furthermore, they both represent the equivalence class of the quaternion algebra $(a, b)$ in $\text{Br}(K) \cong H^2(G_K, \bar{K}^*).$\\
\end{lemma}
Now we prove Theorem \ref{thm: 1}.\\
\end{comment}
\begin{proof}
We know $\operatorname{et}a \in H^1(G_K, J[2])$ corresponds to $(a, b, c, d) \in (K^*/(K^*)^2)^4$ via taking the Weil pairing with $P, Q, R, S$. Hence, $\operatorname{et}a$ is represented by the cocycle $$\sigma \mapsto \tilde{b}_{\sigma}P + \tilde{a}_{\sigma}Q+ \tilde{d}_{\sigma}R+ \tilde{c}_{\sigma}S,$$
where $\sigma \in G_K$ and for each element $x \in K^*/(K^*)^2$, we define $\tilde{x}_{\sigma} \in \{0, 1\}$ such that $(-1)^{\tilde{x}_{\sigma}}=\sigma(\sqrt{x})/\sqrt{x}$. \\
Then the image of $\operatorname{et}a$ in $H^1(G_K, \text{Pic}^0(J_{\epsilon}))$ is represented by the cocycle that sends $\sigma \in G_K$ to
$$\tilde{b}_{\sigma} \phi_{\epsilon}^*[\tau_P^*\Theta-\Theta] + \tilde{a}_{\sigma}\phi_{\epsilon}^*[\tau_Q^*\Theta-\Theta]+ \tilde{d}_{\sigma}\phi_{\epsilon}^*[\tau_R^*\Theta-\Theta]+ \tilde{c}_{\sigma}\phi_{\epsilon}^*[\tau_S^*\Theta-\Theta].$$
By Remark \ref{rem: DP, DQ, DR, DS}, there exist $K$-rational divisors $D_P, D_Q, D_R, D_S$ on $J_{\epsilon}$ such that the above cocycle sends $\sigma \in G_K$ to
$$\tilde{b}_{\sigma} [D_P-D] + \tilde{a}_{\sigma}[D_Q-D] + \tilde{d}_{\sigma}[D_R-D]+ \tilde{c}_{\sigma}[D_S-D].$$
We need to map this element in $H^1(G_K, \text{Pic}^0(J_{\epsilon}))$ to an element in $H^2(G_K, \bar{K}(J_{\epsilon})^*/\bar{K}^*)$ via the connecting map induced by the short exact sequence $$0 \rightarrow \bar{K}(J_{\epsilon})^*/\bar{K}^* \rightarrow \text{Div}^0(J_{\epsilon}) \rightarrow \text{Pic}^0(J_{\epsilon}) \rightarrow 0.$$ Hence, by the formula for the connecting map and the fact that the divisors $D, D_P, D_Q, D_R,$ $ D_S$ are all $K$-rational, we get that the corresponding element in $H^2(G_K, \bar{K}(J_{\epsilon})^*/\bar{K}^*)$ has image in $H^2(G_K, \text{Div}^0(J_{\epsilon}))$ represented by the following cocycle:
\begin{align*}
(\sigma, \tau) \mapsto & (\tilde{b}_{\tau}-\tilde{b}_{\sigma\tau}+ \tilde{b}_{\sigma}) (D_P-D) + (\tilde{a}_{\tau}-\tilde{a}_{\sigma\tau}+ \tilde{a}_{\sigma})(D_Q-D) \\
&+ (\tilde{d}_{\tau}-\tilde{d}_{\sigma\tau}+ \tilde{d}_{\sigma})(D_R-D)+(\tilde{c}_{\tau}-\tilde{c}_{\sigma\tau}+ \tilde{c}_{\sigma})(D_S-D),
\end{align*}
for $\sigma, \tau \in G_K$.\\
It can be checked that, for $x \in K^*/(K^*)^2$ and $\sigma, \tau \in G_K$, we get $\tilde{x}_{\tau}-\tilde{x}_{\sigma\tau}+ \tilde{x}_{\sigma}=2$ if both $\sigma$ and $\tau$ flip $\sqrt{x}$ and otherwise it is equal to zero. Define $\iota_{\sigma, \tau, x} = 1$ if both $\sigma$ and $\tau$ flip $\sqrt{x}$ and otherwise $\iota_{\sigma, \tau, x} = 0$. Note that the map that sends $x \in K^*/(K^*)^2$ to the class of $(\sigma, \tau) \mapsto \iota_{\sigma, \tau, x}$ explicitly realizes the map $K^*/(K^*)^2 \cong H^1(G_K, \frac{1}{2}\mathbb{Z}/\mathbb{Z}) \subset H^1(G_K, \mathbb{Q}/\mathbb{Z}) \rightarrow H^2(G_K, \mathbb{Z})$. Then, for $\sigma, \tau \in G_K$, the cocycle in the last paragraph sends $(\sigma, \tau)$ to
$$
\iota_{\sigma, \tau, b} \cdot 2(D_P-D) + \iota_{\sigma, \tau, a} \cdot 2(D_Q-D) + \iota_{\sigma, \tau, d} \cdot 2(D_R-D)+ \iota_{\sigma, \tau, c} \cdot 2(D_S-D).
$$
Hence, by Remark \ref{rem: DP, DQ, DR, DS}, the corresponding element in $H^2(G_K, \bar{K}(J_{\epsilon})^*/\bar{K}^*)$ is represented by
$$(\sigma, \tau) \mapsto [f_P^{\iota_{\sigma, \tau, b}} \cdot f_Q^{\iota_{\sigma, \tau, a}} \cdot f_R^{\iota_{\sigma, \tau, d}} \cdot f_S^{\iota_{\sigma, \tau, c}}],$$
for all $\sigma, \tau \in G_K$.\\
For each place $v$ of $K$, following the homogeneous space definition of $\langle \epsilon, \operatorname{et}a \rangle_{CT}$ as given in Section \ref{sec:definition-of-the-cassels-tate-pairing}, we obtain an element in $H^2(G_{K_v}, \bar{K_v^*})$ from the long exact sequence induced by the short exact sequence $0 \rightarrow \bar{K_v^*} \rightarrow \bar{K_v}(J_{\epsilon})^* \rightarrow \bar{K_v}(J_{\epsilon})^*/\bar{K_v}^* \rightarrow 0$. The long exact sequence is the local version of \eqref{eqn:def} with $X$ replaced by $J_{\epsilon}$. By Remark \ref{rem:CT_Selmer}(ii), this element in $H^2(G_{K_v}, \bar{K_v}^*)$ can be represented by\\
$$(\sigma, \tau) \mapsto f_P(P_v)^{\iota_{\sigma, \tau, b}} \cdot f_Q(P_v)^{\iota_{\sigma, \tau, a}} \cdot f_R(P_v)^{\iota_{\sigma, \tau, d}} \cdot f_S(P_v)^{\iota_{\sigma, \tau, c}}, $$
for all $\sigma, \tau \in G_K$ and some local point $P_v \in J_{\epsilon}(K_v)$ avoiding the zeros and poles of $f_P, f_Q, f_R, f_S$. \\
Hence, the above element in $\text{Br}(K_v) \cong H^2(G_{K_v}, \bar{K_v}^*)$ is the class of the tensor product of quaternion algebras
$$(f_P(P_v), b) +(f_Q(P_v), a) + (f_R(P_v), d)+ (f_S(P_v), c).$$
Then, we have that
\begin{align*}
\text{inv}&\big((f_P(P_v), b) +(f_Q(P_v), a) + (f_R(P_v), d)+ (f_S(P_v), c)\big)\\
&= (f_P(P_v, b)_v (f_Q(P_v), a)_v(f_R(P_v), d)_v (f_S(P_v), c)_v,
\end{align*}
where $(\;,\; )_v$ denotes the Hilbert symbol: $K_v^* \times K_v^* \rightarrow \{1, -1\}$, as required.\\
\end{proof}
\begin{remark}\label{rem: CTP finite product}
In Section \ref{sec:prime-bound}, we will directly show that the formula for the Cassels-Tate pairing on $\text{Sel}^2(J) \times \text{Sel}^2(J)$ given in Theorem \ref{thm: 1} is a finite product.\\
\end{remark}
\section{Explicit Computation}
In this section, we explain how we explicitly compute the Cassels-Tate pairing on $\text{Sel}^2(J) \times \text{Sel}^2(J)$ using the formula given in Theorem \ref{thm: 1}, under the assumption that all points in $J[2]$ are defined over $K$. We fix $\epsilon \in \text{Sel}^2(J)$ and $(J_{\epsilon}, [2] \circ \phi_{\epsilon})$, the 2-covering of $J$ corresponding to $\epsilon$ with $\phi_{\epsilon}: J_{\epsilon} \subset \mathbb{P}^{15}\rightarrow J \subset \mathbb{P}^{15}$ given in Theorem \ref{theorem: explicit twist of J}. The statement of Theorem \ref{thm: 1} suggests that we need to compute the $K$-rational divisors $D, D_P, D_Q, D_R, D_S$ on $J_{\epsilon}$ and the $K$-rational function $f_P, f_Q, f_R, f_S$ on $J_{\epsilon}$, as in Remark \ref{rem: DP, DQ, DR, DS}.\\
\subsection{Computing the twist of the Kummer surface}\label{sec:computing-the-twist-of--the-kummer-surface}
We describe a practical method for computing a linear isomorphism $\psi_{\epsilon}: \mathcal{K}_{\epsilon} \subset \mathbb{P}^3 \rightarrow \mathcal{K} \subset \mathbb{P}^3$ corresponding to $\epsilon$. More explicitly, we need to compute $\psi_{\epsilon}$ such that $\psi_{\epsilon}(\psi_{\epsilon}^{-1})^{\sigma}$ is the action of translation by $\epsilon_{\sigma} \in J[2]$ on $\mathcal{K}$ and $(\sigma \mapsto \epsilon_{\sigma})$ is a cocycle representing $\epsilon$. Since all points in $J[2]$ are defined over $K$, the coboundaries in $B^1(G_K, J[2])$ are trivial. Therefore, these conditions determine $\psi_{\epsilon}$ uniquely up to a change of linear automorphism of $\mathcal{K}_{\epsilon} \subset \mathbb{P}^3$ over $K$. \\
For each $T \in J[2]$, we have an explicit formula for $M_T \in \text{GL}_4(K)$, given in \cite[Chapter 3 Section 2]{the book}, representing the action of translation by $T\in J[2]$ on the Kummer surface $\mathcal{K} \subset \mathbb{P}^3$. It can be checked that they form a basis of $\text{Mat}_4(K)$ and we suppose $M_P^2 = c_PI, M_Q^2 = c_QI,M_R^2 = c_RI, \; \text{and} \; M_S^2 = c_SI.$
The explicit formulae for $c_P,c_Q,c_R,c_S$ can also be found in \cite[Chapter 3 Section 2]{the book}. Moreover, by \cite[Chapter 3 Section 3]{the book} and the Weil pairing relationship among the generators $P, Q, R, S$ of $J[2]$ specified by \eqref{equation: WP matrix}, we know that $[M_P, M_Q] = [M_R, M_S] = -I$ and the commutators of the other pairs are trivial.\\
Suppose $(a, b, c, d) \in (K^*/(K^*)^2)^4$ represents $\epsilon$. Let $A \in \text{GL}_4(\bar{K})$ represent the linear isomorphism $\psi_{\epsilon}$ and let $M_T'=A^{-1}M_TA \in \text{GL}_4(\bar{K})$ represent the action of $T$ on the twisted Kummer $\mathcal{K}_{\epsilon}$. It can be checked, see \cite[Lemma 3.2.1]{thesis} for details, that the set of matrices in $\text{PGL}_4(\bar{K})$ that commute with $M_T$ in $\text{PGL}_4(\bar{K})$ for all $T \in J[2]$ is $\{[M_T], T \in J[2]\}$. This implies that any $B \in \text{GL}_4(\bar{K})$ such that $[M_T]'=[B^{-1}M_TB] \in \text{PGL}_4(\bar{K})$ for any $T \in J[2]$ is equal to a multiple of $M_T$ composed with $A$ and so can also represents $\psi_{\epsilon}$. Hence, it will suffice to compute the matrices $M_T'$.\\
Consider $[M_T'] \in \text{PGL}_4(\bar{K})$ and $\sigma \in G_K$. We have $$[M_T']([M_T']^{-1})^{\sigma}=[A^{-1}M_TA(A^{-1})^{\sigma}M_T^{-1}A^{\sigma}]\in \text{PGL}_4(\bar{K}).$$ Recall that for each element $x \in K^*/(K^*)^2$, we define $\tilde{x}_{\sigma} \in \{0, 1\}$ such that $(-1)^{\tilde{x}_{\sigma}}=\sigma(\sqrt{x})/\sqrt{x}$. Since $[A(A^{-1})^{\sigma}]=[M_P^{\tilde{b}_{\sigma}}M_Q^{\tilde{a}_{\sigma}}M_R^{\tilde{d}_{\sigma}}M_S^{\tilde{c}_{\sigma}}]$, we have $[M_T']$ is in $\text{PGL}_4(K)$. This means that we can redefine $M_T'=\lambda_TA^{-1}M_TA$ for some $\lambda_T\in \bar{K}$ such that $M_T'\in \text{GL}_4(K)$ by Hilbert Theorem~90. \\
Let $N_P=1/\sqrt{c_P}M_P, N_Q=1/\sqrt{c_Q}M_Q, N_R=1/\sqrt{c_R}M_R, N_S=1/\sqrt{c_S}M_S$. Then $N_T^2=I$ for $T=P, Q, R, S$. Define $N_T'=A^{-1}N_TA\in \text{GL}_4(\bar{K})$ for $T=P, Q, R, S$. We note that $N_T', M_T'$ represent the same element in $\text{PGL}_4(K)$ and $N_T'^2=I$ for each $T=P, Q, R, S$. Suppose $M_P'^2=\alpha_PI, M_Q'^2=\alpha_QI, M_R'^2=\alpha_RI, M_S'^2=\alpha_SI.$ Then $N_T'=1/\sqrt{\alpha_T}M_T'$ for each $T=P, Q, R, S$. Note that there might be some sign issues here but they will not affect the later computation. Since
$$N_P'(N_P'^{-1})^{\sigma}=A^{-1}N_PA(A^{-1})^{\sigma}(N_P^{-1})^{\sigma}A^{\sigma},$$
using $N_P=1/\sqrt{c_P}M_P$ and $N_P'=1/\sqrt{\alpha_P}M_P'$ with $M_P, M_P' \in \text{GL}_4(K)$, we compute that $$\frac{\sigma(\sqrt{\alpha_P})}{\sqrt{\alpha_P}}=\frac{\sigma(\sqrt{a})}{\sqrt{a}}\frac{\sigma(\sqrt{c_P})}{\sqrt{c_P}},$$
and similar equations for $Q, R, S.$\\
This implies that $\alpha_P=c_P a$ up to squares in $K$ and so via rescaling $M_P'$ by elements in $K$, we have $M_P'^2 = c_P aI$. Similarly, $M_Q'^2 = c_Q bI, M_R'^2 = c_R cI, M_S'^2 = c_S dI$. We note that we also have $[M_P', M_Q'] = [M_R', M_S'] = -I$ and the commutators of the other pairs are trivial. This implies that
$$\text{Mat}_4(K) \cong (c_P a, c_Q b) \otimes (c_R c, c_S d)$$
$$M_P' \mapsto i_1 \otimes 1, M_Q' \mapsto j_1 \otimes 1, M_R' \mapsto 1 \otimes i_2, M_S' \mapsto 1\otimes j_2,$$
where $(c_P a, c_Q b)$ and $(c_R c, c_S d)$ are quaternion algebras with generators $i_1, j_1$ and $i_2, j_2$ respectively. In Section5, we will interpret this isomorphism as saying that the image of $\epsilon$ via the obstruction map is trivial.\\
Let $A = (c_P a, c_Q b), B= (c_R c, c_S d)$. By the argument above, we know $A \otimes B$ represents the trivial element in $\text{Br}(K)$ and an explicit isomorphism $A \otimes B \cong \text{Mat}_4(K)$ will give us the explicit matrices $M_P', M_Q', M_R', M_S'$ we seek. Since the classes of $A, B$ are in $\text{Br}[2]$, we have $A, B$ representing the same element in $\text{Br}(K)$. This implies that $A \cong B$ over $K$, by Wedderburn's Theorem. We have the following lemma.\\
\begin{lemma}\label{lem: computation of M_P' etc}
Consider a tensor product of two quaternion algebras $A \otimes B$, where $A =(\alpha, \beta), \; B=(\gamma, \delta),$ with generators $i_1, j_1$ and $i_2, j_2$ respectively. Suppose there is an isomorphism $\psi: B \xrightarrow{\sim} A$ given by
\begin{align*}
i_2 \mapsto a_1 \cdot 1 + b_1 \cdot i_1 + c_1 \cdot j_1 + d_1 \cdot i_1j_1,\\
j_2 \mapsto a_2 \cdot 1 + b_2 \cdot i_1 + c_2 \cdot j_1 + d_2 \cdot i_1j_1.\\
\end{align*}
Then there is an explicit isomorphism $$A \otimes B \cong \text{Mat}_4(K)$$
given by
$$i_1\otimes 1\mapsto M_{i_1} := \begin{bmatrix}
0 & \alpha & 0 &0\\
1 & 0 & 0 & 0\\
0 & 0 & 0 & \alpha\\
0 & 0 & 1 & 0\\
\end{bmatrix}$$
$$j_1 \otimes 1\mapsto M_{j_1} := \begin{bmatrix}
0 & 0 & \beta &0\\
0 & 0 & 0 & -\beta\\
1 & 0 & 0 & 0\\
0 & -1 & 0 & 0\\
\end{bmatrix}$$
$$1 \otimes i_2 \mapsto M_{i_2} := \begin{bmatrix}
a_1 & b_1 \cdot \alpha & c_1 \cdot \beta & - d_1 \cdot \alpha\beta\\
b_1 & a_1 & -d_1 \cdot \beta & c_1 \cdot \beta\\
c_1 & d_1 \cdot \alpha & a_1 & - b_1 \cdot \alpha\\
d_1 & c_1 & - b_1 & a_1\\
\end{bmatrix}$$
$$1 \otimes j_2 \mapsto M_{j_2} := \begin{bmatrix}
a_2 & b_2 \cdot \alpha & c_2 \cdot \beta & - d_2 \cdot \alpha\beta\\
b_2 & a_2 & -d_2 \cdot \beta & c_2 \cdot \beta\\
c_2 & d_2 \cdot \alpha & a_2 & - b_2 \cdot \alpha\\
d_2 & c_2 & - b_2 & a_2\\
\end{bmatrix}$$
\\
\end{lemma}
\begin{proof}
We have that $A \otimes A^{op}$ is isomorphic to a matrix algebra. More specifically, $A \otimes A^{op} \cong \text{End}_{K}(A)$ via $u \otimes v \mapsto (x \mapsto uxv)$, which makes $A \otimes A^{op} \cong \text{Mat}_4(K)$ after picking a basis for $A$. Hence,
$$
\begin{array}{ccc}
A \otimes B^{op} &\cong &\text{Mat}_4(K)\\
u \otimes v & \mapsto &(x \mapsto ux\psi(v)).\\
\end{array}
\;
$$
More explicitly, fixing the basis of $A$ to be $\{1, i_1, j_1, i_1j_1\}$, the isomorphism is as given in the statement of the lemma.\\
\end{proof}
By taking $A = (c_P a, c_Q b), B= (c_R c, c_S d)$ in Lemma \ref{lem: computation of M_P' etc}, we know that the matrices $M_P', M_Q', M_S', M_R'$ and $M_{i_1}, M_{j_1}, M_{i_2}, M_{j_2}$ are equal up to conjugation by a matrix $C \in \text{GL}_4(K)$ via the Noether Skolem Theorem. After a change of coordinates for $\mathcal{K}_{\epsilon}\subset \mathbb{P}^3$ according to $C$, we have that $M_P', M_Q', M_S', M_R'$ are equal to $M_{i_1}, M_{j_1}, M_{i_2}, M_{j_2}$. Lemma \ref{lem: computation of M_P' etc} therefore reduces the problem of computing the $M_T'$ to that of computing an isomorphism between two quaternion algebras. See \cite[Corollary 4.2.3 ]{thesis} for the description of an explicit algorithm. Finally we solve for a matrix $A$ such that $M_T'=\lambda_TA^{-1}M_TA$ for some $\lambda_T \in \bar{K}$ and $T=P, Q, R, S$ by linear algebra.\\
\begin{comment}
We now give an explicit and practical algorithm for finding an isomorphism between two quaternion algebras over $\mathbb{Q}$ that are known to be isomorphic. First we have the following lemma.\\
\begin{lemma}\label{lem: iso between (a, b) (c d)}
Let $A$ be the quaternion algebra $(a,b)$ over $\mathbb{Q}$,
where $a,b \in \mathbb{Q}^*$. Let $w \not= 0,1$ be a
squarefree integer. The following are equivalent.
\begin{enumerate}[label=(\roman*)]
\item The algebra $A$ contains a subalgebra isomorphic to $\mathbb{Q}(\sqrt{w})$.
\item There exist $s,t,u \in \mathbb{Q}$ with $a s^2 + b t^2 - ab u^2 = w$.\\
\end{enumerate}
\end{lemma}
\begin{proof}
Explicitly, $A$ has $\mathbb{Q}$-basis $1,i,j,ij$ and multiplication determined by
$i^2 = a$, $j^2 = b$ and $ij = -ji$. Suppose $x \in A$ is given as $r + s i + t j + u ij$ where $r,s,t,u \in \mathbb{Q}$. We compute
$$x^2 = r^2 + a s^2 + b t^2 - ab u^2 + 2r(si + tj + uij).$$
Then $x^2 = w$ if and only if $r=0$ and the equation in (ii) is satisfied.\\
\end{proof}
\begin{corollary}\label{cor: iso between (a, b) (c d)}
Let $A=(a, b)$ and $B=(c, d)$ be two isomorphic quaternion algebras over $\mathbb{Q}$. There is an explicit and practical algorithm for finding such an isomorphism $A \cong B$.\\
\end{corollary}
\begin{proof}
Let $i_1, j_1$ denote the generators of $A$. We can assume $a, b, c, d$ are squarefree integers after multiplying by suitable squares. If $c=d=1$, then we are done by Corollary \ref{cor: trivializing matrix algebra n=2}. Otherwise, we can assume $c \neq 1$. Since $A \cong B$, we know $A$ has a subalgebra $\mathbb{Q}(\sqrt{c})$. By Lemma \ref{lem: iso between (a, b) (c d)}, there exist $s_1, t_1, u_1 \in \mathbb{Q}$ such that $as_1^2+bt_1^2-abu_1^2=c$. Let $\alpha=s_1i_1+t_1j_1+u_1i_1j_1\in A$. Now we look for $s_2, t_2, u_2 \in \mathbb{Q}$ that satisfies $$\alpha(s_2i_1+t_2j_1+u_2i_1j_1)=-(s_2i_1+t_2j_1+u_2i_1j_1)\alpha.$$
Let $\beta=s_2i_1+t_2j_1+u_2i_1j_1\in A$. Define $e=a s_2^2 + b t_2^2 - ab u_2^2$ and we have $\beta^2=e$. Consider the quaternion algebra $(c, e)$ with generators denoted by $i_2, j_2$. We have an explicit isomorphism $(c, e) \cong A$ such that
$$i_2 \mapsto \alpha, j_2 \mapsto \beta.$$
Then, by Lemma \ref{lem: iso QA (a, b) (a, c)}, we compute an explicit isomorphism $(c, d)\cong (c, e)$. The composition of these two isomorphisms give $B \cong A$, as required.\\
\end{proof}
\end{comment}
\subsection{Explicit computation of $D$}\label{sec:explicit-computation-of-d}
In this section, we explain a method for computing the $K$-rational divisor $D$ on $J_{\epsilon}$ representing the divisor class $\phi_{\epsilon}^*(2\Theta)$. The idea is to compute it via the commutative diagram \eqref{diagram: BS diagram} in Remark \ref{rem: diagram of twited kummer}. \\
By Theorem \ref{theorem: explicit twist of J}, there is an explicit isomorphism $J_{\epsilon} \subset \mathbb{P}^{15} \xrightarrow{\phi_{\epsilon}} J \subset \mathbb{P}^{15}$. We write $u_0, ..., u_9, v_1, ..., v_6$ for the coordinates on the ambient space of $J_{\epsilon} \subset \mathbb{P}^{15}$ and write $k_{11}, k_{12}, ..., k_{44}, b_1, ..., b_6$ for the coordinates on the ambient space of $J \subset \mathbb{P}^{15}$. By the same theorem, $\phi_{\epsilon}$ is represented by a block diagonal matrix consisting of a block of size 10 corresponding to the even basis elements and a block of size 6 corresponding to the odd basis elements. Following Section \ref{sec:computing-the-twist-of--the-kummer-surface}, we can compute an explicit isomorphism $\psi_{\epsilon} : \mathcal{K}_{\epsilon} \subset \mathbb{P}^3 \rightarrow \mathcal{K} \subset \mathbb{P}^3$ corresponding to $\epsilon$. We write $k_1', ..., k_4'$ for the coordinates on the ambient space of $\mathcal{K}_{\epsilon} \subset \mathbb{P}^3$. Recall that since all points in $J[2]$ are defined over $K$, all coboundaries in $B^1(G_K, J[2])$ are trivial. So, we have that $\phi_{\epsilon}(\phi_{\epsilon}^{-1})^{\sigma}$ and $\psi_{\epsilon} (\psi_{\epsilon}^{-1})^{\sigma}$ both give the action of translation by some $\epsilon_{\sigma} \in J[2]$ such that $(\sigma \mapsto \epsilon_{\sigma})$ represents $\epsilon \in \text{Sel}^2(J)$. \\
Define $k_{ij}'=k_i'k_j'$. The isomorphism $\psi_{\epsilon}: \mathcal{K}_{\epsilon} \subset \mathbb{P}_{k_i'}^3 \rightarrow \mathcal{K}\subset \mathbb{P}_{k_i}^3$, induces a natural isomorphism $\tilde{\psi_{\epsilon}}: \mathbb{P}^9_{k_{ij}'} \rightarrow \mathbb{P}^9_{k_{ij}}$. More explicitly, suppose $\psi_{\epsilon}$ is represented by the $4 \times 4$ matrix $A$ where $(k_1':..., k_4') \mapsto (\sum_{i=1}^4A_{1i}k_i':...:\sum_{i=1}^4A_{4i}k_i')$. Then $\tilde{\psi_{\epsilon}}: \mathbb{P}^9_{k_{ij}'} \rightarrow \mathbb{P}^9_{k_{ij}}$ is given by $(k_{11}': k_{12}':... :k_{44}')\mapsto (\sum_{i, j=1}^4A_{1i}A_{1j}k_{ij}':\sum_{i, j=1}^4A_{1i}A_{2j}k_{ij}':...:\sum_{i, j=1}^4A_{4i}A_{4j}k_{ij}')$. \\
\begin{comment}
We have the following commutative diagram which also give embeddings of $\mathcal{K}, \mathcal{K}_{\epsilon}$ in $\mathbb{P}^9$.
$$
\begin{tikzcd}
\mathcal{K}_{\epsilon} \subset \mathbb{P}^9_{k_{ij}'} \arrow[d, "\tilde{\psi_{\epsilon}}"] \arrow[r, "g_2"]&\mathcal{K}_{\epsilon} \subset \mathbb{P}^3_{k_i'} \arrow[d, "\psi_{\epsilon}"]\\
\mathcal{K}\subset \mathbb{P}^9_{k_{ij}} \arrow[r, "g_1"]&\mathcal{K} \subset \mathbb{P}^3_{k_i},
\end{tikzcd}
$$
where $g_1: (k_{11}: ...: k_{44}) \mapsto (k_{11}: ... : k_{14})$ and $g_2: (k_{11}': ...: k_{44}')\mapsto (k_{11}': ... : k_{14}')$ are the projection maps. We observe the natural morphisms $(k_1:...:k_4) \mapsto (k_{11}: k_{12}: ...:k_{44})$ and $(k_1':...:k_4') \mapsto (k_{11}': k_{12}': ...:k_{44}')$ are the inverses of $g_1$ and $ g_2$, when restricted to $\mathcal{K}$ and $\mathcal{K}_{\epsilon}$ respectively. We also note that $\mathcal{K}_{\epsilon}$ and $\mathcal{K}$ do not lie on any hyperplane in $\mathbb{P}^9$. This makes $\tilde{\psi_{\epsilon}}: \mathbb{P}^9_{k_{ij}'} \rightarrow \mathbb{P}^9_{k_{ij}}$ the natural map, as it is the unique extension of the morphism $\mathcal{K}_{\epsilon} \subset \mathbb{P}^9_{k_{ij}'}\rightarrow \mathcal{K}\subset \mathbb{P}^9_{k_{ij}}$ that makes the above diagram commute.\\
\end{comment}
On the other hand, the isomorphism $\phi_{\epsilon}: J_{\epsilon} \subset \mathbb{P}^{15}_{\{u_i, v_i\}} \rightarrow J \subset \mathbb{P}^{15}_{\{k_{ij}, b_i\}}$ induces a natural isomorphism $\tilde{\phi_{\epsilon}}: \mathbb{P}^9_{u_i} \rightarrow \mathbb{P}^9_{k_{ij}}$ represented by the $10\times 10$ block of the matrix representing $\phi_{\epsilon}$. Since $\phi_{\epsilon}(\phi_{\epsilon}^{-1})^{\sigma}$ and $\psi_{\epsilon} (\psi_{\epsilon}^{-1})^{\sigma}$ both give the action of translation by some $\epsilon_{\sigma} \in J[2]$, we get $\tilde{\phi_{\epsilon}}(\tilde{\phi_{\epsilon}}^{-1})^{\sigma}=\tilde{\psi_{\epsilon}}(\tilde{\phi_{\epsilon}}^{-1})^{\sigma}$. Therefore, $\tilde{\psi_{\epsilon}}^{-1}\tilde{\phi_{\epsilon}}$ is defined over $K$ and we obtain the following commutative diagram that decomposes the standard commutative diagram \eqref{diagram: BS diagram}:
\begin{equation}\label{eqn: D}
\begin{tikzcd}
J_{\epsilon} \subset \mathbb{P}^{15}_{\{u_i, v_i\}} \arrow[r, "proj"] \arrow[d, "\phi_{\epsilon}"]& \mathbb{P}^9_{u_i} \arrow[rr, "(\tilde{\psi_{\epsilon}})^{-1}\tilde{\phi_{\epsilon}}"] \arrow[dr, "\tilde{\phi_{\epsilon}}"]&&\mathbb{P}^9_{k_{ij}'} \arrow[dl, "\tilde{\psi_{\epsilon}}"] \arrow[r, "g_2"]&\mathcal{K}_{\epsilon} \subset \mathbb{P}^3_{k_i'} \arrow[d, "\psi_{\epsilon}"]\\
J \subset \mathbb{P}^{15}_{\{k_{ij}, b_i\}}\arrow[rr, "proj"]& & \mathbb{P}^9_{k_{ij}} \arrow[rr, "g_1"]&&\mathcal{K} \subset \mathbb{P}^3_{k_i},
\end{tikzcd}
\end{equation}
where $g_1:(k_{11}:...:k_{44}) \rightarrow (k_1: ...: k_4)$ and $g_2:(k'_{11}:...:k'_{44}) \rightarrow (k'_1: ...: k'_4)$ are the projection maps. The composition of the morphisms on the bottom row gives the standard morphism $J \xrightarrow{|2\Theta|} \mathcal{K}\subset \mathbb{P}^3$ and the composition of the morphisms on the top row gives $J_{\epsilon} \xrightarrow{|\phi_{\epsilon}^*(2\Theta)|} \mathcal{K}_{\epsilon}\subset \mathbb{P}^3$. \\
Let $D$ be the pull back on $J_{\epsilon}$ via $J_{\epsilon} \subset \mathbb{P}^{15}_{\{u_i, v_i\}} \xrightarrow{proj} \mathbb{P}^9_{u_i} \xrightarrow{(\tilde{\psi_{\epsilon}})^{-1}\tilde{\phi_{\epsilon}}} \mathbb{P}^9_{k_{ij}'} \xrightarrow{proj} \mathbb{P}^3_{k_i'}$ of the hyperplane section given by $k_1'=0$. This implies that $D$ is a $K$-rational divisor on $J_{\epsilon}$ representing the class of $\phi_{\epsilon}^*(2\Theta)$. Moreover, the pull back on $J_{\epsilon}$ via $J_{\epsilon} \subset \mathbb{P}^{15}_{\{u_i, v_i\}} \xrightarrow{proj} \mathbb{P}^9_{u_i} \xrightarrow{(\tilde{\psi_{\epsilon}})^{-1}\tilde{\phi_{\epsilon}}} \mathbb{P}^9_{k_{ij}'}$ of the hyperplane section given by $k_{11}'=0$ is $2D$. \\
\begin{comment}
We have the following proposition for computing the divisor $D$.\\
\begin{proposition}\label{prop: horizontal map over K for D}
Let $J$ be the Jacobian variety of a genus two curve. Suppose all points in $J[2]$ are over $K$. Let $J_{\epsilon}$ be the 2-covering of $J$ corresponding to $\epsilon \in \text{Sel}^2(J)$. The morphism $J_{\epsilon} \subset \mathbb{P}^{15} \rightarrow \mathcal{K}_{\epsilon} \subset \mathbb{P}^3$ that makes the following diagram commute is over $K$.
\[
\begin{tikzcd}
J_{\epsilon} \subset \mathbb{P}^{15}\arrow[r] \arrow[d, "\phi_{\epsilon}"]& \mathcal{K}_{\epsilon} \subset \mathbb{P}^3 \arrow[d, "\psi_{\epsilon}"]\\
J \subset \mathbb{P}^{15}\arrow[r, "|2\theta|"]& \mathcal{K} \subset \mathbb{P}^3,\\
\end{tikzcd}
\]
where $\phi_{\epsilon}$ is the isomorphism with explicit formula described in Remark \ref{rem: explicit twist map of J} and $\psi_{\epsilon}$ is the change of coordinates map on $\mathbb{P}^3$ corresponding to $\epsilon$ that is computable with algorithms explained in the beginning of this section. \\
\end{proposition}
\begin{proof}
We know that $\phi_{\epsilon} (\phi_{\epsilon}^{-1})^{\sigma} = \tau_{P_{\epsilon_{\sigma}}}$ with $P_{\epsilon_{\sigma}} \in J[2]$ and $\psi_{\epsilon} (\psi_{\epsilon}^{-1})^{\sigma} = \tau_{P'_{\epsilon_{\sigma}}}$ with $P'_{\epsilon_{\sigma}} \in J[2]$ for each $\sigma \in G_K$. Moreover, $(\sigma \mapsto P_{\epsilon_{\sigma}})$ and $(\sigma \mapsto P'_{\epsilon_{\sigma}})$ both represent the Selmer element $\epsilon \in H^1(G_K, J[2])$. Since we assume that all points in $J[2]$ are over $K$, all elements in $B^1(G_K, J[2])$ are trivial. Hence $P_{\epsilon_{\sigma}}= P'_{\epsilon_{\sigma}}$ for all $\sigma \in G_K$. So we can decompose the above commutative diagram to the following:
\[
\begin{tikzcd}
J_{\epsilon} \subset \mathbb{P}^{15} \arrow[r, "proj"] \arrow[d, "\phi_{\epsilon}"]& \mathbb{P}^9 \arrow[rr, "f"] \arrow[dr, "\phi_{\epsilon}^*"]&&\mathbb{P}^9_{k_{ij}'} \arrow[dl, "\psi_{\epsilon}^*"] \arrow[r, "g_2'"]&\mathcal{K}_{\epsilon} \subset \mathbb{P}^3_{k_i'} \arrow[d, "\psi_{\epsilon}"]\\
J \subset \mathbb{P}^{15}\arrow[rr, "g_1"]& & \mathbb{P}^9_{k_{ij}} \arrow[rr, "g_2"]&&\mathcal{K} \subset \mathbb{P}^3_{k_i}.\\
\end{tikzcd}
\]
Here $\{k_i\}$ denotes the coordinates on $\mathbb{P}^3$ where $\mathcal{K}$ is embedded in and $k_{ij}=k_ik_j$ form a set of 10 even basis elements of $L(4\Theta)$. Similarly $\{k_i'\}$ denotes the coordinates on $\mathbb{P}^3$ where $\mathcal{K}_{\epsilon}$ is embedded in and $k'_{ij}=k_i'k_j'$ form a set of 10 even basis elements of $L(\phi_{\epsilon}^*(4\Theta))$. Note here even functions on $J_{\epsilon}$ is with respect to the induced involution map $Q \mapsto \phi_{\epsilon}^{-1}(-\phi_{\epsilon}(Q))$ for $Q \in J_{\epsilon}$. \\
Also the bottom horizontal morphism $J \subset \mathbb{P}^{15} \xrightarrow{|2\Theta|} \mathcal{K} \subset \mathbb{P}^3$ is now the composition of the map $J \subset \mathbb{P}^{15} \xrightarrow{g_1} \mathbb{P}^9_{k_{ij}}$ which has explicit formula given in [\cite{explicit twist}, Proposition 6.12] as mentioned in Remark \ref{rem: explicit twist map of J} and the natural projection map $\mathbb{P}^9_{k_{ij}} \xrightarrow{g_2} \mathbb{P}^3_{k_i}$ that sends $(k_{ij})$ to $(k_{11}, k_{12}, k_{13}, k_{14})$. The top horizontal morphism $J_{\epsilon} \subset \mathbb{P}^{15} \xrightarrow{|\phi_{\epsilon}^*(2\Theta)|} \mathcal{K}_{\epsilon} \subset \mathbb{P}^3$ is now the composition of $J_{\epsilon} \subset \mathbb{P}^{15} \xrightarrow{proj} \mathbb{P}^9$ which is a projection to the even coordinates of the $\mathbb{P}^{15}$ where $J_{\epsilon}$ is embedded in followed by an isomorphism $f$ between two sets of even basis of $L(\phi_{\epsilon}^*(4\Theta))$ and then the natural projection $\mathbb{P}^9_{k'_{ij}} \xrightarrow{g_2'} \mathcal{K}_{\epsilon} \subset \mathbb{P}_{k_i'}^3$ that sends $(k'_{ij})$ to $(k'_{11}, k'_{12}, k'_{13}, k'_{14}).$\\
Hence it suffices to show that $f$ is over $K$. This is done by considering the vertical maps. Note that $\phi_{\epsilon}^*$ and $\psi_{\epsilon}^*$ are naturally induced from $\phi_{\epsilon}$ and $\psi_{\epsilon}$ respectively. By the argument in the beginning of the proof, we know $\phi_{\epsilon}^*((\phi_{\epsilon}^*)^{-1})^{\sigma}= \psi_{\epsilon}^*((\psi_{\epsilon}^*)^{-1})^{\sigma}$ for every $\sigma \in G_K$. This implies that $f= (\psi_{\epsilon}^*)^{-1} \circ \phi_{\epsilon}^*$ is over $K$ as required. \\
\end{proof}
\begin{remark}\label{rem: explicit D}
Proposition \ref{prop: horizontal map over K for D} in fact gives us a way to compute the $K-$rational morphism $J_{\epsilon} \xrightarrow{|\phi_{\epsilon}^*(2\Theta)|} \mathbb{P}^3$. More explicitly it is computable via the composition $J_{\epsilon} \subset \mathbb{P}^{15} \xrightarrow{proj} \mathbb{P}^9 \xrightarrow{f} \mathbb{P}^9_{k'_{ij}} \xrightarrow{g_2'} \mathbb{P}^3_{k_i'}$. The isomorphism $f$ is computable via $(\psi_{\epsilon}^*)^{-1} \circ \phi_{\epsilon}^*$ which is induced from $\phi_{\epsilon}$ and $\psi_{\epsilon}$. Then pulling back a hyperplane via the composition of these morphisms gives us a $K-$rational divisor $D$ that represents the divisor class of $\phi_{\epsilon}^*(2\Theta)$. \\
\end{remark}
\end{comment}
\subsection{Explicit computation of $D_P, D_Q, D_R, D_S$}\label{sec:explicit-computation-of-dp-dq-dr-ds}
In this section, we explain how to compute the $K$-rational divisors $D_P, D_Q, D_R, D_S$ defined in Remark \ref{rem: DP, DQ, DR, DS}. More explicitly, for $T \in J[2]$, we give a method for computing a $K$-rational divisor $D_T$ on $J_{\epsilon}$ representing the divisor class of $\phi_{\epsilon}^*(\tau_{T_1}^*(2\Theta))$ for some $T_1$ on $J$ such that $2T_1=T$. Recall that we assume all points in $J[2]$ are defined over $K$ and we have an explicit isomorphism $\phi_{\epsilon}: J_{\epsilon} \rightarrow J$ such that $(J_{\epsilon}, [2]\circ \phi_{\epsilon})$ is the 2-covering of $J$ corresponding to $\epsilon\in \text{Sel}^2(J)$. Recall $\delta: J(K) \rightarrow H^1(G_K, J[2])$ in \eqref{eqn: connecting map}. We first prove the following lemma.\\
\begin{lemma}\label{lem: longest vertical map over K}
Let $T \in J(K)$. Suppose $\phi_{\epsilon+\delta(T)}: J_{\epsilon+\delta(T)}\rightarrow J$ is an isomorphism and $(J_{\epsilon+\delta(T)}, [2] \circ \phi_{\epsilon+\delta(T)})$ is the 2-covering of $J$ corresponding to $\epsilon+\delta(T) \in H^1(G_K, J[2])$. Let $T_1 \in J$ such that $2T_1=T$. Then, $\phi_{\epsilon+ \delta(T)}^{-1} \circ \tau_{T_1} \circ \phi_{\epsilon}: J_{\epsilon} \rightarrow J_{\epsilon+\delta(T)}$ is defined over $K$.
\end{lemma}
\begin{proof}
Using the same argument as in the proof of Lemma \ref{lem: existence of rational divisor D_T}(i), we know that $(J_{\epsilon}, [2] \circ \tau_{T_1} \circ \phi_{\epsilon})$ is the 2-covering of $J$ corresponding to $\epsilon+\delta(T) \in H^1(G_K, J[2])$. Since all points in $J[2]$ are defined over $K$, we have $\tau_{T_1} \circ \phi_{\epsilon} \circ ((\tau_{T_1} \circ \phi_{\epsilon} )^{-1})^{\sigma} = \phi_{\epsilon+ \delta(T)} \circ (\phi_{\epsilon+ \delta(T)}^{-1})^{\sigma}$, as required. \\
\end{proof}
Let $T \in J(K)$ with $2T_1=T$. Consider the commutative diagram below which is formed by two copies of the standard diagram \eqref{diagram: BS diagram}. Note that all the horizontal maps are defined over $K$ as is the composition of the vertical map on the left by Lemma \ref{lem: longest vertical map over K}. Hence, the composition of the thick arrows is defined over $K$. Then the pull back on $J_{\epsilon}$ via the thick arrows of a hyperplane section on $\mathcal{K}_{\epsilon+\delta(T)} \subset \mathbb{P}^3$ is a $K$-rational divisor $D_T$ on $J_{\epsilon}$ representing the divisor class $\phi_{\epsilon}^*(\tau_{T_1}^*(2\Theta))$. We note that in the case where $T \in J[2]$, the composition of the vertical maps on the left hand side of the the diagram below is in fact given by a $16 \times 16$ matrix defined over $K$ even though the individual maps are not defined over $K$.\\
\begin{equation}\label{diagram 1}
\begin{tikzcd}[column sep = large]
J_{\epsilon} \subset \mathbb{P}^{15} \arrow[r, "|\phi_{\epsilon}^*(2\Theta)|"] \arrow[d, "\phi_{\epsilon}", very thick] & \mathcal{K}_{\epsilon} \subset \mathbb{P}^3 \arrow[d, "\psi_{\epsilon}"]\\
J\subset \mathbb{P}^{15} \arrow[r, "|2\Theta|"] \arrow[d, "\tau_{T_1}", very thick] & \mathcal{K} \subset \mathbb{P}^3\\
J\subset \mathbb{P}^{15} \arrow[r, "|2\Theta|"] \arrow[d, "\phi_{\epsilon+\delta(T)}^{-1}", very thick]& \mathcal{K} \subset \mathbb{P}^3\\
J_{\epsilon + \delta(T)} \subset \mathbb{P}^{15} \arrow[r, "|\phi^*_{\epsilon+\delta(T)}(2\Theta)|", very thick] & \mathcal{K}_{\epsilon+\delta(T)} \subset \mathbb{P}^3 \arrow[u, "\psi_{\epsilon + \delta(T)}"'].\\
\end{tikzcd}
\end{equation}
The bottom horizontal morphism $J_{\epsilon+\delta(T)} \xrightarrow{|\phi^*_{\epsilon+ \delta(T)}(2\Theta)|} \mathcal{K}_{\epsilon+\delta(T)} \subset \mathbb{P}^3$ can be explicitly computed using the algorithm in Section \ref{sec:explicit-computation-of-d} with the Selmer element $\epsilon$ replaced by $\epsilon+\delta(T)$. Also, by Theorem \ref{theorem: explicit twist of J}, we have explicit formulae for $\phi_{\epsilon}$ and $\phi_{\epsilon+ \delta(T)}.$ Hence, to explicitly compute $D_T$, we need to find a way to deal with $\tau_{T_1}$, for some $T_1$ such that $2T_1=T$.\\
Since we need to apply the above argument to $T=P, Q, R, S$, the basis for $J[2]$, it would suffice to compute the translation maps $\tau_{T_1}: J\subset \mathbb{P}^{15} \rightarrow J \subset \mathbb{P}^{15}$ when $T_1 \in J[4]$. This map is given by a $16 \times 16$ matrix. We show how to compute a $10 \times 16$ matrix in the following proposition. We then explain below why this is sufficient for our purposes.\\
\begin{proposition}\label{prop: partial translation by 4 torsion}
Suppose $T_1 \in J[4]$. Given the coordinates of $T_1 \in J \subset \mathbb{P}^{15}_{\{k_{ij}, b_i\}}$, we can compute the following composition of morphisms:
$$\Psi: J \subset \mathbb{P}^{15}_{\{k_{ij}, b_i\}} \xrightarrow{\tau_{T_1}} J \subset \mathbb{P}^{15}_{\{k_{ij}, b_i\}} \xrightarrow{proj} \mathbb{P}^{9}_{k_{ij}}.$$\\
\end{proposition}
\begin{proof}
Let $T =2T_1 \in J[2]$. Recall that we let $M_T$ denote the action of translation by $T$ on $\mathcal{K} \subset \mathbb{P}^3$. Then for any $P \in J$, we have $k_i(P+T)=\sum_{j=1}^4(M_T)_{ij}k_j(P)$ projectively as a vector of length 4, and projectively as a vector of length 10, $k_{ij}(P+T_1)$ is equal to
$$k_i(P+T_1)k_j(P+T_1)=k_i(P+T_1) \sum_{l=1}^4(M_T)_{jl}k_l(P-T_1)=\sum_{l=1}^4(M_T)_{jl}k_l(P-T_1)k_i(P+T_1).$$
\begin{comment}
\begin{align*}
&k_{ij}(P+T_1)\\
&= k_i(P+T_1)k_j(P+T_1)\\
&=k_i(P+T_1)k_j(\tau_T(P-T_1))\\
&=k_i(P+T_1)\cdot \sum_{l=1}^4(M_T)_{jl}k_l(P-T_1)\\
&=\sum_{l=1}^4(M_T)_{jl}k_l(P-T_1)k_i(P+T_1).
\end{align*}
\end{comment}
By \cite[Theorem 3.2]{the gp law paper}, there exists a $4 \times 4$ matrix of bilinear forms $\phi_{ij}(P, T_1)$, with explicit formula, that is projectively equal to the matrix $k_i(P-T_1)k_j(P+T_1)$. Since we have an explicit formula for $M_T$ in \cite[Chapter 3, Section 2]{the book}, we can partially compute the linear isomorphism $\tau_{T_1}$: $$\Psi: J \subset \mathbb{P}^{15}_{\{k_{ij}, b_i\}} \xrightarrow{\tau_{T_1}} J \subset \mathbb{P}^{15}_{\{k_{ij}, b_i\}} \xrightarrow{proj} \mathbb{P}^{9}_{k_{ij}},$$
as required. \\
\end{proof}
\begin{comment}Hence for any point $P\in J$, the $4 \times 4$ matrix of
\begin{align*}
&k_i(P+T_1)k_j(P-T_1)\\
&= k_i(P+T_1)k_j(P+T_1-T)\\
&=k_i(P+T_1)k_j(\tau_T(P+T_1))\\
&=k_i(P+T_1)(M_T)_{jl}k_l(P+T_1)\\
\end{align*}
is projectively equal to a known $4 \times 4$ matrix of bilinear forms of the coordinates of $P, T_1 \in J \subset \mathbb{P}^{15}_{k_{ij}, b_i}$. Recall, we have an explicit formula for $M_T$ in \cite[Chapter 3, Section 2]{the book}. Also from \cite[Appendix C]{the gp law paper}, we can compute the explicit coordinates of $T_1$ from the coordinates of $T \in J \subset \mathbb{P}^{15}$ that are given.
\end{comment}
\begin{remark}\label{rem: computing T_1}
Suppose $2T_1=T \in J[2]$. From the doubling formula on $\mathcal{K}$ as in \cite[Appendix~C]{the gp law paper}, we can compute the coordinates of the image of $T_1$ on $\mathcal{K} \subset \mathbb{P}^3$ from the coordinates of the image of $T$ on $\mathcal{K} \subset \mathbb{P}^3$. This gives the 10 even coordinates, $k_{ij}(T_1)$ and we can solve for the odd coordinates by the 72 defining equations of $J$ given in Theorem \ref{theorem: 72}. Note that by Lemma \ref{lem: longest vertical map over K}, we know the field of definition of $T_1$ is contained in the composition of the field of definition of $\phi_{\epsilon}$ and $\phi_{\epsilon+\delta(T)}$. Hence, we can compute this field explicitly which helps solving for this point using MAGMA \cite{magma}.\\
\end{remark}
Consider $T \in J[2]$ with $T_1 \in J[4]$ such that $2T_1=T$. We follow the discussion in Section \ref{sec:explicit-computation-of-d} with $\epsilon$ replaced by $\epsilon+\delta(T)$. This gives a diagram analogous to \eqref{eqn: D}. Let $k_{1, T}', ..., k_{4, T}'$ be the coordinates on the ambient space of $\mathcal{K}_{\epsilon+\delta(T)}\subset \mathbb{P}^3$ and let $u_{0, T}, ..., u_{9, T}, v_{1, T}, ..., v_{6, T}$ be the coordinates on the ambient space of $J_{\epsilon+\delta(T)}\subset \mathbb{P}^{15}$. Let $k_{ij, T}'=k_{i, T}'k_{j, T}'$. Decomposing the lower half of the diagram \eqref{diagram 1} gives the commutative diagram below:
\begin{equation}\label{equation: D_T}
\begin{tikzcd}[ampersand replacement=\&,column sep = large]
J_{\epsilon} \subset \mathbb{P}^{15}_{\{u_i, v_i\}} \arrow[rr, "|\phi_{\epsilon}^*(2\Theta)|"] \arrow[d, "\phi_{\epsilon}", very thick]\&\& \mathcal{K}_{\epsilon} \subset \mathbb{P}^3_{k_i'} \arrow[d, "\psi_{\epsilon}"]\\
J\subset \mathbb{P}^{15}_{\{k_{ij}, b_i\}} \arrow[rr, "|2\Theta|"] \arrow[d, "\tau_{T_1}"] \arrow[dr, "\Psi", very thick]\&\& \mathcal{K} \subset \mathbb{P}_{k_i}^3\\
J\subset \mathbb{P}^{15}_{\{k_{ij}, b_i\}} \arrow[r, "proj"]\&\mathbb{P}_{k_{ij}}^9 \arrow[r, "g_1"]\arrow[d, "(\tilde{\psi}_{\epsilon+\delta(T)})^{-1}", very thick]\& \mathcal{K} \subset \mathbb{P}_{k_i}^3\\
J_{\epsilon + \delta(T)} \subset \mathbb{P}^{15}_{\{u_{i, T}, v_{i, T}\}} \arrow[r] \arrow[u, "{\phi}_{\epsilon+\delta(T)}"]\&\mathbb{P}_{k_{ij, T}'}^9 \arrow[r, "g_2", very thick]\& \mathcal{K}_{\epsilon+\delta(T)} \subset \mathbb{P}_{k_{i, T}'}^3 \arrow[u, "\psi_{\epsilon + \delta(T)}"].\\
\end{tikzcd}
\end{equation}
Recall Proposition \ref{prop: partial translation by 4 torsion} explains how $\Psi$ can be explicitly computed and the composition of the thick arrows in \eqref{equation: D_T} is defined over $K$ by Lemma \ref{lem: longest vertical map over K}. Let $D_T$ be the pull back on $J_{\epsilon}$ via the thick arrows in \eqref{equation: D_T} of the hyperplane section given by $k_{1, T}'=0$. This implies that $D_T$ is a $K$-rational divisor on $J_{\epsilon}$ representing the class of $\phi_{\epsilon}^*(\tau_{T_1}^*(2\Theta))$. Moreover, the pull back on $J_{\epsilon}$ via $$J_{\epsilon} \subset \mathbb{P}^{15}_{\{u_i, v_i\}} \xrightarrow{\phi_{\epsilon}} J \subset \mathbb{P}^{15}_{\{k_{ij, b_i}\}}\xrightarrow{\Psi} \mathbb{P}^9_{k{ij}} \xrightarrow{(\tilde{\psi}_{\epsilon+\delta(T)})^{-1}} \mathbb{P}^9_{k_{ij, T}'}$$ of the hyperplane section given by $k_{11, T}'=0$ is $2D_T$. \\
We now apply the above discussion with $T=P, Q, R, S$ and get that the divisors $D_P, D_Q, D_R, D_S$ on $J_{\epsilon}$ described in Remark \ref{rem: DP, DQ, DR, DS} as required.\\
\begin{remark}\label{rem: explicit fp, fq, fr, fs}
From the above discussion and the discussion in Section \ref{sec:explicit-computation-of-d}, the $K$-rational functions $f_P, f_Q, f_R, f_S$ in the formula for the Cassels-Tate pairing in Theorem \ref{thm: 1} are quotients of linear forms in the coordinates of the ambient space of $J_{\epsilon} \subset \mathbb{P}^{15}$. They all have the same denominator, this being the linear form that cuts out the divisor $2D$.\\
\end{remark}
\section{The Obstruction Map}
In this section, we will state and prove an explicit formula for the obstruction map $\text{Ob}: H^1(G_K, J[2]) \rightarrow \text{Br}(K)$. See below for the definition of this map. This generalizes a formula in the elliptic curve case due to O'Neil \cite [Proposition 3.4]{oneil}, and later refined by Clark \cite[Theorem 6]{clark}.
Although this is not needed for the computation of the Cassels-Tate pairing, it explains why we needed to work with quaternion algebras in Section \ref{sec:computing-the-twist-of--the-kummer-surface}.\\
\begin{definition}
The obstruction map $$\text{Ob}: H^1(G_{K}, J[2]) \rightarrow H^2(G_{K}, \bar{K}^*) \cong \text{Br}(K)$$ is the composition of the map $H^1(G_{K}, J[2]) \rightarrow H^1(G_{K}, \text{PGL}_4(\bar{K}))$ induced by the action of translation of $J[2]$ on $\mathcal{K} \subset \mathbb{P}^3$, and the injective map $H^1(G_{K}, \text{PGL}_4(\bar{K})) \rightarrow H^2(G_{K}, \bar{K}^*)$ induced from the short exact sequence $0 \rightarrow \bar{K}^* \rightarrow \text{GL}_4(\bar{K}) \rightarrow \text{PGL}_4(\bar{K}) \rightarrow 0$.\\
\end{definition}
\begin{comment}
In this section, we give the explicit formula for the obstruction map $\text{Ob}: H^1(G_K, J[2]) \rightarrow \text{Br}(K)$ defined in Section \ref{sec:obstruction map 1}. Recall we have explicit formula for $M_T\in \text{GL}_4(K)$ given in \cite[Chapter 3 Section 2]{the book}, which represents the action of translation by $T\in J[2]$ on the Kummer surface $\mathcal{K} \subset \mathbb{P}^3$. In particular, we define $c_P, c_Q, c_R, c_S \in K$ such that
$M_P^2 = c_PI, M_Q^2 = c_QI,M_R^2 = c_RI, \; \text{and} \; M_S^2 = c_SI$ with $P, Q, R, S$ a set of generators for $J[2]$ satisfying the Weil pairing matrix \eqref{equation: WP matrix}. Also, $[M_P, M_Q]=[M_R, M_S]=-I$ and the commutators of the other pairs are trivial. Consider $\epsilon \in \text{Sel}^2(J)$ that corresponds to $(a, b, c, d) \in (K^*/(K^*)^2)^4$ as in Section \ref{sec:choice-of-generators-of-j2}. By Proposition \ref{prop:obstruction map and enveloping algebra}, we know that a representation of $\text{Ob}(\epsilon)$ is the enveloping algebra for $\mathbf{\Theta_{\epsilon}}$ which is naturally given as $(c_P a, c_Q b) \otimes (c_R c, c_S d)$ from the discussion in Section \ref{sec:modified-naive-method}. From this observation and the formula for the obstruction map in the case of elliptic curves, we conjectured that such formula exists for any element in $H^1(G_K, J[2])$. This is proved in the following theorem.\\
\end{comment}
\begin{theorem}\label{thm: ob map}
Let $J$ be the Jacobian variety of a genus two curve defined over a field $K$ with $char(K) \neq 2$. Suppose all points in $J[2]$ are defined over $K$. For $\epsilon \in H^1(G_K, J[2])$, represented by $(a, b, c, d) \in (K^*/(K^*)^2)^4$ as in Section \ref{sec:formula-for-the-cassels-tate-pairing}, the obstruction map $\mathrm{Ob}: H^1(G_K,J[2]) \rightarrow \mathrm{Br}(K)$ sends $\epsilon$ to the class of the tensor product of two quaternion algebras:
$$\mathrm{Ob} (\epsilon) = (c_P a, c_Q b) + (c_Rc, c_Sd),$$
where $c_P, c_Q, c_R, c_S \in K$ are such that $M_P^2 = c_PI, M_Q^2 = c_QI,M_R^2 = c_RI, \; \text{and} \; M_S^2 = c_SI$ as defined in Section \ref{sec:computing-the-twist-of--the-kummer-surface}.\\
\end{theorem}
\begin{proof}
Let $N_P = 1/\sqrt{c_P}M_P, N_Q= 1/\sqrt{c_Q}M_Q, N_R=1/\sqrt{c_R}M_R, N_S =1/\sqrt{c_S}M_S \in \text{GL}_4(\bar{K})$. Then $N_P$ is a normalized representation in $\text{GL}_4(\bar{K})$ of $[M_P] \in \text{PGL}_4(K)$. Similar statements are true for $Q, R, S$. Notice that $N_P^2= N_Q^2 = N_R^2 = N_S^2= I$. So there is a uniform way of picking a representation in $\text{GL}_4(\bar{K})$ for the translation induced by $\alpha_1P + \alpha_2 Q + \alpha_3R + \alpha_4S$ for $\alpha_i \in \mathbb{Z}$, namely $N_P^{\alpha_1} N_Q^{\alpha_2} N_R^{\alpha_3} N_S^{\alpha_4}.$\\
Since $\epsilon \in H^1(K, J[2]) $ is represented by $(a, b, c, d) \in (K^*/{K^*}^2)^4$ and $P, Q, R, S$ satisfy the Weil pairing matrix \eqref{equation: WP matrix}, a cocycle representation of $\epsilon$ is: $$\sigma \mapsto \tilde{b}_{\sigma}P +\tilde{a}_{\sigma}Q +\tilde{d}_{\sigma}R +\tilde{c}_{\sigma}S ,$$ where for each element $x \in K^*/(K^*)^2$, we define $\tilde{x}_{\sigma} \in \{0, 1\}$ such that $(-1)^{\tilde{x}_{\sigma}}=\sigma(\sqrt{x})/\sqrt{x}$.\\% as in the proof of Theorem \ref{thm: 1}. \\
Now consider the following commutative diagram of cochains:
$$
\begin{tikzcd}
C^1(G_K, \bar{K}^*) \arrow[r] \arrow[d, "d"]& C^1(G_K, \text{GL}_4) \arrow[r] \arrow[d, "d"]& C^1(G_K, \text{PGL}_4) \arrow[d, "d"]\\
C^2(G_K, \bar{K}^*) \arrow[r]& C^2(G_K, \text{GL}_4) \arrow[r]& C^2(G_K, \text{PGL}_4).\\
\end{tikzcd}
$$
Defining $N_{\sigma}=N_P^{\tilde{b}_{\sigma}}N_Q^{\tilde{a}_{\sigma}}N_R^{\tilde{d}_{\sigma}}N_S^{\tilde{c}_{\sigma}}$, we have
$$
\begin{array}{ccc}
H^1(K, J[2])&\rightarrow &H^1(G_K, \text{PGL}_4)\\
(a, b, c, d)& \mapsto & (\sigma \mapsto [N_{\sigma}]).
\end{array}
$$
Then $(\sigma \mapsto [N_{ \sigma}]) \in C^1(G_K, \text{PGL}_4)$ lifts to $(\sigma \mapsto N_{ \sigma}) \in C^1(G_K, \text{GL}_4)$ which is then mapped to $$((\sigma, \tau) \mapsto (N_{\tau})^{\sigma}N_{\sigma \tau}^{-1}N_{ \sigma}) \in C^2(G_K, \text{GL}_4).$$
Note that $$N_P^{\sigma}= (\frac{1}{\sqrt{c_P}}M_P)^{\sigma}=\frac{1}{\sigma(\sqrt{c_P})}M_P =\frac{\sqrt{c_P}}{\sigma(\sqrt{c_P})}N_P= (-1)^{(\widetilde{c_P})_{\sigma}}N_P,$$ treating $c_P$ in $K^*/(K^*)^2$. Similar results also hold for $Q, R, S$. Observe that for any $x \in K^*/(K^*)^2$ and $ \sigma, \tau \in G_K$, we have $\tilde{x}_{\sigma}-\tilde{x}_{\sigma\tau}+ \tilde{x}_{\sigma}$ is equal to 0 or 2. Since $N_P^2= N_Q^2 = N_R^2 = N_S^2= I$, $[N_P, N_Q]=[N_R, N_S]=-I$ and the commutators of the other pairs are trivial, we have \\
\begin{align*}
(N_{\tau})^{\sigma}N_{\sigma \tau}^{-1}N_{ \sigma}
=& (N_P^{\tilde{b}_{\tau}}N_Q^{\tilde{a}_{\tau}}N_R^{\tilde{d}_{\tau}}N_S^{\tilde{c}_{\tau}})^{\sigma} \cdot N_S^{-\tilde{c}_{\sigma\tau}}N_R^{-\tilde{d}_{\sigma\tau}}N_Q^{-\tilde{a}_{\sigma\tau}}N_P^{-\tilde{b}_{\sigma\tau}} \cdot N_P^{\tilde{b}_{\sigma}}N_Q^{\tilde{a}_{\sigma}}N_R^{\tilde{d}_{\sigma}}N_S^{\tilde{c}_{\sigma}}\\
=& (-1)^{(\widetilde{c_P})_{\sigma} \cdot \tilde{b}_{\tau}} \cdot (-1)^{(\widetilde{c_Q})_{\sigma} \cdot \tilde{a}_{\tau}} \cdot(-1)^{(\widetilde{c_R})_{\sigma} \cdot \tilde{d}_{\tau}} \cdot(-1)^{(\widetilde{c_S})_{\sigma} \cdot \tilde{c}_{\tau}}\\
& \cdot N_P^{\tilde{b}_{\tau}}N_Q^{\tilde{a}_{\tau}} \cdot N_R^{\tilde{d}_{\tau}}N_S^{\tilde{c}_{\tau}}N_S^{-\tilde{c}_{\sigma\tau}}N_R^{-\tilde{d}_{\sigma\tau}} \cdot N_Q^{-\tilde{a}_{\sigma\tau}}N_P^{-\tilde{b}_{\sigma\tau}}N_P^{\tilde{b}_{\sigma}}N_Q^{\tilde{a}_{\sigma}} \cdot N_R^{\tilde{d}_{\sigma}}N_S^{\tilde{c}_{\sigma}}\\
=& (-1)^{(\widetilde{c_P})_{\sigma} \cdot \tilde{b}_{\tau}} \cdot (-1)^{(\widetilde{c_Q})_{\sigma} \cdot \tilde{a}_{\tau}} \cdot(-1)^{(\widetilde{c_R})_{\sigma} \cdot \tilde{d}_{\tau}} \cdot(-1)^{(\widetilde{c_S})_{\sigma} \cdot \tilde{c}_{\tau}}\\
& \cdot N_P^{\tilde{b}_{\tau}}N_Q^{\tilde{a}_{\tau}}N_Q^{-\tilde{a}_{\sigma\tau}}N_P^{-\tilde{b}_{\sigma\tau}}N_P^{\tilde{b}_{\sigma}}N_Q^{\tilde{a}_{\sigma}} \cdot N_R^{\tilde{d}_{\tau}}N_S^{\tilde{c}_{\tau}}N_S^{-\tilde{c}_{\sigma\tau}}N_R^{-\tilde{d}_{\sigma\tau}}N_R^{\tilde{d}_{\sigma}}N_S^{\tilde{c}_{\sigma}}\\
=& (-1)^{(\widetilde{c_P})_{\sigma} \cdot \tilde{b}_{\tau}} \cdot (-1)^{(\widetilde{c_Q})_{\sigma} \cdot \tilde{a}_{\tau}} \cdot(-1)^{(\widetilde{c_R})_{\sigma} \cdot \tilde{d}_{\tau}} \cdot(-1)^{(\widetilde{c_S})_{\sigma} \cdot \tilde{c}_{\tau}} \cdot (-1)^{ \tilde{a}_{\sigma} \cdot \tilde{b}_{\tau}} \cdot (-1)^{\tilde{c}_{\sigma} \cdot \tilde{d}_{\tau} } \cdot I.\\
\end{align*}
On the other hand, $(c_P, c_Q) \otimes (c_R, c_S)$ is isomorphic to $\langle M_P, M_Q, M_R, M_S\rangle = \text{Mat}_4(K)$ which represents the identity element in the Brauer group. Hence, we have $$(c_P a, c_Q b) + (c_R c, c_S d)= (a, b) + (c, d)+ (c_P, b)+(c_Q, a) +(c_R, d)+(c_S, c),$$
which is precisely represented by a cocycle that sends $(\sigma, \tau)$ to \\
$$(-1)^{(\widetilde{c_P})_{\sigma} \cdot \tilde{b}_{\tau}} \cdot (-1)^{(\widetilde{c_Q})_{\sigma} \cdot \tilde{a}_{\tau}} \cdot(-1)^{(\widetilde{c_R})_{\sigma} \cdot \tilde{d}_{\tau}} \cdot(-1)^{(\widetilde{c_S})_{\sigma} \cdot \tilde{c}_{\tau}} \cdot (-1)^{ \tilde{a}_{\sigma} \cdot \tilde{b}_{\tau}} \cdot (-1)^{\tilde{c}_{\sigma} \cdot \tilde{d}_{\tau} },$$
for all $\sigma, \tau \in G_K$ as required.
\begin{comment}
From Proposition \ref{prop: brauer iso to H^2}, we have $\text{Br}(K) \cong H^2(G_K, \bar{K}^*)$. By Remark \ref{rem: quaternion cocycle}, we know the cocycle representation of the class of a quaternion algebra $(\alpha, \beta)$ in $\text{Br}(K)$ is precisely $(\sigma, \tau) \mapsto (-1)^{{\tilde{\alpha}}_{\sigma} \cdot {\tilde{\beta}}_{\tau}}$, treating $\alpha, \beta \in K^*/(K^*)^2$. Therefore, a cocycle representation of $(a, b) + (c, d) + (c_P, b) + (c_Q, a) +(c_R, d) + (c_S, c) \in \text{Br}(K) \cong H^2(G_K, \bar{K}^*)$ sends $(\sigma, \tau)$ to \\
$$(-1)^{(\tilde{c_P})_{\sigma} \cdot \tilde{b}_{\tau}} \cdot (-1)^{(\tilde{c_Q})_{\sigma} \cdot \tilde{a}_{\tau}} \cdot(-1)^{(\tilde{c_R})_{\sigma} \cdot \tilde{d}_{\tau}} \cdot(-1)^{(\tilde{c_S})_{\sigma} \cdot \tilde{c}_{\tau}} \cdot (-1)^{\tilde{b}_{\tau} \cdot \tilde{a}_{\sigma}} \cdot (-1)^{\tilde{d}_{\tau} \cdot \tilde{c}_{\sigma}},$$
for all $\sigma, \tau \in G_K$ as required. \\
\end{comment}
\end{proof}
\section{Bounding the Set of Primes}\label{sec:prime-bound}
In this section, we directly show that the formula for $\langle \epsilon, \operatorname{et}a \rangle_{CT}$ in Theorem \ref{thm: 1} is actually always a finite product, as mentioned in Remark \ref{rem: CTP finite product}. Since for a local field with odd residue characteristic, the Hilbert symbol between $x$ and $y$ is trivial when the valuations of $x$, $y$ are both 0, it suffices to find a finite set $S$ of places of $K$, such that outside $S$ the first arguments of the Hilbert symbols in the formula for $\langle \epsilon, \operatorname{et}a \rangle_{CT}$ have valuation 0 for some choice of the local point $P_v$.\\
Let $\mathcal{O}_K$ be the ring of integers for the number field $K$. By rescaling the variables, we assume the genus two curve is defined by $y^2=f(x)=f_6x^6 + ...+f_0$ where the $f_i$ are in $\mathcal{O}_K$.\\
The first arguments of the Hilbert symbols in the formula for $\langle \epsilon, \operatorname{et}a \rangle_{CT}$ are $f_P(P_v)$, $f_Q(P_v)$, $f_R(P_v)$ or $f_S(P_v),$ where $f_P, f_Q, f_R, f_S$ can be computed as the quotients of two linear forms in $\mathbb{P}^{15} $ with the denominators being the same, as explained in Remark \ref{rem: explicit fp, fq, fr, fs}. Since we know that the Cassels-Tate pairing is independent of the choice of the local points $P_v$ as long as these are chosen to avoid all the zeros and poles, it suffices to make sure that there exists at least one local point $P_v$ on $J_{\epsilon}$ for which the values of the quotients of the linear forms all have valuation 0 for all $v$ outside $S$. The idea is to first reduce the problem to the residue field. \\
By Theorem \ref{theorem: explicit twist of J} and Remark \ref{rem: explicit twist of J formula}, we have an explicit formula for the linear isomorphism
$$J_{\epsilon}\subset \mathbb{P}^{15} \xrightarrow{\phi_{\epsilon}} J \subset \mathbb{P}^{15},$$
which is defined over $K'=K(\sqrt{a}, \sqrt{b}, \sqrt{c}, \sqrt{d})$ where $\epsilon = (a, b, c, d) \in (K^*/(K^*)^2)^4$. Suppose $\phi_{\epsilon}$ is represented by $M_{\epsilon} \in \text{GL}_{16}(K')$. Note we can assume that all entries of $M_{\epsilon}$ are in $\mathcal{O}_{K'}$, the ring of integers of $K'$.\\
\begin{notation}\label{notation: reduction}
Let $K$ be a local field with valuation ring $\mathcal{O}_K$, uniformizer $\pi$ and residue field $k$. Let $X \subset \mathbb{P}^N$ be a variety defined over $K$ and $I(X) \subset K[x_0,...,x_N]$ be the ideal of $X$. Then the reduction of $X$, denoted by $\bar{X}$, is the variety defined by the polynomials $\{\bar{f} : f \in I(X) \cap \mathcal{O}_K[x_0,...,x_N]\}$. Here $\bar{f}$ is the polynomial obtained by reducing all the coefficients of $f$ modulo $\pi$. Note that this definition of the reduction of a variety $X \subset \mathbb{P}^N$ defined over a local field $K$ is equivalent to taking the special fibre of the closure of $X$ in $\mathbb{P}^{N}_S$, where
$S = \operatorname{Spec} \mathcal {O}_K$.\\
\end{notation}
Let $S_0=\{\text{places of bad reduction for } \mathcal{C}\} \cup \{\text{places dividing 2}\} \cup \{\text{infinite places}\}$. Fix a place $v \notin S_0$ and suppose it is above the prime $p$. We now treat $J, J_{\epsilon}$ and $\mathcal{C}$ as varieties defined over the local field $K_v$. Let $\mathcal{O}_v$ denote the valuation ring of $K_v$ and $\mathbb{F}_{q}$ denote its residue field, where $q$ is some power of $p$. It can be shown that $\bar{J}$ is also an abelian variety as the defining equations of $J$ are defined over $\mathcal{O}_v$ and are derived algebraically in terms of the coefficients of the defining equation of the genus two curve $\mathcal{C}$ by Theorem \ref{theorem: 72}. In fact, $\bar{J}$ is the Jacobian variety of $\bar{\mathcal{C}}$, the reduction of $\mathcal{C}$. \\
Now fix a place $v'$ of $K'$ above the place $v$ of $K$. Let $\mathcal{O}_{v'}$ and $\mathbb{F}_{q^r}$ denote the valuation ring and the residue field of $K'_{v'}$. It can be checked that as long as $v'$ does not divide $\det M_{\epsilon} \in \mathcal{O}_{K'}$, the following diagram commutes and $\bar{M_{\epsilon}}$ is a well defined linear isomorphism defined over the residue field $\mathbb{F}_{q^r}$ between two varieties defined over $\mathbb{F}_q$: \\
\[
\begin{tikzcd}
J_{\epsilon} \subset \mathbb{P}^{15}\arrow[r, "M_{\epsilon}"] \arrow[d, "\text{reduction}"]& J \subset \mathbb{P}^{15} \arrow[d, "\text{reduction}"]\\
\bar{J_{\epsilon}} \subset \mathbb{P}^{15} \arrow[r, "\bar{M_{\epsilon}}"]& \bar{J}\subset \mathbb{P}^{15},\\
\end{tikzcd}
\]
where $\bar{M_{\epsilon}}$ denotes the reduction of the matrix $M_{\epsilon}$ over the residue field $\mathbb{F}_{q^r}$.\\
This linear isomorphism $\bar{M_{\epsilon}}$ implies that $\bar{J_{\epsilon}}$ is smooth whenever $\bar{J}$ is. In this case, $\bar{J_{\epsilon}}$ is a twist of $\bar{J}$ and it in fact a 2-covering of $\bar{J}$. Indeed, the surjectivity of the natural map $\text{Gal}(K'_{v'}/K_v) \rightarrow \text{Gal}(\mathbb{F}_{q^r}/\mathbb{F}_q)$ shows that $M_{\epsilon}(M_{\epsilon}^{-1})^{\sigma}=\tau_{P_{\sigma}}$ for all $\sigma \in \text{Gal}(K'_{v'}/K_v)$ implies that $\bar{M_{\epsilon}}(\bar{M_{\epsilon}}^{-1})^{\bar{\sigma}}=\tau_{\bar{P_{\bar{\sigma}}}}$ for all $\bar{\sigma} \in \text{Gal}(\mathbb{F}_{q^r}/\mathbb{F}_q)$.
We know any principal homogeneous space of $\bar{J}$ over a finite field has a point by \cite[Theorem 2]{lang} and so is trivial by Proposition \ref{prop:2 covering has a point}. Therefore, there exists an isomorphism $\bar{J_{\epsilon}} \xrightarrow{\psi} \bar{J}$ defined over $\mathbb{F}_q$. Hence, as long as $v \notin S_0$ and $v$ does not divide $N_{K'/K} (\det M_{\epsilon})$, $\bar{J_{\epsilon}}$ has the same number of $\mathbb{F}_q$-points as $\bar{J}$. By the Hasse-Weil bound, we know the number of $\mathbb{F}_q$-points on $\mathcal{C}$ is bounded below by $q-1-4\sqrt{q}$. Since we can represent points on $\bar{J}$ by pairs of points on $\bar{\mathcal{C}}$ and this representation is unique other than the identity point on $\bar{J}$. The number of $\mathbb{F}_q$-points on $\bar{J}$ is bounded below by $(q-1-4\sqrt{q})(q-3-4\sqrt{q})/2$.\\
On the other hand, let $l_1, ..., l_5$ be the 5 linear forms that appear as numerator or denominator of $f_P, f_Q, f_R, f_S$. We can assume that the coefficients of $l_i$ are in $\mathcal{O}_K$ by scaling, for all $i = 1, ..., 5$. Fix a place $v$ of $K$ that does not divide all the coefficients of $l_i$, for any $i=1, ..., 5$. Let $H_i$ be the hyperplane defined by the linear form $l_i$ and $\bar{H_i}$ be its reduction, which is a hyperplane defined over the residue field $\mathbb{F}_q$, We need to bound the number of $\mathbb{F}_q$-points of $\bar{J_{\epsilon}}$ that lie on one of the hyperplanes $\bar{H_i}$. Let $r_i$ be the number of irreducible components of $\bar{J_{\epsilon}} \cap \bar{H_i}$. By \cite[Chapter~1, Theorem 7.2 (Projective Dimension Theorem) and Theorem 7.7]{hartshorne}, we know that each irreducible component $C^i_j$ of $\bar{J_{\epsilon}} \cap \bar{H_i}$, where $j =1, ..., r_i$, is a curve and the sum of degrees of all the irreducible components counting intersection multiplicity is $\deg \bar{J_{\epsilon}}=32.$ Leting $d^i_j =\deg C^i_j$, we have $\sum_{j=1}^{r_i}d^i_j \le 32$ for all $i$.
\begin{lemma}
Let $C \subset \mathbb{P}^N$ be a curve of degree $d$. Then
$\# C(\mathbb{F}_q)\le d(q+1)$.
\end{lemma}
\begin{proof}
We may assume that $C$ is contained in no hyperplane. Then projection to the first two coordinates gives a nonconstant morphism $C \rightarrow \mathbb{P}^1$ of degree $\le d$. Since $\#\mathbb{P}^1(\mathbb{F_q})=q+1$, this gives the required bound.\\
\end{proof}
By applying the above lemma to each $C_j^i$, we get the number of $\mathbb{F}_q$-points of $\bar{J_{\epsilon}}$ that lie on one of the hyperplanes $\bar{H_i}, i = 1, ..., 5,$ is no more than $$\sum_{i=1}^{5} \sum_{j=1}^{r_i} d^i_j \cdot (q+1) \le 160(q+1).$$
We compute that for any $x > 500$, we have $(x-1-4\sqrt{x})(x-3-4\sqrt{x})/2>160(x+1)$. Recall $q$ is a power of $p$. Hence, if $v$ is a place of $K$ above the prime $p>500$ such that $v \notin S_0$ and $v$ does not divide $N_{K'/K}(\det M_{\epsilon})$ or all the coefficients of $l_i$ for some $i$, we have a smooth $\mathbb{F}_q$-point on $\bar{J_{\epsilon}}$ which by Hensel's Lemma \cite[Exercise C.9(c)]{hensel} lifts to the point $P_v$ as required. This implies that the first arguments of the Hilbert symbols in the formula for the local Cassels-Tate pairing of $\langle \epsilon, \operatorname{et}a \rangle_{CT}$ have valuation 0. It can be checked that since $v \notin S_0$, the second arguments of these Hilbert symbols also have valuation 0. Hence, the formula for the Cassels-Tate is indeed always a finite product.\\
Note that in the case where $K=\mathbb{Q}$ or more generally if $K$ has class number 1, we can always make the linear forms primitive by scaling. Therefore, in this case, the subset $\{$places dividing all the coefficients of the denominator or the numerator of $f_P, f_Q, f_R \text{ or } f_S\}$ is empty.\\
\section{Worked Example}
Now we demonstrate the algorithm with a worked example computed using MAGMA \cite{magma}. In particular, we will see with this example, that computing the Cassels-Tate pairing on $\text{Sel}^2(J)$ does improve the rank bound obtained via a 2-descent. This genus two curve was kindly provided by my PhD supervisor, Tom Fisher, along with a list of other genus two curves for me to test the algorithm.\\
Consider the following genus two curve
$$\mathcal{C}: y^2=-10x(x+10)(x+5)(x-10)(x-5)(x-1).$$
Its Jacobian variety $J$ has all its two torsion points defined over $\mathbb{Q}$. A set of generators of $J[2]$ compatible with the Weil pairing matrix \eqref{equation: WP matrix} are $P= \{(0, 0), (-10, 0)\}, Q= \{(0, 0), (-5, 0)\},
R= \{(10, 0), (5, 0)\}, S= \{(10, 0), (1, 0)\}.$ We identify $H^1(G_K, J[2])=(\mathbb{Q}^*/(\mathbb{Q}^*)^2)^4$ as in Section \ref{sec:formula-for-the-cassels-tate-pairing}. Consider $\epsilon, \operatorname{et}a \in \text{Sel}^2(J)$ represented by $(-33, 1, -1, -11)$ and $( 11, 1, -1, -11)$ respectively. The images of $[P], [Q], [R], [S]$ via $\delta: J(\mathbb{Q})/2J(\mathbb{Q}) \rightarrow H^1(G_{\mathbb{Q}}, J[2])$, computed via the explicit formula as in \cite[Chapter 6, Section 1]{the book}, are $\delta([P])=(-66, 1, 6, 22), \delta([Q])=(-1, 1, 3, 1),
\delta([R])=( 6, 3, 1, 3 ), \delta([S])=(22, 1, -3, -11 ).$ Now following the discussions in Sections \ref{sec:explicit-computation-of-d} and \ref{sec:explicit-computation-of-dp-dq-dr-ds}, we can compute, using the coordinates $c_0, ..., c_9, d_1, ..., d_6$ for $J_{\epsilon} \in \mathbb{P}^{15}$ as described in Remark \ref{rem: explicit twist of J formula}. we have
\begin{align*}
k_{11}'&= 618874080c_0 - 496218440c_1 - 390547052c_3
+ 205551080c_4\\
&+ 384569291c_6 + 52868640c_8;\\
\\
k_{11, P}'&= -36051078800000c_2 + 8111492730000c_3 + 265237150000c_7\\
& - 196928587500c_8 - 6786529337500c_9 + 22531924250d_2 \\
&- 126449158891d_4 - 117221870375d_5 + 937774963000d_6; \\
\\
k_{11, Q}'&= 134800c_1 + 235600c_3 + 62000c_4 + 52235c_6 + 60016d_1 - 5456d_5;\\
\\
k_{11, R}'&= -30223125c_6 + 4050000c_8 - 49750d_3 + 709236d_4 \\
\\
k_{11, S}' &= 4724524800c_1 + 8557722360c_3 + 13102732800c_4 + 1258642935c_6 \\
&+ 7291944000c_9 -
2709362304d_1 + 97246845d_2 + 8475710d_3\\
& + 30788208d_5.
\end{align*}
Hence, we have explicit formulae for
$$f_P = \frac{k_{11, P}'}{k_{11}'}, f_Q = \frac{k_{11, Q}'}{k_{11}'}, f_R = \frac{k_{11, R}'}{k_{11}'}, f_S = \frac{k_{11, S}'}{k_{11}'}.$$
In particular, they are defined over $\mathbb{Q}$ as claimed. From Section \ref{sec:prime-bound}, we compute that only primes below 500 can potentially contribute to $\langle \epsilon, \operatorname{et}a \rangle_{CT}$. Then, it turns out that the only nontrivial local Cassels-Tate pairings between $\epsilon$ and $\operatorname{et}a$ are at places $11, 19, \infty$ and $\langle \epsilon, \operatorname{et}a \rangle_{CT}=-1$.\\
Under the isomorphism $H^1(G_{\mathbb{Q}}, J[2]) \rightarrow (\mathbb{Q}^*/(\mathbb{Q}^*)^2)^4$, $\text{Sel}^2(J)$ has size $2^6$ and is generated by $(-33, 1, -1, -11), (11, 1, -1, -11 ), (66, 1, 2, 22),(11, 1, 2, 22), (3, 3, 3, 3), (3, 1, 3, 1).$ Since $\mathcal{C}$ has rational points, the Cassels-Tate pairing can be shown to be alternating using \cite[Corollary 7]{poonen stoll}. Since all the two torsion points on $J$ are rational and $\langle \epsilon, \operatorname{et}a \rangle_{CT}=-1$, we get $|\ker \langle \;, \; \rangle_{CT}|=2^4$.\\
Indeed, we verified that the Cassels-Tate pairing matrix, with the generators of $\text{Sel}^2(J)$ listed above, is
$$\begin{bmatrix}
1 &-1& 1& 1& -1& -1\\
-1& 1& 1& -1& 1& -1\\
1 &1 &1 & 1 & 1 & 1\\
1 &-1 &1& 1& -1& -1\\
-1 & 1 & 1 & -1 & 1 &-1\\
-1 &-1 & 1 &-1 &-1 & 1\\
\end{bmatrix},$$
which is a rank 2 matrix.\\
As shown in \cite[Remark 1.9.4(ii)]{thesis}, in the case where all points in $J[2]$ are defined over the base field, computing the Cassels-Tate pairing on $\text{Sel}^2(J)$ gives the same rank bound as obtained from carrying out a $4$-descent, i.e. computing $\text{Sel}^4(J)$, which can potentially give a better rank bound than the one given by a 2-descent. In this example, the rank bound coming from 2-decent was $\operatorname{rank}(J(\mathbb{Q}))\le 2$. Our calculations of the Cassels-Tate pairing on $\text{Sel}^2(J)$ improves this bound and in fact shows that $\operatorname{rank}(J(\mathbb{Q}))=0$.\\
\begin{bibdiv}
\begin{biblist}
\bib{monique}{article}{
author ={van Beek, M.},
title={Computing the Cassels-Tate Pairing},
note={Doctoral Dissertation. University of Cambridge, 2015.}
}
\bib{3 isogeny}{article}{
author ={van Beek, M.},
author ={Fisher, T. A.},
title={Computing the Cassels-Tate pairing on 3-isogeny Selmer groups via cubic norm equations},
journal={Acta Arithmetica},
volume={185},
number={4},
date = {2018},
pages={367-396},
doi={10.4064/AA171108-11-4},
}
\bib{magma}{article}{
author ={Bosma, W.},
author={Cannon, J. },
author={Playoust, C. }
title={The Magma algebra system. I. The user language},
journal={J. Symbolic Comput.}
volume={24}
number={3-4}
date={1997}
pages={235–265}
doi={10.1006/jsco.1996.0125}
}
\bib{cassels1}{article}{
author ={Cassels, J. W. S.},
title={Arithmetic on Curves of Genus 1. I. On a conjecture of Selmer.},
journal={Journal für die reine und angewandte Mathematik},
volume={202},
date = {1959},
pages={52-99},
}
\bib{cassels2}{article}{
author ={Cassels, J. W. S.},
title={J. W. S. Cassels, Arithmetic on curves of genus 1, IV. Proof of the Hauptvermutung.},
journal={Journal für die reine und angewandte Mathematik},
volume={211},
date = {1962},
pages={95-112},
doi={10.1515/crll.1962.211.95 },
}
\bib{cassels98}{article}{
author ={Cassels, J. W. S.},
title={Second Descents for Elliptic Curves},
journal={Journal für die reine und angewandte Mathematik},
volume={494}
date = {1998},
pages={101-127},
doi={10.1515/crll.1998.001 },
}
\bib{the book}{book}{
author ={Cassels, J. W. S. },
author ={Flynn, E. V.},
title={Prolegomena to a MiddleBrow Arithmetic of Curves of Genus 2},
series={London Mathematical Society Lecture Note Series},
volume={230},
publisher={Cambridge University Press},
date= {1996},
isbn={9780511526084},
doi={10.1017/CBO9780511526084},
}
\bib{clark}{article}{
author ={Clark, P. L.},
title={The Period–Index Problem in WC-Groups I: Elliptic Curves},
journal={Journal of Number Theory},
volume={114},
date = {2005},
number={1},
pages={193-208},
doi={10.1016/j.jnt.2004.10.001},
}
\bib{bs hasse}{article}{
author ={Coray, D.},
author ={Manoil, C.},
title={On large Picard groups and the Hasse Principle for curves and K3 surfaces},
journal={Acta Arithmetica},
volume={76},
date = {1996},
number={2},
pages={165-189},
doi={10.4064/aa-76-2-165-189},
}
\bib{steve}{article}{
author ={Donnelly, S.},
title={Algorithms for the Cassels-Tate pairing},
date={2015},
note={preprint}
}
\bib{platonic}{article}{
author ={Fisher, T. A.},
title={The Cassels–Tate pairing and the Platonic solids},
journal={Journal of Number Theory},
volume={98},
number={1}
date = {2003},
pages={105-155},
doi={10.1016/S0022-314X(02)00038-0},
}
\bib{binary quartic}{article}{
author ={Fisher, T. A.},
title={On binary quartics and the Cassels-Tate pairing },
date={2016},
note={preprint},
}
\bib{3 selmer}{article}{
author ={Fisher, T. A.},
author ={Newton, R.},
title={Computing the Cassels-Tate pairing on the 3-Selmer group of an elliptic curve},
journal={Journal of Number Theory},
volume={10},
number={7},
date = {2014},
pages={18811907},
doi={10.1142/S1793042114500602},
notes ={Available at http://centaur.reading.ac.uk/58175/}
}
\bib{72 theorem}{article}{
author ={Flynn, E. V.},
title={The Jacobian and Formal Group of a Curve of Genus 2 over an Arbitrary Ground Field},
journal={Mathematical Proceedings of the Cambridge Philosophical Society},
volume={107},
date = {1990},
number={3},
pages={425-441},
doi={10.1017/S0305004100068729},
}
\bib{the gp law paper}{article}{
author ={Flynn, E. V.},
title={The Group Law on the Jacobian of a Curve of Genus 2},
journal={Journal für die reine und angewandte Mathematik},
volume={439},
date = {1993},
pages={45-70},
isbn={0075-4102},
}
\bib{explicit twist}{article}{
author ={Flynn, E. V.},
author = {Testa, D.},
author = {van Luijk, R.},
title={Two-Coverings of Jacobians of Curves of Genus 2},
journal={Proceedings of the London Mathematical Society},
volume={104},
date = {2012},
number={2},
pages={ 387–429},
doi={10.1112/plms/pdr012},
}
\bib{QA}{book}{
author ={Gille, P.},
author ={Szamuely, T.},
title={Central Simple Algebras and Galois Cohomology},
series={Cambridge Studies in Advanced Mathematics},
volume={101},
publisher={Cambridge University Press},
date= {2006},
isbn={9780521861038},
}
\bib{hartshorne}{book}{
author ={Hartshorne, R.},
title={Algebraic Geometry},
series={Graduate Texts in Mathematics}
volume={52},
publisher={Springer, New York, NY},
date= {1977},
isbn={978-1-4419-2807-8},
doi={10.1007/978-1-4757-3849-0},
}
\bib{hensel}{book}{
author ={Hindry, M.},
author={Silverman, J. H.}
title={Diophantine Geometry: An Introduction},
series={Graduate Texts in Mathematics 201}
volume={52},
publisher={Springer},
date= {2000}
}
\bib{lang}{article}{
author ={Lang, S.},
title={Algebraic Groups Over Finite Fields},
volume={78},
number={3},
year = {1956},
pages = {555-563},
journal = {American Journal of Mathematics},
doi={10.2307/2372673},
}
\bib{abelian varieties}{book}{
author ={Milne, J.S.},
title={Abelian Varieties, Second Edition},
date = {2008},
pages ={166+vi},
note={Available at www.jmilne.org/math/},
}
\bib{theta}{book}{
author ={Mumford,D.},
title={Abelian Varieties},
publisher={Oxford University Press},
series={Tata Institute of Fundamental Research Studies in Mathematics},
date = {1970},
number ={5}
note={Published for the Tata Institute of Fundamental Research, Bombay}
}
\bib{oneil}{article}{
author ={O'Neil, C.},
title={The Period-Index Obstruction for Elliptic Curves},
journal={Journal of Number Theory},
volume={95},
date = {2002},
number={2},
pages={329-339},
doi={10.1006/jnth.2001.2770},
}
\bib{poonen stoll}{article}{
author ={Poonen, B.},
author ={Stoll, M.},
title={The Cassels-Tate pairing on polarized abelian varieties},
journal={Annals of Mathematics},
volume={150},
date = {1999},
number={3},
pages={1109-1149},
review={\MR{1740984}},
doi={10.2307/121064<},
}
\bib{local tate duality}{article}{
author ={Skorobogatov, A.},
title={Abelian varieties over local and global fields},
note={TCC course, Spring 2016}
}
\bib{tate}{article}{
author ={Tate, J.},
title={Duality theorems in Galois cohomology over number fields},
journal={Proc. In- ternat. Congr. Mathematicians (Stockholm)},
date = {1962},
pages={288–295},
note ={Inst. Mittag-Leffler, Djursholm (1963)}
}
\bib{thesis}{article}{
author ={Yan, J.},
title={Computing the Cassels-Tate Pairing for Jacobian Varieties of Genus Two Curves},
note={Doctoral Dissertation. University of Cambridge, 2021.}
}
\begin{comment}
\bib{Amb99}{misc}{
author ={Ambro, Florin},
title={The Adjunction Conjecture and its applications},
year = {1999},
note = {https://arxiv.org/abs/math/9903060v3},
}
\bib{Bir16a}{article}{
author = {Birkar, Caucher},
title = {Anti-pluricanonical systems on Fano varieties},
journal={Ann. Math.},
volume={19},
date={2019},
number={2},
pages={345--463},
}
\end{comment}
\end{biblist}
\end{bibdiv}
\end{document} | math | 116,419 |
\begin{document}
\title{Optimal Partition of a Tree with Social Distance\thanks{This work is supported by JSPS KAKENHI Grant Numbers 17K19960, 17H01698, 18H06469.}}
\author{Masahiro Okubo\inst{1}
\and
Tesshu Hanaka\inst{2}
\and
Hirotaka Ono\inst{3}
}
\authorrunning{M. Okubo et al.}
\institute{Graduate School of Informatics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan \email{[email protected]} \and
Department of Information and System Engineering, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo, Japan \email{[email protected]} \and Graduate School of Informatics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan \email{[email protected]}}
\maketitle
\begin{abstract}
We study the problem to find a partition of \textcolor{black}{a} graph $G$ with maximum social welfare based on social distance between vertices in $G$, called {\sf MaxSWP}.
This problem is known to be NP-hard in general. In this paper, we first give a complete characterization of optimal partitions of trees with small diameters. Then, by utilizing these results, we show that {\sf MaxSWP} can be solved in linear time for trees.
Moreover, we show that {\sf MaxSWP} is NP-hard even for 4-regular graphs.
\keywords{graph algorithm \and tree \and graph partition \and social distance}
\end{abstract}
\section{Introduction}
With the development of \textcolor{black}{Social Networking Services (SNS)} such as Twitter,
Facebook, Instagram and so on, it has become much easier than before to obtain
graphs that represent human relationship, and there are many attempts
to utilize such graphs for extracting useful information.
Among them, grouping people according to the graph structures is
focused and investigated from many
standpoints. For example, if a community consisting of members with a
common interest is found, advertising or promoting some products might be
very effective for members of the community due to the strong
interest.
Here, there are roughly two standpoints how we group communities.
One is \textcolor{black}{context based} grouping, and the other is
based on link structures.
Previous work on community detection and grouping based on graph structure is summarized in \cite{Fortunato2010,Newman2010a,SCHAEFFER2007}, for example.
Basically, these studies formulate network structure identification (community detection, grouping, and partition) as an optimization problem (sometimes it is not explicitly conscious), and design a fast algorithm to (approximately) solve the optimization problem.
\textcolor{black}{Then, network structures to identify are obtained as outputs of the proposed algorithm. Here, network structures to identify are already abstract, e.g., dense subgraphs; the proposed algorithm can be used not only for the original purpose but also other purposes.}
In fact, \cite{Shi2000} is originally about boundary line detection in image data, but the proposed techniques are used for community detentions (e.g., \cite{Newman2010b}), and it is further used
for the detection of industrial clusters in economic networks~\cite{Kagawa2013}.
As above, the versatility of ``optimization problems'' is very useful.
However, we may think that they do not best utilize the features or characteristics
of the target network.
For example, the criteria for group partition, image processing, and detecting industrial clusters in economic networks could be different. In other words, we might expect a better performance by considering an optimization problem specialized for community detection.
From these, we consider the problem for group partition (or simply say partition) in networks (graphs), taking into account the characteristics of SNS. In SNS, people communicate and exchange information with also a person who is not directly acquainted, i.e., followers. That is, in SNS, not only members with direct connections but also members without direct connections are loosely connected, which enables us to share information widely. Here, ``looseness'' is related to the degree of sharing information, and it is natural to define it as the distance (i.e., the length of a shortest path) between the persons on the network.
Based on such observation, Branzei et al. introduced a new grouping scale for human relations networks~\cite{Branzei2011}. The definition of the utility in \cite{Branzei2011} is as follows: given a partition, the utility of an individual is defined as the sum of reciprocal distances to other people in the same coalition divided by the size of the coalition. Based on this, the social welfare of a partition is also defined as the sum of the utilities of all the members.
Unfortunately, finding a partition with maximum social welfare ({\sf MaxSWP}) is known to be NP-hard {even on graphs with maximum degree $6$~\cite{Balliu2017b}.
Also, the characterizations of optimal partitions are known only for trivial cases such as complete graphs and complete bipartite graphs~\cite{Branzei2011}.
Even for trees, it is not known whether {\sf MaxSWP} can be solved in polynomial time.
One of the reasons seems to be the objective function of {\sf MaxSWP}.
In a typical graph optimization problem, the objective function often forms a linear sum of weights, whereas
the one of {\sf MaxSWP} takes the form of a nonlinear function,
which is the sum of the reciprocal distances.
\subsection{Our contribution}
\color{black}
In this paper, we mainly study finding an optimal partition with social distance of a tree, which is
one of the most basic and important structures in graph algorithm design.
In the process of research, we first give a complete characterization of optimal partitions of paths.
Although the argument is simple, it gives an insight about the hardness related to the nonlinearity of the utility and the social welfare.
Next, we give a similar characterization of optimal partitions of trees.
In the characterization, we find out sub-trees with small diameters appeared in optimal partitions of trees.
By using the characterization, we design a linear-time algorithm for computing an optimal partition of a tree.
Finally, we show that {\sf MaxSWP} is NP-hard even for 4-regular graphs. This result strengthens the previous work for graphs with maximum degree $6$~\cite{Balliu2017b}.
\color{black}
\subsection{Related work}\label{sec:relate}
\color{black}
Graph partition is \textcolor{black}{one of} the most basic and important problem in computer science and there are many \textcolor{black}{studies} about graph partition in various contexts, such as image processing and
cluster analysis~\cite{Shi2000,Fortunato2010,Newman2010a,SCHAEFFER2007,Kagawa2013}.
Graph partition with social distance has been studied in the context of
coalition formation games~\cite{Branzei2011}.
In coalition formation games, each player has the utility based on the preference for other players in the same coalition.
Intuitively, a player is happy if the utility is high, that is, there are many players he/she prefer in the same coalition.
In the field of coalition formation games, many researchers study about desirable coalition formations, namely, partitions, in terms of maximum social welfare, stability, and core~\cite{AZIZ2013316,RAHWAN2015139}.
Furthermore, the price of anarchy (PoA) and the price of stability (PoS) are also well-studied for evaluating agents systems~\cite{Balliu2017a}.
The PoA or PoS are more related to this paper because they are defined as the maximum and minimum ratio between a Nash stable solution and the best solution, respectively.
In coalition formation games on graphs, there are many utility functions for agents.
For example, in \cite{AZIZ2013316,SLESS2018217}, the utility of an agent is defined as the sum of edge-weights between him/her and other agents in the same coalition.
The weight of an edge represents the strength of the relationship between agents.
In social distance games, the utility is defined as the harmonic function of the distance between agents.
This is based on the concept of the closeness centrality, which is one of classical measures for network analysis~\cite{Branzei2011,Balliu2017b}.
\textcolor{black}{As mentioned above}, finding the best partition, that is, a partition with maximum social welfare is NP-hard even on graphs with maximum degree $6$~\cite{Balliu2017b}.
On the other hand, there is a 2-approximaiton algorithm for finding such a partition~\cite{Branzei2011}.
\color{black}
The organization of this paper is as follows.
In Section \ref{sec:pre}, we give basic terminologies, notation, and \textcolor{black}{definitions}.
In Section \ref{sec:path}, we give a complete characterization of optimal partitions of paths.
In Section \ref{sec:tree},
we propose a linear-time algorithm for {\sf MaxSWP} on trees.
Finally, we show that {\sf MaxSWP} is NP-hard even on $4$-regular graphs in Section \ref{sec:NPh}.
\color{black}
Due to the space limitation, we out the proofs of propositions lemmas and theorems marked with (*).
The detailed profs can be found in Appendix.
\color{black}
\section{Related work}\label{sec:relate}
Partition with social distance has been resarched as social distances games (SDGs) in coalition forming games \cite{Branzei2011}.
Coalition forming game is one of the coalition structure forming problem which is natural abstraction of one of the most important challenges in multi-agent systems. If you know the coalition structure forming problem more, please read this survey \cite{RAHWAN2015139}.
On the other hand, there are researches where they divide with quality function to partition.
In recent years, researches on modularity proposed by Newman et al. \cite{PhysRevE.69.026113} are particularly popular.
Variability approximation algorithms have been proposed since the NP-hard was shown in 2008 \cite{Brandes:980374}.
Please see this paper for details \cite{Fortunato2010}.
The former performs division under the action of the agent actively.
The latter does not impose such a condition, it divides under the evaluation basis.
Our research follows the former condition as mentioned earlier.
In coalition forming games, there are many researches about partition based on various utilities has been studied.
For example, In \cite{SLESS2018217}, each agent's utility is her sum value of the members of coalition, where value is given integer.
Moreover, There are researches with utility that is her average value of the members of coalition \cite{AzizBBHOP17}.
They are subclass of hedonic games (HGs) which have been considered with various stability concepts \cite{Dreze1980,BOGOMOLNAIA2002201,Elkind:2009}.
Similarly, in SDGs, the various fundamental notions of stability such as core stability \cite{Branzei2011}, Nash stability \cite{Balliu2017a} and Pareto stability \cite{Balliu2017b} have been already investigated.
For any stable concepts, it is important that the ratio between stable solution and maximum solution (ie., Price of Anarchy \cite{Koutsoupias1999}, Price of Pareto Optimality \cite{ElkindFF16}).
Unfortunately, about maximum solution, little is known except that it is NP-hard in the graph of maximum degree $6$ \cite{Balliu2017b} and that the approximate solution is $1/2$ or more \cite{Branzei2011}.
In spite of that approximate solution is found by using a tree, we do not know whether there is a polynomial time algorithm for finding the maximum solution in a tree.
\section{Preliminaries}\label{sec:pre}
\subsection{Terminologies}
\color{black}
We use standard terminologies on graph theory.
Let $G=(V(G), E(G))$ be a simple, connected, and undirected graph.
For simplicity, we may denote $V(G)$ and $E(G)$ by $V$ and $E$, respectively.
We also denote the number of vertices and edges by $n$ and $m$, respectively.
A path from $u$ to $v$ of minimum length is called a {\em shortest path}, and the length is denoted by $dist_G(u,v)$.
In a graph $G$, if there is no path from the vertex $u$ to the vertex $v$, we define $dist_G (u,v) =\infty$.
\color{black}
Let $G[C]$ be the subgraph induced by vertex set $C\subseteq{V}$. We sometimes denote $dist_{G[C]}(u,v)$ of $u,v \in{C}$ in $G[C]$ by $dist_{C}(u,v)$ for simplicity.
For graph $G$, we denote the {\em diameter} of $G$ by $diam(G)=\max_{u,v\in{V}, u\neq{v}}{dist_G (u, v)}$.
For a vertex $v$, we denote the set of neighbors of $v$ by $N(v)= \{u \mid (u,v)\in{E} \} \subseteq{V}$.
We also define the degree of $v$ as $d(v)=|{N(v)}|$.
For $G=(V,E)$, we denote the maximum degree of $G$ by $\Delta(G)=\max_{v\in V}d(v)$.
For simplicity, we sometimes denote it by $\Delta$.
For a positive integer $n$, we define $[n]=\{1, \ldots, n\}$.
A graph $G=(V,E)$ is called a {\em path graph} denoted by $P_n$ if $E=\{(v_i, v_{i +1})\mid{1} \le i<n\}$. We also sometimes simply call it a path.
Moreover, if $E=\{(v_1, v_{i})\mid{2} \le i\le n \}$, $G$ is said to be a {\em star} and denoted by $K_{1,n-1}$.
A graph $G$ is a {\em tree} if $G$ is connected and it has no cycle. We denote an $n$-vertex tree by $T_{n}$.
Moreover, we denote a tree with the diameter $d$ by $T^d_n$.
\color{black}
\subsection{Coalition and Utility}
\color{black}
The definitions here are based on~\cite{Branzei2011}.
\color{black}
Given a graph $G=(V,E)$ and $C\subseteq{V}$, we define the utility $U(v, C)$ of a vertex $v\in{C}$ as follows:
$$U(v,C)=\frac{1}{|{C}|}\sum_{u\in{C\backslash\{v\}}}\frac{1}{dist_{G[C]}(v, u)}.$$
By the definition, it satisfies that $0\leq U(v,C)\leq1$.
In a graph $G=(V,E)$, a {\em partition} of $G$ is defined as the family of sets of vertices ${\mathcal C}=\{C_1,\ldots, C_k\}$, where $C_1 \cup \cdots \cup C_k =V$ and $C_i \cap{C_j}=\emptyset$ for $i\neq{j}\in[k]$.
Moreover, $C\in{\mathcal C}$ is called a {\em coalition} of partition ${\mathcal C}$.
In particular, if ${\mathcal C} = \{V\}$, ${\mathcal C}$ is called the {\em grand} of $G$ and $V$ is called the {\em grand coalition}.
If $\{v\} \in {\mathcal C}$ for a vertex $v\in V$, $v$ is said to be an {\em isolated vertex} of partition ${\mathcal C}$.
We define the utility of an isolated vertex as $U(v,\{v\})=0$.
Next, we define the {\em social welfare} of a partition ${\mathcal C}$ in graph $G$ as follows.
We define the {\em social welfare} $\varphi(G,{\mathcal C})$ of partition ${\mathcal C}$ in $G=(V,E)$ as follows:
$$\varphi(G,{\mathcal C})=\sum_{C\in {\mathcal C}} \sum_{v\in {C}}U(v, C).$$
If $\mathcal{C}$ is the grand of $G$, that is, $\mathcal{C}=\{V\}$, we simply denote $\varphi(G,\{V\})$ by $\varphi(G)$.
\textcolor{black}{We can observe that $\varphi(G,\mathcal{C})$ is bounded by $n-1$}.
Moreover, we define the {\em average social welfare} $\tilde{\varphi}(G, \mathcal{C})$ for partition $\mathcal{C}$ in $G$.
The {\em average social welfare} of partition $\mathcal{C}$ in $G=(V, E)$ is defined as follows:
$$\tilde{\varphi}(G,{\mathcal C})=\frac{\varphi(G,{\mathcal C})}{|{V}|}.$$
If $\mathcal{C}$ is the grand of $G$, that is, $\mathcal{C}=\{V\}$, we simply denote $\tilde{\varphi}(G,\{V\})$ by $\tilde{\varphi}(G)$.
Finally, we define a partition ${\mathcal C}^*$ with {\em maximum} social welfare in graph $G$.
A partition $\mathcal{C^*}$ is {\em maximum} if it satisfies that $\varphi(G,{\mathcal C}^*) \ge \varphi(G,\mathcal{C})$ for any partition $\mathcal{C}$ in $G$.
\color{black}
We call the problem of finding a partition with maximum social welfare {\sf MaxSWP}.
We also call an optimal solution of {\sf MaxSWP} an {\em optimal partition}.
In previous work, it is shown that {\sf MaxSWP} is NP-hard even for graphs with maximum degree $6$~\cite{Balliu2017b}.
\color{black}
On the other hand, it is known that the grand is the only optimal partition of {\sf MaxSWP} on complete graphs and complete bipartite graphs~\cite{Branzei2011}.
\begin{proposition}[\cite{Branzei2011}]\label{opt:perfect}\rm
On complete graphs and complete bipartite graphs, the grand is the only optimal partition of {\sf MaxSWP}.
\end{proposition}
\color{black}
Branzei et al. showed that there exists a partition where the utility of each vertex $v$ attains at least $1/2$ and a polynomial-time algorithm that finds such a partition for any graph~\cite{Branzei2011}.
\begin{proposition}[\cite{Branzei2011}]\label{dia2}
\textcolor{black}{There is a polynomial-time algorithm that finds a partition such that each agent utility is at least $1/2$ for any graph}.
\end{proposition}
From Proposition~\ref{dia2}, it can be easily seen that there exists a partition $\mathcal{C}$ that satisfies $\varphi(G,{\mathcal C}) \ge{n/2}$. Thus, the social welfare of an optimal partition for any graph is also at least $n/2$.
\color{black}
\begin{corollary}\label{opt:partition}
Any optimal partition $\mathcal{C}^*$ of graph $G$ satisfies $\varphi(G,\mathcal{C}^*)\ge{n/2}$.
\end{corollary}
\color{black}
Since for any $G$ and $\mathcal{C}$, $\varphi(G,\mathcal{C})$ is bounded by $n-1$, the algorithm proposed by Branzei et al.~\cite{Branzei2011} is a $2$-approximation algorithm.
In the end of this section, we give another property of an optimal partition of {\sf MaxSWP}.
\begin{proposition}[{\bf *}]\label{connect:coalition}
For each coalition $C\in{\mathcal C}^*$ of optimal partition ${\mathcal C}^*$, $G[C]$ is connected.
\end{proposition}
\section{Optimal partition of a path}\label{sec:path}
In this section, we characterize the optimal partition of a path $P_n$.
In a path, the subgraph induced by a coalition is also a path by \textcolor{black}{Proposition~\ref{connect:coalition}}.
By using this property and examining the average social welfare of $P_n$, we can identify the graph structures of coalitions in the optimal partitions of $P_n$.
In the following, we first examine average social welfare of $P_n$.
Then we give the optimal partition of $P_n$.
Let $h(k)=\sum_{i = 1}^{k}1/i$ be the harmonic function for some positive integer $k$. The social welfare and the average social welfare of $P_n$ can be denoted by $\varphi(P_n)=(2\sum_{k=1}^{n-1}h(k))/n$ and $\tilde{\varphi}(P_n)=(2\sum_{k=1}^{n-1}h(k))/n^2$, respectively.
Then, we obtain the following lemmas.
\begin{lemma}[{\bf *}]\label{lem:pathaverage}
It holds that $\tilde{\varphi}(P_2)<\tilde{\varphi}(P_3)$, and $\tilde{\varphi}(P_n)>\tilde{\varphi}(P_{n+1})$ for $n\ge3$.
\end{lemma}
\color{black}
\begin{lemma}[{\bf *}]\label{path:isolated}
For an optimal partition $\mathcal{C}^*$ of a path $P_n$ and a coalition $C\in\mathcal{C}^*$, $G[C]$ is either $P_2$, $P_3$ or $P_4$.
\end{lemma}
\color{black}
Finally, we give the optimal partition of $P_n$.
\begin{theorem}[{\bf *}]\label{th:pathopt}
The optimal partition of path $P_n$ is
\color{black}
\begin{enumerate}
\item $\mathcal{C}^*=\bigl\{\{v_{3i-2},v_{3i-1},v_{3i}\}\mid{1}\le{i}\le{n/3}\bigr\}$ if $n\equiv0\pmod{3}$,
\item $\mathcal{C}^*=\bigl\{\{v_1,v_2,v_3,v_4\},\{v_{3i+2},v_{3i+3},v_{3i+4}\}\mid{1}\le{i}\le{(n-4)/3}\bigr\}$ if $n\equiv1\pmod{3}$, and
\item $\mathcal{C}^*=\bigl\{\{v_1,v_2\},\{v_{3i},v_{3i+1},v_{3i+2}\}\mid{1}\le{i}\le{(n-2)/3}\bigr\}$ or $\bigl\{\{v_1,v_2,v_3,v_4\}, \break \{v_5,v_6,v_7,v_8\},\{v_{3i+6},v_{3i+7},v_{3i+8}\}\mid1\le{i}\le{(n-8)/3}\bigr\}$ if $n\equiv2\pmod{3}$.
\end{enumerate}
\color{black}
\end{theorem}
\section{Optimal partition of a tree}\label{sec:tree}
In Section \ref{sec:path}, we identified the optimal partition of {\sf MaxSWP} on a path.
In this section, we consider {\sf MaxSWP} on trees.
Since a tree is more general and complicated than a path and the optimal structure of {\sf MaxSWP} is quite different from typical graph optimization problems, {\sf MaxSWP} is non-trivial even on trees.
To solve {\sf MaxSWP}, we design an algorithm based on dynamic programming.
However, we do not know which information we keep track of in dynamic programming since the optimal structure of {\sf MaxSWP} is unknown.
For this, we identify the small coalitions in the optimal partition.
According to Corollary~\ref{opt:partition}, if there is a coalition whose social welfare is less than $n/2$, it is not included in the optimal partition since the social welfare can be increased by dividing the coalition.
Thus, we only keep track of the subgraph structures of coalitions of social welfare at least $n/2$.
We can identify such coalitions by calculating the social welfare of each coalition. By using the subgraph structures of such coalitions, we design a linear-time algorithm for {\sf MaxSWP} on a tree based on dynamic programming.
\subsection{Social welfare of trees with small diameters}\label{sec:opt-smalltree}
Since the utility of an agent is defined as the harmonic function with respect to the distance to all others in the same coalition, the diameter of the subgraph induced by each coalition affects the social welfare.
Intuitively, the social welfare of a subgraph with large diameter is very low in a tree because a tree is quite sparse.
Therefore, we characterize the subgraph structures of coalitions in the optimal partition in terms of small diameters.
We first consider trees $T^{\le2}_n$ with diameter at most 2.
Such graphs are stars denoted by $K_{1,n-1}$.
Since a star is a complete bipartite graph, it satisfies that $\mathcal{C}^*=\{V\}$ by Property~\ref{opt:perfect}.
Here, we investigate the number of vertices of a star that maximizes the average social welfare. This gives the upper bound of the social welfare of $T^{\le2}_n=K_{1,n-1}$.
\begin{lemma}[{\bf *}]\label{max:ASW:diameter2}
For tree $T^{\le2}_n$ with diameter at most 2, $n/2 \le \varphi(T^{\le2}_n) \le 9n/16$ holds.
\end{lemma}
\begin{figure}
\caption{diameter $3$ tree $T^3_n$}
\label{fig:diameter3tree}
\caption{diameter $4$ tree $T^4_n$}
\label{fig:diameter4tree}
\end{figure}
Next, we consider trees with diameter $3$.
Any tree with diameter $3$ can be represented as $T^3_n=(V(T^3_n), E(T^3_n))$ where $V(T^3_n)=\{u_1,u_2,s_1,\ldots,s_k,t_1,\ldots, t_{\ell}\}$ and $E(T^3_n)=(u_1,u_2)\cup \{(u_1, s_i) \mid 1\le i\le k\}\cup \{(u_2, t_j) \mid 1\le j\le \ell\}$ for any $k,\ell \ge1$ (see Figure~\ref{fig:diameter3tree}).
Note that $n=|V(T^3_n)|=k+\ell+2$.
Then, the following lemma holds.
\begin{lemma}[{\bf *}]\label{SW:diameter3tree}
For tree $T^3_n$ with diameter 3, if $k=2$ and $\ell\ge 7$, $k \ge 7$ and $\ell=2$, $k>3$ and $\ell\ge 3$, or $k\ge 3$ and $\ell>3$, $\varphi(T^3_n)<n/2$ holds, and otherwise $\varphi(T^3_n)\ge n/2$.
\end{lemma}
As with trees with diameter 3, we identify the types of trees with diameter 4 that satisfy $\varphi(T^4_n)\ge n/2$.
Any tree with diameter 4 can be represented as $T^4_n=(V(T^4_n), E(T^4_n))$ where $V(T^4_n)=\{v, u_1,\ldots,u_k,w_{1,1},\ldots,w_{1, \ell_1},\ldots, w_{k,1},\ldots, w_{k, \ell_k}\}$ for $k\ge 2$ and $\ell_1, \ell_2\ge 1$, and $E(T^4_n)=\{(v, u_i)\mid 1\le i\le k\}\cup \{(u_i, w_{i, j})\mid 1\le i\le k, 1\le j \le \ell_k \}$ (see Figure~\ref{fig:diameter4tree}).
For each $i$, $\ell_i$ represents the number of leaves of $u_i$.
We denote the total number of leaves of $T^4_n$ by $\alpha_k=\sum_{i=1}^k\ell_i$ and then the number of vertices of $T^4_n$ is represented as $n = |V(T^4_n)|=k+\alpha_k+1$.
Then, we obtain the following lemma.
\begin{lemma}[{\bf *}]\label{tree:diameter4}
For tree $T^4_n$ with diameter 4, $\varphi(T^4_n)\ge {n}/{2}$ holds if $(k,\alpha_k)=(2,2),(2,3),(3,2),(4,2)$, and otherwise $\varphi(T^4_n)<{n}/{2}$ holds.
\end{lemma}
Finally, we show that the social welfare of the grand coalition of a tree with diameter at least 5 is less than $n/2$.
\begin{lemma}[{\bf *}]\label{tree:diameter6}
For tree $T^\mu_n$ with diameter $\mu\ge5$, $\varphi(T^\mu_n)<n/2$ holds.
\end{lemma}
By the above discussion, the optimal partition of a tree does not contain not only coalitions with large diameters but also particular coalitions with small diameters.
In the following, we further refine the candidates for coalitions in the optimal partition of a tree.
First, we show that any optimal partition does not contain a coalition that consists of a tree with diameter 4. Thus, there is no coalition that consists of a tree with diameter at least 4 in the optimal solution by Lemma~\ref{tree:diameter6}.
\begin{lemma}[{\bf *}]\label{opt:diameter4tree}
Any optimal partition of a tree $T$ does not contain a coalition that consists of a tree $T^{\ge4}$ with diameter at least 4.
\end{lemma}
Moreover, we prove that the candidates for coalitions in the optimal partition are only three types.
\begin{lemma}[{\bf *}]\label{lem:treemaxSW}
Let $\mathcal{C}^*$ be the optimal partition of tree $T_n$.
Then, the subgraph $G[C]$ induced by $C\in{\mathcal C}^*$ is one of the following:
\color{black}
$(1)$ a star $K_{1,|C|-1}$ (see Figure~\ref{fig:star}), $(2)$ a path $P_4$ of length 3 (see Figure~\ref{fig:P_4}), and $(3)$ a tree $T^3_5$ of size five with diameter 3 (see Figure~\ref{fig:P'_4}).
\end{lemma}
\color{black}
\begin{figure}
\caption{$K_{1,|C|-1}
\label{fig:star}
\caption{$P_4$}
\label{fig:P_4}
\caption{$T^3_5$}
\label{fig:P'_4}
\end{figure}
\subsection{Algorithm}\label{sec:algo-opttree}
In this section, we propose an algorithm that finds an optimal partition of a tree in linear time.
This algorithm is based on dynamic programing with keeping track of the candidates identified by Lemma~\ref{lem:treemaxSW}.
First, we introduce some notations to design our algorithm.
Given a tree, we root it at arbitrary vertex $r$.
We denote a subtree whose root is $v\in{V}$ by $T_v$ and its partition by $\mathcal{C}_v$.
For subtree $T_v$, we also denote the coalition including $v$ by $C_v\in \mathcal{C}_v$.
By Lemma~\ref{lem:treemaxSW}, the subgraph $G[C]$ induced by coalition $C\in {\mathcal C}^*$ is $K_{1, |C|-1}$, $P_4$ or $T^3_5$.
The algorithm recursively computes a partition of $T_v$ which attains the maximum social welfare for each $v$ from the leaves of $T$.
Intuitively, our algorithm constructs coalitions in each step of dynamic programming. For example, a vertex $u$ is added a coalition as an isolated vertex in $T_u$. In next step, vertex $v$ must be added to the same coalition in $T_v$ since the optimal solution does not contain an isolated vertex.
Here, we keep track of not only coalition $C_v$, but also the position of $v$ in the coalition since we compute a coalition with maximum social welfare by combining sub-coalitions in subtrees of $T_v$.
For example, if $v$ is positioned at a leaf of $K_{1, f}$, it is combined with a coalition that consists of $K_{1,f-1}$. On the other hand, if $v$ is positioned at the center vertex of $K_{1, f}$, it is combined with $f$ coalitions that consist of isolated vertices.
Then we compute each coalition and sub-coalition with maximum social welfare including a new vertex in next step again.
Since the optimal partition contains only coalitions of $K_{1, |C|-1}$, $P_4$ and $T^3_5$, we keep track of coalition $C_v$ with the position of $v$ that consists of them.
Let $H$ be a subgraph induced by a coalition with the position of $v$.
Then we consider the following types of $H$:
\begin{enumerate}
\item $H$ is an isolated vertex of $v$, denoted by $H=(\{v\},\emptyset)$,
\item $H$ is a star $K_{1,f}$ and
\begin{enumerate}
\item $v$ is the center of $K_{1,f}$, denoted by $H=K^{mid}_{1,f}$,
\item $v$ is a leaf of $K_{1,f}$, denoted by $H=K^{leaf}_{1,f}$,
\end{enumerate}
\item $H$ is $P_4$ and
\begin{enumerate}
\item $v$ is a leaf of $P_4$, denoted $H=P^{leaf}_4$,
\item root $v$ is not a leaf of $P_4$, denoted by $H=P^{mid}_4$,
\end{enumerate}
\item $H$ is $T^3_5$ and
\begin{enumerate}
\item $v$ is $s_1$, denoted by $H=T^3_5(s_1)$,
\item $v$ is $t_1$ or $t_2$, denoted by $H=T^3_5(t)$,
\item $v$ is $u_1$, denoted by $H=T^3_5(u_1)$,
\item $v$ is $u_2$, denoted by $H=T^3_5(u_2)$.
\end{enumerate}
\end{enumerate}
We can observe that connected proper subgraphs of $T^3_5$ are subgraphs of $P_4$ and star $K_{1,3}$. Also connected proper subgraphs of $P_4$ are subgraphs of star $K_{1,2}$. Thus, by only keeping track of stars, we can treat $P_4$ and so $T^3_5$ as seen later.
Let $\mathcal{G}_v$ be the set of above subgraphs with the position of $v$ in $T_v$.
For $T_v$, we define the recursive formula $\rho(v,H)=\max_{\mathcal{C}_v\ni{C_v}:G[C_v]=H}\varphi(T_v,\mathcal{C}_v)$
as the maximum social welfare of the partition of $T_v$ such that the subgraph induced by coalition $C_v$ including $v$ is $H\in {\mathcal{G}_v}$.
We also define $\rho(v)=\max_{H\in\mathcal{G}_v}\rho(v,H)$ as the social welfare of the optimal partition of $T_v$.
Then, the social welfare of the optimal partition of $T$ with root $r$ is denoted by $\rho(r)=\max_{H\in\mathcal{G}_r}\rho(r,H)$.
\color{black}
Let $w_j$ be the children of $v$ where $1\le j\le d(v)-1$ in $T_v$.
Then, we define the recursive formulas of $\rho(v,H)$ for $H\in {\mathcal{G}_v}$ to compute $\rho(r)$ as follows.
\begin{enumerate}
\item {\bf $H$ is an isolated vertex $(\{v\},\emptyset)$}
If $H=(\{v\},\emptyset)$, $\varphi(H)=0$.
Since $v$ separates trees $T_{w_j}$, $\rho(v,(\{v\},\emptyset))$ is the sum of the social welfare of the optimal partition in $T_{w_j}$.
Thus, $\rho(v,(\{v\},\emptyset))$ is defined as $\rho(v,(\{v\},\emptyset))=\sum_{j=1}^{d(v)-1}\rho(w_j).$
\item {\bf $H$ is a star $K_{1,f}$}
\begin{enumerate}
\item {\bf $H=K^{mid}_{1,f}$ with center $v$}
In this case, we include $f$ children of $v$ in coalition $C_v$.
Note that $f$ children are isolated vertices in subtrees of $T_v$ since $C_v$ forms $K_{1,f}$ in $T_v$.
Let $\delta_j=\rho(w_j)-\rho(w_j,(\{w_j\},\emptyset))$. Then $\delta_j$ means the difference between the maximum social welfare in $T_{w_j}$ and the maximum social welfare of the partition such that $w_j$ is an isolated vertex in $T_{w_j}$.
In other words, $\delta_j$ is the cost to include $w_j$ in $C_v$.
Thus, choosing the smallest $f$ children of $\delta_j$ maximizes $\rho(v,{K}^{mid}_{1,f}(v))$ since it consists of the social welfare of $C_v$, $f$ optimal partitions of $T_{w_j}$ such that $w_j$ is an isolated vertex in $T_{w_j}$, and $d(v)-1-f$ optimal partitions of $T_{w_j}$.
Let $w_1, w_2, \ldots, w_f$ be such children, where the indices are sorted in ascending order.
Then, $\rho(v,{K}^{mid}_{1,f}(v))$ is defined as
$\rho(v,{K}^{mid}_{1,f}(v))=\varphi(K_{1,f})+\sum_{j=1}^{f}\rho(w_j, (\{w_j\}, \emptyset))+\sum_{j=f+1}^{d(v)}\rho(w_j)$.
\item {\bf $H=K^{leaf}_{1,f}$ with leaf $v$}
Since $C_v$ forms a star $K_{1,f}$ and $v$ is a leaf of it in $T_v$, we include vertex $v$ in a coalition of $K_{1,f-1}$ with center $w_k$ that is a child of $v$ in a subtree $T_{k}$.
Thus, we need to choose such child $w_k$ that maximize the social welfare of $T_v$.
In this case, the maximum social welfare of $T_v$ is the sum of the social welfare of the optimal partition of subtrees $T_{w_j}$ except for $T_{w_k}$, $\varphi(K_{1,f})$, and the social welfare of the partition of $T_{w_k}$ such that $w_k$ is the center of $K_{1,f-1}$ of coalition $C_{w_k}$ minus $\varphi(K_{1,f-1})$.
Thus, $\rho(v,K^{leaf}_{1,f})$ is defined as follows:
{\small
\begin{flalign*}
\hspace{-1.0cm}\rho(v,K^{leaf}_{1,f})
&=\max_{k\in [d(v)]} \bigl\{ \sum_{j\in[d(v)]\setminus \{k\}}\rho(w_j)+\varphi(K_{1,f})+\rho(w_k,K^{mid}_{1,f-1}(w_k))-\varphi(K_{1,f-1})\bigr\}\\
&= \max_{k\in [d(v)]} \bigl\{ \sum_{j=1}^{d(v)} \rho(w_j)-\rho(w_k)+\varphi(K_{1,f})+\rho(w_k,K^{mid}_{1,f-1})-\varphi(K_{1,f-1})\bigr\}\\
&= \sum_{j=1}^{d(v)} \rho(w_j)+\varphi(K_{1,f})-\varphi(K_{1,f-1})+\max_{k\in [d(v)]} \bigl\{\rho(w_k,K^{mid}_{1,f-1})-\rho(w_k)\bigr\}.
\end{flalign*}}
\end{enumerate}
We define $\rho(v,H)$ for the rest of $H\in {\mathcal G}_v$ in the same way.
\item {\bf $H$ is a path $P_4$}
\begin{enumerate}
\item {\bf $H=P^{leaf}_4$ with leaf $v$}
A path whose one of leaves is $v$ consists of one $P_3=K_{1,2}$ and $v$.
Thus we choose one child of $v$ whose coalition is $K_{1,2}$ and maximizes $\rho(v,P^{leaf}_4)$.
{\small
\begin{flalign*}
\hspace{-1.0cm}\rho(v,P^{leaf}_4)=&\varphi(P_4)-\varphi(P_3)+\sum_{j=1}^{d(v)}\rho(w_j)+\max_{k\in[d(v)]}\left\{\rho(w_k,K^{leaf}_{1,2})-\rho(w_k)\right\}.
\end{flalign*}
}
\item {\bf $H=P^{mid}_4$ with non-leaf $v$}
A path whose one of non-leaf vertices is $v$ consists of one $P_2=K_{1,1}$, $v$, and one isolated vertex.
Thus we choose two children of $v$ such that each coalitions that includes them is $K_{1,1}$ and an isolated vertex, respectively, and they maximize $\rho(v,P^{mid}_4)$.
{\small
\begin{flalign*}
\hspace{-1.0cm}\rho(v,P^{mid}_4)=&\varphi(P_4)-\varphi(P_2)+\sum_{j=1}^{d(v)}\rho(w_j)\\
&+\max_{a,b\in[d(v)],a\neq{b}}\bigl\{\rho(w_a,K_{1,1}^{leaf})+\rho(w_b,(\{w_b\},\emptyset))-\rho(w_a)-\rho(w_b)\bigr\}.
\end{flalign*}
}
\end{enumerate}
\item {\bf $H$ is a tree $T^3_5$}
\begin{enumerate}
\item {\bf $H=T^3_5(s_1)$ with $v=s_1$}
Since $v$ is $s_1$ of $T^3_5$ in $T_v$, we combine $K^{leaf}_{1,3}$ whose leaf is a child $w_j$ of $v$ with $v$.
Thus, we choose such a child of $v$ that maximizes $\rho(v,T^3_5(s_1))$.
Then, $\rho(v,T^3_5(s_1))$ is defined as follows:
{\small
\begin{flalign*}
\hspace{-1.0cm}\rho(v,T^3_5(s_1))=\varphi(T^3_5)-\varphi(K_{1,3})+\sum_{j=1}^{d(v)}\rho(w_j)+\max_{k\in[d(v)]}\left\{\rho(w_k,K^{leaf}_{1,3})-\rho(w_k)\right\}.
\end{flalign*}
}
\item {\bf $H=T^3_5(t)$ with $v=t_1$ or $t_2$}
Since $v$ is $t_1$ or $t_2$ of $T^3_5$ in $T_v$, we combine $P^{mid}_4$ in a subtree $T_{w_j}$ with $v$.
Thus, we choose such a child of $v$ that maximizes $\rho(v,T^3_5(t))$.
Then, $\rho(v,T^3_5(t))$ is defined as follows:
{\small
\begin{flalign*}
\hspace{-1.0cm}\rho(v,T^3_5(t))=\varphi(T^3_5)-\varphi(P_4)+\sum_{j=1}^{d(v)}\rho(w_j)+\max_{k\in[d(v)]}\left\{\rho(w_k,P^{mid}_4)-\rho(w_k)\right\}.
\end{flalign*}
}
\item {\bf $H=T^3_5(u_1)$ with $v=u_1$}
Since $v$ is $u_1$ of $T^3_5$ in $T_v$, we combine one coalition of $K_{1,1}$ whose center is a child of $v$, one coalition of an isolated vertex, and $v$ to construct coalition $C_v$.
Thus, we choose two such children of $v$ that maximizes $\rho(v,T^3_5(u_1))$.
Then, $\rho(v,T^3_5(u_1))$ is defined as follows:
{\small
\begin{flalign*}
\hspace{-1.0cm}\rho(v,T^3_5(u_1))=&\varphi(T^3_5)-\varphi(P_3)+\sum_{j=1}^{d(v)}\rho(w_j)\\
&+\max_{a,b\in[d(v)],a\neq{b}}\bigl\{\rho(w_a,K^{mid}_{1,2}(w_a))+\rho(w_b,(\{w_b\},\emptyset))-\rho(w_a)-\rho(w_b)\bigr\}.
\end{flalign*}
}
\item {\bf $H=T^3_5(u_2)$ with $v=u_2$}
Since $v$ is $u_2$ of $T^3_5$ in $T_v$, we combine one coalition of $P_2$ whose leaf is a child of $v$, two coalitions of isolated vertices, and $v$ to construct coalition $C_v$.
Note that such $P_2$ is a star $K^{leaf}_{1,1}$.
Thus, we choose three such children of $v$ that maximizes $\rho(v,T^3_5(u_2))$.
Then $\rho(v,T^3_5(u_2))$ is defined as follows:
{\small
\begin{flalign*}
\hspace{-1.0cm}\rho(v,T^3_5(u_2))=&\varphi(T^3_5)-\varphi(K^{leaf}_{1,1})+\sum_{j=1}^{d(v)}\rho(w_j)\\
&+\max_{a,b,c\in[d(v)],a\neq{b}\neq{c}}\bigl\{\rho(w_a,K^{leaf}_{1,1})+\rho(w_b,(\{w_b\},\emptyset))+\rho(w_c,(\{w_c\},\emptyset))\\
&\hspace{3cm}-\rho(w_a)-\rho(w_b)-\rho(w_c)\bigr\}.
\end{flalign*}
}
\end{enumerate}
\end{enumerate}
\begin{figure}
\caption{Computing $\rho(r_0,H)$ for $H=T^3_5(u_1)$}
\label{fig:dpalgorithm3}
\end{figure}
Figure~\ref{fig:dpalgorithm3} shows an example of computing $\rho(r_0,H)$ where $H=T^3_5(u_1)$.
To compute $\rho(r_0,H)$, we use the $\rho$'s values of its subtrees.
The pattern $H=T^3_5(u_1)$ contains one subtree with $H=K^{mid}_{1,2}$ and one with $H=\{\{v\},\emptyset \}$.
The best combination of these can be computed by the DP procedure $(c)$ explained above.
Finally, we evaluate the running time of our algorithm.
In Case 1, we can compute the recursive formula in time $O(d(v))$.
In Case 2, for case (a), we need to compute largest $\rho(v,{K}^{mid}_{1,f}(v))$ among $f=1,2,\ldots,d(v)-1$. This can be done by a binary search with SELECT,
since $\delta_i$ is increasing and $w_i$'s utility in $K_{1,f}$ is decreasing.
We can find the optimal $f$ in $O(d(v)+d(v)/2 + d(v)/4 \cdots)=O(d(v))$.
Case (b) is also computable in the same running time. In Case 3, both cases can be computed in time $O(d(v))$ by memorizing the best score among its children.
Finally, in Case 4, all the cases can be computed in $O(d(v))$ by a similar manner of Case 3.
Thus the total running time of this algorithm is $\sum_{v\in{V}}{O(d(v))}=O(n)$, since $\sum_{v\in{V}}d(v)=2|E|=2(n-1)$ holds for a tree by the handshaking lemma.
Hence, we obtain the following theorem.
\begin{theorem}\label{th:treeoptDP}
{\sf MaxSWP} for a tree can be solved in linear time.
\end{theorem}
\color{black}
\section{Hardness result of {\sf MaxSWP} for 4-regular graphs}~\label{sec:NPh}
\color{black}
It is mentioned in \cite{Balliu2017b} that {\sf MaxSWP} is NP-hard for graphs with maximum degree $6$, though the proof is omitted in the conference paper.
Actually, we can show a stronger result, that is, {\sf MaxSWP} is NP-hard even for $4$-regular graphs.
The proof is based on a reduction from a restricted variant of $3$-SAT problem called {\sf M3XSAT(3L)}, which is shown to be NP-complete in \cite[Lemma 5]{Porschen2014}.
\color{black}
\begin{theorem}[{\bf *}]\label{nphard:4-regular}
{\sf MaxSWP} is NP-hard even for 4-regular graphs.
\end{theorem}
\section{Conclusion}\label{sec:conc}
In this paper, we examined the structure of optimal social welfare partition of trees and designed an algorithm to find optimal solution in trees.
For path graphs which are subclass of trees, we gave complete characterization of optimal partition.
For general trees, by estimating the upper bound for the social welfare, we showed that each coalition has the diameter is three or less in optimal partition of a tree.
By using this property, we designed an algorithm for finding optimal partition of trees based on dynamic programming method. The computational complexity of this algorithm is $O(\Delta^2n)$ time.
A similar polynomial time algorithm can be expected as long as it is a graph class that is limited to a constant number of coalition constituting the optimal partition and to which dynamic programming can be applied.
In this paper, it became clear that the optimal partition of trees can be obtained with polynomial time, but it is unknown whether similar results can be obtained in the superior class of trees.
On the other hand, the currently known NP difficulty is limited to graphs with degree restrictions, and the relationship with the structural features of the graph is often unknown in many cases.
From the above, it is a future task to clarify the boundary between NP difficulty and polynomial time solvable from the viewpoint of graph class.
\section{Omitted proof in Section~\ref{sec:pre}}
\let\temp\ref{connect:coalition}
\renewcommand{\ref{connect:coalition}}{\ref{connect:coalition}}
\begin{proposition}
For each coalition $C\in{\mathcal C}^*$ of optimal partition ${\mathcal C}^*$, $G[C]$ is connected.
\end{proposition}
\let\ref{connect:coalition}\temp
\begin{proof}\rm
Let $\mathcal {C}^*$ be an optimal partition of $G$.
Suppose that there is a coalition $C\in \mathcal{C}^*$ such that $G[C]$ has at least two connected components.
Note that $1\leq|C_1|, |C_2|<|C|$.
We also set $\mathcal{C}'=\mathcal{C}^* \setminus \{C\} \cup \{C_1,C_2\}$.
\textcolor{black}{By using the fact that $|C_1|<|C|$, for any $v\in C_1\cup C$} it satisfies that
\begin{flalign*}
U(v,C)=\frac{1}{|{C}|}\sum_{u\in{C\setminus \{v\}}}\frac{1}{dist_{G[C]}(v, u)}&=\frac{1}{|{C}|}\sum_{u\in{C_1\setminus \{v\}}}\frac{1}{dist_{G[C]}(v, u)}\\
&<\frac{1}{|{C_1}|}\sum_{u\in{C_1\setminus \{v\}}}\frac{1}{dist_{G[C_1]}(v, u)}=U(v,C_1).
\end{flalign*}
In the same way, we obtain $U(v,C)<U(v,C_2)$ for any $v\in C_2\cup C_1$.
Thus,
\begin{align*}
\sum_{v\in C} U(v,C)=\sum_{v\in C_1} U(v,C)+\sum_{v\in C_2} U(v,C)< \sum_{v\in C_1} U(v,C_1) +\sum_{v\in C_2} U(v,C_2).
\end{align*}
Let $\varphi(G,\mathcal{C^*})$ be the social welfare of $\mathcal{C^*}$ in $G$. Because the social welfare of $\mathcal{C'}$ in $G$ is $\varphi(G,{\mathcal C}')=\varphi(G,{\mathcal C}^*)-\sum_{v\in C}U(v,C)$ +\textcolor{black}{$\sum_{v\in C_1} U(v,C_1) +\sum_{v\in C_2} U(v,C_2)$} , it holds that $\varphi(G,{\mathcal C}')>\varphi(G,{\mathcal C}^*)$.
This contradicts that ${\mathcal C}^*$ is optimal.
\qed \end{proof}
\section{Omitted proof in Section~\ref{sec:path}}
\let\temp\ref{lem:treemaxSW}
\renewcommand{\ref{lem:treemaxSW}}{\ref{lem:pathaverage}}
\begin{lemma}
It holds that $\tilde{\varphi}(P_2)<\tilde{\varphi}(P_3)$, and $\tilde{\varphi}(P_n)>\tilde{\varphi}(P_{n+1})$ for $n\ge3$.
\end{lemma}
\let\ref{lem:treemaxSW}\temp
\addtocounter{lemma}{-1}
\begin{proof}\rm
First, we compute the average social welfare of $P_n$.
\small
\color{black}
\begin{flalign*}
\tilde{\varphi}(P_n)=\frac{2}{n^2}\sum_{k=1}^{n-1}h(k)&=\frac{2}{n^2}\left(\frac{1}{n-1}+\frac{2}{n-2}+\cdots+\frac{n-1}{1}\right)\\
&=\frac{2}{n^2}\left(\frac{n-(n-1)}{n-1}+\frac{n-(n-2)}{n-2}+\cdots+\frac{n-1}{1}\right)\\
&=\frac{2}{n^2}\left(\frac{n}{n-1}+\frac{n}{n-2}+\cdots+\frac{n}{1}-1\cdot(n-1)\right)\\
&=\frac{2}{n^2}\left(n\cdot h(n-1)+1-n\right).
\end{flalign*}
\normalsize
Next, we consider the difference between $\tilde{\varphi}(P_{n+1})$ and $\tilde{\varphi}(P_n)$.
{\small
\begin{flalign*}
&\tilde{\varphi}(P_{n+1})-\tilde{\varphi}(P_n)\\
&=\frac{2}{n^2(n+1)^2}\bigl(n^2(n\cdot h(n)+h(n)-n)-(n+1)^2(n\cdot h(n-1)+1-n)\bigr)\\
&=\frac{2}{n^2(n+1)^2}\bigl((n^3\cdot h(n)+n^2\cdot h(n)-n^3)-(n^3\cdot h(n-1)+n^2-n^3)\\
&\hspace{5.0cm}-(2n+1)(n\cdot h(n-1)+1-n)\bigr)\\
&=\frac{2}{n^2(n+1)^2}\left(n^2\cdot h(n)-(2n+1)(n\cdot h(n-1)+1-n)\right)\\
&=\frac{2}{n^2(n+1)^2}\left(2n^2-n^2\cdot h(n-1)-n\cdot h(n-1)-1\right).
\end{flalign*}}
Let $q_n=n^2(2-h(n-1))-n\cdot h(n-1)-1$ for $n$. Then the sign of $q_n$ and the sign of $\tilde{\varphi}(P_{n+1})-\tilde{\varphi}(P_n)$ are the same.
\color{black}
We also note that $q_n$ is always negative for $n\ge3$ and positive for $n=2$. Thus, we obtain $\tilde{\varphi}(P_2)<\tilde{\varphi}(P_3)$, and $\tilde{\varphi}(P_n)>\tilde{\varphi}(P_{n+1})$ for $n\ge3$.
\qed \end{proof}
\let\temp\ref{lem:treemaxSW}
\renewcommand{\ref{lem:treemaxSW}}{\ref{path:isolated}}
\begin{lemma}
For the optimal partition $\mathcal{C}^*$ of a path $P_n$ and coalition $C\in\mathcal{C}^*$, $G[C]$ is either $P_2$, $P_3$ or $P_4$.
\end{lemma}
\let\ref{lem:treemaxSW}\temp
\addtocounter{lemma}{-1}
\begin{proof}\rm
We first observe that the candidates for $G[C]$ are only $P_2, P_3, P_4, P_5 $ or an isolated vertex.
This is because we can increase the social welfare by dividing $P_n$ for $n>5$ to them from Corollary~\ref{opt:partition}, Lemma~\ref{lem:pathaverage}, and $\varphi(P_n)<n/2$ for $\textcolor{black}{n>5}$.
Next, we show that $P_5$ is not included in the candidate.
Since we have $\varphi(P_5)=77/30<8/3=\varphi(P_2)+\varphi(P_3)$, we can increase the social welfare by dividing $P_5$ into two coalition $V(P_2)$ and $V(P_3)$, if there is a coalition such that $G[C]=P_5$.
Therefore $P_5$ is not included in the candidate for $G[C]$.
Finally, we show that an isolated vertex is not the candidate.
Suppose that an isolated vertex is included in the optimal partition of $P_n$.
Because a graph is a path graph, any isolated vertex is adjacent to at least one vertex.
Now, we suppose that an isolated vertex is adjacent to a coalition that consists of another isolated vertex, $P_2$, $P_3$, or $P_4$.
In the following, we show the contradiction for these four cases.
\begin{enumerate}
\item If an isolated vertex $v$ is adjacent to an isolated vertex $u$, we set $\varphi(P_n,\mathcal{C}^*\setminus \{\{u\},\{v\}\} \cup \{\{u,v\}\})=\varphi(P_n,\mathcal{C}^*)-0+1$. This contradicts the optimality of $\varphi(P_n,\mathcal{C}^*)$.
\item If an isolated vertex $v$ is adjacent to a vertex of the coalition $V (P_2)$, we set $\varphi(P_n,\mathcal{C}^*\setminus \{V(P_2),\{v\}\}\cup \{V(P_3\})=\varphi(P_n,\mathcal{C}^*)-1+5/3$. This contradicts the optimality of $\varphi(P_n,\mathcal{C}^*)$.
\item If an isolated vertex $v$ is adjacent to a vertex of the coalition $V(P_3)$, we set $\varphi(P_n,\mathcal{C}^*\setminus \{V(P_3),\{v\}\}\cup \{V(P_4\})=\varphi(P_n,\mathcal{C}^*)-5/3+13/6$. This contradicts the optimality of $\varphi(P_n,\mathcal{C}^*)$.
\item If an isolated vertex $v$ is adjacent to a vertex of the coalition $V(P_4)$, we set $\varphi(P_n,\mathcal{C}^*\setminus \{V(P_4),\{v\}\}\cup \{V(P_2),V(P_3)\})=\varphi(P_n,\mathcal{C}^*)-13/6+8/3$. This contradicts the optimality of $\varphi(P_n,\mathcal{C}^*)$.
\end{enumerate}
In any case, an isolated vertex is not included in the optimal partition.
Therefore, $G[C]$ is $P_2,P_3$ or $P_4$ in the optimal partition of $P_n$.
\qed \end{proof}
\let\temp\ref{nphard:4-regular}
\renewcommand{\ref{nphard:4-regular}}{\ref{th:pathopt}}
\begin{theorem}
The following partitions are optimal partitions of path $P_n$:
\color{black}
\begin{enumerate}
\item $\mathcal{C}=\bigl\{\{v_{3i-2},v_{3i-1},v_{3i}\}\mid{1}\le{i}\le{n/3}\bigr\}$ if $n\equiv0\pmod{3}$,
\item $\mathcal{C}=\bigl\{\{v_1,v_2,v_3,v_4\},\{v_{3i+2},v_{3i+3},v_{3i+4}\}\mid{1}\le{i}\le{(n-4)/3}\bigr\}$ if $n\equiv1\pmod{3}$, and
\item $\mathcal{C}=\bigl\{\{v_1,v_2\},\{v_{3i},v_{3i+1},v_{3i+2}\}\mid{1}\le{i}\le{(n-2)/3}\bigr\}$ or $\bigl\{\{v_1,v_2,v_3,v_4\}, \break\{v_5,v_6,v_7,v_8\},\{v_{3i+6},v_{3i+7},v_{3i+8}\} \mid1\le{i}\le{(n-8)/3}\bigr\}$ if $n\equiv2\pmod{3}$.
\end{enumerate}
\color{black}
\end{theorem}
\let\ref{nphard:4-regular}\temp
\addtocounter{theorem}{-1}
\begin{proof}\rm
From Lemma~\ref{path:isolated}, we only have to consider partitions that consist of $P_2$,$P_3$ or $P_4$.
Note that the order of coalitions in a path graph does not affect.
Thus, we can rearrange the order of coalitions in a path graph without changing the social welfare.
Now, we show that the optimal solution of $P_n$ contains at most two coalitions of $V(P_2)$.
If three or more coalitions of $V(P_2)$ are included in the optimal partition, we first rearrange three coalitions of $V(P_2)$ so that they are consecutive without changing the social welfare. Next, we replace them by consecutive two coalitions of $V(P_3)$. Because $\varphi(P_2)=1$, $\varphi(P_3)=5/3$ and $3\cdot \varphi(P_2)=3<10/3=2\cdot \varphi(P_3)$, the social welfare increases.
Similarly, we show that the optimal partiton of $P_n$ contains at most two coalitions of $V(P_4)$. If three or more coalitions of $V(P_4)$ are included in the optimal partition, we first rearrange three coalitions of $V(P_4)$ so that they are consecutive. Then, we change them to consecutive four coalitions of $V(P_3)$. Because $\varphi(P_4)=13/6$, $\varphi(P_3)=5/3$ and $3\cdot \varphi(P_4)=13/2<20/3=4\cdot \varphi(P_3)$, the social welfare also increases.
Thus, the optimal partition of $P_n$ contains at most two coalitions of $V(P_2)$ and $V(P_4)$, respectively.
Moreover, if both $V(P_2)$ and $V(P_4)$ are included in the optimal partition, we rearrange one $V(P_2)$ and one $V(P_4)$ so that they are consecutive. Then, we replace them by two consecutive consists of $V(P_3)$. Because $\varphi(P_2)+\varphi(P_4)=19/6<10/3=2\cdot \varphi(P_3)$, the social welfare increases.
Thus, both $V(P_2)$ and $V(P_4)$ are not included in the optimal partition.
Finally, if the optimal partiton of $P_n$ contains two coalitions of $V(P_2)$, we rearrange them so that so that they are consecutive.
Next, we change two $V(P_2)$ into one $V(P_4)$. Because $2\cdot \varphi(P_2)=2<13/6=\varphi(P_4)$, the social welfare increases.
Thus, the optimal partition of $P_n$ contains at most one coalition of $V(P_2)$.
Therefore, the optimal partition of $P_n$ contains at most one coalition of $V(P_2)$, at most two coalitions of $V(P_4)$. Moreover, it does not contain both $V(P_2)$ and $V(P_4)$.
From the above discussion, we can identify the optimal partition of a path graph $P_n$.
\begin{enumerate}
\item If $n\equiv0\pmod{3}$,\\
A partition $\mathcal{C}=\{V(P_3), \ldots, V(P_3) \}$ satisfies the above conditions.
That is, the partition that consists of coalitions $V(P_3)$'s is the optimal partition.
\item If $n\equiv1\pmod{3}$,\\
A partition $\mathcal{C}=\{V(P_4),V(P_3),\ldots,V(P_3)\}$ is the only partition that satisfies the above condition.
\item If $n\equiv2\pmod{3}$,\\
Partitions $\mathcal{C}=\{V(P_2),V(P_3),\ldots,V(P_3)\}$ and $\mathcal{C}=\{V(P_4), V(P_4), V(P_3),\break\ldots, V(P_3)\}$ satisfy the above conditions.
The social welfare of the former is $1+(5/3)\cdot(n-2)/3=(5n-1)/9$ and the latter is $(13/6)\cdot2+(5/3)\cdot(n-8)/3=(5n-1)/9$.
Thus, both of them are optimal partitions.
\end{enumerate}
\qed \end{proof}
\section{Omitted proof in Section~\ref{sec:tree}}
\let\temp\ref{lem:treemaxSW}
\renewcommand{\ref{lem:treemaxSW}}{\ref{max:ASW:diameter2}}
\begin{lemma}
For tree $T^{\le2}_n$ with diameter at most 2, $n/2 \le \varphi(T^{\le2}_n) \le 9n/16$ holds.
\end{lemma}
\let\ref{lem:treemaxSW}\temp
\addtocounter{lemma}{-1}
\begin{proof}
Let $\varphi (K_{1,n-1})$ and $\tilde{\varphi}(K_{1,n-1})$ be the social welfare and the average social welfare of the grand coalition of $K_{1,n-1}$, respectively. Then we can express them as
$\varphi(K_{1,n-1})={(n-1)(n+2)}/{2n}$ and $\tilde{\varphi}(K_{1,n-1})={(n-1)(n+2)}/{2n^2}$.
Moreover, we differentiate the average social welfare by $n$: ${d\tilde{\varphi}(K_{1,n-1})}/{d{n}}=({4-n})/{2n^3}$.
Therefore, the average social welfare is maximum when $n=4$.
By combining Corollary~\ref{opt:partition}, $n/2 \le \varphi(K_{1,n-1}) \le 9n/16$ holds.
\qed \end{proof}
\let\temp\ref{lem:treemaxSW}
\renewcommand{\ref{lem:treemaxSW}}{\ref{SW:diameter3tree}}
\begin{lemma}
For tree $T^3_n$ with diameter 3, if $k=2$ and $\ell\ge 7$, $k \ge 7$ and $\ell=2$, $k>3$ and $\ell\ge 3$, or $k\ge 3$ and $\ell>3$, $\varphi(T^3_n)<n/2$ holds, and otherwise $\varphi(T^3_n)\ge n/2$.
\end{lemma}
\let\ref{lem:treemaxSW}\temp
\addtocounter{lemma}{-1}
\begin{proof}
For $T^3_n$,
since the utilities of $s_i$, $t_j$, $u_1$ and $u_2$ are $U(s_i, V(T^3_n))=(1+k/2+\ell/3)/n$, $U(t_j, V(T^3_n))=(1+\ell/2+k/3)/n$, $U(u_1, V(T^3_n))=((k+1)+\ell/2)/n$ and $U(u_2, V(T^3_n))=((\ell+1)+k/2)/n$, respectively,
the social welfare of the grand coalition of $T^3_n$ $\varphi(T^3_n)$ is as follows:
\begin{flalign*}
\varphi(T^3_n)&=\sum_{i=1}^k U(s_i, V(T^3_n))+\sum_{j=1}^{\ell} U(t_j, V(T^3_n))+U(u_1, V(T^3_n))+U(u_2, V(T^3_n))\\
&=\frac{1}{n} \bigl(\ell(1+\frac{\ell}{2}+\frac{k}{3})+k(1+\frac{k}{2}+\frac{\ell}{3})+((k+1)+\frac{\ell}{2})+((\ell+1)+\frac{k}{2})\bigr)\\
&=\frac{1}{n} \bigl (\frac{k^2}{2}+\frac{5k}{2}+\frac{\ell^2}{2}+\frac{5\ell}{2}+\frac{2k\ell}{3}+2 \bigr).
\end{flalign*}
Since $n=k+\ell+2$,
\begin{flalign*}
\varphi(T^3_n)-\frac{n}{2}&=\frac{\frac{k^2}{2}+\frac{5k}{2}+\frac{\ell^2}{2}+\frac{5\ell}{2}+\frac{2k\ell}{3}+2}{k+\ell+2}-\frac{k+\ell+2}{2}\\
&=\frac{k^2+5k+\ell^2+5\ell+\frac{4k\ell}{3}+4}{2(k+\ell+2)}-\frac{(k+\ell+2)^2}{2(k+\ell+2)}\\
&=\frac{1}{2(k+\ell+2)}(k+\ell-\frac{2k\ell}{3})\\
&=\frac{1}{3(k+\ell+2)}\left(\frac{9}{4}-(k-\frac{3}{2})(\ell-\frac{3}{2})\right).
\end{flalign*}
Since $k$ and $\ell$ are positive integers, if $k=2$ and $\ell\ge 7$, $k \ge 7$ and $\ell=2$, $k>3$ and $\ell\ge 3$, or $k\ge 3$ and $\ell>3$, it satisfies that $\varphi(T^3_n)-n/2<0$, and otherwise $\varphi(T^3_n)-n/2\ge0$.
\qed \end{proof}
\let\temp\ref{lem:treemaxSW}
\renewcommand{\ref{lem:treemaxSW}}{\ref{tree:diameter4}}
\begin{lemma}
For tree $T^4_n$ with diameter 4, $\varphi(T^4_n)\ge{n}/{2}$ holds if \textcolor{black}{$(k,\alpha_k)=(2,2),\break(2,3),(3,2),(4,2)$}, and otherwise $\varphi(T^4_n)<{n}/{2}$ holds.
\end{lemma}
\let\ref{lem:treemaxSW}\temp
\addtocounter{lemma}{-1}
\begin{proof}
For $T^4_n$, since $\alpha_k=\sum_{i=1}^k\ell_i$, the utilities of $v$, $u_i$ and $w_{i, j}$ are denoted by $U(v, V(T^4_n))=k+\alpha_k/2$, $U(u_i, V(T^4_n))=(\ell_i+1)+(k-1)/2+(\alpha_k-\ell_i)/3$ and $U(w_{i, j}, V(T^4_n))=1+\ell_i/2+(k-1)/3+(\alpha_k-\ell_i)/4$, respectively.
Here, let $\beta_k=\sum^{k}_{i=1}\ell_i^2$, then the utilities of $u_i$ and $w_{i, j}$ can be expressed as follows:
{\small
\begin{flalign*}
\sum^k_{i=1}U(u_i, V(T^4_n))&= \sum^k_{i=1}\left((\ell_i+1)+\frac{k-1}{2}+\frac{\alpha_k-\ell_i}{3}\right)=\frac{k\alpha_k}{3}+\frac{2\alpha_k}{3}+\frac{k^2}{2}+\frac{k}{2},\\
\sum^k_{i=1}\sum^{\ell_i}_{j=1}U(w_{i, j},V(T^4_n))&=\sum^k_{i=1}\ell_i \left(1+\frac{\ell_i}{2}+\frac{k-1}{3}+\frac{\alpha_k-\ell_i}{4}\right)=\frac{\alpha_k^2}{4}+\frac{\beta_k}{4}+\frac{k\alpha_k}{3}+\frac{2\alpha_k}{3}.
\end{flalign*}}
Therefore, the social welfare of the grand coalition of $T^4_n$ $\varphi(T^4_n)$ is represented as:
\begin{flalign*}
\varphi(T^4_n)&=\frac{1}{n}\sum_{v\in{V_k}}U(v,V)\\
&=\frac{1}{n}(\sum^k_{i=1}U(u_i, V(T^4_n))+\sum^k_{i=1}\sum^{\ell_i}_{j=1}U(w_{i, j},V(T^4_n))+\textcolor{black}{k+\alpha_{k}/2})\\
&=\frac{1}{n}\left(\frac{\alpha_k^2}{4}+\frac{\beta_k}{4}+\frac{2k\alpha_k}{3}+\frac{11\alpha_k}{6}+\frac{k^2}{2}+\frac{3k}{2}\right).
\end{flalign*}
Since $n=k+\alpha_k+1$,
{\small
\begin{flalign*}
\varphi(T^4_n)-\frac{n}{2}&=\frac{1}{2(k+\alpha_k+1)}\left(\frac{\alpha_k^2}{2}+\frac{\beta_k}{2}+\frac{4k\alpha_k}{3}+\frac{11\alpha_k}{3}+k^2+3k \right)-\frac{(k+\alpha_k+1)^2}{2(k+\alpha_k+1)}\\
&=\frac{1}{2(k+\alpha_k+1)} \left(\frac{\beta_k}{2}-\frac{\alpha_k^2}{2}-\frac{2k\alpha_k}{3}+\frac{5\alpha_k}{3}+k-1\right).
\end{flalign*}}
Let ${\boldsymbol \ell}=(\ell_1, \ldots, \ell_k)$.
Since $\beta_k-\alpha_k^2=-\sum_{i,j\in[k],i\neq{j}}2\ell_i \ell_j$ and \textcolor{black}{$1/(2(k+\alpha+1))>0$}, if we define $f(k,{\boldsymbol \ell}) =-\sum_{i,j\in[k],i\neq{j}}\ell_i \ell_j-2k\alpha_k/3+5\alpha_k/3+k-1$, then the sign of $f(k,{\boldsymbol \ell})$ and the sign of $\varphi(T^4_n)-n/2$ are the same.
Because $\alpha_k\ge2$ and $\ell_1, \ell_2\ge 1$,
\begin{flalign*}
\frac{\partial f(k,{\boldsymbol \ell})}{\partial{k}}=1-\frac{2\alpha_k}{3}<0.
\end{flalign*}
Moreover, since $k\ge2$ and $\sum_{j\in[k]\setminus \{i\} }\ell_j\ge1$,
\begin{flalign*}
\frac{\partial f(k,{\boldsymbol \ell})}{\partial{\ell_i}}=\frac{5}{3}-\frac{2k}{3}-\sum_{j\in[k]\setminus \{i\} }\ell_j<0.
\end{flalign*}
Thus, $f(k,{\boldsymbol \ell})$ is a decreasing function related to $k, \ell_1, \ldots, \ell_k$. Therefore, the case that $f(k, \boldsymbol{\ell})\ge 0$ for $k\ge 2$ and $\ell_1, \ell_2\ge 1$ is \textcolor{black}{$(k,\ell_1, \ell_2, \ell_3, \ldots, \ell_k)=(2,1,1,0, \ldots, 0)$, $(2,1,1,1,\ldots, 0)$, $(3,1,1,0,\ldots, 0)$, $(4,1,1,0,\ldots, 0)$}.
Thus,
\begin{flalign*}
\varphi(T^4_n)-\frac{n}{2}
\begin{cases}
\ge0
&(k,\alpha_k)=(2,2),(2,3),(3,2),(4,2). \\
<0
&\mbox{otherwise}.
\end{cases}
\end{flalign*}
\qed \end{proof}
\let\temp\ref{lem:treemaxSW}
\renewcommand{\ref{lem:treemaxSW}}{\ref{tree:diameter6}}
\begin{lemma}
For tree $T^\mu_n$ with diameter $\mu\ge5$, $\varphi(T^\mu_n)<n/2$ holds.
\end{lemma}
\let\ref{lem:treemaxSW}\temp
\addtocounter{lemma}{-1}
\begin{proof}
We show that $\varphi(T^\mu_n)<n/2$ by using mathematical induction.
We observe that there always exists a path $P_{\mu+1}$ with diameter $\mu$ in $T_n^\mu$.
By adding vertex and edges to $P_{\mu+1}$ without changing the diameter, we can express any $T^\mu_n$. Then we show that $\varphi(T^\mu_n)<n/2$ in the process of adding vertices and edges.
By Lemma~\ref{lem:pathaverage} and $\varphi(P_6)-(\mu+1)/2=29/10-6/2<0$, the difference between $\varphi(P_{\mu+1})$ and \textcolor{black}{$(\mu+1)/2$} is $\varphi(P_{\mu+1})-(\mu+1)/2<0$ for $P_{\mu+1}$.
We assume that $T^\mu_{k-1}<(k-1)/2$ holds.
Let $u\in{V(T^\mu_{k-1})}$ be a vertex that does not change the diameter of $T^\mu_{k-1}$ even if we connect a new vertex $w$ to $u$ in $T^\mu_{k-1}$.
Moreover, let $T^\mu_k$ be a tree such that a new vertex $w$ is connected to such vertex $u$ and $(w,u)$ are added to $T^\mu_{k-1}$.
\textcolor{black}{
Here, the social welfare of $T^\mu_k$ consists of the social welfare with respect to a new vertex $w$ and the social welfare with in $k-1$ vertices in $T^\mu_{k-1}$.
The later can be represented as $(k-1)\varphi(T^\mu_{k-1})/k$.
Let $p_k$ be the former one. Then we have $\varphi(T^\mu_k)=(k-1)\varphi(T^\mu_{k-1})/k+p_k$.}
Let $n_2$ and $n_3$ be the number of verticies at distance two and three from $w$ in $T^\mu_k$, respectively, and $n_{\ge4}$ be the number of verticies at distance four and more.
Then $p_k$ is represented as
\begin{flalign}
p_k=\frac{2+n_2+\frac{2n_3}{3}+\cdots}{k}<\frac{2+n_2+\frac{2n_3}{3}+\frac{n_{\ge4}}{2}}{k}.
\end{flalign}
Moreover, since $k=n_2+n_3+n_{\ge4}+2$ and \textcolor{black}{$\varphi(T^\mu_{k-1})<(k-1)/2$}, it holds that
\begin{flalign}
\frac{k-1}{k}\varphi(T^\mu_{k-1})<\frac{(k-1)^2}{2k}=\frac{(n_2+n_3+n_{\ge4}+1)^2}{2(n_2+n_3+n_{\ge4}+2)}.
\end{flalign}
From \textcolor{black}{inequalities $(1)$ and $(2)$}, it holds that
\begin{flalign*}
\varphi(T^\mu_k)=\frac{k-1}{k}\varphi(T^\mu_{k-1})+p_k&<\frac{(n_2+n_3+n_{\ge4}+1)^2}{2(n_2+n_3+n_{\ge4}+2)}+\frac{2+n_2+\frac{2n_3}{3}+\frac{n_{\ge4}}{2}}{n_2+n_3+n_{\ge4}+2}\\
&=\frac{(n_2+n_3+n_{\ge4}+1)^2+4+2n_2+\frac{4n_3}{3}+n_{\ge4}}{2(n_2+n_3+n_{\ge4}+2)}.
\end{flalign*}
Finally,
{\small
\begin{flalign*}
\varphi(T^\mu_k)-\frac{k}{2}&<\frac{(n_2+n_3+n_{\ge4}+1)^2+4+2n_2+\frac{4n_3}{3}+n_{\ge4}}{2(n_2+n_3+n_{\ge4}+2)}-\frac{(n_2+n_3+n_{\ge4}+2)^2}{2(n_2+n_3+n_{\ge4}+2)}\nonumber\\
&=\frac{4+2n_2+\frac{4n_3}{3}+n_{\ge4}-2n_2-2n_3-2n_{\ge4}-3}{2(n_2+n_3+n_{\ge4}+2)}\nonumber\\
&=\frac{1-\frac{2n_3}{3}-n_{\ge4}}{2(n_2+n_3+n_{\ge4}+2)}\\
&<0.
\end{flalign*}}
Note that since the diameter of $T^\mu_k$ is $\mu\geq5$, it holds that $n_3, n_{\ge4} \geq1$.
\qed \end{proof}
\let\temp\ref{lem:treemaxSW}
\renewcommand{\ref{lem:treemaxSW}}{\ref{opt:diameter4tree}}
\begin{lemma}
Any optimal partition of a tree $T$ does not contain a coalition that consists of a tree $T^{\ge4}$ with diameter at least 4.
\end{lemma}
\let\ref{lem:treemaxSW}\temp
\addtocounter{lemma}{-1}
\begin{proof}
We consider coalitions with $(1)$ diameter 4 and $(2)$ diameter at least 5.
\begin{enumerate}
\item
From Lemma~\ref{tree:diameter4}, if \textcolor{black}{$(k,\alpha_k)=(2,2),(2,3),(3,2),(4,2)$}, it holds that $\varphi(T^4)\ge{n/2}$.
For each case, we check whether there is a partition $\mathcal{C}$ which satisfies $\varphi(T^4)<\varphi(T^4,\mathcal{C})$.
\begin{enumerate}
\item If $(k,\alpha_k)=(2,2)$,
we set $\mathcal{C}=\{\{v,u_1,w_{1,1}\},\{u_2,w_{2,1}\}\}$. Since $\varphi(T^4)=77/30<8/3=\varphi(T^4,\mathcal{C})$, the optimal solution of $T$ does not contain $T^4$ as a coalition.
\item \textcolor{black}{If $(k,\alpha_k)=(2,3)$,
we set $\mathcal{C}=\{\{v,u_1,w_{1,1},w_{1,2}\},\{u_2,w_{2,1}\}\}$. Since $\varphi(T^4)=3<10/3=\varphi(T^4,\mathcal{C})$, the optimal solution of $T$ does not contain $T^4$ as a coalition.}
\item If $(k,\alpha_k)=(3,2)$,
we set $\mathcal{C}=\{\{v,u_1,u_2,w_{1,1}\},\{u_3,w_{3,1}\}\}$. Since $\varphi(T^4)=109/36<10/3=\varphi(T^4,\mathcal{C})$, any optimal solution of $T$ does not contain $T^4$ as a coalition.
\item \textcolor{black}{If $(k,\alpha_k)=(4,2)$,
we set $\mathcal{C}=\{\{v,u_3,u_4\},\{u_1,w_{1,1}\},\{u_2,w_{2,1}\}\}$. Since $\varphi(T^4)=7/2<11/3=\varphi(T^4,\mathcal{C})$, the optimal solution of $T$ does not contain $T^4$ as a coalition.}
\end{enumerate}
\item
Since there exists a partition ${\mathcal C}$ such that ${\varphi(T^{\ge5},\mathcal{C})}\ge n/2$ by Proposition~\ref{dia2} and $\varphi(T^{\ge5})<n/2$ by Lemma~\ref{tree:diameter6},
any optimal solution of $T$ does not contain $T^{\ge5}$ as a coalition.
\end{enumerate}
\qed \end{proof}
\let\temp\ref{lem:treemaxSW}
\renewcommand{\ref{lem:treemaxSW}}{\ref{lem:treemaxSW}}
\begin{lemma}
Let $\mathcal{C}^*$ be the optimal partition of tree $T_n$.
Then, the subgraph $G[C]$ induced by $C\in{\mathcal C}^*$ is one of the following:
\color{black}
$(1)$ a star $K_{1,|C|-1}$ (see Figure~\ref{fig:star}), $(2)$a path $P_4$ of length 3 (see Figure~\ref{fig:P_4}), and $(3)$a tree $T^3_5$ of size five with diameter 3 (see Figure~\ref{fig:P'_4}).
\color{black}
\end{lemma}
\let\ref{lem:treemaxSW}\temp
\addtocounter{lemma}{-1}
\begin{proof}
From Lemma~\ref{opt:diameter4tree}, there is no coalition $C$ in the optimal partition such that the diameter of $G[C]$ is at least $4$.
Thus, we consider diameter at most $2$, and diameter $3$.
\begin{description}
\item[diameter at most 2.]
A tree with diameter at most $2$ is a star.
\item[diameter 3.]
From Corollary~\ref{opt:partition}, we only have to consider a tree of social welfare more than $n/2$.
By Lemma~\ref{SW:diameter3tree}, if $(k,\ell)=(1,\ell'),(2,2),(2,3)$, $(2,4),(2,5),(2,6)$, $(3,3),(3,2)$, $(4,2),(5,2),(6,2)$, $(k',1)$, a tree $T^3$ with diameter 3 satisfies $\varphi(T^3)\ge{|V(T^3)|/2}$.
Here, by using the fact that $k$ and $\ell$ are symmetric, we only consider $(1,\ell')$, $(2,2)$, $(2,3)$, $(2,4)$, $(2,5)$, $(2,6)$ and $(3,3)$.
\begin{description}
\item[Case:$(k,\ell)=(1,\ell')$.]
\textcolor{black}{From the fact that $\varphi(P_2)=1$ and $\varphi(K_{1,\ell'})=\ell'(\ell'+3)/2(\ell'+1)$}, if $\mathcal{C}=\{\{s_1,u_1\},\{u_2,t_1,\ldots,t_{\ell'}\}\}$, the social welfare $\varphi(T)$ and $\varphi(T,\mathcal{C})$ are respectively represented as
\begin{flalign*}
\varphi(T)=\frac{\ell'^2+\frac{19\ell'}{3}+10}{2(\ell'+3)} \mbox{\textcolor{black}{ \ and \ }} \varphi(T,\mathcal{C})=\frac{\ell'^2+5\ell'+2}{2(\ell'+1)}.
\end{flalign*}
Thus,
\textcolor{black}{
\begin{flalign*}
\varphi(T)-\varphi(T,\mathcal{C})&=\frac{(\ell'+1)(\ell'^2+\frac{19\ell'}{3}+10)-(\ell'+3)(\ell'^2+5\ell'+2)}{2(\ell'+1)(\ell'+3)}\\
&=-\frac{(\ell'-2)}{3(\ell'+1)}.
\end{flalign*}}
\textcolor{black}{If $\ell' \in\{1,2\}$, the social welfare of the grand coalition is actually at least the one of $\mathcal{C}$}. Moreover, if $\ell'\ge3$, the social welfare of $\mathcal{C}$ is more than the grand coalition.
Therefore, candidates for a coalition in the optimal partition are only $T^3_5$ and $P_4$.
\item [Case:$(k,\ell)=(2,2)$.]
If $\mathcal{C}=\{\{s_1,s_2,u_1\},\{u_2,t_1,t_2\}\}$, $\varphi(T^3)=28/9<10/3\break=\varphi(T^3,\mathcal{C})$.
Thus, a tree with diameter three and $(k,\ell)=(2,2)$ is not candidate for coalition in the optimal partition.
\item[Case:$(k,\ell)=(2,3)$.]
If we set $\mathcal{C}=\{\{s_1,s_2,u_1\},\{u_2,t_1,t_2,t_3\}\}$, $\varphi(T^3)=25/7<47/12=\varphi(T^3,\mathcal{C})$ holds.
\item[Case:$(k,\ell)=(2,4)$.]
If we set $\mathcal{C}=\{\{s_1,s_2,u_1\},\{u_2,t_1,t_2,t_3,t_4\}\}$, $\varphi(T^3)=97/24<67/15=\varphi(T^3,\mathcal{C})$ holds.
\item[Case:$(k,\ell)=(2,5)$.]
If we set $\mathcal{C}=\{\{s_1,s_2,u_1\}$, $\{u_2,t_1,t_2,t_3,t_4,t_5\}\}$, \break$\varphi(T^3)=122/27<5=\varphi(T^3,\mathcal{C})$ holds.
\item[Case:$(k,\ell)=(2,6)$.]
If we set $\mathcal{C}=\{\{s_1,s_2,u_1\}$, $\{u_2,t_1,t_2,t_3,t_4,t_5,t_6\}\}$, \break$\varphi(T^3)=5<116/21=\varphi(T^3,\mathcal{C})$ holds.
\item[Case:$(k,\ell)=(3,3)$.]
If we set $\mathcal{C}=\{\{s_1,s_2,s_3,u_1\},\{u_2,t_1,t_2,t_3\}\}$, \break$\varphi(T^3)=4<9/2=\varphi(T^3,\mathcal{C})$ holds.
\end{description}
Since a tree with diameter 3 is $P_4$ if $(k,\ell)=(1,1)$ and $T^3_5$ if $(k,\ell)=(2,1), (1,2)$, candidates for a coalition in the optimal partition are $P_4$ and $T^3_5$
\end{description}
Now, we know that $G[C]$ is one of $(\{v\},\emptyset)$, $K_{1,|C|-1}$, $P_4$ and $T^3_5$ for $C\in\mathcal{C^*}$.
Finally, we show that optimal partition does not contain an isolated vertex $(\{v\},\emptyset)$ as a coalition in a tree.
As with Proposition~\ref{path:isolated}, we assume that an isolated vertex $v$ exists in optimal partition ${\mathcal C}^*$ and lead to contradiction.
In the following, we assume that an isolated vertex $v$ is adjacent to another isolated vertex, the center of a star, one of leaves of a star, a vertex in $P_4$, and a vertex in $T^3_5$.
\begin{description}
\item [An isolated vertex $v$ is adjacent to another isolated vertex $u$.]
It is inconsistent as the social welfare increases by setting partition to ${\mathcal C}^*\setminus \{\{u\}, \{v\}\} \cup \{\{u,v\}\}$.
\item [An isolated vertex $v$ is adjacent to the center of star $K_{1,f}$.]
Since \break$\varphi(K_{1,f+1})=f/2+3/2-1/(f+2)>f/2+1-1/(f+1)=\varphi(K_{1,f})$, the social welfare of ${\mathcal C}^*\setminus (\{V(K_{1,f}),\{v\}\}) \cup (\{V(K_{1,f+1})\})$ is larger than $\mathcal{C}^*$. This is contradiction.
\item [An isolated vertex $v$ is adjacent to one of leaves of star $K_{1,f}$.]
In the case of $f=1, 2$, it is the same as the case of a path. Thus, this is contradiction.
For $f\ge3$, we chose one leaf $u$ in $K_{1,f}$.
Since $\varphi(K_{1,f-1})+\varphi(P_2)=(f^2+3f-2)/2f>(f^2+3f)/2(f+1)=\varphi(K_{1,f})$, if we set $\mathcal{C'}={\mathcal C}^*\setminus (\{V(K_{1,f}),\{v\}\}) \cup (\{V(K_{1,f})\setminus \{u\}, \{u,v\}\})$, $\varphi(G,\mathcal{C'})>\varphi(G,\mathcal{C^*}).$ This is contradiction.
\item [An isolated vertex $v$ is adjacent to $P_4$.]
If $v$ is adjacent to a leaf in $P_4$, this is contradiction by Lemma~\ref{path:isolated}.
If $v$ is adjacent not to a leaf, since $\varphi(T^3_5)=8/3>13/6=\varphi(P_4)$, if we set $\mathcal{C'}={\mathcal C}^*\setminus (\{\{V(P_4),\{v\}\}) \cup \{T^3_5\}$, $\varphi(G,\mathcal{C'})>\varphi(G,\mathcal{C^*}).$ This is contradiction.
\item [An isolated vertex $v$ is adjacent to $T^3_5$.]
We consider the following four cases.
\begin{description}
\item[An isolated vertex $v$ is adjacent to $s_1$.]
Since $\varphi(P_3)+\varphi(P_3)=10/3>8/3=\varphi(T^3_5)$, if we set $\mathcal{C'}={\mathcal C}^*\setminus (\{V(T^3_5),\{v\}\}) \cup (\{V(P_3), V(P_3)\})$, $\varphi(G,\mathcal{C'})>\varphi(G,\mathcal{C^*}).$ This is contradiction.
\item[An isolated vertex $v$ is adjacent to $u_1$.]
Since $\varphi(P_3)+\varphi(P_3)=10/3>8/3=\varphi(T^3_5)$, if we set $\mathcal{C'}={\mathcal C}^*\setminus (\{V(T^3_5),\{v\}\}) \cup (\{V(P_3), V(P_3)\})$, $\varphi(G,\mathcal{C'})>\varphi(G,\mathcal{C^*}).$ This is contradiction.
\item[An isolated vertex $v$ is adjacent to $u_2$.]
Since $\varphi(K_{1,3})+\varphi(P_2)=13/4>8/3=\varphi(T^3_5)$, if we set $\mathcal{C'}={\mathcal C}^*\setminus (\{V(T^3_5),\{v\}\}) \cup (\{V(K_{1,3}), V(P_2)\})$, $\varphi(G,\mathcal{C'})>\varphi(G,\mathcal{C^*}).$ This is contradiction.
\item[An isolated vertex $v$ is adjacent to $t_1$ or $t_2$.]
Since $\varphi(P_2)+\varphi(P_4)\break=19/6>8/3=\varphi(T^3_5)$, if we set $\mathcal{C'}={\mathcal C}^*\setminus (\{V(T^3_5),\{v\}\}) \cup (\{V(P_2), \break V(P_4)\})$, $\varphi(G,\mathcal{C'})>\varphi(G,\mathcal{C^*}).$ This is contradiction.
\end{description}
\end{description}
All the cases contradict the optimality of $\mathcal{C}^*$. Therefore, optimal partition $\mathcal{C}^*$ does not include an isolated vertex.
This completes the proof.
\qed \end{proof}
\section{Omitted proof in Section~\ref{sec:NPh}}~\label{sec:appendix:NPh}
\color{black}
In this section, we prove Theorem~\ref{nphard:4-regular}.
\color{black}
\let\temp\ref{nphard:4-regular}
\renewcommand{\ref{nphard:4-regular}}{\ref{nphard:4-regular}}
\begin{theorem}
{\sf MaxSWP} is NP-hard even for 4-regular graphs.
\end{theorem}
\let\ref{nphard:4-regular}\temp
\addtocounter{theorem}{-1}
\subsection{Reduction}
In this subsection, we give a reduction from {\sf M3XSAT(3L)}.
An instance $\psi$ of {\sf M3XSAT(3L)} forms a set of clause, say $S_1,\ldots,S_m$, where each clause consists of three literals from $x_1,x_2,\ldots,x_n$, and each $x_i$ appears three times in $S_j$'s. Note that we can regard a clause as just a $3$-set because of the monotonicity. Since $\sum_{j=1}^m |S_j| = 3m$ and the number of the total occurrences of literals is $3n$, $n=m$ holds.
From $\psi$, we construct
a 4-regular graph $G_{\psi}=(V(G_{\psi}), E(G_{\psi}))$ from a given instance of {\sf M3XSAT(3L)}.
First, we prepare a vertex set $V_{x_i} = \{x^{(1)}_i, x^{(2)}_i, x^{(3)}_i\}$ that corresponds to literal $x_i$. We call these {\em literal vertices}. We also prepare a vertex set $V_{S_j}=\{S^{(1)}_j, S^{(2)}_j\}$ that corresponds to $3$-set $S_j$. These are called {\em clause vertices}. Here, $x^{(1)}_i$, $x^{(2)}_i$ and $x^{(3)}_i$ correspond to the appearances of $x_i$. For example, if $x_4$ appears in $C_2$, $C_3$ and $C_5$, $x^{(1)}_4$, $x^{(2)}_4$ and $x^{(3)}_4$ are associated with $C_2$, $C_3$ and $C_5$, respectively. The vertex set of $G_{\psi}$ consists of $V_{x_i}$'s and $V_{S_j}$'s, that is, $V(G_{\psi}) = \bigcup_{i=1}^n V_{x_i} \cup \bigcup_{j=1}^m V_{S_j}$.
Thus $|V(G_{\psi})|=3n+2m=5n$ holds.
We connect these vertices by the following manner: (1) $V_{x_i}$ forms a triangle, that is, there are three edges $(x^{(1)}_i, x^{(2)}_i)$, $(x^{(2)}_i, x^{(3)}_i)$ and $(x^{(3)}_i, x^{(1)}_i)$, (2) vertices in $V_{S_j}$ have an edge, that is, there is an edge $(S^{(1)}_j, S^{(2)}_j)$, and (3) vertices in $V_{x_i}$ are connected with vertices in their corresponding $V_{S_j}$. For example, if $x^{(2)}_i$ appears in $S_j$, there are edges $(x^{(2)}_i, S^{(1)}_j)$ and
$(x^{(2)}_i, S^{(2)}_j)$ (see Figure \ref{fig:reducedgraph}). We call a subgraph consists of $V_{S_j}$ and its corresponding three literal vertices a {\em clause gadget}. It is easy to see that the degree of every vertex of $G_{\psi}$ is $4$.
\begin{figure}
\caption{Graph $G_{\psi}
\label{fig:reducedgraph}
\caption{(single) triangle, double triangle, and triple triangle}
\label{fig:maxasw}
\end{figure}
\color{black}
We can show the following important property of this $G_{\psi}$. A {\em triangle} is a cycle with $3$ vertices, and a {\em double triangle} is a graph that consists of two triangles sharing an edge, a {\em triple triangle} is a graph that consists of three triangles sharing an edge. See Figure \ref{fig:maxasw}.
\color{black}
\begin{lemma}\label{lem:more2/3}
If a coalition $C$ of $G_{\psi}$ satisfies $\tilde{\varphi}(G[C]) \ge 2/3$, $C$ forms one of the following: triangle, double triangle and triple triangle. The average social welfare of triangle, double triangle and triple triangle are $2/3=0.666\cdots$, $11/16=0.6875$ and $17/25=0.68$, respectively.
\end{lemma}
To this end, we show several sub-lemmas. The first one is very simple, but it gives an insight of a general property of average social welfare.
\begin{lemma}\label{lem:k}
For a graph $G$ of $k$ vertices and $\ell$ edges, $\tilde{\varphi}(G)\leq(k^2-k+2\ell)/2k^2$ holds.
\end{lemma}
\begin{proof}\rm
For a vertex $v$, we partition the vertex set $V$ of $G$ as $(V_0,V_1,V_2, \ldots, V_{k})$, where $V_i$ is the set of vertices $i$ distant from $v$.
Let $d_v$ denote the degree of vertex $v$. Then, $V_0=\{v\}$ and $|V_1|=d_v$.
By the definition of the utility, we have
\[
U(v,G) = \frac{1}{k}\sum_{i=1}\frac{|V_i|}{i} \le \frac{1}{k}(|V_1| + \sum_{i=2}\frac{|V_i|}{2}) = \frac{1}{k}\left(d_v + \frac{k-d_v -1}{2}\right) = \frac{d_v + k -1}{2k},
\]
and thus the following holds:
\[
\tilde{\varphi}(G) \le \frac{1}{2k^2}\sum_{v\in V}(d_v + k -1) = \frac{2\ell + k(k-1)}{2k^2}
\]
The last equality comes from the handshaking lemma, $\sum_{v\in V}d_v = 2\ell$.
\qed \end{proof}
By Lemma \ref{lem:k}, we obtain the following.
\begin{lemma}\label{ASW:degree4}
For a 4-regular graph $G=(V,E)$ with $|V|\geq 9$, the average social welfare of the grand coalition is at most $2/3$, i.e., $\tilde{\varphi}(G)\leq 2/3$ holds.
Furthermore, for a coalition $C$ that is not a grand coalition of $G$,
if $|C| \ge 9$, $\tilde{\varphi}(G[C])< 2/3$ holds.
\end{lemma}
\begin{proof}\rm
By the regularity of $G$, $4|V|=2|E|$ holds. Then by Lemma \ref{lem:k},
we have
\[
\tilde{\varphi}(G) \le \frac{|V|^2-|V|+4|V|}{2|V|^2} = \frac{1}{2}+\frac{3}{2k} \le \frac{2}{3}.
\]
For a coalition $C$ that is not a grand, suppose that $G[C]$ is a graph with $k$ vertices and $\ell$ edges. Since
$G[C]$ contains a vertex whose degree is at most $4$, we have $4k > 2\ell$.
Thus, by the similar arguement, we have
\[
\tilde{\varphi}(G[C]) \le \frac{k^2-k+2\ell}{2k^2}<\frac{2}{3}.
\]
\qed \end{proof}
The next one is not about a general graph but about $G_{\psi}$.
\begin{lemma}\label{ASW:edge6}
For a coalition $C$ in $G_{\psi}$, $\tilde{\varphi}(G_{\psi}[C])<2/3$ holds if $|C|=6,7,8$.
\end{lemma}
\begin{proof}\rm
Throughout the proof, we denote $|C|$ by $k$ and the number of edges of $G_{\psi}[C]$ by $\ell$.
($k=6$)
From Lemma~\ref{lem:k}, \textcolor{black}{if} $k=6$ and $\ell < 9$, $\tilde{\varphi}(G_{\psi}[C])< 2/3$ holds.
We prove the statement by showing that $G_{\psi}[C]$ contains at most $8$ edges.
We show this by contradiction; we assume that $G_{\psi}[C]$ has $9$ or more edges. Then, the average degree is $\ell \cdot 2/k \ge 3$; it forms a 3-regular graph or a graph with maximum degree is $4$.
\begin{description}
\item[Case:3-regular graph.]
Due to $k=6$, $G_{\psi}[C]$ contains at least one clause vertex. If two clause vertices in a clause gadget are contained in $G_{\psi}[C]$, exactly two literal vertices in the same clause gadget should be contained, otherwise it violates $3$-regularity. However, \textcolor{black}{if the degrees of these literal vertices are $3$, the degrees of two extra vertices cannot be $3$}. Thus only one clause vertex in a clause gadget is contained in $G_{\psi}[C]$. Then the three literal vertices in the clause gadget should be contained, but again the degrees of these literal vertices cannot be $3$ by adding two extra vertices. From these, $G_{\psi}[C]$ cannot be a 3-regular graph.
\item[Case:graph with a vertex of degree 4.]
We consider the case where $G_{\psi}[C]$ contains a clause vertex of degree $4$ \textcolor{black}{or does not include} a clause vertex of degree $4$.
\begin{description}
\item[Case:including a clause vertex of degree 4.]
It contains all the vertices of a clause gadget, and the remaining one vertex chosen outside the clause gadget.
However, $\ell \leq8$ is obtained for any case.
\item[Case:not including a clause vertex of degree 4.]
The induced subgraph contains three or four vertices in a clause gadget.
Here, if we do not include two clause vertices of the clause gadget, there is no vertex of degree $4$.
For this reason, the induced subgraph contains two clause vertices in the clause gadget.
Here, if the induced subgraph include three literal vertices in the clause gadget, a literal vertex can be of degree $4$.
Without lose generality, let $v_{x_i}^{(1)}$ be such a literal vertex.
Since the degree of $v_{x_i}^{(1)}$ is 4, other two vertices $v_{x_i}^{(2)}, v_{x_i}^{(3)}\in V_{x_i}$ must be included in the same induced subgraph.
Now, the induced subgraph contains five vertices: two clause vertices and three literal vertices, we only consider the remaining one vertex.
If we choose a clause vertex adjacent to either $v_{x_i}^{(2)}$ or $v_{x_i}^{(3)}$, the number of edges is at most $7$.
If we do not choose a clause vertex adjacent to either $v_{x_i}^{(2)}$ or $v_{x_i}^{(3)}$, the number of edges is at most $8$.
\end{description}
\end{description}
($k=7$)
From Lemma~\ref{lem:k}, if $k=7$ and $\ell\ge12$, $\tilde{\varphi}(G_{\psi}[C])\ge2/3$ holds, but if $k=7$, we show $\ell<12$.
Here we assume that there is a induced subgraph, where $\ell \ge12$.
The induced subgraph contains a vertex of degree 4 from \textcolor{black}{$12 \cdot 2/7 > 3.4$}.
In the following, we will investigate separately when the induced subgraph contains a clause vertex of degree $4$ or a clause vertex of degree \textcolor{black}{at most} $3$.
\begin{description}
\item [Case:graph with a vertex of degree 4.]
Since the induced subgraph contains all the vertices of a clause gadget, the number of edges is $7$.
Therefore, we consider whether there are vertices where the number of edges is $12$ by including the remaining $2$ vertices.
There are three ways to include vertices, but none of them will exceed the number of edges by more than $12$.
\item [Case:graph with a vertex of degree 3.]
The induced subgraph contains three or four vertices in a clause gadget.
Here, if the induced subgraph does not include two clause vertices of a clause gadget, there is no vertex of degree $4$.
For this reason, the induced subgraph includes two clause vertices within a clause gadget.
If a coalition contains three vertices in a clause gadget, the vertex of degree $4$ is the literal vertex in a clause gadget.
That is, since the coalition contains all vertices adjacent to the literal vertex, we consider only the remaining two vertices.
It may be possible to include two vertices in other gadget adjacent to literal vertices of degree $4$, but the number of edges is at most $9$.
If the induced subgraph includes four vertices in a clause gadget, two clause vertices are included as well as including the three vertices in a clause gadget.
Therefore, we only consider the remaining one vertex.
In this case, there are two combinations, but in either case the number of edges is at most $9$.
\end{description}
($k=8$)
From Lemma~\ref{lem:k}, if $k=8$ and $\ell\ge15$, $\tilde{\varphi}(G_{\psi}[C])\ge2/3$ holds, but if $k=8$, we show $\ell<15$.
we assume that the induced subgraph satisfy $\ell\ge15$.
Since the $G_{\psi}$ is two-connected four-regular, the degree of the subgraph is at most $4$ and includes at least two vertices of the degree of \textcolor{black}{at most} $3$.
Therefore, the maximum edges of the subgraph with the number of eight vertices is $15$ from $((8-2) \cdot 4+2\cdot3)/2=15$, but we show that there is no subgraph that realizes this number of edges.
Suppose that $G[C]$ includes six vertices of degree $4$ and two vertices of degree $3$.
Here, we consider the case where the induced subgraph contains a clause vertex of degree $4$, or a clause vertex of degree \textcolor{black}{at most} $3$.
\begin{description}
\item [Case:graph with a vertex of degree 4.]
The induced subgraph contains all the vertices in a clause gadget.
Therefore, we think about the remaining three vertices.
In order to make the degree of literal vertex in a clause gadget be equal to or lager than $3$, the induced subgraph includes literal vertex of other gadget, but we can not make degree of literal vertex of other gadget $3$ or more.
This is contradiction.
\item [Case:graph with a vertex of degree 3.]
The induced subgraph includes two clause vertices and two literal vertices in a clause gadget.
Here, the degree of the clause vertex is $3$, that is, the vertices other than the clause vertices are degree $4$.
However, in order to make degree of literal vertex in a clause gadget $4$, the induced subgraph need include literal vertex of other gadget.
Moreover, in order to make degree of literal vertex in a clause gadget $4$, the induced subgraph need include clause vertex.
This is contradiction.
\end{description}\qed \end{proof}
\begin{lemma}
For any coalition $C\in\mathcal{C}$ of $G_{\psi}$, $\tilde{\varphi}(G_{\psi}[C])\leq11/16$ holds.
\end{lemma}
\begin{proof}
From Lemmas \ref{ASW:degree4} and \ref{ASW:edge6}, if $k\ge6$, $\tilde{\varphi}(G_{\psi}[C])\leq2/3$ holds.
We consider the average social \textcolor{black}{welfare} for $k=2,3,4,5$.
If $k=2$, the upper bound of the average utility is $1/2$.
Moreover, if $k=3$, the average social welfare of a complete graph of size $3$ is maximum and $2/3$.
If $k=4$, the average social welfare of a clique of size 4 minus one edge is maximum in $G_{\psi}$ since it is two-connected four-regular and it does not contain a clique of size 4.
In this case, the average social welfare of a clique of size 4 minus one edge is $11/16$.
\textcolor{black}{Since the average social welfare on other subgraphs in $G_{\psi}$ of size $4$ is at most $11/16$, maximum average social welfare of subgraph in $G_{\psi}$ of size $4$ is also $11/16$.}
If $k=5$, there are 7 induced subgraphs of $G_{\psi}$.
Since the upper bound of the average utility is proportional to the number of edges from Lemma~\ref{lem:more2/3}, we only consider the case of the largest number of edges.
Such an induced subgraph is a triple triangle corresponding to a clause gadget.
The average social welfare of a triple triangle is $17/25$.
\textcolor{black}{Since the average social welfare on other subgraphs in $G_{\psi}$ of size $5$ is at most $17/25$, maximum average social welfare of subgraph in in $G_{\psi}$ of size $5$ is also $17/25$.}
Therefore, for any coalition $C\in\mathcal{C}$ of $G_{\psi}$, $\tilde{\varphi}(G_{\psi}[C])\leq11/16$ holds.
\qed \end{proof}
\subsection{NP-hardness of {\sf MaxSWP}}
From Lemma~\ref{lem:more2/3}, we prove the following lemma.
\begin{lemma}\label{NPhard}
An instance $\psi$ of {\sf M3XSAT(3L)} is a yes-instance if and only if
$G_{\psi}$ has a partition $\mathcal{C}$ such that $\varphi(G_{\psi},\mathcal{C})=41n/12$.
\end{lemma}
\begin{proof}
($\Rightarrow$) Assume that there is a truth assignment, and we construct a partition of $G_{\psi}$ from the assignment. In the partition, each $V_{x_i}$ with true $x_i$ forms a coalition \textcolor{black}{as a triangle}, and each $V_{S_j}$ \textcolor{black}{together with vertices of false literals} forms a coalition \textcolor{black}{as a double triangle}. For example, suppose that $S_j=\{x^{(1)}_1,x^{(1)}_2,x^{(1)}_3\}$ and $S_j$ is satisfied by $x^{(1)}_1$ (actually $x_i$). We then consider a coalition $\{S^{(1)}_j,S^{(2)}_j,x^{(1)}_2, x^{(1)}_3\}$ for $V_{S_j}$ which forms a double triangle.
Since the truth assignment satisfies all the clauses, each $V_{S_j}$ is a part of coalition of size $4$. Literal vertices $V_{x_i}$ themselves form a coalition if $x_i=1$, and otherwise they are included in $C_j$'s.
Since the utility of the coalition of a double triangle is $11/4$ and the utility of a triangle is $2$, the utility of the reduced graph is $\varphi(G_{\psi},\mathcal{C})$=$11n/4+2\cdot(n/3)=41n/12$.
\noindent
($\Leftarrow$) Assume that there exists a partition $\mathcal{C}$ of $G_{\psi}$ whose social welfare is at least $41n/12$, and the average social welfare is at least $(41n/12)/(5n)=41/60=0.683\cdots$. This and Lemma \ref{lem:more2/3} imply that $\mathcal{C}$ contains at least one double triangle.
Notice that a coalition can form a double triangle only in a clause gadget. If a clause gadget contains a coalition of a double triangle, one literal vertex is left. \textcolor{black}{Since such a literal vertex belongs to a coalition of a subgraph of a triangle, its utility is at most $2/3$.} Again, by Lemma \ref{lem:more2/3}, the average social welfare of a vertex not included in double triangle is at most $17/25$. From these, if $\mathcal{C}$ contains $p$ double triangles, the social welfare is at most
\[
\frac{11}{16}\cdot 4p + \frac{2}{3} \cdot p + \frac{17}{25}\cdot 5(n-p) = \frac{17}{5} n + \left(\frac{41}{12} - \frac{17}{5} \right)p.
\]
Since the social welfare of $\mathcal{C}$ is at least $41n/12$ by the assumption, we have
\[
\frac{17}{5} n + \left(\frac{41}{12} - \frac{17}{5} \right)p \ge \frac{41n}{12},
\]
and thus $p\ge n$ holds. On the other hand, $p$ is the number of double triangles, and it is at most $n$. It follows that $p=n$ and every clause gadget contains a double triangle as a coalition in $\mathcal{C}$. Furthermore, a literal vertex not included in a double triangle belongs to a coalition of a triangle, which corresponds to a literal $x_i$. Then by assigning true to such $x_i$, every clause gadget includes exactly one true literal, which is a solution of {\sf M3XSAT(3L)}.
\qed \end{proof}
\color{black}
By this Lemma~\ref{NPhard} and the NP-hardness of {\sf M3XSAT(3L)}, we complete the proof of Theorem~\ref{nphard:4-regular}.
\color{black}
\end{document} | math | 83,495 |
\begin{document}
\centerline{\bf On Generalized Integrals and Ramanujan-Jacobi Special Functions}
\vskip .4in
\centerline{N.D. Bagis}
\centerline{Department of Informatics, Aristotele University}
\centerline{Thessaloniki, Greece}
\centerline{[email protected]}
\vskip .2in
\[
\]
\textbf{Keywords}: Integrals; Elliptic Functions; Ramanujan; Special Functions; Continued Fractions; Generalization; Evaluations
\[
\]
\centerline{\bf Abstract}
\begin{quote}
In this article we consider new generalized functions for evaluating integrals and roots of functions. The construction of these generalized functions is based on Rogers-Ramanujan continued fraction, the Ramanujan-Dedekind eta, the elliptic singular modulus and other similar functions. We also provide modular equations of these new generalized functions and remark some interesting properties.
\end{quote}
\section{Introduction}
Let
\begin{equation}
\eta(\tau)=e^{\pi i\tau/12} \prod^{\infty}_{n=1}(1-e^{2\pi in\tau})
\end{equation}
denotes the Dedekind eta function which is defined in the upper half complex plane. It not defined for real $\tau$.\\
Let for $|q|<1$ the Ramanujan eta function be
\begin{equation}
f(-q)=\prod^{\infty}_{n=1}(1-q^n).
\end{equation}
The following evaluation holds (see [12]):
\begin{equation}
f(-q)=2^{1/3}\pi^{-1/2}q^{-1/24}k^{1/12}k^{{*}{1/3}}K(k)^{1/2},
\end{equation}
where $k=k_r$ is the elliptic singular modulus, $k^{*}=\sqrt{1-k^2}$ and $K(x)$ is the complete elliptic integral of the first kind.\\
The Rogers-Ramanujan continued fraction is (see [8],[9],[14]):
\begin{equation}
R(q):=\frac{q^{1/5}}{1+}\frac{q^1}{1+}\frac{q^2}{1+}\frac{q^3}{1+}... ,
\end{equation}
which have first derivative (see [5]):
\begin{equation}
R'(q)=5^{-1}q^{-5/6}f(-q)^4R(q)\sqrt[6]{R(q)^{-5}-11-R(q)^5}
\end{equation}
and also we can write
\begin{equation}
\frac{dR(q)}{dk}=5^{-1}\cdot2^{1/3}(kk^{*})^{-2/3}R(q)\sqrt[6]{R(q)^{-5}-11-R(q)^5}.
\end{equation}
Ramanujan have proven that
\begin{equation}
R(q)^{-5}-11-R(q)^5=\frac{f(-q)^6}{qf(-q^5)^6}.
\end{equation}
Also holds the following interesting identity
\begin{equation}
\frac{dk}{dq}=\frac{-2k(k^{*})^2K(k)^2}{q\pi^2},
\end{equation}
which is a result of Ramanujan (see [7] and [8]) for the first derivative of $k=k_r$ with respect to $q=e^{-\pi\sqrt{r}}$, $r>0$.
\section{Propositions}
\textbf{Definition 1.}\\
For any smooth function $G$, we define $m_G(A)$ to be such that
\begin{equation}
A=\pi\int^{+\infty}_{\sqrt{m_G(A)}}\eta\left(it/2\right)^4G\left(R\left(e^{-\pi t}\right)\right)dt.
\end{equation}
\\
\textbf{Theorem 1.}\\
If
\begin{equation}
5\int^{y}_{0}\frac{G(t)}{t\sqrt[6]{t^{-5}-11-t^5}}dt=A,
\end{equation}
then
\begin{equation}
y=R\left(e^{-\pi\sqrt{m_G(A)}}\right).
\end{equation}
\\
\textbf{Proof.}\\
From [5] it is known that if $a,b\in (0,1)$ then
\begin{equation}
\int^{a}_{b}f(-q)^4q^{-5/6}G(R(q))dq=5\int^{R(a)}_{R(b)}\frac{G(x)}{x\sqrt[6]{x^{-5}-11-x^5}}dx ,
\end{equation}
which is equivalent to write
$$
-\pi \int^{a_1}_{b_1}f\left(-e^{-t\pi}\right)^4e^{-t\pi/6}G\left(R\left(
e^{-\pi t}\right)\right)dt=5\int^{R\left(e^{-\pi a_1}\right)}_{R\left(e^{-\pi b_1}\right)}\frac{G(x)}{x\sqrt[6]{x^{-5}-11-x^5}}dx .
$$
Setting $b_1=+\infty$, $a_1=\sqrt{m_G(A)}$ and then using Definition 1 and the Dedekind eta expansion (1) we get the result.\\
Also differentiating (8) one can get
\begin{equation}
\frac{dm_G^{(-1)}(r)}{dr}=-\frac{\pi\eta\left(i\sqrt{r}/2\right)^4G(R(q))}{2\sqrt{r}}\textrm{, }q=e^{-\pi\sqrt{r}}\textrm{, }r>0
\end{equation}
and if
\begin{equation}
\phi(r):=-\frac{2\sqrt{r}}{\pi \eta\left(\frac{i\sqrt{r}}{2}\right)^4G\left(R\left(q\right)\right)},
\end{equation}
then\\
\\
\textbf{Proposition 1.}\\
If $A>0$, then
\begin{equation}
\frac{d}{dA}m_G(A)=\phi\left(m_G(A)\right).
\end{equation}
\\
Derivating (10) with respect to $q_{m_G}=e^{-\pi\sqrt{m_G}}$ we get
$$
\frac{5G(R(q_{m_G}))}{R(q_{m_G})\sqrt[6]{R(q_{m_G})^{-5}-11-R(q_{m_G})^5}}\frac{dR(q_{m_G})}{dq_{m_G}}=\frac{dA}{dk_{m_G}}\frac{1}{\frac{dq_{m_G}}{dk_{m_G}}},
$$
or the equivalent, using (8),(5) and (3):\\
\\
\textbf{Theorem 2.}
\begin{equation}
\frac{dA}{dk_{m_G}}=\sqrt[3]{2}\frac{G\left(y(A)\right)}{\left(k_{m_G}k^{*}_{m_G}\right)^{2/3}}.
\end{equation}
\\
Also with integration of (16) we have
\begin{equation}
3\sqrt[3]{2k_{m_G}}\cdot{}_2F_{1}\left[\frac{1}{6},\frac{1}{3};\frac{7}{6};k_{m_G}^2\right]=\int^{A}_{0}\frac{1}{G\left( R\left(e^{-\pi\sqrt{m_G(t)}}\right)\right)}dt.
\end{equation}
Hence\\
\\
\textbf{Definition 2.}\\
We define the function $h$ as
\begin{equation}
x=\int^{h(x)}_{0}\frac{dt}{G\left(R\left(e^{-\pi\sqrt{m_G(t)}}\right)\right)}=\int^{h(x)}_{0}\frac{dt}{G\left(y(t)\right)}
\end{equation}
\textbf{Theorem 3.}\\
Set $m_G(A)=r$, then
\begin{equation}
A=h\left(3\sqrt[3]{2k_r}\cdot{}_2F_{1}\left[\frac{1}{6},\frac{1}{3};\frac{7}{6};k_r^2\right]\right),
\end{equation}
or beter
\begin{equation}
m_G^{(-1)}(r)=h\left(3\sqrt[3]{2k_r}\cdot{}_2F_{1}\left[\frac{1}{6},\frac{1}{3};\frac{7}{6};k_r^2\right]\right).
\end{equation}
\\
From relation (10) differentiating we have
$$
5\frac{G(y(A))y'(A)}{y(A)\sqrt[6]{y(A)^{-5}-11-y(A)^5}}=1.
$$
Hence
\begin{equation}
5\int^{y(A)}_{0}\frac{dt}{t\sqrt[6]{t^{-5}-11-t^5}}=\int^{A}_{0}\frac{dt}{G(y(t))}=b_{m_G},
\end{equation}
where
\begin{equation}
b_r=3\sqrt[3]{2k_r}\cdot{}_2F_{1}\left[\frac{1}{6},\frac{1}{3};\frac{7}{6};k_r^2\right].
\end{equation}
Also if $m(x)$ is the function defined as (see [13])
\begin{equation}
\pi\int^{+\infty}_{\sqrt{m(r)}}\eta(it/2)^4dt=r.
\end{equation}
Then we have also
\begin{equation}
b^{(-1)}_r=m(r).
\end{equation}
Hence
\begin{equation}
y(A)=R\left(e^{-\pi\sqrt{m_G}}\right)=F_1\left(b_{m_G}\right),
\end{equation}
where we have define $F_1$ by
\begin{equation}
x=5\int^{F_1(x)}_{0}\frac{dt}{t\sqrt[6]{t^{-5}-11-t^5}}.
\end{equation}
\textbf{Theorem 4.}
\begin{equation}
5\int^A_0\frac{G(t)dt}{t\sqrt[6]{t^{-5}-11-t^5}}=h\left(5\int^{A}_{0}\frac{dt}{t\sqrt[6]{t^{-5}-11-t^5}}\right).
\end{equation}
\textbf{Proof.}\\
From relations (21) and (19) and (10) we have
\begin{equation}
h\left(5\int^{y(A)}_{0}\frac{dt}{t\sqrt[6]{t^{-5}-11-t^5}}\right)=h(b_{m_{G}})=A=5\int^{y(A)}_{0}\frac{G(t)dt}{t\sqrt[6]{t^{-5}-11-t^5}},
\end{equation}
which by inversion of $y(A)$ we get the desired result.\\
Also from relation (20) is
\begin{equation}
m^{(-1)}_G(A)=h(b_A).
\end{equation}
Hence knowing the function $h$ we know almost everything. For this, for a given function $P(x)$ we can set
\begin{equation}
G(x)=5^{-1}xP'(x)\sqrt[6]{x^{-5}-11-x^5},
\end{equation}
then
\begin{equation}
m_G^{(-1)}(A)=P\left(R\left(e^{-\pi\sqrt{A}}\right)\right)=h(b_A).
\end{equation}
Hence inverting $b_A$ we get the function $h$. Note also that $P(A)=y^{(-1)}(A)$ and
\begin{equation}
m_G(A)=v_i(y(A)),
\end{equation}
where $v_i(x)$ is the inverse function of $R(e^{-\pi\sqrt{x}})$.\\
\\
\textbf{Remarks.}\\
1) Note by definition that when we know $G$ the $m_G^{(-1)}(A)$ is a closed form formula and $m_G(A)$ is not (it needs inversion).\\
2) Continuing from relation (30) and $y(x)=P^{(-1)}(x)$ we have
$$
G(y(x))=5^{-1}P^{(-1)}(x)P'\left(P^{(-1)}(x)\right)\sqrt[6]{\left(P^{(-1)}(x)\right)^{-5}-11-\left(P^{(-1)}(x)\right)^5}.
$$
From equation (28) we get also\\
\\
\textbf{Corollary 1.}\\
\begin{equation}
h^{(-1)}(A)=5\int^{y(A)}_{0}\frac{dt}{t\sqrt[6]{t^{-5}-11-t^5}}.
\end{equation}
\\
\textbf{Corollary 2.}
\begin{equation}
y'(A)=5^{-1}h'_i(A)y(A)\sqrt[6]{y(A)^{-5}-11-y(A)^5},
\end{equation}
with
\begin{equation}
h_i(A)=h^{(-1)}(A)=b_{m_G(A)}=\left(b\circ m_G\right)(A)
\end{equation}
and also
\begin{equation}
\frac{dy(A)}{dk_{m_G}}=5^{-1}\cdot 2^{1/3}\left(k_{m_G}k^{*}_{m_G}\right)^{-2/3}y(A)\sqrt[6]{y(A)^{-5}-11-y(A)^5}.
\end{equation}
\\
Using the above one can show with differentiation the following\\
\\
\textbf{Theorem 5.}
\begin{equation}
h_i(A)=\pi\int^{\infty}_{\sqrt{m_G(A)}}\eta\left(it/2\right)^4dt=b_{m_G(A)}=5\int^{y(A)}_{0}\frac{dt}{t\sqrt[6]{t^{-5}-11-t^5}}.
\end{equation}
\\
Also another interesting theorem arises from the definition of $h(x)$. By setting
\begin{equation}
h_1(t):=\left(\frac{1}{h_i'(x)}\right)^{(-1)}(t),
\end{equation}
then\\
\\
\textbf{Theorem 6.}
\begin{equation}
5\int^{G_i(x)}_{0}\frac{dt}{t\sqrt[6]{t^{-5}-11-t^5}}=\int^{x}_{c}\frac{h'_1(t)}{t}dt.
\end{equation}
\\
\textbf{Proof.}\\
From relation (18) we have
$$
G\left(y(t)\right)=\frac{1}{h_i'(t)},
$$
hence inverting
$$
y_i\left(G_i(t)\right)=h_1(t).
$$
Taking the derivatives in both parts we get
$$
5\frac{G_i'(t)}{G_i(t)\sqrt[6]{G_i(t)^{-5}-11-G_i(t)^5}}=\frac{h_1'(t)}{t}.
$$
Lastly integrating the above relation in both parts we get the result.\\
Equation (39) can also written in the next form using (26)
\begin{equation}
G_i(x)=F_1\left(\int^{x}_{c}\frac{h_1'(t)}{t}dt\right).
\end{equation}
Suppose the $n$-th order modular equation of $y(A)$ is
\begin{equation}
P_n(A)=y\left(n\cdot y^{(-1)}(A)\right).
\end{equation}
Then we can find $P_n(x)$ by solving
\begin{equation}
\int^{P_n(A)}_{0}\frac{G(t)}{t\sqrt[6]{t^{-5}-11-t^5}}dt=n\int^{A}_{0}\frac{G(t)}{t\sqrt[6]{t^{-5}-11-t^5}}dt,
\end{equation}
with respect to $P_n(A)$. Setting $Q_n(A)$ to be
\begin{equation}
Q_n(A):=m_G\left(n^2\cdot m_G^{(-1)}\left(A\right)\right),
\end{equation}
the $n$-nth degree modular equation of $m_G(A)$ and using (15) we have
\begin{equation}
m_G^{(-1)}(A)=\int^{A}_{0}\frac{dt}{\phi(t)}
\end{equation}
and
$$
\int^{Q_n(A)}_{0}\frac{dt}{\phi(t)}=n\int^{A}_{0}\frac{dt}{\phi(t)} .
$$
Also
$$
Q^{(-1)}_n\left(A\right)=Q_{1/n}\left(A\right)
$$
and
$$
\int^{Q_{nm}(A)}_{0}\frac{dt}{\phi(t)}=n\int^{Q_m(A)}_{0}\frac{dt}{\phi(t)}=nm\int^{A}_{0}\frac{dt}{\phi(t)}
$$
and
$$
\int^{Q_{n}\left(Q_{m}\left(A\right)\right)}_{0}\frac{dt}{\phi(t)}=n\int^{Q_m(A)}_{0}\frac{dt}{\phi(t)}=nm\int^{A}_{0}\frac{dt}{\phi(t)}.
$$
Hence
$$
\int^{Q_{n}\left(Q_{m}\left(A\right)\right)}_{0}\frac{dt}{\phi(t)}=\int^{Q_{nm}(A)}_{0}\frac{dt}{\phi(t)}
$$
and consequently
\begin{equation}
Q_n\left(Q_m\left(A\right)\right)=Q_m\left(Q_n\left(A\right)\right)=Q_{nm}(A),
\end{equation}
$Q_1(A)=A$
and if $n=p_1^{a_1}p_2^{a_2}\ldots p_s^{a_s}$
\begin{equation}
Q_n(x)=\left(Q_{p_1}\circ\ldots\circ Q_{p_1}\right)\circ \left(Q_{p_2}\circ \ldots \circ Q_{p_2}\right)\circ\ldots \circ\left( Q_{p_s}\circ\ldots \circ Q_{p_s}\right),
\end{equation}
respectively iterated $a_1,a_2,\ldots,a_s$ times.\\
Hence
$$
y\left(m^{(-1)}_G\circ m_G^{(-1)}\circ Q_{n}\circ m_G\circ m_G(A)\right)=R\left(e^{-\pi\sqrt{m^{(-1)}_G\circ Q_{n}\circ m_G\circ m_G(A)}}\right)=
$$
$$
=R\left(e^{-\pi n\sqrt{m_G(A)}}\right)=\Omega_n\left(y(A)\right).
$$
But
\begin{equation}
m^{(-1)}_G\circ m_G^{(-1)}\circ Q_{n}\circ m_G\circ m_G(A)=m_G^{(-1)}\left(n^2\cdot m_G(A)\right).
=Q^{*}_{n}(A)
\end{equation}
\\
By this we lead to the following\\
\\
\textbf{Theorem 7.}
$$
y\left(Q^{*}_{n^2}(A)\right)=\Omega_n\left(y(A)\right),
$$
where $\Omega_n(A)$ is the $n$-th modular equation of the Rogers-Ramanujan continued fraction and $Q_{n^2}^{*}(A)$ is the $n$-th order modular equation of $m_G^{(-1)}(A)$ i.e $Q^{*}_{n^2}(A)=m_G^{(-1)}\left(n^2\cdot m_G\left(A\right)\right)$.\\
\\
\textbf{Theorem 8.}\\If we know $f_0=y$ and $y_i=f^{(-1)}_0$, then
$$
m_G(A)=b^{(-1)}\circ F_1^{(-1)}\circ f_0(A)\eqno{:(\nu1)}
$$
and for all $n$ positive real:
$$
m_G^{(-1)}(n^2)=f^{(-1)}_0\circ\Omega_{n}\circ f_0\circ m_G^{(-1)}(1)=f^{(-1)}_0\left(R\left(e^{-\pi n}\right)\right),\eqno{:(\nu 2)}
$$
where $n$ can take and positive real values as long as
$$
\Omega_{n}(x)=v\left(n^2\cdot v^{(-1)}(x)\right)\eqno{:(\nu 3)}
$$
and $v(x)=R\left(e^{-\pi\sqrt{x}}\right)$, i.e. $v(x)$ is the Rogers-Ramanujan continued fraction. The constant function $\Omega_n(x)$ is the $n$-th degree modular equation of the $v(x)$. Hence knowing $y(x)$ and $y^{(-1)}(x)=y_i(x)$ we know respectively $m_G(x)$ and $m_G^{(-1)}(x)$.\\
\\
\textbf{Examples.}\\
\textbf{1)} If $f(x)=x^2+2x$ then $f^{(-1)}(x)=-1+\sqrt{1+x}$ and
$$
m_G(x)=b^{(-1)}\circ F_1^{(-1)}\left(-1+\sqrt{1+x}\right).
$$
\\
\textbf{2)} If
$$
G(t)=\frac{t\sqrt[6]{t^{-5}-11-t^5}}{t+1},
$$
then $y(x)=-1+e^{x/5}$ and $y_i(x)=5\log(x+1)$. Hence exist constant $c_1$ such that
$$
m_G^{(-1)}(n^2)=5\log\left(1+\Omega_{n}(c_1)\right)\textrm{, }\forall n>0
$$
\\
\textbf{3)}
If
$$
G(x)=\frac{x\sqrt[6]{x^{-5}-11-x^5}}{5\sqrt{1-k \sin(x)^2}},
$$
then
$$
y(x)=E(x,k)\textrm{ and }y_i(x)=\textrm{am}(x,k),
$$
where $\textrm{am}$ is the Jacobi amplitude i.e the inverse of the incomplete elliptic integral of the first kind $E[x,k]$, with $k$ a parameter. Hence
$$
m_G^{(-1)}(4x)=Q^{*}_4\left(m_G^{(-1)}(x)\right)\textrm{, }Q^{*}_4(x)=E\left[\Omega_2\left(\textrm{am}(x,k)\right),k\right].
$$
If $k=1$
\begin{equation}
m_G^{(-1)}\left(n^2\right)
=\log\left[\sec\left(\Omega_n\left(t\right)\right)+\tan\left(\Omega_n\left(t\right)\right)\right],
\end{equation}
where
\begin{equation}
t=-\frac{\pi}{2}+2\arctan\left(e^{m_G^{(-1)}(1)}\right),
\end{equation}
for every $n\in \bf R^{*}_{+}\rm$, ($t$ is a constant). Also for $k=1/2$
\begin{equation}
m_G^{(-1)}\left(4\right)=E\left[\frac{1}{2} \left(-1-\sqrt{5}+\sqrt{2 \left(5+\sqrt{5}\right)}\right),\frac{1}{2}\right].
\end{equation}
\\
\textbf{4)} If $n\in \bf N\rm$ and $y(x)=BR(x)$ (the Bring Radical function), then
$$
m^{(-1)}_G(n^2)=\left(\Omega_n\circ F_1\circ b_1\right)^5+\Omega_n\circ F_1\circ b_1
$$
$$
b_1=3\sqrt[6]{2}\cdot{}_2F_{1}\left[\frac{1}{6},\frac{1}{3};\frac{7}{6};\frac{1}{2}\right]
$$
and $F_1$ defined from (26).
\section{Further transformations}
The function $k_i(A)$ is the inverse function of the elliptic singular modulus $k_A=k(A)$. We have
\begin{equation}
k_i(x)=\left(\frac{K\left(\sqrt{1-x^2}\right)}{K(x)}\right)^2\textrm{, }0<x<1.
\end{equation}
Also we define
\begin{equation}
Q_G(x):=m_{G}^{(-1)}\left(k_i(x)\right),
\end{equation}
then
\begin{equation}
m^{(-1)}_G(r)=Q_G(k_r).
\end{equation}
From (11) we have
$$
y\left(Q_G(k_r)\right)=R\left(e^{-\pi\sqrt{r}}\right).
$$
We are interested to find an expresion for $G$. From (10) we get
$$
5\int^{y\left(Q_G(k_r)\right)}_{0}\frac{G(t)}{t\sqrt[6]{t^{-5}-11-t^5}}dt=Q_G(k_r),
$$
or equivalently
\begin{equation}
5\int^{R(q)}_{0}\frac{G(t)dt}{t\sqrt[6]{t^{-5}-11-t^5}}dt=Q_G(k_r)
\end{equation}
and differentiating the last relation we get
$$
5\frac{G\left(R(q)\right)R'(q)}{R(q)\sqrt[6]{R(q)^{-5}-11-R(q)^{5}}}=Q'_G(k_r)\frac{dk}{dq}.
$$
Or, using (3),(5),(8) we arive to
\begin{equation}
Q'_G(k_r)=\frac{2^{1/3}}{\left(k_rk^{*}_r\right)^{2/3}}G\left(R(q)\right).
\end{equation}
But from the fact that Rogers-Ramanujan's continued fraction is algebraic function of elliptic singular moduli we have $R(q)=F(k_r)$. Hence
\begin{equation}
Q'_G(A)=\frac{2^{1/3}}{\left(A\sqrt{1-A^2}\right)^{2/3}}G\left(F(A)\right).
\end{equation}
Inverting $F$ we get\\
\begin{equation}
G(A)=2^{-1/3}\left(F_i(A)\sqrt{1-F_i(A)^2}\right)^{2/3}Q'_G\left(F_i(A)\right).
\end{equation}
The study of $F(A)$ has to reveal some interesting properties of the general function
\begin{equation}
y_s(A)=R\left(e^{-\pi\sqrt{k_i\left(s(A)\right)}}\right),
\end{equation}
where $s(A)$ is ''arbitrary'' function. Allong with $s(A)$, we attach the function $\sigma(A)$, which satisfies the condition $s'(A)=\sigma(s(A))$ and
\begin{equation}
m_G(A)=k_i\left(s(A)\right).
\end{equation}
It holds from the definition of $y_s$ and Theorem 5 and (22):
$$
\frac{1}{G_s\left(y_s(A)\right)}=\frac{1}{G\left(y(A)\right)}=h'_i(A)=\frac{d}{dA}\left(3\sqrt[3]{2s(A)}\cdot{}_2F_1\left[\frac{1}{3},\frac{1}{6};\frac{7}{6};s(A)^2\right]\right)=
$$
\begin{equation}
=\frac{2^{1/3}}{s(A)^{2/3}\left(1-s(A)^2\right)^{1/3}}s'(A).
\end{equation}
Hence
\begin{equation}
\frac{1}{G\left(R\left(e^{-\pi\sqrt{k_i(A)}}\right)\right)}=\frac{2^{1/3}}{A^{2/3}\left(1-A^2\right)^{1/3}}\sigma\left(A\right).
\end{equation}
Inverting the $F(x)$ function (see and Apendix) we get\\
\\
\textbf{Theorem 9.}
\begin{equation}
Q(A)=s^{(-1)}(A)
\end{equation}
and
\begin{equation}
G(A)=G_s(A)=\frac{2^{-1/3}\left(F_i(A)\sqrt{1-F_i(A)^2}\right)^{2/3}}{\sigma\left(F_i(A)\right)}.
\end{equation}
\\
\textbf{Theorem 10.}\\
Suppose $G_0(x)$ is that of (155),(65) below and $F(x)=R\left(e^{-\pi\sqrt{k_i(x)}}\right)$. Both functions are ''constant'' and algebraic. If
\begin{equation}
5\int^{y_s(A)}_{0}\frac{G_0(t)}{\sigma\left(F_i(t)\right)}\frac{dt}{t\sqrt[6]{t^{-5}-11-t^5}}=A,
\end{equation}
then $y_s(A)=y(A)$ is that of (58), with $s'(A)=\sigma(s(A))$ and
$$
G(t)=\frac{G_0(t)}{\sigma\left(F_i(t)\right)}.\eqno{(64.0)}
$$
\\
\textbf{Proof.}\\
Inverting $s(A)$ we get
$$
5\int^{R\left(e^{-\pi\sqrt{k_i(A)}}\right)}_{0}\frac{G_0(t)}{\sigma\left(F_i(t)\right)t\sqrt[6]{t^{-5}-11-t^5}}dt=s_i(A).
$$
Then differentiating (the $h$ is refering to $R\left(e^{-\pi\sqrt{k_i(A)}}\right)$ function)
$$
\frac{G_0\left(R\left(e^{-\pi\sqrt{k_i(A)}}\right)\right)}{\sigma(A)}h_i'(A)=s_i'(A).
$$
Using now Corollary 2 and (60),(61), we get
$$
\frac{G_0\left(R\left(e^{-\pi\sqrt{k_i(A)}}\right)\right)}{\sigma(A)}\frac{2^{1/3}}{A^{2/3}(1-A^2)^{1/3}}=s_i'(A).
$$
Equivalently
$$
\frac{2^{-1/3}A^{2/3}(1-A^2)^{1/3}}{\sigma(A)}\frac{2^{1/3}}{A^{2/3}(1-A^2)^{1/3}}=s_i'(A)
$$
and finaly
$$
\sigma(A)s_i'(A)=1, \eqno{(64.1)}
$$
which is true hence we get (64).\\
\\
\textbf{Note.}\\
For the function $G_0(x)$ holds
\begin{equation}
G_0\left(F(x)\right)=\frac{\left(x\sqrt{1-x^2}\right)^{2/3}}{\sqrt[3]{2}}
\end{equation}
and hence $G_0$ is algebraic (see also Appendix for $G_0$).\\
\\
\textbf{Theorem 11.}\\
We set $c(A)=3\sqrt[3]{2A}\cdot{}_2F_{1}\left[\frac{1}{6},\frac{1}{3};\frac{7}{6};A^2\right]$, then
\begin{equation}
Q_G^{(-1)}\left(h(A)\right)=c_i(A).
\end{equation}
Also
\begin{equation}
\phi(A)=\frac{\sigma\left(k(A)\right)}{k'(A)},
\end{equation}
where $\phi(x)$ is that of (14). Also
$$
m^{(-1)}_G(A)=\int^{k(A)}_{0}\frac{dt}{\sigma(t)}.\eqno{(67.1)}
$$
\\
\textbf{Proof.}\\
From Theorem 5 and (35) we have $h_i\left(m_G^{(-1)}(A)\right)=b_A$ or $h_i\left(Q(A)\right)=c(A)$. Inverting we get the first result. For the second result we have
$$
m'_G(A)=\phi(m_G(A)),
$$
or from (59)
$$
k'_i(s(A))s'(A)=\phi(m_G(A)),
$$
or
$$
k'_i(s(A))s'(A)=\phi(k_i(s(A))),
$$
or
$$
\frac{k'_i(A)}{s'_i(A)}=\phi(k_i(A)),
$$
or
$$
s'_i(A)=\frac{k'_i(A)}{\phi(k_i(A))},
$$
or
\begin{equation}
s_i(A)=\int^{k_i(A)}_{0}\frac{dt}{\phi(t)}=\int^{A}_{0}\frac{dt}{\sigma(t)}.
\end{equation}
Differentiating the last relation and inverting $k_A=k(A)$ we get the result.\\
\\
\textbf{Note.}\\
One can see imediately that
\begin{equation}
h\left(3\sqrt[3]{2A}\cdot{}_2F_1\left[\frac{1}{3},\frac{1}{6};\frac{7}{6};A^2\right]\right)=s_i(A),
\end{equation}
which means that $h$ and $s_i$ ''generalized'' functions are esentialy the same. We are going to describe this kind of relation between generalized functions.\\
\\
\textbf{Definition 3.}\\
We say that a function $f$ is generalized, if it is not ''constant'' function.\\
\\
\textbf{Definition 4.}\\
We say that two invertible generalized functions $f,g$ are equivalent $f\equiv g$, if exist constant functions $\alpha_1(x),\beta_1(x),\gamma_1(x)$ such that
\begin{equation}
f(x)=\frac{\alpha_1\left(g\left(\beta_1(x)\right)\right)}{\gamma_1(x)}.
\end{equation}
\\
\textbf{Proposition 2.}\\
The notation $\equiv$ is an equivalence relation i.e. it has the following properties\\
i. \textbf{Reflection}: $f\equiv f$\\
ii. \textbf{Symmetry}: If $f\equiv g$ then $g\equiv f$\\
iii. \textbf{Transition}: If $f\equiv g$ and $g\equiv h$, then $f\equiv h$.\\
\\
\textbf{Proposition 3.}\\
We have the following equivalences of functions
\begin{equation}
y\equiv m_G\equiv h_i\equiv Q_G^{(-1)} \equiv s
\end{equation}
\begin{equation}
y_i\equiv m_G^{(-1)}\equiv h\equiv Q_G\equiv s_i
\end{equation}
\begin{equation}
y'_i\equiv G\equiv h'\equiv\phi\equiv m_G^{(-1)}{'}\equiv Q'_G\equiv \sigma\equiv s'_i
\end{equation}
\\
\textbf{Theorem 12.}
\begin{equation}
y'(A)=5^{-1}\sqrt[3]{2}\left(s(A)\sqrt{1-s(A)^2}\right)^{-2/3}s'(A)y(A)\sqrt[6]{y(A)^{-5}-11-y(A)^5}.
\end{equation}
\\
Set now
\begin{equation}
U(x):=256\frac{\left(1-x^2+x^4\right)^3}{x^4\left(1-x^2\right)^2}
\end{equation}
and
\begin{equation}
U_2(x)=\frac{1-\sqrt{1-U(x)^2}}{1+\sqrt{1-U(x)^2}}=16\frac{(1+14 x^2+x^4)^3}{x^2(1-x^2)^4},
\end{equation}
then we have the next\\
\\
\textbf{Theorem 13.}\\For an arbitrary $G(x)$ the value of $\sigma(x)$ is given from
\begin{equation}
\sigma(x)=\frac{\left(x\sqrt{1-x^2}\right)^{2/3}}{\sqrt[3]{2}\cdot G\left(F(x)\right)}
\end{equation}
and the value of $s(x)$ from
\begin{equation}
\int^{s(x)}_{0}\frac{dt}{\sigma(t)}=x.
\end{equation}
Then the value of $y(x)$ at $x=A$ can evaluated from
\begin{equation}
U_2\left(s\left(A\right)\right)^{1/3}Y^{5/3}=Y^2+250Y+3125,
\end{equation}
where
\begin{equation}
Y=y\left(A\right)^{-5}-11-y\left(A\right)^5.
\end{equation}
\\
\textbf{Notes.}\\
\textbf{I.} Assume that $G(A)=G^{*}\left(F_i(A)\right)$, where $G^{*}(A)$ known. Then from (77) we have
\begin{equation}
\sigma(A)=2^{-1/3}\frac{\left(A\sqrt{1-A^2}\right)^{2/3}}{G^{*}(A)}.
\end{equation}
Hence
\begin{equation}
\int^{A}_{0}\frac{dt}{\sigma(t)}=\sqrt[3]{2}\int^{A}_{0}\frac{G^{*}(t)dt}{(t\sqrt{1-t^2})^{2/3}}=s_i(A).
\end{equation}
\\
Let $G(t)=\sqrt[3]{1-F_i(t)^2}$, then $s(A)=\frac{A^3}{54}$.\\
\\
\textbf{i.} If $A_0=3\sqrt[3]{6-4\sqrt{2}}$ we have
$$
j^{1/3}=U_2\left(s\left(A_0\right)\right)^{1/3}=12.
$$
Then equation (79) becomes
$$
12Y^{5/3}=Y^2+250Y+3125
$$
and
$$
Y=125(2+\sqrt{5}).
$$
Hence
$$
y(A_0)=y\left(\sqrt[3]{6-4\sqrt{2}}\right)=\frac{2}{\sqrt{2 \left(5+\sqrt{5}\right)}+\sqrt{5}+1}.
$$
Finaly the function $y(A)$ which is a solution of
$$
5\int^{y(A)}_{0}\frac{\sqrt[3]{1-F_i(t)^2}}{t\sqrt[6]{t^{-5}-11-t^5}}dt=A,
$$
is
$$
y(A)=R\left(e^{-\pi\sqrt{k_i\left(A^3/54\right)}}\right)
$$
and can be determined in closed form up to a 6th degree poynomial equation (that of (79)).\\
\textbf{ii.} If
$$
A_0=3 \sqrt[3]{66+48 \sqrt{2}-8 \sqrt{140+99 \sqrt{2}}},
$$
then
$$
j^{1/3}=U_2\left(s(A_0)\right)^{1/3}=66
$$
and equation (79) becomes
$$
66Y^{5/3}=Y^2+250 Y+3125,
$$
with solution
$$
Y=\frac{125}{2} \left(1147+513\sqrt{5}+\sqrt{2630810+1176534 \sqrt{5}}\right).
$$
Hence
$$
y\left(3 \sqrt[3]{66+48 \sqrt{2}-8 \sqrt{140+99 \sqrt{2}}}\right)=\sqrt[5]{\frac{-11-Y+\sqrt{125+22Y+Y^2}}{2}}.
$$
\\
\textbf{II.} If for a function $y(A)$ we know $G$, then from (77) we have
$$
\int^{A}_{0}\frac{dt}{\sigma(t)}=s_i(A).\eqno{:(eq1)}
$$
Knowing $s(A)$ we solve
$$
\sqrt[3]{16}\frac{1+14 s(A)^2+s(A)^4}{s(A)^{2/3} \left(1-s(A)^2\right)^{4/3}}\cdot Y^{5/3}=Y^2+250Y+3125\eqno{:(eq2)}
$$
and we get that
$$
y(A)=y_s(A)=\sqrt[5]{\frac{-11-Y+\sqrt{125+22Y+Y^2}}{2}}.\eqno{:(eq3)}
$$
Hence we find the closed form of $y(A)$ in (64) (and hence to the problem (10)) up to the inverting of the integral of $(eq1)$ and solving the sextic equation $(eq2)$.\\
\\
\textbf{i)} Suppose that $\sigma(A)=A+1$, then $s(A)=e^A-1$. This case coresponds to
$$
5\int^{y(x)}_{0}\frac{G_0(t)}{F_i(t)+1}\frac{dt}{t\sqrt[6]{t^{-5}-11-t^5}}=x
$$
and is solvable up to the sextic equation $(eq2)$. Also
$$
y(A)=R\left(e^{-\pi\sqrt{k_i(e^A-1)}}\right).
$$
\\
\textbf{ii)} Another example is with $\sigma(A)=1$. This leads to $s(A)=A$ and coresponds to
$$
5\int^{y(x)}_{0}\frac{G_0(t)}{t\sqrt[6]{t^{-5}-11-t^5}}dt=x.
$$
The first derivative according to Theorem 12 is
$$
y'(A)=5^{-1}\sqrt[3]{2}\left(A\sqrt{1-A^2}\right)^{-2/3}y(A)\sqrt[6]{y(A)^{-5}-11-y(A)^5}
$$
and
$$
y(A)=R\left(e^{-\pi\sqrt{k_i(A)}}\right).
$$
For more details see section Applications.\\
\\
\textbf{iii)} For $\sigma(A)=1/A$ we get $s(A)=\sqrt{2A}$, hence
$$
5\int^{y(x)}_{0}\frac{G_0(t)F_i(t)dt}{t\sqrt[6]{t^{-5}-11-t^5}}=x
$$
and
$$
y'(A)=\frac{5^{-1}}{\sqrt{2}\left(1-2A\right)^{1/3}A^{5/6}}y(A)\sqrt[6]{y(A)^{-5}-11-y(A)^5},
$$
with
$$
y(A)=R\left(e^{-\pi\sqrt{k_i\left(\sqrt{2A}\right)}}\right).
$$
\\
\textbf{iv)} If $\sigma(A)=\sqrt{1-A^2}\sqrt{1-k A^2}$, then $s(A)=\textrm{sn}(A,k)$ and the solution $y(x)$ of
$$
5\int^{y(x)}_{0}\frac{G_0(t)}{\sqrt{1-F_i(t)^2}\sqrt{1-k F_i(t)^2}}\frac{dt}{t\sqrt[6]{t^{-5}-11-t^5}}=x,
$$
is given from $(eq2)$ and $(eq3)$. Esentialy the function $y(x)=y_k(x)=y(x,k)$ is algebraic function of $s(A)=\textrm{sn}(A,k)$ and hence double periodic elliptic function. Also
$$
\frac{y'_k(A)^2}{1-kF_i\left(y_k(A)\right)^2}=\frac{y'_l(A)^2}{1-lF_i\left(y_l(A)\right)^2}=C(A)
$$
and
$$
y'_k(A)=5^{-1}\sqrt[3]{2}\frac{\textrm{cn}(A,k)^{1/3}\textrm{dn}(A,k)}{\textrm{sn}(A,k)^{2/3}}y_k(A)\sqrt[6]{y_k(A)^{-5}-11-y_k(A)^5},
$$
with
$$
y(A)=y_k(A)=R\left(e^{-\pi\sqrt{k_i\left(sn(A,k)\right)}}\right).
$$
\textbf{III.} Another notation but not so detailed can found using (58),(77),(78) and relation $R(q)=F(k_r)$. We have
\begin{equation}
y_i(A)=\sqrt[3]{2}\int^{F_i(A)}_{0}\frac{G(F(t))}{\left(t\sqrt{1-t^2}\right)^{2/3}}dt.
\end{equation}
As application we set $B(x,a,b)=\int^{x}_{0}t^{a-1}(1-t)^{b-1}dt$ to be the incoplete beta function. Then if
$$
G\left(F(A)\right)=\frac{1}{\sqrt[3]{2}}\left(A\sqrt{1-A^2}\right)^{2/3}\left(A-A^2\right)^{a-1},
$$
we have
\begin{equation}
y\left(B(x,a,a)\right)=F(x)
\end{equation}
and (see [13]) the solution of
\begin{equation}
\frac{B\left(1-\beta_r,a,a\right)}{B\left(\beta_r,a,a\right)}=r\textrm{, }r>0\textrm{, }0<\beta_r<1,
\end{equation}
is equivalent to
\begin{equation}
B\left(\beta_r ,a,a\right)=\frac{\Gamma(a)^2}{\Gamma(2a)(r+1)}.
\end{equation}
Hence we get
\begin{equation}
y\left(\frac{\Gamma(a)^2}{\Gamma(2a)(r+1)}\right)=F\left(\beta_r\right).
\end{equation}
\\
\textbf{IV.}
Also from
$$
\int^{A}_{0}\frac{dt}{\sigma(t)}=s_i(A),
$$
we have
$$
\int^{k_r}_{0}\frac{dt}{\sigma(t)}=s_i\left(k_r\right).
$$
Hence from (59): $s_i(A)=m_G^{(-1)}(k_i(A))$ or equivalently $s_i(k_r)=m_G^{(-1)}(r)$ and we get
\begin{equation}
\int^{k_r}_0\frac{dt}{\sigma(t)}=m_G^{(-1)}(r).
\end{equation}
\textbf{V.} If $G(F(A))$ is polynomial
\begin{equation}
G\left(F(x)\right)=\sum^{M}_{n=0}c_nx^n,
\end{equation}
then using
\begin{equation}
\int^{A}_{0}\frac{t^n}{(t\sqrt{1-t^2})^{2/3}}dt=3\frac{A^{n+1/3}}{3n+1}{}_2F_{1}\left[\frac{1}{3},\frac{3n+1}{6};\frac{3n+7}{6};A^2\right],
\end{equation}
we get
\begin{equation}
y\left(3\sqrt[3]{2}\sum^{M}_{n=0}c_n\frac{A^{n+1/3}}{3n+1}\cdot {}_2F_{1}\left[\frac{1}{3},\frac{3n+1}{6};\frac{3n+7}{6};A^2\right]\right)=F(A),
\end{equation}
or if someone preferes
\begin{equation}
3\sqrt[3]{2}\sum^{M}_{n=0}c_n\frac{A^{n+1/3}}{3n+1}\cdot {}_2F_{1}\left[\frac{1}{3},\frac{3n+1}{6};\frac{3n+7}{6};A^2\right]+c=y_i\left(F(A)\right).
\end{equation}
and concequently:\\
\\
\textbf{Theorem 14.}\\
If
\begin{equation}
\sigma(A)=\frac{(A\sqrt{1-A^2})^{2/3}}{\sqrt[3]{2}\sum^{M}_{m=0}c_mA^m},
\end{equation}
then
\begin{equation}
y_i(A)=3\sqrt[3]{2}\sum^{M}_{n=0}c_n\frac{F_i(A)^{n+1/3}}{3n+1}\cdot {}_2F_{1}\left[\frac{1}{3},\frac{3n+1}{6};\frac{3n+7}{6};F_i(A)^2\right]+c.
\end{equation}
\\
Assume function $G$ as in (89), then from the fact that $F$ is algebraic, there exists coefficients $a_{n,l}$ such that
\begin{equation}
\sum^{N}_{n,l=0}a_{kl}G(x)^nx^l=0.
\end{equation}
But then also
$$
\sum^{N}_{n,l=0}a_{nl}G\left(F(x)\right)^nF(x)^l=0
$$
and
\begin{equation}
\sum^{N}_{n,l=0}a_{nl}\left(\sum^{M}_{m=0}c_mx^m\right)^nF(x)^l=0.
\end{equation}
Finaly
\begin{equation}
\sum^{N}_{n,l=0}a_{nl}\left(\sum^{M}_{m=0}c_m(k_r)^m\right)^nR(q)^l=0.
\end{equation}
Hence knowing $c_m$ we find $a_{nl}$ from relation (97) and equation
$$
x^2 \left(1-x^2\right)^4 \left(y^{20}-228 y^{15}+494 y^{10}+228 y^5+1\right)^3+
$$
\begin{equation}
+16 \left(x^4+14 x^2+1\right)^3 y^5 \left(y^{10}+11 y^5-1\right)^5=0,
\end{equation}
(which is Klein formula for the icosahedron $y=F(x)$). That is the $a_{nl}$ of (95) can be found from that of $c_m$, by equating coefficients of the identity:
$$
\sum^{N}_{n,l=0}a_{nl}\left(\sum^{M}_{m=0}c_mx^m\right)^ny^l=x^2 \left(1-x^2\right)^4 \left(y^{20}-228 y^{15}+494 y^{10}+228 y^5+1\right)^3+
$$
\begin{equation}
+16 \left(x^4+14 x^2+1\right)^3 y^5 \left(y^{10}+11 y^5-1\right)^5.
\end{equation}
If hapens $G(F(x))=\psi(x)$ be more complicated, for example algebraic, then we solve the equation $G=\psi(x)$ with respect to $x$, $x=\psi^{(-1)}(G)=\psi_i(G)$ and
\begin{equation}
\sum^{60}_{n,l=0}A_{nl}\left(\psi_i(G(x))\right)^nx^l=0,
\end{equation}
where the $A_{nl}$ are that of Klein's equation
$$
\sum^{60}_{n,l=0}A_{nl}x^ny^l=x^2 \left(1-x^2\right)^4 \left(y^{20}-228 y^{15}+494 y^{10}+228 y^5+1\right)^3+
$$
\begin{equation}
+16 \left(x^4+14 x^2+1\right)^3 y^5 \left(y^{10}+11 y^5-1\right)^5.
\end{equation}
Hence in general:\\
\\
\textbf{Theorem 15.}\\
If $G(F(x))=\psi(x)$ is known resonable function (polynomial algebraic etc...), then the minimal equation for $G$ is (100), with coefficients $A_{nl}$ that of (101).\\
\\
\textbf{Example.}\\
For $G(F(x))=2^{-1/3}(x^2-x^4)^{1/3}$, we get $Q'_G(x)=1$, hence $s_i(x)=x$ and $y(x)=F(x)$, where $G(x)=G_0(x)$ is solution of
\begin{equation}
\sum^{60}_{n,l=0}A_{nl}\left(\sqrt{\frac{1-\sqrt{1-8G(x)^3}}{2}}\right)^nx^l=0.
\end{equation}
The coefficients $A_{nl}$ are that of (101).
\section{Solution of General Equations and Inversion}
Consider a polynomial $w(x)$ and the equation
\begin{equation}
w(x)=\lambda
\end{equation}
Set $G(x)=5^{-1}xw'(x)\sqrt[6]{x^{-5}-11-x^5}$, then
\begin{equation}
m_G^{(-1)}(r)=w\left(R\left(e^{-\pi\sqrt{r}}\right)\right)
\end{equation}
and the solution of (103) is $x=R\left(e^{-\pi\sqrt{r}}\right)$, where $r=m_{G}(\lambda)$.\\
Taking the derivatives with respect to $A$ in $w(f^{(-1)}(A))=A$, we lead to
\begin{equation}
w{'}\left(f^{(-1)}\left(A\right)\right)f^{(-1)}{'}(A)=1,
\end{equation}
or
\begin{equation}
f^{(-1)}{'}\left(A\right)=\frac{1}{w{'}\left(f^{(-1)}\left(A\right)\right)}.
\end{equation}
Hence if we set $W(A)=\frac{1}{w{'}\left(A\right)}$,
then clearly
\begin{equation}
f^{(-1)}{'}\left(A\right)=W\left(f^{(-1)}\left(A\right)\right).
\end{equation}
Setting where $f^{(-1)}(A)=y(A)$, with $y(A)$ that of (9) and (10) and taking the derivatives with respect to $A$ we have
$$
5\frac{G\left(f^{(-1)}(A)\right)f^{(-1)}{'}(A)}{f^{(-1)}(A)\sqrt[6]{\left(f^{(-1)}(A)\right)^{-5}-11-\left(f^{(-1)}(A)\right)^{5}}}=
$$
\begin{equation}
=5\frac{G\left(f^{(-1)}(A)\right)W\left(f^{(-1)}(A)\right)}{f^{(-1)}(A)\sqrt[6]{\left(f^{(-1)}(A)\right)^{-5}-11-\left(f^{(-1)}(A)\right)^{5}}}=1.
\end{equation}
After inverting $f^{(-1)}(x)$ we have
$$
G(x)W(x)=5^{-1}x\sqrt[6]{x^{-5}-11-x^5},
$$
or equivalently
\begin{equation}
G(x)=5^{-1}xw'(x)\sqrt[6]{x^{-5}-11-x^5}
\end{equation}
and it will be
\begin{equation}
f^{(-1)}(A)=R\left(e^{-\pi\sqrt{m_G(A)}}\right).
\end{equation}
From the above we can state the following theorem\\
\\
\textbf{Theorem 16.}\\
The equation $w(y(x))=x$ have solution
\begin{equation}
y(x)=R\left(e^{-\pi\sqrt{m_G(x)}}\right),
\end{equation}
with $G$ that of relation (109) and $m_G(A)$ as defined in (9).\\
\\
\textbf{Example.}\\
Let $\rho_1=\frac{1}{2} \left(11-5 \sqrt{5}\right)$, $\rho_2=\frac{1}{2} \left(11+5 \sqrt{5}\right)$ and
consider the equation
\begin{equation}
6\frac{x^{a+\frac{11}{6}}}{6 a+11} F_{Ap}\left(\frac{6 a+11}{30},\frac{1}{6},\frac{1}{6},\frac{6 a+41}{30},\rho_1 x^5,\rho_2 x^5\right)=A,
\end{equation}
where $a$ is parameter. The function $G$ is $G(x)=\frac{x^{a+1}}{5}$, and
\begin{equation}
x=R\left(e^{-\pi\sqrt{m_G(A)}}\right),
\end{equation}
where the $m_G(x)$ is given from
\begin{equation}
x=\frac{\pi}{5}\int^{\infty}_{\sqrt{m_G(x)}}\eta\left(it/2\right)^4R\left(e^{-\pi t}\right)^{a+1}dt.
\end{equation}
Hence if
$$
g_a(x)=6 \frac{x^{a+\frac{11}{6}}}{6 a+11} F_{Ap}\left(\frac{6 a+11}{30},\frac{1}{6},\frac{1}{6},\frac{6 a+41}{30},\rho_1 x^5,\rho_2 x^5\right),
$$
then
\begin{equation}
g_a\left(R\left(e^{-\pi A}\right)\right)=\frac{\pi}{5}\int^{\infty}_{A}\eta\left(it/2\right)^4R\left(e^{-\pi t}\right)^{a+1}dt.
\end{equation}
\\
\textbf{Theorem 17.}\\
Given the equation $P(x)=a:(\epsilon)$, the inverse of $P(x)$ is $y(x)$, then $(\epsilon)$ is equivalent to
\begin{equation}
\int^{F_i(x)}_{0}\frac{dt}{\sigma(t)}=a.
\end{equation}
\\
\textbf{Proof.}\\
Easy\\
\\
\textbf{Example.}\\
If $\sigma(x)=x+1$, then
$$
P(x)=5\int^{x}_{0}\frac{G_0(t)}{F_i(t)+1}\frac{dt}{t\sqrt[6]{t^{-5}-11-t^5}}
$$
and the equation $P(x)=a$ have solution $x$ such that
$$
\int^{F_i(x)}_{0}\frac{dt}{t+1}=a,
$$
or equivalently $x=F\left(e^a-1\right)$.\\
\\
In general we have the next formula\\
\\
If $|x|<1$ then
\begin{equation}
\int^{x}_{0}\frac{f(-q)^5R(q)g'\left(R(q)\right)}{f(-q^5)}dq=5g\left(R\left(x\right)\right),
\end{equation}
which is consequence of the next identity
\begin{equation}
\frac{R'(q)}{R(q)}=\frac{f(-q)^5}{5qf(-q^5)}.
\end{equation}
Relation (118) was given by Ramanujan (see [3]). We know that
$$
\int^{q_1}_{q_2}f(-q)^4q^{-5/6}R(q)^{5\nu}dq=-\pi\int^{\sqrt{r_1}}_{\sqrt{r_2}}\eta(it/2)^4R(e^{-\pi t})^{5\nu}dt
$$
and (see [13]):
$$
C(\nu):=\int^{1}_{0}f(-q)^4q^{-5/6}R(q)^{5\nu}dq=
$$
\begin{equation}
=\Gamma\left(\frac{5}{6}\right)\left(\frac{11+5 \sqrt{5}}{2}\right)^{-\frac{1}{6}-\nu} \frac{\Gamma\left(\frac{1}{6}+\nu\right)}{\Gamma(1+\nu)} {}_2F_1\left(\frac{1}{6},\frac{1}{6}+\nu;1+\nu;\frac{11-5 \sqrt{5}}{11+5\sqrt{5}}\right),
\end{equation}
where $\nu\geq0$. Hence\\
\\
\textbf{Theorem 18.}\\
If $G(x)$ is a polynomial (or analytic function when $n\rightarrow+\infty$) of the form
\begin{equation}
G(x)=\sum^{n}_{m=0}a_nx^{p_{m}},
\end{equation}
with $p_m-$positive reals, ($\lim p_m=+\infty$) and $R\left(1\right)=\frac{\sqrt{5}-1}{2}$, then\\
\textbf{i)}
\begin{equation}
m_G^{(-1)}(0)=\pi\int^{\infty}_{0}\eta(it/2)^4G\left(R(e^{-\pi t})\right)dt=\sum^{n}_{m=0}a_m C\left(5^{-1}p_m\right).
\end{equation}
\textbf{ii)} If $y$ is a smooth function and $G$ is of the form (120) then the equation
\begin{equation}
y(x)=\frac{\sqrt{5}-1}{2},
\end{equation}
have a solution
\begin{equation}
x=x_0=\sum^{n}_{m=0}a_mC(5^{-1}p_m).
\end{equation}
\\
\textbf{Example.}\\
Let $G(x)=e^{-x}-1$. Then $a_m=\frac{(-1)^m}{m!}$, $m=1,2,\ldots$ and
\begin{equation}
5\int^{y(x)}_{0}\frac{e^{-t}-1}{t\sqrt[6]{t^{-5}-11-t^5}}dt=x
\end{equation}
and the solution of $y(x)=\frac{\sqrt{5}-1}{2}$ is
\begin{equation}
x=\sum^{\infty}_{m=1}\frac{(-1)^m}{m!}C\left(\frac{m}{5}\right).
\end{equation}
Note here that if $G$ has finite expansion (120) the result (123) becomes more meaningful since the evaluation of the root of (122) by hypergeometric functions is beter than the integral:
\begin{equation}
5\int^{\frac{\sqrt{5}-1}{2}}_{0}\frac{G(t)}{t\sqrt[6]{t^{-5}-11-t^5}}dt.
\end{equation}
\\
Assume now
$$
G^{*}(t):=G(t\alpha)\sqrt[6]{\frac{t^{-5}-11-t^5}{(\alpha t)^{-5}-11-(\alpha t)^5}}.\eqno{(126.1)}
$$
If $y^{*}(A)$ coresponds to $G^{*}(t)$, one can easily see that
$$
y^{*}(A)=\frac{y(A)}{\alpha}.\eqno{(126.2)}
$$
Hence we have the next theorem which is generalization of Theorem 18:\\
\\
\textbf{Theorem 18.1}\\
Assume the function $G(t)$ is given near the origin by (under certain converging conditions):
$$
G(t)=\sum^{\infty}_{m=0}a_mt^{p_m},\eqno{(126.3)}
$$
where $p_m$ is any increasing sequence of positive real numbers with $\lim p_m=+\infty$. Then the function $y(A)$ defined as
$$
5\int^{y(A)}_{0}\frac{G(t)}{t\sqrt[6]{t^{-5}-11-t^5}}dt=A,\eqno{(126.4)}
$$
have the following property: Every equation of the form
$$
y(x)=\alpha \frac{\sqrt{5}-1}{2},\eqno{(126.5)}
$$
have solution $x$ such that
$$
x=\sum^{\infty}_{m=0}a_m^{*}(\alpha)C\left(5^{-1}p^{*}_m\right),\eqno{(126.6)}
$$
where $a^{*}_m(\alpha)$ is such that
$$
G(x\alpha)\sqrt[6]{\frac{x^{-5}-11-x^5}{(\alpha x)^{-5}-11-(\alpha x)^5}}=\sum^{\infty}_{m=0}a^{*}_{m}(\alpha)x^{p^{*}_m}.\eqno{(126.7)}
$$
Hence if we set $\xi^{-1}=\frac{\sqrt{5}-1}{2}$, then the inverse of $y(A)$ is
$$
y_i(A)=\sum^{\infty}_{m=0}a^{*}_m(\xi A)C\left(5^{-1}p^{*}_m\right),\eqno{(126.8)}
$$
provited the convergence of (126.1),(126.3),(126.4),(126.6),(126.7),(126.8).\\
\\
Set
$$
G=G_1(t)=\frac{t\sqrt[6]{t^{-5}-11-t^5}}{5\sqrt{1-t^2}\sqrt{1-k^2t^2}}.
$$
Then
\begin{equation}
5\int^{y}_{0}\frac{G_1(t)}{t\sqrt[6]{t^{-5}-11-t^5}}dt=\int^{y}_{0}\frac{1}{\sqrt{1-t^2}\sqrt{1-k^2 t^2}}dt=A.
\end{equation}
Hence (see [4]):
$$
y=\textrm{sn}(A,k)=R\left(e^{-\pi\sqrt{m_G}}\right)\textrm{ and }A=h\left(b_{m_G}\right),
$$
where
$$
b_{m_G}=3\sqrt[3]{2k_{m_G}}\cdot{}_2F_{1}\left[\frac{1}{6},\frac{1}{3};\frac{7}{6};k_{m_G}^2\right]
$$
and
$$
h^{(-1)}(x)=\int^{x}_{0}\frac{du}{G_1\left(\textrm{sn}\left(u,k_r\right)\right)}=5\int^{x}_{0}\frac{\textrm{dn}(u)\textrm{cn}(u)}{\textrm{sn}(u)\sqrt[6]{\textrm{sn}(u)^{-5}-11-\textrm{sn}(u)^5}}du.
$$
Hence if $y(A)$ is defined by (9),(10) and $\textrm{sn}(u)=\textrm{sn}(u,k)$, $\textrm{dn}(u)=\textrm{dn}(u,k)$, then\\
\\
\textbf{Theorem 19.}
$$
5\int^{E\left[\arcsin(y(A)),k\right]}_{0}\frac{\textrm{dn}(u)\textrm{cn}(u)}{\textrm{sn}(u)\sqrt[6]{\textrm{sn}(u)^{-5}-11-\textrm{sn}(u)^5}}du=
$$
\begin{equation}
=3\sqrt[3]{2k_{m_G}}\cdot{}_2F_{1}\left[\frac{1}{6},\frac{1}{3};\frac{7}{6};k_{m_G}^2\right],
\end{equation}
where $k$ is independent parameter, $m_G=m_G(A)$ and $E$ denotes the incomplete elliptic integral of the first kind, $y$ is the function defined in (9),(10),(11).\\
\\
Inverting the above integral we get a formula for the Rogers-Ramanujan continued fraction:\\
Set
\begin{equation}
5\int^{H_o(x)}_{0}\frac{\textrm{dn}(u)\textrm{cn}(u)}{\textrm{sn}(u)\sqrt[6]{\textrm{sn}(u)^{-5}-11-\textrm{sn}(u)^5}}du=x,
\end{equation}
then
\begin{equation}
R(q)=\textrm{sn}\left(H_o(b_r),k\right).
\end{equation}
For the function $\textrm{sn}$ we have $G(t)=G_1(t)$ and if the equation
$$
\textrm{sn}(x,k)=a,
$$
have $m_1$ such that $R\left(e^{-\pi\sqrt{m_1}}\right)=a$.
The solution is $x=x_1$:
$$
x_1=E\left[\arcsin(a),k\right]=\pi\int^{+\infty}_{\sqrt{m_1}}\eta\left(it/2\right)^4G_1\left(R\left(e^{-\pi t}\right)\right)dt.
$$
Taking derivatives in (129) we get
\begin{equation}
H_o{'}\left(H_o^{(-1)}(x)\right)=\frac{1}{H_o^{(-1)}{'}(x)}=\frac{\textrm{sn}(x)\sqrt[6]{\textrm{sn}(x)^{-5}-11-\textrm{sn}(x)^5}}{5\textrm{cn}(x)\textrm{dn}(x)}.
\end{equation}
Hence
\begin{equation}
H_o^{(-1)}{'}\left(E\left[\arcsin\left(x\right),k^2\right]\right)=\frac{5\sqrt{1-k^2x^2}\sqrt{1-x^2}}{x\sqrt[6]{x^{-5}-11-x^5}}.
\end{equation}
According to (129) and (130), the function $H_o=H_o(x,m)$, takes special values
$$
H_o(b_r)=H_o(b_r,m)=E(\arcsin(R(q)),m),
$$
where $q=e^{-\pi\sqrt{r}}$, $r>0$ for all $0<m<1$.\\
By this way $H_o$ can evaluated with known functions
$$
H_o(A,k)=E\left(\arcsin\left(R\left(e^{-\pi\sqrt{m(A)}}\right)\right),k\right),\eqno{(132.1)}
$$
with $m(A)$ that of (23),(24).
\section{More integrals}
Let $F_1$ be the function introduced in the above sections i.e. $F_1(x)=R\left(e^{-\pi\sqrt{m(x)}}\right)$, then $F_1$ is such that
\begin{equation}
F'_1(x)=5^{-1}F_1\left(x\right)\sqrt[6]{\left(F_1(x)\right)^{-5}-11-\left(F_1(x)\right)^5}.
\end{equation}
Also $F_1^{(-1)}$ is a specific Appell function
\begin{equation}
F_{Ap}[a,b_1,b_2,c,x,y]:=\sum^{\infty}_{m=0}\sum^{\infty}_{n=0}\frac{(a)_{m+n}}{(c)_{m+n}}\frac{(b_1)_m (b_2)_{n}}{m! n!}x^m y^n .
\end{equation}
More precicely
\begin{equation}
F_1^{(-1)}(x)=6x^{5/6}F_{Ap}\left[\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{7}{6},\frac{-2x^5}{11+5\sqrt{5}},\frac{-2x^5}{11-5\sqrt{5}}\right].
\end{equation}
Set also $t=t(w)$ such that
\begin{equation}
t(w)=F_1\left[(-1)^{m+1}a^{m-1}D^{-m+1/2}B\left(\frac{-b+\sqrt{D}-2aw}{2\sqrt{D}},1-m,1-m\right)\right],
\end{equation}
where $B(x,a,b)=\int^{x}_{0}t^{a-1}(1-t)^{b-1}dt$, $D=b^2-4ac$.\\
Also let
\begin{equation}
U(a,b,c;m;x):=(-1)^{m+1}a^{m-1}D^{-m+1/2}B\left(\frac{-b+\sqrt{D}-2ax}{2\sqrt{D}},1-m,1-m\right).
\end{equation}
Then
$$
\frac{d}{dx}U(a,b,c;m;x)=\frac{1}{(ax^2+bx+c)^m}\Leftrightarrow
$$
\begin{equation} \int^{B}_{A}\frac{dt}{(at^2+bt+c)^m}=U(a,b,c;m;B)-U(a,b,c;m;A).
\end{equation}
Hence we get
$$
5\int^{t_1}_{t_0}\frac{G(t)dt}{t\sqrt[6]{t^{-5}-11-t^5}}=
$$
\begin{equation}
=\int^{w_1}_{w_0} \frac{G\left(F_1\left[(-1)^{m+1}a^{m-1}D^{-m+1/2}B\left(\frac{-b+\sqrt{D}-2aw}{2\sqrt{D}},1-m,1-m\right)\right]\right)}{(aw^2+bw+c)^m}dw ,
\end{equation}
where $w_1,w_0,t_0,t_1$ are such that
\begin{equation}
U_i\left(a,b,c;m;F_1^{(-1)}\left(t_1\right)\right)=w_1\textrm{ and }U_i\left(a,b,c;m;F_1^{(-1)}\left(t_0\right)\right)=w_0.
\end{equation}
Suppose that we wish to evaluate the integral
\begin{equation}
I:=\int^{w_1}_{w_0}\frac{f(w)}{\sqrt{aw^2+bw+c}}dw.
\end{equation}
Then easily from (139) with $m=1/2$
$$
I=5\int^{F_1\left(U\left(w_1\right)\right)}_{F_1\left(U\left(w_0\right)\right)}\frac{f\left(U_i\left(F_1^{(-1)}(t)\right)\right)}{t\sqrt[6]{t^{-5}-11-t^5}}dt=
$$
\begin{equation}
=y_i\left(F_1\left(U\left(w_1\right)\right)\right)-y_i\left(F_1\left(U\left(w_0\right)\right)\right)=h\left(U(w_1)\right)-h\left(U(w_0)\right),
\end{equation}
where $y(x)=R\left(e^{-\pi\sqrt{m_G(x)}}\right)$ and
\begin{equation}
G(x)=f\left(U_i\left(F_1^{(-1)}(x)\right)\right)
\end{equation}
and
\begin{equation}
U_i(x):=\frac{1}{2a} \left(\sqrt{b^2-4 a c}\cdot \sinh \left(\sqrt{a} x\right)-b\right).
\end{equation}
Until now we have evaluated $G(x)$. From Theorem 4 we have
$$
h'\left(5\int^{x}_{0}\frac{dt}{t\sqrt[6]{t^{-5}-11-t^5}}\right)5\frac{1}{\sqrt[6]{x^{-5}-11-x^5}}=5\frac{G(x)}{x\sqrt[6]{x^{-5}-11-x^5}}.
$$
Hence
\begin{equation}
h'(x)=G\left(F_1(x)\right)=f\left(U_i(x)\right),
\end{equation}
which is a resonable equation to find $h$. In general we can state the following\\
\\
\textbf{Theorem 20.}\\
We have
\begin{equation}
\int^{w_1}_{w_0}\frac{f(w)}{(aw^2+bw+c)^m}dw=h\left(U\left(w_1\right)\right)-h\left(U\left(w_0\right)\right),
\end{equation}
where
\begin{equation}
G(x)=f\left(U_i\left(F_1^{(-1)}(x)\right)\right).
\end{equation}
\\
\textbf{Theorem 20.1}\\
Assume that
$$
G(x)=\sum^{\infty}_{n=0}\frac{G^{(n)}(0)}{n!}x^n.\eqno{(147.1)}
$$
Then
$$
h(A)=Q_G(c_i(A))=5F_1(A)^{5/6}\times
$$
$$
\times\sum^{\infty}_{n=0}\frac{G^{(n)}(0)}{n!}\frac{F_1(A)^{n}}{5/6+n}F_{Ap}\left[\frac{1}{6}+\frac{n}{5},\frac{1}{6},\frac{1}{6},\frac{7}{6}+\frac{n}{5},\rho_1 F_1(A)^5,\rho_2 F_1(A)^5\right].\eqno{(147.2)}
$$
\\
\textbf{Proof.}\\
See section Applications paragraph 6.2.\\
\\
\textbf{Notes.}\\
More general if
$$
G(x)=\sum^{\infty}_{n=0}G_nx^{p_n},\eqno{(147.3)}
$$
where $p_n$ is increasing sequence of positive real numbers, with $\lim p_n=+\infty$, then
$$
h(A)=Q_G(c_i(A))=5F_1(A)^{5/6}\times
$$
$$
\times\sum^{\infty}_{n=0}G_n\frac{F_1(A)^{p_n}}{5/6+p_n}F_{Ap}\left[\frac{1}{6}+\frac{p_n}{5},\frac{1}{6},\frac{1}{6},\frac{7}{6}+\frac{p_n}{5},\rho_1 F_1(A)^5,\rho_2 F_1(A)^5\right].\eqno{(147.4)}
$$
\\
\textbf{Theorem 21.}\\
It is $G\left(F_1(x)\right)=f\left(U_i(x)\right)$ and
\begin{equation}
Q_G(A)=h(c_A)=\int^{c_A}_{0}f(U_i(t))dt=\int^{c_A}_{0}G(F_1(t))dt.
\end{equation}
\\
\textbf{Proof.}\\
From (145) we have $h'(b_A)=G(R(q))=f(U_i(b_A))$ inverting $k(A)$ we get $h'(c_A)=G(F(A))=f(U_i(c_A))$. Hence $h'(c_A)c'_A=f(U_i(c_A))c'_A$ and integrating, $h(c_A)=Q_G
(A)=\int^{c_A}_{0}f(U_i(t))dt=\int^{c_A}_{0}G(F_1(t))dt$. Hence we get the result.\\
\\
\textbf{Example.}\\
If $G(t)=\sqrt[6]{t}$, then from (167) below
$$
Q_G(A)=\int^{c_A}_{0}\sqrt[6]{F_1(t)}dt=
$$
$$
=5F_1(c_A)F_{Ap}\left[\frac{1}{5},\frac{1}{6},\frac{1}{6},\frac{6}{5},\frac{11-5\sqrt{5}}{2}F_1(c_A)^5,\frac{11+5\sqrt{5}}{2}F_1(c_A)^5\right].
$$
Hence $y(A)=R\left(e^{-\pi\sqrt{k_i\left(Q_G^{(-1)}(A)\right)}}\right)$ and holds the following semi-algebraic relation for the function $y(x)$:
$$
y\left(5F_1(c_A)F_{Ap}\left[\frac{1}{5},\frac{1}{6},\frac{1}{6},\frac{6}{5},\frac{11-5\sqrt{5}}{2}F_1(c_A)^5,\frac{11+5\sqrt{5}}{2}F_1(c_A)^5\right]\right)=F(A).
$$
\\
\textbf{Theorem 22.}\\
Given $G$ there holds the relation
\begin{equation}
y_i\left(G_i(x)\right)=xF_1^{(-1)}(G_i(x))-\int^{x}_{0}F_1^{(-1)}\left(G_i(t)\right)dt,
\end{equation}
where $F_1^{(-1)}$ is the Appell function of (135).\\
\\
\textbf{Proof.}\\
Integration by parts.\\
\\
\textbf{Example.}\\
Suppose $G(x)=\log\left(F_1^{(-1)}(x)+1\right)$, then $F_1^{(-1)}(G_i(x))=e^x-1$ hence
$$
y_i\left(F_1\left(e^x-1\right)\right)=e^x(x-1)
$$
and $h(x)=\int^{x}_{0}\log(t+1)dt=(x+1)\log(x+1)-x$.\\
Also from Theorems 17 and 19 we get\\
\\
\textbf{Theorem 23.}\\
Let $\xi^{-1}=\sqrt[5]{\frac{-11+5\sqrt{5}}{2}}$ and
\begin{equation}
U_i\left(a,b,c;m;F_1^{(-1)}\left(\xi^{-1}\right)\right)=p_1\textrm{ and }U_i\left(a,b,c;m;F_1^{(-1)}(0)\right)=p_0,
\end{equation}
then
\begin{equation}
\int^{p_1}_{p_0} \frac{G\left(F_1\left[U(a,b,c;m;w)\right]\right)}{(aw^2+bw+c)^m}dw=\sum^{\infty}_{n=0}\frac{G^{(n)}(0)}{n!}C\left(\frac{n}{5}\right).
\end{equation}
\\
\textbf{Corollary 1.}
$$
5\int^{p_1}_{p_0} \frac{F_1\left[U\left(a,b,c;m;w\right)\right]}{(aw^2+bw+c)^m}dw=C\left(\frac{1}{5}\right)=
$$
\begin{equation}
=\left(\frac{2}{11+5 \sqrt{5}}\right)^{11/30} \frac{\Gamma \left(\frac{11}{30}\right) \Gamma \left(\frac{5}{6}\right)}{\Gamma \left(\frac{6}{5}\right)} \, _2F_1\left(\frac{1}{6},\frac{11}{30};\frac{6}{5};-\frac{123}{2}+\frac{55 \sqrt{5}}{2}\right).
\end{equation}
\section{Applications}
\subsection{\textbf{The study of $y(x)=F(x)=R\left(e^{-\pi\sqrt{k_i(x)}}\right)$ function}}
Assume that $y(x)=F(x)=R\left(e^{-\pi\sqrt{k_i(x)}}\right)$, then from Theorem's 1,5 we have
$m_{G_0}(x)=k_i(x)$ and
$$
5\int^{y(x)}_{0}\frac{dt}{t\sqrt[6]{t^{-5}-11-t^5}}=3\sqrt[3]{2x}\cdot {}_2F_1\left[\frac{1}{6},\frac{1}{3};\frac{7}{6};x^2\right]=\int^{x}_{0}\frac{dt}{G_0(y(t))}.
$$
Differentiating the above equation we get
\begin{equation}
h'_i(x)=\frac{1}{G_0(y(x))}=\frac{2^{1/3}}{x^{2/3}(1-x^2)^{1/3}}.
\end{equation}
Hence
\begin{equation}
y_i(x)=\sqrt{\frac{1-\sqrt{1-8G_0(x)^3}}{2}}
\end{equation}
and
$$
5\int^{x}_{0}\frac{G_0(t)}{t\sqrt[6]{t^{-5}-11-t^5}}dt=y_i(x).
$$
Hence the final equation for evaluating the $G-$function is
$$
\sqrt{\frac{1-\sqrt{1-8G_0(x)^3}}{2}}=5\int^{x}_{0}\frac{G_0(t)}{t\sqrt[6]{t^{-5}-11-t^5}}dt,
$$
which under differentiation becomes
$$
\frac{5}{x \sqrt[6]{-x^5+\frac{1}{x^5}-11}}=\frac{3 G_0(x) G'_0(x)}{\sqrt{\frac{1}{2}-4 G_0(x)^3} \sqrt{1-\sqrt{1-8 G_0(x)^3}}}.
$$
Solving the last differential equation, we get the following relation for $G_0(x)$:
$$
\frac{3\cdot 2^{2/3}\sqrt{G_0(x)}}{\sqrt[6]{1+\sqrt{1-8 G_0(x)^3}}}\cdot {}_2F_1\left[\frac{1}{6},\frac{1}{3};\frac{7}{6};\frac{1}{2}\left(1-\sqrt{1-8 G_0(x)^3}\right)\right]=
$$
\begin{equation}
=6x^{5/6} F_{Ap}\left[\frac{1}{6};\frac{1}{6},\frac{1}{6};\frac{7}{6};\frac{11-5 \sqrt{5}}{2}x^5,\frac{11+5 \sqrt{5}}{2}x^5\right]=F^{(-1)}_1(x).
\end{equation}
From (155) the partial evaluation of $G_0(x)$ follows.
\subsection{\bf The case of $G(x)=1$ function}
In case $G(x)=1$, then from relation (26)
\begin{equation}
y(x)=F_1(x).
\end{equation}
From $h(x)=x$, we have
\begin{equation}
F'_1(x)=5^{-1}F_1(x)\sqrt[6]{F_1(x)^{-5}-11-F_1(x)^5}
\end{equation}
and from (56)
\begin{equation}
Q'_G(x)=s'_i(x)=\frac{1}{\sigma(x)}=\frac{2^{1/3}}{\left(x\sqrt{1-x^2}\right)^{2/3}}.
\end{equation}
Hence
\begin{equation}
Q_G(x)=2^{1/3}\int\frac{dx}{\left(x\sqrt{1-x^2}\right)^{2/3}}=3\sqrt[3]{2x}\cdot{}_2F_1\left[\frac{1}{6},\frac{1}{3};\frac{7}{6};x^2\right]+c.
\end{equation}
The modular equation for $y_i(x)$ is $P_n(x)$ and
\begin{equation}
P_n(x)=5\int^{n^2F_1(x)}_{0}\frac{dt}{t\sqrt[6]{t^{-5}-11-t^5}}.
\end{equation}
Also
\begin{equation}
m_G(x)=b^{(-1)}_x=b^{(-1)}(x).
\end{equation}
Avoiding the inverse of $b_r$, which is a hypergeometric function (more precicely, Beta function) we can use the function $m_G(x)=m(x)$ defined by (see relation (23))
\begin{equation}
\pi\int^{+\infty}_{\sqrt{m(x)}}\eta\left(it/2\right)^4dt=x.
\end{equation}
Hence
\begin{equation}
y(x)=F_1(x)=R\left(e^{-\pi\sqrt{m(x)}}\right)
\end{equation}
and
$$
m(x)=k_i\left(s(x)\right).
$$
Hence
\begin{equation}
Q\left(k_r\right)=\pi\int^{+\infty}_{\sqrt{r}}\eta\left(it/2\right)^4dt=b^{(-1)}_r=m(r).
\end{equation}
\begin{equation}
F_1^{(-1)}{'}(x)=\frac{5}{t\sqrt[6]{t^{-5}-11-t^5}}.
\end{equation}
$$
\int F_1(t)^{\nu}dt=\int \frac{x^{\nu}}{x\sqrt[6]{x^{-5}-11-x^5}}dx=
$$
\begin{equation}
=\frac{30x^{5/6+\nu}}{5+6\nu}F_{Ap}\left[\frac{1}{6}+\frac{\nu}{5},\frac{1}{6},\frac{1}{6},\frac{7}{6}+\frac{\nu}{5},\frac{11-5\sqrt{5}}{2}x^5,\frac{11+5\sqrt{5}}{2}x^5\right],
\end{equation}
where we have make the change of variable $x=F_1(t)$. Hence seting $\rho_1=\frac{11-5\sqrt{5}}{2}$, $\rho_2=\frac{11+5\sqrt{5}}{2}$ we get
\begin{equation}
\int F_1(t)^{\nu}dt=\frac{5F_1(t)^{5/6+\nu}}{5/6+\nu}F_{Ap}\left[\frac{1}{6}+\frac{\nu}{5},\frac{1}{6},\frac{1}{6},\frac{7}{6}+\frac{\nu}{5},\rho_1 F_1(t)^5,\rho_2 F_1(t)^5\right]+c.
\end{equation}
Hence if $G(t)=t^{\nu}$, then
$$
y\left(\frac{5F_1(c_A)^{5/6+\nu}}{5/6+\nu}F_{Ap}\left[\frac{1}{6}+\frac{\nu}{5},\frac{1}{6},\frac{1}{6},\frac{7}{6}+\frac{\nu}{5},\rho_1 F_1(c_A)^5,\rho_2 F_1(c_A)^5\right]\right)=F(A).\eqno{(167.1)}
$$
More general we have: If
$$
G(t)=\sum^{\infty}_{n=1}\frac{G^{(n)}(0)}{n!}t^n,
$$
then
$$
\int G(F_1(t))dt=5F_1(c_A)^{5/6}\times
$$
$$
\times\sum^{\infty}_{n=0}\frac{G^{(n)}(0)}{n!}\frac{F_1(c_A)^{n}}{5/6+n}F_{Ap}\left[\frac{1}{6}+\frac{n}{5},\frac{1}{6},\frac{1}{6},\frac{7}{6}+\frac{n}{5},\rho_1 F_1(c_A)^5,\rho_2 F_1(c_A)^5\right]
$$
and
$$
y_i(F(A))=5F_1(c_A)^{5/6}\times
$$
$$
\sum^{\infty}_{n=0}\frac{G^{(n)}(0)}{n!}\frac{F_1(c_A)^{n}}{5/6+n}F_{Ap}\left[\frac{1}{6}+\frac{n}{5},\frac{1}{6},\frac{1}{6},\frac{7}{6}+\frac{n}{5},\rho_1 F_1(c_A)^5,\rho_2 F_1(c_A)^5\right].\eqno{(167.2)}
$$
Also
$$
Q_{G}(A)=5F_1(c_A)^{5/6}\times
$$
$$
\times\sum^{\infty}_{n=0}\frac{G^{(n)}(0)}{n!}\frac{F_1(c_A)^{n}}{5/6+n}F_{Ap}\left[\frac{1}{6}+\frac{n}{5},\frac{1}{6},\frac{1}{6},\frac{7}{6}+\frac{n}{5},\rho_1 F_1(c_A)^5,\rho_2 F_1(c_A)^5\right].
$$
Hence
$$
h(A)=Q_G(c_i(A))=5F_1(A)^{5/6}\times
$$
$$
\times\sum^{\infty}_{n=0}\frac{G^{(n)}(0)}{n!}\frac{F_1(A)^{n}}{5/6+n}F_{Ap}\left[\frac{1}{6}+\frac{n}{5},\frac{1}{6},\frac{1}{6},\frac{7}{6}+\frac{n}{5},\rho_1 F_1(A)^5,\rho_2 F_1(A)^5\right].\eqno{(167.3)}
$$
This can be seen as the evaluation of Theorem 4 (evaluation with inverse integrals). However now we have infinite series expansion. The above equations can also help find $y(A)$ in case $G(A)$ is a polynomial.
\subsection{\bf The Case of $G(F(x))=x$ function}
From (77) we have
\begin{equation}
\sigma(x)=\frac{\left(x\sqrt{1-x^2}\right)^{2/3}}{\sqrt[3]{2}\cdot x}
\end{equation}
and
\begin{equation}
Q_G(x)=s_i(x)=\frac{3\sqrt[3]{x^4}}{2\sqrt[3]{4}}\cdot {}_2F_{1}\left[\frac{1}{3},\frac{2}{3};\frac{5}{3};x^2\right]+c.
\end{equation}
Hence
\begin{equation}
y\left(\frac{3\sqrt[3]{x^4}}{2\sqrt[3]{4}}\cdot {}_2F_{1}\left[\frac{1}{3},\frac{2}{3};\frac{5}{3};x^2\right]+c\right)=F(x).
\end{equation}
$$
h_1^{(-1)}(x)=G\left(y(x)\right)=G\left(F(k_{m_G})\right)=k(m_G(x))=Q_G^{(-1)}(x).
$$
Hence $h_1(x)=Q_G(x)$
$$
h'_1(x)=s'_i(x)=\frac{1}{\sigma(x)}=\frac{\sqrt[3]{2}\cdot x}{(x\sqrt{1-x^2})^{2/3}}.
$$
\subsection{\bf The Case of Jacobi Theta Functions}
For an extended version of this section one can see [15].\\
For $a,p$ positive rationals with $a<p$ and $|q|<1$ we define the forms
\begin{equation}
\left[a,p;q\right]:=\prod^{\infty}_{n=0}\left(1-q^{np+a}\right)\left(1-q^{np+p-a}\right)
\end{equation}
and
\begin{equation}
\theta(a,p;q):=\left[a,p;q\right]^{*}:=q^{C_0}\left[a,p;q\right],
\end{equation}
where $C_0:=p/12-a/2+a^2/(2p)$.\\
With the help of Jacobi triple product identity it can be shown that
\begin{equation}
\vartheta\left(\frac{p}{2},\frac{p}{2}-a;q\right)=q^{C_0}\eta_1(p\tau)\left[a,p;q\right],
\end{equation}
where
\begin{equation}
\vartheta\left(a,b;q\right):=\sum^{\infty}_{n=-\infty}(-1)^nq^{an^2+bn}\textrm{, }|q|<1,
\end{equation}
is a theta function and
\begin{equation}
\eta_1(\tau):=\prod^{\infty}_{n=1}\left(1-q^n\right)\textrm{, }q^{i\pi\tau}\textrm{, }\tau=i\sqrt{r}\textrm{, }r>0,
\end{equation}
is the Ramanujan-Dedekind eta function.\\
We also define
\begin{equation}
Q_{\{a,p\}}(x):=\left[a,p;e^{-\pi\sqrt{k_i(x)}}\right]^{*}.
\end{equation}
Here $Q_{\{a,p\}}(x)$ is the $Q_G(x)$ function defined in Section 3 relation (52) above. In the case of Jacobi theta functions describes their algebraic part (conjecture).\\
\\
We will try to characterize these functions $Q_{\{a,p\}}(x)$. For this, assume that $P_{n}$ is the $n$-th modular equation of $\theta(a,p;q)$, then
\begin{equation}
\theta(a,p;q^n)=P_{n}\left(\theta(a,p;q)\right).
\end{equation}
Also assume that our conjecture holds, then
$$
Q_{\{a,p\}}\left(k_{n^2r}\right)=P_{n}\left(Q_{\{a,p\}}(k_r)\right).
$$
By inverting $k_r$, we get
$$
Q_{\{a,p\}}\left(k_{n^2k_i(x)}\right)=P_n\left(Q_{\{a,p\}}(x)\right).
$$
Setting
\begin{equation}
S_n(x):=k_{n^2k_i(x)},
\end{equation}
we lead to the next\\
\\
\textbf{Theorem 24.}\\
If the $n$-th modular equation of $\theta(a,p;q)$ is that of (177), then
\begin{equation}
k_{n^2k_i(x)}=S_n(x)=Q_{\{a,p\}}^{(-1)}\left(P_{n}\left(Q_{\{a,p\}}(x)\right)\right),\textrm{ }n=2,3,4,...
\end{equation}
If one manages to solve equation (179) with respect to $Q_{\{a,p\}}(x)$ for given $a,p$, then
\begin{equation}
\sum^{\infty}_{n=-\infty}(-1)^nq^{pn^2/2+(p-2a)n/2}=q^{-\frac{p}{12}+\frac{a}{2}-\frac{a^2}{2p}}\eta(q^p)Q_{\{a,p\}}(k_r), \forall r>0
\end{equation}
and $Q_{\{a,p\}}(x)$ will be a root of a minimal polynomial of degree $\nu=\nu(a,p,x)$.\\
Note that in case of rational $x\in(0,1)$ and $a,p$ rational with $0<a,p$, then the degree $\nu$ is independent of $x$ and the minimal polynomial of $Q_{\{a,p\}}(x)$ will have integer coefficients.\\
\\
\textbf{Example.}\\
The 2nd degree modular equation of $A(1,4;q)$ is
\begin{equation}
16 u^8+u^{16}v^8-v^{16}=0.
\end{equation}
If we solve with respect to $v$ we get $v=P_2(u)$, where $v=A(1,4;q^2)$ and $u=A(1,4;q)$. Moreover
\begin{equation}
P_2(w)=\frac{\left(w^{16}+w^4\sqrt{64+w^{24}}\right)^{1/8}}{2^{1/8}}.
\end{equation}
It is $n=2$ then hold (see [9])
\begin{equation}
k_{4r}=\frac{1-\sqrt{1-k_r^2}}{1+\sqrt{1-k_r^2}}.
\end{equation}
Hence
\begin{equation}
S_2(x)=k_{4k_i(x)}=\frac{1-\sqrt{1-x^2}}{1+\sqrt{1-x^2}}.
\end{equation}
Finally we get from the relation (179) of Theorem 24:
\begin{equation}
\frac{\sqrt[8]{Q_{\{1,4\}}(x)^{16}+Q_{\{1,4\}}(x)^4\sqrt{Q_{\{1,4\}}(x)^{24}+64} }}{\sqrt[8]{2}}=Q_{\{1,4\}}\left(\frac{1-\sqrt{1-x^2}}{1+\sqrt{1-x^2}}\right),
\end{equation}
which has indeed a solution
$$
Q_{\{1,4\}}(x)=\sqrt[12]{\frac{4(1-x^2)}{x}}.
$$
\\
\textbf{Note.}\\
We note that function $m(q)=k_r^2$ is implemented in program Mathematica. However a useful expansion is
\begin{equation}
k_r=\sqrt{m(q)}=4q^{1/2}\exp\left(-4\sum^{\infty}_{n=1}q^n\sum_{d|n}\frac{(-1)^{d+n/d}}{d}\right),
\end{equation}
where $q=e^{-\pi\sqrt{r}}$, $r>0$.\\
Continuing we denote
\begin{equation}
\theta(q):=\theta_{\{a,p\}}(q)=q^{p/12-a/2+a^2/(2p)}\frac{\vartheta\left(\frac{p}{2},\frac{p-2a}{2};q\right)}
{f\left(-q^p\right)}\textrm{, }q=e^{-\pi\sqrt{r}}.
\end{equation}
In the case of Jacobi theta functions we set
\begin{equation}
Q(x):=Q_{\{a,p\}}(x)=\theta_{\{a,p\}}\left(e^{-\pi\sqrt{k_i(x)}}\right).
\end{equation}
But it holds $y(x)=F\left(k\left(m_G(x)\right)\right)$ and
$m_G(x)=k_i\left(Q_i(x)\right)$, hence $y(x)=F\left(Q_i(x)\right)$, inverting
\begin{equation}
y_i(x)=\theta_{\{a,p\}}\circ k_i\circ F_i(x).
\end{equation}
Hence we get the following\\
\\
\textbf{Theorem 25.}\\
If $q=e^{-\pi\sqrt{r}}$, $r>0$, then
\begin{equation}
y\left(\theta(q)\right)=R\left(q\right)\textrm{ and }m_G^{(-1)}(r)=\theta(q).
\end{equation}
\\
\textbf{Theorem 26.}\\
The $n-$th modular equation of $\theta(q)=\theta_{\{a,p\}}(q)$ is
\begin{equation}
P_n(x)=Q_{\{a,p\}}\left(k\left(n^2k_i\left(Q_{\{a,p\}}^{(-1)}(x)\right)\right)\right).
\end{equation}
Also $m_G(x)=k_i\left(Q_{\{a,p\}}^{(-1)}(x)\right)$, $m_G^{(-1)}(n)=P_{\sqrt{n}}\left(Q_{\{a,p\}}(1)\right)$, $n>0$
and
\begin{equation}
y(x)=R\left(e^{-\pi\sqrt{k_i\left(Q^{(-1)}_{\{a,p\}}(x)\right)}}\right).
\end{equation}
\\
\textbf{Theorem 27.}\\
If $q=e^{-\pi\sqrt{r}}$, $r>0$, then
\begin{equation}
\frac{d\theta(q)}{dr}=\frac{1}{\phi(r)}.
\end{equation}
\\
\textbf{Proof.}\\
From $m_G^{(-1)}(A)=\theta(q)$, $q=e^{-\pi\sqrt{A}}$ we get
\begin{equation}
5\int^{R(q)}_{0}\frac{G(t)dt}{t\sqrt[6]{t^{-5}-11-t^5}}=\theta(q).
\end{equation}
After derivating the above relation and using (5), we get
\begin{equation}
G\left(R(q)\right)q^{-5/6}f(-q)^4=\theta'(q).
\end{equation}
Using (14), we get the result.\\
\\
\textbf{Notes.}\\
Assuming the above we have
$$
\theta'(q)=q^{-1}\eta(z)^4G\left(R(q)\right),
$$
where
$$
\eta(z):=q^{1/24}f(-q)\textrm{, }q=e^{2\pi i z}\textrm{, }Im(z)>0,
$$
is the Dedekind eta function (see also relation (1)).\\
\\
\textbf{Conjecture 1.}\\
In the case $Q_G(x)=Q_{\{a,p\}}(x)$ we have that $G(R(q))$ is root of polynomial with integer coefficients.\\
\\
\textbf{Definition 4.}\\
We call theta function of the $G-$transformation of Theorem 1 the function $m_G^{(-1)}(A)$, where $q=e^{-\pi\sqrt{A}}$, $A>0$. By this way the definition of the theta functions is generalized and related with the $G-$transform.\\
Therefore for the function $m_G(A)$ holds
\begin{equation}
\theta\left(e^{-\pi\sqrt{m_G(A)}}\right)=A
\end{equation}
and
\begin{equation}
R\left(e^{-\pi\sqrt{m_G(A)}}\right)=y(A).
\end{equation}
\\
\textbf{Theorem 28.}\\For the modularity of $m^{(-1)}_G(A)$ we have the next relation
\begin{equation}
m^{(-1)}_{G}\left(\frac{1}{A}\right)=Q_G\left(\sqrt{1-Q^{(-1)}_G\left(m_G^{(-1)}(A)\right)^2}\right).
\end{equation}
\\
\textbf{Proof.}\\
From relation (53) and the idenity $k_{1/r}=k'_r$, we get the result.\\
\\
\textbf{Example.}\\
Suppose $a=1$, $p=4$, then
\begin{equation}
\sum^{\infty}_{n=-\infty}(-1)^nq^{2n^2+n}=q^{1/24} \eta(4\tau)Q_{\{1,4\}}(k_r).
\end{equation}
Then $Q_{\{1,4\}}(x)$ will be
$$
Q(x)=\sqrt[6]{2}\sqrt[12]{\frac{1-x^2}{x}}.
$$
For a certain $G$ we have from (62) and (64.1):
$$
\sigma(x)=-6\cdot2^{5/6}x^{13/12}(1-x^2)^{11/12}(1+x^2)^{-1}.
$$
Hence
$$
y(x)=R\left(e^{-\pi\sqrt{k_i\left(\frac{1}{8}\left(-x^{12}+\sqrt{64+x^{24}}\right)\right)}}\right).
$$
\[
\]
\centerline{\bf References}\vskip .2in
\noindent
[1]: M. Abramowitz and I.A. Stegun. 'Handbook of Mathematical Functions'. Dover Publications, New York.(1972)\\
[2]: G.E. Andrews. 'Number Theory'. Dover Publications, New York.(1994)\\
[3]: G.E. Andrews, Amer. Math. Monthly, 86, 89-108.(1979)\\
[4]: J.V. Armitage, W.F. Eberlein. 'Elliptic Functions'. Cambridge University Press.(2006)\\
[5]: N.D. Bagis and M.L. Glasser. 'Integrals related with Rogers Ramanujan continued fraction and $q$-products'. arXiv:0904.1641v1 [math.NT].(2009)\\
[6]: B.C. Berndt. 'Ramanujan`s Notebooks Part I'. Springer Verlag, New York.(1985)\\
[7]: B.C. Berndt. 'Ramanujan`s Notebooks Part II'. Springer Verlag, New York.(1989)\\
[8]: B.C. Berndt. 'Ramanujan`s Notebooks Part III'. Springer Verlag, New York.(1991)\\
[9]: B.C. Berndt. 'Ramanujan`s Notebooks Part V'. Springer Verlag, New York.(1998)\\
[10]: I.S. Gradshteyn and I.M. Ryzhik. 'Table of Integrals, Series and Products'. Academic Press.(1980)\\
[11]: N.N. Lebedev. 'Special Functions and their Applications'. Dover Pub. New York.(1972)\\
[12]: E.T. Whittaker and G.N. Watson. 'A course on Modern Analysis'. Cambridge U.P.(1927)\\
[13]: N.D. Bagis. 'Generalized Elliptic Integrals and Applications'.\\arXiv:1304.2315v2 [math.GM].(2013)\\
[14]: Bruce C. Berndt, Heng Huat Chan, Sen-Shan Huang, Soon-Yi Kang, Jaebum Sohn and Seung Hwan Son. 'The Rogers-Ramanujan Continued Fraction'. J. Comput. Appl. Math., 105 (1999), 9-24.\\
[15]: N.D. Bagis. 'On the Complete Evaluation of Theta Functions'. arXiv:1503.01141v4 [math.GM] 10 Mar 2021.\\
\end{document} | math | 59,208 |
\begin{document}
\title{Randomized Incremental Construction of Net-Trees}
\setcounter{page}{0}
\begin{abstract}
Net-trees are a general purpose data structure for metric data that have been used to solve a wide range of algorithmic problems.
We give a simple randomized algorithm to construct net-trees on doubling metrics using $O(n\log n)$ time in expectation.
Along the way, we define a new, linear-size net-tree variant that simplifies the analyses and algorithms.
We show a connection between these trees and approximate Voronoi diagrams and use this to simplify the point location necessary in net-tree construction.
Our analysis uses a novel backwards analysis that may be of independent interest.
\varepsilonnd{abstract}
\thispagestyle{empty}
\section{Introduction}
\label{sec:Introduction}
Har-Peled \& Mendel introduced the net-tree as a linear-size data structure that efficiently solves a variety of (geo)metric problems such as approximate nearest neighbor search, well-separated pair decomposition, spanner construction, and others~\cite{har-peled06fast}.
More recently, such data structures have been used in efficient constructions for topological data analysis (TDA)~\cite{sheehy13linear}.
Net-trees are similar to several other data structures that store points in hierarchies of metric nets (subsets satisfying some packing and covering constraints) arranged into a tree or DAG.
Examples include navigating nets~\cite{krauthgamer04navigating}, cover trees~\cite{beygelzimer06cover}, dynamic hierarchical spanners~\cite{cole06searching,gottlieb08optimal}, and deformable spanners~\cite{gao06deformable}.
The extensive literature on such data structures can be partitioned into two disjoint groups: those that are easy to implement and those that can be constructed in $O(n\log n)$ time for doubling metrics (see Section~\ref{sec:preliminaries} for the definition).
In this paper, we present an algorithm that is both simple and asymptotically efficient.
We combine several ideas already present in the literature with a randomized incremental approach.
The challenge is relegated to the analysis, where the usual tricks for randomized incremental algorithms do not apply to net-trees, mostly because they are not canonically defined by a point set.
There are two known algorithms for building a net-tree~\cite{har-peled06fast} or a closely related structure~\cite{gottlieb08optimal} in $O(n\log n)$ time for doubling metrics.
Both are quite complex and are primarily of theoretical interest.
The algorithm of Har-Peled \& Mendel~\cite{har-peled06fast} requires a complex sequence of approximating data structures.
Cole \& Gottlieb~\cite{cole06searching} proposed a similar data structure that supports dynamic insertions and deletions in $O(\log n)$ time.
Their data structure maintains a so-called centroid path decomposition of a spanning tree of the hierarchy into a collection of paths that are each represented by a biased skip list.
A much simpler algorithm due to Clarkson~\cite{clarkson02nearest} can be combined with an algorithm of Har-Peled \& Mendel~\cite{har-peled06fast} to run in $O(n\log \Delta)$ time, where $\Delta$ is the the \varepsilonmph{spread} of the input, i.e.\ the ratio of the largest to smallest pairwise distances.
Most of the complications of the theoretical algorithm are to eliminate this dependence on the spread.
The goal of this paper is to combine the conceptual simplicity of Clarkson's idea with a simple randomized incremental algorithm to achieve the same $O(n\log n)$ running time of the best theoretical algorithms.
The main improvement over the related data structures~\cite{cole06searching,har-peled06fast} that can be computed in $O(n\log n)$ time is the increased simplicity.
In particular, the point location structure, the primary bottleneck in all related work is just a dictionary mapping uninserted points to nodes in the tree and is very easy to update.
As testament to the simplicity, a readable implementation in python is only $\sim200$ lines of code and is available online~\cite{githubcode}.
A second improvement is in the tighter bounds on the so-called relative constant.
This is the constant factor that bounds the ratio of distances between relative links, the edges stored between nearby nodes in the same level of the tree.
These relative links form a hierarchical spanner, and their number dominates the space complexity of the data structure.
In similar constructions used in TDA, it was found that although a relative constant of $10$ was needed for a particular algorithm, the space blowup required using a constant closer to $3$ in practice, sacrificing theoretical guarantees~\cite{oudot14zigzag}.
In previous work the relative constant was $13$ or more.
In this work, we show that the relative constant can be pushed towards $2$ as a function of the difference in scales between adjacent levels in the tree.
A value of $6$ is easily achievable in practice.
\paragraph*{Related Work}
Uhlmann~\cite{uhlmann91satisfying} proposed \varepsilonmph{metric trees} to solve range searches in general metric spaces, but there are no performance guarantees on the queries.
Yianilos~\cite{yianilos93data} devised a similar data structure, called the \varepsilonmph{vp-tree}, and he showed that when the search radii are very small, queries can be run in $O(\log n)$ expected time.
These structures are balanced binary search trees and they can be constructed recursively in $O(n\log n)$ time by partitioning the points into two subsets according to their distance to the median.
Clarkson~\cite{clarkson99nearest} proposed two randomized data structures to answer approximate nearest neighbor queries in metric spaces satisfying a sphere packing property, which is equivalent to having constant doubling dimension.
The first data structure assumes that the query points have the same distribution as the given input points and it may fail to return a correct answer.
The second one always returns a correct answer, but it requires more time and space.
Roughly speaking, both of these data structures can be constructed in $O(n\log\Delta)$ time, with $O(n\log\Delta)$ size, and they answer queries in $O(\log\Delta)$ time.
Karger \& Ruhl~\cite{karger02finding} proposed \varepsilonmph{metric skip lists} for the so-called growth restricted metrics.
Their data structure can be constructed in $O(n\log n\log\log n)$ time and has $O(n\log n)$ size.
Krauthgamer \& Lee~\cite{krauthgamer04navigating} presented a dynamic, deterministic data structure called \varepsilonmph{navigating nets} to address proximity searches in doubling spaces.
Navigating nets are comprised of hierarchies of nested metric nets connected as a DAG.
In more detail, the points at some scale $i$ are of distance $2^i$ and the balls centered at those points with radii $2^i$ contain all points of scale $i-1$.
They showed that navigating nets have linear size and can be constructed in $O(n\log\Delta)$ time.
Gao et al.~\cite{gao06deformable} independently devised a very similar data structure called the \varepsilonmph{deformable spanner} as a dynamic $(1+\varepsilonpsilon)$-spanner in Euclidean spaces that can be maintained under continuous motion.
Beygelzimer et al.~\cite{beygelzimer06cover} proposed the \varepsilonmph{cover tree}, a spanning tree of a navigating net, to make the space independent of the doubling dimension.
Their experimental results showed that cover trees have good performance in practice.
Besides the space complexity, cover trees do not theoretically outperform navigating nets.
Recently, we~\cite{jahanseir16transform} showed that cover trees with slight modifications also satisfy the stricter properties of net-trees, and a net-tree can be constructed from a cover tree in linear time assuming the space may depend on the doubling dimension.
Refer to~\cite{clarkson06nearest,chavez01searching} for surveys on proximity searches in metric spaces.
\paragraph*{Overview of the Algorithm and its Analysis}
Our approach is to construct a net-tree incrementally.
Each new point is added by first attaching it as a leaf of the tree.
Then, new nodes are added for that point by propagating up the tree one level at a time using just local updates until the covering property is satisfied.
We call this the \varepsilonmph{bottom-up insertion}.
By making only local updates, we can show that the total work of updating the tree is linear in the size and thus, linear in the number of points.
This algorithm relies on a point location data structure that associates each uninserted point with a node in the tree called its \varepsilonmph{center}, which provides a starting point for the bottom-up insertion.
Each time a new node is added, a local search is required to see if any of the uninserted points should have the new node as their center.
The challenge is to show that the expected total number of distance computations for the point location data structure is only $O(n\log n)$ for points inserted in random order.
Our approach to point location is similar in spirit to that proposed by Clarkson~\cite{clarkson02nearest}.
The difference is that instead of associating uninserted points with Voronoi cells of inserted points, we associate them with nodes in the tree.
This allows us to work with arbitrary (and random) orderings of the points rather than just greedy permutations.
Our analysis includes a novel backwards analysis on random orderings to bound the expected running time.
The main difficulty is that the tree structure is far from canonical for a given set of points.
Instead of looking at the tree construction backwards, we define a set of random events that can occur in a permutation of metric points and then charge the work of point location to these events.
We show that each event is charged only a constant number of times and there are at most $O(n\log n)$ events in expectation.
The resulting expected running time of $O(n\log n)$ matches the best theoretical algorithms.
\section{Preliminaries}
\label{sec:preliminaries}
\subsection{Doubling Metrics and Packing}
\label{sub:doubling_metrics}
The input is a set of $n$ points $P$ in a metric space.
The \varepsilonmph{closed metric ball} centered at $p$ with radius $r$ is denoted $\mathbf{B}(p,r):=\{q\in P \mid \mathrm{\mathbf{d}}(p,q)\le r\}$.
The \varepsilonmph{doubling constant} $\rho$ of $P$ is the minimum $\rho$ such that every ball $\mathbf{B}(p,r)$ can be covered by $\rho$ balls of radius $r/2$.
We assume $\rho$ is constant.
The \varepsilonmph{doubling dimension} is defined as $\lg\rho$.
A metric space with a constant doubling dimension is called a \varepsilonmph{doubling metric}.
Throughout, we assume that the input metric is doubling.
The doubling dimension was first introduced by Assouad~\cite{assouad83plongements} and has since found many uses
in algorithm design and analysis.
Other notions of dimension for general metric spaces have also been proposed.
The notion of growth-restricted metrics of Karger \& Ruhl~\cite{karger02finding} is similar to doubling
metrics, though it is more restrictive.
Gupta et al.~\cite{gupta03bounded} showed that the dimension of a growth restricted metric is upper bounded by its doubling dimension.
Perhaps the most useful property of doubling metrics is that they allow for the use of packing and covering arguments similar to those used in Euclidean space to carry over to a more general class of metrics.
The following lemma is at the heart of all the packing arguments in this paper.
\begin{lemma}[Packing Lemma]\label{lem:packing}
If $X\subseteq\mathbf{B}(p,r)$ and for every two distinct points $x,y\in X$, $\mathrm{\mathbf{d}}(x,y)>r'$, where $r>r'$, then $|X|\le\rho^{\lfloor\lg (r/r')\rfloor+1}$.
\varepsilonnd{lemma}
\begin{proof}
By the definition of the doubling constant, $X$ can be covered by $\rho$ balls of radius $r/2$.
These balls can each be covered by $\rho$ balls of radius $r/4$.
Repeating this $\lfloor\lg (r/r')\rfloor+1$ times results in at most $\rho^{\lfloor\lg (r/r')\rfloor+1}$ balls of radius less than $r'$, and these balls contain at most one point of $X$ each, so $X$ has at most $\rho^{\lfloor\lg (r/r')\rfloor+1}$ points.
\varepsilonnd{proof}
The spread of a point set $P$ often plays a role in running times of metric data structures.
The following lemma captures the relationship between the spread and the cardinality of $P$ and follows directly from the Packing Lemma.
\begin{lemma}
A finite metric $P$ has at most $\rho^{O(\log\Delta)}$ points.
\varepsilonnd{lemma}
The distance from a point $p$ to a compact set $Q$ is defined as $\mathrm{\mathbf{d}}(p, Q) \coloneqq \min_{q\in Q} \mathrm{\mathbf{d}}(p,q)$.
The \varepsilonmph{Hausdorff distance} between two sets $P$ and $Q$ is $\mathrm{\mathbf{d}}_H(P,Q)\coloneqq\max\{\max_{p\in P} \mathrm{\mathbf{d}}(p,Q),\max_{q\in Q} \mathrm{\mathbf{d}}(q,P)\}$.
\subsection{Metric Nets and Net-Trees}
\label{sub:net_trees}
A metric net or a \varepsilonmph{Delone set} is a subset of points satisfying some packing and covering properties.
More formally, an $(\alpha,\beta)$-net is a subset $Q\subseteq P$ such that: for all distinct points $p,q\in Q$, $\mathrm{\mathbf{d}}(p,q)>\alpha$ (packing), and for all $p\in P$, $\mathrm{\mathbf{d}}(p, Q)\le \beta$ (covering).
\begin{figure}[htb]
\centering
\includegraphics[width=\columnwidth]{figures/nets.jpg}
\caption{Nets at three different scales are shown from the left and the corresponding net-tree is illustrated on the right.
White dots represent a net and the circles show the covering balls.}
\label{fig:nets_and_nettree}
\varepsilonnd{figure}
A net-tree is a tree $T$ in which each level represents a metric net at some scale, see Fig.~\ref{fig:nets_and_nettree}.
In net-trees, points are leaves in level $-\infty$ and each point can be associated with many internal nodes.
Each node is uniquely identified by its associated point and an integer called its \varepsilonmph{level}.
The node in level $\varepsilonll$ associated with a point $p$ is denoted $p^\varepsilonll$.
We assume that the root is in level $+\infty$.
For a node $p^\varepsilonll\in T$, we define $\mathrm{par}(p^\varepsilonll)$ and $\mathrm{ch}(p^\varepsilonll)$ to be the parent and the set of children of that node, respectively.
Let $P_{p^\varepsilonll}$ denote leaves of the subtree rooted at $p^\varepsilonll$.
For each node $p^\varepsilonll$ in a net-tree, the following properties hold.
\begin{itemize}
\item \textbf{Packing:} $\mathbf{B}(p, c_p \tau^\varepsilonll)\bigcap P \subseteq P_{p^\varepsilonll}$.
\item \textbf{Covering:} $P_{p^\varepsilonll}\subset \mathbf{B}(p, c_c\tau^{\varepsilonll})$.
\item \textbf{Nesting:} If $\varepsilonll>-\infty$, then $p^\varepsilonll$ has a child with the same associated point $p$.
\varepsilonnd{itemize}
The constant $\tau>1$, called the \varepsilonmph{scale factor}, determines the change in scale between levels.
We call $c_p$ and $c_c$ the \varepsilonmph{packing constant} and the \varepsilonmph{covering constant}, respectively, and $c_c\ge c_p>0$.
We represent all net-trees with the same scale factor, packing constant, and covering constant with $\mathrm{NT}(\tau,c_p,c_c)$.
From the above definition, Har-Peled \& Mendel showed that each level of a net-tree is a metric net~\cite{har-peled06fast}.
There are two different representations for net-trees.
In the \varepsilonmph{uncompressed} representation, every root to leaf path has a node in every level down to the scale of the smallest pairwise distance.
The size complexity of this representation is $O(n\log\Delta)$, because there are $O(\log\Delta)$ explicit levels between $-\infty$ and $+\infty$.
The $compressed$ representation is obtained from the uncompressed one by removing the nodes that are the only child of their parents and they have only one child and merging the two adjacent edges as a long edge, see Fig.~\ref{fig:SCNT}.
We call such long edges \varepsilonmph{jumps}.
It is not hard to see that this representation has size of $O(n)$.
Note that compressed net-trees are similar to compressed quadtrees.
A net-tree can be augmented to maintain a list of nearby nodes called relatives.
We define relatives of a node $p^\varepsilonll\in T$ to be
\begin{align*}
\mathrm{rel}(p^\varepsilonll)\coloneqq\{x^f\in T \text{ with } y^g\coloneqq\mathrm{par}(x^f) \mid f \le \varepsilonll < g, \text{ and } \mathrm{\mathbf{d}}(p,x)\le c_r\tau^\varepsilonll\},
\varepsilonnd{align*}
see Fig.~\ref{fig:SCNT}.
We call $c_r$ the \varepsilonmph{relative constant}, and it is a function of the other parameters of a net-tree.
In this paper, we assume that net-trees are always equipped with relatives.
Har-Peled \& Mendel defined compressed net-trees in the class of $\mathrm{NT}(\tau=11,\frac{\tau-5}{2(\tau-1)},\frac{2\tau}{\tau-1})$ with $c_r=13$.
The following easy to prove lemma uses the Packing Lemma and the definition of net-trees. It implies that a compressed net-tree on a doubling metric has $\rho^{O(1)}n$ size.
\begin{lemma}
\label{lem:ntsize}
For each node $p^\varepsilonll$ in $T\in\mathrm{NT}(\tau,c_p,c_c)$, we have $|\mathrm{ch}(p^\varepsilonll)|\le\rho^{\lfloor\lg (c_c\tau/c_p)\rfloor+1}$ and $|\mathrm{rel}(p^\varepsilonll)|\le\rho^{\lfloor\lg (c_r/c_p)\rfloor+1}$.
\varepsilonnd{lemma}
We defined $\mathrm{ch}(\cdot)$, $\mathrm{par}(\cdot)$, and $\mathrm{rel}(\cdot)$ for a node of a tree; however, we abuse notation slightly and apply them to set of nodes.
In such cases, the result will be the union of output for each node.
Furthermore, the distance between nodes of a net-tree is the distance between their corresponding points.
\section{Net-Tree Variants}
\label{sec:variations_of_nettrees}
In this section, we introduce two natural modifications to net-trees that simplify both construction and analysis.
In the first variant, we replace the global packing and covering conditions of a net-tree with local ones that are easier to check, and we show that these local conditions imply the global conditions.
In the second variant, we show how a less aggressive compression criterion still results in a linear-size data structure while guaranteeing that relatives are on the same level in the tree, are symmetric, and are consistent up the tree (i.e.\ parents of relatives are relatives).
This makes it much simpler to reason about local neighborhoods by local search among relatives.
\input{local_nettrees}
\input{semi_compressed}
\section{Approximate Voronoi Diagrams from Net-Trees}
\label{sec:approx_vd}
Many metric data structures naturally induce a partition of the search space.
The use of hierarchies of partitions at different scales is a fundamental idea in the \varepsilonmph{approximate near neighbor} problem (also known as \varepsilonmph{point location in equal balls} (PLEB)) which is at the heart of many \varepsilonmph{approximate nearest neighbor} algorithms, including high dimensional approaches using locality-sensitive hashing~\cite{indyk98approximate,har-peled01replacement,indyk04nearest,sabharwal06nearest,har-peled12approximate}.
Given a set of points $P$ and a query $q$, the \varepsilonmph{nearest neighbor} of $q$ in $P$ is the point $p\in P$ such that for all $p'\in P$, we have $\mathrm{\mathbf{d}}(q,p)\le \mathrm{\mathbf{d}}(q,p')$.
Relaxing this notion, $p$ is a \varepsilonmph{$c$-approximate nearest neighbor} (or $c$-ANN) of $q$ if for all $p'\in P$, we have $\mathrm{\mathbf{d}}(q,p)\le c\mathrm{\mathbf{d}}(q,p')$.
The Voronoi diagram of a set of points $P$ is a decomposition of space into cells, one per point $p\in P$ containing all points for which $p$ is the nearest neighbor.
The nearest neighbor search problem can be viewed as point location in a Voronoi diagram, though it is not necessary to represent the Voronoi diagram explicitly.
In this section we give a particular decomposition of space, an approximate Voronoi diagram from a net-tree.
The purpose is not to introduce a new approximate Voronoi diagram (there are several already~\cite{har-peled01replacement,sabharwal06nearest,arya02linear}), but rather to provide a clear description of the point location problem at the heart of our construction.
Just as in Clarkson's \textbf{sb} data structure~\cite{clarkson02nearest}, we will keep track of what ``cell'' contains each uninserted point.
However, instead of using the Voronoi cells, we will use the approximate cells described below.
Moreover, instead of having one cell per point, we have one cell per node, thus we can simulate having a Voronoi diagram of a net at each scale.
We want to associate points with the closest node in the tree that is close enough to be a relative.
Ties are broken between nodes associated to the same point by always choosing the one that is lowest in the tree.
Formally, we define the following function mapping a point of the metric space $\mathcal{M}$ and a node to a pair of numbers.
\[
f(x, p^\varepsilonll) :=
\begin{cases}
(\mathrm{\mathbf{d}}(x,p), \varepsilonll) & \text{if $\mathrm{\mathbf{d}}(x,p) \le c_r \tau^\varepsilonll$}\\
(\infty, \infty) & \text{otherwise}
\varepsilonnd{cases}
\]
The Voronoi cell of a node $p^\varepsilonll$ is then defined as
\begin{align*}
\mathrm{Vor}(p^\varepsilonll) \coloneqq \{x\in \mathcal{M}\mid f(x,p^\varepsilonll) \le f(x, q^m) \text{ for all } q^m\in T\},
\varepsilonnd{align*}
where ordering on pairs is lexicographical.
For a point $q\notin P$, the \varepsilonmph{center} for $q$ in $T$, denoted $\mathcal{C}(q)$, is the node $p^\varepsilonll\in T$ such that $q\in \mathrm{Vor}(p^\varepsilonll)$.
As we will see in Section~\ref{sec:the_bottomup_construction}, finding the center of a point is the basic point location operation required to insert it into the net-tree.
Fig.~\ref{fig:approx_vd} illustrates the construction.
\begin{figure}[htb]
\centering
\includegraphics[trim= 0 7in 0 0.5in,width=0.5\textwidth]{figures/approx_vd.pdf}
\caption{The net-tree on the left induces the approximate Voronoi diagram on the right.}
\label{fig:approx_vd}
\varepsilonnd{figure}
The union of Voronoi cells $p^\varepsilonll$ for all $\varepsilonll$ gives an approximate Voronoi cell for the point $p$.
The following lemma makes this precise.
\begin{lemma}
\label{lem:NN_center}
Let $T$ be a net-tree in $\mathrm{LNT}(\tau,c_p,c_c)$ with $c_r>\frac{c_c\tau}{\tau-1}$ on a point set $P\subset \mathcal{M}$.
For any point $q\in\mathcal{M}$, if $\mathcal{C}(q)=p^\varepsilonll$, then $p$ is a $(\frac{c_r\tau(\tau-1)}{c_r(\tau-1)-c_c\tau})$-ANN of $q$ in $P$.
\varepsilonnd{lemma}
\begin{proof}
Let $m\coloneqq\lceil\log_\tau (\mathrm{\mathbf{d}}(p,q)/c_r)\rceil$.
Then, $m\le\varepsilonll$ and $c_r\tau^{m-1}<\mathrm{\mathbf{d}}(p,q)\le c_r\tau^m$.
Since $p\in N_m$ and $p^\varepsilonll$ is the center of $q$, $\mathrm{\mathbf{d}}(q,N_m)>c_r\tau^{m-1}$.
Furthermore, $\mathrm{\mathbf{d}}(q,N_{m-1})>c_r\tau^{m-1}$, because otherwise $\mathcal{C}(q)$ should be a node other than $p^\varepsilonll$ so that the corresponding point belongs to $N_{m-1}$, which contradicts the assumption.
Also note that each node associated to a point in $P\setminus N_{m-1}$ has an ancestor in a level at least $m-1$.
If the lowest ancestor in a level at least $m-1$ is above $m-1$, then it is the top of a jump, and the bottom node with the same associated point is in a level less than $m-1$.
Therefore, using Lemma~\ref{lem:covering}, $\mathrm{\mathbf{d}}_H(N_{m-1},P)\le c_c\tau^m/(\tau-1)$.
Now, using the triangle inequality,
\begin{align*}
\mathrm{\mathbf{d}}(q,P)
&\ge\mathrm{\mathbf{d}}(q,N_{m-1})-\mathrm{\mathbf{d}}_H(N_{m-1},P)
> c_r\tau^{m-1}-\frac{c_c}{\tau-1}\tau^{m}
>\left(\frac{1}{\tau}-\frac{c_c}{c_r(\tau-1)}\right)\mathrm{\mathbf{d}}(p,q).
\varepsilonnd{align*}
Therefore, $\mathrm{\mathbf{d}}(p,q)<\frac{c_r\tau(\tau-1)}{c_r(\tau-1)-c_c\tau}\mathrm{\mathbf{d}}(q,P)$.
\varepsilonnd{proof}
\section{Bottom-up Construction of a Net-Tree}
\label{sec:the_bottomup_construction}
Constructing a net-tree one point at a time has three phases.
First, one finds the center (as defined in Section~\ref{sec:approx_vd}) of the new point.
Second, the new point is inserted as a relative of its center, with its parent, children, and relatives computed by a constant-time local search.
Third, new nodes associated with the point are added up the tree until the parent satisfies the covering property.
In principle, this promotion phase can propagate all the way to the root.
Along the way, it is sometimes necessary to split a compressed edge to create a node that now has a relative (our new point) or remove an existing node that now has no relatives.
In the original work on net-trees, the difficult part of the algorithm finds not only the centers (or its equivalent), but also finds an ordering that avoids the propagation phase.
Other algorithms have used the tree itself as the search structure to find the centers when needed~\cite{gao06deformable}, but this can lead to linear time insertions if the tree is deep.
In this section, we will give the construction assuming the center of each new point is known, and we will describe the point location data structure in Section~\ref{sec:point_location}.
\subsection{Insertion}
\label{sub:insertion}
Once the center is found, $p$ is added to the tree as follows.
Let $q^\varepsilonll\coloneqq\mathcal{C}(p)$.
We find the lowest level $h$ in $T$ that $p$ has a relative (not itself).
By the definition of relatives, $h\coloneqq\lceil\log_\tau (\mathrm{\mathbf{d}}(q,p)/c_r)\rceil$.
If $p$ does not satisfy the packing property at level $h$, that is $\mathrm{\mathbf{d}}(p,q)\le c_p\tau^h$, then set $h\coloneqq h-1$.
Next, we create node $p^h$.
If $q^h$ is not already in the tree, then we add it to the tree.
If the parent of $q^h$ is a node associated to point $q$ and $q^{h+1}\notin T$, then we create $q^{h+1}$ and add it to the tree.
We also set the parent of $p^h$ to $\mathrm{par}(q^h)$.
To ensure that the parent, children, and relatives of the new node $p^h$ are correct, an update procedure will be executed.
In this procedure, we find relatives and children of $p^h$ from $\mathrm{ch}(\mathrm{rel}(\mathrm{par}(p^h)))$ and $\mathrm{ch}(\mathrm{rel}(p^h))$, respectively.
Also, the parent of $p^h$ will be the closest node to $p$ among $\mathrm{rel}(\mathrm{par}(p^h))$.
Note that when node $p^h$ receives a new child, say $x^{h-1}$, we check the previous parent of $x^{h-1}$ against the semi-compressed condition to determine whether that node should be removed from the tree or not.
The following lemma proves the correctness of the insertion algorithm.
\begin{lemma}
\label{lem:insert}
Given a semi-compressed tree $T\in\mathrm{LNT}(\tau,c_p,c_c)$ with $c_r\ge\frac{2c_c\tau}{\tau-2}$ and an uninserted point $p$ with $q^\varepsilonll\coloneqq\mathcal{C}(p)$.
The insertion algorithm adds $p$ into $T$ and results a semi-compressed local net-tree $T'\in\mathrm{LNT}(\tau,c_p,c_c+\frac{c_r}{\tau})$.
\varepsilonnd{lemma}
\begin{proof}
We need to show that the resulted tree satisfies the covering, the packing, and the parent invariants, also relatives are correct and the output is semi-compressed.
Let $x^{h+1}$ be the closest node to $p^h$ at level $h+1$.
By the parent property, $\mathrm{\mathbf{d}}(p,x)\le\mathrm{\mathbf{d}}(p^h,\mathrm{par}(q^h))$.
By the triangle inequality,
\begin{align*}
\mathrm{\mathbf{d}}(p^h,\mathrm{par}(q^h))\le\mathrm{\mathbf{d}}(p^h,q^h)+\mathrm{\mathbf{d}}(q^h,\mathrm{par}(q^h))\le c_r\tau^h+c_c\tau^{h+1}=(c_c+\frac{c_r}{\tau})\tau^{h+1}.
\varepsilonnd{align*}
Therefore, $\mathrm{\mathbf{d}}(p,x)<(c_c+c_r/\tau)\tau^{h+1}$, which implies that the covering constant of $T'$ is $c_c+\frac{c_r}{\tau}$.
Note that the distance of any node in any level $\varepsilonll$ in $T'$ except $p^h$ to its parent is at most $c_c\tau^{\varepsilonll+1}$.
Let $h$ be the minimum value so that $\mathrm{\mathbf{d}}(p,q)\le c_r\tau^h$.
Then $c_r\tau^{h-1}<\mathrm{\mathbf{d}}(p,q)\le c_r\tau^h$, as such $h\coloneqq\lceil\log_\tau(\mathrm{\mathbf{d}}(p,q)/c_r)\rceil$.
Insertion of $p$ at level $h$ should preserve the packing property, i.e. $\mathrm{\mathbf{d}}(p,q)>c_p\tau^h$.
Since $c_p\le c_c$ and $c_r\ge \frac{2c_c\tau}{\tau-2}$, we have $c_p\tau^h\le c_c\tau^h<\frac{2c_c\tau}{\tau-2}\tau^h\le c_r\tau^h$.
However, $c_p\tau^h<c_r\tau^{h-1}$ does not necessarily hold, so if $p$ is inserted at level $h$, it may violate the packing property.
Furthermore, we have $\mathrm{\mathbf{d}}(p,q)>c_r\tau^{h-1}\ge \frac{2c_c\tau}{\tau-2}\tau^{h-1}>c_c\tau^{h-1}\ge c_p\tau^{h-1}$, which implies that the insertion of $p$ at level $h-1$ satisfies the packing property.
Therefore, the insertion algorithm correctly maintains the packing property
To prove the parent property, we need to show that the parent and children of $p^h$ in $T'$ are correct.
Since $p^h\sim q^h$, Lemma~\ref{lem:functorial} implies $\mathrm{par}(p^h)\sim\mathrm{par}(q^h)$, so the algorithm correctly finds the parent of $p^h$.
To show that the children of $p^h$ in $T'$ are correct, we first prove that $p^h$ cannot serve as the parent of any node with a level less than $h-1$, then we show that the algorithm correctly finds its children in level $h-1$.
Consider a node $s^g\in T$, where $g\le h-2$.
We have $\mathrm{\mathbf{d}}(p,\mathrm{par}(s^{g}))>c_r\tau^{h-1}$, otherwise $\mathrm{par}(s^{g})$ should have been the center of $p$.
By the triangle inequality,
\begin{align*}
\mathrm{\mathbf{d}}(p^h,s^g)
&\ge \mathrm{\mathbf{d}}(p^h,\mathrm{par}(s^{g}))-\mathrm{\mathbf{d}}(\mathrm{par}(s^{g}),s^g)
>c_r\tau^{h-1}-c_c\tau^{g+1} \\
&> \frac{2c_c\tau}{\tau-2}\tau^{h-1}-c_c\tau^{h-1}=c_c\frac{\tau+2}{\tau-2}\tau^{h-1}>c_c\tau^{h-1}.
\varepsilonnd{align*}
Therefore, $p$ cannot cover $s^g$, which implies that we only need to check the nodes at level $h-1$ to find children of $p^h$.
Furthermore, we show that if $p^h$ is the closest node at level $h$ to a node $s^{h-1}$, then $p^h\sim\mathrm{par}(s^{h-1})$, which implies that the algorithm correctly finds children of $p^h$.
By the parent property, $\mathrm{\mathbf{d}}(p,s)<\mathrm{\mathbf{d}}(s^{h-1},\mathrm{par}(s^{h-1}))$.
By the triangle inequality,
\begin{align*}
\mathrm{\mathbf{d}}(p^h,\mathrm{par}(s^{h-1}))
&\le \mathrm{\mathbf{d}}(p^h,s^{h-1})+\mathrm{\mathbf{d}}(s^{h-1},\mathrm{par}(s^{h-1}))
<2\mathrm{\mathbf{d}}(s^{h-1},\mathrm{par}(s^{h-1}))
<2c_c\tau^h
<c_r\tau^h.
\varepsilonnd{align*}
Lemma~\ref{lem:functorial} implies that the algorithm correctly finds the relatives of $p^h$ (nodes that have $p^h$ as their relative will be updated too).
If $q^h$ is added to the tree, we do not need to find its relatives separately because $p^h$ is its only relative.
Also, if $q^{h+1}$ is inserted to the tree, we do not need to update $\mathrm{rel}(q^{h+1})$ because it does not have any relatives other than itself.
Therefore, the algorithm correctly updates relatives after each insertion.
Eventually, $T'$ is semi-compressed because the algorithm removes those nodes that do not satisfy the semi-compressed condition (while updating children) and the created nodes have more than one relative or more than one child.
\varepsilonnd{proof}
\subsection{Bottom-Up Propagation}
\label{sub:bottom_up_propagation}
If the insertion of a new point $p$ violates the local covering property (change the covering constant from $c_c$ to $c_c+c_r/\tau$), then the bottom-up propagation algorithm restores the covering property by promoting $p^\varepsilonll$ to higher levels of the tree as follows.
Let $q^{\varepsilonll+1}\coloneqq\mathrm{par}(p^\varepsilonll)$.
First, we create node $p^{\varepsilonll+1}$ and make it as the parent of $p^\varepsilonll$.
Then, we make the closest node among $\mathrm{rel}(\mathrm{par}(q^{\varepsilonll+1}))$ to $p$ as the parent of $p^{\varepsilonll+1}$.
Finally, we find relatives and children of $p^{\varepsilonll+1}$ in a way similar to the insertion algorithm (we also remove the nodes that do not satisfy the semi-compressed condition).
If node $p^{\varepsilonll+1}$ still violates the covering property, we use the same procedure to promote it to a higher level.
Here, we use iteration $i$ to indicate promotion of point $p$ to level $\varepsilonll+i$.
\begin{lemma}
\label{lem:dist_violatingnode_to_parent}
Given $c_r\ge \frac{2c_c\tau}{\tau-2}$ and a violating node $p^\varepsilonll$, in the $i$-th iteration of the bottom-up propagation algorithm, $\mathrm{\mathbf{d}}(p^{\varepsilonll+i},\mathrm{par}(p^{\varepsilonll+i}))\le (c_c+\frac{c_r}{\tau})\tau^{\varepsilonll+i+1}< c_r\tau^{\varepsilonll+i+1}$.
\varepsilonnd{lemma}
\begin{proof}
We prove this lemma by induction.
For the base case $i=0$, Lemma~\ref{lem:insert} implies $\mathrm{\mathbf{d}}(p,\mathrm{par}(p^\varepsilonll))\le (c_c+\frac{c_r}{\tau})\tau^{\varepsilonll+1}$.
Also, for $c_r\ge \frac{2c_c\tau}{\tau-2}$, $c_c+\frac{c_r}{\tau}\le \frac{c_r(\tau-2)}{2\tau}+\frac{c_r}{\tau}=\frac{c_r}{2}<c_r$.
Assume that the lemma holds for some $i-1\ge 0$, and we show that it is also true for $i$.
In other words, the distance between $p^{\varepsilonll+i-1}$ to $q^{\varepsilonll+i}\coloneqq\mathrm{par}(p^{\varepsilonll+i-1})$ is greater than $c_c\tau^{\varepsilonll+i}$, as such $p$ should be promoted to level $\varepsilonll+i$.
The algorithm finds the parent of $p^{\varepsilonll+i}$ among the relatives of $\mathrm{par}(q^{\varepsilonll+i})$.
Therefore, $\mathrm{par}(p^{\varepsilonll+i})$ is a node in level $\varepsilonll+i+1$ so that $\mathrm{\mathbf{d}}(p^{\varepsilonll+i},\mathrm{par}(p^{\varepsilonll+i}))\le \mathrm{\mathbf{d}}(p^{\varepsilonll+i},\mathrm{par}(q^{\varepsilonll+i}))$.
By the triangle inequality,
\begin{align*}
\mathrm{\mathbf{d}}(p^{\varepsilonll+i},\mathrm{par}(p^{\varepsilonll+i}))
&\le \mathrm{\mathbf{d}}(p^{\varepsilonll+i},\mathrm{par}(q^{\varepsilonll+i}))
\le \mathrm{\mathbf{d}}(p^{\varepsilonll+i},q^{\varepsilonll+i})+\mathrm{\mathbf{d}}(q^{\varepsilonll+i},\mathrm{par}(q^{\varepsilonll+i}))\\
&\le (c_c+\frac{c_r}{\tau})\tau^{\varepsilonll+i}+c_c\tau^{\varepsilonll+i+1}=(\frac{c_r}{\tau^2}+\frac{c_c}{\tau}+c_c)\tau^{\varepsilonll+i+1}\\
&<(c_c+\frac{c_r}{\tau})\tau^{\varepsilonll+i+1}<c_r\tau^{\varepsilonll+i+1}. \qedhere
\varepsilonnd{align*}
\varepsilonnd{proof}
The following lemma states that the bottom-up propagation algorithm correctly restores the covering property, and its proof is similar to the proof of Lemma~\ref{lem:insert}
\begin{lemma}
\label{lem:promotion}
Given a semi-compressed tree $T\in\mathrm{LNT}(\tau, c_p, c_c+\frac{c_r}{\tau})$ with $c_r\ge\frac{2c_c\tau}{\tau-2}$.
Let for all nodes $x^m\in T$ except $p^\varepsilonll$, $\mathrm{\mathbf{d}}(x^m,\mathrm{par}(x^m))\le c_c\tau^{m+1}$ and for $p^\varepsilonll$, $c_c\tau^{\varepsilonll+1}<\mathrm{\mathbf{d}}(p^\varepsilonll,\mathrm{par}(p^\varepsilonll))\le (c_c+\frac{c_r}{\tau})\tau^{\varepsilonll+1}$.
Then, the bottom-up propagation algorithm results a semi-compressed tree $T'\in\mathrm{LNT}(\tau,c_p,c_c)$.
\varepsilonnd{lemma}
\begin{proof}
First, we prove that the local packing, covering, and parent properties are mintained.
Since $\mathrm{\mathbf{d}}(p^{\varepsilonll+i},\mathrm{par}(p^{\varepsilonll+i}))>c_c\tau^{\varepsilonll+i+1}\ge c_p\tau^{\varepsilonll+i+1}$, the promotion does not modify the packing constant.
Also, the violating node can be promoted up to the root (at level $+\infty$), so the algorithm results the covering constant of $c_c$.
Using~\Cref{lem:dist_violatingnode_to_parent,lem:functorial}, $\mathrm{par}(p^{\varepsilonll+i})\sim\mathrm{par}(q^{\varepsilonll+i})$, so the parent is in $\mathrm{rel}(\mathrm{par}(q^{\varepsilonll+i}))$.
The proof of correctness of $\mathrm{ch}(p^{\varepsilonll+i})$ is similar to Lemma~\ref{lem:insert}.
It is easy to see that the relatives of $p^{\varepsilonll+i}$ are among $\mathrm{rel}(\mathrm{par}(q^{\varepsilonll+i}))$ and the algorithm correctly finds the relatives.
Finally, we need to show that $T'$ is semi-compressed.
In other words, we should prove that all the created nodes for $p$ are required in $T'$.
Lemma~\ref{lem:dist_violatingnode_to_parent} implies that node $p^{\varepsilonll+i}$ has at least one relative besides itself, i.e. $q^{\varepsilonll+i}$, which is the old parent of $p^{\varepsilonll+i-1}$ in iteration $i-1$.
So, it is always necessary to create node $p^{\varepsilonll+i}$ in the $i$-th iteration.
\varepsilonnd{proof}
\subsection{Analysis}
\label{sub:analysis}
In the following theorem, we analyze the running time of the bottom-up construction algorithm without considering the point location cost which will be handled in Section~\ref{sec:point_location}.
\begin{theorem}
\label{thm:construction_without_PL}
Not counting the PL step, the bottom-up construction runs in $O(\rho^{O(1)}n)$ time.
\varepsilonnd{theorem}
\begin{proof}
To prove this theorem, we use an amortized analysis which imposes the cost of each iteration on a node in the output.
In the promotion phase, Lemma~\ref{lem:dist_violatingnode_to_parent} implies that every node of $p^{\varepsilonll+i}$ has at least one relative besides itself, namely $q^{\varepsilonll+i}$.
So, we can make $q^{\varepsilonll+i}$ responsible to pay the cost of iteration $i$ for $p$.
Note that a node $q^{\varepsilonll+i}$ will not be removed by any points that will be processed next, because $p^{\varepsilonll+i}\sim q^{\varepsilonll+i}$ satisfies the semi-compressed condition.
In other words, there is always a node in the output that pays the cost of promotion.
By Lemma~\ref{lem:ntsize}, the cost of each iteration is $\rho^{O(1)}$ and $q^{\varepsilonll+i}$ has $\rho^{O(1)}$ relatives, as such $q^{\varepsilonll+i}$ receives $\rho^{O(1)}$ cost in total.
Therefore, to pay the cost of all promotions for all $n$ points, each node in the output requires $\rho^{O(1)}$ charge.
By Lemma~\ref{lem:promotion}, the output is semi-compressed and Theorem~\ref{thm:semi-compressed_linear_size} implies that it has $O(\rho^{O(1)}n)$ size.
Thus, the total cost of all promotions for all $n$ points does not exceed $O(\rho^{O(1)}n)$.
Notice that when a point is inserted to the tree for the first time, it does not necessarily have any other relatives.
However, the insertion occurs only once for each point and it requires $\rho^{O(1)}$ time.
Therefore, all insertions can be done in $O(\rho^{O(1)}n)$ time.
\varepsilonnd{proof}
\section{Randomized Incremental Construction}
\label{sec:point_location}
In this section, we show how to eagerly compute the centers of all uninserted points.
The centers are updated each time either a new node is added or an existing node is deleted by doing a local search among parents, children, and relatives of the node.
We show that the following invariant is satisfied after each insertion or deletion.
\begin{invariant*}
The centers of all uninserted points are correctly maintained.
\varepsilonnd{invariant*}
In Section~\ref{sec:pl_algorithm}, we present the point location algorithm.
Then, in Section~\ref{sec:pl_events}, we show that for a random ordering of points, the point location takes $O(n\log n)$ time in expectation.
As this point location work is the main bottleneck in the algorithm, the following theorem is main contribution of this paper.
\begin{theorem}
Given a random permutation $\pi=\langle p_1,\ldots,p_n\rangle$.
A net-tree $T\in \mathrm{NT}(\tau,\frac{c_p(\tau-1)-2c_c}{2(\tau-1)},\frac{c_c\tau}{\tau-1})$ with $c_r=\frac{2c_c\tau}{\tau-4}$ can be constructed from $\pi$ in $O(\rho^{O(1)}n\log n)$ expected time, where $\tau\ge \max\{5,\frac{2c_c}{c_p}+2\}$ and $0<c_p\le c_c<\frac{c_p(\tau-1)}{2}$ are constants.
\varepsilonnd{theorem}
\input{pl_algorithm}
\input{pl_events}
\section{Conclusion}
\label{sec:conclusion}
In this paper, we proposed local net-trees as a variation of net-trees with much easier to maintain properties.
We proved that local net-trees are also net-trees with a slightly different parameters.
Then, we presented a simple algorithm to construct local net-trees incrementally from an arbitrary permutation of points in a doubling metric space.
We relegated the challenge of achieving $O(n\log n)$ time complexity to the analysis part.
To analyze our algorithm, we defined a notion of touches corresponding to the number of distance computations, and proved that the total expected number of touches in a permutation is $O(n\log n)$.
\varepsilonnd{document} | math | 41,669 |
\begin{document}
\title{Wiener-Hopf factorization for time-inhomogeneous Markov chains and its application}
{\footnotesize
\begin{tabular}{l@{} p{350pt}}
\hline \\[-.2em]
\textsc{Abstract}: \ & In this paper we derive the Wiener-Hopf factorization for a finite-state time-inhomogeneous Markov chain. To the best of our knowledge, this study is the first attempt to investigate the Wiener-Hopf factorization for time-inhomogeneous Markov chains. In this work we only deal with a special class of time-inhomogeneous Markovian generators, namely piece-wise constant, which allows to use an appropriately tailored randomization technique. Besides the mathematical importance of the Wiener-Hopf factorization methodology, there is also an important computational aspect: it allows for efficient computation of important functionals of Markov chains. \\[0.5em]
\textsc{Keywords:} \ & Wiener-Hopf factorization, inhomogeneous Markov chain, fluctuation theory, randomization method, additive functional. \\
\textsc{MSC2010:} \ & 60J27, 60J28, 60K25 \\[1em]
\hline
\end{tabular}
}
\section{Introduction}
In this paper we derive the Wiener-Hopf factorization (WHf) for a finite-state time-inhomogeneous Markov chain. As far as we know, our study is the first attempt to investigate the Wiener-Hopf factorization for time-inhomogeneous Markov chains. In this pioneering study we only deal with a special class of time-inhomogeneous Markovian generators, namely piece-wise constant. This allows to use an appropriately tailored randomization technique. Besides the mathematical importance of the WHf, there is an important computational aspect: this methodology allows for very efficient computation of important functionals of Markov chains.
The Wiener-Hopf factorization for finite-state Markov chains was originally derived in \cite{Barlow1980} in the time-homogeneous case; see also \cite{London1982} and \cite{Williams1991}. For the WHf in case of time-homogeneous Feller Markov processes we refer to \cite{Williams2008}. For some related applied work we refer to \cite{Avram2003}, which deals with the ruin problem, and to \cite{Asmussen1995,rogers1994fluid,rogers_shi_1994} that study fluid models.
In addition, \cite{Kennedy1990} studies the so called ``noisy'' Wiener-Hopf factorizations; for applications see \cite{Asmussen1995,rogers1994fluid,rogers_shi_1994,Jobert2006,Jiang2008,Mijatovic2011,Jiang2012,Hieber2014,Hainaut2016}.
It needs to be stressed that even though the classical WHf of \cite{Barlow1980} can be applied to the generator matrix, say $\mathsf{G}_t$, of a time-inhomogeneous Markov chain $X$ at every time $t$, these factorizations do not have any probabilistic meaning with regard to the process $X$. In particular, they are of no use for computing functionals such as \eqref{eq:tau0+}-\eqref{eq:taut-} below. So, a relevant WHf for a time-inhomogeneous Markov chain requires a different approach than the one that would just directly apply the results of \cite{Barlow1980} to each $\mathsf{G}_t$, $t\geq 0$.
The paper is organized as follows. In Section \ref{sec:setup} we provide a motivation and the setup up for our problem. In Section \ref{subsec:th} we introduce a randomization method and we give the main results of the paper. Section \ref{sec:Numerical} provides a numerical algorithm for computing our version of the WHf and its application in a specific example. Finally, we give some supporting results in the Appendix.
\section{Motivation and problem set-up}\label{sec:setup}
Let $\mathbf{E}$ be a finite set, $(\Omega,\mathscr{F},\mathbb{P})$ be a complete probability space, and $X:=(X_{t})_{t\geq 0}$ be a {\it time-inhomogeneous} Markov chain on $(\Omega,\mathscr{F},\mathbb{P})$ with state space $\mathbf{E}$ and generator function $\mathsf{G}=\{\mathsf{G}_{t},t\geq 0\}$. In particular, each $\mathsf{G}_{t}$ is a $|\mathbf{E}|\times |\mathbf{E}|$ matrix. We assume that $\mathbb{P}\left(X_{0}=i\right)>0$ for each $i\in\mathbf{E}$ and we let $\mathbb{P}^{i}$ be the probability measure on $(\Omega,\mathscr{F})$ defined by
\begin{align*}
\mathbb{P}^{i}(A):=\mathbb{P}\left(A\,|\,X_{0}=i\right),\quad A\in\mathscr{F},
\end{align*}
with $\mathbb{E}^{i}$ denoting the associated expectation.
In this paper we assume that the generator $\mathsf{G}$ is piecewise constant, namely we assume that
\begin{align*}
\mathsf{G}_{t}=\left\{\begin{array}{ll} \mathsf{G}_{1},\quad &\text{if }\,s_{0}\leq t<s_{1},\\ \mathsf{G}_{2},\quad &\text{if }\,s_{1}\leq t<s_{2},\\ \,\,\vdots & \\ \mathsf{G}_{n},\quad &\text{if }\,s_{n-1}\leq t<s_{n},\\ \mathsf{G}_{n+1},\quad &\text{if }\,t\geq s_{n}, \end{array}\right.
\end{align*}
for some $n\in\mathbb{N}$ and $0=s_{0}<s_{1}<\ldots<s_{n}$.
Without loss of generality we assume that $\mathsf{G}_{1},\ldots,\mathsf{G}_{n+1}$ are not sub-Markovian. That is, the sums of row elements of $\mathsf{G}_{k}$ are all zero, for any $k=1,\ldots,n+1$. The results of this paper carry over to the sub-Markovian case by the standard augmentation of the state space.
Next, we consider a function $v:\mathbf{E}\rightarrow\mathbb{R}\setminus\{0\}$ and we put
\begin{align*}
\mathbf{E}^{+}:=\left\{\left.i\in\mathbf{E}\,\right|v(i)>0\right\}\quad\text{and}\quad\mathbf{E}^{-}:=\left\{\left.i\in\mathbf{E}\,\right|v(i)<0\right\}.
\end{align*}
We also define the additive functional
\begin{align*}
\varphi_{t}:=\int_{0}^{t}v(X_{u})\dif u,\quad t\geq 0,
\end{align*}
and the first passage times
\begin{align*}
\tau_{t}^{+}:=\inf\left\{\left.r\geq 0\,\right|\varphi_{r}>t\right\}\quad\text{and}\quad\tau_{t}^{-}:=\inf\left\{\left.r\geq 0\,\right|\varphi_{r}<-t\right\}.
\end{align*}
The main goal of this paper is to apply the Wiener-Hopf factorization technique, which we work out in Section \ref{subsec:th}, to compute the following expectations,
\begin{align}\label{eq:tau0+}
\Pi_{c}^{+}(i,j;s_{1},\ldots,s_{n})&:=\mathbb{E}\left(e^{-c\tau_{0}^{+}}\mathbbm{1}_{\{X_{\tau_{0}^{+}}=j\}}|X_{0}=i\right),\quad i\in\mathbf{E}^{-},\,j\in\mathbf{E}^{+},\\
\label{eq:taut+}
\Psi_{c}^{+}(\ell,i,j;s_{1},\ldots,s_{n})&:=\mathbb{E}\left(e^{-c\tau_{\ell}^{+}}\mathbbm{1}_{\{X_{\tau_{\ell}^{+}}=j\}}|X_{0}=i\right),\quad i\in\mathbf{E}^{+},\,j\in\mathbf{E}^{+},\,\ell>0, \\
\label{eq:tau0-}
\Pi_{c}^{-}(i,j;s_{1},\ldots,s_{n})&:=\mathbb{E}\left(e^{-c\tau_{0}^{-}}\mathbbm{1}_{\{X_{\tau_{0}^{-}}=j\}}|X_{0}=i\right),\quad i\in\mathbf{E}^{+},\,j\in\mathbf{E}^{-},\\
\label{eq:taut-}
\Psi_{c}^{-}(\ell,i,j;s_{1},\ldots,s_{n})&:=\mathbb{E}\left(e^{-c\tau_{\ell}^{-}}\mathbbm{1}_{\{X_{\tau_{\ell}^{-}}=j\}}|X_{0}=i\right),\quad i\in\mathbf{E}^{-},\,j\in\mathbf{E}^{-},\,\ell>0.
\end{align}
We will focus on the computation of $\Pi_{c}^{+}(i,j;s_{1},\ldots,s_{n})$ and $\Psi_{c}^{+}(\ell,i,j;s_{1},\ldots,s_{n})$. By symmetry, analogous results can be obtained for $\Pi_{c}^{-}(i,j;s_{1},\ldots,s_{n})$ and $\Psi_{c}^{-}(\ell,i,j;s_{1},\ldots,s_{n})$.
To simplify the notations, we will frequently write $\Pi_{c}^{+}(i,j)$ and $\Psi_{c}^{+}(\ell,i,j)$ in place of $\Pi_{c}^{+}(i,j;s_{1},\ldots,s_{n})$ and $\Psi_{c}^{+}(\ell,i,j;s_{1},\ldots,s_{n})$, respectively.
\section{A randomization method and the Wiener-Hopf factorization}\label{subsec:th}
In this section we construct a {\it time-homogeneous} Markov chain associated to $X$, by randomizing the discontinuity times $s_{1},\ldots,s_{n}$ of the generator $\mathsf{G}$. This key construction will allow us to compute the expectations \eqref{eq:tau0+} and \eqref{eq:taut+} using analogous expectations corresponding to this time-homogeneous chain. The latter expectations can be computed using Wiener-Hopf factorization theory of \cite{Barlow1980}.
Define $\mathbb{N}_{n}:={\{0,\ldots,n\}}$, $\widetilde{\mathbf{E}}:=\mathbb{N}_{n}\times\mathbf{E}$ and let $(\widetilde{\Omega},\widetilde{\mathcal{F}},\widetilde{\mathbb{P}})$ be a complete probability space. Next, let us consider a {\it time-homogeneous} Markov chain, say $Z=(N,Y):=(N_{t},Y_{t})_{t\geq 0}$, defined on $(\widetilde{\Omega},\widetilde{\mathcal{F}},\widetilde{\mathbb{P}})$, taking values in $\widetilde{\mathbf{E}}$ and with generator matrix $\widetilde{\mathsf{G}}\left((n_{1},j_{1}),(n_{2},j_{2}) \right)_{(n_{1},j_{1}),(n_{2},j_{2})\in \widetilde{\mathbf{E}}}$ given as
\begin{align*}
\widetilde{\mathsf{G}}=\kbordermatrix{ & \{0\}\times\mathbf{E} & \{1\}\times\mathbf{E} & \cdots & \{n-1\}\times\mathbf{E} & \{n\}\times\mathbf{E} \\ \{0\}\times\mathbf{E} & \mathsf{G}_{1}-q_{1}\mathsf{I} & q_{1}\mathsf{I} & \cdots & 0 & 0 \\ \{1\}\times\mathbf{E} & 0 & \mathsf{G}_{2}-q_{2}\mathsf{I} & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ \{n-1\}\times\mathbf{E} & 0 & 0 & \cdots & \mathsf{G}_{n}-q_{n}\mathsf{I} & q_{n}\mathsf{I} \\ \{n\}\times\mathbf{E} & 0 & 0 & \cdots & 0 & \mathsf{G}_{n+1}},
\end{align*}
where $q_{1},\ldots,q_{n}$ are positive constants and $\mathsf{I}$ is the identity matrix. For each $i\in\mathbf{E}$, we define the probability measure $\widetilde{\mathbb{P}}^{i}$ on $(\widetilde{\Omega},\widetilde{\mathcal{F}})$ by
\begin{align}\label{eq:TildePi}
\widetilde{\mathbb{P}}^{i}(A):=\widetilde{\mathbb{P}}\left(\left.A\,\right|Z_{0}=(0,i)\right),\quad A\in\widetilde{\mathcal{F}}.
\end{align}
The next result regards the Markov property of process $N$.
\begin{proposition}\label{prop:NMC}
For any $i\in\mathbf{E}$, the process $N$ is a time-homogeneous Markov chain under $\widetilde{\mathbb{P}}^{i}$, with generator matrix given by
\begin{align*}
\widetilde{\mathsf{G}}_{N}=\kbordermatrix{ & 0 & 1 & \cdots & n-1 & n \\ 0 & -q_{1} & q_{1} & \cdots & 0 & 0 \\ 1 & 0 & -q_{2} & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ n-1 & 0 & 0 & \cdots & -q_{n} & q_{n} \\ n & 0 & 0 & \cdots & 0 & 0}.
\end{align*}
\end{proposition}
\begin{proof} We will proceed in three steps.
\noindent
\textsf{Step 1.} We start by showing that
\begin{align}\label{eq:TildeGtoGN}
\sum_{j_{2}\in\mathbf{E}}\left(\widetilde{\mathsf{G}}^{k}\right)\left((n_{1},j_{1}),(n_{2},j_{2}) \right)=\left(\widetilde{\mathsf{G}}_{N}^{k}\right)(n_{1},n_{2}),
\end{align}
for any $j_{1}\in\mathbf{E}$, $k\in\mathbb{N}$, and $0\leq n_{1},n_{2}\leq n$,
In particular, note that the left-hand-side of \eqref{eq:TildeGtoGN} does not depend on $j_{1}$.
We will prove \eqref{eq:TildeGtoGN} by induction in $k$. Clearly \eqref{eq:TildeGtoGN} holds true for $k=1$.
Next, assume that \eqref{eq:TildeGtoGN} holds for some $k=\ell\in\mathbb{N}$. Now, for $\ell+1$,
\begin{align*}
\sum_{j_{2}\in\mathbf{E}}\left(\widetilde{\mathsf{G}}^{\ell+1}\right)\left((n_{1},j_{1}),(n_{2},j_{2})\right)&=\sum_{j_{2}\in\mathbf{E}}\sum_{m=0}^{n}\sum_{j\in\mathbf{E}}\left(\widetilde{\mathsf{G}}^{\ell}\right)\left((n_{1},j_{1}),(m,j)\right)\widetilde{\mathsf{G}}\left((m,j),(n_{2},j_{2})\right)\\
&=\sum_{m=0}^{n}\sum_{j\in\mathbf{E}}\left(\widetilde{\mathsf{G}}^{\ell}\right)\left((n_{1},j_{1}),(m,j)\right)\sum_{j_{2}\in\mathbf{E}}\widetilde{\mathsf{G}}\left((m,j),(n_{2},j_{2})\right)\\
&=\sum_{m=0}^{n}\sum_{j\in\mathbf{E}}\left(\widetilde{\mathsf{G}}^{\ell}\right)\left((n_{1},j_{1}),(m,j)\right)\widetilde{\mathsf{G}}_{N}(m,n_{2})\\
&=\sum_{m=0}^{n}\left(\widetilde{\mathsf{G}}_{N}^{\ell}\right)(n_{1},m)\widetilde{\mathsf{G}}_{N}(m,n_{2})=\left(\widetilde{\mathsf{G}}_{N}^{\ell+1}\right)(n_{1},n_{2}),
\end{align*}
where we used the inductive assumptions for $k=1$ and $k=\ell$ in the third and the fourth equalities, respectively. Hence \eqref{eq:TildeGtoGN} is established.
\noindent
\textsf{Step 2.} We will show that
\begin{align}\label{eq:NNotDependonY}
\widetilde{\mathbb{P}}^{i}\left(\left.N_{t+s}=n_{2}\,\right|N_{t}=n_{1}\right)=\widetilde{\mathbb{P}}^{i}\!
\left(\left.N_{t+s}=n_{2}\,\right|N_{t}=n_{1},Y_{t}=j\right)=e^{s\widetilde{\mathsf{G}}_{N}}(n_{1},n_{2}),
\end{align}
for any $t,s\geq 0$, $j\in\mathbf{E}$, and $0\leq n_{1}\leq n_{2}\leq n$.
In particular, note that the left-hand side of \eqref{eq:NNotDependonY}, and thus $\widetilde{\mathbb{P}}^{i}(N_{t+s}=n_{2}|N_{t}=n_{1})$, does not depend on $t$.
We start by checking the second equality in \eqref{eq:NNotDependonY}. For any $t,s\geq 0$, $j\in\mathbf{E}$, and $0\leq n_{1}\leq n_{2}\leq n$,
\begin{align*}
\widetilde{\mathbb{P}}^{i}\left(\left.N_{t+s}=n_{2}\,\right|N_{t}=n_{1},Y_{t}=j\right)&=\sum_{k\in\mathbf{E}}\widetilde{\mathbb{P}}^{i}\left(\left.N_{t+s}=n_{2},Y_{t+s}=k\,\right|N_{t}=n_{1},Y_{t}=j\right)\\
&=\sum_{k\in\mathbf{E}}e^{s\,\widetilde{\mathsf{G}}}\left((n_{1},j),(n_{2},k)\right)=\sum_{k\in\mathbf{E}}\sum_{\ell=0}^{\infty}\frac{s^{\ell}}{\ell!}\,\widetilde{\mathsf{G}}^{\ell}\left((n_{1},j),(n_{2},k)\right)\\
&=\sum_{\ell=0}^{\infty}\frac{s^{\ell}}{\ell!}\sum_{k\in\mathbf{E}}\widetilde{\mathsf{G}}^{\ell}\left((n_{1},j),(n_{2},k)\right)\\
&=\sum_{\ell=0}^{\infty}\frac{s^{\ell}}{\ell!}\,\widetilde{\mathsf{G}}^{\ell}_{N}(n_{1},n_{2})=e^{s\widetilde{\mathsf{G}}_{N}}(n_{1},n_{2}),
\end{align*}
where we used the result of Step 1 in the last two equalities. In particular, $\widetilde{\mathbb{P}}^{i}(N_{t+s}=n_{2}|N_{t}=n_{1},Y_{t}=j)$ does not depend on the choice of $j\in\mathbf{E}$.
As far as the first equality in \eqref{eq:TildeGtoGN}, for any $t,s\geq 0$ and $0\leq n_{1}\leq n_{2}\leq n$,
\begin{align*}
\widetilde{\mathbb{P}}^{i}\left(\left.N_{t+s}=n_{2}\,\right|N_{t}=n_{1}\right)&=\frac{\widetilde{\mathbb{P}}^{i}\left(N_{t+s}=n_{2},N_{t}=n_{1}\right)}{\widetilde{\mathbb{P}}^{i}\left(N_{t}=n_{1}\right)}=\frac{\sum_{\ell\in\mathbf{E}}\widetilde{\mathbb{P}}^{i}\left(N_{t+s}=n_{2},N_{t}=n_{1},Y_{t}=j\right)}{\sum_{j\in\mathbf{E}}\widetilde{\mathbb{P}}^{i}\left(N_{t}=n_{1},Y_{t}=j\right)}\\
&=\frac{\sum_{j\in\mathbf{E}}\widetilde{\mathbb{P}}^{i}\left(\left.N_{t+s}=n_{2}\,\right|N_{t}=n_{1},Y_{t}=j\right)\widetilde{\mathbb{P}}^{i}\left(N_{t}=n_{1},Y_{t}=j\right)}{\sum_{j\in\mathbf{E}}\widetilde{\mathbb{P}}^{i}\left(N_{t}=n_{1},Y_{t}=j\right)}\\
&=\frac{\sum_{j\in\mathbf{E}}\widetilde{\mathbb{P}}^{i}\left(N_{t}=n_{1},Y_{t}=j\right)}{\sum_{j\in\mathbf{E}}\widetilde{\mathbb{P}}^{i}\left(N_{t}=n_{1},Y_{t}=j\right)}\,e^{s\widetilde{\mathsf{G}}_{N}}(n_{1},n_{2})=e^{s\widetilde{\mathsf{G}}_{N}}(n_{1},n_{2}).
\end{align*}
\noindent
\textsf{Step 3.} We are ready to complete the proof of the proposition. Towards this end we observe that, for any $m\in\mathbb{N}$, $0=t_{0}\leq t_{1}<\ldots<t_{m}$, and any $0\leq n_{1}\leq\ldots\leq n_{m}\leq n$,
\begin{align*}
\widetilde{\mathbb{P}}^{i}(N_{t_{m}}=n_{m}\,\;|\; N_{t_{m-1}} = & \, n_{m-1},\ldots,N_{t_{1}}=n_{1})=\frac{\widetilde{\mathbb{P}}^{i}\left(N_{t_{1}}=n_{1},\ldots,N_{t_{m}}=n_{m}\right)}{\widetilde{\mathbb{P}}^{i}\left(N_{t_{1}}=n_{1},\ldots,N_{t_{m-1}}=n_{m-1}\right)}\\
& =\frac{\sum_{j_{1},\ldots,j_{m}\in\mathbf{E}}\widetilde{\mathbb{P}}^{i}\left(N_{t_{1}}=n_{1},Y_{t_{1}}=j_{1};\ldots;N_{t_{m}}=n_{m},Y_{t_{m}}=j_{m}\right)}{\sum_{j_{1},\ldots,j_{m-1}\in\mathbf{E}}\widetilde{\mathbb{P}}^{i}\left(N_{t_{1}}=n_{1},Y_{t_{1}}=j_{1};\ldots;N_{t_{m-1}}=n_{m-1},Y_{t_{m}}=j_{m-1}\right)}\\
& =\frac{\sum_{j_{1},\ldots,j_{m}\in\mathbf{E}}\prod_{k=1}^{m}\widetilde{\mathbb{P}}^{i}\left(\left.N_{t_{k}}=n_{k},Y_{t_{k}}=j_{k}\,\right|N_{t_{k-1}}=n_{k-1},Y_{t_{k-1}}=j_{k-1}\right)}{\sum_{j_{1},\ldots,j_{m-1}\in\mathbf{E}}\prod_{k=1}^{m-1}\widetilde{\mathbb{P}}^{i}\left(\left.N_{t_{k}}=n_{k},Y_{t_{k}}=j_{k}\,\right|N_{t_{k-1}}=n_{k-1},Y_{t_{k-1}}=j_{k-1}\right)}\\
&=\sum_{j_{m}\in\mathbf{E}}\widetilde{\mathbb{P}}^{i}\left(\left.N_{t_{m}}=n_{m},Y_{t_{m}}=j_{m}\,\right|N_{t_{m-1}}=n_{m-1},Y_{t_{m-1}}=j_{m-1}\right)\\
&=\widetilde{\mathbb{P}}^{i}\left(\left.N_{t_{m}}=n_{m}\,\right|N_{t_{m-1}}=n_{m-1},Y_{t_{m-1}}=j_{m-1}\right)\\
&=\widetilde{\mathbb{P}}^{i}\left(\left.N_{t_{m}}=n_{m}\,\right|N_{t_{m-1}}=n_{m-1}\right)=e^{(t_{m}-t_{m-1})\widetilde{\mathsf{G}}_{N}}(n_{m-1},n_{m}),
\end{align*}
where we used the Markov property of $Z=(N,Y)$ under $\widetilde{\mathbb{P}}^{i}$ in the third equality, and the result of Step 2 in the last two equalities.
The proof is complete.
\end{proof}
Let $\widetilde \mathbb{F}^Y=(\widetilde {\mathcal{F}}_{t}^{Y})_{t\geq 0}$ be the filtration generated by $Y$, and let $\widetilde{\mathcal{F}}_{\infty}^{Y}=\sigma(\bigcup_{t\geq 0}\widetilde{\mathcal{F}}_{t}^{Y})$. For each $i\in\mathbf{E}$, we will construct a probability measure $\overline{\mathbb{P}}^{i}$ on $(\widetilde{\Omega},\widetilde{\mathcal{F}}_{\infty}^{Y})$ such that, the law of $Y$ under $\overline{\mathbb{P}}^{i}$ is the same as the law of $X$ under $\mathbb{P}^{i}$. Moreover, we will establish a connection between $\overline{\mathbb{P}}^{i}$ and $\widetilde{\mathbb{P}}^{i}$. For this purpose, we first let
\begin{align*}
S_{k}:=\inf\left\{t\geq 0\,|\,N_{t}=k\right\},\quad k=1,\ldots,n.
\end{align*}
We will now derive the joint density of $N$, and $(S_{1},\ldots,S_{n})$ under $\widetilde{\mathbb{P}}^{i}$. For that, we set
\begin{align}\label{eq:StoT}
T_{1}:=S_{1},\quad T_{k}:=S_{k}-S_{k-1},\quad k=2,\ldots,n.
\end{align}
It is shown in \cite[Section 1.1.4]{Syski1992} that $T_{k}$'s are independent and that
\begin{align*}
\widetilde{\mathbb{P}}^{i}\left(T_{1}>t_{1},\ldots,T_{n}>t_{n} \right)=\prod_{k=1}^{n}\,e^{-q_{k}t_{k}},\quad t_{1},\ldots,t_{n}>0,
\end{align*}
which implies that the joint density of $(T_{1},\ldots,T_{n})$ is given by
\begin{align}\label{eq:densityT}
f_{T_{1},\ldots,T_{n}}(t_{1},\ldots,t_{n})=\prod_{k=1}^{n}q_{k}\,e^{-q_{k}t_{k}},\quad t_{1},\ldots,t_{n}>0.
\end{align}
Combining \eqref{eq:StoT} and \eqref{eq:densityT}, we deduce that
\begin{align*}
f_{S_{1},\ldots,S_{n}}(s_{1},\ldots,s_{n})=\prod_{k=1}^{n}q_{k}\,e^{-q_{k}(s_{k}-s_{k-1})},\quad 0=s_{0}<s_{1}<\ldots<s_{n}.
\end{align*}
\begin{theorem}\label{thm:ConstBarPi}
For any $i\in\mathbf{E}$, any $0<s_{1}<\ldots<s_{n}$, and any cylinder set $A\in\widetilde{\mathcal{F}}_{\infty}^{Y}$ of the form
\begin{align*}
A=\left\{\left(Y_{u_{1}},\ldots,Y_{u_{m}}\right)\in B\right\},\quad 0\leq u_{1}<u_{2}<\ldots<u_{m},\quad B\subseteq\mathbf{E}^{m},\quad m\in\mathbb{N},
\end{align*}
the limit
\begin{align}\label{eq:DefBarPiCylSet}
\overline{\mathbb{P}}^{i}\left(A;s_{1},\ldots,s_{n}\right):=\lim_{\Delta s_{k}\rightarrow 0,\,k=1,\ldots,n}\frac{\widetilde{\mathbb{P}}^{i}\left(A,\,s_{k}<S_{k}\leq s_{k}+\Delta s_{k},\,k=1,\ldots,n\right)}{\widetilde{\mathbb{P}}^{i}\left(s_{k}<S_{k}\leq s_{k}+\Delta s_{k},\,k=1,\ldots,n\right)},
\end{align}
exists, and can be extended to a probability measure $\overline{\mathbb{P}}^{i}(\cdot\,;s_{1},\ldots,s_{n})$ on $(\widetilde{\Omega},\widetilde{\mathcal{F}}_{\infty}^{Y})$. Moreover, for any $A\in\widetilde{\mathcal{F}}_{\infty}^{Y}$, the function $\overline{\mathbb{P}}^{i}(A;\,\ldots)$ is Borel measurable on $\{(s_{1},\ldots,s_{n})\in\mathbb{R}^{n}\,|\,0<s_{1}<\ldots<s_{n}\}$, and
\begin{align}\label{eq:TildePiBarPi}
\widetilde{\mathbb{P}}^{i}(A)=\int_{0}^{\infty}\int_{s_{1}}^{\infty}\cdots\int_{s_{n-1}}^{\infty}\overline{\mathbb{P}}^{i}(A;s_{1},\ldots,s_{n})\prod_{k=1}^{n}\left(q_{k}\,e^{-q_{k}(s_{k}-s_{k-1})}\right)ds_{n}\cdots ds_{2}\,ds_{1}.
\end{align}
\end{theorem}
In the proof of the theorem we will use the following lemma.
\begin{lemma}\label{lem:ConstBarPiCylSet}
Let us fix $i\in\mathbf{E}$, $0<s_{1}<\ldots<s_{n}$, and let $0=k_{0}<k_{1}<\ldots<k_{n+1}$ be positive integers. In addition, let $0=u_0< u_{1}<\ldots<u_{k_{1}}\leq s_{1}<u_{k_{1}+1}<\ldots<u_{k_{2}}\leq s_{2}<\ldots\leq s_{n}<u_{k_{n}+1}<\ldots<u_{k_{n+1}}$, $i_0=i$ and $i_{1},\ldots,i_{k_{n+1}}\in\mathbf{E}$. Then, for any
cylinder set $A\in\widetilde{\mathcal{F}}_{\infty}^{Y}$ of the form
\begin{align}\label{eq:CylSet}
A=\bigcap_{j=0}^{n}\left\{Y_{u_{k_{j}+1}}=i_{k_{j}+1},\ldots,Y_{u_{k_{j+1}}}\!=i_{k_{j+1}}\right\}
\end{align}
we have
\begin{align}
\lim_{\Delta s_{\ell}\rightarrow 0,\,\ell=1,\ldots,n}&\frac{\widetilde{\mathbb{P}}^{i}\left(A,\,s_{\ell}<S_{\ell}\leq s_{\ell}+\Delta s_{\ell},\,\ell=1,\ldots,n\right)}{\widetilde{\mathbb{P}}^{i}\left(s_{\ell}<S_{\ell}\leq s_{\ell}+\Delta s_{\ell},\,\ell=1,\ldots,n\right)}
=\!\prod_{\ell=0}^{n}\!\left(\!\prod_{m=k_{\ell}+1}^{k_{\ell+1}}\!\!\!\!e^{(u_{m}-u_{m-1})
\mathsf{G}_{\ell}}\!(i_{m-1},i_{m})\!\right) \nonumber \\
& \qquad\quad \cdot\sum_{j_{1},\ldots,j_{n}\in\mathbf{E}}\prod_{\ell=1}^{n}e^{(s_{\ell}-u_{k_{\ell}})
\mathsf{G}_{\ell-1}}\!(i_{k_{\ell}},j_{\ell})e^{(u_{k_{\ell}+1}-s_{\ell})\mathsf{G}_{\ell}}\!(j_{\ell},i_{k_{\ell}+1}).
\label{eq:BarPiSimCylSet}
\end{align}
In particular, for any $A\in\widetilde{\mathcal{F}}_{\infty}^{Y}$ of the form \eqref{eq:CylSet}, the above limit is Borel measurable with respect to $(s_{1},\ldots,s_{n})$ in $\Delta_n:=\{(s_{1},\ldots,s_{n})\in\mathbb{R}^{n}\,|\,0<s_{1}<\ldots<s_{n}\}$.
\end{lemma}
\begin{proof}
For $\ell=1,\ldots,n$ choose $\Delta s_{\ell}>0$ so that, $s_{\ell}+\Delta s_{\ell}\leq u_{k_{\ell}+1}$. Then,
\begin{align*}
&\widetilde{\mathbb{P}}^{i}\left(A,\,s_{\ell}<S_{\ell}\leq s_{\ell}+\Delta s_{\ell},\,\ell=1,\ldots,n\right)\\
&\quad =\widetilde{\mathbb{P}}^{i}\left(Y_{u_{k_{\ell}+1}}=i_{k_{\ell}+1},\ldots,Y_{u_{k_{\ell+1}}}=i_{k_{\ell+1}},\,\ell=0,\ldots,n;\,N_{s_{\ell}}=\ell-1,N_{s_{\ell}+\Delta s_{\ell}}=\ell,\,\ell=1,\ldots,n\right)\\
&\quad =\sum_{j_{1},\ldots,j_{n},\,j_{1}',\ldots,j_{n}'\in\mathbf{E}}\widetilde{\mathbb{P}}^{i}\left(Z_{u_{k_{\ell}+1}}=(\ell,i_{k_{\ell}+1}),\,\ldots,\,Z_{u_{k_{\ell+1}}}=(\ell,i_{k_{\ell+1}}),\,\ell=0,\ldots,n;\right.\\
&\qquad\qquad\qquad\qquad\qquad\qquad\,\,\,Z_{s_{\ell}}=(\ell-1,j_{\ell}),\,Z_{s_{\ell}+\Delta s_{\ell}}=(\ell,j_{\ell}'),\,\,\ell=1,\ldots,n\Big)\\
&\quad =\sum_{j_{1},\ldots,j_{n},\,j_{1}',\ldots,j_{n}'\in\mathbf{E}}\!\left[\prod_{\ell=0}^{n}\left(\prod_{m=k_{\ell}+1}^{k_{\ell+1}}\!e^{(u_{m}-u_{m-1})\widetilde{\mathsf{G}}}\!\left((\ell,i_{m-1}),(\ell,i_{m})\right)\right)\right]\left(\prod_{\ell=1}^{n}e^{\Delta s_{\ell}\widetilde{\mathsf{G}}}\!\left((\ell-1,j_{\ell}),(\ell,j_{\ell}')\right)\right)\\
&\qquad\qquad\qquad\qquad\quad\cdot\left(\prod_{\ell=1}^{n}e^{(s_{\ell}-u_{k_{\ell}})\widetilde{\mathsf{G}}}\left((\ell-1,i_{k_{\ell}}),(\ell-1,j_{\ell})\right)e^{(u_{k_{\ell}+1}-s_{\ell}-\Delta s_{\ell})\widetilde{\mathsf{G}}}\left((\ell,j_{\ell}'),(\ell,i_{k_{\ell}+1})\right)\right).
\end{align*}
In the above summation, the first product in the brackets provides the transition probabilities of the evolutions of $Z$ between the times $u_{k_{\ell}}$ and $u_{k_{\ell+1}}$, $\ell=0,\ldots,n$, the second product gives the transition probabilities of the evolutions of $Z$ between the times $s_{\ell}$ and $s_{\ell}+\Delta s_{\ell}$, for each $\ell=1,\ldots,n$, and the third product denotes the transition probabilities of the evolutions of $Z$ between the times $u_{k_{\ell}}$ and $s_{\ell}$, and between the times $s_{\ell}+\Delta s_{\ell}$ and $u_{k_{\ell}+1}$, for each $\ell=1,\ldots,n$.
Next, for each $\ell=1,\ldots,n$,
\begin{align*}
\lim_{\Delta s_{\ell}\rightarrow 0}\frac{1}{\Delta s_{\ell}}\,e^{\Delta s_{\ell}\widetilde{\mathsf{G}}}\left((\ell-1,j_{\ell}),(\ell,j_{\ell}')\right)=\widetilde{\mathsf{G}}\left((\ell-1,j_{\ell}),(\ell,j_{\ell}')\right)=\left\{\begin{array}{ll} q_{\ell}, &\text{if }\,j_{\ell}=j_{\ell}', \\ 0, &\text{otherwise}. \end{array}\right.
\end{align*}
Hence,
\begin{align}
&\lim_{\Delta s_{\ell}\rightarrow 0,\,\ell=1,\ldots,n}\frac{1}{\Delta s_{1}\cdots\Delta s_{n}}\,\widetilde{\mathbb{P}}^{i}\left(A,\,s_{\ell}<S_{\ell}\leq s_{\ell}+\Delta s_{\ell},\,\ell=1,\ldots,n\right)\nonumber\\
&\quad =\prod_{\ell=0}^{n}\left(\prod_{m=k_{\ell}+1}^{k_{\ell+1}}e^{(u_{m}-u_{m-1})\widetilde{\mathsf{G}}}\!\left((\ell,i_{m-1}),(\ell,i_{m})\right)\right)\nonumber\\
\label{eq:LimProbASell} &\qquad\cdot\sum_{j_{1},\ldots,j_{n}\in\mathbf{E}}\,\prod_{\ell=1}^{n}\left(q_{\ell}\,e^{(s_{\ell}-u_{k_{\ell}})\widetilde{\mathsf{G}}}\left((\ell-1,i_{k_{\ell}}),(\ell-1,j_{\ell})\right)e^{(u_{k_{\ell}+1}-s_{\ell})\widetilde{\mathsf{G}}}\left((\ell,j_{\ell}),(\ell,i_{k_{\ell}+1})\right)\right).
\end{align}
Note that, for any $j_{1},j_{2}\in\mathbf{E}$, and any $k\in\mathbb{N}$,
\begin{align*}
\widetilde{\mathsf{G}}^{k}\left((\ell,j_{1}),(\ell,j_{2})\right)&=(\mathsf{G}_{\ell}-q_{\ell+1}\mathsf{I})^{k}(j_{1},j_{2}),\quad\ell=0,\ldots,n-1,\\
\widetilde{\mathsf{G}}^{k}\left((n,j_{1}),(n,j_{2})\right)&=\mathsf{G}_{n}^{k}(j_{1},j_{2}),
\end{align*}
so that, for $t\geq 0$, we have
\begin{align*}
e^{t\,\widetilde{\mathsf{G}}}\left((\ell,j_{1}),(\ell,j_{2})\right)&=e^{t\,(\mathsf{G}_{\ell}-q_{\ell+1}\mathsf{I})}(j_{1},j_{2})=e^{-q_{\ell+1}t}\,e^{t\,\mathsf{G}_{\ell}}(j_{1},j_{2}),\quad\ell=0,\ldots,n-1,\\
e^{t\,\widetilde{\mathsf{G}}}\left((n,j_{1}),(n,j_{2})\right)&=e^{t\,\mathsf{G}_{n}}(j_{1},j_{2}).
\end{align*}
This, together with \eqref{eq:LimProbASell}, implies that
\begin{align*}
&\lim_{\Delta s_{\ell}\rightarrow 0,\,\ell=1,\ldots,n}\frac{1}{\Delta s_{1}\cdots\Delta s_{n}}\,\widetilde{\mathbb{P}}^{i}\left(A,\,s_{\ell}<S_{\ell}\leq s_{\ell}+\Delta s_{\ell},\,\ell=1,\ldots,n\right)\\
&\quad =e^{-\sum_{\ell=1}^{n}q_{\ell}(u_{k_{\ell}}-u_{k_{\ell-1}})}\cdot\prod_{\ell=0}^{n}\left(\prod_{m=k_{\ell}+1}^{k_{\ell+1}}e^{(u_{m}-u_{m-1})\mathsf{G}_{\ell}}(i_{m-1},i_{m})\right)\\ &\qquad\,\,e^{-\sum_{\ell=1}^{n}q_{\ell}(s_{\ell}-u_{k_{\ell}})}e^{-\sum_{\ell=1}^{n-1}q_{\ell}(u_{k_{\ell}+1}-s_{\ell})}\!\!\!\sum_{j_{1},\ldots,j_{n}\in\mathbf{E}}\prod_{\ell=1}^{n}\!\left(q_{\ell}e^{(s_{\ell}-u_{k_{\ell}})\mathsf{G}_{\ell-1}}(i_{k_{\ell}},j_{\ell})e^{(u_{k_{\ell}+1}-s_{\ell})\mathsf{G}_{\ell}}(j_{\ell},i_{k_{\ell}+1})\right)\\
&\quad =e^{-\sum_{\ell=1}^{n}q_{\ell}(s_{\ell}-s_{\ell-1})}\cdot\prod_{\ell=0}^{n}\left(\prod_{m=k_{\ell}+1}^{k_{\ell+1}}e^{(u_{m}-u_{m-1})\mathsf{G}_{\ell}}(i_{m-1},i_{m})\right)\\
&\qquad\,\cdot\sum_{j_{1},\ldots,j_{n}\in\mathbf{E}}\,\prod_{\ell=1}^{n}\left(q_{\ell}\,e^{(s_{\ell}-u_{k_{\ell}})\mathsf{G}_{\ell-1}}(i_{k_{\ell}},j_{\ell})\,e^{(u_{k_{\ell}+1}-s_{\ell})\mathsf{G}_{\ell}}(j_{\ell},i_{k_{\ell}+1})\right)\\
&\quad =\left(\prod_{\ell=1}^{n}q_{\ell}\,e^{-q_{\ell}(s_{\ell}-s_{\ell-1})}\right)\cdot\left[\prod_{\ell=0}^{n}\left(\prod_{m=k_{\ell}+1}^{k_{\ell+1}}e^{(u_{m}-u_{m-1})\mathsf{G}_{\ell}}(i_{m-1},i_{m})\right)\right]\\
&\qquad\,\,\cdot\sum_{j_{1},\ldots,j_{n}\in\mathbf{E}}\,\prod_{\ell=1}^{n}\left(e^{(s_{\ell}-u_{k_{\ell}})\mathsf{G}_{\ell-1}}(i_{k_{\ell}},j_{\ell})\,e^{(u_{k_{\ell}+1}-s_{\ell})\mathsf{G}_{\ell}}(j_{\ell},i_{k_{\ell}+1})\right).
\end{align*}
Finally, in view of the above and the fact that
\begin{align}\label{eq:nice-fact}
\lim_{\Delta s_{\ell}\rightarrow 0,\,\ell=1,\ldots,n}\frac{1}{\Delta s_{1}\cdots\Delta s_{n}}\,\widetilde{\mathbb{P}}^{i}\left(s_{\ell}<S_{\ell}\leq s_{\ell}+\Delta s_{\ell},\,\ell=1,\ldots,n\right)=\prod_{\ell=1}^{n}q_{\ell}\,e^{-q_{\ell}(s_{\ell}-s_{\ell-1})},
\end{align}
we obtain \eqref{eq:BarPiSimCylSet}. The proof is complete.
\end{proof}
We are now ready to prove Theorem \ref{thm:ConstBarPi}.
\begin{proof}[Proof of Theorem \ref{thm:ConstBarPi}]
Let $\mathcal{C}$ be the collection of all cylinder sets in $\widetilde{\mathcal{F}}_{\infty}^{Y}$ of the form
\begin{align*}
C=\left\{\left(Y_{u_{1}},\ldots,Y_{u_{m}}\right)\in B\right\},\quad 0\leq u_{1}<u_{2}<\ldots<u_{m},\quad B\subseteq\mathbf{E}^{m},\quad m\in\mathbb{N}.
\end{align*}
Clearly, $\mathcal{C}$ is an algebra.
We first show that for any $C\in \mathcal{C}$ the limit in \eqref{eq:DefBarPiCylSet} exists and that an explicit formula for it can be derived. In fact, Lemma \ref{lem:ConstBarPiCylSet} shows that the limit in \eqref{eq:DefBarPiCylSet} exists, and belongs to $[0,1]$, for all the cylinder sets of the form \eqref{eq:CylSet}. Thus, for a cylinder set $C\in \mathcal{C}$ an explicit formula for the limit on the right-hand side of \eqref{eq:DefBarPiCylSet} can be obtained as follows. First, we refine the partition $0\leq u_{1}<u_{2}<\ldots<u_{m}$ so that each subinterval of the partition $0<s_{1}<\ldots<s_{n}$ contains at least one of the $u_i$'s. Clearly, since $B_m$ is finite, $A$ can be decomposed into a finite union of disjoint cylinder sets of the form \eqref{eq:CylSet} on the refined partition. Moreover, \eqref{eq:BarPiSimCylSet} provides an explicit formula for the limit in \eqref{eq:DefBarPiCylSet} for each of those cylinder sets of the form \eqref{eq:CylSet} on the refined partition. Finally, taking the finite sum over all those limits, we obtain the limit in \eqref{eq:DefBarPiCylSet} for $C$. In particular, for every cylinder set $C$, the limit in \eqref{eq:DefBarPiCylSet} is Borel measurable with respect to $(s_{1},\ldots,s_{n})$ in $\Delta_{n}$.
In the second step we will demonstrate that the limit in \eqref{eq:DefBarPiCylSet} can be extended to a probability measure on $\sigma(\mathcal{C})=\widetilde{\mathcal{F}}_{\infty}^{Y}$.
We start from verifying the countable additivity of $\overline{\mathbb{P}}^{i}(\cdot\,;s_{1},\ldots,s_{n})$ on $\mathcal{C}$ for any fixed $0<s_{1}<\ldots<s_{n}$.
Since $\mathbf{E}$ is a finite set, if $(C_{k})_{k\in\mathbb{N}}$ is a sequence of disjoint cylinder sets in $\mathcal{C}$ such that their union also belongs to $\mathcal{C}$, then only finite many of them are non-empty. Therefore, it suffices to verify the finite additivity of $\overline{\mathbb{P}}^{i}(\cdot\,;s_{1},\ldots,s_{n})$ on $\mathcal{C}$. Let $C_{1},\ldots,C_{k}\in\mathcal{C}$ be disjoint cylinder sets, then there exists $m\in\mathbb{N}$ and $0\leq u_{1}<u_{2}<\ldots<u_{m}$, such that
\begin{align*}
C_{\ell}=\left\{\left(Y_{u_{1}},\ldots,Y_{u_{m}}\right)\in B_{\ell}\right\}\,\,\,\,\text{for some }\,B_{\ell}\subseteq\mathbf{E}^{m},\quad\ell=1,\ldots,k.
\end{align*}
Each $\overline{\mathbb{P}}^{i}(C_{\ell}\,;s_{1},\ldots,s_{n})$ can be represented as
\begin{align*}
\overline{\mathbb{P}}^{i}(C_{\ell}\,;s_{1},\ldots,s_{n})=\sum_{A_{\ell}\in\mathcal{C}_{\ell}}\overline{\mathbb{P}}^{i}(A_{\ell}\,;s_{1},\ldots,s_{n}),\quad j=1,\ldots,k,
\end{align*}
where $\mathcal{C}_{\ell}$, $\ell=1,\ldots,k$, are disjoint classes of disjoint simple cylinder sets. Therefore, we have
\begin{align*}
\sum_{\ell=1}^{k}\overline{\mathbb{P}}^{i}(C_{\ell}\,;s_{1},\ldots,s_{n})&=\sum_{\ell=1}^{k}\sum_{A_{\ell}\in\mathcal{C}_{\ell}}\overline{\mathbb{P}}^{i}(A_{\ell}\,;s_{1},\ldots,s_{n})\\
&=\sum_{A\in\mathcal{C}_{1}\cup\cdots\cup\mathcal{C}_{k}}\overline{\mathbb{P}}^{i}(A\,;s_{1},\ldots,s_{n})=\overline{\mathbb{P}}^{i}\left(\bigcup_{\ell=1}^{k}C_{\ell}\,;s_{1},\ldots,s_{n}\right).
\end{align*}
Note that for any $0<s_{1}<\ldots<s_{n}$, $\overline{\mathbb{P}}^{i}(C\,;s_{1},\ldots,s_{n})\leq 1$ for all $C\in\mathcal{C}$. By the Carath\'{e}odory extension theorem, for any $0<s_{1}<\ldots<s_{n}$, $\overline{\mathbb{P}}^{i}(\cdot\,;s_{1},\ldots,s_{n})$ can be uniquely extended to a probability measure on $(\widetilde{\Omega},\widetilde{\mathcal{F}}_{\infty}^{Y})$.
Let $\Delta_{n}:=\{(s_{1},\ldots,s_{n})\in\mathbb{R}^{n}\,|\,0<s_{1}<\ldots<s_{n}\}$
and
\begin{align*}
\mathcal{D}_{1}:=\left\{\left.A\in\widetilde{\mathcal{F}}_{\infty}^{Y}\,\right|\overline{\mathbb{P}}^{i}(A\,;\,\cdot,\cdots,\cdot)\,\,\text{is Borel measurable on }\Delta_{n}\right\}.
\end{align*}
We will show that $\mathcal{D}_{1}=\widetilde{\mathcal{F}}_{\infty}^{Y}$. Towards this end, we first observe that \eqref{eq:DefBarPiCylSet} and \eqref{eq:BarPiSimCylSet} imply that, for any $A\in \mathcal{C}$, $\overline{\mathbb{P}}^{i}(A\,;\,\cdot,\cdots,\cdot)$ is Borel measurable with respect to $(s_{1},\ldots,s_{n})$ on $\Delta_{n}$, and thus $\mathcal{D}_{1}\supset\mathcal{C}$.
Next, we will show that $\mathcal{D}_{1}$ is a monotone class. For this, let $(A_{k})_{k\in\mathbb{N}}\subset\mathcal{D}_{1}$ be an increasing sequence of events, so that, for any $0<s_{1}<\ldots<s_{n}$, we have
\begin{align*}
\overline{\mathbb{P}}^{i}\left(\bigcup_{k=1}^{\infty}A_{k}\,;\,s_{1},\ldots,s_{n}\right)=\lim_{m\rightarrow\infty}\overline{\mathbb{P}}^{i}\left(A_{m}\,;\,s_{1},\ldots,s_{n}\right).
\end{align*}
Thus, $\overline{\mathbb{P}}^{i}(\cup_{k}A_{k}\,;\cdot,\cdots,\cdot)$, being a limit of a sequence of Borel measurable functions on $\Delta_{n}$, is Borel measurable on $\Delta_{n}$, and hence $\cup_{k}A_{k}\in\mathcal{D}_{1}$.
Similarly, one can show that if $(A_{k})_{k\in\mathbb{N}}\subset\mathcal{D}_{1}$ is a decreasing sequence of events, then $\cap_{k}A_{k}\in\mathcal{D}_{1}$. Therefore, $\mathcal{D}_{1}$ is a monotone class, and by the monotone class theorem $\mathcal{D}_{1}=\sigma(\mathcal{C})=\widetilde{\mathcal{F}}_{\infty}^{Y}$.
It remains to show that \eqref{eq:TildePiBarPi} holds true. In view of \eqref{eq:DefBarPiCylSet} and \eqref{eq:nice-fact}, for any cylinder set $A\in\mathcal{C}$,
\begin{align*}
\overline{\mathbb{P}}^{i}(A\,;s_{1},\ldots,s_{n})&=\lim_{\Delta s_{k}\rightarrow 0,\,k=1,\ldots,n}\frac{\widetilde{\mathbb{P}}^{i}\left(A,\,s_{k}<S_{k}\leq s_{k}+\Delta s_{k},\,k=1,\ldots,n\right)}{\widetilde{\mathbb{P}}^{i}\left(s_{k}<S_{k}\leq s_{k}+\Delta s_{k},\,k=1,\ldots,n\right)}\\
&=\frac{\lim_{\Delta s_{k}\rightarrow 0,\,k=1,\ldots,n}(\Delta s_{1}\cdots\Delta s_{n})^{-1}\,\widetilde{\mathbb{P}}^{i}\left(A,\,s_{k}<S_{k}\leq s_{k}+\Delta s_{k},\,k=1,\ldots,n\right)}{\lim_{\Delta s_{k}\rightarrow 0,\,k=1,\ldots,n}(\Delta s_{1}\cdots\Delta s_{n})^{-1}\,\widetilde{\mathbb{P}}^{i}\left(s_{k}<S_{k}\leq s_{k}+\Delta s_{k},\,k=1,\ldots,n\right)}\\
&=\frac{\partial^{n}}{\partial s_{1}\cdots\partial s_{n}}\widetilde{\mathbb{P}}^{i}\left(A,\,S_{k}\leq s_{k},\,k=1,\ldots,n\right)\cdot\left(\prod_{k=1}^{n}q_{k}\,e^{-q_{k}(s_{k}-s_{k-1})}\right)^{-1}.
\end{align*}
Hence, for any $A\in\mathcal{C}$,
\begin{align*}
&\int_{0}^{\infty}\int_{s_{1}}^{\infty}\cdots\int_{s_{n-1}}^{\infty}\overline{\mathbb{P}}^{i}(A;s_{1},\ldots,s_{n})\prod_{k=1}^{n}q_{k}\,e^{-q_{k}(s_{k}-s_{k-1})}\dif s_{1}\cdots \dif s_{n}\\
&\quad=\int_{0}^{\infty}\int_{s_{1}}^{\infty}\cdots\int_{s_{n-1}}^{\infty}\frac{\partial^{n}}{\partial s_{1}\cdots\partial s_{n}}\widetilde{\mathbb{P}}^{i}\left(A,\,S_{k}\leq s_{k},\,k=1,\ldots,n\right)\dif s_{1}\cdots \dif s_{n}=\widetilde{\mathbb{P}}^{i}(A),
\end{align*}
and thus $\mathcal{C} \subset \mathcal{D}_{2}$, where $\mathcal{D}_{2}:=\left\{\left.A\in\widetilde{\mathcal{F}}_{\infty}^{Y}\,\right|\text{\eqref{eq:TildePiBarPi} holds for }A\right\}$. Next, for any increasing sequence of events $(A_{k})_{k\in\mathbb{N}}\subset\mathcal{D}_{2}$, we have that
\begin{align*}
\widetilde{\mathbb{P}}^{i}\left(\bigcup_{k=1}^{\infty}A_{k}\right)=\lim_{k\rightarrow\infty}\widetilde{\mathbb{P}}^{i}(A_{k})&=\lim_{k\rightarrow\infty}\int_{0}^{\infty}\!\!\!\int_{s_{1}}^{\infty}\!\!\cdots\!\int_{s_{n-1}}^{\infty}\!\overline{\mathbb{P}}^{i}(A_{k};s_{1},\ldots,s_{n})\prod_{\ell=1}^{n}q_{\ell}\,e^{-q_{\ell}(s_{\ell}-s_{\ell-1})}\dif s_{1}\cdots \dif s_{n}\\
&=\int_{0}^{\infty}\!\!\!\int_{s_{1}}^{\infty}\!\!\cdots\!\int_{s_{n-1}}^{\infty}\!\overline{\mathbb{P}}^{i}\!\left(\bigcup_{k=1}^{\infty}A_{k};s_{1},\ldots,s_{n}\right)\!\prod_{\ell=1}^{n}q_{\ell}\,e^{-q_{\ell}(s_{\ell}-s_{\ell-1})}\dif s_{1}\cdots \dif s_{n},
\end{align*}
where the last equality follows from the dominated convergence theorem as well as the fact that $\overline{\mathbb{P}}^{i}(A_{k};s_{1},\ldots,s_{n})\leq 1$, for all $k\in\mathbb{N}$ and $0<s_{1}<\ldots<s_{n}$. Hence, $\cup_{k}A_{k}\in\mathcal{D}_{2}$. Similarly, one can show that if $(A_{k})_{k\in\mathbb{N}}\subset\mathcal{D}_{2}$ is a decreasing sequence, then $\cap_{k}A_{k}\in\mathcal{D}_{2}$. Therefore, $\mathcal{D}_{2}$ is a monotone class, and by the monotone class theorem $\mathcal{D}_{2}=\sigma(\mathcal{C})=\widetilde{\mathcal{F}}_{\infty}^{Y}$.
This completes the proof.
\end{proof}
Next, we will prove that the law of $Y$ under $\overline{\mathbb{P}}^{i}$ is the same as that of $X$ under $\mathbb{P}^{i}$.
As usual, $\overline{\mathbb{E}}^{i}(\cdot\,;s_{1},\ldots,s_{n})$ will denote the expectation associated with $\overline{\mathbb{P}}^{i}(\cdot\,;s_{1},\ldots,s_{n})$, for $i\in\mathbf{E}$ and $0<s_{1}<\ldots<s_{n}$. In the sequel, if there is no ambiguity, we will omit the parameters $s_{1},\ldots,s_{n}$ in $\overline{\mathbb{P}}^{i}$ and $\overline{\mathbb{E}}^{i}$.
\begin{theorem}\label{thm:YBarPiXPi}
For any $i\in\mathbf{E}$ and $0<s_{1}<\ldots<s_{n}$, under $\overline{\mathbb{P}}^{i}$, $Y$ is a time-inhomogeneous Markov chain with generator $\mathsf{G}=\{\mathsf{G}_{t},t\geq 0\}$. In particular, $X$ and $Y$ have the same law under respective probability measures $\mathbb{P}^{i}$ and $\overline{\mathbb{P}}^{i}$.
\end{theorem}
\begin{proof}
Let $u_0,u_{1},\ldots,u_{m}$ be such that
\begin{align*}
0=u_{0}\leq u_{1}<\ldots<u_{k_{1}}\leq s_{1}<u_{k_{1}+1}<\ldots<u_{k_{2}}\leq s_{2}<\ldots\leq s_{n}<u_{k_{n}+1}<\ldots<u_{k_{n+1}}=u_{m}.
\end{align*}
By \eqref{eq:BarPiSimCylSet}, for any $i_{1},\ldots,i_{m}\in\mathbf{E}$,
\begin{align*}
\overline{\mathbb{P}}^{i}(Y_{u_{m}}=i_{m}\;|\; & Y_{u_{m-1}}=i_{m-1},\ldots,Y_{u_{1}}=i_{1})=\frac{\overline{\mathbb{P}}^{i}\left(Y_{u_{1}}=i_{1},\ldots,Y_{u_{m}}=i_{m}\right)}
{\overline{\mathbb{P}}^{i}\left(Y_{u_{1}}=i_{1},\ldots,Y_{u_{m-1}}=i_{m-1}\right)}\\
&=\frac{\prod_{\ell=0}^{n}\left(\prod_{p=k_{\ell}+1}^{k_{\ell+1}}\!e^{(u_{p}-u_{p-1})
\mathsf{G}_{\ell}}(i_{p-1},i_{p})\right)}{\left[\prod_{\ell=0}^{n-1}\left(\prod_{p=k_{\ell}+1}^{k_{\ell+1}}\!e^{(u_{p}-u_{p-1})\mathsf{G}_{\ell}}(i_{p-1},i_{p})\right)\right]\left(\prod_{p=k_{n}+1}^{k_{n+1}-1}\!e^{(u_{p}-u_{p-1})\mathsf{G}_{\ell}}(i_{p-1},i_{p})\right)}\\
&=e^{(u_{m}-u_{m-1})\mathsf{G}_{n}}(i_{m-1},i_{m}).
\end{align*}
On the other hand, by \eqref{eq:BarPiSimCylSet} again,
\begin{align*}
\overline{\mathbb{P}}^{i} (Y_{u_{m}}&=i_{m}\;|\; Y_{u_{m-1}}=i_{m-1})=\frac{\overline{\mathbb{P}}^{i}\left(Y_{u_{m}}=i_{m}\,Y_{u_{m-1}}=i_{m-1}\right)}{\overline{\mathbb{P}}^{i}\left(Y_{u_{m-1}}=i_{m-1}\right)}\\
&=\frac{\sum_{i_{1},\ldots,i_{m-2}\in\mathbf{E}}\overline{\mathbb{P}}^{i}\left(Y_{u_{1}}=i_{1},\ldots,Y_{u_{m}}=i_{m}\right)}{\sum_{i_{1},\ldots,i_{m-2}\in\mathbf{E}}\overline{\mathbb{P}}^{i}\left(Y_{u_{1}}=i_{1},\ldots,Y_{u_{m-1}}=i_{m-1}\right)}\\
&=\frac{\sum_{i_{1},\ldots,i_{m-2}\in\mathbf{E}}\prod_{\ell=0}^{n}\left(\prod_{p=k_{\ell}+1}^{k_{\ell+1}}\!e^{(u_{p}-u_{p-1})\mathsf{G}_{\ell}}(i_{p-1},i_{p})\right)}{\sum_{i_{1},\ldots,i_{m-2}\in\mathbf{E}}\left[\prod_{\ell=0}^{n-1}\left(\prod_{p=k_{\ell}+1}^{k_{\ell+1}}\!e^{(u_{p}-u_{p-1})\mathsf{G}_{\ell}}(i_{p-1},i_{p})\right)\right]\left(\prod_{p=k_{n}+1}^{k_{n+1}-1}\!e^{(u_{p}-u_{p-1})\mathsf{G}_{\ell}}(i_{p-1},i_{p})\right)}\\
&=e^{(u_{m}-u_{m-1})\mathsf{G}_{n}}(i_{m-1},i_{m}).
\end{align*}
Analogous argument carries for any $u_0<u_{1}<\ldots<u_{m}$, which completes the proof.
\end{proof}
In analogy to $\varphi_t$ and $\tau_t^+$ we now define an additive functional $\psi$ given as
$\psi_{t}:=\int_{0}^{t}v(Y_{u})\dif u,\ t\geq 0$, and we consider the following first passage time $\rho_{t}^{+}:=\inf\left\{\left.r\geq 0\,\right|\psi_{r}>t\right\}, \ t\geq 0$.
We end this part of this section with the following corollary to Theorem \ref{thm:YBarPiXPi}.
\begin{corollary}\label{thm:XYWHRelation}
For any $(s_{1},\ldots,s_{n})$ in $\Delta_{n}$, $c>0$, and $t>0$,
\begin{align}\label{eq:rho0+}
\Pi_{c}^{+}(i,j;s_{1},\ldots,s_{n})&=\overline{\mathbb{E}}^{i}\left(e^{-c\rho_{0}^{+}}\mathbbm{1}_{\{Y_{\rho_{0}^{+}}=j\}};s_{1},\ldots,s_{n}\right),\quad i\in\mathbf{E}^{-},\,j\in\mathbf{E}^{+},\\
\label{eq:rhot+} \Psi_{c}^{+}(t,i,j;s_{1},\ldots,s_{n})&=\overline{\mathbb{E}}^{i}\left(e^{-c\rho_{t}^{+}}\mathbbm{1}_{\{Y_{\rho_{t}^{+}}=j\}};s_{1},\ldots,s_{n}\right),\quad i\in\mathbf{E}^{+},\,j\in\mathbf{E}^{+}.
\end{align}
In particular, $\Pi_{c}^{+}(i,j;s_{1},\ldots,s_{n})$ and $\Psi_{c}^{+}(t,i,j;s_{1},\ldots,s_{n})$ are Borel measurable with respect to $(s_{1},\ldots,s_{n})$ in $\Delta_{n}$.
\end{corollary}
\subsection{Wiener-Hopf Factorization for $Z=(N,Y)$}
This subsection is devoted to computing the expectations on the right-hand side in \eqref{eq:rho0+} and \eqref{eq:rhot+}. This will be done by computing the corresponding expectations related to the \textit{time-homogeneous} Markov chain $Z=(N,Y)$. The latter computation will be done using the classical Wiener-Hopf factorization results for finite state time-homogeneous Markov chains, originally derived in \cite{Barlow1980}.
We begin with a restatement of the classical Wiener-Hopf factorization applied to $Z$. Towards this end, we let $\widetilde{\mathbf{E}}^{+}:=\mathbb{N}_{n}\times\mathbf{E}^{+}$ and $\widetilde{\mathbf{E}}^{-}:=\mathbb{N}_{n}\times\mathbf{E}^{-}$, and $\widetilde{v}:\widetilde{\mathbf{E}}\rightarrow\mathbb{R}\setminus\{0\}$ be a function on $\widetilde{\mathbf{E}}$ such that $\widetilde{v}(k,i)=v(i)$, for all $(k,i)\in\widetilde{\mathbf{E}}$. Next, we define the additive functional $\widetilde{\varphi}$ and the corresponding first passage times as
\begin{align*}
\widetilde{\varphi}_{t}:=\int_{0}^{t}\widetilde{v}(Z_{u})\dif u
,\quad
\widetilde{\tau}_{t}^{\pm}:=\inf\left\{\left.r\geq 0\,\right|\pm\widetilde{\varphi}_{r} > t\right\},\quad t\geq 0.
\end{align*}
Let $\widetilde{V}:=diag\{\widetilde{v}(k,i):(k,i)\in\widetilde{\mathbf{E}}\}$ (a diagonal matrix). We denote by $\widetilde{\mathsf{I}}^{\pm}$ the identity matrix of dimension $|\widetilde{\mathbf{E}}^{\pm}|$. Finally, $\mathcal{Q}(m)$ will stand for the set of $m\times m$ generator matrices (i.e., matrices with non-negative off-diagonal entries and non-positive row sums), and $\mathcal{P}(m,\ell)$ will be the set of $m\times\ell$ matrices whose rows are sub-probability vectors.
\begin{theorem}\label{thm:WHZ} \cite[Theorem 1 $\&$ 2]{Barlow1980}$\,\,$
Fix $c>0$. Then,
\begin{itemize}
\item [(i)] there exists a unique quadruple of matrices $(\widetilde{\Lambda}_{c}^{+},\widetilde{\Lambda}_{c}^{-},\widetilde{\mathsf{G}}_{c}^{+},\widetilde{\mathsf{G}}_{c}^{-})$, where $\widetilde{\Lambda}_{c}^{+}\in\mathcal{P}(|\widetilde{\mathbf{E}}^{-}|,|\widetilde{\mathbf{E}}^{+}|)$, $\widetilde{\Lambda}_{c}^{-}\in\mathcal{P}(|\widetilde{\mathbb{E}}^{+}|,|\widetilde{\mathbf{E}}^{-}|)$, $\widetilde{\mathsf{G}}_{c}^{+}\in\mathcal{Q}(|\widetilde{\mathbf{E}}^{+}|)$, and $\widetilde{\mathsf{G}}_{c}^{-}\in\mathcal{Q}(|\widetilde{\mathbf{E}}^{-}|)$, such that
\begin{align}\label{eq:WHZ}
\widetilde{V}^{-1}\left(\widetilde{\mathsf{G}}-c\,\widetilde{\mathsf{I}}\right)\left(\begin{array}{cc} \widetilde{\mathsf{I}}^{+} & \widetilde{\Lambda}_{c}^{-} \\ \widetilde{\Lambda}_{c}^{+} & \widetilde{\mathsf{I}}^{-} \end{array}\right)=\left(\begin{array}{cc} \widetilde{\mathsf{I}}^{+} & \widetilde{\Lambda}_{c}^{-} \\ \widetilde{\Lambda}_{c}^{+} & \widetilde{\mathsf{I}}^{-} \end{array}\right)\left(\begin{array}{cc} \widetilde{\mathsf{G}}_{c}^{+} & 0 \\ 0 & -\widetilde{\mathsf{G}}_{c}^{-} \end{array}\right);
\end{align}
\item [(ii)] the matrices $\widetilde{\Lambda}_{c}^{+}$, $\widetilde{\Lambda}_{c}^{-}$, $\widetilde{\mathsf{G}}_{c}^{+}$, and $\widetilde{\mathsf{G}}_{c}^{-}$, admit the following probabilistic representations, \begin{align}\label{eq:TildeTau0+}
\widetilde{\Lambda}_{c}^{+}((k,i),(\ell,j))&=\widetilde{\mathbb{E}}\left(\left.e^{-c\widetilde{\tau}_{0}^{+}}\mathbf{1}_{\{Z_{\widetilde{\tau}_{0}^{+}}=(\ell,j)\}}\,\right|Z_{0}=(k,i)\right),\quad (k,i)\in\widetilde{\mathbf{E}}^{-},\,(\ell,j)\in\widetilde{\mathbf{E}}^{+},\\
\label{eq:TildeTau0-} \widetilde{\Lambda}_{c}^{-}((k,i),(\ell,j))&=\widetilde{\mathbb{E}}\left(\left.e^{-c\widetilde{\tau}_{0}^{-}}\mathbf{1}_{\{Z_{\widetilde{\tau}_{0}^{-}}=(\ell,j)\}}\,\right|Z_{0}=(k,i)\right),\quad (k,i)\in\widetilde{\mathbf{E}}^{+},\,(\ell,j)\in\widetilde{\mathbf{E}}^{-},\\
\label{eq:TildeTaut+}
e^{t\,\widetilde{\mathsf{G}}_{c}^{+}}((k,i),(\ell,j))&=\widetilde{\mathbb{E}}\left(\left.e^{-c\widetilde{\tau}_{t}^{+}}\mathbf{1}_{\{Z_{\widetilde{\tau}_{t}^{+}}=(\ell,j)\}}\,\right|Z_{0}=(k,i)\right),\quad (k,i)\in\widetilde{\mathbf{E}}^{+},\,(\ell,j)\in\widetilde{\mathbf{E}}^{+},\\
\label{eq:TildeTaut-}
e^{t\,\widetilde{\mathsf{G}}_{c}^{-}}((k,i),(\ell,j))&=\widetilde{\mathbb{E}}\left(\left.e^{-c\widetilde{\tau}_{t}^{-}}\mathbf{1}_{\{Z_{\widetilde{\tau}_{t}^{-}}=(\ell,j)\}}\,\right|Z_{0}=(k,i)\right),\quad (k,i)\in\widetilde{\mathbf{E}}^{-},\,(\ell,j)\in\widetilde{\mathbf{E}}^{-},
\end{align}
for any $t\geq 0$.
\end{itemize}
\end{theorem}
In what follows we will use the ``+'' part of the above formulae and only for $k=0$. Accordingly, we define (recall \eqref{eq:TildePi})
\begin{align}\label{eq:TildeTau0+0}
\widetilde{\Pi}_{c}^{+}(i,j,\ell)&:=\widetilde{\Lambda}_{c}^{+}((0,i),(\ell,j))=\widetilde{\mathbb{E}}^{i}\left(e^{-c\widetilde{\tau}_{0}^{+}}\mathbf{1}_{\{Z_{\widetilde{\tau}_{0}^{+}}=(\ell,j)\}}\right),\quad i\in\mathbf{E}^{-},\,j\in\mathbf{E}^{+},\,\ell\in\mathbb{N},\\
\label{eq:TildeTaut+0}
\widetilde{\Psi}_{c}^{+}(t,i,j,\ell)&:=e^{t\,\widetilde{\mathsf{G}}_{c}^{+}}((0,i),(\ell,j))=\widetilde{\mathbb{E}}^{i}\left(e^{-c\widetilde{\tau}_{t}^{+}}\mathbf{1}_{\{Z_{\widetilde{\tau}_{t}^{+}}=(\ell,j)\}}\right),\quad i,j\in\mathbf{E}^{+},\,\ell\in\mathbb{N},\,t\geq 0.
\end{align}
Note that, for any $t\geq 0$, $\widetilde{v}(Z_{t})=v(Y_{t})$, which implies that $\widetilde{\varphi}_{t}=\psi_{t}$, and so $\rho_{t}^{+}=\widetilde{\tau}_{t}^{+}$, $\rho_{t}^{-}=\widetilde{\tau}_{t}^{-}$.
Hence, by taking summations over all $\ell\in\mathbb{N}$ in \eqref{eq:TildeTau0+0} and \eqref{eq:TildeTaut+0}, we obtain that
\begin{align}\label{eq:SumTildeTau0+}
\widetilde{\mathbb{E}}^{i}\left(e^{-c\rho_{0}^{+}}\mathbbm{1}_{\{Y_{\rho_{0}^{+}}=j\}}\right)&=\sum_{\ell=0}^{n}\,\widetilde{\Pi}_{c}^{+}(i,j,\ell),\quad i\in\mathbf{E}^{-},\,j\in\mathbf{E}^{+},\\
\label{eq:SumTildeTaut+}
\widetilde{\mathbb{E}}^{i}\left(e^{-c\rho_{t}^{+}}\mathbbm{1}_{\{Y_{\rho_{t}^{+}}=j\}}\right)&=\sum_{\ell=0}^{n}\,\widetilde{\Psi}_{c}^{+}(t,i,j,\ell),\quad i,j\in\mathbf{E}^{+},\,t\geq 0.
\end{align}
Observe that, in view of \eqref{eq:TildePiBarPi}, if $U:\widetilde{\Omega}\rightarrow\mathbb{R}$ is an $\widetilde{\mathcal{F}}_{\infty}^{Y}$-measurable bounded random variable, then for any $i\in\mathbf{E}$,
\begin{align*}
\widetilde{\mathbb{E}}^{i}(U)=\int_{0}^{\infty}\int_{s_{1}}^{\infty}\cdots\int_{s_{n-1}}^{\infty}\overline{\mathbb{E}}^{i}(U;s_{1},\ldots,s_{n})\prod_{k=1}^{n}\left(q_{k}\,e^{-q_{k}(s_{k}-s_{k-1})}\right)ds_{n}\cdots ds_{2}ds_{1}.
\end{align*}
Therefore, in light of Corollary~\ref{thm:XYWHRelation}, \eqref{eq:SumTildeTau0+} and \eqref{eq:SumTildeTaut+}, we have that
\begin{align*}
\widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n)&:=\sum_{\ell=0}^{n}\,\widetilde{\Pi}_{c}^{+}(i,j,\ell)\\
&=\int_{0}^{\infty}\int_{s_{1}}^{\infty}\ldots\int_{s_{n-1}}^{\infty}\Pi_{c}^{+}(i,j;s_{1},\ldots,s_{n})\prod_{k=1}^{n}\left(q_{k}e^{-q_{k}(s_{k}-s_{k-1})}\right)ds_{n}\cdots ds_{2}\,ds_{1}.
\end{align*}
\begin{align*}
\widehat{\Psi}_{c}^{+}(t,i,j;q_1,\ldots,q_n)&:=\sum_{\ell=0}^{n}\,\widetilde{\Psi}_{c}^{+}(t,i,j,\ell)\\
&=\int_{0}^{\infty}\!\!\!\int_{s_{1}}^{\infty}\!\!\ldots\!\int_{s_{n-1}}^{\infty}\!\Psi_{c}^{+}(t,i,j;s_{1},\ldots,s_{n})\prod_{k=1}^{n}\left(q_{k}e^{-q_{k}(s_{k}-s_{k-1})}\right)ds_{n}\!\cdots ds_{2}\,ds_{1}.
\end{align*}
By change of variables, we obtain
\begin{align*}
\widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n)
&=\int_{0}^{\infty}\int_{0}^{\infty}\ldots\int_{0}^{\infty}\Pi_{c}^{+}(i,j;t_{1},\ldots,t_{1}+\ldots+t_{n})\prod_{k=1}^{n}\left(q_{k}e^{-q_{k}t_{k}}\right)dt_{1}\cdots dt_{n},\\
\widehat{\Psi}_{c}^{+}(i,j;q_1,\ldots,q_n)
&=\int_{0}^{\infty}\int_{0}^{\infty}\ldots\int_{0}^{\infty}\Psi_{c}^{+}(i,j;t_{1},\ldots,t_{1}+\ldots+t_{n})\prod_{k=1}^{n}\left(q_{k}e^{-q_{k}t_{k}}\right)dt_{1}\cdots dt_{n}.
\end{align*}
The above two equalities together with the argument in Section~\ref{sec:inverLaplaceTransform}, implies that
\begin{align*}
q_{1}^{-1}\cdots q_{n}^{-1}\widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n), \qquad q_{1}^{-1}\cdots q_{n}^{-1}\widehat{\Psi}_{c}^{+}(i,j;q_1,\ldots,q_n)
\end{align*}
are well-defined for $q_{k}\in\mathbb{C}^+:= \set{z\in\mathbb{C} \;|\; \Re(z)>0},k=1,\ldots,n$, as being the Laplace transforms of $\Pi_{c}^{+}(i,j;t_{1},\ldots,t_{1}+\ldots+t_{n})$ and $\Psi_{c}^{+}(i,j;t_{1},\ldots,t_{1}+\ldots+t_{n})$, respectively.
All the above leads to the following result, which is our main theorem, and where we make use of the inverse multivariate Laplace transform. We refer to the Appendix for the definition and the properties of the inverse multivariate Laplace transform relevant to our set-up.
\begin{theorem}\label{th:main} We have that
\begin{align}\label{eq:invLaplacePi+}
\Pi_{c}^{+}(i,j;s_{1},\ldots,s_{n})=\mathcal{L}^{-1}\left(q_{1}^{-1}\cdots q_{n}^{-1}\widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n)\right)(s_{1},s_{2}-s_{1},\ldots,s_{n}-s_{n-1}),
\end{align}
for any $i\in\mathbf{E}^{-}$, $j\in\mathbf{E}^{+}$, and
\begin{align}\label{eq:invLaplacePsi+}
\Psi_{c}^{+}(t,i,j;s_{1},\ldots,s_{n})=\mathcal{L}^{-1}\left(q_{1}^{-1}\cdots q_{n}^{-1}\widehat{\Psi}_{c}^{+}(t,i,j;q_1,\ldots,q_n)\right)(s_{1},s_{2}-s_{1},\ldots,s_{n}-s_{n-1}),
\end{align}
for any $t>0$, $i,j\in\mathbf{E}^{+}$, where $\mathcal{L}^{-1}$ is the inverse multivariate Laplace transform.
\end{theorem}
\begin{remark}
It needs to be stressed that we can compute the values of $\widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n)$ and $\widehat{\Psi}_{c}^{+}(t,i,j;q_1,\ldots,q_n)$ only for positive values of $q_{i}$'s. Thus, Theorem \ref{th:main} may not be directly applied to compute $\Pi_{c}^{+}(i,j;s_{1},\ldots,s_{n})$ and $\Psi_{c}^{+}(t,i,j;s_{1},\ldots,s_{n})$. However, we can approximate these functions, as explained in Section \ref{subsubsec:specialcase} by using only the values of $\widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n)$ and $\widehat{\Psi}_{c}^{+}(t,i,j;q_1,\ldots,q_n)$ for positive values of $q_{i}$'s.
\end{remark}
\section{Numerical Example}\label{sec:Numerical}
In this section we will illustrate our theoretical results with a simple, but telling example. We first describe a numerical method to approximate $\Pi_{c}^{+}$ and $\Psi_{c}^{+}$, and then we proceed with its application to a concrete example.
\subsection{Numerical Procedure to approximate $\Pi_{c}^{+}$ and $\Psi_{c}^{+}$}
We only consider $\Pi_{c}^{+}$. The procedure to approximate $\Psi_{c}^{+}$ is analogous.
According to Theorem \ref{th:main} and Section \ref{subsubsec:specialcase}, to approximate $\Pi_{c}^{+}$, we need to compute
$
\widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n)
$
for any $q_{1},\ldots,q_{n}>0$, and then to use the Gaver-Stehfest algorithm. Note that $\widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n)$ can be computed by solving \eqref{eq:WHZ} directly using the diagonalization method of \cite{rogers_shi_1994}. However, because of the special structure of $\widetilde{\mathsf{G}}$, we can simplify the calculation by working on matrices of smaller dimensions. Towards this end we observe that matrices in \eqref{eq:WHZ} can be written the block form as follows,
\begin{align}
\widetilde{\mathsf{G}}=
\kbordermatrix{
& (0,\mathbf{E}^{+}) & (1,\mathbf{E}^{+}) & \cdots & (n,\mathbf{E}^{+}) & (0,\mathbf{E}^{-}) & (1,\mathbf{E}^{-}) & \cdots & (n,\mathbf{E}^{-}) \\
(0,\mathbf{E}^{+}) & \mathsf{A}_{1}-q_{1}\mathsf{I}^{+} & q_{1}\mathsf{I}^{+} & \cdots & 0 & \mathsf{B}_{1} & 0 & \cdots & 0\\
(1,\mathbf{E}^{+}) & 0 & \mathsf{A}_{2}-q_{2}\mathsf{I}^{+} & \cdots & 0 & 0 & \mathsf{B}_{2} & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\
(n-1,\mathbf{E}^{+}) & 0 & 0 & \cdots & q_{n}\mathsf{I}^{+} & 0 & 0 & \cdots & 0\\
(n,\mathbf{E}^{+}) & 0 & 0 & \cdots & \mathsf{A}_{n+1} & 0 & 0 & \cdots & \mathsf{B}_{n+1}\\
(0,\mathbf{E}^{-}) & \mathsf{C}_{1} & 0 & \cdots & 0 & \mathsf{D}_{1}-q_{1}\mathsf{I}^{-} & q_{1}\mathsf{I}^{-} & \cdots& 0\\
(1,\mathbf{E}^{-}) & 0 & \mathsf{C}_{2} & \cdots & 0 & 0 & \mathsf{D}_{2}-q_{2}\mathsf{I}^{-} & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\
(n-1,\mathbf{E}^{-}) & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & q_{n}\mathsf{I}^{-}\\
(n,\mathbf{E}^{-}) & 0 & 0 & \cdots & \mathsf{C}_{n+1} & 0 & 0 & \cdots & \mathsf{D}_{n+1}
},
\end{align}
\begin{align}\label{eq:blockV}
\widetilde{\mathsf{V}}=
\kbordermatrix{
& (0,\mathbf{E}^{+}) & (1,\mathbf{E}^{+}) & \cdots & (n,\mathbf{E}^{+}) & (0,\mathbf{E}^{-}) & (1,\mathbf{E}^{-}) & \cdots & (n,\mathbf{E}^{-}) \\
(0,\mathbf{E}^{+}) & \mathsf{V}^{+} & 0 & \cdots & 0 & 0 & 0 & \cdots& 0\\
(1,\mathbf{E}^{+}) & 0 & \mathsf{V}^{+} & \cdots & 0 & 0 & 0 & \cdots& 0\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\
(n-1,\mathbf{E}^{+}) & 0 & 0 & \cdots &0 & 0 & 0 & \cdots & 0\\
(n,\mathbf{E}^{+}) & 0 & 0 & \cdots & \mathsf{V}^{+} & 0 & 0 & \cdots& 0\\
(0,\mathbf{E}^{-}) & 0 & 0 & \cdots & 0 & \mathsf{V}^{-} & 0 & \cdots & 0\\
(1,\mathbf{E}^{-}) & 0 & 0 & \cdots& 0 & 0 & \mathsf{V}^{-} & \cdots& 0\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\
(n-1,\mathbf{E}^{-}) & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0\\
(n,\mathbf{E}^{-}) & 0 & 0 & \cdots& 0 & 0 & 0 & \cdots & \mathsf{V}^{-}
},
\end{align}
\begin{align}\label{eq:blockPi+}
\widetilde{\Lambda}_{c}^{+}=
\kbordermatrix{
& (0,\mathbf{E}^{+}) & (1,\mathbf{E}^{+}) & \cdots & (n-1,\mathbf{E}^{+}) & (n,\mathbf{E}^{+}) \\
(0,\mathbf{E}^{-}) &\widetilde{\Lambda}_{c,00}^{+} & \widetilde{\Lambda}_{c,01}^{+} & \cdots & \widetilde{\Lambda}_{c,0,n-1}^{+} & \widetilde{\Lambda}_{c,0n}^{+} \\
(1,\mathbf{E}^{-}) & 0 & \widetilde{\Lambda}_{c,11}^{+} & \cdots & \widetilde{\Lambda}_{c,1,n-1}^{+} & \widetilde{\Lambda}_{c,1n}^{+} \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
(n-1,\mathbf{E}^{-}) & 0 & 0 & \cdots & \widetilde{\Lambda}_{c,n-1,n-1}^{+} & \widetilde{\Lambda}_{c,n-1,n}^{+} \\
(n,\mathbf{E}^{-}) & 0 & 0 & \cdots & 0 & \widetilde{\Lambda}_{c,nn}^{+}
}
\end{align}
\begin{align}\label{eq:blockPi-}
\widetilde{\Lambda}_{c}^{-}=
\kbordermatrix{
& (0,\mathbf{E}^{-}) & (1,\mathbf{E}^{-}) & \cdots & (n-1,\mathbf{E}^{-}) & (n,\mathbf{E}^{-}) \\
(0,\mathbf{E}^{+}) & \widetilde{\Lambda}_{c,00}^{-} & \widetilde{\Lambda}_{c,01}^{-} & \cdots & \widetilde{\Lambda}_{c,0,n-1}^{-} & \widetilde{\Lambda}_{c,0n}^{-} \\
(1,\mathbf{E}^{+}) & 0 & \widetilde{\Lambda}_{c,11}^{-} & \cdots & \widetilde{\Lambda}_{c,1,n-1}^{-} & \widetilde{\Lambda}_{c,1n}^{-} \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
(n-1,\mathbf{E}^{+}) & 0 &0 & \cdots & \widetilde{\Lambda}_{c,n-1,n-1}^{-} & \widetilde{\Lambda}_{c,n-1,n}^{-} \\
(n,\mathbf{E}^{+}) & 0 & 0 & \cdots & 0 & \widetilde{\Lambda}_{c,nn}^{-}
},
\end{align}
\begin{align}\label{eq:blockG+}
\widetilde{\mathsf{G}}_{c}^{+}=
\kbordermatrix{
& (0,\mathbf{E}^{+}) & (1,\mathbf{E}^{+}) & \cdots & (n-1,\mathbf{E}^{+}) & (n,\mathbf{E}^{+}) \\
(0,\mathbf{E}^{+}) & \widetilde{\mathsf{G}}_{c,00}^{+} & \widetilde{\mathsf{G}}_{c,01}^{+} & \cdots & \widetilde{\mathsf{G}}_{c,0,n-1}^{+} & \widetilde{\mathsf{G}}_{c,0n}^{+} \\
(1,\mathbf{E}^{+}) & 0 & \widetilde{\mathsf{G}}_{c,11}^{+} & \cdots & \widetilde{\mathsf{G}}_{c,1,n-1}^{+} & \widetilde{\mathsf{G}}_{c,1n}^{+} \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
(n-1,\mathbf{E}^{+}) & 0 & 0 & \cdots & \widetilde{\mathsf{G}}_{c,n-1,n-1}^{+} & \widetilde{\mathsf{G}}_{c,n-1,n}^{+} \\
(n,\mathbf{E}^{+}) & 0 & 0 & \cdots & 0 & \widetilde{\mathsf{G}}_{c,nn}^{+}
},
\end{align}
and
\begin{align}\label{eq:blockG-}
\widetilde{\mathsf{G}}_{c}^{-}=
\kbordermatrix{
& (0,\mathbf{E}^{-}) & (1,\mathbf{E}^{-}) & \cdots & (n-1,\mathbf{E}^{-}) & (n,\mathbf{E}^{-}) \\
(0,\mathbf{E}^{-}) & \widetilde{\mathsf{G}}_{c,00}^{-} & \widetilde{\mathsf{G}}_{c,01}^{-} & \cdots & \widetilde{\mathsf{G}}_{c,0,n-1}^{-} & \widetilde{\mathsf{G}}_{c,0n}^{-} \\
(1,\mathbf{E}^{-}) & 0 & \widetilde{\mathsf{G}}_{c,11}^{-} & \cdots & \widetilde{\mathsf{G}}_{c,1,n-1}^{-} & \widetilde{\mathsf{G}}_{c,1n}^{-} \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
(n-1,\mathbf{E}^{-}) & 0 & 0 & \cdots & \widetilde{\mathsf{G}}_{c,n-1,n-1}^{-} & \widetilde{\mathsf{G}}_{c,n-1,n}^{-} \\
(n,\mathbf{E}^{-}) & 0 & 0 & \cdots &0 & \widetilde{\mathsf{G}}_{c,nn}^{-}
}.
\end{align}
Plugging \eqref{eq:blockV}--\eqref{eq:blockG-} into \eqref{eq:WHZ} and then comparing all the block entries on both sides, we end up with the following procedure to compute the factorization recursively.
In accordance to Theorem \ref{thm:WHZ}, for any generator matrix $\mathsf{H}$ and any constant $c>0$, we denote by
\begin{align}
(\Lambda_{c}^{+}(\mathsf{H}), \Lambda_{c}^{-}(\mathsf{H}),\mathsf{G}_{c}^{+}(\mathsf{H}), \mathsf{G}_{c}^{-}(\mathsf{H}))
\end{align}
the unique quadruple constituting the classical Wiener-Hopf factorization (cf. \cite{Barlow1980}) corresponding to $\mathsf{H}$ with killing rate $c$. In order to proceed, we let $c_{k}=q_{k}+c,\quad k\geq 1$.
We are now ready to describe the algorithm to compute $q_{1}^{-1}\cdots q_{n}^{-1}\widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n)$.
\begin{enumerate}[Step 1.]
\item \textbf{Compute the first diagonal:} for $k=1,\ldots,n+1$, compute
\begin{align}
\widetilde{\Lambda}_{c,k-1,k-1}^{+}=\Lambda_{c_{k}}^{+}(\mathsf{G}_{k}),
\end{align}
using the diagonalization method in \cite{rogers_shi_1994}.
\item \textbf{Compute the second diagonal:} for $k=1,\ldots,n$, solve the following linear system
\begin{align}
q_{k}\mathsf{I}^{+}+\mathsf{B}_{k}\widetilde{\Lambda}_{c,k-1,k}^{+}&=\mathsf{V}^{+}\widetilde{\mathsf{G}}_{c,k-1,k}^{+},\\
[\mathsf{D}_{k}-c_{k}\mathsf{I}^{-}]\widetilde{\Lambda}_{c,k-1,k}^{+}+q_{k}\widetilde{\Lambda}_{c,kk}^{+}&=\mathsf{V}^{-}\widetilde{\Lambda}_{c,k-1,k-1}^{+}\widetilde{\mathsf{G}}_{c,k-1,k}^{+}+\mathsf{V}^{-}\widetilde{\Lambda}_{c,k-1,k}^{+}\widetilde{\mathsf{G}}_{c,kk}^{+},
\end{align}
for $\widetilde{\Lambda}_{c,k-1,k}^{+}$ and $\widetilde{\mathsf{G}}_{c,k-1,k}^{+}$.
\item \textbf{Compute the other diagonals:} for $r=2,\ldots,n$, $k=0,\ldots,n-r$, solve the linear system
\begin{align}
\mathsf{B}_{k+1}\widetilde{\Lambda}_{c,k,k+r}^{+}&=\mathsf{V}^{+}\widetilde{\mathsf{G}}_{c,k,k+r}^{+},\\
[\mathsf{D}_{k+1}-c_{k+1}\mathsf{I}^{-}]\widetilde{\Lambda}_{c,k,k+r}^{+}+q_{k+1}\widetilde{\Lambda}_{c,k+1,k+r}^{+}&=\mathsf{V}^{-}\sum_{j=0}^{r}\widetilde{\Lambda}_{c,k,k+j}^{+}\widetilde{\mathsf{G}}_{c,k+j,k+r}^{+},
\end{align}
for $\widetilde{\Lambda}_{c,k,k+r}^{+}$ and $\widetilde{\mathsf{G}}_{c,k,k+r}^{+}$.
\item \textbf{Compute}
\begin{align*}
P^{+}(q_{1},\ldots,q_{n}):=q_{1}^{-1}\cdots q_{n}^{-1}\widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n)=q_{1}^{-1}\cdots q_{n}^{-1}\sum_{\ell=0}^{n}\,\widetilde{\Lambda}_{c,0\ell}.
\end{align*}
for $q_1,\ldots,q_n>0$.
\item \textbf{Compute the approximate inverse Laplace transform of $P^{+}(q_{1},\ldots,q_{n})$:} use the method discussed in Section \ref{subsubsec:specialcase}.
\end{enumerate}
\begin{remark}
If $|\mathbf{E}^{+}|=|\mathbf{E}^{-}|=1$, then the matrices in Steps 1-3 become numbers. Step 1 reduces to solving $n+1$ quadratic equations for a root in $[0,1]$. In Step 2 and 3, for each loop, the system reduces to a system of two linear equations of two unknowns. Moreover, in this case, $P^{+}$ has a closed-form representation for $q_1,\ldots,q_n>0$, and hence, for any $q_1,\ldots,q_n\in\mathbb{C}^+$, as mentioned in the previous section. This allows to use general numerical inverse Laplace transform methods, not necessary the Gaver-Stehfest formula from Section~\ref{subsubsec:specialcase}. In particular, one can use Talbot approximation formula \eqref{eq:talbotInversion} presented in Section~\ref{sec:inverLaplaceTransform}, which is more efficient than the Gaver-Stehfest under fairly general assumptions (cf. \cite{abate2006unified}).
\end{remark}
\subsection{Application in Fluid flow problems}
The Wiener-Hopf factorization for a time-homogeneous finite Markov chain was applied in \cite{rogers1994fluid} in the context of fluid models of queues. In this section, we will apply our results to a time-inhomogeneous Markov chain fluid flow problem.
First, we briefly review the classical fluid flow problem (cf \cite{mitra1988} and \cite{rogers1994fluid} for detailed discussion). Suppose we have a large water tank with capacity $a\in (0,\infty]$. On the top of the tank, there are $I_{t}\in \mathcal{I}$ pipes open at time $t$, with each pipe pouring water into the tank at rate $r^{+}$. At the bottom of the tank, there are $O_{t}\in \mathcal{O}$ taps open at time $t$, with each tap allowing water to flow out at rate $r^{-}$. We assume that $\mathcal{I}$ and $\mathcal{O}$ are finite sets.
Then, the volume $\xi_{t}$ of water in the tank at time $t$ satisfies
\begin{align*}
\frac{\dif \xi_{t}}{\dif t}=r^{+}I_{t}-r^{-}O_{t}, \quad\text{if }0<\xi_{t}<a.
\end{align*}
Moreover, if $\xi_{t}=0$, i.e. if the tank is empty, then the outflow ceases. If $\xi_{t}=a$, i.e. if the tank is full, then water flows over the top.
Let $f$ be a real valued function on $\mathcal{I}\times \mathcal{O}$. We assume $X_{t}:=f(I_{t},O_{t}),\ t\geq 0,$ is a (finite state) time-inhomogeneous Markov chain, and we denote by $\mathbf{E}$ the state space of $X$. Let
\begin{align*}
v(x):=V(r^{+},r^{-},x),\quad x\in \mathbf{E},
\end{align*}
model the water outflow/inflow, in terms of the states of $X$, so that
\begin{align*}
v(X_{t})=V(r^{+},r^-,f(I_{t},O_{t})),\quad t\ge 0
\end{align*}
represents the water outflow/inflow at time $t$.
Let $\mathbf{E}^{+}$ be the set of states of $X$ such that the water tank has greater water inflow than outflow, and let $\mathbf{E}^{-}$ be the set of states of $X$ such that the water tank has greater water outflow than inflow. The integral
\begin{align*}
\varphi_{t}=\int_{0}^{t}v(X_{u})\dif u
\end{align*}
is not exactly the water content at time $t$, because we should take into account those periods when the tank is full or empty. However, as noted in \cite{rogers1994fluid}, understanding $\varphi_{t}$, and the corresponding $\tau_{t}^{\pm}$ and $X_{\tau_{t}^{\pm}}$ allows us to easily express the quantities of interest for $\xi_{t}$ in terms of Wiener-Hopf factorization, and to further compute these quantities once we compute the Wiener-Hopf factorization numerically.
We now assume that the tank has infinite capacity, $a=\infty$, and that it contains $\ell$ amount of water at time $t=0$. Thus, $\tau_{\ell}^{-}$ represents the first time after $t=0$ that the tank goes empty. We will compute the quantity
\begin{align}\label{eq:computeTarget}
\Pi_{c}^{-}(i,j)=\mathbb{E}^{i}\left(e^{-c\tau_{0}^{-}}\mathbbm{1}_{\{X_{\tau_{0}^{-}}=j\}}\right),\quad i\in\mathbf{E}^{+},\, j\in\mathbf{E}^{-}.
\end{align}
Towards this end, we further assume that the tank has either an aggregate water inflow at rate $v^{+}$ or an aggregate water outflow at rate $v^{-}$. In other words,
\begin{align*}
\mathbf{E}^{+}=\{e_+\},\quad\mathbf{E}^{-}=\{e_-\},\quad v(e_+)=v^{+},\quad\text{and}\quad v(e_-)=v^{-}.
\end{align*}
Moreover, we assume that the time-inhomogeneous Markov chain $X$ has the generator
\begin{align*}
\mathsf{G}_{t}=\left\{\begin{array}{lll}
& \mathsf{G}_{1},\quad & s_{0}\le t <s_{1},\\
& \mathsf{G}_{2},\quad &s_{1}\le t <s_{2},\\
& \mathsf{G}_{3},\quad &t\ge s_{2},
\end{array}\right.
\end{align*}
where $0<s_{1}<s_{2}$.
We take the following inputs: $c=0.5, v(e_+)=2, v(e_-)=-3,s_{1}=2,s_{2}=8$,
\begin{align*}
\mathsf{G}_{0}=
\kbordermatrix{
& e_+ & e_- \\
e_+ &-2 & 2\\
e_- &1 & -1
}, \quad
\mathsf{G}_{1}=
\kbordermatrix{
& e_+ & e_- \\
e_+ &-3 & 3\\
e_- &2 & -2
},\quad
\mathsf{G}_{2}=
\kbordermatrix{
& e_+ & e_- \\
e_+ &-5 & 5\\
e_- &3 & -3
}.
\end{align*}
The following table compares our result and execution time with Monte-Carlo simulation (10000 paths).\\
\begin{center}
\begin{tabular}{llr}
\hline
\multicolumn{3}{c}{Numerical Results} \\
\cline{1-3}
Method & Wiener-Hopf & Monte-Carlo \\
\hline
$\Pi_{c}^{-}(e_+,e_-)$ & $0.6501$ & $0.6462$ \\
Execution time & $0.15\,s$ & $3.12\,s$ \\
\hline
\end{tabular}
\end{center}
\begin{remark}
One can also compute $\Pi_{c}^{+}(e_-,e_+)$, if it is the quantity of interest in the model. Note that if we change the labels of the states from $\{e_+,e_-\}$ to $\{e_-,e_+\}$ and modify the inputs accordingly, we can compute $\Pi_{c}^{+}(e_-,e_+)$ using the same algorithm that computes $\Pi_{c}^{-}(e_+,e_-)$.
\end{remark}
\section{Appendix: Approximation of Multivariate Inverse Laplace Transform}\label{sec:inverLaplaceTransform}
For the convenience of the reader, we will briefly recall the basics of Laplace transform and its inverse. Then, we will proceed with an important result regarding the approximation of the multivariate inverse Laplace transform.
Let $f:[0,\infty)^{n}\rightarrow [0,\infty)$ be a Borel-measurable function such that
\begin{align*}
\int_{0}^{\infty}\cdots\int_{0}^{\infty}f(t_{1},\ldots,t_{n})\prod_{k=1}^{n}e^{-q_{k}t_{k}}\dif t_{1}\cdots\dif t_{n}
\end{align*}
exists for any $q_{1},\ldots,q_{n}>0$. Then, the multivariate Laplace transform $\widehat{f}$ of $f$, defined by
\begin{align*}
\widehat{f}(q_{1},\ldots,q_{n})=\mathcal{L}(f)(q_{1},\ldots,q_{n}):=\int_{0}^{\infty}\cdots\int_{0}^{\infty}f(t_{1},\ldots,t_{n})\prod_{k=1}^{n}e^{-q_{k}t_{k}}\dif t_{1}\cdots\dif t_{n},
\end{align*}
is well-defined for any $q_{k} \in \mathbb{C}^+$, $k=1,\ldots,n$, where\footnote{We will denote by $\Re(z)$ the real part of $z\in\mathbb{C}$, and $\mathrm{i}=\sqrt{-1}$ will be used to denote the imaginary unit.} $\mathbb{C}^+:= \set{z\in\mathbb{C} \;|\; \Re(z)>0}$ with $\Re(z)$ denoting the real part of $z\in\mathbb{C}$.
The inverse multivariate Laplace transform of function $g:(\mathbb{C}^+)^n \to\mathbb{C}$, is the function $\check{g}$, such that $\mathcal{L}(\check{g}) = g$. We will also write
$\check{g} = \mathcal{L}^{-1}(g)$. The existence and uniqueness of the inverse Laplace transform is a well understood subject (cf. \cite{Widder1941}). Although there are explicit formulas of the inverse Laplace transform for many functions, generally speaking, in many practical situations the inverse Laplace transform of a function is computed by numerical approximation technics. We refer the reader to \cite{abate2006unified}, and the references therein, for a unified framework for numerically inverting the Laplace transform. For sake of completeness, we present here one such method - the Talbot inversion formula - for one and two dimensional case; the multidimensional case is done by analogy.
Assume that $\widehat{f}$ is the Laplace transform of a function $f:(0,+\infty)\to\mathbb{C}$. The Talbot inversion formula to approximate $f$ is given by
\begin{align}\label{eq:talbotInversion}
f^{b}_M(t)=\frac2{5t}\sum_{k=0}^{M-1}\Re\left(\gamma_{k}\widehat{f}(\frac{\delta_{k}}{t})\right),
\end{align}
where
\begin{align}
\delta_{0}&=\frac{2M}{5},\quad\delta_{k}=\frac{22k\pi}{5}(\cot(\frac{k\pi}{M})+\mathrm{i}),\quad 0<k<M,\nonumber\\
\gamma_{0}&=\frac12 e^{\delta_{0}},\quad\gamma_{k}=\left(1+\mathrm{i}\frac{k\pi}{M}(1+\cot^{2}(\frac{k\pi}{M}))- \mathrm{i}\cot(\frac{k\pi}{M}) \right)e^{\delta_{k}},\quad 0<k<M.\label{eq:talbotGamma}
\end{align}
Analogously, given a Laplace transform $\widehat{g}$ of a complex-valued function $g$ of two non-negative real variables,
the Talbot inversion formula to compute $g(t_{1},t_{2})$ numerically is given by
\begin{align*}
g^{b}_M(t_{1},t_{2})=\frac{2}{25t_{1}t_{2}}\sum_{k_{1}=0}^{M-1}\Re\left\{\gamma_{k_{1}}\sum_{k_{2}=0}^{M-1}\left[\gamma_{k_{2}}\widehat{g}\left(\frac{\delta_{k_{1}}}{t_{1}},\frac{\delta_{k_{2}}}{t_{2}} \right) +\bar{\gamma}_{k_{2}}\widehat{g}\left(\frac{\delta_{k_{1}}}{t_{1}},\frac{\bar{\delta}_{k_{2}}}{t_{2}} \right)\right] \right\},
\end{align*}
where $\delta_{k},\gamma_{k},0\le k<M,$ are given in \eqref{eq:talbotGamma}.
\subsection{A Special Case of Numerical Inverse Laplace Transform}\label{subsubsec:specialcase}
Let us consider a function $f:[0,\infty)\rightarrow [0,\infty)$ and its Laplace transform $\widehat{f}(q)$, for $q\in\mathbb{C}^+$. It turns out that the inverse Laplace transform of $f$ can be approximated numerically by using only values of the function $\widehat{f}$ on the positive real line. One such approximation is the Gaver-Stehfest formula
\begin{align}\label{eq:GaverStehfest}
f_{n}(t)=\frac{n\log 2}{t}\binom{2n}{n}\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}\widehat{f}\left(\frac{(n+k)\log 2}{t} \right).
\end{align}
For other methods and the comparison of their speeds of convergence we refer to \cite{abate2006unified}. Consecutive application of \eqref{eq:GaverStehfest} leads to the multivariate Gaver-Stehfest formula.
{\small
\def$'${$'$}
}
\end{document} | math | 68,540 |
\begin{document}
\author{Gijs M. Tuynman}
\title{The Super Orbit Challenge}
\address{Laboratoire Paul Painlev{\'e}, U.M.R. CNRS 8524 et D{\'e}partement de Math{\'e}matiques, Facult{\'e} des Sciences et Technologies, Universit{\'e} de Lille, 59655 Villeneuve d'Ascq Cedex, France}
\email{FirstName[dot]LastName[at]univ-lille[dot]fr}
\begin{abstract}
When using the generally adopted definition of a super unitary representation, there are lots of super Lie groups for which the regular representation is not super unitary.
I propose a new definition of a super unitary representation for which all regular representations are super unitary.
I then choose a particular super Lie group (of Heisenberg type) for which I provide a list of super unitary representations in my new sense, obtained by a heuristic super orbit method.
The super orbit challenge is to find a well defined {super orbit method} that will reproduces more or less my list of super unitary representations (or explains why they should not appear).
\end{abstract}
\thanks{This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01).}
\keywords{Super unitary representation, super orbit method}
\subjclass{58A50, 22E99, 57S20}
\maketitle
\section{Introduction}
In order to make this paper as easily accessible as possible, I will interpret a super Lie group as a super Harish-Chandra pair $(G_o,\Liealg g)$, even though I prefer to interpret them as a supermanifold $G$ with a compatible group structure (in the sense of $\CA$-manifolds \cite{Tu04}).
In a super Harish-Chandra pair $(G_o,\Liealg g)$, $\Liealg g=\Liealg g_0 \oplus \Liealg g_1$ is a super Lie algebra (over $\RR$) and $G_o$ an ordinary Lie group acting on $\Liealg g$ such that:
\begin{enumerate}[{\labelSHCP}1.]
\item
the Lie algebra of $G_o$ is (isomorphic to) $\Liealg g _0$;
\item
the action of $G_o$ preserves each $\Liealg g_\alpha$ (the action is \myquote{even});
\item
the restriction of the $G_o$ action to $\Liealg g_0$ is (isomorphic to) the adjoint action of $G_o$ on it Lie algebra.
\end{enumerate}
The generally accepted definition of a super unitary representation of a super Lie group $(G_o,\Liealg g)$ is the one that can be found (among others) in \cite[Def. 2, \S2.3]{CCTV:2006} and \cite{AllHilLau:2013}.
One defines a super Hilbert space $(\Hilbert,\inprodsym, \supinprsym)$,as a graded Hilbert space $\Hilbert = \Hilbert_0\oplus \Hilbert_1$ with scalar product $\inprodsym$ and super scalar product $\supinprsym$ (a graded symmetric non-degenerate sesquilinear form) satisfying the following conditions:
\begin{enumerate}[\labelIntroSHS1.]
\item\label{IntroSHS1label}
$\inprod{\Hilbert_0}{\Hilbert_1}=0$;
\item\label{IntroSHS2label}
for all homogeneous $x,y\in \Hilbert$ we have $\superinprod xy = i^{\parity x}\cdot \inprod xy$.
\end{enumerate}
With these ingredients a super unitary representation of $(G_o,\Liealg g)$ on the super Hilbert space $(\Hilbert,\inprodsym, \supinprsym)$ then is a couple $(\rho_o,\tau)$ in which $\rho_o$ is an ordinary unitary representation of $G_o$ on the Hilbert space $\Hilbert$ and $\tau:\Liealg g\to \End\bigl(C^\infty(\rho_o)\bigr)$ an even super Lie algebra representation of $\Liealg g$ on $C^\infty(\rho_o)$, the space of smooth vectors for $\rho_o$ defined by
$$
C^\infty(\rho_o) = \{\,\psi\in \Hilbert \mid g\mapsto \rho(g)\psi \text{ is a smooth map }G\to\Hilbert\,\}
\mapob,
$$
satisfying the conditions:
\begin{enumerate}[\labelIntroSUR1.]
\item
for each $g\in G_o$ the map $\rho_o(g)$ preserves each $\Hilbert_\alpha$ (the representation is \myquote{even});
\item
for each $X\in \Liealg g_0$ (the Lie algebra of $G_o$!) the map $\tau(X)$ is the restriction of the infinitesimal generator of $\rho_o\bigl(\exp(tX)\bigr)$ to $C^\infty(\rho_o)$;
\item
for each $X\in \Liealg g_\alpha$ the map $\tau(X)$ is graded skew-symmetric with respect to $\supinprsym$;
\item
for all $g\in G_o$ and all $X\in \Liealg g_1$ we have
$$
\tau(g\cdot X) = \rho_o(g)\scirc \tau(X) \scirc \rho_o(g\mo)
\mapob,
$$
where on the left we denote by $g\cdot X$ the action of $G_o$ on $\Liealg g$.
\end{enumerate}
Unfortunately, already for the most simple super Lie group $\RR^{0\vert 1}$, the $0\vert1$-dimensional abelian super Lie group for which the super Harish-Chandra pair is $(\{e\}, \{0\}\oplus\RR)$, the (left-) regular representation is not super unitary in the above sense.
The representation space is the space of (smooth) functions $C^\infty(\RR^{0\vert1})$ of a single odd variable, \ie, isomorphic to $\CC^2$ via $f(\xi) = a_0+a_1\xi$ and the infinitesimal action is given by the operator $\partial_\xi$, \ie, by the matrix $(\begin{smallmatrix} 0 & 1 \\ 0 & 0\end{smallmatrix})$. As $\CC^2=\CC\oplus\CC$ is $1\vert1$-dimensional, there is no possible choice for a super Hilbert space structure on $\CC^2$ for which the regular representation is super unitary.
In \cite{Tuynman:2017}\footnote{As \cite{Tuynman:2017} was too long for most journals, a shortened version without the sections on Berezin-Fourier decomposition will appear as \cite{Tuynman:2018}.} I proposed to change the definition of a super Hilbert space to a triple $(\Hilbert, \inprodsym, \supinprsym)$ by changing the condition \refmetnaam{\labelIntroSHS}{IntroSHS2label} to
\begin{enumerate}[\labelIntroSHS'1.]
\setcounter{enumi}{1}
\item
$\supinprsym$ is continuous with respect to the topology of $\Hilbert$ defined by $\inprodsym$.
\end{enumerate}
But remember, $\supinprsym$ is just a non-degenerate graded symmetric sesquilinear form, not necessarily even nor homogeneous.
And then I proposed to change the definition of a super unitary representation on a super Hilbert space $(\Hilbert, \inprodsym, \supinprsym)$ as a triple $(\rho_o,\Dense,\tau)$ in which $\rho_o$ is an ordinary unitary representation of $G_o$ on the Hilbert space $\Hilbert$ and $\tau:\Liealg g\to \End\bigl(\Dense\bigr)$ an even super Lie algebra representation of $\Liealg g$ on $\Dense \subset C^\infty(\rho_o) \subset \Hilbert$, a dense graded subspace of $\Hilbert$ contained in the set of smooth vectors of the unitary representation $\rho_o$, satisfying the conditions:
\begin{enumerate}[\labelIntroSUR'1.]
\item
for each $g\in G_o$ the map $\rho_o(g)$ preserves each $\Hilbert_\alpha$ (the representation is \myquote{even});
\item
for each $X\in \Liealg g_0$ (the Lie algebra of $G_o$!) the map $\tau(X)$ is the restriction of the infinitesimal generator of $\rho_o\bigl(\exp(tX)\bigr)$ to $\Dense$;
\item
for each $X\in \Liealg g_\alpha$ the map $\tau(X)$ is graded skew-symmetric with respect to $\supinprsym$;
\item
for all $g\in G_o$ and all $X\in \Liealg g_1$ we have
$$
\tau(g\cdot X) = \rho_o(g)\scirc \tau(X) \scirc \rho_o(g\mo)
\mapob;
$$
\item
$\Dense\subset \Hilbert$ is maximal with respect to the four conditions above.
\end{enumerate}
I then showed that the left-regular representation of any connected super Lie group is super unitary in this new sense.\footnote{A slightly less far going modification of the notion of a super Hilbert space and an associated notion of a super unitary representation is proposed in \cite
{deGoursacMichel:2015}.}
In particular for the simplest example of the $0\vert1$-dimensional super Lie group cited above, it suffices to take an odd super scalar product $\supinprsym$ instead of an even one as imposed by the standard definition.
Now I think that rendering all regular representations super unitary is sufficient reason to justify my change of the definition of a super unitary representation, but my initial motivation comes from a heuristic super version of the orbit method.
In \cite{Tuynman:2010}\footnote{The timeline of the official publications is different from the production timeline as can be seen from the arXiv dates.} I introduced the notion of a mixed symplectic form and I showed that coadjoint orbits of a super Lie group carry in a natural way such a mixed symplectic form. In \cite{Tuynman:2009} (see also \cite{Tuynman:2010/2}) I then showed that representations associated to orbits with a non-homogeneous symplectic form appear in the (Fourier-Berezin) decomposition\footnote{The same Fourier-Berezin decomposition technique was used in \cite{AllHilWur:2016} to decompose the regular representation of the $0\vert1$-dimensional super Lie group described above, using a family of representations depending on an odd parameter.} of the regular representation of an explicit example of dimension $4\vert4$, justifying the introduction of non-homogeneous symplectic forms.
Now there seems to be a certain reluctance to accept the notion of non-even symplectic forms (see for instance \cite{AllHilWur:2016}) and my \myquote{justifying} paper \cite{Tuynman:2009} has a serious drawback: half of the used procedure is heuristic and no (super) Hilbert spaces are mentioned.
I still have no satisfactory way to produce (by means of super geometric quantization of super symplectic manifolds with a non-even symplectic form) structures that might lead to super Hilbert spaces; for super Lie groups, I only have a systematic way to produce representations (essentially on spaces of smooth functions) associated to coadjoint orbits and polarizations, and then I have to invent by hand the (super) Hilbert space structure adapted to such a representation and I have to adapt by hand the dependence on odd parameters linked to the specific orbit.
But now that I have a convenient notion of a super unitary representation, I will give, for a particular (Heisenberg like) super Lie group of dimension $3\vert3$, a list of super unitary representations in my new sense.
I am convinced this will be the complete list of all inequivalent irreducible super unitary representations of this group, but (of course) I have no proof and I might be wrong.
And then the challenge is to find a systematic way to obtain them via a well-defined orbit method.
The interested reader will find another example in \cite{Tuynman:2009} (for which I have the same conviction) to test any super orbit method, although in that example no mention is made of any kind of notion of super unitary representation.
\section{A super Lie group and a list of super unitary representations}
As a super Lie group of dimension $3\vert3$ our example $G$ \myquote{is} $\RR^{3\vert3}$ with three global even coordinates $a,b,c$ and three global odd coordinates $\alpha, \beta, \gamma$ and group law given by the multiplication
\begin{align*}
(a, b, \alpha, \beta,c,\gamma) \cdot (\ah, \bh, \alphah, \betah, \ch, \gammah)
&
=
\bigl(a+\ah, b+\bh, \alpha+\alphah, \beta+\betah,
\\&
\kern4em
c+\ch + \tfrac12(a\bh -b\ah-\alpha\betah-\beta\alphah),
\\&
\kern4em
\gamma + \gammah + \tfrac12(a\betah - \beta\ah+b\alphah - \alpha\bh)\bigr)
\mapob.
\end{align*}
As a super Harish-Chandra pair $(G_o,\Liealg g)$ it is given by the standard Heisenberg group $G_o = \RR^3$ of dimension $3$ with group law
\begin{moneq}
(a, b, c) \cdot (\ah, \bh, \ch)
=
\bigl(a+\ah, b+\bh,
c+\ch + \tfrac12(a\bh -b\ah)
\bigr)
\mapob.
\end{moneq}
The super Lie algebra $\Liealg g = \Liealg g_0 \oplus \Liealg g_1$ of dimension $3\vert 3$ with three even basis vectors $e_0,e_1,e_2$ and three odd basis vectors $f_0,f_1,f_2$ is described by the commutators
\begin{moneq}
{}
[e_1,e_2]=e_0 = [f_1,f_2]
\qquad,\qquad
[e_1,f_2] = f_0 = [e_2,f_1]
\mapob,
\end{moneq}
all others either $0$ or determined by graded skew-symmetry.
It is a central extension of the abelian super group of dimension $2\vert2$ by a $1\vert1$-dimensional center; at the algebra level the center is generated by the vectors $e_0,f_0$.
And finally the (adjoint) action of $G_o$ on $\Liealg g$ is given by
\begin{moneq}
\begin{aligned}
(a,b,c)\cdot e_0 &= e_0
\quad,&
(a,b,c)\cdot e_1 &= e_1 - b\,e_0
\quad,&
(a,b,c)\cdot e_2 &= e_2 + a\,e_0
\\
(a,b,c)\cdot f_0 &= f_0
\quad,&
(a,b,c)\cdot f_1 &= f_1 + b\,f_0
\quad,&
(a,b,c)\cdot f_2 &= f_2 + a\,f_0
\mapob.
\end{aligned}
\end{moneq}
Once we have the description of our super Lie group, we can provide our list of seven families of super unitary representations.
However, instead of providing the unitary representation $\rho_o$ of $G_o$ and the infinitesimal representation $\tau$, I will give the integrated version $\rho$, which is a bona fide representation of the full super group $G$. The unitary representation $\rho_o$ is directly obtained by putting $\alpha=\beta=\gamma=0$ in the expression for $\rho$, and $\tau$ is obtained by computing the derivatives of $\rho$ with respect to the six variables $a,b,c,\alpha, \beta, \gamma$ at the point $(a,b,c,\alpha,\beta,\gamma) = \mathbf0$.
For the third family this will be done explicitly.
\nextfamily
We start with a family of $1$-dimensional representations depending on two real parameters $k,\ell$ and two odd parameters $\kappa, \lambda$.
Our graded Hilbert space is given by $\Hilbert = \CC \oplus \{0\}$ with scalar product and super scalar product $\superinprod\chi\psi = \inprod{\chi}{\psi} = \overline\chi\cdot \psi$.
And then the representation $\rho$ is given by
\begin{moneq}
\rho(a,b,\alpha,\beta,c,\gamma)\psi
=
\eexp^{i(a k+b \ell + \alpha\kappa + \beta\lambda)}\,\psi
\mapob.
\end{moneq}
\nextfamily
For this family the Hilbert space is $\Hilbert = L^2(\RR^2)\oplus \{0\}$ with its standard scalar product and super scalar product given by
\begin{moneq}
\inprod\chi\psi = \superinprod\chi\psi
=
\int \overline{\chi(x,y)}\,\psi(x,y)\ \extder x \,\extder y
\mapob.
\end{moneq}
On this Hilbert space we define a one-parameter family of representations $\rho$ depending on a nonzero odd parameter $\kappa$ by
\begin{moneq}
\bigl(\rho(a,b,\alpha,\beta,c,\gamma)\psi\bigr)(x, y)
=
\psi(x + b, y + a) \, \eexp^{i\alpha\kappa x} \, \eexp^{i\beta\kappa y}
\,\eexp^{i (\gamma + \frac12(\beta a + b \alpha))\kappa}
\mapob.
\end{moneq}
\nextfamily
Here the graded Hilbert space is $\Hilbert$ is given by $\Hilbert = L^2(\RR)\oplus L^2(\RR)$, which I interpret as functions of one even variable $x$ and one odd variable $\xi$ according to
\begin{moneq}
(\psi_0, \psi_1)\in L^2(\RR) \oplus L^2(\RR)
\qquad\cong\qquad
\psi(x,\xi)=\psi_0(x)+\xi\psi_1(x)
\mapob.
\end{moneq}
The scalar product $\inprodsym$ and the (odd) super scalar product $\supinprsym$ are given by
\begin{align*}
\inprod\chi\psi
&
=
\int \overline{\chi_0(x)}\,{\psi_0(x)} + \overline{\chi_1(x)}\, {\psi_1(x)} \ \extder x
\\
\superinprod\chi\psi
&
=
\int \overline{\chi_0(x)}\,{\psi_1(x)} + \overline{\chi_1(x)}\, {\psi_0(x)} \ \extder x
\mapob.
\end{align*}
On this Hilbert space we define a $1$-parameter family of representations $\rho$ depending on a nonzero real parameter $k$ by
\begin{moneq}
\bigl(\rho(a,b,\alpha,\beta,c,\gamma)\psi\bigr)(x,\xi)
=
\psi(x+k a,\xi-k \alpha) \, \eexp^{ibx} \, \eexp^{-i\beta\xi}
\, \eexp^{i k(c + \frac12(ab-\alpha\beta))}
\mapob.
\end{moneq}
This means that the unitary representation $\rho_o$ is given by
\begin{align*}
\bigl(\rho_o(a,b,c)\psi\bigr)(x,\xi)
&
=
\psi(x+k a,\xi)\, \eexp^{ix b}
\,
\eexp^{i k(c + \frac12ab)}
\end{align*}
and the super Lie algebra representation is given by
\begin{moneq}
\begin{aligned}
\tau(e_0)\psi
&
= i k \,\psi
&\quad
\tau(e_1)\psi
&
=
k\,\fracp\psi{x}
&\quad
\tau(e_2)\psi
&
=
ix\,\psi
\\
\tau(f_0)\psi
&
=
0
&
\tau(f_1)\psi
&
=
-k\,\fracp\psi{\xi}
&\quad
\tau(f_2)\psi
&
=
-i\xi\,\psi
\mapob.
\end{aligned}
\end{moneq}
\nextfamily
For this family the Hilbert space is the same as for the third family.
On it we define a two-parameter family of representations $\rho$ depending on a real parameter $k$ and a nonzero odd parameter $\kappa$ by
\begin{moneq}
\bigl(\rho(a,b,\alpha,\beta,c,\gamma)\psi\bigr)(x, \xi)
=
\psi(x + a, \xi - \alpha) \, \eexp^{ib(k+\xi\kappa)} \, \eexp^{i\beta\kappa x}
\,\eexp^{i (\gamma + \frac12(\beta a - b \alpha))\kappa}
\mapob.
\end{moneq}
\nextfamily
For this family the Hilbert space is again the same as for the third family.
On it we define a two-parameter family of representations $\rho$ depending on a nonzero real parameter $k$ and a nonzero odd parameter $\kappa$ by
\begin{moneq}
\begin{aligned}
\bigl(\rho(a,b,\alpha,\beta,c,\gamma)\psi\bigr)(x,\xi)
&
=
\psi(x + a,\xi - \alpha) \, \eexp^{ib(xk+\xi\kappa)} \, \eexp^{i\beta(x\kappa - \xi k)}
\\&
\kern5em
\eexp^{i (\gamma + \frac12(\beta a - b \alpha))\kappa} \,
\eexp^{ik (c + \frac12(a b +\beta \alpha))}
\mapob.
\end{aligned}
\end{moneq}
\nextfamily
Here the graded Hilbert space is $\Hilbert = \CC^2 \oplus \CC^2$, which I interpret as functions of two odd variables $\xi$ and $\eta$ according to
\begin{moneq}
\bigl((\psi_0,\psi_{12})\oplus (\psi_1,\psi_2)\bigr) \in \CC^2 \oplus \CC^2
\quad\cong\quad
\psi(\xi,\eta) = \psi_0 + \xi\,\psi_1 + \eta\,\psi_2 + \xi\eta\,\psi_{12}
\mapob.
\end{moneq}
The standard scalar product $\inprodsym$ and the super scalar product $\supinprsym$ are given by
\begin{align*}
\inprod\chi\psi
&
=
\overline{\chi_0}\,\psi_0 + \overline{\chi_{12}}\,\psi_{12} + \overline{\chi_1}\,\psi_1 + \overline{\chi_2}\,\psi_2
\\
\superinprod\chi\psi
&
=
\overline{\chi_0}\,\psi_{12} + \overline{\chi_{12}}\,\psi_{0} + \overline{\chi_1}\,\psi_2 - \overline{\chi_2}\,\psi_1
\mapob.
\end{align*}
On this Hilbert space we define a three-parameter family of representations $\rho$ depending on two real parameters $k,\ell$ and a nonzero odd parameter $\kappa$ by
\begin{moneq}
\bigl(\rho(a,b,\alpha,\beta,c,\gamma)\psi\bigr)(\xi, \eta)
=
\psi(\xi - \beta,\eta - \alpha) \, \eexp^{ia(\xi \kappa + k)} \, \eexp^{ib(\eta \kappa + \ell)}
\,\eexp^{i(\gamma - \frac12(\beta a + b \alpha))\kappa }
\mapob.
\end{moneq}
\nextfamily
For this family the Hilbert space is given by $\Hilbert = L^2(\RR)^2 \oplus L^2(\RR)^2$, which I interpret as functions of one even variable $x$ and two odd variables $\xi,\eta$ according to
\begin{multline*}
\bigl((\psi_0,\psi_{12})\oplus (\psi_1,\psi_2)\bigr) \in L^2(\RR)^2 \oplus L^2(\RR)^2
\\
\qquad\cong\qquad
\psi(x,\xi,\eta) = \psi_0(x) + \xi\,\psi_1(x) + \eta\,\psi_2(x) + \xi\eta\,\psi_{12}(x)
\mapob.
\end{multline*}
The scalar product $\inprodsym$ and the super scalar product $\supinprsym$ are given by
\begin{align*}
\inprod\chi\psi
&
=
\int
\overline{\chi_0(x)}\,\psi_0(x) + \overline{\chi_{12}(x)}\,\psi_{12}(x) + \overline{\chi_1}(x)\,\psi_1(x) + \overline{\chi_2(x)}\,\psi_2(x) \ \extder x
\\
\superinprod\chi\psi
&
=
\int
\overline{\chi_0(x)}\,\psi_{12}(x) + \overline{\chi_{12}(x)}\,\psi_{0}(x) + \overline{\chi_1(x)}\,\psi_2(x) - \overline{\chi_2(x)}\,\psi_1(x) \ \extder x
\mapob.
\end{align*}
On this Hilbert space we define a three parameter family of representations $\rho$ depending on two nonzero real parameters $k,p$ and a nonzero odd parameter $\kappa$ by
\begin{moneq}
\begin{aligned}
\bigl(\rho(a,b,\alpha,\beta,c,\gamma)\psi\bigr)(x,\xi,\eta)
&
=
\psi(x + a - p b, \xi- \alpha, \eta - \beta)
\eexp^{ib( xk + \xi \kappa + p\eta \kappa)} \,
\\&
\kern4em
\eexp^{i\beta (x \kappa - \xi k)} \,
\eexp^{i(\gamma -p b \beta
+ \frac12 (\beta a - b \alpha))\kappa}\,
\\&
\kern6em
\eexp^{ ik(c + \frac12(a b +\beta \alpha - pb^2))}
\mapob.
\end{aligned}
\end{moneq}
\section{Concluding remarks}
$\bullet$
All (super) Hilbert spaces are interpreted as spaces of functions on super spaces of the form $\RR^{p\vert q}$, or more precisely as spaces whose elements we can interpret as (smooth) functions of $q$ odd variables with values in the space of square integrable functions of $p$ real variables, \ie, $\Hilbert = C^\infty\bigl(\RR^{0\vert q}; L^2(\RR^p)\bigr)$.
It then turns out that in all cases the super scalar product $\superinprod\chi\psi$ is realised as the (translation invariant) Berezin-Lebesgue integral $\int_{\RR^{p\vert q}} \overline{\chi(m)}\,\psi(m)\ \extder m$.
$\bullet$
All (infinitesimal) representations $\tau$ act by differentiation or multiplication, and as such these operators act on the space of smooth functions $C^\infty(\RR^{p\vert q})$.
In all cases the unmentioned dense subspace $\Dense$ then is given by
\begin{moneq}
\Dense = \{\,\psi\in C^\infty(\RR^{p\vert q}) \mid \forall k\in \NN\ \forall X_1, \dots, X_k\in \Liealg g : \tau(X_1) \scirc \cdots \scirc \tau(X_k)\psi \in \Hilbert\,\}
\ .
\end{moneq}
$\bullet$
In most of the families of representations we have required some of the parameters to be nonzero.
Not because these representations do not exist when they take the value zero, but because in those cases the representation will certainly not be irreducible.
$\bullet$
In my way of thinking, the first family is associated to (coadjoint) orbits of dimension $0\vert0$, the third family is associated to orbits of dimension $2\vert 2$ with an even symplectic form, the families $2$, $4$ and $6$ are associated to orbits of dimension $2\vert 2$ with an odd symplectic form, and the families $5$ and $7$ are associated to orbits of dimension $2\vert 2$ with a non-homogeneous symplectic form.
The seventh family is atypical as it is obtained by a polarization that is not of \myquote{maximal} dimension.
For $p=0$ we recover (apart from the term $\partial_\eta$ in $\tau(f_2)$) the fifth family with the additional variable $\eta$.
$\bullet$
The attentive reader will have noticed that I have not been completely honest, as the families 4--7 do not fit my description of a super unitary representation.
In particular $\rho_o$ is not an ordinary unitary representation of the ordinary Lie group $G_o$, due to the presence of the (supposedly nonzero) odd parameter $\kappa$.
On the other hand, apart from the fact that some of the parameters are odd, all these representations definitely have a \myquote{unitary} look, especially when one realises that the super scalar product is defined by integration of the product $\overline\chi\,\psi$ with respect to a translation invariant \myquote{measure.}
As moreover all these families are obtained in the same way, I am sorely tempted to want to enlarge the definition of a super unitary representation even more in order to include all these families (e.g., linking the scalar product $\inprodsym$ directly to the super scalar product $\supinprsym$ in the spirit of a Krein space as in \cite{deGoursacMichel:2015}, but by dropping the condition that $\rho_o$ should preserve $\inprodsym$).
Unfortunately I have not (as yet) a satisfactory way to do so.
$\bullet$ And last but not least: all feed back will be appreciated.
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\end{document} | math | 22,604 |
\begin{document}
\thispagestyle{empty}
\title[The max-characteristic function]{An offspring of multivariate extreme value theory: the max-characteristic function}
\author{Michael Falk$^{(1)}$ \& Gilles Stupfler$^{(2)}$}
\let\thefootnote\relax\footnote{\small $^{(1)}$ Universit\"at W\"urzburg, 97074 W\"urzburg, Germany \\
$^{(2)}$ School of Mathematical Sciences, The University of Nottingham, University Park, \\ Nottingham NG7 2RD, United Kingdom}
\email{[email protected], [email protected]}
\begin{abstract}
This paper introduces max-characteristic functions (max-CFs), which are an offspring of multivariate extreme-value theory. A max-CF characterizes the distribution of a random vector in $\mathbb{R}^d$, whose components are nonnegative and have finite expectation. Pointwise convergence of max-CFs is shown to be equivalent to convergence with respect to the Wasserstein metric. The space of max-CFs is not closed in the sense of pointwise convergence. An inversion formula for max-CFs is established.
\end{abstract}
\keywords{Multivariate extreme-value theory, max-characteristic function, Wasserstein metric, convergence}
\subjclass[2010]{Primary 60E10, secondary 60F99, 60G70}
\maketitle
\section{Introduction}
\label{intro}
Multivariate extreme-value theory (MEVT) is the proper toolbox for analyzing several extremal events simul\-taneous\-ly. Its practical relevance in particular for risk assessment is, consequently, obvious. But on the other hand MEVT is by no means easy to access; its key results are formulated in a measure theoretic setup; a common thread is not visible.
Writing the `angular measure' in MEVT in terms of a random vector, however, provides the missing common thread: Every result in MEVT, every relevant probability distribution, be it a max-stable one or a generalized Pareto distribution, every relevant copula, every tail dependence coefficient etc. can be formulated using a particular kind of norm on multivariate Euclidean space, called $D$-norm; see below. For a summary of MEVT and $D$-norms we refer to \citet{fahure10, aulbf11, aulfaho11b,aulfaho11, aulfahozo14, aulfazo14, falk15}. For a review of copulas in the context of extreme-value theory, see, e.g., \citet{gennes12}.
A norm $\norm\cdot_D$ on $\mathbb{R}^d$ is a \textit{$D$-norm}, if there exists a random vector (rv) $\bm{Z}=(Z_1,\dots,Z_d)$ with $Z_i\ge 0$, ${\rm E}(Z_i)=1$, $1\le i\le d$, such that
\begin{equation*}
\norm{\bm{x}}_D={\rm E}\left\{\max_{1\le i\le d}\left(\abs{x_i}Z_i\right)\right\},\qquad \bm{x}=(x_1,\dots,x_d)\in\mathbb{R}^d.
\end{equation*}
In this case the rv $\bm{Z}$ is called \textit{generator} of $\norm\cdot_D$. Here is a list of $D$-norms and their generators:
\begin{itemize}
\item $\norm{\bm{x}}_\infty = \max_{1\le i\le d}\abs{x_i}$ is
generated by $\bm{Z}=(1,\dots,1)$,
\item $\norm{\bm{x}}_1=\sum_{i=1}^d\abs{x_i}$ is
generated by $\bm{Z}=$ random permutation of $(d,0,\dots,0)\in\mathbb{R}^d$ with equal probability $1/d$,
\item $\norm{\bm{x}}_\lambda=\left(\sum_{i=1}^d\abs{x_i}^\lambda\right)^{1/\lambda}$,
$1<\lambda<\infty$. Let $X_1,\dots,X_d$ be independent and identically Fr\'{e}chet-distributed random variables, i.e., $\Pr(X_i\le x)=$ $\exp(-x^{-\lambda})$, $x>0$, $\lambda>1$. Then $\bm{Z}=(Z_1,\dots,Z_d)$ with
\begin{equation*}\label{eq:frechet_generator}
Z_i=\frac{X_i}{\Gamma(1-1/\lambda)},\quad i=1,\dots,d,
\end{equation*}
generates $\norm\cdot_{\lambda}$. By $\Gamma(p)=\int_0^\infty x^{p-1}e^{-x}\,dx$, $p>0$, we denote the usual Gamma function.
\end{itemize}
$D$-norms are a powerful tool when analyzing dependence in MEVT. The first letter of the word ``dependence'' is, therefore, the reason for the index $D$.
The generator of a $D$-norm is not uniquely determined, even its distribution is not. Let, for example, $X\ge 0$ be a random variable with ${\rm E}(X)=1$ and put $\bm{Z}=(X,\dots,X)$. Then $\bm{Z}$ generates $\norm\cdot_\infty$ as well. However, we can, given a generator $\bm{Z}$ of a $D$-norm, design a $D$-norm in a simple fashion so that it characterizes the distribution of $\bm{Z}$: consider the $D$-norm on $\mathbb{R}^{d+1}$
$$
(t,\bm{x})\mapsto {\rm E}\left\{\max(\abs t,\abs{x_1}Z_1,\dots ,\abs{x_d}Z_d)\right\}.
$$
Then it turns out that the knowledge of this $D$-norm fully identifies the distribution of $\bm{Z}$; it is actually enough to know this $D$-norm when $t=1$, as Lemma~\ref{lem:uniqueness_of_max-cf} below shows, and this shall be the basis for our definition of a max-characteristic function.
\begin{lemma}\label{lem:uniqueness_of_max-cf}
Let $ \bm{X} = (X_1, \dots, X_d) \geq \bm{0}$, $\bm{Y} = (Y_1, \dots, Y_d) \geq \bm{0}$ be random vectors with
$ {\rm E}(X_i), {\rm E}(Y_i) < \infty$ for all $i \in \{ 1, \ldots , d\}$.
If we have for each $\bm{x} > \bm{0} \in \mathbb{R}^d$
\[
{\rm E}\left\{\max(1,x_1X_1,\dots ,x_dX_d)\right\} = {\rm E}\left\{\max(1,x_1Y_1, \dots, x_dY_d)\right\},
\]
then $\bm{X}=_d\bm{Y}$, where ``$=_d$'' denotes equality in distribution.
\end{lemma}
\begin{proof}
Fubini's theorem implies ${\rm E}(X)=\int_0^\infty \Pr(X>t)\,dt$ for any random variable $X\ge 0$. consequently, we
have for $\bm{x} > \bm{0}$ and $c > 0$
\begin{align*}
{\rm E}\left\{\max\left(1,\frac{X_1}{cx_1},\dots ,\frac{X_d}{cx_d}\right)\right\}
&= \int_{0}^{\infty} 1-\Pr\left\{\max\left(1,\frac{X_1}{cx_1},\dots ,\frac{X_d}{cx_d}\right) \leq t \right\} \, dt\\
&= \int_{0}^{\infty}1-\Pr(1 \leq t, X_i \leq tcx_i,\, 1 \leq i \leq d) \, dt\\
&= 1+\int_{1}^{\infty}1-\Pr\left(X_i \leq tcx_i, \,1 \leq i \leq d \right) \, dt.
\end{align*}
The substitution $t \mapsto t/c $ yields that the right-hand side above equals
\[
1+\frac{1}{c}\int_{c}^{\infty}1-\Pr(X_i\leq tx_i, 1 \leq i \leq d) \, dt.
\]
Repeating the preceding arguments with $Y_i$ in place of $X_i$, we obtain for all $c>0$ from the assumption the equality
$$\int_{c}^{\infty}1-\Pr(X_i\leq tx_i, 1 \leq i \leq d) \, dt
= \int_{c}^{\infty}1-\Pr(Y_i\leq tx_i, 1 \leq i \leq d)\, dt.
$$
Taking right derivatives with respect to $c$ we obtain for $c>0$
\begin{align*}
1-\Pr(X_i\leq cx_i,\, 1 \leq i \leq d) = 1 - \Pr(Y_i \leq cx_i,\, 1\leq i \leq d),
\end{align*}
and, thus, the assertion.
\end{proof}
Let $\bm{Z}=(Z_1,\dots,Z_d)$ be a random vector, whose components are nonnegative and integrable. Then we call
\[
\varphi_{\bm{Z}}(\bm{x})={\rm E}\left\{\max\left(1,x_1Z_1,\dots,x_dZ_d\right)\right\}, \quad \bm{x}=(x_1,\dots,x_d)\ge\bm{0}\in\mathbb{R}^d,
\]
the \textit{max-characteristic function} (max-CF) pertaining to $\bm{Z}$. Lemma \ref{lem:uniqueness_of_max-cf} shows that the distribution of a nonnegative and integrable random vector $\bm{Z}$ is uniquely determined by its max-CF.
Some obvious properties of $\varphi_{\bm{Z}}$ are $\varphi_{\bm{Z}}(\bm{0})=1$, $\varphi_{\bm{Z}}(\bm{x})\ge 1$ for all $\bm{x}$ and
\[
\varphi_{\bm{Z}}(r\bm{x})\begin{cases}
\le r \varphi_{\bm{Z}}(\bm{x})& \mbox{if } r\ge 1,\\
\ge r \varphi_{\bm{Z}}(\bm{x})& \mbox{if } 0 \le r\le 1.
\end{cases}
\]
It is straightforward to show that any max-CF is a convex function and, thus, it is continuous and almost everywhere differentiable; besides, its derivative from the right exists everywhere. This fact will be used in Section \ref{subsec:an_inversion_formula}, where we will establish an inversion formula for max-CFs.
When $\bm{Z}$ has bounded components, we have $\varphi_{\bm{Z}}(\bm{x})=1$ in a neighborhood of the origin. Finally, the max-CF of $\max(\bm{Z}_1,\bm{Z}_2)$ (where the max is taken componentwise) evaluated at $\bm{x}$ is equal to the max-CF of the vector $(\bm{Z}_1,\bm{Z}_2)$ evaluated at the point $(\bm{x},\bm{x})$.
\begin{rem} \upshape When $d=1$, the max-CF of a nonnegative and integrable random variable $Z$ is
\begin{eqnarray*}
\varphi_Z(x) = {\rm E}\left\{\max\left(1,xZ\right)\right\} &=& 1+\int_{1}^{\infty} \Pr(xZ > t) \, dt \\
&=& 1+x\int_{1/x}^{\infty} \Pr\left(Z > z \right)\, dz \\
&=& 1+x {\rm E}\{(Z-1/x)\mathbf{1}_{\{ Z>1/x\}}\}.
\end{eqnarray*}
The latter expression is connected to the expected shortfall of $Z$; see \citet{embkm97}. Indeed, if $q_Z$ is the quantile function of $Z$ then the expected shortfall of $Z$ is defined, for all $\alpha \in (0,1)$, by
$$
\mathrm{ES}_Z(\alpha)=\frac{1}{1-\alpha}\int_{\alpha}^1 q_Z(\beta)\, d\beta.
$$
When the distribution function (df) of $Z$ is continuous, defining
$$
g(\beta)=\min\left(\frac{\beta}{1-\alpha},1 \right) = \left\{
\begin{array}{ll}
\dfrac{\beta}{1-\alpha} & \mbox{if} \quad \beta \leq 1-\alpha, \\[5pt]
1 & \mbox{otherwise,}
\end{array}
\right.
$$
for all $\beta\in (0,1)$, then
$$
\mathrm{ES}_Z(\alpha)=\frac{1}{1-\alpha}\int_0^{1-\alpha} q_Z(1-\beta)\, d\beta=\int_0^1 q_Z(1-\beta)\, dg(\beta).
$$
An integration by parts and the change of variables $\beta=\Pr(Z>z)$ give
\begin{eqnarray*}
\mathrm{ES}_Z(\alpha)=\int_0^1 g(\beta) dq_Z(1-\beta) &=& \int_0^{\infty} g\{\Pr(Z>z)\} dz \\
&=& q_Z(\alpha)+\frac{1}{1-\alpha} \int_{q_Z(\alpha)}^{\infty} \Pr\left(Z > z \right) \, dz.
\end{eqnarray*}
A similar argument in the more general context of Wang distortion risk measures is given in \citet{elmstu2016}. Letting $x=x_{\alpha}=1/q_Z(\alpha)$, $\alpha\in (0,1)$, we obtain
$$
\varphi_Z(x_{\alpha}) = 1+x_{\alpha} (1-\alpha) \{\mathrm{ES}_Z(\alpha) - q_Z(\alpha)\}.
$$
If the stop-loss premium risk measure of $Z$ is defined as
$$
\mathrm{SP}_Z(\alpha) = (1-\alpha)\{\mathrm{ES}_Z(\alpha)- q_Z(\alpha)\} = \int_{q_Z(\alpha)}^{\infty} \Pr\left(Z > z \right) \, dz,
$$
see \citet{embkm97}, then
$$
\varphi_Z(x_{\alpha}) = 1+x_{\alpha} \mathrm{SP}_Z(\alpha).
$$
\end{rem}
This remark suggests that max-CFs are closely connected to well-known elementary objects such as conditional expectations and risk measures; a particular consequence of it is that computing a max-CF is, in certain cases, much easier than computing a standard CF, i.e., a {\it Fourier transform}. The following example illustrates this idea.
\begin{exam}\upshape Let $Z$ be a random variable having the generalized Pareto distribution with location parameter $\mu\geq 0$, scale parameter $\sigma>0$ and shape parameter $\xi\in (0,1)$, whose distribution function is
$$
\Pr(Z\leq z)=1-\left( 1+\xi\frac{z-\mu}{\sigma} \right)^{-1/\xi},\qquad z\ge \mu.
$$
The expression of the characteristic function of this distribution is a fairly involved one which depends on the Gamma function evaluated in the complex plane. However, it is straightforward that, for all $x>0$,
$$
\int_x^{\infty} \Pr\left(Z > z \right)\, dz = \begin{cases} {\rm E}(Z)-x=\mu-x+\dfrac{\sigma}{1-\xi} & \mbox{if } x <\mu, \\[10pt] \dfrac{\sigma}{1-\xi} \left( 1+\xi\dfrac{x-\mu}{\sigma} \right)^{1-1/\xi} & \mbox{if } x \geq\mu. \end{cases}
$$
Hence the max-CF of $Z$ is
$$
\varphi_Z(x)=\begin{cases} x{\rm E}(Z)=x\left( \mu+\dfrac{\sigma}{1-\xi} \right) & \mbox{if } x >\dfrac{1}{\mu}, \\[10pt] 1+\dfrac{\sigma x}{1-\xi} \left( 1+\xi\dfrac{1-\mu x}{\sigma x} \right)^{1-1/\xi} & \mbox{if } x \leq\dfrac{1}{\mu}. \end{cases}
$$
\end{exam}
The following example is a consequence of the Pickands--de Haan--Resnick representation of a max-stable distribution function; see, e.g., \citet[Theorems~4.2.5, 4.3.1]{fahure10}. In this paper, all operations on vectors $\bm{x},\bm{y}\in\mathbb{R}^d$ such as $\bm{x}+\bm{y}$, $\bm{x}/\bm{y}$, $\bm{x}\le \bm{y}$, $\max(\bm{x},\bm{y})$ etc. are always meant componentwise.
\begin{exam}\upshape
Let $G$ be a $d$-dimensional max-stable distribution function with identical univariate Fr\'{e}chet-margins $G_i(x)=\exp(-x^{-\alpha})$, $x>0$, $\alpha >1$. Then there exists a $D$-norm $\norm\cdot_D$ on $\mathbb{R}^d$ such that $G(\bm{x})=\exp\left(-\norm{1/\bm{x}^\alpha}_D\right)$, $\bm{x}>\bm{0}\in\mathbb{R}^d$. Let the random vector $\bm{x}i$ have distribution function $G$. Its max-CF is
\begin{align*}
\varphi_{\bm{x}i}(\bm{x})&= 1 + \int_1^\infty 1-\exp\left(-\frac{\norm{\bm{x}^\alpha}_D}{y^\alpha}\right)\,dy\\
&=1+ \norm{\bm{x}^\alpha}_D^{1/\alpha} \int_{1/\norm{\bm{x}^\alpha}_D^{1/\alpha}} 1-\exp(-y^{-\alpha})\,dy,\qquad \bm{x}\ge \bm{0}\in\mathbb{R}^d.
\end{align*}
\end{exam}
This paper is organized as follows. In Section \ref{sec:convergence_of_max-cf} we establish among others the fact that pointwise convergence of max-CFs is equivalent to convergence with respect to the Wasserstein distance. In Section \ref{subsec:general_remarks_on_max-cf} we list some general remarks on max-CFs. In particular, it is shown that the space of max-CFs is not closed in the sense of pointwise convergence.
An inversion formula for max-CF, by which the distribution function of a nonnegative and integrable random variable can be restored by knowing its max-CF, is established in Section \ref{subsec:an_inversion_formula}.
\section{Convergence of max-characteristic functions}\label{sec:convergence_of_max-cf}
Denote by $d_W(P,Q)$ the Wasserstein metric between two probability
distributions on $\mathbb{R}^d$ with finite first moments, i.e.,
\begin{equation*}
d_W(P,Q)\\
=\inf\set{{\rm E}\left(\norm{ \bm{X}- \bm{Y}}_1\right):\, \bm{X}\mathrm{\ has\ distribution\ }P,\, \bm{Y} \mathrm{\ has\ distribution\ }Q}.
\end{equation*}
It is well known that convergence of probability measures $P_n$ to $P_0$ with respect to the
Wasserstein metric is equivalent to weak convergence together with
convergence of the sequence of moments
\[
\int_{\mathbb{R}^d} \norm{\bm{x}}_1\,P_n(d \bm{x}) \to_{n\to\infty} \int_{\mathbb{R}^d}\norm{\bm{x}}_1\,P_0(d \bm{x});
\]
see, for example, Definition 6.8 of \citet{villani09}.
Let $\bm{X},\bm{Y}$ be integrable random vectors in $\mathbb{R}^d$ with distributions $P$ and $Q$. By $d_W(\bm{X},\bm{Y})=d_W(P,Q)$ we denote the Wasserstein distance between $\bm{X}$ and $\bm{Y}$. The next result states that pointwise convergence of max-CFs is equivalent to convergence with respect to the Wasserstein metric.
\begin{theorem}\label{theo:characterization_of_pointwise_convergence_of_cf}
Let $\bm{Z}$, $\bm{Z}^{(n)}$, $n\in\mathbb{N}$, be nonnegative and integrable random vectors in $\mathbb{R}^d$ with corresponding max-CF $\varphi_{\bm{Z}}$, $\varphi_{\bm{Z}^{(n)}}$, $n\in\mathbb{N}$. Then $\varphi_{\bm{Z}^{(n)}}\to_{n\to\infty}\varphi_{\bm{Z}}$ pointwise $\Leftrightarrow$ $d_W\left(\bm{Z}^{(n)},\bm{Z}\right)\to_{n\to\infty}0$.
\end{theorem}
\begin{proof}
Suppose that $d_W(\bm{Z}^{(n)},\bm{Z})\to_{n\to\infty}0$. Then we can find versions $\bm{Z}^{(n)}$, $\bm{Z}$ such that ${\rm E}\left(\norm{\bm{Z}^{(n)}-\bm{Z}}_1\right)\to_{n\to\infty}0$. This implies, for $\bm{x}=(x_1,\dots,x_d)\ge 0$,
\begin{align*}
\varphi_{\bm{Z}^{(n)}}(\bm{x}) &= {\rm E}\left(\max [1,x_1 \{Z_1+(Z_1^{(n)}-Z_1) \}, \dots, x_d \{Z_d+(Z_d^{(n)}-Z_d) \} ]\right)\\
&\begin{cases}
\le {\rm E}\{\max(1,x_1Z_1,\dots,x_dZ_d) \} + \norm{\bm{x}}_\infty {\rm E}\left(\norm{\bm{Z}^{n}-\bm{Z}}_1\right)\\
\ge {\rm E}\left\{\max(1,x_1Z_1,\dots,x_dZ_d) \right\} - \norm{\bm{x}}_\infty {\rm E}\left(\norm{\bm{Z}^{n}-\bm{Z}}_1\right)
\end{cases}\\
&=\varphi_{\bm{Z}}(\bm{x})+ o(1).
\end{align*}
Suppose next that $\varphi_{\bm{Z}^{(n)}}\to_{n\to\infty}\varphi_{\bm{Z}}$ pointwise. We have for $t>0$ and $\bm{x}=(x_1,\dots,x_d)\ge \bm{0}$
$$
t\varphi_{\bm{Z}^{(n)}} \left (\frac \bm{x} t \right) = {\rm E} \{ \max ( t,x_1 Z_1^{(n)},\ldots,x_d Z_d^{(n)}) \}.
$$
This gives
\begin{eqnarray*}
t\varphi_{\bm{Z}^{(n)}}\left( \frac \bm{x} t \right) &=& \int_0^{+\infty} \Pr \{ \max ( t,x_1 Z_1^{(n)},\ldots,x_d Z_d^{(n)} )>y \} dy \\
&=& t+\int_t^{+\infty} \Pr \{\max ( x_1 Z_1^{(n)},\ldots,x_d Z_d^{(n)} )>y \} dy
\end{eqnarray*}
so that
\begin{equation}\label{eqn:integral_representation_of_max_cf}
t\varphi_{\bm{Z}^{(n)}}\left(\frac \bm{x} t\right) = t + \int_t^\infty 1- \Pr (x_iZ_i^{(n)}\le y,1\le i\le d)\,dy.
\end{equation}
Now, for $\varepsilon > 0$ and $1\le i\le d$
\begin{align*}
{\rm E} (Z_i^{(n)} ) - {\rm E}\left(Z_i\right)&= \int_0^\infty 1- \Pr (Z_i^{(n)}\le y )\,dy - \int_0^\infty 1- \Pr\left(Z_i\le y\right)\,dy\\
&= \int_{\varepsilon/2}^\infty 1- \Pr (Z_i^{(n)}\le y )\,dy - \int_{\varepsilon/2}^\infty 1- \Pr\left(Z_i\le y\right)\,dy + R_{n,i}(\varepsilon)
\end{align*}
where
$$
|R_{n,i}(\varepsilon)| =\left| \int_0^{\varepsilon/2} \Pr (Z_i^{(n)}\le y ) - \Pr\left(Z_i\le y\right) \,dy \right| \leq \varepsilon/2.
$$
Equation~(\ref{eqn:integral_representation_of_max_cf}) then gives
$$
\left| {\rm E} (Z_i^{(n)} ) - {\rm E}\left(Z_i\right) \right| \leq \varepsilon \ \mbox{ for large enough } n,
$$
which entails convergence of ${\rm E} (Z_i^{(n)})$ to ${\rm E}\left(Z_i\right)$. Consequently, we have to establish weak convergence of $\bm{Z}^{(n)}$ to $\bm{Z}$. From Equation \eqref{eqn:integral_representation_of_max_cf} we obtain for $0<s<t$ and $\bm{x}=(x_1,\dots,x_d)\ge 0$
\begin{align}\label{eqn:limit_of_difference_of_mcf}
t \varphi_{\bm{Z}^{(n)}}\left(\frac \bm{x} t\right) - s \varphi_{\bm{Z}^{(n)}}\left(\frac \bm{x} s\right) &= \int_s^t \Pr (x_iZ_i^{(n)}\le y, 1\le i\le d )\,dy\nonumber\\
&\to_{n\to\infty} t \varphi_{\bm{Z}}\left(\frac \bm{x} t\right) - s \varphi_{\bm{Z}}\left(\frac \bm{x} s\right)\nonumber\\
&= \int_s^t \Pr\left(x_iZ_i\le y, 1\le i\le d\right)\,dy.
\end{align}
Let $\bm{x}=(x_1,\dots,x_d)\ge 0$ be a point of continuity of the distribution function of $\bm{Z}$. Suppose first that $\bm{x}>0$. Then we have
\[
\Pr (\bm{Z}^{(n)}\le \bm{x} ) = \Pr\left(\frac 1{x_i}Z_i^{(n)}\le 1,1\le i\le d\right).
\]
If
\[
\limsup_{n\to\infty} \Pr\left(\frac 1{x_i}Z_i^{(n)}\le 1,1\le i\le d\right) > \Pr\left(\frac 1 {x_i}Z_i\le 1,1\le i\le d\right)
\]
or
\[
\liminf_{n\to\infty} \Pr\left(\frac 1{x_i}Z_i^{(n)}\le 1,1\le i\le d\right) < \Pr\left(\frac 1 {x_i}Z_i\le 1,1\le i\le d\right),
\]
then Equation \eqref{eqn:limit_of_difference_of_mcf} readily produces a contradiction by putting $s=1$ and $t=1+\varepsilon$ or $t=1$ and $s=1-\varepsilon$ with a small $\varepsilon >0$. We, thus, have
\begin{equation}\label{eqn:weak_convergence_in_the_positive_case}
\Pr (\bm{Z}^{(n)}\le \bm{x} )\to_{n\to\infty} \Pr\left(\bm{Z}\le \bm{x}\right)
\end{equation}
for each point of continuity $\bm{x}=(x_1,\dots,x_d)$ of the distribution function of $\bm{Z}$ with strictly positive components.
Suppose next that $x_j=0$ for $j\in T\subset \set{1,\dots,d}$, $x_i>0$ for $i\not\in T$, $T\not=\emptyset$. In this case we have
\[
\Pr(\bm{Z}\le \bm{x})= \Pr(Z_i\le x_i,i\not\in T, Z_j\le 0, j\in T)=0
\]
by the continuity from the left of the distribution function of $\bm{Z}$ at $\bm{x}$. We thus have to establish
\[
\limsup_{n\to\infty}\Pr (\bm{Z}^{(n)}\le \bm{x} ) =\limsup_{n\to\infty}\Pr\left(Z_i^{(n)}\le x_i,i\not\in T, Z_j^{(n)}\le 0, j\in T\right)=0.
\]
Suppose that
\[
\limsup_{n\to\infty}\Pr\left(Z_i^{(n)}\le x_i,i\not\in T, Z_j^{(n)}\le 0, j\in T\right) =c >0.
\]
Choose a point of continuity $\bm{y}>\bm{x}$. Then we obtain
\[
0<c\le \limsup_{n\to\infty}\Pr (\bm{Z}^{(n)}\le \bm{y} )=\Pr(\bm{Z}\le \bm{y})
\]
by Equation \eqref{eqn:weak_convergence_in_the_positive_case}. Letting $\bm{y}$ converge to $\bm{x}$ we obtain $\Pr(\bm{Z}\le \bm{x})\ge c>0$ and, thus, a contradiction. This completes the proof of Theorem \ref{theo:characterization_of_pointwise_convergence_of_cf}. \end{proof}
Convergence of a sequence of max-CFs is therefore stronger than the convergence of standard CFs: the example of a sequence of real-valued random variables $(Z_n)$ such that
$$
\Pr(Z_n=e^n)=\frac{1}{n} \ \mbox{ and } \ \Pr(Z_n=0)=1-\frac{1}{n}
$$
is such that $Z_n\to 0$ in distribution, as can be seen from computing the related sequence of CFs, but ${\rm E}(Z_n)=e^n/n\to \infty \neq 0$.
Corollary~\ref{corconv} below, which is obtained by simply rewriting Theorem~\ref{theo:characterization_of_pointwise_convergence_of_cf}, is tailored to applications to MEVT.
\begin{cor}\label{corconv}
Let $\bm{X}^{(n)}$, $n\in\mathbb{N}$, be independent copies of a random vector $\bm{X}$ in $\mathbb{R}^d$ that is nonnegative and integrable in each component. Let $\bm{x}i=(\xi_1,\dots,\xi_d)$ be a max-stable random vector with Fr\'{e}chet margins $\Pr(\xi_i\le x)=\exp\left(-1/x^{\alpha_i}\right)$, $x>0$, $\alpha_i>1$, $1\le i\le d$. Then we obtain from Theorem \ref{theo:characterization_of_pointwise_convergence_of_cf} the equivalence
\[
d_W\left(\frac{\max_{1\le i\le n}\bm{X}^{(i)}}{\bm{a}^{(n)}},\bm{x}i\right)\to_{n\to\infty} 0
\]
for some norming sequence $\bm{0}<\bm{a}^{(n)}\in\mathbb{R}^d$ if and only if
\[
\varphi_n\to_{n\to\infty}\varphi_{\bm{x}i} \qquad \mbox{pointwise},
\]
where $\varphi_n$ denotes the max-CF of $\max_{1\le i\le n}\bm{X}^{(i)}/\bm{a}^{(n)}$, $n\in\mathbb{N}$.
\end{cor}
The following example shows a nice application of the use of max-CFs to the convergence of the componentwise maxima of independent generalized Pareto random vector in the total variation distance.
\begin{exam}\upshape
Let $U$ be a random variable that is uniformly distributed on $(0,1)$ and let $\bm{Z}=(Z_1,\dots,Z_d)$ be the generator of a $D$-norm $\norm\cdot_D$ with the additional property that each $Z_i$ is bounded, i.e., $Z_i\le c$, $1\le i\le d$, for some constant $c\ge 1$. We require that $U$ and $\bm{Z}$ are independent.
Then the random vector
\[
\bm{V}=(V_1,\dots,V_d)= \frac 1 {U^{1/\alpha}} (Z_1^{1/\alpha},\dots,Z_d^{1/\alpha} )
\]
with $\alpha>0$ follows a \textit{multivariate generalized Pareto distribution}; see, e.g., \citet{buihz08} or \citet[Chapter 5]{fahure10}. Precisely, we have for $\bm{x}\ge (c^{1/\alpha},\dots,c^{1/\alpha})\in\mathbb{R}^d$
\begin{equation*}
\Pr(\bm{V}\le \bm{x})= \Pr\left(U\ge \max_{1\le i\le d}\frac{Z_i}{x_i^{\alpha}}\right)
= 1- {\rm E}\left(\max_{1\le i\le d}\frac{Z_i}{x_i^{\alpha}}\right)
= 1 - \norm{\frac 1{\bm{x}^\alpha}}_D.
\end{equation*}
Let now $\bm{V}^{(1)}, \bm{V}^{(2)},\dots$ be independent copies of $\bm{V}$ and put
\[
\bm{Y}^{(n)}= \frac{\max_{1\le i\le n}\bm{V}^{(i)}}{n^{1/\alpha}}.
\]
Then we have for $\bm{x}>\bm{0}\in\mathbb{R}^d$ and $n$ large
\begin{equation}\label{eqn:convergence_of_gpd}
\Pr (\bm{Y}^{(n)}\le \bm{x} ) = \left(1- \norm{\frac 1{n\bm{x}^\alpha}}_D\right)^n \to_{n\to\infty}\exp\left(-\norm{\frac 1{\bm{x}^\alpha}}_D \right) = \Pr(\bm{x}i\le \bm{x}),
\end{equation}
where $\bm{x}i$ is a max-stable random vector with identical Fr\'{e}chet margins $\Pr(\xi_i\le x)=\exp(-1/x^\alpha)$, $x>0$.
Choose $\alpha>1$; in this case the components of $\bm{V}$ and $\bm{x}i$ have finite expectations. By
writing
\[
\varphi_{\bm{Y}^{(n)}}(\bm{x}) = 1 +\int_1^\infty 1- \Pr (\bm{Y}^{(n)}\le t/\bm{x} )\,dt
\]
and using Equation \eqref{eqn:convergence_of_gpd}, elementary arguments such as a Taylor expansion make it possible to show that the sequence of max-CF $\varphi_{\bm{Y}^{(n)}}$ converges pointwise to the max-CF $\varphi_{\bm{x}i}$ of $\bm{x}i$. Since convergence with respect to the Wasserstein metric is equivalent to convergence in distribution, denoted by $\to_d$, together with convergence of the moments, we obtain from Theorem \ref{theo:characterization_of_pointwise_convergence_of_cf} that in this example we actually have both $\bm{Y}^{(n)}\to_d\bm{x}i$ and ${\rm E} (Y^{(n)}_i )\to_{n\to\infty} {\rm E}(\xi_i)=\Gamma(1-1/\alpha)$ for $1\le i\le d$.
\end{exam}
\begin{exam}\upshape
Let $\bm{U}^{(1)},\bm{U}^{(2)},\dots$ be independent copies of the random vector $\bm{U}=(U_1,\dots,U_d)$, which follows a copula $C$ on $\mathbb{R}^d$, i.e., each $U_i$ is uniformly distributed on $(0,1)$. It is well-known (see, e.g., \citet[Section 5.2]{fahure10}) that there exists a non-degenerate random vector $\bm{e}ta=(\eta_1,\dots,\eta_d)$ on $(-\infty,0]^d$ such that
\begin{equation*}
\bm{V}^{(n)}= n\left(\max_{1\le j\le n}\bm{U}^{(j)}-\bm{1}\right) \to_d\bm{e}ta
\end{equation*}
if and only if there exists a $D$-norm $\norm\cdot_D$ on $\mathbb{R}^d$ such that, for all $\bm{x}\le\bm{0}\in\mathbb{R}^d$,
\begin{equation*}
\Pr (\bm{V}^{(n)}\le \bm{x} )\to_{n\to\infty} \exp\left(-\norm{\bm{x}}_D\right)= G(\bm{x}),
\end{equation*}
or if and only if there exists a $D$-norm $\norm\cdot_D$ on $\mathbb{R}^d$ such that
\begin{equation*}
C(\bm{u}) = 1-\norm{\bm{1}-\bm{u}}_D + o\left(\norm{\bm{1}-\bm{u}}\right)
\end{equation*}
as $\bm{u}\to\bm{1}$, uniformly for $\bm{u}\in[0,1]^d$.
We have for $1\le i\le d$
\begin{equation*}
{\rm E}\left\{n\left(1-\max_{1\le j\le n}U_i^{(j)} \right)\right\}=\frac n{n+1}\to_{n\to\infty} 1
\end{equation*}
and, thus, we obtain from Theorem \ref{theo:characterization_of_pointwise_convergence_of_cf} the characterization
\begin{equation*}
\bm{V}^{(n)}\to_d\bm{e}ta \Leftrightarrow d_W\left(\bm{V}^{(n)},\bm{e}ta\right)\to_{n\to\infty} 0 \Leftrightarrow\varphi_{-\bm{V}^{(n)}}\to_{n\to\infty}\varphi_{-\bm{e}ta}\mbox{ pointwise}.
\end{equation*}
For instance, when $d=2$, straightforward computations yield that $-{\bm{e}ta}$ arises as a weak limit above if and only if it has a max-CF of the form
$$
\varphi_{{-\bm{e}ta}}(\bm{x})= 1+x_1 \exp(-1/x_1)+x_2 \exp(-1/x_2)- \frac 1{\norm{1/\bm{x}}_D} \exp(-\norm{1/\bm{x}}_D).
$$
\end{exam}
\begin{cor}
Let $\bm{Z}$, $\bm{Z}^{(n)}$, $n\in\mathbb{N}$, be generators of $D$-norms on $\mathbb{R}^d$. Then $\varphi_{\bm{Z}^{(n)}}\to_{n\to\infty}\varphi_{\bm{Z}}$ pointwise $\Leftrightarrow$ $\bm{Z}^{(n)}\to_d \bm{Z}$.
\end{cor}
Interestingly, the convergence of a sequence of max-CFs of generators of $D$-norms also implies pointwise convergence of the related $D$-norms. We denote by $\norm\cdot_{D,\bm{Z}}$ that $D$-norm, which is generated by $\bm{Z}$.
\begin{cor}\label{coro:convergence_of_max-CF_implies_convergence_of_D-norms}
Let $\bm{Z}$, $\bm{Z}^{(n)}$, $n\in\mathbb{N}$, be generators of $D$-norms in $\mathbb{R}^d$ with respective max-CF $\varphi_{\bm{Z}}$, $\varphi_{\bm{Z}^{(n)}}$, $n\in\mathbb{N}$. Then the pointwise convergence $\varphi_{\bm{Z}^{(n)}}\to_{n\to\infty}\varphi_{\bm{Z}}$ implies $\| \cdot \|_{D,\bm{Z}^{(n)}}\to_{n\to\infty}\| \cdot \|_{D,\bm{Z}}$ pointwise.
\end{cor}
\begin{proof}
We have for $\bm{x}=(x_1,\dots,x_d)\ge \bm{0}$
\begin{align*}
\norm{\bm{x}}_{D^{(n)}}&= {\rm E}\left\{\max_{1\le i\le d}\left(x_iZ_i^{(n)}\right) \right\} = {\rm E}\left[\max_{1\le i\le d}\left\{x_iZ_i + x_i\left(Z_i^{(n)}-Z_i\right)\right\} \right]\\
&= {\rm E}\left\{\max_{1\le i\le d}(x_iZ_i)\right\} + O\left\{{\rm E}\left(\norm{\bm{Z}^{(n)}-\bm{Z}}_1 \right)\right\}\\
&\to_{n\to\infty} {\rm E}\mathbb{B}igl\{\max_{1\le i\le d}(x_iZ_i)\mathbb{B}igr\} = \norm{\bm{x}}_D
\end{align*}
with proper versions of $\bm{Z}^{(n)}$ and $\bm{Z}$.
\end{proof}
\subsection{Some general remarks on max-characteristic functions}\label{subsec:general_remarks_on_max-cf}
The goal of this section is to give a few elements about the structure of the set of max-characteristic functions. This is done by constructing a particular functional mapping between max-CFs for generators of $D$-norms, and then iterating this mapping to draw our conclusions. Specifically, in what follows we let, for any $p\in (0,1]$, $T_p$ be the functional mapping which sends any function $f:\mathbb{R}^d\to\mathbb{R}$ to
$$
T_p(f)=1-p+p f\left( \frac{\cdot}{p} \right).
$$
\begin{lemma}
\label{maxCFtrans}
If $\varphi$ is the max-CF of a generator of a $D$-norm then, for any $p\in (0,1]$, so is the function $T_p(\varphi)$.
\end{lemma}
\begin{proof}
Let $\bm{Z}$ be a generator of the max-CF $\varphi$. Pick a Bernoulli random variable $U$ having expectation $p$ and independent of $\bm{Z}$, and set
$$
\psi(\bm{x})={\rm E}\left\{\max\left( 1,x_1 \frac{U}{p} Z_1,\ldots,x_d \frac{U}{p} Z_d \right) \right\}.
$$
Then clearly $\psi$ is the max-CF of the generator of a $D$-norm, and
\begin{eqnarray*}
\psi(\bm{x}) &=& {\rm E}\left\{ \mathbf{1}_{\{ U=0 \}} + \max\left( 1,\frac{x_1}{p} Z_1,\ldots,\frac{x_d}{p} Z_d \right) \mathbf{1}_{\{ U=1 \}} \right\} \\
&=& \Pr(U=0) + \Pr(U=1) {\rm E}\left\{ \max\left( 1,\frac{x_1}{p} Z_1,\ldots,\frac{x_d}{p} Z_d \right) \right\}
\end{eqnarray*}
by the independence of $U$ and $Z$. The result follows because of the right-hand side being exactly $1-p+p\varphi(\bm{x}/p)$.
\end{proof}
\begin{lemma}
\label{maxCFiter}
For any integer $k\geq 1$, the $k$th iterate of the functional $T_p$ is
$$
f\mapsto T_p^{(k)}(f)=\Pr(X\leq k)+ p^k f\left( \frac{\cdot}{p^k} \right),
$$
where $X$ is a geometric random variable having parameter $1-p$.
\end{lemma}
\begin{proof}
The result is clearly true for $k=1$. That the conclusion holds for every integer $k$ follows by straightforward induction because
$$
(1-p)+p \Pr(X\leq k)=(1-p)+p\sum_{j=1}^k p^{j-1} (1-p) = \sum_{j=1}^{k+1} p^{j-1} (1-p)= \Pr(X\leq k+1)
$$
whenever $X$ has a geometric distribution with parameter $1-p$.
\end{proof}
In the following lemma, the phrase ``$\bm{x}\to \infty \mbox{ in } \mathbb{R}R_+^d$'' means $\norm{\bm{x}}_\infty\to\infty$ and $\bm{x}\in \mathbb{R}R_+^d$.
\begin{lemma}
\label{maxCFbound}
If $\varphi_{\bm{Z}}$ is the max-CF of a generator $\bm{Z}$ of a $D$-norm, then
$$
\max(1,\| \bm{x} \|_{D,Z})\leq \varphi_{\bm{Z}}(\bm{x})\leq 1+\| \bm{x} \|_{D,\bm{Z}} \ \mbox{ for all } \ \bm{x}=(x_1,\ldots,x_d)\in \mathbb{R}R_+^d.
$$
Especially, if $\mathcal{G}$ denotes the set of all generators of $D$-norms,
$$
\sup_{\bm{Z}\in \mathcal{G}} \left| \frac{\varphi_{\bm{Z}}(\bm{x})}{\| \bm{x} \|_{D,\bm{Z}}} - 1 \right| \to 0 \ \mbox{ as } \ \bm{x}\to \infty \mbox{ in } \mathbb{R}R_+^d.
$$
\end{lemma}
\begin{proof}
The lower bound is obtained by noting that
\begin{equation}
1\leq \max(1,x_1 Z_1,\ldots,x_d Z_d), \; \ \max(x_1 Z_1,\ldots,x_d Z_d)\leq \max(1,x_1 Z_1,\ldots,x_d Z_d)
\end{equation}
and taking expectations. The upper bound is a consequence of the inequality $\max(a,b)\leq a+b$, valid when $a,b\geq 0$. Finally, the uniform convergence result is obtained by writing
$$
1\leq \frac{\varphi_{\bm{Z}}(\bm{x})}{\| \bm{x} \|_{D,\bm{Z}}}\leq 1+\frac{1}{\| \bm{x} \|_{D,\bm{Z}}} \ \mbox{ for all } \ \bm{Z}\in \mathcal{G} \mbox{ and } \bm{x}\in \mathbb{R}R_+^d\setminus\{ \bm{0} \}.
$$
Because $\| \cdot \|_{D,\bm{Z}} \geq \| \cdot \|_{\infty}$, this entails
$$
\sup_{\bm{Z}\in \mathcal{G}} \left| \frac{\varphi_{\bm{Z}}(\bm{x})}{\| \bm{x} \|_{D,\bm{Z}}} - 1 \right| \leq \frac{1}{\| \bm{x} \|_{\infty}} \ \mbox{ for all } \ \bm{x}\in \mathbb{R}R_+^d\setminus\{ \bm{0} \}
$$
from which the conclusion follows.
\end{proof}
It is noteworthy that the inequalities of Lemma \ref{maxCFbound} are sharp, in the sense that for $\bm{Z}=(1,\ldots,1)$, $\varphi_{\bm{Z}}(\bm{x}) = \max(1,\| \bm{x} \|_{\infty}) =\max(1,\| \bm{x} \|_{D,Z})$ and therefore the leftmost inequality is in fact an equality in this case, while the rightmost inequality $\varphi_{\bm{Z}}(\bm{x})\leq a+b \| \bm{x} \|_{D,\bm{Z}}$ can only be true if $a,b\geq 1$ because of the leftmost inequality again.
Lemma \ref{maxCFbound} has the following corollary, which can also be obtained as a consequence of the monotone convergence theorem.
\begin{cor}
\label{maxCFrangecst}
No constant function can be the max-CF of a generator of a $D$-norm.
\end{cor}
Such a result is of course not true for standard CFs, since the CF of the constant random variable 0 is the constant function 1.
The next result looks at what can be said when examining the pointwise limit of iterates of the functional $T_p$ on the set of max-CFs.
\begin{prop}
\label{maxCFlim}
If $\varphi_{\bm{Z}}$ is the max-CF of a generator $\bm{Z}$ of a $D$-norm, then for any $p\in (0,1)$, the sequence of mappings $\{T_p^{(k)}(\varphi_{\bm{Z}})\}$ has a pointwise limit which is independent of $p$ and equal to
$$
T(\varphi_{\bm{Z}})=1+\| \cdot \|_{D,\bm{Z}}.
$$
\end{prop}
\begin{proof}
By Lemma~\ref{maxCFiter}, we have for any $\bm{x}\in \mathbb{R}R_+^d$, $\bm{x}\not=\bm{0}\in\mathbb{R}^d$, and any $k\geq 1$ that
$$
T_p^{(k)}(\varphi)(\bm{x})=\Pr(X\leq k)+ p^k \varphi_{\bm{Z}}\left( \frac{\bm{x}}{p^k} \right).
$$
On one hand, when $k\to\infty$, the first term on the right-hand side converges to 1; on the other hand, because $p\in (0,1)$, we have $\bm{x}/p^k\to\infty$ in $\mathbb{R}R_+^d$ and therefore
$$
\lim_{k\to\infty} p^k \varphi_{\bm{Z}}\left( \frac{\bm{x}}{p^k} \right) = \| \bm{x} \|_{D,\bm{Z}} \lim_{k\to\infty} \frac{\varphi_{\bm{Z}}(\bm{x}/p^k)}{\| \bm{x}/p^k \|_{D,\bm{Z}}} = \| \bm{x} \|_{D,\bm{Z}}
$$
by Lemma~\ref{maxCFbound}. The conclusion follows by adding these limits.
\end{proof}
\begin{cor}
\label{maxCFcoro1}
If $\| \cdot \|_{D,\bm{Z}}$ is any $D-$norm then there is an explicit, iterative way to realize the function $1+\| \cdot \|_{D,\bm{Z}}$ as a limit of max-CFs. In particular, the expression of a $D-$norm is explicitly determined by the knowledge of the max-CF of any of its generators.
\end{cor}
Note that this result certainly cannot be true the other way around, since a single $D-$norm can in general be generated by different generators.
The next result looks a bit further into the range of the map $\bm{Z}\mapsto \varphi_{\bm{Z}}$. By considering the generator $(1,\ldots,1)\in \mathbb{R}R^d$ that generates the $D$-norm $\norm\cdot_\infty$, it is obvious that $\max(1,\| \cdot \|_{\infty})$ is actually the max-CF of a generator of a $D$-norm. Looking at Lemma~\ref{maxCFbound}, one may wonder if this remains true if $\| \cdot \|_{\infty}$ is replaced by some other $D-$norm, or, in other words, if the lower bound $\max(1,\| \cdot \|_{D,Z})$ in Lemma~\ref{maxCFbound} can be achieved as a $D-$norm, and similarly for the upper bound $1+\| \cdot \|_{D,Z}$. The next result says that this is not the case.
\begin{prop}
\label{maxCFrange}
Let $\bm{Z}$ be a generator of a $D$-norm.
\begin{enumerate}
\item[(i)] The mapping $1+\| \cdot \|_{D,\bm{Z}}$ cannot be the max-CF of a generator of a $D$-norm.
\item[(ii)] If moreover $\| \cdot \|_{D,\bm{Z}}\neq \| \cdot \|_{\infty}$, then $\max(1,\| \cdot \|_{D,\bm{Z}})$ cannot be the max-CF of a generator of a $D$-norm.
\end{enumerate}
\end{prop}
\begin{proof}
We start by proving (i). Suppose there is a generator of a $D$-norm $\bm{Y}$ such that $\varphi_{\bm{Y}}=1+\| \cdot \|_{D,\bm{Z}}$. By Proposition~\ref{maxCFlim}, the sequence of mappings $T_p^{(k)}(\varphi_{\bm{Y}})$, $k\geq 1$, has the pointwise limit
$$
T(\varphi_{\bm{Y}})=1+\| \cdot \|_{D,\bm{Y}}.
$$
Besides, if $X$ is a geometric random variable with parameter $1-p$, then for all $\bm{x}\in \mathbb{R}R_+^d$
$$
T_p^{(k)}(\varphi_{\bm{Y}})(\bm{x})=T_p^{(k)}(1+\| \cdot \|_{D,\bm{Z}})(\bm{x})=\Pr(X\leq k)+ p^k (1+\| \bm{x}/p^k \|_{D,\bm{Z}})\to 1+\| \bm{x} \|_{D,\bm{Z}}
$$
as $k\to\infty$, so that $\| \cdot \|_{D,\bm{Y}} = \| \cdot \|_{D,\bm{Z}}$.
We now conclude by using Theorem~\ref{theo:characterization_of_pointwise_convergence_of_cf}: the random vector
$$
\bm{Y}^{(n)} = \frac{U_1 \cdots U_n}{p^n} \bf Y
$$
where $U_1,\ldots,U_n$ are independent Bernoulli random variables with mean $p$ which are independent of $\bf Y$, is the generator of a $D$-norm, with max-CF
$$
\varphi_{\bm{Y}^{(n)}} = T_p^{(n)}(\varphi_{\bm{Y}});
$$
see the proof of Lemma~\ref{maxCFtrans} and Lemma~\ref{maxCFiter}. By Proposition~\ref{maxCFlim}, $\varphi_{\bm{Y}^{(n)}}\to_{n\to\infty}1+\| \cdot \|_{D,\bm{Y}}=\varphi_{\bm{Y}}$ pointwise, and thus Theorem~\ref{theo:characterization_of_pointwise_convergence_of_cf} yields $d_W\left(\bm{Y}^{(n)},\bm{Y}\right)\to_{n\to\infty}0$.
But
$$
\Pr (\bm{Y}^{(n)} \neq {\bf 0} )=p^n\to_{n\to\infty}0
$$
which shows that $\bm{Y}^{(n)}$ converges in distribution to $\bm{0}$. This is a contradiction and (i) is proven.
We turn to the proof of (ii). Again, suppose there is a generator of a $D$-norm $\bm{Y}$ such that $\varphi_{\bm{Y}}=\max(1,\| \cdot \|_{D,\bm{Z}})$. We shall prove that $\| \cdot \|_{D,\bm{Z}}=\| \cdot \|_{\infty}$. The sequence of mappings $T_p^{(k)}(\varphi_{\bm{Y}})$, $k\geq 1$, has the pointwise limit
$$
T(\varphi_{\bm{Y}})=1+\| \cdot \|_{D,\bm{Y}},
$$
and if $X$ is a geometric random variable with parameter $1-p$ then for all $\bm{x}\in \mathbb{R}R_+^d$,
\begin{align*}
T_p^{(k)}(\varphi_{\bm{Y}})(\bm{x})&=T_p^{(k)}\{\max(1,\| \cdot \|_{D,\bm{Z}})\}(\bm{x})\\
&= \Pr(X\leq k)+ p^k \max(1,\| \bm{x}/p^k \|_{D,\bm{Z}}) \\
&= \Pr(X\leq k)+ \max(p^k,\| \bm{x} \|_{D,\bm{Z}}) \\
&\to 1+\| \bm{x} \|_{D,\bm{Z}}
\end{align*}
as $k\to\infty$, so that $\| \cdot \|_{D,\bm{Y}} = \| \cdot \|_{D,\bm{Z}}$. Consequently:
$$
\varphi_{\bm{Y}}(\bm{x})= {\rm E}\{\max(1,x_1 Y_1,\ldots,x_d Y_d)\} =\max[1,{\rm E}\{\max(x_1 Y_1,\ldots,x_d Y_d)\}]
$$
for all $\bm{x}=(x_1,\ldots,x_d)\in \mathbb{R}R_+^d$. For all $i\in \{ 1,\ldots,d\}$, specializing $x_j=0$ for $j\neq i$ and $x_i=1$ gives
$$
{\rm E}\{\max(1,Y_i)\} =\max\{1,{\rm E}(Y_i)\} = 1 \ \mbox{ for all } \ i\in \{ 1,\ldots,d\}.
$$
Because $Y_i$ has expectation 1, this implies that the random variables $\max(1,Y_i)-1$ and $\max(1,Y_i)-Y_i$, being nonnegative and having expectation zero, must be almost surely zero. In other words, $Y_i\leq 1$ and $Y_i\geq 1$ almost surely, and thus $Y_i=1$ almost surely for all $i$. But then $Y=(1,\ldots,1)$ is a generator of the norm $\| \cdot \|_{\infty}$, so that $\| \cdot \|_{D,\bm{Z}}=\| \cdot \|_{D,\bm{Y}}=\| \cdot \|_{\infty}$. The proof is complete.
\end{proof}
Combining Propositions~\ref{maxCFlim} and~\ref{maxCFrange}(i), we get the following corollary.
\begin{cor}
\label{maxCFcoro2}
The set of max-CFs of generators of $D$-norms is not closed in the sense of pointwise convergence.
\end{cor}
It should be noted that Corollary~\ref{maxCFcoro2} is also true for usual characteristic functions, as we can see with the example of a sequence of random variables $(X_n)$ such that for every $n$, $X_n$ is normally distributed, centered, and has variance $n^2$. Then
$$
\varphi_n(t)={\rm E}\left(e^{itX_n} \right) = e^{-n^2 t^2/2} \ \mbox{ for all } \ t\in\mathbb{R}R
$$
so that the sequence $(\varphi_n)$ converges pointwise to the indicator function of $\{ 0 \}$, which is not a characteristic function because it is not continuous.
\subsection{An inversion formula for max-characteristic functions}\label{subsec:an_inversion_formula}
As mentioned in the Introduction, any max-CF is a convex function and thus it is continuous and almost everywhere differentiable; furthermore, its derivative from the right exists everywhere.
Recall that for a vector $\bm{x}\in \mathbb{R}R^d$, the notation $\bm{x}>\bm{0}$ means that $\bm{x}$ has strictly positive components. The next result contains both an inversion formula for max-CFs and a criterion for a function to be a max-CF.
\begin{prop}
\label{invmaxCF}
Let $\bm{Z}$ be a nonnegative and integrable random vector with max-CF $\varphi_{\bm{Z}}$.
\begin{enumerate}
\item[(i)] We have, for all $\bm{x}=(x_1,\ldots,x_d)>\bm{0}$,
$$
\Pr(Z_j\leq x_j, \ 1\leq j\leq d)= \frac{\partial_{+}}{\partial t} \left\{ t\varphi_{\bm{Z}}\left( \frac{1}{t\bm{x}} \right) \right\}\mathbb{B}igr|_{\substack{t=1}}
$$
where $\partial_{+}/\partial t$ denotes the right derivative with respect to the univariate variable $t$.
\item[(ii)] If $\psi$ is a continuously differentiable function such that
\begin{eqnarray*}
\frac{\partial}{\partial t} \left\{ t\psi\left( \frac{1}{t\bm{x}} \right) \right\}\mathbb{B}igr|_{\substack{t=1}} &=& \Pr(Z_j\leq x_j, \ 1\leq j\leq d) \\
\mbox{and } \ \lim_{t\to\infty} t\left\{ \psi\left( \frac{1}{t\bm{x}} \right) -1 \right\} &=& 0
\end{eqnarray*}
for all $\bm{x}=(x_1,\ldots,x_d)>\bm{0}$, then $\psi=\varphi_{\bm{Z}}$ on $(0,\infty)^d$.
\end{enumerate}
\end{prop}
\begin{proof}
Notice first that, similarly to equation~(\ref{eqn:integral_representation_of_max_cf}), we have
$$
t\varphi_{\bm{Z}}\left( \frac{1}{t\bm{x}} \right) = t+\int_t^{+\infty} 1-\Pr(Z_j\leq yx_j, \ 1\leq j \leq d)\, dy.
$$
Note that the above representation yields $\lim_{t\to\infty}t[\varphi_{\bm{Z}}\{1/(t\bm{x})\}-1] =0$.
To show (i), notice that taking right derivatives with respect to $t$ yields
$$
\frac{\partial_{+}}{\partial t} \left\{t\varphi_{\bm{Z}}\left( \frac{1}{t\bm{x}} \right) \right\} = \Pr(Z_j\leq tx_j, \ 1\leq j\leq d).
$$
Setting $t=1$ concludes the proof of (i). To prove (ii), remark that
$$
\frac{\partial}{\partial t} \left\{ t\psi\left( \frac{1}{t\bm{x}} \right) \right\} = \psi\left( \frac{1}{t\bm{x}} \right) - \frac{1}{t}\sum_{i=1}^d \frac{1}{x_j} \partial_j \psi\left( \frac{1}{t\bm{x}} \right) \ \mbox{ for all } \ t>0,
$$
where $\partial_j \psi$ denotes the partial derivative of $\psi$ with respect to its $j$th component. In particular, because
$$
\Pr(Z_j\leq x_j, \ 1\leq j\leq d) = \frac{\partial}{\partial t} \left\{ t\psi\left( \frac{1}{t\bm{x}} \right) \right\}\mathbb{B}igr|_{\substack{t=1}} = \psi\left( \frac{1}{\bm{x}} \right) - \sum_{i=1}^d \frac{1}{x_j} \partial_j \psi\left( \frac{1}{\bm{x}} \right)
$$
we obtain by replacing $\bm{x}$ with $t\bm{x}$ that for all $t>0$,
$$
\frac{\partial}{\partial t} \left\{ t\psi\left( \frac{1}{t\bm{x}} \right) \right\} = \Pr(Z_j\leq t x_j, \ 1\leq j\leq d).
$$
Write now
\begin{eqnarray*}
t\psi\left( \frac{1}{t\bm{x}} \right) &=& t -\int_t^{\infty} \frac{\partial}{\partial y} \left[ y\left\{ \psi\left( \frac{1}{y\bm{x}} \right) -1 \right\} \right] dy \\
&=& t +\int_t^{\infty} 1-\Pr(Z_j\leq yx_j, \ 1\leq j \leq d)\, dy \\
&=& t\varphi_{\bm{Z}}\left( \frac{1}{t\bm{x}} \right)
\end{eqnarray*}
to conclude the proof of (ii).
\end{proof}
\begin{rem}\upshape
This result makes it possible to improve upon the result of Proposition~\ref{maxCFrange}(i). Assume that $\varphi_{\bm{Z}}$ is the max-CF of a nonnegative and integrable random vector such that
$$
\varphi_{\bm{Z}}(\bm{x})=\psi(1,\| \bm{x} \|),
$$
where $\psi:\mathbb{R}R_+^2\to \mathbb{R}R_+$ is a 1-homogeneous function and $\| \cdot \|$ is a norm on $\mathbb{R}R^d$. Then informally,
$$
\frac{\partial_{+}}{\partial t} \left\{ t\varphi_{\bm{Z}}\left( \frac{1}{t\bm{x}} \right) \right\}\mathbb{B}igr|_{\substack{t=1}} = \frac{\partial_{+}}{\partial t}\left\{ \psi(t,\| 1/\bm{x} \|) \right\}\mathbb{B}igr|_{\substack{t=1}} = \partial_{1,+}\psi(1,\| 1/\bm{x} \|)
$$
if $\partial_{1,+}$ denotes the right derivative with respect to the first component. In particular,
$$
\frac{\partial_{+}}{\partial t} \left\{ t\varphi_{\bm{Z}}\left( \frac{1}{t\bm{x}} \right) \right\}\mathbb{B}igr|_{\substack{t=1}} \to \begin{cases} \partial_{1,+}\psi(1,0) & \mbox{if } \bm{x}\to \bm{\infty}, \\[5pt] \partial_{1,+}\psi(1,\infty) & \mbox{if } \bm{x}\to \bf0. \end{cases}
$$
In other words, by Proposition~\ref{invmaxCF}, unless $\partial_{1,+}\psi(1,y)$ both converges to 1 as $y\to 0$ and to 0 as $y\to\infty$, the function $\psi(1,\| \cdot \|)$ cannot be a max-CF. Applying this to the example $\psi(x,y)=x+y$, we find the result of Proposition~\ref{maxCFrange}(i) again.
\end{rem}
\section*{Acknowledgment} The authors are indebted to Professor Chen Zhou for stimulating discussions, which led to Lemma \ref{lem:uniqueness_of_max-cf} and the definition of a max-CF. The authors are also grateful to two anonymous reviewers for their careful reading of the manuscript and their constructive remarks.
\end{document} | math | 42,911 |
\begin{document}
\title{The Tur\'an number for the edge blow-up of trees: the missing case}
\author{
Cheng Chi\thanks{School of Mathematical Sciences, East China Normal University, 500 Dongchuan Road, Shanghai 200240, China.
Email: [email protected].}\qquad Long-Tu Yuan\thanks{School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai 200240, P.R. China.
Email: [email protected]. Supported in part by National Natural Science Foundation of China grant 11901554 and Science and Technology Commission
of Shanghai Municipality (No. 18dz2271000).}
}
\date{}
\maketitle
\begin{abstract}
The edge blow-up of a graph is the graph obtained from replacing each edge of it by a clique of the same size where the new vertices of the cliques are all different.
Wang, Hou, Liu and Ma determined the Tur\'{a}n number of the edge blow-up of trees except one particular case.
Answering an problem posed by them, we determined the Tur\'{a}n number of this particular case.
\end{abstract}
\section{Introduction}
Given a family of graphs $\mathcal{H}$, a graph $G$ is said to be $\mathcal{H}$-free ($H$-free if $\mathcal{H}=\set{H}$) if $G$ does not contain any copy of $H\in\mathcal{H}$ as a subgraph.
A typical problem in extremal combinatorics is the following Tur\'an-type problem: what is the maximum number of edges in an $\mathcal{H}$-free graph on $n$ vertices?
The aforementioned number is called the {\it extremal number} for $\mathcal{H}$ and denoted by $\ex(n,\mathcal{H})$.
Denote by $\EX(n,\mathcal{H})$ the set of $\mathcal{H}$-free graphs on $n$ vertices with $\ex(n,\mathcal{H})$ edges and call a graph in $\EX(n,\mathcal{H})$ an {\it extremal graph} for $\mathcal{H}$.
We use $\ex(n,H)$ and $\EX(n,H)$ instead of $\ex(n,\mathcal{H})$ and $\EX(n,\mathcal{H})$ respectively when $\mathcal{H}=\set{H}$.
Much interests has been attracted to this problem during the last few decades.
In 1907, Mantal \cite{mantel1907} determined the extremal number for triangle for all $n\ge3$.
Tur\'an \cite{turan1941} extended Mantel's result to complete graph with any given order in 1941.
Our notations are standard, see \cite{Bondy}.
Given a graph $H$ and a set of vertices $A\subseteq V(H)$, we denote $\min\{\deg_H(x)\text{ ; }x\in A\}$ by $\delta_H(A)$.
Given a graph $H$ and a positive integer $p\ge2$, the {\it edge blow-up} of $H$, denoted by $H^{p+1}$,
is the graph obtained from $H$ by replacing each edge of $H$ by a clique of size $p+1$ where the new vertices of the cliques are all distinct.
In \cite{liu2013,ni2020} and \cite{yuan2019}, $\ex(n,H^{p+1})$ has been investgated for a large family of graphs $H$.
In \cite{wang2021}, Wang, Hou, Liu and Ma determined the extremal number when $H$ is tree satisfies some conditions and $p\ge3$.
Furthermore, the authors of \cite{wang2021} posed the following question.
\begin{question}\label{question1}
Give $p\ge3$ and a tree $T$ such that its two coloring classes $A$ and $B$ satisfying $|A|\le|B|$,
determine $\ex(n,T^{p+1})$ when $\delta_T(A)=1$ and $\alpha(T)>|B|$.
\end{question}
We solve this question.
First we introduce some notations.
Given two disjoint graphs $G$ and $H$, the {\it disjoint union} of $G$ and $H$, denoted by $G\cup H$,
is the graph with vertex set $V(G)\cup V(H)$ and edge set $E(G)\cup E(H)$.
We use $kG$ to denote the disjoint union of $k$ copies of $G$.
The {\it join} of $G$ and $H$, denoted by $G+H$, is the graph obtained from $G\cup H$ by adding all edges of the form $gh$, where $g\in V(G)$ and $h\in V(H)$.
Denoted by $P_n$ a path on $n$ vertices, $S_n$ a star on $n$ vertices, $C_n$ a cycle on $n$ vertices, $M_n$ a matching on $n$ vertices and
$K_{n_1,\cdots,n_p}$ the complete $p$-partite graph with the size of $i$-partite class $n_i$.
A $p$-partite Tur\'an graph on $n$ vertices, denoted by $T(n,p)$, is a $K_{n_1,\cdots,n_p}$ with $\sum_{i=1}^p n_i=n$ and $|n_i-n_j|\le1$ for $1\le i,j\le p$.
Let $H'(n,p,q)=\overline{K}_{q-1}+T(n-q+1,p)$ and $h'(n,p,q)=e(H'(n,p,q))$.
Let $e(T(n,p))=t(n,p)$.
\begin{definition}[Simonovits \cite{simonovits1974}]
Given a family of graphs $\mathcal{L}$ with $p(\mathcal{L})=p \ge 3$, let $\mathcal{M}:=\mathcal{M}(L)$ be a family of minimal graphs $M$ up to subgraph senses such that
there exist a large constant $t=t(\mathcal{L})$ depending on $\mathcal{L}$ such that there exists a graph $L\in\mathcal{L}$ such that
$L$ is a subgraph of $M\cup I_v + T$, where $T=T(t,p-2)$ and $I_v$ is an independent set on $v$ vertices.
We call $\mathcal{M}(\mathcal{L})$ the {\it decomposition family} of $\mathcal{L}$.
\end{definition}
A {\it covering} of a graph is a set of vertices $U$ such that every edge of this graph meets at least one vertices of $U$.
An {\it independent covering} of a bipartite graph is a covering $U$ such that no two vertices of $U$ are adjacent.
The {\it covering number} $\beta(T)$ of a graph $T$ is the minimum order of a covering of $T$.
The {\it independent covering number} $q(T)$ of a bipartite graph $T$ is the minimum order of an independent covering of $T$.
The {\it independent number} $\alpha(T)$ of a graph $T$ is the maximum order of a set of vertices such that no two of which are adjacent.
For a family of graphs $\mathcal{F}$ which contains at least one bipartite graph, the {\it independent covering number} of $\mathcal{F}$ is defined by
$$q(\mathcal{F})=\min\{q(F)\text{ ; $F\in\mathcal{F}$ and $F$ is bipartite.}\}$$
\begin{theorem}[Liu \cite{liu2013}]
Let $p\ge3$ be an integer and $T$ be a tree.
Let coloring classes of $T$ be $A$ and $B$, where $|A|\le|B|$.
When $n$ is sufficiently large, we have that
\begin{itemize}
\item if $\delta_T(A)=1$ and $\alpha(T)=|B|$,
then $\ex(n,T^{p+1})=h(n,p,|A|)$;
\item if $\delta_T(A)\ge2$,
then $\ex(n,T^{p+1})=h(n,p,|A|)+1$.
\end{itemize}
Furthermore, extremal graphs are characterized.
\end{theorem}
Wang, Hou, Liu and Ma \cite{wang2021} extended Liu's result to a larger family of trees very recently.
Before stating their results, we need follow definitions. \\
Define
\begin{equation*}
g_1(k)=
\begin{cases}
k^2-\frac{3}{2}k & \text{$k$ is even;} \\
k^2-\frac{3k-1}{2} & \text{$k$ is odd,}
\end{cases}
\text{ and }
g_2(k)=
\begin{cases}
k^2-\frac{3}{2}k & \text{$k$ is even;} \\
k^2-k & \text{$k$ is odd.}
\end{cases}
\end{equation*}
\begin{theorem}[Wang, Hou, Liu and Ma \cite{wang2021}]
Let $p\ge3$ be an integer and $T$ be a tree.
Let coloring classes of $T$ be $A$ and $B$, where $|A|\le|B|$.
Let $A_0=\{x\in A\text{ ; }\deg_T(x)=\delta_T(A)\}$ and $B_0=\{y\in B\text{ ; }|N(y)\cap A_0|\ge2\}$.
Denote by $q=|A|$, $k=\delta_T(A)$ and $b+2=\delta(B_0)$.
If $k\ge2$, then for sufficiently large $n$, we have $\ex(n,T^{p+1})=$
\begin{equation*}
\begin{cases}
h(n,p,q)+g_1(k) & \text{$k$ is even;} \\
h(n,p,q)+g_2(k) & \text{$k$ is odd and $B_0=\emptyset$;} \\
h(n,p,q)+g_1(k) & \text{$k$ is odd and $0\le b\le q-1-\ceil{\frac{k-1}{q-1}}$}; \\
h'(n,p,q)+g_2(k)+ \lfloor(q-1)(b-1)/2\rfloor & \text{$k$ is odd and $b\ge\max\left\{1,q-1-\ceil{\frac{k-1}{q-1}}\right\}$}. \\
\end{cases}
\end{equation*}
Furthermore, all extremal graphs are characterized.
\end{theorem}
Now we set $\mathcal{M}=\mathcal{M}(T^{p+1})$ and $q=q(\mathcal{M})$.
If there exists a graph $T'\in\mathcal{M}$ such that $\beta(T')\le q-1$, then
we let $\mathcal{B}:=\mathcal{B}(T)$ be the family of graph $T'[A_{T'}]$, where $T'\in\mathcal{M}$ and $A_{T'}$ is a covering set with size at most $q-1$ of $T'$.
If $\beta(T')\ge q$ for every $T'\in\mathcal{M}$, then we set $\mathcal{B}=\{K_q\}$.
\begin{theorem}\label{THM:main result 1}
Let $p\ge3$ be an integer and $T$ be a tree.
Let coloring classes of $T$ be $A$ and $B$, where $|A|\le|B|$.
If $\delta_T(A)=1$ and $\alpha(T)>|B|$,
then for sufficiently large $n$, we have
\begin{equation}
\ex(n,T^{p+1})=h'(n,p,q)+\ex(q-1,\mathcal{B})
\end{equation}
where $q$ and $\mathcal{B}$ are defined as above.
Furthermore, all extremal graphs are characterized.
\end{theorem}
\noindent {\bf Remark.} Combining with the results in \cite{liu2013} and \cite{ni2020}, the extremal number for $T^{p+1}$ is determined, where $T$ is an arbitrary tree and $p\ge3$.
A double broom $B(\ell,s,t)$ is a tree obtained from a path $P_{\ell}$
by attaching $s$ pendant edges to one end vertex of $P_{\ell}$
and $t$ pendant edges to the other end vertex of $P_{\ell}$, where $\ell,s,t\ge2$.
The double broom $B(7,5,3)$ is as in Figure~\ref{fig:double broom}.
\begin{figure}
\caption{Double broom $B(7,5,3)$}
\label{fig:double broom}
\end{figure}
\begin{corollary}
Let $p\ge3$ be an integer and $T=B(2k,s,t)$ be a double broom satisfying $k,s,t\ge2$.
Then for suffciently large $n$, we have
$$
\ex(n,T^{p+1})=h(n,p,k+1)
$$
Furthermore, $H(n,p,k+1)$ is the unique extremal graph.
\end{corollary}
\begin{proof}
It can be easily checked that $q(\mathcal{M}(T^{p+1}))=k+1$ and $\beta(T')\ge k+1$ holds for every $T'\in\mathcal{M}(T^{p+1})$.
Furthermore, we have $\alpha(T)=k-1+s+t>k+\min\{s,t\}=|B|$ and $\delta_T(A)=1$.
Therefore, the result holds by applying Theorem~\ref{THM:main result 1} with $q=k+1$ and $\mathcal{B}=\{K_{q}\}$.
\end{proof}
\section{Preliminaries}
\subsection{Technical lemmas}
Given a graph $T$, a {\it vertex split} on some vertex $v\in V(T)$ is defined by replacing $v$ by an independent set of size $\deg_T(v)$
in which each vertex is adjacent to exactly one distinct vertex in $N_T(v)$.
The family of graphs that can be obtained by applying vertex split on some $U\subseteq V(T)$ is denoted by $\mathcal{H}(T)$.
The following lemma can help us to determine the graphs in $\mathcal{M}(T^{p+1})$.
\begin{lemma}[Liu \cite{liu2013}]\label{lem:decomposition family after splitting}
Given $p\ge3$ and any graph with $\chi(H)\le p-1$, we have $\mathcal{M}(H^{p+1})=\mathcal{H}(H)$.
In particular, a matching of size $e(H)$ is in $\mathcal{M}(H^{p+1})$.
\end{lemma}
\begin{theorem}[Erd\'{o}s and Stone \cite{erdos1946}]\label{THM:turan graph}
For all integers $p\ge1$, $N\ge1$, and every $\varepsilon>0$, there exists an integer $n_0(\varepsilon,N,p+1)$ such that every graph with $n\ge n_0$ vertices and at least $t(n,p)+\varepsilon n^2$ edges contains $T(N,p+1)$ as a subgraph.
\end{theorem}
\section{Proof of Theorem~\ref{THM:main result 1}}
Given a tree $T$ with coloring classes $A$ and $B$ satisfying $|A|\le|B|$, $\delta_T(A)=1$ and $\alpha(T)>|B|$.
Now we set $\mathcal{M}$ be the decomposition family of $T^{p+1}$.
It follows from Lemma~\ref{lem:decomposition family after splitting} that $\mathcal{M}=\mathcal{H}(T)$.
Furthermore, $\mathcal{M}$ contains a matching of size $t$, where $t=e(T)$.
Let $\mathfrak{U_n}$ be the family of graphs obtained from $H'(n,p,q)$ by embedding a copy of $Q\in\EX(q-1,\mathcal{B})$ in $\overline{K}_{q-1}$ in $H'(n,p,q)$.
The definition of $\mathcal{B}$ implies every $H_n\in\mathfrak{U_n}$ is $T^{p+1}$-free, and hence we have
\begin{equation}\label{eq:ex lower bound}
\ex(n,T^{p+1})\ge h'(n,p,q)+\ex(q-1,\mathcal{B})
\end{equation}
Let $\phi(n):=\ex(n,T^{p+1})-h'(n,p,q)-\ex(q-1,\mathcal{B})$ and $K=\max\{\phi(n):n\le n_0\}$, where $n_0$ is a large constant depending on $p$ and $T$.
Clearly, $\phi(n)$ is a non-negative integer.
For the upper bound, we will show that if $n>n_0$ and $\phi(n)>0$, then there exists an $n_4$ depending on $p$ and $T$ such that $\phi(n)<\phi(n-n_4p)$.
This would imply that if $n= n_0+mn_4p$, then $\phi(n)<K-m$, and hence the theorem holds for $n\ge n_0+Kn_4p$.
Let $n_1$ be a sufficiently large constant.
Let $n_0=n_0(\varepsilon,n_1p,p)$ be the constant from Theorem~\ref{THM:turan graph}, where $\varepsilon=1/(2p(p-1))$.
Let $L_n$ be a $T^{p+1}$-free graph with $\ex(n,T^{p+1})$ edges, where $n\ge n_0$.
Equation~\ref{eq:ex lower bound} and Theorem~\ref{THM:turan graph} imply that
$L_n$ contains a $T=T(n_1p,p)$ with partite class $\widetilde{B}_1^0,\cdots,\widetilde{B}_p^0$ as a subgraph.
Note that $M_{2t}\in\mathcal{M}$.
It follows from the definition of decomposition family and the fact that $L_n$ is $T^{p+1}$-free that $\nu(L_n[\widetilde{B}_i^0])\le t$ for $i\in\brac{p}$.
Let the maximum matching in $L_n[\widetilde{B}_i^0]$ be $\{x_1y_1,\cdots,x_{t_i},y_{t_i}\}$ with $t_i\le t$.
Let $B_i^0=\widetilde{B}_i^0 \setminus\{x_1y_1,\cdots,x_{t_i},y_{t_i}\}$.
By the definition of $B_i^0$, there is no edge in $L_n[B_i^0]$.
Hence there is an induced subgraph $T_0=T(n_2p,p)$ of $L_n$ with partite class $B_1^0,\cdots,B_p^0$ obtained by deleting $2t$ vertices from each $\widetilde{B}_i$, where $n_2=n_1-2t$.
Let $c<1/(1+t)$ be a sufficiently small constant.
If there exists a vertex $x_1\in L_n-T_0$ such that $x_1$ is adjacent to at least $c^2n_2$ vertices of each partite class of $T_0$,
then $T_0$ contains a $T_1=T(c^2n_2p,p)$ such that each vertex of which is joint to $x_1$.
Generally, if there exists a vertex $x_i\in L_n-T_{i-1}-\{x_1,\cdots,x_{i-1}\}$ such that $u$ is adjacent to at least $c^{2i}n_2$ vertices of each partite class of $T_{i-1}$,
then $T_{i-1}$ contains $T_i=T(c^{2i}n_2p,p)$ such that each vertex of which is joint to $x_1,\cdots,x_i$.
Thus we can define a sequence of graphs recursively.
However, it follows from the definition of $L_n$ and $q$ that the above process stops at last after $T_{q-1}$.
Suppose to the contrary, let $V(T_q)=B_1^q\cup\cdots\cup B_p^q$.
Note that the graph induced by $B_1^q\cup\{x_1,\cdots,x_q\}$ contains some element of $\mathcal{M}$ by the definition of $q$.
Then $L_n$ contains a copy of $T^{p+1}$ by the definition of decomposition family, a contradiction.
Now suppose that the above process ends with $T_s$ with $s\le q-1$.
Let $E=\{x_1,\cdots,x_s\}$ and the partite class of $T_s$ be $B_1^s,\cdots,B_p^s$.
Denote $|B_1^s|$ by $n_3$ for convenience.
We can partition the remaining vertices into following set: Let $x\in V(L_n)\setminus(T_s\cup E)$.
If there exists an $i\in\brac{p}$ such that $x$ is adjacent to less than $c^2n_3$ vertices of $B_i^s$ and is adjacent to at least $(1-c)n_3$ vertices of $B_j^s$ for all $j\ne i$, then let $x\in C_i$.
If there exists an $i\in\brac{p}$ such that $x$ is adjacent to less than $c^2n_3$ vertices of $B_i^s$ and is adjacent to less than $(1-c)n_3$ vertices of $B_j^s$ for some $j\ne i$, then let $x\in D$.
It follows from the definition of $T_s$ that $C_1\cup\cdots\cup C_p\cup D$ is a partition of $V(L_n)\setminus(T_s\cup E)$.
Note that for a $S\subset B_i^s\cup C_i$ with $|S|\le2t$, the common neighbourhoods of $S$ in $B_j^s$ is at least $(1-2tc)n_3\ge n_3/2$, where $j\ne i$.
It follows from the definition of decomposition family and Lemma~\ref{lem:decomposition family after splitting} that $\nu(L_n[B_i^s\cup C_i])\le t$.
Now consider the edges joining $B_i^s$ and $C_i$ and select a maximum matching, say $y_1z_1,\cdots,y_{t_i}z_{t_i}$ with $y_{i'}\in B_i^s$, $z_{i'}\in C_i$ and $1\le i'\le t_i\le t$.
Let $X_i=\cup_{i'=1}^{t_i} (N_{L_n}(z_{i'})\cap B_i^s)$. Then $|X_i|\le tc^2n_3$ by the definition of $C_i$.
Let $C_i'=C_i\cup X_i$ and $B_i'=B_i^s\setminus X_i$, then $L_n[B_i'\cup C_i']$ contains no edge by the maximality of $y_1z_1,\cdots,y_{t_i}z_{t_i}$.
Hence it is possible to move $tc^2n_3$ vertices from $B_i^s$ to $C_i$ to obtain $B_i'$ and $C_i'$ such that $B_i'\subset B_i^s$ and $C_i\subset C_i'$.
Let $n_4=(1-tc^2)n_3=|B_i'|$, $T_s'=T(n_4p,p)$ and $\widehat{L}=L_n-T_s'$.
Then $T_s'$ is an induced subgraph of $L_n$ and the vertices of $\widehat{L}$ can be partitioned into $p+2$ sets $C_1',\cdots,C_p',D$ and $E$ such that
\begin{itemize}
\item every $x\in E$ is adjacent to each vertex of $T_s'$ and $|E|=s$,
\item every $x\in C_i'$ is adjacent to no vertex of $B_i'$ and
is adjacent to at least $(1-c-tc^2)n_3$ vertices of $B_j'$ for all $j\ne i$.
\item every $x\in D$ is adjacent to at most $c^2n_3$ vertices of $B_i'$ and
is adjacent to at most $(1-c)n_3$ vertices of $B_j'$ for some $i,j\in \brac{p}$ with $i\ne j$.
\end{itemize}
\noindent Let the number of edges joining $T_s'$ and $\widehat{L}$ in graph $L_n$ denoted by $e_L$.
Then we have
\begin{equation*}
e(L_n)=e(\widehat{L})+e_L+e(T_s')
\end{equation*}
Let $H_n\in \mathfrak{U}_n$ and $T_s''$ be an induced copy of $T(n_4p,p)$ in $H_n$.
Let $H_{n-n_4p}=H_n-T_s''$ and $e_H$ be the number of edges joining $T_s''$ and $H_{n-n_4p}$ in graph $H_n$.
Then
\begin{equation*}
e(H_n)=e(H_{n-n_4p})+e_H+e(T_s'')
\end{equation*}
Since $\widehat{L}$ contains no copy of $T^{p+1}$, we have $e(\widehat{L})\le e(L_{n-n_4p})$, where $L_{n-n_4p}\in\EX(n-n_4p,T^{p+1})$.
Obviously we have $e(T_s')=e(T_s'')$.
Simple calculation show that
\begin{equation}\label{equ:e_H}
\begin{aligned}
e_H
&=(q-1)n_4p+(n-n_4p-q+1)n_4(p-1)\\
&=(q-1)n_4+(n-n_4p)n_4(p-1)\\
\end{aligned}
\end{equation}
It follows from the definition of $C_i'$, $D$ and $E$ that
\begin{equation}\label{equ:e_L}
\begin{aligned}
e_L
&\le sn_4p+(n-n_4p-s-|D|)n_4(p-1)+|D|((p-2)n_4+(1-c+c^2)n_3) \\
&=sn_4+(n-n_4p)n_4(p-1)-|D|(n_4-(1-c+c^2)n_3) \\
&\le(q-1)n_4+(n-n_4p)n_4(p-1)-|D|n_3(c-(t+1)c^2) \\
&=e_H-|D|n_3(c-(t+1)c^2)
\end{aligned}
\end{equation}
Hence we have
\begin{equation*}
\begin{aligned}
\phi(n)
&=e(L_n)-e(H_n) \\
&\le e(L_{n-n_4p})-e(H_{n-n_4p})+e_L-e_H \\
&=\phi(n-n_4p)+e_L-e_H
\end{aligned}
\end{equation*}
If $e_L-e_H<0$, then we have $\phi(n)<\phi(n-n_4p)$, where $n_4\le n_2$.
Hence we suppose that $e_L-e_H\ge 0$.
Combined with Equation~\ref{equ:e_H} and~\ref{equ:e_L}, we conclude that $e_L=e_H$. (Note that $c<1/(1+t)$ is sufficiently small.)
Note that $e_L=e_H$ holds if and only if $|D|=0$, $s=q-1$ and $C_i'$ is complete to $B_j'$ for $i\in\brac{p}$ and $j\ne i$.
If $e(L_n[E])$ contains some copy of $B'\in\mathcal{B}$, then $L_n$ contains a copy of $T^{p+1}$ by the definition of $\mathcal{B}$.
Hence we conclude that $L_n[E]$ is $\mathcal{B}$-free and $e(L_n[E])\le\ex(q-1,\mathcal{B})$.
The rest of the proof will be divided into two cases.
\noindent{\bf Case 1. $ q=q(T)$.}
In this case, note that $T$ is a tree.
Clearly, $T$ admits a unique proper $2$-coloring and hence $q(T)=|A|$ holds.
Note that $\delta_T(A)=\min\{\deg_T(x)\text{ ; }x\in A\}=1$ by assumptions.
Hence there exists a vertex $u\in A$ such that $N_T(u)=\{v\}$.
Since $|A|=q(T)=q$, we can find a copy of $(T-\{u\})^{p+1}$ using vertices in $E\cup B_1'\cup\cdots\cup B_p'$ in $L_n$
Let $\phi$ be an embedding from $(T-\{u\})^{p+1}$ to $L_n$ such that
$\image \psi \subseteq E\cup B_1'\cup\cdots\cup B_p'$ and $\psi(A\setminus\{u\})=E$.
Now we will show that $B_i'\cup C_i'$ is an independent set of $L_n$ for each $i$.
It suffices to show $C_i'$ is an independent set for each $i$ since there is no edge incident with $B_i'$.
We assume that $\psi(v)\in B_\ell$, where $\ell\ne i$.
\footnote{This is possible since $E$ is joint to every vertex in $B_1'\cup\cdots\cup B_p'$.}
In fact, if there is an edge $u'u''$ in $L_n[C_i']$,
then we can choose $u_{j}$ in $B_j'$ such that $u_j\notin \image \psi$ for $j\in\brac{p}\setminus\{i,\ell\}$.
It can be seen immediately from the definition of $B_i'$, $C_i'$ and $E$ that
$u'$, $u''$ and $\psi(v)$ togerther with all $u_j$ forms a copy of $K_{p+1}$.
Furthermore, it can be verified that the mapping constructed above is an embedding from $T^{p+1}$ to $L_n$.
This completes the proof for this case.
\noindent{\bf Case 2. $ q<\sigma(T)$.}
Let $F\in\mathcal{M}$ such that $q(F)=q$. Let $A_F$ and $B_F$ be coloring classes of $F$ such that $q(F)=|A_F|$.
Now we show that $\min\{\deg_F(x)\text{ ; }x\in A_F\}=1$.
It follows from the definition of decomposition family that $\min\{\deg_F(x)\text{ ; }x\in A_F\}\ge1$.
If $A_F$ contains a vertex $u$ which is obtained by splitting a vertex in $T$, then the result follows since $\deg_F(u)=1$.
Now we assume that every $u\in A_F$ is not a vertex obtained by splitting a vertex in $T$.
Then we have $u\in V(T)$ for every $u\in A_F$.
By lemma~\ref{lem:decomposition family after splitting}, we may assume $F$ is obtained by splitting $X\subseteq V(T)$.
It is easy to see that $X\cap A_F=\emptyset$.
Otherwise, we can find a vertex obtained by splitting a vertex in $T$, a contradiction.
Let the vertices obtained by splitting $X$ in $T$ be $Y$ and $Z=B_F\setminus X$.
It is clear that $V(T)$ is the disjoint union of $A_F$, $X$ and $Z$.
Furthermore, $V(F)$ is the disjoint union of $A_F$, $Y$ and $Z$.
Note that we have $E_T(A_F,Z)=E_F(A_F,Z)$ by the definition of $Z$.
It follows from the definition that $Y\cup Z$ is an independent set of $F$.
Note that $\delta(F)\ge1$ since $F$ is obtained by splitting $X\subseteq V(T)$.
Hence every $y\in Y$ is adjacent to some $v\in A_F$ in graph $F$.
Therefore, $A_F$ is a independent covering of $T$.
Then we have $q(T)\le|A_F|=q<q(T)$, a contradiction.
Hence we have $\min\{\deg_F(x)\text{ ; }x\in A_F\}=1$.
Then similar arguments in Case 1 show that $B_i'\cup C_i'$ is an independent set of $L_n$.
Note that $B_i'\cup C_i'$ is an independent set of $L_n$ for each $i$ in both cases, then we have
\begin{equation}
\begin{aligned}
e(L_n)
&\le L_n[E]+e_{L_n}(E,V(L_n)\setminus E)+\sum_{1\le i<j\le p}e_{L_n}(B_i'\cup C_i',B_j'\cup C_j' ) \\
&\le \ex(q-1,\mathcal{B})+(q-1)(n-q+1)+\sum_{1\le i<j\le p}|B_i'\cup C_i'||B_j'\cup C_j'| \\
&\le \ex(q-1,\mathcal{B})+(q-1)(n-q+1)+t(n-q+1,p) \\
&= \ex(q-1,\mathcal{B})+h'(n,p,q)
\end{aligned}
\end{equation}
which contradicts the fact that $\phi(n)>0$. The theorem follows.
\end{document} | math | 21,467 |
\begin{document}
\jname{Biometrika}
\markboth{L. Lei et~al.}{FDR ~control under structural constraints}
\title{STAR: A general interactive framework for FDR ~control under structural constraints}
\author{Lihua Lei}
\affil{Departments of Statistics, Stanford University \\ 202 Sequoia Hall, Stanford, CA94305, U.S.A. \email{[email protected]}}
\author{Aaditya Ramdas}
\affil{Department of Statistics and Data Science, Carnegie Mellon University \\ 132H Baker Hall, CMU, Pittsburgh, PA15232, U.S.A. \email{[email protected]}}
\author{William Fithian}
\affil{Departments of Statistics, University of California, Berkeley \\ 301 Evans Hall, UC Berkeley, Berkeley, CA94720, U.S.A. \email{[email protected]}}
\maketitle
\begin{abstract}
We propose a general framework based on \emph{selectively traversed accumulation rules} (star) ~for interactive multiple testing with generic structural constraints on the rejection set. It combines accumulation tests from ordered multiple testing with data-carving ideas from post-selection inference, allowing for highly flexible adaptation to generic structural information. Our procedure defines an interactive protocol for gradually pruning a candidate rejection set, beginning with the set of all hypotheses and shrinking with each step. By restricting the information at each step
via a technique we call masking, our protocol enables interaction while controlling the false discovery rate (FDR) in finite samples for any data-adaptive update rule that the analyst may choose. We suggest update rules for a variety of applications with complex structural constraints, show that star ~performs well for problems ranging from convex region detection to FDR ~control on directed acyclic graphs, and show how to extend it to regression problems where knockoff statistics are available in lieu of $p$-values.
\end{abstract}
\begin{keywords}
interactive multiple testing, data carving, masking, knockoffs, false discovery rate, accumulation test
\end{keywords}
section{Introduction}
A classical statistical perspective divides data analysis into two distinct types: exploratory analysis is a flexible and iterative process of searching the data for interesting patterns while only pursuing a loose error guarantee, or none at all, while confirmatory analysis involves performing targeted inferences on questions that were pre-selected for focused study. Selective inference blends exploratory and confirmatory analysis by allowing for inference on questions that may be selected in a data-adaptive way, but most selective inference methods still require the analyst to pre-commit to selection rules before observing the data, falling short of the freewheeling nature of true exploratory analysis. By contrast, interactive methods are a subset of selective inference methods allowing the analyst to react to data, consult her own internal judgment, and revise her models and research plans to adapt to patterns she may not have expected to find, while still achieving valid inferences.
We consider the problem of multiple hypothesis testing with $p$-values $p_1,\ldots,p_n$, with each $p_i$ corresponding to a different null hypothesis $H_i$. A multiple testing method examines the $p$-values, possibly along with additional data, and decides which null hypotheses to reject. Let $\mathcal{H}_0 = \{i:\; H_i \text{ is true}\}$ denote the set of truly null hypotheses and let $\mathcal{R} = \{i:\; H_i \text{ is rejected}\}$ denote the rejection set. Then $R = \left|\mathcal{R}\right|$ is the number of rejections and $V = \left|\mathcal{R} \cap \mathcal{H}_0 \right|$ is the number of erroneous rejections. \citet{bh95} defined the false discovery proportion (FDP) as $V/\max\{1,R\}$ and famously proposed controlling its expectation, the false discovery rate (FDR), at some pre-specified level $\alpha$.
This work proposes a new framework for interactive multiple testing in structured settings with FDR ~control, called selectively traversed accumulation rules (star). Our method is especially well-suited to settings where we wish to impose structural constraints on the set of rejected hypotheses --- for example, to enforce a hierarchy principle for interactions in a regression or analysis of variance (ANOVA) problem, or to detect a convex spatial region where the signal exceeds a certain level in a signal processing application. In certain cases, enforcing a structural constraint is important for logical coherence or interpretability. For instance, in the case of hierarchical testing \citep[e.g.][]{yekutieli08} where hypotheses are represented as nodes on a tree, the parent of a rejected hypothesis must be rejected as well. In other cases, the structural constraint serves as a type of side information which reflects the scientific domain knowledge. By using this information judiciously, the statistical power may be boosted if the true signals meet the constraint, exactly or at least approximately.
More formally, our procedure controls the FDR ~in finite samples while guaranteeing that the rejection set $\mathcal{R}\in \mathcal{K}$ where $\mathcal{K}$ is a collection of subsets satisfying the constraint. For instance, in the case of hierarchical testing, $\mathcal{K}$ includes all subsets that correspond to rooted subtrees. The notion of structure is meant in a rather general way: $\mathcal{K}$ might also depend on auxiliary covariate information $x_i$ about the hypothesis $H_i$. For instance, in the spatial testing setting, $x_i$ gives the geographic location in $\mathbb{R}^{2}$ and $\mathcal{K}$ may include all subsets that can be written as the intersection of a convex set on $\mathbb{R}^{2}$ with $(x_i)_{i=1}^{n}$.
STAR ~generalizes the notion of masking used by the adaptive p-value thresholding (AdaPT) method of \citet{lei2018adapt}, which achieves its error control guarantee by judiciously limiting the analyst's knowledge about the data. As such, it is natural to view star ~as an iterative interaction between two agents: the analyst, who drives the search for discoveries based on partial observation of the data, and a hypothetical oracle, who observes the full data set and gradually reveals information to the analyst based on her actions. Typically the oracle is a computer program and the analyst is either a human or an automated adaptive search algorithm based on pre-defined modeling assumptions. In Section~\ref{sec:implementation} we discuss generic strategies for defining good automated rules.
A key difference between star ~and most earlier works is that they give the analyst power to enforce structural constraints on the final rejection set.
Previous works such as the Benjamini-Hochberg (BH) procedure \citep{bh95}, independent hypothesis weighting (IHW) \citep{ignatiadis2016data}, structure-adaptive Benjamini-Hochberg algorithm (SABHA) \citep{li2019multiple} and AdaPT ~\citep{lei2018adapt} all produce potentially heterogeneous thresholds for $p$-values and then reject all hypotheses whose $p$-values are below their corresponding thresholds. Because any given $p$-value could be above the chosen threshold, none of these methods can enforce structural constraints.
On the other hand, there are various other algorithms that are tailored to particular structural constraints, e.g. \cite{yekutieli08, lynch16} for hierarchical testing on trees and \cite{lynch16, ramdas2017dagger} for testing on directed acyclic graphs. However, these methods are non-adaptive and non-interactive in the sense that they can neither learn the structural information from data nor incorporate extra side information, and they apply to very specific types of constraints. To our knowledge, our method is the first data adaptive and interactive multiple testing framework that can impose generic structural constraints on the rejection set.
section{Selectively Traversed Accumulation Rules}\label{sec:STAR}
subsection{The framework}
We assume that for each hypothesis $H_i$, $i=1,\ldots,n$, the oracle observes a generic covariate $x_i\in \mathcal{X}$ and a $p$-value $p_i\in [0,1]$; Section~\ref{sec:disc} discusses a generalization to the setting where we have knockoff statistics instead of $p$-values. In addition, the analyst may impose a generic structural constraint $\mathcal{K} subseteq 2^{[n]}$ denoting the allowable rejection sets, where $subseteq$ denotes a subset. We require that $\emptyset \in \mathcal{K}$.
star ~proceeds by adaptively generating a sequence of candidate rejection sets $[n] = \mathcal{R}_0 supsetneq \mathcal{R}_1 supsetneq \cdots$, where $supsetneq$ denotes a strict superset. At step $t$, the oracle estimates the FDP ~of the current rejection set as
\begin{equation}\label{eq:accumFDP}
\widehat{\textnormal{FDP}}_t = \frac{h(1) + sum_{i\in\mathcal{R}_t}h(p_i)}{1 + |\mathcal{R}_t|},
\end{equation}
where $h:\; [0,1] \to [0,\infty)$ is non-decreasing and bounded, with $\int_0^1 h(p)\,dp = 1$, for example, $h(p) = 2\cdot \mathbf{1}\{p \geq 0.5\}$. The function $h$ is called an accumulation function, and the estimator~\eqref{eq:accumFDP} is based on the accumulation test of \citet{li2016accumulation}, itself a generalization of procedures proposed by \citet{barber15} and \citet{gsell2016sequential}. Informally, $sum_{i\in\mathcal{R}_t}h(p_i)$ plays the role of estimating $V_t = \left|\mathcal{R}_t \cap \mathcal{H}_0\right|$.
To allow for the analyst to make data-dependent choices without inflating Type I error, we generalize a technique by \citet{lei2018adapt} called masking. Specifically, for any choice of accumulation function $h$ as described above, we show how to derive a masking function $g$, constructed so that $h(p)$ is mean-independent of $g(p)$ when $p sim U[0,1]$. \begin{equation}~\label{eq:masking-condition}
\mathbb{E}_{u sim U[0,1]}[h(u) \mid g(u)] \eqAS \mathbb{E}_{u sim U[0,1]}[h(u)] = 1.
\end{equation}
For example, if we choose $h(p) = 2\cdot \mathbf{1}\{p \geq 0.5\}$, then we may choose $g(p) = \min\{p,1-p\}$, by observing that for a null uniform $p$-value $p$, we have $\mathbb{E}[h(p)|g(p)]=\mathbb{E}[h(p)]=1$. Section~\ref{subsec:masking} describes a general recipe for constructing such a function $g$. Even though the masking function $g$ is constructed using condition \eqref{eq:masking-condition}, we next show that when null $p$-values are not exactly uniform, the same $g,h$ pairs satisfy a more general property that will be crucial in the proof of FDR ~control. The proof is presented in Appendix \ref{subapp:proof_proposition_density}.
\begin{proposition}\label{prop:density}
If the density of a null $p$-value $p$ is non-decreasing and functions $h,g$ are chosen such that condition~\eqref{eq:masking-condition} holds, then we have
\[\mathbb{E}[h(p) \mid g(p)] \geq 1 \,\,\mbox{ almost surely}.\]
\end{proposition}
The star ~protocol works by first revealing all masked $p$-values $g(p_i)$s to the analyst. Then, as the analyst shrinks the rejection set,
$p$-values that can no longer be rejected get unmasked, meaning that the analyst observes all of the $p$-values $p_i$ for $i \notin \mathcal{R}_t$. Revealing $g(p)$ to the analyst is an example of data carving \citep{fithian2014optimal}, where a part of a random variable, $g(p_i)$, is used for selection, while the remainder $h(p_i)$ is used for inference. Because these two views of the data are designed to be orthogonal to each other under the null, the masked $p$-values and covariates together, along with prior information, provide guidance to the analyst on how to adaptively shrink the rejection set. Unlike other approaches \citep[e.g.][]{dwork2015reusable}, masking does not introduce extra randomness. This is desirable in scientific research to prevent cheating by specifying a favorable random seed.
At each step $t$, the oracle reports $\widehat{\textnormal{FDP}}_t$ to the analyst. If $\widehat{\textnormal{FDP}}_t \leq \alpha$, then the entire procedure halts and $\mathcal{R}_t$ is rejected. Otherwise, the analyst is responsible to select a smaller rejection set $\mathcal{R}_{t+1} subsetneq \mathcal{R}_t$ using covariates, intuition, and any desired statistical model or procedure, along with the oracle's revealed information, subject to the constraint that $\mathcal{R}_{t+1} \in \mathcal{K}$. After the analyst chooses $\mathcal{R}_{t+1}$, the oracle then re-estimates the FDP ~and reveals $p_i$ for $i\in \mathcal{R}_{t}setminus \mathcal{R}_{t+1}$. The analyst may then update their model, prior, constraints or intuition, and the process repeats until $\widehat{\textnormal{FDP}}_t\le \alpha$.
The update rule from $\mathcal{R}_{t}$ to $\mathcal{R}_{t+1}$ is a user-specified sub-routine and should only exploit the information contained in the $sigma$-field representing all information the analyst is allowed to observe by time $t$:
\begin{equation}\label{eq:Ft}
\mathcal{F}_{t} = sigma\left( \{ x_{i}, g(p_{i})\}_{i=1}^{n},\; (p_{i})_{i\notin \mathcal{R}_{t}},\; sum_{i\in \mathcal{R}_{t}}h(p_{i})\right).
\end{equation}
For instance, at step $t=0$, the analyst is free to use $\{x_{i}, g(p_{i})\}_{i=1}^{n}$ to train an arbitrary model to estimate which hypotheses are most likely to be true, and choose $\mathcal{R}_1$ by eliminating the most likely hypothesis subject to the constraint $\mathcal{R}_1 \in \mathcal{K}$. We will discuss specific rules tailored to different problems in Sections \ref{sec:convex} and \ref{sec:DAG}. For now, the update rule is any sub-routine that produces $\mathcal{R}_{t+1}subsetneq \mathcal{R}_{t}$, with $\mathcal{R}_{t+1}\in \mathcal{K}$, and with its outcome $\mathcal{F}_{t}$-measurable. Because $\mathcal{R}_t$ shrinks with each step, revealing more information to the analyst, the $sigma$-fields form a filtration with $\mathcal{F}_0 subseteq \mathcal{F}_1 subseteq \cdots$. This can be easily proved using induction; see Lemma \ref{lem:Ft_filtration} in Appendix \ref{subapp:proof_FDR} for details. As a consequence, the information available to the analyst accrues over the time. Algorithm \ref{algo:STAR} summarizes the procedure.
\begin{algo}\label{algo:STAR}
STAR
\begin{tabbing}
\quad \textbf{Input: }Predictors and $p$-values $(x_{i}, p_{i})_{i=1}^{n}$, constraint set $\mathcal{K}$, target FDR ~level $\alpha$.\\
\quad $\mathcal{R}_{0} = [n]$\\
\quad While $\mathcal{R}_{t}\not= \emptyset$\\
\quad \qquad\enspace $\widehat{\textnormal{FDP}}_{t}\gets \frac{1}{1 + |\mathcal{R}_{t}|}\left\{ h(1) + sum_{i\in \mathcal{R}_{t}}h(p_{i})\right\}$\\
\quad \qquad \enspace If $\widehat{\textnormal{FDP}}_{t} \le \alpha$ or $\mathcal{R}_t = \emptyset$\\
\quad \qquad \qquad\enspace Stop and return $\mathcal{R}_t$, and reject $\{H_{i}: i\in \mathcal{R}_{t}\}$\\
\quad \qquad\enspace Select $\mathcal{F}_t$-measurable $\mathcal{R}_{t+1}subsetneq\mathcal{R}_{t}$ with $\mathcal{R}_{t+1}\in \mathcal{K}$\\
\quad Output $\mathcal{R}_{t}$ as the rejection set
\end{tabbing}
\end{algo}
\begin{remark}
In some applications, the analyst may want to choose the structural constraint $\mathcal{K}$ after looking at data. Indeed, star ~allows $\mathcal{K}$ to be chosen based on $\mathcal{F}_{0}$ and even to vary with $t$. Likewise, there is no requirement that $\mathcal{R}_t \in \mathcal{K}$ for every $t$, provided we require $\mathcal{R}_t \in \mathcal{K}$ as an additional requirement for stopping the algorithm. We discuss these details in Appendix \ref{subapp:Kt} to avoid extra complication.
\end{remark}
The mechanisms used by star ~to enforce structural constraints and to learn structural information are different. The former is enabled by the fact that $\mathcal{R}_{t+1}$ can be updated without relying on the size of $p$-values, in contrast to BH -type algorithms, while the latter is enabled by masking functions which provide a partial view of data without sacrificing validity.
STAR ~can be viewed as a generalization of the ordered multiple testing setting of \citet{li2016accumulation}, in which a full pre-ordering of hypotheses based on outside data or prior knowledge must be supplied as an input to the analysis, and with a low-quality pre-ordering the method may be powerless, as shown in \citet{li2016accumulation} and \citet{lei2016power}.
By contrast, for our method, a pre-ordering is just one potential source of side information that may or may not be available in any given case; other possibilities include spatial structure, covariate information, or a partial ordering from a directed acyclic graph (DAG). Our method then determines a data-adaptive ordering using interactive guidance from the scientist, or an algorithm acting on their behalf. This interactive ordering respects any constraints the analyst has imposed, and using both the side information and the masked $p$-values $g(p_i)$. This interactive ordering, which is the counterpart to accumulation tests' pre-ordering, is simply the order in which the scientist/algorithm decides to peel off unpromising hypotheses from the candidate rejection set, based on masked $p$-values and prior information. If the ordering is pre-specified before seeing the data and masked $p$-values are ignored entirely, and there is no interaction, then our method reduces to an accumulation test, with $H_n$ peeled off first, then $H_{n-1}$, and so on.
The flexibility of our method is enabled by carving the $p$-value into two parts $g(p)$ and $h(p)$, where the first is used to adaptively determine the ordering, and the second part is used for controlling the FDR. Our use of the masked $p$-values is a selective-inference free lunch: compared to accumulation tests, we are using the same $h(p_i)$ values in the same way to estimate the FDP; we have brought more information $g(p_i)$ to bear on guiding our methodology without inflating the FDR ~or requiring any further correction. In other words, accumulation tests can be thought of as also calculating $g(p_i)$ values and then simply discarding them. On the other hand, the masked $p$-values can be highly informative about which hypotheses are non-null. For example, to observe that $g(p_{i}) = \min\{p_{i}, 1 - p_{i}\} = 10^{-8}$ is extremely suggestive (a) that $H_{i}$ is most likely false, (b) that $h(p_{i}) = 2I(p_{i} > 0.5)$ is most likely 0, and possibly (c) that other hypotheses $H_{j}$ with nearby spatial locations or covariate values to $H_{i}$ are more likely to be false as well and have $h(p_{j}) \approx 0$. More generally, by examining in aggregate all of the masked $p$-values, and most of the unmasked ones too in later stages of the procedure, we can learn areas of the covariate space with many non-null hypotheses and focus the power on those by placing them closer to the front of the list, that is, by peeling them off last. When the true signals do not meet the constraint, the enforced constraint narrows down potential rejection sets and may affect power negatively compared to algorithms which do not enforce the constraint; however, the data-adaptive exploration made possible by masking allows users to learn the structure that could compensate for the power loss at no cost of false discoveries. In fact, the theory and algorithm would be exactly identical if the prespecified $\mathcal{K}$ is replaced by a data-dependent constraint $\mathcal{K}_t$, as long as $\mathcal{K}_t$ is predictable, that is $\mathcal{K}_t$ is $\mathcal{F}_{t-1}$-measurable.
In Appendix \ref{app:asymptotics}, we conduct an asymptotic analysis under a slightly more general framework of \cite{li2016accumulation} to quantify the benefit of using masking functions. In a nutshell, in the absence of an informative pre-ordering, the accumulation test is powerless while the masking functions have some power in most practical cases. Moreover, even when an informative pre-ordering is available, we can still improve the power further by combining it with the masked $p$-values to obtain an even better ordering.
subsection{False discovery rate control}
Intuitively, the quantity $sum_{i\in\mathcal{R}_t} h(p_i)$ is a conservative estimator of $V_t = |\mathcal{H}_0 \cap \mathcal{R}_t|$, the number of false rejections we would incur if we rejected the set $\mathcal{R}_t$: each null hypothesis in $\mathcal{R}_t$ contributes at least 1 in expectation, and the non-null hypotheses contribute a non-negative amount. Hence $|\mathcal{R}_t|^{-1}sum_{i\in\mathcal{R}_t}h(p_i)$ can be interpreted as an upwardly biased estimator of the FDP ~of rejection set $\mathcal{R}_t$. With the correction term in equation \eqref{eq:accumFDP}, we can prove that Algorithm~\ref{algo:STAR} controls FDR ~in finite samples however the analyst update the rejection set $\mathcal{R}_{t}$, under certain regularity conditions on the joint distribution of $p$-values.
\begin{theorem}\label{thm:main}
Assume that
\begin{enumerate}[\textbf{A}1]
\item the null $p$-values $(p_i)_{i\in \mathcal{H}_0}$ are mutually independent, and independent of the non-nulls $(p_i)_{i\notin \mathcal{H}_0}$, conditional on the covariates $(x_i)_{i=1}^n$;
\item each null $p$-value has a non-decreasing density, which may differ across $p$-values.
\end{enumerate}
If the accumulation function $h$ is non-decreasing, and the masking function $g$ satisfies condition~\eqref{eq:masking-condition}, then, conditional on $\{x_i, g(p_i)\}_{i=1}^n$, star ~controls the FDR ~at level $\alpha$. Hence, unconditionally, the set $\mathcal{R}_\tau$ chosen using algorithm \ref{algo:STAR} satisfies $\mathbb{E}\{\textnormal{FDP}(\mathcal{R}_\tau)\} \leq \alpha$.
\end{theorem}
Assumption 1 is common in the multiple testing literature. Assumption 2 strengthens the usual assumption that a null $p$-value is stochastically larger than uniform, but is significantly weaker than assuming exact uniformity. It also strengthens the mirror-conservatism proposed in \citep{lei2018adapt}. This theorem is proved via a martingale argument, which is elaborated in Appendix \ref{subapp:proof_FDR}.
\begin{remark}\label{rem:bounded-h}
Following \citet{li2016accumulation}, we could allow for $h$ to be unbounded and replace $h(1)$ in \eqref{eq:accumFDP} with a constant $C>0$, and halt when $\widehat{\textnormal{FDP}}_t$ is below a corrected level:
\[
\frac{C + sum_{i\in\mathcal{R}_t}h(p_i)}{1 + |\mathcal{R}_t|} ~\leq~ \alpha\int_0^1 \{h(p) \wedge C\}\,dp.
\]
However, one can show that replacing such an unbounded accumulation function by its bounded counterpart $h^C(p) = \{h(p) \wedge C\}/\int_0^1 \{h(p) \wedge C\}\,dp$ results in a strictly more powerful procedure. Thus, we assume without loss of generality that $h$ is bounded.
\end{remark}
subsection{Masking functions}\label{subsec:masking}
We now give a recipe to construct a masking function $g$ for a generic accumulation rule $h$:
\begin{theorem}\label{thm:gp}
Let $h:\;[0,1]\to [0,\infty)$ be any non-negative non-decreasing function with $\int_{0}^{1} h(p) = 1$, and let
\[H(p) = \int_{0}^{p}\{h(x) - 1\}\, dx.\]
Then, we have the following two conclusions:
\begin{enumerate}[(i)]
\item There exists a continuous and strictly decreasing function $s(p)$ which is also differentiable except on a set of zero measure, namely a Lebesgue null set, such that
\begin{equation}\label{eq:integral_equation}
H(s(p)) = H(p), \quad s(0) = 1, s(1) = 0.
\end{equation}
\item The masking function
$g(p) = \min\{p, s(p)\}$
satisfies condition~\eqref{eq:masking-condition}.
\end{enumerate}
\end{theorem}
This theorem is proved in Appendix~\ref{subapp:proof_theorem_gp}. The aforementioned function $s(p)$ can be obtained numerically by a quick binary search whenever $H(\cdot)$ can be computed efficiently. All accumulation functions used in this paper satisfy the conditions of the above theorem and thus their associated $s(p)$ is almost surely differentiable. Letting $p_{*}$ denote the unique solution to $s(p_{*})=p_{*}$, for any $q<p_{*}$ the set $\{p: g(p) = q\}$ contains exactly two points and hence such a $g(p)$ is very informative because $g(p)$ only masks $1$ bit of information. For instance, if $h(p) = 2I(p\ge 0.5)$, then it is easy to show from Theorem \ref{thm:gp} that $g(p) = \min\{p, 1 - p\}$. Then $g(p) = 0.01$ implies that $p = 0.01$ or $p = 0.99$ and the analyst just needs to make a guess from two candidate values.
For brevity, we will focus throughout the main text on the simple accumulation function $h(p) = 2I(p\ge 0.5)$ with masking function $g(p) = \min\{p, 1-p\}$; we have found that this simple choice works reasonably well in all settings we have tried. We provide several other examples in Appendix \ref{subapp:masking_functions} and examine their performance in Appendix \ref{sec:more}.
section{Implementation}\label{sec:implementation}
subsection{Guidance to update rejection sets}\label{subsec:guidance}
Theorem~\ref{thm:main} showed that the FDR ~is controlled no matter how the analyst chooses $\mathcal{R}_{t+1}$ based on $\mathcal{F}_t$; however, having a good update rule will be vital to operationalizing star ~in any given context. Still, we recommend that any human-in-the-loop interactions be grounded in principled data analysis. For example, our method is to some degree susceptible to the same free-rider dynamic as other data-adaptive multiple testing procedures like AdaPT and knockoffs: namely, if we find 100 strong signals in one part of the data set, we might be able to throw in two or three more favorite hypotheses from elsewhere without threatening false discovery rate control. Well-chosen and principled constraints on the rejection set can serve as a safeguard against this behavior, which is epistemically problematic even if not formally disallowed. In addition, too-frequent looks at the data may tempt users to look back and modify their previous rejection sets; such actions are prohibited by Theorem~\ref{thm:main} and would break the theoretical guarantee.
In general, we can describe an update rule in three steps: (1) find all candidate sets of hypotheses that we can peel off from $\mathcal{R}_t$ without leaving the constraint set $\mathcal{K}$; (2) compute a score using all the information in $\mathcal{F}_{t}$ that measures the likelihood that each candidate set has non-nulls; and (3) delete the candidate set with the worst score. We provide a flowchart in Section \ref{subapp:flowchart} summarizing the pipeline schematically. The inclusion of candidate set marks the fundamental difference between star ~and AdaPT ~because the essential candidate sets of the latter are simply all remaining hypotheses while the former operates in a more greedy way.
As a concrete example, suppose that the hypotheses correspond to vertices of a tree and one aims at detecting a rooted subtree of signals. Then the deletion candidates are all hypotheses on leaf nodes of the subtree given by $\mathcal{R}_{t}$ because deletion of any leaf node does not change the rooted subtree structure of the rejection set.
The next step is to compute a score for each candidate. Heuristically, the score should be highly correlated to the $p$-values. As discussed in Section \ref{sec:STAR}, the most straightforward score for the $i$-th hypothesis is $g(p_{i})$. We refer to it as the canonical score. For candidate sets that contains multiple hypotheses, we define the canonical score as the average of $g(p_i)$'s. A larger canonical score gives stronger evidence that the candidate set is mostly null. Although the canonical score is straightforward to use, the user is allowed to fit any model using the covariates and the partially-masked $p$-values. Thus, one can estimate the signal strength, or a posterior probability of being null, as the score: we call this a model-assisted score.
Finally, given the score, it is natural to remove the least favorable candidate. For instance, when using the canonical score, the candidate with largest score will be removed. On the other hand, if the score measures the signal strength or the likelihood of being non-null, the candidate with smallest score will be removed.
Our principle here can be summarized as follows: given a working model or belief about the data generating process, we have a generic EM-algorithm based pipeline that incorporates it into our method to produce scores yielding the adaptive ordering. The researcher can always incorporate their favorite data-generating model into our method, improving power if their model is correct/good, but never violating the FDR ~control if their model is inaccurate.
subsection{Conditional one-group model as the working model}\label{subsec:one-group}
Although the canonical scores are effective in many cases as will be shown in Section \ref{sec:DAG}, they do not fully exploit covariate information, apart from enforcing the constraint. For instance, when the hypotheses are arranged spatially, we may expect the non-nulls will concentrate on a few clusters, and/or that the signal strength will be smooth on the underlying space. This prior knowledge may neither be reflected directly from the $p$-values nor be explicitly used to strengthen the $p$-values; instead, we can use a working model to assist calculating the scores. We emphasize that no matter how misspecified our working model is, the FDR~is still controlled.
\cite{lei2018adapt} proposed a conditional two-group model and used the estimated local FDR ~as model-assisted scores. They proved in their Theorem 2 that the local FDR ~gives the optimal score to order hypotheses. The model can be fitted by an expectation-maximization (EM) algorithm \citep[Appendix A of ][]{lei2018adapt}.
Despite the approximate optimality of the above approach, the EM ~algorithm for conditional two-group models is computationally intensive because it fits two separate models for the proportion and the signal strength of non-nulls at each iteration. Additionally, \cite{lei2018adapt} pointed out in their Appendix A.3 some instability issues of the algorithm. For these reasons, we proposed a conditional \textit{one}-group model as an alternative:
\begin{align}
p_i sim f(p; \mu_{i})\quad\text{with } \eta(\mu_i) = \beta'\phi(x_i),\label{eq:onegroup_GLM}
\end{align}
where $f$ is the density function, $\eta$ is the link function and $\phi$ is an arbitrary featurization. As will be detailed in Appendix \ref{subapp:EM}, $\mu_{i}$ can be estimated via an EM ~algorithm. The model-assisted scores are then given by the estimates $\hat{\mu}_{i}$. Although $\hat{\mu}_{i}$'s lose the optimality guarantee, they provide good proxy of how promising each hypothesis is. On the other hand, \eqref{eq:onegroup_GLM} is easier to fit as it only involves one set of parameters. At step $t$, the algorithm only needs to impute the masked $p$-values. Thus it is computationally more efficient and partly solves the issue raised in \cite{lei2018adapt}.
Finally, as discussed in \cite{lei2018adapt}, $p$-values may not be the objects that are most amenable to modeling, in which case one can either model transformed $p$-values or directly model the data used to produce them. For instance, in many applications, z-values are available and one-sided $p$-values are obtained by the transformation $p_{i} = 1 - \mathbb{P}hi(z_{i})$, where $\mathbb{P}hi$ is the distribution function of a standard Gaussian. In this case, we can directly model $z_{i}$ as $z_{i}sim N(\mu_{i}, 1)$.
section{Example 1: convex region detection}\label{sec:convex}
subsection{Problem Setup}
In some applications, the $p$-values may be associated with features $x_{i}\in \mathcal{X} subseteq \mathbb{R}^d$, which encode some contextual information such as predictor variables or a spatial location, and which may be associated with the underlying signal. We may wish to use this feature information to discover regions of the feature space where the signal is relatively strong; for example, \citet{drevelegas10} seek a convex region to locate the boundary of tumors.
As a concrete mathematical example, suppose that for each point $x_i$ on a regular spatial grid, we observe an independent observation $z_i sim N(f(x_i), 1)$ for some non-negative function $f$, and we hope to discover the region $\mathcal{C} = \{x:\; f(x) > 0\}$, where we have some prior belief that $\mathcal{C}$ is convex; for example, if $f$ is known to be a concave function, then $\mathcal{C}$ is a superlevel set and is hence convex. Since we cannot expect to perfectly find $\mathcal{C}$, we may hope to discover a smaller region $\widehat{\mathcal{C}}$ which is mostly contained within $\mathcal{C}$, that is,
\begin{equation}\label{eq:convFDR}
\frac{\text{Vol}(\widehat{\mathcal{C}} \cap \mathcal{C}^c)}{\text{Vol}(\widehat{\mathcal{C}})} \leq \alpha.
\end{equation}
We can frame the above as a multiple testing problem by computing a one-sided $p$-value $p_i = 1-\mathbb{P}hi(z_i)$ for each $H_i$ and constraining the rejection set to be of the form $\mathcal{R} = \{i:\; x_i \in \widehat{\mathcal{C}}\}$, for some convex set $\widehat{\mathcal{C}}$, leading to a constraint $\mathcal{K}$ on the allowable rejection sets. If the grid is relatively fine, then FDR ~is a natural error criterion to control since the FDP ~of $\mathcal{R}$ approximates criterion \eqref{eq:convFDR}.
As another application in supervised learning, we may observe a pair of features $x_i$ and response $y_i$ for $i=1,\ldots,n$, and hope to find a subregion of the feature space $\mathcal{X}$ where the $y$ values tend to be relatively large. In bump-hunting \citep{friedman99}, we seek to discover a rectangle in predictor space; Appendix~\ref{sec:bump} discusses this application.
More generally, we may want to discover a set of hypotheses $\mathcal{R} = \{i:\; i\in \widehat{\mathcal{C}}\}$ where $\widehat{\mathcal{C}}$ is convex, or is a rectangle, or satisfies some other geometric property. To the best of our knowledge, no previously existing procedure can solve the above problems while guaranteeing FDR ~control. To implement these goals using the star ~framework, we need only define the procedure-specific functions elaborated in Section \ref{sec:implementation}. To illustrate, we first consider an example where $x_{i}\in \mathbb{R}^{2}$.
subsection{Procedure}\label{subsubsec:convex_proc}
To preserve convexity, we consider an automated procedure that gradually peels off the boundaries of the point cloud $\{x_{i}\}$. At each iteration, we choose a direction $\theta\in [0, 2\pi)$ and a small constant $\delta$, and peel off a proportion $\delta$ of points that are farthest along this direction.
Specifically, for each angle $\theta$, we define a candidate set $C(\theta; \delta)$ to be observed as the set of indices corresponding to the $\delta$-proportion of points that are farthest along the direction $\theta$.
If the goal is to detect an axis-parallel box, $\theta$ can be restricted to only take values in $\{0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\}$; otherwise we set $\theta$ to an equi-spaced grid on $[0, 2\pi]$ of length 100. Given a score $S_{i}$ for each hypothesis, which we soon define, we may evaluate the signal strength of each candidate set by the average of $S_{i}$. Then we may update $\mathcal{R}_{t}$ as
\[\mathcal{R}_{t} := \mathcal{R}_{t-1}setminus C(\hat{\theta}; \delta), \quad\mbox{with}\quad \hat{\theta} := \argmin_{\theta}\mathrm{Avg}\{S_{i}: i\in C(\theta; \delta)\}.
\]
Finally, to define the score $S_{i}$'s, we can directly use the canonical score $g(p_{i})$. However, in most problems of this type where $x_{i}$ represents the location in some continuous space, it is reasonable to assume that the distributions of $p$-values are smoothly varying. In particular, we use the conditional one-group model on $p$-values with Beta distribution, i.e. $p_i sim \mathrm{Beta}(1 / \mu_i, 1)$, as the working model and fit a generalized additive model \citep{hastie90} using the the smooth spline basis of $x_{i}$ as the featurization $\phi$ defined in \eqref{eq:onegroup_GLM}. This working model is motivated by the conditional two-group Gamma generalized linear model in \cite{lei2018adapt}.
subsection{Simulation results}\label{subsubsec:convex_simulation}
We consider an artificial dataset where the predictors form an equi-spaced $50\times 50$ grid in the area $[-100, 100]\times [-100, 100]$.
Let $\mathcal{C}_{0}$ be a convex set on $\mathbb{R}^{2}$ and set $\mathcal{H}_{0}^{c} = \{x_{i}: x_{i}\in \mathcal{C}_{0}\}$. We generate $p$-values i.i.d. from a one-sided normal test, i.e.
\begin{equation}\label{eq:simul_pvals}
p_{i} = 1 - \mathbb{P}hi(z_{i}), \quad \mbox{and} \quad z_{i}sim N(\mu, 1),
\end{equation}
where $\mathbb{P}hi$ is the cumulative distribution function of $N(0, 1)$. For $i\in \mathcal{H}_{0}$ we set $\mu = 0$ and for $i\not\in \mathcal{H}_{0}$ we set $\mu = 2$. Figure~\ref{fig:convex_expr_truth} shows three types of $\mathcal{C}_{0}$ that we conduct tests on.
\begin{figure}
\caption{True underlying signal for our convex region detection simulation. Each point represents a hypothesis, 2500 in total, with black representing non-nulls with $\mu = 2$.}
\label{fig:convex_expr_truth}
\end{figure}
Although our procedure is the only one that is able to enforce the convexity, it is still illuminating to compare it with other procedures to examine the power. In particular, we consider the BH ~procedure \citep{bh95} and AdaPT ~\citep{lei2018adapt}. We implement star ~with the model-assisted score based on the generalized additive model.
For each procedure and level $\alpha$, we calculate the FDP ~and the power as
\begin{align}\label{eq:defs}
\mathrm{FDP}(\alpha) &= \frac{|\mathcal{R}(\alpha)\cap \mathcal{H}_{0}|}{|\mathcal{R}(\alpha)|}, \quad \mathrm{power}(\alpha) ~= \frac{|\mathcal{R}(\alpha)\cap \mathcal{H}_{0}^{c}|}{|\mathcal{H}_{0}^{c}|},
\end{align}
where $\mathcal{R}(\alpha)$ is the rejection set at level $\alpha$. We then estimate the FDR ~and the power by the average of $\mathrm{FDP}(\alpha)$ and $\mathrm{power}(\alpha)$ over 100 sets of independently generated $p$-values. The results are plotted in Figure \ref{fig:rej_convex_methods} for a list of $\alpha$'s from 0.01 to 0.3. We see that our method is comparable to AdaPT, and more powerful than the BH ~procedure, neither of which enforce the convexity constraint. Furthermore, it is worth mentioning that the model-assisted star ~achieves high power despite using a generic and misspecified generalized additive beta model for the $p$-values.
\begin{figure}
\caption{Comparing our method (black solid), BH ~procedure (red dashed) and AdaPT ~(blue dotted) for convex region detection.}
\label{fig:rej_convex_methods}
\end{figure}
The power gain over the BH ~procedure is in part from the fact that the convexity constraint reflects the truth and our method implicitly builds it into the selection. It is also driven by the effective learning of underlying spatial structure. To see that,
we plot $\hat{\mu}(x)$ in Figure \ref{fig:convex_expr_score}. The top panel shows the initial score that only uses partially masked $p$-values $g(p_i)$ and $x_i$. The bottom panel shows the oracle result when fitting the model to fully observed $p$-values. It is surprising that even the initial estimate is good enough to clearly show the contour of the non-nulls, and is nearly as good as the final estimate using fully observed $p$-values; in fact, the correlation between the two estimates is above 0.98 in all three cases. This explains why our method can accurately pinpoint the non-nulls and hence enhance the power.
\begin{figure}
\caption{Model-assisted score of star. Darker pixels represents higher score.}
\label{fig:convex_expr_score}
\end{figure}
section{Example 2: testing on directed acyclic graphs}\label{sec:DAG}
subsection{Setup and Procedure}\label{subsubsec:DAG_proc}
Another case involves hypotheses arranged on a DAG ~and the non-nulls are known apriori to satisfy the heredity principle. The strong or weak heredity principle, also referred to as effect hierarchical principle~\citep{wu00}, states that an effect is significant only if all or one of its parent effects are significant. An application on variable selection in factorial experiments under heredity principle is discussed in Section \ref{sec:factorial}.
Multiple testing on DAG s has extensive applications in genomics \citep[e.g.,][]{goeman08, saunders14, meijer15} and clinical trials \citep[e.g.,][]{dmitrienko13}. However, most prior work deals with the family-wise error rate (FWER) control, but the setting of FDR ~control is relatively under-studied. To the best of our knowledge, the only existing FDR ~control procedures for DAG s were proposed in Gavin Lynch's thesis \citep{lynch14} and in the sequential setting~\citep{ramdas2017dagger} and the multi-layer setting~\citep{ramdas2017unified}. All aforementioned works were designed for the strong heredity principle.
It is straightforward to apply our method to guarantee FDR ~control under both the strong and the weak heredity principle. For the strong heredity principle, we select the candidates as all the leaf nodes. For the weak heredity principle, we select the candidates as all nodes by removing which the remaining graph satisfies the principle. We also present an application to a factorial experiment in Appendix \ref{sec:factorial}.
subsection{Simulation results}
\begin{figure}
\caption{Three types of graphs in simulation studies, with fewer nodes for illustration. Each node represents a hypothesis, 1000 in total, with black ones being the nulls.}
\label{fig:DAG_expr_truth}
\end{figure}
\begin{figure}
\caption{Comparing our method (black solid), \cite{lynch16}
\label{fig:rej_DAG_methods}
\end{figure}
We focus on the strong heredity principle in order to compare star ~with existing methods. To account for the structure, we consider three types of DAG s: shallow regular graph, with 4 layers and 250 nodes in each layer, deep regular graph, with 10 layers and 100 nodes in each layer, and triangular graph, with 5 layers with 50, 100, 200, 300, 350 nodes in each layer respectively. For each graph, we set 50 nodes which satisfy SHP to be non-null and generate $p$-values using equation~\eqref{eq:simul_pvals} with $\mu = 2$ for non-nulls. The settings are illustrated in Figure~\ref{fig:DAG_expr_truth}, with fewer nodes for readability.
We compare our method with canonical scores described in Section \ref{subsec:guidance}, to the method proposed in \cite{lynch14}, referred to as Self-Consistent Rejection procedure, as well as a recently developed sequential method, referred to as Greedily Evolving Rejections, a generalization of Lynch's hierarchical test by \cite{ramdas2017dagger}. The results are plotted in Figure \ref{fig:rej_DAG_methods}. It is clear that in all cases our method is more powerful than other methods when $\alpha$ is not too small. When $\alpha$ is small, our method is powerless in this setting because of the finite sample correction constant $h(1)$ in \eqref{eq:accumFDP}, which requires at least $h(1) / \alpha - 1$ rejections to get non-empty rejection set at level $\alpha$. In our case, $h(1) = 2$ but we can reduce $h(1)$ by choosing other accumulation functions. We provide a few examples that resolve this issue in Appendix \ref{sec:more}.
section{Discussion}
\label{sec:disc}
Using knockoff statistics instead of $p$-values.
We have focused on the case where the test statistics are independent. For more general settings, our framework dovetails naturally with the knockoff framework proposed by \citet{barber15} and extended by \citet{candes2016panning}, which convert complex regression problems into independent one-bit $p$-values for each variable. When running sparse regression algorithms, the underlying variables often have some structure---for example, a tree structure with wavelet coefficients in compressed sensing---and we may want to use the knockoff procedure to select a structured subset of variables.
Keeping this motivation in mind, let $W_i$ denote the knockoff statistic for hypothesis $i$ and define $p_i = (1+\mathbf{1}\{W_i>0\})/2$. Then, knockoff constructions guarantee that $(p_i)_{i\in \mathcal{H}_0}$ are independent $p$-values, conditional on $(|W_i|)_{i=1}^n$ and $(p_i)_{i\notin \mathcal{H}_0}$. The absolute value $|W_i|$ may be viewed as the free side-information $g(p_i)$ in Algorithm~\ref{algo:STAR}, while the location of the variable on the tree would be the structural information $x_i$. Although the constructed $p_i$ does not technically have a decreasing density, the accumulation function $h(p_i) = 2 I\{p_i > 1/2\} = \text{sign}(W_i) + 1$ nevertheless has expectation 1 under the null. The analyst may then use star ~to interactively pick a structured subset of variables.
Like AdaPT, knockoffs as defined in \citet{barber15} do not enforce constraints on the rejection set, nor do they allow for interactive adaptation as more $p$-values are unmasked. Combining our method with knockoff statistics allows for controlling the FDR ~in many interesting regression problems with constraints on the rejection set, such as hierarchy constraints for interactions in a regression.
{\varepsilon}space{0.05in}
Handling dependence.
When the underlying problem is a sparse regression problem, the aforementioned knockoff procedure can be used to construct independent $p$-values even when the covariates are not orthogonal.
However, it is not yet known how to construct knockoffs in most settings, but we can often still construct dependent $p$-values, and it is an important open problem to provide guarantees for such settings.
As we show in Appendix \ref{sec:sensitivity}, the experiments under dependence are encouraging, especially under negative dependence, but we do not currently know how to prove any results about robustness to deviations from independence. Such results would immediately be applicable in several other settings, such as ordered testing \citep{li2016accumulation}, and knockoffs \citep{barber15}.
\allowdisplaybreaks
\appendix
\begin{center}
\begin{Large}
\textbf{Supplementary Materials}
\end{Large}
\end{center}
The supplementary materials include all technical proofs (Section \ref{sec:proofs}), a more detailed description of star ~(Section \ref{sec:star}), four more applications on bump hunting, hierarchical testing, wavelet thresholding and interaction selection for factorial experiments (Section \ref{sec:bump} - \ref{sec:factorial}), comparison of star ~with different masking functions (Section \ref{sec:more}), a sensitivity analysis for star ~under dependent p-values (Section \ref{sec:sensitivity}) and an asymptotic power analysis for a subclass of star ~(Section \ref{app:asymptotics}). The \texttt{R} code to replicate all results in the paper are available at \texttt{https://github.com/lihualei71/STAR}. The sub-folder \texttt{movies} includes GIFs illustrating our method on several applications.
section{Technical Proofs}\label{sec:proofs}
subsection{Proof of Proposition \ref{prop:density}}\label{subapp:proof_proposition_density}
\begin{proof}
Fix any $x\in [0, 1]$ and a set $A$ with non-zero Lebesgue measure. Let $P_0$ denote the true distribution of $p$, and $P_u$ denote the uniform distribution on $[0, 1]$. Then, we have
\begin{align*}
P_0(p\in [0, x] \mid g(p)\in A) &= \frac{P_0([0, x] \cap g^{-1}(A))}{P_0(g^{-1}(A))
}\\
& = \frac{P_0([0, x] \cap g^{-1}(A))}{P_0([0, x]\cap g^{-1}(A)) + P_0([x, 1]\cap g^{-1}(A))}\\
& = \frac{1}{1 + \frac{P_0([x, 1]\cap g^{-1}(A))}{P_0([0, x]\cap g^{-1}(A))}}.
\end{align*}
Let $f$ be the true density of $p$. Then, we may write
\begin{align*}
\frac{P_0([x, 1]\cap g^{-1}(A))}{P_0([0, x]\cap g^{-1}(A))} = \frac{\int_{x}^{1} I(y\in g^{-1}(A)) f(y)dy}{\int_{0}^{x} I(y\in g^{-1}(A)) f(y)dy} &\ge \frac{f(x) \cdot \int_{x}^{1} I(y\in g^{-1}(A)) dy}{f(x) \cdot \int_{0}^{x}I(y \in g^{-1}(A))dy}\\
& = \frac{P_u([x, 1]\cap g^{-1}(A))}{P_u([0, x]\cap g^{-1}(A))}.
\end{align*}
As a consequence, we conclude that for all $x$, we have
\[P_0(p\in [0, x] \mid g(p)\in A)\le P_u(p\in [0, x] \mid g(p)\in A),\]
and consequently, since $P_0([0,1]) = P_u([0,1]) = 1$, we also have
\[P_0(p\in [x,1] \mid g(p)\in A) \ge P_u(p\in [x,1] \mid g(p)\in A),\]
This entails that $P_u$ stochastically dominates $P_0$, when conditioning on $\{g(p)\in A\}$. Since accumulation functions $h$ are non-decreasing, they are larger on $[x,1]$ than on $[0,1]$, and hence we have
\begin{equation}\label{eq:density1}
\mathbb{E}_0[h(p) \mid g(p)\in A] \ge \mathbb{E}_u[h(p) \mid g(p)\in A].
\end{equation}
Note that equation~\eqref{eq:density1} holds for all sets $A$ with nonzero Lebesgue measure. This immediately yields the theorem. To see this more formally, fix any ${\varepsilon}arepsilonsilon > 0$, and let
\[B_{{\varepsilon}arepsilonsilon} = \{x: \mathbb{E}_u[h(p) \mid g(p)=x] > \mathbb{E}_0[h(p) \mid g(p)=x] + {\varepsilon}arepsilonsilon\}.\]
Then, $B_{{\varepsilon}arepsilonsilonilon}$ must be a Lebesgue null set, in order to not contradict equation \eqref{eq:density1}.
Therefore,
\[P_u(B_{0}) = P_u(\cup_{n\ge 1}B_{1/n})= 0.\]
As a result,
\[\mathbb{E}_0[h(p) \mid g(p)]stackrel{a.s.}{\ge} \mathbb{E}_u[h(p) \mid g(p)]\ge 1,\]
as desired.
\end{proof}
subsection{Proof of Theorem \ref{thm:main}}\label{subapp:proof_FDR}
We start from a lemma that shows $\mathcal{F}_{t}$ defined in \eqref{eq:Ft} is a filtration.
\begin{lemma}\label{lem:Ft_filtration}
Let
\[\mathcal{F}_{t} = sigma\left( \{x_{i}, g(p_{i})\}_{i=1}^{n},\; (p_{i})_{i\notin \mathcal{R}_{t}},\; sum_{i\in \mathcal{R}_{t}}h(p_{i})\right).\]
Then $\mathcal{F}_{t}$ is a filtration in the sense that for all $t\ge 0$,
\[\mathcal{F}_{t}subset \mathcal{F}_{t+1}.\]
\end{lemma}
\begin{proof}
The proof is completed by observing that (1) $\mathcal{F}_{0} subset \mathcal{F}_{t}$ for any $t$; (2) $\mathcal{R}_{u+1}\in \mathcal{F}_{u}subset \mathcal{F}_{t}$ implies $\mathcal{F}_{u+1}subset \mathcal{F}_{t}$.
\end{proof}
The proof of Theorem \ref{thm:main} is based on an optional stopping argument, generalizing the one presented in \citet{lei2016power}, which in turn generalized arguments from \citet{li2016accumulation} and \citet{barber2016knockoff}.
\begin{lemma}\label{lem:bernoulli}[Lemma 1 of \citet{lei2018adapt}]
Suppose that, conditionally on the $sigma$-field $\mathcal{G}_{-1}$, $b_1,\ldots,b_n$ are independent Bernoulli random variables with
\[\mathbb{P}(b_i = 1 \mid \mathcal{G}_{-1}) = \rho_i \geq \rho > 0, \mbox{almost surely}.\]
Let $(\mathcal{G}_{t})_{t=0}^{\infty}$ be a filtration with $\mathcal{G}_{0}subseteq \mathcal{G}_{1}subseteq \cdots$ and suppose that $[n] supseteq \mathcal{C}_0 supseteq \mathcal{C}_1 supseteq \cdots$, with each subset $\mathcal{C}_{t+1}$ measurable with respect to $\mathcal{G}_{t}$. If we have
\[
\mathcal{G}_t supset sigma\left(\mathcal{G}_{-1}, \mathcal{C}_t, (b_i)_{i \notin \mathcal{C}_t}, sum_{i \in \mathcal{C}_t} b_i\right),
\]
and $\tau$ is an almost-surely finite stopping time with respect to the filtration $(\mathcal{G}_t)_{t \geq 0}$, then
\[
\mathbb{E}\left[\frac{1 + |\mathcal{C}_{\tau}|}{1 + sum_{i\in \mathcal{C}_{\tau}} b_i} \right] \leq \rho^{-1}.
\]
\end{lemma}
\begin{proof}[\textbf{of Theorem~\ref{thm:main}}]
\label{sec:proof-thm1}
By Proposition \ref{prop:density},
\begin{equation}\label{eq:conditional_expectation_1}
\mathbb{E}[h(p_i)\mid g(p_i)]stackrel{a.s.}{\ge} 1,\quad \forall i\in \mathcal{H}_{0}.
\end{equation}
Since $h$ is non-decreasing, \eqref{eq:conditional_expectation_1} implies that $h(1)\ge 1$. Generate $(V_{i})_{i\in \mathcal{H}_{0}}stackrel{i.i.d.}{sim} U([0, 1])$, which are also independent of $(x_{i}, p_{i})_{i=1}^{n}$ and all operational randomness involved in the procedure. Let $b_i = I(V_i\le \frac{h(p_{i})}{h(1)})$ and recall that $\tau$ is the smallest $t$ such that $\widehat{\textnormal{FDP}}_{t}\le \alpha$, then
\begin{align*}
\textnormal{FDR} & = \mathbb{E} [\textnormal{FDP}_\tau ] = \mathbb{E}\left[ \frac{|\mathcal{R}_\tau \cap \mathcal{H}_{0}|}{1{\varepsilon}ee |\mathcal{R}_\tau |}\right]\le \mathbb{E}\left[ \frac{1 + |\mathcal{R}_\tau \cap \mathcal{H}_{0}|}{1 + |\mathcal{R}_\tau |}\right]\\
& stackrel{(\mathrm{i})}{\le} \frac{\alpha}{h(1)}\cdot \mathbb{E}\left[ \frac{1 + |\mathcal{R}_\tau \cap \mathcal{H}_{0}|}{1 + sum_{i\in \mathcal{R}_{\tau}}\frac{h(p_{i})}{h(1)}}\right]\\
& stackrel{(\mathrm{ii})}{\le} \frac{\alpha}{h(1)}\cdot \mathbb{E}\left[ \frac{1 + |\mathcal{R}_{\tau}\cap \mathcal{H}_{0}|}{1 + sum_{i\in \mathcal{R}_{\tau}\cap\mathcal{H}_{0}}\mathbb{E}[b_{i} \mid p_{i}]}\right]\\
& stackrel{(\mathrm{iii})}{\le} \frac{\alpha}{h(1)}\cdot \mathbb{E}\left[ \frac{1 + |\mathcal{R}_{\tau}\cap \mathcal{H}_{0}|}{1 + sum_{i\in \mathcal{R}_{\tau}\cap\mathcal{H}_{0}}\mathbb{E}[b_{i} \mid (p_{i})_{i\in \mathcal{H}_{0}}]}\right]\\
& stackrel{(\mathrm{iv})}{\le} \frac{\alpha}{h(1)} \cdot \mathbb{E}\left[\mathbb{E}\left[ \frac{1 + |\mathcal{R}_{\tau}\cap \mathcal{H}_{0}|}{1 + sum_{i\in \mathcal{R}_{\tau}\cap\mathcal{H}_{0}}b_{i}} \mid (p_{i})_{i\in \mathcal{H}_{0}}\right]\right]\\
& = \frac{\alpha}{h(1)} \cdot \mathbb{E}\left[ \frac{1 + |\mathcal{R}_{\tau}\cap \mathcal{H}_{0}|}{1 + sum_{i\in \mathcal{R}_{\tau}\cap\mathcal{H}_{0}}b_{i}} \right],
\end{align*}
where (i) follows because at time $\tau$, we have $\frac{h(1) + sum_{i \in \mathcal{R}_{\tau}} h(p_i)}{1 + |\mathcal{R}_\tau|} \leq \alpha$, (ii) follows by substituting the definition of $b_i$ and restricting the indices of the denominator summation to just the rejected nulls, (iii) follows because of the independence of null p-values, while (iv) uses Jensen's inequality and the convexity of the mapping $y\mapsto \frac{1}{1 + y}$. Define the initial $sigma$-field as
\[\mathcal{G}_{-1} = sigma\bigg( \{x_{i}, g(p_{i})\}_{i=1}^{n}, (p_{i})_{i\not\in \mathcal{H}_{0}}\bigg).\]
Then
\begin{equation}\label{eq:bi_cond}
\mathbb{E} [b_{i} | \mathcal{G}_{-1}] = \mathbb{E} \left[\frac{h(p_{i})}{h(1)}\mid \mathcal{G}_{-1}\right] = \mathbb{E} \left[\frac{h(p_{i})}{h(1)}\mid g(p_{i})\right]\ge \frac{1}{h(1)} \quad \text{ for all } i \in \mathcal{H}_{0}.
\end{equation}
Recall that $\mathcal{R}_{0} = [n]$ and define the filtration $(\mathcal{G}_{t})_{t\ge 0}$ as
\begin{align*}
\mathcal{G}_{t} &= sigma\bigg( \mathcal{G}_{-1}, (p_{i}, V_{i})_{i\notin \mathcal{R}_{t}\cap \mathcal{H}_{0}}, \{(p_{i}, V_{i}) : {i\in \mathcal{R}_{t}\cap \mathcal{H}_{0}}\}\bigg),
\end{align*}
where $\{\cdot\}$ denotes the unordered set. Then we have the following observations:
\begin{enumerate}[(a)]
\item Since $\mathcal{R}_{0}supset \mathcal{R}_{1}supset\cdots$, we necessarily have $\mathcal{G}_{0}subset \mathcal{G}_{1}subset\cdots$.
\item By definition \eqref{eq:Ft}, note that we have
\[\mathcal{F}_{t} = sigma\bigg(\mathcal{G}_{-1}, (p_{i})_{i\notin \mathcal{R}_{t}}, sum_{i\in\mathcal{R}_{t}}h(p_{i})\bigg)subseteq sigma\bigg(\mathcal{G}_{-1}, (p_{i})_{i\notin \mathcal{R}_{t}}, \{p_{i}: i\in \mathcal{R}_{t}\}\bigg) subseteq \mathcal{G}_t.\]
As a consequence, $\tau \leq n$ is also a finite stopping time with respect to filtration $(\mathcal{G}_{t})_{t\ge 0}$.
\item Since $b_{i}$ is a function of $(p_{i}, V_{i})$ and $(p_{i}, V_{i})_{i\not\in \mathcal{R}_{t}\cap \mathcal{H}_{0}} \in \mathcal{G}_{t}$, we have
\begin{equation}\label{eq:accept_bi}
(b_{i})_{i\not\in \mathcal{R}_{t}\cap \mathcal{H}_{0}}\in \mathcal{G}_{t}.
\end{equation}
\item Lastly, observe that
\[
sum_{i\in \mathcal{R}_{t}\cap \mathcal{H}_{0}}b_{i} \in sigma\bigg(\{b_{i}: i\in \mathcal{R}_{t}\cap \mathcal{H}_{0}\}\bigg)subseteq sigma\bigg(\{(p_{i}, V_{i}): i\in \mathcal{R}_{t}\cap \mathcal{H}_{0}\}\bigg) subseteq \mathcal{G}_t.
\]
\end{enumerate}
Putting the pieces together and applying Lemma \ref{lem:bernoulli} with $\mathcal{C}_{t} = \mathcal{R}_{t}\cap \mathcal{H}_{0}$, we conclude that
\begin{equation}\label{eq:FDR}
\mathbb{E}\left[ \frac{1 + |\mathcal{R}_{\tau}\cap \mathcal{H}_{0}|}{1 + sum_{i\in \mathcal{R}_{\tau}\cap\mathcal{H}_{0}}b_{i}}\right]\le h(1).
\end{equation}
As a result, we may conclude that
\[\textnormal{FDR} ~\le~ \frac{\alpha}{h(1)} \cdot \mathbb{E}\left[ \frac{1 + |\mathcal{R}_{\tau}\cap \mathcal{H}_{0}|}{1 + sum_{i\in \mathcal{R}_{\tau}\cap\mathcal{H}_{0}}b_{i}}\right] ~\le~ \alpha,\]
as claimed by the theorem.
\end{proof}
subsection{Proof of Theorem \ref{thm:gp}}\label{subapp:proof_theorem_gp}
\label{sec:proof-thm2}
\begin{proof}
We first prove statement (i). We start by assuming that $\{p: h(p) = 1\}$ is a Lebesgue null set. Since $h$ is non-decreasing and $\int_{0}^{1}h(p)\, dp = 1$, we must have $h(0) < 1 < h(1)$. Let $p_{*} = sup\{p: h(p)\le 1\}$, then
\[\lim_{p\uparrow p_{*}} h(p)\le 1 \le \lim_{p\downarrow p_{*}}h(p).\]
As a consequence, $H(x)$ is strictly decreasing on $[0, p_{*}]$ and strictly increasing on $[p_{*}, 1]$ and $H(0) = H(1) = 0$. First we define the function $s(p)$ on $[p_{*}, 1]$: for any $p\in [p_{*}, 1]$, let $s(p)$ be the unique solution on $[0, p_{*}]$ such that $H(s(p)) = H(p)$. Then, it is easy to see that $s(\cdot)$ is strictly decreasing on $[p_{*}, 1]$ with $s(1) = 0$ and $s(p_{*}) = p_{*}$. Since the function $H$ is continuous and strictly decreasing on $[0, p_{*}]$, we know that $s(\cdot)$ is continuous on $[p_{*}, 1]$. Similarly we can define $s(p)$ on $[0, p_{*}]$. The continuity is guaranteed at $p_{*}$ since $s(p_{*}) = p_{*}$.
~\\
Next we prove that $s(\cdot)$ is differentiable except on a Lebesgue null set. Let
\[\mathcal{D} = \{p: h(\cdot)\mbox{ is continuous at both }p\mbox{ and }s(p)\}.\]
Since $h(\cdot)$ is increasing on $[0, 1]$, the standard argument in real analysis (e.g. \cite{rudin}) implies that $\mathcal{D}$ is countable and hence a Lebesgue null set. It is left to prove that $s(\cdot)$ is differentiable on $\mathcal{D}^{c}$. By the definition of $s(\cdot)$, for any $0\le p_{1}\le p_{2}\le 1$,
\begin{equation}\label{eq:differentiability}
\frac{H(s(p_{2})) - H(s(p_{1}))}{s(p_{2}) - s(p_{1})} = \frac{H(p_{2}) - H(p_{1})}{s(p_{2}) - s(p_{1})} = \frac{H(p_{2}) - H(p_{1})}{p_{2} - p_{1}}\cdot \frac{p_{2} - p_{1}}{s(p_{2}) - s(p_{1})}.
\end{equation}
Take any $p\in \mathcal{D}^{c}$ and by definition we know that $h(\cdot)$ is continuous on both $p$ and $s(p)$. By Newton-Leibniz theorem \citep{rudin},
\[H'(p) = h(p) - 1, \quad \text{ and ~}~ H'(s(p)) = h(s(p)) - 1.\]
Now, letting $p_{1} = p$ and $p_{2}\rightarrow p$ in \eqref{eq:differentiability}, the continuity of $s(\cdot)$ implies that $s(p_{2})\rightarrow s(p)$ and the differentiability of $H(\cdot)$ at $p$ implies that
\[h(s(p)) - 1 = H'(s(p)) = H'(p)\cdot \lim_{p_{2}\rightarrow p}\frac{p_{2} - p}{s(p_{2}) - s(p)} = (h(p) - 1)\cdot \lim_{p_{2}\rightarrow p}\frac{p_{2} - p}{s(p_{2}) - s(p)}.\]
This entails that the derivative of $s(p)$ can be written as
\begin{equation}\label{eq:sderiv}
s'(p)\triangleq \lim_{p_{2}\rightarrow p}\frac{s(p_{2}) - s(p)}{p_{2} - p} = \frac{h(p) - 1}{h(s(p)) - 1}.
\end{equation}
Now suppose $\{p: h(p) = 1\}$ is not a Lebesgue null set. Since $h$ is non-decreasing, it must be an interval. Let $[p_{1}, p_{2}]$ be the closure of $\{p: h(p) = 1\}$. Then $H$ is strictly decreasing on $[0, p_{1}]$, strictly increasing on $[p_{2}, 1]$ and is flat on $[p_{1}, p_{2}]$. For $p\in [0, p_{1})\cup (p_{2}, 1]$, we can define $s(p)$ is the same way as above. By construction, $s(p_{2}) = p_{1}, s(p_{1}) = p_{2}$ On $[p_{1}, p_{2}]$, we simply define $s(p)$ as the linear interpolation between $p_{1}$ and $p_{2}$, i.e. $s(p) = p_{1} + p_{2} - p$. It is easy to see that $s(p)$ is continuous, strictly decreasing and differentiable almost everywhere.
Now we prove theorem statement (ii). Take any $q\not\in s(\mathcal{D})$ and write $s^{-1}(q)$ as $\tilde{q}$ for short. Note that $\{p: g(p) = q\}$ only contains two points $\{q, \tilde{q}\}$. If $g^{-1}(q)subseteq \{p: h(p) = 1\}$, then
\[\mathbb{E}_{psim U([0, 1])} [ h(p) \mid g(p) = q] = 1.\]
Otherwise, by equation~\eqref{eq:sderiv} and the fact that $q = s(\tilde{q})$, we infer that
\begin{align}
\mathbb{E}_{psim U([0, 1])} [ h(p) \mid g(p) = q] &= \frac{h(q) - h(\tilde{q}) / s'(\tilde{q})}{1 - 1 / s'(\tilde{q})}.\nonumber\\
& = \frac{h(q) - h(\tilde{q}) (h(q) - 1) / (h(\tilde{q}) - 1)}{1 - (h(q) - 1) / (h(\tilde{q}) - 1)} ~=~ 1.\label{eq:ode_pre}
\end{align}
On the other hand, since $s(\cdot)$ is strictly decreasing, $\{p: s(p) = q\}$ contains at most two points for any $q$. As a result,
\begin{align*}
P_{psim U([0,1])}\left( \mathbb{E}_{psim U([0, 1])} [ h(p) \mid g(p) = q] = 1\right) &\ge 1 - P_{psim U([0, 1])}\left( s^{-1}(\mathcal{D})\right)\\ &\ge 1 - 2P_{psim U([0, 1])}(\mathcal{D}) ~=~ 1.
\end{align*}
Hence, we have proved that our choice of $g$ satisfies condition~\eqref{eq:masking-condition}, and this concludes the proof of the theorem.
\end{proof}
section{More Details About Selectively Traversed Accumulation Rules}\label{sec:star}
subsection{Flowchart of the framework}\label{subapp:flowchart}
The scheme in Section \ref{subsec:guidance} is presented explicitly in Figure~\ref{fig:flowchart}, and the three steps of a generic update rule are highlighted in red.
\begin{figure}
\caption{Flowchart of the framework}
\label{fig:flowchart}
\end{figure}
subsection{Data adaptive structural constraint}\label{subapp:Kt}
Let $\mathcal{K}_{0}, \mathcal{K}_{1}, \ldots$ be a sequence of structural constraints with $\mathcal{K}_{t}\in \mathcal{F}_{t}$. We can then generalizes Algorithm \ref{algo:STAR} by incorporating time-varying structural constraints and allowing the rejection set to temporarily leave the constraint. For example, if the analyst had started by wanting to find a convex set, but the masked p-values very clearly reveal a banana shape, or two circles in opposite corners of the grid, then she can change her mind and update $\mathcal{K}$.
\begin{algo}\label{algo:STARKt}
{\em STAR}
\begin{tabbing}
\quad \textbf{Input: }Predictors and $p$-values $(x_{i}, p_{i})_{i=1}^{n}$, constraint set $\mathcal{K}$, target FDR ~level $\alpha$.\\
\quad $\mathcal{R}_{0} = [n]$\\
\quad While $\mathcal{R}_{t}\not= \emptyset$\\
\quad \qquad\enspace $\widehat{\textnormal{FDP}}_{t}\gets \frac{1}{1 + |\mathcal{R}_{t}|}\left( h(1) + sum_{i\in \mathcal{R}_{t}}h(p_{i})\right)$\\
\quad \qquad \enspace If ($\widehat{\textnormal{FDP}}_{t} \le \alpha$ and $\mathcal{R}_t \in \mathcal{K}_t$) or $\mathcal{R}_t = \emptyset$\\
\quad \qquad \qquad\enspace Stop and return $\mathcal{R}_t$, and reject $\{H_{i}: i\in \mathcal{R}_{t}\}$\\
\quad \qquad\enspace Select $\mathcal{R}_{t+1}subseteq\mathcal{R}_{t}$ with $\mathcal{R}_{t+1}\in \mathcal{K}_{t}\cap \mathcal{F}_{t}$\\
\quad \qquad\enspace Select $\mathcal{K}_{t+1}\in\mathcal{F}_{t+1}$\\
\quad Output $\mathcal{R}_{t}$ as the rejection set\\
\end{tabbing}
\end{algo}
subsection{Examples of masking functions}\label{subapp:masking_functions}
We show masking functions of several accumulation functions that are used in literature.
\begin{enumerate}
\item (SeqStep, \cite{barber15}) When $h(p) = \frac{1}{1-p_{*}} 1\{ p > p_{*}\}$, one may derive
\[s(p) = \frac{p_{*}}{1-p_{*}}(1 - p).\]
\item (ForwardStop, \cite{gsell2016sequential}) For the unbounded accumulation function $h(p) = -\log(1 - p)$, we can obtain a bounded function $h^C(p)$ by truncating at $C>0$ and renormalizing as in Remark~\ref{rem:bounded-h}; in order to avoid a large renormalization (corresponding to a large correction of the FDR ~level), we fix $C = -\log(0.01) = 4.605$, in which case $\int_0^1(h(p)\wedge C)\,dp = 0.99$. For any $C>0$, one can derive
\[H(p; C) = \left\{
\begin{array}{ll}
e^{-C}p + (1 - p)\log (1 - p) & \text{ if } p\le 1 - e^{-C},\\
(1 - p)(1 - C - e^{-C}) & \text{ if } p > 1 - e^{-C}.
\end{array}
\right.\]
and solve for $s(p)$ numerically as shown in Figure~\ref{fig:masking_fun}.
\item (HingeExp, \cite{li2016accumulation}) ForwardStop may be generalized to obtain the unbounded accumulation function $h(p) = \frac{1}{1 - p_{*}}\log \frac{1 - p_{*}}{1 - p}1\{p\ge p_{*}\}$ for some $p_{*}\in (0, 1)$ ($p_{*}=0$ gives ForwardStop after reparametrization).
Using a similar reasoning to ForwardStop, for each $p_{*}$ we recommend truncating $h(p)$ at $C = \frac{-\log (0.01)}{1 - p_{*}}$ so that $\int_0^1(h(p)\wedge C)\,dp = 0.99$.
After truncating and renormalizing using any $C>0$, we have
\[
H(p; C) = \left\{
\begin{array}{ll}
-p\int_0^1(h(p)\wedge C)\,dp & \text{ if } p < p_{*},\\
e^{-C(1-p_{*})}p + \frac{1-p}{1-p_{*}}\log\frac{1-p}{1-p_{*}} - \frac{p_{*}}{1-p_{*}}(1-p) & \text{ if } p_{*}\le p\le 1 - (1 - p_{*})e^{-C(1 - p_{*})},\\
(1 - p)(1 - C - e^{-C(1-p_{*})}) & \text{ if } p > 1 - (1 - p_{*})e^{-C(1 - p_{*})}.
\end{array}
\right.
\]
Once more we can calculate $s(p)$ numerically, as shown in Figure~\ref{fig:masking_fun} below.
\end{enumerate}
\begin{figure}
\caption{Masking functions for different accumulation functions.}
\label{fig:masking_fun}
\end{figure}
subsection{Details of EM ~algorithm}\label{subapp:EM}
Consider the working model \eqref{eq:onegroup_GLM}. At step $t$, let
\[\tilde{p}_{t, i} = p_{i}I(i\not\in \mathcal{R}_{t}) + g(p_{i})I(i\in \mathcal{R}_{t}),\]
where $g(p) = \min\{p, s(p)\}$. For simplicity, we assume that $f(p; \theta)$ is the model of the original $p$-values. Note that the following derivation directly carries over to the transformed $p$-values.
Define a sequence of hypothetical labels $w_{t, i} = I(\tilde{p}_{t, i} = p_{i})$. Note that for unmasked p-values, $w_{t, i} = 1$. Then the joint log-likelihood of $\{\tilde{p}_{t, i}\}$ and $\{w_{t, i}\}$ is
\begin{align}
&\ell(\{\tilde{p}_{t, i}, w_{t, i}\}) = sum_{i\not\in \mathcal{R}_{t}}\log f(\tilde{p}_{t, i}; \mu(x_{i}))\nonumber\\
& \quad + sum_{i\in \mathcal{R}_{t}}w_{t, i}\log f(\tilde{p}_{t, i}; \mu(x_{i})) + sum_{i\in \mathcal{R}_{t}} (1 - w_{t, i}) \log f(s^{-1}(\tilde{p}_{t, i}); \mu(x_{i})).\nonumber
\end{align}
The standard EM ~algorithm replaces $w_{t, i}$ by its conditional mean $\mathbb{E} (w_{t, i} \mid \tilde{p}_{t, i})$ in the E-step. Using a similar argument as equation~\eqref{eq:ode_pre}, we have
\begin{align}
\tilde{w}_{t, i}\triangleq \mathbb{E} (w_{t, i} \mid \tilde{p}_{t, i}, \theta_{\mathrm{old}}(x_{i}))& = \mathbb{P} (p_{i} = \tilde{p}_{t, i} \mid g(p_{i}) = \tilde{p}_{t, i})\nonumber\\
& = \frac{f(\tilde{p}_{t, i}; \theta_{\mathrm{old}}(x_{i}))}{f(\tilde{p}_{t, i}; \theta_{\mathrm{old}}(x_{i})) - (s^{-1})'(\tilde{p}_{t, i})\cdot f(s^{-1}(\tilde{p}_{t, i}); \theta_{\mathrm{old}}(x_{i}))}.\label{eq:Estep}
\end{align}
where $\theta_{\mathrm{old}}(\cdot)$ is from the last iteration. Here $(s^{-1})'(\cdot) = 1 / s'(s^{-1}(\cdot))$ is known to exist almost everywhere by Theorem \ref{thm:gp}. Then in the M-step, we replace $\theta_{\mathrm{old}}(\cdot)$ by
\begin{align}
& \theta_{\mathrm{new}}(\cdot) = \argmax_{\theta(\cdot)\in { \mathrm{\scriptscriptstyle T} }heta} sum_{i\not\in \mathcal{R}_{t}}\log f(\tilde{p}_{t, i}; \theta(x_{i}))\nonumber\\
& \quad + sum_{i\in \mathcal{R}_{t}}\tilde{w}_{t, i}\log f(\tilde{p}_{t, i}; \theta(x_{i})) + sum_{i\in \mathcal{R}_{t}} (1 - \tilde{w}_{t, i}) \log f(s^{-1}(\tilde{p}_{t, i}); \theta(x_{i})).\label{eq:Mstep}
\end{align}
The above optimization problem is equivalent to solving a weighted MLE on an artificial dataset $\{(\tilde{p}_{t, i})_{i=1}^{n}, (s^{-1}(\tilde{p}_{t, i}))_{i\in \mathcal{R}_{t}}\}$. Therefore any algorithm that solves the weighted MLE can be embedded into this framework.
We provide two instantiations below, which will be used in later sections. For illustration, we only consider the accumulation function $h(p) = 2 I(p\ge 0.5)$ with $s(p) = 1 - p$.
{\varepsilon}space{0.3em}
\noindent \textbf{Example 1: Beta family for p-values.} Consider the model \eqref{eq:onegroup_GLM} with $h(p; \mu) = \frac{1}{\mu}p^{\frac{1}{\mu} - 1}$. The E-step \eqref{eq:Estep} simplifies to\begin{equation*}
\tilde{w}_{t, i} = \frac{\tilde{p}_{t,i}^{\frac{1}{\beta_{\mathrm{old}}'\phi(x_{i})} - 1}}{\tilde{p}_{t,i}^{\frac{1}{\beta_{\mathrm{old}}'\phi(x_{i})} - 1} + (1 - \tilde{p}_{t, i})^{\frac{1}{\beta_{\mathrm{old}}'\phi(x_{i})} - 1}},
\end{equation*}
and the M-step \eqref{eq:Mstep} can be calculated as
\begin{align}
\beta_{\mathrm{new}} &= \argmax_{\beta\in \mathbb{R}^{m}} sum_{i\not\in \mathcal{R}_{t}}\{(\log \tilde{p}_{t, i})\beta_{\mathrm{old}}'\phi(x_{i}) + \log (\beta_{\mathrm{old}}'\phi(x_{i}))\}\nonumber\\
& \quad + sum_{i\in \mathcal{R}_{t}}\left\{\left(\tilde{w}_{t, i}\log \tilde{p}_{t, i} + (1 - \tilde{w}_{t,i})\log \left( 1 - \tilde{p}_{t,i}\right)\right)\beta_{\mathrm{old}}'\phi(x_{i}) + \log (\beta_{\mathrm{old}}'\phi(x_{i}))\right\}.\nonumber
\end{align}
Define $y_{t, i}$ as
\[y_{t, i} = -\log(\tilde{p}_{t, i})I(i\not\in \mathcal{R}_{t}) - \left(\tilde{w}_{t, i}\log \tilde{p}_{t, i} + (1 - \tilde{w}_{t,i})\log \left( 1 - \tilde{p}_{t,i}\right)\right) I(i\in \mathcal{R}_{t}).\]
Then, we have
\begin{equation*}
\beta_{\mathrm{new}} = \argmax_{\beta\in \mathbb{R}^{m}} sum_{i=1}^{n}\{-y_{t, i}\beta_{\mathrm{old}}'\phi(x_{i}) + \log (\beta_{\mathrm{old}}'\phi(x_{i}))\},
\end{equation*}
which is equivalent to the solution of an \emph{unweighted} Gamma generalized linear model with a inverse link function on data $\{y_{t, i}\}$ with covariate $\phi(x_{i})$.
{\varepsilon}space{0.3em}
\noindent \textbf{Example 2: Gaussian family for z-values.} Consider the model $p_{i} = 1 - \mathbb{P}hi(z_{i})$ with $z_{i}sim N(\mu_{i}, 1)$. Define the partially-masked z-values as
\[\tilde{z}_{t, i} = \max\{z_{i}, \mathbb{P}hi^{-1}(1 - (1 - p_{i}))\} = \max\{z_{i}, -z_{i}\} = |z_{i}|.\]
Thus the E-step \eqref{eq:Estep}, replacing the $p$-values by z-values, can be simplified as
\[\tilde{w}_{t, i} = \frac{\exps{-\frac{(|z_i| - \mu_{i,\mathrm{old}})^2}{2}}}{\exps{-\frac{(z_i - \mu_{i,\mathrm{old}})^2}{2}} + \exps{-\frac{(-z_i - \mu_{i,\mathrm{old}})^2}{2}}} = \frac{1}{1 + \exps{-2\mu_{i, \mathrm{old}}\cdot |z_i|}}.\]
In the M-step, $\mu_{i}$'s are updated by
\begin{align}
(\mu_{i, \mathrm{new}}) &= \argmin ~sum_{i\not\in \mathcal{R}_{t}}(z_{i} - \mu_{i})^2 + sum_{i\in \mathcal{R}_{t}}\tilde{w}_{t, i}(z_{i} - \mu_{i})^{2} + (1 - \tilde{w}_{t, i})(-z_{i} - \mu_{i})^2\nonumber\\
& = \argmin ~ sum_{i\not\in \mathcal{R}_{t}}(z_{i} - \mu_{i})^2 + sum_{i\in \mathcal{R}_{t}}((2\tilde{w}_{t, i} - 1)z_{i} - \mu_{i})^{2},\nonumber
\end{align}
which reduces to an \emph{unweighted} least-squares problem on a pseudo-dataset $\{\tilde{z}_{t, i}: i = 1, \ldots, n\}$ where $\tilde{z}_{t, i} = z_{i}$ for unmasked hypotheses and $\tilde{z}_{t, i} = (2\tilde{w}_{t, i} - 1)z_{i}$ for masked hypotheses. Note that we can solve it as a non-parametric least-squares problem if $\phi(x)$ corresponds to some basis functions, or as a constrained problem with $\mu_{i}$'s lying in an isotonic cone.
section{Example 3: bump hunting}\label{sec:bump}
Bump hunting is widely applied in areas such as astronomy~\citep{good80}, risk management~\citep{becker01}, bioinformatics~\citep{jiang06}, and epidemiology~\citep{jaffe12}. In these areas, one collects a response $y$ together with a possibly high dimensional vector $x$ of predictors and aims at obtaining knowledge of $f(x) = \mathbb{E} [y | x]$. In many applications, it is not necessary to estimate $f(x)$ uniformly over the domain but simply detect a scientifically interesting subregion of the predictor space instead. In bump hunting, we usually aim to detect a subregion within which the average of $y$ is larger than that on the entire space. However, most existing procedures lack formal statistical guarantees.
We can cast the problem as a nonparametric multiple testing problem by defining the null hypothesis $H_i$ that the conditional response distribution at the $i$th data point is
\[
H_i:\; \mathcal{L}(y_i | x_{i})\preceq \mathcal{L}(y_i + B),
\]
where $\preceq$ denotes stochastic dominance and $\mathcal{L}$ denotes the marginal or conditional distribution of $y$. Informally, we wish to find a clustered set of non-nulls, corresponding to a rectangular region of the feature space where the response is unusually large, by some fixed location offset $B\geq 0$.
Let $F_{0}$ denote the marginal distribution function of $y$. If $F_{0}$ is known, one can define the $p$-value as $p_{i} = 1 - F_{0}(y_{i}-B)$ (if $F_0$ is not continuous, we may use a randomized version instead). To discover a rectangular region, we can apply the convex region detection algorithm of Section~\ref{subsubsec:convex_proc} with the restriction that we always peel off an axis-parallel rectangle in the form of $\{i: x_{ij}\ge v_{j}\}$ or $\{i: x_{ij}\le v_{j}\}$. More precisely, given a patience parameter $\delta\in (0, 1)$, the candidate sets are given by $\{C(j, b; \delta): j \in \{1,\ldots, p\}, b \in \{-1, 1\}\}$, where
\[C(j, -1; \delta) = \{i: x_{ij}\le v_{j}\}, \quad C(j, 1; \delta) = \{i: x_{ij}\ge v_{j}\},\]
and $v_{j}$ is set to be the minimal value such that $|C(j, b; \delta)| \ge \lceil n \delta\rceil$.
\begin{figure}
\caption{Comparison of income distributions before and after selection by star.}
\label{fig:income_dist}
\end{figure}
For illustration, we consider a moderately sized demographics dataset, that contains questions from $n = 9409$ questionnaires filled out by shopping mall customers; see Section 14.2.3 of \cite{ESL} for details. The goal is to predict the income using the first 13 questions, listed in the first column of Table~\ref{tab:bump_hunting}, that provide the basic demographics. All variables are either binary or ordinal. We use the empirical distribution of $y_{i}$, the income, as a proxy for $F_{0}$, and use $B=0$ for the location offset. Since the $y_{i}$'s are discrete, the $p$-values are made continuous by randomization; to account for the effect of randomization, we repeat the entire experiment 100 times.
We find that the box produced by star ~is quite stable across experiments and the target FDR ~level $\alpha$. The results are reported in Table~\ref{tab:bump_hunting}. The last column details the interval for each variable of the most frequent box among 100 repetitions for $\alpha \in \{0.05, 0.1, 0.2\}$. The middle three columns contain the frequency of this particular box among 100 repetitions. Because the box is quite stable for most predictors, we conclude the randomization of the $p$-values does not substantially destabilize the discovered region. We also plot the income distribution of this sub-population and that of the overall population in Figure \ref{fig:income_dist}; thus, we see that our method has detected a subpopulation with significantly higher income than the overall population. Compared to other bump hunting algorithms, star ~has statistical guarantees (FDR ~control).
\begin{table}
small
\textsc{na}tering
\begin{tabular}{lllll}
\hline
Attributes & box20 freq. & box10 freq. & box5 freq. & box \\
\hline
sex & 1.00 & 1.00 & 1.00 & male/female \\
marital status & 1.00 & 1.00 & 1.00 & married/single \\
age & 0.92 & 0.92 & 0.54 & [18, 54] \\
education & 0.99 & 0.99 & 0.99 & $>=$ high school \\
occupation & 0.92 & 0.75 & 0.53 & professional/manager/student \\
years in bay area & 0.63 & 0.63 & 0.63 & $>$10 \\
dual incomes & 0.92 & 0.92 & 0.92 & not married/yes \\
number in household & 1.00 & 1.00 & 1.00 & [2,4] \\
number of children & 0.59 & 0.59 & 0.59 & $<=$2 \\
householder status & 1.00 & 1.00 & 1.00 & own \\
type of home & 1.00 & 1.00 & 1.00 & house \\
ethnic classification & 1.00 & 1.00 & 1.00 & white \\
language in home & 1.00 & 1.00 & 1.00 & english \\
\hline
\end{tabular}
\caption{Results of bump hunting on the income dataset: the first column reports the variable names; the last column reports the selected interval of each variable in the detected box; the second to the fourth colums report the frequency of the box listed in the last column among 100 randomizations}\label{tab:bump_hunting}
\end{table}
section{Example 4: hierarchical testing}\label{sec:tree}
subsection{Problem Setup}
A well-studied case of structured multiple testing is that of hierarchical testing where the hypotheses have an intrinsic rooted tree structure and the non-null hypotheses form a rooted subtree. Most earlier works focus on FWER ~control~\citep[e.g.,][]{dmitrienko06, meinshausen08, huque08, brechenmacher11, goeman12}. However FWER ~controlling procedures are often quite conservative, having low power.
In contrast, \cite{yekutieli06,yekutieli08} proposed a novel procedure in microarray analysis that guarantees FDR ~control under independence.
FDR ~controlling methods have since been applied to other areas including genomics~\citep[e.g.,][]{heller09, guo10, benjamini14, li14, lynch16} and neural image analysis~\citep[e.g.,][]{benjamini07, singh10, schildknecht16}. New procedures have also been recently introduced for multi-layer or multi-resolution FDR ~guarantees~\citep{barber16,peterson16, katsevich17, bogomolov17}.
Note that in many hierarchical testing problems, the $p$-value for a given node is derived from the $p$-values of the nodes descending from it (using, for example, the Simes test); in such problems, the $p$-values would be dependent and star ~would not be applicable. However, when it is applicable, it is quite straightforward to be applied to hierarchical testing problems. Similar to Section \ref{sec:convex}, we consider an artificial example to describe the procedure and compare the performance of star ~with other existing methods.
subsection{Procedure}\label{subsec:tree_proc}
In order to maintain a subtree structure of the rejection set, at any step of the algorithm, we can simply set the candidates to be observed as all leaf nodes of the subtree of still masked $p$-values. Equivalently, star ~will peel off the leaf nodes that have least favorable scores at each step.
In this article, we only consider the canonical score. However, it is worth mentioning that there are various reasonable model-assisted scores that can be applied in hierarchical testing. For a certain class of problems such as wavelet-based image-denoising, it is common to assume that the signal strength has an isotonic ordering on the tree under which the signal strength of the parent node is higher than that of the child node. With this prior knowledge, we can combine the EM ~algorithm and isotonic regression \citep[e.g.,][]{best90, mair09, stout13}, with a tree ordering, to compute the model-assisted score; See Example 2 in Section \ref{subapp:EM} for implementation details.
subsection{Simulations}
\begin{figure}
\caption{Hierarchical testing problems with non-nulls arranged with breadth-first-search ordering and depth-first-search ordering. Each node represents a hypothesis (1000 in total) with black ones being the nulls and white ones being non-nulls.}
\label{fig:tree_expr_truth}
\end{figure}
To illustrate, we construct a balanced binary tree with $n = 1000$ nodes and set 50 nodes as non-nulls. We place the non-nulls as the first 50 nodes either in breath-first-search ordering or in depth-first-search ordering. The settings are plotted in Figure~\ref{fig:tree_expr_truth}. Heuristically, the methods by \cite{yekutieli08} and \cite{lynch16} may prefer the breath-first-search ordering since they are top-down algorithms that proceed layer-by-layer, and only proceed to child nodes when the parent node is rejected. When the non-nulls are placed in the DFS ordering, as shown in the right panel of Figure \ref{fig:tree_expr_truth}, those methods run the risk of stopping early in the long chain of $p$-values, and may therefore be less powerful. By contrast, star ~proceeds adaptively in a bottom-up manner from leaves to the root, and we may expect it to be more robust to the layout of non-nulls.
Another important factor that affects the power is the pattern of signal strength along the tree. The top-down procedures should be more favorable if the signal strength is in an isotonic ordering on the tree where the root node has the strongest signal. However, when the signals in top nodes are weak, these procedures risk being powerless. To account for this effect, we generate $p$-values by
\begin{equation}\label{eq:simul_pvals}
p_{i} = 1 - \mathbb{P}hi(z_{i}), \quad \mbox{and} \quad z_{i}sim N(\mu_{i}, 1).
\end{equation}
where the null $\mu_{i}$'s equal 0 and the non-null $\mu_{i}$'s are set in one of the following three ways:
\begin{enumerate}[{Case} 1:]
\item $\mu_{i}\equiv 2, \,\, \forall i\in \mathcal{H}_{0}^{c}$;
\item $\displaystyle \mu_{i} = \left\{\begin{array}{ll}
2.5 & i\in \{25 \mbox{ nodes with smallest indices in }\mathcal{H}_{0}^{c}\},\\
1.5 & \mbox{otherwise;}
\end{array}\right.$
\item $\displaystyle \mu_{i} = \left\{\begin{array}{ll}
1.5 & i\in \{25 \mbox{ nodes with smallest indices in }\mathcal{H}_{0}^{c}\},\\
2.5 & \mbox{otherwise.}
\end{array}\right.$
\end{enumerate}
In summary, we consider six cases: the non-nulls are placed in breath-first-search ordering or depth-first-search ordering and the $p$-values are set in one of the above three cases. For each setting, we apply star, with $h(p) = 2I(p\ge 0.5)$ and canonical scores, as well as \cite{yekutieli08}'s procedure and \cite{lynch16}'s procedures in their sections 4.1 and 4.3. We plot the results in Figures~\ref{fig:rej_tree_BFS_methods} and~\ref{fig:rej_tree_DFS_methods}.
\begin{figure}
\caption{Comparison of star ~with canonical score (black solid line), \cite{yekutieli08}
\label{fig:rej_tree_BFS_methods}
\end{figure}
From Figure~\ref{fig:rej_tree_BFS_methods}, we see that all methods control FDR ~exactly. In cases 1 and 2, star ~has lower power than \cite{lynch16}'s second procedure, but is competitive with other procedures. When the top non-nulls are weak as in Case 3, the forward procedures lose power remarkably while star ~gains power as expected. It is clearly shown that the power of our method is quite stable across the different layouts as opposed to top-down procedures.
\begin{figure}
\caption{Comparison of star ~with canonical score (black solid line), \cite{yekutieli08}
\label{fig:rej_tree_DFS_methods}
\end{figure}
From Figure~\ref{fig:rej_tree_DFS_methods}, we see that star ~is most powerful even when the non-nulls are placed in a DFS ordering. Comparing to Figure~\ref{fig:rej_tree_BFS_methods}, the performance of our method does not degrade much. However, the top-down procedures lose power considerably and even become powerless in Case 1 and Case 3.
section{Example 5: wavelet thresholding}\label{sec:wavelet}
\begin{figure}
\caption{Schematic representation of the 2-dimensional discrete wavelet decomposition: panels (a) - (c) gives the schematic representation of first three levels of decomposition and the rightmost panel gives the description of the hierarchical structure. The figure is copied from \texttt{http://www.debugmode.com/imagecmp/classify.htm}
\label{fig:pyramid}
\end{figure}
Wavelet decomposition has been an efficient tool in signal processing for decades \citep[e.g.,][and references therein]{mallat99}.
It provides an efficient and elegant methodology that represents signals at different scales, ranging from "backgrounds/trends'' to "edges/anomalies".
Due to the hierarchical nature of wavelet decomposition, the wavelet coefficients can be described by a balanced tree. Figure~\ref{fig:pyramid} gives a schematic description of the 2-dimensional discrete wavelet decomposition which is widely used in image processing. Given an image with size $2^{k}\times 2^{k}$, a high-pass filter and a low-pass filter are applied to the rows and columns to decompose the image into four sub-bands, LL, LH, HL and HH, where LL contains all information in lower frequencies and the last three contain the high-frequency information in different orientations. The procedure then proceeds recursively on LL to decompose the low-frequency sub-band as illustrated in panels (a) - (c) of Figure \ref{fig:pyramid} until LL only contains one pixel. The wavelet coefficients can be arranged in a quadtree; See the rightmost panel of Figure \ref{fig:pyramid} for illustration. We refer the readers to \cite{mallat99} for details.
Often, natural signals can be represented by a small subset of wavelet coefficients; equivalently, the wavelet coefficient vector is sparse. Under the standard assumption that the signal is multivariate normal and homoscedastic, the wavelet coefficients are independent normal variables since the transformation is unitary. Denote by $\hat{d}_{jk}$ the $j$-th wavelet coefficient in the $k$-th level, then $\hat{d}_{jk}sim N(\mu_{jk}, sigma^{2})$ for some common variance $sigma^{2} > 0$. The problem of detecting ``large'' wavelet coefficients can be formalized as a selection problem that aims at detecting nonzero $\mu_{jk}$'s. The classic procedures, such as hard thresholding \citep{donoho94} and soft thresholding \citep{donoho95}, are proved to be minimax optimal from the estimation viewpoint. However, for most images it is reasonable to assume that the large coefficients form a subtree \citep[e.g.,][]{shapiro93, hegde15}. This tree structure has been exploited since \cite{shapiro93}'s Embedded Zerotrees of Wavelet transforms algorithm for efficient encoding of images.
On the other hand, \citep{abramovich96} formalized the problem in terms of multiple hypothesis testing with $H_{jk}: \mu_{jk} = 0$ and applied the BH ~procedure on the $p$-values calculated as $p_{jk} = 1 - \mathbb{P}hi(\hat{d}_{jk} / \hat{sigma})$, where $\hat{sigma}$ is estimated from the coefficients at the finest scale. This idea is exploited further using Bayesian FDR ~control methods \citep[e.g.,][]{tadesse05, lavrik08}.
However, these methods also do not take the structured sparsity into consideration. This motivates us to apply star ~with a tree constraint that is discussed in Section~\ref{subsec:tree_proc}. To illustrate we compare our method with other methods on 48 standard gray-scale images of size $512\times 512$, available at \texttt{http://decsai.ugr.es/cvg/CG/images/base/$X$.gif}, where $X$ is an integer $\in \{1,\dots,48\}$. For each figure we add Gaussian white noise with SNR = 0.5dB, where SNR (signal-to-noise ratio) is defined as $10\log_{10}\left( s_{\mathrm{image}}^{2} / s_{\mathrm{noise}}^{2}\right)$ with unit dB (decibel). The two panels in the left column of Figure \ref{fig:image_cat} show one original image and its contaminated version.
\begin{figure}
\caption{A sample image of cat. The left column show the original image and the contaminated version. The other four panels display the recovered images of star, the BH ~procedure, hard thresholding (H.T.) and soft thresholding (S.T.), respectively. The signal-to-noise ratio and the compression ratio are reported in the title.}
\label{fig:image_cat}
\end{figure}
We compare star ~with the BH ~procedure \citep{abramovich96}, hard thresholding \citep{donoho94} and soft thresholding \citep{donoho95}. We estimate the variance $\hat{sigma}^{2}$ separately for LH, HL and HH sub-bands using the normalized median of the coefficients at the finest scale; See Chapter 11 of \cite{mallat99} for details. For star ~and the BH ~procedure, we calculate $p$-values by $p_{jk} = 1 - \mathbb{P}hi(\hat{d}_{jk} / \hat{sigma}_{w})$ where $w\in \{LH, HL, HH\}$ depending on the location of $\hat{d}_{jk}$; for hard/soft thresholding, the threshold is chosen as $sqrt{2\hat{sigma}_{w}^{2}\log N}$ where $N$ denotes the total number of coefficients. For each method, we record the signal-to-noise ratio (SNR) and compression ratio (CR), defined as the ratio of the total number of wavelet coefficients and number of selected coefficients. To illustrate, we report the SNRs and CRs on the top of four panels in Figure \ref{fig:image_cat}. We observe that star ~has the largest SNR and a more compact representation than the BH ~procedure. Note that although the BH ~procedure produces a visually clearer image, the compression ratio is much smaller than other algorithms. For thorough comparison, we compute the ratio of SNR and CR between star ~and other methods and provide the boxplot in Figure \ref{fig:SNR_CR}. It is clearly shown that star ~has larger SNR than other methods and provide a more parsimonious representation than the BH ~procedure for most figures. We conclude that star ~has reasonable performance in wavelet-based image denoising.
\begin{figure}
\caption{Comparison of SNR and CR between star ~(normalized to the red line at unity) and other methods (the BH ~procedure, hard thresholding, soft thresholding) on 48 gray-scale images.}
\label{fig:SNR_CR}
\end{figure}
section{Example 6: interaction selection in factorial experiments}\label{sec:factorial}
The heredity principle dates back to early work of \cite{yates37} on factorial experiments. The term ``heredity'' was coined by \cite{hamada92} in the context of experimental design and originally used to ensure the compatibility of the selected model in the presence of complex aliasing. On the other hand, \cite{nelder77} introduced the marginality principle, an equivalent version of the strong heredity principle, driven by interpretability. In recent years, this topic has been revisited under the high dimensional settings \citep[e.g.,][]{yuan09, choi10, bien13}. However, none of these works provides error measures, either FWER ~or FDR, regarding the selected variables.
All aforementioned works consider the linear model with all main effects and second-order interaction effects:
\[
y = \beta_{0} + sum_{j=1}^{p}\beta_{j}X_{j} + sum_{j,k=1}^{p}\beta_{jk}X_{j}X_{k} + {\varepsilon}arepsilonsilon,
\]
where $y\in \mathbb{R}^{n}$ is the response variable and $(X_{1}, \ldots, X_{p})\in \mathbb{R}^{n\times p}$ are $p$ factors. This induces a two-layer DAG ~with the main effects $\{X_{j}\}$ in the first layer and the interaction effects $\{X_{j}X_{k}\}$ in the second layer. Write $Z$ for the design matrix including the intercept term, all main effects and interaction effects, i.e. $Z = (\textbf{1},\, X_{1}, X_{2}, \ldots, X_{p}, X_{1}X_{2}, \ldots, X_{p-1}X_{p})\in \mathbb{R}^{n\times \left( \frac{p^2 + p + 2}{2}\right)}$ and $\beta$ for the coefficients, i.e. $\beta = (\beta_{0}, \beta_{1}, \ldots, \beta_{p}, \beta_{12}, \ldots, \beta_{p-1,p})$. Then the model may be succinctly represented as
\[
y = Z\beta + {\varepsilon}arepsilonsilon.
\]
If one can construct independent $p$-values for each entry of $\beta$, star ~can be applied to guarantee the heredity principle and the FDR ~control simultaneously.
For illustration we consider a pharmaceutical dataset from \cite{jaynes13}. Their work aims at investigating the effect of six anti-viral drugs, namely Interferon-alpha (A), Interferon-beta (B), Interferon-gamma (C), Ribavirin (D), Acyclovir (E), and TNF-alpha (F), to Herpes simplex virus type 1 (HSV-1). They applied a $2^{6-1}$ fractional factorial design with $32$ runs and encode all factors by $+1$ and $-1$ (in fact, they have 35 runs with the last 3 runs being the replicated center points to evaluate the lack-of-fit). The minimal word of the half-fraction design is $ABCDEF$ and hence it has resolution VI; see \cite{wu00} for the terminology and details. In other words, the main effects and the second-order interaction effects are not aliased with each other. This means that we can estimate all main effects and all two-factor interactions assuming that fourth-order and higher interactions are negligible, which is quite a reasonable assumption in practice \citep{jaynes13}. The response variable is set to be the logarithm of the viral infection load.
Under the standard assumption that ${\varepsilon}arepsilonsilonsim N(0, sigma^{2}I_{n\times n})$, the least-squares estimator is
\[\hat{\beta}\triangleq (\hat{\beta}_{0}, \hat{\beta}_{1}, \ldots, \hat{\beta}_{p}, \hat{\beta}_{1,2}, \ldots, \hat{\beta}_{p-1,p}) = (Z^{T}Z)^{-1}Z^{T}ysim N(\beta, sigma^{2}(Z^{T}Z)^{-1}).\]
Due to the nature of fractional factorial designs, $Z$ is an orthogonal matrix with
\[Z^{T}Z = K\cdot I_{K\times K}, \quad K = \frac{p^2 + p + 2}{2} = 22.\]
Thus the entries of $\hat{\beta}$ are independent. Here we simply replace $sigma$ by $\hat{sigma}$, obtained from the regression residuals, i.e.
\[\hat{sigma}^{2} = \frac{1}{n - \frac{p^{2} + p + 2}{2}}\|y- Z\hat{\beta}\|^{2} = \frac{1}{10}\|y- Z\hat{\beta}\|^{2}.\]
Then we can construct the $p$-values by
\begin{equation}\label{eq:DAG_pvals}
p_{j} = 1 - \mathbb{P}hi\left(\frac{\hat{\beta}_{j}}{\hat{sigma}}\right), \quad p_{jk} = 1 - \mathbb{P}hi\left(\frac{\hat{\beta}_{jk}}{\hat{sigma}}\right).
\end{equation}
Note that the constructed $p$-values may have some dependence due to sharing $\hatsigma$. However as demonstrated experimentally in Section \ref{sec:sensitivity}, star ~still seems to control the FDR ~when correlations are not too large.
Finally, we apply star ~on the $p$-values defined in \eqref{eq:DAG_pvals} with $\alpha = 0.2$ using the accumulation function $h(p) = 2I(p\ge 0.5)$ and the canonical scores. The selected variables include all main effects and three interaction effects: $A\times B, A\times D, C\times D$. This model identifies more effects than those in \cite{jaynes13}. To illustrate the performance of the selection procedure, we refit a linear model using these variables and find that all selected variables are marginally significant except $C$. The estimate and the $p$-values for both full model and refitted model are reported in Table~\ref{tab:DAG_pvals}. This suggests that star ~may have successfully identified the important effects with the guarantee that the FDR ~is controlled at level $0.2$.
\begin{table}[h]
\textsc{na}tering
\begin{tabular}{c|ccccccccc}
\toprule
& $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $A\times B$ & $A\times D$ & $C\times D$\\
\midrule
Estimate & 0.04 & 0.07 & 0.02 & -0.32 & 0.11 & 0.05 & -0.05 & 0.04 & 0.05\\
Orig. $p$-value & 0.118 & 0.012 & 0.458 & 0.000 & 0.001 & 0.037 & 0.053 & 0.086 & 0.038\\
Refit. $p$-value & 0.060 & 0.002 & 0.381 & 0.000 & 0.000 & 0.011 & 0.018 & 0.038 & 0.011\\
\bottomrule
\end{tabular}
\caption{Regression results for the pharmaceutical data using variables selected by star.}\label{tab:DAG_pvals}
\end{table}
section{More experimental results}\label{sec:more}
In this section, we provide more simulation results on the comparison of star ~with SeqStep accumulation function with different cutoffs and the comparison of star ~with different accumulation functions. The settings are the same as their counterpart in Sections \ref{sec:convex}, \ref{sec:tree} and \ref{sec:DAG}.
subsection{Convex region detection}\label{app:experiment_convex}
\begin{figure}
\caption{Comparison of star ~with the SeqStep accumulation function with $p_{*}
\label{fig:rej_convex_STARSS}
\end{figure}
\begin{figure}
\caption{Comparison of star ~with the SeqStep accumulation function (written as SS) with $p_{*}
\label{fig:rej_convex_h}
\end{figure}
subsection{Hierarchical testing}\label{app:experiment_tree}
\begin{figure}
\caption{Comparison of star ~with the SeqStep accumulation function with $p_{*}
\label{fig:rej_tree_BFS_STARSS}
\end{figure}
\begin{figure}
\caption{Comparison of star ~with the SeqStep accumulation function (written as SS) with $p_{*}
\label{fig:rej_tree_h}
\end{figure}
\begin{figure}
\caption{Comparison of star ~with the SeqStep accumulation function with $p_{*}
\label{fig:rej_tree_depth-first-search_STARSS}
\end{figure}
\begin{figure}
\caption{Comparison of star ~with the SeqStep accumulation function (written as SS) with $p_{*}
\label{fig:rej_tree_h}
\end{figure}
subsection{Selection under heredity principle}\label{app:experiment_DAG}
\begin{figure}
\caption{Comparison of star ~with the SeqStep accumulation function with $p_{*}
\label{fig:rej_DAG_STARSS}
\end{figure}
\begin{figure}
\caption{Comparison of star ~with the SeqStep accumulation function (written as SS) with $p_{*}
\label{fig:rej_DAG_h}
\end{figure}
section{Sensitivity analysis}\label{sec:sensitivity}
In this Section, we examine the performance of star ~in the presense of correlated $p$-values in all three cases considered in Section \ref{sec:convex}, \ref{sec:tree} and \ref{sec:DAG}. We generate $p$-values from one-sided normal test with
\[p_{j} = 1 - \mathbb{P}hi(z_{i}), \quad \mbox{where }z = (z_{1}, \ldots, z_{n})sim N(\mu, \Sigma).\]
where $\mu = (\mu_{1}, \ldots, \mu_{n})$ is set to the same as in each section. Instead of letting $\Sigma = I_{n\times n}$ in the main text, we set $\Sigma$ as an equi-correlated matrix, i.e.
\[\Sigma = \left(
\begin{array}{cccc}
1 & \rho & \cdots & \rho\\
\rho & 1 & \cdots & \rho\\
{\varepsilon}dots & {\varepsilon}dots & {\varepsilon}dots & {\varepsilon}dots\\
\rho & \rho & \cdots & 1
\end{array}
\right).\]
In the following analysis, we consider both the positive correlated case where $\rho = 0.5$ and the negative correlated case where $\rho = -0.5 / n$; in the latter case, we set the coefficient proportional to $1/n$ in order to guarantee that $\Sigma$ is positive semi-definite.
It turns out that in all cases, the FDR ~is still controlled at the target level and the power remains high compared to other competitors. The results are plotted in the following subsections. Therefore, we conclude that star ~is not sensitive to the correlation of $p$-values and can be used safely when the correlation between the $p$-values is not high.
subsection{Convex region detection}
\begin{figure}
\caption{Comparison of star ~(black solid), the BH ~procedure (red dashed) and AdaPT ~(blue dotted) in the positive correlated case $\rho = 0.5$. This is a counterpart of Figure \ref{fig:rej_convex_methods}
\label{fig:rej_convex_methods_5}
\end{figure}
\begin{figure}
\caption{Comparison of star ~(black solid), the BH ~procedure (red dashed) and AdaPT ~(blue dotted) in the negative correlated case $\rho = -0.5/n$. This is a counterpart of Figure \ref{fig:rej_convex_methods}
\label{fig:rej_convex_methods_-5}
\end{figure}
subsection{Hierarchical testing}
\begin{figure}
\caption{Comparison of star ~(black solid), \cite{yekutieli08}
\label{fig:rej_tree_BFS_methods_5}
\end{figure}
\begin{figure}
\caption{Comparison of star ~(black solid), \cite{yekutieli08}
\label{fig:rej_tree_BFS_methods_-5}
\end{figure}
\begin{figure}
\caption{Comparison of star ~(black solid), \cite{yekutieli08}
\label{fig:rej_tree_DFS_methods_5}
\end{figure}
\begin{figure}
\caption{Comparison of star ~(black solid), \cite{yekutieli08}
\label{fig:rej_tree_DFS_methods_-5}
\end{figure}
subsection{Selection under heredity principle}
\begin{figure}
\caption{Comparison of star ~(black solid), \cite{lynch16}
\label{fig:rej_DAG_methods_5}
\end{figure}
\begin{figure}
\caption{Comparison of star ~(black solid), \cite{lynch16}
\label{fig:rej_DAG_methods_-5}
\end{figure}
section{Benefit of Using Masking Functions}\label{app:asymptotics}
subsection{Asymptotic false discovery rate and power}
In this section, we investigate the performance of our method under certain asymptotic regimes. The goal of this section is to characterize the benefit of using the masking function through the comparison between our method and the plain accumulation test. For illustration, we focus on the cases without structural constraints, namely $\mathcal{K} = 2^{[n]}$. Additionally, we restrict the attention into the non-interactive version of our method that computes a score $T_{i}$ for each p-value, described in Section \ref{sec:implementation}, only in the initial stage using $\{x_{i}, g(p_{i})\}_{i=1}^{n}$ and never updates it. This is equivalent to the accumulation test with p-values sorted by $T_{i}$. It is worth emphasizing that even in this basic setting where our method certainly loses many advantageous features, we can still observe the gain of using masking functions. The power analysis for interactive versions under general structural constraints is left to the future research.
The aforementioned non-interactive version can be equivalently formulated as rejecting all p-values less than or equal to $\hat{t}_{n}$, where
\begin{equation}
\label{eq:hatt}
\hat{t}_{n} = sup\left\{t: \frac{h(1) + sum_{i=1}^{n}h(p_{i})I(T_{i}\le t)}{1 + sum_{i=1}^{n}I(T_{i}\le t)}\le \alpha\right\}.
\end{equation}
Since the rejection set only depends on the ordering of $T_{i}$'s, we can assume without loss of generality that $T_{i}\in
(0, 1]$; otherwise we can transform $T_{i}$ by $\mathrm{arctan}(1 / T_{i})/\pi + 1 / 2$. The accumulation test is a special case with $T_{i} = i / n$. As in previous works \citep{li2016accumulation, lei2016power}, assume that for each $t\in [0, 1]$,
\begin{equation}
\label{eq:Fn}
F_{n}(t)\triangleq \frac{1}{n}sum_{i=1}^{n}I(T_{i} \le t)stackrel{p}{\rightarrow} F(t)
\end{equation}
and
\begin{equation}
\label{eq:Fn}
H_{n}(t)\triangleq \frac{1}{n}sum_{i=1}^{n}h(p_{i})I(T_{i} \le t)stackrel{p}{\rightarrow} H(t),
\end{equation}
for some functions $F(t)$ and $H(t)$, which are not necessarily continuous. Note that both $F(t)$ and $H(t)$ are non-decreasing with $F(0) = H(0) = 0$ and we can assume $F(t) > 0$ for all $t\in (0, 1]$ without loss of generality; otherwise if $t_{0} = sup\{t: F(t) = 0\}$ we can tranform $T_{i}$ to $(T_{i} - t_{0}) / (1 - t_{0})$. Intuitively,
\[\frac{h(1) + sum_{i=1}^{n}h(p_{i})I(T_{i}\le t)}{1 + sum_{i=1}^{n}I(T_{i}\le t)} = \frac{h(1) + nH_{n}(t)}{1 + nF_{n}(t)}\approx \frac{H(t)}{F(t)}.\]
This motivates us to define $t^{*}$ as
\begin{equation}
\label{eq:tstar}
t^{*} = sup\{t\in [0, 1]: H(t)\le \alpha F(t)\}.
\end{equation}
Note that $t^{*}$ is always well-defined because $H(0) = 0 \le \alpha F(0)$. The following lemma justifies the above heuristic argument that $t^{*}$ is the limit of $\hat{t}_{n}$. The proof is relegated to Section \ref{subapp:power_proof}.
\begin{lemma}\label{lem:thattstar}
Let $H_{\alpha}(t) = H(t) - \alpha F(t)$. Assume that
\begin{equation}
\label{eq:uniform_convergence}
sup_{t\in [0, 1]}|F_{n}(t) - F(t)|stackrel{p}{\rightarrow} 0, \quad sup_{t\in [0, 1]}|H_{n}(t) - H(t)|stackrel{p}{\rightarrow} 0.
\end{equation}
Then $\hat{t}_{n}stackrel{p}{\rightarrow} t^{*}$ if either of the following conditions hold:
\begin{enumerate}
\item $t^{*} = 0$ and
\begin{equation}
\label{eq:tstar0}
\inf_{t' \ge t}H_{\alpha}(t') > 0, \quad \mbox{for any }t > t^{*};
\end{equation}
\item $t^{*} = 1$ and there exists a sequence $t_{m}\uparrow t^{*}$ such that
\begin{equation}
\label{eq:tstar1}
H_{\alpha}(t_{m}) < 0, \quad \mbox{for any }m;
\end{equation}
\item $t^{*}\in (0, 1)$ and both \eqref{eq:tstar0} and \eqref{eq:tstar1} hold.
\end{enumerate}
\end{lemma}
\begin{remark}
If $H_{\alpha}(t)$ is continuous, then the condition \eqref{eq:tstar0} can be removed from part 1 and part 3. This is because if there exists $t > t^{*}$ such that
\[\inf_{t' \ge t}H_{\alpha}(t')\le 0,\]
then the continuity of $H_{\alpha}$ implies the existence of $\tilde{t}\ge t > t^{*}$ with $H_{\alpha}(\tilde{t})\le 0$. This contradicts the definition of $t^{*}$.
\end{remark}
Lemma \ref{lem:thattstar} enables us to compute the asymptotic false discovery rate and power. To be precise, the \emph{false discovery proportion} and the \emph{true positive rate} are defined as
\[\textnormal{FDP}_{n} \triangleq \frac{sum_{i=1}^{n}I(h_{i} = 0, T_{i}\le \hat{t}_{n})}{sum_{i=1}^{n}I(T_{i}\le \hat{t}_{n})}, \quad \textnormal{TPR}_{n} = \frac{sum_{i=1}^{n}I(h_{i} = 1, T_{i}\le \hat{t}_{n})}{sum_{i=1}^{n}I(h_{i} = 1)},\]
where $h_{i} = 1$ iff $H_{i}$ is false. Then by definition the \emph{false discovery rate} and the \emph{power} can be written as
\[\textnormal{FDR}_{n} = \mathbb{E} [\textnormal{FDP}_{n}], \quad \textnormal{Pow}_{n} = \mathbb{E} [\textnormal{TPR}_{n}].\]
Assume that for all $t \in [0, 1]$,
\begin{equation}
\label{eq:Fn01}
F_{n0}(t)\triangleq \frac{1}{n}sum_{i=1}^{n}I(h_{i} = 0, T_{i}\le t)stackrel{p}{\rightarrow} F_{0}(t), \quad F_{n1}(t)\triangleq \frac{1}{n}sum_{i=1}^{n}I(h_{i} = 1, T_{i}\le t)stackrel{p}{\rightarrow} F_{1}(t),
\end{equation}
for some functions $F_{0}(t)$ and $F_{1}(t)$. Note that
\[F_{n}(t) = F_{n0}(t) + F_{n1}(t), \quad F(t) = F_{0}(t) + F_{1}(t).\]
The following theorem establishes the asymptotic false discovery rate and power for procedures in the form of \eqref{eq:hatt}.
\begin{theorem}\label{thm:FDR_power}
Assume that $F_{n0}(t)$ and $F_{n1}(t)$ are continuous at $t = t^{*}$. Then under the assumptions of Lemma \ref{lem:thattstar},
\[\textnormal{FDR}_{n}\rightarrow \frac{F_{0}(t^{*})}{F(t^{*})}, \quad \mbox{if }t^{*} > 0,\]
and
\[\textnormal{Pow}_{n}\rightarrow \frac{F_{1}(t^{*})}{F_{1}(1)},\]
where $t^{*}$ is defined in \eqref{eq:tstar}. Note that the asymptotic power does not require $t^{*} > 0$.
\end{theorem}
As in \cite{li2016accumulation}, we assume there exists a continuous function $f(t)$ such that
\begin{equation}
\label{eq:ft}
sup_{k\in [n]}\bigg|\frac{1}{k}sum_{i=1}^{k}I(h_{i} = 1) - f\left(\frac{k}{n}\right)\bigg|\rightarrow 0,
\end{equation}
where $h_{i}$'s are treated as fixed and
\[p_{i}\mid h_{i} = 0sim \mathbb{P}_{0}, \quad p_{i}\mid h_{i} = 1sim \mathbb{P}_{1},\]
where $\mathbb{P}_{0}$ has a non-decreasing density. Recall Proposition \ref{prop:density} that
\[\mathbb{E}_{0} [h(p_{i})\mid g(p_{i})]\le 1\,\, \mbox{almost surely}.\]
Denote by $E_{0}$ (resp. $G_{0}$) and $\mathbb{E}_{1}$ (resp. $G_{1}$) the expectation (resp. distribution) given $h_{i} = 0$ and $h_{i} = 1$ respectively. The following lemma yields the form of $F(t), F_{0}(t), F_{1}(t), H(t)$.
\begin{lemma}\label{lem:uniform}
Assume that $p_{i}$'s are independent. Then
\[F_{0}(t) = \lim_{n\rightarrow \infty}\frac{1}{n}sum_{i=1}^{n}I(h_{i} = 0)\mathbb{P}_{0}(T_{i}\le t), \quad F_{1}(t) = \lim_{n\rightarrow \infty}\frac{1}{n}sum_{i=1}^{n}I(h_{i} = 1)\mathbb{P}_{1}(T_{i}\le t)\]
and
\[F(t) = F_{0}(t) + F_{1}(t), \quad H(t) = F_{0}(t) + \lim_{n\rightarrow \infty}\frac{1}{n}sum_{i=1}^{n}I(h_{i} = 1)\mathbb{E}_{1}h(p_{i})I(T_{i}\le t).\]
\end{lemma}
subsection{Re-analysis of accumulation tests}
The accumulation test of \citet{li2016accumulation} corresponds to the choice $T_{i} = i / n$. In this case,
\[F_{1}(t) = \lim_{n\rightarrow\infty}\frac{1}{n}sum_{i=1}^{\lfloor nt\rfloor}I(h_{i} = 1) = \lim_{n\rightarrow\infty}\frac{\lfloor nt\rfloor}{n}\frac{1}{\lfloor nt\rfloor}sum_{i=1}^{\lfloor nt\rfloor}I(h_{i} = 1) = tf(t).\]
Similarly,
\[F_{0}(t) = t(1 - f(t)), \quad F(t) = t.\]
Let $\mu = \mathbb{E}_{1}h(p_{i})$,
\[H(t) = F_{0}(t) + \lim_{n\rightarrow\infty}\frac{1}{n}sum_{i=1}^{\lfloor nt\rfloor}I(h_{i} = 1)\mu = t(1 - f(t)) + \mu tf(t) = t(1 - (1 - \mu)f(t)).\]
By definition \eqref{eq:tstar},
\begin{align}
t_{\mathrm{AT}}^{*} &= sup\{t\in [0, 1]: H(t)\le \alpha F(t)\} = sup\{t\in [0, 1]: t(1 - (1 - \mu)f(t))\le \alpha t\}\nonumber\\
& = sup\left\{t\in [0, 1]: f(t) \ge \frac{1 - \alpha}{1 - \mu}\right\}\label{eq:tstarAT}.
\end{align}
Note that $t_{\mathrm{AT}}^{*} > 0$ only if $\mu\ge \alpha$ since $f(t)\le 1$ for any $t\in [0,1]$. If $f(t)$ is non-increasing and $tf(t)$ is non-decreasing, as considered in \cite{li2016accumulation}, it is easy to verify the conditions \eqref{eq:tstar0} and \eqref{eq:tstar1} in Lemma \ref{lem:thattstar}. Thus, by Theorem \ref{thm:FDR_power}, if $t_{\mathrm{AT}}^{*} > 0$,
\begin{align}
\textnormal{FDR}_{n}&\rightarrow \frac{F_{0}(t_{\mathrm{AT}}^{*})}{F(t_{\mathrm{AT}}^{*})} = \alpha \frac{F_{0}(t_{\mathrm{AT}}^{*})}{H(t_{\mathrm{AT}}^{*})} = \alpha \frac{t_{\mathrm{AT}}^{*}(1 - f(t_{\mathrm{AT}}^{*}))}{t_{\mathrm{AT}}^{*}(1 - f(t_{\mathrm{AT}}^{*})) + \mu t_{\mathrm{AT}}^{*}f(t_{\mathrm{AT}}^{*})}\nonumber\\
& = \alpha \frac{1 - f(t_{\mathrm{AT}}^{*})}{1 - (1 - \mu)f(t_{\mathrm{AT}}^{*})} = \frac{\alpha - \mu}{1 - \mu} = \alpha - \frac{1 - \alpha}{1 - \mu}\mu. \label{eq:FDR_AT}
\end{align}
Thus the conservatism of false discovery rate is $\frac{1 - \alpha}{1 - \mu}\mu$, which is decreasing in $\mu$. Similarly,
\begin{equation}
\label{eq:power_AT}
\textnormal{Pow}_{n}\rightarrow \frac{F_{1}(t_{\mathrm{AT}}^{*})}{F_{1}(1)} = \frac{t_{\mathrm{AT}}^{*}f(t_{\mathrm{AT}}^{*})}{f(1)}.
\end{equation}
This recovers Theorem 3 of \cite{li2016accumulation}. Since $tf(t)$ is non-decreasing, the asymptotic power is non-decreasing in $t_{\mathrm{AT}}^{*}$ and is thus non-increasing in $\mu$. Therefore, when $\mu = \mathbb{E}_{1}[h(p_{i})]$ becomes smaller, the conservatism of false discovery rate is reduced while the power is enhanced. By Lemma 2 of \cite{li2016accumulation}, $h(t) = I(p > p_{*}) / (1 - p_{*})$ yields the smallest $\mu$ among all functions bounded by $1 / (1 - p_{*})$.
subsection{Analysis of our method without informative pre-ordering}
In most applications, an informative pre-ordering is not available. A typical two-group model assumes that $h_{i}stackrel{i.i.d.}{sim}\mathrm{Ber}(\pi_{1})$. In this case,
\[f(t)\equiv \pi_{1}.\]
Therefore, as observed by \cite{lei2016power}, the plain accumulation test either has zero power or full power since
\[t_{\mathrm{AT}}^{*} = I\left(\pi_{1}\ge \frac{1 - \alpha}{1 - \mu}\right).\]
Now we investigate our method with canonical score $T_{i} = g(p_{i})$. Without loss of generality we assume $g([0,1]) subseteq [0,1]$; otherwise we can compose it with a strictly monotone transformation to map it to the unit interval. It is not hard to see that
\[F_{1}(t) = \pi_{1}\tau_{g1}(t), \quad F_{0}(t) = (1 - \pi_{1})\tau_{g0}(t), \quad F(t) = F_{0}(t) + F_{1}(t)\]
where
\[\tau_{g1}(t) = \mathbb{P}_{1}(g(p_{i})\le t), \quad \tau_{g0}(t) = \mathbb{P}_{0}(g(p_{i})\le t),\]
and
\[H(t) = F_{0}(t) + \pi_{1}\mu_{g}(t)\]
where
\[\mu_{g}(t) = \mathbb{E}_{1}h(p_{i})I(g(p_{i})\le t).\]
By definition \eqref{eq:tstar},
\begin{align*}
t^{*} &= sup\{t\in [0, 1]: (1 - \pi_{1})\tau_{g0}(t) + \pi_{1}\mu_{g}(t)\le \alpha ((1 - \pi_{1})\tau_{g0}(t) + \pi_{1}\tau_{g1}(t))\}\\
& = sup\{t\in [0, 1]: (1 - \alpha)(1 - \pi_{1})\tau_{g0}(t)+ \pi_{1}\mu_{g}(t) \le \alpha\pi_{1}\tau_{g1}(t)\}\\
& = sup\left\{t\in [0, 1]: \pi_{1}\ge \frac{(1 - \alpha)\tau_{g0}(t)}{(1 - \alpha)\tau_{g0}(t) + \alpha\tau_{g1}(t) - \mu_{g}(t)}\right\}
\end{align*}
In the regime where the plain accumulation test has full power, i.e. $\pi_{1}\ge (1 - \alpha) / (1 - \mu)$ and $t_{\mathrm{AT}}^{*} = 1$, it is easy to see that our method has full power as well because $\tau_{g0}(1) = \tau_{g1}(1) = 1$ and $\mu_{g}(1) = \mathbb{E}_{1}h(p_{i}) = \mu$. In the regime where the plain accumulation test has zero power, i.e. $\pi_{1} < (1 - \alpha) / (1 - \mu)$ and $t_{\mathrm{AT}}^{*} = 0$, our method may still have non-zero power if $\tau_{g1}(t) >\!\!> \tau_{g0}(t), \mu_{g}(t)$ for some $t$. For instance, when $h(p) = I(p > p_{*}) / (1 - p_{*})$ and $g(p) = \min\{p, p_{*} (1 - p) / (1 - p_{*})\}$ as described in Subsection \ref{subapp:masking_functions},
\[\tau_{g0}(t) = \mathbb{P}_{0}(p_{i}\le t) + \mathbb{P}_{0}(p_{i}\ge 1 - (1 - p_{*}) t / p_{*} ) = t + \frac{1 - p_{*}}{p_{*}}t = \frac{t}p_{*}\]
and
\[\tau_{g1}(t) = \mathbb{P}_{1}(p_{i}\le t) + \mathbb{P}_{1}(p_{i}\ge 1 - (1 - p_{*}) t / p_{*} ) = G_{1}(t) + 1 - G_{1}\left( 1 - \frac{1 - p_{*}}{p_{*}}t\right).\]
On the other hand, since $g(p)\le p_{*}$ for all $p\in [0, 1]$, we can assume $t\le p_{*}$ as well. Then
\begin{align*}
\mu_{g}(t) &= \mathbb{E}_{1}h(p_{i})I(p_{i}\le t) + \mathbb{E}_{1}h(p_{i})I\left( p_{i}\ge 1 - \frac{1 - p_{*}}{p_{*}}t\right)\\
& = \mathbb{E}_{1}h(p_{i})I\left( p_{i}\ge 1 - \frac{1 - p_{*}}{p_{*}}t\right) = \frac{1 - G_{1}\left( 1 - \frac{1 - p_{*}}{p_{*}}t\right)}{1 - p_{*}}.
\end{align*}
Assume that $G_{1}$ has density $g_{1}$. Then
\begin{equation}
\label{eq:preorderinglimit}
\lim_{t\rightarrow 0}\frac{\tau_{g0}(t)}{t} = \frac{1}{p_{*}}, \quad \lim_{t\rightarrow 0}\frac{\tau_{g1}(t)}{t} = g_{1}(0) + \frac{1 - p_{*}}{p_{*}}g_{1}(1), \quad \lim_{t\rightarrow 0}\frac{\mu_{g}(t)}{t} = \frac{g_{1}(1)}{p_{*}}.
\end{equation}
Therefore,
\begin{align*}
\lim_{t\rightarrow 0}\frac{(1 - \alpha)\tau_{g0}(t)}{(1 - \alpha)\tau_{g0}(t) + \alpha\tau_{g1}(t) - \mu_{g}(t)} &= \frac{1 - \alpha}{1 - \alpha + \alpha (p_{*} g_{1}(0) + (1 - p_{*})g_{1}(1)) - g_{1}(1)}.
\end{align*}
Thus if $g_{1}(0)$ is sufficiently large, this limit is below $\pi_{1}$, implying that $t^{*} > 0$ and hence a non-zero asymptotic power by Theorem \ref{thm:FDR_power}. In fact, $g_{1}(0) = \infty$ for both examples in Section \ref{subapp:EM}.
subsection{Analysis of our method with informative pre-ordering}
When the pre-ordering is indeed informative in the sense that $f(t)$ is decreasing with $f(1) < (1 - \alpha)/ (1 - \mu)$ so that $t_{\mathrm{AT}}^{*} > 0$. We show that using the masking function $g(p_{i})$ can still improve the power. For illustration, we consider the score
\begin{equation}
\label{eq:AT_dominate}
T_{i} = \min\left\{\frac{i}{n}, \frac{g(p_{i})}{b}\right\}
\end{equation}
for some $b \ge 0$. The plain accumulation test is a special case with $b = 0$. Intuitively, this method not only rejects the first $\lfloor n t^{*}\rfloor$ hypotheses as in accumulation tests but also the remaining ones with tiny p-values.
Before analyzing this procedure rigorously in theory, we illustrate it using a simple simulation. Suppose the p-values are computed from one-sided z-tests, i.e.
\[p_{i} = 1 - \mathbb{P}hi(z_{i})\]
where $\mathbb{P}hi$ is the cumulative distribution function of a standard normal distribution, and $z_{i}$'s are independently generated from normal mixture models with unit variance, i.e.
\[z_{i}sim \pi_{i1}N(0, 1) + (1 - \pi_{i1})N(3, 1).\]
We consider the case where
\[\pi_{i1} = Ci^{-1/2},\]
where $C$ is a constant governing the proportion of non-nulls. In this case, the ordering is fully informative as $\pi_{i1}$ is strictly decreasing. We simulate the FDP ~and the power for both accumulation tests with $h(p) = 2I(p\ge 0.5)$ and our method with the same accumulation function and scores \eqref{eq:AT_dominate} with $b = 0.01$. The number of hypotheses is chosen as $20000$ and the constant $C$ is chosen from $\{0.5, 1, 2, 5\}$. For each setting the FDP ~and the power are recorded for $10000$ independent replicates. The box-plots are displayed in Figure \ref{fig:AT_dominate}. The advantage of using masking functions is clear: it reduces the variability of FDP ~while enhances the power significantly.
\begin{figure}
\caption{Simulation study comparing accumulation tests and our method using the score \eqref{eq:AT_dominate}
\label{fig:AT_dominate}
\end{figure}
Now we show that for appropriately chosen positive $b$, the asymptotic power is higher. By Lemma \ref{lem:uniform},
\begin{align*}
F_{1}(t) &= \lim_{n\rightarrow \infty}\frac{1}{n}sum_{i=1}^{\lfloor nt\rfloor}I(h_{i} = 1) + \frac{1}{n}sum_{i > \lfloor nt \rfloor}I(h_{i} = 1)\mathbb{P}_{1}(g(p_{i})\le bt)\\
& = tf(t) + (f(1) - tf(t))\tau_{g1}(bt),
\end{align*}
\begin{align*}
F_{0}(t) &= \lim_{n\rightarrow \infty}\frac{1}{n}sum_{i=1}^{\lfloor nt\rfloor}I(h_{i} = 0) + \frac{1}{n}sum_{i > \lfloor nt \rfloor}I(h_{i} = 0)\mathbb{P}_{0}(g(p_{i})\le bt)\\
& = t(1 - f(t)) + (1 - f(1) - t(1 - f(t)))\tau_{g0}(bt),
\end{align*}
and
\begin{align*}
H(t)& = \lim_{n\rightarrow \infty}\frac{1}{n}sum_{i=1}^{\lfloor nt\rfloor}I(h_{i} = 0)\mathbb{E}_{0}h(p_{i}) + \frac{1}{n}sum_{i=1}^{\lfloor nt\rfloor}I(h_{i} = 1)\mathbb{E}_{1}h(p_{i})\\
&\quad + \frac{1}{n}sum_{i > \lfloor nt \rfloor}I(h_{i} = 0)\mathbb{E}_{0}h(p_{i})I(g(p_{i})\le bt) + \frac{1}{n}sum_{i > \lfloor nt \rfloor}I(h_{i} = 1)\mathbb{E}_{1}h(p_{i})I(g(p_{i})\le bt)\\
& = \lim_{n\rightarrow \infty}\frac{1}{n}sum_{i=1}^{\lfloor nt\rfloor}I(h_{i} = 0) + \frac{1}{n}sum_{i=1}^{\lfloor nt\rfloor}I(h_{i} = 1)\mu\\
&\quad + \frac{1}{n}sum_{i > \lfloor nt \rfloor}I(h_{i} = 0)\mathbb{P}_{0}(g(p_{i})\le bt) + \frac{1}{n}sum_{i > \lfloor nt \rfloor}I(h_{i} = 1)\mathbb{E}_{1}h(p_{i})I(g(p_{i})\le bt)\\
& = F_{0}(t) + tf(t)\mu + (f(1) - tf(t))\mu_{g}(bt),
\end{align*}
where the second equality uses the fact that $\mathbb{E}_{0} h(p_{i}) = 1, \mathbb{E}_{1} h(p_{i}) = \mu$ and
\[\mathbb{E}_{0}h(p_{i})I(g(p_{i})\le bt) = \mathbb{E}_{0}[I(g(p_{i})\le bt)\mathbb{E}_{0}[h(p_{i})\mid g(p_{i})]] = \mathbb{E}_{0}[I(g(p_{i})\le bt)] \mathbb{P}_{0}(g(p_{i})\le bt).\]
By definition \eqref{eq:tstar},
\[t^{*} = sup\left\{t: H(t) \le \alpha (F_{0}(t) + F_{1}(t))\right\}.\]
Note that for any given $t$, by definition this method rejects no less than the plain accumulation test with same $t$. For this reason, this method is more powerful asymptotically if $t^{*} > t_{\mathrm{AT}}^{*}$. If $\tau_{g1}, \tau_{g0}, \mu_{g}$ are all continuous, then it is left to show that
\[H(t_{\mathrm{AT}}^{*}) < \alpha (F_{0}(t_{\mathrm{AT}}^{*}) + F_{1}(t_{\mathrm{AT}}^{*})).\]
By some algebra and the fact that $f(t_{\mathrm{AT}}^{*}) = (1 - \alpha) / (1 - \mu)$, this is equivalent to
\[\frac{(1 - t_{\mathrm{AT}}^{*})\tau_{g0}(bt_{\mathrm{AT}}^{*}) + (f(1) - t_{\mathrm{AT}}^{*}f(t_{\mathrm{AT}}^{*}))(\mu_{g}(bt_{\mathrm{AT}}^{*}) - \tau_{g0}(bt_{\mathrm{AT}}^{*}))}{(1 - t_{\mathrm{AT}}^{*})\tau_{g0}(bt_{\mathrm{AT}}^{*}) + (f(1) - t_{\mathrm{AT}}^{*}f(t_{\mathrm{AT}}^{*}))(\tau_{g1}(bt_{\mathrm{AT}}^{*}) - \tau_{g0}(bt_{\mathrm{AT}}^{*}))} < \alpha.\]
For instance, when $h(p) = I(p > p_{*}) / (1 - p_{*})$ and $g(p) = \min\{p, p_{*} (1 - p) / (1 - p_{*})\}$ as in last subsection, by \eqref{eq:preorderinglimit},
\begin{align*}
&\lim_{b\rightarrow 0}\frac{(1 - t_{\mathrm{AT}}^{*})\tau_{g0}(bt_{\mathrm{AT}}^{*}) + (f(1) - t_{\mathrm{AT}}^{*}f(t_{\mathrm{AT}}^{*}))(\mu_{g}(bt_{\mathrm{AT}}^{*}) - \tau_{g0}(bt_{\mathrm{AT}}^{*}))}{(1 - t_{\mathrm{AT}}^{*})\tau_{g0}(bt_{\mathrm{AT}}^{*}) + (f(1) - t_{\mathrm{AT}}^{*}f(t_{\mathrm{AT}}^{*}))(\tau_{g1}(bt_{\mathrm{AT}}^{*}) - \tau_{g0}(bt_{\mathrm{AT}}^{*}))} \\
& = \frac{(1 - t_{\mathrm{AT}}^{*}) + (f(1) - t_{\mathrm{AT}}^{*}f(t_{\mathrm{AT}}^{*}))(g_{1}(1) - 1)}{(1 - t_{\mathrm{AT}}^{*}) + (f(1) - t_{\mathrm{AT}}^{*}f(t_{\mathrm{AT}}^{*}))(p_{*} g_{1}(0) + (1 - p_{*})g_{1}(1) - 1)}.
\end{align*}
Thus the limit is strictly below $\alpha$ if $g_{1}(0)$ is sufficiently large. As commented at the end of last subsection, $g_{1}(0) = \infty$ in many applications.
Therefore, even with an informative pre-ordering, using the masking function may further improve the power. This inspires an interesting question on how to combine the pre-ordering and the masked p-values in an optimal way to enhance power. However, this is beyond the main focus of this paper and we leave it to future research.
subsection{Proofs}\label{subapp:power_proof}
\begin{proof}[of Lemma \ref{lem:thattstar}]
Let $H_{\alpha n}(t) = H_{n}(t) - \alpha F_{n}(t)$ and
\[\delta_{n} = sup_{t\in [0, 1]}|H_{\alpha n}(t) - H_{\alpha}(t)|.\]
By definition,
\[\hat{t}_{n} = sup\{t: H_{\alpha n}(t)\le 0\}, \quad t^{*} = sup\{t: H_{\alpha}(t)\le 0\}.\]
Then \eqref{eq:uniform_convergence} implies that $\delta_{n}stackrel{p}{\rightarrow}0$. We prove for each case separately.
\begin{enumerate}
\item For any ${\varepsilon}arepsilonsilon > 0$,
\[\inf_{t > {\varepsilon}arepsilonsilon}H_{\alpha n}(t) \ge\inf_{t > {\varepsilon}arepsilonsilon} H_{\alpha}(t) - \delta_{n}.\]
By \eqref{eq:tstar0},
\[\mathbb{P}\left( \inf_{t > {\varepsilon}arepsilonsilon}H_{\alpha n}(t) > 0\right)\rightarrow 1.\]
As a result,
\[\mathbb{P}(\hat{t}_{n}\le {\varepsilon}arepsilonsilon)\rightarrow 1.\]
Since this holds for any ${\varepsilon}arepsilonsilon$, we conclude that $\hat{t}_{n}stackrel{p}{\rightarrow} 0$.
\item By \eqref{eq:uniform_convergence}, for any $m$
\[H_{\alpha n}(t_{m}) \le H_{\alpha}(t_{m}) + \delta_{n}.\]
By \eqref{eq:tstar1},
\[\mathbb{P}(H_{\alpha n}(t_{m})\le 0)\rightarrow 1.\]
This entails that
\[\mathbb{P}(\hat{t}_{n}\ge t_{m})\rightarrow 1.\]
Since this holds for all $m$ and $t_{m}\uparrow 1$, we arrive at $\hat{t}_{n}stackrel{p}{\rightarrow} 1$.
\item Let $t_{m}'$ be any sequence that $t_{m}'\downarrow t^{*}$. For any $m$,
\[H_{\alpha n}(t_{m})\le H_{\alpha}(t_{m}) + \delta_{n}\]
and
\[\inf_{t > t_{m}'}H_{\alpha n}(t) > \inf_{t > t_{m}'}H_{\alpha}(t) - \delta_{n}.\]
By \eqref{eq:tstar0} and \eqref{eq:tstar1}, we have
\[\mathbb{P}\left( H_{\alpha n}(t_{m}) < 0, \,\, \inf_{t > t_{m}'}H_{\alpha n}(t) > 0\right)\rightarrow 1.\]
This implies that
\[\mathbb{P}(\hat{t}_{n}\in [t_{m}, t_{m}'])\rightarrow 1.\]
Since $t_{m}\uparrow t^{*}$ and $t_{m}'\downarrow t^{*}$, we conclude that $\hat{t}_{n}stackrel{p}{\rightarrow}t^{*}$.
\end{enumerate}
\end{proof}
\begin{proof}[of Theorem \ref{thm:FDR_power}]
Note that $F_{n0}(t), F_{n1}(t), F_{n}(t), F_{0}(t), F_{1}(t), F(t)$ are all non-decreasing functions. Let $t_{m}$ and $t_{m}'$ be any two sequences such that $t_{m}\uparrow t^{*}, t_{m}'\downarrow t^{*}$. Let $\mathcal{E}_{n, m}$ denotes the event that $\hat{t}_{n}\in [t_{m}, t_{m}']$. Then Lemma \ref{lem:thattstar} implies that $\mathbb{P}(\mathcal{E}_{n, m})\rightarrow 1$ for each $m$.
First we prove that $\textnormal{FDP}_{n}stackrel{p}{\rightarrow} F_{0}(t^{*}) / F(t^{*})$ when $t^{*} > 0$. Without loss of generality we assume that $t_{m} > 0$. On event $\mathcal{E}_{n, m}$,
\[\textnormal{FDP}_{n}\in \left[\frac{F_{n0}(t_{m})}{F_{n}(t_{m}')}, \min\left\{1, \frac{F_{n0}(t_{m}')}{F_{n}(t_{m})}\right\}\right], \]
By \eqref{eq:Fn01}, Slusky's theorem and the fact that $F(t_{m}) > 0$,
\[\frac{F_{n0}(t_{m})}{F_{n}(t_{m}')}stackrel{p}{\rightarrow} \frac{F_{0}(t_{m})}{F(t_{m}')}, \quad \frac{F_{n0}(t_{m}')}{F_{n}(t_{m})}stackrel{p}{\rightarrow} \frac{F_{0}(t_{m}')}{F(t_{m})}.\]
Thus, for any ${\varepsilon}arepsilonsilon > 0$,
\[\mathbb{P}\left( \textnormal{FDP}_{n}I_{\mathcal{E}_{n, m}}\in \left[\frac{F_{0}(t_{m})}{F(t_{m}')} - {\varepsilon}arepsilonsilon, \frac{F_{0}(t_{m}')}{F(t_{m})} + {\varepsilon}arepsilonsilon\right]\right)\rightarrow 1.\]
Since $\mathbb{P}(\mathcal{E}_{n, m})\rightarrow 1$,
\[\mathbb{P}\left( \textnormal{FDP}_{n}\in \left[\frac{F_{0}(t_{m})}{F(t_{m}')} - {\varepsilon}arepsilonsilon, \frac{F_{0}(t_{m}')}{F(t_{m})} + {\varepsilon}arepsilonsilon\right]\right)\rightarrow 1.\]
Since $t_{m}\rightarrow t^{*}, t_{m}' \rightarrow t^{*}$ and $F_{0}, F$ are both continuous at $t^{*}$,
\[\mathbb{P}\left( \bigg|\textnormal{FDP}_{n} - \frac{F_{0}(t^{*})}{F(t^{*})}\bigg|\le {\varepsilon}arepsilonsilon\right)\rightarrow 0.\]
This holds for arbitrary ${\varepsilon}arepsilonsilon > 0$. Thus,
\begin{equation}
\label{eq:FDPn}
\textnormal{FDP}_{n}stackrel{p}{\rightarrow} \frac{F_{0}(t^{*})}{F(t^{*})}.
\end{equation}
Since $\textnormal{FDP}_{n}\in [0, 1]$, \eqref{eq:FDPn} implies the convergence in $L_{1}$, i.e.
\[\textnormal{FDR}_{n} = \mathbb{E} [\textnormal{FDP}_{n}]\rightarrow \frac{F_{0}(t^{*})}{F(t^{*})}.\]
For the asymptotic power, the monotonicity of $F_{n1}$ implies that
\[\textnormal{TPR}_{n} \in \left[\frac{F_{n1}(t_{m})}{F_{n1}(1)}, \frac{F_{n1}(t_{m}')}{F_{n1}(1)}\right].\]
Using the same argument as above, we have
\[\textnormal{TPR}_{n}stackrel{p}{\rightarrow} \frac{F_{1}(t^{*})}{F_{1}(1)}.\]
Since $\textnormal{TPR}_{n}\in [0, 1]$, this implies
\[\textnormal{Pow}_{n}\rightarrow \frac{F_{1}(t^{*})}{F_{1}(1)}.\]
\end{proof}
\begin{proof}[of Lemma \ref{lem:uniform}]
Since $p_{i}$'s are independent, $T_{i}$'s are independent. Note that $F_{n0}(t), F_{n1}(t), F_{n}(t), H_{n}(t)$ can be in the form of
\[\frac{1}{n}sum_{i=1}^{n}m_{i}(p_{i})I(T_{i}\le t).\]
for some deterministic bounded functions $m_{1}, \ldots, m_{n}$. Let $B$ denote the bound of $m_{i}$'s. Then $B = 1$ for $F_{n0}(t), F_{n1}(t), F_{n}(t)$ and $B = h(1)$ for $H_{n}(t)$. Let $f_{i}(p_{i}; t) = m_{i}(p_{i})I(T_{i}\le t)$. Then $F_{i}\equiv B$ is an upper envelop of $f_{i}$. Also, for any given $(p_{1}, \ldots, p_{n})$, by Sauer's lemma,
\[\#\{(f_{1}(p_{1}; t), \ldots, f_{n}(p_{n}; t)): t\in [0, 1]\} = \#\{(I(T_{1}\le t), \ldots, I(T_{n}\le t)): t\in [0, 1]\}\le n + 1.\]
This implies that
\[\log \#\{(f_{1}(p_{1}; t), \ldots, f_{n}(p_{n}; t)): t\in [0, 1]\} = O(\log n) = o(n).\]
By Theorem 8.2 of \cite{pollard1990empirical},
\[sup_{t\in [0, 1]}\bigg|\frac{1}{n}sum_{i=1}^{n}(f_{i}(p_{i}, t) - \mathbb{E} f_{i}(p_{i}, t))\bigg| = o_{p}(1).\]
It is easy to compute $\mathbb{E} F_{n0}(t), \mathbb{E} F_{n1}(t)$ and $\mathbb{E} F_{n}(t)$. For $\mathbb{E} H_{n}(t)$, recalling that $\mathbb{E}_{0} [h(p_{i})\mid g(p_{i})] \le 1$ almost surely and $T_{i}$ depends on $p_{i}$ through $g(p_{i})$,
\begin{align*}
\mathbb{E} H_{n}(t) &= \frac{1}{n}sum_{i=1}^{n}\mathbb{E} [h(p_{i})I(T_{i}\le t)]\\
& = \frac{1}{n}sum_{i=1}^{n}I(h_{i} = 0)\mathbb{E}_{0} [h(p_{i})I(T_{i}\le t)] + \frac{1}{n}sum_{i=1}^{n}I(h_{i} = 1)\mathbb{E}_{1} [h(p_{i})I(T_{i}\le t)]\\
& = \frac{1}{n}sum_{i=1}^{n}I(h_{i} = 0)\mathbb{P}_{0}(T_{i}\le t) + \frac{1}{n}sum_{i=1}^{n}I(h_{i} = 1)\mathbb{E}_{1} [h(p_{i})I(T_{i}\le t)]
\end{align*}
\end{proof}
\end{document} | math | 124,745 |
\begin{document}
\title{Singularity of sparse Bernoulli matrices}
\begin{abstract}
Let $M_n$ be an $n\times n$ random matrix with i.i.d.\ Bernoulli($p$) entries. We show that there is a universal constant
$C\geq 1$ such that, whenever $p$ and $n$ satisfy $C\log n/n\leq p\leq C^{-1}$,
\begin{align*}
{\mathbb P}\big\{\mbox{$M_n$ is singular}\big\}&=(1+o_n(1)){\mathbb P}\big\{\mbox{$M_n$ contains a zero row or column}\big\}\\
&=(2+o_n(1))n\,(1-p)^n,
\mathcal{E}_{n-1}d{align*}
where $o_n(1)$ denotes a quantity which converges to zero as $n\to\infty$.
We provide the corresponding upper and lower bounds on the smallest singular value of $M_n$ as well.
\mathcal{E}_{n-1}d{abstract}
{\small
\|\cdot\|indent{\bf AMS 2010 Classification:}
primary: 60B20, 15B52;
secondary: 46B06, 60C05.\\
\|\cdot\|indent
{\bf Keywords: }
Littlewood--Offord theory,
Bernoulli matrices, sparse matrices, smallest singular value, invertibility}
\tableofcontents
\section{Introduction}
\left\langlebel{s:intro}
Invertibility of discrete random matrices attracts considerable attention in the literature.
The classical problem in this direction --- estimating the singularity probability of a square
random matrix $B_n$ with i.i.d.\ $\mathbb{P}m 1$ entries --- was first addressed by Koml\'os
in the 1960-es. Koml\'os \cite{Komlos} showed that ${\mathbb P}\{\mbox{$B_n$ is singular}\}$
decays to zero as the dimension grows to infinity. A breakthrough result of Kahn--Koml\' os--Szemer\' edi
\cite{KKS95} confirmed that the singularity probability of $B_n$ is exponentially small in dimension.
Further improvements on the singularity probability were obtained by Tao--Vu \cite{TV disc1, TV bernoulli}
and Bourgain--Vu--Wood \cite{BVW}.
An old conjecture states that ${\mathbb P}\{\mbox{$B_n$ is singular}\}=\big({\mathcal F}rac{1}{2}+o_n(1)\big)^n$.
The conjecture was resolved in \cite{KT-last}.
Other models of non-symmetric discrete random matrices considered in the literature include
adjacency matrices of $d$-regular digraphs,
as well as the closely related model of sums of independent uniform permutation matrices
\cite{LSY, Cook adjacency, Cook circular, LLTTY-cras, LLTTY:15, LLTTY first part, LLTTY third part,
LLTTY rank n-1, BCZ}.
In particular, the recent breakthrough works \cite{Huang, Mez, NW} confirmed that the adjacency
matrix of a uniform random $d$--regular digraph of a constant degree $d\geq 3$ is non-singular with probability
decaying to zero as the number of vertices of the graph grows to infinity.
A closely related line of research deals with the rank of random matrices over finite fields.
We refer to \cite{LMN} for some recent results and further references.
The development of the {\it Littlewood--Offord theory}
and a set of techniques of geometric functional analysis reworked in the random matrix context,
produced strong invertibility results for a broad class of distributions. Following works \cite{TV ann math,R-ann}
of Tao--Vu and Rudelson,
the paper \cite{RV} of Rudelson and Vershynin established optimal small ball probability estimates for the
smallest singular value in the class of square matrices with i.i.d.\ subgaussian entries,
namely, it was shown that any $n\times n$ matrix $A$ with i.i.d.\ subgaussian entries
of zero mean and unit variance satisfies ${\mathbb P}\{s_{\min}(A)\leq t\,n^{-1/2}\}\leq Ct+2\mathcal{E}xp(-cn)$
for all $t>0$ and some $C,c>0$ depending only on the subgaussian moment.
The assumptions of identical distribution of entries and of bounded subgaussian moment
were removed in subsequent works \cite{RT,Livshyts,LTV}.
This line of research lead to positive solution of the Bernoulli matrix conjecture mentioned in the first paragraph.
Let us state the result of \cite{KT-last} for future reference.
\begin{theor*}[Invertibility of dense Bernoulli matrices, \cite{KT-last}]\hspace{0cm}
\begin{itemize}
\item For each $n$, let $B_n$ be the $n\times n$ random matrix with i.i.d.\ $\mathbb{P}m 1$ entries.
Then for any $\varepsilon>0$ there is $C$ depending only on $\varepsilon$ such that
the smallest singular value $s_{\min}(B_n)$ satisfies
$$
{\mathbb P}\big\{s_{\min}(B_n)\leq t n^{-1/2}\big\}\leq Ct+C(1/2+\varepsilon)^n,\quad t>0.
$$
In particular, ${\mathbb P}\big\{\mbox{$B_n$ is singular}\big\}=(1/2+o_n(1))^n$,
where the quantity $o_n(1)$ tends to zero as $n$ grows to infinity.
\item
For each $\varepsilon>0$ and $p\in(0,1/2]$ there is $C>0$ depending on $\varepsilon$ and $p$
such that for any $n$ and for random $n\times n$ matrix $M_n$ with i.i.d.\ Bernoulli($p$) entries,
$$
{\mathbb P}\big\{s_{\min}(M_n)\leq t n^{-1/2}\big\}\leq Ct+C(1-p+\varepsilon)^n,\quad t>0.
$$
In particular, for a fixed $p\in(0,1/2]$, we have ${\mathbb P}\big\{\mbox{$M_n$ is singular}\big\}=(1-p+o_n(1))^n$.
\mathcal{E}_{n-1}d{itemize}
\mathcal{E}_{n-1}d{theor*}
Sparse analogs of the Rudelson--Vershynin invertibility theorem \cite{RV} were obtained, in particular, in works
\cite{TV sparse, GT, LiRi, BasRud, BasRud circ, BasRud-sharp}, with the strongest small ball probability estimates
in the i.i.d.\ subgaussian setting available in \cite{BasRud, BasRud circ, BasRud-sharp}.
Here, we state a result of Basak--Rudelson \cite{BasRud} for
Bernoulli($p_n$) random matrices.
\begin{theor*}[Invertibility of sparse Bernoulli matrices, \cite{BasRud}]
There are universal constants $C,c>0$ with the following property. Let $n\in{\mathbb N}$ and let
$p_n\in(0,1)$ satisfy $C\log n/n\leq p_n\leq 1/2$. Further, let $M_n$ be the
random $n\times n$ matrix with i.i.d. Bernoulli($p_n$) entries (that is, $0/1$ random variables
with expectation $p$). Then
\begin{align*}
{\mathbb P}\big\{s_{\min}(M_n)\leq t\,\mathcal{E}xp\big(-C\log(1/p_n)/\log(np_n)\big)\,\sqrt{p_n/n}\big\}\leq Ct+2\mathcal{E}xp(-cnp_n),\quad t>0.
\mathcal{E}_{n-1}d{align*}
\mathcal{E}_{n-1}d{theor*}
The singularity probabilities implied by the results \cite{KT-last,BasRud}
may be regarded as suboptimal
in a certain respect. Indeed, while \cite{KT-last} produced an
asymptotically sharp base of the power in the singularity probability
of $B_n$,
the estimate of \cite{KT-last}
is off by a factor $(1+o_n(1))^n$ which may (and in fact does, as analysis of the proof shows)
grow to infinity with $n$ superpolynomially fast.
Further, the upper bound on the singularity probability of sparse Bernoulli matrices implied by \cite{BasRud}
captures an exponential dependence on $n p_n$, but does not recover an asymptotically optimal base of the power.
A folklore conjecture for matrices $B_n$ asserts that
${\mathbb P}\{\mbox{$B_n$ is singular}\}=(1+o_n(1))n^2 2^{1-n}$,
where the right hand side of the expression is the probability that two rows or two columns
of the matrix $B_n$ are equal up to a sign (see, for example, \cite{KKS95}).
This conjecture can be naturally extended to the model with Bernoulli($p_n$) ($0/1$) entries as follows.
\begin{conj}[Stronger singularity conjecture for Bernoulli matrices]
For each $n$, let $p_n\in (0,1/2]$, and let $M_n$ be the $n\times n$
matrix with i.i.d.\ Bernoulli($p_n$) entries. Then
\begin{align*}
{\mathbb P}\{&\mbox{$M_n$ is singular}\}\\
&=(1+o_n(1))
{\mathbb P}\big\{\mbox{a row or a column of $M_n$ equals zero, or two rows or columns are equal}\big\}.
\mathcal{E}_{n-1}d{align*}
In particular, if $\limsup p_n<1/2$ then
\begin{align*}
{\mathbb P}\{\mbox{$M_n$ is singular}\}&=(1+o_n(1))
{\mathbb P}\big\{\mbox{either a row or a column of $M_n$ equals zero}\big\}.
\mathcal{E}_{n-1}d{align*}
\mathcal{E}_{n-1}d{conj}
Conceptually, the above conjecture asserts that the main causes for singularity are local in the sense
that the linear dependencies typically appear within small subsets of rows or columns.
In a special regime $n p_n\leq \ln n+o_n(\ln \ln n)$, the conjecture was positively
resolved in \cite{BasRud-sharp} (note that if $n p_n\leq \ln n$ then the matrix
has a zero row with probability at least $1-1/e-o_n(1)$). However,
the regime $\liminf (np_n/\log n)> 1$ was not covered in \cite{BasRud-sharp}.
The main purpose of our paper is to develop methods capable of capturing the singularity probability with a
sufficient precision to answer the above question. Interestingly, this appears to be more accessible
in the sparse regime, when $p_n$ is bounded above by a small universal constant
(we discuss this in the next section in more detail).
It is not difficult to show that when $\liminf (np_n/\ln n)>1$,
the events that a given row or a given column equals zero, almost do not intersect, so that
$$
{\mathbb P}\big\{\mbox{either a row or a column of $M_n$ equals zero}\big\}=(2+o_n(1))n\,(1-p_n)^n.
$$
Our main result can be formulated as follows.
\begin{theor}\left\langlebel{th: main}
There are universal constants $C,\widetilde C\geq 1$ with the following property.
Let $n\geq 1$ and let $M_n$ be an $n\times n$ random matrix such that
\begin{equation}\tag{{\bf{}A}}\left\langlebel{eq: assumptions}
\mbox{The entries of $M_n$ are i.i.d.\ Bernoulli($p$) , with $p=p_n$ satisfying $C\ln n\leq np\leq C^{-1}$.}
\mathcal{E}_{n-1}d{equation}
Then
$${\mathbb P}\big\{\mbox{$M_n$ is singular}\big\}=(2+o_n(1))n\,(1-p)^n, $$
where $o_n(1)$ is a quantity which tends to zero as $n\to\infty$.
Moreover, for every $t>0$,
$$
{\mathbb P}\big\{s_{\min}(M_n)\leq t\, \mathcal{E}xp(-3\ln^2(2n))\big\}\leq t+(1+o_n(1)){\mathbb P}\big\{\mbox{$M_n$ is singular}\big\}
=t+(2+o_n(1))n\,(1-p)^n.
$$
\mathcal{E}_{n-1}d{theor}
In fact, our approach gives much better estimates on $s_{\min}$ in the regime when $p_n$ is constant,
see Theorem~\rightef{const-p-th} below.
At the same time, we note that obtaining small ball probability estimates for $s_{\min}$
was not the main objective of this paper, and the argument was not fully optimized in that respect.
Geometrically, the main result of our work asserts that (under appropriate assumptions on $p_n$)
the probability that
a collection of $n$ independent random vectors $X_1^{(n)},{\rm dist}ots,X_n^{(n)}$ in ${\mathbb R}^n$,
with i.i.d Bernoulli($p_n$) components is linearly dependent is equal (up to $(1+o_n(1))$ factor)
to probability of the event that either $X_i^{(n)}$ is zero for some $i\leq n$ or $X_1^{(n)},{\rm dist}ots,X_n^{(n)}$ are contained
in the same coordinate hyperplane:
\begin{align*}
&{\mathbb P}\big\{\mbox{$X_1^{(n)},{\rm dist}ots,X_n^{(n)}$ are linearly dependent}\big\}
=(1+o_n(1))\,{\mathbb P}\big\{X_i^{(n)}={\bf 0}\mbox{ for some }i\leq n\big\}\\
&\hspace{1cm}+(1+o_n(1))\,{\mathbb P}\big\{\mathcal{E}xists\,
\mbox{ a coordinate hyperplane $H$ such that }X_i^{(n)}\in H\mbox{ for all }i\leq n\big\}.
\mathcal{E}_{n-1}d{align*}
Thus, the linear dependencies between the vectors, when they appear,
typically have the prescribed structure, falling into one of the two categories described above with the (conditional)
probability ${\mathcal F}rac{1}{2}+o_n(1)$.
The paper is organized as follows.
In the next section, we give an overview of the proof of the main result.
In Section~\rightef{preliminaries}, we gather some preliminary facts and important notions to be used later.
In Section~\rightef{s: unstructured}, we consider new anti-concentration inequalities for random $0/1$ vectors with prescribed
number of non-zero components, and introduce a functional ({\it u-degree} of a vector)
which enables us to classify vectors on the sphere according to anti-concentration properties of
inner products with the random $0/1$ vectors.
In the same section, we prove a key technical result --- Theorem~\rightef{th: gradual} --- which states, roughly speaking,
that with very high probability a random unit vector orthogonal to $n-1$ columns of $M_n$ is either close
to being sparse or to being a constant multiple of $(1,1,{\rm dist}ots,1)$, or the vector is {\it very unstructured,}
i.e., has a very large u-degree.
In Section~\rightef{steep:constant p}, we consider a special regime of constant probability of success $p$.
In this regime, estimating the event that $M_n$ has an almost null vector which is either close to sparse or almost constant,
is relatively simple. The reader who is interested only in the regime of constant $p$ can thus skip
the more technical Section~\rightef{s: steep} and have the proof of the main result as a combination of
the theorems in Sections~\rightef{s: unstructured} and~\rightef{steep:constant p}. In Section~\rightef{s: steep}, we consider the entire range for $p$.
Here, the treatment of almost constant and close to sparse null vectors is much more challenging and involves
a careful analysis of multiple cases. Finally, in Section~\rightef{s: main th} we establish an {\it invertibility via distance}
lemma and prove the main result of the paper. Some open questions are discussed in Section~\rightef{s: further}.
\section{Overview of the proof}\left\langlebel{s: overview}
In this section, we provide a high-level overview of the proof; technical details will be discussed further in the text.
The proof utilizes some known approaches to the matrix invertibility, which involve, in particular,
a decomposition of the space into {\it structured}
and {\it unstructured} parts, a form of {\it invertibility via distance} argument, small ball probability estimates based on the Esseen
lemma, and various forms of the $\varepsilon$--net argument.
The novel elements of the proof are anti-concentration inequalities for random vectors with a prescribed
cardinality of the support, a structural theorem for normals to random hyperplanes spanned by vectors with
i.i.d. Bernoulli($p$) components, and a sharp analysis of the matrix invertibility over the set of structured vectors.
We will start the description with our use of the partitioning trick, followed by a modified
invertibility-via-distance lemma, and then consider the anti-concentration inequality
and the theorem for normals (Subsection~\rightef{ss: anti-c})
as well as invertibility over the structured vectors (Subsection~\rightef{ss: steep overview}).
The use of decompositions of the
space ${\mathbb R}^n$ into structured and unstructured vectors has become rather standard in the literature.
A common idea behind such partitions is to apply the Littlewood--Offord theory to analyse the unstructured vectors
and to construct a form of the $\varepsilon$--net argument to treat the structured part.
Various definitions of structured and unstructured have been used in works dealing with the matrix invertibility.
One of such decomposition was introduced in \cite{LPRT} and further developed in \cite{RV}. In this splitting
the structured vectors are {\it compressible}, having a relatively small
Euclidean distance to the set of {\it sparse} vectors, while the vectors
in the complement are {\it incompressible}, having a large distance to sparse vectors and, as a consequence, many components
of roughly comparable magnitudes.
In our work, the decomposition of ${\mathbb R}^n$ is closer to the one introduced in \cite{LLTTY first part,LLTTY-TAMS}.
Let $x^*$ denote a non-increasing rearrangement of absolute values of components of
a vector $x$, and let $r,{\rm dist}eltalta,\rightho\in(0,1)$ be some parameters.
Further, let ${\bf g}$ be a non-decreasing function from $[1,\infty)$ into $[1,\infty)$;
we shall call it {\it the growth function}. At this moment, the choice of the growth function is not important;
we can assume that ${\bf g}(t)$ grows roughly as $t^{\ln t}$.
Define the set of {\it gradual non-constant vectors} as
\begin{align}
\left\langlebel{eq: gnc def}
{\mathcal V}_n={\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)
&:=\big\{x\in{\mathbb R}^n\, :\, x^*_{\lfloor rn\rightfloor}=1,
\;x^*_i\leq {\bf g}(n/i)\mbox{ for all $i\leq n$}, \,\,\, \mbox{ and } \|\cdot\|number
\\&\mathcal{E}xists\,
Q_1,Q_2\subset[n] \quad \mbox{ such that } \quad
|Q_1|,|Q_2|\geq {\rm dist}eltalta n \quad \mbox{ and } \quad \max\limits_{i\in Q_2} x_i\leq \min\limits_{i\in Q_1}x_i- \rightho
\big\}.
\mathcal{E}_{n-1}d{align}
In a sense, constant multiples of the gradual non-constant vectors occupy most of the space ${\mathbb R}^n$,
they play role of the unstructured vectors in our argument. By negation, the structured vectors,
\begin{equation}\left\langlebel{strvect}
{\mathcal S}_n={\mathcal S}_n(r,{\bf g},{\rm dist}eltalta,\rightho):={\mathbb R}^n\setminus \bigcup_{\left\langlembdabda\geq 0}(\left\langlembdabda \,
{\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)),
\mathcal{E}_{n-1}d{equation}
are either almost constant (with most of components nearly equal)
or have a very large ratio of $x^*_i$ and $x^*_{\lfloor rn\rightfloor}$ for some $i< rn$.
For simplicity, we only discuss the problem of singularity at this moment.
As $M_n$ and $M_n^\top$ are equidistributed, to show that
${\mathbb P}\big\{\mbox{$M_n$ is singular}\big\}=(2+o_n(1))n\,(1-p)^n,$ it is sufficient to verify that
\begin{equation}\left\langlebel{eq: aux 209582705982}
\begin{split}
{\mathbb P}\Big(&\big\{M_n x=0\mbox{ for some }x\in {\mathcal V}_n\big\}
\cap \Big\{M_n^\top x\neq 0\mbox{ for all }x\in {\mathcal S}_n \Big\}\Big)=o_n(n)\,(1-p)^n,
\mathcal{E}_{n-1}d{split}
\mathcal{E}_{n-1}d{equation}
and
$$
{\mathbb P}\Big\{M_n x=0\mbox{ for some }x\in {\mathcal S}_n
\Big\}=(1+o_n(1))n\,(1-p)^n.
$$
The first relation is dealt with by using a variation of the {\it invertibility via distance} argument
which was introduced in \cite{RV} to obtain sharp small ball probability estimates for the smallest singular value.
In the form given in \cite{RV}, the argument reduces the problem of invertibility over unstructured
vectors to estimating distances of the form ${\rm dist}({\bf C}_i(M_n),H_i(M_n))$, where
${\bf C}_i(M_n)$ is the $i$--th column of $M_n$, and
$H_i(M_n)$ is the linear
span of columns of $M_n$ except for the $i$--th. In our setting, however, the argument needs to be modified to pass to
estimating the distance {\it conditioned} on the size of the support of the column,
as this allows using much stronger anti-concentration inequalities (see the following subsection).
By the invariance of the distribution of $M_n$ under permutation of columns, it can be shown that in order to prove the
relation \mathcal{E}qref{eq: aux 209582705982}, it is enough to verify that
\begin{equation}\left\langlebel{eq: aux -8572059873}
{\mathbb P}\big\{|{\rm supp\, }{\bf C}_1(M_n)|\in [\mbox{${\mathcal F}rac{pn}{8}$}, 8pn]\mbox{ and }
\left\langlengle{\bf Y},{\bf C}_1(M_n)\rightangle=0\mbox{ and }
{\bf Y}/{\bf Y}^*_{\lfloor r n\rightfloor}\in {\mathcal V}_n
\big\}=o_n(n)\,(1-p)^n,
\mathcal{E}_{n-1}d{equation}
where ${\bf Y}$ is a non-zero random vector orthogonal to and measurable with respect to $H_1(M_n)$
(see Lemma~\rightef{l: inv via dist} and the beginning of the proof of Theorem~\rightef{th: main}).
In this form, the question can be reduced to studying the anti-concentration of the linear
combinations $\sum_{i=1}^n{\bf Y}_i b_i$, where the Bernoulli random variables $b_1,{\rm dist}ots,b_n$ are mutually independent with ${\bf Y}$
and conditioned to sum up to a fixed number in $[pn/8, 8pn]$.
This intermediate problem is discussed in the next subsection.
\subsection{New anti-concentration inequalities for random vectors with prescribed support cardinality}
\left\langlebel{ss: anti-c}
The {\it Littlewood--Offord theory} --- the study of anti-concentration properties of random variables
--- has been a crucial ingredient of many recent results on invertibility of random matrices, starting
with the work of Tao--Vu \cite{TV ann math}.
In particular, the breakthrough result \cite{RV} of Rudelson--Vershynin mentioned in the introduction, is largely based on studying
the L\'evy function
$\mathcal{Q}(\left\langlengle {\bf C}_1(A),{\bf Y}\rightangle,t)$, with ${\bf C}_1(A)$ being the first column of the random matrix $A$ and
${\bf Y}$ --- a random unit vector orthogonal to the remaining columns of $A$.
We recall that given a random vector $X$ taking values in ${\mathbb R}^n$, the {\it L\' evy concentration function}
$\mathcal{Q}(X,t)$ is defined by
$$
\mathcal{Q}(X,t):=\sup\limits_{y\in{\mathbb R}^n}{\mathbb P}\big\{\|X-y\|\leq t\big\},\quad t\geq 0;
$$
in particular for a scalar random variable $\xi$ we have $\mathcal{Q}(\xi,t):=\sup\limits_{\left\langlembdabda\in{\mathbb R}}{\mathbb P}\{|\xi-\left\langlembdabda|\leq t\}$.
A common approach is to determine structural properties of a fixed vector which would
imply desired upper bounds on the L\'evy function of its scalar product with a random vector (say, a matrix' column).
The classical result of Erd\H os--Littlewood--Offord \cite{Erdos,LO lemma} asserts that whenever $X$
is a vector in ${\mathbb R}^n$ with i.i.d.\ $\mathbb{P}m 1$ components, and $y=(y_1,{\rm dist}ots,y_n)\in{\mathbb R}^n$ is such that
$|y_i|\geq 1$ for all $i$, we have
$$
\mathcal{Q}(\left\langlengle X,y\rightangle,t)\leq Ct\,n^{-1/2}+Cn^{-1/2},
$$
where $C>0$ is a universal constant.
It can be further deduced from the L\'evy--Kolmogorov--Rogozin inequality \cite{Rog} that the above assertion remains true
whenever $X$ is a random vector with independent components $X_i$ satisfying $\mathcal{Q}(X_i,c)\leq 1-c$
for some constant $c>0$.
More delicate structural properties, based on whether components of $y$ can be embedded into a generalized
arithmetic progression with prescribed parameters were employed in \cite{TV ann math} to prove
superpolynomially small upper bounds on the singularity probability of discrete random matrices.
The Least Common Denominator (LCD) of a unit vector introduced in \cite{RV} played a central role in establishing
the exponential upper bounds on the matrix singularity under more general assumptions on the entries' distributions.
We recall that the LCD of a unit vector $y$ in ${\mathbb R}^n$ can be defined as
\begin{equation}\left\langlebel{eq: LCD overview}
{\rightm LCD}(y):=\inf\big\{\thetaeta>0:\;{\rm dist}(\thetaeta y,{\mathbb Z}^n)\leq\min(c_1\|\thetaeta y\|,c_2\sqrt{n})\big\}
\mathcal{E}_{n-1}d{equation}
for some parameters $c_1,c_2\in(0,1)$. The small ball probability theorem of Rudelson and Vershynin \cite{RV}
states that given a vector $X$ with i.i.d.\ components of zero mean and unit variance satisfying some additional mild assumptions,
$$
\mathcal{Q}(\left\langlengle X,y\rightangle,t)\leq Ct+{\mathcal F}rac{C'}{{\rightm LCD}(y)}+2e^{-c' n}
$$
for some constants $C,C',c'>0$ (see \cite{RV09} for a generalization of the statement).
The LCD, or its relatives, were subsequently used in studying
invertibility of non-Hermitian square matrices under broader assumptions \cite{RT,Livshyts,LTV},
and delocalization of eigenvectors of non-Hermitian random matrices
\cite{RV no gaps,LOR,LyTi}, among many other works.
Anti-concentration properties of random linear combinations naturally play a central role in the current work, however,
the measures of unstructuredness of vectors existing in the literature do not allow to obtain the precise
estimates we are aiming for. Here, we develop a new functional for dealing with linear combinations of
{\it dependent} Bernoulli variables.
Given $n\in{\mathbb N}$, $1\leq m\leq n/2$, a vector $y\in{\mathbb R}^n$ and parameters $K_1,K_2\geq 1$, we define the {\it degree
of unstructuredness (u-degree)} of vector $y$ by
\begin{align}\left\langlebel{udeg}
{\bf UD}_n(y,m,K_1,K_2):=\sup\bigg\{&t>0:\; A_{nm}\sum\limits_{S_1,{\rm dist}ots,S_m}\;
\int\limits_{-t}^t \mathbb{P}rod\limits_{i=1}^{m}\mathbb{P}si_{K_2}\big(
\big|{\mathbb E}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,y_{\mathcal{E}ta[S_i]}\,m^{-1/2} s\big)\big|\big)\,ds\leq K_1\bigg\},
\mathcal{E}_{n-1}d{align}
where the sum is taken over all sequences $(S_i)_{i=1}^m$ of disjoint subsets $S_1,{\rm dist}ots,S_m\subset[n]$,
each of cardinality $\lfloor n/m\rightfloor$ and
\begin{align}\left\langlebel{anm}
A_{nm} = {\mathcal F}rac{\big((\lfloor n/m\rightfloor)!\big)^m\,(n-m\lfloor n/m\rightfloor)!}{n!}\cdot
\mathcal{E}_{n-1}d{align}
Here $\mathcal{E}ta[S_i]$, $i\leq m$, denote mutually independent integer random variables uniformly distributed on respective $S_i$'s.
The function $\mathbb{P}si_{K_2}$ in the definition acts as a smoothing of $\max({\mathcal F}rac{1}{K_2},t)$,
with $\mathbb{P}si_{K_2}(t)={\mathcal F}rac{1}{K_2}$ for all $t\leq {\mathcal F}rac{1}{2K_2}$ and $\mathbb{P}si_{K_2}(t)= t$ for all $t\geq {\mathcal F}rac{1}{K_2}$
(we prefer to skip discussion of this purely technical element of the proof in this section, and
refer to the beginning of Section~\rightef{s: unstructured} for the full list of conditions imposed on $\mathbb{P}si_{K_2}$).
The functional ${\bf UD}_n(y,m,K_1,K_2)$ can be understood as follows.
The expression inside the supremum is the average value of the integral
$$
\int\limits_{-t}^t \mathbb{P}rod\limits_{i=1}^{m}\mathbb{P}si_{K_2}\big(
\big|{\mathbb E}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,y_{\mathcal{E}ta[S_i]}\,m^{-1/2} s\big)\big|\big)\,ds,
$$
with the average taken over all choices of sequences $(S_i)_{i=1}^m$.
The function under the integral, disregarding the smoothing $\mathbb{P}si_{K_2}$, is the
absolute value of the characteristic function of the random variable $\left\langlengle y,Z\rightangle$, where
$Z$ is a random $0/1$--vector with exactly $m$ ones, and with the $i$-th one distributed uniformly on
$S_i$. A relation between the magnitude of the characteristic function and anti-concentration properties of a random variable (the Esseen lemma (Lemma~\rightef{ess} below)
has been commonly used in works on the matrix invertibility (see, for example, \cite{Rud13}),
and determines the shape of the functional ${\bf UD}_n(\cdot)$.
The definition of the u-degree is designed
specifically to work with random $0/1$--vectors having a fixed sum (equal to $m$).
The next statement follows from the definition of ${\bf UD}_n(\cdot)$
and the Esseen lemma.
\begin{theor}[A Littlewood--Offord-type inequality in terms of the u-degree]\left\langlebel{p: cf est}
Let $m,n$ be positive integers with $m\leq n/2$, and let $K_1,K_2\geq 1$.
Further, let $v\in{\mathbb R}^n$, and let $X=(X_1,{\rm dist}ots,X_n)$ be a random $0/1$--vector in ${\mathbb R}^n$
uniformly distributed on the set of vectors with $m$ ones and $n-m$ zeros.
Then
$$
\mathcal{Q}\Big(\sum\limits_{i=1}^n v_i X_i,\sqrt{m}\,\tau\Big)
\leq C_{\text{\tiny\rightef{p: cf est}}}\,\big(\tau+{\bf UD}_n(v,m,K_1,K_2)^{-1}\big)\quad \mbox{for all $\tau>0$},
$$
where $C_{\text{\tiny\rightef{p: cf est}}}>0$ may only depend on $K_1$.
\mathcal{E}_{n-1}d{theor}
The principal difference of the u-degree and the above
theorem from the notion of the LCD and \mathcal{E}qref{eq: LCD overview}
is that the former allow to obtain stronger anti-concentration inequalities in the same regime of sparsity,
assuming that the coefficient vector $y$ is sufficiently unstructured.
In fact, under certain conditions, {\it sparse random $0/1$ vectors with prescribed support cardinality
admit stronger anti-concentration inequalities compared to the i.i.d.\ model.}
The last principle can be illustrated by taking the coefficient vector $y$
as a ``typical'' vector on the sphere $S^{n-1}$.
First, assume that $b_1,{\rm dist}ots,b_n$ are i.i.d.\ Bernoulli($p$) , with $p< 1/2$. Then it is easy to see that
for almost all (with respect to normalized Lebesgue measure) vectors $y\in S^{n-1}$,
$$
\mathcal{Q}\Big(\sum_{i=1}^n y_i b_i,0\Big)=(1-p)^n.
$$
In words, for a typical coefficient vector $y$ on the sphere, the linear combination $\sum_{i=1}^n y_i b_i$
takes distinct values for any two distinct realizations of $(b_1,{\rm dist}ots,b_n)$, and thus the L\'evy
function at zero is equal to the probability measure of the largest atom of the distribution of $\sum_{i=1}^n y_i b_i$
which corresponds to all $b_i$ equal to zero.
In contrast, if the vector $(b_1,{\rm dist}ots,b_n)$ is uniformly distributed on the
set of $0/1$--vectors with support of size $d=pn$, then for almost all $y\in S^{n-1}$,
the random sum $\sum_{i=1}^n y_i b_i$ takes ${n\choose d}$ distinct values.
Thus,
$$
\mathcal{Q}\Big(\sum_{i=1}^n v_i b_i,0\Big)={n\choose np}^{-1},
$$
where ${n\choose np}^{-1}\ll (1-p)^n$ for small $p$.
The above example provides only qualitative estimates
and does not give an information on the location of the atoms of the distribution of $\sum_{i=1}^n y_i b_i$.
The notion of the u-degree addresses this problem.
The following theorem, which is the main result of Section~\rightef{s: unstructured},
asserts that with a very large probability the normal vector to the (say, last) $n-1$ columns of our matrix $M_n$
is either very structured or has a very large u-degree, much greater than
the critical value $(1-p)^{-n}$.
\begin{theor}
\left\langlebel{th: gradual}
Let $r,{\rm dist}eltalta,\rightho\in(0,1)$, $s>0$, $R\geq 1$, and let $K_3\geq 1$. Then there are $n_0\in{\mathbb N}$, $C\geq 1$
and $K_1\geq 1$, $K_2\geq 4$ depending on $r,{\rm dist}eltalta,\rightho,R,s,K_3$ such that the following holds.
Let $n\geq n_0$, $p\leq C^{-1}$, and $s\ln n\leq pn$.
Let ${\bf g} \, : [1,\infty) \to [1,\infty)$ be an increasing (growth) function satisfying
\begin{align}\left\langlebel{gfncond}
{\mathcal F}orall a\geq 2\,\,{\mathcal F}orall t\geq 1:\,\,\,\, {\bf g}(a\,t)\geq {\bf g}(t)+a
\quad \quad \quad \mbox{ and } \quad \quad\quad
\mathbb{P}rod_{j=1}^\infty {\bf g}(2^j)^{j\,2^{-j}}\leq K_3.
\mathcal{E}_{n-1}d{align}
Assume that $M_n$ is
an $n\times n$ Bernoulli($p$) random matrix. Then
with probability at least $1-\mathcal{E}xp(-R pn)$ one has
\begin{align*}
&\{\mbox{Set of normal vectors to ${\bf C}_2(M_n),{\rm dist}ots,{\bf C}_{n}(M_n)\}\cap{\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)\subset$}\\
&\hspace{3cm}\mbox{$\{x\in{\mathbb R}^n:\;x^*_{\lfloor rn\rightfloor}=1,\,\, \,
{\bf UD}_n(x,m,K_1,K_2)\geq \mathcal{E}xp(Rpn)\,\,$ for all $\,\, pn/8\leq m\leq 8pn\}$.}
\mathcal{E}_{n-1}d{align*}
\mathcal{E}_{n-1}d{theor}
We would like to emphasize that the parameter $s$ in this theorem
can take values less than one, in the regime when the matrix $M_n$ typically has null rows and columns.
In this respect, the restriction $p\geq C\ln n/n$ in the main theorem comes from the treatment of structured vectors.
The proof of Theorem~\rightef{th: gradual} is rather involved, and is based on a double counting argument and
specially constructed lattice approximations of the normal vectors.
We refer to Section~\rightef{s: unstructured} for details.
Here, we only note that, by taking $R$ as a sufficiently large constant, the theorem implies the
relation \mathcal{E}qref{eq: aux -8572059873}, hence, accomplishes the treatment of unstructured vectors.
\subsection{Almost constant, steep and ${\mathcal R}$-vectors}
\left\langlebel{ss: steep overview}
In this subsection we discuss our treatment of the set of structured vectors, ${\mathcal S}_n$.
In the proof we partition the set ${\mathcal S}_n$ into several subsets and work with them separately.
In a simplistic form, the structured vectors are dealt with in two ways:
either by constructing discretizations and taking the union bound (variations of the $\varepsilon$--net argument),
or via deterministic estimates
in the case when there are very few very large components in the vector.
We note here that the discretization procedure has to take into account the non-centeredness
of our random matrix model: while in case of centered matrices with i.i.d. components (and under appropriate moment conditions)
the norm of the matrix is typically of order $\sqrt{n}$ times the standard deviation
of an entry, for our Bernoulli($p$) model it has order $pn$ (i.e., roughly $\sqrt{p}n$ times the standard deviation
of an entry), which makes a direct application of the $\varepsilon$--net
argument impossible. Fortunately, this
large norm is attained only in one direction --- the direction of the vector ${\bf 1}=(1, 1, {\rm dist}ots, 1)$ while
on the orthogonal complement of ${\bf 1}$ the typical norm is $\sqrt{pn}$. Therefore it is enough
to take a standard net in the Euclidean norm and to make it denser in that one direction, which almost does not
affect the cardinality of the net.
We refer to Section~\rightef{net} for details.
Let us first describe our approach in the (simpler) case when $p\in (q, c)$, where $c$ is a small enough absolute
constant and $q\in (0,c)$ is a fixed parameter (independent of $n$). We introduce four auxiliary sets and
show that the set of unit structured vectors, ${\mathcal S}_n\cap S^{n-1}$, is contained in the closure of their union.
The first set, $\mathcal{T}t_1$, consists of unit vectors close to vectors of
the canonical basis, specifically, unit vectors $x$ satisfying $x_1^* > 6pn x_2^*$,
where $x^*$ denotes the non-inreasing rearrangement of the vector $(|x_i|)_{i\leq n}$. For any such vector $x$
the individual bound is rather straightforward --- conditioned on the event that there are no zero columns in our matrix $M$,
and that the Euclidean norms of the matrix rows are not too large, we get
$Mx\neq 0$. This class is the main contributor
to the bound $(1+o_n(1))n(1-p)^n$ for non-invertibility over the structured vectors ${\mathcal S}_n$.
For the other three sets we
use anti-concentration probability estimates and discretizations. An application of Rogozin's lemma (Proposition~\rightef{rog})
implies that probability of having small inner product of a given row of our matrix with $x$ is small,
provided that there is a subset $A\subset [n]$ such that the maximal coordinate of $P_A x$ is
bounded above by $c\sqrt{p}\|P_A x\|$, where $\|\cdot\|$ denotes the
standard Euclidean norm and $P_A$ is the coordinate projection onto ${\mathbb R}^A$.
Combined with tensorization Lemma~\rightef{tens} this implies exponentially (in $n$)
small probability of the event that $\|Mx\|$ is close to zero --- see Proposition~\rightef{rogozin} below.
Specifically, we define
$\mathcal{T}t_2$ as the set of unit vectors satisfying the above condition with $A=[n]$, that is,
satisfying $x_1^*\leq c\sqrt{p}$, and for $\mathcal{T}t_3$ we take all unit vectors satisfying the condition
with $A={\sigma}ma_x ([2, n])$, that is, satisfying $x_2^*\leq c\sqrt{p} \|P_{{\sigma}ma_x ([2, n])}x\|$,
where
${\sigma}ma_x$ is a permutation satisfying $x^*_i=|x_{{\sigma}ma_x(i)}|$, $i\leq n$.
For vectors from these two sets we have very good individual probability estimates, but, unfortunately,
the complexity of both sets is large --- they don't admit nets of small cardinality. To overcome this issue, we
have to redefine these sets by intersecting them with specially chosen sets of vectors having many almost equal coordinates.
For the precise definition of such sets, denoted by $U(m, \gammamma)$, see Subsection~\rightef{net}. A set $U(m, \gammamma)$ is a variant
of the class of {\it almost constant} vectors, $\mathcal{AC}(\rightho)$ (see (\rightef{acv}) below), introduced to deal with general $p$.
Having a large part of coordinates of a vector almost equal to each other reduces the complexity of the set making
possible to construct a net of small cardinality. This resolves the problem and allows us to deal with these
two classes of sets. The remaining class of vectors, $\mathcal{T}t_4$, consists of vectors $x$
with $x_1^*\geq x_2^*\geq c\sqrt{p} \|P_{{\sigma}ma_x ([2, n])}x\|$,
i.e., vectors with relatively big two largest components.
For such vectors we produce needed anti-concentration estimates for the matrix-vector products
by using only those two components, i.e.,
we consider anti-concentration for the vector $P_A x$, where $A={\sigma}ma_x(\{1, 2\})$. Since the Rogozin lemma is
not suitable for this case, we compute the anti-concentration directly in Proposition~\rightef{anti2}.
As for the classes $\mathcal{T}t_2,\mathcal{T}t_3$, we
actually intersect the fourth class with appropriately chosen sets of almost constant vectors
in order to control cardinalities of the nets.
The final step is to show that the set ${\mathcal S}_n$ is contained in
the union of four sets described here. Careful analysis of this approach shows that the result can be proved with
all constants and parameters $r, {\rm dist}eltalta, \rightho$ depending only on $q$. Thus, it works for $p$ being between the two constants
$q$ and $c$.
The case of small $p$, that is, the case $C(\ln n)/n \leq p \leq c$, requires a more sophisticated splitting of ${\mathcal S}_n$ ---
we split it into {\it steep vectors} and {\it ${\mathcal R}$-vectors}. The definition and the treatment of steep vectors
essentially follows \cite{LLTTY first part, LLTTY-TAMS}, with corresponding adjustments for our model. The set
of {\it steep} vectors consists of vectors having a large jump between order statistics measured at certain indices.
The first subclass of steep
vectors, $\mathcal{T}_0$, is the same as the class $\mathcal{T}t_1$ described above --- vectors having very large maximal coordinate --- and is
treated as $\mathcal{T}t_1$. Similarly to the case of constant $p$, this class is the main contributor
to the bound $(1+o_n(1))n(1-p)^n$ for non-invertibility
over structured vectors. Next we fix certain $ma_1pprox 1/p$ and consider a sequence
$n_0=2$, $n_{j+1}/n_j = \mathcal{E}ll_0$, $j\leq s_0-1$, $n_{s_0+1}=m$ for some
specially chosen parameters $\mathcal{E}ll_0$ and $s_0$ depending on $p$ and $n$.
The class $\mathcal{T}_1$ will be defined as the class of vectors such that there exists $j$ with
$x^*_{n_{j+1}}> 6pn x^*_{n_j}$. To work with vectors from this class, we first show that for a given
$j$ the event that for every choice of two disjoint sets $|J_1|=n_j$ and $|J_2|=n_{j+1}-n_j$,
a random Bernoulli($p$) matrix has a row with exactly one $1$ in components indexed by $J_1$ and no $1$'s among components
indexed by $J_2$, holds with a very high probability. Then, conditioned on this event, for every $x\in \mathcal{T}_1$,
we choose $J_1$ corresponding to $x_i^*$, $i\leq n_j$, and $J_2$ corresponding to
$x_i^*$, $n_{j} \leq i\leq n_{j+1}$, and the corresponding row. Then the inner product of this row
with $x$ will be large in absolute value due to the jump (see Lemma~\rightef{l:T0} for the details).
Thus, conditioned on the described event, for every $x\in \mathcal{T}_1$ we have a good lower bound on $\|Mx\|$.
Then next two classes of steep vectors, $\mathcal{T}_2$ and $\mathcal{T}_3$,
consist of vectors having a jump of order $C\sqrt{pn}$, namely, vectors in $\mathcal{T}_2$ satisfy
$x_m^* > C\sqrt{pn} x_k^*$ and vectors in $\mathcal{T}_3$ satisfy
$x_k^* > C\sqrt{pn} x_\mathcal{E}ll^*$, where $ka_1pprox \sqrt{n/p}$ and $\mathcal{E}ll = \lfloor rn\rightfloor$ ($r$ is the parameter
from the definition of ${\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)$). Trying to apply the same idea for these two subclasses
one sees that the size of corresponding sets $J_1$ and $J_2$ is too large to
have exactly one $1$ among a row's components indexed by $J_1\cup J_2$ with a high probability.
Therefore the proof of individual probability bounds is more delicate and technical as
well as a construction of corresponding nets for $\mathcal{T}_2,\mathcal{T}_3$. We discuss the details in Subsection~\rightef{subs: nets}.
The class of ${\mathcal R}$-vectors consists of non-steep vectors to which Rogozin's lemma (Proposition~\rightef{rog}) can be applied
when we project a vector on $n-k$ smallest coordinates with $m<k\leq n/\ln^2(pn)$, thus vectors from this class
satisfy $\|P_A x\|\leq c\sqrt{p}\|P_A x\|_\infty$ for $A={\sigma}ma_x([k, n]$ (we will take union over all choices of
integer $k$ in the interval $(m, n/\ln^2(pn)]$). Thus, the individual probability bounds for ${\mathcal R}$-vectors
will follow from Rogozin's lemma together with tensorization lemma as for classes $\mathcal{T}t_2$, $\mathcal{T}t_3$, described above.
Thus the remaining part is to construct a good net for ${\mathcal R}$-vectors. For simplicity, dealing with such vectors, we fix
the normalization $x^*_{\lfloor rn\rightfloor}=1$. Since vectors are non-steep, we have a certain control of largest coordinates and, thus,
on the Euclidean norm of a vector. The upper bound on $k$ is chosen in such a way that the cardinality
on a net corresponding to largest coordinates of a vector is relatively small (it lies in $n/\ln^2(pn)$-dimensional
subspace). For the purpose of constructing of a net of small cardinality, we need to control the Euclidean norm of $P_A x$
for an ${\mathcal R}$-vector. Therefore we split ${\mathcal R}$-vectors into level sets according to the value of $\|P_A x\|$. There will be
two different types of level sets --- vectors with relatively large Euclidean norm of $P_A x$ and vectors with small $\|P_Ax\|$.
A net for level sets with large $\|P_A x\|$ is easier to construct, since we can zero all coordinates starting with
$x^*_{\lfloor rn\rightfloor}=1$. If the Euclidean norm is small, we cannot do this, so we intersect this subclass with almost
constant vectors (in fact we incorporate this intersection into the definition of ${\mathcal R}$-vectors), defined by
\begin{equation}
\left\langlebel{acv}
\mathcal{AC}(\rightho) :=\{ x\in {\mathbb R}^n\, : \, \mathcal{E}xists \left\langlembdabda \in {\mathbb R} \, \mbox{ s. t. }\,
|\left\langlembda|= x^*_{\lfloor rn\rightfloor} \, \mbox{ and }\, |\{ i\leq n \, : \,
|x_i - \left\langlembdabda|\leq \rightho|\left\langlembda|\}| >n- \lfloor rn\rightfloor\}.
\mathcal{E}_{n-1}d{equation}
As in the
case of constant $p$, this essentially reduces the dimension corresponding to almost constant part to
one and therefore reduce the cardinality of a net. The rather technical construction of nets is presented in
Subsection~\rightef{sub-nets}. In some aspects the construction follows ideas developed in \cite{LLTTY first part}.
\section{Preliminaries}
\left\langlebel{preliminaries}
\subsection{General notation}
\left\langlebel{gen-not}
By {\it universal} or {\it absolute} constants we always mean numbers independent of all involved parameters,
in particular independent of $p$ and $n$. Given positive integers $\mathcal{E}ll<k$ we denote sets
$\{1, 2, \ldots , \mathcal{E}ll\}$ and $\{\mathcal{E}ll, \mathcal{E}ll + 1, \ldots , k\}$ by $[\mathcal{E}ll]$ and $[\mathcal{E}ll, k]$ correspondingly.
Having two functions $f$ and $g$ we write $fa_1pprox g$ if there are two absolute positive constants
$c$ and $C$ such that $cf\le g\le Cf$. As usual, $\Pi_n$ denotes the permutation group
on $[n]$.
For every vector $x=(x_i)_{i=1}^n\in {\mathbb R}^n$,
by $(x_i^*)_{i=1}^n$ we denote the non-increasing rearrangement of the sequence $(|x_i|)_{i=1}^n$
and we fix one permutation ${\sigma}ma_x$ satisfying $|x_{{\sigma}ma_x(i)}|= x_i^*$, $i\leq n$.
We use $\left\langle\cdot, \cdot \righta$ for the standard inner product on ${\mathbb R}^n$, that is
$\left\langle x, y \righta = \sum _{i= 1}^{n} x_i y_i$.
Further, we write $\|x\|_{\infty}=\max_i |x_i|$ for the $\mathcal{E}ll_{\infty}$-norm of $x$.
We also denote ${\bf 1}=(1, 1, {\rm dist}ots, 1)$.
\subsection{Lower bound on the singularity probability}\left\langlebel{subs: lower b}
Here, we provide a simple argument showing that for the sequence of random Bernoulli($p_n$) matrices $(M_n)$,
with $p_n$ satisfying $(n p_n-\ln n)\longrightarrow \infty$ as $n\to \infty$, we have
$$
{\mathbb P}\big\{\mbox{$M_n$ contains a zero row or column}\big\}\geq (2-o_n(1))n\,(1-p)^n.
$$
Our approach is similar to that applied in \cite{BasRud-sharp} in the related context.
Fix $n> 1$ and write $p=p_n$.
Let ${\bf 1}_R$ be the indicator of the event that there is zero row in the matrix $M_n$, and, similarly,
let ${\bf 1}_C$ be the indicator of the event that $M_n$ has a zero column.
Then, obviously,
$$
{\mathbb E}\,{\bf 1}_R={\mathbb E}\,{\bf 1}_C=1-\big(1-(1-p)^n\big)^n,
$$
hence,
$$
{\mathbb E}({\bf 1}_R+{\bf 1}_C)^2\geq 2-2\big(1-(1-p)^n\big)^n.
$$
On the other hand,
$$
{\mathbb E} \,{\bf 1}_R\,{\bf 1}_C\leq \sum_{i=1}^n\sum_{j=1}^n{\mathbb P}\big\{\mbox{$i$--th row and
$j$--th column of $M_n$ are zero}\big\}= n^2(1-p)^{2n-1},
$$
implying
$$
{\mathbb E}({\bf 1}_R+{\bf 1}_C)^2={\mathbb P}\big\{{\bf 1}_R+{\bf 1}_C=1\big\}+4\,{\mathbb P}\big\{{\bf 1}_R\,{\bf 1}_C=1\big\}
\leq {\mathbb P}\big\{{\bf 1}_R+{\bf 1}_C=1\big\}+4n^2(1-p)^{2n-1}.
$$
Therefore,
\begin{align*}
{\mathbb P}\big\{\mbox{$M_n$ contains a zero row or column}\big\}&\geq
{\mathbb P}\big\{{\bf 1}_R+{\bf 1}_C=1\big\}\\
&\geq {\mathbb E}({\bf 1}_R+{\bf 1}_C)^2-4n^2(1-p)^{2n-1}\\
&\geq 2-2\big(1-(1-p)^n\big)^n-4n^2(1-p)^{2n-1}.
\mathcal{E}_{n-1}d{align*}
It remains to note that, with our assumption on the growth rate of $p=p_n$, we have
$n(1-p)^n\longrightarrow 0$, which implies
$$
{\mathcal F}rac{1}{n(1-p)^n}\big(2-2\big(1-(1-p)^n\big)^n-4n^2(1-p)^{2n-1}\big)\longrightarrow 2.
$$
\subsection{Gradual non-constant vectors}
\left\langlebel{gradnac}
For any $r\in(0,1)$, we define ${\Upsilon}_n(r)$ as the set of all vectors $x$ in ${\mathbb R}^n$ with $x^*_{\lfloor rn\rightfloor}=1$.
We will call these vectors {\it $r$-normalized}.
By a {\it growth function} ${\bf g}$ we mean any non-decreasing function from $[1,\infty)$ into $[1,\infty)$.
Let ${\bf g}$ be an arbitrary growth function.
We will say that a vector $x\in {\Upsilon}_n(r)$ is {\it gradual} (with respect to the function ${\bf g}$) if $x^*_i\leq {\bf g}(n/i)$
for all $i\leq n$.
Further, if $x\in {\Upsilon}_n(r)$ satisfies
\begin{align}
\left\langlebel{cond2}
\mathcal{E}xists\,
Q_1,Q_2\subset[n] \quad \mbox{ such that } \quad
|Q_1|,|Q_2|\geq {\rm dist}eltalta n \quad \mbox{ and } \quad \max\limits_{i\in Q_2} x_i\leq \min\limits_{i\in Q_1}x_i- \rightho
\mathcal{E}_{n-1}d{align}
then we say that the vector $x$ is {\it essentially non-constant} or just {\it non-constant} (with parameters ${\rm dist}eltalta,\rightho$).
Recall that the set ${\mathcal V}_n ={\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)$ was defined in (\rightef{eq: gnc def}) as
$$
\big\{x\in{\Upsilon}_n(r):\,\, x \, \mbox{ is gradual with ${\bf g}$ and satisfies \mathcal{E}qref{cond2}}\big\}.
$$
Vectors from this set we call {\it gradual non-constant vectors}.
Recall that the set ${\mathcal S}_n={\mathcal S}_n(r,{\bf g},{\rm dist}eltalta,\rightho)$ of structured vectors was defined in (\rightef{strvect})
as the complement of scalar multiples of ${\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)$.
The next simple lemma will allow us to reduce analysis of $\{x/\|x\|:\;x\in{\mathcal S}_n\}$ to the treatment of the set
$\{x/\|x\|:\;x\in {\Upsilon}_n(r)\setminus {\mathcal V}_n\}$.
\begin{lemma}\left\langlebel{l:closure}
For any choice of parameters $r,{\bf g},{\rm dist}eltalta,\rightho$,
the set $\{x/\|x\|:\;x\in{\mathcal S}_n\}$ is contained in the closure of the set $\{x/\|x\|:\;x\in {\Upsilon}_n(r)\setminus {\mathcal V}_n\}$.
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Let $y$ be a unit vector such that $y=x/\|x\|$ for some $x\in {\mathcal S}_n$.
If $x^*_{\lfloor rn\rightfloor}\ne 0$ then $y=z/\|z\|$, where
$z=x/x^*_{\lfloor rn\rightfloor}\in {\Upsilon}_n(r)\setminus {\mathcal V}_n$.
If $x^*_{\lfloor rn\rightfloor}= 0$, we can consider a sequence of vectors $(x(j))_{j\geq 1}$ in ${\mathbb R}^n$
defined by $x(j)_i=x_j$ for $i\ne \lfloor rn\rightfloor$ and $x(j)_{\lfloor rn\rightfloor}=1/j$. Let
$$
y(j):=x(j)/x(j)^*_{\lfloor rn\rightfloor}\in {\Upsilon}_n(r),\quad j\geq 1.
$$
Clearly, $y(j)^*_1\longrightarrow\infty$, so for all sufficiently large $j$ we have $y(j)\|\cdot\|tin {\mathcal V}_n$.
Thus, for all large $j$,
$$
y(j)/\|y(j)\|\in \{x'/\|x'\|:\;x'\in {\Upsilon}_n(r)\setminus {\mathcal V}_n\},
$$
whereas $y(j)/\|y(j)\|=x(j)/\|x(j)\|\longrightarrow x/\|x\|$. This implies the desired result.
\mathcal{E}_{n-1}d{proof}
We will need two following lemmas. The first one states that vectors which do not satisfy
(\rightef{cond2}) are almost constant (that is, have large part of coordinates nearly equal to each other).
The second one is a simple combinatorial estimate, so we omit
its proof.
\begin{lemma}
\left\langlebel{a-c-cond2}
Let $n\geq 1$, ${\rm dist}eltalta,\rightho, r\in(0,1)$.
Denote $k=\lceileil {\rm dist}eltalta n\rightceileil$ and $m=\lfloor rn \rightfloor$ and assume
$n\geq 2m> 4k$. Assume $x\in{\Upsilon}_n(r)$ does not satisfy (\rightef{cond2}).
Then there exist $A\subset [n]$ of cardinality $|A|> n-m$ and $\left\langlembdabda$ with
$|\left\langlembdabda|=1$ such that $|x_i-\left\langlembda|<\rightho$ for every $i\in A$.
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
By $(x_i^{\#})_i$ denote the non-increasing rearrangement
of $(x_i)_i$ (we would like to emphasize that we do not take absolute values).
Note that there are two subsets $Q_1, Q_2\subset[n]$ with $|Q_1|,|Q_2|\geq k$
satisfying $\max_{i\in Q_2} x_i\leq \min_{i\in Q_1}x_i- \rightho$ if and only if
$x_k^{\#}-x_{n-k+1}^{\#}\geq \rightho$. Therefore, using that $x$ does not satisfy (\rightef{cond2}),
we observe $x_k^{\#}-x_{n-k+1}^{\#}< \rightho$. Next consider the set
$$A:=\{x_i^{\#}\,\, :\,\, k<i\leq n-k\}.$$
Then $|A|=n-2k> n-m$.
Since $x^*_m=1$ we obtain that
$$
|\{i\, :\, |x_i|>1\}| <m\leq n-m \quad \mbox{ and } \quad |\{i\, :\, |x_i|<1\}|\leq n-m.
$$
Therefore, there exists an index $j\in A$ such that $|x_j|=1$. Taking $\left\langlembda=x_j$,
we observe that for every $i\in A$, $|x_i-\left\langlembda|<\rightho$. This completes the proof.
\mathcal{E}_{n-1}d{proof}
\begin{lemma}\left\langlebel{l: aux 2498276098059385-}
For any ${\rm dist}eltalta\in(0,1]$ there are $n_{\rm dist}eltalta\in{\mathbb N}$, $c_{\rm dist}eltalta>0$ and $C_{\rm dist}eltalta\geq 1$ depending only on ${\rm dist}eltalta$ with the following property.
Let $n\geq n_{\rm dist}eltalta$ and let $m\in{\mathbb N}$ satisfy $n/m\geq C_{\rm dist}eltalta$.
Denote by $\mathcal S$ the collection of sequences $(S_1,{\rm dist}ots,S_m)\subset[n]$
with $|S_i|=\lfloor n/m\rightfloor$ and $S_i\cap S_j=\mathcal{E}mptyset\mbox{ for all }i\neq j$.
Let $A_{nm}$ be as in (\rightef{anm}).
Then for any pair $Q_1,Q_2$ of disjoint subsets of $[n]$ of cardinality at least ${\rm dist}eltalta n$ each,
one has
\begin{align*}
\Big|\Big\{&(S_1,{\rm dist}ots,S_m)\in \mathcal S:\;
\min(|S_i\cap Q_1|,|S_i\cap Q_2|)\geq {\mathcal F}rac{{\rm dist}eltalta}{2}\lfloor n/m\rightfloor
\mbox{ for at most $c_{\rm dist}eltalta m$ indices $i$}
\Big\}\Big|
\leq e^{-c_{\rm dist}eltalta n} A_{nm}^{-1}.
\mathcal{E}_{n-1}d{align*}
\mathcal{E}_{n-1}d{lemma}
\subsection{Auxiliary results for Bernoulli r.v. and random matrices }
\left\langlebel{ben-ineq}
Let $p\in (0,1)$, ${\rm dist}eltalta$ is Bernoulli random variable taking
value $1$ with probability $p$ and $0$ with probability $1-p$.
We say that ${\rm dist}eltalta$ is a Bernoulli($p$) random variable.
A random matrix with i.i.d.\ entries distributed as ${\rm dist}eltalta$ will be
called {\it Bernoulli($p$) random matrix}.
Here we provide four lemmas needed below. We start with
notations for random matrices used throughout the paper.
The class of all $n\times n$ matrices having $0/1$ entries we
denote by ${\mathcal{M}_{n}}$.
We will consider a probability measure on ${\mathcal{M}_{n}}$ induced by the distribution of an $n\times n$
Bernoulli($p$) random matrix. We will use the same notation ${\mathbb P}$ for this probability measure;
the parameter $p$ will always be clear from the context.
Let $M=\{\mu_{ij}\}\in {\mathcal{M}_{n}}$. By $\rightow_i=\rightow_i(M)$
we denote the $i$-th row of $M$, and by ${\bf C}_i(M)$ --- the $i$-th column, $i\leq n$. By $\|M\|$ we always denote
the operator norm of $M$ acting as an operator $\mathcal{E}ll_2\to \mathcal{E}ll_2$. This
norm is also called spectral norm and equals the largest singular number.
We will need the following form of Bennett's inequality.
\begin{lemma}
\left\langlebel{bennett}
Let $n\geq 1$, $0<q< 1$, and ${\rm dist}eltalta$ be a Bernoulli($p$) q random variable.
Let ${\rm dist}eltalta_i$ and ${\rm dist}eltalta_{ij}$, $i,j\leq n$, be independent copies of ${\rm dist}eltalta$.
Define the function $h(u):=(1+u)\ln (1+u) -u$, $u\geq 0$.
Then for every $t>0$,
\begin{align*}
\max\left(\mathbb{P}\left(\sum_{i=1}^n {\rm dist}eltalta_i >qn+t \rightight),
\mathbb{P}\left(\sum_{i=1}^n {\rm dist}eltalta_i < qn-t \rightight) \rightight)
&\leq
\mathcal{E}xp\left(-{\mathcal F}rac{nq(1-q)}{\max^2(q, 1-q)} \, h\left({\mathcal F}rac{t\max(q, 1-q) }{nq(1-q)}\rightight)
\rightight).
\mathcal{E}_{n-1}d{align*}
In particular, for $0<\varepsilon \leq q\leq 1/2$,
\begin{align*}
\max\left(\mathbb{P}\left(\sum_{i=1}^n {\rm dist}eltalta_i >(q+ \varepsilon) n \rightight),
\mathbb{P}\left(\sum_{i=1}^n {\rm dist}eltalta_i <(q- \varepsilon) n \rightight) \rightight)
&\leq
\mathcal{E}xp\left(-{\mathcal F}rac{n\varepsilon^2}{2q(1-q)} \,
\left(1-{\mathcal F}rac{\varepsilon}{3q}\rightight)\rightight),
\mathcal{E}_{n-1}d{align*}
and for $q\leq 1/2$, $\tau >e$,
\begin{align*}
\mathbb{P}\left(\sum_{i=1}^n {\rm dist}eltalta_i >(\tau +1) qn \rightight)
&\leq
\mathcal{E}xp\left(- \tau \ln(\tau/e)qn \rightight).
\mathcal{E}_{n-1}d{align*}
Furthermore, for $50/n\leq q\leq 0.1$,
\begin{align*}
\mathbb{P}\Big(qn/8\leq \sum_{i=1}^n {\rm dist}eltalta_i \leq 8 qn \Big)
&\geq 1- (1-q)^{n/2}.
\mathcal{E}_{n-1}d{align*}
Moreover, if $n\geq 30$ and $p=q \geq (4\ln n)/n$ then denoting
$$
{\mathbb E}vent_{sum} :=\Big\{M=\{{\rm dist}eltalta_{ij}\}_{i,j\leq n}\in {\mathcal{M}_{n}} \, :\,
\sum _{j=1}^n {\rm dist}eltalta_{ij} \leq 3.5 pn \quad
\mbox{ for every }\,\,\, i\leq n\Big\}
$$
we have $\mathbb{P}({\mathbb E}vent_{sum} )\geq 1- \mathcal{E}xp(-1.5 np)$.
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Recall that Bennett's
inequality states that for
mean zero independent random variables $\xi_1$, {\rm dist}ots, $\xi_n$
satisfying $\xi_i \leq \rightho$ (for a certain fixed $\rightho>0$)
almost surely for $i\leq n$, one has for every $t>0$,
$$
\mathbb{P}\left(\sum_{i=1}^n \xi_i > t \rightight) \leq \mathcal{E}xp\left(-{\mathcal F}rac{{\sigma}ma^2}{\rightho^2}\,
h\left({\mathcal F}rac{\rightho t}{{\sigma}ma^2}\rightight)\rightight),
$$
where ${\sigma}ma^2 = \sum_{i=1}^n{\mathbb E} \xi_i^2$
(see e.g. Theorem 1.2.1 on p. 28 in \cite{lama} or Exercise 2.2
on p. 11 in \cite{DevLug} or Theorem 2.9 in \cite{BLM}).
Take $\xi_i = {\rm dist}eltalta_i -q$, $\xi_i'=-\xi_i$, $i\leq n$. Then for every $i\leq n$,
$\xi_i'$ and $\xi_i$ are
centered, $|\xi _i'|= |\xi _i|=\max(q, 1-q)$, and ${\sigma}ma^2= n q(1-q)$. Applying the Bennett inequality
with $\rightho=\max(q, 1-q)$ twice --- to $\xi_i$ and $\xi _i'$, we observe the first inequality.
To prove the second inequality, we take $t=\varepsilon n$ and use that
$h(\cdot)$ is an increasing function satisfying $h(u)\geq u^2/2-u^3/6$
on ${\mathbb R}^+$. The third inequality follows by taking $t=\tau qn$ and using
$h(u)\geq u\ln (u/e)$.
For the ``furthermore" part, we apply the third inequality with $\tau=7$, to get
$$
{\mathbb P}\Big\{\sum_{i=1}^n {\rm dist}eltalta_i> 8qn\Big\}\leq \mathcal{E}xp(-6qn).
$$
On the other hand, using $q\leq 0.1$,
\begin{align*}
{\mathbb P}\Big\{\sum_{i=1}^n {\rm dist}eltalta_i< qn/8\Big\}
&= \sum_{i=0}^{\lfloor qn/8\rightfloor}{n\choose i}q^i(1-q)^{n-i}
\leq (1-q)^n+\sum_{i=1}^{\lfloor qn/8\rightfloor}\bigg({\mathcal F}rac{enq}{i(1-q)}\bigg)^i\,(1-q)^n\\
&\leq (1-q)^n+{\mathcal F}rac{qn}{8}\bigg({\mathcal F}rac{8e}{1-q}\bigg)^{qn/8}\,(1-q)^n\leq
(1-q)^n+{\mathcal F}rac{qn}{8}\bigg({\mathcal F}rac{80e}{9}\bigg)^{qn/8}\,(1-q)^n
\mathcal{E}_{n-1}d{align*}
Since $(80 e/9)^{1/8}\leq e^{0.4}$, $(1-q)^n\leq \mathcal{E}xp(-qn)$, $qn\geq 50$, and $\ln x \leq x/e$ on $[0, \infty)$,
this implies
\begin{align*}
\mathbb{P}\Big(qn/8\leq \sum_{i=1}^n {\rm dist}eltalta_i < qn /8\Big)
&\leq \mathcal{E}xp(-6qn)+ (1+\mathcal{E}xp(0.45 qn)) (1-q)^n \leq (1-q)^{n/2} .
\mathcal{E}_{n-1}d{align*}
Finally, to get the last inequality, we take $t=2.5 qn=2.5pn$, then
$$
\mathbb{P}\left(\sum _{j=1}^n {\rm dist}eltalta_{ij} > 3.5 p n\rightight) \leq
\mathcal{E}xp\left(-{\mathcal F}rac{n p}{1-p}\,
h\left(2.5\rightight)\rightight) \leq \mathcal{E}xp\left(- n p \,
\left(3.5 \ln 3.5 -2.5 \rightight)\rightight)\leq \mathcal{E}xp\left(- 1.8 n p \right).
$$
Since under our assumptions, $n\mathcal{E}xp\left(- 1.8 n p \right)\leq \mathcal{E}xp\left(- 1.5 n p \right)$,
the bound on $\mathbb{P}({\mathbb E}vent_{sum} )$ follows by the union bound.
\mathcal{E}_{n-1}d{proof}
We need the following simple corollary of the Bennet's lemma.
\begin{lemma}\left\langlebel{l: column supports}
For any $R\geq 1$ there is $C_{\text{\tiny\rightef{l: column supports}}}
=C_{\text{\tiny\rightef{l: column supports}}}(R)\geq 1$ with the following property.
Let $n\geq 1$ and $p\in(0,1)$ satisfy $C_{\text{\tiny\rightef{l: column supports}}}p\leq 1$
and $C_{\text{\tiny\rightef{l: column supports}}}\leq p n$.
Further, let $M$ be an $n\times n$ be Bernoulli($p$) random matrix.
Then with probability at least $1-\mathcal{E}xp(-n/C_{\text{\tiny\rightef{l: column supports}}})$
one has
$$
8pn\geq |{\rm supp\, } {\bf C}_i(M)|\geq pn/8 \quad \mbox{for all but $\,\,\, \lfloor(pR)^{-1}\rightfloor\,\,\,$
indices $\, i\in [n]\setminus\{1\}$.}
$$
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
For each $i\in [n]\setminus\{1\}$, let $\xi_i$ be the indicator of the event
$$\big\{8pn< |{\rm supp\, } {\bf C}_i(M)|\quad\quad \mbox{ or }\quad \quad|{\rm supp\, } {\bf C}_i(M)|< pn/8\big\}.$$
By Lemma~\rightef{bennett}, ${\mathbb E}\,\xi_i\leq e^{-pn/2}$. Since $\xi_i$'s are independent, by the Markov inequality,
\begin{align*}
{\mathbb P}\Big\{\sum_{i=2}^n \xi_i\geq {\mathcal F}rac{1}{p R}\Big\}
\leq {n-1 \choose \lfloor(p R)^{-1}\rightfloor}\big(e^{-pn/2}\big)^{\lfloor(p R)^{-1}\rightfloor}
\leq {n-1 \choose \lfloor(p R)^{-1}\rightfloor}\,e^{- n/(4R)}.
\mathcal{E}_{n-1}d{align*}
The result follows.
\mathcal{E}_{n-1}d{proof}
The following lemma provides a bound on the norm of a random Bernoulli
matrix. It is similar to \cite[Theorem~1.14]{BasRud-sharp}, where the case of
symmetric matrices was treated. For the sake of completeness
we sketch its proof.
\begin{lemma}
\left\langlebel{mnorm}
Let $n$ be large enough and
$(4\ln n)/n \leq p\leq 1/4$. Let
$M=({\rm dist}eltalta_{ij})_{i,j}$ be a Bernoulli($p$) random matrix.
Then for every $t\geq 30$ one has
\begin{equation*}\left\langlebel{bdd}
\mathbb{P}\big\{\|M - {\mathbb E} M\|\geq 2 t \sqrt{n p} \big\}\leq 4 e^{-t^2 pn/4}
\quad \quad \mbox{ and } \quad \quad
\mathbb{P}\big\{\|M\|\geq 2 t \sqrt{n p} + pn \big\}\leq 4 e^{-t^2 pn/4}.
\mathcal{E}_{n-1}d{equation*}
In particular, taking $t=\sqrt{pn}$,
\begin{equation}\left\langlebel{normofone}
\mathbb{P}\left( \|M {\bf 1}\| \geq 3 p n^{3/2} \right)\leq 4 \mathcal{E}xp(- n^2 p^2/4).
\mathcal{E}_{n-1}d{equation}
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Given an $n\times n$ random matrix $T=(t_{ij})_{i,j}$ with independent
entries taking values in $[0,1]$.
we consider it as a vector in
${\mathbb R}^m$ with $m={n^2}$. Then the Hilbert--Schmidt norm of $T$ is the standard
Euclidean norm on ${\mathbb R}^m$. Let $f$ be any function in ${\mathbb R}^m$
which is convex and is $1$-Lipschitz with respect to the standard Euclidean norm.
Then the Talagrand inequality (see e.g. Corollary~4.10 and Proposition~1.8
in \cite{Ledoux}) gives that for every $s>0$,
$$
\mathbb{P}\left( f(T) \geq {\mathbb E} f(T) + s +4\sqrt{\mathbb{P}i} \right) \leq 4\mathcal{E}xp (-s^2/4) .
$$
We apply this inequality twice, first with the function $f(T):=\|T\|$ to the matrix
$T:=M-{\mathbb E} M$. At the end of this proof we show that ${\mathbb E}\|M-{\mathbb E} M\|\leq 20\sqrt{pn}$.
Therefore, taking $s=t\sqrt{pn}$ with $t\geq 30$, we obtain the first bound.
For the second bound, note that all entries of ${\mathbb E} M$ equal $p$,
hence $\|{\mathbb E} M\| = pn$. Thus, the second bound follows by the triangle inequality.
It remains to prove that ${\mathbb E}\|M-{\mathbb E} M\|\leq 20 \sqrt{pn}$.
Recall that ${\rm dist}eltalta_{ij}$ are the entries of $M$.
Let ${\rm dist}eltalta_{ij}'$, $i, j\leq n$ be independent
copies of ${\rm dist}eltalta_{ij}$ and set $M':= ({\rm dist}eltalta_{ij}')_{i,j}$.
Denote by $r_{ij}$ independent Rademacher random variables
and by $g_{ij}$ independent standard Gaussian random variables.
We assume that all our variables are mutually independent and set
$\xi_{ij}:= {\rm dist}eltalta_{ij}-{\rm dist}eltalta_{ij}'$.
Since for every $i,j\leq n$,
$\xi_{ij}$ is symmetric, it has the same distribution as
$|\xi_{ij}| r_{ij}$ and the same as $\sqrt{2/\mathbb{P}i}|\xi_{ij}| r_{ij} {\mathbb E} |g_{ij}|$.
Then we have
$$
{\mathbb E}_{{\rm dist}eltalta} \|M-{\mathbb E} M\| = {\mathbb E}_{{\rm dist}eltalta}\|M-{\mathbb E}_{{\rm dist}eltalta'} M' \|\leq
{\mathbb E}_{{\rm dist}eltalta} {\mathbb E}_{{\rm dist}eltalta'}\|M- M' \|=
{\mathbb E}_{\xi} \| (\xi_{ij})_{i,j}\| =
\sqrt{2/\mathbb{P}i}\, {\mathbb E}_{\xi, r} \| (\xi_{ij}r_{ij} {\mathbb E}_g |g_{ij}|)_{i,j}\|
$$
$$
\leq\sqrt{2/\mathbb{P}i}\, {\mathbb E}_{\xi, r, g} \| (\xi_{ij}r_{ij} |g_{ij}|)_{i,j}\| =
\sqrt{2/\mathbb{P}i}\, {\mathbb E}_{\xi} {\mathbb E}_g \| (\xi_{ij} |g_{ij}|)_{i,j}\| .
$$
Applying a result of Bandeira and Van Handel (see the beginning of Section~3.1 in \cite{BVH}),
we obtain
$$
{\mathbb E}_{{\rm dist}eltalta} \|M-{\mathbb E} M\| \leq {\mathbb E}_{\xi} (4 \max({\sigma}ma_1, {\sigma}ma_2) + 15{\sigma}ma_* \sqrt{\ln (2n)}),
$$
where
$$
{\sigma}ma_1 =\max_{i\leq n} \sqrt{\sum_{j=1}^n\xi_{ij}^2}, \quad
{\sigma}ma_2 =\max_{j\leq n} \sqrt{\sum_{i=1}^n\xi_{ij}^2}, \quad \mbox{ and } \quad
{\sigma}ma_*=\max_{i, j\leq n} |\xi_{ij}|\leq 1.
$$
Note that $\xi_{ij}^2$ are Bernoulli($p$) q\, random variables with $q=2p(1-p)$. Therefore,
using $(4\ln n)/n\leq p\leq 1/2$ and
applying the ``moreover part" of Lemma~\rightef{bennett}, we obtain that
$\max({\sigma}ma_1, {\sigma}ma_2)> 2\sqrt{pn}$ with probability at most
$2\mathcal{E}xp(-1.5 nq)\leq 2/n^6$.
Moreover, since $\xi_{ij}^2\leq 1$, we have $\max({\sigma}ma_1, {\sigma}ma_2)\leq \sqrt n$.
Therefore,
$$
{\mathbb E}_{\xi} (4 \max({\sigma}ma_1, {\sigma}ma_2) + 15{\sigma}ma_* \sqrt{\ln (2n)})
\leq 8\sqrt{pn} + 4/n^5 + 15\sqrt{\ln (2n)}\leq 20\sqrt{pn}.
$$
\mathcal{E}_{n-1}d{proof}
As an elementary corollary of the above lemma, we have the following statement where the restriction $pn\geq 4\ln n$ is removed.
\begin{cor}\left\langlebel{cor: norm of centered}
For every $s>0$ and $R\geq 1$ there is $C_{\text{\tiny\rightef{cor: norm of centered}}}\geq 1$
depending on $s,R$ with the following property.
Let $n\geq 16/s$ be large enough and let $p\in(0,1/4]$ satisfy $s\ln n\leq pn$.
Let $M_n$ be an $n\times n$ Bernoulli($p$) random matrix.
Then
$$
{\mathbb P}\big\{\|M_n-{\mathbb E} M_n\|\leq C_{\text{\tiny\rightef{cor: norm of centered}}}\sqrt{pn}\big\}\geq 1-\mathcal{E}xp(-Rpn).
$$
\mathcal{E}_{n-1}d{cor}
\begin{proof}
Let $w:=\max(1,\lceileil 8/s\rightceileil)$, $\widetilde n:=w\,n$, and let $\widetilde M_n$ be $\widetilde n\times \widetilde n$
Bernoulli($p$) matrix. Assuming that $n$ is sufficiently large, we get
$$
p\,\widetilde n = wpn \geq s\max(1,\lceileil 8/s\rightceileil) \ln n \geq 4\ln \widetilde n.
$$
Thus, the previous lemma is applicable, and we get
$$
{\mathbb P}\big\{\|\widetilde M_n-{\mathbb E} \widetilde M_n\|\leq C_{\text{\tiny\rightef{cor: norm of centered}}}\sqrt{pn}\big\}\geq 1-\mathcal{E}xp(-Rpn),
$$
for some $C_{\text{\tiny\rightef{cor: norm of centered}}}>0$ depending only on $s,R$.
Since the norm of a matrix is not less than the norm of any of its submatrices, and because any $n\times n$ submatrix of
$\widetilde M_n$ is equidistributed with $M_n$, we get the result.
\mathcal{E}_{n-1}d{proof}
\subsection{Anti-concentration}
In this subsection we combine anti-concentration inequalities with the
following tensorization lemma (see Lemma~3.2 in \cite{KT-last}, Lemma~2.2
in \cite{RV} and Lemma 5.4 in \cite{R-ann}).
We also provide Esseen's lemma.
\begin{lemma}[{Tensorization lemma}]
\left\langlebel{tens}
\left\langlebel{l: tensor}
Let $\left\langlembda, \gammamma>0$.
Let $\xi_1, \xi_2, \ldots, \xi_m$ be independent random variables.
Assume that for all $j\leq m$, $\mathbb{P}(|\xi_j|\leq \left\langlembda)\leq \gammamma$. Then for every $\varepsilon\in (0,1)$
one has
$$
\mathbb{P}(\|(\xi_1, \xi_2, ...,\xi_m)\|\leq \left\langlembda\sqrt{\varepsilon m} )\leq (e/\varepsilon)^{\varepsilon m} \gammamma^{m(1-\varepsilon)}.
$$
Moreover, if there exists $\varepsilon_0>0$ and $K>0$ such that for every $\varepsilon\geq \varepsilon_0$ and
for all $j\leq m$ one has $\mathbb{P}(|\xi_j|\leq \varepsilon)\leq K \varepsilon$ then
there exists an absolute constant $C_{\text{\tiny\rightef{l: tensor}}}>0$ such that for every $\varepsilon\geq \varepsilon_0$,
$$
\mathbb{P}(\|(\xi_1, \xi_2, ...,\xi_m)\|\leq \varepsilon\sqrt{m} )\leq (C_{\text{\tiny\rightef{l: tensor}}} K\varepsilon)^{ m}.
$$
\mathcal{E}_{n-1}d{lemma}
Recall that for a real-valued random variable $\xi$ its {\it L\'evy concentration function} $\mathcal{Q}(\xi,t)$ is defined as
$$
\mathcal{Q}(\xi,t):=\sup\limits_{\left\langlembdabda\in{\mathbb R}}{\mathbb P}\bigl\{|\xi-\left\langlembdabda|\leq t\bigr\},\;\;t>0.
$$
We will need bounds on the L\'evy concentration function of sums of independent random variables.
Such inequalities were investigated in many works, starting with L\'evi, Doeblin, Kolmogorov, Rogozin.
We quote here a result due to Kesten \cite{kesten}, who improved Rogozin's estimate \cite{Rog}.
\begin{prop}\left\langlebel{rog}\left\langlebel{prop: esseen}
Let $\xi_1, \xi_2, \ldots, \xi_m$ be independent random variables and $\left\langlembda, \left\langlembda _1, ...,\left\langlembda _m>0$
satisfy $\left\langlembda \geq \max_{i\leq m} \left\langlembda _i$. Then there exists an absolute positive constant $C$ such that
$$
\mathcal{Q}\Bigl(\sum_{i=1}^m \xi_i, \left\langlembda \Bigr)\leq {\mathcal F}rac{C\, \left\langlembda \, \max_{i\leq m} \mathcal{Q}(\xi_i, \left\langlembda)
}{ \sqrt{\sum_{i=1}^m \left\langlembda_i^2(1-\mathcal{Q}(\xi_i, \left\langlembda _i))}} .
$$
\mathcal{E}_{n-1}d{prop}
This proposition together with Lemma~\rightef{tens} immediately implies the following consequence,
in which, given $A\subset [m]$ and $x\in {\mathbb R}^m$,
$x_A$ denotes coordinate projection of $x$ on ${\mathbb R}^A$.
\begin{prop}\left\langlebel{rogozin}
There exists and absolute constant $C_0\geq 1$ such that the following holds.
Let $p\in (0, 1/2)$. Let ${\rm dist}eltalta$ be a Bernoulli($p$) random variable. Let
${\rm dist}eltalta_j$, $j\leq n$,
and ${\rm dist}eltalta_{ij}$, $i,j\leq n$,
be independent copies of ${\rm dist}eltalta$. Let $M=({\rm dist}eltalta_{ij})_{ij}$.
Let $A\subset [n]$ and $x\in {\mathbb R}^n$ be such that $\|x_A\|_\infty \leq C_0^{-1}\sqrt{p}\, \|x_A\|$.
Then
$$
\mathbb{P}\Bigl(\|Mx\| \leq {\mathcal F}rac{\sqrt{pn}}{3\sqrt{2} C_0} \|x_A\| \Bigr)\leq e^{-3n}.
$$
Moreover, if $\left\langlembda := {\mathcal F}rac{\sqrt{p}\, \|x_A\|}{ 3 C_0} \leq 1/3$ then
$
\mathcal{Q}\Bigl(\sum_{j=1}^n {\rm dist}eltalta_j x_j, \left\langlembda \Bigr)\leq e^{-8}.
$
\mathcal{E}_{n-1}d{prop}
\begin{proof} We start with the ``moreover" part. Assume ${\sqrt{p}\, \|x_A\|} \leq C_0$.
Let $\left\langlembda _j=|x_j|/3$. Clearly, for every $j\leq n$, $\mathcal{Q}(x_j {\rm dist}eltalta _j,|x_j|/3)= \mathcal{Q}({\rm dist}eltalta _j, 1/3) = 1-p$.
Proposition~\rightef{rog} implies that for every $\left\langlembda$ satisfying $\max_{j\in A} \left\langlembda _j\leq \left\langlembda \leq 1/3$
one has
$$
\mathcal{Q}\Bigl(\sum_{j=1}^n x_j{\rm dist}eltalta_j, \left\langlembda \Bigr)\leq
\mathcal{Q}\Bigl(\sum_{j\in A} x_j{\rm dist}eltalta_j, \left\langlembda \Bigr)\leq
{\mathcal F}rac{C\, \left\langlembda
}{ \sqrt{\sum_{j\in A} \left\langlembda_j^2 \, p}} =
{\mathcal F}rac{3 C\, \left\langlembda
}{ \sqrt{ p}\, \|x_A\|}.
$$
Choosing $C_0=C e^8$ and $\left\langlembda = \sqrt{ p}\, \|x_A\|/ (3C_0)$
(note that the assumption on $\|x_A\|_\infty$
ensures that $\left\langlembdabda\geq \left\langlembdabda_j$ for all $j\in A$) we obtain the ``moreover" part.
Now apply Lemma~\rightef{tens} with $\xi _i= (Mx)_i = \sum_{j=1}^n x_j{\rm dist}eltalta_{ij}$, $\varepsilon =1/2$,
$\gammamma = e^{-8}$, $m=n$. We have
$$
\mathbb{P}\Bigl(\|Mx\| \leq \left\langlembda \sqrt{n/2}
\Bigr)\leq (2e)^{n/2} \mathcal{E}xp(-4 n )\leq \mathcal{E}xp(-3 n ).
$$
This implies the bound under assumption ${\sqrt{p}\, \|x_A\|} \leq C_0$, which can be removed
by normalizing $x$.
\mathcal{E}_{n-1}d{proof}
We also will need the following combination of a simple anti-concentration fact
with Lemma~\rightef{tens}.
\begin{prop}\left\langlebel{anti2}
Let $p\in (0, 1/20)$ and $a_1lphapha >0$. Let ${\rm dist}eltalta$ be a Bernoulli($p$) random variable. Let
${\rm dist}eltalta_j$, $j\leq n$,
and ${\rm dist}eltalta_{ij}$, $i,j\leq n$,
be independent copies of ${\rm dist}eltalta$. Let $M=({\rm dist}eltalta_{ij})_{ij}$.
Let $x\in {\mathbb R}^n$ be such that $x_2^*\geq a_1lphapha$.
Then
$$
\mathcal{Q}\Bigl(\sum_{j=1}^n x_j {\rm dist}eltalta_j, a_1lphapha/2.1 \Bigr)\leq \mathcal{E}xp(-1.9 p)
\quad \quad \mbox{ and } \quad \quad
\mathbb{P}\Bigl(\|Mx\| \leq {\mathcal F}rac{a_1lphapha \sqrt{pn}}{7\sqrt{\ln(e/p)}} \Bigr)\leq \mathcal{E}xp( - 1.6np).
$$
\mathcal{E}_{n-1}d{prop}
\begin{proof}
Without loss of generality we assume that $x_1^* = |x_1|$ and $x_2^*=|x_2|$. Note that
$x_1{\rm dist}eltalta_1+x_2 {\rm dist}eltalta_2$ takes value in $E_1:=\{0, x_1+x_2\}$ with probability $(1-p)^2+p^2\leq 1-1.9p$
and in $E_2:=\{x_1, x_2\}$ with probability $2p(1-p)\leq 1-1.9p$. Since the distance between $E_1$ and $E_2$
is $\min(|x_1|, |x_2|) =|x_2|$ and since $\mathcal{Q}\bigl(\sum_{j=1}^n x_j{\rm dist}eltalta_j, \left\langlembda \bigr)\leq
\mathcal{Q}\bigl(\sum_{j=1}^2 x_j{\rm dist}eltalta_j, \left\langlembda \bigr)$, the first inequality follows.
Now apply Lemma~\rightef{tens} with $\xi _i= (Mx)_i = \sum_{j=1}^n x_j{\rm dist}eltalta_{ij}$, $\varepsilon =p/(10\ln (e/p))$,
$\gammamma = e^{-1.9 p}$, $m=n$. We note that then $\varepsilon\ln (e/\varepsilon) \leq p/4$ and therefore we have
$$
\mathbb{P}\Bigl(\|Mx\| \leq {\mathcal F}rac{a_1lphapha \sqrt{pn}}{2.1\sqrt{10\ln(e/p)}}
\Bigr)\leq (e/\varepsilon)^{\varepsilon n} \mathcal{E}xp(- 1.9 p n(1-\varepsilon) )
$$
$$
\leq
\mathcal{E}xp(pn/4 - 1.9np(1-\varepsilon)) \leq \mathcal{E}xp( - 1.6np),
$$
which completes the proof.
\mathcal{E}_{n-1}d{proof}
Finally we provide Esseen's lemma \cite{Esseen-type}, needed to prove Theorem~\rightef{p: cf est}.
\begin{lemma}[Esseen] \left\langlebel{ess}
There exists an absolute constant $C>0$ such that the following holds.
Let $\xi_i$, $i\leq m$ be independent random variables. Then for every $\tau>0$,
\begin{align*}
\mathcal{Q}\Big(\sum\limits_{i=1}^m \xi_i,\tau\Big)
&\leq C\int\limits_{-1}^1 \mathbb{P}rod\limits_{i=1}^m|{\mathbb E}\mathcal{E}xp(2\mathbb{P}i{\bf i}\xi_i s/\tau)|\,ds.
\mathcal{E}_{n-1}d{align*}
\mathcal{E}_{n-1}d{lemma}
\subsection{Net argument}
\left\langlebel{net}
Here we discuss special nets that will be used and corresponding approximations.
We fix the following notations. Let $\mathcal{E}dv={\bf 1} /\sqrt{n}$ be the unit vector in the direction
of ${\bf 1}$. Let $P_\mathcal{E}dv$ be the projection on $\mathcal{E}dv^\mathbb{P}erp$ and
$P_\mathcal{E}dv^\mathbb{P}erp$ be the projection on $\mathcal{E}dv$, that is $P_\mathcal{E}dv^\mathbb{P}erp = \left\langle \cdot , \mathcal{E}dv\righta \mathcal{E}dv$.
Similarly, for $j\leq n$, let $P_j$ be the projection on $e_j^\mathbb{P}erp$ and
$P_j^\mathbb{P}erp$ be the projection on $e_j$. Recall that for $x\in {\mathbb R}^n$, the permutation ${\sigma}ma_x$
satisfies $|x_{{\sigma}ma_x(i)}|= x_i^*$, $i\leq n$. Define a (non-linear)
operator $Q:{\mathbb R}^n\to {\mathbb R}^n$ by $Qx = P_{F(x)} x$ --- the coordinate projection on ${\mathbb R}^{F(x)}$,
where $F(x)={\sigma}ma _x ([2, n])$, in other words $Q$ annihilates the largest coordinate
of a vector.
Consider the triple norm on ${\mathbb R}^n$ defined by
$$
||| x |||^2 := \|P_\mathcal{E}dv x\|^2 + p n \|P_\mathcal{E}dv^\mathbb{P}erp x\|^2
$$
(note that $\|P_\mathcal{E}dv^\mathbb{P}erp x\| = |\left\langle x , \mathcal{E}dv\righta|$).
We will use the following notion of shifted sparse vectors
(recall here that ${\sigma}ma_x$ is the permutation responsible for
the non-increasing rearrangement). Given $m\leq n$ and a parameter $\gammamma >0$, define
$$
U(m, \gammamma):=\Big\{ x\in {\mathbb R}^n \,\, : \, \, \mathcal{E}xists A\subset [n], |A|=n- m,\,\, \mathcal{E}xists |\left\langlembda|\leq {\mathcal F}rac{2}{\sqrt m} \, \,
{\mathcal F}orall i \in A
\,\, \mbox{ one has }
\, \, |x_i- \left\langlembda | \leq {\mathcal F}rac{\gammamma}{\sqrt n} \Big\}.
$$
Further, given another parameter $\beta >0$, define the set
$$
V(\beta):=\{ x\in {\mathbb R}^n \, : \, \|x\|_\infty \leq 1 \, \,
\mbox{ and } \, \, \|Qx\|\leq \beta \}.
$$
\begin{lemma}\left\langlebel{cardnet}
Let $0<8\gammamma\leq \varepsilon \leq \beta$ and $1\leq m\leq n$.
Then there exists an $\varepsilon$-net
in $V(\beta)\cap U(m, \gammamma)$
with respect to $|||\cdot|||$ of cardinality at most
$$
{\mathcal F}rac{ 2^{10} \sqrt{p}\, n^2 }{\varepsilon^2\, \sqrt{m}} \left({\mathcal F}rac{9 \beta}{\varepsilon} \right) ^{m} {n\choose m}.
$$
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Denote $V:=V(\beta)\cap U(m, \gammamma)$.
For each $x\in V$ let $A(x)$ be a set from the definition of $U(m, \gammamma)$
(if the choice of $A(x)$ is not unique, we fix one of them).
Fix $E\subset [n]$ of cardinality $m$.
We first consider vectors $x\in V$ satisfying $A(x)=E^c$.
Fix $j\leq n$ and denote
$$
V_j=V_j(E):= \{ x\in V \, : \, j={\sigma}ma_x(1)\,\, \mbox{ and } \, \, A(x)=E^c\}
$$
(thus $x_1^*=|x_j|$ on $V_j$).
We now construct a net for $V_j$. It will be obtained as the sum of four nets, where the first one
deals with just one coordinate, $j$, ``killing" the maximal coordinate; the second one deals with
non-constant part of the vector, consisting of at most $m$ coordinates (excluding $x_1^*$);
the third one deals with almost constant coordinates (corresponding to $A(x)$); and the fourth net deals with the direction of the constant vector.
This way, three
of our four nets are $1$-dimensional.
Let $P_W$ be the coordinate projection onto ${\mathbb R}^{W}$, where $W=E\setminus\{j\}$.
Note that the definition of $V(\beta)$ implies that $\|P_W(x)\|\leq \beta$ for every $x\in V_j$.
Let, as before, $P_j^\mathbb{P}erp$ be the projection onto $e_j$.
Let $\mathcal{N}_1$ be an $\varepsilon/4$-net in
$P_j^\mathbb{P}erp (V_j)\subset [-1, 1]e_j$
of cardinality at most $8/\varepsilon$.
Let $\mathcal{N}_2$ be an $\varepsilon/4$-net (with respect to the Euclidean metric)
in $P_F( V_j)$ of cardinality
at most
$
\left(1+8 \beta/\varepsilon\right)^{m}.
$
Further, let $\mathcal{N}_3$ be an $\varepsilon/(8\sqrt{n})$-net in the segment
$[-2/\sqrt{m}, 2/\sqrt{m}]\sum_{i\in E^c\setminus\{j\}} e_i$
with cardinality at most $16\sqrt{n}/(\varepsilon\sqrt{m})$.
Then by the construction of the nets and by the definition of $U(m, \gammamma)$
for every $x\in V_j$ there exist $y^i_x\in \mathcal{N}_i$, $i\leq 3$,
such that for $y_x=y^1_x+y^2_x+y^3_x$,
$$
\| x - y_x \| ^2\leq {\mathcal F}rac{\varepsilon^2}{16}+{\mathcal F}rac{\varepsilon^2}{16}+\sum_{i\in E^c\setminus\{j\}}
\left({\mathcal F}rac{\gammamma}{\sqrt{n}} + {\mathcal F}rac{\varepsilon}{8\sqrt{n}}\right)^2\leq
{\mathcal F}rac{3\varepsilon^2}{16};
$$
in particular, $\|P_\mathcal{E}dv (x-y_x)\|\leq \sqrt{3/16}\varepsilon$.
Finally, let $\mathcal{N}_4$ be an $\varepsilon/(4\sqrt{pn})$-net in the segment
$(\varepsilon/2)[-\mathcal{E}dv, \mathcal{E}dv]$ with cardinality at most $8\sqrt{pn}$.
Then for every $x\in V_j$ there exists $y_x$ as above and $y_x^4\in \mathcal{N}_4$
with
$$
||| x - y_x -y^4_x||| ^2=
||| P_\mathcal{E}dv (x - y_x) +P_\mathcal{E}dv^\mathbb{P}erp (x-y_x) -y^4_x|||^2=
\| P_\mathcal{E}dv (x - y_x)\|^2 +pn \|P_\mathcal{E}dv^\mathbb{P}erp (x-y_x) -y^4_x\|^2\leq \varepsilon^2/4.
$$
Thus the set
$
\mathcal{N}_{E,j} = \mathcal{N}_1 + \mathcal{N}_2 + \mathcal{N}_3+ \mathcal{N}_4
$
is an $(\varepsilon/2)$-net for $V_j$ with respect to $|||\cdot|||$ and its cardinality is bounded by
$$
{\mathcal F}rac{2^{10} \sqrt{p}\, n}{\varepsilon^2\sqrt{m}}
\left(1+{\mathcal F}rac{8 \beta}{\varepsilon}\right)^{m}.
$$
Taking union of such nets over all choices of $E\subset [n]$ and all $j\leq n$ we obtain an $(\varepsilon/2)$-net $\mathcal{N}_0$
in $|||\cdot|||$ for $V$ of desired cardinality.
Using standard argument, we pass to an $\varepsilon$-net $\mathcal{N} \subset V$ for $V$.
\mathcal{E}_{n-1}d{proof}
Later we apply Lemma~\rightef{cardnet} with the following proposition.
\begin{prop}\left\langlebel{nettri}
Let $n$ be large enough and $(4\ln n)/n\leq p<1/2$, and $\varepsilon >0$.
Denote
$$
{\mathbb E}vent_{nrm}:= \{ M\in {\mathcal{M}_{n}} \, : \, \|M - p {\bf 1}{\bf 1}^\top\|\leq 60 \sqrt{n p} \quad \mbox{ and } \quad
\|M {\bf 1}\| \leq 3 p n^{3/2} \}.
$$
Then for every $x\in {\mathbb R}^n$ satisfying $|||x|||\leq \varepsilon$ and every $M\in {\mathbb E}vent_{nrm}$ one has
$ \|M x\| \leq 100\sqrt{ p n} \varepsilon .$
\mathcal{E}_{n-1}d{prop}
\begin{proof}
Let $w=P_\mathcal{E}dv^\mathbb{P}erp x$. Then, by the definition of the triple norm,
$\|w\|\leq |||x|||/\sqrt{pn}\leq \varepsilon /\sqrt{pn}$.
Clearly,
$$
( p{\bf 1}{\bf 1}^\top) (x-w)= ( p{\bf 1}{\bf 1}^\top)P_\mathcal{E}dv x = 0.
$$
Therefore, using that $M\in {\mathbb E}vent_{nrm}$, we get
$$
\| M (x-w)\| =\| (M-p{\bf 1}{\bf 1}^\top) (x-w)\|\leq 60 \sqrt{p n}\|x-w\|\leq 70 \sqrt{p n} \varepsilon.
$$
Since $w={\bf 1} \|w\|/\sqrt{n}$ and $\|w\|\leq \varepsilon /\sqrt{pn}$, using again that
$M\in {\mathbb E}vent_{nrm}$,
we observe that
$$
\|Mw\| \leq {\mathcal F}rac{\varepsilon}{\sqrt{p}\, n} \|M {\bf 1} \| \leq
3 \sqrt{pn} \varepsilon .
$$
The proposition follows by the triangle inequality.
\mathcal{E}_{n-1}d{proof}
\section{Unstructured vectors}\left\langlebel{s: unstructured}
The goal of this section is to prove Theorem~\rightef{th: gradual}.
Recall that given growth function ${\bf g}$ and parameters $r,{\rm dist}eltalta,\rightho\in (0,1)$,
the set of vectors ${\mathcal V}_n={\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)$ was defined in (\rightef{eq: gnc def}).
In the next two sections (dealing with invertibility over structured vectors) we work with two
different growth functions; one will be applied to the case of constant $p$ and the other
one (giving a worse final estimate) is
suitable in the general case. For this reason, and to increase flexibility of our argument,
rather than fixing a specific growth function here, we will work with an arbitrary
non-decreasing function ${\bf g}\, : \, [1,\infty)\to [1,\infty)$
satisfying the additional assumption (\rightef{gfncond})
with a ``global'' parameter $K_3\geq 1$.
\subsection{Degree of unstructuredness: definition and basic properties}
Below, for any non-empty finite integer subset $S$, we denote by $\mathcal{E}ta[S]$ a random variable uniformly
distributed on $S$.
Additionally, for any $K_2\geq 1$, we fix a smooth version of $\max({\mathcal F}rac{1}{K_2},t)$.
More precisely, let us fix a function $\mathbb{P}si_{K_2}:{\mathbb R}_+\to{\mathbb R}_+$ satisfying
\begin{itemize}
\item The function $\mathbb{P}si_{K_2}$ is twice continuously differentiable, with
$\|\mathbb{P}si_{K_2}'\|_\infty =1$ and $\|\mathbb{P}si_{K_2}''\|_\infty <\infty$;
\item $\mathbb{P}si_{K_2}(t)={\mathcal F}rac{1}{K_2}$ for all $t\leq {\mathcal F}rac{1}{2K_2}$;
\item ${\mathcal F}rac{1}{K_2}\geq \mathbb{P}si_{K_2}(t)\geq t$ for all ${\mathcal F}rac{1}{K_2}\geq t\geq {\mathcal F}rac{1}{2K_2}$;
\item $\mathbb{P}si_{K_2}(t)= t$ for all $t\geq {\mathcal F}rac{1}{K_2}$.
\mathcal{E}_{n-1}d{itemize}
In what follows, we view the maximum of the second derivative of $\mathbb{P}si_{K_2}$
as a function of $K_2$ (the nature of this function is completely irrelevant as we do not attempt to track
magnitudes of constants involved in our arguments).
Fix an integer $n\geq 1$ and an integer $m\leq n/2$. Recall that
given a vector $v\in{\mathbb R}^n$ and parameters $K_1,K_2\geq 1$, the {\it degree of unstructuredness (u-degree)}
${\bf UD}_n = {\bf UD}_n(v,m,K_1,K_2)$ of $v$ was defined in (\rightef{udeg}).
The quantity ${\bf UD}_n$ will serve as a measure of unstructuredness of the vector $v$
and in its spirit is similar to the notion of the essential least common denominator introduced
earlier by Rudelson and Vershynin \cite{RV}.
Here {\it unstructuredness} refers to the uniformity in the locations of components of $v$
on the real line.
The larger the degree is, the better
anti-concentration properties of an associated random linear combination are.
The functions $\mathbb{P}si_{K_2}$ employed in the definition will be important when discussing certain
stability properties of ${\bf UD}_n$.
We start with a proof of Theorem~\rightef{p: cf est} which connects the definition of the u-degree with
anti-concentra\-tion properties.
\begin{proof}[Proof of Theorem~\rightef{p: cf est}]
For any sequence of disjoint subsets $S_1,{\rm dist}ots,S_m$ of $[n]$ of cardinality $\lfloor n/m\rightfloor$ each, set
$$
{\mathbb E}vent_{S_1,{\rm dist}ots,S_m}:=\big\{{\rm supp\, } X\cap S_i=1\mbox{ for all $i\leq m$}\big\}.
$$
Note that each point $\omega$ of the probability space belongs to the same number of events from the collection
$\{{\mathbb E}vent_{S_1,{\rm dist}ots,S_m}\}_{S_1,{\rm dist}ots,S_m}$, therefore, for $A_{nm}$ defined in (\rightef{anm}) we have
for any $\left\langlembdabda\in{\mathbb R}$ and $\tau>0$,
\begin{equation}\left\langlebel{eq: aux-92-502}
\begin{split}
{\mathbb P}\Big\{&\Big|\sum\limits_{i=1}^n v_i X_i-\left\langlembdabda\Big|\leq\tau\Big\}=
A_{nm} \,
\sum\limits_{S_1,{\rm dist}ots,S_m}{\mathbb P}\Big\{\Big|\sum\limits_{i=1}^n v_i X_i-\left\langlembdabda\Big|\leq\tau\;\big|\;{\mathbb E}vent_{S_1,{\rm dist}ots,S_m}\Big\}.
\mathcal{E}_{n-1}d{split}
\mathcal{E}_{n-1}d{equation}
Further, conditioned on an event ${\mathbb E}vent_{S_1,{\rm dist}ots,S_m}$, the random sum $\sum\limits_{i=1}^n v_i X_i$
is equidistributed with $\sum\limits_{i=1}^m v_{\mathcal{E}ta[S_i]}$
(where we assume that $\mathcal{E}ta[S_1],{\rm dist}ots,\mathcal{E}ta[S_m]$ are jointly independent with ${\mathbb E}vent_{S_1,{\rm dist}ots,S_m}$).
On the other hand, applying Lemma~\rightef{ess}, we observe that for every $\tau>0$,
\begin{align*}
\mathcal{Q}\Big(\sum\limits_{i=1}^m v_{\mathcal{E}ta[S_i]},\tau\Big)
&\leq C'\int\limits_{-1}^1 \mathbb{P}rod\limits_{i=1}^m|{\mathbb E}\mathcal{E}xp(2\mathbb{P}i{\bf i}v_{\mathcal{E}ta[S_i]} s/\tau)|\,ds\\
&= C'\,m^{-1/2}\,\tau\int\limits_{-\sqrt{m}/\tau}^{\sqrt{m}/\tau} \mathbb{P}rod\limits_{i=1}^m|{\mathbb E}\mathcal{E}xp(2\mathbb{P}i{\bf i}v_{\mathcal{E}ta[S_i]}\,m^{-1/2} s)|\,ds,
\mathcal{E}_{n-1}d{align*}
for a universal constant $C'>0$.
Combining this with \mathcal{E}qref{eq: aux-92-502}, we get for every $\tau>0$,
\begin{align*}
\mathcal{Q}\Big(\sum\limits_{i=1}^n v_i X_i,\tau\Big)
&\leq A_{nm}\,
\sum\limits_{S_1,{\rm dist}ots,S_m}\mathcal{Q}\Big(\sum\limits_{i=1}^n v_i X_i,\tau\;\big\vert\;{\mathbb E}vent_{S_1,{\rm dist}ots,S_m}\Big)\\
&\leq {\mathcal F}rac{C'\tau
A_{nm}}{\sqrt{m}}
\sum\limits_{S_1,{\rm dist}ots,S_m}
\int\limits_{-\sqrt{m}/\tau}^{\sqrt{m}/\tau} \mathbb{P}rod\limits_{i=1}^m|{\mathbb E}\mathcal{E}xp(2\mathbb{P}i{\bf i}v_{\mathcal{E}ta[S_i]} \,m^{-1/2}s)|\,ds.
\mathcal{E}_{n-1}d{align*}
Setting $\tau:=\sqrt{m}/{\bf UD}_n$, where ${\bf UD}_n={\bf UD}_n(v,m,K_1,K_2)$, we obtain
\begin{align*}
\mathcal{Q}&\Big(\sum\limits_{i=1}^n v_i X_i,\sqrt{m}/{\bf UD}_n\Big)
\leq {\mathcal F}rac{C'
A_{nm}}{{\bf UD}_n}\,\sum\limits_{S_1,{\rm dist}ots,S_m}
\int\limits_{-{\bf UD}_n}^{{\bf UD}_n} \mathbb{P}rod\limits_{i=1}^m|{\mathbb E}\mathcal{E}xp(2\mathbb{P}i{\bf i}v_{\mathcal{E}ta[S_i]}\,m^{-1/2} s)|\,ds
\leq {\mathcal F}rac{C' K_1}{{\bf UD}_n},
\mathcal{E}_{n-1}d{align*}
in view of the definition of ${\bf UD}_n(v,m,K_1,K_2)$.
The result follows.
\mathcal{E}_{n-1}d{proof}
For the future use we state an immediate consequence
of Theorem~\rightef{p: cf est} and Lemma~\rightef{l: tensor}.
\begin{cor}\left\langlebel{cor: cf tensor}
Let $n,\mathcal{E}ll\in{\mathbb N}$, let $m_1,{\rm dist}ots,m_\mathcal{E}ll$ be integers with $m_i\leq n/2$ for all $i$,
and let $K_1,K_2\geq 1$.
Further, let $v\in{\mathbb R}^n$, and let $B$ be an $\mathcal{E}ll\times n$ random matrix with independent
rows such that the $i$-th row is
uniformly distributed on the set of vectors with $m_i$ ones and $n-m_i$ zeros.
Then for any non-random vector $Z\in{\mathbb R}^\mathcal{E}ll$ we have
$$
{\mathbb P}\big\{\| Bv-Z\|\leq \sqrt{\mathcal{E}ll}\, t\big\}\leq
\Big(2C_{\text{\tiny\rightef{l: tensor}}}C_{\text{\tiny\rightef{p: cf est}}} t/\sqrt{\min\limits_i m_i}\Big)^\mathcal{E}ll
\quad \mbox{for all }t\geq \max\limits_i{\mathcal F}rac{\sqrt{m_i}}{{\bf UD}_n(v,m_i,K_1,K_2)}.
$$
\mathcal{E}_{n-1}d{cor}
The parameter $K_2$
which did not participate in any way in the proof of Theorem~\rightef{p: cf est} is needed to guarantee
a certain stability property of ${\bf UD}_n(v,m,K_1,K_2)$.
We would like to emphasize that the use of functions $\mathbb{P}si_{K_2}$ is a technical element of the argument.
\begin{prop}[Stability of the u-degree]\left\langlebel{l: stability of bal}
For any $K_2\geq 1$ there are $c_{\text{\tiny\rightef{l: stability of bal}}},c_{\text{\tiny\rightef{l: stability of bal}}}'>0$
depending only on $K_2$ with the following property.
Let $K_1\geq 1$, $v\in{\mathbb R}^n$, $k\in{\mathbb N}$, $m\leq n/2$, and assume that ${\bf UD}_n(v,m,K_1,K_2)\leq c_{\text{\tiny\rightef{l: stability of bal}}}' k$.
Then there is a vector $y\in \big({\mathcal F}rac{1}{k}{\mathbb Z}\big)^n$ such that $\|v-y\|_\infty\leq {\mathcal F}rac{1}{k}$, and such that
$$
{\bf UD}_n(y,m,c_{\text{\tiny\rightef{l: stability of bal}}} K_1,K_2)\leq {\bf UD}_n(v,m,K_1,K_2)\leq
{\bf UD}_n(y,m,c_{\text{\tiny\rightef{l: stability of bal}}}^{-1} K_1,K_2)
$$
\mathcal{E}_{n-1}d{prop}
To prove the proposition we need two auxiliary lemmas.
$u>0$, $\varepsilon\in(0,u/2]$,
\begin{lemma}\left\langlebel{l: aux 0948523059873205}
Let $0\ne z\in \mathbb{C}$, $\varepsilon \in [0, |z|/2]$ and let $W$ be a random vector in $\mathbb{C}$ with
${\mathbb E} W=0$ and with $|W|\leq \varepsilon$ everywhere on the probability space.
Then
$$
\big|{\mathbb E} |z+W|-|z|\big|\leq {\mathcal F}rac{\varepsilon^2}{|z|}.
$$
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
We can view both $z$ and $W$ as vectors in ${\mathbb R}^2$, and can assume without loss of generality that $z=(z_1,0)$, with $z_1=|z|$. Then $|z_1 + W_1|= z_1 + W_1$ and
$$
z_1+W_1 \leq |z+W|=\sqrt{(z_1+W_1)^2+W_2^2}
\leq (z_1+W_1)+{\mathcal F}rac{W_2^2}{2|z_1+W_1|}\leq (z_1+W_1)+{\mathcal F}rac{\varepsilon^2}{2(|z|-\varepsilon)}.
$$
Hence,
$$
|z|=z_1 = {\mathbb E} (z_1+W_1) \leq {\mathbb E} |z+W| \leq {\mathbb E} (z_1+W_1) + {\mathcal F}rac{\varepsilon^2}{|z|}
=|z|+{\mathcal F}rac{\varepsilon^2}{|z|},
$$
which implies the desired estimate.
\mathcal{E}_{n-1}d{proof}
\begin{lemma}\left\langlebel{l: aux 29853205983475938}
Let $\left\langlembda, \mu\in{\mathbb R}$, and let $\xi$ be a random variable in ${\mathbb R}$ with ${\mathbb E}\xi=\mu$ and with
$|\xi-\mu|\leq \left\langlembda$ everywhere on the probability space.
Then for any $s\in{\mathbb R}$ we have
$$
\big|{\mathbb E}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,\xi\,s\big)-\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,\mu\,s\big)\big|\leq (2\mathbb{P}i \left\langlembda s)^2.
$$
\mathcal{E}_{n-1}d{lemma}
\begin{proof} Denote $\xi'=\xi-\mu$. Then ${\mathbb E}\xi=0$ and $|\xi'|\leq \left\langlembda$. Therefore, using that
$|\sin x|\leq |x|$ and $|\sin x - x |\leq x^2/2$ for every $x\in {\mathbb R}$, we obtain
\begin{align*}
\big|{\mathbb E}\mathcal{E}xp&\big(2\mathbb{P}i{\bf i}\,\xi\,s\big)-\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,\mu\,s\big)\big|=
\big|{\mathbb E}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,\xi' s\big)-1\big|=
\big|{\mathbb E} \cos\big(2\mathbb{P}i\xi's\big)-1 + {\bf i}\, {\mathbb E} \sin\big(2\mathbb{P}i\xi' s\big)\big|
\\ &=
\big|-2 {\mathbb E} \sin^2 \big(\mathbb{P}i\xi's\big) + {\bf i}\, {\mathbb E} \big(\sin\big(2\mathbb{P}i\xi's\big)- 2\mathbb{P}i\xi's\big)\big|
\leq 2 (\mathbb{P}i\left\langlembda s)^2 + (2\mathbb{P}i\left\langlembda s)^2/2 = (2\mathbb{P}i\left\langlembda s)^2.
\mathcal{E}_{n-1}d{align*}
\mathcal{E}_{n-1}d{proof}
\begin{proof}[Proof of Proposition~\rightef{l: stability of bal}]
To prove the proposition, we will use the {\it randomized rounding} which is a well known
notion in computer science, and was recently applied in the random matrix context in
\cite{Livshyts} (see also \cite{KT-last,LTV}).
Define a random vector $Y$ in $\big({\mathcal F}rac{1}{k}{\mathbb Z}\big)^n$ with independent components $Y_1,{\rm dist}ots,Y_n$
such that each component $Y_i$ has distribution
$$
Y_i=\begin{cases}
{\mathcal F}rac{1}{k}\lfloor k v_i\rightfloor,&\mbox{ with probability $\lfloor k v_i\rightfloor-kv_i+1$},\\
{\mathcal F}rac{1}{k}\lfloor k v_i\rightfloor+{\mathcal F}rac{1}{k},&\mbox{ with probability $kv_i-\lfloor k v_i\rightfloor$}.
\mathcal{E}_{n-1}d{cases}
$$
Then ${\mathbb E} Y_i=v_i$, $i\leq n$ and, deterministically, $\|v-Y\|_\infty\leq1/k$.
Fix for a moment a number $s\in(0,k/(14\mathbb{P}i K_2)]$ and a subset $S\subset[n]$ of cardinality $\lfloor n/m\rightfloor$.
Our intermediate goal is to estimate the quantity
$$
{\mathbb E}\,\mathbb{P}si_{K_2}\Big(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}\sum_{j\in S}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,Y_{j} \,s\big)\Big|
\Big).
$$
Denote
$$
V=V_S:=\left|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}\sum_{j\in S}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,v_j \,s\big)\right|
= \big|{\mathbb E}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,v_{\mathcal{E}ta[S]} \,s\big)\big|
$$
and consider two cases.
\|\cdot\|indent
{\it Case 1. }
$V\leq {\mathcal F}rac{1}{2K_2}-{\mathcal F}rac{2\mathbb{P}i\,s}{k}$.
Using that $|e^{{\bf i} x}-1|\leq |x|$ for every $x\in {\mathbb R}$, we observe that deterministically
\begin{equation}\left\langlebel{ineqktwo}
|\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,v_j \,s\big)-\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,Y_j \,s\big)|\leq 2\mathbb{P}i s/k.
\mathcal{E}_{n-1}d{equation}
Therefore, by the definition of the function $\mathbb{P}si_{K_2}$, in this case we have on the entire probability space
$$
\mathbb{P}si_{K_2}\Big(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}\sum_{j\in S}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,Y_{j} \,s\big)\Big|
\Big)=\mathbb{P}si_{K_2}(V)={\mathcal F}rac{1}{K_2}.
$$
\|\cdot\|indent
{\it Case 2. }
$V> {\mathcal F}rac{1}{2K_2}-{\mathcal F}rac{2\mathbb{P}i\,s}{k} \geq {\mathcal F}rac{1}{4K_2}$.
Set
$$
z:={\mathcal F}rac{1}{\lfloor n/m\rightfloor}{\mathbb E}\,\sum_{j\in S}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,Y_{j} \,s\big) \quad \mbox{ and } \quad
W:={\mathcal F}rac{1}{\lfloor n/m\rightfloor}\sum_{j\in S}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,Y_{j} \,s\big)-z.
$$
Then ${\mathbb E} W=0$ and, using again $|e^{{\bf i} x}-1|\leq |x|$, we see that $|W|\leq 2\mathbb{P}i s/k$ everywhere.
By Lemma~\rightef{l: aux 29853205983475938},
$|z-V|\leq (2\mathbb{P}i s /k)^2$, in particular, $z\geq V-(2\mathbb{P}i s /k)^2\geq 1/(3K_2) \geq 4\mathbb{P}i s /k \geq |W|/2$.
Therefore we may apply Lemma~\rightef{l: aux 0948523059873205}, to obtain
$$
\big|{\mathbb E}|W+z|-|z|\big|\leq {\mathcal F}rac{4\mathbb{P}i^2 s^2}{|z|k^2}\leq {\mathcal F}rac{12\mathbb{P}i^2 K_2 s^2}{k^2}.
$$
This implies,
\begin{align}
\Big|{\mathbb E}\Big|&{\mathcal F}rac{1}{\lfloor n/m\rightfloor}\sum_{j\in S}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,Y_{j} \,s\big)\Big|
-V\Big|
=
\Big|{\mathbb E}|W+z|-|z|+|z| -V\Big|
\leq {\mathcal F}rac{16\mathbb{P}i^2 K_2 s^2}{k^2}.\left\langlebel{eq: aux 5987325938}
\mathcal{E}_{n-1}d{align}
To convert the last relation to estimating $\mathbb{P}si_{K_2}(\cdot)$, we will use the
assumption that the second derivative of $\mathbb{P}si_{K_2}$ is uniformly bounded.
Applying Taylor's expansion around the point $V$,
we get
\begin{align*}
{\mathbb E}\,\mathbb{P}si_{K_2}\Big(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}\sum_{j\in S}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,Y_{j} \,s\big)\Big|
\Big)
=
\mathbb{P}si_{K_2}\big(V\big) &+{\mathbb E}\Big(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}\sum_{j\in S}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,Y_{j} \,s\big)\Big|-V\Big)\,\mathbb{P}si_{K_2}'(V)
\\&+C''\Big\|\,\,\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}\sum_{j\in S}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,Y_{j} \,s\big)\Big|-V\Big\|_\infty^2,
\mathcal{E}_{n-1}d{align*}
for some $C''>0$ which may only depend on $K_2$.
Here, $\|\cdot\|_\infty$ denotes the essential supremum of the random variable,
and is bounded above by $2\mathbb{P}i s/k$ by \rightef{ineqktwo}.
Together with \mathcal{E}qref{eq: aux 5987325938} and with $\|\mathbb{P}si_{K_2}'\|_\infty\leq 1$, this gives
$$
\Big|{\mathbb E}\,\mathbb{P}si_{K_2}\Big(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}\sum_{j\in S}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,Y_{j} \,s\big)\Big|\Big)-\mathbb{P}si_{K_2}(V)
\Big|\leq {\mathcal F}rac{\bar C\,s^2}{k^2},
$$
where $\bar C$ depends only on $K_2$.
Since $\mathbb{P}si_{K_2}'\geq 1/(2K_2)$, in both cases we obtain for some $\hat C>0$ depending only on $K_2$,
\begin{align*}
\Big|&{\mathbb E}\,\mathbb{P}si_{K_2}\Big(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}\sum_{j\in S}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,Y_{j} \,s\big)\Big|\Big)
-\mathbb{P}si_{K_2}(V)\Big| \leq {\mathcal F}rac{\hat C\,s^2}{k^2}\mathbb{P}si_{K_2}(V).
\mathcal{E}_{n-1}d{align*}
Using this inequality together with definition of $V=V_S$, integrating over $s$, and summing over all choices of disjoint subsets $S_1,{\rm dist}ots,S_m$ of cardinality $\lfloor n/m\rightfloor$, for every $t\in(0,k/(14\mathbb{P}i K_2)]$ we get the relation
\begin{align*}
\sum\limits_{S_1,{\rm dist}ots,S_m}\;
&\int\limits_{-t}^t \max\bigg(0,1-{\mathcal F}rac{c_0\,s^2}{k^2}\bigg)^m\mathbb{P}rod\limits_{i=1}^{m}\mathbb{P}si_{K_2}\big(
\big|{\mathbb E}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,v_{\mathcal{E}ta[S_i]} \,s\big)\big|\big)\,ds\\
&\leq
\sum\limits_{S_1,{\rm dist}ots,S_m}\;
\int\limits_{-t}^t \mathbb{P}rod\limits_{i=1}^{m}
{\mathbb E}_Y\,\mathbb{P}si_{K_2}\Big(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}\sum_{j\in S_i}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,Y_{j} \,s\big)\Big|\Big)\,ds\\
&\leq
\sum\limits_{S_1,{\rm dist}ots,S_m}\;
\int\limits_{-t}^t \bigg(1+{\mathcal F}rac{C_0\,s^2}{k^2}\bigg)^m\mathbb{P}rod\limits_{i=1}^{m}\mathbb{P}si_{K_2}\big(
\big|{\mathbb E}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,v_{\mathcal{E}ta[S_i]} \,s\big)\big|\big)\,ds,
\mathcal{E}_{n-1}d{align*}
where $C_0,c_0>7\mathbb{P}i K_2$ are constants that may only depend on $K_2$.
Using independence of the components of $Y$, we can take the expectation with respect to $Y$ out of the integral.
Given a vector $Q=(q_1,{\rm dist}ots,q_n)\in{\mathbb R}^n$ and $t\in(0,k/(14\mathbb{P}i K_2)]$, denote
$$
g_t(Q):=
\sum\limits_{S_1,{\rm dist}ots,S_m}\;
\int\limits_{-t}^t \mathbb{P}rod\limits_{i=1}^{m}
\,\mathbb{P}si_{K_2}\Big(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}\sum_{j\in S_i}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,q_{j} \,s\big)\Big|\Big)\,ds.
$$
The above relation implies that there are
two (non-random) realizations $Y'$ and $Y''$ of $Y$ such that for
\begin{align*}
g_t(Y') &\geq I_1:= \max\bigg(0,1-{\mathcal F}rac{c_0\,t^2}{k^2}\bigg)^m\sum\limits_{S_1,{\rm dist}ots,S_m}\;
\int\limits_{-t}^t\mathbb{P}rod\limits_{i=1}^{m}\mathbb{P}si_{K_2}\big(
\big|{\mathbb E}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,v_{\mathcal{E}ta[S_i]} \,s\big)\big|\big)\,ds
\mathcal{E}_{n-1}d{align*}
and
\begin{align*}
g_t(Y'')\leq I_2:=\bigg(1+{\mathcal F}rac{C_0\,t^2}{k^2}\bigg)^m\sum\limits_{S_1,{\rm dist}ots,S_m}\;
\int\limits_{-t}^t\mathbb{P}rod\limits_{i=1}^{m}\mathbb{P}si_{K_2}\big(
\big|{\mathbb E}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,v_{\mathcal{E}ta[S_i]} \,s\big)\big|\big)\,ds.
\mathcal{E}_{n-1}d{align*}
Using properties of the function $\mathbb{P}si_{K_2}$, we note
that for any two non-random vectors $\widetilde Y$ and $\hat Y$ in the range of $Y$
such that they differ on a single coordinate, one has
$
g_t(\widetilde Y)\leq 4K_2\, g_t(\hat Y).
$
Consider a path $Y^{(1)}=Y',Y^{(2)},Y^{(3)},{\rm dist}ots,Y''$
from $Y'$ to $Y''$ consisting of a sequence of non-random vectors in
the range of $Y$ such that each adjacent pair $Y^{(i)},Y^{(i+1)}$ differs on a single coordinate and
let
$$S := \{i\, : \, g_t(Y^{(i)})> 4K_2 I_2\}\subset [1, n-1].$$
If
$S=\mathcal{E}mptyset$, take ${\bf Y}=Y^{(1)}$. Otherwise, let
$\mathcal{E}ll = \max\{i\, : \, g_t(Y^{(i)})> 4K_2 I_2\}$. Then take ${\bf Y}=Y^{(\mathcal{E}ll+1)}$ and note
$g_t(Y^{(\mathcal{E}ll+1)}) \geq g_t(Y^{(\mathcal{E}ll)})/(4K_2)\geq I_2\geq I_1$.
Thus the vector ${\bf Y}$ is in the range of $Y$ and
\begin{align*}
I_1\leq g_t({\bf Y})&\leq 4K_2 I_2.
\mathcal{E}_{n-1}d{align*}
Making substitutions $s'=\sqrt{m} s$, $t'=\sqrt{m} t$ in the integrals in $I_1, I_2$, and assuming that
$t'\leq k/\max(2C_0,2c_0)$ (in this case the condition $t\leq k/(14\mathbb{P}i K_2)$ is satisfied),
we can rewrite the last inequalities as
\begin{multline*}
{\mathcal F}rac{1}{2}\sum\limits_{S_1,{\rm dist}ots,S_m}\;
\int\limits_{-t'}^{t'} \mathbb{P}rod\limits_{i=1}^{m}\mathbb{P}si_{K_2}\big(
\big|{\mathbb E}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,v_{\mathcal{E}ta[S_i]} m^{-1/2}\,s\big)\big|\big)\,ds\\
\leq
\sum\limits_{S_1,{\rm dist}ots,S_m}\;
\int\limits_{-t'}^{t'} \mathbb{P}rod\limits_{i=1}^{m}
\,\mathbb{P}si_{K_2}\Big(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}\sum_{j\in S_i}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,{\bf Y}_{j} m^{-1/2}\,s\big)\Big|\Big)\,ds\\
\leq 6K_2 \,\sum\limits_{S_1,{\rm dist}ots,S_m}\;
\int\limits_{-t'}^{t'} \mathbb{P}rod\limits_{i=1}^{m}\mathbb{P}si_{K_2}\big(
\big|{\mathbb E}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,v_{\mathcal{E}ta[S_i]} m^{-1/2}\,s\big)\big|\big)\,ds.
\mathcal{E}_{n-1}d{multline*}
The result follows by the definition of ${\bf UD}_n(\cdot)$.
\mathcal{E}_{n-1}d{proof}
The last statement to be considered in this subsection asserts that the u-degree of any
vector from ${\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)$ is at least of order $\sqrt{m}$.
\begin{prop}[Lower bound on the u-degree]\left\langlebel{p: low bound on bal}
For any $r,{\rm dist}eltalta,\rightho$ there is
$C_{\text{\tiny\rightef{p: low bound on bal}}}>0$
depending only on $r,{\rm dist}eltalta,\rightho$ with the following property.
Let $K_2\geq 2$, $1\leq m\leq n/C_{\text{\tiny\rightef{p: low bound on bal}}}$,
$K_1\geq C_{\text{\tiny\rightef{p: low bound on bal}}}$
and let $x\in {\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)$.
Then $${\bf UD}_n(x,m,K_1,K_2)\geq \sqrt{m}.$$
\mathcal{E}_{n-1}d{prop}
\begin{lemma}\left\langlebel{l: bal lower aux}
For any $\rightho>0$ and $\kappa\in (0, 1/2]$ there is a constant $\widetilde C>0$ depending only on $\rightho$ and $\kappa$
with the following property. Let $S\neq \mathcal{E}mptyset$ be a finite subset of ${\mathbb Z}$, and let
$(y_w)_{w\in S}$ be a real vector (indexed by $S$).
Assume further that $S_1,S_2$ are two disjoint subsets of $S$, each of cardinality
at least $\kappa|S|$ such that $\min\limits_{w\in S_1}y_w\geq \max\limits_{w\in S_2}y_w+\rightho$. Let $K_2\geq 2$ and
$f$ be a function on $[0,1]$ defined by
$$
f(t):=\mathbb{P}si_{K_2}\Big(\Big|{\mathcal F}rac{1}{|S|}\sum_{w\in S}\mathcal{E}xp(2\mathbb{P}i{\bf i}\,y_w\,t)\Big|\Big),\quad t\in[0,1].
$$
Then for every $b>0$ one has
$$
\big|\big\{t\in[0,1]:\;f(t)\geq 1-b^2\big\}\big|\leq \widetilde C b.
$$
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Clearly we may assume that $b\leq 1/\sqrt{2}$.
Denote $m= \lceileil \kappa|S|\rightceileil$ and
$$
g(t):=\Big|\sum_{w\in S}\mathcal{E}xp(2\mathbb{P}i{\bf i}\,y_w\,t)\Big|,\quad t\in{\mathbb R}.
$$
Let $T\subset S_1\times S_2$ be of cardinality $T=m$ and such that for all $(q,j),(q',j')\in T$ with
$(q,j)\neq (q',j')$ one has $q\neq q'$ and $j\neq j'$.
Then for all $t\in{\mathbb R}$,
$$
g(t)= \Big|\sum_{w\in S_1\cup S_2}\mathcal{E}xp(2\mathbb{P}i{\bf i}\,y_w\,t)+\sum_{w\|\cdot\|tin S_1\cup S_2}\mathcal{E}xp(2\mathbb{P}i{\bf i}\,y_w\,t)
\Big| \leq \sum_{(q,j)\in T}\big|1+\mathcal{E}xp(2\mathbb{P}i{\bf i}\,(y_j-y_q)\,t)\big|+|S|-2m.
$$
Further, take any $u\in (0, 1/\sqrt{2\kappa})$ and observe that for each $(q,j)\in T$, since $|y_j-y_q|\geq \rightho$,
we have
$$
\big|\big\{t\in[0,1]:\;\big|1+\mathcal{E}xp(2\mathbb{P}i{\bf i}\,(y_j-y_q)\,t)\big|\geq 2-2u^2\big\}\big|\leq C'u,
$$
where $C'>0$ may only depend on $\rightho$.
This implies that
$$
\Big|\Big\{t\in[0,1]:\;\big|1+\mathcal{E}xp(2\mathbb{P}i{\bf i}\,(y_j-y_q)\,t)\big|\geq 2-2u^2\mbox{ for at least $m/2$
pairs $(q,j)\in T$}\Big\}\Big|\leq 2C'u.
$$
On the other hand, whenever $t\in[0,1]$ is such that $\big|1+\mathcal{E}xp(2\mathbb{P}i{\bf i}\,(y_j-y_q)\,t)\big|\geq 2-2u^2$
for at most $m/2$ pairs $(q,j)\in T$, we have
$$
g(t)\leq {\mathcal F}rac{m}{2}(2-2u^2)+{\mathcal F}rac{m}{2}\cdot 2+|S|-2m
= |S|-m u^2 \leq |S|(1-\kappa u^2),
$$
whence $f(t)\leq \max\big({\mathcal F}rac{1}{K_2}, 1-\kappa u^2\big) = 1-\kappa u^2$.
Taking $u={\mathcal F}rac{b}{\sqrt{\kappa}}$ we obtain the
desired result with $\widetilde C= {\mathcal F}rac{2C'}{\sqrt{\kappa}}$.
\mathcal{E}_{n-1}d{proof}
\begin{proof}[Proof of Proposition~\rightef{p: low bound on bal}]
Let $A_{nm}$ be defined as in (\rightef{anm}) and $n_{\rm dist}eltalta$, $C_{\rm dist}eltalta$, $\mathcal S$ be from
Lemma~\rightef{l: aux 2498276098059385-}. We assume that $n\geq n_{\rm dist}eltalta$
and $n/m\geq C_{\rm dist}eltalta$. For every $i\leq m$ denote
$$
f_i (s) = \mathbb{P}si_{K_2}\big(
\big|{\mathbb E}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,x_{\mathcal{E}ta[S_i]}\,m^{-1/2} s\big)\big|\big).
$$
Further, let subsets $Q_1$ and $Q_2$ be taken from the definition of non-constant vectors applied to $x$.
Then by Lemma~\rightef{l: aux 2498276098059385-} and since $\mathbb{P}si_{K_2}(1)\leq 1$,
\begin{align*}
&A_{nm}\,
\sum\limits_{(S_1,{\rm dist}ots,S_m)\in\mathcal S}\;
\int\limits_{-\sqrt{m}}^{\sqrt{m}} \mathbb{P}rod\limits_{i=1}^{m}f_i\,ds
\leq \,e^{-c_{\rm dist}eltalta n}\,2\sqrt{m}
+
A_{nm}\,
\sum\limits_{(S_1,{\rm dist}ots,S_m)\in\mathcal S'}\;
\int\limits_{-\sqrt{m}}^{\sqrt{m}} \mathbb{P}rod\limits_{i=1}^{m}f_i\,ds,
\mathcal{E}_{n-1}d{align*}
where $\mathcal S'$ is set of all sequences $(S_1,{\rm dist}ots,S_m)\in\mathcal S$ such that
is the subset of $S$ such that
\begin{equation}\left\langlebel{manyind}
\min(|S_i\cap Q_1|,|S_i\cap Q_2|)\geq {\mathcal F}rac{{\rm dist}eltalta}{2}\lfloor n/m\rightfloor\,\,
\mbox{ for at least $\,\, c_{\rm dist}eltalta m\,\,$ indices $\,i$.}
\mathcal{E}_{n-1}d{equation}
Take any $(S_1,{\rm dist}ots,S_m)\in\mathcal S'$ and denote $m_0:=\lceileil c_{\rm dist}eltalta m\rightceileil$.
Without loss of generality
we assume that (\rightef{manyind}) holds for all $i\leq m_0$.
Applying Lemma~\rightef{l: bal lower aux}
with $\kappa:={\rm dist}eltalta/2$ and $b=\sqrt{1-u}$, we get for all $u\in (0,1]$ and $i\leq m_0$,
$$
\mu (u):= \Big|\Big\{s\in[-\sqrt{m},\sqrt{m}]:\;
f_i \geq u\Big\}\Big|\leq \widetilde C\sqrt{m}\sqrt{1-u},
$$
where $\widetilde C>0$ depends only on ${\rm dist}eltalta$ and $\rightho$.
This estimate implies that for $i\leq m_0$,
$$
\int\limits_{-\sqrt{m}}^{\sqrt{m}}
(f_i(s))^{m_0}\,ds = \int\limits_{0}^{1} m_0\, u^{m_0-1}\, \mu_u \,ds \leq \widetilde C\sqrt{m}\, m_0\,
B(3/2, m_0 )\leq C_2,
$$
where $B$ denotes the Beta-function and $C_2>0$ is a constant depending only on $\rightho$ and ${\rm dist}eltalta$.
Applying H\"older's inequality, we obtain
\begin{align*}
\int\limits_{-\sqrt{m}}^{\sqrt{m}} \mathbb{P}rod\limits_{i=1}^{m}\mathbb{P}si_{K_2}\big(
\big|{\mathbb E}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,x_{\mathcal{E}ta[S_i]}\,m^{-1/2} s\big)\big|\big)\,ds
&\leq \int\limits_{-\sqrt{m}}^{\sqrt{m}} \mathbb{P}rod\limits_{i=1}^{ m_0}\mathbb{P}si_{K_2}\big(
\big|{\mathbb E}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,x_{\mathcal{E}ta[S_i]}\,m^{-1/2} s\big)\big|\big)\,ds
\leq C_2,
\mathcal{E}_{n-1}d{align*}
which impies the desired result.
\mathcal{E}_{n-1}d{proof}
\subsection{No moderately unstructured normal vectors}\left\langlebel{s: no mod}
Let $M_n$ be an $n\times n$ Bernoulli($p$) random matrix..
For each $i\leq n$, denote by $H_i=H_i(M_n)$ the span of columns ${\bf C}_j(M_n)$, $j\neq i$.
The goal of this subsection is to prove Theorem~\rightef{th: gradual}, which asserts that
under appropriate restrictions on $n$ and $p$
with a very large probability (say, at least $1-2e^{-2pn}$), the subspace $H_i^\mathbb{P}erp$ is either
structured or very unstructured.
The main ingredient of the proof --- Proposition~\rightef{prop: 09582593852} --- will be considered in the next subsection.
Here, we will only state the proposition to be used as a blackbox
and for this we need to introduce an additional product structure, which, in a sense, replaces
the set ${\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)$.
Fix a permutation ${\sigma}ma\in\Pi_n$, two disjoint subsets $Q_1,Q_2$ of cardinality $\lceileil{\rm dist}eltalta n\rightceileil$ each,
and a number $h\in{\mathbb R}$ such that
\begin{equation}\left\langlebel{eq: h admissible}
{\mathcal F}orall i\in Q_1: \, \, h+2 \leq {\bf g}(n/{\sigma}ma^{-1}(i)) \quad \quad \mbox{ and } \quad \quad
{\mathcal F}orall i\in Q_2: \, \, -{\bf g}(n/{\sigma}ma^{-1}(i))\leq h-\rightho-2.
\mathcal{E}_{n-1}d{equation}
Define the sets $\Lambda_n=\Lambda_n(k,{\bf g},Q_1,Q_2,\rightho,{\sigma}ma,h)$ by
\begin{equation}\left\langlebel{eq: param l def}
\begin{split}
\Lambda_n :=\bigg\{x\in{\mathcal F}rac{1}{k}{\mathbb Z}^n:\;&|x_{{\sigma}ma(i)}|\leq {\bf g}(n/i)
\;\;\mbox{for all }i\leq n,\;
\;\;\min\limits_{i\in Q_1}x_i\geq h,\,\,\, \mbox{ and }\,\,\,\max\limits_{i\in Q_2} x_i\leq h-\rightho\bigg\}.
\mathcal{E}_{n-1}d{split}
\mathcal{E}_{n-1}d{equation}
In what follows, we adopt the convention that $\Lambda_n=\mathcal{E}mptyset$
whenever $h$ does not satisfy \mathcal{E}qref{eq: h admissible}.
\begin{lemma}\left\langlebel{l: permut}
There exists an absolute constant $C_{\text{\tiny\rightef{l: permut}}}\geq 1$ such that for every
$n\geq 1$ there is a subset $\bar \Pi_n\subset\Pi_n$ of cardinality at most
$\mathcal{E}xp({C_{\text{\tiny\rightef{l: permut}}}n})$ with the following property.
For any two partitions $(S_i)_{i=1}^m$ and $(S_i')_{i=1}^m$ of $[n]$ with $2^{-i+1}n\geq |S_i|=|S_i'|$, $i\leq m$,
there is ${\sigma}ma\in \bar \Pi_n$ such that ${\sigma}ma(S_i)=S_i'$, $i\leq m$.
\mathcal{E}_{n-1}d{lemma}
This lemma immediately follows from the fact that the total number of partitions $(S_i)_{i=1}^m$ of $[n]$
satisfying $2^{-i+1}n\geq |S_i|$, $i\leq m$, is exponential in $n$ (one can take $C_{\text{\tiny\rightef{l: permut}}}=23$).
Using Lemma~\rightef{l: permut}, we provide an efficient approximation of ${\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)$.
\begin{lemma}\left\langlebel{l: disc with l}
For any $x\in{\mathcal V}_n={\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)$, $k\geq 4/\rightho$,
and any $y\in{\mathcal F}rac{1}{k}{\mathbb Z}^n$ with $\|x-y\|_\infty\leq 1/k$ one has
$$
y \in
\bigcup\limits_{q=\lfloor-4{\bf g}(6 n)/\rightho\rightfloor}^{
\lceileil4{\bf g}(6 n)/\rightho \rightceileil}\;
\bigcup\limits_{\bar{\sigma}ma\in\bar\Pi_n}\bigcup\limits_{|Q_1|,|Q_2|=\lceileil{\rm dist}eltalta n\rightceileil}
\Lambda_n(k,{\bf g}(6\, \cdot),Q_1,Q_2,\rightho/4,\bar{\sigma}ma,\rightho q/4),
$$
where the set of permutations $\bar\Pi_n$ is taken from Lemma~\rightef{l: permut}.
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Let $x\in {\mathcal V}_n$, and assume that $y\in{\mathcal F}rac{1}{k}{\mathbb Z}^n$ satisfies
$\|x-y\|_\infty\leq 1/k$.
Then, by the definition of ${\mathcal V}_n$, there exist sets $Q_1,Q_2\subset[n]$, each of cardinality
$\lceileil {\rm dist}eltalta n\rightceileil$, satisfying
$$\max\limits_{i\in Q_2}y_i-{\mathcal F}rac{1}{k}\leq \max\limits_{i\in Q_2}x_i
\leq \min\limits_{i\in Q_1}x_i-\rightho\leq \min\limits_{i\in Q_1}y_i-\rightho+{\mathcal F}rac{1}{k}.$$
Then
$
\max\limits_{i\in Q_2}y_i\leq \min\limits_{i\in Q_1}y_i-{\mathcal F}rac{\rightho}{2},
$
hence we can find a number $h\in {\mathcal F}rac{\rightho}{4}{\mathbb Z}$ such that
$$\min\limits_{i\in Q_1}y_i\geq h \quad \quad \mbox{ and } \quad \quad \max\limits_{i\in Q_2}y_i\leq h-{\mathcal F}rac{\rightho}{4}.$$
By the definition of ${\mathcal V}_n$ we also have $|x_{{\sigma}ma_x(i)}|\leq {\bf g}(n/i)$ for all $i\in[n]$.
By the definition of $\bar\Pi_n$, we can find a permutation $\bar{\sigma}ma\in \bar\Pi_n$ such that
$${\sigma}ma_x\big(\{\lfloor n/2^{\mathcal{E}ll}\rightfloor+1,{\rm dist}ots,\lfloor n/2^{\mathcal{E}ll-1}\rightfloor\}\big)
=\bar{\sigma}ma\big(\{\lfloor n/2^{\mathcal{E}ll}\rightfloor+1,{\rm dist}ots,\lfloor n/2^{\mathcal{E}ll-1}\rightfloor\}\big)\quad\mbox{ for all }\mathcal{E}ll\geq 1.$$
Clearly for such a permutation we have
$|x_{\bar{\sigma}ma(i)}|\leq {\bf g}(2 n/i)$ for every $i\leq n$.
Using (\rightef{gfncond}), we obtain
$$
|y_{\bar{\sigma}ma(i)}|\leq |x_{\bar{\sigma}ma(i)}|+{\mathcal F}rac{1}{k}\leq
{\bf g}(2n/i)+{\mathcal F}rac{1}{k}\leq {\bf g}(6n/i)-2.
$$
Thus
$$
{\mathcal F}orall i\in\bar{\sigma}ma^{-1}(Q_1): \, \, h\leq \min\limits_{i\in Q_1}y_i \leq {\bf g}(6n/i)-2\quad \mbox{ and } \quad
{\mathcal F}orall i\in\bar{\sigma}ma^{-1}(Q_2): \, \, h-{\mathcal F}rac{\rightho}{4}\geq \max\limits_{i\in Q_2}y_i\geq 2-{\bf g}(6n/i).
$$
Since $h=\rightho q/4$ for some $q\in {\mathbb Z}$, this implies the desired result.
\mathcal{E}_{n-1}d{proof}
The following statment, together with Theorem~\rightef{p: cf est} and Proposition~\rightef{l: stability of bal},
is the main ingredient of the proof of Theorem~\rightef{th: gradual}.
\begin{prop}\left\langlebel{prop: 09582593852}
Let $\varepsilon\in(0,1/8]$, $\rightho,{\rm dist}eltalta\in(0,1/4]$ and let the growth function ${\bf g}$
satisfies \mathcal{E}qref{gfncond}. There exist
$K_{\text{\tiny\rightef{prop: 09582593852}}}=K_{\text{\tiny\rightef{prop: 09582593852}}}({\rm dist}eltalta,\rightho)\geq 1$,
$n_{\text{\tiny\rightef{prop: 09582593852}}}=n_{\text{\tiny\rightef{prop: 09582593852}}}(\varepsilon,{\rm dist}eltalta,\rightho,K_3)$, and
$C_{\text{\tiny\rightef{prop: 09582593852}}}=C_{\text{\tiny\rightef{prop: 09582593852}}}(\varepsilon,{\rm dist}eltalta,\rightho,K_3)\in{\mathbb N}$
with the following property.
Let ${\sigma}ma\in\Pi_n$, $h\in{\mathbb R}$, and let $Q_1,Q_2\subset[n]$ be
such that $|Q_1|,|Q_2|=\lceileil {\rm dist}eltalta n\rightceileil$.
Let $8\leq K_2\leq 1/\varepsilon$,
$n\geq n_{\text{\tiny\rightef{prop: 09582593852}}}$, $m\geq C_{\text{\tiny\rightef{prop: 09582593852}}}$ with
$n/m\geq C_{\text{\tiny\rightef{prop: 09582593852}}}$, $1\leq k\leq\min\big(
(K_2/8)^{m/2},2^{n/C_{\text{\tiny\rightef{prop: 09582593852}}}}\big)$,
and let $X=(X_1,{\rm dist}ots,X_n)$ be a random vector
uniformly distributed on $\Lambda_n(k,{\bf g},Q_1,Q_2,\rightho,{\sigma}ma,h)$.
Then
$$
{\mathbb P}\big\{{\bf UD}_n(X,m,K_{\text{\tiny\rightef{prop: 09582593852}}},
K_2)< k m^{1/2}/C_{\text{\tiny\rightef{prop: 09582593852}}}\big\}\leq \varepsilon^n.
$$
\mathcal{E}_{n-1}d{prop}
Let us describe the proof of Theorem~\rightef{th: gradual} informally.
Assume that the hyperplane $H_1$ admits a normal vector $X$ which belongs to ${\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)$.
We need to show that with a large probability the u-degree ${\bf UD}_n(X,m,K_1,K_2)$ of $X$ is very large,
say, at least $\varepsilon^{-m}$ for a small $\varepsilon>0$.
The idea is to split the collection ${\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)$ into about $\log_2(\varepsilon^{-m})$ subsets according
to the magnitude of the u-degree (that is, each subset
$\mathcal T_N$ will have a form
$\mathcal T_N=\big\{x\in {\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho):\;{\bf UD}_n(x,m,K_1,K_2)\in [N,2N)\big\}$
for an appropriate $N$).
To show that for each $N\ll \varepsilon^{-m}$ the probability of $X\in \mathcal T_N$
is very small, we define a discrete approximation ${\mathcal A}_N$ of $\mathcal T_N$ consisting of all vectors $y\in {\mathcal F}rac{1}{k}{\mathbb Z}^n$
such that $\|y-x\|_\infty\leq 1/k$ for some $x\in \mathcal T_N$ and additionally,
in view of Proposition~\rightef{l: stability of bal},
${\bf UD}_n(y,m,c_{\text{\tiny\rightef{l: stability of bal}}} K_1,K_2)\leq 2N$ and
${\bf UD}_n(y,m,c_{\text{\tiny\rightef{l: stability of bal}}}^{-1} K_1,K_2)\geq N$.
We can bound the cardinality of such set ${\mathcal A}_N$ by $(\tilde\varepsilon\,k)^n$, for a small $\tilde\varepsilon>0$,
by combining Proposition~\rightef{prop: 09582593852} with Lemma~\rightef{l: disc with l} and with the following simple fact.
\begin{lemma}\left\langlebel{l: Lambda_n card}
Let $k\geq 1$, $h\in{\mathbb R}$, $\rightho,{\rm dist}eltalta\in(0,1)$, $Q_1,Q_2\subset[n]$ with
$|Q_1|,|Q_2|=\lceileil {\rm dist}eltalta n\rightceileil$, and ${\bf g}$
satisfies \mathcal{E}qref{gfncond} with some $K_3\geq 1$. Then
$|\Lambda_n(k,{\bf g},Q_1,Q_2,\rightho,{\sigma}ma,h)|\leq \big(C_{\text{\tiny\rightef{l: Lambda_n card}}}k\big)^n$,
where $C_{\text{\tiny\rightef{l: Lambda_n card}}}\geq 1$ depends only on $K_3$.
\mathcal{E}_{n-1}d{lemma}
On the other hand, for each fixed vector $y$ in the set ${\mathcal A}_N$
we can estimate the probability that it ``approximates'' a normal vector to $H_1$
by using Corollary~\rightef{cor: cf tensor}:
$$
{\mathbb P}\big\{\mbox{$y$ is an ``approximate'' normal vector to $H_1$}\big\}\leq (C'/k)^n\quad \mbox{for every }y\in {\mathcal A}_N,
$$
for some constant $C'\ll \tilde\varepsilon^{-1}$.
Taking the union bound, we obtain
$$
{\mathbb P}\big\{X\in \mathcal T_N\big\}\leq {\mathbb P}\big\{\mbox{${\mathcal A}_N$ contains an ``approximate''
normal vector to $H_1$}\big\}\leq (C'/k)^n\,(\tilde\varepsilon\,k)^n\ll 1.
$$
Below, we make this argument rigorous.
\begin{proof}[Proof of Theorem~\rightef{th: gradual}]
We start by defining parameters. We always assume that $n$ is large enough, so
all statements used below work for our $n$.
Fix any $R\geq 1$, $r>0$ and $s>0$, and set $b:=\lfloor(2p R)^{-1}\rightfloor$.
Let $K_2= 32 \mathcal{E}xp(16R).$
Note that the function ${\bf g}(6\, \cdot)$ is a growth function that satisfies condition \mathcal{E}qref{gfncond}
with parameter $K_3'=(K_3)^8$. In particular, choosing $j$ so taht $2^{j-1}\leq 6n\leq 2^j$, we have
$${\bf g}(6n)\leq {\bf g}(2^j)\leq (K_3')^{2^j/j}\leq (K_3')^{12n/\log_2(6n)}\leq K_3^n.$$
For brevity, we denote $$C_{\text{\tiny\rightef{cor: norm of centered}}}:=C_{\text{\tiny\rightef{cor: norm of centered}}}(s,2R),\,\,
C_{\text{\tiny\rightef{l: column supports}}}:=C_{\text{\tiny\rightef{l: column supports}}}(2R),\,\,
c_{\text{\tiny\rightef{l: stability of bal}}}':=c_{\text{\tiny\rightef{l: stability of bal}}}'(K_2),\,\,
c_{\text{\tiny\rightef{l: stability of bal}}}:=c_{\text{\tiny\rightef{l: stability of bal}}}(K_2),\,\,
C_{\text{\tiny\rightef{l: Lambda_n card}}}
=C_{\text{\tiny\rightef{l: Lambda_n card}}}(K_3').$$
Set
$$
K_1:=\max\big(K_{\text{\tiny\rightef{prop: 09582593852}}}({\rm dist}eltalta,\rightho/4)/c_{\text{\tiny\rightef{l: stability of bal}}},
C_{\text{\tiny\rightef{p: low bound on bal}}}(r,{\rm dist}eltalta,\rightho)\big),
$$
and
$$
\varepsilon:= \min\Big(K_2^{-1},\, c_{\text{\tiny\rightef{l: stability of bal}}}' \big( 384e K_3\, \mathcal{E}xp({C_{\text{\tiny\rightef{l: permut}}}})\,
C_{\text{\tiny\rightef{l: Lambda_n card}}}
C_{\text{\tiny\rightef{l: tensor}}}C_{\text{\tiny\rightef{p: cf est}}}C_{\text{\tiny\rightef{cor: norm of centered}}}
\big)^{-1}\mathcal{E}xp(-3R)\Big)
$$
We will assume that $pn$ is sufficiently large so that
$$5\mathcal{E}xp(-2Rpn)\leq\mathcal{E}xp(-Rpn)\quad \mbox{
and }\quad \mathcal{E}xp(-3Rpn)\leq{\mathcal F}rac{1}{2Rpn}\mathcal{E}xp(-2Rpn).$$
Moreover, we will assume that
\begin{equation}\left\langlebel{eq: aux 2-9582-598}
\mbox{$2R C_{\text{\tiny\rightef{l: column supports}}}p\leq 1\quad $
and $\quad C_{\text{\tiny\rightef{l: column supports}}}\leq p n$}
\mathcal{E}_{n-1}d{equation}
and
\begin{align*}
&{\mathcal F}rac{1}{8p}\geq \max(C_{\text{\tiny\rightef{prop: 09582593852}}}(\varepsilon,{\rm dist}eltalta,\rightho/4,K_3'),
C_{\text{\tiny\rightef{p: low bound on bal}}}(r,{\rm dist}eltalta,\rightho));\quad
pn\geq 16C_{\text{\tiny\rightef{prop: 09582593852}}}(\varepsilon,{\rm dist}eltalta,\rightho/4,K_3')^2;\\
&e^{2Rp}\leq 2^{1/C_{\text{\tiny\rightef{prop: 09582593852}}}(\varepsilon,{\rm dist}eltalta,\rightho/4,K_3')};\quad
c_{\text{\tiny\rightef{l: stability of bal}}}'/3\geq \mathcal{E}xp(-Rpn);\quad \lfloor \mathcal{E}xp(Rpn)/c_{\text{\tiny\rightef{l: stability of bal}}}'\rightfloor\,n\leq 2^n.
\mathcal{E}_{n-1}d{align*}
Define two auxiliary random objects as follows.
Set
$$Z:=\mbox{$\{x\in{\mathbb R}^n:\;x^*_{\lfloor rn\rightfloor}=1,\,\, \,
{\bf UD}_n(x,m,K_1,K_2)\geq \mathcal{E}xp(Rpn)\,\,$ for all $\,\, pn/8\leq m\leq 8pn\}$,}$$
and let $X$ be a random vector measurable with respect to $H_1$ and such that
\begin{itemize}
\item $X\in \big({\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)\cap H_1^\mathbb{P}erp\big)\setminus Z\quad\mbox{whenever }\;
\big({\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)\cap H_1^\mathbb{P}erp\big)\setminus Z\neq\mathcal{E}mptyset$;
\item $X\in \big({\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)\cap H_1^\mathbb{P}erp\big)\cap Z\quad\mbox{whenever }\;
\big({\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)\cap H_1^\mathbb{P}erp\big)\setminus Z=\mathcal{E}mptyset$ and
${\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)\cap H_1^\mathbb{P}erp\neq \mathcal{E}mptyset$;
\item $X={\bf 0}\quad\mbox{whenever }\quad
{\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)\cap H_1^\mathbb{P}erp= \mathcal{E}mptyset$.
\mathcal{E}_{n-1}d{itemize}
(Note that $H_1^\mathbb{P}erp$ may have dimension larger than one with non-zero probability,
and thus $\mathbb{P}m X$ is not uniquely defined). Note that to prove the theorem, it is sufficient to show that
with probability at least $1-\mathcal{E}xp(-R pn)$ one has either $X={\bf 0}$ or $X\in Z$.
Next, we denote
$$
\xi:=\begin{cases}\min\limits_{8pn \geq m\geq pn/8}{\bf UD}_n(X,m,K_1,K_2),&\mbox{whenever }\;X\neq {\bf 0};\\
+\infty,&\mbox{otherwise.}\mathcal{E}_{n-1}d{cases}
$$
Then, proving the theorem amounts to showing that $\xi<\mathcal{E}xp(R pn)$ with probability at most $\mathcal{E}xp(-R pn)$.
We say that a collection of indices $I\subset[n]$ is admissible if $1\|\cdot\|tin I$ and $|I|\geq n-b-1$.
For admissible sets $I$ consider disjoint collection of events $\{{\mathbb E}vent_I\}_I$ defined by
\begin{align*}
{\mathbb E}vent_I:=\big\{{\mathcal F}orall i\in I :\, \, |{\rm supp\, } {\bf C}_i(M_n)|\in [pn/8,8pn]
\quad \mbox{ and } \quad
{\mathcal F}orall i\|\cdot\|tin I :\, \, |{\rm supp\, } {\bf C}_i(M_n)|\|\cdot\|tin [pn/8,8pn]
\big\}.
\mathcal{E}_{n-1}d{align*}
Further, denote
$$
\widetilde {\mathbb E}vent:=\big\{\|M_n-{\mathbb E} M_n\|\leq C_{\text{\tiny\rightef{cor: norm of centered}}}\sqrt{pn}\big\}.
$$
According to Corollary~\rightef{cor: norm of centered}, ${\mathbb P}(\widetilde {\mathbb E}vent)\geq 1-\mathcal{E}xp(-2Rpn)$, while
by Lemma~\rightef{l: column supports} and \mathcal{E}qref{eq: aux 2-9582-598},
$$
{\mathbb P}\Big(\bigcup_{I}{\mathbb E}vent_I\Big)\geq 1-\mathcal{E}xp(-n/C_{\text{\tiny\rightef{l: column supports}}})\geq 1-\mathcal{E}xp(-2Rpn).
$$
Denote by $\mathcal I$ the collection of all admissible $I$ satisfying
$2{\mathbb P}({\mathbb E}vent_I\cap\widetilde{\mathbb E}vent)\geq {\mathbb P}({\mathbb E}vent_I)$. Then for $I\in \mathcal I$, we have
${\mathbb P}({\mathbb E}vent_I)\geq 2{\mathbb P}({\mathbb E}vent_I\cap\widetilde{\mathbb E}vent^c)$, and, using that events ${\mathbb E}vent_I$ are disjoint,
$$
{\mathbb P}\Big(\bigcup_{I\in \mathcal I}{\mathbb E}vent_I\Big)\geq 1-\mathcal{E}xp(-2Rpn)-2{\mathbb P}(\widetilde{\mathbb E}vent^c)\geq 1-3\mathcal{E}xp(-2Rpn).
$$
Hence,
\begin{align*}
{\mathbb P}\big\{\xi<\mathcal{E}xp(Rpn)\big\}
&\leq \sum\limits_{I\in \mathcal I}{\mathbb P}\big(\big\{\xi<\mathcal{E}xp(Rpn)\big\}\cap {\mathbb E}vent_I\cap \widetilde {\mathbb E}vent\big)
+{\mathbb P}\Big(\bigcap_{I\in \mathcal I}{\mathbb E}vent_I^c\Big)+{\mathbb P}(\widetilde {\mathbb E}vent^c)\\
&\leq \sum\limits_{I\in \mathcal I}{\mathbb P}\big(\big\{\xi<\mathcal{E}xp(Rpn)\big\}\;|\;{\mathbb E}vent_I\cap \widetilde {\mathbb E}vent\big)
{\mathbb P}({\mathbb E}vent_I\cap \widetilde {\mathbb E}vent)+4\mathcal{E}xp(-2Rpn).
\mathcal{E}_{n-1}d{align*}
Therefore, to prove the theorem it is sufficient to show that for any $I\in\mathcal I$,
$$
{\mathbb P}\big(\big\{\xi<\mathcal{E}xp(Rpn)\big\}\;|\;{\mathbb E}vent_I\cap \widetilde {\mathbb E}vent\big)\leq \mathcal{E}xp(-2Rpn).
$$
Fix an admissible $I\in\mathcal I$,
denote by $B_I$ the $|I|\times n$ matrix obtained by transposing
columns ${\bf C}_i(M_n)$, $i\in I$, and let $\widetilde B_I$ be the non-random $|I|\times n$ matrix
with all elements equal to $p$.
Note that, in view of our definition of $K_1$, the assumptions on $p$
and Proposition~\rightef{p: low bound on bal}, we have a {\it deterministic} relation
$$
\xi\geq \sqrt{pn/8}
$$
everywhere on the probability space.
For each real number $N\in J_p:=[\sqrt{pn/8},\mathcal{E}xp(Rpn)/2]$, denote by ${\mathbb E}vent_{N,I}$
the event
$$
{\mathbb E}vent_{N,I}:=\big\{\xi\in[N,2N)\big\}\cap {\mathbb E}vent_I\cap \widetilde {\mathbb E}vent.
$$
Splitting the interval $J_p$ into subintervals, we observe that it is sufficient to show that for every
$N\in J_p$ we have
$$
{\mathbb P}\big({\mathbb E}vent_{N,I}\;|\;{\mathbb E}vent_I\cap \widetilde {\mathbb E}vent\big)\leq \mathcal{E}xp(-3Rpn)\leq{\mathcal F}rac{1}{2Rpn}\mathcal{E}xp(-2Rpn).
$$
The rest of the argument is devoted to estimating probability of ${\mathbb E}vent_{N,I}$ for fixed $N\in J_p$ and
fixed $I\in \mathcal I$.
Set $k:=\lceileil 2N/c_{\text{\tiny\rightef{l: stability of bal}}}'\rightceileil$.
Let ${\bf m}:{\mathbb E}vent_{N,I}\to [pn/8,8pn]$ be a (random) integer such that
$${\bf UD}_n(X,{\bf m},K_1,K_2)\in [N,2N)\;\,\,\, \mbox{ everywhere on }\,\,\, \;{\mathbb E}vent_{N,I}.$$
Since on ${\mathbb E}vent_{N,I}$ we have
${\bf UD}_n(X,{\bf m},K_1,K_2)\leq 2N\leq c_{\text{\tiny\rightef{l: stability of bal}}}'k$, applying
Proposition~\rightef{l: stability of bal}, we can construct a random vector ${\bf Y}:{\mathbb E}vent_{N,I}\to {\mathcal F}rac{1}{k}{\mathbb Z}^n$
having the following properties:
\begin{itemize}
\item $\|{\bf Y}-X\|_\infty\leq 1/k$ everywhere on ${\mathbb E}vent_{N,I}$,
\item ${\bf UD}_n({\bf Y},{\bf m},c_{\text{\tiny\rightef{l: stability of bal}}} K_1,K_2)\leq 2N$ everywhere on ${\mathbb E}vent_{N,I}$,
\item ${\bf UD}_n({\bf Y},m,c_{\text{\tiny\rightef{l: stability of bal}}}^{-1} K_1,K_2)\geq N$
for all $m\in [pn/8,8pn]$ and everywhere on ${\mathbb E}vent_{N,I}$.
\mathcal{E}_{n-1}d{itemize}
The first condition together with the inclusion ${\mathbb E}vent_{N,I}\subset \widetilde {\mathbb E}vent$ implies that
$$
\|(B_I-\widetilde B_I)({\bf Y}-X)\|\leq C_{\text{\tiny\rightef{cor: norm of centered}}}\sqrt{p}n/k.
$$
Using that $B_I X=0$ and that $\widetilde B_I ({\bf Y}-X)=p(\sum_{i=1}^n ({\bf Y}_i-X_i))\,{\bf 1}_I$,
we observe that there is a random number ${\bf z}:{\mathbb E}vent_{N,I}\to [-pn/k, pn/k]\cap {\mathcal F}rac{\sqrt{pn}}{k}{\mathbb Z}$
such that everywhere on ${\mathbb E}vent_{N,I}$ one has
$$
\|B_I {\bf Y}-{\bf z}\,{\bf 1}_I\|\leq 2C_{\text{\tiny\rightef{cor: norm of centered}}}\sqrt{p}n/k.
$$
Let $\Lambda$ be a subset of
$$
\bigcup\limits_{q=\lfloor-4{\bf g}(6 n)/\rightho \rightfloor}^{
\lceileil 4{\bf g}(6 n)/\rightho \rightceileil}\;
\bigcup\limits_{\bar{\sigma}ma\in\bar\Pi_n}\bigcup\limits_{|Q_1|,|Q_2|=\lceileil{\rm dist}eltalta n\rightceileil}
\Lambda_n(k,{\bf g}(6\, \cdot),Q_1,Q_2,\rightho/4,\bar{\sigma}ma,\rightho q/4),
$$
consisting of all vectors $y$ such that
\begin{itemize}
\item ${\bf UD}_n(y,m,c_{\text{\tiny\rightef{l: stability of bal}}} K_1,K_2)\leq 2N$ for {\it some } $m\in[pn/8,8pn]$;
\item ${\bf UD}_n(y,m,c_{\text{\tiny\rightef{l: stability of bal}}}^{-1} K_1,K_2)\geq N$
for all $m\in [pn/8,8pn]$.
\mathcal{E}_{n-1}d{itemize}
Note that by Lemma~\rightef{l: disc with l} the entire range of ${\bf Y}$
on ${\mathbb E}vent_{N,I}$ falls into $\Lambda$.
Combining the above observations,
$${\mathbb E}vent_{N,I}\subset
\big\{\|B_I y-z{\bf 1}_I\|\leq 2C_{\text{\tiny\rightef{cor: norm of centered}}}\sqrt{p}n/k\;\mbox{ for some }y\in\Lambda,
\mbox{ $z\in [-pn/k, pn/k]\cap {\mathcal F}rac{\sqrt{pn}}{k}{\mathbb Z}$}\big\},
$$
whence, using that $2{\mathbb P}({\mathbb E}vent_I\cap \widetilde{\mathbb E}vent)\geq {\mathbb P}({\mathbb E}vent_I)$ by the definition of $\mathcal I$,
\begin{align*}
{\mathbb P}({\mathbb E}vent_{N,I}\,|\,{\mathbb E}vent_I\cap \widetilde{\mathbb E}vent)
&\leq 2{\mathbb P}\big\{\|B_I y-z{\bf 1}_I\|\leq 2C_{\text{\tiny\rightef{cor: norm of centered}}}\sqrt{p}n/k\;\mbox{ for some }y\in\Lambda,
\mbox{ $z\in [-pn/k, pn/k]\cap {\mathcal F}rac{\sqrt{pn}}{k}{\mathbb Z}$}\;|\;{\mathbb E}vent_I\big\}\\
&\leq 6|\Lambda|\sqrt{pn}\,\max\limits_{z\in {\mathcal F}rac{\sqrt{pn}}{k}{\mathbb Z}}\,\,\max\limits_{y\in \Lambda}{\mathbb P}\big\{
\|B_I y-z{\bf 1}_I\|\leq 2C_{\text{\tiny\rightef{cor: norm of centered}}}\sqrt{p}n/k\;|\;{\mathbb E}vent_I\big\}.
\mathcal{E}_{n-1}d{align*}
To estimate the last probability, we apply Corollary~\rightef{cor: cf tensor}
with $t:= C_{\text{\tiny\rightef{cor: norm of centered}}} \sqrt{8pn}/{N}$ (note that $k\geq 2N$, $2|I|\geq n$, and
that $t$ satisfies the assumption of the corollary).
We obtain that for all admissible $y$ and $z$,
\begin{align*}
{\mathbb P}\big\{
\|B_I y-z{\bf 1}_I\|\leq 2C_{\text{\tiny\rightef{cor: norm of centered}}}\sqrt{p}n/k\;|\;{\mathbb E}vent_I\big\}
&\leq{\mathbb P}\bigg\{
\|B_I y-z{\bf 1}_I\|\leq {\mathcal F}rac{C_{\text{\tiny\rightef{cor: norm of centered}}}\sqrt{8pn}}{N}\,\sqrt{|I|}\;|\;{\mathbb E}vent_I\bigg\}\\
&\leq (16C_{\text{\tiny\rightef{l: tensor}}}C_{\text{\tiny\rightef{p: cf est}}}C_{\text{\tiny\rightef{cor: norm of centered}}}/N)^{|I|}.
\mathcal{E}_{n-1}d{align*}
On the other hand, the cardinality of $\Lambda$ can be estimated by combining
Lemma~\rightef{l: Lambda_n card}, Lemma~\rightef{l: permut}
and Proposition~\rightef{prop: 09582593852}
(note that our choice of parameters guarantees applicability of these statements):
$$
|\Lambda|\leq 8pn \varepsilon^n\, (9{\bf g}(6n)/\rightho ) \mathcal{E}xp({C_{\text{\tiny\rightef{l: permut}}}n})\,2^{2n}
(C_{\text{\tiny\rightef{l: Lambda_n card}}}k)^n
\leq (72pn/\rightho ) \varepsilon^n\, K_3^n\, \mathcal{E}xp({C_{\text{\tiny\rightef{l: permut}}}n})\,2^{2n}
(C_{\text{\tiny\rightef{l: Lambda_n card}}}k)^n,
$$
where $C_{\text{\tiny\rightef{l: Lambda_n card}}}
=C_{\text{\tiny\rightef{l: Lambda_n card}}}(K_3')$.
Thus, using our choice of parameters and assuming in addition that $2^n\geq 72pn/\rightho $
\begin{align*}
{\mathbb P}({\mathbb E}vent_{N,I}\,|\,{\mathbb E}vent_I\cap \widetilde{\mathbb E}vent)
&\leq \varepsilon^n\, (8 K_3\, \mathcal{E}xp({C_{\text{\tiny\rightef{l: permut}}}})\,
C_{\text{\tiny\rightef{l: Lambda_n card}}}k)^n
\,(16C_{\text{\tiny\rightef{l: tensor}}}C_{\text{\tiny\rightef{p: cf est}}}C_{\text{\tiny\rightef{cor: norm of centered}}}/N)^{|I|}\\
&\leq
\varepsilon^n\, (8 K_3\, \mathcal{E}xp({C_{\text{\tiny\rightef{l: permut}}}})\,
C_{\text{\tiny\rightef{l: Lambda_n card}}}k)^n\,
(48C_{\text{\tiny\rightef{l: tensor}}}C_{\text{\tiny\rightef{p: cf est}}}C_{\text{\tiny\rightef{cor: norm of centered}}}
/(c_{\text{\tiny\rightef{l: stability of bal}}}'k))^{n}N^{1+\lfloor (2pR)^{-1}\rightfloor}\\
&\leq \varepsilon^n\, (384 K_3\, \mathcal{E}xp({C_{\text{\tiny\rightef{l: permut}}}})\,
C_{\text{\tiny\rightef{l: Lambda_n card}}}
C_{\text{\tiny\rightef{l: tensor}}}C_{\text{\tiny\rightef{p: cf est}}}C_{\text{\tiny\rightef{cor: norm of centered}}}
/(c_{\text{\tiny\rightef{l: stability of bal}}}'))^{n}\, e^n\\
&\leq \mathcal{E}xp(-3Rn),
\mathcal{E}_{n-1}d{align*}
by our choice of parameters. The result follows.
\mathcal{E}_{n-1}d{proof}
\subsection{Anti-concentration on a lattice}
The goal of this subsection is to prove Proposition~\rightef{prop: 09582593852}. Thus, in this subsection,
we fix $\rightho,{\rm dist}eltalta\in(0,1/4]$, a growth function ${\bf g}$ satisfying \mathcal{E}qref{gfncond}, which in particular
means that ${\bf g}(n)\leq K_3 ^{2n/\log_2 n}$,
a permutation ${\sigma}ma\in\Pi_n$, a number $h\in{\mathbb R}$, two sets $Q_1,Q_2\subset[n]$
such that $|Q_1|,|Q_2|=\lceileil {\rm dist}eltalta n\rightceileil$, and we do not repeat these assumptions in lemmas below.
We also always use short notation $\Lambda_n$ for the set $\Lambda_n(k,{\bf g},Q_1,Q_2,\rightho,{\sigma}ma,h)$
defined in \mathcal{E}qref{eq: param l def}.
We start with auxiliary probabilistic statements which are just special forms of Markov's inequality.
\begin{lemma}[Integral form of Markov's inequality, I]\left\langlebel{l: int markov}
For each $s\in[a,b]$, let $\xi(s)$ be a non-negative random variable with $\xi(s)\leq 1$ a.e.
Assume that the random function $\xi(s)$ is integrable on $[a,b]$ with probability one.
Assume further that for some integrable function $\mathbb{P}hi(s):\, [a,b]\to{\mathbb R}_+$ and some $\varepsilon>0$ we have
$$
{\mathbb P}\big\{\xi(s)\leq \mathbb{P}hi(s)\big\}\geq 1-\varepsilon
$$
for all $s\in[a,b]$. Then for all $t>0$,
$$
{\mathbb P}\bigg\{\int_a^b \xi(s)\,ds\geq \int_a^b \mathbb{P}hi(s)\,ds+t(b-a)\bigg\}\leq \varepsilon/t.
$$
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Consider a random set
$$
I:=\big\{s\in[a,b]:\;\xi(s)> \mathbb{P}hi(s)\big\}.
$$
Since ${\mathbb P}\{s\in I\}\leq \varepsilon$ for any $s\in[a,b]$,
we have
$
{\mathbb E}|I|\leq \varepsilon(b-a).
$
Therefore, by the Markov inequality,
$
{\mathbb P}\big\{|I|\geq t(b-a)\big\}\leq \varepsilon/t
$
for all $t>0$.
The result follows by noting that
$$
\int_a^b \xi(s)\,ds\leq |I|+\int_a^b \mathbb{P}hi(s)\,ds.
$$
\mathcal{E}_{n-1}d{proof}
\begin{lemma}[Integral form of Markov's inequality, II]\left\langlebel{l: sum markov}
Let $I$ be a finite set, and for each $i\in I$, let $\xi_i$ be a non-negative random variable with $\xi_i\leq 1$ a.e.
Assume further that for some $\mathbb{P}hi(i):I\to{\mathbb R}_+$ and some $\varepsilon>0$ we have
$$
{\mathbb P}\big\{\xi_i\leq \mathbb{P}hi(i)\big\}\geq 1-\varepsilon
$$
for all $i\in I$. Then for all $t>0$,
$$
{\mathbb P}\bigg\{{\mathcal F}rac{1}{|I|}\sum_{i\in I} \xi_i\geq {\mathcal F}rac{1}{|I|}\sum_{i\in I} \mathbb{P}hi(i)+t\bigg\}\leq \varepsilon/t.
$$
\mathcal{E}_{n-1}d{lemma}
The proof of Lemma~\rightef{l: sum markov} is almost identical to that of Lemma~\rightef{l: int markov}, and we omit it.
Our next statement will be important in an approximation (discretization) argument used later in the proof.
\begin{lemma}[Lipschitzness of the product $\mathbb{P}rod\mathbb{P}si_{K_2}(\cdot)$]\left\langlebel{l: lip of prod}
Let $y_1,{\rm dist}ots,y_n\in {\mathbb R}$ and set $y:=\max\limits_{w\leq n}|y_w|$.
Further, let $S_1,{\rm dist}ots,S_m$ be some non-empty subsets of $[n]$.
For $i\leq m$ denote
$$
f_i(s):=\mathbb{P}si_{K_2}\bigg(\Big|{\mathcal F}rac{1}{|S_i|}
\sum_{w\in S_i}\mathcal{E}xp(2\mathbb{P}i{\bf i}\,y_w s)\Big|\bigg)\quad \mbox{ and let } \quad
f(s):=\mathbb{P}rod\limits_{i=1}^m f_i(s).
$$
Then $f$ (viewed as a function of $s$) is $(8 K_2\mathbb{P}i y\,m)$-Lipschitz.
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
By our definition, $\mathbb{P}si_{K_2}$ is $1$-Lipschitz for any $K_2\geq 1$,
hence
$f_i$ (viewed as a function of $s$) is
$2\mathbb{P}i y$-Lipschitz.
Since $\big|\sum_{w\in S_i}\mathcal{E}xp(2\mathbb{P}i{\bf i}\,y_w s)\big|\leq |S_i|$,
by the definition of the function $\mathbb{P}si_{K_2}$, we have
$1/(2K_2)\leq f_i \leq 1$, hence, for all $s,{\mathcal{D}_{n,d}}elta s\in{\mathbb R}$,
$$
{\mathcal F}rac{f_i(s)}{f_i(s+{\mathcal{D}_{n,d}}elta s)} = 1 + {\mathcal F}rac{f_i(s)-f_i(s+{\mathcal{D}_{n,d}}elta s)}{f_i(s+{\mathcal{D}_{n,d}}elta s)}
\leq 1+4 K_2\mathbb{P}i y\,|{\mathcal{D}_{n,d}}elta s|.
$$
Taking the product, we obtain that
$$
{\mathcal F}rac{f(s)}{f(s+{\mathcal{D}_{n,d}}elta s)} \leq \big(1+4 K_2\mathbb{P}i y\,|{\mathcal{D}_{n,d}}elta s|\big)^m
\leq 1+8 K_2\mathbb{P}i y\,m\,|{\mathcal{D}_{n,d}}elta s|
$$
whenever $8 K_2\mathbb{P}i y\,m\,|{\mathcal{D}_{n,d}}elta s|\leq 1/2$.
This, together with the bound $f\leq 1$ implies for all $s,{\mathcal{D}_{n,d}}elta s\in{\mathbb R}$,
$$
f(s) - f(s+{\mathcal{D}_{n,d}}elta s) \leq 8 K_2\mathbb{P}i y\,m\,|{\mathcal{D}_{n,d}}elta s|,
$$
which completes the proof.
\mathcal{E}_{n-1}d{proof}
In the next two lemmas we initiate the study of random variables $\mathcal{E}xp(2\mathbb{P}i{\bf i}\,\mathcal{E}ta[I_w]\,s_j/k)$,
more specifically, we will be interested in the property that, under appropriate assumptions on $s_j$'s,
the sum of such variables is close to zero on average.
\begin{lemma}\left\langlebel{l: aux 43098275}
Let $\varepsilon\in(0,1]$, $k\geq 1$, $\mathcal{E}ll\geq 2/\varepsilon$. Let $I$ be an integer interval
and let $s_1,{\rm dist}ots,s_\mathcal{E}ll$ be real numbers such that for all $j\neq u$,
$$ {\mathcal F}rac{k}{\varepsilon |I|} \leq |s_j-s_u|\leq {\mathcal F}rac{k}{2}.$$
Then
\begin{align*}
{\mathbb E}\,\Big|\sum_{j=1}^\mathcal{E}ll \mathcal{E}xp(2\mathbb{P}i{\bf i}\,\mathcal{E}ta[I]\,s_j/k)\Big|^2\leq\varepsilon\mathcal{E}ll^2.
\mathcal{E}_{n-1}d{align*}
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
We will determine the restrictions on parameter $R$ at the end of the proof.
We have
\begin{equation}\left\langlebel{eq: aux 976120975}
\begin{split}
{\mathbb E}\,\Big|\sum_{j=1}^\mathcal{E}ll \mathcal{E}xp(2\mathbb{P}i{\bf i}\,\mathcal{E}ta[I]\,s_j/k)\Big|^2
&=\sum_{j=1}^{\mathcal{E}ll}\sum_{u=1}^{\mathcal{E}ll}{\mathbb E} \mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,\mathcal{E}ta[I]\,(s_j-s_u)/k\big)\\
&\leq \mathcal{E}ll+\Big|\sum\limits_{j\neq u}{\mathbb E} \mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,\mathcal{E}ta[I]\,(s_j-s_u)/k\big)\Big|.
\mathcal{E}_{n-1}d{split}
\mathcal{E}_{n-1}d{equation}
Further, denoting $a=\min I$ and $b=\min I$, we observe for any $j\neq u$,
\begin{align*}
{\mathbb E} &\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,\mathcal{E}ta[I]\,(s_j-s_u)/k\big)\\
&={\mathcal F}rac{1}{|I|}\sum_{v=a}^{b}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,v\,(s_j-s_u)/k\big)\\
&={\mathcal F}rac{1}{|I|}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,a\,(s_j-s_u)/k\big)\cdot
{\mathcal F}rac{1-\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,(b-a+1)\,(s_j-s_u)/k\big)}{1-\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,(s_j-s_u)/k\big)}.
\mathcal{E}_{n-1}d{align*}
In view of assumptions on $|s_j-s_u|$
$$\big|1-\mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,(s_j-s_u)/k\big)\big|= \big|2\sin(\mathbb{P}i\,(s_j-s_u)/k)
\big| \geq {\mathcal F}rac{4|s_j-s_u|}{k}\geq {\mathcal F}rac{4}{\varepsilon |I|}.$$
Therefore,
$$
\big|{\mathbb E} \mathcal{E}xp\big(2\mathbb{P}i{\bf i}\,\mathcal{E}ta[I]\,(s_j-s_u)/k\big)\big|\leq {\mathcal F}rac{\varepsilon}{2}.
$$
Using \mathcal{E}qref{eq: aux 976120975}, we complete the proof.
\mathcal{E}_{n-1}d{proof}
\begin{lemma}\left\langlebel{l: aux 9871039481}
For every $\varepsilon\in(0, 1/2]$ there are $R_{\text{\tiny\rightef{l: aux 9871039481}}}
=R_{\text{\tiny\rightef{l: aux 9871039481}}}(\varepsilon)>0$
and $\mathcal{E}ll:=\mathcal{E}ll_{\text{\tiny\rightef{l: aux 9871039481}}}(\varepsilon)\in{\mathbb N}$, $\mathcal{E}ll\geq 1000$,
with the following property.
Let $k\geq 1$, $u\geq \mathcal{E}ll$, let $I_w$ ($w=1,2,{\rm dist}ots,u$) be integer intervals, and let
$s_1,{\rm dist}ots,s_\mathcal{E}ll$ be real numbers such that $|I_w|\,|s_j-s_q|\geq R_{\text{\tiny\rightef{l: aux 9871039481}}}k$,
and $|s_j-s_q|\leq k/2$ for all $j\neq q$ and $w\leq u$.
Then, assuming that random variables $\mathcal{E}ta[I_w]$, $w\leq u$, are mutually independent, one has
$$
{\mathbb P}\Big\{\Big|{\mathcal F}rac{1}{u}\sum_{w=1}^u
\mathcal{E}xp(2\mathbb{P}i{\bf i}\,\mathcal{E}ta[I_w]\,s_j/k)\Big|\geq \varepsilon\;\mbox{ for at least $\varepsilon\mathcal{E}ll$ indices }j\Big\}
\leq \varepsilon^{u}.
$$
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Fix any $\varepsilon\in(0,1/2]$, and set $\varepsilon_1:=2^{-10}e^{-6}\varepsilon^{4+9/\varepsilon}$.
Set $R:=1/\varepsilon_1$
and $\mathcal{E}ll:=\lceileil 2/\varepsilon_1\rightceileil$.
Assume that $u\geq \mathcal{E}ll$, and let numbers $s_j$ and integer intervals $I_w$ satisfy the assumptions of the lemma.
Denote the event
$$
\Big\{\Big|{\mathcal F}rac{1}{u}\sum_{w=1}^u
\mathcal{E}xp(2\mathbb{P}i{\bf i}\,\mathcal{E}ta[I_w]\,s_j/k)\Big|\geq \varepsilon\;\mbox{ for at least $\varepsilon\mathcal{E}ll$ indices }j\Big\}
$$
by ${\mathbb E}vent$, and additionally, for any subset $Q\subset[\mathcal{E}ll]$ of cardinality $\lfloor \varepsilon\mathcal{E}ll/4\rightfloor$
and any vector $z\in\{-1,1\}^2$, set
$$
{\mathbb E}vent_{Q,z}:=\Big\{\Big\left\langlengle\Big({\mathcal F}rac{1}{u}\sum_{w=1}^u
\cos(2\mathbb{P}i\,\mathcal{E}ta[I_w]\,s_j/k),
{\mathcal F}rac{1}{u}\sum_{w=1}^u
\sin(2\mathbb{P}i\,\mathcal{E}ta[I_w]\,s_j/k)\Big),z\Big\rightangle\geq \varepsilon\;\mbox{ for all }j\in Q\Big\}.
$$
It is not difficult to see that
$$
{\mathbb E}vent\subset\bigcup\limits_{Q,z}{\mathbb E}vent_{Q,z},
$$
whence it is sufficient to show that for any admissible $Q,z$,
\begin{equation}\left\langlebel{entosh}
{\mathbb P}({\mathbb E}vent_{Q,z})\leq {\mathcal F}rac{1}{4}{\mathcal{E}ll\choose \lfloor \varepsilon\mathcal{E}ll/4\rightfloor}^{-1}\varepsilon^u.
\mathcal{E}_{n-1}d{equation}
Without loss of generality, we can consider $Q=Q_0:=\big[\lfloor \varepsilon\mathcal{E}ll/4\rightfloor\big]$.
Event ${\mathbb E}vent_{Q_0,z}$ is contained inside the event
$$
\Big\{\Big|\sum_{j\in Q_0}\sum_{w=1}^u
\mathcal{E}xp(2\mathbb{P}i{\bf i}\,\mathcal{E}ta[I_w]\,s_j/k)\Big|\geq 2^{-1/2}\varepsilon u\,\lfloor \varepsilon\mathcal{E}ll/4\rightfloor\Big\},
$$
while the latter is contained inside the event
$$
\Big\{\Big|\sum_{j\in Q_0}
\mathcal{E}xp(2\mathbb{P}i{\bf i}\,\mathcal{E}ta[I_w]\,s_j/k)\Big|\geq {\mathcal F}rac{\varepsilon}{4}\,\lfloor \varepsilon\mathcal{E}ll/4\rightfloor
\mbox{ for at least $\varepsilon u/4$ indices $w$}\Big\}.
$$
Thus, taking the union over all admissible choices of $\lceileil\varepsilon u/4\rightceileil$ indices $w\in[u]$, we get
$$
{\mathbb P}({\mathbb E}vent_{Q_0,z})
\leq {u\choose \lceileil\varepsilon u/4\rightceileil}\max\limits_{F\subset[u],\,|F|=\lceileil\varepsilon u/4\rightceileil}{\mathbb P}\Big\{
\Big|\sum_{j\in Q_0}
\mathcal{E}xp(2\mathbb{P}i{\bf i}\,\mathcal{E}ta[I_w]\,s_j/k)\Big|\geq {\mathcal F}rac{\varepsilon}{4}\,\lfloor \varepsilon\mathcal{E}ll/4\rightfloor\mbox{ for all $w\in F$}
\Big\}.
$$
To estimate the last probability, we apply Markov's inequality, together with the bound for the second moment from
Lemma~\rightef{l: aux 43098275} (applied with $\varepsilon_1$), and using independence of $\mathcal{E}ta[I_w]$,
$w\leq u$.
We then get
$$
\max\limits_{F\subset[u]a_1top |F|=\lceileil\varepsilon u/4\rightceileil}{\mathbb P}\Big\{
\Big|\sum_{j\in Q_0}
\mathcal{E}xp(2\mathbb{P}i{\bf i}\,\mathcal{E}ta[I_w]\,s_j/k)\Big|\geq {\mathcal F}rac{\varepsilon}{4}\,\lfloor \varepsilon\mathcal{E}ll/4\rightfloor\mbox{ for all $w\in F$}
\Big\}\leq
\bigg({\mathcal F}rac{\varepsilon_1 \mathcal{E}ll^2}{(\varepsilon^2 \mathcal{E}ll/32)^2}\bigg)^{\lceileil \varepsilon u/4\rightceileil}
\leq
e^{-3\varepsilon u/2} \varepsilon^{2u}.
$$
In view of (\rightef{entosh}) this implies the result, since using $8\leq \mathcal{E}ll \leq u$ and $\varepsilon<1/2$,
we have
$$
4 {\mathcal{E}ll\choose \lfloor \varepsilon\mathcal{E}ll/4\rightfloor}\varepsilon^{-u} {u\choose \lceileil\varepsilon u/4\rightceileil} e^{-3\varepsilon u/2} \varepsilon^{2u}
\leq 4e^{-3\varepsilon u/2} \bigg({\mathcal F}rac{4e}{\varepsilon }\bigg)^{\varepsilon \mathcal{E}ll/4}
\bigg({\mathcal F}rac{2e}{\varepsilon }\bigg)^{\varepsilon u/2}\varepsilon^{u} \leq 4 (16e^{-3})^{\varepsilon u/4}\,
\varepsilon^{u/4}\leq 1.
$$
\mathcal{E}_{n-1}d{proof}
Our next step is to show that for the vector $X=(X_1,{\rm dist}ots,X_n)$ uniformly distributed on $\Lambda_n$
the random product $\mathbb{P}rod\limits_{i=1}^m\mathbb{P}si_{K_2}\big(\big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}
\sum_{w\in S_i}\mathcal{E}xp(2\mathbb{P}i{\bf i}X_w s)\big|\big)$ is, in a certain sense,
typically small (for most choices of $s$). To do this we first show that given a collection
of distinct numbers $s_1,{\rm dist}ots,s_\mathcal{E}ll$ which are pairwise well separated, the above product is small
for at least one $s_j$ with very high probability.
\begin{lemma}\left\langlebel{l: aux 0876958237}
For any $\varepsilon\in(0,1/2]$ there are $R_{\text{\tiny\rightef{l: aux 0876958237}}}
=R_{\text{\tiny\rightef{l: aux 0876958237}}}(\varepsilon)\geq 1$
and $\mathcal{E}ll:=\mathcal{E}ll_{\text{\tiny\rightef{l: aux 0876958237}}}(\varepsilon)\in{\mathbb N}$
with the following property.
Let $k,m,n\in{\mathbb N}$ be with $n/m\geq \mathcal{E}ll$. Let $1\leq K_2\leq 2/\varepsilon$,
$X=(X_1,{\rm dist}ots,X_n)$ be a random vector
uniformly distributed on $\Lambda_n$, and
let $s_1,{\rm dist}ots,s_\mathcal{E}ll$ be real numbers in $[0,k/2]$ such that $|s_j-s_q|\geq R_{\text{\tiny\rightef{l: aux 0876958237}}}$ for all $j\neq q$.
Fix disjoint subsets $S_1,{\rm dist}ots,S_m$ of $[n]$, each of cardinality $\lfloor n/m\rightfloor$.
Then
$$
{\mathbb P}\Big\{{\mathcal F}orall j\leq \mathcal{E}ll\, : \, \, \mathbb{P}rod\limits_{i=1}^m\mathbb{P}si_{K_2}\bigg(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}
\sum_{w\in S_i}\mathcal{E}xp(2\mathbb{P}i{\bf i}X_w s_j)\Big|\bigg)\geq (K_2/2)^{-m/2}\Big\}
\leq \varepsilon^n.
$$
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Fix any $\varepsilon\in(0,1/2]$ and set $\mathcal{E}ll:=\mathcal{E}ll_{\text{\tiny\rightef{l: aux 9871039481}}}(\varepsilon^5)\geq 1000$
and $R:=R_{\text{\tiny\rightef{l: aux 9871039481}}}(\varepsilon^5)$.
Assume that $n/m\geq \mathcal{E}ll$.
Note that, by our definition of $\Lambda_n$, the coordinates
of $X$ are independent and, moreover,
each variable $kX_w$ is distributed on an integer interval of cardinality at least $k$.
Thus, it is sufficient to prove that for any collection of integer intervals $I_j$, $j\leq n$, such that
$|I_j|\geq k$, the event
$$
{\mathbb E}vent:=\Big\{{\mathcal F}orall j\leq \mathcal{E}ll\, : \, \, \mathbb{P}rod\limits_{i=1}^m\mathbb{P}si_{K_2}\bigg(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}
\sum_{w\in S_i}\mathcal{E}xp(2\mathbb{P}i{\bf i}\,\mathcal{E}ta[I_w]\, s_j/k)\Big|\bigg)\geq (K_2/2)^{-m/2}\Big\}.
$$
has probability at most $\varepsilon^n$, where, as usual, we assume that the variables $\mathcal{E}ta[I_w]$, $w\in S_i$, $i\leq m$,
are jointly independent.
Observe that, as $\mathbb{P}si_{K_2}(t)\leq 1$ for all $t\leq 1$, the event ${\mathbb E}vent$ is contained inside the event
$$
{\mathbb E}vent':=\Big\{{\mathcal F}orall j\leq \mathcal{E}ll\, : \, \,
a_{ij}\geq 2/K_2\mbox{ for at least $m/2$ indices $i$}\Big\},
$$
where $a_{ij}:=\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor} \sum_{w\in S_i}\mathcal{E}xp(2\mathbb{P}i{\bf i}\,\mathcal{E}ta[I_w]\, s_j/k)\Big|$,
$i\leq m$, $j\leq \mathcal{E}ll$. Denoting $b_{ij}=1$ if $a_{ij}\geq 2/K_2$ and $b_{ij}=0$ otherwise and using
a simple counting argument for the matrix $\{b_{ij}\}_{ij}$, we obtain that
$$
{\mathbb E}vent\subset{\mathbb E}vent'\subset {\mathbb E}vent'':=
\Big\{\Big|\Big\{i\,:\,\,\, a_{ij}\geq 2/K_2\,\,\, \mbox{ for at least $\mathcal{E}ll/4$ indices $j$}\Big\}\Big|\geq m/4\Big\}.
$$
To estimate ${\mathbb P}({\mathbb E}vent'')$ we use Lemma~\rightef{l: aux 9871039481} with $\varepsilon^5$.
Note that $\varepsilon^5\leq \min(2/K_2, 1/2)$, and that by our choice of $R$, for any $j\neq q$ we have
$ |I_w|\,|s_j-s_q|\geq k\,|s_j-s_q|\geq R_{\text{\tiny\rightef{l: aux 9871039481}}}(\varepsilon^5)k$,
while $|s_j-s_q|\leq k/2$.
Thus,
$$
{\mathcal F}orall i\leq m \, : \quad {\mathbb P}\Big\{ a_{ij} \geq 2/K_2\,\,\, \mbox{ for at least $\mathcal{E}ll/4$ indices $j$}
\Big\}\leq \varepsilon^{5\lfloor n/m\rightfloor}.
$$
Hence,
$$
{\mathbb P}({\mathbb E}vent'')\leq {m\choose \lceileil m/4\rightceileil}\varepsilon^{5\lfloor n/m\rightfloor\,m/4}
\leq 2^m\varepsilon^{5\lfloor n/m\rightfloor\,m/4}\leq \varepsilon^n,
$$
which completes the proof.
\mathcal{E}_{n-1}d{proof}
\begin{lemma}[Very small product everywhere except for a set of measure $O(1)$]\left\langlebel{l: aux 2398205987305}
For any $\varepsilon\in(0,1/2]$ there are
$R_{\text{\tiny\rightef{l: aux 2398205987305}}}
=R_{\text{\tiny\rightef{l: aux 2398205987305}}}(\varepsilon)\geq 1$,
$\mathcal{E}ll=\mathcal{E}ll_{\text{\tiny\rightef{l: aux 2398205987305}}}(\varepsilon)\in{\mathbb N}$
and $n_{\text{\tiny\rightef{l: aux 2398205987305}}}=n_{\text{\tiny\rightef{l: aux 2398205987305}}}(\varepsilon,K_3)\in{\mathbb N}$ with the following property.
Let $k,m,n\in{\mathbb N}$, $n\geq n_{\text{\tiny\rightef{l: aux 2398205987305}}}$, $k\leq 2^{n/\mathcal{E}ll}$,
$n/m\geq \mathcal{E}ll$, and $4\leq K_2\leq 2/\varepsilon$.
Let $X=(X_1,{\rm dist}ots,X_n)$ be a random vector
uniformly distributed on $\Lambda_n$.
Fix disjoint subsets $S_1,{\rm dist}ots,S_m$ of $[n]$, each of cardinality $\lfloor n/m\rightfloor$.
Then
$$
{\mathbb P}\bigg\{\Big|\Big\{s\in[0,k/2]:\;\mathbb{P}rod\limits_{i=1}^m\mathbb{P}si_{K_2}\bigg(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}
\sum_{w\in S_i}\mathcal{E}xp(2\mathbb{P}i{\bf i}X_w s)\Big|\bigg)\geq (K_2/4)^{-m/2}\Big\}\Big|\leq R_{\text{\tiny\rightef{l: aux 2398205987305}}}\bigg\}
\geq 1-(\varepsilon/2)^n.
$$
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Fix any $\varepsilon\in(0,1/2]$, and define $\widetilde\varepsilon:=\varepsilon^{3/2}/32$,
$\widetilde\mathcal{E}ll:=\mathcal{E}ll_{\text{\tiny\rightef{l: aux 0876958237}}}(\widetilde\varepsilon)$,
$\mathcal{E}ll:=2\widetilde\mathcal{E}ll$,
and $R:=4 R_{\text{\tiny\rightef{l: aux 0876958237}}}(\widetilde\varepsilon)
\mathcal{E}ll_{\text{\tiny\rightef{l: aux 0876958237}}}(\widetilde\varepsilon)> 1$.
Assume that the parameters $k,m,n$ and $S_1,{\rm dist}ots,S_m$ satisfy the assumptions of the lemma.
In particular, we assume that $n$ is large enough so that $(8K_2\mathbb{P}i n)^{\widetilde \mathcal{E}ll}\leq 2^n$ and ${\bf g}(n)^{\widetilde\mathcal{E}ll}\leq 2^n$.
Denote
$$
\beta:=(8K_2\mathbb{P}i m{\bf g}(n))^{-1}(2K_2)^{-m/2} \quad \mbox{ and } \quad
a_{ij}:=\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor} \sum_{w\in S_i}\mathcal{E}xp(2\mathbb{P}i{\bf i}\,\mathcal{E}ta[I_w]\, s_j/k)\Big|,
\, \, i\leq m,\, j\leq \widetilde\mathcal{E}ll .
$$
Let $T:=[0,k/2]\cap\,\beta{\mathbb Z}$. By Lemma~\rightef{l: aux 0876958237}
for any collection $s_1,{\rm dist}ots,s_{\widetilde\mathcal{E}ll}$ of points from $T$ satisfying $|s_j-s_q|\geq
R_{\text{\tiny\rightef{l: aux 0876958237}}}(\widetilde\varepsilon)$
for all $j\neq q$, we have
$$
{\mathbb P}\bigg\{{\mathcal F}orall j\leq \widetilde\mathcal{E}ll \,:\,\, \, \mathbb{P}rod\limits_{i=1}^m\mathbb{P}si_{K_2}(a_{ij})
\geq (K_2/2)^{-m/2}\bigg\}
\leq \widetilde\varepsilon^{\,n}.
$$
Taking the union bound over all possible choices of $s_1,{\rm dist}ots,s_{\widetilde\mathcal{E}ll}$ from $T$, we get
\begin{equation}\left\langlebel{eq: aux 2058032958}
\begin{split}
{\mathbb P}\bigg\{\mathbb{P}rod\limits_{i=1}^m\mathbb{P}si_{K_2}(a_{ij})
&\geq (K_2/2)^{-m/2}\mbox{ for all $j\leq {\widetilde\mathcal{E}ll}$ and for
some $s_1,{\rm dist}ots,s_{\widetilde\mathcal{E}ll}\in T$}\\
&\mbox{with $|s_p-s_q|\geq R_{\text{\tiny\rightef{l: aux 0876958237}}}(\widetilde\varepsilon)$
for all $p\neq q$}\bigg\}
\leq \widetilde\varepsilon^{\,n}|T|^{\widetilde\mathcal{E}ll}.
\mathcal{E}_{n-1}d{split}
\mathcal{E}_{n-1}d{equation}
Further, in view of Lemma~\rightef{l: lip of prod},
for any realization of $X_w$'s
the product
$$
f(s):=\mathbb{P}rod\limits_{i=1}^m \mathbb{P}si_{K_2}\bigg(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}
\sum_{w\in S_i}\mathcal{E}xp(2\mathbb{P}i{\bf i}X_w s)\Big|\bigg),
$$
viewed as a function of $s$, is $(8 K_2\mathbb{P}i {\bf g}(n)m)$-Lipschitz.
This implies that for any pair $(s,s')\in{\mathbb R}_+^2$,
satisfying $|s-s'|\leq \beta$, we have
$$
f(s)\geq (K_2/2)^{-m/2}\quad \quad \mbox{whenever}\quad
f(s')\geq (K_2/4)^{-m/2}.
$$
Moreover, for any collection $s_1',{\rm dist}ots,s_{\widetilde\mathcal{E}ll}'$ of numbers from $[0,k/2]$
satisfying $|s_p'-s_q'|\geq 2R_{\text{\tiny\rightef{l: aux 0876958237}}}(\widetilde\varepsilon)$ for all
$p\neq q$ there are numbers $s_1,{\rm dist}ots,s_{\widetilde\mathcal{E}ll}\in T$ with $|s_q-s_q'|\leq \beta$
$|s_p-s_q|\geq R_{\text{\tiny\rightef{l: aux 0876958237}}}(\widetilde\varepsilon)$ for all $p\neq q$
(we used also $2\beta\leq 1\leq R_{\text{\tiny\rightef{l: aux 0876958237}}}(\widetilde\varepsilon)$).
This, together with \mathcal{E}qref{eq: aux 2058032958}, yields
\begin{align*}
{\mathbb P}\bigg\{&\mathbb{P}rod\limits_{i=1}^m\mathbb{P}si_{K_2}\bigg(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}
\sum_{w\in S_i}\mathcal{E}xp(2\mathbb{P}i{\bf i}X_w s_j')\Big|\bigg)\geq (K_2/4)^{-m/2}\mbox{ for all
$j\leq {\widetilde\mathcal{E}ll}$ and some $s_1',{\rm dist}ots,s_{\widetilde\mathcal{E}ll}'\in [0,k/2]$}\\
&\mbox{with $|s_p'-s_q'|\geq 2R_{\text{\tiny\rightef{l: aux 0876958237}}}(\widetilde\varepsilon)$
for all $p\neq q$}\bigg\}\\
&\hspace{1cm}\leq \widetilde\varepsilon^{\,n}|T|^{\widetilde\mathcal{E}ll}\leq \widetilde\varepsilon^{\,n}\,
(k/\beta)^{\widetilde\mathcal{E}ll}
\leq \widetilde\varepsilon^{\,n} \,2^n\, (8K_2\mathbb{P}i m{\bf g}(n))^{\widetilde\mathcal{E}ll} (2K_2)^{m\widetilde\mathcal{E}ll/2}\\
&\hspace{1cm}\leq \widetilde\varepsilon^{\,n}\, 8^n\,(4/\varepsilon)^{m\widetilde\mathcal{E}ll/2}\,
\leq \widetilde\varepsilon^{\,n}\, \varepsilon^{-n/2} \,16^n\,
\leq (\varepsilon/2)^n.
\mathcal{E}_{n-1}d{align*}
The event whose probability is estimated above, clearly contains the event in question ---
$$
\bigg\{\Big|\Big\{s\in[0,k/2]:\;\mathbb{P}rod\limits_{i=1}^m\mathbb{P}si_{K_2}\Big(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}
\sum_{w\in S_i}\mathcal{E}xp(2\mathbb{P}i{\bf i}X_w s)\Big|\Big)\geq (K_2/4)^{-m/2}\Big\}\Big|
\geq 4 R_{\text{\tiny\rightef{l: aux 0876958237}}}(\widetilde\varepsilon){\widetilde\mathcal{E}ll}\bigg\}.
$$
This, and our choice of parameters, implies the result.
\mathcal{E}_{n-1}d{proof}
\begin{lemma}[Moderately small product for almost all $s$]\left\langlebel{l: aux 20985059837}
For any $\varepsilon\in(0,1]$ and $z\in(0,1)$ there are $\varepsilon'=\varepsilon'(\varepsilon)\in(0,1/2]$,
$n_{\text{\tiny\rightef{l: aux 20985059837}}}
=n_{\text{\tiny\rightef{l: aux 20985059837}}}(\varepsilon,z)\geq 10$,
and $C_{\text{\tiny\rightef{l: aux 20985059837}}}=C_{\text{\tiny\rightef{l: aux 20985059837}}}(\varepsilon,z)\geq 1$
with the following property.
Let $n\geq n_{\text{\tiny\rightef{l: aux 20985059837}}}$, $2^n\geq k\geq 1$,
$C_{\text{\tiny\rightef{l: aux 20985059837}}}\leq m\leq n/4$, and $4\leq K_2\leq 1/\varepsilon$.
Let $X=(X_1,{\rm dist}ots,X_n)$ be a random vector
uniformly distributed on $\Lambda_n$.
Fix disjoint subsets $S_1,{\rm dist}ots,S_m$ of $[n]$, each of cardinality $\lfloor n/m\rightfloor$.
Then
$$
{\mathbb P}\bigg\{ {\mathcal F}orall s\in [z,\varepsilon' k] \, : \, \, \,\mathbb{P}rod\limits_{i=1}^m\mathbb{P}si_{K_2}\bigg(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}
\sum_{w\in S_i}\mathcal{E}xp(2\mathbb{P}i{\bf i}X_w s)\Big|\bigg)\leq e^{-\sqrt{m}}\bigg\}
\geq 1-(\varepsilon/2)^n.
$$
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Let $\varepsilon'>0$ will be chosen later. Fix any $s\in [z,\varepsilon ' k]$.
Assume $m\geq (\varepsilon' z)^{-4}\geq 10$.
For $i\leq m$ denote
$$
\gammamma _i(s):=\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}\sum_{w\in S_i}\mathcal{E}xp(2\mathbb{P}i{\bf i}X_w s)\Big|,
\quad \quad f_i(s):=\mathbb{P}si_{K_2}\big( \gammamma _i(s)\big),
\quad \mbox{ and } \quad
f(s):=\mathbb{P}rod\limits_{i=1}^m f_i(s)
$$
Observe that by the definition of $\mathbb{P}si_{K_2}$ for each $i\leq m$ we have
$f_i(s)=\gammamma _i(s)$,
provided $\gammamma _i(s)\geq 1/K_2$.
Next note that if for some complex unit numbers $z_1 ,..., z_N$ their average
$v:=\sum _{i=1}^N z_i /N$ has length $1-a_1lphapha>0$ then, taking the unit complex number
$z_0$ satisfying $\left\langle z_0, v\righta =|v|$ we have
$$N(1-a_1lphapha) \leq \sum _{i=1}^N Re \left\langle z_i, v\righta \leq N,$$
therefore there are at least $N/2+1$ indices $i$ such that $Re \left\langle z_i, v\righta \geq 1-4a_1lphapha$.
This in turn implies that there exists an index $j$ such that there are at least $N/2$
indices $i$ with $Re \left\langle z_i, \bar z_j\righta \geq 1-16a_1lphapha$. Thus that the event
$\big\{f_i(s)\geq 1-{\mathcal F}rac{2}{\sqrt{m}}\big\}$ is contained in the event
$$
\Big\{\mathcal{E}xists\;\; w'\in S_i :\;\;\; \cos(2\mathbb{P}i s(X_w- X_{w'}))\geq
1-{\mathcal F}rac{32}{\sqrt{m}}\;\;\mbox{for at least } \, {\mathcal F}rac{n}{2m}\,
\mbox{ indices \, }w\in S_i\setminus\{w'\}\Big\}.
$$
To estimate the probability of the later event, we take the union bound over all choices of
$n/(2m)$ indices from $S_i$, and over all choices of $w'$. We then get
\begin{align*}
{\mathbb P}\bigg\{ f_i(s)
\geq 1-{\mathcal F}rac{2}{\sqrt{m}}\bigg\}
&\leq {\mathcal F}rac{n}{m}\,2^{\lfloor n/m\rightfloor}\,
\max\limits_{w'\in S_i,\,F\subset S_i\setminus\{w'\},a_1top |F|\geq n/(2m)}
{\mathbb P}\bigg\{{\mathcal F}orall w\in F :\,\, {\rm dist}(s(X_w-X_{w'}),{\mathbb Z})\leq {\mathcal F}rac{2}{m^{1/4}}\bigg\}
\mathcal{E}_{n-1}d{align*}
To estimate the probability under maximum we use the definition of $\Lambda_n$ and
independence of coordinates of the vector $X$. Note that for each fixed $w$ there
is an integer interval $I_w$ of the length at least $2k$ such that $X_w$ is uniformly
distributed on $I_w/k$. Therefore, fixing a realization $X_{w'}=b/k$, $b\in {\mathbb Z}$,
we need to count how many $a\in I_w$ are such that $s(a-b)/k$ is close to an integer.
This can be done by splitting $I_w$ into subintervals of length $k$ and considering cases
$z\leq s\leq 1$, $1<s\leq C'k/m^{1/4}$ (this case can be empty), and
$C'k/m^{1/4}<s\leq \varepsilon' k$.
This leads to the following bound with an absolute constant $C''>0$,
\begin{align*}
{\mathbb P}\bigg\{
f_i(s)
\geq 1-{\mathcal F}rac{2}{\sqrt{m}}\bigg\}
&\leq
{\mathcal F}rac{n}{m}\,2^{n/m}\,\bigg(\max\Big({\mathcal F}rac{C''}{z\,m^{1/4}},C''\varepsilon'\Big)\bigg)^{n/(2m)}
\leq {\mathcal F}rac{n}{m}\,\big(4 C''\varepsilon'\big)^{n/(2m)}
\mathcal{E}_{n-1}d{align*}
Using this estimate and the fact that
$\mathbb{P}si_{K_2}(t)\leq 1$ for
$t\leq 1$ (so, each $f_i (s)\leq 1$), we obtain
\begin{align*}
{\mathbb P}\bigg\{ f(s)
\geq \Big(1-{\mathcal F}rac{2}{\sqrt{m}}\Big)^{3m/4}\bigg\}
&\leq
{\mathbb P}\bigg\{ f_i(s)
\geq 1-{\mathcal F}rac{2}{\sqrt{m}}\;\;\mbox{for at least $m/4$ indices $i$}
\bigg\}\\
&\leq 2^m \bigg({\mathcal F}rac{n}{m}\,\big(4 C''\varepsilon'\big)^{n/(2m)}\bigg)^{m/4}=
\bigg({\mathcal F}rac{16n}{m}\bigg)^{m/4}
\big(4 C''\varepsilon'\big)^{n/8}.
\mathcal{E}_{n-1}d{align*}
The last step of the proof is somewhat similar to the one used in the proof of
Lemma~\rightef{l: aux 2398205987305} --- we discretize the interval $[z,\varepsilon ' k]$
and use the Lipschitzness $f(s)$. Recall that ${\bf g}(n)\leq 2^n$ and thus, by
Lemma~\rightef{l: lip of prod}, $f(s)$ is $(8 K_2\mathbb{P}i 2^n\,m)$-Lipschitz.
Let
$$
\beta:= \big(1-2/\sqrt{m}\big)^{3m/4}\big(8 K_2\mathbb{P}i \,2^n m \big)^{-1}
\quad \quad \mbox{ and } \quad \quad T:=[z,\varepsilon ' k]\cap \beta{\mathbb Z}.
$$
Then for any $s,s'\in [z,\varepsilon ' k]$
satisfying $|s-s'|\leq \beta$ we have
$|f(s)-f(s')|\leq \big(1-2/\sqrt{m}\big)^{3m/4}$
deterministically.
This implies that
\begin{align*}
{\mathbb P}\bigg\{{\mathcal F}orall s \in[z,\varepsilon'k]\, :\,\,\,
f(s)&\leq 2\Big(1-{\mathcal F}rac{2}{\sqrt{m}}\Big)^{3m/4}\bigg\}
\geq{\mathbb P}\bigg\{{\mathcal F}orall s \in T\, :\,\,\,
f(s)\leq \Big(1-{\mathcal F}rac{2}{\sqrt{m}}\Big)^{3m/4}\bigg\}
\\&\geq 1-{\mathcal F}rac{k}{\beta}\bigg({\mathcal F}rac{16n}{m}\bigg)^{m/4}
\big(4 C''\varepsilon'\big)^{n/8}
\geq 1-(\varepsilon/2)^n,
\mathcal{E}_{n-1}d{align*}
provided that $\varepsilon':=c''\varepsilon^{8}$ for a sufficiently small universal constant $c''>0$.
\mathcal{E}_{n-1}d{proof}
\begin{lemma}\left\langlebel{l: aux 98508746104921-4}
Let $\rightho, \varepsilon\in(0,1]$, $k\geq 1$, $h\in{\mathbb R}$, $a_1\geq h+1$, $a_2\leq h-\rightho-1$.
Let $Y_1,Y_2$ be independent random variables,
with $Y_1$ uniformly distributed on $[h,a_1]\cap {\mathcal F}rac{1}{k}{\mathbb Z}$ and
$Y_2$ uniformly distributed on $[a_2,h-\rightho]\cap {\mathcal F}rac{1}{k}{\mathbb Z}$.
Then for every $s\in [-\varepsilon/8, \varepsilon/8]$ one has
$$
{\mathbb P}\big\{\big|\mathcal{E}xp\big(2\mathbb{P}i{\bf i}Y_{1} s\big)+\mathcal{E}xp\big(2\mathbb{P}i{\bf i}Y_{2} s\big)\big|> 2
- 2\mathbb{P}i \rightho^2 s^2\big\}\leq \varepsilon.
$$
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Clearly, it is enough to consider $0<s<\varepsilon/8$ only. Note that
$$
\big|\mathcal{E}xp\big(2\mathbb{P}i{\bf i}Y_{1} s\big)+\mathcal{E}xp\big(2\mathbb{P}i{\bf i}Y_{2} s\big)\big|=
\big|1+\mathcal{E}xp\big(2\mathbb{P}i{\bf i}(Y_1-Y_{2}) s\big)\big|= 2\big|\cos\big(\mathbb{P}i{\bf i}(Y_1-Y_{2}) s\big)\big|.
$$
We consider two cases.
\|\cdot\|indent
{\it Case 1. } $a_1\leq h+2\varepsilon^{-1}$ and $a_2\geq h-2\varepsilon^{-1}$.
In this case, deterministically, $\rightho \leq Y_1-Y_{2}\leq 4/\varepsilon$, therefore,
using that $\cos t\leq 1-t^2/\mathbb{P}i$ on $[-\mathbb{P}i/2, \mathbb{P}i/2]$, we have
for every $s\in(0,\varepsilon/8]$,
$$
\big|\mathcal{E}xp\big(2\mathbb{P}i{\bf i}Y_{1} s\big)+\mathcal{E}xp\big(2\mathbb{P}i{\bf i}Y_{2} s\big)\big|\leq 2-2\mathbb{P}i \rightho^2s^2.
$$
\|\cdot\|indent
{\it Case 2. } Either $a_1> h+2\varepsilon^{-1}$ or $a_2< h-2\varepsilon^{-1}$.
Without loss of generality, we will assume the first inequality holds.
We condition on a realization $\widetilde Y_2$ of $Y_2$
(further in the proof, we compute conditional probabilities given $Y_2=\widetilde Y_2$).
For any $s\leq \varepsilon/8$, the event
$$
\big\{\big|1+\mathcal{E}xp\big(2\mathbb{P}i{\bf i}(Y_1-\widetilde Y_{2}) s\big)\big|\geq 2-s^2\big\}
$$
is contained inside the event
$$
\big\{{\rm dist}\big((Y_1-\widetilde Y_{2}) s,{\mathbb Z}\big)\leq s\big\}.
$$
On the other hand, since $(Y_1-\widetilde Y_{2})s$ is uniformly distributed on a set
$[b_1,b_2]\cap {\mathcal F}rac{s}{k}{\mathbb Z}$, for some $b_2\geq b_1+2\varepsilon^{-1} s$,
the probability of the last event is less than $\varepsilon$. The result follows.
\mathcal{E}_{n-1}d{proof}
\begin{lemma}[Integration for small $s$]\left\langlebel{l: aux -29802609872}
For any $\widetilde\varepsilon\in(0,1]$, $\rightho\in(0,1/4]$ and ${\rm dist}eltalta\in (0,1/2]$ there are
$n_{\text{\tiny\rightef{l: aux -29802609872}}}
=n_{\text{\tiny\rightef{l: aux -29802609872}}}(\widetilde\varepsilon,{\rm dist}eltalta,\rightho)$,
$C_{\text{\tiny\rightef{l: aux -29802609872}}}=C_{\text{\tiny\rightef{l: aux -29802609872}}}(\widetilde\varepsilon,{\rm dist}eltalta,\rightho)\geq 1$, and
$K_{\text{\tiny\rightef{l: aux -29802609872}}}=K_{\text{\tiny\rightef{l: aux -29802609872}}}({\rm dist}eltalta,\rightho)\geq 1$
with the following property.
Let $A_{nm}$ be defined as in (\rightef{anm}), $n\geq n_{\text{\tiny\rightef{l: aux 20985059837}}}$, $k\geq 1$,
$m\in{\mathbb N}$ with $n/m\geq C_{\text{\tiny\rightef{l: aux -29802609872}}}$ and $m\geq 2$, and let $X=(X_1,{\rm dist}ots,X_n)$ be a random vector
uniformly distributed on $\Lambda_n$.
Then for every $K_2\geq 4$,
\begin{align*}
{\mathbb P}\bigg\{&A_{nm} \,
\sum\limits_{S_1,{\rm dist}ots,S_m}\;
\int\limits_{-\sqrt{m}/C_{\text{\tiny\rightef{l: aux -29802609872}}}}^{\sqrt{m}/C_{\text{\tiny\rightef{l: aux -29802609872}}}}
\mathbb{P}rod\limits_{i=1}^{m}\mathbb{P}si_{K_2}\bigg(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}
\sum_{w\in S_i}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}X_{w} m^{-1/2}\,s\big)\Big|\bigg)\,ds\geq K_{\text{\tiny\rightef{l: aux -29802609872}}}
\bigg\}\leq (\widetilde\varepsilon/2)^n,
\mathcal{E}_{n-1}d{align*}
where the sum is taken over all disjoint subsets $S_1,{\rm dist}ots,S_m\subset[n]$ of cardinality $\lfloor n/m\rightfloor$ each.
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Let $n_{\rm dist}eltalta,C_{\rm dist}eltalta,c_{\rm dist}eltalta$, and $\mathcal S$ be as in Lemma~\rightef{l: aux 2498276098059385-}).
A given choice of subsets $(S_1,{\rm dist}ots,S_m)\in\mathcal S$ denote
$$
\gammamma _i(s):=\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}\sum_{w\in S_i}\mathcal{E}xp(2\mathbb{P}i{\bf i}X_w s)\Big|,
\quad \quad f_i(s):=\mathbb{P}si_{K_2}\big( \gammamma _i(s)\big),
\quad \mbox{ and } \quad
f(s):=\mathbb{P}rod\limits_{i=1}^m f_i(s)
$$
(note that functions $\gammamma _i(s)$, $f_i(s)$, $f(s)$ depend on the
choice of subsets $S_i$).
First, we study the distribution of the variable $f(s)$
for a given choice of subsets $S_i$.
We assume that $n\geq n_{\rm dist}eltalta$ and $n/m\geq C_{\rm dist}eltalta$.
We also denote $\varepsilon:=2^{-10/{\rm dist}eltalta}\, \widetilde\varepsilon^{\, 16/{\rm dist}eltalta c_{\rm dist}eltalta}$
and
\begin{align*}
\mathcal S':=
\Big\{&(S_1,{\rm dist}ots,S_m)\in \mathcal S:\;
\min(|S_i\cap Q_1|,|S_i\cap Q_2|)\geq {\rm dist}eltalta \lfloor n/m\rightfloor/2
\mbox{ for at least $c_{\rm dist}eltalta m$ indices $i$}
\Big\}.
\mathcal{E}_{n-1}d{align*}
Fix a sequence $(S_1,{\rm dist}ots,S_m)\in \mathcal S'$, and $J\subset[m]$ be a subset of cardinality $\lceileil c_{\rm dist}eltalta m\rightceileil$
such that
$$
{\mathcal F}orall i\in J\, : \,\,\min(|S_i\cap Q_1|,|S_i\cap Q_2|)\geq {\rm dist}eltalta \lfloor n/m\rightfloor/2 .
$$
For any $i\in J$, $w_1\in S_i\cap Q_1$, and $w_2\in S_i\cap Q_2$ by Lemma~\rightef{l: aux 98508746104921-4}
we have for $s\in [-\varepsilon/8, \varepsilon/8]$,
$$
{\mathbb P}\big\{\big|\mathcal{E}xp\big(2\mathbb{P}i{\bf i}X_{w_1} s\big)+\mathcal{E}xp\big(2\mathbb{P}i{\bf i}X_{w_2} s\big)\big|\geq 2
-2\mathbb{P}i\rightho^2 s^2\big\}\leq \varepsilon.
$$
Within $S_i$, we can find at least ${\mathcal F}rac{{\rm dist}eltalta}{2}\lfloor n/m\rightfloor$ disjoint pairs
of indices $(w_1,w_2)\in Q_1\times Q_2$
satisfying the above condition. Let $T$ be a set of such pairs with $|T|={\mathcal F}rac{{\rm dist}eltalta}{2}\lfloor n/m\rightfloor$.
Using the independence of coordinates of $X$, and denoting $z:=\min\big(\sqrt{1/(\mathbb{P}i\rightho^2{\rm dist}eltalta)},
\varepsilon/8\big)$, we obtain for every $s\in[-z,z]$,
\begin{align*}
{\mathbb P}\bigg\{&\gammamma_i(s)\geq 1-{\mathcal F}rac{\mathbb{P}i\rightho^2{\rm dist}eltalta s^2}{2}\bigg\}\\
&\leq
{\mathbb P}\big\{\big|\mathcal{E}xp\big(2\mathbb{P}i{\bf i}X_{w_1} s\big)+\mathcal{E}xp\big(2\mathbb{P}i{\bf i}X_{w_2} s\big)\big|\geq 2
-2\mathbb{P}i\rightho^2 s^2\mbox{ for at least ${\mathcal F}rac{{\rm dist}eltalta}{4}\lfloor n/m\rightfloor$ pairs $(w_1,w_2)\in T$}\big\}\\
&\leq 2^{{\rm dist}eltalta\lfloor n/m\rightfloor/2}\,\varepsilon^{{\rm dist}eltalta\lfloor n/m\rightfloor/4}
\leq (4\varepsilon) ^{{\rm dist}eltalta n/(4m)}.
\mathcal{E}_{n-1}d{align*}
Applying this for all $i\in J$ together with observations $f(s)\leq 1$ and $f_i(s)=\gammamma_i(s)$
(when $\gammamma_i(s)\geq 1/K_2$), we conclude that for every $s\in[-z,z]$,
\begin{align*}
{\mathbb P}\bigg\{f(s)\geq \big(1-\mathbb{P}i\rightho^2{\rm dist}eltalta s^2/2\big)^{|J|/2}\bigg\}
&\leq {\mathbb P}\bigg\{
f_i(s)\geq 1-\mathbb{P}i\rightho^2{\rm dist}eltalta s^2/2\, \,
\mbox{ for at least $|J|/2$ indices $i\in J$}\bigg\}\\
&\leq 2^{|J|}\,(4\varepsilon) ^{{\rm dist}eltalta |J|n/(8m)}
\mathcal{E}_{n-1}d{align*}
At the next step, we apply the Lemma~\rightef{l: int markov} with $\xi(s)=f(s)$ to obtain
from the previous relation
$$
{\mathbb P}\bigg\{\int\limits_{-z}^{z}
f(s)\,ds
\leq \int\limits_{-z}^{z}
\bigg(1-{\mathcal F}rac{\mathbb{P}i\rightho^2{\rm dist}eltalta s^2}{2}\bigg)^{|J|/2}\,ds
+m^{-1/2}\bigg\}\geq 1-2z m^{1/2}\,2^{|J|}\,(4\varepsilon) ^{{\rm dist}eltalta |J|n/(8m)}.
$$
Next we apply Lemma~\rightef{l: sum markov}) with $I=\mathcal S'$, $\xi_i=f(s)$
(recall that $f(s)$ depends also on the choice of $(S_1,{\rm dist}ots,S_m)\in\mathcal S$).
We obtain
\begin{align*}
{\mathbb P}\bigg\{A_{nm}\,
\sum\limits_{(S_1,{\rm dist}ots,S_m)\in\mathcal S'}\;\int\limits_{-z}^{z}
f(s)\,ds
\leq \int\limits_{-z}^{z}
\bigg(1-{\mathcal F}rac{\mathbb{P}i\rightho^2{\rm dist}eltalta s^2}{2}\bigg)^{|J|/2}\,ds+2m^{-1/2}\bigg\}
\geq 1-2z m\,2^{|J|}\,(4\varepsilon) ^{{\rm dist}eltalta |J|n/(8m)}.
\mathcal{E}_{n-1}d{align*}
Further, since by Lemma~\rightef{l: aux 2498276098059385-} we have
$|\mathcal S'|\geq (1-e^{-c_{\rm dist}eltalta n})|\mathcal S|$ and since $f(s)\leq 1$, we observe
that
\begin{align*}
&A_{nm}\,
\sum\limits_{(S_1,{\rm dist}ots,S_m)\in \mathcal S\setminus\mathcal S'}\;\int\limits_{-z}^{z}
f(s)\, ds
\leq 2z\,e^{-c_{\rm dist}eltalta n}
\mathcal{E}_{n-1}d{align*}
deterministically.
Recalling that $|J|=\lceileil c_{\rm dist}eltalta m\rightceileil$,
we obtain
\begin{align*}
{\mathbb P}\bigg\{&
A_{nm}\,
\sum\limits_{(S_1,{\rm dist}ots,S_m)\in\mathcal S}\;\int\limits_{-z}^{z}
f(s)\, ds
\leq C'' m^{-1/2} \bigg\}
\geq 1-2z m\,2^{|J|}\,(4\varepsilon) ^{{\rm dist}eltalta |J|n/(8m)}
\geq 1-(\widetilde \varepsilon/2)^n,
\mathcal{E}_{n-1}d{align*}
for some $C''\geq 1$ depending only on ${\rm dist}eltalta$ and $\rightho$, provided that
$n\geq n_0(\widetilde\varepsilon,{\rm dist}eltalta,\rightho)$.
The result follows by the substitution $s= m^{-1/2}u$ in the integral.
\mathcal{E}_{n-1}d{proof}
As a combination of Lemmas~\rightef{l: aux 2398205987305},~\rightef{l: aux 20985059837}
and~\rightef{l: aux -29802609872}, we obtain Proposition~\rightef{prop: 09582593852}.
\begin{proof}[Proof of Proposition~\rightef{prop: 09582593852}]
As we mentioned at the beginning of this subsection, we
fix $\rightho,{\rm dist}eltalta\in(0,1/4]$, a growth function ${\bf g}$ satisfying \mathcal{E}qref{gfncond},
a permutation ${\sigma}ma\in\Pi_n$, a number $h\in{\mathbb R}$, two sets $Q_1,Q_2\subset[n]$
such that $|Q_1|,|Q_2|=\lceileil {\rm dist}eltalta n\rightceileil$, and we use
$\Lambda_n$ for the set $\Lambda_n(k,{\bf g},Q_1,Q_2,\rightho,{\sigma}ma,h)$
defined in \mathcal{E}qref{eq: param l def}. We also fix $\varepsilon\in (0,1/4]$.
We start by selecting the parameters.
Assume that $n$ is large enough.
Set $\mathcal{E}ll:=\mathcal{E}ll_{\text{\tiny\rightef{l: aux 2398205987305}}}(\varepsilon)$.
Let $\varepsilon'=\varepsilon'(\varepsilon)$ be taken from Lemma~\rightef{l: aux 20985059837}.
Set $z:=1/C_{\text{\tiny\rightef{l: aux -29802609872}}}(\varepsilon,{\rm dist}eltalta,\rightho)$.
Fix an integer $m\in [C_{\text{\tiny\rightef{l: aux 20985059837}}}(\varepsilon,z),
n/\max(\mathcal{E}ll,C_{\text{\tiny\rightef{l: aux -29802609872}}})]$ satisfying the condition
$R_{\text{\tiny\rightef{l: aux 2398205987305}}}\sqrt{m}\,e^{-\sqrt{m}}\leq 1$,
and take $1\leq k\leq \min\big(2^{n/\mathcal{E}ll},(K_2/8)^{m/2}\big)$.
Let $A_{nm}$ be defined as in (\rightef{anm}).
We assume that $h$ is chosen in such a way that the set $\Lambda_n$
is non-empty. As before $X$ denotes the random vector uniformly distributed on
$\Lambda_n$.
Let $\mathcal S$ be as in Lemma~\rightef{l: aux 2498276098059385-}).
A given choice of subsets $(S_1,{\rm dist}ots,S_m)\in\mathcal S$ denote
$$
f(s)=f_{S_1,{\rm dist}ots,S_m}(s):=\mathbb{P}rod\limits_{i=1}^{m}\mathbb{P}si_{K_2}\bigg(\Big|{\mathcal F}rac{1}{\lfloor n/m\rightfloor}
\sum_{w\in S_i}\mathcal{E}xp\big(2\mathbb{P}i{\bf i}X_{w} m^{-1/2}\,s\big)\Big|\bigg).
$$
We have
\begin{align*}
A_{nm}\sum\limits_{S_1,{\rm dist}ots,S_m}\;
&\int\limits_{-\varepsilon' m^{1/2}k}^{\varepsilon' m^{1/2}k}
f(s)\, ds
=
A_{nm}\sum\limits_{S_1,{\rm dist}ots,S_m}\;
\int\limits_{-z \sqrt{m}}^{z \sqrt{m}} f(s)\, ds
+2 A_{nm}\sum\limits_{S_1,{\rm dist}ots,S_m}\;
\int\limits_{z \sqrt{m}}^{\varepsilon'k\sqrt{m}}
f(s)\, ds
\mathcal{E}_{n-1}d{align*}
In view of Lemma~\rightef{l: aux -29802609872},
with probability at least $1-(\varepsilon/2)^n$ the first summand is bounded above by
$K_{\text{\tiny\rightef{l: aux -29802609872}}}$.
To estimate the second summand, we combine Lemmas~\rightef{l: aux 2398205987305} and~\rightef{l: aux 20985059837}
(we assume that $z \leq\varepsilon' k$ as otherwise there is no second summand).
Fix for a moment a collection $(S_1,{\rm dist}ots,S_m)\in\mathcal S$.
By Lemma~\rightef{l: aux 2398205987305}, with probability at least $1-(\varepsilon/2)^n$
the function $f$ on $[0,k\sqrt{m}/2]$ is bounded above by $(K_2/4)^{-m/2}$
for all points $s$ outside of some set of measure at most $R_{\text{\tiny\rightef{l: aux 2398205987305}}}\sqrt{m}$
(note that we apply variable transformation $s\to m^{-1/2}s$ to use the lemma here).
Further, by Lemma~\rightef{l: aux 20985059837}, with probability at least $1-(\varepsilon/2)^n$ we have
that $f$ is bounded above by $e^{-\sqrt{m}}$ for all $s\in[z \sqrt{m},\varepsilon' k\sqrt{m}]$.
Thus, with probability at least $1-2(\varepsilon/2)^n$,
$$
\int\limits_{z \sqrt{m}}^{\varepsilon' k\sqrt{m}}f(s)\,ds\leq \sqrt{m}k\,\Big({\mathcal F}rac{K_2}{4}\Big)^{-m/2}+
R_{\text{\tiny\rightef{l: aux 2398205987305}}}\sqrt{m}\,e^{-\sqrt{m}}.
$$
Applying Lemma~\rightef{l: sum markov} with $I=\mathcal S$ and $\xi_i=f(s)$, we obtain that
\begin{align*}
& A_{nm}\sum\limits_{S_1,{\rm dist}ots,S_m}\;
\int\limits_{z \sqrt{m}}^{\varepsilon'k\sqrt{m}}
f(s)\, ds \leq \sqrt{m}k\,\Big({\mathcal F}rac{K_2}{4}\Big)^{-m/2}+
R_{\text{\tiny\rightef{l: aux 2398205987305}}}\sqrt{m}\,e^{-\sqrt{m}}+1\leq 3
\mathcal{E}_{n-1}d{align*}
with probability at least $1-2(\varepsilon/2)^n$.
Thus,
taking $K_1:=K_{\text{\tiny\rightef{l: aux -29802609872}}}+3$, we obtain
$${\mathbb P}\{{\bf UD}_n(X,m,K_1,K_2)\geq \varepsilon' m^{1/2}k\}\geq 1-3( \varepsilon/2)^n\geq 1-3 \varepsilon^n.$$
\mathcal{E}_{n-1}d{proof}
\section{Complement of gradual non-constant vectors: constant $p$}
\left\langlebel{steep:constant p}
In this section, we study the problem of invertibility of the Bernoulli($p$) matrix $M$ over the set
${\mathcal S}_n$ defined by (\rightef{strvect}) in the case when the parameter $p$ is a small constant.
This setting turns out to be much simpler
than treatment of the general case $C\ln n/n\leq p\leq c$ given in the next section.
Although the results of Section~\rightef{s: steep} essentially absorb the statements of this section,
we prefer to include analysis of the constant $p$ in our work, first, because it provides
a short and relatively simple illustration of our method and, second, because the estimates obtained here
allow to derive better quantitative bounds\ for the smallest singular value of $M$.
\subsection{Spliting of ${\mathbb R}^n$ and main statements}
\left\langlebel{subs: steep vectors}
We define the following four classes of vectors $\mathcal{T}t_1, {\rm dist}ots, \mathcal{T}t_{4}$. For simplicity,
we normalize vectors with respect to the Euclidean norm.
The first class is the set of vectors with one coordinate much larger than the others, namely,
$$
\mathcal{T}t _{1}= \mathcal{T}t _{1}(p): =\{x\in S^{n-1}\,:\, x_{1}^{*}> 6 pn\, x_{2}^{*}\}.
$$
For the next sets we fix a parameter $\beta_p = \sqrt{p}/C_0$, where $C_0$ is the
absolute constant from Proposition~\rightef{rogozin}. Recall also that the operator
$Q$ (which annihilates the maximal coordinate of a given vector) and the set
$U(m, \gammamma)$ were introduced in Subsection~\rightef{net}.
We also fix a small enough absolute positive constant $c_0$. We don't try to compute
the actual value of $c_0$, the conditions on how small $c_0$ is can be obtained
from the proofs. We further fix an integer $1\leq m\leq n$.
The second class of vectors consist of those vectors for which
the Euclidean norm dominates the maximal coordinate.
To control cardinalities of nets (discretizations) we intersect this
class with $U(m, c_0)$, specifically, we set
$$
\mathcal{T}t_2 =\mathcal{T}t_2(p,m):= \mathcal{T}t_2' \cap U(m, c_0), \quad \mbox{ where }\quad
\mathcal{T}t_2' := \left\{x\in S^{n-1}\,:\, x\|\cdot\|t\in \mathcal{T}t_{1}
\,\, \mbox{ and } \, \, x_{1}^{*}\leq \beta_p \right\}.
$$
The next set is similar to $\mathcal{T}t _2$, but instead of comparing $x_1^*$ with
the Euclidean norm of the entire vector, we compare $x_2^*$ with $\|Qx\|$.
For a technical reason,
we need to control the magnitude of $\|Qx\|$ precisely; thus we partition the third set into subsets.
Let numbers $\left\langlembda_k$, $k\leq \mathcal{E}ll$, be defined by
\begin{equation}\left\langlebel{eq: 0495205965029385}
\left\langlembda_1 = {\mathcal F}rac{1}{6pn},\quad \left\langlembda_{k+1}= 3 \left\langlembda _k, \, \, k<\mathcal{E}ll -1,
\quad 1/3\leq \left\langlembda_{\mathcal{E}ll-1} <1 \quad \mbox{ and } \quad \left\langlembda _\mathcal{E}ll =1.
\mathcal{E}_{n-1}d{equation}
Clearly, $\mathcal{E}ll \leq \ln n$.
Then for each $k\leq \mathcal{E}ll-1$ we define
\begin{align*}
\mathcal{T}t _{3,k}=\mathcal{T}t _{3,k}(p,m) &: =\left\{x\in S^{n-1}\,:\, x\|\cdot\|t\in \mathcal{T}t_{1} \cup \mathcal{T}t_{2}', \, \,
x_{2}^{*}\leq \beta_p \|Qx\|
\,\, \mbox{ and } \, \, \left\langlembda_k \leq \|Qx\| < \left\langlembda_{k+1}\right\}\cap U(m, c_0 \left\langlembda_k) .
\mathcal{E}_{n-1}d{align*}
To explain the choice of $\left\langlembda_1$, note that if $x\|\cdot\|t\in \mathcal{T}t_{1} \cup \mathcal{T}t_{2}'$
and $\|x\|=1$, then
$x_2^*\geq x_1^*/(6pn)\geq \beta_p/(6pn)$. Thus, if in addition
$\beta_p\|Qx\| \geq x_{2}^{*}$, then $\|Qx\| \geq 1/(6pn)=\left\langlembda_1$.
We set $$\mathcal{T}t _{3}=\mathcal{T}t _{3}(p,m):= \bigcup_{k=1}^{\mathcal{E}ll-1} \mathcal{T}t _{3,k}.$$
The fourth set covers the remaining options for vectors having a large almost constant part.
Let numbers $\mu_k$, $k\leq s$, be defined by
\begin{equation}\left\langlebel{eq: 9635194-9580-98}
\mu_1 = {\mathcal F}rac{\beta_p}{6pn},\quad \mu_{k+1}= 3 \mu _k, \, \, k<s -1,
\quad 1/3\leq \mu_{s-1} <1
\quad \mbox{ and } \quad \mu _s =1.
\mathcal{E}_{n-1}d{equation}
Clearly, $s \leq \ln n$.
Then for each $k\leq s-1$ define the set $\mathcal{T}t _{4,k} =\mathcal{T}t _{4,k}(p,m)$ as
$$
\left\{x\in S^{n-1}\,:\, x\|\cdot\|t\in \mathcal{T}t_{1} \cup \mathcal{T}t_{2}', \, \,
x_{2}^{*}> \beta_p \|Qx\| \,\, \mbox{ and } \, \,
\mu_k \leq x^*_2 < \mu_{k+1}\right\}\cap U(m, c_0 \mu_k /\sqrt{\ln(e/p)}).
$$
Note that if $x\|\cdot\|t\in \mathcal{T}t_{1} \cup \mathcal{T}t_{2}'$
and $\|x\|=1$, then $x_2^*\geq x_1^*/(6pn)\geq \beta_p/(6pn)$, justifying the choice of $\mu_1$.
We set $$\mathcal{T}t _{4}=\mathcal{T}t _{4}(p,m)= \bigcup_{k=1}^{\mathcal{E}ll-1} \mathcal{T}t _{4,k}.$$
Finally define $\mathcal{T}t$ as the union of these four classes,
$
\mathcal{T}t=\mathcal{T}t(p,m):=\bigcup_{j=1}^{4} \mathcal{T}t_{j}.
$
In this section we prove two following theorems.
\begin{theor} \left\langlebel{t:steep}
There exists positive absolute constants $c, C$ such that the following holds.
Let $n$ be large enough, let $m\leq cpn/\ln(e/p)$, and $(30\ln n)/n\leq p<1/20$.
Let $M$ be an $n\times n$ Bernoulli($p$) random matrix. Then
\begin{equation*}
\left\langlebel{Psteep}
{\mathbb P} \Big\{\mathcal{E}xists\;x\in \mathcal{T}t\, \, \, \mbox{ such that }
\, \, \,
\|M x\| < {\mathcal F}rac{ 1}{C \sqrt{n \ln(e/p)}} \, \, \|x\|
\Big\} \leq n(1-p)^n + 4e^{-1.5np},
\mathcal{E}_{n-1}d{equation*}
where the set $\mathcal{T}t=\mathcal{T}t(p,m)$ is defined above.
\mathcal{E}_{n-1}d{theor}
Recall that the set
${\mathcal V}_n$ was introduced in Subsection~\rightef{gradnac}.
The next theorem shows that, after a proper normalization,
the complement of ${\mathcal V}_n$ (taken in ${\Upsilon}_n(r)$) is contained in $\mathcal{T}t$
for some choice of $r, {\rm dist}eltalta, \rightho$ and for the growth function
${\bf g}(t)=(2t)^{3/2}$ (clearly, satisfying (\rightef{gfncond})).
\begin{theor}
\left\langlebel{compl-1}
There exists an absolute (small) positive constant $c_1$ such that the following holds.
Let $q\in (0, c_1)$ be a parameter. Then there exist $n_q\geq 1$,
$r=r(q), \rightho=\rightho (q)\in (0,1)$ such that for $n\geq n_q$, $p\in (q, c_1)$, ${\rm dist}eltalta = r/3$,
${\bf g}(t)=(2t)^{3/2}$, and $m=\lfloor rn \rightfloor$ one has
$$
\Big\{x/\|x\|\, \, : \, \, x\in {\Upsilon}_n(r)\setminus {\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)\Big\}\subset \mathcal{T}t(p,m).
$$
\mathcal{E}_{n-1}d{theor}
\subsection{Proof of Theorem~\rightef{t:steep}}
Theorem~\rightef{t:steep} is a consequence of four
lemmas that we prove in this section. Each lemma treats
one of the classes $\mathcal{T}t_i$, $i\leq 4$, and Theorem~\rightef{t:steep}
follows by the union bound.
Recall that $U(m, \gammamma)$ was introduced in Subsection~\rightef{net} and
that given $x$, we fixed one permutation, ${\sigma}ma_x$,
such that $x_i^*=|x_{{\sigma}ma_x(i)}|$ for $i\le n$. Recall also that the event
${\mathbb E}vent_{nrm}$ was introduced in Proposition~\rightef{nettri}.
\begin{lemma} \left\langlebel{st1}
Let $n\geq 1$ and $p\in (0, 1/2]$.
Let ${\mathbb E}vent_{sum}$ (with $q=p$)
be the event introduced in Lemma~\rightef{bennett} and by ${\mathbb E}vent_{col}\subset {\mathcal{M}_{n}}$
denote the subset of $0/1$ matrices with no zero columns.
Then for every $M\in {\mathbb E}vent_{sum}\cap {\mathbb E}vent_{col}$ and every $x\in \mathcal{T}t_1$,
$$
\|M x\| \geq {\mathcal F}rac{1}{3\sqrt{n}}\, \|x\|.
$$
In particular,
$$
\mathbb{P}\Bigl\{M\in{\mathcal{M}_{n}}:\;\mathcal{E}xists x\in\mathcal{T}t_1 \,\, \mbox{ with }\,\,
\|Mx\| \leq {\mathcal F}rac{1}{3\sqrt{n}}
\Bigr\}\leq n (1-p)^n + e^{-1.5np}.
$$
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Let ${\rm dist}eltalta_{ij}$, $i,j\leq n$ be entries of $M\in {\mathbb E}vent_{sum}\cap {\mathbb E}vent_{col}$.
Let ${\sigma}ma={\sigma}ma_x$.
Denote, $\mathcal{E}ll={\sigma}ma(1)$.
Since $M\in {\mathbb E}vent_{col}$, there exists $s\leq n$ such that
${\rm dist}eltalta_{s\mathcal{E}ll}=1$. Then
\begin{align*}
|\left\langlengle R_{s} (M),\, x \rightangle|
&=\Big| x_{\mathcal{E}ll}
+ \sum_{j\ne\mathcal{E}ll} {\rm dist}eltalta _{sj} x_j
\Big|
\geq|x_{\mathcal{E}ll}|- \sum_{j\ne\mathcal{E}ll} {\rm dist}eltalta _{sj} \, x_{j}\geq
|x_{\mathcal{E}ll}|- \sum_{j=1}^n {\rm dist}eltalta _{sj} \,x_{n_2}^* .
\mathcal{E}_{n-1}d{align*}
Using that $M\in {\mathbb E}vent_{sum}$ we observe that
$\sum_{j=1}^n {\rm dist}eltalta _{sj} \leq 3.5 pn$. Thus,
$$
\|Mx\| \geq |\left\langlengle R_{s} (M),\, x \rightangle| \geq x_{1}^* - 3.5 pn x_{n_2}^* \geq x_{1}^*/3.
$$
The trivial bound $\|x\|\leq \sqrt{n} \, x_{1}^*$ completes the first estimate.
The ``in particular" part follows by the ``moreover" part of Lemma~\rightef{bennett} and since
$\mathbb{P}({\mathbb E}vent_{col})\leq n(1-p)^n$.
\mathcal{E}_{n-1}d{proof}
\begin{lemma} \left\langlebel{st2}
There exists a (small) absolute positive constant $c$ such that the following holds.
Let $n$ be large enough and $m\leq cn$.
Let $(4\ln n)/n\leq p<1/2$
and $M$ be Bernoulli($p$) matrix.
Then
$$
\mathbb{P}\Bigl(M\in {\mathbb E}vent_{nrm} \quad \mbox{ and } \quad \mathcal{E}xists x\in\mathcal{T}t_2 \,\, \mbox{ with }\,\,
\|Mx\| \leq {\mathcal F}rac{\sqrt{pn}}{5C_0}
\Bigr)\leq e^{-2n}.
$$
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
By Lemma~\rightef{cardnet} for $\varepsilon\in [8c_0, 1)$ there exists an $(\varepsilon/2)$--net
in
$V(1)\cap U(m, c_0)$ with respect to the triple norm $|||\cdot|||$, with cardinality at most
$$
{\mathcal F}rac{C n^{2}}{\varepsilon^2} \left({\mathcal F}rac{18 e n }{\varepsilon m}\right)^m.
$$
Since $\mathcal{T}t_2\subset V(1)\cap U(m, c_0)$,
by a standard ``projection'' trick, we can obtain from it an $\varepsilon$--net $\mathcal{N}$
in $\mathcal{T}t_2$ of the same cardinality.
Let $x\in \mathcal{T}t_2$.
Let $z\in \mathcal{N}$ be such that $|||x-z|||\leq \varepsilon$.
Since on $\mathcal{T}t_2$ we
have $z_1^*\leq \beta_p \|z\|=\beta _p$, Proposition~\rightef{rogozin} implies that
with probability at least $1-e^{-3n}$,
\begin{equation}\left\langlebel{eq: 49820598207492740329}
\|Mz\| \geq {\mathcal F}rac{\sqrt{pn}}{3\sqrt{2} C_0} .
\mathcal{E}_{n-1}d{equation}
Further, in view of Proposition~\rightef{nettri}, conditioned on \mathcal{E}qref{eq: 49820598207492740329}
and on $\{M\in {\mathbb E}vent_{nrm}\}$, we have
$$
\|Mx\| \geq \|Mz\| -\|M(x-z)\| \geq {\mathcal F}rac{\sqrt{pn}}{3\sqrt{2} C_0}
- 100 \sqrt{pn} \varepsilon \geq {\mathcal F}rac{\sqrt{pn}}{5 C_0} ,
$$
where
we have chosen $\varepsilon = 1/(5000 C_0 )$.
Using the union bound and our choice of $\varepsilon$, we obtain that
$$
\mathbb{P}\Bigl(M\in {\mathbb E}vent_{nrm} \quad \mbox{ and } \quad \mathcal{E}xists x\in\mathcal{T}t_2 \,\, \mbox{ with }\,\,
\|Mx\| \leq {\mathcal F}rac{\sqrt{pn}}{5C_0}
\Bigr)\leq e^{-3n}|\mathcal{N}| \leq e^{-2n}
$$
for sufficiently large $n$ and provided that $c_0\leq 1/(40000 C_0)$
and $m\leq cn$ for
small enough absolute positive constant $c$. This completes the proof.
\mathcal{E}_{n-1}d{proof}
\begin{rem}
Note that we used Proposition~\rightef{rogozin} with the set $A=[n]$. In this
case we could use slightly easier construction for nets than the one in
Lemma~\rightef{cardnet} --- we don't need to distinguish the first
coordinate in the net construction, in other words we could have only one
special direction, not two. However this would not lead
to a better estimate and in the remaining lemmas we will need
the fult strength of our construction.
\mathcal{E}_{n-1}d{rem}
Next we threat the case of vectors in $\mathcal{T}t_3$. The proof is similar to
the proof of Lemma~\rightef{st2}, but we need to remove the maximal coordinate
and to deal with remaining part of the vector. Recall that the operator $Q$
serves this purpose.
\begin{lemma} \left\langlebel{st4}
There exists a (small) absolute positive constant $c$ such that the following holds.
Let $n$ be large enough, and $m\leq cpn/\ln(e/p)$, $(4\ln n)/n\leq p<1/2$.
Let $M$ be a random Bernoulli($p$) matrix.
Then
$$
\mathbb{P}\Bigl(M\in {\mathbb E}vent_{nrm} \quad \mbox{ and } \quad \mathcal{E}xists x\in\mathcal{T}t_3 \mbox{ with }\,\,
\|Mx\| \leq {\mathcal F}rac{1}{30C_0\sqrt{pn}}
\Bigr)\leq e^{-2 n}.
$$
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Fix $1\leq k \leq \mathcal{E}ll - 1$.
By Lemma~\rightef{cardnet} for $\varepsilon\in [8c_0 \left\langlembda_k, \left\langlembda_{k+1})$
there exists an $(\varepsilon/2)$--net in
$V(\left\langlembda_{k+1})\cap U(m, c_0\left\langlembda_k)$ with respect to $|||\cdot|||$, with cardinality at most
$$
{\mathcal F}rac{C n^{2}}{\varepsilon^2} \left({\mathcal F}rac{18 e \left\langlembda_{k+1} n }{\varepsilon m}\right)^m
\leq {\mathcal F}rac{C n^{2}}{\varepsilon^2} \left({\mathcal F}rac{54 e \left\langlembda_{k} n }{\varepsilon m}\right)^m.
$$
Again using a ``projection'' trick, we can construct an $\varepsilon$--net $\mathcal{N}_k$ in $\mathcal{T}t_{3,k}$
of the same cardinality.
Let $x\in \mathcal{T}t_{3,k}$.
Let $z\in \mathcal{N}_k$ be such that $|||x-z|||\leq \varepsilon$.
Since on $\mathcal{T}t_{3,k}$ we
have $z_2^*\leq \beta_p \|Qz\|$, Proposition~\rightef{rogozin} applied
with $A={\sigma}ma_z([2, n])$ implies that
with probability at least $1-e^{-3n}$,
$$
\|Mz\| \geq {\mathcal F}rac{\sqrt{pn}\, \|Qz\|}{3\sqrt{2} C_0}
\geq {\mathcal F}rac{\sqrt{pn}\, \left\langlembda_k}{3\sqrt{2} C_0} .
$$
Conditioned on the above inequality and on the event $\{M\in {\mathbb E}vent_{nrm}\}$,
Proposition~\rightef{nettri} implies that
$$
\|Mx\| \geq \|Mz\| -\|M(x-z)\| \geq {\mathcal F}rac{\sqrt{pn}\, \left\langlembda_k}{3\sqrt{2} C_0}
- 100 \sqrt{pn} \varepsilon \geq {\mathcal F}rac{\sqrt{pn}\,\left\langlembda_k}{5 C_0} ,
$$
where
we have chosen $\varepsilon = \left\langlembda_k /(5000C_0)$. Using the union bound, our choice
of $\varepsilon$ and $\left\langlembda_k\geq 1/(6pn)$, we obtain that
$$
P_k:=\mathbb{P}\Bigl(\mathcal{E}xists x\in\mathcal{T}t_{3,k} \,\, \mbox{ with }\,\,
\|Mx\| \leq {\mathcal F}rac{\sqrt{pn}\,\left\langlembda_k}{5C_0}
\Bigr)\leq e^{-3n}|\mathcal{N}_k | \leq e^{-2.5 n}
$$
for large enough $n$ and for $m\leq cn$, where $c>0$ is a
small enough absolute constant (we also assume $c_0\leq 1/(40000 C_0)$).
Since $\mathcal{E}ll \leq \ln n$ and $\left\langlembda_k\geq \left\langlembda_1\geq 1/(6pn)$,
we obtain
$$
\mathbb{P}\Bigl(\mathcal{E}xists x\in\mathcal{T}t_{3} \,\, \mbox{ with }\,\,
\|Mx\| \leq {\mathcal F}rac{1}{30C_0\sqrt{pn}}
\Bigr)\leq \sum _{k=1}^{\mathcal{E}ll-1 } P_k\leq e^{-2p n}.
$$
This completes the proof.
\mathcal{E}_{n-1}d{proof}
Finally we threat the case of vectors in $\mathcal{T}t_4$.
\begin{lemma} \left\langlebel{st5}
There exists a (small) absolute positive constant $c$ such that the following holds.
Let $n$ be large enough and let $m\leq cpn/\ln(e/p)$, $(30\ln n)/n\leq p<1/20$.
Let $M$ be a Bernoulli($p$) random matrix.
Then
$$
\mathbb{P}\Bigl(M\in {\mathbb E}vent_{nrm} \quad \mbox{ and } \quad \mathcal{E}xists x\in\mathcal{T}t_4 \,\, \mbox{ with }\,\,
\|Mx\| \leq {\mathcal F}rac{ 1}{60C_0 \sqrt{n \ln(e/p)}}
\Bigr)\leq e^{-1.5pn}.
$$
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Fix $1\leq k \leq s-1$.
By Lemma~\rightef{cardnet} for $\varepsilon\in [8c_0 \mu_k/\sqrt{\ln(e/p)}, \mu_{k+1})$
there exists an $(\varepsilon/2)$--net in
$$V(\mu_{k+1}/\beta_p)\cap U(m, c_0 \mu_k /\sqrt{\ln(e/p)})$$ with respect to $|||\cdot|||$ with cardinality at most
$$
{\mathcal F}rac{C n^{2}}{\varepsilon^2} \left({\mathcal F}rac{18 e \mu_{k+1} n }{\varepsilon m \beta_p}\right)^m
\leq {\mathcal F}rac{C n^{2}}{\varepsilon^2} \left({\mathcal F}rac{54 e \mu_{k} n }{\varepsilon m \beta_p}\right)^m.
$$
By the projection trick, we get an $\varepsilon$--net $\mathcal{N}_k$ in $\mathcal{T}t_{4,k}\subset V(\mu_{k+1}/\beta_p)\cap U(m, c_0 \mu_k /\sqrt{\ln(e/p)})$.
Let $x\in \mathcal{T}t_{4,k}$.
Let $z\in \mathcal{N}_k$ be such that $|||x-z|||\leq \varepsilon$.
Since on $\mathcal{T}t_4$ we
have $z_1^*\geq z_2^*\geq \mu_k$, Proposition~\rightef{anti2} implies that
with probability at least $1-e^{-1.6np}$,
$$
\|Mz\| \geq {\mathcal F}rac{ \mu_k \sqrt{pn}}{7\sqrt{\ln(e/p)}} .
$$
Conditioned on the above and on $\{M\in {\mathbb E}vent_{nrm}\}$, Proposition~\rightef{nettri} implies that
$$
\|Mx\| \geq \|Mz\| -\|M(x-z)\| \geq {\mathcal F}rac{\mu_k \sqrt{pn}}{7\sqrt{\ln(e/p)}}
- C_1 \sqrt{pn} \varepsilon \geq {\mathcal F}rac{\mu_k \sqrt{pn}}{10\sqrt{\ln(e/p)}} ,
$$
where
we have chosen
$$
\varepsilon = \mu_k/(50C_1\sqrt{\ln(e/p)})
\geq 8 c_0 \mu_k/ \sqrt{\ln(e/p)} ,
$$
provided that $c_0\leq 1/40000$.
Using the union bound and our choice
of $\varepsilon$ we obtain that
$$
P_k:= \mathbb{P}\Bigl(M\in {\mathbb E}vent_{nrm} \quad \mbox{ and } \quad \mathcal{E}xists x\in\mathcal{T}t_{4,k} \,\, \mbox{ with }\,\,
\|Mx\| \leq {\mathcal F}rac{ \mu_k \sqrt{pn}}{10\sqrt{\ln(e/p)}}
\Bigr)\leq e^{-1.6pn}|\mathcal{N}_k| \leq e^{-1.55 p n}
$$
for large enough $n$ and for $m\leq cpn/\ln(e/p)$, where $c>0$ is a
small enough absolute constant.
Since $s \leq \ln n$ and
$\mu_k\geq \mu_1\geq \beta_p/(6pn)= 1/(6C_0 n\sqrt{p})$,
we obtain
$$
\mathbb{P}\Bigl(M\in {\mathbb E}vent_{nrm} \quad \mbox{ and } \quad \mathcal{E}xists x\in\mathcal{T}t_{4} \,\, \mbox{ with }\,\,
\|Mx\| \leq {\mathcal F}rac{ 1}{60C_0 \sqrt{n \ln(e/p)}}
\Bigr)\leq \sum _{k=1}^{s-1} P_k\leq e^{-1.5p n}.
$$
This completes the proof.
\mathcal{E}_{n-1}d{proof}
\begin{proof}[Proof of Theorem \rightef{t:steep}.]
Lemmas~\rightef{st1}, \rightef{st2}, \rightef{st4}, and \rightef{st5} imply that
$$
{\mathbb P}({\mathbb E}vent) \leq n(1-p)^n + 3e^{-1.5np} +\mathbb{P}({\mathbb E}vent^c_{nrm}),
$$
where ${\mathbb E}vent$ denotes the event from Theorem \rightef{t:steep}.
Lemma~\rightef{mnorm} applied with $t=30$ and (\rightef{normofone}) imply that
$
\mathbb{P}({\mathbb E}vent^c_{nrm})\leq e^{-10pn},
$
provided that $pn$ is large enough. This completes the proof.
\mathcal{E}_{n-1}d{proof}
\subsection{Proof of Theorem \rightef{compl-1}}
\begin{proof}
We prove the statement with $r=r(q)= cq/\ln(e/q)$, where $c$ is the constant from Theorem~\rightef{t:steep},
and $\rightho= \rightho(q)= c_0 \sqrt{r}\beta_q /(6\sqrt{\ln(e/q)})$.
Note that under our choice of parameters (and assuming $c_1$ is small), $9{\rm dist}eltalta/2 \leq c_0\beta_q /\sqrt{\ln(e/q)}
\leq c_0\beta_p /\sqrt{\ln(e/p)}$.
Assume that $x\in {\Upsilon}_n(r)\setminus {\mathcal V}_n$.
By $(x_i^{\#})_i$ denote the non-increasing rearrangement
of $(x_i)_i$ (we would like to emphasize that we do not take absolute values).
Note that for any $t>0$ there are two subsets $Q_1, Q_2\subset[n]$ with $|Q_1|,|Q_2|\geq \lceileil{\rm dist}eltalta n\rightceileil$
satisfying $\max\limits_{i\in Q_2} x_i\leq \min\limits_{i\in Q_1}x_i- t$ if and only if
$x_{\lceileil{\rm dist}eltalta n\rightceileil}^{\#}-x_{n-\lceileil{\rm dist}eltalta n\rightceileil+1}^{\#}\geq t$. This leads to the two following cases.
\|\cdot\|indent
{\it Case 1. $x_{\lceileil{\rm dist}eltalta n\rightceileil}^{\#}-x_{n-\lceileil{\rm dist}eltalta n\rightceileil+1}^{\#}\geq \rightho$. }
Since $x\|\cdot\|tin {\mathcal V}_n$, in this case there exists an index $j\leq n$ with $x_j^*>(2n/j)^{3/2}$. Note that since
$x^*_{\lfloor rn\rightfloor}=1$, we have $j<rn=3{\rm dist}eltalta n$.
\|\cdot\|indent
{\it Subcase 1a. $1< j< 3{\rm dist}eltalta n$. }
Since $x_j^*>(2n/j)^{3/2}$ we get
$$
\|Qx\|^2 \geq \sum _{i=2}^j (x_i^*)^2\geq \sum _{i=2}^j (2n/i)^{3} \geq {\mathcal F}rac{j}{2}\, (2n/j)^{3}
=n (2n/j)^{2}.
$$
Therefore,
$$
{\mathcal F}rac{x^*_{\lfloor rn\rightfloor+1}}{\|Qx\|}\leq
{\mathcal F}rac{1}{\sqrt{n}} \, {\mathcal F}rac{j}{2n}
\leq
{\mathcal F}rac{(3{\rm dist}eltalta/2) }{\sqrt{n}} .
$$
Now let $y=x/\|x\|$. Then
\begin{equation}\left\langlebel{estym}
y^*_{\lfloor rn\rightfloor+1} = {\mathcal F}rac{x^*_{\lfloor rn\rightfloor+1}}{\|x\|}\leq {\mathcal F}rac{3{\rm dist}eltalta/2}{\sqrt{n}} \,
{\mathcal F}rac{\|Qx\|}{\|x\|} =
{\mathcal F}rac{3{\rm dist}eltalta/2}{\sqrt{n}} \, \|Qy\| .
\mathcal{E}_{n-1}d{equation}
Our goal is to show that $y\in \mathcal{T}t(p,m)$ (with $m=\lfloor rn\rightfloor$).
If $y\in \mathcal{T}t_1(p)$, we are done.
Otherwise, if $y\in \mathcal{T}t_2'$, then (\rightef{estym}) implies
that $y^*_{\lfloor rn\rightfloor+1}\leq c_0/\sqrt{n}$, that is, there are at least $n-m$ coordinates
at the distance at most $c_0/\sqrt n$ from zero. Thus $y\in U(m, c_0)$ and
hence $y\in \mathcal{T}t_{2}$.
If
$y\|\cdot\|t\in \mathcal{T}t_1\cup \mathcal{T}t_2'$ and
$y^*_2\leq \beta_p\|Qy\|$, then necessarily $\left\langlembda _k\leq \|Qy\|<\left\langlembda_{k+1}\leq 3\left\langlembda_k$ for some $k$, where
$\left\langlembda_k,\left\langlembda_{k+1}$ are defined according to \mathcal{E}qref{eq: 0495205965029385}. Then
(\rightef{estym}) implies that $y^*_{\lfloor rn\rightfloor+1}\leq c_0 \left\langlembda_k/\sqrt{n}$, that is,
there are at least $n-m$ coordinates at the distance at most $c_0\left\langlembda_k/\sqrt n$ from zero.
Thus $y\in U(m, c_0\left\langlembda _k)$ and hence $y\in \mathcal{T}t_{3,k}$.
If $y\|\cdot\|t\in \mathcal{T}t_1\cup \mathcal{T}t_2'$ and
$y^*_2> \beta_p\|Qy\|$ then necessarily $\mu _k\leq y_2^*<\mu_{k+1}\leq 3\mu_k$,
where $\mu_k,\mu_{k+1}$ are given by \mathcal{E}qref{eq: 9635194-9580-98}.
Then,
similarly,
$$
y^*_{\lfloor rn\rightfloor+1}
\leq {\mathcal F}rac{3{\rm dist}eltalta/2}{\sqrt{n}} \, \|Qy\|
\leq {\mathcal F}rac{3{\rm dist}eltalta/2}{\sqrt{n}} \, {\mathcal F}rac{y_2^*}{\beta_p}
\leq {\mathcal F}rac{9{\rm dist}eltalta/2}{\beta_p\sqrt{n}} \,
\mu_k \leq {\mathcal F}rac{c_0 \mu_k}{\sqrt{\ln(e/p)} \sqrt{n}} .
$$
This implies that $y\in U(m, c_0\mu _k/\sqrt{\ln(e/p)})$ and, thus, $y\in \mathcal{T}t_{4,k}$.
\|\cdot\|indent
{\it Subcase 1b. $j=1$. } In this case $x_1^*\geq (2n)^{3/2}$. Assume $x\|\cdot\|t\in \mathcal{T}t_1$, that is
$x_1^*<6pn x_2^*$. Then
$$
{\mathcal F}rac{x^*_{\lfloor rn\rightfloor+1}}{\|Qx\|}\leq {\mathcal F}rac{1}{x_2^*}\leq {\mathcal F}rac{6pn}{(2n)^{3/2}}=
{\mathcal F}rac{6p}{2^{3/2} \sqrt{n}}.
$$
We can now define $y:=x/\|x\|$ and, having noted that $y^*_{\lfloor rn\rightfloor+1} \leq
{\mathcal F}rac{6p}{2^{3/2}\sqrt{n}} \, \|Qy\|$, proceed similarly to the Subcase~1a. We will need to
use the condition
$18 p \leq 2^{3/2} c_0 \beta_p /\sqrt{\ln(e/p)}$, which holds for small enough $p$.
\|\cdot\|indent
{\it Case 2. $x_{\lceileil{\rm dist}eltalta n\rightceileil}^{\#}-x_{n-\lceileil{\rm dist}eltalta n\rightceileil+1}^{\#}<\rightho$. }
Set ${\sigma}ma$ be a permutation of $[n]$ such that $x_{i}^{\#}=x_{{\sigma}ma(i)}$, $i\leq n$
(note that ${\sigma}ma$ is in general different from the permutation ${\sigma}ma_x$ defined in connection with the
non-increasing rearrangement of the absolute values $|x_i|$).
Define the following set, which will play the role of the set in the definition
of $U(m, \gammamma)$ (see Subsection~\rightef{net}),
$$A:=\{{\sigma}ma(i) :\,\, \lceileil{\rm dist}eltalta n\rightceileil<i\leq n-\lceileil{\rm dist}eltalta n\rightceileil\}.$$
Then $|A|=n-2\lceileil{\rm dist}eltalta n\rightceileil$, and $m> 2\lceileil{\rm dist}eltalta n\rightceileil=2\lceileil r n/3\rightceileil$.
Since $x^*_m=1$, we observe
that either $x_{\lceileil{\rm dist}eltalta n\rightceileil+1}^{\#}\geq 1$ or $x_{n-\lceileil{\rm dist}eltalta n\rightceileil}^{\#}\leq -1$ (or both).
Moreover, since $r<1/2$, we necessarily have that $|x^{\#}_i|\leq 1$ for some $\lceileil{\rm dist}eltalta n\rightceileil<i\leq n-\lceileil{\rm dist}eltalta n\rightceileil$.
Therefore, there exists an index $j\in A$ such that $|x_j|=1$. Taking $b=x_j$,
we observe that for every
$i\in A$, $|x_i-b|<\rightho$.
On the other hand we have
$$
\|x\|^2\geq \|Qx\|^2\geq \sum_{i=2}^{m}
x_i^*\geq m -1 \geq m/2 \quad \mbox{ and } \quad
{\mathcal F}orall i\in A\, : \,
{\mathcal F}rac{|x_i - b|}{\|Q x\|}\leq {\mathcal F}rac{\sqrt{2}\, \rightho}{\sqrt{m}}\leq
{\mathcal F}rac{1}{\sqrt{n}}\,\, {\mathcal F}rac{2 \rightho}{\sqrt{r}}.
$$
Now let $y=x/\|x\|$. Then
\begin{equation}\left\langlebel{setam}
{\mathcal F}orall i\in A\, : \,
\left|y_i - {\mathcal F}rac{b}{\|x\|}\rightight| =
{\mathcal F}rac{|x_i - b|}{\|Q x\|}\, {\mathcal F}rac{\|Qx\|}{\|x\|} \leq
{\mathcal F}rac{1}{\sqrt{n}}\,\, {\mathcal F}rac{2 \rightho}{\sqrt{r}} \, \|Qy\|.
\mathcal{E}_{n-1}d{equation}
The end of the proof is similar to the end of the proof of Case~1.
If $y\in \mathcal{T}t_1$, we are done.
If $y\in \mathcal{T}t_2'$, then using (\rightef{setam}), $\|Qy\|\leq \|y\|=1$,
and $6\rightho/\sqrt{r} \leq c_0$ we obtain
that $y\in U(m, c_0)$ and, thus, $y\in \mathcal{T}t_{2}$.
If $y\|\cdot\|t\in \mathcal{T}t_1\cup \mathcal{T}t_2'$,
$y^*_2\leq \beta_p\|Qy\|$, and $\left\langlembda _k\leq \|Qy\|<\left\langlembda_{k+1}\leq 3\left\langlembda_k$ then,
using (\rightef{setam}) and $6\rightho/\sqrt{r} \leq c_0$ we obtain
that $y\in U(m, c_0\left\langlembda _k)$ and, thus, $y\in \mathcal{T}t_{3,k}$.
If $y\|\cdot\|t\in \mathcal{T}t_1\cup \mathcal{T}t_2'$,
$y^*_2\geq \beta_p\|Qy\|$, and $\mu _k\leq y_2^*<\mu_{k+1}\leq 3\mu_k$ then,
similarly, using (\rightef{setam}) and $6\rightho/\sqrt{r} \leq c_0\beta_p /\sqrt{\ln(e/p)}$,
we obtain that $y\in U(m, c_0\mu _k/\sqrt{\ln(e/p)})$ and, thus, $y\in \mathcal{T}t_{4,k}$.
This completes the proof.
\mathcal{E}_{n-1}d{proof}
\section{Complement of gradual non-constant vectors: general case}
\left\langlebel{s: steep}
We split ${\mathbb R}^n$ into two classes of vectors. The first class, the class of {\it steep} vectors
$\mathcal{T}$, is constructed in essentially the same way as in
\cite{LLTTY first part} and \cite{LLTTY-TAMS}. The proof of bound for this class resembles
corresponding proofs in \cite{LLTTY first part} and \cite{LLTTY-TAMS}, however, due to the
differences of the models of randomness, there are important modifications.
The second class ${\mathcal R}$, which we call ${\mathcal R}$-vectors, will consist
of vectors to which Proposition~\rightef{rogozin} can be applied, therefore dealing with this class is simpler.
To control the cardinality of nets, part of this class will be intersected with
the almost constant vectors.
Then we show that the complement of ${\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)$ in ${\Upsilon}_n(r)$
is contained in $\mathcal{T}\cup {\mathcal R}$.
We now introduce the following parameters, which will be used throughout this section.
It will be convenient to denote $d=pn$. We always assume that $p\leq 0.0001$ and
$n$ is large enough (that is, larger than a certain positive absolute constant).
We also always assume that the ``average degree'' $d=pn\geq 200\ln n$.
Fix a sufficiently small absolute positive constant $a_1aa$ and sufficiently
large absolute positive constant $C_\tau$
(we do not try to estimate the actual values of $a_1aa$ and $C_\tau$, the conditions
on how small $r$ can be extracted from the proofs, in particular, the condition on $C_\tau$
comes for (\rightef{ctau})).
We also fix two positive integers $\mathcal{E}ll_0$ and $s_0$ such that
\begin{equation}\left\langlebel{eq: l0s0 def}
\mathcal{E}ll_0 = \left\lfloor {\mathcal F}rac{pn}{4\ln (1/p)}\right\rightfloor
\quad \mbox{ and } \quad \mathcal{E}ll_0^{s_0-1} \leq {\mathcal F}rac{1}{64 p}={\mathcal F}rac{n}{64 d}< \mathcal{E}ll_0 ^{s_0} .
\mathcal{E}_{n-1}d{equation}
Note that $\mathcal{E}ll_0 \geq 50$ and that $s_0> 1$ implies $p\leq c\sqrt{(\ln n)/n}$.
For $1\leq j\leq s_0$ we set
$$
n_0:=2, \,\, \,\, \,\, \,\, n_{j}:= 30 \mathcal{E}ll_0^{j-1}, \,\,\,\, \,\, \,\,
\,\,\,\, \,\, \,\, n_{s_0+2} :=\left\lfloor \sqrt{n/p} \right\rightfloor
=\left\lfloor {\mathcal F}rac{n}{\sqrt{d} } \right\rightfloor ,
\,\,\,\, \,\, \,\, \mbox{ and } \,\, \,\, \,\, \,\, n_{s_0+3} :=\lfloor a_1aa n \rightfloor.
$$
Then, in the case $\left\lfloor 1/(64p) \right\rightfloor \geq 15 n_{s_0}$ we set
$n_{s_0+1}=\left\lfloor 1/(64p) \right\rightfloor$. Otherwise,
let $n_{s_0+1}=n_{s_0}$.
Note that with this definition we always have $n_{s_0+2}>n_{s_0+1}$.
{\bf The indices $n_j$, $j\leq s_0+3$, are global parameters which will be used throughout the section.}
Below we provide the proof only for the case
$\left\lfloor 1/(64p) \right\rightfloor=n_{s_0+1}\geq 15 n_{s_0}$, the other case is treated similarly (in particular,
in that other case the set $\mathcal{T}_{1 (s_0+1)}$ defined below, will be empty).
We also will use another parameter,
\begin{equation}\left\langlebel{eq: kappa def}
\kappa = \kappa(p):= {\mathcal F}rac{\ln (6pn)}{\ln \mathcal{E}ll_0}.
\mathcal{E}_{n-1}d{equation}
Note that the function $f(p)=\ln (6pn)/(4\ln (1/p))$
is a decreasing function on $(0,1)$, therefore
for $p\geq (100 \ln n)/n$ and sufficiently larg $n$ we have
$1<\kappa\leq \ln \ln n$. Moreover, it is easy to see that if
$p\geq (100 \ln^2 n)/n$, then $\kappa\leq 2$.
We also notice that if $pn\geq 6(5\ln n)^{1+\gammamma}$ for some $\gammamma\in (0,1)$ then
$\kappa \leq 1+1/\gammamma$ and, using the definition of $\mathcal{E}ll_0$ and $s_0$,
\begin{equation}\left\langlebel{kappain}
(6d)^{s_0-1} =
\mathcal{E}ll_0^{(s_0-1)\kappa}\leq 1/(64p)^{\kappa}.
\mathcal{E}_{n-1}d{equation}
\subsection{Two classes of vectors and main results}
\left\langlebel{subs: steep vectors}
We first introduce the class of steep vectors. It will be constructed as a union of four subclasses.
Set
$$
\mathcal{T} _{0} : =\{x\in {\mathbb R}^n\,:\, x_{1}^{*}> 6 d\, x_{2}^{*}\}
\quad \mbox{ and } \quad \mathcal{T} _{11} : =\{x\in {\mathbb R}^n\,:\, x\|\cdot\|t\in \mathcal{T}_{0}
\,\, \mbox{ and } \, \, x_{2}^{*}> 6 d\, x_{n_1}^{*}\} .
$$
Then for $2\leq j\leq s_0+1$,
$$
\mathcal{T} _{1 j} : =\left\{x\in {\mathbb R}^n\,:\, x\|\cdot\|t\in \mathcal{T}_{0} \cup \bigcup_{i=1}^{j-1} \mathcal{T}_{1 i}
\,\, \mbox{ and } \, \, x_{n_{j-1}}^{*}> 6 d\, x_{n_{j}}^{*}\right\}
\quad \mbox{ and } \quad \mathcal{T} _{1} : = \bigcup_{i=1}^{s_0+1} \mathcal{T}_{1 i}.
$$
Finally, for $k=2,3$ set $j=j(k)= s_0+k$ and define
$$
\mathcal{T} _{k} : =\left\{x\in {\mathbb R}^n\,:\, x\|\cdot\|t\in \bigcup_{i=0}^{k-1} \mathcal{T}_{i}
\,\, \mbox{ and } \, \, x_{n_{j-1}}^{*}> C_\tau\sqrt{d}\,x_{n_{j}}^{*}\right\}.
$$
The set of steep vectors is $\mathcal{T}:=\mathcal{T}_0\cup\mathcal{T}_1\cup\mathcal{T}_2 \cup\mathcal{T}_3$.
The ``rules'' of the partition are summarized in the diagram.
\includegraphics[width=1.0\textwidth]{Tdiagram.png}
For this class we prove the following bound.
\begin{theor} \left\langlebel{steep}
There exist positive absolute constants $c,C>0$ such that the following holds.
Let $n\geq C$, and let $0<p<c$ satisfy $pn\geq C \ln n$.
Let $M$ be a Bernoulli($p$) random matrix and denote
$$
{\mathbb E}vent_{steep}:=\bigg\{\mathcal{E}xists\;x\in \mathcal{T}\, \, \, \mbox{ such that }
\, \, \,
\|M x\| < {\mathcal F}rac{c(64p)^{\kappa}}{(pn)^2}\, \min\left( 1, {\mathcal F}rac{1}{p^{1.5}n} \right) \, \, \|x\|
\bigg\},
$$
where as before $\kappa = \kappa(p):= (\ln (6pn))/\ln \mathcal{E}ll_0.$
Then
\begin{equation*}
\left\langlebel{Psteep}
{\mathbb P}({\mathbb E}vent_{steep}) \leq n(1-p)^n + 2 e^{-1.4 pn}.
\mathcal{E}_{n-1}d{equation*}
\mathcal{E}_{n-1}d{theor}
Next we introduce the class of ${\mathcal R}$-vectors, denoted by ${\mathcal R}$.
Let $C_0$ be the constant from Proposition~\rightef{rogozin} and recall that the class $\mathcal{AC}(\rightho)$
of almost constant vectors was defined by (\rightef{acv}) in Subsection~\rightef{ss: steep overview}.
Given $n_{s_0+1}<k\leq n/\ln^2 d$ denote $A=A(k):=[k, n]$ and consider the sets
$$
{\mathcal R}_k^1 :=\left\{x\in \big({\Upsilon}_n(r)\setminus \mathcal{T}\big) \cap \mathcal{AC}(\rightho) \, \, : \, \,
{\mathcal F}rac{\|x_{{\sigma}ma_x(A)}\|}{\|x_{{\sigma}ma_x(A)}\|_\infty} \geq {\mathcal F}rac{C_0}{\sqrt{p}}
\quad \mbox{ and } \quad \sqrt{n/2}\leq \|x_{{\sigma}ma_x(A)}\| \leq C_\tau \sqrt{dn}\right\},
$$
and
$$
{\mathcal R}_k^2 :=\left\{x\in{\Upsilon}_n(r)\setminus \mathcal{T}
\, \, : \, \, {\mathcal F}rac{\|x_{{\sigma}ma_x(A)}\|}{\|x_{{\sigma}ma_x(A)}\|_\infty} \geq {\mathcal F}rac{C_0}{\sqrt{p}}
\quad \mbox{ and } \quad {\mathcal F}rac{2\sqrt{n}}{r}\leq \|x_{{\sigma}ma_x(A)}\| \leq C_\tau^2 d \sqrt{n}\right\}.
$$
Define
$
{\mathcal R}:=\bigcup _{n_{s_0+1}<k\leq n/\ln^2 d}\, ({\mathcal R}_k^1 \cup {\mathcal R}_k^2).
$
The class ${\mathcal R}$ should be thought of as the class of {\it sufficiently spread}
vectors, not steep, but possibly without having two subsets of coordinates of size proportional to $n$,
which are separated by $\rightho$ (which would allow us to treat those vectors as part of the set ${\mathcal V}_n$).
Crucially, the sets ${\mathcal R}_k^1$ and ${\mathcal R}_k^2$ are ``low complexity'' sets because they admit
$\varepsilon$--nets of relatively small cardinalities (see Subsection~\rightef{sub-nets}).
For the class ${\mathcal R}$ we prove the following bound.
\begin{theor}
\left\langlebel{classB}
There are absolute constants $r_0,\rightho_0,C$ with the following property.
Let $0<r\leq r_0$, $0<\rightho\leq \rightho_0$, let
$n\geq 1$ and $p\in (0, 0.001]$ be such that
$d=pn\geq C\ln n$.
Then
$$
\mathbb{P}\left(\left\{\mathcal{E}xists x\in {\mathcal R} \, : \,
\|Mx\|\leq {\mathcal F}rac{\sqrt{p} n}{12 C_0}\right\}\rightight) \leq e^{-2n} + e^{-200pn}.
$$
\mathcal{E}_{n-1}d{theor}
Finally we show that together with ${\mathcal V}_n$ classes $\mathcal{T}$ and ${\mathcal R}$ cover
all (properly normalized) vectors for the growth function defined by
\begin{equation}\left\langlebel{gfn-str}
{\bf g}(t)=(2t)^{3/2}\,\,\, \mbox{ for }\, 1\leq t<64pn\quad \quad \mbox{ and } \quad\quad
{\bf g}(t)=\mathcal{E}xp(\ln^2(2t))\,\,\, \mbox{ for }\, t\geq 64pn.
\mathcal{E}_{n-1}d{equation}
It is straightforward to check that ${\bf g}$ satisfies (\rightef{gfncond}) with some absolute constant $K_3$.
\begin{theor}
\left\langlebel{complement}
There are universal constants $c,C>0$ with the following property.
Let $n\geq C$, $p\in (0, c)$, and assume that $d=pn\geq 100 \ln n$.
Let $r\in(0,1/2)$, ${\rm dist}eltalta \in(0,r/3)$, $\rightho\in(0,1)$, and let ${\bf g}$ be as in (\rightef{gfn-str}).
Then
\left\langlebel{compl}
$$
{\Upsilon}_n(r)\setminus {\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho) \subset {\mathcal R}\cup \mathcal{T}.
$$
\mathcal{E}_{n-1}d{theor}
\subsection{Auxiliary lemmas}
\left\langlebel{twolemmas}
In the following lemma we provide a simple bound on the Euclidean norms of vectors
in the class $\mathcal{T}$ and its complement in terms of their order statistics.
\begin{lemma}\left\langlebel{euclnorm}
Let $n$ be large enough and $(200 \ln n)/n<p<0.001$.
Consider the vectors $x\in \mathcal{T}_{1j}$ for some $1\leq j\leq s_0+1$,
$y\in \mathcal{T}_2$, $z\in \mathcal{T}_3$ and $w\in \mathcal{T}^c$. Then
$$
{\mathcal F}rac{\|x\|}{ x_{n_{j-1}}^*}\leq {\mathcal F}rac{ 64(pn)^2}{(64p)^{\kappa}}, \quad
{\mathcal F}rac{\|y\|}{ y_{n_{s_0+1}}^*}\leq {\mathcal F}rac{ 384(pn)^3}{(64p)^{\kappa}}, \quad
{\mathcal F}rac{\|z\|}{ z_{n_{s_0+2}}^*}\leq {\mathcal F}rac{ 384C_\tau (pn)^{3.5}}{(64p)^{\kappa}},
\quad \mbox{ and } \quad {\mathcal F}rac{\|w\|}{ w_{n_{s_0+3}}^*}\leq {\mathcal F}rac{ 384C_\tau^2 (pn)^{4}}{(64p)^{\kappa}}.
$$
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Let $d=pn$. Since $x\in \mathcal{T}_{1j}$, denoting $m=n_{j-1}$, we have
$$
x_1^*\leq (6d) x^*_{2}\leq (6d)^2 x^*_{n_1}\leq \ldots \leq
(6d)^j x_{n_{j-1}}^* = (6d)^j x_m^*.
$$
Since $n_i=30\mathcal{E}ll_0^{i-1} \leq 30d^{i-1}$, $i\leq s_0$, since $\kappa>1$, and in view of \mathcal{E}qref{kappain},
we obtain
\begin{align*}
\|x\|^2 &= (x_{1}^*)^2+ (x_2^*+ \ldots + (x_{n_1}^*)^2) + ((x_{n_1+1}^*)^2 + {\rm dist}ots + (x_{n_2}^*)^2)
+ \ldots
\\&\leq
((6d)^{2 j} + n_1 (6d)^{2(j-1)} + n_2 (6d)^{2(j-2)}\ldots + n_{j-1} (6d)^{2} + n) (x_m^*)^2
\\&\leq
\Big((6d)^{2 j} + 5(6d)^{2 j-2}\sum_{i\geq 0}(6d)^{-i} +n \Big) (x_m^*)^2 \leq
\left(2 (6d)^{2 (s_0+1)} +n \right) (x_m^*)^2
\\&\leq
\left(2 (6d)^4 /(64p)^{2\kappa}+n \right) (x_m^*)^2 \leq \left(3 (6d)^4 /(64p)^{2\kappa} \right) (x_m^*)^2 .
\mathcal{E}_{n-1}d{align*}
This implies the first bound. The bounds for $y,z, w$ are obtained similarly.
\mathcal{E}_{n-1}d{proof}
The next two Lemmas~\rightef{col} and~\rightef{c:SJ} will be used to bound from below the norm of the matrix-vector
product $M x$ for vectors $x$ with a ``too large'' almost constant part which does
not allow to directly apply the L\'evy--Kolmogorov--Rogozin anti-concentration inequality together with the
tensorization argument. Lemma~\rightef{col} will be used to bound $\|Mx\|$ by a single inner product
$|\left\langlengle \rightow_i(M),x\rightangle|$ for a specially chosen index $i$, while Lemma~\rightef{c:SJ}
will allow to extract a subset of ``good'' rows having large inner products with $x$.
\begin{lemma}
\left\langlebel{col}
Let $n\geq 30$ and $0<p<0.001$ satisfy $pn\geq 200 \ln n$.
Let $m, \mathcal{E}ll=\mathcal{E}ll(m) \geq 2$ be such that either
$$m=2\mbox{ and }\mathcal{E}ll = 15,$$
or
$$
m\geq 30, \quad \mathcal{E}ll m\leq {\mathcal F}rac{1}{64 p} \quad \quad \mbox{and}
\quad \quad \mathcal{E}ll \leq {\mathcal F}rac{np}{4\ln {\mathcal F}rac{1}{pm}}.
$$
Let $M$ be an $n\times n$ Bernoulli($p$) random matrix.
By ${\mathbb E}vent_{col}={\mathbb E}vent_{col}(\mathcal{E}ll, m)$ denote the event that
for any choice of two disjoint sets $J_1, J_2\subset [n]$ of cardinality
$|J_1|=m$, $|J_2|=\mathcal{E}ll m - m$ there exists a row of $M$ with exactly one $1$ among components
indexed by $J_1$ and no $1$s among components indexed by $J_2$.
Then $\mathbb{P}({\mathbb E}vent_{col} )\geq 1-\mathcal{E}xp(-1.5 pn).$
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
We first treat the case $m\geq 30$.
Fix two disjoint sets $J_1, J_2\subset [n]$ of required cardinality. The probability that
a fixed row has exactly one $1$ among components
indexed by $J_1$ and no $1$s among components indexed by $J_2$ equals
$$
q:=m p(1-p)^{\mathcal{E}ll m - 1}\geq m p \mathcal{E}xp(- 2p\mathcal{E}ll m ) \geq 29mp/30,
$$
where we used $\mathcal{E}ll m p\leq 1/64$.
Since the rows are independent, the probability that $M$ does not have such a row
is
$$
(1-q)^n \leq \mathcal{E}xp(-nq) \leq \mathcal{E}xp(-29 m p n/30).
$$
Note that the number of all choices of $J_1$ and $J_2$ satisfying the conditions of the lemma is
$$
{n \choose \mathcal{E}ll m - m} {n-\mathcal{E}ll m +m \choose m} \leq
\left( {\mathcal F}rac{en}{(\mathcal{E}ll-1) m}\right)^{\mathcal{E}ll m -m} \left({\mathcal F}rac{en}{m}\right)^m
\leq \left( {\mathcal F}rac{3n}{\mathcal{E}ll m}\right)^{\mathcal{E}ll m } (2\mathcal{E}ll) ^m .
$$
Thus union bound over all choices of $J_1$ and $J_2$ implies
$$
\mathbb{P}(({\mathbb E}vent_{col})^c )\leq \left( {\mathcal F}rac{3n}{\mathcal{E}ll m}\right)^{\mathcal{E}ll m } (2\mathcal{E}ll) ^m \mathcal{E}xp(-29 m p n/30).
$$
Using that $m\leq 1/(64p)$ and $\mathcal{E}ll \leq {\mathcal F}rac{np}{4\ln (1/(pm))}$, we observe
$
\left( {\mathcal F}rac{3n}{\mathcal{E}ll m}\right)^{\mathcal{E}ll m } \leq \mathcal{E}xp( m p n/2).
$
Since $np\geq 200 \ln n$, we have $(2\mathcal{E}ll) ^m \leq \mathcal{E}xp( 2m p n/5)$.
Thus,
$$
\mathbb{P}(({\mathbb E}vent_{col} )^c )\leq \mathcal{E}xp(- m p n/15) \leq \mathcal{E}xp(- 2 p n),
$$
which proves this case.
The case $m=2$, $\mathcal{E}ll = 15$ is similar.
Fixing two disjoint sets $J_1, J_2\subset [n]$ of the required cardinality, the probability that
a fixed row has exactly one $1$ among components
indexed by $J_1$ and no $1$s among components indexed by $J_2$ equals
$$
q:=2p(1-p)^{29}\geq 2p \mathcal{E}xp(-29 p).
$$
Since rows are independent, the probability that $M$ does not have such a row
is
$$
(1-q)^n \leq (1- 2p \mathcal{E}xp(-29 p))^n \leq \mathcal{E}xp(-2pn \mathcal{E}xp(-29 p))
\leq \mathcal{E}xp(-1.8 pn ).
$$
Using union bound over all choices of $J_1$ and $J_2$ we obtain
$$
\mathbb{P}({\mathbb E}vent_{sum}^c )\leq {\mathcal F}rac{n^{30}}{2\cdot 28!}\mathcal{E}xp(-1.8p n)\leq
\mathcal{E}xp(-1.5p n),
$$
which proves the lemma.
\mathcal{E}_{n-1}d{proof}
In the next lemma we restrict a matrix to a certain set of columns
and estimate the cardinality of a set of rows having exactly one $1$.
To be more precise, for any $J\subset [n]$ and a $0/1$ matrix $M$
denote
$$
I_J=I(J,M) := \{i\le n :\, |{\rm supp\, } \rightow_i(M)\cap J|=1\}.
$$
The following statement is similar to Lemma 2.7 from \cite{LLTTY first part} and
Lemma~3.6 in \cite{LLTTY-TAMS}.
\begin{lemma}\left\langlebel{c:SJ}
Let $\mathcal{E}ll \geq 1$ be an integer and $p\in (0, 1/2]$ be such that $p\mathcal{E}ll \leq 1/32$.
Let $M$ be a Bernoulli($p$) random matrix.
Then with probability at least
$$
1- 2{n \choose \mathcal{E}ll}\mathcal{E}xp\left(-n\mathcal{E}ll p/4\right)
$$
for every $J\subset [n]$ of cardinality $\mathcal{E}ll$ one has
$$
\mathcal{E}ll pn/16 \leq |I(J,M)| \leq 2\mathcal{E}ll n p.
$$
In particular, if $\mathcal{E}ll = 2\lfloor 1/(64p)\rightfloor\leq n$, $n\geq 10^5$, and $p\in [100/n, 0.001]$ then,
denoting
$$
{\mathbb E}vent _{card} ={\mathbb E}vent _{card} (\mathcal{E}ll) :=\{M\in {\mathcal{M}_{n}} \, :\, {\mathcal F}orall J\subset [n]\,\,\, \mbox{with}\,\,\, |J|=\mathcal{E}ll
\,\,\, \mbox{one has}\,\,\, |I(J,M)|\in [\mathcal{E}ll p n/16, 2 \mathcal{E}ll p n]\},
$$
we have
$$
\mathbb{P}\left({\mathbb E}vent _{card} \right)\geq 1- 2\mathcal{E}xp\left(-n/500\rightight).
$$
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Fix $J\subset [n]$ of cardinality $\mathcal{E}ll$. Denote $q=\mathcal{E}ll p (1-p)^{\mathcal{E}ll -1}$. Since $\mathcal{E}ll p\leq 1/32$,
$$
15\mathcal{E}ll p/16\leq \mathcal{E}ll p(1-2p\mathcal{E}ll )\leq \mathcal{E}ll p \mathcal{E}xp(-2 p\mathcal{E}ll)\leq q\leq \mathcal{E}ll p \leq 1/2.
$$
For every $i\leq n$, let $\xi_i$ be the indicator of the event $\{i\in I(J,M)\}$.
Clearly, $\xi_i$'s are independent Bernoulli($p$) q random variables and
$|I(J,M)| = \sum_{i=1}^n \xi_i$. Applying Lemma~\rightef{bennett}, we observe
that for every $0<\varepsilon<q$
$$
\mathbb{P}\left(|I(J,M)|\in [(q-\varepsilon)n, (q+\varepsilon)n]\right)\geq 1- 2\mathcal{E}xp\left(-{\mathcal F}rac{n\varepsilon^2}{2q(1-q)} \,
\left(1-{\mathcal F}rac{\varepsilon}{3q}\rightight)\rightight).
$$
Taking $\varepsilon = 14q/15$ we obtain that
$$
(q-\varepsilon)n= qn/15\geq \mathcal{E}ll pn/16 \quad \quad \mbox{and} \quad \quad
(q+\varepsilon)n\leq 2 qn\leq 2\mathcal{E}ll pn,
$$
and
$$
{\mathcal F}rac{n\varepsilon^2}{2q(1-q)} \, \left(1-{\mathcal F}rac{\varepsilon}{3q}\right)
\geq {\mathcal F}rac{98 \cdot 31 nq}{225 \cdot 45} \geq
0.3 n\mathcal{E}ll p (1-2\mathcal{E}ll p) \geq n\mathcal{E}ll p/4.
$$
This implies the bound for a fixed $J$. The lemma follows by the union bound.
\mathcal{E}_{n-1}d{proof}
\subsection{Cardinality estimates for $\varepsilon$--nets}
\left\langlebel{sub-nets}
In this subsection we provide bounds on cardinality of certain discretizations of the sets of vectors introduced earlier.
Recall that $\mathcal{E}dv$ denotes the vector ${\bf 1} /\sqrt{n}$,
$P_\mathcal{E}dv$ denotes the projection on $\mathcal{E}dv^\mathbb{P}erp$, and
$P_\mathcal{E}dv^\mathbb{P}erp$ is the projection on $\mathcal{E}dv$, that is $P_\mathcal{E}dv^\mathbb{P}erp = \left\langle \cdot , \mathcal{E}dv\righta \mathcal{E}dv$.
We recall also that given $A\subset [n]$, $x_A$ denotes coordinate projection of $x$ on ${\mathbb R}^A$,
and that given $x\in {\mathbb R}^n$, ${\sigma}ma_x$ is a (fixed) permutation corresponding to
non-increasing rearrangement of $\{|x_i|\}_{i=1}^n$.
Our first lemma deals with nets for $\mathcal{T}_2$ and $\mathcal{T}_3$.
We will consider the following normalization:
$$
\mathcal{T}'_2=\{ x\in \mathcal{T}_2\,:\, x_{n_{s_0+1}}^{*}=1 \}, \quad \mbox{ and } \quad
\mathcal{T}'_3=\{ x\in \mathcal{T}_3\,:\, x_{n_{s_0+2}}^{*}=1 \}.
$$
The triple norm is defined by
$
||| x |||^2 := \|P_\mathcal{E}dv x\|^2 + p n \|P_\mathcal{E}dv^\mathbb{P}erp x\|^2.
$
\begin{lemma}
\left\langlebel{l:nets}
Let $n\geq 1$, $p\in (0, 0.001]$, and assume that $d=pn$ is sufficiently large. Let $i\in\{2,3\}$.
Then there exists a set ${\mathbb N}et _i= {\mathbb N}et_i' + {\mathbb N}et_i''$, ${\mathbb N}et_i'\subset {\mathbb R}^n$, ${\mathbb N}et_i''\subset {\rm span}\,\{{\bf{}1}\}$, with the following properties:
\begin{itemize}
\item $
|{\mathbb N}et_i| \le \mathcal{E}xp\left( 2 n_{s_0+i}\ln d \right).
$
\item For every $u\in {\mathbb N}et_i'$ one has $u_j^*=0$ for all $j\geq n_{s_0+i}$.
\item For every $x\in \mathcal{T}_i'$ there are $u\in {\mathbb N}et_i'$ and $w\in {\mathbb N}et_i''$
satisfying
$$
\|x-u\|_\infty \leq {\mathcal F}rac{1}{C_\tau \sqrt{d}},
\quad \|w\|_\infty \leq{\mathcal F}rac{1}{C_\tau \sqrt{d}}, \quad \mbox{ and } \quad
|||x-u-w|||\leq {\mathcal F}rac{\sqrt{2n}}{C_\tau \sqrt{d}}.
$$
\mathcal{E}_{n-1}d{itemize}
\mathcal{E}_{n-1}d{lemma}
Since the proof of this lemma in many parts repeats the proofs
of Lemma~3.8 from \cite{LLTTY first part} and of Lemma~\rightef{newnet} below,
we only sketch it.
\begin{proof} Fix $\mu =1/(C_\tau\sqrt{d}$) and $i\in \{2, 3\}$.
We first repeat the proof of Lemma~3.8 from \cite{LLTTY first part} with our choice of parameters.
See also the beginning of the proof of Lemma~\rightef{newnet} below --- many definitions, constructions, and
calculations are exactly the same, however note that the normalization is slightly different.
In particular,
the definitions of sets $B_1(x)$, $B_2(x)$ (with $k-1=n_{s_0+i-1}$),
$B_3(x)$ are the same (we do not need the sets $B_0(x)$ and $B_4(x)$).
This will show (for large enough $d$) the existence of a $\mu$-net $\mathcal{N}_i'$ (in the $\mathcal{E}ll_\infty$ metric)
for $\mathcal{T}_i'$ such that for every $u\in {\mathbb N}et_i'$ one has $u_j^*=0$ for all $j\geq n_{s_0+i}$ and
$
|\mathcal{N}_i'|\leq \mathcal{E}xp\left( 1.1 n_{s_0+i}\ln d \right)$.
Next given $x\in \mathcal{T}_i'$ let $u=u(x)\in \mathcal{N}_i'$
be such that $\|x-u\|_\infty \leq \mu$. Then $\|P_\mathcal{E}dv^\mathbb{P}erp (x-u)\|\leq \mu \sqrt{n}$.
Let $\mathcal{N}_i''$ be a $(\mu\sqrt{n/d})$-net in the segment $\mu \sqrt{n} \, [-\mathcal{E}dv, \mathcal{E}dv]$ of cardinality at most
$2\sqrt{d}$ (note, we are in the one-dimensional setting). Note that every $w\in \mathcal{N}_i''$ is of the form
$w=a\, \mathcal{E}dv=a\, {\bf 1}/\sqrt{n}$, $|a|\leq \mu \sqrt{n}$, in particular, $\|w\|_\infty\leq \mu$.
Then for $x$ (and the corresponding $u=u(x)$),
there exists $w\in \mathcal{N}_i''$ such that
$$
|||x-u-w|||^2=
\| P_\mathcal{E}dv (x - u-w)\|^2 + d\| P_\mathcal{E}dv^\mathbb{P}erp (x - u -w) \|^2=
\| P_\mathcal{E}dv (x - u)\|^2 + d\| P_\mathcal{E}dv^\mathbb{P}erp (x - u) -w \|^2\leq 2 \mu^2 n .
$$
Finally, note that $|\mathcal{N}_i' + \mathcal{N}_i''|\leq 2\sqrt{d} \mathcal{E}xp\left( 1.1 n_{s_0+i}\ln d \right)\leq \mathcal{E}xp\left( 2 n_{s_0+i}\ln d \right)$.
This completes the proof.
\mathcal{E}_{n-1}d{proof}
Let ${\mathcal R}_{k}^1$, ${\mathcal R}_{k}^2$ be the vector subsets introduced in Subsection~\rightef{subs: steep vectors}.
Consider the increasing sequence $\left\langlembda _1<\left\langlembda_2<\ldots <\left\langlembda _m$, $m\geq 1$, defined by
\begin{equation}\left\langlebel{eq 2498520598207560}
\mbox{$\left\langlembda _1 = 1/\sqrt{2}$, $\,\, \, \left\langlembda _{i+1}=3\left\langlembda _i\,\,\, $ for $1<i<m,\quad $ and
$\quad \left\langlembda _{m-1} <\left\langlembda _m=C_\tau^2 d \leq 3\left\langlembda _{m-1}$.}
\mathcal{E}_{n-1}d{equation}
Clearly $m\leq n$.
For $s\in\{1,2\}$, $n_{s_0+1}<k\leq n/\ln^2 d$ and $i\leq m$ set
$$
{\mathcal R}_{k i}^s :=
\left\{x\in{\mathcal R}_k^s \, \, : \, \, \left\langlembda _i \sqrt{n}\leq \|x_{{\sigma}ma_x([k,n])}\| \leq \left\langlembda _{i+1} \sqrt{n}\right\}.
$$
It is not difficult to see that the union of ${\mathcal R}_{k i}^s$'s over admissible $i$ gives ${\mathcal R}_k^s$.
The sets ${\mathcal R}_{k i}^s$ are ``low complexity'' sets in the sense that they admit
efficient $\varepsilon$-nets. For $s=1$, the low complexity is a consequence of the condition that
${\mathcal R}_{k i}^1\subset \mathcal{AC}(\rightho)$, i.e., the vectors have a very large almost constant part.
For the sets ${\mathcal R}_{k i}^2$, we do not assume the almost constant behavior, but instead rely on the assumption
that $\|x_{{\sigma}ma_x([k,n])}\|$ is large (much larger than $\sqrt{n}$). This will allow us to pick
$\varepsilon$ much larger than $\sqrt{n}$, and thus construct a net of small cardinality.
\begin{lemma}
\left\langlebel{newnet}
Let $R\geq 40$ be a (large) constant.
Then there is $r_0>0$
depending on $R$ with the following property.
Let $0<r\leq r_0$, $0<\rightho\leq 1/(2R)$, let
$n\geq 1$ and $p\in (0, 0.001]$ so that $d=pn$ is sufficiently large (larger than a constant depending on $R,r$).
Let $s\in \{1,2\}$, $n_{s_0+1}<k\leq n/\ln^2 d$, $t\leq m$, and
$40 \left\langlembda_t \sqrt{n}/R\leq \varepsilon \leq \left\langlembda_t\sqrt{n}$,
where $\left\langlembda_t$ and $m$ are defined according to relation \mathcal{E}qref{eq 2498520598207560}.
Then there exists an $\varepsilon$-net
$\mathcal{N}_{k t}^s\subset {\mathcal R}_{k t}^s$ for ${\mathcal R}_{k t}^s$ with respect to $|||\cdot|||$
of cardinality at most $(e/r)^{3rn}$.
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Note that in case of $s=2$ the set ${\mathcal R}_{k t}^2$ is empty whenever $3\left\langlembdabda_t<{\mathcal F}rac{2}{r}$.
So, in the course of the proof we will implicitly assume that $3\left\langlembdabda_t\geq{\mathcal F}rac{2}{r}$ whenever $s=2$.
We follow ideas of the proof of Lemma~3.8 from \cite{LLTTY first part}.
We split a given vector from ${\mathcal R}_{k t}^s$ into few parts according to magnitudes of its coordinates
and approximate each part separately.
Then we construct nets for vectors with the
same splitting and take the union over all nets. We now discuss the splitting. For each $x\in {\mathcal R}_{k t}^s$
consider the following (depending on $x$) partition of $[n]$.
If $s=2$, set $B_0'(x)=\mathcal{E}mptyset$. If $s=1$ then $x\in \mathcal{AC}(\rightho)$ and we set
$$
B_0'(x) :={\sigma}ma _x(\{ j\leq n \, : \, |x_j-\left\langlembda _x|\leq \rightho \}),
$$
where $\left\langlembda _x=\mathbb{P}m 1$ is from the definition of $\mathcal{AC}(\rightho)$ (note that under the normalization
in ${\Upsilon}_n(r)$ we have $x^*_{n_{s_0+3}}=1$). Then $|B_0'(x)|> n - n_{s_0+3}$ for $s=1$.
Next, we set
\begin{align*}
B_1(x)&={\sigma}ma_x([n_{s_0+1}]);\\
B_2(x)&= {\sigma}ma_x([k-1])\setminus B_1(x);\\
B_3(x)&= {\sigma}ma_x([n_{s_0+3}])\setminus (B_1(x)\cup B_2(x));\\
B_0(x) &= B_0'(x)\setminus (B_1(x)\cup B_2(x)\cup B_3(x));\\
B_4(x) &=[n]\setminus (B_0(x)\cup B_1(x)\cup B_2(x)\cup B_3(x) )
\mathcal{E}_{n-1}d{align*}
(one of the sets $B_0(x)$, $B_4(x)$ could be empty).
Denote $\mathcal{E}ll _x:=|B_0(x)|$.
Note that the definition of $B_3(x)$ and $B_4(x)$ imply that $\mathcal{E}ll_x \leq n-n_{s_0+3}$,
while the condition $k-1\leq n_{s_0+3}$ and the above observation for $B_0'(x)$ give
$n-2n_{s_0+3}< \mathcal{E}ll_x$ for $s=1$. Clearly, $\mathcal{E}ll_x=0$ for $s=2$.
Moreover, we have both for $s=1$ and $s=2$:
\begin{equation}\left\langlebel{cardpart}
|B_1(x)|=n_{s_0+1}, \quad |B_2(x)|= k-1 - n_{s_0+1},
\quad |B_3(x)|= n_{s_0+3} - k+1,
\quad |B_4(x)|= n- \mathcal{E}ll_x - n_{s_0+3}.
\mathcal{E}_{n-1}d{equation}
Thus, given $\mathcal{E}ll \in \{0\}\cup [n-n_{s_0+3}-k+1, n-k+1]$ and a partition of $[n]$ into five sets
$B_i$, $0\leq i\leq 4$, with cardinalities as in (\rightef{cardpart}),
it is enough to construct a net for vectors $x\in {\mathcal R}_{k t}^s$ with
$B_i(x)=B_i$, $0\leq i\leq 4$, $\mathcal{E}ll_x=\mathcal{E}ll$, and then to take the union of nets over all possible realizations of $\mathcal{E}ll$ and all
such partitions $\{B_0,B_1,B_2, B_3, B_4\}$ of $[n]$.
Now we describe our construction. Fix $\mathcal{E}ll$ as above and fix two parameters $\mu= 1/(C_\tau \sqrt{d})$,
and
$\nu=9\left\langlembda_t \sqrt{n}/R$.
We would like to emphasize that for the actual calculations in this lemma, taking $\mu$ to be a small constant multiple of
$R^{-1}$ would be sufficient, however, we would like to run the proof with the above choice of $\mu$ because this
corresponds to the parameter choice in the previous Lemma~\rightef{l:nets} whose proof we only sketched.
Note that for $x\in {\mathcal R}_{k t}^s$ we have $x\|\cdot\|t\in \mathcal{T}$, hence
$x^*_{n_{s_0+1}}\leq C_\tau \sqrt{d}x^*_{n_{s_0+2}}\leq C_\tau^2 d$ and
\begin{equation}\left\langlebel{decr2}
x_1^*\leq (6d) x^*_{2}\leq (6d)^2 x^*_{n_1}\leq \ldots \leq (6d)^{s_0+2} x_{n_{s_0+1}}^*
\leq C_\tau^2 d (6d)^{s_0+2}.
\mathcal{E}_{n-1}d{equation}
Fix $I_0\subset [n]$ with $|I_0|=n_{s_0+1}$ (which will play the role of $B_1$). We shall construct
a $\mu$-net ${\mathbb N}et_{I_0}$ (in the $\mathcal{E}ll_\infty$-metric) for the set
\begin{align*}
\mathcal{T} _{I_0}:=\big\{P_{B_1(x)}x:\;x\in{\mathcal R}_{k t}^s,\;B_1(x)=I_0\big\}.
\mathcal{E}_{n-1}d{align*}
Clearly, the nets ${\mathbb N}et_{I_0}$ for various $I_0$'s can be related by appropriate permutations, so
without loss of generality we can assume for now that $I_0=[n_{s_0+1}]$.
First, consider the partition of $I_0$ into sets $I_1, \ldots, I_{s_0+2}$ defined by
$$
I_1=[2] \quad \mbox{ and } \quad \, \, \, I_j=[n_{j-1}]\setminus [n_{j-2}], \, \, \mbox{ for }\,\, 2\leq j \leq s_0+2.
$$
Consider the set
$$\mathcal{T}^*:=\big\{x\in\mathcal{T}_{[n_{s_0+1}]}:\,{\sigma}ma_x(I_j)=I_j,\;\;j=1,2,{\rm dist}ots,s_0+2\big\}.$$
By the definition of $\mathcal{T} _{I_0}$, for every $x\in \mathcal{T}^*$, one has $\|P_{I_j}x\|_\infty\le b_j:=C_\tau^2 d (6d)^{s_0+3-j}$ for every
$j\le s_0+2$ (where as usual $P_I$ denotes the coordinate projection onto ${\mathbb R}^I$). Define
a $\mu$--net (in the $\mathcal{E}ll_\infty$-metric) for $\mathcal{T}^*$ by setting
$$
{\mathbb N}et^*:={\mathbb N}et_{1}\oplus{\mathbb N}et_{2}\oplus\cdots\oplus{\mathbb N}et_{s_0+2},
$$
where ${\mathbb N}et_{j}$ is a $\mu$-net (in the $\mathcal{E}ll_\infty$-metric) of cardinality at most
$$
(3 b_j/\mu )^{|I_j|} \leq (C_\tau^3 d^{3/2} (6d)^{s_0+3-j})^{n_{j-1}} \leq (C_\tau^3 (6d)^{s_0+5-j})^{n_{j-1}
}
$$
in the coordinate projection of the cube $P_{I_{j}}(b_j B_\infty^n)$.
Recall that $n_0=2$, $n_j=30\mathcal{E}ll_0^{j-1}$, $1\leq j\leq s_0$, where $\mathcal{E}ll_0$ and $s_0$ are given by \mathcal{E}qref{eq: l0s0 def}.
Since $d$ is large enough,
\begin{align*}
2s_0 +8 + 30 \sum_{j=2}^{s_0+1} (s_0+5-j) \mathcal{E}ll_0^{j-2} &=
2s_0 +8 + 30 \sum_{m=1}^{s_0-1} (m+3) \mathcal{E}ll_0^{s_0-m}
\leq 121 \mathcal{E}ll_0^{s_0-1} \leq 4.1 n_{s_0+1},
\mathcal{E}_{n-1}d{align*}
which implies
$$
|{\mathbb N}et^*|\le\mathbb{P}rod_{j=1}^{s_0+2}|{\mathbb N}et_{j}| \le \mathcal{E}xp( 7.1 n_{s_0+1} \ln (6 C_\tau^2 d)).
$$
To pass from the net for $\mathcal{T}^*$ to the net for $\mathcal{T}_{[n_{s_0+1}]}$, let ${\mathbb N}et_{[n_{s_0+1}]}$ be the union
of nets constructed as ${\mathbb N}et^*$ but for arbitrary partitions $I_1',{\rm dist}ots, I_{s_0+2}'$ of $[n_{s_0+1}]$ with $|I_j'|=|I_j|$.
Using that
$$
\sum _{j=1}^{s_0+1} {n_{j-1}} \le 2 + 30 \sum _{j=0}^{s_0-1} \mathcal{E}ll_0^j \leq 2 + 30 \mathcal{E}ll_0^{s_0-1}/(1-1/\mathcal{E}ll_0) \leq 2 n_{s_0+1}
$$
and $e\mathcal{E}ll_0\leq d$ we obtain that the cardinality of ${\mathbb N}et_{[n_{s_0+1}]}$ is at most
\begin{align*}
|{\mathbb N}et^*|\, \mathbb{P}rod_{j=1}^{s_0+1} { n_{j} \choose n_{j-1} }
& \le |{\mathbb N}et^*|\, \mathbb{P}rod_{j=1}^{s_0+1} \Big({\mathcal F}rac{e n_{j}}{n_{j-1}}\Big)^{n_{j-1}} \le
|{\mathbb N}et^*|\, \mathbb{P}rod_{j=1}^{s_0+1} (e \mathcal{E}ll_0)^{n_{j-1}}
\le
\mathcal{E}xp(9.1 n_{s_0+1} \ln (6 C_\tau^2 d)) .
\mathcal{E}_{n-1}d{align*}
Next we construct a net for the parts of the vectors corresponding to $B_2$. Fix
$J_0\subset [n]$ with $|J_0|= k-1 - n_{s_0+1}$ (it will play the role
of $B_2$). We construct a $\mu$-net (in the $\mathcal{E}ll_\infty$-metric) for the set
$$
\mathcal{T}^2 _{J_0}:=\{ P_{B_2(x)}x \, : \, x\in{\Upsilon}_n(r)\setminus \mathcal{T},\,\, B_2(x)=J_0 \}.
$$
Since by \mathcal{E}qref{decr2}, we have $x^*_{n_{s_0+1}}\leq C_\tau^2 d$ for every $x\in {\Upsilon}_n(r)\setminus \mathcal{T}$,
it is enough to take a $\mu$-net ${\mathcal{K}}_{J_0}$ of cardinality at most
$$
|{\mathcal{K}}_{J_0}| \leq (3 C_\tau^2 d/\mu) ^{|J_0|} \leq
(3 C_\tau^3 d^{3/2}) ^{k}
$$
in the coordinate projection of the cube $P_{J_{0}}(C_\tau^2 d B_\infty^n)$.
Now we turn to the part of the vectors corresponding to $B_3$. Fix $D_0\subset [n]$ with
$|D_0|= n_{s_0+3} - k+1$ (it will play the role of $B_3$).
For this part we use $\mathcal{E}ll_2$-metric and construct a $\nu$-net (in the {\it Euclidean metric} this time) for the set
$$
\mathcal{T}^3 _{D_0}:=\{ P_{B_3(x)} x \, : \, x\in {\mathcal R}_{k t}^s,\,\, B_3(x)=D_0 \}.
$$
Since for $x\in {\mathcal R}_{k t}^s$ we have $\|x_{B_3(x)}\|\leq \|x_{{\sigma}ma_x([k,n])}\|\le 3\left\langlembda _t \sqrt{n}$, there exists
a corresponding $\nu$-net ${\mathcal{L}}_{D_0}$ in the coordinate projection of the Euclidean ball
$P_{D_{0}}(3\left\langlembda_t \sqrt{n} B_2^n)$
of cardinality at most
$$
|{\mathcal{L}}_{D_0}| \leq (9\left\langlembda _t \sqrt{n}/\nu) ^{|D_0|} \leq
R^{n_{s_0+3}}\leq R^{r n}.
$$
Next we approximate the almost constant part of a vector (corresponding to $B_0$),
provided that it is not empty (otherwise we skip this step). Fix $A_0\subset [n]$ with
$|A_0|= \mathcal{E}ll$ (it will play the role of $B_0$) and denote
$$
\mathcal{T}^0 _{A_0}:=\{ P_{B_0(x)} x \, : \, x\in\big({\Upsilon}_n(r)\setminus \mathcal{T}\big) \cap \mathcal{AC}(\rightho),\,\, B_0(x)=A_0 \}.
$$
Let ${\mathcal{K}}^0_{A_0} :=\{ \mathbb{P}m P_{A_0} {\bf 1}\}$. Since for every $x\in{\Upsilon}_n(r)$ we have
either $\left\langlembda_x=1$ or $\left\langlembda_x=-1$, by the definition of $B_0(x)$, every $z\in \mathcal{T}^0 _{A_0}$
is approximated by one of $\mathbb{P}m P_{A_0} {\bf 1}$ within error $\rightho$ in the $\mathcal{E}ll_\infty$-metric.
The last part of the vector, corresponding to $B_4$ we just approximate by $0$.
Note that for any $x\in{\mathcal R}_{k t}^1$ we have $\|P_{B_4(x)}x\|\leq \sqrt{rn}\leq \sqrt{2r}\left\langlembdabda_t\sqrt{n}$,
in view of the condition $x\in \mathcal{AC}(\rightho)$.
On the other hand, for $x\in{\mathcal R}_{k t}^2$ we have $\|P_{B_4(x)}x\|\leq \sqrt{n}\leq {\mathcal F}rac{3r}{2}\left\langlembdabda_t\sqrt{n}$.
Now we combine our nets. Consider the net
$$
{\mathbb N}et_0 :=\bigcup\limits_{\mathcal{E}ll,I_0,J_0,D_0,A_0}\big\{y=y_1+y_2+y_3+y_0:\,y_1\in{\mathbb N}et_{I_0},\,y_2\in\mathcal{K}_{J_0},\,
y_3\in \mathcal{L}_{D_0}, y_0\in \mathcal{K}^0_{A_0}\big\},
$$
where the union is taken over all $\mathcal{E}ll\in \{0\}\cup [n-2n_{s_0+3}, n-n_{s_0+3}]$ and all partitions of $[n]$ into $I_0, J_0, D_0, A_0, B$ with
$|I_0|=n_{s_0+1}$, $|J_0|=k-1 - n_{s_0+1}$, $|D_0|= n_{s_0+3} - k+1$, $|A_0|=\mathcal{E}ll$, and
$B=[n]\setminus(I_0\cup J_0 \cup D_0 \cup A_0)$.
Then the cardinality of ${\mathbb N}et_0$,
\begin{align*}
|{\mathbb N}et_0|&\le n {n\choose n_{s_0+1}} {n- n_{s_0+1}\choose k-1 - n_{s_0+1}}
{n- k+1 \choose n_{s_0+3} - k+1} {n- n_{s_0+3} \choose \mathcal{E}ll} \max\limits_{I_0}|{\mathbb N}et_{I_0}| \max\limits_{J_0}|\mathcal{K}_{J_0}|
\max\limits_{D_0}|\mathcal{L}_{D_0}| \max\limits_{A_0}|\mathcal{K}^0_{A_0}|.
\mathcal{E}_{n-1}d{align*}
Using that $n_{s_0+1}\leq n/(64d)$, $k\leq n/\ln^2 d$, $n_{s_0+3}\leq rn$, $\mathcal{E}ll =0$ or $\mathcal{E}ll\geq n-2n_{s_0+3}$,
the obtained bounds on nets, as well as that $d$ is large enough and $r$ is small enough
(smaller than a constant depending on $R$),
we observe that the cardinality of ${\mathbb N}et_0$ is bounded by
$$
n \left(e d\right)^{n/d}\, \left(2e \ln ^2 d\right)^{n/\ln^2 d}\, \left(2e /r\right)^{rn}\,
\left(2e /r\right)^{rn}\, \mathcal{E}xp(9.1n \ln (6 C_\tau^2 d)/(64d))\, (3 C_\tau^3 d^{3/2}) ^{n/\ln ^2 d}
R^{rn}\, \cdot 2 \leq \left(e /r\right)^{2.5 rn}.
$$
By construction, for every $x\in {\mathcal R}_{k t}^s$ there exists $y=y_1+y_2+y_3+y_0\in {\mathbb N}et_0$ such that
\begin{align*}
\|x-y\|&\leq \| P_{B_1(x)}x-y_1\|+\| P_{B_2(x)}x-y_2\| +\| P_{B_3(x)}x-y_3\| + \| P_{B_4(x)}x\|
+\| P_{B_0(x)}x-y_0\|
\\&\leq
\mu \sqrt{n_{s_0+1}} + \mu \sqrt{k-1- n_{s_0+1}} + \nu +\sqrt{2r}\left\langlembdabda_t\sqrt{n} + \rightho \sqrt{n}\leq {\mathcal F}rac{2\sqrt{n}}{C_\tau \sqrt{d}}
+ \rightho \sqrt{n}+ {\mathcal F}rac{9\left\langlembda_t\sqrt{n}}{R}\leq {\mathcal F}rac{10\left\langlembda_t\sqrt{n}}{R},
\mathcal{E}_{n-1}d{align*}
where we used that $\rightho \leq 1/(2R)\leq \left\langlembda_1 /(\sqrt{2} R)\leq \left\langlembda_t /(\sqrt{2} R)$ and that $r$ is sufficiently small.
Finally we adjust our net to $|||\cdot|||$. Note that by Lemma~\rightef{euclnorm} for every $x\in {\Upsilon}_n(r)\setminus \mathcal{T}$,
$$
|\left\langle x , \mathcal{E}dv\righta|= \left|\sum _{i=1}^n {\mathcal F}rac{x_i}{\sqrt{n}}\right|
\leq \|x\|\leq
{\mathcal F}rac{ 384C_\tau^2 d^{4}}{(64p)^{\ln (6d)}}\leq e^{rn}.
$$
Therefore, there exists an $\varepsilon/(4\sqrt{pn})$-net $\mathcal{N}_*$ in $P_\mathcal{E}dv^\mathbb{P}erp {\mathcal R}_{k t}^s$ of
cardinality $8\sqrt{pn}e^{rn}/\varepsilon$ (note, the rank of $P_\mathcal{E}dv^\mathbb{P}erp$ is one).
Then, by the constructions of nets,
for every $x\in {\mathcal R}_{k t}^s$ there exist $y\in \mathcal{N}_0$ and $y_*\in \mathcal{N}_*$ such that
$$
||| x - P_\mathcal{E}dv y -y_*||| ^2=
\| P_\mathcal{E}dv (x - y)\|^2 + pn \|P_\mathcal{E}dv^\mathbb{P}erp x -y_*\|^2\leq {\mathcal F}rac{100\left\langlembda_t^2n}{R^2} + \varepsilon^2/16\leq \varepsilon^2/8.
$$
Thus the set $\mathcal{N} = P_\mathcal{E}dv(\mathcal{N}_0) + \mathcal{N}_*$
is an $(\varepsilon/2)$-net for ${\mathcal R}_{k t}^s$ with respect to $|||\cdot|||$ and its cardinality is bounded by $(e/r)^{3rn}$.
Using standard argument we pass to an $\varepsilon$-net $\mathcal{N}_{k t}^s \subset {\mathcal R}_{k t}^s$ for ${\mathcal R}_{k t}^s$.
\mathcal{E}_{n-1}d{proof}
\subsection{Proof of Theorem~\rightef{classB}}
\begin{proof}
Recall that the sets ${\mathcal R}_{ki}^s$ were introduced just before Lemma~\rightef{newnet} and
the event ${\mathbb E}vent_{nrm}$ was defined in Proposition~\rightef{nettri}.
Fix $s\in \{1, 2\}$, $k\leq n/\ln^2 d$, $A:=[k, n]$, $i\leq m$.
Set $\varepsilon :=\left\langlembda _i \sqrt{n}/(600\sqrt{2} C_0)$, where $\left\langlembda_i$ and $m$ are defined according to \mathcal{E}qref{eq 2498520598207560}.
Applying Lemma~\rightef{newnet} with $R= 24000 \sqrt{2} C_0$, we
find an $\varepsilon$-net (in the $|||\cdot|||$--norm) $\mathcal{N}_{k i}^s\subset {\mathcal R}_{k i}^s$ for ${\mathcal R}_{k i}^s$
of cardinality at most $(e/r)^{3rn}$.
Take for a moment any $y\in \mathcal{N}_{k i}^s$. Note that
$\|y_{{\sigma}ma(A)}\| \geq C_0\|y_{{\sigma}ma(A)}\|_\infty/\sqrt{p}$, $\|y_{\sigma}ma(A)\|\geq \left\langlembda_i \sqrt{n}$
(where ${\sigma}ma={\sigma}ma_y$).
Then Proposition~\rightef{rogozin} implies $\mathbb{P}({\mathbb E}vent_y^c)\leq e^{-3n}$, where
$$
{\mathbb E}vent_y=\left\{ \|My\|> {\mathcal F}rac{\sqrt{pn}}{3\sqrt{2} C_0} \, \|y_{{\sigma}ma(A)}\| \right\}.
$$
Let us condition on the event ${\mathbb E}vent_{nrm}\cap \bigcap\limits_{y\in \mathcal{N}_{k i}^s}{\mathbb E}vent_y$.
Using the definition of $\mathcal{N}_{k i}^s$ and ${\mathcal R}_{k i}^s$, the triangle inequality, and
the definition of ${\mathbb E}vent_{nrm}$ from
Proposition~\rightef{nettri}, we get that for any $x\in {\mathcal R}_{k i}^s$ there is $y\in \mathcal{N}_{k i}^s$ such that
$|||x-y|||\leq \varepsilon$, and hence
$$
\|Mx\|\geq \|My\| - \|M(x-y)\| > {\mathcal F}rac{\sqrt{pn}}{3\sqrt{2} C_0} \, \|y_{{\sigma}ma(A)}\| - 100\sqrt{pn} \varepsilon \geq
{\mathcal F}rac{\sqrt{p}\left\langlembda _i n }{6\sqrt{2} C_0} .
$$
Using that $|\mathcal{N}_{k i}^s|\leq (e/r)^{3rn}$, that $\left\langlembda_i\geq 1/\sqrt{2}$, and the union bound, we obtain
$$
\mathbb{P}\left({\mathbb E}vent_{nrm} \cap \left\{\mathcal{E}xists x\in {\mathcal R}_{ki}^s \, : \,
\|Mx\|\leq {\mathcal F}rac{\sqrt{p} n}{12 C_0}\right\}\rightight)
\leq {\mathbb P}\Big({\mathbb E}vent_{nrm}\cap \bigcup\limits_{y\in \mathcal{N}_{k i}^s}{\mathbb E}vent_y^c\Big)
\leq e^{-3(1-r\ln(e/r))n}.
$$
Since ${\mathcal R}=\bigcup_{k,i}\, ({\mathcal R}_{ki}^1 \cup {\mathcal R}_{ki}^2)$ and $r$ is small enough,
the result follows by the union bound and by Lemma~\rightef{bdd} applied with $t=30$ in order to estimate ${\mathbb P}({\mathbb E}vent_{nrm})$.
\mathcal{E}_{n-1}d{proof}
\subsection{Lower bounds on $\|Mx\|$ for vectors from $\mathcal{T}_0\cup \mathcal{T}_1$}
The following lemma provides a lower bound on the ratio $\|Mx\|/\| x \| _2$ for vectors $x$ from $\mathcal{T}_0\cup \mathcal{T}_1$.
\begin{lemma} \left\langlebel{l:T0}
Let $n\geq 1$, $0<p<0.001$, and assume that $d=pn\geq 200 \ln n$. Then
$$
\mathbb{P}\left(\Big\{\mathcal{E}xists\;x\in \mathcal{T}_0 \cup \mathcal{T}_1 \, \, \, \mbox{ such that }
\, \, \, \|M x\| \leq {\mathcal F}rac{(64p)^{\kappa}}{ 192(pn)^2}\, \|x\| \Big\}\right)
\leq n(1-p)^n+e^{-1.4np},
$$
where $\kappa$ is defined by \mathcal{E}qref{eq: kappa def}.
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
Let ${\rm dist}eltalta_{ij}$, $i,j\leq n$ be entries of $M$.
Let ${\mathbb E}vent$ be the event that there are no zero columns in $M$.
Clearly, $\mathbb{P}({\mathbb E}vent)\geq 1-n(1-p)^n$.
Also, for each $1\leq j\leq s_0+1$, let ${\mathbb E}vent_j= {\mathbb E}vent_{col} (\mathcal{E}ll_0, n_{j-1})$ be the event
introduced in Lemma~\rightef{col} (with $s_0,\mathcal{E}ll_0$ defined in \mathcal{E}qref{eq: l0s0 def}), and observe that, according to Lemma~\rightef{col},
$\mathbb{P}({\mathbb E}vent_j)\geq 1-e^{-1.5np}$ for every $j$.
Recall that ${\sigma}ma_x$ denotes a permutation $[n]$ such that $x_i^*=|x_{{\sigma}ma(i)}|$ for $i\le n$.
Pick any $x\in \mathcal{T}_0 \cup \mathcal{T}_1$.
In the case $x\in \mathcal{T}_{0}$ set $m=m_1=1$ and $m_2=2$.
In the case $x\in \mathcal{T}_{1j}$ for some $1\leq j\leq s_0+1$ set $m=m_1=n_{j-1}$ and $m_2=n_j$.
Then by the definition of sets $\mathcal{T}_{0}, \mathcal{T}_1$ we have
$x^*_m>6d x^*_{m_2}$.
Let
$$
J^\mathcal{E}ll=J^\mathcal{E}ll(x)={\sigma}ma_x([m]), \quad J^r=J^r(x)={\sigma}ma_x([m_2-1]\setminus[m]), \quad \mbox{ and } \quad
J(x)=(J^\mathcal{E}ll\cup J^r)^c
$$
(if $x\in \mathcal{T}_{0}$ then $J^r=\mathcal{E}mptyset$).
Note that by our definition we have $|x_i|>6d |x_u| $ for any $i\in J^\mathcal{E}ll(x)$ and $u\in J(x)$, and that
$\max_{i\in J(x)}|x_i|\le x^*_{m_2}$.
Denote by $I^\ell(x)$ the (random) set of rows of $M$ having exactly one 1 in $J^\mathcal{E}ll(x)$ and no 1's in $J^r(x)$.
Now we recall that the event ${\mathbb E}vent_{sum}$ was introduced in Lemma~\rightef{bennett}
(we use it with $q=p$) and set
$$
{\mathbb E}vent':= {\mathbb E}vent\cap {\mathbb E}vent_{sum}\cap \bigcap_{j=1}^{s_0+1} {\mathbb E}vent_j.
$$
Clearly, conditioned on ${\mathbb E}vent'$, the set $I^\ell(x)$ is not empty for any $x\in \mathcal{T}_0 \cup \mathcal{T}_1$.
By definition, for every $s\in I^\ell(x)$ there exists
$j(s)\in J^\mathcal{E}ll(x)$ such that
$$
{\rm supp\, } R_{s}(M)\cap J^\mathcal{E}ll(x)=\{j(s)\},\quad {\rm supp\, } R_{s}(M)\cap J^r(x)= \mathcal{E}mptyset.
$$
Since
$j(s)\in J^{\mathcal{E}ll}(x)$
(which implies $|x_{j(s)}|\geq x^*_m> 6 d x^*_{m_2}$), we obtain
\begin{align*}
|\left\langlengle R_{s} (M),\, x \rightangle|
&=\Big| x_{j(s)}
+ \sum_{j\in J(x)} {\rm dist}eltalta _{sj} x_j \Big|
\geq|x_{j(s)}|- x_{m_2}^* \sum_{j\in J(x)} {\rm dist}eltalta _{sj}
\geq x_{m}^* - {\mathcal F}rac{x_{m}^*}{6d} \sum_{j\in J(x)} {\rm dist}eltalta _{sj} .
\mathcal{E}_{n-1}d{align*}
Observe that conditioned on ${\mathbb E}vent_{sum}$ we have
$\sum_{j\in J(x)} {\rm dist}eltalta _{sj} \leq \sum_{j=1}^n {\rm dist}eltalta _{sj} \leq 3.5 pn=3.5 d$. Thus,
everywhere on ${\mathbb E}vent'$ we have for all $x\in \mathcal{T}_0 \cup \mathcal{T}_1$,
$$
\|M x \| \geq |\left\langlengle R_{s} (M),\, x\rightangle|
\geq x_{m}^*/3,\quad s\in I^\ell(x).
$$
Finally, in the case $x\in \mathcal{T}_{0}$ we have $m=1$ and $\|x\|\leq \sqrt{n}x^*_1$.
In the case $x\in \mathcal{T}_{1j}$ by Lemma~\rightef{euclnorm} we have
$$
\|x\|\leq {\mathcal F}rac{ 64(pn)^2}{(64p)^{\kappa}}\, x^*_{m},
$$
This proves the lower bound on $\|Mx\|/\|x\|$ conditioned on ${\mathbb E}vent'$.
The probability bound
follows by the union bound, Lemmas~\rightef{bennett} and \rightef{col}, and since $s_0\leq \ln n$,
indeed
$$
\mathbb{P}\left({\mathbb E}vent\cap {\mathbb E}vent_{sum}\cap \bigcap_{j=1}^{s_0+1} {\mathbb E}vent_j\right) \geq
1- n(1-p)^n - (s_0+2)e^{-1.5np} \leq 1- n(1-p)^n - e^{-1.4 np} .
$$
\mathcal{E}_{n-1}d{proof}
\subsection{Individual bounds for vectors from $\mathcal{T}_2 \cup \mathcal{T}_3$}
\left\langlebel{subs: nets}
In this section we provide individual probability bounds for vectors from the nets constructed in
Lemma~\rightef{l:nets}.
To obtain the lower bounds on $\|M x\|$, we consider
the behavior of the inner products $\left\langle \rightow_i(M), x \righta$, more specifically, of the
L\'evy concentration function for $\left\langle \rightow_i(M), x\righta$. To estimate this
function, we will consider $2m$ columns of $M$ corresponding
to the $m$ biggest and $m$ smallest (in absolute value) coordinates of $x$, where $m=n_{s_0+1}$ or $m=n_{s_0+2}$.
In a sense, our anti-concentration estimates will appear in the process of swapping $1$'s and $0$'s within a specially
chosen subset of the matrix rows. A crucial element in this process is to extract a pair of subsets of indices on which the chosen
matrix rows have only one non-zero component. This will allow to get anti-concentration bounds by ``sending'' the
non-zero component into the other index subset from the pair.
The main difficulty in this scheme comes from the restriction $2m p \leq 1/32$ from
Lemma~\rightef{c:SJ}, which guarantees existence of sufficiently many required subsets (and rows)
but which cannot be directly applied to $m=n_{s_0+2}$. To resolve this problem we use idea from
\cite{LLTTY-TAMS}. We split
the initially fixed set of $2m$ columns into smaller subsets of columns of size at most $1/(64 p)$ each,
and create independent random variables corresponding to this splitting.
Then we apply Proposition~\rightef{prop: esseen},
allowing to deal with the L\'evy concentration function for sums of independent random variables.
We first describe subdivisions of ${\mathcal{M}_{n}}$ used in \cite{LLTTY-TAMS}. Recall that ${\mathcal{M}_{n}}$ denotes
the class of all $n\times n$ matrices with $0/1$ entries.
We recall also that the probability measure ${\mathbb P}$ on ${\mathcal{M}_{n}}$ is always assumed to be induced by a
Bernoulli($p$) random matrix.
Given $J\subset [n]$ and $M\in {\mathcal{M}_{n}}$
denote
$$
I (J, M) = \{i \leq n \, : \, |{\rm supp\, } \rightow_i(M) \cap J | =1\}.
$$
By ${\mathcal M} _J$ we denote
the set of $n\times |J|$ matrices with $0/1$ entries and with columns indexed
by $J$.
Fix $q_0\leq n$ and a partition $J_0$, $J_1$, ..., $J_{q_0}$ of $[n]$.
Given subsets $I_1, {\rm dist}ots,I_{q_0}$ of $[n]$ and $V=(v_{ij})\in {\mathcal M} _{J_0}$, denote
${\mathcal I} = (I_1, \ldots, I_{q_0})$ and consider the class
$$
{\mathcal F} ({\mathcal I}, V) = \left\{M=(\mu_{ij})\in {\mathcal{M}_{n}} \, :\, {\mathcal F}orall q\in [q_0] \quad I (J_q, M) = I_q \,\,
\mbox{ and } \,\,{\mathcal F}orall i\leq n\, {\mathcal F}orall j\in J_0 \,\,\, \mu_{ij}= v_{ij} \rightight\}.
$$
In words, we fix the columns
indexed by $J_0$ and for each $q\in [q_0]$ we fix the row indices having exactly one $1$ in columns indexed by $J_q$.
Then, for any fixed partition $J_0$, $J_1$, ..., $J_{q_0}$, ${\mathcal{M}_{n}}$ is the disjoint union of classes ${\mathcal F} ({\mathcal I}, V)$ over all $V\in {\mathcal M} _{J_0}$ and
all ${\mathcal I} \in (\mathcal{P} ([n]))^{q_0}$, where $\mathcal{P} (\cdot)$ denotes the power set.
The following is an important, but simple observation.
\begin{lemma}\left\langlebel{l: indep 20598}
Let ${\mathcal F} ({\mathcal I}, V)$ be a non-empty class (defined as above), and denote by ${\mathbb P}_{{\mathcal F}}$ the induced probability measure on
${\mathcal F} ({\mathcal I}, V)$, i.e., let
$$
{\mathbb P}_{{\mathcal F}}(B):={\mathcal F}rac{{\mathbb P}(B)}{{\mathbb P}({\mathcal F} ({\mathcal I}, V))},\quad B\subset {\mathcal F} ({\mathcal I}, V).
$$
Then the matrix rows for matrices in ${\mathcal F} ({\mathcal I}, V)$ are mutually independent with respect to ${\mathbb P}_{{\mathcal F}}$,
in other words, a random matrix distributed according to ${\mathbb P}_{{\mathcal F}}$ has mutually independent rows.
\mathcal{E}_{n-1}d{lemma}
Finally, given a vector $v\in {\mathbb R}^n$, a class ${\mathcal F} ({\mathcal I}, V)$, indices $i\leq n$, $q\leq q_0$,
define
\begin{equation}\left\langlebel{xiq}
\xi _q(i) = \xi_q (M,v,i) := \sum _{j\in J_q} {\rm dist}eltalta _{ij} v_j,\quad M=({\rm dist}eltalta_{ij})\in {\mathcal F} ({\mathcal I}, V).
\mathcal{E}_{n-1}d{equation}
We will view $\xi_q(i)$ as random variables on ${\mathcal F} ({\mathcal I}, V)$ (with respect to the measure ${\mathbb P}_{{\mathcal F}}$).
It is not difficult to see that for every fixed $i$, the variables $\xi_1(i),{\rm dist}ots\xi_{q_0}(i)$ are mutually independent,
and, moreover, whenever $i\in I_q$, the variable $\xi_q(i)$ is uniformly distributed on the multiset $\{v_j\}_{j\in J_q}$.
Thus, we may apply Proposition~\rightef{prop: esseen} to
$$
\left|\left\langle \rightow_i(M), v \righta\rightight| = \Big| \sum _{q=0}^{q_0} \xi_q(i)\Big|
$$
with some $a_1lphapha >0$ satisfying $\mathcal{Q} (\xi_q(i) , 1/3)\leq a_1lphapha$ for every $i\in I_q$. This gives
\begin{equation}\left\langlebel{conc-inner}
{\mathbb P}_{{\mathcal F}} \left\{\left|\left\langle \rightow_i(M), x+y \righta \rightight| \leq 1/3\rightight\} \leq {\mathcal F}rac{C_0 a_1lphapha }{\sqrt{(1-a_1lphapha) |\{q\geq 1:\,i\in I_q\}|}},
\mathcal{E}_{n-1}d{equation}
where $C_0$ is a positive absolute constant.
We are ready now to estimate individual probabilities.
\begin{lemma}[Individual probabilities]
\left\langlebel{individual}
There exist absolute constants $C, C'>1>c_1>0$ such that the following holds.
Let $p\in (0, 1/64]$, $d=pn\geq 2$,
Set $m_0= \lfloor 1/(64 p)\rightfloor$ and
let $m_1$ and $m_2$ be such that
$$1\leq m_1<m_2\leq n-m_1.$$
Let $y\in {\rm span}\,\{{\bf{1}}\}$ and assume that
$x\in {\mathbb R}^n$ satisfies
$$
x^*_{m_1}> 2/3 \quad \mbox{ and } \quad x^*_i = 0 \, \, \, \mbox{ for every }\, \, i> m_2.
$$
Denote $m=\min(m_0, m_1)$ and consider the event
$$
E(x, y) = \left\{
M \in {\mathcal{M}_{n}}\, :\, \|M (x+y) \|\leq \sqrt{ c_1 m d} \right\}.
$$
Then in the case $m_1 \leq m_0$ one has
$$
{\mathbb P}(E(x, y)\cap {\mathbb E}vent_{card})\leq 2^{-m d/20},
$$
and in the case $m_1> C' m_0$ one has
$$
{\mathbb P}(E(x, y)\cap{\mathbb E}vent_{card} )\leq \left({\mathcal F}rac{C n}{m_1 d}\right)
^{m d/20},
$$
where ${\mathbb E}vent_{card}$ is the event introduced in Lemma~\rightef{c:SJ} with $\mathcal{E}ll=2m$.
\mathcal{E}_{n-1}d{lemma}
\begin{rem}\left\langlebel{rem-ind}
We apply this lemma below for sets $\mathcal{T}_i$ with the following choice of parameters.
For $i=2$ we set
$$
m_1 =m_0= n_{s_0+1}=\max(30 \mathcal{E}ll_0^{s_0-1}, \left\lfloor 1/(64p) \right\rightfloor),
\quad m_2=n_{s_0+2}, \quad \mbox{and} \quad
p\leq 0.001,
$$
obtaining
\begin{equation*} \left\langlebel{individual-one}
{\mathbb P}(E(x, y)\cap {\mathbb E}vent_{card} )\leq 2^{-n_{s_0+1} d/20}.
\mathcal{E}_{n-1}d{equation*}
For $i=3$, we set
$$
m_1=n_{s_0+2}=\lfloor n/\sqrt{d} \rightfloor > m_0=n_{s_0+1}, \quad
m_2=n_{s_0+3}, \quad \mbox{and} \quad p\leq 0.001,
$$
obtaining for large enough $d$,
\begin{equation*}\left\langlebel{individual-two}
{\mathbb P}(E(x, y)\cap {\mathbb E}vent_{card} )\leq \left({\mathcal F}rac{C n}{n_{s_0+2} d}\right)
^{n_{s_0+1} d/20} \leq \left( \sqrt{d}/(2C)\right)
^{-n_{s_0+1} d/20} .
\mathcal{E}_{n-1}d{equation*}
\mathcal{E}_{n-1}d{rem}
To prove Lemma~\rightef{individual} it will be convenient to use the same notation as in
Lemma~\rightef{l:T0}.
Given two disjoint subsets $J^\mathcal{E}ll$, $J^r\subset[n]$
and a matrix $M\in {\mathcal{M}_{n}}$, denote
$$
I^\ell=I^\ell(M):=\{i\le n :\,|{\rm supp\, } \rightow_i(M)\cap J^\mathcal{E}ll|=1 \, \, \text{ and }\, \,{\rm supp\, } \rightow_i(M)\cap J^r=\mathcal{E}mptyset\},
$$
and
$$
I^r=I^r(M):=\{i\le n :\,{\rm supp\, } \rightow_i(M)\cap J^\mathcal{E}ll=\mathcal{E}mptyset\,\,\text{ and }\,\,|{\rm supp\, } \rightow_i(M)\cap J^r|=1\}.
$$
Here the upper indices $\mathcal{E}ll$ and $r$ refer to {\it left} and {\it right}.
\begin{proof}
Let $d=pn$ and fix $\gammamma = mp/72= md/(72n)$.
Fix $x\in {\mathbb R}^n$ and $y\in {\rm span}\,\{{\bf 1}\}$ satisfying the conditions of the lemma. Let ${\sigma}ma={\sigma}ma_x$, that is,
a permutation of $[n]$ such that
$x_i^*=|x_{{\sigma}ma(i)}|$ for all $i\le n$. Denote $q_0=m_1/m$ and without loss of generality
assume that either $q_0=1$ or that $q_0$ is a large enough integer. Let
$J^{\mathcal{E}ll}_1, J_2^\mathcal{E}ll, \ldots, J^\mathcal{E}ll_{q_0}$ be a partition
of ${\sigma}ma ([m_1])$ into sets of cardinality $m$ each, and
let $J^{r}_1, J_2^r, \ldots, J^r_{q_0}$ be a partition
of ${\sigma}ma ([n-m_1+1, n])$ into sets of cardinality $m$ each. Denote
$$
J_q:=J^\mathcal{E}ll_q\cup J^r_q \, \, \, \mbox{ for }\,\,\, q\in [q_0] \quad \mbox{ and } \quad
J_0:= [n]\setminus \bigcup _{q=1}^{q_0} J_q.
$$
Then $J_0$, $J_1$, ..., $J_{q_0}$ is a partition of $[n]$, which we fix in this proof.
Let $M$ be a $0/1$ $n\times n$ matrix.
For every pair $J^\mathcal{E}ll_q$, $J^r_q$, let the sets $I^\ell_q(M)$ and $I^r_q(M)$
be defined as
after Remark~\rightef{rem-ind} and let $I_q(M)= I^\ell_q(M) \cup I^r_q(M)$.
Since
$$
|J_q|=2m \le 2m_0\le 1/(32p),
$$
and by the definition of the event ${\mathbb E}vent_{card}$ (see Lemma~\rightef{c:SJ} with $\mathcal{E}ll=2m$), we have
\begin{equation}\left\langlebel{cond-card}
|I_q(M)|\in[md/8,\,4md]
\mathcal{E}_{n-1}d{equation}
everywhere on ${\mathbb E}vent_{card}$.
Now we represent ${\mathcal{M}_{n}}$ as a disjoint union of classes ${\mathcal F} ({\mathcal I}, V)$ defined at the beginning of this subsection
with $V\in {\mathcal M} _{J_0}$ and ${\mathcal I} =(I_1, \ldots, I_q)$.
Since it is enough to prove a uniform upper bound for classes ${\mathcal F} ({\mathcal I}, V)\cap {\mathbb E}vent_{card}$
and since for every such non-empty class ${\mathcal I}$ must satisfy (\rightef{cond-card}) for every $q\leq q_0$,
we have
$$
{\mathbb P}(E(x, y)\cap {\mathbb E}vent_{card} )\leq \max \mathbb{P} (E(x, y) \cap {\mathbb E}vent_{card}\,|\, {\mathcal F} ({\mathcal I}, V))
\leq \max \mathbb{P} (E(x, y) |\, {\mathcal F} ({\mathcal I}, V)),
$$
where the first maximum is taken over all ${\mathcal F} ({\mathcal I}, V)$
with ${\mathcal F} ({\mathcal I}, V) \cap {\mathbb E}vent_{card} \ne \mathcal{E}mptyset$
and
the second maximum is taken over all ${\mathcal F} ({\mathcal I}, V)$ with $I_q$'s satisfying
condition (\rightef{cond-card}).
Fix any class ${\mathcal F} ({\mathcal I}, V)$, where ${\mathcal I}$ satisfies \mathcal{E}qref{cond-card}, and denote the corresponding induced probability measure on the class
by $\mathbb{P}_{\mathcal F}$, that is
$$
\mathbb{P}_{\mathcal F} (\cdot) = \mathbb{P}( \cdot \, | \, {\mathcal F} ({\mathcal I}, V)).
$$
Let
$$
I: = \bigcup _{q=1}^{q_0} I_q.
$$
Note that $|I|\leq 4 q_0 md$.
We first show that the set of $i$'s which belongs to many $I_q$'s is large.
More precisely, denote
$$
A_i = \{ q\in[q_0]\, : \, i\in I_q\},\;\;i\in[n], \quad \quad \mbox{ and }\quad\quad
I_{0}=\{i\leq n\, : \, |A_i|\ge \gammamma q_0\}.
$$
Then, using bounds on cardinalities of $I_q$'s, one has
$$
m d q_0 /8 \leq \sum_{q=1}^{q_0} |I_q| = \sum_{i=1}^n |A_i| \leq |I_{0}| q_0 + (n-|I_{0}|) \gammamma q_0
\leq |I_{0}| q_0 + n \gammamma q_0.
$$
Thus,
$$
|I_0|\geq m d/8 - n\gammamma \geq md/9.
$$
Without loss of generality
we assume that $I_0=\{1, 2, \ldots |I_0|\}$ and only consider the first
$k:=\lceil md/9 \rightceil$ indices from it. Then $[k]\subset I_0$.
Now, by definition, for matrices $M\in E(x, y)$ we have
$$
\|M(x+y)\|^2 = \sum _{i=1}^n | \left\langle \rightow_i(M), x+y \righta|^2 \le c_1\, md .
$$
Therefore there are at most $9 c_1 md$ rows with
$| \left\langlengle \rightow_i(M), x+y) \rightangle|\ge 1/3$. Hence,
$$
|\{i\leq k\, : \,| \left\langlengle \rightow_i(M), x+y \rightangle|< 1/3\}|\ge
md/9 - 9 c_1 md \ge (1/9- 9c_1)md.
$$
Let
$
k_0:= \lceil (1/9- 9c_1) md\rightceil
$
and for every $i\leq k$ denote
$$
\Omega_i:=\{M\in{\mathcal F} ({\mathcal I}, V) \, :\, |\left\langle \rightow_i(M), x+y \righta|< 1/3 \}
\quad \mbox{ and } \quad \Omega_0= {\mathcal F} ({\mathcal I}, V) .
$$
Then
\begin{align*}
{\mathbb P}_{{\mathcal F}}(E(x,y)) &\le \sum _{B\subset [k]a_1top |B|=k_0 } \,
{\mathbb P}_{{\mathcal F}}\Big(\bigcap_{i\in B}\Omega_i\Big)
\le {k \choose k_0 }\, \max _{B\subset [k]a_1top |B|=k_0 } \, {\mathbb P}_{{\mathcal F}}\Big(\bigcap_{i\in B} \Omega_i\Big).
\mathcal{E}_{n-1}d{align*}
Without loss of generality we assume that the maximum above is attained at $B=[k_0]$. Then
\begin{equation} \left\langlebel{ptensor}
{\mathbb P}_{{\mathcal F}}(E(x, y)) \le
\left(e/(81 c_1)\right)^{9c_1 md} \,\, \, \mathbb{P}rod_{i=1}^{k_0} \, {\mathbb P}_{{\mathcal F}}(\Omega_{i}|\,\Omega_1\cap\ldots\cap \Omega_{i-1})
=\left(e/(81 c_1)\right)^{9c_1 md} \,\, \, \mathbb{P}rod_{i=1}^{k_0} \, {\mathbb P}_{{\mathcal F}}(\Omega_{i}),
\mathcal{E}_{n-1}d{equation}
where at the last step we used mutual independence of the events $\Omega_{i}$ (with respect to measure ${\mathbb P}_{{\mathcal F}}$),
see Lemma~\rightef{l: indep 20598}.
Next we estimate the factors in the product. Fix $i\leq k_0$ and $A_i= \{ q\, : \, i\in I_q\}$. Since, by our assumptions,
$i\in I_0$, we have
$|A_i|\geq \gammamma q_0$.
Consider the random variables $\xi_q(i)=\xi_q(M,x+y,i)$, $q\in A_i$,
defined in (\rightef{xiq}). Then by
(\rightef{conc-inner}) we have
\begin{align*}
{\mathbb P}_{{\mathcal F}}(\Omega_{i}) &= {\mathbb P}_{{\mathcal F}}\big\{|\left\langle \rightow_i(M), x+y \righta | < 1/3\big\}
\leq \mathcal{Q}_{{\mathcal F}}\Big(\sum _{q=0}^{q_0} \xi_q(i),1/3\Big)\\
&\leq\mathcal{Q}_{{\mathcal F}}\Big(\sum _{q\in A_i} \xi_q(i),1/3\Big)
\leq {\mathcal F}rac{C_0 a_1lphapha}{\sqrt{(1-a_1lphapha) |A_i|}}
\leq{\mathcal F}rac{C_0 a_1lphapha}{\sqrt{(1-a_1lphapha) \gammamma q_0}}
\mathcal{E}_{n-1}d{align*}
where
$a_1lphapha = \max _{q\in A_i} \mathcal{Q}_{{\mathcal F}} (\xi_q (i), 1/3)$.
Moreover, in the case $q_0=1$ we just have
$$
{\mathbb P}_{{\mathcal F}}(\Omega_{i}) \leq a_1lphapha = \mathcal{Q} (\xi_1 (i), 1/3).
$$
Thus it remains to estimate $\mathcal{Q}_{{\mathcal F}} (\xi_q(i) , 1/3)$ for $q\in A_i$. Fix $q\in A_i$, so that $i\in I_q$.
Recall that, by construction, the intersection of the support of $\rightow_i(M)$ with $J_q$
is a singleton everywhere on ${\mathcal F} ({\mathcal I}, V)$. Denote the corresponding index by $j(q,M)=j(q,M,i)$. Then
$$
\xi _q(i) = \xi_q(M, x+y,i) = \sum _{j\in J_q} {\rm dist}eltalta _{ij} (x_j+y_1) = x_{j(q,M)}+y_1,
$$
and note that $|x_{j(q,M)}|>2/3$ whenever $j(q,M)\in J^\mathcal{E}ll_q$ and
$x_{j(q,M)} =0$ whenever $j(q,M)\in J^r_q$.
Observe further that ${\mathbb P}_{{\mathcal F}}\big\{j(q,M)\in J^r_q\big\}={\mathbb P}_{{\mathcal F}}\big\{j(q,M)\in J^\mathcal{E}ll_q\big\}=1/2$.
Hence, we obtain
$$
\mathcal{Q}_{{\mathcal F}} (\xi_q (i), 1/3) \leq 1/2:=a_1lphapha.
$$
Combining the probability estimates starting with (\rightef{ptensor}) and
using that $\gammamma = md/(72n)$,
we obtain in the case $q_0=m_1/m\geq C'$,
\begin{align*}
{\mathbb P}_{{\mathcal F}}(E(x, y))&\leq \left( {\mathcal F}rac{e}{81 c_1}\right)^{9 c_1 md} \,\, \, \left({\mathcal F}rac {C_0}{\sqrt{2\gammamma q_0}}\right)^{(1/9- 9c_1) md}
\\& =
\left({\mathcal F}rac{e}{81c_1}\right)^{9 c_1 md} \,\, \, \left({\mathcal F}rac{6 C_0 \sqrt{n}}{\sqrt{ m_1 d}}\right)^{(1/9-9 c_1)md} \leq
\left({\mathcal F}rac{C_1 n}{m_1 d}\right)^{md/20},
\mathcal{E}_{n-1}d{align*}
provided that $c_1$ is small enough and $C_1=36C_0^2$. Note that the bound is
meaningful only if $C'$ is large enough.
In the case $q_0=1$ we have
$$
{\mathbb P}_{{\mathcal F}}(E(x,y))\leq \left( {\mathcal F}rac{e}{81 c_1}\right)^{9 c_1 md} \,\, \, \left( {\mathcal F}rac{1}{2}\right)^{(1/9-9 c_1) md} \leq
\left( {\mathcal F}rac{1}{2}\right)^{md/20},
$$
provided that $c_1$ is small enough.
This completes the proof.
\mathcal{E}_{n-1}d{proof}
\subsection{Proof of Theorem \rightef{steep}}
We are ready to complete the proof.
Denote
$$m= m_0=n_{s_0+1}:=\max(30 \mathcal{E}ll_0^{s_0-1}, \left\lfloor 1/(64p) \right\rightfloor)\in [n/(64 d), n/(2d)].$$
Lemma~\rightef{l:T0} implies that
$$
\mathbb{P}\left(\Big\{\mathcal{E}xists\;x\in \mathcal{T}_0 \cup \mathcal{T}_1 \, \, \, \mbox{ such that }
\, \, \, \|M x\| \leq {\mathcal F}rac{(64p)^{\kappa}}{ 192(pn)^2}\, \|x\|\Big\}\right)
\leq n(1-p)^n+e^{-1.4np}.
$$
We now turn to the remaining cases. Fix $j\in \{2, 3\}$.
Let
\begin{align}
&{\mathbb E}vent_{j}:=\Big\{M\in{\mathcal{M}_{n}}\,:\,\mathcal{E}xists\, x\in \mathcal{T}_j \,\, \, \mbox{such that}\,\,\,\|M x\|\le
{\mathcal F}rac{\sqrt{ c_1 m d}}{2 \, b_j }\, \|x\|\Big\},\|\cdot\|tag
\mathcal{E}_{n-1}d{align}
where $c_1$ is the constant from Lemma~\rightef{individual},
and $b_{2}= 384(pn)^3/(64p)^{\kappa}$, $b_{3}=384C_\tau (pn)^{3.5}/(64p)^{\kappa}$.
Recall that ${\mathbb E}vent_{nrm}$ was defined in Proposition~\rightef{nettri}.
For any matrix $M\in {\mathbb E}vent_{j}\cap {\mathbb E}vent_{nrm}$ there exists $x=x(M)\in \mathcal{T}_j$ satisfying
$$
\|M x\|\le {\mathcal F}rac{\sqrt{ c_1 m d}}{2 \, b_j }\, \|x\|.
$$
Normalize $x$ so that $x_{n_{s_0+j-1}}^{*}=1$, that is, $x\in \mathcal{T}_j'$.
By Lemma~\rightef{euclnorm} we have $\|x\|\leq b_j$.
Let ${\mathbb N}et_j={\mathbb N}et_j'+{\mathbb N}et_j''$ be the net constructed in Lemma~\rightef{l:nets}. Then there exist
$u\in {\mathbb N}et_j'$ with
$$u_{{s_0+j-1}}^{*}\geq 1-1/(C_\tau\sqrt{d})>2/3$$
and $u_{\mathcal{E}ll}^{*}=0$ for $\mathcal{E}ll> n_{s_0+j}$, and $w\in {\mathbb N}et_j''\subset {\rm span}\,\{{\bf 1}\}$, such that
$|||x-(u+w)|||\leq \sqrt{2n}/(C_\tau \sqrt{d}).$
Applying Proposition~\rightef{nettri} (where ${\mathbb E}vent_{nrm}$ was introduced), and using that $C_\tau$ is large enough,
we obtain that
for every matrix $M\in {\mathbb E}vent_{j}\cap {\mathbb E}vent_{nrm}$ there exist
$u=u(M)\in {\mathbb N}et_j'$ and $w=w(M)\in {\mathbb N}et_j''\subset {\rm span}\,\{{\bf 1}\}$ with
\begin{equation}
\left\langlebel{ctau}
\|M (u+w)\|\le \|Mx\| + \|M (x-u-w)\| \leq
\sqrt{ c_1 m d}/2 + 200 \sqrt{2n}/C_\tau \leq
\sqrt{ c_1 m d}.
\mathcal{E}_{n-1}d{equation}
Using our choice of $n_{s_0+1}$, $n_{s_0+2}$, $n_{s_0+3}$, Lemma~\rightef{l:nets}, and
Lemma~\rightef{individual} twice ---
first with $m_1=m_0=n_{s_0+1}$, $m_2=n_{s_0+2}$,
then with $m_1=n_{s_0+2}>m_0=n_{s_0+1}$, $m_2=n_{s_0+3}$ (see Remark~\rightef{rem-ind}),
we obtain that for small enough $a_1aa$ and large enough $d$ the probability
$\mathbb{P}\left({\mathbb E}vent_{2} \cap {\mathbb E}vent _{nrm}\cap {\mathbb E}vent_{card}\right)$ is bounded by
$$
\mathcal{E}xp \left( 2 n_{s_0+2} \ln d\right) 2^{-n_{s_0+1} d/20} \leq
\mathcal{E}xp \left(- n_{s_0+1} d/30 \right)\leq \mathcal{E}xp \left(- n /2000 \right)
$$
and that the probability
$\mathbb{P}\left({\mathbb E}vent_{3} \cap {\mathbb E}vent _{nrm}\cap {\mathbb E}vent_{card}\right)$ is bounded by
$$
\mathcal{E}xp \left(2 n_{s_0+3} \ln d\right) \left(\sqrt{d}/(2C)\right)
^{-n_{s_0+1} d/20} \leq
\mathcal{E}xp \left(- n \ln d/10000\right),
$$
where ${\mathbb E}vent_{card}$ is the event
introduced in Lemma~\rightef{c:SJ} with $\mathcal{E}ll=2m$.
Combining all three cases we obtain that the desired bound holds
for all $x\in \mathcal{T}$ with
probability at most
$$
2\mathcal{E}xp \left(- n /2000 \right)
+ \mathbb{P}\left( {\mathbb E}vent _{norm}^c\right)
+ \mathbb{P}\left( {\mathbb E}vent_{card}^c\right).
$$
It remains to note that since $np$ is large, by Lemma~\rightef{bdd} (applied with $t=30$)
and by Lemma~\rightef{c:SJ},
$$
\mathbb{P}\left( {\mathbb E}vent _{nrm}^c\right)
+ \mathbb{P}\left( {\mathbb E}vent_{card}^c\right)\leq 4 e^{-225n p}+2\mathcal{E}xp(-n/500) \leq \mathcal{E}xp(-10pn) .
$$
\null
$\Box $ \\
\subsection{Proof of Theorem~\rightef{complement}}
\begin{proof}
Clearly, it is enough to show that
${\Upsilon}_n(r)\setminus ({\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho) \cup \mathcal{T}) \subset {\mathcal R}.$
Let $x\in{\Upsilon}_n(r)\setminus \mathcal{T}$ and set ${\sigma}ma:={\sigma}ma_x$. Note that $|x_{n_{s_0+2}}|\leq C_\tau \sqrt{d}$,
where $s_0$ was defined in \mathcal{E}qref{eq: l0s0 def}.
Denote $m_0=\lfloor n/\ln^2 d\rightfloor> 2{n_{s_0+2}}$.
Assume first that $x$ does not satisfy (\rightef{cond2}).
Then by Lemma~\rightef{a-c-cond2},
$x\in \mathcal{AC}(\rightho)$.
If $x_{m_0}^*\leq \ln^2 d$ then denoting
$k=m_0$, $A=[k, n]$, and using the definition of $\mathcal{AC}(\rightho)$, we observe
$$
\|x_{{\sigma}ma(A)}\|\geq\sqrt{(n-n_{s_0+3}-k)(1-\rightho)}\geq \sqrt{n/2},
$$
whence
$$
{\mathcal F}rac{\|x_{{\sigma}ma(A)}\|}{\|x_{{\sigma}ma(A)}\|_\infty} \geq {\mathcal F}rac{\sqrt{n/2}}{\ln^2 d}\geq {\mathcal F}rac{C_0}{\sqrt{p}}.
$$
On the other hand, $x_{m_{0}}^*\leq |x_{n_{s_0+2}}|\leq C_\tau \sqrt{d}$, hence $\|x_{{\sigma}ma(A)}\|\leq C_\tau \sqrt{dn}$.
This implies that $x\in {\mathcal R}_k^1\subset {\mathcal R}$.
Now, if $x_{m_0}^*> \ln^2 d$ then denoting
$k=n_{s_0+2}$, $A=[k, n]$, we get
$$
\|x_{{\sigma}ma(A)}\|^2\geq \sum _{i=n_{s_0+2}}^{m_{0}} (x_i^*)^2 \geq
(m_0/2)\ln^4 d \geq (n/4)\ln^2 d,
$$
whence
$$
{\mathcal F}rac{\|x_{{\sigma}ma(A)}\|}{\|x_{{\sigma}ma(A)}\|_\infty} \geq {\mathcal F}rac{\sqrt{n}\ln d}{2C_\tau \sqrt{d}}\geq {\mathcal F}rac{C_0}{\sqrt{p}}.
$$
As in the previous case we have $\|x_{{\sigma}ma(A)}\|\leq C_\tau \sqrt{dn}$, which
implies that $x\in {\mathcal R}_k^1\subset {\mathcal R}$.
Next we assume that $x$ does satisfy (\rightef{cond2}). Then, by the definition of the set ${\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)$
and our function ${\bf g}$,
$x$ does not satisfy the following condition:
\begin{align*}
\left\langlebel{cond1}
{\mathcal F}orall i\leq {\mathcal F}rac{1}{64p} : \, \, \, x^*_i\leq \mathcal{E}xp (\ln ^2 (2n/i))
\quad \mbox{ and } \quad
{\mathcal F}orall {\mathcal F}rac{1}{64p} < i\leq n : \, \, \, x^*_i\leq (2n/i)^{3/2}.
\mathcal{E}_{n-1}d{align*}
We fix the smallest value of $j\geq 1$
which breaks this condition and consider several cases.
Note that since $x\in {\Upsilon}_n(r)$, we must have $j\leq rn$.
\|\cdot\|indent
{\it Case 1. } $2 m_0 \leq j\leq rn$. In this case by the conditions and by
minimality of $j$, we have
$x_{m_0}^*\leq (2n/m_0)^{3/2}$ and $x_j^*\geq (2n/j)^{3/2}$. Take $k=m_0$ and $A=[k, n]$.
Then we have
$$
\|x_{{\sigma}ma(A)}\|\geq \sqrt{j - m_0+1 } \, x_{j}^* \geq
\sqrt{j/2} \, (2n/j)^{3/2}\geq \sqrt{rn/2} \, (2/r)^{3/2} = 2\sqrt{n} /r ,
$$
hence
$$
{\mathcal F}rac{\|x_{{\sigma}ma(A)}\|}{\|x_{{\sigma}ma(A)}\|_\infty} \geq \left({\mathcal F}rac{2}{r}\right)\,
{\mathcal F}rac{ \sqrt{n}}{(2n/m_0)^{3/2}}\geq\left({\mathcal F}rac{2}{r}\right) {\mathcal F}rac{ \sqrt{n}}{(2\ln d)^{3}}
\geq {\mathcal F}rac{C_0}{\sqrt{p}}.
$$
As above we have $\|x_{{\sigma}ma(A)}\|\leq C_\tau \sqrt{dn}$, which
implies that $x\in {\mathcal R}_k^2\subset {\mathcal R}$.
\|\cdot\|indent
{\it Case 2. } $16 C_0^2 n/d \leq j\leq 2 m_0$. Take $k=\lceileil j/2\rightceileil$ and $A=[k, n]$.
Then we have $x_{k}^*\leq (2n/k)^{3/2}\leq (4n/j)^{3/2}$, $x_j\geq (2n/j)^{3/2}$, and
$$
\|x_{{\sigma}ma(A)}\|\geq \sqrt{j - k +1 } \, x_{j}^* \geq
\sqrt{j/2} \, (2n/j)^{3/2} \geq (2/r)\, \sqrt{n}.
$$
Therefore,
$$
{\mathcal F}rac{\|x_{{\sigma}ma(A)}\|}{\|x_{{\sigma}ma(A)}\|_\infty} \geq \left({\mathcal F}rac{j}{2}\right)^{1/2}{\mathcal F}rac{ (2n/j)^{3/2}}{(4n/j)^{3/2}}\geq {\mathcal F}rac{C_0}{\sqrt{p}}.
$$
Since $x\|\cdot\|t\in \mathcal{T}$, we observe $x_k^*\leq C_\tau^2 d$, hence $\|x_{{\sigma}ma(A)}\|\leq C_\tau^2 d\sqrt{n}$ and
$x\in {\mathcal R}_k^2\subset {\mathcal R}$.
In the rest of the proof we show that we must necessarily have $j\geq 16 C_0^2 n/d$.
\|\cdot\|indent
{\it Case 3. } $n_{s_0+1}\leq j <C_1 n/d $, where $C_1=16 C_0^2$.
Using that $x\|\cdot\|t\in \mathcal{T}$,
in this case we have
$$
C_\tau^2 d \geq x_j^*\geq \left({\mathcal F}rac{2n}{j}\right)^{3/2}\geq \left({\mathcal F}rac{2 d}{C_1}\right)^{3/2},
$$
which is impossible for large enough $d$.
\|\cdot\|indent
{\it Case 4. } $n_{s_0}\leq j < n_{s_0+1}$.
Using that $x\|\cdot\|t\in \mathcal{T}$ and that
$n_{s_0+1}=\left\lfloor 1/(64p) \right\rightfloor=\left\lfloor n/(64d) \right\rightfloor$, in this case we have
$$
(6d) C_\tau^2 d \geq x_j^*\geq
\mathcal{E}xp (\ln ^2 (2n/j))\geq \mathcal{E}xp (\ln ^2 (2n/n_{s_0+1}))\geq
\mathcal{E}xp (\ln ^2 (128 d))
$$
which is impossible for large enough $d$.
\|\cdot\|indent
{\it Case 5. } $n_k\leq j < n_{k+1}$ for some $1\leq k\leq s_0-1$.
Recall that $n_k=30\mathcal{E}ll _0^{k-1}$ and recall also that if $s_0>1$
(as in this case) then $p\leq c\sqrt{n\ln n}$.
Using that $x\|\cdot\|t\in \mathcal{T}$, in this case we have
\begin{equation*}\left\langlebel{case5in}
(C_\tau^2 d) (6d)^{s_0-k+1} \geq x_j^*\geq \mathcal{E}xp (\ln ^2 (2n/j))\geq
\mathcal{E}xp (\ln ^2 (2n/(30\mathcal{E}ll_0^k))),
\mathcal{E}_{n-1}d{equation*}
hence
\begin{equation}\left\langlebel{case5in}
(C_\tau^2 d) (6d)^{s_0+1} \geq (6d)^{k} \mathcal{E}xp (\ln ^2 (2n/(30\mathcal{E}ll_0^k))).
\mathcal{E}_{n-1}d{equation}
Considering the function $f(k):= k \ln (6d) + \ln ^2 (2n/(30\mathcal{E}ll_0^k)$,
we observe that its derivative is linear in $k$, therefore $f$ attains its maximum
either at $k=1$ or at $k=s_0-1$. Thus, to show that (\rightef{case5in})
is impossible it is enough to consider $k=1, s_0-1$ only. Let $k=1$.
By (\rightef{kappain}),
$(6d)^{s_0}\leq (6d) \,1/(64p)^{\kappa}$, where $\kappa = {\mathcal F}rac{\ln (6d)}{\ln \mathcal{E}ll_0}$.
Therefore, the logarithm of the left hand side of (\rightef{case5in}) is
\begin{equation}\left\langlebel{boundln}
\ln ((C_\tau^2 d) (6d)^{s_0+1})\leq 4\ln d + {\mathcal F}rac{\ln (6d)}{ \ln \mathcal{E}ll_0}\, \ln (1/64p) .
\mathcal{E}_{n-1}d{equation}
On the other hand, $n/\mathcal{E}ll_0 \geq (4\ln (1/p) )/p$, therefore the
logarithm of the left hand side of (\rightef{case5in}) is larger than
$\ln ^2 (\ln (1/p)/(4p ))$. Thus,
it is enough to check that
$$
(1/2) \ln ^2 (\ln (1/p)/(4p )) \geq 4 \ln d
\quad \mbox{ and } \quad
(1/2) \ln ^2 (\ln (1/p)/(4p )) \ln \mathcal{E}ll_0 \geq \ln (6d)\, \ln (1/64p) .
$$
Both inequalities follows since $p\leq c\sqrt{n\ln n}$, $d=pn$, $d$ and $n$
are large enough, and since $\mathcal{E}ll_0\geq 25$.
Next assume that $k=s_0-1$. Note that in this case $\mathcal{E}ll_0^k\leq n/(64 d)$.
Thus, to disprove (\rightef{case5in}) it is enough to show that
$$
\ln ^2 (64d/15) \geq \ln (36 C_\tau^2 d^3),
$$
which clearly holds for large enough $d$.
\|\cdot\|indent
{\it Case 6. } $2\leq j< 30$. In this case we have
$$
(C_\tau^2 d) (6d)^{s_0+1} \geq x_j^* \geq \mathcal{E}xp (\ln ^2 (2n/j))\geq
\mathcal{E}xp (\ln ^2 (2n/30)),
$$
By (\rightef{boundln}) this implies
$$
4\ln d + {\mathcal F}rac{\ln (6d)}{ \ln \mathcal{E}ll_0}\, \ln (1/64p)\geq \ln ^2 (2n/30),
$$
which is impossible.
\|\cdot\|indent
{\it Case 7. } $j=1$. In this case we have
$
(C_\tau^2 d) (6d)^{s_0+2} \geq x_1^*\geq \mathcal{E}xp (\ln ^2 (2n))
$
and we proceed as in Case~6.
\mathcal{E}_{n-1}d{proof}
\section{Proof of the main theorem}\left\langlebel{s: main th}
In this section, we combine the results of Sections~\rightef{s: unstructured}, \rightef{steep:constant p},
and \rightef{s: steep}, as well as Subsection~\rightef{subs: lower b}
to prove the main theorem, Theorems~\rightef{th: main}, and the following improvement
for the case of constant $p$:
\begin{theor}\left\langlebel{const-p-th}
There exists an absolute positive constant $c$ with the following property.
Let $q\in (0, c)$ be a parameter (independent of $n$).
Then there exist $C_q$ and $n_q\geq 1$ (both depend only on $q$),
such that for every $n\geq n_q$ and every $p\in (q, c)$ a
Bernoulli($p$) $n\times n$ random matrix $M_n$ satisfies
$${\mathbb P}\big\{\mbox{$M_n$ is singular}\big\}=(2+o_n(1))n\,(1-p)^n,$$
and, moreover, for every $t>0$,
$$
{\mathbb P}\big\{s_{\min}(M_n)\leq C_q\, n^{-2.5}\, t \big\}\leq t+(1+o_n(1)){\mathbb P}\big\{\mbox{$M_n$ is singular}\big\}
=t+(2+o_n(1))n\,(1-p)^n.
$$
\mathcal{E}_{n-1}d{theor}
At this stage, the scheme of the proof to a large extent follows the approach of Rudelson and Vershynin
developed in \cite{RV}.
However, a crucial part of their argument --- ``invertibility via distance'' (see \cite[Lemma~3.5]{RV}) ---
will be reworked in order to keep sharp probability estimates
for the matrix singularity and to be able to bind this part of the argument with the previous
sections, where we essentially condition on row- and column-sums of our matrix.
We start by restating main results of Sections~\rightef{steep:constant p} and \rightef{s: steep}
using the vector class ${\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)$ defined by \mathcal{E}qref{eq: gnc def},
together with Lemma~\rightef{l:closure}.
\begin{cor}\left\langlebel{cor: steep}
There are universal constants $C\geq 1$,
${\rm dist}eltalta,\rightho\in(0,1)$ and $r\in(0,1)$ with the following property. Let $M_n$
be a random matrix satisfying \mathcal{E}qref{eq: assumptions} with $C$
and let the growth function ${\bf g}$ be given by \mathcal{E}qref{gfn-str}.
Then
\begin{equation}\left\langlebel{singval}
{\mathbb P}\Big\{\|M_n x\|\leq a_n^{-1}
\|x\|\,\,\, \mbox{ for some }\,\,\, x\|\cdot\|tin \bigcup\limits_{\left\langlembdabda\geq 0}\big(\left\langlembdabda\,{\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)\big)\Big\}
=(1+o_n(1))n\,(1-p)^n,
\mathcal{E}_{n-1}d{equation}
where $a_n={\mathcal F}rac{(pn)^2}{c(64p)^{\kappa}}\, \max\left(1, p^{1.5} n\right)$,
$\kappa = \kappa(p):= (\ln (6pn))/\ln \big\lfloor{\mathcal F}rac{pn}{4\ln (1/p)}\big\rightfloor.$
\mathcal{E}_{n-1}d{cor}
Further, Theorems~\rightef{t:steep},~\rightef{compl-1} and Lemma~\rightef{l:closure} are combined as follows.
\begin{cor}\left\langlebel{cor: steep2}
There are universal positive constants $c, C$ with the following property.
Let $q\in (0, c)$ be a parameter. Then there exist $n_0=n_0(q)\geq 1$,
$r=r(q), \rightho=\rightho (q)\in (0,1)$ such that for $n\geq n_0$, $p\in (q, c)$, ${\rm dist}eltalta = r/3$,
${\bf g}(t)=(2t)^{3/2}$,
the random Bernoulli($p$) $n\times n$ matrix $M_n$ satisfies (\rightef{singval}) with
$a_n=C \sqrt{n \ln(e/p)}$.
\mathcal{E}_{n-1}d{cor}
Below is our version of ``invertibility via distance,'' which deals with {\it pairs} of columns.
\begin{lemma}[Invertibility via distance]\left\langlebel{l: inv via dist}
Let $r,{\rm dist}eltalta,\rightho\in(0,1)$, and let ${\bf g}$ be a growth function.
Further, let $n\geq 6/r$ and let $A$ be an $n\times n$ random matrix. Then for any $t>0$ we have
\begin{align*}
{\mathbb P}\big\{&\|A x\|\leq t\,\|x\|\quad \mbox{ for some }\quad x\in {\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)\big\}\\
&\leq {\mathcal F}rac{2}{(rn)^2}\sum\limits_{i\neq j}{\mathbb P}\big\{{\rm dist}(H_i(A),{\bf C}_i(A))\leq t\, b_n
\quad \mbox{ and } \quad {\rm dist}(H_j(A),{\bf C}_j(A))\leq t\, b_n \big\},
\mathcal{E}_{n-1}d{align*}
where the sum is taken over all ordered pairs $(i,j)$ with $i\neq j$ and $b_n=\sum_{i=1}^n {\bf g}(i)$.
\mathcal{E}_{n-1}d{lemma}
\begin{proof}
For every $i\neq j$, denote by ${\bf 1}_{ij}$ the indicator of the event
$$
{\mathbb E}vent_{ij}:= \big\{{\rm dist}(H_i(A),{\bf C}_i(A))\leq t\, b_n\quad \mbox{ and } \quad {\rm dist}(H_j(A),{\bf C}_j(A))\leq t\, b_n\big\}.
$$
The condition
$$
\|A x\|\leq t\,\|x\|
$$
for some $x\in {\mathcal V}_n={\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)$
implies that for every $i\leq n$,
$$
|x_i|\, {\rm dist}(H_i(A),{\bf C}_i(A))\leq\|Ax\|\leq t\,b_n,
$$
where the last inequality follows from the definition of ${\mathcal V}_n$.
Since $x^*_{\lfloor rn\rightfloor }=1$, we get that everywhere on the event
$\{\|A x\|\leq t\,\|x\|\mbox{ for some }x\in {\mathcal V}_n\}$ there are at least
$\lfloor rn\rightfloor\,(\lfloor rn\rightfloor-1)\geq (rn)^2/2$ ordered pairs of indices $(i,j)$ such that for each pair
the event ${\mathbb E}vent_{ij}$ occurs.
Rewriting this assertion in terms of indicators, we observe
$$
\{\|A x\|\leq t\,\|x\|\mbox{ for some }x\in {\mathcal V}_n\}
\subset\Big\{\sum\limits_{i\neq j}{\bf 1}_{ij}\geq (rn)^2/2\Big\}.
$$
Applying Markov's inequality in order to estimate probability of the event on the right hand side, we obtain
the desired result.
\mathcal{E}_{n-1}d{proof}
\begin{proof}[Proof of Theorems~\rightef{th: main} and \rightef{const-p-th}]
The proofs of both theorems are almost the same, the only difference is that
Theorem~\rightef{th: main} uses Corollary~\rightef{cor: steep2} while Theorem~\rightef{th: main}
uses Corollary~\rightef{cor: steep}.
Let parameters ${\rm dist}eltalta,\rightho,r,{\bf g}, a_n$ be taken from Corollary~\rightef{cor: steep} or
from Corollary~\rightef{cor: steep2} correspondingly. We always
write ${\mathcal V}_n$ for ${\mathcal V}_n(r,{\bf g},{\rm dist}eltalta,\rightho)$.
Let $b_n=\sum_{i=1}^n {\bf g}(i)$.
Without loss of generality, we can assume that $n\geq 6/r$.
Fix $t\in (0, 1]$, and denote by ${\mathbb E}vent$ the complement of the event
$$
\Big\{\|M_n x\|\leq a_n^{-1}\|x\|\;
\mbox{ or }\;\|M_n^\top x\|\leq a_n^{-1} \|x\|\quad
\mbox{ for some }\quad x\|\cdot\|tin\bigcup\limits_{\left\langlembdabda\geq 0}\big(\left\langlembdabda\,{\mathcal V}_n\big)\Big\}.
$$
For $i=1,2$ denote
$$
{\mathbb E}vent_i:= \big\{{\rm dist}(H_i(M_n),{\bf C}_i(M_n))\leq a_n^{-1} \, t\big\}.
$$
Applying Corollary~\rightef{cor: steep} (or Corollary~\rightef{cor: steep2}), Lemma~\rightef{l: inv via dist} and
the invariance of the conditional distribution of $M_n$ given ${\mathbb E}vent$
under permutation of columns, we obtain
\begin{align*}
{\mathbb P}&\big\{s_{\min}(M_n)\leq (a_n b_n)^{-1} t\big\}
\\&\leq (2+o_n(1))n\,(1-p)^n
+{\mathbb P}\big(\big\{\|M_n x\|
\leq (a_n b_n)^{-1}\, t\|x\|\quad \mbox{ for some }\quad x\in {\mathcal V}_n\big\}\cap {\mathbb E}vent\big)
\\&\leq (2+o_n(1))n\,(1-p)^n+ {\mathcal F}rac{2}{r^2}\, {\mathbb P}\big({\mathbb E}vent\cap {\mathbb E}vent_1\cap {\mathbb E}vent_2\big) .
\mathcal{E}_{n-1}d{align*}
At the next step, we consider events
$$
\Omega_i:= \big\{|{\rm supp\, }{\bf C}_i(M_n)|\in [pn/8, 8pn]\big\},\, i=1,2,\quad \mbox{ and } \quad
\Omega:= \Omega_1\cup \Omega_2.
$$
Since columns of $M$ are independent and consist of i.i.d.
Bernoulli($p$) variables, applying Lemma~\rightef{bennett}, we observe
$$
{\mathbb P}\big(\Omega^c\big)= {\mathbb P}\big(\Omega_1^c\big){\mathbb P}\big(\Omega_2^c\big)
\leq (1-p)^{n}.
$$
Therefore, in view of equidistribution of the first two columns, we get
\begin{align*}
{\mathbb P}&\big({\mathbb E}vent\cap {\mathbb E}vent_1\cap {\mathbb E}vent_2\big)
\leq (1-p)^n+{\mathbb P}\big({\mathbb E}vent\cap {\mathbb E}vent_1\cap {\mathbb E}vent_2\cap \Omega \big)
\leq (1-p)^n+2{\mathbb P}\big({\mathbb E}vent\cap {\mathbb E}vent_1\cap \Omega_1\big).
\mathcal{E}_{n-1}d{align*}
Denote by ${\bf Y}$ a random unit vector orthogonal to (and measurable with respect to) $H_1(M_n)$.
Note that on the event ${\mathbb E}vent_1$
the vector ${\bf Y}$ satisfies
$$
|\left\langlengle{\bf Y},{\bf C}_1(M_n)\rightangle|= \|M_n^\top {\bf Y}\|\leq a_n^{-1}\, t \, \|{\bf Y}\|,
$$
which implies that on the event ${\mathbb E}vent\cap {\mathbb E}vent_1$ we also have
${\bf Y}^*_{\lfloor r n\rightfloor}\neq 0$, and
${\bf Z}:={\bf Y}/{\bf Y}^*_{\lfloor r n\rightfloor}\in {\mathcal V}_n$.
By the definition of ${\mathcal V}_n$, we have $\|{\bf Z}\|\leq b_n$,
therefore,
\begin{align*}
&P_0:={\mathbb P}\big({\mathbb E}vent\cap {\mathbb E}vent_1\cap \Omega_1\big)\leq {\mathbb P}\big(\Omega_1 \cap \big\{\mbox{There is $Z\in H_1(M_n)^\mathbb{P}erp\cap {\mathcal V}_n$}:\;
|\left\langlengle Z,{\bf C}_1(M_n)\rightangle|\leq a_n^{-1}\, b_n\, t\big\}\big).
\mathcal{E}_{n-1}d{align*}
On the other hand, applying Theorem~\rightef{th: gradual} with $R=2$, we get that
for some constants $K_1\geq 1$ and $K_2\geq 4$, with probability at least $1-\mathcal{E}xp(-2pn)$,
\begin{align*}
&H_1(M_n)^\mathbb{P}erp\cap {\mathcal V}_n
\subset\big\{
x\in{\Upsilon}_n(r):\;{\bf UD}_n(x,m,K_1,K_2)\geq \mathcal{E}xp(2pn)\,\,\, \mbox{ for any }\,\,\, m\in [pn/8, 8pn]
\big\}.
\mathcal{E}_{n-1}d{align*}
Combining the last two assertions and applying Theorem~\rightef{p: cf est}, we observe
\begin{align*}
P_0\leq \mathcal{E}xp(-2pn)+{\mathbb P}\big(&\Omega_1\cap\big\{\mbox{There is $Z\in H_1(M_n)^\mathbb{P}erp\cap {\mathcal V}_n$}:\;
|\left\langlengle Z,{\bf C}_1(M_n)\rightangle|\leq a_n^{-1}\, b_n\, t,\mbox{ and }\\
&{\bf UD}_n(Z,m,K_1,K_2)\geq \mathcal{E}xp(2pn)\mbox{ for any }m\in [pn/8, 8pn]\big\}\big)\\
&\hspace{-3.6cm}\leq
\mathcal{E}xp(-2pn)+\sup\limits_{\substack{m\in [pn/8, 8pn],\,y\in {\Upsilon}_n(r),\\
{\bf UD}_n(y,m,K_1,K_2)\geq \mathcal{E}xp(2pn)}}{\mathbb P}\big\{|\left\langlengle y,{\bf C}_1(M_n)\rightangle|
\leq a_n^{-1}b_n\, t\,\,\big|\,\, |{\rm supp\, }{\bf C}_1(M_n)|=m\big\}
\\&\hspace{-3.6cm}\leq
(1+C_{\text{\tiny\rightef{p: cf est}}})\mathcal{E}xp(-2pn)+ {\mathcal F}rac{C_{\text{\tiny\rightef{p: cf est}}}b_n}{ a_n\sqrt{pn/8}} \, t.
\mathcal{E}_{n-1}d{align*}
Thus
$$
{\mathbb P}\big\{s_{\min}(M_n)\leq (a_n b_n)^{-1} t\big\}
\leq (2+o_n(1))n\,(1-p)^n + {\mathcal F}rac{8 C_{\text{\tiny\rightef{p: cf est}}}b_n}{ r^2 \, a_n\sqrt{pn}} \, t.
$$
By rescaling $t$ we obtain
$$
{\mathbb P}\Big\{s_{\min}(M_n)\leq {\mathcal F}rac{r^2 \,\sqrt{pn}}{(8 C_{\text{\tiny\rightef{p: cf est}}}b_n^2)}\, t\Big\}
\leq (2+o_n(1))n\,(1-p)^n+t,\quad 0\leq t\leq {\mathcal F}rac{8 C_{\text{\tiny\rightef{p: cf est}}}b_n}{ r^2 \, a_n\sqrt{pn}}.
$$
In the case of constant $p$ (applying Corollary~\rightef{cor: steep2}) we have $a_n=C \sqrt{n \ln(e/p)}$ and $b_n\leq 2\sqrt{3}n^{3/2}$,
and we get the small ball probability estimate of Theorem~\rightef{const-p-th}.
In the case of ``general'' $p$ (with the application of Corollary~\rightef{cor: steep}) we have
$a_n={\mathcal F}rac{(pn)^2}{c(64p)^{\kappa}}\, \max\left(1, p^{1.5} n\right)$ and $b_n\leq \mathcal{E}xp(1.5\ln^2(2n))$.
Therefore, $${\mathcal F}rac{r^2 \,\sqrt{pn}}{(8 C_{\text{\tiny\rightef{p: cf est}}}b_n^2)}\geq \mathcal{E}xp(-3\ln^2(2n))$$ for large enough $n$,
and the $s_{\min}$ estimate follows.
In both cases the upper bound on $t$, ${\mathcal F}rac{8 C_{\text{\tiny\rightef{p: cf est}}}b_n}{ r^2 \, a_n\sqrt{pn}}$,
is greater than $1$, so we may omit it.
Finally, applying the argument of Subsection~\rightef{subs: lower b}, we get the matching lower bound for the singularity probability.
This completes the proof.
\mathcal{E}_{n-1}d{proof}
\section{Open questions}\left\langlebel{s: further}
The result of this paper leaves open the problem of estimating the singularity probability
for Bernoulli matrices in two regimes: when $n p_n$ is logarithmic in $n$ and when $p_n$ is larger
than the constant $C^{-1}$ from Theorem~\rightef{th: main}.
For the first regime, we recall that the singularity probability of $M_n$, with $n p_n$ in a (small) neighborhood of $\ln n$,
was determined up to the $1+o_n(1)$ multiple in the work of Basak--Rudelson \cite{BasRud-sharp}.
Definitely, it would be of interest to bridge that result and the main theorem of this paper.
\begin{Problem}[A brigde: Theorem~\rightef{th: main} to Basak--Rudelson]
Let $p_n$ satisfy
$$1\leq \liminf np_n/\ln n\leq \limsup np_n/\ln n<\infty,$$
and for each $n$ let $M_n$
be the $n\times n$ matrix with i.i.d.\ Bernoulli($p_n$) entries.
Show that
$$
{\mathbb P}\big\{
M_n\mbox{ is singular}
\big\}=(1+o_n(1)){\mathbb P}\big\{M_n\mbox{ has a zero row or a zero column}\big\}.
$$
\mathcal{E}_{n-1}d{Problem}
Note that the main technical result for unstructured (gradual non-constant) vectors,
Theorem~\rightef{th: gradual} proved in Section~\rightef{s: unstructured},
remains valid for these values of $p_n$. It may be therefore expected that the above problem can be positively resolved
by finding an efficient treatment for the structured vectors (the complement of gradual non-constant vectors),
which would replace (or augment) the argument from Section~\rightef{s: steep}.
On the contrary, the second problem --- singularity of random Bernoulli matrices with large values of $p_n$
--- seem to require
essential new arguments for working with the unstructured vectors as the
basic idea of Section~\rightef{s: unstructured} --- gaining on anti-concentration estimates by grouping
together several components of a random vector --- does not seem to be applicable in this regime.
\begin{Problem}[Optimal singularity probability for dense Bernoulli matrices below the $1/2$ threshold]
Let the sequence $p_n$ satisfy
$$0< \liminf p_n\leq \limsup p_n<1/2.$$ Show that
\begin{align*}
{\mathbb P}\big\{
M_n\mbox{ is singular}
\big\}
&=(1+o_n(1)){\mathbb P}\big\{M_n\mbox{ has a zero row or a zero column}\big\}
=(2+o_n(1))n\,(1-p_n)^n.
\mathcal{E}_{n-1}d{align*}
\mathcal{E}_{n-1}d{Problem}
\subsection*{Acknowledgments}
K.T.\ was partially supported by the Sloan Research Fellowship.
\|\cdot\|cite{*}
\begin{thebibliography}{99}
\bibitem{BVH}
A.S. Bandeira, R. van Handel,
{ Sharp nonasymptotic bounds on the norm of random matrices with independent entries.}
Ann. Probab. 44 (2016), 2479--2506.
\bibitem{BCZ}
{
A. Basak,\ N. Cook and\ O. Zeitouni, Circular law for the sum of random permutation matrices, Electronic Journal of Probability, {\bf 23} (2018), Paper No. 33, 51 pp.
}
\bibitem{BasRud}
{
A. Basak\ and\ M. Rudelson, Invertibility of sparse non-Hermitian matrices, Adv. Math. {\bf 310} (2017), 426--483. MR3620692
}
\bibitem{BasRud circ}
{
A. Basak\ and\ M. Rudelson, {The circular law for sparse non-Hermitian matrices}, arXiv:1707.03675
}
\bibitem{BasRud-sharp}
{
A. Basak\ and\ M. Rudelson,
Sharp transition of the invertibility of the adjacency matrices of random graphs,
arXiv:1809.08454
}
\bibitem{BLM}
{
S. Boucheron, G. Lugosi, P. Massart,
Concentration inequalities. A nonasymptotic theory of independence.
With a foreword by Michel Ledoux. Oxford University Press, Oxford, 2013.
}
\bibitem{BVW}
{
J. Bourgain, V. H. Vu\ and\ P. M. Wood, On the singularity probability of discrete random matrices,
J. Funct. Anal. {\bf 258} (2010), no.~2, 559--603. MR2557947
}
\bibitem{lama}
{D. Chafa{\"i}, O. Gu{\'e}don, G. Lecu{\'e}, A. Pajor,
Interactions between compressed sensing random matrices and high
dimensional geometry,
Panoramas et Synth{\`e}ses [Panoramas and Syntheses], {\bf 37}.
Soc. Math. de France, Paris, 2012.
}
\bibitem{Cook adjacency}
{
N. A. Cook, On the singularity of adjacency matrices for random regular digraphs,
Probab. Theory Related Fields {\bf 167} (2017), no.~1-2, 143--200. MR3602844
}
\bibitem{Cook circular}
{
N. Cook, The circular law for random regular digraphs,
Ann. Inst. Henri Poincare Probab. Stat., to appear,
arXiv:1703.05839
}
\bibitem{DevLug}
{
L. Devroye; G. Lugosi,
Combinatorial methods in density estimation. Springer Series in Statistics. Springer-Verlag, New York, 2001.
}
\bibitem{Erdos}
{
P. Erd\"{o}s, On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc. {\bf 51} (1945), 898--902. MR0014608
}
\bibitem{Esseen-type}
{
C. G. Esseen, On the Kolmogorov-Rogozin inequality for the concentration function,
Z. Wahrsch. Verw. Gebiete {\bf 5} (1966), 210--216. MR0205297
}
\bibitem{GT}
{
F. G\"otze\ and\ A. Tikhomirov, The circular law for random matrices, Ann. Probab. {\bf 38} (2010), no.~4, 1444--1491. MR2663633
}
\bibitem{Huang}
{
J. Huang, Invertibility of adjacency matrices for random $d$-regular graphs, preprint,
arXiv:1807.06465, 2018.
}
\bibitem{KKS95}
{
J. Kahn, J. Koml\' os\ and\ E. Szemer\' edi,
On the probability that a random $\mathbb{P}m 1$-matrix is singular, J. Amer. Math. Soc. {\bf 8} (1995), no.~1, 223--240. MR1260107
}
\bibitem{kesten}
{
H. Kesten, A sharper form of the Doeblin-L\'evy-Kolmogorov-Rogozin inequality for concentration functions,
Math. Scand. {\bf 25} (1969), 133--144. MR0258095
}
\bibitem{Komlos}
{
J. Koml\'{o}s, On the determinant of $(0,\,1)$ matrices, Studia Sci. Math. Hungar {\bf 2} (1967), 7--21. MR0221962
}
\bibitem{LSY}
{
B. Landon, P. Sosoe, H. Yau, Fixed energy universality of Dyson Brownian motion, Adv. Math. {\bf 346} (2019),
1137--1332.
}
\bibitem{Ledoux}
{
M. Ledoux, The concentration of measure phenomenon. Mathematical Surveys and Monographs,
{\bf 89}. American Mathematical Society, Providence, RI, 2001.
}
\bibitem{LO lemma}
{
J. E. Littlewood\ and\ A. C. Offord, On the number of real roots of a random algebraic equation. III,
Rec. Math. [Mat. Sbornik] N.S. {\bf 12(54)} (1943), 277--286. MR0009656
}
\bibitem{LLTTY-cras}
{
A. E. Litvak, A.~Lytova, K.~Tikhomirov, N.~Tomczak-Jaegermann\ and\ P.~Youssef, Anti-concentration property for random digraphs and invertibility of their adjacency matrices, C. R. Math. Acad. Sci. Paris {\bf 354} (2016), no.~2, 121--124. MR3456885
}
\bibitem{LLTTY:15}
{
A. E. Litvak, A.~Lytova, K.~Tikhomirov, N.~Tomczak-Jaegermann\ and\ P.~Youssef, Adjacency matrices of random digraphs: singularity and anti-concentration, J. Math. Anal. Appl. {\bf 445} (2017), no.~2, 1447--1491. MR3545253
}
\bibitem{LLTTY first part}
{
A. E. Litvak, A.~Lytova, K.~Tikhomirov, N.~Tomczak-Jaegermann\ and\ P.~Youssef,
{The smallest singular value of a shifted $d$-regular random square matrix},
Probab. Theory Related Fields, {\bf 173} (2019), 1301--1347.
}
\bibitem{LLTTY third part}
{
A. E. Litvak, A.~Lytova, K.~Tikhomirov, N.~Tomczak-Jaegermann\ and\ P.~Youssef,
{The circular law for sparse random regular digraphs}, J. European Math. Soc., to appear.
}
\bibitem{LLTTY rank n-1}
{
A. E. Litvak, A.~Lytova, K.~Tikhomirov, N.~Tomczak-Jaegermann\ and\ P.~Youssef,
The rank of random regular digraphs of constant degree, J. of Complexity, {\bf 48} (2018), 103--110.
}
\bibitem{LLTTY-TAMS}
{
A. E. Litvak, A.~Lytova, K.~Tikhomirov, N.~Tomczak-Jaegermann\ and\ P.~Youssef,
Structure of eigenvectors of random regular digraphs, Trans. Amer. Math. Soc., {\bf 371} (2019), 8097--8172.
}
\bibitem{LPRT}
{
A. E.~Litvak, A.~Pajor, M.~Rudelson\ and\ N.~Tomczak-Jaegermann,
Smallest singular value of random matrices and geometry of random polytopes, Adv. Math. {\bf 195} (2005), no.~2, 491--523. MR2146352
}
\bibitem{LiRi}
{
A. E. Litvak\ and\ O. Rivasplata, Smallest singular value of sparse random matrices, Studia Math. {\bf 212} (2012), no.~3, 195--218. MR3009072
}
\bibitem{Livshyts}
{
G. V. Livshyts, The smallest singular value of heavy-tailed not necessarily i.i.d.\ random matrices via random rounding, arXiv:1811.07038.
}
\bibitem{LTV}
{
G. V. Livshyts, K. Tikhomirov, R. Vershynin,
The smallest singular value of inhomogeneous square random matrices, arXiv:1909.04219
}
\bibitem{LyTi}
{
A. Lytova, K. Tikhomirov,
On delocalization of eigenvectors of random non-Hermitian matrices, Probab. Theor. Rel. Fields, to appear.
}
\bibitem{LMN}
{
K. Luh, S. Meehan, H.H.\ Nguyen,
Some new results in random matrices over finite fields, arXiv:1907.02575
}
\bibitem{LOR}
{
K. Luh, S. O'Rourke, Eigenvector Delocalization for Non-Hermitian Random Matrices and Applications, arXiv:1810.00489
}
\bibitem{Mez}
A. M\'esz\'aros, The distribution of sandpile groups of random regular graphs, preprint, arXiv: 1806.03736, 2018.
\bibitem{NW}
H.H. Nguyen and M.M. Wood, Cokernels of adjacency matrices of random r-regular graphs,
preprint, arXiv: 1806.10068, 2018.
\bibitem{RT}
{
E. Rebrova, K. Tikhomirov, Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries,
Israel J. Math., to appear.
}
\bibitem{Rog}
{
B. A. Rogozin, On the increase of dispersion of sums of independent random variables, Teor. Verojatnost. i Primenen {\bf 6} (1961), 106--108. MR0131894
}
\bibitem{R-ann}
{
M. Rudelson, Invertibility of random matrices: norm of the inverse,
Ann. of Math. {\bf 168} (2008), 575--600.
}
\bibitem{Rud13}
{
M. Rudelson, Recent developments in non-asymptotic theory of random matrices, in {\it Modern aspects of random matrix theory}, 83--120, Proc. Sympos. Appl. Math., 72, Amer. Math. Soc., Providence, RI. MR3288229
}
\bibitem{RV}
{
M. Rudelson\ and\ R. Vershynin, The Littlewood--Offord problem and invertibility of random matrices, Adv. Math. {\bf 218} (2008), no.~2, 600--633. MR2407948
}
\bibitem{RV09}
{
M. Rudelson\ and\ R. Vershynin, Smallest singular value of a random rectangular matrix,
Comm. Pure Appl. Math. {\bf 62} (2009), no.~12, 1707--1739. MR2569075
}
\bibitem{RV no gaps}
{
M. Rudelson\ and\ R. Vershynin, No-gaps delocalization for general random matrices, Geom. Funct. Anal. {\bf 26} (2016), no.~6, 1716--1776. MR3579707
}
\bibitem{TV disc1}
{
T. Tao\ and\ V. Vu, On random $\mathbb{P}m1$ matrices: singularity and determinant,
Random Structures Algorithms {\bf 28} (2006), no.~1, 1--23. MR2187480
}
\bibitem{TV bernoulli}
{
T. Tao\ and\ V. Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. {\bf 20} (2007), no.~3, 603--628. MR2291914
}
\bibitem{TV sparse}
{
T. Tao\ and\ V. Vu, Random matrices: the circular law, Commun. Contemp. Math. {\bf 10} (2008), 261--307. MR2409368
}
\bibitem{TV ann math}
{
T. Tao\ and\ V. H. Vu, Inverse Littlewood-Offord theorems and the condition number of random discrete matrices,
Ann. of Math. {\bf 169} (2009), 595--632. MR2480613
}
\bibitem{KT-last}
{
K.~Tikhomirov, Singularity of random Bernoulli matrices, Annals of Math., to appear. arXiv:1812.09016
}
\mathcal{E}_{n-1}d{thebibliography}
a_1ddress
\mathcal{E}_{n-1}d{document} | math | 288,141 |
\begin{document}
\title{AABC: approximate approximate Bayesian computation when simulating a large number of data sets is computationally infeasible}
\author{Erkan O. Buzbas \\
Department of Biology \\ Stanford University, Stanford, CA 94305-5020 USA \\
and\\
Department of Statistical Science \\ University of Idaho, Moscow, ID 84844-1104 USA\\
email: \texttt{[email protected]}\\
and\\
Noah A. Rosenberg \\
Department of Biology \\ Stanford University, Stanford, CA 94305-5020 USA \\
email: \texttt{[email protected]}
}
\maketitle
\begin{center}
\textbf{Abstract}
\end{center}
Approximate Bayesian computation (ABC) methods perform inference on model-specific parameters of mechanistically motivated parametric statistical models
when evaluating likelihoods is difficult.
Central to the success of ABC methods is computationally inexpensive simulation of data sets from the parametric model of interest.
However, when simulating data sets from a model is so computationally expensive that the posterior distribution of parameters cannot be adequately sampled by ABC, inference is not straightforward.
We present ``approximate approximate Bayesian computation'' (AABC), a class of methods that extends simulation-based inference by ABC to models in which simulating data is expensive.
In AABC, we first simulate a \emph{limited number} of data sets that is computationally feasible to simulate from the parametric model.
We use these data sets as fixed background information to inform a non-mechanistic statistical model that approximates the correct parametric model and enables efficient simulation of a large number of data sets by Bayesian resampling methods.
We show that under mild assumptions, the posterior distribution obtained by AABC converges to the posterior distribution obtained by ABC, as the number of data sets simulated from the parametric model and the sample size of the observed data set increase simultaneously.
We illustrate the performance of AABC on a population-genetic model of natural selection, as well as on a model of the admixture history of hybrid populations.
\vspace*{.3in}
\noindent\textsc{Keywords}: {Approximate Bayesian computation, likelihood-free methods, nonparametrics, posterior distribution}
\section{Introduction}\label{Intro}
Stochastic processes motivated by mechanistic considerations enable investigators to capture salient phenomena in modeling natural systems.
Statistical models resulting from these stochastic processes are often parametric, and estimating model-specific parameters---which often have a natural interpretation---is a major aim of data analysis.
Contemporary mechanistic models tend to involve complex stochastic processes, however, and parametric statistical models resulting from these processes lead to computationally intractable likelihood functions.
When likelihood functions are computationally intractable, likelihood-based inference is a challenging problem that has received considerable attention
in the literature \citep{RobertCasella2004, Liu2008}.
\par
When statistical models are known only at the level of the stochastic mechanism generating the data---such as in implicit statistical models \citep{DiggleGratton1984}---explicit evaluation of likelihoods might be impossible.
In these models, standard computational methods that require evaluation of likelihoods up to a proportionality constant (e.g., rejection methods) cannot be used to sample distributions of interest.
However, data sets simulated from the model under a range of parameter values
can be used to assess parameter likelihoods without explicit evaluation \citep{Rubin1984}.
Approximate Bayesian computation (ABC) methods \citep{Tavareetal1997,Beaumontetal2002, Marjorametal2003}
implement this idea in a Bayesian context to sample an {\em approximate} posterior distribution of the parameters.
Intuitively, parameter values producing simulated data sets similar to the observed data set
arise in approximate proportion to their likelihood, and hence, when weighted by prior probabilities, to their posterior probabilities.
\par
\subsection{The ABC literature}
ABC methods have been based on rejection algorithms
\citep{Tavareetal1997, Beaumontetal2002, BlumFrancois2010},
Markov chain Monte Carlo \citep{Beaumont2003, Marjorametal2003, Bortotetal2007, Wegmannetal2009}, and
sequential Monte Carlo \citep{Sissonetal2007, Sissonetal2009, Beaumontetal2009, Tonietal2009}.
Model selection using ABC \citep{Pritchardetal1999, Fagundesetal2007, Grelaudetal2009,
BlumJakobsson2010, Robertetal2011}, the choice of summary statistics when the likelihood is based on
summary statistics instead of the full data \citep{JoyceMarjoram2008, Wegmannetal2009, NunesBalding2010, FearnheadPrangle2012},
and the equivalence of posterior distributions targeted in different ABC methods \citep{Wilkinson2008, Sissonetal2010} have also
been investigated.
\par
ABC methods have had a considerable effect on model-based inference in disciplines that rely on genetic data, particularly data
shaped by diverse evolutionary, demographic, and environmental forces.
Example applications have included problems in the demographic history of populations \citep{Pritchardetal1999, Francoisetal2008, Verduetal2009, BlumJakobsson2010} and species
\citep{Estoupetal2004, PlagnolTavare2004, BecquetPrzeworski2007, Fagundesetal2007, Wilkinsonetal2010},
as well as problems in the evolution of cancer cell lineages \citep{Tavare2005, Siegmundetal2008} and the evolution of protein networks \citep{Ratmannetal2009}.
Other applications outside of genetics have included inference on the physics of stereological extremes \citep{Bortotetal2007},
the ecology of tropical forests \citep{JabotChave2009}, dynamical systems in biology
\citep{Tonietal2009}, and small-world network disease models \citep{Walkeretal2010}.
ABC methods have been reviewed by \citet{MarjoramTavare2006}, \citet{Cornuetetal2008}, \citet{Beaumontetal2009}, \citet{Beaumont2010},
\citet{Csilleryetal2010}, and \citet{Marinetal2011}.
\par
\subsection{A limitation of ABC methods}
An informal categorization of the information available about the likelihood function is helpful
to illustrate the class of models in which ABC methods are most useful.
First, exact inference on the posterior distribution of the parameters is possible only if the likelihood function is analytically available.
Second, if the likelihood function is not analytically available but can be evaluated up to a constant given a parameter value, then standard computational methods such as rejection algorithms can sample the posterior distribution.
In this case, inference is exact up to a Monte Carlo error due to sampling from the posterior.
Third, if the likelihood function cannot be evaluated, but data sets can feasibly be simulated from the model, then ABC methods sample the posterior distribution using approximations on the {\em data space} in addition to a Monte Carlo error due to sampling.
\par
Although ABC methods sample the posterior distribution of parameters without evaluating the likelihood function, they are computationally intensive.
Adequately sampling a posterior distribution of a parameter by ABC requires many random realizations from the prior distribution
of the parameter and the sampling distribution of the data.
Simulating from the prior is straightforward, but the computational cost of simulating a data set from the mechanistic model increases quickly with the complexity and number of stochastic processes involved.
Henceforth, we refer to statistical models in which not only evaluating the likelihoods is difficult but also simulating a large number of data sets is computationally infeasible as {\em limited-generative} models.
When a model is limited-generative and only a small number of data sets can be simulated from the model, likelihoods cannot be assessed using ABC and hence, the posterior distribution of parameters cannot be adequately sampled.
\par
\subsection{Our contribution}
In this article, we introduce {\em approximate} approximate Bayesian computation (AABC), a class of methods that perform inference on model-specific parameters of limited-generative models when standard ABC methods are computationally infeasible to apply.
In AABC, the idea of assessing the likelihoods approximately using simulated data sets is taken one step further than in ABC.
AABC methods make approximations on the {\em parameter space} and the {\em model space} in addition to standard ABC approximations on the data space.
In conjunction with Bayesian resampling methods, these approximations help us overcome the computational intractability associated with simulating data from a limited-generative model (Figure \ref{fig:1}).
\par
Our key innovation is to condition on a limited number of data sets that can be feasibly simulated from the limited-generative model and to employ a non-mechanistic statistical model to simulate a large number of data sets.
We set up the non-mechanistic model based on empirical distributions of the limited number of data sets simulated from the mechanistic model.
Since the data values from the limited number of simulated data sets are used to construct new random data sets by resampling methods, it is computationally inexpensive to simulate a large number of data sets in AABC.
The AABC approach allows a researcher to allocate a fixed computer time to simulating a limited number of data sets from the limited-generative model,
thus making otherwise challenging likelihood-based inference attainable.
\par
Intuitively, the information conditioned upon by the non-mechanistic model increases with the number of data sets simulated from the mechanistic model, and the expected accuracy of inference obtained by AABC methods increases.
We formalize this intuition by showing that the posterior distribution of parameters obtained by AABC converges to the corresponding posterior distribution obtained by standard ABC, as the sample size of the observed data set and the number of data sets simulated from the limited-generative model increase simultaneously.
\par
\begin{figure}\label{fig:1}
\end{figure}
\par
AABC methods utilize the established machinery of ABC methods in sampling the posterior distribution of the parameters.
Therefore, standard approximations on the data space involved in an ABC method---which facilitate the sampling of the posterior distribution---apply to AABC methods as well.
We now briefly review these approximations in the context of ABC by rejection algorithms.
\section{Review of ABC by rejection algorithms}\label{sec:ABCreview}
To more formally set up the class of problems in which ABC methods are useful, we assume that a parametric model generates observations conditional on parameter $\theta \in \Theta \equiv \mathbf{R}^p,\;p\geq1.$
We let $P_{\theta}$ be the sampling distribution of a data set of $n$ observations independent and identically distributed (IID) from this model.
We denote a random data set by ${\bf x}=(x_1,x_2,...,x_n) \in \mathcal{X},$ where $\mathcal{X}$ is the space in which the data set sits, and the observed data set by ${\bf x}obs.$
In the genetics context, a data point $x_i$
might be a vector denoting the allelic types of a genetic locus at genomic position $i$ in a group of individuals;
the data matrix ${\bf x}$ might then contain genotypes from these individuals in a sample of $n$ independent genetic loci.
\par
Suppose that $P_{\theta}$ is available to the extent that the likelihood function $p({\bf x}obs|\theta)$ can be evaluated up to a constant whose value does not depend on the parameters.
Given a prior distribution $\pi(\theta)$ on parameter $\theta,$ the posterior
distribution of $\theta$ given the observed data ${\bf x}obs$ under the model $P_{\theta}$ is
$\pi(\theta|{\bf x}obs, P_{\theta}).$
Then $\pi(\theta|{\bf x}obs, P_{\theta})$ can be sampled by standard rejection sampling from $p({\bf x}obs|\theta)\pi(\theta),$ a quantity that is proportional to
$\pi(\theta|{\bf x}obs, P_{\theta})$ by Bayes' Theorem.
In principle, sampling $\pi(\theta|{\bf x}obs, P_{\theta})$ without evaluating the
likelihood function $p({\bf x}obs|\theta)$ is possible, if simulating the data from the model $P_{\theta}$ is feasible.
An early example due to Tavar\'e {\em et al.} (1997) samples $\pi(\theta|{\bf x}obs,P_{\theta})$
by accepting a value $\theta_i$ simulated from the prior $\pi(\theta)$
only if the data set ${\bf x}_i$ simulated from $P_{\theta}sub_{\theta_i}$ satisfies ${\bf x}_i={\bf x}obs.$
By standard rejection algorithm arguments, the $\theta_i$ sampled in this fashion are from the correct posterior distribution.
However, the acceptance condition ${\bf x}_i={\bf x}obs$ is rarely satisfied with high-dimensional data.
A first approximation in ABC methods is dimension reduction by substituting the data set ${\bf x}$ with a low-dimensional set of summary statistics $\textrm{\boldmath$s$}.$
The observed data ${\bf x}obs$ and the simulated data ${\bf x}_i$ are substituted by
$\textrm{\boldmath$s$}obs$ and $\textrm{\boldmath$s$}_i,$ calculated from their respective data sets.
This is equivalent to substituting the likelihood function of the data $p({\bf x}|\theta)$ with the likelihood function of the summary statistics $p(\textrm{\boldmath$s$}|\theta).$
Since ABC is most useful in statistical models that do not admit sufficient statistics, dimension
reduction to summary statistics often entails information loss about the parameters.
The choice of summary statistics minimizing this information loss
is an active research area \citep{JoyceMarjoram2008, Wegmannetal2009, Robertetal2011, Aeschbacheretal2012, FearnheadPrangle2012}.
\par
When the data are substituted with summary statistics, the acceptance condition ${\bf x}_i={\bf x}obs$ is substituted by $\textrm{\boldmath$s$}_i=\textrm{\boldmath$s$}obs,$ but exact equality may still be too stringent a condition to be satisfied with simulated data.
A second approximation in ABC is to relax the exact acceptance condition with a tolerance acceptance condition.
For example, \citet{Pritchardetal1999} used the Euclidean distance $||\cdot||$ and a small tuning parameter $\epsilon$
to accept a value $\theta_i$ from an approximate posterior distribution if the data set ${\bf x}_i$ simulated from $P_{\theta}sub_{\theta_i}$ produced $\textrm{\boldmath$s$}_i$ satisfying
\begin{equation}\label{eq:euclideandistance}
||\textrm{\boldmath$s$}_i-\textrm{\boldmath$s$}obs||=\left[\sum_{j=1}^{k}(s_{ij}-s_{oj})^2\right]^{1/2}\leq\epsilon,
\end{equation}
where $\textrm{\boldmath$s$}$ is a $k$-dimensional statistic, and $s_{ij}$ and $s_{oj}$ are the $j$th components of $\textrm{\boldmath$s$}_i$ and $\textrm{\boldmath$s$}obs,$ respectively
(see also \citet{WeissVonHaeseler1998} for an application in a pure likelihood inference context).
Distance metrics other than the Euclidean distance, such as the total variation distance \citep{Tavareetal2002}, have also been used.
\par
Substituting the binary accept/reject step in the rejection sampling by weighting $\textrm{\boldmath$s$}_i$ smoothly according to its distance from $\textrm{\boldmath$s$}obs$ using a kernel density ${\rm K}_{\epsilon}(\textrm{\boldmath$s$}_i,\textrm{\boldmath$s$}obs)$ with bandwidth $\epsilon$ leads to importance sampling \citep{Wilkinson2008}.
The tolerance condition $||\textrm{\boldmath$s$}_i-\textrm{\boldmath$s$}obs||\leq\epsilon$ in the rejection algorithm of \citet{Pritchardetal1999} then corresponds to using a uniform kernel on an $\epsilon$-ball around $\textrm{\boldmath$s$}obs.$
Other approaches to kernel choice include Epanechnikov \citep{Beaumontetal2002} and Gaussian \citep{LeuenbergerWegmann2010}
kernels.
\par
When the data likelihood is substituted by the likelihood based on the summary statistics and a
tolerance condition with a uniform kernel and the Euclidean distance is used, the posterior distribution
sampled with ABC by rejection is
\begin{equation}\label{eq:2}
\pi_{\epsilon}(\theta|{\bf x}obs, P_{\theta}) =\frac{1}{C_{P_{\theta}}}\int_{\mathcal{X}} \mathbf{I}_{\{||\textrm{\boldmath$s$}-\textrm{\boldmath$s$}obs||<\epsilon\}} p({\bf x}|\theta)\pi(\theta) \;d{\bf x},
\end{equation}
where $\mathbf{I}_{A}$ is an indicator function that takes a value of 1 on set $A$ and is zero otherwise,
and $C_{P_{\theta}}=\int_{\Theta}\int_{\mathcal{X}} \mathbf{I}_{\{||\textrm{\boldmath$s$}-\textrm{\boldmath$s$}obs||<\epsilon\}} p({\bf x}|\theta)\pi(\theta) \;d{\bf x} \;d\theta$ is the normalizing constant.
A standard ABC algorithm that samples $\pi_{\epsilon}(\theta|{\bf x}obs, P_{\theta})$ appears in Figure \ref{fig:2}.
\par
\begin{figure}\label{fig:2}
\end{figure}
\par
The choice of summary statistics, tolerance parameter $\epsilon,$ distance function, and kernel constitute approximations on the data space in ABC methods.
We assume that these standard ABC approximations work reasonably well, and we focus on new modeling approximations on the parameter and model spaces introduced by AABC (Figure \ref{tablo:1}).
\begin{figure}\label{tablo:1}
\end{figure}
\par
\section{Approximate approximate Bayesian computation (AABC)}\label{sec:theory}
Algorithm 1 returns an adequate sample size from the posterior distribution of a parameter if it is iterated a large number of times, $M$.
The set of realizations simulated from the joint distribution of the parameter and the data by steps 1 and 2 of Algorithm 1 is then $\{({\bf x}_1, \theta_1),({\bf x}_2, \theta_2),...,({\bf x}_M, \theta_M)\}.$
AABC methods seek inference on parameter $\theta$ when the model $P_{\theta}$ is limited-generative, and simulating $M$ data sets under $P_{\theta}$ is therefore computationally infeasible.
We thus assume that only a limited number $m$ of data sets ${\bf x}_1,{\bf x}_2,...,{\bf x}_m$ can be obtained by step 2 of Algorithm 1 $(m \ll M)$.
We denote the set of realizations simulated from the joint distribution of the parameter and the data by $\mathcal{Z}_{n,m}=\{({\bf x}_1, \theta_1),({\bf x}_2, \theta_2),...,({\bf x}_m, \theta_m)\},$
where each data set ${\bf x}_i$ of $n$ IID observations is simulated from the model $P_{\theta}sub_{\theta_i}.$
\par
In AABC, we substitute the joint sampling distribution $P_{\theta}$ of a data set of size $n$ with the joint sampling distribution $Q_{\theta},$ from which simulating
data sets is computationally inexpensive.
In replacing $P_{\theta}$ with $Q_{\theta},$ we require that the posterior distribution $\pi(\theta|{\bf x}obs, Q_{\theta})$ based on the likelihood implied by model $Q_{\theta}$ approximates the posterior distribution $\pi(\theta|{\bf x}obs, P_{\theta})$ based on the likelihood implied by model $P_{\theta}.$
Further, we require that $Q_{\theta}$ can be used with a wide range of $P_{\theta},$ in the sense that $Q_{\theta}$ is constructed without using the details of model $P_{\theta}.$
\par
\subsection{Approximations on the parameter and model spaces due to replacing $P_{\theta}$ with $Q_{\theta}$}\label{subsec:nonparametric}
Two approximations are involved in substituting $P_{\theta}$ with $Q_{\theta}.$
First, $\mathcal{Z}_{n,m}$ includes only $m$ parameter values $\theta_1,\theta_2,...,\theta_m$ under which data sets are simulated from $P_{\theta}$.
After obtaining $\mathcal{Z}_{n,m},$ for any new parameter value $\theta$ from the prior distribution under which we want to simulate a new data set, we substitute $\theta$ with $\tilde{\theta}$ such that
$(\tilde{{\bf x}},\tilde{\theta})\in\mathcal{Z}_{n,m}.$
The value $\tilde{\theta}$ has the minimum Euclidean distance to the value $\theta$ among all parameter values in $\mathcal{Z}_{n,m}.$
More precisely, $\tilde{\theta}=\displaystyle{\mathop{\mbox{arg\;min}}_{\theta_j \in \mathcal{Z}_{n,m}}}||\theta_j-\theta||.$
In essence, this approximation is equivalent to replacing the sampling distribution of the data set $P_{\theta}$ with the sampling distribution $P_{\theta}sub_{\tilde{\theta}}$; we call this an approximation on the parameter space.
However, this parameter space approximation is not sufficient to simulate data sets efficiently, since the model $P_{\theta}sub_{\tilde{\theta}}$ is still limited-generative after this substitution.
\par
As a second approximation, we substitute the model $P_{\theta}sub_{\tilde{\theta}}$ with the empirical distribution of the data set $\tilde{{\bf x}}$ that has already been simulated from
$P_{\theta}sub_{\tilde{\theta}}$ as $(\tilde{{\bf x}},\tilde{\theta})\in\mathcal{Z}_{n,m}.$
Here, we assume a positive probability mass only on the data values observed in the set $\tilde{{\bf x}}.$
We call this an approximation on the model space because the model $P_{\theta}sub_{\tilde{\theta}}$ is substituted with the empirical distribution of a data set simulated from
$P_{\theta}sub_{\tilde{\theta}}.$
\par
To simulate a new data set ${\bf x}$ in AABC, we utilize a vector of positive auxiliary parameters $\textrm{\boldmath$\phi$}=(\phi_1,\phi_2,...,\phi_{n}),$ that satisfy $\sum_{i=1}^{n}\phi_i=1.$
We let $\phi_i$ be the probability that a random data value $x_j\in{\bf x}$ is equal to a given value $\tilde{x}_i$ found in the data set $\tilde{{\bf x}}=(\tilde{x}_1,\tilde{x}_2,...,\tilde{x}_n).$
The premise is that the sample $\tilde{{\bf x}}$ simulated under $\tilde{\theta}$ provides information about the model $P_{\theta}sub_{\tilde{\theta}},$ and by an approximation of $\theta$ to $\tilde{\theta}$ on the parameter space, about $P_{\theta}$.
\par
If we denote the approximate sampling distribution of a data set ${\bf x}=(x_1,x_2,...,x_n)$ by $Q_{\theta},$ its joint probability mass function is
\begin{equation}\label{eq:Q}
\int_{{\Phi}}q({\bf x}|\textrm{\boldmath$\phi$},\tilde{{\bf x}})\pi(\textrm{\boldmath$\phi$}) \;d\textrm{\boldmath$\phi$}\; \mathbf{I}_{\{\theta,\tilde{\theta}\}},
\end{equation}
where $q({\bf x}|\textrm{\boldmath$\phi$},\tilde{{\bf x}})={n \choose n_1 \;n_2\; \cdots\; n_k}\prod_{j=1}^{n}\prod_{i=1}^{n}\phi_i^{\mathbf{I}_{\{x_j=\tilde{x}_i\}}},$ and $\mathbf{I}_{\{\theta,\tilde{\theta}\}}$ is 1 if $\tilde{\theta}\in\mathcal{Z}_{n,m}$ is the closest value to $\theta$ in the Euclidean sense and is 0 otherwise.
Here, $n_i$ is the number of times $\tilde{x}_i$ observed in the new sample ${\bf x},$ $k$ is the number of distinct data values observed in the data set ${\bf x},$ and $\mathbf{I}_{\{x_j=\tilde{x}_i\}}$ is 1 if $x_j=\tilde{x}_i$ and is 0 otherwise.
The distribution $q({\bf x}|\textrm{\boldmath$\phi$},\tilde{{\bf x}})$ is that of an IID sample ${\bf x}=(x_1,x_2,...,x_n),$ where $x_j$ is drawn from the values $(\tilde{x}_1,\tilde{x}_2,...,\tilde{x}_n)$ with probabilities $(\phi_1,\phi_2,...,\phi_n).$
\par
The probability vector $\textrm{\boldmath$\phi$}$ is a parameter of the model conditional on $\tilde{{\bf x}},$ and thus, we need to posit a prior distribution on $\textrm{\boldmath$\phi$}.$
As a natural prior on probabilities, we let the prior distribution $\pi(\textrm{\boldmath$\phi$})$ on $\textrm{\boldmath$\phi$}$ be the symmetric Dirichlet distribution on the $(n-1)${\em-}dimensional simplex $\Phi,$ with hyperparameters (1,1,...,1) and a uniform probability density function proportional to $1.$
This choice assigns equal weight to all distributions placing positive probability mass on the data points $\tilde{x}_i\in\tilde{{\bf x}}.$
Further, it assigns zero posterior probability to data values unobserved in the sample $\tilde{{\bf x}},$ thereby avoiding difficulties created by such values in the likelihood \citep{Rubin1981, Owen1990}.
\par
To distinguish the parameter and data set realizations in $\mathcal{Z}_{n,m}=\{({\bf x}_i,\theta_i)\}_{i=1}^{m}$ from the parameter and data sets simulated using AABC, we use starred versions of each quantity to denote specific values simulated in AABC.
For example, as the sampling distribution $P_{\theta}sub_{\theta_i}$ delivers a data set ${\bf x}_i$ under a given parameter value $\theta_i$ in the ABC procedure of Algorithm 2, the sampling distribution
$Q_{\theta}sub_{\theta^*_i}$ delivers a data set ${\bf x}^*_i$ under a given parameter value $\theta^*_i$ simulated from its prior distribution (see Figure \ref{fig:11} for notation).
\begin{figure}\label{fig:11}
\end{figure}
\par
The sampling distribution $Q_{\theta}$ utilizes the information available in the set of realizations $\mathcal{Z}_{n,m}$ through the parameter $\textrm{\boldmath$\phi$},$ since the prior distribution of $\textrm{\boldmath$\phi$}$ conditions on $(\tilde{{\bf x}},\tilde{\theta})\in \mathcal{Z}_{n,m}$ and thus on the set $\mathcal{Z}_{n,m}.$
In this sense, the available realizations $\mathcal{Z}_{n,m}$ are used as fixed background information about $P_{\theta},$ and inferences using the substitute model $Q_{\theta}$ are conditional on the simulated sets $\mathcal{Z}_{n,m}.$
\par
\subsection{The posterior distribution of $\theta$ sampled by AABC}\label{sec:posterior}
In sampling the approximate posterior distribution of $\theta$ by AABC methods, we use the two ABC approximations described in Section \ref{sec:ABCreview}.
First, we substitute each data instance ${\bf x}$ with summary statistics $\textrm{\boldmath$s$}.$
Second, we use an acceptance condition with tolerance $\epsilon,$ employing the Euclidean distance to measure the proximity of the summary statistics calculated from the observed and simulated data, as in equation \ref{eq:euclideandistance}.
If we let $\theta^*_j$ be a new parameter value simulated from its prior distribution after obtaining the set $\mathcal{Z}_{n,m},$ in AABC we accept
the parameter values $\theta^*_j$ producing summary statistics $\textrm{\boldmath$s$}^*_j$ that satisfy the condition $||\textrm{\boldmath$s$}^*_j-\textrm{\boldmath$s$}obs||<\epsilon$ as being draws from the posterior distribution.
This acceptance condition corresponds to a uniform kernel, which we use throughout this article, although like ABC, AABC can employ other kernels
to obtain smooth weighting of $\textrm{\boldmath$s$}^*_j$ values by their distance from $\textrm{\boldmath$s$}obs.$
Substituting $P_{\theta}$ with $Q_{\theta}$ involves replacing $p({\bf x}|\theta)$ in expression \ref{eq:2} with expression \ref{eq:Q} and adjusting the normalizing constant accordingly.
The approximate posterior distribution sampled by an AABC method is
\begin{equation}\label{eq:4}
\pi_{\epsilon}(\theta|{\bf x}obs,Q_{\theta}) =
\frac{1}{C_{Q_{\theta}}}\int_{\mathcal{X}}\mathbf{I}_{\{||\textrm{\boldmath$s$}-\textrm{\boldmath$s$}obs||<\epsilon\}}\left[\int_{\Phi}q({\bf x}|\textrm{\boldmath$\phi$},\tilde{{\bf x}})\pi(\textrm{\boldmath$\phi$})\;d\textrm{\boldmath$\phi$}\;\mathbf{I}_{\{\theta,\tilde{\theta}\}}\right]\pi(\theta) \;d{\bf x},
\end{equation}
where $C_{Q_{\theta}}=\int_{\Theta}\int_{\mathcal{X}}\mathbf{I}_{\{||\textrm{\boldmath$s$}-\textrm{\boldmath$s$}obs||<\epsilon\}}\left[\int_{\Phi}q({\bf x}|\textrm{\boldmath$\phi$},\tilde{{\bf x}})\pi(\textrm{\boldmath$\phi$})\;d\textrm{\boldmath$\phi$}\;\mathbf{I}_{\{\theta,\tilde{\theta}\}}\right]\pi(\theta) \;d{\bf x} \;d\theta$ is the normalizing constant.
\par
The AABC approach is sensible in that as the limited generative model increasingly permits a larger number of simulated data sets, for large sample sizes the posterior distribution obtained by an AABC method approaches the same distribution as the posterior distribution obtained by an ABC method.
We codify this claim with a theorem.
\par
\vskip 0.5cm
\noindent
{\em Theorem.}
Let $\pi(\theta)$ be a bounded prior on $\theta.$ Let $\pi_{\epsilon}(\theta|{\bf x}obs,P_{\theta})$ and $\pi_{\epsilon}(\theta|{\bf x}obs,Q_{\theta})$ be the posterior distributions sampled by a standard ABC method and an AABC method, respectively. Then
\begin{equation}\label{eq:theorem}
\lim_{m \rightarrow \infty}\lim_{n\rightarrow \infty}\pi_{\epsilon}(\theta|{\bf x}obs,Q_{\theta})=\lim_{n \rightarrow \infty}\pi_{\epsilon}(\theta|{\bf x}obs,P_{\theta}).
\end{equation}
A proof of the theorem is given in Appendix 1.
The convergence of the posterior distribution sampled by AABC is a consequence of the fact that, for each given value of $\theta,$ the sampling distribution $\int_{\Phi}q({\bf x}|\textrm{\boldmath$\phi$},\tilde{{\bf x}})\pi(\textrm{\boldmath$\phi$})\;d\textrm{\boldmath$\phi$}\;\mathbf{I}_{\{\theta,\tilde{\theta}\}}$ converges to the true sampling distribution $p({\bf x}|\theta)$ as the sample size $n$ and the number of simulated samples $m$ from $P_{\theta}$ increase.
The intuition for the double limit in equation \ref{eq:theorem} is as follows.
The standard notion of a distibution converging to a point in the parameter space as the sample size $n$ increases does not directly apply to the posterior distribution $\pi_{\epsilon}(\theta|{\bf x}obs,Q_{\theta}),$ since this posterior depends not only on the sample size $n,$ but also on the number $m$ of simulated data sets from $P_{\theta}.$
Hence, for convergence of the posterior distribution based on the likelihood of $Q_{\theta},$ the requirement is that both $n\rightarrow \infty$ and $m\rightarrow \infty.$
As $n\rightarrow\infty,$ the empirical distribution converges to $P_{\theta}sub_{\tilde{\theta}},$ the correct sampling distribution with the incorrect parameter value $\tilde{\theta}.$
As $m\rightarrow \infty,$ the distance between the parameter value $\theta$ under which we want to simulate a new data set and the parameter value $\tilde{\theta}\in\mathcal{Z}_{n,m}$ closest to $\theta$ approaches zero.
Therefore, taking both limits simultaneously results in convergence to the correct sampling distribution $P_{\theta}.$
\par
\subsection{AABC algorithms}\label{subsec:ABCapproximations}
The structure of AABC algorithms sampling the posterior distribution in expression \ref{eq:4} can be conveniently summarized in three parts, as shown in AABC by a rejection algorithm (Figure \ref{fig:3}).
In Algorithm 2, Part I involves obtaining a limited number of realizations
from the joint distribution of the parameter and the data from the limited-generative model $P_{\theta}.$
Part I simply involves the application of steps 1 and 2 from Algorithm 1, but only for $m$ iterations.
Part II involves simulating a new parameter value $\theta^*_i$ from its prior distribution (step 4) and then simulating a data set ${\bf x}^*_i$
from the model $Q_{\theta}sub_{\theta^*_i}$ (steps 5, 6, 7), conditional on $\mathcal{Z}_{n,m}$ obtained in Part I.
Part III involves comparing the summary statistics $\textrm{\boldmath$s$}^*_i$ calculated from the simulated data set ${\bf x}^*_i$ with the summary statistics $\textrm{\boldmath$s$}obs$ calculated from the observed data set ${\bf x}obs,$ to accept or reject the parameter value $\theta^*_i.$
The calculation and comparison of summary statistics follows the same procedure as in steps 3 and 4 of Algorithm 1.
Hence, Part II of AABC by rejection has the novel steps 5, 6, and 7, whereas Parts I and III use the machinery of ABC by rejection from Algorithm 1.
\par
We can show that Algorithm 2 samples the correct posterior distribution $\pi_\epsilon(\theta|{\bf x}obs,Q_{\theta}).$
The probability of sampling a parameter value $\theta$ in Algorithm 2 is proportional to
\begin{align*}
&\sum_{\textrm{\boldmath$s$}}\sum_{\textrm{\boldmath$\phi$}}\pi(\theta)\mathbf{I}_{\{\theta,\tilde{\theta}\}}\pi(\textrm{\boldmath$\phi$})q({\bf x}|\textrm{\boldmath$\phi$},\tilde{{\bf x}})
\mathbf{I}_{\{||\textrm{\boldmath$s$}-\textrm{\boldmath$s$}obs||<\epsilon\}}\\
& = \sum_{\textrm{\boldmath$s$}}\sum_{\textrm{\boldmath$\phi$}}\pi(\theta,\textrm{\boldmath$\phi$})\mathbf{I}_{\{\theta,\tilde{\theta}\}}q({\bf x}|\textrm{\boldmath$\phi$},\tilde{{\bf x}})\mathbf{I}_{\{||\textrm{\boldmath$s$}-\textrm{\boldmath$s$}obs||<\epsilon\}}\\
&\propto \sum_{\textrm{\boldmath$s$}}\sum_{\textrm{\boldmath$\phi$}} \pi(\theta,\textrm{\boldmath$\phi$}|Q_{\theta})\mathbf{I}_{\{||\textrm{\boldmath$s$}-\textrm{\boldmath$s$}obs||<\epsilon)\}}\\
& \propto \pi_\epsilon(\theta|{\bf x}obs,Q_{\theta}),
\end{align*}
where the third line follows from the fact that the expression on the second line is the product of the likelihood under the model $Q_{\theta}$ and the prior, and therefore it is proportional to the posterior distribution of parameters based on the model $Q_{\theta}.$
\begin{figure}\label{fig:3}
\end{figure}
\section{Applications}
In this section, we investigate the inferential performance of AABC approach with two examples.
The following simulation setup is used in both examples.
\subsection{Simulation study design}\label{sec:simulation}
We simulated a reference set with $M=10^5$ realizations $\{({\bf x}_1,\theta_1),({\bf x}_2,\theta_2),...,({\bf x}_{10^5},\theta_{10^5})\},$ by first generating $\theta_i\sim \pi(\theta)$ and then simulating a data set ${\bf x}_i\sim P_{\theta}sub_{\theta_i}.$
We then sampled 1000 pairs $({\bf x}_i,\theta_i)$ from the reference set, uniformly at random without replacement.
Thus, we selected 1000 ``true'' parameter values $\theta_i,$ along with corresponding test data sets ${\bf x}_i$ generated under each value $\theta_i$ from the model $P_{\theta}sub_{\theta_i}$.
Further, we built the sets $\mathcal{Z}_{n,m},$ with $m=10^2, 5\times 10^2, 10^3, 5\times 10^3, 10^4,5\times 10^4, 10^5$ by sampling the reference set uniformly at random without replacement for $m<10^5,$ and taking all the realizations in the reference set for $m=M=10^5.$
The sample size $n$ of the data is described in each relevant example.
\par
On each test data set, we performed AABC by rejection sampling (Algorithm 2) using each set $\mathcal{Z}_{n,m}.$
In example 1, where our goal is to compare the performance of the AABC and ABC approaches, we performed ABC analyses by rejection sampling (Algorithm 1) using the same sets $\mathcal{Z}_{n,m}.$
For all analyses, we obtained a sample from the joint posterior distribution of the parameter vector $\theta$ by accepting the parameter vector values that generated data whose summary statistics were in the top $1$ percentile with respect to the statistics calculated from the test data set, in the sense of equation \ref{eq:euclideandistance}.
Compared to the approach of fixing the $\epsilon$ cutoff, accepting parameter vectors that generate data whose summary statistics are in a top percentile has the advantage that a desired number of samples from the posterior is always obtained given a total fixed number of proposed parameter values.
This approach is often preferred by ABC practitioners and is convenient in our case for comparing ABC and AABC.
\par
We assessed the accuracy of the posterior samples for each component of the parameter vector $\theta$ separately, using the root sum of squared error for standardized parameter values accepted in the posterior sample.
For a generic scalar parameter $\alpha,$ the root sum of squared errors is given by $\textrm{RSSE}=(1/r)\sqrt{\sum_{j=1}^{r}(\alpha_j-\alpha_T)^2/\textrm{Var}(\mathbf{\alpha})},$
where $\mathbf{\alpha}=(\alpha_1,\alpha_2,...,\alpha_r)$ are $r$ accepted values in the posterior sample, $\alpha_T$ is the true parameter value, and $\textrm{Var}(\mathbf{\alpha})$ is the variance of the set of $r$ values.
We report the mean RSSE over 1000 test data sets as $\textrm{RMSE}=(1/1000)\sum_{i=1}^{1000}\textrm{RSSE}_i$ (see \citet{NunesBalding2010}).
\par
\subsection{Example 1: The strength of balancing selection in a multi-locus $K${\em-}allele model}\label{sec:example1}
In this section, we consider inference from the stationary distribution of allele frequencies in the diffusion approximation
to a Wright-Fisher model with symmetric balancing selection and mutation \citep{Wright1949}.
If we let $a_i>0,$ with $i=1,2,...,K,$ and $\sum_{i=1}^{K}a_i=1,$ and denote the frequency of allelic type $i$ in the population at a genetic locus, the joint probability density function of allele frequencies $x=(a_1,a_2,...,a_K)$ is $f(x|\sigma, \mu)= c(\sigma,\mu)^{-1}\exp(-\sigma\sum_{i=1}^{K}a_i^2)\prod_{i=1}^{K}a_i^{\mu/K-1}.$
Parameters $\sigma$ and $\mu$ determine the population-scaled strength of balancing selection and the mutation rate, respectively.
A data set of observed allele frequencies is a random sample of $n$ draws from the population frequencies $f(x|\sigma,\mu).$
\par
ABC methods are well-suited for inference from this model for three reasons.
First, the statistics $\sum_{j=1}^{K}a_j^2$ and $-\sum_{j=1}^{K}\log a_j$ are jointly sufficient for parameters $\sigma$ and $\mu,$ and no information loss occurs in dimension reduction to the summary statistics.
Second, the parameter-dependent normalizing constant $c(\sigma,\mu)$ is hard to calculate, and performing likelihood-based inference on $\sigma$ and $\mu$ is therefore difficult.
Third, a method specifically designed to simulate data sets from $f(x|\sigma,\mu)$ is readily available \citep{Joyceetal2012}, and performing ABC is therefore straightforward.
For simplicity, we assume 100 loci with the same true parameter values, each with $K=4,$ and that the allele frequencies at each locus are independent of the allele frequencies at other loci.
Thus, the joint probability density function of allele frequencies for 100 loci is equal to the product of probability density functions across loci.
We choose uniform prior distributions, on $(0.1,10)$ for the mutation rate $(\mu),$ and on $(0,50)$ for the selection parameter $(\sigma)$.
\par
{\em Results.}
Posterior samples model parameters $(\sigma,\mu)$ obtained by ABC and AABC using a typical data set are given in Figure \ref{fig:4}.
In analyses with $m=10^2,5\times 10^2, 10^3$ or $5\times 10^3$ simulated data sets, few samples are accepted with ABC, and thus, little mass is observed in ABC histograms (black).
For small $m,$ ABC does not produce an adequate sample size from the posterior distribution of parameters.
AABC, however, produces a posterior sample of size $10^3$ for any $m,$ because $10^5$ data sets are simulated from the non-mechanistic model (Algorithm 2, steps 5, 6, 7) and the top 1 percentile are accepted as belonging to the approximate posterior distribution.
The histograms obtained by AABC recover the true value reasonably well (Figure \ref{fig:4}).
The RMSE values in AABC procedures are approximately constant with increasing $m.$
For $m=10^2,5\times 10^2, 10^3, 5\times 10^3, 10^4, 5 \times 10^4,$ and $10^5$ simulated data sets, the RMSE values for parameter $\mu$ are 5.988, 5.932, 6.012, 6.086, 6.125, 6.078, and 6.088 respectively, close to the RMSE of 5.290 obtained by a standard ABC approach using $M=10^5$ simulated data sets from the mechanistic model.
The RMSE values in the last column of Figure \ref{fig:4} show that an AABC approach produces posterior samples that have on average greater variance than posterior samples obtained from ABC with the same large number of realizations.
Here, greater variance in posterior samples obtained by AABC is a result of simulating data sets in AABC by resampling the observed data values that are found only in the $m$ realizations in $\mathcal{Z}_{n,m}.$
Consider two parameter values $\theta^*_1$ and $\theta^*_2$ for which data sets ${\bf x}^*_1$ and ${\bf x}^*_2$ are simulated in the AABC approach by steps 5, 6, 7 of Algorithm 2 such that the parameter value $\tilde{\theta}\in\mathcal{Z}_{n,m}$ closest to both $\theta^*_1$ and $\theta^*_2$ is the same value.
The data sets ${\bf x}^*_1$ and ${\bf x}^*_2$ can include only the data values observed in $\tilde{{\bf x}}$ of the pair $(\tilde{{\bf x}},\tilde{\theta})\in\mathcal{Z}_{n,m}.$
On average, ${\bf x}^*_1$ and ${\bf x}^*_2$ share more observations in common than two data sets simulated from the respective mechanistic models $P_{\theta}sub_{\theta^*_1}$ and $P_{\theta}sub_{\theta^*_2}.$
Therefore, each data set simulated in the AABC approach using $Q_{\theta}$ is expected to be less able to distinguish between different parameter values than the independent data sets simulated in the ABC approach using $P_{\theta}.$
This situation results in relatively flat likelihoods and hence posterior samples with larger variance.
\par
\begin{figure}\label{fig:4}
\end{figure}
\subsection{Example 2: Admixture rates in hybrid populations}
Models in which hybrid populations are founded by, and receive genetic contributions from, multiple source populations are of interest in describing the demographic history of admixture.
Stochastic models including admixture often result in likelihoods that are difficult to calculate, and statistical methods capable of performing inference on admixture rates have received much attention for their implications on topics ranging from human evolution to conservation ecology \citep{Falushetal2003, Tangetal2005, BuerkleLexer2008}.
Here, we consider inference on admixture rates from a mechanistic model of \citet{VerduRosenberg2011}.
We use reported estimates of individual admixture as data.
\par
We consider a model of admixture for a diploid hybrid population of constant size $N,$ founded at some known $t$ generations in the past with contributions from source populations A and B.
We follow the distribution of admixture fractions of individuals in the hybrid population at a given genetic locus.
Each generation, the admixture fraction for each individual in the hybrid population is obtained as the mean of the admixture fractions of its parents.
The parents are chosen independently of each other, from source population A, source population B, or the hybrid population of the previous generation with probabilities $p_A,p_B,$ and $p_H,$ respectively ($p_A+p_B+p_H=1$).
In the special case of the founding generation, $p_H=0,$ and we assume $p_A=p_B=0.5.$
Individuals from source populations A and B are assigned admixture fractions of $1$ and $0$ respectively.
For example, if both parents of an individual in the hybrid population of the founding generation are from source population A, that individual has admixture fraction
$(1+1)/2=1.$
If both parents are from population 2, the admixture fraction is $(0+0)/2=0,$ and if one parent is from population 1 and the other is from population B, then the admixture fraction is $(1+0)/2=0.5.$
The distribution of the admixture fraction in the hybrid population is propagated in this manner for $t$ generations until the present, in which a sample of $n$ individuals is obtained from the resulting distribution (Figure \ref{fig:5}).
Our goal is to estimate the admixture rates $(p_A,p_B,p_H),$ given the individual admixture fractions estimated from observed genetic data.
\begin{figure}\label{fig:5}
\end{figure}
\par
We apply the AABC approach using individual admixture fractions from $n=604$ individuals from Central African Pygmy populations reported by \citet{Verduetal2009}, with an assumed constant population size of $N=10^4.$
This assumption differs slightly from the original model in \citet{VerduRosenberg2011} in that a finite population size is assumed, so that only $10^4$ admixture fraction values are allowed in the population at any given generation.
We assume that an admixture event with contributions from two ancestral source populations started at the mean estimate of $t=771$ generations ago \citep{Verduetal2009} with a generation time of 25 years, and that it continued until the present.
Source population A refers to an ancestral Pygmy population, and source population B refers to an ancestral non-Pygmy population.
The feature of this model relevant to our method is the computational intractability of simulating data sets.
For each set of parameter values $(p_A,p_B,p_H)$ simulated from the priors, the distribution of admixture fractions is discrete on a support of a number of admixture fraction values that doubles each generation, and this distribution evolves for 771 generations.
A random sample of admixture fraction values comparable to the values calculated from the observed data set is obtained from the distribution of the present generation.
Simulating a large number of data sets under this model with such a large number of generations is computationally infeasible, and standard ABC is impractical.
We thus perform AABC by rejection (Algorithm 2) using $m=10^4$ realizations from this model.
We assume a Dirichlet prior with hyperparameters $(1,1,1)$ on parameters $(p_A,p_B,p_H).$
\par
We also assessed the contribution of the approximations on the parameter and model spaces in the AABC approach to the RMSE separately, with a simulation study using a small number of generations ($t=30$), where simulating data sets from the mechanistic model is feasible.
First, we performed AABC with rejection as in Algorithm 2 with 1000 ``true'' data sets using $m=10^2, 5\times10^2,10^3, 5\times10^3,10^4, 5\times10^4,$ and $10^5$ realizations from the model, and we calculated the RMSE for $p_A,p_B,$ and $p_H$ over 1000 ``true'' data sets as described in Section \ref{sec:simulation}.
This AABC analysis includes error due to approximations on the parameter space and on the model space.
Second, we performed an AABC analysis with the same set of $m$ realizations, by including the error only due to the approximation on the parameter space.
We achieved this by running Algorithm 2 up through step 5, and then simulating data sets from the mechanistic model by substituting steps 6 and 7 of Algorithm 2 with step 2 of Algorithm 1, the standard ABC approach by rejection.
By this substitution, all data sets are simulated from the mechanistic model, but each data set is obtained using a parameter vector $(\tilde{p}_A,\tilde{p}_B,\tilde{p}_H)$ found in step 5 of Algorithm 2.
In this procedure, the error due to the approximation on the model space is eliminated, because data sets are simulated from the correct mechanistic model and not by resampling from the available realizations in $\mathcal{Z}_{n,m}$.
However, this procedure includes error due to the approximation on the parameter space, because each data set is simulated not under the correct proposed parameter value, but under the parameter value $(\tilde{p}_A,\tilde{p}_B,\tilde{p}_H),$ the closest value to the correct proposed value that can be found in $\mathcal{Z}_{n,m}.$
We compared the RMSE of the AABC procedure involving the approximation on both the parameter and model spaces and the RMSE of the AABC procedure involving only the approximation on the parameter space to the RMSE obtained from a standard ABC approach.
For these two AABC procedures, we also compared the percent excess in RMSE, defined as the ratio of the absolute difference in RMSE of the AABC and standard ABC approaches to the RMSE of the standard ABC approach, expressed as a percent.
\par
{\em Results.}
The individual admixture fractions calculated from the Pygmy data carry substantial information about the admixture parameters $p_A,p_B,$ and $p_H,$ since the joint posterior distribution is concentrated in a relatively small region of the 3-dimensional unit simplex on which $(p_A,p_B,p_H)$ sits (Figure \ref{fig:6}A). The marginal posterior distributions (Figure \ref{fig:6}B, \ref{fig:6}C, and \ref{fig:6}D) have means $p_A=0.151,\; p_B=0.132,$ and $p_H=0.717.$
These values are interpreted as contribution of genetic material of 15.1\% from the ancestral Pygmy population (source population A), 13.2\% from the ancestral Non-Pygmy population (source population B), and 71.7\% from the hybrid population to itself at each generation, over $771$ generations of constant admixture.
\begin{figure}\label{fig:6}
\end{figure}
\par
For the simulation study with $t=30$ generations and 1000 ``true data'' sets, the RMSE values from AABC analyses decrease with increasing $m$ (Figure \ref{fig:7}A, \ref{fig:7}B,
\ref{fig:7}C).
Further, as $m$ increases, the error due to the approximation on the parameter space decreases (Figure \ref{fig:7}D last column), due to the fact that for large $m,$ the difference decreases between the closest parameter value chosen at step 5 of Algorithm 2 and the correct parameter value under which we want to simulate a data set.
In fact, the RMSE from the AABC analysis with $m=10^5$ realizations and approximation only on the parameter space and the RMSE from the standard ABC approach are virtually indistinguishable (Figure \ref{fig:7}A, \ref{fig:7}B, \ref{fig:7}C, red star).
For $m=10^3,$ the AABC analysis with approximations on the parameter and model spaces has a percent excess RMSE of 13.81\%, whereas AABC analysis including only the approximation on the parameter space has excess RMSE of 6.61\%.
That is, at $m=10^3,$ approximately half of the excess RMSE in the AABC approach with respect to the standard ABC analysis comes from the error due to the approximation on the parameter space and half arises due to the approximation on the model space.
\begin{figure}\label{fig:7}
\end{figure}
\section{Discussion}
Performing likelihood-based inference from statistical models incorporating a multitude of stochastic processes is often challenging due to computationally intractable likelihoods.
In principle, when stochastic processes are complex but a family of parametric statistical models is well-defined, data can be simulated from the model to assess the parameter likelihoods.
In the last decade, ABC methods have become a standard tool to perform approximate Bayesian inference in subject areas such as ecology and evolution, by exploiting
the idea of simulating many data sets from a model, when such simulations are computationally feasible.
To deliver an adequate sample from the posterior distribution of the parameters, however, ABC requires a large number of simulated data sets, and it might not perform well when only a limited number of data sets can be simulated.
\par
In this article, we introduced an approach that extends simulation-based Bayesian inference methods to model spaces in which only a limited number of data sets can be simulated from the model, at the expense of requiring approximations on the parameter and the model spaces.
Our AABC approaches rely on two statistical approximations.
In our approximation on the parameter space, for each parameter simulated from the prior distribution, we take the closest parameter value available in the set of realizations
$\mathcal{Z}_{n,m}$ obtained from the mechanistic model.
This approach has a uniform kernel smoothing interpretation
in the sense that each parameter value in the set $\mathcal{Z}_{n,m}$ dissects the support of the prior distribution into non-overlapping components such that each interval is mapped to the same parameter value in $\mathcal{Z}_{n,m}.$
Each component then represents the support of a uniform kernel.
Kernel approximations have an operational role in implementing ABC methods, and a natural future direction for AABC is to improve the accuracy of posterior samples using smooth weighting kernels for the approximation on the parameter space.
\par
The approximation on the model space is achieved by assigning Dirichlet probabilities to data points of realizations obtained from the mechanistic model.
This is a variation on the resampling method originally introduced in Rubin's Bayesian bootstrap \citep{Rubin1981}, and therefore, it is an application of Bayesian nonparametric methods.
From this perspective, AABC methods connect standard model-based Bayesian inference on model-specific parameters and Bayesian nonparametric methods within the ABC framework.
\par
Our approach of using a non-mechanistic model and Bayesian resampling methods to help perform inference on model-specific parameters of a mechanistic model is a fundamental difference between AABC and existing ABC methods.
ABC performs inference on model-specific parameters of a mechanistic model using a likelihood based purely on the mechanistic model.
AABC instead performs inference on the same model-specific parameters of the mechanistic model as ABC, using a likelihood based on a non-mechanistic model that incorporates a limited number of data sets simulated from the mechanistic model.
Consequently, the model likelihoods used in ABC and AABC are not exactly the same, and the posterior distributions targeted by the two classes of methods are not exaxctly equivalent for finite sample sizes.
The advantage of AABC methods in contrast to pure non-mechanistic modeling approaches (e.g., nonparametric methods) is that AABC can perform inference on the quantities of interest---the model-specific parameters of the mechanistic model.
\par
Unlike other ABC methods, the AABC approach delivers a posterior sample of desired size from the joint distribution of parameters for any $m>1.$
This is both a strength and a limitation of AABC.
The strength is that in practice, a researcher can fix $m$ and thus the computation time {\em a priori}, to simulate data from the mechanistic model to obtain a reasonable inference by AABC; other ABC methods may fail to produce an adequate posterior sample in equivalent computation time.
In our example, for moderate values of $m$ (e.g., $10^3$ to $10^4$) for which standard ABC approaches were unsatisfactory, AABC adequately sampled an approximate posterior distribution.
The limitation is that when $m$ is too small, the posterior sample obtained by AABC can be a distorted representation of the true posterior distribution.
Although in the limit, AABC and ABC are expected to produce similar results, the posterior distribution sampled by an AABC approach is not the correct posterior distribution, because many parameter values simulated from the prior are tested for acceptance based on repeated use of the data values in $m$ realizations, instead of based on data sets simulated independently of each other.
A future direction is to investigate the relationship between $m$ and the dimensionality of the parameter space to optimize $m$ in producing a given level of accuracy for approximating the true posterior distributions.
\par
\section*{Acknowledgments}
The authors thank Paul Verdu for helpful discussions on the genetics of Central African Pygmy populations. Support for this research is partially provided by NIH grant R01 GM 081441, NSF grant DBI-1146722, and the Burroughs Wellcome Fund.
\section*{Appendix 1}
We let $k\leq n$ be the number of distinct values $\tilde{x}_1,\tilde{x}_2,...,\tilde{x}_k$ in the data set $\tilde{{\bf x}},$ and denote the number of observed $\tilde{x}_i$ by $\tilde{n}_i,$ where $n=\sum_{i=1}^{k}\tilde{n}_i.$
Then the prior distribution for the probabilities of an AABC replicate data set based on the ABC simulated data set $\tilde{{\bf x}}$ is the Dirichlet distribution $\pi(\textrm{\boldmath$\phi$})=[\Gamma(\sum_{i=1}^k \tilde{n}_i)/\prod_{i=1}^k\Gamma(\tilde{n}_i)] \prod_{i=1}^{k}\phi^{\tilde{n}_i-1}$ with parameters $\tilde{n}_1,\tilde{n}_2,...,\tilde{n}_k.$
The special case of the prior proportional to $1$ described in the text is obtained with $k=n,$ when all observations in $\tilde{{\bf x}}$ are distinct $(\tilde{n}_1,=\tilde{n}_2=\;\cdots\;=\tilde{n}_n=1)$.
Our goal is to show that $\lim_{m\rightarrow \infty}\lim_{n\rightarrow \infty}\pi_\epsilon(\theta|{\bf x}obs,Q_{\theta})=\lim_{n\rightarrow\infty}\pi_{\epsilon}(\theta|{\bf x}obs,P_{\theta}).$
\par
Recalling equation \ref{eq:4},
\begin{equation}\label{eq:app1}
\lim_{m\rightarrow \infty}\lim_{n\rightarrow \infty}\pi_\epsilon(\theta|{\bf x}obs,Q_{\theta})= \displaystyle{\lim_{m\rightarrow\infty}\lim_{n\rightarrow \infty}}\frac{1}{C_{Q_{\theta}}}\int_{\mathcal{X}}\mathbf{I}_{\{||\textrm{\boldmath$s$}-\textrm{\boldmath$s$}obs||<\epsilon\}}\left[\int_{\Phi}q({\bf x}|\textrm{\boldmath$\phi$},\tilde{{\bf x}})\pi(\textrm{\boldmath$\phi$})\;d\textrm{\boldmath$\phi$}\;\mathbf{I}_{\{\theta,\tilde{\theta}\}}\right]\pi(\theta)\; d{\bf x}.
\end{equation}
The integral in the brackets is the expectation of $q({\bf x}|\textrm{\boldmath$\phi$},\tilde{{\bf x}}),$ with respect to the prior $\pi(\textrm{\boldmath$\phi$}).$ We let $C={n \choose n_1 \;n_2\; \cdots\; n_k},$ and using the definition of $q({\bf x}|\textrm{\boldmath$\phi$},\tilde{{\bf x}})=C\;\prod_{j=1}^{n}\prod_{i=1}^{n}\phi_i^{\mathbf{I}_{\{x_j=\tilde{x}_i\}}}$ in section \ref{subsec:nonparametric}, and $\pi(\textrm{\boldmath$\phi$})=[\Gamma(\sum_{i=1}^k \tilde{n}_i)/\prod_{i=1}^k\Gamma(\tilde{n}_i)] \prod_{i=1}^{k}\phi^{\tilde{n}_i-1}$ we get
\begin{equation*}
\int_{\Phi}q({\bf x}|\textrm{\boldmath$\phi$},\tilde{{\bf x}})\pi(\textrm{\boldmath$\phi$})\;d\textrm{\boldmath$\phi$}=C\;\frac{\Gamma(\sum_{i=1}^{k}\tilde{n}_i)}{\prod_{i=1}^{k}\Gamma(\tilde{n}_i)} \;\prod_{j=1}^{n}\int_{\Phi}\left(\prod_{i=1}^{n}\phi_i^{\mathbf{I}_{\{x_j=\tilde{x}_i\}}}\right)\left(\prod_{i=1}^{k}\phi_i^{\tilde{n}_i-1}\right)\;d\textrm{\boldmath$\phi$}.
\end{equation*}
Here, we have exchanged the order of the product over $j$ with the integral since the expectation of the product of $n$ IID observations in sample ${\bf x}$ is equal to the the product of the expectations of observations $x_j.$
We label the realized value of the $j$th data point $x_j$ by $(j)$ such that $\prod_{i=1}^{n}\phi_i^{\mathbf{I}_{\{x_j=\tilde{x}_i\}}}=\phi_{(j)},$ and write
\begin{equation}\label{eq:app2}
\int_{\Phi}q({\bf x}|\textrm{\boldmath$\phi$},\tilde{{\bf x}})\pi(\textrm{\boldmath$\phi$})\;d\textrm{\boldmath$\phi$}=C\;\frac{\Gamma(\sum_{i=1}^{k}\tilde{n}_i)}{\prod_{i=1}^{k}\Gamma(\tilde{n}_i)}\;\prod_{j=1}^{n}\int_{\Phi} \left(\prod_{\substack{i=1\\i\neq(j)}}^{k}\phi_i^{\tilde{n}_i-1}\right)\phi_{(j)}^{\tilde{n}_{(j)}}\;d\textrm{\boldmath$\phi$}.
\end{equation}
Using
$\int_{\Phi}\frac{\Gamma[(\sum_{i=1,i\neq (j)}^{k}\tilde{n}_i)+\tilde{n}_{(j)}+1]}{[\prod_{i=1,i\neq (j)}^{k}\Gamma(\tilde{n}_i)]\Gamma(\tilde{n}_{(j)}+1)} \; \left(\prod_{i=1,i\neq (j)}^{k}\phi_i^{\tilde{n}_i-1}\right)\phi_{(j)}^{\tilde{n}_{(j)}}\;d\textrm{\boldmath$\phi$}=1$
(p. 487, \citet{Kotzetal2000}), we substitute the integral in equation (\ref{eq:app2}) with the ratio of the gamma functions to get
\begin{align*}
\int_{\Phi}q({\bf x}|\textrm{\boldmath$\phi$},\tilde{{\bf x}})\pi(\textrm{\boldmath$\phi$})\;d\textrm{\boldmath$\phi$}&=C\;\frac{\Gamma(\sum_{i=1}^{k}\tilde{n}_i)}{\prod_{i=1}^{k}\Gamma(\tilde{n}_i)}\prod_{j=1}^{n}
\frac{\left[\prod_{i=1,i\neq (j)}^{k}\Gamma(\tilde{n}_i)\right]\Gamma(\tilde{n}_{(j)}+1)}{\Gamma[(\sum_{i=1,i\neq (j)}^{k}\tilde{n}_i)+\tilde{n}_{(j)}+1]}\\
&=C\;\prod_{j=1}^{n}\frac{\Gamma(n)}{\Gamma(\tilde{n}_{(j)})}\frac{\Gamma(\tilde{n}_{(j)}+1)}{\Gamma(n+1)}=C\;\prod_{j=1}^{n}\left(\frac{\tilde{n}_{(j)}}{n}\right).
\end{align*}
Substituting $C\;\prod_{j=1}^{n}\left(\frac{\tilde{n}_{(j)}}{n}\right)$ for the integral in brackets in equation (\ref{eq:app1}), we have
\begin{align}
\nonumber
\lim_{m\rightarrow \infty}\lim_{n\rightarrow \infty}\pi_\epsilon(\theta|{\bf x}obs,Q_{\theta})&=\displaystyle{\lim_{m\rightarrow\infty}\lim_{n\rightarrow \infty}}\frac{1}{C_{Q_{\theta}}}\int_{\mathcal{X}}\mathbf{I}_{\{||\textrm{\boldmath$s$}-\textrm{\boldmath$s$}obs||<\epsilon\}}\;C\;\prod_{j=1}^{n}\left(\frac{\tilde{n}_{(j)}}{n}\right)\;\mathbf{I}_{\{\theta,\tilde{\theta}\}}\pi(\theta)\; d{\bf x}\\
\label{eq:exchangelimit}
&=\frac{\displaystyle{\lim_{m\rightarrow\infty}\lim_{n\rightarrow \infty}}\int_{\mathcal{X}}\mathbf{I}_{\{||\textrm{\boldmath$s$}-\textrm{\boldmath$s$}obs||<\epsilon\}}\;C\;\prod_{j=1}^{n}\left(\frac{\tilde{n}_{(j)}}{n}\right)\;\mathbf{I}_{\{\theta,\tilde{\theta}\}}\pi(\theta)\; d{\bf x}}{\displaystyle{\lim_{m \rightarrow \infty }\lim_{n\rightarrow \infty}}C_{Q_{\theta}}}.
\end{align}
\par
We apply the dominated convergence theorem to exchange the limits in $n$ and the integrals in the numerator and denominator of equation (\ref{eq:exchangelimit}).
The assumptions of the theorem are satisfied as follows:
1) The integrand in equation (\ref{eq:exchangelimit}) is bounded: The indicator functions are bounded by 1, the ratios $(\tilde{n}_{(j)}/n),$ where $n_{(j)}\leq n$ are bounded by 1, and the prior $\pi(\theta)$ is bounded by assumption.
2) $\lim_{n\rightarrow \infty}(\tilde{n}_{(j)}/n)$ converges pointwise to the probability of $x_{(j)}$ under $\tilde{\theta}$ and the model $P_{\theta}sub_{\tilde{\theta}},$ given by
$p(x_{(j)}|\tilde{\theta}),$ by the frequency interpretation of probability.
Exchanging the limits in $n$ and the integrals, and using $\lim_{n\rightarrow \infty}(\tilde{n}_{(j)}/n)=p(x_{(j)}|\tilde{\theta}),$
\begin{align}
\nonumber
\lim_{m\rightarrow \infty}\lim_{n\rightarrow \infty}\pi_\epsilon(\theta|{\bf x}obs,Q_{\theta})&=\frac{\displaystyle{\lim_{m\rightarrow\infty}}\int_{\mathcal{X}}\mathbf{I}_{\{||\textrm{\boldmath$s$}-\textrm{\boldmath$s$}obs||<\epsilon\}}\prod_{j=1}^{k}\left[p(x_{(j)}|\tilde{\theta})\right]^{n_{(j)}}\;\mathbf{I}_{\{\theta,\tilde{\theta}\}}\pi(\theta)\;d{\bf x}}{\displaystyle{\lim_{m \rightarrow \infty}}C_{P_{\theta}sub_{\tilde{\theta}}}}\\
\label{eq:app4}
&=\frac{\displaystyle{\lim_{m\rightarrow\infty}}\int_{\mathcal{X}}\mathbf{I}_{\{||\textrm{\boldmath$s$}-\textrm{\boldmath$s$}obs||<\epsilon\}}p({\bf x}|\tilde{\theta})\;\mathbf{I}_{\{\theta,\tilde{\theta}\}}\pi(\theta)\;d{\bf x}}{\displaystyle{\lim_{m \rightarrow \infty}}C_{P_{\theta}sub_{\tilde{\theta}}}},
\end{align}
where (\ref{eq:app4}) follows by the definition of the joint distribution $p({\bf x}|\tilde{\theta})=\prod_{j=1}^{k}\left[p(x_{(j)}|\tilde{\theta})\right]^{n_{(j)}}.$
\par
We now apply the dominated convergence theorem a second time to exchange the limits in $m$ and the integrals on $\mathcal{X}$. Again, the assumptions of the dominated convergence theorem are satisfied since the integrand in (\ref{eq:app4}) is a sequence in $m$ of bounded functions, and as $m\rightarrow \infty,$ $\tilde{\theta}\rightarrow \theta,$ and $p({\bf x}|\tilde{\theta})\rightarrow p({\bf x}|\theta).$
We get
\begin{equation*}
\lim_{m\rightarrow \infty}\lim_{n\rightarrow \infty}\pi_\epsilon(\theta|{\bf x}obs,Q_{\theta})=\frac{1}{C_{P_{\theta}}}\int_{\mathcal{X}}\mathbf{I}_{\{||\textrm{\boldmath$s$}-\textrm{\boldmath$s$}obs||<\epsilon\}}p({\bf x}|\theta)\pi(\theta)\;d{\bf x}=\displaystyle{\lim_{n\rightarrow \infty}}\pi_{\epsilon}(\theta|{\bf x}obs,P_{\theta})
\end{equation*}
which shows that AABC posterior converges to the ABC posterior as the sample size $n$ and the simulated number of data sets $m$ increase.
\end{document} | math | 62,518 |
Subsets and Splits